The NMR spectra of porphyrins—19: 13C and proton NMR spectra of metal-free porphyrins with the Type-IX substituent orientation,and of their zinc(II) complexes

The ¹³C and proton NMR spectra of six porphyrins bearing the substituent orientation characteristic of the natural “Type-IX” arrangement are reported and assigned. Significant concentration effects in the spectra of the free base porphyrins, together with the broadening of the C_α (and occasionally C_β) carbon resonances due to NH tautomerism caused a significant loss of data in these spectra. However, the spectra of the corresponding zinc(II) porphyrins (with addition of excess pyrrolidine) show that both these extraneous effects are completely removed to give well-resolved spectra with accurately reproducible chemical shifts. These spectra are assigned and an analysis of the chemical shifts allows the deduction of substituent chemical shift (SCS) parameters for the peripheral substituents at the beta and meso carbons. There is no global effect of these beta substituents, the beta carbon SCS being confined to the immediate pyrrole ring, and the meso carbon SCS to the two adjacent pyrrole rings. The SCS parameters are analyzed and it is shown how they can be used to predict the peripheral and meso carbon chemical shifts of any porphyrin bearing the substituents discussed.     

Penalty and relaxation methods for the optimal placement and operation of control valves in water supply networks

In this paper, we investigate the application of penalty and relaxation methods to the problem of optimal placement and operation of control valves in water supply networks, where the minimization of average zone pressure is the objective. The optimization framework considers both the location and settings of control valves as decision variables. Hydraulic conservation laws are enforced as nonlinear constraints and binary variables are used to model the placement of control valves, resulting in a mixed-integer nonlinear program. We review and discuss theoretical and algorithmic properties of two solution approaches. These include penalty and relaxation methods that solve a sequence of nonlinear programs whose stationary points converge to a stationary point of the original mixed-integer program. We implement and evaluate the algorithms using a benchmarking water supply network. In addition, the performance of different update strategies for the penalty and relaxation parameters are investigated under multiple initial conditions. Practical recommendations on the numerical implementation are provided.     

Continuity of invariant measures for families of maps

Let {τ_r} be the family of maps from [0,1] into [0,1] with properties similar to those of τ_r(x) = rx(1 ? x), 0 ≤ r ≤ 4. The limiting behaviour of orbits {τ_r^j(x)}_j^∞ = 1 is a complicated and discontinuous function of the parameter r. The stochastic approximation to the difference equation x_{n + 1} = τ_r(x_n), x_{n + 1} = τ_r(x_n) + W, where W is a fixed random variable independent of r and x_n, is considered. It is shown that this Markov process admits a unique absolutely continuous invariant measure μ_r and, furthermore, that the map r → μ_r is continuous. Such a result is important in applications, since slight changes in the shape of τ_r no longer cause discontinuous consequences in the limiting behaviour of the system.     

On exponential stability of second order delay differential equations

We propose a new method for studying stability of second order delay differential equations. Results we obtained are of the form: the exponential stability of ordinary differential equation implies the exponential stability of the corresponding delay differential equation if the delays are small enough. We estimate this smallness through the coefficients of this delay equation. Examples demonstrate that our tests of the exponential stability are essentially better than the known ones. This method works not only for autonomous equations but also for equations with variable coefficients and delays.     

A touch of mathematics: coming to our senses by observing the visually impaired

During the last three decades, the impact of visualization processes on the teaching and learning of mathematics has been extensively researched. However, considerably less work has been devoted to haptic processes. In this paper, we describe and analyze the role of haptic processes from data collected in mathematics lessons taught in a school for blind students. The analysis builds upon studies from perception and also from teaching experiences in order to examine the teacher’s and the students’ hand movements and metaphors when handling solids of revolution and communicating verbally with each other about their insights. We highlight the powerful combination of the visual and the haptic components of their interactions for the conceptualization of mathematical experiences, and we also note the critical role mathematical language plays in supporting the teaching–learning processes in this context. Finally, we consider important educational implications not only for blind people, but for all students and teachers of mathematics.     

Analytical models for supplier selection and order quantity allocation

In the literature, decision models and techniques for supplier selection do not often consider inventory management of the items being purchased as part of the analysis. In this article, two mixed integer nonlinear programming models are proposed to select the best set of suppliers and determine the proper allocation of order quantities while minimizing the annual ordering, inventory holding, and purchasing costs under suppliers’ capacity and quality constraints. The first model allows independent order quantities for each supplier while the second model restricts all order quantities to be of equal size, as it would be required in a multi-stage (supply chain) inventory model. Illustrative examples are used to highlight the advantages of the proposed models over a previous model introduced in the literature.     

Randomly coloring sparse random graphs with fewer colors than the maximum degree

We analyze Markov chains for generating a random k‐coloring of a random graph G_n,d/n. When the average degree d is constant, a random graph has maximum degree Θ(log n/log log n), with high probability. We show that, with high probability, an efficient procedure can generate an almost uniformly random k‐coloring when k = Θ(log log n/log log log n), i.e., with many fewer colors than the maximum degree. Previous results hold for a more general class of graphs, but always require more colors than the maximum degree. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Variable neighborhood search with ejection chains for the antibandwidth problem

In this paper, we address the optimization problem arising in some practical applications in which we want to maximize the minimum difference between the labels of adjacent elements. For example, in the context of location models, the elements can represent sensitive facilities or chemicals and their labels locations, and the objective is to locate (label) them in a way that avoids placing some of them too close together (since it can be risky). This optimization problem is referred to as the antibandwidth maximization problem (AMP) and, modeled in terms of graphs, consists of labeling the vertices with different integers or labels such that the minimum difference between the labels of adjacent vertices is maximized. This optimization problem is the dual of the well-known bandwidth problem and it is also known as the separation problem or directly as the dual bandwidth problem. In this paper, we first review the previous methods for the AMP and then propose a heuristic algorithm based on the variable neighborhood search methodology to obtain high quality solutions. One of our neighborhoods implements ejection chains which have been successfully applied in the context of tabu search. Our extensive experimentation with 236 previously reported instances shows that the proposed procedure outperforms existing methods in terms of solution quality.     

Piecewise finite series solutions of seasonal diseases models using multistage Adomian method

The aim of this paper is to apply the multistage Adomian Decomposition Method MADM to solve systems of nonautonomous nonlinear differential equations that describe several epidemic models with periodic behavior. Here the concept of the MADM is introduced and then it is employed to obtain a piecewise finite series solution. The MADM is used here as a hybrid analytical–numerical technique for approximating the solutions of the epidemic models. In order to show the efficiency of the method, the obtained numerical results are compared with the fourth-order Runge–Kutta method solutions. Numerical comparisons show that the MADM is accurate, easy to apply and the calculated solutions preserve the periodic behavior of the continuous models. Moreover, the method has the advantage of giving a functional form of the solution for any time interval. Furthermore, it is shown that if the truncation order and the time step size are not properly chosen large computational work is required and inaccurate solutions may be obtained.