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Ratio and proportion
Published in John Bird, Bird's Basic Engineering Mathematics, 2021
Two variables, x and y, are in inverse proportion to one another if y is proportional to 1x, i.e. yα1x or y=kx or k=xy where k is a constant, called the coefficient of proportionality.
Cognitive Internet of Things
Published in J P Patra, Gurudatta Verma, Cognitive IoT, 2022
Naive Bayes (NB): NB classifiers are exceedingly scalable, needing numerous parameters linear in the number of variables (features/predictors) in a learning problem. NB is a simple technique for constructing classifiers: models that assign class labels to problem instances, represented as vectors of feature values, where the class labels are drawn from some finite set. It is not a single algorithm for training such classifiers but a family of algorithms based on a common principle: all NB classifiers assume that the value of a particular feature is independent of the value of any other feature, given the class variable. For the mathematical understanding of the NB classifier, we must know the following terms: Conditional Probability: a measure of the probability of event A occurring given that another event has occurred. For example, “what is the probability that it will rain given that it is cloudy?” is an example of conditional probability.Joint Probability: a measure that calculates the likelihood of two or more events occurring at the same time.Proportionality: refers to the relationship between two quantities that are multiplicatively connected to a constant, or in simpler terms, whether their ratio yields a constant.Bayes Theorem: describes the probability of an event (posterior) based on the prior knowledge of conditions that might be related to the event.
Introduction
Published in Roberts Charles, Elementary Differential Equations, 2018
where k is the constant of proportionality. Multiplying equation (1) by dt and dividing by Q, we obtain dQQ=kdt. $$ \begin{aligned} \dfrac{dQ}{Q} = k dt. \end{aligned} $$
Pre-service mathematics teachers’ semiotic transformation of similar triangles: Euclidean geometry
Published in International Journal of Mathematical Education in Science and Technology, 2022
The above dialogue revealed that the participant was unable to visualize the conversion from one semiotic representation to another in a different register. We observe the participant not making sense of the geometric information given about the two triangles; one being an enlargement of the other while the other is a reduction of the other. However, the progress of the interview continued as follows: A: Let us observe the angles of the two triangles to verify if they are equal. Then by illustration through observation, based on geometric rules, P= S, Q = T and R = V. Do you agree that the two triangles are equal?T16: Yes.A: Now what does it mean when the corresponding angles of a triangle are equal and proportional?T16: SilentA: Alright, let us use an example; if the sides of the first triangle are 2, 3, and 4 then if you multiply by a factor of 2, all the three sides will be multiplied by a factor of 2 to get an enlarged triangle, 4, 6 and 8. If you multiply by a factor of 3, then the new triangle could be 6, 9, and 12, which means that if you divide the first triangle by an enlarged triangle you will get the same answer, for instance = ; = and = .A: So, when two triangles are similar, we know that the corresponding sides are proportional.T16: Ok ma.A: Even if we do not know the sizes of the triangles, we can divide we get the same answer as and we get the same answer as.T16: Ok.A: Proportionality means that the ratio of the sides is the same.