Remarks on the Emergence of Weak Euler Solutions in the Vanishing Viscosity Limit

Journal of Nonlinear Science - We prove that if the local second-order structure function exponents in the inertial range remain positive uniformly in viscosity, then any spacetime $$L^2$$ weak...     

The long term impact of a coherence based model for mathematics intervention

This article presents a large-scale longitudinal study of hundreds of students across the state of Kentucky that participated in a dual-focus mathematics intervention initiative when they were in the third grade. Rather than an exclusive focus on intervention, this initiative focused on both (i) high quality pull-out intervention and (ii) coherence between pull-out intervention and classroom instruction. The study found that over half of the third grade intervention students that participated in this initiative were classified as “novice” (the lowest possible performance category) on state standardized mathematics assessments at the end of the third grade. However, over the course of the following four years, the novice reduction rate of these students was significantly (p < .01) greater than other novices in Kentucky that did not participate in the initiative. These findings indicate that when implementing intervention initiatives to help students that are struggling with mathematics, it may be important to establish coherence between pull-out intervention and classroom instruction. The long term impact of this approach among traditionally underrepresented minorities suggest that this publication may provide insight into important equity issues where long-term analyses may sometimes be needed to capture the full impact of intervention initiatives.     

Trigonometric fitted modification of RADAU5

Implicit Runge-Kutta (RK) methods are in common use when addressing stiff initial value problems (IVP). They usually share the property of A-stability that is of crucial importance in solving the latter type of IVP. Radau IIA family of implicit RK methods is among the preferred ones. Especially its fifth-order representative named RADAU5 has received a lot of attention for use with lax accuracies. Here, we try the lesser possible perturbation of its coefficients. Then, we derive a trigonometric fitted modification that is intended to be applied in periodic IVPs. Numerical tests over a variety of problems with oscillatory solutions justify our effort.     

A Districtwide Study of Automaticity When Included in Concept‐Based Elementary School Mathematics Instruction

While conceptual understanding of properties, operations, and the base‐ten number system is certainly associated with the ability to access math facts fluently, the role of math fact memorization to promote conceptual understanding remains contested. In order to gain insight into this question, this study looks at the results when one of three elementary schools in a school district implements mandatory automaticity drills for 10 minutes each day while the remaining two elementary schools, with the same curriculum and very similar demographics, do not. This study looks at (a) the impact that schoolwide implementation of automaticity drills has on schoolwide computational math skills as measured by the ITBS and (b) the relationship between automaticity and conceptual understanding as measured by statewide standardized testing. The results suggest that while there may be an association between automaticity and higher performance on standardized tests, caution should be taken before assuming there are benefits to promoting automaticity drills. These results are consistent with those that support a process‐driven approach to automaticity based on familiarity with properties and strategies associated with the base‐ten number system; they are not consistent with those that support an answer‐driven approach to automaticity based on memorization of answers.     

Equilibrium for a combinatorial Ricci flow with generalized weights on a tetrahedron

Chow and Lou [2] showed in 2003 that under certain conditions the combinatorial analogue of the Hamilton Ricci flow on surfaces converges to Thruston’s circle packing metric of constant curvature. The combinatorial setting includes weights defined for edges of a triangulation. A crucial assumption in [2] was that the weights are nonnegative. We have recently shown that the same statement on convergence can be proved under weaker conditions: some weights can be negative and should satisfy certain inequalities. In this note we show that there are some restrictions for weakening the conditions. Namely, we show that in some situations the combinatorial Ricci flow has no equilibrium or has several points of equilibrium and, in particular, the convergence theorem is no longer valid.     

Increased Water Reduction Efficiency of Polyelectrolyte‐Bound Trimetallic [Ru,Rh,Ru] Photocatalysts in Air‐Saturated Aqueous Solutions

The groundbreaking use of polyelectrolytes to increase the efficiency of supramolecular photocatalysts in solar H₂ production schemes under aqueous aerobic conditions is reported. Supramolecular photocatalysts of the architecture [{(TL)₂Ru(BL)}₂RhX₂]⁵⁺ (BL=bridging ligand, TL=terminal ligand, X=halide) demonstrate high efficiencies in deoxygenated organic solvents but do not function in air‐saturated aqueous solution because of the quenching of the metal‐to‐ligand charge‐transfer (MLCT) excited state under these conditions. The new photocatalytic system incorporates poly(4‐styrenesulfonate) (PSS) into aqueous solutions containing [{(bpy)₂Ru(dpp)}₂RhCl₂]⁵⁺ (bpy=2,2′‐bipyridine, dpp=2,3‐bis(2‐pyridyl)pyrazine). PSS has a profound impact on the photocatalyst efficiency, increasing H₂ production over three times that of deoxygenated aqueous solutions alone. H₂ photocatalysis proceeds even under aerobic conditions for PSS‐containing solutions, an exciting consequence for solar hydrogen‐production research.     

A Simple Approach for Multi‐fidelity Experimentation Applied to Financial Engineering

In practical applications, information about the accuracy or ‘fidelity’ of alternative surrogate systems may be ambiguous and difficult to determine. To address this problem, we propose to treat surrogate system fidelity level as a categorical factor in optimal response surface design. To design the associated experiments, we apply the Expected Integrated Mean Squared Error optimal design criterion, which takes into account both variance and bias errors. The performance of the proposed design was compared using three test cases to four types of alternatives using the Empirical Integrated Squared Error. Because of its ability to foster relatively accurate predictions, the proposed design is recommended in fidelity experimental design, particularly when the experimenters lack sufficient information about the fidelity levels of surrogate systems. The method was applied to the case of intraday trading optimization in which data were collected from the Taiwan Futures Exchange. We also calculated the implied volatility from the Merton's Jump‐diffusion model via the fast Fourier transform algorithm with three different models of varying fidelity levels. Copyright © 2014 John Wiley & Sons, Ltd.     

Bidirectional soliton street

The bidirectional long-wave model introduced by Wu (1994)^[1] and Yih & Wu (1995)^[2] is applied to evaluate interactions between multiple solitary waves progressing in both directions in a uniform channel of rectangular cross-section and undergoing collisions of two classes, one being head-on and the other overtaking collisions between these solitons. For a binary head-on collision, the two interacting solitary waves are shown to merge during a phase-locking period from which they reemerge separated, each asymptotically recovering its own initial identity while both being retarded in phase from their original pathlines. For a binary overtaking collision between a soliton of height α₁ overtaking a weaker one of height α₁, the two solition peaks are shown to either pass through each other or remain separated throughout the encounter according as α₁/α₂ or <3, respectively. With no phase locking during the overtaking, the two solitary waves re-emerge afterwards with their initial forms recovered and with the stronger wave being advanced whereas the weaker one retarded in phase from their original pathlines. By extension, the theory is generalized to apply to uniform channels of arbitrary cross-sectional shape. The Inaugural Pei-Yuan Chou Memorial Lecture, presented at The Sixth Asian Congress of Fluid Mechanics. Singapore, 21–26 May 1995     

Statistical and Numerical Considerations of Backus-Average Product Approximation

In this paper, we examine the applicability of the approximation, \(\overline{f g}\approx \overline{f}\,\overline{g}\), within Backus (J. Geophys. Res. 67(11):4427–4440, 1962) averaging. This approximation is a crucial step in the method proposed by Backus (J. Geophys. Res. 67(11):4427–4440, 1962), which is widely used in studying wave propagation in layered Hookean solids. According to this approximation, the average of the product of a rapidly varying function and a slowly varying function is approximately equal to the product of the averages of those two functions.Considering that the rapidly varying function represents the mechanical properties of layers, we express it as a step function. The slowly varying function is continuous, since it represents the components of the stress or strain tensors. In this paper, beyond the upper bound of the error for that approximation, which is formulated by Bos et al. (J. Elast. 127:179–196, 2017), we provide a statistical analysis of the approximation by allowing the function values to be sampled from general distributions.Even though, according to the upper bound, Backus (J. Geophys. Res. 67(11):4427–4440, 1962) averaging might not appear as a viable approach, we show that—for cases representative of physical scenarios modelled by such an averaging—the approximation is typically quite good. We identify the cases for which there can be a deterioration in its efficacy.In particular, we examine a special case for which the approximation results in spurious values. However, such a case—though physically realizable—is not likely to appear in seismology, where Backus (J. Geophys. Res. 67(11):4427–4440, 1962) averaging is commonly used. Yet, such values might occur in material sciences, in general, for which Backus (J. Geophys. Res. 67(11):4427–4440, 1962) averaging is also considered.