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.
Build a compact overview of an unfamiliar codebase with TLDR. Summarize its file tree, functions, classes, and calls between components, then highlight complex functions while reducing the amount of source text the agent needs to read.
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.
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.
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.
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.
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.
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 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.
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.
Choose a mathematical approach for a problem you do not yet know how to model. Guided questions clarify change, uncertainty, optimization goals, and required precision, leading to a recommended framework, starting equations, tools, and related skills.
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.
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.
Match code investigation questions to the appropriate tldr command. Find ways to inspect files, function calls, branching, variable flow, and line dependencies, helping you explore a project or investigate a bug.
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 abstract algebra problems about fields, number systems supporting addition, subtraction, multiplication, and division by nonzero elements. Follow a decision tree covering field axioms, extensions, characteristic, and algebraic elements, with commands for polynomial calculations.
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.
Work through information theory problems about channel capacity, the maximum information a channel can carry. Organize channel models, calculate mutual information, and choose formulas or iterative methods to examine capacity and input distributions.
Choose numerical methods for approximating integrals based on the interval, function behavior, and number of dimensions. The skill guides SciPy calculations, error tolerance settings, and accuracy checks against known solutions or repeated calculations with tighter tolerances.
Choose methods for differential equations with conditions specified at boundaries. The skill guides problem classification, shooting methods, finite differences, numerical solvers, and eigenvalue problems, with command examples for exploring solutions and checking numerical accuracy.
Work through first-order ordinary differential equations by classifying their form and choosing a solution method. The skill also covers numerical initial-value problems, checking solutions against the equation and conditions, and examining equilibrium stability.
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.
Approach abstract algebra problems involving groups, sets with an operation satisfying specific rules. Follow checks for group axioms, subgroups, structure-preserving maps, and element counts or orders, supported by proof and symbolic calculation commands.
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