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Hierarchical Voronoi Graphs : Spatial Representation and Reasoning for Mobile Robots
What is space? Is there space when there are objects to occupy it or is there space only when there are no objects to occupy it?Can there be space without objects? These are old philosophical questions that concern the ontology of space in the philosophical sense of ‘ontology’ – what is the nature of space?Cognitive science in general and arti?cial intelligence in particular are less c- cerned with the nature of things than with their mental conceptualizations.In spatial cognition research we address questions like What do we know about space?How is space represented? What are the representational entities? What are the rep- sentational structures? Answers to these questions are described in what is called ontologies in arti?cial intelligence.Different tasks require different knowledge, and different representations of knowledge facilitate different ways of solving problems.In this book, Jan Oliver Wallgrün develops and investigates representational structures to support tasks of autonomous mobile robots, from the acquisition of knowledge to the use of this knowledge for navigation.The research presented is concerned with the robot mapping problem, the pr- lem of building a spatial representation of an environment that is perceived by s- sors that only provide incomplete and uncertain information; this information usually needs to be related to other imprecise or uncertain information.The routes a robot can take can be abstractly described in terms of graphs where alternative routes are represented by alternative branches in these route graphs.
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Math For Security : From Graphs and Geometry to Spatial Analysis
Applied Math for Security is one of the first math-based guides specifically geared for information security practitioners.Readers will learn how to use concepts from various fields of mathematics - like graph theory, computational geometry, and statistics - to create and implement ready-to-use security tools.The book is written in a lively, conversational style that engages readers from the get-go.Chapters are enriched with code examples written in Python, and feature hands-on 'proof of concept' projects that involve developing math-based applications to solve real-world problems.Readers are also able to apply the mathematical constructs that they learn to a variety of challenging scenarios, like determining the ideal location for fire stations, disrupting information flow in a social network, building facial recognition software, and designing custom tools for modern security work.
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Charts & Graphs
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Graphs & Digraphs
Graphs & Digraphs, Seventh Edition masterfully employs student-friendly exposition, clear proofs, abundant examples, and numerous exercises to provide an essential understanding of the concepts, theorems, history, and applications of graph theory.This classic text, widely popular among students and instructors alike for decades, is thoroughly streamlined in this new, seventh edition, to present a text consistent with contemporary expectations. Changes and updates to this edition include:A rewrite of four chapters from the ground upStreamlining by over a third for efficient, comprehensive coverage of graph theoryFlexible structure with foundational Chapters 1–6 and customizable topics in Chapters 7–11Incorporation of the latest developments in fundamental graph theoryStatements of recent groundbreaking discoveries, even if proofs are beyond scopeCompletely reorganized chapters on traversability, connectivity, coloring, and extremal graph theory to reflect recent developmentsThe text remains the consummate choice for an advanced undergraduate level or introductory graduate-level course exploring the subject’s fascinating history, while covering a host of interesting problems and diverse applications.Our major objective is to introduce and treat graph theory as the beautiful area of mathematics we have always found it to be.We have striven to produce a reader-friendly, carefully written book that emphasizes the mathematical theory of graphs, in all their forms.While a certain amount of mathematical maturity, including a solid understanding of proof, is required to appreciate the material, with a small number of exceptions this is the only pre-requisite. In addition, owing to the exhilarating pace of progress in the field, there have been countless developments in fundamental graph theory ever since the previous edition, and many of these discoveries have been incorporated into the book.Of course, some of the proofs of these results are beyond the scope of the book, in which cases we have only included their statements.In other cases, however, these new results have led us to completely reorganize our presentation.Two examples are the chapters on coloring and extremal graph theory.
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What is spatial visualization ability?
Spatial visualization ability refers to the capacity to mentally manipulate and comprehend spatial relationships between objects. Individuals with strong spatial visualization skills can easily visualize and understand how objects relate to each other in space, such as rotating or manipulating shapes in their mind. This ability is crucial in various fields such as engineering, architecture, and mathematics, as it allows individuals to solve complex problems and understand spatial concepts more effectively. Improving spatial visualization ability can enhance problem-solving skills and overall cognitive performance.
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Is spatial visualization important for engineers?
Yes, spatial visualization is important for engineers as it allows them to mentally manipulate and understand complex 3D objects and structures. Engineers often need to design and analyze various components and systems, and spatial visualization skills help them to conceptualize and communicate their ideas effectively. Whether it's designing a new product, creating blueprints for a building, or solving complex problems, spatial visualization is a crucial skill that allows engineers to think critically and innovate in their field.
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Are the graphs identical?
No, the graphs are not identical. While they may have similar shapes and patterns, there are differences in the specific data points and values represented on each graph. These differences could be due to variations in the data, different scales or axes used, or other factors that affect the visualization of the information. Therefore, it is important to carefully compare the details of each graph to understand the differences between them.
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Can one improve their spatial visualization skills?
Yes, it is possible to improve spatial visualization skills through practice and training. Engaging in activities such as puzzles, building models, and playing spatial reasoning games can help develop these skills. Additionally, practicing mental rotation exercises and regularly challenging oneself with spatial tasks can also contribute to improvement. With consistent effort and dedication, individuals can enhance their spatial visualization abilities over time.
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Spectra of Graphs
This book gives an elementary treatment of the basic material about graph spectra, both for ordinary, and Laplace and Seidel spectra.The text progresses systematically, by covering standard topics before presenting some new material on trees, strongly regular graphs, two-graphs, association schemes, p-ranks of configurations and similar topics.Exercises at the end of each chapter provide practice and vary from easy yet interesting applications of the treated theory, to little excursions into related topics.Tables, references at the end of the book, an author and subject index enrich the text. Spectra of Graphs is written for researchers, teachers and graduate students interested in graph spectra.The reader is assumed to be familiar with basic linear algebra and eigenvalues, although some more advanced topics in linear algebra, like the Perron-Frobenius theorem and eigenvalue interlacing are included.
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Dynamics on Graphs
This extensive revision of the 2007 book 'Random Graph Dynamics,' covering the current state of mathematical research in the field, is ideal for researchers and graduate students.It considers a small number of types of graphs, primarily the configuration model and inhomogeneous random graphs.However, it investigates a wide variety of dynamics.The author describes results for the convergence to equilibrium for random walks on random graphs as well as topics that have emerged as mature research areas since the publication of the first edition, such as epidemics, the contact process, voter models, and coalescing random walk.Chapter 8 discusses a new challenging and largely uncharted direction: systems in which the graph and the states of their vertices coevolve.
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Polytopes and Graphs
This book introduces convex polytopes and their graphs, alongside the results and methodologies required to study them.It guides the reader from the basics to current research, presenting many open problems to facilitate the transition.The book includes results not previously found in other books, such as: the edge connectivity and linkedness of graphs of polytopes; the characterisation of their cycle space; the Minkowski decomposition of polytopes from the perspective of geometric graphs; Lei Xue's recent lower bound theorem on the number of faces of polytopes with a small number of vertices; and Gil Kalai's rigidity proof of the lower bound theorem for simplicial polytopes.This accessible introduction covers prerequisites from linear algebra, graph theory, and polytope theory.Each chapter concludes with exercises of varying difficulty, designed to help the reader engage with new concepts.These features make the book ideal for students and researchers new to the field.
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The Graphs Gaff
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What are problems with spatial visualization skills?
Some problems with spatial visualization skills include difficulty in understanding and interpreting maps, graphs, and diagrams. Individuals with poor spatial visualization skills may struggle with tasks such as navigating through unfamiliar environments, understanding 3D objects, and mentally rotating objects. This can impact their performance in subjects such as math, science, and engineering, as well as in everyday activities such as driving and assembling furniture. Additionally, poor spatial visualization skills can lead to frustration and decreased confidence in one's abilities.
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How do you read graphs?
When reading graphs, it is important to first identify the axes and the variables being represented. Next, look at the scale of the axes to understand the range of values being shown. Pay attention to the trend or pattern in the data points, such as whether they are increasing, decreasing, or staying constant. Finally, analyze any labels, titles, or legends to fully interpret the information being presented in the graph.
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What are self-complementary graphs?
Self-complementary graphs are graphs that are isomorphic to their own complement. In other words, if you take a graph and replace each edge with a non-edge and each non-edge with an edge, you will get the same graph. Self-complementary graphs have a number of interesting properties and are often used in graph theory to study symmetrical structures and relationships between vertices and edges. Examples of self-complementary graphs include the Petersen graph and the Paley graph.
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How to interpret mathematical graphs?
Mathematical graphs can be interpreted by analyzing the shape, slope, and intersection points of the lines or curves. The x-axis represents one variable, while the y-axis represents another variable. The point where the graph intersects the axes can provide important information, such as the intercepts. Additionally, the overall trend of the graph can indicate relationships between the variables, such as positive or negative correlations.
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