Existence and Ulam type stability for nonlinear Riemann-Liouville fractional differential equations with constant delay

A nonlinear Riemann–Liouville fractional differential equation with constant delay is studied. Initially, some existence results are proved. Three Ulam type stability concepts are defined and studied. Several sufficient conditions are obtained. Some of the obtained results are illustrated on fractional biological models

Quantitative Photo-acoustic Tomography with Partial Data

Photo-acoustic tomography is a newly developed hybrid imaging modality that combines a high-resolution modality with a high-contrast modality. We analyze the reconstruction of diffusion and absorption parameters in an elliptic equation and improve an earlier result of Bal and Uhlmann to the partial date case. We show that the reconstruction can be uniquely determined by the knowledge of 4 internal data based on well-chosen partial boundary conditions. Stability of this reconstruction is ensured if a convexity condition is satisfied. Similar stability result is obtained without this geometric constraint if 4n well-chosen partial boundary conditions are available, where $n$ is the spatial dimension. The set of well-chosen boundary measurements is characterized by some complex geometric optics (CGO) solutions vanishing on a part of the boundary.Comment: arXiv admin note: text overlap with arXiv:0910.250

Thermoacoustic tomography arising in brain imaging

We study the mathematical model of thermoacoustic and photoacoustic tomography when the sound speed has a jump across a smooth surface. This models the change of the sound speed in the skull when trying to image the human brain. We derive an explicit inversion formula in the form of a convergent Neumann series under the assumptions that all singularities from the support of the source reach the boundary

Stability of gene regulatory networks modeled by generalized proportional caputo fractional differential equations

A model of gene regulatory networks with generalized proportional Caputo fractional derivatives is set up, and stability properties are studied. Initially, some properties of absolute value Lyapunov functions and quadratic Lyapunov functions are discussed, and also, their application to fractional order systems and the advantage of quadratic functions are pointed out. The equilibrium of the generalized proportional Caputo fractional model and its generalized exponential stability are defined, and sufficient conditions for the generalized exponential stability and asymptotic stability of the equilibrium are obtained. As a special case, the stability of the equilibrium of the Caputo fractional model is discussed. Several examples are provided to illustrate our theoretical results and the influence of the type of fractional derivative on the stability behavior of the equilibrium.publishe

Quadratic Lyapunov functions for stability of the generalized proportional fractional differential equations with applications to neural networks

A fractional model of the Hopfield neural network is considered in the case of the application of the generalized proportional Caputo fractional derivative. The stability analysis of this model is used to show the reliability of the processed information. An equilibrium is defined, which is generally not a constant (different than the case of ordinary derivatives and Caputo-type fractional derivatives). We define the exponential stability and the Mittag–Leffler stability of the equilibrium. For this, we extend the second method of Lyapunov in the fractional-order case and establish a useful inequality for the generalized proportional Caputo fractional derivative of the quadratic Lyapunov function. Several sufficient conditions are presented to guarantee these types of stability. Finally, two numerical examples are presented to illustrate the effectiveness of our theoretical results.publishe

Measuring urban social diversity using interconnected geo-social networks

Large metropolitan cities bring together diverse individuals, creating opportunities for cultural and intellectual exchanges, which can ultimately lead to social and economic enrichment. In this work, we present a novel network perspective on the interconnected nature of people and places, allowing us to capture the social diversity of urban locations through the social network and mobility patterns of their visitors. We use a dataset of approximately 37K users and 42K venues in London to build a network of Foursquare places and the parallel Twitter social network of visitors through check-ins. We define four metrics of the social diversity of places which relate to their social brokerage role, their entropy, the homogeneity of their visitors and the amount of serendipitous encounters they are able to induce. This allows us to distinguish between places that bring together strangers versus those which tend to bring together friends, as well as places that attract diverse individuals as opposed to those which attract regulars. We correlate these properties with wellbeing indicators for London neighbourhoods and discover signals of gentrification in deprived areas with high entropy and brokerage, where an influx of more affluent and diverse visitors points to an overall improvement of their rank according to the UK Index of Multiple Deprivation for the area over the five-year census period. Our analysis sheds light on the relationship between the prosperity of people and places, distinguishing between different categories and urban geographies of consequence to the development of urban policy and the next generation of socially-aware location-based applications.This work was supported by the Project LASAGNE, Contract No. 318132 (STREP), funded by the European Commission and EPSRC through Grant GALE (EP/K019392).This is the author accepted manuscript. The final version is available from the Association for Computing Machinery via http://dx.doi.org/10.1145/2872427.288306

New and Current Microbiological Tools for Ecosystem Ecologists: Towards a Goal of Linking Structure and Function

Interest in the relationships between soil microbial communities and ecosystem functions is growing with increasing recognition of the key roles microorganisms play in a variety of ecosystems. With a wealth of microbial methods now available, selecting the most appropriate method can be daunting, especially to those new to the field of microbial ecology. In this review, we highlight those methods currently used and most applicable to ecological studies, including assays to study various aspects of the carbon and nitrogen cycles (e.g., pool dilution, acetylene reduction, enzyme analyses, among others), methods to assess microbial community composition (e.g., phospholipid fatty acid analysis (PLFA), denaturing gradient gel electrophoresis (DGGE), terminal restriction fragment length polymorphism analysis (TRFLP), quantitative polymerase chain reaction (qPCR)) and methods to directly link community structure to function (e.g., stable isotope probing (SIP)). In our discussion of these methods, we describe the information each method provides, as well as some of their strengths and weaknesses. Using a case study, we illustrate how these methods can be applied to investigate relationships between microbial communities and the processes they perform in wetland ecosystems. We end our discussion with a series of questions to consider prior to designing experiments, in the hope that these questions will help guide ecologists in selecting the most appropriate method(s) for their research

Thermoacoustic tomography with detectors on an open curve: an efficient reconstruction algorithm

Practical applications of thermoacoustic tomography require numerical inversion of the spherical mean Radon transform with the centers of integration spheres occupying an open surface. Solution of this problem is needed (both in 2-D and 3-D) because frequently the region of interest cannot be completely surrounded by the detectors, as it happens, for example, in breast imaging. We present an efficient numerical algorithm for solving this problem in 2-D (similar methods are applicable in the 3-D case). Our method is based on the numerical approximation of plane waves by certain single layer potentials related to the acquisition geometry. After the densities of these potentials have been precomputed, each subsequent image reconstruction has the complexity of the regular filtration backprojection algorithm for the classical Radon transform. The peformance of the method is demonstrated in several numerical examples: one can see that the algorithm produces very accurate reconstructions if the data are accurate and sufficiently well sampled, on the other hand, it is sufficiently stable with respect to noise in the data

Stability of delay Hopfield neural networks with generalized proportional Riemann-Liouville fractional derivative

The general delay Hopfield neural network is studied. It is considered the case of time-varying delay, continuously distributed delays, time varying coefficients and a special type of a Riemann-Liouville fractional derivative (GPRLFD) with an exponential kernel. The presence of delays and GPRLFD in the model require two special types of initial conditions. The applied GPRLFD also required a special definition of the equilibrium of the model. A constant equilibrium of the model is defined. We use Razumikhin method and Lyapunov functions to study stability properties of the equilibrium of the model. We apply Lyapunov functions defined by absolute values as well as quadratic Lyapunov functions. We prove some comparison results for Lyapunov function connected deeply with the applied GPRLFD and use them to obtain exponential bounds of the solutions. These bounds are satisfied for intervals excluding the initial time. Also, the convergence of any solution of the model to the equilibrium at infinity is proved. An example illustrating the importance of our theoretical results is also included