Degree of Approximation to Functions in a Normed Space

We compute the degree of approximation of functions, in the H$ouml;lder metric,using lower triangular nonnegative matrices. Ourwork generalizes that of Singh [5], and other authors, since we do notassume any monotonicity condition on the rows of the matrix.     

Bitableaux and Zero Sets of Dual Canonical Basis Elements

We state new results concerning the zero sets of polynomials belonging to the dual canonical basis of mathbbC[x₁, 1, . . . , x_n, n]{mathbb{C}[x_1, 1, . . . , x_n, n]} . As an application, we show that this basis is related by a unitriangular transition matrix to the simpler bitableau basis popularized by Désarménien-Kung-Rota. It follows that spaces spanned by certain subsets of the dual canonical basis can be characterized in terms of products of matrix minors, or in terms of their common zero sets.     

α-Synuclein induces both positive mean curvature and negative Gaussian curvature in membranes

Using a combination of X-ray scattering, fluorescence correlation spectroscopy, coarse-grained molecular dynamics (MD) simulations and potential of mean force calculations, we have explored the membrane remodeling effects of monomeric α-synuclein (αS). Our initial findings from multiple approaches are that αS (1) causes a significant thinning of the bilayer and (2) stabilizes positive mean curvature, such that the maximum principle curvature matches that of synaptic vesicles, αS-induced tubules, and the synthetic lipid vesicles to which the protein binds most tightly. This suggests that αS binding to synaptic vesicles likely stabilizes their intrinsic curvature. We then show that αS induces local negative Gaussian curvature, an effect that occurs in regions of αS shown previously via NMR and corroborated by MD simulation to have significant conformational flexibility. The induction of negative Gaussian curvature, which has implications for all curvature-sensing and curvature-generating amphipathic α-helices, supports a hypothesis that connects helix insertion to fusion and fission of vesicles, processes that have recently been linked to αS function. Then, in an effort to explain these biophysical properties of αS, we promote an intrinsic curvature-field model that recasts long-range protein-protein interactions in terms of the interactions between the local curvature fields generated by lipid-protein complexes.     

Necessary conditions for absolute double matrix summability factors

We obtain necessary conditions for a doubly triangular matrix A to have the property that a double series ΣΣ λ _mn b _mn is summable |A| _k whenever the series ΣΣb _mn is bounded |A| _k .     

Promotion and cyclic sieving via webs

We show that Schützenberger’s promotion on two and three row rectangular Young tableaux can be realized as cyclic rotation of certain planar graphs introduced by Kuperberg. Moreover, following work of the third author, we show that this action admits the cyclic sieving phenomenon.     

Parking structures: Fuss analogs

For any irreducible real reflection group W with Coxeter number h, Armstrong, Reiner, and the author introduced a pair of \(W \times {\mathbb {Z}}_{h}\) -modules which deserve to be called W-parking spaces which generalize the type A notion of parking functions and conjectured a relationship between them. In this paper we give a Fuss analog of their constructions. For a Fuss parameter k≥1, we define a pair of \(W \times {\mathbb {Z}}_{kh}\) -modules which deserve to be called k-W-parking spaces and conjecture a relationship between them. We prove the weakest version of our conjectures for each of the infinite families ABCDI of finite reflection groups, together with proofs of stronger versions in special cases. Whenever our weakest conjecture holds for W, we have the following corollaries.

• There is a simple formula for the character of either k-W-parking space.
• We recover a cyclic sieving result due to Krattenthaler and Müller which gives the cycle structure of a generalized rotation action on k-W-noncrossing partitions.
• When W is crystallographic, the restriction of either k-W-parking space to W is isomorphic to the action of W on the finite torus Q/(kh+1)Q, where Q is the root lattice.

     

Differential operators for harmonic weak Maass forms and the vanishing of Hecke eigenvalues

For integers k ≥ 2, we study two differential operators on harmonic weak Maass forms of weight 2 ? k. The operator ξ_2-k (resp. D ^k-1) defines a map to the space of weight k cusp forms (resp. weakly holomorphic modular forms). We leverage these operators to study coefficients of harmonic weak Maass forms. Although generic harmonic weak Maass forms are expected to have transcendental coefficients, we show that those forms which are “dual” under ξ_2-k to newforms with vanishing Hecke eigenvalues (such as CM forms) have algebraic coefficients. Using regularized inner products, we also characterize the image of D ^k-1.