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Integral of Complex-Valued Measurable Function

Keiko Narita, Noboru Endou, Yasunari Shidama (2008)

Formalized Mathematics

In this article, we formalized the notion of the integral of a complex-valued function considered as a sum of its real and imaginary parts. Then we defined the measurability and integrability in this context, and proved the linearity and several other basic properties of complex-valued measurable functions. The set of properties showed in this paper is based on [15], where the case of real-valued measurable functions is considered.MML identifier: MESFUN6C, version: 7.9.01 4.101.1015

Integral of Real-Valued Measurable Function 1

Yasunari Shidama, Noboru Endou (2006)

Formalized Mathematics

Based on [16], authors formalized the integral of an extended real valued measurable function in [12] before. However, the integral argued in [12] cannot be applied to real-valued functions unconditionally. Therefore, in this article we have formalized the integral of a real-value function.

Integral Transforms Method to Solve a Time-Space Fractional Diffusion Equation

Nikolova, Yanka, Boyadjiev, Lyubomir (2010)

Fractional Calculus and Applied Analysis

Mathematical Subject Classification 2010: 35R11, 42A38, 26A33, 33E12.The method of integral transforms based on using a fractional generalization of the Fourier transform and the classical Laplace transform is applied for solving Cauchy-type problem for the time-space fractional diffusion equation expressed in terms of the Caputo time-fractional derivative and a generalized Riemann-Liouville space-fractional derivative.

Integrals and Banach spaces for finite order distributions

Erik Talvila (2012)

Czechoslovak Mathematical Journal

Let ℬ c denote the real-valued functions continuous on the extended real line and vanishing at - ∞ . Let ℬ r denote the functions that are left continuous, have a right limit at each point and vanish at - ∞ . Define 𝒜 c n to be the space of tempered distributions that are the n th distributional derivative of a unique function in ℬ c . Similarly with 𝒜 r n from ℬ r . A type of integral is defined on distributions in 𝒜 c n and 𝒜 r n . The multipliers are iterated integrals of functions of bounded variation. For each n ∈ ℕ , the spaces...

Integration of some very elementary functions

Jan Mařík (1993)

Mathematica Bohemica

Let m be a natural number. Let f , g and Q be real polynomials such that { d e g f , d e g g } ⊂ { 1 , 2 } , d e g Q < m d e g f , g is not a square and f has imaginary roots, if it is not linear. Effective methods for the integration of Q / ( f m g are exhibited.

Invariance identity in the class of generalized quasiarithmetic means

Janusz Matkowski (2014)

Colloquium Mathematicae

An invariance formula in the class of generalized p-variable quasiarithmetic means is provided. An effective form of the limit of the sequence of iterates of mean-type mappings of this type is given. An application to determining functions which are invariant with respect to generalized quasiarithmetic mean-type mappings is presented.

Invariant graphs of functions for the mean-type mappings

Janusz Matkowski (2012)

ESAIM: Proceedings

Let I be a real interval, J a subinterval of I, p ≥ 2 an integer number, and M1, ... , Mp : Ip → I the continuous means. We consider the problem of invariance of the graphs of functions ϕ : Jp−1 → I with respect to the mean-type mapping M = (M1, ... , Mp).Applying a result on the existence and uniqueness of an M -invariant mean [7], we prove that if the graph of a continuous function ϕ : Jp−1 → I ...

Invariant scrambled sets and maximal distributional chaos

Xinxing Wu, Peiyong Zhu (2013)

Annales Polonici Mathematici

For the full shift (Σ₂,σ) on two symbols, we construct an invariant distributionally ϵ-scrambled set for all 0 < ϵ < diam Σ₂ in which each point is transitive, but not weakly almost periodic.

Investigation of smooth functions and analytic sets using fractal dimensions

Emma D'Aniello (2004)

Bollettino dell'Unione Matematica Italiana

We start from the following problem: given a function f : 0 , 1 → 0 , 1 what can be said about the set of points in the range where level sets are «big» according to an opportune definition. This yields the necessity of an analysis of the structure of level sets of C n functions. We investigate the analogous problem for C n , a functions. These are in a certain way intermediate between C n and C n + 1 functions. The results involve a mixture of Real Analysis, Geometric Measure Theory and Classical Descriptive Set Theory.

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