I have developed "tennis elbow" from lugging this book around the past four weeks, but it is worth the pain, the effort, and the aspirin. It is also worth the (relatively speaking) bargain price. Including appendixes, this book contains 894 pages of text. The entire panorama of the neural sciences is surveyed and examined, and it is comprehensive in its scope, from genomes to social behaviors. The editors explicitly state that the book is designed as "an introductory text for students of biology, behavior, and medicine," but it is hard to imagine any audience, interested in any fragment of neuroscience at any level of sophistication, that would not enjoy this book. The editors have done a masterful job of weaving together the biologic, the behavioral, and the clinical sciences into a single tapestry in which everyone from the molecular biologist to the practicing psychiatrist can find and appreciate his or

Principles of Neural Science

This is lecture notes for several courses on Functional Analysis at School of Mathematics of University of Leeds. They are based on the notes of Dr. Matt Daws, Prof. Jonathan R. Partington and Dr. David Salinger used in the previous years. Some sections are borrowed from the textbooks, which I used since being a student myself. However all misprints, omissions, and errors are only my responsibility. I am very grateful to Filipa Soares de Almeida, Eric Borgnet, Pasc Gavruta for pointing out some of them. Please let me know if you find more. The notes are available also for download in PDF. The suggested textbooks are [1,6,8,9]. The other nice books with many interesting problems are [3, 7]. Exercises with stars are not a part of mandatory material but are nevertheless worth to hear about. And they are not necessarily difficult, try to solve them! CONTENTS List of Figures 3 Notations and Assumptions 4 Integrability conditions 4 1. Motivating Example: Fourier Series 4 1.1. Fourier series: basic notions 4 1.2. The vibrating string 8 1.3. Historic: Joseph Fourier 10 2. Basics of Linear Spaces 11 2.1. Banach spaces (basic definitions only) 12 2.2. Hilbert spaces 14 2.3. Subspaces 16 2.4. Linear spans 19 3. Orthogonality 20 3.1. Orthogonal System in Hilbert Space 21 3.2. Bessel’s inequality 23 3.3. The Riesz–Fischer theorem 25 3.4. Construction of Orthonormal Sequences 26 3.5. Orthogonal complements 28 4. Fourier Analysis 29 Date: 16th October 2017. 1

Introduction to Functional Analysis. By Angus E. Taylor. Pp. xvi. 423. $12.50. 1958. (John Wiley and Sons)

In this paper the theory of two-scale convergence (introduced by G. Nguetseng \ref[SIAM J. Math. Anal. 20 (1989), no. 3, 608--623; and developed later by G. Allaire \ref[SIAM J. Math. Anal. 23 (199 ...

Two-scale convergence

In this paper we study a variational problem in the space of functions of bounded Hessian. Our model constitutes a straightforward higher-order extension of the well known ROF functional (total variation minimisation) to which we add a non-smooth second order regulariser. It combines convex functions of the total variation and the total variation of the first derivatives. In what follows, we prove existence and uniqueness of minimisers of the combined model and present the numerical solution of the corresponding discretised problem by employing the split Bregman method. The paper is furnished with applications of our model to image denoising, deblurring as well as image inpainting. The obtained numerical results are compared with results obtained from total generalised variation (TGV), infimal convolution and Euler's elastica, three other state of the art higher-order models. The numerical discussion confirms that the proposed higher-order model competes with models of its kind in avoiding the creation of undesirable artifacts and blocky-like structures in the reconstructed images--a known disadvantage of the ROF model--while being simple and efficiently numerically solvable.

A Combined First and Second Order Variational Approach for Image Reconstruction

Studies of nanosized forms of bismuth (Bi)-containing materials have recently expanded from optical, chemical, electronic, and engineering fields towards biomedicine, as a result of their safety, cost-effective fabrication processes, large surface area, high stability, and high versatility in terms of shape, size, and porosity. Bi, as a nontoxic and inexpensive diamagnetic heavy metal, has been used for the fabrication of various nanoparticles (NPs) with unique structural, physicochemical, and compositional features to combine various properties, such as a favourably high X-ray attenuation coefficient and near-infrared (NIR) absorbance, excellent light-to-heat conversion efficiency, and a long circulation half-life. These features have rendered bismuth-containing nanoparticles (BiNPs) with desirable performance for combined cancer therapy, photothermal and radiation therapy (RT), multimodal imaging, theranostics, drug delivery, biosensing, and tissue engineering. Bismuth oxyhalides (BiOx, where X is Cl, Br or I) and bismuth chalcogenides, including bismuth oxide, bismuth sulfide, bismuth selenide, and bismuth telluride, have been heavily investigated for therapeutic purposes. The pharmacokinetics of these BiNPs can be easily improved via the facile modification of their surfaces with biocompatible polymers and proteins, resulting in enhanced colloidal stability, extended blood circulation, and reduced toxicity. Desirable antibacterial effects, bone regeneration potential, and tumor growth suppression under NIR laser radiation are the main biomedical research areas involving BiNPs that have opened up a new paradigm for their future clinical translation. This review emphasizes the synthesis and state-of-the-art progress related to the biomedical applications of BiNPs with different structures, sizes, and compositions. Furthermore, a comprehensive discussion focusing on challenges and future opportunities is presented.

/pdf/the-versatile-biomedical-applications-of-bismuth-based-4weqqirlrh.pdf

The versatile biomedical applications of bismuth-based nanoparticles and composites: therapeutic, diagnostic, biosensing, and regenerative properties

This paper focus on the properties of boundary layers in periodic homogenization in rectangular domains which are either fixed or have an oscillating boundary. Such boundary layers are highly oscillating near the boundary and decay exponentially fast in the interior to a non-zero limit that we call boundary layer tail. The influence of these boundary layer tails on interior error estimates is emphasized. They mainly have two effects (at first order with respect to the period e): first, they add a dispersive term to the homogenized equation, and second, they yield an effective Fourier boundary condition.

/pdf/boundary-layer-tails-in-periodic-homogenization-3l60xzv2v5.pdf

Boundary layer tails in periodic homogenization

/pdf/a-brief-survey-on-homogenization-with-a-physical-application-lf324krq06.pdf

A brief survey on homogenization with a physical application

Given a function u : Ω ⊆ _ ℝ n → ℝ , we introduce a notion of total variation of u depending on a possibly discontinuous Finsler metric. We prove some integral representation results for this total variation, and we study the connections with the theory of relaxation.

/pdf/a-notion-of-total-variation-depending-on-a-metric-with-17n5ts6g7b.pdf

A notion of total variation depending on a metric with discontinuous coefficients

We study a problem set in a finely mixed periodic medium, modelling electrical conduction in biological tissues The unknown electric potential solves standard elliptic equations set in different conductive regions (the intracellular and extracellular spaces), separated by a dielectric surface (the cell membranes), which exhibits both a capacitive and a nonlinear conductive behaviour Accordingly, dynamical conditions prevail on the membranes, so that the dependence of the solution on the time variable t is not only of parametric character As the spatial period of the medium goes to zero, the electric potential approaches in a suitable sense a homogenization limit u0, which keeps the prescribed boundary data, and solves the equation  This is an elliptic equation containing a term depending on the history of the gradient of u0; the matrices B0, A1 in it depend on the microstructure of the medium More exactly, we have that, in the limit, the current is still divergence-free, but it depends on the history of the potential gradient, so that memory effects explicitly appear The limiting equation also contains a term ℱ keeping trace of the initial data

Evolution and memory effects in the homogenization limit for electrical conduction in biological tissues

In this paper we derive a hierarchy of models for electrical conduction in a biological tissue, which is represented by a periodic array of period e of conducting phases surrounded by dielectric shells of thickness epsilon eta included in a conductive matrix. Such a hierarchy will be obtained from the Maxwell equations by means of a concentration process eta-->0, followed by a homogenization limit with respect to epsilon. These models are then compared with regard to their physical meaning and mathematical issues.

/pdf/on-a-hierarchy-of-models-for-electrical-conduction-in-1bfsmcw1ly.pdf

Micol Amar

Papers

Boundary layer tails in periodic homogenization

A brief survey on homogenization with a physical application

A notion of total variation depending on a metric with discontinuous coefficients

Evolution and memory effects in the homogenization limit for electrical conduction in biological tissues

On a hierarchy of models for electrical conduction in biological tissues