Topological Point Of View Research Articles

Hydrogen bonding and structural complexity in the Cu5(PO4)2(OH)4 polymorphs (pseudomalachite, ludjibaite, reichenbachite): combined experimental and theoretical study

Hydrogen bonding in the Cu5(PO4)2(OH)4 polymorphs pseudomalachite, ludjibaite and reichenbachite has been studied by low-temperature single-crystal X-ray diffraction (XRD; pseudomalachite) and solid-state density functional theory (DFT; pseudomalachite, ludjibaite, reichenbachite) calculations. Pseudomalachite at 100 K is monoclinic, P21/c, a = 4.4436(4), b = 5.7320(5), c = 16.9300(15) A, β = 91.008(8)°, V = 431.15(7) A3 and Z = 2. The structure has been refined to R 1 = 0.025 for 1383 unique observed reflections with |F o| ≥ 4σF. DFT calculations were done with the CRYSTAL14 software package. For pseudomalachite, the difference between the calculated and experimental H sites does not exceed 0.152 A. Structural configurations around hydroxyl groups in all three polymorphs show many similarities. Each OH5 group is involved in a three-center (bifurcated) hydrogen bond with the H···A distances in the range of 2.141–2.460 A and the D–H···A angles in the range of 122.41°–139.30°, whereas each OH6 group forms a four-center (trifurcated) bond (H···A = 2.093–2.593 A; D–H···A = 122.79°–137.71°). The crystal structures of the Cu5(PO4)2(OH)4 polymorphs are based on three-dimensional frameworks of Cu and P polyhedra. The copper-centered octahedra share edges to form two-dimensional layers parallel to (100) in all three structures. The layers have square voids above and beneath PO4 tetrahedra that link adjacent layers by sharing O atoms with two CuO6 octahedra each. From the topological point of view, none of the polymorphs can be obtained from another by a displacive transformation, and therefore pseudomalachite, ludjibaite and reichenbachite can be viewed as combinatorial polymorphs. According to information-based structural complexity considerations, the three phases are very similar in their configurational entropies and preferential crystallization of one phase over another cannot be entropy driven and is probably governed by other mechanisms that may involve such factors as structures of prenucleation clusters, chemical admixtures, etc.

Arene-Alkene Cycloaddition.

Cycloadditions are among the most efficient chemical processes, combining atom economy, stereospecificity, and the ability to generate molecular complexity in a single step. Aromatic rings would in principle be ideal reaction partners, as they contain, at least from the topological point of view, both olefinic and diene subunits; however, the stability of the conjugated aromatic system would be broken by cycloaddition reactions, which are therefore rarely applied, because kinetics and thermodynamics hinder the process. From that aspect, photochemical activation opens interesting perspectives, as one can selectively provide excess energy to one of the reactants but not to the product, thus preventing thermal back reaction. Indeed, aromatic rings show a rich photochemistry, ranging from isomerizations, substitutions, and additions to cycloadditions. In this review, we will focus on cycloadditions, covering literature from early observations up to the present.

A polymeric silver thiosaccarinate complex with a two-dimensional triply entangled mesh and argentophilic interactions.

Silver(I) complexes with sulfur-donor ligands have a broad range of pharmacological applications. One of the most important factors for tuning the biological activity is the type of donor atom and the ease of ligand replacement. Silver thiosaccharinates display a wide range of structures from mono- to polynuclear complexes. We report the synthesis, crystal structure and vibrational spectroscopic analysis of a two-dimensional Ag(I)-thiosaccharinate coordination polymer, namely poly[tris(μ2-4,4'-bipyridine-κ(2)N:N')bis(μ3-1,1-dioxo-1,2-benzisothiazole-3-thiolato-κ(3)N:S(3):S(3))bis(μ2-1,1-dioxo-1,2-benzisothiazole-3-thiolato-κ(2)S(3):S(3))tetrasilver(I)], [Ag2(C7H4NO2S2)2(C10H8N2)1.5]n, with 4,4'-bipyridine acting as a spacer. A relevant feature of the structure is the presence of an unusually short Ag...Ag separation of 2.8859 (10) Å, well within the range of argentophilic interactions and confirmed as such by Raman analysis of the low-frequency spectrum. From a topological point of view, the structure presents interpenetration in the form of a threefold entangled 2D→2D mesh (2D is two-dimensional).

Topological textures and their bifurcation processes in 2D ferromagnetic thin films

In this paper, by the use of the topological current theory, the topological structures and the dynamic processes in thin-film ferromagnetic systems are investigated directly from the viewpoint of topology. It is found that the topological charge of a thin-film ferromagnetic system can be changed by annihilation or creation processes of opposite polarized vortex–antivortex pairs taking place at space–time singularities of the normalized magnetization vector field of the system, the variation of the topological charge is integer and can further be expressed in terms of the Hopf indices and Brouwer degrees of the magnetization vector field around the singularities. Moreover, the change of the topological charge of the system is crucial to vortex core reversal processes in ferromagnetic thin films. With the help of the topological current theory and implicit function theorem, the processes of vortex merging, splitting as well as vortex core reversal are discussed in detail.

Structural Understanding of Superior Battery Properties of Partially Ni-Doped Li2MnO3 as Cathode Material.

We examined the crystal structures of Li2(NixMn1-x)O3(-δ) (x = 0, 1/10, 1/6, and 1/4) to elucidate the relationship between the structure and electrochemical performance of the compounds using neutron and synchrotron X-ray powder diffraction analyses in combination. Our examination revealed that these crystals contain a large number of stacking faults and exhibit significant cation mixing in the transition-metal layers; the cation mixing becomes significant with an increase in the Ni concentration. Charge-discharge measurements showed that the replacement of Mn with Ni lowers the potential of the charge plateau and leads to higher charge-discharge capacities. From a topological point of view with regard to the atomic arrangement in the crystals, it is concluded that substituting Mn in Li2MnO3 with Ni promotes the formation of smooth Li percolation paths, thus increasing the number of active Li ions and improving the charge-discharge capacity.

Bruckner–Garg-Type Results with Respect to Haar Null Sets inC[0, 1

AbstractA setisshyorHaar null(in the sense of Christensen) if there exists a Borel setand a Borel probability measureμonC[0, 1] such thatandfor allf∈C[0, 1]. The complement of a shy set is called aprevalentset. We say that a set isHaar ambivalentif it is neither shy nor prevalent.The main goal of the paper is to answer the following question: what can we say about the topological properties of the level sets of the prevalent/non-shy manyf∈C[0, 1]?The classical Bruckner–Garg theorem characterizes the level sets of the generic (in the sense of Baire category)f∈C[0, 1] from the topological point of view. We prove that the functionsf∈C[0, 1] for which the same characterization holds form a Haar ambivalent set.In an earlier paper, Balkaet al. proved that the functionsf∈C[0, 1] for which positively many level sets with respect to the Lebesgue measure λ are singletons form a non-shy set inC[0, 1]. The above result yields that this set is actually Haar ambivalent. Now we prove that the functionsf∈C[0, 1] for which positively many level sets with respect to the occupation measure λ ◦f–1are not perfect form a Haar ambivalent set inC[0, 1].We show that for the prevalentf∈C[0, 1] for the genericy∈f([0, 1]) the level setf–1(y) is perfect. Finally, we answer a question of Darji and White by showing that the set of functionsf∈C[0, 1] for which there exists a perfect setPf⊂ [0, 1] such thatfʹ(x) = ∞ for allx∈Pfis Haar ambivalent.

Expansive homeomorphisms on non-Hausdorff spaces

We study expansive dynamical systems from the viewpoint of general topology. We introduce the notions of orbit and refinement expansivity on topological spaces extending expansivity in the compact metric setting. Examples are given on non-Hausdorff compact spaces. Topological and dynamical properties are studied in relation to separability axioms, metrizability, uniform expansivity, asymptotic points and positive expansivity. We also prove that every refinement expansive homeomorphism on a T1 space, in particular every expansive homeomorphism in the usual sense, is conjugate to a subshift with a possibly non-Hausdorff topology on the symbols.

Topological charge pump by surface acoustic waves**Project supported by the National Natural Science Foundation of China (Grant No. 11374036) and the National Basic Research Program of China (Grant No. 2012CB821403).

Quantized electron pumping by the surface acoustic wave across barriers created by a sequence of split metal gates is interpreted from the viewpoint of topology. The surface acoustic wave serves as a one-dimensional periodical potential whose energy spectrum possesses the Bloch band structure. The time-dependent phase plays the role of an adiabatic parameter of the Hamiltonian which induces a geometrical phase. The pumping currents are related to the Chern numbers of the filled bands below the Fermi energy. Based on this understanding, we predict a novel effect of quantized but non-monotonous current plateaus simultaneously pumped by two homodromous surface acoustic waves.

Banach fixed point theorem from the viewpoint of digital topology

Banach fixed point theorem from the viewpoint of digital topology

Identification of glycosylphosphatidylinositol-anchored proteins and ω-sites using TiO2-based affinity purification followed by hydrogen fluoride treatment

Glycosylphosphatidylinositol anchored proteins (GPI-APs) in the outer leaflet of the membrane microdomains, commonly referred to as lipid rafts, play important roles in many biological processes such as signal transduction, cell adhesion, protein trafficking, and antigen presentation. From a topological viewpoint, elucidating the presence and localization of GPI-anchor modification sites (ω-sites) is important for the study of the biophysical properties and anchoring mechanisms of these proteins. However, very few reports have actually identified ω-sites of GPI-APs. To enable large-scale site-specific analysis of GPI anchoring, we developed a method for identification of ω-sites by mass spectrometry by combining titanium dioxide-based affinity purification and hydrogen fluoride treatment. This method was able to identify ~3-fold more GPI-APs than our previous method: the new technique identified a total of 73 ω-sites derived from 49 GPI-APs. In 13 of the 49 GPI-APs identified, the GPI-anchor attached to multiple amino acids in the C-terminal site, yielding a variety of different protein species. This method allows us to simultaneously identify many GPI-AP protein species with different ω-sites. We also demonstrated the C-terminal GPI anchor attachment signal peptide, based on information about the GPI anchor binding sites of 49 GPI-APs. Thus, our results provide evidence for new insight into the GPI-anchored proteome and the role of GPI anchoring. Biological significanceGPI-anchored proteins (GPI-APs) are localized to the outer leaflet of the plasma membranes. Because the GPI anchor is a complex structure, the identification of GPI-anchored peptides by mass spectrometry has always been considered difficult. To improve the feasibility of large-scale site-specific analysis of GPI anchoring, we developed a method for identification of GPI-anchored peptides by combining titanium dioxide–based affinity purification with hydrogen fluoride treatment. Using this novel technique, we identified a total of 73 ω-sites derived from 49 GPI-APs. These data may help us to develop a comprehensive understanding of the GPI-anchored proteome and the role of GPI anchoring. Moreover, this method could be used to discover GPI-APs as candidate biomarkers.

New supersymmetric localizations from topological gravity

Supersymmetric field theories can be studied exactly on suitable off-shell supergravity backgrounds. We show that in two dimensions such backgrounds are identifiable with BRST invariant backgrounds of topological gravity coupled to an abelian topological gauge multiplet. This latter background is required for the consistent coupling of the topological `matter' YM theory to topological gravity. We make use of this topological point of view to obtain, in a simple and straightforward way, a complete classification of localizing supersymmetric backgrounds in two dimensions. The BRST invariant topological backgrounds are parametrized by both Killing vectors and $S^1$-equivariant cohomology of the 2-dimensional world-sheet. We reconstruct completely the supergravity backgrounds from the topological data: some of the supergravity fields are twisted versions of the topological backgrounds, but others are "composite", i.e. they are non-linear functionals of them. We recover all the known localizing 2-dimensional backgrounds and (infinitely) many more that have not been explored so far. We show that the supersymmetric $\Omega$-deformation is nothing but the background value of the ghost-for-ghost of topological gravity, a result which holds for other dimensions too. The new localizing backgrounds are characterized by non-trivial fluxes for both the graviphotons of the supergravity multiplet.

Some results on almost square Banach spaces

We study almost square Banach spaces under a topological point of view. Indeed, we prove that the class of Banach spaces which admits an equivalent norm to be ASQ is that of those Banach spaces which contain an isomorphic copy of c0. We also prove that the symmetric projective tensor products of an almost square Banach space have the strong diameter two property.

Chern–Weil theory and the group of strict contactomorphisms

In this paper we study the groups of contactomorphisms of a closed contact manifold from a topological viewpoint. First we construct examples of contact forms on spheres whose Reeb flow has a dense orbit. Then we show that the unitary group [Formula: see text] is homotopically essential in the group of contactomorphisms of the standard contact sphere [Formula: see text] and present a proof of the homotopy equivalence [Formula: see text]. In the second part of the paper we focus on the group of strict contactomorphisms — using the framework of Chern–Weil theory we introduce and study contact characteristic classes analogous to the Reznikov Hamiltonian classes in symplectic topology. We carry out several explicit calculations illustrating the nontriviality of these contact characteristic classes.

A Graph-Theoretic Framework for Assessing the Resilience of Sectorised Water Distribution Networks

Water utilities face a challenge in maintaining a good quality of service under a wide range of operational management and failure conditions. Tools for assessing the resilience of water distribution networks are therefore essential for both operational and maintenance optimization. In this paper, a novel graph-theoretic approach for the assessment of resilience for large scale water distribution networks is presented. This is of great importance for the management of large scale water distribution systems, most models containing up to hundreds of thousands of pipes and nodes. The proposed framework is mainly based on quantifying the redundancy and capacity of all possible routes from demand nodes to their supply sources. This approach works well with large network sizes since it does not rely on precise hydraulic simulations, which require complex calibration processes and computation, while remaining meaningful from a physical and a topological point of view. The proposal is also tailored for the analysis of sectorised networks through a novel multiscale method for analysing connectivity, which is successfully tested in operational utility network models made of more than 100,000 nodes and 110,000 pipes.

Relative Helicity and Jog Densities in Continuum Descriptions of Dislocations

ABSTRACTDislocations are line like crystal defects mediating plasticity in single crystals. In the current contribution we review classical continuum concepts of dislocation theory from a topological view point. Subsequently, we introduce a new measure for the density of jogs mutually impaired on each other by dislocations on different slip systems. This jog density is closely related to a topological measure of the interlinkage of the dislocations on the involved slip systems, known as relative helicity in other branches of physics.

Spectral spaces of semistar operations

We investigate, from a topological point of view, the classes of spectral semistar operations and of eab semistar operations, following methods recently introduced in [11,13]. We show that, in both cases, the subspaces of finite type operations are spectral spaces in the sense of Hochster and, moreover, that there is a distinguished class of overrings strictly connected to each of the two types of collections of semistar operations. We also prove that the space of stable semistar operations is homeomorphic to the space of Gabriel–Popescu localizing systems, endowed with a Zariski-like topology, extending to the topological level a result established in [14]. As a side effect, we obtain that the space of localizing systems of finite type is also a spectral space. Finally, we show that the Zariski topology on the set of semistar operations is the same as the b-topology defined recently by B. Olberding [37,38].

Topological aspects of holomorphic mappings of hyperquadrics from ℂ2to ℂ3

Based on the results in [Rei14a] we deduce some topological results concerning holomorphic mappings of Levi-nondegenerate hyperquadrics under biholomorphic equivalence. We study the class $\mathcal F$ of so-called nondegenerate and transversal holomorphic mappings sending locally the sphere in $\mathbb C^2$ to a Levi-nondegenerate hyperquadric in $\mathbb C^3$, which contains the most interesting mappings. We show that from a topological point of view there is a major difference when the target is the sphere or the hyperquadric with signature $(2,1)$. In the first case $\mathcal F$ modulo the group of automorphisms is discrete in contrast to the second case where this property fails to hold. Furthermore we study some basic properties such as freeness and properness of the action of automorphisms fixing a given point on $\mathcal F$ to obtain a structural result for a particularly interesting subset of $\mathcal F$.

Hausdorff separability of the boundaries for spacetimes and sequential spaces

There are several ideal boundaries and completions in general relativity sharing the topological property of being sequential, i.e., determined by the convergence of its sequences and, so, by some limit operator L. As emphasized in a classical article by Geroch, Liang, and Wald, some of them have the property, commonly regarded as a drawback, that there are points of the spacetime M non-T1-separated from points of the boundary ∂M. Here, we show that this problem can be solved from a general topological viewpoint. In particular, there is a canonical minimum refinement of the topology in the completion M¯ which T2-separates the spacetime M and its boundary ∂M — no matter the type of completion one chooses. Moreover, we analyze the case of sequential spaces and show how the refined T2-separating topology can be constructed from a modification L∗ of the original limit operator L. Finally, we particularize this procedure to the case of the causal boundary and show how the separability of M and ∂M can be introduced as an abstract axiom in its definition.

A two-dimensional Zn(II) coordination polymer with a three-dimensional supramolecular architecture comprising 5-dimethylamino-isophthalic acid and 1,3-bis(4-pyridyl)propane

ABSTRACTA new zinc coordination polymer, [Zn(L)(bpp)·H2O]n, was synthesized using 5-dimethylamino-isophthalic acid (H2L) and 1,3-bis(4-pyridyl)propane (bpp) as organic linkers. In the complex, adjacent Zn2+ ions are linked by L2− anions to form a one-dimensional (1D) [ZnL]n chain. Then bpp ligands expand the 1D chains into a corrugated two-dimensional (2D) layer network by linking neighboring Zn2+ ions. These adjacent layers are further stacked together by direct C−H···π supramolecular interactions, generating a three-dimensional (3D) supramolecular structure. From the viewpoint of topology, the 2D network can be rationalized to a uninodal four-connected non-interpenetrated sql/Shubnikov tetragonal plane net with {44.62} topology. Moreover, the solid state properties such as thermogravimetric analysis and luminescence were also investigated.

Structural vulnerability analysis of electric power distribution grids

Power grid outages cause huge economical and societal costs. Disruptions in the power distribution grid are responsible for a significant fraction of electric power unavailability to customers. The impact of extreme weather conditions, continuously increasing demand, and the over-ageing of assets in the grid, deteriorates the safety of electric power delivery. Measures to analyse the robustness characteristics and to identify vulnerabilities of power grids are of utmost importance. This paper proposes a metric to quantitatively assess the robustness of power distribution grids from a topological point of view. Real-world data is used to demonstrate the applicability of the proposed metric as a tool to assess the criticality of assets in a distribution grid.