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numpy.org. NumPy (pronounced / ˈnʌmpaɪ / NUM-py) is a library for the Python programming language, adding support for large, multi-dimensional arrays and matrices, along with a large collection of high-level mathematical functions to operate on these arrays. [3] The predecessor of NumPy, Numeric, was originally created by Jim Hugunin with ...
The function g must equal the inverse of f on the image of f, but may take any values for elements of Y not in the image. A function f with nonempty domain is injective if and only if it has a left inverse. [21] An elementary proof runs as follows: If g is the left inverse of f, and f(x) = f(y), then g(f(x)) = g(f(y)) = x = y.
Python is a multi-paradigm programming language. Object-oriented programming and structured programming are fully supported, and many of their features support functional programming and aspect-oriented programming (including metaprogramming [73] and metaobjects). [74] Many other paradigms are supported via extensions, including design by ...
ROT13 (Rotate13, " rotate by 13 places ", sometimes hyphenated ROT-13) is a simple letter substitution cipher that replaces a letter with the 13th letter after it in the Latin alphabet. ROT13 is a special case of the Caesar cipher which was developed in ancient Rome. Because there are 26 letters (2×13) in the basic Latin alphabet, ROT13 is its ...
The TPK algorithm is a simple program introduced by Donald Knuth and Luis Trabb Pardo to illustrate the evolution of computer programming languages.In their 1977 work "The Early Development of Programming Languages", Trabb Pardo and Knuth introduced a small program that involved arrays, indexing, mathematical functions, subroutines, I/O, conditionals and iteration.
Python sets are very much like mathematical sets, and support operations like set intersection and union. Python also features a frozenset class for immutable sets, see Collection types. Dictionaries (class dict) are mutable mappings tying keys and corresponding values. Python has special syntax to create dictionaries ({key: value})
For functions of a single variable, the theorem states that if is a continuously differentiable function with nonzero derivative at the point ; then is injective (or bijective onto the image) in a neighborhood of , the inverse is continuously differentiable near = (), and the derivative of the inverse function at is the reciprocal of the derivative of at : ′ = ′ = ′ (()).
v. t. e. In calculus, the inverse function rule is a formula that expresses the derivative of the inverse of a bijective and differentiable function f in terms of the derivative of f. More precisely, if the inverse of is denoted as , where if and only if , then the inverse function rule is, in Lagrange's notation,