Fundamental Formula Research Articles

Approximate Fundamental Frequency Formula for Cantilevers with Weakly Non-Uniform Sections and Verification with Experiments and Other Studies in the Literature

Approximate Fundamental Frequency Formula for Cantilevers with Weakly Non-Uniform Sections and Verification with Experiments and Other Studies in the Literature

Principles and prospects of Bose-Einstein correlation study at BESIII* *This study was supported by the National Key R&D Program of China (2020YFA0406403) and the National Natural Science Foundation of China (11275211, 11335008, 12035013)

One method for determining the characteristic parameters of a hadron production source is to measure the Bose-Einstein correlation functions. In this study, we present fundamental concepts and formulas related to the Bose-Einstein correlations, focusing on the measurement principles and the Lund model from an experimental perspective. We perform Monte Carlo simulations using the Lund model generator in the 2–3 GeV energy range. Through these feasibility studies, we identify key features of the Bose-Einstein correlations that offer valuable insights for experimental measurements. Utilizing data samples collected at BESIII, we perform measurements of the Bose-Einstein correlation functions, with an expected experimental precision of a few percent for the hadron source radius and incoherence parameter.

Some New Algebraic Method Developments in the Characterization of Matrix Equalities

Algebraic expressions and equalities can be constructed arbitrarily in a given algebraic framework according to the operational rules provided, and thus it is a prominent and necessary task in mathematics and applications to construct, classify, and characterize various simple general algebraic expressions and equalities. As an update to this prominent topic in matrix algebra, this article reviews and improves upon the well-known block matrix methodology and matrix rank methodology in the construction and characterization of matrix equalities. We present a collection of fundamental and useful formulas for calculating the ranks of a wide range of block matrices and then derive from these rank formulas various valuable consequences. In particular, we present several groups of equivalent conditions in the characterizations of the Hermitian matrix, the skew-Hermitian matrix, the normal matrix, etc.

A spherical source distribution method for calculating acoustic radiation of elastic underwater structures

In this paper, a spherical source distribution method is established, and two kinds of spherical sound sources of symmetric and antisymmetric, distributed on a line inside the structure are proposed, in order to realize the vibro-acoustic calculation of three-dimensional elastic underwater structure. The spherical source distribution method has strong applicability and is suitable for the case where the shape of the structure is not axisymmetric. This method is a new method, and its fundamental formula is similar to the traditional acoustic boundary integral method, but it also has obvious differences. In numerical calculation, the traditional boundary element method is to divide the surface elements on the wet surface of the object, and transform the three-dimensional acoustic problem into the two-dimensional discrete element problem to solve. However, the spherical source distribution method, whose source points are only distributed on a straight line inside the object, transforms the three-dimensional acoustic problem into the quasi-one-dimensional discrete element problem, which makes the complexity and computation amount of the whole programming significantly reduced. In this paper, the fundamental principle of the spherical source distribution method, the calculation formula and the verification results of several numerical examples are discussed.

Nonlinear thermo-mechanical static stability analysis of FG-TPMS shallow spherical shells

An analytical solution for the nonlinear static stability problem of functionally graded triply periodic minimal surface (FG-TPMS) shallow spherical shells is studied in the current research for the first time. Three common TPMS structures including Primitive (P), Gyroid (G), and I-graph and Wrapped Package-graph (IWP) with three models of functionally graded porosity distribution along the thickness are considered. The shallow spherical shells (shallow SSs) are subjected to combined thermo-mechanical loadings and rested on a nonlinear elastic foundation. The fundamental formulas are expressed based on the higher-order shear deformation theory (HSDT) and von Kármán's geometrical nonlinearities. Employing the Ritz energy minimization method, the explicit relationship between load and deflection is derived. Subsequently, the static stability behavior of FG-TPMS shallow SSs is investigated. Numerical illustrations are investigated to show the superior thermo-mechanical load-carrying performance of the FG-TPMS SSs compared to corresponding isotropic structures of the same weight. The significant effects of geometric parameters, nonlinear elastic foundation parameters, and the type of FG-TPMS structures on the nonlinear static stability behavior of shallow SSs are further considered.

Novel Approach by Shifted Fibonacci Polynomials for Solving the Fractional Burgers Equation

This paper analyzes a novel use of the shifted Fibonacci polynomials (SFPs) to treat the time-fractional Burgers equation (TFBE). We first develop the fundamental formulas of these polynomials, which include their power series representation and the inversion formula. We establish other new formulas for the SFPs, including integer and fractional derivatives, in order to design the collocation approach for treating the TFBE. These derivative formulas serve as tools that aid in constructing the operational metrics for the integer and fractional derivatives of the SFPs. We use these matrices to transform the problem and its underlying conditions into a system of nonlinear equations that can be treated numerically. An error analysis is analyzed in detail. We also present three illustrative numerical examples and comparisons to test our proposed algorithm. These results showed that the proposed algorithm is advantageous since highly accurate approximate solutions can be obtained by choosing a few terms of retained modes of SFPs.

Study on joint effects of modal dispersion, mode-dependent loss and noise by unified density-matrix formalism

We generalize the density-matrix formalism to provide a simple and unified framework for the study of modal dispersion (MD), mode-dependent loss (MDL) and noise in mode-division-multiplexing communication systems. Fundamental concepts and formulas in the original formalism are generalized, while new elements and associated equations are introduced and/or derived. Due to the ability of the density operator to characterize partially-coherent fields, the generalized density-matrix formalism is particularly useful for analyzing the field propagation in presence of noise. Combining the unified density-matrix formalism and the stochastic-process theory, we conduct detailed numerical simulation and analysis on the modal properties of a randomly-perturbed 6-mode fiber with loss and noise. We find that although the MDL only slightly affects the statistics of the group delays, it can cause significant non-orthogonality among the principal modes, which might induce extra crosstalk and temporal spreading in the signal pulses. Furthermore, with the coherency of fields in the fiber being reduced by the noise, the MDL can induce substantial fluctuation in the field coherency even though it does not significantly affect the averaged field coherency.

Ricci Solitons on Spacelike Hypersurfaces of Generalized Robertson–Walker Spacetimes

Ricci Solitons on Spacelike Hypersurfaces of Generalized Robertson–Walker Spacetimes

Topological numbers of fuzzy soft graphs and their applications in globalizing the world by mutual trade

In this research draft, by keeping in view, the importance of Zagreb numbers, first we defined some familiar graph families and their degrees in fuzzy soft environment, and then by converting these two novel topological numbers in fuzzy soft framework, their formulae are calculated for these graphical networks. An application of international trade using two kinds of routes is also discussed in the article, and by comparing the results efficiency of both the numbers is compared and analyzed how topological descriptors are useful in examining the progress and profit in the countries, that are a part of the project. Graphs are used in the modeling of links and processes within physical, biological, social, and information systems. A fuzzy graph is the most adaptable tool in mathematics, which describes the fuzzy relationship between objects. Fuzzy graphs are very helpful to give more accuracy to the system as compared to crisp graphs. A classification of the universe with respect to certain specified parameters is known as a soft set. It is a new method for simulating ambiguity and uncertainty. Soft set theory was created by Molodtsov in 1999 as a method for simulating vagueness and uncertainty. Soft set theory is presently used widely in academia to address decision-making issues. The concept of soft graphs is used to provide a parameterized point of view for graphs. In many areas of mathematics and chemistry, as well as in network theory, spectral graph theory, and molecular chemistry, the Zagreb numbers are crucial graph parameters. Due to the importance and significance of Zagreb numbers, in this research article we define these graph parameters in fuzzy soft environment and then derived the fundamental formula for first and second Zagreb numbers of some popular graph families. Also by keeping in view the importance of graph theory in solving decision making problems in business industry, and the most generic behavior of fuzzy soft graph due to its parameterized nature we use our work in application of decision making in trade among different nations.

Identification, Prioritization and Mapping of Major Constraints and Potentials in the Medo Watershed the Case of West Arsi District in Central Rift Valley Area of Ethiopia

The concentration to characterize the biophysical and socio-economic features in the integrated watersheds is to identify and prioritize production constraints for designing appropriate research and development interventions. The objective of this study was to assess the current biophysical of characteristics of Medo watershed, West Arsi Zone of Oromia, Ethiopia. To do so relevant data and tools were used; ArcGIS, Microsoft Excel sheet and fundamental formulae were applied for the analysis are some. The results of the study indicated that the major land use types in the watershed are agricultural land covering 50%, vegetation (shrubs, forests and, plantations) covering 24% and settlement covering 16% out of the total land use. Slope gradient of Medo watershed ranges from zero to more than 20 and the slope gradient of 2-5 and 5-10 cover the greatest in area coverage representing 201 ha and 170 ha respectively. We can also observe that about 8% out of the total area is subjected to severe erosion. Sandy clay loam was the dominant soil textural classes in the surface soils and Phaeozems, Retinols and Andosols are the dominant soil types, which cover 57%, 36% and 7% of the sub-watershed area for lower, middle, and upper slope positions, respectively. Mean annual rainfall of the area ranges from 960.09 mm at lower part of the watershed to the 1304.93 mm at upper part of the watershed. Accordingly, Medo sub-watershed is laid in majority of at sub humid agro ecological zone. An assessment of the trees within the landscape of watershed showed some remnant natural forests and a wide variety of shrubby vegetation is encountered at all landscapes. About 40 percent of the watershed, is under high to extremely severe soil loss values (>45 ton per hectare per year). The baseline study also identified that those natural resources degradation such as land, and or/ soil fertility, reduction, and recent changes in the areas’ weather condition in line of climate change prevailing in current years are few of the many factors that are contributing to the land and crop productivity reductions in the area. Therefore, prioritizing the identified problem and preparing intervention of different technologies and development plans to solve the problems by taking into consideration the existing opportunities of the watershed.

Development and Implementation of a Microlearning Curriculum of Physiology Formulas for Postgraduate Clinical Fellows in Pediatric Critical Care Medicine.

Postgraduate medical education in clinical settings poses many challenges secondary to the large volume of knowledge to be acquired, competing clinical responsibilities, and fatigue. To address these challenges, a microlearning curriculum using flipped classroom methodologies was created to facilitate the mastering of fundamental physiology formulas by pediatric critical care medicine fellows. Forty physiology formulas were distilled into 5-minute microlearning sessions. Fellows were provided the weekly formula and encouraged to self-study prior to the face-to-face learning. The 5-minute session took place at the beginning of a regularly scheduled clinical care conference where normal values, explanatory diagrams, and board-like questions were discussed. A faculty or fellow facilitator then provided a more in-depth explanation and shared clinical pearls related to the formula. Following the session, an e-mail summarizing the learning points was sent. The curriculum was well received by fellows and faculty. Over 5 years, the curriculum evolved through phases of active development, implementation, minor modifications, transition to a virtual platform, shift to senior fellow-led instruction, and harmonization with other curricular activities. Engagement and sustainability were addressed with a fully flipped classroom, where senior fellows served as teachers to junior fellows. Microlearning in a multimodal manner is an excellent method for teaching busy postgraduate clinical trainees fundamental physiology formulas that underpin pediatric critical care decision-making. The gradual transition from individual learning to a flipped classroom taught by peers with faculty support was well tolerated and consistent with adult learning theories. The transition was essential to ensure the sustainability of the curriculum.

Numerical simulation and experimental data blade investigation of horizontal wind turbine

The article presents the results of numerical simulation of a wind turbine blade. The fundamental formulas of the theory of aerodynamics, on which the calculations are based, are presented. Computer models of air movement near the blade are shown. Data on the distribution of speed have been obtained. The coefficient of wind energy utilization depends on the parameters of both the oncoming flow and the blade profile. This coefficient is different for vertical and horizontal wind turbines. To improve the operation of the wind turbine, it is proposed to use non-network algorithms. The neural network should collect data on the received errors in the system of mechanization and automatic control. Errors in control systems are analyzed, individual parameters are weighted to stabilize the operation of the wind turbine as a whole.

Citation Fall Index (CFI): An Indicator to Measure the Centripetal Nature of Accretion of Citation

That citation attracts citation and fairly-cited items quickly become highly-cited, whereas poorly-cited or uncited items are hardly cited even after a long span is a known phenomenon referred to as the ‘Cumulative Advantage’ process (“Success breeds success”). The citation distribution always shows a highly skewed pattern to a very small number of core groups. This paper has proposed a fundamental formula to measure the change in relative fall in citation count with a corresponding change in the ranking of different kinds of cited items (author, journal, article, etc.). Two new indicators are mathematically developed here, the names given to which are, Relative Citation Fall (RCF) and Citation Fall Index (CFI). These indicators will be tested for different kinds of cited items in different subject domains to quantitatively measure citation skewness in future studies.

Results on the modified degenerate Laplace-type integral associated with applications involving fractional kinetic equations

Abstract Recently, integral transforms are a powerful tool used in many areas of mathematics, physics, engineering, and other fields and disciplines. This article is devoted to the study of one important integral transform, which is called the modified degenerate Laplace transform (MDLT). The fundamental formulas and properties of the MDLT are obtained. Furthermore, as an application of the acquired MDLT, we solved a simple differential equation and fractional-order kinetic equations. The outcomes covered here are general in nature and easily reducible to new and known outcomes.

Fundamental formulas specifying the minimum reinforcement ratio in the draft of standard recommendations amendment

Fundamental formulas specifying the minimum reinforcement ratio in the draft of standard recommendations amendment

Calculation method and mathematical model of projectile echo power of active laser detection target

AbstractTo meet the requirements for parameter testing within the projectile test system, a computational model for calculating the echo power of the projectile passing through the detection screen based on the principle of active laser detection target is established in this paper. On the basis of the fundamental formula of echo energy in laser detection, and by combining the energy distribution function of the detection screen, the light scattering cross‐section formula specific to the projectile, and the illuminated area formula when the projectile intersects the detection screen, the formula for calculating the echo power of projectile is derived and analyzed. Furthermore, taking into account the parameters of the laser detection target, a computational model is constructed to calculate the voltage amplitude associated with the echo power of the projectile. Combining the atmospheric transmission optical characteristics, distance variations, and changes in the projectile's reflection attitude in the test environment, we conducted a simulation analysis to assess the accuracy of the established calculation model. Additionally, experimental analysis of the echo power calculation model was performed. In both simulation and experimental analyses, the trends in the impact of atmospheric transmission optical characteristics, distance conditions, and changes in projectile attitude upon the calculation of projectile echo power were consistent. Moreover, the width of the projectile echo waveform obtained in experiments was equal to the calculated projectile duration time. These research findings can serve as a reference basis for the development of theories in laser detection target.

Политико-правовая идеология И.В. Сталина: общий дискурс

Introduction. The study of the state and law of the USSR, whose successor and legal continuer is the modern Russian Federation, is impossible in isolation from the activities and political outlook of I.V. Stalin, one of its founders and state leader for almost thirty years. The subject of the paper is the discourse analysis of the political and legal ideology of I.V. Stalin by studying the ontological and epistemological prerequisites of its content and identifying on this basis the historical and cultural conditionality of its conceptual idea. Methods. The methodological framework for the work is a set of methods of scientific cognition, among which the main ones are a combination of dialectics and metaphysics, formational and civilizational approaches, and a number of general scientific and specific scientific methods. The main content of the paper is to establish the features of Stalin’s worldview as a politician and theorist and to study the main points of view on the general nature of his political ideology to provide its most adequate scientific description and definition. Results and conclusions. In relation to the general discourse of I.V. Stalin’s political ideology, the most adequate definitions would be “national Bolshevism” for the period of its formation and crystallization and “national-state socialism” for a later period, which characterizes its content in its mature and final version much more accurately. The fundamental conceptual formula of the Russian nationalization of the communist doctrine was the Stalinist idea of the possibility of building socialism in one single country. It became the value foundation of Stalin’s socio-political project, in whose theoretical justification there was a creative synthesis of the teachings of Marxism-Leninism with the traditions of Russian statehood.

Relationship between box-counting fractal dimension and properties of fracture networks

Due to the capacity to quantify the complexity of a fracture network, fractal dimension (D) is widely applied in analyzing fracture network-related issues, such as connectivity and permeability. While the relationship between D and individual properties of a fracture network has been extensively studied, D is influenced by a combination of multiple attributes of the fracture network. Therefore, this work utilizes multivariate analysis to establish an equation for predicting D, taking into account various properties of the fracture network, namely fracture length, number, and orientation. Monte Carlo simulation is employed to generate a substantial number of fracture network models. Subsequently, relationships between the fractal dimension (D) and various properties are derived for three types of fracture networks with (1) invariant fracture length and random orientation, (2) exponential fracture length and random orientation, and (3) exponential fracture length and von-Mises orientation. The initial analysis focuses on the simplest relationship, wherein the fundamental formula of fractional expression is determined. Then the second and third relationships are obtained through replacing the fixed parameter in the first relationship with the distribution parameters of fracture properties. Correlation analyses between the predicted D and the actual values reveal a remarkably high correlation (>0.99). To validate the established relationships, a fracture network obtained from geological outcrops is utilized. The results demonstrate the validity of the derived relationships. The utilization of these equations enhances the efficiency, practicality, and convenience of estimating fractal dimensions from fracture properties. As a result, the analysis of fracture network-related issues becomes more feasible and accessible.

Influences of fatigue stiffness degradation of in-service concrete T-beams on natural vibration frequency

ABSTRACT In order to obtain the fatigue stiffness degradation law of in-service concrete T-beam and carry out fatigue life prediction, a dual-stage stiffness degradation model considering cracks was established for T-beams and formulas for calculating the fundamental frequency of damaged concrete T-beams were deduced based on relevant theories of damage mechanics. The developed model aimed to investigate the influences of the fatigue stiffness degradation of in-service concrete T-beams on the natural vibration frequency of structures. A fatigue test was conducted for full-scale test beams to examine the crack development and fundamental frequency change law of structures during the fatigue process. Based on a full scale model of 10 m reinforced concrete T-beams, static and fatigue failure tests were carried out to obtain the evolution law of fatigue residual stiffness with load times. Fatigue damage coefficient D and stiffness degradation coefficient k were introduced to establish the fatigue life prediction model of T-beams in service. In line with the related test data and fundamental frequency formulas, stiffness updating methods were put forward for the concrete T-beam finite element models. By comparing the calculated value and test value of the natural vibration frequency of structures, the relative error was estimated to be within 6%. This verified the accuracy and reliability of the fundamental frequency calculation formulas and the updated finite element models. The research results can be used as a reference for the precise simulation of the fatigue damage finite element models of concrete T-beams. Finally, a method for life prediction of in-service reinforced concrete beam Bridges is proposed, and relevant research results can provide theoretical support for the study of life prediction of such Bridges.

The uncertainty analysis of bridge displacement in computer vision

Computer vision is commonly employed to measure the displacement of bridge structures. However, changes in environments often lead to poor data reproducibility. The uncertainty is an important parameter associated with the results, representing the dispersion of the data. This paper mainly discusses the uncertainty of measurement utilizing computer vision. The fundamental formula of measurement is derived from the principle of the pinhole camera model, followed by calculations of the uncertainty as well as a detailed analysis and deduction of the components that affect the uncertainty of measurement (e.g. pitch angle, deviation angle, observation distance, temperature, humidity). Both laboratory and on-site experiments have shown that the uncertainty model is more applicable than error estimation. In addition, the variation law of the uncertainty is revealed under the combination of different angles, laying a solid foundation for accurate measurement in the future.