# Mathematical Entertainments: A Collection of Illuminating by M. H Greenblatt

By M. H Greenblatt

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Note that Eq. 7) tells us that the fact that forward sectors point upward is equivalent to the temperature being everywhere positive. To illustrate what is involved here, let us consider a system of N independent spins in a magnetic field, so that each spin has energy either 0 or e. X). The first half of the energy range, 0(º/(Ne)(1/2 has positive temperature while the second half 1/2(º/(Ne)(1 has negative temperature, according to Eq. 7). H. Lieb, J. Yngvason / Physics Reports 310 (1999) 1—96 our formulation of simple systems?

1 rely only on the convexity of and on axioms A1—A7. 1 hold for compounds as well. Proof. t)Z) . 1, tXOt½. By scaling, A4, XO½. ᭿ Our first theorem in this section, about closedness, is crucial because it lies behind many of the more complex theorems. Once again, the seemingly innocuous stability axiom A6 plays a central role. As we said in Section 2, this axiom amounts to some kind of continuity in a setting in which, at first, there is not even any topology on the state spaces. H. Lieb, J. Yngvason / Physics Reports 310 (1999) 1—96 evident and stability reveals its true character in the statement of closedness in the usual topological sense.

Lieb, J. Yngvason / Physics Reports 310 (1999) 1—96 43 a finite slope with respect to the work coordinates and the slope is moreover assumed to be a locally ¸ipschitz continuous function of X. We emphasize that this tangent plane to A is initially assumed to exist only at X itself. 5 will state that this does 6 not occur. The precise meaning of the statements in axiom S2 is the following: The tangent plane at X"(º, ») is, like any plane in RL>, defined by a linear equation. 3) in which the X dependent numbers P (X) are the parameters that define the slope of the plane G passing through X.