On the Connection Between 2d Topological Gravity and the Reduced Hermitian Matrix Model

We discuss how concepts such as geodesic length and the volume of space-time can appear in 2d topological gravity. We then construct a detailed mapping between the reduced Hermitian matrix model and 2d topological gravity at genus zero. This leads to a complete solution of the counting problem for planar graphs with vertices of even coordination number. The connection between multi-critical matrix models and multi-critical topological gravity at genus zero is studied in some detail.Comment: 29 pages, LaTe

Stochastic integration in UMD Banach spaces

In this paper we construct a theory of stochastic integration of processes with values in $\mathcal{L}(H,E)$, where $H$ is a separable Hilbert space and $E$ is a UMD Banach space (i.e., a space in which martingale differences are unconditional). The integrator is an $H$-cylindrical Brownian motion. Our approach is based on a two-sided $L^p$-decoupling inequality for UMD spaces due to Garling, which is combined with the theory of stochastic integration of $\mathcal{L}(H,E)$-valued functions introduced recently by two of the authors. We obtain various characterizations of the stochastic integral and prove versions of the It\^{o} isometry, the Burkholder--Davis--Gundy inequalities, and the representation theorem for Brownian martingales.Comment: Published at http://dx.doi.org/10.1214/009117906000001006 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Stochastic evolution equations in UMD Banach spaces

We discuss existence, uniqueness, and space-time H\"older regularity for solutions of the parabolic stochastic evolution equation dU(t) = (AU(t) + F(t,U(t))) dt + B(t,U(t)) dW_H(t), t\in [0,\Tend], U(0) = u_0, where $A$ generates an analytic $C_0$-semigroup on a UMD Banach space $E$ and $W_H$ is a cylindrical Brownian motion with values in a Hilbert space $H$. We prove that if the mappings $F:[0,T]\times E\to E$ and $B:[0,T]\times E\to \mathscr{L}(H,E)$ satisfy suitable Lipschitz conditions and $u_0$ is \F_0-measurable and bounded, then this problem has a unique mild solution, which has trajectories in C^\l([0,T];\D((-A)^\theta) provided $\lambda\ge 0$ and $\theta\ge 0$ satisfy \l+\theta<\frac12. Various extensions of this result are given and the results are applied to parabolic stochastic partial differential equations.Comment: Accepted for publication in Journal of Functional Analysi

Ito's formula in UMD Banach spaces and regularity of solutions of the Zakai equation

Using the theory of stochastic integration for processes with values in a UMD Banach space developed recently by the authors, an Ito formula is proved which is applied to prove the existence of strong solutions for a class of stochastic evolution equations in UMD Banach spaces. The abstract results are applied to prove regularity in space and time of the solutions of the Zakai equation.Comment: Accepted for publication in Journal of Differential Equation

Barrier-controlled carrier transport in microcrystalline semiconducting materials: Description within a unified model

A recently developed model that unifies the ballistic and diffusive transport mechanisms is applied in a theoretical study of carrier transport across potential barriers at grain boundaries in microcrystalline semiconducting materials. In the unified model, the conductance depends on the detailed structure of the band edge profile and in a nonlinear way on the carrier mean free path. Equilibrium band edge profiles are calculated within the trapping model for samples made up of a linear chain of identical grains. Quantum corrections allowing for tunneling are included in the calculation of electron mobilities. The dependence of the mobilities on carrier mean free path, grain length, number of grains, and temperature is examined, and appreciable departures from the results of the thermionic-field-emission model are found. Specifically, the unified model is applied in an analysis of Hall mobility data for n-type microcrystalline Si thin films in the range of thermally activated transport. Owing mainly to the effect of tunneling, potential barrier heights derived from the data are substantially larger than the activation energies of the Hall mobilities. The specific features of the unified model, however, cannot be resolved within the rather large uncertainties of the analysis.Comment: REVTex, 19 pages, 9 figures; to appear in J. Appl. Phy

Primary radiotherapy in progressive optic nerve sheath meningiomas: a long-term follow-up study

Background/aims: To report the outcome of primary radiotherapy in patients with progressive optic nerve sheath meningioma (ONSM). Methods: The clinical records of all patients were reviewed in a retrospective, observational, multicentre study. Results: Thirty-four consecutive patients were included. Twenty-six women and eight men received conventional or stereotactic fractionated radiotherapy, and were followed for a median 58 (range 51–156) months. Fourteen eyes (41%) showed improved visual acuity of at least two lines on the Snellen chart. In 17 (50%) eyes, the vision stabilised, while deterioration was noted in three eyes (9%). The visual outcome was not associated with age at the time of radiotherapy (p=0.83), sex (p=0.43), visual acuity at the time of presentation (p=0.22) or type of radiotherapy (p=0.35). Optic disc swelling was associated with improved visual acuity (p<0.01) and 4/11 patients with optic atrophy also showed improvement. Long-term complications were dry eyes in five patients, cataracts in three, and mild radiation retinopathy in four. Conclusion: Primary radiotherapy for patients with ONSM is associated with long-term improvement of visual acuity and few adverse effects.Peerooz Saeed, Leo Blank, Dinesh Selva, John G. Wolbers, Peter J.C.M. Nowak, Ronald B. Geskus, Ezekiel Weis, Maarten P. Mourits, Jack Rootma