Numerical Constants Research Articles

Intelligent spectrophotometric resolution platforms for the challenging spectra of ipratropium and fenoterol in their combination inhaler with ecological friendliness assessment.

Asthma and chronic obstructive pulmonary disease (COPD) are the most common diagnoses for adults and children with respiratory tract inflammation. Recently, a novel fixed dose combination consisting of Ipratropium and Fenoterol has been released for the management and control of the symptoms of such disorders. The current research has newly developed andoptimizedthree smart, accurate,simple, cost-effective, and eco-friendly spectrophotometric methods that enabled the simultaneous determination of the drugs under study in their combined inhaler dosage form, without the need for any previous separation steps, using water as a green solvent. The strategy employed was based oncalculating oneor two factors as a numerical spectrum or constant, whichprovided thecomplete removal of any component in the mixture that might overlap and the mathematical filtration of the targeted analyte. The methods developed could be classified into twotypesof spectrophotometric windows. Window I; involved absorption spectrum in their original zero-order forms (°D), which included recently designed methods named induced concentration subtraction (ICS) and induced dual wavelength (IDW). While window III focused on the ratio spectrum as theinduced amplitude modulation (IAM) method. The extremely low absorptivity and lack of distinct absorption maximum in the zero-order absorption spectrum of Ipratropium were two intrinsic challenges that were better overcome by the proposed spectrophotometric methods than by the conventionally used ones. According to ICH guidelines, the proposed methods were validated using unified regression over range 2.0-40.0 µg/mL in the ICS method, while the linearity ranges for the IDW and IAM methods were 5.0-40.0 µg/mL of Ipratropium and 2.0-40.0 µg/mL of Fenoterol. Moreover, the three proposed methods were effectively used to assay the co-formulated marketed inhaler and further expanded to confirm the delivered dose uniformityin compliance with the USP guidelines. Finally, the established methods were evaluated for their greenness and blueness, in comparison to the official and reported analysis methods, using advanced cutting edge software metrics. Furthermore, the suggested techniques adhered well to the white analytical chemistry postulates that were recently published.

Extracting phonon coupling parameters from multi color center photoluminescence

Vibronic coupling of color centers plays a crucial role in their effectiveness for quantum technologies. This study presents a novel approach to extract key parameters, including the Huang–Rhys factor and the one-phonon coupling function, from photoluminescence spectra containing the signal from multiple color centers. Monotonic splines are used to represent the one-phonon coupling function, optimized using a variable-length non-dominated sorting genetic algorithm. Our method is applied to the combination of the well-studied negatively charged nitrogen-vacancy center and its less explored neutral counterpart in diamond, observed through photoluminescence at 14 K. Despite the complexity inherent in the theoretical framework governing phonon coupling in photoluminescence, we achieve robust fits, yielding a normalized root-mean-squared error of approximately 6%. The neutral nitrogen-vacancy center exhibits a Huang–Rhys factor of 3.0, while its negatively charged counterpart showcases a value of 3.95. The quasi-localized vibrations of both charge states are clearly present in the fitted one-phonon spectral function. Furthermore, the consistency of our methodology is bolstered by temperature-dependent photoluminescence, which was calculated up to numerical constants using fit-derived parameters, matching the experimental measurements up to 300 K.

A Combustion Kinetic Model Based on the Fast Chemistry Assumption: Application to Non-Sooty and Sooty Laminar Diffusion Flames

ABSTRACT In this article, a combustion kinetic model based on the fast chemistry (FC) assumption is proposed to compute the finite reaction rate of gaseous species in laminar diffusion flames. The algorithm developed for this purpose is numerically robust and relies on the adiabatic flame temperature to limit the fuel consumption rate. While there is no FC model specifically developed for laminar flames, the Eddy Dissipation Model (EDM) correction scheme for laminar regimes is taken as the reference model. The proposed model addresses the shortcomings of the reference model, i.e. the sensitivity to the transport time scale and the cell size and the need to calibrate a numerical constant repeatedly. The new model formulation does not rely on any numerical constant and completely removes the dependency on the cell size and the species transport time scale, as shown for multiple counterflow and axisymmetric laminar diffusion flames. Furthermore, the proposed model provides consistent temperature and species mass fraction predictions across the mentioned flames. Finally, a sooty axisymmetric flame is simulated using the mentioned models to investigate the numerical interaction between combustion and soot chemistry. The results show that the reference model requires recalibration to capture the temperature and soot volume fraction distributions while the proposed model yields reasonably accurate temperature and soot predictions.

Supersymmetric index for small black holes

Supersymmetric elementary string states in the compactified heterotic string theory are described by small black holes that have zero area event horizon. In this paper we compute the supersymmetric index of such elementary string states using gravitational path integral. The dominant contribution to the path integral comes from an Euclidean rotating black hole solution of the supergravity theory with a finite area event horizon, but the logarithm of the index, computed from the saddle point, vanishes. Nevertheless we show that the solution is singular on certain subspaces of the horizon where higher derivative corrections can be important, and once the higher derivative corrections are taken into account the solution could yield a finite result for the logarithm of the index whose form agrees with the microscopic results up to an overall numerical constant. While the numerical constant is not determined in our analysis, we show that it is independent of the details of the compactification and even the number of non-compact dimensions, in agreement with the microscopic results.

Robust estimation of a regression function in exponential families

We observe n pairs of independent (but not necessarily i.i.d.) random variables X1=(W1,Y1),…,Xn=(Wn,Yn) and tackle the problem of estimating the conditional distributions Qi⋆(wi) of Yi given Wi=wi for all i∈{1,…,n}. Even though these might not be true, we base our estimator on the assumptions that the data are i.i.d. and the conditional distributions of Yi given Wi=wi belong to a one parameter exponential family Q¯ with parameter space given by an interval I. More precisely, we pretend that these conditional distributions take the form Qθ(wi)∈Q¯ for some θ that belongs to a VC-class Θ¯ of functions with values in I. For each i∈{1,…,n}, we estimate Qi⋆(wi) by a distribution of the same form, i.e. Qθ̂(wi)∈Q¯, where θ̂=θ̂(X1,…,Xn) is a well-chosen estimator with values in Θ¯. We establish non-asymptotic exponential inequalities for the upper deviations of a Hellinger-type distance between the true conditional distributions of the data and the estimated one based on the exponential family Q¯ and the class of functions Θ¯ we chose. We show that our estimation strategy is robust to model misspecification, contamination and the presence of outliers. Besides, when the data are truly i.i.d., the exponential family Q¯ is suitably parametrized and the conditional distributions Qi⋆(wi) of the form Qθ⋆(wi)∈Q¯ for some unknown Hölderian function θ⋆ with values in I, we prove that the estimator θ̂ of θ⋆ is minimax (up to a logarithmic factor). Finally, we provide an algorithm for calculating θ̂ when Θ¯ is a VC-class of functions of low or moderate dimension and we carry out a simulation study to compare its performance to that of the MLE and median-based estimators. The proof of our main result relies on an upper bound, with explicit numerical constants, on the expectation of the supremum of an empirical process over a VC-subgraph class. This bound can be of independent interest.

A high-precision result for a full-color three-loop three-point form factor in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM

We perform a high-precision computation of the three-loop three-point form factor of the stress-tensor supermultiplet in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM. Both the leading-color and sub-leading-color form factors are expanded in terms of simple integrals. We compute the complete set of integrals at a special kinematic point with very high precision using AMFlow. The high-precision leading-color result enables us to obtain the analytic form of a numerical constant in the three-loop BDS ansatz, which is previously known only numerically. The high-precision values of the non-leading-color finite remainder as well as all integrals are also presented, which can be valuable for future use.

Optimal Program Synthesis via Abstract Interpretation

We consider the problem of synthesizing programs with numerical constants that optimize a quantitative objective, such as accuracy, over a set of input-output examples. We propose a general framework for optimal synthesis of such programs in a given domain specific language (DSL), with provable optimality guarantees. Our framework enumerates programs in a general search graph, where nodes represent subsets of concrete programs. To improve scalability, it uses A * search in conjunction with a search heuristic based on abstract interpretation; intuitively, this heuristic establishes upper bounds on the value of subtrees in the search graph, enabling the synthesizer to identify and prune subtrees that are provably suboptimal. In addition, we propose a natural strategy for constructing abstract transformers for monotonic semantics, which is a common property for components in DSLs for data classification. Finally, we implement our approach in the context of two such existing DSLs, demonstrating that our algorithm is more scalable than existing optimal synthesizers.

Thermal one-point functions: CFT’s with fermions, large d and large spin

We apply the OPE inversion formula on thermal two-point functions of fermions to obtain thermal one-point function of fermion bi-linears appearing in the corresponding OPE. We primarily focus on the OPE channel which contains the stress tensor of the theory. We apply our formalism to the mean field theory of fermions and verify that the inversion formula reproduces the spectrum as well as their corresponding thermal one-point functions. We then examine the large N critical Gross-Neveu model in d = 2k + 1 dimensions with k even and at finite temperature. We show that stress tensor evaluated from the inversion formula agrees with that evaluated from the partition function at the critical point. We demonstrate the expectation values of 3 different classes of higher spin currents are all related to each other by numerical constants, spin and the thermal mass. We evaluate the ratio of the thermal expectation values of higher spin currents at the critical point to the Gaussian fixed point or the Stefan-Boltzmann result, both for the large N critical O(N) model and the Gross-Neveu model in odd dimensions. This ratio is always less than one and it approaches unity on increasing the spin with the dimension d held fixed. The ratio however approaches zero when the dimension d is increased with the spin held fixed.

Rigorous and simple results on very slow thermalization, or quasi-localization, of the disordered quantum chain

This paper originates from lectures delivered at the summer school “Fundamental Problems in Statistical Physics XV” in Bruneck, Italy, in 2021. We give a brief and limited introduction into ergodicity-breaking induced by disorder. As the title suggests, we include a simple yet rigorous and original result: For a strongly disordered quantum chain, we exhibit a full set of quasi-local quantities whose dynamics is negligible up to times of order exp{c(logW)2−ϵ}, with ϵ<1, c a numerical constant, and W the disorder strength. Such a result, that is often referred to as “quasi-localization”, can in principle be obtained in other systems as well, but for a disordered quantum chain, its proof is relatively short and transparent.

Symbolic Metamodels for Interpreting Black-Boxes Using Primitive Functions

One approach for interpreting black-box machine learning models is to find a global approximation of the model using simple interpretable functions, which is called a metamodel (a model of the model). Approximating the black-box with a metamodel can be used to 1) estimate instance-wise feature importance; 2) understand the functional form of the model; 3) analyze feature interactions. In this work, we propose a new method for finding interpretable metamodels. Our approach utilizes Kolmogorov superposition theorem, which expresses multivariate functions as a composition of univariate functions (our primitive parameterized functions). This composition can be represented in the form of a tree. Inspired by symbolic regression, we use a modified form of genetic programming to search over different tree configurations. Gradient descent (GD) is used to optimize the parameters of a given configuration. Our method is a novel memetic algorithm that uses GD not only for training numerical constants but also for the training of building blocks. Using several experiments, we show that our method outperforms recent metamodeling approaches suggested for interpreting black-boxes.

Ray-tracing images for melting PCM bridges in microgravity

A ray-tracing algorithm is developed to obtain synthetic images for tracking the melting of the phase change material (PCM) n-octadecane in a liquid bridge configuration. The translucent nature of the solid n-octadecane is modelled through a sigmoid function which provides a distance-dependent illumination intensity throughout the solid. The code is validated against two types of experiments: front shapes made of 3D-printed PLA and cast front shapes of solid n-octadecane. Comparison with these experiments allows the numerical constants involved in the model to be tuned to appropriate values. For application to the MarPCM microgravity experiment (Porter et al., 2023), the current study provides the expected images of the solid/liquid front obtained by the optical diagnosis module for three different values of the dimensionless volume (defined with respect to a perfect cylinder), V=0.9, 1, and 1.1, and two applied temperature differences, ΔT=5 and 30K. The solid/liquid front shapes are obtained numerically from a two-dimensional axisymmetric code as in Varas et al. (2021). The results reveal a strong dependence of image quality on the liquid fraction L. Visual inspection leads to the conclusion that, below L≈60%, an accurate determination of the front location may be difficult, particularly for the relatively shallow layer of liquid near the thermocapillary interface.

Exceptional structural and functional stability of Zr-doped REBaCuO tapes with respect to neutron irradiation and overheating

The impact of neutron irradiation on the properties of second-generation REBaCuO tapes was studied. The main aim of the present study was to identify the fast neutrons irradiation fluence threshold, φ th, at which the high-field critical current at low temperatures, I c (10 K, 8 T), starts degrading. Following up on our previous publication for low irradiation fluences, two new irradiation steps were added. Up to the last but one irradiation step, the T c in all samples followed a linear dependence on the fast neutrons irradiation fluence, T c(φ) ≈ T c(0) − τφ (τ being a numerical constant in units of K/1022 m−2 and φ in 1022 m−2). In parallel, I c (10 K, 8 T) continuously grew with φ up to the threshold fluence, φ th ≈ 3× to 4.3 × 1022 m−2. The last gradual irradiation by φ≈ 1.83 × 1022 m−2 brought a severe degradation both in T c and I c (10 K, 8 T) in most samples, irrespective of their irradiation history. This effect was attributed to radiation overheating. Annealing of pristine tapes in dilute Ar confirmed this scenario. Two of the SuperPower tapes, doped by Zr, appeared to be exceptionally stable against both types of overheating. The tape doped with 7.5% Zr, T c(φ) followed the linear dependence up to the highest cumulative fluence φ =6.16 × 1022 m−2, while the I c (10 K, 8 T) drop was several orders of magnitude lower than in the samples without Zr. TEM study found that the Zr-doped tapes survived the overheating during the last irradiation step in a crystalline form, while all other samples were amorphized.

Numerical modeling of hydrogen mixing in a direct-injection engine fueled with gaseous hydrogen

Hydrogen is considered as one of the most promising options to achieve effective decarbonization of the energy and transportation sectors. As such, it has recently been receiving increasing attention because of its promising potential as an energy carrier for advanced energy and propulsion systems. With a focus on internal combustion engines, direct injection (DI) of gaseous hydrogen during the compression stroke offers great potential for high engine efficiency and specific power while reducing the risk of backfiring and pre-ignition issues. Therefore, many experimental and numerical efforts have recently been dedicated to understanding the physical and chemical behaviors of hydrogen in engine during mixing and combustion. This study focuses on computational fluid dynamics (CFD) modeling of the hydrogen DI process in a hydrogen optical research engine. Under the conditions studied, gaseous hydrogen is injected into the combustion chamber via a centrally located single-hole injector at a pressure of 100 bar. Two configurations, namely low- and high-tumble, are investigated to understand the impact of different in-cylinder flow patterns on the fuel-air mixture preparation. Simulations are carried out using the commercial CFD software CONVERGE. The in-cylinder turbulence is modeled with an unsteady Reynolds-averaged Navier-Stokes (URANS) formulation closed by the renormalization group (RNG) k-ε model. Several numerical methods and model constants, including but not limited to turbulent Schmidt number, are evaluated. The numerical results are systematically compared against experimental measurements of velocity and hydrogen concentration fields on the vertical center plane to assess the performance of the CFD model, unveil the physics of hydrogen mixing, and establish best practices for modeling hydrogen DI under relatively high injection pressure conditions.

Integrating genomic prediction and genotype specific parameter estimation in ecophysiological models: overview and perspectives

Abstract The Genome-to-Phenome (G2P) problem is one of the highest-priority challenges in applied biology. Ecophysiological crop models (ECM) and genomic prediction (GP) models are quantitative algorithms, which, when given information on a genotype and environment, can produce an accurate estimate of a phenotype of interest. In this article, we discuss how the GP algorithms can be used to estimate genotype-specific parameters (GSPs) in ECMs to develop robust prediction methods. In this approach, the numerical constants (GSPs) that ECMs use to distinguish and characterize crop cultivars/varieties are treated as quantitative traits to be predicted by genomic prediction models from underlying genetic information. In this article we provide information on which GP methods appear favorable for predicting different types of GSPs, such as vernalization sensitivity or potential radiation use efficiency. For each example GSP, we assess a number of GP methods in terms of their suitability using a set of three criteria grounded in genetic architecture, computational requirements, and the use of prior information. In general, we conclude that the most useful algorithms were dependent on both the nature of the particular GSP and the GP methods considered.

Frictional properties of phase-separated agarose hydrogels in water permeation.

We studied the friction coefficient between the polymer gel network and water f for thermoreversible agarose gels under various conditions of agarose concentration and gelation temperature. Since agarose gels exhibit phase separation below the gelation temperature, f strongly depends on the thermal history. We found that the friction coefficient of the phase-separated agarose gel normalized by the water viscosity, f/η, is expressed as f/η = S/ξνSD where ξSD is the frictional pore size and ν and S are constant parameters. ξSD corresponds to the correlation length of the frozen density fluctuations of the polymers via spinodal decomposition determined from small-angle light scattering. The least-squares analysis of the results shows that the exponent is ν ≃ 2 with the numerical constant of S ≃ 105/2π. The results suggest that the frictional properties of phase-separated agarose gels are dominated by the dilute regions of the bicontinuous gel structure.

A Proposed Algorithm for Determining the Suitable Number of Significant Digits of Force Transducer Calibration Constants

Precision measurement is one of the important goals of measurement and calibration. Positive measures are designed to provide researches with the highest accuracy and precision levels. In force measurements, force measuring instrument manufacturers are striving to achieve the best force measurement resolution to accompany the high level of calibration measurement capabilities achieved by the primary deadweight machines. In the calibration of the force transducers, polynomial equations which correlate, the applied load on the transducer and its corresponding output signal in mV/V are deduced. These equations contain numerical constants A, B and C. This paper introduces a mechanism to determine the suitable number of significant digits for these constants. Error analysis procedure for the calibration polynomial equation is proposed. This proposed algorithm safeguards the most accurate and realistic method for rounding the calibration constants

Evolutionary Approaches to the Identification of Dynamic Processes in the Form of Differential Equations and Their Systems

Evolutionary approaches are widely applied in solving various types of problems. The paper considers the application of EvolODE and EvolODES approaches to the identification of dynamic systems. EvolODE helps to obtain a model in the form of an ordinary differential equation without restrictions on the type of the equation. EvolODES searches for a model in the form of an ordinary differential equation system. The algorithmic basis of these approaches is a modified genetic programming algorithm for finding the structure of ordinary differential equations and differential evolution to optimize the values of numerical constants used in the equation. Testing for these approaches on problems in the form of ordinary differential equations and their systems was conducted. The influence of noise present in the data and the sample size on the model error was considered for each of the approaches. The symbolic accuracy of the resulting equations was studied. The proposed approaches make it possible to obtain models in symbolic form. They will provide opportunities for further interpretation and application.

Influence of Temperature and Pressure on the Lattice Constant of SrTiO3 Perovskite by the Statistical Moment Method with Improved Interatomic Potential

Temperature and pressure dependence of lattice constants of a cubic Strontium Titanate (SrTiO3) has been investigated using the statistical moment method. The lattice constants at various temperatures is derived in closed analytic form by including the anharmonic effects of the lattice vibrations explicitly. The potential with the partial charge model and Morse function is used. The numerical lattice constants at high temperatures by the statistical moment method are in good and reasonable agreement with the other theories and the experimental data. Variations of the lattice parameter of SrTiO3 with the temperature are obtained at 1 atm, 8.2 GPa, and 15.2 GPa. Increasing of the lattice constants with increasing temperature is due to enhancing atomic anharmonic fluctuations in SrTiO3 lattice crystal at higher temperature. A decrease of the lattice constants with increasing pressure can be demonstrated by the reduction of atomic vibrations in SrTiO3 crystal lattice at higher pressures.&#x0D;

Theory of chain walking catalysis: From disordered dendrimers to dendritic bottle-brushes.

The chain walking (CW) polymerization technique has the unique property of a movable catalyst synthesizing its own path by creating branch-on-branch structures. By successive attachment of monomers, the resulting architecture ranges from dendritic to linear growth depending on the walking rate, which is defined by the ratio of walking steps and reaction events of the catalyst. The transition regime is characterized by local dendritic sub-structures (dendritic blobs) and a global linear chain feature forming a dendritic bottle-brush. A scaling model for structures obtained by CW catalysis is presented and validated by computer simulation relating the extensions of CW structures to the catalyst's walking ability. The limiting case of linear (low walking rate) and dendritic growth (high walking rate) is recovered, and the latter is shown to bear analogies to the Barabási-Albert graph and Bernoulli growth random walk. We could quantify the size of the dendritic blob as a function of the walking rate by using spectral properties of the connectivity matrix of the simulated macromolecules. This allows us to fit the numerical constants in the scaling approach. We predict that independent of the underlying chemical process, all CW polymerization syntheses involving a highly mobile catalyst ultimately result in bottle-brush structures whose properties depend on a unique parameter: the walking rate.

A random matrix theory approach to damping in deep learning

We conjecture that the inherent difference in generalisation between adaptive and non-adaptive gradient methods in deep learning stems from the increased estimation noise in the flattest directions of the true loss surface. We demonstrate that typical schedules used for adaptive methods (with low numerical stability or damping constants) serve to bias relative movement towards flat directions relative to sharp directions, effectively amplifying the noise-to-signal ratio and harming generalisation. We further demonstrate that the numerical damping constant used in these methods can be decomposed into a learning rate reduction and linear shrinkage of the estimated curvature matrix. We then demonstrate significant generalisation improvements by increasing the shrinkage coefficient, closing the generalisation gap entirely in both logistic regression and several deep neural network experiments. Extending this line further, we develop a novel random matrix theory based damping learner for second order optimisers inspired by linear shrinkage estimation. We experimentally demonstrate our learner to be very insensitive to the initialised value and to allow for extremely fast convergence in conjunction with continued stable training and competitive generalisation. We also find that our derived method works well with adaptive gradient methods such as Adam.