s stringlengths 1 39 | o stringclasses 996
values | p stringlengths 7 77 | confidence float64 0.4 1 | predicate_class stringclasses 2
values | community int64 0 14 ⌀ | equation_name stringlengths 14 73 ⌀ | equation_latex stringlengths 5 217 ⌀ | equation_sympy stringclasses 617
values | equation_form stringclasses 472
values | variables stringlengths 2 85 ⌀ | domain stringclasses 119
values | scm_name stringclasses 165
values | source stringlengths 9 50 ⌀ | extraction_method stringclasses 8
values |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
q | f | urn:hailstone:ml_ai:structural_equation[39114af362c3] | 1 | structural_equation | 0 | arxiv.cs.IT.2605.10594v1.eq_621 | q=\mathfrak{R}+1,d=μ+2 | arxiv_extracted | ["f", "q", "a", "t", "m", "h"] | arxiv | arxiv | arxiv:cs.IT:2605.10594v1 | cross_domain_amplification | |
r | l | urn:hailstone:ml_ai:structural_equation[db75c13be01f] | 1 | structural_equation | 0 | arxiv.cs.IT.2605.10508v1.eq_622 | (r,\ell)=(2,2) | arxiv_extracted | ["e", "l", "r"] | arxiv | arxiv | arxiv:cs.IT:2605.10508v1 | cross_domain_amplification | |
v | s | urn:hailstone:ml_ai:structural_equation[58240eff8aa4] | 1 | structural_equation | 0 | arxiv.cs.IT.2605.10395v1.eq_623 | \varepsilon^{\rm BO}=Θ(k_c d/n) | arxiv_extracted | ["e", "a", "p", "r", "s", "v"] | arxiv | arxiv | arxiv:cs.IT:2605.10395v1 | cross_domain_amplification | |
b | b | urn:hailstone:ml_ai:structural_equation[f2dda765b5fd] | 1 | structural_equation | 0 | arxiv.cs.IT.2605.09970v1.eq_624 | \bar{m} = q\binom{n}{3} | arxiv_extracted | ["q", "a", "m", "r", "b"] | arxiv | arxiv | arxiv:cs.IT:2605.09970v1 | cross_domain_amplification | |
w | x | urn:hailstone:ml_ai:structural_equation[e2fb31c49a80] | 1 | structural_equation | 0 | arxiv.cs.IT.2605.09561v1.eq_625 | w(x,y)=e^{-λd(x,y)} | arxiv_extracted | ["e", "w", "y", "d", "x"] | arxiv | arxiv | arxiv:cs.IT:2605.09561v1 | cross_domain_amplification | |
w | x | urn:hailstone:ml_ai:structural_equation[5ac8de263dcb] | 1 | structural_equation | 0 | arxiv.cs.IT.2605.09561v1.eq_626 | w(x,y)=e^{-d(x,y)^2/(2σ^2)} | arxiv_extracted | ["e", "w", "y", "d", "x"] | arxiv | arxiv | arxiv:cs.IT:2605.09561v1 | cross_domain_amplification | |
t | E | urn:hailstone:ml_ai:structural_equation[7b3b52e64be8] | 1 | structural_equation | 0 | arxiv.cs.IT.2605.08492v1.eq_627 | \text{FER}=4\cdot10^{-3} | arxiv_extracted | ["e", "F", "t", "E", "x"] | arxiv | arxiv | arxiv:cs.IT:2605.08492v1 | cross_domain_amplification | |
t | a | urn:hailstone:ml_ai:structural_equation[291aeca355c9] | 1 | structural_equation | 0 | arxiv.cs.IT.2605.07698v1.eq_628 | Φ_t(y)=\Pr_p[\mathrm{valid\ completion}\mid y] | arxiv_extracted | ["a", "t", "m", "r_p", "y", "P"] | arxiv | arxiv | arxiv:cs.IT:2605.07698v1 | cross_domain_amplification | |
w | w | urn:hailstone:ml_ai:structural_equation[c552baf786f5] | 1 | structural_equation | 0 | arxiv.cs.IT.2605.06826v1.eq_629 | δ,γ,α=w^{\top} R w | arxiv_extracted | ["w", "t", "o", "p", "R"] | arxiv | arxiv | arxiv:cs.IT:2605.06826v1 | cross_domain_amplification | |
d | e | urn:hailstone:ml_ai:structural_equation[a63462ef797d] | 1 | structural_equation | 0 | arxiv.cs.IT.2605.05168v1.eq_630 | \dot{C}_{\text{DI}}(W) = \frac 12 | arxiv_extracted | ["e", "t", "o", "d", "C"] | arxiv | arxiv | arxiv:cs.IT:2605.05168v1 | cross_domain_amplification | |
s | p_i | urn:hailstone:ml_ai:structural_equation[afd9b3f1e42e] | 1 | structural_equation | 0 | arxiv.cs.IT.2605.03685v1.eq_631 | \sum_{i=1}^n f(p_i) | arxiv_extracted | ["f", "m_{i", "p_i", "u", "s", "n"] | arxiv | arxiv | arxiv:cs.IT:2605.03685v1 | cross_domain_amplification | |
p_i | n | urn:hailstone:ml_ai:structural_equation[d7f4b7a28c99] | 1 | structural_equation | 0 | arxiv.cs.IT.2605.03685v1.eq_632 | (p_i)_{i=1}^n | arxiv_extracted | ["n", "p_i", "i"] | arxiv | arxiv | arxiv:cs.IT:2605.03685v1 | cross_domain_amplification | |
K | m | urn:hailstone:ml_ai:structural_equation[6aa607fc361c] | 1 | structural_equation | 0 | arxiv.cs.IT.2605.03552v1.eq_633 | K(u):=-\log_2 \mathbb{P}(X_1^m=u) | arxiv_extracted | ["g_2", "m", "o", "u", "K", "l"] | arxiv | arxiv | arxiv:cs.IT:2605.03552v1 | cross_domain_amplification | |
m | b | urn:hailstone:ml_ai:structural_equation[49a8ed95a49f] | 1 | structural_equation | 0 | arxiv.cs.IT.2605.03552v1.eq_634 | \mathbb{E}[|C(X_1)|]=\tfrac32, \qquad \mathbb{E}[|C(X_1^n)|]<\tfrac32\,n\quad (n\ge 2). | arxiv_extracted | ["a", "t", "m", "h", "b"] | arxiv | arxiv | arxiv:cs.IT:2605.03552v1 | cross_domain_amplification | |
Z_n | c | urn:hailstone:ml_ai:structural_equation[d2b294f34f2a] | 1 | structural_equation | 0 | arxiv.cs.IT.2605.01839v1.eq_635 | Z_n(β|\mathcal{C}) = \sum_{y^n}[P_{Y^n|\mathcal{C}}(y^n)]^β | arxiv_extracted | ["a", "t", "m", "h", "Z_n", "c"] | arxiv | arxiv | arxiv:cs.IT:2605.01839v1 | cross_domain_amplification | |
R | Y | urn:hailstone:ml_ai:structural_equation[3f05dd0b6050] | 1 | structural_equation | 0 | arxiv.cs.IT.2605.01839v1.eq_636 | (β, R) = (1, I(X;Y)) | arxiv_extracted | ["X", "Y", "R", "I"] | arxiv | arxiv | arxiv:cs.IT:2605.01839v1 | cross_domain_amplification | |
f | x | urn:hailstone:ml_ai:structural_equation[1e389ddd6db4] | 1 | structural_equation | 0 | arxiv.cs.IT.2604.26819v1.eq_637 | f(x) = |x| | arxiv_extracted | ["f", "x"] | arxiv | arxiv | arxiv:cs.IT:2604.26819v1 | cross_domain_amplification | |
k | M | urn:hailstone:ml_ai:structural_equation[aed9e94a4537] | 1 | structural_equation | 0 | arxiv.cs.IT.2604.24918v1.eq_638 | (k,M){=}(20,64) | arxiv_extracted | ["M", "k"] | arxiv | arxiv | arxiv:cs.IT:2604.24918v1 | cross_domain_amplification | |
m | a | urn:hailstone:ml_ai:structural_equation[1b7173053d90] | 1 | structural_equation | 0 | arxiv.cs.IT.2604.23097v1.eq_639 | \mathcal{C}(\boldsymbolα)=\operatorname{im}Φ_{\boldsymbolα} | arxiv_extracted | ["a", "t", "m", "h", "c"] | arxiv | arxiv | arxiv:cs.IT:2604.23097v1 | cross_domain_amplification | |
b | y | urn:hailstone:ml_ai:structural_equation[40adcfffcb55] | 1 | structural_equation | 0 | arxiv.cs.IT.2604.23097v1.eq_640 | Φ_{\boldsymbolα}=\sum_iα_iF_i | arxiv_extracted | ["y", "o", "s", "b", "d", "l"] | arxiv | arxiv | arxiv:cs.IT:2604.23097v1 | cross_domain_amplification | |
d | e | urn:hailstone:ml_ai:structural_equation[b1e2dcb7a38e] | 1 | structural_equation | 0 | arxiv.cs.IT.2604.23097v1.eq_641 | \dim\operatorname{Hull}(\mathcal{C}(\boldsymbolα))=\operatorname{rank}(Φ_{\boldsymbolα})-\operatorname{rank}(G(\boldsymbolα)) | arxiv_extracted | ["e", "m", "o", "i", "p", "d"] | arxiv | arxiv | arxiv:cs.IT:2604.23097v1 | cross_domain_amplification | |
C_{λ | a | urn:hailstone:ml_ai:structural_equation[0c1003c617fe] | 1 | structural_equation | 0 | arxiv.cs.IT.2604.23097v1.eq_642 | C_{λ,μ}=\operatorname{im}(λx+μL(x)) | arxiv_extracted | ["e", "a", "C_{\u03bb", "o", "p", "r"] | arxiv | arxiv | arxiv:cs.IT:2604.23097v1 | cross_domain_amplification | |
m | a | urn:hailstone:ml_ai:structural_equation[bdf21874a0be] | 1 | structural_equation | 0 | arxiv.cs.IT.2604.23097v1.eq_643 | \mathcal{C}=\langle X,F_1,\ldots,F_k\rangle_{\mathbb{F}_{q^m}} | arxiv_extracted | ["a", "t", "m", "h", "c"] | arxiv | arxiv | arxiv:cs.IT:2604.23097v1 | cross_domain_amplification | |
X | d | urn:hailstone:ml_ai:structural_equation[ecbb7bf6fa16] | 1 | structural_equation | 0 | arxiv.cs.IT.2604.20729v1.eq_644 | δ_X(d)=1 | arxiv_extracted | ["X", "d"] | arxiv | arxiv | arxiv:cs.IT:2604.20729v1 | cross_domain_amplification | |
a | i | urn:hailstone:ml_ai:structural_equation[b62af06bf6ef] | 1 | structural_equation | 0 | arxiv.cs.IT.2604.19712v1.eq_645 | α_a=\lim_{s\rightarrow\infty} α_{ult_s} = \lim_{s\rightarrow\infty} \barα{ult_s} = \lim_{r\rightarrow\infty} α_{c}^{(r)} | arxiv_extracted | ["m_{s", "a", "i", "r", "l"] | arxiv | arxiv | arxiv:cs.IT:2604.19712v1 | cross_domain_amplification | |
p | s | urn:hailstone:ml_ai:structural_equation[b551981ccad8] | 1 | structural_equation | 0 | arxiv.cs.IT.2604.17864v1.eq_646 | p \ge t=2s | arxiv_extracted | ["e", "g", "t", "p", "s"] | arxiv | arxiv | arxiv:cs.IT:2604.17864v1 | cross_domain_amplification | |
l | i | urn:hailstone:ml_ai:structural_equation[d49d8dacd1cf] | 1 | structural_equation | 0 | arxiv.cs.IT.2604.17443v1.eq_647 | l^{best}_{i}=i | arxiv_extracted | ["e", "t", "i", "s", "b", "l"] | arxiv | arxiv | arxiv:cs.IT:2604.17443v1 | cross_domain_amplification | |
v | s | urn:hailstone:ml_ai:structural_equation[4a80face72e3] | 1 | structural_equation | 0 | arxiv.cs.IT.2604.14915v1.eq_648 | \varepsilon = A^{-β} | arxiv_extracted | ["e", "a", "p", "r", "s", "v"] | arxiv | arxiv | arxiv:cs.IT:2604.14915v1 | cross_domain_amplification | |
K | p | urn:hailstone:ml_ai:structural_equation[9a5db8288cf0] | 1 | structural_equation | 0 | arxiv.cs.IT.2604.14915v1.eq_649 | K_\varepsilon(A) = Θ(A\sqrt{\log A}) | arxiv_extracted | ["e", "a", "p", "r", "K", "v"] | arxiv | arxiv | arxiv:cs.IT:2604.14915v1 | cross_domain_amplification | |
k | o | urn:hailstone:ml_ai:structural_equation[74ab09066f55] | 1 | structural_equation | 0 | arxiv.cs.IT.2604.14671v1.eq_650 | α(k,q)=λ_{k-1}q^k+o(q^k) | arxiv_extracted | ["q", "k", "o"] | arxiv | arxiv | arxiv:cs.IT:2604.14671v1 | cross_domain_amplification | |
m | f | urn:hailstone:ml_ai:structural_equation[fd6d35446ff1] | 1 | structural_equation | 0 | arxiv.cs.IT.2604.14667v1.eq_651 | \mathbf{A}=(A_0,\cdots,A_{M-1}) | arxiv_extracted | ["f", "a", "t", "m", "h", "b"] | arxiv | arxiv | arxiv:cs.IT:2604.14667v1 | cross_domain_amplification | |
m | f | urn:hailstone:ml_ai:structural_equation[31e23268e7f3] | 1 | structural_equation | 0 | arxiv.cs.IT.2604.14667v1.eq_652 | \mathbf{B}=({B}_0,\cdots,{B}_{M-1}) | arxiv_extracted | ["f", "a", "t", "m", "h", "b"] | arxiv | arxiv | arxiv:cs.IT:2604.14667v1 | cross_domain_amplification | |
F | q | urn:hailstone:ml_ai:structural_equation[f9bfb9026718] | 1 | structural_equation | 0 | arxiv.cs.IT.2604.13967v1.eq_653 | F(x) = x^{3q - 2} | arxiv_extracted | ["q", "F", "x"] | arxiv | arxiv | arxiv:cs.IT:2604.13967v1 | cross_domain_amplification | |
b | x | urn:hailstone:ml_ai:structural_equation[ab8b7ffa98e6] | 1 | structural_equation | 0 | arxiv.cs.IT.2604.13678v1.eq_654 | \bm{X}=\bm{x}\bm{x}^* | arxiv_extracted | ["X", "b", "m", "x"] | arxiv | arxiv | arxiv:cs.IT:2604.13678v1 | cross_domain_amplification | |
b | t | urn:hailstone:ml_ai:structural_equation[ee596300d6c5] | 1 | structural_equation | 0 | arxiv.cs.IT.2604.13678v1.eq_655 | \bm{y}=\mathcal{A}(\bm{X})\in\mathbb{R}^m | arxiv_extracted | ["a", "t", "m", "y", "b"] | arxiv | arxiv | arxiv:cs.IT:2604.13678v1 | cross_domain_amplification | |
m | a | urn:hailstone:ml_ai:structural_equation[f1820df07e56] | 1 | structural_equation | 0 | arxiv.cs.IT.2605.00848v1.eq_656 | \mathcal{N}_G = C_ρ^{-2} | arxiv_extracted | ["a", "t", "m", "h", "c"] | arxiv | arxiv | arxiv:cs.IT:2605.00848v1 | cross_domain_amplification | |
m | m | urn:hailstone:ml_ai:structural_equation[cbdd82f78e51] | 1 | structural_equation | 0 | arxiv.cs.IT.2604.12340v1.eq_657 | λ_{\mathrm{cut}}^{*} = ε | arxiv_extracted | ["a", "t", "m", "h", "r"] | arxiv | arxiv | arxiv:cs.IT:2604.12340v1 | cross_domain_amplification | |
m | a | urn:hailstone:ml_ai:structural_equation[1f45b728b6d9] | 1 | structural_equation | 0 | arxiv.cs.IT.2604.12187v4.eq_658 | \mathcal{H}_0 = L^2(\mathcal{C}, ν_0) | arxiv_extracted | ["a", "t", "m", "h", "c"] | arxiv | arxiv | arxiv:cs.IT:2604.12187v4 | cross_domain_amplification | |
m | a | urn:hailstone:ml_ai:structural_equation[d852f3660d88] | 1 | structural_equation | 0 | arxiv.cs.IT.2604.12186v1.eq_659 | (\mathcal{G}=\mathbb{Z}_q) | arxiv_extracted | ["a", "t", "m", "h", "c"] | arxiv | arxiv | arxiv:cs.IT:2604.12186v1 | cross_domain_amplification | |
d | e | urn:hailstone:ml_ai:structural_equation[b93f60953ab9] | 1 | structural_equation | 0 | arxiv.cs.IT.2604.11782v1.eq_660 | \dot{C}_{\text{DI}}(\mathcal{G})=\frac{1}{2} | arxiv_extracted | ["e", "t", "o", "d", "C"] | arxiv | arxiv | arxiv:cs.IT:2604.11782v1 | cross_domain_amplification | |
v | s | urn:hailstone:ml_ai:structural_equation[262852a97a52] | 1 | structural_equation | 0 | arxiv.cs.IT.2604.09521v1.eq_661 | \varepsilon = O(1/T) | arxiv_extracted | ["e", "a", "p", "r", "s", "v"] | arxiv | arxiv | arxiv:cs.IT:2604.09521v1 | cross_domain_amplification | |
x | e | urn:hailstone:ml_ai:structural_equation[782b5027ded9] | 1 | structural_equation | 0 | arxiv.cs.IT.2604.09477v1.eq_662 | x_{\ell}=A^{\ell}f | arxiv_extracted | ["e", "A", "l", "x"] | arxiv | arxiv | arxiv:cs.IT:2604.09477v1 | cross_domain_amplification | |
p | d | urn:hailstone:ml_ai:structural_equation[e3a055e0a2d7] | 1 | structural_equation | 0 | arxiv.cs.IT.2604.08311v1.eq_663 | , provided that \[ \ell(n):=\min_{γ:~\F_2(γ)=\F_{2^n}} \dim_{\F_2}\F_2[σ]γ>m, \] where | arxiv_extracted | ["i", "o", "p", "r", "d", "v"] | arxiv | arxiv | arxiv:cs.IT:2604.08311v1 | cross_domain_amplification | |
r | l | urn:hailstone:ml_ai:structural_equation[db75c13be01f] | 1 | structural_equation | 0 | arxiv.cs.IT.2604.04519v2.eq_664 | (r,\ell)=(2,2) | arxiv_extracted | ["e", "l", "r"] | arxiv | arxiv | arxiv:cs.IT:2604.04519v2 | cross_domain_amplification | |
k | n | urn:hailstone:ml_ai:structural_equation[7e75d1f84bf8] | 1 | structural_equation | 0 | arxiv.cs.IT.2604.03991v1.eq_665 | ker(φ)=\langle π\rangle | arxiv_extracted | ["e", "k", "a", "r", "n", "l"] | arxiv | arxiv | arxiv:cs.IT:2604.03991v1 | cross_domain_amplification | |
m | m | urn:hailstone:ml_ai:structural_equation[5fa9cf170511] | 1 | structural_equation | 0 | arxiv.cs.IT.2604.00089v1.eq_666 | \mathrm{ci}_\circ(N) = 0 | arxiv_extracted | ["a", "t", "m", "h", "r"] | arxiv | arxiv | arxiv:cs.IT:2604.00089v1 | cross_domain_amplification | |
m | m | urn:hailstone:ml_ai:structural_equation[20972ba365c2] | 1 | structural_equation | 0 | arxiv.cs.IT.2604.00089v1.eq_667 | \mathrm{ci}_\circ(N \otimes \mathrm{id}^c_β) = |X| | arxiv_extracted | ["a", "t", "m", "h", "r"] | arxiv | arxiv | arxiv:cs.IT:2604.00089v1 | cross_domain_amplification | |
K | r | urn:hailstone:ml_ai:structural_equation[e147e69221e7] | 1 | structural_equation | 0 | arxiv.cs.IT.2603.29945v2.eq_668 | (K, KM/N)=(2q+1, 2r) | arxiv_extracted | ["q", "M", "r", "K", "N"] | arxiv | arxiv | arxiv:cs.IT:2603.29945v2 | cross_domain_amplification | |
N | r | urn:hailstone:ml_ai:structural_equation[66f65a2e5672] | 1 | structural_equation | 0 | arxiv.cs.IT.2603.28918v1.eq_669 | N_\mathrm{MC}=400 | arxiv_extracted | ["a", "t", "m", "h", "r", "N"] | arxiv | arxiv | arxiv:cs.IT:2603.28918v1 | cross_domain_amplification | |
N | r | urn:hailstone:ml_ai:structural_equation[f48d3039c664] | 1 | structural_equation | 0 | arxiv.cs.IT.2603.28918v1.eq_670 | N_\mathrm{RF}=8 | arxiv_extracted | ["a", "t", "m", "h", "r", "N"] | arxiv | arxiv | arxiv:cs.IT:2603.28918v1 | cross_domain_amplification | |
s | r_{3 | urn:hailstone:ml_ai:structural_equation[a988c6cd54d5] | 1 | structural_equation | 0 | arxiv.cs.IT.2603.25757v2.eq_671 | σ^\star_{3,5}=0.10 | arxiv_extracted | ["s", "t", "r_{3", "a"] | arxiv | arxiv | arxiv:cs.IT:2603.25757v2 | cross_domain_amplification | |
s | r_{5 | urn:hailstone:ml_ai:structural_equation[dbfaeb109032] | 1 | structural_equation | 0 | arxiv.cs.IT.2603.25757v2.eq_672 | σ^\star_{5,7}=0.1375 | arxiv_extracted | ["s", "r_{5", "t", "a"] | arxiv | arxiv | arxiv:cs.IT:2603.25757v2 | cross_domain_amplification | |
m | m | urn:hailstone:ml_ai:structural_equation[95a0c5a43f72] | 1 | structural_equation | 0 | arxiv.cs.IT.2603.25757v2.eq_673 | Δ_{\mathrm{MWPM-UF}}=-0.00383 | arxiv_extracted | ["a", "t", "m", "h", "r"] | arxiv | arxiv | arxiv:cs.IT:2603.25757v2 | cross_domain_amplification | |
d | q | urn:hailstone:ml_ai:structural_equation[56cc3614af99] | 1 | structural_equation | 0 | arxiv.cs.IT.2603.21391v2.eq_674 | dy/dx = y^q | arxiv_extracted | ["q", "d", "y", "x"] | arxiv | arxiv | arxiv:cs.IT:2603.21391v2 | cross_domain_amplification | |
q_n | n | urn:hailstone:ml_ai:structural_equation[e4b1443cbc8b] | 1 | structural_equation | 0 | arxiv.cs.IT.2603.22358v2.eq_675 | 1-q_n = O(n^{-1}) | arxiv_extracted | ["n", "q_n", "O"] | arxiv | arxiv | arxiv:cs.IT:2603.22358v2 | cross_domain_amplification | |
H_1 | d | urn:hailstone:ml_ai:structural_equation[8fe1fe801aa7] | 1 | structural_equation | 0 | arxiv.cs.IT.2603.20135v1.eq_676 | H_1: θ\in [L]:=\{1,\ldots,L\} | arxiv_extracted | ["H_1", "i", "L", "n", "d", "l"] | arxiv | arxiv | arxiv:cs.IT:2603.20135v1 | cross_domain_amplification | |
w | a | urn:hailstone:ml_ai:structural_equation[dc39b059bc9e] | 1 | structural_equation | 0 | arxiv.cs.IT.2603.19559v1.eq_677 | \widehat C=\widehat A\,\widehat B | arxiv_extracted | ["e", "w", "a", "h", "i", "d"] | arxiv | arxiv | arxiv:cs.IT:2603.19559v1 | cross_domain_amplification | |
u | r | urn:hailstone:ml_ai:structural_equation[660076bd7a65] | 1 | structural_equation | 0 | arxiv.cs.IT.2603.19559v1.eq_678 | u_{\mathrm{peak}}=\pm\sqrt{3-1/ρ^2} | arxiv_extracted | ["a", "t", "m", "h", "u", "r"] | arxiv | arxiv | arxiv:cs.IT:2603.19559v1 | cross_domain_amplification | |
p | m | urn:hailstone:ml_ai:structural_equation[f2fc31a0ed6a] | 1 | structural_equation | 0 | arxiv.cs.IT.2603.18568v1.eq_679 | \prod_{i=1}^{s} \mathbb{F}_{q^{n_i}} | arxiv_extracted | ["m", "o", "p", "r", "s", "d_{i"] | arxiv | arxiv | arxiv:cs.IT:2603.18568v1 | cross_domain_amplification | |
k | n | urn:hailstone:ml_ai:structural_equation[baf23fba0908] | 1 | structural_equation | 0 | arxiv.cs.IT.2603.17703v1.eq_680 | kd^2/n = 8.40 | arxiv_extracted | ["n", "d", "k"] | arxiv | arxiv | arxiv:cs.IT:2603.17703v1 | cross_domain_amplification | |
k | n | urn:hailstone:ml_ai:structural_equation[baf23fba0908] | 1 | structural_equation | 0 | arxiv.cs.IT.2603.17703v1.eq_681 | kd^2/n = 8.40 | arxiv_extracted | ["n", "d", "k"] | arxiv | arxiv | arxiv:cs.IT:2603.17703v1 | cross_domain_amplification | |
k | n | urn:hailstone:ml_ai:structural_equation[6cb6cef2d3dd] | 1 | structural_equation | 0 | arxiv.cs.IT.2603.17703v1.eq_682 | kd^2/n = 2.67 | arxiv_extracted | ["n", "d", "k"] | arxiv | arxiv | arxiv:cs.IT:2603.17703v1 | cross_domain_amplification | |
k | n | urn:hailstone:ml_ai:structural_equation[357371850321] | 1 | structural_equation | 0 | arxiv.cs.IT.2603.17703v1.eq_683 | kd^2/n = 6.48 | arxiv_extracted | ["n", "d", "k"] | arxiv | arxiv | arxiv:cs.IT:2603.17703v1 | cross_domain_amplification | |
k | n | urn:hailstone:ml_ai:structural_equation[2ed32e7e4752] | 1 | structural_equation | 0 | arxiv.cs.IT.2603.17703v1.eq_684 | kd^2/n = 10.0 | arxiv_extracted | ["n", "d", "k"] | arxiv | arxiv | arxiv:cs.IT:2603.17703v1 | cross_domain_amplification | |
F | y | urn:hailstone:ml_ai:structural_equation[8b0ddf3a2323] | 1 | structural_equation | 0 | arxiv.cs.IT.2603.15146v1.eq_685 | . For \[ G_a(x,y,z)=(x^{q+1}+ax^qz+yz^q,\; x^qz+y^{q+1},\; xy^q+ay^qz+z^{q+1}), \] where | arxiv_extracted | ["F", "y", "G_a", "o", "r", "x"] | arxiv | arxiv | arxiv:cs.IT:2603.15146v1 | cross_domain_amplification | |
g | m | urn:hailstone:ml_ai:structural_equation[905d801eb9a6] | 1 | structural_equation | 0 | arxiv.cs.IT.2603.15146v1.eq_686 | \gcd(i,m)=1 | arxiv_extracted | ["g", "m", "i", "c", "d"] | arxiv | arxiv | arxiv:cs.IT:2603.15146v1 | cross_domain_amplification | |
a | q | urn:hailstone:ml_ai:structural_equation[32551bddbcf3] | 1 | structural_equation | 0 | arxiv.cs.IT.2603.15146v1.eq_687 | a^{q^2+q+1}=1 | arxiv_extracted | ["q", "a"] | arxiv | arxiv | arxiv:cs.IT:2603.15146v1 | cross_domain_amplification | |
H | t | urn:hailstone:ml_ai:structural_equation[2cb413e61d94] | 1 | structural_equation | 0 | arxiv.cs.IT.2603.14649v1.eq_688 | H(d_1,\ldots,d_n) = \sum_{v\in V} d_v \log_2(m/d_v) | arxiv_extracted | ["t", "H", "o", "d_1", "d", "l"] | arxiv | arxiv | arxiv:cs.IT:2603.14649v1 | cross_domain_amplification | |
m | m | urn:hailstone:ml_ai:structural_equation[7ddaca6e340a] | 1 | structural_equation | 0 | arxiv.cs.IT.2603.10981v1.eq_689 | \mathrm{d}_\mathrm{L} = \mathrm{d}_\mathrm{P} = 2 | arxiv_extracted | ["a", "t", "m", "h", "r"] | arxiv | arxiv | arxiv:cs.IT:2603.10981v1 | cross_domain_amplification | |
M | K | urn:hailstone:ml_ai:structural_equation[ccd41429adaa] | 1 | structural_equation | 0 | arxiv.cs.IT.2603.10769v1.eq_690 | (M,N,T,K)=(2,4,2,2) | arxiv_extracted | ["K", "M", "N", "T"] | arxiv | arxiv | arxiv:cs.IT:2603.10769v1 | cross_domain_amplification | |
M | K | urn:hailstone:ml_ai:structural_equation[3f0ed0006974] | 1 | structural_equation | 0 | arxiv.cs.IT.2603.10769v1.eq_691 | (M,N,T,K)=(2,N,2,K) | arxiv_extracted | ["K", "M", "N", "T"] | arxiv | arxiv | arxiv:cs.IT:2603.10769v1 | cross_domain_amplification | |
M | K | urn:hailstone:ml_ai:structural_equation[3f0ed0006974] | 1 | structural_equation | 0 | arxiv.cs.IT.2603.10769v1.eq_692 | (M,N,T,K)=(2,N,2,K) | arxiv_extracted | ["K", "M", "N", "T"] | arxiv | arxiv | arxiv:cs.IT:2603.10769v1 | cross_domain_amplification | |
v | s | urn:hailstone:ml_ai:structural_equation[9dad8846ddac] | 1 | structural_equation | 0 | arxiv.cs.IT.2603.10073v1.eq_693 | \varepsilon_0 = \varepsilon_0(n) | arxiv_extracted | ["e", "a", "p", "r", "s", "v"] | arxiv | arxiv | arxiv:cs.IT:2603.10073v1 | cross_domain_amplification | |
L_n | D_C | urn:hailstone:ml_ai:structural_equation[8cc6d342e795] | 1 | structural_equation | 0 | arxiv.cs.IT.2603.08308v3.eq_694 | L_n^* = \exp\{-n D_C^{\mathrm{w}}(\mathbb{P}, \mathbb{Q}) + o(n)\}, \quad n \to \infty, | arxiv_extracted | ["e", "p", "L_n", "D_C", "n", "x"] | arxiv | arxiv | arxiv:cs.IT:2603.08308v3 | cross_domain_amplification | |
v | i | urn:hailstone:ml_ai:structural_equation[c3fbf7539c10] | 1 | structural_equation | 0 | arxiv.cs.IT.2603.08308v3.eq_695 | \varphi(x_1^n) = \prod_{i=1}^n \varphi(x_i) | arxiv_extracted | ["a", "h", "i", "p", "r", "v"] | arxiv | arxiv | arxiv:cs.IT:2603.08308v3 | cross_domain_amplification | |
C | r | urn:hailstone:ml_ai:structural_equation[c55b76ac18bd] | 1 | structural_equation | 0 | arxiv.cs.IT.2603.07415v1.eq_696 | C_\mathrm{ctx} = 0 | arxiv_extracted | ["a", "t", "m", "h", "r", "C"] | arxiv | arxiv | arxiv:cs.IT:2603.07415v1 | cross_domain_amplification | |
s | o | urn:hailstone:ml_ai:structural_equation[9885915f953f] | 1 | structural_equation | 0 | arxiv.cs.IT.2603.06893v1.eq_697 | \sum_{i=1}^{N}(\log_2(1+a_iP_i)-T_i)^2 | arxiv_extracted | ["m_{i", "o", "u", "s", "l", "N"] | arxiv | arxiv | arxiv:cs.IT:2603.06893v1 | cross_domain_amplification | |
U_C | H_C} | urn:hailstone:ml_ai:structural_equation[a2f823638976] | 1 | structural_equation | 0 | arxiv.cs.IT.2603.01809v2.eq_698 | U_C(γ)=e^{-iγH_C} | arxiv_extracted | ["e", "i", "H_C}", "U_C"] | arxiv | arxiv | arxiv:cs.IT:2603.01809v2 | cross_domain_amplification | |
k | w | urn:hailstone:ml_ai:structural_equation[be574892b2e1] | 1 | structural_equation | 0 | arxiv.cs.IT.2603.00910v1.eq_699 | ζ_k^2 = g_k^\top \widetilde{H}_{kk}^{-1} g_k | arxiv_extracted | ["g_k", "w", "t", "o", "p", "k"] | arxiv | arxiv | arxiv:cs.IT:2603.00910v1 | cross_domain_amplification | |
o | i | urn:hailstone:ml_ai:structural_equation[45094d0faa9e] | 1 | structural_equation | 0 | arxiv.cs.IT.2602.21315v2.eq_700 | \overline{p} = (p_0,p_1,p_2,\ldots) | arxiv_extracted | ["e", "o", "i", "r", "l", "v"] | arxiv | arxiv | arxiv:cs.IT:2602.21315v2 | cross_domain_amplification | |
L_x | d | urn:hailstone:ml_ai:structural_equation[9629d88056f5] | 1 | structural_equation | 0 | arxiv.cs.IT.2602.19057v1.eq_701 | (L_x, L_y) = (2d, d) | arxiv_extracted | ["L_y", "L_x", "d"] | arxiv | arxiv | arxiv:cs.IT:2602.19057v1 | cross_domain_amplification | |
m | b | urn:hailstone:ml_ai:structural_equation[a748373de233] | 1 | structural_equation | 0 | arxiv.cs.IT.2602.18255v1.eq_702 | \mathbb F_2[u]/ \langle u^k \rangle= \mathbb F_2+u\mathbb F_2+u^2\mathbb F_2+\cdots+u^{k-1}\mathbb F_2 , | arxiv_extracted | ["a", "t", "m", "h", "b"] | arxiv | arxiv | arxiv:cs.IT:2602.18255v1 | cross_domain_amplification | |
m | a | urn:hailstone:ml_ai:structural_equation[be66cb712a31] | 1 | structural_equation | 0 | arxiv.cs.IT.2602.18255v1.eq_703 | \mathcal{R}=M_4 (\mathbb F_2( u)/ \langle u^k \rangle) | arxiv_extracted | ["a", "t", "m", "h", "c"] | arxiv | arxiv | arxiv:cs.IT:2602.18255v1 | cross_domain_amplification | |
v | v | urn:hailstone:ml_ai:structural_equation[968e4003adad] | 1 | structural_equation | 0 | arxiv.cs.IT.2602.18255v1.eq_704 | v^4=0, u^k=0, u^iv^j=v^ju^i | arxiv_extracted | ["u", "k", "v", "i"] | arxiv | arxiv | arxiv:cs.IT:2602.18255v1 | cross_domain_amplification | |
b | a | urn:hailstone:ml_ai:structural_equation[3f765ae24468] | 1 | structural_equation | 0 | arxiv.cs.IT.2602.18077v1.eq_705 | \bm w_c^\star=\bm 0 | arxiv_extracted | ["a", "t", "m", "s", "b", "w_c"] | arxiv | arxiv | arxiv:cs.IT:2602.18077v1 | cross_domain_amplification | |
n | o | urn:hailstone:ml_ai:structural_equation[c30c7a2acb1c] | 1 | structural_equation | 0 | arxiv.cs.IT.2602.14998v1.eq_706 | n^3 \mathop{\rm tr}^2(κ^3) = 1 | arxiv_extracted | ["a", "t", "m", "h", "o", "n"] | arxiv | arxiv | arxiv:cs.IT:2602.14998v1 | cross_domain_amplification | |
g | L | urn:hailstone:ml_ai:structural_equation[da9e0f868c82] | 1 | structural_equation | 0 | arxiv.cs.IT.2602.12766v1.eq_707 | \gcd(q, L) = 1 | arxiv_extracted | ["g", "q", "L", "c", "d"] | arxiv | arxiv | arxiv:cs.IT:2602.12766v1 | cross_domain_amplification | |
C | m | urn:hailstone:ml_ai:structural_equation[03754f81e0f2] | 1 | structural_equation | 0 | arxiv.cs.IT.2602.11568v1.eq_708 | C^{NS,ca}=\max_{P_{X|S}}I(X;Y\mid S) | arxiv_extracted | ["a", "C", "m", "S", "c", "N"] | arxiv | arxiv | arxiv:cs.IT:2602.11568v1 | cross_domain_amplification | |
C | m | urn:hailstone:ml_ai:structural_equation[13cf4fd9f552] | 1 | structural_equation | 0 | arxiv.cs.IT.2602.11568v1.eq_709 | C^{NS,nc}=\max_{P_{X|S}}I(X;Y\mid S) | arxiv_extracted | ["C", "m", "S", "n", "c", "N"] | arxiv | arxiv | arxiv:cs.IT:2602.11568v1 | cross_domain_amplification | |
h | u | urn:hailstone:ml_ai:structural_equation[d23ec8cf2247] | 1 | structural_equation | 0 | arxiv.cs.IT.2602.10028v1.eq_710 | h(x) = x^m - λ+ \sum_{i=0}^{m-1} h_i x^i | arxiv_extracted | ["m", "h", "u", "s", "x"] | arxiv | arxiv | arxiv:cs.IT:2602.10028v1 | cross_domain_amplification | |
x | d_{i | urn:hailstone:ml_ai:structural_equation[986eddcc4786] | 1 | structural_equation | 0 | arxiv.cs.IT.2602.10028v1.eq_711 | x^m - λ= \prod_{i=1}^{t} f_i(x)^{e_i}, | arxiv_extracted | ["m", "o", "p", "r", "d_{i", "x"] | arxiv | arxiv | arxiv:cs.IT:2602.10028v1 | cross_domain_amplification | |
N | p | urn:hailstone:ml_ai:structural_equation[d504f3c128ec] | 1 | structural_equation | 0 | arxiv.cs.IT.2602.10028v1.eq_712 | N \cdot \prod_{i=1}^{t} \left( q^{°f_i} - 1 \right), | arxiv_extracted | ["t", "d", "o", "p", "c", "N"] | arxiv | arxiv | arxiv:cs.IT:2602.10028v1 | cross_domain_amplification | |
l | e | urn:hailstone:ml_ai:structural_equation[be7a51690d6f] | 1 | structural_equation | 0 | arxiv.cs.IT.2602.10028v1.eq_713 | 0 \leq k < \left\lfloor \frac{m - 1}{r} \right\rfloor + 1 \quad \text{and} \quad \gcd(1 + rk, m) = 1. | arxiv_extracted | ["e", "q", "k", "l"] | arxiv | arxiv | arxiv:cs.IT:2602.10028v1 | cross_domain_amplification | |
R | t | urn:hailstone:ml_ai:structural_equation[9be0877f2a75] | 1 | structural_equation | 0 | arxiv.cs.IT.2602.09176v1.eq_714 | R(n, d, ε)=\mathbb{R}_n(d)+\sqrt{\frac{\mathbb{V}_n(d)}{n}}Q^{-1}(ε)+O \left(\frac{\log n}{n}\right) | arxiv_extracted | ["a", "t", "m", "R", "n", "d"] | arxiv | arxiv | arxiv:cs.IT:2602.09176v1 | cross_domain_amplification | |
D | a | urn:hailstone:ml_ai:structural_equation[b74cdef15a7e] | 1 | structural_equation | 0 | arxiv.cs.IT.2602.08110v1.eq_715 | \Disp_n(\mathbf t)=n^r | arxiv_extracted | ["D", "a", "p_n", "m", "i", "s"] | arxiv | arxiv | arxiv:cs.IT:2602.08110v1 | cross_domain_amplification | |
f | q | urn:hailstone:ml_ai:structural_equation[88f21c0d6de6] | 1 | structural_equation | 0 | arxiv.cs.IT.2602.06361v1.eq_716 | \frac{\sqrt m }{(Δ/ m)^2} = \frac{m^{2.5}}{Δ^2} | arxiv_extracted | ["f", "q", "a", "r", "s", "c"] | arxiv | arxiv | arxiv:cs.IT:2602.06361v1 | cross_domain_amplification | |
t | R | urn:hailstone:ml_ai:structural_equation[4b2d1649e0d8] | 1 | structural_equation | 0 | arxiv.cs.IT.2602.05779v1.eq_717 | \text{CReLU}_{τ,m}(x)=\min(\max(x-τ,0),m) | arxiv_extracted | ["e", "t", "R", "C", "x"] | arxiv | arxiv | arxiv:cs.IT:2602.05779v1 | cross_domain_amplification | |
O_i | m | urn:hailstone:ml_ai:structural_equation[859d532cfb6c] | 1 | structural_equation | 0 | arxiv.cs.IT.2602.04952v1.eq_718 | \{O_i\}_{i=1}^m | arxiv_extracted | ["m", "O_i", "i"] | arxiv | arxiv | arxiv:cs.IT:2602.04952v1 | cross_domain_amplification | |
m | m | urn:hailstone:ml_ai:structural_equation[21318c03116e] | 1 | structural_equation | 0 | arxiv.cs.IT.2602.04952v1.eq_719 | \{\mathrm{tr}(O_iρ)\}_{i=1}^m | arxiv_extracted | ["a", "t", "m", "h", "r"] | arxiv | arxiv | arxiv:cs.IT:2602.04952v1 | cross_domain_amplification | |
O_i | m | urn:hailstone:ml_ai:structural_equation[859d532cfb6c] | 1 | structural_equation | 0 | arxiv.cs.IT.2602.04952v1.eq_720 | \{O_i\}_{i=1}^m | arxiv_extracted | ["m", "O_i", "i"] | arxiv | arxiv | arxiv:cs.IT:2602.04952v1 | cross_domain_amplification |
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