Finiteness results for Abelian tree models

Equivariant tree models are statistical models used in the reconstruction of phylogenetic trees from genetic data. Here equivariant refers to a symmetry group imposed on the root distribution and on the transition matrices in the model. We prove that if that symmetry group is Abelian, then the Zariski closures of these models are defined by polynomial equations of bounded degree, independent of the tree. Moreover, we show that there exists a polynomial-time membership test for that Zariski closure. This generalises earlier results on tensors of bounded rank, which correspond to the case where the group is trivial, and implies a qualitative variant of a quantitative conjecture by Sturmfels and Sullivant in the case where the group and the alphabet coincide. Our proofs exploit the symmetries of an infinite-dimensional projective limit of Abelian star models.Comment: 27 pages. arXiv admin note: substantial text overlap with arXiv:1103.533

Regular maps of high density

A regular map is a surface together with an embedded graph, having properties similar to those of the surface and graph of a platonic solid. We analyze regular maps with reflection symmetry and a graph of density strictly exceeding 1/2, and we conclude that all regular maps of this type belong to a family of maps naturally defined on the Fermat curves x^n+y^n+z^n=0, excepting the one corresponding to the tetrahedron.Comment: 13 pages, 4 figure

Uniform error bounds for smoothing splines

Almost sure bounds are established on the uniform error of smoothing spline estimators in nonparametric regression with random designs. Some results of Einmahl and Mason (2005) are used to derive uniform error bounds for the approximation of the spline smoother by an ``equivalent'' reproducing kernel regression estimator, as well as for proving uniform error bounds on the reproducing kernel regression estimator itself, uniformly in the smoothing parameter over a wide range. This admits data-driven choices of the smoothing parameter.Comment: Published at http://dx.doi.org/10.1214/074921706000000879 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

Topological noetherianity for cubic polynomials

Let $P_3(\mathbf{C}^{\infty})$ be the space of complex cubic polynomials in infinitely many variables. We show that this space is $\mathbf{GL}_{\infty}$-noetherian, meaning that any $\mathbf{GL}_{\infty}$-stable Zariski closed subset is cut out by finitely many orbits of equations. Our method relies on a careful analysis of an invariant of cubics introduced here called q-rank. This result is motivated by recent work in representation stability, especially the theory of twisted commutative algebras. It is also connected to certain stability problems in commutative algebra, such as Stillman's conjecture.Comment: 13 page

Nictaba homologs from Arabidopsis thaliana are involved in plant stress responses

Plants are constantly exposed to a wide range of environmental stresses, but evolved complicated adaptive and defense mechanisms which allow them to survive in unfavorable conditions. These mechanisms protect and defend plants by using different immune receptors located either at the cell surface or in the cytoplasmic compartment. Lectins or carbohydrate-binding proteins are widespread in the plant kingdom and constitute an important part of these immune receptors. In the past years, lectin research has focused on the stress-inducible lectins. The Nicotiana tabacum agglutinin, abbreviated as Nictaba, served as a model for one family of stress-related lectins. Here we focus on three non-chimeric Nictaba homologs from Arabidopsis thaliana, referred to as AN3, AN4, and AN5. Confocal microscopy of ArathNictaba enhanced green fluorescent protein (EGFP) fusion constructs transiently expressed in N. benthamiana or stably expressed in A. thaliana yielded fluorescence for AN4 and AN5 in the nucleus and the cytoplasm of the plant cell, while fluorescence for AN3 was only detected in the cytoplasm. RT-qPCR analysis revealed low expression for all three ArathNictabas in different tissues throughout plant development. Stress application altered the expression levels, but all three ArathNictabas showed a different expression pattern. Pseudomonas syringae infection experiments with AN4 and AN5 overexpression lines demonstrated a significantly higher tolerance of several transgenic lines to P. syringae compared to wild type plants. Finally, AN4 was shown to interact with two enzymes involved in plant defense, namely TGG1 and BGLU23. Taken together, our data suggest that the ArathNictabas represent stress-regulated proteins with a possible role in plant stress responses. On the long term this research can contribute to the development of more stress-resistant plants

Polynomials and tensors of bounded strength

Notions of rank abound in the literature on tensor decomposition. We prove that strength, recently introduced for homogeneous polynomials by Ananyan-Hochster in their proof of Stillman's conjecture and generalised here to other tensors, is universal among these ranks in the following sense: any non-trivial Zariski-closed condition on tensors that is functorial in the underlying vector space implies bounded strength. This generalises a theorem by Derksen-Eggermont-Snowden on cubic polynomials, as well as a theorem by Kazhdan-Ziegler which says that a polynomial all of whose directional derivatives have bounded strength must itself have bounded strength.Comment: Improved the bounds on strength as a function of the dimension of the space where one first sees nontrivial equations for the tensor property

Noetherianity for infinite-dimensional toric varieties

We consider a large class of monomial maps respecting an action of the infinite symmetric group, and prove that the toric ideals arising as their kernels are finitely generated up to symmetry. Our class includes many important examples where Noetherianity was recently proved or conjectured. In particular, our results imply Hillar-Sullivant's Independent Set Theorem and settle several finiteness conjectures due to Aschenbrenner, Martin del Campo, Hillar, and Sullivant. We introduce a matching monoid and show that its monoid ring is Noetherian up to symmetry. Our approach is then to factorize a more general equivariant monomial map into two parts going through this monoid. The kernels of both parts are finitely generated up to symmetry: recent work by Yamaguchi-Ogawa-Takemura on the (generalized) Birkhoff model provides an explicit degree bound for the kernel of the first part, while for the second part the finiteness follows from the Noetherianity of the matching monoid ring.Comment: 20 page