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# Soft filtering edge of tilebase Terrain

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Hi , I'm learning about tile base map with elevation(height).
I succeed in render the map with diamonded tile, or using Ortho-isometric View to make square tile become diamonded (There are 2 methods, the first is mostly use 2d tile texture)!

My current test is like this (2nd Method) (The lighting is a bit wrong, but the biggest problem is the edge) :

So I move to the next level with height. I referenced this, and so wonder how can they do it! what is the technique!

In the old game, I think they dont use texture like this, but draw everything in 2D and add the light shading that make it looked 3d. In this HD version, i think they move all the terrain to full HD, so we can see the wireframe. So what I really want to know is how to achieve this, or what is the technique behind this achievement. I assume that they try to interpolate 1 poly into many small poly for smoothment, but that will reduce the performance, and anyhow i dont know which algrorith let us do this interpolation. Thanks all. Edited by greenpig83

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It doesn't look like there is any LOD here. It's probably just one vertex per point in the height map. Then they triangulate each square along the line between the two vertices that differ least in height.

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LOD you mean Level of Detail. Havent work with LOD before, so I will spend some time research it. So now you mention HeightMap, and I saw many ppl use it for terrain. I will try to make a demo with heightmap. I hope I can get the result I want. I'll post the result right after I finish it . Thanks .

FurtherMore, in the note of the game Development Diary. They say the use "Soft filtering and noise for terrain lighting ', but I don't really understand what it means. This is a specific technique ?

Edited by greenpig83

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I only mentioned LOD because you said "I assume that they try to interpolate 1 poly into many small poly for smoothment". For this type of overhead view, you don't need to bother with LOD, since the terrain is all roughly the same distance from the camera.

"Soft filtering and noise for terrain lighting" -> I don't know what this is either. It just sounds like a bunch of buzzwords put in a sentence. It's difficult to say without more context. Do you have a link? Edited by phil_t

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I only mentioned LOD because you said "I assume that they try to interpolate 1 poly into many small poly for smoothment". For this type of overhead view, you don't need to bother with LOD, since the terrain is all roughly the same distance from the camera.

"Soft filtering and noise for terrain lighting" -> I don't know what this is either. It just sounds like a bunch of buzzwords put in a sentence. It's difficult to say without more context. Do you have a link?

Yes , here is the link. Thanks for the answer.  http://www.ageofempires.com/age2.html  (search for noise)

I'm working with the heightMap solution now. Divide each tile into smaller one. assign the height of it.  Render all of them together.

Trying use this : http://www.nfostergames.com/lessons/TerrainSmoothing.htm  for smoothing. But the result is quite different from what I expected. It only smooth surround the elevated tile. Not the tiles surround it.

Maybe I understand the algorith wrong, or fail to apply!

Edited by greenpig83

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The noise they are referring to is probably a perlin noise texture that is used to determine which of two textures to sample from in the pixel shader. If you look at the grass closely it looks "randomly patchy". So it's likely two textures, and a third texture (the perlin noise) is used to determine how much from each of the two source textures is used in the final output. This is usually referred to as multitexturing.

The soft filtering, I'm guessing, refers to the smooth transition from grass to dirt (or whatever) along the tile edges. This is yet another level of multi-texturing, probably using another "mask" texture that has different "shapes" in it for the different neighbouring patterns of tiles (instead of the noise texture).

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Trying use this : http://www.nfostergames.com/lessons/TerrainSmoothing.htm  for smoothing. But the result is quite different from what I expected. It only smooth surround the elevated tile. Not the tiles surround it.

It looks like you need to do multiple passes in order to get the smoothing to "spread out" farther.

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Here the result that I work with HeightMap (not really HeightMap, i dont use any gray image for height or anything. But I think it's the same in idea, cause I have a NxM height map )

Searching all day for a good example of B-Sline surface, Bezier surface, CoxDeBoor algorithm, NURBS... Hard to find an an easy example work for this case. Most of them the surface not going through the control point. In this case I want my surface to go through the Points. So I decide to go the no-brain way, draw a curves n*(n*n) curve throught points in the map. And it does work as I expected. However with those triangle, I think it will hit the performance a lot (The demo now run at 44 fps, ofcourse there is no optimize at the moment, but my Comp can run Crysis 3 at 60fps ) . Because I will have to render hundred of tile on screen.

For the curve I use D3DXVec3CatmullRom to create a curve go through 4 control points.

Edited by greenpig83

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The noise they are referring to is probably a perlin noise texture that is used to determine which of two textures to sample from in the pixel shader. If you look at the grass closely it looks "randomly patchy". So it's likely two textures, and a third texture (the perlin noise) is used to determine how much from each of the two source textures is used in the final output. This is usually referred to as multitexturing.

The soft filtering, I'm guessing, refers to the smooth transition from grass to dirt (or whatever) along the tile edges. This is yet another level of multi-texturing, probably using another "mask" texture that has different "shapes" in it for the different neighbouring patterns of tiles (instead of the noise texture).

I know about perlin noise, it used to create natural thing like cloudy, grass, fire... even generate random Map. So i think it has nothing to do with smooth.

About the smooth transitions between tiles, It's about blending , and there are tons of artile about it, so I thinks it's not very hard. After working with smooth edge, I'll move to the blending issue. I think it will be alot easier than softing this edge.

Anyway, I'm still not sure this is the best approach, I think there will be a better (in performance, or visual quality). So if any one can enlight me, I will really appreciate it. I will post anything that I found here.

Edited by greenpig83

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How do you calculate the normals for your quads? Just inside the quad or with corresponding neighbours? Your first screen looks like first option.

When you calculate a normal for one heightmap point by all 4 triangles, which use the same point, you should get a smooth shading.

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# Newtons Third Law. How is jumping from a pier different to jumping from a boat?

This is the absolute basic of a physics and yet 2 hours of googling fails to find an answer ! So ignoring all vertical movement and just concentrating on the horizontal movement:-

1. A man who weighs 75 kg jumps off a pier (steps off horizontally) with a force of 150N. Ignoring gravity he accelerates by 2 meters per sec per sec.

2. A man who weighs 75kg jumps off (steps off horizontally) a boat that weighs 75 kg with a force of 150N. Ignoring gravity he accelerates by 2 meters per sec per sec. So does the boat (it accelerates by same amount), but in the opposite direction.

3. What is wrong here? Im sure the boat and the man would end up at half the speed in each direction compared to example 1. Example 2 sounds like free energy compared to example one. What am I missing, apart from a working version of google, and an easy way to format this text?

4. p.s. I think one of my key confusions is in example 2, is the acceleration relative to a point in space or is the acceleration relative to the other body. I.e. do they accelerate away from each other at 2m per sec^2 or 4m per sec^2? I guess my other confusion is if they are accelerating away at twice the rate then I appear to be doing twice the amount of work for the same amount of force applied which seems odd, and feels like free energy.

• What is the question? Jun 12, 2020 at 10:13
• acceleration (force) is not a conserved quantity. Conservation balances should be done with energy and/or momentum Jun 12, 2020 at 10:55
• Force and acceleration are instantaneous quantities. Knowing only the applied force and the resulting acceleration at some particular instant in time will not tell you the final speed of the man or the boat. Here's a clue: The man's legs can only extend so far. What are the implications of that limited extension when the man jumps off the boat, which moves backward because of the applied force, vs. the case where he jumps off of the unyielding pier? Jun 12, 2020 at 11:15
• So in both my examples I am pretty sure the mans legs would be in contact and accelerating for the same length of time, I am pretty sure the force applied would be the same, the impulse the same, the energy used the same and the total momentum the same and the work done the same. So basic questions are is he actually still accelerating at the same rate in example 2? Is that relative to the same point in space, or is it that his acceleration in example 2 is actually an acceleration relative to the boat rather than a general accel in a direction, which is what is shown in all the diagrams ? Jun 12, 2020 at 12:39
• Actually Im not sure if the momentum would be the same, so there is another question would it be the same between the 2 examples ? How would you calculate the end speeds if in both cases if say the force lasted for 1 second ? Ill start you off with the first one it would be pier 0 m/s and man 2 m/s. But what about the second example of man and boat going in opposite directions ? Is it that the length of time available for acceleration is less for some reason ? Jun 12, 2020 at 12:59

Good question, though you have stated it a little imprecisely I think. So first I will state the question more precisely.

Scenario 1. Man leaps from pier, such that the force between himself and the ground, in the horizontal direction, reaches 150 N (for some length of time to be discussed).

Scenario 2. Man leaps from a boat having the same mass as the man, such that the force between himself and the boat, in the horizontal direction, reaches 150 N (for some length of time to be discussed).

You are correct to suppose that in both cases the acceleration of the man, relative to some convenient frame of reference such as planet Earth, reaches the same value in the two scenarios at the moment when the force reaches 150 N. But in the second scenario the man will find it harder to get the force to reach that value, because the boat is being pushed away as his legs get straighter. He could do it, but he would find that he had to expend more energy.

To find the final velocity of the man in the two cases, you can use either energy or momentum. In terms of momentum, what you need to know is the length of time for which any give force is applied. Leaping from a pier, a man can get to some given force pretty quickly, before his body has moved very much, and then maintain that force for, let's say, half a second. So he gets the momentum approximately 75 kg m/s. Leaping from a boat, the force will rise from zero more slowly, so will take longer to get to 150 N. So he will only manage to apply 150 N for some shorter time. His final momentum will therefore be lower than in first case.

Next let's think about energy. Now we have to consider the distance over which the centre of mass of the man moved while the force applied to the centre of mass was 150 N. Leaping from the pier, this distance is approximately the distance by which his legs extended from the crouch to the leap. Leaping from the boat, since the boat moves away, his centre of mass moves by about the half the distance of the first case. Therefore the energy delivered to the man in the second case is about half that of the first case. On this estimate the man does the same work in the two cases, but when leaping from the boat half of the work goes to kinetic energy of the boat, and half to the man, whereas in the first case the man gets all the kinetic energy (the change in motion of the pier being negligible).

• I was kind of thinking in terms of the horizontal acceleration and ignoring all vertical movement ! So like stepping off a pier or a boat. I think the force applied would be pretty instantaneous. I think from what others have said the relative acceleration between the 2 bodies in the second case (stepping off a boat) would be twice that of the first case, so it shines twice as bright for half as long. I.e. in my second example I get twice the accel but for only half the length of time. So the legs would be in contact applying the same force but accelerating twice as quickly for half as long. Jun 13, 2020 at 12:10

To avoid getting sidetracked into questions of how muscles work, let's replace the man with a sack of potatoes with a mass of $$75$$ kg and let's suppose the force is provided by a compressed spring which produces an initial force of $$150$$ N when it is released.

If the sack of potatoes is on a pier (or on the ground) then the spring exerts a force of $$150$$ N on the sack and (by Newton's third law) on the pier. The man has a mass of $$75$$ kg and so initially accelerates at $$2$$ metres per second per second. The pier has (effectively) the mass of the whole Earth, so its acceleration is negligible. If the sack travels a small distance $$d$$ (small enough so that the force exerted by the spring does not change) the work done by the spring is $$150d$$ Joules. This energy was originally potentially energy in the compressed spring, and becomes the kinetic energy of the sack. Note that the pier does not move (or, to be precise, its movement is negligible) so the spring does no work on the pier, and the kinetic energy of the pier is negligible.

If the sack of potatoes is on a boat (or on a trolley) which has a mass of $$75$$ kg then the spring exerts a force of $$150$$ N on the sack and on the boat. Again, the man has a mass of $$75$$ kg and so initially accelerates at $$2$$ metres per second per second. But this time the boat has a mass of $$75$$ kg so its acceleration is $$2$$ metres per second per second in the opposite direction. This time, when the sack travels a small distance $$d$$ (small enough so that the force exerted by the spring does not change) then the boat also travels the same distance $$d$$ in the opposite direction, so the total work done by the spring is now $$300d$$ Joules. So both the sack and the boat accelerate and gain kinetic energy, but the work done by the spring is twice what it was in the first case.

• is acceleration a conserved quantity? Jun 12, 2020 at 10:56
• Isnt this just bolstering the free energy theory ? I would imagine a spring would have a set amount of energy in it ? I know in the case of the spring or a person leaping the acceleration will only be being applied as long as things are making contact. Im still kind of struggling getting it all to add up though ! Jun 12, 2020 at 12:24
• no because in case 1 the other 150d went to earth. but dissipated Jun 12, 2020 at 13:03
• @anna v No, the accelerations of the sack of potatoes and the boat only have the same magnitude in the second example because they have the same mass. If the mass of boat were $150$ kg then its acceleration would only be $1$ metre per second per second. Jun 12, 2020 at 13:54
• @gandalf61 thenks, I did not read it carefully thinking it copied the question. Your example is confusing I would think, as ot is not stating this last clearly. Jun 12, 2020 at 13:57

As @anna_v has addressed several times in the comments, your confusion is caused by acceleration not being a conserved quantity. That it, the sum of accelerations is not necessary the same at different times. The total momentum and energy is, however, conserved at all times. As a small exercise, it might be fruitful for you to compute the total momentum of the system and check that it vanishes at all times.

• I still dont know which bit is wrong though ! Is it that in my second example the 2m /s^2 acceleration of the person is relative to the boat and is not an acceleration in a direction relative to a point ? Is it actually the case that the person and boat accelerate away from each other at 4m /s^2 or 2m/s^2 (Assuming the same force is applied by the person in both examples)?You say total momentum is conserved. I would have thought he would have more momentum when he jumped off the pier than when he was stationary on it ? Jun 12, 2020 at 15:54
• @MajorTom In the second example the boat and the person are accelerating at $2$ m/s^2 in opposite directions, so, yes, the acceleration of the person relative to the boat is $4$ m/s^2. The total momentum of the boat and the person is always $0$ - remember that momentum, like velocity, is a vector quantity, so the boat's momentum plus the person's momentum nets to $0$. This is also true in the first example - the earth's momentum plus the person's momentum nets to $0$ - the earth has a very large mass, and so ends up with a very small velocity. Jun 12, 2020 at 17:02
• You are exactly correct @MajorTom in point 1 and 2 of your original question. However, there is no contradiction between these two statements. Regardless whether you jump off a pier or a boat you gain the same final velocity. The object you jump from does not matter. The only thing that matters is the force acting on you. Newtons equation $F=ma$ only involves quantities associated with you. $F$ is the force acting on your, $m$ is your mass and $a$ is your acceleration. There is no pier/boat in this equation. I hope this clarifies your question :) Jun 12, 2020 at 18:32
• Something is still wrong there because if do have the same acceleration in both, then the relative acceleration between the 2 items will be twice as big in the second example, so the length of time I am accelerating for would be reduced (probably halved in this case?). Assuming your legs only stretch the same amount in both examples ! So the end velocities would probably be different. This would probably make the total momentum between the 2 different too as I think that proportional to v^2. Jun 12, 2020 at 22:47

In the first case the man does a force in the pier and the pier does a force in the man. Both of 150 N. Of course only during the leg's impulse.

In the second case, the man does a force in the boat and the boat does a force in the man. Both of 150 N.

No difference concerning the third law.

And third law is conservation of momentum: $$F_{12} = -F_{21} => \frac{\partial \mathbf p_2}{\partial t} = -\frac{\partial \mathbf p_1}{\partial t}$$

About conservation of momentum, the second case is easy because the objects have the same mass. In the first one, the pier is attached to the Earth, so all the Earth is accelerated backwards by the force. But as its mass is so big, the acceleration is negligible.

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## Global shifted potentials for moduli spaces of sheaves on CY4

Publikation: Working paper/Preprint Working paperForskning

### Dokumenter

• 1908.00651v1

Forlagets udgivne version, 364 KB, PDF-dokument

• Dennis Borisov, Harvard University
• ,
• Artan Sheshmani, Higher School of Economics, Harvard University
• ,
• Shing-Tung Yau, Harvard University, USA
It is shown that any derived scheme over $\mathbb C$ equipped with a $-2$-shifted symplectic structure, and having a Hausdorff space of classical points, admits a globally defined Lagrangian distribution as a dg $\mathbb{C}^{\infty}$-manifold. The main tool for proving this theorem is a strictification result for Lagrangian distribution. It is shown that existence of a global Lagrangian distribution allows us to realize the moduli space of sheaves on Calabi-Yau fourfolds as a derived critical locus of a potential of degree $-1$ on the moduli space of $Spin(7)$ instantons.
Originalsprog Engelsk ArXiv Udgivet - 1 aug. 2019

### Forskningsområder

• math.AG, hep-th, math.DG

Citationsformater

ID: 176974216


text:
On exponential sums with sparse polynomials and rational functions.(English)Zbl 0881.11081

Let $$f(x)= a_1x^{r_1}+\cdots+ a_tx^{r_t}$$, where $$r_1,\dots, r_t$$ are some pairwise distinct nonzero integers, be a sparse polynomial with integral coefficients. Suppose that $S(f,q)=\sum^q_{\substack{ x=1\\ (x,q)=1}} \exp[2\pi if(x)/q],$ where $$q$$ is an integer and $$(a_1,\dots, a_t,q)=1$$, and $$T_m(f)= \sum_{x\in\mathbb{F}_{2^m}} \chi(f(x))$$, where $$\chi$$ is a nontrivial additive character of $$\mathbb{F}_{2^m}$$. In the present paper, the author proves the following: $S(f,q)= O(q^{1-(1/t)+ \varepsilon}) \qquad\text{and}\qquad T_m(f)\leq r2^{m/2}.$

MSC:

 11T23 Exponential sums 11T24 Other character sums and Gauss sums 11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
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# What minerals are found in north korea

###### Question:

what minerals are found in north korea

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# Can electrolyzing ice cause an explosion?

I know electrolyzing water separates it into hydrogen and oxygen. However, I am curious, if ice is electrolyzed, could it explode? My thought process is that oxygen and hydrogen bubbles would build up in the ice, and continued electrolysis would ignite the gasses (forming water again). However, since it is contained within the ice, it could ignite explosively. Is this what would indeed happen? One way to try this might be using a tazer or bug zapper on ice. I believe this could be dangerous, and appropriate precautions would need to be taken.

Bulk electrolysis of ice would likely not get very far. Electrolysis works in liquids because the liquid is able to transport an ion to complete the circuit. In the case of water electrolysis, water is needed to transport protons (or hydroxide, but we'll stick to just one thing).

The half reactions for electrolysis of water are

Oxidation at anode:

$$\ce{2 H2O (l) -> O2(g) + 2H+(aq) + 2e-}$$

The electrons travel through the wires and the protons through solution to the cathode where the following happens:

Reduction at cathode:

$$\ce{2H+(aq) + 2e- -> H2(g)}$$

Electrolysis requires ion mobility. In solid water this mobility is severely decreased. All water molecules are locked in a rigid hydrogen bond network. A proton can propagate slowly through this network by proton exchange, but it will be slow. Thus electrolysis will be slowed.

Also, electrolysis only occurs at the surface of the electrodes, which must provide a means for the gas to escape (whether they are on the surface of the ice or driven into the ice, gaps will exist for gas to escape).

• You're also forgetting the very low autoionisation of water 10^-7 meaning pure water electrolyses slowly and since freezing eliminates impurities from the lattice ice tends to be very pure indeed – MY2K Jul 14 '14 at 4:08


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##### #1. Applications of mesoscopic CLTs in random matrix theory
###### Benjamin Landon, Philippe Sosoe
We present some applications of central limit theorems on mesoscopic scales for random matrices. When combined with the recent theory of "homogenization" for Dyson Brownian Motion, this yields the universality of quantities which depend on the behavior of single eigenvalues of Wigner matrices and $\beta$-ensembles. Among the results we obtain are the Gaussian fluctuations of single eigenvalues for Wigner matrices (without an assumption of 4 matching moments) and classical $\beta$-ensembles ($\beta=1, 2, 4$), Gaussian fluctuations of the eigenvalue counting function, and an asymptotic expansion up to order $o(N^{-1})$ for the expected value of eigenvalues in the bulk of the spectrum. The latter result solves a conjecture of Tao and Vu.
more | pdf | html
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MathPaper: Applications of mesoscopic CLTs in random matrix theory. https://t.co/9QJEd3yvLK
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##### #2. Scaling limits in divisible sandpiles: a Fourier multiplier approach
###### Alessandra Cipriani, Jan de Graaff, Wioletta M. Ruszel
In this paper we complete the investigation of scaling limits of the odometer in divisible sandpiles on $d$-dimensional tori generalising the works Chiarini et al. (2018), Cipriani et al. (2017, 2018). Relaxing the assumption of independence of the weights of the divisible sandpile, we generate generalised Gaussian fields in the limit by specifying the Fourier multiplier of their covariance kernel. In particular, using a Fourier multiplier approach, we can recover fractional Gaussian fields of the form $(-\Delta)^{-(1+s)} W$ for $s>0$ and $W$ a spatial white noise on the $d$-dimensional unit torus.
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###### Tweets
MathPHYPapers: Scaling limits in divisible sandpiles: a Fourier multiplier approach. https://t.co/i0ARLBn4IV
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##### #3. The algorithmic hardness threshold for continuous random energy models
###### Louigi Addario-Berry, Pascal Maillard
We prove an algorithmic hardness result for finding low-energy states in the so-called \emph{continuous random energy model (CREM)}, introduced by Bovier and Kurkova in 2004 as an extension of Derrida's \emph{generalized random energy model}. The CREM is a model of a random energy landscape $(X_v)_{v \in \{0,1\}^N}$ on the discrete hypercube with built-in hierarchical structure, and can be regarded as a toy model for strongly correlated random energy landscapes such as the family of $p$-spin models including the Sherrington--Kirkpatrick model. The CREM is parameterized by an increasing function $A:[0,1]\to[0,1]$, which encodes the correlations between states. We exhibit an \emph{algorithmic hardness threshold} $x_*$, which is explicit in terms of $A$. More precisely, we obtain two results: First, we show that a renormalization procedure combined with a greedy search yields for any $\varepsilon > 0$ a linear-time algorithm which finds states $v \in \{0,1\}^N$ with $X_v \ge (x_*-\varepsilon) N$. Second, we show that the value...
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##### #4. Long term behaviour of a reversible system of interacting random walks
###### Svante Janson, Vadim Shcherbakov, Stanislav Volkov
This paper concerns the long-term behaviour of a system of interacting random walks labeled by vertices of a finite graph. The model is reversible which allows to use the method of electric networks in the study. In addition, examples of alternative proofs not requiring reversibility are provided.
more | pdf | html
None.
###### Tweets
MathPaper: Long term behaviour of a reversible system of interacting random walks. https://t.co/pduHTPUSHA
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Authors: 3
Total Words: 12779
Unqiue Words: 2347

##### #5. Mixing Times and Hitting Times for General Markov Processes
###### Robert M. Anderson, Haosui Duanmu, Aaron Smith
The hitting and mixing times are two fundamental quantities associated with Markov chains. In Peres and Sousi[PS2015] and Oliveira[Oli2012], the authors show that the mixing times and "worst-case" hitting times of reversible Markov chains on finite state spaces are equal up to some universal multiplicative constant. We use tools from nonstandard analysis to extend this result to reversible Markov chains on general state spaces that satisfy the strong Feller property. Finally, we show that this asymptotic equivalence can be used to find bounds on the mixing times of a large class of Markov chains used in MCMC, such as typical Gibbs samplers and Metroplis-Hastings chains, even though they usually do not satisfy the strong Feller property.
more | pdf | html
None.
###### Tweets
roydanroy: Check out Haosui "Kevin" Duanmu's latest application of nonstandard analysis, this time to mixing and hitting times in general Markov processes. https://t.co/JCJc9JFXtA #teamroy https://t.co/ehBeCdyizy https://t.co/B0b6ckMhjZ
MathPaper: Mixing Times and Hitting Times for General Markov Processes. https://t.co/8UopepC0lu
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Sample Sizes : None.
Authors: 3
Total Words: 24921
Unqiue Words: 3593

##### #6. Simultaneous Small Noise Limit for Singularly Perturbed Slow-Fast Coupled Diffusions
We consider a simultaneous small noise limit for a singularly perturbed coupled diffusion described by \begin{eqnarray*} dX^{\varepsilon}_t &=& b(X^{\varepsilon}_t, Y^{\varepsilon}_t)dt + \varepsilon^{\alpha}dB_t, dY^{\varepsilon}_t &=& - \frac{1}{\varepsilon} \nabla_yU(X^{\varepsilon}_t, Y^{\varepsilon}_t)dt + \frac{s(\varepsilon)}{\sqrt{\varepsilon}} dW_t, \end{eqnarray*} where $B_t, W_t$ are independent Brownian motions on ${\mathbb R}^d$ and ${\mathbb R}^m$ respectively, $b : \mathbb{R}^d \times \mathbb{R}^m \rightarrow \mathbb{R}^d$, $U : \mathbb{R}^d \times \mathbb{R}^m \rightarrow \mathbb{R}$ and $s :(0,\infty) \rightarrow (0,\infty)$. We impose regularity assumptions on $b$, $U$ and let $0 < \alpha < 1.$ When $s(\varepsilon)$ goes to zero slower than a prescribed rate as $\varepsilon \rightarrow 0$, we characterize all weak limit points of $X^{\varepsilon}$, as $\varepsilon \rightarrow 0$, as solutions to a differential equation driven by a measurable vector field. Under an additional assumption on the behaviour of $U(x,... more | pdf | html ###### Figures None. ###### Tweets mathPRb: Siva R. Athreya, Vivek S. Borkar, K. Suresh Kumar, Rajesh Sundaresan : Simultaneous Small Noise Limit for Singularly Perturbed Slow-Fast Coupled Diffusions https://t.co/dH9XcUe1Ms MathPaper: Simultaneous Small Noise Limit for Singularly Perturbed Slow-Fast Coupled Diffusions. https://t.co/JFjcP3e1gs ###### Github None. ###### Youtube None. ###### Other stats Sample Sizes : None. Authors: 4 Total Words: 18094 Unqiue Words: 3220 ###### 0.0 Mikeys ##### #7. Busemann functions and Gibbs measures in directed polymer models on$\mathbb{Z}^2$###### Christopher Janjigian, Firas Rassoul-Agha We consider random walk in a space-time random potential, also known as directed random polymer measures, on the planer square lattice with nearest-neighbor steps and general i.i.d. weights on the vertices. We construct covariant cocycles and use them to prove new results on existence, uniqueness/non-uniqueness, and asymptotic directions of semi-infinite polymer measures (solutions to the Dobrushin-Lanford-Ruelle equations). We also prove non-existence of covariant or deterministically directed bi-infinite polymer measures. Along the way, we prove almost sure existence of Busemann function limits in directions where the limiting free energy has some regularity. more | pdf | html ###### Figures None. ###### Tweets mathPRb: Christopher Janjigian, Firas Rassoul-Agha : Busemann functions and Gibbs measures in directed polymer models on$\mathbb{Z}^2$https://t.co/3iZfvxQWQl MathPaper: Busemann functions and Gibbs measures in directed polymer models on$\mathbb{Z}^2$. https://t.co/3ysPUbkGfb gabrielanthonyp: RT @MathPaper: Busemann functions and Gibbs measures in directed polymer models on$\mathbb{Z}^2$. https://t.co/3ysPUbkGfb ###### Github None. ###### Youtube None. ###### Other stats Sample Sizes : None. Authors: 2 Total Words: 26949 Unqiue Words: 3894 ###### 0.0 Mikeys ##### #8. On a gateway between continuous and discrete Bessel and Laguerre processes ###### Laurent Miclo, Pierre Patie By providing instances of approximation of linear diffusions by birth-death processes, Feller [13], has offered an original path from the discrete world to the continuous one. In this paper, by identifying an intertwining relationship between squared Bessel processes and some linear birth-death processes, we show that this connection is in fact more intimate and goes in the two directions. As by-products, we identify some properties enjoyed by the birth-death family that are inherited from squared Bessel processes. For instance, these include a discrete self-similarity property and a discrete analogue of the beta-gamma algebra. We proceed by explaining that the same gateway identity also holds for the corresponding ergodic Laguerre semi-groups. It follows again that the continuous and discrete versions are more closely related than thought before, and this enables to pass information from one semi-group to the other one. more | pdf | html ###### Figures None. ###### Tweets ###### Github None. ###### Youtube None. ###### Other stats Sample Sizes : [0, 0] Authors: 2 Total Words: 17189 Unqiue Words: 2744 ###### 0.0 Mikeys ##### #9. The stochastic geometry of unconstrained one-bit data compression ###### François Baccelli, Eliza O'Reilly A stationary stochastic geometric model is proposed for analyzing the data compression method used in one-bit compressed sensing. The data set is an unconstrained stationary set, for instance all of$\mathbb{R}^n$or a stationary Poisson point process in$\mathbb{R}^n$. It is compressed using a stationary and isotropic Poisson hyperplane tessellation, assumed independent of the data. That is, each data point is compressed using one bit with respect to each hyperplane, which is the side of the hyperplane it lies on. This model allows one to determine how the intensity of the hyperplanes must scale with the dimension$n$to ensure sufficient separation of different data by the hyperplanes as well as sufficient proximity of the data compressed together. The results have direct implications in compressive sensing and in source coding. more | pdf | html ###### Figures None. ###### Tweets MathPaper: The stochastic geometry of unconstrained one-bit data compression. https://t.co/N4kd6punFk ###### Github None. ###### Youtube None. ###### Other stats Sample Sizes : None. Authors: 2 Total Words: 13024 Unqiue Words: 2325 ###### 0.0 Mikeys ##### #10. Strong Feller property for SDEs driven by multiplicative cylindrical stable noise ###### Tadeusz Kulczycki, Michał Ryznar, Paweł Sztonyk We consider the stochastic differential equation$dX_t = A(X_{t-}) \, dZ_t$,$ X_0 = x$, driven by cylindrical$\alpha$-stable process$Z_t$in$R^d$, where$\alpha \in (0,1)$and$d \ge 2$. We assume that the determinant of$A(x) = (a_{ij}(x))$is bounded away from zero, and$a_{ij}(x)$are bounded and Lipschitz continuous. We show that for any fixed$\gamma \in (0,\alpha)$the semigroup$P_t$of the process$X_t$satisfies$|P_t f(x) - P_t f(y)| \le c t^{-\gamma/\alpha} |x - y|^{\gamma} ||f||_\infty$for arbitrary bounded Borel function$f\$. Our approach is based on Levi's method.
more | pdf | html
None.
###### Tweets
MathPaper: Strong Feller property for SDEs driven by multiplicative cylindrical stable noise. https://t.co/Ehur8UYA7m
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None.
###### Other stats
Sample Sizes : None.
Authors: 3
Total Words: 20599
Unqiue Words: 2802

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text:
### Home > CALC > Chapter 3 > Lesson 3.3.2 > Problem3-102

3-102.

Find $f ^\prime(x)$ for each of the following functions.

1. $f ( x ) = \frac { 2 } { 3 } ( x - 5 ) ^ { 3 } + x ^ { 2 }$

$f '\left(x\right) = 2\left(x − 5\right)^{2} + 2x$

1. $f ( x ) = \sqrt [ 3 ] { x }$

Before you find the derivative, rename $f(x)$ using laws of exponent.

1. $f ( x ) = \operatorname { sin } \frac { \pi } { 4 }$

Careful. What is $\operatorname{sin}\frac{\pi }{4}$?
Describe the graph of $y=\text{ sin}\frac{\pi }{4}$.
Describe the behavior of its slope.

1. $f ( x ) = \frac { 1 } { ( x + 1 ) ^ { 2 } }$

Refer the the hint in part (b).


text:
# Trigonometry inequalities

1. Apr 26, 2008

### Physicsissuef

1. The problem statement, all variables and given/known data
Find the values for x.

$$3sin^2x - 3sinxcosx + 2cos^2x > 1$$

2. Relevant equations

3. The attempt at a solution

$$3sin^2x - 3sinxcosx + 2cos^2x > 1$$

$$3sin^2x - 3sinxcosx + 2cos^2x > sin^2x+cos^2x$$

$$2sin^2x - 3sinxcosx + cos^2x > 0$$

What to do next?

2. Apr 26, 2008

### Kurret

You can use the same formula sin^2+cos^2=1 again. Now rewrite the equation in terms of only one trigonometric function.

3. Apr 26, 2008

### Tom Mattson

Staff Emeritus
I think it would be easier to just factor it the way it is.

4. Apr 26, 2008

### Physicsissuef

What to do now?
If I try with $sin^2+cos^2=1[/tex] I would get: $$sin^2x-3sinxcosx>-1$$ Tom Mattson: $$(sinx-cosx)(2sinx-cosx)>0$$ I am wondering why in my book says, I am giving quote: Hm... 5. Apr 26, 2008 ### Tom Mattson Staff Emeritus Your book probably took a different approach. Doing it my way, the two cases are: 1.) [itex]\sin(x)-\cos(x)>0$ and $2\sin(x)-\cos(x)>0$, or

2.) $\sin(x)-\cos(x)<0$ and $2\sin(x)-\cos(x)<0$.

6. Apr 26, 2008

### Physicsissuef

But they have 3 solutions for x.
The first: $$x=\frac{\pi}{2}+k\pi$$
The second: $$tgx<\frac{1}{2}$$
The third: $$tgx>1$$

In your case we have only 2:
$$tgx<\frac{1}{2}$$
$$tgx>1$$

7. Apr 26, 2008

### Tom Mattson

Staff Emeritus
No, Physicsissuef. When you rearranged the inequalities, you divided by $\cos(x)$. That means you obtained results that are valid only for $\cos(x)\neq 0$. To find out what happens when $\cos(x)=0$, you have to examine those points separately.

So, for which $x$ is $\cos(x)=0$, and do the inequalities hold for all such $x$?

8. Apr 26, 2008

### Physicsissuef

But both points for $cosx=0$ and $\cos(x)\neq 0$ are valid.
For pi/2 and -pi/2 cosx=0

9. Apr 26, 2008

### Tom Mattson

Staff Emeritus
That's exactly what I just told you. When you divided by $\cos(x)$, you obtained results that are valid only for $\cos(x)\neq 0$. You then have to go back and find out if the inequalities hold for all $x$ such that $\cos(x)=0$.

10. Apr 26, 2008

### Physicsissuef

but why not sin(x)=0, that's my question, or why not sin(x)=1 or cos(x)=1, etc.etc, how will I know which of all of this values to choose?

11. Apr 26, 2008

### Tom Mattson

Staff Emeritus
You didn't divide by $\sin(x)$, did you?

Question: What did you divide by?

Answer: $\cos(x)$.

Question: So what are the only values of $x$ for which that division is not defined?

Answer: The values of $x$ for which $\cos(x)$, the thing you divided by, is zero.

12. Apr 26, 2008

### Physicsissuef

what means "the division is not defined"?
Do you mean $\frac{sinx}{cosx}=tgx$, so for cosx=0, there will be infinity, but if it is only cosx=0, there will be no infinity right?

13. Apr 27, 2008

### tiny-tim

Hi Physicsissuef!

Solutions 2 and 3 are in terms of tanx, and technically tanx is undefined at multiples of π/2.

For example, just below π/2, it's huge and positive, but just above π/2 it's huge and negative. So at π/2 it's both-infinity-and-minus-infinity … which doesn't help with tanx > 1, does it?

So they had to add Solution 1, just to cover all possibliities!


text:
Current Series

[View Past Series]

Wed Sep 27

PDE Seminar

3:35pm - Vincent Hall 570
The Calderón problem for variable coefficients nonlocal elliptic operators
Lin Yi-Hsuan

Abstract: We introduce an inverse problem of a Schrödinger type
variable nonlocal elliptic operator (-?. (A(x)?)) s + q, for 0 < s < 1.
We determine the unknown bounded potential q from the exterior partial measurements associated with the nonlocal Dirichlet-to-Neumann map for any space dimension n = 2. Our results generalize the recent initiative of introducing and solving inverse problem for fractional Schrödinger operator. We also prove some regularity results of the
direct problem corresponding to the variable coefficients fractional differential operator and the associated degenerate elliptic operator.

Wed Oct 04

PDE Seminar

3:35pm - Vincent Hall 570
Regularity and asymptotic of Keller-Segel models coupled to fluid equations
Kyungkeun Kang, Yonsei University

We consider chemotaxis equations coupled to the Navier-Stokes
equations, which is a mathematical model describing the dynamics of oxygen,
swimming bacteria (Bacillus subtilis), and viscous incompressible fluids. It is not
known for parameters of general chemotactic sensitivity and consumption rate in
two and three dimensions whether or not regular solutions with sufficiently
smooth initial data exist globally in time or develop a singularity in a finite time.
We discuss existence of regular solutions and asymptotic as well as temporal
decays of solutions, under a certain type of conditions of parameters and initial
data, as time tends to infinity.

Wed Oct 18

PDE Seminar

3:35pm - Vincent Hall 570
Stochastic Models for Turbulent Convection
Nathan Glatt-Holtz, Tulane University

Buoyancy driven convection plays a fundamental role in diverse physical settings: from cloud formation to large scale oceanic and atmospheric circulation processes to the internal dynamics of planets and stars. Typically, such fluid systems are driven by heat fluxes acting both through boundaries (i.e. heating from below) and in the bulk (i.e. internal volumetric' heating sources) both of which can have an essentially stochastic nature. In this talk I will discuss some recent mathematical developments concerning ergodicity, singular parameter limits and the onset of instability in the stochastic Boussinesq and Magnetohydrodynamics equations.

Wed Oct 25

PDE Seminar

3:35pm - Vincent Hall 570
Minimal initial data for potential Navier-Stokes singularities in a half-space
Tuan Pham, University of Minnesota

I will talk about the existence of potential minimal blow-up data for the Navier-Stokes equations in the half space. The problem in the whole space has been studied by Rusin - Sverak (2011), Gallagher - Koch - Planchon (2013) and Jia - Sverak (2013). The regularities of weak solutions near a flat boundary studied by Seregin (2002 - 2005) make it possible to reuse much of the work that has been done for the whole-space problem.

Wed Nov 01

PDE Seminar

3:35pm - Vincent Hall 570
Serrin Lecture: Spectral stability of inviscid columnar vortices
Thierry Gallay, Univ. de Grenoble Alpes, France

Columnar vortices are stationary solutions of the three-dimensional
Euler equations with axial symmetry, where the fluid particles spin
around the vertical axis while moving in horizontal planes. Stability
of such flows was first investigated by Rayleigh and by Kelvin in
1880, but until today the only analytical results that can be found in
the literature are necessary conditions for instability under either
planar or axisymmetric perturbations. The goal of this talk is to
show that columnar vortices are spectrally stable for a large class
of velocity profiles, including the most commonly used models in
atmospheric flows and engineering applications. This a joint work
with Didier Smets (Paris).

Wed Nov 08

PDE Seminar

3:35pm - Vincent Hall 570
Patterns and complex singularities of reaction diffusion equations
Hirokazu Ninomiya., Meiji University, Japan

In this talk I will extend the reaction diffusion equation into the complex domain.
Based on this extension, I will discuss the relation between patterns and the complex singularities. This is based on the joint work with K. Ikeda, M. Katsurada and M. Onodera.

Wed Nov 22

PDE Seminar

3:35pm - Vincent Hall 570
Stability of Planar Fronts of the Bidomain Equation
Yoichiro Mori, University of Minnesota

The bidomain model is the standard model describing electrical activity of the heart. In this talk, I will discuss the stability of planar front solutions of the bidomain equation with a bistable nonlinearity (the bidomain AllenCahn equation) in two spatial dimensions. In the bidomain Allen-Cahn equation a Fourier multiplier operator whose symbol is a positive homogeneous rational function of degree two (the bidomain operator) takes the place of the Laplacian in the classical Allen-Cahn equation. Stability of the planar front may depend on the direction of propagation given the anisotropic nature of the bidomain operator. Our analysis reveals that planar fronts can be unstable in the bidomain Allen-Cahn equation in striking contrast to the classical or anisotropic Allen-Cahn equations. We identify two types of instabilities, one with respect to long-wavelength perturbations, the other with respect to medium-wavelength perturbations. Interestingly, whether the front is stable or unstable under long-wavelength perturbations does not depend on the bistable nonlinearity and is fully determined by the convexity properties of a suitably defined Frank diagram. On the other hand, stability under intermediate-wavelength perturbations does depend on the choice of bistable nonlinearity. Intermediate-wavelength instabilities can occur even when the Frank diagram is convex, so long as the bidomain operator does not reduce to the Laplacian. We shall also give a remarkable example in which the planar front is unstable in all directions. This is joint work with Hiroshi Matano of the University of Tokyo.

Wed Nov 29

PDE Seminar

3:35pm - Vincent Hall 570
Regularity Results for a Penalized Boundary Obstacle Problem
Donatella Danielli, Purdue University

n this talk we will discuss a two-penalty boundary obstacle problem of interest in thermics and fluid dynamics. Specifically, our goal is to establish existence, uniqueness and optimal regularity of the solutions, as well as structural properties of the free boundary. The study hinges on the monotone character of a perturbed frequency function of Almgren's? type, and the analysis of the associated blow-ups. This is joint work with Thomas Backing and Rohit Jain.

Wed Dec 06

PDE Seminar

3:35pm - Vincent Hall 570
Two-Bubble Dynamics for the Equivariant Wave Maps Equation
Jacek Jendrej, University of Chicago

I will consider the energy-critical wave maps equation with values in the two-dimensional sphere in the equivariant case, that is for symmetric initial data. It is known that if the initial data has small energy, then the corresponding solution scatters. Moreover, the initial data of any scattering solution has topological degree 0. I try to answer the following question: what are the non-scattering solutions of topological
degree 0 and the least possible energy? Such "threshold" solutions would have to decompose asymptotically into a superposition of two ground states at different scales, with no radiation. It turns out that one can construct non-scattering threshold solutions. I will also describe the dynamical behavior of any threshold solution. Joint work with Andrew Lawrie (MIT).

Wed Dec 13

PDE Seminar

3:35pm - Vincent Hall 570
On boundary estimates for solutions to second order elliptic and parabolic equations
Mikhail Safonov, University of Minnesota

I am going to discuss results and techniques by different authors related to interior and boundary regularity of solutions to second order elliptic and parabolic equations.

Wed Jan 24

PDE Seminar

3:35pm - Vincent Hall 570
Channel of energy inequality and Null concentration of energy for wave maps
Hao Jia, University of Minnesota

The channel of energy inequality for linear wave equations was introduced by Duyckaerts-Kenig-Merle to study soliton resolution for the focusing energy critical wave equations. It has been very useful for understanding long time behavior of semilinear wave equations, in the non-perturbative regime. Sometimes, it is the only available tool to understand dispersion of energy in the presence of solitons. Recently we found a new form of this type of inequality for outgoing waves, that turns out to be useful for studying energy critical wave maps, especially in ruling out the so-called `null concentration of energy". In this talk we will give an outline of the inequality, the history of wave maps, explain why the absence of null energy is important, and why channel of energy inequality seems to be uniquely good for ruling out this type of energy concentration. Joint work with Duyckaerts, Kenig and Merle.

Wed Feb 07

PDE Seminar

3:35pm - Vincent Hall 570
On positive solutions of semi-linear elliptic inequalities on Riemannian manifolds
Alexander Grigor'yan, Bielefeld University, Germany

Plan of the talk:
1. Laplace-Beltrami operator and its Green function on a Riemannian manifold.
2. Semi-linear inequality $\Delta u+ F(x) u^\sigma \le 0$.
Pointwise lower bound for positive solutions $u$ via the Green function.
3. Approach to the proof.
4. Existence of positive solutions of $\Delta u+ u^\sigma \le 0$.
Volume test for non-existence. Green function test.
5. Examples of application.
6. Approach to the proof.

Wed Feb 21

PDE Seminar

3:35pm - Vincent Hall 570
Elasticity and curvature: the elastic energy of non-Euclidean thin bodies
Cy Maor, University of Toronto, Canada

Non-Euclidean, or incompatible elasticity is an elastic theory for bodies that do not have a reference, stress-free configuration. It applies to many systems, in which the elastic body undergoes inhomogeneous growth (e.g. plants, self-assembled molecules). Mathematically, it is a question of finding the "most isometric" immersion of a Riemannian manifold (M,g) into Euclidean space of the same dimension, by minimizing an appropriate energy functional. Much of the research in non-Euclidean elasticity is concerned with elastic bodies that have one or more slender dimensions (such as leaves), and finding appropriate dimensionally-reduced models for them. In this talk I will give an introduction to non-Euclidean elasticity, and then focus on thin bodies and present some recent results on the relations between their elastic behavior and their curvature.
Based on a joint work with Asaf Shachar.

Wed Mar 28

PDE Seminar

3:35pm - Vincent Hall 570
PDE Seminar
Hongjie Dong, Brown University
Wed Apr 04

PDE Seminar

3:35pm - Vincent Hall 570
PDE Seminar
Andrew Lorent
Wed Apr 11

PDE Seminar

3:35pm - Vincent Hall 570
PDE Seminar
Yannick Sire
Wed Apr 18

PDE Seminar

3:35pm - Vincent Hall 570
PDE Seminar
Luca Rossi
Wed Apr 25

PDE Seminar

3:35pm - Vincent Hall 570
PDE Seminar
Jason Murphy


text:
# Reversing 2.4GHz remote control

I have an old project on Github called rf-car for controlling a radio car with HackRF. A few months ago, my daughter received a new RC car made by Dickie Toys:

This car was faster than the previous one and it was more fun to play with. I thought it’d be great if I can add support for it in the rf-car project and the fun began.

## FCC docs

The car works on 2.4GHz which is clearly stated in the product description. The remote control has FCC ID NLB24054TX and from the FCC documents I found that it uses GFSK modulation. From the internal photos of the remote control we can see it pretty much consists of four buttons connected to unlabeled IC:

So the only useful information I got from the FCC docs is that the car uses GFSK modulation.

## Signal analysis

The go-to tool for signal analysis is, of course, inspectrum. I have successfully reverse engineered the ASK/OOK signal of my old car with it but I didn’t have any experience with FSK signals.

This is how the recorded signal looks like when a single button on the remote is pressed:

The remote is sending 9 batches of 16 packets (on the screenshot above only 3 of the 9 batches are shown). The signal is GFSK, so we add a derived frequency plot (right click, add derived plot, add frequency plot):

To analyse the frequency plot, we need to decrease the FFT size and increase the zoom. This is how a single packet looks like:

With the symbol markers we can easily find the symbol period which is 1us:

Finally, we add a derived threshold plot from which we can extract the symbols as 0s and 1s (right click, extract symbols):

There are 146 symbols (bits) in each packet. The packets in each batch are the same.

## Signal generation

Now when we know the symbol rate (1M bits/sec), the modulation and the data being transferred, we can generate the signal ourselves. The following code modulates the input in data using FSK and produces an IQ signal in the out vector:

float phase = 0;
int samples_per_bit = SAMPLE_RATE / SYMBOL_RATE;
for (auto b : data) {
float freq = b ? FREQ1 : FREQ2;
float phase_change_per_sample = (2*M_PI * freq) / SAMPLE_RATE;
for (int i = 0; i < samples_per_bit; i++) {
out.push_back(cos(phase));
out.push_back(sin(phase));
phase += phase_change_per_sample;
if (phase > 2*M_PI) {
phase -= 2*M_PI;
}
}
}

## Two-way communication

One interesting detail is that the communication between the car and the remote control is actually two-way! When the remote is switched on, it starts sending designated packets until a response from the car is received. When response from the car is received, the remote sends a final packet to the car and then the car starts accepting commands from the remote. It is very similar to the TCP handshake but much simpler because the packets are always the same. In my implementation I opted to simply transmit the sync packets from the remote with some delay between them on program start. This works fine as long as the car is already switched on and receives those packets.

## Result

Here is a short demo of how it works:


text:
# IEEE Transactions on Information Theory

## Filter Results

Displaying Results 1 - 25 of 26
• ### Information rates of Wiener processes

Publication Year: 1970, Page(s):134 - 139
Cited by:  Papers (39)
| | PDF (786 KB)

Rate distortion functions are calculated for time discrete and time continuous Wiener processes with respect to the mean squared error criterion. In the time discrete case, we find the interesting result that, for0 \leq D \leq \sigma^2 /4,R(D)for the Wiener process is identical toR(D)for the sequence of zero mean independent normally distributed increments of varia... View full abstract»

• ### On the representation of continuous parameter processes by a sequence of random variables

Publication Year: 1970, Page(s):139 - 141
Cited by:  Papers (8)
| | PDF (435 KB)

This paper examines the question of representing a continuous parameter random process<tex>\{ x_t, t \in T \}</tex>by a sequence of random variables "without loss of information." The principal result is that such a representation by expansion coefficients relative to a basis<tex>\{ \phi_i \}</tex>of<tex>\mathcal{L}_2(T)</tex>is always possible, regardless of th... View full abstract»

• ### On the representation of a continuous stochastic intensity by Poisson shot noise

Publication Year: 1970, Page(s):142 - 147
Cited by:  Papers (4)
| | PDF (954 KB)

In many applications a Poisson shot noise (PSN) process is said to statistically "represent" its intensity process. In this paper an investigation is made of the relationship between a PSN process and its intensity, when the latter is a sample function of a continuous stochastic process. The difference of the moments and the mean-square difference between the two processes are examined. The contin... View full abstract»

• ### Alias-free randomly timed sampling of stochastic processes

Publication Year: 1970, Page(s):147 - 152
Cited by:  Papers (66)  |  Patents (1)
| | PDF (1058 KB)

The notion of alias-free sampling is generalized to apply to random processes<tex>x(t)</tex>sampled at random times<tex>t_n</tex>; sampling is said to be alias free relative to a family of spectra if any spectrum of the family can be recovered by a linear operation on the correlation sequence<tex>\{r(n)\}</tex>, where<tex>r(n) = E[x(l_{m+n}) \overline{x(t_... View full abstract»

• ### Nonlinear estimation with quantized measurements--PCM, predictive quantization, and data compression

Publication Year: 1970, Page(s):152 - 161
Cited by:  Papers (31)
| | PDF (1334 KB)

Statistics conditioned on quantized measurements are considered in the general case. These results are specialized to Gaussian parameters and then extended to discrete-time linear systems. The conditional mean of the system's state vector may be found by passing the conditional mean of the measurement history through the Kalman filter that would be used had the measurements been linear. Repetitive... View full abstract»

• ### Identification of a transition matrix of a Markov chain from noisy measurements of state

Publication Year: 1970, Page(s):161 - 166
Cited by:  Papers (5)
| | PDF (709 KB)

We consider the problem of recursively estimating the transition matrix of a regular Markov chain with a finite number of states using only noisy measurements of the state. The measurement noise sequence is assumed to be independent with known mean and unknown variance. We also discuss the convergence rates and computational aspects of the algorithms and the methods of accelerating them. View full abstract»

• ### Probing linear filters-- Signal design for the detection problem

Publication Year: 1970, Page(s):167 - 171
Cited by:  Papers (11)
| | PDF (711 KB)

The problem of designing an optimum probing signal for discriminating between two known linear filters in the presence of noise is considered. It is shown that in the general case of colored noise, the optimum signal is the solution of two simultaneous integral equations. An expression for the probability of error is given, and some illustrative examples are presented. View full abstract»

• ### Bounds on performance of optimum quantizers

Publication Year: 1970, Page(s):172 - 184
Cited by:  Papers (19)
| | PDF (1620 KB)

A quantizer<tex>Q</tex>divides the range [0, 1] of a random variable<tex>x</tex>into<tex>K</tex>quantizing intervals the<tex>i</tex>th such interval having length<tex>\Delta x_i</tex>. We define the quantization error for a particular value of<tex>x</tex>(unusually) as the length of the quantizing interval in which<tex>x... View full abstract»

• ### The two-armed-bandit problem with time-invariant finite memory

Publication Year: 1970, Page(s):185 - 195
Cited by:  Papers (49)
| | PDF (1669 KB)

This paper solves the classical two-armed-bandit problem under the finite-memory constraint described below. Given are probability densities<tex>p_0</tex>and<tex>p_1</tex>, and two experiments<tex>A</tex>and<tex>B</tex>. It is not known which density is associated with which experiment. Thus the experimental outcome<tex>Y</tex>of experime... View full abstract»

• ### Signals that can be calculated from their ambiguity function

Publication Year: 1970, Page(s):195 - 202
Cited by:  Papers (6)
| | PDF (1057 KB)

A new lemma relates the analytic extensions of two time functions<tex>u(t)</tex>and<tex>v(t)</tex>to the Laplace transform of their ambiguity function<tex>\psi_{uv}</tex>. This lemma is used to derive necessary conditions for<tex>u</tex>and<tex>v</tex>from two bounds on the behavior of<tex>\psi_{uv}</tex>, at infinity. In part... View full abstract»

• ### Geometry of optimum incoherent detection

Publication Year: 1970, Page(s):202 - 205
Cited by:  Papers (2)
| | PDF (702 KB)

The problem of distinguishing between two narrow-band signals in the presence of additive Gaussian noise when the forms of both signals are known except for phase is basically four dimensional, as the lack of phase information turns signal vectors into signal circles. The geometry of signal vectors and of the planes of signal circles is shown to account for the optimality of matched-filter detecti... View full abstract»

• ### Signal design for efficient detection in dispersive channels

Publication Year: 1970, Page(s):206 - 213
Cited by:  Papers (7)
| | PDF (1187 KB)

Signal design for maximizing the efficiency of the Neyman-Pearson detection procedure in randomly dispersive media is investigated. The medium is modeled as a randomly time-varying linear filter; by viewing the filter transfer function as a homogeneous random field on the time-frequency plane, a second-order theory results that relates various second-order measures of the time and frequency struct... View full abstract»

• ### Flicker effect and television compression

Publication Year: 1970, Page(s):214 - 218
Cited by:  Papers (4)
| | PDF (975 KB)

The perception of flicker by the human eye depends upon the picture content, and this paper describes a relation between the critical flicker frequency<tex>f_c</tex>and the spatial frequency<tex>f_x</tex>. Provided that the visual picture is presented in such a way that each spatial component, presented at a frequency<tex>f_p</tex>, obeys the inequality<tex&g... View full abstract»

• ### Error probability bounds for systematic convolutional codes

Publication Year: 1970, Page(s):219 - 224
Cited by:  Papers (11)
| | PDF (920 KB)

Upper and lower bounds on error probability for systematic convolutional codes are obtained. These bounds show that the probability of error for a systematic convolutional code is larger than the probability of error for a nonsystematic convolutional code of the same encoder-shift-register length. Two different interpretations of these results are given. View full abstract»

• ### A note on filtering for independent increment processes (Corresp.)

Publication Year: 1970, Page(s):224 - 226
Cited by:  Papers (2)
| | PDF (468 KB)

Using the equations of Fisher and Stear, a suboptimal filter for estimating the state of a linear system subject to non-Gaussian noise is developed. Some qualitative properties of this filter are derived. View full abstract»

• ### Fitting continuous probability density functions over [0,<tex>infty</tex>) using information theory ideas (Corresp.)

Publication Year: 1970, Page(s):226 - 230
Cited by:  Papers (27)
| | PDF (717 KB)

The problem of estimating a probability density function<tex>p(x)</tex>over<tex>[0, \infty)</tex>when several low-order moments are known is considered. A computational procedure, based on information theory, is used to obtain the maximum entropy estimates of<tex>p(x)</tex>in a number of cases. The situation when the first two moments only are known is considere... View full abstract»

• ### Comments on the Estimation of Distribution Functions [Corresp.]

Publication Year: 1970, Page(s): 226
| PDF (139 KB)
• ### The inversion of covariance matrices by finite Fourier transforms (Corresp.)

Publication Year: 1970, Page(s):230 - 232
Cited by:  Papers (16)
| | PDF (387 KB)

A matrix series is derived that converges to the inverse of a covariance matrix. The members of the series are derived from a circular matrix that can be inverted by taking finite Fourier transforms. An example of the method is presented. View full abstract»

• ### An integral relevant to the detection of a signal in polarized Gaussian noise (Corresp.)

Publication Year: 1970, Page(s): 232
Cited by:  Papers (1)
| | PDF (125 KB)

An integral is calculated that arises while determining the probability of detecting a signal in polarized Gaussian noise and the probability of hitting a particular elliptical target with an imperfect weapon. View full abstract»

• ### Calculation of Fourier transforms on finite Abelian groups (Corresp.)

Publication Year: 1970, Page(s):233 - 234
Cited by:  Papers (15)
| | PDF (329 KB)

A recent paper by Crimmins et al. deals with minimization of mean-square error for group codes by the use of Fourier transforms on groups. In this correspondence a method for representing the groups in a form suitable for machine calculation is shown. An efficient method for calculating the Fourier transform of a group is also proposed and its relationship to the fast Fourier transform is shown. F... View full abstract»

• ### A few useful details about a known technique for factoring 1 +X^{2q-1} (Corresp.)

Publication Year: 1970, Page(s):234 - 235
Cited by:  Papers (2)
| PDF (298 KB)
• ### Nonbinary random error-correcting codes (Corresp.)

Publication Year: 1970, Page(s):236 - 237
Cited by:  Papers (2)
| | PDF (224 KB)

Primitive BCH codes with symbols from<tex>GF(q)</tex>and designed distance<tex>d</tex>have parameter values \begin{align} \text{block length} &= n = q^m - 1 \\ \text{check symbols/block} &= r \leq m(d - 1) \end{align} where<tex>m</tex>is any positive integer. For many nonbinary BCH codes (called maximally redundant codes), the maximum number of c... View full abstract»

• ### Some difference triangles for constructing self-orthogonal codes (Corresp.)

Publication Year: 1970, Page(s):237 - 238
Cited by:  Papers (7)
| PDF (263 KB)
• ### Further results on the synchronization of binary cyclic codes (Corresp.)

Publication Year: 1970, Page(s):238 - 241
Cited by:  Papers (6)  |  Patents (2)
| | PDF (544 KB)

It is shown that certain coset codes derived from binary cyclic codes can determine the magnitude of a synchronization error, as well as its direction by examining only the syndrome of the received<tex>n</tex>tuple. For such coset codes, therefore, the need for a search procedure to recover synchronism is eliminated. In addition, the range of slip that can be detected and corrected for... View full abstract»

• ### Synchronizable error-correcting binary codes (Corresp.)

Publication Year: 1970, Page(s):241 - 242
Cited by:  Papers (1)  |  Patents (1)
| PDF (367 KB)

## Aims & Scope

IEEE Transactions on Information Theory publishes papers concerned with the transmission, processing, and utilization of information.

Full Aims & Scope

## Meet Our Editors

Editor-in-Chief
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Department of Electrical and Computer Engineering and the Institute for Systems Research, University of Maryland

email: abarg-ittrans@ece.umd.edu


text:
# Thread: Four colors, no back-to-backs (recursive)

1. ## Four colors, no back-to-backs (recursive)

(I posted already that for-loop question from our exam, sorry if this comes out as flooding)

So the question was short and clear, but I can't figure out the answer.

We have four colors (blue and three others) of poker chips in a stack of n chips. In how many ways can the stack be formed so that blue chips are not directly on top of each other?

What I understood by myself:
For example with three chips the options are
1) b o b (o = other, b = blue)
2) o b o
3) o o b
4) b o o
5) o o o

Which gives us 3+9+9+9+27=57 ways.

With two chips the options are
1) b o
2) o b
3) o o

Which gives us 3+3+9 = 15 ways.

But I can't figure out any kind of formula for this.

2. I cannot give you a recursive solution. But can give a counting solution.
If we have $N$ chips then maximum number of blue chips is $\max = \left\lceil {\frac{N}{2}} \right\rceil$.
Then the count is $\sum\limits_{k = 0}^{\max } {\binom{N-k+1}{k}3^{N - k} }$.

3. Originally Posted by Plato
I cannot give you a recursive solution. But can give a counting solution.
If we have $N$ chips then maximum number of blue chips is $\max = \left\lceil {\frac{N}{2}} \right\rceil$.
Then the count is $\sum\limits_{k = 0}^{\max } {\binom{N-k+1}{k}3^{N - k} }$.
The max part is clear - if you have more than half of stack blue chips, there has to be blue chips on top of each other. On count, that 3 is the number of other colors, right?

4. Originally Posted by jflurrie
The max part is clear - if you have more than half of stack blue chips, there has to be blue chips on top of each other. On count, that 3 is the number of other colors, right?

5. Hi jlflurrie,

Plato has given you an excellent solution, but if I understand you correctly, you may be interested in a recursive solution, is that right?

If so let's say an arrangement is "acceptable" if no blue chips are adjacent, and let $a_n$ be the number of acceptable arrangements with n poker chips. We know

[1] $a_1 = 4$ and
[2] $a_2 = 15$.

Now suppose we have a stack of n+1 chips, where $n \geq 2$. The chip on top is either blue or one of the other colors. If it is blue, then the chip under it must be one of the other 3 colors, and the remaining n-1 chips can be any acceptable arrangement; so the number of arrangements with a blue chip on top is $3 a_{n-1}$. If the chip on top is one of the other 3 colors, then the remaining n chips can be any acceptable arrangement; so this can be done in $3 a_n$ ways.

Hence
$a_{n+1} = 3 a_n + 3 a_{n-1}$.

Together with [1] and [2], this is enough to find $a_n$ for any n.


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# 线性混合模型 Linear Mixed Models STATS5054_1/STATS4045_1

0

The methods discussed above for the full parameter vector now directly carry over to calculate local influences on the geometric surface defined by $\mathrm{LD}{1}(\omega)$. We now partition $\ddot{L}$ as $$\ddot{L}=\left(\begin{array}{ll} \ddot{L}{11} & \ddot{L}{12} \ \ddot{L}{21} & \ddot{L}{22} \end{array}\right),$$ according to the dimensions of $\theta{1}$ and $\theta_{2}$. Cook (1986) has then shown that the local influence on the estimation of $\theta_{1}$, of perturbing the model in the direction of a normalized vector $\boldsymbol{h}$, is given by
$$C_{h}\left(\boldsymbol{\theta}{1}\right)=2\left|\boldsymbol{h}^{\prime} \Delta^{\prime}\left[\ddot{L}^{-1}-\left(\begin{array}{cc} 0 & 0 \ 0 & \ddot{L}{22}^{-1} \end{array}\right)\right] \Delta \boldsymbol{h}\right|$$
Because all eigenvalues of the malrix
$$\left(\begin{array}{ll} \ddot{L}{11} & \ddot{L}{12} \ \ddot{L}{21} & \ddot{L}{22} \end{array}\right)\left(\begin{array}{cc} 0 & 0 \ 0 & \ddot{L}{22}^{-1} \end{array}\right)=\left(\begin{array}{cc} 0 & \ddot{L}{12} \ddot{L}_{22}^{-1} \ 0 & I \end{array}\right)$$

## MATHS5077_1COURSE NOTES ：

A measure of influence, proposed , is then
$$\rho_{i}=-\left(\hat{\boldsymbol{\theta}}-\hat{\boldsymbol{\theta}}{(i)}^{1}\right)^{\prime} \ddot{L}\left(\hat{\boldsymbol{\theta}}-\hat{\boldsymbol{\theta}}{(i)}^{1}\right)$$$$\boldsymbol{\Delta}{i}=-\sum{j \neq i} \boldsymbol{\Delta}{j}=\ddot{L}{(i)}(\widehat{\boldsymbol{\theta}})\left(\hat{\boldsymbol{\theta}}{(i)}^{1}-\widehat{\boldsymbol{\theta}}\right)$$ such that expression (11.4) becomes $$C{i}=-2\left(\hat{\boldsymbol{\theta}}-\hat{\boldsymbol{\theta}}{(i)}^{1}\right)^{\prime} \ddot{L}{(i)} \ddot{L}^{-1} \ddot{L}{(i)}\left(\hat{\boldsymbol{\theta}}-\hat{\boldsymbol{\theta}}{(i)}^{1}\right)$$


text:
# zbMATH — the first resource for mathematics

Polynomial processes and their applications to mathematical finance. (English) Zbl 1270.60079
The authors introduce a class of Markov processes, called polynomial processes (or $$m$$-polynomial processes), which have the property that the expected value of any polynomial of the random variable $$X_t$$, $$t\geq 0$$, is again a polynomial in the initial value of the process, meaning that moments of all orders of $$X_T$$ can be computed in an easy and efficient way (even without probability distribution or characteristic function).
It is shown that this class of processes contains exponential Lévy processes, affine processes and processes of Pearson diffusion type. These processes can be applied, e.g., to GMM estimation procedures and to new techniques for option pricing and hedging. In particular, it can be also applied to variance reduction techniques in Monte Carlo methods due to the efficient and easy computation of moments.

##### MSC:
 60J25 Continuous-time Markov processes on general state spaces 91B70 Stochastic models in economics
Full Text:
##### References:
 [1] Akhiezer, N.I.: The Classical Moment Problem and Some Related Questions in Analysis. Hafner Publishing Co., New York (1965). Translated by N. Kemmer · Zbl 0135.33803 [2] Bates, D.S.: Post-’87 crash fears in the S&P 500 futures option market. J. Econom. 94, 181–238 (2000) · Zbl 0942.62118 · doi:10.1016/S0304-4076(99)00021-4 [3] Carr, P., Madan, D.: Option valuation using the fast Fourier transform. J. Comput. Finance 2, 61–73 (1998) [4] Chen, L., Filipović, D., Poor, H.V.: Quadratic term structure models for risk-free and defaultable rates. Math. Finance 14, 515–536 (2004) · Zbl 1098.91048 · doi:10.1111/j.0960-1627.2004.00203.x [5] Cheridito, P., Filipović, D., Yor, M.: Equivalent and absolutely continuous measure changes for jump-diffusion processes. Ann. Appl. Probab. 15, 1713–1732 (2005) · Zbl 1082.60034 · doi:10.1214/105051605000000197 [6] Çinlar, E., Jacod, J., Protter, P., Sharpe, M.J.: Semimartingales and Markov processes. Z. Wahrscheinlichkeitstheor. Verw. Geb. 54, 161–219 (1980) · Zbl 0443.60074 · doi:10.1007/BF00531446 [7] Duffie, D., Filipović, D., Schachermayer, W.: Affine processes and applications in finance. Ann. Appl. Probab. 13, 984–1053 (2003) · Zbl 1048.60059 · doi:10.1214/aoap/1060202833 [8] Dunkl, C.F.: Hankel transforms associated to finite reflection groups. In: Beals, R.W. (ed.) Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications, Tampa, FL, 1991. Contemp. Math., vol. 138, pp. 123–138. Amer. Math. Soc., Providence (1992) · Zbl 0789.33008 [9] Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics, vol. 194. Springer, New York (2000). With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli, R. Schnaubelt · Zbl 0952.47036 [10] Forman, J.L., Sørensen, M.: The Pearson diffusions: A class of statistically tractable diffusion processes. Scand. J. Stat. 35, 438–465 (2008) · Zbl 1198.62078 · doi:10.1111/j.1467-9469.2007.00592.x [11] Gallardo, L., Yor, M.: Some remarkable properties of the Dunkl martingales. In: Emery, M., Yor, M. (eds.) In memoriam Paul-André Meyer: Séminaire de Probabilités XXXIX. Lecture Notes in Math., vol. 1874, pp. 337–356. Springer, Berlin (2006) · Zbl 1128.60027 [12] Gatheral, J.: The Volatility Surface: A Practitioner’s Guide. Wiley Finance (2006) [13] Golub, G.H., Van Loan, C.F.: Matrix Computations. Johns Hopkins Studies in the Mathematical Sciences, 3rd edn. Johns Hopkins University Press, Baltimore (1996) [14] Gourieroux, C., Jasiak, J.: Multivariate Jacobi process with application to smooth transitions. J. Econom. 131, 475–505 (2006) · Zbl 1337.62365 · doi:10.1016/j.jeconom.2005.01.014 [15] Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 288. Springer, Berlin (1987) · Zbl 0635.60021 [16] Jacod, J., Kurtz, T.G., Méléard, S., Protter, P.: The approximate Euler method for Lévy driven stochastic differential equations. Ann. Inst. Henri Poincaré Probab. Stat. 41, 523–558 (2005) · Zbl 1071.60046 · doi:10.1016/j.anihpb.2004.01.007 [17] Keller-Ressel, M., Schachermayer, W., Teichmann, J.: Affine processes are regular. Probab. Theory Relat. Fields 151, 591–611 (2011) · Zbl 1235.60093 · doi:10.1007/s00440-010-0309-4 [18] Kunita, H.: Stochastic differential equations based on Lévy processes and stochastic flows of diffeomorphisms. In: Real and Stochastic Analysis. Trends Math., pp. 305–373. Birkhäuser, Boston (2004) · Zbl 1082.60052 [19] Moler, C., Van Loan, C.: Nineteen dubious ways to compute the exponential of a matrix. SIAM Rev. 20, 801–836 (1978) · Zbl 0395.65012 · doi:10.1137/1020098 [20] Reed, M., Simon, B.: Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness. Academic Press [Harcourt Brace Jovanovich Publishers], New York (1975) · Zbl 0308.47002 [21] Reimer, M.: Multivariate Polynomial Approximation. International Series of Numerical Mathematics, vol. 144. Birkhäuser, Basel (2003) · Zbl 1038.41002 [22] Rogers, L.C.G., Williams, D.: Diffusions, Markov Processes, and Martingales. Vol. 1 Foundations. Cambridge Mathematical Library. Cambridge University Press, Cambridge (2000). Reprint of the second (1994) edition · Zbl 0826.60002 [23] Zhou, H.: Itô conditional moment generator and the estimation of short rate processes. J. Financ. Econom. 1, 250–271 (2003) · doi:10.1093/jjfinec/nbg009
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text:
1
JEE Main 2022 (Online) 27th June Morning Shift
Numerical
+4
-1

The rate constant for a first order reaction is given by the following equation:

$$\ln k = 33.24 - {{2.0 \times {{10}^4}\,K} \over T}$$

The activation energy for the reaction is given by ____________ kJ mol$$-$$1. (In nearest integer) (Given : R = 8.3 J K$$-$$1 mol$$-$$1)

2
JEE Main 2022 (Online) 26th June Evening Shift
Numerical
+4
-1

Catalyst A reduces the activation energy for a reaction by 10 kJ mol$$-$$1 at 300 K. The ratio of rate constants, $${{{}^kT,\,Catalysed} \over {{}^kT,\,Uncatalysed}}$$ is ex. The value of x is ___________. [nearest integer]

[Assume that the pre-exponential factor is same in both the cases. Given R = 8.31 J K$$-$$1 mol$$-$$1]

3
JEE Main 2022 (Online) 26th June Morning Shift
Numerical
+4
-1

A flask is filled with equal moles of A and B. The half lives of A and B are 100 s and 50 s respectively and are independent of the initial concentration. The time required for the concentration of A to be four times that of B is ___________ s.

(Given : ln 2 = 0.693)

4
JEE Main 2022 (Online) 25th June Evening Shift
Numerical
+4
-1

At 345 K, the half life for the decomposition of a sample of a gaseous compound initially at 55.5 kPa was 340 s. When the pressure was 27.8 kPa, the half life was found to be 170 s. The order of the reaction is ____________. [integer answer]

JEE Main Subjects
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text:
# Focused information criterion

In statistics, the focused information criterion (FIC) is a method for selecting the most appropriate model among a set of competitors for a given data set. Unlike most other model selection strategies, like the Akaike information criterion (AIC), the Bayesian information criterion (BIC) and the deviance information criterion (DIC), the FIC does not attempt to assess the overall fit of candidate models but focuses attention directly on the parameter of primary interest with the statistical analysis, say ${\displaystyle \mu }$, for which competing models lead to different estimates, say ${\displaystyle {\hat {\mu }}_{j}}$ for model ${\displaystyle j}$. The FIC method consists in first developing an exact or approximate expression for the precision or quality of each estimator, say ${\displaystyle r_{j}}$ for ${\displaystyle {\hat {\mu }}_{j}}$, and then use data to estimate these precision measures, say ${\displaystyle {\hat {r}}_{j}}$. In the end the model with best estimated precision is selected. The FIC methodology was developed by Gerda Claeskens and Nils Lid Hjort, first in two 2003 discussion articles in Journal of the American Statistical Association and later on in other papers and in their 2008 book.

The concrete formulae and implementation for FIC depend firstly on the particular parameter of interest, the choice of which does not depend on mathematics but on the scientific and statistical context. Thus the FIC apparatus may be selecting one model as most appropriate for estimating a quantile of a distribution but preferring another model as best for estimating the mean value. Secondly, the FIC formulae depend on the specifics of the models used for the observed data and also on how precision is to be measured. The clearest case is where precision is taken to be mean squared error, say ${\displaystyle r_{j}=b_{j}^{2}+\tau _{j}^{2}}$ in terms of squared bias and variance for the estimator associated with model ${\displaystyle j}$. FIC formulae are then available in a variety of situations, both for handling parametric, semiparametric and nonparametric situations, involving separate estimation of squared bias and variance, leading to estimated precision ${\displaystyle {\hat {r}}_{j}}$. In the end the FIC selects the model with smallest estimated mean squared error.

Associated with the use of the FIC for selecting a good model is the FIC plot, designed to give a clear and informative picture of all estimates, across all candidate models, and their merit. It displays estimates on the ${\displaystyle y}$ axis along with FIC scores on the ${\displaystyle x}$ axis; thus estimates found to the left in the plot are associated with the better models and those found in the middle and to the right stem from models less or not adequate for the purpose of estimating the focus parameter in question.

Generally speaking, complex models (with many parameters relative to sample size) tend to lead to estimators with small bias but high variance; more parsimonious models (with fewer parameters) typically yield estimators with larger bias but smaller variance. The FIC method balances the two desired data of having small bias and small variance in an optimal fashion. The main difficulty lies with the bias ${\displaystyle b_{j}}$, as it involves the distance from the expected value of the estimator to the true underlying quantity to be estimated, and the true data generating mechanism may lie outside each of the candidate models.

In situations where there is not a unique focus parameter, but rather a family of such, there are versions of average FIC (AFIC or wFIC) that find the best model in terms of suitably weighted performance measures, e.g. when searching for a regression model to perform particularly well in a portion of the covariate space.

It is also possible to keep several of the best models on board, ending the statistical analysis with a data-dicated weighted average of the estimators of the best FIC scores, typically giving highest weight to estimators associated with the best FIC scores. Such schemes of model averaging extend the direct FIC selection method.

The FIC methodology applies in particular to selection of variables in different forms of regression analysis, including the framework of generalised linear models and the semiparametric proportional hazards models (i.e. Cox regression).


text:
# Finding probability distribution of $X$

In the box, there are 3 red balls and 3 blue balls, and from this box, the extraction of the ball is continued until the blue ball comes out. $$X$$ denotes extracted balls until blue ball comes out. Find the probability distribution, and find the mean and variance of $$X$$.

I am trying to find the function for probability distribution $$\frac{\binom{3}{x-1} \times \binom{3}{1}}{\binom{6}{x-1}}$$ , but I don't get the right answer. Which part is wrong?

Note that $$\Pr[X=1]=\frac36$$ because you get a blue ball from a group of $$3$$ blue balls from a total number of $$6$$ balls in the box.

Now $$\Pr[X=2]=\frac36\cdot\frac35$$ because you get first one red ball out of a group of $$3$$ red balls from a total of $$6$$ balls and after you took a blue ball from the group of $$3$$ blue balls from a total of $$5$$ balls in the box.

Continuing this reasoning we find that $$\Pr[X=3]=\frac36\cdot\frac25\cdot\frac34$$ and $$\Pr[X=4]=\frac36\cdot\frac25\cdot\frac14\cdot\frac 33$$. By last note that for $$k>4$$ we have that $$\Pr[X=k]=0$$ by the pigeonhole principle.

The probability mass function can be written compactly as $$f_X(k)=\frac{3^{\underline{k-1}}\cdot3}{6^\underline k}$$ where $$n^\underline k$$ is a falling factorial.

• thankyou! i know it is possible to do that way , but is there a way to find the probability function ? – fiksx May 21 at 13:58
• @fiksx well, you already have now the complete probability distribution – Masacroso May 21 at 14:03
• i was thinking , $\frac{\binom{3}{x-1} \times \binom{3}{1}}{\binom{6}{x-1}}$ , choosing red for $x-1$ and 3 blue , but it is seems not right? – fiksx May 21 at 14:07
• thankyou, i have never learn falling factorial , so its better to solve for every X? – fiksx May 21 at 14:37
• @fiksx for this case it seems easier to solve directly one by one, for other cases maybe not – Masacroso May 21 at 15:58


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# Conjugacy classes in non-abelian group of order $pq$

Suppose I have a non-abelian group $$G$$ of order $$pq$$ ($$\Rightarrow$$ the center of $$G$$ is trivial) such that $$p | (q-1)$$.

The Class Equation gives $$|G| = \sum[G:C(x)]$$, where $$C(x)$$ denotes the centralizer of $$x$$, and the sum ranges over one element $$x$$ from each nontrivial conjugacy class. Since the centralizer of an element $$x$$ of $$G$$ is a subgroup of $$G$$, each of the summands is a divisor of $$|G| = pq$$. Thus, we may only have conjugacy classes of order $$1$$, $$p$$, $$q$$, or $$pq$$.

It's easy to see that, since the center of $$G$$ is trivial (i.e., the identity element $$e$$ is the only member of $$Z(G)$$ ), $$[G:C(e)] = |G|/|C(e)| = (pq)/(pq) = 1$$. Since $$e$$ is the only element of $$G$$ whose centralizer is all of $$G$$, it follows that we only have one conjugacy class of order $$1$$ in $$G$$. As such, we cannot have a conjugacy class of order $$pq$$, as then we would exceed the number of elements in the group. Thus, we only have conjugacy classes of order $$p$$ or $$q$$ remaining.

Upon viewing Number of conjugacy classes of nonabelian group of order $pq$., the user comment states that there are $$p-1$$ conjugacy classes of order $$q$$, and $$(q-1)/p$$ conjugacy classes of order $$p$$.

My question is, how did this user arrive at these values ? I definitely see that, taking the answer for granted for a moment, we then have $$pq = 1 \cdot 1 + (p-1) \cdot q + (q-1)/p \cdot p$$, so the given orders make sense. But, how to arrive at these exact orders $$p-1$$ and $$(q-1)/p$$ ? I see I haven't used the fact that $$p|(q-1)$$ here, so maybe that helps us out.

In general, how can I detect the number of conjugacy classes of a certain order, if we're not given the specific group $$G$$ that we're working with ?

Thanks!

Let $$a$$ and $$b$$ be elements of order $$p$$ and $$q$$ respectively. Then $$a^{-1}ba=b^s$$ where $$s$$ is a non-trivial solution of $$s^p\equiv1\pmod p$$.
As $$\left$$ is normal in $$G$$, there are $$q-1$$ elements of order $$q$$, and any element outside $$\left$$ has order $$p$$. So there are $$pq-q$$ elements of order $$p$$.
No element of order $$p$$ can commute with an element of order $$q$$ since otherwise their product would have order $$pq$$ and so the group would be cyclic. Therefore the centraliser of an element of order $$p$$ has order $$p$$, and each element of order $$p$$ has $$q$$ conjugates. Likewise, the centraliser of an element of order $$q$$ has order $$q$$, and each element of order $$q$$ has $$p$$ conjugates.
Therefore the $$q-1$$ elements of order $$q$$ fall into $$(q-1)/p$$ conjugacy classes (each of order $$p$$). Likewise, the $$pq-q$$ elements of order $$p$$ fall into $$(pq-q)/q=p-1$$ conjugacy classes (each of order $$q$$).


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# ML4Jets2021

Jul 6 – 8, 2021
Europe/Zurich timezone

## Fast Simulation of Jets with VAEs

Jul 7, 2021, 3:40 PM
20m

### Speaker

Mary Touranakou (National and Kapodistrian University of Athens (GR))

### Description

Typically, high-energy physics (HEP) data analysis heavily relies on the production and the storage of large datasets of simulated events. At the LHC, the end-to-end simulation workflow can require up to 50% of the available computing resources of an experiment. Speeding up the simulation process would be crucial to save resources that could be otherwise utilized.
In our study, we investigate the use of Deep Generative Models, and more specifically, of Variational Autoencoders for fast simulation of jets at the LHC. We represent jets as lists of jet constituents (particles) characterized by their momenta. Starting from a simulation of the jet before detector effects, we train a Deep Variational Autoencoder (VAE) to learn a parametric description of the detector response and produce the corresponding list of jet constituents. Doing so, we bypass both the detector simulation and the reconstruction, potentially speeding up significantly the events generation workflow.
For our domain-specific application, we use a permutation-invariant loss function, the Chamfer distance, as the reconstruction loss of the Variational Autoencoder based on the jet constituents. We further modify the reconstruction loss by adding extra penalty terms for the jet mass and the jet transverse momentum $p_T$ to impose physics constraints to the model to learn the jet kinematics. Preliminary results show that jet features like jet pseudorapidity $\eta$, jet polar angle in the transverse plane $\phi$, or jet cartesian momenta $p_{x}$, $p_{y}$, $p_{z}$ are modelled within 10% of accepted accuracy, whereas it is more complicated for the jet mass to be learned.

Academic Rank PhD student National and Kapodistrian University of Athens

### Primary authors

Breno Orzari (UNESP - Universidade Estadual Paulista (BR)) Mary Touranakou (National and Kapodistrian University of Athens (GR)) Javier Mauricio Duarte (Univ. of California San Diego (US)) Jean-Roch Vlimant (California Institute of Technology (US)) Maurizio Pierini (CERN) Raghav Kansal (Univ. of California San Diego (US)) Thiago Tomei Fernandez (UNESP - Universidade Estadual Paulista (BR))

### Presentation materials

 Mary_Touranakou.mp4 ml4jets_070721_MT_f.pdf


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# Math Help - Finding terms for maclaurin series

1. ## Finding terms for maclaurin series

Problem- Find the terms through x^5 in the Maclaurin series for f(x)=e^(-x) * cos(x)

After trying to find f'(x), f''(x), f'''(x), and so on, the derivatives seemed overly complex. I heard that it is possible to use known Maclaurin series (e.g. e^(-x) and cos(x) separately) and then perform multiplications, divisions, etc. How would I combine them?

Thanks!

2. Originally Posted by dyy
Problem- Find the terms through x^5 in the Maclaurin series for f(x)=e^(-x) * cos(x)

After trying to find f'(x), f''(x), f'''(x), and so on, the derivatives seemed overly complex. I heard that it is possible to use known Maclaurin series (e.g. e^(-x) and cos(x) separately) and then perform multiplications, divisions, etc. How would I combine them?

Thanks!
Yes.

$e^x = 1 + x + \frac{x^2}{2!} + ...$
So,
$e^{-x} = 1 - x + \frac{x^2}{2!} - \frac{x^3}{3!}+...$

And

$\cos x = 1 - \frac{x^2}{2!}+ \frac{x^4}{4!}-...$

That means,
$e^{-x} \cos x = \left( 1 - x + \frac{x^2}{2!} - \frac{x^3}{3!}+... \right)\left( 1 - \frac{x^2}{2!}+ \frac{x^4}{4!}-... \right)$

Now preform something called the Cauchy Product.
On the first 5 terms in each one.

3. Originally Posted by dyy
Problem- Find the terms through x^5 in the Maclaurin series for f(x)=e^(-x) * cos(x)

After trying to find f'(x), f''(x), f'''(x), and so on, the derivatives seemed overly complex. I heard that it is possible to use known Maclaurin series (e.g. e^(-x) and cos(x) separately) and then perform multiplications, divisions, etc. How would I combine them?

Thanks!
Another way of doing this is to write:

$
f(x)= Re\left[ e^{x(-1+i)}\right]
$

Then:

$
f(x)= Re\left[
1 + [x(-1+i)] + \frac{[x(-1+i)]^2}{2!} + \frac{[x(-1+i)]^3}{3!}+\frac{[x(-1+i)]^4}{4!}+ ...
\right]
$

The right hand side of this is fairly easy to simplify to find the real part
and so find the required series.

RonL

4. I'm not sure I understand the cauchy product entirely unfortunately-

I'm trying to multiply the terms from the two series- so far I have-

1+(-1)x+(1/2-1/2)x^2+[the third term]<-where I'm stuck, since there's no term in cosx with x^3. What do I multiply the x^3/3! term from e^(-x) with, if anything?

Thanks once again.