Related papers: On Holant Theorem and Its Proof
In theorem proving, the task of selecting useful premises from a large library to unlock the proof of a given conjecture is crucially important. This presents a challenge for all theorem provers, especially the ones based on language…
This paper introduces several new classes of mathematical structures that have close connections with physics and with the theory of dynamical systems. The most general of these structures, called indivisible stochastic processes,…
In the last decades, several objects such as grammars, economical agents, laws of physics... have been defined as algorithms. In particular, after Brouwer, Heyting, and Kolomogorov, mathematical proofs have been defined as algorithms. In…
This is a complete and exhaustive review on the so-called holographic axion model -- a bottom-up holographic system characterized by the presence of a set of shift symmetric scalar bulk fields whose profiles are taken to be linear in the…
Recently the breakthrough method of hypergraph containers, developed independently by Balogh, Morris, and Samotij as well as Saxton and Thomason, has been used to study sparse random analogs of a variety of classical problems from…
We present a new methodology for utilising machine learning technology in symbolic computation research. We explain how a well known human-designed heuristic to make the choice of variable ordering in cylindrical algebraic decomposition may…
For a given asphere the grating equation is used to derive the design for a Computer Generated Hologram (CGH), sometimes referred to as a diffractive null corrector. The CGH converts a spherical wavefront to the shape appropriate for the…
Let $\mathcal C$ be a real plane algebraic curve defined by the resultant of two polynomials (resp. by the discriminant of a polynomial). Geometrically such a curve is the projection of the intersection of the surfaces $P(x,y,z)=Q(x,y,z)=0$…
Section 10.4 of the 1998 Springer-Verlag book {\em Complexity and Real Computation}, by Blum, Cucker, Shub, and Smale, contains a particularly elegant proof of the Fundamental Theorem of Algebra: The central idea of the proof naturally…
We give a simple proof of Kolmogorov's theorem on the persistence of a quasiperiodic invariant torus in Hamiltonian systems. The theorem is first reduced to a well-posed inversion problem (Herman's normal form) by switching the frequency…
Basic concepts of quantum theory of information, principles of quantum calculations and the possibility of creation on this basis unique on calculation power and functioning principle device, named quantum computer, are briefly reviewed.…
Persistent homology is a popular tool in Topological Data Analysis. It provides numerical characteristics of data sets which reflect global geometric properties. In order to be useful in practice, for example for feature generation in…
According to the holographic principle, the information content assigned to a gravitational region is processed by its lower dimensional boundary. As an example setup compatible with this principle, the AdS/CFT correspondence relies on the…
We describe a formal correctness proof of RANKING, an online algorithm for online bipartite matching. An outcome of our formalisation is that it shows that there is a gap in all combinatorial proofs of the algorithm. Filling that gap…
Computers and algorithms play an ever-increasing role in obtaining new results in graph theory. In this survey, we present a broad range of techniques used in computer-assisted graph theory, including the exhaustive generation of all…
"Systems that Explain Themselves" appears a provocative wording, in particular in the context of mathematics education -- it is as provocative as the idea of building educational software upon technology from computer theorem proving. In…
Holography is a technique based on the wave nature of light which allows us to utilize wave interference between the object beam and the coherent background. Holography is usually associated with images being made from light, however, this…
The intrinsic connection between lattice theory and topology is fairly well established, For instance, the collection of open subsets of a topological subspace always forms a distributive lattice. Persistent homology has been one of the…
Hyperdimensional computing (HDC) is an increasingly popular computing paradigm with immense potential for future intelligent applications. Although the main ideas already took form in the 1990s, HDC recently gained significant attention,…
We show how well known tools of algebraic geometry for the study of finite sets can be fruitfully applied to the study of Waring decompositions of symmetric tensors (forms). We mainly focus on the uniqueness of a given decomposition (the…