nt.number theory – Is this number theoretic quantity upper bounded?

It seems my thinking gets organized after posting the question.

The natural logarithm of the quantity $pi(n)!$ is near $pi(n)log(pi(n)/e) + (log(taupi(n)))/2$ (a number-theoretic use for $tau$, the circumference of a unit radius circle). Using an approximation to $pi(n)$ we get that this is less than $An$ for some $A lt 2$. But $An/(n-h)$ is bounded above by $2A$, and gets very close to $A$. So with some work the original quantity should be shown to be less than $e^A$.

Verification is still appreciated.

Gerhard “And Still Worth An Acknowledgement” Paseman, 2020.05.30.

Apps related to the icons on the upper left corner on an Android Samsung

enter image description here

Partial circle with inner circle arrow
and the rectangle with the clock_

Ubuntu (Gnome and 18.04) always hides the upper LH corner close box under a menu – can I change this?

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inequality – Upper bound on x where $2^x leq (ax)^4$

We have some constant $a > 1$ and we know the following inequality:
$$2^x leq (ax)^4$$

And need to find an upper bound on $x$.

I thought of trying to calculate where $2^x$ intersects $(ax)^4$ and then the larger intersection would be an upper bound for $x$.
So this is what I did:I called the value where they intersects $t$ and solved:

$$2^t = (at)^4\
tln2 = 4ln(at)\
atln2=4aln(at)\
frac{ln2}{4a}=(at)^{-1}ln(at)\
-frac{ln2}{4a}=(at)^{-1}ln((at)^{-1})\
-frac{ln2}{4a}=e^{ln((at)^{-1})}ln((at)^{-1})\
Wleft(-frac{ln2}{4a}right)=ln((at)^{-1})\
t=frac{e^{-Wleft(-frac{ln2}{4a}right)}}{a}$$

And therefore:

$$xleq max left{frac{e^{-W_0left(-frac{ln2}{4a}right)}}{a},frac{e^{-W_{-1}left(-frac{ln2}{4a}right)}}{a}right}$$

But I don’t know how to continue from here. How can I bound this expression with $W$?

java – Do not distinguish between upper and lower case

I have this exercise that asks for a word on the screen and outputs the first non-repeating character it finds, but I would like it not to be case sensitive since if it finds an a and an A it thinks they are two different characters.
Code:

System.out.println("Palabra:");
CaracterNoRepetido();
}
public static void CaracterNoRepetido(){
Scanner lector = new Scanner(System.in);
String cadena = lector.nextLine();
char caracter = 0;
for(int i = 0; i < cadena.length(); i++) {

    boolean repetido = false;
    for(int j = 0; !repetido && j < cadena.length(); j++) {
        if(j != i)
            repetido = cadena.charAt(i) == cadena.charAt(j);
    }

    if(!repetido) {
        caracter = cadena.charAt(i);
        break;
    }
}

System.out.println(caracter);

}
}

iPad 2 screen is unresponsive in the upper half

I have an iPad 2 that I got for free that I am going to fix (digitizer) and give to my nephew. Before fixing it, I was figuring out what I would have to do for him, and I noticed that while in portrait mode you couldn't click or touch half of the screen. Would you need another part or just the digitizer? I plan to fix this no matter what, so don't complain about how useless the iPad 2 is. Thanks in advance!

algorithms: upper limit for the recurrence equation $ T (n) = 4T left ( frac {n} {2} right) + n ^ 2 $

It's been a while for me since my undergraduate class in algorithms. Could you help me find an upper limit for this recurrence equation?

$$
T (n) =
begin {cases}
14 & quad text {if} n <1 \
4T left ( frac {n} {2} right) + n ^ 2 & quad text {if} n gt1 \
end {cases}
$$

integration – Upper limit of a complex integral

Working with a class of polynomials, I have found this integral

$$ A_n (x) = frac {n!} {2 pi , i} int_C frac {e ^ {x , (e ^ t-1)}} {t ^ {n + 1}} dt $$
where $ C $ h is a closed circuit described in the positive sense surrounding the
origin.

I wonder if it is possible to find an upper limit of this integral (depending on $ n $)

Complex analysis: exponential decay of the Fourier series of a periodic holomorphic function in the upper half plane

Leave $ f $ be a periodic holomorphic function in the upper half plane $ mathbb H $. Here newspaper means $ f (z + 1) = f (z) $ for all $ z in mathbb H $.

So $ f $ equals your Fourier series
$$
f (z) = sum_ {n in mathbb Z} c_n q ^ n, quad q = e ^ {2 pi i z}.
$$

Suppose that the Fourier series of $ f $ it has the form

$$
sum_ {n geq n_0} c_n q ^ n
$$

for some $ n_0 in mathbb Z $.

It is true that

$$
sum_ {n geq 1} c_n q ^ n ll e ^ {- epsilon y}
$$

for some $ and $, where $ z = x + iy $?

This question originates from the following statement: & # 39; A weakly holomorphic modular form is a harmonic form of mass & # 39 ;. For the original question, see https://mathoverflow.net/questions/357309/a-weakly-holomorphic-modular-form-is-a-harmonic-maass-form

Ordinary differential equations: example of exhaust flow from the lower half-plane to the upper half-plane

There is a full vector field $ v = (v_ {1}, v_ {2}): D rightarrow mathbb {R} ^ 2 $, where $ D $ is an open subset of $ mathbb {R} ^ 2 $, $ v in C ^ 1 $, with the following properties:

1) say $ l: = {x_ {2} = 0 } $, we have $ D cap l neq emptyset $ and $ v_ {2} (x) leq 0 $ for all $ x in l cap D $ and there is a point $ x in l $ S t. $ v_ {2} (x) <0 $;

2) The whole $ {x in l cap D s.t. v_ {2} (x) = 0 } $ has an accumulation point $ A en l cap D $

3) There is a point $ x in D $ with $ x_ {2} <0 $ and a time $ t in mathbb {R} $ S t. $ phi ^ {t} (x) 2> 0 $ (i.e., the integral curve from $ x $ sometime it reaches half of the upper plane).

The motivation for the question lies in the following statement "if the vector field $ v $ never points to the middle of the top plane along $ l $, then each solution from the lower half plane remains in the closed lower half plane, "which I cannot say whether it is true or not. Actually, I know that if the vector field is tangent throughout $ l $ or if the set of points where it is tangent has no accumulation point in $ l $, then the claim is true. This motivates 1) and 2).

Finally let me say that I would take as an answer an example where $ v $ it is not complete.

Thank you for reading.