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  2. Lower envelope - Wikipedia

    en.wikipedia.org/wiki/Lower_envelope

    For an infinite set of functions, the same notions may be defined using the infimum in place of the minimum, and the supremum in place of the maximum. [ 1 ] For continuous functions from a given class, the lower or upper envelope is a piecewise function whose pieces are from the same class.

  3. List of limits - Wikipedia

    en.wikipedia.org/wiki/List_of_limits

    If is expressed in radians: ⁡ = ⁡ ⁡ = ⁡ These limits both follow from the continuity of sin and cos. ⁡ =. [7] [8] Or, in general, ⁡ =, for a not equal to 0. ⁡ = ⁡ =, for b not equal to 0.

  4. List of periodic functions - Wikipedia

    en.wikipedia.org/wiki/List_of_periodic_functions

    This is a list of some well-known periodic functions. The constant function f (x) = c, where c is independent of x, is periodic with any period, but lacks a fundamental period. A definition is given for some of the following functions, though each function may have many equivalent definitions.

  5. Mathematical optimization - Wikipedia

    en.wikipedia.org/wiki/Mathematical_optimization

    The minimum value in this case is 1, occurring at x = 0. Similarly, the notation asks for the maximum value of the objective function 2x, where x may be any real number. In this case, there is no such maximum as the objective function is unbounded, so the answer is "infinity" or "undefined".

  6. Computational complexity of mathematical operations - Wikipedia

    en.wikipedia.org/wiki/Computational_complexity...

    The elementary functions are constructed by composing arithmetic operations, the exponential function (), the natural logarithm (), trigonometric functions (,), and their inverses. The complexity of an elementary function is equivalent to that of its inverse, since all elementary functions are analytic and hence invertible by means of Newton's ...

  7. Special functions - Wikipedia

    en.wikipedia.org/wiki/Special_functions

    Many special functions appear as solutions of differential equations or integrals of elementary functions.Therefore, tables of integrals [1] usually include descriptions of special functions, and tables of special functions [2] include most important integrals; at least, the integral representation of special functions.

  8. Extreme value theorem - Wikipedia

    en.wikipedia.org/wiki/Extreme_value_theorem

    The extreme value theorem was originally proven by Bernard Bolzano in the 1830s in a work Function Theory but the work remained unpublished until 1930. Bolzano's proof consisted of showing that a continuous function on a closed interval was bounded, and then showing that the function attained a maximum and a minimum value.

  9. Activation function - Wikipedia

    en.wikipedia.org/wiki/Activation_function

    The binary step activation function is not differentiable at 0, and it differentiates to 0 for all other values, so gradient-based methods can make no progress with it. [ 7 ] These properties do not decisively influence performance, nor are they the only mathematical properties that may be useful.

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