Homeomorphic Research Articles

CO2-Responsive Hydrophobic Deep Eutectic Solvent Based on Surfactant-Free Microemulsion-Mediated Synthesis of BaF2 Nanoparticles.

A new form of surfactant-free microemulsion (SFME) including hydrophobic deep eutectic solvent (HDES)/ethanol/water was constructed based on its CO2 response, and three regions, that is, HDES-in-water (HDES/W), bicontinuous (B.C.), and water-in-HDES (W/HDES) regions, were recognized. It is anticipated that SFMEs with tunable microstructures have outstanding applications as nanoreactors in reaction processes. The feasibility of preparing nanoparticles from HDES/ethanol/water SFME using barium fluoride (BaF2) as a model nanoparticle was investigated. HDES-based microemulsions benefit from HDES's excellent properties (novel, low toxicity, CO2-responsive, easy availability) and have potential in universal reactions, drug delivery, advanced material fabrication, etc. In this research, HDES-based microemulsions were prepared using HDES as the oil phase. Phase equilibria and microstructure were investigated using a ternary phase diagram, UV spectrophotometry, and the conductivity method. The CO2 switchable characteristics of the HDES-based microemulsions were investigated. HDES-based microemulsions were proposed as nanoreactors for the synthesis of barium fluoride nanomaterials. The microemulsion structure can modulate the size, morphology, and physicochemical properties of the nanoparticles through the CO2 switchable properties. It is argued that nanoreactors constructed with versatile HDES will offer a new direction for creation of cutting-edge scientific applications.

Chaotic Pulse-Shaping Filter Based on Root-Raised-Cosine Division.

Chaotic baseband wireless communication system (CBWCS) suffers bit error rate (BER) degradation due to their intrinsic intersymbol interference (ISI). To this end, an ISI-free chaotic filter based on root-raised-cosine (RRC) division is constructed to generate a chaotic signal. A wireless communication system using this chaotic signal as a baseband waveform is proposed. The chaotic property is proved by the corresponding new hybrid dynamical system with topological conjugation to symbolic sequences and a positive Lyapunov exponent. Simulation results show that under a single-path channel and multi-path channel, the proposed method outperforms CBWCS in both BER performance and computational complexity.

Crystal structure and optical absorption spectra of LiCo(SO4)OH and its remarkable relationship to the Zn-Mn-silicate hodgkinsonite

Crystals of the compound LiCo(SO4)OH were synthesised at low-hydrothermal conditions, and the crystal structure was determined and refined from single crystal X-ray diffraction data. LiCo(SO4)OH crystallises monoclinic, space group P21/c, Z = 4, a = 9.586(2), b = 5.425(1), c = 7.317(1) Å, β = 109.65(1)°, V = 358.3 Å3, wR2 = 0.0485 (2215 unique reflections, 78 variables). The structure is built from chains of edge-sharing, quite strongly bond-length and -angle distorted Co(OH)3O3 octahedra (< Co–O > = 2.126 Å), which are further linked by common corners, hydrogen bonds, and by properly shaped SO4 tetrahedra (< S–O > = 1.476 Å) to sheets parallel (100). These sheets are connected to a three-dimensional framework by sharing corners with distorted LiO4 polyhedra (< Li–O > = 1.956 Å). Apart from the isotypic sulfates of Mn2+ and Fe2+, only the molybdate LiCd(MoO4)OH crystallises isostructural with LiCo(SO4)OH. However, a very close structural relationship exists with the rare mineral hodgkinsonite, Zn2Mn(SiO4)(OH)2, yielding crystal chemically very uncommon topological equivalents of Zn2+ ≡ S6+ and Si4+ ≡ Li+, aside from the expectable substitution Mn2+ ≡ Co2+. Polarised optical absorption spectra of LiCo(SO4)OH reveal that the dominating spin-allowed 4T1(P) band system of Co2+ (d7 configuration) is strongly split up and covers a prominent part (~ 15,500–24,500 cm−1) of the visible spectral range, in accordance with the significant distortion of the Co(OH)3O3 polyhedron. The spectra are interpreted in terms of the Superposition Model of crystal fields, yielding a new set of intrinsic and interelectronic repulsion parameters for Co2+.

Topological equivalence of global attractors for Lipschitz perturbations of the Chafee-Infante equation

In this paper we introduce the notion of spectral decomposition for global attractors of gradient semigroups, which allows us to define the concept of saddles to non-differentiable semigroups. Using this notion, we exhibit necessary and sufficient conditions to obtain the topological equivalence between two global attractors $ \mathcal{A}_1 $ and $ \mathcal{A}_2 $ of gradient semigroups $ T_1 $ and $ T_2 $, respectively, both having a spectral decomposition. As an application, we show that Lipschitz perturbations of the Chafee-Infante equation produce gradient semigroups with global attractors with spectral decomposition, and that are topologically equivalent to the unperturbed attractor.

The orbit spaces $G_{n,2}/T^n$ and the Chow quotients $G_{n,2}//(\pmb{\mathbb{C}}^{\ast})^n$ of the Grassmann manifolds $G_{n,2}$

The complex Grassmann manifolds $G_{n,k}$ appear as one of the fundamental objects in developing an interaction between algebraic geometry and algebraic topology. The case $k=2$ is of special interest on its own as the manifolds $G_{n,2}$ have several remarkable properties which distinguish them from the $G_{n,k}$ for $k&gt;2$. In our paper we obtain results which, essentially using the specifics of the Grassmann manifolds $G_{n,2}$, develop connections between algebraic geometry and equivariant topology. They are related to well-known problems of the canonical action of the algebraic torus $(\mathbb{C}^{\ast})^n$ on $G_{n,2}$ and the induced action of the compact torus $T^n\subset(\mathbb{C}^{\ast})^n$. Kapranov proved that the Deligne-Mumford-Grothendieck-Knudsen compactification $\overline{\mathcal{M}}(0,n)$ of the space of $n$-pointed rational stable curves can be realized as the Chow quotient $G_{n,2}//(\mathbb{C}^{\ast})^{n}$. In recent papers of the authors a constructive description of the orbit space $G_{n,2}/T^n$ was obtained. In deducing this result the notions of the complex of admissible polytopes and the universal space of parameters $\mathcal{F}_{n}$ for the $T^n$-action on $G_{n,2}$ were of essential use. Using the techniques of wonderful compactification, in this paper an explicit construction of the space $\mathcal{F}_{n}$ is presented. In combination with Keel's description of $\overline{\mathcal{M}}(0,n)$, this construction enabled one to obtain an explicit diffeomorphism between $\mathcal{F}_{n}$ and $\overline{\mathcal{M}}(0,n)$. In this way, we give a description of $G_{n,2}//(\mathbb{C}^{\ast})^{n}$ as the space $\mathcal{F}_{n}$ with a structure described in terms of admissible polytopes $P_\sigma$ and spaces $F_\sigma$. Bibliography: 32 titles.

Equivalence of ill-posed dynamical systems

The problem of topological classification is fundamental in the study of dynamical systems. However, when we consider systems without well-posedness, it is unclear how to generalize the notion of equivalence. For example, when a system has trajectories distinguished only by parametrization, we cannot apply the usual definition of equivalence based on the phase space, which presupposes the uniqueness of trajectories. In this study, we formulate a notion of “topological equivalence” using the axiomatic theory of topological dynamics proposed by Yorke [7], where dynamical systems are considered to be shift-invariant subsets of a space of partial maps. In particular, we study how the type of problems can be regarded as invariants under the morphisms between systems and how the usual definition of topological equivalence can be generalized. This article is intended to also serve as a brief introduction to the axiomatic theory of ordinary differential equations (or topological dynamics) based on the formalism presented in [6].

On partial maps derived from flows

The first-return map, or the Poincar\'e map, is a fundamental concept in the theory of flows. However, it can generally be defined only partially, and additional conditions are required to define it globally. Since this partiality reflects the dynamics, the flow can be described by considering the domain and behavior of such maps. In this study, we define the concept of first-exit maps and first-return maps, which are partial maps derived from flows, to enable such analysis. Moreover, we generalize some notions related to the first-return maps. It is shown that the boundary points of an open set can be classified based on the behavior of these maps, and that this classification is invariant under topological equivalence. Further, we show that some dynamical properties of a flow can be described in terms of the types of boundary points. In particular, if the flow is planar and the open set has a Jordan curve as its boundary, a more detailed analysis is possible, and we present results on the conditions which restrict possible forms of the first-exit maps . Finally, as an application of the results obtained, we consider the relationship between flows and a class of hybrid systems.

Topological Conjugacy of the Simplest Nonsingular Three-Dimensional Flows

Topological Conjugacy of the Simplest Nonsingular Three-Dimensional Flows

Knot as a complete invariant of a Morse-Smale 3-diffeomorphism with four fixed points

It is known that the topological conjugacy class of a Morse-Smale flows with unique saddle is defined by the equivalence class of the Hopf knot in $\mathbb S^2\times\mathbb S^1$ that is the projection of the one-dimensional saddle separatrix onto the basin of attraction of the nodal point, and the ambient manifold of a diffeomorphism in this class is the 3-sphere. In the present paper a similar result is obtained for gradient-like diffeomorphisms with exactly two saddle points and unique heteroclinic curve. Bibliography: 11 titles.

Some new results for the smoothness of topological equivalence in uniformly asymptotically stable systems

In this article we revisit a method of topological linearization for nonautonomous and uniformly asymptotically stable ordinary differential equations developed by Kenneth J. Palmer and Faxing Lin. In particular, sufficient conditions are obtained ensuring the smoothness of the above mentioned topological linearization.

Topological Equivalence Theorem and Double-Copy for Chern-Simons Scattering Amplitudes.

We study the mechanism of topological mass generation for 3-dimensional Chern-Simons gauge theories and propose a brand-new topological equivalence theorem to connect scattering amplitudes of the physical gauge boson states to that of the transverse states under high-energy expansion. We prove a general energy cancelation mechanism for N-point physical gauge boson amplitudes, which predicts large cancelations of E4 - L → E(4 - L) - N at any L-loop level (L ⩾ 0). We extend the double-copy approach to construct massive graviton amplitudes and to study their structures. We newly uncovered a series of strikingly large energy cancelations E12 → E1 of the tree-level 4-graviton scattering amplitude under high-energy expansion and establish a new correspondence between the 2 energy cancelations in the topologically massive Yang-Mills gauge theory and the topologically massive gravity theory. We further study the scattering amplitudes of Chern-Simons gauge bosons and gravitons in the nonrelativistic limit.

A combinatorial invariant of gradient-like flows on a connected sum of $\mathbb{S}^{n-1}\times\mathbb{S}^1$

We obtain necessary and sufficient conditions for the topological equivalence of gradient-like flows without heteroclinic intersections defined on the connected sum of a finite number of manifolds homeomorphic to $\mathbb{S}^{n-1}\times \mathbb{S}^1$, $n\geq 3$. For $n&gt;3$, this result extends substantially the class of manifolds such that structurally stable systems on these manifolds admit a topological classification. Bibliography: 36 titles.

Neutrosophic Soft n-Topological Spaces: A Framework for Decision-Making Problems

This article introduces the concept of neutrosophic soft n-topological spaces and their application in decision-making problems. Neutrosophic soft sets are used to define the open sets in these spaces, which allow for greater flexibility and uncertainty in the decision-making process. The concept of neutrosophic soft n-homeomorphism is also introduced, which describes the topological equivalence between two neutrosophic soft n-topological spaces. The article provides examples of how neutrosophic soft n-topological spaces can be used in decision-making problems, such as medical diagnosis and stock market analysis. The theory and applications presented in this article provide a valuable tool for dealing with uncertainty in decision-making problems.

Discrete Gravity in Emergent Space Theory: Hidden Conical Defects and Teleparallel Gravity

In the first part of the paper, we try to identify the presence of gravity, at a microscopic level, by introducing conical defects and maintaining an approach that assumes topological equivalence among the underlying manifolds that form the tissue of the D-brane itself. In the second part, we will present an alternative to the conical defects, considering the theory of teleparallel gravity, in which the presence of the torsion and the construction of the brane always occurs within the context of our new interpretation of emergent spaces.

Equivalence of Collet–Eckmann conditions for slowly recurrent rational maps

In this short note we observe that within the family of slowly recurrent rational maps on the Riemann sphere, the Collet–Eckmann, second Collet–Eckmann, and topological Collet–Eckmann conditions are equivalent and also invariant under topological conjugacy.

Homomorphic encoders of profinite abelian groups I

Let {Gi|i∈N} be a family of finite Abelian groups. We say that a subgroup G≤∏i∈NGi is order controllable if for every i∈N there is ni∈N such that for each c∈G, there exists a∈G satisfying that a|[1,i]=c|[1,i], supp(a)⊆[1,ni], and order(a) divides order(c|[1,ni]). In this paper we investigate the structure of order controllable subgroups. It is proved that every order controllable, profinite, abelian group contains a subset {gn|n∈N} that topologically generates the group and whose elements gn all have finite support. As a consequence, sufficient conditions are obtained that allow us to encode, by means of a topological group isomorphism, order controllable profinite abelian groups. Further applications of these results to group codes will appear subsequently.

Topological equivalence of submersion functions and topological equivalence of their foliations on the plane: The linear-like case

Let f,g:R2→R be two submersion functions and F(f) and F(g) be the regular foliations of R2 whose leaves are the connected components of the levels sets of f and g, respectively. The topological equivalence of f and g implies the topological equivalence of F(f) and F(g), but the converse is not true, in general. In this paper, we introduce the class of linear-like submersion functions, which is wide enough in order to contain non-trivial behaviors, and provide conditions for the validity of the converse implication for functions inside this class. Our results lead us to a complete topological invariant for topological equivalence in a certain subclass of linear-like submersion functions.

A positively invariant attracting set for a predator-prey model with a non-differentiable functional response

In this work, the ecologically well-posedness of a predator-prey model is demonstrated by showing that a region of the first quadrant is a positively invariant attracting set for the solutions of that model. The predator-prey model considers logistic-type growth in both populations and a non-differentiable functional response that generalizes previous ones. Due to non-differentiability, there is no uniqueness of solutions, and the standard methodology cannot be applied directly. Topological equivalences, geometrical arguments, and the Poincare-Bendixson theorem are used to obtain our result.

Iterated Minkowski sums, horoballs and north-south dynamics

Given a finite generating set A for a group \Gamma , we study the map W \mapsto WA as a topological dynamical system – a continuous self-map of the compact metrizable space of subsets of \Gamma . If the set A generates \Gamma as a semigroup and contains the identity, there are precisely two fixed points, one of which is attracting. This supports the initial impression that the dynamics of this map is rather trivial. Indeed, at least when \Gamma= \mathbb{Z}^d and A \subseteq \mathbb{Z}^d is a finite positively generating set containing the identity, the natural invertible extension of the map W \mapsto W+A is always topologically conjugate to the unique “north-south” dynamics on the Cantor set. In contrast to this, we show that various natural “geometric” properties of the finitely generated group (\Gamma,A) can be recovered from the dynamics of this map, in particular, the growth type and amenability of \Gamma . When \Gamma = \mathbb{Z}^d , we show that the volume of the convex hull of the generating set A is also an invariant of topological conjugacy. Our study introduces, utilizes and develops a certain convexity structure on subsets of the group \Gamma , related to a new concept which we call the sheltered hull of a set. We also relate this study to the structure of horoballs in finitely generated groups, focusing on the abelian case.

Envy-free division in the presence of a dragon

Following a novel approach, where the emphasis is on configuration spaces and equivariant topology, we prove several results addressing the envy-free division problem in the presence of an unpredictable (secretive, non-cooperative) player, called the dragon. There are two basic scenarios. (1) There are \(r-1\) players and a dragon. Once the “cake” is divided into r parts, the dragon makes his choice and grabs one of the pieces. After that, the players should be able to share the remaining pieces in an envy-free fashion. (2) There are \(r+1\) players who divide the cake into r pieces. A ferocious dragon comes and swallows one of the players. The players need to cut the cake in advance in such a way that no matter who is the unlucky player swallowed by the dragon, the remaining players can share the tiles in an envy-free manner. We emphasize that in both settings, some players may prefer to choose degenerate pieces of the cake. Moreover, the players construct in advance both a cut of the cake and a decision tree, allowing them to minimize the uncertainty of what pieces can be given to each of the players.