Related papers: Detecting unknots via equational reasoning, I: Exp…
Deep learning is a powerful tool for solving nonlinear differential equations, but usually, only the solution corresponding to the flattest local minimizer can be found due to the implicit regularization of stochastic gradient descent. This…
Form a pure mathematical point of view, common functional forms representing different physical phenomena can be defined. For example, rates of chemical reactions, diffusion and heat transfer are all governed by exponential-type…
Traditional symbolic reasoning engines, while attractive for their precision and explicability, have a few major drawbacks: the use of brittle inference procedures that rely on exact matching (unification) of logical terms, an inability to…
Complex networks have now become integral parts of modern information infrastructures. This paper proposes a user-centric method for detecting anomalies in heterogeneous information networks, in which nodes and/or edges might be from…
Standpoint logic is a recently proposed formalism in the context of knowledge integration, which advocates a multi-perspective approach permitting reasoning with a selection of diverse and possibly conflicting standpoints rather than…
A step-by-step presentation of the code for a small theorem prover introduces theorem-proving techniques. The programming language used is Standard ML. The prover operates on a sequent calculus formulation of first-order logic, which is…
Knot diagrams are among the most common visual tools in topology. Computer programs now make it possible to draw, manipulate and render them digitally, which proves to be useful in knot theory teaching and research. Still, an openly…
We describe a "top down" approach for automated theorem proving (ATP). Researchers might usefully investigate the forms of the theorems mathematicians use in practice, carefully examine how they differ and are proved in practice, and code…
We illustrate the power of Experimental Mathematics and Symbolic Computation to suggest irrationality proofs of natural constants, and the determination of their irrationality measures. Sometimes such proofs can be fully automated, but…
We present two practical and widely applicable methods, including some criteria and a general procedure, for detecting Brunnian property of a link, if each component is known to be unknot. The methods are based on observation and handwork.…
We design a proof system for propositional classical logic that integrates two languages for Boolean functions: standard conjunction-disjunction-negation and binary decision trees. We give two reasons to do so. The first is…
We study symmetric crossing change operations for strongly invertible knots. Our main theorem is that the most natural notion of equivariant unknotting number is not additive under connected sum, in contrast with the longstanding conjecture…
We discuss the possibility of the existence of finite algorithms that may give distinct knot classes. In particular we present two attempts for such algorithms which seem promising, one based on knot projections on a plane, the other on…
We describe an algorithm to find ribbon disks for alternating knots, and the results of a computer implementation of this algorithm. The algorithm is underlain by a slice link obstruction coming from Donaldson's diagonalisation theorem. It…
We focus on determining the separability of an unknown bipartite quantum state $\rho$ by invoking a sufficiently large subset of all possible entanglement witnesses given the expected value of each element of a set of mutually orthogonal…
We show that there is a knot satisfying the property that for each minimal crossing number diagram of the knot and each single crossing of the diagram, changing the crossing results in a diagram for a knot whose unknotting number is at…
Knots are fascinating topological structures that have been observed in various contexts, ranging from micro-worlds to macro-systems, and are conjectured to play a fundamental role in their respective fields. In order to characterize their…
We first investigate properties of M-tensor equations. In particular, we show that if the constant term of the equation is nonnegative, then finding a nonnegative solution of the equation can be done by finding a positive solution of a…
We present an enhanced prime decomposition theorem for knots that gives the isotopy classes of composite knots that can be constructed from a given list of prime factors (allowing for the mirroring and orientation reversing for each…
I briefly discuss a method of obtaining distinct classes of topologically equivalent knots by developing appropriate computer programs.