This paper generalizes the Aleksandrov problem: the Mazur-Ulam theorem on $n-G$-quasi normed spaces. It proves that a one-$n$-distance preserving mapping is an $n$-isometry if and only if it has the zero-$n-G$-quasi preserving property, and two kinds of $n$-isometries on $n-G$-quasi normed space are equivalent; we generalize the Benz theorem to $n$-normed spaces with no restrictions on the dimension of spaces.

Nuij's theorem states that if a polynomial $p\in \mathbb{R}[z]$ is hyperbolic (i.e., has only real roots), then $p+s{{p}^{'}}$ is also hyperbolic for any $s\in \mathbb{R}$. We study other perturbations of hyperbolic polynomials of the form ${{p}_{a}}(z,s)\,\,:=\,\,\,p\,(z)+\,\sum\nolimits_{k=1}^{d}{{{a}_{k}}{{s}^{k}}{{p}^{(k)}}(z)}$. We give a full characterization of those $a=({{a}_{1}},...,{{a}_{d}})\,\in \,{{\mathbb{R}}^{d}}$ for which ${{p}_{a}}(z,s)$ is a pencil of hyperbolic polynomials. We also give a full characterization of those $a=({{a}_{1}},...,{{a}_{d}})\,\in \,{{\mathbb{R}}^{d}}$ for which the associated families $ $ admit universal determinantal representations. In fact, we show that all these sequences come fromspecial symmetric Toeplitz matrices.

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.

We characterize two important notions of amenability and compactness of a locally compact quantum group $\mathbb{G}$ in terms of certain homological properties. For this, we show that $\mathbb{G}$ is character amenable if and only if it is both amenable and co-amenable. We finally apply our results to Arens regularity problems of the quantum group algebra ${{L}^{1}}\left( \mathbb{G} \right)$. In particular, we improve an interesting result by Hu, Neufang, and Ruan.

Let $\mathbf{S}:={{\{S(t)\}}_{t\ge 0}}$ be a ${{\text{C}}_{0}}$-semigroup of quasinilpotent operators (i.e., $\sigma (S(t))=\{0\}$ for each $t>0$). In dynamical systems theory the above quasinilpotency property is equivalent to a very strong concept of stability for the solutions of autonomous systems. This concept is frequently called superstability and weakens the classical finite time extinction property (roughly speaking, disappearing solutions). We show that under some assumptions, the quasinilpotency, or equivalently, the superstability property of a ${{\text{C}}_{0}}$-semigroup is preserved under the perturbations of its infinitesimal generator.

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.

This paper provides a constructive proof of the weak factorization of the classical Hardy space ${{H}^{1}}({{\mathbb{R}}^{n}})$ in terms of multilinear Riesz transforms. As a direct application, we obtain a new proof of the characterization of $BMO({{\mathbb{R}}^{n}})$ (the dual of ${{H}^{1}}({{\mathbb{R}}^{n}})$) via commutators of the multilinear Riesz transforms.

In this paper, we establish $BMO$ estimates for generalized commutators of rough fractional maximal and integral operators on generalized weighted Morrey spaces, respectively.

In this paper we investigate the endpoint regularity properties of the multisublinear fractional maximal operators, which include the multisublinear Hardy–Littlewood maximal operator. We obtain some new bounds for the derivative of the one-dimensional multisublinear fractional maximal operators acting on the vector-valued function $\overrightarrow{f}=({{f}_{1}},...,{{f}_{m}})$ with all ${{f}_{j}}$ being $BV$-functions.

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.

It is shown that the Dirichlet problem for the slab $\left( a,\,b \right)\,\times \,{{\mathbb{R}}^{d}}$ with entire boundary data has an entire solution. The proof is based on a generalized Schwarz reflection principle. Moreover, it is shown that for a given entire harmonic function $g$, the inhomogeneous difference equation $h\left( t\,+\,1,\,y \right)\,-\,h\left( t,\,y \right)\,=\,g\left( t,\,y \right)$ has an entire harmonic solution $h$.

The notion of coaxers is introduced in a pseudo-complemented distributive lattice. Boolean algebras are characterized in terms of coaxer ideals and congruences. The concept of coaxer lattices is introduced in pseudo-complemented distributive lattices and characterized in terms of coaxer ideals and maximal ideals. Finally, the coaxer lattices are also characterized in topological terms.

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 prove Wise's $W$-cycles conjecture. Consider a compact graph $\Gamma '$ immersing into another graph $\Gamma $. For any immersed cycle $\Lambda :{{S}^{1}}\to \Gamma $, we consider the map $\Lambda '$ from the circular components $\mathbb{S}$ of the pullback to $\Gamma '$. Unless $\Lambda '$ is reducible, the degree of the covering map $\mathbb{S}\to {{S}^{1}}$ is bounded above by minus the Euler characteristic of $\Gamma '$. As a corollary, any finitely generated subgroup of a one-relator group has a finitely generated Schur multiplier.

The chromatic functor of a simple graph is a functorization of the chromatic polynomial. M. Yoshinaga showed that two finitely graphs have isomorphic chromatic functors if and only if they have the same chromatic polynomial. The key ingredient in the proof is the use of stable partitions of graphs. The latter is shown to be closely related to chromatic functors. In this note, we further investigate some interesting properties of chromatic functors associated with simple graphs using stable partitions. Our first result is the determination of the group of natural automorphisms of the chromatic functor, which is, in general, a larger group than the automorphism group of the graph. The second result is that the composition of the chromatic functor associated with a finitely graph restricted to the category FI of finitely sets and injections with the free functor into the category of complex vector spaces yields a consistent sequence of representations of symmetric groups that is representation stable in the sense of Church–Farb.

In this paper we give the bifurcation diagram of the family of cubic vector fields $\dot{z}=$${{z}^{3}}+{{\epsilon }_{1}}z+{{\epsilon }_{0}}$ for $z\in \mathbb{C}{{\mathbb{P}}^{1}}$, depending on the values of ${{\epsilon }_{1}},{{\epsilon }_{0}}\in \mathbb{C}$. The bifurcation diagram is in ${{\mathbb{R}}^{^{4}}}$, but its conic structure allows describing it for parameter values in ${{\mathbb{S}}^{3}}$. There are two open simply connected regions of structurally stable vector fields separated by surfaces corresponding to bifurcations of homoclinic connections between two separatrices of the pole at infinity. These branch from the codimension 2 curve of double singular points. We also explain the bifurcation of homoclinic connection in terms of the description of Douady and Sentenac of polynomial vector fields.

Let $G$ be a finite group and let $A\left( G \right)$ denote the Burnside ring of $G$. Then an inverse limit $L\left( G \right)$ of the groups $A\left( H \right)$ for proper subgroups $H$ of $G$ and a homomorphism res from $A\left( G \right)$ to $L\left( G \right)$ are obtained in a natural way. Let $Q\left( G \right)$ denote the cokernel of res. For a prime $p$, let $N\left( p \right)$ be the minimal normal subgroup of $G$ such that the order of ${G}/{N}\;\left( p \right)$ is a power of $p$, possibly 1. In this paper we prove that $Q\left( G \right)$ is isomorphic to the cartesian product of the groups $Q\left( {G}/{N\left( p \right)}\; \right)$, where $p$ ranges over the primes dividing the order of $G$.

The characteristic polynomial ${{P}_{A}}({{x}_{0}},...,{{x}_{r}})$ of an $r$-tuple $A\,:=({{A}_{1}},...,{{A}_{r}})$ of $n\times n$-matrices is defined as

1

$${{P}_{A}}({{x}_{0}},...,{{x}_{r}}):=\det ({{x}_{0}}I+{{x}_{1}}{{A}_{1}}+\ldots +{{x}_{r}}{{A}_{r}}).$$

We show that if $r\,\,3$ and $A\,:=({{A}_{1}},...,{{A}_{r}})$ is an $r$-tuple of $n\times n$-matrices in general position, then up to conjugacy, there are only finitely many $r$-tuples $A'\,:=(A_{1}^{'},...,A_{r}^{'})$ such that ${{p}_{A}}={{p}_{A'}}$. Equivalently, the locus of determinantal hypersurfaces of degree $n$ in ${{\text{P}}^{r}}$ is irreducible of dimension $(r-1){{n}^{2}}+1$.

Let $K$ be an ultraspherical hypergroup associated with a locally compact group $G$ and a spherical projector $\pi$ and let $\text{VN}(K)$ denote the dual of the Fourier algebra $A(K)$ corresponding to $K$. In this note, we show that the set of invariant means on $\text{VN}(K)$ is singleton if and only if $K$ is discrete. Here $K$ need not be second countable. We also study invariant means on the dual of the Fourier algebra ${{A}_{0}}(K)$, the closure of $A(K)$ in the cb-multiplier norm. Finally, we consider generalized translations and generalized invariant means.

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