139,962 research outputs found
Teaching square roots: conceptual complexity in mathematics language
The article presents strategies for communicating the concept of square roots to limited English-proficient students. Instead of using mathematical terminology, mathematics teachers could communicate the concept of square roots by integrating the literal concept of square to the actual meaning of a square root. Such approach addresses the learning difficulty experienced by limited English proficient students who tend to associate square roots to geometric concepts. Several strategies for illustrating the concept of a square root using square figures are presented.<br /
Integrating technologies into mathematics: Comparing the cases of square roots and integrals
Although the term is often used to denote electronic devices, the idea of a 'technology', with its origins in the Greek techne (art or skill), refers in its most general sense to a way of doing things. The development and availability of various technologies for computation over the past forty years or so have influenced what we regard as important in mathematics, and what we teach to students, given the inevitable time pressures on our curriculum. In this note, we compare and contrast current approaches to two important mathematical ideas, those of square roots and of integrals, and how these have changed (or not) over time
Matrix Models, Monopoles and Modified Moduli
Motivated by the Dijkgraaf-Vafa correspondence, we consider the matrix model
duals of N=1 supersymmetric SU(Nc) gauge theories with Nf flavors. We
demonstrate via the matrix model solutions a relation between vacua of theories
with different numbers of colors and flavors. This relation is due to an N=2
nonrenormalization theorem which is inherited by these N=1 theories.
Specializing to the case Nf=Nc, the simplest theory containing baryons, we
demonstrate that the explicit matrix model predictions for the locations on the
Coulomb branch at which monopoles condense are consistent with the quantum
modified constraints on the moduli in the theory. The matrix model solutions
include the case that baryons obtain vacuum expectation values. In specific
cases we check explicitly that these results are also consistent with the
factorization of corresponding Seiberg-Witten curves. Certain results are
easily understood in terms of M5-brane constructions of these gauge theories.Comment: 27 pages, LaTeX, 2 figure
On Soliton-type Solutions of Equations Associated with N-component Systems
The algebraic geometric approach to -component systems of nonlinear
integrable PDE's is used to obtain and analyze explicit solutions of the
coupled KdV and Dym equations. Detailed analysis of soliton fission, kink to
anti-kink transitions and multi-peaked soliton solutions is carried out.
Transformations are used to connect these solutions to several other equations
that model physical phenomena in fluid dynamics and nonlinear optics.Comment: 43 pages, 16 figure
Variable-delay feedback control of unstable steady states in retarded time-delayed systems
We study the stability of unstable steady states in scalar retarded
time-delayed systems subjected to a variable-delay feedback control. The
important aspect of such a control problem is that time-delayed systems are
already infinite-dimensional before the delayed feedback control is turned on.
When the frequency of the modulation is large compared to the system's
dynamics, the analytic approach consists of relating the stability properties
of the resulting variable-delay system with those of an analogous distributed
delay system. Otherwise, the stability domains are obtained by a numerical
integration of the linearized variable-delay system. The analysis shows that
the control domains are significantly larger than those in the usual
time-delayed feedback control, and that the complexity of the domain structure
depends on the form and the frequency of the delay modulation.Comment: 13 pages, 8 figures, RevTeX, accepted for publication in Physical
Review
Weakly nonlinear waves in magnetized plasma with a slightly non-Maxwellian electron distribution. Part 2, Stability of cnoidal waves
We determine the growth rate of linear instabilities resulting from long-wavelength transverse perturbations applied to periodic nonlinear wave solutions to the SchamelâKortewegâde VriesâZakharovâKuznetsov (SKdVZK) equation which governs weakly nonlinear waves in a strongly magnetized cold-ion plasma whose electron distribution is given by two Maxwellians at slightly different temperatures. To obtain the growth rate it is necessary to evaluate non-trivial integrals whose number is kept to a minimum by using recursion relations. It is shown that a key instance of one such relation cannot be used for classes of solution whose minimum value is zero, and an additional integral must be evaluated explicitly instead. The SKdVZK equation contains two nonlinear terms whose ratio b increases as the electron distribution becomes increasingly flat-topped. As b and hence the deviation from electron isothermality increases, it is found that for cnoidal wave solutions that travel faster than long-wavelength linear waves, there is a more pronounced variation of the growth rate with the angle θ at which the perturbation is applied. Solutions whose minimum values are zero and which travel slower than long-wavelength linear waves are found, at first order, to be stable to perpendicular perturbations and have a relatively narrow range of θ for which the first-order growth rate is not zero
Dynamics of viscoelastic snap-through
We study the dynamics of snap-through when viscoelastic effects are present.
To gain analytical insight we analyse a modified form of the Mises truss, a
single-degree-of-freedom structure, which features an `inverted' shape that
snaps to a `natural' shape. Motivated by the anomalously slow snap-through
shown by spherical elastic caps, we consider a thought experiment in which the
truss is first indented to an inverted state and allowed to relax while a
specified displacement is maintained; the constraint of an imposed displacement
is then removed. Focussing on the dynamics for the limit in which the timescale
of viscous relaxation is much larger than the characteristic elastic timescale,
we show that two types of snap-through are possible: the truss either
immediately snaps back over the elastic timescale or it displays
`pseudo-bistability', in which it undergoes a slow creeping motion before
rapidly accelerating. In particular, we demonstrate that accurately determining
when pseudo-bistability occurs requires the consideration of inertial effects
immediately after the indentation force is removed. Our analysis also explains
many basic features of pseudo-bistability that have been observed previously in
experiments and numerical simulations; for example, we show that
pseudo-bistability occurs in a narrow parameter range at the bifurcation
between bistability and monostability, so that the dynamics is naturally
susceptible to critical slowing down. We then study an analogous thought
experiment performed on a continuous arch, showing that the qualitative
features of the snap-through dynamics are well captured by the truss model. In
addition, we analyse experimental and numerical data of viscoelastic
snap-through times reported in the literature. Combining these approaches
suggests that our conclusions may also extend to more complex viscoelastic
structures used in morphing applications.Comment: Main text 37 pages, Appendices 13 page
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