Approximate Formula Research Articles

The Activity of a Solar-Type Star TOI-1422 and Mass Loss of the Atmosphere of the Planet TOI-1422b

The results of the analysis of the activity of a solar type star G2V TOI-1422 are presented and estimates of the loss of atmospheric matter of the planet TOI-1422b are made. The planet can be attributed to the type of hot Neptunes. Compared to other exoplanets of a similar mass range, it is expected that the planet has an extensive gas envelope. According to the long-term photometric survey of Kamogata Wide-field Survey (KWS) the TOI-1422 activity was analyzed and the existence of possible activity cycles of 1650–1680 days and 2450 days was suggested. The probable value of the rotation period P of the star lies in the range of 27 (+19, –8) days, observations of the KWS survey in the V filter indicate the most possible value for the period P = 32 days. Mass loss of the planet’s atmosphere is determined using an approximation formula corresponding to the model of atmospheric loss with an energy restriction. To estimate the flux of XUV photons analytical dependences were used linking the flux values and the parameter log R_{{{text{HK}}}}^{'}. Calculations have shown that the atmosphere matter loss of TOI-1422b is dot {M} = 9.4 times {{10}^{8}} g/s; considering the existing errors in determining the parameters of the atmosphere it lies in the range from 6.8 times {{10}^{8}} to 1.4 times {{10}^{9}} g/s, and considering the uncertainty in the value of the chromospheric activity index—from 8.3 times {{10}^{8}} to 1.1 times {{10}^{9}} g/s.

THE ACTIVITY OF A SOLAR-TYPE STAR TOI-1422 AND MASS LOSS OF THE ATMOSPHERE OF THE PLANET TOI-1422b

The results of the analysis of the activity of a solar type star G2V TOI-1422 are presented and estimates of the loss of atmospheric matter of the planet TOI-1422b are made. The planet can be attributed to the type of hot neptunes. Compared to other exoplanets of a similar mass range it is expected that the planet has an extensive gas envelope. According to the long-term photometric survey of Kamogata Wide-field Survey (KWS) the TOI-1422 activity was analyzed and the existence of possible activity cycles of 1650-1680 days and 2450 days was suggested. The probable value of the rotation period P of the star lies in the range of 27 (+19, –8) days, observations of the KWS survey in the V filter indicate the most possible value for P = 32 days. Mass loss of the planet’s atmosphere is determined using an approximation formula corresponding to the model of atmospheric loss with an energy restriction. To estimate the flux of XUV photons analytical dependences were used linking the flux values and the parameter \(\log R_{{{\text{HK}}}}^{'}\). Calculations have shown that the mass loss of TOI-1422b is \(\dot {M} = 9.4 \times \) 108 g/s and taking into account the existing errors in determining the parameters of the atmosphere it lies in the range from 6.8 × 108 g/s to 1.4 × 109 g/s.

Formulas for Beam Shift and Beam Narrowing in 1-D Leaky-Wave Antennas Due to Element Pattern

The radiation pattern of one-dimensional (1-D) unidirectional leaky-wave antennas (LWAs) is typically derived from the analysis of the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">space factor</i> (SF) only. However, when the antenna aperture is characterized by a longitudinal radiating current, a cosine-type <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">element pattern</i> term has to be included: this factor may considerably affect the beam properties. Specifically, the beam peak is shifted and the beamwidth is narrowed when the element pattern is included. These effects are more prominent as the antenna size decreases and the beam angle increases. Approximate but accurate analytical formulas are here provided and numerically validated to predict the beam angle and the half-power beamwidth for these kinds of 1-D unidirectional LWAs when the element pattern is included.

The timescale of thermal quench during disruptions in EAST

Since 2015, the timescale of thermal quench (TQ) during disruptions on the EAST tokamak has been examined using electron cyclotron emission and soft x-ray diagnostic systems. The database includes both major disruptions (MDs) and hot vertical displacement events (VDEs), where the TQ duration of the former is within 56–788 μs, and the latter is approximately within 100–3000 μs. In particular, for MDs, the lower bound of TQ duration, indicating the minimum durations at different current plateaus, decreases as the plasma current increases. This decrease is due to the connection length shortening and the plasma temperature increasing. For MDs, two typical TQ processes, single-stage TQ and double-stage TQ, are characterized by different magnetic perturbations. In single-stage TQ, a fast-loss stage is triggered by magnetic perturbation exceeding 4.3 × 10−3 T with a fast growth rate of 1.5 × 10−2 μs−1. In contrast, fast quench is triggered by a slightly smaller magnetic perturbation of 3.6 × 10−3 T in double-stage TQ, and the growth rate 5.3 × 10−3 μs−1 is an order of magnitude smaller than single-stage TQ. For hot VDEs, the plasma temperature collapses step by step from the edge to the core, and every progressive collapse corresponds to a magnetic perturbation, whose growth rate is approximately equal to or less than double-stage TQ. The whole process of TQ energy release can be divided into the transport in a stochastic field within the separatrix and in the scrape-off layer and, according to the typical parameters of EAST, TQ duration in MDs is roughly estimated to be 245 μs by an approximate formula, which is consistent with the experimental results.

Analysis of supply chain finance risk assessment based on numerical analysis algorithm

To promote the coordination and stability of supply chain finance and improve the financing environment of small and medium-sized enterprises, this paper designs a supply chain finance risk assessment and analysis platform. Combining the characteristics of a large amount of risk assessment data, a numerical analysis algorithm is introduced in the process of platform design, and the extrapolation method in the numerical analysis algorithm is used to calculate the risk assessment- related data. To make the calculation faster and the data more accurate, the central difference quotient extrapolation is used to accelerate and a downtime mechanism is introduced. Firstly, the approximation formula for the calculation is constructed, followed by the construction of a sequence of variable steps to obtain a sequence of approximations. Finally, the obtained approximate sequence values are used to construct an interpolating polynomial, and the constant term of the polynomial, which is the final risk factor, is obtained through continuous iteration. To verify the effectiveness of the numerical analysis-based algorithm in supply chain financial risk assessment, the simulation results show that the risk assessment accuracy of the numerical analysis-based supply chain financial risk assessment platform is as high as 99% and the time required is 17 seconds higher than other assessment models, which verifies that the numerical analysis algorithm can improve the accuracy and rapidity of risk assessment.

The high‐frequency phenomenon of copper clad aluminum wires which are lower the alternating current resistance than copper solid wires, and the application of copper clad aluminum wires to the electrical equipment

AbstractThe paper is proposed approximate formulas of the ac resistance of lead wires and coils which made by using copper clad aluminum (CCA) wires. There are the frequency range that the ac resistance of CCA wires is lower than it of cupper solid wires for lead wires and coils of an electrical equipment. It is shown that the application of CCA wires is economical as comparing with copper solid wires, if the design is due to successful calculation in the electrical equipment. The measured resistance is compared with the formula. CCA wires is an example of multiple conductors which shall be expected to apply in future.

Approximate Subdifferential of the Difference of Two Vector Convex Mappings

This paper deals with the strong approximate subdifferential formula for the difference of two vector convex mappings in terms of the star difference. This formula is obtained via a scalarization process by using the approximate subdifferential of the difference of two real convex functions established by Martinez-Legaz and Seeger, and the concept of regular subdifferentiability. This formula allows us to establish approximate optimality conditions characterizing the approximate strong efficient solution for a general DC problem and for a multiobjective fractional programming problem.

Simulation and Theoretical Study on the Mutual Inductance of the B-Dot Used for High-Power Transmission Lines

For the B-dot used for high-power transmission lines, due to its special structure, the current method to obtain its mutual inductance is calibration experiments and there lacks an effective method to estimate its mutual inductance at the design stage. In this paper, the mutual inductance of this kind of B-dot is studied by simulation and theoretical analysis. Through simulation, the frequency response of the mutual inductance is obtained and it is found that the mutual inductance at high frequency which is the main working frequency range of the B-dot is lowest and stable. Through theoretical analysis, the approximate calculation formula of mutual inductance at any frequency is derived. Using the experimental results in previous reference, the theoretical results and the simulation results are verified. The influencing factors of the B-dot’s mutual inductance at high frequency are analyzed. The results show that the number of wire loops, the inner radius of wire loops and the radius of hole are positively related to the mutual inductance, and the height of the center of the wire loops is negatively related to the mutual inductance. Among these four factors, increasing the inner radius of wire loops can improve the mutual inductance more significantly. The work in this paper reveals the physical meaning of the mutual inductance change and provides guidance for the design of the B-dot.

Evaluation of modified Hantush’s function: A tool for parametric study with the finite line-source model

A solution to the three-dimensional finite line-source (FLS) model for borehole heat exchangers (BHEs) is identical to Hantush’s M function, commonly used in groundwater pumping modeling. A modified Hantush function E appears in the steady-state solution to the moving finite line-source (MFLS) model for borehole heat exchangers (BHEs) with groundwater flow that takes into account heat conduction and advection by Péclet number. An exact series representation is obtained for this two-parametric function E, and from it two-types of approximate expressions for the E are derived. One is suitable for small values of a parameter Pe and the other for its large values. High accuracy of the approximate formulae is verified by comparison with the exact values provided by modified Hantush’s function E in those validity domains.These analytical expressions can be used for designing complex borehole configurations and may be useful in steady-state analysis of partially penetrated wells (PPWs) in a leaky aquifer.

Large time behavior and optimal decay estimate for solutions to the generalized Kadomtsev–Petviashvili–Burgers equation in 2D

We consider the Cauchy problem for the generalized Kadomtsev–Petviashvili–Burgers equation in 2D. This is one of the nonlinear dispersive-dissipative type equations, which has a spatial anisotropic dissipative term. Under some suitable regularity assumptions on the initial data u0, especially the condition ∂x−1u0∈L1(R2), it is known that the solution to this problem decays at the rate of t−74 in the L∞-sense. In this paper, we investigate the more detailed large time behavior of the solution and construct the approximate formula for the solution at t→∞. Moreover, we obtain a lower bound of the L∞-norm of the solution and prove that the decay rate t−74 of the solution given in the previous work to be optimal.

線形地震応答での構造躯体の層間変位と2次系変位の相関係数

In the seismic design of nonstructural components, inertial forces and relative displacements are usually considered separately. But the two factors should be appropriately combined in some cases of connecting conditions to the structural frames. In this paper, the correlation coefficients between the story drifts of structural frames and the displacements of secondary systems are investigated in linear seismic responses. An approximate evaluation formula for multistory buildings is presented and validated through response history analysis. It is shown that the correlations are relatively high in the range of periods shorter than the first natural periods of structural frames.

Early universe dynamics of PQ field with very small self-coupling and its implications for axion dark matter

Axion-like particles (ALPs) are often considered as good candidates for dark matter. Several mechanisms generating relic abundance of ALP dark matter have been proposed. They may involve processes which take place before, during or after cosmic inflation. In all cases an important role is played by the potential of the corresponding Peccei-Quinn (PQ) field. Quite often this potential is assumed to be dominated by a quartic term with a very small coupling. We show that in such situation it is crucial to take into account different kinds of corrections especially in models in which the PQ field evolves during and after inflation. We investigate how such evolution changes due to radiative, thermal and geometric corrections. In many cases those changes are very important and result in strong modifications of the predictions of a model. They may strongly influence the amount of ALP contributions to cold and warm components of dark matter as well as the power spectrum of associated isocurvature perturbations. Models with a quasi-supersymmetric spectrum of particles to which the PQ field couples seem to be especially interesting.Qualitative features of such models are discussed with the help of approximate analytical formulae. However, the dynamics of the PQ field with the considered corrections taken into account is more complicated than in the case without corrections so dedicated numerical calculations are necessary to obtain precise predictions.We present such results for some characteristic benchmark points in the parameter space.

An Approximation Formula to the Draining Tank Problem

We give an approximation formula to the drain time of the draining tank problem for an incompressible and nonviscous liquid with an unsteady flow effect. One can approximate the unsteady flow effect by elementary functions, and it is suitable for university students to learn Bernoulli’s equation with the unsteady flow effect.

Study on Characteristics of Dynamic Equilibrium Angle of the Spin-Stabilized Projectile

This study calculates and comparatively tests the dynamic equilibrium angle characteristics of the spin-stabilized projectile. Based on the analysis of the spatial relationships of angular motion of the projectile, a six-degrees-of-freedom (6 DoF) motion model is constructed. Through simulation and calculation, the changing law of the main parameters affecting the dynamic equilibrium angle with time is obtained, and then the whole-trajectory dynamic equilibrium angle curve is drawn using the law and the approximate formulas for dynamic equilibrium angle calculation. With the projectile-borne geomagnetic attitude testing technology, a dynamic equilibrium angle measurement method is developed. We combine this method with the trajectory angle curves, and work out the dynamic equilibrium angle values. The accuracy of the geomagnetic measurement is verified by comparing the angle of attack values obtained by geomagnetism and high-speed photogrammetry, and the credibility of the dynamic equilibrium angle measurement method is confirmed by comparing the measured and calculated values of the dynamic equilibrium angle. This study solves the problem of dynamic equilibrium angle measurement of spin-stabilized projectiles, and obtains the dynamic change curve of the dynamic equilibrium angle for the first time, which provids a new method and technology for the research on dynamic equilibrium angles.

Approximate Inversion of a Class of Generalized Radon Transforms

Generalized Radon transforms (GRT) serve, for instance, as linear models for seismic imaging in the acoustic regime. They occur when the corresponding inverse problem is linearized about a known background compression wave speed (Born approximation). The resulting GRT is completely determined by this background velocity. In this work, we present an implementation of approximate inversion formulas for this class of GRTs proposed and analyzed in [Grathwohl et al., Inverse Problems, 34 (2018), 014002] and [Grathwohl et al., Inverse Problems, 34 (2018), 114001], where we restrict ourselves to layered background velocities in two dimensions. In a series of numerical experiments, we intensively test our implementation, reproducing theoretical predictions. Further, we drive the validity of the linearization to its limits.

M3C2 (Cr3C2 Type) carbides formed in steel

As an authigenic alloy phase, M3C2 (Cr3C2 type) carbide, Cr-rich M3C2 carbide with a simple orthorhombic crystal structure, was confirmed to be present in an 11%Cr ferritic/martensitic steel normalized at 1323 K (1050 °C) for 0.5 h and tempered at 873 K (600 °C) for 1.5 h by transmission electron microscopy. The identified Cr-rich M3C2 carbides were found to be the main phase as the Cr-rich M23C6 and M2C carbides according to the present TEM examination, and to have the following characteristics: irregular block, branch crystal-like and needle-like morphologies; larger sizes ranging from approximately 100 to 350 nm in diameter and approximately 200–470 nm for the length of the long axis of the needles; approximate chemical formula of (Cr0.55Fe0.4W0.05)3C2. The formation mechanism of the M3C2 carbides during the tempering is discussed. The M3C2 carbides may be transformed by the reaction of carbon atoms in the matrix with Cr-rich M7C3 carbides formed in situ from Cr-enriched M3C carbides, which were formed from Fe-rich M3C carbides already present in the normalized steel by the diffusion of Cr solute atoms.

Jammed Keplerian gas leads to the formation and disappearance of spiral arms in a coupled map lattice for astronomical objects

Abstract The formation and disappearance of spiral arms are studied by focusing on jammed Keplerian gas in a coupled map lattice (CML) with a minimal set of procedures for simulating diverse patterns in astronomical objects. The CML shows that a spiral arm is a type of traffic jam, and its motion is governed by both a gas inflow into and outflow from the jam. In particular, we present a new approach to simply and directly evaluating the disappearance of spiral arms, called “light-in and heavy-out”. It is based on the gas flow rate difference between the light inflow and heavy outflow leading to the disappearance of traffic jams. Furthermore, we propose an approximate formula for the remaining lifetime of spiral arms, which is immediately derived from the “light-in and heavy-out” approach without calculating their pattern speeds as in conventional differential rotation. The proposed formula is successfully applied to the CML simulations.

Convergence analysis of a novel fractional product integration method for solving the second kind weakly singular Volterra integral equations with non-smooth solutions based on Jacobi polynomials

In this paper, we introduce a new fractional basis function based on Lagrange polynomials. We define the new interpolation formula for approximation of the solutions of the second kind weakly singular Volterra integral equations. The product integration method is used for the numerical solution of these equations based on Jacobi polynomials. It is known that the weakly singular Volterra integral equations typically have solutions whose derivatives are unbounded at the left end-point of the interval of integration. We use the suitable transformations to overcome this non-smooth behaviour. An upper error bound of the proposed method is determined and the convergence analysis is discussed. Finally, some numerical examples with non-smooth solutions are prepared to test the efficiency and accuracy of the method.

PS reflection moveout in a homogeneous anisotropic layer of arbitrary symmetry and tilt

Use of unconverted reflected PP waves leads to moveout formulae, which relate the moveout to only a limited number of parameters specifying anisotropy. Use of converted reflected PS waves extends the number of parameters to, generally, a complete set of twenty-one parameters. It leads, on the other hand, to more complicated formulae and phenomena unknown from studies of unconverted PP waves. In anisotropic media one has to deal with the existence of two S waves and their singularities, acoustic approximation cannot be used. We present and test approximate formulae for the reflection moveout of PS waves converted at a horizontal reflector underlying a homogeneous layer of arbitrary anisotropy symmetry and orientation. For their derivation, we use the combination of weak-anisotropy approximation and A-parameters specifying anisotropy. The complete set of twenty-one A-parameters represents an alternative to twenty-one independent elements of the stiffness tensor. In addition to the moveout formulae for separate PS1 and PS2 waves, we also present moveout formula for a P to common S wave whose traveltimes represent an average of traveltimes of PS1 and PS2 waves. Use of the P to common S wave reflection leads to several simplifications. Presented tests indicate that for most offsets and models with P-wave anisotropy up-to to 26% and S-wave anisotropy over 25%, the relative traveltime errors do not exceed 2.5%.

Mean escape time of switched Riccati differential equations

Riccati differential equations are a class of first-order quadratic ordinary differential equations and have various applications in systems and control theory. In this study, we analyzed a switched Riccati differential equation driven by a Poisson-like stochastic signal. We specifically focused on computing the mean escape time of the switched Riccati differential equation. The contribution of this study is twofold. We first show that, under the assumption that the subsystems described as deterministic Riccati differential equations escape in finite time regardless of their initial state, the mean escape time of the switched Riccati differential equation admits a power series expression. To further expand the applicability of this result, we then present an approximate formula to compute the escape time of deterministic Riccati differential equations. Numerical simulations were performed to illustrate the obtained results.