https://tue.osiris-student.nl/onderwijscatalogus/extern/cursus?cursuscode=2MBA80&collegejaar=2025&taal=enThis course is about “transformations” in applied mathematics. Transformations are, broadly speaking, tools that help us (i) unveil hidden patterns and (ii) exploit symmetries within a mathematical problem. They offer a shift in perspective that can help illuminate solutions to problems that may appear difficult (or cumbersome) to solve at a first glance. Generating functions, for instance, transform sequences into functions. By studying and/or manipulating generating functions, we gain insights into structures within the original sequences. They are, effectively, a bridge between discrete mathematics and continuous analysis. Integral transforms, similarly, transform functions from one domain into functions from another domain. Each allows us to examine the original function using a different perspective; and each is suited to exploit certain techniques from different domains such as algebra, analysis, and probability theory. Examples include the Fourier transform, which can e.g. decompose a function into its constituent frequencies; and the Laplace transform, which can e.g. transform continuity relations into algebraic equations. By completing this course, you will become familiar with overarching concepts behind generating functions and integral transforms; their formal properties (such as linearity, time shifting, frequency shifting, et cetera); their most important theorems (such as convolution, injectivity, uniqueness, et cetera); and their explicit uses in different domains of applied mathematics. You will also have developed skills in circumventing generally difficult computations by passing to a transform instead and exploiting their formal properties together with their injectivity.
This course teaches:
- General understanding of the role of transforms in mathematics: The student should understand the importance and application of transforms in mathematics, including their ability to simplify complex properties and their injective nature.
- Familiarity with Generating Functions and Integral Transforms: The student should become well familiar with the concepts of generating functions and integral transforms, and be able to classify and relate different representations.
- Exact knowledge of formal properties of, and theorems on, transforms: The student should be able to recite formal properties and important theorems of generating functions and integral transforms.
- Problem-solving methods that rely on transforms: The student should demonstrate an ability to simplify complex computations by using transforms and exploiting their formal properties such as injectivity.
- An impression of typical applications to which transforms are applied: The student should be able to understand and/or analyze solution techniques that rely on generating functions and/or integral transforms when applied to an application within applied mathematics.
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Written examination