Theory of calculus

2009

In the 1940s, two different views of the brain and the computer were equally important. One was the analog technology and theory that had emerged before the war. The other was the digital technology and theory that was to become the main paradigm of computation. 1 The outcome of the contest between these two competing views derived from technological and epistemological arguments. While digital technology was improving dramatically, the technology of analog machines had already reached a significant level of development. In particular, digital technology offered a more effective way to control the precision of calculations. But the epistemological discussion was, at the time, equally relevant. For the supporters of the analog computer, the digital model -which can only process information transformed and coded in binary -wouldn't be suitable to represent certain kinds of continuous variation that help determine brain functions. With analog machines, on the contrary, there would be few or no layers between natural objects and the work and structure of computation (cf. [4, 1]). The 1942-52 Macy Conferences in cybernetics helped to validate digital theory and logic as legitimate ways to think about the brain and the machine [4]. In particular, those conferences helped made McCulloch-Pitts' digital model of the brain [3] a very influential paradigm. The descriptive strength of McCulloch-Pitts model led von Neumann, among others, to seek identities between the brain and specific kinds of electrical circuitry [1].

Applied Mathematics and Computation, 2009

In the 1940s, two different views of the brain and the computer were equally important. One was the analog technology and theory that had emerged before the war. The other was the digital technology and theory that was to become the main paradigm of computation. 1 The outcome of the contest between these two competing views derived from technological and epistemological arguments. While digital technology was improving dramatically, the technology of analog machines had already reached a significant level of development. In particular, digital technology offered a more effective way to control the precision of calculations. But the epistemological discussion was, at the time, equally relevant. For the supporters of the analog computer, the digital model -which can only process information transformed and coded in binary -wouldn't be suitable to represent certain kinds of continuous variation that help determine brain functions. With analog machines, on the contrary, there would be few or no layers between natural objects and the work and structure of computation (cf. [4, 1]). The 1942-52 Macy Conferences in cybernetics helped to validate digital theory and logic as legitimate ways to think about the brain and the machine [4]. In particular, those conferences helped made McCulloch-Pitts' digital model of the brain [3] a very influential paradigm. The descriptive strength of McCulloch-Pitts model led von Neumann, among others, to seek identities between the brain and specific kinds of electrical circuitry [1].

2019

In this explanatory work, we make an attempt to briefly discuss the work and technical achievements of twenty female mathematicians. The work may be useful as a historical resource, but there is very little biography or history, and the primary focus is on the mathematics. In fact, the main use will probably be for a student who is coming to a new area of mathematics for the first time and needs an overview of some of the key results and references viewed through the work of one of that area's main contributors.

The Western Ontario Series in Philosophy of Science, 2011

The present book project grew out of the Swiss Society for Logic and Philosophy of Science (SSLPS) annual meeting on "Foundational theories of mathematics" which was held in Freiburg (Switzerland) on October 11/12 2006. John Bell and Gerhard Jäger, both participating in different functions in this meeting, responded to this book project with great enthusiasm and fueled its evolution with recurrent positive feedbacks. I'm happy to have had the opportunity over the years to discuss with them Foundations of Mathematics (FOM) and many other hot topics as well as some less hot ones. I'm grateful to Oxford University Press for permission to reprint Section I.2 of Penelope Maddy's book Naturalism in Mathematics (Oxford: Clarendon Press, 1997) and also to the publishing company Polimetrica for permission to reprint Solomon Feferman's article which first appeared in G. Sica's book What is Category Theory? (Monza: Polimetrica, 2006). I'd like to thank Norman Sieroka for helping me to translate several papers from (ancient) Word to (modern) LaTeX. I'd also like to thank Lucy Fleet, Senior Assistant to the Editorial Director-Humanities of Springer, for her patience and her refreshing confidence that this book would eventually see the light of day. I'm very grateful to the IT Service of the ETH Zurich and in particular to its collaborator Dieter Hennig for his very valuable support in my rather dilettante LaTeX-engineering. Last and most I'd like to thank my wife Andrea Gemma and my little son Beniamino for not having banned me from home before regaining normality, and even more so for their loving care with which they endured and accepted my mixture of mental absence and busy bee existence during the more productive phases of this Contents ix

Chemical Physics, 2006

Herein, we present analytical solutions for the electronic energy eigenvalues of the hydrogen molecular ion H ¡ , namely the one-electron two-fixed-center problem. These are given for the homonuclear case for the countable infinity of discrete states when the magnetic quantum number © is zero i.e. for ¡ states. In this case, these solutions are the roots of a set of two coupled three-term recurrence relations. The eigensolutions are obtained from an application of experimental mathematics using Computer Algebra as its principal tool and are vindicated by numerical and algebraic demonstrations. Finally, the mathematical nature of the eigenenergies is identified.

Physical review, 1990

We present a matrix-eigenvalue algorithm for accurately computing the quasinormal frequencies and modes of charged static black holes. The method is then refined through the introduction of a continued fraction step. We believe the approach will generalize to a variety of non-separable wave equations, including the Kerr-Newman case of charged rotating black holes.

Journal of Logic and Computation, 2007

Computability in Europe (CiE) is an informal network of European scientists working on computability theory, including its foundations, technical development, and applications. Among the aims of the network is to advance our theoretical understanding of what can and cannot be computed, by any means of computation. Its scientific vision is broad: computations may be performed with discrete or continuous data by all kinds of algorithms, programs, and machines. Computations may be made by experimenting with any sort of physical system obeying the laws of a physical theory such as Newtonian mechanics, quantum theory or relativity. Computations may be very general, depending upon the foundations of set theory; or very specific, using the combinatorics of finite structures. CiE also works on subjects intimately related to computation, especially theories of data and information, and methods for formal reasoning about computations. The sources of new ideas and methods include practical developments in areas such as neural networks, quantum computation, natural computation, molecular computation, and computational learning. Applications are everywhere, especially, in algebra, analysis and geometry, or data types and programming.

2011

Programming physicists use, as all programmers, arrays, lists, tuples, records, etc., and this requires some change in their thought patterns while converting their formulae into some code, since the "data structures" operated upon, while elaborating some theory and its consequences, are rather: power series and Pad\'e approximants, differential forms and other instances of differential algebras, functionals (for the variational calculus), trajectories (solutions of differential equations), Young diagrams and Feynman graphs, etc. Such data is often used in a [semi-]numerical setting, not necessarily "symbolic", appropriate for the computer algebra packages. Modules adapted to such data may be "just libraries", but often they become specific, embedded sub-languages, typically mapped into object-oriented frameworks, with overloaded mathematical operations. Here we present a functional approach to this philosophy. We show how the usage of Haskell datat...

The Mathematical Intelligencer, 2005

Foundations of Science, 2021

Some terms identify enigmata of today’s cosmology: “Inflation” is expected to explain the homogeneity and isotropy of the cosmic background. The repulsive force of a “dark energy” shall prevent a re-collapse of the cosmos. The additional gravitational effect of a “dark matter” was originally supposed to explain the deviations of the rotation curves of the galaxies from Kepler’s laws. Adopting a theory founded on the core notion of absolute quantum information–Protyposis–being a cosmological concept from the outset, the observed phenomena can be explained without postulating further unknown specific “particles” or “fields”. Moreover, this theory allows for a rationalization of the fact that huge black holes with their enormous jet structures, acting as “seeds” of the galaxies, are detected ever closer to the big bang. The problem of the rotation curves in the galaxies can be addressed outside of General Relativity within a Newtonian approximation: by an attenuation of the gravitation...