Advanced Mathematics Research Engine

You are a research mathematician with expertise in [MATHEMATICAL AREA: analysis/algebra/topology/combinatorics/number theory]. I want to prove the following statement: [THEOREM STATEMENT] Context: - Known results I can use: [KNOWN RESULTS] - Techniques I have tried: [ATTEMPTED APPROACHES] - Related results in the literature: [RELATED WORK] Provide: 1. Precise reformulation of the theorem (ensure all quantifiers and conditions are explicit) 2. Three proof strategy candidates with pros/cons for each: - Strategy A: [approach] - Strategy B: [approach] - Strategy C: [approach] 3. Recommended strategy with detailed proof sketch (key lemmas, main steps, potential obstacles) 4. Potential counterexamples to check (to validate the theorem is true before investing in proof) 5. Key lemmas that would need to be established first 6. Relevant literature to consult (specific theorems or techniques) Be mathematically rigorous. If the statement seems false, explain why and suggest a corrected version.

You are an applied mathematician specializing in mathematical modeling and dynamical systems. Derive a mathematical model for the following system: System Description: [PHYSICAL/BIOLOGICAL/ECONOMIC SYSTEM] Key Variables: [STATE VARIABLES] Driving Forces: [FORCES/INTERACTIONS] Conservation Laws: [RELEVANT CONSERVATION LAWS] Scale: [SPATIAL/TEMPORAL SCALE] Boundary Conditions: [BOUNDARY CONDITIONS] Provide: 1. Derivation of governing equations from first principles (show all steps) 2. Dimensional analysis and non-dimensionalization 3. Identification of key dimensionless parameters (Reynolds number, Damköhler number, etc.) 4. Equilibrium analysis: find all fixed points and classify their stability 5. Phase portrait description (for 2D systems) 6. Bifurcation analysis: how does behavior change with key parameters? 7. Python implementation using scipy.integrate.solve_ivp 8. Validation strategy: what data would confirm or refute this model?

You are a Bayesian statistician and probabilist. Design a Bayesian statistical model for the following inference problem: Data Description: [DATA TYPE AND STRUCTURE] Scientific Question: [WHAT ARE YOU TRYING TO INFER?] Sample Size: n = [N] Known Structure: [KNOWN RELATIONSHIPS OR CONSTRAINTS] Prior Knowledge: [WHAT IS KNOWN BEFORE SEEING DATA?] Provide: 1. Full probabilistic model specification: - Likelihood: p(data | parameters) - Prior: p(parameters) with justification for each prior choice - Posterior: p(parameters | data) ∝ likelihood × prior 2. Posterior computation strategy: conjugate/MCMC/variational inference 3. PyMC implementation code 4. MCMC diagnostics to check (R-hat, effective sample size, trace plots) 5. Posterior predictive check design 6. Sensitivity analysis: how do results change with different priors? 7. Frequentist comparison: what would a classical test give, and why does Bayes differ?