Quaternions, invented in 1843 by the famous mathematical physicist Hamilton, largely fell out of favor long ago, being replaced by vectors, which are direct descendants of quaternions. Quaternions have become popular recently because rotation operations are simpler with quaternions than with matrices. Therefore, quaternions are widely used in computer applications involving rotations. The relation between rotations and quaternions was established shortly after the invention of the latter, but recently it has been claimed that Hamilton misunderstood the relation, and that credit should be given to Rodrigues, who had published a paper on rotations in 1840. This claim has been repeated widely, but a review of original sources shows that it is based on an incomplete reading of Hamilton's work. The same review shows, however, that Gauss had derived results similar to those of Hamilton and Rodrigues in 1819, but they remained unpublished until 1900, and thus unknown to either of them. These matters are described in a self-contained way in this article.

In this work, some boundedness conditions on the coefficients of some general half-linear quasilinear elliptic equations are used to establish some oscillation criteria for those equations. Emphasis is put on strongly oscillatory conditions. This is obtained by using some comparison methods and Picone-type formulas.

In this paper, we use tools of potential theory to study the existence of positive continuous solutions for some boundary value problems based on the fractional Laplacian $\left( -\Delta \right) ^{\alpha /2},$ $0\lt\alpha \lt 2,$ $% \ $in an exterior domain $D$ in $ \mathbb{R}^{n}$, $n\geq 3$. Our arguments use properties of an appropriate Kato class of functions $K_{\alpha }^{\infty }\left( D\right) .$

We introduce a class of integral operators related to parametric Marcinkiewicz operators. We present a multiplier formula characterizing the $ L^{2}$ boundedness of such class of operators. Also, we prove $\mathcal{L} _{-\beta}^{p}$ (inhomogeneous Sobolev space)$\rightarrow L^{p}$ estimates provided that the kernels are in $L(\log L)(\mathbf{S}^{n-1})$. In fact, we show that the global parts of the introduced operators are bounded on the Lebesgue spaces $L^{p}(1\lt p \lt \infty )$ while the local parts are bounded on certain Sobolev spaces $\mathcal{L}_{-\beta }^{p}$.

In this paper, we establish fixed point theorems for contractive mappings on a metric space endowed with an amorphous binary relation. The presented theorems extend and generalize many existing results on metric and ordered metric spaces. We apply also our main result to derive fixed point theorems for cyclical contractive mappings.

In this paper, we establish an atomic decomposition characterization of weighted weak Hardy spaces $H^{p,\infty}_\omega$ on spaces of homogeneous type. As an application, we prove a interpolation theorem in $H^{p,\infty}_\omega$.

Let $A$ and $\tilde A$ be linear bounded operators in a separable Hilbert space, and $f$ be a function analytic on the closed convex hull of the spectra of $A$ and $\tilde A$. Let $SN_2$ and $SN_1$ be the ideals of Hilbert-Schmidt and nuclear operators, respectively. In the paper, a sharp estimate for the norm of $f(A)-f(\tilde A)$ is established, provided $A$ and $\tilde A$ have the so called Hilbert-Schmidt property. In particular, $A$ has the Hilbert-Schmidt property, if one of the following conditions holds: $A-A^*\in SN_2$, or $AA^*-I\in SN_1$. Here $A^*$ is adjoint to $A$, and $I$ is the unit operator. Our results are new even in the finite dimensional case.

We consider homomorphisms $H:G_1\longrightarrow G_2$ of holomorphic (group or pseudo-group) actions $G_1$ and $G_2$ on domains $\Omega_1$ and $\Omega_2$ respectively in $\bf C$, together with meromorphic functions $f$ that are compatible with these homomorphisms in the sense that \begin{equation} f(g(z))=H(g)(f(z))\nonumber \end{equation} for every $g\in G_1$ and $z\in\Omega_1$. Such situations are rooted in the cases of elliptic and modular functions, modular and automorphic forms, etc... We investigate various aspects of such cases, such as constructions and correspondences between families of functions compatible with different homomorphisms, that transform one family of functions compatible with one homomorphism to another one compatible with a different homomorphism.