Related papers: Encoding and Topological Computation on Textiles
Complex computer codes are often too time expensive to be directly used to perform uncertainty propagation studies, global sensitivity analysis or to solve optimization problems. A well known and widely used method to circumvent this…
We introduce a numerical method for identifying topological order in two-dimensional models based on one-dimensional bulk operators. The idea is to identify approximate symmetries supported on thin strips through the bulk that behave as…
Knot theory is the Mathematical study of knots. In this paper we have studied the Composition of two knots. Knot theory belongs to Mathematical field of Topology, where the topological concepts such as topological spaces, homeomorphisms,…
Every finite graph $G$ can be decomposed in a canonical way that displays its local connectivity-structure [DJKK26]. These decompositions are defined via a suitable more tree-like covering of $G$, whose tangle-tree structure is projected…
Recent progress toward classifying low-dimensional chaos measured from time series data is described. This classification theory assigns a template to the time series once the time series is embedded in three dimensions. The template…
We use Topological Data Analysis tools for studying the inner organization of cells in segmented images of epithelial tissues. More specifically, for each segmented image, we compute different persistence barcodes, which codify lifetime of…
Network coding is studied when an adversary controls a subset of nodes in the network of limited quantity but unknown location. This problem is shown to be more difficult than when the adversary controls a given number of edges in the…
Weighted counting problems are a natural generalization of counting problems where a weight is associated with every computational path of polynomial-time non-deterministic Turing machines and the goal is to compute the sum of the weights…
A {\it stuck knot} is a knot diagram containing designated crossings, called {\it stuck crossings}, whose incident strands are required to remain locally non-separable. These rigidity constraints restrict the allowable ambient isotopies and…
Arithmetic coding is an essential class of coding techniques. One key issue of arithmetic encoding method is to predict the probability of the current coding symbol from its context, i.e., the preceding encoded symbols, which usually can be…
We introduce the Insertion Chain Complex, a higher-dimensional extension of insertion graphs, as a new framework for analyzing finite sets of words. We study its topological and combinatorial properties, in particular its homology groups,…
The guessing number of a directed graph (digraph), equivalent to the entropy of that digraph, was introduced as a direct criterion on the solvability of a network coding instance. This paper makes two contributions on the guessing number.…
Quantification and classification of protein structures, such as knotted proteins, often requires noise-free and complete data. Here we develop a mathematical pipeline that systematically analyzes protein structures. We showcase this…
Graphs are a representation of structured data that captures the relationships between sets of objects. With the ubiquity of available network data, there is increasing industrial and academic need to quickly analyze graphs with billions of…
This paper talk about the complexity of computation by Turing Machine. I take attention to the relation of symmetry and order structure of the data, and I think about the limitation of computation time. First, I make general problem named…
Virtual knot theory is a generalization (discovered by the author in 1996) of knot theory to the study of all oriented Gauss codes. (Classical knot theory is a study of planar Gauss codes.) Graph theory studies non-planar graphs via…
We show that the problem of recognizing that a knot diagram represents a specific torus knot, or any torus knot at all, is in the complexity class ${\sf NP} \cap {\sf co\text{-}NP}$, assuming the generalized Riemann hypothesis. We also show…
Predictive coding has emerged as an influential normative model of neural computation, with numerous extensions and applications. As such, much effort has been put into mapping PC faithfully onto the cortex, but there are issues that remain…
A Gauss diagram (or, more generally, a chord diagram) consists of a circle and some chords inside it. Gauss diagrams are a well-established tool in the study of topology of knots and of planar and spherical curves. Not every Gauss diagram…
We observe that any knot invariant extends to virtual knots. The isotopy classification problem for virtual knots is reduced to an algebraic problem formulated in terms of an algebra of arrow diagrams. We introduce a new notion of finite…