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?

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?

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.

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!!

}

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?

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.

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

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.

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.

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 $?