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.

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.

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Bastion Cael Hammer by Volpin Props

The Cael Hammer was built for Supergiant Games to display at Penny Arcade Expo East 2015. This replica was used in a photobooth by convention attendees, and has been constructed to be both lightweight and durable in order to withstand thousands of pairs of hands wielding it over the course of a three day weekend. Since Bastion is an isomorphic perspective game, a new high-poly render was created by studio artists specifically for this project.

Fully assembled the Cael Hammer is 44 inches long. Being constructed from foam and hollow vacuumformed panels keeps the weight to 3.5lbs. A recessed threaded rod between the handle and the hammer splits the piece into two parts for shipping or future repairs. A display base was also created to showcase the replica at the Supergiant Games booth while not in use.

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.

I think the biggest clue was right there in her name all along:

Garnet is not a single mineral, but a group contains closely related, isomorphous minerals that form a series with each other. The Garnet members form intermediary minerals between each member, and may even intergrow within a single crystal.”

anonymous asked:

Looked this up randomly and... I'm sure you can put two and two together: Garnet is not a single mineral, but a group contains closely related, isomorphous minerals that form a series with each other. The Garnet members form intermediary minerals between each member, and may even intergrow within a single crystal.

👍seriously you guys someone get on a science related Steven Universe tumblr.

I understand SO LITTLE about this stuff it’s almost sad.

Combinatorics and Reasoning: Representing, Justifying and Building Isomorphisms free ebook ,

Combinatorics and Reasoning: Representing, Justifying and Building Isomorphisms

<p>2010 | ISBN-10: 0387981314, 940079293X | 246 pages | PDF | 4 MB</p> <p><br /> Combinatorics and Reasoning: Representing, Justifying and Building Isomorphisms is based on the accomplishments of a cohort group of learners from first grade through high school and beyond, concentrating on their work on a set of combinatorics tasks. By studying these students, the editors gain insight into the foundations of proof building, the tools and environments necessary to make connections, activities to extend and generalize combinatoric learning, and even explore implications of this learning on the undergraduate level.<br /> <br /> This volume underscores the power of attending to basic ideas in building arguments; it shows the importance of providing opportunities for the co-construction of knowledge by groups of learners; and it demonstrates the value of careful construction of appropriate tasks. Moreover, it documents how reasoning that takes the form of proof evolves with young children and discusses the conditions for supporting student reasoning.</p>

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,

what is it about the binary coproducts of finite sets that makes A+B = A+C imply B = C? (note: “=“ means “is isomorphic to” in this post)

it’s easy to see that this isn’t the case for general coproducts, and indeed this isn’t even true for infinite sets, since strict inequalities of infinite sets A>B>C imply A+B = A+C = A (maybe you need choice for that?)


so apparently the property of interest isn’t a consequence of FinSet being a topos, nor the fact that its coproducts are freely generated

what’s so special about coproducts in FinSet, and where else does A+B = A+C imply B = C?

Tag 10 people you want to know more about

I was tagged by: rorysampats whom i love and is the literal greatest!!!

Name: ella
Time and Date: march 26 5:28pm
Average Hours Of Sleep A Night: not enough. never enough.
Last Thing I Googled: strength typography

Nickname: ella is a nickname! ^_^
Birthday: june 18
Gender: female 

Sexual orientation: straight
Height: like 5′4 or 5′5 i believe

Favorite Color: green
One Place That Makes Me Happy: a beach - like literally any beach.
How Many Blankets Do I Sleep Under: 1
Favourite Movie: the lion king
What I’m Wearing Right Now: a layered orange and pink striped tank and my pink jeans
Last Book You Read: fuck i haven’t read in so long i’m so busy but i think it was mr. kiss and tell (which came out in like january?? wtf man)
Most Used Phrase: probably “good morning” bc of my fucking job. (but it’s probably okie dokie tbh)
What I Last Said To A Family Member: my mom and i were talking about these yoga classes at work, i think the last thing i said was “yeah i understand”?
What Is Family?: whatever/whoever you choose for it to be!
Favourite Beverage: lemonade and hot chocolate (but not like together)
Favourite Food: pizza probably or pasta w/ chicken or seafood in it mmmm.
Last Movie I Watched In Theaters: kingsmen i think? or the spongebob movie which came out first?
Dream Wedding: ugh idk probably like a really, really classy outdoor wedding
Dream pet:  i really want a cream chow named nala
Dream Job: like, realistically, editor of instyle. less realistically, editor and founder of my own magazine.

Tagging: not actually ten but: olliversmoak isomorphic rodawn stealthebuttons (if she ever comes back *sob*) kbishops ofprincessesandrebels

anonymous asked:

Oooh! What does that make the TARDIS in this universe? A magic, sentient, blue crystal cave?

The TARDIS is most certainly a cave, an enchanted cave left to rot from many uses of sorcerers and such, if you press on the various gems, a portal will then open and release you to the outer reaches of time and space. It is, of course, highly dangerous for a human alone to travel through the enchanted, electromagnetic portal, the only way for a human to pass is if a being with the isomorphic power to create said portal is travelling with them at that exact time.

3.E
table, th, td { border: 1px solid black; border-collapse: collapse; } th, td { padding: 5px; } Coin Game

We are given 8 moves and the operation * which consists of performing any two moves in succession. Let Mn be abbreviated as n and for this table. Also, let the move “Do not change anything” be represented by 0, because I am afraid “1” and “I” will be difficult to parse otherwise.



Note that this table is not symmetric about the main diagonal and therefore the coin game group is not an abelian.

Check:
  6*2 = (Flip coin at B; then switch)*(Flip over coin at B) = Flip both coins then switch = 7

  2*6 = (Flip over coin at B)*(Flip coin at B; then switch) = 4 = Switch both coins

There are, however, abelian subgroups. For example, \{0, 1, 2, 3}, */, is a subgroup that happens to be isomorphic to the 2 by 2 checkerboard group.