Infinitely many solutions for a class of discrete non-linear boundary value problems

The existence of infinitely many solutions for a discrete non-linear Dirichlet problem involving the p-Laplacian, under appropriate oscillating behaviours of the non-linear term, is established. The approach is based on the critical point theory.     

Existence of subharmonic solutions and homoclinic orbits for a class of even higher order differential equations

Using the variational structure and the Z ₂ group index theory we establish the existence of subharmonic solutions and homoclinic solutions for a class of even higher order differential equation.     

Coupled spring equations

Coupled spring equations for modelling the motion of two springs with weights attached, hung in series from the ceiling are described. For the linear model using Hooke's Law, the motion of each weight is described by a fourth-order linear differential equation. A nonlinear model is also described and damping and external forcing are considered. The model has many features that permit the meaningful introduction of many concepts including: accuracy of numerical algorithms, dependence on parameters and initial conditions, phase and synchronization, periodicity, beats, linear and nonlinear resonance, limit cycles, harmonic and subharmonic solutions. These solutions produce a wide variety of interesting motions and the model is suitable for study as a computer laboratory project in a beginning course on differential equations or as an individual or a small-group undergraduate research project.     

Analyzing Transverse Momentum Spectra of Pions,Kaons and Protons in p–p,p–A and A–A Collisions via the Blast-Wave Model with Fluctuations

The transverse momentum spectra of different types of particles, π±, K±, p and p¯, produced at mid-(pseudo)rapidity in different centrality lead–lead (Pb–Pb) collisions at 2.76 TeV; proton–lead (p–Pb) collisions at 5.02 TeV; xenon–xenon (Xe–Xe) collisions at 5.44 TeV; and proton–proton (p–p) collisions at 0.9, 2.76, 5.02, 7 and 13 TeV, were analyzed by the blast-wave model with fluctuations. With the experimental data measured by the ALICE and CMS Collaborations at the Large Hadron Collider (LHC), the kinetic freeze-out temperature, transverse flow velocity and proper time were extracted from fitting the transverse momentum spectra. In nucleus–nucleus (A–A) and proton–nucleus (p–A) collisions, the three parameters decrease with the decrease of event centrality from central to peripheral, indicating higher degrees of excitation, quicker expansion velocities and longer evolution times for central collisions. In p–p collisions, the kinetic freeze-out temperature is nearly invariant with the increase of energy, though the transverse flow velocity and proper time increase slightly, in the considered energy range.     

Cross-country skiing motion equations,locomotive forces and mass scaling laws

This article presents differential equations for locomotive force and velocity during cross-country skiing. A muscle's work power is modelled. Thereafter, a locomotive force that depends on the skier's velocity is constructed. The external forces aerodynamic drag, friction forces and the force of gravity are incorporated in order to provide the equation of motion. Some allometric mass scaling relations are established and used to analyse the effect of a skier's mass on velocity. The model is tested by using a GPS instrument. We compare analytically and experimentally determined skiing distances and velocities as functions of time, and under different conditions. The article provides tools useful for practising athletes and coaches.