Related papers: Tangle Machines II: Invariants
Complex networks describe a wide range of systems in nature and society, much quoted examples including the cell, a network of chemicals linked by chemical reactions, or the Internet, a network of routers and computers connected by physical…
The first part of this paper consists of lecture notes which summarize the machinery of filtered root invariants. A conceptual notion of "homotopy Greek letter element" is also introduced, and evidence is presented that it may be related to…
A Tangle is a smooth simple closed curve formed from arcs (or ``links'') of circles with fixed radius. Most previous study of Tangles has dealt with the case where these arcs are quarter-circles, but Tangles comprised of thirds and sixths…
This paper proposes a novel perspective on learning, positing it as the pursuit of dynamical invariants -- data combinations that remain constant or exhibit minimal change over time as a system evolves. This concept is underpinned by both…
Framed combinatorial topology is a novel theory describing combinatorial phenomena arising at the intersection of stratified topology, singularity theory, and higher algebra. The theory synthesizes elements of classical combinatorial…
Invariants are a set of properties over program attributes that are expected to be true during the execution of a program. Since developing those invariants manually can be costly and challenging, there are a myriad of approaches that…
It is a recent observation that entanglement classification for qubits is closely related to local $SL(2,\CC)$-invariants including the invariance under qubit permutations, which has been termed $SL^*$ invariance. In order to single out the…
A variant of Turing machines is introduced where the tape is replaced by a single tree which can be manipulated in a style akin to purely functional programming. This yields two benefits: first, the extra structure on the tape can be…
The class of random-cluster models is a unification of a variety of stochastic processes of significance for probability and statistical physics, including percolation, Ising, and Potts models; in addition, their study has impact on the…
We consider parameterized concurrent systems consisting of a finite but unknown number of components, obtained by replicating a given set of finite state automata. Components communicate by executing atomic interactions whose participants…
A difference equation based method of determining two factors of a composite is presented. The feasibility of P-complexity is shown. Presentation of material is non-theoretical; intended to be accessible to a broader audience of non…
Every end of an infinite graph $ G $ defines a tangle of infinite order in $ G $. These tangles indicate a highly cohesive substructure in the graph if and only if they are closed in some natural topology. We characterize, for every finite…
A class of two-dimensional superintegrable systems on a constant curvature surface is considered as the natural generalization of some well known one-dimensional factorized systems. By using standard methods to find the shape-invariant…
We define the topological multiplicity of an invertible topological system $(X,T)$ as the minimal number $k$ of real continuous functions $f_1,\cdots, f_k$ such that the functions $f_i\circ T^n$, $n\in\mathbb Z$, $1\leq i\leq k,$ span a…
This paper is a short introduction to the theory of tangles, both in graphs and general connectivity systems. An emphasis is put on the correspondence between tangles of order k and k-connected components. In particular, we prove that there…
Various topics concerning the entanglement of composite quantum systems are considered with particular emphasis concerning the strict relations of such a problem with the one of attributing objective properties to the constituents. Most of…
Originally, tangles were invented as an abstract tool in mathematical graph theory to prove the famous graph minor theorem. In this paper, we showcase the practical potential of tangles in machine learning applications. Given a collection…
People solve different problems and know that some of them are simple, some are complex and some insoluble. The main goal of this work is to develop a mathematical theory of algorithmic complexity for problems. This theory is aimed at…
We consider the simplest gauge theories given by one- and two- matrix integrals and concentrate on their stringy and geometric properties. We remind general integrable structure behind the matrix integrals and turn to the geometric…
These lecture notes cover 13 sessions and are presented as an e-print, intended to evolve over time. Quantum invariants do more than distinguish topological objects; they build bridges between topology, algebra, number theory and quantum…