J_s
All posts
- Features of Special Relativity due to Lorentz Transformation: Time Dilation
- Momentum and Impulse Relationship
- Uniform Acceleration Linear Motion and its Graphs
- Uniform Motion and Graphs
- Vector Triple Product, BAC-CAB Rule
- Levi-Civita Symbol
- Kronecker Delta
- Einstein Notation
- Parabolic Motion: Horizontal Range and Maximum Height Angle
- Product of Two Levi-Civita Symbols
- The minimum energy of a hydrogen atom in quantum mechanics
- Dirac Delta Function
- Properties of the Dirac Delta Function
- The Period of Simple Pendulum Motion is Independent of the Pendulum's Mass
- 규격화된 파동함수의 상태는 시간의 변화에 무관하다
- Prove that the Product of the Slopes of Two Perpendicular Lines is Always -1
- Finding Internal and External Division Points on a Line
- Gradient of the Magnitude of Separation Vectors
- Separation Vector
- Scalar Triple Product
- Unit Vectors of the Spherical Coordinate System Expressed in Terms of Unit Vectors of the Cartesian Coordinate System
- Velocity and Acceleration in Polar Coordinates
- Velocity and Acceleration in Cylindrical Coordinates
- Multiplication Formula Table
- Velocity and Acceleration in a Coordinate System
- Equation of the Tangent to a Circle with Slope m
- Finding the Equation of the Tangent Line at a Point on a Circle
- Velocity and Acceleration in Cartesian Coordinate System
- Compton scattering
- The rest mass of a photon is zero.
- Electrons Cannot Be Constituents of the Nucleus
- Momentum operator in quantum mechanics
- Commutator of Momentum and Position
- Prove that the expectation value of momentum is always a real number.
- Euclidean Space
- The Importance of the Relative Phase of the Wave Function
- Finding the Speed of Electromagnetic Light from Maxwell's Equations
- The Curl of a Gradient is Always Zero
- Curl of the Curl of Vector Functions
- Inertia Moment and Turning Radius
- Inertia Moment of a Thin Rod
- Parallel Axis Theorem
- Vertical Axis Theorem
- Inertia Moments of Disks and Cylinders
- Moment of Inertia of a Ring and Cylindrical Shell
- Sphere's Moment of Inertia
- The Divergence of Curl is Always Zero
- Lorentz Transformation Derivation
- Relativity Theory and Lorentz Transformation
- World Line and Galilean Transformation
- Matrix Rank, Nullity
- Characteristics of Special Relativity due to Lorentz Transformation: Loss of Simultaneity
- Implications of Special Relativity due to Lorentz Transformation: Length Contraction
- Definition of Vector Space
- Subspace of Vector Space
- Finding the Wave Function Eigenfunctions and Energy Eigenvalues in an Infinite Potential Well
- Linear Combination, Span
- Conversion of Cartesian Coordinate System Unit Vectors to Spherical Coordinate System Unit Vectors
- Properties of Commutators
- Commutation Relations of the Angular Momentum Operator
- Gradient, Divergence, Curl, and Laplacian in Curvilinear Coordinates
- What is an operator in physics (quantum mechanics)
- Dimension of the Vector Space
- What is Dirac Notation?
- Hermitian Operator
- Proof that the expectation eigenvalue of a Hermitian operator is always real
- The Meaning of Eigenvalue Equations in Quantum Mechanics
- Two eigenvectors with different eigenvalues are orthogonal.
- Two operators with simultaneous eigenfunctions are commutative.
- The Relationship Between the Powers of i and the Powers of e
- For any arbitrary operator, always in the Hermitian form
- Ladder Operators for Angular Momentum
- Simultaneous Eigenfunctions of Angular Momentum
- The condition for the product of two Hermitian operators to be a Hermitian operator
- Relationship between simultaneous eigenfunctions of angular momentum and ladder operators
- There is no solution to the time-independent Schrödinger equation when the energy is less than the potential.
- What is Degeneracy of Wave Functions in Quantum Mechanics?
- Energy Levels in an Infinite Potential Well
- Solving Harmonic Oscillator Problems using the Operator Method: Definition of Ladder Operators
- Matrix Representation of Operators in Quantum Mechanics
- Matrix Representation of the Harmonic Oscillator Operator
- Matrix Representation of Angular Momentum Operator
- Conservation of Momentum: An Easy Proof (High School Level)
- Collision of Two Objects and the Coefficient of Restitution
- Perfectly Elastic Collisions and the Conservation of Kinetic Energy
- Definition and Examples of Differential Equations
- Classification of Differential Equations
- Double Angle and Half Angle Formulas of Trigonometric Functions
- Proof that Sine Squared Plus Cosine Squared Equals 1
- Indefinite Integral of the Form e^{x^2}
- Separable First-Order Differential Equations
- Wronskian Definition and Determination of Linear Independence
- Homogeneous Functions and First-Order Differential Equations
- Linear Expansion Coefficient and Volumetric Expansion Coefficient
- Differentiation of Vectors, Dot Product, and Cross Product in Cartesian Coordinates
- Kinetic and Potential Energy Definitions in Physics
- Socks-Shoes Property: The Inverse of ab is Equal to the Product of the Inverse of b and the Inverse of a
- Moment of Inertia of a Spherical Shell
- Definition and Discrimination Method of an Exact Differential Equation
- Integrating Factor Method for First Order Linear Differential Equations
- Solution to the Exact Differential Equation
- Fluid Pressure
- The Formula to Calculate the Pressure of a Fluid Based on Depth
- Pressure of the Fluid Depending on the Depth when an Object is Placed on the Fluid
- Sum and Difference Formulas and Product-to-Sum Formulas of Trigonometric Functions
- Linear Combination of Solutions to Homogeneous Linear Differential Equations is Also a Solution
- Second-Order Linear Homogeneous Differential Equations with Constant Coefficients and Characteristic Equation
- Restoring Force and One-Dimensional Simple Harmonic Oscillator
- Solution to Second Order Homogeneous Differential Equations
- Homogeneous Meaning in Homogeneous Differential Equations
- Solutions to the Homogeneous Second-Order Linear Differential Equation and the Wronskian
- General Solution to Second-Order Linear Nonhomogeneous Differential Equations
- Methods for Finding the Second Solution of Second Order Differential Equations
- Solution to the Bernoulli Differential Equation
- Riccati Differential Equation Solutions
- Solution to Clairaut's Differential Equation
- Gauss's Theorem, Divergence Theorem
- Definition and Test Method of Subgroups
- Distinguishing Between W and Omega in Mathematics and Physics Textbooks
- What is a Pseudovector?
- Simplifying the Exponentiation of Two-digit Numbers Ending in 5
- Properties and Proofs of Surplus Types
- Definition and Criterion of Subrings
- Rings in Abstract Algebra
- Rules for Multiplication in a Ring
- Laplace Transform of Constant Functions
- Laplace Transform of Exponential Functions
- Laplace Transform of Hyperbolic Functions
- Laplace Transform of Polynomial Functions
- Laplace Transform of Trigonometric Functions
- Laplace Transform Table
- Linearity of Laplace Transform
- Definition and Existence Proof of the Laplace Transform
- Laplace Transform of the Step Function
- Staircase Function
- Inverse Laplace Transform of F(as+b)
- Inverse Laplace Transform of F(ks)
- Laplace Transform of f(ct)
- Laplace Transform of the First Order Derivative
- Laplace Transform Translation
- Solving Second-Order Linear Nonhomogeneous Differential Equations Using Laplace Transforms
- The Laplace Transform of the n-th Order Derivative
- Laplace Transform of t^{n}f(t)
- Laplace Transform of Periodic Functions
- Laplace Transform of the Dirac Delta Function
- Partition, Riemann Sum, Riemann Integral
- Riemann-Stieltjes Integral
- Segmentation
- Upper integral is greater than or equal to lower integral.
- Necessary and Sufficient Conditions for Riemann(-Stieltjes) Integrability
- The Fundamental Theorem of Slopes
- Coulomb's Law and Electric Fields
- Divergence of a Separation Vector
- Applications of Gauss's Law in Integral Form
- Divergence of the Electric Field
- Electric Flux and Gauss's Law
- Electric Field Curl
- Potential
- Properties of Potential
- Continuous Functions are Riemann-Stieltjes Integrable
- Monotone Functions are Riemann-Stieltjes Integrable
- Integrability is Preserved in the Composition with Continuous Functions
- Integrability is Preserved in the Multiplication of Two Functions
- Solving Nonhomogeneous Euler Differential Equations Using Substitution
- Solution of the Laplace Equation Independent of Azimuthal Angle in Spherical Coordinates using the Method of Separation of Variables
- Solving the Laplace Equation Independent of the Jet Axis in Cylindrical Coordinates Using the Method of Separation of Variables
- Equations Involving the Laplacian Operator, Second Order Partial Derivatives
- Fourier Series and Bessel's Inequality
- Proof that Differentiating the Heaviside Step Function Yields the Dirac Delta Function
- Proof of Leibniz's Theorem
- Series, Infinite Series
- Series Solution of Legendre Differential Equation: Legendre Polynomial
- Rodrigues Formula for Legendre Polynomial
- Proof that the Length of an Arc and the Length of a Chord are Approximately Equal When the Central Angle is Small
- Rotation of Separation Vector
- Derivation of Fourier Series
- Dirichlet Kernel
- Legendre Polynomials are orthogonal to any lower degree polynomial
- Orthogonal Functions and Orthogonal Sets
- Orthogonality of Legendre Polynomials
- Proof of the Orthogonality of the Set of Trigonometric Functions
- Sum of Trigonometric Functions Orthogonal to Each Other
- Multipole Expansion of Potential and Dipole Moments
- Proof of the Second Cosine Law Using the Definition of Trigonometric Functions
- 리만적분가능한 함수의 푸리에 급수는 수렴한다
- Alignment of Polar Molecules by a Constant External Electric Field
- Alignment of Polar Molecules by a Non-uniform Electric Field
- Ampère's Law and Applications
- Current and Current Density
- Divergence (Divergence) and Curl of Magnetic Fields
- Electric Field Created by a Dipole
- Electric Field Created by Bound Charges and Polarized Objects
- Magnetic Fields Produced by Magnetic Dipoles
- Magnetic Forces Do Not Work
- Magnetic Vector Potential
- Magnetism and the Law of Lorentz Force
- Multipole Expansion of Vector Potentials and Magnetic Dipole Moments
- Polarization density and genomes
- Steady Current and Biot-Savart Law
- Stokes' Theorem
- Probabilistic interpretation and normalization of the wave function in quantum mechanics
- Bound Current Density and the Vector Magnetic Field Created by a Magnetized Object
- Changes in Electron Orbits Due to External Magnetic Fields and Diamagnetism
- Self-Density and Ferromagnets
- The Torque on a Magnetic Dipole in an External Magnetic Field and Paramagnetism
- Chebyshev Differential Equations and Chebyshev Polynomials
- Series Solution of Chebyshev Differential Equation
- Electromotive Force and Kinetic Electromotive Force
- Partial Integration of Expressions Containing the Del Operator
- Work and Energy in Electrostatics
- Wronskian of Two Solutions of a Second Order Linear Differential Equation
- Continuity in Every Piece, Smoothness in Every Segment
- Convergence of Fourier Series at Discontinuities
- Derivative's Fourier Coefficients
- Fourier Series Integration
- Mean of Function Values
- The constant term of the Fourier series is equal to the average of one period of the function.
- The Integral of a Periodic Function Over One Period is Constant Regardless of the Integration Interval
- The Limit of Fourier Coefficients is Zero
- Exterior Unit Normal Vector
- Green's Theorem
- Green's Theorem, Integration by Parts Formula
- Initial Value Problem and Inhomogeneous Problem Solutions for the Transport Equation
- Laplace's Equation
- Proving the Invariance of the Laplace Equation with Respect to Orthogonal Transformations
- Definition of Convolution
- Heat Equation, Diffusion Equation
- Any Function Can Always Be Expressed as the Sum of Odd and Even Functions
- Half-Wave Symmetric Function
- Fourier Cosine Series, Sine Series, Fourier Coefficients of Even and Odd Functions
- Fourier Coefficients of Odd Functions
- Poisson's Equation Fundamental Solution
- Faraday's Law and Lenz's Law
- Mutual Inductance
- Self-Inductance
- Energy in a Magnetic Field
- The sufficient condition for the Fourier series of a function to converge absolutely and uniformly to the function
- Continuity Equations in Electromagnetism
- The Mean Value Theorem for Laplace's Equation
- Maxwell's Equations
- Poynting's Theorem and Poynting Vector
- In Physics, What is a Tensor
- Maxwell's Stress Tensor
- Electromagnetic Force on a Charge inside a Volume
- Maximum Principle of Harmonic Functions
- Uniqueness of the Solution to the Dirichlet Problem for the Poisson Equation
- Angular Momentum of the Electromagnetic Field
- Conservation of Momentum in Electrodynamics
- Derivation of the One-Dimensional Wave Equation
- Mollifiers
- Sine Waves and Complex Wave Functions
- Wave Boundary Conditions: Reflection and Transmission
- Mollification
- Multi Index Notation
- Smoothing Effect of Harmonic Functions
- Coulomb Gauge and Lorentz Gauge
- Gauge Transformation
- Vertical Waves, Parallel Waves, Plane Polarization
- Notation for Nonlinear First-Order Partial Differential Equations
- Characteristics of Nonlinear First-Order Partial Differential Equations
- Solution of Nonlinear First Order PDE Using Characteristic Equations
- Delayed Potential on Continuous Distribution of Delay Times
- Convergence of Norms of Function Sequences
- Bessel's Inequality in L2 Space
- Complete Orthonormal Basis and Complete Orthonormal Set
- Holder Continuous Function Spaces
- Generalized Hölder's Inequality, Corollaries of Hölder's Inequality
- Linearization of Boundaries in Nonlinear First-Order Differential Equations
- Proof of the Karatheodory's Theorem
- Additive and Multiplicative Functions
- Properties of Fourier Transform
- Fourier Transform of Gaussian Functions
- Fourier Transform of Characteristic Functions
- Riemann-Lebesgue Lemma
- Convergence of Mollification
- Exponential Function Set and Trigonometric Function Set are Orthogonal Bases
- Lines Determined by Polar Coordinates
- Fourier Inversion Theorem
- The Relationship between L1 Space and L2 Space
- Properties of Radon Transform
- Solving Differential Equations Using Fourier Transform
- Discrete Fourier Transform
- Predictable Functions
- Properties of Measurable Functions with Real Values
- Methods of Expressing an Arbitrary Function as Two Non-negative Functions
- How to Comment and Uncomment Multiple Lines at Once in MATLAB
- How to Save Data Calculated in MATLAB to an Excel File
- One-Dimensional D'Alembert's Formula
- Lagrangians and Euler-Lagrange Equations in Partial Differential Equations
- Hamilton-Jacobi Equation and Hamiltonian Equation
- Importing Excel Data into MATLAB
- Back Projection: The Dual of the Radon Transform
- Fourier Slice Theorem
- Radon Inverse Transform: Filtered Back Projection (FBP)
- Hilbert Transform
- Hamiltonian Equations Derived from Variational Calculus and Euler-Lagrange Equation
- Laplace Transform Convolution
- Legendre Transformation
- Hamiltonian and Lagrangian Convex Duality
- Mellin Transformation
- Hopf-Lax Formula
- Fundamental Theorem of Algebra
- Proof that the Hopf-Lax Formula Satisfies the Hamilton-Jacobi Equation
- Lagrangian Mechanics and Hamiltons Variational Principle
- Euler-Lagrange Equation in Physics
- List of Special Symbols Available for Graphs in MATLAB
- Derivation of Bessel's Equation
- Uniform Convexity
- Linear Functionals on Lp Spaces
- Interpolation Inequalities in Lebesgue Spaces
- Proof of Clarkson's Inequalities
- Holder's Inequality
- Minkowski's Inequality
- Embeddings in Mathematics, Insertion Mappings
- Sobolev Norm and Sobolev Spaces
- Sobolev Spaces are Banach Spaces: A Proof
- Sobolev Spaces are Separable, Uniformly Convex, and Reflexive: A Proof
- What is a Norm Space?
- Proof that All Isometric Mappings are Embeddings
- The Riesz Representation Theorem for Lp Spaces
- Semi Norm
- Semi-linear Function
- Hahn Banach Theorem for Real, Complex, Seminorm
- Hahn-Banach Extension Theorem
- Natural Embeddings and Reflexive Spaces
- Gram-Schmidt Orthogonalization Process in Quantum Mechanics
- Probability Flow
- Reflection and Transmission of Wave Functions
- Proving that Lp Spaces are Uniformly Convex and Reflexive
- Solution of the Schrödinger Equation for a Step Potential
- How to Output Multiple Figures on One Page in MATLAB
- Borel Sigma-Algebra, Borel Measurable Space
- Extended Real Number System
- Conditions for a Function with Extended Real Values to be Measurable
- Physics Appendix
- Solution of Schrödinger Equation for Finite Square Well Potential
- Solution of the Schrödinger Equation for a Potential Barrier
- Electric Potential and Electromagnetic Fields
- The Zhemengko Equation
- Liénard-Wiechert Potentials
- Gradient of Time Delay
- Definition of a General Parallelepiped
- Finite Cone
- Locally Finite Covers
- Under Cone Conditions
- Weak Cone Condition
- Line Segment Conditions
- General Definitions of Measure
- Signed Measures
- Positive Set, Negative Set, Null Set
- Hahn Decomposition Theorem
- Mutually Singular
- Jordan Decomposition Theorem
- Sets Outside/Inside a Certain Distance from the Boundary of a Set
- Uniform Cone Condition
- Strong Local Lipschitz Condition
- Matrix Size and Related Functions in MATLAB
- Uniform C^m-Regularity Condition
- How to Perform Element-wise Operations on Two Matrices in MATLAB
- Creating Special Matrices in MATLAB
- Rotating an Image in MATLAB
- Fréchet Derivative
- How to Specify Colors, Line Styles, and Marker Types in MATLAB Graphs
- Chain Rule for Fréchet Derivatives
- Matrix Functions, Definition of Matrix Exponential Functions
- Selecting Specific Rows and Columns in a Matrix in MATLAB
- Total Variation
- Absolute Continuity of the Sign Measure
- Lebesgue-Radon-Nikodym Theorem
- Relationship between Absolutely Continuous and Integrable Functions
- How to Create Equally Spaced Row Vectors in MATLAB
- Algebra, Quasi-measure
- Complex Measures, Vector Measures
- Disjoint Union: Disjoint Unions
- Separated Union Topological Space
- Saturation and Definition of Fibers in Mathematics
- Hardy-Littlewood Maximal Function
- Maximal Lemma
- Maximal Theorem
- The Mean Value of Locally Integrable Functions Converges to the Value of the Function at the Center.
- In Topology, What is a Coordinate System?
- Homomorphism Preserves Basis
- Easy Ways to Memorize Surjections, Injections, Ranges, and Domains, Explained
- Several Equivalent Conditions for the Interior in a Topological Space
- Subspace Topology, Relative Topology
- Translating Arrays in Julia
- Various Methods of Creating Vectors in Julia
- Functions for 2D Array Operations in Julia
- How to output and save arrays as heatmap images in Julia
- How to Rotate Image Arrays in Julia
- Resizing Images in MATLAB
- How to Change Image Size in Julia
- Measuring Code Execution Time in MATLAB
- Generating Topology from a Basis
- Properties of the Interior in Topological Spaces and Subspaces
- Derivation of the Gamma Function
- Leibniz Integral Rule
- Factorial, Double Factorial, and Multifactorial
- Various Important Formulas Involving the Gamma Function and Factorials
- Relationship between Beta Function and Gamma Function
- Representation of the Beta Function in the Form of an Improper Integral
- Euler Integrals: Beta Function and Gamma Function
- Ordinary Differential Equation
- Second Derivative, Higher Order Derivative
- Angular Momentum Operator in Spherical Coordinates
- Series Solution of the Bessel Equation: Bessel Functions of the First Kind
- Frobenius Method
- Vectors and Inner Products in Quantum Mechanics
- For an Angle Small Enough
- Velocity and Acceleration of an Object Moving in a Rotating Coordinate System
- Trigonometric Form of the Legendre Differential Equation
- Differential Equations for Physics: Solutions to Commonly Encountered Differential Equations
- Time Derivatives of the Lienard-Wiechert Potentials
- Parity Operator
- Moving Point Charges and the Electric Fields They Create
- Expectation Value in Quantum Mechanics
- de Broglie Equation and Matter Waves
- In Quantum Mechanics, what is a commutator?
- Physics에서의 Del 연산자
- Magnetic Field Created by a Moving Point Charge
- Principles of CT (Computed Tomography)
- Spherical Harmonics: General Solutions for the Polar and Azimuthal Angles in the Spherical Coordinate Laplace's Equation
- General Solution to the Radial Component Equation in the Laplace's Equation in Spherical Coordinates
- General Solution to the Laplace Equation in Spherical Coordinates
- Derivation of the Schrödinger Equation
- Solutions to Euler's Differential Equations
- Associated Legendre Differential Equations and Polynomials
- Recursive Relations of Legendre Polynomials
- Generating Functions of Legendre Polynomials
- Legendre Polynomials
- Associated Legendre Polynomials for Negative Index m
- Orthogonality of Associated Legendre Polynomials
- Normalization of Spherical Harmonic Functions
- Associated Legendre Polynomials
- Schrödinger Equation in Spherical Coordinates
- The Second Series Solution of the Bessel Equation: Bessel Functions of the Second Kind, Neumann Functions, Weber Functions
- Bessel Function's Recursive Relations
- Bessel Functions as Solutions to Differential Equations
- Orthogonality of Bessel Functions
- Bessel Functions
- Hankel Functions, Bessel Functions of the Third Kind
- Modified Bessel Equation and Modified Bessel Function
- Series Solutions to the Airy Differential Equation
- The Airy Function
- The eigenfunctions of the angular momentum operator are spherical harmonics.
- Ladder Operators of Angular Momentum in Spherical Coordinates
- Angular Momentum and Position/Momentum Commutation Relations
- Wave Function and Hilbert Space in Quantum Mechanics
- What is a Differential Operator in Physics?
- Hermite Functions
- Operator Solution of the Differential Equation Satisfied by Hermite Functions
- Hermite Differential Equations and Series Solutions
- Series Solution of Laguerre Differential Equation
- Foehammer Symbol
- Hermite Polynomials
- Hermite Polynomials' Generating Function
- Hermite Polynomials' Recursive Relations
- Orthogonality of Hermite Polynomials
- Laguerre Polynomials' Rodrigues' Formula
- First-Order Linear Differential Equation System
- 지수성장방정식/상수 계수를 갖는 1계 선형 동차 미분 방정식
- Definition of Wavelets
- Linearity of Riemann(-Stieltjes) Iintegral
- Center of Mass and Linear Momentum of a Particle System
- Newton's Laws of Motion
- Physics: The Definition of Mass, Force, and Momentum
- The Reason Why Momentum is Denoted by p
- Angular Momentum and Torque
- Angular Momentum of Particle Systems
- Centrifugal Force
- Kinetic Energy of Particle Systems
- Law of Universal Gravitation: Gravity
- Kepler's Laws of Planetary Motion
- Uniform Sphere Shell and the Gravity of a Separated Particle
- Derivation of the Equation of an Ellipse
- Kepler's Second Law: The Law of Equal Areas
- The Magnitude of the Cross Product of Two Vectors is Equal to the Area of the Parallelogram They Form
- Orbit Equation of a Particle under Central Force
- Ellipse
- Equation of an Ellipse with the Focus at the Origin in Polar Coordinates
- Elliptic Integral of the Second Kind
- Perimeter of an Ellipse
- Kepler's First Law: The Law of Elliptical Orbits
- Kepler's Third Law: The Harmony of the Worlds
- Formula for the Roots of a Cubic Equation
- Relationships between the Roots and Coefficients of Quadratic/Tertiary/nth Degree Equations
- Day, Work-Energy Theorem
- Integrable Functions and Absolute Values
- Riemann-Stieltjes Integrability is Preserved within an Interval
- The Fundamental Theorem of Calculus in Analysis
- The Relationship Between the Size of Integrals Based on the Order of Functions
- Closure and Derived Set in Metric Space
- Neighborhood, Limit Point, Open, Closed in Metric Space
- Properties of Open and Closed Sets in Metric Spaces
- Relatively Open Sets in Metric Spaces
- Closed Subsets of Compact Sets in Metric Spaces are Compact
- Compactness in Metric Space
- Every k Cell is Compact
- Generalized Cantor's Intersection Theorem in Metric Spaces
- Every Non-Empty Perfect Set in Euclidean Space is Uncountable
- Properties of Converging Real Sequences
- Convergence of Cauchy Sequences in Metric Spaces
- Convergence of Sequences in Metric Spaces
- Diameter of a Set in a Metric Space
- Connected Sets in Metric Spaces
- Continuity and Compactness in Metric Spaces
- Limit of Functions in Metric Spaces
- Maximum and Minimum Theorem in Metric Spaces
- Proof that Continuous Functions on Compact Metric Spaces are Uniformly Continuous
- Properties of Continuous Functions in Metric Spaces
- Properties of Limits of Functions in Metric Spaces
- The Composition of Continuous Functions in Metric Spaces Preserves Continuity
- The Importance of Compactness Conditions in Metric Spaces
- The inverse of a continuous bijection on a compact metric space is continuous.
- 거리공간에서 연속함수일 동치 조건
- Damped Harmonic Oscillation
- Principle of the Compton Camera
- What is Tomography?
- Forced Harmonic Vibration and Resonant Frequency
- Hyperbolic Functions' Identities
- Proof of the Addition Formulas for Hyperbolic Functions
- Curl of Vector Functions in 3D Cartesian Coordinates
- Double and Half Angle Formulas for Hyperbolic Functions
- Hyperbolic Functions
- Sum and Difference Formulas and Multiplication Formulas for Hyperbolic Functions
- Differential Volume in Cylindrical Coordinates
- Infinitesimal Area in Polar Coordinates, Infinitesimal Volume in Cylindrical Coordinates
- How to Initialize the Workspace and Remove All Variables in MATLAB
- Multi-Resolution Analysis
- Multi-Resolution Analysis Scaling Equation
- Frequently Used Symbols and Abbreviations in Mathematics
- Physical Pendulum
- Coupled Oscillations
- Fluids and the Definition of Fluid Dynamics
- Multiple Spring Oscillation
- Gradient of Scalar Function in Cartesian Coordinate System
- Why Functional is Named Functional
- The History of the Delta Function and Why Dirac Used the Delta Function
- Convergence in the Space of Test Functions
- Distributions, Generalized Functions
- Locally Integrable Function
- Proving that All Locally Integrable Functions Can Be Extended to Distributions
- Test Functions and Test Function Space
- Translation of Distributions
- The Dirac Delta Function Rigorously Defined through Distributions
- Characteristic Function, Indicator Function
- Dilation of Distributions
- Fourier Transform of the Dirac Delta Function
- Proof that the Dirac Delta Function is Not a Regularized Distribution
- Differentiation of Distributions
- Divergence of Vector Function in Cartesian Cooridenates System
- Reasons Not to Use r, Theta as Variables in Cylindrical Coordinates
- Total Differentiation, Exact Differentiation
- Convergence of Distributions
- Convergence of Sequences in Normed Spaces
- Weak Convergence in Hilbert Spaces
- Curvilinear Coordinates in Three-Dimensional Space
- Derivative Approximation
- Differentiation of the Product of Distributions
- Multiplication of a Distribution with a Smooth Function
- Scaling Factors of Curvilinear Coordinates
- Schwartz Space and Schwartz Functions
- Curved Coordinate Systems: Coordinate Transformations and Jacobians
- What is Overfitting and Regularization in Machine Learning?
- Commonly Used Datasets in Machine Learning
- Paper Review: Do We Need Zero Training Loss After Achieving Zero Training Error?
- Spectroscopy
- Spectrum and Fraunhofer Lines
- Zeeman Effect
- Unit Partition in Mathematics
- Spline, B-Spline in Analysis
- Gradient of a Scalar Function in Curvilinear Coordinates
- Divergence of Vector Functions in Curvilinear Coordinates
- Partial Differential Equations
- Fourier Series in Complex Notation
- Regulating Supersaturation
- If Differentiable, Then Continuous
- Differentiable Function Properties
- Definition and Relationship of Extremum in Analysis and Differential Coefficients
- The Chain Rule of Differentiation in Analysis
- Mean Value Theorem in Analysis
- Monotonic Functions, Increasing Functions, Decreasing Functions
- The Relationship between Derivatives and the Increasing/Decreasing of Functions
- Limits from the Left and the Right Strictly Defined in Analysis
- Classification of Discontinuities
- Mellin Transform Convolution
- Continuous Learning in Deep Learning
- What is Computer Vision
- Relations among Inner Product Spaces, Normed Spaces, and Metric Spaces
- Semi-Linear (Conjugate Linear) Functions
- Inner product spaces
- Inner Product Spaces and the Cauchy-Schwarz Inequality
- Properties of the Norm Associated with the Inner Product Defined in Inner Space
- Prove that Norm is a Continuous Mapping
- Perceptron Definition
- Integral Transformation
- Convolution's General Definition
- Sturm-Liouville Differential Equation
- Eigenvalues and Eigenfunctions in S-L Problems
- Orthogonality of Solutions to the Regular Sturm-Liouville Problem
- The Fundamental Theorem of Calculus in Analysis
- Integration by Parts
- Integration of Vector-Valued Functions
- Measuring Curves: A Guide to Length
- If the Derivative of a Curve is Continuous, the Curve Can Be Measured
- Convolution Convergence Theorem
- Laplacian of a Scalar Function in the Three-Dimensional Cartesian Coordinate System
- Laplacian of a Scalar Function in Curvilinear Coordinates
- Convolution Norm Convergence Theorem
- Inverse Fourier Transform Theorem for Smooth Functions
- Operations and Notation Table of Vectors and Matrices
- Linear Models for Regression in Machine Learning
- Convergence of Distributions to the Dirac Delta Distribution
- Differentiability of Distributions is Continuous with Respect to Weak Convergence
- Necessary and Sufficient Condition for Uniform Convergence
- Convolution of Distributions, Distributions as Functions Defined on Real Numbers
- Distributional Convolution Lemma
- Proof That the Space of Test Functions is a Proper Subset of the Schwartz Space
- Plancherel's Theorem
- Several Definitions and Notations of Fourier Transform
- Properties of Convolution
- Properties of B-Splines
- Orthogonality, Orthogonal Sets, and Orthonormal Sets in Inner Product Spaces
- Fourier Transform of B-Splines
- Explicit Formulas of B-splines
- Regularity of B-Splines
- Central B-spline
- B-spline Scaling Equation
- Properties of Zero in Inner Product Spaces
- Norm Space에서의 Infinite Series Span Total Sequence
- Convex Sets in Vector Spaces
- Properties of Adjoint Operators
- Derivatives of Vectors and Matrices
- Inner Product is a Continuous Mapping
- Generalized Fourier Coefficients and Fourier Series in Hilbert Spaces
- Solution of Differential Equations Using Series Solutions
- Radon Transformation
- Product Rule Involving the Del Operator
- Matrix Definitions
- code summary
- How to Print and Save a 2D Array as a Heatmap Image in MATLAB
- Matrix Operations: Scalar Multiplication, Addition, and Multiplication
- Square Matrix
- Diagonal Matrix
- Identity Matrix, Unit Matrix
- Transpose Matrix
- Inverse Matrix, Reversible Matrix
- Conditions for a Matrix Being Invertible
- Symmetric Matrices, Skew-Symmetric Matrices
- Conjugate Transpose Matrix
- Inner Product with Vector
- Orthogonal Matrix
- What is Reinforcement Learning in Machine Learning
- Properties of Orthogonal Matrices
- 대각합
- Equivalence Conditions for Orthogonal Matrices
- Hermitian Matrix
- Unitary Matrix
- Equivalents Codes in Julia, MATLAB, Python, and R
- Linear Function
- Simultaneous Linear Equations
- Area of a Surface in Differential Geometry
- Augmented Matrices and Elementary Row Operations
- Classification of Surfaces of Revolution According to Gaussian Curvature
- Properties of Determinants
- Properties of Rotational Surfaces
- Linear Forms
- Partial Derivatives
- Referencing Equations in TeX (Hyperlink)
- Various Function Spaces
- Matrix Similarity
- Matrix Transformation
- Basis of Vector Space
- Gaussian-Jordan Elimination
- Translation of Vector Field Translation:
- Wirtinger Derivatives of Complex Functions
- Simultaneous Homogeneous Linear Equations
- Definition of Heat in Physics
- Various Meanings of the Fourier Transform
- Basic Matrix
- Perceptron Convergence Theorem
- Projection Theorem in Linear Algebra
- Inverse Matrices and Systems of Linear Equations
- Necessary and Sufficient Conditions for a Basis in Finite-Dimensional Vector Spaces
- Derivatives of Logarithmic Functions
- Linear Transformation
- What is Inner Product in Real Vector Spaces?
- Composition of Functions
- Basis of Row Space, Column Space, and Null Space
- Relationship Between Orthogonality and Linear Independence
- Differentiation of Exponential Functions
- Basis Addition/Subtraction Theorem
- Orthogonal Basis and Its Coordinates
- The Definition of Euler's Constant, the Natural Number e
- Weighted Lp Spaces
- Convolution of Multivariable Functions
- Integration of Multivariable Functions
- Monotone Sequence and Monotone Convergence Theorem
- Limits of Exponential and Logarithmic Functions
- Sampling Theorem
- Properties of Divergent Real Sequences
- Heisenberg Uncertainty Principle
- Proof of Heisenberg's Uncertainty Principle
- Picard's Theorem
- Convergence in Schwartz Spaces
- Discrete Fourier Transform Properties
- Expectation Value of Speed and Velocity of Gas Molecules
- Derivation of the Ideal Gas Equation through Kinetic Theory of Gases
- Solid Angle of a Sphere
- Dalton's Law
- Gas Flux
- Discrete Fourier Inversion
- Distributional Convolution Convergence Theorem
- The Basis of the Domain Generates the Image of the Linear Transformation
- Kernel and Range of Linear Transformation
- Rank, Nullity, and Dimension Theorems of Linear Transformations
- Necessary and Sufficient Conditions for Linear Transformations to be Surjective and Injective
- Composition of Linear Transformations
- Norm of Linear Transformations
- Properties of the Space of Invertible Linear Transformations
- Back Propagation Algorithm
- Uniform Convergence and Continuity of Function Sequences
- The Matrix Representation of a Linear Transformation
- Every n-dimensional Real Vector Space is Isomorphic to R^n
- Transport Equation
- Laplacian of a Scalar Field
- Partial Derivatives: Derivatives of Multivariable Vector Functions
- Absolute Value Function
- Rotation Transformation, Rotation Matrix
- Smoothness of Boundaries
- Fundamental Solution of the Laplace Equation
- Dirichlet Boundary Conditions
- How to Use Radon Transform in Julia, MATLAB, Python
- How to Use the Shepp-Logan Phantom in Julia, MATLAB, and Python
- What is a State Function in Thermophysics?
- How to Solve the "Fail to create pixmap with TK_GetPixmap in TKImgPhotoInstanceSetSize" Error in Python matplotlib
- Wave Equation
- Cauchy Problem, Initial Value Problem
- Definition of Improper Integrals
- PyTorch RuntimeError: "grad can be implicitly created only for scalar outputs" Solution
- Various Formulas of Vector Integration Involving the Del Operator
- Definition and Properties of Vector Areas
- Inverse Holder's Inequality: A Sufficient Condition for Lp Functions
- L infinity space
- Embedding Theorems in Lp Spaces
- Derivatives of 3D Scalar/Vector Functions
- Conformal Mapping
- How to Implement MLP in PyTorch
- Initializing Weights in PyTorch
- Generalization of Gaussian Integrals
- Formulas Related to Factorials
- Integrals of Trigonometric Functions Table
- Creating and Using Custom Datasets from Numpy Arrays in PyTorch
- Definition of Directional Derivative
- Smooth Functions
- Length of a Curve
- Scalar Field Line Integral
- Line Integrals of Vector Fields
- Saving and Loading Weights, Models, and Optimizers in PyTorch
- Increasing Line Spacing in LaTeX
- Differentiable Manifolds
- Dense Subsets and Closures
- Properties of Bounded Linear Operators
- Extensions of Bounded Linear Operators
- Coordinates of a Three-Dimensional Unit Sphere
- Differential Equations: Fundamental Solutions and Green's Functions
- Helmholtz Equation
- Expansion and Contraction of a Function
- Including Functions
- What is an Inverse Problem?
- Moduli Spaces
- What is Scattering Theory?
- Vector Space
- Sommerfeld Radiation Condition
- Differentiable Functions from a Differentiable Manifold to a Differentiable Manifold
- Scattering Problem of Sound Waves
- Tangent Vector on Differentiable Manifold
- Chain Rule for Multivariable Vector Functions
- How to Find Derivatives in Julia
- Differentiation of Functions Defined on Differentiable Manifolds
- Alternating Function
- Jacobian of Composite Functions
- Inverse Function Theorem in Analysis
- Creating Random Permutations and Shuffling Tensor Order in PyTorch
- Differentiable Homomorphism
- Tangent Vectors on Simple Surfaces
- Points to Note When Slicing in Python
- Eigen Decomposition
- How to Read and Write Greek Characters and Their Meaning in Mathematics and Science
- Definition of Surfaces in Differential Geometry
- Sign function
- First Basic Forms, Riemannian Metrics
- Potential, A General Definition of Potential Energy
- Specific Examples of Calculations Using the Riemann Metric
- Boundary Value Problems in Partial Differential Equations
- Parametric Curves on a Simple Surface
- Neumann Boundary Conditions
- Gauss Curvature and Geodesic Curvature
- Robin Boundary Conditions
- Second Fundamental Form in Differential Geometry
- Christoffel Symbols in Differential Geometry
- Paper Review: Neural Ordinary Differential Equations
- Gauss's Theorem in Differential Geometry
- Silver-Müller Radiation Condition
- Definition of Intrinsic in Differntial Geometry
- Taylor's Theorem for Multivariable Functions
- The Christoffel Symbols are Intrinsic
- The Tangent Space on an n-Dimensional Differentiable Manifold is an n-Dimensional Vector Space
- Geodesic Curvature is Intrinsic
- Identity Function
- Definition of a Straight Line (Geodesic) in Differential Geometry
- Embedding in Topology
- Rotational Surfaces in Differential Geometry
- Emersion and Embedding on a Differential Manifold
- Geodesic on a Surface of Revolution
- How to Define Artificial Neural Network Layers with Lists and Loops in PyTorch
- Definition of Parallel Vector Field along a Curve on Surface
- Differences among \mathrm, \text, and \operatorname in LaTeX
- Uniqueness Theorem of Geodesics
- How to Deep Copy Tensors in PyTorch
- If It's the Shortest Curve, It's a Geodesic
- Writing Multiple for Loops in One Line Using Python
- Directional Derivatives in Differential Geometry
- Solutions When Python npy Files Won't Open
- Geodesic Coordinate Transformation
- How to Obtain the Weight Values of a Model in PyTorch
- Properties of the Second Normal Form
- How to Set Plot Scale Range in Python matplotlib
- Definition of Normal Sections and Menelaus's Theorem
- How to Concatenate or Stack Tensors in PyTorch
- Bingarten Map
- How to Get Column and Row Labels of Data Frame in Python Pandas
- Bingarten Equation
- Using Machine Learning Datasets in Julia
- The Relationship between the Second Normal Form and the Vingarten Map
- The Relationship between the Fundamental Form and Coordinate Transformation
- Curvature of a Principal Curve
- Drawing Vertical and Horizontal Lines in Python matplotlib
- Euler's Theorem in Differential Geometry
- How to Remove Axes in Python matplotlib
- Properties of Vector Fields Parallel to a Curve
- How to Pad PyTorch Tensors
- Gaussian Curvature and Mean Curvature
- Definition and Relationship between the Gaussian Map and Gaussian Curvature
- Methods for Coloring Up to a Certain Value from Curves in Julia / Between Two Curves / Inside a Closed Curve
- Necessary and Sufficient Conditions for a Vector Field to be Parallel Along a Curve
- Handling the Dimensions and Sizes of PyTorch Tensors
- Riemann Curvature Tensor, Gauss Equation, and Codazzi-Mainardi Equation in Differential Geometry
- Performing Operations on Vectors of Different Sizes Component-wise in Julia
- Gauss's Great Theorem
- Handling Hidden Layers in Julia Flux
- Differentiable Functions Between Two Surfaces
- Implementing MLP in Julia Flux and Optimizing with Gradient Descent
- Differential Geometry: Isometric Mappings
- Overlaying Plots on Heatmaps in Julia
- Differentiable Geometry: Local Isometries
- How to load a npy file in Julia
- Fundamental Theorem of Curved Surfaces
- List of Available Commands in Julia Package Management Mode
- Positive Gaussian Curvature Surfaces of Revolution
- How to Perform One-Hot Encoding in Julia Flux
- Two Rotational Surfaces with Positive Curvature are Locally Isometric
- Implementing MLP in Julia Flux and Learning with MNIST
- The Rotational Surface with Zero Curvature
- How to Directly Define Multidimensional Arrays in Julia
- Differentiable Manifolds in Compact Surfaces
- Resolving 'TypeError: can't convert cuda:0 device type tensor to numpy. Use Tensor.cpu() to copy the tensor to host memory first.' with Lists in PyTorch
- Implementing MLP in Julia Flux to Approximate Nonlinear Functions
- Gaussian Curvature
- n-Dimensional Polar Coordinates
- Differentiable Surfaces and Boundaries of Regions in Differential Geometry
- n-Dimensional Radon Transform
- What is a Heatmap?
- Homotopy to Null in Differential Geometry
- What is a Phantom in Tomography?
- Simple Connected Region
- What is a Sinogram in Tomography?
- Gauss-Bonnet Theorem
- Principles of Photoacoustic Tomography
- How to Install TeX and Use It in VS Code
- Euler Characteristics in Geometry
- Writing Korean in LaTeX
- Radon Transform and Product Integration, Convolution
- Convolution Support
- Angular Momentum Operator in Quantum Mechanics
- Hierarchical Clustering
- Cotangent Space and First-Order Differential Forms
- What is a Dendrogram?
- Second-Order Differential Form
- How to Draw a Dendrogram in Julia
- kth Order Differential Forms
- How to Perform Hierarchical Clustering in Julia
- Operations on Differential Forms: Sum and Wedge Product
- Gradient Descent Learning of Linear Regression Models in Machine Learning
- Pull Back in Differential Geometry
- 머신러닝에서 선형회귀모델의 최소제곱법 학습
- Differential Forms of Type k
- Supersaturated and Undersaturated Systems
- Emulsions are locally embedded.
- Lagrange Multiplier Method
- Tangent Bundles on Differentiable Manifolds
- Fundamental Spaces of Matrices
- Vector Field on Differentiable Manifold
- Properties of Full Rank Matrices
- Lie Brackets of Vector Fields
- Adding a New Column to a DataFrame in Julia
- Linear Transformation Trace
- Riemann Metric and Riemann Manifolds
- Linear Transformations Between Finite-Dimensional Vector Spaces
- Isometries and Local Isometries on Riemann Manifolds
- Order Basis and Coordinate Vectors
- Vector Fields Along Curves on Differential Manifolds
- Linear Functional
- Affine Connection
- Linear Transformation Space
- Covariant Derivative of Vector Fields
- Inverse of Linear Transformations
- Parallel Vector Fields on Differential Manifolds
- Homomorphism
- Coexistence Compatible Connection
- Linear Transformation Spaces and Their Matrix Representation Spaces are Isomorphic
- Symmetry of Connection
- Transpose of Linear Transformations Defined by Dual Spaces
- Levi-Civita Connection, Riemannian Connection, Coefficients of Connection, Christoffel Symbols
- Dual Pair Spaces
- Geodesics on a Differentiable Manifold
- The Equivalence Condition When the Range of a Linear Transformation is Smaller than the Kernel
- How to Draw a Histogram in Excel
- Geodesic Flow
- Homogeneity of Geodesics
- Using Tuples for Indexing in Python
- What is a Commutator in Group Theory?
- Differentiable Curves and Minimization
- What is a commutator in field theory?
- Parameterized Surfaces
- Triangular Matrix
- Gauss's Lemma in Riemannian Geometry
- Nilpotent Matrix
- Minimizing Geodesics
- Binomial operation's Jacobi Identity
- Exponential Mapping and Normal Neighborhood
- Methods for Symbolic Computation in Julia
- Poincaré Metric
- Paper Review: Physics-Informed Neural Networks
- Differential Geometry of Curved Manifolds
- Sectional Curvature of Differential Manifolds
- Differences in Array Dimensions in Julia, Python (NumPy, PyTorch)
- Bianchi Identity
- Explicit Runge-Kutta Methods
- Symmetry of the Riemann Curvature Tensor
- Implicit Runge-Kutta Methods
- Coordinate Representation of the Riemann Curvature Tensor
- Expansion and Contraction of the Basis
- If the Sectional Curvature is the Same, the Riemannian Curvature is also the Same.
- Block Matrices
- Ricci Curvature of Differentiable Manifolds
- Matrices of Linear Transformations from a Basis of a Subspace to an Extended Basis
- Scalar Curvature of Differential Manifolds
- Matrix Spaces
- Left Multiplication Transformation (Matrix Transformation)
- Lie Groups
- Convolutional Layer
- Coordinate Transformation of Vectors
- The Relationship between Covariant Derivative and Riemann Curvature Tensor
- Basis Transformation (Coordinate Transformation) of Linear Transformations
- Tensors Defined on Differentiable Manifolds
- Differentiable Vector Fields on a Differentiable Manifold
- Eigenvalues and Eigenvectors of Finite-Dimensional Linear Transformations
- Sets of Differentiable Real-Valued Functions on a Differentiable Manifold
- 딥러닝에서 인공신경망(ANN), 심층신경망(DNN), 순방향신경망(FNN)의 뜻과 차이점
- Characteristics Polynomial of Linear Transformation
- Diagonalizable Linear Transformation
- Eigenvectors Corresponding to Distinct Eigenvalues are Linearly Independent
- Equations of Motion for a Variable Mass System
- Polynomial Vector Spaces
- Tsiolkovsky Rocket Equation
- The Characteristic Polynomial of a Diagonalizable Linear Transformation Is Factorizable
- Graph (Network) Visualization and Analysis Program Gephi
- Multiplicity of Eigenvalues of Linear Transformations
- Graph (Network) Analysis Package NetworkX in Python
- Eigen Spaces of Linear Transformations and Geometric Multiplicity
- Julia's Graph Analysis Package Graphs.jl
- The Union of Linearly Independent Sets from Different Eigenspaces is Linearly Independent
- Reading and Writing GEXF Files in NetworkX
- Invariant Subspaces of Vector Spaces
- Mass Dependent on Position: The Motion of Balls Connected by Chains
- The Relationship between Invariant Subspaces and Eigenvectors
- Position, Velocity, and Acceleration
- 대각화가능한 선형변환의 불변부분공간으로의 축소사상도 대각화가능하다
- Free Fall Motion
- Residual Classes and Quotient Spaces in Linear Algebra
- Ramp Function
- 몫공간의 기저와 차원
- What is ReLU in Machine Learning?
- Mapping to the Quotient Space
- Shannon Entropy in Classical Information Theory
- Linear Transformations on the Quotient Space
- Various Properties of Convex Functions
- Characteristics of the Mapping into the Quotient Space via Linear Transformations between Polynomial Relations
- Gaussian Curvature with Negative Values on Rotational Surfaces
- Transformations on the Quotient Space of Diagonalizable Linear Transformations are also Diagonalizable
- Sum and Difference Identities for Trigonometric Functions
- Sum of Subspaces in a Vector Space
- Hyperbolic Functions Composite Formula
- 직합의 성질
- The Coordinate Patch Mapping of a Torus in Three-Dimensional Space
- The Creation of Unions is Equal to the Sum of Creations
- How to Change Axis Style in Julia Plots `framestyle`e`
- Diagonalizability of Linear Transformations, Multiplicity of Eigenvalues, and the Relationship with Eigenspaces
- Spring 2025 Omakase: Your Name
- Power Series Linear Transformation
- Encoder and Decoder in Machine Learning
- The Eigenvalues of the Null Matrix are Only Zero
- Training/Validation/Test Sets in Machine Learning
- Fully Connected Layer (Linear Layer, Dense Layer)
- Orthogonal Triangular Matrices are Nilpotent
- Null Space of Power Maps
- Matrix
- Cyclic Subspaces of Vector Spaces
- Structural Medical Imaging and Functional Medical Imaging
- Cayley-Hamilton Theorem
- Types of Cross Sections in Medical Imaging
- Direct Sum of Invariant Subspaces and Its Characteristic Polynomial
- Zero Matrix
- Direct Sum of Matrices
- What is a Softplus Function?
- What is a Flag in Linear Algebra?
- 고전정보이론에서 정보량이란?
- How to Check the Device on which the PyTorch Model/Tensor is loaded
- Boolean Functions
- Conjunction, AND Gate
- Disjunction, OR Gate
- Negation, NOT Gate
- Exclusive Disjuction, XOR Gate
- NAND Gate
- NOR Gate
- What is a Functionally Complete Set?
- Replication Function
- Controlled NOT(CNOT) Gate
- Toffoli/CCNOT Gate
- Fredkin/CSWAP Gate
- Shadow and Injection
- Tensor Product of Vector Spaces
- Tensor Product of Product Vectors
- Universal Properties of Tensor Products
- Tensor Product of Linear Transformations
- Kronecker Product of Matrices
- Matrix Representation of Tensor Product
- Bit: Unit of Information Classical Computer
- Qubits: The Basic Unit of Information in Quantum Computers
- Quantum Gates and Quantum Circuits
- Hadamard Gate
- Topological Gates
- Pauli Gates
- Quantum CNOT Gate
- Exchange Gate
- Quantum Topology/CCNOT Gate
- Quantum Fredkin/CSWAP Gate
- Solvay-Kitaev Theorem
- Impossibility of Cloning Theorem
- 줄리아에서 U-net 구현하기
- Hadamard Product of Matrices
- Encoding and Decoding in Information Theory
- What is One-Hot Encoding in Machine Learning?
- How to Change Basic Data Types in Julia
- How to Use Fast Fourier Transform (FFT) in Julia
- Automatic differentiation
- Various Deep Learning Frameworks of Julia
- MNIST Database
- Iris Dataset
- Total Variation in Differential Geometry
- Multilayer Perceptron (MLP), Fully Connected Neural Network (FCNN)
- 딥러닝에서 풀링층이란?
- Convolutional Neural Network (CNN)
- General Linear Group
- Unitary Group
- Special Linear Group
- Orthogonal Group
- Special Unitary Group
- Matrix Representation of the Sum and Scalar Multiplication of Linear Transformations
- Drawing Diagrams of Universal Properties with TikZ
- CSS color name tags
- Writing TeX Code in Gmail (Email)
- Sampling Randomly in Julia
- Sampling Randomly from a Given Distribution in Julia
- What is Joint Entropy in Classical Information Theory?
- What is Conditional Entropy in Classical Information Theory?
- Relative Entropy (Kullback-Leibler Divergence) in Classical Information Theory
- Mutual Information
- Inequalities for the Logarithmic Function 1-1/x < log x < x-1
- Elliptic Partial Differential Equations
- Parabolic Partial Differential Equation
- Coordinate Systems and Coordinates in Physics
- Physics에서 Coordinate Transformation
- Summary of Measure Theory and Probability Theory
- Parabola
- Hyperbola
- Definition of Circle
- Quadratic Curve
- 딥러닝에서 레이어란?
- Sampling Randomly from a Given Distribution in PyTorch
- Solving 'RuntimeError: Boolean value of Tensor with more than one value is ambiguous' Error in PyTorch
- Functions for Tensor Sorting in PyTorch
- Flux-PyTorch-TensorFlow Cheat Sheet
- Specifying the Color of Axes, Axis Names, Ticks, and Tick Values in Julia Plots
- Meaning of Weak and Strong in Mathematics
- The Fast Fourier Transform Algorithm
- Radial Functions
- Finite Difference Method
- Numerical Solutions of Heat Equations: Finite Difference Method
- Egorov's Theorem
- Lusin's Theorem
- Broadcasting of Multivariable Functions in Julia
- How to Create a Meshgrid in Julia
- How to Neatly Print without Axes, Scales, etc. in Julia
- Solutions to 'RuntimeError: Parent directory does not exist' Error When Saving Models in PyTorch
- Python Matplotlib Basics & Custom Line Styles
- What is the Dot Product in Three-Dimensional Space?
- Converting Strings like 'False', 'True' to Bool Type in Python
- Existence of a Sequence of Simple Functions Converging to a Measurable Function
- How to Replace Existing Output with New Output in Python and Display Progress
- How to Insert GIF Animations into a PDF File Using LaTeX
- Drawing Subplots in LaTeX
- Entropy of Normal Distribution
- Hyperbolic Partial Differential Equations
- Quadratic Form
- Bilinear Forms and Hermitian Forms
- Conditional Expectation Minimizes the Sum of Squared Deviations
- Monte Carlo Integration
- 중요도 샘플링
- Bounded Function
- Rejection Sampling
- Square Root Expansion Formula
- Monte Carlo Method
- What is Data Augmentation?
- 자기장의 기호로 B를 사용하는 이유
- How to Insert Code Blocks in LaTeX Documents
- How to Prevent the Date from Appearing when Creating a Title with maketitle in LaTeX
- Online Learning vs. Batch Learning in Machine Learning
- Modular Arithmetic in PyTorch
- Momentum Method in Gradient Descent
- Adaptive Learning Rates: AdaGrad, RMSProp, Adam
- Secant Method: Newton's Method
- Taylor's Theorem Rest Term
- Second Order Necessary/Sufficient Conditions for the Extreme Values of Multivariable Functions
- First-Order Necessary Conditions for Extrema of Multivariable Functions
- Curl of a Vector Function in Curvilinear Coordinates
- Notation for Vectors Entering and Exiting the Plane
- Origin of the arc notation for inverse trigonometric functions
- Origin of Trigonometric Functions
- Homogeneous Function
- Lipschitz Continuity
- Absolutely Continuous Real Function
- 아날로그신호와 디지털신호의 정의
- 신호의 에너지와 평균 전력
- 신호의 자기상관함수
- L² Space
- EasyDict: A Python Package for Convenient Dictionary Use
- 신호의 교차상관함수
- How to Draw Subplots in Python matplotlib
- Drawing Subplots with Complex Layouts in Python matplotlib
- Drawing Subplots at Desired Arbitrary Locations or Overlapping in Python matplotlib
- The Hyperfunctional Derivative of Brownian Motion is White Noise
- How to Draw a Box Plot in Julia
- How to Draw a Box Plot in Python matplotlib
- How to Fix the Random Seed in Julia
- How to Adjust Camera Position for 3D Plots in Julia
- What is a Box Plot?
- Drawing Arrows in Graphics with Julia
- How to Adjust the Size and Resolution of an Image in Julia
- How to Pass Arguments When Executing a Python File
- Python's Command Line Parsing Module argparse
- How to Define and Train MLP with the Sequence Model and Functional API in TensorFlow and Keras
- Dimensionality Reduction in Data Science
- Integrability of 1/x^p
- Sampling with Replacement and without Replacement in Python
- What are Frequency and Oscillation?
- The limit of sin(x)/x
- The Limit of 1-cos(x)/x
- Differentiation of Trigonometric Functions
- Integration of 1/(1+x^2)
- How to k-means cluster in Julia
- Geodesic Coordinate Patch Mapping and Christoffel Symbols
- Geodesic Coordinate Mapping and Gaussian Curvature
- Definition of a Line
- Definition of Coordinate Plane
- Polar Coordinate System
- Coordinate Space, Cartesian Coordinate System
- Coordinates of a Möbius Strip in Three-Dimensional Space
- Sperical Coordinate System
- 파이썬에서 PC 이름 얻는 방법
- 파이썬에서 운영체제 확인하는 방법
- Biharmonic Functions
- Polyharmonic Functions
- Differentiation Operators and Symbols
- What is Degrees of Freedom in Physics?
- What is a Constraint in Physics?
- What are Generalized Coordinates in Physics?
- Latent Variable and Latent Space
- Proximal Operator
- Proximal Minimization Algorithm
- Alternating Optimization
- Subgradient
- Subgradient Method
- Proximal Gradient Method
- Proximal Alternating Linearized Minimization Algorithm (PALM)
- Decorating Text Output with Built-in Functions in Julia
- How to Use Colors in Julia Plots
- Package for Color Processing in Julia
- How to Use RGB Color Codes in Julia
- Specifying the Color of Graph Elements in Julia Plots
- How to Specify Graph Colors for Each Subplot in Julia Plots
- Specifying Background Color in Julia Plots
- Decorating the Background Grid in Julia Plots
- List of Plot Properties in Julia Plots
- How to Plot Two Data Axes of Different Scales in Julia Plots
- How to Use Palettes in Julia Plots
- How to Use Color Gradients in Julia Plots
- Compact Action Spaces
- Necessary and Sufficient Conditions for a Subset of a Normed Space to be Bounded
- What is Sequential Compactness in Topological Spaces?
- Equivalence of Various Compactnesses in Metric Spaces
- In a Metric Space, Compact Implies Closed and Bounded
- The Spectrum and Decomposition Set of Matrices
- Compact Operator Equivalence Conditions
- Integration Operator
- Compact Integral Operators
- Uniform Continuity in Metric Spaces
- Using AdaBelief Optimizer in PyTorch
- What is a Special Function?
- Rodrigues' Formula for Multiple Polynomials
- Solution of Wave Equation with Zero Initial Condition
- What is Skip Connection in Artificial Neural Networks?
- What are Weights in Machine Learning?
- Difference Between torch.nn and torch.nn.functional in PyTorch
- Creating a New Command in LaTeX That Acts Like a Function Using the begin...end Structure
- 프레드홀 적분 방정식
- Customizing Chapter and Section Numbering in LaTeX
- Laguerre Polynomials
- Rodrigues' Formula for Hermite Polynomial
- Quantum Mechanics: The Position Operator
- What is a Ladder Operator in Quantum Mechanics?
- Hamiltonian Operator
- What is Flux in Physics?
- Gradient, Divergence and Curl in Curvilinear Coordinates
- Curvilinear Coordinate System and the Del Operator
- Cylindrical Coordinates에서의 Del 연산자
- How to Change the Default Save Location for Excel, PowerPoint, and Word
- Cylindrical Coordinate System
- Variables Separable Function
- Paper Review: DeepONet
- 양자 조화 진동자의 사다리 연산자
- How to render text, formulas, pictures, and more transparently in LaTeX
- How to Use Quotation Marks in LaTeX
- 줄리아 미분방정식 패키지 DiffetentialEquations 튜토리얼
- Derivatives of Inverse Trigonometric Functions
- Trigonometric Identities
- Derivatives of Hyperbolic Functions
- Inverse Hyperbolic Functions
- Derivative of the Inverse Hyperbolic Functions
- How to Set Training and Testing Modes for Neural Networks in Julia Flux
- Fall 2024 Omakase: Dual Numbers and Automatic Differentiation
- Comparing NaN in Julia
- Differences Between Vectors and Tuples in Julia
- Methods for Calculating Arrays Column-wise in Julia
- Comprehensions in Julia
- How to Define an Array by Specifying Its Type in Julia
- NamedArrays.jl Package in Julia
- 파이토치로 PINN 논문 구현하기
- How to Use Automatic Differentiation in PyTorch
- Comprehensive Guide to File I/O in Julia
- Implementing DeepONet Paper Step by Step (PyTorch)
- Parametric Equation
- How to Pass Multiple Keyword Arguments at Once Using a Dictionary in Julia
- How to Order Alphabetically with enumerate in LaTeX
- How to Use Strikethrough in LaTeX
- ulem: Packages Related to Underlining in LaTeX
- How to Remove Indentation in the Entire Document in LaTeX
- Geometric Series
- Limits of Geometric Sequence
- Divergence Test
- Harmonic Series
- Alternating Harmonic Series
- Alternating Series Test
- Alternating Series
- Limits of Subsequences and Convergence of Sequences
- Subsequence
- Integral Test
- Continued Fraction
- Recurrence Relation
- Applications of Taylor Series
- How to Read Text (txt) Files as Strings in Julia
- Taylor Series and Maclaurin Series
- How to Edit and Save Text (txt) Files in Julia
- Special Angles of Trigonometric Functions
- Uniform Convergence and Differentiation
- Uniform Convergence and Integrability of Function Series
- Notation and Naming Conventions of Hyperbolic Functions
- Radius of Convergence of Power Series
- Root Test
- Ratio Test
- Absolute Convergence and Conditional Convergence
- Comparison Test
- Limit Comparison Test
- p-Series and the p-Series Test
- Convergence Properties of Series
- Differentiation of Power Series
- Convergence of Power Series
- The limit of n^(1/n)
- Integration of Power Series
- Absorbing Boundary Conditions
- Numerical Solution of the Wave Equation: Finite Difference Method (FDM)
- Notation of Expectation Values in Physics
- Numerical Solution of the Wave Equation: K-Space Method
- Difference between Abbreviation and Acronym
- Do not use articles for Figure, Table, and Appendice in English papers
- Additive Group of Integer Modulo n
- How to Define Neural Networks Using Functional API in Julia Flux
- How to Choose Coefficients in the Runge-Kutta Method
- How to Sort Bibliographies Chronologically or Reverse Chronologically in BibTeX
- How to not omit author names in BibTeX
- Dual Numbers
- Automatic Differentiation and Dual Numbers
- Differentiable Real Functions Defined on a Dual Numbers
- Implementing Automatic Differentiation Forward Mode Using Dual Numbers in Julia
- Azimuth and Direction Cosines
- How to Merge Multiple TeX Files into One
- How to Highlight (with a Marker) Text and Equations in LaTeX
- The Relationship between the Dot Product of Two Vectors and the Angle Between Them
- Derivative of a Vector-Valued Function
- Limits and Continuity of Vector-Valued Functions
- A Constant-Magnitude Vector-Valued Function is Orthogonal to Its Derivative
- Angle Between Two Vectors in an n-Dimensional Euclidean Space
- How to Automatically Fill Blank Spaces in LaTeX
- IguanaTeX: How to Input TeX in PowerPoint
- How to Set a LaTeX Document Horizontally
- Customizing the Order of References in BibLaTeX
- Hiding Specific Fields in Biblatex
- Free Online Resources for Artificial Intelligence, Machine Learning, and Deep Learning
- Discrete Fourier Transform Matrix
- Implementing a Discrete Fourier Transform Matrix in Julia
- Jacobi's Formula
- How to Add Hyperlinks to Bibliography Numbers in LaTeX
- How to Make References Appear in Numerical Order When Using \cite{} in LaTeX
- Website for Easily Creating LaTeX Tables
- How to Use Square Brackets in LaTeX Enumerate
- Difference between Noise and Artifacts in Images (Signals, Data)
- How to Assign Numbers Arbitrarily in LaTeX Enumerate
- Autoencoder
- Subgroup Test
- Antiderivatives and Indefinite Integrals
- Differentiation of Constant Functions
- Zero-function
- Approximation, Best Approximation
- Directed Set
- The Net, Moore-Smith Sequence
- What is a Differential Equation Solver?
- Numerical solution of ordinary differential equations with external forces (Julia DifferentialEquations Package)
- Avoid Starting Sentences with And, But, and So When Writing English Papers
- How to Check the PC Name in Python
- How to Interpolate Strings in Python
- Inner Product of Dual Space
- Parity Plot
- Optimizer
- Birthday Problem: Probability of Sharing the Same Birthday
- Graph Embedding, Node Embedding, Edge Embedding
- Graph Laplacian
- gravitational acceleration
- A Comprehensive Compilation of Mathematical and Scientific Terms (Korean, English, Japanese)
- Generative Model
- Paper Review: Denoising Diffusion Probabilistic Models (DDPM)
- Mixture Distributions
- Maximum a Posteriori Estimation for Linear Regression Model in Machine Learning
- Bayesian Inference in Machine Learning
- Maximum Likelihood Estimation for Linear Regression Model in Machine Learning
- Laplace Distribution
- Mean and Variance of Laplace Distribution
- Moment Generating Function of the Laplace Distribution
- Maximum Likelihood Estimator of the Laplace Distribution
- Summary of Python Special Methods
- How to Mute VSCode (Not Audio Cues)
- Differences between Python special methods __str__ and __repr__
- Generalizing Differentiation: Gradient Matrices and Matrix Calculus
- Bernoulli Distribution
- Categorical Distribution
- Moment Generating Function of Bernoulli Distribution
- The Mean and Variance of the Bernoulli Distribution
- Differentiation of the Absolute Value Function
- How to Use Hyperlinks in LaTeX
- List of Available Fonts in LaTeX
- Free Online Available R Textbook (Original Version)
- Professor Choi Byung-sun's Freely Distributed Textbooks on Mathematics, Statistics, and Economics
- How to Automatically Number Figures in LaTeX Beamer
- How to Use Colors in LaTeX: The xcolor Package
- How to Define and Use Your Colors in LaTeX
- How to Use Dice Images in LaTeX
- Energy-Based Model
- Probability Vector
- Matrix Calculus of Quadratic and Bilinear Forms
- Formula for Matrix Power Form
- Outer Product of Two Vectors
- Matrix Calculus of Trace
- Inner Product of Matrices (Frobenius Inner Product)
- Total Differential of Function of a Matrix
- Matrix Differential Dissection
- Trace Trick
- Matrix Differentiation Table for Scalar Functions
- Paper Review: Score Matching
- Geometric Mean
- Radian
- Inverse Function
- How to input physical units in LaTeX
- Relative Entropy (Kullback-Leibler Divergence) between Two Normal Distributions
- Kronecker Delta through Matrices
- Equivariant Map of Group Action
- Representation of Groups
- Differentiation of Polynomial Functions
- Topological Group
- Diagonalization of Matrices
- Orthogonal Similarity and Orthogonal Diagonalization of Matrices
- Comprehensive Summary of Julia String Syntax and Functions
- Rectangle
- Square
- Parallelogram
- Rhombus
- Parallelepiped
- Trapezoid
- Volume Formula of a Parallelepiped
- Summary of PBS Job Commands
- Angstrom
- Meter
- How to Browse Files and Paths in Julia
- How to View Folder Tree Structures in Julia
- Crystal Structure
- Comprehension Syntax in Python
- How to Read and Write JSON Files in Julia (JSON.jl)
- Reading and Writing JSON Files in Julia (JSON3.jl)
- Riesz Isomorphism
- Functions for Calculating Cumulative Sum and Product of Vectors in Julia
- Transposed Convolution
- Solution to TaskFailedException; nested task error when opening CSV files in Julia
- How to Check Version and System Information in Julia REPL
- Young Operator
- Self-adjoint Operator
- How to Use Loops in Linux Shell Scripts
- Operator Norm
- How to Check the Metadata of a File in Julia
- How to Check the Last Modified Time of a File in Julia