## Isomorphism between \$Z[X,Y ]/(2Y−1)\$ and \$Z[frac{1}/{2}][X]\$

Hi I been trying to proof that $$Z(X,Y )/(2Y−1)$$ is isomorphic to $$Z({1}/{2})(X)$$ where$$Z({1}/{2})$$= $$frac{a}{2^n} : a ∈ Z, n ∈ N, n ≥ 0$$

## reference request – Explicit homotopy for Hochschild chains from natural isomorphism

Let $$A,B$$ be $$k$$-linear (possibly, dg-)categories, let $$f,g:Ato B$$ be two linear functors, and let $$T:fRightarrow g$$ be a natural isomorphism.

If one denotes by $$C_bullet(A,A)$$ the standard Hochschild chain complex of $$A$$ with coefficients in itself, then $$f$$ and $$g$$ induce two chain maps $$mathbf{f},mathbf{g}:C_bullet(A,A)to C_bullet(B,B)$$.

One knows from general facts about the homotopy theory of dg-categories that there exists a homotopy $$mathbf{T}$$ between $$mathbf{f}$$ and $$mathbf{g}$$.

Here is my question: Is there a reference where one can find an explicit expression for this homotopy?

Ideally, I’d like a reference where the compatibility with the mixed structure on Hochschild chains (that is, with Connes’ boundary operator) is discussed.

One can of course guess (and, with some work, prove) a formula. I’m almost sure such a formula is already written somewhere, but I couldn’t find a reference.

Note that I’d already be happy to know about a reference for (dg-)algebras (in which case a natural isomorphism $$T:fRightarrow g$$ is just the data of an invertible element $$b$$ of $$B$$ such that for every $$ain A$$, $$bf(a)=g(a)b$$).

## abstract algebra – Second Isomorphism Theorem and Jordan-Holder

In another posting, there was a question about the following:

Let $$G$$ be a finite non-trivial group with the following two composition series:

$${e} = M_0 triangleleft M_1 triangleleft M_2 = G$$

$${e} = N_0 triangleleft N_1 triangleleft cdots triangleleft N_r = G.$$

Prove that $$r=2$$ and that $$G/M_1 cong G/N_1$$ and $$N_1/N_0 cong M_1/M_0$$.

The person posting the question went on to state that “By the second isomorphism theorem I know that $$M_1N_2/N_2 cong M_1/(N_2cap M_1)$$

Here is my question. In order to use the second isomorphism theorem with $$M_1$$ and $$N_2$$ don’t we need to know that $$M_1 leq N_G(N_2)$$? And if so, then how do we know that $$M_1$$ actually is in the normalizer of $$N_2$$ in $$G$$?

If I can get over this hurdle, I understand the remainder of the original posting. Perhaps this is something obvious, but please help me see whatever it is that I am missing.

## linear algebra – Proving that an isomorphism from a vector space \$X\$ to \$X’\$ using symmetric bilinear forms is onto

Here’s the problem:

Let $$f$$ be a symmetric bilinear form over an n-dimensional vector space $$X$$, and $$f$$ is non-degenerate (for all nonzero $$x in X$$, there exists $$y in X$$ so $$f(x,y) neq 0$$). Prove that $$L_f: X to X’$$ defined as $$x mapsto f(x, -)$$ is an isomorphism.

I can easily prove $$L_f$$ is one-to-one, and argue that it’s an isomorphism since we are dealing with finite-dimensional spaces. (If $$L_f(x) = L_f(y)$$ then $$f(x-y, z) = 0$$ for all $$z$$. However, since $$f$$ is non-degenerate, if $$x-y neq 0$$ then there would exist a $$z$$ that would make $$f$$ nonzero. Therefore, $$x=y$$)

However, I am stuck trying to prove that it is onto, to maybe extend this result to infinite dimensional spaces. Is it even possible?

## nt.number theory – Will numerical coincidences remain conjectures forever? Number of groups of order \$2^n\$ up to isomorphism for \$n>3\$ is not a palindrome in base 10

Letting $$a_n$$ be the number of non-isomorphic groups of order $$n,$$ it appears that $$a_n$$ is not a palindrome in base $$10$$ for any $$n>3.$$ But how do we even begin to prove this? We can use heuristics to argue that if a small palindrome hasn’t been found for sequences growing at least exponentially, then we won’t find one.

But how can we ever convert these heuristics to a proof when there’s no relation between the nature of palindromes in base $$10$$ and sequences like $$a_n$$ that aren’t remotely close to having a closed form? It seems like this coincidence and similar such coincidences (consider rings, posets, monoids, and magmas instead) will remain a conjecture forever.

## How do I prove that if T \$in\$ Hom\$_G\$(V, W) is an isomorphism, then T\$^-1\$ \$in\$ Hom\$_G\$(W, V)?

How do I prove the following:

Suppose that V and W are representations of G. Then, if T $$in$$ Hom$$_G$$(V,W) is an isomorphism, we have that T$$^{-1}$$ $$in$$ Hom$$_G$$(W,V).

## fa.functional analysis – Explicit isomorphism between \$L^2(mathbb{R}^2)\$ and \$L^2(mathbb{R})\$?

As Hilbert spaces, $$L^2(mathbb{R}^2)$$ and $$L^2(mathbb{R})$$ are isomorphic. Of course the isomoprhism is vastly not unique. I wonder if there are any particularly nice explicit isomorphisms. E.g. I wonder if there is an integral transform
$$f(x,y) mapsto (K f)(z)=int dx, dy K(x,y,z) f(x,y)$$
with a nice explicit kernel $$K(x,y,z)$$ which maps $$L^2(mathbb{R}^2)$$ isometrically onto $$L^2(mathbb{R})$$? Any example would be appreciated.

## Sub graph isomorphism algorithms

What are good sources to learn about practical algorithms for subgraph isomorphism ? I wish to learn in depth on the latest and efficient ones.

## ct.category theory – Word for two morphisms that are equivalent up to right-composition with isomorphism

Let $$f:Ato C$$, $$g:Bto C$$ be morphisms in some category.
I call $$f,g$$ “equivalent” iff there exists an isomorphism $$h$$ such that $$fcirc h=g$$ (and consequently $$gcirc h^{-1}=f$$).

Question: Is there an established term for this kind of equivalence?

Background: In a paper, I am defining this in a slightly more specific setting, and I would like to add a clarifying sentence such as “note that this is the same as the notion of … for general categories”.

## vector spaces – Isomorphism between \$V^* otimes V\$ and End\$V\$

If $$V$$ is a vector space and End$$V$$ is the set of endomorphisms from $$V$$ to $$V$$.

Defining a map from $$V^*otimes Vrightarrow$$ End $$V$$ by sending some element say, $$fotimes v in V^* otimes V$$ to endomorphism whose value at $$win V$$ is $$f(w)v$$.

I am trying to show this map is an isomorphism between $$V^* otimes V$$ and End $$V$$. I am trying to verify it using dual basis.

If some hint can be provided, it will be a great help!