Despite the variety of possible foundations for mathematics, in practice real mathematicians do not express their work in terms of a foundation. It seems that most modern mathematicians tend to think in terms that are compatible with ZFC (which is ZF extended with the Axiom of Choice), but in...
comparing their performances from differentmathematical programsin terms of correctness of individual tasks is not particularly revealing unless the reviewers explore the thinking and methods that led students to their correct answers. For example, two students may receive ...
Intriguingly, Knuth's explanation of the working principles in terms of boxes and glue, the boxes being the sorts and the glue variable amounts of space, shows similarity to the physical workings of the Monotype machine, with which Knuth was familiar. The fixed-size of a box dictates a ...
of text, often formatted with a single font. In this situation, actions such as contextual substitution or kerning can be done with access to the complete context of the line of text, and the rules can be expressed in terms of known glyph sequences. Math layout is quite different from ...
In human engineering terms this is the equivalent of creating and powering a factory containing 10 million assembly lines, where half a million of the components created every minute are assembled into 7,000 new assembly lines per minute - with worn out parts recycled where practical. And sustain...
with practical problems and then summarise the mathematical problems through analysis. Teachers should not do this for the students but may give them some hints. In addition, teachers should remind students to make a plan to solve the problem, rather than directly starting to discuss some of its...
We first summarize the theoretical results of our study, whose detailed proofs will be the subject of subsequent sections where the un-defined terms will be clarified. The optimal objective value function of the exact -problem is concave, nondecreasing, and piecewise affine with possibly exponentially...
terms of proliferation, migration or oxygen consumption, since all these processes depend as well on the particular conditions of the surrounding environment, but that all cells respond equally when they are subjected to the same local environmental conditions. We consider with this assumption that ...
grammar — influenced by Montague solely with respect to semantics — a natural language grammar is a finite set of rules which assigns structure to sentences, and it is these syntactic structures to which denotational interpretations are assigned, defined in terms of truth with respect to a model...
… it seems clear that Leibniz had conceived the possibility of elaborating a basic science which would be like mathematics in some respects, but would include also traditional logic and some studies as yet undeveloped. Having noticed that logic, with its terms,propositions, and syllogisms, bore ...