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[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 ...
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.
A satirical petition ostensibly aiming to crowdfund a trillion dollars to allow Denmark to buy California has received more than 200,000 signatures.
Storm totals of 4–5 ft (1.2–1.5 m) were expected in Ashtabula and Lake counties in Ohio and 4–6 ft (1.2–1.8 m) in Northern Erie and Southern Erie counties. [25] In preparation of the storm, Interstate 90 shut down over 80 mi (130 km) of highway. Snow also resulted in portions of Interstate 94 and Pennsylvania State Route 5 closing. [26]
Consider a system of n linear equations for n unknowns, represented in matrix multiplication form as follows: = where the n × n matrix A has a nonzero determinant, and the vector = (, …,) is the column vector of the variables.
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.