DOSEN PROFIL LENGKAP

X&Y is the third studio album by the British rock band Coldplay. It was released on 6 June 2005 by Parlophone in the United Kingdom, and a day later by...

X+Y, released in the US as A Brilliant Young Mind, is a 2014 British drama film directed by Morgan Matthews and starring Asa Butterfield, Rafe Spall,...

Pokémon X and Pokémon Y are 2013 role-playing video games developed by Game Freak and published by The Pokémon Company and Nintendo for the Nintendo 3DS...

However, the equation x y = y x {\displaystyle x^{y}=y^{x}} has an infinity of solutions, consisting of the line ⁠ x = y {\displaystyle x=y} ⁠ and a smooth...

X/Y is a 2014 American drama film written and directed by Ryan Piers Williams and starring America Ferrera, Ryan Piers Williams, Melonie Diaz, Jon Paul...

showed that the original X-Gerät was known as Wotan I, and the Y-Gerät as Wotan II. Had he known the name was also associated with X-Gerät it was unlikely...

from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the...

x , + y , + z ) ( − x , + y , + z ) ( + x , − y , + z ) ( + x , + y , − z ) ( + x , − y , − z ) ( − x , + y , − z ) ( − x , − y , + z ) ( − x , − y ...

the question is about an end user's attempted solution (X) rather than the root problem itself (Y). The XY problem obscures the real issues and may even...

⁠x/y⁠: e x y = 1 + 2 x 2 y − x + x 2 6 y + x 2 10 y + x 2 14 y + ⋱ {\displaystyle e^{\frac {x}{y}}=1+{\cfrac {2x}{2y-x+{\cfrac {x^{2}}{6y+{\cfrac {x^{2}}{10y+{\cfrac...

problem in mathematics If A x + B y = C z {\displaystyle A^{x}+B^{y}=C^{z}} where A, B, C, x, y, z are positive integers and x, y, z are > 2, do A, B, and...

ln ⁡ ( x y ) = ln ⁡ x + ln ⁡ y for  x > 0 and  y > 0 , ln ⁡ ( x / y ) = ln ⁡ x − ln ⁡ y for  x > 0 and  y > 0 , ln ⁡ ( x y ) = y ln ⁡ x for  x > 0 , ln...

⁡ x cos ⁡ y = e i x + e − i x 2 ⋅ e i y + e − i y 2 = 1 2 ⋅ e i ( x + y ) + e i ( x − y ) + e i ( − x + y ) + e i ( − x − y ) 2 = 1 2 ( e i ( x + y )...

the Pythagorean theorem, x and y satisfy the equation x 2 + y 2 = 1. {\textstyle x^{2}+y^{2}=1.} Since x2 = (−x)2 for all x, and since the reflection...

X=Prem (2022) X Ray: The Inner Image (2019) X-Ray of a Lie (2004) The X-Rays (1897) X the Unknown (1956) X Videos (2018) X & Y (2018) X to Y (2009) X...

f X , Y ( x , y ) = f Y ∣ X ( y ∣ x ) f X ( x ) = f X ∣ Y ( x ∣ y ) f Y ( y ) {\displaystyle f_{X,Y}(x,y)=f_{Y\mid X}(y\mid x)f_{X}(x)=f_{X\mid Y}(x\mid...

fixed and the x ′ {\displaystyle x'} and y ′ {\displaystyle y'} axes are obtained by rotating the x {\displaystyle x} and y {\displaystyle y} axes counterclockwise...

x 2 + y 2 − 1 = 0 {\displaystyle x^{2}+y^{2}-1=0} of the unit circle defines y as an implicit function of x, y = 1 − x 2 {\displaystyle y={\sqrt {1-x^{2}}}}...

that x 4 + 4 y 4 = ( ( x + y ) 2 + y 2 ) ⋅ ( ( x − y ) 2 + y 2 ) = ( x 2 + 2 x y + 2 y 2 ) ⋅ ( x 2 − 2 x y + 2 y 2 ) . {\displaystyle {\begin{aligned}x^{4}+4y^{4}&={\bigl...

d ( F x d x + F y d y ) = F x x d x 2 + F y x d x d y + F x y d y d x + F y y d y 2 + F x d 2 x + F y d 2 y = 0. {\displaystyle d(F_{x}\,dx+F_{y}\...

efficient alternative to raster data. An X–Y plotter is a plotter that operates in two axes of motion ("X" and "Y") in order to draw continuous vector graphics...

( x , y ) = | x − y | {\displaystyle d(x,y)=\left|x-y\right|} if x {\displaystyle x} and y {\displaystyle y} lie in the same spine, and by d ( x , y )...

( x 1 , x 2 , x 3 , x 4 ) {\displaystyle (x_{1},x_{2},x_{3},x_{4})} and ( y 1 , y 2 , y 3 , y 4 ) {\displaystyle (y_{1},y_{2},y_{3},y_{4})} on the line...

1 12 [ X , [ X , Y ] ] + 1 12 [ Y , [ Y , X ] ] + ⋯ , {\displaystyle Z=X+Y+{\frac {1}{2}}[X,Y]+{\frac {1}{12}}[X,[X,Y]]+{\frac {1}{12}}[Y,[Y,X]]+\cdots...

constant terms: ( y 1 − y 2 ) x + ( x 2 − x 1 ) y + ( x 1 y 2 − x 2 y 1 ) = 0 {\displaystyle (y_{1}-y_{2})x+(x_{2}-x_{1})y+(x_{1}y_{2}-x_{2}y_{1})=0} (exchanging...

[ x , x , x ] + [ x , y , x ] + {\displaystyle [x,y,x]=[x,x,x]+[x,y,x]+} − [ x , x + y , x + y ] = {\displaystyle -[x,x+y,x+y]=} = [ x , x + y , − y ]...

x , y ) ∼ 2 π x x − 1 / 2 y y − 1 / 2 ( x + y ) x + y − 1 / 2 {\displaystyle \mathrm {B} (x,y)\sim {\sqrt {2\pi }}{\frac {x^{x-1/2}y^{y-1/2}}{({x+y})^{x+y-1/2}}}}...

y | x ) log 2 p ( y | x ) = − ∑ x ∈ X , y ∈ Y p ( x ) p ( y | x ) log 2 p ( y | x ) = − ∑ x ∈ X , y ∈ Y p ( x ) p ( y | x ) log 2 ( p ( y | x ) p ( x...

( x ) y + a 1 ( x ) y ′ + a 2 ( x ) y ″ ⋯ + a n ( x ) y ( n ) = b ( x ) {\displaystyle a_{0}(x)y+a_{1}(x)y'+a_{2}(x)y''\cdots +a_{n}(x)y^{(n)}=b(x)} where...

x {\displaystyle x} . Then the value of a harmonic function f {\displaystyle f} at x {\displaystyle x} is equal to f ( x ) = ∫ ∂ D f ( y ) d ν x ( y )...

rules ( x + y ) 2 = x 2 + 2 x y + y 2 {\displaystyle (x+y)^{2}=x^{2}+2xy+y^{2}} ( x + y ) ( x − y ) = x 2 − y 2 {\displaystyle (x+y)(x-y)=x^{2}-y^{2}} when...

integral is an integral operator T ( f ) ( x ) = ∫ K ( x , y ) f ( y ) d y , {\displaystyle T(f)(x)=\int K(x,y)f(y)\,dy,} whose kernel function K : R n ×...

function f x {\displaystyle f_{x}} , such that for any y {\displaystyle y} in Y {\displaystyle Y} , f x ( y ) = f ( x , y ) {\displaystyle f_{x}(y)=f(x,y)} ....

( x ) d x = ∫ R x 2 f ( x ) d x − 2 μ ∫ R x f ( x ) d x + μ 2 ∫ R f ( x ) d x = ∫ R x 2 d F ( x ) − 2 μ ∫ R x d F ( x ) + μ 2 ∫ R d F ( x ) = ∫ R x 2...

form y ′ ( x ) = q 0 ( x ) + q 1 ( x ) y ( x ) + q 2 ( x ) y 2 ( x ) {\displaystyle y'(x)=q_{0}(x)+q_{1}(x)\,y(x)+q_{2}(x)\,y^{2}(x)} where q 0 ( x ) ≠...

y))=(G_{1}(x,y),f(G_{1}(x,y),G_{2}(x,y))),} implying x = G 1 ( x , y ) {\displaystyle x=G_{1}(x,y)} and y = f ( x , G 2 ( x , y ) ) . {\displaystyle y=f(x,G_{2}(x...

identity x y = 2 ( x y + x + 1 ) y mod ( 2 x y + x + 1 − ˙ x ) {\displaystyle x^{y}=2^{(xy+x+1)y}{\bmod {(}}2^{xy+x+1}\mathbin {\dot {-}} x)} allows x y {\displaystyle...

(x,y)} on part of the curve, one has y = f ( x ) {\displaystyle y=f(x)} . An example is the unit circle, whose points ( x , y ) {\displaystyle (x,y)}...

x − x 1 ) ( x − x 2 ) + ( y − y 1 ) ( y − y 2 ) ( y − y 1 ) ( x − x 2 ) − ( y − y 2 ) ( x − x 1 ) = ( x 3 − x 1 ) ( x 3 − x 2 ) + ( y 3 − y 1 ) ( y 3...

R X Y = [ E ⁡ [ X 1 Y 1 ] E ⁡ [ X 1 Y 2 ] ⋯ E ⁡ [ X 1 Y n ] E ⁡ [ X 2 Y 1 ] E ⁡ [ X 2 Y 2 ] ⋯ E ⁡ [ X 2 Y n ] ⋮ ⋮ ⋱ ⋮ E ⁡ [ X m Y 1 ] E ⁡ [ X m Y 2 ]...

≡ y ( mod p ) {\displaystyle p\mid x-y\iff x\equiv y{\pmod {p}}} , The fact that x n − y n = ( x − y ) ( x n − 1 + x n − 2 y + x n − 3 y 2 + ⋯ + y n −...

of X {\displaystyle X} given Y = y {\displaystyle Y=y} is E ⁡ ( X ∣ Y = y ) = ∫ − ∞ ∞ x f X ∣ Y ( x ∣ y ) d x = 1 f Y ( y ) ∫ − ∞ ∞ x f X , Y ( x , y )...

x x − M x x + Q x x Y x x + Q x y Y x y Q x y − M x y + Q x x Y x y + Q x y Y y y Q y x − M y x + Q y x Y x x + Q y y Y x y Q y y − M y y + Q y x Y x...

( x , y ) d y ) d x = ∫ Y ( ∫ X f ( x , y ) d x ) d y . {\displaystyle \,\iint \limits _{X\times Y}f(x,y)\,\mathrm {d} (x,y)=\int _{X}\left(\int _{Y}f(x...

⁡ ( X , Y ) = E ⁡ [ ( X − E ⁡ [ X ] ) ( Y − E ⁡ [ Y ] ) ] = E ⁡ [ X Y − X E ⁡ [ Y ] − E ⁡ [ X ] Y + E ⁡ [ X ] E ⁡ [ Y ] ] = E ⁡ [ X Y ] − E ⁡ [ X ] E...

x ) = G ( x , y ) ,   K y ( 0 ) = 0 {\displaystyle K_{y}'(x)=G(x,y),\ K_{y}(0)=0} i.e. K ( x , y ) = K y ( x ) = ∫ 0 x G ( z , y ) d z = { x , 0 ≤ x <...

X ] = ∫ x Pr [ X = x ]   d x E ⁡ [ X ∣ Y = y ] = ∫ x Pr [ X = x ∣ Y = y ]   d x E ⁡ [ E ⁡ [ X ∣ Y ] ] = ∫ ( ∫ x Pr [ X = x ∣ Y = y ]   d x ) Pr [ Y =...

( x , y )   . {\displaystyle \ g(x,y)~.} To summarize ∇ x , y , λ L ( x , y , λ ) = 0 ⟺ { ∇ x , y f ( x , y ) = − λ ∇ x , y g ( x , y ) g ( x , y ) =...

x + y ) d x d y = ∫ 0 1 d x ∫ x 2 1 ( x + y ) d y = ∫ 0 1 d x   [ x y + y 2 2 ] x 2 1 {\displaystyle \iint _{D}(x+y)\,dx\,dy=\int _{0}^{1}dx\int _{x^{2}}^{1}(x+y)\...

X Y & Zee (also known as Zee and Co. and Zee and Company) is a 1972 British drama film directed by Brian G. Hutton and starring Elizabeth Taylor, Michael...

arctan(y/x). Hence, ∂ ∂ x atan2 ⁡ ( y , x ) = ∂ ∂ x arctan ⁡ ( y x ) = − y x 2 + y 2 , ∂ ∂ y atan2 ⁡ ( y , x ) = ∂ ∂ y arctan ⁡ ( y x ) = x x 2 + y 2 . {\displaystyle...

to d ( x , z ) ≤ max { d ( x , y ) , d ( y , z ) } {\displaystyle d(x,z)\leq \max \left\{d(x,y),d(y,z)\right\}} for all x {\displaystyle x} , y {\displaystyle...

numbers x, y, z, and t, x t ( x − y ) ( x − z ) + y t ( y − z ) ( y − x ) + z t ( z − x ) ( z − y ) ≥ 0 {\displaystyle x^{t}(x-y)(x-z)+y^{t}(y-z)(y-x)+z^{t}(z-x)(z-y)\geq...

specification (binary expression) x ′ {\displaystyle x'} = y {\displaystyle y} +1 ∧ y ′ {\displaystyle y'} = y {\displaystyle y} ∧ z ′ {\displaystyle z'} =...

(x,y,z)} is ψ ( x , y , z ) = A e i k x x e i k y y e i k z z = A e i ( k x x + k y y ) e i k z z = A e i ( k x x + k y y ) e ± i z k 2 − k x 2 − k y 2...

x + x 2 + x 9 ) ( 1 + x 4 + x 9 ) = x + x 2 + x 5 + x 6 + x 9 + x 10 + x 11 + x 13 + x 18 . {\displaystyle {\begin{aligned}(&x+x^{2}+x^{9})(1+x^{4}+x...

part v ( x , y ) {\displaystyle v(x,y)} are u ( x , y ) = x 2 − y 2 v ( x , y ) = 2 x y {\displaystyle {\begin{aligned}u(x,y)&=x^{2}-y^{2}\\v(x,y)&=2xy\end{aligned}}}...

following axioms: x y = y x {\displaystyle xy=yx} (commutative law) ( x x ) ( x y ) = x ( ( x x ) y ) {\displaystyle (xx)(xy)=x((xx)y)} (Jordan identity)...

the parallelogram law: ‖ x + y ‖ 2 + ‖ x − y ‖ 2 = 2 ‖ x ‖ 2 + 2 ‖ y ‖ 2 . {\displaystyle \|x+y\|^{2}+\|x-y\|^{2}=2\|x\|^{2}+2\|y\|^{2}.} In fact, as observed...

where F ( x , y , z ) = (   ( 1 + x y ) 3 z + y 2 ( 1 + x y ) ( 4 + 3 x y ) , y + 3 x ( 1 + x y ) 2 z + 3 x y 2 ( 4 + 3 x y ) , 2 x − 3 x 2 y − x 3 z   )...

x ) = max ( x , y ) ¬ x = 1 − x {\displaystyle {\begin{aligned}x\wedge y&=xy=\min(x,y)\\x\vee y&=x+y-xy=x+y(1-x)=\max(x,y)\\\neg x&=1-x\end{aligned}}}...

locus of all points (x, y, z) such that ( x − x 0 ) 2 + ( y − y 0 ) 2 + ( z − z 0 ) 2 = r 2 . {\displaystyle (x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}=r^{2}...

variables, X {\displaystyle X} , Y {\displaystyle Y} , are said to be uncorrelated if their covariance, cov ⁡ [ X , Y ] = E ⁡ [ X Y ] − E ⁡ [ X ] E ⁡ [ Y ] {\displaystyle...

p X ∣ Y = y ( x ) p Y ( y ) p X ( x ) p Y ( y ) = ∑ y ∈ Y p Y ( y ) ∑ x ∈ X p X ∣ Y = y ( x ) log ⁡ p X ∣ Y = y ( x ) p X ( x ) = ∑ y ∈ Y p Y ( y ) D...

x 2 + y 2 = z 2 {\displaystyle x^{2}+y^{2}=z^{2}} , has an infinite number of positive integer solutions for x {\displaystyle x} , y {\displaystyle y}...

x [ p ( x ) d y d x ] + q ( x ) y = − λ w ( x ) y {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\left[p(x){\frac {\mathrm {d} y}{\mathrm {d} x...

difference of two real numbers x {\displaystyle x} and y {\displaystyle y} is given by | x − y | {\displaystyle |x-y|} , the absolute value of their...

the map Φ : X → C b ( X ) {\displaystyle \Phi :X\rightarrow C_{b}(X)} defined by Φ ( x ) ( y ) = d ( x , y ) − d ( x 0 , y ) for all x , y ∈ X {\displaystyle...

x\cosh y+\cosh x\sinh y\\\cosh(x+y)&=\cosh x\cosh y+\sinh x\sinh y\\\tanh(x+y)&={\frac {\tanh x+\tanh y}{1+\tanh x\tanh y}}\\\sinh(x-y)&=\sinh x\cosh...

description: it sends each element y ∈ Y {\displaystyle y\in Y} to the unique element x ∈ X {\displaystyle x\in X} such that f(x) = y. As an example, consider the...

form: min x ∈ R d x max y ∈ R d y f ( x , y )        such that        g ( x , y ) ≤ 0 {\displaystyle \min _{\mathbf {x} \in \mathbb {R} ^{d_{x}}}\max _{\mathbf...

complex number ⟨ T x , y ⟩ {\displaystyle \langle Tx,y\rangle } is continuous for any vectors x {\displaystyle x} and y {\displaystyle y} in the Hilbert...

Minimize  ∑ x ∈ X , y ∈ Y γ x y c x y + ε γ x y ln ⁡ γ x y subject to:  γ ≥ 0 ∑ y ∈ Y γ x y = μ x , ∀ x ∈ X ∑ x ∈ X γ x y = ν y , ∀ y ∈ Y {\displaystyle...

( x − y ) 2 = x 2 − 2 x y + y 2 = x 2 + 2 x y + y 2 − 4 x y = ( x + y ) 2 − 4 x y . {\displaystyle {\begin{aligned}0&\leq (x-y)^{2}\\&=x^{2}-2xy+y...

such vectors, x ,   y ∈ R n {\displaystyle \mathbf {x} ,\ \mathbf {y} \in \mathbb {R} ^{n}} , we say that x {\displaystyle \mathbf {x} } weakly majorizes...

Theory X and Theory Y are theories of human work motivation and management. They were created by Douglas McGregor while he was working at the MIT Sloan...

X {\displaystyle X} and Y {\displaystyle Y} is a set of ordered pairs ( x , y ) {\displaystyle (x,y)} , where x {\displaystyle x} is an element of X {\displaystyle...

injection. Notationally: ∀ x , x ′ ∈ X , f ( x ) = f ( x ′ ) ⟹ x = x ′ , {\displaystyle \forall x,x'\in X,f(x)=f(x')\implies x=x',} or, equivalently (using...

notation, such as in: f x ′ ( x , y , … ) , ∂ f ∂ x ( x , y , … ) . {\displaystyle f'_{x}(x,y,\ldots ),{\frac {\partial f}{\partial x}}(x,y,\ldots ).} The symbol...

f ( x , y ) d y = g ( x , y ) d x , {\displaystyle f(x,y)\,dy=g(x,y)\,dx,} where f and g are homogeneous functions of the same degree of x and y. In this...

∫ X × Y V ( f ( x → ) , y ) p ( x → , y ) d x → d y = ∫ X ∫ Y ϕ ( y f ( x → ) ) p ( y ∣ x → ) p ( x → ) d y d x → = ∫ X [ ϕ ( f ( x → ) ) p ( 1 ∣ x →...

( x , y ) = y 2 − y y 2 − y 1 f ( x , y 1 ) + y − y 1 y 2 − y 1 f ( x , y 2 ) = y 2 − y y 2 − y 1 ( x 2 − x x 2 − x 1 f ( Q 11 ) + x − x 1 x 2 − x 1 f...

( x , y ) = x 2 + y 2 {\displaystyle f(x,y)=x^{2}+y^{2}} has two arguments, x {\displaystyle x} and y {\displaystyle y} , in an ordered pair ( x , y )...

The Y chromosome is one of two sex chromosomes in therian mammals and other organisms. Along with the X chromosome, it is part of the XY sex-determination...

holds: a n ( x ) f ( x ) n + a n − 1 ( x ) f ( x ) n − 1 + ⋯ + a 1 ( x ) f ( x ) + a 0 ( x ) = 0 {\displaystyle a_{n}(x)f(x)^{n}+a_{n-1}(x)f(x)^{n-1}+\cdots...

X ≠ X ~ {\displaystyle X\neq {\tilde {X}}} , with X ~ = f ( Y ) {\displaystyle {\tilde {X}}=f(Y)} being an approximate version of X {\displaystyle X}...

n ∑ x y = k g ( x ) h ( y ) , {\displaystyle F(n)=\sum _{k=1}^{n}\sum _{xy=k}^{}g(x)h(y),} where the inner sum runs over all ordered pairs (x,y) of positive...

relation R over a set X can be seen as a set of ordered pairs (x,y) of members of X. The relation R holds between x and y if (x,y) is a member of R. For...

M\geq 0} such that for all x ∈ X , {\displaystyle x\in X,} ‖ L x ‖ Y ≤ M ‖ x ‖ X . {\displaystyle \|Lx\|_{Y}\leq M\|x\|_{X}.} The infimum such M {\displaystyle...

= 0 {\displaystyle x=0} , we can see that: b x = y ⟺ b ( 0 ) = y ⟺ 1 = y ⟺ y = 1 {\displaystyle b^{x}=y\iff b^{(0)}=y\iff 1=y\iff y=1} So, substituting...

( x , y ) = ( x 2 + y − 11 ) 2 + ( x + y 2 − 7 ) 2 . {\displaystyle f(x,y)=(x^{2}+y-11)^{2}+(x+y^{2}-7)^{2}.\quad } It has one local maximum at x = −...

first: If x {\displaystyle x} is zero then x − y = − y {\displaystyle x-y=-y} , and if y {\displaystyle y} is zero then x − y = x {\displaystyle x-y=x} , so...

defined by f ( x , y ) = ( a − x ) 2 + b ( y − x 2 ) 2 {\displaystyle f(x,y)=(a-x)^{2}+b(y-x^{2})^{2}} It has a global minimum at ( x , y ) = ( a , a 2...

thought of as "what x can be". More precisely, given a function f : X → Y {\displaystyle f\colon X\to Y} , the domain of f is X. In modern mathematical...

a function ⁠ f : X → Y {\displaystyle f:X\to Y} ⁠ is the set of all ⁠ f ( x ) {\displaystyle f(x)} ⁠ such that ⁠ x {\displaystyle x} ⁠ belongs to the...

{K} } is a triple ( X , Y , b ) {\displaystyle (X,Y,b)} consisting of two vector spaces, X {\displaystyle X} and Y {\displaystyle Y} , over K {\displaystyle...

a > 0 {\displaystyle a>0} : u ( a ⋅ x , a ⋅ y ) = a ⋅ u ( x , y ) {\displaystyle u(a\cdot x,a\cdot y)=a\cdot u(x,y)} In mathematics, a homothetic function...

field X {\displaystyle X} on a manifold, there exist local coordinates y 1 , … , y n {\displaystyle y_{1},\dots ,y_{n}} such that X = ∂ / ∂ y 1 {\displaystyle...

equations: x = x , y = g 1 ( x ) , a ≤ x ≤ b {\displaystyle x=x,y=g_{1}(x),a\leq x\leq b} . Then ∫ C 1 L ( x , y ) d x = ∫ a b L ( x , g 1 ( x ) ) d x . {\displaystyle...

X Y = σ X Y = E [ ( X − μ X ) ( Y − μ Y ) ] {\displaystyle {\text{cov}}_{XY}=\sigma _{XY}=E[(X-\mu _{X})\,(Y-\mu _{Y})]} correlation corr X Y = ρ X Y...

max x ∈ X min y ∈ Y f ( x , y ) = min y ∈ Y max x ∈ X f ( x , y ) {\displaystyle \max _{x\in X}\min _{y\in Y}f(x,y)=\min _{y\in Y}\max _{x\in X}f(x,y)}...

( x + y ) 3 = ( x + y ) ( x + y ) ( x + y ) = x x x + x x y + x y x + x y y _ + y x x + y x y _ + y y x _ + y y y = x 3 + 3 x 2 y + 3 x y 2 _ + y 3 {\displaystyle...

c ( x , y ) γ ( x , y ) d x d y = ∫ c ( x , y ) d γ ( x , y ) . {\displaystyle \iint c(x,y)\gamma (x,y)\,\mathrm {d} x\,\mathrm {d} y=\int c(x,y)\,\mathrm...

sin ⁡ ( x ) ∂ y − y 2 ∂ x ( 2 y d y ∧ d x ∧ d z ) = ( − 2 x y 2 d x + ( − 2 y x 2 − 4 y 3 ) d y ) ∧ d z + ( 2 y sin ⁡ ( x ) d x ∧ d z + 2 y 3 d y ∧ d z...

P(x,y) of real numbers is known to be symmetric in x and y, namely that P(x,y) is equivalent to P(y,x), then in proving that P(x,y) holds for every x and...

⟨ x , y , z ⟩ 3 = ⟨ ( 3 z 2 − x 2 − y 2 ) x ( x 2 − 3 y 2 ) x 2 + y 2 , ( 3 z 2 − x 2 − y 2 ) y ( 3 x 2 − y 2 ) x 2 + y 2 , z ( z 2 − 3 x 2 − 3 y 2 )...

product [ x , y ] = x × y . {\displaystyle [x,y]=x\times y.} This is skew-symmetric since x × y = − y × x {\displaystyle x\times y=-y\times x} , and instead...

S B P A − S C Q B = 1 − x z x + x + 1 − y x y + y + 1 − z y z + z + 1 = ( x y z − 1 ) 2 ( x z + x + 1 ) ( y x + y + 1 ) ( z y + z + 1 ) . {\displaystyle...

x , y ∈ X {\displaystyle x,y\in X} by d ( x , y ) := ‖ y − x ‖ = ‖ x − y ‖ . {\displaystyle d(x,y):=\|y-x\|=\|x-y\|.} This makes X {\displaystyle X}...

( x , y , z ) d x ∧ d y + E ( x , y , z ) d y ∧ d z + F ( x , y , z ) d z ∧ d x . {\displaystyle G(x,y,z)\,dx\wedge dy+E(x,y,z)\,dy\wedge dz+F(x,y,z)\...

polynomial x 3 + x y + y 4 {\displaystyle x^{3}+xy+y^{4}} would become min { x + x + x , x + y , y + y + y + y } {\displaystyle \min\{x+x+x,\;x+y,\;y+y+y+y\}}...

2 y d x 2 = 1 x y 3 / 2 {\displaystyle {\frac {d^{2}y}{dx^{2}}}={\frac {1}{\sqrt {x}}}y^{3/2}} subject to the boundary conditions y ( 0 ) = 1 , { y (...

X , Y , Z ) {\displaystyle F(X,Y,Z)} or by the corresponding polynomial in two variables f ( x , y ) = F ( X , Y , 1 ) . {\displaystyle f(x,y)=F(X,Y,1)...

x , y ) ≥ 0 {\displaystyle d(x,y)\geq 0} d ( x , y ) = 0 ⟺ x = y {\displaystyle d(x,y)=0\iff x=y} d ( x , y ) = d ( y , x ) {\displaystyle d(x,y)=d(y...

2} (because of the monomial 3 x y {\displaystyle 3xy} ) whereas f ( x , y , z ) = x 2 + 4 y {\displaystyle f(x,y,z)=x^{2}+4y} is not. The degree of a...

) = P ⁡ ( X ≤ x ) = ∑ x i ≤ x P ⁡ ( X = x i ) = ∑ x i ≤ x p ( x i ) . {\displaystyle F_{X}(x)=\operatorname {P} (X\leq x)=\sum _{x_{i}\leq x}\operatorname...

( x ) log ⁡ p ( x ) q ( x ) d x = ∫ x a x b p ~ ( y ( x ) ) | d y d x | log ⁡ p ~ ( y ( x ) ) | d y d x | q ~ ( y ( x ) ) | d y d x | d x = ∫ y a y b...

variables ⁠ x {\displaystyle x} ⁠ and ⁠ y {\displaystyle y} ⁠ has the form f ( x , y ) = a x 2 + b x y + c y 2 + d x + e y + f , {\displaystyle f(x,y...

positive integer x, y or z that is part of a solution to the Markov Diophantine equation x 2 + y 2 + z 2 = 3 x y z , {\displaystyle x^{2}+y^{2}+z^{2}=3xyz...

( y ) : y ∈ Y } = { { x ∈ X : f ( x ) = y } : y ∈ Y } {\displaystyle \left\{f^{-1}(y)\mathrel {:} y\in Y\right\}\quad =\quad \left\{\left\{x\in X\mathrel...

X ( x , y ) ≤ d Y ( ϕ ( x ) , ϕ ( y ) ) ≤ C L d X ( x , y ) {\displaystyle Ld_{X}(x,y)\leq d_{Y}(\phi (x),\phi (y))\leq CLd_{X}(x,y)} for every x , y...

groups: Proper Euler angles (z-x-z, x-y-x, y-z-y, z-y-z, x-z-x, y-x-y) Tait–Bryan angles (x-y-z, y-z-x, z-x-y, x-z-y, z-y-x, y-x-z). Tait–Bryan angles are...

⁡ ( Z ( x ) , Z ( y ) ) = E [ ( Z ( x ) − E [ Z ( x ) ] ) ( Z ( y ) − E [ Z ( y ) ] ) ] . {\displaystyle C(x,y):=\operatorname {cov} (Z(x),Z(y))=\mathbb...

H ( x , y ) = [ f x x ( x , y ) f x y ( x , y ) f y x ( x , y ) f y y ( x , y ) ] . {\displaystyle H(x,y)={\begin{bmatrix}f_{xx}(x,y)&f_{xy}(x,y)\\f_{yx}(x...

functional equation is the functional equation: f ( x + y ) = f ( x ) + f ( y ) . {\displaystyle f(x+y)=f(x)+f(y).} A function f {\displaystyle f} that solves...

( x , y ) = { x , if  x = y , x − y ln ⁡ x − ln ⁡ y , otherwise , {\displaystyle L(x,y)=\left\{{\begin{array}{l l}x,&{\text{if }}x=y,\\{\dfrac {x-y}{\ln...

Y   λ x . x ) = ( λ x . ( λ x . x ) ( x   x ) )   ( λ x . ( λ x . x ) ( x   x ) ) {\displaystyle (\mathrm {Y} \ \lambda x.x)=(\lambda x.(\lambda x.x)(x\...

x + y ) + z = x + ( y + z ) = x + y + z ( x y ) z = x ( y z ) = x y z     } for all  x , y , z ∈ R . {\displaystyle \left.{\begin{matrix}(x+y)+z=x+(y+z)=x+y+z\quad...

formula f ( x , y ) = ∫ x 0 x A ( x , y ) d x + ∫ y 0 y B ( x , y ) d y ; {\displaystyle f(x,y)=\int _{x_{0}}^{x}A(x,y)\,dx+\int _{y_{0}}^{y}B(x,y)\,dy;} while...

ϕ ( x , y , z ) = ( x − 2 Δ y − Δ 2 z , y + Δ z , z ) , {\displaystyle \phi (x,y,z)=(x-2\Delta y-\Delta ^{2}z,y+\Delta z,z),} where Δ = x z + y 2 {\displaystyle...

y z = x y ∨ x ¯ z ∨ ( x ∨ x ¯ ) y z = x y ∨ x ¯ z ∨ x y z ∨ x ¯ y z = ( x y ∨ x y z ) ∨ ( x ¯ z ∨ x ¯ y z ) = x y ( 1 ∨ z ) ∨ x ¯ z ( 1 ∨ y ) = x y ∨...

X, Y & Z: The Real Story of How Enigma Was Broken is a 2018 book by Dermot Turing about the Enigma machine, which was used by Nazi Germany in World War...

H ( X , Y ) := max { sup x ∈ X d ( x , Y ) ,   sup y ∈ Y d ( X , y ) } , {\displaystyle d_{\mathrm {H} }(X,Y):=\max \left\{\,\sup _{x\in X}d(x,Y),\ \sup...

syntactically: // while (x == y) { something(); something_else(); } As is this: // for (int i=0; i < x; i++) // while (x == y) if (x == y) { something(); something_else();...

X {\displaystyle X} is compactly embedded in Y {\displaystyle Y} " is X ⊂⊂ Y {\displaystyle X\subset \subset Y} , or X ⋐ Y {\displaystyle X\Subset Y}...

(x = y) is equivalent to the form (x ∧ y) ∨ (¬x ∧ ¬y). ( x = y ) = ¬ ( x ⊕ y ) = ¬ x ⊕ y = x ⊕ ¬ y = ( x ∧ y ) ∨ ( ¬ x ∧ ¬ y ) = ( ¬ x ∨ y ) ∧ ( x ∨...

follows: x 3 + y 3 = ( x + y ) ( x 2 − x y + y 2 ) {\displaystyle x^{3}+y^{3}=(x+y)(x^{2}-xy+y^{2})} x 3 − y 3 = ( x − y ) ( x 2 + x y + y 2 ) {\displaystyle...

Majority ⁡ ( x , y , z ) = ( x ∧ y ) ∨ ( x ∧ z ) ∨ ( y ∧ z ) ; {\displaystyle \operatorname {Majority} (x,y,z)=(x\wedge y)\vee (x\wedge z)\vee (y\wedge z);}...

y 1 ( x 2 − x m i x ) y 2 ( x m i x − x 1 ) = y 1 ( y 2 − y m i x ) y 2 ( y m i x − y 1 ) {\displaystyle {\frac {L_{1}}{L_{2}}}={\frac {y_{1}\left(x_{2}-x_{\mathrm...

identifying and assisting families and children who have X and Y Chromosomal Variations (also called X & Y chromosomal variations), dyslexia and/or developmental...

( x , y ) , z ) = F ( x , F ( y , z ) ) {\displaystyle F(F(x,y),z)=F(x,F(y,z))} for a function F : X × X → X {\displaystyle F\colon X\times X\to X} ....

that d M ( x i , y ) ≤ d M ( x i , x ) + d M ( x , y ) < 1 2 δ x i + δ ≤ δ x i , {\displaystyle d_{M}(x_{i},y)\leq d_{M}(x_{i},x)+d_{M}(x,y)<{\frac {1}{2}}\delta...

called the X chromosome and Y chromosome, contain genes which code for sex. In these species, one or more genes are present on their Y chromosome that...

pullback is X × f , Y , 1 Y Y = { ( x , y ) : f ( x ) = 1 Y ( y ) } = { ( x , y ) : f ( x ) = y } ⊆ X × Y , {\displaystyle X\times _{f,Y,1_{Y}}Y=\{(x,y)\colon...

of the object. For example, the equations x = cos ⁡ t y = sin ⁡ t {\displaystyle {\begin{aligned}x&=\cos t\\y&=\sin t\end{aligned}}} form a parametric...

r , θ , φ ) ∂ ( x , y , z ) = ( x r y r z r x z r 2 x 2 + y 2 y z r 2 x 2 + y 2 − ( x 2 + y 2 ) r 2 x 2 + y 2 − y x 2 + y 2 x x 2 + y 2 0 ) . {\displaystyle...

form of the type A ( x , y , z ) d x + B ( x , y , z ) d y + C ( x , y , z ) d z {\displaystyle A(x,y,z)\,dx+B(x,y,z)\,dy+C(x,y,z)\,dz} is called a differential...

∀ x ∈ X ∀ y ∈ X ( [ x < y ∧ ¬ ( y < x ) ∧ ¬ ( x = y ) ] ∨ [ ¬ ( x < y ) ∧ y < x ∧ ¬ ( x = y ) ] ∨ [ ¬ ( x < y ) ∧ ¬ ( y < x ) ∧ x = y ] ) . {\displaystyle...

x , y ) = F ( u , x , y ) = 0 {\displaystyle F(t,x,y)=F(u,x,y)=0\,} or, equivalently, F ( t , x , y ) = 0     a n d     F ( u , x , y ) − F ( t , x ...

Y , Y ^ ) = ∑ i ( Y i − Y ¯ ) ( Y ^ i − Y ¯ ) ∑ i ( Y i − Y ¯ ) 2 ⋅ ∑ i ( Y ^ i − Y ¯ ) 2 = ∑ i ( Y i − Y ^ i + Y ^ i − Y ¯ ) ( Y ^ i − Y ¯ ) ∑ i ( Y...

\operatorname {NapLog} (x)+\operatorname {NapLog} (y)-161180956} or NapLog ⁡ ( x y / 10 7 ) = NapLog ⁡ ( x ) + NapLog ⁡ ( y ) {\displaystyle \operatorname {NapLog}...

sets X and Y the Tversky index is a number between 0 and 1 given by S ( X , Y ) = | X ∩ Y | | X ∩ Y | + α | X ∖ Y | + β | Y ∖ X | {\displaystyle S(X,Y)={\frac...

sets X , Y , Z {\displaystyle X,Y,Z} such that f : X × Y → Z {\displaystyle \,f\colon X\times Y\rightarrow Z} where X × Y {\displaystyle X\times Y} is...

f ( x , y , z ) d x ∧ d y + g ( x , y , z ) d z ∧ d x + h ( x , y , z ) d y ∧ d z ) . {\displaystyle \int _{S}\left(f(x,y,z)\,dx\wedge dy+g(x,y,z)\,dz\wedge...

specifically linear algebra, a degenerate bilinear form f ( x , y ) {\displaystyle f(x,y)} on a vector space V {\displaystyle V} is a bilinear form such...

surjective if ∀ y ∈ Y , ∃ x ∈ X , f ( x ) = y {\displaystyle \forall y\in Y,\,\exists x\in X,\;\;f(x)=y} . For any set X, the identity function idX on X is surjective...

{p(x,z|y)}{p(x|y)p(z|y)}}\\&{}=\log \left[{\frac {p(x,y)}{p(x)p(y)}}{\frac {p(x,z|y)}{p(x|y)p(z|y)}}\right]\\&{}=\log {\frac {p(x|y)p(y)p(x,z|y)}{p(x...

number components x and y, and is written z = x + y j . {\displaystyle z=x+yj.} The conjugate of z is z ∗ = x − y j . {\displaystyle z^{*}=x-yj.} Since j 2...

the mutual information I ( X ; Y ) {\displaystyle I(X;Y)} as   C = sup p X I ( X ; Y ) {\displaystyle \ C=\sup _{p_{X}}I(X;Y)} has the following property...

fields X {\displaystyle X} and Y {\displaystyle Y} on a smooth manifold M {\displaystyle M} a third vector field denoted [ X , Y ] {\displaystyle [X,Y]} ...

σ 31 σ 32 σ 33 ] ≡ [ σ x x σ x y σ x z σ y x σ y y σ y z σ z x σ z y σ z z ] ≡ [ σ x τ x y τ x z τ y x σ y τ y z τ z x τ z y σ z ] {\displaystyle {\boldsymbol...

y − y 0 y 1 − y 0 = x − x 0 x 1 − x 0 {\displaystyle {\frac {y-y_{0}}{y_{1}-y_{0}}}={\frac {x-x_{0}}{x_{1}-x_{0}}}} . Since we know the column, x, the...

) = ⋃ x , y ∈ V x y ¯ {\displaystyle \operatorname {Sect} (V)=\bigcup _{x,y\in V}{\overline {xy}}} (for x = y {\displaystyle x=y} , the line x y ¯ {\displaystyle...

0\leq k<1} such that for all x and y in M, d ( f ( x ) , f ( y ) ) ≤ k d ( x , y ) . {\displaystyle d(f(x),f(y))\leq k\,d(x,y).} The smallest such value...

A x x x 0 + A x y y 0 + B x , F = A x y x 0 + A y y y 0 + B y , G = B x x 0 + B y y 0 + C . {\displaystyle {\begin{aligned}E&=A_{xx}x_{0}+A_{xy}y_{0}+B_{x}...

f Y ∣ X ( y ∣ x ) f X ( x ) = f X , Y ( x , y ) = f X | Y ( x ∣ y ) f Y ( y ) . {\displaystyle f_{Y\mid X}(y\mid x)f_{X}(x)=f_{X,Y}(x,y)=f_{X|Y}(x\mid...

x , y ∈ R d {\displaystyle x,y\in \mathbb {R} ^{d}} such that x {\displaystyle x} is majorized by y {\displaystyle y} , one has that f ( x ) ≤ f ( y )...

ordered pairs ( x , y ) {\displaystyle (x,y)} , where f ( x ) = y . {\displaystyle f(x)=y.} In the common case where x {\displaystyle x} and f ( x ) {\displaystyle...

x p , x m ) × ( y p , y m ) = ( x p × y p + x m × y m , x p × y m + x m × y p ) . {\displaystyle (x_{p},\,x_{m})\times (y_{p},\,y_{m})=(x_{p}\times y_{p}+x_{m}\times...

strings x z , x y z , x y y z , x y y y z , . . . {\displaystyle xz,xyz,xyyz,xyyyz,...} are also in L {\displaystyle L} . The process of repeating y {\displaystyle...

let ( p 1 ( x ) y ′ ) ′ + q 1 ( x ) y = 0 {\displaystyle (p_{1}(x)y^{\prime })^{\prime }+q_{1}(x)y=0} ( p 2 ( x ) y ′ ) ′ + q 2 ( x ) y = 0 {\displaystyle...

i y ⋅ ξ f ( y ) d y , {\displaystyle ({\mathcal {F}}f)(\xi ):=\int _{\mathbb {R} }e^{-2\pi iy\cdot \xi }\,f(y)\,dy,} then f ( x ) = ∫ R e 2 π i x ⋅ ξ...

For all x and y, the machine M runs in time 2 p ( | x | ) {\displaystyle 2^{p(|x|)}} on input ⁠ ( x , y ) {\displaystyle (x,y)} ⁠ For all x in L, there...

integration; i.e., ( T f ) ( x ) = ∫ f ( y ) K ( x , y ) d y {\displaystyle (Tf)(x)=\int f(y)K(x,y)\,dy} where K ( x , y ) {\displaystyle K(x,y)} is called an integration...

x , y ) {\displaystyle (x,y)} is an integer point on a Mordell curve, then so is ( x , − y ) {\displaystyle (x,-y)} . If ( x , y ) {\displaystyle (x,y)}...

quadratic homogeneous polynomial in two variables q ( x , y ) = a x 2 + b x y + c y 2 , {\displaystyle q(x,y)=ax^{2}+bxy+cy^{2},\,} where a, b, c are the coefficients...

x f x ( 0 , 0 ) Δ x f x ( 1 , 0 ) Δ x f x ( 0 , 1 ) Δ x f x ( 1 , 1 ) Δ y f y ( 0 , 0 ) Δ y f y ( 1 , 0 ) Δ y f y ( 0 , 1 ) Δ y f y ( 1 , 1 ) Δ x Δ y...

X 1 ∨ X 2 ∨ … ∨ X n ) ∧ ( Y 1 ∨ X 2 ∨ … ∨ X n ) ∧ ( X 1 ∨ Y 2 ∨ … ∨ X n ) ∧ ( Y 1 ∨ Y 2 ∨ … ∨ X n ) ∧ … ∧ ( Y 1 ∨ Y 2 ∨ … ∨ Y n ) . {\displaystyle (X_{1}\vee...

a set X {\displaystyle X} is a subset Y {\displaystyle Y} of X {\displaystyle X} whose images under G {\displaystyle G} either coincide with Y {\displaystyle...

i 2 : Y → Y ⊔ X Y {\displaystyle i_{1},i_{2}:Y\to Y\sqcup _{X}Y} (resp. i 1 , i 2 : Y → Y ⊗ X Y {\displaystyle i_{1},i_{2}:Y\to Y\otimes _{X}Y} ). The...

operation, x ∨ y = y ∨ x {\displaystyle x\lor y=y\lor x} , associativity, ( x ∨ y ) ∨ z = x ∨ ( y ∨ z ) {\displaystyle (x\lor y)\lor z=x\lor (y\lor z)} ...

^{3}} such that f ( x , y ) = ( f 1 ( x , y ) , f 2 ( x , y ) , f 3 ( x , y ) ) . {\displaystyle f(x,y)=(f_{1}(x,y),f_{2}(x,y),f_{3}(x,y)).} Then, under the...

morphisms X , Y → X ∗ Y {\displaystyle X,Y\rightarrow X*Y} , which combine into a canonical morphism X + Y → X ∗ Y {\displaystyle X+Y\rightarrow X*Y} through...

F X 1 , X 2 , X 3 ( x 1 , x 2 , x 3 ) = F X 1 ( x 1 ) ⋅ F X 2 ( x 2 ) ⋅ F X 3 ( x 3 ) {\displaystyle F_{X_{1},X_{2},X_{3}}(x_{1},x_{2},x_{3})=F_{X_{1}}(x_{1})\cdot...

u ∂ x + ∂ v ∂ y = 0 {\displaystyle {\dfrac {\partial u}{\partial x}}+{\dfrac {\partial v}{\partial y}}=0} x {\displaystyle x} -Momentum: u ∂ u ∂ x + v...

f_{X\vert Y=y}(x)={\frac {f_{X,Y}(x,y)}{f_{Y}(y)}}} f Y | X = x ( y ) = f X , Y ( x , y ) f X ( x ) {\displaystyle f_{Y\vert X=x}(y)={\frac {f_{X,Y}(x,y)}{f_{X}(x)}}}...

= C ( X , Y ) = X + Y + 1 2 [ X , Y ] + 1 12 [ X , [ X , Y ] ] − 1 12 [ Y , [ X , Y ] ] + ⋯ . {\displaystyle Z=C(X,Y)=X+Y+{\tfrac {1}{2}}[X,Y]+{\tfrac...

{Y} }} or Σ X Y {\displaystyle \Sigma _{\mathbf {X} \mathbf {Y} }} . For random vectors X {\displaystyle \mathbf {X} } and Y {\displaystyle \mathbf {Y}...

2×2-determinants by hand: 2 A = | x 1 x 2 y 1 y 2 | + | x 2 x 3 y 2 y 3 | + ⋯ + | x n x 1 y n y 1 | = | x 1 x 2 x 3 ⋯ x n x 1 y 1 y 2 y 3 ⋯ y n y 1 | {\displaystyle...

R ( x , y ) > 0 iff  y > x R ( x , y ) = 0 iff  y = x R ( x , y ) < 0 iff  y < x . {\displaystyle {\begin{cases}R(x,y)>0&{\text{iff }}y>x\\R(x,y)=0&{\text{iff...

( x , y ) = { x 2 y x 4 + y 2 x 2 + y 2 ( x , y ) ≠ ( 0 , 0 ) 0 ( x , y ) = ( 0 , 0 ) {\displaystyle f(x,y)={\begin{cases}{\frac {x^{2}y}{x^{4}+y^{2}}}{\sqrt...

independent solutions to the differential equation d 2 y d x 2 − x y = 0 , {\displaystyle {\frac {d^{2}y}{dx^{2}}}-xy=0,} known as the Airy equation or the...

distribution). If X , Y ∼ Exponential ( λ ) {\displaystyle X,Y\sim {\textrm {Exponential}}(\lambda )} then X − Y ∼ Laplace ( 0 , λ − 1 ) {\displaystyle X-Y\sim {\textrm...

X, Y, and Z are random variables on the same probability space, and the covariance of X and Y is finite, then cov ⁡ ( X , Y ) = E ⁡ ( cov ⁡ ( X , Y ∣...

homogeneous elements x, y satisfy x y = ( − 1 ) | x | | y | y x , {\displaystyle xy=(-1)^{|x||y|}yx,} where |x| and |y| denote the degrees of x and y. A commutative...

x ∈ A {\displaystyle x\in A} , and y ∈ x {\displaystyle y\in x} , then y ∈ A {\displaystyle y\in A} . whenever x ∈ A {\displaystyle x\in A} , and x {\displaystyle...

Y ∣ X = x ) = ∫ y f ( y ∣ x ) d y = ∫ y f ( x , y ) f ( x ) d y {\displaystyle \operatorname {E} (Y\mid X=x)=\int yf(y\mid x)\,dy=\int y{\frac {f(x,y)}{f(x)}}\...

, y ) {\displaystyle (x,y)} . Associativity is formally expressed as that ( x ⋅ y ) ⋅ z = x ⋅ ( y ⋅ z ) {\displaystyle (x\cdot y)\cdot z=x\cdot (y\cdot...

( f ( x ) , f ( y ) ) = r d ( x , y ) {\displaystyle d(f(x),f(y))=rd(x,y)} for all points x , y ∈ M {\displaystyle x,y\in M} , where d ( x , y ) {\displaystyle...

Z ( z ) ∑ y ∈ Y ∑ x ∈ X p X , Y | Z ( x , y | z ) log ⁡ p X , Y | Z ( x , y | z ) p X | Z ( x | z ) p Y | Z ( y | z ) {\displaystyle I(X;Y|Z)=\sum _{z\in...

if X and Y are CW-complexes, and f : X → Y is a continuous map, then f is said to be cellular if f takes the n-skeleton of X to the n-skeleton of Y for...

( x , y ) {\displaystyle (x,y)} ⁠ in the Cartesian plane with ⁠ y > 0. {\displaystyle y>0.} ⁠ The lower half-plane is the set of points ⁠ ( x , y ) {\displaystyle...

x 4 − x 1 ) ( x 4 − x 2 ) + ( y 4 − y 1 ) ( y 4 − y 2 ) ( y 4 − y 1 ) ( x 4 − x 2 ) − ( y 4 − y 2 ) ( x 4 − x 1 ) = ( x 3 − x 1 ) ( x 3 − x 2 ) + ( y...

x ± y ) = sin ⁡ x cos ⁡ y ± cos ⁡ x sin ⁡ y , cos ⁡ ( x ± y ) = cos ⁡ x cos ⁡ y ∓ sin ⁡ x sin ⁡ y , tan ⁡ ( x ± y ) = tan ⁡ x ± tan ⁡ y 1 ∓ tan ⁡ x tan...

X X ′ ∼ △ M Y Y ′ , ∴ M X ¯ M Y ¯ = X X ′ ¯ Y Y ′ ¯ . △ M X X ″ ∼ △ M Y Y ″ , ∴ M X ¯ M Y ¯ = X X ″ ¯ Y Y ″ ¯ . △ A X X ′ ∼ △ C Y Y ″ , ∴ X X ′ ¯ Y Y...

⟨ x , y , z ∣ ( x y ) 2 = x 2 = y 2 , z x z − 1 = y , z y z − 1 = x y , z 3 k = 1 ⟩ {\displaystyle \langle x,y,z\mid (xy)^{2}=x^{2}=y^{2},zxz^{-1}=y,zyz^{-1}=xy...

x ∈ X : A x  is meager (resp. comeager) in  Y } {\displaystyle \{x\in X:A_{x}{\text{ is meager (resp. comeager) in }}Y\}} is comeager in X, where A x...

11 x + 11 y + 40 y 3 + 32 y 2 + 46 y x + 24 x y 3 + 52 x y 2 + 25 x 2 + 29 y 4 + 15 y 5 + 5 y 6 + 6 y 4 x + 39 y x 2 + 20 x 3 + y 7 + 8 y x 3 + 7 x 4 +...

x + y + z = v 2 + w 2 + x 2 + y 2 + z 2 = v 3 + w 3 + x 3 + y 3 + z 3 = 0. {\displaystyle v+w+x+y+z=v^{2}+w^{2}+x^{2}+y^{2}+z^{2}=v^{3}+w^{3}+x^{3}+y^{3}+z^{3}=0...

( X , Y = y ) {\displaystyle P(X)=\sum _{y}P(X,Y=y)} and P ( Y ) = ∫ x P ( Y , X = x ) {\displaystyle P(Y)=\int _{x}P(Y,X=x)} (considering X as continuous...

by}}\\f(y,x_{1},\dots ,x_{k})&={\begin{cases}g(x_{1},\dots ,x_{k})&{\text{if }}y=0,\\h(y',f(y',x_{1},\dots ,x_{k}),x_{1},\dots ,x_{k})&{\text{if }}y=S(y'){\text{...

{\displaystyle \phi .} ϕ = ∃ x 2 ( x 2 ∈ x 1 ∧ ¬ ∃ x 3 ( x 3 ∈ x 2 ) ) ∧ ∃ x 2 ( x 2 ∈ x 1 ∧ ∃ x 3 ( x 3 ∈ x 2 ∧ ¬ ∃ x 4 ( x 4 ∈ x 3 ) ) ) A = D o m ( E 2...

as: x ≲ y {\displaystyle x\lesssim y} if and only if x < y  or  x = y . {\displaystyle x<y{\text{ or }}x=y.} Two elements x , y ∈ S {\displaystyle x,y\in...

( x + y ) 2 cos 2 ⁡ θ ′ + 4 x y sin 2 ⁡ θ ′ ( x + y ) + ( x − y ) sin 2 ⁡ θ ′   d θ = 2 x ( x + y ) − ( x − y ) sin 2 ⁡ θ ′ ( ( x + y ) + ( x − y ) sin...

lim y → b lim x → a f ( x , y ) = lim y → b ( lim x → a f ( x , y ) ) , {\displaystyle \lim _{y\to b}\lim _{x\to a}f(x,y)=\lim _{y\to b}\left(\lim _{x\to...

{1}{2A}}{\begin{pmatrix}x_{2}y_{3}-x_{3}y_{2}&y_{2}-y_{3}&x_{3}-x_{2}\\x_{3}y_{1}-x_{1}y_{3}&y_{3}-y_{1}&x_{1}-x_{3}\\x_{1}y_{2}-x_{2}y_{1}&y_{1}-y_{2}&x_{2}-x...

d x y + p ( x ) y = 0 {\displaystyle {\frac {d}{dx}}y+p(x)y=0} d y d x = − p ( x ) y {\displaystyle {\frac {dy}{dx}}=-p(x)y} d y y = − p ( x ) d x , {\displaystyle...

(x, y) of the plane A x x x 2 + 2 A x y x y + A y y y 2 + 2 B x x + 2 B y y + C = 0 {\displaystyle A_{xx}x^{2}+2A_{xy}xy+A_{yy}y^{2}+2B_{x}x+2B_{y}y+C=0}...

x , y , x − 1 , y − 1 | x y = y x , x x − 1 = x − 1 x = y y − 1 = y − 1 y = 1 ⟩ . {\displaystyle \langle x,y,x^{-1},y^{-1}\,|\,xy=yx,xx^{-1}=x^{-1}x...

x [ ϕ ( x ) ] → ∃ z ∀ y [ O y z ↔ ∃ x [ ϕ ( x ) ∧ O y x ] ] . {\displaystyle \exists x[\phi (x)]\to \exists z\forall y[Oyz\leftrightarrow \exists x[\phi...

\|X-X'\|-\operatorname {E} \|Y-Y'\|\geq 0,} where (X, X', Y, Y') are independent, the cdf of X and X' is F, the cdf of Y and Y' is G, E {\displaystyle \operatorname...

Y\otimes X} by ε F Y ∘ F ( η Y ) ( y ⊗ x ) = η Y ( y ) ( x ) = y ⊗ x . {\displaystyle \varepsilon _{FY}\circ F(\eta _{Y})(y\otimes x)=\eta _{Y}(y)(x)=y\otimes...

h ( X ) = a , Y = y ∣ Z = z ) = ∑ X : h ( X ) = a P ( X = x , Y = y ∣ Z = z ) {\displaystyle P(h(X)=a,Y=y\mid Z=z)=\sum _{X\colon h(X)=a}P(X=x,Y=y\mid...

The point (x, y, 1) projects to ( x x + y , y x + y , 1 x + y ) . {\displaystyle \left({\frac {x}{x+y}},{\frac {y}{x+y}},{\frac {1}{x+y}}\right).} The...

x , y , u ( x , y ) ) b ( x , y , u ( x , y ) ) c ( x , y , u ( x , y ) ) ] = a ( x , y , u ( x , y ) ) ∂ u ∂ x ( x , y ) + b ( x , y , u ( x , y ) )...

) ( x ) = ∫ − π π f ( y ) D n ( x − y ) d y = 2 π ∑ k = − n n f ^ ( k ) e i k x , {\displaystyle (D_{n}*f)(x)=\int _{-\pi }^{\pi }f(y)D_{n}(x-y)\,dy=2\pi...

states that ( x + y ) ( x 2 − x y + y 2 ) = x 3 + y 3 {\displaystyle (x+y)(x^{2}-xy+y^{2})=x^{3}+y^{3}} . Here x 3 + y 3 {\displaystyle x^{3}+y^{3}} is a...

_{Y}^{2}={\frac {2}{3}}} Therefore: ρ X , Y = 1 σ X σ Y E [ ( X − μ X ) ( Y − μ Y ) ] = 1 σ X σ Y ∑ x , y ( x − μ X ) ( y − μ Y ) P ( X = x , Y = y )...

) ∂ x + ∂ ( φ g ) ∂ y ) d x d y = ∬ D ( ∂ ( φ x ˙ ) ∂ x + ∂ ( φ y ˙ ) ∂ y ) d x d y = ∮ C φ ( − y ˙ d x + x ˙ d y ) = ∮ C φ ( − y ˙ x ˙ + x ˙ y ˙ ) d...

y ″ + p ( x ) y ′ + q ( x ) y = g ( x ) {\displaystyle y''+p(x)y'+q(x)y=g(x)} has a solution expressible by a generalised Frobenius series when p ( x...

for all x , y ∈ L : {\displaystyle x,y\in L:} ( x ∧ y ) ∗ = x ∗ ∨ y ∗ {\displaystyle (x\wedge y)^{*}=x^{*}\vee y^{*}} ; ( x ∨ y ) ∗ ∗ = x ∗ ∗ ∨ y ∗ ∗ {\displaystyle...

Selection Theorem—Let X be a paracompact space and Y be a separable Banach space. Let F : X → Y {\displaystyle F\colon X\to Y} be a lower hemicontinuous...

sets f : X → Y is a subset Γ f   ⊆   X × Y . {\displaystyle \Gamma _{f}\ \subseteq \ X\times Y.} Write f(x) for the set of those y ∈ Y with (x,y) ∈ Γf....

Γ ( x , y ) = ( 2 x + y , x + y ) mod 1 . {\displaystyle \Gamma (x,y)=(2x+y,x+y){\bmod {1}}.} Equivalently, in matrix notation, this is Γ ( [ x y ] )...

coefficients, meaning they satisfy | x | ≤ | b / d | {\displaystyle |x|\leq |b/d|} and | y | < | a / d | {\displaystyle |y|<|a/d|} ; equality occurs only if...

p ( x + y ) ≤ p ( x ) + p ( y ) {\displaystyle p(x+y)\leq p(x)+p(y)} for all x , y ∈ X . {\displaystyle x,y\in X.} Absolute homogeneity: p ( s x ) = |...

( x ) f j ( y ) h j = k n h n k n + 1 f n ( y ) f n + 1 ( x ) − f n + 1 ( y ) f n ( x ) x − y {\displaystyle \sum _{j=0}^{n}{\frac {f_{j}(x)f_{j}(y)}{h_{j}}}={\frac...

One can show that a d x ( y ) = [ x , y ] {\displaystyle \mathrm {ad} _{x}(y)=[x,y]\,} for all x , y ∈ g {\displaystyle x,y\in {\mathfrak {g}}} , where...

X   =   f ( x , y ) , ⋅ X   =   x ∂ f ∂ x = x f x , X ⋅   =   y ∂ f ∂ y = y f y , : X  or  ⋅ ( ⋅ X )   =   x 2 ∂ 2 f ∂ x 2 = x 2 f x x , X :  or  ( X...

Translational envelopes: Moving forward and backward on the X-axis. (Surge) Moving left and right on the Y-axis. (Sway) Moving up and down on the Z-axis. (Heave)...

x ) = ∫ X k ( x , y ) f ( y ) d y {\displaystyle (Tf)(x)=\int _{X}k(x,y)f(y)\,dy} is compact. If k ( x , y ) = k ( y , x ) ¯ , {\displaystyle k(x,y)={\overline...

variables, p Y | X ( y | x ) = P ( Y = y ∣ X = x ) = P ( X = x , Y = y ) P X ( x ) {\displaystyle p_{Y|X}(y|x)=P(Y=y\mid X=x)={\frac {P(X=x,Y=y)}{P_{X}(x)}}} For...

of X and Y is the quotient space of the disjoint union of X and Y by the identification x 0 ∼ y 0 : {\displaystyle x_{0}\sim y_{0}:} X ∨ Y = ( X ⊔ Y )...

of x is just the closure of {x}: ↓ ⁡ x = { y ∈ X : y ≤ x } = cl { x } {\displaystyle \mathop {\downarrow } x=\{y\in X:y\leq x\}={\textrm {cl}}\{x\}} while...

curve defined by { ( x , y ) : y 3 − x 2 = 0 } {\displaystyle \left\{(x,y):y^{3}-x^{2}=0\right\}} in the ( x , y ) {\displaystyle (x,y)} coordinate system...

sequences such that for all sequences x = ( x n ) {\displaystyle x=(x_{n})} , y = ( y n ) {\displaystyle y=(y_{n})} in ℓ ∞ {\displaystyle \ell ^{\infty...

of Y. A dependency FD: X → Y means that the values of Y are determined by the values of X. A functional dependency FD: X → Y is called trivial if Y is...

reactions A → X {\displaystyle A\rightarrow X} 2 X + Y → 3 X {\displaystyle 2X+Y\rightarrow 3X} B + X → Y + D {\displaystyle B+X\rightarrow Y+D} X → E {\displaystyle...

( X ) = ∑ x ∈ X f ( x ) {\displaystyle f(X)=\sum _{x\in X}f(x)} and X ∨ Y = { x ∨ y ∣ x ∈ X , y ∈ Y } {\displaystyle X\vee Y=\{x\vee y\mid x\in X,y\in...

x 1 x 2 , x 1 x 3 , x 2 x 3 , x 1 2 − x 2 2 , 2 x 3 2 − x 1 2 − x 2 2 ) . {\displaystyle \mathbf {A} _{2}=\operatorname {span} _{\mathbb {C} }(x_{1}x_{2}...

{\displaystyle x} - y {\displaystyle y} -plane, described by x ≤ μ , 0 ≤ y ≤ F ( x ) or x ≥ μ , F ( x ) ≤ y ≤ 1 {\displaystyle x\leq \mu ,\;\,0\leq y\leq F(x)\quad...

y)} (or f ⟨ x , y ⟩ {\displaystyle f\;\langle x,y\rangle } with the argument ⟨ x , y ⟩ ∈ X × Y {\displaystyle \langle x,y\rangle \in X\times Y} written with...

function exp ⁡ ( − x 2 ) {\displaystyle \exp(-x^{2})} as E ( x , y ) = ∑ n = 0 ∞ ( ρ / 2 ) n n !   H n ( x ) H n ( y )   . {\displaystyle E(x,y)=\sum _{n=0}^{\infty...

the cross ratio: c log ⁡ Ω x y + Ω x y 2 − Ω x x Ω y y Ω x y − Ω x y 2 − Ω x x Ω y y = 2 i c ⋅ arccos ⁡ Ω x y Ω x x ⋅ Ω y y {\displaystyle c\log {\frac...

x ) × ( ∇ y ) = ( e x ∂ x ∂ x + e y ∂ x ∂ y + e z ∂ x ∂ z ) × ( e x ∂ y ∂ x + e y ∂ y ∂ y + e z ∂ y ∂ z ) = ( e x ⋅ 1 + e y ⋅ 0 + e z ⋅ 0 ) × ( e x ⋅...

⟨ x | y ⟩ = x 1 y 1 + x 2 y 2 + x 3 y 3 − x 4 y 4 . {\displaystyle \langle \mathbf {x} |\mathbf {y} \rangle =x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}-x_{4}y_{4}...

equation: R = Z ″ Y ′ X = ( X Y Z Y − 1 X − 1 ) ( X Y X − 1 ) X = X Y Z Y − 1 ( X − 1 X ) Y ( X − 1 X ) = X Y Z ( Y − 1 Y ) = X Y Z {\displaystyle...

x=X-X_{0}} , y = Y − Y 0 {\displaystyle y=Y-Y_{0}} , z = Z − Z 0 {\displaystyle z=Z-Z_{0}} , and d = D − a X 0 − b Y 0 − c Z 0 {\displaystyle d=D-aX_{0}-bY_{0}-cZ_{0}}...

( x ) ) = x {\displaystyle g(f(x))=x} for each x {\displaystyle x} in X {\displaystyle X} and f ( g ( y ) ) = y {\displaystyle f(g(y))=y} for each y {\displaystyle...

( x − x 1 ) + y 1 {\displaystyle y=m(x-x_{1})+y_{1}} , with m = Δ y Δ x = y 2 − y 1 x 2 − x 1 {\displaystyle \textstyle m={\frac {\Delta y}{\Delta x}}={\frac...

provides the output y from the input x. That is, y = log b ⁡ x {\displaystyle y=\log _{b}x} is equivalent to x = b y {\displaystyle x=b^{y}} if b is a positive...

specification might look like this: (f(x,y) = x ∨ f(x,y) = y) ∧ f(x,y) ≥ x ∧ f(x,y) ≥ y and the grammar might be: <Exp> ::= x | y | 0 | 1 | <Exp> + <Exp> | ite(<Cond>...

( X , Y ) := E ⁡ [ ‖ X − X ′ ‖ ‖ Y − Y ′ ‖ ] + E ⁡ [ ‖ X − X ′ ‖ ] E ⁡ [ ‖ Y − Y ′ ‖ ] − E ⁡ [ ‖ X − X ′ ‖ ‖ Y − Y ″ ‖ ] − E ⁡ [ ‖ X − X ″ ‖ ‖ Y − Y ′...

_{Y}(i)/|Y|} then J W ( x , y ) = | X ∩ Y | | X ∖ Y | + | X | < J ( X , Y ) . {\displaystyle J_{\mathcal {W}}(x,y)={\frac {|X\cap Y|}{|X\setminus Y|+|X|}}<J(X...

a partial function f from a set X to a set Y is a function from a subset S of X (possibly the whole X itself) to Y. The subset S, that is, the domain...

x′y′z′ + x′y′z + x′yz′ + xyz′ = x′y′ + yz′). P → Q, P ⊢ Q i.e.: ( ~(y & z) & (x → y) ) → ( ~ (x & z) ), ( ~(y & z) & (x → y) ) ⊢ ( ~ (x & z) ) i.e.: IF...

U x , U y ⟩ = ⟨ x , y ⟩ ¯ {\displaystyle \langle Ux,Uy\rangle ={\overline {\langle x,y\rangle }}} for all x {\displaystyle x} and y {\displaystyle y} in...

by x, defined as x−1ax. x y = x [ x , y ] . {\displaystyle x^{y}=x[x,y].} [ y , x ] = [ x , y ] − 1 . {\displaystyle [y,x]=[x,y]^{-1}.} [ x , z y ] =...

{\displaystyle x=e^{y}} and y = ln ⁡ x {\displaystyle y=\ln x} , we have: d x d y = x ⇒ d x x = d y ⇒ ∫ d x x = ∫ d y = y + C = ln ⁡ x + C . {\displaystyle...

function y = f ( x ) {\displaystyle y=f(x)} with respect to changes in the independent variable. The differential d y {\displaystyle dy} is defined by d y =...

( x , y ) = { X + Y − X Y if  X , Y > 0 X + Y + X Y if  X , Y < 0 X + Y 1 − min ( | X | , | Y | ) otherwise {\displaystyle CF(x,y)={\begin{cases}X+Y-XY&{\text{if...

if ( x x ) y = x ( x y ) {\displaystyle (xx)y=x(xy)} for all x , y ∈ G {\displaystyle x,y\in G} and right alternative if y ( x x ) = ( y x ) x {\displaystyle...

the variables are ( x 1 , x 2 , x 3 , x 4 , x 5 ) = ( x , y , u , v , λ ) {\displaystyle (x_{1},x_{2},x_{3},x_{4},x_{5})=(x,y,u,v,\lambda )} . The corresponding...

y 1 x y 1 y 1 ′ y 1 + x y 1 ′ | = y 1 ( y 1 + x y 1 ′ ) − x y 1 y 1 ′ = y 1 2 + x y 1 y 1 ′ − x y 1 y 1 ′ = y 1 2 = e − b a x ≠ 0. {\displaystyle W(y_{1}...

distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity. In projective geometry and related contexts...

bound to x. The next step in the process is now ( ν x ) ( 0 | x ( y ) . 0 | x ¯ ⟨ x ⟩ . 0 ) {\displaystyle {\begin{aligned}(\nu x)&\;(\;0\\&\;|\;x(y).\;0\\&\;|\;{\overline...

x , y ∈ P , {\displaystyle x,y\in P,} x R y {\displaystyle xRy} is written if and only if ( x , y ) ∈ R , {\displaystyle (x,y)\in R,} in which case x...

= U ( x , y ) {\displaystyle U=U(x,y)} S = S ( x , y ) {\displaystyle S=S(x,y)} V = V ( x , y ) {\displaystyle V=V(x,y)} d U = ( ∂ U ∂ x ) y d x + ( ∂...

example, { 3 x + 2 y − z = 1 2 x − 2 y + 4 z = − 2 − x + 1 2 y − z = 0 {\displaystyle {\begin{cases}3x+2y-z=1\\2x-2y+4z=-2\\-x+{\frac {1}{2}}y-z=0\end{cases}}}...

D Q = ∫ − ∞ ∞ ∫ − ∞ ∞ P X , Y ( x , y ) ( x − y ) 2 d x d y = ∫ − ∞ ∞ ∫ − ∞ ∞ Q Y ∣ X ( y ∣ x ) P X ( x ) ( x − y ) 2 d x d y . {\displaystyle D_{Q}=\int...

that space according to the rule ⟨ A x , y ⟩ = ⟨ x , A ∗ y ⟩ , {\displaystyle \langle Ax,y\rangle =\langle x,A^{*}y\rangle ,} where ⟨ ⋅ , ⋅ ⟩ {\displaystyle...

{x} } and y {\displaystyle \mathbf {y} } F X , Y ( x , y ) = F X ( x ) ⋅ F Y ( y ) {\displaystyle F_{\mathbf {X,Y} }(\mathbf {x,y} )=F_{\mathbf {X} }(\mathbf...

inverse −1: x y = { X L ∣ X R } { Y L ∣ Y R } = { X L y + x Y L − X L Y L , X R y + x Y R − X R Y R ∣ X L y + x Y R − X L Y R , x Y L + X R y − X R Y L } {\displaystyle...

f ( x ) = y 1 − y 0 x 1 − x 0 ( x − x 0 ) + y 0 {\displaystyle f(x)={\tfrac {y_{1}-y_{0}}{x_{1}-x_{0}}}(x{-}x_{0}\!)+y_{0}} . Its graph y = f ( x ) {\displaystyle...

4 w x d x 4 = q 1 ( x ) − n 1 ( x ) d 2 w x d x 2 − d n 1 d x d w x d x − 1 2 d n 2 d x d θ x d x − n 2 ( x ) 2 d 2 θ x d x 2 b 3 D 12 d 4 θ x d x 4 −...

X ( x ) = N ( x ; μ X , σ X 2 ) = 1 2 π σ X e − ( x − μ X ) 2 / ( 2 σ X 2 ) f Y ( y ) = N ( y ; μ Y , σ Y 2 ) = 1 2 π σ Y e − ( y − μ Y ) 2 / ( 2 σ Y...

x ) = f ( y ) {\displaystyle f(x)=f(y)} implies x ≤ y  and  y ≤ x {\displaystyle x\leq y{\text{ and }}y\leq x} and in turn x = y {\displaystyle x=y}...

{}{p}}}\ X} and Y n   → p   Y {\displaystyle Y_{n}\ {\xrightarrow {\overset {}{p}}}\ Y} , then a X n + b Y n   → p   a X + b Y {\displaystyle aX_{n}+bY_{n}\...

differential equation if it is of the form y ′ + P ( x ) y = Q ( x ) y n , {\displaystyle y'+P(x)y=Q(x)y^{n},} where n {\displaystyle n} is a real number...

x , y , z ) = − ∇ x , y ⋅ [ I ( x , y , z ) ∇ x , y Φ ] , {\displaystyle {\frac {2\pi }{\lambda }}{\frac {\partial }{\partial z}}I(x,y,z)=-\nabla _{x...

f ( x , y , z , w ) = x y + y z + w {\displaystyle f(x,y,z,w)=xy+yz+w} include the terms x y {\displaystyle xy} , x y z {\displaystyle xyz} , x y z w...

( x ) | 2 = | ∫ Y K ( x , y ) f ( y ) d y | 2 ≤ ( ∫ Y K ( x , y ) q ( y ) d y ) ( ∫ Y K ( x , y ) f ( y ) 2 q ( y ) d y ) ≤ α p ( x ) ∫ Y K ( x , y )...

graph of the two-variable function f ( x , y ) = ( 2 x 2 − y ) ( y − x 2 ) . {\displaystyle f(x,y)=(2x^{2}-y)(y-x^{2}).} It was proposed by Giuseppe Peano...

A x ( x + d x ) − A x ( x ) ] d y d z + [ A y ( y + d y ) − A y ( y ) ] d x d z + [ A z ( z + d z ) − A z ( z ) ] d x d y d x d y d z = ∂ A x ∂ x + ∂...

{\displaystyle x} horismos y {\displaystyle y} (often denoted x → y {\displaystyle x\to y} or x ↗ y {\displaystyle x\nearrow y} ) if x = y {\displaystyle x=y} or...

takes any point with coordinates ( x , y ) {\displaystyle (x,y)} to the point ( x + 2 y , y ) {\displaystyle (x+2y,y)} . In this case, the displacement...

equation: x 2 d 2 y d x 2 + x d y d x + ( x 2 − α 2 ) y = 0 , {\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}+\left(x^{2}-\alpha ^{2}\right)y=0...

y 1 ( x − x 0 ) − y 0 ( x − x 0 ) x 1 − x 0 = y 1 x − y 1 x 0 − y 0 x + y 0 x 0 + y 0 x 1 − y 0 x 0 x 1 − x 0 = y 0 ( x 1 − x ) + y 1 ( x − x 0 ) x 1...

more than one variable (for example f ( x , y ) {\displaystyle f(x,y)} or f ( x , y , z ) {\displaystyle f(x,y,z)} ) in such a way that each of the integrals...

− W 0 ( Y e Y ) = Y − W 0 ( Y e Y ) for  Y < − 1 , W 0 ( Y e Y ) − W − 1 ( Y e Y ) = Y − W − 1 ( Y e Y ) for  − 1 < Y < 0. {\displaystyle X(Y...

equation is given by: A ( x , y , z ) = − 1 2 π ∬ − ∞ + ∞ A ′ ( x ′ , y ′ ) e i k r r z r ( i k − 1 r ) d x ′ d y ′ , {\displaystyle A(x,y,z)=-{\frac {1}{2\pi...

( x + y ) = f ( x ) + f ( y )  (additivity)  f ( s x ) = s ¯ f ( x )  (conjugate homogeneity)  {\displaystyle {\begin{alignedat}{9}f(x+y)&=f(x)+f(y)&&\qquad...

Examples are x 1 − 7 x 2 + 3 x 3 = 0 {\displaystyle x_{1}-7x_{2}+3x_{3}=0} and ⁠ 1 4 x − y = 4 {\displaystyle \textstyle {\frac {1}{4}}x-y=4} ⁠. A system...

that the formulas ( x + y ) − y = x {\displaystyle (x+y)-y=x} and ( x + y ) − ( ( x + y ) − y ) = y {\displaystyle (x+y)-((x+y)-y)=y} hold in any abelian...

point ( x , y ) {\textstyle (x,y)} , then it updates to w t + 1 = w t + r ( y − f w t ( x ) ) x {\textstyle w_{t+1}=w_{t}+r(y-f_{w_{t}}(x))x} . If y = 0 {\textstyle...

∂ f ∂ x ( x 0 , y 0 ) ( x − x 0 ) + ∂ f ∂ y ( x 0 , y 0 ) ( y − y 0 ) . {\displaystyle z-z_{0}={\frac {\partial f}{\partial x}}(x_{0},y_{0})(x-x_{0})+{\frac...

\{}\langle x,y\rangle \in \omega \times B\mid (x\in B\land y=x)\lor (x\notin B\land y=y_{0}){\big \}}} i.e. q ( x ) := { x x ∈ B y 0 x ∉ B {\displaystyle q(x...

of points ( x , y ) {\displaystyle (x,y)} in the Cartesian plane that satisfy the implicit equation x 2 − y 2 = 1 {\displaystyle x^{2}-y^{2}=1} . In the...

continuous) is ∂ φ ∂ x = ∂ ∂ x ∫ a , b x , y v ⋅ d r = ∂ ∂ x ∫ a , b x 1 , y v ⋅ d r + ∂ ∂ x ∫ x 1 , y x , y v ⋅ d r = 0 + ∂ ∂ x ∫ x 1 , y x , y v ⋅ d r {\displaystyle...

x + y = 2x – 1 is solved for the unknown x by the expression x = y + 1, because substituting y + 1 for x in the equation results in (y + 1) + y = 2(y...

setting x = X + Y 2 , y = X − Y 2 . {\displaystyle x={{X+Y} \over {\sqrt {2}}},y={{X-Y} \over {\sqrt {2}}}.} In the X , Y {\displaystyle X,Y} plane the...

produce NOT_EQUAL from ( x ⋅ y ) / z = ? ( x / x ) ⋅ y {\displaystyle (x\cdot y)/z\mathrel {\overset {?}{=}} (x/x)\cdot y} . The most direct solution to...

d ( x , y ) = | x − y | . {\displaystyle d(x,y)=|x-y|.} The standard absolute value on the integers, the rationals and the real numbers: | x | = x {\displaystyle...

{\sin(y\pi )}{\sin(x\pi )}}{\binom {-y-1}{-x-1}}={\frac {\sin((x-y)\pi )}{\sin(x\pi )}}{\binom {y-x-1}{y}};} moreover, ( x y ) ⋅ ( y x ) = sin ⁡ ( ( x − y )...

on a set X {\displaystyle X} is said to be dense if, for all x {\displaystyle x} and y {\displaystyle y} in X {\displaystyle X} for which x < y {\displaystyle...

inequality f ( θ x + ( 1 − θ ) y ) ≥ f ( x ) θ f ( y ) 1 − θ {\displaystyle f(\theta x+(1-\theta )y)\geq f(x)^{\theta }f(y)^{1-\theta }} for all x, y ∈ dom f and...

numbers: whenever x > y and y > z, then also x > z whenever x ≥ y and y ≥ z, then also x ≥ z whenever x = y and y = z, then also x = z. More examples...

x j {\displaystyle x_{i}>x_{j}} and y i > y j {\displaystyle y_{i}>y_{j}} or if x i < x j {\displaystyle x_{i}<x_{j}} and y i < y j {\displaystyle y_{i}<y_{j}}...

\left(t-{\frac {vx}{c^{2}}}\right)\\x'&=\gamma \left(x-vt\right)\\y'&=y\\z'&=z\end{aligned}}} where (t, x, y, z) and (t′, x′, y′, z′) are the coordinates of...

x 2 m ) ( 1 + x 2 m − 1 y 2 ) ( 1 + x 2 m − 1 y 2 ) = ∑ n = − ∞ ∞ x n 2 y 2 n , {\displaystyle \prod _{m=1}^{\infty }\left(1-x^{2m}\right)\left(1+x...

Y ⟶ X + P II X + Y ⟶ 2 P III A + X ⟶ 2 X + 2 Z IV 2 X ⟶ A + P V B + Z ⟶ 1 2 f Y d [ X ] d t = k I [ A ] [ Y ] − k I I [ X ] [ Y ] + k I I I [ A ] [ X...

d ( x 2 ) d x cos ⁡ ( x 2 ) − d ( ln ⁡ x ) d x e x − ln ⁡ ( x ) d ( e x ) d x + 0 = 4 x 3 + 2 x cos ⁡ ( x 2 ) − 1 x e x − ln ⁡ ( x ) e x . {\displaystyle...

λ x . x x ) ( λ x . x x ) {\displaystyle \Omega =(\lambda x.xx)(\lambda x.xx)} . Here ( λ x . x x ) ( λ x . x x ) → ( x x ) [ x := λ x . x x ] = ( x [...

1S. f(x +ℝ y) = f(x) +S f(y) and f(x ×ℝ y) = f(x) ×S f(y), for all x and y in R . {\displaystyle \mathbb {R} .} x ≤ℝ y if and only if f(x) ≤S f(y), for...

y\,\forall x\,P(x,y)} ∃ x ∃ y P ( x , y ) ⇔ ∃ y ∃ x P ( x , y ) {\displaystyle \exists x\,\exists y\,P(x,y)\Leftrightarrow \exists y\,\exists x\,P(x,y)}...

connection: R ( X , Y ) Z = ∇ X ∇ Y Z − ∇ Y ∇ X Z − ∇ [ X , Y ] Z {\displaystyle R(X,Y)Z=\nabla _{X}\nabla _{Y}Z-\nabla _{Y}\nabla _{X}Z-\nabla _{[X,Y]}Z} or equivalently...

quartic equation: A x 4 + B y 4 + C x 3 y + D x 2 y 2 + E x y 3 + F x 3 + G y 3 + H x 2 y + I x y 2 + J x 2 + K y 2 + L x y + M x + N y + P = 0 , {\displaystyle...

the Hall resistance Rxy exhibits steps that take on the quantized values R x y = V Hall I channel = h e 2 ν , {\displaystyle R_{xy}={\frac...

factored as ( x + y ) ( x − y ) {\displaystyle (x+y)(x-y)} , so we have: x 2 − y 2 + x − y = ( x + y ) ( x − y ) + x − y = ( x − y ) ( x + y + 1 ) . {\displaystyle...

∂ x ∂ y = q x 2 2 b [ 1 − 2 + cosh ⁡ [ ν b ( x − 2 a ) ] − cosh ⁡ [ ν b x ] 2 cosh 2 ⁡ [ ν b ( x − a ) ] ] Q z x = ∂ M x x ∂ x − ∂ M x y ∂ y = q x 1 b...

laws: Commutativity: x + y = y + x. Associativity: (x + y) + z = x + (y + z). Huntington equation: n(n(x) + y) + n(n(x) + n(y)) = x. Herbert Robbins immediately...

( x ) y + a 1 ( x ) y ′ + a 2 ( x ) y ″ + ⋯ + a n ( x ) y ( n ) + b ( x ) = 0 , {\displaystyle a_{0}(x)y+a_{1}(x)y'+a_{2}(x)y''+\cdots +a_{n}(x)y^{(n)}+b(x)=0...

convolution is [ x 11 x 12 ⋯ x 1 n x 21 x 22 ⋯ x 2 n ⋮ ⋮ ⋱ ⋮ x m 1 x m 2 ⋯ x m n ] ∗ [ y 11 y 12 ⋯ y 1 n y 21 y 22 ⋯ y 2 n ⋮ ⋮ ⋱ ⋮ y m 1 y m 2 ⋯ y m n ] = ∑...

percentage. "X in Y" means that the probability is p = X / Y. "X to Y in favor" and "X to Y on" mean that the probability is p = X / (X + Y). "X to Y against"...

X and Y such that X ≤ m < XY, there are two integers x and y such that a y ≡ x ( mod m ) {\displaystyle ay\equiv x{\pmod {m}}} and | x | < X , 0 < y <...

given by C o v ( x , y ) = ∫ H ⟨ x , z ⟩ ⟨ y , z ⟩ d P ( z ) {\displaystyle \mathrm {Cov} (x,y)=\int _{H}\langle x,z\rangle \langle y,z\rangle \,\mathrm...

{\displaystyle *} on a set S is commutative if x ∗ y = y ∗ x {\displaystyle x*y=y*x} for all x , y ∈ S {\displaystyle x,y\in S} . An operation that is not commutative...

f y = e x 1 + y , f x x = e x ln ⁡ ( 1 + y ) , f y y = − e x ( 1 + y ) 2 , f x y = f y x = e x 1 + y . {\displaystyle {\begin{aligned}f_{x}&=e^{x}\ln(1+y)...

x_{1}y_{2}+x_{2}y_{1},k_{1}k_{2})} and ( x 1 x 2 − N y 1 y 2 , x 1 y 2 − x 2 y 1 , k 1 k 2 ) . {\displaystyle (x_{1}x_{2}-Ny_{1}y_{2},x_{1}y_{2}-x_{2}y_{1},k_{1}k_{2})...

\emptyset } ) {x} {y} {z} {x, y} {x, z} {y, z} {x, y, z} and hence the power set of S is {{}, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}}. If S is a finite...

X {\displaystyle X} and Y {\displaystyle Y} are sets and L ⊆ X × Y {\displaystyle L\subseteq X\times Y} is a relation from X {\displaystyle X} to Y ...

C ( X , Y ) → Y {\displaystyle X\times {\mathcal {C}}(X,Y)\to Y} is continuous, and each continuous map X × Z → Y {\displaystyle X\times Z\to Y} may...

groups of a product space X × Y {\displaystyle X\times Y} and those of the spaces X {\displaystyle X} and Y {\displaystyle Y} . The theorem first appeared...

{\displaystyle K} into itself such that ‖ f ( x ) − f ( y ) ‖ ≤ ‖ x − y ‖ {\displaystyle \|f(x)-f(y)\|\leq \|x-y\|} (i.e. f {\displaystyle f} is non-expansive)...

f : X → Y {\displaystyle f:X\to Y} such that, for each morphism Y ′ → Y {\displaystyle Y'\to Y} , the base change X × Y Y ′ → Y ′ {\displaystyle X\times...

{R} } defined by g ( x , y ) := x + y {\displaystyle g(x,y):=x+y} and h ( x , y ) := 1 x + y {\displaystyle h(x,y):={\frac {1}{x+y}}} are both continuous...

shift operator s ( x 1 , x 2 , x 3 , x 4 , … ) = ( x 2 , x 3 , x 4 , … ) . {\displaystyle s(x_{1},x_{2},x_{3},x_{4},\ldots )=(x_{2},x_{3},x_{4},\ldots ).}...

binary relation R ⊆ X×Y between two sets X and Y is total (or left total) if the source set X equals the domain {x : there is a y with xRy }. Conversely,...

= X 2 = Y 2 = Z 2 = − i X Y Z = I {\displaystyle I^{2}=X^{2}=Y^{2}=Z^{2}=-iXYZ=I} The Pauli matrices also anti-commute, for example Z X = i Y = − X Z...

L = E [ | y − a | | X = x ] = ∫ − ∞ ∞ | y − a | f Y | X ( y ) d y = ∫ − ∞ a ( a − y ) f Y | X ( y ) d y + ∫ a ∞ ( y − a ) f Y | X ( y ) d y . {\displaystyle...

function f ( x , y ) = x 2 + y 2 {\displaystyle f(x,y)=x^{2}+y^{2}} is homogeneous of degree 2: f ( t x , t y ) = ( t x ) 2 + ( t y ) 2 = t 2 ( x 2 + y 2 ) =...

notation: X + 0 = X. X + s(Y) = s(X + Y). i.e. X + (Y + 1) = (X + Y) + 1 X × 0 = 0. X × s(Y) = X + (X × Y). i.e. X × (Y + 1) = X + (X × Y). Here are the same...

form q ′ ( x , y ) = a ′ x 2 + x y + b ′ y 2 {\displaystyle q'(x,y)=a'x^{2}+xy+b'y^{2}} is equivalent to q ( x , y ) {\displaystyle q(x,y)} , then the...

x m + x ⋅ y n + y = y n + y ⋅ x m + x , {\displaystyle x^{m}+x\cdot y^{n}+y=y^{n}+y\cdot x^{m}+x,} or similar forms. Taking logs, m n = ln ⁡ y ln ⁡ x...

x 2 {\displaystyle 3\times x^{2}} is written as 3 x 2 {\displaystyle 3x^{2}} , and 2 × x × y {\displaystyle 2\times x\times y} may be written 2 x y {\displaystyle...

y ′ ε + y x ′ ε + x ′ y ′ ε 2 = x y + ( x y ′ + y x ′ ) ε x + x ′ ε y + y ′ ε = x y + x ′ ε y 1 + y ′ ε y = ( x y + x ′ ε y ) ⋅ ( 1 − y ′ ε y ) = x y...

SSIM ( x , y ) = l ( x , y ) α ⋅ c ( x , y ) β ⋅ s ( x , y ) γ {\displaystyle {\text{SSIM}}(x,y)=l(x,y)^{\alpha }\cdot c(x,y)^{\beta }\cdot s(x,y)^{\gamma...

) [ x − x 0 ] + b ( x 0 , y 0 , z 0 ) [ y − y 0 ] + c ( x 0 , y 0 , z 0 ) [ z − z 0 ] = 0 {\displaystyle a(x_{0},y_{0},z_{0})[x-x_{0}]+b(x_{0},y_{0}...

X&Y is a 2018 Swedish drama film directed by Anna Odell. The film had a gala premiere at the Stockholm International Film Festival on 6 November 2018...

example, if function div(x,y) = x/y, then div with the parameter x fixed at 1 is another function: div1(y) = div(1,y) = 1/y. This is the same as the function...

Frobenius identity is the generally-false equation ( x + y ) n = x n + y n {\displaystyle (x+y)^{n}=x^{n}+y^{n}} . Beginning students commonly make this error...

Evolved" forms, Mega Charizard X and Y, that were likely both designed by Tomohiro Kitakaze, the designer of Mega Charizard X. It also has a Gigantamax form...

1 x y − x if 0 ≤ x < y ≤ 1 1 − x if 0 < x = y 0 everywhere else . {\displaystyle f(x,y)={\begin{cases}{\frac {y}{x}}-y&{\text{if}}\quad 0\leq y<x\leq...

truth: S says "P does not know X and Y." P says "Now I know X and Y." S says "Now I also know X and Y." What are X and Y? The problem is rather easily...

y 1 y 2 y 3 ) = x 1 y 1 + x 2 y 2 + x 3 y 3 . {\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}\cdot {\begin{pmatrix}y_{1}\\y_{2}\\y...

difference of two squares: x 2 − y 2 = ( x + y ) ( x − y ) {\displaystyle x^{2}-y^{2}=(x+y)(x-y)} which is true for all x and y. Trigonometry is an area...

⁡ ( x ) {\displaystyle \operatorname {erf} (x)} is the probability that Y {\displaystyle Y} falls in the range [ − x , x ] {\displaystyle [-x,x]} . Two...

x + y , x + y ⟩ = ⟨ x , x ⟩ + 2 ⟨ x , y ⟩ + ⟨ y , y ⟩ . {\displaystyle \langle x+y,x+y\rangle =\langle x,x\rangle +2\langle x,y\rangle +\langle y,y\rangle...

x , y ) {\displaystyle P(x,y)} and Q ( x , y ) {\displaystyle Q(x,y)} the intersection is the collection of all points ( x , y ) {\displaystyle (x,y)}...

:F_{X}(x)\geq y\}} is considered instead. It is easy to see that x ≥ F X ∗ ( y ) ⟺ F X ( x ) ≥ y ( or equivalently:  x < F X ∗ ( y ) ⟺ F X ( x ) < y ) ...

affine varieties X , Y {\displaystyle X,Y} is a dense regular map which induces isomorphic inclusion k [ Y ] ↪ k [ X ] {\displaystyle k\left[Y\right]\hookrightarrow...

constant M, | f ( x ) − f ( y ) | ≤ M | x − y | α , x , y ∈ [ a , b ] {\displaystyle \left|f(x)-f(y)\right|\leq M\,|x-y|^{\alpha },\qquad x,y\in [a,b]} is...

as p ( x , y , z ) = p ( x ) p ( y | x ) p ( z | y ) = p ( y ) p ( x | y ) p ( z | y ) {\displaystyle p(x,y,z)=p(x)p(y|x)p(z|y)=p(y)p(x|y)p(z|y)} In this...

g > 1, given by an equation of the form y 2 + h ( x ) y = f ( x ) {\displaystyle y^{2}+h(x)y=f(x)} where f(x) is a polynomial of degree n = 2g + 1 > 4...

choosing random points (x, y) in the square −1 < x < 1, −1 < y < 1 until 0 < s = x 2 + y 2 < 1 , {\displaystyle 0<s=x^{2}+y^{2}<1,\,} and then returning...

equations: x 3 + y 3 = ( x + y ) ( ( x − y ) 2 + x y ) ,  and x 3 − y 3 = ( x − y ) ( ( x + y ) 2 − x y ) . {\displaystyle {\begin{aligned}x^{3}+y^{3}&=(x+y...

x\;P(x)} is an abbreviation of ∃ x ( P ( x ) ∧ ¬ ∃ y ( P ( y ) ∧ y ≠ x ) ) {\displaystyle \exists x\,(P(x)\wedge \neg \exists y\,(P(y)\wedge y\neq x))}...

TVSs X and Y is denoted by X ⊗ π Y {\displaystyle X\otimes _{\pi }Y} and the completion of this space will be denoted by X ⊗ ^ π Y {\displaystyle X{\widehat...

x\in X} , B x = { y ∈ Y : ( x , y ) ∈ B } {\displaystyle B_{x}=\{y\in Y:(x,y)\in B\}} is a convex set in Y {\displaystyle Y} . A function f ( x , y )...

Pr ( Y ≤ x ) = Pr ( { | X | ≤ c  and  X ≤ x }  or  { | X | > c  and  − X ≤ x } ) = Pr ( | X | ≤ c  and  X ≤ x ) + Pr ( | X | > c  and  − X ≤ x ) = Pr...

the point (x,x). Now let P = (x, y) be a lattice point on a branch H with x, y > 0 and x ≠ y (as the previous remark covers the case x = y). By symmetry...

If X , Y ∼ N ( 0 , 1 ) X , Y {\displaystyle X,Y\sim {\textrm {N}}(0,1)\,X,Y} independent, then X Y ∼ Cauchy ( 0 , 1 ) {\displaystyle {\tfrac {X}{Y}}\sim...

= x 3 + a 2 x 2 + a 1 x + a 0 {\displaystyle f(x)=x^{3}+a_{2}x^{2}+a_{1}x+a_{0}} such that substituting x = x ( y ) {\displaystyle x=x(y)} yields a new...

x r 2 ( x ) d x , y ¯ = r ( x ) L y , u ¯ = u , v ¯ = L r ( v + r ′ r y u ) , U ¯ = U , {\displaystyle {\begin{array}{l}\displaystyle {\bar {x}}={\frac...

equation in two variables which can be written in the form A x 2 + B x y + C y 2 + D x + E y + F = 0. {\displaystyle Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0.} The geometric...

x , y ,  and  z {\displaystyle x,y,{\text{ and }}z} of S , {\displaystyle S,} x ∗ ( y + z ) = ( x ∗ y ) + ( x ∗ z ) ; {\displaystyle x*(y+z)=(x*y)+(x*z);}...

of the title are X, Y, and Z. Xavier: Renegade Angel (2007-2009) Xavier Riddle and the Secret Museum (2019-) Xchange (2000-2006) X Company (2015-2017)...

monkey saddle is the surface defined by the equation z = x 3 − 3 x y 2 , {\displaystyle z=x^{3}-3xy^{2},\,} or in cylindrical coordinates z = ρ 3 cos...

x y ( t ) d t = − i γ ( M x y ( t ) B z ( t ) − M z ( t ) B x y ( t ) ) − M x y ( t ) T 2 d M z ( t ) d t = i γ 2 ( M x y ( t ) B x y ( t ) ¯ − M x y...

density f X , Y ( x , y ) {\displaystyle f_{X,Y}(x,y)} satisfies f X , Y ( x , y ) = f X ( x ) f Y ( y ) . {\displaystyle f_{X,Y}(x,y)=f_{X}(x)f_{Y}(y).} That...

Float pow2 x = x ** 2 add :: Float -> Float -> Float add x y = x + y pyth :: Float -> Float -> Float pyth x y = sqrt (add (pow2 x) (pow2 y)) To transform...

{x↦y+y} = ((y+y)+y) {x↦y+y} = (y+y)+y, while the substitution { x ↦ x+y } is non-idempotent, e.g. ((x+y) {x↦x+y}) {x↦x+y} = ((x+y)+y) {x↦x+y} = ((x+y)+y)+y...

hyperreal number x in R∗, the monad of x is the set monad ( x ) = { y ∈ R ∗ ∣ x − y  is infinitesimal } . {\displaystyle {\text{monad}}(x)=\{y\in \mathbb {R}...

segment ( x ≠ y ∧ B x y z ∧ B x ′ y ′ z ′ ∧ x y ≡ x ′ y ′ ∧ y z ≡ y ′ z ′ ∧ x u ≡ x ′ u ′ ∧ y u ≡ y ′ u ′ ) → z u ≡ z ′ u ′ . {\displaystyle {(x\neq y\land...

on the set of distinct ordered pairs { ( x , y ) ∈ S × S : x ≠ y } {\displaystyle \{(x,y)\in S\times S:x\neq y\}} . That is, assuming (without a real loss...

denoted by the Roman numeral I, I ( x , y ) = ⟨ x , y ⟩ . {\displaystyle \mathrm {I} (x,y)=\langle x,y\rangle .} Let X(u, v) be a parametric surface. Then...

finite is C [ t ] → C [ t ] [ x , y ] / ( y 2 − x 3 − t ) {\displaystyle \mathbb {C} [t]\to \mathbb {C} [t][x,y]/(y^{2}-x^{3}-t)} . The analogous notion...

spaces X {\displaystyle X} and Y {\displaystyle Y} is the quotient M f = ( ( [ 0 , 1 ] × X ) ⨿ Y ) / ∼ {\displaystyle M_{f}=(([0,1]\times X)\amalg Y)\,/\...

x ~ a ∂ x j ∂ x ~ b ∂ x k ) ( X ~ c ∂ x j ∂ x ~ c ) ( Y ~ d ∂ x k ∂ x ~ d ) = R ~ a b X ~ c Y ~ d ( ∂ x ~ a ∂ x j ∂ x j ∂ x ~ c ) ( ∂ x ~ b ∂ x k ∂ x...

following: For a morphism f : X → Y {\displaystyle f:X\to Y} of ringed spaces, an O X {\displaystyle {\mathcal {O}}_{X}} -module F {\displaystyle {\mathcal...

equations: x = X / Z {\displaystyle x=X/Z} y = Y / Z {\displaystyle y=Y/Z} X X = X ⋅ X {\displaystyle XX=X\cdot X} Y Y = Y ⋅ Y {\displaystyle YY=Y\cdot Y} Z Z...

f : X → Y {\displaystyle f:X\to Y} ⁠, if there is a function g : Y → X {\displaystyle g:Y\to X} such that for every ⁠ x ∈ X {\displaystyle x\in X} ⁠,...

of the form: M ( x , y ) d x + N ( x , y ) d y = 0 {\displaystyle M(x,y)\,dx+N(x,y)\,dy=0} satisfying the condition ∂ M ∂ y ≠ ∂ N ∂ x {\displaystyle {\frac...

conditions' y ( x , 0 ) = f ( x ) , ∂ y ( x , 0 ) ∂ t = g ( x ) . {\displaystyle y(x,0)=f(x),\qquad {\frac {\partial y(x,0)}{\partial t}}=g(x).} Here, f...

diagonal { ( x , x ) : x ∈ X } {\displaystyle \{(x,x):x\in X\}} in X × X {\displaystyle X\times X} be an entourage, and X {\displaystyle X} is a discrete...

function u ( x , y 1 , . . , y n ) = x + θ 1 ( y 1 ) + . . + θ n ( y n ) {\displaystyle u(x,y_{1},..,y_{n})=x+\theta _{1}(y_{1})+..+\theta _{n}(y_{n})} where...

x ( x , y ) ≈ f ( x + h , y ) − f ( x − h , y ) 2 h f y ( x , y ) ≈ f ( x , y + k ) − f ( x , y − k ) 2 k f x x ( x , y ) ≈ f ( x + h , y ) − 2 f ( x...

for every x ∈ V {\displaystyle x\in V} and y ∈ V {\displaystyle y\in V} , ‖ x + y ‖ ≤ ‖ x ‖ + ‖ y ‖ . {\displaystyle \|x+y\|\leq \|x\|+\|y\|.} If V {\displaystyle...

2 ( x , y ) = A 2 y 2 , q 3 ( x , y ) = A 3 x y {\displaystyle q_{0}(x,y)=f(0,0)+\nabla f(0,0)\cdot (x,y),q_{1}(x,y)=A_{1}x^{2},q_{2}(x,y)=A_{2}y^{2}...

x , v y ) {\displaystyle (u,\,v,\,h,\,u^{x},\,v^{y})} , with u , v , h {\displaystyle u,\,v,\,h} random group elements and x , y {\displaystyle x,\,y}...

X ( x ) f Y ( z / x ) 1 x d x − ∫ − ∞ 0 f X ( x ) f Y ( z / x ) 1 x d x = ∫ 0 ∞ f X ( x ) f Y ( z / x ) 1 | x | d x + ∫ − ∞ 0 f X ( x ) f Y ( z / x )...

for all x ∈ ( x 0 − δ , x 0 ) ∪ ( x 0 , x 0 + δ ) {\displaystyle x\in (x_{0}-\delta ,x_{0})\cup (x_{0},x_{0}+\delta )} , f ( x ) − f ( x 0 ) x − x 0 ∈ (...

vectors: d ( x , y ) = 1 − x ⋅ y ‖ x ‖ ‖ y ‖ {\displaystyle d(\mathbf {x} ,\mathbf {y} )=1-{\frac {\mathbf {x} \cdot \mathbf {y} }{\lVert \mathbf {x} \rVert...

x=y)} ∀ x   ( x + 0 = x ) {\displaystyle \forall x\ (x+0=x)} ∀ x , y   ( x + S ( y ) = S ( x + y ) ) {\displaystyle \forall x,y\ (x+S(y)=S(x+y))} ∀ x...

scalar-valued function: div ⁡ F = ∇ ⋅ F = ( ∂ ∂ x , ∂ ∂ y , ∂ ∂ z ) ⋅ ( F x , F y , F z ) = ∂ F x ∂ x + ∂ F y ∂ y + ∂ F z ∂ z . {\displaystyle \operatorname...

false) for any given x and y values: E = { x < y y > x x ≱ y y ≰ x {\displaystyle E={\begin{cases}x<y\\y>x\\x\ngeq y\\y\nleq x\end{cases}}} This relies...

sets x, y there is a set w = x ∪ {y} given by "adjoining" the set y to the set x. It is stated as ∀ x . ∀ y . ∃ w . ∀ z . ( z ∈ w ↔ ( z ∈ x ∨ z = y ) )...

identities x / y ⋅ y = x = y / y ⋅ x {\displaystyle x/y\cdot y=x=y/y\cdot x} v / w ⋅ ( x / y ⋅ z ) = ( v / w ⋅ x ) / y ⋅ z {\displaystyle v/w\cdot (x/y\cdot...

Laguerre's differential equation: x y ″ + ( 1 − x ) y ′ + n y = 0 ,   y = L ( x ) {\displaystyle xy''+(1-x)y'+ny=0,\ y=L(x)} which is a second-order linear...

y ) = ∑ a P ( a , b | x ′ , y ) =: P B ( b | y ) , {\displaystyle \sum _{a}P(a,b|x,y)=\sum _{a}P(a,b|x^{\prime },y)=:P_{B}(b|y),} ∑ b P ( a , b | x ...

operation) that satisfies the identity (x • y) • (u • v) = (x • u) • (y • v), or more simply, xy • uv = xu • yv for all x, y, u and v, using the convention that...

i ( x , y ) = { [ x , x + a ] × [ y , y + a ]  if  i = 1 [ x − a , x ] × [ y , y + a ]  if  i = 2 [ x − a , x ] × [ y − a , y ]  if  i = 3 [ x , x + a...

above equations gives λ ( x , y ) = arctan ⁡ [ sinh ⁡ x k 0 a sec ⁡ y k 0 a ] , φ ( x , y ) = arcsin ⁡ [ sech x k 0 a sin ⁡ y k 0 a ] . {\displaystyle...

properties in common. That is, entities x and y are identical if every predicate possessed by x is also possessed by y and vice versa. It states that no two...

( x ) = ∫ 0 H / 2 ρ ( x , y ) u ( x , y ) ρ e u e ( x ) ( 1 − u ( x , y ) u e ( x ) ) d y . {\displaystyle \delta _{2}(x)=\int _{0}^{H/2}{{\rho (x,y)u(x...

x , y ) = ( x ( t ) , y ( t ) ) {\displaystyle (x,y)=(x(t),y(t))} for specific functions x ( t ) {\displaystyle x(t)} and y ( t ) . {\displaystyle y(t)...

Using the identities e x + y = e x e y {\displaystyle e^{x+y}=e^{x}e^{y}} and e y ln ⁡ x = x y , {\displaystyle e^{y\ln x}=x^{y},} one gets z w = ρ c e...

represented as y = ( x β + u ) 1 [ x β + u > 0 ] {\displaystyle y=(x\beta +u)1[x\beta +u>0]} , where u | x ∼ N ( 0 , σ 2 ) {\displaystyle u|x\sim N(0,\sigma...

Y ∈ p : x ∈ Y ) ∧ ( ∀ Y 1 , Y 2 ∈ p : ( x ∈ Y 1 ∧ x ∈ Y 2 ) → Y 1 = Y 2 ) . {\displaystyle (\exists Y\in p:x\in Y)\land (\forall Y_{1},Y_{2}\in p:(x\in...

( x , y , z ) = ( y z , x z , x y ) . {\displaystyle f(x,y,z)=(yz,xz,xy).} This gives an implicit formula of x 2 y 2 + y 2 z 2 + z 2 x 2 − r 2 x y z =...

x, y, z in L: ( x ∧ y ) ∨ ( y ∧ z ) ∨ ( z ∧ x ) = ( x ∨ y ) ∧ ( y ∨ z ) ∧ ( z ∨ x ) . {\displaystyle (x\wedge y)\vee (y\wedge z)\vee (z\wedge x)=(x\vee...

fields X {\displaystyle X} and Y {\displaystyle Y} we have ∇ X Y − ∇ Y X = [ X , Y ] {\displaystyle \nabla _{X}Y-\nabla _{Y}X=[X,Y]} , where [ X , Y ] {\displaystyle...

x + y = [ x p , x n ] + [ y p , y n ] = x p − x n + y p − y n = ( x p + y p ) − ( x n + y n ) = [ x p + y p , x n + y n ] {\displaystyle x+y=[x_{p},x_{n}]+[y_{p}...

mathematics, the rational points on the unit circle are those points (x, y) such that both x and y are rational numbers ("fractions") and satisfy x2 + y2 = 1. The...

Given two sets X and Y, a binary relation f between X and Y is a function (from X to Y) if for every element x in X there is exactly one y in Y such that f...

μ X ∪ Y = 1 N X ∪ Y ( N X μ X + N Y μ Y − N X ∩ Y μ X ∩ Y ) σ X ∪ Y = 1 N X ∪ Y ( N X [ σ X 2 + μ X 2 ] + N Y [ σ Y 2 + μ Y 2 ] − N X ∩ Y [ σ X ∩ Y 2...

solution in the natural numbers of x a − y b = 1 {\displaystyle x^{a}-y^{b}=1} for a, b > 1, x, y > 0 is x = 3, a = 2, y = 2, b = 3. The history of the problem...

( x , y ) = { x 3 y x 6 + y 2 ( x , y ) ≠ ( 0 , 0 ) 0 ( x , y ) = ( 0 , 0 ) {\displaystyle f(x,y)={\begin{cases}{\frac {x^{3}y}{x^{6}+y^{2}}}&(x,y)\neq...

− ∑ y ∈ Y p ( y ) ∑ x ∈ X p ( x | y ) log ⁡ p ( x | y ) = − ∑ x , y p ( x , y ) log ⁡ p ( x | y ) . {\displaystyle H(X|Y)=\mathbb {E} _{Y}[H(X|y)]=-\sum...

}|y|\,p_{X,Y}(zy,y)\,dy.} If the two variables are independent then p X Y ( x , y ) = p X ( x ) p Y ( y ) {\displaystyle p_{XY}(x,y)=p_{X}(x)p_{Y}(y)}...

h(x)=z(y(x))} for every x, then the chain rule is, in Lagrange's notation, h ′ ( x ) = z ′ ( y ( x ) ) y ′ ( x ) . {\displaystyle h'(x)=z'(y(x))y'(x).}...

the form { ( x , y ) , ( x + h , y ) , ( x , y + h ) } {\displaystyle \{(x,y),(x+h,y),(x,y+h)\}} , where x , y , h ∈ Z {\displaystyle x,y,h\in \mathbb...

means that y − x = y ′ − x ′ {\displaystyle y-x=y'-x'} implies that f ( y ) − f ( x ) = f ( y ′ ) − f ( x ′ ) . {\displaystyle f(y)-f(x)=f(y')-f(x').} If...

ratio of shear stress to shear strain: G := τ x y γ x y = F A Δ x l = F l A Δ x τ x y = F A γ x y = Δ x l {\displaystyle {\begin{aligned}G&:={\frac {\tau...

x , y ) = ∂ f ∂ x ( x , y ) = ∂ f ∂ y ( x , y ) = 0 {\displaystyle f(x,y)={\frac {\partial f}{\partial x}}(x,y)={\frac {\partial f}{\partial y}}(x,y)=0}...

with (x, y) ↦ (f1(x, y), f2(x, y), f3(x, y)), given by f ( [ x y ] ) = [ f 1 ( x , y ) f 2 ( x , y ) f 3 ( x , y ) ] = [ x 2 y 5 x + sin ⁡ y 4 y ] . {\displaystyle...

point y {\displaystyle y} such that | x − y | < δ {\displaystyle |x-y|<\delta } and | f ( x ) − f ( y ) | ≥ ε {\displaystyle |f(x)-f(y)|\geq \varepsilon }...

respectively: log ⁡ ( x × y ) = log ⁡ ( x ) + log ⁡ ( y ) , {\displaystyle \log(x\times y)=\log(x)+\log(y)\,,} log ⁡ ( x / y ) = log ⁡ ( x ) − log ⁡ ( y ) . {\displaystyle...

vectors x {\displaystyle x} and y {\displaystyle y} in R n , {\displaystyle \mathbb {R} ^{n},} the following inequality holds: ( 1 + | x | 2 1 + | y | 2 )...

all sets X , Y ⊆ S {\displaystyle X,Y\subseteq S} Closure operators are determined by their closed sets, i.e., by the sets of the form cl(X), since the...

In mathematics, a real-valued function u ( x , y ) {\displaystyle u(x,y)} defined on a connected open set Ω ⊂ R 2 {\displaystyle \Omega \subset \mathbb...

μ y y < z R ( y ) .     The least   y < z   such that   R ( y ) ,   if   ( ∃ y ) y < z R ( y ) ;   otherwise ,   z . {\displaystyle \mu y_{y<z}R(y).\...

p X ( x )   p Y ( y ) {\displaystyle p_{X,Y}(x,y)=\Pr(X=x,Y=y)=p_{X}(x)\ p_{Y}(y)} The information content of this joint event is: I X , Y ⁡ ( x , y )...

formal definition explains: A binary relation P(x,y), where y is at most polynomially longer than x, is in FNP if and only if there is a deterministic...

x {\displaystyle x} is x 2 − 4 x + 7 {\displaystyle x^{2}-4x+7} . An example with three indeterminates is x 3 + 2 x y z 2 − y z + 1 {\displaystyle x^{3}+2xyz^{2}-yz+1}...

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 1 ∂ y 1 ∂ x 2 ⋯ ∂ y 1 ∂ x n ∂ y 2...

b]\rightarrow \mathbb {R} } where K ( x , y ) = K ( y , x ) {\displaystyle K(x,y)=K(y,x)} for all x , y ∈ [ a , b ] {\displaystyle x,y\in [a,b]} . K is said to be...

^{2}X}{\partial x^{2}}}=(ik_{x})^{2}.} Solving for X(x), X ( x ) = A k x e i k x x + B k x e − i k x x . {\displaystyle X(x)=A_{k_{x}}e^{ik_{x}x}+B_{k_{x}}e^{-ik_{x}x}...

y = x ( 100 − x ) {\displaystyle xy=x(100-x)} The derivative with respect to x {\displaystyle x} is: d d x x y = d d x x ( 100 − x ) = d d x ( 100 x −...

p ( x , y , 1 ) = p ( x , y ) {\displaystyle ^{h}p(x,y,1)=p(x,y)} and, if p is defined by p(x, y) = P(x, y, 1), then h p ( x , y , z ) = P ( x , y , z...

points ( x , y , z ) {\displaystyle (x,y,z)} of an object according to the formula ( x , y , z ) → ( x + Δ x , y + Δ y , z + Δ z ) {\displaystyle (x,y,z)\to...

∑ x ∈ X ∑ y ∈ Y P ( x , y ) log 2 ⁡ [ P ( x , y ) ] {\displaystyle \mathrm {H} (X,Y)=-\sum _{x\in {\mathcal {X}}}\sum _{y\in {\mathcal {Y}}}P(x,y)\log...

y {\displaystyle x/y} : gcd ( x , y ) = gcd ( y , x % y ) {\displaystyle \gcd(x,y)=\gcd(y,x\%y)} if y ≠ 0 {\displaystyle y\neq 0} gcd ( x , 0 ) = x {\displaystyle...

statistics, given two stochastic processes { X t } {\displaystyle \left\{X_{t}\right\}} and { Y t } {\displaystyle \left\{Y_{t}\right\}} , the cross-covariance...

functional form x ′ = x + t x ; y ′ = y + t y {\displaystyle x'=x+t_{x};y'=y+t_{y}} becomes: [ x ′ y ′ 1 ] = [ 1 0 t x 0 1 t y 0 0 1 ] [ x y 1 ] . {\displaystyle...

X → R X = { ( x , y ) ∈ R 2 : x 2 + y 2 ≤ 8 , x ≠ 0 , y ≠ 0 } z ( x , y ) = 1 2 x y x 2 + y 2 {\displaystyle {\begin{aligned}&z:X\to \mathbb {R} \\&X=\left\{(x...

of characteristic p {\displaystyle p} by an equation y p − y = f ( x ) {\displaystyle y^{p}-y=f(x)} for some rational function f {\displaystyle f} over...

That is, a function f : X → Y {\displaystyle f:X\to Y} is open if for any open set U {\displaystyle U} in X , {\displaystyle X,} the image f ( U ) {\displaystyle...

power x n {\displaystyle x^{n}} of x {\displaystyle x} , with n {\displaystyle n} a positive integer. If several variables are considered, say, x , y , z...

x :   ρ ( ∂ t u x + u x ∂ x u x + u y ∂ y u x + u z ∂ z u x ) = − ∂ x p + μ ( ∂ x 2 u x + ∂ y 2 u x + ∂ z 2 u x ) + 1 3 μ   ∂ x ( ∂ x u x + ∂ y u y +...

vector fields X and Y by T ( X , Y ) := ∇ X Y − ∇ Y X − [ X , Y ] {\displaystyle T(X,Y):=\nabla _{X}Y-\nabla _{Y}X-[X,Y]} where [X, Y] is the Lie bracket...

x ) = ∫ R n e i λ S ( x , y ) a ( x , y ) u ( y ) d y , x ∈ R m , y ∈ R n , {\displaystyle T_{\lambda }u(x)=\int _{\mathbb {R} ^{n}}e^{i\lambda S(x,y)}a(x...

x | y ) ≤ K ( x ) ≤ K ( x , y ) ≤ max ( K ( x | y ) + K ( y ) , K ( y | x ) + K ( x ) ) ≤ K ( x ) + K ( y ) {\displaystyle K(x|y)\leq K(x)\leq K(x,y)\leq...

: X → Y {\displaystyle f:X\to Y} of schemes such that (a) it is locally of finite presentation and (b) for each x ∈ X {\displaystyle x\in X} and y = f...

foldl1 f [x] = x foldl1 f (x:y:xs) = foldl1 f (f x y : xs) foldr1 f [x] = x foldr1 f (x:xs) = f x (foldr1 f xs) foldt1 f [x] = x foldt1 f (x:y:xs) = foldt1...

satisfies ⟨ P x , y ⟩ = ⟨ x , P y ⟩ {\displaystyle \langle P\mathbf {x} ,\mathbf {y} \rangle =\langle \mathbf {x} ,P\mathbf {y} \rangle } for all x , y ∈ V {\displaystyle...

x = X ( u , v ) {\displaystyle x=X(u,v)} and y = Y ( u , v ) . {\displaystyle y=Y(u,v).} An equation for ∂ x ∂ u {\displaystyle {\dfrac {\partial x}{\partial...

The exponential function x ↦ e x {\displaystyle x\mapsto e^{x}} satisfies e x + y = e x e y , {\displaystyle e^{x+y}=e^{x}e^{y},} and is thus a homomorphism...

, y ) = 1 {\displaystyle A(x,y)=1} ⁠ if x ∉ L {\displaystyle x\not \in L} , there is no certificate y with | y | = O ( | x | c ) {\displaystyle |y|=O(|x|^{c})}...

functions z 1 ( x , y ) = x 2 − y 2 2 {\displaystyle z_{1}(x,y)={\frac {x^{2}-y^{2}}{2}}} and z 2 ( x , y ) = x y {\displaystyle z_{2}(x,y)=xy} are harmonic...

( x 2 − x 1 ) − t ( x 4 − x 3 ) = x 3 − x 1 , {\displaystyle s(x_{2}-x_{1})-t(x_{4}-x_{3})=x_{3}-x_{1},} s ( y 2 − y 1 ) − t ( y 4 − y 3 ) = y 3 − y 1...

form y ′ = f 3 ( x ) y 3 + f 2 ( x ) y 2 + f 1 ( x ) y + f 0 ( x ) {\displaystyle y'=f_{3}(x)y^{3}+f_{2}(x)y^{2}+f_{1}(x)y+f_{0}(x)\,} where f 3 ( x ) ≠...

( x − y ) = Δ ( x − y ) Θ ( x 0 − y 0 ) {\displaystyle G_{\text{ret}}(x-y)=\Delta (x-y)\Theta (x^{0}-y^{0})} G adv ( x − y ) = − Δ ( x − y ) Θ ( y 0 −...

( u x , u y , u , x , y ) u x x + a 2 ( u x , u y , u , x , y ) u x y + a 3 ( u x , u y , u , x , y ) u y x + a 4 ( u x , u y , u , x , y ) u y y + f...

as the false statements 2 = 1, or x 3 + y 3 = 5 {\displaystyle x^{3}+y^{3}=5} and x 3 + y 3 = 6 {\displaystyle x^{3}+y^{3}=6} (which implies 5 = 6). Both...

x 2 + y 2 ) 2 ( x ± i y ) ℓ − 4 z 0 + − ( ∂ x 2 + ∂ y 2 ) ( ( x 2 + y 2 ) 2 ( x ± i y ) ℓ − 4 ) 1 ⋅ 2 z 2 + ( ∂ x 2 + ∂ y 2 ) 2 ( ( x 2 + y 2 ) 2 ( x...

all x, y, z: x ∗ (y ∗ (x ∗ z)) = ((x ∗ y) ∗ x) ∗ z z ∗ (x ∗ (y ∗ x)) = ((z ∗ x) ∗ y) ∗ x (x ∗ y) ∗ (z ∗ x) = x ∗ ((y ∗ z) ∗ x) (x ∗ y) ∗ (z ∗ x) = (x ∗...

for a regular map f : X → Y {\displaystyle f:X\rightarrow Y} of varieties is a kind of 'large' subvariety of X {\displaystyle X} which is 'crushed' by...

x + y ) ( y − x ) ( x − 1 ) ( x − 3 2 ) − 2 ( y 2 + x ( x − 2 ) ) 2 − k = 0 , {\displaystyle (x+y)(y-x)(x-1)\left(x-{\tfrac {3}{2}}\right)-2\left(y...

= function(x, y) return {x = x, y = y} -- return {["x"] = x, ["y"] = y} end Point.set_x = function(point, x) point.x = x -- point["x"] = x; end Tables...

x ‖ ≤ ‖ y ‖ {\displaystyle {|x|\leq |y|}\Rightarrow {\|x\|\leq \|y\|}} holds, where the absolute value |·| is defined as | x | = x ∨ − x := sup { x ...

x ) = 1 , He 1 ⁡ ( x ) = x , He 2 ⁡ ( x ) = x 2 − 1 , He 3 ⁡ ( x ) = x 3 − 3 x , He 4 ⁡ ( x ) = x 4 − 6 x 2 + 3 , He 5 ⁡ ( x ) = x 5 − 10 x 3 + 15 x ...

Axioms: ∀x ¬ Sx = 0 ∀x ¬ x = 0 → ∃y Sy = x ∀x∀y Sx = Sy → x = y ∀x x + 0 = x ∀x∀y x + Sy = S(x + y) ∀x x × 0 = 0 ∀x∀y x × Sy = (x × y) + x. IΣn is first-order...

}(f^{-1}(y))}}.} y = x 2 {\displaystyle y=x^{2}} (for positive x) has inverse x = y {\displaystyle x={\sqrt {y}}} . d y d x = 2 x         ;         d x d y =...

0 , y 0 , z 0 ) {\displaystyle (x_{0},y_{0},z_{0})} is F x ( x 0 , y 0 , z 0 ) ( x − x 0 ) + F y ( x 0 , y 0 , z 0 ) ( y − y 0 ) + F z ( x 0 , y 0 , z...

algebraic geometry, a morphism f : X → Y {\displaystyle f:X\to Y} between schemes is said to be quasi-compact if Y can be covered by open affine subschemes...

: X → Y {\displaystyle f:X\to Y} and g : Y → X {\displaystyle g:Y\to X} are morphisms whose composition f ∘ g : Y → Y {\displaystyle f\circ g:Y\to Y} is...

defined as P ( E ) = { x ∣ x ⊂ E } = { x ∣ y ∈ x ⟹ y ∈ E } . {\displaystyle {\mathcal {P}}(E)=\{x\mid x\subset E\}=\{x\mid y\in x\implies y\in E\}.} The fact...

y)\mapsto x\otimes y} (from X × Y {\displaystyle X\times Y} to X ⊗ Y {\displaystyle X\otimes Y} ) is continuous. When equipped with this topology, X ⊗ Y {\displaystyle...

hold: ⌊ x ⌋ + ⌊ y ⌋ ≤ ⌊ x + y ⌋ ≤ ⌊ x ⌋ + ⌊ y ⌋ + 1 , ⌈ x ⌉ + ⌈ y ⌉ − 1 ≤ ⌈ x + y ⌉ ≤ ⌈ x ⌉ + ⌈ y ⌉ . {\displaystyle {\begin{aligned}\lfloor x\rfloor +\lfloor...