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home [2023-03-17] – [Mission] Martin Ziegler | home [2023-03-18] (current) – [Mission] Martin Ziegler | ||
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===== Motivation ===== | ===== Motivation ===== | ||
- | Digital Computers naturally process discrete data, such as bits or integers or strings or graphs. From bits to advanced data structures, from a first semi-conducting transistor to billions in wafer-scale integration, | + | Digital Computers naturally process discrete data, such as bits or integers or strings or graphs. From bits to advanced data structures, from a first semi-conducting transistor to billions in wafer-scale integration, |
the success story of digital computing arguably is due to (1) hierarchical layers of abstraction and (2) the ultimate reliability of each layer for the next one to build on ― for processing discrete data. | the success story of digital computing arguably is due to (1) hierarchical layers of abstraction and (2) the ultimate reliability of each layer for the next one to build on ― for processing discrete data. | ||
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Deviations between mathematical structures and their hardware counterparts are common also in the discrete realm, | Deviations between mathematical structures and their hardware counterparts are common also in the discrete realm, | ||
- | such as the “integer” wraparound 255+1=0 occurring in bytes that led to the ”Nuclear Gandhi'' | + | such as the “integer” wraparound 255+1=0 occurring in bytes that led to the // |
- | Similarly, deviations between exact and approximate continuous data underlie infamous failures such as the Ariane 501 flight V88 or the Sleipner-A oil platform. | + | Similarly, deviations between exact and approximate continuous data underlie infamous failures such as the |
- | Nowadays high-level programming languages (such as Java or Python) provide a user data type (called for example BigInt or mpz_t) that fully agrees with mathematical integers, simulated in software using a variable number of hardware bytes. | + | Nowadays high-level programming languages (such as Java or Python) provide a user data type (called for example |
This additional layer of abstraction provides the reliability for advanced discrete data types (such as weighted or labelled graphs) to build on, as mentioned above. | This additional layer of abstraction provides the reliability for advanced discrete data types (such as weighted or labelled graphs) to build on, as mentioned above. | ||
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[[https:// | [[https:// | ||
- | * Infinite sequences of real numbers arise as elements of ℓp spaces; and as coefficients to analytic function germs. | + | * Infinite sequences of real numbers arise as elements of ℓ<sup>p</ |
[[http:// | [[http:// | ||
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see also the [[https:// | see also the [[https:// | ||
+ | * [[https:// | ||
+ | |||
+ | * [[http:// | ||
===== References ===== | ===== References ===== |