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Introduction to modelling
Published in Karthik Raman, An Introduction to Computational Systems Biology, 2021
Models essentially capture some physical system of interest, say a space shuttle, or a bacterial cell, or a human population where a communicable disease is spreading. Mathematical models capture these systems typically by means of mathematical equations. Models are abstractions of real-world systems; in fact, they are abstractions of (key) parts of a real-world system, capturing features deemed to be essential by the modeller. Modelling is thus a very subjective process, driven by the need to answer specific questions about the real-world system. Models can comprise a bunch of mathematical objects or equations, or even a computer program. Every model is characterised by assumptions on the real-world system, as well as approximations. These assumptions cover: variables (things that change),parameters (things that do not change, or are assumed to not change), andfunctional forms (that connect the variables and parameters),
Mathematical Fundamentals of FEM
Published in Özlem Özgün, Mustafa Kuzuoğlu, ®-based Finite Element Programming in Electromagnetic Modeling, 2018
A mathematical structure, defined on a set, is a mathematical object, such that the set now has an additional “meaning”. The three major mathematical structures are: Algebraic structuresTopological structuresOrder structures
Continuous Tracking: Optimal Control
Published in Richard J. Jagacinski, John M. Flach, Control Theory for Humans, 2018
Richard J. Jagacinski, John M. Flach
Modern control theory uses the analytic power of linear algebra to draw inferences about properties of dynamic systems. Models are constructed in the form of matrix equations and then these models can be manipulated as mathematical objects. The properties of the mathematical objects can then be used to make inferences about the dynamic systems they represent. Two important properties of dynamic systems are controllability and observability. A system is controllable if every state in the state space can be reached from any other state in the space with finite duration control inputs. This is sometimes called complete reachability. If there is a state that cannot be reached from another state (i.e., you cannot get there from here), then the system is not completely controllable. A system is observable if the initial state of the system can be determined by observing its outputs over a finite time interval. In other words, it is possible to use the outputs to trace back to the initial state of the system. If the initial state of the system cannot be determined from observation of the outputs (i.e., the behavior) with known control inputs, then the system is not completely observable. The controllability and observability of a dynamic system can be inferred from properties of the matrices used to represent the system (see Luenberger, 1979, for a more complete discussion of this issue). A controllability matrix will show whether the controls are completely “connected” to the states, and an observability matrix will show whether the outputs are unambiguously “linked” to the states of the system. Although the mathematics of modern control theory provide analytic tests of observability and controllability, Luenberger (1979) observed that “it is rare that in specific practical applications controllability or observability represent cloudy issues that must be laboriously resolved. Usually, the context, or one’s intuitive knowledge of the system, makes it clear whether the controllability and observability conditions are satisfied” (p. 289).
Analysing theories of meaning in mathematics education from the onto-semiotic approach
Published in International Journal of Mathematical Education in Science and Technology, 2022
Juan D. Godino, María Burgos, María M. Gea
Symbols, external material representations and manipulatives, are involved in school and professional mathematical activity and, consequently, they are considered mathematical objects, because they intervene in mathematical practices. The concepts of number, fraction, derivative, etc., are mathematical objects of different nature and function than ostensive representations; they are non-ostensive, mental objects (when they intervene in personal, or individual practices), or institutional objects (when they intervene in shared social-cultural practices). In both cases, they are objects that regulate the mathematical activity, while their ostensive representations support or facilitate the performance of the said activity.
Students’ ability to determine the truth value of mathematical propositions in the context of operation meanings
Published in International Journal of Mathematical Education in Science and Technology, 2022
In the previous studies, it was determined that middle school students posed problems with exercise type as well as the formal meanings of multiplication and division and gave wrong problems that were not suitable for the given operation (Graeber & Johnson, 1991; Kılıç, 2013; Tertemiz, 2017). It can be named operational and wrong problems in the understanding of operations that reflect these problems that students posed. In the operational understanding, students focus only on the exercise aspect of the operations. Mathematical objects are used in their understanding, not examples from real life. For example, when students are asked to establish a real-life problem suitable for the 12 × 3 operation, problems such as ‘What is the result if 12 and 3 are multiplied?’ or ‘What is the result of the 12 × 3 process?’ are formed. It can be said that students with this operation structure have difficulty associating operations with real life. For example, Haylock and Cuckborn (2014) stated that multiplication is perceived as numerical operations in mathematics lessons at school for many children, they can not make link between multiplication and real life. It can be said that students who pose wrong problems do not have a correct understanding of operations. It is seen that the problem situations revived in their minds regarding these operations are not in accordance with the logic of the operations. Some of the students in this understanding confuse the operations with each other. For example, when students are asked to pose a problem suitable for the 12 × 3 operation, the problem such as ‘I have 12 pens. I bought 3 more pens. How many pens do I have now?’ can be posed. The reason for this situation is that students accept a rule or concept that belongs to a particular situation as if it works in other situations (Graeber & Johnson, 1991). In addition, some of the students better understand the concepts such as ‘more’ (Glendon et al., 1990) and prefer operations based on these key words in word problems (Varol & Kubanç, 2015). For this reason, students can make preferences for different operations rather than the desired operation. Operational and wrong understandings may reflect some students’ understanding of operations, even though they are not appropriate. Therefore, these two insights were taken into consideration in current study.