Articles of analysis

Law of iterated logarithms for BM

The law of iterated logarithms for the standard Brownian motion asserts that $(\ast) \limsup\limits_{h \downarrow 0} \frac{B(h)}{\sqrt{2h\log\log(\frac{1}{h})}} = 1$ I’m trying to prove the following: if $T_a = \inf\{t \geq 0 \ | B(t) = a\}$ then $\limsup\limits_{h \downarrow 0} \frac{B(T_1)-B(T_1-h)}{\sqrt{2h\log\log(\frac{1}{h})}} \leq 1$ I tried to use $(\ast)$ for the right stopping time and the […]

Why $\lim\limits_{(x,y)\to(0,0)}\frac{\sin(x^2+y^2)}{x^2+y^2}=\lim\limits_{t\to 0}\frac{\sin t}{t}$?

Why $\displaystyle\lim_{(x,y)\to(0,0)}\frac{\sin(x^2+y^2)}{x^2+y^2}=\lim_{t\to 0}\frac{\sin t}{t}$( and hence equals to $1$)? Any rigorous reason? (i.e. not just say by letting $t=x^2+y^2$.)

liminf in terms of the point-to-set distance

Let $\mathcal{X}$ be a normed space and $C\subseteq \mathcal{X}$. We define the point-to-set distance for the set $C$ to be: $$ d_C:\mathcal{X}\ni x \mapsto d_c(x):= \inf_{y\in C}\|x-y\| \in [0,\infty] $$ Additionally, we define the inner limit of a sequence of sets $C_n$ in $\mathcal{X}$ to be: $$ \liminf_n C_n = \bigcup_{n=1}^\infty \bigcap_{m=n}^\infty C_m $$ This […]

Find a conformal map from the disc to the first quadrant.

Find a conformal mapping of the disk $x^2+(y-1)^2\lt 1$ onto the first quadrant $x, y \gt 0$ I did something, which may be false or not, I cannot exactly say anything. I used the composition of a conformal map, which is conformal. Firstly, let’s get a conformal map from the disk to the unit disk, […]

Spivak's Calculus – Chapter 1 Question 23

In Spivak’s Calculus Chapter 1 Question 23: Replace the question marks in the following statement ing $\varepsilon, x_0$ and $y_0$ so that the conclusion will be true: If $y_0\neq 0$ and $$|y-y_0|<? \qquad\text{and}\qquad |x-x_0|<?$$ then $y\neq 0$ and $$ \bigg| \frac{x}{y}-\frac{x_0}{y_0}\bigg|<\varepsilon.$$ The answer in its Solution Manual is $$|x-x_0|<\min\bigg(\frac{\varepsilon}{2(1/|y_0|+1)},1 \bigg)$$ and $$|y-y_0|<\min\bigg( \frac{|y_0|}{2},\frac{\varepsilon|y_0|^2}{4(|x_0|+1)} \bigg).$$ The […]

Is the ring of germs of $C^\infty$ functions at $0$ Noetherian?

I’m considering the property of the ring $R:=C^\infty(\mathbb R)/I$, where $I$ is the ideal of all smooth functions that vanish at a neighborhood of $0$. I find that $R$ is a local ring of which the maximal ideal is exactly $(x)$. I also want to determine if $R$ is a Noetherian ring, but I have […]

$\lim_{x\to 0+}\ln(x)\cdot x = 0$ by boundedness of $\ln(x)\cdot x$

I saw a proof that $$ \lim_{x\to 0} \ln|x|\cdot x = 0 $$ where is is argued that for $x \in (0,1)$ we have $$ | \ln(x) x | = \left| \int_1^x x/t ~\mathrm d t \right| = \left| \int_x^1 x/t ~\mathrm d t \right| \le \left|\int_x^1 1 dt\right| = |1 – x| \le 1 […]

Application of Weierstrass theorem

Let f be a continuously differentiable function on $[a.b]$. Show that there is a sequence of polynomials $\{P_n\}$ such that $P_n(x) \to f$ and $P’_n(x) \to f’ (x)$ uniformly on $[a,b]$ My approach has been as follows. Since f is continuously differentiable, we have $Q_n(x) \ to f'(x)$ on$ [a,b]$ uniformly (Weierstrass) . I’m not […]

Doubling measure is absolutely continuous with respect to Lebesgue

Let $\mu$ be a fixed finite measure on $\mathbb R$. We say that $\mu$ is doubling if there exists a constant $C>0$, such that for any two adjacent intervals $I=[x−h,x]$ and $J=[x,x+h]$, $$C^{−1}\mu(I)≤\mu(J)≤C\mu(I).$$ Assuming that $\mu$ is doubling, show that there exist positive constants $B$ and $a$, such that for every interval $I$, $$\mu(I)≤B[length(I)]^a$$ By […]

Compute $\int_0^{\infty}\frac{\cos(\pi t/2)}{1-t^2}dt$

Compute $$\int_0^{\infty}\frac{\cos(\pi t/2)}{1-t^2}dt$$ The answer is $\pi/2$. The discontinuities at $\pm1$ are removable since the limit exists at those points.