Let Mn be the algebra of n × n matrices over an algebraically closed field of characteristic zero. For A ∊ Mn, denote by the collection of all matrices in Mn that are similar to A. In this paper we characterize those invertible linear operators ϕ on Mn that satisfy , where for some given A1,..., Ak ∊ Mn and denotes the (Zariski) closure of S. Our theorem covers a result of Howard on linear operators mapping the set of matrices annihilated by a given polynomial into itself, and extends a result of Chan and Lim on linear operators commuting with the function f(x) = xk for a given positive integer k ≥ 2. The possibility of weakening the invertibility assumption in our theorem is considered, a partial answer to a conjecture of Howard is given, and some extensions of our result to arbitrary fields are discussed.

In this paper we study left invariant Einstein–Randers metrics on compact Lie groups. First, we give a method to construct left invariant non-Riemannian Einstein–Randers metrics on a compact Lie group, using the Zermelo navigation data. Then we prove that this gives a complete classification of left invariant Einstein–Randers metrics on compact simple Lie groups with the underlying Riemannian metric naturally reductive. Further, we completely determine the identity component of the group of isometries for this type of metrics on simple groups. Finally, we study some geometric properties of such metrics. In particular, we give the formulae of geodesics and flag curvature of such metrics.

In [2], A. Sade gives a construction for quasigroups which he calls the singular direct product. In this paper we generalize Sades' construction. As an application we obtain a recursive construction for quasigroups orthogonal to their transposes. All quasigroups considered in this paper will be finite.

Let $H$ be a complex Hilbert space, and $\mathcal{H}\left( H \right)$ be the real linear space of bounded selfadjoint operators on $H$. We study linear maps $\phi :\,\mathcal{H}(H)\,\to \,\mathcal{H}(H)$ leaving invariant various properties such as invertibility, positive definiteness, numerical range, etc. The maps $\phi$ are not assumed a priori continuous. It is shown that under an appropriate surjective or injective assumption $\phi$ has the form $X\mapsto \xi TX{{T}^{*}}orX\mapsto \xi T{{X}^{t}}{{T}^{*}}$, for a suitable invertible or unitary $T$ and $\xi \,\in \,\{1,\,-1\}$, where ${{X}^{t}}$ stands for the transpose of $X$ relative to some orthonormal basis. Examples are given to show that the surjective or injective assumption cannot be relaxed. The results are extended to complex linear maps on the algebra of bounded linear operators on $H$. Similar results are proved for the (real) linear space of (selfadjoint) operators of the form $\alpha I\,+\,K$, where $\alpha$ is a scalar and $K$ is compact.

Let $(X,\,d)$ be a metric space, and let $\text{Lip(}X\text{)}$ denote the Banach space of all scalar-valued bounded Lipschitz functions $f$ on $X$ endowed with one of the natural norms

$$\left\| f \right\|\,=\,\max \{{{\left\| f \right\|}_{\infty }},\,L(f)\}\,\,\text{or}\,\,\left\| f \right\|\,=\,{{\left\| f \right\|}_{\infty }}\,+\,L(f),$$

where $L(f)$ is the Lipschitz constant of $f$. It is said that the isometry group of $\text{Lip(}X\text{)}$ is canonical if every surjective linear isometry of $\text{Lip(}X\text{)}$ is induced by a surjective isometry of $X$. In this paper we prove that if $X$ is bounded separable and the isometry group of $\text{Lip(}X\text{)}$ is canonical, then every 2-local isometry of $\text{Lip(}X\text{)}$ is a surjective linear isometry. Furthermore, we give a complete description of all 2-local isometries of $\text{Lip(}X\text{)}$ when $X$ is bounded.

Let α be a positive irrational real number, and let fα (n) = [(n + l)α] — [nα] — [α],n > 1, where [x] denotes the greatest integer not exceeding x. It is shown that the sequence fα has a certain 'substitution property' if and only if α is the root of a quadratic equation over the rationals.

We show that under reasonable restrictions on the metric spaces X and Y, every surjective isometric isomorphism between Lip(X) and Lip(Y) arises in a simple manner from an isometry between X and Y. Our result differs from several previous results along these lines in that we do not require X and Y to be compact.

We consider the self-adjoint second-order scalar difference equation (1) Δ(rnΔxn) +pnXn+1 = 0 and the matrix system (2) Δ(RnΔXn) + PnXn+1 = 0, where are seQuences of real numbers (d x d Hermitian matrices) with rn > 0(Rn > 0). The oscillation and nonoscillation criteria for solutions of (1) and (2), obtained in [3, 4, 10], are extended to a much wider class of equations by Riccati and averaging techniques.

In this paper we prove that there is no nontrivial ${{L}^{q}}$-integrably $p$-harmonic 1-form on a complete manifold with nonnegatively Ricci curvature $\left( 0\,<\,q\,<\,\infty \right)$.

We consider two dynamical systems associated with a substitution of Pisot type: the usual $\mathbb{Z}$-action on a sequence space, and the $\mathbb{R}$-action, which can be defined as a tiling dynamical system or as a suspension flow. We describe procedures for checking when these systems have pure discrete spectrum (the “balanced pairs algorithm” and the “overlap algorithm”) and study the relation between them. In particular, we show that pure discrete spectrum for the $\mathbb{R}$-action implies pure discrete spectrum for the $\mathbb{Z}$-action, and obtain a partial result in the other direction. As a corollary, we prove pure discrete spectrum for every $\mathbb{R}$-action associated with a two-symbol substitution of Pisot type (this is conjectured for an arbitrary number of symbols).

By a tournament we mean the outcome of a round-robin tournament in which there are no draws. Such a tournament may be represented by a graph in which the n players are represented by vertices labelled 1, 2, …, n, and the outcomes of the games are represented by directed edges so that every pair of vertices is joined by one directed edge. We call such a graph a complete directed graph. One can also represent such a tournament by an nXn matrix T=(tij) in which tij is 1 if i beats j, and 0 otherwise, so that T is a (0, 1) matrix with tij + tji = 1 for i≠j and (by definition) tii = 0.

We use the Carleson measure-embedding theorem for weighted Bergman spaces to characterize the positive Borel measures $\unicode[STIX]{x1D707}$ on the unit disc such that certain analytic function spaces of Dirichlet type are embedded (compactly embedded) in certain tent spaces associated with a measure $\unicode[STIX]{x1D707}$. We apply these results to study Volterra operators and multipliers acting on the mentioned spaces of Dirichlet type.

Two types of pseudo-bases, σ-disjoint and σ-discrète, are utilized in this note. In the next section, we show that a first countable Hausdorff space has a σ-disjoint pseudo-base if and only if it has a dense metrizable subspace. This result implies that many first countable spaces have dense metrizable subspaces. In section 3, we show that if X is a Hausdorff space that either is quasi-developable or has a base of countable order, then X has a dense metrizable subspace if and only if it has a dense metrizable Gδ subspace. We give an example to show that the conclusion of this theorem is false for semi-metrizable spaces. Finally, in the last section, we investigate when a quasi-developable (resp. semi-metrizable) space can be embedded as a dense subspace of a quasi-developable (resp. semi-metrizable) Baire space.

How many spheres of given diameter can be packed in a cube of given size? Or: What is the maximum diameter of k identical spheres if they can be packed in a cube of given size? These questions are obviously equivalent to the following problem:

Let d(Pi, Pj) denote the distance between the points Pi and Pj, and Γk the set of all configurations of k points Pi (1 ≤ i < j ≤ k) in a closed unit cube C.

In this paper we consider two natural notions of connectivity for hypergraphs: weak and strong. We prove that the strong vertex connectivity of a connected hypergraph is bounded by its weak edge connectivity, thereby extending a theorem of Whitney from graphs to hypergraphs. We find that, while determining a minimum weak vertex cut can be done in polynomial time and is equivalent to finding a minimum vertex cut in the 2-section of the hypergraph in question, determining a minimum strong vertex cut is NP-hard for general hypergraphs. Moreover, the problem of finding minimum strong vertex cuts remains NP-hard when restricted to hypergraphs with maximum edge size at most 3. We also discuss the relationship between strong vertex connectivity and the minimum transversal problem for hypergraphs, showing that there are classes of hypergraphs for which one of the problems is NP-hard, while the other can be solved in polynomial time.

A q-analogue of the orthogonality property of the Bessel functions on the zeros is obtained in terms of a q-integral.

The ${{Q}_{p}}$ spaces coincide with the Bloch space for $p\,>\,1$ and are subspaces of $\text{BMOA}$ for $0\,<\,p\,\le \,1$. We obtain lower and upper estimates for the essential norm of a composition operator from the Bloch space into ${{Q}_{p}}$, in particular from the Bloch space into $\text{BMOA}$.

This paper deals with the projective objects in the category of all Z-frames, where the latter is a common generalization of different types of frames. The main result obtained here is that a Z-frame is E-projective if and only if it is stably Z-continuous, for a naturally arising collection E of morphisms.

We present a new approach to noncommutative real algebraic geometry based on the representation theory of ${{C}^{*}}$-algebras. An important result in commutative real algebraic geometry is Jacobi's representation theorem for archimedean quadratic modules on commutative rings. We show that this theorem is a consequence of the Gelfand–Naimark representation theorem for commutative ${{C}^{*}}$-algebras. A noncommutative version of Gelfand–Naimark theory was studied by I. Fujimoto. We use his results to generalize Jacobi's theorem to associative rings with involution.

The study of general multiplier theorems (Kuhn-Tucker Conditions) for constrained optimization problems has led to extensions of the notion of a differentiable arc. Abadie [1], Varaiya [10], Guignard [5], Zlobec [11] and Massam [12] investigated the so called cone of tangent vectors to a point in a set for optimization purposes.