Partial data inverse problems for the Hodge Laplacian

We prove uniqueness results for a Calderon type inverse problem for the Hodge Laplacian acting on graded forms on certain manifolds in three dimensions. In particular, we show that partial measurements of the relative-to-absolute or absolute-to-relative boundary value maps uniquely determine a zeroth order potential. The method is based on Carleman estimates for the Hodge Laplacian with relative or absolute boundary conditions, and on the construction of complex geometric optics solutions which reduce the Calderon type problem to a tensor tomography problem for 2-tensors. The arguments in this paper allow to establish partial data results for elliptic systems that generalize the scalar results due to Kenig-Sjostrand-Uhlmann.Comment: 54 pages, updated versio

Slowly varying control parameters, delayed bifurcations and the stability of spikes in reaction-diffusion systems

We present three examples of delayed bifurcations for spike solutions of reaction-diffusion systems. The delay effect results as the system passes slowly from a stable to an unstable regime, and was previously analysed in the context of ODE's in [P.Mandel, T.Erneux, J.Stat.Phys, 1987]. It was found that the instability would not be fully realized until the system had entered well into the unstable regime. The bifurcation is said to have been "delayed" relative to the threshold value computed directly from a linear stability analysis. In contrast, we analyze the delay effect in systems of PDE's. In particular, for spike solutions of singularly perturbed generalized Gierer-Meinhardt (GM) and Gray-Scott (GS) models, we analyze three examples of delay resulting from slow passage into regimes of oscillatory and competition instability. In the first example, for the GM model on the infinite real line, we analyze the delay resulting from slowly tuning a control parameter through a Hopf bifurcation. In the second example, we consider a Hopf bifurcation on a finite one-dimensional domain. In this scenario, as opposed to the extrinsic tuning of a system parameter through a bifurcation value, we analyze the delay of a bifurcation triggered by slow intrinsic dynamics of the PDE system. In the third example, we consider competition instabilities of the GS model triggered by the extrinsic tuning of a feed rate parameter. In all cases, we find that the system must pass well into the unstable regime before the onset of instability is fully observed, indicating delay. We also find that delay has an important effect on the eventual dynamics of the system in the unstable regime. We give analytic predictions for the magnitude of the delays as obtained through analysis of certain explicitly solvable nonlocal eigenvalue problems. The theory is confirmed by numerical solutions of the full PDE systems.Comment: 31 pages, 20 figures, submitted to Physica D: Nonlinear Phenomen

Partial data inverse problems for Maxwell equations via Carleman estimates

In this article we consider an inverse boundary value problem for the time-harmonic Maxwell equations. We show that the electromagnetic material parameters are determined by boundary measurements where part of the boundary data is measured on a possibly very small set. This is an extension of earlier scalar results of Bukhgeim-Uhlmann and Kenig-Sjostrand-Uhlmann to the Maxwell system. The main contribution is to show that the Carleman estimate approach to scalar partial data inverse problems introduced in those works can be carried over to the Maxwell system. (C) 2017 Elsevier Masson SAS. All rights reserved.Peer reviewe

Structural Requirements for the Tissue-Specific and Tissue-General Functions of the Caenorhabditis elegans Epidermal Growth Factor LIN-3

Caenorhabditis elegans lin-3 encodes a homolog of the epidermal growth factor (EGF) family of growth factors. LIN-3 is the inductive signal for hermaphrodite vulval differentiation, and it is required for animal viability, hermaphrodite fertility, and the specification of anterior cell fates in the male B cell lineage. We describe the cloning of a lin-3 homolog from C. briggsae, sequence comparison of C. elegans lin-3 with C. briggsae lin-3, and the determination of molecular lesions in alleles of C. elegans lin-3, including three new alleles. We also analyzed the severity of phenotypes caused by the new and existing alleles of lin-3. Correlation of mutant phenotypes and their molecular lesions, as well as sequence comparison between two species, reveal that the EGF motif and the N-terminal portion of the cytoplasmic domain are important for the functions of LIN-3 in all tissues, while the C-terminal portion of the cytoplasmic domain is involved in the tissue-specific functions of lin-3. We discuss how the structure of lin-3 contributes to its functions in multiple developmental processes

Weakly Nonlinear Analysis of Vortex Formation in a Dissipative Variant of the Gross-Pitaevskii Equation

For a dissipative variant of the two-dimensional Gross-Pitaevskii equation with a parabolic trap under rotation, we study a symmetry breaking process that leads to the formation of vortices. The first symmetry breaking leads to the formation of many small vortices distributed uniformly near the Thomas-Fermi radius. The instability occurs as a result of a linear instability of a vortex-free steady state as the rotation is increased above a critical threshold. We focus on the second subsequent symmetry breaking, which occurs in the weakly nonlinear regime. At slightly above threshold, we derive a one dimensional amplitude equation that describes the slow evolution of the envelope of the initial instability. We show that the mechanism responsible for initiating vortex formation is a modulational instability of the amplitude equation. We also illustrate the role of dissipation in the symmetry breaking process. All analyses are confirmed by detailed numerical computations

On the stability of the exact solutions of the dual-phase lagging model of heat conduction

The dual-phase lagging (DPL) model has been considered as one of the most promising theoretical approaches to generalize the classical Fourier law for heat conduction involving short time and space scales. Its applicability, potential, equivalences, and possible drawbacks have been discussed in the current literature. In this study, the implications of solving the exact DPL model of heat conduction in a three-dimensional bounded domain solution are explored. Based on the principle of causality, it is shown that the temperature gradient must be always the cause and the heat flux must be the effect in the process of heat transfer under the dual-phase model. This fact establishes explicitly that the single- and DPL models with different physical origins are mathematically equivalent. In addition, taking into account the properties of the Lambert W function and by requiring that the temperature remains stable, in such a way that it does not go to infinity when the time increases, it is shown that the DPL model in its exact form cannot provide a general description of the heat conduction phenomena

IMECE2005-82282 CONTROL OF CYLINDRICAL SHELL PANELS WITH INPUT SHAPING AND PHASE SHIFT OF SHAPE MEMORY RING SEGMENTS

ABSTRACT The purpose of this study is to investigate the control effect from shape memory alloy (SMA) ring segments placed at the desired positions along the length of a cylindrical shell panel. Equations of motion for an elastic cylindrical shell panel are defined first and then used with the assumed mode shape functions for the appropriate boundary conditions in a free vibration analysis. The results from this are used with the generic shell sensing equation to determine the spatial strain distribution. From this, optimal placement of ring segments for each given mnth mode is determined. Through use of the modal expansion method, the open-loop control force induced by the SMA ring segments applied to a cylindrical shell panel is determined next. This evaluation shows that only the odd modes in the circumferential direction can be controlled. Longitudinal modes are controlled via placing a varying number, depending on the mode, of ring segments along the length of the cylindrical shell panel. To predict control effects of the SMA ring segments, the modal participation factor response is determined for an external harmonic excitation applied to the shell along with SMA control force induced to eliminate the unwanted effects. The results show that with proper choice of waveform function for the applied temperature to the SMA ring segments and minor modifications to frequency and phase, the SMA ring segments can control unwanted external vibration

A New Seismic-Geotechnical Strong Motion Approach

We have developed a new approach to estimate site-specific strong motion due to earthquakes on specific faults or source zones. It combines seismologic and geotechnical studies. It entails obtaining records of small earthquakes at the site, both at the surface and downhole in bedrock, as well as performing geotechnical dynamic site characterization. This new approach has the dual result of providing an optimized definition of the dynamic geotechnical site properties and providing calculated free-field, strong motion estimates. The procedure is demonstrated at the Painter Street Bridge site in Rio Dell, CA, for which we provide a range of surface motions corresponding to an earthquake of magnitude 7 on the subducting plate underlying this region. These calculated motions bracket the records of the Petrolia event (M = 7) measured near the site

Some analytical models of radiating collapsing spheres

We present some analytical solutions to the Einstein equations, describing radiating collapsing spheres in the diffusion approximation. Solutions allow for modeling physical reasonable situations. The temperature is calculated for each solution, using a hyperbolic transport equation, which permits to exhibit the influence of relaxational effects on the dynamics of the system.Comment: 17 pages Late