Key Formulas Research Articles

A discretization-free deep neural network-based approach for advection-dispersion-reaction mechanisms

This study aims to provide insights into new areas of artificial intelligence approaches by examining how these techniques can be applied to predict behaviours for difficult physical processes represented by partial differential equations, particularly equations involving nonlinear dispersive behaviours. The current advection-dispersion-reaction equation is one of the key formulas used to depict natural processes with distinct characteristics. It is composed of a first-order advection component, a third-order dispersion term, and a nonlinear response term. Using the deep neural network approach and accounting for physics-informed neural network awareness, the problem has been elaborately discussed. Initial and boundary conditions are added as constraints when the neural networks are trained by minimizing the loss function. In comparison to the existing results, the approach has produced qualitatively correct kink and anti-kink solutions, with losses often remaining around 0.01%. It has also outperformed several traditional discretization-based methods.

Linguistic and cultural analysis of the concept “politeness”

Abstract The need to study the concept of “politeness” from the point of view of its linguistic and cultural nature is caused by the desire to study the national identity of speech etiquette in different cultural spaces and conditions. The aim of the work was to form an idea about the specifics of the implementation and understanding of the concept of “politeness” in the Uzbek information field. In this study, the following methods were used: contextual, conceptual, communicative, linguocultural, analytical-synthetic, and comparative. This study is focused on the study of key lexical meanings (stylistically neutral and marked, basic and additional) that are within the functional-semantic field of the concept “politeness.” With the help of contextual study of different variants of the use of the lexeme, the meanings were distributed in the conceptual structure (core, near and far periphery). Also, the key etiquette formulas representing the originality and national-specific features of the Uzbek linguocultural tradition were considered. The importance of politeness in the information space of Uzbekistan is assessed, including with the help of both positive and negative associations, that is, from an axiological point of view. Speech etiquette and linguistic formulas were considered from the point of view of the influence of social, cultural, and political values of the Uzbek people. In the process of analyzing the lexical layer, the boundaries of the functional-semantic field of the concept “politeness” were determined: from the principles of communication and a set of rules of etiquette to the strategy of obtaining benefits from communication and insincere attitude. In the future, this work can be used for comparative analysis of the conceptual structure of politeness with models presented in other close and distant languages, comparison of speech etiquette and linguistic formulas in different national cultures.

Demazure crystals and the Schur positivity of Catalan functions

Catalan functions, the graded Euler characteristics of certain vector bundles on the flag variety, are a rich class of symmetric functions which include k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$k$\\end{document}-Schur functions and parabolic Hall-Littlewood polynomials. We prove that Catalan functions indexed by partition weight are the characters of Uq(slˆℓ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$U_{q}(\\widehat{\\mathfrak{sl}}_{\\ell })$\\end{document}-generalized Demazure crystals as studied by Lakshmibai-Littelmann-Magyar and Naoi. We obtain Schur positive formulas for these functions, settling conjectures of Chen-Haiman and Shimozono-Weyman. Our approach more generally gives key positive formulas for graded Euler characteristics of certain vector bundles on Schubert varieties by matching them to characters of generalized Demazure crystals.

Arc entropy of uncertain random variables and its applications

Due to the complexity of the real world, randomness and uncertainty are ubiquitous and interconnected in the real world. In order to measure the research objects that contain both randomness and uncertainty in practical problems, and extend the entropy theory of uncertain random variables, this paper introduces the arc entropy of uncertain random variables and the arc entropy of their functions. On this basis, the mathematical properties of arc entropy and two key formulas for calculating arc entropy are also studied and derived. Finally, two types of the mean variance entropy model with the risk and diversification are established, and the corresponding applications to rare book selection for the rare book market are also introduced.

Isolation of Extracellular Vesicles Using Formulas to Adapt Centrifugation to Different Centrifuges.

Extracellular vesicles (EVs) are small lipid bilayer vesicles released by cells to facilitate cell-to-cell communication. To study their biological roles and functions, they need to be isolated and purified, which can be achieved through a variety of methods. Here, we describe different methods for isolating and purifying EVs, with a focus on calculating the required g-force and centrifugation time with different centrifuges and rotors. We have compiled key formulas and tested predicted parameters for EV acquisitions to provide a comprehensive guide for EV isolation.

CALCULATION OF THE EFFECTIVENESS OF THE CHATBOT IMPLEMENTATION AND ANALYSIS OF THE QUALITY OF ITS WORK

This article evaluates the effectiveness of a chatbot and its operational performance. The concept of a chat bot, its operation principles, types of chat bots that differ in their complexity, as well as the purpose of chat bots and their application in various fields of activity are considered. The article contains key formulas for mathematical calculation, which allow evaluating the effectiveness of chat-bots in a parti cular field of activity, as well as their detailed description. An analysis that allows to make conclusions about the effectiveness of the chatbot use in working process of special areas have been conducted. This article also includes illustrations on the areas of bot use and query statistics through their usage. The conclusion is made according to the results of analysis, which allows to evaluate chatbot usage effectiveness and performance.

A realistic framework for volume-delay function determination

Volume-Delay Function (VDF) is a key input in several tasks including traffic assignment and trip distribution estimation. However, there is no clear framework to identify the VDF formula and parameters. Furthermore, the VDF performance in the oversaturated condition is uncontrolled. In this paper, most VDF-related aspects are investigated, including the key attributes, traffic model and VDF formulas, oversaturated condition assumptions, and optimization techniques. In each aspect, different methods are compared for best performance, while a new procedure is introduced, if necessary. The best traffic model and VDF formulas are nominated to satisfy a set of logic and statistical conditions, while a new optimization technique is presented. Furthermore, a new realistic assumption to control the VDF performance is discussed. This framework is applied on 32 worldwide roads of different categories and characteristics, which indicates accurate and realistic results. The results encourage for further investigation of the VDF performance in future work.

Transforming Islamic entertainment and values in Malaysian films

Building a society with high moral and values is imperative in Islamic teachings. As media technologies have changed the ways of human communication and offered new platforms for media content, it provides many opportunities to produce Islamic entertainment programmes that suit with the nature of various medium. Producing Islamic media content could play a unifying role for representing Muslims’ diverse cultures as well as avoiding a negative picture of Islam. Ideally, Muslim culture is derived from the cultural and communication heritage. Islamic entertainment and values are rich in terms of virtues and can be transformed into various genres of media products including films. This article will examine the debate of the Islamic entertainment particularly by examining al-Ghazali’s views ofal-sama’ (‘entertainment’) and subsequently contextualizing it to the Malaysian Islamic ambiences. Using narrative analysis, two popular melodramas (films) will be examined to manifest the process of transforming Islamic values into entertainment programmes. It is argued that the representation of values is considered essential in Islamic communication perspective. It encompasses social, religious and cultural foundations. Muslims must actively participate in producing media content so that it will strengthen both the form and content for communicating Islamic values across the world. Unity, brotherhood, equality, love and tolerance are part of the key formulas to ensure a better process of disseminating good connotations of Islam via media.

Data evaluation strategy for identification of key molecular formulas in dissolved organic matter as proxies for biogeochemical reactivity based on abundance differences from ultrahigh resolution mass spectrometry

The molecular composition of dissolved organic matter (DOM) is of relevance for global carbon cycling and important for drinking water processing also. The detection of variation of DOM composition as function of time and space from a methodological viewpoint is essential to observe DOM processing and was addressed so far. High resolution concerning DOM quality was achieved with Fourier-transform ion cyclotron resonance mass spectrometry (FTICR-MS).However almost none of the existing FTICR-MS data sets were evaluated addressing the fate of single mass features / molecular formulas (MFs) abundance during experiments. In contrast to former studies we analyze the function of MF abundance of time and space for such MFs which are present in all samples and which were formerly claimed as recalcitrant in not all but a great number of studies. For the first time the reactivity of MFs was directly compared by their abundance differences using a simple equation, the relative intensity difference (δRI). Search strategies to find out the maximum δRI values are introduced. The corresponding MFs will be regarded as key MFs (KEY-MFs). In order to test this new approach data from a recent photo degradation experiment were combined with monitoring surveys conducted in two drinking water reservoirs. The δRI values varied over one order of magnitude (more than five-fold). MFs like C9H12O6 and C10H14O6 revealed high biogeochemical reactivity as photo products. Some of the KEY-MFs were identical with MFs identified as disinfection byproducts precursors in recent studies. Other KEY-MFs were oxygen-rich and relatively unsaturated (poly-phenol-like) and hence relevant to flocculation procedures.

Equilibrium points of heterogeneous small body in finite element method

ABSTRACT This paper presents a finite element method to search for equilibrium points around a heterogeneous small body, which degenerates from the full two-body problem in finite element form. The gravitational potential, acceleration, gravitational gradient matrix, and the linearized perturbation equation, which are key formulas for solving the equilibrium points and discriminating their stabilities, are interpolated by the tetrahedral model nodes of the small body. The finite element method could capture the complex internal structures of small bodies and provide a uniform and simple formula for various configurations. The method is applied to the Kuiper Belt Object (486958) Arrokoth, which is a contact binary asteroid. Three types of heterogeneous structures are hypothesized to investigate the evolutions of equilibrium points, namely density disparity of two sub-lobes, hardcore structure, and cavity structure. Under the condition of constant Arrokoth mass, the density disparity of the two sub-lobes greatly influences the local gravitational field. It thus has a significant influence on the positions of equilibrium points. The hardcore and cavity structures have less impact on the equilibrium points. These three simulation groups verified that the equilibrium points of a heterogeneous small body are quite different from those of a homogeneous small body. Such investigation of the equilibrium points may give an in-depth understanding of the dynamical environment around the heterogeneous small body, which is significant for future deep-space missions.

Dual Leonardo numbers

<abstract><p>This paper introduced the concept of dual Leonardo numbers to generalize the earlier studies in harmony and establish key formulas, including the Binet formula and the generating function. Both were employed to obtain specific elements from the sequence. Moreover, we presented a range of identities that provided deeper insights into the relationships within this numerical family, such as the Cassini and d'Ocagne identities, along with various summation formulas.</p></abstract>

Fast converging Chernoff approximations to the solution of heat equation with variable coefficient of thermal conductivity

This paper is devoted to a new method for constructing approximations to the solution of a parabolic partial differential equation. The Cauchy problem for the heat equation on a straight line with a variable heat conduction coefficient is considered. In this paper, a sequence of functions is constructed that converges to the solution of the Cauchy problem uniformly in the spatial variable and locally uniformly in time. The functions that make up the sequence are explicitly expressed in terms of the initial condition and the thermal conductivity coefficient, i.e. through functions that play the role of parameters. When constructing functions that converge to the solution, ideas and methods of functional analysis are used, namely, Chernoff's theorem on approximation of operator semigroups, which is why the constructed functions are called Chernoff approximations. In most previously published papers, the error (i. e., the norm of the difference between the exact solution and the Chernoff approximation with number n) does not exceed const/n. Therefore, approximations, when using which the error decreases to zero faster than const/n, we call fast convergent. This is exactly what the approximations constructed in this work are, as follows from the recently proved Galkin-Remizov theorem. Key formulas, explicit forms of constructed approximations, and proof schemes are given in the paper. The results obtained in this paper point the way to the construction of fast converging Chernoff approximations for a wider class of equations.

Production and distribution of value

The fundamental contribution of this research is regarding five of Cobb and Douglas’s core questions that we believe the theory of economic growth has not made clear until now. The answers we provide are an attempt to measure relative and total resource contribution to production growth and its distribution between them. It implies paying close attention to the latest refinement of the theory and dealing with new critical assumptions, key formulas, and method of attack. The evaluation of these efforts by Cobb and Douglas’s data support the hypothesis that distribution processes are modeled by those of the production of value. The scope of this research could motivate further progress in the study of economic growth.

Orbital angular momentum spectrum of antenna vortex beam based on loop integration

Due to the divergence characteristics of vortex microwaves, it is difficult to determine orbital angular momentum (OAM) orders through far fields, while the spiral spectrum analysis method based on the area integral of antenna near fields is complicated. In this paper, combined with near-field approximation processing, a spiral spectrum analysis method based on loop integration is proposed. According to the Laguerre–Gaussian vortex beam model, the key parameter estimation formulas suitable for low-order OAM microwave beams are derived, including the position of the reliable observation plane and the size of the annulus area. On this basis, the radius of the integral loop used for the spiral spectrum calculation is provided. Since only field data on the loop are collected in this method, the calculation amount is reduced and the analysis efficiency of the spiral spectrum is greatly improved. Due to the combination of near-field and far-field considerations, this method can fully reflect the spiral spectrum characteristics of antenna vortex waves.

Modified Theory of Physical Optics Approach to Diffraction by an Interface between PEMC and Absorbing Half-Planes

The scattering of electromagnetic plane waves by an interface, located between perfect electromagnetic conductor and absorbing half-planes is investigated. The perfect electromagnetic conductor half-plane is divided into perfect electric conductor and perfect magnetic conductor half-screens. The same decomposition is done for the absorbing surface. Then four separate geometries are defined according to this approach. The scattered fields by the four sub-problems are obtained with the aid of the modified theory of physical optics. The resultant scattering integrals are combined in a single expression by using key formulas, defined for the perfect electromagnetic conductor and absorbing surfaces. The scattering integral is asymptotically evaluated for large values of the wave-number and the diffracted and geometric optics fields are obtained. The behaviors of the derived field expressions are analyzed numerically.

История и поэтика карачаево-балкарской загадки

Introduction. The riddle which reflects the world outlook of the people is a popular genre in the Karachay-Balkar folklore. However, the richest materials collected over decades have not been studied so far. The present study aims at examining the history and poetics of the Karachay-Balkar riddle drawing on the works of leading foreign and Russian theorists. This has involved i) the discussion of the socio-cultural factors that were conducive to the emergence of the genre, ii) the description of the five principal thematic circles, iii) the analysis of the key structural formulas of the Karachay-Balkar riddle, and iv) the description of the expressive language of the puzzle. Data and research methods. Academic collections, including pre-revolutionary sources, were used as the database for the research, which involved comparative-historical, system-structural, semiotic, textological, and analytical methods. Results. The Karachay-Balkar puzzle may be seen as a bilateral artistic subculture, with one side facing the archetypal invariant, and the other characterized by variability, granted the historical and geographical contexts, the landscape of the North Caucasus, the life, and the culture of the highlanders. The conceptualization of the world in riddles involves five main ontological conceptual spheres: anthropocentric, ethnocultural, natural, cosmogonic, mental, and ethical. The riddle images are most often associated with ethnically marked objects. The distinct character of the Karachay-Balkar riddle is due to frequent use of rhymed poetic forms, verse formulas, proper names, as well as to a wealth of their expressive devices (metaphor, comparison, alliteration, antithesis, sarcasm, and humor). Also, the riddles have a great pedagogical potentiality. Conclusions. The analysis of the Karachay-Balkar riddles resulted in identifying the distinct features in the conceptualizations of the world by the North Caucasian highlanders, as well as in drawing a picture of their values in the material and spiritual spheres.

SU3lib: A C++ library for accurate computation of Wigner and Racah coefficients of SU(3)

We present the C++ library SU3lib for accurate computation of SU(3) Wigner coupling and Racah recoupling coefficients. It is built on the efficient mathematical algorithm originally proposed by Draayer and Akiyama [1]. The presented library extends the reach of this algorithm towards large SU(3) irreducible representations and outer multiplicities that were heretofore inaccessible due to floating-point precision errors. As large irreducible representations of SU(3) play an important role in medium- and heavy-mass atomic nuclei, SU3lib expands the scope of approaches to nuclear structure and reactions that rely on available SU(3) coupling-recoupling coefficients. Program summaryProgram Title: SU3libCPC Library link to program files:https://doi.org/10.17632/j977v8v5fp.1Developer's repository link:https://gitlab.com/tdytrych/SU3libLicensing provisions: BSD 2-clauseProgramming language: C++External libraries: WIGXJPF [3], BoostNature of problem: Accurate calculation of SU(3)⊃SO(3) and SU(3)⊃SU(2)×U(1) Wigner coupling and Racah recoupling coefficients for arbitrary couplings and multiplicity.Solution method: We adopt the mathematical procedure proposed by Draayer and Akiyama [1], who also provided its implementation as a FORTRAN library [2]. The challenge is to avoid the loss of precision due to cancellation in sums of large alternating terms in transformation between SU(3)⊃SO(3) and SU(3)⊃SU(2)×U(1) schemes, and to compute SU(3)⊃SU(2)×U(1) Wigner coefficients accurately for large outer multiplicities. The present library tackles these challenges by implementing key formulas and data structures as C++ templates and utilizing floating-point data types with extended precision provided by the Boost.Multiprecision library as template arguments. This permits an efficient and accurate computation of SU(3) coefficients even for large SU(3) irreps and outer multiplicities that were heretofore inaccessible.

Some fractional derivatives of A-function of multivariable

Abstract In the present paper, we study and develop Fractional derivatives of multivariable A – function. We derive two theorems which will act as the key formulas from which can obtain their special cases.

Fast multipole method for 3-D Poisson-Boltzmann equation in layered electrolyte-dielectric media

In this paper, we propose a fast multipole method (FMM) for 3-D linearized Poisson–Boltzmann (PB) equation in layered electrolyte-dielectric media. We will extend our previous work on FMMs for Helmholtz and Laplace equations in layered media [1,2] to the case of electrolyte and dielectric layered media for applications arising from biophysics and colloidal fluids, such as ion channel transport and Helmholtz double layers. Two key mathematical formulas are developed for this purpose: Firstly, a Funk–Hecke formula for purely imaginary wave numbers is derived, which facilitates the derivation of multipole expansions of the potential far fields of charges in layered electrolyte-dielectric media. Secondly, a recurrence formula is constructed for run-time computations of the Sommerfeld-type integrals used in the FMM algorithm. Numerical results show that the proposed FMM for interactions of charges embedded in layered media under screened PB potentials has the same accuracy and the O(N) computational complexity as the classic FMM for charge interactions in the free space.

Dynamic economic growth: A new method of approach

This paper looks at Harrod’s fundamental assumptions and equations in ‘An essay in dynamic theory’ as the foundation for suggesting a better understanding of his postulates on economic growth. The analysis we present attempts to measure the effects of resource composition on the field of economic growth by determining its elasticity and the coefficient of acceleration. By addressing new critical assumptions, key formulas, and methods of approach, some crucial findings provide a strong theoretical model with which to examine dynamic economic growth. Furthermore, this paper offers a brief review of Solow’s and Swan’s critiques of Harrod’s essay. This analysis could inspire further advances in the study of economic growth at the academic level and even in terms of government macroeconomic policies and business decisions.