Returned 94 matches (100 formulae, 64 docs)
    Lookup 8.724 ms, Re-ranking 96312.626 ms
    Found 72637 tuple postings, 34069 formulae, 8849 documents
[ formulas ] [ documents ] [ documents-by-formula ]

Doc 1
1.0000
0.0000
113.0000
1.1774
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000013/Articles/Anisotropic_Network_Model.html
H i j = [ 2 V i j x i x j 2 V i j x i y j 2 V i j x i z j 2 V i j y i x j 2 V i j y i y j 2 V i j y i z j 2 V i j z i x j 2 V i j z i y j 2 V i j z i z j ]
2 V i j x i y j = - γ s i j 2 ( x j - x i ) ( y j - y i )
2 V i j x i 2 = 2 V i j x j 2 = γ s i j 2 ( x j - x i ) 2
Doc 2
0.5350
-100.0000
47.0000
0.5350
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Strategic_complements.html
[ d x 1 * d p 1 d x 2 * d p 1 d y 2 * d p 1 ] = [ 2 U x x 1 x 1 2 U x x 1 x 2 2 U x x 1 y 2 2 U x x 1 x 2 2 U x x 2 x 2 2 U x y 2 x 2 2 U y x 1 y 2 2 U y x 2 y 2 2 U y y 2 y 2 ] - 1 [ - 2 U x p 1 x 1 - 2 U x p 1 x 2 - 2 U y p 1 y 2 ]

Doc 3
0.5328
-13.0000
52.0000
0.6616
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Mean_curvature.html
Hess ( F ) = ( 2 F x 2 2 F x y 2 F x z 2 F x y 2 F y 2 2 F y z 2 F x z 2 F y z 2 F z 2 ) .
F = ( F x , F y , F z )

Doc 4
0.4260
-27.0000
42.0000
0.4260
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000008/Articles/Polar_curve.html
H ( f ) = [ 2 f x 2 2 f x y 2 f x z 2 f y x 2 f y 2 2 f y z 2 f z x 2 f z y 2 f z 2 ] ,

Doc 5
0.3689
-93.0000
32.0000
0.3689
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Curvilinear_coordinates.html
𝐉 = [ x 1 q 1 x 1 q 2 x 1 q 3 x 2 q 1 x 2 q 2 x 2 q 3 x 3 q 1 x 3 q 2 x 3 q 3 ] , 𝐉 - 1 = [ q 1 x 1 q 1 x 2 q 1 x 3 q 2 x 1 q 2 x 2 q 2 x 3 q 3 x 1 q 3 x 2 q 3 x 3 ]

Doc 6
0.3600
-80.0000
31.0000
1.1563
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Jacobian_matrix_and_determinant.html
𝐉 𝐅 ( x 1 , x 2 , x 3 ) = [ y 1 x 1 y 1 x 2 y 1 x 3 y 2 x 1 y 2 x 2 y 2 x 3 y 3 x 1 y 3 x 2 y 3 x 3 y 4 x 1 y 4 x 2 y 4 x 3 ] = [ 1 0 0 0 0 5 0 8 x 2 - 2 x 3 cos x 1 0 sin x 1 ] .
𝐉 𝐅 ( r , θ , φ ) = [ x r x θ x φ y r y θ y φ z r z θ z φ ] = [ sin θ cos φ r cos θ cos φ - r sin θ sin φ sin θ sin φ r cos θ sin φ r sin θ cos φ cos θ - r sin θ 0 ] .
𝐉 = d 𝐟 d 𝐱 = [ 𝐟 x 1 𝐟 x n ] = [ f 1 x 1 f 1 x n f m x 1 f m x n ]
𝐉 𝐟 ( x , y ) = [ f 1 x f 1 y f 2 x f 2 y ] = [ 2 x y x 2 5 cos y ]
𝐉 ( r , φ ) = [ x r x φ y r y φ ] = [ cos φ - r sin φ sin φ r cos φ ]

Doc 7
0.3422
-16.0000
30.0000
0.3422
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Fourier_series.html
[ x 1 x x 1 y x 1 z x 2 x x 2 y x 2 z x 3 x x 3 y x 3 z ]

Doc 8
0.3418
-97.0000
31.0000
0.5766
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Hessian_matrix.html
H ( f , g ) = [ 0 g x 1 g x 2 g x n g x 1 2 f x 1 2 2 f x 1 x 2 2 f x 1 x n g x 2 2 f x 2 x 1 2 f x 2 2 2 f x 2 x n g x n 2 f x n x 1 2 f x n x 2 2 f x n 2 ]
H = [ 2 f x 1 2 2 f x 1 x 2 2 f x 1 x n 2 f x 2 x 1 2 f x 2 2 2 f x 2 x n 2 f x n x 1 2 f x n x 2 2 f x n 2 ] .

Doc 9
0.3333
-53.0000
28.0000
0.3333
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Cartesian_tensor.html
( x ¯ 1 x ¯ 2 x ¯ 3 ) = ( x ¯ 1 x 1 x ¯ 1 x 2 x ¯ 1 x 3 x ¯ 2 x 1 x ¯ 2 x 2 x ¯ 2 x 3 x ¯ 3 x 1 x ¯ 3 x 2 x ¯ 3 x 3 ) ( x 1 x 2 x 3 )

Doc 10
0.2622
-43.0000
21.0000
0.2622
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000007/Articles/Pitchfork_bifurcation.html
f x ( 0 , r o ) = 0 , 2 f x 2 ( 0 , r o ) = 0 , 3 f x 3 ( 0 , r o ) 0 , f r ( 0 , r o ) = 0 , 2 f r x ( 0 , r o ) 0.

Doc 11
0.2575
-6.0000
25.0000
0.3992
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000014/Articles/Invariant_of_a_binary_form.html
H ( f ) = [ 2 f x 2 2 f x y 2 f y x 2 f y 2 ] .
det [ f x f y g x g y ]

Doc 12
0.2336
-71.0000
16.0000
0.2336
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Fisher_information.html
Σ θ m = [ Σ 1 , 1 θ m Σ 1 , 2 θ m Σ 1 , N θ m Σ 2 , 1 θ m Σ 2 , 2 θ m Σ 2 , N θ m Σ N , 1 θ m Σ N , 2 θ m Σ N , N θ m ] .

Doc 13
0.2308
-41.0000
18.0000
0.2308
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Kronecker_delta.html
δ = 1 4 π R s t | x y z x s y s z s x t y t z t | ( x 2 + y 2 + z 2 ) x 2 + y 2 + z 2 d s d t .

Doc 14
0.2066
-85.0000
17.0000
0.2066
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000015/Articles/Linear_seismic_inversion.html
𝐀 = [ F 1 ( q ) p 1 F 1 ( q ) p 2 F 1 ( q ) p n F 2 ( q ) p 1 F 2 ( q ) p n - 1 F 2 ( q ) p n F j ( q ) p i F m ( q ) p 1 F m ( q ) p 2 F m ( q ) p n ]

Doc 15
0.1999
-127.0000
20.0000
0.3821
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Infinitesimal_strain_theory.html
[ ε x x ε x y ε x z ε y x ε y y ε y z ε z x ε z y ε z z ] = [ u x x 1 2 ( u x y + u y x ) 1 2 ( u x z + u z x ) 1 2 ( u y x + u x y ) u y y 1 2 ( u y z + u z y ) 1 2 ( u z x + u x z ) 1 2 ( u z y + u y z ) u z z ]
ε i j = 1 2 ( u i , j + u j , i ) = [ ε 11 ε 12 ε 13 ε 21 ε 22 ε 23 ε 31 ε 32 ε 33 ] = [ u 1 x 1 1 2 ( u 1 x 2 + u 2 x 1 ) 1 2 ( u 1 x 3 + u 3 x 1 ) 1 2 ( u 2 x 1 + u 1 x 2 ) u 2 x 2 1 2 ( u 2 x 3 + u 3 x 2 ) 1 2 ( u 3 x 1 + u 1 x 3 ) 1 2 ( u 3 x 2 + u 2 x 3 ) u 3 x 3 ]

Doc 16
0.1902
-95.0000
18.0000
0.1902
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000010/Articles/Capillary_surface.html
σ i j = - ( p 0 0 0 p 0 0 0 p ) + μ ( 2 u x u y + v x u z + w x v x + u y 2 v y v z + w y w x + u z w y + v z 2 w z ) = - p I + μ ( 𝐯 + ( 𝐯 ) T )

Doc 17
0.1902
-132.0000
21.0000
0.1902
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000013/Articles/Capelli's_identity.html
X = | x 11 x 12 x 13 x 1 n x 12 x 22 x 23 x 2 n x 13 x 23 x 33 x 3 n x 1 n x 2 n x 3 n x n n | , D = | 2 x 11 x 12 x 13 x 1 n x 12 2 x 22 x 23 x 2 n x 13 x 23 2 x 33 x 3 n x 1 n x 2 n x 3 n 2 x n n |

Doc 18
0.1851
-14.0000
14.0000
0.1851
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000012/Articles/Elasticity_coefficient.html
ε = [ v 1 S 1 v 1 S m v n S 1 v n S m ] .

Doc 19
0.1813
-72.0000
18.0000
0.1813
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Observed_information.html
= - ( 2 θ 1 2 2 θ 1 θ 2 2 θ 1 θ n 2 θ 2 θ 1 2 θ 2 2 2 θ 2 θ n 2 θ n θ 1 2 θ n θ 2 2 θ n 2 ) ( θ ) | θ = θ *

Doc 20
0.1774
-53.0000
13.0000
0.1774
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Constraint_algorithm.html
𝐉 = ( σ 1 λ 1 σ 1 λ 2 σ 1 λ n σ 2 λ 1 σ 2 λ 2 σ 2 λ n σ n λ 1 σ n λ 2 σ n λ n ) .

Doc 21
0.1761
-24.0000
14.0000
2.2698
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Matrix_calculus.html
𝐅 ( 𝐗 ) 𝐗 = [ f 1 , 1 𝐗 f 1 , p 𝐗 f m , 1 𝐗 f m , p 𝐗 ]
𝐅 𝐗 = [ 𝐅 X 1 , 1 𝐅 X n , 1 𝐅 X 1 , m 𝐅 X n , m ] ,
y 𝐗 = [ y x 11 y x 12 y x 1 q y x 21 y x 22 y x 2 q y x p 1 y x p 2 y x p q ] .
y 𝐗 = [ y x 11 y x 21 y x p 1 y x 12 y x 22 y x p 2 y x 1 q y x 2 q y x p q ] .
y 𝐗 = [ y x 11 y x 21 y x p 1 y x 12 y x 22 y x p 2 y x 1 q y x 2 q y x p q ] .
𝐘 x = [ y 11 x y 12 x y 1 n x y 21 x y 22 x y 2 n x y m 1 x y m 2 x y m n x ] .
𝐘 x = [ y 11 x y 12 x y 1 n x y 21 x y 22 x y 2 n x y m 1 x y m 2 x y m n x ] .
𝐲 𝐱 = [ y 1 x 1 y 1 x 2 y 1 x n y 2 x 1 y 2 x 2 y 2 x n y m x 1 y m x 2 y m x n ] .
𝐲 𝐱 = [ y 1 x 1 y 1 x 2 y 1 x n y 2 x 1 y 2 x 2 y 2 x n y m x 1 y m x 2 y m x n ] .
𝐲 𝐱 = [ y 1 x 1 y 2 x 1 y m x 1 y 1 x 2 y 2 x 2 y m x 2 y 1 x n y 2 x n y m x n ] .
ϕ ( 𝐗 ) 𝐗 = [ ϕ x 1 , 1 ϕ x 1 , q ϕ x n , 1 ϕ x n , q ]
f = f 𝐱 = [ f x 1 f x 2 f x 3 ] .
y 𝐱 = [ y x 1 y x 2 y x n ] .
𝐲 x = [ y 1 x y 2 x y m x ] .
𝐲 x = [ y 1 x y 2 x y m x ] .
Doc 22
0.1685
-9.0000
13.0000
0.1685
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Power-flow_study.html
J = [ Δ P θ Δ P | V | Δ Q θ Δ Q | V | ]

Doc 23
0.1685
-55.0000
15.0000
0.6025
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Metric_tensor.html
D φ = [ φ 1 x 1 φ 1 x 2 φ 1 x n φ 2 x 1 φ 2 x 2 φ 2 x n φ m x 1 φ m x 2 φ m x n ] .
J = [ u u u v v u v v ] .
[ d u d v ] = [ u u u v v u v v ] [ d u d v ]
[ E F F G ] = [ u u u v v u v v ] T [ E F F G ] [ u u u v v u v v ]

Doc 24
0.1595
-19.0000
14.0000
0.1595
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000016/Articles/Intersection_(Euclidean_geometry).html
( f 1 x f 1 y f 2 x f 2 y ) ( δ x δ y ) = ( - f 1 - f 2 )

Doc 25
0.1595
-34.0000
13.0000
0.1595
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Implicit_function_theorem.html
J = [ x ( R , θ ) R x ( R , θ ) θ y ( R , θ ) R y ( R , θ ) θ ] = [ cos θ - R sin θ sin θ R cos θ ] .

Doc 26
0.1506
-10.0000
15.0000
0.1506
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Cauchy–Riemann_equations.html
D f ( x , y ) = [ u x u y v x v y ]

Doc 27
0.1466
-19.0000
13.0000
0.1466
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Vector_field.html
V = f = ( f x 1 , f x 2 , f x 3 , , f x n ) .

Doc 28
0.1417
-14.0000
14.0000
0.1417
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Probability_density_function.html
| U Y U Z V Y V Z | = | Z Y 0 1 | = | Z | .

Doc 29
0.1417
-35.0000
14.0000
0.1417
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Cramer's_rule.html
x u = | - F u F y - G u G y | | F x F y G x G y | .

Doc 30
0.1377
-30.0000
12.0000
0.1377
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000013/Articles/Experimental_uncertainty_analysis.html
Δ z ( z x 1 z x 2 z x 3 z x p ) ( Δ x 1 Δ x 2 Δ x 3 Δ x p ) 𝐄𝐪 ( 𝟖 )

Doc 31
0.1373
-13.0000
15.0000
0.1373
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000013/Articles/Cayley's_Ω_process.html
Ω = | x 11 x 1 n x n 1 x n n | .

Doc 32
0.1343
-60.0000
10.0000
0.1343
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Divergence.html
div ( ϵ ¯ ¯ ) = [ ϵ x x x + ϵ x y y + ϵ x z z ϵ y x x + ϵ y y y + ϵ y z z ϵ z x x + ϵ z y y + ϵ z z z ]

Doc 33
0.1288
-27.0000
11.0000
0.1288
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Hydraulic_head.html
h = ( h x , h y , h z ) = h x 𝐢 + h y 𝐣 + h z 𝐤

Doc 34
0.1288
-63.0000
10.0000
0.1288
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Vorticity.html
ω = × v = ( x , y , z ) × ( v x , v y , v z ) = ( v z y - v y z , v x z - v z x , v y x - v x y )

Doc 35
0.1276
-21.0000
12.0000
0.1276
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000012/Articles/Mild-slope_equation.html
( Φ x Φ y Φ z ) ( f φ x f φ y f z φ ) .

Doc 36
0.1276
-69.0000
10.0000
0.1276
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000016/Articles/Variance_function.html
D n × p = [ μ 1 β 1 μ 1 β p μ 2 β 1 μ 2 β p μ m β 1 μ m β p ] V n × n = diag ( V ( μ 1 ) , V ( μ 2 ) , , , V ( μ n ) )

Doc 37
0.1020
-6.0000
11.0000
0.1020
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Notation_for_differentiation.html
= ( x , y , z ) ,

Doc 38
0.1020
-29.0000
11.0000
0.1020
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Del.html
= ( x , y , z ) = e x x + e y y + e z z

Doc 39
0.1020
-63.0000
9.0000
0.1020
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Special_relativity.html
( 1 c ϕ t ϕ x ϕ y ϕ z ) = ( 1 c ϕ t ϕ x ϕ y ϕ z ) ( γ - β γ 0 0 - β γ γ 0 0 0 0 1 0 0 0 0 1 ) .

Doc 40
0.1020
-65.0000
11.0000
0.1020
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Trimaximal_mixing.html
U = [ 1 3 1 3 1 3 ω 3 1 3 ω ¯ 3 ω ¯ 3 1 3 ω 3 ] ( | U i α | 2 ) = [ 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 ]

Doc 41
0.0931
-50.0000
8.0000
0.0931
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000008/Articles/List_of_formulas_in_Riemannian_geometry.html
R i k m = 1 2 ( 2 g i m x k x + 2 g k x i x m - 2 g i x k x m - 2 g k m x i x )

Doc 42
0.0931
-51.0000
5.0000
0.0931
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Chain_rule.html
2 y x i x j = k ( y u k 2 u k x i x j ) + k , ( 2 y u k u u k x i u x j ) .

Doc 43
0.0931
-130.0000
10.0000
0.0931
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Acoustic_theory.html
\cdotsymbol τ s i j x i = μ [ x i ( u i x j + u j x i ) ] + λ [ x i ( u k x k ) ] δ i j = μ 2 u i x i x j + μ 2 u j x i x i + λ 2 u k x k x j = ( μ + λ ) 2 u i x i x j + μ 2 u j x i 2 ( μ + λ ) ( 𝐮 ) + μ 2 𝐮 .

Doc 44
0.0842
-12.0000
9.0000
0.0842
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Partial_derivative.html
2 f x i x j = 2 f x j x i .

Doc 45
0.0842
-13.0000
9.0000
0.0842
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000007/Articles/John's_equation.html
2 u x i y j - 2 u y i x j = 0

Doc 46
0.0842
-16.0000
9.0000
0.0842
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Lorentz_covariance.html
a = [ 1 c t , x , y , z ]

Doc 47
0.0842
-17.0000
9.0000
0.0842
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Potential_flow.html
2 Φ x i x i - M i M j 2 Φ x i x j = 0.

Doc 48
0.0842
-28.0000
9.0000
0.0842
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Symmetry_of_second_derivatives.html
2 f x i x j ( a 1 , , a n ) = 2 f x j x i ( a 1 , , a n ) .

Doc 49
0.0842
-30.0000
9.0000
0.0842
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000016/Articles/Discretization_of_Navier–Stokes_equations.html
u i t + u i u j x j = - P x i + ν 2 u i x j x j + f i

Doc 50
0.0842
-31.0000
9.0000
0.4212
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Large_eddy_simulation.html
u i t + u i u j x j = - 1 ρ p x i + ν 2 u i x j x j .
u i ¯ t + u i u j x j ¯ = - 1 ρ p ¯ x i + ν 2 u i ¯ x j x j .
u i t ¯ + u i u j x j ¯ = - 1 ρ p x i ¯ + ν 2 u i x j x j ¯ .
u i ¯ t + u j ¯ u i ¯ x j = - 1 ρ p ¯ x i + ν 2 u i ¯ x j x j - τ i j x j .
u i ¯ t + u i ¯ u j ¯ x j = - 1 ρ p ¯ x i + ν 2 u i ¯ x j x j - ( u i u j x j ¯ - u i ¯ u j ¯ x j ) .

Doc 51
0.0842
-32.0000
9.0000
0.1685
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Reynolds-averaged_Navier–Stokes_equations.html
u i t + u j u i x j = f i - 1 ρ p x i + ν 2 u i x j x j
u i ¯ t + u j ¯ u i ¯ x j + u j u i x j ¯ = f i ¯ - 1 ρ p ¯ x i + ν 2 u i ¯ x j x j .

Doc 52
0.0842
-32.0000
9.0000
0.0842
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000016/Articles/Computational_Fluid_Dynamics_for_Phase_Change_Materials.html
ρ ( u i t + u j u i x j ) = - p x i + μ 2 u i x j x j + f i

Doc 53
0.0842
-38.0000
7.0000
0.4628
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000011/Articles/Stress_functions.html
σ y z = - 2 Φ y z x x - 2 Φ x x y z + 2 Φ z x y x + 2 Φ x y z x
σ z x = - 2 Φ z x y y - 2 Φ y y z x + 2 Φ x y z y + 2 Φ y z x y
σ x y = - 2 Φ x y z z - 2 Φ z z x y + 2 Φ y z x z + 2 Φ z x y z
σ x = 2 Φ y y z z + 2 Φ z z y y - 2 2 Φ y z y z
σ y = 2 Φ x x z z + 2 Φ z z x x - 2 2 Φ z x z x
σ z = 2 Φ y y x x + 2 Φ x x y y - 2 2 Φ x y x y
Doc 54
0.0842
-47.0000
9.0000
0.0842
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Saint-Venant's_compatibility_condition.html
W i j k l = 2 F i j x k x l + 2 F k l x i x j - 2 F i l x j x k - 2 F j k x i x l

Doc 55
0.0842
-49.0000
9.0000
0.0842
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Reynolds_stress.html
ρ [ u i ¯ t + u i ¯ u j ¯ x j + u i ¯ u j ¯ x j ] = - p ¯ x i + μ 2 u i ¯ x j x j .

Doc 56
0.0842
-63.0000
8.0000
0.0842
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000006/Articles/Finite_strain_theory.html
G i j x k = ( 2 X α x i x k X β x j + X α x i 2 X β x j x k ) g α β + X α x i X β x j g α β x k

Doc 57
0.0842
-77.0000
9.0000
0.0842
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Pfaffian.html
1 pf ( A ) 2 pf ( A ) x i x j = 1 2 tr ( A - 1 2 A x i x j ) - 1 2 tr ( A - 1 A x i A - 1 A x j ) + 1 4 tr ( A - 1 A x i ) tr ( A - 1 A x j ) .

Doc 58
0.0842
-95.0000
9.0000
0.0842
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000013/Articles/Method_of_steepest_descent.html
2 S ( s y m b o l ϕ ( y ) ) y i y j = l , k = 1 n 2 S ( z ) z k z l | z = s y m b o l ϕ ( y ) ϕ k y i ϕ l y j + k = 1 n S ( z ) z k | z = s y m b o l ϕ ( y ) 2 ϕ k y i y j

Doc 59
0.0791
-64.0000
10.0000
0.0791
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Tribimaximal_mixing.html
[ | U e 1 | 2 | U e 2 | 2 | U e 3 | 2 | U μ 1 | 2 | U μ 2 | 2 | U μ 3 | 2 | U τ 1 | 2 | U τ 2 | 2 | U τ 3 | 2 ] = [ 2 3 1 3 0 1 6 1 3 1 2 1 6 1 3 1 2 ] .

Doc 60
0.0753
-13.0000
8.0000
0.0753
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000007/Articles/Eikonal_equation.html
2 x k x j H = 2 x j x k H .

Doc 61
0.0753
-30.0000
8.0000
0.2745
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Maxwell_relations.html
( 2 S y x ) = ( 2 S x y ) : ( 2 V y x ) = ( 2 V x y )
y ( z x ) y = x ( z y ) x = 2 z y x = 2 z x y
( 2 U x y ) = ( T x ) y ( S y ) x + T ( 2 S x y ) - ( P x ) y ( V y ) x - P ( 2 V x y )
( 2 U y x ) = ( T y ) x ( S x ) y + T ( 2 S y x ) - ( P y ) x ( V x ) y - P ( 2 V y x )

Doc 62
0.0753
-46.0000
7.0000
0.0753
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Partial_differential_equation.html
f ( x 1 , , x n , u , u x 1 , , u x n , 2 u x 1 x 1 , , 2 u x 1 x n , ) = 0.

Doc 63
0.0664
-20.0000
7.0000
0.0664
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000007/Articles/Second_derivative.html
2 f x y , 2 f x z , and 2 f y z .

Doc 64
0.0664
-42.0000
7.0000
0.0664
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Elliptic_partial_differential_equation.html
a 2 u x 2 + b 2 u x y + c 2 u y 2 + d 2 u y z + e 2 u z 2 + (lower-order terms) = 0 ,