Surjective Map Research Articles

A remark on non-surjective composition operators

Assume that A, B are uniform algebras on compact Hausdorff spaces X and Y, respectively, and ∂A, ∂B are the Šilov boundaries of A, B. Let T : A −1 → B− 1 be a map with T1 = 1. We show that, if there exist constants α, β ≥ 1 such that β− 1‖f·g− 1‖ ≤ ‖Tf·(Tg) − 1∥ ≤ α∥f·g− 1∥ for all f, g ∈ A− 1, then there is a non-empty closed subset Y 0 of ∂B and a surjective continuous map τ : Y 0 → ∂A such that for all f ∈ A− 1 and all y ∈ Y 0. Moreover we give an example which shows that the multiple α 2 β in the above inequality is the best possible.

Factorizations of surjective maps of connected quandles

We consider the problem of when one quandle homomorphism will factor through another, restricting our attention to the case where all quandles involved are connected. We provide a complete solution to the problem for surjective quandle homomorphisms using the structure theorem for connected quandles of Ehrman et al. (2008) and the factorization system for surjective quandle homomorphsims of Bunch et al. (2010) as our primary tools. The paper contains the substantive results obtained by an REU research group consisting of the first four authors under the mentorship of the fifth, and was supported by National Science Foundation, grant DMS-1659123.

On almost Hurewicz property in bitopological spaces

The purpose of this paper is to investigate properties and results of the almost Hurewicz property in bitopological spaces. We characterize this property using $(i, j)$- regular open sets and almost continuous surjective map.

Extension of phase-isometries between the unit spheres of complex lp(Γ)-spaces (p>1)

Let Γ, ∆ be nonempty index sets. For p ∈ (1, ∞), we prove that every surjective mapping f: Slp(Γ) → Slp(∆) satisfying the functional equation {||f(x) + f(y)||, ||f(x)− f(y)||} = {||x+y||, ||x−y||} (x, y ∈ Slp(Γ)), its positive homogeneous extension is a phase-isometry which is phase equivalent a real linear isometry.

Lévy processes on smooth manifolds with a connection

We define a Lévy process on a smooth manifold M with a connection as a projection of a solution of a Marcus stochastic differential equation on a holonomy bundle of M, driven by a holonomy-invariant Lévy process on a Euclidean space. On a Riemannian manifold, our definition (with Levi-Civita connection) generalizes the Eells-Elworthy-Malliavin construction of the Brownian motion and extends the class of isotropic Lévy process introduced in Applebaum and Estrade [3]. On a Lie group with a surjective exponential map, our definition (with left-invariant connection) coincides with the classical definition of a (left) Lévy process given in terms of its increments. Our main theorem characterizes the class of Lévy processes via their generators on M, generalizing the fact that the Laplace-Beltrami operator generates Brownian motion on a Riemannian manifold. Its proof requires a path-wise construction of the stochastic horizontal lift and anti-development of a discontinuous semimartingale, leading to a generalization of Pontier and Estrade [32] to smooth manifolds with non-unique geodesics between distinct points.

Lie maps on alternative rings preserving idempotents

Let $\Re$ and $\Re'$ unital $2$,$3$-torsion free alternative rings and $\varphi: \Re \rightarrow \Re'$ be a surjective Lie multiplicative map that preserves idempotents. Assume that $\Re$ has a nontrivial idempotents. Under certain assumptions on $\Re$, we prove that $\varphi$ is of the form $\psi + \tau$, where $\psi$ is either an isomorphism or the negative of an anti-isomorphism of $\Re$ onto $\Re'$ and $\tau$ is an additive mapping of $\Re$ into the centre of $\Re'$ which maps commutators into zero.

Preservers of partial orders on the set of all variance-covariance matrices

Let H+n(R) be the cone of all positive semidefinite n x n real matrices. Two of the best known partial orders that were mostly studied on subsets of square complex matrices are the L?wner and the minus partial orders. Motivated by applications in statistics we study these partial orders on H+ n (R). We describe the form of all surjective maps on H+ n (R), n > 1, that preserve the L?wner partial order in both directions. We present an equivalent definition of the minus partial order on H+ n (R) and also characterize all surjective, additive maps on H+ n (R), n ? 3, that preserve the minus partial order in both directions.

Extension of phase-isometries between the unit spheres of complex L∞(Γ) spaces

Let Γ be nonempty index set, and X, Y are complex L∞(Γ)-type spaces. f: SX, SY will denote their unit spheres. Give a surjective mapping f: SX → SY satisfying the functional equation {||f(x) + f(y)||, ||f(x)− f(y)||} = {||x+y||, ||x−y||} (x, y ∈ SX ) We show that there exists a function ε: SX → {−1,1} such that ε f is an isometry. Moreover, this isometry is the restriction of a real linear isometry from X to Y.

A Remark for the Hyers–Ulam Stabilities on n-Banach Spaces

In this article, we deal with stabilities of several functional equations in n-Banach spaces. For a surjective mapping f into a n-Banach space, we prove the generalized Hyers–Ulam stabilities of the cubic functional equation and the quartic functional equation for f in n-Banach spaces.

Uniformly positive entropy of induced transformations

AbstractLet $(X,T)$ be a topological dynamical system consisting of a compact metric space X and a continuous surjective map $T : X \to X$ . By using local entropy theory, we prove that $(X,T)$ has uniformly positive entropy if and only if so does the induced system $({\mathcal {M}}(X),\widetilde {T})$ on the space of Borel probability measures endowed with the weak* topology. This result can be seen as a version for the notion of uniformly positive entropy of the corresponding result for topological entropy due to Glasner and Weiss.

Phase-isometries between normed spaces

Let X and Y be real normed spaces and f:X→Y a surjective mapping. Then f satisfies {‖f(x)+f(y)‖,‖f(x)−f(y)‖}={‖x+y‖,‖x−y‖}, x,y∈X, if and only if f is phase equivalent to a surjective linear isometry, that is, f=σU, where U:X→Y is a surjective linear isometry and σ:X→{−1,1}. This is a Wigner's type result for real normed spaces.

Maps preserving the $c$-numerical radius of products for operators in $\mathfrak{B}(H)$

Let 𝔅(ℋ) and 𝔅s(ℋ) be the algebra of all bounded linear operators on a complex Hilbert space ℋ and the real Jordan algebra of all self-adjoint operators on ℋ, respectively. Suppose 𝒲 and 𝒱 are subsets of 𝔅(ℋ) or 𝔅s(ℋ) and F(⋅) is the c-numerical radius or the diameter of an operator. Under an assumption on 𝒲 and 𝒱, characterizations are obtained for surjective maps Φ:𝒲→𝒱 satisfying F ( A B ) = F ( Φ ( A ) Φ ( B ) ) ( A , B ∈ 𝒲 ) . When dimℋ≥3, to establish the proofs, some general results are obtained for functions F:𝔅(ℋ)→[0,+∞) satisfying (P1) F(UAU∗)=F(A) for all A∈𝔅(ℋ) and unitary U on ℋ; (P2) For A∈𝔅(ℋ), F(A)=0 if and only if A is a multiple of the identity; (P3) There are nonnegative real numbers α,β with α2+β2≠0 such that F(T)=α∥T∥+β|tr(T)| for each rank-one T∈𝔅(ℋ).

Sandwiched Rényi relative entropy on density operators

Relative entropies play important roles in classical and quantum information theory. In this paper, we discuss the sandwiched Rényi relative entropy for [Formula: see text] on [Formula: see text] (the cone of positive trace-class operators acting on an infinite-dimensional complex Hilbert space [Formula: see text]) and characterize all surjective maps preserving the sandwiched Rényi relative entropy on [Formula: see text]. Such transformations have the form [Formula: see text] for each [Formula: see text], where [Formula: see text] and [Formula: see text] is either a unitary or an anti-unitary operator on [Formula: see text]. Particularly, all surjective maps preserving sandwiched Rényi relative entropy on [Formula: see text] (the set of all quantum states on [Formula: see text]) are necessarily implemented by either a unitary or an anti-unitary operator.

Types of Expansivity on Bi-Shadowing Property

Let a continuous maps f:X → X, g:X → X be defined on a compact metric space X. We showed that f has bi-shadowing with g is equivalent to f has backwards bi-shadowing with g and f has two-sided bi-shadowing with g when the maps f, g are onto. We used this result to go on prove that, for expansive surjective maps f, g the properties f has bi-shadowing with g, f has two-sided bi-shadowing with g, f has s- limit bi-shadowing with g and f has two-sided s-limit bi-shadowing with g are equivalent. Finally we concluded that if a continuous maps f, g are surjective. Then f has two-sided s-limit bi-shadowing with g if and only if it has L-bi-shadowing with g.

Holomorphic Bundles Trivializable by Proper Surjective Holomorphic Map

AbstractGiven a compact complex manifold $M$, we investigate the holomorphic vector bundles $E$ on $M$ such that $\varphi ^* E$ is holomorphically trivial for some surjective holomorphic map $\varphi $, to $M$, from some compact complex manifold. We prove that these are exactly those holomorphic vector bundles that admit a flat holomorphic connection with finite monodromy homomorphism. A similar result is proved for holomorphic principal $G$-bundles, where $G$ is a connected reductive complex affine algebraic group.

Nonlinear fixed points preservers

Let $${\mathcal {B}}(X)$$ be the algebra of all bounded linear operators on an infinite dimensional complex Banach space X. For $$A\in {\mathcal {B}}(X)$$ , let F(A) be the set of all fixed points of A. For an integer $$k\ge 2$$ , let $$(i_1,\dots ,i_m)$$ be a finite sequence with terms chosen from $$\{1,\dots ,k\}$$ and assume that at least one of the terms in $$(i_1,\dots ,i_m)$$ appears exactly once. The generalized product of k operators $$A_1,\dots ,A_k \in {\mathcal {B}}(X)$$ is defined by $$\begin{aligned} A_1*A_2*\cdots *A_k=A_{i_{1}}A_{i_{2}}\dots A_{i_{m}} \end{aligned}$$ and includes the usual product and the triple product. In this paper we characterize the form of surjective maps from $${\mathcal {B}}(X)$$ into itself satisfying $$\begin{aligned} \dim F(\phi (A_{1})*\dots *\phi (A_{k}))=\dim F(A_{1}*\cdots *A_{k}) \end{aligned}$$ for all $$A_1,\dots ,A_k \in {\mathcal {B}}(X)$$ .

Condition Spectrum of Rank One Operators and Preservers of the Condition Spectrum of Skew Product of Operators

Let $${\mathscr {L}}({\mathscr {H}})$$ be the algebra of all bounded linear operators on a complex Hilbert space $${\mathscr {H}}$$ with $$\dim {\mathscr {H}}\ge 3$$ , and let $$\mathscr {A} $$ and $$\mathscr {B}$$ be two subsets of $${\mathscr {L}}({\mathscr {H}})$$ containing all operators of rank at most one. For $$\varepsilon \in (0,1)$$ the $$\varepsilon $$ -condition spectrum of any $$A\in {\mathscr {L}}({\mathscr {H}})$$ is defined by $$\begin{aligned} \sigma _{\epsilon }(A) := \sigma (A)\cup \left\{ \lambda \in \mathbb {C}\setminus \sigma (A):~\Vert (\lambda I -A)^{-1}\Vert \Vert \lambda I -A\Vert \ge \frac{1}{\varepsilon }\right\} , \end{aligned}$$ where $$\sigma (A)$$ is the spectrum of A. The $$\varepsilon $$ -condition spectral radius of A is given by $$\begin{aligned} r_\varepsilon (A):=\sup \left\{ |z| : z\in \sigma _\varepsilon (A) \right\} . \end{aligned}$$ We compute the $$\varepsilon $$ -condition spectrum of any operator of rank at most one, and give an explicit formula for its $$\varepsilon $$ -condition spectral radius. It is then illustrated that the results can be applied to characterize surjective mappings $$\phi :\mathscr {A} \longrightarrow \mathscr {B}$$ satisfying $$\begin{aligned} \delta (\phi (A)^*\phi (B)) = \delta (A^*B) \quad \text{ for } \text{ all } A,B\in \mathscr {A} \end{aligned}$$ where $$\delta $$ stands for $$\sigma _\varepsilon (\cdot )$$ or $$r_\varepsilon (\cdot ).$$

Superposition of snarks revisited

In this paper we propose a new approach to superposition of snarks, a powerful method of constructing large cubic graphs with no 3-edge-colouring from small ones. The main idea is to use surjective mappings between graphs similar to graph homomorphisms and to control flows induced from the domain graph to the target graph via the mappings. This leads to significant strengthening of the power of the classical superposition, which we illustrate by several examples and two applications. First, we describe a family of cyclically 5-edge-connected snarks Gd of order 3⋅2d−2, with d≥2, each being spanned by the balanced cubic tree of depth d; the family contains the Petersen graph as G2. The existence of such a family was conjectured by Hoffmann-Ostenhof and Jatschka (2017). Second, we construct cyclically 5-edge-connected permutation snarks of order n for each n≡2(mod8) with n≥34. Our construction employs a rather exotic form of superposition where the proof of uncolourability of the resulting graph requires two graphs derived from the target graph to be snarks rather than just the target graph itself.

Сюрреалистический код в литературе: определение и механизм реализации

The paper aims to identify the role of a style code in surrealistic fiction throughout its history. The article analyses attempts to formulate a universal definition of surrealistic trends. Scientific originality of the study lies in the fact that the researcher develops a scheme of interaction of a special style code (surrealistic code), the surrealistic conceptual sphere and literary texts proper correlating with the surreal. The conducted research allows concluding that surrealistic conceptions are largely a surjective function of a code and have minimum two arguments, which ensures the surrealistic style identification.

Maps preserving some spectral domains of operator products

Let X be an infinite-dimensional Banach space and let denote the algebra of all bounded linear operators on X. For the sets and are called, respectively, the semi-Fredholm domain, the Fredholm domain, and the Weyl domain of T in the spectrum, . We characterize surjective maps on (with no additivity assumption) leaving invariant or for all and .