Research, academic papers, education, and exam prep (4.4K skills)
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.
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 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 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.
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 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 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.
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.
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.
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.
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 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.
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.
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 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.
Work through information theory problems involving entropy, a measure of uncertainty. Provides formulas, properties, and calculation examples for discrete and continuous distributions, joint and conditional entropy, and choosing distributions through the maximum entropy principle.
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.
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.
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 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.
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.
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.
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.
Work through category theory problems involving limits and colimits. Identify the construction, check its universal property—the condition defining its unique mapping behavior—and use concrete set calculations or Lean 4 references to guide verification.