e^pi vs pi^e

Hi Prof Lee,

Thanks for the lunch (including the advice part). I came up with some ideas about this brain teaser —

Q: which is bigger e^pi vs pi^e

One solution I can think of is, suppose e has a value close to 2 and pi is much larger.

Suppose e = 2 and pi = 10. Clearly e^pi wins.

Another way is, define 2 functions

f1(x) = 2.718281828^x and find the growth rate when x is slightly above e. This growth rate is e^x,

f2(x) = x^2.718281828 and find the growth rate when x is slightly above e. This grow rate is e/x * x^e, which is smaller, since x is slightly bigger than e.

Therefore, f1 grows faster than f2, over the range of (e , 3.15). Therefore e^pi wins.

pick line passing the most points – solved

Q: Given N points with positive integer coordinates, find the straight line passing through the most points
A: For each of (N*N-N)/2 pairs of points, compute a Line object identified by 2 numbers:
* a slope S = (y2 – y1)/(x2 – x1)
* intercept on y-axis.

So these 2 numbers can be computed easily from any pair.

Save each Line object as key in a hashmap. When a pair gives a Line that’s already seen, increment its count.

Intercept formula y_inter(int, int, int, int) can be assumed to exit. Writing this function isn’t relevant to a coding interview:

Suppose this value is y3, so the incept point is (0,y3), so

(y3-y1)/(0-x1) = S, so y3 = y1 – S x1

2 overlapping rectangles

bloomberg-lp-interview-question.

Q: You have a rectangle a and b. Determine if the two rectangles overlap. That is at least some part of either rectangle should be within the other.

I will assume the coordinates are given.

%%A: find both centers.
– case 1: identify box A’s corner that’s closes to B’s center. Check if this corner is inside B.

– case 2: if they lie on one vertical line, then only the bottom border (of upper box) vs top border (of lower box) matters.

PDE/SDE

I have seen a few PDE/SDE combos. There’s a pattern among them.

The SDE tends to describe a process as a signal-noise description. It is actually rather precise, as precise as it gets — you can compute the exact probability of the “particle” falling into a range at a given time t.

However, the SDE won’t enable us to compute today’s price as an equation can, such as “sin(x) + log(x) – sqrt(x) = pi”. Reason? The dW term is an obstacle. Therefore, we need to somehow get rid of the dW term. We end up with a differential equation, often a PDE. If there are sufficient boundary conditions, we could solve the equation to get a precise time-0 price

PDE solver – phrasebook

[[Numerical methods and optimization in finance]] has simple examples

of a PDE numerical solver. This book is more in-depth than [[basic

black-scholes]]

I think the Daniel Duffy book also covers Finite Difference method

Brute-force — often there's no other solution (like mathematician

tackling a proof) to a PDE. Numerical solution is kinda versatile.

Fwd: comparing exp( (a+b)/2 ) vs 0.5 exp(a) + 0.5 exp(b)

How about Jensen’s inequality?


Hi Richard (Qu Miao),

brain teaser — compare
A = exp( (a+b)/2 )    vs
B = 0.5 exp(a) + 0.5 exp(b)
Here’s my solution. See if it is correct.
Denote f = 2B/A. So f = exp(.5a  – .5b)  +  exp(.5b  – .5a) ….. symmetry
Denote x = .5a – .5b , so f(x) = exp(x) + exp(-x).
Now f(x) curve goes to infinity on both sides. So f(x) has minimum value of 2 occurring at x=0.
That means 2B/A has a minimum value of 2. In other words, B >= A

Combination Sum question – no solution yet

Given a set of candidate numbers (C) and a target number (T), find all unique combinations in C where the candidate numbers sums to T. The same repeated number may be chosen from C unlimited number of times.

Note:
All numbers (including target) will be positive integers. Elements in a combination (a1, a2, … , ak) must be in
non-descending order. (ie, a1 ≤ a2 ≤ … ≤ ak). The solution set must not contain duplicate combinations.

For example, given candidate set 2,3,6,7 and target 7, A solution set is:
[7]
[2, 2, 3]