The forcing number of toroidal polyhexes

The forcing number, denoted by f(G), of a graph G with a perfect matching is the minimum number of independent edges that completely determine the perfect matching of G. In this paper, we consider the forcing number of a toroidal polyhex H(p,q,t) with a torsion t, a cubic graph embedded on torus with every face being a hexagon. We obtain that f(H(p,q,t)) ≥ min{p,q}, and equality holds for p ≤ q or p > q and t∈{ 0,p−q,p−q + 1,..., p−1}. In general, we show that f(H(p,q,t)) is equal to the side length of a maximum triangle on H(p,q,t). Based on this result, we design a linear algorithm to compute the forcing number of H(p,q,t).     

On unicyclic conjugated molecules with minimal energies

The energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph. Let U(k) be the set of all unicyclic graphs with a perfect matching. Let C _g(G) be the unique cycle of G with length g(G), and M(G) be a perfect matching of G. Let U ⁰(k) be the subset of U(k) such that g(G)≡ 0 (mod 4), there are just g/2 independence edges of M(G) in C _g(G) and there are some edges of E(G)\ M(G) in G\ C _g(G) for any G∈U ⁰(k). In this paper, we discuss the graphs with minimal and second minimal energies in U ^*(k) = U(k)\ U ⁰(k), the graph with minimal energy in U ⁰(k), and propose a conjecture on the graph with minimal energy in U(k).      

Embedding on alphabet overlap digraphs

Alphabet overlap digraphs can be viewed as a generalization of directed de Bruijn graphs. Given three integers α ≥ 1, k ≥ 2 and 1 ≤ i < k, the alphabet overlap digraph O(α, k ; i) is a digraph: the set of all words of length k over a certain alphabet with cardinality α is vertex set, and there is an arc from a vertex u to a vertex v if and only if the word of last k−i letters of u coincides with the word of first k−i letters of v. In this paper, we consider whether O(α, k ; i) can be embedded in O(α, k ; j) for given integers 1 ≤ i < j < k. In order to resolve this problem, we give an O(1)-time algorithm to decide whether there exists a permutation on {1, . . . ,k} from O(α, k ; i) to O(α, k ; j). If such a permutation exists, for any vertex of O(α, k ; i), we apply the permutation to change its label’s position and map it to a vertex of O(α, k ; j). Furthermore, we obtain an embedding from O(α, k ; i) to O(α, k ; j). Hence, we solve partly the problem. As a consequence, we show that every directed de Bruijn graph can be embedded in all alphabet overlap digraphs with the same parameters α and k.     

Asymmetric Phase-Transfer Catalysis by Quaternary Ammonium Ions Derived from Cinchona-Alkaloid Analogs Containing 1,1′-Binaphthalene Moieties

The synthesis and catalytic properties of a new type of enantioselective phase-transfer catalysts, incorporating both the quinuclidinemethanol fragment of Cinchona alkaloids and a 1,1′-binaphthalene moiety, are described. Catalyst (+)-(aS,3R,4S,8R,9S)- 4 with the quinuclidine fragment attached to C(7′) in the major groove of the 1,1′-binaphthalene residue was predicted by computer modeling to be an efficient enantioselective catalyst for the unsymmetric alkylation of 6,7-dichloro-5-methoxy-2-phenylindanone ( 1 ; Scheme 1, Fig. 1). Its synthesis involved the selective oxidative cross-coupling of two differently substituted naphthalen-2-ols to afford the asymmetrically substituted 1,1′-binaphthalene derivative (±)- 17 in high yield (Scheme 3). Chromatographic optical resolution via formation of diastereoisomeric camphorsulfonyl esters and functional-group manipulation gave access to the 7-bromo-1,1′-binaphthalene derivative (−)-(aS)- 11 (Scheme 4). Nucleophilic addition of lithiated (−)-(aS)- 11 to the quinuclidine Weinreb amide (+)-(3R,4S,8R)- 8 afforded the two ketones (aS,3R,4S,8R)- 27 and (aS,3R,4S,8S)- 28 as an inseparable mixture of diastereoisomers (Scheme 6). Stereoselective reduction of this mixture with DIBAL-H (diisobutylaluminum hydride; preferred formation of the C(8)−C(9) erythro-pair of diastereoisomers with 18% de) or with NaBH₄ (preferred formation of the threo-pair of diastereoisomers with 50% de) afforded the four separable diastereoisomers (+)-(aS,3R,4S,8S,9S)- 29 , (+)-(aS,3R,4S,8R,9R)- 30 , (−)-(aS,3R,4S,8S,9R)- 31 , and (+)-(aS,3R,4S,8R,9S)- 32 (Scheme 6). A detailed conformational analysis, combining ¹H-NMR spectroscopy and molecular-mechanics computations, revealed that the four diastereoisomers displayed distinctly different conformational preferences (Figs. 2 and 3). These novel Cinchona-alkaloid analogs were quaternized to give (+)-(aS,3R,4S,8R,9S)- 4 , (+)-(aS,3R,4S,8S,9S)- 5 , (+)-(aS,3R,4S,8R,9R)- 6 , and (−)-(aS,3R,4S,8S,9R)- 7 (Scheme 7) which were tested as phase-transfer agents in the asymmetric allylation of phenylindanone 1 . Without any optimization work, (+)-(aS,3R,4S,8R,9S)- 4 was found to catalyze the allylation of 1 yielding the predicted enantiomer (+)-(S)- 3b in 32% ee. The three diastereoisomeric catalysts (+)- 5 , (+)- 6 , and (−)- 7 gave access to lower enantioselectivities (6 to 22% ee's), which could be rationalized by computer modeling (Fig. 4).     

Synthesis,Isolation, and Full Spectroscopic Characterization of Eleven (Z)-Isomers of (3R,3′R)-Zeaxanthin

Developmental efforts to improve the yield of the chemical synthesis of (3R,3′R)-zeaxanthin resulted in the isolation, partly by chromatography from reaction mixtures, and full spectroscopic characterization by ¹H-NMR, UV/VIS, and CD spectrosocpy of eleven (Z/E)-isomers of zeaxanthin: (7Z)-, (9Z), (13Z)-, (15Z)-, (7Z,7′Z)-, (9Z,9′Z)- (7Z,9Z,7Z)-, (7Z,11Z,7′Z)-, (9Z,13Z,9′Z)-, (7Z,9Z,7′Z,9′Z)-, and (7Z,9Z,11Z,7′Z,9′Z)-zeaxanthin. Five of these isomers were obtained by specific synthesis, namely the (7Z)-, (7Z,7′Z)-, (9Z,9′Z)-, (7Z,9Z,7′Z)-, and (7Z,9Z,7′Z-9Z)-isomers.     

A complete characterization for <Emphasis Type="Italic">k</Emphasis>-resonant Klein-bottle polyhexes

A hexagonal tessellation K(p, q, t) on Klein bottle, a non-orientable surface with cross-cap number 2, is a finite-sized elemental benzenoid which can be produced from a p × q-parallelogram of hexagonal lattice with usual identifications of sides and with torsion t. Unlike torus, Klein bottle polyhex K(p, q, t) is not transitive except for some degenerated cases. We shall show, however, that K(p, q, t) does not depend on t. Accordingly, criteria for K(p, q, t) to be k-resonant for every positive integer k will be given. Moreover, we shall show that K(3, q, t) of 3-resonance are fully-benzenoid.      

Rogiolol Acetate: A Novel β-Chamigrene-Type Sesquiterpene Isolated from a Marine Sponge

The marine sponge Spongia zimocca SCHMIDT , 1862, collected in front of the torrent II Rogiolo, south of Livorno, contains the sesquiterpene rogiolol acetate (= (+)-(2R,3S,6R,8R,9R)-2,8-dibromo-9-chloro-1,1,9-trimethyl-5-methylidenespiro[5.5]undec-3-yl acetate; (+)- 3a ), which represents the first chamigrane isolated from a sponge. Although compounds of this class are common in red seaweeds of the genus Laurencia, and our sponge actually contains 9-bromochamigrene and a variety of other metabolites of nearby growing Laurencia sp., (+)- 3a is unique to our sponge.     

The stability condition of phase equilibrium for various concentration variables

The stability conditions of phase equilibrium for various concentration variables are educed according to thermodynamic principle. When a system with k components arrives at stable equilibrium, if the mole number n_i or the mole fraction y_i(=n_i/n_k) or molality m_i[= n_i/(n_kM_k)] of component i(i = 1,2,...,k − 1) are elected as concentration variables, thermodynamic theory is able to confirm that the sign of every order determinant composed of the second-order partial differential of chemical potential with respect to these concentration variables is positive; if the mole fraction x_i(= n_i/n) or mass fraction w_i(= n_iM_i/W are elected as the concentration variables, thermodynamic theory is only able to confirm that the sign of (k−1) order determinant is positive; if molarity c_i(= n_i/V are elected as the concentration variables, thermodynamic theory is not able to confirm the sign of every order determinant.     

Study on Self-assembly of Poly(ethylene glycol)- block-poly(γ-benzyl L -glutamate)-graft-poly(ethylene glycol) Copolymer and Poly(γ-benzyl L -glutamate)- block-poly(ethylene glycol) Copolymer in Ethanol

Poly(ethylene glycol)-block-poly(γ-benzyl L-glutamate)-graft-poly(ethylene glycol) (PEG-b-PBLG-g-PEG) copolymer was synthesized by the ester exchange reaction of poly(γ-benzyl L-glutamate)-block-poly(ethylene glycol) (PBLG-block-PEG) copolymer with PEG chain, and PBLG-block-PEG copolymer was prepared by a standard N-carboxyl-γ-benzyl-L-glutamate anhydride (NCA) method. Nuclear magnetic resonance (NMR) spectroscopy was used to confirm the components of PBLG-block-PEG and PEG-b-PBLG-g-PEG. The self-association behaviors of PBLG-block-PEG and PEG-b-PBLG-g-PEG in ethanol were investigated by transmission electron microscopy (TEM), dynamic laser scattering (DLS), and viscometry. The experimental results revealed that the different molecular structures could exert marked effects on the self-assembly behaviors of PBLG-block-PEG and PEG-b-PBLG-g-PEG in ethanol. PBLG-block-PEG and PEG-b-PBLG-g-PEG could self-assemble to form polymeric micelles with a core-shell structure in the shapes of plump spherical and regular rice-like, respectively. Effects of the introduction of PBLG homopolymer on the average particle diameter of the micelles of PBLG-block-PEG and PEG-b-PBLG-g-PEG and influence of testing temperature on the critical micelle concentration of different copolymers were studied.     

Ordering chemical unicyclic graphs by Wiener polarity index

Suppose G is a chemical graph with vertex set V(G). Define D(G) = {{u, v} ⊆ V (G) | d _G (u, v) = 3}, where d _G (u, v) denotes the length of the shortest path between u and v. The Wiener polarity index of G, W _p (G), is defined as the size of D(G). In this article, an ordering of chemical unicyclic graphs of order n with respect to the Wiener polarity index is given.     

Intershell nl4f electron correlation effects

The pair correlation energies for some nl4f pairs of the ground state of the Yb atom are calculated for the first time. The partial wave (PW ) increments to the second-order pair energies are generated using numerical first-order radial pair functions obtained as the solution of two-dimensional differential equations. The analysis of the PW s contributions shows the dominant role of the df, fg, and gh PW s for the 4d4f pair, of the pf and dg PW s for the 4p4f and 5p4f pairs, and of the sf and pg PW s for the 4s4f, 5s4f, and 6s4f pairs. A discussion of the similarities and differences of the structure of the correlation energy found in this paper with those calculated earlier for smaller atoms is given.     

The spread of unicyclic graphs with given size of maximum matchings

The spread s(G) of a graph G is defined as s(G) = max _i,j |λ _i − λ _j |, where the maximum is taken over all pairs of eigenvalues of G. Let U(n,k) denote the set of all unicyclic graphs on n vertices with a maximum matching of cardinality k, and U ^*(n,k) the set of triangle-free graphs in U(n,k). In this paper, we determine the graphs with the largest and second largest spectral radius in U ^*(n,k), and the graph with the largest spread in U(n,k).      

The PI index of phenylenes

The Padmakar–Ivan (PI) index is a graph invariant defined as the summation of the sums of n _eu (e|G) and n _ev (e|G) over all the edges e = uv of a connected graph G, i.e., PI(G) = ∑ _e∈E(G)[n _eu (e|G) + n _ev (e|G)], where n _eu (e|G) is the number of edges of G lying closer to u than to v and n _ev (e|G) is the number of edges of G lying closer to v than to u. An efficient formula for calculating the PI index of phenylenes is given, and a simple relation is established between the PI index of a phenylene and of the corresponding hexagonal squeeze.     

Diffusion of nonionic azo dyes in nylon-6

Diffusion of 2,4-dinitroaniline and three nonionic azo dyes in Nylon-6 film was studied by analysis of the concentration-distance curves (profiles) of penetrants in the polymer. Actual diffusivities D_(c) of penetrants in polymer, diffusion coefficients as a function of the concentration C_f of penetrant in polymer, were calculated from the profile. It was found that D_(c) is almost constant or decreases gradually with decreasing C_f in the range of high-medium C_f but decreases appreciably with decreasing C_f at low C_f. The change in D_(c) with C_f was explained in terms of the dual-mode sorption-diffusion model. The penetrants diffuse in the polymer as two distinct species, i.e., a dissolved species and an adsorbed species. The former is the penetrant taken up by the polymer by a partition mechanism (dissolved species) and the latter is that taken up by Langmuir sorption (adsorbed species). The actual diffusivity D_P(c) of the dissolved species decreases with decreasing C_f. While the actual diffusivity D_L(c) of the adsorbed species normally increases gradually with decreasing C_f. D_P(c) is usually larger than D_L(c). © 1993 John Wiley & Sons, Inc.     

On p -nilpotency and minimal subgroups of finite groups

We call a subgroup H of a finite group G c-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K ≤ core(H). In this paper it is proved that a finite group G is p-nilpotent if G is S ₄-free and every minimal subgroup of P ∩ G ^N is c-supplemented in N _G(P), and when p = 2 P is quaternion-free, where p is the smallest prime number dividing the order of G, P a Sylow p-subgroup of G. As some applications of this result, some known results are generalized.     

Construction of linearly independent relativistic symmetry orbitals for finite double-point groups including time reversal symmetry

A formalism is developed for the construction of relativistic symmetry-adapted molecular basis functions under consideration of time reversal invariance. The theory is applicable to the finite double point groups C_n, C_nh, S_n, C_nv, D_n, D_nd, D_nh, T, T_h, T_d, O, and O_h. It is based on the LCAO method. A projection operator technique is employed to construct molecular symmetry orbitals from atomic orbitals. The search for linearly independent basis function is simplified by means of group theoretical considerations.     

Dirac equation with a Coulomb plus scalar potential in D + 1 dimensions

We generalize the Dirac equation to D + 1‐dimensional spacetime. The exact solutions of the D‐dimensional radial equations with a Coulomb plus scalar potential taking the form 1/r are analytically presented by studying the Tricomi equations. The energies E(n, l, D) are exactly presented. The dependences of the energies E(n, l, D) on the dimension D are analyzed in some detail. The energies E(n, 0, D) first decrease and then increase when increasing dimension D, but the energies E(n, l, D) (l ≠ 0) increase when increasing dimension D. The energies E(n, 0, D) are symmetric with respect to D = 1 for D ∈ (0, 2). It is shown that the energies E(n, l, D) (l ≠ 0) are almost independent of the quantum number l for large D and are completely independent of it if the Coulomb potential is equal to the scalar one. The energies E(n, l, D) almost overlap for large D. The dependences of the energies E(n, l, v) and E(n, l, s) on the vector potential parameter v and scalar potential one s are also studied for D = 3. All are found to decrease when these parameters are increased. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Syntheses of the Enantiomers of γ-Cyclogeranic Acid, γ-Cyclocitral,and γ-Damascone: Enantioselective protonation of enolates

(R)-and (S)-γ-cyclogeranic acid ((R)-and (S)- 9 , resp.) were obtained by resolution of the racemate, and their absolute configurations determined by chemical correlation. The γ-acids (R)-and (S)- 9 were converted into (R)-and (S)-methyl γ-cyclogeranate ((R)-and (S)- 6 , resp.), and (R)-and (S)-γ-damascone ((R)-and (S)- 5 , resp.). A more direct entry to (R)-and (S)- 9 consisted in the enantioselective protonation of a thiol ester enolate with (?)- or (γ)-N-isopropylephedrine((?)- or (γ)- 20 ) and subsequent hydrolysis of the (R)-and (S)-S-phenyl γ-thiocyclogeranate ((R)- and (S)- 24 , resp.; 97% ee). The esters (R)- and (S)- 24 were also used as precursors of (R)- and (S)-γ-damascone ((R)- and (S)- 5 , resp.). Alternatively, (S)- 5 (75% ee) was obtained by enantioselective protonation of ketone enolate 29 with (?)-N-isopropylephedrine ((?)- 20 ). Organoleptic evaluation demonstrated that the (S)-enantiomers of methyl γ-cyclogeranate and γ-damascone are markedly superior to their (R)-enantiomers.     

Relation between the critical point and the precipitation threshold

The relationship between the critical point and the precipitation threshold is examined in the Flory—Huggins approximation with concentration-independent interaction parameter χ. Approximate explicit expressions for the difference between the critical point and the threshold can be derived by series expansion of threshold conditions. In the first-order approximation, the concentration difference depends only on the chainlength averages x_w, x_z, and x_z+1, in the second-order approximation it depends on x_w, x_z, x_z+1, and x_z+2, etc. For polymers of low polydispersity, the second-order approximation gives a good estimate of the concentration difference; for instance, for polymers with exponential distribution and x_w/x_n < 1.25, the error is lower than ca. 1%. The approximation is not suitable for polymers with x_z+1 ? x_z (including polymers whose cloud-point curve exhibits a triple point). Irrespective of the polydispersity of the polymer, the threshold as well as the whole cloud-point curve depend only on the weight-average and higher averages, x_w, x_z, x_z+1, …, x_z+k, where k → ∞; they are, however, independent of the number average x_n.     

Chiral Carbon Nanorings: Synthesis,Properties and Hierarchical Self-assembly of Chiral Ternary Complexes Featuring a Narcissistic Chiral Self-Recognition for Chiral Amines

We report the synthesis and chiroptical properties of novel chiral carbon nanorings S p-/Rp-[12]PCPP containing a planar chiral [2.2]PCP unit, and demonstrate that S p-/Rp-[12]PCPP can not only host crown ether 18-Crown-6 to form ring-in-ring complexes with a binding constant 3.35×10³ M⁻¹, but also accommodate the complexes of 18-Crown-6 and S/R-protonated amines to form homochiral S @Sp -/ R @Rp - and heterochiral S @Rp -/ R @Sp - ternary complexes, displaying significantly larger binding constants of up to 3.31×10⁵ M⁻¹ depending on the chiral guests. Importantly, homochiral S @Sp -/ R @Rp - ternary complexes exhibit an enhanced CD signal, while the heterochiral S @Rp -/ R @Sp - ones have a constant CD signal compared with the chiral carbon nanorings, respectively, which suggests that homochiral S @Sp -/ R @Rp - ternary complexes display a highly narcissistic chiral self-recognition for S/R-protonated chiral amines, respectively. Finally, the chiral ternary complexes can be further applied to determine the ee values of chiral guests. The findings highlight a new application of carbon nanorings in supramolecular sensors, beyond the common recognition of π-conjugated molecules.