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Tychonoff’s theorem

Tychonoff’s theorem (an arbitrary product of compact sets is compact) is one of the high points of any general topology course.  When I’ve taught this in recent years, I’ve usually given the proof using universal nets, which I think is due to Kelley.

Recently though I read a very nice paper by Wright  which reproduces, and then modifies, Tychonoff’s original proof (otherwise inaccessible to me because of my lack of German).  I thought the original proof was really elegant and though I would try to give an exposition.

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Jost Bürgi’s Method for Calculating Sines

Jost Bürgi (28.2.1552 -31.1.1632) Astronom, Mathematiker, Instrumentenbauer, Entdecker der Logarithmen. aus:7523 (Rar);Frontispiz

Jost Bürgi (28.2.1552 -31.1.1632)
Astronom, Mathematiker, Instrumentenbauer, Entdecker der Logarithmen.
aus:7523 (Rar);Frontispiz

I just learned (via Facebook, no less) of a fascinating paper with the above title by Andreas Thom and coauthors.

Jost Burgi (1552-1632) was a Swiss mathematician, astronomer and clockmaker.  He worked with Johannes Kepler from 1604 and is thought to have arrived at the notion of logarithms independent of Napier.  He was also reputed to have constructed a table of sines by a brand new method, but until now the details of his Kunstweg (“artful method”) for computing sines were thought to have been lost.  The beautiful book of van Brummelen, The Mathematics of the Heavens and the Earth: The Early History of Trigonometry,  (Princeton University Press, 2009) documents how computing trigonometric ratios was  a central theoretical and practical preoccupation of ancient mathematics. We know of Burgi’s Kunstweg via a statement of his colleague and friend Nicolaus Ursus: “the calculation (of a table of sines)… can be done by a special way, by dividing a right angle into as many parts as one wants; and this is arithmetically. This has been found by Justus Burgi from Switzerland, the skilful technician of His Serene Highness, the Prince of Hesse.”  But the details are not clear and apparently nobody, starting with Kepler himself, was ever able to reconstruct Burgi’s method.

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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