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# Tag Info

11

Pisier https://arxiv.org/abs/1908.02705 very recently constructed a non-nuclear $C^\ast$-algebra $A$ with the weak expectation property (WEP) and the local lifting property (LLP). By a celebrated result of Kirchberg (see Corollary 13.2.5 in Brown and Ozawa's book) it follows that if $B$ has WEP and $C$ has LLP, then $B\otimes_{\max{}} C = B\otimes_{\min{}} C$...

9

Yes to both.$\newcommand{\Cst}{{\rm C}^*}$ The standard example for the first is: take a discrete group $G$ and let $A$ be its full $\Cst$-algebra, $B$ its reduced $\Cst$-algebra. There is a canonical homomorphism $q:A\to B$ which is injective when restricted to $\ell^1(G)$; but $q$ is injective if and only if $G$ is amenable. So any non-amenable discrete ...

8

There are even commutative counterexamples. Let $A = C[0,2]$ and let $A_0$ be the $*$-subalgebra of all polynomials in $x$. Then let $p: C[0,2] \to C[0,1]$ be the restriction map. (My first example took $A = C[0,3]$ and let $A_0$ be the set of all polynomials in $x$ with rational coefficients and $p: A \mapsto \mathbb{C}$ the point evaluation at $x = e$. ...

5

Go to the Fourier transform picture. Then these become the operators that commute with multiplication by exponentials, which means they commute with all multiplication operators, which means you have just described the operators which become multiplication operators when you conjugate by the Fourier transform.

4

It's not true. For simplicity, suppose $A$ is unital, so that its spectrum is compact Hausdorff. If $X$ is a compact Hausdorff space and $C(X)$ is separable, then you can show that $X$ is second countable. (Let $\{f_k\}$ be a countable dense subset of $C(X)$, and $\{U_n\}$ a countable basis of open sets in $\mathbb{C}$; then the countable collection of ...

3

I think there are some subtle points here about what the "right action" even means. For a general $*$-algebra $A_0$ and a functional $\phi:A_0\rightarrow\mathbb C$, we first of all have to decide what "positive" means for $\phi$. We could take this as being $\phi(a^*a)\geq0$ for all $a$. Then Cauchy-Schwarz holds and we can form $L^2(A_0,\phi)$. Why, ...

2

The problem is still ill-posed, because the set of nilpotent bounded operators is not a vector space. (The sum of two nilpotents need not be nilpotent.) Let's interpret the question as: which algebras admit faithful representations as algebras of nilpotent bounded operators? Note that "nilpotent" usually means $T^n = 0$ for some $n$, not $T^2 = 0$. But ...

1

For separable von Neumann algebras the Haagerup property for $N$ is equivalent to the existence of a real closable derivation $\delta$ such that $\delta^*\delta$ has compact resolvents in $\mathcal B(L^2(N))$. This is true even in the non-tracial case. See Theorem 7.7 in: Martijn Caspers, Adam Skalski, The Haagerup Approximation Property for von Neumann ...

Only top voted, non community-wiki answers of a minimum length are eligible


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# The Unapologetic Mathematician

## The Implicit Function Theorem I

Let’s consider the function $F(x,y)=x^2+y^2-1$. The collection of points $(x,y)$ so that $F(x,y)=0$ defines a curve in the plane: the unit circle. Unfortunately, this relation is not a function. Neither is $y$ defined as a function of $x$, nor is $x$ defined as a function of $y$ by this curve. However, if we consider a point $(a,b)$ on the curve (that is, with $F(a,b)=0$), then near this point we usually do have a graph of $x$ as a function of $y$ (except for a few isolated points). That is, as we move $y$ near the value $b$ then we have to adjust $x$ to maintain the relation $F(x,y)=0$. There is some function $f(y)$ defined “implicitly” in a neighborhood of $b$ satisfying the relation $F(f(y),y)=0$.

We want to generalize this situation. Given a system of $n$ functions of $n+m$ variables

$\displaystyle f^i(x;t)=f^i(x^1,\dots,x^n;t^1,\dots,t^m)$

we consider the collection of points $(x;t)$ in $n+m$-dimensional space satisfying $f(x;t)=0$.

If this were a linear system, the rank-nullity theorem would tell us that our solution space is (generically) $m$ dimensional. Indeed, we could use Gauss-Jordan elimination to put the system into reduced row echelon form, and (usually) find the resulting matrix starting with an $n\times n$ identity matrix, like

$\displaystyle\begin{pmatrix}1&0&0&2&1\\{0}&1&0&3&0\\{0}&0&1&-1&1\end{pmatrix}$

This makes finding solutions to the system easy. We put our $n+m$ variables into a column vector and write

$\displaystyle\begin{pmatrix}1&0&0&2&1\\{0}&1&0&3&0\\{0}&0&1&-1&1\end{pmatrix}\begin{pmatrix}x^1\\x^2\\x^3\\t^1\\t^2\end{pmatrix}=\begin{pmatrix}x^1+2t^1+t^2\\x^2+3t^1\\x^3-t^1+t^2\end{pmatrix}=\begin{pmatrix}0\\{0}\\{0}\end{pmatrix}$

and from this we find

\displaystyle\begin{aligned}x^1&=-2t^1-t^2\\x^2&=-3t^1\\x^3&=t^1-t^2\end{aligned}

Thus we can use the $m$ variables $t^j$ as parameters on the space of solutions, and define each of the $x^i$ as a function of the $t^j$.

But in general we don’t have a linear system. Still, we want to know some circumstances under which we can do something similar and write each of the $x^i$ as a function of the other variables $t^j$, at least near some known point $(a;b)$.

The key observation is that we can perform the Gauss-Jordan elimination above and get a matrix with rank $n$ if and only if the leading $n\times n$ matrix is invertible. And this is generalized to asking that some Jacobian determinant of our system of functions is nonzero.

Specifically, let’s assume that all of the $f^i$ are continuously differentiable on some region $S$ in $n+m$-dimensional space, and that $(a;b)$ is some point in $S$ where $f(a;b)=0$, and at which the determinant

$\displaystyle\det\left(\frac{\partial f^i}{\partial x^j}\bigg\vert_{(a;t)}\right)\neq0$

where both indices $i$ and $j$ run from $1$ to $n$ to make a square matrix. Then I assert that there is some $k$-dimensional neighborhood $T$ of $b$ and a uniquely defined, continuously differentiable, vector-valued function $g:T\rightarrow\mathbb{R}^n$ so that $g(b)=a$ and $f(g(t);t)=0$.

That is, near $(a;b)$ we can use the variables $t^j$ as parameters on the space of solutions to our system of equations. Near this point, the solution set looks like the graph of the function $x=g(t)$, which is implicitly defined by the need to stay on the solution set as we vary $t$. This is the implicit function theorem, and we will prove it next time.

November 19, 2009 Posted by | Analysis, Calculus | 4 Comments

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# Program to find minimum number of operations required to make one number to another in Python

PythonServer Side ProgrammingProgramming

Suppose we have a number start and another number end (start < end), we have to find the minimum number of operations required to convert start to end using these operations −

• Increment by 1
• Multiply by 2

So, if the input is like start = 5, end = 11, then the output will be 2, as we can multiply 2 to get 10, then add 1 to get 11.

To solve this, we will follow these steps −

• ct:= 0
• while end/2 >= start, do
• if end mod 2 is same as 1, then
• end := end - 1
• end := end/2
• ct := ct + 2
• otherwise,
• end:= end/2
• ct := ct + 1
• ct := ct +(end-start)
• return ct

Let us see the following implementation to get better understanding −

## Example

Live Demo

class Solution:
def solve(self, start, end):
ct=0
while(end/2>=start):
if end%2==1:
end-=1
end=end/2
ct+=2
else:
end=end/2
ct+=1
ct+=(end-start)
return ct
ob = Solution()
print(ob.solve(5,11))

## Input

5,11

## Output

2
Published on 05-Oct-2020 07:31:00


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Time Limit : sec, Memory Limit : KB

Greatest Common Divisor

Write a program which finds the greatest common divisor of two natural numbers a and b

Input

a and b are given in a line sparated by a single space.

Output

Output the greatest common divisor of a and b.

1 ≤ a, b ≤ 109

Hint

You can use the following observation:

For integers x and y, if xy, then gcd(x, y) = gcd(y, x%y)

Sample Input 1

54 20

Sample Output 1

2

Sample Input 2

147 105

Sample Output 2

21


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# Which algorithm can be used to reduce dimension of multiple time series?

In my dataset, a data point is essentially a Time series of 6 feature over a year per month so in all, it results in 6*12=72 features. I need to find class outliers so I perform dimensionality reduction hoping the difference in data is maintained and then apply k-means clustering and compute distance.

For dimensionality reduction I have tried PCA and simple autoencoder to reduce dimension from 72 to 6 but results are unsatisfactory.

Can anyone please suggest any other way to reduce dimension of this type of data.

## 2 Answers

So, to be clear, you observe $$N$$ samples with $$P$$ measurements at $$T$$ times (where $$P = 6$$ and $$T = 12$$).

I would first determine whether or not you expect the joint distribution of your $$P$$ features to change over time. For example, if these features are $$X$$, does $$\mathcal{P}(X) = \mathcal{P}(X | T)$$ for all values of $$T$$? If so, you could do something like stack your data over time (so, you'd have $$NT$$ samples of $$P$$ features) and perform PCA, then use the top $$k$$ loadings to rotate the measurements at individual timepoints, leaving you with $$N$$ $$k$$-dimensional time series of length $$T$$. You could choose $$k = 1$$ if necessary, and use it to visually identify outliers or set a simple cutoff. If not, you'd want to be more careful with this technique - especially if the covariance changes. If it's the variance that's changing but correlation is remaining the same, perhaps some procedures can be done (e.g. perform PCA on the correlation matrix/standardized data) and this technique would still work.

What form do you expect your outliers to take? If it's a temporary deviance from the rest of the sample (e.g. only at one or two measurement times), your technique of PCA on the concatenated data will not be very sensitive. The technique I just described would be more sensitive.

• For some cases, these $P$ features might vary over time. For a few cases, they might even show a somewhat cyclical behavior. To elaborate a little, I am working with a collection of land cover images. The $N$ samples represent N pixels, and $P$ features are the 6 bands-(R,G,B,NIR,SWIR-1,SWIR-2). So yes these values might vary over the timespan of 1 year. – Faiz Kidwai Aug 1 '19 at 5:54
• Perhaps this would be a good use case for an autoencoder with a prior over the latent space to enforce smoothness over time (e.g. a GP prior over each individual dimension of the latent space). If $N$ is too large for this to be effective, then perhaps a small term on the autoencoder loss to encourage neighboring timesteps' latent spaces to be close together (e.g. by Euclidean distance) would be effective. – wn0 Aug 9 '19 at 14:29

You can apply the Wrapper Sequential Feature Selection (SFS) algorithm. SFS is a family of greedy search algorithms that are used to reduce an initial d-dimensional feature space to a k-dimensional feature subspace where k < d. The idea of the algorithm is to run any classification algorithm (parameter) based on k_features (parameter). It will select those feature gives the highest accuracy for corresponding algorithm.

clf = svm.SVC()

# Build step forward feature selection
sfs1 = sfs(clf,
k_features=15,
forward=True,
floating=False,
verbose=2,
scoring='accuracy',
cv=5)

# Perform SFFS
sfs1 = sfs1.fit(X_Train, Y_Train)

feat_cols = list(sfs1.k_feature_idx_)
print(feat_cols)

One downside is- you have to fix the desired number of features, the algorithm will give you which those features better fits in terms of accuracy. For your scenario, you have 72 features. Let's say, you want to use only 15 features but you don't know which 15 features among these 72 features. SFS will tell you the best selection. Here, in the above code, feat_cols is the list of those 15 features.


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# Talk:Multibrot set

WikiProject Mathematics (Rated Start-class, Low-importance)
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
 Start Class
 Low Importance
Field:  Analysis

The following discussion was originally at Talk:Mandelbrot_set

## Image of a mandelbrot generalization

Hi there Cuddlyable3, I would appreciate it if you would check this out a bit before revoking the edit, although it may not look like your standard mandelbrot fractal, this is a quite standard image of a generalized mandelbrot fractal. I have placed it in the generalizations section for good reason - this is a mandelbrot with power = (-1.5,0).

A generalized 'Mandelbrot fractal' with complex power: (-1.5, 0). All points shown are on the "Inside" of the lake. The location shown is (-0.350262915827,-0.111522432559) at 820 times magnification

Please give a better reason for removing it other than 'this appears to be'. Danwills (talk) 14:55, 9 June 2008 (UTC)

This image is not notable in any way. It is not a useful illustration of anything in the article. Why was this function and this location and this magnification chosen ? What colouring algorithm has been used ? What exactly is the reader meant to learn by looking at this image ? We could fill pages with such random pretty pictures. I agree with User:Cuddlyable3 - it should be removed. Gandalf61 (talk) 15:13, 9 June 2008 (UTC)
I have added this image as a basic example of the types of things you see when you explore the generalized mandelbrot formula, the video in this section only shows the positive powers, and only colors the points outside the lake, I added this image because it uses a negative exponent, and colors the inside of the lake, there were no other examples of this in the article.
The image is not a composite, and the coloring method is standard "exponential smoothing", if you try it yourself you'll easily be able to reproduce it.
I didn't upload it just to add another pretty picture, (I have plenty of more complicated renderings of things like this) but to show how the generalized mandelbrot formula is capable of generating images that are very different to the standard mandelbrot image, that is what makes it notable. Danwills (talk) 22:56, 9 June 2008 (UTC)
It's still a random pretty picture. It tells the reader nothing, and it has probably been massaged in various ways to make it look interesting. For example, can you explain exactly how z(-1.5) is calculated when producing this image ? Since zq is a multivalued function for non-integer q, there must be some choice of one root rather than another going on somewhere in this process. This arbitary choice is probably what leads to the straight lines and angles that we see in the image. I imagine these patterns change if you change your bailout value. If so, then they are an artefact of the processing, not a natural part of the fractal. Gandalf61 (talk) 08:48, 10 June 2008 (UTC)
Gandalf61, thanks again your rigor! I assure you that I haven't massaged any settings or done anything tricky here. I initially thought as you do - that this may have been a rendering artifact - but I am now quite sure that that is not the case. Changing the bailout in fact does not alter the structure, it is quite stable, and is still apparent even when using different coloring methods. Changing the number of iterations of course has a significant effect, but even then the structure does not vary wildly, but rather changes in the same way that a (2.0, 0) power setting varies as iteration number is varied: more iterations resolves finer details, and vice versa. What this image tells the reader, is that there is more to the dynamics of the mandelbrot formula than what is apparent for power=(2.0, 0), that in fact the dynamics are quite well behaved for the standard setting, compared to the dynamics that result from using negative (real) exponent settings. Danwills (talk) 12:30, 10 June 2008 (UTC)

Danwills thank you for bringing your image to Talk. I don't doubt that we could obtain the image that you obtained, using your software and your settings. While I don't call (-1.5,0) a complex power the subject of extending Mandelbrot's set to complex or negative powers is interesting, and I think it is worth some SOURCED explanation possibly supported by truly explanatory images. One might show the whole lake, and there is no good reason to quote coordinates with a precision quite incomensurate with the magnification. As you see, there is no consensus to add your image to the Mandelbrot Set page, which is already well loaded with images that Benoit Mandelbrot himself could be expected to endorse. Do look at Fractal art and WP:OR.Cuddlyable3 (talk) 09:52, 10 June 2008 (UTC)
Hi Cuddlyable3, I agree that this image is perhaps not the best way to illustrate the generalization of the Mandelbrot fractal formula, I will take it over to the Fractal art article. I also agree that a whole-lake rendering would be better, so I have swapped the image for a zoom out (of the same settings) that shows the entire fractal. Would you accept this as a valid example instead? You have a point about the power not being complex, since there is no imaginary part the power may really only be said to be negative, but isn't it still relevant that the value for the imaginary part of the power is zero? I'm ok with either way of describing it. Thanks for your comments! Danwills (talk) 12:30, 10 June 2008 (UTC)
Danwills your latest image is indeed interesting and I shall let it stand for now. Well done!
Can you help explain the math behind the image i.e. what are the axes (as we show in the "mathematician's depiction" of the basic MS) ? Is this a case of iterating
${\displaystyle z\mapsto z^{d}+c\,,}$
for d < 0 ?
(I hope you will excuse my previous reverts and my qualification "for now". This is because I don't yet see a clearly defined Mandelbrot set or how far a "generalisation" (negative? complex?) may take us. A new page may evolve for this subject, if we pin down exactly what we are describing Cuddlyable3 (talk) 18:50, 10 June 2008 (UTC)
This [1] implies 0>d>(-2) does not give a fractal. Cuddlyable3 (talk) 18:36, 11 June 2008 (UTC)
I think I can understand a bit about why the formula might be thought not to be a fractal under these settings, for one, all the orbits appear to be convergent, and even despite that fact, the boundary of the 'pseudo-lake' that forms is not very complex for 0>d>(-2) - it's mostly a fairly smooth triangular shape. However, the dynamics inside the lake are certainly interesting, and rather self-similar (upon closer inspection) for these power settings. There is an exception to this for d=(-1), which seems to be a discontinuity that makes the formula break down completely. I'm sure there's an obvious mathematical reason for that, but I have not looked into exactly why it would be.
 A nova fractal with relaxation.real set to 3.0 There is a striking similarity between these images submitted by Danwills. Cuddlyable3 (talk) 11:42, 14 June 2008 (UTC) A generalized 'Mandelbrot fractal' with complex power: (-1.5, 0). Colored using smooth coloring

Wow, that is quite interesting Cuddlyable3, I had not even compared those two images myself! The similarity is certainly striking! This is particularly strange because one is a modified Newton (the Nova fractal, with real relaxation increased) and the other is a negative-power Mandelbrot (I'll send you the UPR's if you want to verify this for yourself?). I looked into stretching out the negMandel to better fit the shape of the nova, but it didn't seem to elucidate very much really :\. I think that perhaps some of the similarity is partially coming from the fact that they both have stripes from the gradient repeating, and that this happens in the same areas of the plane, minus a contraction in the mandelbrot case, but I am a bit out of depth in terms of actually analyzing how the orbit dynamics could be similar, I might look into it a bit and let you know what I see :).

adjusted example showing -1.5 power mandelbrot overlaid with the common 2.0 power 'mandelbrot set'

I noticed that the image has been taken off the main article again, is this because you would like to resolve this discussion first? I am happy with that, I'd just like to know what's happening. Can you think of any other images that would make valid additions to the page? I would be happy to run any renders you would like to see.

I asked about the multivalued issue that Gandalf61 raised (which I agree is an interesting side problem to whether this image is notable), on the UltraFractal mailing list, and the author of the Nova formula, Damien M Jones, gave me a good non-technical response:

my question was:

I wonder if anyone can help to answer the question that gandalf61 asked here, of whether the discontinuities in a neg-mandel like this are coming from sensitive selection of one root over another by the formula?

the response from Damien:

It's a valid question, because as gandalf61 pointed out, the log function for complex values is multivalued. If you think about complex numbers in polar coordinates rather than rectangular coordinates, it's easier to see. Most implementations I've seen (including the one in UF) take the "simplest" log. The actual full multi-valued log function is like a crazy piece of spiral pasta centered at the origin, sticking straight out of the plane; we cut a small section of that pasta so that when we look straight down at the plane, none of the pasta overlaps itself. We choose the piece that's actually touching the origin and has equal amounts above and below it. Choosing any other part of the pasta is "valid" and will produce different fractals; Jim Muth over on the FractInt list has done some exploring with formulas that adjust the "cut" and found some interesting results.
In a sense UF's choice is "arbitrary" but it's arbitrary in the same way that we restrict the range of the arcsine function to the "simple" -pi/2 <= asin(n) <= pi/2 even though there are other "valid" results. And since this is the "most natural" place to make the cut, most implementations make it in the same place. Arguing that it's an artifact is incorrect because similar details exist regardless of where the actual cut is.

I understand what Damien is saying here and I think it makes it clear that the negMandel image is not just a rendering artifact. You have to resolve the multivalued problem somehow, and this is the most common way of doing it. I could imagine a rendering solution that takes the other solutions into account, but I have never seen anything like that in fractal rendering software yet.

Our interpretation of WP:V policy must be to provide definitive information that enables any suitably skilled person to understand and duplicate these plots. Danwills if you are using software that you did not write then it may be difficult to meet those criteria using hearsay. Please inspect the plots and pseudocode that I have added to the article for positive and negative real integer powers of the complex variable, which do not seem problematic.
Difficulty arises with fractional i.e. non-integral real powers, such as -1.5 that you have presented, because of the discontinuity of roots of a complex number. Does your superposition of the classic Mandelbrot set (rotated so the imaginary axis points sideways ! ) mean that your triangular figure is plotted on identical axes ? I must ask because you use the word "adjusted". Your correspondent Damien has, I think, illustrated the problem more than the implementation of a solution.Cuddlyable3 (talk) 15:48, 20 June 2008 (UTC)
You are really welcome to add these ideas/algorithms to the Wikibook about Fractals. [2] Paxinum (talk) 00:31, 17 July 2008 (UTC)

## Generalising to complex power

If we are to generalise Mandelbrot's iteration to powers that are complex numbers, which I don't see done yet, then can someone please clarify the actual coding ? I suppose one defines, for complex a and b, the general complex power ab of a nonzero complex number a as

${\displaystyle a^{b}=e^{\log(a^{b})}=e^{b\cdot \log a}.\,}$.

I see that a can be in polar form (${\displaystyle {r,theta}}$) so

${\displaystyle \log a=\log r+i\theta \,,}$

and thus

${\displaystyle a^{b}=e^{b\log a}=e^{b(\log r+i\theta )}.\,}$

If b is decomposed as c + di, then it seems the formula for ab can be written more explicitly as

${\displaystyle \left(r^{c}e^{-d\theta }\right)e^{i(d\log r+c\theta )}=\left(r^{c}e^{-d\theta }\right)\left[\cos(d\log r+c\theta )+i\sin(d\log r+c\theta )\right].}$

Is there an expert reading this who can put the above into practical software code? Cuddlyable3 (talk) 11:42, 11 June 2008 (UTC)

Problem here is that the complex logarithm "function" is not single valued; log(a) is a set of complex numbers that differ by multiples of 2πi. If b is an integer power then this does not matter because blog(a) is another set of values that differ by multiples of 2πi, and so exp(b log(a)) has a single value, which we define to be ab. But if b is not an integer then ab is not single valued (simple example is 1½ has two values; +1 or −1). If we select one value rather than another then we are introducing an arbitrary branch cut into the function, and ab then takes on very different values for values of a that are close to one another but on different sides of the branch cut. This creates an arbitary discontinuity in the function - I believe this discontinuity is the source of the straight line artefacts that we see in the −1.5 power image in the section above. Take a different branch cut in your iterations process and you get a different image.Gandalf61 (talk) 13:08, 11 June 2008 (UTC)
Indeed. Example (1 + i)½ is even worse (and there are 3 more quadrants in the complex plane to handle. Seeking a complex power of a complex variable may lead us to rush into quarternions where I fear to tread. Cuddlyable3 (talk) 15:59, 20 June 2008 (UTC)
One can visualise the problem with fractional powers this way. Note that positive (negative) integer valus of d in the iteration yields plots that have (d-1) ((d+1)) fold rotational symmetries. A plot for a non-integer value of d would therefore need to have a non-integer fold rotational symmetry which is AFAIK impossible.Cuddlyable3 (talk) 09:30, 17 July 2008 (UTC)
Hey guys, I went ahead and put one in: z^n+c = |z|^n * cis(n*θ) + c, where cis(θ) = cos(θ) + i*sin(θ), |z| is the magnitude of z, sqrt(x^2 + y^2) for z = (x,y), and θ is the angle made with the positive x axis, atan2(y,x). It works in my code. Minime12358 (talk) 23:56, 15 November 2012 (UTC)

## Generalizations: Positive powers illustrations wrong?

Am I mistaken or do the illustrations for the positive powers go a little wrong from z<-- z5 + c onwards? Playing with some of my own code, based on what's in this page, I get something akin to this [3] for z<-- z5 + c. davep (talk) 16:41, 2 July 2008 (UTC)

It is. I left a plea to the author of the images on his talk page for recreating correct images in the same style (http://en.wikipedia.org/wiki/User_talk:Cuddlyable3#Mandelbrot_set_Z.5En_pictures_set). Otherwise we could just fix the descriptions, or redo all the images. BTW, same error is with negative powers. --89.113.78.50 (talk) 15:38, 15 July 2008 (UTC)
Well caught. You are right. Do go ahead and change the picture captions for now. I can provide correct pictures for d = 5, 6, -5 and -6 later. These will be fixed on the new page Multibrot sets. Cuddlyable3 (talk) 16:33, 28 September 2008 (UTC)

## Negative power illustrations are wrong.

The illustrations are wrong. If one increases the bailout, the images change. This is because the negative-integer-power sets are actually the complements of "dusts" of points that, at some point in their orbit, reach 0, at which point they then instantly explode to infinity (or becomes undefined if we do not define division by 0). (Note we should start iterating with z = c, not z = 0, as then points go to infinity on the very first loop.) These dust sets are much more detailed and do not look "blobby". That one "Nova Fractal" image in the thread a little ways above this one (the big one that has the funny 3-pointed thingy on it) shows approximately what it looks like, although the "Nova Fractal"-specific features of it make it ungood for the article. mike4ty4 (talk) 08:02, 29 July 2008 (UTC)

Yes, they are wrong (out of the reasons you mentioned ;-). "Correct" pictures may be created by using the Lyapunov-exponent. Ken Shirriff describes how to do it at http://www.righto.com/papers/frac3.ps . An example of a picture may be found at http://www.righto.com/fractals/neg.html (left side). I can provide such pictures (http://lostindiversity.junetz.de/mintwo.png) but I'm not quite sure if the iteration z->z^-d + c even fits into the topic. Maybe better leave it out or create a new topic "generalized mandelbrot sets". Enyaw ecurb (talk) 21:10, 30 July 2008 (UTC)
This is right. But I'm not sure whether or not it should be left out or simply have the pictures replaced with more accurate ones. 70.101.147.16 (talk) 08:25, 31 July 2008 (UTC)
I am not convinced that the negative power illustrations are wrong or inaccurate. They do however reflect the (arbitrary) bailout threshold chosen which for consistency is the same (=4) for all the examples. It may be more correct to state When d is negative the perimeter of the set is not fractal and tends towards a circle whose radius squared is the chosen threshold ( =4 ).
mike4ty4 I believe you when you point out that there is more detail to be found in the area within the set. (That has been called the lake.) At present the article just calls the behaviour there "interesting complicated".
Enyaw ecurb the d=-3 d=-2 example looks similar to Ken Shirriff's example picture (also "blobby") so I don't see a problem. I agree that we should consider collecting these generalised-power examples into a new topic, not least because they overload the present article. I suggest as a title "Multibrot sets". Cuddlyable3 (talk) 12:30, 31 July 2008 (UTC)
Shiriffs picture is d=-2 (I think you mean that ;-). However you are correct when you say that the pictures aren't too wrong because with increasing treshold they will tend too look more and more like the ones generated with lyapunov-techniques. I'm saying they are wrong because of "my" definition of the sets. If you define M by the points which don't go over a certain threshold the pictures are "correct" but you will get problems because infinty itself is the critical point of a set (in fact a fixed point). Rather define M by the connectedness or by the points going to a stable cycle. Then the perimeter of the set is not the threshold but infinity (and the pictures are (rather bad) approximations of the "real" set). Enyaw ecurb (talk) 14:25, 31 July 2008 (UTC)
Enyaw thank you for the correction d=-2 :-) . I agree that thresholding is the source of the "artificial" perimeter. Changing that approach needs more changes to the basic escape time algorithm than just a substitution in its inner loop, so I think you have provided here more incentive for extracting these cases to use in building a new article. Cuddlyable3 (talk) 19:08, 1 August 2008 (UTC)
And because they are so bad -- much worse than the pictures of the "vanilla" M-set, I propose they be replaced with better ones generated using, say, the Lyapunov technique or other technique (practically any algorithm coloring based on the z-values attained during the iteration will show the underlying shape in some way or another. Although Lyapunov methods seem to provide the clearest pictures.). mike4ty4 (talk) 07:43, 1 August 2008 (UTC)
Mike4ty4 better pictures are always welcome. Are you able to provide any? If so, it's a good idea to bring the picture first to this Talk page. We might need to ask you to explain a Lyapunov method, and whether the result fits best here, at Lyapunov fractal or at Fractal art. Cuddlyable3 (talk) 19:08, 1 August 2008 (UTC)
Here are some (hopefully better) examples. The use of the Lyapunov-Exponent for the generation of Mandelbrot sets is described in the paper I linked above but in short the Lyapunov-Exponent is the error growth-rate of a given sequence. You calculate the iteration sequence with N iterations, then calculate the Exponent with 1/N * sum(ln(|f'(Z)|)) and if the Exponent is negative the sequence is stable. The white pixels in the pictures are the parameters c for which the Exponent is positive aka unstable. The pictures are additionally colored after the periods of the cycles which the orbits are attracted to. All points colored dark-blue (outside) are attracted by a fixedpoint, all points in the middle (lighter blue) are attracted by a period 2 cycle and so on. The pictures have not much to do with Lyapunov fractals (except the calculation of the exponent somewhere in the routine) and are not really beautiful enough to fit into Fractal art. Enyaw ecurb (talk) 12:58, 2 August 2008 (UTC)
Using an on-line postcript viewer I did not find the figures for Shirriff's article at http://www.righto.com/papers/frac3.ps. Enyaw your images are an improvement, thanks. It's a revelation to see at very high magnification the multitude of micro Mandelbrots ( self-similarity par excellence!). I might quibble that bounded (in set) pixels could be coloured black rather than dark blue.Cuddlyable3 (talk) 13:48, 2 August 2008 (UTC)
Not only the dark blue part is in the set, every colored pixel is part of the set as they go to a stable cycle (white = not in the set). I just like to color this periodicity cause it shows the relation between the geometry of the set and its dynamics. However I can color the set black and try to give the white (=outer) part some kind of coloring to resemble the other pictures (d=3,4,5). On the other hand I can make pictures like above for d=3,4,5. If we put them in the article the pictures used should look alike.Enyaw ecurb (talk) 18:32, 2 August 2008 (UTC)

## Pseudocode

Hi. What do you think about moving pseudocode to wikibooks ? --Adam majewski (talk) 20:49, 30 September 2008 (UTC)

The Escape Time Algorithm pseudocode can be the same for Mandelbrot set and Multibrot sets. Provided it is clearly linked from both these articles, I don't think it matters whether it is located in Wikipedia or in the Wikibook on fractals. Cuddlyable3 (talk) 20:27, 1 October 2008 (UTC)
Wikipedia is not a manual, guidebook or textbook --Adam majewski (talk) 21:28, 1 October 2008 (UTC)

There is a page for multibrot sets diffrent then a page for Mandelbrot set for complex quadratic polynomials --Adam majewski (talk) 21:32, 1 October 2008 (UTC)

These sets are defined by a recursion performed on each value in the complex plane i.e. at each pixel of an image. We have to explain what that recursion is. That is why the article shows a simple pseudocode. I hope the Wikibook on fractals will be developed in the form of a tutorial textbook, ideally with examples in a common coding language that the student can compile and run. The pseudocode does not work that way since for transparency it is oblivious to language, variable type, hardware and image size.Cuddlyable3 (talk) 16:29, 3 October 2008 (UTC) You are right. OK. --Adam majewski (talk) 15:39, 6 October 2008 (UTC)

## Is Multibrot set really a Wikipedia article ?

The Mandelbrot set for ${\displaystyle z^{-2}+c}$
The Sea Horse Valley in the Mandelbrot set for${\displaystyle 1/(1-z^{2})+c}$

It seems to me that this article is of no interest to any Wikipedia reader - it ought to be reduced to a few lines and moved back to Mandelbrot set. It contains unverifiable (and wrong) personal observations like "the sets appears ...", an obscure formula (what is Z'?) and all the references are to Internet sites. On this talkpage is advertised for a correct picture of the Mandelbrot set for ${\displaystyle z^{-2}+c}$, and here you have such a picture (drawn by distance estimation). Anybody is welcome to include it in the article, but I will not do that for the following reason: some days ago I made this contribution to the Multibrot article:

The Mandelbrot set for ${\displaystyle z^{2}+c}$ has lots of beautiful localities, but the Mandelbrot set for ${\displaystyle z^{-2}+c}$ has none. The reason is, that a Mandelbrot set must be constructed from two different critical points of the function, and ${\displaystyle z^{2}}$ has two critical points, namely 0 and ∞, but ${\displaystyle z^{-2}}$ has only ∞ as critical point (see the definition in Julia set). We can as well draw a Mandelbrot set from only one critical point, but then it will have empty interior: we see only a colouring and the boundary. And the picture for ${\displaystyle z^{-2}+c}$ has no aesthetic value. However, if we replace ${\displaystyle 1/z^{2}}$ by ${\displaystyle 1/(z^{2}+a)}$, there will be two different critical points, namely 0 and ∞, and we can draw a Mandelbrot set having just as many interesting localities as the usual Mandelbrot set.

But instantly my text and image were deleted (by Cuddlyable3), for the reason that "the section is not the subject of the article". (Gertbuschmann (talk) 19:16, 25 April 2010 (UTC))

The term Mandelbrot set must be reserved for set(s) actually introduced by Benoit Mandelbrot. It is incorrect to call other iterated sets "mandelbrot sets" without reputable sources. Your monochrome figure here for ${\displaystyle z^{-2}+c}$ has a misleading file name Juliasetsdkpict55.jpg and is described as "Mandelbrot set". It would add nothing to the figures already in the Multibrot set article. Its calculation by distance estimation could reveal finer detail than the basic Escape Time algorithm but not in such a small image.
The first sentence of the article shows that z is a "value in the complex plane". You are wrong to call this text "the sets appears[sic]..." an "unverifiable" and "wrong personal observation" because the circular perimeter is objectively visible and must be explained as being a product of the Escape Time algorithm and not of the set itself.
Your quest is to seek images with "interesting localities" and reject others you think have "no aesthetic value". Your image here that enlarges a part of ${\displaystyle 1/(1-z^{2})+c}$ is pretty. The section Mandelbrot set#Other non-analytic mappings could easily be overloaded with such beautiful examples. Your image may be acceptable at Fractal art and/or Wikimedia Commons. There is a Wikibook on fractals that needs more how-to contribution.
The 4th Reference in the article is I believe Ken Shirriff's published paper and not just a website. Cuddlyable3 (talk) 17:01, 26 April 2010 (UTC)
For the use of the name Mandelbrot set, see my comment on the page Talk:Mandelbrot set.
My remark about the difference between ${\displaystyle 1/z^{2}}$ and ${\displaystyle 1/(z^{2}+a)}$ is interesting, because in the same way as the usual Mandelbrot set is the prototype of the Mandelbrot sets where the iterations are towards ∞, the Mandelbrot sets for ${\displaystyle 1/(z^{2}+a)}$ (a <> 0) are the prototype of the Mandelbrot sets where the iterations are towards finite cycles, and I propose this family of Mandelbrot sets studied more closely.
My pictures are nothing, they are produced in a minute and have nothing to do with fractal art, but possibly this picture is the first ever made of these Mandelbrot sets.
Yes, z is a value in the complex plane, but what is Z'? - the derivative of z in the sequence with respect to c? It is not true that the sets for d negative "actually extend in all directions to infinity". The names of my uploaded pictures start with the name of my Internet site (on aesthetics). (Gertbuschmann (talk) 15:46, 28 April 2010 (UTC))
I believe by "critical points" you mean the same as what are usually called attractor and repeller points, but please correct me if I am wrong.
You say your pictures are nothing, they are produced in a minute and have nothing to do with fractal art. I disagree with all 3 statements! (or with 2 of them if you really have such a fast computer).
what is Z'?. It was a misprint for z, no differentiation intended. I have fixed it though the font is not quite right.
You say It is not true that the sets for d negative "actually extend in all directions to infinity" but I find I can enlarge indefinitely the apparent outer boundary of the sets by increasing the bail-out value shown in this part of the pseudocode:
x*x + y*y <= (2*2)

Cuddlyable3 (talk) 07:54, 29 April 2010 (UTC)

The concept of critical point is defined in Julia set, and its significance lies in that each Fatou domain contains a critical point. A Mandelbrot set is constructed from two critical points, and if the family of iterations is of the form ${\displaystyle f(z)+c}$ for ${\displaystyle f(z)}$ rational, then the Mandelbrot set is of bounded extent (easy to prove). A Julia set can reach out to infinity (as the Julia set for a Newton iteration, for instance). (Gertbuschmann (talk) 15:45, 30 April 2010 (UTC))
I don't want to stir up a big discussion here but my two cents: ∞ is a critical point of the iteration z → 1/z^2+c. It also maps to itself. Therefore it is part of the set which is indeed not a "Mandelbrot-set" but I would rather have defined as the set of all points c with stable orbit. Of course the set is bounded but not towards ∞. It contains ∞ and extends to ∞ on an riemannian metric. Btw: I still think my pictures posted above are a quite correct representation ;-) But back to topic: In my opinion half of the article should be moved to wikibooks: pictures and code. Just keep a few notes here: Who coined the name, what it is, etc. Enyaw ecurb 84.177.59.107 (talk) 11:52, 1 May 2010 (UTC)
The Mandelbrot sets for ${\displaystyle 1/z^{p}+c}$ are empty (degenerated), because they are constructed from only one critical point. But you can still colour the plane. In the (correct) pictures above, drawn by the Lyapunov-Exponent method, the plane is coloured after the periods of the attracting cycles. The white is the boundary of the degenerated Mandelbrot set. You could as well have coloured the plane by choosing a small number ${\displaystyle \epsilon }$, and colour after the (real) number of iterations needed to bring the sequence into distance ${\displaystyle \epsilon }$ from the attracting cycle. The radius of the Mandelbrot set is 2 for p = 2, and I think that this radius converges to 1 for p converging to infinity. (Gertbuschmann (talk) 03:02, 3 May 2010 (UTC))
Gertbuschmann you persist in calling these sets Mandelbrot sets and disregard the title of the article which is properly sourced. I don't see any consensus arising for your indiscriminate usage of "Mandelbrot set" nor of "degenerated Mandelbrot". When the exponent is negative the set of values whose iterations are bounded is empty about the origin. The border of the set is simply star-like and not fractal, and the set extends to (a singularity at) infinity. Inside the set border the colour contours in the article are made as you describe. Bail-out radius (your ${\displaystyle \epsilon }$) was arbitrarily made 2 in the code snippet above for convenient computation. I don't know whether you are now convinced that the green outer parts of all the grouped images are only an artifact of that choice and not a characteristic of the actual set, but that is demonstrably true. I understand your last statement, where you choose to change to different symbol p from the article (so p = -d), as referring to the maximum inner radii of the empty part of the set i.e. the star tips. You may be right. Cuddlyable3 (talk) 16:45, 3 May 2010 (UTC)
The word "Mandelbrot set" is so frequently used for a construction with another family than ${\displaystyle z^{2}+c}$, that there is agreement on this use of the word, and Wikipedia must be in accordance with the world. I cannot see any reason for giving the constructions other names than "the Julia set for f(z) = ..." or "the Mandelbrot set for f(z) = ..." (besides I find the name Multibrot ridiculous). Here I will (like Mandelbrot himself) use the letter M for the construction, and say the following:
For the family ${\displaystyle 1/(z^{d}+0.000001)+c}$ (and critical points 0 and ∞), the set M is non-empty, and it looks like my black-and-white picture above. For ${\displaystyle 1/z^{d}+c}$ (and critical point ∞ two times), the set M is empty, but we can colour a point c in the plane by the real iteration number, which is calculated in this way: starting in ∞ we iterate a large number of times (by ${\displaystyle 1/z^{d}+c}$), and if the terminus is an attracting cycle, we perform the same iteration again, but now stopping when the point is within distance ${\displaystyle \epsilon }$ (= 0.00000001, for instance) from the cycle. The number of iterations needed to this is divided by the order of the cycle, and a number in the interval [0, 1[ is subtracted (as described in Julia set). The result is the real iteration number, which is connected with the potential function. If we want a picture in black-and white, we must use the formula for the distance estimation (derived from the potential function), because the sets M are without interior points.
These sets M (for d = 2, 3, 4, ...) are all lying inside a circle of radius 2, and I assert that the radii of the (outer) perimeter-circles of the sets converge to 1 for d converging to ∞.
I understand that you define the Multibrot sets for negative powers as the complement of my sets M, and then of course the Multibrot sets for negative powers (contrary to the case of positive powers) contain the point ∞.
The bail-out radius 2 is without sense in the constructions for negative powers.

(Gertbuschmann (talk) 02:47, 4 May 2010 (UTC))

The term M-set might mean Menshov set and I don't see your usage of it to mean Mandelbrot set noted in any article, though maybe it should be, with its source. Yes indeed, we define the set as comprising those values whose continual iteration remains bounded and that is not your sets M. Yet I see no reason to disagree with your assertion about the radii of the sets for d = 2, 3, 4, ... because we are talking about the same sets for positive d. (I think I was confused by your change to the symbol p. I am further confused now you substitute d for p. Please stick to the sign of d used in the article.).
The names Multibrot and recently Mandelbulb have both arisen in reliable sources so this is where Wikipedia must follow the world, even if it sounds ridiculous. WP:AfD is worth reading if you believe there would be consensus to delete the article; there was a proposal to delete another fractal article but the decision was keep. Your black-and-white picture is for d= -2 but I see that you use a different algorithm to generate it than the Escape Time algorithm described in the pseudocode in the article. That algorithm does not involve arbitrary tiny offsets.
A bail-out radius is essential for operation of the Escape Time algorithm. In constructing the maps for negative exponents one can, in principle, set the bail-out radius as large as one likes with no benefit other than slower computation (or putting the outer border artifact somewhere out of sight). For positive exponents, increasing the bail-out threshold gains greater detail in the fractal border of the set. Cuddlyable3 (talk) 11:48, 4 May 2010 (UTC)
My set M for the family ${\displaystyle 1/z^{d}+c}$ (coloured black) is precisely the same set as the set coloured white in the corresponding picture by Enyaw ecurb. When c is numerically large, the equation ${\displaystyle w=1/w^{d}+c}$ has one and only one solution, namely a w very near c. Therefore, for such a c, the sequence generated by ${\displaystyle 1/z^{d}+c}$ and starting in ∞, converges to this solution w. That is, all the points c outside a circle of sufficiently large radius are contained in the domain coloured dark-blue in the picture. Now I have proved, that the sets in question are lying within a circle (apparently of radius 2), that is, that they do not grow towards infinity for increasing bail-out radius. (Gertbuschmann (talk) 09:32, 5 May 2010 (UTC))
Ah, I think I get it. Your M is contained exactly in the points not converging to infinity, which is why you state the need of a second critical point (which is quite correct as Julia-Sets are defined over this kind of dichotomy). Silly me, I did define (as I pointed out above) the set M' as the set of points with a stable orbit on a riemannian metric (containing ∞), which seems to be in the case of z^2 M=M' (but not proven). As far as I'm concerned I think this is a valid extension of the normal definition (which again is not published and citable knowledge as far as I know). Your "M", the white part in my pictures, black in yours are the points that do not enter a stable orbit and in lack of another critical point _never enter a stable orbit_. It follows that this set C \ M' contains exactly the parameters c for which the "Julia-Set" z->z^-2+c is the entire complex plane and all periodic orbits are repelling... (try it, nice random noise images) Enyaw ecurb 84.177.114.192 (talk) 21:14, 6 May 2010 (UTC)

But well, back to topic... z^d with d<0 are not "Multibrot-Sets" anyway (which by the way is maybe a ridiculous name, but a correct one) and seem to be of low mathematical interest (Shiriffs "paper", two from Wang, thats it). They can not be defined over the connectedness locus of these families (which is afaik "the definition" of Multibrot sets, e.g. [4]). Move them to wikibooks or wherever and stick to the basics: If someone stumbles over the name "Multibrot" he should be able to find out _what it is_. Not what we assume, discuss or fight about it to be. Enyaw ecurb 84.177.114.192 (talk) 21:14, 6 May 2010 (UTC)

The gallery below shows details of the first quadrant of the multibrot set for the iteration z ↦ z−2 + c, all rendered with the Escape Time algorithm. Black is the set defined in the article.

Cuddlyable3 (talk) 23:19, 6 May 2010 (UTC)

That the function has only one critical point is not the essential. Mandelbrot had to use two critical points because he had no other colour than black, and it is best to use two points because you then (usually) get a black domain and more details, but if you use distance estimation or a rich colour scale, you can be content with only one critical point. The first picture below shows the Mandelbrot set constructed from the critical point 0 (in the same way as my set M above - this set (without interior points) is the closure of the set of points c such that the sequence does not converges towards an attracting cycle).
The two pictures show Julia sets for iterations of the form ${\displaystyle 1/z^{2}+c}$. And as you see, these families (Multibrot's for negative powers) are not of "low mathematical interest". Outside the star-like figures, the Julia sets are disconnected, but inside they are connected, and (as it is the case for the Julia sets for positive powers) they divide up the plan in just as many parts as the order of the attracting cycle (in the pictures, the orders are 3 and 4). A Julia set for a rational complex function is always without inner points, and can thus never be the entire plane (for a transcendental function, such as sin(z), a Julia set can be dense in the plane).

(Gertbuschmann (talk) 15:53, 7 May 2010 (UTC))

The pictures of the Julia-Sets are good! However, I suspect the respective parameter c is from "my M'". Try one out of "my white area". For J=C, I will not answer directly, just cite "The beauty of fractals", p. 55: "Let R(x) = ((x-2)/x)^2, then J=C." The discussion follows. If you do not own the book, I refer to Milnors' "Dynamics in one complex variable", where I faintly remember it being an example. May be wrong on that though. With "low mathematical interest" I meant that there is not much published matter on this family of iterations. As far as I know. You may prove me wrong on that. Enyaw...
The Mandelbrot set image above looks like it is made by colouring internal distance estimates and is not for use in this article. The two continuously coloured Julia set images are beautiful. There is no suitable neologism for them like "multijulias". They could illustrate a new section here or as the article Julia set is non specific about the function iterated it might also be a home for them. Cuddlyable3 (talk) 12:46, 8 May 2010 (UTC)
The domain inside the Mandelbrot set is coloured in the way explained above, that is, first an iteration a large number of times, and then the same iteration again, but now set to stop when the distance from the cycle is smaller than ${\displaystyle \epsilon }$. My first Julia set above (for ${\displaystyle 1/z^{2}+c}$) is for the cusp of one of the largest cardioids, that is, one of your white points. For points farther inside the white, the Julia sets become more dense, but they will not expand. It is true that you can construct a rational function such that the Julia set is the entire plane (namely if all the critical points are iterated into repelling points). I have drawn Julia sets for ${\displaystyle ((z-2)/z)^{2}+c}$, for c near 0, and the Julia sets grow in size and complexity when c converges to 0. (Gertbuschmann (talk) 14:28, 8 May 2010 (UTC)) —Preceding unsigned comment added by Gertbuschmann (talkcontribs) 14:27, 8 May 2010 (UTC)

## Questionable escape conditions

The generalizations of the Mandelbrot set used throughout this article use same escape condition as the z->z^2+z_0 case. I am not sure this is appropriate. For exponents >= 2, this works as large numbers will be dominated by the exponent and the added constant will have little effect on the orbit. However, for negative exponents, the sequence may very well enter into a repeating sequence that causes it's magnitude to alternately exceed the threshold and then become small again. Consider the sequence z-> z^-2+z_0 starting at 0.1. we find z_1 = 100.1, z_2 = 0.1001, z_3 = 99.9007, z_4 = 0.1001, and so on. The sequence continues alternating between numbers on the order of the starting value, and numbers that are much larger. It can't be said that the number ever really escapes, that is, becomes dominated by the exponent. All these techniques do is naively apply an incorrect escape condition that is appropriate only to the n>=2 case. If the escape conditions are modified to be more appropriate, you will find that many of these fractals do not really have any significance. —Preceding unsigned comment added by Willyshop (talkcontribs) 16:25, 13 October 2010 (UTC)


text:
# Absorbing element

In algebra, an absorbing element or a zero element for a binary operation has a property similar to that of multiplication by zero.

Formally, let $\star$ be a binary operation on a set X. An element O of X is absorbing for $\star$ if

$O \star x = O = x \star O \,$

holds for all x in X. An absorbing element, if it exists, is unique.


text:
# reinforcement learning and optimal control draft

## 09 Dec reinforcement learning and optimal control draft

Furthermore, its references to the literature are incomplete. A 13-lecture course, Arizona State University, 2019 Videos on Approximate Dynamic Programming. Publisher: Athena Scientific 2019 Number of pages: 276. This is of particular interest in Deep Reinforcement Learning (DRL), specially when considering Actor-Critic algorithms, where it is aimed to train a Neural Network, usually called "Actor", that delivers a function a(s). /Length 15 Initially, the iterate is some random point in the domain; in each iterati… /Filter /FlateDecode /Matrix [1 0 0 1 0 0] They operate in an iterative fashion and maintain some iterate, which is a point in the domain of the objective function. /Subtype /Form We note that soon after our paper appeared, (Andrychowicz et al., 2016) also independently proposed a similar idea. Discounted reinforcement learning is fundamentally incompatible with function approximation for control in continuing tasks. The date of last revision is given below. /Matrix [1 0 0 1 0 0] Reinforce- ... Dr Gordon Cheng reviewed an earlier draft. Videos and slides on Reinforcement Learning and Optimal Control. endobj /Type /XObject Reinforcement Learning and Optimal Control A Selective Overview Dimitri P. Bertsekas Laboratory for Information and Decision Systems Massachusetts Institute of Technology March 2019 Bertsekas (M.I.T.) /FormType 1 Ordering, Home (A “revision” is any version of the chapter that involves the addition or the deletion…, Reinforcement Learning: a Comparison of UCB Versus Alternative Adaptive Policies, A reinforcement learning approach to hybrid control design, A projected primal-dual gradient optimal control method for deep reinforcement learning, A Nonparametric Off-Policy Policy Gradient, Constrained Reinforcement Learning for Dynamic Optimization under Uncertainty, Multiagent Value Iteration Algorithms in Dynamic Programming and Reinforcement Learning, DDPG++: Striving for Simplicity in Continuous-control Off-Policy Reinforcement Learning, Multiagent Reinforcement Learning: Rollout and Policy Iteration, Apprenticeship Learning using Inverse Reinforcement Learning and Gradient Methods, Policy Gradient Methods for Reinforcement Learning with Function Approximation, Reinforcement Learning From State and Temporal Differences, Reinforcement Learning Algorithm for Partially Observable Markov Decision Problems, Analysis of Some Incremental Variants of Policy Iteration: First Steps Toward Understanding Actor-Cr, Theoretical Results on Reinforcement Learning with Temporally Abstract Options, On-line Q-learning using connectionist systems, View 4 excerpts, cites methods and background, Encyclopedia of Machine Learning and Data Mining, By clicking accept or continuing to use the site, you agree to the terms outlined in our. The overall problem of learning from stream REINFORCEMENT LEARNING AND OPTIMAL CONTROL BOOK, Athena Scientific, July 2019. For several topics, the book by Sutton and Barto is an useful reference, in particular, to obtain an intuitive understanding. 34 0 obj /Resources 33 0 R 2019. 32 0 obj To explore thecommon boundarybetween AI and optimal control To provide a bridge that workers with background in either ﬁeld ﬁnd itaccessible (modest math) Textbook: Will be followed closely NEW DRAFT BOOK: Bertsekas, Reinforcement Learning and Optimal Control, 2019, on-line from my website Supplementary references Q-Learning is a method for solving reinforcement learning problems. /Subtype /Form /Type /XObject Semantic Scholar is a free, AI-powered research tool for scientific literature, based at the Allen Institute for AI. x���P(�� �� 38 0 obj /Resources 35 0 R stream >> But on his website all I see is PDFs of selected sections of chapters. PREFACE ix >> ISBN: 978-1-886529-39-7 Publication: 2019, 388 pages, hardcover Price: $89.00 AVAILABLE. REINFORCEMENT LEARNING AND OPTIMAL CONTROL METHODS FOR UNCERTAIN NONLINEAR SYSTEMS By Shubhendu Bhasin August 2011 Chair: Warren E. Dixon Major: Mechanical Engineering Notions of optimal behavior expressed in natural systems led researchers to develop reinforcement learning (RL) as a computational tool in machine learning to learn actions Batch process control represents a challenge given its dynamic operation over a large operating envelope. << This is Chapter 4 of the draft textbook “Reinforcement Learning and Optimal Control.” The chapter represents “work in progress,” and it will be periodically updated. REINFORCEMENT LEARNING AND OPTIMAL CONTROL. Reinforcement Learning and Optimal Control by Dimitri P. Bertsekas Massachusetts Institute of Technology DRAFT TEXTBOOK This is a draft of a textbook that is scheduled to be finalized in 2019, … /Filter /FlateDecode The book and course is on http://web.mit.edu/dimitrib/www/RLbook.html It more than likely contains errors (hopefully not serious ones). These methods have their roots in studies of animal learning and in early learning control work. On the other hand, Reinforcement Learning (RL), which is one of the machine learning tools recently widely utilized in the field of optimal control of fluid flows [18,19,20,21], can automatically discover the optimal control strategies without any prior knowledge. The book is available from the publishing company Athena Scientific, or from Amazon.com.. Click here for an extended lecture/summary of the book: Ten Key Ideas for Reinforcement Learning and Optimal Control. REINFORCEMENT LEARNING AND OPTIMAL CONTROL BOOK, Athena Scientific, July 2019. /FormType 1 R. Sutton and A. Barto, Reinforcement Learning, Second Edition draft, (2016) The properties of an optimal policy are described by ellman’s optimality equation (from Optimal Control theory) Reinforcement Learning: from Vision to Today’s Reality 11 Your comments and suggestions to the author at dimitrib@mit.edu are welcome. Recent work of Werbos, 2009 , Werbos, 2008 , Werbos, 2007 , Werbos, 2004 is pushing further the boundaries and taking the ideas of RL and ADP to ‘understand and replicate’ the functionality of the brain. This is because it is not an optimization problem --- it lacks an objective function. I have appedned contents to the draft textbook and reconginzed the slides of CSE691 of MIT. Errata. /Matrix [1 0 0 1 0 0] endstream endstream /FormType 1 Reinforcement learning is not applied in practice since it needs abundance of data and there are no theoretical garanties like there is for classic control theory. (2018). Reinforcement Learning: An Introduction Second edition, in progress ****Draft**** Richard S. Sutton and Andrew G. Barto c 2014, 2015, 2016 A Bradford Book The MIT Press Cambridge, Massachusetts ... of optimal control and dynamic programming. It more than likely contains errors (hopefully not serious ones). /Length 15 This is Chapter 3 of the draft textbook “Reinforcement Learning and Optimal Control.” The chapter represents “work in progress,” and it will be periodically updated. Reinforcement Learning and Optimal Control. Conventionally,decision making problems formalized as reinforcement learning or optimal control have been cast into a framework that aims to generalize probabilistic models by augmenting them with utilities or rewards, where the reward function is viewed as an extrinsic signal. Recht, B. /Length 875 %���� Adaptive control [1], [2] and optimal control [3] represent different philosophies for designing feedback controllers. 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Introduction This is a summary of the book Reinforcement Learning and Optimal Control which is wirtten by Athena Scientific. by Dimitri P. Bertsekas. stream /Filter /FlateDecode stream Dynamic programming, the model-based analogue of reinforcement learning, has been used to solve the optimal control problem in both of these scenarios. Their discussion ranges from the history of the field's intellectual foundations to the most rece… /Resources 31 0 R Description: The purpose of the book is to consider large and challenging multistage decision problems, which can be solved in principle by dynamic programming and optimal control, but their exact solution is computationally intractable. Reinforcement Learning and Optimal Control by Dimitri P. Bertsekas. Contents, Preface, Selected Sections. Furthermore, its references to the literature are incomplete. /BBox [0 0 5669.291 8] Link - http://web.mit.edu/dimitrib/www/RLbook.html He mentions that the draft of his book is available on his website. Exploration versus exploitation in reinforcement learning: a stochastic control approach Haoran Wangy Thaleia Zariphopoulouz Xun Yu Zhoux First draft: March 2018 This draft: February 2019 Abstract We consider reinforcement learning (RL) in continuous time and study the problem of achieving the best trade-o between exploration and exploitation. %PDF-1.5 This is a draft of a book that is scheduled to be finalized sometime within 2019, and to be published by Athena Scientific. After substantiating these claims, we go on to address some misconceptions about discounting and its connection to the average reward formulation. In Reinforcement Learning, Richard Sutton and Andrew Barto provide a clear and simple account of the key ideas and algorithms of reinforcement learning. Furthermore, its references to the literature are incomplete. >> Abstract: This article describes the use of principles of reinforcement learning to design feedback controllers for discrete- and continuous-time dynamical systems that combine features of adaptive control and optimal control. The book is available from the publishing company Athena Scientific, or from Amazon.com.. Click here for an extended lecture/summary of the book: Ten Key Ideas for Reinforcement Learning and Optimal Control.The purpose of the book is to consider large and challenging multistage decision problems, … ... Reinforcement Learning and Control as Probabilistic Inference: Tutorial and Review. /Filter /FlateDecode endobj The performance of conventional NMPC can be unsatisfactory in the presence of uncertainties. x���P(�� �� /Length 15 endobj The overall problem of learning from interaction to achieve. Reinforcement learning (RL) which can utilize simulation or real operation data is a … >> ArXiv. /BBox [0 0 16 16] << Dimitri P. Bertsekas. Reinforcement Learning and Optimal Control by Dimitri P. Bertsekas Massachusetts Institute of Technology DRAFT TEXTBOOK This is a draft of a textbook that is scheduled to be ﬁna ... D., and Zelinsky, A. Reinforcement Learning and Optimal Control (draft). Reinforcement Learning 1 / 36 Reinforcement learning has gradually become one of the most active research areas in machine learning, arti cial intelligence, and neural net- ... and developing the relationships to the theory of optimal control and dynamic programming. Overall, we have demonstrated the potential for control of multi-species communities using deep reinforcement learning. D. I came across the book and a series of lectures delivered by Prof. Bertsekas at Arizona State University in 2019. This is Chapter 4 of the draft textbook “Reinforcement Learning and Optimal Control.” The chapter represents “work in progress,” and it will be periodically updated. Athena Scientific. /Type /XObject x���P(�� �� Nonlinear model predictive control (NMPC) is the current standard for optimal control of batch processes. This draft was prepared using the LaTeX style le belonging to the Journal of Fluid Mechanics 1 Robust ow control and optimal sensor placement using deep reinforcement learning Romain Paris1y, Samir Beneddine1 and Julien Dandois1 1ONERA DAAA, 8 rue des Vertugadins, 92190 Meudon, France (Received xx; revised xx; accepted xx) James Ashton kept the computers’ wheels turning. In our paper last year (Li & Malik, 2016), we introduced a framework for learning optimization algorithms, known as “Learning to Optimize”. You are currently offline. /Subtype /Form !�T��N�����I�*�#Ɇ���5�����H�����:t���~U�m�ƭ�9x���j�Vn6�b���z�^����x2\ԯ#nؐ��K7�=e�fO�4J!�p^� �h��|�}�-�=�cg?p�K�dݾ���n���y��$�÷)�Ee�i���po�5yk����or�R�)�tZ�6��d�^W��B��-��D�E�u��u��\9�h���'I��M�S��XU1V��C�O��b. A reinforcement learning agent interacts with its environment and uses its experience to make decisions towards solving the problem. Video Course from ASU, and other Related Material. Some features of the site may not work correctly. Reinforcement learning, one of the most active research areas in artificial intelligence, is a computational approach to learning whereby an agent tries to maximize the total amount of reward it receives when interacting with a complex, uncertain environment. 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# Renormalization group for open quantum systems using environment temperature as flow parameter

### Submission summary

 As Contributors: Konstantin Nestmann Arxiv Link: https://arxiv.org/abs/2111.07320v1 (pdf) Date submitted: 2021-11-16 09:11 Submitted by: Nestmann, Konstantin Submitted to: SciPost Physics Academic field: Physics Specialties: Quantum Physics Approach: Theoretical

### Abstract

We present the $T$-flow renormalization group method, which computes the memory kernel for the density-operator evolution of an open quantum system by lowering the physical temperature $T$ of its environment. This has the key advantage that it can be formulated directly in real time, making it particularly suitable for transient dynamics, while automatically accumulating the full temperature dependence of transport quantities. We solve the $T$-flow equations numerically for the example of the single impurity Anderson model. In the stationary limit, readily accessible in real-time for voltages on the order of the coupling or larger, we benchmark using results obtained by the functional renormalization group, density-matrix renormalization group and the quantum Monte Carlo method. Here we find quantitative agreement even in the worst case of strong interactions and low temperatures, indicating the reliability of the method. For transient charge currents we find good agreement with results obtained by the 2PI Green's function approach. Furthermore, we analytically show that the short-time dynamics of both local and non-local observables follow a "universal" temperature-independent behavior when the metallic reservoirs have a flat wide band.

###### Current status:
Editor-in-charge assigned


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# Associations to the word «Total»

## Wiktionary

TOTAL, noun. An amount obtained by the addition of smaller amounts.
TOTAL, noun. (informal) (mathematics) Sum.
TOTAL, adjective. Entire; relating to the whole of something.
TOTAL, adjective. (used as an intensifier) Complete; absolute.
TOTAL, verb. (transitive) To add up; to calculate the sum of.
TOTAL, verb. To equal a total of; to amount to.
TOTAL, verb. (transitive) (US) (slang) to demolish; to wreck completely. (from total loss)
TOTAL, verb. (intransitive) To amount to; to add up to.
TOTAL ALLERGY SYNDROME, noun. A supposed illness with various symptoms caused by sensitivity to various substances in the modern environment.
TOTAL CLEARANCE, noun. (snooker) The potting of all the object balls on the table.
TOTAL CLEARANCE, noun. Used other than as an idiom: see total, clearance.
TOTAL CLEARANCES, noun. Plural of total clearance
TOTAL DEPRAVITY, noun. (religion) Calvinist doctrine that all human action is tainted by original sin
TOTAL ECLIPSE, noun. (astronomy) An eclipse in which the eclipsed body is completely obscured to the viewer.
TOTAL FOOTBALL, noun. (soccer) A football (soccer) tactic in which the outfield players assume different roles during a game, while keeping an organised structure.
TOTAL INTERNAL REFLECTION, noun. (optics) (uncountable) A phenomenon where light, instead of passing into a medium of a lower refractive index and refracting away from the normal, is completely reflected by the boundary between the two media because it approaches at an angle greater than or equal to the critical angle.
TOTAL INTERNAL REFLECTION, noun. (countable) An instance of this phenomenon.
TOTAL INTERNAL REFLECTIONS, noun. Plural of total internal reflection
TOTAL INTERNAL REFLEXION, noun. Alternative form of total internal reflection
TOTAL INTERNAL REFLEXIONS, noun. Plural of total internal reflexion
TOTAL LOSS, noun. (accounting) An insured item of which the entire value is written off
TOTAL ORDER, noun. (set theory) A relation that is reflexive, antisymmetric, and transitive (i.e., that is a partial order), and having the property that for any two elements of its set, one is related to the other.
TOTAL ORDERING RELATION, noun. (mathematics) (rare) A total order.
TOTAL ORDERING RELATIONS, noun. Plural of total ordering relation
TOTAL ORDERS, noun. Plural of total order
TOTAL QUALITY MANAGEMENT, noun. (management) A strategic approach to management aimed at embedding awareness of quality in all organizational processes.
TOTAL RECALL, noun. The ability to remember with great details; having photographic memory.
TOTAL RETURN SWAP, noun. (finance) A financial derivative that transfers both the credit risk and market risk of an underlying asset.
TOTAL REVENUE, noun. (economics) The revenue from each item multiplied by the number of items sold
TOTAL REVENUES, noun. Plural of total revenue
TOTAL SYNTHESES, noun. Plural of total synthesis
TOTAL SYNTHESIS, noun. (chemistry) The synthesis of a complex organic compound (especially a naturally occurring one) from simple, readily available compounds.
TOTAL WAR, noun. Warfare where all of a country's available resources, military as well as civilian, are employed.
TOTAL WARS, noun. Plural of total war

## Dictionary definition

TOTAL, noun. The whole amount.
TOTAL, noun. A quantity obtained by the addition of a group of numbers.
TOTAL, verb. Add up in number or quantity; "The bills amounted to $2,000"; "The bill came to$2,000".
TOTAL, verb. Determine the sum of; "Add all the people in this town to those of the neighboring town".
TOTAL, verb. Damage beyond the point of repair; "My son totaled our new car"; "the rock star totals his guitar at every concert".
TOTAL, adjective. Constituting the full quantity or extent; complete; "an entire town devastated by an earthquake"; "gave full attention"; "a total failure".
TOTAL, adjective. Complete in extent or degree and in every particular; "a full game"; "a total eclipse"; "a total disaster".

## Wise words

The chief difference between words and deeds is that words are always intended for men for their approbation, but deeds can be done only for God.
Leo Tolstoy


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### Where does the runtime fit in?

A SurveyMan workflow looks something like this :

We currently only support csv input; the design of the csv language is such that a human could reasonably write a survey in it. Molly is currently working on a Python library that will streamline some of the more advanced features of SurveyMan and Python proficient users to design surveys in Python and output a JSON equivalent to a csv survey.

The main SurveyMan program is written in Java. It handles all of the static checks on the input and loads the survey into an internal representation. When it is time to send the program to a crowdsourcing platform, the Java program produces a minimal JSON representation of the survey and sends this wrapped in HTML. When the crowd platform is Mechanical Turk, the HTML is sent inside an XML payload and handled by AMT. When the platform is a local server, it just sends the HTML. We can support any crowdsourcing platform that allows us to send arbitrary HTML.

We send the minimal JSON representation for two reasons. The first is that for AMT, we have a maximum payload allowed. The second reason is because not all of the survey data is useful to the runtime. We need sufficient information to determine how to display the survey to the user. We don't do any analysis on the crowdsourcing platform, so we don't need to send over information like the CORRELATED column or any user-provided columns. We do allow survey designers to add custom Javascript to their survey, which can be used to perform computations and return the results of those computations.

### Input

The input to the SurveyMan program is a survey csv. This csv is parsed and loaded into an internal representation, which is then translated into a JSON payload. The resulting JSON representation of the survey is sent to the crowdsourcing platform. This survey JSON is then interpreted by a Javascript program that controls flow through the survey and how individual questions are displayed.

### The Finite State Machine

SurveyMan surveys are interpreted in Javascript on a finite state machine that controls the underlying evaluation and display.

There are two layers to the survey interpreter. The first handles survey logic; it essentially runs in a loop, proffering questions and consuming responses. The second layer handles display information, updating the HTML in response to certain events.

#### User's view of the FSM

A respondent navigates to the webpage displaying the survey. For AMT, we show a consent form in the HIT preview. When an individual accepts a HIT, they begin a survey. The consent form mechanism is not currently implemented in the local server.

The user then sees the first question* and the answer options. When they select some answer (or type in a text box), the next button and a "Submit Early" button appear. If the question is instructional and is not the final question in the survey, only the next button appears. When the respondent reaches the final question, only a

Each user sees a different ordering of questions. However, a single user's questions are presented in the same order; the random number generator is seeded with the user's session or "assignment" id. In AMT, this means that if the user navigates away from the page and returns to the HIT, the question order will be the same upon second viewing.

Only one question is displayed at a time. This design decision is not purely aesthetic; it helps us measure breakoff.

#### Underlying machinery

In addition to the functions that implement the state machine, there are three other underlying data structures:

1. Block Stack : Since our path through the survey is determined by branching over blocks, and since we may only branch forward, we keep the top-level blocks on a stack.
2. Question Stack : Once we know which block we're to execute, we can fetch the appropriate questions for that block.
3. Branch Reference Cell : The branch target is stored here, since we defer executing any branching at least until all of the questions in the block have been seen.

The SurveyMan interpreter is initialized as follows : the survey JSON is parsed into an internal survey representation and then randomized using a seeded random number generator. This randomization preserves necessary invariants (e.g. the partial order over blocks). We then push the top level blocks onto the block stack. Then we pop off the first block and initialize the question stack using getAllQuestions.

The interpreter's execution runs as follows :
 procedure run begin   while blockStack is not empty   do     question <- getNextQuestion()     display question and answer options     answer <- user-provided answer     store answer     if the question is a branch question     then       branchRef <- answer's branch destination     fi     if user selects submit     then       return stored answers     fi   done   return stored answers end

Answers are stored in the HTML forms in the usual way. We have three main divs in the HTML : one for the displayed question, one for the displayed answers, and one for response input data. The response input elements are kept in a hidden div below the answer display div. Each new response is inserted above the previous one, emulating a stack.

The function getNextQuestion handles control flow and is the primary point of interaction with the FSM:
 function getNextQuestion   output : question begin   if questionStack is empty     if branchRef is set     then       while true       do         nextBlock <- peek(blockStack)         if nextBlock = branchRef         then           questionStack <- getAllQuestions(pop(blockStack))           unset branchRef           break         else if nextBlock is a floating block         then           questionStack <- getAllQuestions(pop(blockStack))           break         else           pop(blockStack)         fi       done     else       questionStack <- getAllQuestions(pop(blockStack))   fi   return pop(questionStack) end

### Determinism in the runtime

There are two main sources of nondeterminism in SurveyMan. The first is our random number generator; the second is human behavior.

We had previously generated a new static HTML version of the survey for each respondent. Each question was contained in its own div and branch destinations pointed to these divs. None of the blocking structure was visible to the user. The Javascript was minimal; if the question was a branch, it would hide the current question and display the the div with the appropriate id. If the question did not have branching, it would navigate the DOM and just display the next div.

This prior approach lead to a technical issue that proved to be a nice metaphor for the underlying system. The problem was that, since we had to send over unique HTML for every desired respondent, we ended up with many HITs posted on AMT. AMT groups together HITs that have similar parameters and facilitates workers' acceptance of consecutive HITs in groups. Having many HITs not only attracted bad actors, but it also annoyed honest respondents, who did not always realize that they were not supposed to answer all of the HITs.

While each respondent sees a different version of the survey, the underlying survey is essentially "the same." It made more sense to move the survey logic into the Javascript and post a single HIT. It was also cleaner to group together user behavior with survey execution, rather than flattening the survey and adding an extra (albeit simpler) evaluator that primarily responded to user nondeterminism.

What's key about these two approaches is that the two types of nondeterminism are quite different from each other. Randomization gives us predictable behavior in the limit and, depending upon the task may not even have an impact on the computation. Randomization does not occur in response to user behavior and can be done before executing anything. Once we know which block to execute, the total set of questions is almost deterministic. That is, rather than having to account for nondeterminism and uncertainty at every step in the evaluation, we only have to handle it at known junctures. So long as we can prove that the set of surveys produced at each junction are equivalent, we know that the evaluation is sound.

* The respondent may also see a breakoff notice first. This is just a page stating that they may submit results at any time and that they will be paid bonuses commensurate on the number and quality of questions answered. It can be turned off.


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Linear Independence Dependence of a Set of Functions

# Linear Independence Dependence of a Set of Functions

Recall from the Wronskian Determinants and Higher Order Linear Homogenous Differential Equations page that if we have an $n^{\mathrm{th}}$ order linear homogenous differential equation $\frac{d^ny}{dt^n} + p_1(t) \frac{d^{n-1}y}{dt^{n-1}} + ... + p_{n-1}(t) \frac{dy}{dt} + p_n(t)y = 0$ where $p_1$, $p_2$, …, $p_n$ are continuous on an open interval $I$ and if $y = y_1(t)$, $y = y_2(t)$, …, $y = y_n(t)$ are solutions to this differential equation, then provided that $W(y_1, y_2, ..., y_n) \neq 0$ for at least one point $t \in I$, then $y_1$, $y_2$, …, $y_n$ form a fundamental set of solutions to this differential equation - that is, for constants $C_1$, $C_2$, …, $C_n$, then every solution to this differential equation can be written in the form:

(1)
\begin{align} \quad y = C_1y_1(t) + C_2y_2(t) + ... + C_ny_n(t) \end{align}

We will now look at the connection between the solutions $y_1$, $y_2$, …, $y_n$ forming a fundamental set of solutions and the linear independence/dependence of such solutions. We first define linear independence and linear dependence below.

 Definition: The functions $f_1$, $f_2$, …, $f_n$ are said to be Linearly Independent on an interval $I$ if for constants $k_1$, $k_2$, …, $k_n$ we have that $k_1f_1(t) + k_2f_2(t) + ... + k_nf_n(t) = 0$ implies that $k_1 = k_2 = ... = k_n = 0$ for all $t \in I$. This set of functions is said to be Linearly Dependent if $k_1f_1(t) + k_2f_2(t) + ... + k_nf_n(t) = 0$ where $k_1$, $k_2$, …, $k_n$ are not all zero for all $t \in I$.

Perhaps the simplest linearly independent sets of functions is that set that contains $f_1(t) = 1$, $f_2(t) = t$, and $f_3(t) = t^2$. Let $k_1$, $k_2$, and $k_3$ be constants and consider the following equation:

(2)
\begin{align} \quad k_1f_1(t) + k_2f_2(t) + k_3f_3(t) = 0 \\ \quad k_1 + k_2t + k_3t^2 = 0 \end{align}

It's not hard to see that equation above is satisfied if and only if the constants $k_1 = k_2 = k_3 = 0$.

For another example, consider the functions $f_1(t) = \sin t$ and $f_2(t) = \sin (t + \pi)$ defined on all of $\mathbb{R}$. This set of functions is not linearly independent. To show this, let $k_1$ and $k_2$ be constants and consider the following equation:

(3)
\begin{align} \quad k_1f_1(t) + k_2f_2(t) = 0 \\ \quad k_1 \sin t + k_2 \sin (t + \pi) = 0 \end{align}

Now choose $t = \pi$. Then we have that:

(4)
\begin{align} \quad k_1 \sin \pi + k_2 \sin 2\pi = 0 \\ \end{align}

But the above equation is true for any choice of constants $k_1$ and $k_2$ since $\sin \pi = \sin 2\pi = 0$, and thus $f_1$ and $f_2$ do not form a linearly independent set on all of $\mathbb{R}$.

From the concept of linear independence/dependence, we obtain the following theorem on fundamental sets of solutions for $n^{\mathrm{th}}$ order linear homogenous differential equations.

 Theorem 1: Let $\frac{d^ny}{dt^n} + p_1(t) \frac{d^{n-1}y}{dt^{n-1}} + ... + p_{n-1}(t) \frac{dy}{dt} + p_n(t)y = 0$ be an $n^{\mathrm{th}}$ order linear homogenous differential equation. If $y = y_1(t)$, $y = y_2(t)$, …, $y = y_n(t)$ are solutions to this differential equation then $y_1$, $y_2$, …, $y_n$ form a fundamental set of solutions to this differential equation on the open interval $I$ if and only if $y_1$, $y_2$, …, $y_n$ are linearly dependent on $I$.
• Proof: Consider the following $n^{\mathrm{th}}$ order linear homogenous differential equation:
(5)
\begin{align} \quad \frac{d^ny}{dt^n} + p_1(t) \frac{d^{n-1}y}{dt^{n-1}} + ... + p_{n-1}(t) \frac{dy}{dt} + p_n(t)y = 0 \end{align}
• $\Rightarrow$ Suppose that $y = y_1(t)$, $y = y_2(t)$, …, $y = y_n(t)$ form a fundamental set of solutions to this differential equation on the open interval $I$. Then this implies that for all $t_0 \in I$ we have that :
(6)
\begin{align} \quad W(y_1, y_2, ..., y_n) \biggr \rvert_{t_0} \neq 0 \end{align}
• Thus this implies that following system of equations have only trivial solution $k_1 = k_2 = ... = k_n = 0$:
(7)
• Thus the equation $k_1y_1(t) + k_2y_2(t) + ... + k_ny_n(t) = 0$ implies that $k_1 = k_2 = ... = k_n = 0$. Thus $y_1$, $y_2$, …, $y_n$ are linearly independent on $I$.
• $\Leftarrow$ We will prove the converse of Theorem 1 by contradiction. Suppose that $y_1$, $y_2$, …, $y_n$ are linearly independent on $I$, and assume that instead $y_1$, $y_2$, …, $y_n$ do NOT form a fundamental set of solutions on $I$. Then for some $t_0 \in I$, the Wronskian $W(y_1, y_2, ..., y_n) \biggr \rvert_{t_0} = 0$. Thus the system of equations above does not have only the trivial solution. Let the constants $k_1^*$, $k_2^*$, …, $k_n^*$ be a nontrivial solution to this system. Define $\phi(t)$ as:
• Note that $y = \phi(t)$ satisfies the initial conditions $y(t_0) = 0$, $y'(t_0) = 0$, …, $y^{(n-1)} (t_0) = 0$, and $\phi(t)$ satisfies our $n^{\mathrm{th}}$ order linear homogenous differential equation because $\phi(t)$ is a linear combination of the solutions $y_1$, $y_2$, …, $y_n$.
• Now note that the function $y = 0$ also satisfies the differential equation and the initial conditions. By the existence/uniqueness theorem for $n^{\mathrm{th}}$ order linear homogenous differential equations, this implies that $\phi(t) = 0$ for all $t \in I$, so:
• But $y_1$, $y_2$, …, $y_n$ are linearly independent which implies that $k_1^* = k_2^* = ... = k_n^* = 0$. Thus $k_1^*$, $k_2^*$, …, $k_n^*$ is a trivial solution to the system above, which is a contradiction. Therefore our assumption that $y_1$, $y_2$, …, $y_n$ do not form a fundamental set of solutions was false. $\blacksquare$