real analysis – Please check my solution(Uniformly convergence of the functions)

Here is the question in my real analysis textbook.

$f$ is a continuous on $mathbb{R}$

Say $f_n(x) = {n over 2}int_{x-{1 over n }}^{x+{1 over n }} f(t) dt$

Show the $f_n$ is a uniformly converge to $f(x)$ on $(0,1)$


In fact, the textbook itself solve this by M.V.T for integral.

But I tried a different way like the below.

$(sol)$ For $forall x in (0,1)$, consider the $I_n = (x- {1 over n }, x+{1 over n })$

Since $f$ is a continuous on $I_n$, f is uniformly continuous on $I_n $

Therefore, by the definition of the uniform continuity

$(1)$ $exists {2 over n} leq delta $ s.t. $Vert x-y Vert < delta$ $Rightarrow$ $Vert f(x) – f(y) Vert <{2epsilon over n } $

$(2)$plus, By Archimedes $exists k in mathbb N s.t. n geq k Rightarrow {2epsilon over n } < epsilon$

By $(1)$ and $(2)$
$Vert f_n(x) – f(x) Vert = Vert {n over 2}int_{x-{1 over n }}^{x+{1 over n }} (f(t) – f(x)) dt Vert leq {n over 2}int_{x-{1 over n }}^{x+{1 over n }} Vert f(t) – f(x) Vert dt leq {n over 2 } {2 over n } epsilon =epsilon $

Hence, $exists k s.t. n geq k Rightarrow Vert f_n(x) – f(x) Vert <
epsilon$
(Uniformly convergence.)

I don’t have a confidence my solution is right or not. Please check my idea and solution.

Thank you.

differential equations – PDE of two unknown functions

I want to find two functions $h_1(t), h_2(x,t)$ such that the function
$h(x,t) = frac{1}{2} (x-b)^2 h_2(x,t)+(x-b) h_1(t)+frac{g(t)-beta h_1(t)}{alpha }$
satisfies the PDE
$$h_t = frac{1}{2} sigma^2 x^2 h_{xx} +r x h_x$$ ($alpha,beta,sigma,r$ are all constants and $g(t)$ is a given function).

I proceed by plugging

h(x_, t_) := (-b + x) Subscript(h, 1)(t) + (
  g(t) - (Beta) Subscript(h, 1)(t))/(Alpha) + 
  1/2 (-b + x)^2 Subscript(h, 2)(x, t)

into the PDE:

PDE=FullSimplify( D(h(x, t), t) - 1/2 (Sigma)^2 x^2 D(h(x, t), {x, 2}) - r x D(h(x, t), x))

and then I try to find some functions $h_1(t)$, $h_2(x,t)$ that make it true. I am not interested in numerical solutions. I am trying to find whatever functions $h_1(t)$ and $h_2(x,t)$ that solve the PDE.
As a first try, I pick $h_1(t)=0$ and try to use DSolve to solve for $h_2(x,t)$, but it did not work (it gives me the input back). So I was trying to find something to solve it for both like DSolve(PDE==0,h_2(x,t),h_1(t),{x,t}), but I do not know if the syntax is correct when using DSolve for more than one unknown.
I will truly appreciate any help or suggestion for finding them, thank you very much!

I need your view on functions

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cv.complex variables – What can be concluded from the analyticity in a half-plane of a sum of functions?

If I have two functions $F$ and $G$ for which $F(s)+G(s)$ is analytic in some half-plane ${rm Re}(s)>a$, what can be concluded about these functions individually with respect to their analyticity?

In the setting I’m dealing with, both functions are Dirichlet series in a half-plane ${rm Re}(s)>b>a$.

number theory – How to prove this identity related to multiplicative functions

This is question 2.6 of textbook Apostol introduction to analytic number theory and I am unable to solve it.

Prove that $sum_{d^2 | n} mu(d) = {mu}^2(n) $ .

RHS = r where $n = p_{1} … p_{r} {p_{r+1}}^{a_{r+1}}…{ p_{k}}^{a_k} $ , $a_{r+i}$ such that i>0 .

But i am not able to prove lhs equal to it. So, can you please help! !

Custom Functions not working for Google Sheets Addon

I created a GSuite Editor Addon project in Clasp and Typescript. I can test my code via the Script Editor->Execute->Test as Addon and select a Sheet. That works and the addon menu that I defined in the onOpen function appears.

However, all the Custom Functions that I defined in the project are not available from the Spreadsheet. For example, when I type in =double(2), I simply get an unknown function error. But, when I copy the Custom Function to the Spreadsheets’ local scripts (Tools->Scrip editor), it works just fine.

Is there any issue with Custom Functions inside addons?

dg.differential geometry – Closeness of a family of functions

I am studying chapter 4 of Geometric Measure Theory book by H. Federer and I have some questions from the following part:

Assuming that $X, Y$ are Banach spaces with $operatorname{dim} X<infty$ and that $U$ is an open subset of $X$ we let
$$
mathscr{E}(U, Y)
$$

be the vectorspace of all functions of class $infty$ mapping $U$ into $Y$. For each nonnegative integer $i$ and each compact subset $K$ of $U$ we define the seminorm
$$
v_{K}^{i}(phi)=sup left{left|D^{prime} phi(x)right|: 0 leq j leq i text { and } x in Kright}
$$

whenever $phi in mathscr{E}(U, Y)$.

  1. Why is $v^i _K$ a seminorm? For this, I have to prove that if $v_{K}^{i}(phi)=0$, then the mentioned $sup$ is also $0$, but how can I show this?

The family of all seminorms $v_{K}^{i}$ induces a locally convex, translation invariant Hausdorff topology on $mathscr{E}(U, Y) ;$ basic neighborhood of any $psi in mathscr{E}(U, Y)$ are the sets
$$
mathscr{E}(U, Y) capleft{phi: v_{K}^{i}(phi-psi)<rright}
$$

corresponding to all $i, K$ and all $r>0 .$ We let
$$
mathscr{E}^{prime}(U, Y)
$$

be the vectorspace of all continuous real valued linear functions on $mathscr{E}(U, Y),$ and we endow $mathscr{E}^{prime}(U, Y)$ with the weak topology generated by the sets
$$
mathscr{E}^{prime}(U, Y) cap{T: a<T(phi)<b}
$$

corresponding to all $phi in mathscr{E}(U, Y)$ and all $a, b in mathbf{R} .$ Defining
$$
operatorname{spt} phi=U cap operatorname{Clos}{x: phi(x) neq 0} text { for } phi in mathscr{E}(U, Y)
$$

spt $T=U sim bigcup{W: W$ is open, $T(phi)=0$ whenever
$$
phi in mathscr{E}(U, Y) text { and } operatorname{spt} phi subset W}
$$

for $T in mathscr{E}^{prime}(U, Y),$ we observe that spt $T$ is compact because
$T leq M cdot v_{K}^{i}$ for some $i, K$ and some $M<infty$
Thus we find that $mathscr{E}^{prime}(U, Y)$ is the union of its closed subsets
$$
mathscr{E}_{K}^{prime}(U, Y)=mathscr{E}^{prime}(U, Y) cap{T: operatorname{spt} T subset K}
$$

corresponding to all compact subsets $K$ of $U$. It may also be shown that all members of any convergent sequence in $mathscr{E}^{prime}(U, Y)$ belong to some single set $mathscr{E}_{K}^{prime}(U, Y)$
For each compact subset $K$ of $U$ we define
$$
mathscr{D}_{K}(U, Y)=mathscr{E}(U, Y) cap{phi: operatorname{spt} phi subset K}
$$

and observe that $mathscr{D}_{K}(U, Y)$ is closed in $mathscr{E}(U, Y) .$

  1. How can I observe this? Actually, I don’t understand the topology of these spaces. I read about the topology induced by seminorms, but still don’t understand how to prove that this is close!

Then we consider the vectorspace $mathscr{D}(U, Y)=bigcupleft{mathscr{D}_{K}(U, Y): Kright.$ is a compact subset of $left.Uright}$
with the largest topology such that the inclusion maps from all the sets $mathscr{D}_{K}(U, Y)$ are continuous; accordingly a subset of $mathscr{D}(U, Y)$ is open if and only if its intersection with each $mathscr{D}_{K}(U, Y)$ belongs to the relative topology of $mathscr{D}_{K}(U, Y)$ in $mathscr{E}(U, Y)$.

  1. And finally, why is $mathscr{D}(U, Y)$ a vectorspace? For this, I’m gonna take $f in mathscr{D}_{K_1}(U, Y)$ and $g in mathscr{D}_{K_2}(U, Y)$. Now, to prove that $af+g in mathscr{D}(U, Y)$, can I say that it lies in $K=K_1 cup K_2$?

performance – Opitimizing 2 functions written in Python

I would like to decrease the time complexity of these functions. At first I am calling bits_to_rgb() then the output of it I pass as argument in rgb_to_rgb_binary(). When I have like 1000 bits, it takes a few seconds to complete these functions. Is there any way to reduce that time?

def bits_to_rgb(data):
    bits = (255 if x == "1" else 0 for x in ''.join(data))
    full_bits = ()

    for i in bits:
         full_bits.extend((i, i, i))

    return full_bits


def rgb_to_rgb_binary(data):
    binary = ()
    num = 0
    for i in range(8, len(data) + 1, 8):
        binary.extend(data(num:i))  # append to check how it looks in real
        num = i
    return binary

The example of input is ("1001010", "1000001")

functional programming – Doesn’t “Always test through the public interface” contradict testing of individual composed functions?

I’m currently reading “Composing Software” by Eric Elliott, which is about functional programming in JavaScript. He states that if you compose multiple functions together, and that these functions have been fully tested in isolation – then you don’t need to unit test the composed function (as that can be done using an integrated test that performs all the side effects.)

But isn’t simply testing the individual functions violating the core TDD principle of “always test the public inteface, and not the implementation”? Arguably the composing of the smaller functions is an implementation detail. As long as my service performs the action that it says it will do, we shouldn’t really care about the small “private” functions being composed in order to get the right behaviour.

I’m trying to reconcile the two ideas but I’m struggling. The only way around this (that I can think of) is to mock the dependencies that my service needs so that it can be tested through the public interface, but going functional was my attempt to stop using mocks in the first place.

calculus and analysis – Series of inverse functions, unclear numerical constant

I was answering another question here and came up with this simple illustrative example that should have an analytic solution. Indeed it has, but I do not understand it. In particular, where 85 is coming from?

g[x_]:=BesselJ[0,x]
f[x_]:=Exp[x]
Series[f[InverseFunction[g][y]],{y,0,0}]
Out[1]= E^-BesselJZero[0,85]+O[y]^1