Homogenization of Maxwell's equations in periodic composites

We consider the problem of homogenizing the Maxwell equations for periodic composites. The analysis is based on Bloch-Floquet theory. We calculate explicitly the reflection coefficient for a half-space, and derive and implement a computationally-efficient continued-fraction expansion for the effective permittivity. Our results are illustrated by numerical computations for the case of two-dimensional systems. The homogenization theory of this paper is designed to predict various physically-measurable quantities rather than to simply approximate certain coefficients in a PDE.Comment: Significantly expanded compared to v1. Accepted to Phys.Rev.E. Some color figures in this preprint may be easier to read because here we utilize solid color lines, which are indistinguishable in black-and-white printin

Realizability of metamaterials with prescribed electric permittivity and magnetic permeability tensors

We show that any pair of real symmetric tensors \BGve and \BGm can be realized as the effective electric permittivity and effective magnetic permeability of a metamaterial at a given fixed frequency. The construction starts with two extremely low loss metamaterials, with arbitrarily small microstructure, whose existence is ensured by the work of Bouchitt{\'e} and Bourel and Bouchitt\'e and Schweizer, one having at the given frequency a permittivity tensor with exactly one negative eigenvalue, and a positive permeability tensor, and the other having a positive permittivity tensor, and a permeability tensor having exactly one negative eigenvalue. To achieve the desired effective properties these materials are laminated together in a hierarchical multiple rank laminate structure, with widely separated length scales, and varying directions of lamination, but with the largest length scale still much shorter than the wavelengths and attenuation lengths in the macroscopic effective medium.Comment: 12 pages, no figure

About curvature, conformal metrics and warped products

We consider the curvature of a family of warped products of two pseduo-Riemannian manifolds $(B,g_B)$ and $(F,g_F)$ furnished with metrics of the form $c^{2}g_B \oplus w^2 g_F$ and, in particular, of the type $w^{2 \mu}g_B \oplus w^2 g_F$, where $c, w \colon B \to (0,\infty)$ are smooth functions and $\mu$ is a real parameter. We obtain suitable expressions for the Ricci tensor and scalar curvature of such products that allow us to establish results about the existence of Einstein or constant scalar curvature structures in these categories. If $(B,g_B)$ is Riemannian, the latter question involves nonlinear elliptic partial differential equations with concave-convex nonlinearities and singular partial differential equations of the Lichnerowicz-York type among others.Comment: 32 pages, 3 figure

Towards a unified theory of Sobolev inequalities

We discuss our work on pointwise inequalities for the gradient which are connected with the isoperimetric profile associated to a given geometry. We show how they can be used to unify certain aspects of the theory of Sobolev inequalities. In particular, we discuss our recent papers on fractional order inequalities, Coulhon type inequalities, transference and dimensionless inequalities and our forthcoming work on sharp higher order Sobolev inequalities that can be obtained by iteration.Comment: 39 pages, made some changes to section 1

Multidimensional Conservation Laws: Overview, Problems, and Perspective

Some of recent important developments are overviewed, several longstanding open problems are discussed, and a perspective is presented for the mathematical theory of multidimensional conservation laws. Some basic features and phenomena of multidimensional hyperbolic conservation laws are revealed, and some samples of multidimensional systems/models and related important problems are presented and analyzed with emphasis on the prototypes that have been solved or may be expected to be solved rigorously at least for some cases. In particular, multidimensional steady supersonic problems and transonic problems, shock reflection-diffraction problems, and related effective nonlinear approaches are analyzed. A theory of divergence-measure vector fields and related analytical frameworks for the analysis of entropy solutions are discussed.Comment: 43 pages, 3 figure

Analysis of Oscillations and Defect Measures for the Quasineutral Limit in Plasma Physics

We perform a rigorous analysis of the quasineutral limit for a hydrodynamical model of a viscous plasma represented by the Navier Stokes Poisson system in $3-D$. We show that as $\lambda\to 0$ the velocity field $u^{\lambda}$ strongly converges towards an incompressible velocity vector field $u$ and the density fluctuation $\rho^{\lambda}-1$ weakly converges to zero. In general the limit velocity field cannot be expected to satisfy the incompressible Navier Stokes equation, indeed the presence of high frequency oscillations strongly affects the quadratic nonlinearities and we have to take care of self interacting wave packets. We shall provide a detailed mathematical description of the convergence process by using microlocal defect measures and by developing an explicit correctors analysis. Moreover we will be able to identify an explicit pseudo parabolic pde satisfied by the leading correctors terms. Our results include all the previous results in literature, in particular we show that the formal limit holds rigorously in the case of well prepared data.Comment: Submitted pape

Character of Christ: A Proposal for Excellence in Christian Character Education

Moral teaching programs, such as character education, have been implemented nationwide in order to curb the growing trend of violence, abuse, and moral relativism within schools, both public and private. These programs represent a variety of moral training philosophies, and current research is revealing some best practices within the field. However, these programs do little to address the needs of distinctively Christian educators who seek to train their students toward the character of Jesus Christ. The research in this study promotes the development of a curriculum to meet this need. The following research indicates that character education\u27s premise and many of its practices are worthy of consideration when developing a Christian character curriculum. However, the foundation of the character traits promoted by a Christian character curriculum must not be based on the consensus of a pluralistic society. The foundation must be established solely on the person of Christ. Best practices within the field of character education are emerging through current research. These practices and the theories behind them are also examined in light of the development of a Christian character curriculum. Recommendations and implications for a Christian character curriculum are made in both theory and practice

Supportive care in the acute phase of Stevens-Johnson syndrome and toxic epidermal necrolysis : an international, multidisciplinary Delphi-based consensus

Background Supportive care is the cornerstone of management of adult and paediatric Stevens-Johnson syndrome (SJS) and toxic epidermal necrolysis (TEN). However, consensus on the modalities of supportive care is lacking. Objectives Our aim in this international multicentric Delphi exercise was to establish a multidisciplinary expert consensus to standardize recommendations regarding supportive care in the acute phase of SJS/TEN. Methods Participants were sent a survey via the online tool SurveyMonkey, consisting of 103 statements organized into 11 topics: multidisciplinary team composition, suspect drug management, infection prevention, fluid resuscitation and prevention of hypothermia, nutritional support, pain and psychological distress management, management of acute respiratory failure, local skincare, ophthalmological management, management of other mucosa, and additional measures. Participants evaluated the level of appropriateness of each statement on a scale of 1 (extremely inappropriate) to 9 (extremely appropriate). The results were analysed according to the RAND/UCLA Appropriateness Method. Results Forty-five participants from 13 countries (on three continents) participated. After the first round, a consensus was obtained for 82.5% of the 103 initially proposed statements. After the second round, a final consensus was obtained for 102 statements. Conclusions We have reached an international Delphi-based consensus on best supportive care practice for SJS/TEN. Our expert consensus should help guide physicians in treating patients with SJS/TEN and thereby improve short-term prognosis and the risk of sequelae.Peer reviewe

Histoplasma capsulatum Encodes a Dipeptidyl Peptidase Active against the Mammalian Immunoregulatory Peptide, Substance P

The pathogenic fungus Histoplasma capsulatum secretes dipeptidyl peptidase (Dpp) IV enzyme activity and has two putative DPPIV homologs (HcDPPIVA and HcDPPIVB). We previously showed that HcDPPIVB is the gene responsible for the majority of secreted DppIV activity in H. capsulatum culture supernatant, while we could not detect any functional contribution from HcDPPIVA. In order to determine whether HcDPPIVA encodes a functional DppIV enzyme, we expressed HcDPPIVA in Pichia pastoris and purified the recombinant protein. The recombinant enzyme cleaved synthetic DppIV substrates and had similar biochemical properties to other described DppIV enzymes, with temperature and pH optima of 42°C and 8, respectively. Recombinant HcDppIVA cleaved the host immunoregulatory peptide substance P, indicating the enzyme has the potential to affect the immune response during infection. Expression of HcDPPIVA under heterologous regulatory sequences in H. capsulatum resulted in increased secreted DppIV activity, indicating that the encoded protein can be expressed and secreted by its native organism. However, HcDPPIVA was not required for virulence in a murine model of histoplasmosis. This work reports a fungal enzyme that can function to cleave the immunomodulatory host peptide substance P