Measure functions for frames

This paper addresses the natural question: “How should frames be compared?” We answer this question by quantifying the overcompleteness of all frames with the same index set. We introduce the concept of a frame measure function: a function which maps each frame to a continuous function. The comparison of these functions induces an equivalence and partial order that allows for a meaningful comparison of frames indexed by the same set. We define the ultrafilter measure function, an explicit frame measure function that we show is contained both algebraically and topologically inside all frame measure functions. We explore additional properties of frame measure functions, showing that they are additive on a large class of supersets—those that come from so called non-expansive frames. We apply our results to the Gabor setting, computing the frame measure function of Gabor frames and establishing a new result about supersets of Gabor frames.     

等滴光度法测定钼铁中的钼

根据ΔAp=KΔ（bc)p原理，采用“等滴光度法”测定钼铁中的钼，通过试验确定了最佳测定条件。该方法简便、快速，用于钼铁标准样品中钼的测定，测定结果的精密度为0.196%，与标准值之间的相对误差为0.064%（n=4)。     

Preparation and characterization of fused-silica capillary columns coated with <Emphasis Type="Italic">m</Emphasis>-carborane–siloxane copolymers for gas chromatography

The carborane–siloxane copolymers Dexsil 300, a 34.5% bis(dimethylsilyl)-m-carborane–65.5% dimethylsiloxane copolymer, and Dexsil 400, a 24.9% bis(dimethylsilyl)-m-carborane–50.8% dimethyl, 24.3% methylphenylsiloxane copolymer, were coated on fused silica capillary columns and their gas chromatographic properties were evaluated. Their selectivity was evaluated using both Rohrschneider–McReynolds constants and triacylglycerol indices. The bis(dimethylsilyl)-m-carborane unit turned out to be equivalent to two dimethylsiloxy units and one half of a diphenylsiloxy unit. The m-carborane unit was found to cause a 15–25 K shift in the elution temperature between 120 and 360 °C. The working range was from 20 and 0 °C to 380 °C for Dexsil 300 and Dexsil 400, respectively. The column bleeding levels at 380 °C were below 20 and 15 pA for Dexsil 300 and Dexsil 400, respectively.     

Development of a method for estimating an accurate equivalence point in nickel titration of cyanide ions

In a nickel titration of cyanide ions using murexide as indicator, an accurate equivalence point was determined by a non-linear least-squares curve-fitting for a titration curve. This method was developed to establish a standard solution for cyanide ions. In a curve-fitting procedure, a theoretical titration curve was calculated, assuming that nickel ion formed only a 1:4 Ni²⁺:CN⁻ complex with cyanide ions and formed only a 1:1 complex with murexide. Results of the curve-fitting were reasonable at any pH and any indicator concentration studied. The combined standard uncertainty for a concentration of a 1000 mg kg⁻¹ cyanide solution by this method was 0.079%.     

Neutrino Oscillations in Exotic Geometries and the Equivalence Principle Violation

Taking into account the violation of the equivalence principle, we analyze the propagation of neutrinos in the gravitational field generated by wormholes. The analysis is performed for massive and massless neutrinos. By using the constraints on the (negative) mass of wormholes, which is of the order of stellar-substellar masses, we determine constraints on f which characterizes the degree of violation of the equivalence principle.     

George Boole and the origins of invariant theory

Historians have repeatedly asserted that invariant theory was born in two papers of George Boole (1841 and 1842). Although several themes and techniques of 19th-century invariant theory are enunciated in this work, in reacting to it (and thereby founding the British school of invariant theory), Arthur Cayley shifted Boole's research program.     

Symmetry and transformation properties of iterative ordinary differential equation

Symmetries of linear iterative equations and new conditions on the infinitesimals are obtained. Regarding the expressions of the solutions in terms of the parameters of the source equation, an ansatz is made on the original parameters. We have also obtained an expression for the source parameters of the transformed equation under equivalence transformations. We conducted this work with a special emphasis on second-, third- and fourth-order equations, although some of our results are valid for equations of a general order.     

Symmetries and invariants of the systems of two linear second-order ordinary differential equations

For arbitrary systems of two linear second-order ordinary differential equations, the symmetry Lie algebra is described in terms of invariant theory, resulting in eleven non-equivalent symmetry types. The result is compared with the group classification approach recently obtained by different authors.     

Notions of equivalence through ratios: Students with and without learning disabilities

Students with learning disabilities (LD) specific to mathematics historically underperform in foundational content such as rational number equivalence. This study examined the strategy usage and multiplicative thinking of three third grade children (i.e., Bill, a child identified as having a learning disability specific to mathematics, Carl, a child labeled as low achieving in mathematics, and Albert, a child labeled as typically achieving) before, during, and after participating in tutoring sessions consisting of student-centered pedagogy and equivalence tasks presented through an underutilized interpretation of rational number: namely, the ratio interpretation. Constant comparison analysis of the children's work during the tutoring sessions as well as responses to tasks during two clinical interviews seemed to indicate that all three children increased their use of viable strategies, with notable differences in the sophistication of the strategies as well as the level of multiplicative thinking utilized before and after the ratio-based tutoring sessions. Yet, Bill's continued use of rudimentary strategies reflects a need for continued research to investigate why the use of such strategies persists and how supporting the development of more sophisticated strategies (especially among children with LD) can be achieved.     

Mathematical modeling on the effect of equivalence ratio in emission characteristics of compression ignition engine with hydrogen substitution

The investigation presented in this paper concerns on the computational simulation of emissions characteristics in compression ignition engine with hydrogen substitution. Combustion process has been modeled based on Equilibrium Constants Method (ECM) with MATLAB program to calculate the mole fractions of 18 combustion products when hydrogen is burnt along with diesel fuel at variable equivalence ratios. It can be observed that hydrogen substitution causes significant increase in NH₃, H₂, atom H emissions during rich combustion and OH, NO₂, HNO₃ emissions during lean combustion. As the equivalence ratio increases during rich combustion, mole fractions of HCN, CH₄, CO and atom C decreases with increment of hydrogen substitution. N₂, atom N and CO₂ emissions decrease whereas no significant changes in O₂, NO, O₃ and atom O emissions throughout all equivalence ratios as hydrogen is added to the combustion.