Isomorphisms of lattices of Bures-closed bimodules over Cartan MASAs

For i = 1; 2, let (Mi;Di) be pairs consisting of a Cartan MASA Di in a von Neumann algebra Mi, let atom(Di) be the set of atoms of Di, and let Si be the lattice of Bures-closed Di bimodules in Mi. We show that when Mi have separable preduals, there is a lattice isomorphism between S1 and S2 if and only if the sets f(Q1;Q2) 2 atom(Di) atom(Di) : Q1MiQ2 6= (0)g have the same cardinality. In particular, when Di is nonatomic, Si is isomorphic to the lattice of projections in L1([0; 1];m) where m is Lebesgue measure, regardless of the isomorphism classes of M1 and M2

Mi-1-mediated resistance to Meloidogyne incognita in tomato may not rely on ethylene but hormone perception through ETR3 participates in limiting nematode infection in a susceptible host.

Root-knot nematodes, Meloidogyne spp., are important pests of tomato (Solanum lycopersicum) and resistance to the three most prevalent species of this genus, including Meloidogyne incognita, is mediated by the Mi-1 gene. Mi-1 encodes a nucleotide binding (NB) leucine-rich repeat (LRR) resistance (R) protein. Ethylene (ET) is required for the resistance mediated by a subset of NB-LRR proteins and its role in Mi-1-mediated nematode resistance has not been characterized. Infection of tomato roots with M. incognita differentially induces ET biosynthetic genes in both compatible and incompatible interactions. Analyzing the expression of members of the ET biosynthetic gene families ACC synthase (ACS) and ACC oxidase (ACO), in both compatible and incompatible interactions, shows differences in amplitude and temporal expression of both ACS and ACO genes in these two interactions. Since ET can promote both resistance and susceptibility against microbial pathogens in tomato, we investigated the role of ET in Mi-1-mediated resistance to M. incognita using both genetic and pharmacological approaches. Impairing ET biosynthesis or perception using virus-induced gene silencing (VIGS), the ET-insensitive Never ripe (Nr) mutant, or 1-methylcyclopropene (MCP) treatment, did not attenuate Mi-1-mediated resistance to M. incognita. However, Nr plants compromised in ET perception showed enhanced susceptibility to M. incognita indicating a role for ETR3 in basal resistance to root-knot nematodes

Cortez\u27s Christological anthropology in historical perspective (Book Review)

Cortez, M. (2016). Christological anthropology in historical perspective. Grand Rapids, MI: Zondervan. 264 pp. $ 27.99. ISBN 978031051642

Horton\u27s Core Christianity: Finding yourself in God’s story (Book Review)

Horton, M. (2015). Core Christianity: Finding yourself in God’s story. Grand Rapids, MI: Zondervan. 171 pp. $14.99. ISBN 97803105256

An Approximation Problem in Multiplicatively Invariant Spaces

Let $\mathcal{H}$ be Hilbert space and $(\Omega,\mu)$ a $\sigma$-finite measure space. Multiplicatively invariant (MI) spaces are closed subspaces of $L^2(\Omega, \mathcal{H})$ that are invariant under point-wise multiplication by functions in a fix subset of $L^{\infty}(\Omega).$ Given a finite set of data $\mathcal{F}\subseteq L^2(\Omega, \mathcal{H}),$ in this paper we prove the existence and construct an MI space $M$ that best fits $\mathcal{F}$, in the least squares sense. MI spaces are related to shift invariant (SI) spaces via a fiberization map, which allows us to solve an approximation problem for SI spaces in the context of locally compact abelian groups. On the other hand, we introduce the notion of decomposable MI spaces (MI spaces that can be decomposed into an orthogonal sum of MI subspaces) and solve the approximation problem for the class of these spaces. Since SI spaces having extra invariance are in one-to-one relation to decomposable MI spaces, we also solve our approximation problem for this class of SI spaces. Finally we prove that translation invariant spaces are in correspondence with totally decomposable MI spaces.Comment: 18 pages, To appear in Contemporary Mathematic

Bird\u27s What Christians ought to believe: An introduction to Christian doctrine through the Apostle’s Creed (Book Review)

Bird, M. F. (2016). What Christians ought to believe: An introduction to Christian doctrine through the Apostle’s Creed. Grand Rapids, MI: Zondervan. $24.99. ISBN 9780310520924

Wendell and Miller\u27s Torah ethics and early Christian identity (Book Review)

A review of Wendel, S. J., & Miller, D. M. (Eds.). (2016). Torah ethics and early Christian identity. Grand Rapids, MI: Eerdmans. 271 pp, $35.00. ISBN 978080287319