Real analysis: how to demonstrate the uniqueness of the solution of a problem described?

I have a family of positive, convex, parameterized functions by $ lambda in (0, infty) $. Let us denote the functional corresponding to $ lambda $ how $ C λ (f) $.

Existence and uniqueness of minimizer for each of these functionalities in a set $ S $ It has already been tested. We will denote the minimizer of $ C λ (f) $ on the whole $ S $ how $ f lambda $

Now I don't know what functional you should choose to minimize, since each has its own advantages and disadvantages. For some reason, I choose to minimize the functional corresponding to that $ lambda = lambda_0 $for which $ | f _ { lambda} | $ is maximum over $ lambda in (0, infty) $.

Fortunately, I can show that $ lim limits _ { lambda a 0} | f _ { lambda} | = $ 0 Y $ lim limits _ { lambda to infty} | f _ { lambda} | = $ 0, so that it establishes the existence of $ lambda_0 $.

Now I need to demonstrate the uniqueness of $ lambda_0 $.

I am able to prove that the equation $ frac { partial { | f- nabla C _ { lambda} (f) |}} { partial { lambda}} = 0 $, solving for $ lambda $, has a unique solution in $ (0, infty) $ For any $ f in S $. In fact, this equation is linear in $ lambda $.

where $ nabla C _ { lambda} (f) $ denotes the gradient of the functional $ C _ lambda $ in some $ f in S $. It may be called a functional derivative with an appropriate name, but that is not the subject of interest.

Is there any way to use this fact to conclude that $ lambda_0 $ It's unique?

real analysis – https://mathoverflow.net/questions/344022/zeta-functions-zeros-and-extrema-and-cycle-index-polynomials – Mathematics Stack Exchange

Zeta functions and the cycle index polynomials (CIP, $ p_n ( sigma) $) of symmetric groups are closely related as described in MO-Q "Cycling through the zeta garden". This question examines a particular example in which the generic indeterminates for CIPs presented in OEIS A036039 are assigned the values ​​of the Riemann zeta in positive natural numbers greater than one. The graphs of the first four polynomials reveal that

1) the zeros are different

2) the zeros of $ p_n $ they are the abscissa of the extremes of $ p_ {n + 1} $ since they are Appell polynomials, which satisfy $ frac {d} {d sigma} p_n ( sigma) = n p_ {n-1} ( sigma) $

3) the abscissa of each zero / end of $ p_n $ it is the negative of the sum of the abscissa of the remaining zeros / extremes since the coefficient of $ sigma ^ {n-1} $ disappears, that is, the associated trace disappears.

Questions:

A) With this particular example of Riemann zeta of the CIP, are the zeros / extremes always different and real?

B) What restrictions must be established in the indeterminate ones for generic CIPs so that all zeros are different and real?

C) What restrictions on the undetermined are imposed by having all real and distinct zeros?

D) What additional restrictions are imposed on the undetermined if they must behave like Riemann? $ zeta (n> 1) $ with $ 0 < zeta (n + 1) < zeta (n) $?

E) Given these various restrictions, what happens if information on the nature of $ zeta (n) $ strange $ n $ given the most accurate characterization of those for even $ n $?


The first are (with $ sigma $ replaced by $ x $)

$$ p_1 (x) = x $$
$$ p_2 (x) = x ^ 2- zeta (2) $$
$$ p_3 (x) = x ^ 3-3 zeta (2) x + 2 zeta (3) $$
$$ p_4 (x) = x ^ 4-6 zeta (2) x ^ 2 + 8 zeta (3) x + 3 ( zeta ^ 2 (2) -2 zeta (4)) $$
$$ p_5 = x ^ 5-10 zeta (2) x ^ 3 + 20 zeta (3) x ^ 2 + 15 ( zeta ^ 2 (2) -2 zeta (4)) x + 4 (- 5 zeta (2) zeta (3) +6 zeta (5)) $$

The generic CIPs are just these with each $ zeta (n) $ replaced by the generic undetermined $ x_n $. A recursion relationship is presented in this MO-Q.

Functional analysis of fa – Convergence of Ito integrals

I would like to show that yes

$$ lim_ {n rightarrow infty} int_0 ^ T mathbb E (X_n (t) xi (t)) dt = int_0 ^ T mathbb E (X (t) xi (t)) dt $$

for all $ xi in L ^ 2 _ { text {ad}} ( Omega times (0, T)), $ that is to say $ X_n $ converges weakly in Hilbert's sense of space,

then we also possibly have for some subsequence for the integral Ito the same weak convergence

$$ lim_ {n rightarrow infty} int_0 ^ T mathbb E left ( int_0 ^ t X_n (s) dW (s) xi (t) right) dt = int_0 ^ T mathbb E left ( int_0 ^ t X (s) dW (s) xi (t) right) dt $$
This follows almost (!) From the isometry of Ito, but I am missing a final argument.

First, the isometry of Ito gives that

$$ int_0 ^ T mathbb E left ( int_0 ^ t X_n (s) dW (s) xi (t) right) ^ 2 dt $$

It is uniformly delimited.

Therefore, for weak compactness there is a weak limit $ Y in L ^ 2 _ { text {ad}} ( Omega times (0, T)), $ such that
$$ lim_ {n rightarrow infty} int_0 ^ T mathbb E left ( int_0 ^ t X_n (s) dW (s) xi (t) right) dt = int_0 ^ T mathbb E left (Y (t) xi (t) right) dt. $$

In fact, the isometry of Ito implies that if $ xi $ It is also an integral Ito

$$ xi (t) = int_0 ^ t Z (s) dW (s), $$ then in fact

$$ lim_ {n rightarrow infty} int_0 ^ T mathbb E left ( int_0 ^ t X_n (s) dW (s) int_0 ^ t Z (s) dW (s) right) dt = int_0 ^ T mathbb E left ( int_0 ^ t X (s) dW (s) int_0 ^ t Z (s) dW (s) right) dt. $$

How can you get this in general, that is, in general $ xi $?

real analysis – demonstrating integrability and equality

I have trouble trying the following:

Suppose $ f $ is integrable in $ (0, a) $. So $ g (x) = int_ {x} ^ {a} t ^ {-1} f (t) , dt $ is integrable in $ (0, a) $ Y $ int_ {0} ^ {a} g (x) , dx = int_ {0} ^ {a} f (x) , dx $.

Some clue? What kind of theorem should I use in this case? Thank you.

pdes analysis: can harmonic maps with immersive boundary conditions have singular points?

Leave $ mathbb D ^ 2 $ be the drive disk closed in $ mathbb R ^ 2 $. Leave $ f: mathbb D ^ 2 a mathbb {R} ^ 2 $ be a real analytical orientation preserving immersion, and let $ omega: mathbb D ^ 2 to mathbb {R} ^ 2 $ be the only satisfactory harmonic map $ omega | _ { partial mathbb D ^ 2} = f | _ { partial mathbb D ^ 2} $

Make $ d omega neq 0 $ everywhere in $ mathbb D ^ 2 $?

I have two observations:

  1. There is an open neighborhood of $ partial mathbb D ^ 2 $ where $ d omega neq 0 $ .
  2. $ d omega $ it is invertible outside a set of dimensions of Hausdorff $ le 1 $.

Claim $ (1) $ it follows from the fact that for $ p in partial mathbb D ^ 2 $, we have
$$ text {rank} (d omega_p) ge text {rank} big (d ( omega | _ { partial mathbb D ^ 2}) _ p big) = text {rank} big (d (f | _ { partial mathbb D ^ 2}) _ p big) = 1. $$

Per point $ 2 $, note that $ d omega $ it can't be singular everywhere, since $$ int _ { mathbb D ^ 2} det d omega = int _ { mathbb D ^ 2} det df> 0. $$

So, $ big ( det (d omega) big) ^ {- 1} (0) $ it is the zero set of a real analytical function that is not identically zero, which implies dimension $ le 1 $.


real analysis: show all continuous functions in [0,$pi$] is the uniform limit of function sequences of "polynomials of sin (kx)" (Stone Weierstrass theorem)

The exercise is 26E on Elements of Bartle's real analysis.

Ask to use the fact that each function continues with real value in $ (0, pi) $ It is the uniform limit of a sequence of functions of the form:
$$ sum_ {k = 1} ^ {m} a_kcos (kx) $$
To show that exery continues real value function $ f $ in $ (0, pi) $ with $ f (0) = f ( pi) $ It is the uniform limit of a sequence of functions of the form:
$$ sum_ {i = 1} ^ {n} b_ksin (kx) $$

The book also gives the following innuendo:
Yes $ f (0) = f ( pi) = 0 $, first approximate $ f $ for a function $ g $ disappearing at some intervals $ (0, sigma) $ Y $ ( pi- sigma, pi) $. Then consider $ h (x) = frac {g (x)} {sin x} $ for $ x in (0, pi); h (x) = 0 $ for $ x = 0, pi $

real analysis: find the interval that contains the point

I am supposed to find the interval containing point b, as the functional values ​​of the points have a difference of the value g (b) by a maximum of ε.

$$ g (b) = a2, b = -3, varepsilon = frac {2} {11} $$.

I tried to do it from the definition:

$$ – 3- delta <x <-3+ delta Leftrightarrow 9-1 <a ^ {2} <9 + 1 Leftrightarrow 8 <a ^ {2} <10 $$

and I don't know what to do next, or if I'm right.

Can anybody help me?

Real analysis – Subsequence convergence. Equivalent definitions

Suppose we have a sequence $ {x_n } $ in metric space $ (X, d) $ and its sub sequence $ {x_ {n_k} } $ converges to $ a in X $.

I want to show that the following are equivalent:

I) $ forall epsilon> 0 $ $ there is N $ : $ forall k> N $ $ Rightarrow $ $ d (x_ {n_k}, a) < epsilon $.

ii) $ forall epsilon> 0 $ $ there is N $ : $ forall n_k> N $ $ Rightarrow $ $ d (x_ {n_k}, a) < epsilon $.

I could show that ii) $ Rightarrow $ I)

In fact, take arbitrary $ epsilon> 0 $ then on the part ii) $ there is N $ : $ forall n_k> N $ $ Rightarrow $ $ d (x_ {n_k}, a) < epsilon $. But yes $ k> N $ and considering that $ n_k> k $ We will get it $ d (x_ {n_k}, a) < epsilon $.

However, I cannot show the reverse direction. So I would be very grateful if someone can help me, please?

Here is my approach to I) $ Rightarrow $ ii) Take arbitrary $ epsilon> 0 $ then on the part I) $ there is N $ : $ forall k> N $ $ Rightarrow $ $ d (x_ {n_k}, a) < epsilon $. But yes $ n_k> N $ then we will get $$ d (x_ {n_ {n_k}}, a) < epsilon, $$

that is, the last inequality has a double index that really bothers me.

Singapore reduces its leverage limit – News and analysis

Singapore has reduced its leverage limit by more than half. This means that currency traders in the country will now have access to leverage of 1:20, instead of 1:50 as it was before the changes.

The decision was made by the Monetary Authority of Singapore, however, as with the regulatory changes presented by the European Securities and Markets Authority, there are certain gaps that allow more generous leverage under strict conditions.

Accredited investors in Singapore will have access to the original leverage, however, as in Europe, this would mean meeting strict capital requirements.

Among the requirements is a personal net worth of more than 2 million Singapore dollars ($ 1.5 million) or proof of an annual income of more than 300,000 Singapore dollars. Operators with more than 1 million Singapore dollars in cash can also qualify for the highest leverage.

functional analysis – Characterizations of projection operators in Banach spaces

Leave $ mathbb {E} $ be a Banach space (real or complex) and leave $ mathbb {B} ( mathbb {E}) $ be the Banach space of continuous (bounded) linear operators of $ mathbb {E} $ inside $ mathbb {E} $, with the operator standard.

An operator $ P in mathbb {B} ( mathbb {E}) $ it's called projection yes $ P ^ 2 = P $.

Are there other characterizations of projection operators in Banach Spaces?

Where can such characterizations be found?