Intereting Posts

Antisymmetric Relations
Are all the norms of $L^p$ space equivalent?
polynomials such that $P(k)=Q(l)$ for all integer $k$
counterexample to RH; how big would it have to be?
Homotopy functions
Describe the set of all complex numbers $z$ such that $|z-a |+| z-b |=c$
Closed form of the integral ${\large\int}_0^\infty e^{-x}\prod_{n=1}^\infty\left(1-e^{-24\!\;n\!\;x}\right)dx$
Geometric mean of prime gaps?
Basic Reference material about ODEs such as saparability with calculations and examples?
$f(n) = \int_{1}^{n} n^{x^{-1}}dx$. $\frac{df}{dn}$?
Show that $\frac {\sin x} {\cos 3x} + \frac {\sin 3x} {\cos 9x} + \frac {\sin 9x} {\cos 27x} = \frac 12 (\tan 27x – \tan x)$
How do you find all $n$ such that $\phi(n)|n$
Summation equation for $2^{x-1}$
Generating function of $a_{n}^2$ in terms of GF of $a_{n}$?
$f: X \to Y$ is a smooth map and $\omega$ is a $p$-form on $Y$, what is $\omega$?

On the Banach space $(C([-1,1]), ||\cdot||_\infty ) $ consider the operator given by

$(Tf)(x)= \dfrac{1}{3} \displaystyle\int^1_0txf(t)\ dt + e^x – \dfrac{\pi}{3} $

1) prove that the mapping is a continuous function for all $ f \in (C([-1,1]) $ I.e. that T maps $ (C([-1,1]),||\cdot||_{\infty}) $ to itself.

- Uniformly distributed rationals
- How to check the real analyticity of a function?
- Can the the radius of convergence increase due to composition of two power series?
- Factorial and exponential dual identities
- When is $\mathbb{Z}$ dense in $\mathbb{C}$?
- Find a conformal map from the disc to the first quadrant.

2) Show that T is a contraction mapping on $(C([-1,1]),||\cdot||_{\infty})$

3) Lt $f_0 (x) =1 $ Calculate $f_1$ and $ f_2$ where $f_n:= Tf_{n-1}$

This is a past exam question I’ve come across and don’t know how to solve it

- Riemann zeta function and uniform convergence
- Characterization of Dirac Masses on $C(,\mathbb{R}^d)$
- Rudin assumes $(x^a)^b=x^{ab}$(for real $a$ and $b$) without proof?
- Is a uniformly continuous function vanishing at $0$ bounded by $a|x|+c$?
- For closed subsets $A,B \subseteq X$ with $X = A \cup B$ show that $f \colon X \to Y$ is continuous iff $f|_A$ and $f|_B$ are continuous.
- Investigating the convergence of a series using the comparison limit test
- Question about statement of Rank Theorem in Rudin
- Inverse of the sum $\sum\limits_{j=1}^k (-1)^{k-j}\binom{k}{j} j^{\,k} a_j$
- Convergence of the arithmetic mean
- proof of Poisson formula by T. Tao

A related problem.

1) Continuity, note that $|x|\leq 1$ and $|t|\leq 1$,

$$ |(Tf)(x)-(Tg)(x) |\leq \dfrac{1}{3} \displaystyle\int^1_0 |tx||f(t)-g(t)|\ dt \leq \int^1_0 |f(t)-g(t)|\ dt $$

$$ \implies \sup|(Tf)(x)-(Tg)(x) | \leq \frac{1}{3} \int^{1}_{0} \sup|f(t)-g(t)|\ dt $$

$$ ||Tf-Tg||_{\infty} \leq \frac{1}{3} ||f-g||_{\infty}<\frac{\epsilon}{3}=\delta . $$

2) The operator is a contraction mapping, since

$$ ||Tf-Tg||_{\infty} \leq \frac{1}{3} ||f-g||_{\infty}. $$

3) Define

$$ f_{n+1}(x)=(T f_n)(x) = \dfrac{1}{3} \displaystyle\int^1_0tx f_n(t)\ dt + e^x – \dfrac{\pi}{3} \longrightarrow (*) $$

$$ \implies f_{1}(x)=(T f_0)(x) = \dfrac{1}{3} \displaystyle\int^1_0tx f_0(t)\ dt + e^x – \dfrac{\pi}{3} $$

$$ \implies f_{1}(x) = \dfrac{1}{3} \displaystyle\int^1_0tx \ dt + e^x – \dfrac{\pi}{3}. $$

To find $f_2$, subs $f_1$ in $(*)$ and carry on the calculations. This technique is known as the Picard iteration. A related problem.

**Added:** Here is $f_2(x)$

$$ f_{2}(x)=(T f_1)(x) = \dfrac{1}{3} \displaystyle\int^1_0tx f_1(t)\ dt + e^x – \dfrac{\pi}{3}. $$

- In how many ways 3 numbers can be chosen from a from the set {1, …, 18} so that their sum is divisible by 3?
- Prove one set is a convex hull of another set
- Confusion regarding Russell's paradox
- The mode of the Poisson Distribution
- The simple modules of upper triangular matrix algebras.
- Why is the expression $\underbrace{n\cdot n\cdot \ldots n}_{k \text{ times}}$ bad?
- Calculate probability density function from moment generating function
- 100 coin flips, expect to see 7 heads in a row
- Let $K$ be a field and $f(x)\in K$ be a polynomial of degree $n$. And let $F$ be its splitting field. Show that $$ divides $n!$.
- Determine where a point lies in relation to a circle, is my answer right?
- the equivalence between paracompactness and second countablity in a locally Euclidean and $T_2$ space
- The longest string of none consecutive repeated pattern
- Show $\mathbb{Q}(\sqrt{2},\sqrt{3})$ is a simple extension field of $\mathbb{Q}$.
- explaining the derivative of $x^x$
- Fractional anti-derivatives and derivatives of the logarithm