Returned 97 matches (100 formulae, 52 docs)
    Lookup 193.083 ms, Re-ranking 36111.555 ms
    Found 552996 tuple postings, 174581 formulae, 20949 documents
[ formulas ] [ documents ] [ documents-by-formula ]

0.8590
-13.0000
58.0000
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 )

0.4965
-13.0000
32.0000
u 1 ( 𝐱 , z 1 ) = - V x 𝐱 g x ( 𝐱 ) - k 1 ( z 1 - u x ( 𝐱 ) ) + u x 𝐱 ( f x ( 𝐱 ) + g x ( 𝐱 ) z 1 )

0.4965
-14.0000
32.0000
u a 1 ( 𝐱 , z 1 ) = - V x 𝐱 g x ( 𝐱 ) - k 1 ( z 1 - u x ( 𝐱 ) ) + u x 𝐱 ( f x ( 𝐱 ) + g x ( 𝐱 ) z 1 )

0.4825
-40.0000
31.0000
u 1 ( 𝐱 , z 1 ) = 1 g 1 ( 𝐱 , z 1 ) ( - V x 𝐱 g x ( 𝐱 ) - k 1 ( z 1 - u x ( 𝐱 ) ) + u x 𝐱 ( f x ( 𝐱 ) + g x ( 𝐱 ) z 1 ) u a 1 ( 𝐱 , z 1 ) - f 1 ( 𝐱 , z 1 ) )

0.4545
-57.0000
26.0000
u i ( 𝐱 , z 1 , z 2 , , z i 𝐱 i ) = - V i - 1 𝐱 i - 1 g i - 1 ( 𝐱 i - 1 ) - k i ( z i - u i - 1 ( 𝐱 i - 1 ) ) + u i - 1 𝐱 i - 1 ( f i - 1 ( 𝐱 i - 1 ) + g i - 1 ( 𝐱 i - 1 ) z i )

0.4405
-26.0000
26.0000
u 2 ( 𝐱 , z 1 , z 2 ) = - V 1 𝐱 1 g 1 ( 𝐱 1 ) - k 2 ( z 2 - u 1 ( 𝐱 1 ) ) + u 1 𝐱 1 ( f 1 ( 𝐱 1 ) + g 1 ( 𝐱 1 ) z 2 )

0.4405
-29.0000
26.0000
u 3 ( 𝐱 , z 1 , z 2 , z 3 ) = - V 2 𝐱 2 g 2 ( 𝐱 2 ) - k 3 ( z 3 - u 2 ( 𝐱 2 ) ) + u 2 𝐱 2 ( f 2 ( 𝐱 2 ) + g 2 ( 𝐱 2 ) z 3 )

0.4405
-81.0000
25.0000
u i ( 𝐱 , z 1 , z 2 , , z i 𝐱 i ) = 1 g i ( 𝐱 i ) ( - V i - 1 𝐱 i - 1 g i - 1 ( 𝐱 i - 1 ) - k i ( z i - u i - 1 ( 𝐱 i - 1 ) ) + u i - 1 𝐱 i - 1 ( f i - 1 ( 𝐱 i - 1 ) + g i - 1 ( 𝐱 i - 1 ) z i ) Single-integrator stabilizing control  u a i ( 𝐱 i ) - f i ( 𝐱 i - 1 ) )

0.2306
-91.0000
13.0000
V ˙ 1 = V ˙ x ( 𝐱 ) + 1 2 ( 2 e 1 e ˙ 1 ) = V ˙ x ( 𝐱 ) + e 1 e ˙ 1 = V ˙ x ( 𝐱 ) + e 1 v 1 e ˙ 1 = V x 𝐱 𝐱 ˙ (i.e.,  d 𝐱 d t ) V ˙ x  (i.e., d V x d t ) + e 1 v 1 = V x 𝐱 ( ( f x ( 𝐱 ) + g x ( 𝐱 ) u x ( 𝐱 ) ) + g x ( 𝐱 ) e 1 ) 𝐱 ˙ V ˙ x + e 1 v 1

0.2026
-4.0000
13.0000
v 1 = - V x 𝐱 g x ( 𝐱 ) - k 1 e 1

0.1746
-16.0000
9.0000
V ˙ x = V x 𝐱 ( f x ( 𝐱 ) + g x ( 𝐱 ) u x ( 𝐱 ) ) - W ( 𝐱 )

0.1746
-30.0000
12.0000
V ˙ 1 = - W ( 𝐱 ) + V x 𝐱 g x ( 𝐱 ) e 1 + e 1 ( - V x 𝐱 g x ( 𝐱 ) - k 1 e 1 ) v 1

0.1746
-54.0000
9.0000
V ˙ 1 = V x 𝐱 ( f x ( 𝐱 ) + g x ( 𝐱 ) u x ( 𝐱 ) ) - W ( 𝐱 ) + V x 𝐱 g x ( 𝐱 ) e 1 + e 1 v 1 - W ( 𝐱 ) + V x 𝐱 g x ( 𝐱 ) e 1 + e 1 v 1

0.1668
-20.0000
4.0000
V 3 ( 𝐱 , z 1 , z 2 , z 3 ) = V 2 ( 𝐱 2 ) + 1 2 ( z 3 - u 2 ( 𝐱 2 ) ) 2

0.1668
-43.0000
11.0000
V ˙ 1 = - W ( 𝐱 ) + V x 𝐱 g x ( 𝐱 ) e 1 - e 1 V x 𝐱 g x ( 𝐱 ) 0 - k 1 e 1 2 = - W ( 𝐱 ) - k 1 e 1 2 - W ( 𝐱 ) < 0

0.1582
-12.0000
9.0000
V 1 ( 𝐱 , z 1 ) V x ( 𝐱 ) + 1 2 ( z 1 - u x ( 𝐱 ) ) 2
V 1 ( 𝐱 , z 1 ) = V x ( 𝐱 ) + 1 2 ( z 1 - u x ( 𝐱 ) ) 2

0.1582
-20.0000
7.0000
V i ( 𝐱 i ) = V i - 1 ( 𝐱 i - 1 ) + 1 2 ( z i - u i - 1 ( 𝐱 i - 1 ) ) 2

0.1582
-32.0000
8.0000
u s t + 1 = ( 1 - ε ) f ( u s t ) + ε 2 ( f ( u s + 1 t ) + f ( u s - 1 t ) )     t ,   ε [ 0 , 1 ]

0.1527
-13.0000
6.0000
t w ( t , ξ ) = - ξ w ( t , ξ ) + u ( t ) ,

0.1527
-18.0000
5.0000
V 2 ( 𝐱 , z 1 , z 2 ) = V 1 ( 𝐱 1 ) + 1 2 ( z 2 - u 1 ( 𝐱 1 ) ) 2

0.1527
-22.0000
4.0000
V 2 ( 𝐱 , z 1 , z 2 ) = V 1 ( 𝐱 , z 1 ) + 1 2 ( z 2 - u 1 ( 𝐱 , z 1 ) ) 2

0.1466
-7.0000
8.0000
f y ( 𝐲 ) [ f x ( 𝐱 ) + g x ( 𝐱 ) z 1 0 ]

0.1466
-24.0000
5.0000
σ ˙ = σ 𝐱 𝐱 ˙ d 𝐱 d t = σ 𝐱 ( f ( 𝐱 , t ) + B ( 𝐱 , t ) 𝐮 ) 𝐱 ˙

0.1466
-51.0000
8.0000
{ [ 𝐱 ˙ z ˙ 1 ] 𝐱 ˙ 1 = [ f x ( 𝐱 ) + g x ( 𝐱 ) z 1 0 ] f 1 ( 𝐱 1 ) + [ 𝟎 1 ] g 1 ( 𝐱 1 ) z 2  ( by Lyapunov function  V 1 ,  subsystem stabilized by  u 1 ( 𝐱 1 )  ) z ˙ 2 = u 2

0.1466
-57.0000
7.0000
{ [ 𝐱 ˙ z ˙ 1 z ˙ 2 ] 𝐱 ˙ 2 = [ f x ( 𝐱 ) + g x ( 𝐱 ) z 2 z 2 0 ] f 2 ( 𝐱 2 ) + [ 𝟎 0 1 ] g 2 ( 𝐱 2 ) z 3  ( by Lyapunov function  V 2 ,  subsystem stabilized by  u 2 ( 𝐱 2 )  ) z ˙ 3 = u 3

0.1386
-54.0000
6.0000
x n i ( p , m i ) = - v i ( p , m i ) p n v i ( p , m i ) m i = f i ( p ) p n + g ( p ) p n m - f i ( p ) g ( p )

0.1325
-14.0000
8.0000
{ 𝐱 ˙ = f x ( 𝐱 ) + g x ( 𝐱 ) z 1 z ˙ 1 = u 1

0.1325
-15.0000
8.0000
{ 𝐱 ˙ = f x ( 𝐱 ) + g x ( 𝐱 ) z 1 z ˙ 1 = u a 1

0.1325
-21.0000
8.0000
{ 𝐱 ˙ = f x ( 𝐱 ) + g x ( 𝐱 ) z 1 z ˙ 1 = z 2 z ˙ 2 = u 2

0.1325
-24.0000
7.0000
{ 𝐱 ˙ = ( f x ( 𝐱 ) + g x ( 𝐱 ) u x ( 𝐱 ) ) + g x ( 𝐱 ) e 1 e ˙ 1 = v 1

0.1325
-28.0000
8.0000
{ 𝐱 ˙ = f x ( 𝐱 ) + g x ( 𝐱 ) z 1 z ˙ 1 = z 2 z ˙ 2 = z 3 z ˙ 3 = u 3

0.1325
-28.0000
7.0000
{ 𝐱 ˙ = ( f x ( 𝐱 ) + g x ( 𝐱 ) u x ( 𝐱 ) ) + g x ( 𝐱 ) e 1 e ˙ 1 = u 1 - u ˙ x

0.1325
-28.0000
6.0000
{ 𝐲 ˙ = f y ( 𝐲 ) + g y ( 𝐲 ) z 2 ( where this  𝐲  subsystem is stabilized by  z 2 = u 1 ( 𝐱 , z 1 )  ) z ˙ 2 = u 2 .

0.1325
-29.0000
8.0000
{ 𝐱 ˙ = f x ( 𝐱 ) + g x ( 𝐱 ) z 1 z ˙ 1 = f 1 ( 𝐱 , z 1 ) + g 1 ( 𝐱 , z 1 ) u 1

0.1325
-34.0000
8.0000
{ 𝐱 ˙ = f x ( 𝐱 ) + g x ( 𝐱 ) z 1 + ( g x ( 𝐱 ) u x ( 𝐱 ) - g x ( 𝐱 ) u x ( 𝐱 ) ) 0 z ˙ 1 = u 1

0.1325
-38.0000
7.0000
{ x ˙ = ( f x ( 𝐱 ) + g x ( 𝐱 ) u x ( 𝐱 ) ) F ( 𝐱 ) + g x ( 𝐱 ) ( z 1 - u x ( 𝐱 ) ) z 1  error tracking  u x z ˙ 1 = u 1

0.1325
-52.0000
6.0000
{ [ 𝐱 ˙ 1 z ˙ 2 ] 𝐱 ˙ 2 = [ f 1 ( 𝐱 1 ) + g 1 ( 𝐱 1 ) z 2 0 ] f 2 ( 𝐱 2 ) + [ 𝟎 1 ] g 2 ( 𝐱 2 ) z 3  ( by Lyapunov function  V 2 ,  subsystem stabilized by  u 2 ( 𝐱 2 )  ) z ˙ 3 = u 3

0.1325
-54.0000
8.0000
{ 𝐱 ˙ = f x ( 𝐱 ) + g x ( 𝐱 ) z 1 z ˙ 1 = f 1 ( 𝐱 , z 1 ) + g 1 ( 𝐱 , z 1 ) 1 g 1 ( 𝐱 , z 1 ) ( u a 1 - f 1 ( 𝐱 , z 1 ) ) u 1 ( 𝐱 , z 1 )

0.1297
-27.0000
3.0000
= 𝔼 θ [ V ( x ( θ ) , θ ) - u ¯ ( θ 0 ) - 1 - P ( θ ) p ( θ ) V θ - c ( x ( θ ) ) ]

0.1297
-34.0000
4.0000
2 𝐱 i ( t ) t 2 m i = - 𝐱 i [ V ( 𝐱 i ( t ) ) + k = 1 n λ k σ k ( t ) ] ,  i = 1 N .

0.1297
-38.0000
3.0000
H = ( V ( x , θ ) - u ¯ ( θ 0 ) - 1 - P ( θ ) p ( θ ) V θ ( x , θ ) - c ( x ) ) p ( θ ) + ν ( θ ) x θ

0.1244
-24.0000
3.0000
ψ ( 𝐫 1 ,  𝐫 2 ) = 1 2 ( u A ( 𝐫 1 ) u B ( 𝐫 2 ) + u B ( 𝐫 1 ) u A ( 𝐫 2 ) )

0.1244
-26.0000
6.0000
𝔼 θ [ V ( x ( θ ) , θ ) - u ¯ ( θ 0 ) - θ 0 θ V θ ~ d θ ~ - c ( x ( θ ) ) ]

0.1185
-7.0000
6.0000
𝐱 ˙ = f x ( 𝐱 ) + g x ( 𝐱 ) u x ( 𝐱 )

0.1185
-15.0000
6.0000
𝐱 ˙ = F ( 𝐱 )    where    F ( 𝐱 ) f x ( 𝐱 ) + g x ( 𝐱 ) u x ( 𝐱 )

0.1185
-28.0000
6.0000
{ 𝐱 ˙ 1 = f 1 ( 𝐱 1 ) + g 1 ( 𝐱 1 ) z 2  ( by Lyapunov function  V 1 ,  subsystem stabilized by  u 1 ( 𝐱 1 )  ) z ˙ 2 = u 2
{ 𝐱 ˙ 2 = f 2 ( 𝐱 2 ) + g 2 ( 𝐱 2 ) z 3  ( by Lyapunov function  V 2 ,  subsystem stabilized by  u 2 ( 𝐱 2 )  ) z ˙ 3 = u 3

0.1153
-21.0000
5.0000
d f ( 𝒂 ) ( 𝒗 ) = f x 1 ( 𝒂 ) v 1 + + f x n ( 𝒂 ) v n .

0.1153
-25.0000
4.0000
d d s u ( x ( s ) , t ( s ) ) = u x d x d s + u t d t d s

0.1102
-20.0000
3.0000
u 1 ( 𝐱 , z 1 ) = 1 g 1 ( 𝐱 , z 1 ) ( u a 1 - f 1 ( 𝐱 , z 1 ) )

0.1102
-44.0000
6.0000
ln B ( α , β ) β = - ln Γ ( α + β ) β + ln Γ ( α ) β + ln Γ ( β ) β = - ψ ( α + β ) + 0 + ψ ( β )

0.1102
-44.0000
6.0000
ln B ( α , β ) α = - ln Γ ( α + β ) α + ln Γ ( α ) α + ln Γ ( β ) α = - ψ ( α + β ) + ψ ( α ) + 0

0.1051
-24.0000
4.0000
m 2 x ( t ) t 2 = - 6 π R η x ( t ) t - F l x ( t ) + f ( t )

0.1045
-23.0000
4.0000
g t ( z ) t = g t ( z ) ζ ( t ) + g t ( z ) ζ ( t ) - g t ( z ) .

0.1008
-23.0000
4.0000
Ψ = z u b ( ξ ) - z 3 3 ! u b ′′ ( ξ ) + z 5 5 ! u b iv ( ξ ) + ,

0.1008
-32.0000
6.0000
t ( 2 ψ ) + ψ y x ( 2 ψ ) - ψ x y ( 2 ψ ) = ν 4 ψ

0.1008
-36.0000
4.0000
1 2 σ ( r ) 2 2 P r 2 + [ a ( r ) + σ ( r ) + φ ( r , t ) ] P r + P t - r P = 0

0.0960
-11.0000
5.0000
i f x ( z 0 ) = f y ( z 0 ) ,

0.0960
-12.0000
6.0000
u t = u ( 1 - u ) + 2 u x 2 .

0.0960
-54.0000
5.0000
a b ¯ = ( d x + u x x d x ) 2 + ( u y x d x ) 2 = 1 + 2 u x x + ( u x x ) 2 + ( u y x ) 2 d x

0.0960
-55.0000
5.0000
length ( a b ) = ( d x + u x x d x ) 2 + ( u y x d x ) 2 = d x 1 + 2 u x x + ( u x x ) 2 + ( u y x ) 2

0.0904
-20.0000
3.0000
π ( x , t ) x = ( x p ( x ) - C ( x ) ) x - t = 0

0.0904
-21.0000
5.0000
σ T V σ σ ˙ d σ d t d V d t < 0    (i.e.,  d V d t < 0 )

0.0904
-21.0000
3.0000
2 ( x p ( x ) - C ( x ) ) 2 x = 2 π ( x , t ) x 2 ,

0.0904
-34.0000
4.0000
t Q i ( t ) = - 1 Δ x ( f ( q ( t , x i + 1 / 2 ) ) - f ( q ( t , x i - 1 / 2 ) ) ) ,

0.0901
-51.0000
4.0000
M 2 f = - n P ( f ) + n - 2 x n P ( f ) x n - ( n Q ( f ) - n - 2 x n Q ( f ) x n + n - 2 x n 2 Q ( f ) ) e n

0.0862
-12.0000
4.0000
𝓠 = d d t ( T 𝐪 ˙ ) - T 𝐪 ,

0.0862
-18.0000
2.0000
J ( x , t ) = i 2 m ( ψ ψ * x - ψ x ψ )

0.0862
-19.0000
5.0000
f t ( z ) t = - z f t ( z ) ζ ( t ) + z ζ ( t ) - z

0.0862
-26.0000
6.0000
w - t ( w ˙ ) + 2 x 2 ( w x x ) = 0

0.0862
-27.0000
3.0000
f ( x ) 2 u x 2 + g ( x ) u x + h ( x ) u = u t + k ( t ) u

0.0862
-34.0000
3.0000
( A ) S = - ( S ) A = P C P T ( V P ) T + P ( V T ) P 2 + S ( V T ) P

0.0862
-35.0000
4.0000
( E c ( z ) - z 2 2 m c ( z ) z + 2 𝐤 2 2 m c ( z ) ) f k ( z ) = E f k ( z )

0.0817
-18.0000
3.0000
( U y ) x = T ( S y ) x - P ( V y ) x

0.0817
-22.0000
3.0000
( G ) V = - ( V ) G = - V ( V T ) P - S ( V P ) T

0.0817
-38.0000
3.0000
y ˙ = d h ( x ) d t = d h ( x ) d x x ˙ = d h ( x ) d x f ( x ) + d h ( x ) d x g ( x ) u

0.0817
-44.0000
4.0000
{ 𝐱 ˙ = f x ( 𝟎 𝐱 ) + ( g x ( 𝟎 𝐱 ) ) ( 0 z 1 ) = 0 + ( g x ( 𝟎 ) ) ( 0 ) = 𝟎  (i.e.,  𝐱 = 𝟎  is stationary) z ˙ 1 = 0 u 1  (i.e.,  z 1 = 0  is stationary)

0.0763
-20.0000
3.0000
f ( a ) = ( f x 1 ( a ) , , f x n ( a ) ) .

0.0763
-22.0000
4.0000
t ( 2 ψ ) + ( ψ , 2 ψ ) ( y , x ) = ν 4 ψ .

0.0763
-22.0000
3.0000
( U ) T = - ( T ) U = T ( V T ) P + P ( V P ) T

0.0763
-24.0000
3.0000
1 x x ( x x ) S ( x ) + ( 1 - ν 2 x 2 ) S ( x ) = 0

0.0763
-27.0000
3.0000
u ( u s ) + v ( v y ) = ν ( 2 u y 2 ) + g β ( T - T o )

0.0763
-30.0000
3.0000
g x j ( x ) = - ( f y ( x , g ( x ) ) ) - 1 f x j ( x , g ( x ) )

0.0763
-34.0000
3.0000
2 F ( z , w ) z w = f ( z ) f ( w ) ( f ( z ) - f ( w ) ) 2 - 1 ( z - w ) 2 ,

0.0763
-38.0000
3.0000
f ( 𝒖 ( 𝒙 ) ) x i = k = 1 p f ¯ ( 𝒖 ( 𝒙 ) ) u k u k ( 𝒙 ) x i    i = 1 , , n

0.0748
-27.0000
4.0000
g ( v ) = g ( x ) + k = 1 y k k ! ( x ) k - 1 ( f ( x ) k g ( x ) )

0.0713
-24.0000
4.0000
( S ) V = - ( V ) S = C P T ( V P ) T + ( V T ) P 2

0.0672
-20.0000
3.0000
d V d h = π r 2 3 V h + 2 π r h 3 V r d r d h
d V d r = 2 π r h 3 V r + π r 2 3 V h d h d r

0.0672
-22.0000
4.0000
d f d t = f x d x d t + f y d y d t .

0.0672
-30.0000
4.0000
f t + d q ¯ d t f q ¯ + d p ¯ d t f p ¯ = 0 ,

0.0672
-61.0000
4.0000
L = - ρ { - h ( x , y ) ζ ( x , y , t ) [ Φ t + 1 2 ( ( Φ x ) 2 + ( Φ y ) 2 + ( Φ z ) 2 ) ] d z + 1 2 g ( ζ 2 - h 2 ) } ,

0.0622
-19.0000
3.0000
( U θ ) - d d z ( U ( d θ d z ) ) = 0

0.0622
-19.0000
3.0000
( x y ) z = - ( z y ) x ( z x ) y

0.0622
-20.0000
3.0000
( y x ) z = - ( z x ) y ( z y ) x .

0.0622
-23.0000
3.0000
atan2 ( y , x ) x = arctan ( y / x ) x = - y x 2 + y 2

0.0622
-24.0000
4.0000
( P T ) V = - ( V T ) P ( V P ) T = α β T

0.0622
-29.0000
3.0000
u r = + 1 r 2 sin ( θ ) Ψ θ , u θ = - 1 r sin ( θ ) Ψ r .

0.0622
-31.0000
3.0000
M ( z ) = z 1 - z 1 - z 1 ¯ z ,   φ ( z ) = f ( z 1 ) - z 1 - f ( z 1 ) ¯ z .