12 research outputs found
Toward a structural theory of learning algebraic decompositions
We propose a framework generalizing several variants of Prony's method and explaining their relations. These methods are suitable for determining the support of linear combinations in particular in vector spaces of functions from evaluations. They are based on suitable sequences of linear maps resp. their matrices and include Hankel and Toeplitz variants of Prony's method for the decomposition of multivariate exponential sums, polynomials (w.r.t. the monomial and Chebyshev bases), Gau
fian sums, spherical harmonic sums, taking also into account whether they have their support on an algebraic set
Tropical adic spaces I: The continuous spectrum of a topological semiring
Towards building tropical analogues of adic spaces, we study certain spaces
of prime congruences as a topological semiring replacement for the space of
continuous valuations on a topological ring. This requires building the theory
of topological idempotent semirings, and we consider semirings of convergent
power series as a primary example. We consider the semiring of convergent power
series as a topological space by defining a metric on it. We check that, in
tropical toric cases, the proposed objects carry meaningful geometric
information. In particular, we show that the dimension behaves as expected and
give an explicit characterization of the points in terms of classical
polyhedral geometry. This characterization gives a bijection with a class of
prime filters and relates to flags of polyhedra.Comment: 74 pages, 4 figures. Comments welcome
Computer Science for Continuous Data:Survey, Vision, Theory, and Practice of a Computer Analysis System
Building on George Boole's work, Logic provides a rigorous foundation for the powerful tools in Computer Science that underlie nowadays ubiquitous processing of discrete data, such as strings or graphs. Concerning continuous data, already Alan Turing had applied "his" machines to formalize and study the processing of real numbers: an aspect of his oeuvre that we transform from theory to practice.The present essay surveys the state of the art and envisions the future of Computer Science for continuous data: natively, beyond brute-force discretization, based on and guided by and extending classical discrete Computer Science, as bridge between Pure and Applied Mathematics
Paul Lorenzen -- Mathematician and Logician
This open access book examines the many contributions of Paul Lorenzen, an outstanding philosopher from the latter half of the 20th century. It features papers focused on integrating Lorenzen's original approach into the history of logic and mathematics. The papers also explore how practitioners can implement Lorenzen’s systematical ideas in today’s debates on proof-theoretic semantics, databank management, and stochastics. Coverage details key contributions of Lorenzen to constructive mathematics, Lorenzen’s work on lattice-groups and divisibility theory, and modern set theory and Lorenzen’s critique of actual infinity. The contributors also look at the main problem of Grundlagenforschung and Lorenzen’s consistency proof and Hilbert’s larger program. In addition, the papers offer a constructive examination of a Russell-style Ramified Type Theory and a way out of the circularity puzzle within the operative justification of logic and mathematics. Paul Lorenzen's name is associated with the Erlangen School of Methodical Constructivism, of which the approach in linguistic philosophy and philosophy of science determined philosophical discussions especially in Germany in the 1960s and 1970s. This volume features 10 papers from a meeting that took place at the University of Konstanz
Q(sqrt(-3))-Integral Points on a Mordell Curve
We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum