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

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...

d x = d z d y ⋅ d y d x , {\displaystyle {\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}},} and d z d x | x = d z d y | y ( x ) ⋅ d y d x | x ,...

{\displaystyle y={\frac {x}{\log(x)-1.08366}},} (OEIS: A228211), as giving an approximation of y = π ( x ) {\displaystyle y=\pi (x)} with a "very satisfying...

solutions x, y, z ∈ Z. The equivalence is clear if n is even. If n is odd and all three of x, y, z are negative, then we can replace x, y, z with −x, −y, −z...

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}}}...

Sh(X) → Sh(Y) inverse image f∗ : Sh(Y) → Sh(X) direct image with compact support f! : Sh(X) → Sh(Y) exceptional inverse image Rf! : D(Sh(Y)) → D(Sh(X))...

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)}}}...

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)}...

subschemes i Y : Y → X {\displaystyle i_{Y}:Y\to X} such that for each subscheme Y {\displaystyle Y} of the covering, the sheaf F | Y = i Y ∗ F {\displaystyle...

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...

following relation: Ψ = Ψ ( X 1 , … , X i , Y i + 1 , … Y r ) {\displaystyle \Psi =\Psi {\big (}X_{1},\dots ,X_{i},Y_{i+1},\dots Y_{r}{\big )}\,} where for...

allgemeine Theorie der Kreuzung der Nervenbahnen (German, 1899, Verlag Joh. A. Barth)]. Rev. Trim. Micrográfica (in Spanish). 3: 15–65. Ramón y Cajal, Santiago...

be integers with greatest common divisor d. Then there exist integers x and y such that ax + by = d. Moreover, the integers of the form az + bt are exactly...

^{1+d}:-N-{\frac {1}{2}}\leq x\leq N+{\frac {1}{2}},|\alpha _{i}x-y_{i}|\leq {\frac {1}{N^{1/d}}}\right\}.} In his Essai sur la théorie des nombres (1798), Adrien-Marie...

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\}}...

equations: d y d x = f ( x ) d y d x = f ( x , y ) x 1 ∂ y ∂ x 1 + x 2 ∂ y ∂ x 2 = y {\displaystyle {\begin{aligned}{\frac {dy}{dx}}&=f(x)\\[4pt]{\frac...

{\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...

superellipses, and defined by the equation: | x a | n + | y b | n = 1 {\displaystyle \left|\,{x \over a}\,\right|^{n}+\left|\,{y \over b}\,\right|^{n}=1} where n is...

notation: y ″ = a ( x ) y ′ + b ( x ) y {\displaystyle y''=a(x)y'+b(x)y} where a ( x ) {\displaystyle a(x)} , b ( x ) {\displaystyle b(x)} are known, and y is...

output of the function is constrained to fall. It is the set Y in the notation f: X → Y. The term range is sometimes ambiguously used to refer to either...

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...

in the form: z = ∫ X ( x ) e a x d x  and  z = ∫ X ( x ) x a d x . {\displaystyle z=\int X(x)e^{ax}\,dx{\text{ and }}z=\int X(x)x^{a}\,dx.} The Laplace...

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 ≥ 3, do A, B, and...

x = q ( E x + v y B z − v z B y ) , F y = q ( E y + v z B x − v x B z ) , F z = q ( E z + v x B y − v y B x ) . {\displaystyle {\begin{aligned}F_{x...

2022-03-05 at the Wayback Machine The Comprehensive LaTeX Symbol List Bourbaki, N. (1970). Théorie des ensembles (in French). Paris: Hermann. ISBN 978-3-540-34034-8...

x ) = x + log ⁡ x − 1 + log ⁡ x − 2 x − ( log ⁡ x ) 2 − 6 log ⁡ x + 11 2 x 2 + o ( 1 x 2 ) . {\displaystyle B_{2}(x)=x+\log x-1+{\frac {\log x-2}{x}}-{\frac...

statement about the divisibility of solutions to the equation x p + y p = z p {\displaystyle x^{p}+y^{p}=z^{p}} of Fermat's Last Theorem for odd prime p {\displaystyle...

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...

u(x, 0) = g(x) for −∞ < x < ∞ and 0 < t < ∞ by applying a convolution: u ( x , t ) = ∫ Φ ( x − y , t ) g ( y ) d y . {\displaystyle u(x,t)=\int \Phi (x-y...