The graphical representation of equilibrium for the isothermal and isobaric reactions of ideal gases

The equilibrium for the isothermal and isobaric reactions of ideal gases is investigated in virtue of the intuitionistic figure. The curve is similar to the curve of tangential function which has one inflection and two vertical asymptotes. The equation only has one root ξ _e and it is suitable to find ξ _e by dichotomy. For non-inert substance, when or x_i^0 $$" align="middle" border="0"> , to increase substance i will make an equilibrium shift in the direction to deplete substance i; when {\nu_i} \mathord{\left/ {\vphantom {{\nu_i} {\sum_i {\nu _i}}}} \right. \kern-\nulldelimiterspace} {\sum _i {\nu _i}}> 0$$" align="middle" border="0"> , to increase substance i will make an equilibrium shift in the direction to produce more substance i.     

On the largest eigenvalues of trees with perfect matchings

Let λ₁ (G) and Δ (G), respectively, denote the largest eigenvalue and the maximum degree of a graph G. Let be the set of trees with perfect matchings on 2m vertices, and . Among the trees in , we characterize the tree which alone minimizes the largest eigenvalue, as well as the tree which alone maximizes the largest eigenvalue when . Furthermore, it is proved that, for two trees T ₁ and T ₂ in (m≥ 4), if and Δ (T ₁) > Δ (T ₂), then λ₁ (T ₁) > λ₁ (T ₂).     

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.     

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.     

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.     

Unified treatment of complete orthonormal sets of nonrelativistic,quasirelativistic and relativistic sets of spinor wave functions,and Slater spinor orbitals in coordinate,momentum and four-dimensional spaces

Using the properties of tensor spherical harmonics introduced by the author in previous paper (Guseinov, Phys Lett A 372:44, 2007) and complete orthonormal scalar basis sets of nonrelativistic -exponential type orbitals ( -ETO), - momentum space orbitals ( -MSO) and z ^α-hyperspherical harmonics (z ^α-HSH) for particles with spin s = 0 the new analytical relations for the quasirelativistic and relativistic spinor wave functions and Slater spinor orbitals in coordinate, momentum and four-dimensional spaces are derived, where α = 1, 0, −1, −2,.... The 2-component quasirelativistic and 4-component relativistic spinor wave functions obtained are complete without the inclusion of the continuum. The relativistic spinor wave function sets and Slater spinor orbitals are expressed through the corresponding quasirelativistic spinor wave functions and Slater spinor orbitals, respectively. The analytical formulas for overlap integrals over quasirelativistic and relativistic Slater spinor orbitals with the same screening constants in coordinate space are also derived.     

Recurrence and Differential Relations for Spherical Spinors

We present a comprehensive table of recurrence and differential relations obeyed by spin one-half spherical spinors (spinor spherical harmonics) Ω_{κ μ}(n) used in relativistic atomic, molecular, and solid state physics, as well as in relativistic quantum chemistry. First, we list finite expansions in the spherical spinor basis of the expressions A·B Ω_κμ(n) and A·(B×C) Ω_κμ(n), where A, B, and C are either of the following vectors or vector operators: n=r/r (the radial unit vector), e ₀, e _±1 (the spherical, or cyclic, versors), (the 2×2 Pauli matrix vector), (the dimensionless orbital angular momentum operator; I is the 2×2 unit matrix), (the dimensionless total angular momentum operator). Then, we list finite expansions in the spherical spinor basis of the expressions A·B F(r)Ω_κμ(n) and A·(B×C) F(r)Ω_κμ(n), where at least one of the objects A, B, C is the nabla operator , while the remaining ones are chosen from the set .     

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

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.     

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     

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.     

Unified treatment of complete orthonormal sets for exponential type vector orbitals of particles with spin 1 in coordinate, momentum and four-dimensional spaces

The new formulas are obtained for complete orthonormal sets of exponential type vector orbitals of a particle with spin 1 in coordinate, momentum and four-dimensional spaces using the properties of spherical vectors and complete orthonormal scalar basis sets of -exponential type orbitals ( -ETO), -momentum space orbitals ( -MSO) and -hyperspherical harmonics ( -HSH) introduced by the author for particles with spin s = 0, where These vector orbitals are complete without the inclusion of the continuum and, therefore, their group of transformation is the four-dimensional rotation group of O(4). For overlap integrals over vector Slater orbitals with the same screening constant the analytical relations in coordinate space are also derived. It should be noted that the new idea presented in this study is the combination of spherical vectors with complete orthonormal scalar sets for radial parts of -orbitals.     

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     

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.     

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).     

A Method of Computing the PI Index of Benzenoid Hydrocarbons Using Orthogonal Cuts

The Padmakar–Ivan (PI) index of a graph G is defined as PI , where for edge e=(u,v) are the number of edges of G lying closer to u than v, and is the number of edges of G lying closer to v than u and summation goes over all edges of G. The PI index is a Wiener–Szeged-like topological index developed very recently. In this paper, we describe a method of computing PI index of benzenoid hydrocarbons (H) using orthogonal cuts. The method requires the finding of number of edges in the orthogonal cuts in a benzenoid system (H) and the edge number of H – a task significantly simpler than the calculation of PI index directly from its definition. On the eve of 70th anniversary of both Prof. Padmakar V. Khadikar and his wife Mrs. Kusum Khadikar.     

Unicyclic Graphs Possessing Kekulé Structures with Minimal Energy

Unicyclic graphs possessing Kekulé structures with minimal energy are considered. Let n and l be the numbers of vertices of graph and cycle C _l contained in the graph, respectively; r and j positive integers. It is mathematically verified that for and l = 2r + 1 or has the minimal energy in the graphs exclusive of , where is a graph obtained by attaching one pendant edge to each of any two adjacent vertices of C ₄ and then by attaching n/2 − 3 paths of length 2 to one of the two vertices; is a graph obtained by attaching one pendant edge and n/2 − 2 paths of length 2 to one vertex of C ₃. In addition, we claim that for has the minimal energy among all the graphs considered while for has the minimal energy.      

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.      

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.     

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.