An integral model structure and truncation theory for coherent group actions

In this work we study the homotopy theory of coherent group actions from a global point of view, where we allow both the group and the space acted upon to vary. Using the model of Segal group actions and the model categorical Grothendieck construction we construct a model category encompassing all Segal group actions simultaneously. We then prove a global rectification result in this setting. We proceed to develop a general truncation theory for the model-categorical Grothendieck construction and apply it to the case of Segal group actions. We give a simple characterization of n-truncated Segal group actions and show that every Segal group action admits a convergent Postnikov tower built out of its n-truncations.     

Single-shot gradient-assisted photon echo electronic spectroscopy

Two-dimensional electronic spectroscopy (2D ES) maps the electronic structure of complex systems on a femtosecond time scale. While analogous to multidimensional NMR spectroscopy, 2D optical spectroscopy differs significantly in its implementation. Yet, 2D Fourier spectroscopies still require point-by-point sampling of the time delay between two pulses responsible for creating quantum coherence among states. Unlike NMR, achieving the requisite phase stability at optical frequencies between these pulse pairs remains experimentally challenging. Nonetheless, 2D optical spectroscopy has been successfully demonstrated by combining passive and active phase stabilization along with precise control of optical delays and long-term temperature stability, although the widespread adoption of 2D ES has been significantly hampered by these technical challenges. Here, we exploit an analogy to magnetic resonance imaging (MRI) to demonstrate a single-shot method capable of acquiring the entire 2D spectrum in a single laser shot using only conventional optics. Unlike point-by-point sampling protocols typically used to record 2D spectra, this method, which we call GRadient-Assisted Photon Echo (GRAPE) spectroscopy, largely eliminates phase errors while reducing the acquisition time by orders of magnitude. By incorporating a spatiotemporal encoding of the nonlinear polarization along the excitation frequency axis of the 2D spectrum, GRAPE spectroscopy achieves no loss in signal while simultaneously reducing overall noise. Here, we describe the principles of GRAPE spectroscopy and discuss associated experimental considerations.     

Asymptotic behavior of nonparametric estimators of the two-dimensional and bivariate renewal functions

Statistical Inference for Stochastic Processes - We study the asymptotic behavior of the two-dimensional or bivariate nonparametric estimators of the renewal function associated to a sequence of...     

Weak invariance of the multidimensional rank statistic for nonstationary absolutely regular processes

In this paper, we study the weak invariance of the multidimensional rank statistic when the underlying random variables are nonstationary absolutely regular.     

Attention to meaning by algebra teachers

Non-attendance to meaning by students is a prevalent phenomenon in school mathematics. Our goal is to investigate features of instruction that might account for this phenomenon. Drawing on a case study of two high school algebra teachers, we cite episodes from the classroom to illustrate particular teaching actions that de-emphasize meaning. We categorize these actions as pertaining to (a) purpose of new concepts, (b) distinctions in mathematics, (c) mathematical terminology, and (d) mathematical symbols. The specificity of the actions that we identify allows us to suggest several conjectures as to the impact of the teaching practices observed on student learning: that students will develop the belief that mathematics involves executing standard procedures much more than meaning and reasoning, that students will come to see mathematical definitions and results as coincidental or arbitrary, and that students’ treatment of symbols will be largely non-referential.     

Proton NMR studies of reduced porphyrin compounds

Detailed ¹H NMR studies of ms tetraphenylchlorins (H₂TPC), new amino- and hydroxypyrroline substituted ms tetraphenylchlorins and ms tetraphenylisobacteriochlorins (H₂TPisoB) are presented and discussed. The results obtained are consistent with the general aspects of the ring current models as applied to the parent porphyrins. According to proton chemical shifts a gradual reduction in the magnitude of the ring current is observed in the order ms tetraphenylporphyrin (H₂TPP) > H₂TPC> ms tetraphenylbacteriochlorin (H₂TPB) > H₂TPisoB. The ¹H NMR spectra show chemical non-equivalence of the pyrroline ring protons due to adjacent substituents, and effects of steric constraints on the aminoalkyl substituents due to the close vicinity to the meso phenyl rings. The non-equivalence of the methylene protons of the pyrroline ring produces geminal coupling between the two methylene protons and vicinal coupling with the adjacent pyrroline proton, more pronounced in H₂TPC? OH and in . Restricted configuration of the methylene groups in the ethyl groups of H₂TPC? C(H)(CH₃)N(CH₂CH₃)₂ produces observable geminal coupling between the methylene protons. ¹H NMR reveals the difference between two types of meso phenyls in the chlorins, and three types of meso phenyls in isobacteriochlorins, as reflected in the chemical shifts of the o-phenyl protons.     

Classroom-based interventions in mathematics education: relevance, significance, and applicability

This special issue discusses various pedagogical innovations and myriad of significant findings. This commentary is not a synthesis of these contributions, but a summary of my own reflections on selected aspects of the nine papers comprising the special issue. Four themes subsume these reflections: (1) Gestural Communication (Alibali, Nathan, Church, Wolfgram, Kim and Knuth 2013); (2) Development of Ways of Thinking (Jahnke and Wambach 2013; Lehrer, Kobiela and Weinberg 2013; Mariotti 2013; Roberts and A. Stylianides 2013; Shilling-Traina and G. Stylianides 2013; Tabach, Hershkowitz and Dreyfus 2013); (3) Learning Mathematics through Representation (Saxe, Diakow and Gearhart 2013); and (4) Challenges in Dialogic Teaching (Ruthven and Hofmann 2013).     

Optimizing parametric oscillators with tunable boundary conditions

Parametric excitation or pumping is an effective method to create large oscillations by periodically altering a physical parameter of the governing dynamics. Precisely tuned pumping frequencies can lead to exponentially growing oscillations limited only by nonlinear effects like axial stretching of transversely vibrating string. It is demonstrated that a tuned passive dynamical system amplifies the otherwise limited transverse vibrations amplitudes of a nonlinear string considerably and thus increasing the selectivity of the system. This effect becomes more noticeable for shorter wavelengths where nonlinear stretching limits the obtainable vibration amplitudes severely. The present work analyses a passive dynamical system connected to one end of a taught string which parametrically couples its axial motion to transverse vibration. Analysis shows that a specific selection of parameters can reduce the limiting effect of nonlinear stretching thus allowing one to excite high-order modes with small external forces. The result can possibly affect other disciplines where effective parametric amplification is necessary.     

Quantifying the diffusion of a fluid through membranes by double phase encoded remote detection magnetic resonance imaging

We demonstrate that a position correlation magnetic resonance imaging (MRI) experiment based on two phase encoding steps separated by a delay can be used for quantifying diffusion across a membrane. This method is noninvasive, and no tracer substance or concentration gradient across the membrane is required. Because, in typical membranes, the T1 relaxation time of the fluid spins is usually much longer than the T2 time, we developed and implemented a new position correlation experiment based on a stimulated spin-echo, in which the relaxation attenuation of the signal is dominated by T1 instead of T2. This enables using relatively long delays needed in the diffusion measurements. The sensitivity of the double encoded experiment detected in a conventional way is still low because of the low filling factor of the fluid inside the NMR coil around the sample. We circumvent this problem by using the remote detection technique, which significantly increases the sensitivity, making it possible to do the measurements with gaseous fluids that have a low spin-density compared to liquids. We derive a model that enables us to extract a diffusion constant characterizing the diffusion rate through the membrane from the obtained correlation images. The double phase encoded MRI method is advantageous in any kind of diffusion studies, because the propagator of fluid molecules can directly be seen from the correlation image.