Related papers: If Archimedes would have known functions
In this paper we introduced an arithmetic graph function which associates with every group G the directed graph whose vertices corresponds to the divisors of |G|. With the help of such functions we introduced arithmetic graphs of classes of…
The chapter advances a reformulation of the classical problem of the nature of mathematical objects (if any), here called "Plato's problem," in line with the program of a philosophy of mathematical practice. It then provides a sketch of a…
Turing's (1936) paper on computable numbers has played its role in underpinning different perspectives on the world of information. On the one hand, it encourages a digital ontology, with a perceived flatness of computational structure…
Some Goedel centenary reflections on whether incompleteness is really serious, and whether mathematics should be done somewhat differently, based on using algorithmic complexity measured in bits of information. [Enriques lecture given…
While reading ancient texts one has to be cognizant of the assumptions made about the past. One has to ask: Are these assumptions valid? Are we projecting the present views into the past? A case in point is the dating of Vedanga Jyotisa.…
Social computing encompasses the mechanisms through which people interact with computational systems: crowdsourcing systems, ranking and recommendation systems, online prediction markets, citizen science projects, and collaboratively edited…
In 1837, Dirichlet proved that there are infinitely many primes in any arithmetic progression in which the terms do not all share a common factor. We survey implicit and explicit uses of Dirichlet characters in presentations of Dirichlet's…
Mathematical models and algorithms are an essential part of mathematical research data, as they are epistemically grounding numerical data. In order to represent models and algorithms as well as their relationship semantically to make this…
A number of elite thinkers in Europe during the 16th and 17th centuries pursued an agenda which historian Paolo Rossi calls the "quest for a universal language," a quest which was deeply interwoven with the emergence of the scientific…
Over the past 20 years, Wikipedia has gone from a rather outlandish idea to a major reference work, with more than 60 million articles across all languages, including nearly 7 million in English [Wiki01]. Around 27,000 of these articles…
We extend the signal flow calculus---a compositional account of the classical signal flow graph model of computation---to encompass affine behaviour, and furnish it with a novel operational semantics. The increased expressive power allows…
The highest level of mathematics has traditionally been seen as a solitary endeavour, to produce a proof for review and acceptance by research peers. Mathematics is now at a remarkable inflexion point, with new technology radically…
A series of work [GKK+08, Kal22, KPV24] has shown that asymptotic advantages in space complexity are possible for quantum algorithms over their classical counterparts in the streaming model. We give a simple quantum sketch that encompasses…
This paper presents a series of general properties of the r-Complexity calculus, a complexity measurement for assessing the performance and asymptotic behaviour of real-world algorithms. This research describes characteristics such as…
Several successful strategies in automated reasoning rely on human-supplied guidance about which term or clause shapes are interesting. In this paper we aim to discover interesting term shapes automatically. Specifically, we discover…
The sciences of complexity present some recurrent themes: the emergence of qualitatively new behaviors in dissipative systems out of equilibrium, the aparent tendency of complex system to lie at the border of phase transitions and…
This paper is an essentially self-contained and rigorous description of the fundamental principles of quantum computing from a mathematical perspective. It is intended to help mathematicians who want to get a grasp of this quickly growing…
A diagrammatic logical calculus for the syllogistic reasoning is introduced and discussed. We prove that a syllogism is valid if and only if it is provable in the calculus.
We describe a case study of a problem-solving section, using the "Harkness" discussion method, of an honors multivariable calculus course. Students in the problem-solving section had equivalent outcomes on exams, reported higher ratings in…
In the past decade quantum algorithms have been found which outperform the best classical solutions known for certain classical problems as well as the best classical methods known for simulation of certain quantum systems. This suggests…