*David R. Steward*

- Published in print:
- 2020
- Published Online:
- November 2020
- ISBN:
- 9780198856788
- eISBN:
- 9780191890031
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198856788.003.0005
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

Solutions to interface problems may be developed using analytic elements with mathematical solutions to the Laplace equation developed by singular integral equations. This formulation leads to ...
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Solutions to interface problems may be developed using analytic elements with mathematical solutions to the Laplace equation developed by singular integral equations. This formulation leads to solutions with discontinuities occurring across line segments, where the potential or stream function is discontinuous across double layer elements in Section 5.2, and the normal or tangential component of the vector field is discontinuous across single layer elements in Section 5.3. Examples illustrate a broad range of solutions to interface conditions possible with these elements. Series expansions are used to represent the far-field at larger distances from elements in Section 5.4, which leads to higher-order elements with nearly exact solutions and also provides a simpler representation for contiguous strings of adjacent elements. Such strings of elements are used with polygon elements in 5.5 to solve conditions along the interfaces of heterogeneities, and to provide a common series expansion to represent the far-field for a group of neighboring elements. Methods are extended to analytic elements with curvilinear geometry using conformal mappings (Section 5.6) and to three-dimensional fields in Section 5.7.Less

Solutions to interface problems may be developed using analytic elements with mathematical solutions to the Laplace equation developed by singular integral equations. This formulation leads to solutions with discontinuities occurring across line segments, where the potential or stream function is discontinuous across double layer elements in Section 5.2, and the normal or tangential component of the vector field is discontinuous across single layer elements in Section 5.3. Examples illustrate a broad range of solutions to interface conditions possible with these elements. Series expansions are used to represent the far-field at larger distances from elements in Section 5.4, which leads to higher-order elements with nearly exact solutions and also provides a simpler representation for contiguous strings of adjacent elements. Such strings of elements are used with polygon elements in 5.5 to solve conditions along the interfaces of heterogeneities, and to provide a common series expansion to represent the far-field for a group of neighboring elements. Methods are extended to analytic elements with curvilinear geometry using conformal mappings (Section 5.6) and to three-dimensional fields in Section 5.7.

*Ali Taheri*

- Published in print:
- 2015
- Published Online:
- September 2015
- ISBN:
- 9780198733157
- eISBN:
- 9780191797729
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198733157.003.0019
- Subject:
- Mathematics, Analysis

This chapter includes a thorough and detailed discussion of existence theorems for the four basic boundary-value problems associated with the Laplace operator, i.e., the interior and exterior ...
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This chapter includes a thorough and detailed discussion of existence theorems for the four basic boundary-value problems associated with the Laplace operator, i.e., the interior and exterior Dirichlet problems and the interior and exterior Neumann problems. The theory of integral equations and compact operators play a pivotal role.Less

This chapter includes a thorough and detailed discussion of existence theorems for the four basic boundary-value problems associated with the Laplace operator, i.e., the interior and exterior Dirichlet problems and the interior and exterior Neumann problems. The theory of integral equations and compact operators play a pivotal role.

*Yujie Chen*

- Published in print:
- 2015
- Published Online:
- March 2016
- ISBN:
- 9780198723790
- eISBN:
- 9780191791130
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198723790.003.0004
- Subject:
- Linguistics, Language Families, Syntax and Morphology

This chapter explores the common conceptual categories found in the semantic differentiation of demonstratives based on data from more than 300 Sinitic languages. While two-term systems are very ...
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This chapter explores the common conceptual categories found in the semantic differentiation of demonstratives based on data from more than 300 Sinitic languages. While two-term systems are very common in Sinitic languages, there are also one-term, three-term, four-term, even five-term systems, aligned on the scale of distance and visibility. Notably, neutral demonstratives form a one-term system which can co-occur with other types of systems in the same language. Systems greater than three-term may be either single-layered or multi-layered, which is decided by semantic properties other than distance, such as visibility. Special morpho-phonological processes, such as lengthening, reduplication, and tone change, can be used for systems greater than the three-term type in Sinitic languages to code this new kind of differentiation. As the result of the investigation, distance is found to be the main parameter encoded by demonstrative systems in Sinitic languages.Less

This chapter explores the common conceptual categories found in the semantic differentiation of demonstratives based on data from more than 300 Sinitic languages. While two-term systems are very common in Sinitic languages, there are also one-term, three-term, four-term, even five-term systems, aligned on the scale of distance and visibility. Notably, neutral demonstratives form a one-term system which can co-occur with other types of systems in the same language. Systems greater than three-term may be either single-layered or multi-layered, which is decided by semantic properties other than distance, such as visibility. Special morpho-phonological processes, such as lengthening, reduplication, and tone change, can be used for systems greater than the three-term type in Sinitic languages to code this new kind of differentiation. As the result of the investigation, distance is found to be the main parameter encoded by demonstrative systems in Sinitic languages.

*Ginestra Bianconi*

- Published in print:
- 2018
- Published Online:
- July 2018
- ISBN:
- 9780198753919
- eISBN:
- 9780191815676
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198753919.003.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

Chapter 1 constitutes Part I of the book: ‘Single and Multilayer Networks’. This chapter introduces multilayer networks as an important new development of Network Science that allows a more ...
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Chapter 1 constitutes Part I of the book: ‘Single and Multilayer Networks’. This chapter introduces multilayer networks as an important new development of Network Science that allows a more comprehensive understanding of Complex Systems. It identifies the main motivations driving the research activity in this field of multilayer networks and emphasizes the benefits of taking a multilayer network perpective to characterize network data. The main advantages of a multilayer network approach with respect to the more traditional single layer characterization of complex networks are broadly discussed, focusing on the information gain resulting from the analysis of multilayer networks, the non-reducibility of a multilayer network to a large single network and the rich interplay between structure and function in multilayer networks.Less

Chapter 1 constitutes Part I of the book: ‘Single and Multilayer Networks’. This chapter introduces multilayer networks as an important new development of Network Science that allows a more comprehensive understanding of Complex Systems. It identifies the main motivations driving the research activity in this field of multilayer networks and emphasizes the benefits of taking a multilayer network perpective to characterize network data. The main advantages of a multilayer network approach with respect to the more traditional single layer characterization of complex networks are broadly discussed, focusing on the information gain resulting from the analysis of multilayer networks, the non-reducibility of a multilayer network to a large single network and the rich interplay between structure and function in multilayer networks.