Mathematical theory of signal analysis vs. complex analysis method of harmonic analysis

We present recent work of harmonic and signal analysis based on the complex Hardy space approach.     

Design of a High Throughput 25‐Well Parallel Electrolyzer for the Accelerated Discovery of CO2 Reduction Catalysts via a Combinatorial Approach

A combinatorial approach is described for screening homogeneous CO₂ reduction electrocatalysts by means of a high‐throughput 25‐well parallel electrolyzer. Active catalysts were then screened by three methods: (1) the presence of CO gas in the electrolyzer head space; (2) a pH change for the individual solutions before and after bulk electrolysis due to proton consumption during CO₂ reduction; and (3) other physical attributes. Based on these screening methods, two solutions out of the 25 were selected for further study. One out of two selected solutions, nickel cyclam, a well‐known CO₂ reduction electrocatalyst, was correctly identified to be the most active.     

A steady barycentric Lagrange interpolation method for the 2D higher‐order time‐fractional telegraph equation with nonlocal boundary condition with error analysis

In this paper, we consider the numerical solution of the time‐fractional telegraph equation with a nonlocal boundary condition. A novel barycentric Lagrange interpolation collocation method is developed to solve this equation. Two difficulties have been sorted: the singularity of the integration and the higher accuracy. At the same, we put forward a steady barycentric Lagrange interpolation technique to overcome the new “Runge” phenomenon in computation. Error estimates of the barycentric Lagrange interpolation and the time‐fractional telegraph system for the present method are presented in Sobolev spaces. High convergence rates of the proposed method are obtained and are consisted with the numerical values. Especially in the time dimension, we get the error bound, for h‐refinement and for n_t‐density in the L₂ norms. The numerical results obtained show that the proposed numerical algorithm is accurate and computationally efficient for solving time‐fractional telegraph equation. Experiments demonstrate the high convergence rates of the proposed method are consisted with the theoretical values.     

Reductive Methylation of Homogeneous Primary β-Lauryl/myristyl 7/3 Polyethyleneoxy n = 3–18 Ethylamines under Phase-Transfer Catalysis Conditions

Homogeneous tertiary N,N-dimethyl-N-β-lauryl/myristyl 7/3 polyethyleneoxy n = 3–18 ethylamines, LM(EO)_nAT, are niche intermediates in the synthesis of homogeneous N-alkyl (C₁–C₁₈)-N,N-dimethyl-N-β-lauryl/myristyl 7/3 polyethyleneoxy n = 3–18 ethylammonium chlorides (unitary degree of oligomerization of ethylene oxide in the polyoxyethylene chain). This paper synthetically presents the dependence of the reductive methylation yields of homogeneous primary β-lauryl/myristyl 7/3 polyethyleneoxy n = 3–18 ethylamines, LM(EO)_nAP, on the reaction time (10–90 min), the temperature (70 °C), the molar ratio formic aldehyde /LM(EO)_nAP (1.1/1–2.5/1), the molar ratio HCOOH/LM(EO)_nAP (5/1), the degree of oligomerization of ethylene oxide in the homogeneous polyoxyethylene chain in the 3,6,9,12,18 series, and the structure of the phase-transfer catalysts. The steric effects of hydrophobic groups CH₃ and C₁₈H₃₇ grafted onto the ammonium function, and the micellar phenomena in the vicinity of their critical micellar concentration, directly proportional to the homogeneous degree of oligomerization, were highlighted. In all cases, a steady increase in reductive methylation yields was observed, with even quantitative values obtained. The high purity of the homologous series LM(EO)_nAT will allow their personalization as reference structures for the study of the evolution of basic colloidal characteristics useful in forecasting technological applications. LM(EO)_nAP were obtained either by direct amidoethylation (nucleophilic addition under basic catalysis of homogeneous lauryl/myristyl 7/3 polyethoxylated n = 3, 6, 9, 12, 18 alcohols, LM(EO)_nOH, to acrylamide monomer) or by cyanoethylation of LM(EO)_nOH under basic catalysis at 25–50 °C, in the presence of Fe²⁺ cations as oligomerization/polymerization inhibitor, followed by partial acid hydrolysis of homogeneous β-alkyl (C₁₂H₂₅/C₁₄H₂₉) 7/3 polyethyleneoxy n = 3, 6, 9, 12, 18 propionitriles, LM(EO)_nPN, to β-alkyl (C₁₂H₂₅/C₁₄H₂₉) 7/3 polyethyleneoxy n = 3, 6, 9, 12, 18 propionamides, LM(EO)_nPD, which led to LM(EO)_nAP by Hoffmann degradation. Homogeneous higher tertiary polyetheramines LM(EO)_nAT were structurally characterized.     

Interpreting a graph and constructing its derivative graph: stability and change in students’ conceptions

This present study investigated engineering students’ conceptions and misconceptions related to derivative, particularly interpreting the graph of a function and constructing its derivative graph. Participants were 147 first year engineering students from four universities enrolled in first year undergraduate calculus courses with or without the incorporation of computers for the purposes of seeing the power of visualization, investigating worked examples given in steps and solving various questions related to the worked examples, assisting conceptual understanding, and/or providing feedback besides lectures in the classroom. Students were tested before and after being exposed to instruction on differentiation and integration by a diagnostic test measuring their understanding of major aspects of calculus. Follow-up interviews were conducted with 18 students. Analyses of the results revealed that A-level student's performance was improving more than non-A-level students, particularly in computer groups. The analyses of the students’ written and oral responses in all groups indicated that prototypes, poor understanding of the notion of limit, confusion between the process and the product, and difficulties in using graphical information to give meaning to symbolic representation account for the errors and the misconceptions identified.     

A role for M -matrices in modelling population growth

Adopting a discrete-time cohort-type model to represent the dynamics of a population, the problem of achieving a desired total size of the population under a balanced growth (contraction) and the problem of maintaining the desired size, once achieved, are studied. Properties of positive-time systems and M-matrices are used to develop the results, which are illustrated using simple examples. The material is presented in a format that makes it appropriate as background material for interesting applications based modelling assignments on undergraduate programmes in the mathematical sciences, applied sciences, economics and technology. All relevant properties of both positive-time systems and M-matrices are presented and clarified at the outset.     

Solvability of the ideal of all weight zero elements in bernstein algebras

We use a computer to verify that the ideal N of all weight zero elements of any (not necessarily finite dimensional) Bernstein algebra is solvable of index ≤4. We also use a computer to verify that N ² is nilpotent of index ≤9. We give three examples of Bernstein algebras which show that various hypotheses like finite dimensionality, finitely generatedA ² = A, are separately not enough to force N to be nilpotent.     

Further Theoretical Investigations for Laser Relative Intensity Noise with Fiber Nonlinearities and Higher Order Dispersion

In this article, we theoretically investigate relative intensity noise (RIN) in optical communication systems with fiber nonlinearities due to optical Kerr effects and higher order dispersion. The impact of modulation frequencies, launch power, and laser bias current on RIN has been illustrated. We show that RIN increases with modulating frequencies up to the resonance frequency and launch power, and decreases in the laser bias current. We also show that higher order dispersion terms have no impact on the RIN, but with first order dispersion compensation the higher order dispersion terms have significant impact at high modulating frequencies. The RIN with and without fiber nonlinearities is further investigated. It has been shown that the RIN with fiber nonlinearity is more than the RIN without nonlinearity and the effect of nonlinearity appears at higher modulation frequencies only.