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Functions
Published in Rowan Garnier, John Taylor, Discrete Mathematics, 2020
You are almost certainly familiar with the notion of the graph of a function f: ℝ → ℝ. This is the curve drawn in the plane ℝ2 = ℝ x ℝ consisting of all the points (x, y) such that y = f(x). However, according to definition 5.1, this set of points is the function f itself. In other words the graph of f: ℝ → ℝ is just a pictorial representation of the set f. From our point of view, there is little distinction to be made between the function itself and its graph.
A model for analyzing the explanatory writing of undergraduate students when solving mathematical tasks
Published in International Journal of Mathematical Education in Science and Technology, 2023
Giovannina Albano, Osama Swidan, Anna Pierri
Masino implicitly uses several theories to solve the given task. At the beginning of the excerpt, he uses the technology ‘Looking at the graph of the function, we can get some information about … ’ for justifying the drawing of the parabola by referring implicitly to the theory that connects the derivative sign with the inclination, i.e. if the derivative is positive at a point, then the function increases at that point. By using this theory, Masino implicitly applies the technique of drawing the derivative function graph. Then, he applies another technique to solve systems of equations to find the coefficient of the quadratic function; the use of theory and technology is implicit. As the two equations are not enough to find the three coefficients that Masino looked for (‘The information known at this point is not sufficient to estimate the value of c’), he refers to another theory, concerning the connection between the derivative and the family of functions corresponding to that derivative. Thus, he applies the technique of computing the indefinite integral without explicitly mentioning the technology (i.e. the relation between the graph of the function and the graph of the derivative, which was the starting point). Finally, he seems to substitute two values to identify the correct function among the family of the primitives. Another possible interpretation is that Masino directly substitutes the value of c in the derivative expression he found ‘’.
Spectral graph wavelet regularization and adaptive wavelet for the backward heat conduction problem
Published in Inverse Problems in Science and Engineering, 2021
A non-normalized Laplacian is defined as . For any function , can be written as where the summation is done over all vertices k that are connected to the vertex m. Consider the graph G arising from a regular mesh defined by taking vertices as points on a regular one-dimensional grid, with each points connected to its two neighbours with weight (where h is the distance between adjacent grid points). The graph Laplacian to function defined on the vertices is which corresponds to standard 3-point stencil approximation of the continuous Beltrami operator with a difference in sign.
Minimizing maximum job dependent ergonomic risk
Published in International Journal of Production Research, 2021
Sergey Kovalev, Isabelle Chalamon, Stephane E. Collignon
Let us now consider the case, in which there are two workers and two jobs, that is, n=2. For this case, if the sequence of jobs for worker 1 over periods is , , , then the sequence of jobs for worker 2 must be . Denote by x the number of periods in which worker 1 executes job 1, . Then the ergonomic risk of worker 1 is equal to , the ergonomic risk of worker 2 is equal to and the maximum ergonomic risk is equal to Graph of the function for and is shown in Figure 1.