Differentiation Formula Research Articles

Rules of medication for leucorrhoea in ancient books of traditional chinese medicine based on latent structure model and association rules

Objective: To analyze syndrome differentiation and medication rules for leucorrhoea in ancient books of traditional Chinese medicine (TCM) based on the Chinese Medical Code using the latent structure model. Methods: The formulations for leucorrhoea were searched in the Chinese Medical Code (software), based on which an Excel sheet was created to conduct frequency analysis and construct a matrix of high-frequency medications. The high-frequency medications for leucorrhoea were then analyzed using the Latent 5.0 software and a latent structure model. With SPSS Modeler and Latent 5.0 as the main analysis methods, Chinese medications with a frequency greater than 1.00% were investigated using the latent structure model and association rules, and the medication rules in treating leucorrhoea with TCM were dissected based on the data of frequency. Results: For 373 kinds of Chinese medicines retrieved from 236 TCM prescriptions, the total medication frequency was 2522. Chinese Angelica (Angelicae Sinensis Radix), White Peony Root (Paeoniae Radix Alba), Indian Buead Tuckahoe (Poria), Common Oyster Shell (Ostreae Concha), and Largehead Atractylodes Rhizome (Atractylodis Macrocephalae Rhizoma) were high-frequency medicines for leucorrhoea. Tonic, heat-clearing and qi-regulating medicines had a high frequency of usage. According to a formula-based analysis of the evidence, the main syndromes of leucorrhoea included liver qi affecting the spleen pattern, damp heat accumulation pattern, liver qi stagnation pattern, liver-stomach disharmony pattern, yin deficiency of the liver and kidney pattern, and spleen deficiency with dampness pattern, which are generally in line with the etiology and pathogenesis of leucorrhoea, namely pathogenic dampness as the detrimental factor together with spleen-kidney dysfunction. However, in ancient books, emphasis was laid out on the regulation of the liver. This study determined 31 association rules for treating leucorrhoea, where the primary compatible medicines contained Chinese Angelica (Angelicae Sinensis Radix)-Nutgrass Galingale Rhizome (Cyperus rotundus), Chinese Angelica (Angelicae Sinensis Radix)-Szechwan Lovage Rhizome (Ligusticum chuanxiong Hort.), White Peony Root (Paeoniae Radix Alba)-Szechwan Lovage Rhizome (Ligusticum chuanxiong Hort.), White Peony Root (Paeoniae Radix Alba)-Adhesive Rehmannia Root Tuber (Rehmanniae Radix) and Tangerine Peel (Citri Reticulatae Pericarpium)-Ternate Pinellia (Rhizoma Pinelliae). Conclusion: The medication rules targeting leucorrhoea in the Chinese Medical Code was mainly represented by regulation of qi and blood, as well as the three internal organs—the kidney, spleen, and liver.

Unconditional optimal [formula omitted]-norm error estimate and superconvergence analysis of a linearized nonconforming finite element variable-time-step BDF2 method for the nonlinear complex Ginzburg–Landau equation

In this paper, a linearized fully discrete scheme combining the variable-time-step two-step backward differentiation formula (VSBDF2) in time and the nonconforming finite element methods (FEMs) in space is constructed and analyzed for the nonlinear complex Ginzburg–Landau equation. A novel convergence analysis approach is proposed, which shows that the H1-norm error of this scheme can reach optimal order in space and sharp second-order convergence in time. The key ingredients of our analysis are temporal–spatial error splitting approach, a new positive definiteness property of discrete orthogonal convolution (DOC) kernels with respect to complex sequences, an energy projection operator and a discrete Laplace operator Δh related to the nonconforming element space, and a relationship between Δh and classical L2 projection operator in complex space. Furthermore, through a specific treatment of the error equation and the combination technique of interpolating operator and projection operator, we prove that the proposed scheme has the global superconvergence result of order O(τ2+h2) in H1-norm for the first time, which further improves the efficiency of numerical computation. Here τ and h are the maximum time and spatial step sizes, respectively. It should be pointed out that all these error results are rigorously established under mild assumptions of the time step, and get rid of the usual grid ratio constraints between temporal and spatial step sizes in linearized scheme. At last, numerical experiments are provided to verify the theoretical analysis.

Lagrange Interpolation in Matrix Form for Numerical Differentiation and Integration

Numerical differentiation has been widely applied in engineering practice due to its remarkable simplicity in the approximation of derivatives. Existing formulas rely on only three-point interpolation to compute derivatives when dealing with irregular sampling intervals. However, it is widely recognized that employing five-point interpolation yields a more accurate estimation compared to the three-point method. Thus, the objective of this study is to develop formulas for numerical differentiation using more than three sample points, particularly when the intervals are irregular. Based on Lagrange interpolation in matrix form, formulas for numerical differentiation are developed, which are applicable to both regular and irregular intervals and can use any desired number of points. The method can also be extended for numerical integration and for finding the extremum of a function from its samples. Moreover, in the proposed formulas, the target point does not need to be at a sampling point, as long as it is within the sampling domain. Numerical examples are presented to illustrate the accuracy of the proposed method and its engineering applications. It is demonstrated that the proposed method is versatile, easy to implements, efficient, and accurate in performing numerical differentiation and integration, as well as the determination of extremum of a function.

Lagrange Interpolation in Matrix Form for Numerical Differentiation and Integration

Numerical differentiation has been widely applied in engineering practice due to its remarkable simplicity in the approximation of derivatives. Existing formulas rely on only three-point interpolation to compute derivatives when dealing with irregular sampling intervals. However, it is widely recognized that employing five-point interpolation yields a more accurate estimation compared to the three-point method. Thus, the objective of this study is to develop formulas for numerical differentiation using more than three sample points, particularly when the intervals are irregular. Based on Lagrange interpolation in matrix form, formulas for numerical differentiation are developed, which are applicable to both regular and irregular intervals and can use any desired number of points. The method can also be extended for numerical integration and for finding the extremum of a function from its samples. Moreover, in the proposed formulas, the target point does not need to be at a sampling point, as long as it is within the sampling domain. Numerical examples are presented to illustrate the accuracy of the proposed method and its engineering applications. It is demonstrated that the proposed method is versatile, easy to implements, efficient, and accurate in performing numerical differentiation and integration, as well as the determination of extremum of a function.

MATRICS: The implicit matrix-free Eulerian hydrodynamics solver

Context. There exists a zoo of different astrophysical fluid dynamics solvers, most of which are based on an explicit formulation and hence stability-limited to small time steps dictated by the Courant number expressing the local speed of sound. With this limitation, the modeling of low-Mach-number flows requires small time steps that introduce significant numerical diffusion, and a large amount of computational resources are needed. On the other hand, implicit methods are often developed to exclusively model the fully incompressible or 1D case since they require the construction and solution of one or more large (non)linear systems per time step, for which direct matrix inversion procedures become unacceptably slow in two or more dimensions. Aims. In this work, we present a globally implicit 3D axisymmetric Eulerian solver for the compressible Navier–Stokes equations including the energy equation using conservative formulation and a fully simultaneous approach. We use the second-order-in-time backward differentiation formula for temporal discretization as well as the κ scheme for spatial discretization. We implement different limiter functions to prohibit the occurrence of spurious oscillations in the vicinity of discontinuities. Our method resembles the well-known monotone upwind scheme for conservation laws (MUSCL). We briefly present efficient solution methods for the arising sparse and nonlinear system of equations. Methods. To deal with the nonlinearity of the Navier–Stokes equations we used a Newton iteration procedure in which the required Jacobian matrix-vector product was reconstructed with a first-order finite difference approximation to machine precision in a matrix-free way. The resulting linear system was solved either completely matrix-free with a combination of a sufficient Krylov solver and an approximate Jacobian preconditioner or semi-matrix-free with the incomplete lower upper factorization technique as a preconditioner. The latter was dependent on a standalone approximation of the Jacobian matrix, which was optionally calculated and needed solely for the purpose of preconditioning. Results. We show our method to be capable of damping sound waves and resolving shocks even at Courant numbers larger than one. Furthermore, we prove the method’s ability to solve boundary value problems like the cylindrical Taylor-Couette flow (TC), including viscosity, and to model transition flows. To show the latter, we recover predicted growth rates for the vertical shear instability, while choosing a time step orders of magnitude larger than the explicit one. Finally, we verify that our method is second order in space by simulating a simplistic, stationary solar wind.

Energetic spectral-element time marching methods for phase-field nonlinear gradient systems

We propose two efficient energetic spectral-element methods in time for marching nonlinear gradient systems with the phase-field Allen–Cahn equation as an example: one fully implicit nonlinear method and one semi-implicit linear method. Different from other spectral methods in time using spectral Petrov-Galerkin or weighted Galerkin approximations, the presented implicit method employs an energetic variational Galerkin form that can maintain the mass conservation and energy dissipation property of the continuous dynamical system. Another advantage of this method is its superconvergence. A high-order extrapolation is adopted for the nonlinear term to get the semi-implicit method. The semi-implicit method does not have superconvergence, but can be improved by a few Picard-like iterations to recover the superconvergence of the implicit method. Numerical experiments verify that the method using Legendre elements of degree three outperforms the 4th-order implicit-explicit backward differentiation formula and the 4th-order exponential time difference Runge-Kutta method, which were known to have best performances in solving phase-field equations. In addition to the standard Allen–Cahn equation, we also apply the method to a conservative Allen–Cahn equation, in which the conservation of discrete total mass is verified. The applications of the proposed methods are not limited to phase-field Allen–Cahn equations. They are suitable for solving general, large-scale nonlinear dynamical systems.

Joint multi-Gaussian mixture model and its application to multi-model multi-bernoulli filter

The existing multi-model Gaussian mixture-multi-Bernoulli (MM-GM-MB) filter lacks the ability to maintain parallel state estimates under different models. As a result, the change of model-related likelihoods lags behind target maneuvers. To address this problem, a joint multi-Gaussian mixture MB (JMGM-MB) filter is proposed. We first propose a JMGM model. Each single-target state estimate is expressed as a set of parallel model-related Gaussian functions with model probabilities, and a weight characterizes the probability of this state estimate. Based on the Bayesian rule, we derive the updating methods of the weights, model probabilities and model-related means, covariances that are collectively called JMGM components. Then, we approximate the state densities of MB filter using the JMGM model. Based on the interactive multi-model (IMM) method, we derive the interacting, prediction and estimation methods of JMGM components. Then, we provide matched association and merging methods for JMGM components. In nonlinear tracking scenarios where sensors simultaneously carry out translations and rotations, we derive a method based on the derivative rule for composite functions to compute the linearized observation matrix. Simulations demonstrate that whether in active or passive tracking, the proposed JMGM-MB filter overcomes the likelihood lag problem and owns higher accuracy of uncertain maneuvering target tracking than the existing MM-GM-MB filter, requiring lower computational expense.

FINITE ELEMENT ANALYSIS OF LINEARLY EXTRAPOLATED BLENDED BACKWARD DIFFERENCE FORMULA (BLEBDF) FOR THE NATURAL CONVECTION FLOWS

In this paper, we study the stability and convergence of fully discrete finite element method with grad-div stabilization for the incompressible non-isothermal fluid flows. The proposed scheme uses finite element discretization in space and linearly extrapolated blended Backward Differentiation Formula (BLEBDF) in time. We prove the unconditional stability over finite time interval and optimally convergence of the scheme. We also present numerical experiments to verify our theoretical convergence rates and show the reliability of the scheme.

MODELING STOCHASTIC ADVERSE EFFECTS OF CBN 2023 REDESIGNED NAIRA NOTES POLICY ON RURAL FARMERS IN NIGERIA

The recent Central Bank of Nigeria (CBN) 2023 redesigned naira notes is of good benefits to strengthen the economy of the country by checking counterfeiting and hoarding of large volume of banknotes by the public. Despite all the efforts made by the CBN for citizens to enjoy the benefits of this implementation, most rural farmers were faced with adverse effects of uncertainties in the production and marketing of their agricultural produce due to lack of redesigned new naira notes in circulation. The adverse effects of these uncertainties are modeled as Advanced Stochastic Time-Delay Differential Equation (ASTDDE). The modeled equation is solved using Extended Second Derivative Block Backward Differentiation Formulae Method (ESDBBDFM) without the use of interpolation techniques in the evaluations of the delay term and noise term. In comparing the numerical results of this method with other existing methods in literature, the newly developed mathematical expressions for the evaluations of the delay term and the noise term in solving ASDDEs with the discrete schemes of ESDBBDFM gives better results for step number than step numbers and 3 by producing Least Minimum Absolute Random Error (LMARE) in a Lower Computational Processing Unit Time (LCPUT) faster than other existing methods that applied interpolation techniques in evaluations of the delay term and the noise term.

A new generalized beta function associated with statistical distribution and fractional kinetic equation

Several authors have extensively investigated beta function, hypergeometric function and confluent hypergeometric function, their extensions and generalizations due to their several application in many areas of engineering, probability theory and science. The main purpose of this paper is to present a new generalization of extended beta function, hypergeometric function and confluent hypergeometric function with the help of m-parameter Mittag-Leffler function, as well as examine some important properties like integral representations, differential formula and summation formulas. We also examine generalized Caputo fractional derivative operator with the help of m-parameter Mittag-Leffler function with associated properties using the generalized beta function. We define a new beta distribution involving the new generalized beta function. The mean, variance, coefficient of variance, moment generating function and characteristic function and cumulative distribution are derived. Further, we derive the solution of fractional Kinetic equation involving generalized hypergeometric function.

High-order Lagrange-Galerkin methods for the conservative formulation of the advection-diffusion equation

Abstract We introduce in this paper the numerical analysis of high order both in time and space Lagrange-Galerkin methods for the conservative formulation of the advection-diffusion equation. As time discretization scheme we consider the Backward Differentiation Formulas up to order $q=5$. The development and analysis of the methods are performed in the framework of time evolving finite elements presented in C. M. Elliot and T. Ranner, IMA Journal of Numerical Analysis41, 1696–1845 (2021). The error estimates show through their dependence on the parameters of the equation the existence of different regimes in the behavior of the numerical solution; namely, in the diffusive regime, that is, when the diffusion parameter $\mu $ is large, the error is $O(h^{k+1}+\varDelta t^{q})$, whereas in the advective regime, $\mu \ll 1$, the convergence is $O(\min (h^{k},\frac{h^{k+1} }{\varDelta t})+\varDelta t^{q})$. It is worth remarking that the error constant does not have exponential $\mu ^{-1}$ dependence.

A study of Horn matrix functions and Horn confluent matrix functions

Abstract In this paper, we give the matrix version of Horn’s hypergeometric function and their confluent cases. We also discuss the regions of convergence, system of matrix differential equations of bilateral type, differential formulae and infinite summation formulae satisfied by these hypergeometric matrix functions. With the study of these 23 matrix functions, matrix generalization of Horn’s list of 34 hypergeometric series will be completed.

Temporal error analysis of an unconditionally energy stable second-order BDF scheme for the square phase-field crystal model

In this paper, we first propose and study the second-order time-discrete numerical scheme for the sixth-order nonlinear parabolic problem of the square phase-field crystal model. Then, we demonstrate the two-step backward differentiation formula (BDF-2) scheme with mass conservation and energy dissipation, where the higher order nonlinear term is treated implicitly. Moreover, a rigorous error analysis is presented and we prove the optimal second-order convergence rate O(τ2) in H1- norm, where τ is the time step. Finally, some numerical results are provided to confirm our theoretical analysis.

Generalizations of the Jacobi identity to the case of the Lauricella function F D ( N )

ABSTRACT The paper considers the issue of constructing differential relations for the Lauricella hypergeometric function F D ( N ) . The formulas found give explicit expressions for the derivative of the product of the function F D ( N ) and some binomials through the product of other binomials and a combination of adjacent Lauricella functions whose parameters differ by 1 or − 1 . The derived differential relations generalize the Jacobi identity and other differential formulas known in the theory of the Gauss hypergeometric function on the multidimensional case. With special parameter values F D ( N ) , the right-hand side of such Jacobi-type formulas are simplified and have the form of a product of binomials and a polynomial. Such formulas can be used to transform a Cauchy-type integral to the form of the Schwartz–Christoffel integral and have an application to the Riemann–Hilbert problem with piecewise constant coefficients.

A BDF2 method for a singularly perturbed transport equation

A singularly perturbed transport equation is considered. A variable two-step backward differentiation formulas (BDF2) on a Shishkin-type mesh is used to discrete the first-order derivatives of the singularly perturbed transport equation. The stability and error analysis are derived by using the discrete orthogonal convolution kernels. It is proved that the scheme is second-order uniformly convergent with respect to the small parameter, which improves previous results. Numerical experiments are presented to support the theoretical result.

Computations on fall of the leaning tower with considering air resistance

The free fall of a sphere was studied by considering air buoyancy and resistance. After selecting the reasonable drag coefficient formula recommended by the literature, partial differential formulas on the motion of balls falling in the Leaning Tower of Pisa are solved. The variation process of acceleration, velocity and displacement over time during the falling process of two spheres is obtained. The research results indicate that the kinematics of free fall considering air resistance is different from that neglecting air resistance. Air Resistance must be considered in the free fall of the solid ball after 0.3005 seconds. In the free fall of the leaning tower, air resistance makes the solid ball land at 0.4418 seconds which is earlier than the hollow. The variation of the acceleration of a solid ball with time can be described by a second-order function a=-0.115t2 -0.022t+9.801. Velocity does not satisfy the product of acceleration and time but can be described as a polynomial function of the velocity variation with time v = c1t2 + c2t + c3 . About the relationship between displacement and time, for a solid sphere, displacement is proportional to 1.9526 power of time h=5.4945t1.9526 , while for a hollow sphere, it cannot be expressed by a function. The relationship between air resistance and velocity during the falling process of an iron ball satisfies a polynomial function Fd =c1v2 -c2v+c3 rather than a simple relationship where air resistance is directly proportional to the first or second power of speed, which was used in many papers as a reasonable assumption.

Modeling phase or chemical reaction equilibrium with an equation of state, an unrealistic modeling

The traditional chemical thermodynamics considered the chemical potential then fugacity of a component in a single-phase, open system is just the one of the component in a co-existing phase of an equilibrium, multi-component, multi-phase system or in an equilibrium, chemical reaction system and the fugacity can be calculated with the equation of state of the single-phase, open system. Therefore, the most common method to model phase or chemical reaction equilibrium is with an equation of state. Obviously, modeling phase or chemical reaction equilibrium with an equation of state requires to the single-phase, open system is equivalent to the co-existing phase or the equilibrium, chemical reaction system. However, there are significant differences in the thermodynamics properties between the single-phase, open system and the co-existing phase or the equilibrium, chemical reaction system. Therefore, the chemical potential then fugacity of a component in the single-phase, open system is impossible the one of the component in the co-existing phase or in the equilibrium, chemical reaction system. That is, the calculated fugacity of a component in a single-phase, open system with the equation of state of the single-phase, open system is impossibly the fugacity of the component in a co-existing phase of an equilibrium, multi-component, multi-phase system or in an equilibrium, chemical reaction system. Therefore, it is unrealistic to model phase or chemical reaction equilibrium with an equation of state. For this, we establish the chemical potential differential formula of a component in a co-existing phase and in an equilibrium, chemical reaction system to calculate phase or chemical reaction equilibrium. The calculated values of solubility in molar concentration within the framework of the present theory equals exactly the measured values of solubility in molar concentration by the experiments.

Optimal error estimates of nonconforming Wilson element for nonlinear nonlocal parabolic problems

The penalty backward Euler (BE) and 2-step backward differential formula (BDF2) fully discrete schemes are proposed for the nonlinear nonlocal parabolic problems with nonconforming Wilson element. Optimal convergence results with order O(h2+τ) and order O(h2+τ2) in a modified norm, larger than the usual broken H1-norm, are obtained directly which improve the results of order O(h+τ) and order O(h+τ2) with respect to h in the previous studies, respectively. Here, h and τ denote the mesh size and time step, respectively. Finally, some numerical results are provided to confirm the theoretical analysis.

Stiffly Stable Diagonally Implicit Block Backward Differentiation Formula with Adaptive Step Size Strategy for Stiff Ordinary Differential Equations

Stiffly Stable Diagonally Implicit Block Backward Differentiation Formula with Adaptive Step Size Strategy for Stiff Ordinary Differential Equations

Polymorphic higher-order context-free session types

We present an extension of polymorphic context-free session types that allows passing channels on channels, commonly known as higher-order session types. The mixture of functional types and session types has proven to be a challenge for type equivalence formulation: whereas functional type equivalence is often inductive and presented as a system of derivation rules, session type equivalence is often coinductive and usually presented as a bisimulation. We propose a unifying approach that handles the equivalence of functional and higher-order context-free session types together in the form of a system of rules generating a coinductively defined relation. Decidability of type equivalence is obtained via reduction to bisimulation for simple grammars, for which practical algorithms are known. To bridge the gap between types and simple grammars, we introduce a language of types with canonical names instead of bindings (which we call c-types), and propose a notion of canonical renaming to translate types to c-types.