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Design a Concrete-Pictorial-Abstract learning sequence for a mathematical concept using manipulatives. Use when teaching maths through Singapore method or when students struggle with abstraction.
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Real-world problem formulation, mathematical abstraction, and applied mathematics for translating between practical problems and mathematical frameworks. Covers the modeling cycle (problem identification, assumptions, formulation, analysis, validation, interpretation), Polya's framework adapted for modeling, common model types (linear, exponential, logistic, periodic, power-law), dimensional analysis (Buckingham Pi theorem), optimization (linear programming, gradient descent, constraint satisfaction), probability models (Markov chains, queuing theory, Monte Carlo simulation), statistical modeling (regression, hypothesis testing, model selection), model criticism (overfitting, underfitting, sensitivity analysis), and real-world case studies. Use when formulating mathematical models, performing dimensional analysis, optimizing systems, running simulations, or evaluating model validity.
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Calculus, series, numerical methods, and optimization for computational mathematics. Covers limits (epsilon-delta, L'Hopital, squeeze theorem), derivatives (rules, optimization, related rates), integrals (Riemann sums, Fundamental Theorem of Calculus, integration techniques), series (convergence tests, Taylor and Maclaurin expansions), differential equations (separable, linear, exact), numerical methods (Newton-Raphson, Euler method, Simpson's rule), and computational pitfalls (floating-point arithmetic, catastrophic cancellation, numerical stability). Use when computing derivatives, integrals, limits, series, solving differential equations, or analyzing numerical accuracy.
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Conjecture generation, sequence analysis, and experimental mathematics for mathematical discovery. Covers sequence analysis (OEIS lookups, recurrence relations, generating functions), pattern detection methods (finite differences, ratio analysis, modular patterns), conjecture generation workflow, classical integer sequences (Fibonacci, Catalan, partition numbers, Bernoulli numbers), combinatorial identities, the probabilistic method (Erdos, expected value arguments, Lovasz Local Lemma), AI-assisted discovery (FunSearch, Ramanujan Machine), and confidence calibration with cautionary tales (Mertens conjecture, Polya conjecture). Use when discovering patterns, analyzing sequences, generating conjectures, or working with combinatorial identities.
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Physical applications of mathematics. Constants, quantum mechanics, measurement — where abstract meets embodied.
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Scholze-Clausen condensed mathematics bridge to sheaf neural networks via 6-functor formalism
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Scholze-Clausen condensed math - condensed sets, liquid vector spaces, solid modules, 6-functor formalism, Kunneth exactness
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BDD-Driven Mathematical Content Verification Skill Combines Behavior-Driven Development with mathematical formula extraction, verification, and transformation using: - Cucumber/Gherkin for specification - RSpec for implementation verification - mathpix-gem for LaTeX/mathematical content extraction - Pattern matching on syntax trees for formula validation Enables iterative discovery and verification of mathematical properties through executable specifications.
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ANIMA as limit construction over condensed skill applications. Formalizes prediction markets as belief ANIMAs, structure dishes as condensation media, and impact as equivalence class change. Use for understanding agency at maximum entropy, compositional world modeling, or applying Scholze-Clausen condensed mathematics to AI.
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Derive a theoretical result step-by-step from first principles or established theorems, with every step explicitly justified and special cases checked. Use when deriving a formula or theorem from first principles, proving a mathematical statement by logical deduction, re-deriving a textbook result for verification or adaptation, extending a known result to a more general setting, or producing a self-contained derivation for a paper or thesis.
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Convert an existing prose, flat, or loosely organized mathematical proof into a source-mapped Lamport-style hierarchy without repairing or completing its mathematics. Use when a submitted proof needs a traceable hierarchical rendering before a forward Lamport audit or for clearer presentation. Do not use to invent a proof, certify validity, or restructure an already hierarchical proof merely to audit it.
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Computational mathematics expert for numerical solutions, approximations, error analysis, and scientific computing
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Applied math expert for engineering applications including Fourier analysis, transforms, and practical problem solving
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Expert in formal logic, model theory, computability, and foundations of mathematics
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Expert in combinatorics, graph theory, discrete probability, and algorithms with applications to computer science
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Builds comprehensive design systems and design bibles with production-ready CSS. Expert in design tokens, component libraries, CSS architecture. Use for design system creation, token architecture, component documentation, style guide generation. Activate on "design system", "design tokens", "CSS architecture", "component library", "style guide", "design bible". NOT for typography deep-dives (use typography-expert), color theory mathematics (use color-theory-palette-harmony-expert), brand identity strategy (use web-design-expert), or actual UI implementation (use web-design-expert or native-app-designer).
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Search arXiv preprint repository for papers in physics, mathematics, computer science, quantitative biology, and related fields
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SymPy is a Python library for symbolic mathematics that enables exact computation using mathematical symbols rather than numerical approximations.
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Performs numerical and symbolic mathematics using NumPy, SciPy, SymPy, and mpmath. Use when the user needs linear algebra, arrays, optimization, integration, ODEs, symbolic expressions, equation solving, or arbitrary-precision arithmetic.
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Search and retrieve preprints from arXiv via the Atom API. Use this skill when searching for papers in physics, mathematics, computer science, quantitative biology, quantitative finance, statistics, electrical engineering, or economics by keywords, authors, arXiv IDs, date ranges, or categories.
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Search arXiv physics, math, and computer science preprints using natural language queries. Powered by Valyu semantic search.
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Use this skill when working with symbolic mathematics in Python. This skill should be used for symbolic computation tasks including solving equations algebraically, performing calculus operations (derivatives, integrals, limits), manipulating algebraic expressions, working with matrices symbolically, physics calculations, number theory problems, geometry computations, and generating executable code from mathematical expressions. Apply this skill when the user needs exact symbolic results rather than numerical approximations, or when working with mathematical formulas that contain variables and parameters.
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Builds comprehensive design systems and design bibles with production-ready CSS. Expert in design tokens, component libraries, CSS architecture. Use for design system creation, token architecture, component documentation, style guide generation. Activate on "design system", "design tokens", "CSS architecture", "component library", "style guide", "design bible". NOT for typography deep-dives (use typography-expert), color theory mathematics (use color-theory-palette-harmony-expert), brand identity strategy (use web-design-expert), or actual UI implementation (use web-design-expert or native-app-designer).
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Search academic paper repositories (arXiv, Semantic Scholar) for scholarly articles in physics, mathematics, computer science, quantitative biology, AI/ML, and related fields
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