Derivation Of Formula Research Articles

A unified approach to inversion formulae for vector and tensor ray and radon transforms and the Natterer inequality

Most derivations of inversion formulae for x-ray or Radon transform are based on the projection theorem, where for fixed direction the Fourier transform of x-ray or Radon transform is calculated and compared with the Fourier transform of the searched-for function. In contrast to this we start here off from the searched-for field, calculate its Fourier transform for fixed direction, which is now a vector or tensor field, that we then expand in a suitable direction dependent basis. The expansion coefficients are recognized as the Fourier transform of longitudinal, transversal or mixed ray transforms or vectorial Radon transform respectively. The inverse Fourier transform of the searched-for field then directly leads to inversion formulae for those transforms applying problem adapted backprojections. When considering the Helmholtz decomposition of the field we immediately find inversion formulae for those transversal or longitudinal transforms. First inversion formulae for the longitudinal ray transform, similar to those given by Natterer (1986 The Mathematics of Computerized Tomography (Teubner and Wiley)) for x-ray tomography, were given by Natterer-Wübbeling in 2001, Natterer and Wübbeling (2001 Mathematical Methods in Image Reconstruction (SIAM)), but then not pursued by other authors. In this paper, we present the above described method and derive in a unified way inversion formulae for the ray transforms treated in Louis (2022 Inverse Problems 38 065008) containing the results from Louis (2022 Inverse Problems 38 065008) as special cases. Additionally we present new inversion formulae for the vectorial Radon transform. As a consequence the inversion formulae directly give Plancherel’s formulae for the vectorial or tensorial transforms. Together with the Natterer inequality, which is independent of the ray or Radon transforms, we present the Natterer stability of those vectorial and tensorial transforms.

Research on Mechanism of Non-Uniform In-Situ Stress Induced Casing Damage Based on Finite Element Analysis

Casing damage is a common problem encountered during oil and gas field development due to the complex stress state of the casing. Despite the large number of studies focusing on this problem, the mechanism of non-uniform in-situ stress-induced casing damage in a low-permeability reservoir is still unclear. In this study, casing damage due to non-uniform in-situ stress variations was investigated, and then the tectonic stress coefficients in the study area were determined by an in-situ stress inversion technique, which led to the derivation of formulas for calculating the maximum and minimum horizontal in-situ stresses. Subsequently, finite element numerical simulations were performed to assess the stress distribution during the formation of the casing cement sheath in a G155 block, a typical low-permeability reservoir. The results indicate that casing damage is caused not only by non-uniform in-situ stresses but also by various additional creep-induced loads. Subsequent finite element investigations into casing behavior under mudstone creep conditions indicated that immersion of mudstone in water instigated further shearing and deformation of the casing, culminating in premature well failure prior to water inundation. Notably, Von Mises stress levels exhibited a positive correlation with injection production ratios, with values exceeding critical thresholds leading to distinct modes of mechanical failure including shear-induced deformations, longitudinal tensile stress, and localized yielding near water wells. Maintenance of an optimal injection production ratio is identified as a key strategy for prolonging casing longevity in the region. To this end, recommendations include augmenting the casing wall thickness or enhancing the steel pressure specifications to mitigate casing damage progression, thereby extending the operational lifespan. This research serves as a pivotal theoretical framework for informing future development strategies aimed at mitigating and preempting casing failures in a low-permeability reservoir.

Comparing Weights According to Weighing Cycles with Nonlinear Drift of the Comparator

Cycle weighing is used by metrologists around the world to eliminate drift in comparator readings when comparing the mass of reference weights. Various types of cycles and their descriptions are given in the International Recommendation on legal metrology OIML R111-1-2004 adopted in the Russian Federation as a national standard. They consider cycles under the assumption of linear drift of comparator readings. The article discusses weighing cycles that eliminate nonlinear drift of the comparator. Drift models are proposed in the form of exponential and multinomial laws, which are characterized by a fast dependence of readings at the beginning of the cycle and a slower one at the end of the cycle. The derivation of formulas for the difference between the mass of the verified and the original standard according to four comparator readings at equal time intervals is given. The formulas consist of two components: the first coincides with the formula for linear drift, and the second plays the role of a correction for the drift deviation from linearity. The formulas are also correct for linear drift, since in this case the second components become zero. Estimates of the differences in the mass of the compared weights, taking into account the nonlinearity of the comparator drift, make it possible to estimate the measurement uncertainty for linear and nonlinear drift and become relevant with a constant increase in weighing accuracy, especially when comparing high-precision mass standards.

Pseudospectral analysis for multidimensional fractional Burgers equation based on Caputo fractional derivative

This study presents the Chebyshev pseudospectral approach in time and space to approximate a solution to the time-fractional multidimensional Burgers equation. The suggested approach utilizes Chebyshev–Gauss–Lobatto (CGL) points in both spatial and temporal directions. To figure out the fractional derivative matrix at CGL points, we use the Caputo fractional derivative formula. Further, the Chebyshev fractional derivative matrix is utilized to reduce the given problem in an algebraic system of equations. The numerical approach known as the Newton–Raphson is implemented to get the desired results for the system. Error analysis for the set of values of ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ u $$\\end{document} is done for various model examples of fractional Burgers equations, where ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u $$\\end{document} represents the fractional order. The computed numerical results are in perfect agreement with the exact solutions.

Consistent DCF Methods for Constant-Growth Annuities à la Modigliani & Miller or Miles & Ezzell

Abstract In this paper, we develop two complete discounted-cash flow (DCF) frameworks for the valuation of constant-growth annuities and perpetuities. By ‘complete’ we mean that these frameworks allow the valuation of a firm or project by means of different DCF methods, particularly, the equity method, the free-cash-flow (FCF) method, the adjusted-present-value-method, and the capital-cash-flow method. This also requires the derivation of formulas that allow the translation between different required returns, like the required return on unlevered and levered equity, the discount rate in the FCF method, and the required return on the tax-shield. Our paper departs from the two most advocated and mutually exclusive frameworks when dealing with DCF. The first is based on Modigliani and Miller (M&M), where the FCF at different points in time are independently distributed. The second framework rests on the analysis of Miles and Ezzell (M&E) who presume a first-order autoregressive cash-flow process. Some elements of a ‘complete’ framework exist in the literature, but in our opinion, a complete picture has not been developed yet. The contributions of this paper are the following: (1) We develop (or expand) the set of formulas that are required for the valuation of constant-growth annuities and perpetuities; (2) The formulas we develop in this paper are based on a backward-iteration process, which in itself represents a suitable tool for firm valuation; (3) Using a numerical example, we show that the two mutually exclusive frameworks of M&M or M&E achieve very different valuation results; (4) It turns out that the expected returns and the growth rate of the FCF are partly linked, but this relationship is different in the two frameworks; (5) In our numerical examples, we show how the constant-growth annuity or perpetuity, can be integrated with an explicitly planned FCF.

Motion analysis of and experimental research on a magnetic and elastic coupling oscillation system

To analyze the motion laws of a magnetic and elastic coupling system under the influence of various factors, this paper proposes a magnetic coupling pendulum based on spring pieces and magnets—a magnetic–mechanical oscillator. By fixing spring pieces onto two non-magnetic bases and attaching magnets to their upper ends, which repel each other, the potential energy during oscillation is expanded using Fourier series. Subsequently, Lagrange equations are solved to study the effects of the first two terms of potential energy. spring piece Using algebraic simplification and transformation, as well as model experiments, its dynamic behaviors, such as its motion frequency and phase characteristics, were explored according to different influencing factors, including the magnetic potential energy, the elastic modulus of the spring, and the external environment. We analyzed the impact of the simplified system's main parameters on its vibration modes, deriving the model formulas and the experimental results. The conclusions from the formula derivation indicate that the motion frequency of the simplified system is a superposition of two frequency. Under certain conditions, these frequencies can be separated, simplifying the system's dynamics. Equally, the experimental results corroborated the findings from the formula derivation, providing a reference for the design and study of dynamic behaviors in magnetic coupling systems.

Fluctuations of thermal variables investigated by cross-correlation function.

Fluctuations in conjugate thermodynamic variables are studied using the cross-correlation function. A new procedure is given enabling the derivation of fluctuation formulas for a system in equilibrium. Specifically, the cross-correlation function between heat and temperature is employed for thermal variables. Additionally, fluctuation-dissipation relations involving the frequency-dependent specific heat are established. Moreover, a general relation concerning the average entropy production is also given, which is the microscopic analog of the dissipation formula of the linear response theory. In the case of thermal variables, this formula finds application in various scenarios describing fluctuating thermal systems in equilibrium.

Linearly-coupled sigmoid bistable stochastic resonance for weak signal detection

Abstract The paper focuses on developing a stochastic resonance (SR) system designed for the detection of weak signals under alpha-stable-distributed noises. Initially, in view of the strong impulsive characteristics of noises, a linearly-coupled sigmoid bistable stochastic resonance (LSBSR) system is proposed, which is constructed by potential function and sigmoid function. Through formula derivation, it is theoretically proved that the output signal-to-noise ratio (SNR) of the LSBSR system is superior to that of the classical bistable SR system. Then, a new signal processing strategy based on the LSBSR system is introduced. Simulation experiments have demonstrated that under the input SNR = −20 dB, the detection probability of the LSBSR system exceeds 95% for the alpha-stable-distributed noise with α= 1.5. When α is reduced to 0.1, the detection probability approaches 80%, significantly outperforming other detection methods. Finally, the LSBSR system is applied to detect sea-trial signals with an SNR improvement of 22.5 dB, which further validates the practicability of the proposed system.

Topological derivative based sensitivity analysis for three-dimensional discrete variable topology optimization

This study introduces a novel Topological Derivative-based Sensitivity Analysis (TDSA) methodology for three-dimensional (3D) discrete variable topology optimization. Recently, the authors pointed out that the discrete variable sensitivity can be related to the topological derivative, and thus can be rationally approximated by the specially customized topological derivative for plane stress problems. However, the 3D discrete variable sensitivity requires the 3D topological derivative with element-shaped (e.g., unit cube) domain perturbation, whose analytical solution is not available in the literature. This paper proposes a unified parameter fitting framework to calculate the 3D topological derivative with arbitrary shape 3D domain perturbation. The formula for spherical hole perturbation obtained through this parameter fitting framework is identical to the well-known analytical topological derivative formula. Further, the 3D discrete variable sensitivity can be easily obtained through element-shaped domain perturbation. With the recently proposed Sequential Approximate Integer Programming (SAIP), numerical examples demonstrate that TDSA is precise not only for the self-adjoint 3D minimum compliance problems but also for the non-adjoint 3D compliant mechanism design problems. In sum, numerical examples have been conducted by feeding the derived sensitivity to the SAIP optimization algorithms, and the correctness of the sensitivity information has been well demonstrated. This research further solidifies the foundation of rational discrete variable sensitivity analysis in 3D topology optimization.

Practical derivations of fermion and gauge boson reduction formulae in curved spacetimes

LSZ-type reduction formulae are derived for gauge fields and fermions in curved spacetime. The formulae are derived using a conserved current method applicable also to flat spacetimes. The method generalizes to more general quantum field theories. The formulae are then applied to a couple of basic problems to illustrate their use.

Sunlight optimization in residential area design: Introducing sOSA - A comprehensive indicator for swift assessment of outdoor sunshine exposure

The increasing density and height of residential buildings have heightened the concerns regarding inadequate sunshine exposure in outdoor areas. With multi-objective optimization experiments, existing evaluation measures, such as Sunshine Times (ST), are not able to sufficiently capture the spatial features required to find the best design for outdoor sunlight conditions quickly. Spatial Outdoor Sunshine Autonomy (sOSA) is proposed in this paper as a comprehensive indicator for the rapid assessment of cumulative sunlight exposure in various outdoor scenarios. The definition of the indicator is first presented through formula derivation. Subsequently, the rapid simulation method of the sOSA indicator in Rhino is demonstrated. Utilizing the Maigao Bridge Block in Nanjing as a prototype, the application of this indicator in multi-objective optimization experiments for new residential area planning and design is showcased. Finally, the results are evaluated, and the suggestion is made to combine sOSA with ST maps and wind speed cloud maps for enhanced practical guidance.

VALIO: Visual attention-based linear temporal logic method for explainable out-of-the-loop identification

The phenomenon of being Out-Of-The-Loop (OOTL) can significantly undermine pilots’ performance and pose a threat to aviation safety. Previous attempts to identify OOTL status have primarily utilized “black-box” machine learning techniques, which fail to provide explainable insights into their findings. To address this gap, our study introduces a novel application of Linear Temporal Logic (LTL) methods within a framework named Visual Attention LTLf for Identifying OOTL (VALIO), leveraging eye-tracking technology to non-intrusively capture the pilots’ attentional focus. By encoding Areas of Interest (AOIs) and gaze durations within the cockpit into Visual Attention Traces (VAT), the method captures the spatial and temporal dimensions of visual attention. It enables the LTL methods to generate interpretable formulas that classify pilot behaviors and provide insights into the understanding of the OOTL phenomenon. Through a case study of a simulated flight experiment, we compared the efficacy of this approach using different time windows from 10 s to 75 s. The results demonstrate that VALIO’s performance is stable across all time windows with the best F1 score of 0.815 and the lowest F1 of 0.769. And it significantly outperforms the other machine learning methods when using time windows shorter than 30 s, signifying its ability to detect the OOTL status more in-timely. Moreover, the VALIO elucidates pilot behaviors through the derivation of human-readable LTLf formulas, offering the explainability of the results and insights into OOTL characteristics. Overall, this research proposes the VALIO framework as an improvement for OOTL identification in both performance and explainability.

Derivation of the Ultimate Bearing Capacity Formula for Layered Foundations Based on Meyerhof’s Theory

Derivation of the Ultimate Bearing Capacity Formula for Layered Foundations Based on Meyerhof’s Theory

Pressure Relief Gas Drainage in a Fully Mechanised Mining Face Based on the Comprehensive Determination of ‘Three Zones’ Development Height

The gas emission from the goaf generates gas in the upper corner of the working face, and the return airflow exceeds safe limits. The identification of the reasonable level of the high-long borehole is the key factor to ensure the effectiveness of its extraction. On-site test at the 2308 working face of Licun Coal Industry of Lu’an Chemical Industry Group, the development of coal overlying strata fracture was studied through derivation of theoretical and empirical formulae, physical similarity experiment, and UDEC numerical simulation: this was then combined with in-situ microseismic monitoring to obtain the distribution characteristics of mining overburden “falling zone” and “overbreak zone” under the actual working conditions, and accurately design the high-level long borehole end hole layer. The results show that the height of the falling zone is 17.5 m to 20.5 m, and the overbreak zone is 43.5 m to 49.5 m. When the hole position is between 25 m and 30 m in the middle and lower part of the overbreak zone, the flow and concentration of gas extracted by drilling are high, and the pure amount of gas extracted by a single hole is increased by 53% (on average). The investigation of pressure relief gas extraction shows that throughout the mining period, the average gas concentration in the return airway is maintained below 0.36%, and the average gas concentration in the upper corner is kept within 0.48% (effective gas control). The research proves the rationality of the arrangement of the high-level long borehole horizon in the working face and provides a reference for the design of the borehole horizon in the future gas drainage and control in the goaf.

Radial and Tangential Friction Coefficients: Derivation of Formulas, Calculation and Application in Modeling of the Heavy Ion Collision Process

Radial and Tangential Friction Coefficients: Derivation of Formulas, Calculation and Application in Modeling of the Heavy Ion Collision Process

Line Selection of Single-Phase Fault Grounding System Based on the Deviation Criterion of Composite Zero-Sequence Admittance

Abstract The current methods for line selection of small current fault grounding system faults within distribution networks have some limitations. This paper proposes a method based on the deviation criterion of composite zero-sequence admittance for line selection of single-phase fault grounding systems with small currents. First, the derivation formula of zero-sequence admittance for small current grounding systems considering asymmetrical transmission lines is presented. Then, a method based on the deviation criterion of composite zero-sequence admittance is proposed for accurate fault line selection. Finally, a fault model is built by using simulation software and simulation experiments are conducted to verify that this method can rightly select the fault line by significant feature values.

Exploring the Extended Beta-Logarithmic Function: Matrix Arguments and Properties

The beta-logarithmic function substantially generalizes the standard beta function, which is widely recognized for its significance in many applications. This article is devoted to the study of a generalization of the classical beta-logarithmic function in a matrix setting called the extended beta-logarithmic matrix function. The proofs of some essential properties of this extension, such as convergence, partial derivative formulas, functional relations, integral representations, inequalities, and finite and infinite sums, are established. Moreover, an application of the extended beta-logarithmic function in matrix arguments is proposed in probability theory. Further, numerical examples and graphical presentations of the new generalization are obtained.

A friendly derivation of Stirling's formula

We demonstrate the Stirling formula, approximating the factorial, utilising accessible and elementary methods in an engaging manner.

On Certain Properties of Parametric Kinds of Apostol-Type Frobenius–Euler–Fibonacci Polynomials

This paper presents an overview of cosine and sine Apostol-type Frobenius–Euler–Fibonacci polynomials, as well as several identities that are associated with these polynomials. By applying a partial derivative operator to the generating functions, the authors obtain derivative formulae and finite combinatorial sums involving these polynomials and numbers. Additionally, the paper establishes connections between cosine and sine Apostol-type Frobenius–Euler–Fibonacci polynomials of order α and several other polynomial sequences, such as the Apostol-type Bernoulli–Fibonacci polynomials, the Apostol-type Euler–Fibonacci polynomials, the Apostol-type Genocchi–Fibonacci polynomials, and the Stirling–Fibonacci numbers of the second kind. The authors also provide computational formulae and graphical representations of these polynomials using the Mathematica program.

Threshold Voltage Modulation and Gate Leakage Suppression of Enhancement‐Mode GaN HEMT by Metal/Insulator/p‐GaN Gate Structure

Herein, a metal/insulator/p‐GaN gate HEMT (MIP‐HEMT) with Si3N4 gate dielectric layer is fabricated. Compared to the conventional p‐GaN HEMT, the MIP‐HEMT has a higher threshold voltage (Vth) of 4.8 V and better gate leakage suppression. The mechanism of threshold voltage increase in MIP‐HEMT is elucidated through an analysis of the electric field distribution in the gate region and the proposed gate capacitance model. Furthermore, MIP‐HEMTs with gate dielectric layers of varying dielectric constants and thicknesses are designed and simulated. The obtained results lead to the derivation of an empirical formula for the threshold voltage of the MIP‐HEMT under ideal dielectric/p‐GaN interface conditions, in relation to the dielectric constant and thickness of the gate dielectric layer. Additionally, simulations are conducted on the MIP‐HEMT incorporating fixed charges at the dielectric/p‐GaN interface, and the impact of interface charges is investigated. This refinement enables a more accurate prediction of the MIP‐HEMT's threshold voltage. Conclusively, MIP‐HEMTs can effectively modulate the threshold voltage by manipulating the parameters of the gate dielectric layer, thereby extending the threshold voltage range of conventional enhancement‐mode devices.