An Exploration of a Quantitative Reasoning Instructional Approach to Linear Equations in Two Variables with Community College Students

In this exploratory study, we examined the effects of a quantitative reasoning instructional approach to linear equations in two variables on community college students’ conceptual understanding, procedural fluency, and reasoning ability. This was done in comparison to the use of a traditional procedural approach for instruction on the same topic. Data were gathered from a common unit assessment that included procedural and conceptual questions. Results demonstrate that small changes in instruction focused on quantitative reasoning can lead to significant differences in students’ ability to demonstrate conceptual understanding compared to a procedural approach. The results also indicate that a quantitative reasoning approach does not appear to diminish students’ procedural skills, but that additional work is needed to understand how to best support students’ understanding of linear relationships

Stochastic Light-Cone CTMRG: a new DMRG approach to stochastic models

We develop a new variant of the recently introduced stochastic transfer-matrix DMRG which we call stochastic light-cone corner-transfer-matrix DMRG (LCTMRG). It is a numerical method to compute dynamic properties of one-dimensional stochastic processes. As suggested by its name, the LCTMRG is a modification of the corner-transfer-matrix DMRG (CTMRG), adjusted by an additional causality argument. As an example, two reaction-diffusion models, the diffusion-annihilation process and the branch-fusion process, are studied and compared to exact data and Monte-Carlo simulations to estimate the capability and accuracy of the new method. The number of possible Trotter steps of more than 10^5 shows a considerable improvement to the old stochastic TMRG algorithm.Comment: 15 pages, uses IOP styl

Transfer-matrix DMRG for stochastic models: The Domany-Kinzel cellular automaton

We apply the transfer-matrix DMRG (TMRG) to a stochastic model, the Domany-Kinzel cellular automaton, which exhibits a non-equilibrium phase transition in the directed percolation universality class. Estimates for the stochastic time evolution, phase boundaries and critical exponents can be obtained with high precision. This is possible using only modest numerical effort since the thermodynamic limit can be taken analytically in our approach. We also point out further advantages of the TMRG over other numerical approaches, such as classical DMRG or Monte-Carlo simulations.Comment: 9 pages, 9 figures, uses IOP styl

Stochastic switching in delay-coupled oscillators

A delay is known to induce multistability in periodic systems. Under influence of noise, coupled oscillators can switch between coexistent orbits with different frequencies and different oscillation patterns. For coupled phase oscillators we reduce the delay system to a nondelayed Langevin equation, which allows us to analytically compute the distribution of frequencies and their corresponding residence times. The number of stable periodic orbits scales with the roundtrip delay time and coupling strength, but the noisy system visits only a fraction of the orbits, which scales with the square root of the delay time and is independent of the coupling strength. In contrast, the residence time in the different orbits is mainly determined by the coupling strength and the number of oscillators, and only weakly dependent on the coupling delay. Finally we investigate the effect of a detuning between the oscillators. We demonstrate the generality of our results with delay-coupled FitzHugh-Nagumo oscillators

Nonlocal mechanism for cluster synchronization in neural circuits

The interplay between the topology of cortical circuits and synchronized activity modes in distinct cortical areas is a key enigma in neuroscience. We present a new nonlocal mechanism governing the periodic activity mode: the greatest common divisor (GCD) of network loops. For a stimulus to one node, the network splits into GCD-clusters in which cluster neurons are in zero-lag synchronization. For complex external stimuli, the number of clusters can be any common divisor. The synchronized mode and the transients to synchronization pinpoint the type of external stimuli. The findings, supported by an information mixing argument and simulations of Hodgkin Huxley population dynamic networks with unidirectional connectivity and synaptic noise, call for reexamining sources of correlated activity in cortex and shorter information processing time scales.Comment: 8 pges, 6 figure

Precise Critical Exponents for the Basic Contact Process

We calculated some of the critical exponents of the directed percolation universality class through exact numerical diagonalisations of the master operator of the one-dimensional basic contact process. Perusal of the power method together with finite-size scaling allowed us to achieve a high degree of accuracy in our estimates with relatively little computational effort. A simple reasoning leading to the appropriate choice of the microscopic time scale for time-dependent simulations of Markov chains within the so called quantum chain formulation is discussed. Our approach is applicable to any stochastic process with a finite number of absorbing states.Comment: LaTeX 2.09, 9 pages, 1 figur

Directed Percolation with a Wall or Edge

We examine the effects of introducing a wall or edge into a directed percolation process. Scaling ansatzes are presented for the density and survival probability of a cluster in these geometries, and we make the connection to surface critical phenomena and field theory. The results of previous numerical work for a wall can thus be interpreted in terms of surface exponents satisfying scaling relations generalising those for ordinary directed percolation. New exponents for edge directed percolation are also introduced. They are calculated in mean-field theory and measured numerically in 2+1 dimensions.Comment: 14 pages, submitted to J. Phys.

Analysis of CDMA systems that are characterized by eigenvalue spectrum

An approach by which to analyze the performance of the code division multiple access (CDMA) scheme, which is a core technology used in modern wireless communication systems, is provided. The approach characterizes the objective system by the eigenvalue spectrum of a cross-correlation matrix composed of signature sequences used in CDMA communication, which enables us to handle a wider class of CDMA systems beyond the basic model reported by Tanaka. The utility of the novel scheme is shown by analyzing a system in which the generation of signature sequences is designed for enhancing the orthogonality.Comment: 7 pages, 2 figure

Resonant Effects in Nanoscale Bowtie Apertures

Nanoscale bowtie aperture antennas can be used to focus light well below the diffraction limit with extremely high transmission efficiencies. This paper studies the spectral dependence of the transmission through nanoscale bowtie apertures defined in a silver film. A realistic bowtie aperture is numerically modeled using the Finite Difference Time Domain (FDTD) method. Results show that the transmission spectrum is dominated by Fabry-Pérot (F-P) waveguide modes and plasmonic modes. The F-P resonance is sensitive to the thickness of the film and the plasmonic resonant mode is closely related to the gap distance of the bowtie aperture. Both characteristics significantly affect the transmission spectrum. To verify these numerical results, bowtie apertures are FIB milled in a silver film. Experimental transmission measurements agree with simulation data. Based on this result, nanoscale bowtie apertures can be optimized to realize deep sub-wavelength confinement with high transmission efficiency with applications to nanolithography, data storage, and bio-chemical sensing

Nature of phase transitions in a probabilistic cellular automaton with two absorbing states

We present a probabilistic cellular automaton with two absorbing states, which can be considered a natural extension of the Domany-Kinzel model. Despite its simplicity, it shows a very rich phase diagram, with two second-order and one first-order transition lines that meet at a tricritical point. We study the phase transitions and the critical behavior of the model using mean field approximations, direct numerical simulations and field theory. A closed form for the dynamics of the kinks between the two absorbing phases near the tricritical point is obtained, providing an exact correspondence between the presence of conserved quantities and the symmetry of absorbing states. The second-order critical curves and the kink critical dynamics are found to be in the directed percolation and parity conservation universality classes, respectively. The first order phase transition is put in evidence by examining the hysteresis cycle. We also study the "chaotic" phase, in which two replicas evolving with the same noise diverge, using mean field and numerical techniques. Finally, we show how the shape of the potential of the field-theoretic formulation of the problem can be obtained by direct numerical simulations.Comment: 19 pages with 7 figure