Give new users a practical introduction to what the agent can help with, including coding, memory, research, and planning. Explain natural ways to ask for work, recall earlier lessons, and prepare handoffs for later sessions.
Build understanding of mathematical ideas before calculating. The skill guides you through concrete examples, diagrams, simpler cases, and analogies, then summarizes the intuition and suggests a next step with a check.
Work through operator theory problems in functional analysis with structured steps for boundedness, adjoints, spectra, and compactness. Includes example commands for symbolic calculations and proof checks using SymPy and Z3.
Helps select graph algorithms for shortest paths, minimum spanning trees, network flow, and coloring. Provides decision guides and example commands for symbolic calculations and checking mathematical properties.
Plan solutions to modular arithmetic problems, which involve integer remainders. Choose methods for modular inverses, simultaneous congruences, powers, quadratic residues, and primitive roots, with example commands for symbolic calculations and checking mathematical claims.
Choose approaches for prime number problems, including primality testing, factorization, and prime distribution. The skill outlines relevant algorithms and theorems, with commands for symbolic calculation and checking mathematical claims.
Work through linear algebra problems involving vector spaces, from checking subspaces and linear independence to finding bases, dimensions, and coordinate changes. The skill connects each task to symbolic calculation or proof scripts for checking the results.
Work through matrix problems in linear algebra by identifying matrix types and choosing suitable calculations. The skill covers multiplication, inverses, determinants, linear systems, and decompositions that express a matrix as simpler components.
Check whether infrastructure code actually runs when a user triggers it. Trace calls from hooks or commands through scripts and modules, verify registration and permissions, and test the complete path to uncover unreachable code.
Work through matrix eigenvalue problems, from the characteristic polynomial, whose roots give the eigenvalues, to finding eigenvectors. Follow a structured process to check the results and examine algebraic and geometric multiplicities.
Work through predicate logic problems by examining quantifiers, which express “all” or “some,” organizing formulas, and choosing proof steps. The skill also guides resolution proofs and countermodels that show why an argument is invalid.
Plan and examine proofs in mathematical logic. Choose direct proof, contradiction, cases, or induction, then review proof structure using concepts such as cut elimination, soundness, and completeness, with command examples for checking proof steps.
Analyze propositional formulas and arguments using truth tables, deduction rules, and branching proof trees. Classify formulas as always true, always false, or dependent on their inputs, and check whether an argument follows from its premises.
Work through integration problems in measure theory by matching the problem's conditions to suitable theorems. The skill covers simple functions, exchanging limits and integrals, lower bounds, and changing integration order, with commands for symbolic calculations and proof tools.
Work through Lebesgue measure problems, which concern assigning sizes to sets. The skill organizes approaches to outer measure, measurability criteria, translation invariance, countable additivity, and approximation by compact or open sets, with supporting tool command examples.
Work through convex optimization problems by checking convexity, putting objectives and constraints into a standard form, and choosing a solver. The skill provides SciPy examples and outlines checks for constraint satisfaction and mathematical conditions used to assess optimality.
Work through convergence problems in real analysis by identifying sequence and series types, choosing suitable tests, checking limits and bounds, and calculating sums when a series converges.
Study source coding, the information theory behind data compression. The skill guides problems involving Huffman codes, average code length, prefix-code constraints, arithmetic coding, and the tradeoff between compression rate and distortion, with calculation and proof command examples.
Review the technology choices in an implementation plan using past project outcomes and web research. Produce a report that identifies outdated approaches, deprecated tools, and security concerns, with supporting sources and suggested alternatives.
Choose an approach for second-order ordinary differential equations based on their coefficients and forcing terms. Work through characteristic equations and particular solutions, or use numerical methods for problems defined by initial values or boundary conditions.
Work through natural transformation problems in category theory by checking component maps and compatibility conditions. The skill outlines strategies for natural isomorphisms, composition, and the Yoneda lemma, with references for expressing and checking work in Lean 4.
Investigate whether a complex function is analytic, meaning it has a local power series representation. The skill guides derivative checks, Cauchy–Riemann equations, power series and convergence radii, plus ways to explore extension beyond the original domain.
Organize work by identifying whether the goal is research, planning, building, or fixing. The skill selects specialist agents, checks existing plans, adjusts agent allocation, and presents the proposed workflow for approval before execution.
Work through graduate-level problems using Rudin's Real and Complex Analysis. The skill guides theorem selection for measure theory, integration, and complex analysis, with commands for textbook searches, symbolic calculations, and verification.
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