Localized Modes of the Linear Periodic Schr\"{o}dinger Operator with a Nonlocal Perturbation

We consider the existence of localized modes corresponding to eigenvalues of the periodic Schr\"{o}dinger operator $-\partial_x^2+ V(x)$ with an interface. The interface is modeled by a jump either in the value or the derivative of $V(x)$ and, in general, does not correspond to a localized perturbation of the perfectly periodic operator. The periodic potentials on each side of the interface can, moreover, be different. As we show, eigenvalues can only occur in spectral gaps. We pose the eigenvalue problem as a $C^1$ gluing problem for the fundamental solutions (Bloch functions) of the second order ODEs on each side of the interface. The problem is thus reduced to finding matchings of the ratio functions $R_\pm=\frac{\psi_\pm'(0)}{\psi_\pm(0)}$, where $\psi_\pm$ are those Bloch functions that decay on the respective half-lines. These ratio functions are analyzed with the help of the Pr\"{u}fer transformation. The limit values of $R_\pm$ at band edges depend on the ordering of Dirichlet and Neumann eigenvalues at gap edges. We show that the ordering can be determined in the first two gaps via variational analysis for potentials satisfying certain monotonicity conditions. Numerical computations of interface eigenvalues are presented to corroborate the analysis.Comment: 1. finiteness of the number of additive interface eigenvalues proved in a remark below Corollary 3.6.; 2. small modifications and typo correction

Breathers and rogue waves for semilinear curl-curl wave equations

We consider localized solutions of variants of the semilinear curl-curl wave equation $s(x) \partial_t^2 U +\nabla\times\nabla\times U + q(x) U \pm V(x) |U|^{p-1} U = 0$ for $(x,t)\in \mathbb{R}^3\times\mathbb{R}$ and arbitrary $p>1$. Depending on the coefficients $s, q, V$ we can prove the existence of three types of localized solutions: time-periodic solutions decaying to $0$ at spatial infinity, time-periodic solutions tending to a nontrivial profile at spatial infinity (both types are called breathers), and rogue waves which converge to $0$ both at spatial and temporal infinity. Our solutions are weak solutions and take the form of gradient fields. Thus they belong to the kernel of the curl-operator so that due to the structural assumptions on the coefficients the semilinear wave equation is reduced to an ODE. Since the space dependence in the ODE is just a parametric dependence we can analyze the ODE by phase plane techniques and thus establish the existence of the localized waves described above. Noteworthy side effects of our analysis are the existence of compact support breathers and the fact that one localized wave solution $U(x,t)$ already generates a full continuum of phase-shifted solutions $U(x,t+b(x))$ where the continuous function $b:\mathbb{R}^3\to\mathbb{R}$ belongs to a suitable admissible family

Breathers and rogue waves for semilinear curl-curl wave equations

We consider localized solutions of variants of the semilinear curl-curl wave equation $s(x)\partial_t^2U+\nabla\times\nabla\times+q(x)U\pm V(x)|U|^{p−1}U = 0$ for $(x,t)\in\mathbb{R^3}\times\mathbb{R}$ and arbitrary $p > 1$. Depending on the coefficients $s,q,V$ we can prove the existence of three types of localized solutions: time-periodic solutions decaying to $0$ at spatial infinity, time-periodic solutions tending to a nontrivial profile at spatial infinity (both types are called breathers), and rogue waves which converge to $0$ both at spatial and temporal infinity. Our solutions are weak solutions and take the form of gradient fields. Thus they belong to the kernel of the curl-operator so that due to the structural assumptions on the coefficients the semilinear wave equation is reduced to an ODE. Since the space dependence in the ODE is just a para- metric dependence we can analyze the ODE by phase plane techniques and thus establish the existence of the localized waves described above. Noteworthy side effects of our analysis are the existence of compact support breathers and the fact that one localized wave solution $U(x, t)$ already generates a full continuum of phase-shifted solutions $U(x,t + b(x))$ where the continuous function $b:\mathbb{R}^3\to\mathbb{R}$ belongs to a suitable admissible family

Kohortenanalyse als Methode der Untersuchung von Einflußfaktoren politischen Verhaltens: Sekundäranalyse von Umfragedaten des Zentralarchivs

In einer am Kölner Institut für Sozialforschung und Gesellschaftpolitik durchgeführten Untersuchung wurde die Fragestellung untersucht, ob sich das politische Verhalten mit zunehmendem Lebensalter generell ändert oder ob auch andere, vom demographischen Alter unabhängige Einflußfaktoren, scheinbar altersspezifische Veränderungen hervorrufen. Es wurde eine Kohortenanalyse anhand von sieben im Zentralarchiv für Empirische Sozialforschung gespeicherten Umfragen aus den Jahren 1953 bis 1976 vorgenommen. Die Verfahrensweisen der Kohortenanalyse, die einzelnen Variablen und ersten Ergebnisse werden dargestellt. Die Daten lassen Rückschlüsse sowohl auf Kohorteneffekte als auch auf Alterseffekte zu. (GB

A breather construction for a semilinear curl-curlwave equation with radially symmetric coefficients

Abstract. We consider the semilinear curl-curl wave equation s(x)∂2 U + ∇ × ∇ × U + q(x)U ± V (x)|U |p−1 U = 0 for (x, t) ∈ R3 × R. For any p > 1 we prove the existence of time- periodic spatially localized real-valued solutions (breathers) both for the + and the − case under slightly different hypotheses. Our solutions are classical solutions that are radially symmetric in space and decay exponentially to 0 as |x| → ∞. Our method is based on the fact that gradient fields of radially symmetric functions are annihilated by the curl-curl operator. Consequently, the semilinear wave equation is reduced to an ODE with r = |x| as a parameter. This ODE can be efficiently analyzed in phase space. As a side effect of our analysis, we obtain not only one but a full continuum of phase-shifted breathers U (x, t+a(x)), where U is a particular breather and a : R3 → R an arbitrary radially symmetric C 2 -function

In Situ Monitoring of the Catalytic Activity of Cytochrome c Oxidase in a Biomimetic Architecture

AbstractCytochrome c oxidase (CcO) from Paracoccus denitrificans was immobilized in a strict orientation via a his-tag attached to subunit I on a gold film and reconstituted in situ into a protein-tethered bilayer lipid membrane. In this orientation, the cytochrome c (cyt c) binding site is directed away from the electrode pointing to the outer side of the protein-tethered bilayer lipid membrane architecture. The CcO can thus be activated by cyt c under aerobic conditions. Catalytic activity was monitored by impedance spectroscopy, as well as cyclic voltammetry. Cathodic and anodic currents of the CcO with cyt c added to the bulk solution were shown to increase under aerobic compared to anaerobic conditions. Catalytic activity was considered in terms of repeated electrochemical oxidation/reduction of the CcO/cyt c complex in the presence of oxygen. The communication of cyt c bound to the CcO with the electrode is discussed in terms of a hopping mechanism through the redox sites of the enzyme. Simulations supporting this hypothesis are included

Surface Gap Soliton Ground States for the Nonlinear Schr\"{o}dinger Equation

We consider the nonlinear Schr\"{o}dinger equation $(-\Delta +V(x))u = \Gamma(x) |u|^{p-1}u$, $x\in \R^n$ with $V(x) = V_1(x) \chi_{\{x_1>0\}}(x)+V_2(x) \chi_{\{x_1<0\}}(x)$ and $\Gamma(x) = \Gamma_1(x) \chi_{\{x_1>0\}}(x)+\Gamma_2(x) \chi_{\{x_1<0\}}(x)$ and with $V_1, V_2, \Gamma_1, \Gamma_2$ periodic in each coordinate direction. This problem describes the interface of two periodic media, e.g. photonic crystals. We study the existence of ground state $H^1$ solutions (surface gap soliton ground states) for $0<\min \sigma(-\Delta +V)$. Using a concentration compactness argument, we provide an abstract criterion for the existence based on ground state energies of each periodic problem (with $V\equiv V_1, \Gamma\equiv \Gamma_1$ and $V\equiv V_2, \Gamma\equiv \Gamma_2$) as well as a more practical criterion based on ground states themselves. Examples of interfaces satisfying these criteria are provided. In 1D it is shown that, surprisingly, the criteria can be reduced to conditions on the linear Bloch waves of the operators $-\tfrac{d^2}{dx^2} +V_1(x)$ and $-\tfrac{d^2}{dx^2} +V_2(x)$.Comment: definition of ground and bound states added, assumption (H2) weakened (sign changing nonlinearity is now allowed); 33 pages, 4 figure

2016 State of the Commonwealth Report

This is the second State of the Commonwealth Report produced by the Center for Economic Analysis and Policy at Old Dominion University. It is jointly sponsored by ODU\u27s Strome College of Business and the Virginia Chamber of Commerce. While the report represents the work of many people connected in various ways to the university, it does not constitute an official viewpoint of Old Dominion, or it\u27s president, John R. Broderick. Similarly, it does not represent the views of the Virginia Chamber of Commerce or it\u27s president, Barry DuVal. The report maintains the goal of stimulating thought and discussion that ultimately will make Virginia an even better place to live, work, and do business. We are proud of the Commonwealth\u27s many successes, but realize that it is possible to improve our performance. In order to do so, we must have accurate information about where we are and a sound understanding of the policy options open to us.https://digitalcommons.odu.edu/economics_books/1019/thumbnail.jp