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

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

BMX Access in Skateparks Survey

Freestyle BMX is growing increasingly popular. It appeals to youth, doesn't have to cost much, encourages active recreation, and is suitable for both rural or urban environments. Freestyle BMX riders need a legal place to practice and play, and skateparks are among the safest and best venues. Some skateparks deny access to bike riders, citing concerns such as liability, user conflict, and facility damage. At the same time, numerous skateparks have figured how to integrate biking and skating seamlessly. To see how bikers have successfully gained access to skateparks and how skateparks have safely and effectively managed bikers and skaters, Bikes Belong surveyed nearly 100 skatepark managers from 30 states. Here's what we found.Profiles of the skateparks surveyed77% public18% private5% public/private partnership Profiles of the skateparks surveyed46% don't allow bikes, why not?75% say it's too dangerous mixing bikers and skaters64% say bikes cause too much damage48% cite liability concerns30% say bikers weren't around when the park was built7% say the park is too small5% say they don't know whyAccording to the survey respondents, nearly all of these reasons for denying bike access come back to park design. Often, parks weren't designed for bikers because bikers didn't participate in the planning, fundraising and construction processes. Park managers are often open to reviewing their policy on bikes. But, nothing changes unless the bike community is well organized, professional and engaged. Some parks haven't considered allowing bikes because their insurance or park warranty banned bikes from the beginning. In these cases, it's the insurance carriers and park builders who need to be educated about the importance of making room for bikes

Gas Prices/Bike Sales Survey

Is there an upside to high gas prices? If you're a bicycle retailer, there can be -- particularly in the service department. Bikes Belong completed a survey of more than 150 bicycle retailers from nearly 40 states to see if their summer 2008 sales reflected an increase in the use of bicycles for transportation.The majority of retailers who responded said their sales of transportation-related bicycles, accessories, and service have increased in 2008 compared to 2007:73% said they are selling more bikes. 84% said they are selling more accessories. 88% said they are selling more service.Is this increase in sales because of high gas prices? Most retailers who we surveyed think so:95% of shops said customers cited high gas prices as a reason for their transportation-related purchases.80% of retailers said gas prices were helping them sell more bikes for transportation. 86% thought accessory sales were getting a boost. 89% said they were selling more service because of high gas prices. Many new customers are dusting off old bikes and bringing them in for repair. There appears to be a surge of interest in riding bicycles for short trips, errands, and commuting

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