Properties Of Such Maps Research Articles

Expand-contract plasticity on the real line

The study deals with plastic and non-plastic sub-spaces A of the real-line ℝ with the usual Euclidean metric d. It investigates non-expansive bijections, proves properties of such maps, and demonstrates their relevance by hands of examples. Finally, it is shown that the plasticity property of a sub-space A contains at least two complementary questions, a purely geometric and a topological one. Both contribute essential aspects to the plasticity property and get more critical in higher dimensions and more abstract metric spaces.

Mappings and Finite Product of Pairwise Expandable Spaces

In this work, we intend to study certain mappings in bitopological spaces such as pairwise perfect and pairwise countably perfect mappings that possess the property (E). Some properties of such mappings are provided which have helped us to obtain some finite product theorems concerning with pairwise expandable and almost pairwise expandable spaces. Two illustrative examples are given to demonstrate the effectiveness of some proposed results.

On Coupling Lemma and Stochastic Properties with Unbounded Observables for 1-d Expanding Maps

In this paper, we establish a coupling lemma for standard families in the setting of piecewise expanding interval maps with countably many branches. Our coupling method only requires that the expanding map satisfies three assumptions: (H1) Chernov’s one-step expansion at q-scale; (H2) dynamically Hölder continuity of log Jacobian (on each branch); (H3) eventually covering over a magnet interval, which are much weaker than the standard assumptions for uniformly expanding maps. We further conclude the existence of an absolutely continuous invariant probability measure and establish the regularity of its density function. Moreover, we obtain the exponential decay of correlations and the almost sure invariance principle (which is a functional version of the central limit theorem) with respect to a large class of unbounded observables that we call “dynamically Hölder series". Our approach is particularly powerful for the piecewise expanding maps which do not satisfy the “big image property" and the maps that have inverse Jacobian of low regularity. As few is known on the statistical properties of such maps in the literature, we demonstrate our results for a specific class of piecewise expanding maps of this kind.

Linear Maps as Sufficient Criteria for Entanglement Depth and Compatibility in Many-Body Systems

Physical transformations are described by linear maps that are completely positive and trace preserving (CPTP). However, maps that are positive (P) but not completely positive (CP) are instrumental to derive separability/entanglement criteria. Moreover, the properties of such maps can be linked to entanglement properties of the states they detect. Here, we extend the results presented in [34], where sufficient separability criteria for bipartite systems were derived. In particular, we analyze the entanglement depth of an [Formula: see text]-qubit system by proposing linear maps that, when applied to any state, result in a biseparable state for the [Formula: see text] partitions, i.e., [Formula: see text]-entanglement depth. Furthermore, we derive criteria to detect arbitrary [Formula: see text]-entanglement depth tailored to states in close vicinity of the completely depolarized state (the normalized identity matrix). We also provide separability (or [Formula: see text]-entanglement depth) conditions in the symmetric sector, including the diagonal states. Finally, we suggest how similar map techniques can be used to derive sufficient conditions for a set of expectation values to be compatible with separable states or local-hidden-variable theories. We dedicate this paper to the memory of the late Andrzej Kossakowski, our spiritual and intellectual mentor in the field of linear maps.

Creation of discontinuities in circle maps.

Circle maps frequently arise in mathematical models of physical or biological systems. Motivated by Cherry flows and ‘threshold’ systems such as integrate and fire neuronal models, models of cardiac arrhythmias, and models of sleep/wake regulation, we consider how structural transitions in circle maps occur. In particular, we describe how maps evolve near the creation of a discontinuity. We show that the natural way to create discontinuities in the maps associated with both threshold systems and Cherry flows results in a singularity in the derivative of the map as the discontinuity is approached from either one or both sides. For the threshold systems, the associated maps have square root singularities and we analyse the generic properties of such maps with gaps, showing how border collisions and saddle-node bifurcations are interspersed. This highlights how the Arnold tongue picture for tongues bordered by saddle-node bifurcations is amended once gaps are present. We also show that a loss of injectivity naturally results in the creation of multiple gaps giving rise to a novel codimension two bifurcation.

Approximately order zero maps between C⁎-algebras

We investigate linear operators between C⁎-algebras which approximately preserve involution and orthogonality, the latter meaning that for some ε>0 we have ‖ϕ(x)ϕ(y)‖⩽ε‖x‖‖y‖ for all positive x,y with xy=0. We establish some structural properties of such maps concerning approximate Jordan-like equations and almost commutation relations. In some situations (e.g. when the codomain is finite-dimensional), we show that ϕ can be approximated by an approximate Jordan ⁎-homomorphism, with both errors depending only on ‖ϕ‖ and ε.

Bilinear operator multipliers into the trace class

Given Hilbert spaces H1,H2,H3, we consider bilinear maps defined on the cartesian product S2(H2,H3)×S2(H1,H2) of spaces of Hilbert-Schmidt operators and valued in either the space B(H1,H3) of bounded operators, or in the space S1(H1,H3) of trace class operators. We introduce modular properties of such maps with respect to the commutants of von Neumann algebras Mi⊂B(Hi), i=1,2,3, as well as an appropriate notion of complete boundedness for such maps. We characterize completely bounded module maps u:S2(H2,H3)×S2(H1,H2)→B(H1,H3) by the membership of a natural symbol of u to the von Neumann algebra tensor product M1⊗‾M2op⊗‾M3. In the case when M2 is injective, we characterize completely bounded module maps u:S2(H2,H3)×S2(H1,H2)→S1(H1,H3) by a weak factorization property, which extends to the bilinear setting a famous description of bimodule linear mappings going back to Haagerup, Effros-Kishimoto, Smith and Blecher-Smith. We make crucial use of a theorem of Sinclair-Smith on completely bounded bilinear maps valued in an injective von Neumann algebra, and provide a new proof of it, based on Hilbert C⁎-modules.

Non-Markovian Open Dynamics from Collision Models

Convex combinations of completely positive (CP) as well as CP-divisible, continuous time dynamical maps arising from collision models are investigated. While the individual maps are both CP and Markovian we find that convex combinations remain CP but not necessarily Markovian. Examples of such combinations for qubit dynamical maps arising from collisional models are worked out and the invertibility, CP-divisibility, P-divisibility as well as Markovian properties of such maps are explored.

A continuous selection for optimal portfolios under convex risk measures does not always exist

Risk control is one of the crucial problems in finance. One of the most common ways to mitigate risk of an investor’s financial position is to set up a portfolio of hedging securities whose aim is to absorb unexpected losses and thus provide the investor with an acceptable level of security. In this respect, it is clear that investors will try to reach acceptability at the lowest possible cost. Mathematically, this problem leads naturally to considering set-valued maps that associate to each financial position the corresponding set of optimal hedging portfolios, i.e., of portfolios that ensure acceptability at the cheapest cost. Among other properties of such maps, the ability to ensure lower semicontinuity and continuous selections is key from an operational perspective. It is known that lower semicontinuity generally fails in an infinite-dimensional setting. In this note, we show that neither lower semicontinuity nor, more surprisingly, the existence of continuous selections can be a priori guaranteed even in a finite-dimensional setting. In particular, this failure is possible under arbitrage-free markets and convex risk measures.

Shy maps in topology

There is a concept in digital topology of a shy map. We define an analogous concept for topological spaces: We say a function is shy if it is continuous and the inverse image of every path-connected subset of its image is path-connected. Some basic properties of such maps are presented. For example, every shy map onto a semilocally simply connected space induces a surjection of fundamental groups (but a shy map onto a space that is not semilocally simply connected need not do so).

Skeletal examination in a chick embryo model of fetal alcohol syndrome indicates impaired osteogenesis

In this chapter, we study Riemannian maps from almost Hermitian manifolds to Riemannian manifolds. In section 1, we study holomorphic Riemannian maps as a generalization of holomorphic submersions and obtain a characterization of such maps. In section 2, we investigate anti-invariant Riemannian maps as a generalization of anti-invariant submersion, investigate the geometry of leaves of distributions defined by such maps, and give necessary and sufficient conditions for anti-invariant Riemannian maps to be totally geodesic. We also find necessary and sufficient conditions for the total manifold of anti-invariant Riemannian maps to be an Einstein manifold. In section 3, as a generalization of the semi-invariant submersion, we introduce semi-invariant Riemannian maps, give examples, and obtain the main properties of such maps. In section 4, we introduce generic Riemannian maps from almost Hermitian manifolds to Riemannian manifolds as a generalization of generic submersions and we give examples. We also find new conditions for Riemannian maps to be totally geodesic and harmonic. In section 5, we study slant Riemannian maps as a generalization of slant submersion, give examples, and obtain the harmonicity of such maps. We also find new necessary and sufficient conditions for such maps to be totally geodesic. Moreover, we obtain a decomposition theorem by slant Riemannian maps. In section 6, we define semi-slant Riemannian maps and give examples. In section 7, we introduce hemi-slant Riemannian maps as a generalization of hemi-slant submersions and slant Riemannian maps, give examples, obtain integrability conditions for distribution, and investigate the geometry of these distributions. We also obtain conditions for such maps to be harmonic and totally geodesic, respectively.

Typical dynamics of plane rational maps with equal degrees

Let $f:\mathbb{CP}^2⇢\mathbb{CP}^2$ be a rational map with algebraic and topological degrees both equal to $d\geq 2$. Little is known in general about the ergodic properties of such maps. We show here, however, that for an open set of automorphisms $T:\mathbb{CP}^2\to\mathbb{CP}^2$, the perturbed map $T\circ f$ admits exactly two ergodic measures of maximal entropy $\log d$, one of saddle type and one of repelling type. Neither measure is supported in an algebraic curve, and $f_T$ is 'fully two dimensional' in the sense that it does not preserve any singular holomorphic foliation of $\mathbb{C}\mathbb{P}^2$. In fact, absence of an invariant foliation extends to all $T$ outside a countable union of algebraic subsets of $Aut(\mathbb{P}^2)$. Finally, we illustrate all of our results in a more concrete particular instance connected with a two dimensional version of the well-known quadratic Chebyshev map.

Dirac-harmonic maps with torsion

We study Dirac-harmonic maps from surfaces to manifolds with torsion, which is motivated from the superstring action considered in theoretical physics. We discuss analytic and geometric properties of such maps and outline an existence result for uncoupled solutions.

Moduli of curve families and (quasi-)conformality of power-law entropies

We present aspects of the moduli of curve families on a metric measure space which may prove useful in calculating, or in providing bounds to, non-additive entropies having a power-law functional form. We use as paradigmatic cases the calculations of the moduli of curve families for a cylinder and for an annulus in [Formula: see text]. The underlying motivation for these studies is that the definitions and some properties of the modulus of a curve family resembles those of the Tsallis entropy, when the latter is seen from a micro-canonical viewpoint. We comment on the origin of the conjectured invariance of the Tsallis entropy under Möbius transformations of the non-extensive (entropic) parameter. Needing techniques applicable to both locally Euclidean and fractal classes of spaces, we examine the behavior of the Tsallis functional, via the modulus, under quasi-conformal maps. We comment on properties of such maps and their possible significance for the dynamical foundations of power-law entropies.

On some restrictions of an operator to an invariant subspace

For Banach spaces we consider the bounded linear operators which are surjective and noninjective. We show some general properties of such mappings. We examine whether such operators can be restricted to an involution or a projection. Thus, we will show that there exist many invariant subspaces for those operators. In respect to this, we will understand better the structure of many operators.

On a soft mapping and its fixed points

In the article first we introduce a new notion of soft element of a soft set and establish its natural relation with soft operations and soft objects in soft topological spaces. Next, using the notion of soft element, we define, in a different way than in the literature, a soft mapping transforming a soft set into a soft set and provide basic properties of such mappings. The new approach to soft mappings enables us to obtain the natural first fixed-point results in the soft set theory. Throughout the paper a comprehensive set of examples illustrating the discussed topics is presented. MSC:47H10, 54A05.

<b>On a class of α<sub>γ</sub>-open sets in a topological space</b> - doi: 10.4025/actascitechnol.v35i3.15788

In this paper, we introduce the concept of αγ-open sets as a generalization of γ-open sets in a topological space (X, τ). Using this set, we introduce αγT0, αγ-T½, αγT1, αγT2, αγD0, αγD1 and αγD2 spaces and study some of its properties. Finally we introduce α(γ,γ')-continuous mappings and give some properties of such mappings.

Structure and stability of restrictively metrically regular mappings

Restrictive metric regularity (RMR) was first proposed by B.S. Mordukhovich and B. Wang (2004) in [13]. In this paper we study the structure of RMR mappings, and then establish stability properties of such mappings under perturbations.

Operation on Regular Spaces

In this paper, we introduce the concept of \(R_{\gamma}\)-open sets as a strong of \(\gamma\)-open sets in a topological space \((X,\tau)\). Using this set, we introduce \(R_{\gamma}\)-\(T_0\), \(R_{\gamma}\)-\(T_{\frac{1}{2}}\), \(R_{\gamma}\)-\(T_1\), \(R_{\gamma}\)-\(T_2\), \(R_{\gamma}\)-\(T_3\), \(R_{\gamma}\)-\(T_4\), \(R_{\gamma}\)-\(D_0\), \(R_{\gamma}\)-\(D_1\) and \(R_{\gamma}\)-\(D_2\) spaces and study some of its properties. Finally we introduce \(R_{(\gamma,{\gamma^{'}})}\)-continuous mappings and give some properties of such mappings.

New characterization of strongly starlike mappings on balanced pseudoconvex domains in [formula omitted

In the paper there is continued a concept of strongly starlike mappings of order alpha on bounded balanced pseudoconvex domain with C 1 -plurisubharmonic defining function, introduced by H. Hamada, G. Kohr and M. Kohr. The main theorem contains two new geometric properties of such mappings and generalizes some one-dimensional results of W. Ma, D. Minda and T. Sugawa and an n-dimensional result given by the author.