Mostly notes on algorithms
Most working mathematicians I talk to or observe have the same relationship with AI: they paste a question into ChatGPT, get something proof-shaped back, feel vaguely impressed or vaguely cheated, and go back to work. This makes sense, especially under the common opinion: If you ask ChatGPT a few times and it does not solve the problem, then AI can NEVER solve the problem by itself. This opinion…
In an idealized world, you could borrow at the Effective Federal Funds Rate (EFFR) and also earn roughly the same rate on cash deposits. For simplicity, assume the relevant short rate is fixed for a month. Let’s say it turns out to be 10% per month . Assume the stock will rise by 20% that month. In reality, lenders (and banks taking deposits) need to earn a spread, so borrowing and deposit rates…
Disclaimer: Not an accountant nor a lawyer. For employees in the US, employers (not just the employee) can contribute to the employee’s retirement fund 401(k). There is an maximum limit. It is M = \$61,000 for 2022. If a person works for multiple unrelated employers, each employer, in theory, can contribute M to their 401(k). This is because such limit is based on the plan, and not based on the…
This article considers tools one can use to quickly test conjectures. I work in combinatorial optimization, and often there are cases where one can phrase a problem as a conjecture over a finite domain. Testing conjectures using brute force quickly becomes infeasible due to combinatorial explosion. Here, we walk through an actual example and learn a few techniques. We will use Sage and any fast…
Yotta Savings is a FDIC insured savings account that gives you “interest” in another way. It has a 0.2% APR. It is much better than the APR of many banks, but much worse than actual high APR savings accounts, e.g. Marcus, Ally. However, for each $25, Yotta gives you a ticket weekly to their drawing, which you would have some chance of winning a big amount of money. This is very close to Premium…
I’m using beancount to do accounting. In beancount, you are allowed to have commodities. One can view gift card as commodity, but also as a currency. Say, I bought an $100 Amazon gift card for $90, I would write 100 AMAZON {0.9 USD} , which means 100 units of Amazon with unit cost $0.9. Later on, I used $50 Amazon gift card, then I can write 50 AMAZON {0.9 USD} @ 1 USD . Here @ 1 USD means the…
Credit cards often offers rewards for purchases. For example, a card that gives 5% cash back for grocery stores. Also, credit cards often can be used to buy gift cards for specific stores. For example, one can use Chase Ink Cash (CIC) and go to Staples and buy gift card for Amazon for effectively 7.5% off Amazon (I value UR at 1.5 cents each). However, there is usually a limit. CIC caps the reward…
Consider we have an undirected graph G=(V,E) of n vertices, and there are positive cost c:E\to \R^+ on the edges. We define c(F)=\sum_{e\in F}c(e) , to be the cost (value) of F\subset E . Let \mathcal{P} be a partition of V where each partition class is non-empty. We define E(\mathcal{P}) to be the set of edges with end points in two different partition classes. A set of edges F is called a k -cut…
Problem 1 Given a set of strings W , a cost function c:W\to \R and a string s . Find elements w_1,\ldots,w_k\in W such that s=w_1\ldots w_k , and \sum_{i=1}^k c(w_i) is minimized. This problem is a generalization of the word break problem on leetcode. Many algorithms you see online assumes that each string in W has constant length, checking the hash table takes O(1) time, and obtains an O(n^2)…
I’ve read a lot of articles on Traditional vs Roth IRA. One can do a serious mathematical analysis of which one is better, given that you want to invest a total of x income into the account for a particular year. Money in excess of x will be put in a normal investment account. There is a widely held belief that if your effective tax rate is higher today than when you cash out, it is always better…