The functions operating on multiplier algebras

Let 1 < p < ∞ with p ≠ 2. Let G denote one of the groups $\text{T}$ⁿ, $\text{R}$ⁿ, or $\text{Z}$ⁿ. We show that only entire functions operate in certain algebras of multipliers on L_p(G).     

Dynamo theory,vorticity generation,and exponential stretching

A discussion is given of the analogy between the dynamo equation for the generation of a magnetic field by the motion of an electrically conducting fluid and the equation for the evolution of vorticity of a viscous fluid. In both cases exponential stretching is an important feature of the underlying instability problem. For the "fast" dynamo problem, the existence of exponential stretching (i.e., the positivity of the Lyapunov exponent) somewhere in the flow is a necessary condition when the flow is smooth. An example is presented of a flow with exponential stretching (an Anosov flow) that supports fast dynamo action. A parallel treatment is described for the linearized Navier-Stokes equations for the motion of a viscous fluid. In this problem the analogous necessary condition for "fast vorticity generation" is the existence of some instability in the corresponding Euler (i.e., inviscid) equation. Dynamo theory methods give a second related result, namely a universal geometric estimate from below on the growth rate of a small perturbation in an inviscid fluid. This bound gives an effective sufficient condition for local instability for Eulers equations. In particular, it is proved that a steady flow with a hyperbolic stagnation point is unstable. The growth rate of an infinitesimal perturbation in a metric with derivatives depends on this metric. This dependence is completely described.     

Synthesis and characterization of chromium and molybdenum(0) complexes bearing the (pentafluoroethyl)diphenylphosphine (pfepp) ligand

The synthesis and characterization of two Group VI pentacarbonyl complexes (pfepp)M(CO)₅ (M = Cr 1, Mo 2; pfepp = PPh₂C₂F₅) are reported. Thermolysis of M(CO)₆ and pfepp in refluxing octane afforded 1 and 2 in moderate yields. These complexes were completely characterized by multinuclear NMR, IR and elemental analysis. X-ray structures for these complexes indicated they were isostructural, crystalizing in triclinic unit cells with four molecules per asymmetric unit. A comparison of the bond lengths in 1 and 2 to other (L)M(CO)₅ complexes showed a relationship between the M-C_ax bond length and the electronic influence of the phosphine ligand, and establishes the pfepp ligand as neither electron-rich nor electron-poor. A comparison of IR data with other (L)M(CO)₅ complexes also indicates the pfepp ligand is electronically neutral, with an electronic influence that approximates phosphites.     

A tensor-based dictionary learning approach to tomographic image reconstruction

We consider tomographic reconstruction using priors in the form of a dictionary learned from training images. The reconstruction has two stages: first we construct a tensor dictionary prior from our training data, and then we pose the reconstruction problem in terms of recovering the expansion coefficients in that dictionary. Our approach differs from past approaches in that (a) we use a third-order tensor representation for our images and (b) we recast the reconstruction problem using the tensor formulation. The dictionary learning problem is presented as a non-negative tensor factorization problem with sparsity constraints. The reconstruction problem is formulated in a convex optimization framework by looking for a solution with a sparse representation in the tensor dictionary. Numerical results show that our tensor formulation leads to very sparse representations of both the training images and the reconstructions due to the ability of representing repeated features compactly in the dictionary.     

A parameter-choice method that exploits residual information

Most algorithms for choosing the regularization parameter in a discrete ill-posed problem are based on the norm of the residual vector. In this work we propose a different approach that seeks to use all the information available in the residual vector, and we show how to use statistical tools and fast Fourier transforms to extract this information efficiently. This approach leads to a computationally inexpensive parameter-choice rule based on the normalized cumulative periodogram, which is particularly suited for large-scale problems. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Class of Baker–Akhiezer Arrangements

We study a class of arrangements of lines with multiplicities on the plane which admit the Chalykh–Veselov Baker–Akhiezer function. These arrangements are obtained by adding multiplicity one lines in an invariant way to any dihedral arrangement with invariant multiplicities. We describe all the Baker–Akhiezer arrangements when at most one line has multiplicity higher than 1. We study associated algebras of quasi-invariants which are isomorphic to the commutative algebras of quantum integrals for the generalized Calogero–Moser operators. We compute the Hilbert series of these algebras and we conclude that the algebras are Gorenstein. We also show that there are no other arrangements with Gorenstein algebras of quasi-invariants when at most one line has multiplicity bigger than 1.     

Complex geometry of moment-angle manifolds

Moment-angle manifolds provide a wide class of examples of non-Kähler compact complex manifolds. A complex moment-angle manifold \(\mathcal {Z}\) is constructed via certain combinatorial data, called a complete simplicial fan. In the case of rational fans, the manifold \(\mathcal {Z}\) is the total space of a holomorphic bundle over a toric variety with fibres compact complex tori. In general, a complex moment-angle manifold \(\mathcal {Z}\) is equipped with a canonical holomorphic foliation \({\mathcal {F}}\) which is equivariant with respect to the \(({\mathbb {C}}^\times )^m\)-action. Examples of moment-angle manifolds include Hopf manifolds of Vaisman type, Calabi–Eckmann manifolds, and their deformations. We construct transversely Kähler metrics on moment-angle manifolds, under some restriction on the combinatorial data. We prove that any Kähler submanifold (or, more generally, a Fujiki class \(\mathcal {C}\) subvariety) in such a moment-angle manifold is contained in a leaf of the foliation \({\mathcal {F}}\). For a generic moment-angle manifold \(\mathcal {Z}\) in its combinatorial class, we prove that all subvarieties are moment-angle manifolds of smaller dimension and there are only finitely many of them. This implies, in particular, that the algebraic dimension of \(\mathcal {Z}\) is zero.