Topological Framework Research Articles

Two 3D Coordination Frameworks Based on Nanosized Huge Ln26 (Ln = Dy and Gd) Spherical Clusters

Two novel three-dimensional (3D) coordination polymers Zn1.5Dy26(IN)25(CH3COO)8(CO3)11(OH)26(H2O)29 (1) and Zn1.5Gd26(IN)26(CH3COO)7(CO3)11(OH)26(H2O)28 (2) based on the linkages of large nanosized spherical hydroxo Ln26 clusters and zinc centers by organic ligands have been hydrothermally synthesized. Metal organic framework (MOF) 1 crystallizes in the triclinic space group P1, with a = 21.107(2) A, b = 21.185(2) A, c = 36.323(4) A, α = 89.001(2)°, β = 82.486(2)°, γ = 68.359(2)°, V = 14960(3) A3, Z = 2. MOF 2 crystallizes in the triclinic space group P1, a = 19.6213(15) A, b = 22.1850(17) A, c = 34.654(3) A, α = 88.5340(210)°, β = 85.6520(10)°, γ = 72.9790(19)° V = 14382.5(19) A3, Z = 2. Structural analysis indicates that both 3D polymers can be constructed using the building unit of CO3@Ln26 (Dy for 1, Gd for 2) and exhibit similar topological frameworks. During the synthesis, three ligands were used. CO32− plays a very important role in the formation of the spherical Ln26 cluster. CH3COO− makes the L...

A novel 3D (4, 8)-connected metal-organic framework of scu topology with two types of 1D channel constructed by methylenediisophthalic acid (H4MDIP)

A novel coordination polymer 1 {[Zn5(MDIP)2(H2O)6(μ3-OH)2]·2H2O}n has been synthesized using methylenediisophthalic acid (H4MDIP) under hydrothermal conditions. Single-crystal X-ray diffraction analysis indicates that 1 shows a 3D porous framework with two types of 1D channel. The structure is further simplified by topology analysis which indicates a (4, 8)-connected net with a Schläfli symbol of (44.62)2(416.612) topology. Moreover, compound 1 displays a strong blue photoluminescent property.

Three NEW butane-1,2,3,4-tetracarboxylato-bridged coordination polymers: Syntheses, crystal structures and charaterization

Three new butane-1,2,3,4-tetracarboxylato bridged coordination polymers [Zn2(H2O)4(BTC)]·3H2O 1, [Cd2(H2O)4(BTC)]·2H2O 2 and [Y2(H2O)6(H2BTC)(BTC)]·5H2O 3 (H4BTC = butane-1,2,3,4-tetracarboxylic acid) were synthesized and characterized by IR spectroscopy, TG–DTA analyses, elemental analyses and single-crystal X-ray diffraction, and their fluorescence properties were also investigated. In 1, the trigonal pyramidally coordinated Zn atoms are brigded by both μ6η6 and μ6η8 centrosymmetric BTC4− anions into 2D layers, which are connected by the octahedrally coordinated Zn atoms to build up 3D (44·610·8)(44·62)2(44·68·82·10) topological framework with the lattice water molecules in cavities. In 2, the CdO7 pentagonal bipyramids and the CdO8 4,4′-bicapped triagonal antiprisms are edge-shared to generate the Cd2O13 bipolyhedra, which are further corner-shared into 1D metal-oxide chains. The resulting chains are interconnected by the μ6η8 BTC4− anion to form 2D (43)2(46·66·83) networks, which are assembled via interlayer hydrogen bonds into a 3D (411·611·86)(43)(49·6) topological supramolecular architecture with the lattice H2O molecules located in holes. The distorted 4,4′-bicapped triagonal antiprismatically coordinated Y atoms in 3 are interconnected μ4η4 H2BTC2− and μ4η6 BTC4− anions to from 2D (44·62) topological networks, which are held together via strong interlayer hydrogen bonds into a 3D fsc topological supramolecular architecture with the lattice water molecules residing in cavities.

Four novel topological frameworks based on 4,4′-(hexafluoroisopropylidene)diphthalic acid and 1,1′-(1,4-butanediyl)bis(imidazole) ligand

Four new coordination polymers, constructed from 4,4′-(hexafluoroisopropylidene)diphthalic acid (H4L) and 1,1′-(1,4-butanediyl)bis(imidazole) (biim-4) ligand, have been synthesized: [Cd(HL)(Hbiim-4)(biim-4)0.5(H2O)]·2H2O (1), [Cd2(L)(biim-4)0.5(H2O)2]·0.5H2O (2), [Mn2(L)(biim-4)0.5(H2O)2]·0.5H2O (3), and [Zn2(L)(biim-4)1.5]·(CH3CH2OH) (4), where H4L = 4,4′-(hexafluoroisopropylidene)diphthalic acid, and biim-4 = 1,1′-(1,4-butanediyl)bis(imidazole). Compound 1 displays a 1D chain structure, which is extended by intermolecular hydrogen-bonding interactions to give a novel trinodal (4,5)-connected (4·64·8)(42·63·8)(43·64·83) topology. Compounds 2 and 3 are nearly isostructural with the same uncommon (5,6)-connected frameworks. Compound 4 shows an unprecedented self-penetrating (4,5)-connected net with Schlafli symbol of (42·6·72·8)(62·72·82)(42·63·73·82). The structural and topological differences of these four compounds indicate that the pH values and the central metals have significant effects on producing novel structures and topologies of the coordination complexes. The thermogravimetric and photoluminescent properties were also investigated for the compounds.

Modelling leaf shape dynamics in rice

Modelling leaf shape is a helpful tool for designing optimal plant shape and visualizing plant growth. The objectives of this study were to characterize the changes in patterns of leaf growth characteristics during plant development, and to model the changes in morphology of the different leaves of the rice ( Oryza sativa L.) plant. In four experiments, time course observations were made on length and width of the leaves on the main stem and tillers of rice plants grown under different experimental conditions. Three experiments provided the data for developing a model. The results show that the expansion process of a single leaf on the main stem and tillers against growing degree days (GDD) followed a slow–rapid–slow pattern that was described well by a logistic function. The changes in final length of the regular leaves positioned on the main stem were described by a quadratic function that differed from the function for the flag leaf. Final leaf length on the tillers followed the ratio of the length of the simultaneously expanding leaves on the tiller to the length of the corresponding leaves on the main stem. The changes in leaf width with leaf length in relation to GDD for a single leaf on the main stem and tillers were described by a quadratic function. The general relationships between final leaf length and final leaf width for the first fully expanded leaves and the flag leaves on the main stem and tillers were described by exponential functions, whereas for the other leaves quadratic functions were needed. The effects of nitrogen level and water regime on leaf growth were quantified by using leaf-nitrogen and leaf-water contents. The model, which was validated with the independent data from the fourth experiment, predicted the time course morphological changes of leaves on the main stem and tillers of rice grown with different nitrogen and water levels well. The results of this study can lead to a more accurate description of the rice leaf in a 3D space by integrating the model with the morphogenesis models of sheath, internode and panicle within the framework of plant topology. In addition, the present leaf architecture model may be combined with a leaf emergence model and a leaf curvature model for digital representation of light interception and utilization within the leaf canopy of rice.

Four-dimensional address topology for circuits with stacked multilayer crossbar arrays

We present a topological framework that provides a simple yet powerful electronic circuit architecture for constructing and using multilayer crossbar arrays, allowing a significantly increased integration density of memristive crosspoint devices beyond the scaling limits of lateral feature sizes. The truly remarkable feature of such circuits, which is an extension of the CMOL (Cmos + MOLecular-scale devices) concept for an area-like interface to a three-dimensional system, is that a large-feature-size complimentary metal-oxide-semiconductor (CMOS) substrate can provide high-density interconnects to multiple crossbar layers through a single set of vertical vias. The physical locations of the memristive devices are mapped to a four-dimensional logical address space such that unique access from the CMOS substrate is provided to every device in a stacked array of crossbars. This hybrid architecture is compatible with digital memories, field-programmable gate arrays, and biologically inspired adaptive networks and with state-of-the-art integrated circuit foundries.

Graph exploration with robot swarms

PurposeA simultaneous solution to the localization and mapping problem of a graph‐like environment by a swarm of robots requires solutions to task coordination and map merging. The purpose of this paper is to examine the performance of two different map‐merging strategies.Design/methodology/approachBuilding a representation of the environment is a key problem in robotics where the problem is known as simultaneous localization and mapping (SLAM). When large groups of robots operate within the environment, the SLAM problem becomes complicated by issues related to coordination of the elements of the swarm and integration of the environmental representations obtained by individual swarm elements. This paper considers these issues within the formalism of a group of simulated robots operating within a graph‐like environment. Starting at a common node, the swarm partitions the unknown edges of the known graph and explores the graph for a pre‐arranged period. The swarm elements then meet at a particular time and location to integrate their partial world models. This process is repeated until the entire world has been mapped. A correctness proof of the algorithm is presented, and different coordination strategies are compared via simulation.FindingsThe paper demonstrates that a swarm of identical robots, each equipped with its own marker, and capable of simple sensing and action abilities, can explore and map an unknown graph‐like environment. Moreover, experimental results show that exploration with multiple robots can provide an improvement in exploration effort over a single robot and that this improvement does not scale linearly with the size of the swarm.Research limitations/implicationsThe paper represents efforts toward exploration and mapping in a graph‐like world with robot swarms. The paper suggests several extensions and variations including the development of adaptive partitioning and rendezvous schedule strategies to further improve both overall swarm efficiency and individual robot utilization during exploration.Originality/valueThe novelty associated with this paper is the formal extension of the single robot graph‐like exploration of Dudek et al. to robot swarms. The paper here examines fundamental limits to multiple robot SLAM and does this within a topological framework. Results obtained within this topological formalism can be readily transferred to the more traditional metric representation.

Synthesis and Structures of Silver(I) Adducts with 4-Amino-4H-1,2,4-triazole

Reactions of silver(I) salts with 4-amino-4H-1,2,4-triazole (4-NH2-tz) in a 1 : 1 molar ratio yield four new silver adducts, namely [Ag(4-NH2-tz)](CF3SO3) (1), [Ag(4-NH2-tz)1.75](ClO4) (2), [Ag(4- NH2-tz)1.75](BF4) (3) and [Ag(4-NH2-tz)1.75](NO3) (4). X-Ray analysis shows that 1 exhibits a 3D cationic [Ag(4-NH2-tz)]+ metal-organic framework of srs topology; 2, 3 and 4 are isostructural and display an infinite chain structure containing an Ag4tz6 cluster.

A groupoid approach to discrete inverse semigroup algebras

Let K be a commutative ring with unit and S an inverse semigroup. We show that the semigroup algebra KS can be described as a convolution algebra of functions on the universal étale groupoid associated to S by Paterson. This result is a simultaneous generalization of the author's earlier work on finite inverse semigroups and Paterson's theorem for the universal C ∗ -algebra. It provides a convenient topological framework for understanding the structure of KS, including the center and when it has a unit. In this theory, the role of Gelfand duality is replaced by Stone duality. Using this approach we construct the finite dimensional irreducible representations of an inverse semigroup over an arbitrary field as induced representations from associated groups, generalizing the case of an inverse semigroup with finitely many idempotents. More generally, we describe the irreducible representations of an inverse semigroup S that can be induced from associated groups as precisely those satisfying a certain “finiteness condition.” This “finiteness condition” is satisfied, for instance, by all representations of an inverse semigroup whose image contains a primitive idempotent.

Metal naphthalenedicarboxylates with diverse network architectures: Synthesis, crystal structures and properties

Reactions of 1,4-naphthalenedicarboxylic acid (1,4-H 2nda) and a bent dipyridyl co-ligand 4-amino-3,5-bis(4-pyridyl)-1,2,4-triazole (4bpt) with Co II, Zn II, or Cd II acetate afford four coordination polymers, [Co(1,4-nda)(4bpt)(H 2O)] n ( 1), {[Co(1,4-nda) 2(4bpt)(H 2O) 2][Co(4bpt)(H 2O) 4](H 2O) 1.5} n ( 2), {[Zn(1,4-nda) 2(4bpt)(H 2O) 2][Zn(4bpt)(H 2O) 4](H 2O) 1.5} n ( 3), and {[Cd 2(1,4-nda) 2(4bpt) 2(H 2O) 2](DMF) 1.5(H 2O) 3} n ( 4). In the structure of the Co II complex 1, the polycatenation of inclined 2-D (4,4) coordination layers leads to the formation of a 3-D supramolecular framework, whereas two types of 1-D polymeric chains are observed in another Co II coordination species 2, which are interconnected via H-bonding to result in an unusual 3-D host–guest lattice. Notably, complexes 1 and 2 have been prepared under similar hydrothermal conditions and their structural discrepancy can only be ascribed to a subtle change of basicity for the reaction solution. The Zn II complex 3 is isostructural to 2, and the Cd II complex 4 displays a 2-fold parallel interpenetrating array of undulating (4,4) coordination layers. By using the conventional solvent evaporation method, two Pb II naphthalenedicarboxylates [Pb(1,4-nda)(DMF)] n ( 5) and {[Pb 2(2,6-nda) 2(DMF) 2](DMF)} n ( 6) have also been prepared (2,6-nda = 2,6-naphthalenedicarboxylate). Complex 5 has a unique 5-connected 3-D coordination architecture, whereas 6 represents a 3-fold interpenetrating framework of 4-connected diamond topology. Their structural difference suggests the significant isomeric effect of the naphthalenedicarboxylate tectons on structural assemblies. Thermal stability of these crystalline materials has been investigated by thermogravimetric and differential thermal analysis (TG–DTA) technique and solid-state luminescent properties of the Zn II, Cd II, and Pb II complexes have also been explored.

An investigation into the semantics of English topological prepositions

An investigation into the semantics of English topological prepositions

A new (3,4,7)-connected topology framework based on [Cd2(CO2)3O4N2] second building unit

Abstract A new 3D coordination polymer [Cd 2 (L 3 )(BTC)(H 2 O)] ( 1 ) (HL 3 = 3,5-bis(pyridin-3-ylmethoxy)benzoic acid, H 3 BTC = 1,3,5-benzenetricarboxylic acid), has been isolated under hydrothermal condition and characterized by single-crystal X-ray diffraction. Compound 1 is constructed from Cd 2 -based second building units (SBUs) [Cd 2 (CO 2 ) 3 O 4 N 2 ] and displays a 3D (3,4,7)-connected net with (4 2 ·6)(4 3 ·6 3 )(4 5 ·6 11 ·8 5 ) topology. In addition, the photoluminescent spectra indicate compound 1 may be a good candidate for blue-luminescent materials.

Computational Algebraic Topology

The workshop was conducted jointly with a workshop in statistical learning theory. There was substantial interaction between the two groups, both formally in terms of talks attended by members of both groups, as well as via informal discussions. The intellectual themes which were presented during the workshop are described below. Sensor nets and engineering applications: In the opening talk R. Ghrist spoke about the topology necessary to develop methods for determining intruders have entered a net of sensors, and for counting their number. Ghrist, jointly with V. de Silva and Y. Baryshnikov, has developed techniques based directly on homological calculations as well as on integrals over Euler characteristics which hold promise for implementable algorithms. In order for such algorithms to be maximally useful, one must develop error insensitive methods, which will require more probabilistic methods to be included within the algebraic topological framework. Combinatorial applications: Several presentations at the workshop elaborated on the subject of combinatorial algebraic topology. D. Kozlov has given a survey talk, which has set the accents on the subject, tying together structures, methods, and applications, as these are present at the current state of the development. Talks by R. Jardine and M. Raussen concerned the combinatorial and computational aspects of homotopy theory, finding applications of such abstract notions as Quillen's closed model category. K. Knudson gave an interesting account of connections between persistent homology and discrete morse theory. Finally, the talk of E. Babson dealt with more probabilistic aspects and served as a bridge to the presentations of M. Kahle and P. Bubenik. Dynamical systems: K. Mischaikow and S. Day spoke about the use of algebraic topology to understand the qualitative structure of dynamical systems. Mischaikow introduced his paradigm of building databases of dynamical systems based on choices of parameter values. His methods permit the construction of partitions of parameter space within which the qualitative structure remains the same. In addition, Conley index methods, or rather their computational versions, are used to prove the existence of fixed points, recurrent points, and invariant subsets within a given region in a spatial domain. Data analysis: G. Carlsson and V. de Silva spoke about applications of various kinds of diagrams to understand the qualitative geometric nature of data sets. For example, persistence diagrams allow one to recover Betti numbers of sublevel sets of a probability distribution, multidimensional persistence allows one to study sublevel sets of various functions as well, and the analysis of structure theorems for certain kinds of quivers permits one to extend the bootstrap methods to clustering, Betti numbers, as well as to perform dynamic clustering (i.e. clustering over time). There are now viable computational methods for all of these applications. Probabilistic methods: M. Kahle and P. Bubenik spoke about the beginnings of stochastic algebraic topology. Work at the level of zeroth Betti numbers has already been carried out by M. Penrose, under the heading of “geometric random graphs”. What is now needed is an extension of this work to higher dimensional homology groups, as well as to the barcodes which arise in persistent homology. Ultimately, precise results along these lines will open up the possibility of direct evaluation of significance of various qualitative observations given a null hypothesis. There were also several talks more centered at applications, such as vision recognition (J. Giesen) and material science (R. MacPherson). All things considered, the workshop was a great success in terms of scientific interaction, both within this group, as well as with the researchers in statistical learning theory, as was witnessed by many involved discussions, which often lasted well into the late evenings.

ParTopS: compact topological framework for parallel fragmentation simulations

Cohesive models are used for simulation of fracture, branching and fragmentation phenomena at various scales. Those models require high levels of mesh refinement at the crack tip region so that nonlinear behavior can be captured and physical results obtained. This imposes the use of large meshes that usually result in computational and memory costs prohibitively expensive for a single traditional workstation. If an extrinsic cohesive model is to be used, support for dynamic insertion of cohesive elements is also required. This paper proposes a topological framework for supporting parallel adaptive fragmentation simulations that provides operations for dynamic insertion of cohesive elements, in a uniform way, for both two- and three-dimensional unstructured meshes. Cohesive elements are truly represented and are treated like any other regular element. The framework is built as an extension of a compact adjacency-based serial topological data structure, which can natively handle the representation of cohesive elements. Symmetrical modifications of duplicated entities are used to reduce the communication of topological changes among mesh partitions and also to avoid the use of locks. The correctness and efficiency of the proposed framework are demonstrated by a series of arbitrary insertions of cohesive elements into some sample meshes.

Syntheses, crystal structures and luminescent properties of two silver complexes of N,N,N′,N′-tetra(diphenylphosphanylmethyl)ethylene diamine

Treatment of a suspension of AgNO 3 and AgCl in MeOH with a solution of N,N,N′,N′-tetra(diphenylphosphanylmethyl)ethylene diamine (dppeda) in CHCl 3 afforded a binuclear complex [Ag 2(dppeda)Cl](NO 3)·2MeOH ( 1). The analogous reactions using AgSCN and dppeda in EtOH/CH 2Cl 2 gave rise to a polymeric complex [Ag 2(dppeda)(SCN) 2] n ( 2). Both compounds were fully characterized by elemental analyses, IR spectra, 1H( 31P) NMR, and single-crystal X-ray crystallography. The cation of 1 shows an interesting molecular basket framework in which dppeda adopts a side-by-side coordination mode. Compound 2 possesses an unique 2D (6,3) layer network with 34-membered metallomacrocycles in which dppeda takes a end-to-end coordination mode. The 2D topological framework of 2 is rare in the chemistry of tetraphosphines. The photoluminescent properties of 1 and 2 in solid state at ambient temperature were investigated.

Topological description of the Stone-Wales defect formation energy in carbon nanotubes and graphene

We develop a topological continuum framework to compute the formation energies of Stone-Wales defects in graphene and carbon nanotubes. Our approach makes no a priori assumptions about the analytical form of the dislocation strain fields while explicitly accounting for boundary conditions and defect-defect interactions. The continuum formalism reproduces trends observed in the atomistic simulations remarkably well and demonstrates the necessity of considering long-ranged effects to accurately describe defect energetics in graphene-based systems.

Topology on words

We investigate properties of topologies on sets of finite and infinite words over a finite alphabet. The guiding example is the topology generated by the prefix relation on the set of finite words, considered as a partial order. This partial order extends naturally to the set of infinite words; hence it generates a topology on the union of the sets of finite and infinite words. We consider several partial orders which have similar properties and identify general principles according to which the transition from finite to infinite words is natural. We provide a uniform topological framework for the set of finite and infinite words to handle limits in a general fashion.

Digital Topology on Adaptive Octree Grids

The theory of digital topology is used in many different image processing and computer graphics algorithms. Most of the existing theories apply to uniform cartesian grids, and they are not readily extensible to new algorithms targeting at adaptive cartesian grids. This article provides a rigorous extension of the classical digital topology framework for adaptive octree grids, including the characterization of adjacency, connected components, and simple points. Motivating examples, proofs of the major propositions, and algorithm pseudocodes are provided.

A unified theoretical framework for fluctuating-charge models in atom-space and in bond-space

Our previously introduced QTPIE (charge transfer with polarization current equilibration) model [J. Chen and T. J. Martínez, Chem. Phys. Lett. 438, 315 (2007)] is a fluctuating-charge model with correct asymptotic behavior. Unlike most other fluctuating-charge models, QTPIE is formulated in terms of charge-transfer variables and pairwise electronegativities, not atomic charge variables and electronegativities. The pairwise character of the electronegativities in QTPIE allows us to avoid spurious charge transfer when bonds are broken. However, the increased number of variables leads to considerable computational expense and a rank-deficient set of working equations, which is numerically inconvenient. Here, we show that QTPIE can be exactly reformulated in terms of atomic charge variables, leading to a considerable reduction in computational complexity. The transformation between atomic and bond variables is generally applicable to arbitrary fluctuating charge models and uncovers an underlying topological framework that can be used to understand the relation between fluctuating-charge models and the classical theory of electrical circuits.

Trinuclear Cobalt Based Porous Coordination Polymers Showing Unique Topological and Magnetic Variety upon Different Dicarboxylate-like Ligands

One-pot solvothermal self-assembly is employed to prepare a series of Co3 based porous coordination polymers, viz. 1, [H2N(CH3)2]2[Co3(ip)4]·H2O, 2, [H2N(CH3)2]2[Co3(bdc)4]·(DMF)2, 3, [H2N(CH3)2]2[Co3(bpdc)4]·(CH3OH)2(DMF)1.5, where ip, bdc, bpdc, and DMF are 1,3-benzenedicarboxylate, 1,4-benzenedicarboxylate, 4,4′-biphenyldicarboxylate, and N,N′-dimethylformamide, respectively. Via different organic connectors such as ip, bdc, and bpdc, such Co3 molecular building blocks (MBB), [Co3(CO2)8]2−, are associated together to give a tunable 8-connected topological framework, viz. 42464 for 1, 36414576 for 2, and 36418536 for 3. Upon the geometry and the length of organic connectors, their corresponding porosity occupied by c counterions plus solvent molecules also can be tunable, for example, with the consideration of [H2N(CH3)2]+ counterions, the solvent-accessible volume is 460.2 A3 for 1, equal to 23.9% of the cell volume, 3320.4 A3 for 2, equal to 56.3% of the cell volume, 4107.2 A3 for 3, equal to 50% of t...