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The Millennium Prize Problems are seven well-known complex mathematical problems selected by the Clay Mathematics Institute in 2000. The Clay Institute has pledged a US $1 million prize for the first correct solution to each problem.
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
At the Russian School of Mathematics, students begin multi-step problems as early as the first grade, learning to build on previous results to progress towards the solution. In the 1960s, collections of mathematical exercises were translated from Russian and published by W. H. Freeman and Company : The USSR Olympiad Problem Book (1962), [ 8 ...
Dirac–Kähler equation; Doppler equations; Drake equation (aka Green Bank equation) Einstein's field equations; Euler equations (fluid dynamics) Euler's equations (rigid body dynamics) Relativistic Euler equations; Euler–Lagrange equation; Faraday's law of induction; Fokker–Planck equation; Fresnel equations; Friedmann equations; Gauss's ...
Defining the two Wirtinger derivatives as = (), ¯ = (+), the Cauchy–Riemann equations can then be written as a single equation ¯ =, and the complex derivative of in that case is =. In this form, the Cauchy–Riemann equations can be interpreted as the statement that a complex function f {\textstyle f} of a complex variable z {\textstyle z ...
A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram called an Argand diagram, representing the complex plane. Re is the real axis, Im is the imaginary axis, and i is the "imaginary unit", that satisfies i 2 = −1.
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