Linear algebra – Scope of the null space of A

I apologize for the brief nature of this question, but I do not think it has been clarified in a previous post on this topic – Find a Covering Set for Null Space.

When we say that the null space of the matrix $ A $ is equal to the duration of a set of vectors:

$ N (A) = span ( v with v_1, vec v_2, $ $ … vec v_n) $

Are we actually saying that:

$ A ( vec v_1) + A ( vec v_2) + $ $ … A ( vec v_n) = vec 0 $

abstract algebra – Let $ phi: mathbb {Z} _7 times mathbb {Z} _7 $ be a homomorphism such that $ phi ^ 5 = text {id} $. Show that $ phi $ is the identity.

Let $ phi: mathbb {Z} _7 times mathbb {Z} _7 to mathbb {Z} _7 times mathbb {Z} _7 $ to be a homomorphism such as $ phi ^ 5 = text {id} $. CA watch $ phi $ is the identity.

My attempt: Since $ ker ( phi) subset ker ( phi ^ 2) subset cdots subset ker ( phi ^ 5) = 0 $it follows that $ phi $ is injective. Since $ | mathbb {Z} times mathbb {Z} | $ is finished, $ phi $ is automatically an isomorphism.

then $ phi $ should map a subgroup to a subgroup. Consider $ phi ( mathbb {Z} _7 times 0) $, so he should be associated with $ mathbb {Z} _7 times 0 $ or $ 0 times mathbb {Z} _7 $. Without loss of generality, we consider the first case. In particular, $ phi ($ 1.0) should be a generator for $ mathbb {Z} _7 times 0 $. assume $ phi (1,0) = (a, 0) $. then $ (1,0) = phi ^ 5 (1,0) = phi ^ 4 (a, 0) = phi ^ 3 (a ^ 2,0) = cdots = (a ^ 5,0) $. So, $ a ^ 5 = 7k + 1 $ for some people $ k in mathbb {Z} $. By testing $ a in mathbb {Z} _7 $, we see that the only possibility is that $ a = $ 1. The discussion for $ phi (0,1) $ Is similar.

Could someone help me take a look at my attempt? Is my approach reasonable and is there a better way to solve this problem?

Agalgebraic geometry – Which endomorphisms of Tate's algebra are "algebraic"?

For an abelian variety $ A $ on a field k $ with different characteristic of $ ell $ and the Galois group $ G = Gal ( overline k / k) $there is always an injective map of the form:
$$ mathbb Q_ ell otimes End_k (A) to End_G (V_ ell (A)) $$
or $ V_ ell $ is the rational module of Tate. In specific cases, we know that the map is a bijection (k $ = finite fields, numeric fields).

Suppose now that we take the direct limit on both sides to increase k $. Then the left side is simply $ mathbb Q_ ell otimes End _ { overline k} (A) $ and the map is in $ End (V_ ell (A)) $ but it is (almost?) never a bijection.

For example, for non-super singular curves or in characteristic curves $ 0 and $ A $ elliptic curve, the right side is one or two dimensions, while the right side is four-dimensional.

Can we classify or characterize the endomorphisms of $ V_ ell (A) $ which are "algebraic", ie come from the map above for some k $?

reference query – These two natural structures $ A_ infty $ -structures on the realization of an isomorphic cosimplective commutative algebra?

Given a cosimplicial commutative algebra $ A $ bullet on a characteristic zero field, there are two ways to produce a $ A_ infty $-structure on its realization $ | A ^ bullet | : = int ^ Delta C ^ * ( Delta ^ bullet) otimes A ^ bullet $, or $ C ^ * (-) $ is the simplicial complex cochain, that is to say $ | A ^ bullet | $ is the cochain complex with $ | A ^ bullet | ^ n = A ^ n $ and differential the alternate sum of coface cards:

  • The Alexander-Whitney map equips the realization functor with a lax monoidal structure, so that it sends the monoid $ A $ bullet in simplified groups to a monoid in cochain complexes, that is to say a differential graded algebra. Explicitly, the product of $ a_p in A $ p $ and $ a_q in A ^ q $ is (to sign) $ f_ {p, q} (a_p) b_ {p, q} (a_q) $, or $ f_ {p, q}: A ^ p to A ^ {p + q} $ and $ b_ {p, q}: A ^ q to A ^ {p + q} $ are the cofaces "front and back". Note that this does not use the commutativity of $ A $ bullet, and the product is not commutative in general. Call it (dga, and so in particular) $ A_ infty $-algebra $ | A ^ bullet | _ {CG} $.
  • The coend $ Omega ^ * (A ^ bullet): = int ^ Delta Omega_P ^ * ( Delta ^ bullet) otimes A ^ bullet $, or $ Omega ^ * _ P ( Delta ^ n) = k (t_0, points, t_n, mathrm dt_0, points, mathrm dt_n) / (t_0 + points + t_n-1, mathrm dt_0 + points + mathrm dt_n) $ carries a natural (commutative) dga structure. As explained by Cheng and Getzler, there is a simplicial retraction $ Omega_P ^ * ( Delta ^ bullet) rightleftarrows C ^ * ( Delta ^ bullet) $, giving rise to a retraction $ Omega ^ * (A ^ bullet) rightleftarrows | A ^ bullet | $ along which this structure can be transferred to a $ A_ infty $-structure on $ | A_ bullet | $ (in fact, even a $ C_ infty $-structure). The operations are given by sums on trees. Call it $ A_ infty $-algebra $ | A ^ bullet | _ {CG} $.

Obviously, these two structures are very different: for example, the $ 2 $The secondary functioning of the first is associative, but not commutative, while that of the second is commutative, but not associated.

There is a special case where I know that these $ A_ infty $ the algebras are equivalent: Si $ M $ is a smooth variety and $ A ^ bullet = operatorname {Sing} ^ bullet (M) $ is the commutative algebra cosimplicial functions on smooth simplices, there is a zigzag
$$
| A ^ bullet | _ {CG} rightarrow Omega ^ * (A ^ bullet) rightarrow int ^ Delta Omega ^ * ( Delta ^ bullet) otimes A ^ bullet leftarrow Omega ^ * (M) rightarrow | A ^ bullet | _ {AW}
$$

where are the cards, in the order:

  • the canonical $ A_ infty $-morphism produced by the transfer of homotopy
  • the inclusion (simplex) of polynomial forms in smooth forms
  • the map $ omega mapsto ( sigma mapsto sigma ^ * omega) $
  • an explicit $ A_ infty $-isomorphism obtained by Chen's iterated integrals, see for example here.

Is there a natural $ A_ infty $-isomorphism between $ | A ^ bullet | _ {AW} $ and $ | A ^ bullet | _ {CG} $? Can it be reasonably explicit? If so, does it extend the construction above to $ A ^ bullet = operatorname {Sing} ^ bullet (M) $, or is there at least one natural (and explicit?) $ A_ infty $-Homotopy between them?

Operator Theory – Meaning of Affiliate to a von Neumann Algebra

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Abstract algebra – How to apply the theory of "homology" to the Goldbach conjecture using the result of Helfgott (ternary Goldbach true)?

The result of Helfgott: every odd integer $ n geq $ 7 is the sum of $ 3 $ (odd) bonuses. I saw somewhere in the newspaper the mention of odd prime numbers. It means that every single integer $ n geq $ 10 is the sum of 4 odd prime numbers (you just added a $ 3 $ to each odd number).

Let's first assume something bold, to simplify the discussion, that for each integer $ x geq $ 13, the even number $ 2x $ can be expressed as the sum of $ 4 $ distinct prime numbers: $ 2x = p + q + r + s $. To handle all cases, I'm sure it's possible, because it would take either an enumeration of the small set of cases, or a combinatorial argument. Reason is $ x geq $ 13 is because: $ 3 + 5 + 7 + 11 = 26 = 2 cdot $ 13 is the smallest even number which is the sum of $ 4 $ distinct odd primes.

If you are not familiar with the basics of simplicial homology but you know what the free $ Bbb {Z} $-module on a set of symbols is, then google "introduction to simplicial homology". You basically need limit cards: $ partial_n: C_n to C_ {n-1} $ who take each element of the module $ C_n $ at its "limit" in $ C_ {n-1} $, its border being represented as a formal element in the free module, whatever the border map, it must be a $ Bbb {Z} $homomorphism -module and satisfy also: $ partial_ {n-1} circ partial_n = 0 $.

So, according to Helfgott's result (and the net distinction hypothesis), each integer $ x geq $ 13 is such that the even number $ 2x = p + q + r + s $, or $ {p, q, r, s } $ are distinct odd prime numbers. As a result, there is $ {4 choose 2} = $ 6 ways of forming even smaller numbers (by adding two of the prime numbers involved in the sum of $ x $) $ 2x_i, i in {1, dots, 6 } $ such as $ 2x_i + 2x_j = 2x $ or equivalent $ x_i + x_j = x $. Now, there is another problem with duplicates as $ x_i = x_j $ is a possibility. But for now, suppose that $ x_i neq x_j $, for reasons of discussion. So, to recap, everyone $ x_i = u + v $ or $ u, v in {p, q, r, s } $.

So, if we associate each $ x_i $ with the official symbol $ bar {x_i} $ which might not be an integer (for the moment) then we will call $ bar {x_i} $ the faces of the simplex $ bar {x} $. But $ bar {x} $ could also be a simplex face of another integer! It is therefore possible to obtain a long-chain complex eventually.

So we have at least $ Delta = { bar {x}, bar {x_i}: i in {1, points, 6 } } $ like a complex. This does not necessarily correspond to an abstract simplicial complex in which $ forall sigma in Delta, $ if $ tau subset sigma $then $ tau in Delta $. Since for one, we have not defined the $ bar {x_i} $ to be sets – these are just formal symbols.

Another approach would be?

remember that $ sum limits_ {i = 1} ^ 6 alpha_i x_i = x $ or $ sum limits_ {i = 1} ^ 6 alpha_i = 2, alpha_i in {0, 1 $. It's a bit like a "convex combination". Anway, as you can see, I took elements of simplicial homology and tried to adapt it. Do you have any ideas on how to proceed?

group theory gr. – What are the compacts Aut (A) of an algebra A (G), G finite, which contains the identity?

If we have an algebra A on a finite group G, then if G is non-abelian, we can have a non-trivial set of compact automorphisms of A that map the elements of G to an isomorphic set to G. it only has to contain the identity, but I have not checked that.

For example, the field algebra $ C $ more than $ S_3 $ has such a set of automorphisms that form an isomorphic group to $ SO (3) $. When G is dihedral of size $ 2p $, the A (G) have compact automorphisms meeting the above requirement with at most $ 3 (p-1) / $ 2 generators is $ p $ is strange, or $ 3 (p-2) / $ 2 if $ p $ is same.

Has anyone classified this?

Thank you very much to Nik Weaver for explaining the language with which to ask the question.

algebra precalculus – Real Roots of $ 1+ frac {x} {1!} + frac {x ^ 2} {2!} + frac {x ^ 3} {3!} + cdots + frac {x ^ n} {n!} $

Let $ Q_n (x) $ to be the degree $ n $ polynomial
$$ 1+ frac {x} {1!} + Frac {x ^ 2} {2!} + Frac {x ^ 3} {3!} + Cots + frac {x ^ n} {n !} $$
How many real roots the equation $ Q_n (x) = 0 $ to have?


My attempt:

It's obvious that $ Q_n (x) $ will have all its actual roots in the negative part of the actual line if there are any. In addition, we notice that if $ n $ is odd, then there is at least one real root by the complex conjugate root theorem. So I suppose there is exactly one root for $ n $ strange and there is no root for $ n $ even.

However, I do not know how to analyze $ Q_n (x) $. All I can do is take derivatives, which does not provide more useful information. Any clue is appreciated! Thank you.

linear algebra – Retrieves the translational component of a particular matrix multiplication

Let's say that I have a rotation matrix R 4x and a 4×4 translation matrix T. If I multiply the matrices in this order T * R, the translation component of T will not be affected. But if you multiply them in the R * T order, it will be affected by R.'s rotation values.

How can I describe the position vector of the resulting matrix from the multiplication R * T, in terms of the rotational components of R and the translational component of the matrix T? Can I describe it as the scalar product of the respective orientation component of the matrix R and the translation component of the matrix T?

abstract algebra – $ I $ is a free R-module if and only if $ I = Ra $

Problem: Let $ I $ to be a non-zero ideal of a commutative ring $ R $ with identity. Prove it
$ I $ is a free R-module if and only if $ I = Ra $ for some people $ a in R $ it's not a zero
divider.

My idea is:
($ Rightarrow $) Let $ I $ to be generated by $ {a_1, … a_n } (n> 1) $, then I have $ a_1x_1 + … + a_nx_n = 0 $with $ x_1 = a_2, x_2 = -a_1, … $($ n $ is same); $ x_1 = a_2, x_2 = -a_1-a_3, x_3 = a_2, x_4 = a_5, x_5 = -a_4, … $ ($ n $ is odd).
So, $ a_1 = … = a_n = 0 $. Contradiction. So, $ n = $ 1.

Is it good? Help me! Thank you!