Extension Of Bertrand's Theorem And Factorization Of The Radial Schr\"odinger Equation

The Bertrand's theorem is extended, i.e. closed orbits still may exist for other central potentials than the power law Coulomb potential and isotropic harmonic oscillator. It is shown that for the combined potential $V(r)=W(r)+b/r^2$ ($W(r)=ar^{\nu}$), when (and only when) $W(r)$ is the Coulomb potential or isotropic harmonic oscillator, closed orbits still exist for suitable angular momentum. The correspondence between the closeness of classical orbits and the existence of raising and lowering operators derived from the factorization of the radial Schr\"odinger equation is investigated.Comment: 4 pages, 1 figug

Meso-scale modelling of 3D woven composite T-joints with weave variations

A meso-scale modelling framework is proposed to simulate the 3D woven fibre architectures and the mechanical performance of the composite T-joints, subjected to quasi-static tensile pull-off loading. The proposed method starts with building the realistic reinforcement geometries of the 3D woven T-joints at the mesoscale, of which the modelling strategy is applicable for other types of geometries with weave variations at the T-joint junction. Damage modelling incorporates both interface and constituent material damage, in conjunction with a continuum damage mechanics approach to account for the progressive failure behaviour. With a voxel based cohesive zone model, the proposed method is able to model mode I delamination based on the voxel mesh technique, which has advantages in meshing. Predicted results are in good agreement with experimental data beyond initial failure, in terms of load-displacement responses, failure events, damage initiation and propagation. The significant effect of fibre architecture variations on mechanical behaviour is successfully predicted through this modelling method without any further correlation of input parameters in damage model. This predictive method will facilitate the design and optimisation of 3D woven T-joint preforms

Experimental assessment of the mechanical behaviour of 3D woven composite T-joints

To understand the influence of the fibre architecture of 3D woven composite T-joints on mechanical performance, as well as the benefits that 3D woven T-joints can offer over the equivalent 2D laminates, experimental testing is performed on two types of 3D woven T-joint with only weave variation at the junction, and one type of 2D woven laminate T-joint. A quasi-static tensile pull-off loading is selected in this work as this out-of-plane load case is one of the typical loading conditions for such T-joint structures. The significant advantages of 3D woven composite T-joints in terms of ultimate strength and damage tolerance over the 2D alternative were identified in the testing. More importantly, this work showed that variation in the fibre architecture can considerably enhance properties such as delamination resistance and total energy absorption to failure, as well as increasing slightly the stiffness and initial failure load. This experimental assessment has demonstrated that using 3D woven reinforcements is an effective way to improve the load-bearing capability of composite T-joints over laminates, and also that this improvement could be optimised with regard to fibre architecture

The Majority Illusion in Social Networks

Social behaviors are often contagious, spreading through a population as individuals imitate the decisions and choices of others. A variety of global phenomena, from innovation adoption to the emergence of social norms and political movements, arise as a result of people following a simple local rule, such as copy what others are doing. However, individuals often lack global knowledge of the behaviors of others and must estimate them from the observations of their friends' behaviors. In some cases, the structure of the underlying social network can dramatically skew an individual's local observations, making a behavior appear far more common locally than it is globally. We trace the origins of this phenomenon, which we call "the majority illusion," to the friendship paradox in social networks. As a result of this paradox, a behavior that is globally rare may be systematically overrepresented in the local neighborhoods of many people, i.e., among their friends. Thus, the "majority illusion" may facilitate the spread of social contagions in networks and also explain why systematic biases in social perceptions, for example, of risky behavior, arise. Using synthetic and real-world networks, we explore how the "majority illusion" depends on network structure and develop a statistical model to calculate its magnitude in a network

A data-driven game theoretic strategy for developers in software crowdsourcing: a case study

Crowdsourcing has the advantages of being cost-effective and saving time, which is a typical embodiment of collective wisdom and community workers’ collaborative development. However, this development paradigm of software crowdsourcing has not been used widely. A very important reason is that requesters have limited knowledge about crowd workers’ professional skills and qualities. Another reason is that the crowd workers in the competition cannot get the appropriate reward, which affects their motivation. To solve this problem, this paper proposes a method of maximizing reward based on the crowdsourcing ability of workers, they can choose tasks according to their own abilities to obtain appropriate bonuses. Our method includes two steps: Firstly, it puts forward a method to evaluate the crowd workers’ ability, then it analyzes the intensity of competition for tasks at Topcoder.com—an open community crowdsourcing platform—on the basis of the workers’ crowdsourcing ability; secondly, it follows dynamic programming ideas and builds game models under complete information in different cases, offering a strategy of reward maximization for workers by solving a mixed-strategy Nash equilibrium. This paper employs crowdsourcing data from Topcoder.com to carry out experiments. The experimental results show that the distribution of workers’ crowdsourcing ability is uneven, and to some extent it can show the activity degree of crowdsourcing tasks. Meanwhile, according to the strategy of reward maximization, a crowd worker can get the theoretically maximum reward

Invariant Jordan curves of Sierpiski carpet rational maps

In this paper, we prove that if $R\colon\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ is a postcritically finite rational map with Julia set homeomorphic to the Sierpi\'nski carpet, then there is an integer $n_0$, such that, for any $n\ge n_0$, there exists an $R^n$-invariant Jordan curve $\Gamma$ containing the postcritical set of $R$.Comment: 16 pages, 1 figu