Set theory – events of supremum and infimum of random variables

To let $ Y_n $ Be a sequence of real random variables and let $ c in mathbb {R} $ Be a constant. I know that $ inf_n Y_n $ and $ sup_n Y_n $ are also random variables. I am interested in representing events of these sup and inf random variables as associations or intersections of events from $ Y_n $, I intuitively came up with the following relations, but I'm not 100% sure whether they are true:

  • $ { sup_n Y_n> c } = bigcup_n {Y_n> c } $
  • $ { inf_n Y_n> c } = bigcap_n {Y_n> c } $

and

  • $ { sup_n Y_n <c } = bigcap_n {Y_n <c } $
  • $ { inf_n Y_n <c } = bigcup_n {Y_n <c } $

If so, in which cases can the $> $ or $ <$ to be relaxed in $ geq $ or $ leq $?

cv.complex variables – Regarding the supremum achieved through holomorphic functions

I read the article Characterizations of some domains about Carathéodory Extremals.
To let $ Omega $$ subset mathbb {C} ^ n $ be an openly connected sentence (domain). To let $ T ( Omega) $ denote the tangent bundle of $ Omega $, To let $ mathbb {D} $ be the open unit disk in the complex plane $ mathbb {C} $ , To let $ f: Omega longrightarrow mathbb {D} $ is to be called holomorphic with $ f in H ( Omega, mathbb {D}) $,

The Carathéodory Reiffen Pseudometric is for every point $ (z; xi) $ in the $ T ( Omega) $ is defined as
$ c_ Omega (z, xi) = sup {| f ^ * (z) xi |; f: Omega longrightarrow mathbb {D}, ; f in H ( Omega, mathbb {D}) }
= sup {| sum_ {i = 1} ^ {n} { frac {{ partial f}} {{ partial z_i}} (z) xi_i} |; f: Omega longrightarrow mathbb {D}, ; f in H ( Omega, mathbb {D}) } $

A function $ f_0 in H ( Omega, mathbb {D}) $ is said to be a universal caratheodory extremal function if $ | f_0 ^ * (z) xi | = c_ Omega (z, xi) $ for each $ (z; xi) $ in the $ T ( Omega) $,

I can find the universal set in the case of $ Omega = mathbb {D} $,
However, in the case of the Polydisc and the Euclidean sphere unit, the author of the above article (Example 2.2, i and ii) claimed / mentioned that they are the coordinate functions and compositions of projections onto the planes through the center Unity Euclidean sphere. Can anyone tell how these were calculated or provide a reference for them?

Properties of Supremum and Infimum

Part b) of the following is the problem I am trying to solve:
Enter image description here

I solved part a) and used the property in combination with the note offered in part b) to get the following:
$ sup {f} = sup (f-g + g) leq sup (f-g) + sup {g} $and then I would subtract $ sup {g} $:

$ sup {f} – sup {g} leq sup (f-g) $

But I don't know what to do next. I noticed that I could take the absolute value of both sides, but wonder if the statement will still be true (it seems to be wrong). If not, how can this be solved differently using the information provided?

Calculus – what is the supremum of $ f (z) $?

I know that Supremum is the lowest upper limit and Infimum is the highest lower limit. Even according to the supremum axiom, every non-empty set limited above has a supremum. Put

$$ f (z) = frac {| frac { sin (z) + sin (1) – sin (z + 1)} {2-2 cos (1)} | – frac {1} {2} sinh (| z |) – cosh (| z |) +1} { frac {1} { pi ^ 2-1} cosh ( pi | z |) – frac {1} { pi ^ 2-1} – cosh (| z |) +1}, ; ; z in mathbb {C} setminus {0 }. $$

I know that $ f (z) $ is limited above. Now my question is the supremum of $ f (z) $For this purpose I tried to get a limit, but I could not be successful. I appreciate any help on this matter.

real analysis – proving a supremum of a crowd

The question:

Find the supremum of the set $$ { { sqrt (4) {n ^ 4 + n ^ 3} -n: n in mathbb {N} }} $$
And then it tells us that we need to take large values ​​of n to find an appropriate guess, to show that it is an upper bound, and then to prove that it is the smallest upper bound.

I followed the question and found a suitable guess for s = 1/4, showing that this is a good upper limit. My problem is to prove that there is no lower limit. At this point, my work looks like I'm trying to prove by contradiction:

Suppose h is another upper bound such that h <1/4.
$$ { sqrt (4) {n ^ 4 + n ^ 3} -n <h} $$
$$ n ^ 4 + n ^ 3 <(h + n) ^ 4 $$

But after the expansion I can only quit $ n ^ 4 $ This gives me a lot of unknown powers and a really complicated solution that I can do by hand

$$ n ^ 3 <h ^ 4 + 4h ^ 3n + 6h ^ 2n ^ 2 + 4hn ^ 3 $$

That said, I know that I have taken the wrong path, but I'm not sure which direction to prove it. I adapted a response from another book example, but that was only up to potency 2, so it was much easier to simplify this method.

o.operator algebras – infimum and supremum for normal semi-finished trace

A semi-final track $ tau $ on $ M _ {+} $ (for one of Neumann algebra $ M $) is considered normal if $ tau ( sup x_i) = sup tau (x_i) $ for a limited network of positive operators $ (x_i) _ {i in I} $,

Is it true that $ tau ( inf x_i) = inf tau (x_i) $ for a limited decreasing network of positive operators $ (x_i) _ {i in I} $?

If the trail was finite, it is easy enough to see that the above holds the increasing net $ (x_0 – x_i) _ {i in I} $ and the fact that $ tau (x_0) $ is finally. In case the same strategy works, we have to have the following result in the semi-finite case: if $ tau ( inf x_i) < infty $Then there is an index $ j in I $ so for all $ i j j $, we have $ tau (x_i) < infty $, This seems to be something that should be true. But I can not provide quick proof.

Many Thanks.

Essential Supremum and Supremum in anticipation

Suppose that $ {Z_i } _ {i in I} $ are a family of densities in $ L ^ 2 ( Omega, mathcal {F}, mathbb {P}) $, and $ X = L ^ 2 ( Omega, mathcal {F}, mathbb {P}) $, When is it true?
$$
sup_ {i in I} mathbb {E} left[Z_icdot
(X- mathbb{E}[Z_icdot X|mathcal{G}]) ^ 2
Law]=
sup_ {i in I} mathbb {E} left[Z_icdot
(X- operatorname{esssup}_{i in I}mathbb{E}[Z_icdot X|mathcal{G}]) ^ 2
Law]?
$$

I've seen similar questions, but I have not encountered anything like that here.

Expectation of the supremum of a sequence of random variables

To let $ Omega = [0,1]$, $ mathcal {F} = mathcal {B} (0,1) $, P = Lebesgue measure.

To let $ X_n (w) =
begin {cases}
0 quad frac {1} {n} <w leq 1 \
n-n ^ 2w quad 0 leq w leq frac {1} {n}
end {cases} $

The first part of the exercise proves that $ X_ {n} $ is a martingale I've made.
Now my problem is calculating $ E (sup_ {n geq 1} | X_n |) $ and I do not know how to go about it. I know from the solutions that it has to be the same $ + infty $,

Number theory – $ p $ -adic Supremum of the cyclotomic polynomial

To let $ p $ be a prime and $ Phi_ {n} (T) $ be that $ p ^ {n} $the cyclotomic polynomial, which we consider to be a function $ mathbb {C} _ {p} $, In Robert Pollack's newspaper "About the $ p $-adic L-function of a modular form with a supersingular prime number, it is said that
$$
text {sup} _ { vertz vert <r} vert Phi_ {n} (1 + z) vert_ {p} = frac {r} {p ^ {n-1} (p – 1) )}.
$$

I could not do that. My last attempt was as follows:

The polynomial $ Phi_ {n} (T) $ Has $ p ^ {n-1} (p-1) $ Given by roots $ zeta_ {n} – 1 $, from where $ zeta $ is a primitive $ p ^ {n} $the root of the unit. Each of these roots has a rating
$$
v_ {p} ( zeta_ {n} – 1) = frac {1} {p ^ {n-1} (p-1)}.
$$

If we write $ phi_ {n} (1 + T) = sum_ {i = 0} ^ {p ^ {n-1} (p-1)} a_ {i} T ^ {i} $, then for $ r geq 0 $ We can look at the growth module
$$
M _ { Phi} (r) = text {sup} _ {i} vert a_ {i} vert r.
$$

Since $ Phi_ {n} (1 + T) $ has all zeroes in the same radius $ r _ { Phi} $This is the only critical radius (radius) $ r $ in which $ vert Phi_ {n} (1 + r) vert_ {p} neq M _ { Phi} (r) $). To let $ nu = text {sup} {i: vert a_ {i} vert r _ { Phi} ^ {i} = M _ { Phi} (r _ { Phi}) } $ and $ mu = text {inf} {i: vert a_ {i} vert r _ { Phi} ^ {i} = M _ { Phi} (r _ { Phi}) } $ then it is a sentence that the number of zeros of $ Phi_ {n} (1 + T) $ in the closed disc $ overline {D (0, r _ { Phi})} $ (Center $ 0 $radius $ r _ { Phi} $) corresponds $ nu $ and the number of zeros on the opened disc $ D (0, r _ { Phi}) $ corresponds to $ mu $, There are all zeroes $ r _ { Phi} $ these forces
begin {align *}
nu & = p ^ {n-1} (p-1) \
mu & = 0.
end {align *}

So, if we think about it $ r> r _ { Phi} $ then $ r $ is a regular radius and so on
begin {align *}
vert Phi_ {n} (1 + r) vert_ {p} & = text {sup} vert a_ {i} vert r ^ {i} \
& = M _ { Phi} (r) \
& = vert a_ {p ^ {n – 1} (p – 1)} vert r ^ {p ^ {n – 1} (p – 1)} \
& = r ^ {p ^ {n – 1} (p – 1)}.
end {align *}

So for this choice of $ r $ we should have
$$
text {sup} _ { vertz vert <r} vert phi_ {n} (1 + T) vert_ {p} = r ^ {p ^ {n-1} (p-1)}
$$

That's not what Pollack gets. Where (and how wildly) did I go wrong?