Data Snapshot: “Trump Towns” Swung Democratic in New Hampshire Midterms

New Hampshire municipalities with fewer college-educated residents, which generally offered strong support for Donald Trump two years ago, swung toward the opposing party in the 2018 midterms

Cohomology of the Hilbert scheme of points on a surface with values in representations of tautological bundles

Let $X$ a smooth quasi-projective algebraic surface, $L$ a line bundle on $X$. Let $X^{[n]}$ the Hilbert scheme of $n$ points on $X$ and $L^{[n]}$ the tautological bundle on $X^{[n]}$ naturally associated to the line bundle $L$ on $X$. We explicitely compute the image \bkrh(L^{[n]}) of the tautological bundle $L^{[n]}$ for the Bridgeland-King-Reid equivalence \bkrh : \B{D}^b(X^{[n]}) \ra \B{D}^b_{\perm_n}(X^n) in terms of a complex \comp{\mc{C}}_L of \perm_n-equivariant sheaves in \B{D}^b_{\perm_n}(X^n). We give, moreover, a characterization of the image \bkrh(L^{[n]} \tens ... \tens L^{[n]}) in terms of of the hyperderived spectral sequence $E^{p,q}_1$ associated to the derived $k$-fold tensor power of the complex \comp{\mc{C}}_L. The study of the \perm_n-invariants of this spectral sequence allows to get the derived direct images of the double tensor power and of the general $k$-fold exterior power of the tautological bundle for the Hilbert-Chow morphism, providing Danila-Brion-type formulas in these two cases. This yields easily the computation of the cohomology of $X^{[n]}$ with values in L^{[n]} \tens L^{[n]} and $\Lambda^k L^{[n]}$.Comment: 41 pages; revised version, exposition improve

Computing minimal free resolutions of right modules over noncommutative algebras

In this paper we propose a general method for computing a minimal free right resolution of a finitely presented graded right module over a finitely presented graded noncommutative algebra. In particular, if such module is the base field of the algebra then one obtains its graded homology. The approach is based on the possibility to obtain the resolution via the computation of syzygies for modules over commutative algebras. The method behaves algorithmically if one bounds the degree of the required elements in the resolution. Of course, this implies a complete computation when the resolution is a finite one. Finally, for a monomial right module over a monomial algebra we provide a bound for the degrees of the non-zero Betti numbers of any single homological degree in terms of the maximal degree of the monomial relations of the module and the algebra.Comment: 23 pages, to appear in Journal of Algebr

Changes in New Hampshire’s republican party: evolving footprint in presidential politics, 1960-2008

This brief describes a series of dramatic changes in New Hampshire\u27s political landscape over the past four decades. Examining presidential elections from 1960 to 2008, author Dante Scala uncovers a series of significant shifts in New Hampshire\u27s political geography at the county level. He reports that historically Republican counties Grafton and Merrimack have both tilted Democratic consistently in recent decades and that New Hampshire has become less Republican overall. All of these changes have impacted not just general elections in New Hampshire, but the Republican presidential primary as well

Monomial right ideals and the Hilbert series of noncommutative modules

In this paper we present a procedure for computing the rational sum of the Hilbert series of a finitely generated monomial right module $N$ over the free associative algebra $K\langle x_1,\ldots,x_n \rangle$. We show that such procedure terminates, that is, the rational sum exists, when all the cyclic submodules decomposing $N$ are annihilated by monomial right ideals whose monomials define regular formal languages. The method is based on the iterative application of the colon right ideal operation to monomial ideals which are given by an eventual infinite basis. By using automata theory, we prove that the number of these iterations is a minimal one. In fact, we have experimented efficient computations with an implementation of the procedure in Maple which is the first general one for noncommutative Hilbert series.Comment: 15 pages, to appear in Journal of Symbolic Computatio

Minimal Immersions of Kahler manifolds into Euclidean Spaces

It is proved here that a minimal isometric immersion of a Kähler-Einstein or homogeneous Kähler-manifold into an Euclidean space must be totally geodesic