A quadratic programming approach to the Randić index

Let G(k, n) be the set of connected graphs without multiple edges or loops which have n vertices and the minimum degree of vertices is k. The Randi? index χ = χ(G) of a graph G   is defined by χ(G)=_∑(uv)(_δuδv)^-1/2$\chi (G)={\sum }_{(\mathit{uv})}({\delta }_u{\delta }_v{)}^{-1/2}$, where δ_u is the degree of vertex u and the summation extends over all edges (uv) of G. Caporossi et al. [G. Caporossi, I. Gutman, P. Hansen, Variable neighborhood search for extremal graphs IV: Chemical trees with extremal connectivity index, Computers and Chemistry 23 (1999) 469–477] proposed the use of linear programming as one of the tools for finding the extremal graphs. In this paper we introduce a new approach based on quadratic programming for finding the extremal graphs in G(k, n) for this index. We found the extremal graphs or gave good bounds for this index when the number n_k of vertices of degree k is between n − k and n. We also tried to find the graphs for which the Randi? index attained its minimum value with given k (k ? n/2) and n. We have solved this problem partially, that is, we have showed that the extremal graphs must have the number n_k of vertices of degree k less or equal n − k and the number of vertices of degree n − 1 less or equal k.     

On the difference equationx_{n + 1} = \frac{{a + bx_{n - k} - cx_{n - m} }}{{1 + g(x_{n - 1} )}}

In this paper we consider the difference equation $$x_{n + 1} = \frac{{a + bx_{n - k} - cx_{n - m} }}{{1 + g(x_{n - 1} )}},$$ wherea, b, c are nonegative real numbers,k, l, m are nonnegative integers andg(x) is a nonegative real function. The oscillatory and periodic character, the boundedness and the stability of positive solutions of the equation is investigated. The existence and nonexistence of two-period positive solutions are investigated in details. In the last section of the paper we consider a generalization of the equation.     

Lie Superalgebras and Irreducibility of $$A_1^{(1)}$$ –Modules at the Critical Level

We introduce the infinite-dimensional Lie superalgebra and construct a family of mappings from a certain category of –modules to the category of –modules at the critical level. Using this approach, we prove the irreducibility of a large family of –modules at the critical level parameterized by . As a consequence, we present a new proof of irreducibility of certain Wakimoto modules. We also give natural realizations of irreducible quotients of relaxed Verma modules and calculate characters of these representations. Partially supported by the MZOS grant 0037125 of the Republic of Croatia     

Preparation of an ion-selective electrode by chemical treatment of copper wire for the measurement of copper(II) and iodide by batch and flow-injection potentiometry

The preparation of an ion-selective electrode by chemical treatment of copper wire and its application for the measurements of copper (II) and iodide ions is described. The proposed reaction mechanism at the sensing surface, which explains the response of the electrode to Cu²⁺ and iodide ions, is discussed. The prepared electrode was suitable for direct potentiometric measurements of iodide and copper (II) in batch experiments down to concentrations of 1 × 10^–5 mol L^–1. A tubular electrode, prepared in the same way, may be used as a potentiometric sensor in a flow-injection analysis for Cu (II) and/or iodide determinations.     

Multi-velocity and multi-temperature model of the mixture of polyatomic gases issuing from kinetic theory

In this paper, we consider Euler-like balance laws for mixture components that involve macroscopic velocities and temperatures for each different species. These laws are not conservation laws due to mutual interaction between species. In particular, source terms that describe the rate of change of momentum and energy of the constituents appear. These source terms are computed with the help of kinetic theory for mixtures of polyatomic gases. Moreover, if we restrict the attention to processes which occur in the neighborhood of the average velocity and temperature of the mixture, the phenomenological coefficients of extended thermodynamics can be determined from the computed source terms.     

Contribution to the determination of the global minimum time for the brachistochronic motion of a holonomic mechanical system

The problem of the brachistochronic motion of a holonomic scleronomic mechanical system is analyzed. The system moves in an arbitrary field of known potential forces. The problem is formulated as an optimal control task, where generalized speeds are taken as control variables. The problem considered is reduced to solving the corresponding two-point boundary-value problem (TPBVP). In order to determine the global minimal solution of the TPBVP, an appropriate numerical procedure based on the shooting method is presented. The global minimal solution represents the solution with the minimum time of motion. The procedure is illustrated by an example of determining the brachistochronic motion of a disk that performs plane motion in a vertical plane in a homogeneous field of gravity.     

Brief preliminary design for a method of FMS machine tools subsystem selection

The article presents the main points of methodology of machine tools subsystem selection for a flexible manufacturing system. In particular, it shows the main algorithm and stages of the methodology which is based on computer database systems, algorithms of elimination and method of multicriteria optimisation. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Immobilization of modified penicillin G acylase on Sepabeads carriers

An approach to stable covalent immobilization of chemically modified penicillin G acylase from Escherichia coli on Sepabeads® carriers with high retention of hydrolytic activity and thermal stability is presented. The two amino-activated polymethacrylate particulate polymers with different spacer lengths used in the study were Sepabeads® EC EA and Sepabeads® EC HA. The enzyme was first modified by cross-linking with polyaldehyde derivatives of starch in order to provide it with new useful functions. Such modified enzyme was then covalently immobilized on amino supports. The method seems to provide a possibility to couple the enzyme without risking a reaction at the active site which might cause the loss of activity. Performances of these immobilized biocatalysts were compared with those obtained by the conventional method with respect to activity and thermal stability. The thermal stability study shows that starch-PGA immobilized on Sepabeads EC-EA was almost 4.5-fold more stable than the conventionally immobilized one and 7-fold more stable than free non-modified PGA. Similarly, starch-PGA immobilized on Sepabeads EC-HA was around 1.5- fold more stable than the conventionally immobilized one and almost 9.5-fold more stable than free non-modified enzyme.