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The sign of a Latin square is if it has an odd number of rows and columns that are odd permutations; otherwise, it is . Let and be, respectively, the number of Latin squares of order with sign and . The Alon-Tarsi conjecture asserts that when is even. Drisko showed that for prime and asked if similar congruences hold for orders of the form , , or . In this article we show that if , then only if and is an odd prime, thereby showing that Drisko’s method cannot be extended to encompass any of the three suggested cases. We also extend exact computation to , discuss asymptotics for , and propose a generalization of the Alon-Tarsi conjecture.
We establish a -Titchmarsh-Weyl theory for singular -Sturm-Liouville problems. We define -limit-point and -limit circle singularities, and we give sufficient conditions which guarantee that the singular point is in a limit-point case. The resolvent is constructed in terms of Green’s function of the problem. We derive the eigenfunction expansion in its series form. A detailed worked example involving Jackson -Bessel functions is given. This example leads to the completeness of a wide class of -cylindrical functions.
In this paper we give new lower bounds on the Hilbert-Kunz multiplicity of unmixed nonregular local rings, bounding them uniformly away from 1. Our results improve previous work of Aberbach and Enescu.