We de ne a complex R=J of graded modules on a d-dimensional simplicial complex , so that the top homology module of this complex consists of piecewise polynomial functions (splines) of smoothness r on the cone of . In a series of papers ([4], [5], [6]), Billera and Rose used a similar approach to study the dimension of the spaces of splines on , but with a complex substantially di erent fromR=J...

Let G be a pro nite group. We de ne an S[[G]]-module to be a G-spectrum X that satis es certain conditions, and, given an S[[G]]-module X, we de ne the homotopy orbit spectrum XhG. When G is countably based and X satis es a certain niteness condition, we construct a homotopy orbit spectral sequence whose E2-term is the continuous homology of G with coefcients in the graded pro nite b Z[[G]]-mod...

The mitogen-activated protein kinase (MAPK) pathway is an evolutionarily conserved signaling module that controls important cell fate decisions in a variety of physiological contexts. During Xenopus oocyte maturation, the MAPK cascade converts an increasing progesterone stimulus into a switch-like, all-or-nothing response. While the importance of such switch-like behavior is widely discussed in...

Mitogen-activated protein kinase (MAPK) cascades are evolutionarily conserved signaling pathways that regulate cell fate decisions. They generate a wide range of signal outputs, including graded and digital responses. In T cells, MAPK activation is digital in response to T-cell-receptor stimulation; however, whether other receptors on T cells that lead to MAPK activation are graded or digital i...

Definition 1. Let S be a graded ring, set X = ProjS and letM a graded S-module. We define a sheaf of modulesM ̃ on X as follows. For each p ∈ ProjS we have the local ring S(p) and the S(p)module M(p) (GRM,Definition 4). Let Γ(U,M ̃) be the set of all functions s : U −→ ∐p∈U M(p) with s(p) ∈M(p) for each p, which are locally fractions. That is, for every p ∈ U there is an open neighborhood p ∈ V ⊆...

Let A and A! be dual Koszul algebras. By Positselski a filtered algebra U with grU = A is Koszul dual to a differential graded algebra (A!, d). We relate the module categories of this dual pair by a⊗−Hom adjunction. This descends to give an equivalence of suitable quotient categories and generalizes work of Beilinson, Ginzburg, and Soergel.

We construct Quillen equivalences on the Quillen model categories of rings, modules and algebras over Z-graded chain complexes and HZ-module spectra. A Quillen equivalence of Quillen model categories is the most highly structured notion of equivalence between homotopy theories. We use these equivalences in turn to produce algebraic models for rational stable model categories.

A notion of curvature is introduced in multivariable operator theory and an analogue of the Gauss-Bonnet-Chern theorem is established. Applications are given to the metric structure of graded ideals in C[z1, . . . , zd], and the existence of “inner” sequences for closed submodules of the free Hilbert module H(C).

We construct a family of exact functors from the BGG category O of representations of the Lie algebra sln(C) to the category of finite-dimensional representations of the degenerate (or graded) affine Hecke algebra Hl of GLl. These functors transform Verma modules to standard modules or zero, and simple modules to simple modules or zero. Any simple Hl-module can be thus obtained.