text
stringlengths 256
16.4k
|
|---|
Pixel-Wise Test Correlations of All Decoder Outputs (99% Confidence Interval Values in Parentheses).
Versus True LP
Versus True HP
Versus True
LP ridge (2-bin) 0.975 (0.00016) LP Ridge: 2-bin 0.887 (0.00062)
HP NN 0.360 (0.0032) HP NN
+
LP ridge (2-bin) 0.901 (0.00059)
Combined-deblurred 0.912 (0.00055)
Ridge-deblurred 0.903 (0.00057)
Whole ridge 0.963 (0.00021) HP Ridge 0.282 (0.0028) Whole RIDGE 0.890 (0.00061)
LP NN 0.960 (0.00033) Whole NN 0.874 (0.00076)
LP ridge (50-bin) 0.979 (0.00015)
LP LASSO 0.978 (0.00015)
+
Notes: The best results are in bold.
The 2-bin and 50-bin LP ridge labels represent the two linear ridge decoders trained on the low-pass images. The whole ridge decoder is the 2-bin ridge decoder trained on the true whole images themselves, while the HP ridge decoder is the same decoder trained on the high-pass images only. The LP, HP, and whole NN labels denote the spatially restricted neural network decoder trained on low-pass, high-pass, and whole images, respectively. LP LASSO represents the 2-bin LASSO regression decoder trained on low-pass images. Finally, the combined-deblurred images are the deblurred versions of the sum of the HP NN and LP Ridge (2-bin) decoded images, while the ridge-deblurred images are the deblurred versions of the whole ridge decoder outputs. These final three—combined-deblurred, ridge-deblurred, and HP NN
+
LP Ridge (2-bin)—are in bold because they produced best results. The second, fourth, and sixth columns represent pixel-wise test correlations of each decoder's output versus the true low-pass, high-pass, and whole images, respectively.
|
Pressure-reducing valve in a two-phase fluid network - MATLAB - MathWorks Italia
Pressure-Reducing Valve (2P)
Pressure-reducing valve in a two-phase fluid network
The Pressure-Reducing Valve (2P) block models a pressure-controlling reducing valve in a two-phase fluid network. The valve is open when the pressure at port B is less than the set pressure, and closes when the pressure exceeds that value. The control pressure can be set as a constant in the Set pressure (gauge) parameter or, when you set Set pressure control to Controlled, the set pressure can vary according to the input signal at port Ps.
The valve closes when the pressure in the valve, pcontrol, exceeds the set pressure, pset. The valve is fully closed when the control pressure reaches the end of the Pressure regulation range, prange.
\lambda =1-\left(1-{f}_{leak}\right)\frac{\left({p}_{control}-{p}_{set}\right)}{{p}_{range}},
pcontrol is the control pressure, which is the difference between the pressure at port B and atmospheric pressure.
pset is the Set pressure (gauge).
\lambda =1-\left(1-{f}_{leak}\right)\frac{\left({p}_{control}-{p}_{s}\right)}{{p}_{range}},
where ps is the signal at port Ps. If the control pressure exceeds the valve pressure range, the valve opening fraction is 0.
{\stackrel{˙}{m}}_{A}=\lambda {\stackrel{˙}{m}}_{nom}\left[\sqrt{\frac{{v}_{nom}}{2\Delta {p}_{nom}}}\right]\sqrt{\frac{2}{{v}_{in}}}\frac{\Delta p}{{\left(\Delta {p}^{2}+\Delta {p}_{lam}^{2}\right)}^{0.25}},
\Delta {p}_{lam}=\frac{\left({p}_{A}+{p}_{B}\right)}{2}\left(1-{B}_{lam}\right).
{\stackrel{˙}{m}}_{nom}
{v}_{in}=\left(1-{x}_{dyn}\right){v}_{liq}+{x}_{dyn}{v}_{vap},
\frac{d{x}_{dyn}}{dt}=\frac{{x}_{in}-{x}_{dyn}}{\tau },
xdyn is the dynamic fluid vapor quality.
{\stackrel{˙}{m}}_{A}+{\stackrel{˙}{m}}_{B}=0,
{\stackrel{˙}{m}}_{A}
{\stackrel{˙}{m}}_{B}
{\Phi }_{A}+{\Phi }_{B}=0,
Input port for varying the set pressure signal.
Set pressure control — Varying or constant set pressure
Set pressure (gauge) — Pressure threshold
Valve pressure threshold. When the control pressure, pB ̶ patm, exceeds the set pressure, the valve begins to close.
Valve operational range. The valve begins to close at the set pressure value, and is fully closed at pmax, the end of the pressure regulation range: pmax = pset + prange.
Inlet pressure in typical, design, or rated conditions. The valve inlet specific volume is determined from the fluid properties tabulated data based on the Nominal inlet pressure and the setting of the Nominal inlet condition specification parameters.
Inlet vapor quality of the mixture by mass fraction in open, nominal operating conditions. A value of 0 means that the inlet fluid is subcooled liquid. A value of 1 means that the inlet fluid is superheated vapor.
Nominal inlet vapor void fraction — Inlet vapor quality by volume fraction
|
Mrs. Chen has two brothers. Mark is
7
years older than Mrs. Chen and Eric is
11
years younger than Mrs. Chen. The sum of all three of their ages is
149
. Use the 5-D Process to determine the age of Mrs. Chen.
What are the equations you can write to relate the ages of Mrs. Chen, Mark, and Eric? How can you use these equations to find Mrs. Chen's age?
C=\text{Mrs. Chen′s age}
M=\text{Mark′s age}
E=\text{Eric′s age}
one of the equations is
C+M+E=149
Mrs. Chen is
51
|
Areostationary orbit - WikiMili, The Best Wikipedia Reader
An areostationary orbit or areosynchronous equatorial orbit (abbreviated AEO) is a circular areosynchronous orbit (ASO) in the Martian equatorial plane about 17,032 km (10,583 mi) above the surface, any point on which revolves about Mars in the same direction and with the same period as the Martian surface. Areostationary orbit is a concept similar to Earth's geostationary orbit (GEO). The prefix areo- derives from Ares, the ancient Greek god of war and counterpart to the Roman god Mars, with whom the planet was identified. The modern Greek word for Mars is Άρης (Áris).
To date, no artificial satellites have been placed in this orbit, but it is of interest to some scientists foreseeing a future telecommunications network for the exploration of Mars. [1] An asteroid or station placed in areostationary orbit could also be used to construct a Martian space elevator for use in transfers between the surface of Mars and orbit.[ citation needed ]
Orbital speed (how fast a satellite is moving through space) is calculated by multiplying the angular speed of the satellite by the orbital radius:
{\displaystyle R_{syn}={\sqrt[{3}]{G(m_{2})T^{2} \over 4\pi ^{2}}}}
m2 = Mass of the celestial body
T = rotational period of the body
By this formula one can find the geostationary-analogous orbit of an object in relation to a given body, in this case, Mars (this type of orbit above is referred to as an areostationary orbit if it is above Mars).
The mass of Mars being 6.4171×1023 kg and the sidereal period 88,642 seconds. [3] The synchronous orbit thus has a radius of 20,428 km (12693 mi) from the centre of mass of Mars, [4] and therefore areostationary orbit can be defined as approximately 17,032 km above the surface of the Mars equator.
Any satellites in areostationary orbit will suffer from increased orbital station keeping costs, [5] [6] because the areostationary orbits lie between the orbits of the planet's two natural satellites. Phobos has a semi-major axis of 9,376 km, and Deimos has a semi-major axis of 23,463 km. The close proximity to Phobos' orbit in particular (the larger of the two moons) will cause unwanted orbital resonance effects that will gradually shift the orbit of areostationary satellites.
Areosynchronous orbit
A space elevator is a proposed type of planet-to-space transportation system. The main component would be a cable anchored to the surface and extending into space. The design would permit vehicles to travel along the cable from a planetary surface, such as the Earth's, directly into orbit, without the use of large rockets. An Earth-based space elevator would consist of a cable with one end attached to the surface near the equator and the other end in space beyond geostationary orbit. The competing forces of gravity, which is stronger at the lower end, and the outward/upward centrifugal force, which is stronger at the upper end, would result in the cable being held up, under tension, and stationary over a single position on Earth. With the tether deployed, climbers could repeatedly climb the tether to space by mechanical means, releasing their cargo to orbit. Climbers could also descend the tether to return cargo to the surface from orbit.
Phobos is the innermost and larger of the two natural satellites of Mars, the other being Deimos. Both moons were discovered in 1877 by American astronomer Asaph Hall. Phobos is named after the Greek god Phobos, a son of Ares (Mars) and Aphrodite (Venus) and twin brother of Deimos. Phobos was the god and personification of fear and panic.
The Phobos program was an unmanned space mission consisting of two probes launched by the Soviet Union to study Mars and its moons Phobos and Deimos. Phobos 1 was launched on 7 July 1988, and Phobos 2 on 12 July 1988, each aboard a Proton-K rocket.
Phobos 1 was an uncrewed Soviet space probe of the Phobos Program launched from the Baikonour launch facility on 7 July 1988. Its intended mission was to explore Mars and its moons Phobos and Deimos. The mission failed on 2 September 1988 when a computer malfunction caused the end-of-mission order to be transmitted to the spacecraft. At the time of launch it was the heaviest interplanetary spacecraft ever launched, weighing 6200 kg.
Phobos 2 was the last space probe designed by the Soviet Union. It was designed to explore the moons of Mars, Phobos and Deimos. It was launched on 12 July 1988, and entered orbit on 29 January 1989.
Stickney is the largest crater on Phobos, which is a satellite of Mars. It is 9 km (5.6 mi) in diameter, taking up a substantial proportion of the moon's surface.
The areosynchronous orbits (ASO) are the synchronous orbits for artificial satellites around the planet Mars. They are the martian equivalent of the geosynchronous orbits (GSO) on the Earth. The prefix areo- derives from Ares, the ancient Greek god of war and counterpart to the Roman god Mars, with whom the planet was identified. The modern Greek word for Mars is Άρης (Áris).
The two moons of Mars are Phobos and Deimos. They are irregular in shape. Both were discovered by American astronomer Asaph Hall in August 1877 and are named after the Greek mythological twin characters Phobos (fear) and Deimos who accompanied their father Ares into battle. Ares, god of war, was known to the Romans as Mars.
Mars has two moons, Phobos and Deimos. Due to their small size, both moons were discovered only in 1877, by astronomer Asaph Hall. Nevertheless, they frequently feature in works of science fiction.
Yinghuo-1 was a Chinese Mars-exploration space probe, intended to be the first Chinese planetary space probe and the first Chinese spacecraft to orbit Mars. It was launched from Baikonur Cosmodrome, Kazakhstan, on 8 November 2011, along with the Russian Fobos-Grunt sample return spacecraft, which was intended to visit Mars' moon Phobos. The 115-kg (250-lb) Yinghuo-1 probe was intended by the CNSA to orbit Mars for about two years, studying the planet's surface, atmosphere, ionosphere and magnetic field. Shortly after launch, Fobos-Grunt was expected to perform two burns to depart Earth orbit bound for Mars. However, these burns did not take place, leaving both probes stranded in orbit. On 17 November 2011, Chinese state media reported that Yinghuo-1 had been declared lost by the CNSA. After a period of orbital decay, Yinghuo-1 and Fobos-Grunt underwent destructive re-entry on 15 January 2012, finally disintegrating over the Pacific Ocean.
The gravity of Mars is a natural phenomenon, due to the law of gravity, or gravitation, by which all things with mass around the planet Mars are brought towards it. It is weaker than Earth's gravity due to the planet's smaller mass. The average gravitational acceleration on Mars is 3.72076 ms−2 and it varies. In general, topography-controlled isostasy drives the short wavelength free-air gravity anomalies. At the same time, convective flow and finite strength of the mantle lead to long-wavelength planetary-scale free-air gravity anomalies over the entire planet. Variation in crustal thickness, magmatic and volcanic activities, impact-induced Moho-uplift, seasonal variation of polar ice caps, atmospheric mass variation and variation of porosity of the crust could also correlate to the lateral variations. Over the years models consisting of an increasing but limited number of spherical harmonics have been produced. Maps produced have included free-air gravity anomaly, Bouguer gravity anomaly, and crustal thickness. In some areas of Mars there is a correlation between gravity anomalies and topography. Given the known topography, higher resolution gravity field can be inferred. Tidal deformation of Mars by the Sun or Phobos can be measured by its gravity. This reveals how stiff the interior is, and shows that the core is partially liquid. The study of surface gravity of Mars can therefore yield information about different features and provide beneficial information for future landings.
↑ Lay, N.; C. Cheetum; H. Mojaradi; J. Neal (15 November 2001). "Developing Low-Power Transceiver Technologies for In Situ Communication Applications" (PDF). IPN Progress Report 42-147. 42 (147): 22. Bibcode:2001IPNPR.147A...1L. Archived from the original (PDF) on 4 March 2016. Retrieved 2012-02-09.
↑ "Calculating the Radius of a Geostationary Orbit - Ask Will Online". Ask Will Online. 2012-12-27. Retrieved 2017-11-21.
↑ Lodders, Katharina; Fegley, Bruce (1998). The Planetary Scientist's Companion. Oxford University Press. p. 190. ISBN 0-19-511694-1.
↑ "Stationkeeping in Mars orbit". www.planetary.org. Retrieved 2017-11-21.
↑ Romero, P.; Pablos, B.; Barderas, G. (2017-07-01). "Analysis of orbit determination from Earth-based tracking for relay satellites in a perturbed areostationary orbit". Acta Astronautica. 136: 434–442. Bibcode:2017AcAau.136..434R. doi:10.1016/j.actaastro.2017.04.002. ISSN 0094-5765.
↑ Silva and Romero's paper even includes a graph of acceleration, where a reaction force could be calculated using the mass of desired object: Silva, Juan J.; Romero, Pilar (2013-10-01). "Optimal longitudes determination for the station keeping of areostationary satellites". Planetary and Space Science. 87: 16. Bibcode:2013P&SS...87...14S. doi:10.1016/j.pss.2012.11.013. ISSN 0032-0633.
Mars Network - Marsats - NASA site devoted to future communications infrastructure for Mars exploration
Bandwidth available from an areostationary satellite
This Mars spacecraft- or satellite-related article is a stub. You can help Wikipedia by expanding it.
|
Numerical Analysis/ODE in vector form Exercises - Wikiversity
Numerical Analysis/ODE in vector form Exercises
All of the standard methods for solving ordinary differential equations are intended for first order equations. When you need to solve a higher order differential equation, you first convert it to a system of first order of equations. Then you rewrite as a vector form and solve this ODE using a standard method. On this page we demonstrate how to convert to a system of equations and then apply standard methods in vector form.
1 Reduction to a first order system
2.1 Exercise 1: Convert this second order differential equation to a system of first order equations.
2.2 Exercise 2: Apply the Euler method twice.
2.2.1 I. First apply to get y1
2.2.2 2. Second apply to get y2
2.3 Exercise 3: Apply the Backward Euler method twice.
2.4 Exercise 4: Apply the Midpoint method twice.
2.5 Exercise 5: Using the values from the Midpoint method at t = h in exercise3, apply the Two-step Adams-Bashforth method once.
Reduction to a first order systemEdit
(Based on Reduction of Order and Converting a general higher order equation.)
I want to show how to convert higher order differential equation to a system of the first order differential equation. Any differential equation of order n of the form
{\displaystyle f\left(t,u,u',u'',\ \cdots ,\ u^{(n-1)}\right)=u^{(n)}}
{\displaystyle y_{i}=u^{(i-1)}\quad {\text{for}}\quad i=1,2,...n\,.}
The n-dimensional system of first-order coupled differential equations is then
{\displaystyle {\begin{array}{rclcl}y_{1}&=&u\\y_{2}&=&u'\\y_{3}&=&u''\\&\vdots &\\y_{n}&=&u^{(n-1)}.\\\end{array}}}
{\displaystyle {\begin{array}{rclclcl}y_{1}'&=&u'&=&y_{2}\\y_{2}'&=&u''&=&y_{3}\\y_{3}'&=&u'''&=&y_{4}\\&\vdots &\\y_{n}'&=&u^{(n)}&=&f(t,y_{1},\cdots ,y_{n}).\\\end{array}}}
We can express this more compactly in vector form
{\displaystyle \mathbf {y} '=\mathbf {f} (t,\mathbf {y} )}
{\displaystyle \ y_{i+1}=f_{i}\left(t,\mathbf {y} \right)}
{\displaystyle i<n}
{\displaystyle \ f_{n}\left(t,\mathbf {y} \right)}
{\displaystyle \ f\left(t,y_{1},y_{2},\cdots ,y_{n}\right)\,.}
Consider the second order differential equation
{\displaystyle \ u''+u=0}
{\displaystyle \ u{(0)}=1}
{\displaystyle \ u'{(0)}=0}
. We will use two steps with step size
{\displaystyle \ h={\frac {\pi }{8}}}
and approximate the values of
{\displaystyle \ u{({\frac {\pi }{4}})}}
{\displaystyle \ u'{({\frac {\pi }{4}})}.}
Since the exact solution is
{\displaystyle u(t)=\cos(t)}
{\displaystyle \ u{({\frac {\pi }{4}})}=0.707106781}
{\displaystyle u'{({\frac {\pi }{4}})}=-0.707106781}
Exercise 1: Convert this second order differential equation to a system of first order equations.Edit
We have second order differential equation
{\displaystyle \ u''{(t)}=-u{(t)}}
{\displaystyle \ u{(0)}=1}
{\displaystyle \ u'{(0)}=0}
{\displaystyle y_{1}=u}
{\displaystyle y_{2}=u'}
. Differentiating
{\displaystyle y_{1}}
{\displaystyle y_{2}}
{\displaystyle \ y_{1}'=u'=y_{2}}
{\displaystyle \ y_{2}'=u''=-u=-y_{1}.}
Thus we have a system of first order equation in vector form
{\displaystyle \left[{\begin{array}{c}y_{1}'\\y_{2}'\end{array}}\right]=\left[{\begin{array}{c}y_{2}\\-y_{1}\end{array}}\right]=f\left(t,\left[{\begin{array}{c}y_{1}\\y_{2}\end{array}}\right]\right)}
Exercise 2: Apply the Euler method twice.Edit
By the Euler's method,
{\displaystyle y_{n+1}=y_{n}+hf\left(t_{n},y_{n}\right)}
I. First apply to get y1Edit
{\displaystyle y_{1}=y_{0}+hf\left(t_{0},y_{0}\right)}
Now we convert
{\displaystyle \ y_{1}}
as a vector form, where
{\displaystyle \ t_{0}=0,h={\frac {\pi }{8}}}
{\displaystyle \left[{\begin{array}{c}y_{0,1}\\y_{o,2}\end{array}}\right]=\left[{\begin{array}{c}1\\0\end{array}}\right]}
{\displaystyle {\begin{aligned}\left[{\begin{array}{c}y_{1,1}\\y_{1,2}\end{array}}\right]&=\left[{\begin{array}{c}y_{0,1}\\y_{0,2}\end{array}}\right]+hf\left(t_{0},\left[{\begin{array}{c}y_{0,1}\\y_{0,2}\end{array}}\right]\right)\\&=\left[{\begin{array}{c}y_{0,1}\\y_{0,2}\end{array}}\right]+h\left[{\begin{array}{c}y_{0,2}\\-y_{0,1}\end{array}}\right]\\&=\left[{\begin{array}{c}1\\0\end{array}}\right]+h\left[{\begin{array}{c}0\\-1\end{array}}\right]\\&=\left[{\begin{array}{c}{1+0}\\{0-h}\end{array}}\right]\\&=\left[{\begin{array}{c}1\\{-h}\end{array}}\right]\\&=\left[{\begin{array}{c}1\\{-{\frac {\pi }{8}}}\end{array}}\right]\\&=\left[{\begin{array}{c}1\\{-0.392699082}\end{array}}\right]\,.\end{aligned}}}
2. Second apply to get y2Edit
{\displaystyle y_{2}=y_{1}+hf\left(t_{1},y_{1}\right)}
Similarly, we convert
{\displaystyle y_{2}}
{\displaystyle t_{1}=t_{0}+h=0+h=h}
{\displaystyle h={\frac {\pi }{8}}}
{\displaystyle \left[{\begin{array}{c}y_{1,1}\\y_{1,2}\end{array}}\right]=\left[{\begin{array}{c}1\\-h\end{array}}\right]}
{\displaystyle {\begin{aligned}\left[{\begin{array}{c}y_{2,1}\\y_{2,2}\end{array}}\right]&=\left[{\begin{array}{c}y_{1,1}\\y_{1,2}\end{array}}\right]+hf\left(t_{1},\left[{\begin{array}{c}y_{1,1}\\y_{1,2}\end{array}}\right]\right)\\&=\left[{\begin{array}{c}y_{1,1}\\y_{1,2}\end{array}}\right]+h\left[{\begin{array}{c}y_{2,1}\\-y_{1,1}\end{array}}\right]\\&=\left[{\begin{array}{c}1\\{-h}\end{array}}\right]+h\left[{\begin{array}{c}{-h}\\{-1}\end{array}}\right]\\&=\left[{\begin{array}{c}{1-h^{2}}\\{-h-h}\end{array}}\right]\\&=\left[{\begin{array}{c}{1-h^{2}}\\-2h\end{array}}\right]\\&=\left[{\begin{array}{c}{1-{\frac {\pi ^{2}}{8^{2}}}}\\{-2{\frac {\pi }{8}}}\end{array}}\right]\\&=\left[{\begin{array}{c}0.8457874321\\{-0.785398163}\end{array}}\right]\,.\end{aligned}}}
Exercise 3: Apply the Backward Euler method twice.Edit
By the Backward Euler's method,
{\displaystyle \ y_{n+1}=y_{n}+hf\left(t_{n+1},y_{n+1}\right).}
{\displaystyle \ y_{1}=y_{0}+hf\left(t_{1},y_{1}\right)}
{\displaystyle y_{1}}
{\displaystyle t_{1}=t_{0}+h=0+h=h}
{\displaystyle h={\frac {\pi }{8}}}
{\displaystyle \left[{\begin{array}{c}y_{0,1}\\y_{o,2}\end{array}}\right]=\left[{\begin{array}{c}1\\0\end{array}}\right]}
{\displaystyle {\begin{aligned}\left[{\begin{array}{c}y_{1,1}\\y_{1,2}\end{array}}\right]&=\left[{\begin{array}{c}y_{0,1}\\y_{0,2}\end{array}}\right]+hf\left(t_{1},\left[{\begin{array}{c}y_{1,1}\\y_{1,2}\end{array}}\right]\right)\\&=\left[{\begin{array}{c}y_{0,1}\\y_{0,2}\end{array}}\right]+h\left[{\begin{array}{c}y_{1,2}\\-y_{1,1}\end{array}}\right]\\&=\left[{\begin{array}{c}1\\0\end{array}}\right]+\left[{\begin{array}{c}{hy_{1,2}}\\{-hy_{1,1}}\end{array}}\right]\\&=\left[{\begin{array}{c}{1+hy_{1,2}}\\{-hy_{1,1}}\end{array}}\right]\,.\end{aligned}}}
Now we have to solve a system of linear equations
{\displaystyle \ \left\{{\begin{array}{l l}y_{1,1}=1+hy_{1,2}&\quad \\y_{1,2}=-hy_{1,1}\quad \\\end{array}}\right.}
{\displaystyle \ \left\{{\begin{array}{l l}y_{1,1}-hy_{1,2}=1&\quad \\hy_{1,1}+y_{1,2}=0\quad \\\end{array}}\right.}
Set up augmented matrix to solve this system,
{\displaystyle \left({\begin{array}{cc|c}1&-h&1\\h&1&0\\\end{array}}\right)\Rightarrow \left({\begin{array}{cc|c}1&-h&1\\0&1+h^{2}&-h\\\end{array}}\right)}
{\displaystyle \ y_{1,1}={\frac {1}{1+h^{2}}},y_{1,2}=-{\frac {h}{1+h^{2}}}.}
{\displaystyle \ h={\frac {\pi }{8}}}
{\displaystyle \left[{\begin{array}{c}y_{1,1}\\y_{1,2}\end{array}}\right]=\left[{\begin{array}{c}0.8457874321\\{-0.785398163}\end{array}}\right]\,.}
{\displaystyle \ y_{2}=y_{1}+hf\left(t_{2},y_{2}\right)}
{\displaystyle \ y_{2}}
as a vector form, with
{\displaystyle \ t_{2}=t_{1}+h=h+h=2h}
{\displaystyle \ h={\frac {\pi }{8}}}
{\displaystyle \left[{\begin{array}{c}y_{1,1}\\y_{1,2}\end{array}}\right]=\left[{\begin{array}{c}{\frac {1}{1+h^{2}}}\\{-{\frac {h}{1+h^{2}}}}\end{array}}\right]}
{\displaystyle {\begin{aligned}\left[{\begin{array}{c}y_{2,1}\\y_{2,2}\end{array}}\right]&=\left[{\begin{array}{c}y_{1,1}\\y_{1,2}\end{array}}\right]+hf\left(t_{2},\left[{\begin{array}{c}y_{2,1}\\y_{2,2}\end{array}}\right]\right)\\&=\left[{\begin{array}{c}{\frac {1}{1+h^{2}}}\\{-{\frac {h}{1+h^{2}}}}\end{array}}\right]+hf\left(2h,\left[{\begin{array}{c}y_{2,1}\\y_{2,2}\end{array}}\right]\right)\\&=\left[{\begin{array}{c}{\frac {1}{1+h^{2}}}\\{-{\frac {h}{1+h^{2}}}}\end{array}}\right]+h\left[{\begin{array}{c}y_{2,2}\\{-y_{2,1}}\end{array}}\right]\\&=\left[{\begin{array}{c}{{\frac {1}{1+h^{2}}}+hy_{2,2}}\\{-{\frac {h}{1+h^{2}}}-hy_{2,1}}\end{array}}\right]\,.\end{aligned}}}
{\displaystyle \ \left\{{\begin{array}{l l}y_{2,1}={\frac {1}{1+h^{2}}}+hy_{2,2}&\quad \\y_{2,2}=-{\frac {h}{1+h^{2}}}-hy_{2,1}\quad \\\end{array}}\right.}
{\displaystyle \ \left\{{\begin{array}{l l}y_{2,1}-hy_{2,2}={\frac {1}{1+h^{2}}}&\quad \\hy_{2,1}+y_{2,2}=-{\frac {h}{1+h^{2}}}\quad \\\end{array}}\right.}
{\displaystyle \left({\begin{array}{cc|c}1&-h&{\frac {1}{1+h^{2}}}\\h&1&-{\frac {h}{1+h^{2}}}\\\end{array}}\right)\Rightarrow \left({\begin{array}{cc|c}1&-h&{\frac {1}{1+h^{2}}}\\0&1+h^{2}&-{\frac {2h}{1+h^{2}}}\\\end{array}}\right)}
{\displaystyle \ y_{2,1}={\frac {1-h^{2}}{(1+h^{2})^{2}}},y_{2,2}=-{\frac {2h}{(1+h^{2})^{2}}}.}
{\displaystyle \ h={\frac {\pi }{8}}}
{\displaystyle \left[{\begin{array}{c}y_{2,1}\\y_{2,2}\end{array}}\right]=\left[{\begin{array}{c}0.63487705\\{-0.589546795}\end{array}}\right]\,.}
Exercise 4: Apply the Midpoint method twice.Edit
By the Midpoint method,
{\displaystyle y_{n+1}=y_{n}+hf\left(t_{n}+{\frac {1}{2}}h,y_{n}+{\frac {1}{2}}hf\left(t_{n},y_{n}\right)\right).}
{\displaystyle y_{1}=y_{0}+hf\left(t_{0}+{\frac {1}{2}}h,y_{0}+{\frac {1}{2}}hf\left(t_{0},y_{0}\right)\right).}
{\displaystyle \ y_{1}}
{\displaystyle \ t_{0}=0,h={\frac {\pi }{8}}}
{\displaystyle \left[{\begin{array}{c}y_{0,1}\\y_{0,2}\end{array}}\right]=\left[{\begin{array}{c}1\\0\end{array}}\right]}
{\displaystyle {\begin{aligned}\left[{\begin{array}{c}y_{1,1}\\y_{1,2}\end{array}}\right]&=\left[{\begin{array}{c}y_{0,1}\\y_{0,2}\end{array}}\right]+hf\left(t_{0}+{\frac {1}{2}}h,\left[{\begin{array}{c}y_{0,1}\\y_{0,2}\end{array}}\right]+{\frac {1}{2}}hf\left(t_{0},\left[{\begin{array}{c}y_{0,1}\\y_{0,2}\end{array}}\right]\right)\right)\\&=\left[{\begin{array}{c}y_{0,1}\\y_{0,2}\end{array}}\right]+hf\left(t_{0}+{\frac {1}{2}}h,\left[{\begin{array}{c}y_{0,1}\\y_{0,2}\end{array}}\right]+{\frac {1}{2}}h\left[{\begin{array}{c}y_{0,2}\\{-y_{0,1}}\end{array}}\right]\right)\\&=\left[{\begin{array}{c}1\\0\end{array}}\right]+hf\left(0+{\frac {1}{2}}h,\left[{\begin{array}{c}1\\0\end{array}}\right]+{\frac {1}{2}}h\left[{\begin{array}{c}0\\{-1}\end{array}}\right]\right)\\&=\left[{\begin{array}{c}1\\0\end{array}}\right]+h\left[{\begin{array}{c}{-{\frac {1}{2}}h}\\{-1}\end{array}}\right]\\&=\left[{\begin{array}{c}{1-{\frac {1}{2}}h^{2}}\\{-h}\end{array}}\right]\,.\end{aligned}}}
{\displaystyle \ h={\frac {\pi }{8}}}
{\displaystyle \left[{\begin{array}{c}y_{1,1}\\y_{1,2}\end{array}}\right]=\left[{\begin{array}{c}0.922893716\\{-0.392699082}\end{array}}\right]\,.}
{\displaystyle y_{2}=y_{1}+hf\left(t_{1}+{\frac {1}{2}}h,y_{1}+{\frac {1}{2}}hf\left(t_{1},y_{1}\right)\right).}
{\displaystyle \ y_{1}}
{\displaystyle \ t_{1}=t_{0}+h=0+h=h,h={\frac {\pi }{8}}}
{\displaystyle \left[{\begin{array}{c}y_{1,1}\\y_{1,2}\end{array}}\right]=\left[{\begin{array}{c}{1-{\frac {1}{2}}h^{2}}\\{-h}\end{array}}\right]}
{\displaystyle {\begin{aligned}\left[{\begin{array}{c}y_{2,1}\\y_{2,2}\end{array}}\right]&=\left[{\begin{array}{c}y_{1,1}\\y_{1,2}\end{array}}\right]+hf\left(t_{1}+{\frac {1}{2}}h,\left[{\begin{array}{c}y_{1,1}\\y_{1,2}\end{array}}\right]+{\frac {1}{2}}hf\left(t_{1},\left[{\begin{array}{c}y_{1,1}\\y_{1,2}\end{array}}\right]\right)\right)\\&=\left[{\begin{array}{c}y_{1,1}\\y_{1,2}\end{array}}\right]+hf\left(h+{\frac {1}{2}}h,\left[{\begin{array}{c}y_{1,1}\\y_{1,2}\end{array}}\right]+{\frac {1}{2}}h\left[{\begin{array}{c}y_{1,2}\\{-y_{1,1}}\end{array}}\right]\right)\\&=\left[{\begin{array}{c}{1-{\frac {1}{2}}h^{2}}\\{-h}\end{array}}\right]+hf\left({\frac {3}{2}}h,\left[{\begin{array}{c}{1-{\frac {1}{2}}h^{2}}\\{-h}\end{array}}\right]+{\frac {1}{2}}h\left[{\begin{array}{c}{-h}\\{-1+{\frac {1}{2}}h^{2}}\end{array}}\right]\right)\\&=\left[{\begin{array}{c}{1-{\frac {1}{2}}h^{2}}\\{-h}\end{array}}\right]+hf\left({\frac {3}{2}}h,\left[{\begin{array}{c}{1-{\frac {1}{2}}h^{2}-{\frac {1}{2}}h^{2}}\\{-h-{\frac {1}{2}}h+{\frac {1}{4}}h^{3}}\end{array}}\right]\right)\\&=\left[{\begin{array}{c}{1-{\frac {1}{2}}h^{2}}\\{-h}\end{array}}\right]+hf\left({\frac {3}{2}}h,\left[{\begin{array}{c}{1-h^{2}}\\{-{\frac {2}{3}}h+{\frac {1}{4}}h^{3}}\end{array}}\right]\right)\\&=\left[{\begin{array}{c}{1-{\frac {1}{2}}h^{2}}\\{-h}\end{array}}\right]+h\left[{\begin{array}{c}{-{\frac {2}{3}}h+{\frac {1}{4}}h^{3}}\\{-1+h^{2}}\end{array}}\right]\\&=\left[{\begin{array}{c}{1-{\frac {1}{2}}h^{2}}\\{-h}\end{array}}\right]+\left[{\begin{array}{c}{-{\frac {2}{3}}h^{2}+{\frac {1}{4}}h^{4}}\\{-h+h^{3}}\end{array}}\right]\\&=\left[{\begin{array}{c}{1-{\frac {1}{2}}h^{2}-{\frac {2}{3}}h^{2}+{\frac {1}{4}}h^{4}}\\{-h-h+h^{3}}\end{array}}\right]\\&=\left[{\begin{array}{c}{1-2h^{2}+{\frac {1}{4}}h^{4}}\\{-2h+h^{3}}\end{array}}\right]\,.\end{aligned}}}
{\displaystyle \ h={\frac {\pi }{8}}}
{\displaystyle \left[{\begin{array}{c}y_{2,1}\\y_{2,2}\end{array}}\right]=\left[{\begin{array}{c}0.694547552011673\\{-0.7248390292562377}\end{array}}\right]\,.}
Exercise 5: Using the values from the Midpoint method at t = h in exercise3, apply the Two-step Adams-Bashforth method once.Edit
By the Midpoint method, we found
{\displaystyle \left[{\begin{array}{c}y_{1,1}\\y_{1,2}\end{array}}\right]=\left[{\begin{array}{c}{1-{\frac {1}{2}}h^{2}}\\{-h}\end{array}}\right]\,.}
Now apply the Two-step Adams-Bashforth method when
{\displaystyle \ n=0}
, then by formula
{\displaystyle \ y_{2}=y_{1}+{\frac {3}{2}}hf\left(t_{1},y_{1}\right)-{\frac {1}{2}}hf\left(t_{0},y_{0}\right).}
Vector form is following by
{\displaystyle {\begin{aligned}\left[{\begin{array}{c}y_{2,1}\\y_{2,2}\end{array}}\right]&=\left[{\begin{array}{c}y_{1,1}\\y_{1,2}\end{array}}\right]+{\frac {3}{2}}hf\left(t_{1},\left[{\begin{array}{c}y_{1,1}\\y_{1,2}\end{array}}\right]\right)-{\frac {1}{2}}hf\left(t_{0},\left[{\begin{array}{c}y_{0,1}\\y_{0,2}\end{array}}\right]\right)\\&=\left[{\begin{array}{c}{1-{\frac {1}{2}}h^{2}}\\{-h}\end{array}}\right]+{\frac {3}{2}}hf\left(t_{1},\left[{\begin{array}{c}{1-{\frac {1}{2}}h^{2}}\\{-h}\end{array}}\right]\right)-{\frac {1}{2}}hf\left(t_{0},\left[{\begin{array}{c}1\\0\end{array}}\right]\right)\\&=\left[{\begin{array}{c}{1-{\frac {1}{2}}h^{2}}\\{-h}\end{array}}\right]+{\frac {3}{2}}h\left[{\begin{array}{c}{-h}\\{-1+{\frac {1}{2}}h^{2}}\end{array}}\right]-{\frac {1}{2}}h\left[{\begin{array}{c}0\\{-1}\end{array}}\right]\\&=\left[{\begin{array}{c}{1-2h^{2}}\\{-2h+{\frac {3}{4}}h^{3}}\end{array}}\right]\,.\end{aligned}}}
{\displaystyle h={\frac {\pi }{8}},}
{\displaystyle \left[{\begin{array}{c}y_{2,1}\\y_{2,2}\end{array}}\right]=\left[{\begin{array}{c}0.691574862\\{-0.739978812}\end{array}}\right]\,.}
http://www.math.ohiou.edu/courses/math3600/lecture29.pdf
http://www.ohio.edu/people/mohlenka/20131/4600-5600/hw7.pdf
Retrieved from "https://en.wikiversity.org/w/index.php?title=Numerical_Analysis/ODE_in_vector_form_Exercises&oldid=2381162"
|
Evaluate an expression - Maple Help
Home : Support : Online Help : Getting Started : How Do I... : Evaluate an expression
Evaluate an Expression?
Maple can evaluate any mathematical expression and display the results inline or on a new line. Maple can also evaluate an expression at a point.
Evaluating an Expression Inline
Evaluating an Expression on a New Line
Evaluating at a Point
Follow the steps below to evaluate
\frac{2 x}{5}+\frac{3 x}{7}
\frac{2 x}{5}+\frac{3 x}{7}
. For instructions on entering this expression, see entering an expression.
Press Ctrl + = or Alt + Enter to evaluate the expression inline.
The inline evaluation mechanism, when used with Maple's Context Panel, is one of the fastest methods to perform a string of successive computations.
\frac{2 x}{5}+\frac{3 x}{7}
and display the result on a new line.
\frac{2 x}{5}+\frac{3 x}{7}
Press Enter. Maple displays and labels the result of an executed statement on a new line. By default, if this result is the first label in the document, it will be shown as (1). For more information on labels, see the Equation Labels subsection of the How Do I help page.
Equation labels organize the flow of computations. Keep these tips in mind when working with labels:
References can be made to any labeled result.
Labels are automatically numbered in sequence; if any labeled result is removed, then Maple automatically renumbers all equation labels and updates all equation label references.
The display format for labels can be changed. For more information, see the Equation Labels subsection of the How Do I help page.
Follow the steps to evaluate
\sqrt{{x}^{2}+\mathrm{\pi }}
x=2
\sqrt{{x}^{2}+\mathrm{\pi }}
. For instructions on entering this expression, see the How Do I topic on entering an expression.
Click on the expression. In the context panel select Evaluate at a Point.
A dialog box appears; enter
x=2
The substitution is made.
Note that the result was calculated in symbolic form. If you prefer an approximation, click on the result and, from the context panel, choose Approximate and the desired number of digits.
You can also use the "evaluate at a point" template from the Expression palette.
evaluation at a point
|
Bubble Sort | Brilliant Math & Science Wiki
Karleigh Moore, 刚 王, Jimin Khim, and
Bubble sort is a simple, inefficient sorting algorithm used to sort lists. It is generally one of the first algorithms taught in computer science courses because it is a good algorithm to learn to build intuition about sorting. While sorting is a simple concept, it is a basic principle used in complex computer programs such as file search, data compression, and path finding. Running time is an important thing to consider when selecting a sorting algorithm since efficiency is often thought of in terms of speed. Bubble sort has an average and worst-case running time of
O\big(n^2\big)
, so in most cases, a faster algorithm is more desirable. image1
A
n
orderable elements:
A[0,1,...,n-1]
A
B
B[0] \leq B[1] \leq \cdots \leq B[n-1].
<a_n>
i<j
a_i \leq a_j.
[a,b,c,d]
[1,2,3,4,5]
[5,4,3,2,1]
The Bubble sort algorithm compares each pair of elements in an array and swaps them if they are out of order until the entire array is sorted. For each element in the list, the algorithm compares every pair of elements.
The bubble sort algorithm is as follows:
A[0]
A[1]
A[0]
A[1]
, swap the elements.
Move to the next element,
A[1]
(which might now contain the result of a swap from the previous step), and compare it with
A[2]
A[1]
A[2]
, swap the elements. Do this for every pair of elements until the end of the list.
Do steps 1 and 2
n
The animation below illustrates bubble sort:
\hspace{4cm}
Sort the array
A=[7,3,1,4,2]
using the bubble sort algorithm. Show all of the steps that the algorithm takes.
The steps are summarized in the following table:
A = [4, 8,2,12,15,13,1] A = [4,8,2,12,13,1,15] A = [1,2,4,8,12,13,15] A = [2,12,4,8,1,15,13]
What would the following array look like after one iteration of bubble sort?
A =[12,4,8,2,15,13,1]
Here is pseudo-code describing the algorithm:
1 for i = a.length() to 1
2 for j = 1 to i-1
3 if a[j]>a[j+1]
4 swap(a[j],a[j+1]);
5 end if
First, it goes from
j=1
j=N-1
comparing each element of the list with the next
\big(
(j+1)^\text{th}
\big).
j^\text{th}
element is bigger than the next one, they change places, and so on. This way, in the first iteration, the element with the greatest value goes to the last position
(
i.e. goes to
\text{a[N]}).
Doing the same, in the second iteration of the loop,
j
j=1
j=N-2,
and the element of the second greatest value goes to one position before the last element
(
i.e. it goes to
\text{a[N-1]}).
The program does this process until the array is sorted.
The pseudo-code above sorts the list in an increasing order. What would you modify to make your program sort the elements in decreasing order?
You could simply change the third line of the pseudo-code: instead of using
\text{"if a[j]>a[j+1],"}
you should use
\text{"if a[j]<a[j+1]"}.
_\square
Here is one way to implement bubble sort in Python. There are other ways to implement the algorithm, but all implementations stem from the same ideas. Bubble sort can be used to sort any orderable list.
index = len(array) - 1
To calculate the complexity of the bubble sort algorithm, it is useful to determine how many comparisons each loop performs. For each element in the array, bubble sort does
n-1
comparisons. In big O notation, bubble sort performs
O(n)
comparisons. Because the array contains
n
elements, it has an
O(n)
number of elements. In other words, bubble sort performs
O(n)
operations on an
O(n)
number of elements, leading to a total running time of
O\big(n^2\big)
Another way to analyze the complexity of bubble sort is by determining the recurrence relation that represents it.
i=1,
no comparisons are made by the program. When
i=2,
one comparison is made by the program. When
i=3,
two comparisons are made, and so on. Thus, we can conclude that when
i=m,
m-1
comparisons are made. Hence, in an array of length
n,
it does
1+2+3+4+\cdots+(n-2)+(n-1)
\sum_{q=1}^{p} q = \frac{p(p+1)}{2}.
Using the previous formula to calculate
1+2+3+4+ \cdots +(n-2)+(n-1),
\frac{(n-1)(n-1+1)}{2}=\frac{n(n-1)}{2}.
O\big(n^2\big).
O(n)
is the best-case running time for bubble sort. It is possible to modify bubble sort to keep track of the number of swaps it performs. If an array is already in sorted order, and bubble sort makes no swaps, the algorithm can terminate after one pass. With this modification, if bubble sort encounters a list that is already sorted, it will finish in
O(n)
Though bubble sort is simple and easy to implement, it is highly impractical for solving most problems due to its slow running time. It has an average and worst-case running time of
O\big(n^2\big)
, and can only run in its best-case running time of
O(n)
when the input list is already sorted.
Bubble sort is a stable sort with a space complexity of
O(1)
Cite as: Bubble Sort. Brilliant.org. Retrieved from https://brilliant.org/wiki/bubble-sort/
|
Implement Bayesian Linear Regression - MATLAB & Simulink - MathWorks í•œêµ
Econometrics Toolbox™ includes a self-contained framework that allows you to implement Bayesian linear regression. The framework contains two groups of prior models for the regression coefficients β and the disturbance variance σ2:
Choose a joint prior distribution for (β,σ2). Then, using bayeslm, create the Bayesian linear regression model object that completely specifies your beliefs about the joint prior distribution. This table contains the available prior model objects.
Joint Prior Distribution of (β,σ2)
π(β|σ2) is Gaussian with mean Mu and covariance σ2V.
π(σ2) is inverse gamma with shape A and scale B.
You are fairly confident that the parameters have the corresponding joint prior, and that β depends on σ2.
You want to incorporate your prior knowledge of the prior mean and covariance of β and the shape and scale of σ2.
π(β) is Gaussian with mean Mu and covariance V.
You are fairly confident that the parameters have the corresponding joint prior, and that β and σ2 are independent.
\mathrm{Ï}\left(\mathrm{β},{\mathrm{Ï}}^{2}\right)â\frac{1}{{\mathrm{Ï}}^{2}}.
The joint prior distribution is inversely proportional to σ2 (Jeffreys noninformative prior [2]).
Estimates of the mean and covariance matrix of the marginal posterior π(β|y,x) and the mean and variance of π(σ2|y,x).
Estimate the mean and covariance of the conditional distribution π(β|σ2,y,x), that is, implement linear regression with σ2 held fixed.
Approximate the expected value of a function of the parameters with respect to the joint posterior π(β,σ2|y,x). That is, draw multiple samples of (β,σ2) from their joint posterior, apply a function to each draw, and then compute the average of the transformed draws.
Draw from the conditional posterior distributions π(β|σ2,y,x) and π(σ2|β,y,x). This selection is convenient for running a Markov chain Monte Carlo (MCMC) sampler, such as a Gibbs sampler.
Econometrics Toolbox supports two Bayesian predictor selection algorithms: Bayesian lasso regression and SSVS. Choose a predictor selection algorithm, which implies a joint prior distribution for (β,σ2). Then, using bayeslm, create the Bayesian linear regression prior model object that performs the selected predictor selection algorithm, and optionally specify the tuning parameter value. This table contains the available prior model objects for predictor selection. For details on the forms of the prior distributions, see Posterior Estimation and Inference.
The prior variance of β is a function of σ2.
β and σ2 are independent, a priori.
λ, specified by the 'Lambda' name-value pair argument. You can supply a positive scalar or vector.
Larger values indicate more regularization, which implies that the prior of β is dense around zero.
For lasso prior models, determine a regularization path, that is, a series of values for λ to iterate through during posterior estimation. Values are data dependent.
For more on specifying λ, see [3].
Estimates of the mean (Mean) and covariance matrix (Covariances) of the marginal posterior π(β|y,x), and the mean and variance of π(σ2|y,x).
|
Wideband signal collector - MATLAB - MathWorks í•œêµ
Arrival directions of signals, specified as a real-valued 2-by-L matrix. Each column specifies an arrival direction in the form [AzimuthAngle;ElevationAngle]. The azimuth angle must lie between –180° and 180°, inclusive. The elevation angle must lie between –90° and 90°, inclusive. When the Wavefront property is false, the number of angles must equal the number of array elements, N. Units are in degrees.
Subarray steering angle, specified as a length-2 column vector. The vector has the form [azimuthAngle;elevationAngle]. The azimuth angle must be between –180° and 180°, inclusive. The elevation angle must be between –90° and 90°, inclusive. Units are in degrees.
Use the phased.WidebandCollector System object™ to construct a signal arriving at a single isotropic antenna from 10° azimuth and 30° elevation.
{f}_{m}=\left\{\begin{array}{c}{f}_{c}â\frac{{f}_{s}}{2}+\left(mâ1\right)\mathrm{Î}f\text{, }{N}_{B}\text{ even}\\ {f}_{c}â\frac{\left({N}_{B}â1\right){f}_{s}}{2{N}_{B}}+\left(mâ1\right)\mathrm{Î}f\text{, }{N}_{B}\text{ odd}\end{array},\text{ }m=1,â¦,{N}_{B}
|
Cioabă, Sebastian M.1; Koolen, Jack H.2; Nozaki, Hiroshi3
1 University of Delaware Department of Mathematical Sciences Ewing Hall Newark, DE 19716-2553, USA
2 School of Mathematical Sciences University of Science and Technology of China Wen-Tsun Wu Key Laboratory of the Chinese Academy of Sciences Hefei, Anhui, China
3 Aichi University of Education 1 Hirosawa, Igaya-cho, Kariya Aichi 448-8542, Japan
b\left(k,\theta \right)
be the maximum order of a connected bipartite
k
-regular graph whose second largest eigenvalue is at most
\theta
. In this paper, we obtain a general upper bound for
b\left(k,\theta \right)
0\le \theta <2\sqrt{k-1}
. Our bound gives the exact value of
b\left(k,\theta \right)
whenever there exists a bipartite distance-regular graph of degree
k
, second largest eigenvalue
\theta
d
and girth
g
g\ge 2d-2
. For certain values of
d
, there are infinitely many such graphs of various valencies
k
d=11
d\ge 15
, we prove that there are no bipartite distance-regular graphs with
g\ge 2d-2
Classification: 05B25, 05C35, 05C50, 05E30, 42C05, 94B65
Keywords: second eigenvalue, bipartite regular graph, bipartite distance-regular graph, expander, linear programming bound.
Cioabă, Sebastian M. 1; Koolen, Jack H. 2; Nozaki, Hiroshi 3
author = {Cioab\u{a}, Sebastian M. and Koolen, Jack H. and Nozaki, Hiroshi},
title = {A spectral version of the {Moore} problem for bipartite regular graphs},
TI - A spectral version of the Moore problem for bipartite regular graphs
%T A spectral version of the Moore problem for bipartite regular graphs
Cioabă, Sebastian M.; Koolen, Jack H.; Nozaki, Hiroshi. A spectral version of the Moore problem for bipartite regular graphs. Algebraic Combinatorics, Volume 2 (2019) no. 6, pp. 1219-1238. doi : 10.5802/alco.71. https://alco.centre-mersenne.org/articles/10.5802/alco.71/
[1] Abiad, Aida; van Dam, Edwin R.; Fiol, Miquel Ángel Some spectral and quasi-spectral characterizations of distance-regular graphs, J. Combin. Theory Ser. A, Volume 143 (2016), pp. 1-18 | Article | MR: 3519812 | Zbl: 1342.05031
[2] Alon, Noga Eigenvalues and expanders, Combinatorica, Volume 6 (1986) no. 2, pp. 83-96 Theory of computing (Singer Island, Fla., 1984) | Article | MR: 875835
[3] Alon, Noga; Milman, Vitali D.
{\lambda }_{1},
isoperimetric inequalities for graphs, and superconcentrators, J. Combin. Theory Ser. B, Volume 38 (1985) no. 1, pp. 73-88 | Article | MR: 782626
[4] Bannai, Eiichi; Ito, Tatsuro On the spectra of certain distance-regular graphs, J. Combin. Theory Ser. B, Volume 27 (1979) no. 3, pp. 274-293 | Article | MR: 554295 | Zbl: 0427.15005
[5] Bannai, Eiichi; Ito, Tatsuro On the spectra of certain distance-regular graphs. II, Quart. J. Math. Oxford Ser. (2), Volume 32 (1981) no. 4, pp. 389-411 | Article | MR: 635588 | Zbl: 0475.05059
[6] Bannai, Eiichi; Ito, Tatsuro Algebraic combinatorics. I: Association schemes, Mathematics lecture note series, The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1984, xxiv+425 pages | MR: 882540 | Zbl: 0555.05019
[9] Chung, Fan R. K. Diameters and eigenvalues, J. Amer. Math. Soc., Volume 2 (1989) no. 2, pp. 187-196 | Article | MR: 965008 | Zbl: 0678.05037
[10] Cioabă, Sebastian M.; Koolen, Jack H.; Nozaki, Hiroshi; Vermette, Jason R. Maximizing the order of a regular graph of given valency and second eigenvalue, SIAM J. Discrete Math., Volume 30 (2016) no. 3, pp. 1509-1525 | Article | MR: 3537002 | Zbl: 1344.05086
[11] Cohn, Henry; Kumar, Abhinav Universally optimal distribution of points on spheres, J. Amer. Math. Soc., Volume 20 (2007) no. 1, pp. 99-148 | Article | MR: 2257398 | Zbl: 1198.52009
[12] Damerell, Robert Mark; Georgiacodis, Michael A. On the maximum diameter of a class of distance-regular graphs, Bull. London Math. Soc., Volume 13 (1981) no. 4, pp. 316-322 | Article | MR: 620044 | Zbl: 0457.05055
[13] Davidoff, Giuliana; Sarnak, Peter; Valette, Alain Elementary number theory, group theory, and Ramanujan graphs, London Mathematical Society Student Texts, 55, Cambridge University Press, Cambridge, 2003, x+144 pages | Article | MR: 1989434 | Zbl: 1032.11001
[14] Fuglister, Frederick J. On generalized Moore geometries. I, Discrete Math., Volume 67 (1987) no. 3, pp. 249-258 | Article | MR: 915950 | Zbl: 0626.51006
[15] Fuglister, Frederick J. On generalized Moore geometries. II, Discrete Math., Volume 67 (1987) no. 3, pp. 259-269 | Article | MR: 915950 | Zbl: 0626.51006
[16] Geronimus, James On a set of polynomials, Ann. of Math. (2), Volume 31 (1930) no. 4, pp. 681-686 | Article | MR: 1502972
[17] Harris, Lawrence A. Lagrange polynomials, reproducing kernels and cubature in two dimensions, J. Approx. Theory, Volume 195 (2015), pp. 43-56 | Article | MR: 3339053 | Zbl: 1314.41023
[18] Høholdt, Tom; Janwa, Heeralal Eigenvalues and expansion of bipartite graphs, Des. Codes Cryptogr., Volume 65 (2012) no. 3, pp. 259-273 | Article | MR: 2988185 | Zbl: 1254.05103
[19] Høholdt, Tom; Justesen, Jørn On the sizes of expander graphs and minimum distances of graph codes, Discrete Math., Volume 325 (2014), pp. 38-46 | Article | MR: 3181231 | Zbl: 1288.05070
[20] Hoory, Shlomo; Linial, Nathan; Wigderson, Avi Expander graphs and their applications, Bull. Amer. Math. Soc. (N.S.), Volume 43 (2006) no. 4, pp. 439-561 | Article | MR: 2247919 | Zbl: 1147.68608
[21] Hora, Akihito; Obata, Nobuaki Quantum probability and spectral analysis of graphs, Theoretical and Mathematical Physics, Springer, Berlin, 2007, xviii+371 pages (With a foreword by Luigi Accardi) | MR: 2316893 | Zbl: 1141.81005
[22] Koledin, Tamara; Stanić, Zoran Regular graphs with small second largest eigenvalue, Appl. Anal. Discrete Math., Volume 7 (2013) no. 2, pp. 235-249 | Article | MR: 3135926 | Zbl: 1313.05229
[23] Koledin, Tamara; Stanić, Zoran Reflexive bipartite regular graphs, Linear Algebra Appl., Volume 442 (2014), pp. 145-155 | Article | MR: 3134359 | Zbl: 1282.05125
[24] Li, Wen-Ch’ing Winnie; Solé, Patrick Spectra of regular graphs and hypergraphs and orthogonal polynomials, European J. Combin., Volume 17 (1996) no. 5, pp. 461-477 | Article | MR: 1397154 | Zbl: 0864.05072
[25] Marcus, Adam W.; Spielman, Daniel A.; Srivastava, Nikhil Interlacing families I: Bipartite Ramanujan graphs of all degrees, Ann. of Math. (2), Volume 182 (2015) no. 1, pp. 307-325 | Article | MR: 3374962 | Zbl: 1316.05066
[26] Miller, Mirka; Sirán, Jozef Moore graphs and beyond: A survey of the degree/diameter problem, Electron. J. Combin. (2005), Paper no. DS14–Dec, 61 pages | Zbl: 1079.05043
[27] Nozaki, Hiroshi Linear programming bounds for regular graphs, Graphs Combin., Volume 31 (2015) no. 6, pp. 1973-1984 | Article | MR: 3417208 | Zbl: 1332.90160
[28] Singleton, Robert On minimal graphs of maximum even girth, J. Combinatorial Theory, Volume 1 (1966) no. 3, pp. 306-332 | Article | MR: 0201347 | Zbl: 0168.44703
[29] Teranishi, Yasuo; Yasuno, Fumiko The second largest eigenvalues of regular bipartite graphs, Kyushu J. Math., Volume 54 (2000) no. 1, pp. 39-54 | Article | MR: 1762790 | Zbl: 0990.05095
|
Topic: Resistance and Ohm's Law
In 1827, George Ohm demonstrated through a series of experiments that voltage, current, and resistance are related through a fundamental relationship: Voltage
(V)
is equal to Current
(I)
times resistance
(R)
V=I×R
. This most basic equation in electronics shows that when any two of the three quantities are known, the third can be derived. The Ohm’s law triangle in Fig. 1 below shows how to derive each variable given the other two. An important thing to note about Ohm’s law is that a resistor’s voltage and current are related by its resistance.
Figure 1. Ohm’s Law triangle
Resistance is measured in ohms, represented by the symbol Omega (Ω). According to Ohm’s law, one volt impressed across 1 ohm of resistance will cause 1 amp of current to flow. Similarly, 3.3V impressed across 3.3Ω will cause 1A of current to flow.
Figure 2. Example Schematic
In Fig. 2, the lines leaving the positive and negative sides of the power supply represent conductors with an insignificant amount of resistance. Thus, the voltage delivered by the power supply is present at both sides of the resistor - 3.3V at the left side of the resistor, and 0V (GND) and the right side of the resistor. As current flows through the resistor, collisions occur between the electrons flowing from the power supply and the materials in the resistor. These collisions cause electrons to give up their potential energy, and that energy is dissipated as heat. As with any physical system, we define the time derivative of energy as power; in electric circuits, power (measured in Watts) is defined as voltage times current, or
P=V×I
. The power transferred to the resistor at any given time results in resistor heating. The more power transferred to the resistor, the hotter it gets. For a given voltage, a smaller-valued resistor would allow more current to flow (see Ohm’s law), and therefore more energy would be dissipated as heat. The total energy consumed in an electric circuit is simply the time integral of power, measured in Watts per second, or Joules. Thus, in the circuit above, the electric power delivered to the resistor is
P=3.3V×1A
, or 3.3Watts, and in one second,
3.3W×1second
or 3.3J of energy is dissipated. Many devices can provide only limited current; if your circuit draws too much current from the device, it can malfunction. Increasing the resistance in your circuit may solve this problem.
Ohm’s law states that 1 volt impressed upon 1 ohm of resistance will cause 1 amp of current to flow.
In circuits, power is measured in Watts, which is measured in voltage times current.
The power of an electric circuit is measured in Watts, or Joules per Second.
|
RemoveNonNumeric - Maple Help
Home : Support : Online Help : Statistics and Data Analysis : Statistics Package : Data Manipulation : RemoveNonNumeric
SelectNonNumeric(X, options)
RemoveNonNumeric(X, options)
(optional) equation of the form exclude=value
The SelectNonNumeric function selects non-numeric items from X and returns them in a new data sample. Note, that an expression X is considered non-numeric if
\mathrm{evalf}\left(X\right)
does not return a floating-point number. Numeric items are discarded in the newly created data sample.
The RemoveNonNumeric function does the opposite of SelectNonNumeric. It removes the non-numeric values from X.
exclude=infinity, undefined, or [infinity, undefined] -- By default, undefined and infinity are considered non-numeric. This can be changed by adding infinity or undefined or both to the exclude list. Thus, exclude=undefined means that infinity should still be considered numeric but undefined should not.
\mathrm{with}\left(\mathrm{Statistics}\right):
A≔\mathrm{Array}\left([3,3,1,a,b,a,\mathrm{\pi },\mathrm{sin}\left(1\right),\mathrm{undefined},\mathrm{\infty }]\right)
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cccccccccc}\textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{a}& \textcolor[rgb]{0,0,1}{b}& \textcolor[rgb]{0,0,1}{a}& \textcolor[rgb]{0,0,1}{\mathrm{\pi }}& \textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\right)& \textcolor[rgb]{0,0,1}{\mathrm{undefined}}& \textcolor[rgb]{0,0,1}{\mathrm{\infty }}\end{array}]
\mathrm{RemoveNonNumeric}\left(A\right)
[\begin{array}{ccccc}\textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{\mathrm{\pi }}& \textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\right)\end{array}]
\mathrm{SelectNonNumeric}\left(A,\mathrm{exclude}=[\mathrm{undefined}]\right)
[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{a}& \textcolor[rgb]{0,0,1}{b}& \textcolor[rgb]{0,0,1}{a}& \textcolor[rgb]{0,0,1}{\mathrm{\infty }}\end{array}]
\mathrm{RemoveNonNumeric}\left(A,\mathrm{exclude}=[\mathrm{undefined}]\right)
[\begin{array}{cccccc}\textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{\mathrm{\pi }}& \textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\right)& \textcolor[rgb]{0,0,1}{\mathrm{undefined}}\end{array}]
\mathrm{SelectNonNumeric}\left(A\right)
[\begin{array}{ccccc}\textcolor[rgb]{0,0,1}{a}& \textcolor[rgb]{0,0,1}{b}& \textcolor[rgb]{0,0,1}{a}& \textcolor[rgb]{0,0,1}{\mathrm{undefined}}& \textcolor[rgb]{0,0,1}{\mathrm{\infty }}\end{array}]
B≔\mathrm{Array}\left([1,2,3,4,\mathrm{undefined},\mathrm{\infty }],\mathrm{datatype}=\mathrm{float}[8]\right)
\textcolor[rgb]{0,0,1}{B}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cccccc}\textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{2.}& \textcolor[rgb]{0,0,1}{3.}& \textcolor[rgb]{0,0,1}{4.}& \textcolor[rgb]{0,0,1}{Float}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{undefined}}\right)& \textcolor[rgb]{0,0,1}{Float}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{\infty }}\right)\end{array}]
\mathrm{RemoveNonNumeric}\left(B\right)
[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{2.}& \textcolor[rgb]{0,0,1}{3.}& \textcolor[rgb]{0,0,1}{4.}\end{array}]
\mathrm{RemoveNonNumeric}\left(B,\mathrm{exclude}=[\mathrm{\infty }]\right)
[\begin{array}{ccccc}\textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{2.}& \textcolor[rgb]{0,0,1}{3.}& \textcolor[rgb]{0,0,1}{4.}& \textcolor[rgb]{0,0,1}{Float}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{\infty }}\right)\end{array}]
|
Rosenzweig – Pegasus Power
In Physics class, when considering power, the minds of the clever brony might be graced by the question, “How much power did Rainbow Dash produce when performing her famous Sonic Rainboom?”
Power done by a force can be defined as the work done by that force divided by the time the force acted.
P_a = \frac{W_a}{\Delta t}
In the case of Rainbow Dash’s sonic rainboom, from season 1, episode 16 of My Little Pony: Friendship is Magic, we are specifically interested in the power exerted by Rainbow Dash herself. However, that is not the net work done from her flight; when flying in a nosedive, she is also under the influence of gravity and air resistance. Magic horses don’t exist in a vacuum, you know! Thus, where
W_a
is Rainbow Dash’s work,
W_g
is the work from gravity, and
W_r
is the work from air resistance:
W_{net} = W_a + W_g + W_r
By the work-energy theorem, we know that this net work is equal to the change in Rainbow Dash’s kinetic energy:
W_{net} = \Delta \mathrm{KE} = \frac{1}{2} m (v_f^2 - v_i^2)
However, it is prudent to consider the sapphiric horse’s exact flight pattern. In particular, after ascending to a massive height, coming to a brief halt, she flies directly down. That is, we can begin measuring her path from the beginning of her descent. This has two major consequences for the velocities. First,
v_f
is equal to the speed of sound, mach-1. We’ll come back to this fact later. Second, her initial velocity
v_i
is zero as she begins from rest. Rewriting, we see:
W_a + W_g + W_r = \frac{1}{2} m v_f^2
Next, we consider the work done by gravity,
W_g
. A usual high school textbook tells you that this work is related to Rainbow Dash’s mass, the height from which she flies, and the Equestrian gravitational constant
g_E
W_g = m g_E h
Further, basic kinematics tells us that this height
h
is also related to time as
\frac{1}{2} g_E (\Delta t)^2
Again rewriting, we find:
W_g = \frac{1}{2} m g_E^2 (\Delta t)^2
Finally, to find the net work, we need to compute air resistance. As Francis Sparkle from Friendship is Witchcraft episode “Foaly Matripony” would tell you, computing air resistance of a magical horse is gnarly – specifically, “the bad kind of gnarly, when things are gnarled”!
That said, even without computing such a gnarly number, we know that the sign of the work done by air resistance must be negative. (Proving this is trivial and left as an exercise for the reader.) Thus, the presence of air resistance can only increase the work done by our poor little Dashie – which means, critically, that setting it to zero and proceeding will yield the theoretical lower-bound on her work done, and therefore the lower-bound on her power output.
So, we can now solve for the lower bound on Rainbow Dash’s work
W_a
W_a = \frac{1}{2} m v_f^2 - \frac{1}{2} m g_E^2 (\Delta t)^2
We can substitute this back in to the definition of power:
P_a = \frac{1}{\Delta t} (\frac{1}{2} m v_f^2 - \frac{1}{2} m g_E^2 (\Delta t)^2)
Now, it is a matter of estimating or computing values for
\Delta t
m
v_f
g_E
\Delta t
is the easiest to compute, under the assumption that the time shown on camera to the viewer is identical to the real-Equestrian-world time. Pausing the video when Rainbow Dash turns direction at the top and when the Sonic Rainboom first goes boom, we see that the rainboom takes about twenty seconds (plus or minus one second) to be performed. Thus,
\Delta t = 20 \; \text{s}
m
is up next. This is perhaps one of the most difficult terms to estimate, for the simple reason that there is no clear unit of weight shown within canon. Are ponies teensy, like their namesake show suggests, with masses around 30 kilograms as an adult? Are they like human-world horses, clocking in closer to 300 kilograms? Who knows? We won’t be addressing this question in this paper; instead, defer to this analysis by another Equestrian scientist, who estimates a pegasus (as the lightest race) might weigh around 150 pounds, or about 70 kilograms. Given that there is no better reference available, we may as well set
m
70
. The real number, unfortunately, will vary wildly.
v_f
is trivial, under the crucial assumption that an Equestrian rainboom truly does occur when breaking the speed of sound, mach-1, the dominant theory among contemporary Equestrian physicists. Thus, consulting a standard physics text – okay, okay, I used Wikifoalia – yields that
v_f
343
g_E
is puzzling. One option might be to project human notions of gravity onto the planet of Equus, setting
g_E
9.8
meters per second per second. However, it is prudent to remember the derivation of this constant (from Neighton’s Law of Universal Gravitation) depends, crucially, on the mass of the planet and the elevation. Elevation should not be a huge issue here; while Dashie does fly pie in the sky (as her friend Pinkamena might put it), the difference should be negligible. Further, although the rainboom in question does occur in the sky (in the floating city of Cloudsdale), again there should be negligible difference. The real issue is the mass of the planet – there is no reason to believe that Equus has the same mass as Earth. Indeed, prior to the revelations of My Little Pony: the Movie, there was little reason to believe there was much of anything outside of the small sovereign nation of Equestria. Still, even with the new knowledge of the outside world and refined cartography, it appears likely that the Equus is somewhat less massive than Earth; therefore, like when ignoring air resistance, we can set
g_E
to Earth’s gravitational constant to reveal a more or less decent lower-bound on Rainbow Dash’s power. Indeed, given the new revelations, perhaps coupled with some long-time fanon and integration from the Equestria Girls parallel universe, it is possible that Equus could be nearly identical to Earth in size. In any case, we let
g_E
9.8 \; \frac{\text{m}}{\text{s}^2}
From here, we have enough information to compute a lower bound on Rainbow Dash’s power output. Substituting into our equation for Rainbow’s power, we find:
P_a = \frac{1}{\Delta t} (\frac{1}{2} m v_f^2 - \frac{1}{2} m g_E^2 (\Delta t)^2
= 138657.75 \; \text{W}
= 140 \; \text{kW}
However, like air resistance computation, the kilowatt is a gnarly unit when considering Rainbow Dash’s power. Instead, let’s use the conversion factor of 735.5 watts, to finally yield that Rainbow Dash, one horse, produces…
Special shout out to “My Little Pony Physics Presentation” which examines this scene from another light.
|
Cryptography/One time pads - Wikibooks, open books for an open world
A One Time Pad (OTP) is the only potentially unbreakable encryption method. Plain text encrypted using an OTP cannot be retrieved without the encrypting key. However, there are several key conditions that must be met by the user of a one time pad cipher, or the cipher can be compromised.
The key must be random and generated by a non-deterministic, non-repeatable process. Any key generated by an algorithm will not work. The security of the OTP relies on the randomness of the key. Unfortunately, the randomness of a key cannot be proved.
The key must never be reused. Use of the same key to encrypt different messages, no matter how trivially small, compromises the cipher.
The key must not fall in the hands of the enemy. This may seem obvious, but it points to the weakness of system in that you must be able to transmit large amounts of data to the reader of the pad. Typically, one time pad cipher keys are sent via diplomatic pouch.
A typical one time pad system works like this: Generate a long fresh new random key. XOR the plaintext with the key to create the ciphertext. To decrypt the ciphertext, XOR it with the original key. The system as presented is thus a symmetric and reciprocal cipher. Other functions (e.g., addition modulo n) could be used to combine the key and the plaintext to yield the ciphertext, although the resulting system may not be a reciprocal cipher.
If the key is random and never re-used, an OTP is provably unbreakable. Any ciphertext can be decrypted to any message of the same length by using the appropriate key. Thus, the actual original message cannot be determined from ciphertext alone, as all possible plaintexts are equally likely. This is the only cryptosystem for which such a proof is known.
The OTP is extremely simple to implement.[1]
However, there are limitations. Re-use the key and the system becomes extremely weak; it can be broken with pencil and paper. Try to build a "one-time-pad" using some algorithm to generate the keys and you don't have a one-time-pad, you have a stream cipher. There are some very secure stream ciphers, but people who do not know one from a one-time pad are probably not able to design one. It is unfortunately fairly common to see weak stream ciphers advertised as unbreakable one-time pads.
Also, even if you have a well-implemented OTP system and your key is kept secure, consider an attacker who knows the plaintext of part of a message. He can then recover that part of the key and use it to encrypt a message of his own. If he can deliver that instead of yours, you are in deep trouble.
2 Making one-time pads by hand
2.1 letter tiles
2.2 10-sided dice
2.3 6-sided dice
First, an OTP is selected for the plaintext:
Preshared Random Bits = 1010010010101010111010010000101011110101001110100011
Plain text = 110101010101010010100
Length(Plain Text) = 21
Key(21) = 101001001010101011101
Wikipedia has related information at Padding (cryptography)
The example indicates that the plaintext is not always the same length as the key material. This can be handled by methods such as:
appending a terminator to the plaintext before encryption, and terminating the cyphertext with random bits.
prepending the length and a preamble terminator to the plaintext, and terminating with random bits.
Such signaling systems (and possibly the plaintext encoding method) must be designed so that these terminators are not mistaken for plaintext. For this example, therefore, it is assumed the plaintext already contains endpoint/length signaling.
For increasingly long plaintext/key pair lengths, the cross-correlation gets closer to zero.
Key(21) = 101001001010101011101
Plaintext = 110101010101010010100
{\displaystyle \oplus }
bitwise |||||||||||||||||||||
cyphertext = 011100011111111001001
For increasingly long plaintext/cyphertext pair lengths, the cross-correlation also gets closer to zero.
DecryptionEdit
{\displaystyle \oplus }
An astute reader might observe that the decryptor needs to know the length of the plaintext in actual practice. This is done by decrypting the cyphertext as a bitstream (i.e. xor each bit as it is read), and observing the stream until the end-of-plaintext ruleset is satisfied by the signals prepended/appended to the plaintext.
Making one-time pads by handEdit
A full English-language Scrabble tile set. See Scrabble letter distributions for other languages.
One-time pads were originally made without the use of a computer and this is still possible today. The process can be tedious, but if done correctly and the pad used only once, the result is unbreakable.
There are two components needed to make a one-time pad: a way to generate letters at random and a way to record two copies of the result. The traditional way to do the latter was to use a w:typewriter and w:carbon paper. The carbon paper and w:typewriter ribbon would then be destroyed since it is often possible to recover the pad data from them. As typewriters have become scarce, it is also acceptable to hand write the letters neatly in groups of five on two part w:carbonless copy paper sheets, which can be purchased at office supply stores. Each sheet can given a serial number or some other unique marking.
Historically, the key material for manual one-time pads was distributed as a pad of many small pages of paper. Each small page typically had a series of 5-digit groups, each digit randomly selected from 0 to 9.[2][3][4][5][6][7][8][9]
A one-time pad set consists of two identical pads. Some writers refer to the two as "two identical originals", to emphasize that no copies should ever be made of the key material.[10]
Traditionally two-way communication requires two pad sets (a total of 4 pads): One person gets the "IN" pad of one set, and the "OUT" pad of the other set.[11]
Each small page typically contains 50 groups of 5 random decimal digits 0...9, enough for one normal message, and a unique "page number" of five digits.[11][12]
A conversion table is used to convert the letters of the plaintext message to numbers, and the numbers of the decoded message back to letters.[5] Perhaps the simplest conversion table is A=01, B=01, ... Z=26, but historically some sort of straddling checkerboard was usually used, such as CT-37c,[13] CT-37w, CT-46,[14] etc.[15]
The key material for a one-time pad was sometimes written as 50 groups of 5 random letters A...Z.[12][16] One-time pads where the keys are written as letters are sometimes called letter one-time pad (LOP)[17][18] or one-time letter pad (OTLP).[11]
The key material for cryptographic machines, including one-time pad systems, was often punched in a binary code on long, narrow paper tape—a "one time tape" OTT.[10][12][19]
letter tilesEdit
The simplest way to generate random letters in the Roman alphabet is to obtain 26 identical objects with a different letter of the alphabet marked on each object. Tiles from the game w:Scrabble can be used, as long as only one of each letter is selected. Kits for making name charm bracelets are another possibility. One can also write the letters on 26 otherwise identical coins with a marking pen. The objects are placed in a box or cup and shaken vigorously, then one object is withdrawn and its letter is recorded. The object is returned to the box and the process is repeated.
10-sided diceEdit
Another way to make one time pads is to use w:ten-sided dice. One can generate random number groups by rolling several ten-sided dice at a time and recording a group of decimal digits—one decimal digit from each die—for each roll.[11] This method will generate random code groups much faster than using Scrabble tiles. The plaintext message is converted into numeric values with A =01, B =02 and so on. The resulting numeric values are encrypted by adding digits from the one time pads using non-carrying addition. One can then either transmit the numeric groups as is, or use the straddling checkerboard to convert the numbers back into letters and transmit that result.
6-sided diceEdit
Another way to make one time pads is to use 6-sided dice.[20]
It is possible to generate random decimal digits (to make a traditional decimal one-time pad) using 6-sided dice.[11]
If the message is converted into two digit base-6 numbers, then ordinary six-sided dice can be used to generate the random digits in a one time pad. Digits in the pad would be added modulo-6 to the digits in the plaintext message (again without carry), and subtracted modulo 6 from the ciphertext to decrypt. For example:
Table for converting messages to base-6
PT 0 1 2 3 4 5 6 7 8 9
PT A B C D E F G H I J K L M
CT 14 15 20 21 22 23 24 25 30 31 32 33 34
PT N O P Q R S T U V W X Y Z
1x 6 7 8 9 A B
2x C D E F G H
3x I J K L M N
4x O P Q R S T
5x U V W X Y Z
Using this table, "Wikipedia" would convert to 52 30 32 30 41 22 21 30 14. If the pad digits were 42 26 21 35 32 34 22 62 43, the ciphertext would be 34 50 53 05 13 50 43 32 51. (Note that 6 = 0 modulo 6).
Key ExchangeEdit
In order to use this algorithm, each party must possess the same random key. This typically involves meeting the other party in person or using a trusted courier. Other methods are sometime proposed, such as or both users to have identical devices that generate the same semi-random numbers, however these methods are essentially w:stream ciphers and are not covered by the security proof of one time pads.
↑ Infoanarchy wiki: One-Time Pad Cryptosystem (mirror: [1])
↑ Dirk Rijmenants. "Manual One-time pads".
↑ Marcus J. Ranum. "One-Time-Pad (Vernam's Cipher) Frequently Asked Questions".
↑ "One Time Pads : Cold War Coding."
↑ a b Anonymous PI. "Nothing To See Here: The One-Time Pad".
↑ "One-time pad generator"
↑ "The Artifacts of the CIA"
↑ Hal Abelson, Ken Ledeen, Harry Lewis. "Secret Bits: How Codes Became Unbreakable": "Historical Cryptography".
↑ "The Manual One-time Pad"
↑ a b Crypto Museum: "EROLET Key-tape generator"
↑ a b c d e Dirk Rijmenants. "The Manual One-time Pad"
↑ a b c Crypto Museum: "One-Time Pad (OTP)".
↑ "Cryptographilia" describes CT-37c.
↑ "onetimepad"
↑ "Checkerboard Variations"
↑ "Raspberry Pi Thermal Printer One Time Pads"
↑ Stephen Hewitt. "Manual encryption with a one-time pad revisited". 2019.
↑ Christos T. "SOE codes and Referat Vauck". 2013.
↑ Crypto Museum: "Mixer machines: One-Time Tape machines".
↑ David Shaw. Maths by Email. "Try this: One time pad".
Cryptodox: one time pads
Wikipedia: one-time pad
"Visual Cryptography" typically uses a one-time pad. A special arrangement of polarizing filters can be used to implement XOR.
"What is Visual Cryptography" uses a binary one-time pad. A special pattern of dots can be used to implement XOR.
Retrieved from "https://en.wikibooks.org/w/index.php?title=Cryptography/One_time_pads&oldid=3809530"
|
Inclined orbit - WikiMili, The Free Encyclopedia
Find sources: "Inclined orbit" – news · newspapers · books · scholar · JSTOR (November 2015) (Learn how and when to remove this template message)
Earth is the third planet from the Sun and the only astronomical object known to harbor life. According to radiometric dating and other sources of evidence, Earth formed over 4.5 billion years ago. Earth's gravity interacts with other objects in space, especially the Sun and the Moon, Earth's only natural satellite. Earth orbits around the Sun in 365.26 days, a period known as an Earth year. During this time, Earth rotates about its axis about 366.26 times.
A geosynchronous orbit is an inclined orbit with an altitude of 37,000 km (23,000 mi) that completes one revolution every sidereal day tracing out a small figure-eight shape in the sky. [1] A geostationary orbit is a special case of geosynchronous orbit with no inclination, and therefore no apparent movement across the sky from a fixed observation point on the Earth's surface.
In astronomy, an analemma is a diagram showing the position of the Sun in the sky, as seen from a fixed location on Earth at the same mean solar time, as that position varies over the course of a year. The diagram will resemble the figure 8. Globes of Earth often display an analemma.
A geostationary orbit, often referred to as a geosynchronous equatorial orbit (GEO), is a circular geosynchronous orbit 35,786 km (22,236 mi) above Earth's equator and following the direction of Earth's rotation. An object in such an orbit appears motionless, at a fixed position in the sky, to ground observers. Communications satellites and weather satellites are often placed in geostationary orbits, so that the satellite antennas that communicate with them do not have to rotate to track them, but can be pointed permanently at the position in the sky where the satellites are located. Using this characteristic, ocean-color monitoring satellites with visible and near-infrared light sensors can also be operated in geostationary orbit in order to monitor sensitive changes of ocean environments.
In astrodynamics, the orbital maneuvers made by thruster burns that are needed to keep a spacecraft in a particular assigned orbit are called orbital station-keeping.
{\displaystyle \cos(i)\approx -\left({\frac {T}{3.795{\text{ hr}}}}\right)^{\frac {7}{3}}}
{\displaystyle i}s the orbital inclination, and
{\displaystyle T}
is the oribital period.
A geosynchronous orbit is an orbit around Earth of a satellite with an orbital period that matches Earth's rotation on its axis, which takes one sidereal day. The synchronization of rotation and orbital period means that, for an observer on Earth's surface, an object in geosynchronous orbit returns to exactly the same position in the sky after a period of one sidereal day. Over the course of a day, the object's position in the sky may remain still or trace out a path, typically in a figure-8 form, whose precise characteristics depend on the orbit's inclination and eccentricity. Satellites are typically launched in an eastward direction. A circular geosynchronous orbit is 35,786 km (22,236 mi) above Earth's surface. Those closer to Earth orbit faster than Earth rotates, so from Earth, they appear to move eastward while those that orbit beyond geosynchronous distances appear to move westward.
A geosynchronous transfer orbit or geostationary transfer orbit (GTO) is a Hohmann transfer orbit—an elliptical orbit used to transfer between two circular orbits of different radii in the same plane—used to reach geosynchronous or geostationary orbit using high-thrust chemical engines.
A geocentric orbit or Earth orbit involves any object orbiting Planet Earth, such as the Moon or artificial satellites. In 1997 NASA estimated there were approximately 2,465 artificial satellite payloads orbiting the Earth and 6,216 pieces of space debris as tracked by the Goddard Space Flight Center. Over 16,291 previously launched objects have decayed into the Earth's atmosphere.
A Sun-synchronous orbit is a nearly polar orbit around a planet, in which the satellite passes over any given point of the planet's surface at the same local mean solar time. More technically, it is an orbit arranged so that it precesses through one complete revolution each year, so it always maintains the same relationship with the Sun.
The orbital plane of a revolving body is the geometric plane in which its orbit lies. Three non-collinear points in space suffice to determine an orbital plane. A common example would be the positions of the centers of a massive body (host) and of an orbiting celestial body at two different times/points of its orbit.
A Tundra orbit is a highly elliptical geosynchronous orbit with a high inclination, an orbital period of one sidereal day, and a typical eccentricity between 0.2 and 0.3. A satellite placed in this orbit spends most of its time over a chosen area of the Earth, a phenomenon known as apogee dwell, which makes them particularly well suited for communications satellites serving high latitude regions. The ground track of a satellite in a Tundra orbit is a closed figure 8 with a smaller loop over either the northern or southern hemisphere. This differentiates them from Molniya orbits designed to service high-latitude regions, which have the same inclination but half the period and do not loiter over a single region.
A ground track or ground trace is the path on the surface of a planet directly below an aircraft or satellite. In the case of a satellite, it is the projection of the satellite's orbit onto the surface of the Earth.
Orbital perturbation analysis is the activity of determining why a satellite's orbit differs from the mathematical ideal orbit. A satellite's orbit in an ideal two-body system describes a conic section, usually an ellipse. In reality, there are several factors that cause the conic section to continually change. These deviations from the ideal Kepler's orbit are called perturbations.
In orbital mechanics, a frozen orbit is an orbit for an artificial satellite in which natural drifting due to the central body's shape has been minimized by careful selection of the orbital parameters. Typically, this is an orbit in which, over a long period of time, the satellite's altitude remains constant at the same point in each orbit. Changes in the inclination, position of the lowest point of the orbit, and eccentricity have been minimized by choosing initial values so that their perturbations cancel out. This results in a long-term stable orbit that minimizes the use of station-keeping propellant.
↑ Basics of the Geostationary Orbit By Dr. T.S. Kelso
|
Isotope Knowpia
Isotope vs. nuclideEdit
Radioactive, primordial, and stable isotopesEdit
Stable isotopesEdit
F. W. Aston subsequently discovered multiple stable isotopes for numerous elements using a mass spectrograph. In 1919 Aston studied neon with sufficient resolution to show that the two isotopic masses are very close to the integers 20 and 22 and that neither is equal to the known molar mass (20.2) of neon gas. This is an example of Aston's whole number rule for isotopic masses, which states that large deviations of elemental molar masses from integers are primarily due to the fact that the element is a mixture of isotopes. Aston similarly showed[when?] that the molar mass of chlorine (35.45) is a weighted average of the almost integral masses for the two isotopes 35Cl and 37Cl.[29]
Variation in properties between isotopesEdit
Chemical and molecular propertiesEdit
), because deuterium has twice the mass of protium and tritium has three times the mass of protium.[30] These mass differences also affect the behavior of their respective chemical bonds, by changing the center of gravity (reduced mass) of the atomic systems. However, for heavier elements, the relative mass difference between isotopes is much less so that the mass-difference effects on chemistry are usually negligible. (Heavy elements also have relatively more neutrons than lighter elements, so the ratio of the nuclear mass to the collective electronic mass is slightly greater.) There is also an equilibrium isotope effect.
Nuclear properties and stabilityEdit
Numbers of isotopes per elementEdit
Even and odd nucleon numbersEdit
Even atomic numberEdit
Odd atomic numberEdit
Odd neutron numberEdit
Atomic mass of isotopesEdit
The atomic masses of naturally occurring isotopes of an element determine the atomic mass of the element. When the element contains N isotopes, the expression below is applied for the average atomic mass
{\displaystyle {\overline {m}}_{a}}
{\displaystyle {\overline {m}}_{a}=m_{1}x_{1}+m_{2}x_{2}+...+m_{N}x_{N}}
Applications of isotopesEdit
Purification of isotopesEdit
Use of chemical and biological propertiesEdit
Use of nuclear propertiesEdit
^ a b c Scerri, Eric R. (2007) The Periodic Table Oxford University Press, pp. 176–179 ISBN 0-19-530573-6
^ Scerri, Eric R. (2007) The Periodic Table, Oxford University Press, ISBN 0-19-530573-6, Ch. 6, note 44 (p. 312) citing Alexander Fleck, described as a former student of Soddy's.
^ Laidler, Keith (1987). Chemical Kinetics (3rd ed.). India: Pearson Education. p. 427. ISBN 978-81-317-0972-6.
|
Computer architecture and programming languages - Tales of Science & Data
Computer architecture - just some notes
We'll outline some stuff about computers here, a lot of it being prompted by reading Plantz's book and then digging around.
Bits, representations and bytes
Bit stands for binary digit and it's a switch representation: you got a switch which can take a binary value. 8 bits compose the so called byte, and this is a convention which grew to be agreed upon with time: changing with hardware and operating system. IBM in 1956 established the 8-bit standard, which won over other systems because it could accommodate the 256 ASCII characters: 26 upper chars, 26 lower chars, the 10 digits and the most important punctuation marks. Because with time we need to represent more things though, the choice of only 255 was not enough anymore and so architectures with more than one by came to be (see the Quora page).
Note on the hexadecimal system
In computers, the binary and hexadecimal (base 16) systems for number representation are used.
In base 16, you have
0_{10}
1_{10}
9_{10}
10_{10}
11_{10}
16_{10}
Numbers prefixed with 0x in C/C++ means they are expressed in base 16.
The phrasing random access is rather misleading. It's not a random memory in the sense that stuff is placed in random spots, random access means it uses the same time to access each byte in memory, as opposed to a tape which you'd have to walk for access at a specific location. More specifically, it means that a program accesses variables which have been allocated in some places of the memory and does not look for them by scrolling sequentially.
A short and very non-comprehensive timeline of languages
1949 Assembly: low-level, strong correspondence to machine code
1978 C, low-medium level
About Fortran, and why it still is so common within the scientific programming circles, have a read at this brilliant article on Ars Technica.
Main programming paradigms
A programming paradigm is a style of coding. The main ones are
declarative (of which functional is a subset)
Are two opposing ones. In an imperative paradigm, algorithms are implemented in explicit steps and statements are used to change the state: statements are the smallest standalone instructions. In a declarative paradigm, algorithms are logically expressed without the explicit list of instructions (for example with the use of list comprehensions in Python).
In an object-oriented programming paradigm, objects are declared that contain attributes and methods.
The functional programming paradigm, belonging to the declarative class, treats computation as the evaluation of mathematical functions and is based on lambda calculus, avoiding statements. Examples of functional languages are Clojure, Haskell; other that support the functional paradigm are Python, R, Java, Scala.
For example, in Clojure you'd get the square of integers until 25 as
where take, squares-of and integers are functions, as opposed to an explicit for loop you'd write in other languages.
Scala is a functional programming language that runs on the JVM (Java virtual machine), meaning it gets compiled to Java bytecode, it is statically typed, object-oriented. The name stands for "scalable language" because it has been conceived to grow with demand of its users. The project started in 2001 at the E. Polytechnique de Lausanne by M Odersky and it is taking more and more interest in the data science community.
A compiler is something, like a parser, that transforms the source code the programmers uses into machine code (binary file with low-level instructions the machine understands). Compiled languages are for instance C, C++, Fortran.
An interpreter interprets the language, in that when you write your high-level instructions it goes searching for the corresponding binary code, which is part of itself. The difference with a compiler is that this process is executed at run time, making interpreted code sensibly slower than compiled one. Examples are Python and Ruby.
Statically and dynamically typed languages
In a statically typed language, you cannot change the type of a variable after you've declared it. Python is a dynamically typed one as you can do things like
a = "bla"
R G Plantz, An Introduction to Computer Organisation, 2015
Quora on why are there 8 bits in a byte
Scientific computing’s future: Can any coding language top a 1950s behemoth?, an article about Fortran still being used today in numerical work, Ars Technica, 2014
|
TL;DR: Here we discuss some CSS style changes, MathML insertion changes for equation rendering, external link hinting, some new articles worth checking out, some website statistics and finally some news about moving the server.
We now have a dark mode on the website! It's very basic and simply implemented in CSS using a media query selector, being the least intrusive method of implementation I could think of. It uses a query like so:
0001 body, code{
0002 background-color: #DDD;
0004 font-family: "Computer Modern Serif", serif;
0006 margin: 0px;
0009 @media (prefers-color-scheme: dark){
0010 body, code{
0011 background-color: #222;
0012 color: #DDD;
By default the theme is light. I've seen a few websites use @media (prefers-color-scheme: light) to reduce elements being set too many overriding properties - but this doesn't feel fail-safe encase the media query is not supported.
As you can see, we only need to override the colours for the dark mode, and this is only done if the browser supports this. This appears to be the best fail-safe low-compromise solution to the problem.
By default, this is the view I enjoy in Firefox - personally I find it better. I'm still slightly adjusting the colours, but they are mostly there now. I still need to update Dead Social to also support the dark mode theme.
MathML Insertion
This one is more of a discussion about how the sausage is made, but now we are only inserting the MathML JS when we detect that the rendered output requires it. This removes the need for the 95% of pages that do not include any MathML to require the JS, which is only required for Chrome, Chromium and Safari users currently.
MathM{L}_{support}=true
In general I am currently of the opinion that MathML is the real winner of the browser math libraries, and with the one I'm using weighing in at just 80kB (compare to other much heavier libraries) I'm pretty happy with the compromise.
Just recently I was glad to have implemented the MathML functionality when I was discussing
{R}_{0}
values in an article about the COVID lockdown.
External Link Hinting
NOTE: This is a late addition to this article, apologies. The idea was too good to not implement!
After reading an article by Hund and then an article by Christian Oliff I have decided to also do the same, and use external link hinting. I used to do this quite some years ago, but I used bitmaps and it looks like crap.
In order to implement this, I used the following CSS:
0015 /* Annotate links */
0016 a[href^="http"]::after, a[href^="https://"]::after{
0017 content: "";
0018 width: 11px;
0019 height: 11px;
0020 margin-left: 4px;
0021 background-image: url("data:image/svg+xml,%3Csvg xmlns='http://www.w3.org/2000/svg' width='16' height='16' fill='cyan' viewBox='0 0 16 16'%3E%3Cpath fill-rule='evenodd' d='M8.636 3.5a.5.5 0 0 0-.5-.5H1.5A1.5 1.5 0 0 0 0 4.5v10A1.5 1.5 0 0 0 1.5 16h10a1.5 1.5 0 0 0 1.5-1.5V7.864a.5.5 0 0 0-1 0V14.5a.5.5 0 0 1-.5.5h-10a.5.5 0 0 1-.5-.5v-10a.5.5 0 0 1 .5-.5h6.636a.5.5 0 0 0 .5-.5z'/%3E%3Cpath fill-rule='evenodd' d='M16 .5a.5.5 0 0 0-.5-.5h-5a.5.5 0 0 0 0 1h3.793L6.146 9.146a.5.5 0 1 0 .708.708L15 1.707V5.5a.5.5 0 0 0 1 0v-5z'/%3E%3C/svg%3E");
0022 background-position: center;
0023 background-repeat: no-repeat;
0024 background-size: contain;
0025 display: inline-block;
0027 a[href^="https://coffeespace.org.uk/"]::after, a[href^="https://coffeespace.org.uk/"]::after{ display: none !important; }
It is essentially a clone of Hund's CSS, except I use Christian's data URI to save the browser the additional requests, and therefore limit re-layout of the document once the image is requested. My assumption is also that if your browser supports such advanced CSS, then you will likely support the data URI. For example, it works in Firefox and Chromium, but not NetSurf or Dillo 1.
At some point it would be good to add a special link for RSS feeds too... Perhaps in a nice orange colour!
Finally, regarding Hund's comment:
As for the “dark mode” users, there’s unfortunately no way to override the black icon with a white icon. I don’t know why it doesn’t work, but it’s not something I can do about it.
This is relatively easy using the CSS hack I implemented previously.
I have obviously been uploading quite a bit of content (and continue to do so as I type). Particularly worth checking out in the recent articles (in my opinion) are:
Keto diet updates - A series of updates about my keto diet progress.
COVID lockdown - An important discussion about misconception about vaccination results.
Chunk boi pro - The start of creating a custom laptop to replace my current daily driver.
Polygon experiments - A fun project in creating vector images from bitmaps.
Of course as always, use your favourite RSS feed reader and subscribe to the sections of the website that interest you the most.
A while back I discussed website statistics, specifically those of last year. 6 or so months on we can now discuss how things have changed.
Requests per day (large)
Daily requests are now a little higher, averaging more than 10x more traffic per day. A few changes were made based on my previous ideas:
SEO optimization - This was mostly about forcing search engines to check back more regularly for additional content. This turned out to be the most successful change.
Link preview - This didn't have so much effect in the end, but was a worthwhile exercise nevertheless. If the website does gain some mainstream attention (hopefully positive) at least the website will be prepared.
Location of requests (large)
It appears that most of these new requests were picked up in the Netherlands - I have zero idea for why this is the case. I had a look online and couldn't find out too much information as to why this is the case.
Other than that, views from Germany and the US seem to be quite high also. Bare in mind that I probably generate the majority of the traffic from New Zealand, there is an awful lot of traffic coming from other locations that is not generated by myself.
Looking at a random sample of the Nginx logs (filtered), a lot of the traffic consists of bots accessing /git/*, which is the cgit server. Good job I turned on static caching so heavily, otherwise the server would have melted by now!
0028 185.191.###.### - - [###] "GET /git/linux/diff/.gitignore?h=v2.6.34-rc3 HTTP/1.1" 200 1413 "-" "Mozilla/5.0 (compatible; SemrushBot/7~bl; +http://www.semrush.com/bot.html)"
0029 185.191.###.### - - [###] "GET /git/freerouting/commit/?id=dec00042ca637585ca23d1a97b3e0853f28e8bc5 HTTP/1.1" 200 2271 "-" "Mozilla/5.0 (compatible; SemrushBot/7~bl; +http://www.semrush.com/bot.html)"
0030 167.114.###.### - - [###] "GET /git/free-comment/diff/dat?h=v0.2.0&id=9d79abaa5baa04cf8a486f0e0fcfdc419e009a47&id2=cdeb9f1ba03259a53d0252e7a7b5f984c5d6dcbc HTTP/1.1" 200 1551 "-" "Mozilla/5.0 (compatible; MJ12bot/v1.4.8; http://mj12bot.com/)"
0031 185.191.###.### - - [###] "GET /git/oakwm/plain/res?id=cceeb4b702b3ba7a31837392a6ad9d81c0399f5b HTTP/1.1" 200 9457 "-" "Mozilla/5.0 (compatible; SemrushBot/7~bl; +http://www.semrush.com/bot.html)"
0032 114.119.###.### - - [###] "GET /git/free-comment/patch/res/default.properties?id=2d216293167febd6cc1b1aabffdcd1e00f8cb5f0 HTTP/1.1" 200 875 "-" "Mozilla/5.0 (Linux; Android 7.0;) AppleWebKit/537.36 (KHTML, like Gecko) Mobile Safari/537.36 (compatible; PetalBot;+https://webmaster.petalsearch.com/site/petalbot)"
0033 185.191.###.### - - [###] "GET /git/cgit.git/log/cgitrc?id=b44b02a98253e78334f7fd13d9c4e1eb59562392 HTTP/1.1" 200 2390 "-" "Mozilla/5.0 (compatible; SemrushBot/7~bl; +http://www.semrush.com/bot.html)"
0034 167.114.###.### - - [###] "GET /git/free-comment/diff/dat?h=v0.2.0&id=cdeb9f1ba03259a53d0252e7a7b5f984c5d6dcbc&id2=b8c47c2c099bcc3fb4267169e514c97216cb4503 HTTP/1.1" 200 1550 "-" "Mozilla/5.0 (compatible; MJ12bot/v1.4.8; http://mj12bot.com/)"
Traffic is traffic though - anything that gets search engines looking at my website is generally good for discoverability.
This old Scaleway C1 instance (bare metal) has its days numbered now, despite me still being relatively happy with the service. I'm getting quite aggressive emails now telling me to move the instance somewhere else. This is the current timeline:
June 1, 2021: quotas and limitations
As a C1 instance customer, you will be able to keep all your current C1 instances running, and to create one more new C1 instance. You will be allowed to start, stop and back up C1 instances, so you can assess all possible migration scenarios.
As of June 1, Scaleway Elements customers that do not already have C1 instances will not be able to create new ones.
July 1, 2021: end of support
July 1, 2021 will mark the end of support for C1, meaning no more patches, updates or interventions on software or hardware will be provided. You will not be able to open new support tickets for C1 instances either.
August 1, 2021: instance creation freeze
Scaleway Elements users will no longer be able to create C1 instances. As a C1 instance customer, your current instances will keep running. However in case of an instance stopping (planned or unplanned), you will not be able to restart the instance.
September 1, 2021: end of life
Flexible & dynamic public IPs will be detached from C1 instances to force a service interruption as a final warning. You will have 24 hours to migrate your instances.
Please note, flexible IPs will still be charged if you do not delete them.
As it stands, if the server crashes I'm screwed. I either need to migrate to another of their servers, or find a different one elsewhere. I am currently looking at different services. I have had a very good experience so far with ChicagoVPS and could be convinced to migrate my servers there. $15 per year is pretty competitive and I already have accounts and servers there, although it's not nearly as powerful as what I currently run.
One a side note, Dillo does a great job of screwing up CSS in general, especially CSS it doesn't understand.↩
|
Stephen does not like yogurt very much, but he loves apples. Since both make a good snack, Stephen’s mom makes a deal with Stephen. She will keep the refrigerator stocked with
5
2
3
red apples every day. Each day, Stephen will randomly pick a snack. What is the probability Stephen will not get three yogurts on three consecutive days? Use a tree diagram or area model to show all the possible outcomes in the sample space.
P(\text{getting three yogurts})
P(\text{not getting three yogurts})
is the complement of the hint above.
|
Examine the different infinite series presented below. Your goal is to decide which series converge (have a finite sum). Rewrite each series using sigma notation. As you work through each example, try to figure out what feature of the series is most critical in its convergence or divergence.
1 + \frac { 3 } { 2 } + \frac { 9 } { 4 } + \frac { 27 } { 8 } + \ldots
\frac { 1 } { 2 } + \frac { 1 } { 5 } + \frac { 1 } { 10 } + \frac { 1 } { 17 } + \frac { 1 } { 26 } + \ldots
1 - 2 + 3 - 4 + . ..
\frac { 1 } { 2 } + \frac { 1 } { 6 } + \frac { 1 } { 12 } + \frac { 1 } { 20 } + \ldots
- 2 + 1 - \frac { 2 } { 3 } + \frac { 1 } { 2 } - \frac { 2 } { 5 } + \frac { 1 } { 3 } \dots
\operatorname { ln } \frac { 1 } { 2 } + \operatorname { ln } \frac { 2 } { 3 } + \operatorname { ln } \frac { 3 } { 4 } + \dots
100 - 90 + 81 - 72.9 + \ldots
1 + \frac { 1 } { 8 } + \frac { 1 } { 27 } + \frac { 1 } { 64 } + \ldots
1 + \frac { 1 } { 2 } + \frac { 1 } { 6 } + \frac { 1 } { 24 } + \frac { 1 } { 120 } + \ldots
\frac { 1 } { 2 } + 1 + \frac { 9 } { 8 } + 1 + \frac { 25 } { 32 } + \frac { 36 } { 64 } + \ldots
\frac { 1 } { 2 } + \frac { 2 } { 3 } + \frac { 3 } { 4 } + \frac { 4 } { 5 } + \frac { 5 } { 6 } \dots
1 + \frac { 1 } { \sqrt { 2 } } + \frac { 1 } { \sqrt { 3 } } + \frac { 1 } { 2 } + \frac { 1 } { \sqrt { 5 } } + \ldots
|
Week 12. Shortest Path | Algorithms and Data Structures
Week 12. Shortest Path
13 Week 12. Shortest Path
Learn (1) proofs and (2) complexity for Dijkstra’s Algorithm.
Graph Algorithm Problems
DFT and BFT
13.2.1 Warm-up Exercise
Figure 1: A sample graph
Problem 13.1 Consider the graph in Figure 1. Make a manual breadth first search from node
A
, showing the steps, and the BFS tree you end up with.
Problem 13.2 Consider the graph in Figure 1. Manually calculate the shortest paths from
A
using Dijkstra’s algorithm. Show the calculations and the resulting shortest path weights for each target node.
Problem 13.3 Communication networks are typically modelled as graphs. The edges represent links and the vertices relay messages. Imagine an ad hoc network set up with rather primitive equipment with low bandwidth, such as subsea links or long distance space links.
Video transmission of adequate quality is possible if the path from the sender to the receiver is at most four links. Design an algorithm (pseudo-code) which identify, for any given sender, all the nodes able to receive adequate quality.
Problem 13.4 Concerns over Google’s use of personal information motivates an alternative route planner to Google Map, although not with a global ambition.
Explain how you can model the problem using graphs, and how you can algorithmically calculate the optimal route from A to B. Your answer must include a description of the data structure and pseudo-code for the algorithm.
Furthermore, discuss what we could mean by an optimal route, and how this is modelled in the graph. Discuss briefly a couple of alternative interpretations of «optimal».
Problem 13.5 Consider the graph in Figure 1. Make a manual depth first search from node
A
, showing the steps, and the DFS tree you end up with.
Problem 13.6 Consider the adjacency map model. Draw a possible class diagram for an object-oriented implementation of the data structure and describe the purpose of each class.
Give pseude-code for the removeEdge
\left(e\right)
operation, making sure it runs in
O\left(1\right)
|
Specific_angular_momentum Knowpia
{\displaystyle {\vec {h}}}
{\displaystyle \mathbf {h} }
) of a body is the angular momentum of that body divided by its mass.[1] In the case of two orbiting bodies it is the vector product of their relative position and relative velocity, divided by the mass of the body in question.
{\displaystyle \mathbf {r} }
{\displaystyle \mathbf {v} }
{\displaystyle \mathbf {h} =\mathbf {r} \times \mathbf {v} ={\frac {\mathbf {L} }{m}}}
{\displaystyle \mathbf {L} }
{\displaystyle \mathbf {r} \times m\mathbf {v} }
{\displaystyle \mathbf {h} }
Proof of constancy in the two body caseEdit
{\displaystyle \mathbf {r} }
{\displaystyle \mathbf {v} }
{\displaystyle \theta }
{\displaystyle \phi }
{\displaystyle m_{2}}
{\displaystyle m_{1}}
{\displaystyle \theta }
{\displaystyle \nu }
{\displaystyle m_{1}\gg m_{2}}
{\displaystyle {\ddot {\mathbf {r} }}+{\frac {Gm_{1}}{r^{2}}}{\frac {\mathbf {r} }{r}}=0}
{\displaystyle \mathbf {r} }
{\displaystyle m_{1}}
{\displaystyle m_{2}}
{\displaystyle r}
{\displaystyle {\ddot {\mathbf {r} }}}
{\displaystyle \mathbf {r} }
{\displaystyle G}
{\displaystyle \mathbf {r} \times {\ddot {\mathbf {r} }}+\mathbf {r} \times {\frac {Gm_{1}}{r^{2}}}{\frac {\mathbf {r} }{r}}=0}
{\displaystyle \mathbf {r} \times \mathbf {r} =0}
{\displaystyle \mathbf {r} \times {\ddot {\mathbf {r} }}=0}
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left(\mathbf {r} \times {\dot {\mathbf {r} }}\right)={\dot {\mathbf {r} }}\times {\dot {\mathbf {r} }}+\mathbf {r} \times {\ddot {\mathbf {r} }}=\mathbf {r} \times {\ddot {\mathbf {r} }}}
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left(\mathbf {r} \times {\dot {\mathbf {r} }}\right)=0}
{\displaystyle \mathbf {r} \times {\dot {\mathbf {r} }}}
{\displaystyle \mathbf {v} }
{\displaystyle \mathbf {h} }
{\displaystyle \mathbf {h} =\mathbf {r} \times \mathbf {v} }
{\displaystyle \mathbf {r} \times \mathbf {p} }
Kepler's laws of planetary motionEdit
{\displaystyle {\ddot {\mathbf {r} }}\times \mathbf {h} =-{\frac {\mu }{r^{2}}}{\frac {\mathbf {r} }{r}}\times \mathbf {h} }
{\textstyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\dot {\mathbf {r} }}\times \mathbf {h} \right)}
{\displaystyle -{\frac {\mu }{r^{3}}}\left(\mathbf {r} \times \mathbf {h} \right)=-{\frac {\mu }{r^{3}}}\left(\left(\mathbf {r} \cdot \mathbf {v} \right)\mathbf {r} -r^{2}\mathbf {v} \right)=-\left({\frac {\mu }{r^{2}}}{\dot {r}}\mathbf {r} -{\frac {\mu }{r}}\mathbf {v} \right)=\mu {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\mathbf {r} }{r}}\right)}
{\displaystyle \mathbf {C} }
{\displaystyle {\dot {\mathbf {r} }}\times \mathbf {h} =\mu {\frac {\mathbf {r} }{r}}+\mathbf {C} }
{\displaystyle \mathbf {r} }
{\displaystyle {\begin{aligned}\mathbf {r} \cdot \left({\dot {\mathbf {r} }}\times \mathbf {h} \right)&=\mathbf {r} \cdot \left(\mu {\frac {\mathbf {r} }{r}}+\mathbf {C} \right)\\\Rightarrow \left(\mathbf {r} \times {\dot {\mathbf {r} }}\right)\cdot \mathbf {h} &=\mu r+rC\cos \theta \\\Rightarrow h^{2}&=\mu r+rC\cos \theta \end{aligned}}}
Finally one gets the orbit equation[1]
{\displaystyle r={\frac {\frac {h^{2}}{\mu }}{1+{\frac {C}{\mu }}\cos \theta }}}
{\textstyle p={\frac {h^{2}}{\mu }}}
{\textstyle e={\frac {C}{\mu }}}
{\textstyle \mathrm {d} t={\frac {r^{2}}{h}}\,\mathrm {d} \theta }
{\textstyle \mathrm {d} A={\frac {r^{2}}{2}}\,\mathrm {d} \theta }
{\displaystyle \mathrm {d} \theta }
{\displaystyle \mathrm {d} t={\frac {2}{h}}\,\mathrm {d} A}
Kepler's third is a direct consequence of the second law. Integrating over one revolution gives the orbital period[1]
{\displaystyle T={\frac {2\pi ab}{h}}}
{\displaystyle \pi ab}
{\displaystyle b={\sqrt {ap}}}
{\displaystyle h={\sqrt {\mu p}}}
{\displaystyle T=2\pi {\sqrt {\frac {a^{3}}{\mu }}}}
^ a b c d Vallado, David A. (2001). Fundamentals of astrodynamics and applications (2nd ed.). Dordrecht: Kluwer Academic Publishers. pp. 20–30. ISBN 0-7923-6903-3.
|
Ask Answer - Is Matter Around Us Pure - Expert Answered Questions for School Students
When a banana is left outside for few days there is a change in its colour i.e. there are black patches on its surface . this is a kind of chemical change . since there is a change in physical appearence also does that mean a chemical change can also bring physical change ???
What is the use of sodium metabisulphite
Milk is a ____solution but vinegar is a ____solution
Sujal Kumar Thakur
why an egg becomes solid from liquid by heating
Does water has air or not ?????
Experts please answer my question
Liquid Bulb/LED glows/
doesn't glow Good conductor/
Define colloidal solution in short....no links please I want 1 perfect answer
Moneeshka
Why solder is a homogeneous mixture
2nd last q solution pl
Q. For preparing 0.1 M solution of H2SO4 in one litre, we need H2SO4.
(A) 9.8 g (B) 4.9 g
(C) 49.0 g (D) 0.98 g
Can you solve Q 6,7&8 please fast
Classify the given substances as true solutions, colloidal solution and suspension:
(a) blood (b) pure milk (c) common salt solution (d) syrup
Is tincture of iodine a heterogeneous mixture???
write the methods of separation of each homogenous heterogenous suspension and collodial substaces
Please answer Ques. 73 -
{K}_{2}S{O}_{4}.A{l}_{2}{\left(S{O}_{4}\right)}_{3}.24{H}_{2}O
is known as Potash alum. It's aqueous solution does not show the properties of -
{K}^{+}
A{l}^{3-}
S{{O}_{4}}^{2-}
{O}^{2-}
pl ans q62 with explanation
|
Week 6. The Heap | Algorithms and Data Structures
Week 6. The Heap
7 Week 6. The Heap
Reading 6 Goodrich & Tamassia: Chapter 8
For summary of key concepts, see the lecture notes on OneNote.
7.1.1 Group Work and Warp-Up
Problem 7.1 (Expanded from Textbook R-9.1) We discussed in- and post-order tree traversal last week. There is also pre-order traversal, where the parent node is processed before both subtrees. The pre-order rank of a node, is its number in the sequence of node visits in a pre-order traversal. I.e. you make a pre-order traversal and number the nodes in sequence.
T
be a binary tree where each node has a key equal to the preorder rank. What condition(s) must be satisfied for
T
to be a heap?
Hint. Consider the following questions in order: (1) Draw an arbitrary tree and number the nodes. Is the heap property satisfied? (2) Would it hold for all such trees? (3) Which other properties does a heap have to satisfy?
Problem 7.2 Do Textbook R-9.2.
Problem 7.3 Do Textbook C-9.37.
Problem 7.4 Do Textbook C-9.46–47.
Problem 7.5 Do Textbook R-9.12.
Problem 7.7 (Based on textbook P-9.48) Heapsort can be implemented in two different ways. The most straight-forward high-level implementation uses the Priority Queue ADT. All the elements are first inserted in the queue and then extracted in order, giving the sorted sequence (which can be stored in whatever data structure).
The alternative saves memory by sorting in-place. This requires access to the underlying concrete data structure, so that the same storage (usually an array) can be interpreted partly as a heap and partly as an array.
Implement both approaches and compare their practical running time. Do you see a difference? Is there an asymoptotic difference?
|
icosahedron - Maple Help
Home : Support : Online Help : Graphics : Packages : Plot Tools : icosahedron
generate 3-D plot object for a icosahedron
icosahedron([x, y, z], s, options)
location of the icosahedron
(optional) scale of the icosahedron; default is 1
The icosahedron command creates a three-dimensional plot data object, which when displayed is a scaled icosahedron located at point [x,y,z]. Note that the scale factor is applied in each dimension. This command is an interface to the plots[polyhedraplot] routine.
The plot data object produced by the icosahedron command can be used in a PLOT3D data structure, or displayed using the plots[display] command.
\mathrm{with}\left(\mathrm{plottools}\right):
\mathrm{with}\left(\mathrm{plots}\right):
\mathrm{display}\left(\mathrm{icosahedron}\left([0,0,0],0.8\right),\mathrm{orientation}=[45,0]\right)
\mathrm{display}\left(\mathrm{icosahedron}\left([0,0,0],0.8\right),\mathrm{icosahedron}\left([1,1,1],0.5\right),\mathrm{lightmodel}=\mathrm{light2},\mathrm{orientation}=[45,0]\right)
|
Bi-elliptic transfer - WikiMili, The Best Wikipedia Reader
Maneuver that moves a spacecraft from one orbit to another
Hohmann transfer orbit, labelled 2, from an orbit (1) to a higher orbit (3). This is comparable to a bi-elliptic transfer orbit.
A bi-elliptic transfer from a low circular starting orbit (blue) to a higher circular orbit (red). A boost at 1 makes the craft follow the green half-ellipse. Another boost at 2 brings it to the orange half-ellipse. A negative boost at 3 makes it follow the red orbit.
Delta-v 2
Versatility in combination maneuvers
{\displaystyle r_{b}}
away from the central body. At this point a second burn sends the spacecraft into the second elliptical orbit with periapsis at the radius of the final desired orbit, where a third burn is performed, injecting the spacecraft into the desired orbit. [1]
While they require one more engine burn than a Hohmann transfer and generally require a greater travel time, some bi-elliptic transfers require a lower amount of total delta-v than a Hohmann transfer when the ratio of final to initial semi-major axis is 11.94 or greater, depending on the intermediate semi-major axis chosen. [2]
The idea of the bi-elliptical transfer trajectory was first[ citation needed ] published by Ary Sternfeld in 1934. [3]
{\displaystyle v^{2}=\mu \left({\frac {2}{r}}-{\frac {1}{a}}\right),}
{\displaystyle v}
{\displaystyle \mu =GM}
is the standard gravitational parameter of the primary body,
{\displaystyle r}
is the distance of the orbiting body from the primary, i.e., the radius,
{\displaystyle a}
{\displaystyle r_{1}}
is the radius of the initial circular orbit,
{\displaystyle r_{2}}
is the radius of the final circular orbit,
{\displaystyle r_{b}}
is the common apoapsis radius of the two transfer ellipses and is a free parameter of the maneuver,
{\displaystyle a_{1}}
{\displaystyle a_{2}}
are the semimajor axes of the two elliptical transfer orbits, which are given by
{\displaystyle a_{1}={\frac {r_{1}+r_{b}}{2}},}
{\displaystyle a_{2}={\frac {r_{2}+r_{b}}{2}}.}
Starting from the initial circular orbit with radius
{\displaystyle r_{1}}
(dark blue circle in the figure to the right), a prograde burn (mark 1 in the figure) puts the spacecraft on the first elliptical transfer orbit (aqua half-ellipse). The magnitude of the required delta-v for this burn is
{\displaystyle \Delta v_{1}={\sqrt {{\frac {2\mu }{r_{1}}}-{\frac {\mu }{a_{1}}}}}-{\sqrt {\frac {\mu }{r_{1}}}}.}
When the apoapsis of the first transfer ellipse is reached at a distance
{\displaystyle r_{b}}
from the primary, a second prograde burn (mark 2) raises the periapsis to match the radius of the target circular orbit, putting the spacecraft on a second elliptic trajectory (orange half-ellipse). The magnitude of the required delta-v for the second burn is
{\displaystyle \Delta v_{2}={\sqrt {{\frac {2\mu }{r_{b}}}-{\frac {\mu }{a_{2}}}}}-{\sqrt {{\frac {2\mu }{r_{b}}}-{\frac {\mu }{a_{1}}}}}.}
Lastly, when the final circular orbit with radius
{\displaystyle r_{2}}
is reached, a retrograde burn (mark 3) circularizes the trajectory into the final target orbit (red circle). The final retrograde burn requires a delta-v of magnitude
{\displaystyle \Delta v_{3}={\sqrt {{\frac {2\mu }{r_{2}}}-{\frac {\mu }{a_{2}}}}}-{\sqrt {\frac {\mu }{r_{2}}}}.}
{\displaystyle r_{b}=r_{2}}
, then the maneuver reduces to a Hohmann transfer (in that case
{\displaystyle \Delta v_{3}}
can be verified to become zero). Thus the bi-elliptic transfer constitutes a more general class of orbital transfers, of which the Hohmann transfer is a special two-impulse case.
A bi-parabolic transfer from a low circular starting orbit (dark blue) to a higher circular orbit (red)
The maximal possible savings can be computed by assuming that
{\displaystyle r_{b}=\infty }
, in which case the total
{\displaystyle \Delta v}
{\textstyle {\sqrt {\mu /r_{1}}}\left({\sqrt {2}}-1\right)\left(1+{\sqrt {r_{1}/r_{2}}}\right)}
. In this case, one also speaks of a bi-parabolic transfer because the two transfer trajectories are no longer ellipses but parabolas. The transfer time increases to infinity too.
{\displaystyle T=2\pi {\sqrt {\frac {a^{3}}{\mu }}}.}
The total transfer time
{\displaystyle t}
is the sum of the times required for each half-orbit. Therefore:
{\displaystyle t_{1}=\pi {\sqrt {\frac {a_{1}^{3}}{\mu }}}\quad {\text{and}}\quad t_{2}=\pi {\sqrt {\frac {a_{2}^{3}}{\mu }}},}
{\displaystyle t=t_{1}+t_{2}.}
Delta-v required for Hohmann (thick black curve) and bi-elliptic transfers (colored curves) between two circular orbits as a function of the ratio of their radii
The figure shows the total
{\displaystyle \Delta v}
required to transfer from a circular orbit of radius
{\displaystyle r_{1}}
to another circular orbit of radius
{\displaystyle r_{2}}
{\displaystyle \Delta v}
is shown normalized to the orbital speed in the initial orbit,
{\displaystyle v_{1}}
, and is plotted as a function of the ratio of the radii of the final and initial orbits,
{\displaystyle R\equiv r_{2}/r_{1}}
; this is done so that the comparison is general (i.e. not dependent of the specific values of
{\displaystyle r_{1}}
{\displaystyle r_{2}}
, only on their ratio). [2]
The thick black curve indicates the
{\displaystyle \Delta v}
for the Hohmann transfer, while the thinner colored curves correspond to bi-elliptic transfers with varying values of the parameter
{\displaystyle \alpha \equiv r_{b}/r_{1}}
, defined as the apoapsis radius
{\displaystyle r_{b}}
of the elliptic auxiliary orbit normalized to the radius of the initial orbit, and indicated next to the curves. The inset shows a close-up of the region where the bi-elliptic curves cross the Hohmann curve for the first time.
One sees that the Hohmann transfer is always more efficient if the ratio of radii
{\displaystyle R}
is smaller than 11.94. On the other hand, if the radius of the final orbit is more than 15.58 times larger than the radius of the initial orbit, then any bi-elliptic transfer, regardless of its apoapsis radius (as long as it's larger than the radius of the final orbit), requires less
{\displaystyle \Delta v}
than a Hohmann transfer. Between the ratios of 11.94 and 15.58, which transfer is best depends on the apoapsis distance
{\displaystyle r_{b}}
{\displaystyle R}
in this range, there is a value of
{\displaystyle r_{b}}
above which the bi-elliptic transfer is superior and below which the Hohmann transfer is better. The following table lists the value of
{\displaystyle \alpha \equiv r_{b}/r_{1}}
that results in the bi-elliptic transfer being better for some selected cases. [4]
{\displaystyle \alpha \equiv r_{b}/r_{1}}
such that a bi-elliptic transfer needs less
{\displaystyle \Delta v}
Ratio of radii,
{\displaystyle {\frac {r_{2}}{r_{1}}}}
{\displaystyle \alpha \equiv {\frac {r_{b}}{r_{1}}}}
{\displaystyle \infty }
Bi-parabolic transfer
>15.58
{\displaystyle >{\frac {r_{2}}{r_{1}}}}
Any bi-elliptic transfer is better
{\displaystyle t=\pi {\sqrt {\frac {a_{1}^{3}}{\mu }}}+\pi {\sqrt {\frac {a_{2}^{3}}{\mu }}},}
The Hohmann transfer takes less than half of the time because there is just one transfer half-ellipse. To be precise,
{\displaystyle t=\pi {\sqrt {\frac {a^{3}}{\mu }}}.}
While a bi-elliptic transfer has a small parameter window where it's strictly superior to a Hohmann Transfer in terms of delta V for a planar transfer between circular orbits, the savings is fairly small, and a bi-elliptic transfer is a far greater aid when used in combination with certain other maneuvers.
At apoapsis, the spacecraft is travelling at low orbital velocity, and significant changes in periapsis can be achieved for small delta V cost. Transfers that resemble a bi-elliptic but which incorporate a plane-change maneuver at apoapsis can dramatically save delta-V on missions where the plane needs to be adjusted as well as the altitude, versus making the plane change in low circular orbit on top of a Hohmann transfer.
Likewise, dropping periapsis all the way into the atmosphere of a planetary body for aerobraking is inexpensive in velocity at apoapsis, but permits the use of "free" drag to aid in the final circularization burn to drop apoapsis; though it adds an extra mission stage of periapsis-raising back out of the atmosphere, this may, under some parameters, cost significantly less delta V than simply dropping periapsis in one burn from circular orbit.
To transfer from a circular low Earth orbit with r0 = 6700 km to a new circular orbit with r1 = 93 800 km using a Hohmann transfer orbit requires a Δv of 2825.02 + 1308.70 = 4133.72 m/s. However, because r1 = 14r0> 11.94r0, it is possible to do better with a bi-elliptic transfer. If the spaceship first accelerated 3061.04 m/s, thus achieving an elliptic orbit with apogee at r2 = 40r0 = 268 000 km, then at apogee accelerated another 608.825 m/s to a new orbit with perigee at r1 = 93 800 km, and finally at perigee of this second transfer orbit decelerated by 447.662 m/s, entering the final circular orbit, then the total Δv would be only 4117.53 m/s, which is 16.19 m/s (0.4%) less.
In astrodynamics, the vis-viva equation, also referred to as orbital-energy-invariance law, is one of the equations that model the motion of orbiting bodies. It is the direct result of the principle of conservation of mechanical energy which applies when the only force acting on an object is its own weight.
1 2 Vallado, David Anthony (2001). Fundamentals of Astrodynamics and Applications. Springer. p. 318. ISBN 0-7923-6903-3.
↑ Gobetz, F. W.; Doll, J. R. (May 1969). "A Survey of Impulsive Trajectories". AIAA Journal. American Institute of Aeronautics and Astronautics. 7 (5): 801–834. Bibcode:1969AIAAJ...7..801D. doi:10.2514/3.5231.
↑ Escobal, Pedro R. (1968). Methods of Astrodynamics. New York: John Wiley & Sons. ISBN 978-0-471-24528-5.
|
A-level Mathematics/OCR/C4/Algebra and Graphs - Wikibooks, open books for an open world
A-level Mathematics/OCR/C4/Algebra and Graphs
< A-level Mathematics | OCR | C4
2 Division of Polynomials
Rational Expressions[edit | edit source]
A rational expression has a polynomial in the numerator and a polynomial in the denominator:
{\displaystyle {\frac {f\left(x\right)}{g\left(x\right)}}}
. In some cases rational functions can be simplified, which can make differentiation, integration and just solving the equation easier. There is a simple procedure to follow if you want to simplify a fraction.
Fully factor the numerator and denominator.
{\displaystyle {\frac {ax^{3}+bx^{2}+cx+d}{ax^{2}+bx+c}}={\frac {(ex+f)(x+g)(x+h)}{(ix+j)(x+g)(x+h)}}}
If the same factor is in both the numerator and denominator you can cancel them out.
{\displaystyle {\frac {(ex+f){\cancel {(x+g)}}{\cancel {(x+h)}}}{(ix+j){\cancel {(x+g)}}{\cancel {(x+h)}}}}={\frac {(ex+f)}{(ix+j)}}}
If there multiple factors left you can recombine based on the situation.
{\displaystyle {\frac {5x^{3}+40x^{2}+75x}{x^{2}+12x+35}}}
Factor both the numerator and the denominator notice that a 5x can be removed in the numerator.
{\displaystyle {\frac {5x(x^{2}+8x+15)}{(x+5)(x+7)}}={\frac {5x(x+5)(x+3)}{(x+5)(x+7)}}}
Cancel out the same factors
{\displaystyle {\frac {5x(x+5)(x+3)}{(x+5)(x+7)}}={\frac {5x(x+3)}{(x+7)}}}
Recombine the factors
{\displaystyle {\frac {5x(x+3)}{(x+7)}}={\frac {5x^{2}+15x}{x+7}}}
Division of Polynomials[edit | edit source]
In core four you will be expected to divide polynomials of up to the fourth degree by either a linear or quadratic polynomial. This only works if the numerator is greater than the denominator. The resultant will always be in the form;
{\displaystyle {\frac {f(x)}{g(x)}}=q(x)+{\frac {r(x)}{g(x)}}}
where q(x)is the resulting polynomial and
{\displaystyle {\frac {r(x)}{g(x)}}}
Long division of polynomials is very similar to regular long division. I will use a problem to demonstrate how long division works.
{\displaystyle x^{3}+8x^{2}-4x+10}
{\displaystyle x^{2}+3x-1}
The first step is to set up the equation. Make sure that it is in order from highest power of x to lowest power of x.
Then we divide the first term of the dividend by the first term of the divisor.
{\displaystyle {\frac {x^{3}}{x^{2}}}=x}
We place this resultant on top.
Then we multiply the resultant by the divisor and subtract it from the dividend.
{\displaystyle x\left(x^{2}+3x-1\right)=x^{3}+3x^{2}-x}
What is left becomes the new dividend and we repeat the process again. We continue to do this until the first term has a degree less than the first term of the divisor. What is left is the remainder.
Synthetic Division[edit | edit source]
Example One[edit | edit source]
{\displaystyle 2x^{5}+5x^{2}-10x^{3}-30x-171}
by x-3.
The divisor is c. So in this case it will be 3.
Then we need to arrange our divisor in order from highest to lowest degree, if a degree is missing replace it with zero.
{\displaystyle 2x^{5}+0x^{4}-10x^{3}+5x^{2}-30x-171}
We then only look at the coefficients.
2 + 0 - 10 + 5 - 30 - 171
Now we set up our division equation.
Next we carry up the first term of the divisior.
Then we multiply the resultant by the divisor and add it to the next term.
We continue to do this until we reach the end.
Now we need to read the variables. When we read the variables we go from highest degree -1 to lowest degree. The last number is the remainder.
{\displaystyle 2x^{4}+6x^{3}+8x^{2}+29x+57}
This is the answer to the problem.
Partial Fractions[edit | edit source]
Only proper fractions can be made into partial fractions. A proper fraction is a fraction where the numerator is of a smaller degree than the denominator. Partial Fractions can used to make differentiation, integration or work with series easier. The goal of partial fractions is to take a complex polynomial and make it several simpler polynomials. To create a partial fraction equation you need to:
Factor the denominator and the numerator.
{\displaystyle {\frac {ax^{2}+bx+c}{ax^{3}+bx^{2}+cx+d}}\equiv {\frac {(ix+j)(kx+l)}{(ax+b)(cx+d)(ex+f)}}}
Create the partial fraction by making each factor the denominator, with a unique variable for the numerator.
{\displaystyle {\frac {(ix+j)(kx+l)}{(ax+b)(cx+d)(ex+f)}}\equiv {\frac {A}{(ax+b)}}+{\frac {B}{(cx+d)}}+{\frac {C}{(ex+f)}}}
Multiply every fraction by the denominator of original fraction.
{\displaystyle {\frac {(ix+j)(kx+l)(ax+b)(cx+d)(ex+f)}{(ax+b)(cx+d)(ex+f)}}\equiv }
{\displaystyle {\frac {A(ax+b)(cx+d)(ex+f)}{(ax+b)}}+{\frac {B(ax+b)(cx+d)(ex+f)}{(cx+d)}}+{\frac {C(ax+b)(cx+d)(ex+f)}{(ex+f)}}}
Cancel all out all factors that are the same.
{\displaystyle (ix+j)(kx+l)\equiv A(cx+d)(ex+f)+B(ax+b)(ex+f)+C(ax+b)(cx+d)}
{\displaystyle ikx^{2}+ilx+jlx+jl\equiv }
{\displaystyle (Acex^{2}+Acfx+Adex+Adf)+(Baex^{2}+Bafx+Bbex+Bbf)+(Cacx^{2}+Cadx+Cbdx+Cbd)\,}
Group all terms with x to the same index together.
{\displaystyle ikx^{2}+(il+kj)x+jl\equiv }
{\displaystyle (Ace+Bae+Cac)x^{2}+\left(Acf+Ade+Baf+Bbe+Cad+Cbd\right)x+\left(Adf+Bbf+Cbd\right)}
The co-efficient of x raised to the same index is the same on both sides of the identity. Using this fact solve for the variables in the numerator, using simultaneous equations. This is known as equating coefficients.
{\displaystyle ik\equiv \left(Ace+Bae+Cac\right)}
{\displaystyle \left(il+kj\right)\equiv \left(Acf+Ade+Baf+Bbe+Cad+Cbd\right)}
{\displaystyle jl\equiv \left(Adf+Bbf+Cbd\right)}
In step 2 replace the variables A, B, C with the value of the variable obtained from the previous step.
Rewrite as a partial fraction the following expression:
{\displaystyle {\frac {x^{2}+13x+40}{2x^{3}+27x^{2}+111x+140}}}
Using the knowledge from Further Pure 1: Roots of Polynomial Equations. We can factor the denominator as such.
{\displaystyle {\frac {x^{2}+13x+40}{2x^{3}+27x^{2}+111x+140}}\equiv {\frac {(x+5)(x+8)}{(2x+5)(x+7)(x+4)}}}
Now we start making the partial fraction.
{\displaystyle {\frac {x^{2}+13x+40}{(2x+5)(x+7)(x+4)}}\equiv {\frac {A}{(2x+5)}}+{\frac {B}{(x+7)}}+{\frac {C}{(x+4)}}}
Next we multiply everything by the denominator of the original expression.
{\displaystyle {\frac {x^{2}+13x+40(2x+5)(x+7)(x+4)}{(2x+5)(x+7)(x+4)}}\equiv }
{\displaystyle {\frac {A(2x+5)(x+7)(x+4)}{(2x+5)}}+{\frac {B(2x+5)(x+7)(x+4)}{(x+7}}+{\frac {C(2x+5)(x+7)(x+4)}{(x+4)}}}
Now we cancel out terms.
{\displaystyle (x+7)(x+8)\equiv A(x+7)(x+4)+B(2x+5)(x+4)+C(2x+5)(x+7)}
Then multiply all the factors together.
{\displaystyle x^{2}+13x+40=Ax^{2}+11Ax+28A+2Bx^{2}+13Bx+20B+2Cx^{2}+19Cx+35C\,}
We then group the terms together.
{\displaystyle x^{2}+13x+40=(A+2B+2C)x^{2}+(11A+13B+19C)x+(28A+20B+35C)\,}
Now we equate coefficients. Then we need to solve the simultaneous equations.
{\displaystyle 1\equiv (A+2B+2C)\equiv \,}
{\displaystyle 13\equiv (11A+13B+19C)\,}
{\displaystyle 40\equiv (28A+20B+35C)\,}
First we equate A
{\displaystyle 1\equiv (A+2B+2C)\equiv A=1-2B-2C\,}
{\displaystyle 13\equiv 11(1-2B-2C)+13B+19C\equiv 11-22B-22C+13B+19C\equiv 11-9B-3C}
{\displaystyle 9B\equiv 3C+2}
{\displaystyle B\equiv -{\frac {3C+2}{9}}}
We can further equate A to
{\displaystyle A\equiv 1-2(-{\frac {3C+2}{9}})-2C}
{\displaystyle A\equiv {\frac {13-12C}{9}}}
Now we can solve for C
{\displaystyle 40\equiv (28({\frac {13-12C}{9}})+20(-{\frac {3C+2}{9}})+35C)\,}
{\displaystyle 40\equiv (({\frac {364-336C}{9}})+(-{\frac {60C+40}{9}})+35C)\,}
{\displaystyle 40=36-9C\,}
{\displaystyle C={\frac {-4}{9}}}
Now we can solve for the rest
{\displaystyle A\equiv {\frac {13-12{\frac {-4}{9}}}{9}}={\frac {55}{27}}}
{\displaystyle B\equiv -{\frac {3C+2}{9}}={\frac {-2}{27}}}
{\displaystyle C\equiv {\frac {-4}{9}}}
Now we can write out our partial fraction:
{\displaystyle {\frac {x^{2}+13x+40}{(2x+5)(x+7)(x+4)}}\equiv {\frac {55}{27(2x+5)}}-{\frac {2}{27(x+7)}}-{\frac {4}{9(x+4)}}}
Retrieved from "https://en.wikibooks.org/w/index.php?title=A-level_Mathematics/OCR/C4/Algebra_and_Graphs&oldid=3249485"
Book:A-level Mathematics/OCR/C4
|
Create symbolic functions - MATLAB symfun - MathWorks France
Create and Define Symbolic Functions
Return Body and Arguments of Symbolic Function
Combine Two Symbolic Functions
f(inputs) = formula creates the symbolic function f. For example, f(x,y) = x + y. The symbolic variables in inputs are the input arguments. The symbolic expression formula is the body of the function f.
f = symfun(formula,inputs) is the formal way to create a symbolic function.
Define the symbolic function f(x,y) = x + y. First, create the function by using syms. Then define the function.
x+y
Find the value of f at x = 1 and y = 2.
3
Define the function again by using the formal way.
x+y
Return the body of a symbolic function by using formula. You can use the body for operations such as indexing into the function. Return the arguments of a symbolic function by using argnames.
Index into the symbolic function [x^2, y^4]. Since a symbolic function is a scalar, you cannot directly index into the function. Instead, index into the body of the function.
{x}^{2}
{y}^{4}
Return the arguments of the function.
\left(\begin{array}{cc}x& y\end{array}\right)
Create two symbolic functions.
Combine the two symbolic functions into another symbolic function
h\left(x\right)
with the data type symfun.
\left(\begin{array}{c}2 {x}^{2}-x\\ 3 {x}^{2}+2 x\end{array}\right)
h\left(x\right)
x=1
x=2
\left(\begin{array}{c}1\\ 5\end{array}\right)
\left(\begin{array}{c}6\\ 16\end{array}\right)
You can also combine the two functions into an array of symbolic expressions with the data type sym.
\left(\begin{array}{c}2 {x}^{2}-x\\ 3 {x}^{2}+2 x\end{array}\right)
Index into h_expr to access the first and the second symbolic expressions.
2 {x}^{2}-x
3 {x}^{2}+2 x
formula — Function body
symbolic expression | vector of symbolic expressions | matrix of symbolic expressions
Function body, specified as a symbolic expression, vector of symbolic expressions, or matrix of symbolic expressions that can be converted to sym data type
Example: x + y
inputs — Input argument or arguments of function
Input argument or arguments of a function, specified as a symbolic variable or an array of symbolic variables, respectively.
f — Symbolic function
symfun object
Symbolic function, returned as a symfun object.
While the data type of the function f is symfun, the data type of the evaluated function, such as f(1,2), is sym.
|
Rosenzweig – University Days
In June, I announced in #panfrost that I would be taking a break from Panfrost over the summer in order to focus on my summer classes, particularly during the hectic month of July. Luckily for my little graphics drivers, the summer has now elapsed, and my 3D voodoo has resumed. No command stream shall be left untouched. Hehehehe.
My first class was “Writing about Social and Ethical Issues”, a class I take to heart as a free software advocate. A major course theme was the human mind’s resolution of ethical dilemmas. Our texts argued for intuitionism over rationalism – that is, we judge with our heart, and our brains merely provides confirmation bias – with two major implications. First, my acceptance of intuitionism led me into a downward spiral as I attempted to remember how I came to the free software movement in the first place, if not for reason. Second, once I reevaluated the place of proprietary software in society and came to similar conclusions, we learned about scientifically backed persuasive strategies. For instance, we discussed “moral reframing”, the idea that as rhetoricians, we need to appeal to the values of our audience rather than our own; these sets of values likely will not align, or otherwise our audience would already agree with us, and there would be no persuasion needed.
Moral reframing and its associated persuasive techniques provide an interesting opportunity for free software argumentation. So, for my final paper in this writing class, I wrote a persuasive piece criticising Digital Restrictions Management based on non-technical issues, rather than using free software rhetoric. Was I successful? Maybe. Maybe not. You can read the paper and decide for yourself. For some more background, you can also read the corresponding reflection.
Anyway, back to graphics. My other class, perhaps more in line with my Panfrostian interests, was “Linear Algebra and Differential Equations”. Yes, you read that correctly: my work on 3D graphics to date was conducted by a tonta with no formal education in Linear Algebra, better known as “3D math”. Finally, my fellow Panfrostite Connor Abbott need not wag his metaphorical head at me:
<cwabbott> i'd read through the GL spec first, to understand exactly what's going on
<cwabbott> it sounds like you've taken linear algebra, so the math won't be too hard to understand
<alyssa> cwabbott: just finished multivariate calculus, planning to take linear algebra over the summer
<cwabbott> alyssa: ok... actually, i'd recommend taking them in the opposite order
<alyssa> too late ;P
<cwabbott> oh well
So be it; I have now completed the course, so Connor need not “oh well” me any longer.
Oh well. OpenGL aside, Linear Algebra is a fascinating course in its own right. Also, everything is written vertically, so expect to purchase a lot of notebook paper. My compact note days of Multivariable Calculus are over. I think I should have invested in a CVS loyalty card with the shear number of dead trees involved in matrix algebra.
Although there were sadly no official programming assignments in the class, as part of my exam preparation, I implemented some animation demos with Linear Algebra in Tosh. I focused on the use of 3D mathematics, also known as “Chapter 2” or "geometric linear transformations in
\mathbb{R}^3
. The first, Parabolic Paranoia, creates some psychadelic illusions by subjecting many symmetric copies of the graph of
y = x^2
to numerous linear and the occasional affine transformations. The other, Run!, animates a run cycle on a stick figure, drawn with 2D turtle graphics with lines and circles as the primitives, but modeling and animating the figure in a 3D world, implementing perspective projection to translate from the 3D description to 2D commands. I even threw in a simple cast shadow for the shear amount of puns. (Did I already make that joke last paragraph?
\sqrt{-1}
caramba.)
Animations aside, my third and final (unofficial) project of the semester was perhaps the most interesting. In addition, it is possibly novel1: reconstructing depth information from 2D images according to symmetry-based rules in order to transform the image in 3D space. The implementation is still incomplete and slow – I implemented it as an academic project in the Python scientific computing stack, ¿what do you expect! – but I do have a working prototype. The basic idea is to assign a depth value to each pixel based on the perpendicular distance between that pixel and a particular axis running through the object. That depth is tacked on to each pixel’s
\mathbb{R}^2
coordinates to produce ordered triplets in
\mathbb{R}^3
. These coordinates can then be transformed in
\mathbb{R}^3
using standard linear algebra, like affine transformations and rotations. Finally, the transformed coordinates are orthogonally projected back into
\mathbb{R}^2
, throwing away the transformed
z
-coordinate. Thus, the algorithm gives a (highly nonlinear) mapping from source image coordinates to destination image coordinates, which can then be rendered using standard forward mapping techniques. Once the implementation is fleshed out further, I’ll write up the algorithm more thoroughly. But for now, click for demos.
In other words…. Local Linear Algebra Student Discovers Weird Trick for Transforming 2D Images (3D Modelers Hate Her!)
These two classes ate up most of my academic time – but I did not stop there. As you know if you read my introductory blog post on the Free Software Foundation website and cross-posted here, I am one of the FSF’s summer interns, working with single-board computers, remote server administration, and liberating PayPal’s proprietary JavaScript. At the FSF, I have met a number of passionate, like-minded individuals – yes, that includes Richard Stallman who signed my GPG key! – and have grown my appreciation of the free software movement as I had hoped. My internship will continue remotely until the end of August.
Nevertheless, after my finals in my two classes, I returned home last weekend. (In)conveniently, just as my summer term finished, my fall classes have begun. The upshot is that I am now home with time to play with graphics.
For Panfrost, I have been focusing on supporting the T800 family of GPUs. Back in May, PINE64 generously offered to send me a ROCKPRO64 single-board computer, a development board for the Rockchip RK3399 used in the Samsung Chromebook Plus. The RK3399, succeeding the RK3288 (my prior focus), features a Mali T860 GPU, an upgrade over the RK3288’s Mali T760. Accordingly, the new board provides plenty of opportunities for improving Panfrost device support into the nebulous world of the T800.
Spoiler alert: the board is half-way between the preceding T700 and the succeeding Bifrost architecture. As you know if you hang out in #panfrost and have the opportunity to listen to my GPU thought dumps via the neocortex-to-IRC pipeline, there is some exciting related news in the pipeline…
…but for now, you’ll have to enjoy the cliffhanger!
I couldn’t find anything similar to what I wanted on the web, but I also didn’t look too hard.↩︎
|
G
e
G
M
\mathrm{μ} :G→.M
\mathrm{μ}\left(e, x\right) = x
\mathrm{μ}\left(a*b,x\right) = \mathrm{μ}\left(a, \mathrm{μ}\left(b, x\right)\right)
a, b ∈G
x ∈ M
\mathrm{μ},
{\mathrm{μ}}_{1,a}:M→M
{\mathrm{\mu }}_{1,a}\left(x\right) = \mathrm{μ}\left(a, x\right)
{\mathrm{μ}}_{2,x}:G→M
{\mathrm{\mu }}_{2,x}\left(a\right) = \mathrm{μ}\left(a, x\right)
\mathrm{μ}
{\mathrm{Γ}}_{\mathrm{μ}}
M
{\mathrm{\mu }}_{2,x}
G
{\mathrm{Γ}}_{\mathrm{μ} }
{\mathrm{\mu }}_{2,x}
and evaluating the results at the identity.
\mathrm{μ}
{\mathrm{Γ}}_{\mathrm{μ}} = \mathrm{Γ}
{\mathrm{\mu }}_{1,a}
{\mathrm{\mu }}_{2,x}
\mathrm{μ}
\mathrm{𝔤}
\mathrm{Γ}
\mathrm{𝔤}\mathit{ }
\mathrm{𝔤}
\mathrm{Γ}
\mathrm{𝔤}
\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):
\mathrm{with}\left(\mathrm{GroupActions}\right):
\mathrm{with}\left(\mathrm{LieAlgebras}\right):
\mathrm{with}\left(\mathrm{Library}\right):
\left[x,y\right].
\mathrm{DGsetup}\left([x,y],M\right):
\mathrm{Γ}
\mathrm{Gamma}≔\mathrm{evalDG}\left([\mathrm{D_x},\mathrm{D_y},y\mathrm{D_x}]\right)
\textcolor[rgb]{0,0,1}{\mathrm{Γ}}\textcolor[rgb]{0,0,1}{:=}\left[\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\right]
\mathrm{Γ}
\mathrm{LieAlgebraData}\left(\mathrm{Gamma}\right)
\left[\left[\textcolor[rgb]{0,0,1}{\mathrm{e2}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e3}}\right]\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{e1}}\right]
\mathrm{DGsetup}\left([\mathrm{z1},\mathrm{z2},\mathrm{z3}],G\right):
\mathrm{μ1}≔\mathrm{Action}\left(\mathrm{Gamma},G\right)
\textcolor[rgb]{0,0,1}{\mathrm{μ1}}\textcolor[rgb]{0,0,1}{:=}\left[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{z3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{z2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{z3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{z1}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{z2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\right]
\mathrm{newGamma}≔\mathrm{InfinitesimalTransformation}\left(\mathrm{μ1},[\mathrm{z1},\mathrm{z2},\mathrm{z3}]\right)
\textcolor[rgb]{0,0,1}{\mathrm{newGamma}}\textcolor[rgb]{0,0,1}{:=}\left[\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\right]
\mathrm{Γ2}≔\mathrm{evalDG}\left([y\mathrm{D_x},\mathrm{D_x},\mathrm{D_y}]\right)
\textcolor[rgb]{0,0,1}{\mathrm{Γ2}}\textcolor[rgb]{0,0,1}{:=}\left[\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\right]
\mathrm{L2}≔\mathrm{LieAlgebraData}\left(\mathrm{Γ2},\mathrm{Alg2}\right)
\textcolor[rgb]{0,0,1}{\mathrm{L2}}\textcolor[rgb]{0,0,1}{:=}\left[\left[\textcolor[rgb]{0,0,1}{\mathrm{e1}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e3}}\right]\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{e2}}\right]
\mathrm{DGsetup}\left(\mathrm{L2}\right)
\textcolor[rgb]{0,0,1}{\mathrm{Lie algebra: Alg2}}
\mathrm{Adjoint}\left(\right)
\left[\left[\begin{array}{rrr}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\end{array}\right]\textcolor[rgb]{0,0,1}{,}\left[\begin{array}{rrr}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\end{array}\right]\textcolor[rgb]{0,0,1}{,}\left[\begin{array}{rrr}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\end{array}\right]\right]
\mathrm{μ1},B≔\mathrm{Action}\left(\mathrm{Γ2},G,\mathrm{output}=["ManifoldToManifold","Basis"]\right)
\textcolor[rgb]{0,0,1}{\mathrm{μ1}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{B}\textcolor[rgb]{0,0,1}{:=}\left[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{z2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{z1}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{z3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\right]\textcolor[rgb]{0,0,1}{,}\left[\left[\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}\right]\textcolor[rgb]{0,0,1}{,}\left[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}\right]\textcolor[rgb]{0,0,1}{,}\left[\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}\right]\right]
\mathrm{newGamma}≔\mathrm{InfinitesimalTransformation}\left(\mathrm{μ1},[\mathrm{z1},\mathrm{z2},\mathrm{z3}]\right)
\textcolor[rgb]{0,0,1}{\mathrm{newGamma}}\textcolor[rgb]{0,0,1}{:=}\left[\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\right]
\mathrm{map}\left(\mathrm{DGzip},B,\mathrm{Γ2},"plus"\right)
\left[\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\right]
\mathrm{DGsetup}\left([x,y],M\right):
\mathrm{Γ3}≔\mathrm{Retrieve}\left("Gonzalez-Lopez",1,[22,17],\mathrm{manifold}=M\right)
\textcolor[rgb]{0,0,1}{\mathrm{Γ3}}\textcolor[rgb]{0,0,1}{:=}\left[\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{,}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\right]
\mathrm{DGsetup}\left([\mathrm{z1},\mathrm{z2},\mathrm{z3},\mathrm{z4},\mathrm{z5}],\mathrm{G3}\right)
\textcolor[rgb]{0,0,1}{\mathrm{frame name: G3}}
\mathrm{\mu }≔\mathrm{Action}\left(\mathrm{Γ3},\mathrm{G3}\right)
\textcolor[rgb]{0,0,1}{\mathrm{μ}}\textcolor[rgb]{0,0,1}{:=}\left[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{z5}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{z5}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{z1}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{z2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{z3}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{z4}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\right]
\mathrm{InfinitesimalTransformation}\left(\mathrm{\mu },[\mathrm{z1},\mathrm{z2},\mathrm{z3},\mathrm{z4},\mathrm{z5}]\right)
\left[{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{,}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\right]
\mathrm{DGsetup}\left([x,y,u,v],\mathrm{M4}\right):
\mathrm{Γ4}≔\mathrm{Retrieve}\left("Petrov",1,[32,6],\mathrm{manifold}=\mathrm{M4}\right)
\textcolor[rgb]{0,0,1}{\mathrm{Γ4}}\textcolor[rgb]{0,0,1}{:=}\left[\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_u}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_u}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_u}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\right]
\mathrm{DGsetup}\left([\mathrm{z1},\mathrm{z2},\mathrm{z3},\mathrm{z4}],\mathrm{G4}\right)
\textcolor[rgb]{0,0,1}{\mathrm{frame name: G4}}
\mathrm{\mu }≔\mathrm{Action}\left(\mathrm{Γ4},\mathrm{G4}\right)
\textcolor[rgb]{0,0,1}{\mathrm{μ}}\textcolor[rgb]{0,0,1}{:=}\left[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{z3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{z4}}\right)\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{z3}}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{cos}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{z4}}\right)\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{z3}}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{z1}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{z4}}\right)\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{z3}}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{cos}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{z4}}\right)\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{z3}}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{z2}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{v}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{v}\right]
\mathrm{InfinitesimalTransformation}\left(\mathrm{\mu },[\mathrm{z1},\mathrm{z2},\mathrm{z3},\mathrm{z4}]\right)
\left[\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_u}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_u}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_u}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\right]
|
The other day I came across an open source project named 'triangular'. The program converts an image to a polygon representation and the results are pretty impressive. It runs in Go and apparently is blazingly fast, like another polygon image converter I used years ago (I can't remember the name now - but it was also implemented in Go).
Example of their program output
Hopefully you'll agree, the output of their program is actually quite nice.
I thought: "How hard could it be to write my own version in Java?". I last wrote a Genetic Algorithm perhaps over 10 years ago after having heard about the algorithm whilst hungover and partially asleep in an AI lecture.
My thinking was that in just a few hours I could whip up a new implementation that I could run to build preview images for my blog pages, or to generally generate art. Ideally it should run ultra fast, but given the artistic nature of it, I'm willing to accept that it won't be perfect, as long as it resembles the target result.
As it turns out, it's a little harder than I thought.
This will be telling most people how to suck eggs - and for that I'm sorry. Feel free to skip through to a section better for yourself.
For those still reading, a genetic algorithm in it the simplest form is just an algorithm based on Darwin's evolution - survival of the fittest.
This is implemented in the code like so:
0001 /* Generate fitness levels (we don't want to keep doing this) */
0002 long[] curFit = processFitness();
0003 /* Select best based on fitness */
0004 SVG[] curBest = processBest(curFit);
0005 /* Breed the best partners */
0006 breedPopulation(curBest);
0007 /* Mutate them using the mutation rate */
0008 curPolys = (int)(polys * ((double)curGen / (double)runs)) + 1;
0009 mutatePopulation();
Here you can see the important steps in this algorithm:
Evaluate -- This is encapsulated by processFitness() and processBest(). This takes the population and generates each of their fitness values in curFit. The reason for separating the evaluation process in our implementation is that generating fitness values is really expensive.
Breed -- This consists of the breedPopulation() function and it takes the list of best performers curFit and re-populates the population with their off-spring.
Mutate -- Add some randomness into the genome of the population, allowing for "exploration" of the search space.
We stop doing this when the total error goes down to 0.1% or we have run out of computing time.
That's really it! The exciting stuff is really how each of these parts is implemented.
This is the first step - we need to figure out how each member of the population performs. For this implementation we convert the SVG into a bitmap image and then compare that to the original image. The larger the pixel error, the larger the number.
We do the euclidean distance of the RGB values, such that:
error={\sum }_{r,g,b}^{w\cdot h}\mathrm{log}\sqrt{\Delta {r}^{2}+\Delta {g}^{2}+\Delta {b}^{2}}
. We calculate the maximum error as
\mathrm{log}\sqrt{3×{256}^{2}}\cdot w\cdot h
. The reason for using log-error is to not over-weight large areas and to not over-weight small pixel changes.
Next we need to select a subset of the population for breeding which in the implementation uses a very basic search algorithm. It's not particularly efficient, but compared to our other loops it's very fast.
The breeding implementation is again pretty straight forwards, we randomly select two parents from our best list (ensuring we do not select the same parent twice) and they get down and dirty. We select a random mid-point in their internal polygon layers and we take some number of polygons from one parent, and one number of polygons from another.
We then insert the child into the population and continue until we re-fill the population list.
Next in the code we mutate the population, to allow the search space to be explored and to break out of local min/max optimizations.
In our implementation, we mutate two random layers and the latest layer. Why these? The latest layer will likely require the most adjust, therefore we want to concentrate our mutation efforts here. Meanwhile, we still want to be able to perform mutations to other layers to better optimize how the layers work together. Unfortunately, by mutating all of the layers together at the same time, we aren't able to observe the independent mutations effectively and ultimately we end up with a near-random result. We therefore have to limit the amount of experimentation done in each mutation.
We make a few other shortcuts that slightly speed up the implementation and generally make the result better.
As mentioned previously, we periodically add layers, allowing us to 'settle' each layer and generally find a good location for each added polygon.
We reduce the mutation for existing layers, meaning that they don't move to far once we have found some nice-ish location for them.
The colour is calculated by centroid pixel - meaning that we don't spend signifcant processing power picking a colour. This is also a hinderance, as it means the centroid therefore must be at a location that offers a nice colour for the entire polygon.
"Okay, enough of this, show me the goodies!" Sure thing! The following is the original image:
And for the output, it uses 300 polygons and took 15 minutes or so on a single core:
Bare in mind that this operated on a downsampled version and then was upsampled. I think artistically the results are actually pretty interesting for a few minutes of work. If you process for longer and add more polygons, it simply gets better and better output.
The SVG is here (but may not display well depending on your browser):
SVG polygon image
Next we have a graph displaying the error reduction over time for 600 polygons:
Error reduction over time
Note that it's very similar to that of what you might see from something like a neural network. As you can see, clearly you get diminishing returns as you continue to add polygons. The average error sits always just above the best.
For this project, I have a few things I would like to try:
Multi-thread - We should be able to get massive speed-ups by enabling multiple cores.
Pre-place polygons - This should massively reduce the time to finding a solution - we spend significant time figuring out the fact that we should probably cover holes and covering other polygons is pretty wasteful.
|
(Main) data structures - Tales of Science & Data
A data structure is a higher level form based on top of primitive data types (integers, floats/doubles, characters, booleans). Let's quickly go through the main ones.
Lists (collections) of elements. Note that Python has the concept of array and that of list: they differ both in their nature and what you can do with them and their general purpose. See the article in the references for a nice comparison.
This is a list of elements (called nodes) linked one to the next: a node contains the element and the link to the next node. This structure allows for easy replacement, insertion and deletion of elements as they are not stored in contiguous places in the memory, thanks to the links.
Hash tables are key - value pairs, they are super useful: you can access a value by calling its key so the lookup is straightforward (in O(1) time). Hash tables use a hash function to map the key (which can be of any type) to a numerical value, so that given a key the computer knows where is the value to be accessed and can do in constant time (without the need to scroll).
Dictionaries in Python are hash tables - have a read at the blog in the references about this.
A stack is a data structure where you put elements in one on top of the other and it uses the LIFO philosophy to get data out: "last in first out" means that you access elements in the reverse order than the one used to put them in.
A queue is similar, but uses the FIFO philosophy: "first in first out", basically elements come out from the other end than the one you've inserted them in.
Graphs (and trees, and heaps)
Graphs have nodes and connections between them which determine their relation. There is a whole branch of mathematics devoted to their study (graph theory).
A tree is a special type of graph where there is a clear relation between a parent node and a child node, so no cycles appear (there are hierarchical relations).
There are several subtypes of graphs, identified by their main characteristics, e.g. binary graphs (trees) are those where each node as at most 2 children nodes.
A heap is a type of tree structure where data is stored in a way that a parent node either:
always contains values greater than its children nodes (max heap), so that the root node is the maximum
always contains values smaller than its children nodes (min heap), so that the root node is the minimum
These features make heaps data structures that are partially ordered (the ordering is in the vertical direction, not in the horizontal one). In a binary heap where each node has 2 children nodes, because at level
x
2^x
nodes - this means that the length of a binary heap with
nodes is
\log_2 n
An object is a collection of data and, sometimes, functions that work on this data, put together in a coherent place. Normally, an object is implemented via a class.
For example, a AlarmClock object will represent the alarm clock you have on your bedside table: it will store data for the date and time and will have methods to update the time as it goes and ring based on some criteria (always at the same time daily or with better sophistication).
You can use them to build objects that inherit characteristics from others and specialise. For instance, you can do a general class for Vehicle and one for a Train that inherits from it as it is a subclass of it (the basic features will be inherited and specific ones are implemented for it only).
The Python docs for classes is quite educational!
Gayle Laakmann McDowell, Cracking the Coding Interview, CareerCup
Kateryna Koidan, Array vs. List in Python – What's the Difference?, an article on LearnPython.com
Aaron Meurer's blog, What happens when you mess with hashing in Python
Classes in Python, from the docs
|
Division | Brilliant Math & Science Wiki
Aditya Virani, Mahindra Jain, Ashish Menon, and
Division is a basic algebraic operation where we split a number into equal parts of groups. If we had 12 sweets, and wanted to share it equally amongst 3 people, then each of them will get 4 sweets. This is expressed as
12 \div 3 = 4.
Division is also known as the inverse (or "opposite") of multiplication. For example, since
3 \times 4 = 12
, we can divide by 4 on both sides to get
3 = 12 \div 4
. As such, knowing the multiplication tables can be helpful with division.
36 \div 9?
Repeatedly adding nine, we can see that
\begin{array} { l l l l r } 1 & \times & 9 & = & 9 \\ 2 & \times & 9 & = & 18 \\ 3 & \times & 9 & = & 27 \\ 4 & \times & 9 & = & 36 \\ 5 & \times & 9 & = & 45 \\ &&&\vdots \end{array}
4 \times 9 = 36
4 = 36 \div 9
_\square
56 \div 8?
Repeatedly adding 8, we can see that
\begin{array} { l l l l r } 1 & \times & 8 & = & 8 \\ 2 & \times & 8 & = & 16 \\ 3 & \times & 8 & = & 24 \\ 4 & \times & 8 & = & 32 \\ 5 & \times & 8 & = & 40 \\ 6 & \times & 8 & = & 48 \\ 7 & \times & 8 & = & 56 \\ 8 & \times & 8 & = & 64 \\ &&&\vdots \end{array}
7 \times 8 = 56
7 = 56 \div 8
_\square
10 ÷ 5?
\begin{aligned} 5 × 1 & = 5\\ 5 × 2 & = 10\\ 10 ÷ 5 & = 2.\ _\square \end{aligned}
132 \div 11?
Repeatedly adding 11, we can see that
\begin{array} { l l l l r } 1 & \times & 11 & = & 11 \\ 2 & \times & 11 & = & 22 \\ 3 & \times & 11 & = & 33 \\ &&&\vdots \\ 10 & \times & 11 & = & 110 \\ 11 & \times & 11 & = & 121 \\ 12 & \times & 11 & = & 132 \\ 13 & \times & 11 & = & 143 \\ &&&\vdots \end{array}
12 \times 11 = 132
12 = 132 \div 11
_\square
A teacher distributes 56 pencils to her class, and the students each get an equal number of pencils. If there are 14 children in her class, how many pencils does each student receive?
We are trying to find
56\div14.
\begin{aligned} 1 \times 14 & = 14 \\ 2 \times 14 & = 28 \\ 3 \times 14 & = 42 \\ 4 \times 14 & = 56. \end{aligned}
4 \times14 = 56
56\div14=4
. Thus, each student gets 4 pencils.
_\square
Johnny received his $21 allowance for this week. Johnny thinks, "In order not to run out of money on the weekends, I should use an equal amount of money every day." How much money per day can Johnny spend?
Since there are 7 days in a week, the amount of money Johnny can spend per day (in dollars) is equal to
21\div7.
\begin{aligned} 1 \times 7 & = 7 \\ 2 \times 7 & = 14 \\ 3 \times 7 & = 21. \end{aligned}
3 \times 7 = 21
21\div7=3
. Thus, Johnny can spend $3 every day.
_\square
Cite as: Division. Brilliant.org. Retrieved from https://brilliant.org/wiki/calculation-division/
|
Maximum Leverage on Maker | Ian Macalinao
Maximum Leverage on Maker
First off, this article assumes familiarity with Maker CDP's. (If not, I highly suggest reading up on it!)
Recall that the liquidation ratio of a CDP is the collateral-to-debt ratio of the CDP. For example, if the liquidation ratio is 150% on ETH/USD, if ETH is $100 and you have 1 ETH in the CDP, you may accrue up to $66 of debt (that is, generate up to $66 Dai) as your collateral is worth 150% of your debt. This gives us a leverage ratio of 1.6x. But can we go higher?
Assuming enough liquidity exists, we can then buy 0.66 ETH with this Dai.
We can then put this ETH back into our CDP to draw more Dai. We can draw
\$66.66 * 1/1.5 = \$44.44
more Dai from our CDP. We can keep doing this forever to generate more Dai and thus more leverage; however, we get less Dai each time. But how much?
\lambda
represent the liquidation ratio and
L
represent our maximum leverage ratio. We compute
L
to be the following
L = \frac{1}{l} + \frac{1}{\lambda^2} + \frac{1}{\lambda^3} + \ldots
But this is just a geometric series with
a = 1
r = 1/\lambda
. Using the infinite sum of geometric series formula, we get:
L = \frac{1}{1 - r} = \frac{1}{1 - \frac{1}{\lambda}}
which in the case of
\lambda = 1.5
, 3x leverage.
Theoretical vs Actual Leverage
The above calculation accounts for the maximum leverage in an infinitely liquid ETH/DAI market with an instant ability to fund a CDP. However, there are several factors that come into play:
Finite liquidity on ETH/DAI
CDP funding and trade delays
Finite liquidity
The amount of ETH one can buy with Dai is currently very limited, and the spread is significantly wide. The above calculations make the assumption that one can continue to buy more ETH at the market price, which is very far from the truth.
However, this liquidity will improve over time, as the future of the Maker system requires this liquidity to exist for its auto-liquidation system to work.
There are several actions one must perform to increase the collateral locked in the CDP contract, all taking one block each:
Generating Dai from the CDP
Buying W-ETH from the WETH/DAI market
Converting W-ETH to PETH
Locking the newly generated PETH into the CDP.
One should note that step 3 will not exist in the final version of Maker -- pooled Ether is a workaround to the MKR token generation system not being in place. However, this means that at a minimum, three blocks will pass before the CDP can be refunded.
If one generates the maximum amount of Dai from the CDP, they are at immediate risk of liquidation if the price of Ether drops even one cent. Since the process of refunding the CDP is not atomic, there is a very real risk of this liquidation taking place.
In theory, one could write a smart contract that performs this entire 4-step process in one transaction with a specified minimum accepted price. Before this exists, however, one is subject to the aformentioned risks.
(If you liked this post, join our crypto Discord!)
|
Probability Problem: Digits in a circle - An Phạm | Brilliant
Digits in a circle
Suppose we place
K
distinct digits around a circle such that each pair of adjacent digits (read either clockwise or counterclockwise) forms a number that's divisible by
7.
What's the maximum possible value of
K?
For example, the diagram to the right illustrates a solution for
K=3.
Note that 35, 56, and 63 are all multiples of 7.
Note: Not all pairs of digits need to be read in the same direction. Some pairs may be read clockwise and others may be read counterclockwise.
|
Attractor metagene algorithm for feature engineering using mutual information-based learning - MATLAB metafeatures - MathWorks Nordic
Apply Attractor Metagene Algorithm to Gene Expression Data
ReturnUnique
UniqueTolerance
GSorted
GSortedInd
Attractor Metagene Algorithm
Attractor metagene algorithm for feature engineering using mutual information-based learning
M = metafeatures(X)
[M,W] = metafeatures(X)
[M,W,GSorted] = metafeatures(X,G)
[M,W,GSorted,GSortedInd] = metafeatures(___)
[___] = metafeatures(___,Name,Value)
[___] = metafeatures(T)
[___] = metafeatures(T,Name,Value)
M = metafeatures(X) returns the weighted sums of features M in X using the attractor metagene algorithm described in [1].
M is a r-by-n matrix. r is the number of metafeatures identified during each repetition of the algorithm. The default number of repetitions is 1. By default, only unique metafeatures are returned in M. If multiple repetitions result in the same metafeature, then just one copy is returned in M. n is the number of samples (patients or time points).
X is a p-by-n numeric matrix. p is the number of variables, features, or genes. In other words, rows of X correspond to variables, such as measurements of gene expression for different genes. Columns correspond to different samples, such as patients or time points.
[M,W] = metafeatures(X) returns a p-by-r matrix W containing metafeatures weights. M = W'*X. p is the number of variables. r is the number of unique metafeatures or the number of times the algorithm is repeated (the default is 1).
[M,W,GSorted] = metafeatures(X,G) uses a p-by-1 cell array of character vectors or string vector G containing the variable names and returns a p-by-r cell array of variable names GSorted sorted by the decreasing weight.
The ith column of GSorted lists the feature (variable) names in order of their contributions to the ith metafeature.
[M,W,GSorted,GSortedInd] = metafeatures(___) returns the indices GSortedInd such that GSorted = G(GSortedInd).
[___] = metafeatures(___,Name,Value) uses additional options specified by one or more Name,Value pair arguments.
[___] = metafeatures(T) uses a p-by-n table T. Gene names are the row names of the table. M = W'*T{:,:}.
[___] = metafeatures(T,Name,Value) uses additional options specified by one or more Name,Value pair arguments.
It is possible that the number of metafeatures (r) returned in M can be fewer than the number of replicates (repetitions). Even though you may have set the number of replicates to a positive integer greater than 1, if each repetition returns the same metafeature, then r is 1, and M is 1-by-n. This is because, by default, the function returns only unique metafeatures. If you prefer to get all metafeatures, set 'ReturnUnique' to false. A metafeature is considered unique if the Pearson correlation between it and all previously found metafeatures is less than the 'UniqueTolerance' value (the default value is 0.98).
Load the breast cancer gene expression data. The data was retrieved from the Cancer Genome Atlas (TCGA) on May 20, 2014 and contains gene expression data of 17814 genes for 590 different patients. The expression data is stored in the variable geneExpression. The gene names are stored in the variable geneNames.
load TCGA_Breast_Gene_Expression
The data has several NaN values.
sum(sum(isnan(geneExpression)))
Use the k-nearest neighbor imputation method to replace missing data with the corresponding value from an average of the k columns that are nearest.
geneExpression = knnimpute(geneExpression,3);
There are three common drivers of breast cancer: ERBB2, estrogen, and progestrone. metafeatures allows you to seed the starting weights to focus on the genes of interest. In this case, set the weight for each of these genes to 1 in three different rows of startValues. Each row corresponds to initial values for a different replicate (repetition).
erbb = find(strcmp('ERBB2',geneNames));
estrogen = find(strcmp('ESR1',geneNames));
progestrone = find(strcmp('PGR',geneNames));
startValues = zeros(size(geneExpression,1),3);
startValues(erbb,1) = 1;
startValues(estrogen,2) = 1;
startValues(progestrone,3) = 1;
Apply the attractor metagene algorithm to the imputed data.
[meta, weights, genes_sorted] = metafeatures(geneExpression,geneNames,'start',startValues);
The variable meta has the value of three metagenes discovered for each sample. Plot these three metagenes to gain insight into the nature of gene regulation across different phenotypes of breast cancer.
plot3(meta(1,:),meta(2,:),meta(3,:),'o')
xlabel('ERBB2 metagene')
ylabel('Estrogen metagene')
zlabel('Progestrone metagene')
Based on the plot, observe the following.
There is a group of points clustered together with low values for all three metagenes. Based on mRNA levels, these could be triple-negative or basal type breast cancer.
There is a group of points that have high estrogen receptor metagene expression and span across both high and low progestrone metagene expression. There are no points with high progestrone metagene expression and low estrogen metagene expression. This is consistent with the observation that ER-/PR+ breast cancers are extremely rare [3].
The remaining points are the ERBB2 positive cancers. They have less representation in this data set than the hormone-driven and triple negative cancers.
Data, specified as a numeric matrix. Rows of X correspond to variables, such as measurements of gene expression. Columns correspond to different samples, such as patients or time points.
G — Variable names
Variable names, specified as a cell array of character vectors or string vector.
T — Data
Data, specified as a table. The row names of the table correspond to the names of features or genes, and the columns represent different samples, such as patients or time points.
Example: 'Replicates',5 specifies to repeat the algorithm five times.
Alpha — Tuning parameter for the number of metafeatures
Tuning parameter for the number of metafeatures, specified as the comma-separated pair consisting of 'Alpha' and a positive number. This parameter controls the nonlinearity of the function that calculates the weights as described in the Attractor Metagene Algorithm. As alpha increases, the number of metafeatures tends to increase. This parameter is often the most important parameter to adjust in the analysis of a data set.
Start — Option for choosing initial weights
'random' (default) | 'robust' | matrix
Option for choosing initial weights, specified as the comma-separated pair consisting of 'Start' and a character vector, string, or matrix. This table summarizes the available options.
'random' Initialize the weights to a vector of positive weights chosen uniformly at random and scaled such that they sum to 1. Choose a different initial weight vector for each replicate. This option is the default.
'robust' If X or T has n columns, run the algorithm n times. On the ith evaluation of the algorithm, the weights are initialized to all zeros with the exception of the ith weight, which is set to 1. This option is useful when you are attempting to find all metafeatures of a data set.
matrix n-by-r matrix of initial weights. The algorithm runs r times. The weights in the ith run of the algorithm are initialized to the ith column of the matrix.
Example: 'Start','robust'
Replicates — Number of times to repeat the algorithm
Number of times to repeat the algorithm, specified as the comma-separated pair consisting of 'Replicates' and a positive integer. This option is valid only with the 'random' start option. The default is 1.
ReturnUnique — Unique metafeatures flag
Unique metafeatures flag, specified as the comma-separated pair consisting of 'ReturnUnique' and true or false. If true, then only the unique metafeatures are returned. The default is true.
This option is useful when the algorithm is repeated multiple times. By setting this option to true, you choose to look at just the unique metafeatures since the same set of metafeatures can be discovered for different initializations.
A metafeature is considered unique if the Pearson correlation between it and all previously found metafeatures is less than the 'UniqueTolerance' value (the default value is 0.98).
To run the algorithm multiple times, set the 'Replicates' name-value pair argument or the 'Start' option to 'robust' or a matrix with more than 1 row.
Example: 'ReturnUnique',false
UniqueTolerance — Tolerance for metafeature uniqueness
0.98 (default) | real number between 0 and 1
Tolerance for metafeature uniqueness, specified as the comma-separated pair consisting of 'UniqueTolerance' and a real number between 0 and 1.
A metafeature is considered unique if the Pearson correlation between it and all previously found metafeatures is less than the 'UniqueTolerance' value.
Example: 'UniqueTolerance',0.90
Options — Options for controlling the algorithm
Options for controlling the algorithm, specified as the comma-separated pair consisting of 'Options' and a structure. This table summarizes these options.
Display Level of output display. Choices are 'off' or 'iter'. The default is 'off'.
Tolerance If M changes by less than the tolerance in an iteration, then the algorithm stops. The default is 1e-6.
Streams A RandStream object. If you do not specify any streams, metafeatures uses the default random stream.
UseParallel Logical value indicating whether to perform calculations in parallel if a parallel pool and Parallel Computing Toolbox™ are available. For problems with large data sets relative to the available system memory, running in parallel can degrade performance. The default is false.
Example: 'Options',struct('Display','iter')
M — Metafeatures
Metafeatures, returned as a numeric matrix. It is an r-by-n matrix containing the weighted sums of the features in X. r is the number of replicates performed by the algorithm. n is the number of different samples such as time points or patients.
W — Metafeatures weights
Metafeatures weights, returned as a numeric matrix. It is a p-by-r matrix. p is the number of variables. r is the number of replicates performed by the algorithm.
GSorted — Sorted variable names
Sorted variable names, returned as a cell array of character vectors. It is a p-by-r cell array. The names are sorted by decreasing weight. The ith column of the GSorted lists the variable names in order of their contributions to ith metafeature.
If GSorted is requested without G or if T.Properties.RowNames is empty, then the algorithm names each variable (feature) as Vari, which corresponds to the ith row of X.
GSortedInd — Index to GSorted
Index to GSorted, returned as a matrix of indices. It is a p-by-r matrix. The indices satisfy GSorted = G(GSortedInd) or GSorted = T.Properties.RowNames(GSortedInd).
The attractor metagene algorithm [1] is an iterative algorithm that converges to metagenes with important features. A metagene is defined as any weighted sum of gene expression using a nonlinear distance metric. The distance metric is a nonlinear variant of mutual information using binning and splines as described in [2]. In fact, the use of mutual information as a distance metric is one of major benefits of this algorithm since mutual information is a robust information theoretic approach to determine the statistical dependence between variables. Therefore, it is useful for analyzing relationships among gene expression. Another advantage is that the results of the algorithm tend to be more clearly linked with a phenotype defined by gene expression.
The algorithm is initialized by either random or user-specified weights and proceeds in these steps.
The estimate of a metagene during the ith iteration of the algorithm is
{M}_{i}={W}_{i}*G
, where Wi is a vector of weights of size 1-by-p (number of genes), and G is the gene expression matrix of size p-by-n (number of samples).
Update the weights by
{W}_{j,i+1}=J\left({M}_{i},{G}_{j}\right)
, where Wj,i+1 is the jth element of Wi+1, Gj is the jth row of G, and J is a similarity metric, which is defined as follows.
If the Pearson correlation between Mi and Gj is greater than 0, then
J\left({M}_{i},{G}_{j}\right)=I{\left({M}_{i},{G}_{j}\right)}^{\alpha }
I\left({M}_{i},{G}_{j}\right)
is the measure of mutual information between two genes with minimum value 0 and maximum value 1, and α is any nonnegative number.
If the correlation is less than or equal to 0, then
J\left({M}_{i},{G}_{j}\right)=0
The algorithm iterates until the change in Wi between iterations is less than the defined tolerance, that is,
‖{W}_{i}-{W}_{i-1}‖<tolerance
or the maximum number of iterations is reached.
The Role of α
In the similarity metric of the algorithm, the parameter α controls the degree of nonlinearity. As α increases, the number of metagenes tends to increase. If α is sufficiently large, then each gene approximately becomes an attractor metagene. If α is zero, then all weights remain equal to each other. Therefore, there is only one attractor metagene representing the average of all genes.
Therefore, adjusting α for the data set under consideration is a key step in fine tuning the algorithm. In the case of [1], using the TCGA data from several types of cancer to identify attractor metagenes, α value of 5 resulted in between 50 and 150 attractor metagenes discovered from the data.
[1] Cheng, W-Y., Ou Yang, T-H., and Anastassiou, D. (2013). Biomolecular events in cancer revealed by attractor metagenes. PLoS Computational Biology 9(2): e1002920.
[2] Daub, C., Steuer, R., Selbig, J., and Kloska, S. (2004). Estimating mutual information using B-spline functions – an improved similarity measure for analysing gene expression data. BMC Bioinformatics 5, 118.
[3] Hefti, M.M., Hu, R., Knoblauch, N.W., Collins, L.C., Haibe-Kains, B., Tamimi, R.M., and Beck, A.H. (2013). Estrogen receptor negative/progesterone receptor positive breast cancer is not a reproducible subtype. Breast Cancer Research. 15:R68.
Set the 'UseParallel' field of the options structure to true and specify the 'Options' name-value pair argument in the call to this function.
For example: 'Options',struct('UseParallel',true)
relieff | sequentialfs | rankfeatures | randfeatures
|
MathML Render
This is quick one from me - but thought it was worth documenting based on the fact that there doesn't seem to be much information out there about it. I wanted to show some equations in a recent article about polygon images and thought I would share my solution to this problem.
Essentially I'm just going to explain how to setup pandoc with some 'minimal' JS to render MathML in most browsers. It's not perfect, but it's much lighter.
Mathematical Markup Language (MathML) is a mathematical markup language, an application of XML for describing mathematical notations and capturing both its structure and content. It aims at integrating mathematical formulae into World Wide Web pages and other documents. It is part of HTML5 and an ISO standard ISO/IEC 40314 since 2015.
It essentially is a more standardized version of TeX equations rendered in an XML format, that says "put this here, relative to this". One of the great things about it is that with no JS enabled on the page, Firefox will still render equations!
So this is a pretty awesome thing for displaying math natively in browsers, but currently only gecko-based browsers actually support it. This means that anybody using Chrome or Chromium won't see it, Microsoft browsers, Safari, etc.
Many people use MathJax to render, but using their current version 3 minified JS means adding a whopping 794.4kB per page! Just to render a few equations! Holy balls batman!
So the first place I check out is a little Google-foo (but in DuckDuckGo instead), and there are reccomendations to checkout the pandoc manual. Here are the options we are provided:
--mathjax[=URL] Use MathJax to display embedded TeX math in HTML output.
--mathml Convert TeX math to MathML (in epub3, docbook4, docbook5, jats, html4 and html5).
--webtex[=URL] Convert TeX formulas to <img> tags that link to an external script that converts formulas to images.
--katex[=URL] Use KaTeX to display embedded TeX math in HTML output.
--gladtex Enclose TeX math in <eq> tags in HTML output.
I checked out several of the options and had various criticisms of each:
--mathjax[=URL] - Is a super massive JS library.
--mathml - Only works for Firefox and Safari. There is hope that in the future it will have native support across all browsers though.
--webtex[=URL] - Relies on external services to render equations for you - really against the point of self-hosting my server.
--katex[=URL] - I didn't check out this option, but it is very large and will always depend on JS.
--gladtex - This is the closest to what I actually wanted, but introduces a bunch of problems in itself. It can generate images from equations, but I wanted them to be embedded into the page - otherwise I then have to figure out how to find somewhere to store the images and not mess up the auto-pull script in git. It was a lot more headache than I wanted.
Given --mathml's eventual native support in-browser without requiring JS, it seems like the best time investment and best future compatibility option. Given the big names behind it, I don't really see it going anywhere for now.
I'm using an old project by pshihn to render MathML instead, measuring in at just 75kB for the main script and 1.3kB for the PolyFill. The puts us nicely in a ten times saving.
Sure, the project is old and likely not rendering everything perfectly, but it works good enough for any equations I'm going to be writing on here. I can always upgrade to a beefier version at a later date.
So how does this all work with pandoc?
When I writing a document, I'll have something like this:
0001 We represent this as a half, $\frac{1}{2}$.
In the compile script, you then add --mathml option on the command line for pandoc.
This of course will only work for Firefox et al, so now you need to link your JS to get Chrome et al playing well. At the bottom of the page you add:
0002 <script src="/mathml.js" defer></script>
0003 <script src="/mathml-poly.js" defer></script>
The defer loads the script after the page is loaded. There are a few reasons for this:
Time till initial render is important. I want a read-able web-page ASAP, even if the equations look screwed. If you have potato internet, you want to be able to read the page as soon as possible.
It's unlikely there will even be equations on the page, so don't stall the rendering process for an unlikely use case.
Equations are really not that large in general, I wouldn't expect the web-page to move around tonnes anyway.
We represent this as a half,
\frac{1}{2}
Of course, it's not quite as simple as this. Chromium (and possibly Chrome) like to display annotations, which screws up the look after the rendering process. This can be fixed by adding the following CSS:
0004 annotation{ display: none; }
This just hides annotations, as they should be hidden.
Ideally I want to be able to detect if math is even used on the page at all and only add the resource if required. Almost 100kB for something that is mostly not used is still way too much for my liking.
At some point I will figure out how to statically render equations instead of having to use JS-based solutions - there is no reason at all that I couldn't just produce an SVG and embed it into the page. Unfortunately the support for such a thing is really not great at all at the moment, so I am stuck with this hacky option.
|
Spectral-Directional Emittance of CuO at High Temperatures | J. Heat Transfer | ASME Digital Collection
, Auburn, Alabama 36849
e-mail: pjones@eng.auburn.edu
Jones, P. D., Teodorescu, G., and Overfelt, R. A. (October 24, 2005). "Spectral-Directional Emittance of CuO at High Temperatures." ASME. J. Heat Transfer. April 2006; 128(4): 382–388. https://doi.org/10.1115/1.2165207
Spectral-directional emittance measurements for cupric oxide (CuO) are presented. The data cover polar angles of
0-84deg
from the surface normal, wavelengths between 1.5 and
8μm
, and temperatures between 400 and
700°C
. The data were generated using a radiometric, direct emission measurement method. The oxide was grown on a very clean, smooth, and mirror-like copper surface, heated in air at
700°C
until emission measurements became constant
(270h)
. X-ray diffraction and EDS analyses were performed to characterize the spatial and molecular composition of the copper oxide layer. It is generally found that CuO emittance decreases with increasing polar angle, increases with increasing wavelength, and increases with increasing temperature. Spectral-directional emittance values calculated from the Fresnel relations show good agreement with the measurements up to polar angles of
72deg
copper compounds, blackbody radiation, emissivity, refractive index, X-ray diffraction, chemical analysis
Copper, High temperature, Refractive index, Temperature, Wavelength, X-ray diffraction, Mirrors, Emissivity, Oxidation
Properties of Reactively-Sputtered Copper Oxide Thin Films
Czanderna
Optical Properties of Copper Oxide Films
Optical Properties of Some Metal Oxides in Solar Absorbers
Dorai-Raj
Spectral-Directional Emittance of Oxidized Copper
Oxidation and Reduction of Copper Oxide Thin Films
Oxidation of Copper at 600-1000°C
Waalib Singh
Kinetics of Oxidation of Copper Alloy Leadframes
High Temperature Oxidation of High Purity Copper
J. Jpn. Copper Brass Res. Assoc.
|
How does temperature affect the gain of an SiPM? | Hamamatsu Photonics
A silicon photomultiplier (SiPM) is a solid-state photodetector capable of single-photon detection due to its high 105 − 107 internal gain (μ). Only one other photodetector, a photomultiplier tube (PMT), can achieve gain of such magnitude. Since its introduction in the mid-1990s, the SiPM has been undergoing a rapid development improving the performance. Compared to the first prototype, some undesirable characteristics such as high dark count rate density and high probability of optical crosstalk and of afterpulsing have been substantially reduced. These improvements, together with advantages over a PMT such as, to name a few, lower operating voltage, immunity to magnetic fields, and ruggedness, make a SiPM a viable alternative to a PMT as the detector for applications involving very low light levels. Many applications require stability of the photodetector's gain with respect to temperature changes. Fluctuating temperature can affect the gain of a PMT and SiPM; to prevent this, the photodetector operates either at a controlled constant temperature or at the ambient temperature but employing a gain-compensation circuitry. This technical note discusses the origin of the gain-temperature dependence in a SiPM and methods that the practitioner can use to correct for it.
Origin of the gain
A SiPM is a rectangular array of square microcells. Each microcell is composed of a series combination of an avalanche photodiode (APD) and a quenching resistor RQ. All of the microcells are in parallel; thus, a SiPM has two terminals: anode and cathode. All of the APDs and quenching resistors are identical. To achieve high gain, the reverse bias VBIAS applied to the SiPM exceeds the breakdown voltage VBD of the APDs by up to several volts. Thus, the APDs operate in Geiger mode: once initiated, an avalanche would have been perpetual if it hasn't been quenched by an external (to the APD) circuitry, for example, RQ. A SiPM is a current source: in response to a photon, the output is a current pulse i(t) and the gain equals the area under the pulse or
\mu ={\int }_{0}^{\infty }i\left(t\right)dt
. One can show that to an excellent approximation, μ is given by:
\mu =\frac{\left({V}_{BIAS}-{V}_{BD}\right)\phantom{\rule[-0.1em]{0.4em}{0.5em}}{C}_{J}}{e}=\frac{∆V{C}_{J}}{e}
where ΔV is referred to as an "overvoltage," CJ is the capacitance of the APD's avalanche region and known as "junction capacitance," and e is the elementary charge. Equation 1 contains no explicit dependence on temperature, but there is an implicit dependence: it is well known that VBD is a function of temperature, VBD = VBD(T). Additionally, one may not rule out the possibility that CJ can also vary with T. If VBIAS is held constant, μ = μ(T) because ΔV = ΔV(T) and, possibly, CJ = CJ(T). The total derivative of μ with respect to T is given by:
\frac{d\mu }{dT}=\frac{\partial \mu }{\partial T}+\frac{\partial \mu }{\partial ∆V}\phantom{\rule[-0.1em]{0.05em}{0.5em}}\frac{d∆V}{dT}+\frac{\partial \mu }{\partial {C}_{J}}\phantom{\rule[-0.1em]{0.05em}{0.5em}}\phantom{\rule[-0.1em]{0.05em}{0.5em}}\frac{d{C}_{J}}{dT}
Because there is no explicit dependence of μ on T, the first term on the right hand side of Equation 2 is zero. Using Equation 1 and assuming that VBIAS is independent of T, after some rearranging, Equation 2 becomes:
\frac{d\mu }{dT}=\frac{1}{e}\phantom{\rule[-0.1em]{0.05em}{0.5em}}\left(\frac{d{C}_{J}}{dT}\phantom{\rule[-0.1em]{-0.05em}{0.5em}}\phantom{\rule[-0.1em]{-0.05em}{0.5em}}∆\phantom{\rule[-0.1em]{-0.05em}{0.5em}}\phantom{\rule[-0.1em]{-0.05em}{0.5em}}\phantom{\rule[-0.1em]{-0.05em}{0.5em}}\phantom{\rule[-0.1em]{-0.05em}{0.5em}}V-\frac{d{V}_{BD}}{dT}\phantom{\rule[-0.1em]{0.05em}{0.5em}}{C}_{J}\right)
Both ΔV and CJ are positive; therefore, the overall sign and magnitude of dμ/dT depends on the sign and magnitude of each term in the parentheses. The next paragraph discusses the first term.
Otte et al. (2016) is the most recent study to investigate how the key opto-electronic characteristics of a SiPM depend on T. It finds that CJ has a statistically insignificant correlation with T, as indicated by Figure 1 below.
Figure 1. Dependence of junction capacitance on temperature for Hamamatsu S13360-3050CS SiPM. The figure is from Otte et al. (2016).
The best linear fit yields dCJ/dT = 0.015 ± 0.009 fF/°C. Taking CJ ≈ 111 fF from Figure 1, a one degree °C change in T causes a fractional change in CJ of about 0.014%. This change is small; thus, the first term in the parentheses of Equation 3 can be neglected, and CJ can be assumed constant in the second term. Therefore, the sign and magnitude of dμ/dT depends on how VBD varies with T, as discussed next.
Otte et al. (2016) uses three different methods to determine the relationship between VBD and T. All three methods indicate a linear dependence with nearly identical slopes as shown in Figure 2. The average slope, or dVBD/dT, is about 55.7 mV/°C implying, according to Equation 3, that μ decreases linearly with T. For ΔV = 3 V, the one-degree °C change in T causes ~1.9% fractional change in ΔV — this is significant. The temperature-induced change of ΔV affects not only the gain but also other characteristics of a SiPM such as photon detection efficiency, dark count rate, crosstalk, and afterpulsing probability.
Figure 2. Dependence of breakdown voltage on temperature for Hamamatsu S13360-3050CS SiPM. The figure is from Otte et al. (2016).
Let β ≡ dVBD/dT and express VBD(T) = VBD,0[1 + β(T - T0)], where VBD,0 is the breakdown voltage at the reference temperature T0. The gain can now be written as:
\mu \phantom{\rule[-0.1em]{0.05em}{0.5em}}\left(T\right)=\frac{{C}_{J}}{e}\left\{{V}_{BIAS}-{V}_{BD,0}\left[1+\beta \left(T-{T}_{0}\right)\right]\right\}=\frac{{C}_{J}}{e}\phantom{\rule[-0.1em]{-0.05em}{0.5em}}\phantom{\rule[-0.1em]{-0.05em}{0.5em}}\phantom{\rule[-0.1em]{-0.05em}{0.5em}}∆\phantom{\rule[-0.1em]{-0.05em}{0.5em}}\phantom{\rule[-0.1em]{-0.05em}{0.5em}}\phantom{\rule[-0.1em]{-0.05em}{0.5em}}\phantom{\rule[-0.1em]{-0.05em}{0.5em}}V\phantom{\rule[-0.1em]{-0.05em}{0.5em}}\left(T\right)
To keep μ independent of T, ΔV(T) must be kept constant by adjusting VBIAS to offset the change in VBD. Namely:
{V}_{BIAS}={V}_{BD,0}+{V}_{BD,0}\beta \left(T-{T}_{0}\right)
A practical implementation of Equation 5 requires that β is known, that temperature can be accurately measured, and that a circuit with a temperature-voltage feedback loop can be constructed.
Practical implementation of gain control
Some manufacturers of SiPMs offer power supplies with built-in temperature compensation. Below is a view of a driver circuit (C12332-01) using a temperature-compensation power supply (C11204-01) by Hamamatsu.
Figure 3. View of the Hamamatsu C12332-01 SiPM driver circuit with temperature compensation. The SiPM mounts in the sensor sockets of the sensor board.
Figure 4. Gain variation versus temperature for Hamamatsu S13360-3050CS SiPM without temperature compensation (blue line) and with compensation (red line) provided by the C12332-01 driver circuit.
The SiPM is mounted on the "sensor board" containing the temperature sensor. Temperature information from the sensor goes to the "power supply board" where it is processed, and the feedback circuitry outputs the appropriate VBIAS to the SiPM. Figure 4 shows how the gain of Hamamatsu S13360-3050CS SiPM varies with temperature without temperature-compensation (blue line) and with temperature compensation (red line) provided by C12332-01.
The figure indicates that C12332-01 provides an excellent gain stabilization (gain variation less than about 2%) over at least a 60 °C temperature range.
For a fixed bias voltage, the gain of a SiPM changes linearly with temperature because the breakdown voltage varies linearly with temperature. Adjusting the bias voltage so that the overvoltage remains constant eliminates gain-temperature dependence. Manufacturers of SiPMs offer power supplies with a gain stabilization feature.
Otte, A. N., Garcia, D., Nguyen, T., Purushotham, D., Characterization of Three High Efficiency and Blue Sensitive Silicon Photomultipliers, Nuclear Inst. and Methods in Physics Research A, in press (2016)
|
JSON Ramble
Disclaimer: I've had a few drinks - I'm angry, upset and annoyed. This is me just offloading some random ideas that I'll likely regret in the morning!
Let me start with a simple statement: I like JSON. I've been using it now for perhaps coming up to 10 years, since it was first introduced to be as part of a robotics project my a good mentor I was lucky to have. Back then (and likely still now), he was big on JavaScript and NodeJS - two things I don't share a passion with.
JSON has served me well over the years, I've used it for some large projects, including another robotics team and my PhD. I've even been foolish enough to even write my own JSON parser in two different languages 1. I've come to quite like it.
Generally I am a fan of a few things:
C-style formatting - I love me some curly brackets for scoping, square brackets for arrays, speech marks for strings, etc. Whilst I know many new people may find it annoying to have to remember to put end-brackets in, as a relatively seasoned programmer I appreciate structure and consistency. (None of this 'spaced-indentation for scope' rubbish, I'm looking at you Python.)
Array support - These days I think it is absolutely required for a configuration file to support arrays. I use them exceptionally often. What on earth we did before arrays is actually beyond me. A properties file for example is so basic in comparison.
Massive support - These days pretty much every language can import a JSON configuration file.
Better than alternatives - Compared to things like properties files, YAML, etc, JSON is pretty good. The syntax sugar is minimal. yet easy to understand. People sometimes complain there is too much - yet, even somebody that has never seen JSON before can figure it out (as I have shown a few times).
One thing I also like (but don't use often) is templating. For example, you define something like the following:
0002 "key": { "type": "int", "min": "0", "max": "100", "default": "50" }
Then, when you are parsing your configuration, you have an easy way to check the values are valid and even sane defaults if they are not. Your JSON configuration can then fail-safe. With autonomous robotics, sometimes you want to be able to change configuration as the robot is up and running - and sometimes you accidentally set a bad or insane value. The last thing you want is a powerful humanoid robot trying to kill itself or you!
JSON, I like you and all, but we have some things to talk about.
No comment. Literally. There is zero way to explain why something has been assigned a value as it has been. This is one of the major drawbacks of JSON as a configuration language. In properties files you might do something like:
0004 # This is set like this for reasons
0005 key=value
The most obvious thing to do would have been to use C-style comments in my opinion. I would have literally have gone to:
0007 /* This is set like this for reasons */
0008 "key": "value"
With /* indicating the start of a comment and */ indicating the end of a comment. I would avoid // for comments as they are dependant on line endings, and for a parser things could get a little complicated with Windows/Unix line endings for example. This is also the same approach as per CSS. It's my opinion that support for that type of comment structure would be a mistake.
It's possible to set strings (at least in some parsers) as:
0011 "key": "value",
0012 'another-key': 'another-value'
Mixing and matching " and ' is definitely a mistake. Given that visually ' and backtick can be easily confused, I think it would be best to just use ". This is technically how it's supposed to be, but it isn't.
But then if reducing confusion is the goal, JSON is by default UTF-8. Sounds all good and well, but consider that the unicode character or \uXXX are both valid. So you may or may not need to decode the string depending on your application. I believe by default it should be ASCII with all unicode pre-escaped.
Numbers in JSON can literally be infinite in size - there are no limits at all. Each library implements their own arbitrary numerical parsing. Depending on the library, this may or may not convert to something useful - and may or may not throw an error of some type:
0015 "key": 58962345984235890432756982347652735347594375624938759483574398572349573454398257023495704893
As there is no consensus, personally I would just suggest that everything is a string, and let the implementer parse their own data types. I can imagine that some people may want to put letters around their numbers to indicate their type too, such as:
0018 "binary": "b1100",
0019 /* Common format for addresses */
0020 "hex": "0xC",
0021 /* Common format for general hex */
0022 "also-hex": "Ch",
0023 /* Common format for hex colours */
0024 "again-hex": "#C",
0025 "byte": "12b",
0026 "integer": "12",
0027 "long": "12l",
0028 "float": "12.0f"
As you can see, there are literally tonnes of values - and some will be right for your application, some will not. Parsers should leave this to the implementer and only offer helper functions.
This is especially true when it comes to support for exponents using E and e characters in the numbers. Most people aren't going to be using this functionality and it isn't obvious how it should be supported.
And if it wasn't complex enough already, numbers support signing. Not just negative signing with -, but also positive signing +. Technically that also includes zero too, which can lead to awkward things like this:
0031 "num-1": "+0",
0032 "num-2": "-0"
Some languages implement numbers such that
+0\ne -0
- ouch. Depending on your application, this may or may not be a bug. Do you want your parser to leave this in or not? Personally I recommend dropping + altogether and leaving the zero case to the implementer. Some people may even want
\infty
Okay, okay, that must be it? Nope. Booleans. Depending on the parser, all of these could or could not be 'true':
0035 "a": true,
0036 "b": True,
0037 "c": 1,
0038 "d": "true",
0039 "e": "True",
0040 "f": 3426435,
0041 "g": "random text"
Clearly we have a problem here. Again it makes the most sense to leave this problem to the user of the parser with helper function, with everything as a string. Data types are simple not universal.
I believe in general it was arrogant to not offer some versioning, especially with so many 'arbitrary' implementation details. I believe the perfect place for this would have been at the beginning, like this:
0043 "cool-json-version": {
Failing that, if comments are supported you could even have:
0046 /* cool-json-version */
And now for some general points about implementations:
Stop crashing - Parser implementations crash. All. The. Time. That's the last thing you want it to do. You want your parser to be really hard to crash. Sure, it can indicate errors, but if possible you want it to recover or at least have the ability to carry on somehow.
Unhelpful errors - When your parser does crash, it offers a really terrible error. It doesn't explain where the bracket is missing, or which bracket it expects to see another for.
Return blank - If you try to access a value that isn't there, just return blank "" or null. There is no need to crash the program because a value could not be found. Sure there is exception handling, but this just creates more code that you really want to access a simple configuration file.
I still like JSON and I'm not recommending yet another standard. I don't believe I could do better. But for the most part, I will personally be using a subset of it and encourage others to do the same. It's strings only for me.
This parser is terribly out of date and buggy, so please don't use the current version! If there is interest I will invest some time in writing another with a clearer mindset!↩
|
LeVan, Paul1; Raicu, Claudiu1
1 Department of Mathematics University of Notre Dame 255 Hurley Notre Dame IN 46556, USA
We prove a case of a positivity conjecture of Mihalcea–Singh, concerned with the local Euler obstructions associated to the Schubert stratification of the Lagrangian Grassmannian
LG\left(n,2n\right)
. Combined with work of Aluffi–Mihalcea–Schürmann–Su, this further implies the positivity of the Mather classes for Schubert varieties in
LG\left(n,2n\right)
, which Mihalcea–Singh had verified for the other cominuscule spaces of classical Lie type. Building on the work of Boe and Fu, we give a positive recursion for the local Euler obstructions, and use it to show that they provide a positive count of admissible labelings of certain trees, analogous to the ones describing Kazhdan–Lusztig polynomials. Unlike in the case of the Grassmannians in types A and D, for
LG\left(n,2n\right)
the Euler obstructions
{e}_{y,w}
may vanish for certain pairs
\left(y,w\right)
y\le w
in the Bruhat order. Our combinatorial description allows us to classify all the pairs
\left(y,w\right)
{e}_{y,w}=0
. Restricting to the big opposite cell in
LG\left(n,2n\right)
, which is naturally identified with the space of
n×n
symmetric matrices, we recover the formulas for the local Euler obstructions associated with the matrix rank stratification.
Classification: 14M15, 14M12, 05C05, 32S05, 32S60
Keywords: Local Euler obstructions, Schubert stratification, Lagrangian Grassmannian, tree labelings.
LeVan, Paul 1; Raicu, Claudiu 1
author = {LeVan, Paul and Raicu, Claudiu},
title = {Euler obstructions for the {Lagrangian} {Grassmannian}},
TI - Euler obstructions for the Lagrangian Grassmannian
%T Euler obstructions for the Lagrangian Grassmannian
LeVan, Paul; Raicu, Claudiu. Euler obstructions for the Lagrangian Grassmannian. Algebraic Combinatorics, Volume 5 (2022) no. 2, pp. 299-318. doi : 10.5802/alco.211. https://alco.centre-mersenne.org/articles/10.5802/alco.211/
[1] Aluffi, Paolo; Mihalcea, Leonardo C.; Schürmann, Jörg; Su, Changjian Shadows of characteristic cycles, Verma modules, and positivity of Chern–Schwartz–MacPherson classes of Schubert cells (2017) (https://arxiv.org/abs/1709.08697)
[2] Boe, Brian D. Kazhdan–Lusztig polynomials for Hermitian symmetric spaces, Trans. Amer. Math. Soc., Volume 309 (1988) no. 1, pp. 279-294 | Article | MR: 957071 | Zbl: 0669.17009
[3] Boe, Brian D.; Fu, Joseph H. G. Characteristic cycles in Hermitian symmetric spaces, Canad. J. Math., Volume 49 (1997) no. 3, pp. 417-467 | Article | MR: 1451256 | Zbl: 0915.14030
[4] Bressler, Paul; Finkelberg, Michael; Lunts, Valery Vanishing cycles on Grassmannians, Duke Math. J., Volume 61 (1990) no. 3, pp. 763-777 | Article | MR: 1084458 | Zbl: 0727.14027
[5] Grayson, Daniel R.; Stillman, Michael E. Macaulay 2, a software system for research in algebraic geometry (Available at http://www.math.uiuc.edu/Macaulay2/)
[6] Lakshmibai, Venkatramani; Raghavan, Komaranapuram N. Standard monomial theory. Invariant theoretic approach., Encyclopaedia of Mathematical Sciences, 137, Springer-Verlag, Berlin, 2008, xiv+265 pages | Zbl: 1137.14036
[7] Lascoux, Alain; Schützenberger, Marcel-Paul Polynômes de Kazhdan & Lusztig pour les grassmanniennes, Young tableaux and Schur functors in algebra and geometry (Toruń, 1980) (Astérisque), Volume 87, Soc. Math. France, Paris, 1981, pp. 249-266 | MR: 646823 | Zbl: 0504.20007
[8] Lőrincz, András C; Raicu, Claudiu Local Euler obstructions for determinantal varieties (2021) (https://arxiv.org/abs/2105.00271)
[9] MacPherson, R. D. Chern classes for singular algebraic varieties, Ann. of Math. (2), Volume 100 (1974), pp. 423-432 | Article | MR: 361141 | Zbl: 0311.14001
[10] Mihalcea, Leonardo C.; Singh, Rahul Mather classes and conormal spaces of Schubert varieties in cominuscule spaces (https://arxiv.org/abs/2006.04842)
[11] Zhang, Xiping Geometric invariants of recursive group orbit stratification (https://arxiv.org/abs/2009.09362)
|
Perceptron | Brilliant Math & Science Wiki
Akshay Padmanabha, Jamal Kassimi, Satyabrata Dash, and
The perceptron is a machine learning algorithm used to determine whether an input belongs to one class or another. For example, the perceptron algorithm can determine the AND operator—given binary inputs
x_1
x_2
, is (
x_1
x_2
) equal to 0 or 1?
The AND operation between two numbers. A red dot represents one class
(x_1
x_2 = 0)
and a blue dot represents the other class
(x_1
x_2 = 1).
The line is the result of the perceptron algorithm, which separates all data points of one class from those of the other.
The perceptron algorithm was one of the first artificial neural networks to be produced and is the building block for one of the most commonly used neural networks, the multilayer perceptron.
The perceptron algorithm is frequently used in supervised learning, which is a machine learning task that has the advantage of being trained on labeled data. This is contrasted with unsupervised learning, which is trained on unlabeled data. Specifically, the perceptron algorithm focuses on binary classified data, objects that are either members of one class or another. Additionally, it allows for online learning, which simply means that it processes elements in the training dataset one at a time (which can be useful for large datasets).
An example of binary classified data and decision boundaries used by classifiers [1]
Furthermore, the perceptron algorithm is a type of linear classifier, which classifies data points by using a linear combination of the variables used. As seen in the graph above, a linear classifier uses lines
\big(
H_1, H_2
H_3\big)
to classify data points—any object on one side of the line is part of one class and any object on the other side is part of the other class. In this example, a successful linear classifier could use
H_1
H_2
to discriminate between the two classes, whereas
H_3
would be a poor decision boundary.
An interesting consequence of the perceptron's properties is that it is unable to learn an XOR function! As we see above, OR and AND functions are linearly separable, which means that there exists a line that can separate all data points of one class from all data points of the other. However, the XOR function is not linearly separable, and therefore the perceptron algorithm (a linear classifier) cannot successfully learn the concept. This is a principal reason why the perceptron algorithm by itself is not used for complex machine learning tasks, but is rather a building block for a neural network that can handle linearly inseparable classifications.
The perceptron is an algorithm used to produce a binary classifier. That is, the algorithm takes binary classified input data, along with their class membership, and outputs a line that attempts to separate data of one class from data of the other: data points on one side of the line are of one class and data points on the other side are of the other.
Specifically, given an input with
k
x_1, x_2, ..., x_k
, a line is a linear combination of these variables:
w_1 x_1 + w_2 x_2 + \cdots + w_k x_k + b = 0
w_0, w_1, ..., w_k
b
are constants. Note that this can also be written as
\boldsymbol{w} \cdot \boldsymbol{x} + b = 0
\text{}\cdot\text{}
is the dot product between the two vectors
\boldsymbol{w}
\boldsymbol{x}
w_0, w_1, ..., w_k
b
\boldsymbol{w}
b
are used by the binary classifier in the following way: If
\boldsymbol{w} \cdot \boldsymbol{x} + b > 0
The AND operation between two numbers: A red dot represents one class
(x_1
x_2 = 0)
(x_1
x_2 = 1).
So what do
\boldsymbol{w}
b
\boldsymbol{w}
represents the weights of the
k
variables. Simply, a variable's weight determines how steep the line is relative to that variable. A weight is needed for every variable; otherwise, the line would be flat relative to that variable, which may prevent the line from successfully classifying the data. Furthermore,
b
represents the bias of the data. Essentially, this prevents the line from being dependent on the origin
(
the point (0,0)
—the bias shifts the line up or down to better classify the data.
The perceptron algorithm learns to separate data by changing weights and bias over time, where time is denoted as the number of times the algorithm has been run. As such,
\boldsymbol{w(t)}
represents the value of the weights at time
t
b(t)
represents the value of the bias at time
t
\alpha
represents the learning rate, that is, how quickly the algorithm responds to changes. This value has the bound
0 < \alpha \le 1
\alpha
cannot be 0, as this would mean that no learning occurs. If
\alpha
is a large value, the algorithm has a propensity of oscillating around the solution, as illustrated later.
To better elucidate these concepts, the formal steps of the perceptron algorithm are detailed below. In the following,
d_i
represents the correct output value for input
x_i
; one class is given
d_i = 1
x_i
is a member of that class and
d_i = 0
Begin by setting
\boldsymbol{w(0)}, b(0), t = 0
For each input
\boldsymbol{x_i}
\boldsymbol{w(t)} \cdot \boldsymbol{x_i} + b > 0
y_i
be the output for input
\boldsymbol{x_i}
(1 if true, 0 if false).
The weights and bias are now updated for the next iteration of the algorithm:
\boldsymbol{w(t+1)} = \boldsymbol{w(t)} + \alpha(d_i - y_i)\boldsymbol{x_i}
b(t+1) = b(t) + \alpha(d_i - y_i)
for all inputs.
If the learning is offline (if the inputs can be scanned multiple times), steps 2 and 3 can be repeated until errors are minimized. Note:
t
is incremented on every iteration.
Suppose we are attempting to learn the AND operator for the following input-class pairs
\big((x_1, x_2), d_i\big):
\big((0, 0), 0\big), \big((0, 1), 0\big), \big((1, 0), 0\big),
\big((1, 1), 1\big).
Let us use a learning rate of
\alpha=0.5
and run through the algorithm until we can classify all four points correctly.
w(0) = [0, 0], b(0) = 0
1 w(0) = [0, 0], b(0) = 0 y = [0, 0, 0, 0] w(1) = [0.5, 0.5], b(1) = 0.5
2 w(1) = [0.5, 0.5], b(1) = 0.5 y = [1, 1, 1, 1] w(2) = [0, 0]; b(2) = -1
3 w(2) = [0, 0], b(2) = -1 y = [0, 0, 0, 0] w(3) = [0.5, 0.5], b(3) = -0.5
4 w(3) = [0.5, 0.5], b(3) = -0.5 y = [0, 0, 0, 1] SUCCESS!
The perceptron algorithm over time. The green line represents the result of the perceptron algorithm after the second iteration and the black line represents the final results of the perceptron algorithm (after iteration 4).
In the previous example, the perceptron algorithm terminates to the correct value fairly quickly. One reason this occurs is due to a well-chosen learning rate (
\alpha
). With a smaller
\alpha
, the algorithm would take more iterations to finish, whereas a larger
\alpha
could result in the algorithm oscillating forever.
An implementation of the perceptron algorithm is provided below (in Python):
# Example of AND operator, as described above
input_data = [([0, 0], 0), ([0, 1], 0), ([1, 0], 0), ([1, 1], 1)]
# Begin algorithm
# Repeat until we minimize error
# Start with the weights from t-1
new_weights = [i for i in weights]
new_bias = bias
# For each input data point
for input_datum in input_data:
# Add bias (intercept) to line
comparison = bias
list_of_vars = input_datum[0]
# For each variable, compute the value of the line
for index in range(len(list_of_vars)):
comparison += weights[index] * list_of_vars[index]
# Obtain the correct classification and the classification of the algorithm
correct_value = input_datum[1]
classified_value = int(comparison > 0)
# If the values are different, add an error to the weights and the bias
if classified_value != correct_value:
new_weights[index] += alpha * (correct_value - classified_value) * list_of_vars[index]
bias += alpha * (correct_value - classified_value)
# If there is no change in weights or bias, return
if new_weights == weights and new_bias == bias:
return (new_weights, bias)
The perceptron algorithm is one of the most commonly used machine learning algorithms for binary classification. Some machine learning tasks that use the perceptron include determining gender, low vs. high risk for diseases, and virus detection. Basically, any task that involves classification into two groups can use the perceptron! Furthermore, the multilayer perceptron uses the perceptron algorithm to distinguish classes that are not linearly separable, which increases the number of tasks in which the perceptron can be used!
Overall, the perceptron algorithm (and the ideas behind it) is one of the main building blocks of neural networks, and its understanding is crucial for the development of more complex networks.
cyc, . Graphic showing 3 Hyperplanes in 2D. H3 doesn't separate the 2 classes. H1 does, with a small margin and H2 with the maximum margin.. Retrieved May 26, 2016, from https://en.wikipedia.org/wiki/Linear_classifier#/media/File:Svm_separating_hyperplanes.png
Cite as: Perceptron. Brilliant.org. Retrieved from https://brilliant.org/wiki/perceptron/
|
Bayesian vector autoregression (VAR) model with samples from prior or posterior distribution - MATLAB
empiricalbvarm
Required Draws from Distribution
CoeffDraws
SigmaDraws
VAR Model Parameters Derived from Distribution Draws
Create Empirical Model
Bayesian vector autoregression (VAR) model with samples from prior or posterior distribution
The Bayesian VAR model object empiricalbvarm contains samples from the distributions of the coefficients Λ and innovations covariance matrix Σ of a VAR(p) model, which MATLAB® uses to characterize the corresponding prior or posterior distributions.
For Bayesian VAR model objects that have an intractable posterior, the estimate function returns an empiricalbvarm object representing the empirical posterior distribution. However, if you have random draws from the prior or posterior distributions of the coefficients and innovations covariance matrix, you can create a Bayesian VAR model with an empirical prior directly by using empiricalbvarm.
Mdl = empiricalbvarm(numseries,numlags,'CoeffDraws',CoeffDraws,'SigmaDraws',SigmaDraws)
Mdl = empiricalbvarm(numseries,numlags,'CoeffDraws',CoeffDraws,'SigmaDraws',SigmaDraws,Name,Value)
Mdl = empiricalbvarm(numseries,numlags,'CoeffDraws',CoeffDraws,'SigmaDraws',SigmaDraws) creates a numseries-D Bayesian VAR(numlags) model object Mdl characterized by the random samples from the prior or posterior distributions of
\lambda =\text{vec}\left(\Lambda \right)=\text{vec}\left({\left[\begin{array}{ccccccc}{\Phi }_{1}& {\Phi }_{2}& \cdots & {\Phi }_{p}& c& \delta & Β\end{array}\right]}^{\prime }\right)
and Σ, CoeffDraws and SigmaDraws, respectively.
numseries = m, a positive integer specifying the number of response time series variables.
numlags = p, a nonnegative integer specifying the AR polynomial order (that is, number of numseries-by-numseries AR coefficient matrices in the VAR model).
Mdl = empiricalbvarm(numseries,numlags,'CoeffDraws',CoeffDraws,'SigmaDraws',SigmaDraws,Name,Value) sets writable properties (except NumSeries and P) using name-value pair arguments. Enclose each property name in quotes. For example, empiricalbvarm(3,2,'CoeffDraws',CoeffDraws,'SigmaDraws',SigmaDraws,'SeriesNames',["UnemploymentRate" "CPI" "FEDFUNDS"]) specifies the random samples from the distributions of λ and Σ and the names of the three response variables.
Because the posterior distributions of a semiconjugate prior model (semiconjugatebvarm) are analytically intractable, estimate returns an empiricalbvarm object that characterizes the posteriors and contains the Gibbs sampler draws from the full conditionals.
You can set writable property values when you create the model object by using name-value pair argument syntax, or after you create the model object by using dot notation. For example, to create a 3-D Bayesian VAR(1) model from the coefficient and innovations covariance arrays of draws CoeffDraws and SigmaDraws, respectively, and then label the response variables, enter:
Mdl = empiricalbvarm(3,1,'CoeffDraws',CoeffDraws,'SigmaDraws',SigmaDraws);
Mdl.SeriesNames = ["UnemploymentRate" "CPI" "FEDFUNDS"];
CoeffDraws — Random sample from prior or posterior distribution of λ
Random sample from the prior or posterior distribution of λ, specified as a NumSeries*k-by-numdraws numeric matrix, where k = NumSeries*P + IncludeIntercept + IncludeTrend + NumPredictors (the number of coefficients in a response equation). CoeffDraws represents the empirical distribution of λ based on a size numdraws sample.
Columns correspond to successive draws from the distribution. CoeffDraws(1:k,:) corresponds to all coefficients in the equation of response variable SeriesNames(1), CoeffDraws((k + 1):(2*k),:) corresponds to all coefficients in the equation of response variable SeriesNames(2), and so on. For a set of row indices corresponding to an equation:
This figure shows the row structure of CoeffDraws for a 2-D VAR(3) model that contains a constant vector and four exogenous predictors:
\left[\stackrel{{y}_{1,t}}{\overbrace{\begin{array}{ccccccccccc}{\varphi }_{1,11}& {\varphi }_{1,12}& {\varphi }_{2,11}& {\varphi }_{2,12}& {\varphi }_{3,11}& {\varphi }_{3,12}& {c}_{1}& {\beta }_{11}& {\beta }_{12}& {\beta }_{13}& {\beta }_{14}\end{array}}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\stackrel{{y}_{2,t}}{\overbrace{\begin{array}{ccccccccccc}{\varphi }_{1,21}& {\varphi }_{1,22}& {\varphi }_{2,21}& {\varphi }_{2,22}& {\varphi }_{3,21}& {\varphi }_{3,22}& {c}_{2}& {\beta }_{21}& {\beta }_{22}& {\beta }_{23}& {\beta }_{24}\end{array}}}\right],
CoeffDraws and SigmaDraws must be based on the same number of draws, and both must represent draws from either the prior or posterior distribution.
numdraws should be reasonably large, for example, 1e6.
SigmaDraws — Random sample from prior or posterior distribution of Σ
array of positive definite numeric matrices
Random sample from the prior or posterior distribution of Σ, specified as a NumSeries-by-NumSeries-by-numdraws array of positive definite numeric matrices. SigmaDraws represents the empirical distribution of Σ based on a size numdraws sample.
Rows and columns correspond to innovations in the equations of the response variables ordered by SeriesNames. Columns correspond to successive draws from the distribution.
Response series names, specified as a NumSeries length string vector. The default is ['Y1' 'Y2' ... 'YNumSeries']. empiricalbvarm stores SeriesNames as a string vector.
Number of exogenous predictor variables in the model regression component, specified as a nonnegative integer. empiricalbvarm includes all predictor variables symmetrically in each response equation.
\left[\begin{array}{l}{\text{INFL}}_{t}\\ {\text{UNRATE}}_{t}\\ {\text{FEDFUNDS}}_{t}\end{array}\right]=c+\sum _{j=1}^{4}{\Phi }_{j}\left[\begin{array}{l}{\text{INFL}}_{t-j}\\ {\text{UNRATE}}_{t-j}\\ {\text{FEDFUNDS}}_{t-j}\end{array}\right]+\left[\begin{array}{c}{\epsilon }_{1,t}\\ {\epsilon }_{2,t}\\ {\epsilon }_{3,t}\end{array}\right].
t
{\epsilon }_{t}
\Sigma
You can create an empirical Bayesian VAR model for the coefficients
{\left[{\Phi }_{1},...,{\Phi }_{4},\mathit{c}\right]}^{\prime }
and innovations covariance matrix
\Sigma
Indirectly create an empiricalbvarm model by estimating the posterior distribution of a semiconjugate prior model.
Directly create an empiricalbvarm model by supplying draws from the prior or posterior distribution of the parameters.
Indirect Creation
Assume the following prior distributions:
\mathrm{vec}\left({\left[{\Phi }_{1},...,{\Phi }_{4},\mathit{c}\right]}^{\prime }\right)|\Sigma \sim {Ν}_{39}\left(\mu ,\mathit{V}\right)
\mu
is a 39-by-1 vector of means and
\mathit{V}
is the 39-by-39 covariance matrix.
\Sigma \sim Inverse\phantom{\rule{0.16666666666666666em}{0ex}}Wishart\left(\Omega ,\nu \right)
\Omega
\nu
Create a semiconjugate prior model for the 3-D VAR(4) model parameters.
PriorMdl = semiconjugatebvarm(numseries,numlags)
Mu: [39×1 double]
V: [39×39 double]
Omega: [3×3 double]
AR: {[3×3 double] [3×3 double] [3×3 double] [3×3 double]}
Constant: [3×1 double]
Trend: [3×0 double]
Beta: [3×0 double]
Covariance: [3×3 double]
PriorMdl is a semiconjugatebvarm Bayesian VAR model object representing the prior distribution of the coefficients and innovations covariance of the 3-D VAR(4) model.
PosteriorMdl = estimate(PriorMdl,rmDataTable{:,seriesnames},'Display','off')
PosteriorMdl =
empiricalbvarm with properties:
CoeffDraws: [39×10000 double]
SigmaDraws: [3×3×10000 double]
PosteriorMdl is an empiricalbvarm model representing the empirical posterior distribution of the coefficients and innovations covariance matrix. empiricalbvarm stores the draws from the posteriors of
\lambda \text{\hspace{0.17em}}
\Sigma \text{\hspace{0.17em}}
in the CoeffDraws and SigmaDraws properties, respectively.
Draw a random sample of size 1000 from the prior distribution PriorMdl.
numdraws = 1000;
[CoeffDraws,SigmaDraws] = simulate(PriorMdl,'NumDraws',numdraws);
size(CoeffDraws)
size(SigmaDraws)
Create a Bayesian VAR model characterizing the empirical prior distributions of the parameters.
PriorMdlEmp = empiricalbvarm(numseries,numlags,'CoeffDraws',CoeffDraws,...
'SigmaDraws',SigmaDraws)
PriorMdlEmp =
CoeffDraws: [39×1000 double]
SigmaDraws: [3×3×1000 double]
Display the prior covariance mean matrices of the four AR coefficients by setting each matrix in the cell to a variable.
AR1 = PriorMdlEmp.AR{1}
{y}_{t}={\Phi }_{1}{y}_{t-1}+...+{\Phi }_{p}{y}_{t-p}+c+\delta t+Β{x}_{t}+{\epsilon }_{t}.
{y}_{t}={Z}_{t}\lambda +{\epsilon }_{t}.
{y}_{t}={\Lambda }^{\prime }{z}_{t}^{\prime }+{\epsilon }_{t}.
{z}_{t}=\left[\begin{array}{ccccccc}{y}_{t-1}^{\prime }& {y}_{t-2}^{\prime }& \cdots & {y}_{t-p}^{\prime }& 1& t& {x}_{t}^{\prime }\end{array}\right],
\left[\begin{array}{cccc}{z}_{t}& {0}_{z}& \cdots & {0}_{z}\\ {0}_{z}& {z}_{t}& \cdots & {0}_{z}\\ ⋮& ⋮& \ddots & ⋮\\ {0}_{z}& {0}_{z}& {0}_{z}& {z}_{t}\end{array}\right],
\Lambda ={\left[\begin{array}{ccccccc}{\Phi }_{1}& {\Phi }_{2}& \cdots & {\Phi }_{p}& c& \delta & Β\end{array}\right]}^{\prime }
\ell \left(\Lambda ,\Sigma |y,x\right)=\prod _{t=1}^{T}f\left({y}_{t};\Lambda ,\Sigma ,{z}_{t}\right),
semiconjugatebvarm
|
Milk - Ring of Brodgar
Object(s) Required Aurochs' Milk, Cow's Milk, Goat's Milk, Milk, Sheep's Milk
Produced By Bucket, any drinking vessel
Specific Type of Milk, Generics
Required By Batter, Birthday Cake, Butter, Caviar Canapé, Creamy Cock, Curd, Sacrebleu, Strawberries in Cream, Unbaked Butter Scones, Unbaked Egg Cake
Milk is not an actual in-game-object but a generic term referring to one of the following 5 game-objects: Aurochs' Milk, Cow's Milk, Goat's Milk, Milk, Sheep's Milk.
Milk is used to make Butter, Curds, and Batter, and is therefore an ingredient of several Baked Goods.
Milk can be stored in any Drinking Vessel, a Bucket, Barrel, or a Cistern.
Drinking milk increases the FEP gained and helps reduce satiation from Bread, Forageables and Berrys.
Milk is acquired by selecting a bucket, barrel or any drinking vessel and using it on any milk-able animal via right-click.
Cows produce Milk at a rate of Milk Quantity * 0.01 per 10 minutes.
0.1L/10 minutes at Quantity 10. 0.4L/10 minutes at Quantity 40, 2.0L/10 minutes at Quantity 200.
Cows can store a maximum of 25L of Milk
Sheep can store a maximum of 25L of Milk
Goat can store a maximum of 15+ (25L) of Milk
Only 2L can be milked from an Aurochs. A higher level clover produces higher level milk.
Mixing different types of milk in any container will yield neutral 'Milk' of generic property.
Milk has no preferred container and therefore cannot gain quality boosts.
Quality 10 milk recovers 10% stamina and drains 20% energy per 0.05L sip. Higher quality milk will decrease the energy drain but stamina recovery remains the same at all qualities.
This makes high quality milk useful for performing stamina draining tasks without having to eat as much to recover, thus preserving your hunger bonus. Though energy is always shown as a whole number, it is not simply rounded up or down. For example, Q11 milk will alternate between draining 20% and 19% energy at regular intervals. Other drinkable liquids follow the same rules and formula.
{\displaystyle ({\frac {1}{\sqrt {quality/10}}}+1)*10}
You can use a barrel of milk to feed baby animals if you accidentally kill off their mother or if there is no lactating adult female present.
Grade-A Milk (2016-04-12) >"Added new mechanic for drinks. All drinks restore stamina much like water, but they also buff particular food satiations. ...
Also note that drinks will count as half-quality if you drink them from a non-preffered vessel (i.e. drinking wine from anything but a wine glass). Drinking vessel type does not affect the effective quality for stamina regeneration purposes."
Retrieved from "https://ringofbrodgar.com/w/index.php?title=Milk&oldid=92839"
|
Home : Support : Online Help : Mathematics : Algebra : Expression Manipulation : Simplifying : Wronskian
simplify expressions involving wronskian identities
simplify(expr, wronskian)
literal name; wronskian
The simplify/wronskian function is used to simplify expressions that contain the subexpression
fg'-gf'
\mathrm{wr}\left(f,g\right)
, with f and g special functions. In particular, it recognizes the wronskians of
* Identical functions whose indices sum to zero, for example,
\mathrm{wr}\left(J\left(v,z\right),J\left(-v,z\right)\right)
* Two functions with the same indices, for example,
\mathrm{wr}\left(J\left(v,z\right),Y\left(v,z\right)\right)
* A function with arguments that sum to zero, for example,
\mathrm{wr}\left(\mathrm{D}\left(v,z\right),\mathrm{D}\left(v,-z\right)\right)
\mathrm{simplify}\left(\mathrm{BesselJ}\left(v+1,z\right)\mathrm{BesselJ}\left(-v,z\right)+\mathrm{BesselJ}\left(v,z\right)\mathrm{BesselJ}\left(-v-1,z\right),\mathrm{wronskian}\right)
\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{v}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{\pi }}\right)}{\textcolor[rgb]{0,0,1}{\mathrm{\pi }}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}}
\mathrm{simplify}\left(\mathrm{BesselJ}\left(v+1,z\right)\mathrm{BesselY}\left(v,z\right)-\mathrm{BesselJ}\left(v,z\right)\mathrm{BesselY}\left(v+1,z\right),\mathrm{wronskian}\right)
\frac{\textcolor[rgb]{0,0,1}{2}}{\textcolor[rgb]{0,0,1}{\mathrm{\pi }}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}}
|
Fluorocarbon Knowpia
PerfluoroalkanesEdit
{\displaystyle {\ce {{\overset {\delta+}{C}}-{\overset {\delta-}{F}}}}}
In 1993, 3M considered fluorocarbons as fire extinguishants to replace CFCs.[9] This extinguishing effect has been attributed to their high heat capacity, which takes heat away from the fire. It has been suggested that an atmosphere containing a significant percentage of perfluorocarbons on a space station or similar would prevent fires altogether.[10][11] When combustion does occur, toxic fumes result, including carbonyl fluoride, carbon monoxide, and hydrogen fluoride.
Gas dissolving propertiesEdit
Fowler processEdit
Electrochemical fluorinationEdit
Environmental and health concernsEdit
Fluoroalkenes and fluoroalkynesEdit
Perfluoroaromatic compoundsEdit
|
Special right triangle - Wikipedia
Right triangle with a feature making calculations on the triangle easier
"90-45-45 triangle" redirects here. For the drawing tool, see 90-45-45 set square.
Position of some special triangles in an Euler diagram of types of triangles, using the definition that isosceles triangles have at least two equal sides, i.e. equilateral triangles are isosceles.
1.1 45°–45°–90° triangle
2.4 Sides of regular polygons
Angle-based[edit]
Special angle-based triangles inscribed in a unit circle are handy for visualizing and remembering trigonometric functions of multiples of 30 and 45 degrees.
Special triangles are used to aid in calculating common trigonometric functions, as below:
0° 0 0g 0 √0/2 = 0 √4/2 = 1 0 undefined
30° π/6 33+1/3g 1/12 √1/2 = 1/2 √3/2 1/√3 √3
45° π/4 50g 1/8 √2/2 = 1/√2 √2/2 = 1/√2 1 1
60° π/3 66+2/3g 1/6 √3/2 √1/2 = 1/2 √3 1/√3
90° π/2 100g 1/4 √4/2 = 1 √0/2 = 0 undefined 0
The 45°–45°–90° triangle, the 30°–60°–90° triangle, and the equilateral/equiangular (60°–60°–60°) triangle are the three Möbius triangles in the plane, meaning that they tessellate the plane via reflections in their sides; see Triangle group.
45°–45°–90° triangle[edit]
The side lengths of a 45°–45°–90° triangle
In plane geometry, constructing the diagonal of a square results in a triangle whose three angles are in the ratio 1 : 1 : 2, adding up to 180° or π radians. Hence, the angles respectively measure 45° (π/4), 45° (π/4), and 90° (π/2). The sides in this triangle are in the ratio 1 : 1 : √2, which follows immediately from the Pythagorean theorem.
Of all right triangles, the 45°–45°–90° degree triangle has the smallest ratio of the hypotenuse to the sum of the legs, namely √2/2.[1]: p.282, p.358 and the greatest ratio of the altitude from the hypotenuse to the sum of the legs, namely √2/4.[1]: p.282
Triangles with these angles are the only possible right triangles that are also isosceles triangles in Euclidean geometry. However, in spherical geometry and hyperbolic geometry, there are infinitely many different shapes of right isosceles triangles.
This is a triangle whose three angles are in the ratio 1 : 2 : 3 and respectively measure 30° (π/6), 60° (π/3), and 90° (π/2). The sides are in the ratio 1 : √3 : 2.
The proof of this fact is clear using trigonometry. The geometric proof is:
Draw an equilateral triangle ABC with side length 2 and with point D as the midpoint of segment BC. Draw an altitude line from A to D. Then ABD is a 30°–60°–90° triangle with hypotenuse of length 2, and base BD of length 1.
The fact that the remaining leg AD has length √3 follows immediately from the Pythagorean theorem.
The 30°–60°–90° triangle is the only right triangle whose angles are in an arithmetic progression. The proof of this fact is simple and follows on from the fact that if α, α + δ, α + 2δ are the angles in the progression then the sum of the angles 3α + 3δ = 180°. After dividing by 3, the angle α + δ must be 60°. The right angle is 90°, leaving the remaining angle to be 30°.
Side-based[edit]
Right triangles whose sides are of integer lengths, with the sides collectively known as Pythagorean triples, possess angles that cannot all be rational numbers of degrees.[2] (This follows from Niven's theorem.) They are most useful in that they may be easily remembered and any multiple of the sides produces the same relationship. Using Euclid's formula for generating Pythagorean triples, the sides must be in the ratio
m2 − n2 : 2mn : m2 + n2
where m and n are any positive integers such that m > n.
Common Pythagorean triples[edit]
There are several Pythagorean triples which are well-known, including those with sides in the ratios:
The 3 : 4 : 5 triangles are the only right triangles with edges in arithmetic progression. Triangles based on Pythagorean triples are Heronian, meaning they have integer area as well as integer sides.
The possible use of the 3 : 4 : 5 triangle in Ancient Egypt, with the supposed use of a knotted rope to lay out such a triangle, and the question whether Pythagoras' theorem was known at that time, have been much debated.[3] It was first conjectured by the historian Moritz Cantor in 1882.[3] It is known that right angles were laid out accurately in Ancient Egypt; that their surveyors did use ropes for measurement;[3] that Plutarch recorded in Isis and Osiris (around 100 AD) that the Egyptians admired the 3 : 4 : 5 triangle;[3] and that the Berlin Papyrus 6619 from the Middle Kingdom of Egypt (before 1700 BC) stated that "the area of a square of 100 is equal to that of two smaller squares. The side of one is ½ + ¼ the side of the other."[4] The historian of mathematics Roger L. Cooke observes that "It is hard to imagine anyone being interested in such conditions without knowing the Pythagorean theorem."[3] Against this, Cooke notes that no Egyptian text before 300 BC actually mentions the use of the theorem to find the length of a triangle's sides, and that there are simpler ways to construct a right angle. Cooke concludes that Cantor's conjecture remains uncertain: he guesses that the Ancient Egyptians probably did know the Pythagorean theorem, but that "there is no evidence that they used it to construct right angles".[3]
The following are all the Pythagorean triple ratios expressed in lowest form (beyond the five smallest ones in lowest form in the list above) with both non-hypotenuse sides less than 256:
Isosceles right-angled triangles cannot have sides with integer values, because the ratio of the hypotenuse to either other side is √2 and √2 cannot be expressed as a ratio of two integers. However, infinitely many almost-isosceles right triangles do exist. These are right-angled triangles with integer sides for which the lengths of the non-hypotenuse edges differ by one.[5][6] Such almost-isosceles right-angled triangles can be obtained recursively,
an = 2bn−1 + an−1
bn = 2an + bn−1
an is length of hypotenuse, n = 1, 2, 3, .... Equivalently,
{\displaystyle ({\tfrac {x-1}{2}})^{2}+({\tfrac {x+1}{2}})^{2}=y^{2}}
where {x, y} are solutions to the Pell equation x2 − 2y2 = −1, with the hypotenuse y being the odd terms of the Pell numbers 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378... (sequence A000129 in the OEIS).. The smallest Pythagorean triples resulting are:[7]
Alternatively, the same triangles can be derived from the square triangular numbers.[8]
Arithmetic and geometric progressions[edit]
Main article: Kepler triangle
The Kepler triangle is a right triangle whose sides are in geometric progression. If the sides are formed from the geometric progression a, ar, ar2 then its common ratio r is given by r = √φ where φ is the golden ratio. Its sides are therefore in the ratio 1 : √φ : φ. Thus, the shape of the Kepler triangle is uniquely determined (up to a scale factor) by the requirement that its sides be in geometric progression.
The 3–4–5 triangle is the unique right triangle (up to scaling) whose sides are in arithmetic progression.[9]
Sides of regular polygons[edit]
The sides of a pentagon, hexagon, and decagon, inscribed in congruent circles, form a right triangle
Let a = 2 sin π/10 = −1 + √5/2 = 1/φ be the side length of a regular decagon inscribed in the unit circle, where φ is the golden ratio. Let b = 2 sin π/6 = 1 be the side length of a regular hexagon in the unit circle, and let c = 2 sin π/5 =
{\displaystyle {\sqrt {\tfrac {5-{\sqrt {5}}}{2}}}}
be the side length of a regular pentagon in the unit circle. Then a2 + b2 = c2, so these three lengths form the sides of a right triangle.[10] The same triangle forms half of a golden rectangle. It may also be found within a regular icosahedron of side length c: the shortest line segment from any vertex V to the plane of its five neighbors has length a, and the endpoints of this line segment together with any of the neighbors of V form the vertices of a right triangle with sides a, b, and c.[11]
^ a b Posamentier, Alfred S., and Lehman, Ingmar. The Secrets of Triangles. Prometheus Books, 2012.
^ Weisstein, Eric W. "Rational Triangle". MathWorld.
^ a b c d e f Cooke, Roger L. (2011). The History of Mathematics: A Brief Course (2nd ed.). John Wiley & Sons. pp. 237–238. ISBN 978-1-118-03024-0.
^ Gillings, Richard J. (1982). Mathematics in the Time of the Pharaohs. Dover. p. 161.
^ Forget, T. W.; Larkin, T. A. (1968), "Pythagorean triads of the form x, x + 1, z described by recurrence sequences" (PDF), Fibonacci Quarterly, 6 (3): 94–104 .
^ Chen, C. C.; Peng, T. A. (1995), "Almost-isosceles right-angled triangles" (PDF), The Australasian Journal of Combinatorics, 11: 263–267, MR 1327342 .
^ Nyblom, M. A. (1998), "A note on the set of almost-isosceles right-angled triangles" (PDF), The Fibonacci Quarterly, 36 (4): 319–322, MR 1640364 .
^ Beauregard, Raymond A.; Suryanarayan, E. R. (1997), "Arithmetic triangles", Mathematics Magazine, 70 (2): 105–115, doi:10.2307/2691431, MR 1448883 .
^ Euclid's Elements, Book XIII, Proposition 10.
^ nLab: pentagon decagon hexagon identity.
3 : 4 : 5 triangle
45–45–90 triangle – with interactive animations
Retrieved from "https://en.wikipedia.org/w/index.php?title=Special_right_triangle&oldid=1070876106"
|
Mapping of Aluminous Rich Laterite Depositions through Hyper Spectral Remote Sensing ()
1Department of Mines & Geology, Government of Andhra Pradesh, Vijayawada, India.
2Department of Geology, Andhra University, Visakhapatnam, India.
3Department of Geography, Andhra University, Visakhapatnam, India.
4National Remote Sensing Centre, Hyderabad, India.
Babu, M. , Rao, E. , Kallempudi, L. and Chandra, D. (2018) Mapping of Aluminous Rich Laterite Depositions through Hyper Spectral Remote Sensing. International Journal of Geosciences, 9, 93-105. doi: 10.4236/ijg.2018.92006.
L=\left(A\rho /1-{\rho }_{e}S\right)+\left(B{\rho }_{e}/1-{\rho }_{e}S\right)+{L}_{a}
{L}_{e}=\left(A+B\right){\rho }_{e}/\left(1-{\rho }_{e}S\right)+{L}_{a}
[1] Lillesand, T.M. and Kiefer, R.W. (1999) Remote Sensing and Image Interpretation. John Wiley & Sons, Inc., New York, 9-597.
[2] Dash, B. and Das, J.K. (1969) A Structural Study of Eastern Ghat Tectonic around Tigira, Orissa. Prakruti Utkal University Journal Science, 45, 173-176.
[3] Rao, M.G. and Raman, P.K. (1979) The East Coast Bauxite Deposits of India. Bulletin. Geological Survey of India. Series A, No.46, 24.
[4] Hunt, G.R., Salisbury, J.W. and Lehnoff, C.J. (1971) Visible and near Infrared Spectra of Minerals and Rocks: 111. Oxides and Oxyhydroxides. Modern Geology, 2, 195-205.
[5] Boardman, J.W., Kruse, F.A. and Green, R.O. (1995) Mapping Target Signatures via Partial Unmixing of AVIRIS Data. Summaries, Fifth JPL Airborn Earth Science Workshop, 23-26 January 1995, JPL Publication 95-1, 1, 23-26.
[6] Jupp, D.L.B., Datt, B., Lovell, J., Campbell, S. and King, E. (2004) Backgroung Notes for Hyperion Data User Workshop. CSIRO Office of Space Science and Applications, Earth Observation Centre, Canberra.
[7] Boardman, J.W. and Kruse, F.A. (1994) Automated Spectral Analysis: A Geologic Example Using AVIRIS Data, North Grapevine Mountains, Nevada. Proceedings, Tenth Thematic Conference on Geologic Remote Sensing, Environmental Research Institute of Michigan, Ann Arbor, MI, 407-418.
[8] Green, A.A. and Craig, M.D. (1985) Analysis of Aircraft Spectrometer Data with Logarithmic Residuals (abs). Proceedings, AIS Work Shop, 8-10 April 1985, Jet Propulsion Laboratory, Pasadena, 85-41.
|
A Manual of Prayers for the Use of the Catholic Laity/Brief Statement of Christian Doctrine - Wikisource, the free online library
Days of Obligation and Devotion
A Manual of Prayers for the Use of the Catholic Laity (1889) by Clarence E. Woodman
Brief Statement of Christian Doctrine
2195399A Manual of Prayers for the Use of the Catholic Laity — Brief Statement of Christian Doctrine1889Clarence E. Woodman
Brief Statement of Christian Doctrine.
The Ten Commandvients of God.—Exodus xx.
The Six Commandments of the Church.
1. To hear Mass on Sundays, and Holydays of Obligation.
6. Not to marry persons who are not Catholics, or who are related to us within the fourth degree of kindred, nor privately without witnesses, nor to solemnize marriage at forbidden times.
Faith—Hope—Charity.
Prudence—Justice—Fortitude—Temperance.
The Seven Gifts of the Holy Ghost.—Isa. xi. 2, 3.
To admonish the sinner,
To bear wrongs patiently,
To forgive all injuries,
To ransom the captive,
To harbor the harborless,
The Eight Beatitudes.—Matt. v.
8. Blessed are they that suffer persecution for justice' sake; for theirs is the kingdom of heaven.
The Seven Deadly Sins, and the opposite Virtues.
{\displaystyle \scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}}
Contrary Virtues.
{\displaystyle \scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right.}}
Sins against the Holy Ghost.
Presumption of God's mercy—Despair—Impugning the known truth—Envy at another's spiritual good—Obstinacy in sin—Final impenitence.
Sins Crying to Heaven for Vengeance.
Wilful murder—The sin of Sodom—Oppression of the poor—Defrauding laborers of their wages.
Nine Ways of being Accessory to another's Sin.
By counsel—By command—By consent—By provocation—By praise or flattery—By concealment—By partaking—By silence—By defence of the ill done.
Three Eminent good Works.
Prayer—Fasting—Almsgiving.
The Evangelical Counsels.
Voluntary Poverty—Chastity—Obedience.
The Four Last Things to be Remembered.
Death—Judgment—Hell—Heaven.
Remember, Christian soul, that thou hast this day, and every day of thy life—
Lay Baptism.
PROVIDED an infant is in danger of dying before a Priest can be procured, any other person, whether man, woman, or child, may baptize it in the following manner:
While pouring common water on the head or face of the infant, pronounce the words:
Retrieved from "https://en.wikisource.org/w/index.php?title=A_Manual_of_Prayers_for_the_Use_of_the_Catholic_Laity/Brief_Statement_of_Christian_Doctrine&oldid=6591450"
|
Use a ruler and a protractor to create a triangle with two sides of
3
cm each and an angle measuring
60°
This triangle could be equilateral, meaning all sides are the same length. Additionally, all angles would be the same measure. Try making an equilateral triangle with sides of
3
|
Topic: Binary Decoders, De-Multiplexers
Binary Decoders, De-Multiplexers
Basic Combinational Circuit Blocks
Here, we will examine two different types of decoders: a simple binary decoder, and a seven-segment decoder (section 6.4) that can drive a common numeric data display.
A binary decoder has N inputs and
2^N
outputs. It receives N inputs (often grouped as a binary number on a bus) and then asserts one, and only one, of its
2^N
outputs based on that input. If the N inputs are taken as an N-bit binary number, then only the output that corresponds to the input binary number is asserted. For example, if a binary 5 (or 101) is input to a 3:8 decoder, then only the 5th output of the decoder will be asserted and all other outputs will be de-asserted. Practical decoder circuits are usually built as 2:4 decoders with 2 inputs and 22 (4) outputs, 3:8 decoders with 3 inputs and 23 (8) outputs, or 4:16 decoders with 4 inputs 24 (16) outputs. A decoder circuit requires one AND gate to drive each output, and each AND gate decodes a particular binary number. For example, a 3:8 decoder requires 8 AND gates, with the first AND gate having inputs A’‚ B’‚ C’, the second A’‚ B’‚ C, the third A’‚ B‚ C’, etc. Figure 1 below displays a 3:8 binary decoder.
If a binary decoder larger than 4:16 is needed, it can be built from smaller decoders. Only decoders with an enable input can be used to construct larger decoder circuits. As with the mux, the enable input drives all outputs to ‘0’ when de-asserted, and allows normal decoder operation when asserted.
Figure 1. 3:8 Binary Decoder
As with multiplexers, this most common application of decoders is beyond our current presentation, so instead we will consider a less common, somewhat contrived application. Consider the function of a decoder and the truth table, K-map, or minterm representation of a given function. Each row in a truth table, each cell in a K-map, or each minterm number in an equation represents a particular combination of inputs. Each output of a decoder is uniquely asserted for a particular combination of inputs. Thus, if the inputs to a given logic function are connected to the inputs of a decoder, and those same inputs are used as K-map input logic variables, then a direct one-to- one mapping is created between the K-map cells and the decoder outputs. It follows that any given function represented in a truth table or K-map can be directly implemented using a decoder, by simply by OR’ing the decoder outputs that correspond to a truth table row or K-map cell containing a ‘1’ (decoder outputs that correspond to K-map cells that contain a zero are simply left unconnected). In such a circuit, any input combination with a ‘1’ in the corresponding truth table row or K-map cell will drive the output OR gate to a ‘1’, and any input combination with a ‘0’ in the corresponding K-map cell will allow the OR gate to output a ‘0’. Note that when a decoder is used to implement a circuit directly from a truth table or K-map, no logic minimization is performed. Using a decoder in this fashion saves time, but usually results in a less efficient implementation (here again, a logic synthesizer would remove the inefficiencies before such a circuit was implemented in a programmable device).
A decoder with an enable can be used as a de-multiplexer, whereas a multiplexer selects one on N input to pass through to the output, a de-multiplexer takes a single input and routes it to one of N outputs. Figure 2 to the left illustrates the decoder with enable as a de-multiplexer. A multiplexer/de-multiplexer (or more simply, mux/de-mux) circuit can be used to transmit the state of N signals from one place to another using only
Log_2N+1
signals.
Log_2N
signals are used to select the data input for the mux and to drive the decoder inputs, and the rate at which these signals change define the time-window length.
Figure 2. Use a Decoder With Enable as a De-Multiplexer
The data-out of the mux drives the enable-in of the decoder, so that the same logic levels that appear on the mux inputs also appear on the corresponding decoder outputs, but only for the mux input/decoder output currently selected. In this way, the state of N signals can be sent from one place to another using only
Log_2N+1
signals, but only one signal at a time is valid.
2^N
outputs. It receives N inputs (often grouped as a binary number on a bus) and then asserts one and only one of its 2N outputs based on that input.
|
Probability Problem on Probability - Independent events: Lab Rat Independence - Andy Hayes | Brilliant
Lab Rat Independence
A=\text{The 1st rat receives an extra food pellet for the day}
B=\text{The 1st rat runs in the exercise wheel that day}
C=\text{The 2nd rat runs in the exercise wheel that day}
\begin{array}{lll} P(A)=0.5 & P(B)=0.2 & P(C)=0.1 \\ P(A\cap B)=0.1 & P(A\cap C)=0.05 & P(B\cap C)=0.02 \\ P(A\cap B\cap C)=0.01 \\ \end{array}
|
Javascript Date Time Tips and Trics
In this tutorial, I will give you some tips and tricks about JavaScript Date.Before going to show you, some code let’s get familiar with some Terminology.
What is Unix Epoch
UNIX Epoch is a system for describing a point in time. It is the number of seconds that have elapsed since the Unix epoch, that is the time 00:00:00 UTC on 1 January 1970, minus leap seconds.
This is represented as Date (0) in JavaScript.
The Difference Between GMT and UTC. Greenwich Mean Time (GMT) is often interchanged or confused with Coordinated Universal Time (UTC). But GMT is a time zone, and UTC is a time standard.
ISO 8601 (Data elements and interchange formats)
The purpose of this standard is to provide an unambiguous and well-defined method of representing dates and times, to avoid misinterpretation of numeric representations of dates and times, particularly when data is transferred between countries with different conventions for writing numeric dates and times.
What is the time zone
Time zones are based on the fact that the Earth moves 15 degrees longitude each hour. Since there are 24 hours in a day, there are 24 standard time zones on the globe. (24 hours x 15º = 360º)
Let’s see some example.
The time is centred at Greenwich, UK (UTC +0). So, for example, if
it is 2:30 PM in Greenwich (UTC) now, it will be 10:30 PM in Malaysia
If it is 17:06 in Malaysia (UTC+8) now, it will be 2:36 PM in India (UTC+5:30), and 10:05 AM in the UK (UTC+0).
Now let’s write some code in javascript
const date=new Date()
If you run above code you will get following output
Tue Apr 07 2020 17:11:16 GMT+0800 (Singapore Standard Time)
Tue APR 07 2020 17 11 16 GMT+8
Different Ways to create Date in javascript
Below table describe some ways to create Date in javascript
new Date() Current Date/Time
new Date(timestamp) Milliseconds since Unix Epoch
new Date(string) Date String
How to get yesterday or tomorrow date from given date
Convert given Date to unix epoch
const date=new Date().getTime(); // Unix Epoch
Convert one day to milliseconds
const oneDayInMilliseconds=24*60*60*1000;
Pass current milliseconds + tomarrow to Date constructor as show below
const tomarow=new Date(date+oneDayInMilliseconds)
const yesterday=new Date(date-oneDayInMilliseconds)
You can get milliseconds from these two dates and compare using javascript logical operator like >,<,===.
Let’s suppose you want to compare two dates in javascript.
const date1=new Date('07-03-2020');
function compare(date1,date2){
return date1.getTime()>date2.getTime();
coconsole.log(compare(date1,date2)); // false
If you want to get a difference (number of days) between you can use the same approach as we used in calculating the yesterday and tomorrow.
The formula for calculating days
diff =|day1 -day2|
// in milliseconfs
oneDayInMilliSeconds=86400000
days= | diff |\div oneDayInMilliseconds
Let’s uppose you want to calculate diffrence between following two dates
const date2=new Date('07-05-2020')
//Calculate difference in Unix Timestamp
const diff=date1.getTime()-date2.getTime()
const oneDayInMilliseconds=24*60*60*1000; // 24 hours X 60 mins X 60 secs X 1000 msec
const absDiiff=Math.abs(diff);
const numberOfDays=Math.floor(absDiiff/oneDayInMilliseconds);
console.log(numberOfDays); //
console.log(date2) // 2
|
This problem is a checkpoint for solving multi-step equations. It will be referred to as Checkpoint 8.
24=3x+3
6x+12=-x-2
3x+3-x+2=x+5
5(x-1)=5(4x-3)
Answers and extra practice for the Checkpoint problems are located in the back of your printed textbook or in the Reference Tab of your eBook. If you have an eBook for Core Connections, Course 2, login and then click the following link: Checkpoint 8: Solving Multi-Step Equations
|
Logical equality - Wikipedia
For the corresponding concept in combinational logic, see XNOR gate.
{\displaystyle x=y}
{\displaystyle (1001)}
{\displaystyle x\cdot y+{\overline {x}}\cdot {\overline {y}}}
{\displaystyle ({\overline {x}}+y)\cdot (x+{\overline {y}})}
{\displaystyle 1\oplus x\oplus y}
{\displaystyle {\begin{aligned}x&\leftrightarrow y&x&\Leftrightarrow y&\mathrm {E} xy\\x&\mathrm {~EQ~} y&x&=y\end{aligned}}}
{\displaystyle {\begin{aligned}x&+y&x&\not \equiv y&Jxy\\x&\mathrm {~XOR~} y&x&\neq y\end{aligned}}}
2 Alternative descriptions
{\displaystyle (x=y)=\lnot (x\oplus y)=\lnot x\oplus y=x\oplus \lnot y=(x\land y)\lor (\lnot x\land \lnot y)=(\lnot x\lor y)\land (x\lor \lnot y)}
{\displaystyle x\leftrightarrow y}
Retrieved from "https://en.wikipedia.org/w/index.php?title=Logical_equality&oldid=1062675573"
|
H∞ Preview Control for Discrete-Time Systems | J. Dyn. Sys., Meas., Control. | ASME Digital Collection
H∞ Preview Control for Discrete-Time Systems
Chintae Choi,
Mechanical & Electrical Engineering Team, RIST, Pohang, 790-330, Korea
Department of Mechanical & Aerospace Engineering, University of California, Los Angeles, CA 90095
Contributed by the Dynamic Systems and Control Division for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received by the Dynamic Systems and Control Division June 21, 1999. Associate Editor: T. Kurfess.
Choi, C., and Tsao, T. (June 21, 1999). "H∞ Preview Control for Discrete-Time Systems ." ASME. J. Dyn. Sys., Meas., Control. March 2001; 123(1): 117–124. https://doi.org/10.1115/1.1286869
A preview controller to be able to prepare a plant with the future information for external disturbances will guarantee better performance to suppress their effects. A design approach for the optimal
H∞
preview controller in the discrete-time domain is given. The preview and feedback controller are simultaneously designed to minimize the worst case RMS value of the regulated variables when the bounded unknown disturbances and the previewable disturbances hit the dynamic plants. Thus, a state feedback controller and the related preview controller are derived in this design, even though problem formulation and solving an algebraic Riccati equation are based on the full-information
H∞
controller design scheme. The performance of the proposed preview controller is simulated with a rolling stand of the tandem cold mill in the steel-making works. The objective of the control system for the rolling stand is to minimize thickness error of the exit strip and tension variation between stands simultaneously. The entry strip thickness to the stand and the roll gap variation are considered as previewable disturbances, since they can be measured and estimated. The future informations of these physical variables are utilized in the preview controller to suppress their effects on the exit strip thickness and the inter-stand tension. The simulation results shows that the
H∞
preview controller is effective to satisfy the requirements for the thickness and the tension.
discrete time systems, predictive control, optimal control, control system synthesis, minimisation, state feedback
Control equipment, Control systems, State feedback, Tension, Discrete time systems, Strips, Design, Feedback, Algebra, Errors
An optimal continuous time control strategy for active suspensions with preview
Decentralized control of active vehicle suspensions with preview
Performance enhancement of limited-bandwidth active automotive suspension by road preview
Optimal preview semiactive suspension
Robust control systems design using H∞ optimization theory
J. Guid. Control. Dyn.
Moran, A., Nagai, M., and Hayase, M., 1996, “Design of Active Suspensions with H∞ Preview Control,” Proceedings of International Symposium on Advanced Vehicle Control, Aachen, pp. 215–232.
Linear Discrete-Time H∞-Optimal Tracking with Preview
Kojima, A., and Ishijima, S., 1997, “H∞ Control with Preview Compensation,” Proceedings of the American Control Conference, Albuquerque, pp. 1692–1697.
A Unified Hamiltonian Approach for LQ and H∞ Preview Control Algorithms
State-space approach to discrete-time H∞ control
Green, M., and Limebeer, D. J. N., 1990, “H∞ optimal full information control for discrete time systems,” Proc. of the IEEE Conference on Decision and Control, pp. 1769–1774.
Hitachi Co., Ltd, 1981, “Formula of On-line mathematical Models for Tandem Cold Mills,” Hitachi Technical Report.
Sensitivity Reduction by State Derivative Feedback
|
How Much Does RollerCoin Pay Per Game? | Matthew Miner's Blog
One game gaining some popularity among the cryptocurrency craze is RollerCoin. Its players have been promoting it through its referral links, claiming that it is one of the funnest play-to-earn games. You can alleged earn money just playing for free. At first it seems like a simple little idle game, but the main content is actually playing little minigames. There's Match 3, Breakout, 2048, and other classic arcade games, and you have to play these games (or pay real money) to power up your computer to "mine cryptocurrency" (in the game). This power decays in 1–7 days, depending on how often you've been playing, meaning you have to play these little games over and over and over to earn any cryptocurrency.
However this abstraction makes it very difficult to see how much you're actually getting paid. You might get a dopamine rush for "earning money from video games" but have no idea how much you're earning, so I decided to do the math and see.
The amount of power you have compared to the overall share of fictional mining power among the players determines how much money you'll get every time a "block is mined", which is currently every 10 minutes. The "block reward" is currently 30 RLT, which is conveniently pegged to equal $30 (as long as the cryptocurrency bubble never pops at least).
The games all pay varying amounts, but lets say you're smart and just play the more profitable games like "Cryptonoid" and "Crypto Hamster", which both give about "1 TH/s" of power for 1/3/7 days, depending on how many games you've played. Assuming you play RollerCoin every single day without missing one, you'll eventually get a 7-day expiration date, so lets go with that.
The amounts of the payouts are currently $30 if you're "mining RLT" and $12 if you're "mining Bitcoin". The amount of power all players are contributing to these two are "20 EH/s" and "13 EH/s", so you'll get about 50% more money if you go with the game's coin. Most of the other coins are somewhere in between, and RLT is the highest of them all, so let's go with that.
So then, getting to the math, if you play one of the more valuable games to "mine RLT", you'll get 1 TH/s out of 20,000,000 TH/s total. This means, you'll get ¹⁄₂₀,₀₀₀,₀₀₀ of the reward, or one-and-a-half one-millionths of a dollar every 10 minutes. If you keep your in-game computer all the way leveled-up, it'll last for 7 days, or 10,080 minutes. Dividing this by ten and then by then multiplying by the reward you get:
\frac{10080}{10}×$\frac{1.5}{20000000}=$0.001512
So, in the best-case scenario, you'll earn about one seventh of a penny for every game you play. You could earn $1 in only about 660 games of dedicated play though. At one minute per game, that's 9¢ per hour! If you miss a day and only have the 3-day computer or play a suboptimal game, then you'll make about half that.
Long story short: RollerCoin essentially pays nothing. It would be far better to play a game you find more fun and forget about RollerCoin because you're not going to make money from it. You could earn more mowing your neighbor's lawn or begging on the street corner for an hour than you ever would playing RollerCoin, no matter how dedicated you are.
It's worth noting that RollerCoin does have "miners" you can buy for real money that generate money passively without playing the minigames. These are also horrible and risky deals, but there's no gaming element there, and they're essentially unrelated to the free-to-play minigame side, so that doesn't really factor into this post.
|
Tanning - Ring of Brodgar
Required Hunting
Enabled Rope Twining
Required By (45) Armored Striders, Badger Hide Vest, Bear Coat, Boar Tusk Helmet, Bronze Helm, Candleberry Wax, Chieftain's Hat, Coffer, Cutthroat Cuirass, Exquisite Belt, Exquisite Rot, Fox Hat, Fur Boots, Goat Mask, Hardened Leather, Hunter's Belt, Laddie's Cap, Leather, Leather Armor, Leather Backpack, Leather Ball, Leather Basket, Leather Boots, Leather Coat, Leather Merchant's Hat, Leather Pants, Leather Patch, Leather Purse, Mammoth Guard, Parchment, Poor Man's Belt, Poor Man's Gloves, Ranger's Boots, Reinforced Hem, Snakeskin Belt, Stitched Leather Coaster, Stuffed Bear, Tanner's Fishline, Tanning Fluid, Tanning Tub, Traveller's Sack, Untanned Rot, Waterflask, Whaler's Jacket, Wolverine Boots
Tanning is the process of turning Raw Hides into leather using Tanning Fluid and Tanning Tubs.
In order to make leather, follow the steps below. Leather isn't exactly hard to make, but it is a time-consuming process.
Acquire a Raw Hide from a Wild Beast.
Dry the Raw Hide by placing it in a Drying Frame.
Wait 8-24 real-time hours for the hide to finish drying.
Take the Dried Hide and place it in a Tanning Tub.
Fill Your Tanning Tub with Water.
Gather four pieces of Treebark from a Tree.
Right-click the Treebark into the tanning tub to create Tanning Fluid.
Wait 30 real-time hours for the hide to turn into leather.
Retrieve your finished leather.
Tanning tubs need a minimum 20L of fluid to operate, but can hold up to 40L of fluid.
As leather is produced, the volume of fluid goes down.
Tanning Fluid is consumed upon creation of a unit of leather, not as a function of time.
If the level of tanning fluid becomes too low, The tanning process will stop (but not reset)
This can be seen on the tub graphic as no fluid showing. Add more tanning fluid to finish the process
4 treebark or 1 Exquisite Rot is required to make 5% concentration per Tanning Tub
While the necessary concentration of tanning fluid is 5% (4 treebark per tub), you may increase the tanning fluid concentration to 10% (9 treebark per tub).
This has no effect on tanning speed, but if your treebark quality is higher than your water quality, you can incrase your tanning fluid quality by adding more bark.
Each tub can hold a maximum of 4 inventory spaces of hide at a time, ie 4x Rabbit Fur, 2x Fox Hide, or 1x Aurochs Hide.
Tanning fluid can be made directly in containers such as barrels, buckets, and tanning tubs.
Tanning Tub Q =
{\displaystyle {\frac {_{q}AvgBoard*2+_{q}AvgBlock}{3}}}
Tanning Fluid Q =
{\displaystyle {\frac {_{q}Water+_{q}TreeBark}{2}}}
... (Note: Bucket(10L) only. Formula is missing liquid amount part.)
Leather Q =
{\displaystyle {\frac {_{q}Hide*3+_{q}Fluid*2+_{q}Tub}{6}}}
Retrieved from "https://ringofbrodgar.com/w/index.php?title=Tanning&oldid=94417"
|
Ground track - WikiMili, The Best Wikipedia Reader
Path on the surface of the Earth or another body directly below an aircraft or satellite
Ground track of the International Space Station for approximately two periods. The light and dark regions represent the regions of the Earth in daylight and in the night, respectively.
A ground track or ground trace is the path on the surface of a planet directly below an aircraft's or satellite's trajectory. In the case of satellites, it is also known as a suborbital track, and is the vertical projection of the satellite's orbit onto the surface of the Earth (or whatever body the satellite is orbiting). [1]
Aircraft ground tracks
Effect of orbital period
Effect of argument of perigee
Repeat orbits
A satellite ground track may be thought of as a path along the Earth's surface that traces the movement of an imaginary line between the satellite and the center of the Earth. In other words, the ground track is the set of points at which the satellite will pass directly overhead, or cross the zenith, in the frame of reference of a ground observer. [2]
In air navigation, ground tracks typically approximate an arc of a great circle, this being the shortest distance between two points on the Earth's surface. In order to follow a specified ground track, a pilot must adjust their heading in order to compensate for the effect of wind. Aircraft routes are planned to avoid restricted airspace and dangerous areas, and to pass near navigation beacons.
The ground track of a satellite can take a number of different forms, depending on the values of the orbital elements, parameters that define the size, shape, and orientation of the satellite's orbit. (This article discusses closed orbits, or orbits with eccentricity less than one, and thus excludes parabolic and hyperbolic trajectories.)
Typically, satellites have a roughly sinusoidal ground track. A satellite with an orbital inclination between zero and ninety degrees is said to be in what is called a direct or prograde orbit , meaning that it orbits in the same direction as the planet's rotation. A satellite with an orbital inclination between 90° and 180° (or, equivalently, between 0° and −90°) is said to be in a retrograde orbit . (Direct orbits are by far the most common for artificial satellites, as the initial velocity imparted by the Earth's rotation at launch reduces the delta-v needed to achieve orbit.)
A satellite in a direct orbit with an orbital period less than one day will tend to move from west to east along its ground track. This is called "apparent direct" motion. A satellite in a direct orbit with an orbital period greater than one day will tend to move from east to west along its ground track, in what is called "apparent retrograde" motion. This effect occurs because the satellite orbits more slowly than the speed at which the Earth rotates beneath it. Any satellite in a true retrograde orbit will always move from east to west along its ground track, regardless of the length of its orbital period.
Because a satellite in an eccentric orbit moves faster near perigee and slower near apogee, it is possible for a satellite to track eastward during part of its orbit and westward during another part. This phenomenon allows for ground tracks that cross over themselves in a single orbit, as in the geosynchronous and Molniya orbits discussed below.
A geostationary orbit, as viewed from above the North Pole
A satellite whose orbital period is an integer fraction of a day (e.g., 24 hours, 12 hours, 8 hours, etc.) will follow roughly the same ground track every day. This ground track is shifted east or west depending on the longitude of the ascending node, which can vary over time due to perturbations of the orbit. If the period of the satellite is slightly longer than an integer fraction of a day, the ground track will shift west over time; if it is slightly shorter, the ground track will shift east. [2] [3]
As the orbital period of a satellite increases, approaching the rotational period of the Earth (in other words, as its average orbital speed slows towards the rotational speed of the Earth), its sinusoidal ground track will become compressed longitudinally, meaning that the "nodes" (the points at which it crosses the equator) will become closer together until at geosynchronous orbit they lie directly on top of each other. For orbital periods longer than the Earth's rotational period, an increase in the orbital period corresponds to a longitudinal stretching out of the (apparent retrograde) ground track.
A satellite whose orbital period is equal to the rotational period of the Earth is said to be in a geosynchronous orbit. Its ground track will have a "figure eight" shape over a fixed location on the Earth, crossing the equator twice each day. It will track eastward when it is on the part of its orbit closest to perigee, and westward when it is closest to apogee.
A special case of the geosynchronous orbit, the geostationary orbit, has an eccentricity of zero (meaning the orbit is circular), and an inclination of zero in the Earth-Centered, Earth-Fixed coordinate system (meaning the orbital plane is not tilted relative to the Earth's equator). The "ground track" in this case consists of a single point on the Earth's equator, above which the satellite sits at all times. Note that the satellite is still orbiting the Earth — its apparent lack of motion is due to the fact that the Earth is rotating about its own center of mass at the same rate as the satellite is orbiting.
Orbital inclination is the angle formed between the plane of an orbit and the equatorial plane of the Earth. The geographic latitudes covered by the ground track will range from –i to i, where i is the orbital inclination. [3] In other words, the greater the inclination of a satellite's orbit, the further north and south its ground track will pass. A satellite with an inclination of exactly 90° is said to be in a polar orbit, meaning it passes over the Earth's north and south poles.
Launch sites at lower latitudes are often preferred partly for the flexibility they allow in orbital inclination; the initial inclination of an orbit is constrained to be greater than or equal to the launch latitude. Vehicles launched from Cape Canaveral, for instance, will have an initial orbital inclination of at least 28°27′, the latitude of the launch site—and to achieve this minimum requires launching with a due east azimuth, which may not always be feasible given other launch constraints. At the extremes, a launch site located on the equator can launch directly into any desired inclination, while a hypothetical launch site at the north or south pole would only be able to launch into polar orbits. (While it is possible to perform an orbital inclination change maneuver once on orbit, such maneuvers are typically among the most costly, in terms of fuel, of all orbital maneuvers, and are typically avoided or minimized to the extent possible.)
In addition to providing for a wider range of initial orbit inclinations, low-latitude launch sites offer the benefit of requiring less energy to make orbit (at least for prograde orbits, which comprise the vast majority of launches), due to the initial velocity provided by the Earth's rotation. The desire for equatorial launch sites, coupled with geopolitical and logistical realities, has fostered the development of floating launch platforms, most notably Sea Launch.
The ground track of a Molniya orbit
If the argument of perigee is zero, meaning that perigee and apogee lie in the equatorial plane, then the ground track of the satellite will appear the same above and below the equator (i.e., it will exhibit 180° rotational symmetry about the orbital nodes.) If the argument of perigee is non-zero, however, the satellite will behave differently in the northern and southern hemispheres. The Molniya orbit, with an argument of perigee near −90°, is an example of such a case. In a Molniya orbit, apogee occurs at a high latitude (63°), and the orbit is highly eccentric (e = 0.72). This causes the satellite to "hover" over a region of the northern hemisphere for a long time, while spending very little time over the southern hemisphere. This phenomenon is known as "apogee dwell", and is desirable for communications for high latitude regions. [3]
Plot of repeat ground track solutions at different mean altitudes from 300km to 1000km, for a circular orbit at inclination 97.44 degrees.
As orbital operations are often required to monitor a specific location on Earth, orbits that cover the same ground track periodically are often used. On earth, these orbits are commonly referred to as Earth-repeat orbits, and are often designed with "frozen orbit" parameters to achieve a repeat ground track orbit with stable (minimally time-varying) orbit elements. [4] These orbits use the nodal precession effect to shift the orbit so the ground track coincides with that of a previous orbit, so that this essentially balances out the offset in the revolution of the orbited body. The longitudinal rotation after a certain period of time of a planet is given by:
{\displaystyle \Delta L_{1}=-2\pi {\frac {T}{T_{E}}}}
{\displaystyle T}
is the time elapsed
{\displaystyle T_{E}}
is the time for a full revolution of the orbiting body, in the case of Earth one sidereal day
The effect of the nodal precession can be quantified as:
{\displaystyle \Delta L_{2}=-{\frac {3\pi J_{2}R_{e}^{2}cos(i)}{a^{2}(1-e^{2})^{2}}}}
{\displaystyle J_{2}}
is the body's second dynamic form factor
{\displaystyle R_{e}}
is the body's radius
{\displaystyle i}s the orbital inclination
{\displaystyle a}
is the orbit's semi-major axis
{\displaystyle e}
These two effects must cancel out after a set
{\displaystyle j}
orbital revolutions and
{\displaystyle k}
(sidereal) days. Hence, equating the elapsed time to the orbital period of the satellite and combining the above two equations yields an equation which holds for any orbit that is a repeat orbit:
{\displaystyle j\left|\Delta L_{1}+\Delta L_{2}\right|=j\left|-2\pi {\frac {2\pi {\sqrt {\frac {a^{3}}{\mu }}}}{T_{E}}}-{\frac {3\pi J_{2}R_{e}^{2}cos(i)}{a^{2}(1-e^{2})^{2}}}\right|=k2\pi }
{\displaystyle \mu }
is the standard gravitational parameter for the body being orbited
{\displaystyle j}
is the number of orbital revolutions after which the same ground track is covered
{\displaystyle k}
is the number of sidereal days after which the same ground track is covered
Pass (spaceflight), the period in which a spacecraft is visible above the local horizon
Satellite revisit period, the time elapsed between observations of the same point on Earth by a satellite
Satellite watching, as a hobby
Terminator (solar), the moving line that separates the illuminated day side and the dark night side of a planetary body
The nodal period of a satellite is the time interval between successive passages of the satellite through either of its orbital nodes, typically the ascending node. This type of orbital period applies to artificial satellites, like those that monitor weather on Earth, and natural satellites like the Moon.
↑ "suborbital track". AMetSoc.org Glossary of Meteorology. Retrieved 15 March 2022.
1 2 Curtis, Howard D. (2005), Orbital Mechanics for Engineering Students (1st ed.), Amsterdam: Elsevier Ltd., ISBN 978-0-7506-6169-0 .
1 2 3 Montenbruck, Oliver; Gill, Eberhard (2000), Satellite Orbits (1st ed.), The Netherlands: Springer, ISBN 3-540-67280-X .
↑ Low, Samuel Y. W. (January 2022). "Designing a Reference Trajectory for Frozen Repeat Near-Equatorial Low Earth Orbits". AIAA Journal of Spacecraft and Rockets. 59. doi:10.2514/1.A34934.
Lyle, S. and Capderou, Michel (2006) Satellites: Orbits and Missions Springer ISBN 9782287274695 pp 175–264
Satellite Tracker at eoPortal.org
https://isstracker.pl ISS Tracker
satellite ground track software code at smallsats.org
infosatellites.com
|
3. Case studies of APD/MPPC performance calculations
In this section, we will conduct studies of APD/MPPC performance calculations for product selection. As part of this review, we will learn a number of key techniques to perform such calculations. As discussed in the previous section, please keep in mind that optical power [W], whose dimension is normalized to time, can be converted to units of photons (and vice versa) using equation 2-1.
Let’s suppose that an application demands the following conditions:
Peak wavelength of approx. 450 nm,
10 to 106 photons per pulse,
Pulse rates of 10 kHz to 1 MHz,
Typical pulses having widths of approx. 8 ns, rise times of approx. 3 ns, and decay (fall) times of approx. 5 ns.
And with the following requirements: absolute intensity of each pulse must be measured with nonlinearity below 10% while the optical design allows for a detector photosensitive area or field-of-view (FOV) of 3 mm per channel.
Note that the application conditions are fairly generic; many seemingly important details, such as what source is producing the light signal, do not affect these back-of-the-envelope calculations.
3-1. MPPC S/N
We focus on studying photodetector S/N at the signal’s low end. We use the typical specs of a regular Hamamatsu MPPC like the S13360-3050 (3 × 3 mm, 50 µm pixels):
Total pixel count of 3600
Typical PDE @ 450 nm = 40% at an overvoltage of 3 V
Typical gain of 1.7 × 106 at an overvoltage of 3 V
Typical dark count rate (DCR) = 500 kcps at an overvoltage of 3 V
Terminal capacitance (Ct) = 320 pF
Using equation 2-2, we calculate charge output from a 10-photon input pulse: 10 × 0.4 × (1.7 × 106) × (1.6 × 10-19) = 1090 fC. The node sensitivity of a typical digitizing readout (like a QDC, which is the charge-digitizing equivalent of an oscilloscope with similar node sensitivity) is on the order of low 10s of fC/LSB (like 25 fC/LSB in the case of CAEN V965), and since 1090 fC > 25 fC, we conclude that S13360-3050 will be suitable for applications such as particle or nuclear physics in which use of flash digitizers (a QCD is simply a digitizer with on-board charge-to-voltage conversion) is prevalent; that is commonly the case if pulse shape information is required. However, such readout schemes are highly costly and power-intensive, making them unsuitable for common scientific, industrial, or consumer applications.
A cost-effective alternative readout scheme is use of photon-counting circuitry such as a multi-channel analyzer (MCA), consisting of discriminator/scaler, counter, and other signal processing functions. Towards using equation 2-8 to calculate S/N for this scheme, the desired counting integration time must be determined. For the sake of our discussion, let’s assume that is 1 ms for which S13360-3050’s DCR would yield 500 k × 1 m = 500 dark counts. Using equation 2-8, we set S/N = 1 and solve
\frac{{N}_{photon}×0.4}{\sqrt{\left({N}_{photon}×0.4\right)+\left(500\right)}}=1
{N}_{photon}\approx 57
. At a min. pulse rate of 10 kHz, 10 photons per pulse would yield 100 photons in 1 ms. Alternatively, 100-photon S/N can be calculated using equation 2-8 to be
\frac{100×0.4}{\sqrt{\left(100×0.4\right)+\left(500\right)}}\approx 1.7
We repeat the same for S13360-3025 (3 × 3 mm, 25 μm pixels) with PDE @ 450 nm = 25% and typical DCR = 400 kcps to obtain an incident light level of
{N}_{photon}=82
for S/N = 1 (by solving
\frac{{N}_{photon}×0.25}{\sqrt{\left({N}_{photon}×0.25\right)+\left(400\right)}}=1
) or alternatively calculating S/N for 100 photons as
\frac{100×0.25}{\sqrt{\left(100×0.25\right)+\left(400\right)}}=1.2
It is noteworthy that “excellent” S/N is generally considered to be ≥ 10. Most instrument designers typically have a target performance of S/N > X in mind for which X is greater than 1 (even if less than 10). Thus, it is important to use the value of X that represents the instrument designer’s target S/N in the above calculations. For example, if X = 5, the above calculations would yield Nphoton = 312 for S13360-3050 and Nphoton = 453 for S13360-3025. These results would mean that a portion of the lower range of the expected signal levels could not be detected with the target S/N (= 5) performance in this case.
3-2. MPPC linearity
We now assess how linear a MPPC’s response would be under this application’s conditions:
- MPPC pulse-height linearity: The question we face here is: up to how many 450 nm photons would S13360-3050 or S13360-3025 be able to detect with 10% max. nonlinearity? To answer this, we first obtain the pixel capacitance4 by utilizing the nominal value of the MPPC’s typical gain, assuming a unit of electrons for it and converting it to charge in coulombs, and then dividing the resulting charge by the specified overvoltage corresponding to that gain value. We thus have:
\frac{\left(1.7E6 e-\right)×\left(1.6E-19\frac{C}{e-}\right)}{3}\approx 91fF
. Using the quenching resistor value5 of a 50 μm MPPC pixel, we then calculate S13360-3050’s pixel recovery time to be 63 ns [approx. 4.6 × 91f × 150k where 4.6 = –ln(0.01) corresponds to a 99% MPPC recovery] and compare to the application’s light pulse width. Since the condition PW < Trecovery is met (8 ns < 63 ns), we use equation 2-13 to plot the MPPC’s expected response and compare that to its ideal response as obtained from equation 2-12 in Microsoft® Excel® and look for the point at which the 2 plots diverge by 10%. We perform the comparison by plotting the resulting nonlinearity using the combination of equation 2-10 and equation 2-11.
4 Another approach could also be used to determine pixel capacitance; it consists of dividing the specified MPPC terminal capacitance by pixel count, which yields 88 fF (= 320 pF / 3600 pixels) in the case of S13360-3050. Please note, however, that measurement of MPPC gain is affected by the quenching resistor’s parasitic capacitance while measurement of terminal capacitance is affected by the parasitic capacitances of the quenching resistor and also MPPC package and traces, and thus, either method overestimates the MPPC pixel’s junction capacitance (considering that both parasitic capacitances are in parallel to the junction capacitance). This overestimation becomes particularly significant for MPPCs with smaller pixel sizes (10 μm and 15 μm in Hamamatsu’s lineup) whose junction capacitances are relatively quite small.
5 MPPC Rq values: 1 MΩ for 10 μm pixels, 1 MΩ for 15 μm pixels, 300 kΩ for 25 μm pixels, 150 kΩ for 50 μm pixels
Figure 3-1 Linearity calculation with Microsoft® Excel®
For S13360-3050, the point of 10% nonlinearity is at about 2000 photons:
Figure 3-2 Count of fired pixels vs. photons
Figure 3-3 Nonlinearity vs. photons
For S13360-3025, by repeating the same calculations and plots, we find the point of 10% nonlinearity to be about 12000 photons:
- MPPC pulse-rate linearity: From the application conditions, we see that the shortest inter-pulse time is longer than S13360-3050’s pixel recovery time (92 ns > 63 ns), so this application is within the pulse-rate linearity range of the S13360-3050.
3-3. APD S/N
Now, let’s assess APD’s S/N at those signal levels above which MPPC linearity falls short of the application’s linearity requirement. We choose a blue-enhanced APD of suitable size, like Hamamatsu’s S8664-30K, and utilize its characteristics (
QE @ 450nm=75%, {I}_{d}=1 nA, {M}_{opt}=50, F={50}^{0.2}\approx 2.2
) in equation 2-7 to calculate S/N. Considering that the application is photometric (i.e. measuring the amount of incident light in absolute terms for which a charge amplifier is required), we also use the readout noise spec of a sufficiently fast charge amplifier (one that can resolve a single pulse at the max. pulse rate); we note 993 e- in case of Analog Devices AD8488. Note that the charge amplifier’s bandwidth (as inverse of its integration time) must be at least twice that of the application’s max. expected pulse rate, and hence, we will use 2 MHz as the min. amplifier bandwidth limit. We proceed to calculate
{S}_{dark}=\frac{{I}_{d}}{q×△f}=1n/\left(2M×1.6×{10}^{-19}\right)=3125e-
. Using equation 2-7, we thus have
S/N=\frac{0.75×50×2000}{\sqrt{2.2×\left[\left(0.75×{50}^{2}×2000\right)+\left(50×3125\right)\right]+{993}^{2}}}\approx 24
for 2000 photons and
S/N=\frac{0.75×50×12000}{\sqrt{2.2×\left[\left(0.75×{50}^{2}×12000\right)+\left(50×3125\right)\right]+{993}^{2}}}\approx 63
for 12000 photons.
As mentioned before, excellent S/N is typically considered to be ≥ 10. Therefore, in such a photometric application using a charge amplifier, we conclude that S8664-30K can perform well at those signal levels at which S13360-3050 and S13360-3025 exhibit excessive nonlinearity. Thus, it is imperative to use S13360-3050 (instead of S13360-3025) in photon-counting mode to detect the smaller pulses in this application but then consider using S8664-30K for pulses > 2000 ph.
As a side exercise, we calculate the APD’s S/N for making a relative measurement (requiring a resistive trans-impedance amplifier for readout) under the same conditions. First, let’s explore the case that the APD and its output amplifier circuit are intended for use as a pulse counter, which would make the signal’s highest-frequency component to be the max. expected pulse rate of 1 MHz. Since the output amplifier’s bandwidth must be at least twice that of the measurement frequency (which would be the frequency of the signal’s highest-frequency component that is to be detected), we adopt 2 MHz as our min. amplifier bandwidth limit (or cutoff frequency in other words). Using equation 2-6 along with S8664-30K’s characteristics [
\Phi @ 450 nm\approx 0.3 A/W
[derived from QE by equation 2-5],
{I}_{d}=1 nA, {M}_{opt}=50, F ={50}^{0.2}\approx 2.2
] and Texas Instruments amplifier OPA380 characteristics (readout noise spec of
10\frac{fA}{\sqrt{Hz}}
resulting in an estimated readout noise of approx. 14 pA at 2 MHz), we have:
S/N=\frac{0.3×8.8×{10}^{-16}×1E6×50}{\sqrt{2×1.6E-19×2E6×{50}^{2}×2.2×\left[\left(0.3×8.8E-16×1E6\right)+{10}^{-9}\right]+\left(1.96×{10}^{-22}\right)}}~6.2
for detecting pulses of 2000 photons or 0.88 fJ of incident 450 nm light per pulse at a rate of 1 MHz, and
S/N=\frac{0.3×5.28×{10}^{-15}×1E6×50}{\sqrt{2×1.6E-19×2E6×{50}^{2}×2.2×\left[\left(0.3×5.28E-15×1E6\right)+{10}^{-9}\right]+\left(1.96×{10}^{-22}\right)}}~26
for detecting pulses of 12000 photons or 5.28 fJ of incident 450 nm light per pulse at a rate of 1 MHz.
Now, let’s study the case that the APD and its output amplifier circuit will be used to perform pulse-shape discrimination (PSD) on signal pulses, which would require the signal’s highest-frequency component to be obtained from its rise time (since shorter than the fall time) by the following approximation
\frac{0.35}{3 ns}\approx 117MHz
. Like before, since the output amplifier’s cutoff frequency must be at least twice that of the measurement frequency, we adopt 234 MHz as our min. amplifier bandwidth limit. Using equation 2-6 along with S8664-30K’s characteristics
\left(\Phi @ 450nm\approx 0.3A/W, {I}_{d}=1 nA, {M}_{opt}=50,F={50}^{0.2}\approx 2.2\right)
and Analog Devices amplifier AD8015 specifications (indicating a readout noise spec of
3\frac{pA}{\sqrt{Hz}}
resulting in an estimated readout noise of approx. 46 nA at 234 MHz), we have:
S/N=\frac{0.3×\frac{8.8×{10}^{-16}}{8n}×50}{\sqrt{2×1.6E-19×234E6×{50}^{2}×2.2×\left[\left(0.3×\frac{8.8×{10}^{-16}}{8n}\right)+\left(8n×{10}^{-9}\right)\right]+\left(2.1×{10}^{-15}\right)}}\sim 13
for 2000 photons or 0.88 fJ of incident 450 nm light per pulse with a pulse width of 8 ns, and
S/N=\frac{0.3×\frac{5.28×{10}^{-15}}{8n}×50}{\sqrt{2×1.6E-19×234E6×{50}^{2}×2.2×\left[\left(0.3×\frac{5.28×{10}^{-15}}{8n}\right)+\left(8n×{10}^{-9}\right)\right]+\left(2.1×{10}^{-15}\right)}}\sim 33
for 12000 photons or 5.28 fJ of incident 450 nm light per pulse with a pulse width of 8 ns.
These results show that the above signal amplitudes are too low for relative detection at such high bandwidths using S8664-30K and the aforementioned amplifiers. So, let’s calculate at what signal levels we could attain S/N = 1 and S/N = 10 using S8664-30K and the above amplifiers at the same bandwidths.
For the earlier scenario of detecting and counting 450 nm light pulses at a rate of 1 MHz using S8664-30K, we have:
which yields Sinput ≈ 127 pW or ≈ 288 450 nm photons per pulse at 1MHz of pulse rate.
which yields Sinput ≈ 1.51 nW or ≈ 3412 450 nm photons per pulse at 1MHz of pulse rate.
For the latter scenario of performing PSD on 450 nm light pulses with a rise time of 3 ns (and thus, amplifier bandwidth of 234 MHz) using S8664-30K, we have:
which yields Sinput ≈ 3.6 nW or ≈ 66 450 nm photons per pulse with a width of 8ns.
which yields Sinput ≈ 71 nW or ≈ 1283 450 nm photons per pulse with a width of 8ns.
3-4. APD linearity
We now discuss how linear S8664-30K’s response would be under this application’s conditions:
- Pulse-rate linearity: For this, a calculation of response cutoff frequency using equation 2-15 would be made based on the terminal capacitance values specified in Hamamatsu’s APD datasheets for a load resistance of 50 Ω. However, APD cutoff frequency values (calculated in the same way) are provided in Hamamatsu’s APD datasheets (so no need to calculate!). In the case of S8664-30K, the specified cutoff frequency is 140 MHz, which far exceeds this application’s max. pulse rate of 1 MHz.
- Pulse-height linearity: For calculating the upper limit of pulse-height linearity, one would calculate the APD’s charge storage capacity by using Q = Ct x Vbias in which Vbias is the APD’s reverse bias voltage for the desired gain; please note that plots of APD gain and terminal capacitance vs. reverse voltage are provided in Hamamatsu APD datasheets. For our case study, we will use S8664-30K’s terminal capacitance of 22 pF and the reverse bias voltage of 360 V for the optimal gain of Mopt = 50 to obtain a charge storage capacity of 7.9 nC. Back-calculating from that charge storage capacity by taking S8664-30K’s QE (0.75 @ 450 nm) and optimal gain (Mopt = 50) into account, we arrive at a photon count of 1.3 × 109, which is larger than the max. photon count of 106 per pulse in this application. Thus, this application is within S8664-30K’s pulse-height linearity.
Furthermore, one also needs to take the readout circuitry into account. For example, Hamamatsu’s H4083 charge amplifier has a 2 pF storage capacitor biased up to the rail voltage of 12 V, which by Q = C × V yields a max. charge storage capacity of 24 pC or 1.5 × 108 electrons (or alternatively obtained by max. output voltage of 12 V divided by gain of 0.5 V/pC). Like the previous step, back-calculating from that amount of charge by taking S8664-30K’s QE (0.75 @ 450 nm) and optimal gain (Mopt = 50) into account, we arrive at a photon count of 4 × 106, which is larger than the max. photon count of 106 per pulse in this application. Thus, pulse-height linearity would not be limited by the charge amplifier circuitry if H4083 is utilized in this case.
In conclusion, please note that multiple products could turn out to be suitable for a given set of application conditions. By taking price information into consideration, those options can be trimmed down to one or more candidates for characterization and evaluation. With that, considering their peculiar complexities, we will dedicate the next section to describing methods of measuring MPPC characteristics and discussing their specifics.
|
Characteristic energy - WikiMili, The Best Wikipedia Reader
In astrodynamics, the characteristic energy (
{\displaystyle C_{3}}
) is a measure of the excess specific energy over that required to just barely escape from a massive body. The units are length 2 time −2, i.e. velocity squared, or energy per mass.
Non-escape trajectory
Every object in a 2-body ballistic trajectory has a constant specific orbital energy
{\displaystyle \epsilon }
equal to the sum of its specific kinetic and specific potential energy:
{\displaystyle \epsilon ={\frac {1}{2}}v^{2}-{\frac {\mu }{r}}={\text{constant}}={\frac {1}{2}}C_{3},}
{\displaystyle \mu =GM}
is the standard gravitational parameter of the massive body with mass
{\displaystyle M}
{\displaystyle r}
is the radial distance from its center. As an object in an escape trajectory moves outward, its kinetic energy decreases as its potential energy (which is always negative) increases, maintaining a constant sum.
Note that C3 is twice the specific orbital energy
{\displaystyle \epsilon }
of the escaping object.
A spacecraft with insufficient energy to escape will remain in a closed orbit (unless it intersects the central body), with
{\displaystyle C_{3}=-{\frac {\mu }{a}}<0}
{\displaystyle \mu =GM}
{\displaystyle a}
is the semi-major axis of the orbit's ellipse.
If the orbit is circular, of radius r, then
{\displaystyle C_{3}=-{\frac {\mu }{r}}}
A spacecraft leaving the central body on a parabolic trajectory has exactly the energy needed to escape and no more:
{\displaystyle C_{3}=0}
A spacecraft that is leaving the central body on a hyperbolic trajectory has more than enough energy to escape:
{\displaystyle C_{3}={\frac {\mu }{|a|}}>0}
{\displaystyle \mu =GM}
{\displaystyle a}
is the semi-major axis of the orbit's hyperbola (which may be negative in some convention).
{\displaystyle C_{3}=v_{\infty }^{2}}
{\displaystyle v_{\infty }}
is the asymptotic velocity at infinite distance. Spacecraft's velocity approaches
{\displaystyle v_{\infty }}
as it is further away from the central object's gravity.
MAVEN, a Mars-bound spacecraft, was launched into a trajectory with a characteristic energy of 12.2 km2/s2 with respect to the Earth. [1] When simplified to a two-body problem, this would mean the MAVEN escaped Earth on a hyperbolic trajectory slowly decreasing its speed towards
{\displaystyle {\sqrt {12.2}}{\text{ km/s}}=3.5{\text{ km/s}}}
. However, since the Sun's gravitational field is much stronger than Earth's, the two-body solution is insufficient. The characteristic energy with respect to Sun was negative, and MAVEN – instead of heading to infinity – entered an elliptical orbit around the Sun. But the maximal velocity on the new orbit could be approximated to 33.5 km/s by assuming that it reached practical "infinity" at 3.5 km/s and that such Earth-bound "infinity" also moves with Earth's orbital velocity of about 30 km/s.
The InSight mission to Mars launched with a C3 of 8.19 km2/s2. [2] The Parker Solar Probe (via Venus) plans a maximum C3 of 154 km2/s2. [3]
C3 (km2/s2) to get from Earth to various planets: Mars 12, Jupiter 80, Saturn or Uranus 147. [4] To Pluto (with its orbital inclination) needs about 160–164 km2/s2. [5]
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a planet, moon, asteroid, or Lagrange point. In an atom, electrons follow similar curved paths, or orbits, around a nucleus. Normally, orbit refers to a regularly repeating trajectory, although it may also refer to a non-repeating trajectory. To a close approximation, planets and satellites follow elliptic orbits, with the center of mass being orbited at a focal point of the ellipse, as described by Kepler's laws of planetary motion.
In celestial mechanics, escape velocity or escape speed is the minimum speed needed for a free, non-propelled object to escape from the gravitational influence of a primary body, thus reaching an infinite distance from it. It is typically stated as an ideal speed, ignoring atmospheric friction. Although the term "escape velocity" is common, it is more accurately described as a speed than a velocity because it is independent of direction; the escape speed increases with the mass of the primary body and decreases with the distance from the primary body. The escape speed thus depends on how far the object has already traveled, and its calculation at a given distance takes into account the fact that without new acceleration it will slow down as it travels—due to the massive body's gravity—but it will never quite slow to a stop.
In celestial mechanics, the standard gravitational parameterμ of a celestial body is the product of the gravitational constant G and the mass M of the body.
In orbital mechanics, a porkchop plot (also pork-chop plot) is a chart that shows contours of equal characteristic energy (C3) against combinations of launch date and arrival date for a particular interplanetary flight.
Specific mechanical energy is the mechanical energy of an object per unit of mass. Similar to mechanical energy, the specific mechanical energy of an object in an isolated system subject only to conservative forces will remain constant.
Wie, Bong (1998). "Orbital Dynamics". Space Vehicle Dynamics and Control . AIAA Education Series. Reston, Virginia: American Institute of Aeronautics and Astronautics. ISBN 1-56347-261-9.
↑ Atlas V set to launch MAVEN on Mars mission, nasaspaceflight.com, 17 November 2013.
↑ ULA (2018). "InSight Launch Booklet" (PDF).
↑ JHUAPL. "Parker Solar Probe: The Mission". parkersolarprobe.jhuapl.edu. Retrieved 2018-07-22.
↑ NASA studies for Europa Clipper mission
↑ New Horizons Mission Design
|
You can download the library from github: https://github.com/tigergraph/gsql-graph-algorithms
4) Strongly Connected Components
6) Louvain Method with Parallelism and Refinement
8) Weighted PageRank
9) Personalized PageRank
10) Shortest Path, Single-Source, No Weight
11) Shortest Path, Single-Source, Positive Weight
12) Shortest Path, Single-Source, Any Weight
13) Minimal Spanning Tree (MST)
14) Cycle Detection
15) Triangle Counting(minimal memory)
16) Triangle Counting(fast, more memory)
17) Cosine Neighbor Similarity (single vertex)
18) Cosine Neighbor Similarity (all vertices)
19) Jaccard Neighbor Similarity (single vertex)
20) Jaccard Neighbor Similarity (all vertices)
21) k-Nearest Neighbors (Cosine Neighbor Similarity, single vertex)
22) k-Nearest Neighbors (Cosine Neighbor Similarity, batch)
23) k-Nearest Neighbors Cross Validation (Cosine Neighbor Similarity)
The following algorithms are currently available. The algorithms are grouped into five classes:
Classification (NEW)
Yes, with reverse edges (NEW)
K-Nearest Neighbors (with cosine similarity for "nearness")
d_{avg}(v) = \sum_{u \ne v} dist(v,u)/(n-1)
CC(v) = 1/d_{avg}(v)
The Betweenness Centrality of a vertex is defined as the number of shortest paths which pass through this vertex, divided by the total number of shortest paths. That is
BC(v) =\sum_{s \ne v \ne t}PD_{st}(v)= \sum_{s \ne v \ne t} SP_{st}(v)/SP_{st} ,
PD
is called the pair dependency,
SP_{st}
is the total number of shortest paths from node s to node t and
SP_{st}(v)
is the number of those paths that pass through v.
The TigerGraph implementation is based A Faster Algorithm for Betweenness Centrality by Ulrik Brandes, Journal of Mathematical Sociology 25(2):163-177, (2001). For every vertex s in the graph, the pair dependency starting from vertex s to all other vertices t via all other vertices v is computed first,
PD_{s*}(v) = \sum_{t:s \in V} PD_{st}(v)
Then betweenness centrality is computed as
BC(v) =\sum_{s:s \in V}PD_{s*}(v)/2
According to Brandes, the accumulated pair dependency can be calculated as
PD_{s*}(v) =\sum_{w:v \in P_s(w)} SP_{sv}(v)/SP_{sw} \cdot (1+PD_{s*}(w)) ,
where the set of predecessors of vertex w on shortest paths from s
P_s(w)
P_s(w) = \{u \in V: \{u, w\} \in E, dist(s,w) = dist(s,u)+dist(u,w) \} .
For each single vertex, the algorithm works in two phases. The first phase calculates the number of shortest paths passing through each vertex. Then starting from the vertex on the most outside layer in a non-incremental order with pair dependency initial value of 0, traverse back to the starting vertex.
betweenness_cent(INT maxHops)
betweenness_cent_file(STRING filepath, INT maxHops)
betweenness_cent_attr(INT maxHops)
Computes a Betweenness Centrality value (FLOAT type) for each vertex. The result is available in 3 forms:
maxHops: maximum number of iterations
O(E*V), E = number of edges, V = number of vertices.
Considering the high time cost of running this algorithm on a big graph, the users can set a maximum number of iterations. Parallel processing reduces the time needed for computation.
Undirected edges, Unweighted edges
In the example below, Claire is in the very center of the social graph, and has the highest betweenness centrality. Six shortest paths pass through Sam (i.e. paths from Victor to all other 6 people except for Sam and Victor), so the score of Sam is 6. David also has a score of 6, since Brian has 6 paths to other people that pass through David.
betweenness_cent_attr(10) on a social graph with undirected edges Friend
"@@BC": {
"Claire": 17,
"Sam": 6,
"Brian": 0,
"Victor": 0
In the following example, both Charles and David have 9 shortest paths passing through them. Ellen is in a similar position as Charles, but her centrality is weakened due to the path between Frank and Jack.
"Charles": 9,
"Ellen": 8,
"Jack": 0
A strongly connected component (SCC) is a subgraph such that there is a path from any vertex to every other vertex. A graph can contain more than one separate SCC. An SCC algorithm finds the maximal SCCs within a graph. Our implementation is based on the Divide-and-Conquer Strong Components (DCSC) algorithm[1]. In each iteration, pick a pivot vertex v randomly, and find its descendant and predecessor sets, where descendant set D_v is the vertex reachable from v, and predecessor set P_v is the vertices which can reach v (stated another way, reachable from v through reverse edges). The intersection of these two sets is a strongly connected component SCC_v. The graph can be partitioned to 4 sets: SCC_v, descendants D_v excluding SCC_v, predecessors P_v excluding SCC, and the remainders R_v. It is proved that any SCC is a subset of one of the 4 sets [1]. Thus, we can divide the graph into different subsets and detect the SCCs independently and iteratively.
The problem of this algorithm is unbalanced load and slow convergence when there are a lot of small SCCs, which is often the case in real world use cases [3]. We added two trimming stages to improve the performance: size-1 SCC trimming[2] and weakly connected components[3].
The implementation of this algorithm requires reverse edges for all directed edges considered in the graph.
[1] Fleischer, Lisa K., Bruce Hendrickson, and Ali Pınar. "On identifying strongly connected components in parallel." International Parallel and Distributed Processing Symposium. Springer, Berlin, Heidelberg, 2000.
[2] Mclendon Iii, William, et al. "Finding strongly connected components in distributed graphs." Journal of Parallel and Distributed Computing 65.8 (2005): 901-910.
[3] Hong, Sungpack, Nicole C. Rodia, and Kunle Olukotun. "On fast parallel detection of strongly connected components (SCC) in small-world graphs." Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis. ACM, 2013.
scc(INT iter = 500, INT iter_wcc = 5, INT top_k_dist)
scc_file(INT iter = 500, INT iter_wcc = 5, INT top_k_dist, FILE f)
scc_attr(INT iter = 500, INT iter_wcc = 5, INT top_k_dist)
iter: number of maximum iteration of the algorithm
iter_wcc: find weakly connected components for the active vertices in this iteration, since the largest SCCs are already found after several iterations; usually a small number(3 to 10)
top_k_dist: top k result in SCC distribution
O(iter*d), d = max(diameter of components)
Directed edges with reverse direction edges as well
We ran scc on the social26 graph. A portion of the JSON result is shown below.
"trim_set.size()": 8
"@@cluster_dist_heap": [
The first element "i"=1 means the whole graph is processed in just one iteration. The 5 "trim_set.size()" elements mean there were 5 rounds of size-1 SCC trimming. The final "@@.cluster_dist_heap" object" reports on the size distribution of SCCs.There is one SCC with 9 vertices, and 17 SCCs with only 1 vertex in the graph.
Label Propagation tries t
|
Feature: Adding Collisions
• Feature: Adding Collisions
FeatureAdding Collisions
Using what they know from Bootstrap:Algebra, students write a distance function and collision detection function to handle collisions in their games, this time using the Data Structures and Reactor from their games.
30 minThe Distance Formula
30 minCollision Detection
Students add collision-detection to their games
8.F.1-3: The student defines, evaluates, and compares functions
8.G.6-8: The student uses the Pythagorean Theorem to solve real-world and mathematical problems
explanation of a proof of the Pythagorean Theorem and its converse
application of the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions
application of the Pythagorean Theorem to find the distance between two points in a coordinate system
A-CED.1-4: The student solves equations and inequalities in one or more variable that describe numbers or relationships
A-SSE.1-2: The student interprets the structure of expressions to solve problems in context
F-IF.1-3: The student uses function notation to describe, evaluate, and interpret functions in terms of domain and range
F-LE.5: The student interprets expressions for functions in terms of the situations they model
BS-CE: The student translates between structured expressions as arithmetic, code, and Circles of Evaluation
translating a Circle of Evaluation into its equivalent programming syntax
BS-DR.1: The student is able to translate a word problem into a Contract and Purpose Statement
given a word problem, identify the domain and range of a function
given a word problem, write a Purpose Statement (i.e. - rewrite the problem in their own words)
BS-DR.2: The student can derive test cases for a given contract and purpose statement
given a Contract and a Purpose Statement, write multiple examples or test cases
given multiple examples, identify patterns in order to label and name the variables
BS-DR.3: Given multiple test cases, the student can define a function
given examples and labeled variable(s), define the function
write piecewise functions that consume and produce data structures
identifying which quantities are fixed and which are variable
defining and using functions than involve conditionals
identify how functions work together to create and maintain a complex program
helper function: A small function that handles a specific part of another computation, and gets called from other functions
hypotenuse: the side opposite the 90-degree angle in a right triangle
Cutouts of Cat and Dog images
Cutouts of Pythagorean Theorem packets [1, 2] - 1 per cluster
Student Games or the Pyret Ninja Cat Starter file preloaded on students’ machines
Using the Pythagorean theorem and what they know from Bootstrap:Algebra, students write a distance function for their games
The Distance Formula (Time 30 minutes)
The Distance FormulaSo far, none of the animations we’ve created included any distance or collision-detection functions. However, if you want to make a game where the player has to hit a target, avoid an enemy, jump onto platforms, or reach a specific part of the screen, we’ll need to account for collisions. This is going to require a little math, but luckily it’s exactly the same as it was in Bootstrap:Algebra.
This lesson is part of a series of features meant to come at the end of the Bootstrap:Reactive units. Once students have made a number of simple animations and games, they will have lots of ideas for what they want to make next and add to their existing games. We’ve included a number of the most requested features in these lessons. Because each students’ game will be different, we’ve used a Pyret version of the original Ninja Cat game as an example program, but this lesson can be adapted to add collision detection to any game.
In the image above, how far apart are the cat and dog?
If the cat was moved one space to the right, how far apart would they be?
What if the cat and dog switched positions?
Finding the distance in one dimesion is pretty easy: if the characters are on the same number line, we subtract the smaller coordinate from the larger one, and we have our distance.
When the cat and dog were switched, did you still subtract the dog’s position from the cat’s, or subtract the cat’s position from the dog’s? Why?
Draw a number line on the board, with the cutouts of the cat and dog at the given positions. Ask students to tell you the distance between them, and move the images accordingly. Having students act this out can also work well: draw a number line, have two students stand at different points on the line, using their arms or cutouts to give objects of different sizes. Move students along the number line until they touch, then compute the distance on the number line.
Unfortunately, most distances aren’t only measured in one dimention. We’ll need some code to calculate the distance between two points in two dimentions.
How could you find the distance between the two points shown in this image?
How could you find the length of the C, also called the Hypotenuse?
Let’s start with what we do know: if we treat the x- and y-intercepts of C as lines A and B, we have a right triangle.
What is the line-length of A? Would it be different if the triangle pointed downward, and intercepted the point (0, -4)?
Draw this image on the board, with the lines labeled "A", "B", and "C".
Ancient civilizations had the same problem: they also struggled to find the distance between points in two dimensions. Let’s work through a way to think about this problem: what expression computes the length of the hypotenuse of a right triangle?
This exercise is best done in small groups of students (2-3 per group). Pass out Pythagorean Proof materials [1, 2] to each group, and have them review all of their materials:
A large, white square with a smaller one drawn inside
Four gray triangles, all the same size
Everyone will have a packet with the same materials, but each group’s triangles are a little different. The activity workes with triangles of all sizes, so each pair will get to test it out on their own triangles. Draw the diagram on the board.
For any right triangle, it is possible to draw a picture where the hypoteneuse is used for all four sides of a square. In the diagram shown here, the white square is surrounded by four gray, identical right-triangles, each with sides A and B. The square itself has four identical sides of length C, which are the hypoteneuses for the triangles. If the area of a square is expressed by
side * side
, then the area of the white space is
C^{2}
Have students place their gray triangles onto the paper, to match the diagram.
By moving the gray triangles, it is possible to create two rectangles that fit inside the original square. While the space taken up by the triangles has shifted, it hasn’t gotten any bigger or smaller. Likewise, the white space has been broken into two smaller squares, but in total it remains the same size. By using the side-lengths A and B, one can calculate the area of the two squares.
What is the area of the smaller square? The larger square?
You may need to explicitly point out that the side-lengths of the triangles can be used as the side-lengths of the squares.
The smaller square has an area of
A^{2}
, and the larger square has an area of
B^{2}
. Since these squares are just the original square broken up into two pieces, we know that the sum of these areas must be equal to the area of the original square:
A^{2} + B^{2} = C^{2}
Does the same equation work for any values of A and B?
To get C by itself, we take the square-root of the sum of the areas:
\sqrt{A^{2} + B^{2}} = C
Pythagoras proved that you can get the square of the hypotenuse by adding the squares of the other two sides. In your games, you’re going to use the horizontal and vertical distance between two characters as the two sides of your triangle, and use the Pythagorean theorem to find the length of that third side.
Remind students that A and B are the horizontal and vertical lengths, which are calculated by line-length.
Turn to Page 45 of your workbook - you’ll see the formula written out.
Draw out the circle of evaluation, starting with the simplest expression you can see first.
Once you have the circle of evaluation, translate it into Pyret code at the bottom of the page, starting with
check: distance(4, 2, 0, 5) is... end
Now you’ve got code that tells you the distance between the points (4, 2) and (0, 5). But we want to have it work for any two points. It would be great if we had a function that would just take the x’s and y’s as input, and do the math for us.
Turn to Page 46, and read the problem statement and function header carefully.
Use the Design Recipe to write your distance function. Feel free to use the work from the previous page as your first example, and then come up with a new one of your own.
When finished, type your distance functions into your game, and see what happens.
Does anything happen when things run into each other?
You still need a function to check whether or not two things are colliding.
Pay careful attention to the order in which the coordinates are given to the distance function. The player’s x-coordinate (px) must be given first, followed by the player’s y (py), character’s x (cx), and character’s y (cy). Just like with making data structures, order matters, and the distance function will not work otherwise. Also be sure to check that students are using num-sqr and num-sqrt in the correct places.
Students write a collision detection function, and modify their next-state-tick function to handle collisions in their games
Collision Detection (Time 30 minutes)
Collision DetectionSo what do we want to do with this distance?
How close should your danger and your player be, before they hit each other?
At the top of Page 47 you’ll find the Word Problem for is-collision.
Fill in the Contract, two examples, and then write the code. Remember: you WILL need to make use of the distance function you just wrote!
When you’re done, type it into your game, underneath distance.
Using visual examples, ask students to guess the distance between a danger and a player at different positions. How far apart do they need to be before one has hit the other? Make sure students understand what is going on by asking questions: If the collision distance is small, does that mean the game is hard or easy? What would make it easier?
Now that you have a function which will check whether two things are colliding, you can use it in your game! For extra practice, You can also implement collision detection into this Pyret Ninja Cat game. This is the program we’ll be altering for this lesson, as an example. In Ninja Cat, when the cat collides with the dog, we want to put the dog offscreen so that he can come back to attack again.
Out of the major functions in the game (next-state-tick, draw-state, or next-state-key), which do you think you’ll need to edit to handle collisions, changing the GameState when two characters collide?
We’ll need to make some more if branches for next-state-tick.
Start with the test: how could you check whether the cat and dog are colliding? Have you written a function to check that?
What do the inputs need to be?
How do you get the playery out of the GameState? playerx?
How do you get the dangerx out of the GameState? dangery?
if is-collision( g.playerx, g.playery, g.dangerx, g.dangery): ...result...
Remember that next-state-tick produces a GameState, so what function should come first in our result?
if is-collision( g.playerx, g.playery, g.dangerx, g.dangery): game( ...playerx..., ...playery..., ...dangerx..., ...dangery..., ...dangerspeed... ...targetx... ...targety... ...targetspeed...)
And what should happen when the cat and dog collide? Can you think of a number that puts the dog off the screen on the left side? What about the dog’s y-coordinate? We could choose a number and always place it at the same y-coordinate each time, but then the game would be really easy! To make it more challenging, we’d like the dog to appear at a random y-coordinate each time it collides with the cat. Thankfully, Pyret has a function which produces a random number between zero and its input: # num-random :: Number -> Number
if is-collision( g.playerx, g.playery, g.dangerx, g.dangery): game( g.playerx, 200, num-random(480), 0, 0, g.targetx, g.targety, g.targetspeed)
Collision detection must be part of the next-state-tick function because the game should be checking for a collision each time the GameState is updated, on every tick. Students may assume that draw-state should handle collision detection, but point out that the Range of draw-state is an Image, and their function must return a new GameState in order to set the locations of the characters after a collision.
Once you’ve finished, write another branch to check whether the player and the target have collided. Challenges:
Change your first condition so that the danger gets reset only when the player and danger collide AND the cat is jumping. (What must be true about the player’s y-coordinate for it to be jumping?)
Add another condition to check whether the player has collided with the danger while the player is on the ground. This could be a single expression within next-state-tick, or you can write a helper function called game-over to do this work, and use it in other functions as well (maybe the GameState is drawn differently once the game is over.)
For reference, a complete version of the Pyret Ninja Cat game can be found here.
|
Week 4. Stacks and Queues | Algorithms and Data Structures
Week 4. Stacks and Queues
5 Week 4. Stacks and Queues
Reading 4 Goodrich, Tamassia, & Goldwasser: Chapter 6–7. Java specific sections are cursory only.
5.1.1 Group Work Wednesday
5.1.2 Compulsory Projects
Problem 5.1 Give a simple adapter that implements the stack ADT using an instance of a double-ended queue.
Problem 5.2 Give a simple adapter that implements the queue ADT using an instance of a double-ended queue.
Problem 5.3 Describe how you can implement the stack ADT using a single queue as an instance variable and only constant additional local memory within the operations push(), pop(), and peek()/top(). What is the running time of the operations in your design?
Problem 5.4 Goodrich et al C-6.16.
Problem 5.5 Goodrich et al P-6.31. Present your solution as pseudo-code and diagrams. If you implement it in Java, this is in addition to a human-readable design description. It is not necessary to implement, and if you do, please reflect on why and whether you find that exercise useful.
Problem 5.6 (freely based on Goodrich et al C-7.48) Consider a message transmitted over the network. It is broken into
n
data packets which typically arrive at the receiver out of order.
Describe an efficient scheme for the receiver to assemble the packets in the correct order as they arrive. Discuss any assumptions you make. What is the running time?
Problem 5.8 Describe an implementation of the stack ADT using an array or array list for storage.
Problem 5.9 Describe an implementation of the deque ADT using an array or array list for storage.
|
1 Title: Conservation of mass and flux modeling.
Title: Conservation of mass and flux modeling.Edit
Develop the student's ability to create qualitative elemental model of mass flow process that will allow extrapolation to heat flow process.
Reading material on development of flux model based on conservatio of mass.
Consider modeling the movement of a small quantity of material
{\displaystyle \delta M}
through a incremental surface,
{\displaystyle \delta \Omega }
within a unit of time.
Have student(s) post elemental models using online simulation package.
Provide examples of modeling heat flow.
Provide feedback on a student's mode: critique by "experts", comments by other students, providing student with a document that shows them how to determine the validity of their model, a self grading checklist.
Multipe choice or short answer Quizzes
Our goal is to understand Partial Derivatives and equations that are composed of terms that include PD's. Such equations are called Partial Differential Equations or PDEs. More specifically, our interest is to solve PDEs , that is find a function
{\displaystyle f(x,y)}
{\displaystyle a_{1}(x,y){\frac {\partial ^{2}f}{\partial x^{2}}}+a_{2}(x,y){\frac {\partial ^{2}f}{\partial x\partial y}}+a_{3}(x,y){\frac {\partial ^{2}f}{\partial y^{2}}}+a_{4}(x,y){\frac {\partial f}{\partial x}}+a_{5}(x,y){\frac {\partial f}{\partial y}}+a_{6}(x,y)f+a_{7}(x,y)=0}
Conservation of MassEdit
|
Test Cointegrating Vectors - MATLAB & Simulink - MathWorks América Latina
Tests on B answer questions about the space of cointegrating relations. The column vectors in B, estimated by jcitest, do not uniquely define the cointegrating relations. Rather, they estimate a space of cointegrating relations, given by the span of the vectors. Tests on B allow you to determine if other potentially interesting relations lie in that space. When constructing constraints, interpret the rows and columns of the n-by- r matrix B as follows:
Row i of B contains the coefficients of variable
{y}_{it}
in each of the r cointegrating relations.
Column j of B contains the coefficients of each of the n variables in cointegrating relation j.
One application of jcontest is to pretest variables for their order of integration. At the start of any cointegration analysis, trending variables are typically tested for the presence of a unit root. These pretests can be carried out with combinations of standard unit root and stationarity tests such as adftest, pptest, kpsstest, or lmctest. Alternatively, jcontest lets you carry out stationarity testing within the Johansen framework. To do so, specify a cointegrating vector that is 1 at the variable of interest and 0 elsewhere, and then test to see if that vector is in the space of cointegrating relations. The following tests all of the variables in Y a single call:
[h0,pValue0] = jcontest(Y,1,'BVec',{[1 0 0]',[0 1 0]',[0 0 1]'})
The second input argument specifies a cointegration rank of 1, and the third and fourth input arguments are a parameter/value pair specifying tests of specific vectors in the space of cointegrating relations. The results strongly reject the null of stationarity for each of the variables, returning very small p-values.
Another common test of the space of cointegrating vectors is to see if certain combinations of variables suggested by economic theory are stationary. For example, it may be of interest to see if interest rates are cointegrated with various measures of inflation (and, via the Fisher equation, if real interest rates are stationary). In addition to the interest rates already examined, Data_Canada.mat contains two measures of inflation, based on the CPI and the GDP deflator, respectively. To demonstrate the test procedure (without any presumption of having identified an adequate model), we first run jcitest to determine the rank of B, then test the stationarity of a simple spread between the CPI inflation rate and the short-term interest rate:
% Test if inflation is cointegrated with interest rates:
[h,pValue] = jcitest(YI);
Data: YI
% Test if y1 - y2 is stationary:
[hB,pValueB] = jcontest(YI,1,'BCon',[1 -1 0 0]')
hB = logical
pValueB = 0.0242
The first test provides evidence of cointegration, and fails to reject a cointegration rank r = 1. The second test, assuming r = 1, rejects the hypothesized cointegrating relation. Of course, reliable economic inferences would need to include proper model selection, with corresponding settings for the 'model' and other default parameters.
|
Cunningham, Gabe1; Pellicer, Daniel2; Williams, Gordon3
1 University of Massachusetts Boston Department of Mathematics Boston, Massachusetts, USA
2 Centro de Ciencias Matemáticas, UNAM Morelia, Mexico
3 University of Alaska Fairbanks Department of Mathematics Fairbanks, Alaska, USA
There is an increasingly extensive literature on the problem of describing the connection (monodromy) groups and automorphism groups of families of polytopes and maniplexes that are not regular or reflexible. Many such polytopes and maniplexes arise as the result of constructions such as truncations and products. Here we show that for a wide variety of these constructions, the connection group of the output can be described in a nice way in terms of the connection group of the input. We call such operations stratified. Moreover, we show that, if
F
is a maniplex operation in one of two broad subclasses of stratified operations, and if
ℛ
is the smallest reflexible cover of some maniplex
ℳ
, then the connection group of
F\left(ℛ\right)
is equal to the connection group of
F\left(ℳ\right)
. In particular, we show that this is true for truncations and medials of maps, for products of polytopes (including pyramids and prisms over polytopes), and for the mix of maniplexes. As an application, we determine the smallest reflexible covers of the pyramids over the equivelar toroidal maps.
Classification: 52B15, 05E18, 52B05
Keywords: Polytope, maniplex, connection group, monodromy group, truncation, medial, pyramid, prism
Cunningham, Gabe 1; Pellicer, Daniel 2; Williams, Gordon 3
author = {Cunningham, Gabe and Pellicer, Daniel and Williams, Gordon},
title = {Stratified operations on maniplexes},
TI - Stratified operations on maniplexes
%T Stratified operations on maniplexes
Cunningham, Gabe; Pellicer, Daniel; Williams, Gordon. Stratified operations on maniplexes. Algebraic Combinatorics, Volume 5 (2022) no. 2, pp. 267-287. doi : 10.5802/alco.208. https://alco.centre-mersenne.org/articles/10.5802/alco.208/
[1] Berman, Leah Wrenn; Mixer, Mark; Monson, Barry; Oliveros, Deborah; Williams, Gordon The monodromy group of the
n
-pyramid, Discrete Math., Volume 320 (2014), pp. 55-63 | Article | MR: 3147207 | Zbl: 1285.52005
[2] Berman, Leah Wrenn; Monson, Barry; Oliveros, Deborah; Williams, Gordon The monodromy group of a truncated simplex, J. Algebraic Combin., Volume 42 (2015) no. 3, pp. 745-761 | Article | MR: 3403179 | Zbl: 1331.51020
[3] Brehm, Ulrich; Kühnel, Wolfgang Equivelar maps on the torus, European J. Combin., Volume 29 (2008) no. 8, pp. 1843-1861 | Article | MR: 2463161 | Zbl: 1160.52010
[4] Conder, Marston The smallest regular polytopes of given rank, Adv. Math., Volume 236 (2013), pp. 92-110 | Article | MR: 3019717 | Zbl: 1273.51013
[5] Cunningham, Gabe Self-dual, self-Petrie covers of regular polyhedra, Symmetry, Volume 4 (2012) no. 1, pp. 208-218 | Article | MR: 2909633 | Zbl: 1351.51020
[6] Cunningham, Gabe Flat extensions of abstract polytopes, Art Discrete Appl. Math., Volume 4 (2021) no. 3, Paper no. 3.06, 14 pages | Article | MR: 4312690
[7] Cunningham, Gabe; Del Río-Francos, María; Hubard, Isabel; Toledo, Micael Symmetry type graphs of polytopes and maniplexes, Ann. Comb., Volume 19 (2015) no. 2, pp. 243-268 | Article | MR: 3347382 | Zbl: 1327.52019
[8] Cunningham, Gabe; Pellicer, Daniel Open problems on
k
-orbit polytopes, Discrete Math., Volume 341 (2018) no. 6, pp. 1645-1661 | Article | MR: 3784786 | Zbl: 06861858
[9] Danzer, Ludwig Regular incidence-complexes and dimensionally unbounded sequences of such. I, Convexity and graph theory (Jerusalem, 1981) (North-Holland Math. Stud.), Volume 87, North-Holland, Amsterdam, 1984, pp. 115-127 | Article | MR: 791021 | Zbl: 0559.51012
[10] Douglas, Ian; Hubard, Isabel; Pellicer, Daniel; Wilson, Steve The twist operator on maniplexes, Discrete geometry and symmetry (Springer Proc. Math. Stat.), Volume 234, Springer, Cham, 2018, pp. 127-145 | Article | MR: 3816874 | Zbl: 1420.52013
[11] Drach, Kostiantyn; Mixer, Mark Minimal covers of equivelar toroidal maps, Ars Math. Contemp., Volume 9 (2015) no. 1, pp. 77-91 | Article | MR: 3377093 | Zbl: 1330.05049
[12] Fernandes, Maria Elisa; Leemans, Dimitri; Mixer, Mark Polytopes of high rank for the alternating groups, J. Combin. Theory Ser. A, Volume 119 (2012) no. 1, pp. 42-56 | Article | MR: 2844081 | Zbl: 1235.52021
[13] GAP – Groups, Algorithms, and Programming, Version 4.4.12 (2008) (http://www.gap-system.org)
[14] Garza-Vargas, Jorge; Hubard, Isabel Polytopality of maniplexes, Discrete Math., Volume 341 (2018) no. 7, pp. 2068-2079 | Article | MR: 3802160 | Zbl: 1387.05088
[15] Gleason, Ian; Hubard, Isabel Products of abstract polytopes, J. Combin. Theory Ser. A, Volume 157 (2018), pp. 287-320 | Article | MR: 3780416 | Zbl: 1390.52018
[16] Grünbaum, Branko Convex polytopes, Graduate Texts in Mathematics, 221, Springer-Verlag, New York, 2003, xvi+468 pages (Prepared and with a preface by Volker Kaibel, Victor Klee and Günter M. Ziegler) | Article | MR: 1976856 | Zbl: 1024.52001
[17] Hartley, Michael I.; Hulpke, Alexander Polytopes derived from sporadic simple groups, Contrib. Discrete Math., Volume 5 (2010) no. 2, pp. 106-118 | MR: 2791293 | Zbl: 1320.51021
[18] Hartley, Michael I.; Pellicer, Daniel; Williams, Gordon Minimal covers of the prisms and antiprisms, Discrete Math., Volume 312 (2012) no. 20, pp. 3046-3058 | Article | MR: 2956097 | Zbl: 1261.52006
[19] Hartley, Michael I.; Williams, Gordon Representing the sporadic Archimedean polyhedra as abstract polytopes, Discrete Math., Volume 310 (2010) no. 12, pp. 1835-1844 | Article | MR: 2610288 | Zbl: 1192.52018
[20] Helfand, Ilanit Constructions of k-orbit Abstract Polytopes, ProQuest LLC, Ann Arbor, MI, 2013, 147 pages Thesis (Ph.D.)–Northeastern University | MR: 3153240
[21] Hubard, Isabel Two-orbit polyhedra from groups, European J. Combin., Volume 31 (2010) no. 3, pp. 943-960 | Article | MR: 2587042 | Zbl: 1186.68502
[22] Hubard, Isabel; del Río Francos, María; Orbanić, Alen; Pisanski, Tomaž Medial symmetry type graphs, Electron. J. Combin., Volume 20 (2013) no. 3, Paper no. 29, 28 pages | Article | MR: 3104527 | Zbl: 1301.05170
[23] Hubard, Isabel; Orbanić, Alen; Pellicer, Daniel; Weiss, Asia Ivić Symmetries of equivelar 4-toroids, Discrete Comput. Geom., Volume 48 (2012) no. 4, pp. 1110-1136 | Article | MR: 3000577 | Zbl: 1263.51016
[24] Hubard, Isabel A. From geometry to groups and back: The study of highly symmetric polytopes, ProQuest LLC, Ann Arbor, MI, 2007, 168 pages Thesis (Ph.D.)–York University (Canada) | MR: 2712832
[25] Jones, Gareth A.; Poulton, Andrew Maps admitting trialities but not dualities, European J. Combin., Volume 31 (2010) no. 7, pp. 1805-1818 | Article | MR: 2673020 | Zbl: 1235.05151
[26] Koike, Hiroki; Pellicer, Daniel; Raggi, Miguel; Wilson, Steve Flag bicolorings, pseudo-orientations, and double covers of maps, Electron. J. Combin., Volume 24 (2017) no. 1, Paper no. 1.3, 23 pages | Article | MR: 3609173 | Zbl: 1355.05112
[27] McMullen, Peter; Schulte, Egon Abstract regular polytopes, Encyclopedia of Mathematics and its Applications, 92, Cambridge University Press, Cambridge, 2002, xiv+551 pages | Article | MR: 1965665 | Zbl: 1039.52011
[28] Mixer, Mark; Pellicer, Daniel; Williams, Gordon Minimal covers of the Archimedean tilings, part II, Electron. J. Combin., Volume 20 (2013) no. 2, Paper no. 20, 19 pages | Article | MR: 3066359 | Zbl: 1270.52017
[29] Monson, Barry; Pellicer, Daniel; Williams, Gordon Mixing and monodromy of abstract polytopes, Trans. Amer. Math. Soc., Volume 366 (2014) no. 5, pp. 2651-2681 | Article | MR: 3165650 | Zbl: 1286.51014
[30] Orbanić, Alen
F
-actions and parallel-product decomposition of reflexible maps, J. Algebraic Combin., Volume 26 (2007) no. 4, pp. 507-527 | Article | MR: 2341863 | Zbl: 1172.05324
[31] Orbanić, Alen; Pellicer, Daniel; Weiss, Asia Ivić Map operations and
k
-orbit maps, J. Combin. Theory Ser. A, Volume 117 (2010) no. 4, pp. 411-429 | Article | MR: 2592891 | Zbl: 1197.51005
[32] Pellicer, Daniel Cleaved abstract polytopes, Combinatorica, Volume 38 (2018) no. 3, pp. 709-737 | Article | MR: 3876881 | Zbl: 1438.52029
[33] Pellicer, Daniel; Williams, Gordon Minimal covers of the Archimedean tilings, Part 1, Electron. J. Combin., Volume 19 (2012) no. 3, Paper no. 6, 37 pages | MR: 2948453 | Zbl: 1258.51013
[34] Pellicer, Daniel; Williams, Gordon Pyramids over regular 3-tori, SIAM J. Discrete Math., Volume 32 (2018) no. 1, pp. 249-265 | Article | MR: 3755657 | Zbl: 1384.52010
[35] Richter, R. Bruce; Širáň, Jozef; Wang, Yan Self-dual and self-Petrie-dual regular maps, J. Graph Theory, Volume 69 (2012) no. 2, pp. 152-159 | Article | MR: 2864455 | Zbl: 1242.05076
[36] Schulte, Egon Amalgamation of regular incidence-polytopes, Proc. London Math. Soc. (3), Volume 56 (1988) no. 2, pp. 303-328 | Article | MR: 922658 | Zbl: 0609.51018
[37] Vince, Andrew Regular combinatorial maps, J. Combin. Theory Ser. B, Volume 35 (1983) no. 3, pp. 256-277 | Article | MR: 735194 | Zbl: 0514.05032
[38] Wilson, Stephen E. Parallel products in groups and maps, J. Algebra, Volume 167 (1994) no. 3, pp. 539-546 | Article | MR: 1287058 | Zbl: 0836.20032
[39] Wilson, Steve Maniplexes: Part 1: maps, polytopes, symmetry and operators, Symmetry, Volume 4 (2012) no. 2, pp. 265-275 | Article | MR: 2949129 | Zbl: 1351.52010
|
Integration Methods - Maple Help
Home : Support : Online Help : Mathematics : Calculus : Integration : int : Integration Methods
The method option for Integration
The following definite integration methods can be specified with the method option to int.
method=_DEFAULT forces use of the default integration method. It runs all of the integrators in sequence and returns the first answer found.
method=_UNEVAL causes the integrator to return unevaluated without trying any integration methods.
method=Integrator runs only the named Integrator and returns the result or unevaluated. The integrator names are not case sensitive. The most interesting integrators for users are:
LookUp tries to find the integral in a lookup table.
FTOC applies the fundamental theorem of calculus using indefinite integration and limits. The method FTOCMS does the same, but uses the limit implementation in MultiSeries.
Elliptic applies methods to rewrite an integral in terms of elliptic integrals. See the elliptic_int help page. There is also an EllipticTrig method which applies substitutions to find trig and hyperbolic trig forms of elliptic integrals.
Polynomial directly computes the integral algebraically if it is a polynomial. Ratpoly does the same with rational functions.
MeijerG attempts to integrate by converting the integrand into an expression in terms of MeijerG functions.
Running int with infolevel[IntegrationTools] set to 3 will show the list of integrators run.
method=NoIntegrator runs the default integration method but skips the any integrator with a name prefixed by Integrator. e.g. NoElliptic skips methods Elliptic and EllipticTrig.
method=NoXXIntegrator skips only the named integrator. e.g. NoXXElliptic skips only method Elliptic.
method=[method1, method2, etc] combines methods. If the methods are integrators, then each is tried in sequence. If the methods are all of the form NoIntegrator then they are each removed from the default integration sequence. A list with one method or a list combining Integrator and NoIntegrator methods is not particularly useful, but both are supported. _UNEVAL overrides any other methods it might be combined with and _DEFAULT is overridden by any other methods.
method=_RETURNVERBOSE applies all of the known methods and reports the results for each.
When int is called with the numeric option then the value of the method option is passed to evalf/Int. A comprehensive guide to all the possible method options can be found on the evalf/Int help page.
The indefinite integration polyalgorithm in Maple is not formulated as a single pass through a list of integration methods. However, the method option can be used get access to some of the individual integration algorithms used as part of the integration process. The supported values for indefinite integration are below. They can be given as names or strings and are not case sensitive.
method=_DEFAULT is equivalent to not specifying a method, exactly like definite and numeric integration.
method=_UNEVAL returns without trying anything, exactly like definite integration.
method=algorithm runs exactly the method specified:
DDivides tests if the derivative of the integrand divides the the integrand, and if so does a substitution to compute the integral.
Risch applies a partial implementation of Risch's algorithm for expressions involving elementary but no general algebraic functions functions. If no radical functions are in the integrand, failure of this method indicates there is no integral in terms of elementary functions.
Norman applies just the Risch-Norman stage of the method=Risch.
Trager applies the Risch-Trager algorithm for the integration of a pure algebraic function, given by a single extension. (In terms of a single unnested RootOf, it will try to convert radical expressions to RootOf form.)
MeijerG uses a variation of the definite MeijerG method that can find indefinite integrals. Two variations MeijerG_raw and Meijer_hg return the results of the method in terms of MeijerG functions or hypergeom functions, respectively. (Note that some MeijerG functions already automatically evaluate to hypergeometric functions so the raw output may not actually be presented in terms if MiegerG functions in all cases.
Elliptic applies methods to rewrite an integral in terms of elliptic integrals. See the elliptic_int help page.
LookUp tries to find the integral in a lookup table--this is very limited compared to the lookup tables for definite integration.
Gosper applies Gosper's method to find the integral of hyperexponential functions. It calls DEtools[Gosper].
method=[method1, method2, etc] applies every method in the list in turn and returns the answer from the first one successful.
\mathrm{int}\left(\frac{1}{\mathrm{sqrt}\left(\left(1-{t}^{2}\right)\left(1-2{t}^{2}\right)\right)},t=0..1,\mathrm{method}=\mathrm{FTOC}\right)
\textcolor[rgb]{0,0,1}{\mathrm{EllipticK}}\textcolor[rgb]{0,0,1}{}\left(\sqrt{\textcolor[rgb]{0,0,1}{2}}\right)
\mathrm{int}\left(\frac{1}{\mathrm{sqrt}\left(\left(1-{t}^{2}\right)\left(1-2{t}^{2}\right)\right)},t=0..1,\mathrm{method}=\mathrm{Elliptic}\right)
\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{I}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{EllipticK}}\textcolor[rgb]{0,0,1}{}\left(\frac{\sqrt{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{2}}\right)}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\frac{\sqrt{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{EllipticK}}\textcolor[rgb]{0,0,1}{}\left(\frac{\sqrt{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{2}}\right)}{\textcolor[rgb]{0,0,1}{2}}
\mathrm{int}\left(\frac{1}{\mathrm{sqrt}\left(\left(1-{t}^{2}\right)\left(1-2{t}^{2}\right)\right)},t=0..1,\mathrm{method}=\mathrm{NoElliptic}\right)
\textcolor[rgb]{0,0,1}{\mathrm{EllipticK}}\textcolor[rgb]{0,0,1}{}\left(\sqrt{\textcolor[rgb]{0,0,1}{2}}\right)
\mathrm{int}\left(\frac{1}{\mathrm{sqrt}\left(\left(1-{t}^{2}\right)\left(1-2{t}^{2}\right)\right)},t=0..1,\mathrm{method}=\mathrm{Polynomial}\right)
\textcolor[rgb]{0,0,1}{\mathrm{int}}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{1}}{\sqrt{\left(\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{t}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{t}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{method}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{Polynomial}}\right)
\mathrm{int}\left(\frac{1}{\mathrm{sqrt}\left(\left(1-{t}^{2}\right)\left(1-2{t}^{2}\right)\right)},t=0..1,\mathrm{method}=\mathrm{_UNEVAL}\right)
\textcolor[rgb]{0,0,1}{\mathrm{int}}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{1}}{\sqrt{\left(\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{t}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{t}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{method}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{_UNEVAL}}\right)
\mathrm{infolevel}[\mathrm{IntegrationTools}]≔3:
\mathrm{int}\left(\frac{1}{\mathrm{sqrt}\left(\left(1-{t}^{2}\right)\left(1-2{t}^{2}\right)\right)},t=0..1,\mathrm{method}=\mathrm{_DEFAULT}\right)
Definite Integration: Integrating expression on t=0..1
Definite Integration: Using the integrators [distribution, piecewise, series, o, polynomial, ln, lookup, cook, ratpoly, elliptic, elliptictrig, meijergspecial, improper, asymptotic, ftoc, ftocms, meijerg, contour]
LookUp Integrator: unable to find the specified integral in the table
Definite Integration: Method elliptic succeeded.
Definite Integration: Finished sucessfully.
\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{I}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{EllipticK}}\textcolor[rgb]{0,0,1}{}\left(\frac{\sqrt{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{2}}\right)}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\frac{\sqrt{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{EllipticK}}\textcolor[rgb]{0,0,1}{}\left(\frac{\sqrt{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{2}}\right)}{\textcolor[rgb]{0,0,1}{2}}
\mathrm{int}\left(\frac{\mathrm{sin}\left(\mathrm{exp}\left(\mathrm{abs}\left(x\right)\right)\right)}{1+{x}^{2}},x=-10..10,\mathrm{numeric},\mathrm{method}=\mathrm{_DEFAULT}\right)
\textcolor[rgb]{0,0,1}{1.235076653}
\mathrm{int}\left(\frac{1}{1+\mathrm{ln}\left(1+x\right)},x=0..1,\mathrm{numeric},\mathrm{method}=\mathrm{_Gquad}\right)
\textcolor[rgb]{0,0,1}{0.7371607096}
\mathrm{int}\left(\frac{\mathrm{sin}\left(\mathrm{exp}\left(\mathrm{abs}\left(x\right)\right)\right)}{1+{x}^{2}},x=-10..10,\mathrm{numeric},\mathrm{method}=\mathrm{_NoNAG}\right)
\textcolor[rgb]{0,0,1}{1.235076653}
\mathrm{int}\left(\mathrm{ln}\left(1+\mathrm{exp}\left(x\right)\right),x,\mathrm{method}=\mathrm{Risch}\right)
\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{dilog}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{x}}\right)
When a method fails, an unevaluated integral with the method option included is returned
\mathrm{int}\left(\mathrm{ln}\left(1+\mathrm{exp}\left(x\right)\right),x,\mathrm{method}=\mathrm{MeijerG}\right)
\textcolor[rgb]{0,0,1}{\mathrm{int}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{ln}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{x}}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{method}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{MeijerG}}\right)
\mathrm{int}\left(\mathrm{cos}\left(x+1\right),x,\mathrm{method}=\mathrm{Norman}\right)
\frac{\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{tan}}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{x}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\right)}{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathrm{tan}}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{x}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\right)}^{\textcolor[rgb]{0,0,1}{2}}}
\mathrm{int}\left(\mathrm{cos}\left(x+1\right),x,\mathrm{method}=\mathrm{MeijerG_raw}\right)
\textcolor[rgb]{0,0,1}{\mathrm{cos}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{hypergeom}}\textcolor[rgb]{0,0,1}{}\left([]\textcolor[rgb]{0,0,1}{,}[\frac{\textcolor[rgb]{0,0,1}{3}}{\textcolor[rgb]{0,0,1}{2}}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\frac{{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{4}}\right)\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{hypergeom}}\textcolor[rgb]{0,0,1}{}\left([\textcolor[rgb]{0,0,1}{1}]\textcolor[rgb]{0,0,1}{,}[\frac{\textcolor[rgb]{0,0,1}{3}}{\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}{-}\frac{{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{4}}\right)}{\textcolor[rgb]{0,0,1}{2}}
|
Inaccessible cardinal — Wikipedia Republished // WIKI 2
Type of infinite number in set theory
In set theory, an uncountable cardinal is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic. More precisely, a cardinal κ is strongly inaccessible if it is uncountable, it is not a sum of fewer than κ cardinals smaller than κ, and
{\displaystyle \alpha <\kappa }
{\displaystyle 2^{\alpha }<\kappa }
The term "inaccessible cardinal" is ambiguous. Until about 1950, it meant "weakly inaccessible cardinal", but since then it usually means "strongly inaccessible cardinal". An uncountable cardinal is weakly inaccessible if it is a regular weak limit cardinal. It is strongly inaccessible, or just inaccessible, if it is a regular strong limit cardinal (this is equivalent to the definition given above). Some authors do not require weakly and strongly inaccessible cardinals to be uncountable (in which case
{\displaystyle \aleph _{0}}
is strongly inaccessible). Weakly inaccessible cardinals were introduced by Hausdorff (1908), and strongly inaccessible ones by Sierpiński & Tarski (1930) and Zermelo (1930).
Every strongly inaccessible cardinal is also weakly inaccessible, as every strong limit cardinal is also a weak limit cardinal. If the generalized continuum hypothesis holds, then a cardinal is strongly inaccessible if and only if it is weakly inaccessible.
{\displaystyle \aleph _{0}}
(aleph-null) is a regular strong limit cardinal. Assuming the axiom of choice, every other infinite cardinal number is regular or a (weak) limit. However, only a rather large cardinal number can be both and thus weakly inaccessible.
The assumption of the existence of a strongly inaccessible cardinal is sometimes applied in the form of the assumption that one can work inside a Grothendieck universe, the two ideas being intimately connected.
MM70 - Dima Sinapova - The tree property at the single and double successors of a singular cardinal
1 Models and consistency
2 Existence of a proper class of inaccessibles
3 α-inaccessible cardinals and hyper-inaccessible cardinals
4 Two model-theoretic characterisations of inaccessibility
Models and consistency
Zermelo–Fraenkel set theory with Choice (ZFC) implies that the Vκ is a model of ZFC whenever κ is strongly inaccessible. And ZF implies that the Gödel universe Lκ is a model of ZFC whenever κ is weakly inaccessible. Thus, ZF together with "there exists a weakly inaccessible cardinal" implies that ZFC is consistent. Therefore, inaccessible cardinals are a type of large cardinal.
If V is a standard model of ZFC and κ is an inaccessible in V, then: Vκ is one of the intended models of Zermelo–Fraenkel set theory; and Def(Vκ) is one of the intended models of Mendelson's version of Von Neumann–Bernays–Gödel set theory which excludes global choice, replacing limitation of size by replacement and ordinary choice; and Vκ+1 is one of the intended models of Morse–Kelley set theory. Here Def (X) is the Δ0 definable subsets of X (see constructible universe). However, κ does not need to be inaccessible, or even a cardinal number, in order for Vκ to be a standard model of ZF (see below).
The issue whether ZFC is consistent with the existence of an inaccessible cardinal is more subtle. The proof sketched in the previous paragraph that the consistency of ZFC implies the consistency of ZFC + "there is not an inaccessible cardinal" can be formalized in ZFC. However, assuming that ZFC is consistent, no proof that the consistency of ZFC implies the consistency of ZFC + "there is an inaccessible cardinal" can be formalized in ZFC. This follows from Gödel's second incompleteness theorem, which shows that if ZFC + "there is an inaccessible cardinal" is consistent, then it cannot prove its own consistency. Because ZFC + "there is an inaccessible cardinal" does prove the consistency of ZFC, if ZFC proved that its own consistency implies the consistency of ZFC + "there is an inaccessible cardinal" then this latter theory would be able to prove its own consistency, which is impossible if it is consistent.
There are arguments for the existence of inaccessible cardinals that cannot be formalized in ZFC. One such argument, presented by Hrbáček & Jech (1999, p. 279), is that the class of all ordinals of a particular model M of set theory would itself be an inaccessible cardinal if there was a larger model of set theory extending M and preserving powerset of elements of M.
Existence of a proper class of inaccessibles
There are many important axioms in set theory which assert the existence of a proper class of cardinals which satisfy a predicate of interest. In the case of inaccessibility, the corresponding axiom is the assertion that for every cardinal μ, there is an inaccessible cardinal κ which is strictly larger, μ < κ. Thus, this axiom guarantees the existence of an infinite tower of inaccessible cardinals (and may occasionally be referred to as the inaccessible cardinal axiom). As is the case for the existence of any inaccessible cardinal, the inaccessible cardinal axiom is unprovable from the axioms of ZFC. Assuming ZFC, the inaccessible cardinal axiom is equivalent to the universe axiom of Grothendieck and Verdier: every set is contained in a Grothendieck universe. The axioms of ZFC along with the universe axiom (or equivalently the inaccessible cardinal axiom) are denoted ZFCU (not to be confused with ZFC with urelements). This axiomatic system is useful to prove for example that every category has an appropriate Yoneda embedding.
α-inaccessible cardinals and hyper-inaccessible cardinals
The α-inaccessible cardinals can also be described as fixed points of functions which count the lower inaccessibles. For example, denote by ψ0(λ) the λth inaccessible cardinal, then the fixed points of ψ0 are the 1-inaccessible cardinals. Then letting ψβ(λ) be the λth β-inaccessible cardinal, the fixed points of ψβ are the (β+1)-inaccessible cardinals (the values ψβ+1(λ)). If α is a limit ordinal, an α-inaccessible is a fixed point of every ψβ for β < α (the value ψα(λ) is the λth such cardinal). This process of taking fixed points of functions generating successively larger cardinals is commonly encountered in the study of large cardinal numbers.
The term hyper-inaccessible is ambiguous and has at least three incompatible meanings. Many authors use it to mean a regular limit of strongly inaccessible cardinals (1-inaccessible). Other authors use it to mean that κ is κ-inaccessible. (It can never be κ+1-inaccessible.) It is occasionally used to mean Mahlo cardinal.
Mahlo cardinals are inaccessible, hyper-inaccessible, hyper-hyper-inaccessible, ... and so on.
Two model-theoretic characterisations of inaccessibility
Firstly, a cardinal κ is inaccessible if and only if κ has the following reflection property: for all subsets U ⊂ Vκ, there exists α < κ such that
{\displaystyle (V_{\alpha },\in ,U\cap V_{\alpha })}
is an elementary substructure of
{\displaystyle (V_{\kappa },\in ,U)}
. (In fact, the set of such α is closed unbounded in κ.) Equivalently, κ is
{\displaystyle \Pi _{n}^{0}}
-indescribable for all n ≥ 0.
It is provable in ZF that ∞ satisfies a somewhat weaker reflection property, where the substructure (Vα, ∈, U ∩ Vα) is only required to be 'elementary' with respect to a finite set of formulas. Ultimately, the reason for this weakening is that whereas the model-theoretic satisfaction relation ⊧ can be defined, semantic truth itself (i.e.
{\displaystyle \vDash _{V}}
) cannot, due to Tarski's theorem.
Secondly, under ZFC it can be shown that κ is inaccessible if and only if (Vκ, ∈) is a model of second order ZFC.
In this case, by the reflection property above, there exists α < κ such that (Vα, ∈) is a standard model of (first order) ZFC. Hence, the existence of an inaccessible cardinal is a stronger hypothesis than the existence of a standard model of ZFC.
Mahlo cardinal
Club set
Inner model
Von Neumann universe
Drake, F. R. (1974), Set Theory: An Introduction to Large Cardinals, Studies in Logic and the Foundations of Mathematics, vol. 76, Elsevier Science, ISBN 0-444-10535-2
Hausdorff, Felix (1908), "Grundzüge einer Theorie der geordneten Mengen", Mathematische Annalen, 65 (4): 435–505, doi:10.1007/BF01451165, hdl:10338.dmlcz/100813, ISSN 0025-5831
Hrbáček, Karel; Jech, Thomas (1999), Introduction to set theory (3rd ed.), New York: Dekker, ISBN 978-0-8247-7915-3
Kanamori, Akihiro (2003), The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings (2nd ed.), Springer, ISBN 3-540-00384-3
Sierpiński, Wacław; Tarski, Alfred (1930), "Sur une propriété caractéristique des nombres inaccessibles" (PDF), Fundamenta Mathematicae, 15: 292–300, ISSN 0016-2736
Zermelo, Ernst (1930), "Über Grenzzahlen und Mengenbereiche: neue Untersuchungen über die Grundlagen der Mengenlehre" (PDF), Fundamenta Mathematicae, 16: 29–47, ISSN 0016-2736 . English translation: Ewald, William B. (1996), "On boundary numbers and domains of sets: new investigations in the foundations of set theory", From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics, Oxford University Press, pp. 1208–1233, ISBN 978-0-19-853271-2 .
|
Xiang, Ziqing1
1 Department of Mathematics, University of Georgia
Since the introduction of the notion of spherical designs by Delsarte, Goethals, and Seidel in 1977, finding explicit constructions of spherical designs had been an open problem. Most existence proofs of spherical designs rely on the topology of the spheres, hence their constructive versions are only computable, but not explicit. That is to say that these constructions can only give algorithms that produce approximations of spherical designs up to arbitrary given precision, while they are not able to give any spherical designs explicitly. Inspired by recent work on rational designs, i.e. designs consisting of rational points, we generalize the known construction of spherical designs that uses interval designs with Gegenbauer weights, and give an explicit formula of spherical designs of arbitrary given strength on the real unit sphere of arbitrary given dimension.
Classification: 05B30
Keywords: Explicit construction, rational points, spherical designs.
Xiang, Ziqing 1
author = {Xiang, Ziqing},
title = {Explicit spherical designs},
TI - Explicit spherical designs
%T Explicit spherical designs
Xiang, Ziqing. Explicit spherical designs. Algebraic Combinatorics, Volume 5 (2022) no. 2, pp. 347-369. doi : 10.5802/alco.213. https://alco.centre-mersenne.org/articles/10.5802/alco.213/
[1] Abramowitz, Milton; Stegun, Irene A. Handbook of mathematical functions: with formulas, graphs, and mathematical tables, 55, Courier Corporation, 1964
[2] Bajnok, Bela Construction of spherical
t
-designs, Geom. Dedicata, Volume 43 (1992) no. 2, pp. 167-179 | Article | MR: 1180648 | Zbl: 0765.05032
[3] Bondarenko, Andriy; Radchenko, Danylo; Viazovska, Maryna Optimal asymptotic bounds for spherical designs, Ann. of Math. (2), Volume 178 (2013) no. 2, pp. 443-452 | Article | MR: 3071504 | Zbl: 1270.05026
[4] Bondarenko, Andriy; Radchenko, Danylo; Viazovska, Maryna Well-separated spherical designs, Constr. Approx., Volume 41 (2015) no. 1, pp. 93-112 | Article | MR: 3296175 | Zbl: 1314.52020
[5] Bondarenko, Andriy V.; Viazovska, Maryna S. Spherical designs via Brouwer fixed point theorem, SIAM J. Discrete Math., Volume 24 (2010) no. 1, pp. 207-217 | Article | MR: 2600661 | Zbl: 1229.05057
[6] Chen, Xiaojun; Frommer, Andreas; Lang, Bruno Computational existence proofs for spherical
t
-designs, Numer. Math., Volume 117 (2011) no. 2, pp. 289-305 | Article | MR: 2754852 | Zbl: 1208.65032
[7] Chen, Xiaojun; Womersley, Robert S. Existence of solutions to systems of underdetermined equations and spherical designs, SIAM J. Numer. Anal., Volume 44 (2006) no. 6, pp. 2326-2341 | Article | MR: 2272596 | Zbl: 1129.65035
[8] Cui, Zhen; Xia, Jiacheng; Xiang, Ziqing Rational designs, Adv. Math., Volume 352 (2019), pp. 541-571 | Article | MR: 3964155 | Zbl: 1416.05062
[9] Delsarte, P.; Goethals, J. M.; Seidel, J. J. Spherical codes and designs, Geometriae Dedicata, Volume 6 (1977) no. 3, pp. 363-388 | Article | MR: 485471 | Zbl: 0376.05015
[10] Folland, Gerald B. How to integrate a polynomial over a sphere, Amer. Math. Monthly, Volume 108 (2001) no. 5, pp. 446-448 | Article | MR: 1837866 | Zbl: 1046.26503
[11] Gautschi, Walter On inverses of Vandermonde and confluent Vandermonde matrices, Numer. Math., Volume 4 (1962), pp. 117-123 | Article | MR: 139627 | Zbl: 0108.12501
[12] Hardin, Ronald H.; Sloane, Neil J. A. McLaren’s improved snub cube and other new spherical designs in three dimensions, Discrete Comput. Geom., Volume 15 (1996) no. 4, pp. 429-441 | Article | MR: 1384885 | Zbl: 0858.05024
[13] Korevaar, Jacob; Meyers, J. L. H. Spherical Faraday cage for the case of equal point charges and Chebyshev-type quadrature on the sphere, Integral Transform. Spec. Funct., Volume 1 (1993) no. 2, pp. 105-117 | Article | MR: 1421438 | Zbl: 0823.41026
[14] Kuperberg, Greg Special moments, Adv. in Appl. Math., Volume 34 (2005) no. 4, pp. 853-870 | Article | MR: 2129001 | Zbl: 1077.62007
[15] Rabau, Patrick; Bajnok, Bela Bounds for the number of nodes in Chebyshev type quadrature formulas, J. Approx. Theory, Volume 67 (1991) no. 2, pp. 199-214 | Article | MR: 1133060 | Zbl: 0751.41026
[16] Seymour, Paul D.; Zaslavsky, Thomas Averaging sets: a generalization of mean values and spherical designs, Adv. in Math., Volume 52 (1984) no. 3, pp. 213-240 | Article | MR: 744857 | Zbl: 0596.05012
[17] Stein, Elias M.; Shakarchi, Rami Real analysis: measure theory, integration, and Hilbert spaces, Princeton University Press, 2005 | Article
[18] Venkov, Boris B. Even unimodular extremal lattices, Trudy Mat. Inst. Steklov., Volume 165 (1984), pp. 43-48 | MR: 752931 | Zbl: 0544.10017
[19] Wagner, Gerold On averaging sets, Monatsh. Math., Volume 111 (1991) no. 1, pp. 69-78 | Article | MR: 1089385 | Zbl: 0721.65011
|
3 Ways to Find the X Intercept - wikiHow
How to Find the X Intercept
1 Using a Graph of a Line
2 Using the Equation of the Line
In algebra, 2-dimensional coordinate graphs have a horizontal axis, or x-axis, and a vertical axis, or y-axis. The places where lines representing a range of values cross these axes are called intercepts. The y-intercept is the place where the line crosses the y-axis and the x-intercept where the line crosses the x-axis. For simple problems, it is easy to find the x-intercept by looking at a graph. You can find the exact point of the intercept by solving algebraically using the equation of the line.
Using a Graph of a Line
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/8\/86\/Find-the-X-Intercept-Step-1-Version-2.jpg\/v4-460px-Find-the-X-Intercept-Step-1-Version-2.jpg","bigUrl":"\/images\/thumb\/8\/86\/Find-the-X-Intercept-Step-1-Version-2.jpg\/aid2670010-v4-728px-Find-the-X-Intercept-Step-1-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
Identify the x-axis. A coordinate graph has a y-axis and an x-axis. The x-axis is the horizontal line (the line that goes from left-to-right). The y-axis is the vertical line (the line that goes up and down).[1] X Research source It is important to look at the x-axis when locating the x-intercept.
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/5\/5e\/Find-the-X-Intercept-Step-2-Version-2.jpg\/v4-460px-Find-the-X-Intercept-Step-2-Version-2.jpg","bigUrl":"\/images\/thumb\/5\/5e\/Find-the-X-Intercept-Step-2-Version-2.jpg\/aid2670010-v4-728px-Find-the-X-Intercept-Step-2-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
Find the point where the line crosses the x-axis. The x-intercept is this point.[2] X Expert Source David Jia
Academic Tutor Expert Interview. 14 January 2021. If you are asked to find the x-intercept based on the graph, the point will likely be exact (for example, at point 4). Usually, however, you will have to estimate using this method (for example, the point is somewhere between 4 and 5).
Write the ordered pair for the x-intercept. An ordered pair is written in the form
{\displaystyle (x,y)}
and gives you the coordinates for the point on the line. [3] X Research source The first number of the pair is the point where the line crosses the x-axis (the x-intercept). The second number for will always be 0, since a point on the x-axis will never have a value for y.[4] X Research source
For example, if a line crosses the x-axis at point 4, the ordered pair for the x-intercept is
{\displaystyle (4,0)}
Using the Equation of the Line
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/7\/7b\/Find-the-X-Intercept-Step-4-Version-2.jpg\/v4-460px-Find-the-X-Intercept-Step-4-Version-2.jpg","bigUrl":"\/images\/thumb\/7\/7b\/Find-the-X-Intercept-Step-4-Version-2.jpg\/aid2670010-v4-728px-Find-the-X-Intercept-Step-4-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
Determine that the equation of the line is in standard form. The standard form of a linear equation is
{\displaystyle Ax+By=C}
.[5] X Research source In this form,
{\displaystyle A}
{\displaystyle B}
{\displaystyle C}
{\displaystyle x}
{\displaystyle y}
are the coordinates of a point on the line.
For example, you might be given the equation
{\displaystyle 2x+3y=6}
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/7\/75\/Find-the-X-Intercept-Step-5.jpg\/v4-460px-Find-the-X-Intercept-Step-5.jpg","bigUrl":"\/images\/thumb\/7\/75\/Find-the-X-Intercept-Step-5.jpg\/aid2670010-v4-728px-Find-the-X-Intercept-Step-5.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
Plug in 0 for
<b class="whb">{\displaystyle y}</b>
. The x-intercept is the point on the line where the line crosses the x-axis. At this point, the value for
{\displaystyle y}
will be 0. So, in order to find the x-intercept, you need to set the
{\displaystyle y}
to 0 and solve for
{\displaystyle x}
.[6] X Expert Source David Jia
For example, if you substitute 0 for
{\displaystyle y}
, your equation will look like this:
{\displaystyle 2x+3(0)=6}
{\displaystyle 2x=6}
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/d\/d9\/Find-the-X-Intercept-Step-6.jpg\/v4-460px-Find-the-X-Intercept-Step-6.jpg","bigUrl":"\/images\/thumb\/d\/d9\/Find-the-X-Intercept-Step-6.jpg\/aid2670010-v4-728px-Find-the-X-Intercept-Step-6.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
<b class="whb">{\displaystyle x}</b>
. To do this, you need to isolate the x variable by dividing both sides of the equation by the coefficient. This will give you the value of
{\displaystyle x}
{\displaystyle y=0}
, which is the x-intercept.[7] X Expert Source David Jia
{\displaystyle 2x=6}
{\displaystyle {\frac {2x}{2}}={\frac {6}{2}}}
{\displaystyle x=3}
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/c\/c6\/Find-the-X-Intercept-Step-7.jpg\/v4-460px-Find-the-X-Intercept-Step-7.jpg","bigUrl":"\/images\/thumb\/c\/c6\/Find-the-X-Intercept-Step-7.jpg\/aid2670010-v4-728px-Find-the-X-Intercept-Step-7.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
Write the ordered pair. Remember that an ordered pair is written in the form
{\displaystyle (x,y)}
. For the x-intercept, the value of
{\displaystyle x}
will be the value you calculated previously, and the
{\displaystyle y}
value will be 0, since
{\displaystyle y}
always equals 0 at the x-intercept.[8] X Research source
For example, for the line
{\displaystyle 2x+3y=6}
, the x-intercept is at the point
{\displaystyle (3,0)}
Determine that the equation of the line is a quadratic equation. A quadratic equation is an equation that takes the form
{\displaystyle ax^{2}+bx+c=0}
.[9] X Research source A quadratic equation has two solutions, which means a line written in this form is a parabola and will have two x-intercepts.[10] X Research source
{\displaystyle x^{2}+3x-10=0}
is a quadratic equation, so this line will have two x-intercepts.
Set up the quadratic formula. The formula is
{\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}}
{\displaystyle a}
equals the coefficient of the second-degree term (
{\displaystyle x^{2}}
{\displaystyle b}
equals the coefficient of the first-degree term (
{\displaystyle x}
{\displaystyle c}
equals the constant.[11] X Research source
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/2\/2e\/Find-the-X-Intercept-Step-10.jpg\/v4-460px-Find-the-X-Intercept-Step-10.jpg","bigUrl":"\/images\/thumb\/2\/2e\/Find-the-X-Intercept-Step-10.jpg\/aid2670010-v4-728px-Find-the-X-Intercept-Step-10.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
Plug all of the values into the quadratic formula. Make sure you substitute the correct values for each variable from the equation of the line.
For example, if the equation of your line is
{\displaystyle x^{2}+3x-10=0}
, your quadratic formula will look like this:
{\displaystyle x={\frac {-3\pm {\sqrt {3^{2}-4(1)(-10)}}}{2(1)}}}
Simplify the equation. To do this, first complete all of the multiplication. Make sure you pay close attention to all positive and negative signs.
{\displaystyle x={\frac {-3\pm {\sqrt {3^{2}-4(-10)}}}{2(1)}}}
{\displaystyle x={\frac {-3\pm {\sqrt {3^{2}+40}}}{2}}}
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/7\/71\/Find-the-X-Intercept-Step-12.jpg\/v4-460px-Find-the-X-Intercept-Step-12.jpg","bigUrl":"\/images\/thumb\/7\/71\/Find-the-X-Intercept-Step-12.jpg\/aid2670010-v4-728px-Find-the-X-Intercept-Step-12.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
Calculate the exponent. Square the
{\displaystyle b}
term. Then, add this number to the other number under the square root sign.
{\displaystyle x={\frac {-3\pm {\sqrt {3^{2}+40}}}{2}}}
{\displaystyle x={\frac {-3\pm {\sqrt {9+40}}}{2}}}
{\displaystyle x={\frac {-3\pm {\sqrt {49}}}{2}}}
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/a\/a2\/Find-the-X-Intercept-Step-13.jpg\/v4-460px-Find-the-X-Intercept-Step-13.jpg","bigUrl":"\/images\/thumb\/a\/a2\/Find-the-X-Intercept-Step-13.jpg\/aid2670010-v4-728px-Find-the-X-Intercept-Step-13.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
Solve for the addition formula. Since the quadratic formula has a
{\displaystyle \pm }
, you will solve once by adding, and once by subtracting. Solving by adding will give you your first
{\displaystyle x}
{\displaystyle x={\frac {-3+{\sqrt {49}}}{2}}}
{\displaystyle x={\frac {-3+7}{2}}}
{\displaystyle x={\frac {4}{2}}}
{\displaystyle x=2}
Solve for the subtraction formula. This will give you the second value for
{\displaystyle x}
. First calculate the square root, then find the difference in the numerator. Finally, divide by 2.
{\displaystyle x={\frac {-3-{\sqrt {49}}}{2}}}
{\displaystyle x={\frac {-3-7}{2}}}
{\displaystyle x={\frac {-10}{2}}}
{\displaystyle x=-5}
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/8\/8f\/Find-the-X-Intercept-Step-15.jpg\/v4-460px-Find-the-X-Intercept-Step-15.jpg","bigUrl":"\/images\/thumb\/8\/8f\/Find-the-X-Intercept-Step-15.jpg\/aid2670010-v4-728px-Find-the-X-Intercept-Step-15.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
Find the ordered pairs for the x-intercept. Remember that an ordered pair gives the x-coordinate first, then the y-coordinate
{\displaystyle (x,y)}
{\displaystyle x}
values will be the values you calculated using the quadratic formula. The
{\displaystyle y}
value will be 0, since at the x-intercept,
{\displaystyle y}
always equals 0.[12] X Research source
{\displaystyle x^{2}+3x-10=0}
, the x-intercepts are at points
{\displaystyle (2,0)}
{\displaystyle (-5,0)}
The x intercept is wherever the line or the graph crosses the horizontal x axis.
How do you find the x intercept with a line equation?
If you just have a simple equation like y=mx+b, you would find the x-intercept by substituting 0 for y and solving for x. If it's a quadratic equation, you'd solve by either factoring or using the quadratic equation.
What if the square root isnt perfect?
If you mean the square root is not a whole number, that's OK. It just means that the x-intercept will occur somewhere between two integers on the x-axis.
What is the x-intercept of the equation 2x-y=8?
To find the x-intercept, set every other variable to 0, and solve for x. If 2x - 0 = 8, then 2x/2 = 8/2, and x = 4. So 4 is the x-intercept of that equation.
What is the value of y in the equation y = 2x + 1?
In order to find a numerical value for y, you would have to know the numerical value of x, then double it and add 1. Otherwise, "2x + 1" is the only value y can have.
The graph of y=10x2+bx+c has the x intercept at 1.4 and 1.5. What is c-b?
Remember that x-intercepts are roots (or zeros) of an equation, and that roots are factors. Then y = a(x-1.4)(x-1.5), where a must be 10 in order to have the leading coefficient of 10x^2. Now that you know y = 10(x-1.4)(x-1.5), just multiply it out and see what a and b are. y = 10x^2 -29x + 21. So b = -29, c = 21, and c-b = 50.
If you are working with the equation
{\displaystyle y=mx+b}
, you need to know the slope of the line and the y intercept. In the equation, m = the slope of the line and b = the y-intercept. Set y to equal zero, and solve for x. This will give you your x-intercept.
↑ http://www.virtualnerd.com/pre-algebra/linear-functions-graphing/equations/x-y-intercepts/x-intercept-definition
↑ https://www.mathsisfun.com/definitions/ordered-pair.html
↑ http://mathworld.wolfram.com/QuadraticEquation.html
↑ http://www.csun.edu/~ayk38384/notes/mod11/Parabolas.html
↑ http://jwilson.coe.uga.edu/emt668/EMAT6680.Folders/Barron/unit/Lesson%207/7.html
To find the x intercept using the equation of the line, plug in 0 for the y variable and solve for x. You can also use the graph of the line to find the x intercept. Just look on the graph for the point where the line crosses the x-axis, which is the horizontal axis. That point is the x intercept. To learn more, like how to find the x-intercept in a quadratic equation, keep reading the article!
Español:hallar la intersección con el eje X
Italiano:Trovare la X Intercetta
Русский:найти точку пересечения с осью Х
中文:求X轴的截距
Nederlands:Het snijpunt met de x as vinden
Français:trouver la coordonnée x du point d'intersection d'une équation avec l'axe des x
Bahasa Indonesia:Mencari Titik Potong X
ไทย:หาจุดตัดแกน X
हिन्दी:X खंड प्राप्त करें
Tiếng Việt:Tìm Giao điểm X của Hàm số với Trục Hoành
한국어:x절편 구하는 법
العربية:إيجاد التقاطع إكس
Andile Mzizi
"Everything here is clear. I didn't understand my teacher at all, but when I checked here, I found everything vividly explained. I'm so glad, thanks a lot!"..." more
"I needed to find out how to find the X-intercept for a project that is due. This helped me with that!"
"I did not think to use the quadratic formula, so this was very helpful."
|
Cross-field concepts - Tales of Science & Data
Cross-fields concepts are mathematical concepts shared across multiple fields or study, which are employed in Data Science as well.
To run the code here, you just need some imports:
In thermodynamics, the entropy is defined as
\Delta S = \int \frac{\delta q}{T}
In statistical mechanics, Boltzmann gave the definition as a measure of uncertainty and demonstrated that it is equivalent to the thermodynamics definition:
the entropy quantifies the degree to which the probability of the system is spread over different microstates and is proportional to the logarithm of the number of possible microconfigurations which give rise to the macrostate.
Which, written down, is
S = -k_B \sum_i p_i log \, p_i
(sum over all the possible microstates, where
p_i
is the probability of state i to be occupied). The postulate is that the occupation of every microstate is equiprobable.
In Information Theory, Shannon defined the entropy as a measure of the missing information before the reception of a message:
H = -\sum_i p(x_i) \, log \, p(x_i)
p(x_i)
is the probability that character of type
x_i
in the string of interest. This entropy measures the number of binary (YES/NO) questions needed to determine the content of the message. The link between the statistical mechanics and the information theory concepts is debated.
In Ecology, defining the diversity index D as the number of different types (species) in a dataset among which individuals are distributed, so that it is maximised when all types are equally abundant,
D = e^H
where H is the uncertainty in predicting the species of an individual taken at random from the dataset
\begin{aligned} H &= -\sum_i p_i log \, p_i \\ &= - \sum_i log \, p_i^{p_i} \\ &= - log(\Pi_i p_i^{p_i}) \\ &= - log\left(\frac{1}{\Pi_i p_i^{p_i}}\right) \end{aligned}
which at the denominator has the weighted geometric mean of the
p_i
If all types are equally occupied,
p_i = 1/k \ \forall i
H = log(k)
(H max)
If only one type is present
p_i = 0 \forall i \in {1, \ldots, n-1}
p_n = 1
, then H=0
Cross-entropy and Kullback-Leibler divergence
Given two distributions over the set of events, p and q, the cross entropy between them is calculated as
H(p, q) = \mathbb{E}_p [ - \log q] = \sum_i p_i \log q_i = H(p) + D_{KL} (p || q)
H(p)
is the entropy of the distribution p and
D_{KL}
is the Kullback-Leibler divergence of q from p, also known as the relative entropy of p with respect to q.
The cross entropy measures the average number of bits needed to identify an event drawn from the set if another distribution is assumed; the KL divergence measures the difference between the two probability distributions, or, better, the information gained when the priors $q$ are revised in light of posteriors $p$ (in other words, the amount of information lost when $q$ is used instead of $p$). It is defined as
D_{KL}(p || q) = \sum_i p_i \log{\frac{p_i}{q_i}}
Note that for continuous variables the sums become integrals.
The inverse participation ratio quantifies how many states a particle, or whatever has a distribution, is distributed over, and is defined as
I = \frac{1}{\sum_i p_i^2} \ ,
p_i
is the probability of occupation of state i.
The extreme situations are:
If there is only one state, so that
p_j = 1
p_i = 0 \ \forall i \neq 0
I = 1
If there is an even distribution, so that
p_j = 1/N \ \forall j \in {1, \ldots N}
where N is the number of states, then
I = N
With two states, we have
I = \frac{1}{p^2 + (1-p)^2} = \frac{1}{1 + 2p^2 - 2p} \ ,
which has the shape in figure down here, where you see that the maximum is for p=0.5, equally probable states (a fair coin).
p = np.arange(0,1.05,0.05)
I = 1./(1 + 2*p**2 -2*p)
plt.plot(p, I)
plt.xlabel('$p#x27;)
plt.ylabel('$I#x27;)
The "no free lunch" theorem
It is a concept originated in mathematics (optimisation) but often employed in Machine Learning and it asserts that the computational cost of finding a solution for a problem of a given class, averaged over all problems in the class, is the same for every method employed (see references). In short, you don't get anything for nothing (the "free lunch"). The phrasing seem to have its origins into an old practice of USA saloons where you could get food for free when purchasing drinks.
This means that there is no algorithm which is optimal on all possible problems, as its excellent performance on a problem is counterbalanced by bad performance on another problem.
Also see reference for a deeper explanation.
D H Wolpert, W G Macready No free lunch theorems for optimization, IEEE transactions on evolutionary computation, 1.1 (1997)
No Free Lunch Theorems
The mathematics appendix - Previous
Next - The mathematics appendix
|
Electrochemical Impedance Parameters for the Diagnosis of a Polymer Electrolyte Fuel Cell Poisoned by Carbon Monoxide in Reformed Hydrogen Fuel | J. Electrochem. En. Conv. Stor | ASME Digital Collection
Hironori Nakajima,
Department of Mechanical Engineering Science, Faculty of Engineering,
, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan
e-mail: nakajima@mech.kyushu-u.ac.jp
Toshiaki Konomi,
Tatsumi Kitahara,
Hideaki Tachibana
Department of Mechanical Engineering Science, Graduate School of Engineering,
Nakajima, H., Konomi, T., Kitahara, T., and Tachibana, H. (September 11, 2008). "Electrochemical Impedance Parameters for the Diagnosis of a Polymer Electrolyte Fuel Cell Poisoned by Carbon Monoxide in Reformed Hydrogen Fuel." ASME. J. Fuel Cell Sci. Technol. November 2008; 5(4): 041013. https://doi.org/10.1115/1.2931462
We have investigated the behavior of an operating polymer electrolyte fuel cell (PEFC) with supplying a mixture of carbon monoxide (CO) and hydrogen
(H2)
gases into the anode to develop the PEFC diagnosis method for anode CO poisoning by reformed hydrogen fuel. We analyze the characteristics of the CO poisoned anode of the PEFC at
80°C
including CO adsorption and electro-oxidation behaviors by current-voltage
(I‐V)
measurement and electrochemical impedance spectroscopy (EIS) to find parameters useful for the diagnosis.
I‐V
curves show the dependence of the output voltage on the CO adsorption and electro-oxidation. EIS analyses are performed with an equivalent circuit model consisting of several resistances and capacitances attributed to the activation, diffusion, and adsorption∕desorption processes. As the result, those resistances and capacitances are shown to change with current density and anode overpotential depending on the CO adsorption and electro-oxidation. The characteristic changes of those parameters show that they can be used for the diagnosis of the CO poisoning.
adsorption, anodes, desorption, electrochemical electrodes, electrochemical impedance spectroscopy, oxidation, proton exchange membrane fuel cells, diagnosis, polymer electrolyte fuel cell, PEMFC, CO poisoning, electrochemical impedance spectroscopy, complex plane plot, equivalent circuit
Anodes, Carbon, Circuits, Current density, Electrochemical impedance spectroscopy, Electrodes, Electrolytes, Fuel cells, Hydrogen fuels, Overvoltage, Oxidation, Polymers, Hydrogen, Proton exchange membrane fuel cells
A New Approach to the Problem of Carbon Monoxide Poisoning in Fuel Cells Operating at Low Temperatures
Hydrogen Electro-Oxidation on Platinum Catalysts in the Presence of Trace Carbon Monoxide
Lattice Gas Model for CO Electrooxidation on Pt–Ru Bimetallic Surfaces
Electrochemical Impedance Study of Electrode-Membrane Assemblies in PEM Fuel Cells. I. Electro-Oxidation of H2 and H2∕CO Mixtures on Pt-Based Gas-Diffusion Electrodes
Enbäck
Experimentally Validated Model for CO Oxidation on PtRu∕C in a Porous PEFC Electrode
Research on Diagnosis Technique on PEFC Running Condition (High Speed Analysis by FFT and Feasibility Study of Diagnosis)
Research of Diagnosis Technique on PEFC Running Condition (Overvoltage Analysis and Diagnosis of PEFC by FFT)
CO Tolerance of Pt Alloy Electrocatalysts for Polymer Electrolyte Fuel Cells and the Detoxification Mechanism
High CO Tolerance of N,N-Ethylenebis(salicylideneaminato)oxovanadium(IV) as a Cocatalyst to Pt for the Anode of Reformate Fuel Cells
Impedance Plane Display of a Reaction With an Adsorbed Intermediate
Steady-State and EIS Investigations of Hydrogen Electrodes and Membranes in Polymer Electrolyte Fuel Cells I. Modeling
ac Impedance of Faradic Reactions Involving Electrosorbed Intermediates -I. Kinetic Theory
Research on PEFC Overvoltage Analysis Method by Impedance Technique (1st Report, Estimation of Resistances and Overvoltages)
Effects of Internal Leak Current on PEFC Output Voltage (Internal Leak Current at Open Circuit and Close Circuit in Smaller Current Density Region)
|
Another way of establishing the OLS formula is through the method of moments approach. This method supposedly goes way back to Pearson in 1894. It could be thought of as replacing a population moment with a sample analogue and using it to solve for the parameter of interest.
To find an estimator for the sample mean,
\mu=E[X]
, one replaces the expected value with a sample analogue,
\hat{\mu}=\frac{1}{n}\sum_{i=1}^{n} X_{i} = \bar{X}
X_{1}, X_{2}, ..., X_{n}
be drawn from a normal distribution i.e.
X_{i} \sim N(\mu,\sigma^{2})
The goal is to find an estimator for the two parameters,
\mu
\sigma
. The first and second moment of a normal distribution is given by:
\begin{aligned} E[X] &= \mu \\ E[X^{2}] &= \mu_{2} = \mu^{2} + \sigma^{2} \end{aligned}
An estimator for
\mu
is easy and is simply
\hat{\mu} = \frac{1}{n}\sum_{i=1}^{n} X_{i} = \bar{X}
Replace the moment condition with the sample analogue and substitute in the estimator for
\mu
to find an estimator for
\sigma^2
\begin{aligned} \frac{1}{n}\sum_{i=1}^{n} X_{i}^{2} &= \mu^{2} + \hat{\sigma}^{2} \\ \hat{\sigma}^{2} &= \frac{1}{n}\sum_{i=1}^{n} X_{i}^{2} - \bar{X}^{2} \\ &= \frac{1}{n}\sum_{i=1}^{n}(X_{i}-\bar{X})^2 \end{aligned}
X_{1}, X_{2}, ..., X_{n}
be drawn from a poisson distribution i.e.
X_{i} \sim Poisson(\lambda)
. The poisson distribution is characterised by the following equality:
E[X]=var(X)=\lambda
. This gives rise to two possible estimators for
\lambda
\begin{aligned} \hat{\lambda}_{1} &= \bar{X} \\ \hat{\lambda}_{2} &= \frac{1}{n}\sum_{i=1}^{n}(X_{i}-\bar{X})^2 \end{aligned}
Since there is only one parameter to be estimated but two moment conditions, one would need some way of 'combining' the two conditions. Using only one condition would be not making full use of the information at hand.
Regression - Method of Moments
More generally, one can write the moment conditions as a vector of functions
g(X_{i},\beta)
\mathbf{X}_{i}
is the observed data, including all variables
(y_{i}, X_{i})
and instruments
(\mathbf{Z}_{i})
in the regression model, while
\beta
is the vector of parameters of length
k
. The model is identified if the solution is unique, i.e.
Eg(X_{i},\beta)=0
Eg(X_{i},\hat{\beta})=0
\beta=\hat{\beta}
. This requires that we have at least
k
restrictions for
k
For the OLS regression, one can use the moment condition
E(\mathbf{X}_{i}U_{i})=0
E(\mathbf{X_{i}}(y_{i}-\mathbf{X}_{i}'\beta))=0
to solve for the usual OLS estimator.
The idea can be carried over to other more complicated regression models. For example, in the case where
g(X_{i},\beta)
\beta
g(X_{i},\beta) = \mathbf{Z}_{i}(y_{i} - \mathbf{X}_{i}'\beta)
E(\mathbf{Z}_{i}U_{i})=0
, and the model is perfectly identified
(l=k)
, solving the moment condition yields the formula for the IV regression:
\begin{aligned} 0 &= \sum_{i=1}^{n}\mathbf{Z}_{i}(y_{i} - \mathbf{X}_{i}'\hat{\beta}^{IV}) \\ \hat{\beta}^{IV} &= \Big(\sum_{i=1}^{n} \mathbf{Z}_{i}\mathbf{X}_{i}' \Big)^{-1} \sum_{i=1}^{n}\mathbf{Z}_{i}y_{i} \\ &= (\mathbf{Z}'\mathbf{X})^{-1}\mathbf{Z}'\mathbf{y} \end{aligned}
Hence an IV regression could be thought of as substituting 'problematic' OLS moments for hopefully better moment conditions with the addition of instruments.
Extension - Generalised Method of Moments (GMM)
While it is not possible to identify
\beta
if there are too few restrictions, one could still identify
\beta
l > k
restrictions (overidentified), as seen in the poisson example.1 One might then wonder what is the best way to combine these restrictions. The GMM approach, introduced by Hansen in 1982, finds an estimate of
\beta
that brings the sample moments as close to zero as possible. Note that the moment conditions for all the restrictions are still equal to zero, but the sample approximation, being drawn from a finite sample, may not be equal to zero. In other words, the GMM estimator is defined as the value of
\beta
that minimizes the weighted distance of
\frac{1}{n}\sum_{i=1}^{n}g(X_{i},\beta)
\begin{aligned} \hat{\beta}^{GMM} &= \arg \min_{\beta \in B} \Big\lVert \frac{1}{n}\sum_{i=1}^{n}g(X_{i},\beta) \Big\rVert^{2}_{W} \\ &= \arg \min_{\beta \in B} \Big( \frac{1}{n}\sum_{i=1}^{n}g(X_{i},\beta) \Big)'\mathbf{W} \Big( \frac{1}{n}\sum_{i=1}^{n}g(X_{i},\beta) \Big) \end{aligned}
\mathbf{W}
l \times l
matrix of weights which is used to select the ideal linear combination of instruments. In the case of the regression model where
g(X_{i},\beta)
\beta
but is overidentified, the general GMM formula can be found by minimising the above condition and is given by:
\hat{\beta}^{GMM} = \Big((\mathbf{X}'\mathbf{Z})\mathbf{W}(\mathbf{Z}'\mathbf{X}) \Big)^{-1} (\mathbf{X}'\mathbf{Z})\mathbf{W}(\mathbf{Z}'\mathbf{y})
\mathbf{W}=(\mathbf{Z}'\mathbf{Z})^{-1}
\hat{\beta}^{GMM}=\hat{\beta}^{IV}
.2 Please google efficient GMM, for more information on the optimal choice of the weighting matrix.
In the case of regressions, this happens when there are more instruments than endogenous regressors. ↩
This also shows that the 2SLS estimator is a GMM estimator for the linear model.
\mathbf{W}=(\mathbf{Z}'\mathbf{Z})^{-1}
is also the most efficient estimator if the errors are homoskedastic. In general, there may be other more efficient choices of the weighting matrix. ↩
regressionolsnotes
Mapping the Distribution of Religious Beliefs in Singapore
|
Hunger - Ring of Brodgar
Note that Hunger and Energy are different things. Hunger is only related to FEP bar. Energy is different thing.
There are currently two items in the game that can reduce your Hunger: Burrower Beans and Salt Crystal.
1 Hunger Levels
2 Food Efficiency
3 Effect of Food Quality
4 Variety Bonus
5 Variety math and example
6 Symbel Bonuses
Unlike legacy haven hunger levels are NOT effected by ingame activities other than eating foods. Hunger level only decreases over real time depending on your hunger level.
Each food has a hunger value that adds up when eaten to change your Hunger Level once at 100%. Each subsequent hunger level has a lower "Food efficiency" bonuses, but takes less time to return to a lower hunger level.
The hunger level of food is a flat rate, unaffected by quality.
Approx time to lose 1% hunger
Approx time to lose 100% hunger
Variety Example @15 max stat
Ravenous 300% 2 hours 200 hours, 8 days 8 hours 1.097 28.3%
Famished 200% 1 hour 100 hours, 4 days 4 hours 0.894 23.1%
Content 100% 10 minutes 16.67 hours 0.602 15.5%
Times are in Real/?/Game time. [Verify: Real-time or Game-time.]
This value is effectively the same as the "food event bonus" also given on some Symbel. It multiplies the FEP value of the food you eat to be either positive or negative by the percentage shown. It also positively effects Variety bonus see below
Effect of Food Quality
As of World 10, quality of the food has no effect on the hunger filled. (Section left temporarily for refernce.)
Jorb posted that variety bonus effect by food efficiency here [1]. This bonus works simularly to legacy where each new food type eaten will decrease the amount of FEPs in order to level-up an attribute. It is increased by higher food efficiency, but is also reduced by your highest Base attribute (max stat).
Variety math and example
To work out the variety bonus at Famished's 200% food efficiency, we take the value from the chart and multiply it by the squareroot of our example max stat 15:
{\displaystyle 0.894*{\sqrt {15}}=3.46}
This means that our fep requirement is reduced by 3.46, which is 28.3% of our max fep requirement of 15. If our max stat is 40 the reduction is instead 5.6, which is now only 14.1% of our max.
Symbel Bonuses
Eating at a table and chair provide hunger reductions. Equipping the table with various Symbel items further improve the reduction. Please see the appropriate articles on specific item values.
The match is fairly straightforward of hunger filled times the table's displayed hunger mod (when sitting in a chair, not just having the table open). For example, Johnny Hearthlander's table provides a 75% hunger bonus. The food he eats fills 10% hunger. Modified, it will only be 7.5%
Retrieved from "https://ringofbrodgar.com/w/index.php?title=Hunger&oldid=88682"
|
Acidic_oxide Knowpia
An acidic oxide is an oxide that either produces an acidic solution upon addition to water, or acts as an acceptor of hydroxide ions effectively functioning as a Lewis acid.[1] Acidic oxides will typically have a low pKa and may be inorganic or organic. A commonly encountered acidic oxide, carbon dioxide produces an acidic solution (and the generation of carbonic acid) when dissolved.[2]
The acidity of an oxide can be reasonably assumed by its accompanying constituents. Less electronegative elements tend to form basic oxides such as sodium oxide and magnesium oxide, whereas more electronegative elements tend to produce acidic oxides as seen with carbon dioxide and phosphorus pentoxide. Some oxides like aluminium oxides are amphoteric.[3]
Acidic oxides are of environmental concern. Sulfur and nitrogen oxides are considered air pollutants as they react with atmospheric water vapour to produce acid rain.
Carbonic acid is an illustrative example of the lewis acidity of an acidic oxide.
CO2 + 2OH− ⇌ HCO3− + OH− ⇌ CO32− + H2O
This property is a key reason for keeping alkali chemicals well sealed from the atmosphere, as long-term exposure to carbon dioxide in the air can degrade the material.
Carbon dioxide is also the anhydride of carbonic acid:
{\displaystyle {\ce {H2CO3 -> H2O + CO2}}}
Chromium trioxide, which reacts with water forming chromic acid
Dinitrogen pentoxide, which reacts with water forming nitric acid
Manganese heptoxide, which reacts with water forming permanganic acid
Aluminium oxideEdit
Aluminium oxide (Al2O3) is an amphoteric oxide; it can act as a base or acid. For example, with base different aluminate salts will be formed:
Silicon dioxideEdit
[4]Silicon dioxide is an acidic oxide. It will react with strong bases to form silicate salts.
Silicon dioxide is the anhydride of silicic acid:
{\displaystyle {\ce {Si(OH)4 -> 2H2O + SiO2}}}
Phosphorus oxidesEdit
Phosphorus(III) oxide reacts to form phosphorous acid in water:
Phosphorus(V) oxide reacts with water to give phosphoric (v) acid:
Phosphorus trioxide is the anhydride of phosphorous acid:
{\displaystyle {\ce {2H3PO3 -> 3H2O + P2O3}}}
Phosphorus pentoxide is the anhydride of phosphoric acid:
{\displaystyle {\ce {2H3PO4 -> 3H2O + P2O5}}}
Sulfur dioxide reacts with water to form the weak acid, sulfurous acid:
Sulfur trioxide forms the strong acid sulfuric acid with water:
This reaction is important in the manufacturing of sulfuric acid.
Chlorine oxidesEdit
Chlorine(I) oxide reacts with water to form hypochlorous acid, a very weak acid:
{\displaystyle {\ce {Cl2O + H2O <=> 2 HOCl}}}
Chlorine(VII) oxide reacts with water to form perchloric acid, a strong acid:
Iron oxidesEdit
Iron(II) oxide is the anhydride of the aqueous ferrous ion:
{\displaystyle {\ce {[Fe(H2O)6]^2+ -> FeO + 2H+ + 5H2O}}}
Chromium oxidesEdit
Chromium trioxide is the anhydride of chromic acid:
{\displaystyle {\ce {H2CrO4 -> H2O + CrO3}}}
Vanadium oxidesEdit
Vanadium trioxide is the anhydride of vanadous acid:
{\displaystyle {\ce {2H3VO3 -> 3H2O + V2O3}}}
Vanadium pentoxide is the anhydride of vanadic acid:
{\displaystyle {\ce {2H3VO4 -> 3H2O + V2O5}}}
Organic acid anhydride, similar compounds in organic chemistry
^ John Daintith (February 2008). "acidic". A Dictionary of Chemistry. 3. Describing a compound that forms an acid when dissolved in water. Carbon dioxide, for example, is an acidic oxide.
^ David Oxtoby; H. P. Gillis; Alan Campion. Principles of Modern Chemistry (7th ed.). Cengage Learning. pp. 675–676. ISBN 978-0-8400-4931-5.
^ Chang, Raymond; Overby, Jason (2011). General chemistry: the essential concepts (6th ed.). New York, NY: McGraw-Hill. ISBN 9780073375632. OCLC 435711011.
^ Comprehensive Chemistry Volume 1. 113, Golden House, Daryaganj, New Delhi - 110002, India: Laxmi Publications. 2018. p. 6.13. ISBN 978-81-318-0859-7. {{cite book}}: CS1 maint: location (link)
|
Paula has numbered all squares of the square board above, from left to right and top to bottom, starting with the number 1. The central square received the number 5. If she does the same with another square board divided by 49 squares, which number will be written on the central square?
No: B > A Yes: A = B No: A > B
What is the minimum number of cuts required to divide any cube into 27 smaller equal cubes?
Image Credit: Wikimedia Hexahedron.
2^{6}-1 = 63
2^{6}+1 = 65
2^{6} = 64
2^{6 - 1} = 32
A jigsaw puzzle contains 50 pieces. If joining any 2 pieces is considered as one move, what is the fewest number of moves required to join all fifty pieces?
|
Notes on Regression - Approximation of the Conditional Expectation Function
The final installment in my 'Notes on Regression' series! For a review on ways to derive the Ordinary Least Square formula as well as various algebraic and geometric interpretations, check out the previous 5 posts:
Part 1 - OLS by way of minimising the sum of square errors
Part 2 - Projection and Orthogonality
Part 3 - Method of Moments
Part 4 - Maximum Likelihood
Part 5 - Singular Vector Decomposition
A common argument against the regression approach is that it is too simple. Real world phenomenons follow non-normal distributions, power laws are everywhere and multivariate relationships possibly more complex. The assumption of linearity in the OLS regression seems way out place of reality. However, if we take into consideration that the main aim of a statistical model is not to replicate the real world but to yield useful insights, the simplicity of regression may well turn out to be its biggest strength.
In this set of notes I shall discuss the OLS regression as a way of approximating the conditional expectation function (CEF). To be more precise, regression yields the best linear approximation of the CEF. This mathematical property makes regression a favourite tool among social scientist as it places the emphasis on interpretation of an approximation of reality rather than complicated curve fitting. I came across this method from Angrist and Pischke's nearly harmless econometrics.
What is a Conditional Expectation Function?
Expectation as in the statistics terminology normally refers to the population average of a particular random variable. The conditional expectation as its name suggest is the population average conditional holding certain variables fixed. In the context of regression, the CEF is simply
E[Y_{i}\vert X_{i}]
X_{i}
is random, the CEF is random.1
The picture above is an illustrated example of the CEF plotted on a given dataset. Looking at the relationship of the number of stars obtained by a recipe and the log number of reviews, one can calculate the average star rating for a given number of reviews (indicated by the red dots). The CEF function joins all these red dots together (indicated by the blue line).
Nice Properties of the CEF
What could we infer about the relationship between the dependent variable,
Y_{i}
and the CEF? Let's split the dependent variable into two components:
Y_{i} = E[Y_{i} \vert X_{i}] + \epsilon_{i}
Using the law of iterated expectation, we can show that
E[\epsilon_{i} \vert X_{i}]=0
i.e. mean independence and
\epsilon_{i}
is uncorrelated with any function of
X_{i}
. In other words, we can break the dependent variable into a component that is explained by
X_{i}
and another component that is orthogonal to it. Sounds familiar?
Also, if we were to try to find a function of
X
m(X)
that minimises the squared mean error i.e.
min~ E[(Y_{i} - m(X_{i}))^{2}]
, we would find that the optimum choice of
m(X)
is exactly the CEF! To see that expand the squared error term:
\begin{aligned} (Y_{i} - m(X_{i}))^{2} &= ((Y_{i} - E[Y_{i} \vert X_{i}]) + (E[Y_{i} \vert X_{i}] - m(X_{i})))^{2} \\ &= (Y_{i} - E[Y_{i} \vert X_{i}])^{2} + 2(Y_{i} - E[Y_{i} \vert X_{i}])(E[Y_{i} \vert X_{i}] - m(X_{i})) + (E[Y_{i} \vert X_{i}] - m(X_{i}))^{2} \end{aligned}
The first term on the right does not factor in the arg min problem.
(Y_{i} - E[Y_{i} \vert X_{i}])
in the second term is simply
\epsilon_{i}
and a function of
X
multiplied with
\epsilon_{i}
would still give an expectation of zero. Hence, the problem can be simplified to minimising the last term which is only minimised when
m(X_{i})
= CEF.
Now let's link the regression back to the discussion on the CEF. Recall the example of the number of stars a recipe has and the number of reviews submitted. Log reviews is a continuous variable and there are lots of points to take into consideration. Regression offers a way of approximating the CEF linearly i.e.
\beta = \arg \min_{b}E[ E[Y_{i}\vert X_{i}] - X_{i}'b]
To get this result, one can show that minimising
(Y_{i} -X'_{i}b)^{2}
is equivalent to minimising the above equation.2 Thus, even if the CEF is non-linear as in the recipe and star rating example, the regression line would provide the best linear approximation to it (drawn in green below).
In practice, one obtains a sample of the population data and uses the sample to make an approximation of the population CEF. ↩
just add and subtract
E[Y_{i}\vert X_{i}]
and manipulate the terms in a similar way to the previous proof using
m(X)
|
Data Presentation - Histogram Practice Problems Online | Brilliant
The above histogram shows the height of trees (in feet) in a park. If there are 12 trees that have a height of
85
90
feet, how many trees are there in the park?
The above histogram shows the test scores of students in a class. If 28 students scored from 40 to 60, how many students scored between 70 to 80?
The above histogram shows the distance travelled (in miles) from home to work by the inhabitants of a certain town.
If 284 people travelled between 5 and 10 miles, how many people travelled between 20 and 30 miles?
The above histogram shows the amount of weight gained by members of a gym in the month of January. If there are 208 members in the gym, how many of them lost weight in January?
The above histogram shows the birth weight (in pounds) of new born babies at a hospital. A baby that is under 5 pounds is considered underweight, and a baby that is over 9 pounds is considered overweight.
What percentage of babies are neither overweight nor underweight?
|
m-ary tree - Wikipedia
(Redirected from N-ary tree)
An example of a m-ary tree with m=5
In graph theory, an m-ary tree (also known as k-ary or k-way tree) is a rooted tree in which each node has no more than m children. A binary tree is the special case where m = 2, and a ternary tree is another case with m = 3 that limits its children to three.
1 Types of m-ary trees
2 Properties of m-ary trees
3 Traversal methods for m-ary trees
4 Convert a m-ary tree to binary tree
5 Methods for storing m-ary trees
5.2 Pointer-based
6 Enumeration of m-ary trees
6.1 Loopless enumeration
Types of m-ary trees[edit]
A full m-ary tree is an m-ary tree where within each level every node has either 0 or m children.
A complete m-ary tree is an m-ary tree which is maximally space efficient. It must be completely filled on every level except for the last level. However, if the last level is not complete, then all nodes of the tree must be "as far left as possible".[1]
A perfect m-ary tree is a full[1] m-ary tree in which all leaf nodes are at the same depth.[2]
Properties of m-ary trees[edit]
For an m-ary tree with height h, the upper bound for the maximum number of leaves is
{\displaystyle m^{h}}
The height h of an m-ary tree does not include the root node, with a tree containing only a root node having a height of 0.
The height of a tree is equal to the maximum depth D of any node in the tree.
The total number of nodes
{\displaystyle N}
in a perfect m-ary tree is
{\textstyle \sum _{i=0}^{h}m^{i}={\frac {m^{h+1}-1}{m-1}}}
, while the height h is
{\displaystyle {\begin{aligned}&{\frac {m^{h+1}-1}{m-1}}\geq N>{\frac {m^{h}-1}{m-1}}\\[8pt]&m^{h+1}\geq (m-1)\cdot N+1>m^{h}\\[8pt]&h+1\geq \log _{m}\left((m-1)\cdot N+1\right)>h\\[8pt]&h\geq \left\lceil \log _{m}((m-1)\cdot N+1)-1\right\rceil .\end{aligned}}}
By the definition of Big-Ω, the maximum depth
{\displaystyle D=h\geq \left\lceil \log _{m}((m-1)\cdot N+1)-1\right\rceil =O(\log _{m}n)=O(\log n/\log m)}
The height of a complete m-ary tree with n nodes is
{\textstyle \lfloor \log _{m}((m-1)\cdot n)\rfloor }
The total number of possible m-ary tree with n nodes is
{\textstyle C_{n}={\frac {1}{(m-1)n+1}}\cdot {\binom {m\cdot n}{n}}}
(which is a Catalan number).[3]
Traversal methods for m-ary trees[edit]
Traversing a m-ary tree is very similar to binary tree traversal. The pre-order traversal goes to parent, left subtree and the right subtree, and for traversing post-order it goes by left subtree, right subtree, and parent node. For traversing in-order, since there are more than two children per node for m > 2, one must define the notion of left and right subtrees. One common method to establish left/right subtrees is to divide the list of children nodes into two groups. By defining an order on the m children of a node, the first
{\textstyle \{1,\dots ,\lfloor {\frac {m}{2}}\rfloor \}}
nodes would constitute the left subtree and
{\textstyle \{\lceil {\frac {m}{2}}\rceil ,\dots ,m\}}
nodes would constitute the right subtree.
Convert a m-ary tree to binary tree[edit]
An example of conversion of a m-ary tree to a binary tree.m=6
Using an array for representing a m-ary tree is inefficient, because most of the nodes in practical applications contain less than m children. As a result, this fact leads to a sparse array with large unused space in the memory. Converting an arbitrary m-ary tree to a binary tree would only increase the height of the tree by a constant factor and would not affect the overall worst-case time complexity. In other words,
{\textstyle O(\log _{m}n)\equiv O(\log _{2}n)}
{\textstyle \log _{2}m\cdot \log _{m}n={\frac {\log m}{\log 2}}\cdot {\frac {\log n}{\log m}}=\log _{2}n}
First, we link all the immediate children nodes of a given parent node together in order to form a link list. Then, we keep the link from the parent to the first (i.e., the leftmost) child and remove all the other links to the rest of the children. We repeat this process for all the children (if they have any children) until we have processed all the internal nodes and rotate the tree by 45 degrees clockwise. The tree obtained is the desired binary tree obtained from the given m-ary tree.
Methods for storing m-ary trees[edit]
An example of storing a m-ary tree with m=3 in an array
m-ary trees can also be stored in breadth-first order as an implicit data structure in arrays, and if the tree is a complete m-ary tree, this method wastes no space. In this compact arrangement, if a node has an index i, its c-th child in range {1,…,m} is found at index
{\displaystyle m\cdot i+c}
, while its parent (if any) is found at index
{\textstyle \left\lfloor {\frac {i-1}{m}}\right\rfloor }
(assuming the root has index zero, meaning a 0-based array). This method benefits from more compact storage and better locality of reference, particularly during a preorder traversal. The space complexity of this method is
{\displaystyle O(m^{n})}
Pointer-based[edit]
Each node would have an internal array for storing pointers to each of its
{\displaystyle m}
Pointer-based implementation of m-ary tree where m=4.
Compared to array-based implementation, this implementation method has superior space complexity of
{\displaystyle O(m\cdot n)}
Enumeration of m-ary trees[edit]
Listing all possible m-ary trees are useful in many disciplines as a way of checking hypothesis or theories. Proper representation of m-ary tree objects can greatly simplify the generation process. One can construct a bit sequence representation using the depth-first search of a m-ary tree with n nodes indicating the presence of a node at a given index using binary values. For example, the bit sequence x=1110000100010001000 is representing a 3-ary tree with n=6 nodes as shown below.
The problem with this representation is that listing all bit strings in lexicographic order would mean two successive strings might represent two trees that are lexicographically very different. Therefore, enumeration over binary strings would not necessarily result in an ordered generation of all m-ary trees.[4] A better representation is based on an integer string that indicates the number of zeroes between successive ones, known as Simple Zero Sequence.
{\textstyle S=s_{1},s_{2},\dots ,s_{n-1}}
is a Simple Zero Sequence corresponding to the bit sequence
{\textstyle 10^{s_{1}}10^{s_{2}}\ldots 10^{s_{n-1}}10^{j}}
where j is the number of zeroes needed at the tail end of the sequence to make the string have the appropriate length. For example,
{\displaystyle 1110000100010001000\equiv 10^{0}10^{0}10^{4}10^{4}10^{3}\equiv 00433}
is the simple zero sequence representation of the above figure. A more compact representation of 00433 is
{\displaystyle 0^{2}4^{1}3^{2}}
, which is called zero sequence, which duplicate bases cannot be adjacent. This new representation allows to construct a next valid sequence in
{\displaystyle O(1)}
. A simple zero sequence is valid if
{\displaystyle \sum _{i=1}^{i=j}s_{i}\leq (m-1)j\qquad \forall j\leq n-1.}
That is to say that number of zeros in the bit sequence of a m-ary tree cannot exceed the total number of null pointers (i.e., pointers without any child node attached to them). This summation is putting restriction on
{\displaystyle n-1}
nodes so that there is room for adding the
{\displaystyle n^{t}h}
without creating an invalid structure (i.e. having an available null pointer to attached the last node to it).
The table below shows the list of all valid simple zero sequences of all 3-ary trees with 4 nodes:
Starting from the bottom right of the table (i.e., "000"), there is a backbone template that governs the generation of the possible ordered trees starting from "000" to "006". The backbone template for this group ("00X") is depicted below, where an additional node is added in the positions labeled "x".
Once one has exhausted all possible positions in the backbone template, a new template will be constructed by shifting the 3rd node one position to the right as depicted below, and the same enumeration would occur until all possible positions labeled "X" is exhausted.
Going back to the table of enumeration of all m-ary trees, where
{\displaystyle m=3}
{\displaystyle n=4}
, we can easily observe the apparent jump from "006" to "010" can be explained trivially in an algorithmic fashion as depicted below:
The pseudocode for this enumeration is given below:[4]
Procedure NEXT(s1, s2, …, sn−1)
if si = 0 for all i then
i ← max {i | si > 0}
si ← si − 1
if i < n − 1 then
si ← (i + 1) ⋅ (m − 1) − sum(sj)
for j ← i + 2, i + 3, …, n − 1
sj ← k − 1
Loopless enumeration[edit]
A generation algorithm that takes
{\displaystyle O(1)}
worst-case time are called loopless since the time complexity cannot involve a loop or recursion. Loopless enumeration of m-ary trees is said to be loopless if after initialization, it generates successive tree objects in
{\displaystyle O(1)}
. For a given a m-ary tree T with
{\displaystyle a}
being one of its nodes an{\displaystyle d}
{\displaystyle t}
-th child, a left-t rotation at
{\displaystyle a}
is done by making
{\displaystyle d}
the root node, and making
{\displaystyle b}
and all of its subtrees a child of
{\displaystyle a}
, additionally we assign the
{\displaystyle m-1}
left most children of
{\displaystyle d}
{\displaystyle a}nd the right most child of
{\displaystyle d}
stays attached to it while
{\displaystyle d}
is promoted to root, as shown below:
Convert an m-ary tree to left-tree
for i = 1...n:
for t = 2...m:
while t child of node at depth i ≠ 1:
L-t rotation at nodes at depth i
A right-t rotation at d is the inverse of this operation. The left chain of T is a sequence of
{\displaystyle x_{1},x_{2},\dots ,x_{n}}
nodes such that
{\displaystyle x_{1}}
is the root and all nodes except
{\displaystyle x_{n}}
have one child connected to their left most (i.e.,
{\displaystyle m[1]}
) pointer. Any m-ary tree can be transformed to a left-chain tree using sequence of finite left-t rotations for t from 2 to m. Specifically, this can be done by performing left-t rotations on each node
{\displaystyle x_{i}}
until all of its
{\displaystyle m-1}
sub-tree become null at each depth. Then, the sequence of number of left-t rotations performed at depth i denoted by
{\displaystyle c_{i}}
defines a codeword of a m-ary tree that can be recovered by performing the same sequence of right-t rotations.
{\displaystyle m-1}
tuple of
{\displaystyle c_{1},c_{2},\dots ,c_{m-1}}
represent the number of L-2 rotations, L-3 rotations, ..., L-m rotations that has occurred at the root (i.e., i=1).Then,
{\displaystyle c_{(i-1)(m-1)+t-1}}
is the number of L-t rotations required at depth i.
Capturing counts of left-rotations at each depth is a way of encoding an m-ary tree. Thus, enumerating all possible legal encoding would helps us to generate all the m-ary trees for a given m and n. But, not all
{\displaystyle c_{i}}
sequences of m non-negative integers represent a valid m-ary tree. A sequence of
{\displaystyle (n-1)\cdot (m-1)+1}
non-negative integers is a valid representation of a m-ary tree if and only if[5]
{\displaystyle \sum _{i=j}^{n}\sum _{t=2}^{m}c_{(i-1)(m-1)+t-1}\qquad \leq n-j,\qquad \forall j\in 0\dots n.}
Lexicographically smallest code-word representation of a m-ary with n nodes is all zeros and the largest is n−1 ones followed by m−1 zero on its right.
c[i] to zero for all i from 1 to n⋅(k − 1)
p[i] set to n − 1 for i from 1 to n
j ← m − 1
Terminate when c[1] = n − 1
Procedure NEXT[5]
sum ← sum + 1 − c[j + 1]
c[j] ← c[j] + 1
if p[q[j]] > p[q[j + 1]] + 1 then
p[q[j]] ← p[q[j + 1]] + 1
p[q[j + c[j]]] ← p[q[j]]
c[j + 1] ← 0
if sum = p[q[j]] then
p[n] ← sum
One of the applications of m-ary tree is creating a dictionary for validation of acceptable strings. In order to do that, let m be equal to the number of valid alphabets (e.g., number of letters of the English alphabet) with the root of the tree representing the starting point. Similarly, each of the children can have up to m children representing the next possible character in the string. Thus, characters along the paths can represent valid keys by marking the end character of the keys as "terminal node". For example, in the example below "at" and "and" are valid key strings with "t" and "d" marked as terminal nodes. Terminal nodes can store extra information to be associated with a given key. There are similar ways to building such a dictionary using B-tree, Octree and/or trie.
^ a b "Ordered Trees". Retrieved 19 November 2012.
^ Black, Paul E. (20 April 2011). "perfect k-ary tree". U.S. National Institute of Standards and Technology. Retrieved 10 October 2011.
^ Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (1994). Concrete Mathematics: A Foundation for Computer Science (2nd Edition). AIP.
^ a b Baronaigien, Dominique Roelants van (2000). "Loop Free Generation of K-ary trees". Journal of Algorithms. 35 (1): 100–107. doi:10.1006/jagm.1999.1073.
^ a b Korsh, James F (1994). "Loopless generation of k-ary tree sequences". Information Processing Letters. Elsevier. 52 (5): 243–247. doi:10.1016/0020-0190(94)00149-9.
Storer, James A. (2001). An Introduction to Data Structures and Algorithms. Birkhäuser Boston. ISBN 3-7643-4253-6.
N-ary trees, Bruno R. Preiss, Ph.D, P.Eng.
Retrieved from "https://en.wikipedia.org/w/index.php?title=M-ary_tree&oldid=1076478143"
|
Specific orbital energy - WikiMili, The Best Wikipedia Reader
In the gravitational two-body problem, the specific orbital energy
{\displaystyle \varepsilon }
(or vis-viva energy) of two orbiting bodies is the constant sum of their mutual potential energy (
{\displaystyle \varepsilon _{p}}
) and their total kinetic energy (
{\displaystyle \varepsilon _{k}}
), divided by the reduced mass. According to the orbital energy conservation equation (also referred to as vis-viva equation), it does not vary with time:
{\displaystyle {\begin{aligned}\varepsilon &=\varepsilon _{k}+\varepsilon _{p}\\&={\frac {v^{2}}{2}}-{\frac {\mu }{r}}=-{\frac {1}{2}}{\frac {\mu ^{2}}{h^{2}}}\left(1-e^{2}\right)=-{\frac {\mu }{2a}}\end{aligned}}}
{\displaystyle v}
is the relative orbital speed;
{\displaystyle r}
is the orbital distance between the bodies;
{\displaystyle \mu ={G}(m_{1}+m_{2})}
is the sum of the standard gravitational parameters of the bodies;
{\displaystyle h}
is the specific relative angular momentum in the sense of relative angular momentum divided by the reduced mass;
{\displaystyle e}
is the orbital eccentricity;
{\displaystyle a}
is the semi-major axis.
For an elliptic orbit, the specific orbital energy equation, when combined with conservation of specific angular momentum at one of the orbit's apsides, simplifies to: [1]
{\displaystyle \varepsilon =-{\frac {\mu }{2a}}}
{\displaystyle \mu =G\left(m_{1}+m_{2}\right)}
{\displaystyle a}
is semi-major axis of the orbit.
{\displaystyle h^{2}=\mu p=\mu a\left(1-e^{2}\right)}
{\displaystyle \varepsilon ={\frac {v^{2}}{2}}-{\frac {\mu }{r}}}
{\displaystyle v_{p}^{2}={h^{2} \over r_{p}^{2}}={h^{2} \over a^{2}(1-e)^{2}}={\mu a\left(1-e^{2}\right) \over a^{2}(1-e)^{2}}={\mu \left(1-e^{2}\right) \over a(1-e)^{2}}}
{\displaystyle \varepsilon ={\frac {\mu }{a}}{\left[{1-e^{2} \over 2(1-e)^{2}}-{1 \over 1-e}\right]}={\frac {\mu }{a}}{\left[{(1-e)(1+e) \over 2(1-e)^{2}}-{1 \over 1-e}\right]}={\frac {\mu }{a}}{\left[{1+e \over 2(1-e)}-{2 \over 2(1-e)}\right]}={\frac {\mu }{a}}{\left[{e-1 \over 2(1-e)}\right]}}
{\displaystyle \varepsilon =-{\mu \over 2a}}
{\displaystyle \varepsilon =0.}
{\displaystyle \varepsilon ={\mu \over 2a}.}
In this case the specific orbital energy is also referred to as characteristic energy (or
{\displaystyle C_{3}}
) and is equal to the excess specific energy compared to that for a parabolic orbit.
It is related to the hyperbolic excess velocity
{\displaystyle v_{\infty }}
(the orbital velocity at infinity) by
{\displaystyle 2\varepsilon =C_{3}=v_{\infty }^{2}.}
Thus, if orbital position vector (
{\displaystyle \mathbf {r} }
) and orbital velocity vector (
{\displaystyle \mathbf {v} }
) are known at one position, and
{\displaystyle \mu }
is known, then the energy can be computed and from that, for any other position, the orbital speed.
{\displaystyle {\frac {\mu }{2a^{2}}}}
{\displaystyle \mu ={G}(m_{1}+m_{2})}
{\displaystyle a\,\!}
{\displaystyle -{\frac {\mu }{2a}}+{\frac {\mu }{R}}={\frac {\mu (2a-R)}{2aR}}}
{\displaystyle 2a-R}
is the height the ellipse extends above the surface, plus the periapsis distance (the distance the ellipse extends beyond the center of the Earth). For the Earth and
{\displaystyle a}
just little more than
{\displaystyle R}
the additional specific energy is
{\displaystyle (gR/2)}
; which is the kinetic energy of the horizontal component of the velocity, i.e.
{\textstyle {\frac {1}{2}}V^{2}={\frac {1}{2}}gR}
{\displaystyle V={\sqrt {gR}}}
The International Space Station has an orbital period of 91.74 minutes (5504 s), hence by Kepler's Third Law the semi-major axis of its orbit is 6,738 km.[ citation needed ]
For Voyager 1 , with respect to the Sun:
{\displaystyle \mu =GM}
= 132,712,440,018 km3⋅s−2 is the standard gravitational parameter of the Sun
{\displaystyle \varepsilon =\varepsilon _{k}+\varepsilon _{p}={\frac {v^{2}}{2}}-{\frac {\mu }{r}}=\mathrm {146\,km^{2}s^{-2}} -\mathrm {8\,km^{2}s^{-2}} =\mathrm {138\,km^{2}s^{-2}} }
{\displaystyle v_{\infty }=\mathrm {16.6\,km/s} }
Then the time-rate of change of the specific energy of the rocket is
{\displaystyle \mathbf {v} \cdot \mathbf {a} }
: an amount
{\displaystyle \mathbf {v} \cdot (\mathbf {a} -\mathbf {g} )}
for the kinetic energy and an amount
{\displaystyle \mathbf {v} \cdot \mathbf {g} }
for the potential energy.
{\displaystyle {\frac {\mathbf {v\cdot a} }{|\mathbf {a} |}}}
{\displaystyle \Delta \varepsilon =\int v\,d(\Delta v)=\int v\,adt}
An electric field is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field for a system of charged particles. Electric fields originate from electric charges, or from time-varying magnetic fields. Electric fields and magnetic fields are both manifestations of the electromagnetic force, one of the four fundamental forces of nature.
In theoretical physics, relativistic Lagrangian mechanics is Lagrangian mechanics applied in the context of special relativity and general relativity.
↑ Wie, Bong (1998). "Orbital Dynamics" . Space Vehicle Dynamics and Control. AIAA Education Series. Reston, Virginia: American Institute of Aeronautics and Astronautics. p. 220. ISBN 1-56347-261-9.
|
Neumann boundary condition - Wikipedia
In mathematics, the Neumann (or second-type) boundary condition is a type of boundary condition, named after Carl Neumann.[1] When imposed on an ordinary or a partial differential equation, the condition specifies the values of the derivative applied at the boundary of the domain.
It is possible to describe the problem using other boundary conditions: a Dirichlet boundary condition specifies the values of the solution itself (as opposed to its derivative) on the boundary, whereas the Cauchy boundary condition, mixed boundary condition and Robin boundary condition are all different types of combinations of the Neumann and Dirichlet boundary conditions.
For an ordinary differential equation, for instance,
{\displaystyle y''+y=0,}
the Neumann boundary conditions on the interval [a,b] take the form
{\displaystyle y'(a)=\alpha ,\quad y'(b)=\beta ,}
where α and β are given numbers.
PDE[edit]
For a partial differential equation, for instance,
{\displaystyle \nabla ^{2}y+y=0,}
where ∇2 denotes the Laplace operator, the Neumann boundary conditions on a domain Ω ⊂ Rn take the form
{\displaystyle {\frac {\partial y}{\partial \mathbf {n} }}(\mathbf {x} )=f(\mathbf {x} )\quad \forall \mathbf {x} \in \partial \Omega ,}
where n denotes the (typically exterior) normal to the boundary ∂Ω, and f is a given scalar function.
The normal derivative, which shows up on the left side, is defined as
{\displaystyle {\frac {\partial y}{\partial \mathbf {n} }}(\mathbf {x} )=\nabla y(\mathbf {x} )\cdot \mathbf {\hat {n}} (\mathbf {x} ),}
where ∇y(x) represents the gradient vector of y(x), n̂ is the unit normal, and ⋅ represents the inner product operator.
It becomes clear that the boundary must be sufficiently smooth such that the normal derivative can exist, since, for example, at corner points on the boundary the normal vector is not well defined.
The following applications involve the use of Neumann boundary conditions:
In thermodynamics, a prescribed heat flux from a surface would serve as boundary condition. For example, a perfect insulator would have no flux while an electrical component may be dissipating at a known power.
In magnetostatics, the magnetic field intensity can be prescribed as a boundary condition in order to find the magnetic flux density distribution in a magnet array in space, for example in a permanent magnet motor. Since the problems in magnetostatics involve solving Laplace's equation or Poisson's equation for the magnetic scalar potential, the boundary condition is a Neumann condition.
In spatial ecology, a Neumann boundary condition on a reaction–diffusion system, such as Fisher's equation, can be interpreted as a reflecting boundary, such that all individuals encountering ∂Ω are reflected back onto Ω.[2]
Boundary conditions in fluid dynamics
^ Cheng, A. H.-D.; Cheng, D. T. (2005). "Heritage and early history of the boundary element method". Engineering Analysis with Boundary Elements. 29 (3): 268. doi:10.1016/j.enganabound.2004.12.001.
^ Cantrell, Robert Stephen; Cosner, Chris (2003). Spatial Ecology via Reaction–Diffusion Equations. Wiley. pp. 30–31. ISBN 0-471-49301-5.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Neumann_boundary_condition&oldid=1078491049"
|
William thinks that the hypotenuse must be the longest side of a right triangle, but Chad does not agree. Who is correct? Support your answer with an explanation and a counterexample, if possible.
Draw a right triangle and label the
90º
What do you notice about the side opposite the
90º
Remember this is the hypotenuse.
Is there any situation where the hypotenuse is not the longest side?
Draw a picture if you can come up with one.
|
Some geometry - Tales of Science & Data
Voronoi tasselation
The Voronoi tasselation consists in partitioning the plane into regions based on the distance to points in a specific subset of the plane.
For each point, there is a region of all points closer to it than any other one (Voronoi cells).
A simplex is the generalisation of a triangle/tetrahedron to an arbitrary number of dimensions. The k simplex is a k-dimensional polytope, the convex hull of its k+1 vertices.
u_0, \ldots, u_k \in \mathbb{R}^k
u_1 - u_0, \ldots, u_k - u_0
linearly independent, the simplex determined by them is the set of points
C : {\theta_0 u_0 + \cdots + \theta_k u_k | \theta_i \geq 0, 0 \leq i \leq k, \sum_{i=0}^k \theta_i = 1}
the 2-simplex is a triangle;
the 3-simplex is a tetrahedron;
the 4-simplex is a 5-cell
Wikipedia about the Voronoi diagram
|
Mid-infrared Questions & Answers | Hamamatsu Photonics
Mid-infrared Questions & Answers
How do I choose among an MIR LED, a QCL, and a xenon flash lamp?
What is D* (D-star)?
Give me the short answer to “Why should I choose InAsSb?”
There’s a famous phrase in the spectroscopy world, “fit for purpose.” This is a perfect mantra for selecting components. Selecting the right light source starts with the desired application. What are you trying to achieve with this instrument? What is the proper wavelength and power output? What market is it serving? What is the target final cost? The answers to those questions along with the explanations below should lead to an answer.
Wavelength: 3.3 μm (CH₄), 3.9 μm (reference light), and 4.3 μm (CO₂) are provided.
Higher output, higher reliability, lower power consumption, faster response than lamps
Wavelength: 4 μm to 10 μm band
High resolution, high output, high reliability, high-speed response
Xenon (Xe) flash lamps
Wavelength: 0.2 μm to 5.0 μm (continuous spectrum)
High output pulse emission in the microsecond order
LEDs boast reliable lifetimes as well as low power consumption. They also come at a relatively low cost compared to other MIR light sources. The main tradeoff lies with power output, so these units are not intended for analytical accuracy. Portable gas monitors would be a great example application for these components.
Quantum cascade lasers (QCLs) are the gold standard for generating light anywhere between 4-10 microns. Our DFB (distributed feedback) models provide industry-leading linewidth resolution, enabling possible ppb measurements. QCLs also have reliable lifetimes while providing high power output. All this performance comes at a much higher cost, and power requirements for lasing start at around 500mA. An instrument using a QCL will be cost intensive and require quite a bit of expertise to pull off, but nothing will touch the sensitivity it can achieve.
For wide spectral output and high frequency operation, look no further than the Xe flash lamp. With output ranging from 0.2 microns to 5+ microns, these lamps make it possible to create an instrument that detects multiple gases. However, these lamps should not be considered for measuring very low concentrations due to the wide output and stability. Although Xe flash lamps with emission out to 7 microns have been developed, the relative output past 5 microns remains low. Measurements further into the fingerprint region would be very difficult to achieve.
D* is known as the “detectivity” of a detector, or the photosensitivity per unit active area in a detector. As seen in the equation below, the lower the noise equivalent power (NEP) of a detector, the higher the D* (and vice versa). NEP is the minimum power of signal needed for a detector to overcome its noise floor, or for SNR to equal 1. The lower this value, the higher the sensitivity of a detector. This relationship shows, therefore, that the higher the D*, the higher the sensitivity as well. We can also see from the equation below that the smaller the detector active area (A), the higher the D*.
{D}^{*}=\frac{\sqrt{A}}{NEP}
D* takes into account more than just a detector’s active area, however. It is also a function of the temperature [K] or wavelength [µm] of a radiant source, the chopping frequency [Hz], and the bandwidth [Hz] of a detector—as seen in the expression of detectivity as “D* (A, B, C),” with each letter corresponding to the three characteristics mentioned.
What makes D* so useful is that it allows a comparison of different active area sizes and chemistries. While D* provides a better gauge of sensitivity, detector characteristics such as light wavelength, response time, active area shape, and number of elements, as well as the necessary electronics, should be taken into account when selecting an infrared detector.
When the applications demand more sensitivity, cooling serves the function of lowering the noise floor of a detector without reducing its quantum efficiency (QE). As a result, the lower the temperature, the higher the D* at a certain input power. It’s important to remember that cooling drives up cost and complexity, so it’s best to consider uncooled detectors first. Hamamatsu offers a wide range of uncooled detectors as well as detectors with multi-stage thermoelectric (TEC) cooling and liquid nitrogen cooling.
Photovoltaic operation typically leads to slower measurements. Hamamatsu’s InAsSb detectors mitigate that situation by boasting a rise time on the order of nanoseconds. In addition, many infrared detectors contain materials that are not RoHS compliant (mercury and lead), but InAsSb material is fully RoHS compliant. In uncooled applications, InAsSb is a strong contender for providing big cost advantages as well.
Whether she has a camera in hand, or is working alongside her University Support Group team to solve a problem, Stephanie Butron tries to see things from a different perspective—like seeing the invisible side of mid-infrared. As an Applications Engineer at Hamamatsu Corporation, she enjoys learning more about the variety of projects and applications Hamamatsu’s customers are working on, and understanding better how Hamamatsu can help. When she isn’t helping people focus in on their research, Stephanie enjoys focusing her camera lens on the sights around her. From still lives to portraits, Stephanie tries to find new ways to look at the world.
|
Template:Body data/SOI - Kerbal Space Program Wiki
Template:Body data/SOI
Returns the radius of the sphere of influence of a celestial body in meters. It uses the mass (m, mparent), semi-major axis and parent (mparent) parameters with {{Body data}} for the given body. Returns the second parameter, an empty string when not set, if the body has no parent object (like Kerbol).
Due to latex being unable to update, this graphic uses apoapsis (a) and periapsis (p).
{\displaystyle r_{\text{SOI}}={\frac {a+p}{2}}\cdot \left({\frac {m}{m_{\text{parent}}}}\right)^{\frac {2}{5}}}
Retrieved from "https://wiki.kerbalspaceprogram.com/index.php?title=Template:Body_data/SOI&oldid=82080"
|
SAT Algebraic Manipulations | Brilliant Math & Science Wiki
SAT Algebraic Manipulations
To successfully manipulate algebraic expressions on the SAT, you need to know how to:
apply addition, subtraction, multiplication, and division to algebraic expressions
SAT Tips for Algebraic Manipulation
3m
6m
9m
-36
m
\ \ 2
\ \ 1
\ \ 0
\ \ -1
\ \ -2
Using the given equation, we solve for
m
\begin{array}{l c l l l} 3m + 6m + 9m &=& -36 &\quad \text{original expression} &(1)\\ 18m &=& -36 &\quad \text{combine like terms} &(2)\\ \frac{18m}{18} &=& \frac{-36}{18} &\quad \text{divide both sides by}\ 18 &(3)\\ m &=& -2 &\quad \text{perform division} &(4)\\ \end{array}
We plug the value of each answer choice into the given equation and select the one that doesn't yield a contradiction.
m=2
3m + 6m + 9m = 3 \cdot 2 + 6 \cdot 2 + 9 \cdot 2 = 6 +12 +18 = 36 \neq -36.
This is a contradiction. Eliminate (A).
m=1
3m + 6m + 9m = 3 \cdot 1 + 6 \cdot 1 + 9 \cdot 1 = 3 +6 +9 = 18 \neq -36.
This is a contradiction. Eliminate (B).
m=0
3m + 6m + 9m = 3 \cdot 0 + 6 \cdot 0 + 9 \cdot 0 = 0 \neq -36.
This is a contradiction. Eliminate (C).
m=-1
3m + 6m + 9m = 3 \cdot (-1) + 6 \cdot (-1) + 9 \cdot (-1) = -3 -6 -9 = -18 \neq -36.
This is a contradiction. Eliminate (D).
m=-2
3m + 6m + 9m = 3 \cdot (-2) + 6 \cdot (-2) + 9 \cdot (-2) = -6 -12 -18 = -36.
This is correct and therefore (E) is the answer.
-3(2x-5)+8 = -2x+3
x
\ -5
\ -\frac{5}{4}
\ \ 0
\ \ \frac{5}{4}
\ \ 5
Correct Answser: E
We start with the given equation and we simplify.
\begin{array}{rcll} -3(2x-5)+8 &=& -2x+3 &\text{original equation}&\quad (1)\\ -6x+15+8 &=& -2x+3 &\text{use distributive property}&\quad (2)\\ -6x+23 &=& -2x+3 &\text{simplify}&\quad (3)\\ -6x+23-3&=&-2x+3-3 &\text{subtract} \ 3\ \text{from both sides}&\quad (4)\\ -6x+20 &=&-2x &\text{simplify}&\quad (5)\\ -6x +20+6x &=&-2x +6x&\text{add}\ 6x\ \text{to both sides}&\quad (6)\\ 20 &=& 4x &\text{combine like terms}&\quad (7)\\ \frac{20}{4} &=&\frac{4x}{4}&\text{divide both sides by} \ 4&\quad (8)\\ 5&=&x&\text{simplify the fractions}&\quad (9)\\ \end{array}
We can plug each answer choice into the given equation and check if it yields a true statement. If it does, then the choice is right. In this case, only (E) will work.
Refer to the solution above. The answer should be
5
. Selecting
-5
would be a careless mistake.
Tip: When distributing, be careful with signs!
Refer to Solution 1 above. If in step
(2)
we forget to distribute the negative sign, we will get:
\begin{array}{lcl} \fbox{-}3(2x-5)+8 &=& -2x+3 &\text{original equation}&\quad (1)\\ 6x-15+8 &=& -2x+3 &\text{mistake: didn't distribute}\\ &&&\text{negative sign}&\quad (2)\\ \end{array}
\begin{array}{lcl} -\frac{5}{4}&=&x&\text{simplify} \end{array}
But this is wrong.
Plug in and check. If
x=0
\begin{array}{lcl} -3(2x-5)+8 &=& -2x+3\\ -3(2\cdot0-5)+8&=&-2\cdot 0+3&\text{plug in}\ x=0\\ -3(0-5)+8&=&=0+3&\text{simplify}\\ -3(-5)+8&=&3&\text{simplify parentheses}\\ 15+8&=&3&\text{simplify the left side}\\ 23&=&3&\text{simplify the left side again}\\ \end{array}
23\neq3
. Therefore, this choice is wrong.
It is possible you made a mistake when reducing a fraction. Refer to step
(8)
in the solution above and focus on the fraction on the left side of the equation.
\begin{array}{lcll} \frac{20}{4} &=&\frac{4x}{4}&\text{divide both sides by} \ 4&\quad (8)\\ \end{array}
We must divide both the numerator and denominator by their greatest common factor to obtain the correct reduced fraction.
4
is the greatest number that divides both
20
4
\frac{20/4}{4/4}=\frac{5}{1}=5
. But if we forget to divide the denominator by
4
, we will get this wrong answer.
a ( b-c) = 32
ac = 8
ab
\ \ 4
\ \ 8
\ \ 24
\ \ 32
\ \ 40
5x+2=9
5x-2
\ \ -9
\ \ -5
\ \ \frac{7}{5}
\ \ 5
\ \ 7
We don't need to solve for
x
to find the answer. That's the trick. We realize that
5x-2=5x+2-4=9-4=5
We could solve for
x
\begin{array}{rcl} 5x+2&=&9&\quad\text{given}\\ 5x&=&7&\quad\text{subtract}\ \ 2\ \text{from both sides}\\ x&=&\frac{7}{5}&\quad\text{divide both sides by}\ 5\\ \end{array}
5x-2=5\times\frac{7}{5}-2=7-2=5.
5x-2=5
5x+2=9
5x-2
, and you may think that because the sign between
5x
2
changed, you need to change the sign of
9
also in order to get the answer, like this:
5x-2=-9
. But verify your choice. If
5x-2=-9
5x=-7
2
to both sides of this equation, we get
5x+2 = -7+2=-5\neq 9
. Therefore, (C) is the wrong choice.
You likely got this answer because you solved for
x
5x-2
5x
, not for
5x-2
a(b+c)=ab+ac
Cite as: SAT Algebraic Manipulations. Brilliant.org. Retrieved from https://brilliant.org/wiki/sat-algebraic-manipulations/
|
Bowman, Chris1; Doty, Stephen2; Martin, Stuart3
1 Department of Mathematics University of York Heslington, York, YO10 5DD, UK
2 Department of Mathematics and Statistics Loyola University Chicago Chicago, IL 60660 USA
3 DPMMS, Centre for Mathematical Sciences Wilberforce Road Cambridge, CB3 0WB, UK
\mathbf{V}
be a free module of rank
n
𝕜
. We prove that tensor space
{\mathbf{V}}^{\otimes r}
satisfies Schur–Weyl duality, regarded as a bimodule for the action of the group algebra of the Weyl group of
\mathrm{GL}\left(\mathbf{V}\right)
and the partition algebra
{𝒫}_{r}\left(n\right)
𝕜
. We also prove a similar result for the half partition algebra.
Keywords: Schur–Weyl duality, partition algebras, symmetric groups, invariant theory.
Bowman, Chris 1; Doty, Stephen 2; Martin, Stuart 3
author = {Bowman, Chris and Doty, Stephen and Martin, Stuart},
title = {Integral {Schur{\textendash}Weyl} duality for partition algebras},
TI - Integral Schur–Weyl duality for partition algebras
%T Integral Schur–Weyl duality for partition algebras
Bowman, Chris; Doty, Stephen; Martin, Stuart. Integral Schur–Weyl duality for partition algebras. Algebraic Combinatorics, Volume 5 (2022) no. 2, pp. 371-399. doi : 10.5802/alco.214. https://alco.centre-mersenne.org/articles/10.5802/alco.214/
[1] Benkart, Georgia; Halverson, Tom Partition algebras and the invariant theory of the symmetric group, Recent trends in algebraic combinatorics (Assoc. Women Math. Ser.), Volume 16, Springer, Cham, 2019, pp. 1-41 | Article | MR: 3969570 | Zbl: 1421.05094
[2] Benkart, Georgia; Halverson, Tom Partition algebras
{\mathsf{P}}_{k}\left(n\right)
2k>n
and the fundamental theorems of invariant theory for the symmetric group
{\mathsf{S}}_{n}
, J. Lond. Math. Soc. (2), Volume 99 (2019) no. 1, pp. 194-224 | Article | MR: 3909254 | Zbl: 1411.05274
[3] Benson, David; Doty, Stephen Schur–Weyl duality over finite fields, Arch. Math. (Basel), Volume 93 (2009) no. 5, pp. 425-435 | Article | MR: 2563588 | Zbl: 1210.20039
[4] Bowman, Chris; Doty, Stephen; Martin, Stuart An integral second fundamental theorem of invariant theory for partition algebras (2018) preprint (to appear in Represent. Theory), https://arxiv.org/abs/1804.00916
[5] Brualdi, Richard A. Permanent of the product of doubly stochastic matrices, Proc. Cambridge Philos. Soc., Volume 62 (1966), pp. 643-648 | Article | MR: 201331 | Zbl: 0148.01702
[6] Carter, Roger W.; Lusztig, George On the modular representations of the general linear and symmetric groups, Math. Z., Volume 136 (1974), pp. 193-242 | Article | MR: 354887 | Zbl: 0298.20009
[7] de Concini, Corrado; Procesi, Claudio A characteristic free approach to invariant theory, Advances in Math., Volume 21 (1976) no. 3, pp. 330-354 | Article | MR: 422314 | Zbl: 0347.20025
[8] Dipper, Richard; Doty, Stephen The rational Schur algebra, Represent. Theory, Volume 12 (2008), pp. 58-82 | Article | MR: 2375596 | Zbl: 1185.20052
[9] Dipper, Richard; Doty, Stephen; Hu, Jun Brauer algebras, symplectic Schur algebras and Schur–Weyl duality, Trans. Amer. Math. Soc., Volume 360 (2008) no. 1, pp. 189-213 | Article | MR: 2342000 | Zbl: 1157.16004
[10] Dipper, Richard; Doty, Stephen; Stoll, Friederike Quantized mixed tensor space and Schur-Weyl duality, Algebra Number Theory, Volume 7 (2013) no. 5, pp. 1121-1146 | Article | MR: 3101074 | Zbl: 1290.17012
[11] Dipper, Richard; Doty, Stephen; Stoll, Friederike The quantized walled Brauer algebra and mixed tensor space, Algebr. Represent. Theory, Volume 17 (2014) no. 2, pp. 675-701 | Article | MR: 3181742 | Zbl: 1368.17017
[12] Donkin, Stephen On Schur algebras and related algebras VI: Some remarks on rational and classical Schur algebras, J. Algebra, Volume 405 (2014), pp. 92-121 | Article | MR: 3178262 | Zbl: 1334.20042
[13] Donkin, Stephen Cellularity of endomorphism algebras of Young permutation modules, J. Algebra, Volume 572 (2021), pp. 36-59 | Article | MR: 4192817 | Zbl: 07305005
[14] Donkin, Stephen Double centralisers and annihilator ideals of Young permutation modules, J. Algebra, Volume 591 (2022), pp. 249-288 | Article | MR: 4337801 | Zbl: 07428723
[15] Doty, Stephen; Hu, Jun Schur–Weyl duality for orthogonal groups, Proc. Lond. Math. Soc. (3), Volume 98 (2009) no. 3, pp. 679-713 | Article | MR: 2500869 | Zbl: 1177.20051
[16] Garge, Shripad M.; Nebhani, Anuradha Schur–Weyl duality for special orthogonal groups, J. Lie Theory, Volume 27 (2017) no. 1, pp. 251-270 | MR: 3536546 | Zbl: 1430.20046
[17] Gibson, Peter M. Generalized doubly stochastic and permutation matrices over a ring, Linear Algebra Appl., Volume 30 (1980), pp. 101-107 | Article | MR: 568782 | Zbl: 0437.15006
[18] Graham, John J.; Lehrer, Gustav I. Cellular algebras, Invent. Math., Volume 123 (1996) no. 1, pp. 1-34 | Article | MR: 1376244 | Zbl: 0853.20029
[19] Green, J. A. Polynomial representations of
{\mathrm{GL}}_{n}
. With an appendix on Schensted correspondence and Littelmann paths by K. Erdmann, and M. Schocker, Lecture Notes in Mathematics, 830, Springer, Berlin, 2007, x+161 pages | Article | MR: 2349209 | Zbl: 1108.20044
[20] Halverson, Tom; Ram, Arun Partition algebras, European J. Combin., Volume 26 (2005) no. 6, pp. 869-921 | Article | MR: 2143201 | Zbl: 1112.20010
[21] Johnsen, Eugene C. Essentially doubly stochastic matrices. I. Elements of the theory over arbitrary fields, Linear Algebra Appl., Volume 4 (1971), pp. 255-282 | Article | MR: 294377 | Zbl: 0219.15009
[22] Jones, Vaughan F. R. The Potts model and the symmetric group, Subfactors (Kyuzeso, 1993), World Sci. Publ., River Edge, NJ, 1994, pp. 259-267 | MR: 1317365 | Zbl: 0938.20505
[23] Lai, Hang-Chin On the linear algebra of generalized doubly stochastic matrices and their equivalence relations and permutation basis, Japan J. Appl. Math., Volume 3 (1986) no. 2, pp. 357-379 | Article | MR: 899230 | Zbl: 0618.15014
[24] Martin, Paul P. Potts models and related problems in statistical mechanics, Series on Advances in Statistical Mechanics, 5, World Scientific Publishing Co., Inc., Teaneck, NJ, 1991, xiv+344 pages | Article | MR: 1103994 | Zbl: 0734.17012
[25] Martin, Paul P. Temperley–Lieb algebras for nonplanar statistical mechanics—the partition algebra construction, J. Knot Theory Ramifications, Volume 3 (1994) no. 1, pp. 51-82 | Article | MR: 1265453 | Zbl: 0804.16002
[26] Martin, Paul P. The partition algebra and the Potts model transfer matrix spectrum in high dimensions, J. Phys. A, Volume 33 (2000) no. 19, pp. 3669-3695 | Article | MR: 1768036 | Zbl: 0951.82006
|
Procrustes analysis - MATLAB procrustes - MathWorks España
Find Procrustes Distance and Plot Superimposed Shape
Analyze Procrustes Transformation Including Rotation
Analyze Procrustes Transformation Including Reflection
Apply Procrustes Transformation to Larger Set of Points
Compare Shapes Without Reflection
Compare Shapes Without Scaling
d = procrustes(X,Y)
d = procrustes(X,Y,Name,Value)
[d,Z] = procrustes(___)
[d,Z,transform] = procrustes(___)
d = procrustes(X,Y) returns the Procrustes Distance between the shapes of X and Y, which are represented by configurations of landmark points.
d = procrustes(X,Y,Name,Value) specifies additional options using one or more name-value arguments. For example, you can restrict the Procrustes transformation by disabling reflection and scaling.
[d,Z] = procrustes(___) also returns Z, the shape resulting from performing the Procrustes transformation on Y, using any of the input argument combinations in the previous syntaxes
[d,Z,transform] = procrustes(___) also returns the Procrustes transformation.
Construct matrices containing landmark points for two shapes, and visualize the shapes by plotting their landmark points.
X = [40 88; 51 88; 35 78; 36 75; 39 72; 44 71; 48 71; 52 74; 55 77];
Y = [36 43; 48 42; 31 26; 33 28; 37 30; 40 31; 45 30; 48 28; 51 24];
plot(X(:,1),X(:,2),"x")
plot(Y(:,1),Y(:,2),"o")
legend("Target shape (X)","Comparison shape (Y)")
Compare the shapes and view their Procrustes distance.
[d,Z] = procrustes(X,Y)
Visualize the shape that results from superimposing Y onto X.
plot(Z(:,1),Z(:,2),"s")
legend("Target shape (X)","Comparison shape (Y)", ...
"Transformed shape (Z)")
Use the Procrustes transformation returned by procrustes to analyze how it superimposes the comparison shape onto the target shape.
Generate sample data in two dimensions.
Y = normrnd(0,1,[n 2]);
Create the target shape X by rotating Y 60 degrees (pi/3 in radians), scaling the size of Y by factor 0.5, and then translating the points by adding 2. Also, add some noise to the landmark points in X.
S = [cos(pi/3) -sin(pi/3); sin(pi/3) cos(pi/3)]
X = normrnd(0.5*Y*S+2,0.05,n,2);
Find the Procrustes transformation that can transform Y to X.
[~,Z,transform] = procrustes(X,Y);
Display the components of the Procrustes transformation.
transform = struct with fields:
transform.T
transform.T is similar to the matrix S. Also, the scale component (transform.b) is close to 0.5, and the translation component values (transform.c) are close to 2.
Determine whether transform.T indicates a rotation or reflection by computing the determinant of transform.T. The determinant of a rotation matrix is 1, and the determinant of a reflection matrix is –1.
det(transform.T)
In two-dimensional space, a rotation matrix that rotates a point by an angle of
\theta
degrees about the origin has the form
\left[\begin{array}{cc}\mathrm{cos}\theta & -\mathrm{sin}\theta \\ \mathrm{sin}\theta & \mathrm{cos}\theta \end{array}\right]
If you use either
\mathrm{cos}\theta
\mathrm{sin}\theta
, the rotation angle has two possible values between –180 and 180. Use both the
\mathrm{cos}\theta
\mathrm{sin}\theta
values to determine the rotation angle of the matrix without ambiguity. Using the atan2d function, you can determine the
\mathrm{tan}\theta
value from
\mathrm{cos}\theta
\mathrm{sin}\theta
, and also determine the angle.
theta = atan2d(transform.T(2,1),transform.T(1,1))
theta = 61.1037
transform.T is a rotation matrix of 61 degrees.
Create matrices with landmark points for two separate shapes.
X = [20 13; 20 20; 20, 29; 20 40; 12 36];
Y = [36 7; 36 10; 36 14; 36 20; 39 18];
Plot the landmark points to visualize the shapes.
plot(X(:,1),X(:,2),"-x")
plot(Y(:,1),Y(:,2),"-o")
Obtain the Procrustes transformation by using procrustes.
[d,Z,transform] = procrustes(X,Y)
The scale component of the transformation b indicates that the scale of X is about twice the scale of Y.
Find the determinant of the rotation and reflection component of the transformation.
The determinant is –1, which means that the transformation contains a reflection.
In two-dimensional space, a reflection matrix has the form
\left[\begin{array}{cc}\mathrm{cos2}\theta & \mathrm{sin2}\theta \\ \mathrm{sin2}\theta & -\mathrm{cos2}\theta \end{array}\right]
which indicates a reflection over a line that makes an angle
\theta
\mathrm{cos}2\theta
\mathrm{sin}2\theta
, the angle for the line of reflection has two possible values between –90 and 90. Use both the
\mathrm{cos}2\theta
\mathrm{sin}2\theta
values to determine the angle for the line of reflection without ambiguity. Using the atan2d function, you can determine the
\mathrm{tan}2\theta
\mathrm{cos}2\theta
\mathrm{sin}2\theta
theta = atan2d(transform.T(2,1),transform.T(1,1))/2
transform.T reflects points across a line that makes roughly a –90 degree angle with the x-axis; this line indicates the y-axis. The plots of X and Y show that reflecting across the y-axis is required to superimpose Y onto X.
Find the Procrustes transformation for landmark points, and apply the transformation to more points on the comparison shape than just the landmark points.
Create matrices with landmark points for two triangles X (target shape) and Y (comparison shape).
Create a matrix with more points on the triangle Y.
Y_points = [linspace(Y(1,1),Y(2,1),10)' linspace(Y(1,2),Y(2,2),10)'
linspace(Y(2,1),Y(3,1),10)' linspace(Y(2,2),Y(3,2),10)'
linspace(Y(3,1),Y(1,1),10)' linspace(Y(3,2),Y(1,2),10)'];
Plot both shapes, including the larger set of points for the comparison shape.
plot([X(:,1); X(1,1)],[X(:,2); X(1,2)],"bx-")
plot([Y(:,1); Y(1,1)],[Y(:,2); Y(1,2)],"ro-","MarkerFaceColor","r")
plot(Y_points(:,1),Y_points(:,2),"ro")
"Additional points on Y","Location","northwest")
Call procrustes to obtain the Procrustes transformation from the comparison shape to the target shape.
Use the Procrustes transformation to superimpose the other points (Y_points) on the comparison shape onto the target shape, and then visualize the results.
Z_points = transform.b*Y_points*transform.T + transform.c(1,:);
plot([Z(:,1); Z(1,1)],[Z(:,2); Z(1,2)],"ks-","MarkerFaceColor","k")
plot(Z_points(:,1),Z_points(:,2),"ks")
"Additional points on Y","Transformed shape (Z)", ...
"Transformed additional points","Location","best")
Construct the shapes of the handwritten letters d and b using landmark points, and then plot the points to visualize the letters.
D = [33 93; 33 87; 33 80; 31 72; 32 65; 32 58; 30 72;
28 72; 25 69; 22 64; 23 59; 26 57; 30 57];
B = [48 83; 48 77; 48 70; 48 65; 49 59; 49 56; 50 66;
plot(D(:,1),D(:,2),"x-")
plot(B(:,1),B(:,2),"o-")
legend("Target shape (d)","Comparison shape (b)")
Use procrustes to compare the letters with reflection turned off, because reflection would turn the b into a d and not accurately preserve the shape you want to compare.
d = procrustes(D,B,"reflection",false)
Try using procrustes with reflection on to see how the Procrustes distance differs.
d = procrustes(D,B,"reflection","best")
This reflection setting results in a smaller Procrustes distance because reflecting b better aligns it with d.
Construct two shapes represented by their landmark points, and then plot the points to visualize them.
X = [20 13; 20 20; 20 29; 20 40; 12 36];
Y = [36 7; 36 10; 36 14; 36 20; 39 18];
Compare the two shapes using Procrustes analysis with scaling turned off.
[d,Z] = procrustes(X,Y,"scaling",false)
Visualize the superimposed landmark points.
plot(Z(:,1),Z(:,2),"-s")
The superimposed shape Z does not differ in scale from the original shape Y.
X — Target shape
Target shape, specified as an n-by-p matrix where each of the n rows contains a p-dimensional landmark point. The landmark points represent the shape that is the target of the comparison.
Y — Comparison shape
Comparison shape, specified as an n-by-q matrix where each of the n rows contains a q-dimensional landmark point with q ≤ p. The landmark points represent the shape to be compared with the target shape.
Y must have the same number of points (rows) as X, where each point in Y, Y(i,:) corresponds to the point in the same row in X, X(i,:).
Points in Y can have fewer dimensions (number of columns) than points in X. In this case, procrustes appends columns of zeros to Y to match the dimensions of X.
Example: d = procrustes(X,Y,"Scaling",false,"reflection",false) performs Procrustes analysis without scaling or reflection in the transformation.
Scaling — Flag to enable scaling
Flag to enable scaling in the Procrustes transformation, specified as logical 1 (true) or 0 (false). A value of false prevents scaling in the transformation. A value of true allows scaling if it minimizes the differences between the landmark points in X and Y.
Set Scaling to false to compare Y to X without scaling Y to match the scale of X. This option causes shapes of different scales to have a greater Procrustes distance.
Example: "Scaling",false
Reflection — Flag to enable reflection
"best" (default) | true or 1 | false or 0
Flag to enable reflection in the Procrustes transformation, specified as "best", logical 1 (true), or logical 0 (false).
"best" — Find the optimal Procrustes transformation, regardless of whether or not it contains a reflection.
1 (true) — Force the Procrustes transformation to reflect Y, whether or not the transformation minimizes the differences between the landmark points.
0 (false) — Prevent the Procrustes transformation from reflecting Y. This option does not prevent rotation in the transformation.
Set Reflection to false to compare Y to X without reflecting Y to match the shape of X. This option causes shapes that are reflections of each other to have a greater Procrustes distance.
Example: "Reflection",true
Data Types: logical | string | char
d — Procrustes distance
Procrustes distance, a measure of dissimilarity between two shapes, returned as a numeric scalar in the range [0,1]. If Scaling is set to false, the Procrustes distance can be outside of the range [0,1].
procrustes computes the distance using the sum of squared differences between the corresponding points in X and Z. The function then standardizes the Procrustes distance by the scale of X. The scale of X is sum(sum((X-mean(X)).^2)), which is the sum of squared elements of a centered version of X where the columns of X have mean 0.
Z — Transformed shape
Transformed shape of the landmark points in Y, returned as an n-by-p numeric matrix that is the same size as X. The output Z is the result of applying the Procrustes transformation to Y.
transform — Procrustes transformation
Procrustes transformation, returned as a structure with three fields:
T — Rotation and reflection component, specified by a p-by-p transformation matrix that rotates or reflects Y to match the orientation of the landmark points in X.
If T is a rotation matrix, then det(T) is 1.
If T is a reflection matrix, then det(T) is –1.
b — Scale component, specified by a scalar to stretch (b > 1), conserve (b = 1), or shrink (b < 1) the scale of Y to match the scale of X.
c — Translation component, specified by an n-by-p matrix where each row is the p-dimensional vector to add to the points in Y to shift it onto X.
The Procrustes transformation superimposes Y onto X by performing the following transformation:
Z = bYT + c.
Set the Reflection name-value argument to false to ensure that transform.T does not contain a reflection.
Set the Scaling name-value argument to false to remove the scale component, fixing transform.b to 1.
The Procrustes distance is a measure of dissimilarity between shapes based on Procrustes analysis.
The procrustes function finds the Procrustes transformation, which is the best shape-preserving Euclidean transformation (consisting of rotation, reflection, scaling, and translation) between the two shapes X and Y. The Procrustes transformation is an optimal transformation that minimizes the sum of squared differences between the landmark points in X and Z, where Z is the transformed shape of Y that results from superimposing Y onto X.
The procrustes function returns the Procrustes distance (d), transformed shape (Z), and Procrustes transformation (transform). The Procrustes distance is the sum of squared differences between X and Z.
Procrustes analysis is appropriate when all dimensions in X and Y have similar scales. If the columns of X and Y have different scales, standardize the columns by using zscore or normalize.
Procrustes analysis is useful in conjunction with multidimensional scaling. Two different applications of multidimensional scaling can produce reconstructed points that are similar in principle, but look different because they have different orientations. Also, the reconstructed points can have a different orientation than the original points. The procrustes function transforms one set of points to make them more comparable to the other. For an example, see Classical Multidimensional Scaling Applied to Nonspatial Distances.
[1] Kendall, David G. “A Survey of the Statistical Theory of Shape.” Statistical Science. Vol. 4, No. 2, 1989, pp. 87–99.
cmdscale | factoran
|
Phase diagram - Physics
Phase diagram (14997 views - Physics)
For the use of this term in mathematics and physics, see phase space.A phase diagram in physical chemistry, engineering, mineralogy, and materials science is a type of chart used to show conditions (pressure, temperature, volume, etc.) at which thermodynamically distinct phases (such as solid, liquid or gaseous states) occur and coexist at equilibrium.
For the use of this term in mathematics and physics, see phase space.
In the diagram on the right, the phase boundary between liquid and gas does not continue indefinitely. Instead, it terminates at a point on the phase diagram called the critical point. This reflects the fact that, at extremely high temperatures and pressures, the liquid and gaseous phases become indistinguishable,[2] in what is known as a supercritical fluid. In water, the critical point occurs at around Tc = 647.096 K (373.946 °C), pc = 22.064 MPa (217.75 atm) and ρc = 356 kg/m³.[3]
The existence of the liquid–gas critical point reveals a slight ambiguity in labelling the single phase regions. When going from the liquid to the gaseous phase, one usually crosses the phase boundary, but it is possible to choose a path that never crosses the boundary by going to the right of the critical point. Thus, the liquid and gaseous phases can blend continuously into each other. The solid–liquid phase boundary can only end in a critical point if the solid and liquid phases have the same symmetry group.[citation needed]
Water is an exception which has a solid-liquid boundary with negative slope so that the melting point decreases with pressure. This occurs because ice (solid water) is less dense than liquid water, as shown by the fact that ice floats on water. At a molecular level, ice is less dense because it has a more extensive network of hydrogen bonding which requires a greater separation of water molecules.[4] Other exceptions are antimony and bismuth.[6][7]
The value of the slope dP/dT is given by the Clapeyron equation for fusion (melting)[8]
{\displaystyle {\frac {\mathrm {d} P}{\mathrm {d} T}}={\frac {\Delta _{fus}H}{T\,\Delta _{fus}V}},}
It is possible to envision three-dimensional (3D) graphs showing three thermodynamic quantities.[9][10] For example, for a single component, a 3D Cartesian coordinate type graph can show temperature (T) on one axis, pressure (p) on a second axis, and specific volume (v) on a third. Such a 3D graph is sometimes called a p–v–T diagram. The equilibrium conditions are shown as curves on a curved surface in 3D with areas for solid, liquid, and vapor phases and areas where solid and liquid, solid and vapor, or liquid and vapor coexist in equilibrium. A line on the surface called a triple line is where solid, liquid and vapor can all coexist in equilibrium. The critical point remains a point on the surface even on a 3D phase diagram.
For water, the 3D p–v–T diagram is seen here:[11]
The iron–iron carbide (Fe–Fe3C) phase diagram. The percentage of carbon present and the temperature define the phase of the iron carbon alloy and therefore its physical characteristics and mechanical properties. The percentage of carbon determines the type of the ferrous alloy: iron, steel or cast iron
In addition to the above-mentioned types of phase diagrams, there are thousands of other possible combinations. Some of the major features of phase diagrams include congruent points, where a solid phase transforms directly into a liquid. There is also the peritectoid, a point where two solid phases combine into one solid phase during cooling. The inverse of this, when one solid phase transforms into two solid phases during cooling, is called the eutectoid.
Log-lin pressure–temperature phase diagram of water. The Roman numerals indicate various ice phases. [12]
ChemistryCircuit diagramEngineeringMaterials sciencePhysicsVoronoi diagramWaterPhysical chemistryMacroscopic scaleDiagramChart
This article uses material from the Wikipedia article "Phase diagram", which is released under the Creative Commons Attribution-Share-Alike License 3.0. There is a list of all authors in Wikipedia
|
Is 0.999... = 1? | Brilliant Math & Science Wiki
Andrew Ellinor, Stewart Gordon, Zoe Codrington, and
\large 0.999 \ldots = 1
Why some people say it's true: It's so very very close to 1. In fact, it's like
0.0000000\ldots
away from 1.
Why some people say it's false: It's less than one, since it starts with 0.99 instead of 1.00. So it cannot be equal to 1.
0.999 \ldots = 1
\color{#20A900}{\textbf{true}}
Intuitive explanation: Visualize a number line. If two real numbers
x
y
on the number line are different, then we should see some space between them. In fact, there would be room to place another real number, namely their average
\frac{x + y}{2}
. Since no number exists between
0.999\ldots
1,
it must be that they are the same. More reluctance against the equivalence stems from a perception that a number cannot have two different names. The reasoning often applied here is, "If a number has two different names, then it cannot truly be the same number." However, this argument doesn't seem to hold water when given a number like
0.6,
\frac{3}{5}
as an alternative name.
(In this proof, we will assume that the value exists.)
All decimals of finite length, such as 0.5 and 0.123, and all repeating decimals, such as .333... and .121212... can be easily converted into fractions. This first proof uses a standard technique for converting a repeating decimal into a fraction in order to calculate the 'fraction' that .99999... is equivalent to.
\begin{array} {l r l } \text{Let } & A & = 0. 999 \ldots. \\ \text{Multiplying by 10, we get } & 10 A & = 9. 999 \ldots. \\ \text{Subtracting } A, \text{ we get } & 9 A & = 9. \\ \text{Dividing by 9, we get }& A & = 1. \ _\square \end{array}
Let's evaluate the limit
A = 0. 999\ldots
. We consider the sum of a geometric progression with infinite terms, with initial term 0.9 and common ratio 0.1. We have
\begin{aligned} 0.999 \ldots & = 0.9 + 0.09 + 0.009 + \cdots \\ & = 0.9 \times 0.1^0 + 0.9 \times 0.1^1 + 0.9 \times 0.1^2 + \cdots \\ & = \frac{ 0.9 } { 1 - 0.1 } \\ & = 1. \end{aligned}
Since this summation converges, it tells us that the limit exists and
A = 1
_\square
0.999... \text{ and } 1
are not equal because they're not the same decimal. With the exception of trailing 0's, any two decimals that are written differently are different numbers.
0.999... = 1
is another case when two decimals that are written differently are, in fact, the same number. The fact that there are two different ways to write 1 as a decimal is a result of the role that infinite sums have in defining what non-terminating decimals mean.
The decimal system is just a shorthand for writing a number as a sum of the powers of 10, each scaled by an integer between 0 and 9 inclusive. For example,
0.123
\frac{1}{10} + \frac{2}{100} + \frac{3}{1000}
0.999\ldots
\frac{9}{10} + \frac{9}{100} + \frac{9}{1000}+\cdots
and the value of this infinite sum is equal to 1.
0.999\ldots
only tends to 1. It is not equal to 1. We only have an approximation.
Reply: It is true that for the sequence
a_n = 0. \underbrace{99 \ldots 9}_{n \, 9's }
, it is the limit as
n \rightarrow \infty
that is equal to 1. However, since
0.999 \ldots
is defined to be that limit, it is defined to equal 1.
Without using this definition for infinitely repeating decimals, there would be many numbers, such as
\frac{1}{3} = .333...
, that we wouldn't be able to write out as decimals since there exists no finite sum of tenths, hundredths, thousandths, etc. that exactly equals
\frac{1}{3}
Rebuttal: Infinite sums don't make any sense. It's not possible to add up infinitely many things, so any infinite sum is only an approximate value, not a real value.
Reply: Without using infinite sums, there would be many numbers, such as
\frac{1}{3} = .333...
, that we wouldn't be able to write out as decimals. The definition of such an infinite sum is rigorous, but strange in that the sum is defined to equal the limit approached as many terms are added together, whenever this limit exists.
Not all infinite sums of fractions can be evaluated. For example,
\frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \frac{4}{5} + \cdots
could not have a real numerical value. But the definition of an infinite sum includes the restriction that an infinite sum only has a well-defined valuation when, as we add up terms of the series, the total sum zooms in towards one, specific value. In the case of
\frac{3}{10} + \frac{3}{100} + \frac{3}{1000}+\cdots,
\frac{1}{3}
is the value being approached. There would be no other way to define
\frac{1}{3}
as a decimal otherwise since
\frac{1}{3}
is not equal to any finite sum of tenths, hundredths, thousandths, etc.
\frac{9}{10} + \frac{9}{100} + \frac{9}{1000}+\cdots,
1 is the value being approached as more and more of the terms are added together.
For a more complete explanation of infinite sums, check out the Infinite Sums wiki page.
Rebuttal: In proof 1, we cannot cancel the trailing 9's because there are infinitely many of them. We will always be left with one 9.
Reply: Cancellation doesn't happen "term by term," where we compare the first 9 in
10A
with the first 0 in
A
. We are looking at the difference of these two numbers, and taking it all together. We get a trailing series of 0's, with no "9 at the end."
Rebuttal: In proof 2, we can't just add the digits "term by term."
Reply: This argument is valid in that the perspective that we're adding "term by term" is how we are evaluating the limit.
Try this problem now:
0.49999\ldots = 0.5
Cite as: Is 0.999... = 1?. Brilliant.org. Retrieved from https://brilliant.org/wiki/is-0999-equal-1/
|
tomleslie - Replies - MaplePrimes
These are replies submitted by tomleslie
is to always upload a worksheet which illustrates your problem!
Use the big green up-arrow in the Mapleprimes toolbar
You refuse...
@imparter
to address the basic problem.
The ode you define in your worksheets as 'de1' only contains the dependent variable 'f(x)'. The ode you define in your worksheets as 'de2' contains the dependent variables 'f(x)' and 'g(x)'.
When you process the ode 'de1', to produce 'DE1', the latter contains the dependent variables
{b[0](x), b[1](x), b[2](x), b[3](x), b[4](x), b[5](x), b[6](x), b[7](x), b[8](x), b[9](x), b[10](x)}
However when you process the ode 'de2', to produce 'DE2', the latter contains the dependent variables (check the one highlighted in red)
{f(x), c[0](x), c[1](x), c[2](x), c[3](x), c[4](x), c[5](x), c[6](x), c[7](x), c[8](x), c[9](x), c[10](x)}
The mere existence of the undetermined function 'f(x)' in the definition of 'DE2' is the source of your problem.
See the attached for proof - it makes no attempt to solve anything - it just lists the indeterminates. Until you fix the existence of the indeterminate 'f(x)' in the definition of 'DE2' you are going to get precisely nowhere
de1 := (1 - p)*diff(f(x), x $ 3) + p*(diff(f(x), x $ 3) + 1/2*f(x)*diff(f(x), x $ 2)):
de2 := (1 - p)*diff(g(x), x $ 2)/Pr + p*(diff(g(x), x $ 2)/Pr + 1/2*f(x)*diff(g(x), x)):
ibvc := f(0), D(f)(0), D(f)(5) - 1, g(0) - 1, g(5):
F := unapply(add(b[k](x)*p^k, k = 0 .. n), x):
G := unapply(add(c[k](x)*p^k, k = 0 .. n), x):
DE1 := series(eval(de1, f = F), p = 0, n + 1):
indets(DE1, function(name));
DE2 := series(eval(de2, g = G), p = 0, n + 1):
Download indets.mw
actually looking at the system you are trying to solve?
The final execution group in your latest worksheet - ie this piece of code
IBVC1 := select(has, CT, c[k]);
{coeff(DE2, p, k), op(IBVC1)};
slv1 := dsolve({coeff(DE2, p, k), op(IBVC1)});
c[k] := unapply(rhs(slv1), x);
For k=0, your code generates the ODE system
{c[0](0) - 1, diff(c[0](x), x, x), c[0](5)}
a second order ODE with two boundary conditions - which is fine, and Maple generates the solution
slv1 := c[0](x) = -x/5 + 1
{diff(c[1](x), x, x) - f(x)/10, c[1](0), c[1](5)}
a second-order ODE with two boundary conditions, and a completely unknown function of the same independent variable f(x). What solution do you expect?
Maple returns a "formal" solution containing double integrals of the unknown function f(x)
Not sure what point...
Your original code generates an invalid ODE system, which means that dsolve() will produce an error.
This update to your code produces a valid ODE system, which means that dsolve() has a reasobale chance of coming up with a solution.
These two cases, both for loop index k=0 are illustrated in the attached. One errors, for reasons I explained previously and one doesn't.
# OP's original code asks for a solution of
dsolve({1, diff(c[0](x), x, x)});
# which will produce an error
Error, (in dsolve) found the following equations not depending on the unknowns of the input system: {1}
# OP's new code asks for a solution of
dsolve( {D(b[0])(5) - 1, diff(b[0](x), x, x, x), b[0](0), D(b[0])(0)});
Download diffODEs.mw
Use the ignore=true option...
as in the attached "toy" example
V:=Vector[column](9, [124.0, 130.0, 130.0, 119.0, 136.0, 118.0, undefined, 130.0, 95.0]);
Mean(V, ignore=true);
Download getMean.mw
that you were trying to achieve what is shown in the attached? I don't think this approach can be (easily) extended beyond first differences.
a := 1: b := 5: h := 1: f := 1/x: N := (b-a)/h:
for i from 0 while i <= N do
x[i] := h*i+a;
y[i] := eval(f, x = x[i])
printf("_______________________________________________________________________________\n\n\n");
for i from 0 by 1 while i<=2*N do:
if irem(i,2)=0
then printf("%2d%16.7f%16.7f\n",i,x[i/2],y[i/2]);
else printf("%2d%48.7f\n",i, y[(i+1)/2]-y[(i-1)/2]);
printf("_______________________________________________________________________________\n")
Download printStuff.mw
@ check my original comments on the typos in your equation and fix them properly before doing anything else.
I'd be surprised if an analytic solution can be found for the resulting ODE, so you may have to be satisfied with a numeric one - which means that you will need values for all parameters and a couple of initial/bondary conditions:-(
With a trivial modification...
the attached will show which months in a specified year have a Friday 13th. Output is in the form
[year, [list of months with Friday 13th]]
From which is pretty obvious that in 2022, the only month with a Friday 13th is month 5 - ie May!
with(Calendar):
fri:= yr-> local j:seq(`if`(DayOfWeek(yr, j, 13)=6,j, NULL), j=1..12):
# So which months in the supplied year have a Friday 13th
seq( [j, [fri(j)]], j=2000..2022);
Download fri_2.mw
@Will_iii
convert(360*Unit(degrees), units, radians);
in Maple Flow - it doesn't work?
Because if this is true then all you have demonstrated ist that Maple Flow is unable to execute basic Maple commands - which is somewhat scary!
If you want to restrict the range variable to integers, then not only do you need the 'sample' option, but ialso to set the adaptive=false option. If the intent is to restrict the range variable to integers, why have a 'floor(n)' command in the function definition? Isn't this a bit - well - superfluous?
The point I was trying to make...
@ProfG
was that in a plot() command, the plotting variable is continuous - it does not assume integer values. Kitonum circumvented this problem by plotting only points. This has the drawback that if you draw lines between the points (say by using style=pointline), then you will not get "vertical" lines, since x-values will always differ by one
I gave a (quick+dirty) workaround for this - and you can see the effect of superimposing my plot and Kitonum's in the attached. This gives an apparent "right shift to the curve on my plot. If this is a problem it is relatively easy to fix, see the final figure in the attached
f:=n->ceil(sqrt(4*floor(n)))-floor(sqrt(2*floor(n)))-1:
plot(f, 10..100, gridlines=false);
plots:-display( [ plot(Points, style=point, color=red, symbol=solidcircle),
plot(f, 10..100)
g:=n->ceil(sqrt(4*floor(n+1/2)))-floor(sqrt(2*floor(n+1/2)))-1:
plot(g, 10..100)
Download sqPlot2.mw
Read the wikipedia entry...
@Tamour_Zubair
which states (emphasis added)
A system of linear equations with n variables has a solution if and only if the rank of its coefficient matrix A is equal to the rank of its augmented matrix [A|b].[1] If there are solutions, they form an affine subspace of
{\displaystyle \mathbb {R} ^{n}}
of dimension n − rank(A). In particular:
if n = rank(A), the solution is unique,
otherwise there are infinitely many solutions.
It is trivial to produce the coefficient matrix, the augmented matrix and their ranks, using the code
sys:=[subs(x = xmin, uhat1) = rhs(bcf1),
subs(x = xmax, uhat1) = rhs(bcf2),
subs(p = pmin, uhat1) = rhs(bcf3),
subs(p = pmax, uhat1) = rhs(bcf4)]:
vars:=[Af[0, 1](t), Af[0, 2](t), Af[0, 0](t), Af[0, 3](t)]:
coeffMat,b:=GenerateMatrix(sys, vars):
augMat:=GenerateMatrix(sys, vars, augmented=true):
Rank(coeffMat);
Rank(augMat);
which shows that the coefficient matrix has rank 3, and the augmented matrix has rank 4. Thus the sytem has no solution
Your question states...
ScientificConstants[GetValue] is not working in at least Maple 2021 and Maple 2022
No mention of Maple Flow - if you had mentioned Maple Flow, I wouldn't even have attempted an answer
if you uploaded a usable worksheet someone might investigate this
However, if I try to download from Question#3.mw all I get is the error Page not Found.
Try downloading your own upload - just to see if it works!
You state (emphasis added)
The first ode you see is a screen shot of the maple help page to show that the syntax using series worked. That was not an example of mine. It is just screen shot of the help page to show that the syntax in help page worked but gives error when I used it.
The first worksheet (odetest.mw) attached below is obtained directly from Maple help by using (in the help browser) the menu commands View -> Open Page as Worksheet. This whole worksheet has then been re-executed (using !!! in the Maple toolbar). Everything executes with no errors. So I still cannot replicate the problem implied by your remark
the syntax in help page worked but gives error when I used it.
And since you do not provide a worksheet illustrating the problem whihc you are experiencing - there isn't much I can do to fix it.
The second worksheet (testODE2.mw) below shows some of my attempts to replicate your problem. It generates series solutions for the same ODE from the help page in several slightly different ways, then tests each one with odetest() in a couple of different ways. Every single one of these "works" - so again I cannot replicate your problem
The odetest command checks explicit and implicit solutions for ODEs by making a careful simplification of the ODE with respect to the given solution. If the solution is valid, the returned result will be 0; otherwise, the algebraic remaining expression will be returned. In the case of systems of ODEs, odetest can only test explicit solutions, given either as a set or as a list of sets. (For information on non-composed sets of solutions for nonlinear systems, see dsolve,system .)
To test whether a solution satisfies one or many initial or boundary conditions, pass to odetest the ODE together with the initial or boundary conditions, enclosed as a set or list , as second argument.
If odetest returns a nonzero result, the solution being tested is not necessarily wrong; sometimes further simplifications or manipulations of odetest's output are required to obtain zero, and so verify the solution is correct. If the solution was obtained using the dsolve command, it is recommended that you recompute the solution using one or both of the useInt and implicit options - see dsolve . This may facilitate the verification process. Also, an alternative testing technique, particularly useful with linear ODEs, is to try to recompute the ODE departing from the solution which odetest fails in testing. Examples of both types are found at the end of the next section.
To test series solutions , pass the keyword series as an extra argument. Only one series solution for one ODE (can be a set with initial/boundary conditions) can be tested.
ODE := [diff(y(x),x)=sin(x-y(x)), y(0) = 8];
sol := dsolve(ODE);
odetest( sol, ODE ); # verifies 'sol' solves the ode and satisfies y(0) = 8
ODE := diff(y(x),x,x) + diff(y(x),x) + y(x)=0;
bc := y(0) = 1, y(2*Pi) = 1;
sol := dsolve({ODE, bc});
odetest( sol, [ODE,bc] ); # verifies 'sol' solves the ODE and satisfies the bc given
ODE := [diff(y(x),x,x)+diff(y(x),x)^2=0, y(a)=0, D(y)(a)=1];
sol := dsolve( ODE, y(x), type='series');
odetest(sol, ODE, series);
ODE := diff(diff(y(x),x),x) = (3*x^2+c)*diff(y(x),x)+((3-b)*x-a)*y(x);
sol := y(x) = series(1+(-1/2*a)*x^2+(-1/6*b+1/2-1/6*c*a)*x^3+(1/24*a^2-1/24*c*b+1/8*c-1/24*c^2*a)*x^4+O(x^5),x,5);
An ODE with an arbitrary function of (x, y, dy/dx) and a solution involving nested integrals with a RootOf in the integrand
ODE := diff(y(x),x,x) = 1/x^2*_F1(diff(y(x),x)*x/y(x))*y(x);
ODE := diff(y(x),x)=F((y(x)-x*ln(x))/x) + ln(x);
sol := dsolve(ODE,implicit);
ODE := diff(y(x),x) = x*f(x)^2*(x+2)*y(x)^3+f(x)*(x+3)*y(x)^2-diff(f(x),x)/f(x)*y(x);
sol := dsolve(ODE, y(x), implicit);
odetest(sol,ODE, y(x));
ODE := diff(y(x),x,x) = (diff(y(x),x)-y(x)^3-f(x)+3*x*y(x)^2*diff(y(x),x)+x*diff(f(x),x))/x;
sol := dsolve(ODE, y(x));
sysODE := {diff(y(t),t)=-x(t),diff(x(t),t)=y(t)}, {x,y}(t);
solsys := dsolve(sysODE);
odetest(solsys,sysODE);
sysODE := {diff(y(t),t)=-x(t)^2,diff(x(t),t)=y(t)}, {x,y}(t);
solsys := dsolve(sysODE, explicit);
map(odetest, [solsys], sysODE);
One possible workaround for an example where odetest fails in verifying dsolve 's solution
ODE := diff(y(t),t) = ((b+2+2*t)*y(t)+1)/(1-(1+t)^2);
odetest(sol, ODE); # fails in verifying this solution
sol2 := dsolve(ODE, useInt); # compute 'sol' again with 'useInt'
odetest(sol2, ODE); # this solution is easier to test
ODE := diff(y(x),`$`(x,2)) = ((-a[1]*F+a[0]*E)*x+B*e*a[1])*y(x)/B^2/e^2/x/E^3;
sol := dsolve(ODE, [hyper3]); # 1F1 and KummerU solution
ode := PDEtools[dpolyform](sol, no_Fn); # the ode satisfied by sol
normal( ODE - op([1,1],ode) ); # ode = ODE
sol_W:=convert(sol,Whittaker);
odetest( sol_W, ODE ); # this Whittaker solution is easier to test
Download odetest.mw
# ode from help page for odetest() command
# Solution from help page for odetest() command.
# NB this "solution" is given, not computed, and it
# is 5-th order
sol:= y(x) = series(1+(-1/2*a)*x^2+(-1/6*b+1/2-1/6*c*a)*x^3+(1/24*a^2-1/24*c*b+1/8*c-1/24*c^2*a)*x^4+O(x^5),x,5);
# Use dsolve to get a series solution for the above ODE
# (no boundary conditions). This will be 6-th order (by
sol2_6:= dsolve(ODE, y(x), series);
# Just for completeness (and comparison with sol1 above)
# generate a 5-th order solution
Order:=5:
# Use dsolve to get a series solution (with boundary conditions)
# at both 5-th and 6-th order
sol3_5:= dsolve([ODE, y(0)=1, D(y)(0)=0], y(x), series);
sol3_6:= dsolve([ODE, y(0)=1, D(y)(0)=0], y(x), series)
# Five (slightly different) series solutions above. Check each
# of them (two ways)
odetest(sol, ODE, series, point = 0); # from maple help page
odetest(sol, ODE, 'series', point = 0);
odetest(sol2_5, ODE, series, point = 0);
odetest(sol2_5, ODE, 'series', point = 0);
odetest(sol3_5, [ODE, y(0)=1, D(y)(0)=0], 'series');
odetest(sol3_5, [ODE, y(0)=1, D(y)(0)=0], series);
Download testODE2.mw
|
Unarmed Combat - Ring of Brodgar
Damage can be calculated by the following formula:
{\displaystyle damage=MoveDmg*{\sqrt {(str/10)}}}
The Unarmed Combat ability is a factor of MoveDmg, see combat moves for details.
MoveDmg is the listed damage of the martial art used
str is the strength of the hearthling
The base damage of a basic Punch is 10. For a hearthling with a Strength of 80, their damage would be calculated by:
{\displaystyle damage=10*{\sqrt {(80/10)}}=28}
Strength to unarmed dmg multiplier chart: (see also Quality page)
There are currently four types of moves in Haven & Hearth;
Striking, Backhanded, Sweeping, and Oppressive.
Each attack raises your opponent's opening of it's corresponding color.
Example: Using the green attack Punch (A Striking move) will raise your opponent's green opening (Off-Balance)
Maneuvers are defenses against the attacks of your foe. The higher your corresponding stat, the more defense a maneuver will provide.
The following is an incomplete list of all maneuvers currently in the game.
Bloodlust Unarmed Combat * 50% 10 When attacked: Bloodlust is charged by 25% * ∆. When you attack an opponent, your attack weight will be increased by four times the amount that Bloodlust is charged. Rage
Chin Up Unarmed Combat 10
Combat Meditation Unarmed Combat 10 When attacked: Combat Meditation is charged by 25% * ∆. When you attack an opponent, your cooldown will be decreased by the amount that Combat Meditation is charged. Woodsmanship
Death or Glory Unarmed Combat * 50% 10 When attacked: You gain 0.75 * ∆ Points of Initiative against the opponent. Siegecraft
Oak Stance Unarmed Combat * 150% 10 When attacked: Your greatest opening is reduced by 5% * ∆. While Oak Stance is active, all your attacks will have 50% of their normal attack weight. Forestry
Attacks are moves that inflict damage and/or increase openings on your opponent. Using an attack will cause a cooldown based off the listed cooldown of an ability. The attacker must increase the openings of a matching attack type before actual damage is inflicted on their HP. Having a higher agility than your opponent will make your cooldowns shorter, and your foes' longer. Grievous damage is damage done to your foes HHP. New moves are discovered via green highlighted combat discoveries while fighting certain foes, there also seems to be a level requirement to unlock some abilities.
Punch Unarmed Combat * 0.8 * μ Striking +15% Off-Balance 10 5% 30
Left Hook Unarmed Combat * μ Backhanded +15% Dizzy 15 10% 40
Low Blow Unarmed Combat * μ Backhanded +10% Off-Balance 10 30% 50 Gain 1 IP against your opponent. Fox, Boar
Kick Unarmed Combat * μ Sweeping +17.5% Reeling 15 15% 45 Ants
Haymaker Unarmed Combat * μ Sweeping +15% Reeling 20 15% 50 Ants, Bat
Knock Its Teeth Out Unarmed Combat * μ Oppressive +20% Cornered 30 25% 35 1 IP Badger
Go for the Jugular Unarmed Combat * μ Striking, Oppresive +15% Off-Balance, +10% Cornered 40 30% 45 3+3 IP
Punch 'em Both Unarmed Combat * μ Striking, Sweeping +15% Reeling 10 7.5% 40 Punch 'em Both attacks both your primary target and also one other opponent in range. Ants, Fox
Rip Apart Unarmed Combat * μ Striking, Backhanded, Sweeping, Oppressive +5% Off-Balance, +5% Dizzy, +5% Reeling, +5% Cornered 50 30% 80 6 IP
Steal Thunder Unarmed Combat * μ Backhanded, Sweeping +10% Dizzy None None 40 To the extent that it is unblocked, Steal Thunder will take 3 IP from its target and gain you 2 of them.
Uppercut Unarmed Combat * μ Striking, Backhanded +15% Reeling 30 5% 30
Opportunity Knocks Unarmed Combat 45 Opportunity Knocks increases your opponent's greatest opening by 40% * μ 4 IP
Flex Unarmed Combat +15% Dizzy 30 Reduces 10% * μ Backhanded, 10% * μ Oppressive
Retrieved from "https://ringofbrodgar.com/w/index.php?title=Unarmed_Combat&oldid=86627"
|
f\left(x\right)
f
x
f\left(x\right)
f
f\left(a,b,c\right)
a
b
c
f
f\left(a,b,c,...\right)
f
f
f
a,b,c,\dots
f
f\left(x\right)
f
f
x
x
f\left(x\right)
further populate this remember table.
f
f\left(x\right)
\mathrm{\pi }
\mathrm{sin}\left(\mathrm{\pi }\right)
\textcolor[rgb]{0,0,1}{0}
\mathrm{apply}\left(\mathrm{sin},\mathrm{\pi }\right)
\textcolor[rgb]{0,0,1}{0}
\mathrm{sin}\left(x\right)
\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)
\mathrm{apply}\left(\mathrm{sin},x\right)
\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)
\mathrm{type}\left(\mathrm{sin}\left(x\right),'\mathrm{function}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{type}\left(\mathrm{sin}\left(x\right),'\mathrm{procedure}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{type}\left(\mathrm{sin},'\mathrm{function}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{type}\left(\mathrm{sin},'\mathrm{procedure}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{type}\left('\left(x↦{x}^{2}\right)\left(a\right)','\mathrm{function}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\left(\mathrm{sin},\mathrm{cos}\right)\left(x\right)
\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{cos}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)
\mathrm{type}\left([\mathrm{sin},\mathrm{cos}]\left(x\right),'\mathrm{function}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{type}\left([\mathrm{sin},\mathrm{cos}]\left(x\right),'\mathrm{list}'\left('\mathrm{function}'\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{type}\left('[\mathrm{sin},\mathrm{cos}]\left(x\right)','\mathrm{function}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{pointto}\left(\mathrm{assemble}\left(\mathrm{kernelopts}\left(\mathrm{dagtag}=\mathrm{STOP}\right)\right)\right)\left(s\right)
\textcolor[rgb]{0,0,1}{\mathbf{stop}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{s}\right)
\mathrm{type}\left(\mathrm{pointto}\left(\mathrm{assemble}\left(\mathrm{kernelopts}\left(\mathrm{dagtag}=\mathrm{STOP}\right)\right)\right)\left(s\right),'\mathrm{function}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{interface}\left('\mathrm{verboseproc}'=3\right):
\mathrm{unassign}\left('f'\right):
\mathrm{eval}\left(f\right)
\textcolor[rgb]{0,0,1}{f}
f\left(2\right)≔3
\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2}\right)\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{3}
f
\mathrm{eval}\left(f\right)
\textcolor[rgb]{0,0,1}{\mathbf{proc}}\left(\textcolor[rgb]{0,0,1}{}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\textcolor[rgb]{0,0,1}{\mathbf{option}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\textcolor[rgb]{0,0,1}{\mathrm{remember}}\textcolor[rgb]{0,0,1}{;}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\textcolor[rgb]{0,0,1}{'}\textcolor[rgb]{0,0,1}{\mathrm{procname}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{args}}\right)\textcolor[rgb]{0,0,1}{'}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\textcolor[rgb]{0,0,1}{\mathbf{end proc}}\textcolor[rgb]{0,0,1}{\text{#(2) = 3}}
f\left(3\right)≔1:
f\left(1\right)≔2:
\mathrm{eval}\left(f\right)
\textcolor[rgb]{0,0,1}{\mathbf{proc}}\left(\textcolor[rgb]{0,0,1}{}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\textcolor[rgb]{0,0,1}{\mathbf{option}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\textcolor[rgb]{0,0,1}{\mathrm{remember}}\textcolor[rgb]{0,0,1}{;}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\textcolor[rgb]{0,0,1}{'}\textcolor[rgb]{0,0,1}{\mathrm{procname}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{args}}\right)\textcolor[rgb]{0,0,1}{'}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\textcolor[rgb]{0,0,1}{\mathbf{end proc}}\textcolor[rgb]{0,0,1}{\text{#(1) = 2}}\textcolor[rgb]{0,0,1}{\text{#(2) = 3}}\textcolor[rgb]{0,0,1}{\text{#(3) = 1}}
|
Extract the unwrapped phase of a complex input - Simulink - MathWorks América Latina
Phase Extractor
Unwrap phase only within the frame
Extract the unwrapped phase of a complex input
The Phase Extractor block extracts the unwrapped phase of a complex input.
Specify the input signal as a vector or a matrix. When the input is a matrix, the block treats each column of the signal as an independent channel. The first dimension is the length of the channel. The second dimension is the number of channels. The block treats one dimensional inputs as one channel.
Output 1 — Unwrapped phase
The block returns the unwrapped phase of the signal.
Unwrap phase only within the frame — Unwrap phase only within the frame
When you clear this check box, the block ignores boundaries between the input frames. When you select this check box, the block treats each frame of input data independently, and resets the initial phase value for each new input frame.
Extract the Phase of Sine Wave
Use the Phase Extractor block to extract the phase of a sine wave signal.
Consider an input frame of length N:
\left(\begin{array}{l}{x}_{1}\\ {x}_{2}\\ ⋮\\ {x}_{N}\end{array}\right)
The step method acts on this frame and produces this output:
\left(\begin{array}{l}{\Phi }_{1}\\ {\Phi }_{2}\\ ⋮\\ {\Phi }_{N}\end{array}\right)
{\Phi }_{i}={\Phi }_{i-1}+\text{angle}\left({x}_{i-1}^{*}{x}_{i}\right)
Here, i runs from 1 to N. The angle function returns the phase angle in radians.
If the input signal consists of multiple frames:
If you set TreatFramesIndependently to true, the step method treats each frame independently. Therefore, in each frame, the step method calculates the phase using the preceding formula where:
{\Phi }_{0}
{x}_{0}
If you set TreatFramesIndependently to false, the step method ignores boundaries between frames. Therefore, in each frame, the step method calculates the phase using the preceding formula where:
{\Phi }_{0}
is the last unwrapped phase from the previous frame.
{x}_{0}
is the last sample from the previous frame.
|
Tutorial: Multiplexor, Shifter, Encoder, Decoder
Multiplexor, Shifter, Encoder, Decoder
Introduction to multiplexers, shifters, encoders, and decoders.
Create a project Vivado project as you have done before.
Step 2: Design a 4:1 Multiplexor
This project starts with designing a 4:1 1-bit multiplexer. Four on-board slide switches will be used to provide the data inputs, two push buttons will be used as select signals, and LED 0 will be used to show the output of the multiplexer. The most common way to define a 4:1 mux in Verilog is to use a case statement inside an always block. Note: we renamed led[0] to Y in our constraints (.xdc) file, so the output port name is Y.
Create a new source file named mux_4_1.v and enter the code as follows.
module mux_4_1 (
// we can only assign values to registers
// inside an always block
always @(data, sel) begin
2'b00: tmp <= data[0];
default: tmp <= 1'b0;
assign Y = tmp;
We defined our mux to have 4 data inputs, 2 select inputs, and one output signal.
log_2 (4)
data inputs = 2 select inputs.
Since we are using an always block, we created a temporary 1-bit register called tmp.
The always block has the sensitivity list @(data, sel). This means that tmp will be updated whenever data or sel change.
We wrap the contents of the always block with begin…end. This is analogous to wrapping functions with { } in C or Java.
The case statement checks the current value of sel, and then sets tmp to the corresponding data bit. The default keyword tells tmp what it should be if sel doesn’t equal 00, 01, 10, or 11. Every case statement should include a default case. (Each bit of sel can equal 0, 1, x, or z. In this example, the default case covers all cases where one of the bits of sel is an x or z.)
We assign Y to tmp outside the always block. This completes our mux.
Next, change the constraints (.xdc) file as shown. The constraints file renamed led[0] to Y, sw[3:0] to data[3:0], and btn[1:0] to sel[1:0]. Generate a bitstream and upload it to your Blackboard. Verify the mux works as expected.
# Individual LEDS
set_property -dict { PACKAGE_PIN N20 IOSTANDARD LVCMOS33 } [get_ports { Y }]; #IO_L14P_T2_SRCC_34 Schematic=LD0
# set_property -dict { PACKAGE_PIN P20 IOSTANDARD LVCMOS33 } [get_ports { led[1] }]; #IO_L14N_T2_SRCC_34 Schematic=LD1
# set_property -dict { PACKAGE_PIN R19 IOSTANDARD LVCMOS33 } [get_ports { led[2] }]; #IO_0_34 Schematic=LD2
# set_property -dict { PACKAGE_PIN T20 IOSTANDARD LVCMOS33 } [get_ports { led[3] }]; #IO_L15P_T2_DQS_34 Schematic=LD3
# set_property -dict { PACKAGE_PIN T19 IOSTANDARD LVCMOS33 } [get_ports { led[4] }]; #IO_L3P_T0_DWS_PUDC_B_34 Schematic=LD4
# set_property -dict { PACKAGE_PIN U13 IOSTANDARD LVCMOS33 } [get_ports { led[5] }]; #IO_25_34 Schematic=LD5
# set_property -dict { PACKAGE_PIN V20 IOSTANDARD LVCMOS33 } [get_ports { led[6] }]; #IO_L16N_T2_34 Schematic=LD6
# set_property -dict { PACKAGE_PIN W20 IOSTANDARD LVCMOS33 } [get_ports { led[7] }]; #IO_L17N_T2_34 Schematic=LD7
# set_property -dict { PACKAGE_PIN W19 IOSTANDARD LVCMOS33 } [get_ports { led[8] }]; #IO_L16P_T2_34 Schematic=LD8
# set_property -dict { PACKAGE_PIN Y19 IOSTANDARD LVCMOS33 } [get_ports { led[9] }]; #IO_L22N_T3_34 Schematic=LD9
set_property -dict { PACKAGE_PIN R17 IOSTANDARD LVCMOS33 } [get_ports { data[0] }]; #IO_L19N_T3_VREF_34 Schematic=SW0
set_property -dict { PACKAGE_PIN U20 IOSTANDARD LVCMOS33 } [get_ports { data[1] }]; #IO_L15N_T2_DQS_34 Schematic=SW1
set_property -dict { PACKAGE_PIN R16 IOSTANDARD LVCMOS33 } [get_ports { data[2] }]; #IO_L19P_T3_34 Schematic=SW2
set_property -dict { PACKAGE_PIN N16 IOSTANDARD LVCMOS33 } [get_ports { data[3] }]; #IO_L21N_T3_DQS_AD14N_35 Schematic=SW3
# set_property -dict { PACKAGE_PIN R14 IOSTANDARD LVCMOS33 } [get_ports { sw[4] }]; #IO_L6N_T0_VREF_34 Schematic=SW4
# set_property -dict { PACKAGE_PIN P14 IOSTANDARD LVCMOS33 } [get_ports { sw[5] }]; #IO_L6P_T0_34 Schematic=SW5
# set_property -dict { PACKAGE_PIN L15 IOSTANDARD LVCMOS33 } [get_ports { sw[6] }]; #IO_L22N_T3_AD7N_35 Schematic=SW6
# set_property -dict { PACKAGE_PIN M15 IOSTANDARD LVCMOS33 } [get_ports { sw[7] }]; #IO_L23N_T3_35 Schematic=SW7
# set_property -dict { PACKAGE_PIN T10 IOSTANDARD LVCMOS33 } [get_ports { sw[8] }]; #IO_L10P_T1_34 Sch=VGA_R4_CON
# set_property -dict { PACKAGE_PIN T12 IOSTANDARD LVCMOS33 } [get_ports { sw[9] }]; #IO_L10N_T1_34 Sch=VGA_R5_CON
# set_property -dict { PACKAGE_PIN T11 IOSTANDARD LVCMOS33 } [get_ports { sw[10] }]; #IO_L18P_T2_34 Sch=VGA_R6_CON
# set_property -dict { PACKAGE_PIN T14 IOSTANDARD LVCMOS33 } [get_ports { sw[11] }]; #IO_L18N_T2_AD13N_35 Sch=VGA_R7_CON
set_property -dict { PACKAGE_PIN W14 IOSTANDARD LVCMOS33 } [get_ports { sel[0] }]; #IO_L8P_T1_34 Schematic=BTN0
set_property -dict { PACKAGE_PIN W13 IOSTANDARD LVCMOS33 } [get_ports { sel[1] }]; #IO_L4N_T0_34 Schematic=BTN1
# set_property -dict { PACKAGE_PIN P15 IOSTANDARD LVCMOS33 } [get_ports { btn[2] }]; #IO_L24P_T3_34 Schematic=BTN2
# set_property -dict { PACKAGE_PIN M14 IOSTANDARD LVCMOS33 } [get_ports { btn[3] }]; #IO_L23P_T3_35 Schematic=BTN3
To simulate the mux, add a new simulation source file. Paste the following code, save the file, and then press “Run Simulation.” You may need to right-click on the file in the Sources window and then select “Set as Top” if you have other simulation source files in the project. Become familiar with zooming and panning in the simulation window. Understand how the simulation works and see if you can verify the mux is working correctly from the output waveforms alone.
// simulation file
module mux_tb;
// connect test signals to our mux
mux_4_1 CUT (
.sel(sel),
sel = 2'b00;
for(k=0; k < 16; k=k+1) begin
data = k;
#10; // wait 10ns
sel = 2'b1z;
sel = 2'b1x;
Alternative Ways to Create a 4:1 Mux in Verilog
Since the mux is such a common digital circuit element, it shouldn’t too surprising that there are other ways to create a mux. Two common implementations are shown below.
1. Using the ?: selection operator
The first way to code a mux behaviorally is to use the ?: selection operator. This method is most analogous to the if statement. You can think of this statement as follows: assign data[0] to Y if the statement in the parenthesis is true, else assign whatever is after the colon to Y (and so on). This method is most commonly used with 2-input muxes in practice.
assign Y = (sel == 2'd0) ? data[0] : (
(sel == 2'd1) ? data[1] : (
(sel == 2'd2) ? data[2] : data[3]
2. Using an always Block With an if Statement
The second way to code a mux is by using an always block together with an if-else statement. Note that because tmp is assigned in an always block, it must be declared as register type reg (assign statements cannot be used inside an always block). It is important that there is a closing else statement in Verilog, unlike C or Java.
always @ (sel, data)
if (sel == 2'd0)
tmp <= data[0];
else if (sel == 2'd1)
Step 3: Design a 4:1 2-bit Bus Multiplexor
Now let’s design an 4:1 2-bit bus multiplexer (that is, a multiplexor with four 2-bit bus inputs and a 2-bit bus output). Eight on-board slide switches will be used to provide the data inputs (organized as four 2-bit inputs: I0, I1, I2, I3), two push buttons will be used as select signals, while LED 0 and LED 1 will be used to show the 2-bit output of the bus multiplexer.
We will still need 2 select signals, as this lets us choose between
2^2 = 4
different input signals.
input [1:0] I0, I1, I2, I3,
always @(I0, I1, I2, I3, sel) begin
2'b00: tmp <= I0;
default: tmp <= 2'b00;
Create a new simulation file and run the following code. Make sure you right-click this simulation file and select “Set as Top”. If the simulation stops running after 1000ns, click Settings -> Simulation -> Simulation and type in a new simulation runtime.
module mux_4_2_tb;
reg [1:0] I0, I1, I2, I3;
for(k=0; k < 256; k=k+1) begin
{I3, I2, I1, I0} = k;
Simulate and implement the 4:1 2-bit mux on your Blackboard. To create a bitstream, you will need to modify the constraints file from step 2.
In this section you are going to design a 3:8 binary decoder. The example presented here uses a 3-bit bus I[2:0] for input signals, and an 8-bit bus Y[7:0] for output signals. Three individual input wires and eight indiviudal output wires could have been used instead, but then the Verilog `code would be less compact.
Declare a 3:8 Binary Decoder
Create a Verilog module called decoder_3_8 with inputs I and outputs Y as follows.
Perhaps the most readable way to describe the behavior of a decoder is to use a case statement in an always block as shown.
module decoder_3_8 (
input [2:0] I,
always @ (I)
3'd0: Y <= 8'd1;
3'd4: Y <= 8'd16;
3'd7: Y <= 8'd128;
default: Y <= 8'd0;
In this section, you are going to design a 4-input priority encoder. A 4-bit bus I[3:0] will be used as data inputs, and Ein, will act as the “Enable” signal. A 2-bit output bus Y[1:0] will show the encoded value of the inputs, and two one bit outputs GS and Eout will show the group signal and enable output, respectively.
Declare a 4-Input Priority Encoder
Create another Verilog module called encoder with inputs I, Ein, and outputs Eout, GS, and Y.
The most efficient way to describe the behavior of a priority encoder is to use if-else statement in an always block. The priority encoder has three outputs, and so three always blocks are needed to define the output signals. Notice the use of “nested” if statements.
input Ein,
output reg GS,
output reg Eout
always @ (I, Ein)
if(Ein == 1) begin
if (I[3] == 1)
Y <= 2'd3;
else if (I[2] == 1)
if (Ein == 1 && I == 0)
Eout <= 1'b1;
if (Ein == 1 && I != 0)
GS <= 1'b1;
In this section, you are going to design a 4-input Shifter. A 4-bit bus I[3:0] will be used for data inputs, and four other 1-bit inputs are used for the control signals F (fill), R (rotate/shift), D (direction), and En (enable signal). Bus Y[3:0] will show the output of the shifter.
Declare a Shifter
Similar to previous steps, you will use if-else statement again to implement the shifter.
module shifter (
input F,
always @ (I, En)
Y <= I;
Y <= (D == 0) ? {I[2:0], F} : {F, I[3:1]};
Y <= (D == 0) ? {I[2:0], I[3]} : {I[0], I[3:1]};
In the shifter’s behavioral code, {A,B} is used to concatenate two groups of signals into a bus. For example, Y <= {I[2:0], F} means Y[3:1] <= I[2:0] and Y[0] <= F.
Aside - Declaring Outputs as Registers
It is also possible to declare outputs as registers. For example, a 4:1 1-bit mux can be written as follows:
output reg Y
2'b00: Y <= data[0];
default: Y <= 1'b0;
While this does clean up the code and remove reg tmp from the module, it does have one drawback:
Sometimes a digital design is not working properly, and you have no idea where to start looking for the issue. One approach is to begin checking that each individual module simulates correctly. This approach may identify the broken module, but often will not show why the module is not working correctly. More complex modules will have many internal registers, and checking their values as inputs change can be extremely helpful in solving the problem.
To do this, you can copy and paste the body of the broken module into a simulation file, run some tests, and check how the inputs change the internal register values. If an output is declared as a register in the module, it will have to be renamed to a wire in the simulation, as simulation outputs are wires.
Because of this, some digital engineers choose to always declare their module inputs and outputs as wires (as we have done in all previous examples). It makes simulating a broken module a little easier, at the expense of a little more code. Ultimately, this is a preference choice, and there is not right or wrong side to this issue.
Below is an example of a 4:1 mux in a simulation file. Compare and contrast this file to the simulations above.
module mux_4_1_dev;
// ---- begin inputs and outputs of mux module ----- //
// ---- end inputs and outputs of mux module ----- //
// ----- begin body of mux module ----- //
// this section is copy/pasted from a module file
// ----- end body of mux module ----- //
// TODO: write test code
// Internal registers will appear in simulated waveforms
|
Pinwheel tiling - Wikipedia
Non-periodic tiling in geometry
Wikimedia Commons has media related to Pinwheel tiling.
In geometry, pinwheel tilings are non-periodic tilings defined by Charles Radin and based on a construction due to John Conway. They are the first known non-periodic tilings to each have the property that their tiles appear in infinitely many orientations.
1 Conway's tessellation
2 The pinwheel tilings
Conway's tessellation[edit]
Conway's triangle decomposition into smaller similar triangles.
{\displaystyle T}
be the right triangle with side length
{\displaystyle 1}
{\displaystyle 2}
{\displaystyle {\sqrt {5}}}
. Conway noticed that
{\displaystyle T}
can be divided in five isometric copies of its image by the dilation of factor
{\displaystyle 1/{\sqrt {5}}}
The increasing sequence of triangles which defines Conway's tiling of the plane.
By suitably rescaling and translating/rotating, this operation can be iterated to obtain an infinite increasing sequence of growing triangles all made of isometric copies of
{\displaystyle T}
. The union of all these triangles yields a tiling of the whole plane by isometric copies of
{\displaystyle T}
In this tiling, isometric copies of
{\displaystyle T}
appear in infinitely many orientations (this is due to the angles
{\displaystyle \arctan(1/2)}
{\displaystyle \arctan(2)}
{\displaystyle T}
, both non-commensurable with
{\displaystyle \pi }
). Despite this, all the vertices have rational coordinates.
The pinwheel tilings[edit]
A pinwheel tiling: tiles can be grouped in sets of five (thick lines) to form a new pinwheel tiling (up to rescaling)
Radin relied on the above construction of Conway to define pinwheel tilings. Formally, the pinwheel tilings are the tilings whose tiles are isometric copies of
{\displaystyle T}
, in which a tile may intersect another tile only either on a whole side or on half the length
{\displaystyle 2}
side, and such that the following property holds. Given any pinwheel tiling
{\displaystyle P}
, there is a pinwheel tiling
{\displaystyle P'}
which, once each tile is divided in five following the Conway construction and the result is dilated by a factor
{\displaystyle {\sqrt {5}}}
{\displaystyle P}
. In other words, the tiles of any pinwheel tilings can be grouped in sets of five into homothetic tiles, so that these homothetic tiles form (up to rescaling) a new pinwheel tiling.
The tiling constructed by Conway is a pinwheel tiling, but there are uncountably many other different pinwheel tiling. They are all locally undistinguishable (i.e., they have the same finite patches). They all share with the Conway tiling the property that tiles appear in infinitely many orientations (and vertices have rational coordinates).
The main result proven by Radin is that there is a finite (though very large) set of so-called prototiles, with each being obtained by coloring the sides of
{\displaystyle T}
, so that the pinwheel tilings are exactly the tilings of the plane by isometric copies of these prototiles, with the condition that whenever two copies intersect in a point, they have the same color in this point.[1] In terms of symbolic dynamics, this means that the pinwheel tilings form a sofic subshift.
Radin and Conway proposed a three-dimensional analogue which was dubbed the quaquaversal tiling.[2] There are other variants and generalizations of the original idea.[3]
One gets a fractal by iteratively dividing
{\displaystyle T}
in five isometric copies, following the Conway construction, and discarding the middle triangle (ad infinitum). This "pinwheel fractal" has Hausdorff dimension
{\displaystyle d={\frac {\ln 4}{\ln {\sqrt {5}}}}\approx 1.7227}
Use in architecture[edit]
Federation Square's sandstone façade
Federation Square, a building complex in Melbourne, Australia, features the pinwheel tiling. In the project, the tiling pattern is used to create the structural sub-framing for the facades, allowing for the facades to be fabricated off-site, in a factory and later erected to form the facades. The pinwheel tiling system was based on the single triangular element, composed of zinc, perforated zinc, sandstone or glass (known as a tile), which was joined to 4 other similar tiles on an aluminum frame, to form a "panel". Five panels were affixed to a galvanized steel frame, forming a "mega-panel", which were then hoisted onto support frames for the facade. The rotational positioning of the tiles gives the facades a more random, uncertain compositional quality, even though the process of its construction is based on pre-fabrication and repetition. The same pinwheel tiling system is used in the development of the structural frame and glazing for the "Atrium" at Federation Square, although in this instance, the pin-wheel grid has been made "3-dimensional" to form a portal frame structure.
^ Radin, C. (May 1994). "The Pinwheel Tilings of the Plane". Annals of Mathematics. 139 (3): 661–702. CiteSeerX 10.1.1.44.9723. doi:10.2307/2118575. JSTOR 2118575.
^ Radin, C., Conway, J., Quaquaversal tiling and rotations, preprint, Princeton University Press, 1995
^ Sadun, L. (January 1998). "Some Generalizations of the Pinwheel Tiling". Discrete and Computational Geometry. 20 (1): 79–110. arXiv:math/9712263. CiteSeerX 10.1.1.241.1917. doi:10.1007/pl00009379. S2CID 6890001.
Pinwheel at the Tilings Encyclopedia
Dynamic Pinwheel made in GeoGebra
Retrieved from "https://en.wikipedia.org/w/index.php?title=Pinwheel_tiling&oldid=1066481125"
|
Fault-Tolerant Control for Networked Control Systems with Limited Information in Case of Actuator Fault
Wang Yan-feng, Wang Pei-liang, Li Zu-xin, Chen Hui-ying, "Fault-Tolerant Control for Networked Control Systems with Limited Information in Case of Actuator Fault", Mathematical Problems in Engineering, vol. 2015, Article ID 785289, 7 pages, 2015. https://doi.org/10.1155/2015/785289
Wang Yan-feng,1 Wang Pei-liang,1 Li Zu-xin,1 and Chen Hui-ying1
This paper is concerned with the problem of designing a fault-tolerant controller for uncertain discrete-time networked control systems against actuator possible fault. The step difference between the running step and the time stamp of the used plant state is modeled as a finite state Markov chain of which the transition probabilities matrix information is limited. By introducing actuator fault indicator matrix, the closed-loop system model is obtained by means of state augmentation technique. The sufficient conditions on the stochastic stability of the closed-loop system are given and the fault-tolerant controller is designed by solving a linear matrix inequality. A numerical example is presented to illustrate the effectiveness of the proposed method.
Networked control systems (NCSs) are used in many fields such as remote surgery and unmanned aerial vehicles especially in a number of emerging engineering applications such as arrays of microactuators and even neurobiological and socialeconomical systems [1–3]. Compared with the traditional wiring, the communication channels can simplify the installation and reduce the costs of cables and maintenance of the system. However, the network in the control systems also brings many problems, such as network-induced delay and packet dropout, and makes system analysis more challenging [4, 5]. Network-induced delays can degrade the performance of control systems designed without considering them and even destabilize the system [6, 7].
Because of the complexity caused by network, NCSs are more vulnerable to faults. An effective way to increase the reliability of the NCSs is to introduce fault-tolerant control (FTC). Therefore, the research on fault-tolerant control of NCSs has great theoretical and applied significance; however research on FTC for NCSs is different from that for traditional control systems in many aspects [8, 9]. In [10], a fault estimator was proposed for NCSs with transfer delays, process noise, and model uncertainty. On the basis of the information on fault estimator, a fault-tolerant controller using sliding mode control theory was designed to recover the system performance. In [11], the random packet dropout and the sensor or actuator failure were described as binary random variables; the sufficient condition for asymptotical mean-square stability of the NCSs was derived. By using matrix measure technique, a fault-tolerant controller was designed for NCSs with network-induced delay and model uncertainty in [12]. In [13], a FTC algorithm considering actuator failure of an NCS was presented, and the NCS with data packet dropout was modeled as an asynchronous dynamical system. Based on information scheduling, FTC design methods were proposed for NCSs with communication constrains in [14]. In [15], the problem of fault-tolerant control for NCSs with data packet dropout is studied and the closed-loop system was modeled as Markov jump system. However, elements of transition probabilities matrix are assumed to be completely known and the controller can not be solved by LMIs. To the best of the authors’ knowledge, up to now, very limited efforts have been devoted to studying FTC for uncertain NCSs with uncertain transition probability matrices, which motivates our investigation.
Problems of partial sensors inactivation are equal to problems of data pack dropout which can be solved by common technique; we focus on the problems of reliability when actuators are inactivated in this paper.
In this paper, the step difference between the running step and the time stamp of the used plant state is modeled as a finite state Markov chain. And the information of the transition probabilities matrix is limited; that is, a part of elements of transition probabilities matrix is unknown. The closed-loop system model is obtained by means of state augmentation technique and the mode-dependent fault-tolerant controller is designed which guarantees the stochastic stability of the closed-loop system.
This paper is organized as follows. In Section 2, we formulate the state feedback controller design problem. In Section 3, the sufficient conditions to guarantee the stochastic stability are presented, and the fault-tolerant controller is also given. A simulation example is used to illustrate the effectiveness of the proposed method in Section 4. The conclusion remarks are addressed in Section 5.
Consider the NCSs setup in Figure 1, in which the controllers are placed in a remote location, and both sensor measurement data and control data are transmitted through network.
Structure of networked control system.
By adding a buffer to the actuator, the delay from sensor to controller and the delay from controller to actuator can be lumped together, and the new variable is described as which is modeled as a Markov chain. And denotes the step difference between the running step and the time stamp of the used plant state, and it depends on the random time-delay and the data packet drops on the random communication delay and the data packet dropout [16]. Assume that both time-delay and the data packet dropout are bounded, so is bounded. The step delay takes values in and the transition probability matrix of is . That is, jumps from mode to with probability which is defined by , where , , . The set contains modes of , and the transition probabilities of the jumping process in this paper are considered to be partly accessed; that is, some elements in matrix are unknown. For example, for the time-delay with 3 modes, the transition probabilities matrix may be as follows:where “” represents the inaccessible elements. For notational clarity, , we denote with Moreover, if , it is further described as , , where represents the th known element with the index in the th row of the matrix . And is described as , where represents the th unknown element with the index th in the th row of the matrix .
Assume that the model of the plant is an uncertain discrete-time system as follows: where is state vector and is the control input. and are all real constant matrices. , where is an uncertain time-varying matrix satisfying the bound , where denotes the identity matrix with appropriate dimension.
Considering the effect of the random communication delay and the data packet dropout, we describe the state feedback control law asThe fault indicator matrix is given bywith for and means the actuator experiences a total failure, whereas the actuator is in healthy state when . Since there are actuators, the set of possible related failure modes is finite and is denoted by with elements, where is a particular pattern of matrix .
Consequently, the closed-loop system from (3) and (4) can be expressed asAt sampling time , if we augment the state-variable as , the closed-loop system (6) can be written aswherehas all elements being zeros except for the block being identity.
It can be seen that the closed-loop system (7) is a jump linear system with different modes.
It is noticed that and whereThroughout this paper, we use the following definition.
Definition 1. System (7) is stochastically stable if for every finite and initial mode there exists a finite such that the following holds:
The object of this paper is to construct a fault-tolerant controller with structure as given by (4) which achieves that the closed-loop system (7) is stochastically stable under all actuator failure modes. In the following, and for this paper are denoted as and , respectively.
To proceed, we will need the following two lemmas.
Lemma 2 (see [17]). Given matrices , , and of appropriate dimensions and is symmetric, holds for all satisfying if and only if there exists a scalar such that .
Lemma 3 (see [18]). The matrix is of full-array rank; then there exist two orthogonal matrices and , such that , where , where are nonzero singular values of . If matrix has the following structurethere exists a nonsingular matrix such that , where , .
With Definition 1, the sufficient conditions on the stochastic stability of the closed-loop system (7) can be obtained.
Theorem 4. The closed-loop system (7) with partly unknown transition probabilities (2) is stochastically stable if there exists matrix , such thatwhere .
Proof. For the closed-loop system (7), consider the quadratic function which is given byThen,Hence, if (12) and (13) hold, . One haswhere ; hence one can get . According to Definition 1, system (7) is stochastically stable.
Clearly, no knowledge on , is needed in (12) and (13), which completes the proof.
Theorem 5. Consider system (7) with partly unknown transition probabilities (2). If there exist matrices , and positive scalars , , , and such thatwherethen there exists a mode-dependent controller of the form (4) such that the resulting system (7) is stochastically stable. Furthermore, an admissible controller is given by
Proof. According to Theorem 4, we know that the system (7) is stochastically stable with the partly unknown transition probabilities (2) if inequalities (12) and (13) hold. By Schur complement, inequality (12) is equivalent towhere , .
By Lemma 2, there exists a scalar such that where Using Schur complement and Lemma 2 again, one can getSimilarly, from (13) one can obtainPerforming a congruence transformation to (23) and (24) by , setting , one can obtain (25) and (26), respectively. One hasFor the matrix of full-column rank, there always exist two orthogonal matrices and such that where , are nonzero singular values of . Assume that the matrix has the following structure: according to Lemma 3, there exists matrix such that , setting Since , one can get which implies that Thus, (19) is obtained from (29) and (31), which completes the proof.
In this section, a numerical example is given to show the validity and potential of our developed theoretical results. The dynamics are described as follows:Assume the time-delay takes values from and the transition probabilities matrix is . When the first actuator experiences a total failure, that is, the fault indicator matrix , the fault-tolerant and delay-dependent controller gain is solved from Theorem 5 as follows: When the second actuator experiences a total failure while the first actuator works normally, that is, the fault indicator matrix , the controller gain is solved as follows: When both actuators work normally, that is, the fault indicator matrix , the controller gain is solved asZero-input responses of states , are shown in Figures 2 and 3 when .
Zero-input response of .
The curves of zero-input response states , show that the NCS with partly unknown transition probabilities is stochastically stable against actuator possible fault.
This paper is concerned with the problem of fault-tolerant control for uncertain discrete-time networked systems against actuator possible fault. The time-delay is modeled as a finite state Markov chain and the Markov chain’s transition probabilities the information is limited. The closed-loop system is established through the state augmentation technique and the state feedback controller is designed which guarantees the stability of the resulting closed-loop systems. It is shown that the controller design problem under consideration is solvable if a set of LMIs is feasible. Simulation results show that the closed-loop systems are stochastically stable against actuator fault.
This study is supported by the National Natural Science Foundation of China under Grant no. 61174029, the National Natural Science Foundation of China under Grant no. 61503136, the Zhejiang Provincial Natural Science Foundation of China under Grant no. LY12F03008, and the Huzhou Natural Science Foundation of China under Grant no. 2014YZ07.
{H}_{\infty }
J. Wu, H. R. Karimi, and P. Shi, “Network-based
{H}_{\infty }
output feedback control for uncertain stochastic systems,” Information Sciences, vol. 232, pp. 397–410, 2013. View at: Publisher Site | Google Scholar | MathSciNet
{H}_{\infty }
controller design of networked control systems with markov packet dropouts,” IEEE Transactions on Systems, Man, and Cybernetics Part A: Systems, vol. 43, no. 3, pp. 689–697, 2013. View at: Publisher Site | Google Scholar
X. Li and X. B. Wu, “Guaranteed cost fault-tolerant controller design of networked control systems under variable-period sampling,” Information Technology Journal, vol. 8, no. 4, pp. 537–543, 2009. View at: Publisher Site | Google Scholar
S. X. Ding, P. Zhang, C. I. Chihaia, W. Li, Y. Wang, and E. L. Ding, “Advanced design scheme for fault tolerant distributed networked control systems,” in Proceedings of the 17th IFAC World Congress, pp. 13569–13574, Seoul, Republic of Korea, 2008. View at: Google Scholar
Z. H. Mao and B. Jiang, “Fault estimation and accommodation for networked control systems with transfer delay,” Acta Automatica Sinica, vol. 33, no. 7, pp. 738–743, 2007. View at: Publisher Site | Google Scholar | MathSciNet
X. Qi, C. Zhang, and J. Gu, “Robust fault-tolerant control for uncertain networked control systems with state-delay and random data packet dropout,” Journal of Control Science and Engineering, vol. 2012, Article ID 734758, 7 pages, 2012. View at: Publisher Site | Google Scholar
D. M. Kong and H. J. Fang, “Stable fault-tolerance control for a class of networked control systems,” Acta Automatica Sinica, vol. 31, no. 2, pp. 267–273, 2005. View at: Google Scholar | MathSciNet
Z. H. Huo and Z. X. Zhang, “Research on fault-tolerant control for NCS with data packet dropout,” in Proceedings of the 2nd International Symposium on Systems and Control in Aerospace and Astronautics (ISSCAA '08), pp. 1–4, IEEE, Shenzhen, China, December 2008. View at: Publisher Site | Google Scholar
S. Klinkhieo, C. Kambhampati, and R. J. Patton, “Fault tolerant control in NCS medium access constraints,” in Proceedings of the IEEE International Conference on Networking, Sensing and Control (ICNSC '07), pp. 416–423, IEEE, London, UK, April 2007. View at: Publisher Site | Google Scholar
D. X. Xie, D. F. Zhang, G. Huang, and Z. H. Wang, “Robust fault-tolerant control for a class of uncertain networked control systems,” Information and Control, vol. 39, no. 4, pp. 472–478, 2010. View at: Google Scholar
Y. Wang, L. Xie, and C. E. de Souza, “Robust control of a class of uncertain nonlinear systems,” Systems & Control Letters, vol. 19, no. 2, pp. 139–149, 1992. View at: Publisher Site | Google Scholar
{H}_{\infty }
Copyright © 2015 Wang Yan-feng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
|
Clay Pipe - Ring of Brodgar
Skill(s) Required Pottery
Object(s) Required Unfired Clay Pipe
Produced By Kiln
Craft > Clothes & Equipment > Xtras & Accessories > Clay Pipe
Clay Pipe can be equipped in the alternate head slot. In order to acquire Smoking Buffs Restless Hearthling can fill Clay Pipe with Cured Hemp Buds, Opium or Pipestuff and light up, causing the character's pipe to emit smoke for 5 minutes, as well as turning them into a small light source.
Smoking produces two distinct buffs for your character. The longer the character smokes, the more potent the buff is. each 0.01kg of substance smoked takes 1.5 minutes and adds 10% to your buff meter, however, your character loses roughly 4% buff per minute, so the character must smoke at least 30 minutes and 0.20kg before recieving a full 100% buff.
To get Crazy High on Hempstuff you need three cured hemp buds. You will probably not need to smoke all of the third one. Remove the pipe from your equipment and it will go out and the remainder of the cured hemp bud will stay in the pipe for you to use later.
To make a Clay Pipe, you need to fire an Unfired Clay Pipe in a Kiln:
Use the craft menu to mold the clay, and then fire the unburnt Clay Pipe in a Kiln with 5 branches loaded.
After 21 minutes you will get an empty Pipe.
Discovering both clay and Pipestuff is required in order to learn how to craft the Clay Pipe.
To use a pipe, first you will need Cured Hemp Buds, or Opium or Pipestuff. You will then need to right click the pipe with pipestuff, opium or cured hemp to fill the pipe. Once the pipe is full, equip it in your necklace slot and then light it using a Firebrand or Torch. You can refill the pipe while it is lit, so you do not have to light it multiple times.
You cannot sleep off the effects of Cured Hemp Buds, or Opium. Like stamina you need to be logged in for it to reduce so you will need to wait it out. Be aware that your display will change to produce the experience of being high, with color changes and moving visual abnormalities which will make clicking on objects difficult.
Unfired Clay Pipe Quality =
{\displaystyle {\frac {_{q}Clay*3+_{q}PottersWheel}{4}}}
and is softcapped by Dexterity
Clay Pipe Quality =
{\displaystyle 2*_{q}Unburnt+_{q}Fuel+_{q}Kiln \over 4}
Retrieved from "https://ringofbrodgar.com/w/index.php?title=Clay_Pipe&oldid=89729"
|
Circular orbit - WikiMili, The Best Wikipedia Reader
Orbit with a fixed distance from the barycenter
Find sources: "Circular orbit" – news · newspapers · books · scholar · JSTOR (April 2020)
A circular orbit is depicted in the top-left quadrant of this diagram, where the gravitational potential well of the central mass shows potential energy, and the kinetic energy of the orbital speed is shown in red. The height of the kinetic energy remains constant throughout the constant speed circular orbit.
At the top of the diagram, a satellite in a clockwise circular orbit (yellow spot) launches objects of negligible mass:
(1 - blue) towards Earth,
(2 - red) away from Earth,
(3 - grey) in the direction of travel, and
(4 - black) backwards of the direction of travel.
Dashed ellipses are orbits relative to Earth. Solid curves are perturbations relative to the satellite: in one orbit, (1) and (2) return to the satellite having made a clockwise loop on either side of the satellite. Unintuitively, (3) spirals farther and farther behind whereas (4) spirals ahead.
Listed below is a circular orbit in astrodynamics or celestial mechanics under standard assumptions. Here the centripetal force is the gravitational force, and the axis mentioned above is the line through the center of the central mass perpendicular to the plane of motion.
Transverse acceleration (perpendicular to velocity) causes change in direction. If it is constant in magnitude and changing in direction with the velocity, circular motion ensues. Taking two derivatives of the particle's coordinates with respect to time gives the centripetal acceleration
{\displaystyle a\,={\frac {v^{2}}{r}}\,={\omega ^{2}}{r}}
{\displaystyle v\,}
{\displaystyle r\,}
is radius of the circle
{\displaystyle \omega \ }
is angular speed, measured in radians per unit time.
The formula is dimensionless, describing a ratio true for all units of measure applied uniformly across the formula. If the numerical value of
{\displaystyle \mathbf {a} }
is measured in meters per second per second, then the numerical values for
{\displaystyle v\,}
will be in meters per second,
{\displaystyle r\,}
in meters, and
{\displaystyle \omega \ }
in radians per second.
The speed (or the magnitude of velocity) relative to the central object is constant: [1] :30
{\displaystyle v={\sqrt {GM\! \over {r}}}={\sqrt {\mu \over {r}}}}
{\displaystyle G}
, is the gravitational constant
{\displaystyle M}
, is the mass of both orbiting bodies
{\displaystyle (M_{1}+M_{2})}
, although in common practice, if the greater mass is significantly larger, the lesser mass is often neglected, with minimal change in the result.
{\displaystyle \mu =GM}
, is the standard gravitational parameter.
The orbit equation in polar coordinates, which in general gives r in terms of θ, reduces to:[ clarification needed ][ citation needed ]
{\displaystyle r={{h^{2}} \over {\mu }}}
{\displaystyle h=rv}
is specific angular momentum of the orbiting body.
{\displaystyle \mu =rv^{2}}
{\displaystyle \omega ^{2}r^{3}=\mu }
Hence the orbital period (
{\displaystyle T\,\!}
) can be computed as: [1] :28
{\displaystyle T=2\pi {\sqrt {r^{3} \over {\mu }}}}
{\displaystyle T_{ff}={\frac {\pi }{2{\sqrt {2}}}}{\sqrt {r^{3} \over {\mu }}}}
(17.7% of the orbital period in a circular orbit)
{\displaystyle T_{par}={\frac {\sqrt {2}}{3}}{\sqrt {r^{3} \over {\mu }}}}
(7.5% of the orbital period in a circular orbit)
The fact that the formulas only differ by a constant factor is a priori clear from dimensional analysis.[ citation needed ]
The specific orbital energy (
{\displaystyle \epsilon \,}
) is negative, and
{\displaystyle \epsilon =-{v^{2} \over {2}}}
{\displaystyle \epsilon =-{\mu \over {2r}}}
Thus the virial theorem [1] :72 applies even without taking a time-average:[ citation needed ]
The escape velocity from any distance is √2 times the speed in a circular orbit at that distance: the kinetic energy is twice as much, hence the total energy is zero.[ citation needed ]
In Schwarzschild metric, the orbital velocity for a circular orbit with radius
{\displaystyle r}
{\displaystyle v={\sqrt {\frac {GM}{r-r_{S}}}}}
{\displaystyle \scriptstyle r_{S}={\frac {2GM}{c^{2}}}}
is the Schwarzschild radius of the central body.
For the sake of convenience, the derivation will be written in units in which
{\displaystyle \scriptstyle c=G=1}
{\displaystyle u^{\mu }=({\dot {t}},0,0,{\dot {\phi }})}
{\displaystyle \scriptstyle r}
is constant on a circular orbit, and the coordinates can be chosen so that
{\displaystyle \scriptstyle \theta ={\frac {\pi }{2}}}
). The dot above a variable denotes derivation with respect to proper time
{\displaystyle \scriptstyle \tau }
{\displaystyle \left(1-{\frac {2M}{r}}\right){\dot {t}}^{2}-r^{2}{\dot {\phi }}^{2}=1}
{\displaystyle {\ddot {x}}^{\mu }+\Gamma _{\nu \sigma }^{\mu }{\dot {x}}^{\nu }{\dot {x}}^{\sigma }=0}
The only nontrivial equation is the one for
{\displaystyle \scriptstyle \mu =r}
. It gives:
{\displaystyle {\frac {M}{r^{2}}}\left(1-{\frac {2M}{r}}\right){\dot {t}}^{2}-r\left(1-{\frac {2M}{r}}\right){\dot {\phi }}^{2}=0}
{\displaystyle {\dot {\phi }}^{2}={\frac {M}{r^{3}}}{\dot {t}}^{2}}
{\displaystyle \left(1-{\frac {2M}{r}}\right){\dot {t}}^{2}-{\frac {M}{r}}{\dot {t}}^{2}=1}
{\displaystyle {\dot {t}}^{2}={\frac {r}{r-3M}}}
Assume we have an observer at radius
{\displaystyle \scriptstyle r}
, who is not moving with respect to the central body, that is, their four-velocity is proportional to the vector
{\displaystyle \scriptstyle \partial _{t}}
. The normalization condition implies that it is equal to:
{\displaystyle v^{\mu }=\left({\sqrt {\frac {r}{r-2M}}},0,0,0\right)}
{\displaystyle \gamma =g_{\mu \nu }u^{\mu }v^{\nu }=\left(1-{\frac {2M}{r}}\right){\sqrt {\frac {r}{r-3M}}}{\sqrt {\frac {r}{r-2M}}}={\sqrt {\frac {r-2M}{r-3M}}}}
{\displaystyle v={\sqrt {\frac {M}{r-2M}}}}
{\displaystyle v={\sqrt {\frac {GM}{r-r_{S}}}}}
The Klein–Gordon equation is a relativistic wave equation, related to the Schrödinger equation. It is second-order in space and time and manifestly Lorentz-covariant. It is a quantized version of the relativistic energy–momentum relation. Its solutions include a quantum scalar or pseudoscalar field, a field whose quanta are spinless particles. Its theoretical relevance is similar to that of the Dirac equation. Electromagnetic interactions can be incorporated, forming the topic of scalar electrodynamics, but because common spinless particles like the pions are unstable and also experience the strong interaction the practical utility is limited.
In general relativity, Schwarzschild geodesics describe the motion of test particles in the gravitational field of a central fixed mass that is, motion in the Schwarzschild metric. Schwarzschild geodesics have been pivotal in the validation of Einstein's theory of general relativity. For example, they provide accurate predictions of the anomalous precession of the planets in the Solar System and of the deflection of light by gravity.
The Kerr–Newman metric is the most general asymptotically flat, stationary solution of the Einstein–Maxwell equations in general relativity that describes the spacetime geometry in the region surrounding an electrically charged, rotating mass. It generalizes the Kerr metric by taking into account the field energy of an electromagnetic field, in addition to describing rotation. It is one of a large number of various different electrovacuum solutions, that is, of solutions to the Einstein–Maxwell equations which account for the field energy of an electromagnetic field. Such solutions do not include any electric charges other than that associated with the gravitational field, and are thus termed vacuum solutions.
In general relativity, the metric tensor is the fundamental object of study. It may loosely be thought of as a generalization of the gravitational potential of Newtonian gravitation. The metric captures all the geometric and causal structure of spacetime, being used to define notions such as time, distance, volume, curvature, angle, and separation of the future and the past.
The effective potential combines multiple, perhaps opposing, effects into a single potential. In its basic form, it is the sum of the 'opposing' centrifugal potential energy with the potential energy of a dynamical system. It may be used to determine the orbits of planets and to perform semi-classical atomic calculations, and often allows problems to be reduced to fewer dimensions.
Alternatives to general relativity are physical theories that attempt to describe the phenomenon of gravitation in competition to Einstein's theory of general relativity. There have been many different attempts at constructing an ideal theory of gravity.
In classical mechanics, a Liouville dynamical system is an exactly soluble dynamical system in which the kinetic energy T and potential energy V can be expressed in terms of the s generalized coordinates q as follows:
In general relativity, Lense–Thirring precession or the Lense–Thirring effect is a relativistic correction to the precession of a gyroscope near a large rotating mass such as the Earth. It is a gravitomagnetic frame-dragging effect. It is a prediction of general relativity consisting of secular precessions of the longitude of the ascending node and the argument of pericenter of a test particle freely orbiting a central spinning mass endowed with angular momentum .
f(R) is a type of modified gravity theory which generalizes Einstein's general relativity. f(R) gravity is actually a family of theories, each one defined by a different function, f, of the Ricci scalar, R. The simplest case is just the function being equal to the scalar; this is general relativity. As a consequence of introducing an arbitrary function, there may be freedom to explain the accelerated expansion and structure formation of the Universe without adding unknown forms of dark energy or dark matter. Some functional forms may be inspired by corrections arising from a quantum theory of gravity. f(R) gravity was first proposed in 1970 by Hans Adolph Buchdahl. It has become an active field of research following work by Starobinsky on cosmic inflation. A wide range of phenomena can be produced from this theory by adopting different functions; however, many functional forms can now be ruled out on observational grounds, or because of pathological theoretical problems.
The pressuron is a hypothetical scalar particle which couples to both gravity and matter theorised in 2013. Although originally postulated without self-interaction potential, the pressuron is also a dark energy candidate when it has such a potential. The pressuron takes its name from the fact that it decouples from matter in pressure-less regimes, allowing the scalar-tensor theory of gravity involving it to pass solar system tests, as well as tests on the equivalence principle, even though it is fundamentally coupled to matter. Such a decoupling mechanism could explain why gravitation seems to be well described by general relativity at present epoch, while it could actually be more complex than that. Because of the way it couples to matter, the pressuron is a special case of the hypothetical string dilaton. Therefore, it is one of the possible solutions to the present non-observation of various signals coming from massless or light scalar fields that are generically predicted in string theory.
1 2 3 Lissauer, Jack J.; de Pater, Imke (2019). Fundamental Planetary Sciences : physics, chemistry, and habitability. New York, NY, USA: Cambridge University Press. p. 604. ISBN 9781108411981.
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.