Maximum estimates for oblique derivative problems with right hand side in L p , p<n

If u is a solution of the second order elliptic differential equation Lu=f in some Lipschitz domain in ⁿ , we estimate the maximum of u in terms of some oblique derivative prescribed on the boundary of the domain and in terms of the L ^p norm of f with p<n.     

A novel crystallization-induced diastereomeric transformation based on a reversible carbon-sulfur bond formation. Application to the synthesis of a gamma-secretase inhibitor

This paper describes a remarkably efficient process for the preparation of gamma-secretase inhibitor 1. The target is synthesized in only five steps with an overall yield of 58%. The key operation is a highly selective and practical, crystallization-driven transformation for the conversion of a mixture of tertiary benzylic alcohols into the desired sulfide diastereomer with 94:6 dr. This unprecedented process is based upon a reversible carbon-sulfur bond formation under acidic conditions.     

Hessian‐based model reduction: large‐scale inversion and prediction

Hessian‐based model reduction was previously proposed as an approach in deriving reduced models for the solution of large‐scale linear inverse problems by targeting accuracy in observation outputs. A control‐theoretic view of Hessian‐based model reduction that hinges on the equality between the Hessian and the transient observability gramian of the underlying linear system is presented. The model reduction strategy is applied to a large‐scale ( degrees of freedom) three‐dimensional contaminant transport problem in an urban environment, an application that requires real‐time computation. In addition to the inversion accuracy, the ability of reduced models of varying dimension to make predictions of the contaminant evolution beyond the time horizon of observations is studied. Results indicate that the reduced models have a factor speedup in computing time for the same level of accuracy. Copyright © 2012 John Wiley & Sons, Ltd.     

Study of global solutions of parabolic equations via a priori estimates. Part I: Equations with principal elliptic part equal to the laplacian

We study global solutions of u_t=Δu + f(u, Du) with zero boundary data. Under suitable hypotheses on f, we show that an energy, which is roughly the H¹ norm of u, cannot change too rapidly. When f depends only on u, this estimate implies that all global solutions are bounded.     

Chaotic Expansion of Elements of the Universal Enveloping Algebra of a Lie Algebra Associated with a Quantum Stochastic Calculus

The functional Ito formula, in the form df() = f( + d ) –f(),is formulated and proved in the context of a Lie algebra L associatedwith a quantum (non-commutative) stochastic calculus. Here fis an element of the universal enveloping algebra U of L, andf() + d() – f() is given a meaning using the coproductstructure of U even though the individual terms of this expressionhave no meaning. The Ito formula is equivalent to a chaoticexpansion formula for f() which is found explicitly. 1991 MathematicsSubject Classification: primary 81S25; secondary 60H05; tertiary18B25.     

Time scale to ergodicity in the Fermi-Pasta-Ulam system

We study the approach to near-equipartition in the N-dimensional Fermi-Pasta-Ulam Hamiltonian with quartic (hard spring) nonlinearity. We investigate numerically the time evolution of orbits with initial energy in some few low-frequency linear modes. Our results indicate a transition where, above a critical energy which is independent of N, one can reach equipartition if one waits for a time proportional to N(2). Below this critical energy the time to equipartition is exponentially long. We develop a theory to determine the time evolution and the excitation of the nonlinear modes based on a resonant normal form treatment of the resonances among the oscillators. Our theory predicts the critical energy for equipartition, the time scale to equipartition, and the form of the nonlinear modes below equipartition, in qualitative agreement with the numerical results. (c) 1995 American Institute of Physics.     

Efficient delivery of Cre-recombinase to neurons in vivo and stable transduction of neurons using adeno-associated and lentiviral vectors

Background

As development proceeds the human embryo attains an ever more complex three dimensional (3D) structure. Analyzing the gene expression patterns that underlie these changes and interpreting their significance depends on identifying the anatomical structures to which they map and following these patterns in developing 3D structures over time. The difficulty of this task greatly increases as more gene expression patterns are added, particularly in organs with complex 3D structures such as the brain. Optical Projection Tomography (OPT) is a new technology which has been developed for rapidly generating digital 3D models of intact specimens. We have assessed the resolution of unstained neuronal structures within a Carnegie Stage (CS)17 OPT model and tested its use as a framework onto which anatomical structures can be defined and gene expression data mapped.

Results

Resolution of the OPT models was assessed by comparison of digital sections with physical sections stained, either with haematoxylin and eosin (H&E) or by immunocytochemistry for GAP43 or PAX6, to identify specific anatomical features. Despite the 3D models being of unstained tissue, peripheral nervous system structures from the trigeminal ganglion (~300 μm by ~150 μm) to the rootlets of cranial nerve XII (~20 μm in diameter) were clearly identifiable, as were structures in the developing neural tube such as the zona limitans intrathalamica (core is ~30 μm thick). Fourteen anatomical domains have been identified and visualised within the CS17 model. Two 3D gene expression domains, known to be defined by Pax6 expression in the mouse, were clearly visible when PAX6 data from 2D sections were mapped to the CS17 model. The feasibility of applying the OPT technology to all stages from CS12 to CS23, which encompasses the major period of organogenesis for the human developing central nervous system, was successfully demonstrated.

Conclusion

In the CS17 model considerable detail is visible within the developing nervous system at a minimum resolution of ~20 μm and 3D anatomical and gene expression domains can be defined and visualised successfully. The OPT models and accompanying technologies for manipulating them provide a powerful approach to visualising and analysing gene expression and morphology during early human brain development.