Pentecostal hermeneutical reconsideration of the longer ending of Mark 16:9–20

Many scholars accept that Mark 16:9–20 is a late addition to the gospel of Mark based on the testimony of the manuscript tradition and internal evidence. Within early Pentecostalism, Mark 16:9–20 influenced pentecostal practice and proclamation to an inordinate extent, with ‘these signs shall follow’ (v. 17) serving at the same time as a wake-up call to worldwide mission and a litmus test for the authenticity of early pentecostal experience. Most early Pentecostals used Mark 16:9–20 without giving any consideration to its originality; however, some reacted to the scholarly debate about the longer ending by discussing its relevance in terms of its canonical inclusion and value. The article discusses these canonical considerations to answer the question: If it is accepted that the passage was not part of the original manuscript, what are the implications of it being used extensively throughout the history of the church as a part of the canon, and specifically in terms of its value and prevalent use in pentecostal practice?Intradisciplinary and/or interdisciplinary implications: This article is intradisciplinary by touching issues concerning New Testament studies, hermeneutics and church history. Mark 16:9–20 is by scholarly consensus seen as a late addition to the gospel; however, Pentecostal churches have been and still are influenced by the text. If it is viewed as canonical, it calls for another way of thinking about Scripture.

Uniform-to-proper duality of geometric properties of Banach spaces and their ultrapowers

In this note various geometric properties of a Banach space $\mathrm{X} $ are characterized by means of weaker corresponding geometric properties involving an ultrapower $\mathrm{X} ^\mathcal {U}$. The characterizations do not depend on the particular choice of the free ultrafilter $\mathcal {U}$ on $\mathbb{N}$. For example, a point $x\in \mathbf{S} _\mathrm{X} $ is an MLUR point if and only if $\jmath (x)$ (given by the canonical inclusion $\jmath \colon \mathrm{X} \to \mathrm{X} ^\mathcal {U}$) in $\mathbb{B} _{\mathrm{X} ^\mathcal {U}}$ is an extreme point; a point $x\in \mathbf{S} _\mathrm{X} $ is LUR if and only if $\jmath (x)$ is not contained in any non-degenerate line segment of $\mathbf{S} _{\mathrm{X} ^\mathcal {U}}$; a Banach space $\mathrm{X} $ is URED if and only if there are no $x, y \in \mathbf{S} _{\mathrm{X} ^\mathcal {U}}$, $x \neq y$, with $x-y \in \jmath (\mathrm{X} )$.

Simplicial cohomology of augmentation ideals in l1(G)

Let G be a discrete group.We give a decomposition theorem for the Hochschild cohomology of l1(G) with coefficients in certain G-modules. Using this we show that if G is commutative-transitive, the canonical inclusion of bounded cohomology of G into simplicial cohomology of l1(G) is an isomorphism.

On traced monoidal closed categories

The structure theorem of Joyal, Street and Verity says that every traced monoidal category arises as a monoidal full subcategory of the tortile monoidal category Int. In this paper we focus on a simple observation that a traced monoidal category is closed if and only if the canonical inclusion from into Int has a right adjoint. Thus, every traced monoidal closed category arises as a monoidal co-reflexive full subcategory of a tortile monoidal category. From this, we derive a series of facts for traced models of linear logic, and some for models of fixed-point computation. To make the paper more self-contained, we also include various background results for traced monoidal categories.

Chern Characters of Fourier Modules

Let Aθ denote the rotation algebra—the universal C*-algebra generated by unitaries U, V satisfying VU = e2πiθUV, where θ is a fixed real number. Let σ denote the Fourier automorphism of Aθ defined by U ↦ V, V ↦ U-1, and let denote the associated C*-crossed product. It is shown that there is a canonical inclusion for each θ given by nine canonical modules. The unbounded trace functionals of Bθ (yielding the Chern characters here) are calculated to obtain the cyclic cohomology group of order zero HC0(Bθ) when θ is irrational. The Chern characters of the nine modules—and more importantly, the Fourier module—are computed and shown to involve techniques from the theory of Jacobi’s theta functions. Also derived are explicit equations connecting unbounded traces across strongMorita equivalence, which turn out to be non-commutative extensions of certain theta function equations. These results provide the basis for showing that for a dense Gδ set of values of θ one has and is generated by the nine classes constructed here.