text:
# Use the method of partial fraction decomposition to perform the required integration. \int (x^2 -...

## Question:

Use the method of partial fraction decomposition to perform the required integration.

{eq}\int \frac{x^{3} - 3x}{(x^{2} + 6)^{2}} dx {/eq}

## Partial Fraction Decomposition

The partial fraction decomposition, also called the partial fraction expansion, of a rational function is a way of writing the rational function as a sum of simpler rational functions. The writing of a rational function as its partial fraction decomposition is essentially the reverse procedure of combining multiple rational expressions together by finding a common denominator and adding. To find the partial fraction decomposition, begin by fully factoring the denominator. For each linear factor {eq}ax+b {/eq} in the denominator, there is a corresponding term {eq}\dfrac{A}{ax+b} {/eq} in the partial fraction decomposition. Similarly, for every irreducible quadratic factor in the denominator {eq}ax^2+bx+c {/eq} there is a corresponding term in the partial fraction decomposition of the form {eq}\dfrac{Ax+B}{ax^2+bx+c} {/eq}. Once a rational function is fully decomposed into partial fractions, it may be easier to do other things with the function, such as integration.

To evaluate {eq}\displaystyle\int \dfrac{x^{3} - 3x}{(x^{2} + 6)^{2}} dx {/eq}, begin by finding the partial fraction decomposition of the integrand. The denominator is already factored as a power of an irreducible quadratic factor. So we have

{eq}\dfrac{x^{3} - 3x}{(x^{2} + 6)^{2}} = \dfrac{Ax+B}{x^2+6} + \dfrac{Cx+D}{(x^2+6)^2} {/eq}

Multiplying both sides of the equation by the factored denominator, we have

{eq}x^3 - 3x = (Ax+B)(x^2+6) + Cx+D\\ x^3 - 3x = Ax^3 + Bx^2 + (6A+C)x + (6B+D)\\ {/eq}

Comparing coefficients, we have

{eq}1=A\\ 0=B\\ -3 = 6A+C\\ 0 = 6B+D\\ {/eq}

Solving this system, we have

{eq}A=1, B=0, C=-9, D=0 {/eq}

So the partial fraction decomposition is

{eq}\dfrac{x^{3} - 3x}{(x^{2} + 6)^{2}} = \dfrac{x}{x^2+6} + \dfrac{-9x}{(x^2+6)^2} {/eq}

Rewriting the integral,

{eq}\displaystyle \int \left(\dfrac{x}{x^2+6} + \dfrac{-9x}{(x^2+6)^2}\right) dx = \int \dfrac{x}{x^2+6}dx + \int\dfrac{-9x}{(x^2+6)^2}dx {/eq}

Using substitution for both integrals with {eq}u=x^2+6\\ du=2xdx {/eq} we have

{eq}\displaystyle\dfrac{1}{2}\int \dfrac{1}{u}du - \dfrac{9}{2}\int u^{-2}du = \dfrac{1}{2}\ln|u| +\dfrac{9}{2u} + c = \dfrac{1}{2}\ln|x^2+6| + \dfrac{9}{2(x^2+6)} + c {/eq}


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# Algebraic Identities

Go back to  'Algebra'

## Introduction to Algebraic Identities

Let us say that we are dealing with two equations. The two equations are:

$${\,\,5x \,– 3 = 12}$$

$${10x \,– 6 = 24}$$

If you solve both equations separately, you will see that the value of $${x}$$ is 3 in both cases. If you write the equations in the form $${ax \,– b = c,}$$ then you will see that the two equations are:

$${ax\, – b = c}$$

$${2(ax \,– b) = 2c}$$

What you did there was notice a pattern. Allowing us to observe patterns is how algebra appears in our daily lives.

## Algebraic Identities Explained Visually

### What is an identity in math?

• An identity is an equation that is always true regardless of the values assigned to the variables.
• Identity means that the left-hand side of the equation is identical to the right-hand side, for all values of the variables.

\begin{align} \text{E.g. } (a + b)^2 = a^2 + 2ab + b^2 \text{ is true for all values of } a \text{ and } b\end{align}

Most equations in math work only for certain specific values.

\begin{align} \text{E.g. } 4x + 5 = 17 \text{ is true only if } x = 3\end{align}

Because identities are always true regardless of the value of the variables, they are very useful.

## List of the most important and useful identities

\begin{align} \boxed{(a + b)^2 = a^2 + 2ab + b^2} \end{align}

\begin{align} \boxed{(a - b)^2 = a^2 - 2ab + b^2} \end{align}

\begin{align} \boxed{(a + b)(a − b) = a^2 − b^2} \end{align}

\begin{align} \boxed{(𝑥 + a) (x+ b) = x^2 + x(a+b) + ab} \end{align}

## Geogebra Simulation:

 Page title Algebraic Identities_Ranganath Simulation name ( a + b )^2 Concept name Algebraic identity Brief description This Simulation can show that identities hold true always by taking an example of ( a + b )^2. Here we take different values of a and b and show both sides of the identity are equal. The user controls the value of a and b.

## Dont memorise! You can visualise these identities

### Visualise \begin{align} (a + b)^2 \end{align}

Think of \begin{align} (a + b) \end{align} as a length.
So, \begin{align} (a + b)^2 = (a + b) × (a + b) \end{align} will be length squared.
This can be visualised as an area of a square with length \begin{align} (a + b) \end{align}.
Creating a square with length \begin{align} (a + b) \end{align} gives us area which has 4 parts as shown here:

So, \begin{align} (a + b)^2 \end{align} is simply the area of this square. Adding the 4 parts we get:

\begin{align} (a + b)^2 = a^2 + ab + ab + b^2 \end{align}

\begin{align} (a + b)^2 = a^2 + 2ab + b^2 \end{align}

Now $$(a - b)^2$$ can be done in a similar way. Try it out yourself.(Hint: Start with a square of side length $$a$$.)

### Visualise \begin{align} (a + b) (a − b) \end{align}

To visualise this identity, we start once again by visualising a rectangle.

The rectangle will have a length \begin{align} (a + b) \end{align} and breadth \begin{align} (a - b) \end{align}.

The rectangle is shown by the blue and yellow areas together.

Now visualise the yellow vertical strip moving to the bottom of the blue area and aligning horizontally as shown below.

Since we have simply moved it without stretching it, the area remains unchanged.

When viewed this way, we see that the area is now equal to \begin{align} a^2 - b^2 \end{align}.
\begin{align} (a + b) (a − b) = a^2 - b^2 \end{align}

Video Brief:

 Page Titie Algebraic Identities_Ranganath Video Name Proof of Algebraic Identities Concept name Algebraic Identities Target audience Grade 8 Video script The video shows a student trying to learn the proof of the identities by-heart. Then his friend tells him to look at the video of the geometric proof of the identities and shows it to him. The proof could be ; For (a+b)^2, it can show a big square of length (a+b). and how the terms in the expansion are smaller squares, which fit perfectly in the big square. Like this it shows the proof of all 4 identities.

## Solved Examples

Example 1:

Using algebraic identities, find $$(2x + 3y)^2$$.

Solution:

This is similar to the identity \begin{align} (a + b)^2 = a^2 + 2ab + b^2 \end{align}, where $$( 'a'$$ is $$2x$$ and $$'b'$$ is $$3y)$$

\begin{align} (2x + 3y)^2 &= (2x)^2 + 2(2x)(3y) + (3y)^2 \\ &= 4x^2 + 12xy + 9y^2 \end{align}.

Example 2:

Using identities, evaluate $$297 \times 303$$.

Solution:

The above form can be written as $$( 300 - 3 ) \times ( 300 + 3 )$$.

This is based on the formula : \begin{align} (a + b)(a − b) = a^2 − b^2 \end{align}, where $$( 'a' = 300$$ and $$'b' = 3 )$$.

Putting the values in the above identity, we get:

\begin{align} (300 - 3)(300 + 3) &= 300^2 − 3^2 \\ &= 90000 - 9 \\ &= 89991 \end{align}

Example 3:

Expand $$( pqr + 2 )( pqr - 5 )$$.

Solution:

This is based on the formula \begin{align}(𝑥 + a) (x+ b) = x^2 + x(a+b) + ab \end{align},

where  $$( 'x' = pqr$$ and $$'a' = 2$$ and $$'b' = -5)$$.

So, substituting in the above equation:

\begin{align} ( pqr + 2 )( pqr - 5 ) &= (pqr)^2 + pqr(2 + (-5)) + (2)(-5) \\ &= p^2q^2r^2 -3pqr -10 \end{align}

Example 4:

Simplify $$( 7x +4y )^2 + ( 7x - 4y )^2$$

Solution:

For this, we need to use the following formulas:

\begin{align} (a + b)^2 = a^2 + 2ab + b^2 \end{align}

\begin{align} (a - b)^2 = a^2 - 2ab + b^2 \end{align}

Here  $$( 'a' = 7x$$ and $$'b' = 4y )$$.

So,

\begin{align} ( 7x +4y )^2 + ( 7x - 4y )^2 &= ( 49x^2 + 56xy + 16y^2) + (49x^2 - 56xy + 16y^2) \\ &= 98x^2 + 32y^2 \end{align}

## Practice Questions

1. Expand the following expressions.

•  $$(4x - 3y)^2$$
•  $$(4x - 5)(4x +1)$$

2.  Simplfy.

• $$(3x + 5)^2 - (3x - 5)^2$$
• $$(p^2 - q^2p)^2 + 2p^3q^2$$

3. Using identities, evaluate the following:

• $$398 \times 402$$
• $$204^2$$
• $$998^2$$

## Activity

 Activity name: Sub_AlgebraicIdentities_Activity Item name 1 Sub_AlgebraicIdentities_Item1 Item name 2 Sub_AlgebraicIdentities_Item2 Item name 3 Sub_AlgebraicIdentities_Item3 Item name 4 Sub_AlgebraicIdentities_Item4 Item name 5 Sub_AlgebraicIdentities_Item5

## Tips and Tricks related to Algebraic Identities

• Visualise identities as rectangles. That way even if you forget, you can quickly work out the expressions.
E.g. \begin{align} (a + b)^2 \end{align} can be visualised as a square with sides\begin{align} (a + b) \end{align} giving four parts: \begin{align} a^2,\;b^2 \end{align} and two $$ab$$.

• Remember the factored forms rather than the simplified form. It’s difficult to remember what \begin{align} a^2 - b^2 \end{align} factorises to. But if you have the factored form \begin{align} (a + b) (a - b) \end{align}, you can work out the expression even if you don’t remember it.

## Frequently Asked Questions

### 1. What are the four algebraic identities?

Ans. The four commonly used algebraic identities are given below:

• \begin{align} (a + b)^2 = a^2 + 2ab + b^2 \end{align}
• \begin{align} (a - b)^2 = a^2 - 2ab + b^2 \end{align}
• \begin{align} (a + b)(a − b) = a^2 − b^2 \end{align}
• \begin{align} (𝑥 + a) (x+ b) = x^2 + x(a+b) + ab \end{align}

### 2. How can I learn algebraic identity?

Ans.

• Algebraic identities can be easily memorized by visualising the identities as squares or rectangles.
• It also helps by remembering the factored forms rather than the simplified form.

### 3. What is the formula of an identity?

Ans.

• An identity is an equation that is always true regardless of the value that is assigned to the variable.
• Equations are those which hold true for only specific value of the variable.

### 4. How many identities are there?

Ans.There are many identities, but the most common ones are given below:

• \begin{align} (a + b)^2 = a^2 + 2ab + b^2 \end{align}
• \begin{align} (a - b)^2 = a^2 - 2ab + b^2 \end{align}
• \begin{align} (a + b)(a − b) = a^2 − b^2 \end{align}
• \begin{align} (𝑥 + a) (x+ b) = x^2 + x(a+b) + ab \end{align}

### 5. Are all identities equations?

Ans. Yes, since identities are equation that hold true for any value of the variable.

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Keywords = integral group ring
Number of Articles: 3
##### 1. Torsion units for some projected special linear groups

Volume 6, Issue 1, March 2017, Pages 37-53

Joe Gildea

##### 2. Sylow like theorems for $V(\mathbb{Z}G)$

Volume 4, Issue 4, December 2015, Pages 49-59

Wolfgang Kimmerle

##### 3. The prime graph conjecture for integral group rings of some alternatings groups

Volume 2, Issue 1, March 2013, Pages 175-185

Mohamed Salim


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# Can't find maximum of a function on a region

for h=0.2

I'm trying to find the maximum value of

(-h^2/3)*(-1/x^2)

subject to

8.1 < x < 8.5

I've tried pretty much every combination of:

Functions:

FindMaximum[], MaxValue[]

Region (reg):

x \ [Element] Interval[{8.1,8.5}]
x \ [Element] Interval[8.1,8.5]
{x,8.1,8.5}
x, Interval[{8.1,8.5}]
x, Interval[8.1,8.5]
x \ [Element] Interval[{8.1,8.5}], x
x \ [Element] Interval[8.1,8.5], x

I even tried using 81/10 and 85/10 but that didn't work either..

Expression:

(-h^2/3)*(-1/x^2)
{(-h^2/3)*(-1/x^2) , reg}
Assuming[reg , -h^2/(3*x^2)

I'm getting errors like:

The variable x\[Element]Interval[{81/10,17/2}] cannot be localized so that it can be assigned to numerical values.

The variable x\[Element]Interval[{8.099999999,8.500000001}] cannot be localized so that it can be assigned to numerical values.

Constraints in {x\[Element]Interval[{8.099999999,8.500000001}]} are not all equality or inequality constraints. With the exception of integer domain constraints for linear programming, domain constraints or constraints with Unequal (!=) are not supported.

One time it told me the max value of this function was -7.34642395*10^-35.

Honestly I know I'm doing something really stupid but I can't figure out what.

• With[{h = 0.2}, FindMaximum[{(-h^2/3)*(-1/x^2), 8.1 <= x <= 8.5}, {x, 8.3}]]? Feb 29 '16 at 19:09
• This doesn't seem to be working. I had to tweek it a bit for what I was using it for after I posted the question. It works better than what I have but I have:............... With[{h = 0.2}, FindMaximum[{Abs[(h^2/3)*(-1/x^2)], 8.3 <= x <= 8.7}, x]] = {0.000193497, {x -> 8.30103}} ...............But............... Abs[(-0.2^2/(3*8.3^2))] = 0.0001935452654 Feb 29 '16 at 19:31

You should be aware of the fact that your problem may be somewhat ill-posed. Your function simplifies to the following expression:

Simplify[(-h^2/3)*(-1/x^2)]

(* Out: h^2/(3 x^2) *)

The difference of the function's values at the two extrema of your interval is ca. $0.0005\ h^2$. Since $h^2$ is a positive quantity (for $h\ne0$) and the function is monotonically decreasing (plot it), its maximum is attained at $x=8.1$, i.e. infinitesimally outside the region defined by your constraints; in that case Maximize & co. will return the supremum and the closest specifiable point (from the docs).

Many approaches seem to produce a result when correct syntax is used:

FindMaximum[{(-h^2/3)*(-1/x^2), 8.1 < x < 8.5}, x]
Maximize[{(-h^2/3)*(-1/x^2), 8.1 < x < 8.5}, x]

However, I was intrigued by the idea of using an Interval as a region specification, although technically it is NOT equivalent to your desired constraints, because Interval[{8.1, 8.5}] represents the closed interval that includes both end points (i.e. $8.1\le x\le8.5$, whereas you want them excluded.

Maximize[(-h^2/3)*(-1/x^2), x ∈ Interval[{8.1, 8.5}]]

It turns out that the accepted syntax for specifying that $x$ belongs to that interval is {x} ∈ Interval[{8.1, 8.5}]. Using that form, the optimization works fine:

Maximize[(-h^2/3)*(-1/x^2), {x} ∈ Interval[{8.1, 8.5}]]
(* Out: {0.000203221, {x -> 8.1}} *)

This did not strike me as immediately obvious, nor did I find it referenced in the documentation for Interval or other optimizers that accept geometric regions. I am still wondering whether I am missing something and I should have expected that. Hopefully I'll update if I find out more.

• the requirement for the { } is inconsistent with the docs for Element. Possibly should be labeled as a bug? Feb 29 '16 at 19:54
• @george2079 Use of vectors ({ }) is consistent when you consider Interval to be a geometric region when Element is used with it. Points in geometric regions are always coordinates that are described by vectors, even when those vectors are of only one dimension. Plain x is also valid, but at that case it's considered to be a symbolic vector, which needs to be handled accordingly (for instance, with Indexed[x, 1] in equations, instead of being treated as a number. Feb 29 '16 at 20:12
• For symbolic vector case, consider Maximize[(-h^2/3)*(-1/Indexed[x, 1]^2), x \[Element] Interval[{8.1, 8.5}]] (* {0.000203221, {x -> {8.1}}} *) Feb 29 '16 at 20:15
• Makes sense, but I still find the docs lacking. Feb 29 '16 at 21:16

This is really an extended comment about region-based constrained variables.

When you see Element in the variable section of a function such as Maximize, it always means you are dealing with geometric regions, although it is not necessarily immediately obvious. Values in geometric regions are always vectors, even when that vector has a length of one. So it happens to be in the case of Interval, when it is interpreted to be a region. Like some graphics primitives, Interval has second meaning in this context. This kind of constrained variable is declared in a form like Element[{x}, Interval[{0, 1}]]. Compare this with Element[{x, y}, Disk[]].

So, such declarations should have curly braces, right? Not quite. There are also symbolic vectors which are a feature that isn't documented very well. They don't have curly braces, even if they involve both Element and a geometric region. These are such as Element[v, Interval[{0, 1}]] or Element[v, Disk[]]. Sometimes these symbolic vectors can be used directly also on the equation side; consider Maximize[Norm[v], Element[v, Disk[]]], for instance. (Sadly mixing normal and symbolic vectors doesn't work as nicely as one might hope.)

This form allows for accessing individual components of symbolic vectors using Indexed, which is used like this: Maximize[Indexed[x, 1] - Indexed[x, 1]^2, Element[x, Interval[{0, 1}]]].

Note that in case of list-based coordinates, results list those variables as individual scalars:

Maximize[y + x^2, Element[{x, y}, Disk[]]]

(* {5/4, {x -> -(Sqrt[3]/2), y -> 1/2}} *)

... but in the case of symbolic vectors, as vector values:

Maximize[Indexed[v, 2] + Indexed[v, 1]^2, Element[v, Disk[]]]

(* {5/4, {v -> {-(Sqrt[3]/2), 1/2}}} *)

Interval is just one of these regions that can be used, and its elements are also vectors.

EDIT: A side note: FindMaximum doesn't understand region constraints in the variable declaration part. This is understandable, but unfortunately it doesn't understand that in its constraints either, which saddens me. This does work, though (as it expands to normal inequalities):

With[{h = 0.2},
FindMaximum[{(-h^2/3)*(-1/x^2),
RegionMember[Interval[{8.1, 8.5}], {x}]}, x]]


text:
Change search

BETA
Bobylev, Alexander
##### Publications (10 of 46) Show all publications
Bobylev, A., Pulvirenti, M. & Saffirio, C. (2013). From Particle Systems to the Landau Equations: A Consistency Result. Communications in Mathematical Physics, 319(3), 693-702
Open this publication in new window or tab >>From Particle Systems to the Landau Equations: A Consistency Result
2013 (English)In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 319, no 3, p. 693-702Article in journal (Refereed) Published
##### Abstract [en]

We consider a system of N classical particles, interacting via a smooth, short-range potential, in a weak-coupling regime. This means that N tends to infinity when the interaction is suitably rescaled. The j-particle marginals, which obey to the usual BBGKY hierarchy, are decomposed into two contributions: one small but strongly oscillating, the other hopefully smooth. Eliminating the first, we arrive to establish the dynamical problem in term of a new hierarchy (for the smooth part) involving a memory term. We show that the first order correction to the free flow converges, as N →∞, to the corresponding term associated to the Landau equation. We also show the related propagation of chaos.

Springer, 2013
Mathematics
Mathematics
##### Identifiers
urn:nbn:se:kau:diva-16083 (URN)10.1007/s00220-012-1633-6 (DOI)000318291500003 ()
##### Funder
Swedish Research Council, 621-2009-5751 Available from: 2012-12-04 Created: 2012-12-04 Last updated: 2017-12-07Bibliographically approved
Bobylev, A. & Potapenko, I. (2013). Monte Carlo methods and their analysis for Coulomb collisions in multicomponent plasmas. Journal of Computational Physics, 246, 123-144
Open this publication in new window or tab >>Monte Carlo methods and their analysis for Coulomb collisions in multicomponent plasmas
2013 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 246, p. 123-144Article in journal (Refereed) Published
##### Abstract [en]

A general approach to Monte Carlo methods for Coulomb collisions is proposed. Its key idea is an approximation of Landau-Fokker-Planck equations by Boltzmann equations of quasi-Maxwellian kind. It means that the total collision frequency for the corresponding Boltzmann equation does not depend on the velocities. This allows to make the simulation process very simple since the collision pairs can be chosen arbitrarily, without restriction. It is shown that this approach includes the well-known methods of Takizuka and Abe (1977) [12] and Nanbu (1997) as particular cases, and generalizes the approach of Bobylev and Nanbu (2000). The numerical scheme of this paper is simpler than the schemes by Takizuka and Abe [12] and by Nanbu. We derive it for the general case of multicomponent plasmas and show some numerical tests for the two-component (electrons and ions) case. An optimal choice of parameters for speeding up the computations is also discussed. It is also proved that the order of approximation is not worse than O(root epsilon), where epsilon is a parameter of approximation being equivalent to the time step Delta t in earlier methods. A similar estimate is obtained for the methods of Takizuka and Abe and Nanbu. (C) 2013 Elsevier Inc. All rights reserved.

Elsevier, 2013
##### Keywords
Monte Carlo methods, Coulomb collisions, Landau-Fokker-Planck equations, Boltzmann equations, Error of approximation
##### National Category
Other Mathematics
Mathematics
##### Identifiers
urn:nbn:se:kau:diva-38602 (URN)10.1016/j.jcp.2013.03.024 (DOI)000320604000009 ()
Available from: 2015-11-30 Created: 2015-11-23 Last updated: 2017-12-01Bibliographically approved
Andriash, A. V., Bobylev, A. V., Brantov, A. V., Bychenkov, V. Y., Karpov, S. A. & Potapenko, I. F. (2013). Stochastic simulation of the nonlinear kinetic equation with high-frequency electromagnetic fields. PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY (4), 233-237
Open this publication in new window or tab >>Stochastic simulation of the nonlinear kinetic equation with high-frequency electromagnetic fields
2013 (English)In: PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, ISSN 1562-6016, no 4, p. 233-237Article in journal (Refereed) Published
##### Abstract [en]

A general approach to Monte Carlo methods for Coulomb collisions is proposed. Its key idea is an approximation of Landau-Fokker-Planck (LFP) equations by Boltzmann equations of quasi-Maxwellian kind. High-frequency fields are included into consideration and comparison with the well-known results are given.

##### Place, publisher, year, edition, pages
KHARKOV INST PHYSICS & TECHNOLOGY, NATL SCIENCE CTR, 2013
##### Keywords
BASIC INTERACTIONS, CALCULATION METHODS, DIFFERENTIAL EQUATIONS, ELASTIC SCATTERING, ELECTROMAGNETIC INTERACTIONS, EQUATIONS, FUNCTIONS, INTEGRO-DIFFERENTIAL EQUATIONS, INTERACTIONS, KINETIC EQUATIONS
##### National Category
Mathematics Mathematical Analysis
##### Research subject
Materials Science
##### Identifiers
urn:nbn:se:kau:diva-38666 (URN)000324081600054 ()
Available from: 2015-11-23 Created: 2015-11-23 Last updated: 2016-08-12Bibliographically approved
Bobylev, A. & Esposito, R. (2013). Transport Coefficients in the 2-dimensional Boltzmann Equation. Kinetic and Related Models, 6(4), 789-800
Open this publication in new window or tab >>Transport Coefficients in the 2-dimensional Boltzmann Equation
2013 (English)In: Kinetic and Related Models, ISSN 1937-5093, E-ISSN 1937-5077, Vol. 6, no 4, p. 789-800Article in journal (Refereed) Published
##### Abstract [en]

We show that a rarefied system of hard disks in a plane, described in the Boltzmann-Grad limit by the 2-dimensional Boltzmann equation, has bounded transport coefficients. This is proved by showing opportune compactness properties of the gain part of the linearized Boltzmann operator.

##### Keywords
Boltzmann equation, transport coefficients
Mathematics
Mathematics
##### Identifiers
urn:nbn:se:kau:diva-38569 (URN)10.3934/krm.2013.6.789 (DOI)000327733900008 ()
Available from: 2016-01-22 Created: 2015-11-23 Last updated: 2017-11-30Bibliographically approved
Bobylev, A. & Windfäll, Å. (2012). Boltzmann equation and hydrodynamics at the Burnett level. Kinetic and Related Models, 5(2), 237-260
Open this publication in new window or tab >>Boltzmann equation and hydrodynamics at the Burnett level
2012 (English)In: Kinetic and Related Models, ISSN 1937-5093, Vol. 5, no 2, p. 237-260Article in journal (Refereed) Published
##### Abstract [en]

The hydrodynamics at the Burnett level is discussed in detail. First we explain the shortest way to derive the classical Burnett equations from the Boltzmann equation. Then we sketch all the computations needed for details of these equations. It is well known that the classical Burnett equations are ill-posed. We therefore explain how to make a regularization of these equations and derive the well-posed generalized Burnett equations (GBEs). We discuss briefly an optimal choice of free parameters in GBEs and consider a specific version of these equations. It is remarkable that this version of GBEs is even simpler than the original Burnett equations, it contains only third derivatives of density. Finally we prove a linear stability for GBEs. We also present some numerical results on the sound propagation based on GBEs and compare them with the Navier-Stokes results and experimental data.

##### Place, publisher, year, edition, pages
American Institute of Mathematical Sciences, 2012
##### Keywords
Hydrodynamics, regularized Burnett equations, Stability, sound propagation.
Mathematics
Mathematics
##### Identifiers
urn:nbn:se:kau:diva-8710 (URN)10.3934/krm.2012.5.237 (DOI)000302962700002 ()
Available from: 2011-11-03 Created: 2011-11-03 Last updated: 2012-12-04Bibliographically approved
Bobylev, A., Potapenko,, I. F. & Karpov, S. A. (2012). DSMC Methods for Multicomponent Plasmas (1ed.). In: Michel Mareschal, Andrés Santos (Ed.), DSMC and Related Simulations: 28th International Symposium on Rarefied Gas Dynamics  2012. Paper presented at 28th International Symposium on Rarefied Gas Dynamics, Zaragoza, July 9-13th, 2012 (pp. 541-548). New York: American Institute of Physics (AIP)
Open this publication in new window or tab >>DSMC Methods for Multicomponent Plasmas
2012 (English)In: DSMC and Related Simulations: 28th International Symposium on Rarefied Gas Dynamics  2012 / [ed] Michel Mareschal, Andrés Santos, New York: American Institute of Physics (AIP), 2012, 1, p. 541-548Conference paper, Published paper (Refereed)
##### Abstract [en]

A general approach to Monte Carlo methods for Coulomb collisions is proposed. Its key idea is an approximation of the Landau-Fokker-Planck equations by the Boltzmann equations of a quasi-Maxwellian kind. This means that the total collision frequency for the corresponding Boltzmann equation does not depend on velocities. This allows one to make the simulation process very simple since the collision pairs can be chosen arbitrarily, without restriction. It is shown that this approach includes (as particular cases) the well-known methods of Takizuka & Abe(1977) and Nanbu(1997) and generalizes the approach of Bobylev & Nanbu(2000). The numerical scheme of this paper is simpler than the schemes by Takizuka & Abe and by Nanbu. We derive it for the general case of multicomponent plasmas

##### Place, publisher, year, edition, pages
New York: American Institute of Physics (AIP), 2012 Edition: 1
##### Series
AIP Conference Proceedings, ISSN 0094-243X ; 1501
##### Keywords
Boltzmann equations, Coulomb collisions, Landau-Fokker-Planck equations, Monte Carlo methods
Mathematics
Mathematics
##### Identifiers
urn:nbn:se:kau:diva-16082 (URN)10.1063/1.4769589 (DOI)000312411200070 ()978-0-7354-1115-9 (ISBN)
##### Conference
28th International Symposium on Rarefied Gas Dynamics, Zaragoza, July 9-13th, 2012
##### Funder
Swedish Research Council, 621-2009-5751
##### Note

28th International Symposium on Rarefied Gas Dynamics, Zaragoza, July 9-13th, 2012

Available from: 2012-12-04 Created: 2012-12-04 Last updated: 2016-04-25Bibliographically approved
Bobylev, A., Karpov, S. & Potapenko, I. (2012). Monte-Carlo method for two component plasmas. Matematicheskoe Modelirovanie, 24(9), 35-49
Open this publication in new window or tab >>Monte-Carlo method for two component plasmas
2012 (Russian)In: Matematicheskoe Modelirovanie, ISSN 0234-0879, Vol. 24, no 9, p. 35-49Article in journal (Refereed) Published
##### Abstract [en]

The new direct simulation method of Monte-Carlo type (DSMC) for Coulomb collisions in the case of two component plasma is considered.  A brief literature review and preliminary information concerning the problem are given. Further the idea that lies in the basis of the method is discussed and its scheme is provided. The illustrative numerical simulation of the initial distribution relaxation for one and two sorts of particles in 3D case in the velocity space is performed. Simulation results are compared with the numerical results based on the completely conservative finite difference schemes for the Landau-Fokker-Planck equation. Estimation of calculation accuracy obtained from numerical results is given.

##### Place, publisher, year, edition, pages
Moskva: Steklov Mathematical Institute, Russian Academy of Sciences, 2012
Mathematics
Mathematics
##### Identifiers
urn:nbn:se:kau:diva-16081 (URN)
Available from: 2012-12-03 Created: 2012-12-03 Last updated: 2017-06-30Bibliographically approved
Bobylev, A. & Gamba, I. (2012). Solutions of the linear Boltzmann equation and some Dirichlet series. Forum Mathematicum, 24(2), 239-251
Open this publication in new window or tab >>Solutions of the linear Boltzmann equation and some Dirichlet series
2012 (English)In: Forum Mathematicum, ISSN 1435-5337, Vol. 24, no 2, p. 239-251Article in journal (Refereed) Published
##### Abstract [en]

It is shown that a broad class of generalized Dirichlet series (including the polylogarithm, related to the Riemann zeta-function) can be presented as a classof solutions of the Fourier transformed spatially homogeneous linear Boltzmannequation with a special Maxwell-type collision kernel. The result is based on anexplicit integral representationof solutions to the Cauchy problem for the Boltzmann equation. Possibleapplications to the theory of Dirichlet seriesare briefly discussed.

##### Place, publisher, year, edition, pages
Walter de Gruyter, 2012
##### Keywords
Boltzmann equation, Dirichlet series and functional equations, Riemann Zeta and L\$L\$-functions
Mathematics
Mathematics
##### Identifiers
urn:nbn:se:kau:diva-10556 (URN)10.1515/form.2011.058 (DOI)000303419000002 ()
##### Funder
Swedish Research Council, 2006-3404 Available from: 2012-02-08 Created: 2012-02-08 Last updated: 2012-12-04Bibliographically approved
Bobylev, A. & Vinerean (Bernhoff), M. (2012). Symmetric extensions of normal discrete velocity models (1ed.). In: Michel Mareschal, Andrés Santos (Ed.), Michel Mareschal, Andrés Santos (Ed.), 28th International Symposium on Rarefied Gas Dynamics 2012: . Paper presented at 28th International Symposium on Rarefied Gas Dynamics 2012, July 9 - 13, Zaragoza (pp. 254-261). American Institute of Physics (AIP), 1501(1)
Open this publication in new window or tab >>Symmetric extensions of normal discrete velocity models
2012 (English)In: 28th International Symposium on Rarefied Gas Dynamics 2012 / [ed] Michel Mareschal, Andrés Santos, American Institute of Physics (AIP), 2012, 1, Vol. 1501, no 1, p. 254-261Conference paper, Published paper (Refereed)
##### Abstract [en]

In this paper we discuss a general problem related to spurious conservation laws for discrete velocity models (DVMs) of the classical (elastic) Boltzmann equation. Models with spurious conservation laws appeared already at the early stage of the development of discrete kinetic theory. The well-known theorem of uniqueness of collision invariants for the continuous velocity space very often does not hold for a set of discrete velocities. In our previous works we considered the general problem of the construction of normal DVMs, we found a general algorithm for the construction of all such models and presented a complete classification of normal DVMs with small number n of velocities (n<11). Even if we have a general method to classify all normal discrete kinetic models (and in particular DVMs), the existing method is relatively slow and the amount of possible cases to check increases rapidly with n. We remarked that many of our normal DVMs appear to be axially symmetric. In this paper we consider a connection between symmetric transformations and normal DVMs. We first develop a new inductive method that, starting with a given normal DVM, leads by symmetric extensions to a new normal DVM. This method can produce very fast many new normal DVMs with larger number of velocities, showing that the class of normal DVMs contains a large subclass of symmetric models. We finally apply the method to several normal DVMs and construct new models that are not only normal, but also symmetric relatively to more and more axes. We hope that such symmetric velocitysets can be used for DSMC methods of solving Boltzmann equation.

##### Place, publisher, year, edition, pages
American Institute of Physics (AIP), 2012 Edition: 1
##### Series
AIP Conference Proceedings, ISSN 0094-243X, E-ISSN 1551-7616 ; 1501
##### Keywords
Kinetic theory, discrete kinetic (velocity) models, conservation laws
Mathematics
Mathematics
##### Identifiers
urn:nbn:se:kau:diva-16072 (URN)10.1063/1.4769516 (DOI)000312411200032 ()978-0-7354-1115-9 (ISBN)
##### Conference
28th International Symposium on Rarefied Gas Dynamics 2012, July 9 - 13, Zaragoza
Available from: 2012-12-03 Created: 2012-12-03 Last updated: 2017-10-30Bibliographically approved
Bobylev, A. & Windfäll, Å. (2011). Kinetic modeling of economic games with large number of participants. Kinetic and Related Models, 4(1), 169-185
Open this publication in new window or tab >>Kinetic modeling of economic games with large number of participants
2011 (English)In: Kinetic and Related Models, ISSN 1937-5093, Vol. 4, no 1, p. 169-185Article in journal (Refereed) Published
##### Abstract [en]

We study a Maxwell kinetic model of socio-economic behavior introduced in the paper A. V. Bobylev, C. Cercignani and I. M. Gamba, Commun. Math. Phys., 291 (2009), 599-644. The model depends on three non-negative parameters $\displaystyle{\left\lbrace\gamma,{q},{s}\right\rbrace}$ where $\displaystyle{0}<\gamma\leq{1}$ is the control parameter. Two other parameters are fixed by market conditions. Self-similar solution of the corresponding kinetic equation for distribution of wealth is studied in detail for various sets of parameters. In particular, we investigate the efficiency of control. Some exact solutions and numerical examples are presented. Existence and uniqueness of solutions are also discussed.

##### Place, publisher, year, edition, pages
American Institute of Mathematical Sciences, 2011
##### Keywords
Maxwell models, self-similar solutions, distribution of wealth, market economy
Mathematics
Mathematics
##### Identifiers
urn:nbn:se:kau:diva-8711 (URN)10.3934/krm.2011.4.169 (DOI)000286926200010 ()
Available from: 2011-11-03 Created: 2011-11-03 Last updated: 2012-12-04Bibliographically approved

#### Search in DiVA

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text:
## dissociation energy, $$E_{\text{d}}$$, $$D$$

https://doi.org/10.1351/goldbook.D01802
Energy required to dissociate a molecule into two parts. Subscripts 0 and e, i.e. D0 and De, are used to denote the dissociation from the ground state and potential energy minimum, respectively.


text:
Transducers in PHP

Rich Hickey recently announced that transducers are going to be added to Clojure, and it prompted quite a bit of discussion. After a somewhat brief announcement, Hickey followed up with a couple videos that describe transducers in much more detail: Transducers and Inside Transducers + more.async

Transducers are a very powerful concept that can be utilized in almost any language. In fact, they have been ported to various other languages including JavaScript (1 and 2), Python, Ruby, Java, and Go.

And now…transducers are available in PHP via transducers.php!

So, erm… What’s a transducer?

From the Clojure documentation:

Transducers are composable algorithmic transformations. They are independent from the context of their input and output sources and specify only the essence of the transformation in terms of an individual element. Because transducers are decoupled from input or output sources, they can be used in many different processes - collections, streams, channels, observables, etc. Transducers compose directly, without awareness of input or creation of intermediate aggregates.

This is a fancy way of saying that you can use functions like map and filter on basically any type of data source (not just sequences), and by using a step function, transducers can directly output to any type of data structure like iterators, filling an array, writing to a stream, or even invoking a custom function.

Eager evaluation

Transducers can be applied eagerly.

<?php
use Transducers as t;

$xf = t\map(function ($x) { return $x + 1; };$result = t\xform([1, 2, 3], $xf); //> [2, 3, 4] The above example creates a transformation function in$xf that increments each value passed through the transducer by 1. $xf is then applied to an array [1, 2, 3] using the transducers\xform() function. This returns the same type of supplied data structure with the transducer applied to it (so given an array it returns an array, given a string it returns a string, given a stream it returns a stream with an applied filter, etc.). Eager transformations are kinda like this (you consume and emit all of the data): Lazy evaluation We can also use transducers to lazily transform data. <?php$forever = function () {
$i = 0; while (1) yield$i++;
};

// Create a transformation function that multiplies each
// value by two and takes ony 100 values.
$xf = t\comp( t\map(function ($value) { return $value * 2; }), t\take(100) ); // Lazily transform the given generator using t\xform() foreach (t\xform($forever(), $xf) as$value) {
echo $value; } The above example lazily applied a transducer to a generator by creating another generator that applies a reducing step to each value yielded by the generator. Transducers are composable Transducers are composable, allowing you to create a transformation pipeline of data. Take the following example: <?php use Transducers as t;$xf = t\comp(
t\drop(2),
t\map(function ($x) { return$x + 1; },
t\filter(function ($x) { return$x % 2; },
t\take(3)
);

$data = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10];$result = t\xform($data,$xf); //> [5, 7, 9]

We just used the transducers\comp() to compose multiple transducers together into a single function. This composed function applies each transducer function, in the order they are defined, to a series of data. Each transducer performs its operation before deciding whether and how many times to call the next transducer in the pipeline. This means that the t\drop() transducer skips the first two elements of the input series and does not bother calling the subsequent transducers for the elements it skips.

To be clear, this means the pipeline order is as follows: drop -> map -> filter -> take

Available transducers

transducers.php comes with a number of built-in transducer functions.

function description
map(callable $f) Applies a map function$f to each value in a collection.
filter(callable $pred) Filters values that do not satisfy the predicate function$pred .
remove(callable $pred) Removes anything from a sequence that satisfied$pred .
cat() Transducer that concatenates items from nested lists.
mapcat(callable $f) Applies a map function to a collection and concats them into one less level of nesting. flatten() Takes any nested combination of sequential things and returns their contents as a single, flat sequence. partition($size) Partitions the source into arrays of size $size . partition_by(callable$pred) Split inputs into lists by starting a new list each time the predicate passed in evaluates to a different condition (true/false) than what holds for the present list.
take($n) Takes$n number of values from a collection.
take_while(callable $pred) Takes from a collection while the predicate function$pred returns true.
take_nth($nth) Takes every nth item from a sequence of values. drop($n) Drops $n items from the beginning of the input sequence. drop_while(callable$pred) Drops values from a sequence so long as the predicate function $pred returns true. replace(array$smap) Given a map of replacement pairs and a collection, returns a sequence where any elements equal to a key in $smap are replaced with the corresponding$smap value.
keep(callable $f) Keeps$f items for which $f does not return null. keep_indexed(callable$f) Returns a sequence of the non-null results of $f($index, $input) . dedupe() Removes duplicates that occur in order (keeping the first in a sequence of duplicate values). interpose($separator) Adds a separator between each item in the sequence.
tap(callable $interceptor) Invokes interceptor with each result and item, and then steps through unchanged. compact() Trim out all falsey values. words($maxBuffer = 4096) Splits the input by words.
lines($maxBuffer = 10240000) Splits the input by lines. Please consult the README for a full list of transducers and their descriptions. Applying transducers transducers.php offers several built-in ways of applying transducers to common data structures. • function transduce(callable$xf, array $step,$coll, $init = null) - Transform and reduce$coll by applying $xf($step)['step'] to each value.
• function into($target,$coll, callable $xf) - Transduces items from$coll into the given $target, in essence “pouring” transformed data from one source into another data type. • function to_iter($coll, callable $xf) - Creates an iterator that lazily applies the transducer$xf to the $input iterator. • function to_array($iterable, callable $xf) - Converts a value to an array and applies a transducer function.$iterable is passed through vec() in order to convert the input value into an array.
• function to_assoc($iterable, callable$xf) - Creates an associative array using the provided input while applying $xf to each value. Values are converted to arrays that contain the array key in the first element and the array value in the second. • function to_string($iterable, callable $xf) - Converts a value to a string and applies a transducer function to each character.$iterable is passed through vec() in order to convert the input value into an array.
• function xform($coll, callable$xf) - Returns the same data type passed in as $coll with$xf applied.

Each of these functions apply transducers and provide results in different ways. You can find more information and examples in the README.

Applying to streams of data

Remember when we were talking about how transducers don’t know anything about the data source or the output destination?

Here’s a great example of where this comes into play: transducers can be applied to PHP streams using stream filters.

<?php
use transducers as t;

$f = fopen('php://temp', 'w+'); fwrite($f, 'testing. Can you hear me?');
rewind($f);$xf = t\comp(
// Split by words
t\words(),
// Uppercase/lowercase every other word.
t\keep_indexed(function ($i,$v) {
return $i % 2 ? strtoupper($v) : strtolower($v); }), // Combine words back together into a string separated by ' '. t\interpose(' ') ); // Apply a transducer stream filter.$filter = t\append_stream_filter($f,$xf, STREAM_FILTER_READ);
echo stream_get_contents($f); // Be sure to remove the filter to flush out any buffers. stream_filter_remove($filter);
echo stream_get_contents($f); fclose($f);

// Outputs: "testing. CAN you HEAR me?"

Because transducers don’t know anything about the sequence of data being transformed, you can apply transducers in any way you want.

Creating transducers

To create a transducer, you need to create a function that returns the transducer (a transformation function). This function then returns another function that is the actual transducer. Let’s take a look at how map() is implemented.

<?php

function map(callable $f) { return function (array$xf) use ($f) { return [ 'init' =>$xf['init'],
'result' => $xf['result'], 'step' => function ($result, $input) use ($xf, $f) { return$xf['step']($result,$f($input)); } ]; }; } Notice that map is a function that returns a transducer. A transducer is a function that accepts a reducing function array and returns a new reducing function array with added behavior. The above map function decorates the provided reducing function array step function by calling a map function on each value as it passes through. Instead of using a 3 arity function like Clojure, transducers.php uses an associative array of functions for speed and clarity. Reducing function array Reducing function arrays are PHP associative arrays that contain a ‘init’, ‘step’, and ‘result’ key that maps to a function. key arguments Description init none Invoked to initialize a transducer. This function should call the ‘init’ function on the nested reducing function array$xf , which will eventually call out to the transducing process. This function is only called when an initial value is not provided while transducing.
step $result ,$input This is a standard reduction function but it is expected to call the $xf['step'] function 0 or more times as appropriate in the transducer. For example, filter will choose (based on the predicate) whether to call$xf or not. map will always call it exactly once. cat may call it many times depending on the inputs.
result $result Some processes will not end, but for those that do (like transduce), the ‘result’ function is used to produce a final value and/or flush state. This function must call the$xf['result'] function exactly once.

Transducers vs generators

You might be thinking that generators in PHP offer similar functionality to transducers– and you’d be right! However, transducers offer a number of benefits over just decorating iterators with generators.

Transducers are more composable than generators

When composing generators, you need to wrap iterators from the inside out, which can lead to code that’s hard to read and deeply nested. Let’s compare transducer composition vs generator compositon (using to nikic’s iter library).

The following composed transducer is a function that creates a pipeline for transforming data: it skips the first two elements of a collection, adds 1 to each value, filters out even numbers, then takes 3 elements from the collection. This new transformation function can be used with various transducer application functions, including to_iter.

Assume both examples have access to a $data variable that is traversable. <?php$iter = t\to_iter($data, t\comp( t\drop(2), t\map(function ($x) { return $x + 1; }, t\filter(function ($x) { return $x % 2; }, t\take(3) )); Note: to_iter() uses a generator under the hood to lazily apply the transducer Here’s the same transformation using generators (notice it’s awkwardly composed from the inside out): <?php$iter = iter\take(3, iter\filter(
function($x) { return$x % 2; },
iter\map(
function ($x) { return$x + 1; },
iter\drop(2, \$data)
)
));

Transducers are more generic than generators

Using generators to transform a sequence of data requires that your input data is iterable and that your output is iterable. Generators have knowledge of the traversal, transformation, and building up of a result. Transducers on the other hand are only concerned with transforming data and as such can transform any type of data source including but not limited to iterable data. Transducers can perform an action on each item in a series of data (like writing to a stream) without creating an intermediate sequence whereas a generator would require you to foreach over an iterator and then perform the action. This flexibility is demonstrated in the transducer stream filter.

Try it out!

Transducers are an exciting new innovation from the Clojure community that is now available to the PHP community. Give it a try and let me know what you think.


text:
# domain

A http://planetmath.org/node/4811connected non-empty open set in $\mathbb{C}^{n}$ is called a domain.

The topology considered is the Euclidean one (viewing $\mathbb{C}$ as $\mathbb{R}^{2}$). So we have that for a domain $D$ being connected is equivalent to being path-connected.

Since we have that every component of a region $D$ will be a domain, we have that every region has at most countably many components.

This definition has no particular relationship to the notion of an integral domain (http://planetmath.org/IntegralDomain), used in algebra. In number theory, one sometimes talks about fundamental domains in the upper half-plane, these have a different definition and are not normally open. In set theory, one often talks about the domain (http://planetmath.org/Function) of a function. This is a separate concept. However, when one is interested in complex analysis, it is often reasonable to consider only functions defined on connected open sets in $\mathbb{C}^{n}$, which we have called domains in this entry. In this context, the two notions coincide.

A domain in a metric space (or more generally in a topological space) is a connected open set.

Cf. http://mathworld.wolfram.com/Domain.htmlMathworld, http://en.wikipedia.org/wiki/DomainWikipedia.

 Title domain Canonical name Domain Date of creation 2013-03-22 11:56:17 Last modified on 2013-03-22 11:56:17 Owner drini (3) Last modified by drini (3) Numerical id 13 Author drini (3) Entry type Definition Classification msc 30-00 Classification msc 54A05 Classification msc 54E35 Related topic Region Related topic Topology Related topic ComplexNumber Related topic IntegralDomain


text:
## Run Custom Training Loops on a GPU and in Parallel

You can speed up your custom training loops by running on a GPU, in parallel using multiple GPUs, or on a cluster.

It is recommended to train using a GPU or multiple GPUs. Only use single CPU or multiple CPUs if you do not have a GPU. CPUs are normally much slower that GPUs for both training and inference. Running on a single GPU typically offers much better performance than running on multiple CPU cores.

Note

This topic shows you how to perform custom training on GPUs, in parallel, and on the cloud. To learn about parallel and GPU workflows using the `trainNetwork` function, see:

Using a GPU or parallel options requires Parallel Computing Toolbox™. Using a GPU also requires a supported GPU device. For information on supported devices, see GPU Support by Release (Parallel Computing Toolbox). Using a remote cluster also requires MATLAB® Parallel Server™.

### Train Network on GPU

By default, custom training loops run on the CPU. Automatic differentiation using `dlgradient` and `dlfeval` supports running on the GPU when your data is on the GPU. To run a custom training loop on a GPU, simply convert your data to `gpuArray` (Parallel Computing Toolbox) during training.

You can use `minibatchqueue` to manage your data during training. `minibatchqueue` automatically prepares data for training, including custom preprocessing and converting data to `dlarray` and `gpuArray`. By default, `minibatchqueue` returns all mini-batch variables on the GPU if one is available. You can choose which variables to return on the GPU using the `OutputEnvironment` property.

For an example showing how to use `minibatchqueue` to train on the GPU, see Train Network Using Custom Training Loop.

Alternatively, you can manually convert your data to `gpuArray` within the training loop.

To easily specify the execution environment, create the variable `executionEnvironment` that contains either `"cpu"`, `"gpu"`, or `"auto"`.

`executionEnvironment = "auto"`

During training, after reading a mini-batch, check the execution environment option and convert the data to a `gpuArray` if necessary. The `canUseGPU` function checks for useable GPUs.

```if (executionEnvironment == "auto" && canUseGPU) || executionEnvironment == "gpu" dlX = gpuArray(dlX); end```

### Train Single Network in Parallel

When you train in parallel, each worker trains the network simultaneously using a portion of a mini-batch. This means that you must combine the gradients, loss, and any state parameters after each iteration, according to the proportion of the mini-batch processed by each worker.

You can train in parallel on your local machine, or on a remote cluster, for example, in the cloud. Start a parallel pool in the desired resources and partition your data between the workers. During training, combine the gradients, loss, and state after each iteration so that the learnable parameters on each worker update in synchronization. For an example showing how to perform custom training in parallel, see Train Network in Parallel with Custom Training Loop

#### Set Up Parallel Environment

It is recommended to train using a GPU or multiple GPUs. Only use single CPU or multiple CPUs if you do not have a GPU. CPUs are normally much slower that GPUs for both training and inference. Running on a single GPU typically offers much better performance than running on multiple CPU cores.

Set up the parallel environment that you want to use before training. Start a parallel pool using your desired resources. For training using multiple GPUs, start a parallel pool with as many workers as available GPUs. For best performance, MATLAB automatically assigns a different GPU to each worker.

If you are using your local machine, you can use `canUseGPU` and `gpuDeviceCount` (Parallel Computing Toolbox) to determine if you have GPUs available. For example, to check availabilities of GPUs and start a parallel pool with as many workers as available GPUs, use the following code:

```if canUseGPU executionEnvironment = "gpu"; numberOfGPUs = gpuDeviceCount("available"); pool = parpool(numberOfGPUs); else executionEnvironment = "cpu"; pool = parpool; end```

If you are running using a remote cluster, for example, a cluster in the cloud, start a parallel pool with as many workers as the number of GPUs per machine multiplied by the number of machines.

For more information on selecting specific GPUs, see Select Particular GPUs to Use for Training.

#### Specify Mini-Batch Size and Partition Data

Specify the mini-batch size that you want to use during training. For GPU training, a recommended practice is to scale up the mini-batch size linearly with the number of GPUs, in order to keep the workload on each GPU constant. For example, if you are training on a single GPU using a mini-batch size of 64, and you want to scale up to training with four GPUs of the same type, you can increase the mini-batch size to 256 so that each GPU processes 64 observations per iteration.

You can use the following code to scale up the mini-batch size by the number of workers, where `N` is the number of workers in your parallel pool.

```if executionEnvironment == "gpu" miniBatchSize = miniBatchSize .* N end```

If you want to use a mini-batch size that not exactly divisible by the number of workers in your parallel pool, then distribute the remainder across the workers.

```workerMiniBatchSize = floor(miniBatchSize ./ repmat(N,1,N)); remainder = miniBatchSize - sum(workerMiniBatchSize); workerMiniBatchSize = workerMiniBatchSize + [ones(1,remainder) zeros(1,N-remainder)]```

At the start of training, shuffle your data. Partition your data so that each worker has access to a portion of the mini-batch. To partition a datastore, use the `partition` function.

You can use `minibatchqueue` to manage the data on each worker during training. `minibatchqueue` automatically prepares data for training, including custom preprocessing and converting data to `dlarray` and `gpuArray`. Create a minibatchqueue on each worker using the partitioned datastore. Set the `MiniBatchSize` property using the mini-batch sizes calculated for each worker.

At the start of each training iteration, use the `gop` (Parallel Computing Toolbox) function to check that all worker `minibatchqueue` objects can return data. If any worker runs out of data, training stops. If your overall mini-batch size is not exactly divisible by the number of workers and you do not discard partial mini-batches, some workers might run out of data before others.

Write your training code inside an `spmd` (Parallel Computing Toolbox) block, so that the training loop executes on each worker.

```spmd % Reset and shuffle the datastore. reset(augimdsTrain); augimdsTrain = shuffle(augimdsTrain); % Partition datastore. workerImds = partition(augimdsTrain,N,labindex); % Create minibatchqueue using partitioned datastore on each worker workerMbq = minibatchqueue(workerImds,... "MiniBatchSize",workerMiniBatchSize(labindex),... "MiniBatchFcn",@preprocessMiniBatch); ... for epoch = 1:numEpochs % Reset and shuffle minibatchqueue on each worker. shuffle(workerMbq); % Loop over mini-batches. while gop(@and,hasdata(workerMbq)) % Custom training loop ... end ... end end```

To ensure that the network on each worker learns from all data and not just the data on that worker, aggregate the gradients and use the aggregated gradients to update the network on each worker.

For example, suppose you are training the network `dlnet`, using the model gradients function `modelGradients`. Your training loop contains the following code for evaluating the gradient, loss, and statistics on each worker:

`[workerGradients,dlworkerLoss,workerState] = dlfeval(@modelGradients,dlnet,dlworkerX,workerY);`
`dlworkerX` and `workerY` are the predictor and true response on each worker, respectively.

To aggregate the gradients, use a weighted sum. Define a helper function to sum the gradients.

```function gradients = aggregateGradients(dlgradients,factor) gradients = extractdata(dlgradients); gradients = gplus(factor*gradients); end```

Inside the training loop, use `dlupdate` to apply the function to the gradients of each learnable parameter.

`workerGradients.Value = dlupdate(@aggregateGradients,workerGradients.Value,{workerNormalizationFactor});`

#### Aggregate Loss and Accuracy

To find the network loss and accuracy, for example, to plot them during training to monitor training progress, aggregate the values of the loss and accuracy on all of the workers. Typically, the aggregated value is the sum of the value on each worker, weighted by the proportion of the mini-batch used on each worker. To aggregate the losses and accuracy each iteration, calculate the weight factor for each worker and use `gplus` (Parallel Computing Toolbox) to sum the values on each worker.

```workerNormalizationFactor = workerMiniBatchSize(labindex)./miniBatchSize; loss = gplus(workerNormalizationFactor*extractdata(dlworkerLoss)); accuracy = gplus(workerNormalizationFactor*extractdata(dlworkerAccuracy)); ```

#### Aggregate Statistics

If your network contains layers that track the statistics of your training data, such as batch normalization layers, then you must aggregate the statistics across all workers after each training iteration. Doing so ensures that the network learns statistics that are representative of the entire training set.

You can identify the layers that contain statistics information before training. For example, if you are using a `dlnetwork` with batch normalization layers, you can use the following code to find the relevant layers.

```batchNormLayers = arrayfun(@(l)isa(l,'nnet.cnn.layer.BatchNormalizationLayer'),dlnet.Layers); batchNormLayersNames = string({dlnet.Layers(batchNormLayers).Name}); state = dlnet.State; isBatchNormalizationStateMean = ismember(state.Layer,batchNormLayersNames) & state.Parameter == "TrainedMean"; isBatchNormalizationStateVariance = ismember(state.Layer,batchNormLayersNames) & state.Parameter == "TrainedVariance";```
Define a helper function to aggregate the statistics you are using. Batch normalization layers track the mean and variance of the input data. You can aggregate the mean on all the workers using a weighted average. To calculate the aggregated variance ${s}_{c}^{2}$, use a formula of the following form.

`${s}_{c}^{2}=\frac{1}{M}\sum _{j=1}^{N}{m}_{j}\left({s}_{j}^{2}+{\left({\overline{x}}_{j}-{\overline{x}}_{c}\right)}^{2}\right)$`

N is the total number of workers, M is the total number of observations in a mini-batch, mj is the number of observations processed on the jth worker, ${\overline{x}}_{j}$ and ${s}_{j}^{2}$ are the mean and variance statistics calculated on that worker, and ${\overline{x}}_{c}$ is the aggregated mean across all workers.

```function state = aggregateState(state,factor,... isBatchNormalizationStateMean,isBatchNormalizationStateVariance) stateMeans = state.Value(isBatchNormalizationStateMean); stateVariances = state.Value(isBatchNormalizationStateVariance); for j = 1:numel(stateMeans) meanVal = stateMeans{j}; varVal = stateVariances{j}; % Calculate combined mean combinedMean = gplus(factor*meanVal); % Calculate combined variance terms to sum varTerm = factor.*(varVal + (meanVal - combinedMean).^2); % Update state stateMeans{j} = combinedMean; stateVariances{j} = gplus(varTerm); end state.Value(isBatchNormalizationStateMean) = stateMeans; state.Value(isBatchNormalizationStateVariance) = stateVariances; end```

Inside the training loop, use the helper function to update the state of the batch normalization layers with the combined mean and variance.

```dlnet.State = aggregateState(workerState,workerNormalizationFactor,... isBatchNormalizationStateMean,isBatchNormalizationStateVariance);```

#### Plot Results During Training

If you want to plot results during training, you can send data from the workers to the client using a `DataQueue` object.

To easily specify that the plot should be on or off, create the variable `plots` that contains either `"training-progress"` or `"none"`.

`plots = "training-progress";`

Before training, initialize the `DataQueue` and the animated line using the `animatedline` function.

```if plots == "training-progress" figure lineLossTrain = animatedline('Color',[0.85 0.325 0.098]); ylim([0 inf]) xlabel("Iteration") ylabel("Loss") grid on end```
Create the `DataQueue` object. Use `afterEach` to call the helper function `displayTrainingProgress` each time data is sent from the worker to the client.
```Q = parallel.pool.DataQueue; displayFcn = @(x) displayTrainingProgress(x,lineLossTrain); afterEach(Q,displayFcn);```
The `displayTrainingProgress` helper function contains the code used to add points to the animated line and display the training epoch and duration.
```function displayTrainingProgress (data,line) addpoints(line,double(data(3)),double(data(2))) D = duration(0,0,data(4),'Format','hh:mm:ss'); title("Epoch: " + data(1) + ", Elapsed: " + string(D)) drawnow end```

Inside the training loop, at the end of each epoch, use the `DataQueue` to send the training data from the workers to the client. At the end of each iteration, the aggregated loss is the same on each worker, so you can send data from a single worker.

```% Display training progress information. if labindex == 1 data = [epoch loss iteration toc(start)]; send(Q,gather(data)); end```

### Train Multiple Networks in Parallel

To train multiple networks in parallel, start a parallel pool in your desired resources and use `parfor` (Parallel Computing Toolbox) to train a single network on each worker.

You can run locally or using a remote cluster. Using a remote cluster requires MATLAB Parallel Server. For more information about managing cluster resources, see Discover Clusters and Use Cluster Profiles (Parallel Computing Toolbox). If you have multiple GPUs and want to exclude some from training, you can choose the GPUs you use to train on. For more information on selecting specific GPUs, see Select Particular GPUs to Use for Training.

You can modify the network or training parameters on each worker to perform parameter sweeps in parallel. For example, in `networks` is an array of `dlnetwork` objects, you can use code of the following form to train multiple different networks using the same data.

```parpool ("local",numNetworks); parfor idx = 1:numNetworks iteration = 0; velocity = []; % Allocate one network per worker dlnet = networks(idx) % Loop over epochs. for epoch = 1:numEpochs % Shuffle data. shuffle(mbq); % Loop over mini-batches. while hasdata(mbq) iteration = iteration + 1; % Custom training loop ... end end % Send the trained networks back to the client. trainedNetworks{idx} = dlnet; end```
After `parfor` finishes, `trainedNetworks` contains the resulting networks trained by the workers.

#### Plot Results During Training

To monitor training progress on the workers, you can use a `DataQueue` to send data back from the workers.

To easily specify that the plot should be on or off, create the variable `plots` that contains either `"training-progress"` or `"none"`.

`plots = "training-progress";`

Before training, initialize the `DataQueue` and the animated lines using the `animatedline` function. Create a subplot for each network you are training.

```if plots == "training-progress" f = figure; f.Visible = true; for i=1:numNetworks subplot(numNetworks,1,i) xlabel('Iteration'); ylabel('loss'); lines(i) = animatedline; end end```
Create the `DataQueue` object. Use `afterEach` to call the helper function `displayTrainingProgress` each time data is sent from the worker to the client.
```Q = parallel.pool.DataQueue; displayFcn = @(x) displayTrainingProgress(x,lines); afterEach(Q,displayFcn);```
The `displayTrainingProgress` helper function contains the code used to add points to the animated lines.
```function displayTrainingProgress (data,lines) addpoints(lines(1),double(data(4)),double(data(3))) D = duration(0,0,data(5),'Format','hh:mm:ss'); title("Epoch: " + data(2) + ", Elapsed: " + string(D)) drawnow limitrate nocallbacks end```

Inside the training loop, at the end of each iteration, use the `DataQueue` to send the training data from the workers to the client. Send the `parfor` loop index as well as the training information so that the points are added to the correct line for each worker.

```% Display training progress information. data = [idx epoch loss iteration toc(start)]; send(Q,gather(data)); ```

### Use Experiment Manager to Train in Parallel

You can use Experiment Manager to run your custom training loops in parallel. You can either run multiple trails at the same time, or run a single trial at a time using parallel resources.

To run multiple trials at the same time using one parallel worker for each trial, set up your custom training experiment an enable the Use Parallel option before running your experiment.

To run a single trial at a time using multiple parallel workers, define your parallel environment in your experiment training function and use an `spmd` block to train the network in parallel. For more information on training a single network in parallel with a custom training loop, see Train Single Network in Parallel.

For more information on training in parallel using Experiment Manager, see Use Experiment Manager to Train in Parallel.


text:
The probability interpretation of the wavefunction?

1. Aug 21, 2010

jeebs

Hi,
I'm about to go into my 4th and final year of my undergraduate physics degree and after all the quantum mechanics we've done so far, I still get this nagging feeling that I'm answering homework and exam problems blindly. Apologies for the length of this question by the way.

For example, take the infinite square well problem. The particle is confined in a 1-dimensional region of space in an infinite potential. It's easy enough to mess around with the Schrodinger equation and out pops a nice, simple little wavefunction, say, $$\psi (x) = Asin(n\pi x/a)$$ for a particle confined between x=0 and x=a. Now apparently this can be used to determine how likely you are to find the particle in a specific position x, if you simply square the wavefunction.

I have never understood where this came from. No text book I have read has explained why this is done or why it works. Also, in the infinite square well example that I have used, it seems to me to give a bit of a dubious result.
When you square this wavefunction, you find that at x=0 and x=a, you get $$\psi^2 = 0$$. Assuming that squaring the wavefunction really does give you the probability of the particle's position, then this result of zero probability at the walls of the well makes sense - since how could the particle possibly escape an infinite potential and cross over to this impossible-to-reach region? This much I am happy with.
However, take the n=2 (first excited) eigenstate. At x=a/2 we are going to get $$sin^2(\pi) = 0$$ which means that we have a zero probability of finding the particle right in the middle of the well. How does that make sense? This does not seem like a sensible result to me. What is wrong with being in the middle of the well when you are at this energy level? Does this not suggest that the probability interpretation of the square of the wavefunction is less than perfect?

Oh, another thing I have been wondering about is to do with Heisenburg's uncertainty principle. I have watched a few quantum documentaries (eg. with that Michio Kaku guy) on the internet and they always go on about quantum particles existing in 2 places at once, but they seem to be aimed at a less-educated audience so they just state their results with no explanation.
The understanding I have about this and the uncertainty principle is that, if you have particle sitting still in some position, you have to bounce a photon off it to detect it. When the photon leaves the particle, the particle recoils under momentum conservation, so by measuring the position of the particle you have altered its momentum - hence the position and momentum cannot both be known with absolute certainty at the same time.
My question about all of this is, are these documentaries being inaccurate/misleading when they say that the particle is in two positions at the same time?
Just because our imperfect method of measuring introduces this uncertainty in a particle's position, surely that doesn't mean the particle literally exists in 2 positions at once, right?
Shouldn't these documentaries really be trying to say that there is a probability of finding the particle exclusively in each position?
Is there really an intrinsic uncertainty in a particle's position, or is it just that human methods of collecting position data introduce this uncertainty that in reality is not there?
I'm not sure if I've explained what I mean here (hence the long, rambling question ^_^) but I appreciate any attempt at an answer.

Thanks.

2. Aug 21, 2010

skippy1729

If you view the particle in a potential well as a physical experiment rather than a mathematical exercise the problem of finding the particle at the center of the well disappears. Any classical particle detector at the center of the well will detect particles within a certain small but finite range about the center. Over the course of many trials, you will detect the probability density for that range, never the probability at a point.

Is the particle ever actually at the center of the well? According to the CI (Copenhagen Interpretation) the particle does not have any position until it is observed. The question is meaningless according to CI. Of course, there are other interpretations, the deBroglie-Bohm pilot wave theory maintains that there is an unobservable but definite trajectory for all particles. While I am no authority on this theory it would seem that if the particle was in the left side of the well and was later found to be in the right half then at some instant of time it must have been at the center.

As you can see, in neither of these two interpretations is the particle ever at two places at once. Perhaps someone else knows what Dr. Kaku was alluding to.

You are also interested in the origin of the probability interpretation (the Born rule). The strongest justification for this is Gleason's theorem. Which is covered in "Quantum Theory: Concepts and Methods" by Asher Peres and many other advanced texts. If you are at all interested in the Copenhagen viewpoint, it is the best (non-introductory) quantum text I have read. If you have had one formal course in QM, it is readable and suitable for self study.

Cheers, Skippy

3. Aug 21, 2010

jeebs

The Born rule eh? I'll be sure to look it up, although I'm guessing it's complicated since it's been conveniently ignored in every lecture and textbook I've seen. Thanks man.

One thing though, going back to the infinite potential well, what about for higher excited states that have zero probability positions in places other than the centre of the well? If I'm not mistaken you were talking about having some detector positioned right at the centre of it, right? well what about in the 3rd excited state (n=4) where $$\psi^2 = Asin(4\pi x/a)$$, which has a zero value at x=a/4 and x=3a/4 ?

or is this just one of the mad things physics throws at us that is so different to what we might expect in everyday life (like the falseness of galilean transformation of velocities that relativity reveals) but that we have no choice but to accept?

Last edited: Aug 21, 2010
4. Aug 21, 2010

alxm

There's no "probability interpretation" of quantum mechanics, there's an interpretation of what the probabilities mean. The Born rule and such is a postulate of QM and part of the formalism, not the interpretation. The difference here in interpretation is that this probability distribution is, in the deBB version a statistical distribution of quite definite states, whereas in the Copenhagen interpretation it's truly indeterminate (i.e. the probability does not represent a mere lack of knowledge about the system).

Skippy is wrong. A particle being 'in two places at once' alludes to your basic superposition of states and/or the fact that it does not have a definite location. All interpretations allow for this, and the probability distribution most certainly does have meaning even when the particle is not measured. Unless you perform a specific measurement of a specific property, the wave function does not 'collapse' (if we assume the CI interpretation for the sake of argument). You can perform measurements on a superposition state without destroying the superposition. This is your basic double-slit experiment, where determining 'which slit' will destroy the superposition, but merely observing the interference pattern does not.

5. Aug 22, 2010

Vanadium 50

Staff Emeritus
Check the index of those textbooks. You'll be surprised.

The developers of quantum mechanics knew about waves, and they knew that the intensity is the square of the amplitude. They also knew that the intensity had to integrate to particle number. Born concluded that probability is the way to interpret these facts.

6. Aug 22, 2010

skippy1729

First off, I would like to apologize for my sloppy language. I meant to say something like "the interpretation of the quantum state as representing the probability (amplitudes) of the observer". I never meant to imply that there is some kind "Probability Interpretation of Quantum Mechanics".

Secondly, I qualified my statement about deBroglie-Bohm stating that I am not an expert on this. If they want to maintain that their particles, following definite but unknown trajectories can be in two places at the same time, it is fine with me.

Now we come to CI, where I think you have some confusion. The wave function is the summary of the observers state of knowledge of the system. It is not part of the quantum system. It exists in Hilbert space, not space-time. It does not interfere with slits. It has no ontological content. If my state is psi = alpha * |0> + beta * |1> and I perform a measurement indicating the state represented by |0> then my wave function immediatly becomes psi = |0> because my knowledge of it has changed not due to some sort of physical collapse. Perhaps you are confusing what is sometimes known as the VonNeumann-Copenhagen interpretation.

As to the double-slit example: If the experiment is performed with photons (one at a time), each photon will expose one spot on the film and the overall picture (after many photons) will show the interference pattern. No evidence here of the photon being in two places at once. Performing either the "which slit" or "slit interference" measurement destroys not only the superposition but the entire wave function. When the photon detector clicks or the film is exposed the photon ceases to exist, it has no wave function.

Cheers, Skippy

7. Aug 22, 2010

Staff: Mentor

I'm not an expert on deBroglie-Bohm either, nevertheless I'm pretty sure that it does not maintain that a particle can be "in two places at the same time." I'll leave it to Demystifier et al. to elaborate on this if they see this thread.

8. Aug 22, 2010

unusualname

I'm sure they taught the Born rule in first year back when I studied physics. The best strategy for an undergrad is the "shut up and calculate" one, then once you pass your exams to can contemplate the mystery of it all at leisure.

The wavefunction represents a model of the information available to us about particles, measurements give us values with a probability predicted by the amplitudes of the wave function at those values.

Why the probabilities are distributed like a wave is unknown, but it may fall out of the wavy behaviour of strings and information distributed on a holographic screen, or something even more bizarre, but certainly not mystical or unknowable.

9. Aug 22, 2010

JesseM

Keep in mind that when you talk about position, squaring the wavefunction gives a probability density function, not a normal probability distribution (see here for a quick rundown on discrete vs. continuous probability distributions). Since there are an infinite number of distinct values of x between the walls of the well, there's no way to assign a finite probability to each one and still have all the probabilities add to one! Instead a probability density function is a function that can be integrated between two positions x1 and x2 to give the probability that the particle will be found somewhere in that range (by a measurement that only has a resolution equal to the distance from x1 to x2, for example). So although it is true that the probability density is zero at precisely x=a/2, the density is nonzero at nearby values of x, so there will be some finite probability of finding the particle in any small neighborhood of a/2.

Last edited: Aug 22, 2010
10. Aug 22, 2010

unusualname

Yes, good point, that's true for any continuously valued observable like position. (For other observables, you can have an infinite discrete spectrum, with finite probabilities at each distinct value)

So it doesn't make sense to talk about the probability of finding the particle at a single specific point.

11. Aug 22, 2010

alxm

They do, but their idea of a particle's 'location' is not the same as for other interpretations.

I didn't say any of that or the opposite of any of that. But in any case, the wave function exists in configuration space, not 'space-time' or Hilbert space. The states span a Hilbert space.

Performing a measurement is physical.

Yes there is - the interference pattern. And there are any number of experiments showing quantum superpositions where a particle appears to be in several places at once. Such superpositions were even known well before quantum mechanics (e.g. what chemists called 'resonance')

No, it only determines the state of the wave function, changing it from a superposition to a pure state. Depending on the circumstances, that can evolve back into a pure state. Or be 'erased' and returned to a superposition as in a quantum-eraser experiment.

Yes, a non-existent particle has no wave function. So?

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text:
Measurement of differential cross sections and charge ratios for $t$-channel single top quark production in proton-proton collisions at $\sqrt{s} =$ 13 TeV
Eur.Phys.J. C80 (2020) 370, 2020.
The collaboration

Abstract (data abstract)
A measurement is presented of differential cross sections for $t$-channel single top quark and antiquark production in proton-proton collisions at a centre-of-mass energy of $13~\rm{TeV}$ by the CMS experiment at the LHC. From a data set corresponding to an integrated luminosity of $35.9~\rm{fb}^{-1}$, events containing one muon or electron and two or three jets are analysed. The cross section is measured as a function of the top quark transverse momentum ($p_{\rm{T}}$), rapidity, and polarisation angle, the charged lepton $p_{\rm{T}}$ and rapidity, and the $p_{\rm{T}}$ of the W boson from the top quark decay. In addition, the charge ratio is measured differentially as a function of the top quark, charged lepton, and W boson kinematic observables. The results are found to be in agreement with standard model predictions using various next-to-leading-order event generators and sets of parton distribution functions. Additionally, the spin asymmetry, sensitive to the top quark polarisation, is determined from the differential distribution of the polarisation angle at parton level to be $0.440 \pm 0.070$, in agreement with the standard model prediction.


text:
# What is the difference between preterite and imperfect and the past progressive and present progressive?

###### Question:

What is the difference between preterite and imperfect and the past progressive and present progressive?

### Which part of the scientific method is most likely to lead to changes in a theory? (1 point) O methods o hypothesis O problem 0 results

Which part of the scientific method is most likely to lead to changes in a theory? (1 point) O methods o hypothesis O problem 0 results...

### 23. In which compounds are pairs of electrons shared between atoms? 1 methane 2 lead bromide 3 sodium chloride A 1 only B 2 only с 1 and 3 D 1, 2 and 3

23. In which compounds are pairs of electrons shared between atoms? 1 methane 2 lead bromide 3 sodium chloride A 1 only B 2 only с 1 and 3 D 1, 2 and 3...

### Need Help! Please Due later tonight at 11:59 :))

Need Help! Please Due later tonight at 11:59 :))...

### What does a reader learn from a story's falling action?

What does a reader learn from a story's falling action?...

### I'M HAVING TROUBLE PLEASE ANSWER THE ANSWERS FOR ME PLEASE! You are reporting on a concert that is coming to town tonight. Since the concert is a big event, you have a full minute for your news segment. You have shot all kinds of footage with the band, interviews with the fans, video of the venue and the band’s tour bus arriving and found footage from some of the band’s previous shows. Use the storyboard below to edit all of your footage into a one-minute story. In each box describe what footage

I'M HAVING TROUBLE PLEASE ANSWER THE ANSWERS FOR ME PLEASE! You are reporting on a concert that is coming to town tonight. Since the concert is a big event, you have a full minute for your news segment. You have shot all kinds of footage with the band, interviews with the fans, video of the venue an...

### How did the Korean War change of the Cold War?

How did the Korean War change of the Cold War?...

### What is the recommended daily amount of protein

What is the recommended daily amount of protein...

### What is one negative effect of eating a diet high in fat? A. cravings B. longer life span C. blocked arteries D. decreased VO2 max

What is one negative effect of eating a diet high in fat? A. cravings B. longer life span C. blocked arteries D. decreased VO2 max...

### -2(2a+b-3c)+5(a-2b+c)

-2(2a+b-3c)+5(a-2b+c)...

### Which of the following equation has a graph that has a y-intercept of 4 and is parallel to 3y-9x=24? A.-12x+4y=16 B.9x-3y=-15 C.2y=4x+8 D.7y=14x+7 E.3x-9y=14

which of the following equation has a graph that has a y-intercept of 4 and is parallel to 3y-9x=24? A.-12x+4y=16 B.9x-3y=-15 C.2y=4x+8 D.7y=14x+7 E.3x-9y=14...

### Calculate the molality of a solution formed by adding 4.40 g NH4Cl to 11.0 g of water.

Calculate the molality of a solution formed by adding 4.40 g NH4Cl to 11.0 g of water....

### What are four reasons we need governments?

What are four reasons we need governments?...

### Why was Harry upset at the end of the chapter? He thought his uncle was leaving him. He thought Hans was leaving them. He thought the mountain was collapsing on them. He missed his fiancée.

Why was Harry upset at the end of the chapter? He thought his uncle was leaving him. He thought Hans was leaving them. He thought the mountain was collapsing on them. He missed his fiancée....

### When you multiply 2 decimals, how do you know where to place the comma in the product?

When you multiply 2 decimals, how do you know where to place the comma in the product?...


text:
# Variance of continuous stochastic process

In the paper "Directed Information, Causal Estimation, and Communication in Continuous Time" the author show an example of continuous Gaussian Channel:

Let $\{B_t\}$ be a standard Brownian motion and $A ∼ N(0,1)$ be independent of $\{B_t\}$. Let $X_t ≡ A$ for all t and $dY_t = X_tdt + dB_t$. Letting $J(P, N) = (1/2) ln((P + N)/N)$ denote the mutual information between a Gaussian random variable of variance $P$ and its corrupted version by an independent Gaussian noise of variance $N$, we have for every $t \in [0, T)$:

$I(Y_t^{t+δ};X_0^{t+δ}|Y_0^t) = J(\frac{1/t}{1+1/t},1/δ)$

Question:

It looks that the author have $P(X_t^{t+\delta})=\frac{1/t}{1+1/t}$. How does this come into being?

Some useful backgrounds: in Information Theory,under the case of discrete Gaussian channel $Y = X + n$ where $X ∼ N(0,\sigma_x^2)$ and $n ∼ N(0,\sigma_0^2)$, the mutual information $I(X;Y) = \frac{1}{2} ln(\frac{\sigma_x^2 + \sigma_0^2}{\sigma_0^2})$. The paper mentioned above could be accessible at :http://arxiv.org/abs/1109.0351 (Page 7,example 1). A simplified one could be found at http://circuit.ucsd.edu/~yhk/pdfs/isit2009c.pdf


text:
## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

$z=-\dfrac{16}{3}$
Using the properties of equality, the value of the variable that satisfies the given expression, $\dfrac{2}{3}=-\dfrac{z}{8} ,$ is \begin{array}{l}\require{cancel} \dfrac{z}{8}=-\dfrac{2}{3} \\\\ 8\left( \dfrac{z}{8} \right)=-\dfrac{2}{3}(8) \\\\ z=-\dfrac{16}{3} .\end{array}


text:
## Cryptology ePrint Archive: Report 2004/336

Oblivious Transfer Is Symmetric

Stefan Wolf and Jürg Wullschleger

Abstract: We show that oblivious transfer of bits from $A$ to $B$ can be obtained from a single instance of the same primitive from $B$ to $A$. Our reduction is perfect and shows that oblivious transfer is in fact a symmetric functionality. This solves an open problem posed by Crépeau and Sántha in 1991.

Category / Keywords: foundations / oblivious transfer

Publication Info: Eurocrypt '06

Date: received 1 Dec 2004, last revised 23 Feb 2006

Contact author: wjuerg at inf ethz ch

Available format(s): Postscript (PS) | Compressed Postscript (PS.GZ) | PDF | BibTeX Citation

Short URL: ia.cr/2004/336

[ Cryptology ePrint archive ]


text:
Agda-2.6.1: A dependently typed functional programming language and proof assistant

Agda.Utils.Map

Contents

Synopsis

O(n log n). Rebuilds the map from scratch. Not worse than mapKeys.


text:
# Quantum Gravitational Uncertainty of Transverse Position

Quantum Gravitational Uncertainty of Transverse Position - Descarga este documento en PDF. Documentación en PDF para descargar gratis. Disponible también para leer online.

Descargar gratis o leer online en formato PDF el libro: Quantum Gravitational Uncertainty of Transverse Position
It is argued that holographic bounds on the information content of spacetime might be directly measurable. A new uncertainty principle is conjectured to arise from quantum indeterminacy of nearly flat spacetime: Angular orientations of null trajectories of spatial length L are uncertain, with standard deviation in each transverse direction \Delta \theta \sqrt{l P-L}, where l p denotes the Planck length. It is shown that this angular uncertainty corresponds to the information loss and nonlocality that occur if 3+1-D spacetime has a holographic dual description in terms of Planck-scale waves on a 2+1D screen with encoding close to the Planck diffraction limit, and agrees with covariant holographic entropy bounds on total number of degrees of freedom. The spectrum and spatial structure of predicted quantum-gravitational holographic noise are estimated to be directly measurable over a broad range of frequencies using interferometers with current technology.

Autor: Craig J. Hogan

Fuente: https://archive.org/


text:
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# Ashley's Messageboard!

So here it is, my message board. Let's form a circle, for fun and MATHS! ![]

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LOL:| so how's life? · 1 year, 10 months ago

Filled with hollowness. · 1 year, 10 months ago

Woah why bro¿¿ · 1 year, 10 months ago

How hard is it in 12th grade? ( Since I've just begun 11th grade) · 1 year, 10 months ago

12th is not hard. Infact it is interesting, but the only bad thing is the teacher wants us to memorize each word of the book, which I hate... · 1 year, 10 months ago

Chemistry is never interesting. · 1 year, 10 months ago

That"s true!! · 1 year, 10 months ago

What do you like more :Physics or Math? · 1 year, 10 months ago

I love Maths!! · 1 year, 10 months ago

ohh · 1 year, 10 months ago

You? · 1 year, 10 months ago

Both I like but Math is my favorite:) · 1 year, 10 months ago

How is life in Mumbai? · 1 year, 10 months ago

Actually I don't live in Mumbai live near it here it is quiet good.How's life there? · 1 year, 10 months ago

I live near to the sea. People say it feels great boating in the trisea, But I have done it only once in my lifetime:) · 1 year, 10 months ago

Did you like it? · 1 year, 10 months ago

Your status is cool bro, yeah and your problem WE MEET AT INFINITY is also cool. · 1 year, 10 months ago

Ya and thanx and your's seems fragile as well as strong. · 1 year, 10 months ago

I love it, but its a fact that people don't visit places close to them often, · 1 year, 10 months ago

You mean places close to sea? · 1 year, 10 months ago

Yes! · 1 year, 10 months ago

Contradiction: Mumbai · 1 year, 10 months ago

Ya it is :) · 1 year, 10 months ago

any specific reason? · 1 year, 10 months ago

No. People say like that. For eg: How many times have you visited the tourist spot so close to you? · 1 year, 10 months ago

Many times . · 1 year, 10 months ago

Best of luck for ur boards and iit · 1 year, 10 months ago

Thank you.. Are you preparing for IIT? · 1 year, 10 months ago

Yup · 1 year, 10 months ago

Hi @ashley shamidha I'm Ritu, studying in the XIth grade in Kerala. '' Let's form a circle, for fun and MATHS!'' and physics and chem!! Which stream did you opt? I am studying CS.

How's TN state board compared to CBSE?? · 1 year, 10 months ago

Hi @Ritu Roy I'm studying Maths Biology. The syllabus is almost same for both but I don't like the way of evaluation in TN. They need the exact copy of the book in our answer sheets. · 1 year, 10 months ago

I'm so happy of the friends I got now. I hope this will continue outside this message board also:) · 1 year, 10 months ago

Hey Ashley , you live at the bottom-most town of India! You and your town are pretty cool $$\LARGE \ddot\smile$$. · 1 year, 10 months ago

Thank You.. Have you ever visited Kanyakumari? · 1 year, 10 months ago

Nope ... But I will visit it at least once in my lifetime :) · 1 year, 10 months ago

How many female friends do you have that love mathematics? Please invite them to Brilliant! Staff · 1 year, 11 months ago

Wow.. Thanks for following me!! · 1 year, 10 months ago

Only one.. But she just like maths, don't love it. Why did u ask female? · 1 year, 10 months ago

Share Brilliant with her, and all your friends! Staff · 1 year, 10 months ago

I'm from a rural area. Girls here don't use internet often. · 1 year, 10 months ago

Can we have a proof writing contest ? · 1 year, 10 months ago

What's that? · 1 year, 10 months ago

Calvin sir can give us a difficult problem on proof writing and then we can see whose proof is most accurate and he/she will be declared as the winner. · 1 year, 10 months ago

I accept this:) · 1 year, 10 months ago

He asked about female mathematicians because there are very less girls on brilliant. · 1 year, 10 months ago

So I'm proud that I'm a brilliantian!! · 1 year, 10 months ago

Yaa. Do encourage your friends to join brilliant! · 1 year, 10 months ago

I'm from a rural area. Girls here don't use internet often. · 1 year, 10 months ago

That makes me respect u more! BTW u r in which class?? · 1 year, 10 months ago

+2 · 1 year, 10 months ago

Same here! U aiming for IIT? · 1 year, 10 months ago

Ya, but I'm so scared of it. I don't know whether I have the potential because I'm a Tamil Nadu Board student and most of the students don't clear JEE. · 1 year, 10 months ago

Don't underestimate yourself. If you want u can crack the paper. After all u belong to brilliant. BTW u there on g+? · 1 year, 10 months ago

No... · 1 year, 10 months ago

Then do join g+. There's a group on g+ of brilliant where u can communicate with many people · 1 year, 10 months ago

Comment deleted Sep 10, 2015

Can u give me ur email? · 1 year, 10 months ago

But I have to wait for an year to do all those. But really I'll join! · 1 year, 10 months ago

I can understand that this year is important. All the best for ur future! Goodnight! Felt nice talking to u! · 1 year, 10 months ago

$$\text{Ramanujan}$$$$=1729$$ · 1 year, 11 months ago

Hi, have you ever read, "The Man Who Knew Infinity"? I have been searching for it in internet so long. · 1 year, 11 months ago

You can find it on Scribd · 1 year, 8 months ago

All followers are friends only! Hey, do you want to become a mathematician? · 1 year, 11 months ago

Certainly, thats why I joined brilliant!! · 1 year, 11 months ago

Very nice, so our aims and interests match! · 1 year, 11 months ago

Hey friend , how's life ? · 1 year, 11 months ago

Struggling between board and entrance exams, But luckily I completed my sixteen so sweet last month. How about you? · 1 year, 11 months ago

My life is Like My Status . · 1 year, 10 months ago

You like physics? · 1 year, 10 months ago

Yeah very much .What about you? · 1 year, 10 months ago

Hey! I'm sorry to ask this. You're a boy or a girl? Because I couldn't figure it out with your name. · 1 year, 10 months ago

Of course I am a BOY. · 1 year, 10 months ago

Mine is Mathematics!! · 1 year, 10 months ago

Do you want to give JEE or want to do career in pure Mathematics¿ · 1 year, 10 months ago

Do you like carrots? · 1 year, 11 months ago

What is the motto behind this question, DO YOU LIKE CARROTS¿ · 1 year, 10 months ago

Ofcourse. I'm already in your class! · 1 year, 11 months ago


text:
help-gnu-emacs
[Top][All Lists]

## Re: Hiding references when editing LaTeX?

 From: David Kastrup Subject: Re: Hiding references when editing LaTeX? Date: 27 Feb 2003 17:40:35 +0100 User-agent: Gnus/5.09 (Gnus v5.9.0) Emacs/21.3.50

Oliver Scholz <alkibiades@gmx.de> writes:

> David Kastrup <dak@gnu.org> writes:
>
> > "Felix E. Klee" <felix.klee@inka.de> writes:
> [...]
> >> But font-lock-mode can't hide he "\emph{" and the corresponding "}", right?
> > Well, you could let it use an invisible font or something, but that
> > would probably be not quite convenient for editing.  Perhaps in
> > connection with auto-reveal-mode something could be fudged.
> [...]
>
> Hmm, what would be the cleaner/easier solution: to add that feature to
> preview-latex or to the font-locking in AUCTeX?

preview-latex would need to achieve two different things here: not
display \emph at all, and add text properties to the argument.  The
interface does not currently permit a command and its arguments to be
treated differently, and it might be quite a feat to implement
something like that consistently.  OTOH, it would offer quite numerous
possibilities to mark things up in the Emacs buffer that are hard to
parse correctly otherwise.  For example, if you take the source code
of packages like verbatim.sty, it is almost impossible with all the
catcode changes and whatever going on to let Emacs correctly
determine just what is to be font locked as verbatim and what not.
Using preview-latex would make getting the correct information easy.

> [What do you mean by "invisible font"?]

Text property 'invisible would probably be sufficient.  I am not sure
if font-lock-mode can only select particular faces (in which case one
would use something like invisible-face) or arbitrary text properties.

--
David Kastrup, Kriemhildstr. 15, 44793 Bochum