Let $(X,d)$ be a metric space and let $J\subseteq [0,\infty )$ be nonempty. We study the structure of the arbitrary intersections of Lipschitz algebras and define a special Banach subalgebra of ${{\cap }_{\gamma \in J\,}}\text{Li}{{\text{p}}_{\gamma \,}}X$ , denoted by $\text{ILi}{{\text{p}}_{J}}X$ . Mainly, we investigate the $C$ -character amenability of $\text{ILi}{{\text{p}}_{J}}X$ , in particular Lipschitz algebras. We address a gap in the proof of a recent result in this field. Then we remove this gap and obtain a necessary and sufficient condition for $C$ -character amenability of $\text{ILi}{{\text{p}}_{J}}X$ , specially Lipschitz algebras, under an additional assumption.

In this paper, we show that the Möbius invariant function space ${{\mathcal{Q}}_{p}}$ can be generated by variant Dirichlet type spaces ${{\mathcal{D}}_{\mu ,p}}$ induced by finite positive Borel measures $\mu $ on the open unit disk. A criterion for the equality between the space ${{\mathcal{D}}_{\mu ,p}}$ and the usual Dirichlet type space ${{\mathcal{D}}_{p}}$ is given. We obtain a sufficient condition to construct different ${{\mathcal{D}}_{\mu ,p}}$ spaces and provide examples. We establish decomposition theorems for ${{\mathcal{D}}_{\mu ,p}}$ spaces and prove that the non-Hilbert space ${{\mathcal{Q}}_{p}}$ is equal to the intersection of Hilbert spaces ${{\mathcal{D}}_{\mu ,p}}$ . As an application of the relation between ${{\mathcal{Q}}_{p}}$ and ${{\mathcal{D}}_{\mu ,p}}$ spaces, we also obtain that there exist different ${{\mathcal{D}}_{\mu ,p}}$ spaces; this is a trick to prove the existence without constructing examples.

We use the classical Perron envelope method to show a general existence theorem to degenerate complex Monge–Ampére type equations on compact Kähler manifolds.

Let $1\le p<\infty $ , and let $G$ be a discrete group. We give a sufficient and necessary condition for weighted translation operators on the Lebesgue space ${{\ell }^{p}}(G)$ to be densely disjoint hypercyclic. The characterization for the dual of a weighted translation to be densely disjoint hypercyclic is also obtained.

Let $R$ be a prime ring with extended centroid $\text{C,Q}$ maximal right ring of quotients of $R$ , $RC$ central closure of $R$ such that ${{\dim}_{C}}(RC)>4,f({{X}_{1}},...,{{X}_{n}})$ a multilinear polynomial over $C$ that is not central-valued on $R$ , and $f(R)$ the set of all evaluations of the multilinear polynomial $f({{X}_{1}},...,{{X}_{n}})$ in $R$ . Suppose that $G$ is a nonzero generalized derivation of $R$ such that ${{G}^{2}}(u)u\in C$ for all $u\in f(R)$ . Then one of the following conditions holds:

• (i) there exists $a\in \text{Q}$ such that ${{a}^{2}}=0$ and either $G(x)=ax$ for all $x\in R$ or $G(x)=xa$ for all $x\in R$ ;

• (ii) there exists $a\in \text{Q}$ such that $0\ne {{a}^{2}}\in C$ and either $G(x)=ax$ for all $x\in R$ or $G(x)=xa$ for all $x\in R$ and $f{{({{X}_{1}},...,{{X}_{n}})}^{2}}$ is central-valued on $R$ ;

• (iii) char $(R)=2$ and one of the following holds:

• (a) there exist $a,b,\in \text{Q}$ such that $G(x)=ax+xb$ for all $x\in R$ and ${{a}^{2}}={{b}^{2}}\in C$ ;

• (b) there exist $a,b,\in \text{Q}$ such that $G(x)=ax+xb$ for all $x\in R,\,{{a}^{2}},{{b}^{2}}\in C$ and $f{{({{X}_{1}},...,{{X}_{n}})}^{2}}$ is central-valued on $R$ ;

• (c) there exist $a\in \text{Q}$ and an $X$ -outer derivation $d$ of $R$ such that $G(x)=ax+d(x)$ for all $x\in R,{{d}^{2}}=0$ and ${{a}^{2}}+d(a)=0$ ;

• (d) there exist $a\in \text{Q}$ and an $X$ -outer derivation $d$ of $R$ such that $G(x)=ax+d(x)$ for all $x\in R,\,{{d}^{2}}=0,\,{{a}^{2}}+d(a)\in C$ and $f{{({{X}_{1}},...,{{X}_{n}})}^{2}}$ is central-valued on $R$ .

Moreover, we characterize the form of nonzero generalized derivations $G$ of $R$ satisfying ${{G}^{2}}(x)=\lambda x$ for all $x\in R$ , where $\lambda \in C$ .

Suppose $G$ is a connected complex Lie group and $H$ is a closed complex subgroup. Then there exists a closed complex subgroup $J$ of $G$ containing $H$ such that the fibration $\pi :G/H\to $ $G/J$ is the holomorphic reduction of $G/H$ i.e., $G/J$ is holomorphically separable and $\mathcal{O}(G/H)\cong $ ${{\pi }^{*}}\mathcal{O}(G/J)$ . In this paper we prove that if $G/H$ is pseudoconvex, i.e., if $G/H$ admits a continuous plurisubharmonic exhaustion function, then $G/J$ is Stein and $J/H$ has no non-constant holomorphic functions.

We study linear projections on Plücker space whose restriction to the Grassmannian is a non-trivial branched cover. When an automorphism of the Grassmannian preserves the fibers, we show that the Grassmannian is necessarily of $m$ -dimensional linear subspaces in a symplectic vector space of dimension $2m$ , and the linear map is the Lagrangian involution. The Wronski map for a self-adjoint linear diòerential operator and the pole placement map for symmetric linear systems are natural examples.

Fix an irreducible (finite) root system $R$ and a choice of positive roots. For any algebraically closed field $k$ consider the almost simple, simply connected algebraic group ${{G}_{k}}$ over $k$ with root system $k$ . One associates with any dominant weight $\lambda $ for $R$ two ${{G}_{k}}$ -modules with highest weight $\lambda $ , the Weyl module $V{{(\lambda )}_{k}}$ and its simple quotient $V{{(\lambda )}_{k}}$ . Let $\lambda $ and $\mu $ be dominant weights with $\mu <\lambda $ such that $\mu $ is maximal with this property. Garibaldi, Guralnick, and Nakano have asked under which condition there exists $k$ such that $L{{(\mu )}_{k}}$ is a composition factor of $V{{(\lambda )}_{k}}$ , and they exhibit an example in type ${{E}_{8}}$ where this is not the case. The purpose of this paper is to to show that their example is the only one. It contains two proofs for this fact: one that uses a classiffication of the possible pairs $(\lambda ,\mu )$ , and another that relies only on the classiûcation of root systems.

Alfred Schild has established conditions that Lorentz transformationsmap world-vectors $(ct,x,y,z)$ with integer coordinates onto vectors of the same kind. These transformations are called integral Lorentz transformations.

This paper contains supplements to our earlier work with a new focus on group theory. To relate the results to the familiar matrix group nomenclature, we associate Lorentz transformations with matrices in $\text{SL}(2,\mathbb{C})$ . We consider the lattice of subgroups of the group originated in Schild's paper and obtain generating sets for the full group and its subgroups.

A ${{C}^{*}}$ -algebra Ahas the ideal property if any ideal $I$ of $A$ is generated as a closed two-sided ideal by the projections inside the ideal. Suppose that the limit ${{C}^{*}}$ -algebra $A$ of inductive limit of direct sums of matrix algebras over spaces with uniformly bounded dimension has the ideal property. In this paper we will prove that $A$ can be written as an inductive limit of certain very special subhomogeneous algebras, namely, direct sum of dimension-drop interval algebras and matrix algebras over 2-dimensional spaces with torsion ${{H}^{2}}$ groups.

It is known that every Toeplitz matrix $T$ enjoys a circulant and skew circulant splitting (denoted $\text{CSCS}$ ) i.e., $T=C-S$ a circulant matrix and $S$ a skew circulant matrix. Based on the variant of such a splitting (also referred to as $\text{CSCS}$ ), we first develop classical $\text{CSCS}$ iterative methods and then introduce shifted $\text{CSCS}$ iterative methods for solving hermitian positive definite Toeplitz systems in this paper. The convergence of each method is analyzed. Numerical experiments show that the classical $\text{CSCS}$ iterative methods work slightly better than the Gauss–Seidel $(\text{GS})$ iterative methods if the $\text{CSCS}$ is convergent, and that there is always a constant $\alpha $ such that the shifted $\text{CSCS}$ iteration converges much faster than the Gauss–Seidel iteration, no matter whether the $\text{CSCS}$ itself is convergent or not.

In this paper, we obtain some characterizations of the (strong) Birkhoff–James orthogonality for elements of Hilbert ${{C}^{*}}$ -modules and certain elements of $\mathbb{B}\left( H \right)$ . Moreover, we obtain a kind of Pythagorean relation for bounded linear operators. In addition, for $T\in \mathbb{B}(H)$ we prove that if the norm attaining set ${{\mathbb{M}}_{T}}$ is a unit sphere of some finite dimensional subspace ${{H}_{0}}$ of $H$ and $||T|{{|}_{{{H}_{0}}\bot }}\,<\,\,||T||$ , then for every $S\in \mathbb{B}(H)$ , $T$ is the strong Birkhoff–James orthogonal to $S$ if and only if there exists a unit vector $\xi \in {{H}_{0}}$ such that $||T||\xi =\,|T|\xi $ and ${{S}^{*}}T\xi =0$ . Finally, we introduce a new type of approximate orthogonality and investigate this notion in the setting of inner product ${{C}^{*}}$ -modules.

It is known that a bi-orderable group has no generalized torsion element, but the converse does not hold in general. We conjecture that the converse holds for the fundamental groups of 3-manifolds and verify the conjecture for non-hyperbolic, geometric 3-manifolds. We also confirm the conjecture for some infinite families of closed hyperbolic 3-manifolds. In the course of the proof, we prove that each standard generator of the Fibonacci group $F(2,m)\,(m>2)$ is a generalized torsion element.

Let the measure algebra of a topological group $G$ be equipped with the topology of uniform convergence on bounded right uniformly equicontinuous sets of functions. Convolution is separately continuous on the measure algebra, and it is jointly continuous if and only if $G$ has the $\text{SIN}$ property. On the larger space $\text{LUC}{{(G)}^{*}}$ , which includes the measure algebra, convolution is also jointly continuous if and only if the group has the $\text{SIN}$ property, but not separately continuous for many non-SIN groups.

We investigate the role of the Kottman constant of a Banach space $X$ in the extension of $\alpha $ -Hölder continuous maps for every $\alpha \in (0,1]$ .

Let $R$ be a ring and $b,c\in R$ . In this paper, we give some characterizations of the $(b,c)$ -inverse in terms of the direct sum decomposition, the annihilator, and the invertible elements. Moreover, elements with equal $(b,c)$ -idempotents related to their $(b,c)$ -inverses are characterized, and the reverse order rule for the $(b,c)$ -inverse is considered.

In this paper, we construct two classes of rational function operators using the Poisson integrals of the function on the whole real axis. The convergence rates of the uniform and mean approximation of such rational function operators on the whole real axis are studied.

For any ring $R$ , we show that, in the bounded derived category ${{D}^{b}}(\text{Mod}\,R)$ of left $R$ -modules, the subcategory of complexes with finite Gorenstein projective (resp. injective) dimension modulo the subcategory of complexes with finite projective (resp. injective) dimension is equivalent to the stable category $\underline{\text{GP}}(\text{Mod}\,R)\,(resp.\overline{GI}(Mod\,R))$ of Gorenstein projective (resp. injective) modules. As a consequence, we get that if $R$ is a left and right noetherian ring admitting a dualizing complex, then $\underline{\text{GP}}(\text{Mod}\,R)$ and $\overline{\text{GI}}(\text{Mod}\,R)$ are equivalent.