Search results
Results From The WOW.Com Content Network
Jensen's inequality generalizes the statement that a secant line of a convex function lies above its graph. Visualizing convexity and Jensen's inequality. In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function.
In two dimensions, 2x 1 + 2x 2 is the perimeter of a rectangle with sides of length x 1 and x 2. Similarly, 4 √ x 1 x 2 is the perimeter of a square with the same area, x 1 x 2, as that rectangle. Thus for n = 2 the AM–GM inequality states that a rectangle of a given area has the smallest perimeter if that rectangle is also a square.
[1] It is named after the Russian mathematician Andrey Markov , although it appeared earlier in the work of Pafnuty Chebyshev (Markov's teacher), and many sources, especially in analysis , refer to it as Chebyshev's inequality (sometimes, calling it the first Chebyshev inequality, while referring to Chebyshev's inequality as the second ...
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution.
In the same way, an extension K 2 of K 1 can be constructed, etc. The union of all these extensions is the algebraic closure of K , because any polynomial with coefficients in this new field has its coefficients in some K n with sufficiently large n , and then its roots are in K n +1 , and hence in the union itself.
Because (a + 1) 2 = a, a + 1 is the unique solution of the quadratic equation x 2 + a = 0. On the other hand, the polynomial x 2 + ax + 1 is irreducible over F 4, but it splits over F 16, where it has the two roots ab and ab + a, where b is a root of x 2 + x + a in F 16. This is a special case of Artin–Schreier theory.
But beginning with PowerPoint 2007 and PowerPoint 2008 for Mac (PowerPoint version 12.0), this was the only binary format available for saving; PowerPoint 2007 (version 12.0) no longer supported saving to binary file formats used earlier than PowerPoint 97 (version 8.0), ten years before.