Researchers developed an algorithm to count all graph mosaics.
problem Defining and counting graph mosaics to represent graph diagrams.
method Using a recursion formula of state matrices and sixteen graph mosaic tiles.
result Produced the exact enumeration of all graph mosaics.
Enhances knot counting using mosaic diagrams.
problem Counting and classifying surface-links and knots.
method Marked graph diagrams and mosaic numbers.
result Established bounds on mosaic numbers for surface-links.
A new mosaic system for immersed surface-links is introduced.
problem Constructing a mosaic system for immersed surface-links.
method Using singular marked graph diagrams, the mosaic number for immersed surface-links is defined and discussed.
result A mosaic system for immersed surface-links is established.
New method uses mosaics to study wild knots.
problem Classifying wild knots with infinite knotting behavior.
method Extending knot mosaic theory to represent wild knots with isolated wild points.
result Developed a framework for mosaic tangles and mosaic rigid vertex spatial graphs.
This paper finds all prime knots with mosaic number 6 and their minimal space-efficient mosaics.
problem Finding minimal space-efficient mosaics for prime knots with a specific mosaic number.
method Examined prime knots with mosaic number 6, determined their minimal space-efficient mosaics, and calculated their tile numbers.
result A complete list of prime knots with mosaic number 6 and their minimal space-efficient mosaics were found.
Paper introduces spherical knot mosaics for knot and link invariants.
problem Representing knots on a sphere with tiles.
method Tiling a 2-sphere with 11 knot mosaic tiles to define new invariants.
result New knot invariants derived from spherical mosaic tiling.
Rectangular mosaics extend virtual knot studies to larger polygons.
problem Studying virtual knots using mosaic techniques.
method Introduced rectangular mosaics, modified mosaic moves, and provided invariants.
result Developed algorithms for computing virtual knot invariants.
New types of knot mosaics help count and analyze knots efficiently.
problem Counting and analyzing knots efficiently.
method Introducing period and toroidal knot mosaics and developing algorithms for their enumeration.
result Exact enumeration of period knot mosaics and asymptotics of toroidal knot mosaics.
Researchers found algorithms to construct toric mosaics and set upper bounds for their numbers.
problem Finding efficient methods to construct toric mosaics of torus knots.
method Developed two algorithms for constructing toric mosaics on the surface of a torus.
result Provided upper bounds for the toric mosaic number of torus knots.
Algorithm finds mosaic numbers for knots with 10 or fewer crossings.
problem Efficiency in representing knots as mosaics.
method Algorithmic programming approach to find mosaic and tile numbers.
result Table of knot mosaics and mosaic number for prime knots with 10 or fewer crossings.
Expanding on prime knots with 6 or less mosaic tiles, this paper analyzes those with 7 tiles.
problem Determining the tile number and space-efficiency for prime knots with mosaic number 7.
method Extending the methods of Heap and Knowles (2017) to include prime knots with mosaic number 7.
result Identifying the possible tile numbers and space-efficient layouts for all prime knots with mosaic number 7.
Knot mosaics are used to model physical quantum states. The mosaic number of a knot is the smallest integer m such that the knot can be represented as a knot m-mosaic. In this paper we establish an upper bound for the crossing number of a knot in terms of the mosaic number. Given an m-mosaic and any knot K that…
This paper studies virtual knots using mosaic diagrams.
problem Understanding virtual knots through mosaic diagrams.
method Developed moves to preserve knot type and showed all virtual knots can be represented.
result Any virtual knot can be represented as a virtual mosaic.
Computes bounds on mosaic number of Legendrian knots.
problem Determining the mosaic number of Legendrian knots.
method Using modified mosaic tiles and classical invariants, computed lower bounds and provided examples for sharpness.
result Sharp bounds on mosaic number of Legendrian knots in certain cases.
The paper finds bounds and specific tile numbers for knot mosaics.
problem Determining the minimum number of tiles needed to represent knots.
method Analyzing the relationship between tile number and mosaic number of knots.
result Strict bounds and specific tile numbers for various knots are determined.
This paper calculates the growth constant for quantum knot mosaics.
problem Calculating the growth constant for quantum knot mosaics.
method Introduced knot mosaic system, defined quantum knots, and used enumeration to find the growth constant.
result Existence and bounds of the knot mosaic constant δ. Improved bounds for knot crossings in different mosaic patterns.
problem Finding tighter bounds for knot crossings in rectangular and hexagonal mosaics.
method Extended Howard and Kobin's proof to hexagonal mosaics and shortened the rectangular proof.
result New bounds for hexagonal mosaics with improved efficiency in rectangular mosaics.
We investigate relationships between bounds on the crossing number and the mosaic number of mosaic knots.
KnotMosaics package simplifies knot theory computations in SageMath.
problem Efficiently computing knot mosaic diagrams and their properties.
method Developed a SageMath package for knot mosaic diagrams, implementing validation, strand tracing, and computation algorithms.
result Enabled easy computation of knot mosaic diagrams and their properties.
Lomonaco and Kauffman introduced knot mosaic system to give a definition of quantum knot system. This definition is intended to represent an actual physical quantum system. A knot (m,n)-mosaic is an m×n matrix of mosaic tiles which are T0 through T10 depicted as below, representing a knot or a link b…
Lomonaco and Kauffman developed knot mosaics to give a definition of a quantum knot system. This definition is intended to represent an actual physical quantum system. A knot n-mosaic is an n×n matrix of 11 kinds of specific mosaic tiles representing a knot or a link. The mosaic number m(K) of a knot K i…
GANosaic generates high-resolution mosaic images from texture data.
problem Creating smooth, high-resolution mosaic images from texture data.
method Optimization in latent noise space of a generative texture model.
result Generative mosaic images with high resolution and smooth transitions.
In 2008, Lomonaco and Kauffman introduced a knot mosaic system to define a quantum knot system. A quantum knot is used to describe a physical quantum system such as the topology or status of vortexing that occurs on a small scale can not see. Kuriya and Shehab proved that knot mosaic type is a complete invariant of tam…
In this paper, we work to construct mosaic representations of knots on the torus, rather than in the plane. This consists of a particular choice of the ambient group, as well as different definitions of contiguous and suitably connected. We present conditions under which mosaic numbers might decrease by this projection…
Paper proves corner connection tiles can represent knots with fewer tiles.
problem Finding the minimum number of tiles for knot representation.
method Developed corner connection tiles and proved their efficiency.
result Corner connection tiles can represent knots with fewer tiles than traditional tiles.
New tiles allow efficient knot mosaics for small knots.
problem Efficient representation of small knots on a grid.
method Introducing corner connection tiles for knot mosaics.
result Efficient knot mosaics for knots with crossing number 8 or less.
New knots found that can only fit in non-reduced projections.
problem Finding knots that can only fit in non-reduced projections.
method Infinite family of knots, systematic flype finding tool.
result Knots with hexagonal mosaic number realized only in non-reduced projections.
New tile types for knots and links reduce complexity.
problem Determining the minimum number of tiles needed for knot representations.
method Introduced new tile types and analyzed their impact on knot complexity.
result Corner tile number lies between tile number and 3 times tile number.
Lomonaco and Kauffman introduced a knot mosaic system to give a definition of a quantum knot system which can be viewed as a blueprint for the construction of an actual physical quantum system. A knot n-mosaic is an n×n matrix of 11 kinds of specific mosaic tiles representing a knot or a link by adjoining pr…
Inspired by the paper on quantum knots and knot mosaics [23] and grid diagrams (or arc presentations), used extensively in the computations of Heegaard-Floer knot homology [2,3,7,24], we construct the more concise representation of knot mosaics and grid diagrams via mirror-curves. Tame knot theory is equivalent to knot…
MOSAIC detects change points in dynamic networks with low-rank and sparse changes.
problem Detecting change points in dynamic networks with specific structural properties.
method Eigen-decomposition-based test with screened signals and residual-based adjustment.
result MOSAIC achieves minimax-optimal detection and testing rates.
Causal Mosaic distinguishes cause from effect using nonlinear ICA and ensemble methods.
problem Distinguishing cause from effect in bivariate settings.
method Nonlinear ICA and ensemble framework (Causal Mosaic).
result Causal Mosaic shows state-of-the-art performance on artificial and real-world datasets.
Lomonaco and Kauffman developed a knot mosaic system to introduce a precise and workable definition of a quantum knot system. This definition is intended to represent an actual physical quantum system. A knot (m,n)-mosaic is an m×n matrix of mosaic tiles (T0 through T10 depicted in the introduction) re…
Model learns set representations through optimized permutations.
problem Challenges in learning set representations due to permutation-invariance.
method Proposes a Permutation-Optimisation module to learn set permutations.
result Achieves state-of-the-art results on various set learning tasks.
MOSAIC selects few informative exemplars from high-dimensional data with non-linear structures.
problem Representative selection from high-dimensional data with non-linear structures.
method MOSAIC uses a multi-criteria approach with a quadratic formulation to maximize global representation power, diversity, and outlier detection.
result MOSAIC maximizes data coverage in a transformed space and achieves robustness to various outlier types.
Samuel J. Lomonaco Jr and Louis H. Kauffman conjectured that tame knot theory and knot mosaic theory are equivalent. We give a proof of the Lomonaco-Kauffman conjecture.
AutoML uses MCTS to optimize machine learning algorithms and hyperparameters.
problem Optimizing machine learning algorithms and hyperparameters efficiently.
method MCTS-based approach for hybrid optimization of machine learning portfolios.
result Mosaic outperforms Auto-Sklearn on OpenML 100 benchmark and Scikit-learn portfolio.
We introduce MosAIc, an interactive web app that allows users to find pairs of semantically related artworks that span different cultures, media, and millennia. To create this application, we introduce Conditional Image Retrieval (CIR) which combines visual similarity search with user supplied filters or "conditions". …
Deep learning classifies land use from high-resolution aerial imagery.
problem Variations in land features in aerial imagery due to sensor settings and context.
method Used deep convolutional neural networks to classify land use from VHR orthophoto mosaics.
result Deep learning can accurately classify land use from high-resolution visible band multispectral imagery.
Theory explains creativity in diffusion models generating novel images.
problem Diffusion models generate highly original images far from training data.
method Identified locality and equivariance as inductive biases to prevent optimal score-matching.
result Analytic models predict diffusion model outputs with high accuracy.
We show that the orthogonal separation coordinates on the sphere Sn are naturally parametrised by the real version of the Deligne-Mumford-Knudsen moduli space Mˉ0,n+2(R) of stable curves of genus zero with n+2 marked points. We use the combinatorics of Stasheff polytopes tessellating Mˉ0,n+2(R) t…
In this paper, we give a precise and workable definition of a quantum knot system, the states of which are called quantum knots. This definition can be viewed as a blueprint for the construction of an actual physical quantum system. Moreover, this definition of a quantum knot system is intended to represent the "quantu…
New framework boosts neural network performance and resilience.
problem Susceptibility of compact neural network implementations to system disturbances.
method Realistic crossbar simulations and Mosaics framework to re-use synaptic connections.
result Compact neural networks are noise-immune and perform well under disturbances.
FAMOS combines parametric and non-parametric methods for efficient image stylization.
problem Efficiently stylize images with limited data and compute resources.
method Fully Adversarial Mosaics (FAMOS) that integrates parametric and non-parametric approaches.
result Demonstrates the effectiveness of FAMOS in stylizing images with minimal data and compute resources.
Extends Fisher's Discriminant Analysis for interval-valued data.
problem Classifying entities represented by intervals and histograms.
method Adapts Fisher's Discriminant Analysis using Moore's interval arithmetic and Mallows' distance.
result Discriminant directions for interval-valued data are numerically maximized.
This paper explores how deep learning models can fit data exactly and why this is important.
problem Understanding why deep learning models can fit data exactly and generalize well.
method Interpolation and over-parameterization as key themes to understand deep learning.
result Interpolation and over-parameterization are crucial for deep learning models to fit data exactly and generalize well.
Deep learning improves demosaicing but edge devices struggle.
problem Edge devices struggle with deep learning-based demosaicing.
method Exhaustive search of deep neural network architectures to find the best balance between performance and model complexity.
result Found architectures that outperform state-of-the-art demosaicing models on edge devices.
Quantum knots and knotted zeros linked through complex plane mappings.
problem Understanding knotted zeros in quantum states of hydrogen.
method Classifying maps from 3-space to complex plane, relating to quantum knots and lattice structures.
result Every smooth knot in 3-space has a corresponding smooth map to the complex plane with a knotted inverse image of zero.