Sobolev exponents of Butterworth refinable functions

AbstractThe precise Sobolev exponent s∞(φn) of the Butterworth refinable function φn associated with the Butterworth filter of order n, bn(ξ)≔cos2n(ξ/2)cos2n(ξ/2)+sin2n(ξ/2), is shown to be s∞(φn)=nlog23+log2(1+3−n). This recovers the previously given asymptotic estimate of s∞(φn) of Fan and Sun, and gives more accurate regularity of Butterworth refinable function φn

A potentially pure test of cosmic geometry: galaxy clusters and the real space Alcock-Paczynski test

We investigate the possibility of probing dark energy by measuring the isotropy of the galaxy cluster autocorrelation function (an Alcock-Paczynski test). The correlation function is distorted in redshift space because of the cluster peculiar velocities, but if these are known and can be subtracted, the correlation function measurement becomes in principle a pure test of cosmic geometry. Galaxy cluster peculiar velocities can be measured using the kinetic Sunyaev Zel'dovich (kSZ) effect. Upcoming CMB surveys, e.g., ACT, SPT, Planck, are expected to do this with varying levels of accuracy, dependent on systematic errors due to cluster temperature measurements, radio point sources, and the primary CMB anisotropy. We use the Hubble volume N-body simulation and the hydrodynamic simulation results of Nagai et. al (2003) to simulate various kSZ surveys. We find by model fitting that a measurement of the correlation function distortion can be used to recover the cosmological parameters that have been used to generate the simulation. However, the low space density of galaxy clusters requires larger surveys than are taking place at present to place tight constraints on cosmology. For example, with the SPT and ACT surveys, Omega_Lambda could be measured to within 0.1 and 0.2 respectively at one sigma, but only upper limits on the equation of state parameter w will be possible. Nevertheless, with accurate measurements of the kSZ effect, this test can eventually be used to probe the dark energy equation of state and its evolution with redshift, with different systematic errors than other methods.Comment: 13 pages, 15 figures, Monthly Notices in pres

On the Gabor frame set for compactly supported continuous functions

We identify a class of continuous compactly supported functions for which the known part of the Gabor frame set can be extended. At least for functions with support on an interval of length two, the curve determining the set touches the known obstructions. Easy verifiable sufficient conditions for a function to belong to the class are derived, and it is shown that the B-splines (Formula presented.) , (Formula presented.) , and certain ???continuous and truncated??? versions of several classical functions (e.g., the Gaussian and the two-sided exponential function) belong to the class. The sufficient conditions for the frame property guarantees the existence of a dual window with a prescribed size of the support.ope

Multiple Levels of Metacognition: Circumstances Interfering with Students’ Spontaneous Metacognitive Activities

A theoretical model of metacognition in complex modeling activities has been developed based on existing frameworks, by synthesizing the reconceptualization of metacognition at multiple levels by looking at the three sources that trigger metacognition. Using the theoretical model as a framework, this multiple-case study explores students’ spontaneous metacognitive activities while they collaboratively solve complex mathematical modeling tasks. This study used a series of model-eliciting activities—a type of problem-solving activity in which participants are required to verbalize their thoughts while working within a group—as an authentic method for analyzing verbal metacognitive actions. This study identified the circumstances facilitating or interfering with students’ spontaneous metacognitive activities. The findings of the study enrich our understanding of how to design metacognitive learning environments. The current study has the potential to guide teachers, teacher educators, and curriculum developers to create circumstances that support students’ spontaneous development of metacognitive abilities. It also has the potential to guide the development of effective instructional methods to integrate these circumstances into existing curricula

Creating a Virtual World for Mathematics

A virtual world was created using the popular sandbox game Minecraft to support the development of preservice teachers’ knowledge for teaching mathematics. Preservice teachers explored the virtual world for a geometry activity involving area and volume problems. They then discussed how this integration of technology could support students’ effective learning of mathematics in a meaningful way. The findings of the study demonstrated that to a certain extent the Minecraft activity supported the transfer of knowledge from preservice teachers’ mathematics content knowledge to their mathematics pedagogical and instructional practice knowledge. Preservice teachers appreciated the usefulness and effectiveness of the Minecraft activity in enhancing the teaching and learning of mathematics by visualizing mathematical concepts in the virtual world. This integration of technology also gave them an opportunity for professional growth. Although this study focuses on preservice teachers’ perspectives on the Minecraft activity, the technology integration using Minecraft will also be beneficial for students because it engages them in active and discovery learning

The Persistent Difficulty of Early Fraction Ideas in Early Secondary School Mathematics

This study explored the nature of difficulties of seventh and eighth grade students who struggled building their conceptual understanding of early fraction ideas, in particular ordering fractions. The participants engaged in a sequence of lessons that involved the use of fraction circles. The intervention of four weekly class sessions was adapted from the Rational Number Project (RNP) curriculum that has been created for and refined through teaching experiments in the RNP research since 1979. Pre and post group interviews were conducted with each student group for a sufficient identification of the nature of the students’ difficulties. This study identified the whole number dominance strategies used by the students for ordering fractions before and even after the intervention. The students also revealed minimal use of informal ordering strategies that involve more conceptual than procedural understanding of the concept of initial fraction ideas. Considering the short intervention, there was subtle (but meaningful) evidence for a positive influence of the fraction circle model developed within the RNP on students’ developing understanding of early fraction ideas. This study suggests the need of a remedial intervention for early secondary students showing the persistent difficulty with early fraction ideas. Students need to be given enough time with not only concrete models but also appropriate usage of language to support a complete understanding of how to use the models. Keywords: Rational Number Project (RNP), fraction circle model, initial fraction ideas, ordering fraction

Regularity of Dual Gabor Windows

We present a construction of dual windows associated with Gabor frames with compactly supported windows. The size of the support of the dual windows is comparable to that of the given window. Under certain conditions, we prove that there exist dual windows with higher regularity than the canonical dual window. On the other hand, there are cases where no differentiable dual window exists, even in the overcomplete case. As a special case of our results, we show that there exists a common smooth dual window for an interesting class of Gabor frames. In particular, for any value of K∈ℕ, there is a smooth function h which simultaneously is a dual window for all B-spline generated Gabor frames {EmbTnBN(x/2)}m,n∈ℕ for B-splines BN of order N=1,…,2K+1 with a fixed and sufficiently small value of b