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The discriminant of a polynomial of degree n is homogeneous of degree 2n − 2 in the coefficients. This can be seen in two ways. In terms of the roots-and-leading-term formula, multiplying all the coefficients by λ does not change the roots, but multiplies the leading term by λ.
The arithmetic and geometric means "agree" that computer C is the fastest. However, by presenting appropriately normalized values and using the arithmetic mean, we can show either of the other two computers to be the fastest. Normalizing by A's result gives A as the fastest computer according to the arithmetic mean:
To show that ad − bc is the signed area, one may consider a matrix containing two vectors u ≡ (a, b) and v ≡ (c, d) representing the parallelogram's sides. The signed area can be expressed as | u | | v | sin θ for the angle θ between the vectors, which is simply base times height, the length of one vector times the perpendicular ...
where o(‖x − p‖) is a quantity that approaches zero much faster than the distance between x and p does as x approaches p. This approximation specializes to the approximation of a scalar function of a single variable by its Taylor polynomial of degree one, namely
In some circumstances, mathematicians may calculate a mean of an infinite (or even an uncountable) set of values. This can happen when calculating the mean value of a function (). Intuitively, a mean of a function can be thought of as calculating the area under a section of a curve, and then dividing by the length of that section.
3. Between two groups, may mean that the first one is a proper subgroup of the second one. > (greater-than sign) 1. Strict inequality between two numbers; means and is read as "greater than". 2. Commonly used for denoting any strict order. 3. Between two groups, may mean that the second one is a proper subgroup of the first one. ≤ 1.
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This "part" is the so-called complex Hessian, which is the matrix (¯),. Note that if f {\displaystyle f} is holomorphic, then its complex Hessian matrix is identically zero, so the complex Hessian is used to study smooth but not holomorphic functions, see for example Levi pseudoconvexity .