Isomorphous Complexes Research Articles

Session details: Complexity

In the 1980's Eric Lander, Jin-Yi Cai, Martin Fürer and I began to explore the power of fixed point logic with counting (FPC). This natural class is deeply entwined with the challenge of capturing the class of polynomial-time graph properties and determining the complexity of graph isomorphism - problems that remain open. Despite that, so much more is now known about the remarkable power of FPC. Anuj Dawar explains in the following lovely survey.

Unusual Gd-nitronyl nitroxide antiferromagnetic coupling and slow magnetic relaxation in the corresponding Tb analogue.

The lanthanide-radical approach has been applied to construct three isomorphous mononuclear lanthanide complexes [Ln(hfac)3(NIT2-PyOCH3)] (Ln(III) = Gd , Tb , Er ) in which the nitronyl nitroxide radical NIT2-PyOCH3 (NIT2-PyOCH3 = 2-(3'-methoxy-2'-pyridyl)-4,4,5,5-tetramethylimidazoline-1-oxyl-3-oxide) acts as a tridentate chelating ligand via the pyridyl nitrogen atom, the oxygen atom from the NO group and the oxygen atom from the methoxy group. The central lanthanide ions are nine-coordinated and their coordination geometry can be considered as an intermediate between muffin and spherical capped square antiprisms. Interestingly, an unusual antiferromagnetic interaction was observed between the Gd(iii) and the coordinated nitronyl nitroxide radical in complex . Moreover, ac magnetic susceptibilities display frequency-dependent out-of-phase signals in the case of the anisotropic Tb complex, indicating magnetic relaxation behavior.

Microheterogeneity of crystal growth zones as a result of REE fractionation

Structural features of rhythmically zoned florencite and xenotime crystals from quartz veins of the Au-REE occurrences of the Near-Polar Urals are considered. The crystals were studied in the growth zones 0.01–0.15 mm wide. Rhythms consist of small structural elements: oriented intergrowths of several mineral species, layered-block distribution of Sr and Ca, syntaxic intergrowths of minerals of the same structural type, cocrystallization caverns, and late dissolution cavities with inclusions of argentite and pitchblende. The growth zones of crystals show successive change in lanthanides: from light to heavy REEs in florencite, and, vice versa, in xenotime, which is explained by the Ce and Y activity. The mechanism of element fractionation in the crystals is of dual nature. The rare-earth intergrowths contain four lanthanide groups: (La-Ce-Pr), (Nd-Sm-Eu-Gd), (Gd-Tb-Dy), (Ho-Er-Tm-Yb-Lu). This is caused by complexing ability of the elements, which is expressed in the tetrad effect of REE fractionation. Crystals, in turn, separate isomorphic complexes according to the properties of its facets and concentrations of elements in solutions. The structures of crystals and their growth elements are considered from the crystallochemical viewpoint.

Polymeric lanthanide acetates with peripheral cymantrenecarboxylate groups – Synthesis, magnetism and thermolysis

By means of exchange reactions between Ln acetates and CymCO2H (Cym=(η5-C5H4)Mn(CO)3) in H2O–THF–MeOH media, the following isomorphous complexes, having polymeric structures based on tetradentate acetate groups connecting three Ln3+ ions, were obtained: [Ln(CymCO2)2(OAc)(MeOH)]n (Ln=Nd (1), Gd (2) and Dy (3)). In H2O–THF–i-PrOH media, the same reaction results in polymers containing aqua ligands, [Nd(CymCO2)2(OAc)(H2O)2]n·0.5n(i-PrOH)·nH2O (4), [Gd(CymCO2)2(OAc)(H2O)2]n·2nH2O (5) and [Dy(CymCO2)2(OAc)(H2O)]n·0.55nTHF·0.45n(i-PrOH)·0.45nH2O (6). In all cases, the CymCO2 groups are situated at the periphery of the chain and play chelate, monodentate or bridging roles. The reactions with Ho, Er and Tm acetates give the isomorphous binuclear complexes [Ln2(μ2-O2CCym)2(η2-O2CCym)2(η2-OAc)2(H2O)4]·5H2O (Ln=Ho (7), Er (8) and Tm(9)), regardless of which alcohol is present in the reaction mixture. Crystallization of complex 8 from methanol results in the polymeric complex 10, isomorphous with 1–3 (Ln=Er). Magnetic studies of the Gd complexes indicate weak antiferromagnetic interactions between the Gd3+ ions in 2 (J=−0.0374cm−1) and ferromagnetic ones in 5 (J=0.0148cm−1). Thermolysis of the complexes was studied by means of TG and DSC in the 30–900°C temperature range under air and Ar atmosphere; in the first case it affords LnMn2O5 phases which have interesting magnetic properties.

Isostructurality in six celecoxib co-crystals introduced by solvent inclusion

Three co-crystals of celecoxib (CEL) and bipyridine derivatives were synthesized based on the synthon design approach and structurally characterized. The resulting two-component complexes have different unit cell parameters and noticeably different packing patterns. Surprisingly, when different organic solvents are introduced into the binary-host crystal systems, six isostructural three-component crystals are obtained. These isomorphic complexes are formed by CEL with two different sets of bipyridine co-formers containing various solvent molecules. A robust H-bonding framework between CEL dimer molecules is found to tolerate subtle changes in the structures of the co-formers and guest solvent molecules. The physicochemical properties of these isomorphic co-crystals are fully examined and compared with their corresponding non-solvate forms. The desolvation mechanism and relative thermodynamic stabilities are also discussed.

Redetermination of the Niobium(V) Precursor, [NbCl<sub>5</sub>(NCCH<sub>3</sub>)

The monomeric coordination compound pentachloro(acetonitrile)niobium(V), [NbCl5(NCCH3)], (NCCH3= acetonitrile) has been prepared under aerobic conditions and characterized by single-crystal X-ray diffraction. [NbCl5(NCCH3)] crystallized in the monoclinic spacegroupP21/cwitha= 5.964 (3),b= 9.888 (5),c= 15.448 (9) and β = 98.224 (2). Comparison is made to a previously published isomorphous complex that was prepared under unaerobic conditions.

Yttrium and samarium complexes with a linked 1,4,7-triazacyclononane-aryloxide ancillary ligand

The reaction of yttrium trichloride with the stoichiometric amount of the potassium salt of the 1-(3,5-di-tert-butyl-2-hydroxybenzyl)-4,7-diisopropyl-1,4,7-triazacyclononane ligand (iPr2-TACN-CH2C6H2tBu2OH) (TACN = 1,4,7-triazacyclononane) in DME affords in good yield complex [iPr2-TACN-CH2C6H2tBu2O]YCl2 (1). This complex is characterized by elemental analysis and 1H nuclear magnetic resonance; the solid state structure is determined by the single crystal X-ray diffraction analysis. In complex 1, the metal centre is six-coordinated by the aryloxide oxygen atom, the three nitrogen atoms of the TACN ligand, and two chlorine atoms. The X-ray structure of isomorphic samarium complex [iPr2-TACN-CH2C6H2tBu2O]SmCl2 (2) is also determined.

On the complexity of submap isomorphism and maximum common submap problems

Generalized maps describe the subdivision of objects in cells and are widely used to model 2D and 3D images. In this context, several pattern recognition tasks involve solving submap isomorphism problems (to decide if a map is included in another map) or, more generally, computing maximum common submaps (to measure the distance between two maps). Recently, we have described a polynomial-time algorithm for solving the submap isomorphism problem when the pattern map is connected. In this paper, we show that submap isomorphism is NP-complete when the pattern map is not connected. Then, we characterize the inherent difficulty of submap isomorphism with respect to the number of connected components. We show that it is Fixed-Parameter Tractable (FPT) and we give an FPT algorithm for submap isomorphism. We experimentally compare this algorithm with a state-of-the-art subgraph isomorphism algorithm for searching for patterns in an image and we show that it is both more accurate and more efficient. Finally, we study the complexity of the maximum common submap problem, and we show that it is NP-hard even though we restrict the problem to the search of common connected submaps.

Computability-Theoretic Properties of Injection Structures

We study computability-theoretic properties of computable injection structures and the complexity of isomorphisms between these structures. It is proved that a computable injection structure is computably categorical iff it has finitely many infinite orbits. Again, a computable injection structure is $ \Delta_2^0 $ -categorical iff it has finitely many orbits of type ω or finitely many orbits of type Z. Furthermore, every computably categorical injection structure is relatively computably categorical, and every $ \Delta_2^0 $ - categorical injection structure is relatively $ \Delta_2^0 $ -categorical. Analogs of these results are investigated for $ \Sigma_1^0 $ , $ \Pi_1^0 $ , and n-c.e. injection structures. We study the complexity of the set of elements with orbits of a given type in computable injection structures. For example, it is proved that for every c.e. Turing degree b, there is a computable injection structure A in which the set of all elements with finite orbits has degree b, and for every $ \Sigma_2^0 $ Turing degree c, there is a computable injection structure B in which the set of elements with orbits of type ω has degree c. We also have various index set results for infinite computable injection structures. For example, the index set of infinite computably categorical injection structures is a $ \Sigma_3^0 $ -complete set, and the index set of infinite $ \Delta_2^0 $ -categorical injection structures is a $ \Sigma_4^0 $ -complete set. We explore the connection between the complexity of the character and the first-order theory of a computable injection structure. It is shown that for an injection structure with a computable character, there is a decidable structure isomorphic to it. However, there are computably categorical injection structures with undecidable theories.

Two 3D CdIIand ZnIIComplexes Based on Flexible Dicarboxylate Ligand and Nitrogen-containing Pillar: Synthesis, Structure, and Luminescent Properties

Two 3D isomorphous and isostructural complexes, namely, (1) and (2); (BDOA = Benzene-1,4-dioxyacetic acid, bpy = 4,4'-bipyridine) were synthesized under hydrothermal conditions and characterized by means of elemental analyses, thermogravimetric (TG), infrared spectrometry, and single crystal X-ray diffraction. Complexes 1 and 2 crystallize in the triclinic system, space group P-1 and each metal ion in the complexes are six-coordinated with the same coordination environment. In the as-synthesized complexes, anions link central metal ions to form a 1D zigzag chain , whereas bpy pillars connect metal ions to generate a 1D linear chain . Both infinite chains are interweaved into 2D grid-like layers which are further constructed into a 3D open framework, where hydrogen bonds play as the bridges between the adjacent 2D layers. Luminescent properties of complex 1 showed selectivity for ion.

Isolation of Two Different Ni2Zn Complexes with an Unprecedented Cocrystal Formed by One of Them and a “Coordination Positional Isomer” of the Other

A new homometallic trinuclear Ni(II) complex [(NiL)2Ni(NCS)2] (1) and three heterometallic trinuclear Ni(II)-Zn(II)-Ni(II) complexes [(NiL)2Zn(NCS)2] (2), [(NiL)2Zn(NCS)2(CH3OH)2]·2CH3OH (3) and {[(NiL)2Zn(NCS)2(CH3OH)2]} {[(NiL)2Zn(NCS)2]} (4) have been synthesized by using the "complex as ligand" approach with the "metalloligand" [NiL] (H2L = N,N'-bis(salicylidene)-1,3-propanediamine) and thiocyanate in different ratios. All the complexes have been structurally and magnetically characterized. In the isomorphous complexes 1 and 2, the two terminal square planar Ni atoms and the central octahedral nickel atom (in 1) or zinc atom (in 2) are arranged in a bent structure where two cis κN-SCN(-) thiocyanate ions are coordinated to the central atom. The chemical composition of 3 is very similar to that of 2 but, in 3, the central Zn atom is tetrahedral and the κN-SCN(-) thiocyanate ions occupy an axial position of each terminal nickel atom (which now are octahedral with the sixth position occupied by a methanol molecule). Complex 4 consists of two closely related trinuclear units 4A and 4B. In 4A, the coordination environments of the metals are identical to those of 3 whereas 4B is a "coordination position isomer" of complex 2 with the central square pyramidal Zn and one of the terminal square pyramidal Ni atoms coordinated by two κN-SCN(-) thiocyanate ions. Complex 4 is a unique example of a cocrystal formed by two similar trinuclear units (4A and 4B) where 4A is identical to an existing complex (3) and 4B is a "coordination position isomer" of another existing complex (2).

2D→3D polycatenated and 3D→3D interpenetrated metal–organic frameworks constructed from thiophene-2,5-dicarboxylate and rigid bis(imidazole) ligands

Hydrothermal reactions of rigid 1,4-bis(imidazol-1-yl)benzene (dib) and 1,4-bis(imidazol-1-yl)-2,5-dimethylbenzene (dimb) with deprotonated thiophene-2,5-dicarboxylic acid (H2tdc) in the presence of Zn(II) and Cd(II) salts in H2O produced three new metal–organic frameworks, namely, [Zn(µ-tdc)(H2O)(µ-dib)]n (1), [Cd(µ-tdc)(H2O)(µ-dib)]n (2), and {[Cd2(µ3-tdc)2(µ-dimb)2]·(H2O)}n(3). These MOFs were characterized by FT-IR spectroscopy, elemental, thermal (TG, DTA, DTG and DSC), and single-crystal X-ray diffraction analyses. Isomorphous complexes 1 and 2 reveal polycatenated 2D+2D→3D framework based on an undulated (4,4)-sql layer. Complex 3 exhibits a new 4-fold interpenetrating 3D framework with the point symbol of 66. Molecular simulations were used to assess the potentials of the complexes for H2 storage application. Moreover, these coordination polymers exhibit blue fluorescent emission bands in the solid state at room temperature.

The divalent central metal ion exchange of complexes under the single crystal state

The center metal ion exchange is an important method to synthetic isomorphous complexes. It is of great interest for a wide range of removing toxic heavy metal and changing the properties of complexes, such as pharmacological activity, gas adsorption and catalytic, etc. This review introduced the mechanism of the center metal ion exchange in the single crystal state, explored the effect of the central metal ions on the properties of complexes, and described the existing problems and the future directions of the research briefly.

Two-to-one structures

We investigate computability-theoretic properties of computable structures with single unary functions f such that, for every x in the image, f−1(x) has exactly two elements, which we call 2:1 structures. We also investigate structures for which f−1(x) has either exactly two or zero elements, which we call (2,0):1 structures. In particular, we are interested in the complexity of isomorphisms between these structures. We prove that a computable 2:1 structure A is computably categorical if and only if A has only finitely many Z-chains. We show that every computable 2:1 structure is ∆2-categorical. We further investigate computable and higher level categoricity of various natural subclasses of (2,0):1 structures, including highly computable and locally finite strufctures.

Silver–Organic Frameworks Containing Ethynediide or Ethynide with Ancillary Oligo-α-sulfanylpyrazinyl and Dimethylsulfoxide Ligands

Five new silver(I)–organic frameworks containing ethynediide or alkyl and aryl ethynide, together with newly designed oligo-α-sulfanylpyrazinyl ligands, namely, [(Ag2C2)2(AgCF3COO)11(L1)(μ2-DMSO)3(DMSO)5]·1/4H2O (L1 = 2,6-bis(pyridin-2-ylthio)pyrazine, 1), [(Ag2C2)2(AgCF3COO)11(L2)(μ2-DMSO)5(DMSO)3]·1/4H2O (L2 = 2-(pyrazin-2-ylthio)-6-(pyridin-2-ylthio)pyrazine, 2), [(AgC≡CtBu)2(AgCF3COO)5(L3)(DMSO)3(H2O)] (L3 = 2,6-bis(pyrazin-2-ylthio)pyrazine, 3), [(AgC≡CPh)4(AgCF3COO)2(L3)(DMSO)2]·DMSO·1/2H2O (4), and [(AgC≡CC6H4Cl-3)4(AgCF3COO)2(L3)(DMSO)]·H2O (5), have been synthesized and characterized by X-ray crystallography. Isomorphous complexes 1 and 2 feature an infinite coordination chain composed of (C2)2Ag15 clusters alternatively linked by L1 or L2 ligands; weak intermolecular C–H···F hydrogen bonding between adjacent infinite chains results in a three-dimensional supramolecular network. Complex 3 exhibits a coordination layer structure composed of (tBuC2)2Ag7 aggregates. Compounds 4 and 5 are nearly isos...

Isomorphic chain complexes of Hamiltonian dynamics on tori

In this work, for a given smooth, generic Hamiltonian \({H : \mathbb{S}^{1} \times \mathbb{T}^{2n} \rightarrow \mathbb{R}}\) on the torus \({\mathbb{T}^{2n} = \mathbb{R}^{2n}/\mathbb{Z}^{2n}}\) we construct a chain isomorphism \({\Phi_{*} : (C_{*}(H), \partial^{M}_{*}) \rightarrow (C_{*}(H), \partial^{F}_{*})}\) between the Morse complex of the Hamiltonian action AH on the free loop space of the torus \({\Lambda_{0}(\mathbb{T}^{2n})}\) and the Floer complex. Though both complexes are generated by the critical points of AH, their boundary operators differ. Therefore, the construction of \({\Phi}\) is based on counting the moduli spaces of hybrid-type solutions which involves stating a new non-Lagrangian boundary value problem for Cauchy–Riemann type operators not yet studied in Floer theory. We finally want to note that the problem is completely symmetric. So we also could construct an isomorphism \({\Psi_{*} : (C_{*}(H), \partial^{F}_{*}) \rightarrow (C_{*}(H), \partial^{M}_{*})}\).

Macrogenomic Evidence for the Origin of the Black Fly Simulium suzukii (Diptera: Simuliidae) on Okinawa Island, Japan

To determine the geographic origin of the black fly Simulium suzukii on Okinawa Island, Japan, macrogenomic profiles derived from its polytene chromosomes were compared with those of mainland and other insular populations of S. suzukii and of the isomorphic Simulium tani species complex. The Okinawan population is a chromosomally unique cytoform, designated ‘D,’ which is essentially monomorphic and differs by about 27 fixed rearrangements from the chromosomal standard sequence for the subgenus Simulium and by two fixed differences from its nearest known relative, representing the type of S. suzukii, on the main islands of Japan. Chromosomal band sequences revealed two additional, sympatric cytoforms of S. suzukii, designated ‘A’ and ‘B,’ each with species status, in Korea, and a third cytoform, designated ‘C,’ on Hokkaido, Japan. A new cytoform, ‘K,’ of S. tani from Malaysia, representing the type of S. tani, is more closely related to cytoforms in Thailand, as are populations from Taiwan previously treated as S. suzukii but more closely aligned with S. tani and newly recognized as cytoform ‘L’ of the latter nominal species. Rooting of chromosomal band sequences by outgroup comparisons allowed directionality of chromosomal rearrangements to be established, enabling phylogenetic inference of cytoforms. Of 41 macrogenomic rearrangements discovered in the five new cytoforms, four provide evidence for a stepwise origin of the Okinawan population from populations characteristic of the main islands of Japan. The macrogenomic approach applied to black flies on Okinawa Island illustrates its potential utility in defining source areas for other species of flies including those that might pose medical and veterinary risks.

Two Series of Luminescent Flexible Polycarboxylate Lanthanide Coordination Complexes with Double Layer and Rectangle Metallomacrocycle Structures

Two series of eight polycarboxylate lanthanide coordination complexes, namely, [Ln(cpia)(H2O)2]·4H2O {Ln = Ce (1), Nd (2), and Sm (3)} and [Ln(cpia)(H2O)n]2·2H2O {Ln = Eu (4), Gd (5), Tb (6), Er (7), and Lu (8), n = 5 in 4–7; n = 4 in 8} {H3cpia = N-[4-(carboxymethoxy)phenyl]iminodiacetic acid}, have been synthesized by self-assembly and characterized by IR, thermogravimetric analysis, and X-ray crystallography. Complexes 1–3 are isomorphic featuring two-dimensional double layer structures, while the isomorphic complexes 4–8 exhibit [2 + 2] rectangle metallomacrocycle structures. Further, photoluminescence (PL) spectra of complexes 2, 4, and 6 show the strong characteristic luminescence of Eu(III), Tb(III), and Nd(III) ions, respectively, suggesting that cpia3– is able to efficiently sensitize the luminescence of lanthanide ions.

Dzyaloshinski–Moriya (D–M) oriented weak ferromagnets in isomorphic coordination architectures constructed by flexible 1,2,4-triazole-1-acetate ligands with the assistance of halogen or pseudohalogen anions

Abstract Three isomorphic coordination complexes constructed by flexible 1,2,4-triazole-1-acetate with the assistance of halogen or pseudohalogen anions, and [Mn( L )Cl] n ( 1 ), [Co( L )(N 3 )] n ( 2 ) and [Ni( L )(N 3 )] n ( 3 ) ( L = 1,2,4-triazole-1-acetate) have been synthesized and structurally characterized. It is interesting that the linkages of the halogen anions in 1 are the same as those of the pseudohalogen anions in 2 and 3 . Single crystal structure analysis indicated a novel 4,6-connected binodal net with Schlafli {3;4 2 ;5 3 }{3 2 ;4 3 ;5 5 ;6 3 ;7 2 } was formed in these complexes. Dominating antiferromagnetic interactions between the metal ions were detected in these complexes, furthermore evident spin canting presented in 1 and 3 for Dzyaloshinski–Moriya (D–M) interactions.

Nearly Tight Bounds for Testing Function Isomorphism

We study the problem of testing isomorphism (equivalence up to relabeling of the input variables) between Boolean functions. We prove the following: (1) For most functions $f:\{0,1\}^n \to \{0,1\}$, the query complexity of testing isomorphism to $f$ is $\Omega(n)$. Moreover, the query complexity of testing isomorphism to most $k$-juntas $f:\{0,1\}^n \to \{0,1\}$ is $\Omega(k)$. (2) Isomorphism to any $k$-junta $f:\{0,1\}^n \to \{0,1\}$ can be tested with $O(k \log k)$ queries. (3) For some $k$-juntas $f:\{0,1\}^n \to \{0,1\}$, testing isomorphism to $f$ with one-sided error requires $\Omega(k\log(n/k))$ queries. In particular, testing whether $f:\{0,1\}^n \to \{0,1\}$ is a $k$-parity with one-sided error requires $\Omega(k\log(n/k))$ queries. (4) The query complexity of testing isomorphism between two unknown functions $f,g:\{0,1\}^n \to \{0,1\}$ is $\widetilde{\Theta}(2^{n/2})$. These bounds are tight up to logarithmic factors, and they significantly strengthen the bounds proved by Fischer, Kindler, Ron, Safra, and Samorodnitsky [J. Comput. System Sci., 68 (2004), pp. 753--787] and Blais and O'Donnell [Proceedings of the IEEE Conference on Computational Complexity, 2010, pp. 235--246]. We also obtain results closely related to isomorphism testing, answering a question posed by Diakonikolas, Lee, Matulef, Onak, Rubinfeld, Servedio, and Wan [Proceedings of the IEEE Symposium on Foundations of Computer Science, 2007, pp. 549--558]: testing whether a function $f:\{0,1\}^n \to \{0,1\}$ can be computed by a circuit of size $\le s$ requires $s^{\Omega(1)}$ queries. All of our lower bounds apply to general (adaptive) testers.