On the tractability of the piecewise-linear approximation for general discrete-choice network revenue management

The choice network revenue management (RM) model incorporates customer purchase behavioras customers purchasing products with certain probabilities that are a function of the offeredassortment of products, and is the appropriate model for airline and hotel network revenuemanagement, dynamic sales of bundles, and dynamic assortment optimization. The underlyingstochastic dynamic program is intractable and even its certainty-equivalence approximation, inthe form of a linear program called Choice Deterministic Linear Program (CDLP) is difficultto solve in most cases. The separation problem for CDLP is NP-complete for MNL with justtwo segments when their consideration sets overlap; the affine approximation of the dynamicprogram is NP-complete for even a single-segment MNL. This is in contrast to the independentclass(perfect-segmentation) case where even the piecewise-linear approximation has been shownto be tractable. In this paper we investigate the piecewise-linear approximation for network RMunder a general discrete-choice model of demand. We show that the gap between the CDLP andthe piecewise-linear bounds is within a factor of at most 2. We then show that the piecewiselinearapproximation is polynomially-time solvable for a fixed consideration set size, bringing itinto the realm of tractability for small consideration sets; small consideration sets are a reasonablemodeling tradeoff in many practical applications. Our solution relies on showing that forany discrete-choice model the separation problem for the linear program of the piecewise-linearapproximation can be solved exactly by a Lagrangian relaxation. We give modeling extensionsand show by numerical experiments the improvements from using piecewise-linear approximationfunctions.

Studies on integrating kinematic design method with mechanical systems simulation techniques

Theultimate goal of any research in the mechanism/kinematic/design area may be called predictive design, ie the optimisation of mechanism proportions in the design stage without requiring extensive life and wear testing. This is an ambitious goal and can be realised through development and refinement of numerical (computational) technology in order to facilitate the design analysis and optimisation of complex mechanisms, mechanical components and systems. As a part of the systematic design methodology this thesis concentrates on kinematic synthesis (kinematic design and analysis) methods in the mechanism synthesis process. The main task of kinematic design is to find all possible solutions in the form of structural parameters to accomplish the desired requirements of motion. Main formulations of kinematic design can be broadly divided to exact synthesis and approximate synthesis formulations. The exact synthesis formulation is based in solving n linear or nonlinear equations in n variables and the solutions for the problem areget by adopting closed form classical or modern algebraic solution methods or using numerical solution methods based on the polynomial continuation or homotopy. The approximate synthesis formulations is based on minimising the approximation error by direct optimisation The main drawbacks of exact synthesis formulationare: (ia) limitations of number of design specifications and (iia) failure in handling design constraints- especially inequality constraints. The main drawbacks of approximate synthesis formulations are: (ib) it is difficult to choose a proper initial linkage and (iib) it is hard to find more than one solution. Recentformulations in solving the approximate synthesis problem adopts polynomial continuation providing several solutions, but it can not handle inequality const-raints. Based on the practical design needs the mixed exact-approximate position synthesis with two exact and an unlimited number of approximate positions has also been developed. The solutions space is presented as a ground pivot map but thepole between the exact positions cannot be selected as a ground pivot. In this thesis the exact synthesis problem of planar mechanism is solved by generating all possible solutions for the optimisation process ¿ including solutions in positive dimensional solution sets - within inequality constraints of structural parameters. Through the literature research it is first shown that the algebraic and numerical solution methods ¿ used in the research area of computational kinematics ¿ are capable of solving non-parametric algebraic systems of n equations inn variables and cannot handle the singularities associated with positive-dimensional solution sets. In this thesis the problem of positive-dimensional solutionsets is solved adopting the main principles from mathematical research area of algebraic geometry in solving parametric ( in the mathematical sense that all parameter values are considered ¿ including the degenerate cases ¿ for which the system is solvable ) algebraic systems of n equations and at least n+1 variables.Adopting the developed solution method in solving the dyadic equations in direct polynomial form in two- to three-precision-points it has been algebraically proved and numerically demonstrated that the map of the ground pivots is ambiguousand that the singularities associated with positive-dimensional solution sets can be solved. The positive-dimensional solution sets associated with the poles might contain physically meaningful solutions in the form of optimal defectfree mechanisms. Traditionally the mechanism optimisation of hydraulically driven boommechanisms is done at early state of the design process. This will result in optimal component design rather than optimal system level design. Modern mechanismoptimisation at system level demands integration of kinematic design methods with mechanical system simulation techniques. In this thesis a new kinematic design method for hydraulically driven boom mechanism is developed and integrated in mechanical system simulation techniques. The developed kinematic design method is based on the combinations of two-precision-point formulation and on optimisation ( with mathematical programming techniques or adopting optimisation methods based on probability and statistics ) of substructures using calculated criteria from the system level response of multidegree-of-freedom mechanisms. Eg. by adopting the mixed exact-approximate position synthesis in direct optimisation (using mathematical programming techniques) with two exact positions and an unlimitednumber of approximate positions the drawbacks of (ia)-(iib) has been cancelled.The design principles of the developed method are based on the design-tree -approach of the mechanical systems and the design method ¿ in principle ¿ is capable of capturing the interrelationship between kinematic and dynamic synthesis simultaneously when the developed kinematic design method is integrated with the mechanical system simulation techniques.

Response of fine root respiration to variations in biotic and abiotic factors in a mixed Mediterranean forest affected by drought-induced secondary succession

Understanding the factors controlling fine root respiration (FRR) at different temporal scales will help to improve our knowledge about the spatial and temporal variability of soil respiration (SR) and to improve future predictions of CO2 effluxes to the atmosphere. Here we present a comparative study of how FRR respond to variability in soil temperature and moisture in two widely spread species, Scots pines (Pinus sylvestris L.) and Holm-oaks (HO; Quercus ilex L.). Those two species show contrasting water use strategies during the extreme summer-drought conditions that characterize the Mediterranean climate. The study was carried out on a mixed Mediterranean forest where Scots pines affected by drought induced die-back are slowly being replaced by the more drought resistant HO. FRR was measured in spring and early fall 2013 in excised roots freshly removed from the soil and collected under HO and under Scots pines at three different health stages: dead (D), defoliated (DP) and non-defoliated (NDP). Variations in soil temperature, soil water content and daily mean assimilation per tree were also recorded to evaluate FRR sensibility to abiotic and biotic environmental variations. Our results show that values of FRR were substantially lower under HO (1.26 ± 0.16 microgram CO2 /groot·min) than under living pines (1.89 ± 0.19 microgram CO2 /groot·min) which disagrees with the similar rates of soil respiration previously observed under both canopies and suggest that FRR contribution to total SR varies under different tree species. The similarity of FRR rates under HO and DP furthermore confirms other previous studies suggesting a recent Holm-oak root colonization of the gaps under dead trees. A linear mixed effect model approach indicated that seasonal variations in FRR were best explained by soil temperature (p<0.05) while soil moisture was not exerting any direct control over FRR, despite the low soil moisture values during the summer sampling. Plant assimilation rates were positively related to FRR explaining part of the observed variability (p<0.01). However the positive relations of FRR with plant assimilation occurred mainly during spring, when both soil moisture and plant assimilation rates were higher. Our results finally suggest that plants might be able to maintain relatively high rates of FRR during the sub-optimal abiotic and biotic summer conditions probably thanks to their capacity to re-mobilize carbon reserves and their capacity to passively move water from moister layers to upper layers with lower water potentials (where the FR were collected) by hydraulic lift.

Dinâmica não-linear e reagente limitante: um exemplo de bifurcação transcrítica

The nonlinear analysis of a general mixed second order reaction was performed, aiming to explore some basic tools concerning the mathematics of nonlinear differential equations. Concepts of stability around fixed points based on linear stability analysis are introduced, together with phase plane and integral curves. The main focus is the chemical relationship between changes of limiting reagent and transcritical bifurcation, and the investigation underlying the conclusion.

Object-oriented Programming Applied to the Development of Structural Analysis and Optimization Software

This paper presents the development of a two-dimensional interactive software environment for structural analysis and optimization based on object-oriented programming using the C++ language. The main feature of the software is the effective integration of several computational tools into graphical user interfaces implemented in the Windows-98 and Windows-NT operating systems. The interfaces simplify data specification in the simulation and optimization of two-dimensional linear elastic problems. NURBS have been used in the software modules to represent geometric and graphical data. Extensions to the analysis of three-dimensional problems have been implemented and are also discussed in this paper.

Improving Cost-Efficiency and Reducing Environmental Impacts of Intermodal Transportation with Dry Port Concept – Major Rail Transport Corridor in Baltic Sea Region

Transportation plays a major role in the gross domestic product of various nations. There are, however, many obstacles hindering the transportation sector. Cost-efficiency along with proper delivery times, high frequency and reliability are not a straightforward task. Furthermore, environmental friendliness has increased the importance of the whole transportation sector. This development will change roles inside the transportation sector. Even now, but especially in the future, decisions regarding the transportation sector will be partly based on emission levels and other externalities originating from transportation in addition to pure transportation costs. There are different factors, which could have an impact on the transportation sector. IMO’s sulphur regulation is estimated to increase the costs of short sea shipping in the Baltic Sea. Price development of energy could change the roles of different transport modes. Higher awareness of the environmental impacts originating from transportation could also have an impact on the price level of more polluting transport modes. According to earlier research, increased inland transportation, modal shift and slowsteaming can be possible results of these changes in the transportation sector. Possible changes in the transportation sector and ways to settle potential obstacles are studied in this dissertation. Furthermore, means to improve cost-efficiency and to decrease environmental impacts originating from transportation are researched. Hypothetical Finnish dry port network and Rail Baltica transport corridor are studied in this dissertation. Benefits and disadvantages are studied with different methodologies. These include gravitational models, which were optimized with linear integer programming, discrete-event and system dynamics simulation, an interview study and a case study. Geographical focus is on the Baltic Sea Region, but the results can be adapted to other geographical locations with discretion. Results indicate that the dry port concept has benefits, but optimization regarding the location and the amount of dry ports plays an important role. In addition, the utilization of dry ports for freight transportation should be carefully operated, since only a certain amount of total freight volume can be cost-efficiently transported through dry ports. If dry ports are created and located without proper planning, they could actually increase transportation costs and delivery times of the whole transportation system. With an optimized dry port network, transportation costs can be lowered in Finland with three to five dry ports. Environmental impacts can be lowered with up to nine dry ports. If more dry ports are added to the system, the benefits become very minor, i.e. payback time of investments becomes extremely long. Furthermore, dry port network could support major transport corridors such as Rail Baltica. Based on an analysis of statistics and interview study, there could be enough freight volume available for Rail Baltica, especially, if North-West Russia is part of the Northern end of the corridor. Transit traffic to and from Russia (especially through the Baltic States) plays a large role. It could be possible to increase transit traffic through Finland by connecting the potential Finnish dry port network and the studied transport corridor. Additionally, sulphur emission regulation is assumed to increase the attractiveness of Rail Baltica in the year 2015. Part of the transit traffic could be rerouted along Rail Baltica instead of the Baltic Sea, since the price level of sea transport could increase due to the sulphur regulation. Both, the hypothetical Finnish dry port network and Rail Baltica transport corridor could benefit each other. The dry port network could gain more market share from Russia, but also from Central Europe, which is the other end of Rail Baltica. In addition, further Eastern countries could also be connected to achieve higher potential freight volume by rail.

Individual differences and factors affecting male behaviour in field crickets

Individual differences in male sexual behav~our and the factors influencing calling behaviour were studied in the field crickets Gryllus 2 integer and Q. veletis. In a large (13m) outdoor arena individually numbered adult male ~~ integer started calling at three to five days of age but thereafter the age of individual G. integer males did not affect nightly calling duration. Calling also did not correlate with individual weight. In this study individual male calling was continuously distributed from 0 hrs. per night to 3.5 hrs. per night, on average. A temporal effect on the number of G. integer males calling was observed. The number of males calling through the night was uniform, but a sharp increase in the number calling was observed in the early morning. No difference in calling times was observed between the night and dawn callers. AlsC)' males calling at dawn usually didnotc'all during the preceeding night. Calling and reproductive success in 1979 demonstrated a negative logarithmic relationship while in the 1980(initial) population a negative linear relationship was observed. No relationship was seen in the 1980 high density population. The ratio of non-callers to callers also affected the mating of individuals in the 1979 and1980(initial) densities:-non~callers (males calling .5 hrs. per night, on average, or less) obtained more females when the population contained a high number of callers, this being a negative logarithmic relationship to, No such relationship was observed in the 1980 high density population. Individual displacement varied nightly and was not correlated to amount of calling or reproductive success of individual G. integer males. G. integer males were displa~ed more when in a higher density in the outdoor arena Male G. integer and G. veletis behaviours were also observed in an indoor arena at different densities and, in G. veletis, with respect to female presence. When females were present in the arena, in G. veletis, male calling was reduced. Males of both species called less, on average, when in ~ higher density, than when they were in a lower density. Male displacement of both species increased on average when in a higher density as compared to displacement in a lower density. Aggression was measured by aggressive call-ing and fighting and was studied in regards to density.G. integer demonstrated less aggression in all but one comparison at higher density. No difference was observed in the ratio of aggressive calling to f.ighting comparison in G. integer. G. veletis demonstrated mixed results. No difference in aggression between densities was observed in comparisons. Less.aggression did occur in higher densities when comparisons invol.ved fighting behaviour. Male behaviour represents a competitive strategy against ot~er males, strategy being defined as a genetic (in part) alternative to other strategies. In this sense, the factors of time, density, male-male aggression, and female presence are conditions demonstrated to affect male behaviour in G. integer and G. veletis. Individual male differences and other considerations suggest that alternative male behaviours are represented by at least two conditional strategies. This possibility, and the transient 'or stable nature of genetic polymorphisms in field cricket behaviour are considered.

Mating behaviours of the field cricket (Gryllus integer) at different levels of male density

The reproductive behaviour of the field cricket, Gryllus integer, was systematically observed in indoor arenas to determine the extent of female Choice and male-male competition at different sex ratios representing two male densities (12:6 and 6:6). The costs and benefits to males and females in those two densities were analyzed according to the theory of the evolution o£ leks. Observations were conducted during the dark hours when most calling occurred since hourly rates of courtship song and mating did not fluctuate significantly over a 24 h period. Female mating rates were not significantly different between densities, therefore males at high densities were not advantaged because of increased female tendencies to mate when social stimulation was increased. Mean rates of acoustical signalling (calling and courtin"g) did not differ significantly between densities. Mean rates of fighting by males at the high density were significantly greater than those of males at the low density. Mating benefits associated with callin~courting and fighting were measured. Mating rates did not vary with rates of calling at either density. Calling was not a prerequisite to mating. Courtship song preceded all matings. There was a significant power fit between male mating and courting rates, and male mating and fighting rates at the low, but not at the high, density. Density differences in the benefits associated with increased courting and fighting may relate, in part, to greater economic defensibility and monopoly of females due to reduced male competition at the low density. Dominant males may be preferentially chosen by females or better able to monopolize mating opportunities than subordinate males. Three criteria were used to determine whether dominant males were preferentially chosen by females. The number of matings by males who won fights (within 30 min of mating) was significantly greater than the number of matings by males who were defeated in such fights. Mating rates did not vary significantly with rates of winning at either density. There was a significant power fit between male mating rates and the percentage of fights a male won (irrespective of his fighting-frequency) at the low density. The mean duration a male guarded the female after mating did not vary significantly between densities. There was a significant linear relationship between the duration a spermatophore was retained and the duration a male guarded the female after mating. Courtship song apparently stimulated spermatophore removal. Male guarding involved inter-male aggression and reduced courtship attempts by other males. Males at the high density received no apparent reproductive benefits associated with increased social stimulation. Conclusive evidence for preferential choice of males by females, using the criteria examined here, is lacking. Males at the lower density had fewer competitors and could monopolize females more effectively.

Méthode de génération de colonnes pour les problèmes de conception de réseaux avec coûts d’ajout de capacité

Les problèmes de conception de réseaux ont reçu un intérêt particulier et ont été largement étudiés de par leurs nombreuses applications dans différents domaines, tels que les transports et les télécommunications. Nous nous intéressons dans ce mémoire au problème de conception de réseaux avec coûts d’ajout de capacité. Il s’agit d’installer un ensemble d’équipements sur un réseau en vue de satisfaire la demande, tout en respectant les contraintes de capacité, chaque arc pouvant admettre plusieurs équipements. L’objectif est de minimiser les coûts variables de transport des produits et les coûts fixes d’installation ou d’augmentation de capacité des équipements. La méthode que nous envisageons pour résoudre ce problème est basée sur les techniques utilisées en programmation linéaire en nombres entiers, notamment celles de génération de colonnes et de coupes. Ces méthodes sont introduites dans un algorithme général de branch-and-bound basé sur la relaxation linéaire. Nous avons testé notre méthode sur quatre groupes d’instances de tailles différentes, et nous l’avons comparée à CPLEX, qui constitue un des meilleurs solveurs permettant de résoudre des problèmes d’optimisation, ainsi qu’à une méthode existante dans la littérature combinant des méthodes exactes et heuristiques. Notre méthode a été plus performante que ces deux méthodes, notamment pour les instances de très grandes tailles.

Evolving Distributed Algorithms with Genetic Programming

Distributed systems are one of the most vital components of the economy. The most prominent example is probably the internet, a constituent element of our knowledge society. During the recent years, the number of novel network types has steadily increased. Amongst others, sensor networks, distributed systems composed of tiny computational devices with scarce resources, have emerged. The further development and heterogeneous connection of such systems imposes new requirements on the software development process. Mobile and wireless networks, for instance, have to organize themselves autonomously and must be able to react to changes in the environment and to failing nodes alike. Researching new approaches for the design of distributed algorithms may lead to methods with which these requirements can be met efficiently. In this thesis, one such method is developed, tested, and discussed in respect of its practical utility. Our new design approach for distributed algorithms is based on Genetic Programming, a member of the family of evolutionary algorithms. Evolutionary algorithms are metaheuristic optimization methods which copy principles from natural evolution. They use a population of solution candidates which they try to refine step by step in order to attain optimal values for predefined objective functions. The synthesis of an algorithm with our approach starts with an analysis step in which the wanted global behavior of the distributed system is specified. From this specification, objective functions are derived which steer a Genetic Programming process where the solution candidates are distributed programs. The objective functions rate how close these programs approximate the goal behavior in multiple randomized network simulations. The evolutionary process step by step selects the most promising solution candidates and modifies and combines them with mutation and crossover operators. This way, a description of the global behavior of a distributed system is translated automatically to programs which, if executed locally on the nodes of the system, exhibit this behavior. In our work, we test six different ways for representing distributed programs, comprising adaptations and extensions of well-known Genetic Programming methods (SGP, eSGP, and LGP), one bio-inspired approach (Fraglets), and two new program representations called Rule-based Genetic Programming (RBGP, eRBGP) designed by us. We breed programs in these representations for three well-known example problems in distributed systems: election algorithms, the distributed mutual exclusion at a critical section, and the distributed computation of the greatest common divisor of a set of numbers. Synthesizing distributed programs the evolutionary way does not necessarily lead to the envisaged results. In a detailed analysis, we discuss the problematic features which make this form of Genetic Programming particularly hard. The two Rule-based Genetic Programming approaches have been developed especially in order to mitigate these difficulties. In our experiments, at least one of them (eRBGP) turned out to be a very efficient approach and in most cases, was superior to the other representations.

Convex linear combination processes for compositions

Aitchison and Bacon-Shone (1999) considered convex linear combinations of compositions. In other words, they investigated compositions of compositions, where the mixing composition follows a logistic Normal distribution (or a perturbation process) and the compositions being mixed follow a logistic Normal distribution. In this paper, I investigate the extension to situations where the mixing composition varies with a number of dimensions. Examples would be where the mixing proportions vary with time or distance or a combination of the two. Practical situations include a river where the mixing proportions vary along the river, or across a lake and possibly with a time trend. This is illustrated with a dataset similar to that used in the Aitchison and Bacon-Shone paper, which looked at how pollution in a loch depended on the pollution in the three rivers that feed the loch. Here, I explicitly model the variation in the linear combination across the loch, assuming that the mean of the logistic Normal distribution depends on the river flows and relative distance from the source origins

A mixed H2/H∞-based semiactive control for vibration mitigation in flexible structures

In this paper, we address this problem through the design of a semiactive controller based on the mixed H2/H∞ control theory. The vibrations caused by the seismic motions are mitigated by a semiactive damper installed in the bottom of the structure. It is meant by semiactive damper, a device that absorbs but cannot inject energy into the system. Sufficient conditions for the design of a desired control are given in terms of linear matrix inequalities (LMIs). A controller that guarantees asymptotic stability and a mixed H2/H∞ performance is then developed. An algorithm is proposed to handle the semiactive nature of the actuator. The performance of the controller is experimentally evaluated in a real-time hybrid testing facility that consists of a physical specimen (a small-scale magnetorheological damper) and a numerical model (a large-scale three-story building)

Chemistry of mixed-ligand methoxy bonded oxidovanadium(V) complexes with a family of hydrazone ligands containing VO3+ core and their substituent controlled methoxy-bridged dimeric forms

[(VO)-O-IV(acac)(2)] reacts with an equimolar amount of benzoyl hydrazones of 2-hydroxyacetophenone (H2L1), 2-hydroxy-5-methylacetophenone (H2L2) and 5-chloro-2-hydroxyacetophenone (H2L4) in methanol to afford the penta-coordinated mixed-ligand methoxy bonded oxidovanadium(V) complexes [(VO)-O-V(L-1)-(OCHA(3))](1). [(VO)-O-V(L-2)(OCH3)](2), and [(VO)-O-V(L-4)(OCH3)](4), respectively, whereas, the similar reaction with the benzoyl hydrazone of 2-hydroxy-5-methoxyacetophenone (H2L3) producing only the hexa-coordinated dimethoxy-bridged dimeric complex [(VO)-O-V(L-3)(OCH3)](2) (3A). Similar type of hexa-coordinated dimeric analogue of 1 i.e., [(VO)-O-V(L-1)(OCH3)](2) (1A) was obtained from the reaction of [(VO)-O-IV(acac)(2)] with the equimolar amount of H2L1 in presence of half equivalent 4,4'-bipyridine in methanol while the decomposition of [(VO)-O-IV(L-2)(bipy)] complex in methanol afforded the dimeric analogue of 2 i.e., [(VO)-O-V(L-2)(OCH3)](2) (2A). All these dimeric complexes 1A-3A react with an excess amount of imidazole in methanol producing the respective monomeric complex. The X-ray structural analysis of 1-3 and their dimeric analogues 1A-3A indicates that the geometry around the vanadium center in the monomeric form is distorted square-pyramidal while that of their respective dimeric forms is distorted octahedral, where the ligands are bonded to vanadium meridionally in their fully deprotonated enol forms. Due to the formation of bridge, the V-O(methoxy) bond in the dimeric complexes is lengthened to such an extent that it becomes equal in length with the V-O(phenolate) bond in 3A and even longer in 1A and 2A, which is unprecedented. The H-1 NMR spectra of the complexes 1A-3A in CDCl3 solution, indicates that these dimeric complexes are converted appreciably into their respective monomeric form. Complexes are electro-active displaying one quasi-reversible reduction peak near +0.25 V versus SCE in CH2Cl2 solution. The E-1/2 values of the complexes show linear relationship with the Hammett parameter (sigma) of the substituents. All these VO3+-complexes are converted to the corresponding complexes with V2O34+ motif simply on refluxing them in acetone and to the complexes with VO2+ motif on reaction with 2 KOH in methanol. (C) 2008 Elsevier Ltd. All rights reserved.

Use of a reduced Schiff-Base ligand to prepare novel chloro-bridged chains of rare Cu(II) trinuclear complexes with mixed azido/oxo and chloro/oxo bridges

Two mixed bridged one-dimensional (1D) polynuclear complexes, [Cu3L2(mu(1,1)-N-3)(2)(mu-Cl)Cl](n) (1) and {[Cu3L2(mu-Cl)(3)Cl]center dot 0.46CH(3)OH}(n), (2), have been synthesized using the tridentate reduced Schiff-base ligand HL (2-[(2-dimethylamino-ethylamino)-methyl]-phenol). The complexes have been characterized by X-ray structural analyses and variable-temperature magnetic susceptibility measurements. In both complexes the basic trinuclear angular units are joined together by weak chloro bridges to form a 1D chain. The trinuclear structure of 1 is composed of two terminal square planar [Cu(L)(mu(1,1)-N-3)] units connected by a central Cu(II) atom through bridging nitrogen atoms of end-on azido ligands and the phenoxo oxygen atom of the tridentate ligand. These four coordinating atoms along with a chloride ion form a distorted trigonal bipyramidal geometry around the central Cu(II). The structure of 2 is similar; the only difference being a Cl bridge replacing the mu(1,1)-N-3 bridge in the trinuclear unit. The magnetic properties of both trinuclear complexes can be very well reproduced with a simple linear symmetrical trimer model (H = JS(i)S(i+1)) with only one intracluster exchange coupling (J) including a weak intertrimer interaction (.j) reproduced with the molecular field approximation. This model provides very satisfactory fits for both complexes in the whole temperature range with the following parameters: g = 2.136(3), J = 93.9(3) cm(-1) and zj= -0.90(3) cm(-1) (z = 2) for 1 and g = 2.073(7), J = -44.9(4) cm(-1) and zJ = -1.26(6) cm(-1) (z = 2) for 2.