We perform an opportunistic search for cubic surfaces over small fields of characteristic two. The starting point our work is a list complied by Dickson the field with two elements. consider nonsingular ones arising in Dickson' s larger orders investigate properties such as number lines, singularities and automorphism groups. problem determining possible numbers lines surface $\mathbb{C}, \math...

In Communication theory and Coding, it is expected that certain circulant matrices having k ones + 1 zeros in the first row are nonsingular. We prove such always nonsingular when 2 either a power of prime, or product two distinct primes. For any other integer we construct determinant 0. The smallest singular matrix appears = 45 . possibility for to be rather low, smaller than 10 − 4 this case.

It is believed that strong ferromagnetic orders in some solids are generated by subtle interplay between quantum many-body effects and spin-independent Coulomb interactions between electrons. Here we describe our rigorous and constructive approach to ferromagnetism in the Hubbard model, which is a standard idealized model for strongly interacting electrons in a solid. We present, for the first ...

Let X be a singular real rational surface obtained from a smooth real rational surface by performing weighted blow-ups. Denote by Aut(X) the group of algebraic automorphisms of X into itself. Let n be a natural integer and let e = [e1, . . . , el] be a partition of n. Denote by Xe the set of l-tuples (P1, . . . , Pl) of distinct nonsingular curvilinear infinitely near points of X of orders (e1,...

Our central objects of study are nonsingular curves, which were classically studied via linear series. Limit linear series are a generalization of linear series on nonsingular curves to (certain classes of) reducible curves. Their purpose is to examine how linear series degenerate on a family of smooth curves degenerating to a reducible curve. This degeneration technique can then be used to pro...