리서치·학습 스킬

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.

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• Choosing a mathematical approach for a real-world problem
• Deciding how to model discrete or continuous change
• Selecting methods for uncertainty or optimization

Choose approaches to Hilbert space problems in functional analysis using orthogonal decomposition, projection, and representation theorems. The skill provides a decision tree and command examples for inner product calculations and proof tasks.

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• Choosing an approach to Hilbert space exercises
• Studying orthogonal decomposition and projection
• Working through the Riesz representation theorem

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.

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• Choose a convergence test for a series
• Check sequence limits and inequality bounds
• Calculate the sum of a convergent series

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.

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• Checking whether an objective and its constraints are convex
• Putting a maximization problem into standard minimization form
• Choosing a solver for linear, quadratic, or semidefinite problems

Solve optimization problems with equality constraints, inequalities, or variable bounds. Guides method selection, symbolic equations, and optimality checks, with SciPy examples for finding numerical solutions that respect the stated constraints.

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• Classify constraints before choosing a solution method
• Solve equality-constrained problems with Lagrange multipliers
• Check KKT conditions for inequality constraints

Work through abstract algebra problems about rings by checking their defining rules and properties. The skill organizes questions about ideals and structure-preserving maps, with Z3 and Sympy command examples for proof checks and symbolic calculations.

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• Checking whether operations define a ring
• Examining commutativity, identity, and integral domain conditions
• Checking ideal conditions

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.

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• Calculate capacity for standard channel models
• Organize mutual information calculations
• Explore input distributions that maximize capacity

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.

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• Classifying second-order ordinary differential equations
• Solving constant-coefficient homogeneous equations
• Finding particular solutions for nonhomogeneous equations

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.

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• Checking where a complex function is analytic
• Verifying the Cauchy–Riemann equations
• Finding a power series and its convergence radius

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.

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• Checking whether a set is a vector subspace
• Testing vectors for linear independence
• Finding a basis and dimension

Helps solve contour integral problems by choosing suitable paths, locating singularities, and applying the residue theorem, which relates an integral to values around poles. Includes Sympy commands for finding poles, calculating residues, and checking definite integrals.

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• Choose a contour for an improper real integral
• Evaluate trigonometric integrals using the unit circle
• Plan a contour around branch cuts

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.

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• Choosing a theorem for exchanging limits and integrals
• Checking conditions for changing integration order
• Finding lower bounds with Fatou's lemma

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.

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• Choosing a shortest path algorithm
• Planning a minimum spanning tree solution
• Setting up network flow equations

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.

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• Classifying formulas with truth tables
• Checking whether a formula can be true
• Verifying arguments with deduction rules

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.

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• Choose a strategy for a mathematical proof
• Organize base cases and steps in structural induction
• Study cut elimination and proof normalization

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.

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• Choosing a method for a definite or improper integral
• Computing two- or three-dimensional integrals
• Checking accuracy by tightening error tolerances

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.

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• Choosing a method for a first-order differential equation
• Computing a numerical solution with initial values
• Checking a solution against the equation and conditions

Find numerical solutions where a function equals zero. The skill helps choose a method based on known intervals and available derivatives, then guides SciPy implementation, searches for multiple solutions, and checks accuracy and valid ranges.

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• Finding a solution within a known interval
• Choosing a method based on derivative availability
• Solving systems of nonlinear equations

Solve complex analysis problems by choosing residue calculations for different singularities, points where a function is not analytic. The skill covers limits, derivatives, Laurent series, and contour integrals using the residue theorem, with Sympy and Z3 command examples.

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• Calculating residues at simple and higher-order poles
• Determining the order of a pole
• Finding residues at essential singularities using Laurent series

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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• Choosing theorems for limits and integrals
• Working through contour integrals and residues
• Studying function spaces and linear operators

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.

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• Checking quantifier scope and variable binding
• Converting formulas to prenex normal form
• Planning a resolution proof

Guides work on categories and functors by checking definitions, composition rules, and structure-preserving properties. Explains common functor types and provides examples for verifying proofs with Lean 4 and Mathlib.

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• Checking whether a construction satisfies category axioms
• Verifying that a mapping is a functor
• Distinguishing common types of functors

Work through continuity problems in real analysis by checking function values and limits. The skill outlines separate checks for piecewise functions and provides commands for limit calculations with SymPy and proof checks with Z3.

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• Checking whether a function is continuous at a point
• Comparing left and right limits of piecewise functions
• Comparing a computed limit with the function value

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.

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• Finding a matrix's eigenvalues and eigenvectors
• Computing a characteristic polynomial
• Checking eigenvalue and eigenvector results