Composing dinatural transformations: Towards a calculus of substitution

Dinatural transformations, which generalise the ubiquitous natural transformations to the case where the domain and codomain functors are of mixed variance, fail to compose in general; this has been known since they were discovered by Dubuc and Street in 1970. Many ad hoc solutions to this remarkable shortcoming have been found, but a general theory of compositionality was missing until Petri?, in 2003, introduced the concept of g-dinatural transformations, that is, dinatural transformations together with an appropriate graph: he showed how acyclicity of the composite graph of two arbitrary dinatural transformations is a sufficient and essentially necessary condition for the composite transformation to be in turn dinatural. Here we propose an alternative, semantic rather than syntactic, proof of Petri?'s theorem, which the authors independently rediscovered with no knowledge of its prior existence; we then use it to define a generalised functor category, whose objects are functors of mixed variance in many variables, and whose morphisms are transformations that happen to be dinatural only in some of their variables.We also define a notion of horizontal composition for dinatural transformations, extending the well-known version for natural transformations, and prove it is associative and unitary. Horizontal composition embodies substitution of functors into transformations and vice-versa, and is intuitively reflected from the string-diagram point of view by substitution of graphs into graphs.This work represents the first, fundamental steps towards a substitution calculus for dinatural transformations as sought originally by Kelly, with the intention then to apply it to describe coherence problems abstractly. There are still fundamental difficulties that are yet to be overcome in order to achieve such a calculus, and these will be the subject of future work; however, our contribution places us well in track on the path traced by Kelly towards a calculus of substitution for dinatural transformations.     

First total synthesis of A(2) isoprostane

A stereoselective Julia-Lythgoe olefination has allowed the first total synthesis of A(2) isoprostane (1), a recently discovered member of the growing isoprostane family. This elusive compound opens up numerous new avenues for the molecular biology of cyclopentenone prostaglandins, which are endowed of intriguing biological effects such as antitumor, antiinflammatory, and antiviral activities.     

Uniqueness of solutions for some nonlinear Dirichlet problems

We consider here a class of nonlinear Dirichlet problems, in a bounded domain , of the form

Determination of continuous variable entanglement by purity measurements

We classify the entanglement of two-mode Gaussian states according to their degree of total and partial mixedness. We derive exact bounds that determine maximally and minimally entangled states for fixed global and marginal purities. This characterization allows for an experimentally reliable estimate of continuous variable entanglement based on measurements of purity.     

Boosts in an Arbitrary Direction and Maximal Causal Velocities in a Deformed Minkowski Space

We discuss boosts in a deformed Minkowski space, i.e., a four-dimensional spacetime with metric coefficients depending on nonmetric coordinates (in particular on the energy). The general form of a boost in an arbitrary direction is derived in the case of space anisotropy. Two maximal trivector velocities are mathematically possible, an isotropic and an anisotropic one. However, only the anisotropic velocity has physical meaning, being invariant indeed under deformed boosts.     

Characterization of the first beta-class carbonic anhydrase from an arthropod (Drosophila melanogaster) and phylogenetic analysis of beta-class carbonic anhydrases in invertebrates

Background

The β-carbonic anhydrase (CA, EC 4.2.1.1) enzymes have been reported in a variety of organisms, but their existence in animals has been unclear. The purpose of the present study was to perform extensive sequence analysis to show that the β-CAs are present in invertebrates and to clone and characterize a member of this enzyme family from a representative model organism of the animal kingdom, e.g., Drosophila melanogaster.

Results

The novel β-CA gene, here named DmBCA, was identified from FlyBase, and its orthologs were searched and reconstructed from sequence databases, confirming the presence of β-CA sequences in 55 metazoan species. The corresponding recombinant enzyme was produced in Sf9 insect cells, purified, kinetically characterized, and its inhibition was investigated with a series of simple, inorganic anions. Holoenzyme molecular mass was defined by dynamic light scattering analysis and gel filtration, and the results suggested that the holoenzyme is a dimer. Double immunostaining confirmed predictions based on sequence analysis and localized DmBCA protein to mitochondria. The enzyme showed high CO₂ hydratase activity, with a k_cat of 9.5 × 10⁵ s^-1 and a k_cat/K_M of 1.1 × 10⁸ M^{- 1}s^{- 1}. DmBCA was appreciably inhibited by the clinically-used sulfonamide acetazolamide, with an inhibition constant of 49 nM. It was moderately inhibited by halides, pseudohalides, hydrogen sulfide, bisulfite and sulfate (K_I values of 0.67 - 1.36 mM) and more potently by sulfamide (K_I of 0.15 mM). Bicarbonate, nitrate, nitrite and phenylarsonic/boronic acids were much weaker inhibitors (K_Is of 26.9 - 43.7 mM).

Conclusions

The Drosophila β-CA represents a highly active mitochondrial enzyme that is a potential model enzyme for anti-parasitic drug development.     

X-Ray Crystal Structure of Phyllostin,a Metabolite Produced by <Emphasis Type="Italic">Phyllosticta cirsii</Emphasis>, a Potential Mycoherbicide of <Emphasis Type="Italic">Cirsium arvense</Emphasis>

Abstract  

The X-ray crystal structure of phyllostin (8-hydroxy-3-methyl-2-oxo-2,3,4a,5,8,8a-hexahydro-benzo[1,4]dioxine-6-carboxylic acid methyl ester) is reported. Phyllostin was recently and for the first time isolated, together with phytotoxin phyllostoxin and the phyllostictines A–D, from the culture filtrates of Phyllosticta cirsii, a fungal pathogen isolated from diseased Cirsium arvense leaves and evaluated as biocontrol agent of this noxious perennial weed. Phyllostin crystallizes in the monoclinic space group P2₁ with unit cell parameters: a = 8.371(2), b = 7.026(1), c = 9.918(1) ?, β = 92.013(8)°, Z = 2. The final refinement converged to R1 = 0.0530 for 1026 observed reflections having I > 2σ(I). In the molecule two trans joined six-membered rings are present. The cyclohexene ring adopts a half-chair conformation. The 2-oxo-1,4-dioxan ring adopts a distorted half-chair conformation. The relative stereochemistry of the four chiral centers turns out to be 3R,4aS,8R,8aS. In the crystal packing molecules form infinite chains through O–H···O intermolecular hydrogen bonds. Chains run along the crystallographic a axis forming layers of molecules stacking in the [0 1 0] direction.     

Freudenthal duality and generalized special geometry

Freudenthal duality, introduced in Borsten et al. (2009) [1] and defined as an anti-involution on the dyonic charge vector in d=4 space-time dimensions for those dualities admitting a quartic invariant, is proved to be a symmetry not only of the classical Bekenstein-Hawking entropy but also of the critical points of the black hole potential.Furthermore, Freudenthal duality is extended to any generalized special geometry, thus encompassing all N>2 supergravities, as well as N=2 generic special geometry, not necessarily having a coset space structure.     

Fractal regularity results on optimal irrigation patterns

In this paper the problem of the regularity, i.e. fractal behaviour, of the minima of the branched transport problem is addressed. We show that, under suitable conditions on the irrigated measure, the minima present a fractal regularity, that is on a given branch of length l the number of branches bifurcating from it whose length is comparable with ε   can be estimated both from above and below by l/ε$l/\epsilon$.     

Parabolic Capacity and Soft Measures for Nonlinear Equations

We first introduce, using a functional approach, the notion of capacity related to the parabolic p-Laplace operator. Then we prove a decomposition theorem for measures (in space and time) that do not charge the sets of null capacity. We apply this result to prove existence and uniqueness of renormalized solutions for nonlinear parabolic initial boundary-value problems with such measures as right-hand side.