Stephen Covey and Pope Leo X

The second habit in Stephen Covey’s best-selling book The 7 Habits of Highly Effective People is “Begin with the end in mind.” Pope Leo X would have disagreed.

Portrait of Leo X from Wikipedia

Twyla Tharp tells the story of Pope Leo X and his frustration with Leonardo da Vinci in her book The Creative Habit. da Vinci was experimenting with varnishes and so forth when the Pope thought he should be painting. Leo’s assessment of da Vinci:


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When discoveries stay discovered

In what sense did Christopher Columbus discover America? Obviously he wasn’t the first human to step foot on the New World. Columbus wasn’t even the first European. Norwegian explorer Leif Erikson seems to have arrived 500 years before Columbus. But as Stephen Mills famously stated,

There have been other people before Columbus, but when Columbus discovered the New World, it stayed discovered.

The same principle could be used to resolve debates about priorities in mathematical discoveries.There is some debate over whether John Tukey or Carl Gauss discovered the Fast Fourier Transform (FFT). But there is no doubt that after Tukey discovered it, the FFT stayed discovered. The algorithm is now used in digital signal processing applications everywhere.

Gauss and Tukey were both brilliant mathematicians. Tukey, however, also had an aptitude for creating memorable names. For example, you may have heard of “software,” a term he coined.

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Springs, resistors, and harmonic means

Harmonic mean has come up in a couple posts this week (with numbers and functions). This post will show how harmonic means come up in physical problems involving springs and resistors.

Suppose we have two springs in series with stiffness k1 and k2:

Then the combined stiffness k of the two springs satisfies 1/k = 1/k1 + 1/k2. Think about what this says in the extremes. If one of the springs were infinitely stiff, say k2 = ∞. Then k = k1. It would be as if the second spring were not there. Being infinitely stiff, we could think of it as an extension of the block it is attached to. Now think of one of the springs having no stiffness at all, say k1 = 0. Then k = 0. One mushy spring makes the combination mushy.

Next think of two springs in parallel:

Now the combined stiffness of the two springs is k = k1 + k2. Again think of the two extremes. If one spring is infinitely stiff, say k1 = ∞, then k = ∞ and the combination is infinitely stiff. And if one spring has no stiffness, say k2 = 0, then k = k1. We could imagine the spring with no stiffness isn’t there.

The stiffness of springs in series adds harmonically. The stiffness of the combination is half the harmonic mean of the two individual stiffnesses.

Electrical resistors combine in a way that is the opposite of mechanical springs. Resistors in parallel combine like springs in series, and vice versa.

If two resistors have resistance r1 and r2, the combined resistance r of the two resistors in parallel satisfies 1/r = 1/r1 + 1/r2. If one of the resistors has infinite resistance, say r2 = ∞, then r = r1. It would be as if the second resistor were not there. All electrons would flow through the first resistor.

If the two resistors were in series, then r = r1 + r2. If one resistor has infinite resistance, so does the combination. Electrons cannot flow through the combination if they cannot flow through one of the resistors. And if one resistor has zero resistance, say r2 = 0, then r = r1. Since the second resistor offers no resistance to the flow of electrons, it may as well not be there.

These physical problems illustrate why zeros as handled specially in the definition of means.

Image credit: Wikipedia

Means and inequalities for functions

A post on Monday looked at means an inequalities for a lists of non-negative numbers. This post looks at analogous means and inequalities for non-negative functions. We go from means defined in terms of sums to means defined in terms of integrals.

Let p(x) be a non-negative function that integrates to 1. (That makes p(x) a probability density, but you don’t have to think of this in terms of probability.) Think of p(x) as being a weighting function. The dependence on p(x) will be implicit. For a non-negative function f(x) and real number r ≠ 0, define Mr( f ) by

M_r(f) = left(int p(x) left( f(x) right)^r , dx right)^{1/r}

If r > 0 and the integral defining Mr( f ) diverges, we say Mr( f ) = ∞ and Mr( f ) = 0.

Define the geometric mean of the function f by

G(f) = expleft( int p(x) log f(x), dx right)

There are a few special cases to consider when defining G(f). See Inequalities for details.

First we give several limiting theorems.

  • As r → –∞, Mr( f ) → min( f )
  • As r → +∞, Mr( f ) → max( f )
  • As r → 0+, Mr( f ) → G( f )

And now for the big theorem: If rs, then Mr( f ) ≤ Ms( f ).The conditions under which equality hold are a little complicated. Again, see Inequalities for details.

We could derive analogous results for infinite sums since sums are just a special case of integrals.

The assumption that the weight function p(x) has a finite integral is critical. We could change the definition of Mr( f ) slightly to accommodate the case that the integral of p(x) is finite but not equal to 1, and all the conclusions above would remain true. But if we allowed p(x) to have a divergent interval, the theorems do not hold. Suppose p(x) is constantly 1, and our region of integration is (0, ∞). Then Mr( f ) might be more or less than Ms( f ) depending on f. For example, let f(x) = b exp( – bx ) for some b > 0. M1( f ) = 1, but M( f ) = b. Then M1( f ) is less than or greater than M( f ) depending on whether b is less than or greater than 1.

Redbelt problem solving

In the movie Redbelt, Chiwetel Ejiofor plays Mike Terry, a Jiu Jitsu instructor who will fight but will not compete. He will fight in a real fight if necessary, but he won’t fight in a ring because competitions have arbitrary rules. He is a skilled fighter because he is creative, and competitions take away that creativity. At one point in the movie, someone Terry if he teaches people to win. He says no, he teaches people to prevail. In his mind, you can’t “win” a fight. A fight is a problem to be solved.

Mike Terry’s distinction between fights and contests makes me think of the distinction between practical and academic problem solving. Practical problem solving does not have arbitrary constraints whereas academic problems often do: you can use this technique but not that one, you can use this reference but not that one, etc. These academic limitations serve a purpose in their context, but sometimes we can imagine these constraints are still on us after we leave the classroom.

Sometimes we’ll struggle mightily to solve a problem analytically that could be easily be solved numerically (or vice versa). Or we’ll imagine that a problem must be solved using a particular programming language even though it could be done more easily using a different language. It feels like “cheating” to go for the easier solution. But if you’re not in an academic setting, you can’t “cheat.” (Of course I’m not talking about violating ethical standards to solve a problem, only dismissing artificial restrictions. Where there is no law, there is no sin.)

There may be good reasons for pursuing the more difficult solution, such as entertainment value. But often we do things the hard way for no good reason other than not having examined our self-imposed limitations. Maybe we’re trying to win rather than solve the problem.

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Functional in the small, OO in the large



Means and inequalities

The arithmetic mean of two numbers a and b is (a + b)/2.

The geometric mean of a and b is √(ab).

The harmonic mean of a and b is 2/(1/a + 1/b).

This post will generalize these definitions of means and state a general inequality relating the generalized means.

Let x be a vector of non-negative real numbers, x = (x1, x2, x3…, xn). Define Mr( x ) to be

M_r(x) = left(  frac{1}{n} sum_{i = 1}^n x_i^r right)^{1/r}

unless r = 0 or  r is negative and one of the xi is zero. If r = 0, define Mr( x ) to be the limit of Mr( x ) as r decreases to 0 . And if r is negative and one of the xi is zero, define Mr( x ) to be zero. The arithmetic, geometric, and harmonic means correspond to M1, M0, and M-1 respectively.

Define M( x ) to be the limit of Mr( x ) as r goes to ∞. Similarly, define M-∞( x ) to be the limit of Mr( x ) as r goes to –∞. Then M( x ) equals max(x1, x2, x3…, xn) and M-∞( x ) equals min(x1, x2, x3…, xn).

In summary, the minimum, harmonic mean, geometric mean, arithmetic mean and maximum are all special cases of Mr( x ) corresponding to r = –∞, –1, 0, 1, and ∞ respectively. Of course other values of r are possible; these five are just the most familiar. Another common example is the root-mean-square (RMS) corresponding to r = 2.

A famous theorem says that the geometric mean is never greater than the arithmetic mean. This is a very special case of the following theorem.

If r s then Mr( x ) ≤ Ms( x ).

In fact we can say a little more. If r < s then Mr( x ) < Ms( x ) unless x1 = x2 = x3 = …. = xn or s ≤ 0 and one of the xi is zero.

We could generalize the means Mr a bit more by introducing positive weights pi such that p1 + p2 + p3 + … + pn = 1. We could then define Mr( x ) as

M_r(x) = left( sum_{i = 1}^n p_i x_i^r right)^{1/r}

with the same fine print as in the previous definition. The earlier definition reduces to this new definition with pi = 1/n. The above statements about the means Mr( x ) continue to hold under this more general definition.

For more on means and inequalities, see Inequalities by Hardy, Littlewood, and Pólya.

Update: Analogous results for means of functions, replacing sums with integrals. Also, physical examples of harmonic mean with springs and resistors.

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Unicode function names


PS> √ 81


Here’s another example using the symbol for pounds sterling

for the function to convert British pounds to US dollars.

PS> £ 300.00






The buck stops with the programmer




Sign from the desk of Harry Truman saying the buck stops here




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