The sign of a Latin square is $-1$ if it has an odd number of rows and columns that are odd permutations; otherwise, it is $+1$. Let $L_n^{\mathrm{E}}$ and $L_n^{\mathrm{O}}$ be, respectively, the number of Latin squares of order $n$ with sign $+1$ and $-1$. The Alon-Tarsi conjecture asserts that $L_n^{\mathrm{E}}\ne L_n^{\mathrm{O}}$ when $n$ is even. Drisko showed that $L_{p+1}^{\mathrm{E}}\not\equiv L_{p+1}^{\mathrm{O}}(mod{p}^3)$ for prime $p\ge 3$ and asked if similar congruences hold for orders of the form $p^k+1$, $p+3$, or $pq+1$. In this article we show that if $t\le n$, then $L_{n+1}^{\mathrm{E}}\not\equiv L_{n+1}^{\mathrm{O}}(mod{t}^3)$ only if $t=n$ and $n$ 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 $n\le 9$, discuss asymptotics for $L^{\mathrm{O}}/L^{\mathrm{E}}$, and propose a generalization of the Alon-Tarsi conjecture.

We establish here several “invariance of plurigenera type” theorems for twisted pluricanonical forms and metrics of adjoint $\mathbb{R}$-bundles.

We establish a $q$-Titchmarsh-Weyl theory for singular $q$-Sturm-Liouville problems. We define $q$-limit-point and $q$-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 $q$-Bessel functions is given. This example leads to the completeness of a wide class of $q$-cylindrical functions.

It is proved that the pseudodifferential operators ${\sigma }_t(X,D)$ belong to the Schatten $p$-class $C_p$, $0<p\le 2$, if the symbol $\sigma (x,\omega )$ is in certain modulation spaces on ${\mathbf{R}}_x^d\times {\mathbf{R}}_{\omega }^d$.

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.