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This work demonstrates that two systematic errors, coherent betatron oscillations (CBO) and muon losses, can be reduced through application of radio frequency (RF) electric fields, which ultimately increases the sensitivity of the muon g − 2 experiments. As the ensemble of polarized muons goes around a weak focusing storage ring, their spin precesses, and when they decay through the weak interaction, μ + → e + ν e ν μ ̄ , the decay positrons are detected by electromagnetic calorimeters. In addition to the expected exponential decay in the positron time spectrum, the weak decay asymmetry causes a modulation in the number of positrons in a selected energy range at the difference frequency between the spin and cyclotron frequencies, ω a. This frequency is directly proportional to the magnetic anomaly a μ = (g − 2)/2, where g is the g-factor of the muon, which is slightly greater than 2. The detector acceptance depends on the radial position of the muon decay, so the CBO of the muon bunch following injection into the storage ring modulate the measured muon signal with the frequency ω CBO. In addition, the muon populations at the edge of the beam hit the walls of the vacuum chamber before decaying, which also affects the signal. Thus, reduction of CBO and unwanted muon loss increases the a μ measurement sensitivity. Numerical and experimental studies with RF electric fields yield more than a magnitude reduction of the CBO, with muon losses comparable to the conventional method.

Develop a deep understanding of mathematics by grasping the context and purpose behind various strategies. This user-friendly resource presents high school teachers with a logical progression of pedagogical actions, classroom norms, and collaborative teacher team efforts to increase their knowledge and improve mathematics instruction. Explore strategies and techniques to effectively learn and teach significant mathematics concepts and provide all students with the precise, accurate information they need to achieve academic success. Combine student understanding of functions and algebraic concepts so that they can better decipher the world. Benefits * Dig deep into mathematical modeling and reasoning to improve as both a learner and teacher of mathematics. * Explore how to develop, select, or modify mathematics tasks in order to balance cognitive demand and engage students. * Discover the three important norms to uphold in all mathematics classrooms. * Learn to apply the tasks, questioning, and evidence (TQE) process to ensure mathematics instruction is focused, coherent, and rigorous. * Gain clarity about the most productive progression of mathematical teaching and learning for high school. * Watch short videos that show what classrooms that are developing mathematical understanding should look like. Contents Introduction 1. Equations and Functions 2. Structure of Equations 3. Geometry 4. Types of Functions 5. Function Modeling 6. Statistics and Probability Epilogue: Next Steps Appendix: Weight Loss Study Data References Index

A long in-plane beam polarization can be a desired feature for spin measurement experiments in storage rings. The spin precession of the particles within a beam can be controlled by means of the frozen spin method and beam bunching via RF cavities, eventually yielding a polarization lifetime of 10--100 seconds. Previous studies have shown that it can be further improved by sextupoles, which correct the second order effects related to the chromaticity of the beam. However, sextupoles can require readjustment after slight changes in ring parameters. This work presents a real-time sextupole tuning method that relies on a feedback algorithm. It adjusts the sextupole strength during storage, targeting a zero average radial spin component. Satisfying this condition results in a longer polarization lifetime. Simulation studies show that roughly determined feedback coefficients in this method work effectively for a wide range of ring parameters, with practical field imperfections and measurement errors taken into account. Alternatively, this technique can be used to optimize sextupole strengths in a test run without intervening the measurement.

The present study sought to design calculus tasks to determine students' preference for visual or analytic processing as well as examine the role of preferred mode of processing in calculus performance and its relationship to spatial ability and verbal-logical reasoning ability. Data were collected from 150 high school students who were enrolled in Advanced Placement calculus courses. The measures of preferred mode of processing did not correlate with the measures of spatial ability and verbal-logical reasoning ability, suggesting that cognitive abilities did not predict the students' preference for visual or analytic processing. Multiple regression analysis revealed that spatial visualization ability, verbal-logical reasoning ability, preference for visual processing contributed significantly to the variance in calculus performance. Correlations between calculus performance and the measures of preferred mode processing suggest that the nature and complexity of mathematical tasks might have influenced the students' degree of preference for using visual processing.

This paper presents studies with regards to an asymmetric electrostatic quadrupole profile, which helps suppressing image beam induced vertical electric fields. These fields need to remain orders of magnitude smaller than the radial electric field in storage ring electric dipole moment experiments. Otherwise, the coupling with the magnetic dipole moment dominates the spin precession, eventually leading to a false signal. Suppression performance of an optimized quadrupole profile is investigated for various beam offset and size configurations. With the optimum profile, the vertical electric field can be reduced by two orders of magnitude. Several centimeter range beam size and position offsets have been observed to have small effects on the field reduction. Besides the quadrupole elements, using non-magnetic conductor materials, this kind of a profile could be implemented at other sections of accelerators as well.

We present a non-destructive beam profile imaging concept that utilizes machine learning tools, namely genetic algorithm with a gradient descent-like minimization. Electromagnetic fields around a charged beam carry information about its transverse profile. The electrodes of a stripline-type beam position monitor (with eight probes in this study) can pick up that information for visualization of the beam profile. We use a genetic algorithm to transform an arbitrary Gaussian beam in such a way that it eventually reconstructs the transverse position and the shape of the original beam. The algorithm requires a signal that is picked up by the stripline electrodes, and a (precise or approximate) knowledge of the beam size. It can visualize the profile of fairly distorted beams as well.

A new, hybrid design is proposed to eliminate the main systematic errors in the frozen spin, storage ring measurement of the proton electric dipole moment. In this design, electric bending plates steer the particles, and magnetic focusing replaces electric. The magnetic focusing should permit simultaneous clock-wise and counter-clock-wise storage to cancel systematic errors related to the out-of-plane dipole electric field. Errors related to the quadrupole electric fields can be eliminated by successive runs of magnetic focusing with different strengths.

This work demonstrates that two systematic errors, coherent betatron oscillations (CBO) and muon losses can be reduced through application of radio frequency (RF) electric fields, which ultimately increases the sensitivity of the muon g-2 experiments. As the ensemble of polarized muons goes around a weak focusing storage ring, their spin precesses, and when they decay through the weak interaction, μ⁺ → e⁺ νₑ ν̄_(μ), the decay positrons are detected by electromagnetic calorimeters. In addition to the expected exponential decay in the positron time spectrum, the weak decay asymmetry causes a modulation in the number of positrons in a selected energy range at the difference frequency between the spin and cyclotron frequencies, ω_(\\text{a}{}{.}) This frequency is directly proportional to the magnetic anomaly a_(μ) =(g-2)/2, where g is the g-factor of the muon, which is slightly greater than 2. The detector acceptance depends on the radial position of the muon decay, so the CBO of the muon bunch following injection into the storage ring modulate the measured muon signal with the frequency ω_(\\text{CBO}{}{.}) In addition, the muon populations at the edge of the beam hit the walls of the vacuum chamber before decaying, which also affects the signal. Thus, reduction of CBO and unwanted muon loss increases the a_(μ) measurement sensitivity. Numerical and experimental studies with RF electric fields yield more than a magnitude reduction of the CBO, with muon losses comparable to the conventional method.

The present study investigated the object-spatial visualization and verbal cognitive styles among high school students and related differences in spatial ability, verbal-logical reasoning ability, and mathematical performance of those students. Data were collected from 348 students enrolled in Advanced Placement calculus courses at six high schools. Correlational analysis revealed that spatial ability, verbal-logical reasoning ability, and mathematical performance were significantly correlated with each other. High spatial visualizers had significantly higher spatial ability and mathematical performance scores than high object visualizers. No significant differences were found between verbalizers and high spatial visualizers in their verbal-logical reasoning ability and mathematical performance scores. Results provide support for the existence of two contrasting groups of visualizers with respect to their spatial ability.

We show that proton storage ring experiments designed to search for proton electric dipole moments can also be used to look for the nearly dc spin precession induced by dark energy and ultra-light dark matter. These experiments are sensitive to both axion-like and vector fields. Current technology permits probes of these phenomena up to three orders of magnitude beyond astrophysical limits. The relativistic boost of the protons in these rings allows this scheme to have sensitivities comparable to atomic co-magnetometer experiments that can also probe similar phenomena. These complementary approaches can be used to extract the micro-physics of a signal, allowing us to distinguish between pseudo-scalar, magnetic and electric dipole moment interactions.