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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
s | O_i | urn:hailstone:ml_ai:structural_equation[ec061423fe4c] | 1 | structural_equation | 0 | arxiv.cs.IT.2602.04952v1.eq_721 | \sum_{i=1}^m α_i O_i | arxiv_extracted | ["m_{i", "m", "i", "u", "s", "O_i"] | arxiv | arxiv | arxiv:cs.IT:2602.04952v1 | cross_domain_amplification | |
m | m | urn:hailstone:ml_ai:structural_equation[d39516692936] | 1 | structural_equation | 0 | arxiv.cs.IT.2602.04952v1.eq_722 | Γ^{\mathrm{ob}}_\infty = Γ_\infty | arxiv_extracted | ["a", "t", "m", "h", "r"] | arxiv | arxiv | arxiv:cs.IT:2602.04952v1 | cross_domain_amplification | |
m | f | urn:hailstone:ml_ai:structural_equation[48b8e9e64585] | 1 | structural_equation | 0 | arxiv.cs.IT.2602.04862v1.eq_723 | \mathbf{H} = \mathbf{F} + s \mathbf{G} | arxiv_extracted | ["f", "a", "t", "m", "h", "b"] | arxiv | arxiv | arxiv:cs.IT:2602.04862v1 | cross_domain_amplification | |
b | P | urn:hailstone:ml_ai:structural_equation[4df4eaa2a492] | 1 | structural_equation | 0 | arxiv.cs.IT.2602.03718v1.eq_724 | \|\bm x[m]\|_2^2=P | arxiv_extracted | ["b", "m", "P", "x"] | arxiv | arxiv | arxiv:cs.IT:2602.03718v1 | cross_domain_amplification | |
b | v | urn:hailstone:ml_ai:structural_equation[2917c6f1eb46] | 1 | structural_equation | 0 | arxiv.cs.IT.2602.03718v1.eq_725 | \bm V(\bm\varphi)\bm e_1=\bm c | arxiv_extracted | ["V", "b", "m", "v"] | arxiv | arxiv | arxiv:cs.IT:2602.03718v1 | cross_domain_amplification | |
b | x | urn:hailstone:ml_ai:structural_equation[494e3b62ecf8] | 1 | structural_equation | 0 | arxiv.cs.IT.2602.03718v1.eq_726 | \bm c=\bm x/\sqrt{P} | arxiv_extracted | ["b", "c", "m", "x"] | arxiv | arxiv | arxiv:cs.IT:2602.03718v1 | cross_domain_amplification | |
b | c | urn:hailstone:ml_ai:structural_equation[6f500b6d118b] | 1 | structural_equation | 0 | arxiv.cs.IT.2602.03718v1.eq_727 | \|\bm c\|_2=1 | arxiv_extracted | ["b", "c", "m"] | arxiv | arxiv | arxiv:cs.IT:2602.03718v1 | cross_domain_amplification | |
m | m | urn:hailstone:ml_ai:structural_equation[13c60a27c56f] | 1 | structural_equation | 0 | arxiv.cs.IT.2601.21489v1.eq_728 | Λ_{\mathrm{del}}=\sum_{u\in\mathcal P_{\mathrm{trap}}}ζ(u)π(u) | arxiv_extracted | ["a", "t", "m", "h", "r"] | arxiv | arxiv | arxiv:cs.IT:2601.21489v1 | cross_domain_amplification | |
x | k | urn:hailstone:ml_ai:structural_equation[8dd2f21d834f] | 1 | structural_equation | 0 | arxiv.cs.IT.2601.19327v1.eq_729 | x(1+x)^{k-1}=1 | arxiv_extracted | ["k", "x"] | arxiv | arxiv | arxiv:cs.IT:2601.19327v1 | cross_domain_amplification | |
f | n | urn:hailstone:ml_ai:structural_equation[f2deb716c81b] | 1 | structural_equation | 0 | arxiv.cs.IT.2601.18590v1.eq_730 | \frac{q^{2n-k}-1}{q-1}<\frac{c_δ\sqrt{n}\cdot q^{2n}}{\sum_{i=0}^{d-1}\binom{n}{i}(q^2-1)^i} | arxiv_extracted | ["f", "q", "a", "r", "n", "c"] | arxiv | arxiv | arxiv:cs.IT:2601.18590v1 | cross_domain_amplification | |
l | o | urn:hailstone:ml_ai:structural_equation[ddfb6c9dad25] | 1 | structural_equation | 0 | arxiv.cs.IT.2601.16614v1.eq_731 | \log_n \Sn(Γ)=α+o(1) | arxiv_extracted | ["g_n", "S", "o", "n", "l"] | arxiv | arxiv | arxiv:cs.IT:2601.16614v1 | cross_domain_amplification | |
S | o | urn:hailstone:ml_ai:structural_equation[abde7db6c82c] | 1 | structural_equation | 0 | arxiv.cs.IT.2601.16614v1.eq_732 | \Sn(Γ)=n^{α+o(1)} | arxiv_extracted | ["n", "S", "o"] | arxiv | arxiv | arxiv:cs.IT:2601.16614v1 | cross_domain_amplification | |
m | a | urn:hailstone:ml_ai:structural_equation[d1fb32d2afa8] | 1 | structural_equation | 0 | arxiv.cs.IT.2601.16438v1.eq_733 | \mathcal{L}=\{0,1,\ldots,l\} | arxiv_extracted | ["a", "t", "m", "h", "c"] | arxiv | arxiv | arxiv:cs.IT:2601.16438v1 | cross_domain_amplification | |
m | a | urn:hailstone:ml_ai:structural_equation[9ec084e48ef4] | 1 | structural_equation | 0 | arxiv.cs.IT.2601.16438v1.eq_734 | \mathcal{P}=\{h\} | arxiv_extracted | ["a", "t", "m", "h", "c"] | arxiv | arxiv | arxiv:cs.IT:2601.16438v1 | cross_domain_amplification | |
l | r | urn:hailstone:ml_ai:structural_equation[8cbe79a8ed92] | 1 | structural_equation | 0 | arxiv.cs.IT.2601.16164v1.eq_735 | \left\{x_{ij}\right\}_{i=1,\dots,t}^{j=1,\dots m_i} | arxiv_extracted | ["e", "f", "t", "r", "x_{ij}", "l"] | arxiv | arxiv | arxiv:cs.IT:2601.16164v1 | cross_domain_amplification | |
D | L | urn:hailstone:ml_ai:structural_equation[4fa3b452f1a2] | 1 | structural_equation | 0 | arxiv.cs.IT.2601.14252v6.eq_736 | D^\star(L)=\max(0,1-2^L/a) | arxiv_extracted | ["D", "a", "t", "r", "L", "s"] | arxiv | arxiv | arxiv:cs.IT:2601.14252v6 | cross_domain_amplification | |
n_{1} | k_{1} | urn:hailstone:ml_ai:structural_equation[04a793bc3618] | 1 | structural_equation | 0 | arxiv.cs.IT.2601.12780v1.eq_737 | n_{1}=k_{1} | arxiv_extracted | ["n_{1}", "k_{1}"] | arxiv | arxiv | arxiv:cs.IT:2601.12780v1 | cross_domain_amplification | |
n_{2} | n_{2} | urn:hailstone:ml_ai:structural_equation[2bbd0ba2c188] | 1 | structural_equation | 0 | arxiv.cs.IT.2601.12780v1.eq_738 | n_{2}=m<n_{1}n_{2} | arxiv_extracted | ["n_{2}", "m", "n_{1}"] | arxiv | arxiv | arxiv:cs.IT:2601.12780v1 | cross_domain_amplification | |
d | c | urn:hailstone:ml_ai:structural_equation[9631316bc71a] | 1 | structural_equation | 0 | arxiv.cs.IT.2601.12576v1.eq_739 | dH/dt = c | arxiv_extracted | ["t", "d", "H", "c"] | arxiv | arxiv | arxiv:cs.IT:2601.12576v1 | cross_domain_amplification | |
m | f | urn:hailstone:ml_ai:structural_equation[47463c980221] | 1 | structural_equation | 0 | arxiv.cs.IT.2601.10991v1.eq_740 | \mathbf{s}=s_1s_2\cdots s_n | arxiv_extracted | ["f", "a", "t", "m", "h", "b"] | arxiv | arxiv | arxiv:cs.IT:2601.10991v1 | cross_domain_amplification | |
s_t | o | urn:hailstone:ml_ai:structural_equation[0e45c97e4708] | 1 | structural_equation | 0 | arxiv.cs.IT.2601.10991v1.eq_741 | s_t, t=n, \cdots, 2,1 | arxiv_extracted | ["s_t", "t", "d", "o", "n", "c"] | arxiv | arxiv | arxiv:cs.IT:2601.10991v1 | cross_domain_amplification | |
s_t | t | urn:hailstone:ml_ai:structural_equation[4e8864e04b9d] | 1 | structural_equation | 0 | arxiv.cs.IT.2601.10991v1.eq_742 | s_t, t=1, 2, \cdots, n | arxiv_extracted | ["s_t", "t", "d", "o", "c"] | arxiv | arxiv | arxiv:cs.IT:2601.10991v1 | cross_domain_amplification | |
e | r | urn:hailstone:ml_ai:structural_equation[8427dfd29fbd] | 1 | structural_equation | 0 | arxiv.cs.IT.2601.10685v2.eq_743 | \ell = s \prod_{i=1}^n p_i | arxiv_extracted | ["e", "p", "r", "s", "l"] | arxiv | arxiv | arxiv:cs.IT:2601.10685v2 | cross_domain_amplification | |
d_1 | d | urn:hailstone:ml_ai:structural_equation[52b3845810f7] | 1 | structural_equation | 0 | arxiv.cs.IT.2601.10683v1.eq_744 | d_1=d_2=d | arxiv_extracted | ["d_2", "d", "d_1"] | arxiv | arxiv | arxiv:cs.IT:2601.10683v1 | cross_domain_amplification | |
M | X_i | urn:hailstone:ml_ai:structural_equation[9df2639da00f] | 1 | structural_equation | 0 | arxiv.cs.IT.2601.08929v2.eq_745 | M^{(f)}_{ij} = I_f(X_i;X_j) | arxiv_extracted | ["f", "X_i", "M", "j", "I_f", "i"] | arxiv | arxiv | arxiv:cs.IT:2601.08929v2 | cross_domain_amplification | |
e | l | urn:hailstone:ml_ai:structural_equation[e7af203bd6ff] | 1 | structural_equation | 0 | arxiv.cs.IT.2601.08452v1.eq_746 | \ell = 2 | arxiv_extracted | ["e", "l"] | arxiv | arxiv | arxiv:cs.IT:2601.08452v1 | cross_domain_amplification | |
e | l | urn:hailstone:ml_ai:structural_equation[c621989430a6] | 1 | structural_equation | 0 | arxiv.cs.IT.2601.08452v1.eq_747 | \ell = 4 | arxiv_extracted | ["e", "l"] | arxiv | arxiv | arxiv:cs.IT:2601.08452v1 | cross_domain_amplification | |
e | l | urn:hailstone:ml_ai:structural_equation[c866e6bc3551] | 1 | structural_equation | 0 | arxiv.cs.IT.2601.08452v1.eq_748 | \ell = 8 | arxiv_extracted | ["e", "l"] | arxiv | arxiv | arxiv:cs.IT:2601.08452v1 | cross_domain_amplification | |
e | l | urn:hailstone:ml_ai:structural_equation[e7af203bd6ff] | 1 | structural_equation | 0 | arxiv.cs.IT.2601.08452v1.eq_749 | \ell = 2 | arxiv_extracted | ["e", "l"] | arxiv | arxiv | arxiv:cs.IT:2601.08452v1 | cross_domain_amplification | |
m | f | urn:hailstone:ml_ai:structural_equation[076ae5b942b6] | 1 | structural_equation | 0 | arxiv.cs.IT.2601.08387v2.eq_750 | \mathbf H\mathbf H^\top = \mathbf 0 | arxiv_extracted | ["f", "a", "t", "m", "h", "b"] | arxiv | arxiv | arxiv:cs.IT:2601.08387v2 | cross_domain_amplification | |
s | q | urn:hailstone:ml_ai:structural_equation[44fc4fb95013] | 1 | structural_equation | 0 | arxiv.cs.IT.2601.08144v1.eq_751 | \sum^{s-1}_{i=1}q^{ik+h}+1 | arxiv_extracted | ["q", "m", "i", "u", "s"] | arxiv | arxiv | arxiv:cs.IT:2601.08144v1 | cross_domain_amplification | |
C | r | urn:hailstone:ml_ai:structural_equation[98de8e199d8a] | 1 | structural_equation | 0 | arxiv.cs.IT.2601.07257v1.eq_752 | C_{\mathrm{ip}}+C_{\mathrm{i}}=\mathrm{rank}(Σ_{XX})\le d | arxiv_extracted | ["a", "t", "m", "h", "r", "C"] | arxiv | arxiv | arxiv:cs.IT:2601.07257v1 | cross_domain_amplification | |
X | N_0 | urn:hailstone:ml_ai:structural_equation[afdd926ee626] | 1 | structural_equation | 0 | arxiv.cs.IT.2601.07257v1.eq_753 | Σ_{XX}(T)=S+T N_0 | arxiv_extracted | ["X", "T", "S", "N_0"] | arxiv | arxiv | arxiv:cs.IT:2601.07257v1 | cross_domain_amplification | |
E | l | urn:hailstone:ml_ai:structural_equation[8c9407a40a83] | 1 | structural_equation | 0 | arxiv.cs.IT.2601.06609v1.eq_754 | E= \langle κ,τ\mid 2 κ=2 τ=0,~ κ^2=κ,~ τ^2=τ,~ κτ=κ,~ τκ=τ\rangle | arxiv_extracted | ["g", "a", "n", "E", "l"] | arxiv | arxiv | arxiv:cs.IT:2601.06609v1 | cross_domain_amplification | |
N | n | urn:hailstone:ml_ai:structural_equation[5485e024dd0d] | 1 | structural_equation | 0 | arxiv.cs.IT.2601.06503v2.eq_755 | N(n,3,4)=20n-166 | arxiv_extracted | ["n", "N"] | arxiv | arxiv | arxiv:cs.IT:2601.06503v2 | cross_domain_amplification | |
r | R | urn:hailstone:ml_ai:structural_equation[7449726341d5] | 1 | structural_equation | 0 | arxiv.cs.IT.2601.03734v2.eq_756 | {\rm S}^{\tt R}_α(ρ) = \frac{\ln {\rm Tr}(ρ^α)}{1-α} | arxiv_extracted | ["t", "m", "S", "r", "R"] | arxiv | arxiv | arxiv:cs.IT:2601.03734v2 | cross_domain_amplification | |
r | T | urn:hailstone:ml_ai:structural_equation[6809987c063e] | 1 | structural_equation | 0 | arxiv.cs.IT.2601.03734v2.eq_757 | {\rm S}^{\tt T}_q(ρ) = \frac{1-{\rm Tr}(ρ^q)}{q-1} | arxiv_extracted | ["t", "m", "S", "r", "T"] | arxiv | arxiv | arxiv:cs.IT:2601.03734v2 | cross_domain_amplification | |
R_1 | s | urn:hailstone:ml_ai:structural_equation[0e27ec900e95] | 1 | structural_equation | 0 | arxiv.cs.IT.2601.03241v1.eq_758 | (R_1,\ldots,R_K) = (\mathbf{1}(1 \in \mathcal{I}),\ldots,\mathbf{1}(K\in \mathcal{I})) | arxiv_extracted | ["t", "R_1", "o", "s", "d", "l"] | arxiv | arxiv | arxiv:cs.IT:2601.03241v1 | cross_domain_amplification | |
m | m | urn:hailstone:ml_ai:structural_equation[01886428a953] | 1 | structural_equation | 0 | arxiv.cs.IT.2601.03241v1.eq_759 | \mathrm{rank}\left(\bigl[\mathbf{F}_{\mathcal{I}};\mathbf{G}_{\mathcal{I}}\bigr]\right) =\mathrm{rank}\bigl(\mathbf{F}_{\mathcal{I}}\bigr)+N | arxiv_extracted | ["a", "t", "m", "h", "r"] | arxiv | arxiv | arxiv:cs.IT:2601.03241v1 | cross_domain_amplification | |
n | q | urn:hailstone:ml_ai:structural_equation[674035343f6b] | 1 | structural_equation | 0 | arxiv.cs.IT.2602.09028v1.eq_760 | 1/n + 1/q = 1 | arxiv_extracted | ["n", "q"] | arxiv | arxiv | arxiv:cs.IT:2602.09028v1 | cross_domain_amplification | |
g | t | urn:hailstone:ml_ai:structural_equation[a1bc865a7a91] | 1 | structural_equation | 0 | arxiv.cs.IT.2512.23194v3.eq_761 | \gcd(d, q+1+t)=1 | arxiv_extracted | ["g", "q", "t", "c", "d"] | arxiv | arxiv | arxiv:cs.IT:2512.23194v3 | cross_domain_amplification | |
k | r | urn:hailstone:ml_ai:structural_equation[75140ad1816e] | 1 | structural_equation | 0 | arxiv.cs.IT.2512.21922v2.eq_762 | k_{\mathrm{C}Z}=Θ(n) | arxiv_extracted | ["a", "t", "m", "h", "r", "k"] | arxiv | arxiv | arxiv:cs.IT:2512.21922v2 | cross_domain_amplification | |
m | a | urn:hailstone:ml_ai:structural_equation[600271d4fa16] | 1 | structural_equation | 0 | arxiv.cs.IT.2512.20039v1.eq_763 | |\mathcal{X}| = m < \infty | arxiv_extracted | ["a", "t", "m", "h", "c"] | arxiv | arxiv | arxiv:cs.IT:2512.20039v1 | cross_domain_amplification | |
F_r | r | urn:hailstone:ml_ai:structural_equation[e6696406bf66] | 1 | structural_equation | 0 | arxiv.cs.IT.2512.17603v2.eq_764 | F_r(x)=x^r+x^{r+\frac{q-1}{2}} | arxiv_extracted | ["r", "F_r", "x"] | arxiv | arxiv | arxiv:cs.IT:2512.17603v2 | cross_domain_amplification | |
x | x | urn:hailstone:ml_ai:structural_equation[93e7cca0f75b] | 1 | structural_equation | 0 | arxiv.cs.IT.2512.17603v2.eq_765 | χ(x)=χ(x+1)=1 | arxiv_extracted | ["x"] | arxiv | arxiv | arxiv:cs.IT:2512.17603v2 | cross_domain_amplification | |
x | b | urn:hailstone:ml_ai:structural_equation[7e2a07ead162] | 1 | structural_equation | 0 | arxiv.cs.IT.2512.17603v2.eq_766 | (x+1)^r - x^r = b | arxiv_extracted | ["b", "r", "x"] | arxiv | arxiv | arxiv:cs.IT:2512.17603v2 | cross_domain_amplification | |
m | a | urn:hailstone:ml_ai:structural_equation[86ce3521999e] | 1 | structural_equation | 0 | arxiv.cs.IT.2512.13067v1.eq_767 | (\mathcal{O}_i)_{i=1}^k | arxiv_extracted | ["a", "t", "m", "h", "c"] | arxiv | arxiv | arxiv:cs.IT:2512.13067v1 | cross_domain_amplification | |
m | a | urn:hailstone:ml_ai:structural_equation[86ce3521999e] | 1 | structural_equation | 0 | arxiv.cs.IT.2512.13067v1.eq_768 | (\mathcal{O}_i)_{i=1}^k | arxiv_extracted | ["a", "t", "m", "h", "c"] | arxiv | arxiv | arxiv:cs.IT:2512.13067v1 | cross_domain_amplification | |
E | l | urn:hailstone:ml_ai:structural_equation[8c9407a40a83] | 1 | structural_equation | 0 | arxiv.cs.IT.2512.12335v2.eq_769 | E= \langle κ,τ\mid 2 κ=2 τ=0,~ κ^2=κ,~ τ^2=τ,~ κτ=κ,~ τκ=τ\rangle | arxiv_extracted | ["g", "a", "n", "E", "l"] | arxiv | arxiv | arxiv:cs.IT:2512.12335v2 | cross_domain_amplification | |
C | t | urn:hailstone:ml_ai:structural_equation[3496b16de275] | 1 | structural_equation | 0 | arxiv.cs.IT.2512.07631v1.eq_770 | \Ceff = (\Itotal/\Istep), \Cstep | arxiv_extracted | ["e", "f", "t", "C", "I"] | arxiv | arxiv | arxiv:cs.IT:2512.07631v1 | cross_domain_amplification | |
m | a | urn:hailstone:ml_ai:structural_equation[56db1d0d934b] | 1 | structural_equation | 0 | arxiv.cs.IT.2512.07343v2.eq_771 | \mathcal{R}=\frac{\mathbb{F}_q[u]}{\langle u^2 \rangle}\times \mathbb{F}_q | arxiv_extracted | ["a", "t", "m", "h", "c"] | arxiv | arxiv | arxiv:cs.IT:2512.07343v2 | cross_domain_amplification | |
m | a | urn:hailstone:ml_ai:structural_equation[aea3176b7797] | 1 | structural_equation | 0 | arxiv.cs.IT.2512.04955v1.eq_772 | |\mathcal{X}| = 4 | arxiv_extracted | ["a", "t", "m", "h", "c"] | arxiv | arxiv | arxiv:cs.IT:2512.04955v1 | cross_domain_amplification | |
C_{1} | n | urn:hailstone:ml_ai:structural_equation[97f54853a372] | 1 | structural_equation | 0 | arxiv.cs.IT.2512.03117v3.eq_773 | C_{1}=3+2\ln 2 | arxiv_extracted | ["n", "C_{1}", "l"] | arxiv | arxiv | arxiv:cs.IT:2512.03117v3 | cross_domain_amplification | |
C_{1} | C_{2} | urn:hailstone:ml_ai:structural_equation[a306c62b89e2] | 1 | structural_equation | 0 | arxiv.cs.IT.2512.03117v3.eq_774 | C_{1}=C_{2}=1 | arxiv_extracted | ["C_{1}", "C_{2}"] | arxiv | arxiv | arxiv:cs.IT:2512.03117v3 | cross_domain_amplification | |
f | y | urn:hailstone:ml_ai:structural_equation[cfb64d73b3a2] | 1 | structural_equation | 0 | arxiv.cs.IT.2512.00734v1.eq_775 | f_{\infty} = T(P_{\infty}, Q_{\infty}) | arxiv_extracted | ["f", "t", "y", "i", "n"] | arxiv | arxiv | arxiv:cs.IT:2512.00734v1 | cross_domain_amplification | |
d | t | urn:hailstone:ml_ai:structural_equation[aafa3aeb58de] | 1 | structural_equation | 0 | arxiv.cs.IT.2512.00734v1.eq_776 | dQ_{\infty}(x) = e^{x} dP_{\infty}(x) | arxiv_extracted | ["f", "Q", "t", "i", "n", "d"] | arxiv | arxiv | arxiv:cs.IT:2512.00734v1 | cross_domain_amplification | |
I | T | urn:hailstone:ml_ai:structural_equation[a49e9dea63ee] | 1 | structural_equation | 0 | arxiv.cs.IT.2512.00135v1.eq_777 | I(Y;S|T) =0 | arxiv_extracted | ["Y", "T", "S", "I"] | arxiv | arxiv | arxiv:cs.IT:2512.00135v1 | cross_domain_amplification | |
s | s | urn:hailstone:ml_ai:structural_equation[bb5427285266] | 1 | structural_equation | 0 | arxiv.cs.IT.2511.20316v1.eq_778 | \sqrt{2γ(F_n)} = \sqrt{2(n-1)} | arxiv_extracted | ["q", "t", "F_n", "r", "s"] | arxiv | arxiv | arxiv:cs.IT:2511.20316v1 | cross_domain_amplification | |
S | c | urn:hailstone:ml_ai:structural_equation[992558e9c016] | 1 | structural_equation | 0 | arxiv.cs.IT.2511.19378v1.eq_779 | S(c) = Phi(c) + I | arxiv_extracted | ["h", "P", "S", "i", "c"] | arxiv | arxiv | arxiv:cs.IT:2511.19378v1 | cross_domain_amplification | |
I | B | urn:hailstone:ml_ai:structural_equation[9a00defb0b4c] | 1 | structural_equation | 0 | arxiv.cs.IT.2511.17389v1.eq_780 | (I+B)^{p^m}=I+B^{p^m} | arxiv_extracted | ["p", "m", "B", "I"] | arxiv | arxiv | arxiv:cs.IT:2511.17389v1 | cross_domain_amplification | |
b | b | urn:hailstone:ml_ai:structural_equation[d723c7725a1b] | 1 | structural_equation | 0 | arxiv.cs.IT.2511.17240v2.eq_781 | \bar{k}=q\binom{n}{2} | arxiv_extracted | ["q", "a", "r", "b", "k"] | arxiv | arxiv | arxiv:cs.IT:2511.17240v2 | cross_domain_amplification | |
f | s | urn:hailstone:ml_ai:structural_equation[64ac94bcdb28] | 1 | structural_equation | 0 | arxiv.cs.IT.2511.15120v2.eq_782 | f(\boldsymbol{x})=g(\boldsymbol{U}\boldsymbol{x}) | arxiv_extracted | ["f", "o", "s", "b", "d", "l"] | arxiv | arxiv | arxiv:cs.IT:2511.15120v2 | cross_domain_amplification | |
b | y | urn:hailstone:ml_ai:structural_equation[666a72cbfa57] | 1 | structural_equation | 0 | arxiv.cs.IT.2511.15051v1.eq_783 | \boldsymbol{Z} = (Y_{σ(i)})_i | arxiv_extracted | ["y", "o", "s", "b", "d", "l"] | arxiv | arxiv | arxiv:cs.IT:2511.15051v1 | cross_domain_amplification | |
I | d | urn:hailstone:ml_ai:structural_equation[79b36bad024d] | 1 | structural_equation | 0 | arxiv.cs.IT.2511.15051v1.eq_784 | I(X_1; \boldsymbol{Z}\mid (X_i)_{i=2}^n) \le (e^{\varepsilon_0}-1)/(2n) + O(n^{-3/2}) | arxiv_extracted | ["o", "X_1", "b", "d", "l", "I"] | arxiv | arxiv | arxiv:cs.IT:2511.15051v1 | cross_domain_amplification | |
s | C | urn:hailstone:ml_ai:structural_equation[4418afc3626a] | 1 | structural_equation | 0 | arxiv.cs.IT.2511.06795v1.eq_785 | \sum_i h_i = C | arxiv_extracted | ["h_i", "m_i", "u", "s", "C"] | arxiv | arxiv | arxiv:cs.IT:2511.06795v1 | cross_domain_amplification | |
f | y | urn:hailstone:ml_ai:structural_equation[9770dac1044f] | 1 | structural_equation | 0 | arxiv.cs.IT.2510.22671v2.eq_786 | f(x,y)=1 | arxiv_extracted | ["f", "y", "x"] | arxiv | arxiv | arxiv:cs.IT:2510.22671v2 | cross_domain_amplification | |
f | y | urn:hailstone:ml_ai:structural_equation[cdbf0180acd1] | 1 | structural_equation | 0 | arxiv.cs.IT.2510.22671v2.eq_787 | f(x,y)=0 | arxiv_extracted | ["f", "y", "x"] | arxiv | arxiv | arxiv:cs.IT:2510.22671v2 | cross_domain_amplification | |
t | m | urn:hailstone:ml_ai:structural_equation[be73f0e279ad] | 1 | structural_equation | 0 | arxiv.cs.IT.2510.19095v1.eq_788 | \textrm{G} = (\mathbb{Z}/2\mathbb{Z})^m | arxiv_extracted | ["e", "t", "m", "r", "x"] | arxiv | arxiv | arxiv:cs.IT:2510.19095v1 | cross_domain_amplification | |
r | m | urn:hailstone:ml_ai:structural_equation[d8d8764b4837] | 1 | structural_equation | 0 | arxiv.cs.IT.2510.18186v4.eq_789 | Δ_{\rm sw}(ω)=p_{\mathrm{switch}}(ω)-p_{\mathrm{fixed}}^* | arxiv_extracted | ["w", "m", "p", "r", "s"] | arxiv | arxiv | arxiv:cs.IT:2510.18186v4 | cross_domain_amplification | |
r | t | urn:hailstone:ml_ai:structural_equation[ae53d1fc751c] | 1 | structural_equation | 0 | arxiv.cs.IT.2510.18186v4.eq_790 | Δ_{\rm test}(ω)=p_{\mathrm{test}}(ω)-p_{\mathrm{fixed}}^* | arxiv_extracted | ["e", "t", "m", "r", "s"] | arxiv | arxiv | arxiv:cs.IT:2510.18186v4 | cross_domain_amplification | |
r | r | urn:hailstone:ml_ai:structural_equation[970161013f86] | 1 | structural_equation | 0 | arxiv.cs.IT.2510.18186v4.eq_791 | Δ_{\rm int}(ω):=Δ_{\rm test}(ω)-Δ_{\rm sw}(ω)=p_{\rm test}(ω)-p_{\rm switch}(ω) | arxiv_extracted | ["t", "m", "i", "r", "n"] | arxiv | arxiv | arxiv:cs.IT:2510.18186v4 | cross_domain_amplification | |
y_i | g | urn:hailstone:ml_ai:structural_equation[f7d211570f18] | 1 | structural_equation | 0 | arxiv.cs.IT.2510.16965v1.eq_792 | \{y_i=f_i(\langle\mathbf{a}_i,\mathbf{x}\rangle)\}_{i=1}^m | arxiv_extracted | ["g", "a", "y_i", "f_i", "n", "l"] | arxiv | arxiv | arxiv:cs.IT:2510.16965v1 | cross_domain_amplification | |
m | m | urn:hailstone:ml_ai:structural_equation[90d6155bcc55] | 1 | structural_equation | 0 | arxiv.cs.IT.2510.14029v2.eq_793 | \mathrm{R}^{[\mathbf{m} _{r},\mathbf{n}_{r}]}=\mathcal{R}^{[m_{r},n_{r}]}[\mathsf{G}^{[n_{g}]}] | arxiv_extracted | ["a", "t", "m", "h", "r"] | arxiv | arxiv | arxiv:cs.IT:2510.14029v2 | cross_domain_amplification | |
S | d | urn:hailstone:ml_ai:structural_equation[30d5f1adaa65] | 1 | structural_equation | 0 | arxiv.cs.IT.2510.12112v1.eq_794 | ST^2 = \tildeΩ(N) | arxiv_extracted | ["t", "d", "S", "i", "T", "l"] | arxiv | arxiv | arxiv:cs.IT:2510.12112v1 | cross_domain_amplification | |
S | N | urn:hailstone:ml_ai:structural_equation[a3ea161d7b60] | 1 | structural_equation | 0 | arxiv.cs.IT.2510.12112v1.eq_795 | ST + T^2 = Ω(N) | arxiv_extracted | ["N", "T", "S"] | arxiv | arxiv | arxiv:cs.IT:2510.12112v1 | cross_domain_amplification | |
i | g | urn:hailstone:ml_ai:structural_equation[d84e9b7c17fe] | 1 | structural_equation | 0 | arxiv.cs.IT.2510.10300v3.eq_796 | , i.e., \[ Δ= K\big(O_{W,\varnothing}\big) - K\big(O_{W,R}\big) > 0. \] We then prove that the larger | arxiv_extracted | ["e", "g", "i", "b", "K"] | arxiv | arxiv | arxiv:cs.IT:2510.10300v3 | cross_domain_amplification | |
m | a | urn:hailstone:ml_ai:structural_equation[6310201747f5] | 1 | structural_equation | 0 | arxiv.cs.IT.2510.09057v1.eq_797 | \mathcal{R}:=\mathbb{F}_2[x,y]/\langle x^2, y^2, xy-yx\rangle | arxiv_extracted | ["a", "t", "m", "h", "c"] | arxiv | arxiv | arxiv:cs.IT:2510.09057v1 | cross_domain_amplification | |
t | f | urn:hailstone:ml_ai:structural_equation[8a43d2261bf8] | 1 | structural_equation | 0 | arxiv.cs.IT.2510.06622v1.eq_798 | \tilde{f}(t) = \exp(\frac{α-1}αt) | arxiv_extracted | ["e", "f", "t", "i", "d", "l"] | arxiv | arxiv | arxiv:cs.IT:2510.06622v1 | cross_domain_amplification | |
T | t | urn:hailstone:ml_ai:structural_equation[447b2263c11d] | 1 | structural_equation | 0 | arxiv.cs.IT.2510.06082v2.eq_799 | Tor_i(\mathscr{D}_e)=\mathcal{C}^{(i)} | arxiv_extracted | ["a", "t", "m", "o", "r_i", "T"] | arxiv | arxiv | arxiv:cs.IT:2510.06082v2 | cross_domain_amplification | |
d | t | urn:hailstone:ml_ai:structural_equation[a7230d13c228] | 1 | structural_equation | 0 | arxiv.cs.IT.2510.06069v2.eq_800 | \dim \mathcal{D}^{(i)}=λ_1+λ_2+\cdots+λ_i | arxiv_extracted | ["a", "t", "m", "i", "d"] | arxiv | arxiv | arxiv:cs.IT:2510.06069v2 | cross_domain_amplification | |
T | t | urn:hailstone:ml_ai:structural_equation[793e3500ec7a] | 1 | structural_equation | 0 | arxiv.cs.IT.2510.06069v2.eq_801 | Tor_i(\mathcal{C}_e)=\mathcal{D}^{(i)} | arxiv_extracted | ["a", "t", "m", "o", "r_i", "T"] | arxiv | arxiv | arxiv:cs.IT:2510.06069v2 | cross_domain_amplification | |
s | x_i | urn:hailstone:ml_ai:structural_equation[060357a69f64] | 1 | structural_equation | 0 | arxiv.cs.IT.2510.04095v1.eq_802 | \sum_{i=1}^n φ_j(x_i) \le n Γ_j | arxiv_extracted | ["m_{i", "j", "x_i", "u", "s", "n"] | arxiv | arxiv | arxiv:cs.IT:2510.04095v1 | cross_domain_amplification | |
s | j | urn:hailstone:ml_ai:structural_equation[68235d9e9c69] | 1 | structural_equation | 0 | arxiv.cs.IT.2510.04095v1.eq_803 | \sum_{i=m}^n φ_j(x_i, x_{i-1}, \ldots, x_{i-m+1}) \le n Γ_j | arxiv_extracted | ["m_{i", "m", "j", "u", "s", "n"] | arxiv | arxiv | arxiv:cs.IT:2510.04095v1 | cross_domain_amplification | |
m | f | urn:hailstone:ml_ai:structural_equation[bf3e98f6fb54] | 1 | structural_equation | 0 | arxiv.cs.IT.2509.24699v1.eq_804 | \mathbf{y}=|\mathcal{A}(\mathbf{X}_0)|+\boldsymbolη | arxiv_extracted | ["f", "a", "t", "m", "h", "b"] | arxiv | arxiv | arxiv:cs.IT:2509.24699v1 | cross_domain_amplification | |
m | a | urn:hailstone:ml_ai:structural_equation[8ff08090ae09] | 1 | structural_equation | 0 | arxiv.cs.IT.2509.23090v1.eq_805 | \mathcal{G} = \mathbb{Z}_2^m | arxiv_extracted | ["a", "t", "m", "h", "c"] | arxiv | arxiv | arxiv:cs.IT:2509.23090v1 | cross_domain_amplification | |
y_i | r | urn:hailstone:ml_ai:structural_equation[330be8aba045] | 1 | structural_equation | 0 | arxiv.cs.IT.2509.20880v1.eq_806 | y_i=x_{i+mk} \prod_{\substack{j=1,\,\, m \nmid j}}^{mk-1} \left(x_{i+j}+1\right), \,\,{\rm for }\,\, i\in \{0,1,\ldots,n-1\} | arxiv_extracted | ["y_i", "m", "x_{i", "p", "r", "k"] | arxiv | arxiv | arxiv:cs.IT:2509.20880v1 | cross_domain_amplification | |
s | t | urn:hailstone:ml_ai:structural_equation[aac8b42ae233] | 1 | structural_equation | 0 | arxiv.cs.IT.2509.17778v1.eq_807 | μ(γ) = Θ(1/\sqrtγ) | arxiv_extracted | ["s", "t", "q", "r"] | arxiv | arxiv | arxiv:cs.IT:2509.17778v1 | cross_domain_amplification | |
s | t | urn:hailstone:ml_ai:structural_equation[78aa904bb73b] | 1 | structural_equation | 0 | arxiv.cs.IT.2509.15348v1.eq_808 | {\sf M}:=(\mathcal{E},r) | arxiv_extracted | ["f", "a", "t", "M", "m", "s"] | arxiv | arxiv | arxiv:cs.IT:2509.15348v1 | cross_domain_amplification | |
P | f | urn:hailstone:ml_ai:structural_equation[7a181662df0c] | 1 | structural_equation | 0 | arxiv.cs.IT.2509.14305v1.eq_809 | \Pr[M=χ] \le \frac{1}{2} + s(N)\exp(-α_d m^{c/d}) | arxiv_extracted | ["e", "f", "M", "P", "r", "l"] | arxiv | arxiv | arxiv:cs.IT:2509.14305v1 | cross_domain_amplification | |
h | f | urn:hailstone:ml_ai:structural_equation[7651dcea4674] | 1 | structural_equation | 0 | arxiv.cs.IT.2509.14021v2.eq_810 | h(X+Y) = \infty | arxiv_extracted | ["f", "h", "i", "X", "Y", "n"] | arxiv | arxiv | arxiv:cs.IT:2509.14021v2 | cross_domain_amplification | |
b | y | urn:hailstone:ml_ai:structural_equation[bb6d6ffbf4df] | 1 | structural_equation | 0 | arxiv.cs.IT.2509.08526v1.eq_811 | \boldsymbol{a}=(a_{0},a_{1},\cdots,a_{n-k-1})\in\mathbb{F}_{q}^{n-k} | arxiv_extracted | ["y", "o", "s", "b", "d", "l"] | arxiv | arxiv | arxiv:cs.IT:2509.08526v1 | cross_domain_amplification | |
n | I | urn:hailstone:ml_ai:structural_equation[a2ce1af611b4] | 1 | structural_equation | 0 | arxiv.cs.IT.2509.01809v1.eq_812 | n_{\text{INF}}^{\text{SP}} = Θ\left(s\log\left(p/s\right)/\log\left(ds/p\right)\right) | arxiv_extracted | ["e", "x", "t", "n", "I"] | arxiv | arxiv | arxiv:cs.IT:2509.01809v1 | cross_domain_amplification | |
n | S | urn:hailstone:ml_ai:structural_equation[cf49e2247f9d] | 1 | structural_equation | 0 | arxiv.cs.IT.2509.01809v1.eq_813 | n^{\text{Sp-ified}}_{\text{INF}} = Θ\left(p / ψ^2\right) | arxiv_extracted | ["e", "t", "S", "n", "x"] | arxiv | arxiv | arxiv:cs.IT:2509.01809v1 | cross_domain_amplification | |
o | f | urn:hailstone:ml_ai:structural_equation[a5f84faab002] | 1 | structural_equation | 0 | arxiv.cs.IT.2508.18503v3.eq_814 | of the form \[ {\boldsymbol y}_l = A_l X_o {\boldsymbol w}_l + {\boldsymbol z}_l, \quad l = 1, \ldots, L, \] where | arxiv_extracted | ["f", "e", "t", "h", "o"] | arxiv | arxiv | arxiv:cs.IT:2508.18503v3 | cross_domain_amplification | |
n | a | urn:hailstone:ml_ai:structural_equation[7454703c3a05] | 1 | structural_equation | 0 | arxiv.cs.IT.2508.14386v1.eq_815 | ρ(n,q,\tfrac{2(q-1)^{t-1}}{q^{t-1}(t-1)!}n^{t-1}+O(n^{t-2});\mathcal{D}_t)=O(1) | arxiv_extracted | ["f", "q", "a", "t", "r", "n"] | arxiv | arxiv | arxiv:cs.IT:2508.14386v1 | cross_domain_amplification | |
n | a | urn:hailstone:ml_ai:structural_equation[47ef66d96ae9] | 1 | structural_equation | 0 | arxiv.cs.IT.2508.14386v1.eq_816 | ρ(n,q,\tfrac{(q-1)^{t-1}}{(t-1)!}n^{t-1}+O(n^{t-2});\mathcal{I}_t)=\log\log n + O(1) | arxiv_extracted | ["f", "q", "a", "t", "r", "n"] | arxiv | arxiv | arxiv:cs.IT:2508.14386v1 | cross_domain_amplification | |
m | c | urn:hailstone:ml_ai:structural_equation[f39ce5be2513] | 1 | structural_equation | 0 | arxiv.cs.IT.2508.07461v1.eq_817 | \mathscr{S} = S[G] | arxiv_extracted | ["a", "t", "m", "h", "s", "c"] | arxiv | arxiv | arxiv:cs.IT:2508.07461v1 | cross_domain_amplification | |
m | c | urn:hailstone:ml_ai:structural_equation[ab16aab12ec0] | 1 | structural_equation | 0 | arxiv.cs.IT.2508.07461v1.eq_818 | \mathscr{R} = R[G] | arxiv_extracted | ["a", "t", "m", "h", "s", "c"] | arxiv | arxiv | arxiv:cs.IT:2508.07461v1 | cross_domain_amplification | |
m | a | urn:hailstone:ml_ai:structural_equation[cb5afcd0d7a8] | 1 | structural_equation | 0 | arxiv.cs.IT.2508.07461v1.eq_819 | \mathcal{C} + \mathcal{D} = \mathscr{S} | arxiv_extracted | ["a", "t", "m", "h", "c"] | arxiv | arxiv | arxiv:cs.IT:2508.07461v1 | cross_domain_amplification | |
m | a | urn:hailstone:ml_ai:structural_equation[d6b766c25d2b] | 1 | structural_equation | 0 | arxiv.cs.IT.2508.07461v1.eq_820 | \mathcal{C} \cap \mathcal{D} = \{0\} | arxiv_extracted | ["a", "t", "m", "h", "c"] | arxiv | arxiv | arxiv:cs.IT:2508.07461v1 | cross_domain_amplification |
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