Schaum 39;s Outline Introduction To Mathematical Economics Pdf Free Download

[Melissa Russian]

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This module is aimed at students who are considering a Masters and/or PhD in Economics. The module will cover a range of relevant mathematical tools and techniques that are typically required for further study in economics; the aim is to deepen and extend the mathematical preparation of 3rd year undergraduates by exposing them to rigorous higher level mathematics and providing them with the opportunity to develop proofs and apply new mathematical tools. Knowledge of elementary matrix theory and calculus is assumed.

The level of rigor will vary. Parts 1 and 2 will aim for thoroughness rather than for covering a huge range of material. Students should develop a feel for when a proof is complete and rigorous and when arguments are missing. In Part 3, emphasis is placed on learning how to quickly understand mathematical tools and be able to apply them, rather than on theoretical formalism. This is achieved sometimes at the expense of rigor. Where proofs are beyond the level of this course, references are given.

Employability

By solving statistical mathematical problems and exercises, students are equipped with practical problem-solving skills, theoretical skills, and an understanding of mathematical relationships. All of these are highly valuable to employers.

The assessment structure on this module is subject to review and may change before the start of the new academic year. Any changes will be clearly communicated to you before the start of the term and if you wish to change module as a result of this you can do so in the module change window.

In Part 1 we will provide the basics for the following material. The important properties of numbers and how they are described axiomatically (in particular, their order structure and completeness) will be discussed. Central notions of set theory will be developed and illustrated. Important methods of proofs (indirect proof, inductive proof) are illustrated in examples.

In Part 2 we will give a rigorous introduction to basic topological concepts (limit points, neighbourhoods, compact spaces, metric spaces etc.) and provide many examples of topological spaces. The emphasis is on teaching how to do rigorous proofs using abstract concepts.

In Part 3 we will introduce the most elementary notions of first and second differential equations. We will subsequently study systems of linear differential equations, their solutions in the various cases, stability conditions, phase diagrams and some economic applications. For this part only elementary matrix theory and basic calculus are needed.

A key aspect of mathematics is its ability to unify and generalise disparate situations exhibiting similar properties by developing the concepts and language to describe the common features abstractly and reason about them rigorously. In this module, you will be introduced to the language of logic, sets, and functions which underpins of all modern pure mathematics, and will learn how to use it to construct clear and logically correct mathematical proofs. The content goes beyond mathematics taught at A-level: you will learn and use methods to prove rigorous general results about the convergence of sequences and series, justifying the techniques developed in MTH1002 and laying the foundations for a deeper study of Analysis in MTH2008. You will also learn the definitions and properties of abstract algebraic structures such as groups and vector spaces. These ideas are developed further in MTH2010 and MTH2011. The material in this module is fundamental to many other modules in the mathematics degree programmes. It underpins the topics you will see in more advanced modules in fundemental mathematics and enables a deeper understanding and rigorous justification of the mathematical tools you will meet in more applied mathematics modules and which are widely used in physics, economics, and many other disciplines.

The purpose of this module is to provide you with an introduction to axiomatic reasoning in mathematics, particularly in relation to the perspective adopted by modern algebra and analysis. The building blocks of mathematics will be developed, from logic, sets and functions through to proving key properties of the standard number systems. We will introduce and explore the abstract definition of a group, and rigorously prove standard results in the theory of groups, before progressing to consider vector spaces, both in the abstract and with a specific focus on finite-dimensional vector spaces over the real numbers. The ideas and techniques of this module are essential to the further development of these themes in the two second-year streams Analysis and Algebra, and subsequent fundamental mathematics modules in years 3 and 4.

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