B. H. Lei

Index Notation and Embedded Surfaces

2026-08-23T01:00:00.000Z

Index Notation and Embedded Surfaces

Index notation and differential geometry provide

Fourier Multipliers as Operators

2026-08-23T01:00:00.000Z

Fourier Multipliers as Operators

Fourier analysis is especially useful for linear, transla

What does $x^{\prime}$ mean in a Green’s function? — a linear algebra and operator view

2026-04-30T00:00:00.000Z

In a Green's function $G(x,x')$, the primed variable $x'$ marks the source point — the location of a unit impulse — while $x$ is the field point where the response is measured. This can be seen as a matrix element $\langle x | L^{-1} | x' \rangle$ of the inverse operator. For the 1D Poisson equation $-\frac{d^2 u}{dx^2}=f$, we obtain $G(x,x')=-\frac12|x-x'|$ using $\frac{d^2}{dx^2}|x-x'|=2\delta(x-x')$. Additional examples illustrate the same concept.

人应该怎样度过一生一

2026-03-25T00:00:00.000Z

浓雾氤氲的森林，分不清是清晨还是黄昏，天边应该是有丝丝破碎的晨曦亦或是夕阳，还未走进你眼眸，便融化进了昏霭，染成一片模糊的光晕。你的思绪也仿若虚无之地，白茫茫一片，犹豫，抬起脚，犹豫，犹豫，犹豫，向前？向后？是追忆还是邂逅迷茫？

From the Spectral Dirichlet-Neumann Symbol to the Physical-Space Operator Identity

2026-02-05T00:00:00.000Z

We show how the spectral relation $\partial_z \widehat{\varphi}(k, 0)=k \tanh (k h) \widehat{\varphi}(k, 0)$ implies the physical-space operator identity $\varphi_z(\cdot, \cdot, 0)=(|D| \tanh (h|D|)) \varphi(\cdot, \cdot, 0)$, by inverse Fourier transform and the commutation of $\partial_z$ with the horizontal Fourier transform.

Hankel Diagonalization of the Radial Bessel Operator

2026-01-29T21:30:00.000Z

We show that the order- $n$ Hankel transform in $r$ diagonalizes the Bessel-type radial operator $\mathscr{L}_n$, mapping the PDE $\left(\mathscr{L}_n+\partial_z^2\right) \varphi_n=0$ to the transformed equation $\left(\partial_z^2-k^2\right) \hat{\varphi}_n=0$. The key ingredients are the Sturm-Liouville form and self-adjointness of $\mathscr{L}_n$, together with the Bessel differential equation.

Fourier Coefficients of Quadratic Forms in $\cos \gamma$ for Steady Capillary-Gravity Waves

2026-01-29T20:30:00.000Z

We factor the quadratic denominator $a+b \cos \gamma+c \cos ^2 \gamma$ and reduce the computation of its Fourier coefficients to those of $1 /(\cos \gamma-\lambda)$. Solving the associated second-order recurrence for the Fourier coefficients yields a closed-form expression in terms of $\lambda_{ \pm}$and $q_{ \pm}=\lambda_{ \pm}+i \sqrt{1-\lambda_{ \pm}^2}$, leading to the final explicit formula for $c_m(k)$ in the steady ship-wave setting.

Diagonalizing Azimuthal Mode Coupling via a Discrete Fourier Transform

2026-01-29T19:30:00.000Z

We recast the azimuthal mode-coupled surface equations as a block-Toeplitz convolution in the mode index and diagonalize this coupling via a discrete Fourier transform in $n$, then briefly review the basic definition and core properties of the DFT used in this step.

Cartesian Derivatives in Azimuthal Fourier–Hankel Space

2026-01-29T18:30:00.000Z

claim:

\[ \begin{aligned} & {\widehat{\left(\partial_x f\right)_n}}(k)=\frac{k}{2}\left(\hat{f}_{n-1}(k)-\hat{f}_{n+1}(k)\right) \\ & {\widehat{\left(\partial_y f\right)_n}}(k)=\frac{k}{2 i}\left(\hat{f}_{n-1}(k)+\hat{f}_{n+1}(k)\right) \end{aligned}\tag{1} \]

DFT Method Main

2026-01-29T17:30:00.000Z

We consider the governing equations for water waves in the context of an inviscid fluid of infinite depth. The fundamental system of equations are

\[ \begin{array}{c} \partial_x^2 \varphi+\partial_y^2 \varphi+\partial_z^2 \varphi=0 , \quad z \leq 0 \\ \left.\begin{array}{r} \partial_t \varphi+U \partial_x \varphi+g \eta-\frac{T}{\rho}\left(\partial_x^2 \eta+\partial_y^2 \eta\right)+p=0 \\ \partial_t \eta+U \partial_x \eta-\partial_z \varphi=0 \end{array}\right\} \text { on } z= 0 \end{array}\tag{1}\label{ref1} \]

Axisymmetric Angular Fourier Modes and Hankel Transform of a Radial Heaviside Pressure Field

2026-01-29T15:30:00.000Z

We consider an axisymmetric pressure distribution $p(r, \theta)=\bar{p} H(r-l)$ (radial dependence only) and show that its angular Fourier series contains only the $m=0$ mode. Consequently, the polar Fourier representation $\tilde{p}(k, \beta)$ in wavenumber space is independent of the angular variable $\beta$, and the Hankel transform reduces to the order-zero Bessel transform.

UK Driving School & Car Words

2026-01-13T00:00:00.000Z

Knowledge related to the UK driving test

Add Two Numbers - References in Python (with C++ Comparisons)

2025-12-10T00:00:00.000Z

Python variables do not store raw values directly but instead hold references to objects on the heap. This note builds a practical mental model for how references work, how they compare to C++ pointers, and how this affects assignment, mutation, function parameters, copying, and common pitfalls. We include small typed examples and a focused discussion of shallow copies such as a.copy() vs m1.copy() in both flat and nested list cases.

two sum decorators

2025-11-13T00:00:00.000Z

A decorator is just a callable that takes a function and returns a new one, often a wrapper, letting us extend behavior while keeping functions clean and following the Open–Closed Principle.

two sum type annotation

2025-11-10T00:00:00.000Z

Several examples are presented to clarify the definition and application of type annotations in Python.

two sum

2025-11-07T00:00:00.000Z

Given an array of integers nums and an integer target, return indices of the two numbers such that they add up to target.

无名树

2025-11-03T00:00:00.000Z

院角那株不知名的树，自漫长阴雨时节便浸在阴翳里。我总在窗前半掩着帘，看它瘦削的枝桠将天色裁成淡墨的写意，恍若故纸堆里褪了色的水墨小品。

Self Adjoint Operators

2025-10-21T00:00:00.000Z

We consider the radial differential operator for a fixed angular mode $n$,

\[ L_n=\frac{d^2}{d r^2}+\frac{1}{r} \frac{d}{d r}-\frac{n^2}{r^2}, \quad r \in(0, \infty) \]

Azimuthal Fourier Analysis

2025-10-09T16:30:00.000Z

We introduce azimuthal Fourier expansion for functions with circular symmetry: we write $f(r, \phi)$ as a sum of angular modes $\hat{f}_n(r) e^{i n \phi}$, where $\hat{f}_n(r)$ is obtained by integrating over $\phi$ and measures the strength of mode $n$ at radius $r$. We then contrast this discrete angular Fourier series on $[0,2 \pi]$ with the usual continuous Fourier transform on $(-\infty, \infty)$, and we briefly note that this separation of radial and angular dependence is especially useful in optics, quantum mechanics, fluid dynamics, and image processing.

Operator Ladder Relations for Hankel Transforms of Bessel Type

2025-10-04T00:00:00.000Z

Hankel–Bessel Ladder Identities

Claim. For suitable $f_m(r)$ and $k>0$, \[ \mathscr{H}_{m+1}\!\left[\left(\partial_r-\frac{m}{r}\right) f_m\right](k) =-\,k\,\hat{f}_m(k). \tag{1} \] Here $\hat f_m(k):=\mathscr H_m[f_m](k)=\displaystyle\int_0^\infty f_m(r)\,J_m(kr)\,r\,dr$.

浮云记

2025-08-25T00:00:00.000Z

settings for PINN

2025-08-04T00:00:00.000Z

basic concept

ninja

ninja is a build system—a tool that actually run

Surface Waves Generated by A Travelling Pressure Point

2025-05-05T00:00:00.000Z

We study the free-surface response to a moving pressure point over finite depth using 2D Fourier transforms, contour deformation, and a radiation condition. By analysing poles, branch points, and stationary points of the dispersion curve $G(\alpha, \beta)=0$, we derive an asymptotic representation of $\eta$ via residues and stationary-phase contributions and clarify the relevant complex-analytic singularity structure.

Ray Geometry, and Wave Action in Uniform Currents

2025-05-03T00:00:00.000Z

We summarize the ray-theoretic description of Kelvin ship waves in deep and finite depth, deriving the relationship between wavevector and wake angle, the Kelvin wedge, and the scaling of amplitude along rays via wave action conservation, and we connect these to Froude numbers and interference effects between bow and stern waves.

Phase Geometry of Ship Waves: Deriving $p=a \cos ^2 \theta$

2025-04-17T00:00:00.000Z

We derive the ship-wave crest relation $p=a \cos ^2 \theta$ in deep water from the Doppler-shifted dis

c++ debug notes

2025-04-14T00:00:00.000Z

Link Against the GSL Libraries

When we compile code with GSL, we need to tell the linker to

note for python I

2025-04-14T00:00:00.000Z

Lambda function

Lambda functions in Python are small, anonymous functions defined with the lambda
One Dimension Ship Wave With Surface Tension 2025-04-12T00:00:00.000Z The article develops a mathematical model for one-dimensional gravity-capillary ship waves, using residue theory to analyze wave componen
mathjax align vs align* vs aligned 2025-04-10T00:00:00.000Z 在 MathJax 中，aligned、align 和 align* 是用于对齐多行数学公式的环境，但它们在功能、使用场景和编号规则上有明确区别：功能与 English in Paper 2025-03-18T00:00:00.000Z 在英文数学论文中，推导公式时连接前后式子的常用连接词和短语可以分为以下几类，以确保逻辑清晰、推导严谨： Relations among Rays, Group Velocity, Phase Velocity, and the Wave Number Vector in Water Waves 2025-03-18T00:00:00.000Z In dispersive water waves (e.g., surface gravity waves on deep or shallow water), several concepts describe how waves travel, how energy moves, and how the wave patterns evolve. Below is an overview of wave fronts, wave crests, the wave number vector $\mathbf{k}$, phase velocity $\mathbf{c}_p$, group velocity $\mathbf{c}_g$, and rays, along with how their directions compare. Pressure Forcing on a Free Surface and Far-Field Waves in Lamb’s Hydrodynamics 2025-02-12T00:00:00.000Z We summarise Lamb’s treatment of a pressure disturbance on a free surface in a uniform stream, emphasising the contour‐integral evaluation of the far-field wavetrain, the underlying simple-harmonic free-surface condition, a Bessel–delta normalisation, and a geometric construction for envelopes of straight lines. Wave Patterns from Ray Theory: Centered Waves, Wake Angles, and Phase Geometry 2025-02-10T00:00:00.000Z We collect the ray-theoretic relations for wave patterns generated by a point source, derive the slope of characteristics and their connection to the group-velocity symbol $G$, and clarify the geometric roles of wavefronts, wave vectors, and rays, culminating in a phase-distance relation $\theta=k \cos \mu r$ for capillarygravity wakes. Gaussian Integrals and Related Fourier–Bessel Identities 2025-01-29T00:00:00.000Z We collect closed-form formulas for one- and $n$-dimensional Gaussian integrals with linear and oscill 我的小院 2025-01-27T00:00:00.000Z 原想写一些剖析自我的文字，却不知如何下笔，也罢，就写写最近的杂绪吧。早春，院子里的不知名的树依然光秃着树干，纹丝不动地点缀着视野里的红墙，肃杀而冷淡。又是一个冬去了，除却偶或光临的故乡的喜鹊，在小院凌乱的大戏台，唱着故园情、天涯愁。素常，小院是安静的，百无聊回乡偶书二首 2025-01-14T00:00:00.000Z 回乡偶书二首贺知章 < 归园田居其一 2025-01-13T00:00:00.000Z 归园田居其一陶渊明 Hankel Transform of the Bi-Laplacian and an Axisymmetric Free-Surface Problem 2025-01-11T00:00:00.000Z We show how the zeroth-order Hankel transform diagonalises the radial bi-Laplacian, clarify the sign convention $\nabla^4 \leftrightarrow k^4$, and then apply the same transform machinery to an axisymmetric linear free-surface problem to obtain explicit representations for $\eta(r,t)$ and $\hat w(s,z,t)$. Gamma, Polygamma, Hurwitz Zeta, and Bessel Function Identities 2024-06-29T00:00:00.000Z We collect a few basic identities for the gamma and polygamma functions, their link to the Hurwitz zeta function, standard asymptotics for Bessel functions, and a useful Bessel product expansion and inequality for $J_n(z)$. Continuity and Momentum Equations in Eulerian Form and Their Link to Hydraulic Jumps 2024-06-24T00:00:00.000Z We review the Eulerian continuity and Euler momentum equations, clarify the relationship between Eulerian and Lagrangian descriptions via the material derivative, and briefly connect these fundamentals to the physics of hydraulic jumps as transitions between supercritical and subcritical flow. casting operators 2024-06-23T16:05:31.000Z the c++ casting operators the four c++ casting operators are static_cast dyna operator type and overloading -part III 2024-06-22T16:05:31.000Z move constructor and move assignment operator Take a look at the addition operator type and overloading -part II 2024-06-21T16:05:31.000Z binary operators operators that function on two operands are called binary operators. operator type and overloading -part I 2024-06-20T16:05:31.000Z on a board level, operators in c++ can be classified into two type: unary operators and binary operators. un polymorphism -3 2024-06-19T16:05:31.000Z specifier override to indicate intention to override Assume that you w polymorphism -2 2024-06-18T16:05:31.000Z using virtual inheritance to solve the diamond problem how many inst polymorphism -1 2024-06-17T16:05:31.000Z basics of polymorphism e.g. invoking methods using an instance of Base class which belo implementing Inheritance - Q&A 2024-06-16T16:05:31.000Z Q&A Q : class D2 inherits from class D1 Inheritance 2024-06-15T16:05:31.000Z private inheritance private is used in the line where the derived class declares its inher basics of inheritance - part 2 2024-06-14T16:05:31.000Z invoking method of a base class in a derived class Typically, if your sp basics of inheritance - part 1 2024-06-13T16:05:31.000Z C++ syntax of derivation using aggregate initialization on classes and union and struct 2024-06-11T16:05:31.000Z sizeof() a char* pointer's size is fixed at 4 bytes (on 32-bit system, and double it on 64 constructor -3 2024-06-10T16:05:31.000Z move constructor 1< constructor -2 2024-06-09T16:05:31.000Z the difference of & used in reference and store address constructor and shadow copying 2024-06-08T16:05:31.000Z strlen vs sizeof strlen method is used to find the l function 2024-06-07T16:05:31.000Z return用来退出当前函数模块。注意有返回值，即：return (num) 中(num)是某个返回值，则调用函数的时候可以赋 Arrays and Strings 2024-06-06T16:05:31.000Z 1 2 Introduction -2 2024-06-05T16:05:31.000Z 使用变量和常量 std::cin不要用endl e.g. 下面写法是错误的使用 std::cout 显示一行消息，std::endl 命令cout换行,作用就相当于\n ++i vs i++ Beginner's Guide to C++ Basics 2024-06-03T16:05:31.000Z We get to that point, c++, this chapter mainly introduce parts of c++, and understand them with many code example. Eulerian and Lagrangian Forms of the Continuity and Momentum Equations 2024-06-01T00:00:00.000Z We recall the continuity and Euler momentum equations in Eulerian form, derive the material derivative $\mathrm{d} \mathbf{v} / \mathrm{d} t=\partial_t \mathbf{v}+(\mathbf{v} \cdot \nabla) \mathbf{v}$, and explain how Eulerian field descriptions relate to Lagrangian particlebased descriptions via $\rho(\mathbf{x}(t), t)$. The Equations for Water Waves (Linear and Nonlinear Waves) 2024-05-31T00:00:00.000Z This note primarily delves into incompressible flow momentum equation and Surface of water waves and boundary condition and Variational Formulation. stellar+twikoo配置最新评论以及配置live2d 2024-05-18T00:00:00.000Z 博客侧边栏最新评论修改自「星日语」大佬的原文链接emoji

B. H. Lei

Index Notation and Embedded Surfaces

Index Notation and Embedded Surfaces

Fourier Multipliers as Operators

Fourier Multipliers as Operators

What does $x^{\prime}$ mean in a Green’s function? — a linear algebra and operator view

人应该怎样度过一生 一

From the Spectral Dirichlet-Neumann Symbol to the Physical-Space Operator Identity

Hankel Diagonalization of the Radial Bessel Operator

Fourier Coefficients of Quadratic Forms in $\cos \gamma$ for Steady Capillary-Gravity Waves

Diagonalizing Azimuthal Mode Coupling via a Discrete Fourier Transform

Cartesian Derivatives in Azimuthal Fourier–Hankel Space

DFT Method Main

Axisymmetric Angular Fourier Modes and Hankel Transform of a Radial Heaviside Pressure Field

UK Driving School & Car Words

Add Two Numbers - References in Python (with C++ Comparisons)

two sum decorators

two sum type annotation

two sum

无名树

Self Adjoint Operators

Azimuthal Fourier Analysis

Operator Ladder Relations for Hankel Transforms of Bessel Type

Hankel–Bessel Ladder Identities

浮云记

settings for PINN

basic concept

ninja

Surface Waves Generated by A Travelling Pressure Point

Ray Geometry, and Wave Action in Uniform Currents

Phase Geometry of Ship Waves: Deriving $p=a \cos ^2 \theta$

c++ debug notes

Link Against the GSL Libraries

note for python I

Lambda function

One Dimension Ship Wave With Surface Tension

mathjax align vs align* vs aligned

功能与

English in Paper

Relations among Rays, Group Velocity, Phase Velocity, and the Wave Number Vector in Water Waves

Pressure Forcing on a Free Surface and Far-Field Waves in Lamb’s Hydrodynamics

Wave Patterns from Ray Theory: Centered Waves, Wake Angles, and Phase Geometry

Gaussian Integrals and Related Fourier–Bessel Identities

我的小院

回乡偶书二首

归园田居 其一

Hankel Transform of the Bi-Laplacian and an Axisymmetric Free-Surface Problem

Gamma, Polygamma, Hurwitz Zeta, and Bessel Function Identities

Continuity and Momentum Equations in Eulerian Form and Their Link to Hydraulic Jumps

casting operators

the c++ casting operators

operator type and overloading -part III

move constructor and move assignment operator

operator type and overloading -part II

binary operators

operator type and overloading -part I

un

polymorphism -3

specifier override to indicate intention to override

polymorphism -2

using virtual inheritance to solve the diamond problem

polymorphism -1

basics of polymorphism

implementing Inheritance - Q&A

Q&A

Inheritance

private inheritance

basics of inheritance - part 2

invoking method of a base class in a derived class

basics of inheritance - part 1

C++ syntax of derivation

union and struct

sizeof()

constructor -3

move constructor

constructor -2

the difference of & used in reference and store address

constructor and shadow copying

strlen vs sizeof

function

人应该怎样度过一生一

归园田居其一

`sizeof()`

the difference of `&` used in reference and store address

`strlen` vs `sizeof`