Extremal Polyomino Chains on k-matchings and k-independent Sets

Denote by the set of polyomino chains with n squares. For any , let m _k (T _n ) and i _k (T _n ) be the number of k-matchings and k-independent sets of T _n , respectively. In this paper, we show that for any polyomino chain and any , and , with the left equalities holding for all k only if T _n =L _n , and the right equalities holding for all k only if T _n =Z _n , where L _n and Z _n are the linear chain and the zig-zag chain, respectively. This work is supported by NNSFC (10371102).     

Interaction of a finite quantum system with an infinite quantum system that contains a single one-parameter eigenvalue band

Interaction of a finite quantum system that contains ρ eigenvalues and eigenstates with an infinite quantum system that contains a single one-parameter eigenvalue band is considered. A new approach for the treatment of the combined system is developed. This system contains embedded eigenstates with continuous eigenvalues , and, in addition, it may contain isolated eigenstates with discrete eigenvalues . Two ρ × ρ eigenvalue equations, a generic eigenvalue equation and a fractional shift eigenvalue equation are derived. It is shown that all properties of the system that interacts with the system can be expressed in terms of the solutions to those two equations. The suggested method produces correct results, however strong the interaction between quantum systems and . In the case of the weak interaction this method reproduces results that are usually obtained within the formalism of the perturbation expansion approach. However, if the interaction is strong one may encounter new phenomena with much more complex behavior. This is also the region where standard perturbation expansion fails. The method is illustrated with an example of a two-dimensional system that interacts with the infinite system that contains a single one-parameter eigenvalue band. It is shown that all relevant completeness relations are satisfied, however strong the interaction between those two systems. This provides a strong verification of the suggested method.     

On the Laplacian spectral radii of trees with perfect matchings

Denote by the set of trees of order 2k with perfect matchings. GUO [Guo, Linear Algebra Appl. 368:379–385, 2003.] determined the largest value of Laplacian spectral radii μ(T) of the trees T in and gave the corresponding tree T in whose μ(T) reaches this largest value. In this paper, we determine the second to the sixth largest values of μ(T) of the trees T in and also give the corresponding trees T in whose μ(T) reach these values.     

On the derivative of the associated Legendre function of the first kind of integer degree with respect to its order (with applications to the construction of the associated Legendre function of the second kind of integer degree and order)

The derivative of the associated Legendre function of the first kind of integer degree with respect to its order, , is studied. After deriving and investigating general formulas for μ arbitrary complex, a detailed discussion of , where m is a non-negative integer, is carried out. The results are applied to obtain several explicit expressions for the associated Legendre function of the second kind of integer degree and order, . In particular, we arrive at formulas which generalize to the case of (0 ≤ m ≤ n) the well-known Christoffel’s representation of the Legendre function of the second kind, Q _n (z). The derivatives and , all with m > n, are also evaluated.     

Unicyclic Hückel molecular graphs with minimal energy

The minimal energy of unicyclic Hückel molecular graphs with Kekulé structures, i.e., unicyclic graphs with perfect matchings, of which all vertices have degrees less than four in graph theory, is investigated. The set of these graphs is denoted by such that for any graph in , n is the number of vertices of the graph and l the number of vertices of the cycle contained in the graph. For a given n(n ≥ 6), the graphs with minimal energy of have been discussed. MSC 2000: 05C17, 05C35     

Some conjectures on S n -derived nuclear permutational(or NMR) spin encodings

On existence of limiting ‐module decompositional sets for weak ‐branching at high n; on Voronoi polyhedral dual as geometric analogues to Cayley’s SU2 embedding theorem; and on SU dual group with retention of self‐associacy over subduced irrep set, as being the sufficient further condition to ensure the determinacy of SU embeddings. In the context of structures arising from nuclear permutation (NP) or NMR dual‐group spin algebras, the first conjecture sets out the high‐index n, and thus weak‐branching limit (WBL), aspects of module decompositions as giving rise to a set of numerical values for the associated Kostka coefficients which are invariant to further incrementation in the index; the existence of such combinatorial limit properties, implicit in sst tableaux enumerations, has not been addressed in the mathematics literature to date. Conjectures 2 and 3 are concerned with the questions of geometric and sufficient algebraic realisations of the determinacy of natural finite group embeddings in specific SU permutation groups. In conjecture 2, the Voronoi dual‐structures to the regular polyhedra for NP/NMR automorphic SU2 embedded spin symmetries hold the key to physical insight. Specifically, they provide a novel combinatorial geometric view of Cayley’s theorem; the mid‐face intersecting ‐axes of the initial NMR automorphic solids become (vertex) body‐diagonal axes of the specific‐dual Voronoi polyhedra, where a distinctness condition from the spin‐sites gives raise to a geometric statement of Cayley’s theorem. Conjecture 3 is concerned with SU embeddings for which the simple Cayley criterion alone is an insufficient condition to guarantee determinacy. The self‐associacy property and its retention over the subduced irrep‐subset(s) (i.e., comparable to studies of system‐invariants via Yamanouchi–Gel’fand subduction chains) is now seen as indicative of retention of determinacy for such SU group embeddings through the above sufficiency condition. This revised version was published online in July 2006 with corrections to the Cover Date.     

On minimal energies of trees of a prescribed diameter

The energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph. Let denote the set of trees on n vertices and diameter d, . Yan and Ye [Appl. Math. Lett. 18 (2005) 1046–1052] have recently determined the unique tree in with minimal energy. In this article, the trees in with second-minimal energy are characterizedAMS Subject Classification: 05C50, 05C35     

Vibrational isotope effect by the low rank perturbation method

Mathematical formalism of the Low Rank Perturbation method (LRP) is applied to the vibrational isotope effect in the harmonic approximation with a standard assumption that force field does not change under isotopic substitutions. A pair of two n-atom isotopic molecules A and B which are identical except for isotopic substitutions at ρ atomic sites is considered. In the LRP approach vibrational frequencies ω _k and normal modes of the isotopomer B are expressed in terms of the vibrational frequencies ν _i and normal modes of the parent molecule A. In those relations complete specification of the normal modes is not required. Only amplitudes at sites τ affected by the isotopic substitutions and in the coordinate direction s (s = x, y, z) are needed. Out-of-plane vibrations of the (H,D)-benzene isotopomers are considered. Standard error of the LRP frequencies with respect to the DFT frequencies is on average . This error is due to the uncertainty of the input data (± 0.5 cm⁻¹) and in the absence of those uncertainties and in the harmonic approximation it should disappear. In comparing with experiment, one finds that LRP frequencies reproduces experimental frequencies of (H,D)-benzene isotopomers better () than scaled DFT frequencies () which are designed to minimize (by frequency scaling technique) this error. In addition, LRP is conceptually and numerically simple and it also provides a new insight in the vibrational isotope effect in the harmonic approximation.     

On the special values of monic polynomials of hypergeometric type

Special values of monic polynomials y _n (s), with leading coefficients of unity, satisfying the equation of hypergeometric type

Interaction of glycine with cationic, anionic, and nonionic surfactants at different temperatures: a volumetric, viscometric, refractive index, conductometric, and fluorescence probe study

A number of thermodynamic parameters viz. apparent molar volumes, ϕ _v , partial molar volumes, , transfer volumes, , Falkenhagen coefficients, A, Jones–Dole coefficients, B, free energies per mole of solute, , and per mole of solvent, , molar refraction, R _D , and limiting molar conductivity, , have been calculated by using the experimentally measured densities, ρ, viscosities, η, refractive indices, n _D , and specific conductivities, κ, data of glycine (0.02–0.10 m) in 0.01 m aqueous sodium dodecyl sulphate, cetyltrimethylammonium bromide, and triton X-100 (TX-100) solutions at 298.15, 303.15, 308.15, and 313.15 K. The above calculated parameters were found to be sensitive towards the interactions prevailing in the studied amino acid–surfactant–water systems. Moreover, fluorescence study using pyrene as a photophysical probe has also been carried out, the results of which support the conclusions obtained from other techniques.     

Sharp upper bounds for total π-electron energy of alternant hydrocarbons

The energy E(G) of a graph G is defined as the sum of the absolute values of all the eigenvalues of the adjacency matrix of the graph G. This quantity is used in chemistry to approximate the total π-electron energy of molecules and in particular, in case G is bipartite, alternant hydrocarbons. In this paper, we show that if G = (V ₁, V ₂; E) is a bipartite graph with edges and , then

Unicycle Graphs with Maximum Generalized Topological Indices

Let G be an unicycle graph and d _v the degree of the vertex v. In this paper, we investigate the following topological indices for an unicycle graph , , where m ≥ 2 is an integer. All unicycle graphs with the largest values of the three topological indices are characterized. This research is supported by the National Natural Science Foundation of China(10471037)and the Education Committee of Hunan Province(02C210)(04B047).     

Estimation of partial linear error-in-variables models for ρ<Superscript>−</Superscript>-mixing dependence data

Consider the partly linear regression model Y = xβ + g(t) + e where the explanatory x is erroneously measured, and both t and the response Y are measured exactly, the random error e is ρ⁻-mixing. Let be a surrogate variable observed instead of the true x in the primary survey data. Assume that in addition to the primary data set containing N observations of , which is ρ⁻-mixing data sets, an independent validation data containing n observations of is available. The exact observations on x may be obtained by some expensive or diffcult procedures for only a small subset of subjects enrolled in the study. In this paper, inspired by Berberan-Santos et al. [J. Math. Chem. 37 (2005)101], a semiparametric method with the primary data is employed to obtain the estimators of β and g(·) based on the least squares criterion with the help of validata. The proposed estimators are proved to be strongly consistent.      

Unicycle graphs with extremal Merrifield–Simmons Index

The Merrifield–Simmons index f(G) of a (molecular) graph G is defined as the number of subsets of the vertex set, in which any two vertices are non-adjacent, i.e., the number of independent-vertex sets of G. By we denote the set of unicycle graphs in which the length of its unique cycle is k. In this paper, we investigate the Merrifield–Simmons index f(G) for an unicycle graph G in . Unicycle graphs with the largest or smallest Merrifield–Simmons index are uniquely determined.     

Mechanism of the catalyzed by cysteine electroreduction of the insoluble in water cobalt(II) salt as a component of carbon paste electrode

The mechanism of the Co(II) catalytic electroreduction of water insoluble CoR₂ salt in the presence of cysteine was developed. CoR₂ = cobalt(II) cyclohexylbutyrate is the component of a carbon paste electrode. Electrode surface consecutive reactions are: (a) fast (equilibrium) reaction of the complex formation, (b) rate-determining reversible reaction of the promoting process of CoR(Ac⁺) complex formation, (c) rate-determining irreversible reaction of the electroactive complex formation with ligand-induced adsorption, and (d) fast irreversible reaction of the electroreduction. Reactions (a,b) connected with CoR₂ dissolution and reactions (c,d) connected with CoR₂ electroreduction are catalyzed by . Regeneration of (reactions “b,d”) and accumulation of atomic Co(0) (reaction “d”) take place. Experimental data [Sugawara et al., Bioelectrochem Bioenergetics 26:469, 1991]: i _a vs E (i _a is anodic peak, E is cathodic accumulation potential), i _a vs , and i _a vs pH have been quantitatively explained.     

Effect of Ammonium Salts on the Volumetric and Viscometric Behavior of D(+)-Glucose,D(−)-Fructose and Sucrose in Aqueous Solutions at 25°C

Apparent molar volumes, V _φ, and viscosity, η, of D(+)-glucose, D(−)-fructose and sucrose in water and in 0.02, 0.05, 0.5, 1.0 and 2.0 mol·kg⁻¹ aqueous solutions of ammonium bromide, tetraethylammonium bromide and tetra-n-butylammonium bromide have been determined at 25 °C from density and efflux time measurements by using a vibrating-tube digital densimeter and a capillary viscometer, respectively. Partial molar volumes, , at infinite dilution that were extrapolated from the V _φ data were used to obtain the corresponding transfer volumes, , for saccharides from water to different aqueous solutions of co-solutes. The Jones-Dole equation viscosity B-coefficients were obtained from the viscosity data. Positive values of were obtained for the saccharides in the presence of ammonium bromide, whereas both positive and negative values were obtained in the presence of tetraethylammonium and tetra-n-butylammonium bromides. The negative values at very low concentrations have small magnitudes. Volumetric interaction coefficients have been calculated by using the McMillan-Mayer theory and Gibbs energies of activation of viscous flow have been calculated by using Feakin’s transition-state theory equation. The parameters obtained from the volumetric and viscometric studies were used to understand various mixing effects due to the interactions between saccharides and ammonium salts in aqueous solutions.     

Mathematical relations between codon bias indexes and their applications

Codon Adaptation Index (CAI), Effective Number of Codons as well as its modifications , can be used to measure gene codon bias. In this article, we prove is more efficient and unbiased than and by revisiting correlations of them with CAI in the level of individual amino acid’s codon bias. Correlations are studied by mathematical expressions rather than statistic methods, because the latter unavoidably depend on the data set used. Additionally, the immediate cause of correlations of with CAI (as well as those of RSCU with CAI) are also described in mathematical language. Perhaps, mathematics provides us a new way to study correlations between biological indexes.     

On zeroth-order general Randić index of conjugated unicyclic graphs

Let G be a graph and d _v denote the degree of the vertex v in G. The zeroth-order general Randić index of a graph is defined as where α is an arbitrary real number. In this paper, we investigate the zeroth-order general Randić index of conjugated unicyclic graphs G (i.e., unicyclic graphs with a perfect matching) and sharp lower and upper bounds are obtained for depending on α in different intervals.     

Interaction of an isolated state with an infinite quantum system containing several one-parameter eigenvalue bands: I. Time-independent case

A new method for the exact solution of the interaction of an isolated state with an infinite dimensional quantum system §_∞ ^b containing several one-parameter eigenvalue bands is developed. Unlike standard perturbation expansion approach, this method produces correct results however strong the interaction between the state and the system . It is shown that in the case of the weak interaction this method correctly reproduces standard results obtained within the formalism of the perturbation expansion method. In particular, due to the interaction with the system , eigenvalue E of the state shifts to a new position. In addition, if this eigenvalue is embedded inside the range of the unperturbed eigenvalues, this shifted eigenvalue broadens and spectral distribution of the state has the shape of the universal resonance curve. However, if the interaction is strong, one finds much more complex and much more complicated behavior     

Interaction of an isolated state with an infinite quantum system containing several one-parameter eigenvalue bands: II. time-dependent case

Time-dependent properties of a state that interacts with an infinite dimensional quantum system containing several one-parameter eigenvalue bands are considered. This is done by a new mathematical method that produces correct results, however strong the interaction between the state and the system . It is shown that in the case of the weak interaction one obtains standard results that are usually obtained within the formalism of the perturbation expansion method. In particular, if the eigenvalue E of the state is embedded inside the range of the unperturbed eigenvalues, time evolution of the state that is initially prepared in the state has typical exponential decay behavior. One also reproduces standard results concerning probabilities of the transition of the state at infinite time (t=∞) into various eigenvalue bands. However, if the interaction is strong, one finds much more complex and much more complicated behavior.