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The theorem is often given as this special case: If P is an injective polynomial function from an n-dimensional complex vector space to itself then P is bijective. That is, if P always maps distinct arguments to distinct values, then the values of P cover all of C n .
In mathematics, Macdonald polynomials Pλ ( x; t, q) are a family of orthogonal symmetric polynomials in several variables, introduced by Macdonald in 1987. He later introduced a non-symmetric generalization in 1995. Macdonald originally associated his polynomials with weights λ of finite root systems and used just one variable t, but later ...
A trigonometric polynomial can be considered a periodic function on the real line, with period some divisor of 2 π, or as a function on the unit circle.. A basic result is that the trigonometric polynomials are dense in the space of continuous functions on the unit circle, with the uniform norm; this is a special case of the Stone–Weierstrass theorem.
Note that this will be true for all the orthogonal polynomials above, because each p n is constructed to be orthogonal to the other polynomials p j for j<n, and x k is in the span of that set. If we pick the n nodes x i to be the zeros of p n , then there exist n weights w i which make the Gaussian quadrature computed integral exact for all ...
Valuation (algebra) In algebra (in particular in algebraic geometry or algebraic number theory ), a valuation is a function on a field that provides a measure of the size or multiplicity of elements of the field. It generalizes to commutative algebra the notion of size inherent in consideration of the degree of a pole or multiplicity of a zero ...
The output of the transform is a complex -valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function.
The Nuremberg trials were held by the Allies against representatives of the defeated Nazi Germany for plotting and carrying out invasions of other countries across Europe and atrocities against their citizens in World War II . Between 1939 and 1945, Nazi Germany invaded many countries across Europe, inflicting 27 million deaths in the Soviet ...
The Tschebyscheff polynomials for regular polygons inscribed in the unit circle of the complex plane [195] 1970 (M) Rufus Grier Pettis Oklahoma State University: An analysis of the methods being used to make provisions for outstanding high school mathematics students in North Carolina [196] 1973 (F) Therese Hance Braithwaite
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