Riesz transforms on a solvable Lie group of polynomial growth

We study Riesz transforms associated with a sublaplacian H on a solvable Lie group G, where G has polynomial volume growth. It is known that the standard second order Riesz transforms corresponding to H are generally unbounded in L^p(G). In this paper, we establish boundedness in L^p for modified second order Riesz transforms, which are defined using derivatives on a nilpotent group G_N associated with G. Our method utilizes a new algebraic approach which associates a distinguished choice of Cartan subalgebra with the sublaplacian H. We also obtain estimates for higher derivatives of the heat kernel of H, and give a new proof (without the use of homogenization theory) of the boundedness of first order Riesz transforms. Our results can be generalized to an arbitrary (possibly non-solvable) Lie group of polynomial growth.     

Coal Handleability — A good or a bad coal?

The avalanching behaviour of two coal types was determined, one of good and the other of poor handleability characteristics. This revealed significant differences in the nature of flow of the coals. Analysis of the strange attractors of the weight series of avalanches enabled quanitification of this difference, and the establishment of practical, relative criteria for the measurement of coal handleability. These criteria will now enable a structured study of the variables that affect handleability, for example proportion of fines and moisture content, and industrial consumers of coal to specify minimum acceptable limits for its transport characteristics.     

Testing set proportionality and the Ádám isomorphism of circulant graphs

Given two k element subsets , we give a quasi-linear algorithm to either find such that S=λT or prove that no such λ exists.This question is closely related to isomorphism testing of circulant graphs and has recently been studied in the literature.     

Comments on ultra-high-energy photons and D-foam models

We revisit the constraints that the non-observation of ultra-high-energy photons due to the GZK cutoff can impose on models of Lorentz violation in photon propagation, following recent work by Maccione, Liberati and Sigl (2010) [arXiv:1003.5468] that carries further an earlier analysis by the present authors [J. Ellis et al., Phys. Rev. D 63 (2001) 12402, hep-th/0012216]. We argue that the GZK cutoff constraint is naturally evaded in the D-brane model of space–time foam presented recently by the present authors [J. Ellis et al., Phys. Lett. B 665 (2008) 412, arXiv:0804.3566], in which Lorentz-violating effects on photon propagation are independent of possible effects during interactions. We also note a novel absorption mechanism that could provide a GZK-like cutoff for photons in low-scale string models.     

Gaiotto’s Lagrangian Subvarieties via Derived Symplectic Geometry

Let Bun_G be the moduli space of G-bundles on a smooth complex projective curve. Motivated by a study of boundary conditions in mirror symmetry, Gaiotto (2016) associated to any symplectic representation of G a Lagrangian subvariety of T^?Bun_G. We give a simple interpretation of (a generalization of) Gaiotto’s construction in terms of derived symplectic geometry. This allows to consider a more general setting where symplectic G-representations are replaced by arbitrary symplectic manifolds equipped with a Hamiltonian G-action and with an action of the multiplicative group that rescales the symplectic form with positive weight.     

Second-Order Optimality and Beyond: Characterization and Evaluation Complexity in Convexly Constrained Nonlinear Optimization

High-order optimality conditions for convexly constrained nonlinear optimization problems are analysed. A corresponding (expensive) measure of criticality for arbitrary order is proposed and extended to define high-order \(\epsilon \)-approximate critical points. This new measure is then used within a conceptual trust-region algorithm to show that if derivatives of the objective function up to order \(q \ge 1\) can be evaluated and are Lipschitz continuous, then this algorithm applied to the convexly constrained problem needs at most \(O(\epsilon ^{-(q+1)})\) evaluations of f and its derivatives to compute an \(\epsilon \)-approximate qth-order critical point. This provides the first evaluation complexity result for critical points of arbitrary order in nonlinear optimization. An example is discussed, showing that the obtained evaluation complexity bounds are essentially sharp.     

Reversible maps and composites of involutions in groups of piecewise linear homeomorphisms of the real line

An element of a group is reversible if it is conjugate to its own inverse, and it is strongly reversible if it is conjugate to its inverse by an involution. A group element is strongly reversible if and only if it can be expressed as a composite of two involutions. In this paper the reversible maps, the strongly reversible maps, and those maps that can be expressed as a composite of involutions are determined in certain groups of piecewise linear homeomorphisms of the real line.     

Total synthesis of the marine natural product (-)-clavosolide A

[Structure: see text] The total synthesis of the marine metabolite clavosolide A is reported which confirms the structure and absolute configuration of the natural product as the symmetrical diolide glycosylated by permethylated D-xylose moieties, 2.     

Template-directed synthesis of mechanically interlocked molecular bundles using dynamic covalent chemistry

Mixing the dipyrido[24]crown-8 derivatives carrying one or two formyl group(s) on the 4 position(s) of their pyridine ring(s) with a 3-fold symmetrical trisammonium ion template in a 3:1 ratio in CD3NO2 results in the formation of thermodynamically stable [4]pseudorotaxanes which, upon addition of a 1,3,5-trisaminobenzene cap, form mechanically interlocked molecular bundles with one and two caps, respectively, by virtue of dynamic imine bond formation.