Terms Of Multiple Functions Research Articles

A Variant of d’Alembert Functional Equation on Monoids

In this paper, we determine the complex-valued solutions of the functional equation $$\begin{aligned} f(x\sigma (y))+f(\tau (y)x)=2f(x)f(y) \end{aligned}$$for all \(x,y \in M\), where M is a monoid, \(\sigma \): \(M\longrightarrow M\) is an involutive automorphism and \(\tau \): \(M\longrightarrow M\) is an involutive anti-automorphism. The solutions are expressed in terms of multiplicative functions, and characters of 2-dimensional irreducible representations of M.

The small dimension lemma and d’Alembert’s equation on semigroups

We derive an algebraic version over an algebraically closed field k with characteristic $$\ne 2$$ of Yang’s Small Dimension Lemma. With its help we describe the k-valued solutions of d’Alembert’s functional equation on semigroups S in terms of multiplicative functions and irreducible, 2-dimensional representations of S.

The cosine addition law with an additional term

We find the solutions of the cosine addition law with an additional term, including the case of the cosine addition law on semigroups. They are expressed in terms of multiplicative functions and solutions of the sine addition law, apart from an obvious exception. On groups multiplicative and additive functions suffice. As an application we solve extensions of Butler’s problem on semigroups.

Van Vleck’s functional equation for the sine

We solve Van Vleck’s functional equation on semigroups with an involution in terms of multiplicative functions.

Crosslinking of 4.5S RNA to the Escherichia coli ribosome in the presence or absence of the protein Ffh.

Radioactively labeled 4.5S RNA containing statistically distributed 4-thiouridine residues in place of normal uridine was prepared by T7 transcription. The ability of this modified 4.5S RNA to form a complex with the protein Ffh was demonstrated by a gel shift assay. The modified 4.5S RNA, with or without Ffh, was added to Escherichia coli ribosomes under various conditions, and crosslinking from the thiouridine residues was induced by irradiation at 350 nm. The crosslinked ribosomal components were analyzed by our standard procedures. Two clearly defined types of crosslinking were observed. The first was a crosslink to 23S rRNA, which was entirely dependent both on the presence of Ffh and a nascent protein chain in the 50S subunit. This crosslink was localized to nt approximately 2828-2837 of the 23S rRNA, from position 84 of the 4.5S molecule. The second type of crosslinking, to the 30S ribosomal subunit, was independent of the presence of Ffh, and was found both with vacant 70S ribosomes or isolated 30S subunits. Here the crosslink was localized to the 3'-terminal region of the 16S rRNA, from positions 29-50 of the 4.5S RNA. Cross-linking to ribosomal protein S1 was also observed. The known crystal structure of the protein Ffh/4.5S RNA fragment complex was extrapolated by computer modeling so as to include the whole 4.5S molecule, and this was docked onto the ribosome using the crosslinking data. The results are discussed in terms of multiple functions and binding sites of the 4.5S RNA.