isomorphs

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Pyrargyrite, Ag3SbS3 (top) and proustite, Ag3AsS3 (bottom).

These are isomorphous minerals, meaning the atoms have the same arrangement in the two compounds, but where pyrargyrite has antimony, proustite has arsenic.

Antimony and arsenic are in the same column of the periodic table, so we would correctly expect them to make this swap without much fuss.

This little gem is dedicated to Frédéric Vanhove, who passed away yesterday and was our assistent for graph theory. He had a passion for graphs, so I hope he’d like this beautiful theorem - in my opinion one of the most elegant in whole mathematics.

Draw n points and connect them without creating any “loops”; in the result, every point should be accessible from every other point by exactly one path. Such a configuration (a graph) is called a tree. You can see all possible trees on four points in the image. Lots of these trees are essentially the same (isomorphic), but we label the points to distinguish between them.

Carl Wilhelm Borchardt found an elegant formula to express the total number of possible spanning trees on n labeled points, but nowadays the result is named after Arthur Cayley. In our example (with n=4) we find 16 trees, but in general, the formula states that this number is exactly nn-2.

Shortly, nn-2 is the number of spanning trees of a complete graph Kn.

youtube

The Unbearable Lightness of Being - Isomorphism Soundtrack

http://music-producer.net/

Ammonites are a group of extinct cephalopod mollusks with ribbed spiral shells. They are exceptionally diverse and well known to fossil lovers. Researchers have developed the first biomechanical model explaining how these shells form and why they are so diverse. Their approach provides new paths for interpreting the evolution of ammonites and nautili, their smooth-shelled distant “cousins” that still populate the Indian and Pacific oceans.

Group Theory, Isomorphisms & Permutations

I’m not feeling social interaction today so I learned a bunch of group theory, and it just got too cool, so i had to post some of it on here!

Now, before I go on to talk about the main things which are Isomorphisms and Permutations, I should properly define groups.

Taking the axiomatic approach, groups are sets with an operation for combining elements which pass the following tests

Groups are closed under one operation, meaning if you combine any two elements using an operation of your choice (the symbols +and * are often used), you will never be able to create an element which is not in the group.

The operation is associative meaning A*(B*C)=(A*B)*C=A*B*C as long as you combine the elements as shown. This effect can be extended to any string of n elements. 

There exists an element “e” such that x*e=x and e*x=x. This element is called the unit, and if we are talking about multiplication with ordinary numbers, this number would correspond to one.

For each number in a group, there will be an inverse. When we are using multiplication-like operations like we are above, we usually write the number like this a-1, and call it “a inverse” for addition we use -a and call it “the negative of a”. The inverse of the unit is the unit again, and any element combined with its inverse becomes the unit.

You can see that the commutative axiom is not listed, where A*B=B*A this is not always true, as we will see in a second. Take the group consisting of the elements

 
 

 

under the operation of matrix multiplication which works like this.

  Now, if we do the operation A*B, we get the matrix C, if we do the operation B*A, we get the matrix K, meaning that this operation is not commutative. This is not the only operation which behaves this way, so in general we have to say that the operation does not have to commute. If it does, we call the group an Abelian group.

Now that that’s out of the way, we can define an Isomorphism. 

An Isomorphism is a one-to-one function which takes one group and converts it into another while preserving the operation. Basically, if G  is a group under the operation * denoted <G,*> and <H,+> is a group, and 
then an Isomorphism f(x) will take f(a)=a’ and f(b)=b’ and make, f(a*b)=a’+b’ or, entirely in function notation, f(a*b)=f(a)+f(b).

To prove that two groups are isomorphic, you need to prove that the function you are using is one-to-one, that is proving that the function is both injective and surjective. To do this, you have to prove that each element of the range is the image of at most 1 element of the domain, which can be done by showing that it fits this definition  
You then have to prove that the function is surjective by showing that for some element y in he range, there exists an element who’s image is y. Together, these two things prove that the function is bijective, which means that you can turn elements from G into elements in H and vice versa.

From there, all you have to prove is that f(a*b)=f(a)+f(b) and you’re done.

Now why are isomorphisms important? Because they can be used to prove things like this Cayley’s theorem which says that every group is isomorphic to a group of permutations.  Quite a bit is known about permutations, which makes this theorem very strong in terms of its use.

We can prove this in a second, but first, we have to look at permutations. 

A permutation is a function on a set S which takes the elements of S and rearranges them. Now, this must by definition be a bijective function, which takes S and maps it to its self. Taking individual permutations to be elements of a set, we can create a set of n! elements, where n is the number of numbers being used in the permutation. For instance, the set of all permutations on three numbers is:

   

  

and has 3!=6 elements. To combine the elements, we define the permutation of a permutation to be a function which takes the elements as they are arranged by the inner permutation and then rearranges them under the outer permutation called the composition. For instance:

To show that this combining of permutations is closed under the operation above, we can make a chart like the one below, where the first column is multiplied on the left by the first row,

but there are an infinite number of groups of permutations, so we must prove it. We can actually take it a step further into abstraction, and prove the composite of any two functions which are both bijective will yield another bijective function. Why are we proving the combinations of permutations are bijective? Because we started with the assumption that we had made a list of all the possible permutations for n objects, so showing that this new function is also bijective permutation means that it was on this list in the first place. 

Assume a f(x) and g(x) are injective functions, then we have to prove that [f*g][x]=[f*g][y] implies that x=y. Well suppose  [f*g][x]=[f*g][y], then f(g(x))=f(g(y)) because f(x) is injective, g(x)=g(y), and because g(x) is injective, x=y, so [f*g][x] is injective

Assume g:A ->B and f:B ->C are surjective, then we have to prove that every element of C is f*g of some element in a. Assume z is an element of C, then because f is surjective,  f(y)=z for some y in B. Because g is surjective,f(g(x))=z so, for any element of C, there is an element for which z is its image under f*g, so [f*g][x] is surjective.

Now, assuming the functions f and g are both bijective, it follows from what was just proven that f*g is bijective as well.

By this, the group of permutations under composition is closed.

There exist inverses for each element, which return each permutation to the permutation

which also happens to be identity element. Finally, combining permutations is associative, so we have a group.

Now we can prove Cayley’s theorem which again states that each group is isomorphic to a group of permutations.  

Proof

Begin by assuming we have a group G, we wish to prove that it is isomorphic to a set of permutations. Well, permutations require a set, so why not use the set that G is built off of? The only set we have around is the one used to make G, so we have no other choice but to use it. We then define a function

which is defined as:

And gives all the permutations of G, changing each element a of G into ax, and then ranging X over all elements of G to give one permutation per element. 

This function is injective, that is 

and surjective, as for some y in g, 
 
so y is the image of some element a^(-1)*y.

So we have a bijective function which turns an element a of G into a permutation which turns x_1 into a*x_1, x_2 into a*x_2 and so on. So we let G* represent the set of permutations of G created by pi_a(x) as a ranges through the elements of G. 

Notice now that the set of permutations created is not necessarily the set of all permutations of G, but a permutation corresponding to each element. We now have to prove that it is a group, that is

and that for any permutation contained by G*
 
Where pi_e is the identity permutation given by multiplying each x in G by the identity. The associative property can be assumed due to the fact that the operation is simple non-commutative multiplication. 

To begin, we have to prove that 

We can say that 

Because ab is a member of G, we can say that G* is closed with respect to composition. 

Finally, because G is a group with inverses, 

So we have proved that G* is in fact a group.

Now all we have to do is prove that G is isomorphic to G*. We already have the function ready, so we use it as the isomorphism. 

It is injective,

and surjective, every element pi_a of G* is the image of some f(a)

Lastly, 

Thus, f is an isomorphism, and

That’s about all I have left in me for today, but I’ll try to get more up on here tomorrow! I thought Cayley’s theorem would be a good start!

A mathematician has developed a new way to uncover simple patterns that might underlie apparently complex systems, such as clouds, cracks in materials or the movement of the stockmarket. The method, named fractal Fourier analysis, is based on new branch of mathematics called fractal geometry. The method could help scientists better understand the complicated signals that the body gives out, such as nerve impulses or brain waves.

Freud’s greatest discovery, the one which lies at the root of psychodynamics, is that the great cause of much psychological illness is the fear of knowledge of oneself—of one’s emotions, impulses, memories, capacities, potentialities, of one’s destiny. We have discovered that fear of knowledge of oneself is very often isomorphic with, fear of the outside world.
—  Abraham Maslow

the only cool thing in the homework was the bit where we show that the unit sphere of the quaternions (as a topological group) is isomorphic to SU(2)

defilerwyrm replied to your postI swear, every time I hear Led Zeppelin, all I…

no shut up share your thoughts TALK ABOUT THE THING

I can’t right now, because I had 2 whole glasses of wine and now I’m really drunk!  But I have always found it so fascinating the way in which music is used as an identifier of class identity and somehow (I know, anecdotally, that this holds very true in the area I grew up in) Led Zeppelin very much came to be associated with more blue collar, working class lifestyle (obviously though, this relationships is not an isomorphism).

And it always struck me as something that was related to the way in which blue-collar people and those in lower socio-economical stratus are statistically more inclined towards conservative/Republican ideology (again, not an isomorphism), and the way in which conservative/Republician ideologies are associated with a retroactive opinions (i.e. “Things were better in the past!!”).  And somehow, this winds up being also associated with music.  And of course, music is one of those things where young people get into just as they are trying to establish their identity and how they start to determine who they are similar to in their circle of friends (ASK ME ABOUT MY LOVE OF GREEN DAY AND NIRVANA!!!).  So the examination of music choices becomes a microcosm for larger issues of political viewpoints, social class, etc (and once again, obviously does not hold for all people), but it also is a short-hand for those who want to project a certain image on screen to quickly signal the audience to things like class and so on.

However, that tendency to embrace the past with respect to music often leaves music like my beloved David Bowie and Marc Bolan out, since they don’t fit that narrative of classic rock, as they sang about sex, sexuality, in a way that was transgressive then and continues to be transgressive to this day (particularly with more conservative segments of the population).

So the fact that Dean chose to bear the name of Agent Bolan in honour of Marc Bolan IN AN EPISODE WHERE HE FLIRTS BADLY LIKE A SILLY FLUSTERED SCHOOLGIRL WITH A GUY continues to make me completely agog!  Bolan, as the lead singer of T Rex, was a fucking bisexual glam rocker, and though his music has shown endurance, it doesn’t have the same socio-economic connotations that one may see with Led Zeppelin (or various other musical acts Dean has been associated with).  Oddly enough, this was something my father made me aware of.  He was going through my record collection and teased me about having Marc Bolan in my collection because he had listened to him in university.  Somehow, we wound up in a discussion about the use of music as a class signifier and how those in my father’s social class (particularly in the 70s/80s) used their being fans of T Rex and Bowie as a way of expressing their class.

So Dean has shown some awareness of music that transcends the narrow confines of what was considered appropriate for his image/SES/class/self-constructed sense of identity, and the fact that he embraced it during that episode?  FUUUUUUUCK, have I mentioned how much I love Ben Edlund recently?

But, I mean, this is frankly nothing but obviousness and deriativeness.  See what I mean?  There’s nothing to be said and I can hardly be thought to be adding anything of particular interest or substance to such well-tilled soil.

Counting the Isomorphism Classes of the Generalized Petersen Graphs

This was a short talk given by Sarah Hanusch, a graduate student at Texas State, who as far as I could tell does not have a professional website.

———

The generalized Petersen graph GP$(n,k)$ is the graph on $2n$ vertices $\{a_1,\dots a_n,b_1,\dots,b_n\}$ with three types of edges: $a_ib_i$, $a_ia_{i+1}$, and $b_ib_{i+k}$ (addition is taken mod $n$). In this notation, the ordinary Petersen graph is $G(5,2)$, and the prism graphs are $G(n,1)$.

Her research focused on the case when $n$ and $k$ are relatively prime; that is, they have greatest common divisor $1$. Of course, if two generalized Petersen graphs are to be isomorphic, they must have the same $n$, so the issue is to determine, for fixed $n$, which $k$ produce isomorphic graphs.

She determined that the equivalence classes must be of size at most 4, corresponding to the equations $\alpha=\pm\alpha^{\pm 1}$. Two of these equations are automatically thrown out by the relatively prime condition: $\alpha=\pm\alpha$. Therefore, one must check the number of solutions to $\alpha^2=\pm1$. So, this graph theory question has been reduced to a number-theoretic question about the multiplicity of quadratic residues.

She went into some detail explaining exactly how to answer this question and what it meant on the graph theory side. In conclusion, she said that there were some basic obstructions to applying these techniques for $n$ and $k$ not relatively prime, but she seemed optimistic about getting around these difficulties.

ima geek about algebra for a quick sec

Isomorphic is the adjective I’ve been looking for my entire life. We just covered isomorphisms in my Algebraic Structures class (which, by the way, is the most enlightening class I have ever taken) and Ima lay it down for y’all. An isomorphism is essentially a perfect analogy between two structures. You look at them both and you go, “Oh, these are the same thing, the only difference is that the names/symbols have been changed.” 

Like clocks (restricted to counting hours) and the chromatic scale and the set of symmetries of a regular hexagon. Or the English alphabet and the integers 1 - 26. Or, to get mathier, the set of complex numbers and the set of arrows in a plane.

Now to get more poetic and daydreamy.

You could create a map from particular notes in a melody to particular motions of your body and preserve the structure of the song through your dance; your body would be, idealistically of course, structurally indistinguishable from the music.

I dunno, this shit’s just making me thing about creativity and expression in general. If you can recognize a structure, you can replicate it any number of ways. You could recognize the structure behind your emotional cycles throughout the course of a relationship or something (assuming of course that your emotions are well-defined, which I suppose is up for debate), and preserve the structure through a song. You could map every word used in a book to a unique brushstroke/paint-color combination and paint the entire damn book on a canvas. The structure would be preserved.

Or even like, deja vu. You say, “I’ve experienced this before,” but it’s more accurate to say your mind created a perfect analogy between a structure you just recognized in a situation with a situation you’d previously experienced that exhibited the same structure. The two scenarios are just isomorphic.

I wish I could communicate more effectively the emotional high I’m getting from this idea. You’re always going to be able to see something through something else, ya know? 

Come join the team at Yahoo for a morning talk on Fluxible, by Reza Akhavan (@jedireza on Github and Twitter) at our SF JS Breakfast - March 5th at 8:30am

Reza will talk about how Yahoo brought Flux to the server and how Fluxible helps you create isomorphic web apps

Check out the examples here: https://github.com/yahoo/flux-examples

This meetup will start at 8:30am, and we’ll be providing coffee and donuts for those of us not used to an early morning start.