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In Riemannian geometry, a branch of mathematics, the prescribed scalar curvature problem is as follows: given a closed, smooth manifold M and a smooth, real-valued function ƒ on M, construct a Riemannian metric on M whose scalar curvature equals ƒ. Due primarily to the work of J. Kazdan and F. Warner in the 1970s, this problem is well understood.
In the first case the line y = mx + n is an oblique asymptote of ƒ(x) when x tends to +∞, and in the second case the line y = mx + n is an oblique asymptote of ƒ(x) when x tends to −∞. An example is ƒ(x) = x + 1/x, which has the oblique asymptote y = x (that is m = 1, n = 0) as seen in the limits
where is the Euler–Mascheroni constant which equals the value of a number of definite integrals. Finally, a well known result, ∫ 0 2 π e i ( m − n ) ϕ d ϕ = 2 π δ m , n for m , n ∈ Z {\displaystyle \int _{0}^{2\pi }e^{i(m-n)\phi }d\phi =2\pi \delta _{m,n}\qquad {\text{for }}m,n\in \mathbb {Z} } where δ m , n {\displaystyle \delta ...
Horner's method evaluates a polynomial using repeated bracketing: + + + + + = + (+ (+ (+ + (+)))). This method reduces the number of multiplications and additions to just Horner's method is so common that a computer instruction "multiply–accumulate operation" has been added to many computer processors, which allow doing the addition and multiplication operations in one combined step.
In the equation 7x − 5 = 2, the sides of the equation are expressions.. In mathematics, an expression is a written arrangement of symbols following the context-dependent, syntactic conventions of mathematical notation.
Even if it was not obvious, the initialized condition ƒ'(0) = C, ƒ''(0) = D, etc. could be used. If we neglected those initialization terms, the last equation would show the composition of integration, and differentiation (and vice versa) would not hold.
In particular, if Y is a Banach space, then C(X, Y) is itself a Banach space under the uniform norm. The uniform limit theorem also holds if continuity is replaced by uniform continuity. That is, if X and Y are metric spaces and ƒ n : X → Y is a sequence of uniformly continuous functions converging uniformly to a function ƒ, then ƒ must be ...
Applied to a function ƒ, the q-differintegral of f, here denoted by is the fractional derivative (if q > 0) or fractional integral (if q < 0). If q = 0, then the q-th differintegral of a function is the function itself. In the context of fractional integration and differentiation, there are several definitions of the differintegral.