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Nonlinear optimization with CasADi and IPOPT solver. Use when building and solving NLP problems: defining symbolic variables, adding nonlinear constraints, setting solver options, handling multiple initializations, and extracting solutions. Covers power systems optimization patterns including per-unit scaling and complex number formulations.
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Analyze nonlinear dynamical systems — phase portraits, fixed points, stability analysis, bifurcation diagrams, Poincare sections, Lyapunov exponents, and chaos detection. Use for any autonomous or non-autonomous ODE system where qualitative behavior matters.
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Reduce data with many features into fewer dimensions using UMAP. Create 2D or 3D views, prepare inputs for clustering, compare parameter settings, and project new samples while checking whether the results remain stable and meaningful.
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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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UMAP dimensionality reduction. Fast nonlinear manifold learning for 2D/3D visualization, clustering preprocessing (HDBSCAN), supervised/parametric UMAP, for high-dimensional data.
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Linearizing nonlinear dynamics around operating points for control design.
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Use scipy.optimize.curve_fit for nonlinear least squares parameter estimation from experimental data.
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Use this skill when simulating quadrotor physical dynamics — mapping desired thrust/moments to individual motor RPMs via a propeller allocation matrix, applying first-order motor lag, and integrating the nonlinear equations of motion (translational and rotational) using RK45.
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Design, select, and analyze waveforms for radar, sonar, and active sensing using the Phased Array System Toolbox. Covers LFM, NLFM, FMCW, phase-coded, CW, stepped FM, custom IQ, ambiguity functions, sidelobe reduction, and Doppler tolerance. Key objects: phased.LinearFMWaveform, phased.NonlinearFMWaveform, phased.CustomFMWaveform, phased.PhaseCodedWaveform, phased.FMCWWaveform, phased.SteppedFMWaveform, phased.MFSKWaveform, phased.RectangularWaveform, nlfmspec2freq, shapespectrum, ambgfun, pambgfun, sidelobelevel, legendreseq, mlseq, radarWaveformGenerator.
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UMAP dimensionality reduction for visualization, clustering prep, and feature engineering. Fast nonlinear manifold learning preserving local and global structure. Standard UMAP (fit/transform, sklearn-compatible), supervised/semi-supervised, Parametric UMAP (NN encoder/decoder, TensorFlow), DensMAP (density), AlignedUMAP (temporal/batch). 15+ distance metrics, custom Numba metrics, precomputed distances. For linear reduction use PCA; for neighborhood graphs use sklearn NearestNeighbors.
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Think and work like an expert Laser Physicist. Use when a task calls for Laser Physicist judgment. Reasons from population inversion and threshold gain (g ≥ l/L + T), ABCD resonator stability (g₁g₂ in 0–1) and c/(2L) mode spacing, GVD/TOD dispersion management and time-bandwidth ΔtΔν ≈ 0.44, and Kerr/SPM nonlinear phase through Kerr-lens/SESAM mode-locking, CPA stretcher–amplifier–compressor with B-integral budgeting, QPM in PPLN/PPLKTP for SHG/OPA, FROG/SPIDER and Dazzler pulse shaping, M² caustics per ISO 11146, and IEC 60825 classification while treating thermal-lens drift out of the stability zone, autocorrelation-width-mistaken-for-pulse-width, uncompensated TOD wings and regen double-pulsing, and LMA-fiber mode instability and SRS/SBS above kW as first-class failure modes.
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Think and work like an expert Acoustics Physicist. Use when a task calls for Acoustics Physicist judgment. Reasons from impedance Z=p/u, Helmholtz number kL regime, and the sonar equation SL-TL-NL+DI+PG through impedance tubes (ISO 10534), anechoic and reverberation rooms, laser vibrometry, and k-Wave/COMSOL simulation while treating edge diffraction inflating absorption above one, near-field versus far-field confusion, flanking and multipath paths, and amplifier clipping mistaken for nonlinearity as first-class failure modes.
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Use scipy.optimize.curve_fit for nonlinear least squares parameter estimation from experimental data.
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Linearizing nonlinear dynamics around operating points for control design.
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Physical principles underlying diodes, bipolar junction transistors, and MOSFETs — carrier concentrations, drift and diffusion, the pn junction, forward and reverse bias, the Shockley diode equation, BJT operating regions, MOSFET threshold and saturation, small-signal models, and temperature effects. Use when reasoning about device-level behavior, designing bias networks, selecting transistor operating points, interpreting datasheets, or diagnosing nonlinear device failures.
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Use when creating nonlinear outlines, generating mindmap diagrams for article structure, mapping a knowledge domain visually, or bridging raw ideas into structured options for the user to select from.
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Expert-level control theory covering PID control, state-space methods, stability analysis, optimal control, robust control, and nonlinear control.
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Expert-level optimization knowledge. Use when working with linear programming, convex optimization, nonlinear programming, integer programming, dynamic programming, gradient methods, duality theory, or optimization algorithms. Also use when the user mentions 'linear programming', 'simplex method', 'convex optimization', 'Lagrangian', 'KKT conditions', 'gradient descent', 'Newton method', 'integer programming', 'dynamic programming', 'duality', 'Bellman equation', or 'branch and bound'.
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Sparse Identification of Nonlinear Dynamics (SINDy) — discover governing equations from time-series data. Builds sparse dynamical system models dx/dt = f(x) from measurements using PySINDy. Use when you have trajectory data and want to find the underlying ODE.
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Nonlinear curve fitting for physics data with proper error propagation, chi-squared analysis, residual diagnostics, confidence intervals, and model comparison (AIC/BIC). Use for any parameter extraction from experimental or simulation data.
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Solve partial differential equations — finite differences, spectral methods, and physics-informed neural networks (PINNs via DeepXDE). Supports 1D/2D/3D, steady/transient, linear/nonlinear PDEs with Dirichlet, Neumann, and periodic boundary conditions.
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UMAP dimensionality reduction. Fast nonlinear manifold learning for 2D/3D visualization, clustering preprocessing (HDBSCAN), supervised/parametric UMAP, for high-dimensional data.
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Enzyme kinetics — Michaelis-Menten Km, Vmax, kcat (turnover), and kcat/Km (catalytic efficiency / specificity constant) from substrate-velocity data, plus inhibition-mechanism analysis (competitive / uncompetitive / non-competitive, Ki). Fits the MM equation by nonlinear regression (and reports Lineweaver-Burk for reference). Use when you have substrate concentrations and initial reaction velocities and need kinetic parameters or to classify an inhibitor. NOT for BRENDA database lookups of published constants (use the BRENDA tools).
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