We first introduce the weighted averaged projection sequence in $\text{CAT}(\unicode[STIX]{x1D705})$ spaces and then we establish some inequalities for the weighted averaged projection sequence. Using the inequalities, we prove the asymptotic regularity and the $\unicode[STIX]{x1D6E5}$-convergence of the weighted averaged projection sequence. Furthermore, we prove the strong convergence of the sequence under certain regularity or compactness conditions on $\text{CAT}(\unicode[STIX]{x1D705})$ spaces.

We investigate the assumptions under which a subclass of flat quasicoherent sheaves on a quasicompact and semiseparated scheme allows us to ‘mock’ the homotopy category of projective modules. Our methods are based on module-theoretic properties of the subclass of flat modules involved as well as their behaviour with respect to Zariski localizations. As a consequence we get that, for such schemes, the derived category of flat quasicoherent sheaves is equivalent to the derived category of very flat quasicoherent sheaves. If, in addition, the scheme satisfies the resolution property then both derived categories are equivalent to the derived category of infinite-dimensional vector bundles. The equivalences are inferred from a Quillen equivalence between the corresponding models.

Let ${\mathcal{A}}$ be a semisimple Banach algebra with minimal left ideals and $\text{soc}({\mathcal{A}})$ be the socle of ${\mathcal{A}}$. We prove that if $\text{soc}({\mathcal{A}})$ is an essential ideal of ${\mathcal{A}}$, then every 2-local derivation on ${\mathcal{A}}$ is a derivation. As applications of this result, we can easily show that every 2-local derivation on some algebras, such as semisimple modular annihilator Banach algebras, strongly double triangle subspace lattice algebras and ${\mathcal{J}}$-subspace lattice algebras, is a derivation.

The Douglas–Rachford method is a splitting method frequently employed for finding zeros of sums of maximally monotone operators. When the operators in question are normal cone operators, the iterated process may be used to solve feasibility problems of the following form: Find $x\in \bigcap _{k=1}^{N}S_{k}$. The success of the method in the context of closed, convex, nonempty sets $S_{1},\ldots ,S_{N}$ is well known and understood from a theoretical standpoint. However, its performance in the nonconvex context is less well understood, yet it is surprisingly impressive. This was particularly compelling to Jonathan M. Borwein who, intrigued by Elser, Rankenburg and Thibault’s success in applying the method to solving sudoku puzzles, began an investigation of his own. We survey the current body of literature on the subject, and we summarize its history. We especially commemorate Professor Borwein’s celebrated contributions to the area.

Our aim in this paper is to establish a generalization of Sobolev’s inequality for Riesz potentials $J_{\unicode[STIX]{x1D6FC}(\cdot )}^{\unicode[STIX]{x1D70E}}f$ of functions $f$ in Musielak–Orlicz–Morrey spaces $L^{\unicode[STIX]{x1D6F7},\unicode[STIX]{x1D705}}(X)$. As a corollary we obtain Sobolev’s inequality for double phase functionals with variable exponents.

This paper is devoted to the study of the normal (tangential) regularity of a closed set and the subdifferential (directional) regularity of its distance function in the context of Riemannian manifolds. The Clarke, Fréchet and proximal subdifferentials of the distance function from a closed subset in a Riemannian manifold are represented by corresponding normal cones of the set.

Let $F_{n}$ and $L_{n}$ be the Fibonacci and Lucas numbers, respectively. Four corresponding zeta functions in $s$ are defined by

$$\begin{eqnarray}\unicode[STIX]{x1D701}_{F}(s):=\mathop{\sum }_{n=1}^{\infty }{\displaystyle \frac{1}{F_{n}^{s}}},\quad \unicode[STIX]{x1D701}_{F}^{\ast }(s):=\mathop{\sum }_{n=1}^{\infty }{\displaystyle \frac{(-1)^{n+1}}{F_{n}^{s}}},\quad \unicode[STIX]{x1D701}_{L}(s):=\mathop{\sum }_{n=1}^{\infty }{\displaystyle \frac{1}{L_{n}^{s}}},\quad \unicode[STIX]{x1D701}_{L}^{\ast }(s):=\mathop{\sum }_{n=1}^{\infty }{\displaystyle \frac{(-1)^{n+1}}{L_{n}^{s}}}.\end{eqnarray}$$

$R(\unicode[STIX]{x1D70C}),Q(\unicode[STIX]{x1D70C}),$

$P(\unicode[STIX]{x1D70C})$

$\unicode[STIX]{x1D70C}$

$0<\unicode[STIX]{x1D70C}<1$

$s$

$m$

$$\begin{eqnarray}\mathop{\sum }_{s=1}^{m}(t_{s}\unicode[STIX]{x1D701}_{F}(2s)+u_{s}\unicode[STIX]{x1D701}_{F}^{\ast }(2s)+v_{s}\unicode[STIX]{x1D701}_{L}(2s)+w_{s}\unicode[STIX]{x1D701}_{L}^{\ast }(2s))=0\end{eqnarray}$$

$t_{s},u_{s},v_{s},w_{s}\in \mathbb{Q}$

$(1\leq s\leq m)$

$\mathbb{Q}$

$m$

$-2\unicode[STIX]{x1D701}_{F}(2)+\unicode[STIX]{x1D701}_{F}^{\ast }(2)+5\unicode[STIX]{x1D701}_{L}^{\ast }(2)=0$

$2s$