embed_injective

"α(s) = ξ(s) t(s) + μ(s) b(s) (rectifying condition)"

"Lemma 2. The minimum size of a channel decomposition of G is equal to the width w_G of G."

"the continuous extensions of tensor networks maintain ... expressiveness ... invariance under gauge transformations"

"Our contract eliminates the ability of a producer to predict or control the generated random numbers, including the stored history of random numbers. ... Merlin chain ... Vx = SHA3(Vx+1)"

"Minimal Nonfaces as Informative Attribute Release Sequences... Lemma 1"

"the deterministic entropy Sd... dSd ≡ DN dJ/J... the entropy of a dynamical system is its adiabatic invariant."

"the cost of mismatch vanishes if and only if ν lies on the tilted family of the true distribution μ"

"Theorem 1.1 … a3(3^{2α}n + (3^{2α}-1)/4) ≡ 0 (mod 3^α) … proved via the matrix recurrence (2.19) and valuation bounds (4.1)–(4.5)"

"Z = ⊕_{i=0}^∞ (A+BK)^i B W, ˜p ∈ KZ ⊕ W, Jury stability triangle for q1,q2"

"mailbox theory ... encapsulating the information as a mailbox ... cognition is continuously implemented during the transmission process"

"f_G(n)=p a(n)^2 with p depending on parity of n; Chebyshev roots of ∑(2T_{s_j}(w)−2)=1"

"ALIVE unifies problem posing, solving, and judging within a single policy model to internalize the logic of correctness. By coupling adversarial learning with instructive verbal feedback..."

"LGenEvolve = LGRPO + λSDL LSDL with importance-weighted sampled-token reverse-KL"

"Random Shuffling... Random Dropout... approximates sampling from the attribute power set... Attribute Replacement... hard antagonist from the same dimension"

"the r-value estimates how likely a candidate’s latent score belongs to the top-ranked group after accounting for both its mean score and its uncertainty"

"current VLMs rely heavily on shallow spatial cues, struggle to build stable coordinate-aware representations"

"τ+m,j = 0 if y+m,j ∈ y− else 1; τ−m,j = 0 if y−m,j ∈ y+ else −1 (token-level sign flips that leave shared tokens untouched)"

"We use both of these formulations in our framework depending on whether actionable attributes are known or unknown, where the latter uses the latent representations as our attributes (§3.1)"

"Theorem 2.1 (Böcherer–Nagaoka) and application of Θ operator to Leech lattice theta series"

"the vorticity vector Ω... super-Poynting vector Pα = ηαβγδ Eβρ Hγρ uδ... non-vanishing component of the latter associated to a state of gravitational radiation"

"Theorem 1 (Observability of X at S by R): ... π+ : Si +Xi→ Rj, π− : Rj −Xj→ Si, Xj ⊆ Xi"

"While these worst-case elevation angles are correlated for neighboring BSs due to the structure of the PV tessellation, it does not explicitly appear in our analysis because of our focus on the average interference."

"van der Corput estimates on {x/k−a}−{x/k−b}"

"Proposition 1.1 … any set of distinct nonzero multiplicative functions is linearly independent"

"Theorem 1. For n ≥ 1, assume p(x) = ∑ cj Tj(x) ... there exists a σ2 neural network with at most ⌊log2 n⌋ + 1 hidden layers ... O(n) neurons and total non-zero weights."

"The arithmetic of the p-adic increments is proposed... Δv_p(n!)=(1-Δs_p(n))/(p-1)... Δs_p(n)=1-(p-1)v_p(n+1)"

"graph-induced bandwidth ... inverse of a graph-induced banded matrix (Theorem 3.6)"

"A(ℓ) = F²_ℓ |λ(ℓ)–μ*/(1+λ(ℓ))|² and Tc(ℓ)=θ exp(–1–λ(ℓ))/(λ(ℓ)–μ*)"

"The first order for the vertex in particle-hole (PH) channel is given by the diagram shown in Fig.7 ˜Σ(1)12 = −∑s−QMC ∑3,4 γd1234(s) ˜G034"

"Theorem 4 … E(f listD) ≤ E(f pairD)"