Extension of explicit formulas in Poissonian white noise analysis using harmonic analysis on configuration spaces

Harmonic analysis on configuration spaces is used in order to extend explicit expressions for the images of creation, annihilation, and second quantization operators in L2-spaces with respect to Poisson point processes to a set of functions larger than the space obtained by directly using chaos expansion. This permits, in particular, to derive an explicit expression for the generator of the second quantization of a sub-Markovian contraction semigroup on a set of functions which forms a core of the generator.

Time weighted portfolio optimisation

In estimating the inputs into the Modern Portfolio Theory (MPT) portfolio optimisation problem, it is usual to use equal weighted historic data. Equal weighting of the data, however, does not take account of the current state of the market. Consequently this approach is unlikely to perform well in any subsequent period as the data is still reflecting market conditions that are no longer valid. The need for some return-weighting scheme that gives greater weight to the most recent data would seem desirable. Therefore, this study uses returns data which are weighted to give greater weight to the most recent observations to see if such a weighting scheme can offer improved ex-ante performance over that based on un-weighted data.

Producing leisure spaces out of thin air: the case of the Countryside Stewardship Scheme

This paper examines the growing dysfunction between the apparently increasing significance of diverse leisure practices in the countryside and the largely unchanging official response towards them. Although there is recognition in the recent rural White Paper (DOE and MAFF, 1995) that access is essential to enjoying the countryside, the construction of this term is dubious, since paid access agreements, based on producer requirements, are favoured over any form of demand-driven freedom to roam. Using the Countryside Stewardship Scheme (CSS) as an example of the incentive structure developed to promote this policy, the paper applies Plato's simulacrum as a reading of how this process is being utilised to underpin the dominant rights associated with rural property interests. In particular, the paper makes the point that rather than representing the corollary of a market situation, as its supporters claim, the CSS involves government grant for the eclectic provision of short term licences over ground which remains unmapped as anything other than its continued agricultural use. In concluding, the paper asserts that rather than representing an increase in the availability of leisure sites in the countryside, the CSS and other schemes represent a diversion from the wider and deeper socio-cultural process of continued wealth and power redistribution.

Vekua theory for the Helmholtz operator

Vekua operators map harmonic functions defined on domain in \mathbb R2R2 to solutions of elliptic partial differential equations on the same domain and vice versa. In this paper, following the original work of I. Vekua (Ilja Vekua (1907–1977), Soviet-Georgian mathematician), we define Vekua operators in the case of the Helmholtz equation in a completely explicit fashion, in any space dimension N ≥ 2. We prove (i) that they actually transform harmonic functions and Helmholtz solutions into each other; (ii) that they are inverse to each other; and (iii) that they are continuous in any Sobolev norm in star-shaped Lipschitz domains. Finally, we define and compute the generalized harmonic polynomials as the Vekua transforms of harmonic polynomials. These results are instrumental in proving approximation estimates for solutions of the Helmholtz equation in spaces of circular, spherical, and plane waves.

On the operator space structure of Hilbert spaces

Operator spaces of Hilbertian JC∗ -triples E are considered in the light of the universal ternary ring of operators (TRO) introduced in recent work. For these operator spaces, it is shown that their triple envelope (in the sense of Hamana) is the TRO they generate, that a complete isometry between any two of them is always the restriction of a TRO isomorphism and that distinct operator space structures on a fixed E are never completely isometric. In the infinite-dimensional cases, operator space structure is shown to be characterized by severe and definite restrictions upon finite-dimensional subspaces. Injective envelopes are explicitly computed.

New results and open problems on Toeplitz operators in Bergman spaces

We discuss some of the recent progress in the field of Toeplitz operators acting on Bergman spaces of the unit disk, formulate some new results, and describe a list of open problems -- concerning boundedness, compactness and Fredholm properties -- which was presented at the conference "Recent Advances in Function Related Operator Theory'' in Puerto Rico in March 2010.

Toeplitz operators on Bergman spaces with locally integrable symbols

We study the boundedness of Toeplitz operators $T_a$ with locally integrable symbols on Bergman spaces $A^p(\mathbb{D})$, $1 < p < \infty$. Our main result gives a sufficient condition for the boundedness of $T_a$ in terms of some ``averages'' (related to hyperbolic rectangles) of its symbol. If the averages satisfy an ${o}$-type condition on the boundary of $\mathbb{D}$, we show that the corresponding Toeplitz operator is compact on $A^p$. Both conditions coincide with the known necessary conditions in the case of nonnegative symbols and $p=2$. We also show that Toeplitz operators with symbols of vanishing mean oscillation are Fredholm on $A^p$ provided that the averages are bounded away from zero, and derive an index formula for these operators.

Universally reversible JC*-triples and operator spaces

Abstract. We prove that the vast majority of JC∗-triples satisfy the condition of universal reversibility. Our characterisation is that a JC∗-triple is universally reversible if and only if it has no triple homomorphisms onto Hilbert spaces of dimension greater than two nor onto spin factors of dimension greater than four. We establish corresponding characterisations in the cases of JW∗-triples and of TROs (regarded as JC∗-triples). We show that the distinct natural operator space structures on a universally reversible JC∗-triple E are in bijective correspondence with a distinguished class of ideals in its universal TRO, identify the Shilov boundaries of these operator spaces and prove that E has a unique natural operator space structure precisely when E contains no ideal isometric to a nonabelian TRO. We deduce some decomposition and completely contractive properties of triple homomorphisms on TROs.

Rigid body trajectories in different 6D spaces

The objective of this paper is to show that the group SE(3) with an imposed Lie-Poisson structure can be used to determine the trajectory in a spatial frame of a rigid body in Euclidean space. Identical results for the trajectory are obtained in spherical and hyperbolic space by scaling the linear displacements appropriately since the influence of the moments of inertia on the trajectories tends to zero as the scaling factor increases. The semidirect product of the linear and rotational motions gives the trajectory from a body frame perspective. It is shown that this cannot be used to determine the trajectory in the spatial frame. The body frame trajectory is thus independent of the velocity coupling. In addition, it is shown that the analysis can be greatly simplified by aligning the axes of the spatial frame with the axis of symmetry which is unchanging for a natural system with no forces and rotation about an axis of symmetry.