调研、论文、教育、考证学习（4425 个）
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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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.
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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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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.
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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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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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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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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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Plan solutions to Banach space problems in functional analysis. Work through completeness, meaning Cauchy sequences converge, and select major theorems to guide arguments about extending linear functionals, open mappings, and bounded operators.
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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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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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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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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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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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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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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.
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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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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.
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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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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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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 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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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.
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Work through measure theory problems involving sigma algebras, collections of sets closed under complements and countable unions. The skill outlines approaches to checking axioms, generating these collections, testing measurability, and studying products, with supporting tool commands.
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