Functional analysis of fa – Numeric range of the tensor product of two matrices

Leave $ T in M_n $. Is the following true?
$$ bigcap limits_ {B in M_2 \ text {tr} (B) = 0} left {X in M_2: W (X) subseteq W (T) text {y} W ( B otimes X) subseteq W (B otimes T) right } subseteq bigcap limits_ {B in M_2} ​​ left {X in M_2: W (B otimes X) subseteq W (B otimes T) right }? $$ where $ W (S): = { langle Sx, x rangle: Vert x Vert = 1 } $ it's called numerical range of $ S $.

Comments: I have checked the above in MATLAB using some particular option of $ T, B $ which is affirmative for those cases. So, I have tried to prove the above as follows:

  • We know $ W (X) subseteq W (T) $ if a map $ varphi: text {span} {I, T, T ^ * } rightarrow text {span} {I, X, X ^ * } $ S t. $ varphi (aI + bT + cT ^ *) = aI + bX + cX ^ * $ where $ a, b, c in mathbb {C} $ is positive. So, by hypothesis, we have a positive map that says $ psi: text {span} {I_2 otimes I, text {span} {B _ { circ}, B _ { circ} ^ * } otimes text {span} {T , T ^ * } } rightarrow M_4 $ S t.$ psi (I_2 otimes I + B otimes T + B ^ * otimes T ^ *) = I_2 otimes I_2 + B otimes X + B ^ * otimes X ^ * $ it's positive where $ B in M_2 $ S t $ tr (B) = 0 $ Y $ B _ circ}
    begin {pmatrix}
    0 and 1 \
    0 and 0 \
    end {pmatrix}.
    $
    Now, I have no idea if the map $ varphi $ can extend positively to $ text {span} {I, B _ { circ}, B _ { circ} ^ * } otimes text {span} {I, T, T ^ * } $ to get the required result.

It may be false, but I still don't have any counterexample. Thanks in advance. Any comments are greatly appreciated.

Functional analysis: are there no true functions with real values ​​of two or more variables?

I see the following theorem somewhere, without complete proof.

For each irrational number $ lambda $, there are continuous functions $ phi_k: (0,1) to mathbb R, k = 1,2,3,4,5 $, so that for all continuous functions $ f: (0,1) ^ 2 to mathbb R $, there is a continuous function $ g: (0,1) a mathbb R $such that $ f (x, y) = sum_ {k = 1} ^ 5 g ( phi_k (x) + lambda phi_k (y)) $.

There are some mysterious things about this result. Where does the number come from? $ 5 $ come and why $ lambda notin mathbb Q $?

Could anyone offer a reference for proof of this?

Functional analysis of fa – Higher order functional derivatives

Leave $ E, F $ Be Banach spaces. A continuous bilinear function $ langle cdot, cdot rangle: E times F to mathbb {R} $ is named $ E $-do-degenerate yes $ langle x, y rangle = 0 $ for all $ and in F $ it implies $ x = 0 $ (Similarly for $ F $-no-degenerate). Equivalently, the two maps of $ E $ to $ F * Y $ F $ to $ E * defined by $ x mapsto langle x, cdot rangle $ Y $ y mapsto langle cdot, and rangle $, respectively, are one to one. If they are isomorphisms (*), $ langle cdot, cdot rangle $ is named $ E $ or $ F $-Strongly not degenerate. We say that $ E $ Y $ F $ they are in duality if there is a non-degenerated bilinear function $ langle cdot, cdot rangle: E times F to mathbb {R} $, also called pairing from $ E $ with $ F $. If the functional is strongly non-degenerated, we say that the duality is strong.

Consider the following definition.

Definition: Leave $ E $ Y $ F $ be regulated spaces and $ langle cdot, cdot rangle $ a $ E $– non degenerate matching. Leave $ f: F to mathbb {R} $ be Fréchet differentiable at the point $ alpha in F $ (denote this derivative as $ Df (α) $) The functional derivative $ delta f / delta alpha $ from $ f $ with respect to $ alpha $ It is the unique element in $ E $, if it exists, so that:
begin {eqnarray}
Df ( alpha) ( gamma) = langle frac { delta f} { delta alpha}, gamma rangle tag {1} label {1}
end {eqnarray}

for all $ gamma in F $.

Now, I would like to know how to define higher order derivatives of functional derivatives. In other words, suppose Fréchet's derivative of $ f $ to $ alpha $, $ Df (α) $ is Fréchet differentiable in $ beta in F $. Is it possible to define $ delta2 f / delta beta delta alpha $?

calculation and analysis: possible error related to the BesselI derivative

In Mathematica 12.0, I execute the following code:

f(x_) = BesselI(0, x);
f'(x)

that comes back BesselI(1, x) as expected. But if I enter

f(x_) = BesselI(0, 1.0 x);
f'(x)

I get

0.5 (BesselI(1, 1. x) + BesselI(1, 1. System`Private`DerivativeX(1.)))

Further, D(f(x),x) Returns the expected result. I tried to exit the kernel without changes. Is this a mistake or something is wrong with my installation?

real analysis – Search: $ lim limits_ {n to + infty} frac { left (1+ frac {1} {n ^ 3} right) ^ {n ^ 4}} { left ( 1+ frac {1} {(n + 1) ^ 3} right) ^ {(n + 1) ^ 4}} $

Find :

$$ lim limits_ {n to + infty} frac { left (1+ frac {1} {n ^ 3} right) ^ {n ^ 4}} { left (1+ frac {1} {(n + 1) ^ 3} right) ^ {(n + 1) ^ 4}} $$

My attempt: I don't know if it's right or not!
I use this rule:

$$ lim limits_ {n to + infty} (f (x)) ^ {g (x)} = 1 ^ { infty} $$

Then :

$$ lim limits_ {n to + infty} (f (x)) ^ {g (x)} = lim limits_ {n to + infty} e ^ {g (x) (f ( x) -1)} $$

So :

$$ lim limits_ {n to + infty} frac { left (1+ frac {1} {n ^ 3} right) ^ {n ^ 4}} { left (1+ frac {1} {(n + 1) ^ 3} right) ^ {(n + 1) ^ 4}} $$

$$ = lim limits_ {n to + infty} frac {e ^ { frac {n ^ {4}} {n ^ {3}}}} {e ^ { frac {(n + 1 ) 4} {(n + 1) 3}} $$

$$ = lim limits_ {n to + infty} frac {e ^ {n}} {e ^ {n + 1}} $$

$$ = frac {1} {e} $$

Is my approach wrong?

Is this called partial limit calculation?

Functional analysis – How to calculate the Fréchet derivative of the $ varepsilon function: H_ {per} 1 ([0,L]) times L_ {by} 2 ([0,L]) longrightarrow mathbb {R} $

Yes $ varepsilon: H_ {per} ^ ((0, L)) times L_ {per} ^ {((0, L)) longrightarrow mathbb {R} $ is given by
$$ varepsilon (u, v) = frac {1} {2} int_ {0} ^ {L} big ({u_x} ^ 2 + v ^ 2 + 2 cos (u) big) ; dx, : forall : (u, v) in H_ {per} ^ {1} ((0, L)) times L_ {per} ^ 2 ((0, L)), $ $
where $ L> 0 $ is the period of all functions of $ L_ {per} 2 ((0, L)) $ Y $ H_ {per} 1 ((0, L)) subset L_ {per} 2 ((0, L)) $. So what is the derivative of Fréchet or Gateaux of $ varepsilon $? And how to calculate?

Stochastic analysis: proof that a simulated random space from an SDE is a Hilbert space L2

I have an infinite set of simulated values ​​using a stochastic differential equation (similar to Ornstein-Uhlenbeck, but it can be any SDE). I need to use these values ​​to forecast future values ​​(similar to Longtsaff-Schwartz)

I want to show that these generated values ​​belong to an L2 space, so I can make an orthogonal projection (that is, regression).

Question:

1- I don't know where to start to show that some values ​​I got from a simulation are of bounded variance. Any pointer? book? video?

2- Now we apply some continuous functions to these values. Are they still Hilbert?

In formal terms:

I have:

$ mathbb {E} left ( Delta_x ( omega, t, t + delta) ^ 2 mid mathcal {F} _t right) $

where $ Delta_x $ is a $ mathcal {F} _t $adapted process calculated as the difference between two values ​​of the simulated vectors over time (I have $ n $ values ​​of these):

$ { Delta ^ 2 ( omega, t, t + delta)} _ n = {V ( omega, t + delta)} _ n- {V ( omega, t)} _ n $

The norms that I want to use for the projection of these random variables are:

  1. Simple Euclidean norm (to make a polynomial regression)
  2. Weighted standard (to do Kernel regression):

$ langle Weight. ( Delta ^ 2 (.) – X_t), X_t) rangle $

I need to show that the simulated variables belong to L2 (under these two standards) in order to justify an orthopedic projection.

Thank you!

Technical analysis

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Uniform Convergence in Analysis – Mathematics Stack Exchange

Consider the series 1 + x + x2 + x3 + …. We know that this series converges promptly to 1 / (1 – x) for x ∈ (−1,1). Show that
(a) Convergence is not uniform in (−1,1).
(b) The convergence is uniform in each closed and bounded interval (−a, a) where 0 <a <1. Show this using both the definition of uniform convergence and the Weierstrass-M test.

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