Mathematical Treatment Research Articles

Measurement Uncertainty and Risk of False Compliance Assessment Applied to Carbon Isotopic Analyses in Natural Gas Exploratory Evaluation.

The concept of uncertainty in an isotopic analysis is not uniform in the scientific community worldwide and can compromise the risk of false compliance assessment applied to carbon isotopic analyses in natural gas exploratory evaluation. In this work, we demonstrated a way to calculate one of the main sources of this uncertainty, which is underestimated in most studies focusing on gas analysis: the δ13C calculation itself is primarily based on the raw analytical data. The carbon isotopic composition of methane, ethane, propane, and CO2 was measured. After a detailed mathematical treatment, the corresponding expanded uncertainties for each analyte were calculated. Next, for the systematic isotopic characterization of the two gas standards, we calculated the standard uncertainty, intermediary precision, combined standard uncertainty, and finally, the expanded uncertainty for methane, ethane, propane, and CO2. We have found an expanded uncertainty value of 1.8‰ for all compounds, except for propane, where a value of 1.6‰ was obtained. The expanded uncertainty values calculated with the approach shown in this study reveal that the error arising from the application of delta calculation algorithms cannot be neglected, and the obtained values are higher than 0.5‰, usually considered as the accepted uncertainty associated with the GC-IRMS analyses. Finally, based on the use of uncertainty information to evaluate the risk of false compliance, the lower and upper acceptance limits for the carbon isotopic analysis of methane in natural gas are calculated, considering the exploratory limits between -55‰ and -50‰: (i) for the underestimated current uncertainty of 0.5‰, the lower and upper acceptance limits, respectively, are -54.6‰ and -50.4‰; and (ii) for the proposed realistic uncertainty of 1.8‰, the lower and upper acceptance limits would be more restrictive; i.e., -53.5‰ and -51.5‰, respectively.

Geometryczne dane niwelacyjne i niektóre błędy systematyczne w ich przetwarzaniu

The aim of this article is to illuminate some latent systematic faults in the mathematical treatment of precise levelling data. The first one is associated with the use of the average of both measurements of the height differences between the terminal benchmarks in levelling lines. Another weak point in the classical treatment of levelling data is the incomplete minimization of the impact of the spatial network configuration on the produced mean standard errors of the nodal benchmarks from the adjustment. Generating sixty random paired samples of size 1000, derived from three continuous distributions, e.g. Normal (0, 1), Uniform (-1.732, 1.732) and Gamma (1, 1), it was found that the average of two same distributed and ordered observations is very nearby to the theoretical expectation, in comparison to both observations, only in approximately 27-30% of all cases. Contrary, in other 70-74% of cases, either the “first” or the “second” observation is in close proximity to the expectation. The miss of this fact leads to a statistically significant deterioration of the final accuracy of the levelling networks. In the current study, it is also shown that the minimization of the standard errors of the adjusted normal heights of the nodal benchmarks in the Bulgarian Levelling Network 1980 cannot be achieved with the weights w=const.L-1, which are the most popular and used type of weights in the adjustment of geometric levelling networks. Finally, it is illustrated that taking into account the above marks and applying an appropriate adjustment algorithm, the mean of the standard errors of the adjusted heights of the nodal benchmarks in the analysed network is possible to be less than 1mm. The standard error of the adjusted height of the most remoted benchmark “Pushkarov”, which is 598 km far away from the datum point located in Varna, is equal to 1.40mm. The obtained from the adjustment mean standard error for the weight unit is estimated to be 0.164 mm/√km. In comparison, the adjustment mean standard error for the weight unit, but yielded by the classical approach of adjustment of the analysed network, is 1.289 mm/√km or almost 9 times higher. Despite being tedious and time-consuming, it is not on point of discarding the precise geometric levelling as a main geodetic method for solving of a couple of scientific and engineering tasks, where differences in heights have to be determined with the highest accuracy.

Mathematical processing of RGB data in microfluidic paper-based analytical devices

Microfluidic paper-based analytical devices often are combined with scanners as detectors. In this work, different scanning options offered by scanners: resolution, scanning mode, exposure to radiation, colour restoration, and saving format were tested. Moreover, different attempts to mathematical data treatment based on intensities of three channels—Red, Green and Blue, were studied. All measurements presented in this article were conducted for a model dye—bromothymol blue and a model analyte—zinc(II) ion (complexed with xylenol orange in a paper matrix). The article summarizes the scanning options and possibilities of mathematical calculations. Nevertheless, it is suggested that the best option is to use the prior prepared calculation file to paste obtained intensities and compare all presented in this article (and the most frequently used) equations to process intensities and decide which one should be used in the particular analysis.

Complex-Valued FastICA Estimator with a Weighted Unitary Constraint: A Robust and Equivariant Estimator

Independent component analysis (ICA), as a statistical and computational approach, has been successfully applied to digital signal processing. Performance analysis for the ICA approach is perceived as a challenging task to work on. This contribution concerns the complex-valued FastICA algorithm in the range of ICA over the complex number domain. The focus is on the robust and equivariant behavior analysis of the complex-valued FastICA estimator. Although the complex-valued FastICA algorithm as well as its derivatives have been widely used methods for approaching the complex blind signal separation problem, rigorous mathematical treatments of the robust measurement and equivariance for the complex-valued FastICA estimator are still missing. This paper strictly analyzes the robustness against outliers and separation performance depending on the global system. We begin with defining the influence function (IF) of complex-valued FastICA functional and followed by deriving its closed-form expression. Then, we prove that the complex-valued FastICA algorithm based on the optimizing cost function is linear-equivariant, depending only on the source signals.

Development and Validation of Near-Infrared Reflectance Spectroscopy Prediction Modeling for the Rapid Estimation of Biochemical Traits in Potato.

Potato is a globally significant crop, crucial for food security and nutrition. Assessing vital nutritional traits is pivotal for enhancing nutritional value. However, traditional wet lab methods for the screening of large germplasms are time- and resource-intensive. To address this challenge, we used near-infrared reflectance spectroscopy (NIRS) for rapid trait estimation in diverse potato germplasms. It employs molecular absorption principles that use near-infrared sections of the electromagnetic spectrum for the precise and rapid determination of biochemical parameters and is non-destructive, enabling trait monitoring without sample compromise. We focused on modified partial least squares (MPLS)-based NIRS prediction models to assess eight key nutritional traits. Various mathematical treatments were executed by permutation and combinations for model calibration. The external validation prediction accuracy was based on the coefficient of determination (RSQexternal), the ratio of performance to deviation (RPD), and the low standard error of performance (SEP). Higher RSQexternal values of 0.937, 0.892, and 0.759 were obtained for protein, dry matter, and total phenols, respectively. Higher RPD values were found for protein (3.982), followed by dry matter (3.041) and total phenolics (2.000), which indicates the excellent predictability of the models. A paired t-test confirmed that the differences between laboratory and predicted values are non-significant. This study presents the first multi-trait NIRS prediction model for Indian potato germplasm. The developed NIRS model effectively predicted the remaining genotypes in this study, demonstrating its broad applicability. This work highlights the rapid screening potential of NIRS for potato germplasm, a valuable tool for identifying trait variations and refining breeding strategies, to ensure sustainable potato production in the face of climate change.

Mathematical Formulation of Stoichiometric Subspaces Associated with Chemical Reaction Networks: The Linear Algebraic Foundations and Fundamentals

The article presents a consolidated review of the mathematical framework for construction of Stoichiometric subspaces associated with Chemical Reaction Networks, based on theoretically imposed constraints, mechanistic considerations and data driven mathematical modeling; A rigorous mathematical treatment of the problem is presented and is illustrated using an appropriate numerical case study example.

Optimization of resolution as a simultaneous function of solvent composition, pH and temperature in reversed-phase liquid chromatography

BackgroundThe latest chromatographic retention models are capable of accurately describe the dependencies of retention over a wide range of experimental conditions. By using a suitable conversion, these models can be transformed into equations expressing the optimization criteria as function of multiples variables. Even though that theoretical models significantly reduce the experimental requirements for optimizations, these models have been barely used. Instead, most optimizations rely on empirical exploration of the relationships between criterions and variables.There is a need for a strategy to reduce the required number of experiments in multivariated optimization of separations, and Fundamental Models offer a clear opportunity for addressing it. ResultsA Fundamental Model is used to give the simultaneous dependence of chromatographic retention of seven ionizable pesticides on the three variables: solvent composition, temperature and pH (w, T, pH). Based on few experiments, the 10 parameters required to predict the chromatographic retention of those compounds, taken as model analytes, can be obtained. Two mathematical treatments to convert retentions into resolutions between pairs are used: one considering extracolumn dispersions and other neglecting these contributions. Using the Overlapped Resolutions Maps, extended to four dimensions, two optimal conditions can be found for the two different mathematical conversions. Chromatographic conditions were empirically evaluated obtaining the best results for the optimization considering extracolumn dispersions, proving that this condition is a true optimal. It was demonstrated that any small shift in any of the variables from this true optimal leads to a loss in resolution. SignificanceFundamental Models describing chromatographic retention as a simultaneous function of multiple variables are nowadays very accurate. In this work is demonstrated that these models are useful not only to predict retentions, but also to optimize separations, even in the more challenging mode: isocratic, isothermal and iso-pH. However, the success in the optimization procedure depends also on the proper definition of the mathematical conversion of the Fundamental Models into optimization criteria.

The extended persistent homology transform of manifolds with boundary

A shortcoming of persistent homology is that when two domains have different numbers of components or holes the persistence diagrams of any filtration will have an infinite distance between them. We address this issue by revisiting the theory of extended persistence, initially developed by Cohen-Steiner, Edelsbrunner and Harer in 2009 to quantify the support of the essential homology classes for Morse functions on manifolds. We simplify the mathematical treatment of extended persistence by formulating it as a persistence module derived from a sequence of relative homology groups for pairs of spaces. Then, for n-manifolds with boundary embedded in Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^n$$\\end{document}, we use Morse theory to show that the extended persistent homology of a height function over M can be deduced from the extended persistent homology of the same height function restricted to ∂M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial M$$\\end{document}. As an application, we describe the extended persistent homology transform (XPHT); a topological transform which takes as input a shape embedded in Euclidean space, and to each unit vector assigns the extended persistence module of the height function over that shape with respect to that direction. We define a distance between two shapes by integrating over the sphere the distance between the respective extended persistence modules. By using extended persistence we get finite distances between shapes even when they have different numbers of essential classes. We study the application of the XPHT to binary images; outlining an algorithm for efficient calculation of the XPHT exploiting relationships between the PHT of the boundary curves to the extended persistence of the foreground.

Optimal enzyme allocation leads to the constrained enzyme hypothesis: the Soil Enzyme Steady Allocation Model (SESAM; v3.1)

Abstract. Describing the coupling of nitrogen (N), phosphorus (P), and carbon (C) cycles of land ecosystems requires understanding microbial element use efficiencies of soil organic matter (SOM) decomposition. These efficiencies are studied by the Soil Enzyme Steady Allocation Model (SESAM) at the decadal scale. The model assumes that the soil microbial communities and their element use efficiencies develop towards an optimum where the growth of the entire community is maximized. Specifically, SESAM approximated this growth optimization by allocating resources to several SOM-degrading enzymes proportional to the revenue of these enzymes, called the Relative approach. However, a rigorous mathematical treatment of this approximation has been lacking so far. Therefore, in this study we derive explicit formulas of enzyme allocation that maximize the total return from enzymatic processing, called the Optimal approach. Further, we derive another heuristic approach that prescribes the change of allocation without the need of deriving a formulation for the optimal allocation, called the Derivative approach. When comparing predictions across these approaches, we found that the Relative approach was a special case of the Optimal approach valid at sufficiently high microbial biomass. However, at low microbial biomass, it overestimated allocation to the enzymes having lower revenues compared to the Optimal approach. The Derivative-based allocation closely tracked the Optimal allocation. These findings increase our confidence in conclusions drawn from SESAM studies. Moreover, the new developments extend the range of conditions at which valid conclusions can be drawn. Further, based on these findings we formulated the constrained enzyme hypothesis. This hypothesis provides a complementary explanation why some substrates in soil are preserved over decades, although they are often decomposed within a few years in incubation experiments. This study shows how optimality considerations lead to simplified models, new insights, and new hypotheses. This is another step in deriving a simple representation of an adaptive microbial community, which is required for coupled stoichiometric C–N–P dynamic models that are aimed to study decadal processes beyond the ecosystem scale.

A Comparison of Spectral Preprocessing Methods and Their Effects on Nutritional Traits in Cowpea Germplasm

ABSTRACTCowpea (Vigna unguiculata L. (Walp)) is a multipurpose legume, which has good nutritional properties. Nutritional parameters assessed conventionally can be labour intensive, costly and time taking for germplasm screening. Near‐infrared reflectance spectroscopy (NIRS) is a rapid and nondestructive method, which can facilitate high‐throughput germplasm screening. In our study, estimation of amylose and sugars has been done using NIRS. Two preprocessing methods, that is, SNV‐DT (standard normal variate with detrending) and MSC (multiplicative scatter correction), were performed for optimization of the original spectra. Subsequently, MPLS (modified partial least square) regression method was employed to construct the prediction models. In amylose, the best RSQexternal (coefficient of determination) (0.962) was found in SNV‐DT with mathematical treatment 3,8,8,2. The same result was shown in sugar where the best RSQexternal (0.914) was found in SNV‐DT with mathematical treatment 3,4,4,1. Overall, in the case of amylose and sugars, SNV‐DT was found to be a good preprocessing treatment than MSC. Paired t‐test values in all the treatments for both the preprocessing methods were > 0.05 indicating their reliability. High RSQexternal values for both the traits imply the applicability of the prediction models. Thus, these models can facilitate high‐throughput germplasm screening in different national and international crop improvement programmes focusing on quality traits.

A closed-form Timoshenko beam element with multiple localized singularities for nonlinear material behavior using the strong discontinuity approach

The development of a novel closed-form beam element for nonlinear material behavior in structures is described, which is formulated according to Timoshenko beam theory and the embedded strong discontinuity approach. Multiple strong transverse and rotation discontinuities are considered to model several dissipative mechanisms due to localized singularities in the shear strain and curvature fields. Issues of dependence on numerical techniques, as occurs in displacement- and forced-based beam elements with the finite element method, are overcome since the formulated element is derived from a pure mathematical treatment. Theoretical principles are stemming from a Lagrangian variational model, from which singularities are naturally derived. The proposed formulation is solved by the conventional methods of calculus of variations, resulting in the boundary value problem. Then, closed-form expressions for any boundary conditions are obtained. By particularizing closed-form expressions for a basic system, a symmetric closed-form flexibility matrix is obtained. This matrix naturally leads to a symmetric closed-form stiffness matrix for any type of loading and unloading process, in which the nodeless degrees of freedom of the multiple embedded strong discontinuities are naturally condensed. Since Gauss integration methods are not used, the multiple localized singularities are not influenced by predefined positions of the integration points. However, the accuracy depends on the chosen number of singularities and their positions. Then, the proposed element is mesh independent when sufficient discontinuities are considered, which does not imply an increase in the degrees of freedom. Softening discontinuities are modeled at arbitrary positions in a single element, which overcomes the drawbacks of other embedded beam theories where more than one element is necessary to identify failure since strong discontinuities are generally located at the midpoint of the beam element. The obtained closed-form stiffness matrix is not always positive definite for strain-softening problems. Nevertheless, there are no major convergence issues as it is a symmetric matrix that does not require a constraint equation, so even the complex snap-back behavior in structures is adequately modeled. The capability of the formulated beam element for nonlinear material behavior in structures is validated with representative examples.

Radiological and morphological features of vanishing lung syndrome development in patients with COVID-19 community-acquired viral pneumonia

BACKGROUND. Presently actively new direction develops in digital treatment of CТ images – radiomics, that presents the result of co-operation on verge of different sciences (radiology, computer sciences and mathematical statistics). Inaccessible for the unarmed eye additional information of CT images can be got by means of their mathematical treatment and creation of the segmented histograms. Last it is possible to compare and analyse both isolated and with regard to the dynamics of physiopathology descriptions of organs and fabrics at the different human diseases.
 OBJECTIVE. To define the roentgenologic and morphological features of development of vanishing lung syndrome for patients with non-hospital viral pneumonia.
 MATERIALS AND METHODS. Data of CТ are analysed in a dynamics for patients with non-hospital viral pneumonia of COVID-19, that were on treatment in SI “National institute of phthisiology and pulmonology named after F.G. Yanovsky of the NAMS of Ukraine” or were directed from other medical establishments. The Dragonfly program from Object Research Systems (Montreal, Canada), which performs micro-X-ray structural analysis of the examined tissues, was used to analyze CT images of chest. Pathomorphological examination was performed in the laboratory of pathomorphology of the institute.
 RESULTS. Monitoring of CT is conducted in the group, that consisted of 90 patients with non-hospital viral pneumonia of COVID-19. 27 (30,0 %) patients (18 men and 9 women in age from 23 to 68) are educed with the roentgenologic signs of vanishing lung syndrome. 12 from them (9 men and 3 women in age from 23 to 56) were on treatment in the institute in an acute period of disease. Other 15 patients (9 men and 6 women in age from 26 to 68) directed from other curative establishments, where they treated oneself 3-4 months ago.
 CONCLUSIONS. Micro-X-ray structural analysis of data of CT allows to educe the features of changes of parenchima at development of vanishing lung syndrome. These changes are confirmed by the educed changes at pathomorphological research of postoperative preparations of lungs.

The Expected Loss of Preconditioned Langevin Dynamics Reveals the Hessian Rank

Langevin dynamics (LD) is widely used for sampling from distributions and for optimization. In this work, we derive a closed-form expression for the expected loss of preconditioned LD near stationary points of the objective function. We use the fact that at the vicinity of such points, LD reduces to an Ornstein–Uhlenbeck process, which is amenable to convenient mathematical treatment. Our analysis reveals that when the preconditioning matrix satisfies a particular relation with respect to the noise covariance, LD's expected loss becomes proportional to the rank of the objective's Hessian. We illustrate the applicability of this result in the context of neural networks, where the Hessian rank has been shown to capture the complexity of the predictor function but is usually computationally hard to probe. Finally, we use our analysis to compare SGD-like and Adam-like preconditioners and identify the regimes under which each of them leads to a lower expected loss.

Numerical analysis of thermal improvement in hydrogen‐based host fluids under buoyancy force and EPNM effects: Study for vertical cylinder

AbstractThe nanofluids became much of interest due to their superior heat mechanism over the conventional fluids. The addition of nanoparticles in the host solvent enhances the internal ability of the liquid to store and transmit heat. Therefore, these fluids are widely used in biomedical engineering, detergents, medication, applied thermal engineering, mechanical, and chemical engineering and so forth. Keeping in mind the significance of nanofluids, this study is conducted to investigate the comparative heat transmission in and under the Effective Prandtl Number Model (EPNM) effects over a vertical permeable cylinder. Formulation of the model is carried out over a vertical cylinder about the stagnation point and the heat transport model is achieved after the successful implementation of the cylindrical stream function, nanofluids effective characteristics, and cylindrical similarity equations. The mathematical treatment was done via RK technique and furnished the results. The nanofluids have the capacity to control the fluid motion more effectively than ordinary fluid and combined convection is better where rapid fluid movement is desired. In the existence of EPNM, higher heat transmission is achieved and is prominent for based on their thermophysical values. Further, the skin friction and Nusselt number are optimum for against .

A novel green spectrofluorimetric method for simultaneous determination of antazoline and tetryzoline in their ophthalmic formulation.

A novel spectrofluorimetric method has been developed for determination of antazoline (ANT) and tetryzoline (TET) in their pharmaceutical formulation. A combined application of synchronous spectrofluorimetry and second derivative mathematical treatment was developed. The proposed method depends on reacting the cited drugs with dansyl chloride (DNS-Cl) being a suitable derivatizing agent generating highly fluorescent derivatives measured at emission wavelengths of 703.0 and 642.0nm after excitation wavelengths of 350.0 and 320.0nm for ANT and TET, respectively. The joint use of synchronous spectrofluorimetry with second derivative mathematical treatment is for the first time to be developed and optimized in aid of using fluorescence data manager software generating second derivative peak amplitudes at 556.5nm for ANT and 516.7nm for TET. Linear responses have been represented over a wide range of concentration (0.5-12.0μg/mL for ANT and 0.5-10.0μg/mL for TET). Additionally, statistical comparison of the developed method with the official ones has been carried out where no significant difference was found. Additionally, greenness profile assessment was accomplished by means of four metric tools. Indeed, the method developed is found to be precise, sensitive, and discriminating to assess the cited drugs for regular analysis.

Building a mathematical model for a simple harmonic oscillator that uses educational methods found in both physics education research and in the language disciplines that make it accessible to undergraduate students in an introductory musical acoustics course

At the basis of any course in acoustics is the fundamental idea of the simple harmonic oscillator. The term alone is confusing to students with little to no background in physics or math. For courses in musical acoustics at the undergraduate level, this topic is often minimized due to the lack of preparation. This, in turn, results in a more superficial approach to the advanced topics. While deriving the mathematical treatment from first principles is out of reach to these students, approaching the math itself as a language where they are building a description of a simple mass and spring system in their new, yet somewhat familiar, language can be accomplished through pictures, graphs and hands on activities. Students begin to build a vocabulary of “words” that can be strung together in “sentences” that tell the story of how the motion of a mass on a spring is produced. Emphasis is placed on the analogous comparison of physical properties by relating variables such as amplitude and frequency to that of volume and pitch. This model can then be used as a building block to the understanding of how sound is produced, propagated and perceived.

The Need for Editorial Reorganization of Problem Statements on the Logarithmic Function Proposed in School Textbooks to Improve Their Comprehension

We present elements of reflection that allow us to approach the role of the editorial organization in understanding certain problem statements modeled by a logarithmic function proposed in school texts. The main objective is to analyze the role that editorial reorganization plays in problem statements related to the logarithmic function, semiotic elements that promote the articulation of the semiotic representation registers that students can use to provide solutions to the proposed questions, since the understanding of problem statements lies in the coordination of at least two representation registers, but part of this is within the Natural Language register. The methodological design was a descriptive qualitative case study in which the editorial reorganization of four problem statements about the logarithmic function, selected from three school texts, was carried out. These statements were applied to seven university students in order to analyze their productions, mathematical treatments, and the way in which they articulated the different semiotic registers to provide a solution to the presented problem statement. In relation to the development of the sequence of statements, the need to reformulate problem statements that lead to the discrimination of all visual values in the graphic register in correspondence with variations in the significant units of the algebraic register was confirmed.

Didactic experiment for voltametric determination the complexing capacity of natural waters

The titration method can be applied to determine the complexation capacity between metal ions and ligands in natural water samples. The process involves the incremental addition of the metal to the sample containing a specific ligand, resulting in a titration curve that reflects the formation of complexes. The change in the curves slope indicates the saturation of the complexation capacity. In this study, the copper (II) ion was used due to its stability with organic ligands. The titration curve was analyzed to determine the complexation capacity (CC) and conditional formation constant (K) of the complex. Different mathematical treatments are proposed for this analysis. In the described experiment, differential pulse anodic stripping voltammetry was used to determine the complexation capacity of humic substances in natural waters. Experimental parameters such as pH, supporting electrolyte concentration, and volume of additions were previously optimized for the voltammetry technique. The obtained data were processed and analyzed, resulting in titration curves and voltammograms. The analysis of the curves allowed for the calculation of the complexation capacity and conditional stability constant of the formed complex. The results demonstrate the feasibility of the method as a didactic experiment for the voltammetric determination of natural water complexation capacity.

Using spherical harmonics to solve the Boltzmann equation: an operator-based approach

ABSTRACT The transport of charged particles or photons in a scattering medium can be modelled with a Boltzmann equation. The mathematical treatment for scattering in such scenarios is often simplified if evaluated in a frame where the scattering centres are, on average, at rest. It is common therefore to use a mixed coordinate system, wherein space and time are measured in a fixed inertial frame, while momenta are measured in a ‘co-moving’ frame. To facilitate analytical and numerical solutions, the momentum dependence of the phase-space density may be expanded as a series of spherical harmonics, typically truncated at low order. A method for deriving the system of equations for the expansion coefficients of the spherical harmonics to arbitrary order is presented in the limit of isotropic, small-angle scattering. The method of derivation takes advantage of operators acting on the space of spherical harmonics. The matrix representations of these operators are employed to compute the system of equations. The computation of matrix representations is detailed and subsequently simplified with the aid of rotations of the coordinate system. The eigenvalues and eigenvectors of the matrix representations are investigated to prepare the application of standard numerical techniques, e.g. the finite volume method or the discontinuous Galerkin method, to solve the system.

Development of a novel fluorimetric sensor based on polypyrrole doped with 4-(methylsulfonyl)-2-nitrobenzoic acid (MSNBA) for sensitive detection of copper (Cu2+)

In this study, we successfully electrochemically synthesized a polypyrrole film doped with 4-methylsulfonyl-2-nitrobenzoic acid (MSNBA) within an aqueous medium containing 0.01 M MSNBA, 0.003 M NaOH, and 0.01 M pyrrole. Our investigation encompassed a thorough exploration of both electrochemical and spectroscopic properties, confirming the effective incorporation of MSNBA as a dopant within the electroactive PPy chain. The morphological characterization of the film was undertaken using scanning electron microscopy (SEM). The optical properties, particularly UV–visible absorption, and fluorescence spectroscopy, were scrutinized in a dilute DMF solution. The UV–visible absorption spectra revealed a broad absorption band centered around 292 nm, attributed to the π→π* 1B - 1La/1Lb transitions of the oligo-benzenes in the film. In parallel, fluorescence spectra exhibited remarkable consistency between the solid state and solution, featuring an excitation peak at 345 nm (346 nm in solution) and an emission peak around 401 nm (403 nm in solution). Exploring the impact of copper (II) ions on absorption and fluorescence spectra unveiled a notable quenching effect along two distinct linearity domains. Various mathematical treatments, including the Stern-Volmer equation (R1 = 0.98, Ksv1 = 5.104 and R2 = 0.99, Ksv2 = 1.54.106), the Perrin equation, and the polynomial equation, were applied to elucidate the nature of this fluorescence quenching. Our findings indicated that the fluorescence extinction, as a function of Cu2+ ion concentration, aligns well with the polynomial equation, signifying a combination of dynamic and static quenching (Io/I = 1.08(±0.04) + 1.46(±0.44)105[Q]- 1.15(±0.75)1010 [Q]2). Moreover, we explored the potential application of this method for the determination of Cu2+. Calibration graphs demonstrated excellent linearity between fluorescence intensity and Cu2+ concentration in the range of 0–5.4 μM. Notably, the detection limit of 1.2 μM surpassed the LOD for Cu2+ (20 μM) in drinking water set by the EPA, suggesting the viability of this approach for sensitive detection in environmental monitoring.