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In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced / ʃ ə ˈ l ɛ s k i / shə-LES-kee) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g., Monte Carlo simulations.
The 1998 book Proofs from THE BOOK, inspired by Erdős, is a collection of particularly succinct and revelatory mathematical arguments. Some examples of particularly elegant results included are Euclid's proof that there are infinitely many prime numbers and the fast Fourier transform for harmonic analysis .
A number of solutions have been proposed, including supersymmetry, conformal solutions and solutions via extra dimensions such as braneworld models. There are also issues of quantum triviality , which suggests that it may not be possible to create a consistent quantum field theory involving elementary scalar particles. [ 158 ]
Advanced Placement (AP) United States Government and Politics (often shortened to AP Gov or AP GoPo and sometimes referred to as AP American Government or simply AP Government) is a college-level course and examination offered to high school students through the College Board's Advanced Placement Program.
Notational abuse to be found below includes e X for the exponential map exp given an argument, writing g for the element (g, e H) in a direct product G × H (e H is the identity in H), and analogously for Lie algebra direct sums (where also g + h and (g, h) are used interchangeably).
Any real number can be written in the form m × 10 ^ n in many ways: for example, 350 can be written as 3.5 × 10 2 or 35 × 10 1 or 350 × 10 0. In normalized scientific notation (called "standard form" in the United Kingdom), the exponent n is chosen so that the absolute value of m remains at least one but less than ten ( 1 ≤ | m | < 10 ).
Computer engineers typically describe a modern computer's operation in terms of classical electrodynamics.Within these "classical" computers, some components (such as semiconductors and random number generators) may rely on quantum behavior, but these components are not isolated from their environment, so any quantum information quickly decoheres.
[1] [2] This is slightly more than one ninth of the world's population. The population density of Europe (the number of people per area) is the second highest of any continent, behind Asia. The population of Europe is currently slowly decreasing, by about 0.2% per year, [284] because there are fewer births than deaths.