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}...

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 ,...

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}}}...

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...

function y(x), ∫ y ( x ) δ ( x − a ) d x = y ( a ) H ( x − a ) + c , {\displaystyle \displaystyle {\int }y(x)\delta (x-a)dx=y(a)H(x-a)+c,} where H(x) is the...

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)}}}...

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...

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...

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\}}...

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)}...

{\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...

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...

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...

since the asymptotic normality fails for X n = Y n − Y n − 1 {\textstyle X_{n}=Y_{n}-Y_{n-1}} where Y n {\textstyle Y_{n}} are another stationary sequence...

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

2 {\displaystyle {\frac {1}{\sqrt {2}}}} , erf(x) is the probability that Y falls in the range [−x, x]. Two closely related functions are the complementary...

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...

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 )...

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...

nodes or points); E ⊆ { { x , y } ∣ x , y ∈ V and x ≠ y } {\displaystyle E\subseteq \{\{x,y\}\mid x,y\in V\;{\textrm {and}}\;x\neq y\}} , a set of edges (also...

( x 1 , x 2 , x 3 ) = g ( x 1 + a 1 , x 2 , x 3 ) = g ( x 1 , x 2 + a 2 , x 3 ) = g ( x 1 , x 2 , x 3 + a 3 ) ∀ x 1 , x 2 , x 3 . {\displaystyle g(x_{1}...

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...

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...

X {\textstyle X} and ⁠ Y {\displaystyle Y} ⁠ are independent and X + Y {\textstyle X+Y} and X − Y {\textstyle X-Y} are also independent, then both X and...

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...

^{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...

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...

x ) {\displaystyle B(x)} by π ( x ) = x log ⁡ ( x ) − B ( x ) , {\displaystyle \pi (x)={\frac {x}{\log(x)-B(x)}},} and if B ( x ) {\displaystyle B(x)}...

following: If a complex function ⁠ f ( x + i y ) = u ( x , y ) + i v ( x , y ) {\displaystyle f(x+iy)=u(x,y)+i\,v(x,y)} ⁠ is holomorphic, then ⁠ u {\displaystyle...