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Paul Celan (/ ˈ s ɛ l æ n /; [1] German: [ˈtseːlaːn]), born Paul Antschel, (23 November 1920 – c. 20 April 1970) was a Romanian-born French poet, Holocaust survivor, and literary translator. Celan is regarded as one of the most important figures in German-language literature of the post- World War II era and a poet whose verse has ...
In probability theory, Lévy’s continuity theorem, or Lévy's convergence theorem, [1] named after the French mathematician Paul Lévy, connects convergence in distribution of the sequence of random variables with pointwise convergence of their characteristic functions.
where u(x, y) and v(x, y) are real differentiable bivariate functions. Typically, u and v are respectively the real and imaginary parts of a complex-valued function f(x + iy) = f(x, y) = u(x, y) + iv(x, y) of a single complex variable z = x + iy where x and y are real variables; u and v are real differentiable functions of the real variables.
Cauchy's functional equation is the functional equation: (+) = + (). A function that solves this equation is called an additive function.Over the rational numbers, it can be shown using elementary algebra that there is a single family of solutions, namely : for any rational constant .
An arbitrary function φ : R n → C is the characteristic function of some random variable if and only if φ is positive definite, continuous at the origin, and if φ(0) = 1. Khinchine’s criterion. A complex-valued, absolutely continuous function φ, with φ(0) = 1, is a characteristic function if and only if it admits the representation
As in complex analysis of functions of one variable, which is the case n = 1, the functions studied are holomorphic or complex analytic so that, locally, they are power series in the variables z i. Equivalently, they are locally uniform limits of polynomials; or locally square-integrable solutions to the n-dimensional Cauchy–Riemann equations.
The k in the above definition is named the "depth" of a MZV, and the n = s 1 + ... + s k is known as the "weight". [3] The standard shorthand for writing multiple zeta functions is to place repeating strings of the argument within braces and use a superscript to indicate the number of repetitions. For example,
In probability theory and related fields, Malliavin calculus is a set of mathematical techniques and ideas that extend the mathematical field of calculus of variations from deterministic functions to stochastic processes. In particular, it allows the computation of derivatives of random variables.