Books on the topic 'Topological features'

The monograph introduces the most general type of dynamical systems with shock interactions. The concept of local features of these systems is given. Local features of dynamical systems with shock interactions of the first six types are studied. Their topological equivalence, i.e. their sameness, is proved. Two simplest vibration shock systems are studied, which are of fundamental importance for practice. It is intended for researchers, engineers, students and postgraduates interested in the study of dynamic systems and processes.

A network is specified by its links and nodes. However, it can be described by a much wider range of interesting and important topological features. This chapter introduces how a network can be characterized by its microscopic topological features and macroscopic topological features. Microscopic features introduced are degree and clustering coefficients. Macroscopic topological features introduced are the degree distribution; correlation between degrees of connected nodes; modularity; and, the eigenvalue spectrum (which counts the number of closed paths in the graph).

An important application of spin dynamics is the response of a magnetic material subjected to an external stimuli. In the previous chapter we discussed the response of primarily ferromagnets to temperature fluctuations that manifest itself to spin excitations and magnons. In this chapter, we are concerned about magnetic materials with more complicated magnetic texture, such as spin spirals and topological magnetic structures, in particular magnetic skyrmions. Magnetic skyrmions has many appealing and intriguing features that make them interesting both for possible applications but also from a pure theoretical point of view.

This chapter moves beyond viewing nodes as homogeneous dots set on a plane. To introduce more complicated underlying space, multiplex networks (which are defined with layers of interaction on the same underlying node set) and temporal (time-dependent) networks are discussed. It shown that despite the much more complicated underlying space, many of the techniques developed in earlier chapters can be applied. Heterogeneous nodes are introduced as an extension of the stochastic block model for community structure, then extended using methods developed in earlier chapters to more general (continuous) node attributes such as fitness. The chapter closes with a discussion of the intersections and similarities between the many alternative models for capturing topological features that have been presented in the book.

The introductory chapter provides important background information for the readers of the volume. It describes the aims of the volume, which is the first comprehensive and concise generative historical syntax of German. However, the contributions are not only aimed at researchers in the field, but in giving a basic overview of the respective topics and relating them to more descriptive and traditional accounts, the book is also suited for academic teaching, e.g. as a central text book in courses on historical German syntax. The chapter then gives an overview of the syntax of German, introducing the topological model widely used in more traditional accounts and the generative analysis of German clause structure. Finally, it provides an overview of the history of High and Low German including information on basic grammatical features, the textual evidence and central reference books and digital corpora for each period.

This chapter reviews graph generation techniques in the context of applications. The first case study is power grids, where proposed strategies to prevent blackouts have been tested on tailored random graphs. The second case study is in social networks. Applications of random graphs to social networks are extremely wide ranging – the particular aspect looked at here is modelling the spread of disease on a social network – and how a particular construction based on projecting from a bipartite graph successfully captures some of the clustering observed in real social networks. The third case study is on null models of food webs, discussing the specific constraints relevant to this application, and the topological features which may contribute to the stability of an ecosystem. The final case study is taken from molecular biology, discussing the importance of unbiased graph sampling when considering if motifs are over-represented in a protein–protein interaction network.

The book presents a multidisciplinary analysis of the context of quantum physics experiments and the function of the human mind that makes it possible to demonstrate that an object-based model of reality formed at the level of the unconscious is the basis of our worldview. The consciousness experiences a time flow because of the specific features of perception in the form of a model with a sequential fixation of events. Together with the need to relate objects in terms of the model, this generates a space-time representation of the world around us. Acceptance of a mental character of our construct of reality allows for resolution of the problems in quantum physics and its paradoxes, thereby opening the way to an insight into reality. The presented material is organized in a specific order to facilitate the reader`s understanding. First, the fact that if there are no objects in the area of quantum mechanics, then they belong to the corresponding model rather than the reality is proved by case studies of the most discussed and relevant paradoxes of quantum physics. The authors consider a topological variant in constructing an object-based space that describes the physical properties of an object that are the most verified in science and describable with mathematical relations. The functionality of the proposed construct is tested by deriving the laws of conservation of energy and momentum in a relativistic form. The book is oriented towards experts in physics and psychology, advanced students, and readers interested in state-of-the-art science and the philosophy connected to it.

This chapter introduces the space unit vector V of stably dominated types on a definable set V. It first endows unit vector V with a canonical structure of a (strict) pro-definable set before providing some examples of stably dominated types. It then endows unit vector V with the structure of a definable topological space, and the properties of this definable topology are discussed. It also examines the canonical embedding of V in unit vector V as the set of simple points. An essential feature in the approach used in this chapter is the existence of a canonical extension for a definable function on V to unit vector V. This is considered in the next section where continuity criteria are given. The chapter concludes by describing basic notions of (generalized) paths and homotopies, along with good metrics, Zariski topology, and schematic distance.

This text reviews fundametals and incorporates key themes of quantum physics. One theme contrasts boson condensation and fermion exclusivity. Bose–Einstein condensation is basic to superconductivity, superfluidity and gaseous BEC. Fermion exclusivity leads to compact stars and to atomic structure, and thence to the band structure of metals and semiconductors with applications in material science, modern optics and electronics. A second theme is that a wavefunction at a point, and in particular its phase is unique (ignoring a global phase change). If there are symmetries, conservation laws follow and quantum states which are eigenfunctions of the conserved quantities. By contrast with no particular symmetry topological effects occur such as the Bohm–Aharonov effect: also stable vortex formation in superfluids, superconductors and BEC, all these having quantized circulation of some sort. The quantum Hall effect and quantum spin Hall effect are ab initio topological. A third theme is entanglement: a feature that distinguishes the quantum world from the classical world. This property led Einstein, Podolsky and Rosen to the view that quantum mechanics is an incomplete physical theory. Bell proposed the way that any underlying local hidden variable theory could be, and was experimentally rejected. Powerful tools in quantum optics, including near-term secure communications, rely on entanglement. It was exploited in the the measurement of CP violation in the decay of beauty mesons. A fourth theme is the limitations on measurement precision set by quantum mechanics. These can be circumvented by quantum non-demolition techniques and by squeezing phase space so that the uncertainty is moved to a variable conjugate to that being measured. The boundaries of precision are explored in the measurement of g-2 for the electron, and in the detection of gravitational waves by LIGO; the latter achievement has opened a new window on the Universe. The fifth and last theme is quantum field theory. This is based on local conservation of charges. It reaches its most impressive form in the quantum gauge theories of the strong, electromagnetic and weak interactions, culminating in the discovery of the Higgs. Where particle physics has particles condensed matter has a galaxy of pseudoparticles that exist only in matter and are always in some sense special to particular states of matter. Emergent phenomena in matter are successfully modelled and analysed using quasiparticles and quantum theory. Lessons learned in that way on spontaneous symmetry breaking in superconductivity were the key to constructing a consistent quantum gauge theory of electroweak processes in particle physics.