Secondary Options and Post‐secondary Expectations: Standards‐based Mathematics Programs and Student Achievement on College Mathematics Placement Exams

Research on student achievement within the University of Chicago School Mathematics Project (UCSMP) and Core‐Plus Mathematics Project (CPMP) at the secondary level is beginning to accumulate, however, much less is known about how prepared these students are for post‐secondary education. Therefore this study involving students within one tracked school district used multiple linear regression to examine the role of differential experience within two secondary Standards‐based mathematics programs, gender, and prior mathematics achievement on college algebra and calculus readiness placement test scores. Results show that there are no significant differences between students who had completed three and four years of the CPMP curriculum. UCSMP students with four or five years of experience significantly outperformed CPMP students on both assessments. Prior achievement was a significant predictor of student achievement on both examinations. Male students outperformed female students on the algebra placement exam. Students who had studied from both CPMP and UCSMP significantly outperformed students who had studied from CPMP for four years on the calculus readiness examination.     

Minkowski iteration of sets

The Minkowski addition, ⊕, is a natural generalization of vector addition. However, ⊕ and scalar multiplication do not follow all the usual laws of vector space operations. This is reflected upon the properties of a new operation, °, which maps vector sets into vector sets. The study of a linear iterative process (with ° acting recurrently on vector sets) brings out the outstanding value of vector balls and convex hulls for obtaining explicit solutions or bounds.     

An infinite family of convex Brunnian links in $${mathbb{R}^n}$$

This paper proves that convex Brunnian links exist for every dimension n ≥ 3 by constructing explicit examples. These examples are three-component links which are higher-dimensional generalizations of the Borromean rings.     

Low Order Modelling of Freely Vibrating Flexible Cables

A highly efficient, low-order model for vortex-induced vibration of flexible cables at low Reynolds number has been developed. A coupled map lattice (CML) wake model, based on circle maps, is combined with a cable model (linear wave equation). With this model, both the self-excited response of the cable oscillation in the crossflow direction and the resulting wake structures are examined at Re = 100. This represents an improvement over earlier CML models capable of only modelling wakes behind externally forced cables. For comparison, spectral-element based numerical simulations, in addition to experiments on a freely vibrating cable flow, were conducted. The freely vibrating CML model predicted lace-like structures consistent with these numerical simulations and experimental results. Little difference was noted between the vortex shedding wake patterns determined for forced cables (from previous CML models) and freely vibrating cables. The freely vibrating CML matched experimental data for maximum cable amplitude reasonably well over a range of mass-damping parameter employed in offshore structural applications.     

Synchronization of chaotic mode hopping

We propose and demonstrate a scheme for generating synchronized chaotic mode hopping in two wavelength-tunable lasers whose tuning range covers multiple longitudinal modes. Chaotic mode hopping resulting in large hops in wavelength is induced by delayed electrical feedback. We show that, by coupling part of the output of one laser into another, one can synchronize the chaotic mode hopping of two separate lasers and obtain synchronized chaotic on-off modulation patterns in multiple corresponding wavelength bands.     

In vitro comparative study of vibro-acoustography versus pulse-echo ultrasound in imaging permanent prostate brachytherapy seeds

Background

Permanent prostate brachytherapy (PPB) is a common treatment for early stage prostate cancer. While the modern approach using trans-rectal ultrasound guidance has demonstrated excellent outcome, the efficacy of PPB depends on achieving complete radiation dose coverage of the prostate by obtaining a proper radiation source (seed) distribution. Currently, brachytherapy seed placement is guided by trans-rectal ultrasound imaging and fluoroscopy. A significant percentage of seeds are not detected by trans-rectal ultrasound because certain seed orientations are invisible making accurate intra-operative feedback of radiation dosimetry very difficult, if not impossible. Therefore, intra-operative correction of suboptimal seed distributions cannot easily be done with current methods. Vibro-acoustography (VA) is an imaging modality that is capable of imaging solids at any orientation, and the resulting images are speckle free.

Objective and methods

The purpose of this study is to compare the capabilities of VA and pulse-echo ultrasound in imaging PPB seeds at various angles and show the sensitivity of detection to seed orientation. In the VA experiment, two intersecting ultrasound beams driven at f₁ = 3.00 MHz and f₂ = 3.020 MHz respectively were focused on the seeds attached to a latex membrane while the amplitude of the acoustic emission produced at the difference frequency 20 kHz was detected by a low frequency hydrophone.

Results

Finite element simulations and results of experiments conducted under well-controlled conditions in a water tank on a series of seeds indicate that the seeds can be detected at any orientation with VA, whereas pulse-echo ultrasound is very sensitive to the seed orientation.

Conclusion

It is concluded that vibro-acoustography is superior to pulse-echo ultrasound for detection of PPB seeds.     

Newtonian Dynamics from the Principle of Maximum Caliber

The foundations of Statistical Mechanics can be recovered almost in their entirety from the principle of maximum entropy. In this work we show that its non-equilibrium generalization, the principle of maximum caliber (Jaynes, Phys Rev 106:620–630, 1957), when applied to the unknown trajectory followed by a particle, leads to Newton’s second law under two quite intuitive assumptions (both the expected square displacement in one step and the spatial probability distribution of the particle are known at all times). Our derivation explicitly highlights the role of mass as an emergent measure of the fluctuations in velocity (inertia) and the origin of potential energy as a manifestation of spatial correlations. According to our findings, the application of Newton’s equations is not limited to mechanical systems, and therefore could be used in modelling ecological, financial and biological systems, among others.