The Hasse Norm Principle For Biquadratic Extensions

We give an asymptotic formula for the number of biquadratic extensions of the rationals of bounded discriminant that fail the Hasse norm principle.Comment: 19 pages. Proof of Theorem 1 improved/simplified. Accepted by Journal de Th\'eorie des Nombres de Bordeau

A Positive Proportion of Hasse Principle Failures in a Family of Ch\^atelet Surfaces

We investigate the family of surfaces defined by the affine equation $$Y^2 + Z^2 = (aT^2 + b)(cT^2 +d)$$ where $\vert ad-bc \vert=1$ and develop an asymptotic formula for the frequency of Hasse principle failures. We show that a positive proportion (roughly 23.7%) of such surfaces fail the Hasse principle, by building on previous work of la Bret\`{e}che and Browning.Comment: 13 pages, comments welcome. To appear in International Journal of Number Theor

Evidence that an RGD-dependent receptor mediates the binding of oligodendrocytes to a novel ligand in a glial-derived matrix.

A simple adhesion assay was used to measure the interaction between rat oligodendrocytes and various substrata, including a matrix secreted by glial cells. Oligodendrocytes bound to surfaces coated with fibronectin, vitronectin and a protein component of the glial matrix. The binding of cells to all of these substrates was inhibited by a synthetic peptide (GRGDSP) modeled after the cell-binding domain of fibronectin. The component of the glial matrix responsible for the oligodendrocyte interaction is a protein which is either secreted by the glial cells or removed from serum by products of these cultures; serum alone does not promote adhesion to the same extent as the glial-derived matrix. The interaction of cells with this glial-derived matrix requires divalent cations and is not mediated by several known RGD-containing extracellular proteins, including fibronectin, vitronectin, thrombospondin, type I and type IV collagen, and tenascin

Average Bateman--Horn for Kummer polynomials

For any $r \in \mathbb{N}$ and almost all $k \in \mathbb{N}$ smaller than $x^r$, we show that the polynomial $f(n) = n^r + k$ takes the expected number of prime values as $n$ ranges from 1 to $x$. As a consequence, we deduce statements concerning variants of the Hasse principle and of the integral Hasse principle for certain open varieties defined by equations of the form $N_{K/\mathbb{Q}}(\mathbf{z}) = t^r +k \neq 0$ where $K/\mathbb{Q}$ is a quadratic extension. A key ingredient in our proof is a new large sieve inequality for Dirichlet characters of exact order $r$.Comment: V2: Minor correction