Topological Version Research Articles

An Abramov-Rokhlin type theorem for induced topological pressure

In this paper, we define a new type of induced pressure for a family of continuous dynamical systems, parameterized by a probability space and controlled by a noise map. Then we connect this new quantity to the induced pressure of the skew product and the topological entropy of the noise map. This connection may be considered as a topological version of the Abramov-Rokhlin theorem for induced pressure. We also present a variational principle.

Topological Levinson’s theorem in presence of embedded thresholds and discontinuities of the scattering matrix: A quasi-1D example

A family of quasi-[Formula: see text]D Schrödinger operators is investigated through scattering theory. The continuous spectrum of these operators exhibits changes of multiplicity, and some of these operators possess resonances at thresholds. It is shown that the corresponding wave operators belong to an explicitly constructed [Formula: see text]-algebra. The quotient of this algebra by the ideal of compact operators is studied, and an index theorem is deduced from these investigations. This result corresponds to a topological version of Levinson’s theorem in the presence of embedded thresholds, resonances, and changes of multiplicity of the scattering matrices. In the last two sections of the paper, the [Formula: see text]-theory of the main [Formula: see text]-algebra and the dependence on an external parameter are carefully analyzed. In particular, a surface of resonances is exhibited, probably for the first time. The contents of these two sections are of independent interest, and the main result does not depend on them.

On procongruence curve complexes and their automorphisms

This paper launches the exploration of the procongruence completions for three varieties of curve complexes attached to hyperbolic surfaces, as well as their automorphisms groups. The discrete counterparts of these objects, especially the curve complex and the so-called pants complex were defined long ago and have been the subject of numerous studies. Introducing some form of completions is natural and indeed necessary to lay the ground for a topological version of the Grothendieck–Teichmüller theory. Here several results of foundational nature are stated and proved, among which reconstruction theorems in the discrete and complete settings, which give a graph theoretic characterization of versions of the curve complex as well as a rigidity theorem for the complete pants complex, in sharp contrast with the case of the (complete) curve complex, whose automorphisms actually define a version of the Grothendieck–Teichmüller group, to be studied elsewhere. We work all along with the procongruence completions — and for good reasons — recalling however that the so-called congruence conjecture predicts that this completion should coincide with the full profinite completion.

Quasi-shadowing property on compact Hausdorff spaces

In this paper, two topological versions of quasi-shadowing property defined in terms of finite open covers and uniformities are introduced, which coincide on compact Hausdorff spaces and are equivalent to the standard quasi-shadowing property stated by means of a metric on compact metric spaces. At the same time, some general properties about uniform quasi-shadowing are investigated. Also, a map has uniform quasi-shadowing if and only if the induced map on the hyperspace of compact subsets of a compact Hausdorff space does.

Topological spaces satisfying a closed graph theorem

We discuss topological versions of the closed graph theorem, where continuity is inferred from near continuity in tandem with suitable conditions on source or target spaces. We seek internal characterizations of spaces satisfying a closed graph theorem, and we compare closed graph and open mapping spaces.

The entropy formula of shrinking target problem in nonautonomous dynamical systems

This paper is devoted to studying the topological entropy of shrinking target problem in nonautonomous dynamical systems (NDSs). After providing the shrinking target problem of the topological version in NDSs, we obtain a general formula for computing the topological entropy of it. More precisely, we show that the entropy distribution principle still holds in NDSs and apply it to estimate the lower bound of the formula. In addition to the specification property, the concept of control non-decreasing property about the Bowen metric proposed by us is taken as a supplementary condition. Furthermore, we get the connection about the topological entropy between the shrinking target problem and the whole system. This extends the result of Ma et al. [Discrete Contin. Dyn. Syst. 39(5) (2019), pp. 2455–2471], corresponding to the case when the maps are the same as the more general case of different maps.

Tropical graph curves

We study tropical line arrangements associated to a three-regular graph $G$ that we refer to as tropical graph curves. Roughly speaking, the tropical graph curve associated to $G$ , whose genus is $g$ , is an arrangement of $2g-2$ lines in tropical projective space that contains $G$ (more precisely, the topological space associated to $G$ ) as a deformation retract. We show the existence of tropical graph curves when the underlying graph is a three-regular, three-vertex-connected planar graph. Our method involves explicitly constructing an arrangement of lines in projective space, i.e. a graph curve whose tropicalization yields the corresponding tropical graph curve and in this case, solves a topological version of the tropical lifting problem associated to canonically embedded graph curves. We also show that the set of tropical graph curves that we construct are connected via certain local operations. These local operations are inspired by Steinitz’ theorem in polytope theory.

Balanced capacities

We consider capacity (fuzzy measure, non-additive probability) on a compactum as a monotone cooperative normed game. Then it is natural to consider probability measures as elements of core of such game. We prove a topological version of the Bondareva-Shapley theorem that non-emptiness of the core is equivalent to balancedness of the capacity. We investigate categorical properties of balanced capacities and give characterizations of some fuzzy integrals of balanced capacities.

Primal Structure with Closure Operators and Their Applications

Acharjee et al. have created a new structure in mathematics called a primal. Therefore, the primary goal of this research was to introduce and explore more primal space features. Additionally, we studied some of the fundamental characteristics of two novel operators that we define using primal spaces. Using these new operators, we were able to create a weaker version of the original topology. Finally, we provide some examples to further illustrate our discussion of some of their characteristics.

Connectomes and properties of quantum entanglement

Topological quantum field theories (TQFT) encode properties of quantum states in the topological features of abstract manifolds. One can use the topological avatars of quantum states to develop intuition about different concepts and phenomena of quantum mechanics. In this paper we focus on the class of simplest topologies provided by a specific TQFT and investigate what the corresponding states teach us about entanglement. These “planar connectome” states are defined by graphs of simplest topology for a given adjacency matrix. In the case of bipartite systems the connectomes classify different types of entanglement matching the classification of stochastic local operations and classical communication (SLOCC). The topological realization makes explicit the nature of entanglement as a resource and makes apparent a number of its properties, including monogamy and characteristic inequalities for the entanglement entropy. It also provides tools and hints to engineer new measures of entanglement and other applications. Here the approach is used to construct purely topological versions of the dense coding and quantum teleportation protocols, giving diagrammatic interpretation of the role of entanglement in quantum computation and communication. Finally, the topological concepts of entanglement and quantum teleportation are employed in a simple model of information retrieval from a causally disconnected region, similar to the interior of an evaporating black hole.

Three-Phase Multiport DC–AC Inverter for Interfacing Photovoltaic and Energy Storage Systems to the Electric Grid

Distributed renewable energy sources (RES) in combination with hybrid energy storage systems are capable to smooth electric power supply and provide ancillary services to the electric grid. In such applications, multiple separate DC-DC and DC-AC converters are utilized, which are configured in complex and costly architectures. In this paper, a new non-isolated multiport DC-AC power inverter is presented, which comprises less passive components and less high-frequency power semiconductors. The proposed grid-connected multiport converter (MPC) enables the integrated power management of a photovoltaic (PV) array, a battery unit, a supercapacitor bank and the battery of an electric vehicle. The power circuit of the proposed MPC inverter is based on a new version of a split-source inverter topology to support bidirectional power flow and enables to connect the PV source directly to the DC-link. The proposed design is accompanied by a specifically developed control method which enables to implement the maximum power point tracking (MPPT) process without the need of any extra power converter, regulate the power flow at each port independently, as well as to control the power flow between its ports. An experimental prototype of the proposed MPC inverter has been constructed and its operation has been validated experimentally under various power flow scenarios.

On singular generalizations of the Singer–Hopf conjecture

AbstractThe Singer–Hopf conjecture predicts the sign of the topological Euler characteristic of a closed aspherical manifold. In this note, we propose singular generalizations of the Singer–Hopf conjecture, formulated in terms of the Euler–Mather characteristic, intersection homology Euler characteristic and, resp., virtual Euler characteristic of a closed irreducible subvariety of an aspherical complex projective manifold. We prove these new conjectures under the assumption that the cotangent bundle of the ambient variety is numerically effective (nef), or, more generally, when the ambient manifold admits a finite morphism to a complex projective manifold with a nef cotangent bundle. The main ingredients in the proof are the semi‐positivity properties of nef vector bundles together with a topological version of the Riemann–Roch theorem, proved by Kashiwara.

Multilabeled and topological versions of the Hex theorem

The purpose of our paper is to give a purely combinatorial and algorithmic proof of the Hex theorem with multiple labels. Moreover we extend this result for a new class of complexes, called n-essential. We also present a topological version of the Hex theorem and explain its relation to the fixed point property, the Poincaré-Miranda theorem and the covering (Lebesgue) dimension.

COMPARING COMPUTABILITY IN TWO TOPOLOGIES

Abstract Computable analysis provides ways of representing points in a topological space, and therefore of defining a notion of computable points of the space. In this article, we investigate when two topologies on the same space induce different sets of computable points. We first study a purely topological version of the problem, which is to understand when two topologies are not $\sigma $ -homeomorphic. We obtain a characterization leading to an effective version, and we prove that two topologies satisfying this condition induce different sets of computable points. Along the way, we propose an effective version of the Baire category theorem which captures the construction technique, and enables one to build points satisfying properties that are co-meager with respect to a topology, and are computable with respect to another topology. Finally, we generalize the result to three topologies and give an application to prove that certain sets do not have computable type, which means that they have a homeomorphic copy that is semicomputable but not computable.

Computing Riemann–Roch polynomials and classifying hyper-Kähler fourfolds

We prove that a hyper-Kähler fourfold satisfying a mild topological assumption is of K3 [ 2 ] ^{[2]} deformation type. This proves in particular a conjecture of O’Grady stating that hyper-Kähler fourfolds of K3 [ 2 ] ^{[2]} numerical type are of K3 [ 2 ] ^{[2]} deformation type. Our topological assumption concerns the existence of two integral degree-2 cohomology classes satisfying certain numerical intersection conditions. There are two main ingredients in the proof. We first prove a topological version of the statement, by showing that our topological assumption forces the Betti numbers, the Fujiki constant, and the Huybrechts–Riemann–Roch polynomial of the hyper-Kähler fourfold to be the same as those of K3 [ 2 ] ^{[2]} hyper-Kähler fourfolds. The key part of the article is then to prove the hyper-Kähler SYZ conjecture for hyper-Kähler fourfolds for divisor classes satisfying the numerical condition mentioned above.

Topological Explorations on the Fundamental Theorem of Arithmetic

In an interplay between the Fundamental Theorem of Arithmetic and topology, this paper presents material for a capstone seminar that expands on ideas from number theory, analysis, and linear algebra. It is designed to generate an immersive way of learning in which students discover new connections between familiar concepts, create definitions, and investigate non-standard questions. The ideas are arranged as a series of questions for students to explore, with expositions to the instructor on how to guide them in looking for answers. The investigations culminate in a satisfying outcome: that partition topologies are the same as unique factorization topologies, those that satisfy our topological version of the Fundamental Theorem of Arithmetic.

On various entropies of set-valued maps

In this paper, the complexity of an upper semi-continuous set-valued map F on a compact metric space is considered via entropy-like invariants from various perspectives. Several entropies of topological version, including pointwise entropies hp(F) and hm(F), branch entropy hi(F) and tree entropy ht(F), are introduced and investigated. Some basic properties about them are given and the relations among them and the classical entropy htop(F) are considered. The calculation or estimation of these entropies for certain finitely-generated set-valued maps on intervals or graphs are discussed. As the counterparts of ht(F), the measure-theoretic lower and upper tree entropies h_tμ(F) and h‾tμ(F) are introduced for any Borel probability measure μ in a way resembling Hausdorff dimension. A variational inequality ht(F)≥sup⁡{h‾tμ(F)|μ∈P(X)}, where P(X) is the set of Borel probability measures on X, is obtained, and as its applications, the lower bounds of ht(F) are given for certain set-valued maps respectively.

Variational principles for topological pressures on subsets

In this paper, we investigate the relations between various types of topological pressures and different versions of measure-theoretical pressures. We extend Feng–Huang’s variational principle for packing entropy to packing pressure and obtain two new variational principles for Pesin–Pitskel and packing pressures respectively. We show that various types of Katok pressures for an ergodic measure with respect to a potential function are equal to the sum of measure-theoretic entropy of this measure and the integral of the potential function. Moreover, we obtain Billingsley type theorem for packing pressure, which indicates that packing pressure can be determined by measure-theoretic upper local pressure of measures, and a variational principle for packing pressure of the set of generic points for any invariant ergodic Borel probability measure.

On the topological locality of antisymmetric connectedness

The theory of antisymmetric connectedness for a T0-quasi-metric space was established in terms of graph theory lately, as corresponding counterpart of the connectedness for the complement of a graph. Following that in the current study, a topological localized version of the antisymmetrically connected spaces is described and studied through a variety of approaches in the context of T0-quasi-metrics. Within the framework of this, we examine the cases under which conditions a T0-quasi-metric space would become locally antisymmetrically connected as well as some topological characterizations of locally antisymmetrically connected T0-quasi-metric spaces are presented, especially via metrics.

Topological equicontinuity and topological uniform rigidity for dynamical system

In this paper, we study topological equicontinuity, topological uniform rigidity and their properties. For a dynamical system, on a compact, T3 space, we study relations among the set of recurrent points of the map, the set of non-wandering points of the map and the intersection of the range sets of all iterations of the map. We define topological version of uniform rigidity and show that on a compact and T3 space any dynamical system is topologically uniformly rigid if it is first countable, almost topologically equicontinuous and transitive or it is second countable, topologically equicontinuous and has a dense set of periodic points. We show that a topologically uniformly rigid dynamical system, on a compact, Hausdorff space, has zero topological entropy. Moreover, we provide necessary examples and counterexamples.