Skip to main content

22/2/22 22:22

Two years into the pandemic, many continue to suffer from the disease. Even two or three vaccine doses later, our lifestyle remains hostage to the risk of a sudden outbreak that could cripple our liberties, as does the recent surge in cases of infections of the Omicron strain. Twenty-twenty-two brings no guarantee of complete relief and restoration of our freedoms stolen by COVID-19.

But, in spite of (and sometimes, because of) the pandemic, I've built new genuine friendships, gained godchildren, and witnessed blissful weddings of my best friends. I feel hopeful that more good things are to come in 2022. I look forward to being free to travel again, to getting rid of the fear of the virus and getting used to living with it, and to beginning a new life chapter (after I finish writing those chapters I've been working on for the last three years).

It feels awful not being able to celebrate with the people I love. But finding ways to make up despite the distance is itself rewarding and fun. It reminds me just how much they matter to me, and me to them. Difficulties are real, but so are the opportunities for joy and hope, and I am convinced that we are wired to seek and find them. 

Only hope keeps us moving forward—hope for a new beginning, hope for a better life, hope for salvation. Let this blog be a sign of my commitment to continue seeking the True, the Good, and the Beautiful through a life full of hope.

Signed JT on 22-2-22 22:22.

Labels:

Comments

Post a Comment

Popular posts from this blog

Compute $\sin(\sin(...\sin(x)))$

Problem Find the limit $\sin(\sin(...\sin(x)))$ for any real number $x$. Solution The problem can be precisely formulated as follows: Find the limit $L$ such that $$ L \equiv\lim_{n\rightarrow\infty}{f_n(x)}, $$ where $x\in\mathbb{R}$, $n\in\mathbb{N}$, and $$\begin{align} f_1(x) &= \sin(x)\\ f_n(x) &= \sin(f_{n-1}(x)). \end{align}$$ We observe that, if the limit $L$ exists, $$\begin{align} \lim_{n\rightarrow\infty} f_{n-1}(x) &= L\\ \lim_{n\rightarrow\infty} f_n(x) &= L, \end{align}$$ and thus, $$ L = \sin{L}, $$ which we show to be $L=0$ in the following. First, we show that the limit exists. We accept the Axiom of Continuity of the Real Line, which says that any bounded monotonic sequence converges. Thus, to show that $L$ exists, we need only show that (1) $f_n(x)$ is a bounded sequence of $n$, and that (2) $f_n(x)$ is a monotonic sequence of $n$. $f_n(x)$ is clearly bounded because $f_n(...
Post a Comment
Read more

Find the limit of the sequence. $a_1=1, a_{n+1} = \frac{3a_n+4}{2a_n+3}$

We take the limits of sequences defined recursively by first showing the existence of the limits then, actually computing them. Existence is demonstrated by showing that the sequence is both (1) monotonic and (2) bounded. Problem Find the limit of the following sequence. $$\begin{align} $a_1&=1\\ a_{n+1} &= \frac{3a_n+4}{2a_n+3}$ \end{align}$$ Solution Monotonicity : We use mathematical induction to show that for all $n\in\mathbb{N}$, $a_{n+1} - a_n \geq 0$ and therefore, $\{a_n\}$ is monotonic increasing. For the base case at $n=1$, it is easy to see that $a_{n+1}-a_n = \frac{7}{5} - 1 = \frac{2}{5} >0.$ For the inductive case at $n=k$, we assume that $$\begin{align} a_{k+1} - a_{k} & = \frac{3a_k+4}{2a_k+3} - \frac{3a_{k-1}+4}{2a_{k-1}+3}\\ & = \frac{a_k - a_{k-1}}{(2a_k+3)(2a_{k-1}+3)} \geq 0. \end{align}$$ For $n=k...
Post a Comment
Read more

Reasonable Majority Rule: Wise Decision-making in a Democracy

Let the people decide what is good for them. On the one hand, this principle is at the core of democracy and manifests in the phrase "majority rule." On the other hand, the protection of the voice of the minority is another principle a functioning democracy must enforce. Democracy requires both. The fundamental assumption is that, after free and informed debate, the majority will be reasonable enough to judge ideas based on their merits (and not on their emotional relationship with the proponents). Because in a democracy, the Government's wisdom is an extension of the people's wisdom, wise decision-making is the duty of every citizen. The "goodness" of a decision or an idea does not necessarily depend on the number of its proponents but on the independent practical assessment of available information. The difficulty in balancing the interests of opposing sides is highlighted on occasions when the minority is unwilling to accept the decision of the ...
Post a Comment
Read more