Does it matter?

Shuard, Hilary

doi:10.2307/3616802

Cited by 6 publications

(6 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In fact, as explained in [4], a codomain refers to the set Y in a triplet \langle f, X, Y \rangle , which reflects a different function concept altogether. In calculus, codomains cause problems as noted in [15]. Also, for functions with codomains, g \circ f is defined only if the domain of g equals the codomain of f , which would be severely limiting in calculus.…”

Section: Termmentioning

confidence: 99%

A Precise and Reliable Multivariable Chain Rule

Boute¹

2021

SIAM Rev.

View full text Add to dashboard Cite

The multivariable chain rule is often challenging to students because it is usually presented with ambiguities and other defects that hamper systematic and reliable application. A very simple formulation combines the derivation operators for functions and for expressions in a manner not found elsewhere due to common confusion between them. Some issues are rooted more deeply than others and are discussed in a broader perspective, starting with the function concept. The approach is illustrated using various applications including the transport equation, partial derivatives of a definite integral, and the distortionless (but not lossless) transmission line. This note is suitable for a lecture in any first-year course covering partial derivatives, as a complement to the other course material.

show abstract

Section: Termmentioning

confidence: 99%

A Precise and Reliable Multivariable Chain Rule

Boute¹

2021

SIAM Rev.

View full text Add to dashboard Cite

show abstract

“…This forces defining relation backwards, as a relation from X to Y for some X, Y . Also, using Cartesian products at this stage may be an educational burden [54].…”

Section: Definition 1 (Relation)mentioning

confidence: 99%

“…Also, using Cartesian products at this stage may be an educational burden [54]. A more serious problem is that disregarding separation of concerns carries a heavy price in understanding, apparently even for the authors.…”

Section: Definition 2 (Domain Range)mentioning

confidence: 99%

“…Myth #0 (a meta-myth actually) holds that divergences in definitions just reflect different needs in various disciplines [54]. However, our samples come from algebra, analysis/calculus, discrete math, logic, set theory, and reveal that nearly all of them use the same function concept, differing only in the care devoted to design and formulation.…”

Section: Case Study B -Functionsmentioning

confidence: 99%

“…Shuard [54] published perhaps the only paper evaluating the function-withcodomain variant. Her single (!)…”

Section: Design Considerations Variant Concepts and Evaluationmentioning

confidence: 99%

See 2 more Smart Citations

Why mathematics needs engineering

Boute

2016

Journal of Logical and Algebraic Methods in Programming

View full text Add to dashboard Cite

Engineering needs mathematics, but the converse is also increasingly evident. Indeed, mathematics is still recovering from the drawbacks of several "reforms". Encouraging is the revived interest in proofs indicated by various recent introduction to proof-type textbooks. Yet, many of these texts defeat their own purpose by self-conflicting definitions. Most affected are fundamental concepts such as relations and functions, despite flawless accounts 50 years ago. We take the viewpoint that definitions and theorems are tools for capturing, analyzing and understanding mathematical concepts and hence, like any tools, require diligent engineering. This is illustrated for relations and functions, their algebraic properties and their relation to category theory, with the Halmos principle for definitions and the Arnold principle for axiomatization as design guidelines.

show abstract

Pointfree expression and calculation: from quantification to temporal logic

Boute

2010

Form Methods Syst Des

View full text Add to dashboard Cite

Pointfree formulation means suppressing domain variables to focus on higherlevel objects (functions, relations). Advantages are algebraic-style calculation and abstraction as formal logics pursue by axiomatization. Various specific uses are considered, starting with quantification in the wider sense (∀, ∃, , etc.). Pointfree style is achieved by suitable functionals that prove superior to pointwise conventions such as the Eindhoven notation. Pointwise calculations from the literature are reworked in pointfree form. The second use considered is in describing systems, with generic functionals capturing signal flow patterns. An illustration is the mathematics behind a neat magician's trick, whose implementation allows comparing the pointfree style in Funmath, LabVIEW, TLA + , Haskell and Maple. The third use is making temporal logic calculational, with a simple generic Functional Temporal Calculus (FTC) for unification. Specific temporal logics are then captured via endosemantic functions. The example is TLA + . Calculation is illustrated by deriving various theorems, most related to liveness issues, and discovering results by calculation rather than proving them afterwards. To conclude, various ramifications, style and abstraction issues are discussed, in relation to engineering mathematics in general and to categorical formulations.

show abstract

Does it matter?

Cited by 6 publications

References 2 publications

A Precise and Reliable Multivariable Chain Rule

A Precise and Reliable Multivariable Chain Rule

Why mathematics needs engineering

Pointfree expression and calculation: from quantification to temporal logic

Contact Info

Product

Resources

About