On the Invalidity of Fourier Series Expansions of Fractional Order

The purpose of this short paper is to show the invalidity of a Fourier series expansion of fractional order as derived by G. Jumarie in a series of papers. In his work the exponential functions $e^{in\omega x}$ are replaced by the Mittag-Leffler functions $E_\alpha \left (i (n\omega x)^\alpha\right) ,$ over the interval $[0, M_\alpha/ \omega]$ where $0< \omega<\infty$ and $M_\alpha$ is the period of the function $E_\alpha \left( ix^\alpha\right),$ i.e., $E_\alpha \left( ix^\alpha\right)=E_\alpha \left( i(x+M_\alpha)^\alpha\right).

The automorphism groups and derivation algebras of two-dimensional algebras

The automorphisms groups and derivation algebras of all two-dimensional algebras over algebraically closed fields are described.Comment: A type-mistake in $A_{3,2}(\mathbf{c})$ case, page 4, and its consequences are are correcte

Complete Classification of Two-Dimensional Algebras

A complete classification of two-dimensional algebras over algebraically closed fields is providedComment: A type-mistake in $A_{3,2}(\mathbf{c})$ case, page 7, is correcte