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The first Microsoft application to debut VBA was Microsoft Excel 5.0 in 1993, based on Microsoft Visual Basic 3.0. This spurred the development of numerous custom business applications, and the decision was made to release VBA in a range of products.
Spreadsheet risk is the risk associated with deriving a materially incorrect value from a spreadsheet application that will be utilized in making a related (usually numerically based) decision. Examples include the valuation of an asset, the determination of financial accounts, the calculation of medicinal doses, or the size of a load-bearing ...
Multi-objective optimization or Pareto optimization (also known as multi-objective programming, vector optimization, multicriteria optimization, or multiattribute optimization) is an area of multiple-criteria decision making that is concerned with mathematical optimization problems involving more than one objective function to be optimized simultaneously.
The sounding range [3] refers to the pitches produced by an instrument, while the written range [3] refers to the compass (span) of notes written in the sheet music, where the part is sometimes transposed for convenience. A piccolo, for example, typically has a sounding range one octave higher than its written range. [4]
Just as the definite integral of a positive function of one variable represents the area of the region between the graph of the function and the x-axis, the double integral of a positive function of two variables represents the volume of the region between the surface defined by the function (on the three-dimensional Cartesian plane where z = f(x, y)) and the plane which contains its domain. [1]
Since the graph of an affine(*) function is a line, the graph of a piecewise linear function consists of line segments and rays. The x values (in the above example −3, 0, and 3) where the slope changes are typically called breakpoints, changepoints, threshold values or knots. As in many applications, this function is also continuous.
An example where convolutions of generating functions are useful allows us to solve for a specific closed-form function representing the ordinary generating function for the Catalan numbers, C n. In particular, this sequence has the combinatorial interpretation as being the number of ways to insert parentheses into the product x 0 · x 1 ·⋯ ...
Sometimes "range" refers to the image and sometimes to the codomain. In mathematics, the range of a function may refer to either of two closely related concepts: the codomain of the function, or; the image of the function. In some cases the codomain and the image of a function are the same set; such a function is called surjective or onto.