abstract algebra – Properties of a subgroup $ H leq G $ with $ H cap (x ^ {- 1} Hx = {e } $ for all $ x in G, x not in H $

The question is: say $ G $ is finite and $ H $ is a subgroup of $ G $, Accept that $ forall x in G $With $ x not in H $, we have $ H cap (x ^ {- 1} Hx) = {e } $, Show that
$$ | G | – left | bigcup_ {x in G} (x ^ {- 1} Hx) right | = (G: H) = 1 $$

I do not really understand where to start, though I think I understand conceptually. Every single part is pretty clear to me, but I'm lost when it comes to combining those ideas. Any help would be appreciated, thanks!

Abstract Algebra – Linear transforms that are not invertible in a vector space V of real-valued sequences

Find a vector space $ V $ and cards $ F, G in L (V, V) $ so that $ F cdot G = I $ but none $ F $ Yet $ G $ are invertible cards.

I tried to use the vector space of all real sequences as V. I have to find two linear transformations as above. I have a transformation, $ G: V rightarrow V $ so that $ G (a_1, a_2, …) = (a_2, a_3, …) $, But I do not know how to get there $ F $ so that $ F cdot G = I $, Please help.

linear algebra – Formulation of a partial derivative in the form of matrices

https://www.tensorflow.org/api_docs/python/tf/gradients

I use the tensorflow.gradients function that uses the tensor $ y $and tensor $ x $and calculates the partial derivative with respect to $ x $, For several $ x $ s returns the sum of all partial derivatives $ dy / dx $,

But I only have one matrix $ P $ the form (time_n, freq_n). Here, the first dimension is for temporal information, and the second dimension stores the frequency information. My question is whether I want to calculate the partial derivative $ dP / dt $ (in terms of the first dimension (time)), what will mine be? $ y $ and $ x $?

I think, $ y = P $but what will be $ x $ and it is dimension?

Linear algebra – A question to solve a differential equation.

$ mathbf {The Problem is}: $ To let, $ operatorname {u (t) = (u_1 (t), u_2 (t))} $ Where $ t gt 0 $ be the unique solution of the differential equation $
operatorname {du / dt} = Au (t) $
Where $ operatorname {u (0)} = (1,1) $ and $ A = $$
begin {bmatrix}
a & c \
c & b \
end {bmatrix}
$$ is a $
2 × 2$ symmetric matrix with $tr A lt 0$ and $ operatorname {det} A gt 0$, then evaluate $lim_ {t to infty} operatorname {u_1 (t)}. $

$ mathbf {my approach} $ : I could only think that by the given information both eigenvalues ​​of $ A $ are real and negative and therefore $ A ^ (- 1) $ exists and
$$
begin {bmatrix}
operatorname {du_1 (t) / dt} \
operatorname {du_2 (t) / dt} \
end {bmatrix}
=
begin {bmatrix}
au_1 (t) + cu_2 (t) \
cu_1 (t) + bu_2 (t) \
end {bmatrix}
$$
but I can not get any further.

abstract algebra – Toric variety associated with the cone over a polytope

To let $ P = [0, m] subset M _ { mathbb {R} ^ 2} $ be the line segment and look over the cone $ P $,

What is the toric variety of the cone over? $ P $?

The thing is, I'm not quite sure how to build a cone $ sigma $ from the polytope $ P $, It should be only the cone of $ me_1 $? In that case we would not receive the variety $ mathbb {C} times T ^ 1 $, Where $ T $ designates the torus?

What does the "affine variety of the cone of a polytope" actually mean? I'm sorry if this question looks too trivial, but I could not find a reference that answered that question explicitly.

linear algebra eigenvalues ​​and eigenvectors

I learn eigenvalues ​​/ vectors. I wanted to know it since then $ Av = lambda v $ then that means $ lambda_1 v_1 = lambda_2 v_2 $, I have worked through some beginner problems and this does not seem to be the case. Why is that? For example
$$ begin {bmatrix} 1 & 1 \ 4 & 1 end {bmatrix} begin {bmatrix} x \ y end {bmatrix} = lambda v $$

The answers I get are:
$$ lambda_1 v_1 = -1 begin {bmatrix} x \ -2x end {bmatrix} \
lambda_2 v_2 = 3 begin {bmatrix} x \ 2x end {bmatrix} \ $$

Linear Algebra – Understanding the solution of the homework problem

Greetings to all for the following problem

Let x be an n-vector and define y as the non negative vector (i.e., the vector with non negative entries) closest to x. Give an expression for the elements of y.

I have some problems understanding the solution, where solution

I've found that you need to take y-x by comparing the values ​​of y with x. But I do not understand the main purpose of the power 2. Is it to remove the negative value or is it the square root of the || to remove y-x || ?

Linear algebra – Calculation of the position of a camera of a third person from their right vector

I have a 3rd-person camera that always watches the player from a certain distance. It updates its x and z position around the player with respect to the yaw value received from the x-axis mouse input.

I'm trying to add clashes to the 3rd person camera. The collision algorithm accesses the speed of the object to update its position. For the collision to work on the third person's camera, the position can not be easily updated, otherwise the collision will not work.

So I have a "working" implementation in which I update the correct vector of the camera while it is pointed at the player and gives the desired circulating effect. But I scale this right vector by the mouse x offset to recalculate the new camera position, and if I move the mouse fast, it will cut corners in orbit. Here's a diagram of what I mean, I hope it makes sense.

3rd person camera

But I'd like to suggest how to get a smooth orbit and accelerate the orbit as the x offset increases due to the correct vector.

And here is the code:

right = cross(camera.front, camera.up);
right = normalize(right);
right = scale(right, 2.0f * deltaTime * xOffset);
cameraCollider.velocity -= right;

Thanks for your time.

Homologous algebra – an obstacle to splitting an object in a derived category into a sum of two-term complexes

To let $ mathcal {A} $ be an abelian category and $ D $ his limited derived class. An object $ M in D $ can be described as a list of cohomology objects $ H ^ i = H ^ i (M) $ along with some complicated glued data.

I am only interested in the case when $ mathcal {A} $ has homological dimension two. For example, $ mathcal {A} $ can be a category of contiguous discs on a smooth surface. In this case, the glued data is a collection of classes $ xi_i in mathrm {Ext} ^ 2 (H ^ i, H ^ {i-1}) $ between each pair of adjacent cohomology objects, with no restriction on selection.

By definition, an object $ M in D $ is quasi-isomorphic to a direct sum of complexes concentrated in a single degree (i.e., shifts of objects of $ mathcal {A} $) if and only if everyone $ xi_i $ disappears.

Likewise some objects in $ D $ are quasi-isomorphic to direct sums of complexes concentrated in two neighboring degrees. Is it possible to characterize this property by the disappearance of some obstacles resulting from the presentation of an object as a collection? $ {(H_i, xi_i) } _ {i in mathbb {Z}} $ above?

Abstract Algebra – Why should I care if $ V $ is an open subset of $ R ^ n $ with a volume of $> 1 $, then there are 2 rational points $ p, q in V $ s.t. $ p-q in Z ^ n $?

Lemma. If $ V $ is a limited open subset of $ R ^ n $ with volume $> 1 $there are 2 rational points $ p, q in V, p neq q $ s.t. $ p-q in Z ^ n $?

$ textbf {Q:} $ Why should I be interested in rational points? $ p, q $ Here? Most of the time I've seen evidence of using the Minkowski theorem $ p, q in V $ without knowing $ p, q $ rational. This is a well-established version of the statement used to prove Minkowski's theorem, but it seems very plausible if you do not wiggle rationally $ p, q $ to get a bit rational $ p, q $, What are possible application of the above-mentioned lemma?

Ref. Siegel, Lectures on the Geometry of Numbers, Lec II, para. 2.