## Abstract algebra – Could fibers partition a group like cosets?

I have read evidence that cosets $$uN$$, or $$N$$ is the core of some homomorphism, partitioning a group. However, it seems that only the fibers of any homomorphism seem to share a group.

Say a little homomorphism $$varphi: G rightarrow H$$ maps $$u$$, $$v$$, $$p$$, and $$q$$ to exactly one element of the coded domain $$H$$: $$U$$, $$V$$, $$P$$, and $$P$$ respectively (otherwise it is not a function). $$UP = VP$$ it is only when $$U = V$$. Since homomorphism only brings domain values ​​to exactly one value in the coding domain, $$uP = vP$$ it is only when $$u$$ and $$v$$ are in the same homomorphism fiber, so the fibers are shared $$G$$. I said in the same fiber because also note that in this proof all the fiber elements above $$F$$ is not even necessarily a subgroup of $$G$$.

Is the proof correct?

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## abstract algebra – functions in \$ mathbb {C}[x_1, x_2, x_3, x_4]\$ which disappears for all \$ (u ^ 3, u ^ 2v, uv ^ 2, v ^ 3) \$ where \$ u, v in mathbb {C} \$

I want to show that Z = {$$(u ^ 3, u ^ 2v, uv ^ 2, v ^ 3) | u, v in mathbb {C}$$} is an algebraic set.
So i need functions in $$mathbb {C} (x_1, x_2, x_3, x_4)$$ who disappear for all $$(u ^ 3, u ^ 2v, uv ^ 2, v ^ 3)$$ or $$u, v in mathbb {C}$$.

I have already discovered that the function $$f_ {uv} (x_1, x_2, x_3, x_4) = -x_1 – 3x_2 – 3x_3 – x_4 + (u + v) ^ 3$$ disappears at $$(u ^ 3, u ^ 2v, uv ^ 2, v ^ 3)$$. But how can I find one that disappears for all the complex numbers u, v?

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## Abstract algebra – The stabilizing monoid counterparts \$ M to M \$ form a subgroup of \$ text {End} (M) \$.

Let $$M =$$ the free monoid on all symbols $$S$$ and consider the semiring $$text {End} (M)$$. Addition is concatenation and multiplication is the composition of functions. Let $$F subset M$$ to be a finished set. Let $$s: S to F$$ to be a given surjective overview map such that $$s ^ 2 (m) = s (m)$$ for everyone $$m in S$$. Then during the extension $$s$$ homomorphically to all $$m$$ we have $$s ^ 2 (m) = s ^ 2 (g_1) cdots s ^ 2 (g_k) = s (g_1) cdots s (g_k) = s (m)$$ or $$m = g_1 cdots g_k$$ and $$g_i in S$$.
So this stability under iteration goes to all $$m in M ​​$$.

Prop 1. L & # 39; together $$N$$ of all $$f in text {End} (M)$$ so that its iterations $$f ^ k (m)$$ stabilize at $$s (m)$$ is closed under composition. So $$N$$ is a multiplicative subgroup of $$text {End} (M)$$.

Evidence. Is the question.

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## computer algebra – Need to fill in missing parts C ++ language

model
void hashT :: insert (int hashIndex, const CType2 & rec)
{

// Fill this part!

}

model
void hashT :: search (int & hashIndex, const CType2 & rec,
bool & found) const
{

// Fill this part!

}

model
void hashT :: retrieve (int hashIndex, CType1 & rec) const
{
// Complete this part!

}

model
void hashT :: remove (int hashIndex, const CType2 & rec)
{

``````//  Fill up this part!
``````

}

// to print all BSTs for the entire hash table
// For each BST, you must print the stored stateData information
model
void hashT :: print () const
{

``````//  Fill up this part!
``````

}

model
hashT :: hashT (int size)
{

``````//  Fill up this part!!
``````

}

model
hashT :: ~ hashT ()
{

``````//  Fill up this part!!
``````

}

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## Linear algebra – Formula for a completely positive map

Is there a family of cards that are completely positive $$L (A, B)$$ continuously dependent on two non-zero symmetric positive semifinished $$n times n$$ matrices $$A$$ and $$B$$, such as $$L (A, B)$$ maps $$A$$ at $$B$$ and reduces to the identity when $$A = B$$?

Can we give a closed formula for $$L (A, B)$$?

If such a family does not exist, what are the conditions $$A$$ and $$B$$ for such a family exists?

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## linear algebra – Logarithm of a unitary matrix

Let $$U$$ be a $$n times n$$ complex unitary matrix. CA watch $$U$$ has a logarithm. C & # 39; $$U = e ^ V, text {where} V text {is a} n times n text {complex matrix}$$.

My approach
I've used the theorem that says if $$A, B$$ are $$n times n$$ complex matrices, and $$B$$ is then invertible $$e ^ {BAB ^ {- 1}} = Be ^ AB ^ {- 1}$$

After using the theorem above, I ended up getting:
$$e ^ U = Ue ^ UU ^ {- 1} = V ( text {let)}$$

I do not get $$e ^ U = V$$.

I do not know how to use the condition $$U$$ is unitary.

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## Relational algebra _fullouterjoin

I do not know how to approach it:

Either prove or give a counter-example for the following formula

R fullouterjoin (S ∪ K) = (R fullouterjoin S) ∪ (R fullouterjoin K)

Given: T = R fullouterjoin S = (R Leftouterjoin S) ∪ (Rightouterjoin S)

I really need help !! has been trying to solve this problem for 3 hours and none of the books has helped me

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## Linear Algebra – Do teachers write their own questions or do they usually find them somewhere?

I am preparing for the final exams and wish to obtain the possible grade. But I find that previous articles because the course has been taught by different people over the years are of no help because the questions vary greatly.

If you are a speaker, do you write your own questions? Specifically, my exam is on linear algebra and another one on actual analysis, so I want to prepare myself for the better. If someone knows books where I can find exam style questions, that would be great.

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## Linear algebra – Do complex eigenvalues ​​for planar systems with a real-valued matrix always have eigenvectors of complex value?

sorry if that's a silly question, but I was doing my homework and I noticed that every time I solved a planar system that had complex eigenvalues, I always found myself with complex eigenvectors. I was wondering if I could ever get a totally real clean vector from a complex eigenvalue.

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## Abstract algebra – Attach a root to a field that already contains this element

I know that we can take an irreducible polynomial $$f$$ on a field $$K$$ and make an extension containing a root of $$f$$ by construction $$K (x) / (f)$$. What happens if $$f$$ is actually reducible? For example, what is $$mathbb Q ( sqrt2) (x) / (x ^ 2 – 2)$$ look like?

My apologies for the vague question. I guess I ask what kind of structure it is (in particular, I can not say if it's still a field). Which parts depend on the nature of $$f$$ or $$K$$?

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