“Fuel” and “Engines” in mathematics

Note: First blog post in a while, been a crazy few months.

There are many analogies mathematicians use to try to explain the abstract realm they work in to laymen of various levels. For example, Freeman Dyson had tried to categorize mathematicians as “frogs” or “birds” and Tim Gowers tried to categorize them as “problem solvers” or “theory crafters”. In a post on MathOverflow, I tried to categorize certain theorems as ‘workhorses’.

Another useful way I recently thought of to think about various theorems and technique in mathematics is to think of them as “fuel” or “engines”. Fuel and engines are both parts of “machines”, which comprise theories and programs in mathematics. Fuel is very versatile; an improvement in fuel will almost certainly mean that all engines which can use said fuel can operate better. Engines are more specific; they are designed to perform a specific function. A car engine for example is useless if you try to make it chop vegetables, unless you design an entire system which chops vegetables while being powered by a car engine.

This is a good way to explain just how the approaches Yitang Zhang and James Maynard differ. One can think of the original sieve mechanism devised by Goldston, Pintz, and Yildirim, now known as the GPY-sieve. In a sense they constructed a ‘machine’ which allows us to turn churn out small gaps between primes. The engine of the machine is the GPY sieve mechanism; while the fuel (or the raw input) is the Bombieri-Vinogradov theorem. The machine works quite efficiently in that it produces gaps that are arbitrarily small compared to the average gap; but the engine just can’t put out enough power to establish bounded gaps between primes with the current quality of the fuel. This situation was thought to be as far as one can go back in 2013.

Yitang Zhang came along and showed that while it is not possible to develop an improved ‘all purpose’ fuel, i.e. an improvement on the level of distribution present in the Bombieri-Vinogradov theorem, one can devise a specialized and more efficient fuel that works specifically for the GPY engine. This extra boost in power was sufficient to establish bounded gaps.

Maynard, however, showed that one can rip out the GPY engine and replace it with a much more efficient and powerful engine that requires much less fuel and can output much more power. This is Maynard’s multi-dimensional sieve; something thought to be impossible previously.

The problem is that Maynard’s engine doesn’t run on Zhang’s fuel; because the latter is specialized for the GYP engine. Modifying the Maynard engine so that it runs on the Zhang fuel would be the next step forward towards the bounded gap problem.

Bhargavaology Learning Seminar First Talk – 12 Jan 2015

I am doing a learning seminar on ‘Bhargavaology’, which is the collection of techniques and results surrounding the recent work of Manjul Bhargava, a Fields Medalist in 2014. The introductory talk is scheduled on Monday, Jan 12 in MC 5413, from 2:30 PM to 3:30 PM. The abstract is indicated below:

Title: Introduction to Bhargavaology
Speaker: Stanley Yao Xiao
Abstract: In this introductory talk I aim to give a sampling of the vast web of important theorems related to Manjul Bhargava, one of the four Fields Medalists of 2014, and describe their importance in the context of modern number theory. There are no technical details in the talk, making it very accessible.

A cute problem that illustrates the power of integers

The following problem was introduced to me as a ‘homework problem’ for the child of a certain genius-level mathematician, which is likely apocryphal. Nevertheless, the problem itself is interesting.

Problem: Three farmers brought 10, 16, and 26 chickens to the market to sell. To avoid a price war, they agreed to sell each chicken for the same price. By midday, none of them had sold all of their chickens. They then agreed to lower their prices by the same amount in order to unload all of their chickens. By the end of the day, each farmer sold all of the chickens they brought to the market and each made 35 dollars. What is the difference of the morning price and the afternoon price?

This question seems like it does not provide enough information. Indeed it does not, if one fails to consider that certain quantities (namely the amount of chickens) are integers. To wit, let us establish the solution as follows.

Let x,y,z respectively denote the number of chickens sold by farmer 1, 2, and 3 in the morning. Note that x,y,z are necessarily integers. Let C denote the price of a chicken in the morning, and C' denote the price of a chicken in the afternoon. Let D = C'/(C - C'). We then obtain the following equations:

Cx + (10 - x)C' = 35,

Cy + (16 - y)C' = 35, and

Cz + (26-z)C' = 35.

Subtracting the third equation from the first and second and re-arranging, we obtain

x - z = 16D, y - z = 10D.

Now the fact that x,y,z are positive integers really come in handy. In particular, we must have 16D, 10D are both integers. The first condition implies that the denominator of D must be a power of 2, and the second implies that it must in fact be exactly 2. Now we write D = G/2, where G is a positive odd integer. Now we examine the equation

x = 8G + z.

Note that since Farmer 1 did not sell all of his chickens in the morning, we must have

1 \leq x \leq 9.

However, G \geq 1 and z \geq 1, whence 8G + z \geq 9, and so x = 9 is the only possibility. Thus, z = 1 and G = 1. This shows that

y = 5G + 1 = 6.

With this information in hand, we go back to examine the equation

9C + C' = 35, 6C + 10C' = 35.

Substituting, we see that C = 15/4, C' = 5/4. So it would appear that C - C' = 15/4 - 5/4 = 5/2.

But wait! We had only shown that the values of C, C' above satisfy two of the three required equations, so for due diligence we must examine the final equation

C + 25C' = 35.

But

15/4 + 25(5/4) = 140/4 = 35,

so indeed C = 15/4, C' = 5/4 is an acceptable solution.

Note that the solution really required the fact that x,y,z were positive integers and constrained in a specific way. Without these (seemingly unimportant) tidbits of information, the problem would be intractable.

An innocent linear algebra problem and an application of the probabilistic method

The following question was posed to me by a friend:

Suppose that A \subset M_{n \times n}(\mathbb{R}) is a symmetric matrix with non-negative trace, and B \subset M_{n \times n}(\mathbb{R}) is a symmetric matrix with negative trace. Prove that there exists a vector \textbf{w} \in \mathbb{R}^n such that \textbf{w}^T A \textbf{w} \geq 0 and \textbf{w}^T B \textbf{w} < 0.

After fruitlessly trying to give an explicit construction and trying to work out an inductive argument, the hint provided by my friend proved to be ingenious: one should consider a probabilistic argument.

In the end, one can argue that the probabilistic method is not really needed as the argument used below is equivalent to a counting argument. Nevertheless, we give the following proof.

First, we may suppose that A is diagonalized, as the matrix P used to diagonalize A when applied to B gives PBP^{-1}, which remains a symmetric matrix with negative trace as trace is preserved under similarity. Now consider the random vector \textbf{x} = (x_1, \cdots, x_n) where the x_i are independent Bernoulli random variables with mean zero and variance 1. It is immediate that for any such vector \textbf{x}, we have

\textbf{x} A \textbf{x}^T \geq 0.

Now, we compute the expectation of \textbf{x} B \textbf{x}^T. By the linearity of expectation and the fact that the x_i‘s are independent, we have

\displaystyle E(\textbf{x} B \textbf{x}^T) = \sum_{j=1}^n b_{jj} \text{Var}(x_j) = \text{Tr}(B) < 0.

Hence there exists one particular vector \textbf{x}' for which $\textbf{x}’ B (\textbf{x}’)^T < 0$, and we are done.

As remarked earlier, this argument is equivalent to considering the set of vectors \textbf{x} = (\pm 1, \cdots, \pm1) and adding all of them and concluding that at least one of them must give the desired result.

An interesting problem on a function in two variables

One of the starkest differences between dealing with differentiation of a multi-variable function and a single variable function is that there are now an infinite number of ways to take a limit, and it becomes much more difficult to show that the derivative exists at a point by checking that derivatives exist along all possible paths. With functions in two variables, for example, one can usually play around with x and y separately to try to get different values at a limit point. Recently I ran across a problem that I found striking.

Let a,b,c,d be positive real numbers and let

\displaystyle f(x,y) = \frac{|x|^a |y|^b}{|x|^c + |y|^d}.

Prove that \lim_{(x,y) \rightarrow (0,0)} f(x,y) = 0 if \displaystyle \frac{a}{c} + \frac{b}{d} > 1 and that the limit does not exist if \frac{a}{c} + \frac{b}{d} \leq 1.

I found this particularly interesting because it is not at all clear why the quantity \displaystyle \frac{a}{c} + \frac{b}{d} should control the existence of the limit at (0,0). The proof, I thought, was quite clever.

In retrospect the case a/c + b/d \leq 1 is easier, as we just have to exhibit a path for which the limit as (x,y) approaches (0,0) along the path is non-zero. Indeed, trivially we have \lim_{(x,0) \rightarrow (0,0)} f(x,0) = 0 for any positive values of a,b,c,d, and so this suffices. Now we choose the path where x,y > 0, y = x^{c/d}. Then we have

\displaystyle f(x,y) = \frac{x^a x^{bc/d}}{x^c + x^c} = \frac{1}{2}x^{c\left(\frac{a}{c} + \frac{b}{d} - 1\right)}

and this limit is plainly 1/2 if a/c + b/d = 1 and + \infty if a/c + b/d < 1.

The other situation is much trickier. We will require two ad-hoc inequalities.

\displaystyle |x| = (|x|^c|)^{1/c} \leq (|x|^c + |y|^d)^{1/c},

\displaystyle |y| = (|y|^d)^{1/d} \leq (|x|^c + |y|^d)^{1/d}.

Now we have

\displaystyle |x|^a |y|^b \leq (|x|^c + |y|^d)^{a/c} (|x|^c + |y|^d)^{b/d},

whence

0 \leq f(x,y) \leq (|x|^c + |y|^d)^{a/c + b/d - 1},

and if a/c + b/d - 1 > 0 then the right hand side tends to 0 as (x,y) \rightarrow (0,0), thus establishing our proposition.

along this path.

Upcoming TV series about Emperor Wu to be the most expensive TV series made in Chinese history

An upcoming series chronicling the life of China’s only female emperor, Emperor Wu Zetian of the Zhou Dynasty (she was the only emperor of that dynasty, the emperor to come before and after her were all emperors of the more famous Tang Dynasty), is set to become the most expensive TV series ever made in China (http://en.wikipedia.org/wiki/The_Empress_of_China). The TV series will star Chinese megastar Fan Bingbing, and is also produced by her company.

The story of Emperor Wu is one of the most famous in the history of China. She seemed to have done the impossible: she assailed the entrenched sexism and misogyny embedded in feudal China and ascended to the title of Emperor, making her both the nominal and actual leader of China. It is noted that only three women in Chinese history has wielded as much power, Wu Zetian being the second. The other two were Empress Lü Zhi of the Han Dynasty, wife of Emperor Gaozu, the founder of the Han Dynasty and Empress Dowager Cixi, the wife of Emperor Xianfeng and mother of Emperor Tongzhi of the Qing Dynasty. However, the other two never managed to claim the title of Emperor for themselves, instead having to rule from ‘behind the curtain’. 

Much of China’s history, understandably, is written by men and for men. It was noted earlier that in fact Chinese historians have a habit of writing towards a very specific audience: future rulers. Thus Chinese history is written as a warning to future emperors of the challenges they must conquer in order to hold onto power. The most deleterious of these challenges is undoubtedly guarding against the women you trust most. It is no surprise then that all of the above strong women who managed to climb to the peak of power were later demonized in the annals of Chinese history. Both Lü Zhi and Wu Zetian were ‘renowned’ for their immense cruelty. For instance, it was said that Lü Zhi turned Gaozu’s favored concubine, Mistress Zhao, into a human ‘stump’ by cutting off her limbs and mutilating her face and then blinding her, but kept her alive. Wu Zetian was said to employ ‘cruel ministers’ to administer extraordinary acts of cruelty and torture against political opponents and ruled by a campaign of terror. Cixi is widely regarded by modern Chinese history as the principal architect of the Qing Empire’s downfall, and the subsequent humiliation that the Chinese civilization suffered in the 20th century.

However, in more recent times, some familiar narratives of history have been challenged and re-imagined. One such case is that of the Qin Dynasty. Historically noted for their immense cruelty and extreme legal system and anti-Confucianism, the recent work 大秦帝国 has sought to re-tell the story of the rise of the Qin Empire, previously the Qin Kingdom in the Warring States Period, in a positive light. It sought boldly to re-imagine the Chinese civilization as having been born from a warrior nation (Qin), as opposed to a relatively meek and passive nation (context: after uniting China and establishing the Han Dynasty, Emperor Gaozu and his successors Wendi and Jingdi sought a policy of appeasement and compromise against the barbarians to the north, which was finally reversed by Emperor Wudi, who ordered several major military campaigns against China’s northern neighbors). The same has happened for the story of Wu Zetian, where more recent TV series have depicted her in a much more positive light, as a strong and wise leader far outweighing the weak Tang Emperors she replaced.