Introduction to Mathematical Modelling
A mathematical model is a mathematical representation of a system used to make predictions and provide insight about a real-world scenario, and mathematical modelling is the process of constructing, simulating and evaluating mathematical models.
Why do we construct mathematical models? It can often be costly (or impossible!) to conduct experiments to study a real-world problem and so a mathematical model is a way to describe the behaviour of a system and predict outcomes using mathematical equations and computer simulations .
Check out the following resources to get started with mathematical modelling:
Chapter 1: What is Mathematical Modelling? in Principles of Mathematical Modeling
What is Math Modeling?
Wikipedia: Mathematical Model
Mathematical modelling involves observing some real-world phenomenon and formulating a mathematical representation of the system. But how do we even know where to start? Or how to find a solution? The modelling process is a systematic approach:
Clearly state the problem
Identify variables and parameters
Make assumptions and identify constraints
Build solutions
Analyze and assess
Report the results
Models can have a wide range of complexity ! More complex does not necessarily mean better and we can sometimes work with more simplistic models to achieve good results. In many instances, we often start with a simple model and then build-up the complexity by iterating through the steps in modelling process until the model accurately describes the real-world application.
Check out Math Modeling: Getting Started and Getting Solutions to read more about the modelling process.
There are many different types of mathematical models! In this course we focus on the following:
Deterministic models predict future based on current information and do not include randomness. These kinds of models often take the from of systems of differential equations which describe the evolution of a system over time.
Stochastic models include randomness and are based on probability distributions and stochastic processes .
Data-driven models look for patterns in observed data to predict the output of a system. These kinds of models often take the form of functions with parameters computed to fit observed data.
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Representation (mathematics) In mathematics, a representation is a very general relationship that expresses similarities (or equivalences) between mathematical objects or structures. Roughly speaking, a collection Y of mathematical objects may be said to represent another collection X of objects, provided that the properties and relationships ...
For example, in mathematics a group representation is a precisely defined notion: a homomorphism from a given, abstract mathematical group to a group of linear operators acting on a vector space. However the mathematical representations that occur in educational contexts, even when conventional, are extremely varied.
Very roughly speaking, representation theory studies symmetry in linear spaces. It is a beautiful mathematical subject which has many applications, ranging from number theory and combinatorics to ge-ometry, probability theory, quantum mechanics, and quantum eld theory. Representation theory was born in 1896 in the work of the Ger-
mathematical subject which has many applications, ranging from number theory and combinatorics to geometry, probability theory, quantum mechanics and quantum field theory. Representation theory was born in 1896 in the work of the German mathematician F. G. Frobenius. This work was triggered by a letter to Frobenius by R. Dedekind.
Representations are considered to be mathematically conventional, or standard, when they are based on assumptions and conventions shared by the wider mathematical community.Examples of such conventional mathematical representations include configurations of base ten numerals, abaci, number lines, Cartesian graphs, and algebraic equations written using standard notation.
Today, representation theory is a central tool in many mathematical fields: algebra, topology, geometry, mathematical physics and number theory — including the sweeping Langlands program. "This philosophy of representation theory has gone on to gobble vast tracts of mathematics in the second half of the 20th century," Williamson told me ...
Representation theory studies how algebraic structures "act" on objects. A simple example is how the symmetries of regular polygons, consisting of reflections and rotations, transform the polygon. Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations ...
Definition 1.9. A representation of an algebra A (also called a left A-module) is a vector space. V together with a homomorphism of algebras δ : A ⊃ EndV . Similarly, a right A-module is a space V equipped with an antihomomorphism δ : A ⊃ EndV ; i.e., δ satisfies δ(ab) = δ(b)δ(a) and δ(1) = 1.
Representation theory is fundamental in the study of objects with symmetry. It arises in contexts as diverse as card shuffling and quantum mechanics. An early success was the work of Schur and Weyl, who computed the representation theory of the symmetric and unitary groups; the answer is closely related to the classical theory of symmetric functions and deeper study leads to intricate ...
Representation Theory. Published by the American Mathematical Society since 1997, this electronic-only journal is devoted to research in representation theory and seeks to maintain a high standard for exposition as well as for mathematical content. All articles are freely available to all readers and with no publishing fees for authors.
This article explores the role and importance of representation in mathematics education, especially in geometry. It discusses the four modes of representation (verbal, graphic, algebraic, and numeric) and the translation of representation among them.
Learn how to plan and use various representations (e.g. manipulatives, drawings, equations) to enhance students' mathematical understanding and problem solving. Find out the benefits, challenges, and assessment strategies of using representations in math instruction.
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Many said that mathematical representation serves the bridge role for people to understand and express mathematical ideas. Representation consists of internal and external representation. However, representation term that researchers used mostly refers to only external part. Real-world problems can be represented using formula, visual, concrete ...
Very roughly speaking, representation theory studies symmetry in linear spaces. It is a beautiful mathematical subject which has many applications, ranging from number theory and combinatorics to ge-ometry, probability theory, quantum mechanics, and quantum eld theory. Representation theory was born in 1896 in the work of the Ger-
Mathematical representation ability is an essential skill for students to understand mathematical concepts. Many studies have been conducted regarding this ability, but it is necessary to map ...
What is a Mathematical Model? #. A mathematical model is a mathematical representation of a system used to make predictions and provide insight about a real-world scenario, and mathematical modelling is the process of constructing, simulating and evaluating mathematical models. Why do we construct mathematical models?
17 As most commonly interpreted in education, 18 mathematical representations are visible or tangi-. 19 ble productions -such as diagrams, number lines, 20 graphs, arrangements of concrete ...
A representation created by a student is a window into their mind. For a teacher, a student's representation is an invitation to learn about how that student is reasoning about a problem they are working to solve. If a teacher values students' thinking, they need to consider how to make it possible for all students to represent that thinking.
Use and Connect Mathematical Representations. Effective teaching of mathematics engages students in making connections among mathematical representations to deepen understanding of mathematics concepts and procedures and as tools for problem solving. Physical: Use concrete or gestures to show, act upon, or manipulate mathematical ideas (e.g ...
among mathematical representations can be leveraged - specifically in geometry instruction - will also be discussed in this paper. Some effective uses of mathematical representations include connecting instruction with students' experiences and interests (NCTM, 2018). Teaching geometry is crucial in facilitating student
Designing ways to elicit and assess students' abilities to use representations meaningfully to solve problems. Using multiple forms of representations to make sense of and understand mathematics. Describing and justifying their mathematical understanding and reasoning with drawings, diagrams, and other representations.
Mathematical Representations. Establish mathematics goals to focus learning. Implement tasks that promote reasoning and problem solving. Use and connect mathematical representations. Facilitate meaningful mathematical discourse. Pose purposeful questions. Build procedural fluency from conceptual understanding.
In our tests, the next model update performs similarly to PhD students on challenging benchmark tasks in physics, chemistry, and biology. We also found that it excels in math and coding. In a qualifying exam for the International Mathematics Olympiad (IMO), GPT-4o correctly solved only 13% of problems, while the reasoning model scored 83%.