Bifurcations of nontrivial solutions of a cubic Helmholtz system

This paper presents local and global bifurcation results for radially symmetric solutions of the cubic Helmholtz system \begin{equation*} \begin{cases} -\Delta u - \mu u = \left( u^2 + b \: v^2 \right) u &\text{ on } \mathbb{R}^3, \\ -\Delta v - \nu v = \left( v^2 + b \: u^2 \right) v &\text{ on } \mathbb{R}^3. \end{cases} \end{equation*} It is shown that every point along any given branch of radial semitrivial solutions $(u_0, 0, b)$ or diagonal solutions $(u_b, u_b, b)$ (for $\mu = \nu$) is a bifurcation point. Our analysis is based on a detailed investigation of the oscillatory behavior of solutions at infinity that are shown to decay like $\frac{1}{|x|}$ as $|x|\to\infty$.Comment: 31 page

Dual Variational Methods for a nonlinear Helmholtz system

This paper considers a pair of coupled nonlinear Helmholtz equations \begin{align*} -\Delta u - \mu u = a(x) \left( |u|^\frac{p}{2} + b(x) |v|^\frac{p}{2} \right)|u|^{\frac{p}{2} - 2}u, \end{align*} \begin{align*} -\Delta v - \nu v = a(x) \left( |v|^\frac{p}{2} + b(x) |u|^\frac{p}{2} \right)|v|^{\frac{p}{2} - 2}v \end{align*} on $\mathbb{R}^N$ where $\frac{2(N+1)}{N-1} < p < 2^\ast$. The existence of nontrivial strong solutions in $W^{2, p}(\mathbb{R}^N)$ is established using dual variational methods. The focus lies on necessary and sufficient conditions on the parameters deciding whether or not both components of such solutions are nontrivial.Comment: Published version. Contains minor revisions: Quote added, explanations on p.12 concerning F_{\mu\nu} = \infty, correction of exponent on p.1

Bicycle Safety Campaign Review

What do successful bicycle safety campaigns have in common, and what tactics should be used in the future to achieve success? To help answer this, Bikes Belong conducted a review of campaigns, primarily used in the U.S. Important conclusions include: Emotional campaigns are more effective at increasing safety than informational campaigns.Safety campaigns that personalize and humanize cyclists without creating fear are ideal.Messages should be targeted at wide audiences that include both motorists and cyclists

REAL-TIME MEASUREMENT OF DIELECTRIC RELAXATION OF BIOMOLECULES: KINETICS OF A PROTEIN-LIGAND BINDING REACTION *

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/73332/1/j.1749-6632.1977.tb55918.x.pd

Die Perzeption Brasiliens durch deutsche Reisende des 19. Jahrhunderts : Maximilian Prinz Wied zu Neuwied und Johann Moritz Rugendas

This article focuses the expeditions of Maximilian Prinz Wied zu Neuwied and Johann Moritz Rugendas to Brazil. It discusses initially basic aspects of perception from the early colonial period up to the 19th century. It will then analyze the pictorial characterization of Brazil by both travelers and the reception of their work in Europ

The de Rham realization of the elliptic polylogarithm in families

This thesis establishes a geometric approach to the de Rham realization of the polylogarithm. As a central result we construct the logarithm sheaves of rational abelian schemes in terms of the birigidified Poincar\'e bundle with universal integrable connection on the product of the abelian scheme and the universal vectorial extension of its dual. This is achieved essentially by restricting the mentioned data of the Poincar\'e bundle along the infinitesimal neighborhoods of the zero section of the universal extension. We also clarify how these constructions naturally express within the language of the Fourier-Mukai transformation for $\mathcal D$-modules on abelian schemes. Our geometric perspective moreover permits an interpretation of fundamental formal properties of the logarithm sheaves within the standard theory of the Poincar\'e bundle. For a relative elliptic curve we additionally present a related viewpoint on the first logarithm extension via $1$-motives. Having developed in detail the outlined geometric understanding of the logarithm sheaves, we then exploit it systematically for an investigation of the polylogarithm for the universal family of elliptic curves with level $N$ structure. A main theorem of the work gives an explicit analytic description for a variant of the small elliptic polylogarithm via the coefficient functions appearing in the Laurent expansion of a meromorphic Jacobi form defined by Kronecker in the 19th century. Furthermore, using the previous result, we determine the specialization of the modified polylogarithm along torsion sections concretely in terms of certain algebraic Eisenstein series. From this we regain in particular the known expressions of the de Rham Eisenstein classes by algebraic modular forms.Comment: Doctoral thesis, University of Regensbur

Breather solutions of the cubic Klein–Gordon equation

We obtain real-valued, time-periodic and radially symmetric solutions of the cubic Klein–Gordon equation ${\partial }_{t}^{2}U-{\Delta}U+{m}^{2}U={\Gamma}\left(x\right){U}^{3}\;\text{on}\;\mathbb{R}{\times}{\mathbb{R}}^{3},$ which are weakly localized in space. Various families of such \u27breather\u27 solutions are shown to bifurcate from any given nontrivial stationary solution. The construction of weakly localized breathers in three space dimensions is, to the author\u27s knowledge, a new concept and based on the reformulation of the cubic Klein–Gordon equation as a system of coupled nonlinear Helmholtz equations involving suitable conditions on the far field behavior

The Effect of Summer Recess on the Reading Achievement of Title I Students at L.C. Curry School, Bowling Green, Kentucky

The purpose of the study was to determine if there were significant differences between spring reading achievement scores and fall reading achievement scores in the Title I students of L. C. Curry School, Bowling Green, Kentucky, and if significant differences did occur, were these differences related to grade level, IQ, sex, or reading achievement level