During the 2016 U.S. presidential election season, the voting behavior of working-class Americans received a great deal of media attention. Reporters gave us vivid examples of working-class Trump supporters at campaign rallies, with many attributing working-class support for Trump to racism and other problematic attitudes (Smith et al., 2019: 192).

On September 1, 1967, Rev. Martin Luther King, Jr. addressed the annual convention of the American Psychological Association (APA) for the first and only time. Before his talk could be published a few months later, King was assassinated in Memphis, Tennessee, where he had gone to support striking sanitation workers. King began his APA address by noting how rewarding it was for him to take a break from the daily demands of the struggle in which he was engaged, and to speak about that struggle with “concerned friends of good will” (King, 1968: 1). He noted, however, how rarely psychologists and other behavioral scientists had used their knowledge and position to illuminate the terrible realities of the segregated South or to study issues of pressing concern to the civil rights movement. He urged us to “tell it like it is” (King, 1968: 2) and to seek answers to these urgent questions.

This book is designed to serve as a textbook for thefirst course on partial differential equations.After the introduction of the semester system invarious universities of India, normally there aretwo courses of differential equations atundergraduate levels, one is on ordinarydifferential equations (ODEs), whereas the other isexclusively on partial differential equations(PDEs). Generally, most of the books used at theundergraduate level include ODEs as well as PDEs,and all such books emphasise more on ODEs, which,consequently, affect the coverage of the PDEs. Onthe other hand, all the standard textbooks on PDEscover the contents of graduate as well aspostgraduate levels, wherein often undergraduatematerials are discussed in haste. Overall, it isnecessary to write a separate book exclusively forundergraduate students of Indian as well as foreignuniversities. With a view to have a detaileddiscussion on elementary topics of PDEs, weendeavour to write this book.

In fact, this book is based on our lecture notes, whichwe have given to our students at Aligarh MuslimUniversity (AMU) over the last several years. Wehope that this book will meet the requirement aswell as expectations of undergraduate students.While preparing this book, we have examined deeplythe syllabi of all such courses of B.Sc.(Mathematics) of all Indian universities. The coursecontents covered by this book can be described asthe union of all syllabi prescribed by variousIndian universities as well as UGC curriculum up toundergraduate level. The course contents arepresented in such a manner that they can be equallyuseful in various competitive examinations such asUPSC, UGC-CSIR NET, and GATE. On the other hand,from the application point of view, the book alsocontains the relevant contents of MathematicalPhysics/Engineering Mathematics/Applied Mathematics,which are the parts of course contents in B.Sc.(Physics) as well as B.E./B.Tech. We attempt tostrike a balance between theory and problems.Consequently, our book remains equally useful toboth pure as well as applied mathematicians. We havemade every attempt to have a simpler and lucidpresentation without scarifying theoretical rigour.The book provides different types of examples,updated references, and applications in diversefields. All methods of solution and necessaryconcepts are arranged in the form of theorems. Theproofs of theorems may be omitted for anundergraduate course or a course in otherdisciplines.

The previous two chapters have focused on the problem of graph partitioning, which has seen enormous interest and research work in recent years. We continue that aspect of network analysis by introducing the notion of spectral clustering. The main tool of this chapter is the graph Laplacian, which can be unnormalized or normalized. Also discussed is a regularized version of the adjacency matrix.

In California, a 9-year-old boy anonymously used $74.50 in allowance money that he had saved to pay off his third-grade classmates’ school lunch debt (Pitofsky, 2019). In Pennsylvania, a cafeteria worker quit her job after being required to throw away a first grader’s lunch when he did not have the money to pay for it (O’Connor & Humphries, 2019). What prompted these caring actions? Both were responses to lunch shaming, a term used to describe practices and policies that single out and embarrass students whose families cannot pay their school lunch bills.

In recent years, the science of “networks” has become a very popular research topic and a growth area in many different disciplines. Two journals, Social Networks (first published in 1978) and Network Science (first published in 2013), have appeared that focus on network theory and applications. In its inaugural issue, the journal Network Science defined network science as the “study of the collection, management, analysis, interpretation, and presentation of relational data,” and noted that the more one learns about networks, the more one sees networks everywhere.

In this chapter, we recognize that the configurations of almost all networks vary with time. We define dynamic networks, which can be observed in discrete or continuous time. Discrete-time dynamic networks can be visualized as a sequence of snapshots of the network taken at different points in time. Continuous-time dynamic networks are more complicated, both visually and theoretically, and assume that edges can appear and disappear continuously through time. We discuss the idea of dynamic community discovery in which community detection strategies are applied to dynamic networks.