Venn diagrams are a graphical way to represent a set system. Each of the n sets is represented by a simple closed curve. The n curves subdivide the plane into 2^n open connected regions, each of which represents the intersection of its containing curves' sets. For example, two overlapping circles can divide the plane i…
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48 results for “Venn Diagram”
The purpose of this paper is to discuss how topology and geometry provide, in many instances, the connective tissue that enables logical comprehension. We illustrate this theme with many examples including Venn diagrams, knot diagrams, knot-logical diagrams and an arrow of reference that elucidates self-reference and G…
Study recovers neuron assemblies from cognitive data using tensor decomposition.
problem Recovering neuron assemblies from cognitive data.
method Linear independence analysis of rank one tensors, polynomial measurements.
result Reconstructs Venn diagram of neuron assemblies from intersection sizes.
This paper continues study, both theoretical and empirical, of the method of Venn prediction, concentrating on binary prediction problems. Venn predictors produce probability-type predictions for the labels of test objects which are guaranteed to be well calibrated under the standard assumption that the observations ar…
Unified framework for generalized Venn and Venn-Abers calibration for reliable prediction.
problem Asymptotic guarantees of popular distribution-free methods in model calibration.
method Unified framework extending Vovk's approach to generic loss functions, transforming predictors into set-valued predictions.
result Finite-sample set predictions shrink to a single conditionally calibrated prediction, capturing epistemic uncertainty.
Venn GAN models multiple distributions, discovering shared and unique aspects.
problem Modeling and understanding multiple data distributions effectively.
method A GAN design with shared and non-shared generator distributions.
result Effective modeling of various datasets (MNIST, Fashion MNIST, CIFAR-10, Omniglot, CelebA).
Study compares various calibration methods for binary classification tasks.
problem Improving probabilistic predictions in binary classification models.
method Benchmarked 21 classifiers using 5 calibration methods on real data.
result Venn-Abers predictors and Beta calibration show the largest log-loss reductions.
Self-calibrating conformal prediction improves interval efficiency and offers a practical alternative.
problem Improving the reliability and uncertainty quantification of machine learning predictions.
method Combines Venn-Abers calibration and conformal prediction for binary and regression problems.
result Improves interval efficiency through model calibration and offers practical alternatives.
The paper refines transformations of lattice diagrams and introduces dotted diagrams.
problem Investigating transformations and deformations of lattice diagrams and their associated dotted diagrams.
method Introducing dotted diagrams and investigating deformations of these diagrams, relating them to transformations of lattice diagrams.
result Refined results on the relation between deformations of admissible dotted diagrams and transformations of lattice diagrams.
This note explains how to transform Heegaard diagrams into framed link diagrams.
problem No specific problem stated; transformation of diagrams is the focus.
method Explains a procedure to transform Heegaard diagrams into framed link diagrams.
result Demonstrates a method to transform Heegaard diagrams into framed link diagrams.
New minimal link diagrams found, including torus links and homogeneous ones.
problem Finding minimal link diagrams with new classes.
method Morton-Franks-Williams inequality approach.
result New classes of minimal link diagrams, including previously unproven ones.
Algorithm converts Kirby diagrams to trisection diagrams for 4-manifolds.
problem Creating efficient trisection diagrams for 4-manifolds.
method Algorithm converting Kirby diagrams to trisection diagrams.
result Provides examples of trisection diagrams for 4-manifolds.
Kernelized Taylor diagram visualizes data populations with fewer assumptions.
problem Limitations of Taylor diagram in capturing non-linear relationships and sensitivity to outliers.
method Proposes a kernelized version of the Taylor diagram that uses maximum mean discrepancy and kernel mean embedding.
result Kernelized Taylor diagram visualizes data populations with minimal assumptions of data distributions.
Study categorizes knots and links as rigid or shaky based on Reidemeister moves.
problem Classifying knots and links as rigid or shaky based on adaptability to Reidemeister moves.
method Categorization of hard diagrams as rigid or shaky, investigation of rigid and shaky hard diagrams for specific knots and links.
result Every link has a rigid hard diagram, and there is an upper limit for the number of crossings in such diagrams.
A virtual link diagram is called normal if the associated abstract link diagram is checkerboard colorable, and a virtual link is normal if it has a normal diagram as a representative.In this paper, we introduce a method of converting a virtual link diagram to a normal virtual link diagram by use of the double covering …
Twisted graph diagrams are virtual graph diagrams with bars on edges. A bijection between abstract graph diagrams and twisted graph diagrams is constructed. Then a polynomial invariant of Yamada-type is developed which provides a lower bound for the virtual crossing number of virtual graph diagrams.
Persistence diagrams are important descriptors in Topological Data Analysis. Due to the nonlinearity of the space of persistence diagrams equipped with their {\em diagram distances}, most of the recent attempts at using persistence diagrams in machine learning have been done through kernel methods, i.e., embeddings of …
A virtual link diagram is called normal if the associated abstract link diagram is checkerboard colorable, and a virtual link is normal if it has a normal diagram as a representative. Normal virtual links have some properties similar to classical links.In this paper, we introduce a method of converting a virtual link d…
Problems on region choices for knot and link diagrams solved using Alexander numbering.
problem Existence of solutions for region choice problems on knot and link diagrams.
method Alexander numbering for regions, alternative proofs, necessary and sufficient conditions.
result Existence of solutions for region choice problems on link diagrams.
Table of symmetric diagrams for knots up to 10 crossings.
problem Finding symmetric diagrams for strongly invertible knots.
method Compilation of symmetric diagrams for knots up to 10 crossings.
result Similarity of transversal diagrams to symmetric union diagrams for strongly invertible knots.
The presence of slipknots in configurations of proteins and DNA has been shown to affect their functionality, or alter it entirely. Historically, polymers are modeled as polygonal chains in space. As an alternative to space curves, we provide a framework for working with subknots inside of knot diagrams via knotoid dia…
Proves minimal crossing diagrams for specific spatial graphs.
problem Proving minimal crossing diagrams for spatial graphs.
method Analyzing adequate diagrams and replacing vertices and edges.
result All 1-vertex spatial graphs with adequate diagrams have minimal crossing number.
Rectangular diagrams help analyze foliations in 3-sphere.
problem Analyzing foliations in 3-sphere complements.
method Introduced rectangular diagrams for foliations and links.
result Any co-orientable finite depth foliation can be presented by a compatible rectangular diagram.
Bankwitz characterized an alternating diagram representing the trivial knot. A non-alternating diagram is called almost alternating if one crossing change makes the diagram alternating. We characterize an almost alternaing diagram representing the trivial knot. As a corollary we determine an unknotting number one alter…
The paper explores when specific knot operations simplify diagrams.
problem Understanding when arc crossing changes simplify knot diagrams.
method Examined two types of arc crossing changes on link diagrams and determined when they are unknotting operations.
result Any two crossing points in an alternating knot diagram are arc crossing change admissible.
Abstract lists known link diagrams for all principal congruence link complements.
problem Identifying all known link diagrams for principal congruence link complements.
method Compilation of known link diagrams.
result Compilation of known link diagrams for principal congruence link complements.
Gauss diagrams' properties can change with Hamiltonian cycle choice.
problem The impact of Hamiltonian cycle choice on Gauss diagrams.
method Examined realizable and unrealizable Gauss diagrams, and proved preservation of realizability under certain Hamiltonian cycle changes.
result Properties of Gauss diagrams can vary with Hamiltonian cycle choice.
New estimate of semimeander complexity for knots with more than 10 crossings.
problem Estimating the complexity of semimeander diagrams of knots.
method Proved a new upper bound on the number of crossings for semimeander diagrams of knots with more than 10 crossings.
result For knots with more than 10 crossings, semimeander diagrams have no more than crossings.
Paper proves link diagrams can be realized for some but not all types of links.
problem Realizing link diagrams for different types of links.
method Analyzing link diagrams for welded and virtual links.
result Similar results hold for welded links but not for virtual links.
GridPyM handles grid diagrams for knot theory.
problem Handling grid diagrams for knot theory.
method Generates and simplifies grids, models local transformations.
result Models local transformations between grid diagrams.
There is a well-known way to describe a link diagram as a (signed) plane graph, called its Tait graph. This concept was recently extended, providing a way to associate a set of embedded graphs (or ribbon graphs) to a link diagram. While every plane graph arises as a Tait graph of a unique link diagram, not every embedd…
The study improves inequalities for link diagrams and introduces weak rectangular diagrams.
problem Improving inequalities for link diagrams and understanding their properties.
method Introducing weak rectangular diagrams and proving new inequalities.
result Generalizes and subsumes many known inequalities related to multi-crossing numbers.
In this paper, a link diagram is said to be minimal if no Reidemeister move I or II can be applied to it to reduce the number of crossings. We show that for an arbitrary diagram D of a link without a trivial split component, a minimal diagram obtained by applying Reidemeister moves I and II to D is unique. The proof al…
By using the cohomology theory of quandles, quandle cocycle invariants and shadow quandle cocycle invariants are defined for oriented links and surface-links via broken surface diagrams. By using symmetric quandles, symmetric quandle cocycle invariants are also defined for unoriented links and surface-links via broken …
Standard trisection diagrams found for Mazur type 4-manifolds.
problem Finding standard trisection diagrams for Mazur type 4-manifolds.
method Doubling a relative trisection diagram and using an algorithm from Kirby diagrams to trisection diagrams.
result Certain trisection diagrams of Mazur type 4-manifolds are standard.
Algorithm for recognizing and performing Reidemeister moves in Gauss diagrams.
problem Recognizing and performing Reidemeister moves in Gauss diagrams.
method Simple algorithm for recognizing and performing Reidemeister moves in Gauss diagrams.
result Simple algorithm for recognizing and performing Reidemeister moves in Gauss diagrams.
Paper constructs cyclic coverings of virtual link diagrams, proving equivalence of certain maps.
problem Constructing and proving equivalence of cyclic coverings of virtual link diagrams.
method Introducing -fold cyclic covering diagrams and proving their equivalence to original diagrams.
result Well-defined map from virtual links to mod almost classical virtual links.
Minimal grid diagrams for 12-crossing prime knots identified.
problem Identifying minimal grid diagrams for prime knots.
method Listed minimal grid diagrams for 12-crossing prime knots.
result Provided a list of minimal grid diagrams for 12-crossing prime knots.
The paper introduces triple grid diagrams to construct Lagrangian surfaces in complex projective space.
problem Constructing Lagrangian surfaces in complex projective space.
method Defining and analyzing triple grid diagrams to determine Lagrangian caps and surfaces.
result Triple grid diagrams can determine closed Lagrangian surfaces in under certain conditions.
Introduces a new space of Radon measures for better understanding persistence diagrams.
problem Lack of optimal transport-based formalism for persistence diagrams.
method Formalizes persistence diagrams as Radon measures on the upper half plane via optimal partial transport.
result Characterizes convergence and barycenters of persistence diagrams.
Complete criterion for VoI in multi-decision influence diagrams established.
problem Analyzing safety and fairness properties of AI systems using influence diagrams.
method Introduced ID homomorphisms and Tree of Systems to prove properties of multi-decision influence diagrams.
result First complete graphical criterion for VoI in influence diagrams with multiple decisions.
Immersed flat ribbon knotsmath.GT
Study minimum ribbonlength of immersed flat knots and links.
problem Finding the minimum ribbonlength for immersed planar knots and links.
method Embedding into disk diagram space to find length minimizers.
result Computed minimal ribbonlength for some knot and link diagrams.
Algorithm constructs Kirby diagrams for 4D open books.
problem Constructing Kirby diagrams for 4D open books.
method Algorithm using Heegaard diagrams of pages.
result Diffeomorphic open books constructed with different pages and monodromies.
For an oriented virtual link, L.H. Kauffman defined the f-polynomial (Jones polynomial). The supporting genus of a virtual link diagram is the minimal genus of a surface in which the diagram can be embedded. In this paper we show that the span of the f-polynomial of an alternating virtual link L is determined by the nu…
Hard diagrams of split linksmath.GT
Study shows that splitting links requires an arbitrarily large number of extra crossings.
problem The problem is to determine the minimum number of extra crossings needed to transform a diagram of a split link into a split diagram.
method The approach uses Reidemeister moves and the framework of bubble tangles, along with techniques from Riemannian geometry.
result There exist split links with diagrams requiring an arbitrarily large number of extra crossings.
The paper examines circle graphs of Gauss diagrams and finds counterexamples to previous descriptions.
problem Problems with previous descriptions of realizable Gauss diagrams.
method Experimental checking and formulation of new descriptions of realizable circle graphs.
result New descriptions of realizable circle graphs and an algorithm for checking realizability.
Parallelization technique for welded links preserves equivalence and yields specific decompositions.
problem Defining and proving equivalence of parallel welded link diagrams.
method Introduced a parallelization construction for welded link diagrams and showed its well-definedness.
result Parallel diagrams maintain equivalence under specific orientations and yield decompositions.
We consider framed chord diagrams, i.e. chord diagrams with chords of two types. It is well known that chord diagrams modulo 4T-relations admit Hopf algebra structure, where the multiplication is given by any connected sum with respect to the orientation. But in the case of framed chord diagrams a natural way to define…