embed_injective

"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"

"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)"

"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)"

"strange-loop theory... self-referential feedback loop... two-way circulation... interaction (high-level) ... self-reinforcement (low-level)"

"Theorem 7.2: complete Banakh space with G+ subset d[X^2] for non-cyclic G subset Q is isometric to R."

"Theorem D... µn(L′)=µn(L) iff (n+1)-basing... Stallings-Dwyer Theorem"

"Theorem 3.6 ... (k−1−|E|)-knight is a solution to T(D)"

"Theorem 2.14. We have that K₀(Arith_{S¹}(R)) ≃ (⊕_{p∈Spec(R)(0)} Z) ⊕ Z, where the extra copy of Z is the image of a canonical splitting K₀(C(S¹,F)) → K₀(Arith_{S¹}(R)). Each copy of Z indexed by a height 1 prime p ... generated by the class of R/p."

"Theorem 1.1. ... β_k(X^R_B) = sum_{I in [n+1] choose 2k} #alternating B|I-permutations"

"Sim-CLIP adopts a Siamese training architecture with a cosine similarity objective … without requiring large batch sizes or additional momentum encoders"

"dθ/dt = ... dϕ/dt = ... θs = arccos(−1/(2N+1)) ... all states evolve onto a cycle"

"compact-phase quantum geometry... nonuniform first-Chern matrix... non-identical anyons"

"Theorem 1 (Mutual Information Non-Increasing Theorem)... I(sa1; sa2) ≥ I(sb1; sb2)"

"points (a_n,b_n) on parabola P constructed by successive automorphisms ... (ηζ)^n ... Martin harmonic functions h_α = sum κ_m(α) exp(z0·(a_m,b_m))"

"Properties: [0,1]-valued, transformation invariance under isometries of (P_2,W_2), vanishing at infinity, maximality at spatial median"

"Our evaluation shows that models are prone to hallucinations on object existence, and even more so on fine-grained attributes."

"excess EPR vanishes in steady state; housekeeping quantifies nonconservative forces; dual form min D(j'∥˜j) s.t. ∇^T j'=ẋ"

"Lemma 1 ... continuous multivariate function f that ... is permutation-invariant ... approximated using Dot-attention"