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⧼math-wmc-Queries⧽

# fId #Var #matches query reference
1 Van_der_Waerden's_theorem-3 0 1 W ⁢ ( 2 , k ) > 2 k / k ε W 2 k superscript 2 k superscript k ε {\displaystyle W(2,k)>2^{{k}}/k^{{\varepsilon}}} W ( 2 , k ) > 2 k / k ε {\displaystyle W(2,k)>2^{k}/k^{\varepsilon }}
2 Bounded_variation-189 0 9 ( X , Σ ) {\displaystyle (X,\Sigma )} ( X , Σ ) {\displaystyle (X,\Sigma )}
3 Lindemann–Weierstrass_theorem-50 0 1 ( p - 1 ) ! n superscript p 1 n {\displaystyle(p-1)!^{{n}}} ( p − 1 ) ! n {\displaystyle (p-1)!^{n}}
4 Orbit_portrait-68 1 2 f x0 ⁢ ( z ) = z 2 + c subscript f x0 z superscript z 2 c {\displaystyle f_{\qvar@construct{x0}}(z)=z^{2}+c} f c ( z ) = z 2 + c {\displaystyle f_{c}(z)=z^{2}+c}
5 First-order_logic-114 0 1 ∀ x ⁢ ∀ y ⁢ P ⁢ ( x , y ) ⇔ ∀ y ⁢ ∀ x ⁢ P ⁢ ( x , y ) normal-⇔ for-all x for-all y P x y for-all y for-all x P x y {\displaystyle\forall x\,\forall y\,P(x,y)\Leftrightarrow\forall y\,\forall x% \,P(x,y)} ∀ x ∀ y P ( x , y ) ⇔ ∀ y ∀ x P ( x , y ) {\displaystyle \forall x\,\forall y\,P(x,y)\Leftrightarrow \forall y\,\forall x\,P(x,y)}
6 Clenshaw_algorithm-5 0 8 α ( x ) {\displaystyle \alpha (x)} α ( x ) {\displaystyle \alpha (x)}
7 Isolation_lemma-24 0 8 α ( x ) {\displaystyle \alpha (x)} α ( x ) {\displaystyle \alpha (x)}
8 Singularity_spectrum-4 0 8 α ( x ) {\displaystyle \alpha (x)} α ( x ) {\displaystyle \alpha (x)}
9 Time-evolving_block_decimation-36 7 1 | Ψ ⟩ = ∑ x0 Γ x1 [ 1 ] ⁢ i 1 ⁢ λ α 1 [ 1 ] ⁢ Γ α x2 ⁢ α 2 [ 2 ] ⁢ i x3 ⁢ λ x4 x5 ⁢ | i 1 ⁢ i 2 ⟩ ⁢ | x6 ⟩ ket normal-Ψ subscript x0 superscript subscript normal-Γ x1 delimited-[] 1 subscript i 1 superscript subscript λ subscript α 1 delimited-[] 1 superscript subscript normal-Γ subscript α x2 subscript α 2 delimited-[] 2 subscript i x3 superscript subscript λ x4 x5 ket subscript i 1 subscript i 2 ket x6 {\displaystyle|{\Psi}\rangle=\sum_{\qvar@construct{x0}}\Gamma_{{% \qvar@construct{x1}}}^{{[1]i_{1}}}\lambda_{{\alpha_{1}}}^{{[1]}}\Gamma_{{% \alpha_{\qvar@construct{x2}}\alpha_{2}}}^{{[2]i_{\qvar@construct{x3}}}}\lambda% _{\qvar@construct{x4}}^{{\qvar@construct{x5}}}|{i_{1}i_{2}}\rangle|{% \qvar@construct{x6}}\rangle} | Ψ ⟩ = ∑ i 1 , i 2 , α 1 , α 2 Γ α 1 [ 1 ] i 1 λ α 1 [ 1 ] Γ α 1 α 2 [ 2 ] i 2 λ α 2 [ 2 ] | i 1 i 2 ⟩ | Φ α 2 [ 3.. N ] ⟩ {\displaystyle |{\Psi }\rangle =\sum _{i_{1},i_{2},\alpha _{1},\alpha _{2}}\Gamma _{\alpha _{1}}^{[1]i_{1}}\lambda _{\alpha _{1}}^{[1]}\Gamma _{\alpha _{1}\alpha _{2}}^{[2]i_{2}}\lambda _{{\alpha }_{2}}^{[2]}|{i_{1}i_{2}}\rangle |{\Phi _{\alpha _{2}}^{[3..N]}}\rangle }
10 Monoidal_t-norm_logic-38 0 1 z * x ≤ y z x y {\displaystyle z*x\leq y} z ∗ x ≤ y {\displaystyle z*x\leq y}
11 Differentiation_rules-20 0 1 d d ⁢ x ⁢ ( log c ⁡ x ) = 1 x ⁢ ln ⁡ c , c > 0 , c ≠ 1 formulae-sequence d d x subscript c x 1 x c formulae-sequence c 0 c 1 {\displaystyle{{\frac{d}{dx}}}\left(\log_{{c}}x\right)={1\over x\ln c},\qquad c% >0,c\neq 1} d d x ( log c ⁡ x ) = 1 x ln ⁡ c , c > 0 , c ≠ 1 {\displaystyle {\frac {d}{dx}}\left(\log _{c}x\right)={1 \over x\ln c},\qquad c>0,c\neq 1}
12 Fermat's_spiral-2 0 1 θ = n × 137.508 ∘ , θ n superscript 137.508 {\displaystyle\theta=n\times 137.508^{{\circ}},} θ = n × 137.508 ∘ , {\displaystyle \theta =n\times 137.508^{\circ },}
13 Consistency_criterion-2 0 1 s V ⁢ ( ℛ ) subscript s V ℛ {\displaystyle s_{{V}}({{\mathcal{R}}})} s V ( R ) {\displaystyle s_{V}({\mathcal {R}})}
14 Basis_(universal_algebra)-43 0 2 ℓ ⁢ ( m ) normal-ℓ m {\displaystyle\ell(m)} ℓ ( m ) {\displaystyle \ell (m)}
15 Adequality-4 0 1 b ⁢ x - x 2 b x superscript x 2 {\displaystyle bx-x^{{2}}} b x − x 2 {\displaystyle bx-x^{2}}
16 Mason–Weaver_equation-85 0 6 ω k subscript ω k {\displaystyle\omega_{{{k}}}} ω k {\displaystyle \omega _{k}}
17 Kernel_Fisher_discriminant_analysis-2 0 1 𝐦 1 subscript 𝐦 1 {\displaystyle{{\mathbf{m}}}_{{1}}} m 1 {\displaystyle \mathbf {m} _{1}}
18 Implicit_solvation-17 0 16 r i ⁢ j subscript r i j {\displaystyle r_{{{ij}}}} r i j {\displaystyle r_{ij}}
19 Partition_function_(statistical_mechanics)-2 1 1 Z = ∑ j g j ⋅ e x0 Z subscript j normal-⋅ subscript g j superscript normal-e x0 {\displaystyle Z=\sum_{{j}}g_{j}\cdot{\mathrm{e}}^{{\qvar@construct{x0}}}} Z = ∑ j g j ⋅ e − β E j {\displaystyle Z=\sum _{j}g_{j}\cdot \mathrm {e} ^{-\beta E_{j}}}
20 Color_balance-39 0 48 S ′ superscript S normal-′ {\displaystyle S^{{\prime}}} S ′ {\displaystyle S'}
21 Hilbert's_theorem_(differential_geometry)-30 0 48 S ′ superscript S normal-′ {\displaystyle S^{{\prime}}} S ′ {\displaystyle S'}
22 Mayer–Vietoris_sequence-4 1 1 Ker ⁢ ( k * - l * ) ≅ ⁢ x0 ⁢ ( i * , j * ) . Ker subscript k subscript l x0 subscript i subscript j {\displaystyle{\text{Ker}}(k_{*}-l_{*})\cong{\text{\qvar@construct{x0}}}(i_{*}% ,j_{*}).} Ker ( k ∗ − l ∗ ) ≅ Im ( i ∗ , j ∗ ) . {\displaystyle {\text{Ker}}(k_{*}-l_{*})\cong {\text{Im}}(i_{*},j_{*}).}
23 Graphlets-3 2 1 D ⁢ ( G , H ) = ∑ i = 1 x0 | F i ⁢ ( G ) - F x1 ⁢ ( H ) | D G H superscript subscript i 1 x0 subscript F i G subscript F x1 H {\displaystyle D(G,H)=\sum_{{i=1}}^{{\qvar@construct{x0}}}|F_{i}(G)-F_{% \qvar@construct{x1}}(H)|} D ( G , H ) = ∑ i = 1 29 | F i ( G ) − F i ( H ) | {\displaystyle D(G,H)=\sum _{i=1}^{29}|F_{i}(G)-F_{i}(H)|}
24 Kinetic_energy-28 2 1 E ⁢ x0 = E t + E x1 subscript E x0 subscript E t subscript E x1 {\displaystyle E_{{\text{\qvar@construct{x0}}}}=E_{t}+E_{{\qvar@construct{x1}}% }\,} E k = E t + E r {\displaystyle E_{\text{k}}=E_{t}+E_{\text{r}}\,}
25 Lattice_problem-22 0 1 λ ⁢ ( L ⁢ ( B ) ) ≤ d λ L B d {\displaystyle\lambda(L(B))\leq d} λ ( L ( B ) ) ≤ d {\displaystyle \lambda (L(B))\leq d}
26 Huffman_coding-10 0 1 L ⁢ ( C ) ≤ L ⁢ ( T ) L C L T {\displaystyle L\left(C\right)\leq L\left(T\right)} L ( C ) ≤ L ( T ) {\displaystyle L\left(C\right)\leq L\left(T\right)}
27 Dispersion_(optics)-0 0 1 v = c n v c n {\displaystyle v={{\frac{c}{n}}}} v = c n {\displaystyle v={\frac {c}{n}}}
28 Allan_variance-93 0 1 σ y 2 ⁢ ( τ ) = 2 ⁢ π 2 ⁢ τ 3 ⁢ h - 2 superscript subscript σ y 2 τ 2 superscript π 2 τ 3 subscript h 2 {\displaystyle\sigma_{{y}}^{{2}}(\tau)={{\frac{2\pi^{{2}}\tau}{3}}}h_{{{-2}}}} σ y 2 ( τ ) = 2 π 2 τ 3 h − 2 {\displaystyle \sigma _{y}^{2}(\tau )={\frac {2\pi ^{2}\tau }{3}}h_{-2}}
29 Superconducting_radio_frequency-5 1 6 R s ⁢ n ⁢ o ⁢ r ⁢ m ⁢ a ⁢ l = x0 subscript R s n o r m a l x0 {\displaystyle R_{{s\ normal}}={\sqrt{\qvar@construct{x0}}}} R s   n o r m a l = ω μ 0 2 σ {\displaystyle R_{s\ normal}={\sqrt {\frac {\omega \mu _{0}}{2\sigma }}}}
30 Vienna_rectifier-5 1 1 ϕ 1 = - 30 ∘ ⁢ … + 30 x0 subscript ϕ 1 superscript 30 normal-… superscript 30 x0 {\displaystyle\phi_{1}=-30^{\circ}...+30^{\qvar@construct{x0}}} ϕ 1 = − 30 ∘ . . . + 30 ∘ {\displaystyle \phi _{1}=-30^{\circ }...+30^{\circ }}
31 Modigliani–Miller_theorem-19 0 54 T c {\displaystyle T_{c}} T c {\displaystyle T_{c}}
32 Proximity_effect_(superconductivity)-2 0 54 T c {\displaystyle T_{c}} T c {\displaystyle T_{c}}
33 Multicritical_point-9 1 54 T x0 subscript T x0 {\displaystyle T_{\qvar@construct{x0}}} T c {\displaystyle T_{c}}
34 Jenkins–Traub_algorithm-55 1 1 P 1 ⁢ ( X ) = P ⁢ ( X ) / ( X - α x0 ) subscript P 1 X P X X subscript α x0 {\displaystyle P_{1}(X)=P(X)/(X-\alpha_{\qvar@construct{x0}})} P 1 ( X ) = P ( X ) / ( X − α 1 ) {\displaystyle P_{1}(X)=P(X)/(X-\alpha _{1})}
35 Doomsday_argument-4 0 1 = k n . absent k n {\displaystyle={{\frac{k}{n}}}.} = k n . {\displaystyle ={\frac {k}{n}}.}
36 Divisor_function-12 1 1 n = ∏ i = 1 r p i x0 n superscript subscript product i 1 r superscript subscript p i x0 {\displaystyle n=\prod_{{i=1}}^{r}p_{i}^{{\qvar@construct{x0}}}} n = ∏ i = 1 r p i a i {\displaystyle n=\prod _{i=1}^{r}p_{i}^{a_{i}}}
37 LTI_system_theory-87 0 1 H ⁢ ( j ⁢ ω ) = ℱ ⁢ { h ⁢ ( t ) } H j ω ℱ h t {\displaystyle H(j\omega)={{\mathcal{F}}}\{h(t)\}} H ( j ω ) = F { h ( t ) } {\displaystyle H(j\omega )={\mathcal {F}}\{h(t)\}}
38 Phase-shift_keying-31 0 10 π / 4 {\displaystyle \pi /4} π / 4 {\displaystyle \pi /4}
39 Binomial_theorem-6 3 1 ( x + y ) n = ∑ x0 n ( n k ) ⁢ x n - k ⁢ y k = ∑ x1 n ( n k ) ⁢ x x2 ⁢ y n - k . superscript x y n superscript subscript x0 n binomial n k superscript x n k superscript y k superscript subscript x1 n binomial n k superscript x x2 superscript y n k {\displaystyle(x+y)^{n}=\sum_{{\qvar@construct{x0}}}^{n}{n\choose k}x^{{n-k}}y% ^{k}=\sum_{{\qvar@construct{x1}}}^{n}{n\choose k}x^{{\qvar@construct{x2}}}y^{{% n-k}}.} ( x + y ) n = ∑ k = 0 n ( n k ) x n − k y k = ∑ k = 0 n ( n k ) x k y n − k . {\displaystyle (x+y)^{n}=\sum _{k=0}^{n}{n \choose k}x^{n-k}y^{k}=\sum _{k=0}^{n}{n \choose k}x^{k}y^{n-k}.}
40 Rate_equation-33 0 1 [ A ] t = - k ⁢ t + [ A ] 0 subscript delimited-[] A t k t subscript delimited-[] A 0 {\displaystyle\ [A]_{{t}}=-kt+[A]_{{0}}}   [ A ] t = − k t + [ A ] 0 {\displaystyle \ [A]_{t}=-kt+[A]_{0}}
41 Martingale_(betting_system)-1 1 1 q x0 superscript q x0 {\displaystyle q^{{\qvar@construct{x0}}}} q 42 {\displaystyle q^{42}}
42 Borsuk's_conjecture-5 0 1 α ⁢ ( d ) ≤ ( 3 / 2 + ε ) d α d superscript 3 2 ε d {\displaystyle\alpha(d)\leq\left({{\sqrt{3/2}}}+\varepsilon\right)^{{d}}} α ( d ) ≤ ( 3 / 2 + ε ) d {\displaystyle \alpha (d)\leq \left({\sqrt {3/2}}+\varepsilon \right)^{d}}
43 Theoretical_motivation_for_general_relativity-96 0 1 f μ = - 8 ⁢ π ⁢ G 3 ⁢ c 4 ⁢ ( A 2 ⁢ T α ⁢ β + B 2 ⁢ T ⁢ η α ⁢ β ) ⁢ ( δ ν μ + u μ ⁢ u ν ) ⁢ u α ⁢ x ν ⁢ u β superscript f μ 8 π G 3 superscript c 4 A 2 subscript T α β B 2 T subscript η α β superscript subscript δ ν μ superscript u μ subscript u ν superscript u α superscript x ν superscript u β {\displaystyle f^{{\mu}}=-8\pi{G\over{3c^{4}}}\left({A\over 2}T_{{\alpha\beta}% }+{B\over 2}T\eta_{{\alpha\beta}}\right)\left(\delta_{{\nu}}^{{\mu}}+u^{{\mu}}% u_{{\nu}}\right)u^{{\alpha}}x^{{\nu}}u^{{\beta}}} f μ = − 8 π G 3 c 4 ( A 2 T α β + B 2 T η α β ) ( δ ν μ + u μ u ν ) u α x ν u β {\displaystyle f^{\mu }=-8\pi {G \over {3c^{4}}}\left({A \over 2}T_{\alpha \beta }+{B \over 2}T\eta _{\alpha \beta }\right)\left(\delta _{\nu }^{\mu }+u^{\mu }u_{\nu }\right)u^{\alpha }x^{\nu }u^{\beta }}
44 Q-Vectors-0 1 1 D g ⁢ u x0 D ⁢ t - f 0 ⁢ v a - β ⁢ y ⁢ v g = 0 subscript D g subscript u x0 D t subscript f 0 subscript v a β y subscript v g 0 {\displaystyle{\frac{D_{g}u_{\qvar@construct{x0}}}{Dt}}-f_{{0}}v_{a}-\beta yv_% {g}=0} D g u g D t − f 0 v a − β y v g = 0 {\displaystyle {\frac {D_{g}u_{g}}{Dt}}-f_{0}v_{a}-\beta yv_{g}=0}
45 Stencil_code-10 0 8 I c subscript I c {\displaystyle I_{{c}}} I c {\displaystyle I_{c}}
46 Geometric_algebra-81 0 1 A ↦ M ⁢ α ⁢ ( A ) ⁢ M - 1 , maps-to A M α A superscript M 1 {\displaystyle\,A\mapsto M\alpha(A)M^{{{-1}}},} A ↦ M α ( A ) M − 1 , {\displaystyle \,A\mapsto M\alpha (A)M^{-1},}
47 Stochastic_game-53 0 2 Γ ∞ subscript normal-Γ {\displaystyle\Gamma_{{{\infty}}}} Γ ∞ {\displaystyle \Gamma _{\infty }}
48 331_model-3 0 1 Y = β ⁢ T 8 + I ⁢ X Y β subscript T 8 I X {\displaystyle Y=\beta T_{{8}}+IX} Y = β T 8 + I X {\displaystyle Y=\beta T_{8}+IX}
49 Sigma_additivity-12 0 1 μ ⁢ ( A ) = { 1  if  ⁢ 0 ∈ A 0  if  ⁢ 0 ∉ A . μ A cases 1 if 0 A 0 if 0 A {\displaystyle\mu(A)={{\begin{cases}1&{{\mbox{ if }}}0\in A\\ 0&{{\mbox{ if }}}0\notin A.\end{cases}}}} μ ( A ) = { 1  if  0 ∈ A 0  if  0 ∉ A . {\displaystyle \mu (A)={\begin{cases}1&{\mbox{ if }}0\in A\\0&{\mbox{ if }}0\notin A.\end{cases}}}
50 Marcus_theory-56 0 1 λ i ⁢ n subscript λ i n {\displaystyle\lambda_{{{in}}}} λ i n {\displaystyle \lambda _{in}}
51 Centrifugal_fan-2 1 1 r ⁢ p ⁢ m x0 r p subscript m x0 {\displaystyle rpm_{{\qvar@construct{x0}}}} r p m m o t o r {\displaystyle rpm_{motor}}
52 Backstepping-157 16 1 u 1 ⁢ ( 𝐱𝟎 , z 1 ) = v 1 + u ˙ x ⏟ x1 = - ∂ ⁡ V x ∂ ⁡ x2 ⁢ g x ⁢ ( 𝐱𝟑 ) - k 1 ⁢ ( z 1 - u x ⁢ ( 𝐱 ) ⏟ x4 ) ⏞ v 1 + x5 ∂ ⁡ x6 ⁢ ( f x7 ⁢ ( x8 ) + g x ⁢ ( 𝐱𝟗 ) ⁢ z 1 ⏟ x10 ⁢  (i.e.,  ⁢ d ⁡ 𝐱𝟏𝟏 x12 ⁡ t ⁢ ⁢ x13 ) ⏞ x14 x ⁢  (i.e.,  ⁢ d ⁡ u x15 d ⁡ t ⁢ ) subscript normal-⏟ subscript u 1 x0 subscript z 1 subscript v 1 subscript normal-˙ u x x1 superscript normal-⏞ subscript V x x2 subscript g x x3 subscript k 1 subscript normal-⏟ subscript z 1 subscript u x 𝐱 x4 subscript v 1 superscript normal-⏞ x5 x6 subscript normal-⏟ subscript f x7 x8 subscript g x x9 subscript z 1 x10 (i.e., normal-d x11 x12 t x13 subscript x14 x (i.e., normal-d subscript u x15 normal-d t ) {\displaystyle\underbrace{u_{1}({\mathbf{\qvar@construct{x0}}},z_{1})=v_{1}+{% \dot{u}}_{x}}_{\qvar@construct{x1}}=\overbrace{-{\frac{\partial V_{x}}{% \partial{\qvar@construct{x2}}}}g_{x}({\mathbf{\qvar@construct{x3}}})-k_{1}(% \underbrace{z_{1}-u_{x}({\mathbf{x}})}_{\qvar@construct{x4}})}^{{v_{1}}}\,+\,% \overbrace{{\frac{\qvar@construct{x5}}{\partial{\qvar@construct{x6}}}}(% \underbrace{f_{\qvar@construct{x7}}({\qvar@construct{x8}})+g_{x}({\mathbf{% \qvar@construct{x9}}})z_{1}}_{{{\qvar@construct{x10}}{\text{ (i.e., }}{\frac{% \operatorname{d}{\mathbf{\qvar@construct{x11}}}}{\operatorname{\qvar@construct% {x12}}t}}{\text{\qvar@construct{x13}}}}})}^{{{\qvar@construct{x14}}_{x}{\text{% (i.e., }}{\frac{\operatorname{d}u_{\qvar@construct{x15}}}{\operatorname{d}t}}% {\text{)}}}}} u 1 ( x , z 1 ) = v 1 + u ˙ x ⏟ By definition of  v 1 = − ∂ V x ∂ x g x ( x ) − k 1 ( z 1 − u x ( x ) ⏟ e 1 ) ⏞ v 1 + ∂ u x ∂ x ( f x ( x ) + g x ( x ) z 1 ⏟ x ˙  (i.e.,  d ⁡ x d ⁡ t ) ) ⏞ u ˙ x  (i.e.,  d ⁡ u x d ⁡ t ) {\displaystyle \underbrace {u_{1}(\mathbf {x} ,z_{1})=v_{1}+{\dot {u}}_{x}} _{{\text{By definition of }}v_{1}}=\overbrace {-{\frac {\partial V_{x}}{\partial \mathbf {x} }}g_{x}(\mathbf {x} )-k_{1}(\underbrace {z_{1}-u_{x}(\mathbf {x} )} _{e_{1}})} ^{v_{1}}\,+\,\overbrace {{\frac {\partial u_{x}}{\partial \mathbf {x} }}(\underbrace {f_{x}(\mathbf {x} )+g_{x}(\mathbf {x} )z_{1}} _{{\dot {\mathbf {x} }}{\text{ (i.e., }}{\frac {\operatorname {d} \mathbf {x} }{\operatorname {d} t}}{\text{)}}})} ^{{\dot {u}}_{x}{\text{ (i.e., }}{\frac {\operatorname {d} u_{x}}{\operatorname {d} t}}{\text{)}}}}
53 Stochastic_volatility-10 0 1 E ⁢ [ σ ^ 2 ] = n - 1 n ⁢ σ 2 E delimited-[] superscript normal-^ σ 2 n 1 n superscript σ 2 {\displaystyle E\left[{{\hat{\sigma}}}^{{2}}\right]={{\frac{n-1}{n}}}\sigma^{{% 2}}} E [ σ ^ 2 ] = n − 1 n σ 2 {\displaystyle E\left[{\hat {\sigma }}^{2}\right]={\frac {n-1}{n}}\sigma ^{2}}
54 Separation_logic-63 0 1 𝖿𝗏 𝖿𝗏 {\displaystyle{{\mathsf{fv}}}} f v {\displaystyle {\mathsf {fv}}}
55 Image_moment-2 0 1 ∑ x ∑ y I ⁢ ( x , y ) subscript x subscript y I x y {\displaystyle\sum_{{x}}\sum_{{y}}I(x,y)\,\!} ∑ x ∑ y I ( x , y ) {\displaystyle \sum _{x}\sum _{y}I(x,y)\,\!}
56 Geostrophic_wind-5 0 1 𝑭 r subscript 𝑭 r {\displaystyle{{\boldsymbol{F}}}_{{r}}} F r {\displaystyle {\boldsymbol {F}}_{r}}
57 Ext_functor-9 0 1 0 → B → A ⊕ B → A → 0. normal-→ 0 B normal-→ direct-sum A B normal-→ A normal-→ 0. {\displaystyle 0\rightarrow B\rightarrow A\oplus B\rightarrow A\rightarrow 0.} 0 → B → A ⊕ B → A → 0. {\displaystyle 0\rightarrow B\rightarrow A\oplus B\rightarrow A\rightarrow 0.}
58 Covariant_derivative-50 1 1 ( ∇ Y ⁡ T ) ⁢ ( α 1 , α 2 , … , X 1 , X 2 , … ) = Y ⁢ ( T ⁢ ( α 1 , α 2 , … , X x0 , X 2 , … ) ) subscript normal-∇ Y T subscript α 1 subscript α 2 normal-… subscript X 1 subscript X 2 normal-… Y T subscript α 1 subscript α 2 normal-… subscript X x0 subscript X 2 normal-… {\displaystyle(\nabla_{Y}T)(\alpha_{1},\alpha_{2},\ldots,X_{1},X_{2},\ldots)=Y% (T(\alpha_{1},\alpha_{2},\ldots,X_{\qvar@construct{x0}},X_{2},\ldots))} ( ∇ Y T ) ( α 1 , α 2 , … , X 1 , X 2 , … ) = Y ( T ( α 1 , α 2 , … , X 1 , X 2 , … ) ) {\displaystyle (\nabla _{Y}T)(\alpha _{1},\alpha _{2},\ldots ,X_{1},X_{2},\ldots )=Y(T(\alpha _{1},\alpha _{2},\ldots ,X_{1},X_{2},\ldots ))}
59 Anderson_localization-4 1 1 ∑ n ∈ x0 d | ψ ⁢ ( t , n ) | 2 ⁢ | n | ≤ C subscript n superscript x0 d superscript ψ t n 2 n C {\displaystyle\sum_{{n\in{\qvar@construct{x0}}^{d}}}|\psi(t,n)|^{2}|n|\leq C} ∑ n ∈ Z d | ψ ( t , n ) | 2 | n | ≤ C {\displaystyle \sum _{n\in \mathbb {Z} ^{d}}|\psi (t,n)|^{2}|n|\leq C}
60 Trajectory_of_a_projectile-26 5 1 p = - x ± x x0 - 4 ⁢ ( x1 ) ⁢ ( x2 x3 - y ) 2 ⁢ ( - g ⁢ x 2 x4 ) p plus-or-minus x superscript x x0 4 x1 x2 x3 y 2 g superscript x 2 x4 {\displaystyle p={{\frac{-x\pm{\sqrt{x^{\qvar@construct{x0}}-4({% \qvar@construct{x1}})({\frac{\qvar@construct{x2}}{\qvar@construct{x3}}}-y)}}}{% 2({\frac{-gx^{2}}{\qvar@construct{x4}}})}}}} p = − x ± x 2 − 4 ( − g x 2 2 v 2 ) ( − g x 2 2 v 2 − y ) 2 ( − g x 2 2 v 2 ) {\displaystyle p={\frac {-x\pm {\sqrt {x^{2}-4({\frac {-gx^{2}}{2v^{2}}})({\frac {-gx^{2}}{2v^{2}}}-y)}}}{2({\frac {-gx^{2}}{2v^{2}}})}}}
61 Cusp_neighborhood-7 1 1 { z ∈ H : | z | > 1 , | x0 ⁢ ( z ) | < 1 2 } conditional-set z H formulae-sequence z 1 x0 z 1 2 {\displaystyle\left\{z\in H:\left|z\right|>1,\,\left|\,{\mbox{\qvar@construct{% x0}}}(z)\,\right|<{\frac{1}{2}}\right\}} { z ∈ H : | z | > 1 , | Re ( z ) | < 1 2 } {\displaystyle \left\{z\in H:\left|z\right|>1,\,\left|\,{\mbox{Re}}(z)\,\right|<{\frac {1}{2}}\right\}}
62 Spectrum_(functional_analysis)-17 0 4 T − λ I {\displaystyle T-\lambda I} T − λ I {\displaystyle T-\lambda I}
63 Multivariate_normal_distribution-20 4 1 y ⁢ ( x ) = x0 ( ρ ) ⁢ x1 x2 ⁢ ( x - x3 ) + μ y . y x x0 ρ x1 x2 x x3 subscript μ y {\displaystyle y\left(x\right)={{\mathop{\qvar@construct{x0}}}}\left({{\rho}}% \right){\frac{{{\qvar@construct{x1}}}}{{{\qvar@construct{x2}}}}}\left({x-{% \qvar@construct{x3}}}\right)+{\mu_{y}}.} y ( x ) = s g n ( ρ ) σ y σ x ( x − μ x ) + μ y . {\displaystyle y\left(x\right)={\mathop {\rm {sgn}} }\left({\rho }\right){\frac {\sigma _{y}}{\sigma _{x}}}\left({x-{\mu _{x}}}\right)+{\mu _{y}}.}
64 Algebraically_closed_group-56 0 1 x = b x b {\displaystyle x=b\ } x = b   {\displaystyle x=b\ }
65 Analytic_capacity-10 1 1 H 1 ⁢ ( K ) = x0 superscript H 1 K x0 {\displaystyle H^{1}(K)={\qvar@construct{x0}}} H 1 ( K ) = 2 {\displaystyle H^{1}(K)={\sqrt {2}}}
66 Block_cipher_mode_of_operation-8 2 1 P x0 = x1 ⁢ ( E K ⁢ ( S i - 1 ) , x ) ⊕ C i subscript P x0 direct-sum x1 subscript E K subscript S i 1 x subscript C i {\displaystyle P_{\qvar@construct{x0}}={\mbox{\qvar@construct{x1}}}(E_{K}(S_{{% i-1}}),x)\oplus C_{i}} P i = head ( E K ( S i − 1 ) , x ) ⊕ C i {\displaystyle P_{i}={\mbox{head}}(E_{K}(S_{i-1}),x)\oplus C_{i}}
67 Partial_derivative-30 0 1 ∂ ⁡ f ∂ ⁡ x = f x = ∂ x ⁡ f . f x subscript f x subscript x f {\displaystyle{{\frac{\partial f}{\partial x}}}=f_{{x}}=\partial_{{x}}f.} ∂ f ∂ x = f x = ∂ x f . {\displaystyle {\frac {\partial f}{\partial x}}=f_{x}=\partial _{x}f.}
68 Poset_game-0 0 1 P x = P - { a ∣ a ≥ x } subscript P x P conditional-set a a x {\displaystyle P_{{x}}=P-\{a\mid a\geq x\}} P x = P − { a ∣ a ≥ x } {\displaystyle P_{x}=P-\{a\mid a\geq x\}}
69 Engine_efficiency-0 0 1 η = w ⁢ o ⁢ r ⁢ k ⁢ d ⁢ o ⁢ n ⁢ e h ⁢ e ⁢ a ⁢ t ⁢ a ⁢ b ⁢ s ⁢ o ⁢ r ⁢ b ⁢ e ⁢ d = Q ⁢ 1 - Q ⁢ 2 Q ⁢ 1 η w o r k d o n e h e a t a b s o r b e d Q 1 Q 2 Q 1 {\displaystyle\eta={{\frac{work\ done}{heat\ absorbed}}}={{\frac{Q1-Q2}{Q1}}}} η = w o r k   d o n e h e a t   a b s o r b e d = Q 1 − Q 2 Q 1 {\displaystyle \eta ={\frac {work\ done}{heat\ absorbed}}={\frac {Q1-Q2}{Q1}}}
70 Legendre_transformation-29 0 1 d ⁢ f = ∂ ⁡ f ∂ ⁡ x ⁢ d ⁢ x + ∂ ⁡ f ∂ ⁡ y ⁢ d ⁢ y = p ⁢ d ⁢ x + v ⁢ d ⁢ y d f f x d x f y d y p d x v d y {\displaystyle df={\partial f\over\partial x}dx+{\partial f\over\partial y}dy=% pdx+vdy} d f = ∂ f ∂ x d x + ∂ f ∂ y d y = p d x + v d y {\displaystyle df={\partial f \over \partial x}dx+{\partial f \over \partial y}dy=pdx+vdy}
71 Cooperative_diversity-4 1 2 h x0 subscript h x0 {\displaystyle h_{{\qvar@construct{x0}}}} h r , s {\displaystyle h_{r,s}}
72 Algebraic_K-theory-20 2 2 K x0 x1 ⁢ ( k ) := T * ⁢ ( k × ) / ( a ⊗ ( 1 - a ) ) assign superscript subscript K x0 x1 k superscript T superscript k tensor-product a 1 a {\displaystyle K_{\qvar@construct{x0}}^{\qvar@construct{x1}}(k):=T^{*}(k^{% \times})/(a\otimes(1-a))} K ∗ M ( k ) := T ∗ ( k × ) / ( a ⊗ ( 1 − a ) ) {\displaystyle K_{*}^{M}(k):=T^{*}(k^{\times })/(a\otimes (1-a))}
73 Cone_of_curves-45 0 1 { C : K X ⋅ C = 0 } conditional-set C normal-⋅ subscript K X C 0 {\displaystyle\{C:K_{{X}}\cdot C=0\}} { C : K X ⋅ C = 0 } {\displaystyle \{C:K_{X}\cdot C=0\}}
74 Geometrothermodynamics-8 1 1 Θ ∧ ( d ⁢ Θ ) x0 ≠ 0 normal-Θ superscript d normal-Θ x0 0 {\displaystyle\Theta\wedge(d\Theta)^{\qvar@construct{x0}}\neq 0} Θ ∧ ( d Θ ) n ≠ 0 {\displaystyle \Theta \wedge (d\Theta )^{n}\neq 0}
75 Darcy's_law-5 0 1 D ⁢ ( ρ ⁢ u i ) / D ⁢ t ≈ 0 D ρ subscript u i D t 0 {\displaystyle D\left(\rho u_{{i}}\right)/Dt\approx 0} D ( ρ u i ) / D t ≈ 0 {\displaystyle D\left(\rho u_{i}\right)/Dt\approx 0}
76 Unit_root-25 0 1 z t = λ 1 ⁢ z t - 1 + ε t subscript z t subscript λ 1 subscript z t 1 subscript ε t {\displaystyle z_{{{t}}}=\lambda_{{{1}}}z_{{{t-1}}}+\varepsilon_{{{t}}}} z t = λ 1 z t − 1 + ε t {\displaystyle z_{t}=\lambda _{1}z_{t-1}+\varepsilon _{t}}
77 Drucker–Prager_yield_criterion-83 0 4 b 3 {\displaystyle b_{3}} b 3 {\displaystyle b_{3}}
78 Antiplane_shear-9 0 4 b 3 {\displaystyle b_{3}} b 3 {\displaystyle b_{3}}
79 Table_of_thermodynamic_equations-32 2 1 Δ ⁢ W = ∫ V x0 V 2 p ⁢ x1 ⁢ V normal-Δ W superscript subscript subscript V x0 subscript V 2 p x1 V {\displaystyle\Delta W=\int_{{V_{\qvar@construct{x0}}}}^{{V_{2}}}p{% \qvar@construct{x1}}V\,\!} Δ W = ∫ V 1 V 2 p d V {\displaystyle \Delta W=\int _{V_{1}}^{V_{2}}p\mathrm {d} V\,\!}
80 Fulton–Hansen_connectedness_theorem-2 0 1 dim ⁡ f ⁢ ( Z ) > n dimension f Z n {\displaystyle\dim f(Z)>n} dim ⁡ f ( Z ) > n {\displaystyle \dim f(Z)>n}
81 E_(mathematical_constant)-16 2 1 d d ⁢ t ⁢ log e ⁡ t = x0 x1 . d d t subscript e t x0 x1 {\displaystyle{\frac{d}{dt}}\log_{e}t={\frac{\qvar@construct{x0}}{% \qvar@construct{x1}}}.} d d t log e ⁡ t = 1 t . {\displaystyle {\frac {d}{dt}}\log _{e}t={\frac {1}{t}}.}
82 BrownBoost-32 1 1 h i : X → { x0 } normal-: subscript h i normal-→ X x0 {\displaystyle h_{i}:X\to\{\qvar@construct{x0}\}} h i : X → { − 1 , + 1 } {\displaystyle h_{i}:X\to \{-1,+1\}}
83 List_of_RNA_structure_prediction_software-2 0 1 2 ≤ s ⁢ e ⁢ q ⁢ s ≤ 6 2 s e q s 6 {\displaystyle 2\leq seqs\leq 6} 2 ≤ s e q s ≤ 6 {\displaystyle 2\leq seqs\leq 6}
84 Projections_onto_convex_sets-6 1 1 F = { x0 } . F x0 {\displaystyle F=\{\qvar@construct{x0}\}.} F = { ( x , y ) : x ∈ R b , y ∈ R n , x = y } . {\displaystyle F=\{(x,y):x\in {\mathcal {R}}^{b},\,y\in {\mathcal {R}}^{n},\;x=y\}.}
85 Almost_surely-21 0 1 X i ⁢ ( ω ) = ω i subscript X i ω subscript ω i {\displaystyle X_{{i}}(\omega)=\omega_{{i}}} X i ( ω ) = ω i {\displaystyle X_{i}(\omega )=\omega _{i}}
86 Lagrangian_mechanics-67 3 1 ∂ ⁡ L ∂ ⁡ q i = x0 ⁢ ∂ ⁡ x1 ∂ ⁡ x2 . L subscript q i x0 x1 x2 {\displaystyle{\partial{L}\over\partial q_{i}}={\qvar@construct{x0}}{\partial{% \qvar@construct{x1}}\over\partial{{\qvar@construct{x2}}}}.} ∂ L ∂ q i = d d t ∂ L ∂ q i ˙ . {\displaystyle {\partial {L} \over \partial q_{i}}={\mathrm {d} \over \mathrm {d} t}{\partial {L} \over \partial {\dot {q_{i}}}}.}
87 Near_sets-100 1 2 x x0 subscript x x0 {\displaystyle x_{\qvar@construct{x0}}} x 7 {\displaystyle x_{7}}
88 Polynomial_interpolation-30 0 6 Π n subscript normal-Π n {\displaystyle\Pi_{{n}}} Π n {\displaystyle \Pi _{n}}
89 Mutual_fund_separation_theorem-10 0 1 σ 2 = X T ⁢ V ⁢ X , superscript σ 2 superscript X T V X {\displaystyle\sigma^{{2}}=X^{{T}}VX,} σ 2 = X T V X , {\displaystyle \sigma ^{2}=X^{T}VX,}
90 Coarea_formula-3 1 1 ∫ ℝ n f ⁢ d ⁢ x = ∫ 0 ∞ { ∫ x0 f ⁢ d ⁢ S } ⁢ d ⁢ r . subscript superscript ℝ n f d x superscript subscript 0 subscript x0 f d S d r {\displaystyle\int_{{{\mathbb{R}}^{n}}}f\,dx=\int_{0}^{\infty}\left\{\int_{% \qvar@construct{x0}}f\,dS\right\}\,dr.} ∫ R n f d x = ∫ 0 ∞ { ∫ ∂ B ( x 0 ; r ) f d S } d r . {\displaystyle \int _{\mathbb {R} ^{n}}f\,dx=\int _{0}^{\infty }\left\{\int _{\partial B(x_{0};r)}f\,dS\right\}\,dr.}
91 Dirac_bracket-74 0 1 { x , p x } D ⁢ B = { y , p y } D ⁢ B = 1 2 subscript x subscript p x D B subscript y subscript p y D B 1 2 {\displaystyle\{x,p_{{x}}\}_{{{DB}}}=\{y,p_{{y}}\}_{{{DB}}}={{\frac{1}{2}}}} { x , p x } D B = { y , p y } D B = 1 2 {\displaystyle \{x,p_{x}\}_{DB}=\{y,p_{y}\}_{DB}={\frac {1}{2}}}
92 Pickands–Balkema–de_Haan_theorem-17 0 1 G k , σ ⁢ ( y ) = 1 - ( 1 + k ⁢ y / σ ) - 1 / k subscript G k σ y 1 superscript 1 k y σ 1 k {\displaystyle G_{{{k,\sigma}}}(y)=1-(1+ky/\sigma)^{{{-1/k}}}} G k , σ ( y ) = 1 − ( 1 + k y / σ ) − 1 / k {\displaystyle G_{k,\sigma }(y)=1-(1+ky/\sigma )^{-1/k}}
93 Quantum_channel-15 0 1 L ⁢ ( H B ) ⊗ C ⁢ ( X ) tensor-product L subscript H B C X {\displaystyle L(H_{{B}})\otimes C(X)} L ( H B ) ⊗ C ( X ) {\displaystyle L(H_{B})\otimes C(X)}
94 Ehrenfest_model-2 1 1 π i = 2 x0 ⁢ ( N i ) subscript π i superscript 2 x0 binomial N i {\displaystyle\pi_{i}=2^{\qvar@construct{x0}}{{\textstyle\left({{N}\atop{i}}% \right)}}} π i = 2 − N ( N i ) {\displaystyle \pi _{i}=2^{-N}{\tbinom {N}{i}}}
95 Fidelity_of_quantum_states-1 0 1 ( p 1 , ⋯ , p n ) subscript p 1 normal-⋯ subscript p n {\displaystyle({{\sqrt{p_{{1}}}}},\cdots,{{\sqrt{p_{{n}}}}})} ( p 1 , ⋯ , p n ) {\displaystyle ({\sqrt {p_{1}}},\cdots ,{\sqrt {p_{n}}})}
96 Yield_surface-27 0 5 𝒔 𝒔 {\displaystyle{{\boldsymbol{s}}}} s {\displaystyle {\boldsymbol {s}}}
97 Non-linear_least_squares-67 0 1 𝐉 𝐓 ⁢ 𝐖 ⁢ 𝚫 ⁢ 𝐲 superscript 𝐉 𝐓 𝐖 𝚫 𝐲 {\displaystyle{{\mathbf{J^{{T}}W\ \Delta y}}}} J T W   Δ y {\displaystyle \mathbf {J^{T}W\ \Delta y} }
98 Markov_decision_process-87 0 2 V ¯ * superscript normal-¯ V {\displaystyle{{\bar{V}}}^{{*}}} V ¯ ∗ {\displaystyle {\bar {V}}^{*}}
99 Hilbert_series_and_Hilbert_polynomial-9 2 1 x0 x1 x0 x1 {\displaystyle\;{\frac{\qvar@construct{x0}}{\qvar@construct{x1}}}\;} ( n + δ − 1 ) ( n + δ − 2 ) ⋯ n ( δ − 1 ) ! {\displaystyle \;{\frac {(n+\delta -1)(n+\delta -2)\cdots n}{(\delta -1)!}}\;}
100 Overlap–add_method-15 0 3 y k ⁢ [ n ] subscript y k delimited-[] n {\displaystyle y_{{k}}[n]} y k [ n ] {\displaystyle y_{k}[n]}
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