N,N,N^{\prime },N^{\prime }-Tetrabromobenzene-1,3-disulfonamide and poly(N-bromo-N-ethylbenzene-1,3-disulfonamide) as new and efficient catalysts for the synthesis of highly substituted 1,6-dihydropyrazine-2,3-dicarbonitrile derivatives

Various mono and bis-1,6-dihydropyrazine-2,3-dicarbonitrile derivatives were efficiently synthesized by reacting 2,3-diaminomaleonitrile (DAMN), isocyanides and ketones in the presence of a catalytic amount of $N,N{,}N^{\prime }{,}N^{\prime }$ -tetrabromobenzene-1,3-disulfonamide [TBBDA] and poly( $N$ -bromo- $N$ -ethylbenzene-1,3-disulfonamide) [PBBS] in EtOH/H $_{2}$ O at ambient temperature. Graphic abstract $N,N,N^{\prime },N^{\prime }$ -Tetrabromobenzene-1,3-disulfonamide and poly( $N$ -bromo- $N$ -ethylbenzene-1,3-disulfonamide) as new and efficient catalysts for the synthesis of highly substituted 1,6-dihydropyrazine-2,3-dicarbonitrile derivatives.      

Quinazoline-2,4(1H,3H)-diones inhibit the growth of multiple human tumor cell lines

Quinazoline-2,4( $1H,3H$ )-diones exhibit a wealth of biological activities including antitumor proliferation. We established an improved method for the synthesis of quinazoline-2,4( $1H,3H$ )-dione derivatives with three points of molecular diversity. Data indicate that compounds 60 (average $\text{ logGI}_{50} \!=\! -6.1$ ), 65 (average $\text{ logGI}_{50} \!=\! -6.13$ ), 69 (average $\text{ logGI}_{50} \!=\! -6.44$ ), 72 (average $\text{ logGI}_{50} \!=\! -6.39$ ), and 86 (average $\text{ logGI}_{50} = -6.45$ ) significantly inhibited the in vitro growth of 60 human tumor cell lines tested. Structure–activity relationship analyses indicate that chlorophenethylureido is the necessary substituent at the $\text{ D}_{3}$ diversity point (7-position of quinazoline-2,4( $1H,3H$ )-dione), in particular, $o$ -chlorophenethylurea (69) achieved optimal activity. $o$ - or $m$ -Chlorophenethyl substitutions (69 and 72) at the $\text{ D}_{2}$ diversity point (3-position of quinazo line-2,4( $1H,3H$ )-dione) gave the most potent compounds. Methoxyl and 4-methylpiperazin-1-yl substitution at the $\text{ D}_{1}$ diversity point (6-position of quinazoline-2,4( $1H,3H$ )-dione skeleton) may yield better activity than other groups. The quinazoline-2,4( $1H,3H$ )-dione scaffold can be effectively replaced by 2 $H$ -benzo[b][1,4]thiazin-3(4 $H$ )-one.     

Analysis of the \frac{1} {2}^ - and \frac{3} {2}^ - heavy and doubly heavy baryon states with QCD sum rules

In this article, we study the $\frac{1} {2}^ -$ and $\frac{3} {2}^ -$ heavy and doubly heavy baryon states $\Sigma _Q \left( {\frac{1} {2}^ - } \right)$ , $\Xi '_Q \left( {\frac{1} {2}^ - } \right)$ , $\Omega _Q \left( {\frac{1} {2}^ - } \right)$ , $\Xi _{QQ} \left( {\frac{1} {2}^ - } \right)$ , $\Omega _{QQ} \left( {\frac{1} {2}^ - } \right)$ , $\Sigma _Q^* \left( {\frac{3} {2}^ - } \right)$ , $\Xi _Q^* \left( {\frac{3} {2}^ - } \right)$ , $\Omega _Q^* \left( {\frac{3} {2}^ - } \right)$ , $\Xi _{QQ}^* \left( {\frac{3} {2}^ - } \right)$ and $\Omega _{QQ}^* \left( {\frac{3} {2}^ - } \right)$ by subtracting the contributions from the corresponding $\frac{1} {2}^ +$ and $\frac{3} {2}^ +$ heavy and doubly heavy baryon states with the QCD sum rules in a systematic way, and make reasonable predictions for their masses.     

Influence of assortativity and degree-preserving rewiring on the spectra of networks

Newman’s measure for (dis)assortativity, the linear degree correlation coefficient $\rho _{D}$ , is reformulated in terms of the total number N _k of walks in the graph with k hops. This reformulation allows us to derive a new formula from which a degree-preserving rewiring algorithm is deduced, that, in each rewiring step, either increases or decreases $\rho _{D}$ conform our desired objective. Spectral metrics (eigenvalues of graph-related matrices), especially, the largest eigenvalue $\lambda _{1}$ of the adjacency matrix and the algebraic connectivity $\mu _{N-1}$ (second-smallest eigenvalue of the Laplacian) are powerful characterizers of dynamic processes on networks such as virus spreading and synchronization processes. We present various lower bounds for the largest eigenvalue $\lambda _{1}$ of the adjacency matrix and we show, apart from some classes of graphs such as regular graphs or bipartite graphs, that the lower bounds for $\lambda _{1}$ increase with $\rho _{D}$ . A new upper bound for the algebraic connectivity $\mu _{N-1}$ decreases with $\rho _{D}$ . Applying the degree-preserving rewiring algorithm to various real-world networks illustrates that (a) assortative degree-preserving rewiring increases $\lambda _{1}$ , but decreases $\mu _{N-1}$ , even leading to disconnectivity of the networks in many disjoint clusters and that (b) disassortative degree-preserving rewiring decreases $\lambda _{1}$ , but increases the algebraic connectivity, at least in the initial rewirings.     

Local structure of Fe-doped In 2 O 3 films investigated by X-ray absorption fine structure spectroscopy

$(\mathrm{In}_{1-x}\mathrm{Fe}_{x})_{2}\mathrm{O}_{3}$ $(x=0.07, 0.09, 0.16, 0.22, 0.31)$ films were deposited on Si (100) substrates by RF-magnetron sputtering technique. The influence of Fe doping on the local structure of films was investigated by X-ray absorption spectroscopy (XAS) at Fe K-edge and L-edge. For the $(\mathrm{In}_{1-x}\mathrm{Fe}_{x})_{2}\mathrm{O}_{3}$ films with $x=0.07, 0.09 \mbox{ and } 0.16$ , Fe ions dissolve into $\mathrm{In}_{2}\mathrm{O}_{3}$ and substitute for $\mathrm{In}^{3+}$ sites with a mixed-valence state ( $\mathrm{Fe}^{2+}/\mathrm{Fe}^{3+}$ ) of Fe ions. However, a secondary phase of Fe metal clusters is formed in the $(\mathrm{In}_{1-x}\mathrm{Fe}_{x})_{2}\mathrm{O}_{3}$ films with $x=0.22 \mbox{ and } 0.31$ . The qualitative analyses of Fe-K edge extended X-ray absorption fine structure (EXAFS) reveal that the Fe–O bond length shortens and the corresponding Debye–Waller factor ( $\sigma^{2}$ ) increases with the increase of Fe concentration, indicating the relaxation of oxygen environment of Fe ions upon substitution. The anomalously large structural disorder and very short Fe–O distance are also observed in the films with high Fe concentration. Linear combination fittings at Fe L-edge further confirm the coexistence of $\mathrm{Fe}^{2+}$ and $\mathrm{Fe}^{3+}$ with a ratio of ${\sim}3:2$ ( $\mathrm{Fe}^{2+}: \mathrm{Fe}^{3+}$ ) for the $(\mathrm{In}_{1-x}\mathrm{Fe}_{x})_{2}\mathrm{O}_{3}$ film with $x=0.16$ . However, a significant fraction ( ${\sim}40~\mbox{at\%}$ ) of the Fe metal clusters is found in the $(\mathrm{In}_{1-x}\mathrm{Fe}_{x})_{2}\mathrm{O}_{3}$ film with $x=0.31$ .     

Fougerite FeII − III oxyhydroxycarbonate in environmental chemistry and nitrate reduction

The identification of the fougerite mineral responsible for the bluish-green shade of gleysols in aquifers as being the Fe^II???III oxyhydroxycarbonate $\text{GR}(\text{CO}_{3}^{2-})^*$ of formula, $[\text{Fe}^{\rm II}_{6x}\text{Fe}^{\rm III}_{6(1 - x)}\text{O}_{12}\text{H}_{2(7-3x)}]^{2+}\bullet[\text{CO}_{3}^{2-}\bullet3\text{H}_{2}\text{O}]^{2-}$ where the ferric molar ratio x = [Fe^III/Fe_total] is restricted to the domain [1/3–2/3] induces to study the reactivity of the synthetic green rust for reducing some major pollutants. The oxidation within the solid compound $\text{GR}(\text{CO}_{3}^{2-})^*$ in the presence of nitrates is followed by miniaturized Mössbauer spectrometer (MIMOS). Ratio x = [Fe^III/Fe_total] increases up to 0.67 where $\text{GR}(\text{CO}_{3}^{2-})^*$ transforms gradually into magnetite. This could well explain the composition variability of fougerite occurrences.     

Effect of Y3+ substitution on the structural,dielectric, and electrical properties of nanosized ZnAl 2 O 4 spinel

In the present work, we have studied the structural, dielectric, and electrical properties of a series of nanosized $\mathrm{ZnAl}_{2-2x}\mathrm{Y}_{2x}\mathrm{O}_{4}$ ( $x = 0.00$ , 0.01, 0.02, 0.03, 0.04, 0.05, 0.07, and 0.10) system prepared by chemical coprecipitation method. Powder X-ray diffraction (XRD) was carried out to study the influence of $\mathrm{Y}^{3+}$ substitution on the crystal structure of these samples. High Resolution Transmission Electron Microscopy (HRTEM) images reveal the nanocrystalline nature of the samples. The Fourier Transform Infrared (FTIR) spectra confirmed the preference of $\mathrm{Y}^{3+}$ ions at the octahedral B site. The variation of dielectric constant and loss tangent (1 kHz to 1 MHz) at room temperature for all the samples show the normal behavior of spinel compounds. AC conductivity study reveals that the conduction is due to small polaron hopping. The electrical modulus analysis shows that nanocrystalline $\mathrm{ZnAl}_{2-2x}\mathrm{Y}_{2x}\mathrm{O}_{4}$ system exhibits non-Debye-type relaxation. The DC electrical resistivity measured in the temperature range 303–373 K was found to increase with temperature and yttrium content.     

Analysis of the {1\over2}^{\pm} flavor antitriplet heavy baryon states with QCD sum rules

In this article, we study the masses and pole residues of the ${1\over2}^{\pm}$ flavor antitriplet heavy baryon states ( $\varLambda _{c}^{+}$ , $\varXi _{c}^{+},\varXi _{c}^{0})$ and ( $\varLambda _{b}^{0}$ , $\varXi _{b}^{0},\varXi _{b}^{-})$ by subtracting the contributions from the corresponding ${1\over2}^{\mp}$ heavy baryon states with the QCD sum rules, and observe that the masses are in good agreement with the experimental data and make reasonable predictions for the unobserved ${1\over2}^{-}$ bottom baryon states. Once reasonable values of the pole residues λ _Λ and λ _Ξ are obtained, we can take them as basic parameters to study the relevant hadronic processes with the QCD sum rules.     

Preliminary studies of the Raman spectra of \hbox {Ag}_{2}\hbox {Te}\hbox { and }\hbox {Ag}_{5}\hbox {Te}_{3}

The theoretical calculations indicated that the monoclinic low-temperature phase of silver telluride $(\upbeta \hbox {-Ag}_{2}\hbox {Te})$ is a new binary topological insulator with highly anisotropic single Dirac cone surface. We obtained $\upbeta \hbox {-Ag}_{2}\hbox {Te}$ crystal ingots containing few grains by the Bridgman method. We also deposited thin films of tellurium, $\hbox {Ag}_{5}\hbox {Te}_{3}\hbox { and }(\hbox {Te+Ag}_{5}\hbox {Te}_{3})$ by thermal evaporation method. The Raman spectra of $\upbeta \hbox {-Ag}_{2}\hbox {Te}$ , tellurium and $\hbox {Ag}_{5}\hbox {Te}_{3}$ were measured at three excitation wave lengths: 633, 515 and 488 nm. The Raman active modes of $\upbeta \hbox {-Ag}_{2}\hbox {Te}$ , tellurium and $\hbox {Ag}_{5}\hbox {Te}_{3}$ are situated at frequencies below 300  $\hbox {cm}^{-1}$ while vibrations of other phases appear at higher frequencies.     

Structure of some FeII − III hydroxysalt green rusts (carbonate, oxalate, methanoate) from Mössbauer spectroscopy

The abundances of Fe^II and Fe^III environments within green rusts one, GR1s, that intercalate carbonate, oxalate and methanoate (formate) anions are found from Mössbauer spectra for compositions corresponding to [Fe $^{\rm II}_{6}$ Fe $^{\rm III}_{2}$ (OH)₁₆]^2?+??[CO $_{3}^{2-}$ ?5H₂O]^2???, [Fe $^{\rm II}_{4}$ Fe $^{\rm III}_{2}$ (OH)₁₂]^2?+??[CO $_{3}^{2-}$ ?3H₂O]^2???, [Fe $^{\rm II}_{6}$ Fe $^{\rm III}_{2}$ (OH)₁₆]^2?+??[C₂O $_{4}^{2-}$ ?4H₂O]^2??? and [Fe $^{\rm II}_{5}$ Fe $^{\rm III}_{2}$ (OH)₁₄]^2?+??[2HCOO^????3H₂O]^2???. These formulae correspond to orders α, β and γ where cation distances are (2 × a ₀), ( $\surd 3$ × a ₀) or a mixture of both leading to (7 × a ₀), where ratio x = {[Fe^III]/[Fe_total]} = 1/4, 1/3 and 2/7, respectively. Anion distributions within interlayers are also devised and long-range orders determined accordingly.     

Insect growth regulatory activity of carvacrol-based 1,3,4-thiadiazoles and 1,3,4-oxadiazoles

Abstract 

In designing of novel insect growth regulators (IGRs), biologically occurring carvacrol has been structurally modified to thiadiazole and oxadiazole moieties. Two series of carvacrol analogs containing 1,3,4-thiadiazole (VIIIa–e) and 1,3,4-oxadiazole (IXa–e) derivatives are designed and synthesized. Their structures are confirmed by FT-IR, \(^{1}\text {H}\) NMR, \(^{13}\)C NMR and LC-MS. IGR activity is tested against Spodoptera litura. Several analogs displayed IGR activity against this insect pest. Compounds VIIIe and IXe displayed relatively good IGR activity with \(\text {GI}_{50 }\)values 117.43 and 108.83 ppm against Spodoptera litura, respectively.

Graphical Abstract

Synthesis, characterization and evaluation of carvacrol-based 1,3,4-thiadiazole and 1,3,4-oxadiazole derivatives as potent insect growth regulators (IGRs).

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Time decay of solutions to the cauchy problem for time-dependent Schrödinger-Hartree equations

We consider the time-dependent Schrödinger-Hartree equation (1) $$iu_t + \Delta u = \left( {\frac{1}{r}*|u|^2 } \right)u + \lambda \frac{u}{r},(t, x) \in \mathbb{R} \times \mathbb{R}^3 ,$$ (2) $$u(0,x) = \phi (x) \in \Sigma ^{2,2} ,x \in \mathbb{R}^3 ,$$ where λ≧0 and \(\Sigma ^{2,2} = \{ g \in L^2 ;\parallel g\parallel _{\Sigma ^{2,2} }^2 = \sum\limits_{|a| \leqq 2} {\parallel D^a g\parallel _2^2 + \sum\limits_{|\beta | \leqq 2} {\parallel x^\beta g\parallel _2^2< \infty } } \} \) . We show that there exists a unique global solutionu of (1) and (2) such that $$u \in C(\mathbb{R};H^{1,2} ) \cap L^\infty (\mathbb{R};H^{2,2} ) \cap L_{loc}^\infty (\mathbb{R};\Sigma ^{2,2} )$$ with $$u \in L^\infty (\mathbb{R};L^2 ).$$ Furthermore, we show thatu has the following estimates: $$\parallel u(t)\parallel _{2,2} \leqq C,a.c. t \in \mathbb{R},$$ and $$\parallel u(t)\parallel _\infty \leqq C(1 + |t|)^{ - 1/2} ,a.e. t \in \mathbb{R}.$$      

Structural, morphological and optical properties of \hbox {In}_{2}\hbox {S}_{3} thin films obtained by SILAR method

$\hbox {In}_{2}\hbox {S}_{3}$ thin films have been elaborated onto glass substrate by SILAR method at room temperature using different immersion time in the solution of cation and anion and fixing the rinsing time. The film composition, morphology and structure were investigated using energy dispersive X-ray analysis (EDAX), scanning electron microscopy (SEM) and X-ray diffraction techniques. Optical properties, such transmission and band gap have been also analyzed. The effects of annealing on the morphological structure thin films are also described. The x-rays diffraction spectra indicated that the formed compounds are $\upbeta $ - $\hbox {In}_{2}\hbox {S}_{3}$ polycrystalline thin films with $\hbox {In}_{6}\hbox {S}_{7 }$ as second phase in sample S1 and sample S2 and no another phase in sample 3. SEM revealed homogeneous and relatively uniform films and EDAX shows sample 3 with S/In=1.44. For sample 1 and sample 2, we noted an increase of band gap when rinsing time increases.     

Quasi solid state dye-sensitized solar cells based on polyvinyl alcohol (PVA) electrolytes containing \mathbf{I}^{\mathbf{-}}/\mathbf{I}_{\mathbf{3}}^{\mathbf{-}} redox couple

Quasi solid state dye-sensitized solar cells (DSSCs) have been fabricated with electrolytes containing $\text{ I }^{-}/\text{ I }_{3}^{-}$ redox couple using 80 % hydrolyzed polyvinyl alcohol (PVA) doped with potassium iodide (KI) and a mixture of potassium iodide and tetrapropyl ammonium iodide ( $\text{ Pr }_{4}\text{ NI }$ ) salts. The quasi solid state gel polymer electrolytes were prepared using 1:1 ethylene carbonate (EC):propylene carbonate (PC) mixture. The solar cells have the structure of ITO/ $\text{ TiO }_{2}$ /N3-Dye/electrolyte/Pt/ITO. The conductivity of the electrolytes has been calculated from the bulk resistance value determined using the electrochemical impedance spectroscopy. The performance of the DSSCs has been studied by varying the concentration of the doping salts in the electrolyte and incident light intensity. The DSSC fabricated with the KI salt electrolyte containing 9.9 wt% PVA, 39.6 wt% EC, 39.6 wt% PC, 10.9 wt% KI $(+\text{ I }_{2})$ exhibited the best power conversion efficiency of 1.97 %. However, the DSSC with a double-salt electrolyte containing 9.9 wt% PVA: 39.6 wt% EC: 39.6 wt% PC: (6.5 wt% KI: 4.4 wt% $\text{ Pr }_{4}\text{ NI }$ ) ( $+\text{ I }_{2}$ ) exhibited a higher efficiency of 3.27% under $100 \text{ mW/cm }^{2}$ light intensity. The efficiency of this cell increased to 4.59 % under dimmer light of intensity of $54 \text{ mW/cm }^{2}$ .     

Study of the ion exchange equilibrium of Cl−, NO_3^{ - } , and SO_4^{{2 - }} ions on the AMX membrane

Equilibrium between the ion exchange membrane and solutions of anions at various valences has been the subject of this investigation. Competitive ion exchange reactions were studied on a strong base anion exchange membrane AMX manufactured by Tokuyama, commercialized by Eurodia, involving Cl^?, $ {\text NO}_3^{ - } $ and $ {\text SO}_4^{{2 - }} $ ions. Solution concentrations studied were 0.05 and 0.1 M for all the systems reported. Experiments were performed with sodium as the counter ion, and the temperature was kept constant (T?=?298 K). Ionic exchange isotherms for the binary systems— $ {{\text Cl}^{ - }}/{\text NO}_3^{ - } $ , $ {{\text Cl}^{ - }}/{\text SO}_4^{{2 - }} $ , and $ {\text NO}_3^{ - }/{\text SO}_4^{{2 - }} $ —were established. The obtained results show that the sulfate was the most strongly sorbed, and the selectivity order is $ {\text SO}_4^{{2 - }} > {\text NO}_3^{ - } > {{\text Cl}^{ - }} $ at 0.05 M and $ {\text NO}_3^{ - } > {\text SO}_4^{{2 - }} > {{\text Cl}^{ - }} $ at 0.1 M under the experimental conditions. Selectivity coefficients $ K_{{{{{\text Cl} }^{ - }}}}^{{{\text NO}_3^{ - }}} $ , $ K_{{2{{{\text Cl} }^{ - }}}}^{{{\text SO}_4^{{2 - }}}} $ , and $ K_{{2{\text NO}_3^{ - }}}^{{{\text SO}_4^{{2 - }}}} $ for the three binary systems were determined. All the results given by this membrane were compared with those obtained, in the same conditions, with the RPA membrane (produced by RHONE POULENC). Ternary equilibrium data were taken for $ {{\text Cl}^{ - }}/{\text NO}_3^{ - }/{\text SO}_4^{{2 - }} $ . The prediction of the ternary system based only on the binary data was consistent with the experimental data obtained for this system. The good agreement between the experimental and the predicted data showed that the proposed framework can be considered as an effective method to predict many ternary systems from binary systems.     

Indecomposable representations with invariant inner product

Consequences of the existence of an invariant (necessarily indefinite) non-degenerate inner product for an indecomposable representation π of a groupG on a space \(\mathfrak{H}\) are studied. If π has an irreducible subrepresentation π₁ on a subspace \(\mathfrak{H}_1 \) , it is shown that there exists an invariant subspace \(\mathfrak{H}_2 \) of \(\mathfrak{H}\) containing \(\mathfrak{H}_1 \) and satisfying the following conditions: (1) the representation π ₁ ^# =π mod \(\mathfrak{H}_2 \) on \(\mathfrak{H}\) mod \(\mathfrak{H}_2 \) is conjugate to the representation (π₁, \(\mathfrak{H}_1 \) ), (2) \(\mathfrak{H}_1 \) is a null space for the inner product, and (3) the induced inner product on \(\mathfrak{H}_2 \) mod \(\mathfrak{H}_1 \) is non-degenerate and invariant for the representation $$\pi _2 = (\pi _2 |_{\mathfrak{H}_2 } )\bmod \mathfrak{H}_1 ,$$ a special example being the Gupta-Bleuler triplet for the one-particle space of the free classical electromagnetic field with \(\mathfrak{H}_1 \) =space of longitudinal photons and \(\mathfrak{H}_2 \) =the space defined by the subsidiary condition.     

Terahertz generation by nonlinear mixing of laser and its second harmonic in a rippled density plasma

Terahertz radiation generation by second-order nonlinear mixing of laser $ (\omega_{1} ,\,\vec{k}_{1} ) $ and its frequency shifted second harmonic $ \omega_{2} = 2\omega_{1} - \omega ,\,\,\vec{k}_{2} \, $ $ (\omega \ll \omega_{1} ) $ in a plasma, in the presence of an obliquely inclined density ripple of wave number $ \vec{q} $ , are investigated. The lasers exert ponderomotive force on electrons and drive density perturbations at $ (2\omega_{1} ,\,2\vec{k}_{1} - \vec{q}) $ and $ (\omega_{1} - \omega_{2} ,\,\vec{k}_{1} - \vec{k}_{2} - \vec{q}) $ . These perturbations beat with the electron oscillatory velocities due to the lasers to produce a nonlinear current at $ \omega ,\,\vec{k} = 2\vec{k}_{1} - \vec{k}_{2} - \vec{q} $ , resonantly driving the terahertz radiation when $ \vec{q} $ satisfies the phase matching condition. The radiated THz intensity depends on the relative polarization of the lasers and scales as the square of intensity of the fundamental laser and linearly with the square root of the intensity of the second harmonic. The THz emission is maximized when the polarization of the lasers is aligned. These results are consistent with the recent experimental results.     

Open Gromov-Witten Invariants of Toric Calabi-Yau 3-Folds

We present a proof of the mirror conjecture of Aganagic and Vafa (Mirror Symmetry, D-Branes and Counting Holomorphic Discs. http://arxiv.org/abs/hep-th/0012041v1, 2000) and Aganagic et al. (Z Naturforsch A 57(1–2):128, 2002) on disk enumeration in toric Calabi-Yau 3-folds for all smooth semi-projective toric Calabi-Yau 3-folds. We consider both inner and outer branes, at arbitrary framing. In particular, we recover previous results on the conjecture for (i) an inner brane at zero framing in ${K_{\mathbb{P}^2}}$ K P 2 (Graber-Zaslow, Contemp Math 310:107–121, 2002), (ii) an outer brane at arbitrary framing in the resolved conifold ${\mathcal{O}_{\mathbb{P}^1}(-1)\oplus \mathcal{O}_{\mathbb{P}^1}(-1)}$ O P 1 ( - 1 ) ⊕ O P 1 ( - 1 ) (Zhou, Open string invariants and mirror curve of the resolved conifold. http://arxiv.org/abs/1001.0447v1 [math.AG], 2010), and (iii) an outer brane at zero framing in ${K_{\mathbb{P}^2}}$ K P 2 (Brini, Open topological strings and integrable hierarchies: Remodeling the A-model. http://arxiv.org/abs/1102.0281 [hep-th], 2011).     

Finitely-Generated Projective Modules over the θ-Deformed 4-Sphere

We investigate the “θ-deformed spheres” ${C(S^{3}_{\theta})}$ and ${C(S^{4}_{\theta})}$ , where θ is any real number. We show that all finitely-generated projective modules over ${C(S^{3}_{\theta})}$ are free, and that ${C(S^{4}_{\theta})}$ has the cancellation property. We classify and construct all finitely-generated projective modules over ${C(S^{4}_{\theta})}$ up to isomorphism. An interesting feature is that if θ is irrational then there are nontrivial “rank-1” modules over ${C(S^{4}_{\theta})}$ . In that case, every finitely-generated projective module over ${C(S^{4}_{\theta})}$ is a sum of a rank-1 module and a free module. If θ is rational, the situation mirrors that for the commutative case θ = 0.