embed_strictMono_of_one_lt

"Sem-Path ... order-preserving minimum-cost alignment ... 1/(1+J_d)"

"temporal heterogeneity... intralayer redundancy... overlap coefficient O(K(X),K(Y))... Greedy Star Clustering"

"TRA is non-dispersive ... TDA controls the expected number of spurious survivors per row to O(1)"

"Gaussian approximation (8) and (16) equating power of arithmetic-mean and winsorized max-tests to P(‖Σ_0^{1/2}Z + √n D^{-1}μ‖_∞ > c_{n,1-α})"

"The rhythmic dynamics of Clapping Music find a connection with recent research on aperiodic tilings, such as that presented by Fujita and Niizeki"

"the number of uncovered patterns decays at least geometrically fast with index, and greedy algorithms suffice"

"nebular drag … ρ = ρ_c (r/r_c)^d, c = c_c (r/r_c)^{s/2}; resulting K_Drag (25)"

"T_m = 10(2^m - 1) ... m_k = 2^{k-1}(8n + 16) - 16, s_k = 8(n - k) ... doubling flow ... control flow"

"optimal bandgap ladders … DP optimizer yields the ladder Eg(N=5) ≈ (2.10, 1.78, 1.50, 1.24, 1.00) eV"

"R(˜f)−R(f)=1/4(E[f(x)y]−E[fA(x)y])+1/8(E[f(x)fA(x)]−1) (accuracy gap + diversity term)"

"This local approach avoids global feedback chains... TEP maintains stable performance across all scales through local optimization."

"Theorem 3.2 … Rényi-2 entropy … effective rank reff(Σ) = (tr Σ)² / tr(Σ²)"

"diffusion kinetics... reaction-diffusion model... effective diffusivity... swelling-induced modulation of molecular transport"

"Proposition 4.2 ... non-expansiveness of Ψ w.r.t. ∥·∥_V1"

"When preference probabilities follow the Bradley-Terry model P(yw ≻ yl|x) = σ(U(x, yw)−U(x, yl)), the DPO optimal policy satisfies rθ(x, yi)−rθ(x, yj) = U(x, yi)−U(x, yj)"

"For any x1 ≤ x2, the activation vectors satisfy a(x1) ≤ a(x2) element-wise by construction. Since w+_j = ReLU(w_j) ≥ 0, the dot product preserves this ordering"

"ΠG[s;f](x,y) := ∫_G s(f(φ_{g^{-1}}(x)), ψ_{g^{-1}}(y)) dμ(g) ... E[ν(S_{f^G})] ≤ E[ν(S_f)] for increasing convex ν"

"the relation between intrinsic dimension and brain predictivity arises over model pre-training... finetuning models to better predict the brain causally increases both representations' intrinsic dimension and their semantic content"

"Proposition 3.1 one-step sampling error W₂²(pBayes,p) ≤ E Var(xt|xt−1) via temporal coupling"

"GNN-CLS ... Ltotal = λtask · Ltask + λtopo · (Ltopo(U) + Ltopo(W))"

"Krylov spread complexity CS = ℓy/λ with Lanczos coefficients minimizing cost function (3.31),(3.38)"

"the empirical distribution µt tends toward a multi-atomic structure µt ⇀ ∑ πk δLk"

"dspec = dbase + (at · Mf · γ) · ϕload · ϕtput"

"For monoids, we establish a trichotomy: the space complexity is either Θ(1), Θ(log n), or Θ(n)... constant-space solution for commutative monoids... FL∨Com... all remaining monoids require Ω(n) space."

"Restrict to maximal convex subset K* = conv(K_m) away from origin to bound α away from 0 (Corollary 6, Remark 10)"

"interacting particle system that minimizes a Riesz energy functional ... min |F(Ω_i) − F(Ω_j)|^{-s}"

"The Fexxv and Fexxvi emission lines exhibit remarkably large velocity widths, of several thousand km s−1"

"ρ(v) = 2^m √det(A diag(v²) Aᵀ) as fiber density / dynamic manipulability"

"temporal window centered on speech onset … 600 ms width … performance peaks … near speech onset (-200 ms to +400 ms)"

"The faces of A define a 6-regular graph G_A ... contour γ is a maximal connected component ... |Γ_m_l| = O(l^2 10^l)"