The term "frustration" in the context of magnetism was originally used by P. W. Anderson and quickly adopted for application to the description of spin glasses and later to very special lattice types, such as the kagomé. The original use of the term was to describe systems with competing antiferromagnetic interactions and is important in current condensed matter physics in areas such as the description of emergent magnetic monopoles in spin ice. Within molecular magnetism, at least two very different definitions of frustration are used. Here we report the synthesis and characterization of unusual ninemetal rings, using magnetic measurements and inelastic neutron scattering, supported by density functional theory calculations. These compounds show different electronic/magnetic structures caused by frustration, and the findings lead us to propose a classification for frustration within molecular magnets that encompasses and clarifies all previous definitions. ,4,7-triazacyclononane). One area where molecular magnetism might contribute is in spin frustration (3), which is important in condensed matter physics in areas such as spin ice (4) and spin liquids (5). Potential model systems would be odd-numbered metal rings where the metal array forms equilateral triangles, or regular pentagons, heptagons, and larger 2D rings. Alternatively 3D solids made from regular odd-numbered rings should also be frustrated, e.g., regular tetrahedra and octahedra contain triangular faces and regular icosahedra contain pentagonal faces.Studies of spin frustration in molecular systems are largely restricted to that of triangular trimetallic species (6, 7). There are a few reports of five-metal rings but no detailed magnetic studies of such compounds (8), with the exception of a {Cu 5 } cage (9). The only significant examples of larger frustrated molecules are studies of the {M 30 } Keplerates (10) and a recent report of a heptametallic {VO} 7 ring (11), where the studies are limited to low temperature magnetization. There is also a report of an {Fe 9 } complex that can, very approximately, be described as a nine-metal ring (12).A further difficulty is that precisely what is meant by frustration varies in this area. The strictest definition used in molecular magnetism, from Kahn (13), states that frustration must result in a spin degenerate ground state, typically a degenerate pair of S = 1/2 states. Surprisingly, this definition excludes all possible models built from integer spins, as in such cases (for example a regular pentagon of s = 1 spins), the ground total spin state is unique and has S = 0. This definition thus seems prohibitively strict. At the other extreme, the term frustration has been used to describe cases where there are competing antiferromagnetic interactions, even where those interactions result in a unique spin ground state that can be described using a classical picture, with spins represented as arrows pointing alternately up and down. This definition seems, if anything, too permissive.We have reported studies of...

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Mixed-valence polyoxovanadates(IV/V) have emerged as one of the most intricate class of supramolecular all-inorganic host species, able to encapsulate a wide variety of smaller guest templates during their self-assembly formation process. As showcased herein, the incorporation of guests, though governed solely by ultra-weak electrostatic and van der Waals interactions, can cause drastic effects on the electronic and magnetic characteristics of the shell complex of the polyoxovanadate. We address the question of methodology for the magnetochemical analysis of virtually isostructural {V(IV/V) 22 O54 }-type polyoxoanions of D2d symmetry enclosing diamagnetic VO2 F2 (-) (C2v ), SCN(-) (C∞v ), or ClO4 (-) (Td ) template anions. These induce different polarization effects related to differences in their geometric structures, symmetry, ion radii, and valence shells, eventually resulting in a supramolecular modulation of magnetic exchange between the V(3d) electrons that are partly delocalized over the {V22 O54 } shells. We also include the synthesis and characterization of the novel [V(V) O2 F2 @HV(IV) 8 V(V) 14 O54 ](6-) system that comprises the rarely encountered discrete difluorovanadate anion as a quasi-isolated guest species.

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An Ashkin-Teller neural network, allowing for two types of neurons is considered in the case of low loading as a function of the strength of the respective couplings between these neurons. The storage and retrieval of embedded patterns built from the two types of neurons, with different degrees of (in)dependence is studied. In particular, thermodynamic properties including the existence and stability of Mattis states are discussed. Furthermore, the dynamic behaviour is examined by deriving flow equations for the macroscopic overlap. It is found that for linked patterns the model shows better retrieval properties than a corresponding Hopfield model.One of the best known physical models for neural networks is the Hopfield model [1]. In theoretical investigations of network properties, e.g., the retrieval of learned patterns, it plays a similar role as the Ising model does in the theory of magnetism. Extensions of this model to multi-state neurons have received a lot of attention recently (see, e.g., [2] -[5] and the references cited therein). Thereby the ability to store and retrieve so-called grey-toned and coloured patterns has been investigated.In this work we consider another extension of the Hopfield model to allow for multi-functional neurons. The specific model we have in mind is the neural network version of the Ashkin-Teller spin-glass ([6]-[9]). Indeed, on the one hand the Ashkin-Teller model has two different kinds of neurons (spins) at each site interacting with each other. This allows us to interprete this model as a neural network with two types of neurons having different functions. On the other hand, this Ashkin-Teller neural network (ATNN) can be considered as a model consisting out of two interacting Hopfield models.We expect the behaviour of the ATNN to be different from the one of the Hopfield model in a non trivial way. One of the things we want to find out, e.g., is whether this (four-neuron) interaction between the two types of neurons can improve the retrieval process for embedded patterns built from these two types of neurons. We will see, indeed, that for a particular choice of this interaction term the retrieval quality of the embedded patterns is very high in comparison with a corresponding Hopfield model. Therefore, independent of the possible biological relevance of this model, if any, such a study is interesting from the pure physical point of view.In this work we consider both the thermodynamic and dynamic properties of this model in the case of loading of a finite number of patterns.The rest of this paper is organized as follows. In section 2 the ATNN model is introduced. Section 3 discusses the methods used for analyzing both the equilibrium properties and the dynamics of the model. In particular, fixed-point equations as well as flow equations for the relevant macroscopic overlap order parameters are derived. In section 4 numerical solutions of these equations are discussed for a representative set of network parameters. The retrieval properties of embedded patterns with different...

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