95 research outputs found
Proof of the Ergodic Theorem and the H-Theorem in Quantum Mechanics
It is shown how to resolve the apparent contradiction between the macroscopic
approach of phase space and the validity of the uncertainty relations. The main
notions of statistical mechanics are re-interpreted in a quantum-mechanical
way, the ergodic theorem and the H-theorem are formulated and proven (without
"assumptions of disorder"), followed by a discussion of the physical meaning of
the mathematical conditions characterizing their domain of validity.Comment: English translation by Roderich Tumulka of J. von Neumann: Beweis des
Ergodensatzes und des H-Theorems. 41 pages LaTeX, no figures; v2: typos
corrected. See also the accompanying commentary by S. Goldstein, J. L.
Lebowitz, R. Tumulka, N. Zanghi, arXiv:1003.212
Cyclotomic integers, fusion categories, and subfactors
Dimensions of objects in fusion categories are cyclotomic integers, hence
number theoretic results have implications in the study of fusion categories
and finite depth subfactors. We give two such applications. The first
application is determining a complete list of numbers in the interval (2,
76/33) which can occur as the Frobenius-Perron dimension of an object in a
fusion category. The smallest number on this list is realized in a new fusion
category which is constructed in the appendix written by V. Ostrik, while the
others are all realized by known examples. The second application proves that
in any family of graphs obtained by adding a 2-valent tree to a fixed graph,
either only finitely many graphs are principal graphs of subfactors or the
family consists of the A_n or D_n Dynkin diagrams. This result is effective,
and we apply it to several families arising in the classification of subfactors
of index less then 5.Comment: 47 pages, with an appendix by Victor Ostri
Undecidability in number theory
These lecture notes cover classical undecidability results in number theory,
Hilbert's 10th problem and recent developments around it, also for rings other
than the integers. It also contains a sketch of the authors result that the
integers are universally definable in the rationals.Comment: 48 pages. arXiv admin note: text overlap with arXiv:1011.342
Ueber den Gebrauch verschluckbarer, kugelförmiger Maximalthermometer, sowie neuer cylinderförmiger, welche geeignet sind, im Blutgefässsystem lebender Thiere zu circuliren
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