Instantaneous Bethe-Salpeter Equation and Its Analog： Breit-like Equation

We take the (μ士e干) system as an example, but restrict ourselves to highlight the states with quantum number JP = 0-, to explore the different contents of the instantaneous Bethe-Salpeter (BS) equation and its analog,the relativistic version of Breit equation, by solving them exactly. The results show that the two equations are not equivalent, although they are analogous. Furthermore, since the Breit equation contains extra un-physical solutions,so we point out that it should be abandoned if one wishes to have an accurate description of the bound states for the instantaneous interacting binding systems.

Ground State Mass Spectrum for Scalar Diquarks with Bethe-Salpeter Equation

在这篇文章，我们学习假分级的介子π， K 和分级的 diquarks U~a 的结构， D~a，在有限制的联合彩虹 Schwinger-Dyson 方程和梯子 Bethe-Salpeter 方程的框架的 S~a 有效潜力。u， d， s 夸克有小当前的群众，和重正常化很大，在象区域一样的时间的集体杆是不在的它自然地实现监禁。 thepseudoscalar 介子π， K ，和分级的 diquarks U~a 的 Bethe-Salpeter 波浪功能， D~a ， S~a 有一样的类型( Gaussian 类型)动量依赖，集中约零动量并且关于 q~2 =延长到精力规模 1 GeV~2 ，它碰巧是为碎的 chiral 对称的精力规模，在红外线的区域的强壮的相互作用导致界限(或伪界限)状态。群众和π和 K 介子的腐烂常数的数字结果能复制试验性的价值，并且扎根的状态分级的 diquarks U~a 集中， D~a， S~a 与存在一致理论计算。我们建议可以解释分级的 diquarks 的群众的不确定性的新 Lagrangian。

New Closed Expression of Interaction Kernel in Bethe-Salpeter Equation for Quark-Antiquark Bound States

The interaction kernel in the Bethe-Salpeter equation for quark-antiquark bound states is derived newly from QCD in the case where the quark and the antiquark are of different flavors. The technique of the derivation is the usage of the irreducible decomposition of the Green's functions involved in the Bethe-Salpeter equation satisfied by the quark-antiquark four-point Green's function. The interaction kernel derived is given a closed and explicit expression which shows a specific structure of the kernel since the kernel is represented in terms of the quark, antiquark and gluon propagators and some kinds of quark, antiquark and/or gluon three, four, five and six-point vertices. Therefore,the expression of the kernel is not only convenient for perturbative calculations, but also suitable for nonperturbative investigations.

Instantaneous Bethe-Salpeter Equation and Its Exact Solution

We present an approach to solve Bethe-Salpeter (BS) equations exactly without any approximation if the kernel of the BS equations exactly is instantaneous, and take positronium as an example to illustrate the general features of the exact solutions. The key step for the approach is from the BS equations to derive a set of coupled and welldetermined integration equations in linear eigenvalue for the components of the BS wave functions equivalently, which may be solvable numerically under a controlled accuracy, even though there is no analytic solution. For positronium,the exact solutions precisely present corrections to those of the corresponding Schrodinger equation in order v1 (v is the relative velocity) for eigenfunctions, in order v2 for eigenvalues, and the mixing between S and D components in JPC = 1- states etc., quantitatively. Moreover, we also point out that there is a questionable step in some existent derivations for the instantaneous BS equations if one is pursuing the exact solutions. Finally, we emphasize that one should take the O(v) corrections emerging in the exact solutions into account accordingly if one is interested in the relativistic corrections for relevant problems to the bound states.

Bethe-Salpeter equations: mesons beyond the rainbow-ladder truncation

We investigate masses of light mesons from a coupled system of Dyson–Schwinger （DSE） and Bethe– Salpeter equations （BSE）,taking into account dominant non-Abelian,sub-leading Abelian,and dominant pion cloud contributions to the dressed quark-gluon vertex.The axial-vector Ward-Takahashi identity preserving Bethe-Salpeter kernel is constructed and the spectrum of light mesons calculated.Our model goes significantly beyond the rainbow-ladder.We find that sub-leading Abelian corrections are further dynamically suppressed,and that our results supersede early qualitative predictions from simple truncation schemes.

Forward-backward asymmetries inΛ_(b)→Λl^(+)l^(-)in the Bethe-Salpeter equation approach

Using the Bethe-Salpeter equation(BSE),we investigate the forward-backward asymmetries(AFB)inΛb→Λl^(+)l^(-)(l=e,μ,τ)in the quark-diquark model.This approach provides precise form factors that are different from those of quantum chromodynamics(QCD)sum rules.We calculate the rare decay form factors forΛb→Λl^(+)l^(-)b and investigate the(integrated)forward-backward asymmetries in these decay channels.We observe the integrated A l FB,ˉA l FB(Λb→Λe^(+)e^(-)=-0.1371,ˉA l FB(Λb→Λμ^(+)μ^(-))=-0.1376,andˉA l FB(Λb→Λτ^(+)τ^(-))=-0.1053;the hadron side asymmetriesˉA h FB(Λb→Λμ^(+)μ^(-))=-0.2315;the lepton-hadron side asymmetriesˉA lh FB(Λb→Λμ^(+)μ^(-))^(-)0.0827;and the longitudinal polarization fractionsˉF L(Λb→Λμ^(+)μ^(-))^(-)0.5681.

Some Modified Equations of the Sine-Hilbert Type

Three modified sine-Hilbert(sH)-type equations, i.e., the modified sH equation, the modified damped sH equation, and the modified nonlinear dissipative system, are proposed, and their bilinear forms are provided.Based on these bilinear equations, some exact solutions to the three modified equations are derived.

Data-Driven Ai-and Bi-Soliton of the Cylindrical Korteweg-de Vries Equation via Prior-Information Physics-Informed Neural Networks

By the modifying loss function MSE and training area of physics-informed neural networks(PINNs),we propose a neural networks model,namely prior-information PINNs(PIPINNs).We demonstrate the advantages of PIPINNs by simulating Ai-and Bi-soliton solutions of the cylindrical Korteweg-de Vries(cKdV)equation.

Matrix Riccati Equations in Optimal Control

In this paper, the matrix Riccati equation is considered. There is no general way for solving the matrix Riccati equation despite the many fields to which it applies. While scalar Riccati equation has been studied thoroughly, matrix Riccati equation of which scalar Riccati equations is a particular case, is much less investigated. This article proposes a change of variable that allows to find explicit solution of the Matrix Riccati equation. We then apply this solution to Optimal Control.

THE STABILITY OF BOUSSINESQ EQUATIONS WITH PARTIAL DISSIPATION AROUND THE HYDROSTATIC BALANCE

This paper is devoted to understanding the stability of perturbations around the hydrostatic equilibrium of the Boussinesq system in order to gain insight into certain atmospheric and oceanographic phenomena.The Boussinesq system focused on here is anisotropic,and involves only horizontal dissipation and thermal damping.In the 2D case R^(2),due to the lack of vertical dissipation,the stability and large-time behavior problems have remained open in a Sobolev setting.For the spatial domain T×R,this paper solves the stability problem and gives the precise large-time behavior of the perturbation.By decomposing the velocity u and temperatureθinto the horizontal average(ū,θ)and the corresponding oscillation(ū,θ),we can derive the global stability in H~2 and the exponential decay of(ū,θ)to zero in H^(1).Moreover,we also obtain that(ū_(2),θ)decays exponentially to zero in H^(1),and thatū_(1)decays exponentially toū_(1)(∞)in H^(1)as well;this reflects a strongly stratified phenomenon of buoyancy-driven fluids.In addition,we establish the global stability in H^(3)for the 3D case R^(3).

The Maxwell-Heaviside Equations Explained by the Theory of Informatons

In the articles “Newtons Law of Universal Gravitation Explained by the Theory of Informatons” and “The Gravitational Interaction between Moving Mass Particles Explained by the Theory of Informatons” the gravitational interaction has been explained by the hypothesis that information carried by informatons is the substance of gravitational fields, i.e. the medium that the interaction in question makes possible. From the idea that “information carried by informatons” is its substance, it has been deduced that—on the macroscopic level—a gravitational field manifests itself as a dual entity, always having a field- and an induction component (Egand Bg) simultaneously created by their common sources. In this article we will mathematically deduce the Maxwell-Heaviside equations from the kinematics of the informatons. These relations describe on the macroscopic level how a gravitational field (Eg, Bg) is generated by whether or not moving masses and how spatial and temporal changes of Egand Bgare related. We show that there is no causal link between Egand Bg.

THE SMOOTHING EFFECT IN SHARP GEVREY SPACE FOR THE SPATIALLY HOMOGENEOUS NON-CUTOFF BOLTZMANN EQUATIONS WITH A HARDPOTENTIAL

In this article, we study the smoothing effect of the Cauchy problem for the spatially homogeneous non-cutoff Boltzmann equation for hard potentials. It has long been suspected that the non-cutoff Boltzmann equation enjoys similar regularity properties as to whose of the fractional heat equation. We prove that any solution with mild regularity will become smooth in Gevrey class at positive time, with a sharp Gevrey index, depending on the angular singularity. Our proof relies on the elementary L^(2) weighted estimates.

On entire solutions of some Fermat type differential-difference equations

On one hand,we study the existence of transcendental entire solutions with finite order of the Fermat type difference equations.On the other hand,we also investigate the existence and growth of solutions of nonlinear differential-difference equations.These results extend and improve some previous in[5,14].

Equivalence between the internal observability and equation

This paper is concerned with a third order in time linear Moore-Gibson-Thompson equation which describes the acoustic velocity potential in ultrasound wave program.Influenced by the work of Kaltenbacher,Lasiecka and Marchand(Control Cybernet.2011,40:971-988),we establish an observability inequality of the conservative problem,and then discuss the equivalence between the exponential stabilization of a dissipative system and the internal observational inequality of the corresponding conservative system.

Multi-soliton solutions, breather-like and bound-state solitons for complex modified Korteweg–de Vries equation in optical fibers

Under investigation in this paper is a complex modified Korteweg–de Vries(KdV) equation, which describes the propagation of short pulses in optical fibers. Bilinear forms and multi-soliton solutions are obtained through the Hirota method and symbolic computation. Breather-like and bound-state solitons are constructed in which the signs of the imaginary parts of the complex wave numbers and the initial separations of the two parallel solitons are important factors for the interaction patterns. The periodic structures and position-induced phase shift of some solutions are introduced.

ON THE STABILITY OF PERIODIC SOLUTIONS OF PIECEWISE SMOOTH PERIODIC DIFFERENTIAL EQUATIONS

In this paper,we address the stability of periodic solutions of piecewise smooth periodic differential equations.By studying the Poincarémap,we give a sufficient condition to judge the stability of a periodic solution.We also present examples of some applications.

Exact and heuristic formulae to compute the geodetic height from the ellipse equation

The conversion of the cartesian coordinates of a point to its geodetic equivalent coordinates in reference to the geodetic ellipsoid is one of the main challenges in geodesy.The ellipse equation in the meridian plane significantly influences the value of the geodetic coordinates.This research analyzes this influence and how it can contribute to their solutions.The study investigates the mathematical relation between them and presents an exact formula relating to the geodetic height and the ellipse equation.In addition,a heuristic formula for the relation between the geodetic height and the ellipse equation is proposed,which is independent of the geodetic latitude and has a relative accuracy better than 99.9 %.The calculation is stable,and the cost is low.

A deep learning method based on prior knowledge with dual training for solving FPK equation

The evolution of the probability density function of a stochastic dynamical system over time can be described by a Fokker–Planck–Kolmogorov(FPK) equation, the solution of which determines the distribution of macroscopic variables in the stochastic dynamic system. Traditional methods for solving these equations often struggle with computational efficiency and scalability, particularly in high-dimensional contexts. To address these challenges, this paper proposes a novel deep learning method based on prior knowledge with dual training to solve the stationary FPK equations. Initially, the neural network is pre-trained through the prior knowledge obtained by Monte Carlo simulation(MCS). Subsequently, the second training phase incorporates the FPK differential operator into the loss function, while a supervisory term consisting of local maximum points is specifically included to mitigate the generation of zero solutions. This dual-training strategy not only expedites convergence but also enhances computational efficiency, making the method well-suited for high-dimensional systems. Numerical examples, including two different two-dimensional(2D), six-dimensional(6D), and eight-dimensional(8D) systems, are conducted to assess the efficacy of the proposed method. The results demonstrate robust performance in terms of both computational speed and accuracy for solving FPK equations in the first three systems. While the method is also applicable to high-dimensional systems, such as 8D, it should be noted that computational efficiency may be marginally compromised due to data volume constraints.

Sparse-Grid Implementation of Fixed-Point Fast Sweeping WENO Schemes for Eikonal Equations

Fixed-point fast sweeping methods are a class of explicit iterative methods developed in the literature to efficiently solve steady-state solutions of hyperbolic partial differential equations(PDEs).As other types of fast sweeping schemes,fixed-point fast sweeping methods use the Gauss-Seidel iterations and alternating sweeping strategy to cover characteristics of hyperbolic PDEs in a certain direction simultaneously in each sweeping order.The resulting iterative schemes have a fast convergence rate to steady-state solutions.Moreover,an advantage of fixed-point fast sweeping methods over other types of fast sweeping methods is that they are explicit and do not involve the inverse operation of any nonlinear local system.Hence,they are robust and flexible,and have been combined with high-order accurate weighted essentially non-oscillatory(WENO)schemes to solve various hyperbolic PDEs in the literature.For multidimensional nonlinear problems,high-order fixed-point fast sweeping WENO methods still require quite a large amount of computational costs.In this technical note,we apply sparse-grid techniques,an effective approximation tool for multidimensional problems,to fixed-point fast sweeping WENO methods for reducing their computational costs.Here,we focus on fixed-point fast sweeping WENO schemes with third-order accuracy(Zhang et al.2006[41]),for solving Eikonal equations,an important class of static Hamilton-Jacobi(H-J)equations.Numerical experiments on solving multidimensional Eikonal equations and a more general static H-J equation are performed to show that the sparse-grid computations of the fixed-point fast sweeping WENO schemes achieve large savings of CPU times on refined meshes,and at the same time maintain comparable accuracy and resolution with those on corresponding regular single grids.

On Two Types of Stability of Solutions to a Pair of Damped Coupled Nonlinear Evolution Equations

The stability of a set of spatially constant plane wave solutions to a pair of damped coupled nonlinear Schrödinger evolution equations is considered. The equations could model physical phenomena arising in fluid dynamics, fibre optics or electron plasmas. The main result is that any small perturbation to the solution remains small for all time. Here small is interpreted as being both in the supremum sense and the square integrable sense.