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# What does the notation $^{10} C_3$ mean? Or: I am confused by a birthday card…

I've just seen a gift card on the internet that is supposedly for a mathematician's 21st birthday. It says

$$\text{Happy } ^{10} C_3 - 11\ln(e)-\frac{289}{3}+\left(\int_{\pi/6}^{\pi/3}\sec^2(x)dx\right)^2+7$$

I don't think I have ever seen the notation $$^{10} C_3$$ before. What does it mean?

Well,

$$\left(\int_{\pi/6}^{\pi/3}\sec^2(x)dx\right)^2= \left(\tan(\pi/3)-\tan(\pi/6)\right)^2=\left(\sqrt{3}-\sqrt{1/3}\right)^2=4/3,$$

so I can simplify the expression that is wished to be happy to

$$^{10} C_3 - 11-\frac{289}{3}+4/3+7 = {}^{10} C_3 -99.$$

I guess the result is supposed to be 21, so (if I have not made any mistakes) I would expect that $${}^{10} C_3=120,$$ but I have absolutely no idea how to make sense of this...

• $10$ choose $3= \dfrac{10!}{7!\, 3!}=120$ – J. W. Tanner Jul 4 at 0:08
• $^nC_r$ is an alternate notation for the binomial symbol $\binom nr=\frac {n!}{r!(n-r)!}$. – lulu Jul 4 at 0:09

$${}^{10} C_3$$ means "$$10$$ choose $$3$$" and is more frequently denoted by the binomial coefficient $$\binom{10}{3}$$.
• Instead of "more frequently" say "in more advanced math". That ${}^nC_r$ notation is probably in common use in the US for ages 12 to 18 or so. Along with a corresponding ${}^nP_r$ notation. Called "combinations and permutations". – GEdgar Jul 4 at 0:12
• though in the Britannica example, $n$ is subscript rather than superscript, @GEdgar – J. W. Tanner Jul 4 at 0:20
$$^{10}C_3 = {10 \choose 3} = \cfrac{10!}{7! 3!} = \cfrac{ \Gamma(11)}{\Gamma(8) \Gamma(4)} = 120$$ You’ve guessed the right value.


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The radius of a spherical balloon is increasing at a rate of 2 centimeters per minute. How fast is the volume changing when the radius is 12 centimeters?
Note: The volume of a sphere is given by $V=(4/3)\pi r^3$.

Rate of change of volume =

format: plain


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# How do you use Heron's formula to determine the area of a triangle with sides of that are 14, 9, and 13 units in length?

Jan 23, 2016

$18 \sqrt{10} s q . u n i t s$

#### Explanation:

Use Heron's formula:-
sqrt(s(s-a)(s-b)(s-c) Where $S = \frac{a + b + c}{2}$

So
$a = 14 , b = 9 , c = 13 , S = 18 =$ $\frac{9 + 13 + 14}{2}$

$\rightarrow \sqrt{18 \left(18 - 14\right) \left(18 - 13\right) \left(18 - 9\right)}$

$\rightarrow \sqrt{18 \left(4\right) \left(5\right) \left(9\right)}$

$\rightarrow \sqrt{18 \left(180\right)}$

$\rightarrow \sqrt{3240}$

$\sqrt{3240} = 18 \sqrt{10}$


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# Tips on making missile smoke trails?

## Recommended Posts

I'm trying to make nice looking missile smoke trails for my 3d space game, similar to the ones in the battlestar galactica series, or in one of the x3 reunion trailers. So far I've got some billboarded smoke textures with alpha blending and testing on it which gives a nice see through smoke particle, but when i try to make a smoke trail it ends up looking like the snake in nibbles. So any tips?

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You will probably want to make them sorta come out in different directions instead of just coming straight out, so its kinda like:
  ^  |   - missle  |  ^ /|\   - smoke trails/ | \

Also, make the particles fade out and get bigger the longer they have been alive.

Hope this helps :)

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When trying to make good particles effect it can be frustrating to re-compile the code every time you change a minor variable to tweak the thing. I would recommand you using one of many already existing particle editor of any engine, import approximatly your camera settings and your textures, create something looking good and then parsing the values (accel, velocity, randomness, lifetime, size) of the emitters in your code. Well that's only a suggestion :)

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Realistically, the smoke particles would under-go some changes, whether your particles system can handle these depends on your implementation:

* The particles increase in size as they age
* The particles alpha increases, from an initial high opacity fading to totally transparent.
* They would be propelled at some velocity away from the rocket, this might be an important distinction from simply spawning a particle at the end of the rocket and allowing the rocket simply move away from it.
* The particles speed would slow down due to air resistance.
* The particles would rotate, likely in all kinds of direction and this rotation would slow down too.
* As silothesuper said the particles wouldnt be produced in a single stream there would be some particles that come out a slight an angle too... for added realism these angled particles would probably be smaller in size although this may be a bit over the top [smile]
* Each particle would be different, so some would die earlier, some would rotate more in one direction to that of another, but all particles would fade to nothingness just before they die to avoid that nasty popping effect.

Hope that helps

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Hmm, so much work to do!

Although I am using textures at the moment, what do you think about using pixels and lines as particles, like is it possible to make decent looking smoke that way?

eg point1 color(rgba) would be (1,1,1,0) and point 2 would be (1,1,1,1) with alpha blending, so then the line would fade into nothing.

I got the idea from Jeff Lander's The Ocean Spray in Your Face, which mentions using the particle's position and last position to draw a line for the particle.

btw my games graphics style is basic polygons eg a missile looks like an untextured coloured tetrahedron, so I thought lines and points for particles would be more fitting than using textures.

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Have multiple textures for the particles. Thats all I really can think of that hasn't already been said.

Quote:
 Original quote by DungeWhen trying to make good particles effect it can be frustrating to re-compile the code every time you change a minor variable to tweak the thing.

Ditto, you could create your particle system in a way that you could adjust variables like particle life, rate of change in size, random jitter, ect... while it is running. That way it would be alot easier to adjust the way the system looks to get it to look right.

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I've worked with 2d smoke trails a lot. I've found that for missiles the coolest effect I'd gotten is to use a rectangular particle. Every time the missile turns you create a new one and have it follow the missile, changing the length as it goes. For 3d you could use 2 textures that cross each other like "+" so that it looks good. The particles at the end of the trail also shorten and when their length = 0 they are removed and you begin shortening the next smoke trail piece. I've found that method along with some other particle emitter for the fire looks very impressive.

here's a picture of the 2 textures or solid colored quads that create the particle.

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I've started playing homeworld 2, how do they manage to make those seemless thrust streams?

Quote:
 Original post by SirisianI've found that for missiles the coolest effect I'd gotten is to use a rectangular particle.

Do you have any screenshots or maybe the executive still around that I could have a look at?

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I only work in 2d, but in my old game engine (3 years old...)
http://img180.imageshack.us/img180/6439/image3dq4.png

I've never touched 3d programming, but for the idea as I think it's just an array of objects.

class smokePoint
{
float x,y,z;
float orientationX, orientationY, orientationZ;
Vector3d TexturePoints[4];//4 points around the main point.
};

std::vector<smokePoint*> smokeTrails;

okay it's kind of hard to explain since I've never worked with 3d.

On the tip of the wing you could create a smoke point with the x,y,z and orientation set the 4 points around that point to form a + sign. Then after a few frames create another one. Plot 4 points around each of them. Then connect the first 4 texture points to their respective points in the second smoke point. Keep doing that for all the smoke points in the trail and render them.

This is completely theorhetical, but sounds like it could work for smoke trails. Maybe someone with better 3d skillz could clearify a better way.

[Edited by - Sirisian on August 22, 2006 5:44:23 PM]

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If you want long trails, dont rely on particles because you will end up having to draw too many to fill the gaps that can form. A bit of smart thinking and you should be able to think of a way to use a textured geometry strip that shows the path that the rockets been travelling in. Then you can play with the alpha/width etc at any stage along that path.

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zbMATH — the first resource for mathematics

Blow up of solutions to semilinear wave equations. (English) Zbl 1032.35137
Summary: This work shows the absence of global solutions to the equation $u_{tt} = \Delta u + p^{-k} u ^m,$ in the Minkowski space $$\mathbb {M}_0=\mathbb {R}\times\mathbb {R}^N$$, where $$m > 1$$, $$(N-1)m < N+1$$, and $$p$$ is a conformal factor approaching zero at infinity. Using a modification of the method of conformal compactification, we prove that any solution develops a singularity at a finite time.
MSC:
 35L70 Second-order nonlinear hyperbolic equations 35B40 Asymptotic behavior of solutions to PDEs 35L15 Initial value problems for second-order hyperbolic equations
Full Text:


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• ### Near-side azimuthal and pseudorapidity correlations using neutral strange baryons and mesons in d+Au, Cu+Cu and Au+Au collisions at $\sqrt{s_{NN}}$ = 200 GeV(1603.05477)

Aug. 1, 2016 nucl-ex
We present measurements of the near-side of triggered di-hadron correlations using neutral strange baryons ($\Lambda$, $\bar{\Lambda}$) and mesons ($K^0_S$) at intermediate transverse momentum (3 $<$ $p_T$ $<$ 6 GeV/$c$) to look for possible flavor and baryon/meson dependence. This study is performed in $d$+Au, Cu+Cu and Au+Au collisions at $\sqrt{s_{{NN}}}$ = 200 GeV measured by the STAR experiment at RHIC. The near-side di-hadron correlation contains two structures, a peak which is narrow in azimuth and pseudorapidity consistent with correlations due to jet fragmentation, and a correlation in azimuth which is broad in pseudorapidity. The particle composition of the jet-like correlation is determined using identified associated particles. The dependence of the conditional yield of the jet-like correlation on the trigger particle momentum, associated particle momentum, and centrality for correlations with unidentified trigger particles are presented. The neutral strange particle composition in jet-like correlations with unidentified charged particle triggers is not well described by PYTHIA. However, the yield of unidentified particles in jet-like correlations with neutral strange particle triggers is described reasonably well by the same model.
• We report new STAR measurements of mid-rapidity yields for the $\Lambda$, $\bar{\Lambda}$, $K^{0}_{S}$, $\Xi^{-}$, $\bar{\Xi}^{+}$, $\Omega^{-}$, $\bar{\Omega}^{+}$ particles in Cu+Cu collisions at \sNN{200}, and mid-rapidity yields for the $\Lambda$, $\bar{\Lambda}$, $K^{0}_{S}$ particles in Au+Au at \sNN{200}. We show that at a given number of participating nucleons, the production of strange hadrons is higher in Cu+Cu collisions than in Au+Au collisions at the same center-of-mass energy. We find that aspects of the enhancement factors for all particles can be described by a parameterization based on the fraction of participants that undergo multiple collisions.
• Two-particle azimuthal ($\Delta\phi$) and pseudorapidity ($\Delta\eta$) correlations using a trigger particle with large transverse momentum ($p_T$) in $d$+Au, Cu+Cu and Au+Au collisions at $\sqrt{s_{{NN}}}$ =\xspace 62.4 GeV and 200~GeV from the STAR experiment at RHIC are presented. The \ns correlation is separated into a jet-like component, narrow in both $\Delta\phi$ and $\Delta\eta$, and the ridge, narrow in $\Delta\phi$ but broad in $\Delta\eta$. Both components are studied as a function of collision centrality, and the jet-like correlation is studied as a function of the trigger and associated $p_T$. The behavior of the jet-like component is remarkably consistent for different collision systems, suggesting it is produced by fragmentation. The width of the jet-like correlation is found to increase with the system size. The ridge, previously observed in Au+Au collisions at $\sqrt{s_{{NN}}}$ = 200 GeV, is also found in Cu+Cu collisions and in collisions at $\sqrt{s_{{NN}}}$ =\xspace 62.4 GeV, but is found to be substantially smaller at $\sqrt{s_{{NN}}}$ =\xspace 62.4 GeV than at $\sqrt{s_{{NN}}}$ = 200 GeV for the same average number of participants ($\langle N_{\mathrm{part}}\rangle$). Measurements of the ridge are compared to models.
• ### Strange and identified hadron production at the LHC with ALICE(1110.4240)

Oct. 20, 2011 nucl-ex
The ALICE detector was designed to identify hadrons over a wide range of transverse momentum at mid-rapidity. Here measurements of light charged ({\pi}, K, p) and neutral ({\Lambda}, K0S) hadrons in Pb-Pb collisions at sqrt(s_NN) = 2.76 TeV are presented with additional data from a pp reference at sqrt(s) = 7 TeV. Such measurements are crucial for understanding the properties of the fireball produced in heavy-ion collisions at the LHC. The particle-type dependence of the spectra and the yields of particles extracted give information on the expansion dynamics and chemical composition respectively. In addition studying the ratio of baryons to mesons may help in understanding the mechanisms by which hadronisation takes place. We find that, when comparing to data at sqrt(s_NN) = 200 GeV Au+Au collisions at RHIC, a more strongly expanding system is created with a similar relative population of hadron species. We also see that collective effects or complex mechanisms responsible for a relative enhancement of baryons have an influence at a much higher pT than was previously seen.
• The STAR Collaboration at RHIC has measured two-pion correlation functions from p+p collisions at sqrt(s)=200 GeV. Spatial scales are extracted via a femtoscopic analysis of the correlations, though this analysis is complicated by the presence of strong non-femtoscopic effects. Our results are put into the context of the world dataset of femtoscopy in hadron-hadron collisions. We present the first direct comparison of femtoscopy in p+p and heavy ion collisions, under identical analysis and detector conditions.
• We report the first measurements of the kurtosis (\kappa), skewness (S) and variance (\sigma^2) of net-proton multiplicity (N_p - N_pbar) distributions at midrapidity for Au+Au collisions at \sqrt(s_NN) = 19.6, 62.4, and 200 GeV corresponding to baryon chemical potentials (\mu_B) between 200 - 20 MeV. Our measurements of the products \kappa \sigma^2 and S \sigma, which can be related to theoretical calculations sensitive to baryon number susceptibilities and long range correlations, are constant as functions of collision centrality. We compare these products with results from lattice QCD and various models without a critical point and study the \sqrt(s_NN) dependence of \kappa \sigma^2. From the measurements at the three beam energies, we find no evidence for a critical point in the QCD phase diagram for \mu_B below 200 MeV.
• We report on a measurement of the Upsilon(1S+2S+3S) -> e+e- cross section at midrapidity in p+p collisions at sqrt(s)=200 GeV. We find the cross section to be 114 +/- 38 (stat.) +23,-24 (syst.) pb. Perturbative QCD calculations at next-to-leading order in the Color Evaporation Model are in agreement with our measurement, while calculations in the Color Singlet Model underestimate it by 2 sigma. Our result is consistent with the trend seen in world data as a function of the center-of-mass energy of the collision and extends the availability of Upsilon data to RHIC energies. The dielectron continuum in the invariant mass range near the Upsilon is also studied to obtain a combined cross section of Drell-Yan plus (b b-bar) -> e+e-.
• We present measurements of net charge fluctuations in $Au + Au$ collisions at $\sqrt{s_{NN}} =$ 19.6, 62.4, 130, and 200 GeV, $Cu + Cu$ collisions at $\sqrt{s_{NN}} =$ 62.4, 200 GeV, and $p + p$ collisions at $\sqrt{s} =$ 200 GeV using the dynamical net charge fluctuations measure $\nu_{+-{\rm,dyn}}$. We observe that the dynamical fluctuations are non-zero at all energies and exhibit a modest dependence on beam energy. A weak system size dependence is also observed. We examine the collision centrality dependence of the net charge fluctuations and find that dynamical net charge fluctuations violate $1/N_{ch}$ scaling, but display approximate $1/N_{part}$ scaling. We also study the azimuthal and rapidity dependence of the net charge correlation strength and observe strong dependence on the azimuthal angular range and pseudorapidity widths integrated to measure the correlation.
• ### Identified Particle Dependence of Nuclear Modification Factors in d+Au Collisions at RHIC(nucl-ex/0404027)

April 22, 2004 nucl-ex
We present transverse momentum spectra from d+Au collisions at sqrt(s_NN) = 200 GeV for identified hadrons at mid-rapidity and unidentified hadrons in the forward rapidity region using the Solenoidal Tracker at RHIC. We scale these spectra by the mean number of binary collisions to form a transverse momentum dependent ratio to spectra from p+p collisions, showing that the Cronin effect measured at lower centre-of-mass energies is also present at this higher energy. The data also suggest a dependence of the Cronin effect on the hadron species. We also compare central d+Au collisions to more peripheral ones, contrasting the behaviour to that observed in Au+Au collisions and remark on unidentified hadrons at forward rapidities which show a qualitatively different behaviour.


text:
Henry went fishing… - QuestionCove
OpenStudy (anonymous):

Henry went fishing and caught 4 fish weighing 3 1/2, 5 1/4, 2 1/2, and 6 1/4. What is the average weight of the 4 fish henry caught?

6 years ago
OpenStudy (owlfred):

Hoot! You just asked your first question! Hang tight while I find people to answer it for you. You can thank people who give you good answers by clicking the 'Good Answer' button on the right!

6 years ago
OpenStudy (angela210793):

u add all those and then divide them by 4

6 years ago
OpenStudy (anonymous):

17.5/4

6 years ago
OpenStudy (anonymous):

well they only give the choices of 5 1/4, 5, 4 3/8, or 4 1/8

6 years ago
OpenStudy (anonymous):

i think brackett forgot to divide by 4

6 years ago
OpenStudy (anonymous):

mixed numbers are a drag

6 years ago
OpenStudy (anonymous):

lol i did it and got 4 2/8 but its not an answer choice

6 years ago
OpenStudy (anonymous):

oh he wrote $\frac{17.5}{4}$

6 years ago
OpenStudy (angela210793):

7/2+21/4+5/2+25/4=12/2+46/4=6+11.5 =17.5 then the average=17.5/4

6 years ago
OpenStudy (anonymous):

think it is 3/8 not 2/8

6 years ago
OpenStudy (anonymous):

$\frac{17.5}{4}=\frac{35}{8}=4\tfrac{3}{8}$

6 years ago
OpenStudy (anonymous):

ok ty

6 years ago
OpenStudy (anonymous):

whew!

6 years ago


text:
# Toronto Math Forum

## MAT244--2018F => MAT244--Tests => Final Exam => Topic started by: Victor Ivrii on December 14, 2018, 08:06:54 AM

Title: FE-P6
Post by: Victor Ivrii on December 14, 2018, 08:06:54 AM
Typed solutions only. Upload only one picture (a general phase portrait; for general one can use computer generated)
For the system of ODEs
\begin{equation*}
\left\{\begin{aligned}
&x'  = 2y(x^2+y^2+4)\, , \\
&y'  = -2x (x^2+y^2-16)
\end{aligned}\right.
\end{equation*}

(a) Find stationary points.

(b) Linearize the system at stationary points and sketch the phase portrait of this linear system.

(c) Find the equation of the form $H(x,y) = C$, satisfied by the trajectories of the nonlinear system.

(d)  Sketch the full phase portrait.

Hint: avoid redundancy: asymptotically (un)stable node, unstable node, stable center
Title: Re: FE-P6
Post by: Jingze Wang on December 14, 2018, 09:02:51 AM
part (a)
Find critical points
Let $x'=0, y'=0$
Then $2y(x^2+y^2+4)=0, -2x(x^2+y^2-16)=0$
When $y=0, x=0, x=4, x=-4$
So (0,0) (4,0) (-4,0) are critical points.

Part (b)
$F(x,y)=2y(x^2+y^2+4), G(x,y)=-2x(x^2+y^2-16)$
$F_x=4xy, F_y=2x^2+6y^2+8$
$G_x=-6x^2-2y^2+32, G_y=-4xy$
Then plug in to find J matrix
$$J={ \left[\begin{array}{ccc} 4xy & 2x^2+6y^2+8 \\ -6x^2-2y^2+32 & -4xy \end{array} \right ]},$$
$When (0,0)$
$$J={ \left[\begin{array}{ccc} 0 & 8 \\ 32 & 0 \end{array} \right ]},$$
This is a saddle
$When (4,0)$
$$J={ \left[\begin{array}{ccc} 0 & 40 \\ -64 & 0 \end{array} \right ]},$$
This one is a center
$When (-4,0)$
$$J={ \left[\begin{array}{ccc} 0 & 40 \\ -64 & 0 \end{array} \right ]},$$
This one is also a center
Also, the phase portraits are attached in picture 2

Part (c)
$2x(x^2+y^2-16)dx+2y(x^2+y^2+4)dy=0$
Find this equation is exact, then
$H(x,y)=0.5 x^4+x^2y^2-16x^2+h(y)$
$H_y=2x^2 y +h'(y)$
$h'(y)=2y^3+8y$
$h(y)=0.5 y^4+4y^2$
$H(x,y)=0.5 x^4+x^2y^2-16x^2+0.5 y^4+4y^2=c$

Part (d)
Since it is integrable system
Then center is still center in nonlinear system.
See picture 1.
Title: Re: FE-P6
Post by: Jerry Qinghui Yu on December 14, 2018, 10:49:09 AM
clear picture for part d
Title: Re: FE-P6
Post by: Victor Ivrii on December 15, 2018, 06:46:33 PM
I wonder, nobody sees multiple errors?
Title: Re: FE-P6
Post by: Jingze Wang on December 15, 2018, 07:12:23 PM
Corrected, what about now?
Title: Re: FE-P6
Post by: Victor Ivrii on December 16, 2018, 01:44:02 AM
Corrected, what about now?
Still ... :(
Title: Re: FE-P6
Post by: Tzu-Ching Yen on December 16, 2018, 10:50:13 AM
$G_y = -4xy$
Title: Re: FE-P6
Post by: Jingze Wang on December 16, 2018, 01:29:15 PM
Thanks, all corrected, finally :)
Title: Re: FE-P6
Post by: Tzu-Ching Yen on December 16, 2018, 02:22:20 PM
The term in $H(x, y)$ should be $\frac{1}{2}x^4$ as well. Perhaps an extra constant. Otherwise I couldn't find anything else.
Title: Re: FE-P6
Post by: Victor Ivrii on December 16, 2018, 06:50:30 PM
Can anyone write a correct final version of $H(x,y)$?
Title: Re: FE-P6
Post by: Joyce Ye on December 16, 2018, 07:37:50 PM
dx/dt=2x2y+2y3+8y
dy/dt=-2x3-2xy2+32x
so,dx/dy=2x2y+2y3+8y/-2x3-2xy2+32x
so,(2x3+2xy2-32x)dx+(2x2y+2y3+8y)dy=0
Let M=2x3+2xy2-32x,N=2x2y+2y3+8y
My=4xy,Nx=4xy
My=Nx, so Exact.
There exists a 𝐻(𝑥,𝑦) s.t. 𝐻x(𝑥,𝑦)=M,𝐻y(𝑥,𝑦)=N
𝐻(𝑥,𝑦)=1/2x4+x2y2-16x2+h(y)
𝐻y(𝑥,𝑦)=2x2y+ℎ′(𝑦)
so,ℎ′(𝑦)=2y3+8y
h(y)=1/2y4+4y2+C
Therefore, 𝐻(𝑥,𝑦)=1/2x4+x2y2-16x21/2y4+4y2=C
Title: Re: FE-P6
Post by: Tzu-Ching Yen on December 16, 2018, 07:40:28 PM
Therefore, 𝐻(𝑥,𝑦)=1/2x4+x2y2-16x21/2y4+4y2=C

$+$ missing
Title: Re: FE-P6
Post by: Jingze Wang on December 16, 2018, 08:11:04 PM
Can anyone tell me why Professor said my answer is wrong? Also, to avoid confusion, I use 0.5 instead of 1/2.
Title: Re: FE-P6
Post by: Tzu-Ching Yen on December 16, 2018, 08:15:54 PM
Can anyone tell me why Professor said my answer is wrong? Also, to avoid confusion, I use 0.5 instead of 1/2.
No idea tbh. Joyce provided exactly the same answer as your corrected version.
Title: Re: FE-P6
Post by: Victor Ivrii on December 16, 2018, 10:30:25 PM
Now it is correct, but before it was not so. Or, may be, because the correct expression was not articulated in the special line, or like this:
$$\boxed{H(x,y)=\frac{1}{2}\bigl(x^4 +2x^2y^2 +y^4-32 x^2 +8y^2\bigr).}$$
BTW, making this problem I started from $H(x,y)$ in the form
\begin{align*}
H(x,y)=&\bigl( (x-a)^2 +y^2-b^2\bigr)\bigl( (x+a)^2 +y^2-b^2\bigr)=\\
&\bigl( x^2 +y^2-b^2+a^2 -2ax\bigr)\bigl( x^2 +y^2-b^2+a^2 +2ax\bigr)=\\
&\bigl( x^2 +y^2-b^2+a^2\bigr)^2 -4 a^2x^2
\end{align*}
with $b> a>0$.

Question: For $H(x,y)$ what are points $(\pm 4,0)$ and $(0,0)$?
Title: Re: FE-P6
Post by: Jingze Wang on December 17, 2018, 05:44:04 AM
Since system is integrable, so just saddle and center as in linear system :)
Title: Re: FE-P6
Post by: Jingze Wang on December 17, 2018, 05:50:10 AM
I think (-4,0) and (4,0) are maximum and (0,0) is minimum
Title: Re: FE-P6
Post by: Victor Ivrii on December 17, 2018, 07:11:00 AM
There cannot be spiral in the integrable case.

However my question is different and Calculus II related: function $H(x,y)$ has the same critical points as the system. What types they are: maximum, minimum or saddle for each of three points found.
Title: Re: FE-P6
Post by: Qinger Zhang on December 17, 2018, 09:08:12 PM
critical points
Title: Re: FE-P6
Post by: Qinger Zhang on December 17, 2018, 09:09:38 PM
I think (0,0) is a saddle. why (4,0) and (-4,0) are not centre?
Title: Re: FE-P6
Post by: Qinger Zhang on December 17, 2018, 09:10:17 PM
Question: For $H(x,y)$ what are points $(\pm 4,0)$ and $(0,0)$?
From the grape, I don't think any of them is max or min.
Title: FE-P6 Official
Post by: Victor Ivrii on December 17, 2018, 09:18:11 PM
(a)  Since $x^2+y^2+4>0$  all stationary points are at $y=0$, and either $x=0$ or $x^2+y^2-16=0$; thus we have
$A_1=(0,0)$ and $A_{2,3}=(\pm 4,0)$.

(b) Let $f=2y(x^2+y^2+4)$, $g=-2x(x^2+y^2-16)$.

At $A_1$ we have $f_x=0$, $f_y=8$, $g_x=32$, $g_y=0$ and linearization is
$$\begin{pmatrix}X\\ Y\end{pmatrix}'=\begin{pmatrix} 0 &8\\32 &0\end{pmatrix}\begin{pmatrix}X\\Y\end{pmatrix}$$
with eigenvalues $\pm 16$ and eigenvectors $\begin{pmatrix}1 \\\pm2\end{pmatrix}$ so  it is a saddle:

At points $A_{2,3}$ we have $f_x=0$, $f_y=16$, $g_x=-16$, $g_y=0$ with eigenvalues $16i$ and $-16i$; in virtue of (c) it is a center, and since the bottom left element is negative it is clockwise.

Rewriting the system as
$$2x(x^2+y^2-16)\,dx + 2y(x^2+y^2+4)\,dy=0$$
one can check that the form on the left is exact; $H_x= 2(x^3+xy^2-16x)\implies H(x,y)=\frac{1}{2}(x^4 +2x^2y^2 -32 x^2)+h(y)$ and plugging to $H_y=2x^2y +h'(y)= 2yx^2+2y^3 +8y\implies h'(y)=2y^3 +8y\implies h(y)=\frac{1}{2}y^4+4y^2$ (we take a constant equal $0$). Then
$$H(x,y)=\frac{1}{2}\bigl(x^4 +2x^2y^2 +y^4-32 x^2 +8y^2\bigr).$$

Points $(\pm 4,0)$ are minima of $H(x,y)$ and $(0,0)$ is the saddle point. So called symmetric two-well 2D potential.
See 3D plot generated by WolframAlpha


text:
## Seminar information archive

### 2010/03/12

#### Colloquium

15:00-17:30   Room #050 (Graduate School of Math. Sci. Bldg.)

ガルニエ系の数理
[ Abstract ]
ガルニエ系は,パンルヴェ方程式の拡張であり,完全積分可能な多時間ハミルトン系として与えられる。これは2階線型常微分方程式のホロノミック変形を与える非線型完全積分可能な偏微分方程式系であり,講演の対象である2次元系では,8つのタイプの基本形がある。ガルニエ系の研究は,初期値空間やソリトン方程式系の相似簡約などの立場から行われているが,材料が揃ってくれば,一般リーマン・ヒルベルト対応を経由して考察することが自然であるし,数学的であるだろう。パンルヴェ方程式の場合もそのような方向に進んでいる。一方,パンルヴェ方程式については,そのハミルトニアンの満足する非線型常微分方程式が,アフィンワイル群の対称性など数学的な材料を与える上で一定の役割を果たした。ガルニエ系についても,そのハミルトニアンについての非線型偏微分方程式系を具体的に書き下すことは,意味のあることと信じているが,未完である。この話題について,部分的な結果を紹介する。

[ Abstract ]
40年近くの間,さまざまな幾何構造に関する特性類と不変量の研究を続けてきた.葉層構造やリーマン面のモジュライ空間の特性類,そして3次元多様体の位相不変量等である.これらについて振り返りつつ,これからの目標をいくつかの予想を交えてお話ししたい.

### 2010/03/09

#### PDE Real Analysis Seminar

10:30-11:30   Room #123 (Graduate School of Math. Sci. Bldg.)
Joachim Escher (Leibniz University of Hanover)
Shallow water waves with singularities
[ Abstract ]
The Degasperis-Procesi equation is a recently derived shallow water wave equation, which is - similar as the Camassa-Holm equation - embedded in a family of spatially periodic third order dispersive conservation laws.
The coexistence of globally in time defined classical solutions, wave breaking solutions, and spatially periodic peakons and shock waves is evidenced in the talk, and the precise blow-up scenario, including blow-up rates and blow-up sets, is discussed in detail. Finally several conditions on the initial profile are presented, which ensure the occurence of a breaking wave.

### 2010/02/24

#### Lectures

15:00-16:30   Room #370 (Graduate School of Math. Sci. Bldg.)
Robert Penner (Aarhus University / University of Southern California)
Protein Moduli Space
[ Abstract ]
Recent joint works with J. E. Andersen and others
provide explicit discrete and continuous models
of protein geometry. These models are inspired
by corresponding constructions in the study of moduli
spaces of flat G-connections on surfaces, in particular,
for G=PSL(2,R) and G=SO(3). These models can be used
for protein classification as well as for folding prediction,
and computer experiments towards these ends will
be discussed.

### 2010/02/23

#### Lectures

14:00-15:00   Room #122 (Graduate School of Math. Sci. Bldg.)
Bendong LOU (同済大学)
Homogenization Limit and Singular Limit of the Allen-Cahn equation
[ Abstract ]
We consider the Allen-Cahn equation in a cylinder with periodic undulating boundaries in the plane. Our problem involves two small parameters $\\delta$ and $\\epsilon$, where $\\delta$ appears in the equation to denote the scale of the singular limit and $\\epsilon$ appears in the boundary conditions to denote the scale of the homogenization limit. We consider the following two limiting processes:
(I): taking homogenization limit first and then taking singular limit;
(II) taking singular limit first and then taking homogenization limit.

We formally show that they both result in the same mean curvature flow equation, but with different boundary conditions.

### 2010/02/19

#### Lie Groups and Representation Theory

16:30-18:00   Room #126 (Graduate School of Math. Sci. Bldg.)
Yves Benoist (Orsay)
Discrete groups acting on homogeneous spaces V
[ Abstract ]
I will focus on recent advances on our understanding of discrete subgroups of Lie groups.
I will first survey how ideas from semisimple algebraic groups, ergodic theory and representation theory help us to understand properties of these discrete subgroups.

I will then focus on a joint work with Jean-Francois Quint studying the dynamics of these discrete subgroups on finite volume homogeneous spaces and proving the following result:

We fix two integral matrices A and B of size d, of determinant 1, and such that no finite union of vector subspaces is invariant by A and B. We fix also an irrational point on the d-dimensional torus. We will then prove that for n large the set of images of this point by the words in A and B of length at most n becomes equidistributed in the torus.

### 2010/02/18

#### GCOE lecture series

10:30-17:00   Room #126 (Graduate School of Math. Sci. Bldg.)
Yves Benoist (Pars Sud) 10:30-11:30
Discrete groups acting on homogeneous spaces III
[ Abstract ]
In this course I will focus on recent advances
on our understanding of discrete subgroups of Lie groups.

I will first survey how ideas from semisimple algebraic groups,
ergodic theory and representation theory help us to understand properties of these discrete subgroups.

I will then focus on a joint work with Jean-Francois Quint
studying the dynamics of these discrete subgroups on finite volume homogeneous spaces and proving the following result:

We fix two integral matrices A and B of size d, of determinant 1,
and such that no finite union of vector subspaces is invariant by A and B.
We fix also an irrational point on the d-dimensional torus. We will then prove that for n large the set of images of this point by the words in A and B of length at most n becomes equidistributed in the torus.
Yves Benoist (Paris Sud) 15:00-16:00
Discrete groups acting on homogeneous spaces IV

#### Operator Algebra Seminars

16:30-18:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Roberto Longo (University of Rome, Tor Vergata)
Von Neumann Algebras and Boundary Quantum Field Theory

#### Applied Analysis

16:00-17:30   Room #002 (Graduate School of Math. Sci. Bldg.)
Bendong LOU (同済大学)
Homogenization limit of a parabolic equation with nonlinear boundary conditions
[ Abstract ]
We consider a quasilinear parabolic equation with the following nonlinear Neumann boundary condition:
"the slope of the solution on the boundary is a function $g$ of the value of the solution". Here $g$ takes values near its supremum with the frequency of $\\epsilon$. We show that the homogenization limit of the solution, as $\\epsilon$ tends to 0, is the solution satisfying the linear Neumann boundary condition: "the slope of the solution on the boundary is the supremum of $g$".

#### GCOE Seminars

10:10-11:00   Room #122 (Graduate School of Math. Sci. Bldg.)

#### GCOE Seminars

11:00-11:50   Room #122 (Graduate School of Math. Sci. Bldg.)

#### GCOE Seminars

13:20-14:10   Room #122 (Graduate School of Math. Sci. Bldg.)

#### GCOE Seminars

14:10-14:40   Room #122 (Graduate School of Math. Sci. Bldg.)

#### GCOE Seminars

14:50-15:40   Room #122 (Graduate School of Math. Sci. Bldg.)
O. Iliev (フラウンホーファー産業数学研究所、ドイツ)
Flow and material simulation for industrial purposes

### 2010/02/17

#### GCOE lecture series

10:30-16:00   Room #126 (Graduate School of Math. Sci. Bldg.)
Yves Benoist (Paris Sud) 10:30-11:30
Discrete groups acting on homogeneous spaces I
[ Abstract ]
In this course I will focus on recent advances
on our understanding of discrete subgroups of Lie groups.

I will first survey how ideas from semisimple algebraic groups,
ergodic theory and representation theory help us to understand properties of these discrete subgroups.

I will then focus on a joint work with Jean-Francois Quint
studying the dynamics of these discrete subgroups on finite volume homogeneous spaces and proving the following result:

We fix two integral matrices A and B of size d, of determinant 1,
and such that no finite union of vector subspaces is invariant by A and B.
We fix also an irrational point on the d-dimensional torus. We will then prove that for n large the set of images of this point by the words in A and B of length at most n becomes equidistributed in the torus.
Yves Benoist (Paris Sud) 15:00-16:00
Discrete groups acting on homogeneous spaces II

#### Seminar on Probability and Statistics

15:00-16:10   Room #128 (Graduate School of Math. Sci. Bldg.)

[ Abstract ]

[ Reference URL ]
http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2009/14.html

### 2010/02/16

#### Tuesday Seminar on Topology

17:30-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Dieter Kotschick (Univ. M\"unchen)
Characteristic numbers of algebraic varieties
[ Abstract ]
The Chern numbers of n-dimensional smooth projective varieties span a vector space whose dimension is the number of partitions of n. This vector space has many natural subspaces, some of which are quite small, for example the subspace spanned by Hirzebruch--Todd numbers, the subspace of topologically invariant combinations of Chern numbers, the subspace determined by the Hodge numbers, and the subspace of Chern numbers that can be bounded in terms of Betti numbers. I shall explain the relation between these subspaces, and characterize them in several ways. This leads in particular to the solution of a long-standing open problem originally formulated by Hirzebruch in the 1950s.

### 2010/02/05

#### thesis presentations

09:45-11:00   Room #118 (Graduate School of Math. Sci. Bldg.)
Takahiro Tsushima (University of Tokyo)
Elementary computation of ramified components of Jacobi sum Hecke characters (JAPANESE)

#### thesis presentations

11:00-12:15   Room #118 (Graduate School of Math. Sci. Bldg.)
Tomoyuki Abe (University of Tokyo)
Comparison between Swan conductors and characteristic cycles (JAPANESE)

#### thesis presentations

13:00-14:15   Room #118 (Graduate School of Math. Sci. Bldg.)

The structures of generalized principal series representations of SL(3,R) and related Whittaker functions (SL(3,R)の一般主系列表現の構造と関連するWhittaker関数)

#### thesis presentations

14:15-15:30   Room #118 (Graduate School of Math. Sci. Bldg.)

PRINCIPAL SERIES AND GENERALIZED PRINCIPAL SERIES WHITTAKER FUNCTIONS WITH PERIPHERAL K-TYPES ON THE REAL SYMPLECTIC GROUP OF RANK 2 (実二次シンプレクティック群上の主系列表現及び一般主系列表現の周辺的K-TYPEを持つWHITTAKER 関数)

#### thesis presentations

09:45-11:00   Room #122 (Graduate School of Math. Sci. Bldg.)

On the generalized suspension theorem for directed Fukaya categories (有向深谷圏の懸垂定理の一般化について)

#### thesis presentations

11:00-12:15   Room #122 (Graduate School of Math. Sci. Bldg.)

On the Runge theorem for instantons (インスタントンに対するRungeの近似定理について)

#### thesis presentations

11:00-12:15   Room #126 (Graduate School of Math. Sci. Bldg.)

Solvability and irreducibility of difference equations (差分方程式の可解性と既約性)

#### thesis presentations

13:00-14:15   Room #126 (Graduate School of Math. Sci. Bldg.)

Weak Amenability for a Group Acting on a Finite Dimensional CAT(0) Cube Complex (有限次元CAT(0)方体複体に作用する群の弱従順性)

#### thesis presentations

14:15-15:30   Room #126 (Graduate School of Math. Sci. Bldg.)

Stone-Čech boundaries of discrete groups and measure equivalence theory (離散群のストーン-チェック境界と測度同値理論)


text:
1. ## algebra

hi everyone

z-1 =2z+3
_________
z+3= 2z-3

2. Originally Posted by anton5
hi everyone

z-1 =2z+3
/ /
z+3= 2z-3

z-1=2z+3

-1-3=2z-z

z=-4

z+3=2z-3

3+3=2z-z

z=6

3. Thank you mathaddid for your help but I don't understand how you set the equation up. could you write it with the "____" please


text:
# noor /

Filename Size Date modified Message
contrib
docs
noor
454 B
332 B
34.2 KB
244 B
9.6 KB
1.0 KB
16.1 KB
1019 B
2.0 KB
1.2 KB
1.4 KB

# Noor, A python Quran viewer

## Overview

Noor is a program written in Python for viewing quran. You can specify translation files; If specified the translation of each aya is shown after it. These are the interfaces that come with noor:

Interface Description
noorgtk Uses PyGtk library
noorhtmls Outputs quran in html files in the output directory
noorcgi Can be used as a CGI script in web servers
noortxt Outputs a sura to a text file

## New Features

• searching
• bookmarks
• fullscreen
• gtk.ignore_case for ignoring case in searches
• gtk.ignore_diacritics for ignoring diacritics in searches
• gtk.scrollbar for controlling whether to show the scroll bar
• gtk.shadow_options, if enabled, darkens all but the current aya

## Getting Started

### noorgtk

For using the GTK interface you need to install Python and PyGTK (they are installed by default in some linux distributions like Ubuntu). You can start it by running noorgtk.py script either from the command line or by double clicking it in the extracted folder. That is, no installation is necessary.

But if you like, you can run python setup.py install to install Noor. After that you'll be able to run noorgtk.py in the terminal from any location.

### noorhtmls

The noorhtmls interface can be used to output quran in html files in a directory. All you should do is to run:

noorhtmls.py output_folder

The index.html file will contain the list of suras and 001.html through 114.html files will contain the suras. Note that the configuration specified in the ~/.noor file is considered when generating the pages.

### noorcgi

Installing the CGI script, noorcgi.py, is like installing any other CGI script. You probably have to copy it to the cgi-bin directory of your web server. But before that you should either install noor as described in the noorgtk section or change the line in noorcgi.py file which looks like:

# Add noor to the python path if its not installed:
sys.path.append('/noor/extracted/folder/')

to point to the folder in which you've extracted noor. Note that if you want to use the CGI interface you no longer need to install PyGTK.

## Noorgtk Tutorial

After starting noorgtk as described above, you'll be asked to select a sura to view. (Use gtk.ask_sura_at_start config or give a command line argument to skip that).

Now you see the sura you've chosen. Use up and down keys to move the text. Press a key; you'll be asked to enter an aya number. After pressing enter you'll be moved to that aya.

Now press c. A dialog pops up in which you can select a sura.

Press s this time; Noor will ask you to insert a sura number to go to. Also J (that is capital j) moves you to the start of a juz.

Now press C-M-n or C-j (whichever you're most comfortable with). Noor will show the next sura. Likewise C-M-p or C-k can be used for previous sura.

Some commands need an integer argument, such as goto aya, goto the page before aya, goto sura and goto juz. If you type a number before using these commands, that number will be assumed to be the argument and Noor won't ask the number in a dialog.

Note that many of Noor commands can be done in more than one way and have more than one key-binding. See the noorgtk keys and try its commands.

Also you can tell noorgtk to start from a sura and aya using command line arguments. For instance noorgtk.py 100 3 goes to the third aya of the 100th sura.

You can edit the ~/.noor file:

[common]
translation = /path/to/translation/file

The translation file can be in two formats. It can be either a zip file in the same format as Zekr translation files which can be obtained from http://siahe.com/zekr/resources.html or it can be a utf-8 text file with translation of each aya in one line.

## Changing Font

You can change the GTK or html fonts in ~/.noor file:

[gtk]
font = ArabeyesQr 23
translation_font = irannastaliq size:14 fgcolor:#001100 fgcolor:#DDDDFF

[html]
font = quran_me 6 fgcolor:554444
translation_font = nazli 4 fgcolor:444455

A font has face, size, fgcolor and bgcolor attributes. These attributes can be set using attr:value format where attr is attribute name. The old format still works; that is myfont 16 is the same as myfont size:16 and also the same as face:myfont size:16.

## Changing The Language

You can change the language of noorgtk dialogs and noorhtmls pages; for instance:

[common]
lang = en

will use English. The value of common.lang config should be a two letter symbol of a language. Currently Arabic (ar), English (en), Farsi (fa) and Pashto (ps) are supported. If you like, you can translate noor to other languages (ask the mailing list if you need help).

Sura and aya hooks can be specified in ~/.noor file and are run when a new sura or aya is shown in noorgtk.

The first argument of a sura hook is the sura number. You can use noor's sura hook to start playing quran recitation when noorgtk shows a sura. Hook can be specified like:

[gtk]
sura_hook = my_sura_hook

Aya hook is executed when a new aya is shown (see key-binding section; next/previous/current aya). The first argument of an aya hook is sura number and the second one is aya number. An aya hook can be specified in gtk section of the config file:

[gtk]
aya_hook = my_aya_hook

See contrib/playsura and contrib/playaya scripts as examples.

## Hiding Noorgtk Parts

You can hide some parts of noor gtk. For instance:

[gtk]
status_bar = False
toolbar = False
scrollbar = False

will hide menu bar, status bar and toolbar.

## Noorgtk Keys

These keys can be used in the GTK interface. Note that C- stands for control key, M- for alt key, S- for shift key and C-M for control and alt key pressed together. For instance C-n means pressing n key while holding the control key.

Action Keys
move down down, j, C-n
move up up, k, C-p
next page page-down, space, C-v
previous page page-up, S-space, M-v
sura start home, C-a
sura end end, C-e
next sura C-M-n, C-j
choose sura c
previous sura C-M-p, C-k
next aya n
previous aya p
current aya .
append digit to number digits; [0-9]
clear number escape, C-g
goto n-th aya a, return, M-g
goto the page before n-th aya b, S-return, M-G
goto n-th sura s, C-return, C-M-g, s
goto n-th juz start J, C-M-return, C-M-G
back C-[, M-left
forward C-], M-right
fullscreen f, F11
search forward /, C-s, C-f
search backwards ?, C-r
quit q

Note n here means the number typed so far. Actually by pressing digits you prefix commands. For instance pressing 2 followed by 1 and followed by C-M-return, you tell noor to go to the start of the 21st juz.

If no prefix is typed for prefixed commands, noor will ask it in a dialog.

## Configuration File

You can specify configuration options in ~/.noor file. See the end of noor/uihelpers.py file for the default contents of ~/.noor file (the default ~/.noor will be created if it does not exist).

Note that html section of the config file is applied to both noorcgi and noorhtmls interfaces.

## Bookmarks

Noor supports aya bookmarks. When you add a bookmark, the location and its name is saved so that you can jump to that location from the list bookmarks dialog.

Noor saves bookmarks in a ~/.noor.bmk by default. You can change that by using gtk.bookmark_file option. Each line of this file can contain a bookmark like:

sura_number:aya_number bookmark_name

You can edit this file manually to change them.

This program is under the terms of GNU GPL (GNU General Public License). Have a look at COPYING file for more information.

## Source Repository

Noor uses Mercurial CMS:

## Contributing

Patches to noor's code are welcome.

Patch style:

• use four spaces for indentation.


text:
3 Tutor System
Starting just at 265/hour

# In figure, if PQ is perpendicular to PS , PQ || SR , $$\angle{SQR}$$ = $$28^\circ$$ and $$\angle{QRT}$$ = $$65^\circ$$ , then find the values of x and y.

Here, as PQ || SR, by alternate angles axiom, we get,
$$\angle{PQR}$$ = $$\angle{QRT}$$

It is given that,
$$\angle{PQS}$$ = x, $$\angle{SQR}$$ = $$28^\circ$$ and $$\angle{QRT}$$ = $$65^\circ$$
$$\therefore$$ $$\angle{PQS}$$ + $$\angle{SQR}$$ = $$\angle{QRT}$$
$$\Rightarrow$$ x + $$28^\circ$$ = $$65^\circ$$
$$\Rightarrow$$ x = $$37^\circ$$ ...(i)

Now, considering right angled triangle PQS,
$$\angle{SPQ}$$ = $$90^\circ$$
$$\angle{SPQ}$$ + x + y = $$180^\circ$$ ....(Since Sum of all angles of a triangle is equal to 180)

$$\therefore$$ $$90^\circ$$ + $$37^\circ$$ + y = $$180^\circ$$
$$\Rightarrow$$ y = $$180^\circ$$ - $$127^\circ$$
$$\Rightarrow$$ y = $$53^\circ$$


text:
# zbMATH — the first resource for mathematics

The topological classification of cascades on closed two-dimensional manifolds. (English. Russian original) Zbl 0705.58038
Russ. Math. Surv. 45, No. 1, 1-35 (1990); translation from Usp. Mat. Nauk 45, No. 1(271), 3-32 (1990).
This is a review paper on the resuls, mainly by the authors, of the theory of dynamical systems with discrete times (cascades) on two- dimensional manifolds (for the review of the respective results for the systems with continuous time (flows) see e.g. the same authors in Russ. Math. Surv. 41, No.1, 183-208 (1986); translation from Usp. Mat. Nauk 41, No.1, 149-169 (1986; Zbl 0615.58015). The authors discuss mainly the problem of the topological equivalence of A-cascades [see e.g. S. Smale, Bull. Am. Math. Soc. 73, 747-817 (1967; Zbl 0202.552) for definitions].
This problem consists of two parts: the description of the complete system of topological invariants of diffeomorphisms and the construction of the standard diffeomorphism in each class of the topologically equivalent ones. The former problem is discussed in the first chapter of the paper. The authors show that this problem can be reduced to the similar problem of basis sets of nonwandering sets of diffeomorphisms. The topological classification of a diffeomorphism restriction to the basis set is given in terms of the invariants of so-called supports of the basis set. As for the second part (the construction of the standard representative), authors give its solutions basing on the Nielsen theory of the homotopic classification of the homeomorphisms of the two- dimensional manifolds (for the exposition of the Nielsen theory see e.g. J. Gilman, Adv. Math. 40, 68-96 (1981; Zbl 0474.57005).
Reviewer: L.Pastur

##### MSC:
 37D99 Dynamical systems with hyperbolic behavior 37D15 Morse-Smale systems 54H20 Topological dynamics (MSC2010) 37C85 Dynamics induced by group actions other than $$\mathbb{Z}$$ and $$\mathbb{R}$$, and $$\mathbb{C}$$ 28D05 Measure-preserving transformations
Full Text: