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"x^y = y^x - commuting powers". Arithmetical and Analytical Puzzles. Torsten Sillke. Archived from the original on 2015-12-28. dborkovitz (2012-01-29). "Parametric Graph of x^y=y^x". GeoGebra. OEIS sequence A073084 (Decimal expansion of −x, where x is the negative solution to the equation 2^x = x^2)
A point P has coordinates (x, y) with respect to the original system and coordinates (x′, y′) with respect to the new system. [1] In the new coordinate system, the point P will appear to have been rotated in the opposite direction, that is, clockwise through the angle . A rotation of axes in more than two dimensions is defined similarly.
Let ad X be the linear operator on g defined by ad X Y = [X,Y] = XY − YX for some fixed X ∈ g. (The adjoint endomorphism encountered above.) Denote with Ad A for fixed A ∈ G the linear transformation of g given by Ad A Y = AYA −1. A standard combinatorial lemma which is utilized [18] in producing the above explicit expansions is given ...
Vertical line of equation x = a Horizontal line of equation y = b. Each solution (x, y) of a linear equation + + = may be viewed as the Cartesian coordinates of a point in the Euclidean plane. With this interpretation, all solutions of the equation form a line, provided that a and b are not both zero. Conversely, every line is the set of all ...
Given a ring R, let Z(R) denote the set of all elements x in R such that x commutes with every element in R: xy = yx for any y in R. Then Z( R ) is a subring of R , called the center of R . More generally, given a subset X of R , let S be the set of all elements in R that commute with every element in X .
If we condense the skew entries into a vector, (x,y,z), then we produce a 90° rotation around the x-axis for (1, 0, 0), around the y-axis for (0, 1, 0), and around the z-axis for (0, 0, 1). The 180° rotations are just out of reach; for, in the limit as x → ∞ , ( x , 0, 0) does approach a 180° rotation around the x axis, and similarly for ...
The equation = means that the slope of ... If xy = yx, then e x + y = e x e y ... This formula also converges, though more slowly, for z > 2.
For matrix-matrix exponentials, there is a distinction between the left exponential Y X and the right exponential X Y, because the multiplication operator for matrix-to-matrix is not commutative. Moreover, If X is normal and non-singular, then X Y and Y X have the same set of eigenvalues. If X is normal and non-singular, Y is normal, and XY ...