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Form Factor:Ellipsoid of revolution - GISAXS
Form Factor:Ellipsoid of revolution
From GISAXS
An ellipsoid of revolution is a 'squashed' or 'stretched' sphere; technically an oblate or prolate spheroid, respectively.
1.1 Form Factor Amplitude
1.2 Isotropic Form Factor Intensity
2.1 NCNR
2.2 Pedersen
2.3 IsGISAXS
2.4 Sjoberg Monte Carlo Study
4 Approximating by a Sphere
For an ellipsoid of revolution, the size ('radius') along one direction will be distinct, whereas the other two directions will be identical. Assume an ellipsoid aligned along the z-direction (rotation about z-axis, i.e. sweeping the
{\displaystyle \phi }
angle in spherical coordinates), such that the size in the xy-plane is
{\displaystyle R_{r}}
and along z is
{\displaystyle R_{z}=\epsilon R_{r}}
. A useful quantity is
{\displaystyle R_{\theta }}
, which is the distance from the origin to the surface of the ellipsoid for a line titled at angle
{\displaystyle \theta }
with respect to the z-axis. This is thus the half-distance between tangent planes orthogonal to a vector pointing at the given
{\displaystyle \theta }
angle, and provides the 'effective size' of the scattering object as seen by a q-vector pointing in that direction.
{\displaystyle {\begin{alignedat}{2}R_{\theta }&={\sqrt {R_{z}^{2}\cos ^{2}\theta +R_{r}^{2}(1-\cos ^{2}\theta )}}\\&=R_{r}{\sqrt {1+(\epsilon ^{2}-1)\cos ^{2}\theta }}\\&=R_{r}{\sqrt {\sin ^{2}\theta +\epsilon ^{2}\cos ^{2}\theta }}\end{alignedat}}}
The ellipsoid is also characterized by:
{\displaystyle V_{ell}={\frac {4\pi }{3}}R_{z}R_{r}^{2}={\frac {4\pi }{3}}\epsilon R_{r}^{3}}
Form Factor Amplitude
{\displaystyle F_{ell}(\mathbf {q} )=\left\{{\begin{array}{c l}3\Delta \rho V_{ell}{\frac {\sin(qR_{\theta })-qR_{\theta }\cos(qR_{\theta })}{(qR_{\theta })^{3}}}&\mathrm {when} \,\,q\neq 0\\\Delta \rho V_{ell}&\mathrm {when} \,\,q=0\\\end{array}}\right.}
Isotropic Form Factor Intensity
{\displaystyle P_{ell}(q)=\left\{{\begin{array}{c l}18\pi \Delta \rho ^{2}V_{ell}^{2}\int _{0}^{\pi }\left({\frac {\sin(qR_{\theta })-qR_{\theta }\cos(qR_{\theta })}{(qR_{\theta })^{3}}}\right)^{2}\sin \theta \mathrm {d} \theta &\mathrm {when} \,\,q\neq 0\\4\pi \Delta \rho ^{2}V_{ell}^{2}&\mathrm {when} \,\,q=0\\\end{array}}\right.}
From NCNR SANS Models documentation:
{\displaystyle {\begin{alignedat}{2}P(q)&={\frac {\rm {scale}}{V_{ell}}}(\rho _{ell}-\rho _{solv})^{2}\int _{0}^{1}f^{2}[qr_{b}(1+x^{2}(v^{2}-1))^{1/2}]dx+bkg\\f(z)&=3V_{ell}{\frac {(\sin z-z\cos z)}{z^{3}}}\\V_{ell}&={\frac {4\pi }{3}}r_{a}r_{b}^{2}\\v&={\frac {r_{a}}{r_{b}}}\\\end{alignedat}}}
{\displaystyle {\rm {scale}}}
: Intensity scaling
{\displaystyle r_{a}}
: rotation axis (Å)
{\displaystyle r_{b}}
: orthogonal axis (Å)
{\displaystyle \rho _{ell}-\rho _{solv}}
: scattering contrast (Å−2)
{\displaystyle {\rm {background}}}
: incoherent background (cm−1)
From Pedersen review, Analysis of small-angle scattering data from colloids and polymer solutions: modeling and least-squares fitting Jan Skov Pedersen, Advances in Colloid and Interface Science 1997, 70, 171. doi: 10.1016/S0001-8686(97)00312-6
{\displaystyle {\begin{alignedat}{2}&P(q,R,\epsilon )=\int _{0}^{\pi /2}F_{sphere}^{2}[q,r(R,\epsilon ,\alpha )]\sin \alpha d\alpha \\&r(R,\epsilon ,\alpha )=R\left(\sin ^{2}\alpha +\epsilon ^{2}\cos ^{2}\alpha \right)^{1/2}\end{alignedat}}}
{\displaystyle F_{sphere}={\frac {3\left[\sin(qr)-qr\cos(qr)\right]}{(qr)^{3}}}}
{\displaystyle R}
: radius (Å)
{\displaystyle \epsilon R}
: orthogonal size (Å)
IsGISAXS
From IsGISAXS, Born form factors:
{\displaystyle F_{ell}(\mathbf {q} ,R,W,H,\alpha )=2\pi RWH{\frac {J_{1}(\gamma )}{\gamma }}\sin _{c}(q_{z}H/2)\exp(iq_{z}H/2)}
{\displaystyle \gamma ={\sqrt {(q_{x}R)^{2}+(q_{y}W)^{2}}}}
{\displaystyle V_{ell}=\pi RWH,\,S_{anpy}=\pi RW,\,R_{anpy}=Max(R,W)}
Where J is a Bessel function:
{\displaystyle J_{1}(\gamma )={\frac {1}{\pi }}\int _{0}^{\pi }\cos(\tau -x\sin \tau )\,\mathrm {d} \tau }
Sjoberg Monte Carlo Study
From Small-angle scattering from collections of interacting hard ellipsoids of revolution studied by Monte Carlo simulations and other methods of statistical thermodynamics, Bo Sjöberg, J.Appl. Cryst. (1999), 32, 917-923. doi 10.1107/S0021889899006640
{\displaystyle F(\mathbf {q} )=3{\frac {\sin(qs)-qs\cos(qs)}{(qs)^{3}}}}
{\displaystyle s=\left[a^{2}\cos ^{2}\gamma +b^{2}(1-\cos ^{2}\gamma )\right]^{1/2}}
{\displaystyle \gamma }
{\displaystyle \mathbf {q} }
and the a-axis vector of the ellipsoid of revolution (which also has axes b = c);
{\displaystyle \cos \gamma }
is the inner product of unit vectors parallel to
{\displaystyle \mathbf {q} }
and the a-axis. In some sense, s is the 'equivalent size' of a sphere that would lead to the scattering for a particular
{\displaystyle \mathbf {q} }
: it is half the distance between the tangential planes that bound the ellipsoid, perpendicular to the
{\displaystyle \mathbf {q} }
{\displaystyle a=\epsilon b}
{\displaystyle {\begin{alignedat}{2}s&=\left[a^{2}\cos ^{2}\gamma +b^{2}(1-\cos ^{2}\gamma )\right]^{1/2}\\&=\left[b^{2}\epsilon ^{2}\cos ^{2}\gamma +b^{2}(1-\cos ^{2}\gamma )\right]^{1/2}\\&=b\left[\epsilon ^{2}\cos ^{2}\gamma +(1-\cos ^{2}\gamma )\right]^{1/2}\\&=b\left[\epsilon ^{2}\cos ^{2}\gamma +\sin ^{2}\gamma \right]^{1/2}\\&=b\left[1+(\epsilon ^{2}-1)\cos ^{2}\gamma \right]^{1/2}\end{alignedat}}}
For an ellipsoid oriented along the z-axis, we denote the size in-plane (in x and y) as
{\displaystyle R_{r}}
and the size along z as
{\displaystyle R_{z}=\epsilon R_{r}}
{\displaystyle \epsilon }
denotes the shape of the ellipsoid:
{\displaystyle \epsilon =1}
{\displaystyle \epsilon <1}
for an oblate spheroid and
{\displaystyle \epsilon >1}
for a prolate spheroid. The volume is thus:
{\displaystyle V_{ell}={\frac {4\pi }{3}}R_{z}R_{r}^{2}={\frac {4\pi }{3}}\epsilon R_{r}^{3}}
We also note that the cross-section of the ellipsoid (an ellipse) will have coordinates
{\displaystyle (r_{xy},z)}
{\displaystyle r_{xy}}
is a distance in the xy-plane):
{\displaystyle {\begin{alignedat}{2}r_{xy}&=R_{r}\sin \theta \\z&=R_{z}\cos \theta =\epsilon R_{r}\cos \theta \end{alignedat}}}
{\displaystyle \theta }
is the angle with the z-axis. This lets us define a useful quantity,
{\displaystyle R_{\theta }}
, which is the distance to the point from the origin:
{\displaystyle {\begin{alignedat}{2}R_{\theta }&={\sqrt {(R_{r}\sin \theta )^{2}+(R_{z}\cos \theta )^{2}}}\\&={\sqrt {R_{r}^{2}\sin ^{2}\theta +\epsilon ^{2}R_{r}^{2}\cos ^{2}\theta }}\\&=R_{r}{\sqrt {\sin ^{2}\theta +\epsilon ^{2}\cos ^{2}\theta }}\\\end{alignedat}}}
The form factor is:
{\displaystyle {\begin{alignedat}{2}F_{ell}(\mathbf {q} )&=\int \limits _{V}e^{i\mathbf {q} \cdot \mathbf {r} }\mathrm {d} \mathbf {r} \\&=\int _{\phi =0}^{2\pi }\int _{\theta =0}^{\pi }\int _{r=0}^{R_{\theta }}e^{i\mathbf {q} \cdot \mathbf {r} }r^{2}\mathrm {d} r\sin \theta \mathrm {d} \theta \mathrm {d} \phi \\&=2\pi \int _{0}^{\pi }\left[\int _{0}^{R_{\theta }}e^{i\mathbf {q} \cdot \mathbf {r} }r^{2}\mathrm {d} r\right]\sin \theta \mathrm {d} \theta \\\end{alignedat}}}
Imagine instead that we compress/stretch the z dimension so that the ellipsoid becomes a sphere:
{\displaystyle {\begin{alignedat}{2}x^{\prime }&=x\\y^{\prime }&=y\\z^{\prime }&=zR_{r}/R_{z}=z/\epsilon \\r^{\prime }&=\left|\mathbf {r} ^{\prime }\right|=r{\frac {R_{r}}{R_{\gamma }}}\\\mathrm {d} V&=\mathrm {d} x\mathrm {d} y\mathrm {d} z=\mathrm {d} x^{\prime }\mathrm {d} y^{\prime }\epsilon \mathrm {d} z^{\prime }=\epsilon \mathrm {d} V^{\prime }\end{alignedat}}}
This implies a coordinate transformation for the
{\displaystyle \mathbf {q} }
-vector of:
{\displaystyle {\begin{alignedat}{2}q_{x}^{\prime }&=q_{x}\\q_{y}^{\prime }&=q_{y}\\q_{z}^{\prime }&=q_{z}R_{z}/R_{r}=q_{z}\epsilon \\q^{\prime }&=\left|\mathbf {q} ^{\prime }\right|=q{\frac {R_{\gamma }}{R_{r}}}\end{alignedat}}}
{\displaystyle R_{\gamma }}
{\displaystyle R_{\theta }}
relation for a q-vector tilted at angle
{\displaystyle \gamma }
with respect to the z axis. Considered in this way, the integral reduces to the form factor for a sphere. In effect, a particular
{\displaystyle \mathbf {q} }
vector sees a sphere-like scatterer with size (length-scale) given by
{\displaystyle R_{\gamma }}
{\displaystyle {\begin{alignedat}{2}F_{ell}(\mathbf {q} )&=\epsilon \int _{\phi =0}^{2\pi }\int _{\theta =0}^{\pi }\int _{r^{\prime }=0}^{R_{r}}e^{i\mathbf {q} ^{\prime }\cdot \mathbf {r} ^{\prime }}r^{\prime 2}\mathrm {d} r^{\prime }\sin \theta \mathrm {d} \theta \mathrm {d} \phi \\&=3\left({\frac {4\pi }{3}}\epsilon R_{r}^{3}\right){\frac {\sin(q^{\prime }R_{r})-q^{\prime }R_{r}\cos(q^{\prime }R_{r})}{(q^{\prime }R_{r})^{3}}}\end{alignedat}}}
We can then convert back:
{\displaystyle {\begin{alignedat}{2}F_{ell}(\mathbf {q} )&=3V_{ell}{\frac {\sin(qR_{\gamma })-qR_{\gamma }\cos(qR_{\gamma })}{(qR_{\gamma })^{3}}}\end{alignedat}}}
To average over all possible orientations, we use:
{\displaystyle {\begin{alignedat}{2}P_{ell}(q)&=\int _{\phi =0}^{2\pi }\int _{\theta =0}^{\pi }|F_{ell}(\mathbf {q} )|^{2}\sin \theta \mathrm {d} \theta \mathrm {d} \phi \\&=\int _{0}^{2\pi }\int _{0}^{\pi }\left|3V_{ell}{\frac {\sin(qR_{\theta })-qR_{\theta }\cos(qR_{\theta })}{(qR_{\theta })^{3}}}\right|^{2}\sin \theta \mathrm {d} \theta \mathrm {d} \phi \\&=9V_{ell}^{2}\int _{0}^{2\pi }\mathrm {d} \phi \int _{0}^{\pi }\left({\frac {\sin(qR_{\theta })-qR_{\theta }\cos(qR_{\theta })}{(qR_{\theta })^{3}}}\right)^{2}\sin \theta \mathrm {d} \theta \\&=18\pi V_{ell}^{2}\int _{0}^{\pi }\left({\frac {\sin(qR_{\theta })-qR_{\theta }\cos(qR_{\theta })}{(qR_{\theta })^{3}}}\right)^{2}\sin \theta \mathrm {d} \theta \end{alignedat}}}
Approximating by a Sphere
One can approximate a spheroid using an isovolumic sphere of radius Reffective:
{\displaystyle V_{ell}={\frac {4\pi }{3}}R_{z}R_{r}^{2}}
{\displaystyle {\begin{alignedat}{2}R_{\mathrm {effective} }&=\left({\frac {3V_{ell}}{4\pi }}\right)^{1/3}\\&=(R_{z}R_{r}^{2})^{1/3}\\\end{alignedat}}}
Retrieved from "http://gisaxs.com/index.php?title=Form_Factor:Ellipsoid_of_revolution&oldid=6028"
|
The dynamics of $\operatorname{Aut}(F_n)$ on redundant representations | EMS Press
\operatorname{Aut}(F_n)
on redundant representations
Tsachik Gelander
We study some dynamical properties of the canonical
\mathrm{Aut}(F_n)
-action on the space
\mathcal{R}_n(G)
of redundant representations of the free group
F_n
G
G
is the group of rational points of a simple algebraic group over a local field. We show that this action is always minimal and ergodic, confirming a conjecture of A. Lubotzky. On the other hand for the classical cases where
G=\mathrm{SL}_2(\mathbb{R})
\mathrm{SL}_2(\mathbb{C})
we show that the action is not weak mixing, in the sense that the diagonal action on
\mathcal{R}_n(G)^2
is not ergodic.
Tsachik Gelander, Yair N. Minsky, The dynamics of
\operatorname{Aut}(F_n)
on redundant representations. Groups Geom. Dyn. 7 (2013), no. 3, pp. 557–576
|
Stepped-Coupon Bonds - MATLAB & Simulink - MathWorks América Latina
Cash Flows from Stepped-Coupon Bonds
Price and Yield of Stepped-Coupon Bonds
A stepped-coupon bond has a fixed schedule of changing coupon amounts. Like fixed coupon bonds, stepped-coupon bonds could have different periodic payments and accrual bases.
The functions stepcpnprice and stepcpnyield compute prices and yields of such bonds. An accompanying function stepcpncfamounts produces the cash flow schedules pertaining to these bonds.
Consider a bond that has a schedule of two coupons. Suppose that the bond starts out with a 2% coupon that steps up to 4% in 2 years and onward to maturity. Assume that the issue and settlement dates are both March 15, 2003. The bond has a 5-year maturity. Use stepcpncfamounts to generate the cash flow schedule and times.
Settle = datenum('15-Mar-2003');
Maturity = datenum('15-Mar-2008');
ConvDates = [datenum('15-Mar-2005')];
CouponRates = [0.02, 0.04];
[CFlows, CDates, CTimes] = stepcpncfamounts(Settle, Maturity, ...
ConvDates, CouponRates)
CFlows =
0 1 1 1 1 2 2 2 2 2 102
CDates =
CTimes =
Notably, ConvDates has one less element than CouponRates because MATLAB® software assumes that the first element of CouponRates indicates the coupon schedule between Settle (March 15, 2003) and the first element of ConvDates (March 15, 2005), shown diagrammatically below.
Pay 2% from March 15, 2003
Effective 2% on March 15, 2003
Semiannual Coupon Payment
The payment on March 15, 2005 is still a 2% coupon. Payment of the 4% coupon starts with the next payment, September 15, 2005. March 15, 2005 is the end of first coupon schedule, not to be confused with the beginning of the second.
In summary, MATLAB takes user input as the end dates of coupon schedules and computes the next coupon dates automatically.
The payment due on settlement (zero in this case) represents the accrued interest due on that day. It is negative if such amount is nonzero. Comparison with cfamounts in Financial Toolbox™ shows that the two functions operate identically.
The toolbox provides two basic analytical functions to compute price and yield for stepped-coupon bonds. Using the above bond as an example, you can compute the price when the yield is known.
You can estimate the yield to maturity as a number-of-year weighted average of coupon rates. For this bond, the estimated yield is:
\frac{\left(2×2\right)+\left(4×3\right)}{5}
or 3.33%. While definitely not exact (due to nonlinear relation of price and yield), this estimate suggests close to par valuation and serves as a quick first check on the function.
Yield = 0.0333;
[Price, AccruedInterest] = stepcpnprice(Yield, Settle, ...
Maturity, ConvDates, CouponRates)
AccruedInterest =
The price returned is 99.2237 (per $100 notional), and the accrued interest is zero, consistent with our earlier assertions.
To validate that there is consistency among the stepped-coupon functions, you can use the above price and see if indeed it implies a 3.33% yield by using stepcpnyield.
YTM = stepcpnyield(Price, Settle, Maturity, ConvDates, ...
CouponRates)
stepcpnprice | stepcpnyield | stepcpncfamounts
|
Childhood Game | Toph
About twenty years ago, when we had only 16MB RAMs and 5GB hard disks, playing games might seem like a difficult thing to do. But luckily we did not require very high end hardware back then to play some amazingly engaging games. One of these games are called, the "Brick Buster".
The idea behind Brick Buster or similar games was like this: You have a bouncing ball and a tray inside of a rectangular grid. The grid is closed on the three sides: left, right and top. The bottom side is empty. When the ball keeps bouncing inside the grid, it hits different walls on the left, right or top. After hitting the walls, the ball just changes it's direction, but keeps moving at it's initial speed. But when it reaches the bottom of the grid, it falls off and the game ends. A tray floats near the bottom of the grid, which a player can move to left or right of the screen. The purpose of the player is to move his tray in such a way, that the ball does not fall off the grid. Additionally, there are some bricks near the top of the board, you have to break those bricks by hitting the bricks with the bouncing balls.
Figure: Picture of a brick breaking game named "Breakout". Image credit: Wilinckx
Your are cranking up a new company and want to reproduce this game for portable devices. You thought to yourself, well, this is easy. But soon you figured, it is not. Time is running out, and meeting with your investor is coming near. As a result, you have decided to create an easier version of the game.
In your "easier" game, there are no bricks, no trays, not even an empty side at the bottom of the grid. You have a 500 * 500 board, where the upper-left corner has the coordinate (0,0). There are just some moving balls, that keeps bouncing all over the board. Initially, every ball has a specific position and a fixed velocity. Whenever a ball hits the wall, it just changes its direction by the following rules:
If the ball hits the left or the right wall, then only the direction towards the x axis changes to the opposite.
If the ball hits the top or the bottom wall, then only the direction towards the y axis changes to the opposite.
A collision between two balls takes place when the distance between the two balls is less than or equal to the sum of their radius. When two balls collide, they change their direction by following these rules:
If two balls have been moving in the same direction in terms of x-axis, their movement towards the x-axis will not change after the collision. Otherwise, both of the balls movement towards the x-axis will change to the opposite.
If two balls have been moving in the same direction in terms of y-axis, their movement towards the y-axis will not change after the collision. Otherwise, both of the balls' movement towards the y-axis will change to the opposite.
Now, given some balls and their initial velocity, you have to determine the position of the balls after some certain number of seconds.
You can apply the following assumptions for your calculations:
The game only renders one frame every second. The transition from one frame to the next is discrete and the information between two consecutive frames are lost.
If more than two balls get collided with each other at the end of a second, then the collisions must be processed sequentially in the lexicographical order of the balls. If three balls B1, B3 and B5 collide together at the end of a second, then the collisions must be processed in this order: (B1,B3), (B1,B5), (B3,B5). Here, 1, 3 and 5 are the indexes of the balls in their input sequence.
If a ball is in collision with the other balls and walls at the same time, then the collision with the balls will be calculated first.
Every ball starts moving at the 0-th second. After the end of a second, collisions will be calculated.
If a ball touches or exceeds the boundary of a wall at the end of a second, then the ball can be said to be collided with the wall.
You can safely assume that initially no ball is in collision with other balls or the wall.
Input starts with
B (
1 ≤ B ≤ 50
1≤B≤50), the number of balls. Then follow
B lines, each having two integers
X and
Y (
6 ≤ X, Y ≤ 494
6≤X,Y≤494), the initial position of the center of the balls. No balls will be at the same place in the beginning. For simplicity, assume that each of the balls is a circle with a radius of 5. Then follow another
B lines, each with two numbers
V_x
Vx and
V_y
Vy (
|V_x|=1
∣Vx∣=1,
|V_y|=1
∣Vy∣=1), the original velocity towards the
x axis and the
y axis for each of the ball.
In the last line of input, you will be given an integer
T (
T≤10000), the number of seconds from the start of the game, where you need to report the position of the balls.
For each case, output
B lines, one for each ball. In each line, you will have two real numbers
X_i
Y_i
Yi, the position of the
i-th ball (according to the input sequence) after
T seconds. Your answer needs to be accurate for 2 decimal places.
moiddaEarliest, May '19
OnikJahanSagorFastest, 0.0s
moiddaLightest, 131 kB
SuperShubhroShortest, 1169B
Replay of SUST Inter University Programming Contest 2019
SUST Inter University Programming Contest 2019
|
Ground Motion from M 1.5 to 3.8 Induced Earthquakes at Hypocentral Distance < 45 km in the Montney Play of Northeast British Columbia, Canada | Seismological Research Letters | GeoScienceWorld
Pacific Geoscience Center, Geological Survey of Canada, 9860 West Saanich Road, Sidney, British Columbia, Canada V8L 4B2
Alireza Babaie Mahani, Honn Kao; Ground Motion from
M
1.5 to 3.8 Induced Earthquakes at Hypocentral Distance < 45 km in the Montney Play of Northeast British Columbia, Canada. Seismological Research Letters 2017;; 89 (1): 22–34. doi: https://doi.org/10.1785/0220170119
Ground‐motion amplitudes from small potentially induced earthquakes in the Montney Play of northeast British Columbia (MP BC) are used to evaluate the site condition, attenuation, and strong‐motion duration. The dataset includes waveforms from 219 events with local magnitude ranging from 1.5 to 3.8 recorded at hypocentral distance
<45 km
by the local seismographic stations operated by energy companies, complemented with some waveforms from the regional seismographic stations of the Canadian National Seismic Network. Horizontal‐to‐vertical (H/V) spectral ratio of ground‐motion results in amplification factors of 2.5–7.9 (0.4–0.9 in log unit) and 1.6–12.6 (0.2–1.1 in log unit) for seismographic stations in the Graham and Septimus areas, respectively. Following the Hassani and Atkinson (2016) methodology,
VS30
values are estimated for each seismographic station using the fundamental frequency associated with the peak H/V ratios. In this study, ground‐motion prediction equations (GMPEs) are obtained for the Graham and Septimus areas for the entire magnitude range (1.5–3.8) and for the combined dataset for magnitudes above 2.5. The obtained geometrical spreading coefficients for the Graham area suggest higher decay in ground‐motion amplitudes than those in the Septimus area for the peak ground velocity (PGV), peak ground acceleration (PGA), and response spectral acceleration (PSA) at frequency of 10 Hz, whereas the difference is insignificant at lower frequencies. Although, the geometrical attenuation of ground‐motion amplitudes for magnitudes above 2.5 is higher than Atkinson (2015) for PGV and PGA, it is comparable for PSA at frequencies 1, 2, 3.3, 5, and 10 Hz. Duration of the strong ground motion for waveforms with
PGA>1 cm/s2
is also calculated from two widely used methods (bracketed [
Db
] and significant [
Ds
] durations). Both
Db
Ds
are strongly correlated with PGA (
Db
increases while
Ds
decreases with PGA). Moreover,
Db
appears to be correlated with magnitude (increases with
M
), whereas
Ds
is correlated with distance (increases with hypocentral distance). Although the database in this study includes waveforms with relatively large ground acceleration (PGA of
∼115 cm/s2
M
3 event), duration of the strong ground motion is, however, very short at higher acceleration levels (e.g.,
50 cm/s2
) with a typical value of a fraction of a second.
κ0
and Broadband Site Spectra in Southern California from Source Model‐Constrained Inversion
A Ground‐Motion Prediction Equation for Vertical Spectra of the Strong‐Motion Records from the Subduction Slab Events in Japan Using Site Class as the Site Term
Using Simulated Ground Motions to Constrain Near‐Source Ground‐Motion Prediction Equations in Areas Experiencing Induced Seismicity
Procedures from International Guidelines for Assessing Seismic Risk to Flood-Control Levees
|
Pairing function - Wikipedia
Find sources: "Pairing function" – news · newspapers · books · scholar · JSTOR (August 2021) (Learn how and when to remove this template message)
This article may be confusing or unclear to readers. In particular, the notation for pairing functions is inconsistent throughout the article. Please help clarify the article. There might be a discussion about this on the talk page. (August 2021) (Learn how and when to remove this template message)
In mathematics, a pairing function is a process to uniquely encode two natural numbers into a single natural number.[1][2]
Any pairing function can be used in set theory to prove that integers and rational numbers have the same cardinality as natural numbers.[1]
2 Hopcroft and Ullman pairing function
3 Cantor pairing function
3.1 Inverting the Cantor pairing function
4 Other pairing functions
A pairing function is a bijection[verification needed]
{\displaystyle \pi :\mathbb {N} \times \mathbb {N} \to \mathbb {N} .}
More generally, a pairing function on a set A is a function that maps each pair of elements from A into an element of A, such that any two pairs of elements of A are associated with different elements of A,[3] or a bijection from
{\displaystyle A^{2}}
to A.[4]
Hopcroft and Ullman pairing function[edit]
Hopcroft and Ullman (1979) define the following pairing function:
{\displaystyle \langle i,j\rangle :={\frac {1}{2}}(i+j-2)(i+j-1)+i}
{\displaystyle i,j\in \{1,2,3,\dots \}}
Cantor pairing function[edit]
The Cantor pairing function is a primitive recursive pairing function
{\displaystyle \pi :\mathbb {N} \times \mathbb {N} \to \mathbb {N} }
{\displaystyle \pi (k_{1},k_{2}):={\frac {1}{2}}(k_{1}+k_{2})(k_{1}+k_{2}+1)+k_{2}}
[1][verification needed]
{\displaystyle k_{1},k_{2}\in \{0,1,2,3,\dots \}}
It can also be expressed as
{\displaystyle Pair[x,y]:={\frac {x^{2}+3x+2xy+y+y^{2}}{2}}}
It is also strictly monotonic w.r.t. each argument, that is, for all
{\displaystyle k_{1},k_{1}',k_{2},k_{2}'\in \mathbb {N} }
{\displaystyle k_{1}<k_{1}'}
{\displaystyle \pi (k_{1},k_{2})<\pi (k_{1}',k_{2})}
; similarly, if
{\displaystyle k_{2}<k_{2}'}
{\displaystyle \pi (k_{1},k_{2})<\pi (k_{1},k_{2}')}
The statement that this is the only quadratic pairing function is known as the Fueter–Pólya theorem.[1][verification needed] Whether this is the only polynomial pairing function is still an open question. When we apply the pairing function to k1 and k2 we often denote the resulting number as ⟨k1, k2⟩.[citation needed]
This definition can be inductively generalized to the Cantor tuple function[citation needed]
{\displaystyle \pi ^{(n)}:\mathbb {N} ^{n}\to \mathbb {N} }
{\displaystyle n>2}
{\displaystyle \pi ^{(n)}(k_{1},\ldots ,k_{n-1},k_{n}):=\pi (\pi ^{(n-1)}(k_{1},\ldots ,k_{n-1}),k_{n})}
with the base case defined above for a pair:
{\displaystyle \pi ^{(2)}(k_{1},k_{2}):=\pi (k_{1},k_{2}).}
Inverting the Cantor pairing function[edit]
{\displaystyle z\in \mathbb {N} }
be an arbitrary natural number. We will show that there exist unique values
{\displaystyle x,y\in \mathbb {N} }
{\displaystyle z=\pi (x,y)={\frac {(x+y+1)(x+y)}{2}}+y}
and hence that π is invertible. It is helpful to define some intermediate values in the calculation:
{\displaystyle w=x+y\!}
{\displaystyle t={\frac {1}{2}}w(w+1)={\frac {w^{2}+w}{2}}}
{\displaystyle z=t+y\!}
where t is the triangle number of w. If we solve the quadratic equation
{\displaystyle w^{2}+w-2t=0\!}
for w as a function of t, we get
{\displaystyle w={\frac {{\sqrt {8t+1}}-1}{2}}}
which is a strictly increasing and continuous function when t is non-negative real. Since
{\displaystyle t\leq z=t+y<t+(w+1)={\frac {(w+1)^{2}+(w+1)}{2}}}
{\displaystyle w\leq {\frac {{\sqrt {8z+1}}-1}{2}}<w+1}
{\displaystyle w=\left\lfloor {\frac {{\sqrt {8z+1}}-1}{2}}\right\rfloor .}
where ⌊ ⌋ is the floor function. So to calculate x and y from z, we do:
{\displaystyle w=\left\lfloor {\frac {{\sqrt {8z+1}}-1}{2}}\right\rfloor }
{\displaystyle t={\frac {w^{2}+w}{2}}}
{\displaystyle y=z-t\!}
{\displaystyle x=w-y.\!}
Since the Cantor pairing function is invertible, it must be one-to-one and onto.[3][additional citation(s) needed]
To calculate π(47, 32):
6320 ÷ 2 = 3160,
3160 + 32 = 3192,
so π(47, 32) = 3192.
To find x and y such that π(x, y) = 1432:
8 × 1432 = 11456,
11456 + 1 = 11457,
√11457 = 107.037,
107.037 − 1 = 106.037,
106.037 ÷ 2 = 53.019,
⌊53.019⌋ = 53,
so w = 53;
so t = 1431;
1432 − 1431 = 1,
so y = 1;
53 − 1 = 52,
so x = 52; thus π(52, 1) = 1432.[citation needed]
A diagonally incrementing "snaking" function, from same principles as Cantor's pairing function, is often used to demonstrate the countability of the rational numbers.
The graphical shape of Cantor's pairing function, a diagonal progression, is a standard trick in working with infinite sequences and countability.[a] The algebraic rules of this diagonal-shaped function can verify its validity for a range of polynomials, of which a quadratic will turn out to be the simplest, using the method of induction. Indeed, this same technique can also be followed to try and derive any number of other functions for any variety of schemes for enumerating the plane.
A pairing function can usually be defined inductively – that is, given the nth pair, what is the (n+1)th pair? The way Cantor's function progresses diagonally across the plane can be expressed as
{\displaystyle \pi (x,y)+1=\pi (x-1,y+1)}
The function must also define what to do when it hits the boundaries of the 1st quadrant – Cantor's pairing function resets back to the x-axis to resume its diagonal progression one step further out, or algebraically:
{\displaystyle \pi (0,k)+1=\pi (k+1,0)}
Also we need to define the starting point, what will be the initial step in our induction method: π(0, 0) = 0.
Assume that there is a quadratic 2-dimensional polynomial that can fit these conditions (if there were not, one could just repeat by trying a higher-degree polynomial). The general form is then
{\displaystyle \pi (x,y)=ax^{2}+by^{2}+cxy+dx+ey+f}
Plug in our initial and boundary conditions to get f = 0 and:
{\displaystyle bk^{2}+ek+1=a(k+1)^{2}+d(k+1)}
so we can match our k terms to get
d = 1-a
e = 1+a.
So every parameter can be written in terms of a except for c, and we have a final equation, our diagonal step, that will relate them:
{\displaystyle {\begin{aligned}\pi (x,y)+1&=a(x^{2}+y^{2})+cxy+(1-a)x+(1+a)y+1\\&=a((x-1)^{2}+(y+1)^{2})+c(x-1)(y+1)+(1-a)(x-1)+(1+a)(y+1).\end{aligned}}}
Expand and match terms again to get fixed values for a and c, and thus all parameters:
a = 1/2 = b = d
{\displaystyle {\begin{aligned}\pi (x,y)&={\frac {1}{2}}(x^{2}+y^{2})+xy+{\frac {1}{2}}x+{\frac {3}{2}}y\\&={\frac {1}{2}}(x+y)(x+y+1)+y,\end{aligned}}}
is the Cantor pairing function, and we also demonstrated through the derivation that this satisfies all the conditions of induction.[citation needed]
Other pairing functions[edit]
{\displaystyle P_{2}(x,y):=2^{x}(2y+1)-1}
is a pairing function.[2]
In 1990, Regan proposed the first known pairing function that is computable in linear time and with constant space (as the previously known examples can only be computed in linear time iff multiplication can be too, which is doubtful).[5] In fact, both this pairing function and its inverse can be computed with finite-state transducers that run in real time.[5][clarification needed] In the same paper, the author proposed two more monotone pairing functions that can be computed online in linear time and with logarithmic space; the first can also be computed offline with zero space.[5][clarification needed]
In 2001, Pigeon proposed a pairing function based on bit-interleaving, defined recursively as:
{\displaystyle <i,j>_{P}={\begin{cases}T{\text{ if }}i=j=0;\\<\lfloor i/2\rfloor ,\lfloor j/2\rfloor >_{P}:i_{0}:j_{0}{\text{ otherwise,}}\end{cases}}}
{\displaystyle i_{0}}
{\displaystyle j_{0}}
are the least significant bits of i and j respectively.[1][verification needed]
In 2006, Szudzik proposed a "more elegant" pairing function defined by the expression
{\displaystyle ElegantPair[x,y]:={\begin{cases}y^{2}+x{\text{ if }}x\neq \max\{x,y\}\\x^{2}+x+y{\text{ if }}x=\max\{x,y\}\\\end{cases}},}
which can be unpaired using the expression
{\displaystyle ElegantUnpair[z]:={\begin{cases}\{z-\lfloor {\sqrt {z}}\rfloor ^{2},\lfloor {\sqrt {z}}\rfloor \}{\text{ if }}z-\lfloor {\sqrt {z}}\rfloor ^{2}<\lfloor {\sqrt {z}}\rfloor \\\{\lfloor {\sqrt {z}}\rfloor ,z-\lfloor {\sqrt {z}}\rfloor ^{2}-\lfloor {\sqrt {z}}\rfloor {\text{ if }}z-\lfloor {\sqrt {z}}\rfloor ^{2}\geq \lfloor {\sqrt {z}}\rfloor \}\end{cases}}}
.[3] (Qualitatively, it assigns consecutive numbers to pairs along the edges of squares.[3]) This pairing function orders SK combinator calculus expressions by depth.[3][clarification needed] This method is the mere application to
{\displaystyle \mathbb {N} }
of the idea, found in most textbooks on Set Theory,[6] used to establish
{\displaystyle \kappa ^{2}=\kappa }
for any infinite cardinal
{\displaystyle \kappa }
in ZFC. Define on
{\displaystyle \kappa \times \kappa }
the binary relation
{\displaystyle (\alpha ,\beta )\preccurlyeq (\gamma ,\delta )}
{\displaystyle {\begin{cases}{\text{either}}\ (\alpha ,\beta )=(\gamma ,\delta ),\\[4pt]{\text{or}}\ \max(\alpha ,\beta )<\max(\gamma ,\delta ),\\[4pt]{\text{or}}\ \max(\alpha ,\beta )=\max(\gamma ,\delta )\ {\text{and}}\ \alpha <\gamma ,\\[4pt]{\text{or}}\ \max(\alpha ,\beta )=\max(\gamma ,\delta )\ {\text{and}}\ \alpha =\gamma \ {\text{and}}\ \beta <\delta .\end{cases}}}
{\displaystyle \preccurlyeq }
is then shown to be a well-ordering such that every element has
{\displaystyle {}<\kappa }
predecessors, which implies that
{\displaystyle \kappa ^{2}=\kappa }
{\displaystyle (\mathbb {N} \times \mathbb {N} ,\preccurlyeq )}
{\displaystyle (\mathbb {N} ,\leqslant )}
and the pairing function above is nothing more than the enumeration of integer couples in increasing order. (See also Talk:Tarski's theorem about choice#Proof of the converse.)
^ The term "diagonal argument" is sometimes used to refer to this type of enumeration, but it is not directly related to Cantor's diagonal argument.[citation needed]
^ a b c d e f g h i Steven Pigeon. "Pairing function". MathWorld. Retrieved 16 August 2021.
^ a b c d "pairing function". planetmath.org. 22 March 2013. Retrieved 16 August 2021. {{cite web}}: CS1 maint: url-status (link)
^ a b c d e f Szudzik, Matthew (2006). "An Elegant Pairing Function" (PDF). szudzik.com. Archived (PDF) from the original on 25 November 2011. Retrieved 16 August 2021.
^ Szudzik, Matthew P. (2017-06-01). "The Rosenberg-Strong Pairing Function". arXiv:1706.04129 [cs.DM].
^ a b c Regan, Kenneth W. (1992-12-01). "Minimum-complexity pairing functions". Journal of Computer and System Sciences. 45 (3): 285–295. doi:10.1016/0022-0000(92)90027-G. ISSN 0022-0000.
^ See for instance Thomas, Jech (2006). Set theory: the third millennium edition. Springer Monographs in Mathematics. Springer-Verlag. p. 30. doi:10.1007/3-540-44761-X. ISBN 3-540-44085-2.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Pairing_function&oldid=1068818803"
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Calculate Generalized Optimal Subpattern Assignment Metric - Simulink - MathWorks España
Generalized Optimal Subpattern Assignment Metric
GOSPA Metric Without Switching
Missed Target Error
False Track Error
Custom distance function
Switching penalty
GOSPA metric without switching error
Truth bus
Track extractor function
Truth extractor function
Calculate Generalized Optimal Subpattern Assignment Metric
Sensor Fusion and Tracking Toolbox / Track Metrics
The Generalized Optimal Subpattern Assignment Metric block evaluates the performance of a tracking algorithm by computing the generalized optimal subpattern assignment (GOSPA) metric between tracks and known truths. The metric is comprised of the switching error, localization error, missed target error, and false track error components. You can also select each individual error components as a block output.
Track list, specified as a Simulink bus containing a MATLAB structure.
If you specify the Track bus parameter on the Port Setting tab to objectTrack, the structure must use this form:
Each track structure must contain TrackID and State fields. Additionally, if you specify an NEES-based distance (posnees or velnees) in the Distance type parameter, each structure must contain a StateCovariance field.
TrackID Unique track identifier used to distinguish multiple tracks, specified as a nonnegative integer.
Value of state vector at the update time, specified as an N-element vector, where N is the dimension of the state.
Uncertainty covariance matrix, specified as an N-by-N matrix, where N is the dimension of the state.
If you specify an NEES-based distance (posnees or velnees) in the Distance type parameter, then the structure must contain a StateCovariance field.
If you specify the Track bus parameter to custom, then you can use your own track bus format. In this case, you must define a track extractor function using the Track extractor function parameter. The function must use this syntax:
tracks = trackExtractorFcn(trackInputFromBus)
where trackInputFromBus is the input from the track bus and tracks must return as an array of structures with TrackID and State fields.
Truths — Truth list
Truth list, specified as a Simulink bus containing a MATLAB structure.
If you specify the Truth bus parameter on the Port Setting tab to Platform, the structure must use this form:
NumPlatforms Number of truth platforms
Platforms Array of truth platform structures
Each platform structure has these fields:
PlatformID Unique identifier used to distinguish platforms, specified as a nonnegative integer.
Position of the platform, specified as an M-element vector, where M is the dimension of the position state. For example, M = 3 for 3-D position.
Velocity of the platform, specified as an M-element vector, where M is the dimension of the velocity state. For example, M = 3 for 3-D velocity.
If you specify the Truth bus parameter as Actor, the structure must use this form:
NumActors Number of truth actors
Actors Array of truth actor structures
Each actor structure has these fields:
ActorID Unique identifier used to distinguish actors, specified as a nonnegative integer.
Position of the actor, specified as an M-element vector, where M is the dimension of the position state. For example, M = 3 for 3-D position.
Velocity of the actor, specified as an M-element vector, where M is the dimension of the velocity state. For example, M = 3 for 3-D velocity.
If you specify the Truth bus parameter to custom, then you can define your own truth bus format. In this case, you must define a truth extractor function using the Truth extractor function parameter. The function must use this syntax:
truths = truthExtractorFcn(truthInputFromBus)
where truthInputFromBus is the input from the truth bus and truths must return as an array of structures with PlatformID, Position, and Velocity fields.
Assignments — Known assignment
K-by-2 matrix of nonnegative integers
Known assignment, specified as an K-by-2 matrix of nonnegative integers. K is the number of assignment pairs. The first column elements are track IDs, and the second column elements are truth IDs. The IDs in the same row are assigned to each other. If a track or truth is not assigned, specify 0 as the same row element.
The assignment must be a unique assignment between tracks and truths. Redundant or false tracks should be treated as unassigned tracks by assigning them to the "0" TruthID.
To enable this port, on the Port Setting tab, select Assignments.
GOSPA Metric — GOSPA metric including switching error component
GOSPA metric including switching error component, returned as a nonnegative real scalar.
GOSPA Metric Without Switching — GOSPA metric without switching error component
GOSPA metric without switching error component, returned as a nonnegative real scalar.
To enable this port, on the Port Setting tab, select GOSPA metric without switching error component.
Switching Error — Switching error component
Switching error component, returned as a nonnegative real scalar.
To enable this port, on the Port Setting tab, select Switching error.
Localization Error — Localization error component
Localization error component, returned as a nonnegative real scalar.
To enable this port, on the Port Setting tab, select Localization error.
Missed Target Error — Missed target error component
Missed target error component, returned as a nonnegative real scalar.
To enable this port, on the Port Setting tab, select Missed target error.
False Track Error — False track error component
False track error component, returned as a nonnegative real scalar.
To enable this port, on the Port Setting tab, select False track error.
Cutoff distance — Threshold for cutoff distance between track and truth
Threshold for cutoff distance between track and truth, specified as a real positive scalar. If the computed distance between a track and the assigned truth is higher than the threshold, the actual distance incorporated in the metric is reduced to the threshold.
Order — Order of GOSPA metric
Order of GOSPA metric, specified as a positive integer.
Alpha — Alpha parameter of GOSPA metric
2 (default) | positive scalar in range [0, 2]
Alpha parameter of the GOSPA metric, specified as a positive scalar in the range [0, 2].
Distance type — Distance type
posnees (default) | velnees | posabserr | velabserr | custom
Distance type, specified as posnees, velnees, posabserr, or velabserr. The distance type specifies the physical quantity used for distance calculations:
posnees – Normalized estimation error squared (NEES) of track position
velnees – NEES error of track velocity
posabserr – Absolute error of track position
velabserr – Absolute error of track velocity
custom – Custom distance error
If you specify it as custom, you must also specify the distance function in the Custom distance function parameter.
Custom distance function — Custom distance function
Custom distance function, specified as a function handle. The function must support the following syntax:
d = myCustomFcn(Track,Truth)
where Track is a structure for track information, Truth is a structure of truth information, and d is the distance between the truth and the track. See objectTrack for an example on how to organize track information.
Example: @myCustomFcn
To enable this property, set the Distance type parameter to custom.
Motion model — Desired platform motion model
constvel (default) | constacc | constturn | singer
Desired platform motion model, specified as constvel, constacc, constturn, or singer. This property selects the motion model used by the Tracks input port.
The motion models expect the State field of the track structure to have a column vector containing these values:
constvel — Position is in elements [1 3 5], and velocity is in elements [2 4 6].
constacc — Position is in elements [1 4 7], velocity is in elements [2 5 8], and acceleration is in elements [3 6 9].
constturn — Position is in elements [1 3 6], velocity is in elements [2 4 7], and yaw rate is in element 5.
singer — Position is in elements [1 4 7], velocity is in elements [2 5 8], and acceleration is in elements [3 6 9].
The StateCovariance field of the track structure input must have position, velocity, and turn-rate covariances in the rows and columns corresponding to the position, velocity, and turn rate of the State field of the track structure.
Switching penalty — Penalty for assignment switching
Penalty for assignment switching, specified as a nonnegative real scalar.
Assignments — Enable assignment input
Select this parameter to enable the input of know assignments through the Assignments input port.
GOSPA metric without switching error — Enable GOSPA metric without switching error output
Select this parameter to enable the output of the GOSPA metric without the switching error component through the GOSPA Metric Without Switching output port.
Switching error — Enable switching error component output
Select this parameter to enable the output of the switching error component through the Switching Error output port.
Localization error — Enable localization error component output
Select this parameter to enable the output of the localization error component through the Localization Error output port.
Missed target error — Enable missed target error component output
Select this parameter to enable the output of the missed target error component through the Missed Target Error output port.
False track error — Enable false track error component output
Select this parameter to enable the output of the false track error component through the False Track Error output port.
Track bus — Track bus selection
objectTrack (default) | custom
Track bus selection, specified as objectTrack or custom. See the description of the Tracks input port for more details about each selection.
Truth bus — Truth bus selection
Platform (default) | Actor | custom
Truth bus selection, specified as Platform, Actor, or custom. See the description of the Truths input port for more details about each selection.
Track extractor function — Track extractor function
Track extractor function, specified as a function handle. The function must support this syntax:
where trackInputFromBus is the input from the track bus and tracks must return as an array of structures with TrackID and State fields. If you specify an NEES-based distance (posnees or velnees) in the Distance type parameter, then the structure must contain a StateCovariance field.
To enable this property, set the Track bus parameter to custom.
Truth extractor function — Truth extractor function
Truth extractor function, specified as a function handle. The function must support this syntax:
where truthInputFromBus is the input from the track bus and truths must return as an array of structures with PlatformID, Position, and Velocity as field names.
To enable this property, set the Truth bus parameter to custom.
At time tk, a list of truths is:
X=\left[{x}_{1},{x}_{2},\dots ,{x}_{m}\right]
At time tk, a tracker obtains a list of tracks:
Y=\left[{y}_{1},{y}_{2},\dots ,{y}_{n}\right]
In general, the GOSPA metric including the switching component (SGOSPA) is:
SGOSPA={\left(GOSP{A}^{p}+S{C}^{p}\right)}^{1/p}
where p is the order of the metric, SC is the switching component, and GOSPA is the basic GOSPA metric.
Assuming m ≤ n, GOSPA is:
GOSPA={\left[\sum _{i=1}^{m}{d}_{c}^{p}\left({x}_{i},{y}_{\pi \left(i\right)}\right)+\frac{{c}^{p}}{\alpha }\left(n-m\right)\right]}^{1/p}
where dc is the cutoff-based distance and yπ(i) represents the track assigned to truth xi. The cutoff-based distance dc is defined as:
{d}_{c}\left(x,y\right)=\mathrm{min}\left\{{d}_{b}\left(x,y\right),c\right\}
where c is the cutoff distance threshold, and db(x,y) is the base distance between track x and truth y calculated by the distance function. The cutoff based distance dc is the smaller value of db and c. α is the alpha parameter.
The switching component SC is:
SC=SP×{n}_{s}^{1/p}
where SP is the switching penalty and ns is the number of switches. When a track switches assignment from one truth to another truth, the number of switching is counted as 1. When a track switches from assigned to unassigned or switches from unassigned to assigned, the number of switching is counted as 0.5. For example, as shown in the table, Tracks 1 and 2 both switched to different truths, whereas Track 3 switched from assigned to unassigned. Therefore, the total number of switching is 2.5.
Track Switching Scenario
Tracks Truths Tracks Truths
When α = 2, the GOSPA metric can reduce to three components:
GOSPA={\left[lo{c}^{p}+mis{s}^{p}+fals{e}^{p}\right]}^{1/p}
The localization component (loc) is calculated as:
loc={\left[\sum _{i=1}^{h}{d}_{b}^{p}\left({x}_{i},{y}_{\pi \left(i\right)}\right)\right]}^{1/p}
where h is the number of nontrivial assignments. A trivial assignment is when a track is assigned to no truth. The missed target component is calculated as:
miss=\frac{c}{{2}^{1/p}}{\left({n}_{miss}\right)}^{1/p}
where nmiss is the number of missed targets. The false track component is calculated as:
false=\frac{c}{{2}^{1/p}}{\left({n}_{false}\right)}^{1/p}
where nfalse is the number of false tracks.
If m > n, simply exchange m and n in the formulation to obtain the GOSPA metric.
trackAssignmentMetrics | trackErrorMetrics | trackOSPAMetric | trackGOSPAMetric | Optimal Subpattern Assignment Metric
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article->A shell script for converting percentage opacity to hexadecimal
A nest of chickens 2021-08-05 17:00:52
shell script converting percentage opacity
Let's have a script for the first Nuggets ~
The main function of this script is to convert the percentage into 16 Base number .
reason ： The design leader of our company submits to lanhu The color of your design is controlled by percentage opacity , and Android It uses 16 Base number ARGB. So there's this script ~
for example ：tran_calc.sh. Then copy and paste the following ：
# Calculation 0%~100%
tran_table(){
while((tran <= 100))
tran_to_hex $tran;let "tran++"
# Calculate and format the output hex
tran_to_hex(){
# for example ：tran_to_hex FF here $1 refer to FF . It also corresponds to the following 【 Parameters 2】
tran=$1
temp=$((255*$tran/100))
# Calculation hex value , And supplement the preamble 0
hexStr=$(echo "obase=16;$temp"|bc|awk '{ len = (2 - length % 2) % 2; printf "%.*s%s\n", len, "00000000", $0}')
printf " transparency %3s percentage %3s%% Hex %s \n" $temp $tran $hexStr
# When the parameter is greater than two perform then Internal code
# The first parameter is zero hex When it comes to execution then Internal code
if test $1 = "hex"
# call function , And pass in 【 Parameters 2】
tran_to_hex $2
tran_table
# There are less than two parameters , Output cross reference table
Add executable rights
chmod a+x tran_calc.sh
The terminal executes the following instructions
ln -s `pwd`/tran_calc.sh /usr/local/bin/tran_calc
Now you can execute... At the terminal
Output 0%~100% Transparency comparison table
tran_calc
Convert values to 16 Base number
tran_calc hex [ The number ]
for example tran_calc hex 50 That is, calculate 50% Diaphaneity hex value
Shell Pass parameters . Look at the comparison table below , come from Novice tutorial -—— Pass parameter section .
$# The number of parameters passed to the script
$* Display all the parameters passed to the script in a single string . Such as "
*" use 「"」 In a nutshell 、 With "
2 …
n" Output all parameters in the form of .
$$ The current process the script runs ID Number
$! Of the last process running in the background ID Number
[email protected] And
* identical , But use quotation marks , And return each parameter in quotation marks . Such as "
@" use 「"」 In a nutshell 、 With "
1" "
2" … "$n" Output all parameters in the form of .
$- Show Shell Current options used , And set command Function the same .
$? Display the exit status of the last command .0 No mistakes , Any other value indicates an error .
I wanted to see if I had a chance to participate in 8 The month's Gengwen activity . The number of words is too small to participate in 、 A large piece of code can't participate in 、 Those who have a lot of references cannot participate in . Ha ha ha . I took it all . Come on come on ~
Make do with . Welcome to correct ~~
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On $x$-Analytic Solutions to the Cauchy Problem for Partial Differential Equations with Retarded Variables | EMS Press
x
-Analytic Solutions to the Cauchy Problem for Partial Differential Equations with Retarded Variables
A. Augustynowicz
We prove some existence results for solutions analytic with respect to the spatial variables to first-order equations with a delay and some deviations not only at the function, but also at its derivative. We construct a natural Banach space and a norm which make an adequate integral operator contractive. Due to a useful relation of partial order in this space the main problem is also placed in the theory of monotone iterative techniques.
A. Augustynowicz, H. Leszczyński, On
x
-Analytic Solutions to the Cauchy Problem for Partial Differential Equations with Retarded Variables. Z. Anal. Anwend. 15 (1996), no. 2, pp. 345–356
|
Reliability, Kuder-Richardson Formula[Page 1418]
By: Kevin Wombacher
The Kuder-Richarson Formula 20, often abbreviated as KR-20, and the Kuder-Richardson Formula 21, abbreviated KR-21, are measures of internal consistency for measures that feature dichotomous items. As these are measures of internal consistency, they measure the extent to which all the items measure the same characteristic. This entry covers the formulas for both the KR-20 and KR-21, as well as when to use them, what they are used for, how to interpret their outputs, the limitations of these procedures, and some possible alternatives.
The formula for KR20 is
\mathrm{KR}20=n/n-1\phantom{\rule{0.25em}{0ex}}×\phantom{\rule{0.25em}{0ex}}\left[1-\left(\left(pq\right)/\mathrm{Var}\right)\right]
In this formula, KR20 is the output and is the score that serves as an estimation of the internal reliability of the measure. n is simply the number of items that the researcher has; p is the proportion of ...
Wombacher, K. (2017). Reliability, kuder-richardson formula. In M. Allen (Ed.), The sage encyclopedia of communication research methods (Vol. 3, pp. 1418-1420). SAGE Publications, Inc, https://dx.doi.org/10.4135/9781483381411.n493
Wombacher, Kevin. "Reliability, Kuder-Richardson Formula[Page 1418]." In The SAGE Encyclopedia of Communication Research Methods, edited by Allen, Mike, 1418-20. Thousand Oaks, CA: SAGE Publications, Inc, 2017. https://dx.doi.org/10.4135/9781483381411.n493.
Wombacher, K. 2017. Reliability, Kuder-Richardson Formula[Page 1418]. In: Mike Allen Editor, 2017. The SAGE Encyclopedia of Communication Research Methods, Thousand Oaks, CA: SAGE Publications, Inc. pp. 1418-1420 Available at: <https://dx.doi.org/10.4135/9781483381411.n493> [Accessed 22 May 2022].
Wombacher, Kevin. "Reliability, Kuder-Richardson Formula[Page 1418]." The SAGE Encyclopedia of Communication Research Methods. Edited by Mike Allen. Vol. 3. Thousand Oaks: SAGE Publications, Inc, 2017, pp. 1418-20. SAGE Knowledge. 22 May 2022, doi: https://dx.doi.org/10.4135/9781483381411.n493.
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The Pressure Equation in the Fast Diffusion Range | EMS Press
The Pressure Equation in the Fast Diffusion Range
We consider the following degenerate parabolic equation
v_{t}=v\Delta v-\gamma|\nabla v|^{2}\quad\mbox{in $\mathbb{R}^{N} \times(0,\infty)$,}
whose behaviour depends strongly on the parameter
\gamma
. While the range
\gamma < 0
is well understood, qualitative and analytical novelties appear for
\gamma>0
. Thus, the standard concepts of weak or viscosity solution do not produce uniqueness. Here we show that for
\gamma>\max\{N/2,1\}
the initial value problem is well posed in a precisely defined setting: the solutions are chosen in a class
\mathcal{W}_s
of local weak solutions with constant support; initial data can be any nonnegative measurable function
v_{0}
(infinite values also accepted); uniqueness is only obtained using a special concept of initial trace, the
p
-trace with
p=-\gamma < 0
, since the standard concepts of initial trace do not produce uniqueness. Here are some additional properties: the solutions turn out to be classical for
t>0
, the support is constant in time, and not all of them can be obtained by the vanishing viscosity method. We also show that singular measures are not admissible as initial data, and study the asymptotic behaviour as
t\to \infty
Emmanuel Chasseigne, Juan Luis Vázquez, The Pressure Equation in the Fast Diffusion Range. Rev. Mat. Iberoam. 19 (2003), no. 3, pp. 873–917
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2\sin\left(x\right)\cos\left(x\right)+\cos\left(x\right)=0
. Give your answers in degrees or radians.
cos\left(x\right)
from the equation.
\cos(x)(2\sin(x)+1)\ =\ 0
x
\text{cos}(x)\ =\ 0 \ \text{or 2sin}(x) + 1\ =\ 0
Find all the solutions to these statements either using a unit circle or by graphing the functions on a graphing calculator. Do not just take this inverse on your calculator, it will only give you one possible solution. Remember, each of the
4
solutions repeats every
2πn
|
On the Discrete Spectrum of Ordinary Differential Operators in Weighted Function Spaces | EMS Press
On the Discrete Spectrum of Ordinary Differential Operators in Weighted Function Spaces
Inner Mongolia University, Huhhot, China
The investigations deal with the spectrum of ordinary differential equations of order
2n
where the underlying Hubert space is weighted. Especially, conditions on the coefficients are given such that the spectra are discrete.
Erich Müller-Pfeiffer, Sun Jiong, On the Discrete Spectrum of Ordinary Differential Operators in Weighted Function Spaces. Z. Anal. Anwend. 14 (1995), no. 3, pp. 637–646
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Harry and Hermione - 1 | Toph
Harry and Hermione - 1
By mahdi.hasnat · Limits 1s, 512 MB
[1 and 2 are completely different problems. Consider reading both.]
Harry and Hermione are in a fight with the monsters. But Hermione is currently busy with Ron. So Harry is fighting alone.
Harry has initial health point
h. There are
n monsters. For
i-th monster(
1 \le i \le n
1≤i≤n), it has damage health point
a_i
ai and reward health point
b_i
bi.
If Harry chooses to fight the
i-th monster (
1 \le i \le n
1≤i≤n ), his health point will decrease by
a_i
ai.
Harry will win against a monster if and only if, his (Harry) health point is non-negative during the fight. Once Harry loses a fight, then he cannot fight any other monster.
If Harry wins against
i-th monster, that monster releases a potion that increases Harry’s health point by
b_i
Harry can fight the monsters in any order.
Calculate the maximum number of fights Harry can win.
Note That: for all
i (
1 \le i \le n
1≤i≤n ),
a_i \le b_i
ai≤bi
t (1\le t \le 10 )
t(1≤t≤10) — the number of test cases.
The first line of each test case contains two integers
n and
h
(1 \le n \le 2 \times 10^5 ; 1 \le h \le 10^9)
(1≤n≤2×105;1≤h≤109)— the number of monsters and initial health point of Harry respectively.
n integers
a_1,a_2,\ldots,a_n
(1 \le a_i \le 10^9)
(1≤ai≤109) — damage health point for each monster.
n integers
b_1,b_2,\ldots,b_n
b1,b2,…,bn
(1 \le b_i \le 10^9)
(1≤bi≤109) — reward health point for each monster.
It is guaranteed that for all
i (
1 \le i \le n
1≤i≤n),
a_i \le b_i
n over all test cases does not exceed
2 \times 10^5
2×105.
For each test case, print a single integer — maximum number of fights Harry can win.
Explanation of test case 1:
Initial health point of Harry is 10.
Harry chooses to fight with the 1st monster. During the fight his health point becomes 4. So Harry win that fight. Then 1st monster release potion and Harry’s health point becomes 11.
Then Harry chooses to fight with the 2nd monster. In that fight Harry wins and his health point becomes 13.
Then Harry chooses to fight with the 3rd monster. In that fight Harry wins and his health point becomes 17.
So Harry can win all three fights.
Initial health point of Harry is 3.
Harry chooses to fight the 2nd monster. Harry wins in that fight and his health point becomes 5.
Then Harry chooses to fight the 3rd monster. Harry wins in that fight and his health point becomes 7.
Then Harry chooses to fight the 1st monster. Unfortunately Harry losses that fight because Harry’s health point becomes -1 during the fight.
So Harry can win at most two fights.
Harry chooses to fight the 1st monster. Harry wins in that fight and his health point becomes 4.
Then Harry chooses to fight the 2nd monster. Unfortunately Harry losses that fight because Harry’s health point becomes -1 during the fight.
So Harry can win at most one fight.
s_semicolonEarliest, 2M ago
prodip_bsmrstuLightest, 1.7 MB
hattiMattimTimShortest, 555B
|
Three thieves in ancient times stole three bags of gold. The gold in the heaviest bag had three times the value of the gold in the lightest bag and twice the value of the gold in the medium bag. All together, the gold amounted to
330
florins. How much gold was in each bag? Write a system of equations and use matrices to solve the system.
h = 3l
h = 2m
h + m + l = 330
|
PSSCH demodulation reference signal - MATLAB ltePSSCHDRS - MathWorks í•œêµ
ltePSSCHDRS
Generate PSSCH DM-RS Sequence
Generate a PSSCH DM-RS Sequence for V2X
PSSCH Demodulation Reference Signal Processing
PSSCH Demodulation Reference Signal Indexing
PSSCH demodulation reference signal
[seq,info] = ltePSSCHDRS(ue)
[seq,info] = ltePSSCHDRS(ue) returns a complex column vector sequence containing PSSCH demodulation reference signal (DM-RS) values and an associated information structure for the specified UE settings structure. For more information, see PSSCH Demodulation Reference Signal Processing.
Generate a PSSCH DM-RS sequence associated with both DM-RS SC-FDMA symbols in a subframe. Plot the constellation of the sequence.
Create a user equipment settings structure.
ue = [];
ue.NSAID = 34;
ue.PRBSet = (1:10)';
Generate a PSSCH DM-RS sequence. Plot the constellation.
[psschDrsSeq,info] = ltePSSCHDRS(ue);
plot(psschDrsSeq,'o')
Generate a PSSCH DM-RS sequence for V2X using the format 1 SCI PSCCH CRC.
Generate the format 1 SCI PSCCH CRC and assign it to the UE V2X scrambling identity.
sciinfo = lteSCIInfo(ue);
scibits = ones(1,sciinfo.Format1);
[cw,crc] = lteSCIEncode(ue,scibits);
ue.NXID = crc;
plot(psschDrsSeq,'or')
{n}_{\text{ssf}}^{\text{PSSCH}}
seq — PSSCH DM-RS values
PSSCH DM-RS values, returned as a N × 12 × NPRB-by-1 column vector. For more information, see PSSCH Demodulation Reference Signal Processing.
info — PSSCH DM-RS information
PSSCH DM-RS information about the intermediate variables used to create the DM-RS, returned as a parameter structure containing these fields:
Alpha — Reference signal cyclic shift for each slot
Reference signal cyclic shift for each slot, returned as a two-column vector. (α)
Alpha is proportional to NCS, where
\mathrm{α}=\frac{2\mathrm{Ï}{n}_{cs,\mathrm{λ}}}{12}
SeqGroup — Base sequence group number for each slot
Base sequence group number for each slot, returned as a two-column vector. (u)
SeqIdx — Base sequence number for each slot
Base sequence number for each slot, returned as a two-column vector. (v)
RootSeq — Root Zadoff-Chu sequence index for each slot
Root Zadoff-Chu sequence index for each slot, returned as a two-column vector. (q)
Cyclic shift values for each slot, returned as a two-column vector. (
{n}_{\text{cs,}\mathrm{λ}}
{N}_{ZC}^{RS}
OrthSeq — Orthogonal cover value for each slot
Orthogonal cover value for each slot, returned as a matrix. (
\stackrel{¯}{w}
The PSSCH demodulation reference signal (DM-RS) sequence is transmitted alongside the ltePSSCH values using the two SC-FDMA symbols allocated to DM-RS in a PSSCH subframe. The output vector is the repetition of a 12-element sequence and specified in TS 36.211, Section 9.8. The vector is mapped onto the 12 DM-RS SC-FDMA symbol subcarriers in each subframe slot for each PSSCH physical resource block (PRB) transmission on antenna port 1000.
The output PSSCH DM-RS sequence is the concatenation of the two sequences to be mapped onto the DM-RS SC-FDMA symbol subcarriers in each subframe slot carrying a ltePSSCH transmission. Its length is N × 12 × NPRB, where NPRB is the number of PRBs associated with the PSSCH. For D2D sidelink, there is one DM-RS symbol per slot and therefore N=2, and for V2X sidelink, there are two symbols per slot and N=4.
Use the ltePSSCHDRSIndices indexing function and the corresponding ltePSCCHDRS sequence function to populate the resource grid for any PSSCH subframe. The PSSCH DM-RS is transmitted in the available SC-FDMA symbols in a PSSCH subframe, using a single layer on antenna port 1000.
The indices are ordered as the PSSCH DM-RS QPSK modulation symbols should be, applying frequency-first mapping. One-based linear indexing is the default return format, but alternative indexing formats can also be generated.
The resource elements in the last SC-FDMA symbol within a subframe are counted in the mapping process but should not be transmitted. The sidelink-specific SC-FDMA modulation creates the last symbol, which serves as a guard symbol.
For D2D sidelink, when indexing is zero-based, the SC-FDMA symbol indices used are {3,10} for normal cyclic prefix and {2,8} for extended cyclic prefix. The same symbols are used by the ltePUSCHDRSIndices function. For V2X sidelink, there are four DM-RS SC-FDMA symbols with indices {2,5,8,11} for normal cyclic prefix only.
The indicated symbol indices are based on TS 36.211, Section 9.8. However, to align with the LTE Toolbox™ subframe orientation, these indices are expanded from symbol index per slot to symbol index per subframe.
ltePSSCHDRSIndices | ltePSSCH | ltePSSCHDecode | ltePSSCHIndices
|
Uniqueness, symmetry and full regularity of free boundary in optimization problems with volume constraint | EMS Press
Uniqueness, symmetry and full regularity of free boundary in optimization problems with volume constraint
In this paper we study qualitative geometric properties of optimal configurations to a variational problem with free boundary, under suitable assumptions on a fixed boundary. More specifically, we study the problem of minimizing the flow of heat given by
\int_{\partial D} \Gamma (u_\mu) d\sigma
D
is a fixed domain and
u
is the potential of a domain
\Omega \supset \partial D
, with a prescribed volume on
\Omega \setminus D
. Our main goal is to establish uniqueness and symmetry results when
\partial D
has a given geometric property. Full regularity of the free boundary is obtained under these symmetry conditions imposed on the fixed boundary.
Eduardo V. Teixeira, Uniqueness, symmetry and full regularity of free boundary in optimization problems with volume constraint. Interfaces Free Bound. 9 (2007), no. 1, pp. 133–148
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Strong Convergence Theorems for a Finite Family of λi-Strict Pseudocontractions in 2-Uniformly Smooth Banach Spaces by Metric Projections
2012 Strong Convergence Theorems for a Finite Family of
{\lambda }_{i}
-Strict Pseudocontractions in 2-Uniformly Smooth Banach Spaces by Metric Projections
Xin-dong Liu, Shih-sen Chang, Xiong-rui Wang
A new hybrid projection algorithm is considered for a finite family of
{\lambda }_{i}
-strict pseudocontractions. Using the metric projection, some strong convergence theorems of common elements are obtained in a uniformly convex and 2-uniformly smooth Banach space. The results presented in this paper improve and extend the corresponding results of Matsushita and Takahshi, 2008, Kang and Wang, 2011, and many others.
Xin-dong Liu. Shih-sen Chang. Xiong-rui Wang. "Strong Convergence Theorems for a Finite Family of
{\lambda }_{i}
-Strict Pseudocontractions in 2-Uniformly Smooth Banach Spaces by Metric Projections." J. Appl. Math. 2012 (SI15) 1 - 10, 2012. https://doi.org/10.1155/2012/759718
Xin-dong Liu, Shih-sen Chang, Xiong-rui Wang "Strong Convergence Theorems for a Finite Family of
{\lambda }_{i}
-Strict Pseudocontractions in 2-Uniformly Smooth Banach Spaces by Metric Projections," Journal of Applied Mathematics, J. Appl. Math. 2012(SI15), 1-10, (2012)
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Interval Oscillation Criteria for Super-Half-Linear Impulsive Differential Equations with Delay
2012 Interval Oscillation Criteria for Super-Half-Linear Impulsive Differential Equations with Delay
Zhonghai Guo, Xiaoliang Zhou, Wu-Sheng Wang
We study the following second-order super-half-linear impulsive differential equations with delay
\left[r\left(t\right){\phi }_{\gamma }\left(x\prime \left(t\right)\right)\right]\prime +p\left(t\right){\phi }_{\gamma }\left(x\left(t-\sigma \right)\right)+q\left(t\right)f\left(x\left(t-\sigma \right)\right)=e\left(t\right)
t\ne {\tau }_{k}
x\left({t}^{+}\right)={a}_{k}x\left(t\right),x\prime \left({t}^{+}\right)={b}_{k}x\prime \left(t\right)
t={\tau }_{k}
t\ge {t}_{0}\in ℝ
{\phi }_{*}\left(u\right)=|u{|}^{*-1}u
\sigma
is a nonnegative constant,
\left\{{\tau }_{k}\right\}
denotes the impulsive moments sequence with
{\tau }_{1}<{\tau }_{2}<\cdots <{\tau }_{k}<\cdots
{\mathrm{lim} }_{k\to \infty }{\tau }_{k}=\infty
{\tau }_{k+1}-{\tau }_{k}>\sigma
. By some classical inequalities, Riccati transformation, and two classes of functions, we give several interval oscillation criteria which generalize and improve some known results. Moreover, we also give two examples to illustrate the effectiveness and nonemptiness of our results.
Zhonghai Guo. Xiaoliang Zhou. Wu-Sheng Wang. "Interval Oscillation Criteria for Super-Half-Linear Impulsive Differential Equations with Delay." J. Appl. Math. 2012 1 - 22, 2012. https://doi.org/10.1155/2012/285051
Zhonghai Guo, Xiaoliang Zhou, Wu-Sheng Wang "Interval Oscillation Criteria for Super-Half-Linear Impulsive Differential Equations with Delay," Journal of Applied Mathematics, J. Appl. Math. 2012(none), 1-22, (2012)
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Omnidirectional antenna - Wikipedia
Radio antenna that sends signals in every direction
Example of omnidirectional antenna; a whip antenna on a walkie-talkie
In radio communication, an omnidirectional antenna is a class of antenna which radiates equal radio power in all directions perpendicular to an axis (azimuthal directions), with power varying with angle to the axis (elevation angle), declining to zero on the axis.[1][2] When graphed in three dimensions (see graph) this radiation pattern is often described as doughnut-shaped. Note that this is different from an isotropic antenna, which radiates equal power in all directions, having a spherical radiation pattern. Omnidirectional antennas oriented vertically are widely used for nondirectional antennas on the surface of the Earth because they radiate equally in all horizontal directions, while the power radiated drops off with elevation angle so little radio energy is aimed into the sky or down toward the earth and wasted. Omnidirectional antennas are widely used for radio broadcasting antennas, and in mobile devices that use radio such as cell phones, FM radios, walkie-talkies, wireless computer networks, cordless phones, GPS, as well as for base stations that communicate with mobile radios, such as police and taxi dispatchers and aircraft communications.
The radiation pattern of a simple omnidirectional antenna, a vertical half-wave dipole antenna. In this graph the antenna is at the center of the "donut," or torus. Radial distance from the center represents the power radiated in that direction. The power radiated is maximum in horizontal directions, dropping to zero directly above and below the antenna.
Animation of an omnidirectional half-wave dipole antenna transmitting radio waves. The antenna in the center is two vertical metal rods, with an alternating current applied at its center from a radio transmitter (not shown). Loops of electric field (black lines) leave the antenna and travel away at the speed of light; these are the radio waves. This diagram only shows the radiation in a single plane through the axis, the radiation is symmetrical about the antenna's vertical axis.
Radiation pattern of a 3λ/2 monopole antenna. Although the radiation of an omnidirectional antenna is symmetrical in azimuthal directions, it may vary in a complicated way with elevation angle, having lobes and nulls at different angles.
The most common omnidirectional antenna designs are the monopole antenna, consisting of a vertical rod conductor mounted over a conducting ground plane, and vertical dipole antenna, consisting of two collinear vertical rods. The quarter-wave monopole and half-wave dipole both have vertical radiation patterns consisting of a single broad lobe with maximum radiation in horizontal directions, so they are popular. The quarter-wave monopole, the most compact resonant antenna, may be the most widely-used antenna in the world. The five-eighth wave monopole, with a length of
{\displaystyle 5/8=.625}
of a wavelength, is also popular because at that length a monopole radiates maximum power in horizontal directions.
Common types of low-gain omnidirectional antennas are the whip antenna, "Rubber Ducky" antenna, ground plane antenna, vertically oriented dipole antenna, discone antenna, mast radiator, horizontal loop antenna (sometimes known colloquially as a 'circular aerial' because of the shape) and the halo antenna.
Higher-gain omnidirectional antennas can also be built. "Higher gain" in this case means that the antenna radiates less energy at higher and lower elevation angles and more in the horizontal directions. High-gain omnidirectional antennas are generally realized using collinear dipole arrays. These consist of multiple half-wave dipoles mounted collinearly (in a line), fed in phase.[3] The coaxial collinear (COCO) antenna uses transposed coaxial sections to produce in-phase half-wavelength radiators.[4] A Franklin Array uses short U-shaped half-wavelength sections whose radiation cancels in the far-field to bring each half-wavelength dipole section into equal phase. Another type is the Omnidirectional Microstrip Antenna (OMA).[5]
Vertical polarized VHF-UHF biconical antenna 170–1100 MHz with omnidirectional H-plane pattern
{\displaystyle G=eD}
{\displaystyle \sin(b\theta )/{b\theta }}
pattern shape is:[6]
{\displaystyle D\approx 10\log _{10}\left({\frac {101.5}{{\text{HPBW}}-0.00272\,{\text{HPBW}}^{2}}}\right)\ {\text{ dB}}.}
^ Balanis, Constantine A.; Ioannides, Panayiotis I. (2007). Introduction to Smart Antennas. Morgan and Claypool. p. 22. ISBN 978-1598291766.
^ National Telecommunication Information Administration (1997). Federal Standard 1037C: Glossary of Telecommunications Terms. US General Services Administration. pp. O-3. ISBN 1461732328.
^ Johnson, R; Jasik, H, eds. (1984). Antenna Engineering Handbook. McGraw Hill. pp. 27–14.
^ Judasz, T., Balsley, B. (March 1989). "Improved Theoretical and Experimental Models for the Coaxial Colinear Antenna". IEEE Transactions on Antennas and Propagation. 37 (3): 289–296. doi:10.1109/8.18724. {{cite journal}}: CS1 maint: multiple names: authors list (link)
^ Bancroft R (December 5, 2005). "Design Parameters of an Omnidirectional Planar Microstrip Antenna". Microwave and Optical Technology Letters. 47 (5): 414–8. doi:10.1002/mop.21187.
^ McDonald, Noel (April 1999). "Omnidirectional Pattern Directivity in the Presence of Minor Lobes: Revisited". IEEE Transactions on Antennas and Propagation. 41 (2): 63–8. doi:10.1109/74.769693.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Omnidirectional_antenna&oldid=1070874230"
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This problem is a checkpoint for conic sections.
Identify each conic and sketch the graph. Also include all of the key information, such as the locations of the foci and/or the equations of any asymptotes.
\frac { ( x - 4 ) ^ { 2 } } { 50 }+ (y − 3)^2 = 1
\frac { ( y + 2 ) ^ { 2 } } { 9 } - \frac { ( x - 3 ) ^ { 2 } } { 16 }= 1
2y^{2} − x + 4y + 2 = 0
9x^{2} − 16y^{2} + 54x − 32y + 29 = 0
Ideally at this point you are comfortable working with these types of problems and can complete them correctly. If you feel that you need more confidence when completing these types of problems, then review the Conic Sections Checkpoint materials available at cpm.org and try the practice problems provided. From this point on, you will be expected to do problems like these correctly and with confidence.
Click the following link: Conic Sections
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Birthday Present?! | Toph
By sayeef006 · Limits 1s, 256 MB
As everyone gifts graphs or strings to people on their birthdays, Oyshee’s friends decided to do something different and gave her a grid of size
N \times M
N×M on her birthday. Some of the cells of the grid are blocked but others are usable.
Overwhelmed with happiness, she decided to play a game on the grid. She placed a token on the cell
(1,1). Now she wants to move the token to the cell
(N,M).
To do this, she can build some directed bridges. If the token is currently at cell
A(x_1,y_1)
A(x1,y1) and she builds a directed bridge from
A to
B(x_2,y_2)
B(x2,y2), she can immediately move the token to cell
B. But she can’t move the token from
B to
A. Moving the token happens instantaneously. So, it doesn’t take any time. But building a bridge does.
She can build a directed bridge from
A to
B if one of the following two conditions is satisfied:
x_1=x_2
x1=x2 and
y_1<y_2
y1<y2, this takes her
(T+bridge\_length) seconds. Here,
bridge\_length =
(y_2-y_1)
(y2−y1).
y_1=y_2
y1=y2 and
x_1<x_2
x1<x2, this takes her
(T+bridge\_length) seconds. Here,
bridge\_length =
(x_2-x_1)
(x2−x1).
But she can not build a bridge of length more than
L. She can not build a bridge to the outside of the grid. She also can not build a bridge over a blocked cell or place the token on a blocked cell. Two or more bridges may overlap each other. She can not move the token to other cells without bridges.
Now she’s wondering, what is the minimum time needed to move the token from cell
(1,1) to cell
(N,M)?
N and
M (2 \leq N,M \leq 3⋅10^3)
M(2≤N,M≤3⋅103) - the size of the grid.
The next line contains two integers
L and
T (1 \leq L \leq 3⋅10^3, 1 \leq T \leq 10^9)
T(1≤L≤3⋅103,1≤T≤109) - the maximum length of a bridge Oyshee can build, and the time added when building a bridge, respectively.
N lines contain a string of
M characters each. Each character is either . or #. A . represents a usable cell and a # represents a blocked cell. Cell
(1,1) is guaranteed to be usable.
-1
−1 if Oyshee can’t move the token to the cell
(N,M). Else, print the minimum possible time to do so.
Oyshee can move the token to cell
(3,4) in three steps:
Build a bridge from cell
(1,1) to
(1,3) and move the token to cell
(1,3), requiring time 2+2 = 4 seconds.
(1,3) to
(3,3) and move the token to cell
(3,3), requiring time 2+2 = 4 seconds.
(3,3) to
(3,4) and move the token to cell
(3,4), requiring time 2+1 = 3 seconds.
So the total time is (4+4+3) seconds = 11 seconds. This is the minimum time possible.
Here, it is not possible to move the token to cell
(3,3).
Soumya1Earliest, 1M ago
borisalFastest, 0.0s
MrBrionixLightest, 14 MB
MrBrionixShortest, 1341B
We can use Dynamic Programming to solve this problem. Here dpijdp_{ij}dpij denotes the minimum poss...
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z_1 = 6\Big[\cos(\frac { 7 \pi } { 12 }) + i \sin(\frac { 7 \pi } { 12 })\Big]
z_2 = 3 \Big[\cos(\frac { \pi } { 4 }) + i \sin (\frac { \pi } { 4 })\Big]
compute each of the following values. Write your answers in
a + bi
z_{1}z_{2}
Given two complex numbers
z_1=r_1(\cos(a)+i\sin(a))
z_2=r_2(\cos(b)+i\sin(b))
, the product of the two numbers is
z _ { 1 } z _ { 2 } = r _ { 1 } r _ { 2 } ( \operatorname { cos } ( a + b ) + i \operatorname { sin } ( a + b ) )
\frac { z 1 } { z _ { 2 } }
z_1=r_1(\cos(a)+i\sin(a))
z_2=r_2(\cos(b)+i\sin(b))
, the quotient of the two numbers is
\frac { z _ { 1 } } { z _ { 2 } } = \frac { r _ { 1 } } { r _ { 2 } } ( \operatorname { cos } ( a - b ) + i \operatorname { sin } ( a - b ) )
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Calorie - Uncyclopedia, the content-free encyclopedia
In gastronomics, the calorie is the SI unit of energetic deliciousness.
1 Food-Energy Equivalence
2 The Great calorie Disaster of 1903
3 Table of Metricized calorie Units
4 Today's Modern Definition of the calorie
Food-Energy Equivalence[edit]
For the religious among us who choose to believe lies, the so-called experts at Wikipedia have an article very remotely related to Calorie.
In 1812, the ancient mechano-alchemist (and part-time raving environmentalist) James "G" Watt hypothesized that food and energy were not necessarily conserved separately, but could actually be converted into each other, according to the LaTeX formula:
{\displaystyle E=food*c^{2}}
In creationist belief, the calorie was invented by god to punish those who choose to eat awesome food (e.g. McDonald's, candy, corn dogs, soda, chocolates, Reese's, etc.) to make people suffer much like anorexics and vegans.
However, all attempts to accelerate roast beef sandwiches to the speed of light squared made each sandwich travel sideways in time, causing it to transform into Meryl Streep instead of energy. This ridiculous state of affairs was finally explained to everybody's satisfaction in 1976 by Stephen Hawking in his Nobel Prize-winning Unified Phlogiston-Aether-Élan-Vital Field Theory.
The Great calorie Disaster of 1903[edit]
In 1903, due to a bizarre mixup between English and metric units, the British calorie was accidentally defined to be 1,024 times larger than itself. This unfortunate mistake has been determined to be the primary cause of the sinking of the Titanic, the curse of the Bambino, a greater prevailence of obesity in the United States, and the Great Minnesota Incident of 1903. The resulting public relations disaster has caused no end of grief amongst the perfectly innocent members of the scientific community, and has led to the immediate abolishment of the entire God-given English measurement system worldwide, marking the final triumph of secular humanist metricism.
Meanwhile, not far away, Oprah Winfrey and her legion of loyal minions were capitalizing on the rampant confusion by using a capital "C" for the bigger calorie, thereby screwing the American taxpayer out of billions and billions of dollars.
Table of Metricized calorie Units[edit]
Unit Metric equivalent calories How big
calorie calorie 1 1 lick of the outer surface of an unpeeled cucumber
Calorie kilocalorie 1,024 1 Diet Pepsi
CAlorie megacalorie 1,048,576 1 Lean Cuisine
CALorie gigacalorie 1,073,741,824 1 Economy-sized Tub'o'Lard
CALOrie teracalorie 1,099,511,627,776 1 satisfying meal for Oprah Winfrey
CALORie petacalorie 1,125,899,906,842,624 1,099,511,627,776 Diet Pepsis
CALORIe exacalorie 1,152,921,504,606,846,976 1000 megaton nuclear explosion [1]
CALORIE zetacalorie 1,180,591,620,717,411,303,424 god's daily intake!!!
Today's Modern Definition of the calorie[edit]
In today's modern scientific community, there are more kinds of calorie than you can shake a pointed stick at. Every nation on the planet is required to have its own specific definition of the calorie (per UN Resolution 473.29, section 3, subparagraph 62).
The British calorie: energy sufficient to raise 1 gram of warm ale from Ottery St Mary to Basingstoke
The French calorie: like the British calorie but 7.5% tastier
The Canadian calorie: like the French calorie but 7.5% less rude and obnoxious
The American calorie: sufficient food energy to power 1 leggy supermodel
The Mexican calorie: amount of deadly radiation emitted by one (1) medium-sized jalapeño pepper per second
The Russian calorie: amount of life-giving heat from burning a 1000 ruble note in a stove on a cold winter's day in Vostok, Antarctica
The Porn Studio calorie: heat expended by 0.5776 of a pelvic thrust
Retrieved from "https://uncyclopedia.com/w/index.php?title=Calorie&oldid=6130706"
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Tangent lines, inflections, and vertices of closed curves
November 2013 Tangent lines, inflections, and vertices of closed curves
We show that every smooth closed curve
\Gamma
immersed in Euclidean space
{\mathbf{R}}^{3}
satisfies the sharp inequality
2\left(\mathcal{P}+\mathcal{I}\right)+\mathcal{V}\ge 6
which relates the numbers
\mathcal{P}
of pairs of parallel tangent lines,
\mathcal{I}
of inflections (or points of vanishing curvature), and
\mathcal{V}
of vertices (or points of vanishing torsion) of
\Gamma
2\left({\mathcal{P}}^{+}+\mathcal{I}\right)+\mathcal{V}\ge 4
{\mathcal{P}}^{+}
is the number of pairs of concordant parallel tangent lines. The proofs, which employ curve-shortening flow with surgery, are based on corresponding inequalities for the numbers of double points, singularities, and inflections of closed curves in the real projective plane
{\mathbf{RP}}^{2}
{\mathbf{S}}^{2}
which intersect every closed geodesic. These findings extend some classical results in curve theory from works of Möbius, Fenchel, and Segre, including Arnold’s “tennis ball theorem.”
Mohammad Ghomi. "Tangent lines, inflections, and vertices of closed curves." Duke Math. J. 162 (14) 2691 - 2730, November 2013. https://doi.org/10.1215/00127094-2381038
Received: 16 January 2012; Revised: 16 February 2013; Published: November 2013
Secondary: 57R45 , 58E10
Mohammad Ghomi "Tangent lines, inflections, and vertices of closed curves," Duke Mathematical Journal, Duke Math. J. 162(14), 2691-2730, (November 2013)
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PHP Ellipsoid 3D XYZ-Coordinates calculator - NeoProgrammics.com
Ellipsoid 3D XYZ−Coordinates Calculator
Given the Geographic Coordinates and Elevation of Any Point
Relative to the Surface of Any of Several Reference Ellipsoid Models
Latitude (−Neg = South) , Longitude (−Neg = West)
±Deg.dddddddddd or ±Deg min sec.ssss Sea-Level Elevation
in meters or feet (m, ft)
Select an ellipsoid model to compute the rectangular 3D XYZ-coordinates of the point based
on the geographic coordinates and elevation given above and the equations following below.
GRS 80 | WGS 84 Ellipsoid Model Parameters Used worldwide by the ITRS - International Terrestrial Reference System and the GPS - Global Positioning System Equatorial radius a = 6378137.0000 m = 3963.190592 mi Polar radius b = 6356752.3142 m = 3949.902764 mi Flattening f = 0.003352810665 = 1/298.2572235605 Eccentricity e = 0.081819190843 e² = 0.006694379990 Equatorial Quarter = 10018754.1713 m = 6225.3652 mi Meridional Quarter = 10001965.7293 m = 6214.9333 mi Circumference at Equator = 40075016.6856 m = 24901.460897 mi Circumference of Meridian = 40007862.9173 m = 24859.733480 mi Circumference Difference = 67153.7683 m = 41.727417 mi __________________________________________________________________ Location Coordinates and Sea Level Elevation of Point Latitude = +053° 36' 43.1653" N = +53.6119903611° Longitude = -001° 39' 51.9920" W = -1.6644422222° Height/Elev = +299.8000 m = +983.60 ft = +983 ft and 7.1 in __________________________________________________________________ Corresponding Rectangular 3D XYZ-Coordinates of Point at Location x = +3790644.8998 m = +2355.397541 mi y = -110149.2097 m = -68.443546 mi z = +5111482.9706 m = +3176.128268 mi _________________________________________________________________
BASIC ELLIPSOID FORMULAS AND RELATIONSHIPS APPLIED HERE
Given the general static ellipsoid parameters:
a = Equatorial radius of ellipsoid = Length of semi-major axis
b = Polar radius of ellipsoid = Length of semi-minor axis
where, a > b
If ((a = b)), then the form is a perfect sphere, otherwise it is an ellipsoid or oblate
spheroid to some degree, generally wider between two opposite equatorial points
due to a slight equatorial bulge, than between the opposite polar points.
In the equations, any units of measure may be used for (a, b), such as feet, meters,
kilometers, miles, etc., just as long as the same units are used consistently throughout.
The values of (x, y, z) will be expressed in the same units as the parameters (a, b).
Computing 3D Rectangular XYZ-Coordinates For a Point on a Reference Ellipsoid Surface
The reference ellipsoid parameters, location and elevation of the point are:
= Equatorial radius.
= Polar radius.
= Geographic latitude of point on ellipsoid surface.
= Geographic longtitude of point on ellipsoid surface (Negative = West).
= Height or elevation of point relative to ellipsoid surface.
From the given parameters, the 3D XYZ-coordinates of the point, relative to the surface
of the reference ellipsoid, are computed from the equations given below.
= Polar flattening factor of ellipsoid.
e=\sqrt{1-{\left(\frac{b}{a}\right)}^{2}}
= Radius vector from ellipsoid center to surface at latitude .
The Earth Ellipsoid
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Solenoid valve - 3D CAD Models & 2D Drawings
Solenoid valve (25338 views - Mechanical Engineering)
PARTcloud - Solenoid valve
Licensed under Creative Commons Attribution-Share Alike 3.0 (Hammelmann).
{\displaystyle F_{s}=PA=P\pi d^{2}/4}
Where d is the orifice diameter. A typical solenoid force might be 15 N (3.4 lbf). An application might be a low pressure (e.g., 10 psi (69 kPa)) gas with a small orifice diameter (e.g., 3⁄8 in (9.5 mm) for an orifice area of 0.11 in2 (7.1×10−5 m2) and approximate force of 1.1 lbf (4.9 N)).
When high pressures and large orifices are encountered, then high forces are required. To generate those forces, an internally piloted solenoid valve design may be possible.[1] In such a design, the line pressure is used to generate the high valve forces; a small solenoid controls how the line pressure is used. Internally piloted valves are used in dishwashers and irrigation systems where the fluid is water, the pressure might be 80 psi (550 kPa) and the orifice diameter might be 3⁄4 in (19 mm).
Power consumption and supply requirements of the solenoid vary with application, being primarily determined by fluid pressure and line diameter. For example, a popular 3/4" 150 psi sprinkler valve, intended for 24 VAC (50 - 60 Hz) residential systems, has a momentary inrush of 7.2 VA, and a holding power requirement of 4.6 VA.[5] Comparatively, an industrial 1/2" 10000 psi valve, intended for 12, 24, or 120 VAC systems in high pressure fluid and cryogenic applications, has an inrush of 300 VA and a holding power of 22 VA.[6] Neither valve lists a minimum pressure required to remain closed in the un-powered state.
Common components of a solenoid valve:[7][8][9][10]
The core tube contains and guides the core. It also retains the plugnut and may seal the fluid. To optimize the movement of the core, the core tube needs to be nonmagnetic. If the core tube were magnetic, then it would offer a shunt path for the field lines.[11] In some designs, the core tube is an enclosed metal shell produced by deep drawing. Such a design simplifies the sealing problems because the fluid cannot escape from the enclosure, but the design also increases the magnetic path resistance because the magnetic path must traverse the thickness of the core tube twice: once near the plugnut and once near the core. In some other designs, the core tube is not closed but rather an open tube that slips over one end of the plugnut. To retain the plugnut, the tube might be crimped to the plugnut. An O-ring seal between the tube and the plugnut will prevent the fluid from escaping.
Solenoid valves are used in fluid power pneumatic and hydraulic systems, to control cylinders, fluid power motors or larger industrial valves. Automatic irrigation sprinkler systems also use solenoid valves with an automatic controller. Domestic washing machines and dishwashers use solenoid valves to control water entry into the machine. They are also often used in paintball gun triggers to actuate the CO2 hammer valve. Solenoid valves are usually referred to simply as "solenoids."
Solenoid valves can be used for a wide array of industrial applications, including general on-off control, calibration and test stands, pilot plant control loops, process control systems, and various original equipment manufacturer applications. [16]
Butterfly valveSafety valvePinch valveTesla valve밸브Flow control valve기계공학솔레노이드Relief valveBlowoff valveWastegate수격작용
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Help Lazy Tareq | Toph
Help Lazy Tareq
By Tareq_Abrar · Limits 1s, 512 MB
There is a boy named Tareq Abrar who is very lazy. He likes sleeping. He also likes eating. His mother is annoyed with him for his so much laziness. But she loves her son very much. This is why she can't punish him. She hit upon a plan. She gave Tareq a task to solve. She thought that Tareq would not remain lazy while solving the task. Tareq told his friend Issac Shimanto to solve this task. But Shimanto was busy with his newly married wife. Now, Tareq became sad finding no way for solving the task. Tareq knew that you are a famous programmer of the world. He wants your help solve the task. If you solve the task, Tareq Abrar will get opportunity to sleep again.
Tareq was given a positive integer which had a non-zero leading digit & all other digits were zero. He had to find the sum of all positive even divisors of this number.
You will be given the first digit of this number & the number of zeroes of this integer. You have to output the sum of all positive even divisors of this number.
The answer may be a big number. So, give output modulo
10000007.
There are several test cases. The first line of input contains a positive integer
T (
1 \le T \le 10^5
1≤T≤105), which denotes the number of test cases.
In each line of next
T lines, there will be given two positive integers
N (
1 \le N \le 9
1≤N≤9) and
X (
0 \le X \le 10^{17}
0≤X≤1017) separated by space.
N denotes the first digit of the number.
X denotes the number of zeroes.
Output will contain
T lines. In each line, you have to print a non-negative integer which denotes the answer.
For the first one, the leading digit is 2, it has one 'zero'. So, it is 20. 2+4+10+20=36
BinarySearch, Math, NumberTheory
ShafinEarliest, May '20
fayedanikFastest, 0.1s
habijabiLightest, 918 kB
HMSPLC: কোডshock_ 2020
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Chapter 5: Balance: ฿856 – saltedpeaches
Kongpob doesn’t think he’s ever been at school this early, especially on a Monday. As Arthit had told him they would be, the gates are already open, but the entire campus is still with the grey morning, the only movement coming from a few dragonflies circling the garden pond.
It’s almost 7:00 as Kongpob enters the equally silent library. The librarian herself isn’t at her desk yet, and the only person guarding the place is a rather reluctantly breathing member of custodial staff.
Most of the library isn’t even lit, many rows of bookcases stand in the relative darkness, the only light coming from the windows.
One corner, however, towards the cluster of tables in the study area, is illuminated. Kongpob sees Arthit already there, although his head rests on his backpack on the table, and he’s asleep. He moves closer and looks at his friend, who looks oddly at peace, a contrast to his usual tense and slightly agitated demeanour.
Kongpob doesn’t know if it’s normal to feel this, but he finds himself staring at the boy and admiring his soft features – snowy white skin, a strong, pronounced nose, long lashes and small, pink lips hanging slightly open. He smiles to himself and tries to push down the foreign fluttering in his stomach as he sits adjacent to Arthit, wondering if he should wake him up or not.
His decision is made for him, though, when Arthit’s phone alarm goes off, buzzing on the table to jolt both Arthit awake and Kongpob out of his daze. Arthit rubs at his eyes and stretches like a cat, tilting his head side to side to work out the cricks in his neck, before noticing Kongpob, at which he startles a little.
“How long have you been there?” he says, his cheeks slightly flushed.
“Should’ve woken me up…” he mutters, unzipping his backpack to take out his pencil case and maths textbook.
“You seemed tired. Not an early riser?”
Kongpob smiles, taking out his own things and placing them on the table. He also pulls out a linen bag and sets it on the table, taking out a large thermos.
“Mae made breakfast for us when I told her you were tutoring me.”
Arthit eyes the blue thermos for a moment before biting his lip and absentmindedly flipping through the pages of his book to find the right chapter.
“Uh…that’s okay. I already ate at home.”
“Come on, just have some. My mother makes really good congee,” Kongpob unscrews the lid, and Arthit can already smell the fragrant waft of soupy comfort food, which awakens his empty insides. It does look very appetising.
“It’s fine,” he says. “I’m still full. Maybe later.”
He clears his throat loudly over the sound of his growling stomach. Kongpob eyes him strangely, but nods and screws the lid back on.
“So…algebra.”
“Yeah, I didn’t do so well on the last homework.”
Arthit nods and writes something on a blank page of his open notebook, then slides it over to Kongpob.
“Okay, have a look at this question.”
\left(2x+3\right)\left(x-2\right)
Kongpob nods, waiting for him to go on.
“So in order to find the answer, we need to multiply whatever is in the first set of brackets with whatever is in the second set of brackets. He takes a highlighter out of his pencil case and highlights 2x, 3, x, and -2 separately.
“But you don’t multiply within the same set of brackets. So we multiply the first number in the first set with the first number in the second set. What is 2x times x?”
Arthit taps the blank space below the question, gesturing for Kongpob to write his answer. Kongpob thinks for a second before he writes 2x².
“Right. Then you multiply 2x with -2, which is…?”
4x, Kongpob writes. Arthit draws an invisible circle with the back of his pencil around the minus sign in the question.
“When you multiply a positive number with a negative number, the answer becomes negative, so the answer wouldn’t be 4x, but…”
“-4x?” Kongpob raises an eyebrow.
As Arthit continues explaining the question, Kongpob just stares at the boy in front of him, watching as he speaks with ease about the question, explaining it step by step. It’s probably the most at ease that he’s ever seen Arthit be in his presence, and Kongpob is slightly in awe of the way his voice is quite animated, clarifying his points with his hands and circling different parts of the question as he speaks.
“Do…I have something on my face?”
Kongpob is jolted out of his thoughts, and realises he’s been staring at Arthit for the past minute or two. Suddenly, he feels slightly embarrassed and shakes his head rapidly.
“No,” he says. “I was just thinking.”
“Um…okay. So the final answer would be…?”
Kongpob ponders Arthit’s arrows and scribbles for a moment, before putting all the different parts of the equation together.
2x² – x – 6, he writes.
Arthit nods.
“Good,” he says, before flipping the textbook around and pointing to a set of exercises. “Now try these.”
Kongpob gets through about four questions before his attention is drawn back to Arthit, who’s just…watching him work. He suddenly feels slightly self-conscious and puts his pencil down, pondering his next words.
“Arthit,” he starts. “Do you want to have lunch together today?”
Arthit just looks at him, eyes narrowed.
“I mean like, not with just me. With M and Oak as well,” Kongpob feels the need to clarify.
“I told you, I’m busy during lunch.”
“I’m just asking. Who do you usually eat with, then?”
Arthit sighs through his nostrils, pressing his lips together into a tight line.
“What is this? Studying or interrogation?” Arthit snaps, and looks anywhere but at Kongpob. “Finish the set and I’ll have a look.”
Kongpob relents with a sigh and works through the remaining questions before sliding the notebook over to Arthit, who is noticeably stiffer than before. Arthit scans each of his answers with his pencil and nods, circling one or two parts.
“Two negative numbers multiplied make a positive number, so this should be a plus sign instead,” he points to one of Kongpob’s answers. “Otherwise, you seem to have the idea.”
“You explained it well,” Kongpob smiles softly, at which Arthit grimaces a little, closing the book.
“Anyway, we should get back to the classroom.” he says, hurriedly packing his things, as though he wants to get away as quickly as possible. Which makes no sense, of course, seeing as they’re in the same class and therefore heading to the same classroom.
Kongpob just nods, putting his own things away as Arthit practically bolts out of the library and past the sleeping custodian.
“And then she gave me detention again because she said the homework I turned in wasn’t ‘up to standard’! Are teachers allowed to punish us for being bad at English now?”
Oak is complaining yet again about his supposedly unfair treatment. Kongpob just shakes his head and tries to open his lunchbox, struggling a little with the lid as the suction on the air-tight container is particularly strong. Today’s lunch consists of flat rice noodles with a chicken gravy and gai lan.
“No, but you can be punished for only writing idk lol I don’t get the question when the topic is asking you to write 300 words about a childhood memory,” M rolls his eyes.
“Fine. Kong, you can help me with English homework, right? You got the top score in the grade last year, didn’t you?”
“Knowing you, by ‘help’ you mean ‘let you copy’. Therefore, no — ah, crabsticks!”
He finally gets the lid off of the box, and a splatter of the brown gravy sauce plops onto the front of his crisp white uniform. He tuts and puts down the box.
“I’m going to the bathroom,” he shakes his head, getting up from the table.
All the toilets in the main school building are closed off due to drainage issues on the first floor caused by a group of seniors who’d thought it would be funny to flush an entire roll of toilet paper that morning. As a result, Kongpob has to make his way to the side building, where students usually only go when the school holds community service activities and supplies for annual events are kept. There’s a small toilet with only two stalls and three urinals, and Kongpob can only recall ever having used it once before.
As suspected, the building is whisper quiet and probably the source of many ghost stories told throughout the year, but Kongpob isn’t fazed by such ridiculous tales. As he approaches the washroom, he hears shuffling inside, and the click of a stall door’s lock. Just as he’s about to push open the door, it swings open, and he comes face to face with none other than Arthit, who stumbles back in shock, dropping what looks like an empty food container and some eating utensils.
“Arthit, wha—” Kongpob says in surprise, trying to help pick up his things. Arthit shakes his head and holds a trembling hand out to stop him. His entire face goes pale, and his mouth is gaping open and closed like a stunned fish. His eyes dart around in panic for a moment, before he hurriedly wipes his mouth, grabbing the fallen items and hiding them in his arms before darting out before Kongpob can get another word in.
Kongpob blinks after him, trying to parse what had just happened, his breath caught in his throat. He considers going after Arthit, but the boy seems traumatised enough by Kongpob’s discovery that it probably wouldn’t be wise to bombard him with questions at this precise moment.
“Kids used to pick on him all the time…he almost switched schools…lost all the weight before our freshman year…now he just never talks to anyone…”
Suddenly, M’s comments from last week creep into his thoughts, and his throat clenches. He wouldn’t….would he?
Fearing the worst, Kongpob peers into the large bin next to the sink counter, praying not to find what he suspects, and breathes a sigh of muted relief when he sees only one scrunched up paper towel.
Turning around, he eyes the fallen spoon at the door of one of the stalls. He picks it up – it’s definitely been used, but his mind still reaches into the darkest possibilities. Biting his bottom lip, he slowly pushes open the stall door. Wincing a little, he lifts the lid off of the water closet and finds it still full, an indication that it hasn’t been flushed in a while.
He doesn’t know if he should be relieved or not. He recalls his mother doing all of these things almost eight years ago when his sister had gone through some difficult times with her relationship to food as a teenager.
So why was Arthit eating in the toilets, then? And in such an inconspicuous location, too.
His entire mind is plagued with questions as he works at removing the gravy stain from his shirt. When he returns to the courtyard, M and Oak have already finished eating, and are bickering over their DoTA stats. He rejoins them, quietly eating as they continue talking, but not really enjoying what is usually his favourite dish.
“Hey, M,” Kongpob says, after a while. “Can you tell Coach Pak I can’t make it to practice today?”
“I have to handle something at home after school.”
M eyes him suspiciously, but nods.
“Okay. Everything alright?”
“Yeah, just…I have to be there.” he forces a small smile.
Kongpob doesn’t rush after Arthit when his friend scurries from his seat and out the door as soon as their teacher dismisses them. He’d stolen a few glances at Arthit during their afternoon classes, only to see the guy with his head ducked down, refusing to look at anything other than whatever is on his desk. He can’t focus on anything during the lesson, and ends up being scolded by their teacher for spacing out when asked a question.
He sighs, looking over at the empty desk after the bell rings. Tilting his head, he can see that Arthit has left something behind. He goes over to the desk to see what it is, and pauses when he sees that it’s the same eraser he’d lent to Arthit a while back. It has the same tattered paper casing held in place only by a shabby piece of clear tape.
It looks like Arthit has used it religiously, the small pink piece of rubber half the size it was when he’d lent it to him. He almost decides to just keep it, when the casing slips off and Kongpob notices that the letters “Kongp” are scrawled on the side in blue ball pen, slightly smudged and faded.
That’s strange, he thinks. Why would he write half my name on my eraser? Did he run out of space?
Still, he pockets the eraser and heads straight for the front gate.
Despite his suspicions that Arthit would still be avoiding him, Kongpob sees his friend at the cart as usual, busying himself with grilling and stirring despite the small Monday crowd.
Neither of them say a word when Kongpob approaches the cart, Arthit only looking up briefly before turning his attention back to stirring the bucket of marinade.
Silence hangs between them, both daring the other to speak first. Arthit doesn’t ask what he wants to eat, and Kongpob doesn’t push with his usual playful comments.
Eventually, Kongpob picks up the pen and notepad from the side of the worktop and quickly writes something, filling almost the entire page of the small page. Arthit glances at what he’s doing from the corner of his eye, but says nothing. Kongpob tears the entire page out and carefully tears off the bottom strip, pushing over to Arthit.
2 moo-ping, it reads.
Wordlessly, Arthit takes two skewers off the grill and places them in a container before setting it down next to the notepad.
Kongpob picks up the skewers and leaves the remainder of the page he’d written on where the container was, folded up on the worktop. Then, he takes the eraser out of his pocket and places it on top of the paper. Arthit eyes it briefly, but says nothing.
Kongpob just gives him a small smile, before walking down the street towards the bus stop.
As soon as Kongpob is far away enough that Arthit can no longer see the outline of his fading back, Arthit grabs for the eraser and paper, heart beating out of his chest as he unfolds the note.
Arthit,
I just want you to know that you don’t have to tell me anything if you don’t want to, but I’m here if you do want to talk. I hope you’ll still tutor me and be my friend. By the way, I never gave you my contact information, so here’s my Line ID. Hope I’ll see you tomorrow.
Kongp 🙂
Arthit reads the note over and over, biting his lip in a subtle smile before pocketing both the note and eraser. He pulls out his phone, punching in Kongpob’s Line ID and immediately finding his classmate’s profile picture. He stares at it, finger hovering over the ‘Add friend’ button, but pockets his phone again.
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dDAFI Claiming - DAFI Protocol
Get your hands on some.
Fragmented timeframes
As the weight accumulates, when a user decides to claim their rewards, we can extract it from the weight.
Where DPS = Distribute per second, US = User Stake, TS = Total Network Stake, UR = User Reward, and n = Time in seconds
UR1 = (DPS1 * US1 / TS1) + (DPS2 * US1 / TS2) + (DPSn * US1 / TSn)
We can calculate the user reward within a timeframe, as a to b where b > a
Where n = 1 to a,
UR1 = US * ∑DPSn / TSn
UR = US1 * (DPS1 / TS1 + DPS2 / TS2 + DPSx / TSx)
Where n = 1 to b,
UR2 = US * ∑DPSn / TSn
When a user initiates an action in the network, where a = n > 1, we calculate n from 1 to a. When the same user initiates an action again, we calculate from 1 to b, and subtract the previous 1 to a. This can be represented as :
UR = US * ∑ DPSn / TSn, (n = 1 : ∞)
A user's reward between timeframe a and b can be calculated accurately as:
URab = UR2 - UR1 = (US * ∑DPSn / TSn)
where n = 1 to b, and
- (US * ∑DPSn / TSn)
where n = 1 to a
= US * ((∑DPSn / TSn, (n=1:b)) - (∑DPSn / TSn, (n = 1:a)))
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Sheet metal (18434 views - Mechanical Engineering)
Sheet metal is metal formed by an industrial process into thin, flat pieces. It is one of the fundamental forms used in metalworking and it can be cut and bent into a variety of shapes. Countless everyday objects are fabricated from sheet metal. Thicknesses can vary significantly; extremely thin sheets are considered foil or leaf, and pieces thicker than 6 mm (0.25 in) are considered plate. Sheet metal is available in flat pieces or coiled strips. The coils are formed by running a continuous sheet of metal through a roll slitter. In most of the world, sheet metal thickness is consistently specified in millimeters. In the US, the thickness of sheet metal is commonly specified by a traditional, non-linear measure known as its gauge. The larger the gauge number, the thinner the metal. Commonly used steel sheet metal ranges from 30 gauge to about 7 gauge. Gauge differs between ferrous (iron based) metals and nonferrous metals such as aluminum or copper; copper thickness, for example is measured in ounces, which represents the weight of copper contained in an area of one square foot. There are many different metals that can be made into sheet metal, such as aluminium, brass, copper, steel, tin, nickel and titanium. For decorative uses, important sheet metals include silver, gold, and platinum (platinum sheet metal is also utilized as a catalyst.) Sheet metal is used in automobile and truck (lorry) bodies, airplane fuselages and wings, medical tables, roofs for buildings (architecture) and many other applications. Sheet metal of iron and other materials with high magnetic permeability, also known as laminated steel cores, has applications in transformers and electric machines. Historically, an important use of sheet metal was in plate armor worn by cavalry, and sheet metal continues to have many decorative uses, including in horse tack. Sheet metal workers are also known as "tin bashers" (or "tin knockers"), a name derived from the hammering of panel seams when installing tin roofs.
PARTcloud - Sheetmetal
Licensed under Creative Commons Attribution-Share Alike 3.0 (Postdlf at the English language Wikipedia).
Sheet metal is metal formed by an industrial process into thin, flat pieces. It is one of the fundamental forms used in metalworking and it can be cut and bent into a variety of shapes. Countless everyday objects are fabricated from sheet metal. Thicknesses can vary significantly; extremely thin sheets are considered foil or leaf, and pieces thicker than 6 mm (0.25 in) are considered plate.
In most of the world, sheet metal thickness is consistently specified in millimeters. In the US, the thickness of sheet metal is commonly specified by a traditional, non-linear measure known as its gauge. The larger the gauge number, the thinner the metal. Commonly used steel sheet metal ranges from 30 gauge to about 7 gauge. Gauge differs between ferrous (iron based) metals and nonferrous metals such as aluminum or copper; copper thickness, for example is measured in ounces, which represents the weight of copper contained in an area of one square foot.
There are many different metals that can be made into sheet metal, such as aluminium, brass, copper, steel, tin, nickel and titanium. For decorative uses, important sheet metals include silver, gold, and platinum (platinum sheet metal is also utilized as a catalyst.)
3.3 Decambering
3.6 Hemming and seaming
3.8 Incremental sheet forming
3.11 Photochemical machining
3.12 Perforating
3.13 Press brake forming
3.15 Roll forming
3.18 Stamping
3.19 Water jet cutting
3.20 Wheeling
Grade 430 is popular grade, low cost alternative to series 300's grades. Used when high corrosion resistance is not a primary criteria. Common grade for appliance products, often with a brushed finish.
{\displaystyle F_{Max}=k{\frac {TLt^{2}}{W}}}
Clecos[18]
금속 탐지기기계공학Bending (metalworking)Honing (metalworking)Heat treatingMetalworkingSemi-finished casting productsShear (sheet metal)Indicator (distance amplifying instrument)Shearing (manufacturing)Blanking and piercingWeb (manufacturing)Roll slittingDie (manufacturing)
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Reminiscence of the Future... : Salvo Warfare-I.
People who think that I am on some sort of a crusade against the political "science" or what passes today for "history", or rather what it becomes once some "historian" begins to offer "the range of interpretations"... they are absolutely right. These two fields of human "academic" activity--and this is not my definition, many other people used and continue to use it way before me--are the fields in which credentials are bestowed upon primarily interpretations and personal (however "justified" with sources) opinions. But in history, at least, there is some inherent knowable truth which could be found, once layer upon layer of "interpretations" will be peeled off, especially when it is done by professionals who know the subject which constitutes this layer. This is not the case with political "science" which for the last decades produced a dearth of BS and failed to predict just about anything.
It is not surprising. Just take a look at the political "science" courses, say in Columbia University, and you will find there a hodgepodge collection of mostly "current events" theoretical BS which anyone with IQ higher than room temperature can get from media. Here is one "unit" which has some relevance to real world: DATA ANALYSIS & STATS-POL RES.
This course examines the basic methods data analysis and statistics that political scientists use in quantitative research that attempts to make causal inferences about how the political world works. The same methods apply to other kinds of problems about cause and effect relationships more generally. The course will provide students with extensive experience in analyzing data and in writing (and thus reading) research papers about testable theories and hypotheses. It will cover basic data analysis and statistical methods, from univariate and bivariate descriptive and inferential statistics through multivariate regression analysis. Computer applications will be emphasized. The course will focus largely on observational data used in cross-sectional statistical analysis, but it will consider issues of research design more broadly as well. It will assume that students have no mathematical background beyond high school algebra and no experience using computers for data analysis.
As you can see yourself--they give them a very basic math, which later finds its other incidence, buried in the pile of purely story-telling topics such as "ISRAELI NATIONAL SECURITY STRATEGY, POLICY AND DECISION MAKING", such as, and you have guessed it--Game Theory. Among all this disjoint collection of "stories" about politics the most remarkable is this: THEORIES OF WAR AND PEACE.
In this course we undertake a comprehensive review of the literature on the causes of war and the conditions of peace, with a primary focus on interstate war. We focus primarily on theory and empirical research in political science but give some attention to work in other disciplines. We examine the leading theories, their key concepts and causal variables, the causal paths leading to war or to peace, and the conditions under which various outcomes are most likely to occur. We also give some attention to the degree of empirical support for various theories and hypotheses, and we look at some of the major empirical research programs on the origins and expansion of war. Our survey includes research utilizing qualitative methods, large-N quantitative methods, formal modeling, and experimental approaches. We also give considerable attention to methodological questions relating to epistemology and research design. Our primary focus, however, is on the logical coherence and analytic limitations of the theories and the kinds of research designs that might be useful in testing them. This course is designed primarily for graduate students who want to understand and contribute to the theoretical and empirical literature in political science on war, peace, and security. Students with different interests and students from other departments can also benefit from the seminar and are also welcome. Ideally, members of the seminar will have some familiarity with basic issues in international relations theory, philosophy of science, research design, and statistical methods.
Wow! So, as you can see yourself it is a feeble attempt to provide some degree of legitimacy for political "science" graduates' opinions on war, by skipping every single subject which constitutes the foundation of modern war and, as I am on record ad nauseam here, it is higher math, physics and fundamental engineering and military courses ranging from radio-electronics to weapon systems integration, to combat applications, tactics, operational art and research and many other things of which political "scientists" never heard about, not to mention have no clue that such subjects do even exist. How about the theory of survivability of the ship or structure of combat communication networks? Don't hold your breath. Those graduates get a glimpse of Theory of Operations through some statistical methods and basic probabilities course, and then move on to study what anyone with a half-brain can read up on internet in several hours.
Yet, guess from three times who dominates in the modern West (especially in the US) the "discussion" on crucial issues of war and peace, military strategies and geopolitics? You bet, political "scientists" who, as I often use this expression, will not know the difference between LGBT and BTG, which is Battalion (or Brigade) Tactical Group. These are the people who continue to not only spread mostly incompetent sophomoric BS on warfare, they are the MAIN force behind shaping a discussion in the US on geopolitics and strategy, having zero competencies in what defines humanity's main tool of group-against-group survival which is groups' power (capability) and warfare. You can bet your ass also on the fact that this contingent of institutionalized ignoramuses, together with lawyers, constitute the main body of the US legislature and government officials. Recall utter embarrassing failure of all those "scientists" in 2016. How did your political "science" and "statistics" work out, eh?
And here is the main point--modern warfare is complex. Extremely complex. By modern I mean already highly motorized warfare of WW II, with massive mechanized armies, supported by the vast combat aviation fleets, massive naval armadas equipped with radar and sonar clashing on different theaters of operations and producing not only catastrophic destruction and human loses but gigantic volumes of combat data and correlates which not only contributed immensely to a development of tactical and operational models but accelerated technological development of war and its deadly instruments to a breakneck speed. In 1942 a graduate of the Soviet high school or lower college could get into the accelerated artillery officer program, complete it in a few months and be sent to the front line to face Wehrmacht and its panzers. In 1985 the study of missile-artillery officer in academy (officer school) would take full 5 years (6 academic years) with graduate degree in engineering and under-graduate in military science and would involve the study of weapons systems of immense complexity. Same was and is true for naval and air force officers.
Today, the same is done based on immensely complex and state-of-the-art academic facilities which unify in themselves latest in weapon systems of an immense power even with conventional explosives, and warfare today is defined by extremely complex combat networks, computers, some really mind-boggling sensors, instant propagation of information, neural networks, robotics, materials which even 20 years ago seemed inconceivable. Ranges of even what would be considered tactical weapons 40 years ago grew into thousands of kilometers, decision making is assisted by AI elements and battle management systems provide not only sensor fusion but probabilistic analysis of operations. How do you fight such a war. By studying the politics of Japan and basic Game Theory with Statistics? Of course not. Political "science" is simply outclassed by several orders of magnitude by warfare and its instruments, as well as applying "the lessons" from history to modern war and geopolitics is a fool's errand because at the Battle of Lepanto they didn't have to resolve the issue of uncertainties when developing firing solution by long-range supersonic missile salvo against Carrier Battle Group at 500 kilometers.
So, tactics and operations take the front seats and this is what constitutes the most important element of modern day geopolitics as we observe it through the lens of actions by nations-states or their alliances and is manifested in statements by leaders of the states, their ministers of foreign affairs, parliaments and, 99% full of shit, media. It all rests on military-economic power, period. The rest is merely an addition or iteration of what is known as Composite Index of National Capability and if I can build a better weapon and kill you with less damage to myself--this is exactly what constitutes the real national power and, as Den Xiaoping used to play with Clausewitz' famous dictum: "Diplomacy is a continuation of war by other means." You either have a weapon or you become an object (not a subject) of history and your only hope is that you don't become a meal for a hungry aggressive superpower.
Late legendary Captain Wayne Hughes understood it clearly and developed an incredible sense for both strategy and evolution of weapons. Not surprisingly, Hughes was a graduate of the US Naval Academy in 1952 in the times of naval titans the scale of Chester Nimitz, Arleigh Burke and, inevitably, later Elmo Zumwalt, who recognized, unlike many of his contemporaries, the changing nature of the (naval) warfare and was a keen observer of Admiral Gorshkov who built the Soviet Navy around missile weapon systems and that changed everything. Hughes wrote extensively about it and applied his very own Salvo Model to a new paradigm of the naval, and not only, warfare which he presented to a wider public in his famous treatise.
Salvo Equations', unlike Osipov-Lanchester model, deal with discrete values. That is the things which you can actually count in exchange, they are not continuous, such as, for example an infantry battalion under the artillery barrage where it is possible but extremely difficult to model losses because not only the barrage could be continuous (for an hour with unknown number of shells) but depend dramatically on the design of defensive positions capable to take some degree of damage and thus save lives. In the end, even awareness and running skills of soldiers could be a factor in such an ordeal, which makes it extremely difficult to predict. Here is how Lanchester's model looks like for combat:
This is not a nice looking set of differential equations where coefficients a and e define the rate of NON-combat losses, b and f define the rate of losses due to fire impact on areas, c and g define the rate of losses at the front line (immediate contact) and d and h are the numbers of arriving or withdrawing reserves. You see, a hot mess. And then, of course, you cannot shoot down every single artillery shell. Not so with missiles, which you can shoot down and which are discrete by their very nature, as are ships. If you have 5 ships in your task group--that is it. This is 5 ships and that is what gives Salvo Model an elegant and easily understood form. Not in embellished form, I underscore. Embellished Salvo Equations are a bit different animal and require a serious understanding of weapons, but I will touch upon those later. Here is basic Salvo Model for two hostile forces (fleets) A and B.
The beauty of this model is in the fact that it is not necessarily just a naval one. Missile exchange exists not only at the sea and between fleets. One can apply this model to exchange between defended base and combat air component attacking it. It is a classic missile exchange between discrete forces. We also will look into that, but for now it is clear that at this level of basic Salvo Equations one can easily "play" with them based on some assumptions and get a feel of how they work. Mathematically it is very simple and I did present some examples before elsewhere but let's play a bit. But to cool down your enthusiasm a little bit because of a seemingly simple math in this model, the math behind it is actually quite complex and salvo model is basically a tip of the iceberg and its application requires a serious tactical and operational (and engineering) knowledge which, of course, is beyond the grasp of political "scientists".
Here is a simple illustration of the damage analysis for a ship. In any naval academy the course on Theory of Ship Design (Construction) and Survivability is a two full years course, which involves not only a truckload of math and naval architecture but such earthly and prole things like closing damn holes in the hull being under the attack of incoming and roaring water, or extinguishing fires while shit around you explodes and burns--believe me, this is not fun. It looks good only in the movies.
Here is a general solution for a salvo by a submarine:
Or here is maneuvering on board for taking salvo position:
So, this is just a minuscule part of what is needed to fully grasp what is this all about in Salvo Model, not to speak of Embellished Salvo Equations. So, don't get cocky just yet. You will have the chance to get cocky once you will follow my blog and, of course, support (those well-off among you) me on Patreon.
Now to basic play. Let's assume that two forces A and B have equal number of ships, say 5. Thus: A=5 and B=5. Let's continue with other coefficients. Say force A's
\mathrm{\beta =4.}
\mathrm{Now, the following:}
a1 and b1 are what is generally known as omega(
\mathrm{needed\; to\; take\; a\; ship\; out\; of\; action.\; Let\text{'}s\; say\quad }
a1=2 and b1=1 (I use deliberately whole numbers to simplify the task) and now to
a3 which is, basically, effectiveness of A's air defense, which is the number of missiles fired by B destroyed by A's air defense. So, say a3=2 and for B we say same effectiveness: b3=2. Now we are ready to calculate. Let's start with A's losses:
\mathrm{\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \; \Delta }A=\frac{4\times 5-2\times 5}{2}=\frac{20-10}{2}=\frac{10}{2}=5
As you can see yourself A doesn\text{'}t fare that well--it gets completely destroyed, and loses all 5 ships. But what about B.
a1=2 andb1=2, that is being the same, force B could have won this exchange over A and would have retained 2.5 ships afloat. We round it and it is 3 ships--this is victory, a bloody one, but victory nonetheless. So, here we are, with some example of how simple this basic model is. But, of course, as you may have guessed it already, the devil is in those pesky details which define modern missile combat and that is a hell of a topic, which I intent on discussing...
Labels: Basic Salvo Model, Captain Wayne Hughes, Geopolitics, Missile exchange., political "science", warfare
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CoCalc – Online LaTeX Editor
Focus on writing LaTeX. CoCalc takes care of everything else.
\LaTeX
No software install required: 100% online
\LaTeX
editor supports
side-by-side preview with forward and inverse search,
compiles upon saving and marks errors in the source file,
periodically backups all your files,
runs embedded calculations right inside your document,
multi-file support that discovers included files automatically, and
every change is recorded while you type.
Working with
\LaTeX
Tired of sending changes back and forth with your colleagues?
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\LaTeX
environment?
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\LaTeX
\LaTeX
CoCalc makes sure that your desired
\LaTeX
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Collaborative editing without limits
Privately share your project with an unlimited number of collaborators. Simultaneous modifications of your document are synchronized in real time. You see the cursors of others while they edit the document and also see the presence of watching collaborators.
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\LaTeX
Consult the Available Software page or look at our Jupyter Notebook page for more information.
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\LaTeX
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\LaTeX
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\LaTeX
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\LaTeX
source to the corresponding part in the rendered preview. This saves you time looking for the output.
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Hecke algebras, finite general linear groups, and Heisenberg categorification | EMS Press
We define a category of planar diagrams whose Grothendieck group contains an integral version of the infinite rank Heisenberg algebra, thus yielding a categorification of this algebra. Our category, which is a
q
-deformation of one defined by Khovanov, acts naturally on the categories of modules for Hecke algebras of type
A
and finite general linear groups. In this way, we obtain a categorification of the bosonic Fock space. We also develop the theory of parabolic induction and restriction functors for finite groups and prove general results on biadjointness and cyclicity in this setting.
Erratum: Due to a technical error, many lines in the diagrams were printed without indication of the corresponding direction (i.e. as simple lines instead of arrows). A corrected electronic version of the article can be downloaded here.
Anthony Licata, Alistair Savage, Hecke algebras, finite general linear groups, and Heisenberg categorification. Quantum Topol. 4 (2013), no. 2, pp. 125–185
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Analysis of the free boundary for the $p$-parabolic variational problem $(p\ge 2)$ | EMS Press
Analysis of the free boundary for the
p
-parabolic variational problem
(p\ge 2)
Variational inequalities (free boundaries), governed by the
p
-parabolic equation (
p\geq 2
), are the objects of investigation in this paper. Using intrinsic scaling we establish the behavior of solutions near the free boundary. A consequence of this is that the time levels of the free boundary are porous (in
N
-dimension) and therefore its Hausdorff dimension is less than
N
. In particular the
N
-Lebesgue measure of the free boundary is zero for each
t
-level.
Henrik Shahgholian, Analysis of the free boundary for the
p
(p\ge 2)
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Demand Factors - DAFI Protocol
The mathematical adoption of a network
dDAFI = DAFI * demandFactor
However, in this first v1 dDAFI launch, we have chosen Price and TVL as the demand metrics. The chosen target levels are $0.18 and 500M, for Price and TVL respectively. The TVL metric will only be added in Q4 after launching.
It’s important to note that these levels can be extended and modified periodically, as a network becomes more adopted.
demandFactor = ((currentPrice / baselinePrice) * 0.75) + ((currentTVL / baselineTVL) *0.25)
demandFactor => 0.10, or =< 1.00
quantityChange = dDAFI * (newDemandFactor / oldDemandFactor)
Onchain and Market data is provided through oracle feeds, sourced from Chainlink, DIA and Band as close partners.
Each time a user initiates an action within the Staking Network (e.g. Stakes, Unstakes, Claims) it acts as a trigger, which calculates the latest demandFactor, and updates the demandFactor across the whole network automatically.
Also, the demandFactor cannot decline below 0.10 or rise above 1.00. The former condition is to ensure sharp declines in any market do not lead to zero rewards. The latter is to ensure that the maxTokens allocated are never exceeded.
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The probability of long cycles in interchange processes
15 June 2013 The probability of long cycles in interchange processes
Gil Alon, Gady Kozma
We examine the number of cycles of length
k
in a permutation as a function on the symmetric group. We write it explicitly as a combination of characters of irreducible representations. This allows us to study the formation of long cycles in the interchange process, including a precise formula for the probability that the permutation is one long cycle at a given time
t
, and estimates for the cases of shorter cycles.
Gil Alon. Gady Kozma. "The probability of long cycles in interchange processes." Duke Math. J. 162 (9) 1567 - 1585, 15 June 2013. https://doi.org/10.1215/00127094-2266018
Secondary: 20B30 , 43A65 , 60B15
Gil Alon, Gady Kozma "The probability of long cycles in interchange processes," Duke Mathematical Journal, Duke Math. J. 162(9), 1567-1585, (15 June 2013)
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Home : Support : Online Help : Education : Student Packages : Statistics : ShapiroWilkWTest
ShapiroWilkWTest(X, level_option, output_option)
The ShapiroWilkWTest function computes Shapiro and Wilk's W-test applied to a data sample X. This tests whether X comes from a normally distributed population.
The first parameter X is the data sample to use in the analysis. It should contain between
3
2000
\mathrm{with}\left(\mathrm{Student}[\mathrm{Statistics}]\right):
S≔\mathrm{Sample}\left(\mathrm{NormalRandomVariable}\left(5,2\right),10\right):
T≔\mathrm{Sample}\left(\mathrm{UniformRandomVariable}\left(4,6\right),10\right):
\mathrm{ShapiroWilkWTest}\left(S,\mathrm{level}=0.05\right)
[\textcolor[rgb]{0,0,1}{\mathrm{hypothesis}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{true}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{pvalue}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0.856735713777028}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{statistic}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0.967478742211855}]
\mathrm{ShapiroWilkWTest}\left(T,\mathrm{level}=0.05\right)
[\textcolor[rgb]{0,0,1}{\mathrm{hypothesis}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{false}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{pvalue}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0.0351513590317937}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{statistic}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0.832591474899495}]
\mathrm{ShapiroWilkWTest}\left(S,\mathrm{level}=0.05,\mathrm{output}=\mathrm{plot}\right)
\mathrm{report},\mathrm{graph}≔\mathrm{ShapiroWilkWTest}\left(T,\mathrm{level}=0.05,\mathrm{output}=\mathrm{both}\right):
Data Range: 4.0095669690037 .. 5.99292266316546
Bin Width: .0661118564720588
\mathrm{report}
[\textcolor[rgb]{0,0,1}{\mathrm{hypothesis}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{false}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{pvalue}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0.0351513590317937}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{statistic}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0.832591474899495}]
\mathrm{graph}
The Student[Statistics][ShapiroWilkWTest] command was introduced in Maple 18.
Student/Statistics/ShapiroWilkWTest/overview
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Applications of Hofer's geometry to Hamiltonian dynamics | EMS Press
Applications of Hofer's geometry to Hamiltonian dynamics
We prove that for every subset
A
of a tame symplectic manifold
(W,\omega)
meeting a semi-positivity condition, the
\pi_1
-sensitive Hofer--Zehnder capacity of
A
is not greater than four times the stable displacement energy of
A
c_{HZ}^\circ(A,W)\le 4e (A\times S^1, W\times T^*S^1).
This estimate yields almost existence of periodic orbits near stably displaceable energy levels of time-independent Hamiltonian systems. Our main applications are:
\bullet
The Weinstein conjecture holds true for every stably displaceable hypersurface of contact type in
(W,\omega)
\bullet
The flow describing the motion of a charge on a closed Riemannian manifold subject to a non-vanishing magnetic field and a conservative force field has contractible periodic orbits at almost all sufficiently small energies. The proof of the above energy-capacity inequality combines a curve shortening procedure in Hofer geometry with the following detection mechanism for periodic orbits: If the ray
\{\varphi_F^t \}
t \ge 0
, of Hamiltonian diffeomorphisms generated by a compactly supported time-independent Hamiltonian stops to be a minimal geodesic in its homotopy class, then a non-constant contractible periodic orbit must appear.
Felix Schlenk, Applications of Hofer's geometry to Hamiltonian dynamics. Comment. Math. Helv. 81 (2006), no. 1, pp. 105–121
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Jamie was given the problem, “Find the result when the factors of
65x^2+212x-133
are multiplied together.” Before she could answer, her sister, Lauren, said, “I know the answer without factoring or multiplying!” What was Lauren's answer and how did she know?
The result must be the original expression because multiplying and factoring are opposite processes.
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Explosion - Uncyclopedia, the content-free encyclopedia
Whoops! Maybe you were looking for Awesomeness, or Michael Bay?
~ Captain Obvious on explosions
Small children are especially fond of explosions.
An explosion is the rapid combustion of megahurtz, and can be recognised by such things as smoke, fire and the rather unpleasant smell of CGI. Explosions are mostly caused by trying to flush a glock down the toilet while dropping a lit match. Despite their regular appearances in such countries as Iraq and Movieland, explosions can technically be classified as "rare". Zoologist Alan Shepard explains this: "Explosions are penis because, uumm, you know your kitchen? You see explosions less than your kitchen. Yeah." Explosions are the mortal enemies of lasers, and some people think that years ago, long before the birth of mankind and the sticky placenta that followed, lasers and explosions duked it out in a vicious battle for supremacy to see who could be totally awesomer. The life cycle of explosions is peculiar, as it starts of with a sort of larvae called "bombs" or "explosives", or "everything" if you are in a high-budget action flick. Likewise, the cycle ends with remenants of the explosion, named "dead henchmen", strewn across the location of detonation.
1 In mathematical terms
6 The Truce (aka, the pussy years)
6.2 Explosions in Modern Society
In mathematical terms[edit]
Ummm, whoops.
Several penises have offered partial explanations into the maths of explosions, with the most popular theory being that when an object (a) interacts with another object (b), there is a fractional part-chance that either a or b will explode. This potentiality is factorising of a number of aspectations, utmost of which is the quantum mass of each object divided by y-14. Y is a situational coefficient known as the Willis Number. This number ranges from -14 to 14, approaching 14 with the increasing wit in one's dialogue.
If qa+qb/(y-14)>36.24, the combusting energies are sufficient to cause an explosion if both objects are experiencing local gravity equal to or less than that of sea-level. The velocity of the new smaller objects (c, d,e and f) can then be penised by the following:
{\displaystyle x(c+d+e+f)-(qa/qb)/cz.}
Of course, a number of factors can affect the calculation, such as mb (Michael Bay) or n (nuclear weapons). Such factors can do everything from cancelling out c through to f (assuming that they automatically include the airborne alteration) and lives of unsuspecting henchmen, to "screwing the maths and blowing everything the fuck up already".
The most common effect of an explosion is the general amusement of pyromaniacs (see idiot) who were smart enough to view from a safe distance. However there are various others effects of explosions, including the incredibly short lifespan of organisms within, reports on CNN and the general creation of smaller pieces of nonliving objects. The production of smaller objects from larger ones is not always the result of an explosion however. A simple way to tell is to measure how far apart the new objects are, as explosions tend to cause these newer objects to fly about all over the place.
Other signs include the mean temperature of the objects (which scientists think is higher after an explosion, probably due to magic) and how much radiation you absorbed while measuring them (this itself can be measured with a geiger counted, or a friend with a camera and a kilogram of Froot Loops). If the level of radiation causes sudden hair loss and/or an embarrassing rash, it is generally safe to assume that the new objects were created by a thermonuclear explosion, much like the one which occurred at Gillgan's Island, or the ones at Hiroshima. Note that visiting these areas for comparable results is not a good idea.
Fish tank water has been known to explode without warning.
There are many ways to bring about an explosion, such as being highly flammable, but the altogether most common way is to pop a virgin (sometimes called Cherry Bombs), although retrieval of these berries can be dangerous. In such a situation, it is best ask an explosion-proof faerie to pop them for you. In addition, 90% of BFGs produce some form of explosion.
More difficult methods include digitally creating one with a computer and a $200 million budget. Another solution is to join Al-Qaida, although this is maily frowned upon by law enforcers.
Despite what many stupid people believe, explosions are actually sentient creatures. This theory has been met with much argument from many scientists woldwide, but it is the only theory anyway. Explosions are well known as large yet timid creatures and can often be found hanging out with gasoline, dynamite, C4, whales, and human heads. When angered, they unleash a fearsome defense mechanism consisting of totally ripping the shit out of anything in their path. Much like bees, explosions die shortly after their attack.
The only other cause of explosions is the presence of Chuck Norris, while women melt in the path of other, fine specimens of men, women have been known to have their vital organs ripped out of their body in an instant by the absolute power of Chuck. This sometimes has a similar effect on gay men, but straight men have never been known to explode in the presence of Chuck, a thought of Chuck must be provided to instantly sever their head. Explosions may also be caused by Chuck Norris' pee. This is, in fact, one of his favorite passtimes, and as you can guess his bathroom has no roof.
Throughout the war, the two sides fought viciously and often recruited outside forces for help. Lasers are known for their deadliness when combined with sharks, while the explosions saw a short-term alliance with Oprah Winfrey. The latter was not generally accepted by the majority of explosions, but their government told them all to shut the fuck up and stop bitching, so they did.
A rarely seen glimpse into the ferocious war between lasers and explosions. Cameras sucked back then.
Lots of stuff happened and stuff and there were all sorts of weapons and fights and stuff that would just totally blow your mind if you were to ever find out but you can never find out because your puny brain will never be able to handle such vast levels of awesomeness.
The Truce (aka, the pussy years)[edit]
After the many centuries of chaos and destruction, the lasers decided they had had enough. Supplies were scarce and morale was low among the troops. They conceded in 212402 B.C. to the explosions, who celebrated their newfound victory with pie and a nude party. It was hot.
Because of their victory, explosions were discovered long before lasers by humanity. The lasers actually accepted this without a fight because it was generally agreed upon that the explosions were more fit to blow stupid people's limbs and faces off. The discovery of explosions in 4 B.C. allowed them to be commercially available to Jesus Christ upon his birth. Christ, a now-known exposions fanatic, commonly blew many things up, including birds, bushes, and homeless people. Christ would later comment that "Dis shit be tight."
Explosions in Modern Society[edit]
Explosions nowadays are commonly hated upon by such racist groups as hippies and anti-war protesters. They are feared and hated for their power, but are really just misunderstood, like sharks and Paul W.S. Anderson. George Bush is a well-known advocate for explosions and adopted one as his child in 2003.It's name? What else? Karl Rove. Says Bush, "That sumbitch Kerry thinks explosions should be put in a concentration camp. Also, he punches babies."
Some terrorists at a picnic.
Explosions are used extensively by such demographics as the military, construction and demolition companies, and that really creepy guy at the office who's always muttering to himself and has the AK-47 poster up on his cubicle wall beside the KA-BAR with "Vengeance" etched in the blade.
Toy company "Mattel" recently released their "Barbie hostage crisis" playset, complete with radical militant soldiers and actual explosive ordnance. Interestingly, statistics show a 76% increase in injuries related to slipping on the bloodstains of small children and bruising from falling while trying to clean little Susan off the ceiling. Researchers are baffled as to the cause of this or whether these statistics are in any way related to the toy company, though FOX News suggests, "It was probably those liberal faggots again." A spokesman for Mattel claimed that "there was nothing to worry about" and that "you mortals can go about your meaningless lives" shortly before being sucked back into the pit of despair.
Oprah and Her Evil
Aweplosion
Oh, and-- never mind.
Retrieved from "https://uncyclopedia.com/w/index.php?title=Explosion&oldid=6026438"
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Rational Numbers, Popular Questions: CBSE Class 7 MATH, Integers - Meritnation
Harine Anand & 2 others asked a question
What is difference between rational number and fraction
Find all rational numbers whose absolute value is less than 4
Represent the following rational numbers on number line.
i) -3/5.
ii) 7/4.
FIND THE SUM OF 1/2 + 1/6 + 1/12+ 1/20 + 1/30 + 1/42 + 1/56 + 1/72 + 1/90 + 1/110 + 1/132
-2/5-(-3/10)-(-4/15)
Sir with Method Please
Rebecca & 1 other asked a question
Fawaz Abdulla Kutty asked a question
-3 2/7 or -3 4/5
explain how to do
Thanushree.e Thanu & 1 other asked a question
Samir Kumar asked a question
FIND FIVE RATIONAL NUMBERS BETWEEN 6/7 AND 8/9. FIND 10 RATIONAL NUMBERS BETWEEN -1 AND 1.
Pranabesh Bhattacharya & 1 other asked a question
Tanmayi Vaishnavi asked a question
25 bags of wheat each weighing 40kg cost rupess2750.find the cost of 35 bags of whea,if each weighing 50kg
Calculate the following. 4/5 of 3/7 + 4/5 of 3/7 using property
Parth Mishra asked a question
A submarine is at a depth of 2 1/3km below the sea level and if it starts ascending at the rate of 800m/her then find its position after two hours?
Uthra Muthusubramanian asked a question
find the absolute value of the following rational numbers:
(a)1/-5 (b)7/9 (c)0/-4 (d)-3/-2
Advaith Narayan asked a question
What value do you learn from rohan
Abhimanyu Singh asked a question
mahi spends 1/7 of her income on rent 1/5 on groceries and save the rest. If RS 3450,what is her income
Arya A. Doshi & 1 other asked a question
cheena spent 3/4 of her pocket money. She spends 1/2 of it on a book 1/6 on a movie and rest for a dress. whap part of her pocket money did she spend on the dress.
a car covers a distance of 189 1/3 kilometres in 4 4/9 hours find the distance covered in one hour
Rishita Rathi asked a question
list four rational numbers between 5/6 and 7/8.
___ ? 3/7 = -1
What is the answer ???
Smiruti asked a question
Samarth Sharma asked a question
Find the product of 8/35 x 21/-35
Frame an equation and solve: The sum of 13 and an unknown number is 46. What is the number?
Anusha Shishir Paranjpe asked a question
A man sold 1/2 of his land. He gave 1/2 of the remaining portion to his son and 1/2 of the remaining to his daughter. What fraction of his land is left with him
Calculate the following -4/5 of 3/7 + 4/5 of 3/7 using property
Angkuran Dey asked a question
This question is about Ratio. If A : B = 5 : 6 and B : C = 8 : 9 then find A : B : C?
How do you solve this sum??
Sai Sandhya asked a question
what is the moral value behind the concept rational numbers?
Vijay Nandwani asked a question
How to find rational numbers between:
-1 and 0(five)=
-2 and -1(five)=
Q.)find five rational numbers between 4/5 and 6/7 & I-7/18I AND 4/9.
Kousik Satish asked a question
Chirag Jain asked a question
1. By what rational nos should -7/85 be multiplied to obtain 1/17
2.divide the sum of 3/8 and -5/18 by the reciprocal of -15/8*16/27
3. the perimeter of an isocles triangle is 10 3/4cm if one of its equal sides is 2 5/6cm find the 3rd side.
Julie asked a question
Raghav Galgali asked a question
how many rational numbers lie between -1/3 and -1/2
Roshin George asked a question
Subtract the sum of -36/11 and 49/22 from the sum of 33/8 and -19/4.
Vaishnavi Pawar asked a question
Ansh Jaitly asked a question
Do both -5/7 and 5/-7 represent the same negative rational number?
Ishita Saxena asked a question
what is standard form in rational numbers?explain briefly
Anubhav Verma asked a question
how to get standard form with both negative integer like -4 and -3
S.vaishnav Ghautham asked a question
Roshani reads 1/6 hours on the first day 1/4 hours on the second day and 1/3 hours on the third day.if the pattern continues,how long will she read on 5thh day?please fast tomorrow is my exam!
asadz36 asked a question
a person infected with bacteria was prescribed medication. The number of bacteria in his body was 100. After 3 days of medication , the number reduced to 40. Find the rate of bacteria reduction in terms of per hour. ( Hint will be negative in the case.)
if 25,35,x are in continued proportion,the value of x is
Sarthak Manojkumar Jaiswal asked a question
How to represent -11/3 on the number line?
Divide Rs. 2500 into two parts such that the simple interest on one at 4% for 5 years in double that on the other at 5 % for 3 years.
Simplify(-7/6) + 3/8+ 1/4
Pushpa asked a question
find five rational numbers between 2.5 and 2.7
Express the 4/7 in rational denominator
I. 21. Il.-35
Nandeesh asked a question
who is the father of rational numbers
Q). Is the following in a standard form
\frac{-4}{-15}
Charul asked a question
By what rational number should we divided 22/7 , so to get the number -11/24
Please tell me about The History of Rational Numbers. Quick!!
Harsh Modi asked a question
list 5 rational numbers between -1 and 0
Mohammad Faiz asked a question
express the following rational numbers in standard form:
Isfar Syed & 1 other asked a question
Q: Cathy spent 3/4 of her pocket money. If she spent 1/3 of it for a movie, 1/6 for books and the rest for a dress, what part of her pocket money did she spent on the dress?
Can you please assist me to get it done?
A number of bacteria are placed in a glass. One second later, each bacteria divides two and so on. After one minute the glass is full. When was the glass half empty????
the product of two rational number is -8 by nine .if one of the number is -4 by 15. find the other number
Arnav Saini asked a question
give a creative slogan or poem on rational number, fraction ,decimal ,integer
Jishnu Bayan asked a question
Find a rational nmber which in its standard form is equal to 4/5 and the sum of its numerator and denominator is 27.
find the decimal form of 9/8
Express 4/-7 as a equivalent rational number with denomenator.
(1)84 (2) -28
By what rational numbers should -8/39 be multiplied to obtain 5/26 ?
Ruchika Jawahar asked a question
Wins Brar asked a question
Represent the speeds at different intervals in the form of a rational numbers on different number lines. please solve it?
Divide the sum of 12/5 and 21/25 by their difference
divide rs 60 in the ratio of 1:2 between kirti and kamal
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Upsampling - Imaging Artifacts - MATLAB & Simulink - MathWorks Benelux
This example shows how to upsample a signal and how upsampling can result in images. Upsampling a signal contracts the spectrum. For example, upsampling a signal by 2 results in a contraction of the spectrum by a factor of 2. Because the spectrum of a discrete-time signal is
2\pi
-periodic, contraction can cause replicas of the spectrum normally outside of the baseband to appear inside the interval
\left[-\pi ,\pi \right]
\left[-\pi ,\pi \right]
Upsample the signal by 2. Plot the spectrum of the upsampled signal. The contraction of the spectrum has drawn subsequent periods of the spectrum into the interval
\left[-\pi ,\pi \right]
y = upsample(b,2);
text(0.65*[-1 1],0.45*[1 1],["\leftarrow Imaging" "Imaging \rightarrow"], ...
fir2 | freqz | upsample
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Pool Weights - DAFI Protocol
Super Staking Weights
Like most Staking models, the program can indeed run for certain time-periods. For v1 dDAFI we will likely run the program for 1 month, before extending it further.
To create a lightweight architecture, we’re able to track users rewards through weights.
currentPoolWeight = (distributePerSecond * currentTime – lastCalculatedTime) / totalStaked
It calculates how much dDAFI is being distributed within the elapsed time per staked DAFI across the entire pool. It is then added to an ongoing pool weight.
accumulatedPoolWeight = lastAccumulatedWeight + newWeight
Each time any user behaves within the protocol, by staking, unstaking or claiming, the weight is recalculated for the whole network. One user simply acts, and everyone is synchronised.
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Erratum: “Stability Analysis of an Inflatable Vacuum Chamber” [Journal of Applied Mechanics, 2008, 75(4), p. 041010] | J. Appl. Mech. | ASME Digital Collection
Erratum: “Stability Analysis of an Inflatable Vacuum Chamber” [Journal of Applied Mechanics, 2008, 75(4), p. 041010]
This is a correction to: Stability Analysis of an Inflatable Vacuum Chamber
(November 12, 2008). "Erratum: “Stability Analysis of an Inflatable Vacuum Chamber” [Journal of Applied Mechanics, 2008, 75(4), p. 041010]." ASME. J. Appl. Mech. January 2009; 76(1): 017001. https://doi.org/10.1115/1.2966273
mechanical stability, vacuum apparatus
Engineering mechanics, Stability, Vacuum, Mechanical stability
Page 3, 2nd column, 9th line: “
E≡cos(θ∕2)
” should read “
E≡(cosθ)∕2
Page 3, 2nd column, 10th line: “
ε≡sin(θ∕2)
ε≡(sinθ)∕2
−(SC+2εRQ)
(−SC+2εRQ)
Page 4, 1st column, 2nd line: “the dagger denotes” should read “where the dagger denotes.”
Page 7, last line of “Nomenclature”: “
E≡cos(θ∕2)
E≡(cosθ)∕2
Biped RCCR Mechanism
The Stability of Veins Under Torsion
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Dubois torsion, <var>A</var>-polynomial and quantum invariants | EMS Press
Dubois torsion, <var>A</var>-polynomial and quantum invariants
National Science Foundation, Arlington, USA
It is shown that for knots with a sufficiently regular character variety the Dubois torsion detects the
A
-polynomial of the knot. A global formula for the integral of the Dubois torsion is given. The formula looks like the heat kernel regularization of the formula for the Witten–Reshetikhin–Turaev invariant of the double of the knot complement. The Dubois torsion is recognized as the pushforward of a measure on the character variety of the double of the knot complement coming from the square root of Reidemeister torsion. This is used to motivate a conjecture about quantum invariants detecting the
A
-polynomial.
Charles D. Frohman, Joanna Kania-Bartoszynska, Dubois torsion, <var>A</var>-polynomial and quantum invariants. Quantum Topol. 4 (2013), no. 2, pp. 187–227
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Poissonian products of random weights: Uniform convergence and related measures | EMS Press
Poissonian products of random weights: Uniform convergence and related measures
Domaine de Voluceau, Le Chesnay, France
The random multiplicative measures on
\mathbb{R}
introduced in Mandelbrot ([Mandelbrot 1996]) are a fundamental particular case of a larger class we deal with in this paper. An element
\mu
of this class is the vague limit of a continuous time measure-valued martingale
\mu _{t}
, generated by multiplying i.i.d. non-negative random weights, the
(W_M)_{M\in S}
, attached to the points
M
of a Poisson point process
S
, in the strip
H=\{(x,y)\in \mathbb{R}\times\mathbb{R}_+ ; 0 < y\leq 1\}
of the upper half-plane. We are interested in giving estimates for the dimension of such a measure. Our results give these estimates almost surely for uncountable families
(\mu ^{\lambda})_{\lambda \in U}
of such measures constructed simultaneously, when every measure
\mu^{\lambda}
is obtained from a family of random weights
(W_M(\lambda))_{M\in S}
W_M(\lambda)
depends smoothly upon the parameter
\lambda\in U\subset\mathbb{R}
. This problem leads to study in several sense the convergence, for every
s\geq 0
, of the functions valued martingale
Z^{(s)}_t: \lambda \mapsto \mu_{t}^{\lambda }([0,s])
. The study includes the case of analytic versions of
Z^{(s)}_t(\lambda)
\lambda\in\mathbb{C}^n
. The results make it possible to show in certain cases that the dimension of
\mu^{\lambda}
depends smoothly upon the parameter. When the Poisson point process is statistically invariant by horizontal translations, this construction provides the new non-decreasing multifractal processes with stationary increments
s\mapsto \mu ([0,s])
for which we derive limit theorems, with uniform versions when
\mu
\lambda
Julien Barral, Poissonian products of random weights: Uniform convergence and related measures. Rev. Mat. Iberoam. 19 (2003), no. 3, pp. 813–856
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Close Location Value (CLV) Definition
What Is Close Location Value?
What Does CLV Tell You?
Example of How to Use CLV
Difference Between CLV and A/D
Limitations of Using CLV
Formula for CLV
How Do You Identify Accumulation and Distribution?
How Do You Read an Accumulation/Distribution Line?
What Is Accumulation in Technical Analysis?
What Is Close Location Value (CLV)?
Close location value (CLV) is a metric utilized in technical analysis to assess where the closing price of a security falls relative to its day's high and low prices.
Close location value values range from +1.0 to -1.0; a higher positive value indicates the closing price is nearer to the day's high price and a greater negative value indicates that the closing price is nearer to the day's low price.
Close location value (CLV) indicates an asset's closing price relative to its intraday high and low.
A positive value means the closing price is closer to the day's high price, whereas a negative value means the closing price is closer to the day's low.
CLV values of +1 would mean the closing price is the same as the day's high, and -1 the day's low.
CLV is used in conjunction with other indicators.
What Does Close Location Value (CLV) Tell You?
Close location value (CLV) is a technical analysis tool that measures the location of the price in relation to the high-low range. It moves in the range from -1 to +1, or, if multiplied by 100, in the range from -100% to +100%.
Close location value (CLV) readings close to 1 (or 100%) indicate that the closing price is near its high and would be considered a bullish sign. CLV readings close to -1 (or -100%) reveal that the closing price is near its low and might be considered a bearish sign. CLV readings that are close to zero are considered neutral.
Example of How to Use Close Location Value (CLV)
On its own, the close location value (CLV) is not considered to be very important by most traders. This indicator is primarily used as a variable in other technical equations.
The CLV features prominently, for example, in the calculation for the accumulation/distribution (A/D) line (also called the accumulation/distribution indicator [A/D] ):
\begin{aligned} &\text{Accumulation/Distribution} = \text{CLV} \times \text{Period's Volume} \\ \end{aligned}
Accumulation/Distribution=CLV×Period’s Volume
When not part of another equation, the close location value (CLV) can also be used to confirm or reject possible divergences. Keep in mind: It is advisable for any traders using this strategy to use an intermediate or long-time period for their CLV (because this allows enough of a historical perspective to prevent overreacting to every fluctuation).
The Difference Between Close Location Value (CLV) and the Accumulation/Distribution Indicator (A/D)
In general, the goal of technical analysis is to predict future price movements; this is accomplished by studying the activity of price and volume in a given security.
The accumulation/distribution indicator (A/D) is one of many different volume indicators. Volume indicators are mathematical formulas that are visually represented in some of the most commonly used charting platforms. Volume is often viewed as an indicator of liquidity because stocks or markets with the most volume are the most liquid and considered the best for short-term trading. (In other words, there are many buyers and sellers ready to trade at various prices.)
The accumulation/distribution indicator (A/D) uses volume and price to assess whether a stock is being accumulated or distributed and seeks to identify divergences between the stock price and the volume flow.
Close location value (CLV) indicates an asset's closing price relative to its intraday high and low. Close location value (CLV) is used in the calculation for the accumulation/distribution (A/D) line.
Limitations of Using Close Location Value (CLV)
One reason why the close location value (CLV) is not considered useful on its own is that it is extremely sensitive to random spikes or drops in prices. This heightened volatility renders it almost useless in many circumstances. (As a reliable high-low relationship metric, stochastics are often preferred—instead of close location value (CLV)—because they are less choppy and rely on a different formula to determine price location in the high-low range.
The Formula for Close Location Value (CLV) Is
\begin{aligned} &\text{CLV} = \frac { ( \text{Close} - \text{Low} ) - ( \text{High} - \text{Close} ) }{ \text{High} - \text{Low} } \\ \end{aligned}
CLV=High−Low(Close−Low)−(High−Close)
In technical analysis, the accumulation/distribution (A/D) is an indicator that creates a relationship between changes in price and volume. The more volume that accompanies a price move, for example, the more significant (it can be assumed) that price move is. Analysts use accumulation/distribution (A/D) to confirm changes in prices by comparing the volume associated with prices.
When analyzing volume patterns, accumulation is (essentially) buying, and distribution is (essentially) selling. If there is a high level of demand for a stock, it is being accumulated. Distribution refers to when a stock shows more supply than demand in the form of distribution.
The purpose of an accumulation/distribution (A/D) line is to help assess price trends and potentially spot forthcoming reversals.
If the price of a security is in a downtrend while the A/D line is in an uptrend, for example, the indicator shows there may be buying pressure; the security’s price may reverse to the upside. Conversely, if the price of a security is in an uptrend while the A/D line is in a downtrend, then the indicator shows there may be selling pressure (or higher distribution). This warns that the price may be due for a decline.
In general, a rising accumulation/distribution (A/D) line helps confirm a rising price trend, while a falling A/D line helps confirm a price downtrend.
Traders often use volume as a way to assess the significance of changes in a security's price. Volume refers to the number of shares traded during a particular time period.
When analyzing volume patterns, accumulation is (basically) when a share is being bought, and distribution is (basically) when a specific share is being sold.
On-balance volume (OBV) is a momentum indicator that uses volume flow to predict changes in stock price.
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A genericity theorem for algebraic stacks and essential dimension of hypersurfaces | EMS Press
We compute the essential dimension of the functors Forms
_{n,d}
and Hypersurf
_{n, d}
of equivalence classes of homogeneous polynomials in
variables and hypersurfaces in
\mathbb P^{n-1}
, respectively, over any base field
k
0
. Here two polynomials (or hypersurfaces) over
K
are considered equivalent if they are related by a linear change of coordinates with coefficients in
K
. Our proof is based on a new Genericity Theorem for algebraic stacks, which is of independent interest. As another application of the Genericity Theorem, we prove a new result on the essential dimension of the stack of (not necessarily smooth) local complete intersection curves.
Zinovy Reichstein, Angelo Vistoli, A genericity theorem for algebraic stacks and essential dimension of hypersurfaces. J. Eur. Math. Soc. 15 (2013), no. 6, pp. 1999–2026
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The torsion index of the spin groups
15 August 2005 The torsion index of the spin groups
1Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, University of Cambridge
We compute Grothendieck's torsion index of the compact Lie group
\mathrm{Spin}\left(n\right)
n
. We explain the applications of the torsion index to topology and to the study of splitting fields for quadratic forms.
Burt Totaro. "The torsion index of the spin groups." Duke Math. J. 129 (2) 249 - 290, 15 August 2005. https://doi.org/10.1215/S0012-7094-05-12923-4
Burt Totaro "The torsion index of the spin groups," Duke Mathematical Journal, Duke Math. J. 129(2), 249-290, (15 August 2005)
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CoCalc – Use SageMath Online
Use SageMath Online
The goal of SageMath is to create a viable free open source alternative to Magma, Maple, Mathematica and Matlab by building on top of many existing open-source packages, including NumPy, SciPy, matplotlib, SymPy, Maxima, GAP, FLINT, and R.
Try SageMath Now
Run SageMath on CoCalc
Use CoCalc's own realtime collaborative Jupyter Notebooks.
Use SageMath from the collaborative, Linux Terminal or virtual X11 graphical Linux desktop.
Use Sage optimized Worksheets, which provide a single document experience that can be more friendly than the Jupyter notebook "multiple cells" approach.
Easily embed Sage code in your
\LaTeX
documents.
Install almost any Python package for use with Sage:
sage --pip install package_name
Benefits of working with SageMath online
You no longer have to install and maintain SageMath, which can be challenging since Sage is large. When you're teaching a class, students just have to sign in to CoCalc to get started!
You can still easily run older versions of Sage since many are all preinstalled in every CoCalc project.
All your files are private, stored persistently, snapshotted and backed up; moreover, you can rsync them to your computer or push them to GitHub.
You can invite collaborators to your project to simultaneously edit the same notebooks or code files.
Everything runs remotely, which means you do not have to worry about messing up your own computer.
There are many ways to use SageMath online via CoCalc.
Teach using SageMath and Nbgrader
CoCalc's integrated course management system fully supports using nbgrader together with SageMath Jupyter Notebooks.
We provide custom Python templates for all the nbgrader cell types.
Tests run in the student's project by default, so malicious code won't impact anybody except the student.
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Reward Distributions - DAFI Protocol
Pools, not the kind for swimming.
The poolBalance is distributed to users staking. It is initially supplied as native-tokens (DAFI) which is locked in the distributionContract, this is defined as maxTokens.
Pool Balance = (maxTokens – totalDistributed + withdrawalFees) * demandFactor
This is later used, to understand how much dDAFI will be distributed per second, to the entire network.
distributePerSecond = poolBalance / programDuration
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Pressure-maintaining valve for external component in an isothermal system - MATLAB
Pressure Compensator Valve (IL)
Valve Opening Parametrization
Pressure-maintaining valve for external component in an isothermal system
Simscape / Fluids / Isothermal Liquid / Valves & Orifices / Pressure Control Valves
The Pressure Compensator Valve (IL) block represents an isothermal liquid pressure compensator, such as a pressure relief valve or pressure-reducing valve. Use this valve when you would like to maintain the pressure at the valve based on signals from another part of the system.
When the pressure differential between ports X and Y (the control pressure) meets or exceeds the set pressure, the valve area opens (for normally closed valves) or closes (for normally open valves) in order to maintain the pressure in the valve. The pressure regulation range begins at the set pressure. Pset is constant in the case of a Constant valve, or varying in the case of a Controlled valve. A physical sign at port Ps provides a varying set pressure.
Pressure regulation occurs when the sensed pressure, Px – PY, or Pcontrol, exceeds a specified pressure, Pset. The Pressure Compensator Valve (IL) block supports two modes of regulation:
When Set pressure control is set to Controlled, connect a pressure signal to port Ps and set the constant Pressure regulation range. pressure regulation is triggered when Pcontrol is greater than Pset, the Set pressure differential, and below Pmax, the sum of the set pressure and the user-defined Pressure regulation range.
When Set pressure control is set to Constant, the valve opening is continuously regulated between Pset and Pmax by either a linear or tabular parametrization. When Opening parametrization is set to Tabular data, Pset and Pmax are the first and last parameters of the Pressure differential vector, respectively.
Momentum is conserved through the valve:
{\stackrel{˙}{m}}_{A}+{\stackrel{˙}{m}}_{B}=0.
\stackrel{˙}{m}=\frac{{C}_{d}{A}_{valve}\sqrt{2\overline{\rho }}}{\sqrt{P{R}_{loss}\left(1-{\left(\frac{{A}_{valve}}{{A}_{port}}\right)}^{2}\right)}}\frac{\Delta p}{{\left[\Delta {p}^{2}+\Delta {p}_{crit}^{2}\right]}^{1/4}},
\overline{\rho }
\Delta {p}_{crit}=\frac{\pi \overline{\rho }}{8{A}_{valve}}{\left(\frac{\nu {\mathrm{Re}}_{crit}}{{C}_{d}}\right)}^{2}.
P{R}_{loss}=\frac{\sqrt{1-{\left(\frac{{A}_{valve}}{{A}_{port}}\right)}^{2}\left(1-{C}_{d}^{2}\right)}-{C}_{d}\frac{{A}_{valve}}{{A}_{port}}}{\sqrt{1-{\left(\frac{{A}_{valve}}{{A}_{port}}\right)}^{2}\left(1-{C}_{d}^{2}\right)}+{C}_{d}\frac{{A}_{valve}}{{A}_{port}}}.
Pressure recovery describes the positive pressure change in the valve due to an increase in area. If Pressure recovery is set to Off, PRloss is 1.
The opening area, Avalve, is determined by the opening parametrization (for Constant valves only) and the valve opening dynamics.
The linear parametrization of the valve area for Normally open valves is:
{A}_{valve}=\stackrel{^}{p}\left({A}_{leak}-{A}_{\mathrm{max}}\right)+{A}_{\mathrm{max}},
{A}_{valve}=\stackrel{^}{p}\left({A}_{\mathrm{max}}-{A}_{leak}\right)+{A}_{leak}.
For tabular parametrization of the valve area in its operating range, Aleak and Amax are the first and last parameters of the Opening area vector, respectively.
The normalized pressure,
\stackrel{^}{p}
\stackrel{^}{p}=\frac{{p}_{control}-{p}_{set}}{{p}_{\mathrm{max}}-{p}_{set}}.
At the extremes of the valve pressure range, you can maintain numerical robustness in your simulation by adjusting the block Smoothing factor. With a nonzero smoothing factor, a smoothing function is applied to all calculated valve pressures, but primarily influences the simulation at the extremes of these ranges.
When the Smoothing factor, f, is nonzero, a smoothed, normalized pressure is instead applied to the valve area:
{\stackrel{^}{p}}_{smoothed}=\frac{1}{2}+\frac{1}{2}\sqrt{{\stackrel{^}{p}}_{}^{2}+{\left(\frac{f}{4}\right)}^{2}}-\frac{1}{2}\sqrt{{\left(\stackrel{^}{p}-1\right)}^{2}+{\left(\frac{f}{4}\right)}^{2}}.
{\stackrel{˙}{p}}_{dyn}=\frac{{p}_{control}-{p}_{dyn}}{\tau }.
By default, Opening dynamics is set to Off. A nonzero Smoothing factor can provide additional numerical stability when the orifice is in near-closed or near-open position.
Steady-state dynamics are set by the same parametrization as the valve opening, and are based on the control pressure, pcontrol.
When faults are enabled, the valve open area becomes stuck at a specified value in response to one of these triggers:
Closed — The valve freezes at its smallest value, depending on the Opening parameterization:
When Opening parameterization is set to Linear, the valve area freezes at the Leakage area.
When Opening parameterization is set to Tabulated data, the valve area freezes at the first element of the Opening area vector.
Open — The valve freezes at its largest value, depending on the Opening parameterization:
When Opening parameterization is set to Linear, the valve area freezes at the Maximum opening area.
When Orifice parameterization is set to Tabulated data, the valve area freezes at the last element of the Opening area vector.
Maintain last value — The valve area freezes at the valve open area when the trigger occurred.
Due to numerical smoothing at the extremes of the valve area, the minimum area applied is larger than the Leakage area, and the maximum is smaller than the Maximum orifice area, in proportion to the Smoothing factor value.
Entry or exit port of the liquid to or from the valve.
X — Pressure reference, MPa
Absolute pressure in units of MPa, denoted Px.
Y — Pressure reference, MPa
Absolute pressure in units of MPa, denoted Py.
Ps — Controlled pressure differential
Pressure differential for controlled valve operation, specified as a physical signal.
To enable this parameter, set Set pressure control to Controlled.
Valve specification — Valve bias
Normally open (default) | Normally closed
Normal operating condition of the pressure compensator valve. A reducing valve is a Normally open valve and a relief valve is a Normally closed valve.
Set pressure control — Constant or controlled valve operation
Valve operation method. The valve can operate according to a specified pressure regulation range or Opening parametrization.
Opening parameterization — Method of modeling constant valve opening or closing
Method of modeling the valve opening or closing. The valve opening is either parametrized linearly or by a table of values correlating the area to pressure differential.
To enable this parameter, set Set pressure control to Constant.
Set pressure differential — Range outside of which constant valve operation is triggered
0 MPa (default) | scalar
Magnitude of pressure differential that triggers operation of a constant valve when pressure exceeds (Normally closed) or falls below (Normally open).
To enable this parameter, set Set pressure control to Constant and Opening parametrization to Linear.
Pressure regulation range — Valve operational pressure range
Operational pressure range of the valve. The pressure regulation range lies between the Set pressure differential and the maximum valve operating pressure.
Set pressure control to Controlled
Set pressure control to Constant and Opening parametrization to Linear
Maximum opening area — Area of fully-opened valve
Cross-sectional area of the valve in its fully open position.
Sum of all gaps when the valve is in fully closed position. Any area smaller than this value is saturated to the specified leakage area. This contributes to numerical stability by maintaining continuity in the flow.
Pressure differential vector — Vector of differential pressure values for tabular parametrization
[0.2 : 0.2 : 1.2] MPa (default) | 1-byn vector
Vector of pressure differential values for the tabular parameterization of opening area. The vector elements must correspond one-to-one to the values in the Opening area vector parameter. The pressures must be greater than 0 and are listed in ascending order.
To enable this parameter, set Set pressure control to Constant and Opening parametrization to Tabulated data.
Opening area vector — Vector of valve opening areas for tabular parametrization
[1e-05, 8e-06, 6e-06, 4e-06, 2e-06, 1e-10] m^2 (default) | 1-byn vector
Vector of opening area values for the tabular parameterization of opening area. The vector elements must correspond one-to-one to the values in the Pressure differential vector parameter. Area vectors for normally open valves list elements in descending order. Area vectors for normally closed valves list elements in ascending order.
The opening area vector must have the same number of elements as the Pressure differential vector. Linear interpolation is employed between table data points.
Cross-sectional area at ports A and B — Area at valve entry or exit
Cross-sectional area at the entry and exit ports A and B. These areas are used in the pressure-flow rate equation that determines mass flow rate through the valve.
Correction factor accounting for discharge losses in theoretical flows. The default discharge coefficient for a valve in Simscape™ Fluids™ is 0.64.
To enable this parameter, set Opening parameterization to Linear.
Opening dynamics — Whether to account for flow response to valve opening
Select to account for transient effects to the fluid system due to the valve opening. When Opening dynamics is set to On, the block approximates the opening conditions by introducing a first-order lag in the flow response.
Opening time constant — Valve opening time constant
Constant that captures the time required for the fluid to reach steady-state when opening or closing the valve from one position to another. This parameter impacts the modeled opening dynamics.
Opening area when faulted — Set faulted characteristics
Closed (default) | Open | Maintain last value
Sets the faulted valve type. You can choose for the valve to seize when the valve is opened, closed, or at the area when faulting is triggered.
When set to Temporal, when the Simulation time for fault event is reached, the valve area will be set to the value specified in the Opening area when faulted parameter.
When the Simulation time for fault event is reached, the valve area is set to the value specified in the Opening area when faulted parameter.
Pressure-Compensated Flow Control Valve (IL) | Pressure-Reducing 3-Way Valve (IL) | Pressure-Reducing Valve (IL) | Pressure Relief Valve (IL)
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Dictionary:First time derivative of the trace envelope - SEG Wiki
Revision as of 21:04, 12 August 2018 by Jdromerom (talk | contribs) (Marked this version for translation)
The first time derivative of the trace envelope is a seismic attribute. Calculated as the time rate of change of the envelope and shows the variation of the energy of the reflected events. The derivative computed at the onset of the events shows the absorption effects, a slower rise indicates larger absorption effects. A slower rise indicates larger absorption effects. The first time derivative of the envelope can be calculated by:[1]
{\displaystyle {\frac {dE(t)}{dt}}=E(t)*diff(t)}
{\displaystyle E(t)}
{\displaystyle *}
{\displaystyle diff(t)}
is the differential operator. Events with a sharp relative rise also imply a wider bandwidth, hence less absorption effects. This is a physical attribute and points to acknowledge when working with the rate of change of the envelope are:
Sharpness of the rise time relates to absorption in an inversely proportional manner
It is affected by the slope, rather than envelope magnitude
Lateral variation shows discontinuities.
Retrieved from "https://wiki.seg.org/index.php?title=Dictionary:First_time_derivative_of_the_trace_envelope&oldid=114756"
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Phased array ultrasonics - Wikipedia
Find sources: "Phased array ultrasonics" – news · newspapers · books · scholar · JSTOR (February 2013) (Learn how and when to remove this template message)
Animation showing the principle of an ultrasonic scanner used in medical ultrasonic imaging. It consists of a beamforming oscillator (TX) that produces an electronic signal consisting of pulses of sine waves oscillating at an ultrasonic frequency, which is applied to an array of ultrasonic transducers (T) in contact with the skin surface that convert the electric signal into ultrasonic waves traveling through the tissue. The timing of the pulses emitted by each transducer is controlled by programmable delay units (φ) that are controlled by a microprocessor control system (C). The moving red lines are the wavefronts of the ultrasonic waves from each transducer. The wavefronts are spherical, but they combine (superpose) to form plane waves, creating a beam of sound traveling in a specific direction. Since the pulse from each transducer is progressively delayed going up the line, each transducer emits its pulse after the one below it. This results in a beam of sound waves emitted at an angle (θ) to the array. By changing the pulse delays, the computer can scan the beam of ultrasound in a raster pattern across the tissue. Echoes reflected by different density tissue, received by the transducers, build up an image of the underlying structures.
Weld examination by phased array. TOP: The phased array probe emits a series of beams to flood the weld with sound. BOTTOM: The flaw in the weld appears as a red indication on the instrument screen.
Phased array ultrasonics (PA) is an advanced method of ultrasonic testing that has applications in medical imaging and industrial nondestructive testing. Common applications are to noninvasively examine the heart or to find flaws in manufactured materials such as welds. Single-element (non-phased array) probes, known technically as monolithic probes, emit a beam in a fixed direction. To test or interrogate a large volume of material, a conventional probe must be physically scanned (moved or turned) to sweep the beam through the area of interest. In contrast, the beam from a phased array probe can be focused and swept electronically without moving the probe. The beam is controllable because a phased array probe is made up of multiple small elements, each of which can be pulsed individually at a computer-calculated timing. The term phased refers to the timing, and the term array refers to the multiple elements. Phased array ultrasonic testing is based on principles of wave physics, which also have applications in fields such as optics and electromagnetic antennae.
2 Phased array use in the industry
3 Features of phased array
The PA probe consists of many small ultrasonic transducers, each of which can be pulsed independently. By varying the timing, for instance by making the pulse from each transducer progressively delayed going up the line, a pattern of constructive interference is set up that results in radiating a quasi-plane ultrasonic beam at a set angle depending on the progressive time delay. In other words, by changing the progressive time delay the beam can be steered electronically. It can be swept like a search-light through the tissue or object being examined, and the data from multiple beams are put together to make a visual image showing a slice through the object.
Phased array use in the industry[edit]
Phased array is widely used for nondestructive testing (NDT) in several industrial sectors, such as construction, pipelines, and power generation. This method is an advanced NDT method that is used to detect discontinuities i.e. cracks or flaws and thereby determine component quality. Due to the possibility to control parameters such as beam angle and focal distance, this method is very efficient regarding the defect detection and speed of testing.[1] Apart from detecting flaws in components, phased array can also be used for wall thickness measurements in conjunction with corrosion testing.[2][3] Phased array can be used for the following industrial purposes:
Inspection of welds[4]
PAUT Validation/Demonstration Blocks[5]
PAUT & TOFD Standard Calibration Blocks
Features of phased array[edit]
Multiple probe elements produce a steerable and focused beam.[6]
Focal spot size depends on probe active aperture (A), wavelength (λ) and focal length (F).[7] Focusing is limited to the near field of the phased array probe.
{\displaystyle {\text{Focal spot size}}=F\lambda /A}
{\displaystyle {\text{Near Field}}=A^{2}/4\lambda }
prEN 16018, Non destructive testing - Terminology - Terms used in ultrasonic testing with phased arrays
ISO/WD 13588, Non-destructive testing of welds – Ultrasonic testing – Use of (semi-) automated phased array technology
Phased array (general theory and electromagnetic telecommunications).
^ Corrosion under pipe support inspection. Retrieved on July 13, 2012.
^ Phased Array (PA). Retrieved on July 13, 2012
^ Corrosion Mapping with Phased Array Ultrasonics. 2011 API Inspection Summit and Expo. Retrieved on July 13, 2012.[permanent dead link]
^ ASTM, E2700 (2012). Nondestructive Testing, vol 3.03, Contact Ultrasonic Testing of Welds using Phased Arrays. Conshohocken, PA: American Society for Testing of Materials. pp. 1536–44. ISBN 978-0-8031-8729-0.
^ http://www.ndeflawtechnologies.com/paut_validation_block.html
^ ASTM, E2491 (2012). Nondestructive Testing vol 3.03, Evaluating Phased Array Characteristics of Phased Array Ultrasonic Testing Instruments and Systems. Conshohocken, PA: American Society for Testing of Materials. pp. 1358–75. ISBN 978-0-8031-8729-0.
^ Birring, Anmol (September 2008). "Selection of Phased Array Parameters for Weld Testing". Materials Evaluation.
ASME Boiler and Pressure Vessel Code. American Society Of Mechanical Engineers, 2013. Section V — Nondestructive Examination. [See Article 4 — Ultrasonic Examination Methods for Welds. Para E-474 UT-Phased Array Technique]
FOCUS - Fast Object-oriented C++ Ultrasound Simulator [MATLAB routines for creating and simulating phased arrays]
Phased array animated simulator [registration required after 5 minutes' use]
Phased Array Tutorial Education resource from Olympus
Retrieved from "https://en.wikipedia.org/w/index.php?title=Phased_array_ultrasonics&oldid=1077311039"
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On the ℓp-norm of the discrete Hilbert transform
15 February 2019 On the
{\ell }^{p}
-norm of the discrete Hilbert transform
Rodrigo Bañuelos, Mateusz Kwaśnicki
Using a representation of the discrete Hilbert transform in terms of martingales arising from Doob
h
-processes, we prove that its
{\ell }^{p}
-norm,
1<p<\infty
, is bounded above by the
{L}^{p}
-norm of the continuous Hilbert transform. Together with the already known lower bound, this resolves the long-standing conjecture that the norms of these operators are equal.
Rodrigo Bañuelos. Mateusz Kwaśnicki. "On the
{\ell }^{p}
-norm of the discrete Hilbert transform." Duke Math. J. 168 (3) 471 - 504, 15 February 2019. https://doi.org/10.1215/00127094-2018-0047
Received: 3 November 2017; Revised: 25 June 2018; Published: 15 February 2019
Keywords: Burkholder inequalities , discrete Hilbert transform , Gundy–Varopoulos representation , martingale transform
Rodrigo Bañuelos, Mateusz Kwaśnicki "On the
{\ell }^{p}
-norm of the discrete Hilbert transform," Duke Mathematical Journal, Duke Math. J. 168(3), 471-504, (15 February 2019)
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Understanding beta binomial regression (using baseball statistics) | R-bloggers
Posted on May 31, 2016 by David Robinson in R bloggers | 0 Comments
Understanding the beta distribution
Understanding empirical Bayes estimation
Understanding credible intervals
Understanding the Bayesian approach to false discovery rates
Understanding Bayesian A/B testing
In this series we’ve been using the empirical Bayes method to estimate batting averages of baseball players. Empirical Bayes is useful here because when we don’t have a lot of information about a batter, they’re “shrunken” towards the average across all players, as a natural consequence of the beta prior.
But there’s a complication with this approach. When players are better, they are given more chances to bat! (Hat tip to Hadley Wickham to pointing this complication out to me). That means there’s a relationship between the number of at-bats (AB) and the true batting average. For reasons I explain below, this makes our estimates systematically inaccurate.
In this post, we change our model where all batters have the same prior to one where each batter has his own prior, using a method called beta-binomial regression. We show how this new model lets us adjust for the confounding factor while still relying on the empirical Bayes philosophy. We also note that this gives us a general framework for allowing a prior to depend on known information, which will become important in future posts.
As usual, I’ll start with some code you can use to catch up if you want to follow along in R. If you want to understand what it does in more depth, check out the previous posts in this series. (As always, all the code in this post can be found here).
dplyr::select(playerID, nameFirst, nameLast) %>%
prior_mu <- alpha0 / (alpha0 + beta0)
Recall that the eb_estimate column gives us estimates about each player’s batting average, estimated from a combination of each player’s record with the beta prior parameters estimated from everyone (
alpha_0
beta_0
). For example, a player with only a single at-bat and a single hit (
H = 1; AB = 1; H / AB = 1
) will have an empirical Bayes estimate of
Now, here’s the complication. Let’s compare at-bats (on a log scale) to the raw batting average:
career %>%
filter(AB >= 20) %>%
ggplot(aes(AB, average)) +
We notice that batters with low ABs have more variance in our estimates- that’s a familiar pattern because we have less information about them. But notice a second trend: as the number of at-bats increases, the batting average also increases. Unlike the variance, this is not an artifact of our measurement: it’s a result of the choices of baseball managers! Better batters get played more: they’re more likely to be in the starting lineup and to spend more years playing professionally.
Now, there are many other factors that are correlated with a player’s batting average (year, position, team, etc). But this one is particularly important, because it confounds our ability to perform empirical Bayes estimation:
That horizontal red line shows the prior mean that we’re “shrinking” towards (
frac{alpha_0}{alpha_0 + beta_0} = 0.259
). Notice that it is too high for the low-AB players. For example, the median batting average for players with 5-20 at-bats is 0.167, and they get shrunk way towards the overall average! The high-AB crowd basically stays where they are, because each has a lot of evidence.
So since low-AB batters are getting overestimated, and high-AB batters are staying where they are, we’re working with a biased estimate that is systematically overestimating batter ability. If we were working for a baseball manager (like in Moneyball), that’s the kind of mistake we could get fired for!
Accounting for AB in the model
How can we fix our model? We’ll need to have AB somehow influence our priors, particularly affecting the mean batting average. In particular, we want the typical batting average to be linearly affected by
log(mbox{AB})
First we should write out what our current model is, in the form of a generative process, in terms of how each of our variables is generated from particular distributions. Defining
p_i
to be the true probability of hitting for batter
i
(that is, the “true average” we’re trying to estimate), we’re assuming
(We’re letting the totals
mbox{AB}_i
be fixed and known per player). We made up this model in one of the first posts in this series and have been using it since.
I’ll point out that there’s another way to write the
p_i
calculation, by re-parameterizing the beta distribution. Instead of parameters
alpha_0
beta_0
, let’s write it in terms of
mu_0
sigma_0
mu_0
represents the mean batting average, while
sigma
represents how spread out the distribution is (note that
sigma = frac{1}{alpha+beta}
sigma
is high, the beta distribution is very wide (a less informative prior), and when
sigma
is low, it’s narrow (a more informative prior). Way back in my first post about the beta distribution, this is basically how I chose parameters: I wanted
mu = .27
, and then I chose a
sigma
that would give the desired distribution that mostly lay between .210 and .350, our expected range of batting averages.
Now that we’ve written our model in terms of
mu
sigma
, it becomes easier to see how a model could take AB into consideration. We simply define
mu
so that it includes
log(mbox{AB})
as a linear term1:
Then we define the batting average
p_i
and the observed
H_i
just like before:
This particular model is called beta-binomial regression. We already had each player represented with a binomial whose parameter was drawn from a beta, but now we’re allowing the expected value of the beta to be influenced.
Step 1: Fit the model across all players
Going back to the basics of empirical Bayes, our first step is to fit these prior parameters:
mu_0
mu_{mbox{AB}}
sigma_0
. When doing so, it’s ok to momentarily “forget” we’re Bayesians- we picked our
alpha_0
beta_0
using maximum likelihood, so it’s OK to fit these using a maximum likelihood approach as well. You can use the gamlss package for fitting beta-binomial regression using maximum likelihood.
fit <- gamlss(cbind(H, AB - H) ~ log(AB),
data = career_eb,
We can pull out the coefficients with the broom package (see ?gamlss_tidiers):
td <- tidy(fit)
## parameter term estimate std.error statistic p.value
## 1 mu (Intercept) 0.1426 0.001577 90.4 0
## 2 mu log(AB) 0.0153 0.000215 71.2 0
## 3 sigma (Intercept) -6.2935 0.023052 -273.0 0
This gives us our three parameters:
mu_0 = 0.143
mu_mbox{AB} = 0.015
, and (since sigma has a log-link)
sigma_0 = exp(-6.294) = 0.002
This means that our new prior beta distribution for a player depends on the value of AB. For example, here are our prior distributions for several values:
Notice that there is still uncertainty in our prior- a player with 10,000 at-bats could have a batting average ranging from about .22 to .35. But the range of that uncertainty changes greatly depending on the number of at-bats- any player with AB = 10,000 is almost certainly better than one with AB = 10.
Step 2: Estimate each player’s average using this prior
Now that we’ve fit our overall model, we repeat our second step of the empirical Bayes method. Instead of using a single
alpha_0
beta_0
values as the prior, we choose the prior for each player based on their AB. We then update using their
H
AB
just like before.
Here, all we need to calculate are the mu (that is,
mu = mu_0 + mu_{log(mbox{AB})}
) and sigma (
sigma
) parameters for each person. (Here, sigma will be the same for everyone, but that may not be true in more complex models). This can be done using the fitted method on the gamlss object (see here):
mu <- fitted(fit, parameter = "mu")
sigma <- fitted(fit, parameter = "sigma")
head(sigma)
Now we can calculate
alpha_0
beta_0
parameters for each player, according to
alpha_{0,i}=mu_i / sigma_0
beta_{0,i}=(1-mu_i) / sigma_0
. From that, we can update based on
H
AB
to calculate new
alpha_{1,i}
beta_{1,i}
for each player.
career_eb_wAB <- career_eb %>%
dplyr::select(name, H, AB, original_eb = eb_estimate) %>%
mutate(mu = mu,
alpha0 = mu / sigma,
beta0 = (1 - mu) / sigma,
alpha1 = alpha0 + H,
beta1 = beta0 + AB - H,
new_eb = alpha1 / (alpha1 + beta1))
How does this change our estimates?
Notice that relative to the previous empirical Bayes estimate, this one is lower for batters with low AB and about the same for high-AB batters. This fits with our earlier description- we’ve been systematically over-estimating batting averages.
Here’s another way of comparing the estimation methods:
Notice that we used to shrink batters towards the overall average (red line), but now we are shrinking them towards the overall trend- that red slope.2
Don’t forget that this change in the posteriors won’t just affect shrunken estimates. It will affect all the ways we’ve used posterior distributions in this series: credible intervals, posterior error probabilities, and A/B comparisons. Improving the model by taking AB into account will help all these results more accurately reflect reality.
What’s Next: Bayesian hierarchical modeling
This problem is in fact a simple and specific form of a Bayesian hierarchical model, where the parameters of one distribution (like
alpha_0
beta_0
) are generated based on other distributions and parameters. While these models are often approached using more precise Bayesian methods (such as Markov chain Monte Carlo), we’ve seen that empirical Bayes can be a powerful and practical approach that helped us deal with our confounding factor.
Beta-binomial regression, and the gamlss package in particular, offers a way to fit parameters to predict “success / total” data. In this post, we’ve used a very simple model-
mu
linearly predicted by AB. But there’s no reason we can’t include other information that we expect to influence batting average. For example, left-handed batters tend to have a slight advantage over right-handed batters- can we include that information in our model? In the next post, we’ll bring in additional information to build a more sophisticated hierarchical model. We’ll also consider some of the limitations of empirical Bayes for these situations.
If you have some experience with regressions, you might notice a problem with this model: $mu$ can theoretically go below 0 or above 1, which is impossible for a $beta$ distribution (and will lead to illegal $alpha$ and $beta$ parameters). Thus in a real model we would use a “link function”, such as the logistic function, to keep $mu$ between 0 and 1. I used a linear model (and mu.link = "identity" in the gamlss call) to make the math in this introduction simpler, and because for this particular data it leads to almost exactly the same answer (try it). In our next post we’ll include the logistic link. ↩
If you work in in my old field of gene expression, you may be interested to know that empirical Bayes shrinkage towards a trend is exactly what some differential expression packages such as edgeR do with per-gene dispersion estimates. It’s a powerful concept that allows a balance between individual observations and overall expectations. ↩
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Al notices that the ratio of the perimeters of two similar polygons is equal to the ratio of their side lengths. “What about their areas? Are they related in the same way?” he wonders.
What are the perimeter and area of
∆ALT
P = 7 + 5 + 11.4 m
A = (1/2)(7)(3) m^2
Test Al’s question by enlarging
∆ALT
by a scale factor of
3
. What are the perimeter and area of the image triangle?
Use the new side lengths to compute the area and perimeter of the enlarged triangle.
Does the perimeter increase by a scale factor of
3
\frac{70.2}{23.4} = ?
Answer Al’s question. Does the area increase by a scale factor of
3
? Explain what happened.
\frac{94.5}{10.5} = ?
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Standard Deviation and standard deviation after data changing | ePractice - HKDSE 試題導向練習平台
Question Sample Titled 'Standard Deviation and standard deviation after data changing'
The standard deviation of the project scores obtained by a group of students in a Liberal Education project is
{11}
\text{marks}
. The teacher thinks the scores are too low, so she adjusts the project score of each student such that each score is increased by
{20}\%
and extra
{9}
\text{marks}
are added.
Find the standard deviation of the project scores after the score adjustment.
Is there any change in the standard score of each student due to the score adjustment? Explain your answer.
={11}{\left({1}+{20}\%\right)}
={13.2}
\text{marks}
{x}
be the project score and
\overline{{x}}
be the mean of the project scores before the adjustment.
The standard score before the score adjustment.
=\dfrac{{{x}-\overline{{x}}}}{{11}}
The standard score after the score adjustment
=\dfrac{{{\left({x}{\left({1}+{20}\%\right)}+{9}\right)}-{\left(\overline{{x}}{\left({1}+{20}\%\right)}+{9}\right)}}}{{13.2}}
=\dfrac{{{1.2}{\left({x}-\overline{{x}}\right)}}}{{13.2}}
=\dfrac{{{x}-\overline{{x}}}}{{11}}
Thus, there is no change in the standard score of each student due to the score adjustment.
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tensor(deprecated)/Jacobian - Maple Help
Home : Support : Online Help : tensor(deprecated)/Jacobian
compute the Jacobian and its inverse of a coordinate transformation
transform a tensor given the backward transformation, the Jacobian of the backward transformation and its inverse.
Jacobian(Y, Finv, yJx, xJy)
transform(Tx, Finv, yJx, xJy)
list of n names of coordinate variables
list of n functions in the y's that gives the transformation from the y-coordinate system to another one, say, the x-coordinate system
output parameter of Jacobian(), but an input to transform(); tensor_type of character
[1,-1]
, and having the Jacobian matrix of the transformation from the x's to y's as components, yJx must have its components expressed in terms of the y's
[-1,1]
, and having the transpose of the Jacobian matrix of the transformation from the y's to x's (namely, Finv) as components, xJy must have its components expressed in terms of the y's
any tensor_type of rank-1 or higher
Important: The tensor package has been deprecated. Use the superseding commands DifferentialGeometry[Transformation] and Physics[TransformCoordinates] instead.
Given a tensor_type Tx, that represents a tensor T in its x-coordinate representation, transform() constructs Ty, the representation of the same tensor in another coordinate system, say, the y-coordinate system.
transform() needs as input parameters, apart from T of course, the inverse transformation Finv, (that is, the x's expressed as functions of the y's), the jacobian (tensor_type) of Finv expressed in terms of the y's and its inverse (tensor_type).
yJx must have character
[1,-1]
, and more visually, components:
\mathrm{yJx}[\mathrm{compts}][a,i]={\left(\frac{∂y}{∂x}\left(y\right)\right)}_{i}^{a}
The partial of
{y}^{a}
{x}^{i}
, expressed in terms of the
y
xJy MUST have character
[-1,1]
, and components:
\mathrm{xJy}[\mathrm{compts}][a,i]={\left(\frac{∂x}{∂y}\left(y\right)\right)}_{a}^{i}
{x}^{i}
{y}^{a}
y
Note: Note the transposition here in xJy! This is due to the fact xJy (and also yJx) is primarily used by tensor[transform], and tensor[transform] uses tensor[change_basis] to perform component manipulations, and tensor[change_basis] requires that the transformation matrix that acts on covariant indices (where xJy indeed does, since in tensor[transform], the transformation is from the x's to the y's) have this transposed arrangement.
The function Jacobian() can thus be used to generate the two Jacobians needed by transform(). Note that the xJy produced by Jacobian() is already in that transposed arrangement required by transform(). You need to be concerned about that fact only when they are to construct the jacobian(s) themselves.
Jacobian: Each of its components is merely a partial derivative of one of the x's w.r.t. one of the y's and Jacobian() uses the simplifier `tensor/partial_diff/simp` to simplify each of these derivatives.
transform: When Tx is of rank zero, only substitutions (of the x's with the y's) are needed. In this case, transform() uses the simplifier `tensor/raise/simp` to simplify the (scalar) component of the result after the substitutions.
When Tx is of rank one or higher, transform() first makes the substitutions and then invokes tensor[change_basis] to carry out the transformation according to the tensorial rule, and hence it, in case of a nonzero rank tensor, transform() uses also `tensor/raise/simp` as simplifier; `tensor/raise/simp` is the simplifier used by change_basis().
These two functions are part of the tensor package, and can be used in the form Jacobian(..), and transform(..) only after performing the command with(tensor), or with(tensor, Jacobian) or with(tensor, transform). The functions can always be accessed in the long form tensor[Jacobian] or tensor[transform].
\mathrm{with}\left(\mathrm{tensor}\right):
Declare the coordinate variables in the two systems. Note that declaration of Polar is not necessary in the following computations.
\mathrm{dim}≔4:
\mathrm{Polar1}≔[t,R,\mathrm{\phi },\mathrm{\theta }]:
\mathrm{Polar2}≔[t,r,\mathrm{\phi },\mathrm{\theta }]:
Express the coordinates in the first system in terms of the those in the second.
\mathrm{rtoR}≔[t=t,R=\frac{r}{1+\frac{K{r}^{2}}{4}},\mathrm{\phi }=\mathrm{\phi },\mathrm{\theta }=\mathrm{\theta }]:
Define the Robertson-Walker covariant metric in the first system.
\mathrm{gIJ_compts}≔\mathrm{array}\left(\mathrm{symmetric},1..\mathrm{dim},1..\mathrm{dim}\right):
\mathbf{for}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}i\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{to}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dim}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\mathbf{for}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}j\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{from}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}i+1\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{to}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dim}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\mathrm{gIJ_compts}[i,j]≔0\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\mathbf{end}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathbf{end}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}:
\mathrm{gIJ_compts}[1,1]≔-1:
\mathrm{gIJ_compts}[2,2]≔\frac{{S}^{2}}{1-K{R}^{2}}:
\mathrm{gIJ_compts}[3,3]≔{S}^{2}{R}^{2}{\mathrm{sin}\left(\mathrm{\theta }\right)}^{2}:
\mathrm{gIJ_compts}[4,4]≔{S}^{2}{R}^{2}:
\mathrm{gIJ}≔\mathrm{create}\left([-1,-1],\mathrm{op}\left(\mathrm{gIJ_compts}\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{gIJ}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{table}\textcolor[rgb]{0,0,1}{}\left([\textcolor[rgb]{0,0,1}{\mathrm{compts}}\textcolor[rgb]{0,0,1}{=}[\begin{array}{cccc}\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}& \frac{{\textcolor[rgb]{0,0,1}{S}}^{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{K}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{R}}^{\textcolor[rgb]{0,0,1}{2}}\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}& \textcolor[rgb]{0,0,1}{0}& {\textcolor[rgb]{0,0,1}{S}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{R}}^{\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}{\mathrm{\theta }}\right)}^{\textcolor[rgb]{0,0,1}{2}}& \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}{S}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{R}}^{\textcolor[rgb]{0,0,1}{2}}\end{array}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{index_char}}\textcolor[rgb]{0,0,1}{=}[\textcolor[rgb]{0,0,1}{-1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-1}]]\right)
Form the two jacobians.
\mathrm{Jacobian}\left(\mathrm{Polar2},\mathrm{rtoR},\mathrm{yJx},\mathrm{xJy}\right):
\mathrm{op}\left(\mathrm{yJx}\right)
\textcolor[rgb]{0,0,1}{table}\textcolor[rgb]{0,0,1}{}\left([\textcolor[rgb]{0,0,1}{\mathrm{compts}}\textcolor[rgb]{0,0,1}{=}[\begin{array}{cccc}\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}{-}\frac{{\left(\textcolor[rgb]{0,0,1}{K}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{r}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}\right)}^{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{K}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{r}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{4}\right)}& \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}{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}{1}\end{array}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{index_char}}\textcolor[rgb]{0,0,1}{=}[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-1}]]\right)
\mathrm{op}\left(\mathrm{xJy}\right)
\textcolor[rgb]{0,0,1}{table}\textcolor[rgb]{0,0,1}{}\left([\textcolor[rgb]{0,0,1}{\mathrm{compts}}\textcolor[rgb]{0,0,1}{=}[\begin{array}{cccc}\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}{-}\frac{\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{K}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{r}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{4}\right)}{{\left(\textcolor[rgb]{0,0,1}{K}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{r}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}\right)}^{\textcolor[rgb]{0,0,1}{2}}}& \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}{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}{1}\end{array}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{index_char}}\textcolor[rgb]{0,0,1}{=}[\textcolor[rgb]{0,0,1}{-1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]]\right)
Now compute the transformed covariant metric in the second system.
\mathrm{gAB}≔\mathrm{transform}\left(\mathrm{gIJ},\mathrm{rtoR},\mathrm{yJx},\mathrm{xJy}\right)
\textcolor[rgb]{0,0,1}{\mathrm{gAB}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{table}\textcolor[rgb]{0,0,1}{}\left([\textcolor[rgb]{0,0,1}{\mathrm{compts}}\textcolor[rgb]{0,0,1}{=}[\begin{array}{cccc}\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}& \frac{\textcolor[rgb]{0,0,1}{16}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{S}}^{\textcolor[rgb]{0,0,1}{2}}}{{\left(\textcolor[rgb]{0,0,1}{K}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{r}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}\right)}^{\textcolor[rgb]{0,0,1}{2}}}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \frac{\textcolor[rgb]{0,0,1}{16}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{S}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{r}}^{\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}{\mathrm{\theta }}\right)}^{\textcolor[rgb]{0,0,1}{2}}}{{\left(\textcolor[rgb]{0,0,1}{K}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{r}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}\right)}^{\textcolor[rgb]{0,0,1}{2}}}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \frac{\textcolor[rgb]{0,0,1}{16}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{S}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{r}}^{\textcolor[rgb]{0,0,1}{2}}}{{\left(\textcolor[rgb]{0,0,1}{K}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{r}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}\right)}^{\textcolor[rgb]{0,0,1}{2}}}\end{array}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{index_char}}\textcolor[rgb]{0,0,1}{=}[\textcolor[rgb]{0,0,1}{-1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-1}]]\right)
\mathrm{gAB}≔\mathrm{transform}\left(\mathrm{gIJ},\mathrm{rtoR},\mathrm{yJx},\mathrm{xJy}\right)
\textcolor[rgb]{0,0,1}{\mathrm{gAB}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{table}\textcolor[rgb]{0,0,1}{}\left([\textcolor[rgb]{0,0,1}{\mathrm{compts}}\textcolor[rgb]{0,0,1}{=}[\begin{array}{cccc}\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}& \frac{\textcolor[rgb]{0,0,1}{16}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{S}}^{\textcolor[rgb]{0,0,1}{2}}}{{\left(\textcolor[rgb]{0,0,1}{K}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{r}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}\right)}^{\textcolor[rgb]{0,0,1}{2}}}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \frac{\textcolor[rgb]{0,0,1}{16}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{S}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{r}}^{\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}{\mathrm{\theta }}\right)}^{\textcolor[rgb]{0,0,1}{2}}}{{\left(\textcolor[rgb]{0,0,1}{K}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{r}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}\right)}^{\textcolor[rgb]{0,0,1}{2}}}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \frac{\textcolor[rgb]{0,0,1}{16}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{S}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{r}}^{\textcolor[rgb]{0,0,1}{2}}}{{\left(\textcolor[rgb]{0,0,1}{K}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{r}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}\right)}^{\textcolor[rgb]{0,0,1}{2}}}\end{array}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{index_char}}\textcolor[rgb]{0,0,1}{=}[\textcolor[rgb]{0,0,1}{-1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-1}]]\right)
DifferentialGeometry[Transformation]
|
Technoeconomic Analysis of Microalgae Cofiring Process for Fossil Fuel-Fired Power Plants | J. Energy Resour. Technol. | ASME Digital Collection
Department of Mechanical Engineering and Harry Reid Center for Environmental Studies,
, 4505 Maryland Parkway, Box 4009, Las Vegas, NV 89154
e-mail: jian.ma@unlv.edu
Harry Reid Center for Environmental Studies,
e-mail: oliver.hemmers@unlv.edu
Ma, J., and Hemmers, O. (March 29, 2011). "Technoeconomic Analysis of Microalgae Cofiring Process for Fossil Fuel-Fired Power Plants." ASME. J. Energy Resour. Technol. March 2011; 133(1): 011801. https://doi.org/10.1115/1.4003729
The concept of cofiring (algal biomass burned together with coal or natural gas in existing utility power boilers) includes the utilization of
CO2
from power plant for algal biomass culture and oxycombustion of using oxygen generated by biomass to enhance the combustion efficiency. As it reduces
CO2
emission by recycling it and uses less fossil fuel, there are concomitant benefits of reduced greenhouse gas (GHG) emissions. The by-products (oxygen) of microalgal biomass can be mixed with air or recycled flue gas prior to combustion, which will have the benefits of lower nitrogen oxide concentration in flue gas, higher efficiency of combustion, and not too high temperature (avoided by available construction materials) resulting from coal combustion in pure oxygen. A technoeconomic analysis of microalgae cofiring process for fossil fuel-fired power plants is studied. A process with closed photobioreactor and artificial illumination is evaluated for microalgae cultivation, due to its simplicity with less influence from climate variations. The results from this process would contribute to further estimation of process performance and investment. Two case studies show that there are average savings about $0.264 million/MW/yr and $0.203 million/MW/yr for coal-fired and natural gas-fired power plants, respectively. These cost savings are economically attractive and demonstrate the promise of microalgae technology for reducing GHG emission from fossil fuel-fired power plants.
air pollution control, coal, fossil fuels, natural gas technology, power generation economics, renewable energy sources, steam power stations, microalgae, biomass, cofiring process, technoeconomic analysis
Biomass, Carbon dioxide, Coal, Co-firing, Power stations, Water, Fuels, Solar energy, Natural gas, Emissions
Climate Change 2007: Synthesis Report—Contribution of Working Group III to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change
Modeling of CO2-Hydrate Formation in Geological Reservoirs by Injection of CO2 Gas
Plasynski
Mcllvried
Advances in CO2 Capture Technology—The U.S. Department of Energy’s Carbon Sequestration Program
The Use of Microalgae for Assimilation and Utilization of Carbon Dioxide From Fossil Fuel-Fired Power Plant Flue Gas
CO2 Mitigation With Microalgae Systems
Pulverized Coal Combustion in O2/CO2 Mixtures on a Power Plant for CO2 Recovery
Carbon Dioxide Sequestering Using Microalgal Systems—Final Report
Environmental Implications of Power Generation via Coal-Microalgae Cofiring
,” http://en.wikipedia.org/wiki/Photosynthesishttp://en.wikipedia.org/wiki/Photosynthesis.
,” www.britannica.comwww.britannica.com.
Function of the Two Cytochrome Components in Chloroplasts—A Working Hypothesis
,” http://en.wikipedia.org/wiki/Sunlighthttp://en.wikipedia.org/wiki/Sunlight.
Algae—Anatomy, Biochemistry, and Biotechnology
Biological Energy Production
Agricultural Consumer Protection
TexLoc Refractive Index of Polymers
,” http://www.texloc.com/c.oset/co_refractiveindex.htmlhttp://www.texloc.com/c.oset/co_refractiveindex.html.
A Look Back at the U.S. Department of Energy’s Aquatic Species Program—Biodiesel From Algae
,” U.S. D.O.E. Office of Fuels Development, Report No. NREL/TP-580-24190.
The Biological CO2 Fixation and Utilization Project by Rite (2)—Screening and Breeding of Microalgae With High Capability in Fixing CO2
Algal Capture of Carbon Dioxide; Biomass Generation as a Tool for Greenhouse Gas Mitigation With Reference to New Zealand Energy Strategy and Policy
Prediction of Volumetric Productivity of an Outdoor Photobioreactor
Resource Assessment of Algae Biomass for Potential Bioenergy Production in New Zealand
,” Bioenergy Options Program.
Biodiversity and Application of Microalgae
Shear Stress Tolerance and Biochemical Characterization of Phaeodactylum Tricornutum in Quasi Steady-State Continuous Culture in Outdoor Photobioreactors
Life-Cycle Assessment of Biodiesel Production From Microalgae
Environmental Life Cycle Comparison of Algae to Other Bioenergy Feedstocks
,” http://en.wikipedia.org/wiki/Heat_of_combustionhttp://en.wikipedia.org/wiki/Heat_of_combustion.
,” http://me.queensu.ca/courses/MECH430/Assets/Files/Recommended%20Reading/How%20Coal%20Works.pdfhttp://me.queensu.ca/courses/MECH430/Assets/Files/Recommended%20Reading/How%20Coal%20Works.pdf.
Electric Generation Using Natural Gas
,” http://www.naturalgas.org/overview/uses_eletrical.asphttp://www.naturalgas.org/overview/uses_eletrical.asp.
,” http://tonto.eia.doe.gov/oog/info/ngw/ngupdate.asphttp://tonto.eia.doe.gov/oog/info/ngw/ngupdate.asp.
Coal News and Markets
,” http://www.eia.doe.gov/cneaf/coal/page/coalnews/coalmar.htmlhttp://www.eia.doe.gov/cneaf/coal/page/coalnews/coalmar.html.
Power Plant Flue Gas as a Source of CO2 for Microalgae Cultivation: Economic Impact of Different Process Options
Achieves 161 Lumens per Watt From a High-Power LED
,” http://www.cree.com/press/press_detail.asp?i=1227101620851http://www.cree.com/press/press_detail.asp?i=1227101620851.
,” http://en.wikipedia.org/wiki/Light-emitting_diodehttp://en.wikipedia.org/wiki/Light-emitting_diode.
Department of Energy Environmental Protection Agency
Carbon Dioxide Emissions From the Generation of Electric Power in the United States
Department of Energy and Environmental Protection Agency
Evaluation of Biomass and Torrefied Coal Co-Firing in Large Utilities Boilers
|
Sandra wants to park her car so that she optimizes the distance she has to walk to the art museum and the library. That is, she wants to park so that her total distance directly to each building is the shortest. Determine where she should park.
Reflect the
50
feet line across the street. Added to diagram, orange horizontal line segment, right of the vertical street line segment, at the same height as the art museum.
Draw a straight line from the library to point
C
. Added to diagram, point c, at end of horizontal segment, & diagonal line segment connecting point, c, to library.
F
is where Sandra should park her car. Added to diagram, point labeled, f, where the diagonal segment, crosses the vertical street line segment.
Write an equation based on similar triangles to calculate the length
x
. If the length of the entire segment is
80
and one piece is
x
, how can you label the other piece? Added to diagram, the label, x, along the vertical street segment, between point, f, & the horizontal segment from the art museum.
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Predictive Least Squares - Maple Help
Home : Support : Online Help : Statistics and Data Analysis : Statistics Package : Regression : Predictive Least Squares
fit a linear model to data
PredictiveLeastSquares(A, B, v, options)
Matrix; values of independent variables
(optional) equation(s) of the form option=value where option is one of samplesize, tolerance or numtrials
The PredictiveLeastSquares command returns a list, P, and a vector, V that best satisfies the equation A[..,P] . x is approximately equal to B according to random trials using a subset of the data for fitting and the remaining data to test the goodness of the fit.
This command works best in situations where the problem is underspecified. That is, the number of variables, or columns in A is of the same order of magnitude or less than the number of observations or rows of A and B. The returned list, P contains the column index for the variables that have been tested to be most relevant, thus minimizing the effect of outliers and overfitting when using the model to predict new results.
A and B must contain numeric entries. A is a m x n Matrix, and B is a m x 1 vector.
numtrials= integer -- Specify how many random subsamples should be used to determine which variables to drop at each phase of regression. After each sweep that causes one or more variables/columns to be removed, another phase consisting of the specified number of trials is performed. The default is numtrials=15.
tolerance = realcons(nonnegative) -- Set the tolerance that determines whether a fit coefficient can be considered insignificant, and therefore should be removed. This is a relative tolerance, compared to the largest coefficient. The default is 1e-10.
samplesize= realcons(nonnegative) -- Provide the fraction of data that will be used for building the model. A setting of samplesize=.7, will cause snapshots using 70% of the data to be used for fitting, and the remaining 30% to be used for testing. This must be a number between 0 and 1. The default is .55.
\mathrm{with}\left(\mathrm{Statistics}\right):
In this first example, we have a matrix, A, with 100 columns of data, but the data in B only really depends on the first 4 of those columns.
A≔\mathrm{LinearAlgebra}:-\mathrm{RandomMatrix}\left(100,100,\mathrm{datatype}=\mathrm{float}[8]\right):
B≔\mathrm{Vector}\left(100,i↦A[i,1]+0.1\cdot A[i,2]+0.5\cdot A[i,3]-0.3\cdot A[i,4]\right):
The permutation vector computed shows the first 4 entries are relevant, and the coefficient vector, LSP, exactly matches the terms used to build B. All other columns not referenced by p can be discarded.
p,\mathrm{LSP}≔\mathrm{PredictiveLeastSquares}\left(A,B\right)
\textcolor[rgb]{0,0,1}{p}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{LSP}}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{c}\textcolor[rgb]{0,0,1}{1.00000000000000}\\ \textcolor[rgb]{0,0,1}{0.0999999999999998}\\ \textcolor[rgb]{0,0,1}{0.500000000000000}\\ \textcolor[rgb]{0,0,1}{-0.300000000000000}\end{array}]
In this second example, we will create a result vector that depends on 10 variables, of which only 5 of them are measured in the matrix, A (along with 95 other measurements of irrelevant/random properties).
\mathrm{numsamples}≔50:
\mathrm{numvariables}≔100:
Z≔\mathrm{LinearAlgebra}:-\mathrm{RandomMatrix}\left(\mathrm{numsamples},10,\mathrm{datatype}=\mathrm{float}[8]\right):
A≔\mathrm{LinearAlgebra}:-\mathrm{RandomMatrix}\left(\mathrm{numsamples},\mathrm{numvariables},\mathrm{datatype}=\mathrm{float}[8]\right):
A[..,1..5]≔Z[..,1..5]:
B≔\mathrm{Vector}\left(\mathrm{numsamples},i↦\mathrm{add}\left(\frac{Z[i,j]}{j},j=1..10\right)\right):
p,\mathrm{LSP}≔\mathrm{PredictiveLeastSquares}\left(A,B\right):
The notation A[..,p] will select all the rows of A and only the column indices found in the list p. This is the reduced matrix. Note the correlation of B and (A[..,p].LSP)
\mathrm{Correlation}\left(B,A[..,p]·\mathrm{LSP}\right)
\textcolor[rgb]{0,0,1}{0.992800745975950}
Compare this with the standard least squares fit.
\mathrm{LS}≔\mathrm{LinearAlgebra}:-\mathrm{LeastSquares}\left(A,B\right):
\mathrm{Correlation}\left(B,A·\mathrm{LS}\right)
\textcolor[rgb]{0,0,1}{1.00000000000000}
The correlation with the training data is a closer match using standard least squares, but let's see what happens when we use these models to predict results using new data.
\mathrm{Z2}≔\mathrm{LinearAlgebra}:-\mathrm{RandomMatrix}\left(\mathrm{numsamples},10,\mathrm{datatype}=\mathrm{float}[8]\right):
\mathrm{A2}≔\mathrm{LinearAlgebra}:-\mathrm{RandomMatrix}\left(\mathrm{numsamples},\mathrm{numvariables},\mathrm{datatype}=\mathrm{float}[8]\right):
\mathrm{A2}[..,1..5]≔\mathrm{Z2}[..,1..5]:
\mathrm{GuessLS}≔\mathrm{A2}·\mathrm{LS}:
\mathrm{GuessLSP}≔\mathrm{A2}[..,p]·\mathrm{LSP}:
\mathrm{Actual}≔\mathrm{Vector}\left(\mathrm{numsamples},i↦\mathrm{add}\left(\frac{\mathrm{Z2}[i,j]}{j},j=1..10\right)\right):
Note how the correlation of the new data is much better using the predictive model. The standard model suffers from overfitting.
\mathrm{Correlation}\left(\mathrm{GuessLS},\mathrm{Actual}\right)
\textcolor[rgb]{0,0,1}{0.727378412013161}
\mathrm{Correlation}\left(\mathrm{GuessLSP},\mathrm{Actual}\right)
\textcolor[rgb]{0,0,1}{0.959754787645569}
The Statistics[PredictiveLeastSquares] command was introduced in Maple 17.
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Scattering matrix and functions of self-adjoint operators | EMS Press
Scattering matrix and functions of self-adjoint operators
In the scattering theory framework, we consider a pair of operators
H_0
H
. For a continuous function
\varphi
vanishing at infinity, we set
\varphi_\delta(\cdot)=\varphi(\cdot/\delta)
and study the spectrum of the difference
\varphi_\delta(H-\lambda)-\varphi_\delta(H_0-\lambda)
\delta\to0
\lambda
is in the absolutely continuous spectrum of
H_0
H
, then the spectrum of this difference converges to a set that can be explicitly described in terms of (i) the eigenvalues of the scattering matrix
S(\lambda)
H_0
H
and (ii) the singular values of the Hankel operator
H_\varphi
\varphi
Alexander Pushnitski, Scattering matrix and functions of self-adjoint operators. J. Spectr. Theory 1 (2011), no. 2, pp. 221–236
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Maximal and Area Integral Characterizations of Hardy-Sobolev Spaces in the Unit Ball of $\mathbb C^n$ | EMS Press
\mathbb C^n
In this paper we deal with several characterizations of the Hardy-Sobolev spaces in the unit ball of
\mathbb C^n
, that is, spaces of holomorphie functions in the ball whose derivatives up to a certain order belong to the classical Hardy spaces. Some of our characterizations are in terms of maximal functions, area functions or Littlewood-Paley functions involving only complex-tangential derivatives. A special case of our results is a characterization of
H^p
itself involving only complex-tangential derivatives.
Patrick Ahern, Joaquim Bruna, Maximal and Area Integral Characterizations of Hardy-Sobolev Spaces in the Unit Ball of
\mathbb C^n
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Eleventy Threeve » Illogicopedia - The nonsensical encyclopedia anyone can mess up
Eleventy Threeve
Oscarwildium as its "Banana" isotope
Eleventy Threeve is a wondrous number used that is used very frequently in both The Vatican and Jamaica, but more often by oil rig workers, human-banana mutants and fish.
Actual Numerical Value[edit]
Eleventy Threeve is the conjunction of Eleventy (110) and Three + Five (8) resulting in the number 118.
{\displaystyle 110+(3+5)}
As is immediately apparent, 118 is an extremely large number. If someone says to you, "You mean the world to me", what they are really saying you mean Elevnty Threeve Jamaican Dollars to me. Since every one knows that Jamaica has the world's strongest economy, this is a very high compliment.
Eleventy Threeve as a Heavy Element[edit]
Some leading scientologists have speculated that Eleventy Threeve is in fact the Atomic Weight of the Super Heavy Element Oscarwildium. This theory, if proven right could bring about a new era in mankind.
Tenteen
Retrieved from "http://en.illogicopedia.org/w/index.php?title=Eleventy_Threeve&oldid=272906"
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Earth mover's distance - Wikipedia
In statistics, the earth mover's distance (EMD) is a measure of the distance between two probability distributions over a region D. In mathematics, this is known as the Wasserstein metric. Informally, if the distributions are interpreted as two different ways of piling up a certain amount of earth (dirt) over the region D, the EMD is the minimum cost of turning one pile into the other; where the cost is assumed to be the amount of dirt moved times the distance by which it is moved.[1]
The above definition is valid only if the two distributions have the same integral (informally, if the two piles have the same amount of dirt), as in normalized histograms or probability density functions. In that case, the EMD is equivalent to the 1st Mallows distance or 1st Wasserstein distance between the two distributions.[2][3]
1.1 Equivalent Form
1.2 Kantorovich-Rubinstein duality
3 Computing the EMD
4 EMD-based similarity analysis
Assume that we have a set of points in
{\textstyle \mathbb {R} ^{d}}
(dimension
{\textstyle d}
). Instead of assigning one distribution to the set of points, we can cluster them and represent the point set in terms of the clusters. Thus, each cluster is a single point in
{\textstyle \mathbb {R} ^{d}}
and the weight of the cluster is decided by the fraction of the distribution present in that cluster. This representation of a distribution by a set of clusters is called the signature. Two signatures can have different sizes, for example, a bimodal distribution has shorter signature (2 clusters) than complex ones. One cluster representation (mean or mode in
{\textstyle \mathbb {R} ^{d}}
) can be thought of as a single feature in a signature. The distance between each of the features is called as ground distance.
The Earth Mover's Distance can be formulated and solved as a transportation problem. Suppose that several suppliers, each with a given amount of goods, are required to supply several consumers, each with a given limited capacity. For each supplier-consumer pair, the cost of transporting a single unit of goods is given. The transportation problem is then to find a least-expensive flow of goods from the suppliers to the consumers that satisfies the consumers' demand. Similarly, here the problem is transforming one signature(
{\textstyle P}
) to another(
{\textstyle Q}
) with minimum work done.
Assume that signature
{\textstyle P}
{\textstyle m}
clusters with
{\textstyle P=\{(p_{1},w_{p1}),(p_{2},w_{p2}),...,(p_{m},w_{pm})\}}
{\textstyle p_{i}}
is the cluster representative and
{\textstyle w_{pi}>0}
is the weight of the cluster. Similarly another signature
{\textstyle Q=\{(q_{1},w_{q1}),(q_{2},w_{q2}),...,(q_{n},w_{qn})\}}
{\textstyle n}
clusters. Let
{\textstyle D=[d_{i,j}]}
be the ground distance between clusters
{\textstyle p_{i}}
{\textstyle q_{j}}
We want to find a flow
{\textstyle F=[f_{i,j}]}
{\textstyle f_{i,j}}
the flow between
{\textstyle p_{i}}
{\textstyle q_{j}}
, that minimizes the overall cost.
{\displaystyle \min \limits _{F}{\sum _{i=1}^{m}\sum _{j=1}^{n}f_{i,j}d_{i,j}}}
{\displaystyle f_{i,j}\geq 0,1\leq i\leq m,1\leq j\leq n}
{\displaystyle \sum _{j=1}^{n}{f_{i,j}}\leq w_{pi},1\leq i\leq m}
{\displaystyle \sum _{i=1}^{m}{f_{i,j}}\leq w_{qj},1\leq j\leq n}
{\displaystyle \sum _{i=1}^{m}\sum _{j=1}^{n}f_{i,j}=\sum _{i=1}^{m}w_{pi}=\sum _{j=1}^{n}w_{qj}}
The optimal flow
{\textstyle F}
is found by solving this linear optimization problem. The earth mover's distance is defined as the work normalized by the total flow:
{\displaystyle {\text{EMD}}(P,Q)={\frac {\sum _{i=1}^{m}\sum _{j=1}^{n}f_{i,j}d_{i,j}}{\sum _{i=1}^{m}\sum _{j=1}^{n}f_{i,j}}}}
Equivalent Form[edit]
{\displaystyle {\text{EMD}}(P,Q)=\inf \limits _{\gamma \in \Pi (P,Q)}\mathbb {E} _{(x,y)\sim \gamma }[\,\|x-y\|]\,}
{\displaystyle \Pi (P,Q)}
is the set of all joint distributions whose marginals are
{\displaystyle P}
{\displaystyle Q}
Kantorovich-Rubinstein duality[edit]
By Kantorovich-Rubinstein duality, the following is an equivalent expression:
{\displaystyle {\text{EMD}}(P,Q)=\sup \limits _{\|f\|_{L}\leq 1}\mathbb {E} _{x\sim P}[\,f(x)]\,-\mathbb {E} _{x\sim Q}[\,f(x)]\,}
that is, to supremum over all 1-Lipschitz continuous functions, i.e.
{\displaystyle \|\nabla f(x)\|\leq 1\quad \forall x}
Some applications may require the comparison of distributions with different total masses. One approach is to allow for a partial match, where dirt from the most massive distribution is rearranged to make the least massive, and any leftover "dirt" is discarded at no cost. Under this approach, the EMD is no longer a true distance between distributions.
Another approach is to allow for mass to be created or destroyed, on a global and/or local level, as an alternative to transportation, but with a cost penalty. In that case one must specify a real parameter σ, the ratio between the cost of creating or destroying one unit of "dirt", and the cost of transporting it by a unit distance. This is equivalent to minimizing the sum of the earth moving cost plus σ times the L1 distance between the rearranged pile and the second distribution.
Notationally, if
{\displaystyle \pi :P\to Q}
is a partial function which is a bijection on subsets
{\displaystyle P'\subset P}
{\displaystyle Q'\subset Q}
, then one is interested in the distance function
{\displaystyle |P-Q|_{\pi }=|P\setminus P'|+|Q\setminus Q'|}
{\displaystyle P\setminus P'}
denotes set minus. Here,
{\displaystyle P'}
would be the portion of the earth that was moved; thus
{\displaystyle P\setminus P'}
would be the portion not moved, and
{\displaystyle |P\setminus P'|}
the size of the pile not moved. By symmetry, one contemplates
{\displaystyle Q'\subset Q}
as the pile at the destination that 'got there' from P, as compared to the total Q that we want to have there. Formally, this distance indicates how much an injective correspondence differs from an isomorphism.
The EMD can be extended naturally to the case where more than two distributions are compared. In this case, the "distance" between the many distributions is defined as the optimal value of a linear program. This generalized EMD may be computed exactly using a greedy algorithm, and the resulting functional has been shown to be Minkowski additive and convex monotone.[4]
Computing the EMD[edit]
The EMD can be computed by solving an instance of transportation problem, using any algorithm for minimum-cost flow problem, e.g. the network simplex algorithm.
The Hungarian algorithm can be used to get the solution if the domain D is the set {0, 1}. If the domain is integral, it can be translated for the same algorithm by representing integral bins as multiple binary bins.
As a special case, if D is a one-dimensional array of "bins" of size n, the EMD can be efficiently computed by scanning the array and keeping track of how much dirt needs to be transported between consecutive bins. Here the bins are zero-indexed:
{\displaystyle {\begin{aligned}{\text{EMD}}_{0}&=0\\{\text{EMD}}_{i+1}&=P_{i}+{\text{EMD}}_{i}-Q_{i}\\{\text{Total Distance}}&=\sum _{i=0}^{n}|{\text{EMD}}_{i}|\end{aligned}}}
EMD-based similarity analysis[edit]
EMD-based similarity analysis (EMDSA) is an important and effective tool in many multimedia information retrieval[5] and pattern recognition[6] applications. However, the computational cost of EMD is super-cubic to the number of the "bins" given an arbitrary "D". Efficient and scalable EMD computation techniques for large scale data have been investigated using MapReduce,[7][8] as well as bulk synchronous parallel and resilient distributed dataset.[9]
An early application of the EMD in computer science was to compare two grayscale images that may differ due to dithering, blurring, or local deformations.[10] In this case, the region is the image's domain, and the total amount of light (or ink) is the "dirt" to be rearranged.
The EMD is widely used in content-based image retrieval to compute distances between the color histograms of two digital images.[citation needed] In this case, the region is the RGB color cube, and each image pixel is a parcel of "dirt". The same technique can be used for any other quantitative pixel attribute, such as luminance, gradient, apparent motion in a video frame, etc..
More generally, the EMD is used in pattern recognition to compare generic summaries or surrogates of data records called signatures.[11] A typical signature consists of list of pairs ((x1,m1), ... (xn,mn)), where each xi is a certain "feature" (e.g., color in an image, letter in a text, etc.), and mi is "mass" (how many times that feature occurs in the record). Alternatively, xi may be the centroid of a data cluster, and mi the number of entities in that cluster. To compare two such signatures with the EMD, one must define a distance between features, which is interpreted as the cost of turning a unit mass of one feature into a unit mass of the other. The EMD between two signatures is then the minimum cost of turning one of them into the other.
EMD analysis has been used for quantitating multivariate changes in biomarkers measured by flow cytometry, with potential applications to other technologies that report distributions of measurements.[12]
The concept was first introduced by Gaspard Monge in 1781,[13] in the context of transportation theory. The use of the EMD as a distance measure for monochromatic images was described in 1989 by S. Peleg, M. Werman and H. Rom.[10] The name "earth movers' distance" was proposed by J. Stolfi in 1994,[14] and was used in print in 1998 by Y. Rubner, C. Tomasi and L. G. Guibas.[15]
^ Formal definition
^ Elizaveta Levina; Peter Bickel (2001). "The EarthMover's Distance is the Mallows Distance: Some Insights from Statistics". Proceedings of ICCV 2001. Vancouver, Canada: 251–256.
^ C. L. Mallows (1972). "A note on asymptotic joint normality". Annals of Mathematical Statistics. 43 (2): 508–515. doi:10.1214/aoms/1177692631.
^ Kline, Jeffery (2019). "Properties of the d-dimensional earth mover's problem". Discrete Applied Mathematics. 265: 128–141. doi:10.1016/j.dam.2019.02.042.
^ Mark A. Ruzon; Carlo Tomasi (2001). "Edge, Junction, and Corner Detection Using Color Distributions". IEEE Transactions on Pattern Analysis and Machine Intelligence.
^ Kristen Grauman; Trevor Darrel (2004). "Fast contour matching using approximate earth mover's distance". Proceedings of CVPR 2004.
^ Jin Huang; Rui Zhang; Rajkumar Buyya; Jian Chen (2014). "MELODY-Join: Efficient Earth Mover's Distance Similarity Joins Using MapReduce". Proceedings of IEEE International Conference on Data Engineering.
^ Jia Xu; Bin Lei; Yu Gu; Winslett, M.; Ge Yu; Zhenjie Zhang (2015). "Efficient Similarity Join Based on Earth Mover's Distance Using MapReduce". IEEE Transactions on Knowledge and Data Engineering.
^ Jin Huang; Rui Zhang; Rajkumar Buyya; Jian Chen, M.; Yongwei Wu (2015). "Heads-Join: Efficient Earth Mover's Distance Join on Hadoop". IEEE Transactions on Parallel and Distributed Systems.
^ a b S. Peleg; M. Werman; H. Rom (1989). "A unified approach to the change of resolution: Space and gray-level". IEEE Transactions on Pattern Analysis and Machine Intelligence. 11 (7): 739–742. doi:10.1109/34.192468.
^ Rubner, Y.; Tomasi, C.; Guibas, L.J. "A metric for distributions with applications to image databases". Sixth International Conference on Computer Vision (IEEE Cat. No.98CH36271). Narosa Publishing House. doi:10.1109/iccv.1998.710701.
^ Orlova, DY; Zimmerman, N; Meehan, C; Meehan, S; Waters, J; Ghosn, EEB (23 March 2016). "Earth Mover's Distance (EMD): A True Metric for Comparing Biomarker Expression Levels in Cell Populations". PLOS One. 11 (3): e0151859. doi:10.1371/journal.pone.0151859. PMC 4805242. PMID 27008164.
^ "Mémoire sur la théorie des déblais et des remblais". Histoire de l'Académie Royale des Science, Année 1781, avec les Mémoires de Mathématique et de Physique. 1781.
^ J. Stolfi, personal communication to L. J. Guibas, 1994, as cited by Rubner, Yossi; Tomasi, Carlo; Guibas, Leonidas J. (2000). "The earth mover's distance as a metric for image retrieval" (PDF). International Journal of Computer Vision. 40 (2): 99–121. doi:10.1023/A:1026543900054. S2CID 14106275.
^ Yossi Rubner; Carlo Tomasi; Leonidas J. Guibas (1998). "A Metric for Distributions with Applications to Image Databases". Proceedings ICCV 1998: 59–66. doi:10.1109/ICCV.1998.710701. ISBN 81-7319-221-9. S2CID 18648233.
C code for the Earth Mover's Distance
Python implementation with references
Python2 wrapper for the C implementation of the Earth Mover's Distance
C++ and Matlab and Java wrappers code for the Earth Mover's Distance, especially efficient for thresholded ground distances
Java implementation of a generic generator for evaluating large-scale Earth Mover's Distance based similarity analysis
Retrieved from "https://en.wikipedia.org/w/index.php?title=Earth_mover%27s_distance&oldid=1080303860"
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Non-random perturbations of the Anderson Hamiltonian | EMS Press
Non-random perturbations of the Anderson Hamiltonian
University of North Carolina-Charlotte, United States
The Anderson Hamiltonian
H_0=-\Delta+V(x,\omega)
is considered, where
V
is a random potential of Bernoulli type. The operator
H_0
is perturbed by a non-random, continuous potential
-v(x) \leq 0
, decaying at infinity. It will be shown that the borderline between finitely, and infinitely many negative eigenvalues of the perturbed operator, is achieved with a decay of the potential
-v(x)
O(\ln^{-2/d} |x|)
Stanislav Molchanov, Boris Vainberg, Non-random perturbations of the Anderson Hamiltonian. J. Spectr. Theory 1 (2011), no. 2, pp. 179–195
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Robust H∞ performance optimized by musyn - MATLAB musynperf - MathWorks 日本
The robust H∞ performance of an uncertain system is the smallest value γ such that the I/O gain of the system stays below γ for all modeled uncertainty up to size 1/γ (in normalized units). The musyn function synthesizes a robust controller by minimizing this quantity for the closed-loop system over all possible choices of controller. musynperf computes this quantity for a specified uncertain model. For a detailed discussion of robust H∞ performance and how it is computed, see Robust Performance Measure for Mu Synthesis.
[gamma,wcu] = musynperf(clp) calculates the robust H∞ performance for an uncertain closed-loop system clp. The robust H∞ performance is the smallest value γ for which the peak I/O gain stays below γ for all modeled uncertainty up to 1/γ, in normalized units. For example, a value of γ = 1.125 implies the following:
The peak I/O gain is the maximum I/O gain over all inputs, which is also the peak of the largest singular value over all frequencies and uncertainties. In other words, if Δ represents all possible values of the uncertain parameters in the closed-loop transfer function CLP(jω), then
\mathrm{γ}=\underset{\mathrm{Î}}{\mathrm{max}}\underset{\mathrm{Ï}}{\mathrm{max}}{\mathrm{Ï}}_{max}\left(CLP\left(j\mathrm{Ï}\right)\right).
[gamma,wcu,info] = musynperf(___) returns a structure with additional information about the H∞ performance values and the perturbations that drive the I/O gain to γ. See info for details about this structure. You can use this syntax with any of the previous input-argument combinations.
{\mathit{H}}_{\infty }
{\mathit{H}}_{\infty }
{\mathit{H}}_{\infty }
Lower bound on the actual robust H∞ performance γ, returned as a scalar value. The exact value of γ is guaranteed to be no smaller than LowerBound. In other words, some uncertain-element values of magnitude 1/LowerBound exist for which the I/O gain of clp reaches LowerBound. The function returns one such instance in wcu.
info — Additional information about γ values
Additional information about the γ values, returned as a structure with the following fields.
Frequency points at which musynperf returns γ values, returned as a vector.
If the 'VaryFrequency' option of robOptions is 'off', then info.Frequency is the critical frequency, the frequency at which the I/O gain reaches gamma.LowerBound. If the smallest lower bound and the smallest upper bound on γ occur at different frequencies, then info.Frequency is a vector containing these two frequencies.
If you specify a vector of frequencies w at which to compute γ, then info.Frequency = w. When you specify a frequency vector, these frequencies are not guaranteed to include the frequency at which the peak gain occurs.
Lower and upper bounds on the actual γ values, returned as an array. info.Bounds(:,1) contains the lower bound at each corresponding frequency in info.Frequency, and info.Bounds(:,2) contains the corresponding upper bounds.
Sensitivity of γ to each uncertain element, returned as a structure when the 'Sensitivity' option of robOptions is 'on'. The fields of info.Sensitivity are the names of the uncertain elements in clp. Each field contains a percentage that measures how much the uncertainty in the corresponding element affects γ. For example, if info.Sensitivity.p is 50, then a given fractional change in the uncertainty range of p causes half as much fractional change in γ.
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A couple of job opportunities to work with ATLAS in 2017 | Borborigmi di un fisico renitente
I rarely discuss job administrative details on these pages, but I thought it could be interesting to advertise a couple of job openings here. At the beginning of 2016 me and a group of collaborators from the ATLAS experiment were awarded fundings by the French National Research Agency (ANR) for a project called PhotonPortal. The goal of the project, which I am the Principal Investigator of, is to search for phenomena beyond the Standard Model using event topologies with photon in the final state, and with precision measurement of the Higgs boson properties with the
H\to\gamma\gamma
decay. If you are interested in the project details, you can find a 30-pages writeup on the project website.
The PhotonPortal fundings will be mostly used to hire three 2-years post-docs, to be respectively based in LAL Orsay, LPNHE Paris, and at my lab in LAPP Annecy. We are now opening the first position to work in LAL, so if you have a PhD in particle physics and are interested to join the team, consider to apply. The opening has a formal deadline at December 1st, but the position will stay opened until filled. The opening for the position based in Paris will happen during Fall 2017, while that to work with my team in LAPP in Fall 2018. I'll publicise the future openings when the time will come.
At LAPP we are also looking for a new PhD student, to work with me on the measurement of the Higgs boson differential cross section with the
H\to\gamma\gamma
decay. The student is invited to join the team during Spring 2017 for an undergraduate internship that could last from a minimum of six week to four months. This pre-graduate-school internship is strongly advised: it will help to see whether the student is a good match for the team, and the other way around. Upon mutual satisfaction, the student could start the PhD in October 2017, based at LAPP with frequent presence at CERN (or the other way around, if needed). The reference university that will award the PhD title is the University of Grenoble. If interested, do not hesitate to contact me, all my work contact are on the project description.
A note for the Italian students: the undergraduate internship is something one can do between the end of your Master (laurea specialistica) and the beginning of the PhD, or as a part of the Master itself, upon agreement with your University. This is actually what the French student usually do, as part of the compulsory internship requested to complete their second-level Master and graduate.
P.S. The PhotonPortal website is my first attempt to work with a static site generator. I used Jekyll, and I actually while liked it. One day or another I'll discuss the details a bit more.
(That's another entry among my attempts to blog in English. Un esperimento, appunto, da mescolarsi con la consueta scrittura in lingua italica: che lettori non anglofoni non disperano. Gli studenti di fisica di qualunque nazionalità, invece, dovrebbero sapere leggere l'inglese!)
OT. OT. OT.
Non rientro nel target di questo post, ma non posso trattenermi da chiederti un commento.
Io l'ho letto solo su Wired, Repubblica e altri siti.
Non ho conoscenze sufficienti per giudicare, ma l'energia e la materia oscura mi fanno sempre pensare all'etere pre-Michelson e Morley.
D'altro lato sono sempre in attesa di assistere (finché campo: e ho fretta) alla nuova rivoluzione che apra nuovi orizzonti alla fisica.
Scusa la mia attuale iperpresenza, ma mi sto curando con il cortisone e questo mi determina iperattività. 🙂
Oh, paper di Verlinde. Che dire? Questa vignetta l'hai vista? http://xkcd.com/1758/ La dice lunga su quello che molti pensano (it sounds nice but is does not really fit the data)...
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GCD and Euclid's Algorithm - Maple Help
Greatest Common Divisor and the Euclidean Algorithm
The greatest common divisor (GCD) of two integers (not both 0) is the largest positive integer which divides both of them.
There are four simple rules which allow us to compute the GCD of two integers:
\mathrm{GCD}\left(-a, b\right) = \mathrm{GCD}\left(a,b\right)
\mathrm{GCD}\left(a,b\right)=\mathrm{GCD}\left(b,a\right)
\mathrm{GCD}\left(a+b\cdot c, b\right) = \mathrm{GCD}\left(a,b\right)
c
\mathrm{GCD}\left(a, 0\right) = a
The Euclidean Algorithm is a sequence of steps that use the above rules to find the GCD for any two integers
and
b
and
b
are both non-negative and
a≥b
(otherwise we can use rules 1 and 2 above).
{r}_{1}=a
{r}_{2}=b
n=2
{r}_{n}>0
{r}_{n+1}
be the remainder of dividing
{r}_{n-1}
{r}_{n}
n
{r}_{n-1}
Input two integers in the boxes below. Click "Find GCD" and then "Next Step" to follow the steps of the Euclidean Algorithm to find the greatest common divisor of the two integers. Click "Zoom" if the image gets too small to see. The animation starts with a rectangle with the dimensions of
and
b
, and repeatedly subtracts squares, until what remains is a square. That square has sides of length the GCD of
and
b
The GCD of:
& is:
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Wikizero - Capsid
Protein shell of a virus
Illustration of geometric model changing between two possible capsids. A similar change of size has been observed as the result of a single amino-acid mutation[1]
A capsid is the protein shell of a virus, enclosing its genetic material. It consists of several oligomeric (repeating) structural subunits made of protein called protomers. The observable 3-dimensional morphological subunits, which may or may not correspond to individual proteins, are called capsomeres. The proteins making up the capsid are called capsid proteins or viral coat proteins (VCP). The capsid and inner genome is called the nucleocapsid.
Capsids are broadly classified according to their structure. The majority of the viruses have capsids with either helical or icosahedral[2][3] structure. Some viruses, such as bacteriophages, have developed more complicated structures due to constraints of elasticity and electrostatics.[4] The icosahedral shape, which has 20 equilateral triangular faces, approximates a sphere, while the helical shape resembles the shape of a spring, taking the space of a cylinder but not being a cylinder itself.[5] The capsid faces may consist of one or more proteins. For example, the foot-and-mouth disease virus capsid has faces consisting of three proteins named VP1–3.[6]
Some viruses are enveloped, meaning that the capsid is coated with a lipid membrane known as the viral envelope. The envelope is acquired by the capsid from an intracellular membrane in the virus' host; examples include the inner nuclear membrane, the Golgi membrane, and the cell's outer membrane.[7]
Once the virus has infected a cell and begins replicating itself, new capsid subunits are synthesized using the protein biosynthesis mechanism of the cell. In some viruses, including those with helical capsids and especially those with RNA genomes, the capsid proteins co-assemble with their genomes. In other viruses, especially more complex viruses with double-stranded DNA genomes, the capsid proteins assemble into empty precursor procapsids that include a specialized portal structure at one vertex. Through this portal, viral DNA is translocated into the capsid.[8]
Structural analyses of major capsid protein (MCP) architectures have been used to categorise viruses into lineages. For example, the bacteriophage PRD1, the algal virus Paramecium bursaria Chlorella virus-1 (PBCV-1), mimivirus and the mammalian adenovirus have been placed in the same lineage, whereas tailed, double-stranded DNA bacteriophages (Caudovirales) and herpesvirus belong to a second lineage.[9][10][11][12]
1 Specific shapes
1.1 Icosahedral
1.2 Prolate
Specific shapes[edit]
Icosahedral[edit]
Icosahedral capsid of an adenovirus
Virus capsid T-numbers
The icosahedral structure is extremely common among viruses. The icosahedron consists of 20 triangular faces delimited by 12 fivefold vertexes and consists of 60 asymmetric units. Thus, an icosahedral virus is made of 60N protein subunits. The number and arrangement of capsomeres in an icosahedral capsid can be classified using the "quasi-equivalence principle" proposed by Donald Caspar and Aaron Klug.[13] Like the Goldberg polyhedra, an icosahedral structure can be regarded as being constructed from pentamers and hexamers. The structures can be indexed by two integers h and k, with
{\displaystyle h\geq 1}
{\displaystyle k\geq 0}
; the structure can be thought of as taking h steps from the edge of a pentamer, turning 60 degrees counterclockwise, then taking k steps to get to the next pentamer. The triangulation number T for the capsid is defined as:
{\displaystyle T=h^{2}+h\cdot k+k^{2}}
In this scheme, icosahedral capsids contain 12 pentamers plus 10(T − 1) hexamers.[14][15] The T-number is representative of the size and complexity of the capsids.[16] Geometric examples for many values of h, k, and T can be found at List of geodesic polyhedra and Goldberg polyhedra.
Many exceptions to this rule exist: For example, the polyomaviruses and papillomaviruses have pentamers instead of hexamers in hexavalent positions on a quasi T = 7 lattice. Members of the double-stranded RNA virus lineage, including reovirus, rotavirus and bacteriophage φ6 have capsids built of 120 copies of capsid protein, corresponding to a T = 2 capsid, or arguably a T = 1 capsid with a dimer in the asymmetric unit. Similarly, many small viruses have a pseudo T = 3 (or P = 3) capsid, which is organized according to a T = 3 lattice, but with distinct polypeptides occupying the three quasi-equivalent positions [17]
T-numbers can be represented in different ways, for example T = 1 can only be represented as an icosahedron or a dodecahedron and, depending on the type of quasi-symmetry, T = 3 can be presented as a truncated dodecahedron, an icosidodecahedron, or a truncated icosahedron and their respective duals a triakis icosahedron, a rhombic triacontahedron, or a pentakis dodecahedron.[18][clarification needed]
Prolate[edit]
The prolate structure of a typical head on a bacteriophage
An elongated icosahedron is a common shape for the heads of bacteriophages. Such a structure is composed of a cylinder with a cap at either end. The cylinder is composed of 10 elongated triangular faces. The Q number (or Tmid), which can be any positive integer,[19] specifies the number of triangles, composed of asymmetric subunits, that make up the 10 triangles of the cylinder. The caps are classified by the T (or Tend) number.[20]
The bacterium E. coli is the host for bacteriophage T4 that has a prolate head structure. The bacteriophage encoded gp31 protein appears to be functionally homologous to E. coli chaperone protein GroES and able to substitute for it in the assembly of bacteriophage T4 virions during infection.[21] Like GroES, gp31 forms a stable complex with GroEL chaperonin that is absolutely necessary for the folding and assembly in vivo of the bacteriophage T4 major capsid protein gp23.[21]
Helical[edit]
3D model of a helical capsid structure of a virus
Many rod-shaped and filamentous plant viruses have capsids with helical symmetry.[22] The helical structure can be described as a set of n 1-D molecular helices related by an n-fold axial symmetry.[23] The helical transformation are classified into two categories: one-dimensional and two-dimensional helical systems.[23] Creating an entire helical structure relies on a set of translational and rotational matrices which are coded in the protein data bank.[23] Helical symmetry is given by the formula P = μ x ρ, where μ is the number of structural units per turn of the helix, ρ is the axial rise per unit and P is the pitch of the helix. The structure is said to be open due to the characteristic that any volume can be enclosed by varying the length of the helix.[24] The most understood helical virus is the tobacco mosaic virus.[22] The virus is a single molecule of (+) strand RNA. Each coat protein on the interior of the helix bind three nucleotides of the RNA genome. Influenza A viruses differ by comprising multiple ribonucleoproteins, the viral NP protein organizes the RNA into a helical structure. The size is also different; the tobacco mosaic virus has a 16.33 protein subunits per helical turn,[22] while the influenza A virus has a 28 amino acid tail loop.[25]
The functions of the capsid are to:
protect the genome,
deliver the genome, and
interact with the host.
The virus must assemble a stable, protective protein shell to protect the genome from lethal chemical and physical agents. These include extremes of pH or temperature and proteolytic and nucleolytic enzymes. For non-enveloped viruses, the capsid itself may be involved in interaction with receptors on the host cell, leading to penetration of the host cell membrane and internalization of the capsid. Delivery of the genome occurs by subsequent uncoating or disassembly of the capsid and release of the genome into the cytoplasm, or by ejection of the genome through a specialized portal structure directly into the host cell nucleus.
It has been suggested that many viral capsid proteins have evolved on multiple occasions from functionally diverse cellular proteins.[26] The recruitment of cellular proteins appears to have occurred at different stages of evolution so that some cellular proteins were captured and refunctionalized prior to the divergence of cellular organisms into the three contemporary domains of life, whereas others were hijacked relatively recently. As a result, some capsid proteins are widespread in viruses infecting distantly related organisms (e.g., capsid proteins with the jelly-roll fold), whereas others are restricted to a particular group of viruses (e.g., capsid proteins of alphaviruses).[26][27]
A computational model (2015) has shown that capsids may have originated before viruses and that they served as a means of horizontal transfer between replicator communities since these communities could not survive if the number of gene parasites increased, with certain genes being responsible for the formation of these structures and those that favored the survival of self-replicating communities.[28] The displacement of these ancestral genes between cellular organisms could favor the appearance of new viruses during evolution.[27]
Fullerene#Other buckyballs
^ Asensio MA, Morella NM, Jakobson CM, Hartman EC, Glasgow JE, Sankaran B, et al. (September 2016). "A Selection for Assembly Reveals That a Single Amino Acid Mutant of the Bacteriophage MS2 Coat Protein Forms a Smaller Virus-like Particle". Nano Letters. 16 (9): 5944–50. Bibcode:2016NanoL..16.5944A. doi:10.1021/acs.nanolett.6b02948. OSTI 1532201. PMID 27549001.
^ Lidmar J, Mirny L, Nelson DR (November 2003). "Virus shapes and buckling transitions in spherical shells". Physical Review E. 68 (5 Pt 1): 051910. arXiv:cond-mat/0306741. Bibcode:2003PhRvE..68e1910L. doi:10.1103/PhysRevE.68.051910. PMID 14682823. S2CID 6023873.
^ Vernizzi G, Olvera de la Cruz M (November 2007). "Faceting ionic shells into icosahedra via electrostatics". Proceedings of the National Academy of Sciences of the United States of America. 104 (47): 18382–6. Bibcode:2007PNAS..10418382V. doi:10.1073/pnas.0703431104. PMC 2141786. PMID 18003933.
^ Vernizzi G, Sknepnek R, Olvera de la Cruz M (March 2011). "Platonic and Archimedean geometries in multicomponent elastic membranes". Proceedings of the National Academy of Sciences of the United States of America. 108 (11): 4292–6. Bibcode:2011PNAS..108.4292V. doi:10.1073/pnas.1012872108. PMC 3060260. PMID 21368184.
^ Branden C, Tooze J (1991). Introduction to Protein Structure. New York: Garland. pp. 161–162. ISBN 978-0-8153-0270-4.
^ Alberts B, Bray D, Lewis J, Raff M, Roberts K, Watson JD (1994). Molecular Biology of the Cell (4th ed.). p. 280.
^ Newcomb WW, Homa FL, Brown JC (August 2005). "Involvement of the portal at an early step in herpes simplex virus capsid assembly". Journal of Virology. 79 (16): 10540–6. doi:10.1128/JVI.79.16.10540-10546.2005. PMC 1182615. PMID 16051846.
^ Krupovic M, Bamford DH (December 2008). "Virus evolution: how far does the double beta-barrel viral lineage extend?". Nature Reviews. Microbiology. 6 (12): 941–8. doi:10.1038/nrmicro2033. PMID 19008892. S2CID 31542714.
^ Forterre P (March 2006). "Three RNA cells for ribosomal lineages and three DNA viruses to replicate their genomes: a hypothesis for the origin of cellular domain". Proceedings of the National Academy of Sciences of the United States of America. 103 (10): 3669–74. Bibcode:2006PNAS..103.3669F. doi:10.1073/pnas.0510333103. PMC 1450140. PMID 16505372.
^ Khayat R, Tang L, Larson ET, Lawrence CM, Young M, Johnson JE (December 2005). "Structure of an archaeal virus capsid protein reveals a common ancestry to eukaryotic and bacterial viruses". Proceedings of the National Academy of Sciences of the United States of America. 102 (52): 18944–9. doi:10.1073/pnas.0506383102. PMC 1323162. PMID 16357204.
^ Laurinmäki PA, Huiskonen JT, Bamford DH, Butcher SJ (December 2005). "Membrane proteins modulate the bilayer curvature in the bacterial virus Bam35". Structure. 13 (12): 1819–28. doi:10.1016/j.str.2005.08.020. PMID 16338410.
^ Carrillo-Tripp M, Shepherd CM, Borelli IA, Venkataraman S, Lander G, Natarajan P, et al. (January 2009). "VIPERdb2: an enhanced and web API enabled relational database for structural virology". Nucleic Acids Research. 37 (Database issue): D436-42. doi:10.1093/nar/gkn840. PMC 2686430. PMID 18981051. Archived from the original on 2018-02-11. Retrieved 2011-03-18.
^ Johnson JE, Speir JA (2009). Desk Encyclopedia of General Virology. Boston: Academic Press. pp. 115–123. ISBN 978-0-12-375146-1.
^ Mannige RV, Brooks CL (March 2010). "Periodic table of virus capsids: implications for natural selection and design". PLOS ONE. 5 (3): e9423. Bibcode:2010PLoSO...5.9423M. doi:10.1371/journal.pone.0009423. PMC 2831995. PMID 20209096.
^ Sgro J. "Virusworld". Institute for Molecular Virology. University of Wisconsin-Madison.
^ Damodaran KV, Reddy VS, Johnson JE, Brooks CL (December 2002). "A general method to quantify quasi-equivalence in icosahedral viruses". Journal of Molecular Biology. 324 (4): 723–37. doi:10.1016/S0022-2836(02)01138-5. PMID 12460573.
^ Luque A, Reguera D (June 2010). "The structure of elongated viral capsids". Biophysical Journal. 98 (12): 2993–3003. Bibcode:2010BpJ....98.2993L. doi:10.1016/j.bpj.2010.02.051. PMC 2884239. PMID 20550912.
^ Casjens S (2009). Desk Encyclopedia of General Virology. Boston: Academic Press. pp. 167–174. ISBN 978-0-12-375146-1.
^ a b Marusich EI, Kurochkina LP, Mesyanzhinov VV. Chaperones in bacteriophage T4 assembly. Biochemistry (Mosc). 1998;63(4):399-406
^ a b c Yamada S, Matsuzawa T, Yamada K, Yoshioka S, Ono S, Hishinuma T (December 1986). "Modified inversion recovery method for nuclear magnetic resonance imaging". The Science Reports of the Research Institutes, Tohoku University. Ser. C, Medicine. Tohoku Daigaku. 33 (1–4): 9–15. PMID 3629216.
^ a b c Aldrich RA (February 1987). "Children in cities--Seattle's KidsPlace program". Acta Paediatrica Japonica. 29 (1): 84–90. doi:10.1111/j.1442-200x.1987.tb00013.x. PMID 3144854. S2CID 33065417.
^ Racaniello VR, Enquist LW (2008). Principles of Virology, Vol. 1: Molecular Biology. Washington, D.C: ASM Press. ISBN 978-1-55581-479-3.
^ Ye Q, Guu TS, Mata DA, Kuo RL, Smith B, Krug RM, Tao YJ (26 December 2012). "Biochemical and structural evidence in support of a coherent model for the formation of the double-helical influenza A virus ribonucleoprotein". mBio. 4 (1): e00467–12. doi:10.1128/mBio.00467-12. PMC 3531806. PMID 23269829.
^ a b Krupovic M, Koonin EV (March 2017). "Multiple origins of viral capsid proteins from cellular ancestors". Proceedings of the National Academy of Sciences of the United States of America. 114 (12): E2401–E2410. doi:10.1073/pnas.1621061114. PMC 5373398. PMID 28265094.
^ a b Krupovic M, Dolja VV, Koonin EV (July 2019). "Origin of viruses: primordial replicators recruiting capsids from hosts" (PDF). Nature Reviews. Microbiology. 17 (7): 449–458. doi:10.1038/s41579-019-0205-6. PMID 31142823. S2CID 169035711.
^ Jalasvuori M, Mattila S, Hoikkala V (2015). "Chasing the Origin of Viruses: Capsid-Forming Genes as a Life-Saving Preadaptation within a Community of Early Replicators". PLOS ONE. 10 (5): e0126094. Bibcode:2015PLoSO..1026094J. doi:10.1371/journal.pone.0126094. PMC 4425637. PMID 25955384.
Williams R (1 June 1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. pp. 142–144, Figures 4-49, 50, 51: Custers of 12 spheres, 42 spheres, 92 spheres. ISBN 978-0-486-23729-9.
Pugh A (1 September 1976). Polyhedra: A Visual Approach. Chapter 6. The Geodesic Polyhedra of R. Buckminster Fuller and Related Polyhedra. ISBN 978-0-520-02926-2.
Almansour I, Alhagri M, Alfares R, Alshehri M, Bakhashwain R, Maarouf A (January 2019). "IRAM: virus capsid database and analysis resource". Database: The Journal of Biological Databases and Curation. 2019. doi:10.1093/database/baz079. PMC 6637973. PMID 31318422.
IRAM-Virus Capsid Database and Analysis Resource
Wikimedia Commons has media related to Capsid.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Capsid&oldid=1088621219"
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To W. B. Carpenter 3 December [1859]
Ilkley Wells House | Otley Yorkshire
My dear Carpenter
I am perfectly delighted at your letter.1 It is a great thing to have got a great physiologist on our side. I say “our” for we are now a good & compact body of really good men & mostly not old men.— In the long run we shall conquer. I do not like being abused but I feel that I can now bear it; &, as I told Lyell, I am well convinced that it is the first offender who reaps the rich harvest of abuse.— You have done me an essential kindness in checking the odium theologicum in the E.R.— It much pains all one’s female relations & injures the cause.—2
I look at it as immaterial whether we go quite the same lengths; & I suspect, judging from myself, that you will go further, by thinking of a population of forms like Ornithorhynchus & by thinking of the common homological & embryological structure of the several Vertebrate orders. But this is immaterial; I quite agree that the principle is everything. In my fuller M.S. I have discussed a good many instincts; but there will surely be more unfilled gaps here than with corporeal structure; for we have no fossil instincts & know scarcely any except of European animals.— When I reflect how very slowly I came round myself, I am in truth astonished at the candour shown by Lyell, Hooker, Huxley & yourself. In my opinion it is grand. I thank you cordially for taking the trouble of writing a Review for the National:3 God knows I shall have few enough in any degree favourable.—
I am very much obliged for your kind invitation; but I am always knocked up by a Journey & must sleep at my Brothers.— I go to London next Wednesday & have agreed to call on Lyell at 10 oclock on Thursday morning & shall stay there about one hour, or hour &
\frac{1}{2}
. Are you free on Thursday morning from 11
\frac{1}{2}
to 12 oclock? if so, I would call on you. But you must not think of staying in on my account, for I am so subject to headach, that I am never certain of my movements. But I would assuredly come, if possible. If you cannot have me, will you send a line to “57 Queen Anne St. (W.)”—4 if I do not hear I will understand that you will be at home, & come I will if I can.— I must return home that afternoon.—
Yours most sincerely obliged | C. Darwin
Carpenter’s letter has not been located.
Carpenter had probably advised Henry Reeve, editor of the Edinburgh Review, on appropriate reviewers for Origin. CD probably had in mind Adam Sedgwick’s severe review of Vestiges of creation in the Edinburgh Review ([Sedgwick] 1845), particularly after having received Sedgwick’s response to Origin in his letter of 24 November 1859. Henrietta Darwin later wrote that Emma Darwin would not show her ‘Professor Sedgwick’s horrified reprobation’ of Origin (Emma Darwin (1915) 2: 172). See also letter to C. S. Wedgwood, [after 21 November 1859].
Carpenter’s review appeared in National Review 10 (1860): 188–214.
The address of Erasmus Alvey Darwin’s house in London.
Delighted by WBC’s letter about Origin. There is now "a great physiologist on our side". "You have done me an essential kindness in checking the odium theologicum in the E[dinburgh] R[eview] … immaterial whether we go quite the same lengths … the principle is everything."
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Constant gamma clutter simulation - Simulink - MathWorks í•œêµ
\mathrm{γ}
\mathrm{γ}
\mathrm{γ}
Specify the direction of radar platform motion as a 2-by-1 real vector in the form [AzimuthAngle;ElevationAngle]. Units are in degrees. Both azimuth and elevation angle are measured in the local coordinate system of the radar antenna or antenna array. Azimuth angle must be between –180° and 180°. Elevation angle must be between –90° and 90°.
Select this check box to baffle the back response of the element. When back baffled, the responses at all azimuth angles beyond ±90° from broadside are set to zero. The broadside direction is defined as 0° azimuth angle and 0° elevation angle.
Specify the azimuth angles at which to calculate the antenna radiation pattern as a 1-by-P row vector. P must be greater than 2. Azimuth angles must lie between –180° and 180°, inclusive, and be in strictly increasing order.
Specify the elevation angles at which to compute the radiation pattern as a 1-by-Q vector. Q must be greater than 2. Angle units are in degrees. Elevation angles must lie between –90° and 90°, inclusive, and be in strictly increasing order.
Phi angles of points at which to specify the antenna radiation pattern, specify as a real-valued 1-by-P row vector. P must be greater than 2. Angle units are in degrees. Phi angles must lie between 0° and 360° and be in strictly increasing order.
Theta angles of points at which to specify the antenna radiation pattern, specify as a real-valued 1-by-Q row vector. Q must be greater than 2. Angle units are in degrees. Theta angles must lie between 0° and 360° and be in strictly increasing order.
Specify the polar pattern response angles, as a 1-by-P vector. The angles are measured from the central pickup axis of the microphone and must be between –180° and 180°, inclusive.
Specify the magnitude of the custom microphone element polar patterns as an L-by-P matrix. L is the number of frequencies specified in Polar pattern frequencies (Hz). P is the number of angles specified in Polar pattern angles (deg). Each row of the matrix represents the magnitude of the polar pattern measured at the corresponding frequency specified in Polar pattern frequencies (Hz) and all angles specified in Polar pattern angles (deg). The pattern is measured in the azimuth plane. In the azimuth plane, the elevation angle is 0° and the central pickup axis is 0° degrees azimuth and 0° degrees elevation. The polar pattern is symmetric around the central axis. You can construct the microphone response pattern in 3-D space from the polar pattern.
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Asymptotic Behavior of the 3D Compressible Euler Equations with Nonlinear Damping and Slip Boundary Condition
2012 Asymptotic Behavior of the 3D Compressible Euler Equations with Nonlinear Damping and Slip Boundary Condition
The asymptotic behavior (as well as the global existence) of classical solutions to the 3D compressible Euler equations are considered. For polytropic perfect gas
\left(P\left(\rho \right)={P}_{0}{\rho }^{\gamma }\right)
, time asymptotically, it has been proved by Pan and Zhao (2009) that linear damping and slip boundary effect make the density satisfying the porous medium equation and the momentum obeying the classical Darcy's law. In this paper, we use a more general method and extend this result to the 3D compressible Euler equations with nonlinear damping and a more general pressure term. Comparing with linear damping, nonlinear damping can be ignored under small initial data.
Huimin Yu. "Asymptotic Behavior of the 3D Compressible Euler Equations with Nonlinear Damping and Slip Boundary Condition." J. Appl. Math. 2012 1 - 16, 2012. https://doi.org/10.1155/2012/584680
Huimin Yu "Asymptotic Behavior of the 3D Compressible Euler Equations with Nonlinear Damping and Slip Boundary Condition," Journal of Applied Mathematics, J. Appl. Math. 2012(none), 1-16, (2012)
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Countable set - Simple English Wikipedia, the free encyclopedia
set with the same cardinality as the set of natural numbers
In mathematics (particularly set theory), a countable set is a set whose elements can be counted. A set with one thing in it is countable, and so is a set with one hundred things in it. A set with all the natural numbers (counting numbers) in it is countable too. This is because even if it is infinite, someone who counts forever would not miss any of the numbers. The sets which has the same size as the natural numbers are therefore called countably infinite. The size of these sets are then written as
{\displaystyle \aleph _{0}}
(aleph-null)—the first of the aleph numbers.[1] Sometimes when people say 'countable set' they mean countable and infinite.[2] Georg Cantor coined the term.
1 Examples of countable sets
1.1 A more-complex example
1.2 Not all counting schemes will yield a countable list
Examples of countable setsEdit
a diagram illustrating the countability of the rationals
Countable sets include all sets with a finite number of members, no matter how many.
Countable sets also include some infinite sets, such as the natural numbers. Since it is impossible to actually count infinite sets, we consider them countable if we can find a way to list them all without missing any. The natural numbers have been nicknamed "the counting numbers", since they are what we usually use to count things with.
You can count the natural numbers with
{\displaystyle ({1,2,3...})\,\!}
You can count the whole numbers with
{\displaystyle ({0,1,2,3...})\,\!}
You can count the integers with
{\displaystyle ({0,1,-1,2,-2,3,-3...})\,\!}
You can count the square integer numbers with
{\displaystyle ({0,1,4,9,16,25,36,...})\,\!}
A more-complex exampleEdit
The rational numbers are also countable, but its counting becomes tricky. We can't just list the fractions as (0/1, 1/1, -1/1, 2/1, -2/1...) since we will never get to 1/2 this way. However, we can list all of the rational numbers in the form of a table. The horizontal rows are the numerators, while the vertical columns are the denominators. We can then zigzag through the list, starting at 1/1, then 2/1, then 1/2, then 1/3, then 2/2, then 3/1, then 4/1, then 3/2, then 2/3, etc. If we keep this up, we will get to all of the numbers on the table in due time. Each time we hit a new rational number that is new, we add it to the list of counted numbers. We don't want to count numbers twice, so when we hit 3/6, for example, we can skip it, because we already counted the rational number 1/2. In this way, we produce an infinite list with all the rational numbers. Therefore, the rational numbers are countable.
Not all counting schemes will yield a countable listEdit
Suppose the integers are counted differently, as
{\displaystyle ({0,1,2,3,....-1,-2,-3,...})\,\!}
This does not prove that the integers are uncountable, but it does illustrate what one might call a "failed" effort to count a set, because it is never possible to reach
{\displaystyle -1}
. Not all sets are countable. For example, the interval
{\displaystyle (0,1)}
and the set of real numbers can be proven to be uncountable.[3]
↑ Weisstein, Eric W. "Countable Set". mathworld.wolfram.com. Retrieved 2020-09-06.
↑ "9.3: Uncountable Sets". Mathematics LibreTexts. 2017-09-20. Retrieved 2020-09-06.
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Countable_set&oldid=7099137"
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Demand - Course Hero
Microeconomics/Supply and Demand/Demand
Learn all about demand in just a few minutes! Professor Jadrian Wooten of Penn State University explains demand, along with all its related concepts, such as demand schedules, demand supplied, and the law of demand.
Demand is the desire and ability of consumers for a good at a range of prices. Demand describes the behavior of individual consumers and markets made up of many consumers. A market is a group of buyers and sellers of a particular good or service. One person may buy a large quantity of an item when it is inexpensive and less of that item when the price is higher. A demand schedule is a table showing the quantity demanded of a good at different prices and can also be used to show the maximum price that a consumer would be willing to pay. Other people (e.g., persons with more or less disposable income or persons with different shopping priorities) may buy different quantities of an item at the same prices. Taken together, all consumers interested in purchasing the same item constitute a market, and the quantity of items they demand constitutes market demand.
Consider a market made up of three individuals: Christina, Michael, and Maria. For any particular price that donuts might have in that market, each person will buy a different quantity of donuts. Suppose donuts range in price from $1 to $5. Suppose also that Michael has an insatiable desire for donuts and buys three times as many as Christina does at every price point. Maria enjoys donuts, but she is also quite frugal, and she buys half as many as Christina at every price point. If Christina will buy 10 donuts at $1 per donut, then Michael will buy 30 at that price point and Maria will buy 5. The number of items demanded by all three persons taken together is the market demand, and at $1 each, the market quantity demanded is 45 donuts.
30+10+5=45\text{ Donuts}
If Christina is only willing to buy 2 donuts at $5 per donut, then Michael will buy 6 and Maria only 1. Therefore, the market demand at $5 per donut is 9 donuts.
6+2+1=9\text{ Donuts}
Demand for Donuts
Christina's Demand
Michael's Demand
Maria's Demand
The demand schedule for donuts shows the quantity of donuts demanded at different prices for each consumer.
Market demand can be represented by a demand curve, which is a graph that shows the relationship between the quantity of a good or service consumers are willing and able to buy at various prices. (The word curve here means a line on a graph showing the relationship between two variables.) The demand curve illustrates the relationship between quantity demanded for a good and the price set for that good in the market, much like a demand schedule, but over a continuous span instead of a set of particular price points. A demand curve shows the quantity of a product consumers are willing to purchase (the market demand) at each possible price. The vertical axis indicates price, and the horizontal axis indicates the quantity demanded. There is an inverse relationship between price and quantity demanded, thus the demand curve has a downward or inverse slope.
To find a point on the curve, draw a horizontal line from a price point on the vertical axis to the curve. Draw a vertical line from that point on the curve down to the horizontal axis to find the quantity demanded. This method can be used to estimate the quantity that will be demanded at prices anywhere along the curve, not just at the prices listed in a demand schedule. For example, a demand curve could be used to estimate how a certain price increase will affect the market quantity demanded. Constructing estimated demand curves is important for firms (business entities that produce a good or service) when they are setting prices for their products.
The demand curve (D) shows the quantity of a product consumers demand at each possible price. The vertical axis indicates price, and the horizontal axis indicates the quantity demanded. The curve shows the inverse relationship between price and quantity demanded: when price is higher, the quantity demanded is lower.
Quantity demanded is the quantity of a good that consumers are willing and able to buy at a specific price.
Demand and quantity demanded are not the same. Demand refers to the overall buying behavior of consumers, whereas quantity demanded is the quantity of a good or service that consumers are willing and able to buy at a specific price. This can differ from an individual consumer to the market as a whole. If a loaf of wheat bread is $4, some consumers will buy two. If the price of the same loaf goes down to $2, the same consumers may buy three loaves. The quantity demanded is different at each price point.
A change in quantity demanded is shown as a movement along the demand curve; for example, when the price decreases for a loaf of bread there is a movement from one point to another along the demand curve for bread. All other factors that influence consumers' purchasing decisions are held constant. This is called the ceteris paribus assumption. Ceteris paribus is a Latin phrase meaning "all other things held equal; all other factors or variables held constant."
A change in demand, in contrast, is shown by a shift of the whole curve. In this case, if the market changes so do purchasing plans; for example, if consumer's income, tastes, or expectations change these can cause shifts of the entire demand curve. During times of economic expansion, as in the United States in the 1990s, the quantity demanded increased. If demand increases, the curve shifts to the right, representing a new combination of prices and quantities demanded. If demand decreases, the curve shifts to the left. Changes in demand occur when some factor other than price, such as a fragile political climate, influences consumers' purchasing decisions.
The law of demand, an economic law stating that as the price of a good decreases the quantity demanded by consumers will increase, all other things being equal, explains why the demand curve slopes downward. When the price of a good is increased, consumers will buy less; when the price is lowered, consumers will buy more. The relationship is inversely related. If avocados rise in price from $2 to $3 each, the quantity demanded will decrease because consumers may not being willing or able to buy as much when prices rise.
Consumers prioritize goods differently. For example, someone who is an enthusiastic collector of new artwork or antique cars will be prepared to buy several pieces of art or antique cars at higher prices, while another consumer might only require a single piece of artwork or one antique car for occasional use. When prices fall, consumers who attach less importance to a good will be tempted to buy it. For many goods, price increases are a form of rationing; thus, only consumers who have a strong desire for the product purchase it.
<Factors That Shift Supply>Factors That Shift Demand
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To Susan Darwin [14 September 1831]
I arrived here yesterday evening: after a very prosperous sail of three days from London.— I suppose breathing the same air as a sea Captain is a sort of a preventive: for I scarcely ever spent three pleasanter days.— of course there were a few moments of giddiness, as for sickness I utterly scorn the very name of it.— There were 5 or 6 very agreeable people on board, & we formed a table & stuck together, & most jolly dinners they were.— Cap. Fitz. took a little Midshipman (who by the way knows Sir F. Darwin, his name is Musters)1 & you cannot imagine anything more kind & good humoured than the Captains manners were to him.— Perhaps you thought I admired my beau ideal of a Captain in my former letters: all that is quite a joke to what I now feel.— Every body praises him, (whether or no they know my connection with him) & indeed, judging from the little I have seen of him, he well deserves it.— Not that I suppose it is likely that such violent admiration—as I feel for him—can possibly last.— No man is a hero to his valet, as the old saying goes.—& I certainly shall be in much the same predicament as one.—
The vessel is a very small one; three masted; & carrying 10 guns: but every body says it is the best sort for our work, & of its class it is an excellent vessel: new, but well tried, &
\frac{1}{2}
again the usual strength.— The want of room is very bad, but we must make the best of it.—2 I like the officers, (as Cap. F. says they would not do for St. James, but they are evidently very intelligent, active determined set of young fellows.— I keep on ballancing accounts; there are several contra’s, which I did not expect, but on the other hand the pro’s far outweigh them.—
The time of sailing keeps on receding in a greater ratio, than the present time draws on: I do not believe we shall sail till the 20th of October.— I am exceedingly glad of this, as the number of things I have got to do is quite frightful.— I do not think I can stay in Shrewsbury more than 4 days.— I leave Plymouth on Friday and shall be in Cam: at the end of next week.—
I found the money at the Bank, & am much obliged to my Father for it.— My spirits about the voyage are like the tide, which runs one way & that is in favor of it, but it does so by a number of little waves, which may represent all the doubts & hopes that are continually changing in my mind. After such a wonderful high wrought simile I will write no more. So good bye, my dear Susan | Yours C. Darwin
Charles Musters, listed as ‘Volunteer 1st Class’ by FitzRoy (Narrative 2: 20).
For a detailed description of the Beagle see Darling 1978.
Darling, L. 1978. HMS Beagle: further research, or twenty years a-Beagling. Mariner’s Mirror 64: 315–25.
Pleasant three-day voyage to Plymouth has increased CD’s admiration for FitzRoy. Describes the Beagle as an excellent vessel, but the want of room is very bad. He likes the officers.
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Ideal point - Wikipedia
Point at infinity in hyperbolic geometry
This article is about ideal points in hyperbolic geometry. For similar points in other geometries, see Point at infinity.
Three ideal triangles in the Poincaré disk model; the vertices are ideal points
In hyperbolic geometry, an ideal point, omega point[1] or point at infinity is a well-defined point outside the hyperbolic plane or space. Given a line l and a point P not on l, right- and left-limiting parallels to l through P converge to l at ideal points.
Unlike the projective case, ideal points form a boundary, not a submanifold. So, these lines do not intersect at an ideal point and such points, although well-defined, do not belong to the hyperbolic space itself.
The ideal points together form the Cayley absolute or boundary of a hyperbolic geometry. For instance, the unit circle forms the Cayley absolute of the Poincaré disk model and the Klein disk model. While the real line forms the Cayley absolute of the Poincaré half-plane model .[2]
Pasch's axiom and the exterior angle theorem still hold for an omega triangle, defined by two points in hyperbolic space and an omega point.[3]
2 Polygons with ideal vertices
2.1 Ideal triangles
2.2 Ideal quadrilaterals
2.3 Ideal square
2.4 Ideal n-gons
3 Representations in models of hyperbolic geometry
3.1 Klein disk model
3.2 Poincaré disk model
3.3 Poincaré half-plane model
3.4 Hyperboloid model
The hyperbolic distance between an ideal point and any other point or ideal point is infinite.
The centres of horocycles and horoballs are ideal points; two horocycles are concentric when they have the same centre.
Polygons with ideal vertices[edit]
Ideal triangles[edit]
Main article: Ideal triangle
if all vertices of a triangle are ideal points the triangle is an ideal triangle.
Some properties of ideal triangles include:
All ideal triangles are congruent.
Any ideal triangle has an infinite perimeter.
Any ideal triangle has area
{\displaystyle \pi /-K}
where K is the (negative) curvature of the plane.[4]
Ideal quadrilaterals[edit]
if all vertices of a quadrilateral are ideal points, the quadrilateral is an ideal quadrilateral.
While all ideal triangles are congruent, not all quadrilaterals are; the diagonals can make different angles with each other resulting in noncongruent quadrilaterals. Having said this:[clarification needed]
The interior angles of an ideal quadrilateral are all zero.
Any ideal quadrilateral has an infinite perimeter.
Any ideal (convex non intersecting) quadrilateral has area
{\displaystyle 2\pi /-K}
where K is the (negative) curvature of the plane.
Ideal square[edit]
The ideal quadrilateral where the two diagonals are perpendicular to each other form an ideal square.
It was used by Ferdinand Karl Schweikart in his memorandum on what he called "astral geometry", one of the first publications acknowledging the possibility of hyperbolic geometry.[5]
Ideal n-gons[edit]
An ideal n-gon can be subdivided into (n − 2) ideal triangles, with area (n − 2) times the area of an ideal triangle.
Representations in models of hyperbolic geometry[edit]
In the Klein disk model and the Poincaré disk model of the hyperbolic plane the ideal points are on the unit circle (hyperbolic plane) or unit sphere (higher dimensions) which is the unreachable boundary of the hyperbolic plane.
When projecting the same hyperbolic line to the Klein disk model and the Poincaré disk model both lines go through the same two ideal points (the ideal points in both models are on the same spot).
Klein disk model[edit]
Given two distinct points p and q in the open unit disk the unique straight line connecting them intersects the unit circle in two ideal points, a and b, labeled so that the points are, in order, a, p, q, b so that |aq| > |ap| and |pb| > |qb|. Then the hyperbolic distance between p and q is expressed as
{\displaystyle d(p,q)={\frac {1}{2}}\log {\frac {\left|qa\right|\left|bp\right|}{\left|pa\right|\left|bq\right|}},}
Poincaré disk model[edit]
Given two distinct points p and q in the open unit disk then the unique circle arc orthogonal to the boundary connecting them intersects the unit circle in two ideal points, a and b, labeled so that the points are, in order, a, p, q, b so that |aq| > |ap| and |pb| > |qb|. Then the hyperbolic distance between p and q is expressed as
{\displaystyle d(p,q)=\log {\frac {\left|qa\right|\left|bp\right|}{\left|pa\right|\left|bq\right|}},}
Where the distances are measured along the (straight line) segments aq, ap, pb and qb.
Poincaré half-plane model[edit]
In the Poincaré half-plane model the ideal points are the points on the boundary axis. There is also another ideal point that is not represented in the half-plane model (but rays parallel to the positive y-axis approach it).
Hyperboloid model[edit]
In the hyperboloid model there are no ideal points.
Points at infinity for uses in other geometries.
^ Sibley, Thomas Q. (1998). The geometric viewpoint : a survey of geometries. Reading, Mass.: Addison-Wesley. p. 109. ISBN 0-201-87450-4.
^ Struve, Horst; Struve, Rolf (2010), "Non-euclidean geometries: the Cayley-Klein approach", Journal of Geometry, 89 (1): 151–170, doi:10.1007/s00022-010-0053-z, ISSN 0047-2468, MR 2739193
^ Hvidsten, Michael (2005). Geometry with Geometry Explorer. New York, NY: McGraw-Hill. pp. 276–283. ISBN 0-07-312990-9.
^ Thurston, Dylan (Fall 2012). "274 Curves on Surfaces, Lecture 5" (PDF). Retrieved 23 July 2013.
^ Bonola, Roberto (1955). Non-Euclidean geometry : a critical and historical study of its developments (Unabridged and unaltered republ. of the 1. English translation 1912. ed.). New York, NY: Dover. pp. 75–77. ISBN 0486600270.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Ideal_point&oldid=1071330623"
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Depth of focus — Wikipedia Republished // WIKI 2
For the seismology term, see Depth of focus (tectonics).
1 Depth of focus versus depth of field
2 Determining factors
The phrase depth of focus is sometimes erroneously used to refer to the depth of field (DOF), which is the area in front of the lens in acceptable focus, whereas the true meaning of depth of focus refers to the zone behind the lens wherein the film plane or sensor is placed to produce an in-focus image.
The same factors that determine depth of field also determine depth of focus, but these factors can have different effects than they have in depth of field. Both depth of field and depth of focus increase with smaller apertures. For distant subjects (beyond macro range), depth of focus is relatively insensitive to focal length and subject distance, for a fixed f-number. In the macro region, depth of focus increases with longer focal length or closer subject distance, while depth of field decreases.
In small-format cameras, the smaller circle of confusion limit yields a proportionately smaller depth of focus. In motion-picture cameras, different lens mount and camera gate combinations have exact flange focal distance measurements to which lenses are calibrated.
The choice to place gels or other filters behind the lens becomes a much more critical decision when dealing with smaller formats. Placement of items behind the lens will alter the optics pathway, shifting the focal plane. Therefore, often this insertion must be done in concert with stopping down the lens in order to compensate enough to make any shift negligible given a greater depth of focus. It is often advised in 35 mm motion-picture filmmaking not to use filters behind the lens if the lens is wider than 25 mm.
{\displaystyle t=2Nc{\frac {v}{f}},}
where t is the total depth of focus, N is the lens f-number, c is the circle of confusion, v is the image distance, and f is the lens focal length. In most cases, the image distance (not to be confused with subject distance) is not easily determined; the depth of focus can also be given in terms of magnification m:
{\displaystyle t=2Nc(1+m).}
{\displaystyle t\approx 2Nc.}
In astronomy, the depth of focus
{\displaystyle \Delta f}
is the amount of defocus that introduces a
{\displaystyle \pm \lambda /4}
wavefront error. It can be calculated as[4][5]
{\displaystyle \Delta f=\pm 2\lambda N^{2}}
Lipson, Stephen G., Ariel Lipson, and Henry Lipson. 2010. Optical Physics. 4th ed. Cambridge: Cambridge University Press. ISBN 978-0-521-49345-1 (scheduled release October 2010)
McLean, Ian S. (2008). Electronic Imaging in Astronomy: Detectors and Instrumentation (2nd ed.). Chichester, UK: Praxis Publishing Ltd. ISBN 3-540-76582-4.
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論文雑誌「Carbon」のカバーアートを制作しました［豊田中央研究所］ | 株式会社アートアクション
弊社で制作しました豊田中央研究所 熊野尚美様のカバーアートが、
オランダの出版社Elsevierが発行する学術雑誌 Carbonの
2020年10月号 Front Coverに選ばれました。
スラリー特任研究室
熊野 尚美 様
December 2020, Volume 170 pp. 1–688 Link
Because the agglomerate structures of carbon-black-supported platinum nanoparticles (Pt/C) in catalyst inks affect their productivity and performance, identifying the structural parameters that control the dispersion state and stability of Pt/C agglomerates is important. This study aimed to identify agglomerate structures of Pt/C in high-solidity catalyst inks by examining the rheological behavior, fractal dimension, and electrical conductivity of the inks. In inks containing 48–75% water, the amount of adsorbed ionomers decreased with decreasing water content, resulting in increases in the shear viscosity (η), storage modulus (G′), and electrical conductivity (
\Sigma
). The fractal dimension of the agglomerates (df) was 1.7–2.1, suggesting that Pt/C agglomerates form a rigid three-dimensional network via a diffusion-limited cluster aggregation process. Conversely, in inks containing 89% water, the amount of adsorbed ionomers increased, leading to the lowest values for η, G′, and
\Sigma
. The df value was 2.6, suggesting that Pt/C agglomerates formed a weak-link network via a reaction-limited cluster aggregation process, which was inhibited by the steric and electrostatic repulsion effects of the adsorbed ionomers. The results of this study demonstrate that agglomeration can be controlled through ionomer adsorption by Pt/C, leading to changes in the rheological behavior and stability of catalyst inks.
Influence of ionomer adsorption on agglomerate structures in high-solid catalyst inks
Naomi Kumano*, Kenji Kudo, Yusuke Akimoto, Masahiko Ishii, Hiroshi Nakamura
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Specify Custom Layer Backward Function - MATLAB & Simulink - MathWorks India
Create Custom Layer
Create Backward Function
If Deep Learning Toolbox™ does not provide the layer you require for your classification or regression problem, then you can define your own custom layer. For a list of built-in layers, see List of Deep Learning Layers.
The example Define Custom Deep Learning Layer with Learnable Parameters shows how to create a custom PreLU layer and goes through the following steps:
If the forward function only uses functions that support dlarray objects, then creating a backward function is optional. In this case, the software determines the derivatives automatically using automatic differentiation. For a list of functions that support dlarray objects, see List of Functions with dlarray Support. If you want to use functions that do not support dlarray objects, or want to use a specific algorithm for the backward function, then you can define a custom backward function using this example as a guide.
The example Define Custom Deep Learning Layer with Learnable Parameters shows how to create a PReLU layer. A PReLU layer performs a threshold operation, where for each channel, any input value less than zero is multiplied by a scalar learned at training time.[1] For values less than zero, a PReLU layer applies scaling coefficients
{\alpha }_{i}
f\left({x}_{i}\right)=\left\{\begin{array}{cc}{x}_{i}& \text{if }{x}_{i}>0\\ {\alpha }_{i}{x}_{i}& \text{if }{x}_{i}\le 0\end{array}
{x}_{i}
{\alpha }_{i}
{\alpha }_{i}
View the layer created in the example Define Custom Deep Learning Layer with Learnable Parameters. This layer does not have a backward function.
If the layer has a custom backward function, then you can still inherit from nnet.layer.Formattable.
Implement the backward function that returns the derivatives of the loss with respect to the input data and the learnable parameters.
dlnetwork objects do not support custom layers that require a memory value in a custom backward function. To use a custom layer with a custom backward function in a dlnetwork object, the memory input of the backward function definition must be ~.
Because a PReLU layer has only one input, one output, one learnable parameter, and does not require the outputs of the layer forward function or a memory value, the syntax for backward for a PReLU layer is [dLdX,dLdAlpha] = backward(layer,X,~,dLdZ,~). The dimensions of X are the same as in the forward function. The dimensions of dLdZ are the same as the dimensions of the output Z of the forward function. The dimensions and data type of dLdX are the same as the dimensions and data type of X. The dimension and data type of dLdAlpha is the same as the dimension and data type of the learnable parameter Alpha.
During the backward pass, the layer automatically updates the learnable parameters using the corresponding derivatives.
To include a custom layer in a network, the layer forward functions must accept the outputs of the previous layer and forward propagate arrays with the size expected by the next layer. Similarly, when backward is specified, the backward function must accept inputs with the same size as the corresponding output of the forward function and backward propagate derivatives with the same size.
The derivative of the loss with respect to the input data is
\frac{\partial L}{\partial {x}_{i}}=\frac{\partial L}{\partial f\left({x}_{i}\right)}\frac{\partial f\left({x}_{i}\right)}{\partial {x}_{i}}
\partial L/\partial f\left({x}_{i}\right)
is the gradient propagated from the next layer, and the derivative of the activation is
\frac{\partial f\left({x}_{i}\right)}{\partial {x}_{i}}=\left\{\begin{array}{cc}1& \text{if }{x}_{i}\ge 0\\ {\alpha }_{i}& {\text{if x}}_{i}<0\end{array}.
The derivative of the loss with respect to the learnable parameters is
\frac{\partial L}{\partial {\alpha }_{i}}=\sum _{j}^{}\frac{\partial L}{\partial f\left({x}_{ij}\right)}\frac{\partial f\left({x}_{ij}\right)}{\partial {\alpha }_{i}}
where i indexes the channels, j indexes the elements over height, width, and observations, and the gradient of the activation is
\frac{\partial f\left({x}_{i}\right)}{\partial {\alpha }_{i}}=\left\{\begin{array}{cc}0& \text{if }{x}_{i}\ge 0\\ {x}_{i}& \text{if }{x}_{i}<0\end{array}.
Create the backward function that returns these derivatives.
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A quasilinear parabolic singular perturbation problem | EMS Press
A quasilinear parabolic singular perturbation problem
We study a singular perturbation problem for a quasilinear uniformly parabolic operator of interest in combustion theory. We obtain uniform estimates, we pass to the limit and we show that, under suitable assumptions, the limit function
u
is a solution to the free boundary problem
{\rm div } F(\nabla u)-\partial_{t}u=0
\{ u>0 \}
u_\nu=\alpha(\nu, M)
\partial\{ u>0 \}
, in a pointwise sense and in a viscosity sense. Here
\nu
is the inward unit spatial normal to the free boundary
\partial\{ u>0 \}
M
is a positive constant. Some of the results obtained are new even when the operator under consideration is linear.
Claudia Lederman, Dietmar Oelz, A quasilinear parabolic singular perturbation problem. Interfaces Free Bound. 10 (2008), no. 4, pp. 447–482
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FISS Control Program Manual (v 1.1)
Writer: Juhyung Kang
Original Markdwon File download: obs_guide.md
PDF File Download: obs_guide.pdf
Info: This manual is for the upgraded FISS control program developed after 2020.
The currently available FISS control software is written in Python, unlike the previous one written in CVI.
The program is compatible with the Python verison 3.6 or higher, and some of the progroam tools previously written in CVI are loaded in to the dynamic link library (dll). Camera control tool is in CCD.dll, grating and camera focus control tool is in gratfocus.dll and scanner control with observation mode control is in scanobs.dll. Main interface such as GUI program is developed with the PyQt5 pacakge in Python, and 4 to 5 threads will be run until observation. The initial version of the program was successfully debugged at July 20, 2020. This inital version was written by the one of the solar group member in Seoul National University, Juhyung Kang.
The FISS control program is mannaged with the GitHub and, that program is located in the "C:\Control_Program\fiss_control\new" directory in the BBSO local descktop. Please contact with the software manager (Heesu Yang or Juhyung Kang) when you modify the control program.
Danger: As of July 2020, the main GUI is stuck for 30 seconds to 1 minute sometimes, but that situation may not affect the observation. Please do not force quit the control program.
2. Initialization and Observation
Remove the cover (Grating/ Scanner)
Turn on the FISS control box and computer in server room (connet to the desktop with the chrome remote desktop, Teamviewer or VNC)
Execute the FISS program link file located in the desktop ( icon)
Check the temperature parameter of the CCD in Setting Parameter tab and then click the "Apply" button in Instrument Setting tab.
Click the On button in Power tab to turn on the FISS (The FISS is successfully turned on if the CCD, Scanner and Grating and Focus connection lights are turned on after 5 to 10 seconds.) Since the cammera cooling process takes about 30 minutes until cooling at -30 degree celsius, please click the power on button before start to align the optics.
Danger: The program is set not to run observation before the CCD cooling process is finished.
Check the FISS slit cover is opened in Coode room.
Danger: Since 2020, slit cover motor malfunction occurs. For that reason, you have to open the slit cover maually.
Check the alignment in FISS instrument. Check the incident light enters vertically as you screen the light with the paper gradually (The light should be enters to the top inside the FISS instrument) and also the incident light goes to grating (Please contact with the BBSO in advance when you use FISS because the most of case use VIS instrument before).
Change the filters with wavelength bands to be observed
\alpha
Replace the filter in front of the CCD. Filter is in the FISS toolbox. The glass surface is outside the CCD, and the mirror surface face to the insde.
After the CCD temperature is stabilized, click the open button to open the observatioal parameter file (.par) in the Observation setting tab. Following figure shows the FISS program after open the parameter file. You can lean how to write the parameter file in 3. Setting Observation Parameter section.
Information: If you turn on the power, the sample paramter files is created at C:\Data\YYYY\MM\DD\par directory.
Focusing. Screen the some portion of slit and check wheter the boundary is clear in display. The second way is to check wheter the brightness in slit (y) direction varies ahrply in spectrogram. (It is quite useful to run the video focus mode to check the spectrogram and line profile simultaneously. Set the AcqMode equal to 0 (video mode) and then click the Apply button in Obseravtion Setting tab then click the run button in Observation Run tab.)
History: you may be focusing with the flat mirror infront of the scan mirror, focus position is 23 at 2017.
When the Video mode, the video image viewer windows is popped up as like the following figure.
Take Flat image. We recommand to take a flat image at least 2 (before and after the observation) and also take the flat if you changed the filter set. When you take the flat image, please require the operator then operator move the telescope to the Disk center without AO. Set the AcqMode equal to 10 (Take Flat). If modify this parameter click the Apply button and then click the Run botton. The flat image will be saved in C:\Data\YYYY\MM\DD\cal directory.
Information: The flat image is not shown at realtime because of the system structure, but you can check the running the taking flat from the the Take(nScan) number shown in status tab.
History: Old program average the 100 number of flat images, but it may not be good because the last frame of the scan is not be loaded somtimes. For that reaseon, the new program take 101 image and dump the last frame to avoid that problem.
Set the observational parameters. You can use the paramter files directly also modify the parameters in the Setting Parameter table (Please see the 3. Setting Observation Parameter section if you are not familiar with this). You can check the general observational parameters is in sample_files.zip. We highly recommand that double check the PGain, Gain and exptime parameters before run observation. The common settinf parameters for each filter sets are shown in following table.
\alpha
Danger: One of the most important parameters in the parameter file is a "Target". The "Target" parameter is equal to the directory name of observed data. For that reason, we highly recommand to change that parameter when you change the observational target.
After modifing the parameters, click the Apply then the temporary applied parameter will be saved at "applied.par" file in C:\Data\YYYY\MM\DD\par directory. After clicked the Apply button, the Run button will be enabled. Then click the run button to run observation. If you stop the observation click the Stop button.
Before starting the observation loop, you have to check that the target of observation is located at the center of FOV. To check this, we recommand that Scan the large FOV at first and move the scanner. At this sequence you set the nScan equal to 1 to take only one raster image. If scan is done, insert the value to move scanner in Move Scanner tab then click the Go button. After reach the scanner at that position click the Home button to set this point to the zero (home) position. Repeat this sequence after reduce the FOV to the observed size until the target will be located in the correct position.
History: The moderate scanner position is about -3500 at 2017.
Set the observational parmeter to fit the your observation loop. It is the good way to set the nScan value to the high value and then stop the observation to run observation continuously. Again, the one of the most important parameters is Target since that parameter is the directory name of the observed data. Please set this value to fit the each observation target. The observed data will be saved in C:\Data\YYYY\MM\DD\raw\ {Target} directory. The data filename is scan start time for example FISS_YYYYMMDD_HHMMSS.SSS_A.fts. The datails of the parameters and the cadence for each number of nSteps is in 3. Setting Observation Parameter
When you run observation the image viewer will be popped up like follows:
Information: This Image viewer will be closed after 15 seconds as click the stop button or done observation. You can run next observation after close this viewer window.
Take a Flat imgae after final observation end.
Turn off the power by click the Power Off button. When you click Power Off button, the program will be closed automatically after heating the CCD at 5 degree celsius. Since It takes a lot of time, we recommand to cover the grating and scanner during the heating time.
Danger: Do not force quit the program until heating process running.
Download the observed data on your portable hard driver. If control program is turned off, than power off the control box except for the desktop and go to lodge (you download the data remotely).
3. Setting Observation Parameter
Unlike the previous control program, the new program use the parameter file (.par). You can download the (sample_files.zip). Check these files to learn how to write the parameter file. You can also download the program which generate the parameter files (mk_obspar.exe). You can also modify the parameter in FISS control program, but you can not define the number of observation loops (nRun). The number of observation loops is only set or modified in parameter file by set the nRun value or in the parameter generating program.
The following figures show an example parameter file and the parameter generator.
Each observation loops is distinguished by "begin" and "endbegin". For the first loop, you have to define all parameters, but for the following loop, you just define the parameters to be modified. You can check what I mean in the example_nloop.par file from the (sample_files.zip).
Observational parameters in .par file are follows:
nRun: The number of observation loop.
CameraMode: Set the camera to be used, Default is 2 (both).
wvIndex: Set the wavelength band to observe. 1 means Set1 (A: H
\alpha
/ B: Ca II 8542 Å) and 2 means Set2 (A: Na I D2 5890 Å / B: Fe I 5434 Å). If you want to set the grating angle manually, set this value as 0 and should set the gWv, gOrder, gStep and gAngle.
nSteps: is the number of step to take frame (nFrame). In case of nStep equal to 1, the scanner is stopped at origin.
nScan: The number of repeat to scan.
CCD1_exptime: exposure time of the CCD camera in second.
CCD1_PGain: Set PGain Value. Default is 2. Refer to table in Step 12. (0 - 2, integer)
CCD1_Gain: Set Gain Value. (0 - 255, integer)
Target: Target name. We recommand to change this when you change target.
경고: This parameter is the directory name of the observed data. For that reason you have not to use space and special characters.
Observer: Name of observer.
TelXpos: Position of the pointing of the telescope.
stepSize: Step size to move scanner in um.
HBin: Horizontal (spectral direction) Binning.
VBin: Vertical (slit direction) Binning.
AcqMode: Acqusition Mode. In general observation, this value is 1 (Single Step). 0: Video Focus / 1: Single Step / 5: Scan Stop / 6: Frame Transfer / 8: Scan Continuous / 10: Take Flat. Among these value, you can use 0, 1 and 10, 5 and 8 will be deleted, and 6 will be added.
ScanMode: Mode of scanning. Currently 0 is only available value.
Trigger: Camera trigger metnod. Default is 0 (internal) and external trigger will be added soon
ReadMode: Camera redout method . 1: full vertical binning, 4: imaging. In general observation, use 4.
ShutterMode: Camera shutter open/close. In general observation, use 1: open. 0: auto, 1: open, 2: close
The cadences of the FISS for each number of steps are follows (that comes from the test observation in 2020):
100 steps (frames, 16 arcsec): 13.4 sec
Set to the pre-setting grating angle (gAngle), wavelength (gWv) and diffraction order (gOrder) values corresponding to grating motor step (gStep). If you set this parameter to 0, you have to set the gWv, gOrder, gStep, gAngle parameters manually. The gStep is the step count of the motor. ↩︎
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What will be the mass of 6.03 x 1023 molecules of carbon monoxide ?
What will be the mass of 6.03 x
{}^{{10}^{23}}
molecules of carbon monoxide ?
Upper urinary tract trauma MCQ Indian History MCQ Statement & Assumption MCQ General Biology MCQ Fluid Mechanics MCQ Fill in the blanks MCQ Senile enlargement of the prostate MCQ World History MCQ Design of Steel Structures MCQ Digital Electronics MCQ Electrostatics MCQ Mechanical Engineering MCQ
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Bode plot of frequency response, or magnitude and phase data - MATLAB bode - MathWorks India
H\left(s\right)=\frac{{s}^{2}+0.1s+7.5}{{s}^{4}+0.12{s}^{3}+9{s}^{2}}.
Identified LTI models, such as idtf, idss, or idproc models. For such models, the function can also plot confidence intervals and return standard deviations of the frequency response. See Bode Plot of Identified Model.
When you need additional plot customization options, use bodeplot (Control System Toolbox) instead.
z={e}^{j\omega {T}_{s}},\text{ }0\le \omega \le {\omega }_{N}=\frac{\pi }{{T}_{s}},
H\left({e}^{j\omega {T}_{s}}\right)
bodeplot | freqresp | nyquist | spectrum | step
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Cauchy - Maple Help
Home : Support : Online Help : Statistics and Data Analysis : Statistics Package : Distributions : Cauchy
Cauchy(a, b)
CauchyDistribution(a, b)
The Cauchy distribution is a continuous probability distribution with probability density function given by:
f\left(t\right)=\frac{1}{\mathrm{\pi }b\left(1+\frac{{\left(t-a\right)}^{2}}{{b}^{2}}\right)}
a::\mathrm{real},0<b
The Cauchy distribution does not have any defined moments or cumulants.
The Cauchy variate Cauchy(a,b) is related to the standardized variate Cauchy(0,1) by Cauchy(a,b) ~ a + b * Cauchy(0,1).
The ratio of two independent unit Normal variates
N
M
is distributed according to the standard Cauchy variate: Cauchy(0,1) ~ N / M
The standard Cauchy variate Cauchy(0,1) is a special case of the StudentT variate with one degree of freedom: Cauchy(0,1) ~ StudentT(1).
Note that the Cauchy command is inert and should be used in combination with the RandomVariable command.
\mathrm{with}\left(\mathrm{Statistics}\right):
X≔\mathrm{RandomVariable}\left(\mathrm{Cauchy}\left(a,b\right)\right):
\mathrm{PDF}\left(X,u\right)
\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{\mathrm{\pi }}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{+}\frac{{\left(\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{a}\right)}^{\textcolor[rgb]{0,0,1}{2}}}{{\textcolor[rgb]{0,0,1}{b}}^{\textcolor[rgb]{0,0,1}{2}}}\right)}
\mathrm{PDF}\left(X,0.5\right)
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Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics.6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.
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Physics/Essays/Fedosin/Selfconsistent gravitational constants - Wikiversity
Physics/Essays/Fedosin/Selfconsistent gravitational constants
< Physics | Essays | Fedosin
Self-consistent gravitational constants are complete sets of fundamental constants, which are self-consistent and define various physical quantities associated with gravitation. These constants are calculated in the same way as electromagnetic constants in electrodynamics. This is possible because in the weak field equations of general relativity are simplified into equations of gravitoelectromagnetism, similar in form to Maxwell's Equations. Similarly, in the weak field approximation equations of covariant theory of gravitation [1] turn into equations of Lorentz-invariant theory of gravitation (LITG). LITG equations are Maxwell-like gravitational equations, which are similar to equations of gravitoelectromagnetism. If these equations are written with the help of self-consistent gravitational constants, there is the best similarity of equations of gravitational and electromagnetic fields. Since in 19-th century there was no International System of Units, the first mention of gravitational constants was possibly due to Forward (1961).[2]
2 Connection with Planck mass and Stoney mass
3 Connection with fine structure constant
4 Strong gravitational torsion flux quantum
Primary set of gravitational constants is:
1. First gravitational constant:
{\displaystyle ~c_{g}}
, which is the speed of gravitational waves in vacuum; [3]
2. Second gravitational constant:
{\displaystyle ~\rho _{g}}
, which is the gravitational characteristic impedance of free space.
Secondary set of gravitational constants is:
1. Gravitoelectric gravitational constant (like electric constant):
{\displaystyle ~\varepsilon _{g}={\frac {1}{4\pi G}}=1.192708\cdot 10^{9}\mathrm {kg\cdot s^{2}\cdot m^{-3}} ,}
{\displaystyle ~G}
2. Gravitomagnetic gravitational constant (like vacuum permeability):
{\displaystyle ~\mu _{g}={\frac {4\pi G}{c_{g}^{2}}}.}
If the speed of gravitation is equal to the speed of light,
{\displaystyle ~c_{g}=c,}
then [4]
{\displaystyle ~\mu _{g0}=9.328772\cdot 10^{-27}\mathrm {m/kg} .}
Both, primary and secondary sets of gravitational constants are selfconsistent, because they are connected by the following relationships:
{\displaystyle ~{\frac {1}{\sqrt {\mu _{g}\varepsilon _{g}}}}=c_{g},}
{\displaystyle ~{\sqrt {\frac {\mu _{g}}{\varepsilon _{g}}}}=\rho _{g}={\frac {4\pi G}{c_{g}}}.}
{\displaystyle ~c_{g}=c,}
then gravitational characteristic impedance of free space be equal to: [5] [6]
{\displaystyle ~\rho _{g0}={\frac {4\pi G}{c}}=2.796696\cdot 10^{-18}\mathrm {m^{2}/(s\cdot kg)} .}
In Lorentz-invariant theory of gravitation the constant
{\displaystyle ~\rho _{g}}
is contained in formula for vector energy flux density of gravitational field (Heaviside vector): [3]
{\displaystyle ~\mathbf {H} =-{\frac {c_{g}^{2}}{4\pi G}}\mathbf {\Gamma } \times \mathbf {\Omega } =-{\frac {c_{g}}{\rho _{g}}}\mathbf {\Gamma } \times \mathbf {\Omega } ,}
{\displaystyle ~\mathbf {\Gamma } }
is gravitational field strength or gravitational acceleration,
{\displaystyle ~\mathbf {\Omega } }
is gravitational torsion field or simply torsion field.
For plane transverse uniform gravitational wave, in which for amplitudes of field strengths holds
{\displaystyle ~\Gamma =c_{g}\Omega }
, may be written:
{\displaystyle ~H={\frac {\Gamma ^{2}}{\rho _{g}}}.}
A similar relation in electrodynamics for amplitude of flux density of electromagnetic energy of a plane electromagnetic wave in vacuum, in which
{\displaystyle ~E=cB}
, is as follows: [7]
{\displaystyle ~S={\frac {E^{2}}{Z_{0}}},}
{\displaystyle ~\mathbf {S} ={\frac {\mathbf {E} \times \mathbf {B} }{\mu _{0}}}={\frac {c}{Z_{0}}}\mathbf {E} \times \mathbf {B} }
– Poynting vector,
{\displaystyle ~E}
– electric field strength,
{\displaystyle ~B}
– magnetic flux density,
{\displaystyle ~\mu _{0}}
– vacuum permeability,
{\displaystyle ~Z_{0}=c\mu _{0}}
– impedance of free space.
Gravitational impedance of free space
{\displaystyle ~\rho _{g0}}
was used in paper [8] to evaluate the interaction section of gravitons with the matter.
Connection with Planck mass and Stoney mass[edit | edit source]
Since gravitational constant and speed of light are included in Planck mass
{\displaystyle m_{P}={\sqrt {\frac {\hbar c}{G}}}\ }
{\displaystyle ~\hbar }
– reduced Planck constant or Dirac constant, then gravitational characteristic impedance of free space can be represented as:
{\displaystyle ~\rho _{g0}={\frac {2h}{m_{P}^{2}}}}
{\displaystyle ~h}
– Planck constant.
There is Stoney mass, related to elementary charge
{\displaystyle ~e}
and electric constant
{\displaystyle ~\varepsilon _{0}}
{\displaystyle ~m_{S}=e{\sqrt {\frac {\varepsilon _{g}}{\varepsilon _{0}}}}={\frac {e}{\sqrt {4\pi G\varepsilon _{0}}}}}
Stoney mass can be expressed through the Planck mass:
{\displaystyle ~m_{S}={\sqrt {\alpha }}\cdot m_{P}}
{\displaystyle ~\alpha }
is the electric fine structure constant.
This implies another expression for gravitational characteristic impedance of free space:
{\displaystyle ~\rho _{g0}=\alpha \cdot {\frac {2h}{m_{S}^{2}}}}
Newton law for gravitational force between two Stoney masses can be written as:
{\displaystyle ~F_{g}={\frac {1}{4\pi \varepsilon _{g}}}\cdot {\frac {m_{S}^{2}}{r^{2}}}=\alpha _{g}\cdot {\frac {\hbar c}{r^{2}}}.}
Coulomb's law for electric force between two elementary charges is:
{\displaystyle ~F_{e}={\frac {1}{4\pi \varepsilon _{0}}}\cdot {\frac {e^{2}}{r^{2}}}=\alpha \cdot {\frac {\hbar c}{r^{2}}}.}
{\displaystyle ~F_{g}}
{\displaystyle ~F_{e}}
leads to equation for the Stoney mass
{\displaystyle ~m_{S}=e{\sqrt {\frac {\varepsilon _{g}}{\varepsilon _{0}}}},}
that was stated above. Hence the Stony mass may be determined from the condition that two such masses interact via gravitation with the same force as if these masses had the charges equal to the elementary charge and only interact through electromagnetic forces.
Connection with fine structure constant[edit | edit source]
The electric fine structure constant is:
{\displaystyle ~\alpha ={\frac {e^{2}}{2\varepsilon _{0}hc}}.}
We can determine the same value for gravitation so:
{\displaystyle ~\alpha _{g}={\frac {m_{S}^{2}}{2\varepsilon _{g}hc}}=\alpha ,}
with the equality of the fine structure constants for both fields.
On the other hand, the gravitational fine structure constant for hydrogen system at the atomic level and at the level of star is also equal to fine structure constant:
{\displaystyle ~\alpha ={\frac {G_{s}M_{p}M_{e}}{\hbar c}}={\frac {GM_{ps}M_{\Pi }}{\hbar _{s}C_{s}}}={\frac {1}{137.036}}}
{\displaystyle ~G_{s}}
– strong gravitational constant,
{\displaystyle ~M_{p}}
{\displaystyle ~M_{e}}
– the mass of proton and electron,
{\displaystyle ~M_{ps}}
{\displaystyle ~M_{\Pi }}
– mass of the star-analogue of proton and the planet-analogue of electron, respectively,
{\displaystyle ~\hbar _{s}}
– stellar Dirac constant,
{\displaystyle ~C_{s}}
– characteristic speed of stars matter.
Strong gravitational torsion flux quantum[edit | edit source]
The magnetic force between two fictitious elementary magnetic charges is:
{\displaystyle F_{m}={\frac {1}{4\pi \mu _{0}}}\cdot {\frac {q_{m}^{2}}{r^{2}}}=\beta \cdot {\frac {\hbar c}{r^{2}}},\ }
{\displaystyle q_{m}={\frac {h}{e}}\ }
is the magnetic charge,
{\displaystyle \beta ={\frac {\varepsilon _{0}hc}{2e^{2}}}={\frac {\pi \hbar }{c\mu _{0}e^{2}}}}
is the magnetic coupling constant for fictitious magnetic charges. [9]
The force of gravitational torsion field between two fictitious elementary torsion masses is:
{\displaystyle F_{\Omega }={\frac {1}{4\pi \mu _{g0}}}\cdot {\frac {m_{\Omega }^{2}}{r^{2}}}=\beta _{g}\cdot {\frac {\hbar c}{r^{2}}},\ }
{\displaystyle \beta _{g}={\frac {\varepsilon _{g}hc}{2m_{S}^{2}}}={\frac {\pi \hbar }{c\mu _{g0}m_{S}^{2}}}\ }
is the gravitational torsion coupling constant for the gravitational torsion mass
{\displaystyle m_{\Omega }\ }
In the case of equality of the above forces, we shall get the equality of the coupling constants for magnetic field and gravitational torsion field:
{\displaystyle \beta =\beta _{g}={\frac {1}{4\alpha }},\ }
from which the Stoney mass
{\displaystyle m_{S}\ }
and the gravitational torsion mass could be derived:
{\displaystyle m_{S}=e\cdot {\sqrt {\frac {\mu _{o}}{\mu _{g0}}}}={\frac {e}{\sqrt {4\pi \varepsilon _{0}G}}}.\ }
{\displaystyle m_{\Omega }=q_{m}\cdot {\sqrt {\frac {\mu _{g0}}{\mu _{o}}}}={\frac {h{\sqrt {4\pi \varepsilon _{0}G}}}{e}}={\frac {h}{m_{S}}}.\ }
Instead of fictitious magnetic charge
{\displaystyle q_{m}=h/e}
the single magnetic flux quantum Φ0 = h/(2e) ≈ 2.067833758(46)×10−15
Wb [10] has the real meaning in quantum mechanics. On the other hand at the level of atoms the strong gravitation operates and we must use the strong gravitational constant. So we believe that the strong gravitational torsion flux quantum there should be important:
{\displaystyle \Phi _{\Gamma }={\frac {h}{2e}}{\sqrt {\frac {4\pi \varepsilon _{0}G_{s}M_{e}}{M_{p}}}}={\frac {h}{2M_{p}}}=1.98\cdot 10^{-7}}
m2/s,
which is related to proton with its mass
{\displaystyle M_{p}}
and to its velocity circulation quantum.
Physics/Essays/Fedosin/Velocity circulation quantum
↑ Fedosin S.G. Fizicheskie teorii i beskonechnaia vlozhennost’ materii. – Perm, 2009, 844 pages, Tabl. 21, Pic. 41, Ref. 289. ISBN 978-5-9901951-1-0. (in Russian).
↑ R. L. Forward, Proc. IRE 49, 892 (1961).
↑ 3.0 3.1 Fedosin S.G. (1999), Fizika i filosofiia podobiia ot preonov do metagalaktik, Perm, pages 544, ISBN 5-8131-0012-1
↑ Kiefer, C.; Weber, C. On the interaction of mesoscopic quantum systems with gravity. Annalen der Physik, 2005, Vol. 14, Issue 4, Pages 253 – 278.
↑ J. D. Kraus, IEEE Antennas and Propagation. Magazine 33, 21 (1991).
↑ Raymond Y. Chiao. "New directions for gravitational wave physics via “Millikan oil drops”, arXiv:gr-qc/0610146v16 (2007).PDF
↑ Иродов И.Е. Основные законы электромагнетизма. Учебное пособие для студентов вузов. 2- издание. М.: Высшая школа, 1991.
↑ Fedosin S.G. The graviton field as the source of mass and gravitational force in the modernized Le Sage’s model. Physical Science International Journal, ISSN: 2348-0130, Vol. 8, Issue 4, P. 1-18 (2015). http://dx.doi.org/10.9734/PSIJ/2015/22197.
↑ Yakymakha O.L.(1989). High Temperature Quantum Galvanomagnetic Effects in the Two- Dimensional Inversion Layers of MOSFET's (In Russian). Kiev: Vyscha Shkola. p.91. ISBN 5-11-002309-3. djvu.
↑ "magnetic flux quantum Φ0". 2010 CODATA recommended values. Retrieved 10 January 2012.
Selfconsistent gravitational constants in Russian
Retrieved from "https://en.wikiversity.org/w/index.php?title=Physics/Essays/Fedosin/Selfconsistent_gravitational_constants&oldid=2242572"
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Free_entropy Knowpia
A thermodynamic free entropy is an entropic thermodynamic potential analogous to the free energy. Also known as a Massieu, Planck, or Massieu–Planck potentials (or functions), or (rarely) free information. In statistical mechanics, free entropies frequently appear as the logarithm of a partition function. The Onsager reciprocal relations in particular, are developed in terms of entropic potentials. In mathematics, free entropy means something quite different: it is a generalization of entropy defined in the subject of free probability.
A free entropy is generated by a Legendre transformation of the entropy. The different potentials correspond to different constraints to which the system may be subjected.
Name Function Alt. function Natural variables
{\displaystyle S={\frac {1}{T}}U+{\frac {P}{T}}V-\sum _{i=1}^{s}{\frac {\mu _{i}}{T}}N_{i}\,}
{\displaystyle ~~~~~U,V,\{N_{i}\}\,}
Massieu potential \ Helmholtz free entropy
{\displaystyle \Phi =S-{\frac {1}{T}}U}
{\displaystyle =-{\frac {A}{T}}}
{\displaystyle ~~~~~{\frac {1}{T}},V,\{N_{i}\}\,}
Planck potential \ Gibbs free entropy
{\displaystyle \Xi =\Phi -{\frac {P}{T}}V}
{\displaystyle =-{\frac {G}{T}}}
{\displaystyle ~~~~~{\frac {1}{T}},{\frac {P}{T}},\{N_{i}\}\,}
{\displaystyle S}
{\displaystyle \Phi }
is the Massieu potential[1][2]
{\displaystyle \Xi }
is the Planck potential[1]
{\displaystyle U}
is internal energy
{\displaystyle T}
is temperature
{\displaystyle P}
{\displaystyle V}
{\displaystyle A}
is Helmholtz free energy
{\displaystyle G}
is Gibbs free energy
{\displaystyle N_{i}}
is number of particles (or number of moles) composing the i-th chemical component
{\displaystyle \mu _{i}}
is the chemical potential of the i-th chemical component
{\displaystyle s}
is the total number of components
{\displaystyle i}s the
{\displaystyle i}
th components.
Note that the use of the terms "Massieu" and "Planck" for explicit Massieu-Planck potentials are somewhat obscure and ambiguous. In particular "Planck potential" has alternative meanings. The most standard notation for an entropic potential is
{\displaystyle \psi }
, used by both Planck and Schrödinger. (Note that Gibbs used
{\displaystyle \psi }
to denote the free energy.) Free entropies where invented by French engineer François Massieu in 1869, and actually predate Gibbs's free energy (1875).
Dependence of the potentials on the natural variablesEdit
{\displaystyle S=S(U,V,\{N_{i}\})}
By the definition of a total differential,
{\displaystyle dS={\frac {\partial S}{\partial U}}dU+{\frac {\partial S}{\partial V}}dV+\sum _{i=1}^{s}{\frac {\partial S}{\partial N_{i}}}dN_{i}.}
From the equations of state,
{\displaystyle dS={\frac {1}{T}}dU+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}.}
The differentials in the above equation are all of extensive variables, so they may be integrated to yield
{\displaystyle S={\frac {U}{T}}+{\frac {PV}{T}}+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}N}{T}}\right).}
Massieu potential / Helmholtz free entropyEdit
{\displaystyle \Phi =S-{\frac {U}{T}}}
{\displaystyle \Phi ={\frac {U}{T}}+{\frac {PV}{T}}+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}N}{T}}\right)-{\frac {U}{T}}}
{\displaystyle \Phi ={\frac {PV}{T}}+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}N}{T}}\right)}
Starting over at the definition of
{\displaystyle \Phi }
and taking the total differential, we have via a Legendre transform (and the chain rule)
{\displaystyle d\Phi =dS-{\frac {1}{T}}dU-Ud{\frac {1}{T}},}
{\displaystyle d\Phi ={\frac {1}{T}}dU+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}-{\frac {1}{T}}dU-Ud{\frac {1}{T}},}
{\displaystyle d\Phi =-Ud{\frac {1}{T}}+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}.}
The above differentials are not all of extensive variables, so the equation may not be directly integrated. From
{\displaystyle d\Phi }
{\displaystyle \Phi =\Phi ({\frac {1}{T}},V,\{N_{i}\}).}
If reciprocal variables are not desired,[3]: 222
{\displaystyle d\Phi =dS-{\frac {TdU-UdT}{T^{2}}},}
{\displaystyle d\Phi =dS-{\frac {1}{T}}dU+{\frac {U}{T^{2}}}dT,}
{\displaystyle d\Phi ={\frac {1}{T}}dU+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}-{\frac {1}{T}}dU+{\frac {U}{T^{2}}}dT,}
{\displaystyle d\Phi ={\frac {U}{T^{2}}}dT+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i},}
{\displaystyle \Phi =\Phi (T,V,\{N_{i}\}).}
Planck potential / Gibbs free entropyEdit
{\displaystyle \Xi =\Phi -{\frac {PV}{T}}}
{\displaystyle \Xi ={\frac {PV}{T}}+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}N}{T}}\right)-{\frac {PV}{T}}}
{\displaystyle \Xi =\sum _{i=1}^{s}\left(-{\frac {\mu _{i}N}{T}}\right)}
{\displaystyle \Xi }
{\displaystyle d\Xi =d\Phi -{\frac {P}{T}}dV-Vd{\frac {P}{T}}}
{\displaystyle d\Xi =-Ud{\frac {2}{T}}+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}-{\frac {P}{T}}dV-Vd{\frac {P}{T}}}
{\displaystyle d\Xi =-Ud{\frac {1}{T}}-Vd{\frac {P}{T}}+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}.}
{\displaystyle d\Xi }
{\displaystyle \Xi =\Xi \left({\frac {1}{T}},{\frac {P}{T}},\{N_{i}\}\right).}
{\displaystyle d\Xi =d\Phi -{\frac {T(PdV+VdP)-PVdT}{T^{2}}},}
{\displaystyle d\Xi =d\Phi -{\frac {P}{T}}dV-{\frac {V}{T}}dP+{\frac {PV}{T^{2}}}dT,}
{\displaystyle d\Xi ={\frac {U}{T^{2}}}dT+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}-{\frac {P}{T}}dV-{\frac {V}{T}}dP+{\frac {PV}{T^{2}}}dT,}
{\displaystyle d\Xi ={\frac {U+PV}{T^{2}}}dT-{\frac {V}{T}}dP+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i},}
{\displaystyle \Xi =\Xi (T,P,\{N_{i}\}).}
^ a b Antoni Planes; Eduard Vives (2000-10-24). "Entropic variables and Massieu-Planck functions". Entropic Formulation of Statistical Mechanics. Universitat de Barcelona. Archived from the original on 2008-10-11. Retrieved 2007-09-18.
^ T. Wada; A.M. Scarfone (December 2004). "Connections between Tsallis' formalisms employing the standard linear average energy and ones employing the normalized q-average energy". Physics Letters A. 335 (5–6): 351–362. arXiv:cond-mat/0410527. Bibcode:2005PhLA..335..351W. doi:10.1016/j.physleta.2004.12.054. S2CID 17101164.
^ a b The Collected Papers of Peter J. W. Debye. New York, New York: Interscience Publishers, Inc. 1954.
Massieu, M.F. (1869). "Compt. Rend". 69 (858): 1057. {{cite journal}}: Cite journal requires |journal= (help)
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TestFailures - Maple Help
Home : Support : Online Help : Programming : CodeTools : TestFailures
TestFailures
find which, if any, tests have failed
TestFailures()
The TestFailures command will return a list of the labels for all of the calls to Test that have failed in the current session (that is, since the last restart command).
\mathrm{with}\left(\mathrm{CodeTools}\right):
Consider the following test suite. We would expect the second and third test case to fail.
\mathrm{Test}\left(\mathrm{sin}\left(\mathrm{\pi }\right),0,\mathrm{label}="sine of pi"\right)
sine of pi passed
\mathrm{Test}\left(\mathrm{cos}\left(\mathrm{\pi }\right),\frac{1}{2},\mathrm{label}="cosine of pi"\right)
cosine of pi failed
Expected result : 1/2
Evaluated result: -1
Error, (in CodeTools:-Test) TEST FAILED: cosine of pi
\mathrm{Test}\left(\mathrm{tan}\left(\mathrm{\pi }\right),1\right)
Test 1 failed
Expected result : 1
Evaluated result: 0
Error, (in CodeTools:-Test) TEST FAILED
We now obtain the list of failures. Note the label for the third test case has been generated.
\mathrm{TestFailures}\left(\right)
[\textcolor[rgb]{0,0,1}{"cosine of pi"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"Test 1"}]
The CodeTools[TestFailures] command was introduced in Maple 2021.
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Elliptic Self-Similar Stochastic Processes | EMS Press
Elliptic Self-Similar Stochastic Processes
Albert Benassi
M
be a random measure and
L
be an elliptic pseudo-differential operator on
\mathbb{R}^d
. We study the solution of the stochastic problem
LX=M
X(0)=0
when some homogeneity and integrability conditions are assumed. If
M
is a Gaussian measure the process
X
belongs to the class of Elliptic Gaussian Processes which has already been studied. Here the law of
M
is not necessarily Gaussian. We characterize the solutions
X
which are self-similar and with stationary increments in terms of the driving measure
M
. Then we use appropriate wavelet bases to expand these solutions and we give regularity results. In the last section it is shown how a percolation forest can help with constructing a self-similar Elliptic Process with non stable law.
Albert Benassi, Daniel Roux, Elliptic Self-Similar Stochastic Processes. Rev. Mat. Iberoam. 19 (2003), no. 3, pp. 767–796
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Pythagorean theorem - Simple English Wikipedia, the free encyclopedia
The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).
In mathematics, the Pythagorean theorem or Pythagoras's theorem is a statement about the sides of a right triangle.
One of the angles of a right triangle is always equal to 90 degrees. This angle is the right angle. The two sides next to the right angle are called the legs and the other side is called the hypotenuse. The hypotenuse is the side opposite to the right angle, and it is always the longest side.
1 Claim of the theory
2 Types of proofs
3.1 Proof using similar triangles
Claim of the theory[change | change source]
The Pythagorean theorem says that the area of a square on the hypotenuse is equal to the sum of the areas of the squares on the legs. In this picture, the area of the blue square added to the area of the red square makes the area of the purple square. It was named after the Greek mathematician Pythagoras:
If the lengths of the legs are a and b, and the length of the hypotenuse is c, then,
{\displaystyle a^{2}+b^{2}=c^{2}}
Types of proofs[change | change source]
There are many different proofs of this theorem. They fall into four categories:
Those based on linear relations: the algebraic proofs.
Those based upon comparison of areas: the geometric proofs.
Those based upon the vector operation.
Those based on mass and velocity: the dynamic proofs.[1]
One proof of the Pythagorean theorem was found by a Greek mathematician, Eudoxus of Cnidus.
The proof uses three lemmas:
Triangles with the same base and height have the same area.
A triangle which has the same base and height as a side of a square has the same area as a half of the square.
Triangles with two congruent sides and one congruent angle are congruent and have the same area.
The blue triangle has the same area as the green triangle, because it has the same base and height (lemma 1).
Green and red triangles both have two sides equal to sides of the same squares, and an angle equal to a straight angle (an angle of 90 degrees) plus an angle of a triangle, so they are congruent and have the same area (lemma 3).
Red and yellow triangles' areas are equal because they have the same heights and bases (lemma 1).
Blue triangle's area equals area of yellow triangle's area, because
{\displaystyle {\color {blue}A_{blue}}={\color {green}A_{green}}={\color {red}A_{red}}={\color {yellow}A_{yellow}}}
The brown triangles have the same area for the same reasons.
Blue and brown each have a half of the area of a smaller square. The sum of their areas equals half of the area of the bigger square. Because of this, halves of the areas of small squares are the same as a half of the area of the bigger square, so their area is the same as the area of the bigger square.
Proof using similar triangles[change | change source]
We can get another proof of the Pythagorean theorem by using similar triangles.
{\displaystyle {\frac {d}{a}}={\frac {a}{c}}\quad \Rightarrow \quad d={\frac {a^{2}}{c}}\quad (1)}
e/b = b/c => e = b^2/c (2)
From the image, we know that
{\displaystyle c=d+e\,\!}
. And by replacing equations (1) and (2):
{\displaystyle c={\frac {a^{2}}{c}}+{\frac {b^{2}}{c}}}
Multiplying by c:
{\displaystyle c^{2}=a^{2}+b^{2}\,\!.}
Pythagorean triples[change | change source]
Pythagorean triples or triplets are three whole numbers which fit the equation
{\displaystyle a^{2}+b^{2}=c^{2}}
The triangle with sides of 3, 4, and 5 is a well known example. If a=3 and b=4, then
{\displaystyle 3^{2}+4^{2}=5^{2}}
{\displaystyle 9+16=25}
. This can also be shown as
{\displaystyle {\sqrt {3^{2}+4^{2}}}=5.}
The three-four-five triangle works for all multiples of 3, 4, and 5. In other words, numbers such as 6, 8, 10 or 30, 40 and 50 are also Pythagorean triples. Another example of a triple is the 12-5-13 triangle, because
{\displaystyle {\sqrt {12^{2}+5^{2}}}=13}
A Pythagorean triple that is not a multiple of other triples is called a primitive Pythagorean triple. Any primitive Pythagorean triple can be found using the expression
{\displaystyle (2mn,m^{2}-n^{2},m^{2}+n^{2})}
, but the following conditions must be satisfied. They place restrictions on the values of
{\displaystyle m}
{\displaystyle n}
{\displaystyle m}
{\displaystyle n}
are positive whole numbers
{\displaystyle m}
{\displaystyle n}
have no common factors except 1
{\displaystyle m}
{\displaystyle n}
have opposite parity.
{\displaystyle m}
{\displaystyle n}
have opposite parity when
{\displaystyle m}
{\displaystyle n}
is odd, or
{\displaystyle m}
{\displaystyle n}
{\displaystyle m>n}
If all four conditions are satisfied, then the values of
{\displaystyle m}
{\displaystyle n}
create a primitive Pythagorean triple.
{\displaystyle m=2}
{\displaystyle n=1}
create a primitive Pythagorean triple. The values satisfy all four conditions.
{\displaystyle 2mn=2\times 2\times 1=4}
{\displaystyle m^{2}-n^{2}=2^{2}-1^{2}=4-1=3}
{\displaystyle m^{2}+n^{2}=2^{2}+1^{2}=4+1=5}
, so the triple
{\displaystyle (3,4,5)}
is created.
↑ Loomis, Elisha S. 1927. The Pythagorean proposition: its proofs analysed and classified and bibliography of sources. Cleveland, Ohio.
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Pythagorean_theorem&oldid=8162361"
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homogeneousD - Maple Help
Home : Support : Online Help : Mathematics : Differential Equations : Classifying ODEs : First Order : homogeneousD
Solving Homogeneous ODEs of Class D
The general form of the homogeneous equation of class D is given by the following:
homogeneousD_ode := diff(y(x),x)= y(x)/x+g(x)*f(y(x)/x);
\textcolor[rgb]{0,0,1}{\mathrm{homogeneousD_ode}}\textcolor[rgb]{0,0,1}{≔}\frac{\textcolor[rgb]{0,0,1}{ⅆ}}{\textcolor[rgb]{0,0,1}{ⅆ}\textcolor[rgb]{0,0,1}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{=}\frac{\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)}{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)}{\textcolor[rgb]{0,0,1}{x}}\right)
where f(y(x)/x) and g(x) are arbitrary functions of their arguments. See Differentialgleichungen, by E. Kamke, p. 20. This type of ODE can be solved in a general manner by dsolve and the coefficients of the infinitesimal symmetry generator are also found by symgen.
\mathrm{with}\left(\mathrm{DEtools},\mathrm{odeadvisor},\mathrm{symgen},\mathrm{symtest}\right)
[\textcolor[rgb]{0,0,1}{\mathrm{odeadvisor}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{symgen}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{symtest}}]
\mathrm{odeadvisor}\left(\mathrm{homogeneousD_ode}\right)
[[\textcolor[rgb]{0,0,1}{\mathrm{_homogeneous}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{class D}}]]
A pair of infinitesimals for homogeneousD_ode
\mathrm{symgen}\left(\mathrm{homogeneousD_ode}\right)
[\textcolor[rgb]{0,0,1}{\mathrm{_ξ}}\textcolor[rgb]{0,0,1}{=}\frac{\textcolor[rgb]{0,0,1}{x}}{\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{_η}}\textcolor[rgb]{0,0,1}{=}\frac{\textcolor[rgb]{0,0,1}{y}}{\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)}]
\mathrm{ans}≔\mathrm{dsolve}\left(\mathrm{homogeneousD_ode}\right)
\textcolor[rgb]{0,0,1}{\mathrm{ans}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{RootOf}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{-}\left({\textcolor[rgb]{0,0,1}{\int }}_{\textcolor[rgb]{0,0,1}{}}^{\textcolor[rgb]{0,0,1}{\mathrm{_Z}}}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{_a}}\right)}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0,0,1}{ⅆ}\textcolor[rgb]{0,0,1}{\mathrm{_a}}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\int }\frac{\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)}{\textcolor[rgb]{0,0,1}{x}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{ⅆ}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{_C1}}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}
Answers can be tested using odetest
\mathrm{odetest}\left(\mathrm{ans},\mathrm{homogeneousD_ode}\right)
\textcolor[rgb]{0,0,1}{0}
Let's see how the answer above works when turning f into an explicit function; f is the identity mapping.
f≔u↦u
\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{↦}\textcolor[rgb]{0,0,1}{u}
\mathrm{allvalues}\left(\mathrm{value}\left(\mathrm{ans}\right)\right)
\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\int }\frac{\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)}{\textcolor[rgb]{0,0,1}{x}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0,0,1}{ⅆ}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{_C1}}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}
\mathrm{odetest}\left(\mathrm{ans},\mathrm{homogeneousD_ode}\right)
\textcolor[rgb]{0,0,1}{0}
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Propagate signal in LOS channel - MATLAB - MathWorks Deutschland
Propagate Signal in LOS Channel
prop_sig = step(sLOS,sig,origin_pos,dest_pos,origin_vel,dest_vel) returns the resulting signal, prop_sig, when a narrowband signal, sig, propagates through a line-of-sight (LOS) channel from a source located at the origin_pos position to a destination at the dest_pos position. Only one of the origin_pos or dest_pos arguments can specify multiple positions. The other must contain a single position. The velocity of the signal origin is specified in origin_vel and the velocity of the signal destination is specified in dest_vel. The dimensions of origin_vel and dest_vel must match the dimensions of origin_pos and dest_pos, respectively.
sig — Narrowband signal
Narrowband signal, specified as a matrix or struct array, depending on whether is signal or polarized or nonpolarized. The quantity M is the number of samples in the signal, and N is the number of LOS channels. Each channel corresponds to a source-destination pair.
Narrowband nonpolarized scalar signal. Specify sig as an M-by-N complex-valued matrix. Each column contains one signal propagated along the line-of-sight path.
Narrowband polarized signal. Specify sig as a 1-by-N struct array containing complex-valued fields. Each struct represents a polarized signal propagated along the line-of-sight path. Each struct element contains three M-by-1 complex-valued column vectors, sig.X, sig.Y, and sig.Z. These vectors represent the x, y, and z Cartesian components of the polarized signal.
prop_sig — Narrowband propagated signal
Narrowband signal, returned as a matrix or struct array, depending on whether signal is polarized or nonpolarized. The quantity M is the number of samples in the signal and N is the number of narrowband LOS channels. Each channel corresponds to a source-destination pair.
Narrowband nonpolarized scalar signal. prop_sig is an M-by-N complex-valued matrix.
Narrowband polarized scalar signal. prop_sig is a 1-by-N struct array containing complex-valued fields. Each struct element contains three M-by-1 complex-valued column vectors, sig.X, sig.Y, and sig.Z. These vectors represent the x, y, and z Cartesian components of the polarized signal.
Propagate a sinusoidal signal in a line of sight (LOS) channel from a radar at (1000,0,0) meters to a target at (10000,4000,500) meters. Assume the signal propagates in medium fog specified by a liquid water density of 0.05
g/{m}^{3}
. Assume that the radar and the target are stationary. The signal carrier frequency is 10 GHz. The signal frequency is 500 Hz and the sample rate is 8.0 kHz.
Set up the transmitted signal.
fsig = 500.0;
t = [0:dt:.01];
sig = sin(2*pi*fsig*t);
Set the liquid water density and specify the LOS channel System object™.
channel = phased.LOSChannel('SampleRate',fs,'SpecifyAtmosphere',true,...
'LiquidWaterDensity',lwd,'OperatingFrequency',fc);
Set the origin and destination of the signal.
xradar = [1000,0,0].';
xtgt = [10000,4000,500].';
vtgt = [0,0,0].';
Propagate the signal from origin to destination and plot the result.
prog_sig = channel(sig.',xradar,xtgt,vradar,vtgt);
plot(t*1000,real(prog_sig))
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Paige traveled to Australia and is making her favorite bread recipe. She usually bakes the bread in a
350º\text{F}
oven. She is surprised to learn that the oven temperatures in Australia are measured in degrees Celsius. In the formula at right,
F
stands for Fahrenheit and C stands for Celsius. Use the formula and the Order of Operations to help Paige determine how she should set the oven in degrees Celsius.
C=\frac{5(F-32)}{9}
You are given degrees Fahrenheit and you need degrees Celsius.
\Large{C=\frac{5\left(\mathbf{350}-32\right)}{9}}
350º\text{F}
\text{F}
. Then solve within the parentheses.
\Large{C=\frac{\mathbf{5}\left(\mathbf{318}\right)}{9}}
Simplify the numerator by multiplying.
\Large{C=\mathbf{\frac{1590}{9}}}
Divide to get your answer.
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Modulate frequency-domain OFDM subcarriers to time-domain samples for custom communication protocols - Simulink - MathWorks France
X\left(k\right)=F\left[x\left(n\right)\right]=\sum _{n=0}^{N-1}x\left(n\right){e}^{-\frac{j2\pi nk}{N}}
k=k-\frac{N}{2}
X\left(k-\frac{N}{2}\right)=\sum _{n=0}^{N-1}x\left(n\right){e}^{-\frac{j2\pi n\left(k-\frac{N}{2}\right)}{N}}
X\left(k-\frac{N}{2}\right)=\sum _{n=0}^{N-1}{e}^{j\pi n}x\left(n\right){e}^{-\frac{j2\pi nk}{N}}
\sum _{n=0}^{N-1}x\left(n\right){e}^{-\frac{j2\pi nk}{N}}
F\left[x\left(n\right)\right]
{e}^{j\pi }=-1
X\left(k-\frac{N}{2}\right)=F\left[{\left(-1\right)}^{n}x\left(n\right)\right]
|
CORDIC-based absolute value - MATLAB cordicabs - MathWorks í•œêµ
CORDIC-based absolute value
r = cordicabs(c)
r = cordicabs(c,niters)
r = cordicabs(c,niters,'ScaleOutput',b)
r = cordicabs(c,'ScaleOutput',b)
r = cordicabs(c) returns the magnitude of the complex elements of C.
r = cordicabs(c,niters) performs niters iterations of the algorithm.
r = cordicabs(c,niters,'ScaleOutput',b) specifies both the number of iterations and, depending on the Boolean value of b, whether to scale the output by the inverse CORDIC gain value.
r = cordicabs(c,'ScaleOutput',b) scales the output depending on the Boolean value of b.
c is a vector of complex values.
r contains the magnitude values of the complex input values. If the inputs are fixed-point values, r is also fixed point (and is always signed, with binary point scaling). All input values must have the same data type. If the inputs are signed, then the word length of r is the input word length + 2. If the inputs are unsigned, then the word length of r is the input word length + 3. The fraction length of r is always the same as the fraction length of the inputs.
Compare cordicabs and abs of double values.
dblValues = complex(rand(5,4),rand(5,4));
r_dbl_ref = abs(dblValues)
r_dbl_cdc = cordicabs(dblValues)
Compute absolute values of fixed-point inputs.
fxpValues = fi(dblValues);
r_fxp_cdc = cordicabs(fxpValues)
\begin{array}{l}{x}_{0}\text{ is initialized to the }x\text{ input value}\\ {y}_{0}\text{ is initialized to the }y\text{ input value}\\ {z}_{0}\text{ is initialized to }0\end{array}
cordiccart2pol | cordicangle | abs
|
Does lim (t to infty) E(x-x hat) = 0 imply that there will be less disturbance over time? - Murray Wiki
Does lim (t to infty) E(x-x hat) = 0 imply that there will be less disturbance over time?
{\displaystyle E(x)\,}
here denotes the expected value of
{\displaystyle x\,}
{\displaystyle x\,}
is a random variable.
{\displaystyle \lim _{t\to \infty }E(x-{\hat {x}})=0\,}
only implies that the mean estimation error will converge to zero over time. Disturbance, however, will affect the variance of the error, which is given by
{\displaystyle E(x-{\hat {x}})^{2}\,}
. In CDS 110b, we will learn how to design observers that minimize this estimation variance if the disturbance can be modeled as Gaussian noise.
Retrieved from "https://murray.cds.caltech.edu/index.php?title=Does_lim_(t_to_infty)_E(x-x_hat)_%3D_0_imply_that_there_will_be_less_disturbance_over_time%3F&oldid=8335"
|
q
05A40 Umbral calculus
\left(-1\right)
4-core partitions and class numbers
Ken Ono, Lawrence Sze (1997)
A
-analogue of some binomial coefficient identities of Y. Sun.
Guo, Victor J.W., Yang, Dan-Mei (2011)
A bijective proof of Garsia's
q
-Lagrange inversion theorem.
Singer, Dan W. (1998)
A bijective proof of the hook-content formula for super Schur functions and a modified jeu de taquin.
Krattenthaler, C. (1996)
Paolo Massazza, Roberto Radicioni (2010)
We present a CAT (constant amortized time) algorithm for generating those partitions of n that are in the ice pile model
{\text{IPM}}_{k}
(n), a generalization of the sand pile model
\text{SPM}
(n). More precisely, for any fixed integer k, we show that the negative lexicographic ordering naturally identifies a tree structure on the lattice
{\text{IPM}}_{k}
(n): this lets us design an algorithm which generates all the ice piles of
{\text{IPM}}_{k}
(n) in amortized time O(1) and in space O(
\sqrt{n}
{\text{IPM}}_{k}
\text{SPM}
{\text{IPM}}_{k}
{\text{IPM}}_{k}
\sqrt{n}
A characterization of determinacy for Turing degree games
E. Kleinberg (1973)
A combinatorial approach to partitions with parts in the gaps
Dennis Eichhorn (1998)
Many links exist between ordinary partitions and partitions with parts in the “gaps”. In this paper, we explore combinatorial explanations for some of these links, along with some natural generalizations. In particular, if we let
{p}_{k,m}\left(j,n\right)
be the number of partitions of n into j parts where each part is ≡ k (mod m), 1 ≤ k ≤ m, and we let
p{*}_{k,m}\left(j,n\right)
be the number of partitions of n into j parts where each part is ≡ k (mod m) with parts of size k in the gaps, then
p{*}_{k,m}\left(j,n\right)={p}_{k,m}\left(j,n\right)
A combinatorial proof of a partition identity of Andrews and Stanley.
Sills, Andrew V. (2004)
A combinatorial proof of Andrews' smallest parts partition function.
Ji, Kathy Qing (2008)
A combinatorial proof of the sum of
q
-cubes.
Garrett, Kristina C., Hummel, Kristen (2004)
A complete categorization of when generalized Tribonacci sequences can be avoided by additive partitions.
Develin, Mike (2000)
A de Bruijn-type formula for enumerating patterns of partitions.
Dennis E. White (1977)
Bender, Edward A., Canfield, E.Rodney, Richmond, L.Bruce, Wilf, Herbert S. (2003)
A fast algorithm for MacMahon's partition analysis.
Xin, Guoce (2004)
A generalized formula of Hardy.
Campbell, Geoffrey B. (1994)
A left weighted Catalan extension.
Schumacher, Paul R.F. (2010)
A new class of infinite products, and Euler's totient.
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EUDML | An abstract characterization of ...-conditional expectations. EuDML | An abstract characterization of ...-conditional expectations.
An abstract characterization of ...-conditional expectations.
Cecchini, Carlo. "An abstract characterization of ...-conditional expectations.." Mathematica Scandinavica 66.1 (1990): 155-160. <http://eudml.org/doc/167101>.
@article{Cecchini1990,
author = {Cecchini, Carlo},
keywords = {-conditional expectations of von Neumann algebras; canonical state extensions; partial isometries},
title = {An abstract characterization of ...-conditional expectations.},
AU - Cecchini, Carlo
TI - An abstract characterization of ...-conditional expectations.
KW - -conditional expectations of von Neumann algebras; canonical state extensions; partial isometries
Carlo Cecchini, Canonical functional extensions on von Neumann algebras
\omega
-conditional expectations of von Neumann algebras, canonical state extensions, partial isometries
{C}^{*}
{W}^{*}
Articles by Carlo Cecchini
|
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