Discussion on the geometric structure of strange attractor

We discuss the transversal heteroclinic cycle formed by hyperbolic periodic pointes of diffeomorphism on the differential manifold. We point out that every possible kind of transversal heteroclinic cycle has the Smalehorse property and the unstable manifolds of hyperbolic periodic points have the closure relation mutually. Therefore the strange attractor may be the closure of unstable manifolds of a countable number of hyperbolic periodic points. The Henon mapping is used as an example to show that the conclusion is reasonable.

The geometric structure and dynamic properties of lauwerier attractor

 Introduction The strange chaotic attractor (ACS) is an important subject in the nonlinear field. On the basis of the theory of transversal heteroclinic cycles, it is suggested that the strange attractor is the closure of the unstable manifolds of countable infinite hyperbolic periodic points. From this point of view some nonlinear phenomena are explained reasonably. 更多还原

The optimized geometric structure of the (0001) surface of α-Fe 2O3

A tridimensional model of α-Fe₂O₃ and models of (0001) and (1102) surfaces on it were built. Then the structural optimization of the (0001) surface was presented which explored the influence of the system scale and the terminal surface configuration. Four different models including two different system scale structures (MODEL□ and MODEL□) and two different terminal structures (MODEL□ and MODEL□) were analyzed in this paper. It was concluded that the boundary effect was more important in a smaller system in the structure optimization. And the Fe-terminated was more stable than the O-terminated structure which was agreed with the experiences, this structural model can be used in further work including the monatomic adsorption/desorption and the chemical reactions on this surface.

The geometric structure of the brain fiber pathways.

The structure of the brain as a product of morphogenesis is difficult to reconcile with the observed complexity of cerebral connectivity. We therefore analyzed relationships of adjacency and crossing between cerebral fiber pathways in four nonhuman primate species and in humans by using diffusion magnetic resonance imaging. The cerebral fiber pathways formed a rectilinear three-dimensional grid continuous with the three principal axes of development. Cortico-cortical pathways formed parallel sheets of interwoven paths in the longitudinal and medio-lateral axes, in which major pathways were local condensations. Cross-species homology was strong and showed emergence of complex gyral connectivity by continuous elaboration of this grid structure. This architecture naturally supports functional spatio-temporal coherence, developmental path-finding, and incremental rewiring with correlated adaptation of structure and function in cerebral plasticity and evolution.

Geometric Structure of the Adaptive Controller of the Human Arm

The objects with which the hand interacts with may significantly change the dynamics of the arm. How does the brain adapt control of arm movements to this new dynamic? We show that adaptation is via composition of a model of the task's dynamics. By exploring generalization capabilities of this adaptation we infer some of the properties of the computational elements with which the brain formed this model: the elements have broad receptive fields and encode the learned dynamics as a map structured in an intrinsic coordinate system closely related to the geometry of the skeletomusculature. The low--level nature of these elements suggests that they may represent asset of primitives with which a movement is represented in the CNS.

A closer look at chaotic advection in the stratosphere: part I: geometric structure

The relevance of chaotic advection to stratospheric mixing and transport is addressed in the context of (i) a numerical model of forced shallow-water flow on the sphere, and (ii) a middle-atmosphere general circulation model. It is argued that chaotic advection applies to both these models if there is suitable large-scale spatial structure in the velocity field and if the velocity field is temporally quasi-regular. This spatial structure is manifested in the form of “cat’s eyes” in the surf zone, such as are commonly seen in numerical simulations of Rossby wave critical layers; by analogy with the heteroclinic structure of a temporally aperiodic chaotic system the cat’s eyes may be thought of as an “organizing structure” for mixing and transport in the surf zone. When this organizing structure exists, Eulerian and Lagrangian autocorrelations of the velocity derivatives indicate that velocity derivatives decorrelate more rapidly along particle trajectories than at fixed spatial locations (i.e., the velocity field is temporally quasi-regular). This phenomenon is referred to as Lagrangian random strain.

More maximal arcs in Desarguesian projective planes and their geometric structure

In a previous paper R. Mathon gave a new construction method for maximal arcs in finite Desarguesian projective planes via closed sets of conics, as well as giving many new examples of maximal arcs. In the current paper, new classes of maximal arcs are constructed, and it is shown that every maximal arc so constructed gives rise to an infinite class of maximal arcs. Apart from when they are of Denniston type or dual hyperovals, closed sets of conics are shown to give maximal arcs that are not isomorphic to the known constructions. An easy characterisation of when a closed set of conics is of Denniston type is given. Results on the geometric structure of the maximal arcs and their duals are proved, as well as on elements of their collineation stabilisers.

A Geometric Algorithm for Learning Oblique Decision Trees

In this paper we present a novel algorithm for learning oblique decision trees. Most of the current decision tree algorithms rely on impurity measures to assess goodness of hyperplanes at each node. These impurity measures do not properly capture the geometric structures in the data. Motivated by this, our algorithm uses a strategy, based on some recent variants of SVM, to assess the hyperplanes in such a way that the geometric structure in the data is taken into account. We show through empirical studies that our method is effective.

Geometric Decision Tree

In this paper, we present a new algorithm for learning oblique decision trees. Most of the current decision tree algorithms rely on impurity measures to assess the goodness of hyperplanes at each node while learning a decision tree in top-down fashion. These impurity measures do not properly capture the geometric structures in the data. Motivated by this, our algorithm uses a strategy for assessing the hyperplanes in such a way that the geometric structure in the data is taken into account. At each node of the decision tree, we find the clustering hyperplanes for both the classes and use their angle bisectors as the split rule at that node. We show through empirical studies that this idea leads to small decision trees and better performance. We also present some analysis to show that the angle bisectors of clustering hyperplanes that we use as the split rules at each node are solutions of an interesting optimization problem and hence argue that this is a principled method of learning a decision tree.

A geometric approach to cosmological boundary conditions

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Synthesis, structure, and stability of metal-organic frameworks

Metal-organic frameworks (MOFs) obtained much attention because of their unusual structures and properties as well as their potential applications. This dissertation research was focused on (1) the effects of synthesis conditions on the structures of MOFs, (2) the thermal stability of MOFs, (3) pressure-induced amorphization, and (4) the effect of high-valent ions on the structure of a MOF. This research demonstrated that the crystal structure of MOF-5 could be controlled by drying solvents. If the vacuum solvent is dimethylformamide (DMF), the crystal structure of MOF-5 is tetragonal. In contrast, if the DMF is displaced by CH2Cl2 before the vacuum, the obtained MOF-5 occupies a cubic structure. Furthermore, it was found that the tetragonal MOF-5 exhibited a mediate surface area (300-1000 m2/g). The surface area of tetragonal MOF-5 is also dependent on Zn(NO3)2/H2BDC (H2BDC: terephthalic acid) molar ratios used for its synthesis. The optimum ratio is 1.38, at which synthesized tetragonal MOF-5 exhibits the highest crystallinity and surface area (1297 m2/g). The thermal stability and decomposition of MOF-5 were systematically investigated. The thermal decomposition of cubic and tetragonal MOF-5s resulted in the same products: CO2, benzene, amorphous carbon, and crystal ZnO. The thermal decomposition is due to breaking carboxylic bridges between benzene rings and Zn4O clusters. Identifying structural relationships between crystalline and noncrystalline states is of fundamental interest in materials research. Currently, amorphization of solid materials at ambient temperature requires an ultra-high pressure (several GPa). However, this research demonstrated that MOF-5 and IRMOF-8 can be irreversibly amorphized at ambient temperature by employing a low compressing pressure of 3.5 MPa, which is 100 times lower than that required for amorphization of other solids. Furthermore, the pressure-induced amorphization (PIA) of MOFs is strongly dependent on the changeability of bond angles. If the geometric structure of a MOF can allow bond angles to be changed without breaking bonds, it can easily be amorphized by compression. This can explain why MOF-5 and IRMOF-8 can easily be amorphized via compression than Cu-BTC. It is generally recognized that zeolitic imidazolate frameworks (ZIFs) occupy much higher stability than other types of MOFs. The representative of ZIFs is Zn(2-methylimidazole)2 (ZIF-8) exhibiting high-decomposition temperature and high chemical resistance to various solvents. However, so far, it is still unknown whether the high stability of ZIF-8 can be challenged by ions, which is important for its modification by doping ions. In this research, we performed aqueous salt solution treatment on ZIF-8, and the results showed that anions (Cl¯ and NO3¯) in a solution exhibited no effect on the crystal structure of ZIF-8. However, the effect of cations (in a solution) on structure of ZIF-8 strongly depends on the cation valences. The univalent metal cations showed no effect on the structure of ZIF-8, whereas the bivalent or higher-valent metal cations caused the collapse of ZIF-8 crystal structure. Therefore, structure stability of ZIF-8 is considered when it is subjected to the application, in which high-valent metal cations are involved.

The quantum inspired modelling of changing attitudes and self-organising societies

We utilise the well-developed quantum decision models known to the QI community to create a higher order social decision making model. A simple Agent Based Model (ABM) of a society of agents with changing attitudes towards a social issue is presented, where the private attitudes of individuals in the system are represented using a geometric structure inspired by quantum theory. We track the changing attitudes of the members of that society, and their resulting propensities to act, or not, in a given social context. A number of new issues surrounding this "scaling up" of quantum decision theories are discussed, as well as new directions and opportunities.

Mutually unbiased bases as submodules and subspaces

Mutually unbiased bases (MUBs) have been used in several cryptographic and communications applications. There has been much speculation regarding connections between MUBs and finite geometries. Most of which has focused on a connection with projective and affine planes. We propose a connection with higher dimensional projective geometries and projective Hjelmslev geometries. We show that this proposed geometric structure is present in several constructions of MUBs.

An Introduction to Constrained Control

An ubiquitous problem in control system design is that the system must operate subject to various constraints. Although the topic of constrained control has a long history in practice, there have been recent significant advances in the supporting theory. In this chapter, we give an introduction to constrained control. In particular, we describe contemporary work which shows that the constrained optimal control problem for discrete-time systems has an interesting geometric structure and a simple local solution. We also discuss issues associated with the output feedback solution to this class of problems, and the implication of these results in the closely related problem of anti-windup. As an application, we address the problem of rudder roll stabilization for ships.