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Mathematics
\[\int_a^b f'(x)\; dx = f(b) - f(a),\quad\quad\quad\frac{d}{dx}\left(\int_a^x f(t)\; dt\right)=f(x)\]The integration on forms concept is of fundamental importance in differential topology, geometry, and physics, and also yields one of the most important examples of cohomology, namely de Rahm cohomology, which (roughly speaking) measures precisely the extend to which the fundamental theorem of calculus fails in higher and on general manifolds.
(EM-KDS) Abstract Algebra I
Topics: Cyclic Group, Classification of Cyclic Group, Lagrange’s Theorem, Converge of Lagrange’s Theorem
(EM-KDS) Abstract Algebra II
Topics: Group Action, Cayley Theorem, Normal Subgroups, Normality of the Kernel
(EM-KDS) Abstract Algebra III
Topics: First Isomorphism Theorem, Second Isomorphism Theorem, Third Isomorphism Theorem
(EM-KDS) Abstract Algebra to Linear Algebra
Topics: TBA
(EM-KDS) Advanced Calculus I
Topics: Boundedness, Supremum and Infimum, Least Upper Bound Property, Well-Ordering Principle and Mathematical Induction,Archimedean Property
(EM-KDS) Advanced Calculus II
Topics: Convergence of Sequences, Inequality Rule for Absolute Values, Limit Theorem (Algebraic Property of Limit of Sequence)
(EM-KDS) Advanced Calculus III
Topics: Limit of a Function (ε − δ), Continuity of a Function, Monotone Convergent Theorem (MCT), Nested Interval Property (NIP), Bolzano-Weierstrass Theorem, Limit Superior and Limit Inferior
(EM-KDS) Algebraic Structures
Topics: Group, Ring, Field, Module, Vector Space, Algebra
(EM-KDS) Linear Algebra I
Topics: Basis, Hamel Basis Theorem (Existence of Basis), Invariance of Basis Cardinality
(EM-KDS) Linear Algebra II
Topics: Linear Transformation, Classification of Vector Space (up to Isomorphism), Matrix Representation of a Linear Transformation
(EM-KDS) Linear Algebra III
Topics: TBA
(EM-KDS) Linear Algebra IV
Topics: TBA
(EM-KDS) Linear Algebra to Abstract Algebra
Topics: Subgroup,Homomorphism, Isomorphism, Kernel and Image
(EM-KDS) Set Theory I
Topics: Set, Subset, Power set, Cartesian product, Union, Intersection, Function, Injectivity, Surjectivity, Axiom of Choice
(EM-KDS) Set Theory II
Topics: Binary relation, Equivalence relation, Union and Intersection of an Indexed Family, Partitions, Equivalence Class
(EM-KDS) Topology I
Topics: Topology and Topological Space, Open Set, Continuous Mapping, Distance Function and Metric Space, Convergence of Sequences; Continuity of Functions
One-form
Topics: 1-form, integral, FTC
Riemann–Roch
Topics: riemann-roch, riemann-surfaces, algebraic-geometry, divisors, line-bundles, sheaf-cohomology, serre-duality
Cryptography
\[\operatorname{Adv}^{\mathrm{IND\text{-}CPA}}_{\mathcal{A}}(\Pi) \;:=\; \left|\Pr\!\left[\mathsf{Game}^{\mathrm{IND\text{-}CPA}}_{\Pi,\mathcal{A}}(1^\lambda)=1\right]-\tfrac12\right|.\]In modern security definitions, a construction is considered secure if every efficient adversary’s advantage in a well-specified experiment is negligible as a function of the security parameter. This viewpoint makes cryptographic claims precise, composable, and testable against explicit threat models.
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Number Theoretic Transform (NTT)
Category:
Date: 2025-12-23 00:00:00 +0000
Tags: cryptographymathematics
Programming
\[\text{Program} = \text{Algorithms} + \text{Data Structures}\]"Programs must be written for people to read, and only incidentally for machines to execute."