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Hybrid functional calculations are presented for defects in LiGaO 2 with the fraction of non-local exchange adjusted to reproduce the recently reported exciton gap of 6.0 eV. We study how the defect transition levels of the main native defects change with respect to the band edges compared to earlier calculations which assumed a smaller bandgap near 5.1 eV. In addition, we consider defect complexes formed by combining the main native donor Ga Li with the main acceptors, V Li and Li Ga antisites as a function of their relative position. These results are used to tentatively identify the photoluminescence bands previously assigned to donor–acceptor-pair recombination. [ABSTRACT FROM AUTHOR]

Let G be a finite group, p a prime, and k a field of characteristic p. We introduce the notion of an endotrivial chain complex of p -permutation kG -modules, and study the corresponding group E k (G) of endotrivial complexes. Such complexes are shown to induce splendid Rickard autoequivalences of kG. The elements of E k (G) are determined uniquely by integral invariants arising from the Brauer construction and a degree one character G → k ×. Using ideas from Bouc's theory of biset functors, we provide a canonical decomposition of E k (G) , and as an application, give complete descriptions of E k (G) for abelian groups and p -groups of normal p -rank 1. Taking Lefschetz invariants of endotrivial complexes induces a group homomorphism Λ : E k (G) → O (T (k G)) , where O (T (k G)) is the orthogonal unit group of the trivial source ring. Using recent results of Boltje and Carman, we give a Frobenius stability condition elements in the image of Λ must satisfy. [ABSTRACT FROM AUTHOR]

Let G be a finite group and X be a subgroup of G. We investigate the topological properties of the poset C X (G) of cosets Hx in G with X ≤ H < G. We show that C X (G) is non-contractible if G is solvable or N G (X) contains a Sylow 2-subgroup and a Sylow 3-subgroup of G. This result follows J. Shareshian and R. Woodroofe's work in [8] (2016). We also give some divisibility properties of the Euler characteristic of C X (G) when X is a p -group, which follows K. S. Brown's classical result in [3] (2000). [ABSTRACT FROM AUTHOR]

This is the first of a series of papers that develop a systematic bridge between constructions in discrete mathematics and the corresponding continuous analogs. In this paper, we establish an equivalence between Forman's discrete Morse theory on a simplicial complex and the continuous Morse theory (in the sense of any known non-smooth Morse theory) on the associated order complex via the Lovász extension. Furthermore, we propose a new version of the Lusternik–Schnirelman category on abstract simplicial complexes to bridge the classical Lusternik–Schnirelman theorem and its discrete analog on finite complexes. More generally, we can suggest a discrete Morse theory on hypergraphs by employing piecewise-linear (PL) Morse theory and Lovász extension, hoping to provide new tools for exploring the structure of hypergraphs. [ABSTRACT FROM AUTHOR]

We study the procedure of regularization in the context of the Lipschitz version of de Rham calculus on metric simplicial complexes with bounded geometry. It provides us with the machinery to handle the de Rham homomorphism for Lπ-cohomologies. In this respect, we obtain the condition resolving the question of triviality of the kernel for de Rham homomorphism. In particular, we specify the nontrivial cohomology classes explicitly for a sequence of parameters π = missing nonincreasing monotonicity. [ABSTRACT FROM AUTHOR]

To any simple graph G , the clique graph operator K assigns the graph K (G) , which is the intersection graph of the maximal complete subgraphs of G . The iterated clique graphs are defined by K 0 (G) = G and K n (G) = K (K n - 1 (G)) for n ≥ 1 . We associate topological concepts to graphs by means of the simplicial complex Cl (G) of complete subgraphs of G . Hence, we say that the graphs G 1 and G 2 are homotopic whenever Cl (G 1) and Cl (G 2) are. A graph G such that K n (G) ≃ G for all n ≥ 1 is called K -homotopy permanent. A graph is Helly if the collection of maximal complete subgraphs of G has the Helly property. Let G be a Helly graph. Escalante (1973) proved that K (G) is Helly, and Prisner (1992) proved that G ≃ K (G) , and so Helly graphs are K -homotopy permanent. We conjecture that if a graph G satisfies that K m (G) is Helly for some m ≥ 1 , then G is K -homotopy permanent. If a connected graph has maximum degree at most four and is different from the octahedral graph, we say that it is a low degree graph. It was recently proven that all low-degree graphs G satisfy that K 2 (G) is Helly. In this paper, we show that all low-degree graphs have the homotopy type of a wedge or circumferences, and that they are K -homotopy permanent. [ABSTRACT FROM AUTHOR]

This paper is a commutative algebra introduction to the homological theory of quasi-coherent sheaves and contraherent cosheaves over quasi-compact semi-separated schemes. Antilocality is an alternative way in which global properties are locally controlled in a finite affine open covering. For example, injectivity of modules over non-Noetherian commutative rings is not preserved by localizations, while homotopy injectivity of complexes of modules is not preserved by localizations even for Noetherian rings. The latter also applies to the contraadjustedness and cotorsion properties. All the mentioned properties of modules or complexes over commutative rings are actually antilocal. They are also colocal, if one presumes contraadjustedness. Generally, if the left class in a (hereditary complete) cotorsion theory for modules or complexes of modules over commutative rings is local and preserved by direct images with respect to affine open immersions, then the right class is antilocal. If the right class in a cotorsion theory for contraadjusted modules or complexes of contraadjusted modules is colocal and preserved by such direct images, then the left class is antilocal. As further examples, the class of flat contraadjusted modules is antilocal, and so are the classes of acyclic, Becker-coacyclic, or Becker-contraacyclic complexes of contraadjusted modules. The same applies to the classes of homotopy flat complexes of flat contraadjusted modules and acyclic complexes of flat contraadjusted modules with flat modules of cocycles. [ABSTRACT FROM AUTHOR]

Here, we combined magnetometry, multi‐frequency electronic paramagnetic resonance, and wave function based ab initio calculations to investigate magnetic properties of two high spin Co(II) complexes Co(BDPRP) (BDPRP=2,6‐bis((2‐(S)‐di(4‐R)phenylhydroxylmethyl‐1‐pyrrolidi‐nyl)methyl)pyridine, R=H for 8; R=tBu for 9). Complexes 8 and 9 featuring effective D3h symmetry were found to possess D=24.0 and 32.0 cm−1, respectively, in their S=3/2 ground states of 1e''dxz/yz41e'dxy/x2-y221a1'dz21 ${{\left(1{{\rm e}}^{{\rm { {^\prime}}}{\rm { {^\prime}}}}\right({d}_{xz/yz}\left)\right)}^{4}{\left(1{{\rm e}}^{{\rm { {^\prime}}}}\right({d}_{{xy/{x}^{2}-y}^{2}}\left)\right)}^{2}{\left(1{{\rm a}}_{1}^{{\rm { {^\prime}}}}\right({d}_{{z}^{2}}\left)\right)}^{1}}$. Ligand field analyses revealed that the low‐lying d‐d excited states make either positive or vanishing contributions to D. Hence, total positive D values were measured for 8 and 9, as well as related D3h high spin Co(II) complexes. In contrast, negative D values are usually observed for C3v congeners. In‐depth analyses suggested that lowering symmetry from D3h to C3v induces orbital mixing between 1edxz/yz ${1{\rm e}\left({d}_{xz/yz}\right)}$ and 2edxy/x2-y2 ${2{\rm e}\left({d}_{{xy/{x}^{2}-y}^{2}}\right)}$ and admixes excited state 4A21e→2e ${{}^{4}{{\rm A}}_{2}\left(1e\to 2e\right)}$ into the ground state. Both factors turn the total D value progressively negative with the increasing distance (δ) of the Co(II) center out of the equatorial plane. Therefore, δ determines the sign and magnitude of final D values of five‐coordinate trigonal bipyramidal S=3/2 Co(II) complexes as measured for a series of such species with varying δ. [ABSTRACT FROM AUTHOR]

This study explores the utilization of 1,4‐diisocyanobenzene as a linear bridging ligand for the assembly of organometallic CpCo or Cp*Co complexes. Overcoming solubility challenges, the synthesis of organometallic Cp* cobalt complexes incorporating bridging 1,4‐diisocyanobenzene ligands was successfully achieved, while similar CpCo complexes were not attainable. The presented approach encompasses investigations into both stepwise and self‐assembly synthetic pathways, leading to the construction of a rectangular tetranuclear complex [(Cp*CoI)4(μ‐1,4‐CNC6H4NC)4](OTf)4 (3). Through single‐crystal x‐ray diffraction and spectroscopic techniques such as infrared, nuclear magnetic resonance, and electrospray ionization‐mass spectrometry, the cobalt‐isocyanides complexes were characterized. [ABSTRACT FROM AUTHOR]

A new arsenato-polyoxovanadate with the composition {[Ni(dien)2]3[V6As8O26]}2+·2 Cl– has been crystallized under solvothermal conditions as turquoise block-like crystals. The central structural feature is the [V6IVAs8IIIO26]4– cluster anion, which is composed of two trinuclear {V3O11} groups consisting of three edge-sharing VO5 polyhedra. Pairs of pyramidal AsO3 moieties share a common corner forming As2O5 units, which interconnect the {V3O11} groups. One of the [Ni(dien)2]2+ complexes adopts the s-fac (Ni1) and the second complex the mer configuration (Ni2). The Cl− anion is involved in strong hydrogen bonding interactions and links the [Ni(dien)2]2+ complexes to form twelve-membered rings which host the [V6As8O26]4– cluster anions. The Hirshfeld surface analysis yields a detailed picture of the intermolecular interactions revealing clear differences for the two unique [Ni(dien)2]2+ complexes. Intermolecular contacts also include As⋯H, O⋯H and H⋯H interactions. In the electronic spectrum the bands of d–d transitions of the vanadyl group and of the Ni2+ cations overlap, preventing a detailed analysis. [ABSTRACT FROM AUTHOR]

Quantum chemical calculations at the M06L-GD3/def2-TZVPP level were done to investigate hydrogen bonds between amino acid metal complexes and a free water molecule. Octahedral nickel(II) and square planar palladium(II) complexes of glycine, cysteine, phenylalanine, and serine with different charges of metal complexes (+1, 0, and −1) were investigated. The following hydrogen bonds were considered: NH/O (amino acid is a H-donor), O1/HO (coordinated O1 oxygen from amino acid is a H-acceptor), and O2/HO (non-coordinated O2 oxygen from amino acid is a H-acceptor). Amino acid type has a small influence on interaction energies, both for octahedral nickel(II) and square planar palladium(II) complexes. The influence is the largest for NH/O and the smallest for O2/HO hydrogen bonds. For NH/O interaction, palladium(II) complexes showed stronger hydrogen bonds than nickel(II), up to −11.8 kcal mol−1 for singly positively charged complexes. Nickel(II) complexes demonstrated higher O1/HO hydrogen bond strength than palladium(II) with interaction energies up to −8.9 kcal mol−1 for singly negative complexes. With up to −9.0 kcal mol−1 interaction energy for singly negative complexes, O2/HO interactions were also stronger for nickel(II) complexes than palladium(II). [ABSTRACT FROM AUTHOR]

Non-side-looking airborne radar provides important applications on account of its all-round multi-angle airspace coverage. However, it suffers clutter range dependence that makes the samples fail to satisfy the condition of being independent and identically distributed (IID), and it severely degrades traditional approaches to clutter suppression and target detection. In this paper, a novel reweighted extreme learning machine (ELM)-based clutter suppression and range compensation algorithm is proposed for non-side-looking airborne radar. The proposed method involves first designing the pre-processing stage, the special reweighted complex-valued activation function containing an unknown range compensation matrix, and two new objective outputs for constructing an initial reweighted ELM-based network with its training. Then, two other objective outputs, a new loss function, and a reverse feedback framework driven by the specifically designed objectives are proposed for the unknown range compensation matrix. Finally, aiming to estimate and reconstruct the unknown compensation matrix, special processes of the complex-valued structures and the theoretical derivations are designed and analyzed in detail. Consequently, with the updated and compensated samples, further processing including space–time adaptive processing (STAP) can be performed for clutter suppression and target detection. Compared with the classic relevant methods, the proposed algorithm achieves significantly superior performance with reasonable computation time. It provides an obviously higher detection probability and better improvement factor (IF). The simulation results verify that the proposed algorithm is effective and has many advantages. [ABSTRACT FROM AUTHOR]

This paper concerns locally finite 2‐complexes Xm,n$X_{m,n}$ that are combinatorial models for the Baumslag–Solitar groups BS(m,n)$BS(m,n)$. We show that, in many cases, the locally compact group Aut(Xm,n)$\operatorname{Aut}(X_{m,n})$ contains incommensurable uniform lattices. The lattices we construct also admit isomorphic Cayley graphs and are finitely presented, torsion‐free, and coherent. [ABSTRACT FROM AUTHOR]

The reaction of Co(NCS)2 with 4-methylpyridine N-oxide (C6H7NO) leads to the formation of two compounds, namely, tetrakis(4-methylpyridine N-oxide-O)bis(thiocyanato-N)cobalt(II), [Co(NCS)2(C6H7NO)4] (1), and tris(4-methylpyridine N-oxide-O)bis(thiocyanato-N)cobalt(II), [Co(NCS)2(C6H7NO)3] (2). The asymmetric unit of 1 consists of one CoII cation located on a centre of inversion, as well as one thiocyanate anion and two 4-methylpyridine N-oxide coligands in general positions. The CoII cations are octahedrally coordinated by two terminal N-bonding thiocyanate anions in trans positions and four 4-methylpyridine N-oxide ligands. In the extended structure, these complexes are linked by C--H...O and C--H...S interactions. In compound 2, two crystallographically independent complexes are present, which occupy general positions. In each of these complexes, the CoII cations are coordinated in a trigonal-bipyramidal manner by two terminal N-bonding thiocyanate anions in axial positions and by three 4-methylpyridine N-oxide ligands in equatorial positions. In the crystal, these complex molecules are linked by C--H...S interactions. For compound 2, a nonmerohedral twin refinement was performed. Powder X-ray diffraction (PXRD) reveals that 2 was nearly obtained as a pure phase, which is not possible for compound 1. Differential thermoanalysis and thermogravimetry data (DTA-TG) show that compound 2 start to decompose at about 518K. [ABSTRACT FROM AUTHOR]

Let I and J be two ideals of a commutative ring R. We introduce the concepts of the C ˇ ech complex and C ˇ ech cocomplex with respect to (I , J) and investigate their homological properties. In addition, we show that local cohomology and local homology with respect to (I , J) are expressed by the above complexes. Moreover, we provide a proof for the Matlis–Greenless–May equivalence with respect to (I , J) , which is an equivalence between the category of derived (I , J) -torsion complexes and the category of derived (I , J) -completion complexes. As an application, we use local cohomology and the C ˇ ech complex with respect to (I , J) to prove Grothendieck's local duality theorem for unbounded complexes. [ABSTRACT FROM AUTHOR]

Let 퐴 and 퐵 be rings and 푈 a (B , A) -bimodule. The Gorenstein projective, injective and flat complexes over the formal triangular matrix ring T = ( A 0 U B ) are explicitly described. These statements extend some earlier results of Enochs, Cortés-Izurdiaga, Torrecillas, Zhang and Mao in this direction. [ABSTRACT FROM AUTHOR]

This paper presents a construction of local C r interpolations on simplicial grids in R d by using piecewise polynomials of degree k ≥ 2 d r + 1. To this end, an interpolative decomposition of the set of corresponding multi-indices is introduced. It is shown that the piecewise polynomial interpolation is of C r continuity provided that the corresponding function is of C 2 d − 1 r continuity. [ABSTRACT FROM AUTHOR]

Finite element de Rham complexes and finite element Stokes complexes with varying degrees of smoothness in three dimensions are systematically constructed in this paper. Smooth scalar finite elements in three dimensions are derived through a non-overlapping decomposition of the simplicial lattice. H(\operatorname {div})-conforming finite elements and H(\operatorname {curl})-conforming finite elements with varying degrees of smoothness are devised based on these smooth scalar finite elements. The finite element de Rham complexes with corresponding smoothness and commutative diagrams are induced by these elements. The div stability of the H(\operatorname {div})-conforming finite elements is established, and the exactness of these finite element complexes is proven. [ABSTRACT FROM AUTHOR]

We show that the categories of compact Lie groups and complex reductive groups (not necessarily connected) are homotopy equivalent topological categories. In other words, the corresponding categories enriched in the homotopy category of topological spaces are equivalent. This can also be interpreted as an equivalence of infinity categories. [ABSTRACT FROM AUTHOR]

Flavodiiron nitric oxide reductases (FNORs) equip pathogens with resistance to nitric oxide (NO), an important immune defense agent in mammals, allowing these pathogens to proliferate in the human body, potentially causing chronic infections. Understanding the mechanism of how FNORs mediate the reduction of NO contributes to the greater goal of developing new therapeutic approaches against drug-resistant strains. Recent density functional theory calculations suggest that a second coordination sphere (SCS) tyrosine residue provides a hydrogen bond that is critical for the reduction of NO to N2O at the active site of FNORs [J. Lu, B. Bi, W. Lai and H. Chen, Origin of Nitric Oxide Reduction Activity in Flavo-Diiron NO Reductase: Key Roles of the Second Coordination Sphere, Angew. Chem., Int. Ed., 2019, 58, 3795–3799]. Specifically, this H-bond stabilizes the hyponitrite intermediate and reduces the energetic barrier for the N–N coupling step. At the same time, the role of the Fe⋯Fe distance and its effect on the N–N coupling step has not been fully investigated. In this study, we equipped the H[BPMP] (= 2,6-bis[[bis(2-pyridylmethyl)amino]methyl]-4-methylphenol) ligand with SCS amide groups and investigated the corresponding diiron complexes with 0–2 bridging acetate ligands. These amide groups can form hydrogen bonds with the bridging acetate ligand(s) and potentially the coordinated NO groups in these model complexes. At the same time, by changing the number of bridging acetate ligands, we can systematically vary the Fe⋯Fe distance. The reactivity of these complexes with NO was then investigated, and the formation of stable iron(II)–NO complexes was observed. Upon one-electron reduction, these NO complexes form Dinitrosyl Iron Complexes (DNICs), which were further characterized using IR and EPR spectroscopy. [ABSTRACT FROM AUTHOR]

With the aim of constructing hydrogen‐bonding networks in synthetic complexes, two new ligands derived from cis,cis‐1,3,5‐triaminocyclohexane (TACH) have been prepared that feature pendant pyrrole or indole rings as outer‐sphere H‐bond donors. The TACH framework offers a facial arrangement of three N‐donors, thereby mimicking common coordination motifs in the active sites of nonheme Fe and Cu enzymes. X‐ray structural characterization of a series of CuI‐X complexes (X=F, Cl, Br, NCS) revealed that these neutral ligands (H3LR, R=pyrrole or indole) coordinate in the intended facial N3 manner, yielding four‐coordinate complexes with idealized C3 symmetry. The N−H units of the outer‐sphere heterocycles form a hydrogen‐bonding cavity around the axial (pseudo)halide ligand, as verified by crystallographic, spectroscopic, and computational analyses. Treatment of H3Lpyrrole and H3Lindole with divalent transition metal chlorides (MIICl2, M=Fe, Cu, Zn) causes one heterocycle to deprotonate and coordinate to the M(II) center, giving rise to tetradentate ligands with two remaining outer‐sphere H‐bond donors. Further ligand deprotonation is observed upon reaction with Ni(II) and Cu(II) salts with weakly coordinating counteranions. The reported complexes highlight the versatility of TACH‐based ligands with pendant H‐bond donors, as the resulting scaffolds can support multiple protonation states, coordination geometries, and H‐bonding interactions. [ABSTRACT FROM AUTHOR]

Resonance Raman optical activity (RROA) spectra with high sensitivity reveal details on molecular structure, chirality, and excited electronic properties. Despite the difficulty of the measurements, the recorded data for the Co(III) complex with S,S‐N,N‐ethylenediaminedisuccinic acid are of exceptional quality and, coupled with the theory, spectacularly document the molecular behavior in resonance. This includes a huge enhancement of the chiral scattering, contribution of the antisymmetric polarizabilities to the signal, and the Herzberg‐Teller effect significantly shaping the spectra. The chiral component is by about one order of magnitude bigger than for an analogous aluminum complex. The band assignment and intensity profile were confirmed by simulations based on density functional and vibronic theories. The resonance was attributed to the S0→S3 transition, with the strongest signal enhancement of Raman and ROA spectral bands below about 800 cm−1. For higher wavenumbers, other excited electronic states contribute to the scattering in a less resonant way. RROA spectroscopy thus appears as a unique tool to study the structure and electronic states of absorbing molecules in analytical chemistry, biology, and material science. [ABSTRACT FROM AUTHOR]

Resonance Raman optical activity (RROA) spectra with high sensitivity reveal details on molecular structure, chirality, and excited electronic properties. Despite the difficulty of the measurements, the recorded data for the Co(III) complex with S,S‐N,N‐ethylenediaminedisuccinic acid are of exceptional quality and, coupled with the theory, spectacularly document the molecular behavior in resonance. This includes a huge enhancement of the chiral scattering, contribution of the antisymmetric polarizabilities to the signal, and the Herzberg‐Teller effect significantly shaping the spectra. The chiral component is by about one order of magnitude bigger than for an analogous aluminum complex. The band assignment and intensity profile were confirmed by simulations based on density functional and vibronic theories. The resonance was attributed to the S0→S3 transition, with the strongest signal enhancement of Raman and ROA spectral bands below about 800 cm−1. For higher wavenumbers, other excited electronic states contribute to the scattering in a less resonant way. RROA spectroscopy thus appears as a unique tool to study the structure and electronic states of absorbing molecules in analytical chemistry, biology, and material science. [ABSTRACT FROM AUTHOR]

Pabellón de Pica, Tarapaca, Chile, is a former guano mine exploited in the late 19th and early 20th centuries. Guano is in a direct contact with Cu sulfide mineralization leading to the formation of minerals with a highly unusual chemical composition, especially Cu‐organic complexes and nitrogen‐bearing phases. In this work, we report results of the Raman spectroscopic investigation of several unique Cu‐bearing minerals that are the result of supergene processes at the Pabellón da Pica site. We focus on several minerals with new and interesting structure types which were described from this locality for the first time, namely, antipinite (K–Na–Cu oxalate), ammineite (Cu ammine complex), chanabayaite (Cu ammine triazolate complex), and joanneumite (Cu isocyanurate). The Raman spectra contain specific bands attributed to the characteristic vibrations of the molecular groups present in the minerals (oxalate, ammine, triazolate, and isocyanurate groups) and vibrations within Cu‐centered polyhedra. A comparison was made with the vibrational spectra of related minerals and synthetic phases, which revealed substantial proliferation and shifting of numerous bands, which is in agreement with the complex composition of the investigated minerals. The results obtained appropriately complement known structural and spectroscopic data published previously. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the dimension of dual modules of local cohomology of Stanley–Reisner rings to obtain a new vector. This vector contains important information on the Serre's condition (S r) and the CM t property as well as the depth of Stanley–Reisner rings. We prove some results in this regard including lower bounds for the depth of Stanley–Reisner rings. Further, we give a characterization of (d − 1) -dimensional simplicial complexes with codimension two which are (S d − 3) but they are not Cohen–Macaulay. By using this characterization, we obtain a condition to equality of projective dimension of the Stanley–Reisner rings and the arithmetical rank of their Stanley–Reisner ideals. Moreover, our characterization allows us to compute the h -vectors and give a negative answer to a known question regarding these vectors. [ABSTRACT FROM AUTHOR]

Accurately predicting the binding affinity between proteins and ligands is crucial for drug discovery. Recent advances in graph neural networks (GNNs) have made significant progress in learning representations of protein-ligand complexes to estimate binding affinities. To improve the performance of GNNs, there frequently needs to look into protein-ligand complexes from geometric perspectives. While the "off-the-shelf" GNNs could incorporate some basic geometric structures of molecules, such as distances and angles, through modeling the complexes as homophilic graphs, these solutions seldom take into account the higher-level geometric attributes like curvatures and homology, and also heterophilic interactions.To address these limitations, we introduce the Curvature-based Adaptive Graph Neural Network (CurvAGN). This GNN comprises two components: a curvature block and an adaptive attention guided neural block (AGN). The curvature block encodes multiscale curvature informaton, then the AGN, based on an adaptive graph attention mechanism, incorporates geometry structure including angle, distance, and multiscale curvature, long-range molecular interactions, and heterophily of the graph into the protein-ligand complex representation. We demonstrate the superiority of our proposed model through experiments conducted on the PDBbind-V2016 core dataset. [ABSTRACT FROM AUTHOR]

We examine virtual resolutions of Stanley–Reisner ideals for a product of projective spaces by providing sufficient conditions for a simplicial complex to be virtually Cohen–Macaulay (to have a virtual resolution with length equal to its codimension). In particular, we show that all balanced simplicial complexes are virtually Cohen–Macaulay. [ABSTRACT FROM AUTHOR]

Two thione‐coordinated dinuclear bismuth(III) halides were isolated with a rare geometry and characterized using FT‐IR, NMR, and single‐crystal X‐ray diffraction techniques. The sterically less crowded N‐heterocyclic thione ligand gave bismuth(III) chloride‐thione complex in cis isomeric form, while the sterically more crowded N‐heterocyclic thione ligand gave dinuclear bismuth(III) bromide‐thione complex in facial isomeric form. Continuous shape measurement (CShM) revealed that the bismuth centres in cis‐bismuth(III) chloride‐thione complex adopts distorted octahedral geometry with the CShM values of 2.103, which is remarkably different compared to the facial‐dinuclear bismuth(III) bromide‐thione complex (1.071 and 0.391). [ABSTRACT FROM AUTHOR]

A reaction between CoCl2 and L3-(CO2−)2 (2 : 1 stoichiometry) in CH3OH affords a discrete complex [Co II4- {L3-(CO2−)2}2(μ3-OCH3)2(CH3OH)2(H2O)2Cl2] (1) [L3-(CO2−)2 = 3-[N-{2-(pyridin-2-yl)methyl}amino]-bis(propionate)]. The structure of 1 reveals two terminal mononuclear CoII{L3-(CO2−)2}Cl units connected by a dimeric Co II2 (μ3-OCH3)2(CH3OH)2(H2O) unit present in the centre through two methoxo (μ3-OCH3)- and two carboxylate (μ-1,1-OCO-) bridges affording a tetranuclear coordination cluster of Co(II) with a defective dicubane topology. In 1, Co1 (terminal) has distorted octahedral CoIIN2O3Cl and the central Co2 has CoIIO6 coordination. Such coordination arrangements afford the observed topology. Variable-temperature magnetic studies reveal anti-ferromagnetic coupling in 1. Three magnetic exchange interactions (one anti-ferromagnetic and two ferromagnetic: J1 = +3.3 cm−1 (Co⋯Co 3.176 Å; μ-1,1-OCO- and μ3-OCH3 bridges), J2 = −2.5 cm−1 (Co⋯Co 3.228 Å; μ-1-OCO- and μ3-OCH3 bridges) and J3 = +10.6 cm−1 (Co⋯Co 3.084 Å; two μ3-OCH3 bridges)) have been identified, with the inclusion of the orbital reduction parameter (α = Aκ = 1.38), spin–orbit coupling (λ = −158 cm−1) and axial distortion (energy gap Δ = −975 cm−1 between singlet and doublet levels), rationalized by density functional theory (DFT) calculations. [ABSTRACT FROM AUTHOR]

New complexes I–IV of di-tert-butyltin with ligands based on heterocyclic thioamides (2-mercaptobenzoxazole, 2-mercaptobenzothiazole, 2-mercaptobenzimidazole) and 2,6-di-tert-butyl-4-mercaptophenol are synthesized and studied by X-ray diffraction (XRD). The XRD results for single crystals of I, II, and IV are presented (CIF files CCDC nos. 2251495, 2251493, and 2251494, respectively). Specific features of the synthesized crystal structures are discussed. Complexes I and II contain the expected Sn–C and Sn–S bonds and an additional coordination with the nitrogen atom in the heterocycles, which indicates the octahedral environment of the Sn(IV) atom (coordination number 6). The coordination polyhedron in complex IV can be described as a distorted tetrahedron (coordination number 4). The proposed compounds are studied as antiproliferative agents. Their antiproliferative activity is determined using the human cancer cell lines (PC3, MCF-7, НСT116, A549, and normal cells WI38). A dependence of the activity on the ligand structure is found. A comparative evaluation of the activity shows that the introduction of the antioxidant 2,6-di-tert-butyl-4-mercaptophenol fragment into complex IV substantially decreases the cytotoxicity. [ABSTRACT FROM AUTHOR]

The study of algebraic structures endowed with the concept of symmetry is made possible by the link between Lie algebras and symmetric monoidal categories. This relationship between Lie algebras and symmetric monoidal categories is useful and has resulted in many areas, including algebraic topology, representation theory, and quantum physics. In this paper, we present analogous definitions for Lie algebras within the framework of whiskered structures, bimorphisms, crossed complexes, crossed differential graded algebras, and tensor products. These definitions, given for groupoids in existing literature, have been adapted to establish a direct correspondence between these algebraic structures and Lie algebras. We show that a 2-truncation of the crossed differential graded Lie algebra, obtained from our adapted definitions, gives rise to a braided crossed module of Lie algebras. We also construct a functor to simplicial Lie algebras, enabling a systematic mapping between different Lie algebraic categories, which supports the validity of our adapted definitions and establishes their compatibility with established categories. [ABSTRACT FROM AUTHOR]

NiO-based films and nanostructured materials have received increasing attention for a variety of technological applications. Among the possible strategies for their fabrication, atomic layer deposition (ALD) and chemical vapor deposition (CVD), featuring manifold advantages of technological interest, represent appealing molecule-to-material routes for which a rational precursor design is a critical step. In this context, the present study is focused on the coordination sphere engineering of three heteroleptic Ni(II) β-diketonate–diamine adducts of general formula [NiL2TMEDA] [L = 1,1,1-trifluoro-2,4-pentanedionate (tfa), 2,2-dimethyl-6,6,7,7,8,8,8-heptafluoro-3,5-octanedionate (fod) or 2,2,6,6-tetramethyl-3,5-heptanedionate (thd), and TMEDA = N,N,N′,N′-tetramethylethylenediamine]. Controlled variations in the diketonate structure are pursued to investigate the influence of steric hindrance and fluorination degree on the chemico-physical characteristics of the compounds. A multi-technique investigation supported by density functional calculations highlights that all complexes are air-insensitive and monomeric and that their thermal properties and fragmentation patterns are directly dependent on functional groups in the diketonate ligands. Preliminary thermal CVD experiments demonstrate the precursors' suitability for the obtainment of NiO films endowed with flat and homogeneous surfaces, paving the way to future implementation for CVD end-uses. [ABSTRACT FROM AUTHOR]

Let Δ be a one‐dimensional simplicial complex. Let IΔ$I_\Delta$ be the Stanley–Reisner ideal of Δ. We prove that for all s≥1$s \ge 1$ and all intermediate ideals J generated by IΔs$I_\Delta ^s$ and some minimal generators of IΔ(s)$I_\Delta ^{(s)}$, we have regJ=regIΔs=regIΔ(s).$$\begin{equation*} \hspace*{120pt}\operatorname{reg}J = \operatorname{reg}I_\Delta ^s = \operatorname{reg}I_\Delta ^{(s)}.\hspace*{-120pt} \end{equation*}$$ [ABSTRACT FROM AUTHOR]

A comprehensive analysis of the intermolecular interaction energy and anharmonic vibrations of 41 structures of the HXeY⋯HX (X, Y = F, Cl, Br, I) family of noble-gas-compound complexes for all possible combinations of Y and X was conducted. New structures were identified, and their interaction energies were studied by means of symmetry-adapted perturbation theory, up to second-order corrections: this provided insight into the physical nature of the interaction in the complexes. The energy components were discussed, in connection to anharmonic frequency analysis. The results show that the induction and dispersion corrections were the main driving forces of the interaction, and that their relative contributions correlated with the complexation effects seen in the vibrational stretching modes of Xe–H and H–X. Reasonably clear patterns of interaction were found for different structures. Our findings corroborate previous findings with better methods, and provide new data. These results suggest that the entire group of the studied complexes can be labelled as "naturally blueshifting", except for the complexes with HI. [ABSTRACT FROM AUTHOR]

Single-crystal-to-single-crystal metalation of organic ligands represents a novel method to prepare metal–organic complexes, but remains challenging. Herein, a hierarchical self-assembly {(H12L8)·([N(C2H5)4]+)3·(ClO4−)15·(H2O)32} (1) (L = tris(2-benzimidazolylmethyl) amine) consisting of π-stacked cubes which are assembled from eight partially protonated L ligands is obtained. By soaking the crystals of compound 1 in the aqueous solution of Co(SCN)2, the ligands coordinate with Co2+ ions stoichiometrically and ClO4− exchange with SCN− via single-crystal-to-single-crystal transformation, leading to {([CoSCNL]+)8·([NC8H20]+)3·(SCN)11·(H2O)13} (2). [ABSTRACT FROM AUTHOR]

New thioetheramide ligands (L, PhS(CH2)nCONRR') co-extract Pd(II) and Pt(IV) from acidic chloride solutions. The Pd is transferred to a water-immiscible phase as a [Pd(L)2Cl2] complex with thioether groups in the inner sphere whilst Pt is extracted in an outer-sphere assembly, [(LH)2·PtCl6], containing protonated reagent molecules LH+ that charge-balance the chloridoplatinate dianion, [PtCl6]2-. The much higher kinetic and thermodynamic stability of the Pd(II) complex makes it possible to strip the Pt into a weakly acidic aqueous phase before recovering the Pd by back-extraction into aqueous ammonia to form [Pd(NH3)4]Cl2, thereby separating the two elements. An alkyl spacer group with two methylene units between the thioether (S) and amide (C) atoms is a stronger extractant for both metals than those with one or three methylene units. The extractants reject trianionic chloridometalates with higher hydration energies such as [IrCl6]3-. X-ray structures of two [Pd(L)2Cl2] complexes (L, PhSCH2CONH-n-C4H9 or PhS(CH2)2CONH-n-C4H9) have planar coordination with a transarrangement of the thioether groups and geometries very similar to those predicted by DFT calculations. These calculations show that addition of a proton to the proligands L generates a pseudochelate with the added H+ located between the S atom and the carbonyl O atom. In contrast to related ether- and amino-amide extractants, this pseudochelate ring is broken in the [(LH)2.PtCl6] assemblies formed by the thioetheramides and the OH+ and NH units make the close contacts to the PtCl62- ion. [ABSTRACT FROM AUTHOR]

Strong spin-dependent delocalization (double exchange) was previously demonstrated for the complexes, NN-Bridge-SQ-Coiii(py)2Cat-Bridge-NN (where NN = S = 1 2 nitronylnitroxide, Bridge = 1,4-phenylene and single bond, SQ = S = 1 2 orthobenzosemiquinone, Coiii = low-spin d6 cobalt 3+, and Cat = diamagnetic catecholate). The mixed-valent S = 1 2 SQ-Coiii-Cat triad results in ferromagnetic alignment of localized (pinned) NN spins which are ∼22 Å apart (Bridge = Ph). Herein, we report similar ferromagnetic coupling of localized verdazyl (Vdz) radical spins. The origin of the magnetic exchange results from a second order vibronic effect (pseudo Jahn-Teller effect) in [Vdz-diox-Ru(py)2-diox-Vdz]0, which possesses a diamagnetic [diox-Ru-diox]0 triad by virtue of strong antiferromagnetic SQ-Ruiii exchange. [ABSTRACT FROM AUTHOR]

Complete-active-space self-consistent field (CASSCF) calculations combined with different active spaces have been performed on four novel two-coordinate Co(II) imido complexes with the [CoN]+ core for the short Co＝N bond. The computational demand of the selected CAS (11, 8) for all complexes is acceptable, and it also gives the most accurate results compared with the experiments. CAS (11, 8) calculations show that shortening the Co＝N bond would result in the decrease of the total orbital angular momentum L, and then the weakening of the spin–orbit coupling interaction. However, the energy level splittings for different Co＝N bonds are almost close to each other due to the continuously increasing crystal field around Co(II). Thus, the decrease of the axial symmetry is the most important factor on the lower energy barriers for two-coordinate Co(II) imido complexes with shorter Co＝N bonds. Breaking the [CoN]+ core through lengthening the Co＝N bonds for two-coordinate Co-based complexes is a good choice to enhance the energy barrier. [ABSTRACT FROM AUTHOR]

Matroid models are used to approximate complex systems that can be used to solve problems in the real world. The main goal of this paper is to show how matrices and rough sets on simplicial complexes can be used to create new types of matroids called simplicial matroids. We will look at some of their material properties. Because of these results, we are interested in learning about circuit and base axioms, rank functions, and closure operators. We also give more equivalent relations that can be used to make other equivalent simplicial matroids, such as 2-circuit simplicial matroids. [ABSTRACT FROM AUTHOR]

We propose a novel mathematical paradigm for the study of genetic variation in sequence alignments. This framework originates from extending the notion of pairwise relations, upon which current analysis is based on, to k-ary dissimilarity. This dissimilarity naturally leads to a generalization of simplicial complexes by endowing simplices with weights, compatible with the boundary operator. We introduce the notion of k-stances and dissimilarity complex, the former encapsulating arithmetic as well as topological structure expressing these k-ary relations. We study basic mathematical properties of dissimilarity complexes and show how this approach captures watershed moments of viral dynamics in the context of SARS-CoV-2 and H1N1 flu genomic data. [ABSTRACT FROM AUTHOR]

Sphere-bases are constructed for the ℤ 2 vector space formed by the k -dimensional subcomplexes, of n -simplex (or n -cube), for which every (k − 1) -face is contained in a positive even number of k -cells; addition is symmetric difference of the corresponding sets of k -cells. The bases consist of the boundaries of an algorithmically-specified family of k + 1 -simplexes or k + 1 -cubes. Properties of the bases are investigated. [ABSTRACT FROM AUTHOR]

Signals with a sparse representation in a given basis as well as Laplacian eigenvectors of graphs play a big role in signal processing and machine learning. We put these topics together and look at signals on graphs that have a sparse representation in the basis of eigenvectors of the Laplacian matrix, which may appear after convolution with an unknown sparse filter. We give explicit algorithms to recover those sums by sampling the signal only on few vertices, i.e., the number of required samples is independent of the total size of the graph and takes only local properties of the graph into account. We generalize these methods to simplicial complexes. [ABSTRACT FROM AUTHOR]

Two novel Co–Dy heterometallic complexes, namely, [Co III2 Dy III2 (HL)L{(py)2CO2}{(py)2C(OCH3)O}(NO3)4CH3OH]·2CH3CN·2.5H2O (1, H3L = 2-((2-hydroxy-benzylidene)amino)propane-1,3-diol, (py)2C(OH)2 = the gem-diol form of di-2-pyridyl ketone (dpk) and (py)2C(OCH3)OH = the hemiacetal form of dpk) and [CoIII{(py)2C(OH)O}2][Co II2 Dy III4 DyIII(HL)4(L)4]·3CH3CN·15H2O (2), have been obtained by employing H3L, dpk and Dy(NO3)3·5H2O reacting with Co(NO3)2·6H2O or anhydrous Co(CH3CO2)2 in the solutions of CH3OH and CH3CN (V/V = 1/20) containing N(Me)4OH·5H2O or triethylamine. Structural analysis indicated that two DyIII and two CoIII in 1 formed a star-like topology, while the metal ions in [Co II2 Dy III4 DyIII(HL)4(L)4]− of 2 formed a cross-shaped topology, in which one DyIII and two CoII formed a linear Co II2 DyIII paramagnetic unit. Interestingly, the paramagnetic Co II2 DyIII units in 2 were isolated not only by four diamagnetic CoIII in the [Co II2 Dy III4 DyIII(HL)4(L)4]−, but also by the diamagnetic [CoIII{(py)2C(OH)O}2]+. More interestingly, the distortion of the square antiprism (SAP-8) coordination geometry of DyIII in 2 can be ascribed to the presence of four long Dy–O bonds and four short Dy–O bonds, which could result from the discrepancies of the polarization of CoII and CoIII. Magnetic studies indicated that 1 exhibit two types of magnetic relaxation behavior, in which the magnetic relaxation of the fast relaxation (FR) phase is mainly governed by QTM, while that of the slow relaxation (SR) phase is mainly governed by Orbach and Raman processes with an effective energy barrier (Ueff) value of 59.4(9) K. To our knowledge, the Ueff value is relatively high among the reported CoIII–DyIII heterometallic complexes. For 2, it only shows slow relaxation behavior with a Ueff of 6.43 K. Ab initio calculations revealed that the SR phase and FR phase are governed by the magnetic relaxation of Dy2 and Dy1, respectively, though the axial symmetry of Dy2 (muffin, Cs) is lower than that of Dy1 (triangular dodecahedron, D2d). However, for 2, the theoretical calculations indicated that the severely compressed SAP-8 geometry of DyIII leads to a relatively low Ueff value. [ABSTRACT FROM AUTHOR]

The propensity of 4-hydroxybenzhydrazone-related ligands derived from 3-methoxysalicylaldehyde (H2L3OMe), 4-methoxysalicylaldehyde (H2L4OMe), and salicylaldehyde (H2LH) to act as chelating and/or bridging ligands in Ni(II) complexes was investigated. Three clusters of different nuclearities, [Ni3(L3OMe)2(OAc)2(MeOH)2]∙2MeOH∙MeCN (1∙2MeOH∙MeCN), [Ni2(HL4OMe)(L4OMe)(OAc)(MeOH)2]∙4.7MeOH (2∙4.7MeOH), and [Ni4(HLH)2(LH)2(OAc)2]∙4MeOH·0.63H2O·0.5MeCN·HOAc (3∙4MeOH·0.63H2O·0.5MeCN·HOAc), were prepared from Ni(OAc)2∙4H2O and the corresponding ligand in the presence of Et3N. The hydrazones in these acetato- and phenoxido-bridged clusters acted as singly or doubly deprotonated ligands. When pyridine was used, mononuclear complexes with the square-planar geometry seemed to be favoured, as found for complexes [Ni(L3OMe)(py)] (4), [Ni(L4Ome)(py)] (5) and [Ni(LH)(py)] (6). Ligand substituent effects and the stability of square-planar complexes were investigated and quantified by extensive quantum chemical analysis. Obtained results showed that standard Gibbs energies of binding were lower for square-planar than for octahedral complexes. Starting from [MoO2(L)(EtOH)] complexes as precursors and applying the metal-exchange procedure, the mononuclear complexes [Ni(HL3OMe)2]∙MeOH (7∙MeOH) and [Ni(HLH)]∙2MeOH (9∙2MeOH) and hybrid organic–inorganic compound [Ni2(HL4OMe)2(CH3OH)4][Mo4O10(OCH3)6] (10) were achieved. The octahedral complexes [Ni(HL)2] (7–9) can also be obtained by the direct synthesis from Ni(Oac)2∙4H2O and the appropriate ligand under specific reaction conditions. Crystal and molecular structures of 1∙2MeOH∙MeCN, 2∙4.7MeOH, 3∙4MeOH∙0.63H2O∙0.5MeCN∙HOAc, 4, 5, 9∙2MeOH, and 10 were determined by the single-crystal X-ray diffraction method. [ABSTRACT FROM AUTHOR]

The new heterometallic compounds, [{Cu(pn)2}2Ni(NCS)6]n·2nH2O (1), [{CuII(trien)}2Ni(NCS)6CuI(NCS)]n (2) and [Cu(tren)(NCS)]4[Ni(NCS)6] (3) (pn = 1,2-diaminopropane, trien = triethylenetetramine and tren = tris(2-aminoethylo)amine), were obtained and characterized by X-ray analysis, IR spectra, XAS and magnetic measurements. Compounds 1, 2 and 3 show the structural diversity of 2D, 1D and 0D compounds, respectively. Depending on the polyamine used, different coordination polyhedron for Cu(II) was found, i.e., distorted octahedral (1), square pyramidal (2) and trigonal bipyramidal (3), whereas coordination polyhedron for nickel(II) was always octahedral. It provides an approach for tailoring magnetic properties by proper selection of auxiliary ligands determining the topology. In 1, thiocyanate ligands form bridges between the copper and nickel ions, creating 2D layers of sql topology with weak ferromagnetic interactions. Compound 2 is a mixed-valence copper coordination polymer and shows the rare ladder topology of 1D chains decorated with [CuII(tren)]2+ antennas as the side chains attached to nickel(II). The ladder rails are formed by alternately arranged Ni(II) and Cu(I) ions connected by N2 thiocyanate anions and rungs made by N3 thiocyanate. For the Cu(I) ions, the tetrahedral thiocyanate environment mixed N/S donor atoms was found, confirming significant coordination spheres rearrangement occurring at the copper precursor together with the reduction in some Cu(II) to Cu(I). Such topology enables significant simplification of the magnetic properties modeling by assuming magnetic coupling inside {NiIICuII2} trinuclear units separated by diamagnetic [Cu(NCS)(SCN)3]3− linkers. Compound 3 shows three discrete mononuclear units connected by N-H...N and N-H...S hydrogen bonds. Analysis of XAS proves that the average ligand character and the covalency of the unoccupied metal d-based orbitals for copper(II) and nickel(II) increase in the following order: 1 → 2 → 3. In 1 and 2, a weak ferromagnetic coupling between copper(II) and nickel(II) was found, but in 2, additional and stronger antiferromagnetic interaction between copper(II) ions prevailed. Compound 3, as an ionic pair, shows, as expected, a spin-only magnetic moment. [ABSTRACT FROM AUTHOR]

Metal-containing clusters have attracted increasing attention over the past 2–3 decades. This intense interest can be attributed to the fact that these discrete metal aggregates, whose atomically precise structures are resolved by single-crystal X-ray diffraction (SCXRD), often possess intriguing geometrical features (high symmetry, aesthetically pleasing shapes and architectures) and fascinating physical properties, providing invaluable opportunities for the intersection of different disciplines including chemistry, physics, mathematical geometry and materials science. In this review, we attempt to reinterpret and connect these fascinating clusters from the perspective of Platonic and Archimedean solid characteristics, focusing on highly symmetrical and complex metal-containing (metal = Al, Ti, V, Mo, W, U, Mn, Fe, Co, Ni, Pd, Pt, Cu, Ag, Au, lanthanoids (Ln), and actinoids) high-nuclearity clusters, including metal-oxo/hydroxide/chalcogenide clusters and metal clusters (with metal–metal binding) protected by surface organic ligands, such as thiolate, phosphine, alkynyl, carbonyl and nitrogen/oxygen donor ligands. Furthermore, we present the symmetrical beauty of metal cluster structures and the geometrical similarity of different types of clusters and provide a large number of examples to show how to accurately describe the metal clusters from the perspective of highly symmetrical polyhedra. Finally, knowledge and further insights into the design and synthesis of unknown metal clusters are put forward by summarizing these "star" molecules. [ABSTRACT FROM AUTHOR]

Fractal structures with self‐similarity are of fundamental importance in the fields of aesthetic, chemistry and mathematics. Here, by taking advantage of constructs the rational geometry‐directed precursor design, we report the construction of two fascinating Platonic solids, the Sierpiński tetrahedron ST‐T and the Sierpiński octahedron ST‐O, in which each possesses a fractal Sierpiński triangle on their independent faces. These two discrete complexes are formed in near‐quantitative yield from the multi‐component self‐assembly of truncated Sierpiński triangular kernel L1 with tribenzotriquinacene‐based hexatopic and anthracene‐based tetratopic terpyridine ligands (L3 and L4) in the presence of metal ions, respectively. The enhanced stabilities of the 3D discrete structures were investigated by gradient tandem mass spectrometry (gMS2). This work provides new constructs for the imitation of complex virus assemblies and for the molecular encapsulation of giant guest molecules. [ABSTRACT FROM AUTHOR]

Sodium4‐hydroxy‐3‐([2‐picolinoylhydrazineylidene]methyl)benzenesulfonate (NaH2PH) was synthesized as a novel water‐soluble ligand, by the condensation of picolinohydrazide with sodium 3‐formyl‐4‐hydroxybenzenesulfonate. The (NaH2PH) ligand and its isolated Co (II), Fe (III), Hg (II), and Pd (II) complexes were analyzed by elemental analysis and characterized by spectroscopic (Fourier transform infrared spectroscopy, UV–visible, powder XRD, 1H NMR,13C NMR, MS) and magnetic measurements. By comparing IR spectra of both ligand and the metal complexes, one can assume that the (NaH2PH) ligand behaves as a bi‐negative tetradentate (ONNO) in [Co (NaPH)(H2O)2].3H2O, and a mono‐negative tridentate (ONO) in [Fe (NaPH)Cl2(H2O)] complex, whereas in [Hg2(NaPH)Cl2(H2O)] complex, (NaH2PH) coordinates as a bi‐negative pentadentate (ONNNO) ligand via deprotonated OH group of phenolic ring (C=N)Py and (C=N*) coordinated to one of Hg (II) ion and the oxygen atom of enolic group and (C=N)az group with the another Hg (II) ion. Moreover, (NaH2PH) acts as bi‐negative tridentate (ONO) ligand in [Pd (NaPH)(H2O)].2H2O complex. The geometries of complexes were suggested based on the UV–visible spectra, magnetic measurements and confirmed by applying discrete Fourier transform (DFT) optimization studies. The thermal fragmentation of both [Pd (NaPH)(H2O)].2H2O and [Co (NaPH)(H2O)2].3H2O complexes was performed, and the kinetic and thermodynamic parameters were computed using the Coats–Redfern and Horowitz–Metzger methods. The redox behavior of divalent ions of cobalt and mercury were discussed by the cyclic voltammetry technique in the presence and absence of (NaH2PH) ligand. Biological potencies of the ligand and its metal complexes were evaluated as antioxidants (ABTS and DPPH), anticancer, DNA, and antimicrobial (Staphylococcus aureus and Bacillus subtilis as Gram (+) bacteria, Escherichia coli and Pseudomonas aeruginosa as Gram (−) bacteria, and Candida albicans as fungi). [ABSTRACT FROM AUTHOR]

We present a short proof of the adek–Král–Matouek–Vokřínek–Wagner result from the title (in the following form due to Filakovský–Wagner–Zhechev). For any fixed evenlthere is no algorithm recognizing the extendability of the identity map of S l to a PL map X → S l of given 2l-dimensional simplicial complexXcontaining a subdivision of S l as a given subcomplex. We also exhibit a gap in the Filakovský–Wagner–Zhechev proof that embeddability of complexes is undecidable in codimension > 1 . [ABSTRACT FROM AUTHOR]