AC. classical analysis and odes – Alternative characterization of floquet multipliers: Floquet theory

Given an autonomous ode $ dot {x} = f (x) $ in $ mathbb {R} ^ n $ which has a periodic solution period-p $ bar x (t) $, the so-called variational equation can be used on $ bar x $ to study its stability. This equation satisfies begin {align} dot {y} (t) = A (t) y end {align}, where $ A (t) = D_xf ( bar x (t)) $.

Now the typical definition of Floquet multipliers $ lambda $ is that they are the values ​​e. of the time flow map-p U (p) of the variational equation, that is, of the matrix / operator U (p) where U (t) satisfies the variational equation, with $ U (0) = I_ {n times n} $. The exponents of the floquet $ mu $ then they are defined as $ frac {1} {p} log { lambda} $.

Apparently, an alternative definition can be given starting with the exponents. Here one defines $ mu $ as the eigenvalues ​​of the operator $ L (x) (t) = ( frac {d} {dt} -A (t)) x (t) $ in the space of the p-periodic functions. And then multipliers are defined analogously as $ lambda = e ^ {- µp} $.

I am looking for proof that both definitions are the same (mod some signs if necessary).

harmonic analysis – classical singular integral operator

I am working on a problem related to the Biot-Savart law in fluid dynamics. I found a singular integral theorem that is closely related to my research.

Assume that $ K (x) $ is a Calderón-Zygmund nucleus in $ mathbb {R} ^ {3} $ Y $ f $ it's a satisfying smooth function
begin {equation}
| f (x) | leq frac {1} {(1 + r) ^ {a}} quad text {para} quad 0 <a <2,
end {equation}

Y
begin {equation}
| nabla f (x) | leq frac {1} {(1 + r) ^ {b}} quad text {para} quad 0 <b <2,
end {equation}

where $ r = sqrt {x_ {1} ^ {2} + x_ {2} ^ {2}} $. It decomposes only in horizontal directions.

Define $ Tf (x) = int K (x-y) f (y) dy $. So there is a constant $ c_ {0} $ such that
begin {equation}
| Tf (x) | leq frac {c_ {0}} {(1 + r) ^ {d}},
end {equation}

where $ d = min {a, b } $.

My questions:
1. Someone suggested to me that this theorem could be improved. Since it is a point or $ L ^ { infty} $ estimate the singular integral that was not covered by any existing theory. Is it possible to demonstrate something like this? $ d = frac {a + b} {2} $ which will help my project a lot.

  1. One can change the second condition or add something new like decomposition of $ nabla ^ {2} f $.

I am not an expert in harmonic analysis. Any advice is welcome. Thank you!

classical analysis and classical odes: limit of a Fourier coefficient of a non-negative periodic function in terms of its form $ L ^ 2 $

This question is motivated by the previous MO question: Show that $ ( sum_ {k = 1} ^ {n} x_ {k} cos {k}) ^ 2 + ( sum_ {k = 1} ^ {n} x_ {k} sin {k}) ^ 2 le (2+ frac {n} {4}) sum_ {k = 1} ^ {n} x ^ 2_ {k} $.
It is a clean asymptotic version of that question.

Leave $ f $ be a non-negative, periodic function with period $ 1 $and square integrable in $ { Bbb R} / { Bbb Z} $. It is true that
$$
| { widehat f} (1) | ^ 2 = Big | int_0 ^ 1 f (x) e ^ {- 2 pi ix} dx Big | ^ 2 le frac 14 int_0 ^ 1 f (x) ^ 2 dx ?
$$

Equality is achieved, for example, when $ f (x) = max (0, cos (2 pi x)) $.

Note that $ | widehat f (1) | = | widehat f (-1) | $ and from $ f $ it is not negative $ | widehat f (1) | le widehat f (0) $. Thus
$$
int_0 ^ 1 f (x) ^ 2 dx = sum_n | widehat f (n) | ^ 2 ge 3 | widehat f (1) | ^ 2,
$$

so that the estimate is maintained with $ 1/3 $ instead of $ 1/4 $. There is a lot of room to improve this argument, and with a more careful application of Bessel's inequality I could get the constant $ 1/4 + 1/4 pi $. But the alleged inequality looks very clean, and I wonder if (i) is true!, (Ii) is known in some context, and (iii) (hopefully) has an elegant proof?

gr.group theory – Conjugation in simple finite classical groups

Leave $ G_n (q) = mathrm {PSL} _n (q), mathrm {PSU} _n (q), mathrm {PSp} _ {2n} (q) $, … be a simple classic group. Consider the natural inlay $ G_ {n-1} (q) subset G_n (q) $. I would greatly appreciate it if you have an argument or a counterexample for the following:

Claim: yes $ x, y in G_ {n-1} (q) $ such that $ x ^ T = y $ for some $ T in mathrm {Aut} (G_n (q)) $, then there $ t in mathrm {Aut} (G_ {n-1} (q)) $ such that $ x ^ t = y $.

Predictors of linear neuronal branches, exponential classical predictors, in resources?

Wikipedia says:

The main advantage of the neural predictor is its ability to exploit long stories while only requiring linear growth of resources. Classic predictors require exponential growth of resources.

What is the reason for this?

classical mechanics – Stretch elasticity in terms of main stretches

Suppose we have an indifferent isotropic function
$ W: GL _ + (3) a (0, infty) $, where $ GL _ + (3) $ denotes the set of all real $ (3 for 3) $-matrices with positive determinant.
We can write $ W (F) $ in terms of the main tranches (i.e. singular values) of $ F $. More specifically, there is a function
$ f: (0, infty) ^ 3 a (0, infty) $
such that
$$
W (F) = f ( lambda_1 (F), lambda_2 (F), lambda_3 (F)),
$$

where $ lambda_1 (F) geq lambda_2 (F) geq lambda_3 (F) geq 0 $ are the singular values ​​of $ F $.
Assume that $ W $ Y $ f $ they are soft I want to write the elastic tensor. $ frac { partial ^ 2 W} { partial F ^ 2} $ in terms of $ f $ Y $ lambda_1, lambda_2, lambda_3 $.

My ansatz is as follows: symmetry of $ frac { partial ^ 2 W} { partial F ^ 2} $ reduce the degrees of freedom from 81 to 45. In addition, each rotation $ R in SO (3) $ can be written exclusively as a product
$$
R = R_z ( alpha_3) R_y ( alpha_2) R_x ( alpha_1),
$$

where $ R_w ( alpha_i) in SO (3) $ with $ w in {x, y, z }, i in {1,2,3 }, $ denotes rotation around the $ w $-axis by angle $ alpha_i $. By framework of indifference and isotropy we have for each matrix $ F in GL _ + (3) $
$$
W (F) = W (R_z ( alpha_6) R_y ( alpha_5) R_x ( alpha_4) F R_z ( alpha_3) R_y ( alpha_2) R_x ( alpha_1))
$$

for each $ alpha_1, dots, alpha_6 in (0.2 pi) $.
In particular,
$$
0 = frac { partial} { partial alpha_i}
W (R_z ( alpha_6) R_y ( alpha_5) R_x ( alpha_4) F R_z ( alpha_3) R_y ( alpha_2) R_x ( alpha_1))
$$

for each $ i in {1, points, 6 } $. This relationship produces an additional reduction in the degrees of freedom of $ frac { partial ^ 2 W} { partial F ^ 2} $. However, I am not sure how to continue from here. The resulting linear system seems to be very complicated and I suspect there is a more efficient way to determine all the inputs of $ frac { partial ^ 2 W} { partial F ^ 2} $ in terms of $ f, lambda_1, lambda_2, lambda_3 $.

by $ (3 for 2) $-Matrices the answer to my question was presented by Pipkin in this document: https://academic.oup.com/imamat/article-abstract/36/1/85/762621?redirectedFrom=fulltext
The result is a very compact expression. Unfortunately, there are no details of the calculation provided.

Hence my question: Can anyone tell me how to calculate $ frac { partial ^ 2 W} { partial F ^ 2} $ in terms of $ f, lambda_1, lambda_2, lambda_3 $ (or how to improve / simplify my previous ansatz)? Or does anyone know any literature where you can find the desired result?

Note: I posted this question on math.stackexchange.com a few months ago, but it received very little attention:
https://math.stackexchange.com/questions/3421016/elasticity-tensor-in-terms-of-principal-stretches

Does Abstrac Algebra use classical logic?

I didn't know there was something called classical logic. If there is a monetary logic, does the book "Dummit's Abstract Algebra, Foote" use classical logic?

Process: What would be the difference between classical and quantum computers?

I'm not sure that this is the right place for this question (that's why I think it's for tags), but if not, can anyone say which site is?

Recently, as any of you know, Google achieved quantum supremacy. That is, they built a quantum computer that performed the work given approximately 1.5 billion so fast.

Although they are millions or more;) times faster in the execution of programs (the programming is done here in principle in the same way as in the classic computers, or not?) The condition under which you can use a quality control it's very different. You (maybe very far in the future, if Nature still exists) cannot make them very small and portable, for example.

Will the type of programs, in other words, the jobs that will be sorted, for example, be different? I guess so due to the fact that quality control is superfast in testing millions (or even more) different combinations of any type of object (such as a random combination of bits, as in the case of the Google experiment). I heard that quality controls can be used in meteorology, but (as a secondary question) how is there a question about the combinations? In more general terms, for what type of work are quality controls designed or programmed?

Does anyone have more thoughts on this?

ag algebraic geometry: derived categories and classical theorems in homological algebra

So far I have studied a fundamental part of the theory of derived categories, for example, the existence of derived functors, the "composition of derived functors", etc.

Now I have some questions about derived functors in the sense of derived category theory.

(one) Are categorical derivative functors universal derivatives in the classical sense?
Leave $ mathscr {A, B} $ be two abelian categories, $ f: mathscr {A} to mathscr {B} $ an exact left functor, and suppose that $ mathscr {A} $ You have enough injection.
Then there is "the" right derived functor $ mathbb {R} ^ + f: D ^ + ( mathscr {A}) to D ( mathscr {B}) $ from $ f $,
And it is $ i $-th cohomology of an object $ X $ in $ mathscr {A} $ (considering as the complex that has $ X $ to $ 0 $-th grade) is the classic $ R ^ yes (X) $.
So $ {H ^ i ( mathbb {R} ^ + f (X)) } _ i $ it is universal, in the sense of Hartshorne's AG, chapter III, that is, for each $ delta $-functors $ {g ^ i } _ i $ from $ mathscr {A} $ to $ mathscr {B} $ (i.e., a collection of additive functors that satisfy the following condition: for each exact sequence it cuts $ 0 a X a Y a Z a 0 $ in $ mathscr {A} $, there is the "connection map" $ g ^ iZ to g i + 1} X $, doing the exact long sequence.), if we have a natural transformation $ f to g ^ 0 $, there is a unique natural transformation $ R ^ if to g ^ i $.
Is there now a categorical interruption derived from this phenomenon?
That is, for such $ {g ^ i } $ Y $ f to g ^ 0 $there is $ delta $-functor $ g: D ^ + ( mathscr {A}) a D ( mathscr {B}) $ (such that for each $ X in mathscr {A} $, $ H ^ i g (X) = g ^ i (X) $) Y $ mathscr {Q} f to g mathscr {Q} $?
($ mathscr {Q} $ is the location functor).
If so, then by universality (in the derived categorical sense), we have $ mathbb {R} ^ + f to g $.
(I read this post, but it seems to be a little different from my question).

(2) Can we display the "Grothendieck Tohoku" easily using the derived category?
This theorem says that, in particular, for a $ delta $-functor $ {g ^ i } $Yeah $ g ^ i (I) = 0 $ for each $ i gt 0 $ and each injectable object $ I $, so this is universal, that is, $ g ^ i cong R ^ i g ^ 0 $.
If (1) is true, then I believe that this theorem can be translated as follows:

Leave $ {g ^ i } $ be a $ delta $-functor and $ g: D ^ + ( mathscr {A}) a D ( mathscr {B}) $ be "morphism" as in (1).
Yes $ g ^ i (I) = 0 $ for each $ i $ and injective $ I in mathscr {A} $, then $ g $ is the right derived functor of $ g ^ 0: K ^ + ( mathscr {A}) a K ( mathscr {B}) $.

Is this true?

(3) How can we use derived derived categorical functors to show propositions around spectral sequences?
I am studying categories derived from "Waste and duality" of Hartshorne.
In this text, the author says "What used to be a spectral sequence is now simply a composition of functors. And, of course, one can retrieve the old spectral sequence …" (see observations after Proposition 5.4.)
I think the author means that we can show "all" the propositions that the spectral sequence argument used to show using derived categories.
For example:

leave $ mathscr {A, B, C} $ be abelian categories, $ f: mathscr {A} a mathscr {B}, g: mathscr {B} a mathscr {C} $ left exact functors, and suppose that $ mathscr {A, B} $ has enough injectables and that $ f $ take each injectable object of $ mathscr {A} $ yet $ g $– acyclic object.
Leave $ X in mathscr {A} $.
Yes $ R ^ if (X) = 0 $ for each $ i gt 0 $, then $ R ^ n (gf) (X) cong (R ^ ng) (f (X)) $.
(This is obvious. I could show it.)

b) More generally, in the situation of (a), if $ R ^ ifX = 0 $ for $ 0 lt i lt q $, then there is an exact sequence $ 0 a R ^ qg (f (X)) a R ^ q (gf) X a g R ^ q f X $.

c) Leave $ f: X a Y $ be a morphism of appropriate schemes on a field, $ mathscr {F} $ a coherent sheaf in $ X $.
So $ chi ( mathscr {F}) = sum_p (-1) ^ p chi (R ^ p f_ * mathscr {F}) $.

d) Leave $ f: X a Y $ be a morpshim of schemes, and $ mathscr {F} $ a bundle of modules in $ X $.
Suppose for everyone $ q $, $ dim operatorname {supp} R ^ qf _ * mathscr {F} = 0 $.
So $ H ^ 0 (Y, R ^ nf _ * mathscr {F}) cong H ^ n (X, mathscr {F}) $.
(This proposition is used in Mumford's Abelian varieties, $ S $8, theorem1.)

(All propositions are trivial if we use the Grothendieck spectral sequence).

I think there are many of these propositions related to the Grothendieck spectral sequence, but if I understand these (a) ~ (d), it seems similar to show any other type proposition.

Related publication: this

Finally,
(4)Easier test of classical homological propositions
I have heard that we can show Kunneth's formula more easily using derived categories.
I want to know such propositions.
Would you give me references?
I also want references to categorical evidence derived from cohomology and base change theorems (see III.12 of Hartshorne & # 39; s AG or here.
The subsequent statement is too abstract for me.
I prefer Hartshorne's concrete statement).

Thank you!

Is there any non-relativized separation between a class of quantum complexity and a classical one?

I am familiar with the results of the relativized separation for BPP-BQP, BQP-PH and NPC-BQP. I am also aware that, p. The factoring is not believed to be in BPP, it has not been tested and, therefore, we are not close to demonstrating BQP! = BPP.

My question is, do we have any true, theoretical and non-relativized proof that quantum computers are "superior" to classical ones in any specific problem? Where here, I define superior to include any type of superiority of runtime complexity, for example, a quadratic improvement would qualify in this regard.

I would expect Grover's algorithm to qualify for this criterion, but despite being supposedly "obvious", I am not sure if it has been shown that classical computers cannot solve it also in sqrt (n) time, in a case model black (that is, not relativized).