In this chapter, we extend our study of linear algebraic structures to multilinear ones that have broad and profound applications beyond those covered by linear structures. First, we give some remarks on the rich applications of multilinear algebra and consider multilinear forms in a general setting as a starting point that directly generates bilinear forms already studied. Next, we specialize our discussion to consider tensors and their classifications. Then, we elaborate on symmetric and antisymmetric tensors and investigate their properties and characterizations. Finally, we discuss exterior algebras and the Hodge dual correspondence.

Readers will understand the physical laws that form the basis of the fluid equations of motion, and will learn how to obtain the equations of fluid motion in both derivative and integral form. Presentations are included to show how to apply the equations of motion to calculate properties of fluid flows. Readers will understand dynamic similarity and how to calculate Mach number and Reynolds number, including descriptions of the various Mach and Reynolds number regimes and their distinguishing characteristics.

In this chapter, we extend our study on real quadratic forms and self-adjoint mappings to the complex situation. We begin by a discussion on the complex version of bilinear forms and the Hermitian structures. We will relate the Hermitian structure of a bilinear form with representing it by a unique self-adjoint mapping. Then we establish the main spectrum theorem for self-adjoint mappings. Next we focus again on the positive definiteness of self-adjoint mappings. We explore the commutativity of self-adjoint mappings and apply it to obtain the main spectrum theorem for normal mappings. We also show how to use self-adjoint mappings to study a mapping between two spaces.

Technology has become central to both the personal and social aspects of our lives. In the classroom, digital literacy is the pupils’ ability to discern quality sources and evaluate the appropriateness of online content as it relates to the task or activity they are undertaking, while respecting the intellectual property rights of the content owners. The chapter discusses online safety and the use of social media in a considerate and respectful manner, and examines what these issues mean for the student teacher in a modern foreign languages classroom. In addition, it looks at the benefits of technology in modern foreign languages learning and teaching, and highlights important caveats and common pitfalls.

This chapter highlights the knowledge required to work with diverse students who communicate using the different varieties of English that exist in Australia. In line with the ‘Language variation and change’ sub-strand of the Australian Curriculum: English, we discuss linguistic and cultural diversity through the concept of plurilingualism, and the transcultural and sociolinguistic competence and knowledge required by teachers working with culturally and linguistically diverse (CALD) learners. We highlight the challenges and rewards associated with instructing students from varying linguistic and cultural backgrounds. We also stress the crucial role teachers play in nurturing learners of English as an additional language or dialect (EAL/D) students.

Understanding the causes of intrastate armed conflict and civil wars – whether as individual cases or in a more general sense – is the most compelling but perhaps also the most elusive challenge in the study of such conflicts. In this field, causal relationships are complex and difficult to establish beyond doubt, and discrete direct causes rarely exist. This chapter explores the methodological challenges that arise when seeking to identify direct or indirect causes of civil wars, in particular across multiple cases. It presents key theories of civil war onset in relation to political, economic, social, institutional, ecological, identity, and governance conditions. It gives particular attention to “greed” and “grievance” as key concepts for understanding why intrastate armed conflicts occur, the association between democratization and increased risk of violent conflict, and the concept of “ethnic conflict” as a cause of civil war. The question of whether it is more helpful to focus on enabling factors – the conditions that allow violent uprisings to occur – or motivations for participating in armed conflict to understand the causes of civil war is also discussed. The chapter concludes by considering the implications these debates raise for policies designed to prevent violent conflict and build sustainable peace.

Spoken language consists of a complex system of sounds that infants first learn to perceive and produce through social interaction in their linguistic community. This initial exposure before school is critical in preparing them for school, where they learn to encode their spoken language into written language. Once children can sufficiently speak and understand most language at around the age of five, it becomes the role of educators, and parents/care providers to assist them in the development of their explicit phonological awareness. Phonological awareness, which refers to the broad understanding of the sound structure of language, plays a critical role in the development of reading and writing. For instance, before children can effectively start learning to read, they need to develop an awareness of the sound structure of words, including syllables, rhymes and individual speech sounds called phonemes, as well as the ability to segment and blend phonemes.

In this chapter, we present two important and related problems in data analysis: the low-rank approximation and principal component analysis (PCA), both based on singular value decomposition. First, we consider the low-rank approximation problem for mappings between two vector spaces. Next we specialize on the low-rank approximation problem for matrices in both induced norm and the Frobenius norm, which are of independent interest for applications. Then we consider PCA. These results are also useful in machine learning. Furthermore, as an extension of the ideas and methods, we present a study of some related matrix nearness problems.

Consolidate what you learned in Chapters 7–10; revisit the ideas you came across in Chapters 7–10 in a cross-cultural context.

The concept of circulation is presented, including the physical and mathematical concepts of circulation and lift. A description of how potential flow theory is used to model flow for airfoils, including the predictions of lift. Readers are presented with the concept of the Kutta condition, including how it impacts the development of airfoil theory. Thin-airfoil theory is developed for symmetric and cambered airfoils and methods for prediction lift and pitching moment are presented. The accuracy and limitations of thin-airfoil theory is also presented. Descriptions are presented for why laminar flow airfoils have different geometries than airfoils used at higher Reynolds numbers. Finally, high-lift systems are discussed, including why they are important for aircraft design.