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In mathematics, a square-integrable function, also called a quadratically integrable function or function or square-summable function, [1] is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite.
Continuous functions, monotone functions, step functions, semicontinuous functions, Riemann-integrable functions, and functions of bounded variation are all Lebesgue measurable. [2] A function f : X → C {\displaystyle f:X\to \mathbb {C} } is measurable if and only if the real and imaginary parts are measurable.
The eukaryotic cell cycle consists of four distinct phases: G 1 phase, S phase (synthesis), G 2 phase (collectively known as interphase) and M phase (mitosis and cytokinesis). M phase is itself composed of two tightly coupled processes: mitosis, in which the cell's nucleus divides, and cytokinesis, in which the cell's cytoplasm and cell membrane divides forming two daughter cells.
The symmetrized process is a Rademacher process, conditionally on the data . Therefore, it is a sub-Gaussian process by Hoeffding's inequality. Lemma (Symmetrization). For every nondecreasing, convex Φ: R → R and class of measurable functions ,
The integral of a non-negative general measurable function is then defined as an appropriate supremum of approximations by simple functions, and the integral of a (not necessarily positive) measurable function is the difference of two integrals of non-negative measurable functions.
As the eukaryotic cell cycle is a complex process, eukaryotes have evolved a network of regulatory proteins, known as the cell cycle control system, which monitors and dictates the progression of the cell through the cell cycle. [5]
In general, any measurable function can be pushed forward. The push-forward then becomes a linear operator, known as the transfer operator or Frobenius–Perron operator.In finite spaces this operator typically satisfies the requirements of the Frobenius–Perron theorem, and the maximal eigenvalue of the operator corresponds to the invariant measure.
A progressively measurable process, while defined quite technically, is important because it implies the stopped process is measurable. Being progressively measurable is a strictly stronger property than the notion of being an adapted process. [1] Progressively measurable processes are important in the theory of Itô integrals.