Computing with infinite networks

For neural networks with a wide class of weight-priors, it can be shown that in the limit of an infinite number of hidden units the prior over functions tends to a Gaussian process. In this paper analytic forms are derived for the covariance function of the Gaussian processes corresponding to networks with sigmoidal and Gaussian hidden units. This allows predictions to be made efficiently using networks with an infinite number of hidden units, and shows that, somewhat paradoxically, it may be easier to compute with infinite networks than finite ones.

Computation with infinite neural networks

For neural networks with a wide class of weight priors, it can be shown that in the limit of an infinite number of hidden units, the prior over functions tends to a gaussian process. In this article, analytic forms are derived for the covariance function of the gaussian processes corresponding to networks with sigmoidal and gaussian hidden units. This allows predictions to be made efficiently using networks with an infinite number of hidden units and shows, somewhat paradoxically, that it may be easier to carry out Bayesian prediction with infinite networks rather than finite ones.

An alternating boundary integral based method for a Cauchy problem for the Laplace equation in semi-infinite regions

We consider a Cauchy problem for the Laplace equation in a two-dimensional semi-infinite region with a bounded inclusion, i.e. the region is the intersection between a half-plane and the exterior of a bounded closed curve contained in the half-plane. The Cauchy data are given on the unbounded part of the boundary of the region and the aim is to construct the solution on the boundary of the inclusion. In 1989, Kozlov and Maz'ya [10] proposed an alternating iterative method for solving Cauchy problems for general strongly elliptic and formally self-adjoint systems in bounded domains. We extend their approach to our setting and in each iteration step mixed boundary value problems for the Laplace equation in the semi-infinite region are solved. Well-posedness of these mixed problems are investigated and convergence of the alternating procedure is examined. For the numerical implementation an efficient boundary integral equation method is proposed, based on the indirect variant of the boundary integral equation approach. The mixed problems are reduced to integral equations over the (bounded) boundary of the inclusion. Numerical examples are included showing the feasibility of the proposed method.

On an indirect integral equation approach for stationary heat transfer in semi-infinite layered domains in R3 with cavities

Regions containing internal boundaries such as composite materials arise in many applications.We consider a situation of a layered domain in IR3 containing a nite number of bounded cavities. The model is stationary heat transfer given by the Laplace equation with piecewise constant conductivity. The heat ux (a Neumann condition) is imposed on the bottom of the layered region and various boundary conditions are imposed on the cavities. The usual transmission (interface) conditions are satised at the interface layer, that is continuity of the solution and its normal derivative. To eciently calculate the stationary temperature eld in the semi-innite region, we employ a Green's matrix technique and reduce the problem to boundary integral equations (weakly singular) over the bounded surfaces of the cavities. For the numerical solution of these integral equations, we use Wienert's approach [20]. Assuming that each cavity is homeomorphic with the unit sphere, a fully discrete projection method with super-algebraic convergence order is proposed. A proof of an error estimate for the approximation is given as well. Numerical examples are presented that further highlights the eciency and accuracy of the proposed method.

An iterative method based on boundary integrals for elliptic Cauchy problems in semi-infinite domains

In this study, we investigate the problem of reconstruction of a stationary temperature field from given temperature and heat flux on a part of the boundary of a semi-infinite region containing an inclusion. This situation can be modelled as a Cauchy problem for the Laplace operator and it is an ill-posed problem in the sense of Hadamard. We propose and investigate a Landweber-Fridman type iterative method, which preserve the (stationary) heat operator, for the stable reconstruction of the temperature field on the boundary of the inclusion. In each iteration step, mixed boundary value problems for the Laplace operator are solved in the semi-infinite region. Well-posedness of these problems is investigated and convergence of the procedures is discussed. For the numerical implementation of these mixed problems an efficient boundary integral method is proposed which is based on the indirect variant of the boundary integral approach. Using this approach the mixed problems are reduced to integral equations over the (bounded) boundary of the inclusion. Numerical examples are included showing that stable and accurate reconstructions of the temperature field on the boundary of the inclusion can be obtained also in the case of noisy data. These results are compared with those obtained with the alternating iterative method.

Transition of planar Couette flow at infinite Reynolds numbers

An outline of the state space of planar Couette flow at high Reynolds numbers (Re<105) is investigated via a variety of efficient numerical techniques. It is verified from nonlinear analysis that the lower branch of the hairpin vortex state (HVS) asymptotically approaches the primary (laminar) state with increasing Re. It is also predicted that the lower branch of the HVS at high Re belongs to the stability boundary that initiates a transition to turbulence, and that one of the unstable manifolds of the lower branch of HVS lies on the boundary. These facts suggest HVS may provide a criterion to estimate a minimum perturbation arising transition to turbulent states at the infinite Re limit. © 2013 American Physical Society.

Synthesis and characterization of some Mn(II) and Mn(III) complexes of N,N′-o-phenylenebis(salicylideneimine)(LH2) and N,N′-o-phenylenebis(5-bromosalicylideneimine)(L′H2). Crystal structures of [Mn(L)(H2O)(ClO4)], [Mn(L)(NCS)] and an infinite linear chain of [Mn(L)(OAc)]

A series of manganese(II) [Mn(L)] and manganese(III) [Mn(L)(X)] (X = ClO4, OAc, NCS, N3, Cl, Br and I) complexes have been synthesized from Schiff base ligands N,N′-o- phenylenebis(salicylideneimine)(LH2) and N,N′-o-phenylenebis(5- bromosalicylideneimine)(L′H2) obtained by condensation of salicylaldehyde or 5-Br salicylaldehyde with o-phenylene-diamine. The complexes have been characterized by the combination of IR, UV-Vis spectroscopy, magnetic measurements and electrochemical studies. Three manganese(III) complexes 3 [Mn(L)(ClO4)(H2O)], 5 [Mn(L)(OAc)] and 13 [Mn(L)(NCS)] have been characterized by X-ray crystallography. The X-ray structures show that the manganese(III) is hexa-coordinated in 3, it is penta-coordinated in 13, while in 5 there is an infinite chain where the MnL moieties are connected by acetate ions acting as bridging bidentate ligand. The cyclic voltammograms of all the manganese(III) complexes exhibit two reversible/quasi-reversible/ irreversible responses assignable to Mn(III)/Mn(II) and Mn(IV)/Mn(III) couples. It was observed that the ligand L′H2 containing the 5-bromosal moiety always stabilizes the lower oxidation states compared to the corresponding unsubstituted LH2. Cyclic voltammograms of the manganese(II) complexes (1 and 2) exhibit a quasi-reversible Mn(III)/Mn(II) couple at E1/2 -0.08 V for 1 and 0.054 V for 2. © 2005 Elsevier B.V. All rights reserved.

Flt1 acts as a negative regulator of tip cell formation and branching morphogenesis in the zebrafish embryo

Endothelial tip cells guide angiogenic sprouts by exploring the local environment for guidance cues such as vascular endothelial growth factor (VegfA). Here we present Flt1 (Vegf receptor 1) loss- and gain-of-function data in zebrafish showing that Flt1 regulates tip cell formation and arterial branching morphogenesis. Zebrafish embryos expressed soluble Flt1 (sFlt1) and membrane-bound Flt1 (mFlt1). In Tg(flt1(BAC):yfp) × Tg(kdrl:ras-cherry)(s916) embryos, flt1:yfp was expressed in tip, stalk and base cells of segmental artery sprouts and overlapped with kdrl:cherry expression in these domains. flt1 morphants showed increased tip cell numbers, enhanced angiogenic behavior and hyperbranching of segmental artery sprouts. The additional arterial branches developed into functional vessels carrying blood flow. In support of a functional role for the extracellular VEGF-binding domain of Flt1, overexpression of sflt1 or mflt1 rescued aberrant branching in flt1 morphants, and overexpression of sflt1 or mflt1 in controls resulted in short arterial sprouts with reduced numbers of filopodia. flt1 morphants showed reduced expression of Notch receptors and of the Notch downstream target efnb2a, and ectopic expression of flt4 in arteries, consistent with loss of Notch signaling. Conditional overexpression of the notch1a intracellular cleaved domain in flt1 morphants restored segmental artery patterning. The developing nervous system of the trunk contributed to the distribution of Flt1, and the loss of flt1 affected neurons. Thus, Flt1 acts in a Notch-dependent manner as a negative regulator of tip cell differentiation and branching. Flt1 distribution may be fine-tuned, involving interactions with the developing nervous system.