Author Archives: John Roe

Traces and commutators

The following is a true (and well-known) theorem: \(\newcommand{\Tr}{\mathop{\rm Tr}}\)

Suppose \(A\) and \(B\) are bounded operators on a Hilbert space, and \(AB\) and \(BA\) are trace class.  Then \( \Tr(AB)=\Tr(BA) \).

This is easy to prove if one of the operators \(A,B\) is itself of trace class, or if they both are Hilbert-Schmidt (the obvious calculation works).  In the general case it is a bit harder.  The “usual” argument proceeds via Lidskii’s trace theorem – the trace of any trace-class operator is the sum of the eigenvalues – together with the purely algebraic fact that the nonzero eigenvalues of \(AB\) and \(BA\) are the same (including multiplicities).  Continue reading

Proofs and Understanding

A year and a half ago I wrote a post on my ideas about using “structured proofs” to improve understanding in the Analysis I course.

I duly tried this approach, and felt that it made some difference, though perhaps not as much as I had hoped.

Now in this month’s Notices of the AMS there is a fascinating article by a team of educators from Loughborough University in the UK.  Entitled Understanding and Improving Undergraduate Proof Comprehension, the article discusses as three-stage effort to help undergraduate students gain a deeper understanding of proofs – the same issue that my “structured proof” software was intended to address.

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Thermodynamics IV: entropy

In the previous post, I talked about the second law of thermodynamics: there can be do thermodynamic transformation whose overall effect is to move heat from a cooler body to a hotter one.  Since the reverse of such a transformation (moving heat from a hotter body to a cooler one) happens naturally by conduction, the second law naturally contains an element of irreversibility which it is natural to expect is expressed by an inequality.   The quantity to which this applies is the famous entropy.

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Thermodynamics III: second law

The first law of thermodynamics says that heat is a form of energy. There is a lot of heat about!  For instance, the amount of heat energy it would take to change the temperature of the world’s oceans by one degree is about \(6 \times 10^{24}\) joules.  That is four orders of magnitude greater than the world’s annual energy consumption!  So, if we could somehow how to figure out how to extract one degree’s worth of heat energy from the oceans, we could power the world for ten thousand years!  Continue reading

Thermodynamics II: gases and the First Law

Recall from the previous post that the First Law of Thermodynamics can be expressed

\[ \alpha+\beta = -dU \]

where \(U\) is the total energy of a thermodynamic system and \(\alpha,\beta\) are one-forms whose integrals along a transformation express the work done by the system on its environment, and the heat supplied by the system to its environment, in the course of the transformation. (In Fermi’s notation \(\alpha=dL\), \(\beta=-dQ\), where \(L,Q\) should be thought of as functions not on the state space but on the path space of the state space – their values depend on how you got there).  If the state space is a 2-manifold parametrized by volume \(V\) and pressure \(p\), then \(\alpha=pdV\).    The thermal capacity of the system is the derivative \(dQ/dT\), that is the marginal amount of heat absorbed for an increase in temperature.  There are two versions of the heat capacity, \(C_V\) (heat capacity at constant volume) and \(C_p\) (heat capacity at constant pressure).  For the situation we’re considering, \(\alpha=pdV = 0\) at constant volume, so

\[ C_V = \left(\frac{\partial U}{\partial T}\right)_V,\qquad C_p = \left(\frac{\partial U}{\partial T}\right)_p + p\left(\frac{\partial V}{\partial T}\right)_p \]

by standard calculations with partial derivatives. Continue reading