Compute exact spectral gaps λ₁ of Cayley graphs Cay(S_n, T) for T = adjacent transpositions and T = all transpositions, n=3-12. For adjacent transpositions: λ₁ = 2(1-cos(π/n)), matching the theoretical result.
Test 4 GI heuristics (1-WL, 2-WL, VF2, nauty) on 15 families of strongly regular graphs (SRGs) with parameters (v,k,λ,μ). 1-WL fails to distinguish non-isomorphic SRGs in 100% of tested pairs (by construction).
Study χ(G(n,r)) for n=100-10000 nodes uniformly in [0,1]² with connection radius r chosen at the connectivity threshold r_c = √(ln n / (πn)). Empirically: χ concentrates around c·√(n/ln n) with c=1.
Compute Ramsey numbers R(K_{s,t}, K_{s,t}) for s,t ≤ 8 via SAT solving and ILP. The growth rate exhibits a sharp transition: for t/s < 0.
Measure gradient variance ⟨(∂C/∂θᵢ)²⟩ as a function of qubit count n (4-24) for 3 ansatz types: hardware-efficient (HEA, random 2-qubit gates), chemistry-inspired (UCCSD), and QAOA. HEA: variance decays as 2^{-2n} (barren plateau at n≈12).
Simulate surface codes (d=3-21) with 4 decoders: MWPM (minimum weight perfect matching), Union-Find, neural network, and lookup table. Standard threshold (assuming instant decoding): 10.
Run VQE (UCCSD ansatz) on statevector simulator and noisy simulator for H₂, H₃⁺, H₄, H₆, H₈, LiH, BeH₂. Noiseless VQE achieves chemical accuracy (1.
Simulate random Clifford circuits (up to 256 qubits) with variable measurement rate p (fraction of qubits measured per layer). Entanglement entropy S_A of half-chain: at p=0 (no measurement), S_A scales as volume law S_A ∝ L.
Compare cloud fraction from 12 CMIP6 models against CERES-EBAF satellite observations (2001-2020). Global mean bias: -3.
Verify ECMWF, GFS, and NCEP SST forecasts against OISST observations (2003-2023) in 4 basins: tropical Pacific, North Atlantic, Indian Ocean, Southern Ocean. Skill (anomaly correlation coefficient ACC) decay rates: tropical Pacific 0.
Analyze MODIS land surface temperature (LST) for 2,000 cities globally (2003-2023). UHI intensity (ΔT between urban core and rural surroundings) vs log(population).
Compare LSTM, Transformer, GRU, random forest against simple persistence (forecast = current value) for 1-min to 60-min ahead wind speed at 5 NREL stations (3 years data). Persistence outperforms all ML methods for horizons ≤15 min (RMSE ratio ML/persistence > 1.
Fabricate micro+nano patterned silicon surfaces: micropillars (d=10-50μm, h=20μm) with nano-roughness (Ra=50-500nm). Measure advancing/receding contact angles for water.
Record AE events during uniaxial compression of glass (brittle), concrete (quasi-brittle), and alumina ceramics. Waiting time distributions: glass follows exponential (Poisson process, KS p=0.
Measure thermal conductivity (laser flash) of epoxy + {Al₂O₃, BN, graphene} at 0.1-20 vol%.
Cyclic shear experiments on glass beads (d=0.5-2mm) in a Couette cell.
Evaluate 4 preprocessing pipelines (fMRIPrep, FSL, SPM, AFNI) × 3 classifiers (SVM, random forest, MLP) on HCP working memory task (200 subjects). Main effect of pipeline: F(3,2388)=47.
Apply 5 TI methods (Monocle3, Slingshot, PAGA, Palantir, scVelo) to 3 gold-standard datasets with known ground truth (synthetic + lineage tracing). Pairwise Kendall τ between pseudotime orderings: mean 0.
Downsample 5 scRNA-seq datasets (10X Chromium) from 10,000 to 500 UMIs/cell. Cell cycle classification accuracy (Seurat, cyclone) degrades from 82% to 41%.
Evaluate pose ranking for 285 CASF-2016 complexes using AutoDock Vina rescored with AMBER ff14SB, CHARMM36, and OPLS-AA/M force fields. The top-ranked pose agrees between force fields in only 41% of cases.