Wave propagation across interfaces induced by different interaction exponents in ordered and disordered Hertz-like granular chains

We study solitary wave propagation in 1D granular crystals with Hertz-like interaction potentials. We consider interfaces between media with different exponents in the interaction potential. For an interface with increasing interaction potential exponent along the propagation direction we obtain mainly transmission with delayed secondary transmitted and reflected pulses. For interfaces with decreasing interaction potential exponent we observe both significant reflection and transmission of the solitary wave, where the transmitted part of the wave forms a multipulse structure. We also investigate impurities consisting of beads with different interaction exponents compared to the media they are embedded in, and we find that the impurities cause both reflection and transmission, including the formation of multipulse structures, independent of whether the exponent in the impurities is smaller than in the surrounding media. We explain wave propagation effects at interfaces and impurities in terms of quasi-particle collisions. Next we consider wave propagation along Hertz-like granular chains of beads in the presence of disorder and periodicity in the interaction exponents present in the Hertz-like potential, modelling, for instance, inhomogeneity in the contact geometry between beads in the granular chain. We find that solitary waves in media with randomised interaction exponents (which models disorder in the contact geometry) experience exponential decay, where the dependence of the decay rate is similar to the case of randomised bead masses. In the periodic case of chains with interaction exponents alternating between two fixed values, we find qualitatively different propagation properties depending on the choice of the two exponents. In particular, we find regimes with either exponential decay or stable solitary wave propagation with pairwise collective behaviour.Comment: 33 pages, 28 figure

Reduction of dimension for nonlinear dynamical systems

We consider reduction of dimension for nonlinear dynamical systems. We demonstrate that in some cases, one can reduce a nonlinear system of equations into a single equation for one of the state variables, and this can be useful for computing the solution when using a variety of analytical approaches. In the case where this reduction is possible, we employ differential elimination to obtain the reduced system. While analytical, the approach is algorithmic, and is implemented in symbolic software such as {\sc MAPLE} or {\sc SageMath}. In other cases, the reduction cannot be performed strictly in terms of differential operators, and one obtains integro-differential operators, which may still be useful. In either case, one can use the reduced equation to both approximate solutions for the state variables and perform chaos diagnostics more efficiently than could be done for the original higher-dimensional system, as well as to construct Lyapunov functions which help in the large-time study of the state variables. A number of chaotic and hyperchaotic dynamical systems are used as examples in order to motivate the approach.Comment: 16 pages, no figure

An Image\u27s Processor

This research experiments with a new method of visual data analysis. Through this method, which images receive greater processing attention depends on what is important and interesting to the viewer’s mind, instead of how the data is presented. From the standpoint of this study, what visual input is important and interesting for the mind depends upon memory structure. Based on current cognition theory, memory structure is comprised of what specific concepts are wired together and fired together when a certain input is received. In analyzing visual input that should activate many different memory associations, this study seeks to investigate the importance of a viewer’s specific memory in the process of understanding what they see

Paulo Freire’s Pedagogy for the Children of the Oppressors: Educating for Social Justice among the World’s Privileged

University-level scholars in North American and European often claim that they are committed to helping their students become active in social justice concerns. Paulo Freire’s book, The Pedagogy of the Oppressed, claims however that most educational models among the world’s privileged and elite share some complicity in the maintenance of an unjust status-quo of oppression. This article will focus on how Freirean ideas can be applied to students of privilege in order to move them from ignorance, guilty remorse, or paternalistic activism vis-à-vis the world’s oppressed towards a respectful, dialogical inter-relationship