Logbook

5 Sep 2025

• Updates: Nelson’s Counterexample, Nelson’s Commutator Theorem.

4 Sep 2025

• Updates: Universal Enveloping Algebra, Lie Algebra Representations.

3 Sep 2025

• New: Central Extensions of Lie Algebras, Unitary Groups.
• Updates: Lie Algebra Representations, Hopf Algebra.

2 Sep 2025

• Updates: Products of Objects, Non-Commutative Rings, Hopf Algebra, Free Groups and Relations.

1 Sep 2025

• Updates: Real Analysis, Number Systems.

31 Aug 2025

• Updates: Multiple View Geometry, Groups and Categories, Fields and Field Extensions.

30 Aug 2025

• Updates: Nelson’s Counterexample.

25 Aug 2025

• Updates: The Galois Correspondence Theorem, Deck Transformations and Normal Covers, Hash Table.

24 Aug 2025

• Updates: Shannon Coding.

21 Aug 2025

• Updates: Covering Spaces, Covering Space Actions. The Galois Correspondence Theorem.

20 Aug 2025

• Updates: Covering Spaces, Fundamental Theorem of Covering Spaces, The Galois Correspondence Theorem.

11 Aug 2025

• Updates: Covering Spaces, The Fundamental Group, Cauchy Integral Formula for Jordan Domains, Normed and Banach Algebras, C*-Algebras, CW Complexes.

8 Aug 2025

• Updates: Cauchy Integral Formula for Jordan Domains.

6 Aug 2025

• Updates: van Kampen’s Theorem, The Fundamental Group.

4 Aug 2025

• Updates: Stone’s Theorem, Symmetric and Self-Adjoint Operators, The Einstein Field Equation, Time Independent Schrödinger’s Equation, The Spacetime.
• New page: van Kampen’s Theorem.

3 Aug 2025

• Updates: The Fundamental Group.

2 Aug 2025

• Updates: The Spectral Theorem of Unbounded Operators, Quantum State Space, Quantum Mechanics.

1 Aug 2025

• Updates: Unbounded Operators, Garside Normal Form.
• New page: Stone’s Theorem.

29 July 2025

• Updates: Adjoints and Riesz Representation Theorem, The Argument Principle.
• New page: The Exponential Map.

28 July 2025

• Updates: The Fundamental Group, Connectedness and Paths, Fields Construction, Finite Fields,Fields Construction.

27 July 2025

• Updates: Connectedness and Paths, Normed and Banach Algebras, The Homotopy Invariance Theorem.

25 July 2025

• New page: Normed and Banach Algebras.
• Updates: Diffeomorphisms of the Circle.

23 July 2025

• Updates: The Fundamental Group, Covering Spaces, Diffeomorphisms of the Circle.

22 July 2025

• Updates: The Fundamental Group, Homotopy Types

17 Jun 2025

• Updates: The Schwarzschild Solution, Toric Symplectic Manifold, Duality.

16 Jun 2025

• Updates: Duality, Functoriality and Naturality.

10 Jun 2025

• Updates: Construction of Sets, Set Theory, Orthogonality and Bounded Linear Maps, Hilbert Spaces.

4 Jun 2025

• Updates: Duistermaat-Heckman Theorems.

3 Jun 2025

• Updates: Cubic and Quartic Equations, Toric Symplectic Manifold.

15 Apr 2025

• Updates: Factorization.

13 Apr 2025

• Updates: Differential Forms, Galois Theory.

11 Apr 2025

• Updates: Symmetric Group.

10 Apr 2025

• Updates: Ring, Field and Integral Domain.

8 Apr 2025

• Updates: Vector Bundles, Lie Groups, Differential Geometry, Tangent Vectors and Spaces, Principle Ideal Domains, Relations and Maximal Ideals.

26 Mar 2025

• Updates: Lebesgue Integration, Lebesgue Measurability.

23 Mar 2025

• Updates: Image Filtering, The Space of Integrable Functions, Fourier Transform.

22 Mar 2025

• New page: Lebesgue Integration, The Space of Integrable Functions.
• Updates: Measure Theory.

21 Mar 2025

• Updates: Vector Fields and Lie Algebra.

18 Mar 2025

• Updates: Phase Space & Chaos.

8 Mar 2025

• Updates: Manifolds and Atlases, Fourier Series, Variational Calculus.
• New page: Lagrangian Dynamical System.

7 Mar 2025

• Updates: Lebesgue Measurability, The Exterior Measure, Limit Points and Closure.

5 Mar 2025

• Updates: Variational Calculus, Tangent Vectors and Spaces, Principle of Least Action.

2 Mar 2025

• Updates: Domain Adaptation, Waves in Fluids, Fields and Field Extensions.

1 Mar 2025

• Updates: Factorization.

21 Feb 2025

• Updates: The Exterior Measure.

20 Feb 2025

• Updates: Division and Prime, Polynomial Rings.

17 Feb 2025

• New pages: Symplectic Geometry, Newton’s Laws, The Principles of Relativity and Determinacy.
• Updates: Newtonian Mechanics, Group Actions.

16 Feb 2025

• New page: Factorization.
• Updates: Complex Numbers, Ring, Field and Integral Domain, Homomorphisms and Ideals.

9 Feb 2025

• New page: Fractal Geometry, The Exterior Measure.
• Updates: Singular Knots and Vassiliev Invariants, Equinumerous and Countability.

8 Feb 2025

• New pages:Singular Knots and Vassiliev Invariants.
• Updates: Tangles and Categories of Tangles, Grading and Filtration.

2 Feb 2025

• Updates: Symmetric Group, Universal Quantum Invariant, Grading and Filtration.

1 Feb 2025

• Updates: Morphisms.

20 Jan 2025

• Updates: Galois Theory, Grading and Filtration, Domain Adaptation, Convex Sets, Convex Functions.

19 Jan 2025

• Updates: The Alexander and Markov Theorems, Knot and Link Invariants, Grading and Filtration.

14 Jan 2025

• Updates: Free Groups and Relations, Quotients, Homomorphisms, Normal Subgroup & Conjugation.

13 Jan 2025

• Updates: Knots and Links, Monoidal Categories.

11 Jan 2025

• Updates: add definition of an algebra in Ring, Field and Integral Domain.

10 Jan 2025

• Updates: Braids and Braids Group.

24 Nov 2024

• New page: Braids and Braids Group, The Alexander and Markov Theorems.
• Updates: Free Groups and Relations.

13 Nov 2024

• New page: Differentiable Curves, Regular Surfaces.
• Updates: Tangent Vectors and Spaces.

2 Nov 2024

• Updates: Cauchy Integral Formula for Jordan Domains, Conformal Transformations of The Disk, Relations and Functions.

1 Nov 2024

• Updates: Cauchy Integral Formula for Disks, Cauchy Integral Formula for Jordan Domains, The Homotopy Invariance Theorem, Complex Analysis.

31 Oct 2024

• Updates: Möbius Transformations, Integration and Primitives.

30 Oct 2024

• Updates: Möbius Transformations.

29 Oct 2024

• Updates: Complex Numbers, Homomorphisms, Normal Subgroup & Conjugation, Quantum Waves, The Riemann Sphere.

20 Oct 2024

• Updates: Divergence and Bregman Geometry, Convex Functions.

19 Oct 2024

• Updates: The Open Mapping Theorem, The Argument Principle, Holomorphic and Conformal Maps.

17 Oct 2024

• Updates: Conjugate Connections, Divergence and Bregman Geometry, Fisher Information Metric.

15 Oct 2024

• Updates: Residue, Cauchy Integral Formula for Jordan Domains, Laurent Series, Integration and Primitives, Isolated Singularities, Divergence and Bregman Geometry.

14 Oct 2024

• New page: The Argument Principle, Residue.
• Updates: The Homotopy Invariance Theorem, Information Content and Entropy, Laurent Series.

3 Oct 2024

• Updates: Fisher Information Metric.

2 Oct 2024

• Updates: Laurent Series, Division and Prime, Riemann Zeta-Function.

1 Oct 2024

• Updates: Series of Complex Functions, Series Expansion of Holomorphic Functions, Dynamic Programming, Shannon-Fano-Elias and Interval Coding.

26 Sep 2024

• Updates: Topological Spaces, Series Expansion of Holomorphic Functions, Connectedness and Paths, Series of Complex Functions, Shannon-Fano-Elias and Interval Coding.

25 Sep 2024

• Updates: Series of Complex Functions, Uniform Convergence of Functions, Complete Metric Space

24 Sep 2024

• Updates: Random Variable and Probability Space, Tangent Vectors and Spaces, Tensors and Tensor Products.

23 Sep 2024

• New page: Series Expansion of Holomorphic Functions

19 Sep 2024

• Updates: Convex Sets

18 Sep 2024

• Update: Shannon-Fano-Elias and Interval Coding, Tensors and Tensor Products, Fisher Information Metric.

17 Sep 2024

• New page: Shannon-Fano-Elias and Interval Coding.
• Updates: Huffman Coding, Uniform Convergence of Functions, Compactness of Metric Space, Normal Convergence of Holomorphisms, Convex Functions, Complex Numbers, Relations and Functions.

16 Sep 2024

• New page: Normal Convergence of Holomorphisms.

15 Sep 2024

• Update: Fisher Information.

14 Sep 2024

• Now last updated time will show on pages, right below the title.
• Updates: Information Content and Entropy, Basics of Information, Lagrangian and Lagrange Dual.

13 Sep 2024

• Updates: Integration and Primitives, Compactness of Topological Space, Coding and Compression, Shannon Coding, Huffman Coding.

4 Sep 2024

• Updates: Tensors and Tensor Products.

23 Aug 2024

• Updates: Torsion and Curvature, Complex Differentiability, Exponential Family, Random Variable and Probability Space, Compactness of Topological Space, Tensors and Tensor Products, Riemannian Metrics.

21 Aug 2024

• New pages: The Homotopy Invariance Theorem
• Updates: Integration and Primitives.

17 Aug 2024

• New pages: Measures, Measurable Spaces and Functions.
• Updates: Probability Theory.

16 Aug 2024

• Updates: Divergence and Bregman Geometry, Connections.

14 Aug 2024

• New pages: Divergence and Bregman Geometry, Integration and Primitives.
• Updates: Connections.

7 Aug 2024

• New pages: Möbius Transformations, Statistical Manifold.
• Updates: Vector Bundles, The Levi-Civita Connection, Connections, Divergence and Bregman Geometry, Möbius Transformations, Holomorphic and Conformal Maps.

31 July 2024

• New page: Vector Bundles.
• Updates: Vector Spaces, Tangent Vectors and Spaces.

28 July 2024

• New page: Holomorphic and Conformal Maps.
• Updates: Complex Differentiability, Complex Numbers, Complex Analysis.

26 July 2024

• New pages: Connections.
• Updates: Tangent Vectors and Spaces.

25 July 2024

• New page: Lie Groups, Smooth Functions and Maps, Tangent Vectors and Spaces, Riemannian Metrics, Tensor Fields, Tensors and Tensor Products.
• Updates: Construction of Sets.

22 July 2024

• New page: Manifolds and Atlases, Complex Analysis, Complex Numbers.

16 July 2024

• Update: Coding and Compression.

22 June 2024

• New page: Coding and Compression.

9 June 2024

• Updates: Angular Momentum.
• New page: Spin.

30 May 2024

• Updates: Convex Sets, Connectedness and Paths, Compactness of Metric Space, Isometries and Homeomorphisms, Order, Morphisms.

29 May 2024

• New pages: Multiple View Geometry
• Updates: Two View Geometry, Convolutional Neural Network, Image Filtering, Image Processing, Digital Image Representation.

27 May 2024

• Update: Angular Momentum, The Harmonic Oscillator, Totally Bounbdedness, Compactness of Metric Space, Open and Closed Sets.
• New pages: Radiant Intensity, Connectedness and Paths.

23 May 2024

• Update: The Harmonic Oscillator.

21 May 2024

• Updates: Compound Systems and Entanglement, Pure States and Measurements.

18 May 2024

• New pages: Quantum Information Theory, Tensors and Tensor Products, State Spaces, Compound Systems and Entanglement, Pure States and Measurements.

16 May 2024

• Updates: Two View Geometry.

15 May 2024

• Updates: Angular Momentum, Compactness of Metric Space, Totally Bounbdedness, Time Dependent Schrödinger’s Equation.
• New page: The Harmonic Oscillator, Hermite Polynomials.

13 May 2024

• Updates: Bicategories, 2-Categories.

6 May 2024

• Updates: Constructions on Categories, Adjunction, Limits and Colimits,Functoriality and Naturality, Subobjects and Pullback, Free Groups and Relations.

5 May 2024

• New page: Two View Geometry.

3 May 2024

• Updates: Topoi, Compactness of Metric Space, Ordinary Differential Equations, Subobjects and Pullback.
• New page: Preorder and Posets, Totally Bounbdedness.

2 May 2024

• Updates: Structure of Categories, Topoi.
• New pages: Monoidal Categories, Bicategories, Single View Geometry.

1 May 2024

• Updates: Limit Points and Closure, Vector Spaces, Functoriality and Naturality, Homomorphisms, Normal Subgroup & Conjugation.
• New page: 2-Categories.

30 Apr 2024

• Updates: Integral, Uncertainty.

29 Apr 2024

• Updates: Line Fitting, Adjunction, Linear Groups, Geometric Image Formation, Lagrangian and Lagrange Dual, Information Content and Entropy, Uncertainty, Normed Spaces.

28 Apr 2024

• New page: Basics of Information.
• Update: Dispersion, Metric Spaces, Continuity on Metric Space, Adjunction.

27 Apr 2024

• New page: Continuity on Metric Space.
• Update: Complete Metric Space, Compactness of Metric Space, Metric Spaces, Image Processing, Isomorphism Theorems.
• Merge Colour Spaces into Digital Image Representation.
• Merge Corner Detection into Image Filtering.
• Interface updates: Now every future definition/proposition is a callout block. Hover preview can now show the whole block:

26 Apr 2024

• New page: Uniform Convergence of Functions, Newtonian Mechanics, Ordinary Differential Equations, Line Fitting.
• Update: Ring, Field and Integral Domain, Integral.

25 Apr 2024

• Update: Compactness of Metric Space, General Topology.
• New page: Complete Metric Space.
• Add sample code for inverse warping algorithm.

24 Apr 2024

• Add section Matrix Representation in Hilbert Space in Quantum State Space.
• Completed definition of Dirac Notation.
• Update: Operators and Observables, Geometric Image Formation.
• New page: Uncertainty.

23 Apr 2024

• New pages: Integral, Adjunction, Topoi.
• Update: Domain Adaptation, Transfer Learning.

21 Apr 2024

• Update: Groups, Order and Subgroups, Free Groups and Relations.

18 Apr 2024

• New pages: Quantum State Space, Operators and Observables, Measurement.
• Update: Time Dependent Schrödinger’s Equation.

15 Apr 2024

• New page: Yoneda Lemma.

3 Apr 2024

• Update: Functoriality and Naturality.

31 Mar 2024

• New pages: Order Theory, Order, Completeness.
• Add definition of homogeneous binary relation, reflexive, antisymmetric and transitive.
• New section of ordered field.

27 Mar 2024

• Update: Interior, Exterior and Boundary, Limit Points and Closure, Sequence and Convergence, Quantum Mechanics, Set Theory.
• Add Cauchy-Schwarz Inequality.

26 Mar 2024

• Update: Construction of Sets, Equinumerous and Countability, Relations and Functions, Cardinality.

24 Mar 2024

• New page: Exponential.

21 Mar 2024

• Add The Finite Square Well in Time Independent Schrödinger’s Equation.

20 Mar 2024

• Update: Time Independent Schrödinger’s Equation.

12 Mar 2024

• Update: Quantum Waves.

11 Mar 2024

• Add definition of a relation.
• Rearrange equivalence relation.
• Merge definition of a functor into Structure of Categories.
• Merge Generalized Convexity, Epigraph and Sublevel Set and Quasiconvex into Convex Functions.
• New pages: Duality, Groups and Categories, Preorder Category, Relation Category.

10 Mar 2024

• Rename Function to Relations and Functions.
• Move definition of equivalence relation to Relations and Functions.
• Replace the terminology semigroup with monoid.
• Update: Cardinality.

9 Mar 2024

• New page: Image Filtering.

8 Mar 2024

• New page: Inner Products.
• Update: Metric Spaces, Normed Spaces, Vector Spaces.
• Remove definition of norm in Matrix and Vector Calculus.

5 Mar 2024

• New pages: Colour Spaces, Geometric Image Formation.

4 Mar 2024

• Updated Quantum Waves.
• Merge Wave Nature of Matter into Quantum Waves.
• New page: Number Systems.

3 Mar 2024

• Finished Products part in Objects and Elements.
• Added part about Hom-Set in Objects and Elements.
• Fixed Input-Output (IO) Architectures and Microarchitecture.

2 Mar 2024

• New page: Objects and Elements.
• Rearrange the content of Category Theory.

25 Feb 2024

• New page: Constructions on Categories, Wave Nature of Matter, Directed Graphs, Foundations, Large and Small.
• Rearrange the content of Category Theory.

20 Feb 2024

• New page: Functoriality, Finite Category.

19 Feb 2024

• Update Category Theory.
• New page: Structure of Categories.

12 Feb 2024

• Update Classification of Divergences.
• New page: Reversible Jump MCMC.

10 Feb2024

• Update Importance Sampling, add the concept of weights variability.
• New pages: Posterior Asymptotics, Gibbs Sampling, Metropolis-Hastings Algorithm.

7 Feb 2024

• Update Random Variable and Probability Space.
• Add definition of cdf in Probability Distributions.
• Update Inversion Sampling, add an example.

5 Feb 2024

• Add links for logbook.
• Update Fisher Information and Priors.
• Add Effective Rank.

4 Feb 2024

• Update Entropy page in Information Theory.
• Accomplish definition of knots.
• Update the Classical Mechanics page.

30 Jan 2024

• Managed to make SDR online.
• Add computer organization page.
• Add acknowledgement in convex optimization.