Topological Analogue Research Articles

Minimal and proximal examples of -stable and -approachable shift spaces

Abstract We study shift spaces over a finite alphabet that can be approximated by mixing shifts of finite type in the sense of (pseudo)metrics connected to Ornstein’s $\bar {d}$ metric ( $\bar {d}$ -approachable shift spaces). The class of $\bar {d}$ -approachable shifts can be considered as a topological analog of measure-theoretical Bernoulli systems. The notion of $\bar {d}$ -approachability, together with a closely connected notion of $\bar {d}$ -shadowing, was introduced by Konieczny, Kupsa, and Kwietniak [Ergod. Th. & Dynam. Sys.43(3) (2023), 943–970]. These notions were developed with the aim of significantly generalizing specification properties. Indeed, many popular variants of the specification property, including the classic one and the almost/weak specification property, ensure $\bar {d}$ -approachability and $\bar {d}$ -shadowing. Here, we study further properties and connections between $\bar {d}$ -shadowing and $\bar {d}$ -approachability. We prove that $\bar {d}$ -shadowing implies $\bar {d}$ -stability (a notion recently introduced by Tim Austin). We show that for surjective shift spaces with the $\bar {d}$ -shadowing property the Hausdorff pseudodistance ${\bar d}^{\mathrm {H}}$ between shift spaces induced by $\bar {d}$ is the same as the Hausdorff distance between their simplices of invariant measures with respect to the Hausdorff distance induced by Ornstein’s metric $\bar {d}$ between measures. We prove that without $\bar {d}$ -shadowing this need not to be true (it is known that the former distance always bounds the latter). We provide examples illustrating these results, including minimal examples and proximal examples of shift spaces with the $\bar {d}$ -shadowing property. The existence of such shift spaces was announced in the earlier paper mentioned above. It shows that $\bar {d}$ -shadowing indeed generalizes the specification property.

Ethanol-regulated gelation of chitosan dissolved in an alkali/urea aqueous solution: Morphological characteristics for capturing particulate species

Gelation of chitosan dissolved in an alkali/urea aqueous solution promotes the mechanistic diversity of forming a membrane via aqueous phase separation (APS), which is conceptualized to highlight the advantages of using environmentally friendly solvents. When aimed at deepening the knowledge of regulating the structural formation, the alkaline dope of chitosan was employed in this study to investigate how the presence of ethanol in the coagulant could regulate the gelation behavior in APS and thereby change the particle-trapping mechanism. Both the gelling films and the particle-filtering membranes were in-situ characterized by optical coherence tomography (OCT) to compare the film-formation kinetics and the morphological variations in the resulting porous substructures, respectively. It was revealed that the ethanol-containing coagulant could induce the shrinkage of the gelling film and thereby favor the transverse coalescence of polymer-rich phases; that is, increasing the content of ethanol in the coagulant (over a range less than a critical value to avoid film deformation) would give rise to more tortuous substructures (for effectively capturing particles) primarily owing to the lower connectivity in the transverse direction. The theoretical analysis was supported by numerically evaluating the topological analog. This study not only confirms effectiveness of the ethanol-based regulation, but also establishes a paradigm for fabricating membranes with tailored topological characteristics.

Between Soft Complete Continuity and Soft Somewhat-Continuity

This paper introduces two novel concepts of mappings over soft topological spaces: “soft somewhat-r-continuity” and “soft somewhat-r-openness”. We provide characterizations and discuss soft composition and soft subspaces. With the use of examples, we offer numerous connections between these two notions and their accompanying concepts. We also offer extension theorems for them. Finally, we investigated a symmetry between our new concepts with their topological analogs.

Neuropeptide cyclo-L-prolylglycine counteracts scopolamine-induced long-term memory impairment in rats in the novel object recognition test

Background. Cyclo-L-prolylglycine (CPG) was designed and synthesized at the V.V. Zakusov as a topological analogue of the classical nootrop piracetam and was further identified as an endogenous compound. Previously, the nootropic effect of CPG was revealed in a model of retrograde amnesia in rats induced by electroconvulsive shock in the passive avoidance test (PAT).Objective. The aim of the present study was to investigate the nootropic effect of CPG under more physiological conditions in the absence of strong stressors.Methods. Amnesia in rats was modeled by intraperitoneal (ip) administration of scopolamine at a dose of 2 mg/kg. CPG was administered ip at doses of 0.1 and 1.0 mg/kg 15 minutes after scopolamine. Short- and long-term memory were recorded in the novel object recognition test.Results. It was found that scopolamine disrupted only the long-term memory of rats. CPG at a dose of 0.1 mg/ kg almost completely counteracted this impairment. CPG by itself had no effect on memory at both doses studied.Conclusion. Thus, CPG exhibits nootropic activity not only in the aversive conditions of the PAT and electroconvulsive shock-induced amnesia, but also in the neutral situation in the novel object recognition test, when the amnesia was caused by the administration of scopolamine.

Non-variant phenomena in heterogeneous systems. New type of solubility diagrams points

The article gives a general classification of non-invariant points in phase equilibrium diagrams of all possible types. The complete topological isomorphism of the diagrams of fusibility, solubility, and liquid-vapor equilibria in various sets of variables is demonstrated. The stability of mono-variant equilibria near the non-variant points is investigated. Recurrent formulas for calculating the number of topological elements of phase diagrams are given. A previously undescribed type of non-invariant points and phase processes in the solubility diagrams is described and characterized. The last ones have no topological analogs in other types of diagrams. Thus, we have carried out, as far as is available to the authors, a complete classification of invariant points and invariant processes in phase equilibrium diagrams of an arbitrary type and with an arbitrary number of components.

Homotopy groups of cubical sets

We define and study homotopy groups of cubical sets. To this end, we give four definitions of homotopy groups of a cubical set, prove that they are equivalent, and further that they agree with their topological analogues via the geometric realization functor. We also provide purely combinatorial proofs of several classical theorems, including: product preservation, commutativity of higher homotopy groups, the long exact sequence of a fibration, and Whitehead’s theorem.This is a companion paper to our “Cubical setting for discrete homotopy theory, revisited” in which we apply these results to study the homotopy theory of simple graphs.

Soft ωs-irresoluteness and soft pre-ωs-openness insoft topological spaces

We use soft ωs-open sets to define soft ωs-irresoluteness, soft ωs-openness, and soft pre-ωs-openness as three new classes of soft mappings. We give several characterizations for each of them, specially via soft ωs-closure and soft ωs-interior soft operators. With the help of examples, we study several relationships regarding these three notions and their related known notions. In particular, we show that soft ωs-irresoluteness is strictly weaker than soft ωs-continuity, soft ωs-openness lies strictly between soft openness and soft semi-openness, pre-ωs-openness is strictly weaker than ωs-openness, soft ωs-irresoluteness is independent of each of soft continuity and soft irresoluteness, soft pre-ωs-openness is independent of each of soft openness and soft pre-semi-openness, soft ωs-irresoluteness and soft continuity (resp. soft irresoluteness) are equivalent for soft mappings between soft locally countable (resp. soft anti-locally countable) soft topological spaces, and soft pre-ωs-openness and soft pre-semi-continuity are equivalent for soft mappings between soft locally countable soft topological spaces. Moreover, we study the relationship between our new concepts in soft topological spaces and their topological analog.

A π-Bridge Spacer Embedded Electron Donor-Acceptor Polymer for Flexible Electrochromic Zn-Ion Batteries.

Zinc-ion batteries (ZIBs) have drawn much attention for next-generation energy storage for smart and wearable electronics due to their high theoretical gravimetric/volumetric energy capacities, safety from explosive hazards, and cost-effectiveness. However, current state-of-the-art ZIBs lack the energy capacity necessary to facilitate smart functionalities for intelligent electronics. In this work, a "π-bridge spacer"-embedded electron donor-acceptor polymer cathode combined with a Zn2+ -ion-conducting electrolyte is proposed for a smart and flexible ZIB to provide high electrochromic-electrochemical performances. The π-bridge spacer endows the polymeric skeleton with improved physical ion accessibility and sensitive charge transfer throughthe cycles, providing extremely stable cyclability with high specific capacity (110 mAhg-1 ) at very fast rates (8 Ag-1 ) and large coloration efficiency (79.8 cm2 C-1 ) under severe mechanical deformation over a long period. These results are markedly outstanding compared to the topological analogue without the π-bridge spacer (80 mAhg-1 at current density of 8 Ag-1 , 63.0 cm2 C-1 ). The design to incorporate a π-bridge spacer realizes notable electrochromism behaviors and high electrochemical performance, which sheds light on the rational development of multifunctional flexible-ZIBs with color visualization properties for widespread usage in powering smart electronics.

Topological Metamaterials.

The topological properties of an object, associated with an integer called the topological invariant, are global features that cannot change continuously but only through abrupt variations, hence granting them intrinsic robustness. Engineered metamaterials (MMs) can be tailored to support highly nontrivial topological properties of their band structure, relative to their electronic, electromagnetic, acoustic and mechanical response, representing one of the major breakthroughs in physics over the past decade. Here, we review the foundations and the latest advances of topological photonic and phononic MMs, whose nontrivial wave interactions have become of great interest to a broad range of science disciplines, such as classical and quantum chemistry. We first introduce the basic concepts, including the notion of topological charge and geometric phase. We then discuss the topology of natural electronic materials, before reviewing their photonic/phononic topological MM analogues, including 2D topological MMs with and without time-reversal symmetry, Floquet topological insulators, 3D, higher-order, non-Hermitian and nonlinear topological MMs. We also discuss the topological aspects of scattering anomalies, chemical reactions and polaritons. This work aims at connecting the recent advances of topological concepts throughout a broad range of scientific areas and it highlights opportunities offered by topological MMs for the chemistry community and beyond.

Soft C-continuity and soft almost C-continuity between soft topological spaces

We present the notions of soft c-continuity and soft nearly c-continuity, which are weaker versions of soft continuity and soft almost continuity, respectively. We obtain several characterizations for these two concepts. We show that soft c-continuity and soft almost c-continuity are preserved under soft restrictions. In addition, we investigate the conditions under which the composition of two soft c-continuous and soft almost c-continuous functions is soft c-continuous and soft almost c-continuous. Moreover, via soft N-open sets, we give a characterization of the soft compactness of soft topological spaces over finite sets of parameters. In addition, we show that for a given soft topological space (L,Ω,F), the collection of soft regular open sets with soft compact complements forms a soft base for some coarser soft topology. We demonstrate that (L,Ω⁎,F) are all soft compacts. Furthermore, we demonstrate that Ω=Ω⁎ if (L,Ω,F) or (L,Ω⁎,F) is soft compact and soft Hausdorff. Finally, we investigate the correspondences between the novel notions in soft topology and their general topological analogs.

Interpreting mereotopological connection

This paper examines ten possible topological interpretations of connection and for each interpretation, identifies sufficient conditions under which a significant class of topological spaces provides models of General Extensional Mereotopology with Closure Conditions (GEMTC) in which some key mereotopological ideas align with their topological analogues. In particular, there is an interpretation under which the non-empty sets of any symmetric topology are a model of GEMTC with alignment between the mereotopological and topological definitions of (self-)connection, open and closed entities, interior, exterior, closure, and boundary.

Robust elastic wave transport in zone-folding induced topological hierarchical metamaterials

The topologically protected elastic wave transport in hierarchical metamaterials for the in-plane mode and out-of-plane mode is investigated in this work. First of all, the influence of two key parameters (the hierarchical order and length ratio) on dispersion relation is studied in detail. It can be found that the introduction of structural hierarchy is beneficial for the formation of low-frequency Dirac cones (DCs). But the number of DCs is independent of the hierarchical order. Moreover, the frequency of DCs and band gaps can be tuned effectively by the length ratio. Tuning the lower-order length ratio is more effective to form low-frequency DCs. Based on above findings, we proposed novel topological hierarchical metamaterials analogue to quantum spin Hall effects (QSHEs) by zone-folding method. By shrinking or expanding the supercell, two types of supercells with distinct topological property are built. In addition, through the dispersion analysis and full-field numerical simulation, topological interface states for both wave modes are obtained successfully, and the robustness is verified also. To overcome the limited frequency range of topological interface states, we try to introduce the temperature-sensitive epoxy into the topological hierarchical metamaterial. Fortunately, the frequency of topological interface states can be tuned in a wide range, and the energy concentration cannot be affected by the temperature. This work will broaden the applied range of hierarchical metamaterials, and is expected to meet the tremendous needs for multifunctional materials in engineering practice.

Re12C80: A pentagonal hexecontahedron molecule

Based on density-functional theory calculations, a polyhedron molecule, Re12C80 cage, was proposed and showed topological analogy to Catalan pentagonal hexecontahedron. The geometric topology of Re12C80 molecule retains integrity when the effective temperature is up to 782 K. Its electronic structure shows that for the carbon atom there is the feature of sp2-like hybrid orbital, and for the rhenium atom there presents d orbital characteristics. The large hollow space for Re12C80 cage can be useful to encapsulate other atoms or small molecules. The proposal of Re12C80 molecular structure can help to enrich and expand the types of highly symmetric polyhedral molecules.

The soft topology of soft ω ∗ -open sets and soft almost Lindelofness

In this paper, We use the soft closure operator to introduce soft \(\omega ^{\ast }\)-open sets as a new class of soft sets. We prove that this class of soft sets forms a soft topology that lies strictly between the soft topology of soft \(\theta \)-open sets and the soft topology of soft \(\omega \)-open sets. Also, we show that the soft topology of soft \(% \omega ^{\ast }\)-open sets contain the soft co-countable topology and is independent of the topology of soft open sets. Furthermore, several results regarding soft almost Lindelofness are given. In addition to these, we investigate the correspondences between the novel notions in soft topology and their general topological analogs.

Residual surface charge mediated near-field radiative energy transfer: A topological insulator analog

We study the modifications of near-field radiative energy transfer (NFRET) caused by residual surface charges, which are common in micro- and nano-systems like NEMS/MEMS. The host object with the residual surface charges and the inherent bulk state can be treated as an analog of the real three-dimensional topological insulator, which is inherent of also both surface states and bulk states and is promising to modulate NFRET. Through constructing such a topological insulator analog, we aim to modulate NFRET concerning only common trivial materials. Besides the well-known resonant modes (surface polariton and localized surface polariton) supported by the bulk state, the residual surface charges give rise to an additional temperature-dependent mode providing a new heat flux channel. For low temperatures we find a giant surface-charge-induced enhancement of the NFRET due to a good match between the surface-charge-induced resonance and the Planck window. However, for relative high temperatures where the Fröhlich resonance dominates the heat transfer rather the surface-charge-induced resonance, the residual charges result in a weakening of the NFRET. This work paves way for understanding and modulating the near-field radiative energy transfer for micro- and nano-systems.

Somewhat omega continuity and somewhat omega openness in soft topological spaces

In this paper, we introduce soft somewhat ω-continuous soft mappings and soft somewhat ω-open soft mappings as two new classes of soft mappings. We characterize these two concepts. Also, we prove that the class of soft somewhat ω-continuous (resp. soft somewhat ω-open) soft mappings contains the class of soft somewhat continuous (resp. soft somewhat open) soft mappings. Moreover, we obtain some sufficient conditions for the composition of two soft somewhat ω-continuous (resp. soft somewhat ω-open) soft mappings to be a soft somewhat ω-continuous (resp. a soft somewhat ω-open) soft mapping. Furthermore, we introduce some sufficient conditions for restricting a soft somewhat ω-continuous (resp. soft somewhat ω-open) soft mapping to being a soft somewhat ω-continuous (resp. soft somewhat ω-open) soft mapping. In addition to these, we introduce extension theorems regarding soft somewhat ω-continuity and soft somewhat ω-openness. Finally, we investigate the correspondences between the novel notions in soft topology and their general topological analogs.

Abelian Duality in Topological Field Theory

We prove a topological version of abelian duality where the gauge groups are finite abelian. The theories are finite homotopy TFTs, topological analogues of the $p$-form $U(1)$ gauge theories. Using Brown-Comenetz duality, we extend the duality to finite homotopy TFTs of $\pi$-finite spectra.

On topological models of zero entropy loosely Bernoulli systems

We provide a purely topological characterisation of uniquely ergodic topological dynamical systems (TDSs) whose unique invariant measure is zero entropy loosely Bernoulli (following Ratner, we call such measures loosely Kronecker). At the heart of our proofs lies Feldman-Katok continuity (FK-continuity for short), that is, continuity with respect to the change of metric to the Feldman-Katok pseudometric. Feldman-Katok pseudometric is a topological analog of f-bar (edit) metric for symbolic systems. We also study an opposite of FK-continuity, coined FK-sensitivity. We obtain a version of Auslander-Yorke dichotomies: minimal TDSs are either FK-continuous or FK-sensitive, and transitive TDSs are either almost FK-continuous or FK-sensitive.

A Lyndon’s identity theorem for one-relator monoids

For every one-relator monoid M = langle A mid u=v rangle with u, v in A^* we construct a contractible M-CW complex and use it to build a projective resolution of the trivial module which is finitely generated in all dimensions. This proves that all one-relator monoids are of type mathrm{FP}_{infty }, answering positively a problem posed by Kobayashi in 2000. We also apply our results to classify the one-relator monoids of cohomological dimension at most 2, and to describe the relation module, in the sense of Ivanov, of a torsion-free one-relator monoid presentation as an explicitly given principal left ideal of the monoid ring. In addition, we prove the topological analogues of these results by showing that all one-relator monoids satisfy the topological finiteness property mathrm{F}_infty , and classifying the one-relator monoids with geometric dimension at most 2. These results give a natural monoid analogue of Lyndon’s Identity Theorem for one-relator groups.

Enhanced Bounds for rho-invariants for both general and spherical 3-manifolds

We establish enhanced bounds on Cheeger–Gromov [Formula: see text]-invariants for general 3-manifolds and yet stronger bounds for special classes of 3-manifold. As key ingredients, we construct chain null-homotopies whose complexity is linearly bounded by its boundary. This result can be regarded as an algebraic topological analogue of Gromov’s conjecture for quantitative topology. The author hopes for applications to various fields including the smooth knot concordance group, quantitative topology and complexity theory.