Sums and differences of four k-th powers

We prove an upper bound for the number of representations of a positive integer $N$ as the sum of four $k$-th powers of integers of size at most $B$, using a new version of the Determinant method developed by Heath-Brown, along with recent results by Salberger on the density of integral points on affine surfaces. More generally we consider representations by any integral diagonal form. The upper bound has the form $O_{N}(B^{c/\sqrt{k}})$, whereas earlier versions of the Determinant method would produce an exponent for $B$ of order $k^{-1/3}$ in this case. Furthermore, we prove that the number of representations of a positive integer $N$ as a sum of four $k$-th powers of non-negative integers is at most $O_{\epsilon}(N^{1/k+2/k^{3/2}+\epsilon})$ for $k \geq 3$, improving upon bounds by Wisdom.Comment: 18 pages. Mistake corrected in the statement of Theorem 1.2. To appear in Monatsh. Mat

The role of chaotic resonances in the solar system

Our understanding of the Solar System has been revolutionized over the past decade by the finding that the orbits of the planets are inherently chaotic. In extreme cases, chaotic motions can change the relative positions of the planets around stars, and even eject a planet from a system. Moreover, the spin axis of a planet-Earth's spin axis regulates our seasons-may evolve chaotically, with adverse effects on the climates of otherwise biologically interesting planets. Some of the recently discovered extrasolar planetary systems contain multiple planets, and it is likely that some of these are chaotic as well.Comment: 28 pages, 9 figure

On the observability of resonant structures in planetesimal disks due to planetary migration

We present a thorough study of the impact of a migrating planet on a planetesimal disk, by exploring a broad range of masses and eccentricities for the planet. We discuss the sensitivity of the structures generated in debris disks to the basic planet parameters. We perform many N-body numerical simulations, using the symplectic integrator SWIFT, taking into account the gravitational influence of the star and the planet on massless test particles. A constant migration rate is assumed for the planet. The effect of planetary migration on the trapping of particles in mean motion resonances is found to be very sensitive to the initial eccentricity of the planet and of the planetesimals. A planetary eccentricity as low as 0.05 is enough to smear out all the resonant structures, except for the most massive planets. The planetesimals also initially have to be on orbits with a mean eccentricity of less than than 0.1 in order to keep the resonant clumps visible. This numerical work extends previous analytical studies and provides a collection of disk images that may help in interpreting the observations of structures in debris disks. Overall, it shows that stringent conditions must be fulfilled to obtain observable resonant structures in debris disks. Theoretical models of the origin of planetary migration will therefore have to explain how planetary systems remain in a suitable configuration to reproduce the observed structures.Comment: 16 pages, 13 figures. Accepted for publication in A&

Validity of Brainstorming Strategy on Students’ Prior Knowledge and Academic Performance in Chemistry in Selected Secondary Schools in South-South Nigeria

This study examined the effects of brainstorming strategy on students’ prior knowledge and academic performance in chemistry. A sample of 148 participants (made up of 71 female and 77 male students) in their intact classes were drawn from four selected coeducational secondary schools in two states in south-south Nigeria. The schools were assigned into two instructional groups- the experimental (brainstorming strategy) group with 73 and control (lecture method) group with 75 students. Data were obtained through the administration of Pretest followed with a seven week treatment, and then the administration of posttest. Data were analyzed using mean and analysis of covariance (ANCOVA). Results showed statistically significant difference in both mean knowledge and mean academic performance in favour of the brainstorming group. The result further showed no statistically significant difference in the mean academic performance of male and female students in the experimental group. More so, the study revealed no significant interaction effect of instructional strategies and sex on academic performance. From the results of this study it was recommended that science curriculum designers and teachers should respectively design and use appropriate instructional strategies and packages that can evoke students’ prior knowledge to promote learning. 

Mechanical Systems with Symmetry, Variational Principles, and Integration Algorithms

This paper studies variational principles for mechanical systems with symmetry and their applications to integration algorithms. We recall some general features of how to reduce variational principles in the presence of a symmetry group along with general features of integration algorithms for mechanical systems. Then we describe some integration algorithms based directly on variational principles using a discretization technique of Veselov. The general idea for these variational integrators is to directly discretize Hamilton’s principle rather than the equations of motion in a way that preserves the original systems invariants, notably the symplectic form and, via a discrete version of Noether’s theorem, the momentum map. The resulting mechanical integrators are second-order accurate, implicit, symplectic-momentum algorithms. We apply these integrators to the rigid body and the double spherical pendulum to show that the techniques are competitive with existing integrators

Stability of Terrestrial Planets in the Habitable Zone of Gl 777 A, HD 72659, Gl 614, 47 Uma and HD 4208

We have undertaken a thorough dynamical investigation of five extrasolar planetary systems using extensive numerical experiments. The systems Gl 777 A, HD 72659, Gl 614, 47 Uma and HD 4208 were examined concerning the question of whether they could host terrestrial like planets in their habitable zones (=HZ). First we investigated the mean motion resonances between fictitious terrestrial planets and the existing gas giants in these five extrasolar systems. Then a fine grid of initial conditions for a potential terrestrial planet within the HZ was chosen for each system, from which the stability of orbits was then assessed by direct integrations over a time interval of 1 million years. The computations were carried out using a Lie-series integration method with an adaptive step size control. This integration method achieves machine precision accuracy in a highly efficient and robust way, requiring no special adjustments when the orbits have large eccentricities. The stability of orbits was examined with a determination of the Renyi entropy, estimated from recurrence plots, and with a more straight forward method based on the maximum eccentricity achieved by the planet over the 1 million year integration. Additionally, the eccentricity is an indication of the habitability of a terrestrial planet in the HZ; any value of e>0.2 produces a significant temperature difference on a planet's surface between apoapse and periapse. The results for possible stable orbits for terrestrial planets in habitable zones for the five systems are summarized as follows: for Gl 777 A nearly the entire HZ is stable, for 47 Uma, HD 72659 and HD 4208 terrestrial planets can survive for a sufficiently long time, while for Gl 614 our results exclude terrestrial planets moving in stable orbits within the HZ.Comment: 14 pages, 18 figures submitted to A&

Triggers of self-conscious emotions in the sexually transmitted infection testing process

<p>Abstract</p> <p>Background</p> <p>Self-conscious emotions (shame, guilt and embarrassment) are part of many individuals' experiences of seeking STI testing. These emotions can have negative impacts on individuals' interpretations of the STI testing process, their willingness to seek treatment and their willingness to inform sexual partners in light of positive STI diagnoses. Because of these impacts, researchers have called for more work to be completed on the connections between shame, guilt, embarrassment and STI testing. We examine the specific events in the STI testing process that trigger self-conscious emotions in young adults who seek STI testing; and to understand what it is about these events that triggers these emotions.</p> <p>Semi-structured interviews with 30 adults (21 women, 9 men) in the Republic of Ireland.</p> <p>Findings</p> <p>Seven specific triggers of self-conscious emotions were identified. These were: having unprotected sex, associated with the initial reason for seeking STI testing; talking to partners and peers about the intention to seek STI testing; the experience of accessing STI testing facilities and sitting in clinic waiting rooms; negative interactions with healthcare professionals; receiving a positive diagnosis of an STI; having to notify sexual partners in light of a positive STI diagnosis; and accessing healthcare settings for treatment for an STI. Self-conscious emotions were triggered in each case by a perceived threat to respondents' social identities.</p> <p>Conclusion</p> <p>There are multiple triggers of self-conscious emotions in the STI testing process, ranging from the initial decision to seek testing, right through to the experience of accessing treatment. The role of self-conscious emotions needs to be considered in each component of service design from health promotion approaches, through facility layout to the training of all professionals involved in the STI testing process.</p

A model for collisions in granular gases

We propose a model for collisions between particles of a granular material and calculate the restitution coefficients for the normal and tangential motion as functions of the impact velocity from considerations of dissipative viscoelastic collisions. Existing models of impact with dissipation as well as the classical Hertz impact theory are included in the present model as special cases. We find that the type of collision (smooth, reflecting or sticky) is determined by the impact velocity and by the surface properties of the colliding grains. We observe a rather nontrivial dependence of the tangential restitution coefficient on the impact velocity.Comment: 11 pages, 2 figure

On an asymptotic method for computing the modified energy for symplectic methods

We revisit an algorithm by Skeel et al. [5,16] for computing the modified, or shadow, energy associated with symplectic discretizations of Hamiltonian systems. We amend the algorithm to use Richardson extrapolation in order to obtain arbitrarily high order of accuracy. Error estimates show that the new method captures the exponentially small drift associated with such discretizations. Several numerical examples illustrate the theory