Displaying 241 – 260 of 2163

Showing per page

An Expansion Formula for Fractional Derivatives and its Application

Atanackovic, T., Stankovic, B. (2004)

Fractional Calculus and Applied Analysis

An expansion formula for fractional derivatives given as in form of a series involving function and moments of its k-th derivative is derived. The convergence of the series is proved and an estimate of the reminder is given. The form of the fractional derivative given here is especially suitable in deriving restrictions, in a form of internal variable theory, following from the second law of thermodynamics, when applied to linear viscoelasticity of fractional derivative type.

An extension of a theorem of Marcinkiewicz and Zygmund on differentiability

S. Mukhopadhyay, S. Mitra (1996)

Fundamenta Mathematicae

Let f be a measurable function such that Δ k ( x , h ; f ) = O ( | h | λ ) at each point x of a set E, where k is a positive integer, λ > 0 and Δ k ( x , h ; f ) is the symmetric difference of f at x of order k. Marcinkiewicz and Zygmund [5] proved that if λ = k and if E is measurable then the Peano derivative f ( k ) exists a.e. on E. Here we prove that if λ > k-1 then the Peano derivative f ( [ λ ] ) exists a.e. on E and that the result is false if λ = k-1; it is further proved that if λ is any positive integer and if the approximate Peano derivative...

An inconsistency equation involving means

Roman Ger, Tomasz Kochanek (2009)

Colloquium Mathematicae

We show that any quasi-arithmetic mean A φ and any non-quasi-arithmetic mean M (reasonably regular) are inconsistent in the sense that the only solutions f of both equations f ( M ( x , y ) ) = A φ ( f ( x ) , f ( y ) ) and f ( A φ ( x , y ) ) = M ( f ( x ) , f ( y ) ) are the constant ones.

Currently displaying 241 – 260 of 2163