Homeomorphic Research Articles

Entanglement classification from a topological perspective

Classification of entanglement is an important problem in quantum resource theory. In this paper we discuss an embedding of this problem in the context of topological quantum field theories (TQFTs). This approach allows classifying entanglement patterns in terms of topological equivalence classes. In the bipartite case a classification equivalent to the one by stochastic local operations and classical communication (SLOCC) is constructed by restricting to a simple class of connectivity diagrams. Such diagrams characterize quantum states of TQFT up to braiding and tangling of the ``connectome.'' In the multipartite case the same restricted topological classification only captures a part of the SLOCC classes, in particular, it does not see the $W$ entanglement of three qubits. Nonlocal braiding of connections may solve the problem, but no finite classification is attempted in this case. Despite incompleteness, the connectome classification has a straightforward generalization to any number and dimension of parties and has a very intuitive interpretation, which might be useful for understanding specific properties of entanglement and for design of new quantum resources.

On critical renormalization of complex polynomials

Holomorphic renormalization plays an important role in complex polynomial dynamics. We consider invariant continua that are not polynomial-like Julia sets because of extra critical points. However, under certain assumptions, these invariant continua can be identified with Julia sets of lower degree polynomials up to a topological conjugacy. Thus we extend the concept of renormalization.

One-sided topological conjugacy of topological Markov shifts, continuous full groups and Cuntz–Krieger algebras

ABSTRACT We will characterize topologically conjugate one-sided topological Markov shifts in terms of their subgroups of continuous full groups and subalgebras of Cuntz–Krieger algebras.

Topological Isomorphism of Liquid–Vapor, Fusibility, and Solubility Diagrams: Analogues of Gibbs–Konovalov and Gibbs–Roozeboom Laws for Solubility Diagrams

The comprehensive topological isomorphism of liquid–vapor, fusibility, and solubility diagrams in the proper sets of variables is proven with the aid of van der Waals equations of the shift in phase equilibrium. Analogues of Gibbs–Konovalov and Gibbs–Roozeboom laws are demonstrated in solubility diagrams of ternary and quaternary systems under crystallization of different types of solid solutions. For the demonstration, the quaternary reciprocal system K+,NH4+||Cl−,Br−−H2O and its ternary subsystems with modeling of the liquid phase within the framework of the classical Pitzer formalism are mainly used. An algorithm for calculating solubility equilibria in these systems is given.

Dynamics of iteration operators on self-maps of locally compact Hausdorff spaces

AbstractIn this paper, we prove the continuity of iteration operators $\mathcal {J}_n$ on the space of all continuous self-maps of a locally compact Hausdorff space X and generally discuss dynamical behaviors of them. We characterize their fixed points and periodic points for $X=\mathbb {R}$ and the unit circle $S^1$ . Then we indicate that all orbits of $\mathcal {J}_n$ are bounded; however, we prove that for $X=\mathbb {R}$ and $S^1$ , every fixed point of $\mathcal {J}_n$ which is non-constant and equals the identity on its range is not Lyapunov stable. The boundedness and the instability exhibit the complexity of the system, but we show that the complicated behavior is not Devaney chaotic. We give a sufficient condition to classify the systems generated by iteration operators up to topological conjugacy.

NIRDye 812: A molecular platform tailored for multimodal bioimaging applications of targeted fluorescence- and photoacoustic-guided surgery.

The primary treatment for malignant tumors remains to be surgical removal of the diseased tissue. The presence or absence of residual diseased tissue at the tumor margin is the strongest predictor of postoperative prognosis and recurrence. Accordingly, reliance on the ability of surgeons to visually distinguish diseased tissue from healthy tissue unambiguously in real time is crucial. Near infrared-I (NIRI) fluorescence-emitting targeting biomolecular constructs such as anticancer antibody-fluorophore conjugates, namely cetuximab-IRDye® 800CW (CTB-IRDye® 800CW), are FDA-approved for clinical trial usage in the fluorescence-guided resection of diseased tissue due to affording improved direct visualization of tumor tissue when compared to the use of either the unaided eye under standard white light illumination (WLI) surgical techniques or non-targeting fluorophores. Unfortunately, though helpful, CTB-IRDye® 800CW affords limited (i) identification of diseased tissue and (ii) tumor margin delineation, because the immunoconjugate generates suboptimal tumor-to-background ratios (TBRs) as a result of its spectral/photophysical profiles poorly aligning with the fixed optical windows of pre-/clinical setups. As such, CTB-IRDye® 800CW is more prone to affording incomplete resection compared to if TBRs were higher due to otherwise. To aid in accurately identifying deep-seated diseased tissue, photoacoustic (PA) tomography has been implemented alongside CTB-IRDye® 800CW to achieve PA signals that could result in higher TBRs. However, in clinical trial practice, using IRDye® 800CW for PA imaging also yields subpar TBRs due to it affording low PA signals. To overcome such limitations, we developed NIRDye 812, a structurally-modified topological equivalent of IRDye® 800CW, to confer it the capability to yield both higher TBRs and superior PA signal than that of the equivalent CTB-conjugate and fluorophore IRDye® 800CW itself, respectively. To do so, we substituted the oxygen atom at its meso-position with a sulfur atom. CTB-NIRDye 812 demonstrated a red-shifted absorption wavelength at 796nm and a peak NIR-I fluorescence emission wavelength at 820nm, which better dovetails with the fixed windows of preinstalled fixed emission filters within commercial pre-/clinical NIR-I fluorescence imaging instruments. Overall, CTB-NIRDye 812 provided a∼2-fold increase in TBRs compared to those of CTB-IRDye® 800CW in vivo. Also, NIRDye 812 displayed an ∼60% higher PA signal than that of IRDye® 800CW. Collectively, we achieved our goal of improving upon the spectral/photophysical and PA properties of IRDye® 800CW via introducing a subtle modification to its electronic core such that its CTB immunoconjugate could potentially allow for fast track or breakthrough designation by the FDA due to its near-identical structure displaying considerably improved efficacy.

Stone space partitions indexed by a poset

Stone space partitions $\{X_{p}\mid p\in P\}$ satisfying conditions like $\bar{X_{p}}=\bigcup_{q\leqslant p}X_{q}$ for all $p\in P$, where $P$ is a poset or PO system (poset with a distinguished subset), arise naturally in the study both of primitive Boolean algebras and of $\omega$-categorical structures. A key concept for studying such partitions is that of a $p$-trim open set which meets precisely those $X_{q}$ for which $q\geqslant p$; for Stone spaces, this is the topological equivalent of a pseudo-indecomposable set. This paper develops the theory of infinite partitions of Stone spaces indexed by a poset or PO system where the trim sets form a neighbourhood base for the topology. We study the interplay between order properties of the poset/PO system and topological properties of the partition, examine extensions and completions of such partitions, and derive necessary and sufficient conditions on the poset/PO system for the existence of the various types of partition studied. We also identify circumstances in which a second countable Stone space with a trim partition indexed by a given PO system is unique up to homeomorphism, subject to choices on the isolated point structure and boundedness of the partition elements. One corollary of our results is that there is a partition $\{X_{r}\mid r\in[0,1]\}$ of the Cantor set such that $\bar{X_{r}}=\bigcup_{s\leqslant r}X_{s}\text{ for all }r\in[0,1]$.

Topological Equivalence of Stripy States and Skyrmion Crystals

Stripy states, consisting of a collection of stripy spin textures, are the precursors of skyrmion crystals (SkXs). The common belief is that stripy states and SkXs are topologically unconnected and that transitions between SkXs and stripy states are phase transitions. Here, we show that both stripy states and SkXs are skyrmion condensates and they are topologically equivalent. By gradually tuning the stripe whose width goes from smaller than to larger than skyrmion-skyrmion separation, the structure of a skyrmion condensate transforms smoothly and continuously from various stripy phases, including helical states and mazes, to crystals, showing that stripy states are topologically connected to SkXs.

Topological equivalence of linear time-varying control systems

Abstract In this paper, we mainly studied the topological equivalence of linear time-varying (LTV) control system $\dot{x}\left ( t\right )=A(t)x(t)+B(t)u(t)$ defined on an interval $I \subset \mathbb{R}^{+}$. After giving a new definition of the topological equivalence, we investigated the local equivalence of LTV control systems under two new hypotheses. These hypotheses were made by the local behavior of Krylov indices (which turned out to be controllability indices for the linear time-invariant (LTI) control systems). It was found out that Krylov indices play an important role in the classification problem of LTV control systems. Compared with our former work on the topological equivalence of LTI control systems, new methods and techniques were taken to deal with new difficulties occurred for LTV control systems.

A kind of characterization of homeomorphism and homeomorphic spaces by Core fundamental groupoid: a good invariant

In this paper, we give a new topological invariant and a kind of characterization to homeomorphisms. We introduce this topological invariant by constructing an algebraic kind of groupoid structure to each topological space, which is an extension of the concept of fundamental groups. This construction we call it “Core fundamental groupoid” and it is different from the fundamental groupoid as in the book and articles of Ronald Brown [30, 31]. Moreover, both groupoid notions are not equivalent, but, it will be a wide subgroupoid of the fundamental groupoid in category-theoretic. In the entire paper, we consider groupoid in the algebraic sense. First, we discuss basic properties of the Core fundamental groupoid and their significant importance in topological invariants. Further, we separately present the Core fundamental groupoid as an algebraic structure and a topo-algebraic structure and investigate their properties separately. We have an explicit description of both the algebraic structure of the groupoid and a unique topological structure of Core fundamental groupoid. Besides, we give a kind of characterization of homeomorphisms in terms of an invariant that we have obtained, i.e., “f : M → N is a homeomorphism if and only if f# 1M → 1N is a topological groupoid isomorphism”. This result addresses the open problem stated in [1, 19, 24, 26] and also gives a kind of characterization for homeomorphic spaces, but, computationally it seems difficult and it becomes sometimes trivial when intact with topology. Induced groupoid homomorphism, induced base map, topological and smooth structure on the Core fundamental groupoid, homotopic properties of induced groupoid homomorphisms of continuous maps and topological groupoid homomorphism-related properties of induced groupoid homomorphisms are discussed. We also present the relation of homotopy type on Core fundamental groupoid.

Equivalent dimensionality reduction method of doubly‐fed induction generator model and discrimination of Hopf bifurcation type

AbstractThe strong coupling, multivariate, and nonlinear characteristics of the mathematical model of a doubly‐fed induction generator (DFIG) make it challenging to analyze the bifurcation category of the DFIG. Therefore, in this study, we developed a method for analyzing the DFIG model via topologically equivalent dimensionality reduction in the neighborhood of bifurcation values based on the central manifold theorem and realized the discrimination of the Hopf bifurcation type of the DFIG. We established a complex frequency domain model of the DFIG and then calculated the critical open‐loop gain point using the root trajectory method to obtain the relationship between the critical open‐loop gain point and Hopf bifurcation. Thereafter, we performed topological equivalence dimensionality reduction in the neighborhood of the DFIG bifurcation values based on the central manifold theorem. Subsequently, the Hopf bifurcation type of the system was determined using the stability index (first Lyapunov exponent) of the Hopf bifurcation. Finally, a nonlinear time‐domain simulation was used to verify that the four and two‐dimensional DFIGs generate a stable limit cycle by supercritical Hopf bifurcation under the tuning parameter. The results indicated that the proposed method was feasible and effective in the neighborhood of bifurcation values in the state parameter space of DFIG, which provides a theoretical basis for the control of Hopf bifurcation of the DFIG.

Link as a complete invariant of Morse-Smale 3-diffeomorphisms

In this paper we consider gradient-like Morse-Smale diffeomorphisms defined on the three-dimensional sphere S3 . For such diffeomorphisms, a complete invariant of topological conjugacy was obtained in the works of C. Bonatti, V. Grines, V. Medvedev, E. Pecu. It is an equivalence class of a set of homotopically non-trivially embedded tori and Klein bottles embedded in some closed 3-manifold whose fundamental group admits an epimorphism to the group Z . Such an invariant is called the scheme of the gradient-like diffeomorphism f:S3→S3 . We single out a class G of diffeomorphisms whose complete invariant is a topologically simpler object, namely, the link of essential knots in the manifold S2×S1 . The diffeomorphisms under consideration are determined by the fact that their non-wandering set contains a unique source, and the closures of stable saddle point manifolds bound three-dimensional balls with pairwise disjoint interiors. We prove that, in addition to the closure of these balls, a diffeomorphism of the class G contains exactly one nonwandering point, which is a fixed sink. It is established that the total invariant of topological conjugacy of class G diffeomorphisms is the space of orbits of unstable saddle separatrices in the basin of this sink. It is shown that the space of orbits is a link of non-contractible knots in the manifold S2×S1 and that the equivalence of links is tantamount to the equivalence of schemes. We also provide a realization of diffeomorphisms of the considered class along an arbitrary link consisting of essential nodes in the manifold S2×S1 .

Comparison of some dynamical systems on the quotient space of the Sierpinski tetrahedron

In this paper, it is aimed to construct two different dynamical systems on the Sierpinski tetrahedron. To this end, we consider the dynamical systems on a quotient space of $\{ 0,1,2,3 \}^{\mathbb{N}}$ by using the code representations of the points on the Sierpinski tetrahedron. Finally, we compare the periodic points to investigate topological conjugacy of these dynamical systems and we conclude that they are not topologically equivalent.

Combining rational maps and Kleinian groups via orbit equivalence

AbstractWe develop a new orbit equivalence framework for holomorphically mating the dynamics of complex polynomials with that of Kleinian surface groups. We show that the only torsion‐free Fuchsian groups that can be thus mated are punctured sphere groups. We describe a new class of maps that are orbit equivalent to Fuchsian punctured sphere groups. We call these higher Bowen–Series maps. The existence of this class ensures that the Teichmüller space of matings has one component corresponding to Bowen–Series maps and one corresponding to higher Bowen–Series maps. We also show that, unlike in higher dimensions, topological orbit equivalence rigidity fails for Fuchsian groups acting on the circle. We also classify the collection of Kleinian Bers boundary groups that are mateable in our framework.

On the “traveling pulses” of the limit of the FitzHugh–Nagumo equation when [formula omitted

A solution (u(s),v(s)) of the differential system u′=v,v′=−cv−u(u−a)(1−u)+w,w′=−(ɛ/c)(u−γw).with a,c,ɛ∈R such that (u(s),v(s))→(0,0) when s→±∞ is a traveling pulse of the FitzHugh–Nagumo equation. The limit of this differential system when ɛ→0 gives rise to the polynomial differential system u′=v,v′=−cv−u(u−a)(1−u)+w,where now a,c,w∈R. We give the complete description of its phase portraits in the Poincaré disc (i.e. in the compactification of R2 adding the circle S1 of the infinity) modulo topological equivalence.

Envy-free division via configuration spaces

The classical approach to envy-free division and equilibrium problems arising in mathematical economics typically relies on Knaster-Kuratowski-Mazurkiewicz theorem, Sperner's lemma or some extension involving mapping degree. We propose a different and relatively novel approach where the emphasis is on configuration spaces and equivariant topology, originally developed for applications in discrete and computational geometry (Tverberg type problems, necklace splitting problem, etc.). We illustrate the method by proving several counterparts and extensions of the classical envy-free division theorem of David Gale, where the emphasis is on preferences allowing the players to choose degenerate pieces of the cake.

Generation of various vortex beams based on transformation materials

A flexible vortex beam generator (VBG) is proposed based on transformation optics (TO) to convert input plane waves to various output vortex beams. The methodology is a geometrical one based on the topological equivalence between the vortex beam cross section disc and a belt, turning the in-plane curvilinear distributed phase or polarization structures in the disc to be redistributed around the belt. Consequently, abundant vortex beams can be exported by simply configuring the incidence angles and polarization directions of the plane waves. The topology and wave propagation characteristics behind the system are investigated, and TO is used to obtain the desired materials by adopting a specific space mapping. Some sample numerical simulations are described to verify the feasibility of the proposed design, including several complex orbital angular momentum and cylindrical vector beams. Due to its flexibility and simplicity, the proposed VBG is likely to have extensive application prospects.

On the Milnor number of non‐isolated singularities of holomorphic foliations and its topological invariance

AbstractWe define the Milnor number of a one‐dimensional holomorphic foliation as the intersection number of two holomorphic sections with respect to a compact connected component of its singular set. Under certain conditions, we prove that the Milnor number of on a three‐dimensional manifold with respect to is invariant by topological equivalences.

Unraveling the bond structure, porosity, and mechanical properties amorphous ZIF-4 and its topological equivalents: Large scale ab initio calculations

Four large-scale amorphous zeolite imidazolate frameworks (a-ZIFs) have been constructed based on the continuous random network model. Structural properties correlation (SPC) between the mechanical properties and structural order (short-range, mid-range, or long-range orders) of ZIFs samples have been investigated by well-defined density functional theory calculations. The results demonstrated the effect of short-range and mid-range local order on the mechanical properties of the a-ZIFs, respectively. By combining different metal nodes and organic ligands, the porosity of a-ZIFs can be dramatically changed, leading to changes in Young’s modulus and shear modulus. The most important, SPC between structural order and mechanical properties of a-ZIFs is established, which will pave the way to designing high-strength and toughness ZIF-based glass.

Triple-meron crystal in high-spin Kitaev magnets

Skyrmions hold great promise in future spintronics applications since they are robust against local deformations. The meron, due to its topological equivalence to a half skyrmion, has been widely found to appear in pairs. Motivated by recent progresses in high-spin Kitaev magnets, here we investigate numerically a classical Kitaev-Γ model with a single-ion anisotropy. An exotic spin texture consisting of three merons is discovered. Such a state features a peculiar property with an odd number of merons in one magnetic unit cell. Therefore, these merons cannot be dissociated from skyrmions as reported in the literature and their origin is briefly discussed. Moreover, we find that these three merons contribute a finite topological number and thus it can induce the topological Hall effect (THE). Experimentally this spin texture can be observed by the Lorentz transmission electron microscopy and the THE can be used to identify the finite topological number. Our work demonstrates that high-spin Kitaev magnets can host robust unconventional spin textures and thus they offer a versatile platform for exploring exotic spin textures as well as their applications in spintronics.