We define winding numbers of regular closed curves on surfaces with a nice euclidean or hyperbolic geometry. We prove that two regular closed curves are regularly homotopic if and only if they are freely homotopic and have the same winding number.
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48 results for “Winding Numbers”
Windings of prime geodesicsmath.NT
New method calculates winding of geodesics on surfaces.
problem Understanding the distribution of geodesics on surfaces.
method Introducing a new construction of winding numbers for geodesics on cusped hyperbolic orbifolds.
result Winding numbers can be expressed by Rademacher symbols for various arithmetic families of surfaces.
We prove that for any winding number pattern and winding number pattern , there exist knots such that the minimal genus of a cobordism between and is arbitrarily large. This answers a question posed by Cochran-Harvey [CH17] and generalizes a result of Kim-Livingston [KL05].
We study periodic wind-tree models, billiards in the plane endowed with -periodically located identical connected symmetric right-angled obstacles. We show asymptotic formulas for the number of (isotopy classes of) closed billiard trajectories (up to -translations) on the wind-tree billiard.…
Satellite operations with winding number ≠ 1 are not homomorphisms.
problem Characterizing homomorphisms in satellite operations.
method Using -invariants of branched covers and Torelli group properties.
result Satellite operations with winding number ≠ 1 are not homomorphisms.
The paper explores winding numbers of almost embeddings of a 4-vertex graph in the plane.
problem Understanding the winding numbers of almost embeddings of a 4-vertex graph in the plane.
method Constructing examples to show the only relation between the winding numbers of cycles in the graph.
result The sum of winding numbers is odd, and this is the only relation between them.
Geodesics near singularities either hit or wind around, with winding number dependent on singularity type.
problem Understanding geodesic behavior near singularities in Riemannian manifolds.
method Analytical study of geodesics on Riemannian manifolds near singularities.
result The winding number of geodesics around a singularity depends on the singularity type and approaches infinity as the singularity becomes cuspidal.
A graph's winding numbers around two non-adjacent vertices differ by ±1.
problem Understanding the winding numbers of a specific graph configuration in the plane.
method Analyzing continuous maps from a graph to the plane, focusing on the winding numbers of specific cycles.
result The difference in winding numbers of a cycle around two non-adjacent vertices is ±1.
In an earlier paper, I defined a new winding number of regular closed curves on complete euclidean/hyperbolic surfaces and showed that this winding number, together with the free homotopy class, determines the regular homotopy class. In this paper, I give a Whitney-type formula for the winding number of non-null-homoto…
Let P be a knot in a solid torus, K a knot in 3-space and P(K) the satellite knot of K with pattern P. This defines an operator on the set of knot types and induces a satellite operator P:C--> C on the set of smooth concordance classes of knots. There has been considerable interest in whether certain such functions are…
Study minimal annuli in a slab, estimating their area.
problem Estimating the area of minimal annuli in a slab.
method Organized minimal annuli based on winding number, deduced convexity of length function, compared to catenoid waist area.
result Deduced convexity of length function and estimated area of minimal annuli.
New findings on diffusion rates in wind-tree model with rational parameters.
problem Understanding diffusion rates in the wind-tree model with rational parameters.
method Analyzing real numbers in [0,1) as diffusion rates and providing a criterion for Lyapunov spectrum.
result Exhibit an infinite family of wind-tree billiards with the interior of the Lyapunov spectrum being the full square (0,1)^2.
We construct a new grading on the Goldman Lie algebra of a closed oriented surface by the winding number. This grading induces a grading on the HOMFLY-PT skein algebra and related algebras. Our work supports the conjectures of B. Cooper and P. Samuelson
New formula for rotation number without needing a base point.
problem Calculating rotation number without a base point.
method Formula derived from winding numbers of regions and crossing points.
result Does not require a base point for calculation.
Method reconstructs missing wind farm data using graph theory and nearest neighbors.
problem Missing data in wind farm records due to sensor failures.
method Combines spectral graph theory and k-Nearest Neighbors to estimate missing data.
result Significant improvement in data reconstruction over existing methods.
Study symmetry groups and curves from sums of exponentials.
problem Understanding the geometry and symmetry of curves from sums of exponentials.
method Analysis of symmetry groups, winding numbers, and parametrization of the unit circle.
result Unified method for constructing curves with specific properties.
Wind farm layout optimisation tackles space constraints with Bayesian multi-objective approach.
problem Optimizing wind farm layout due to limited space and conflicting objectives.
method Set-based multi-objective Bayesian optimisation using Gaussian process.
result Demonstrates potential of set-based Bayesian multi-objective optimisation for wind farm layout.
Necessary and sufficient conditions are given for a satellite knot to be fibered. Any knot embeds in an unknotted solid torus with arbitrary winding number in such a way that no satellite knot with pattern is fibered. In particular, there exist nonfibered satellite knots wit…
Numerous studies have been carried out to measure wind pressures around circular cylinders since the early 20th century due to its engineering significance. Consequently, a large amount of wind pressure data sets have accumulated, which presents an excellent opportunity for using machine learning (ML) techniques to tra…
Let be a two-periodic braid and let be its quotient. In this paper we show there is a spectral sequence from the next-to-top winding number grading of the sutured annular Khovanov homology of the closure of to the next-to-top winding number grading of the sutured annular Khovanov hom…
New method predicts wind farm power and wakes using weather patterns.
problem Inefficient and computationally intensive wind energy resource assessment.
method Unsupervised clustering of ERA5 data on wind velocity, WRF simulations at cluster centers, and post-processing.
result Accurate long-term predictions of power and wakes with reduced computational time.
Study the symmetry and winding numbers of curves defined by sums of exponentials.
problem Understanding the geometry and topology of curves defined by sums of exponentials.
method Analyzing the continuous transition of the graph of the curve as a parameter changes.
result Determined winding numbers and cusp points for curves defined by sums of exponentials.
Divides state space into regions with identical term structure shapes.
problem Classifying term structure shapes in the two-factor Vasicek model.
method Using envelopes and winding numbers to divide and classify the state space.
result Nearly complete classification of parameter space regarding term structure shapes.
We prove that twisted correction terms in Heegaard Floer homology provide lower bounds on the Thurston norm of certain cohomology classes determined by the strong concordance class of a 2-component link in . We then specialise this procedure to knots in , and obtain a lower bound on their geomet…
Methodology extrapolates wind fields from sparse data with uncertainty quantification.
problem Extrapolating wind fields from limited measurements with uncertainty.
method Nonparametric Bayesian dictionary learning for sparse/incomplete data.
result Enhanced extrapolation accuracy, even in high-dimensional data.
New measure shows how links can be untangled as twists increase.
problem Understanding how links can be simplified through repeated twists.
method Introduced the stable unknotting number to analyze links in a twist family.
result The stable unknotting number depends only on the winding number of the link, not the wrapping number.
Interference effects of tall buildings have attracted numerous studies due to the boom of clusters of tall buildings in megacities. To fully understand the interference effects of buildings, it often requires a substantial amount of wind tunnel tests. Limited wind tunnel tests that only cover part of interference scena…
New geometric proof for rational tangles links-quivers correspondence.
problem Recovering symmetric/antisymmetric colored HOMFLY-PT polynomials from a quiver.
method Geometric approach using winding numbers in punctured plane and its second configuration space.
result Explicit description of quivers for rational tangles.
Concordance of knots in math.GT
We establish a number of results about smooth and topological concordance of knots in . The winding number of a knot in is defined to be its class in . We show that there is a unique smooth concordance class of knots with winding number one. …
We study the knot invariant called trunk, as defined by Ozawa, and the relation of the trunk of a satellite knot with the trunk of its companion knot. Our first result is where denotes the trunk of a knot, is a satellite knot with companion , and …
Most of the 50-year history of the study of the set of knot concordance classes, C, has focused on its structure as an abelian group. Here we take a different approach, namely we study C as a metric space admitting many natural geometric operators, especially satellite operators. We consider several knot concordance sp…
Study on parity of singular set components of maps to surfaces.
problem Parity of the number of components of singular set of maps to oriented surfaces.
method Cumulative winding number and invariant I(f) defined to study parity under homotopy.
result Parity of the number of components of singular set does not change under homotopy under certain conditions.
The paper defines invariants for almost graph embeddings and explores their properties.
problem Understanding the properties and limitations of almost graph embeddings in the plane.
method Introducing and analyzing integer invariants (winding number, Wu numbers) for almost embeddings.
result Some values of invariants are realizable for almost embeddings but not for embeddings.
DeepMIDE: A Multi-Output Spatio-Temporal Method for Ultra-Scale Offshore Wind Energy Forecastingcs.LG
DeepMIDE forecasts wind speeds across space, time, and height for offshore wind energy.
problem Forecasting wind speeds across multiple heights for large offshore wind turbines.
method Statistical deep learning model that jointly models wind speeds at different heights using a multi-output integro-difference equation.
result DeepMIDE forecasts outperform traditional methods in real-world offshore wind energy data.
Wind energy resource quantification, air pollution monitoring, and weather forecasting all rely on rapid, accurate measurement of local wind conditions. Visual observations of the effects of wind---the swaying of trees and flapping of flags, for example---encode information regarding local wind conditions that can pote…
On the square peg problemmath.GT
Square can fit inside curves close to smooth ones.
problem Finding inscribed squares in nearly smooth curves.
method Using curvature and a map to relate curves, proving existence of inscribed squares.
result Curves close to smooth ones contain inscribed squares.
Study short-term wind power and speed predictions using machine learning.
problem Accurate short-term wind power and speed predictions for energy systems.
method Combining numerical weather prediction models with local observations, using machine learning for variable selection and forecasting.
result Improved wind power and speed predictions for 4-hour ahead using machine learning.
An extreme wind speed estimation method that considers wind hazard climate types is critical for design wind load calculation for building structures affected by mixed climates. However, it is very difficult to obtain wind hazard climate types from meteorological data records, because they restrict the application of e…
Let be an oriented manifold, let be an oriented closed manifold, and let be a point in . For a smooth map we introduce an invariant that can be regarded as a generalization of the classical winding number of a planar curve around a point. We show…
WindDragon forecasts wind power with deep learning.
problem Accurate short-term wind power forecasting is crucial for grid operation.
method Automated Deep Learning combined with Numerical Weather Predictions.
result Automated Deep Learning improves wind power forecasting accuracy.
Precisely forecasting wind speed is essential for wind power producers and grid operators. However, this task is challenging due to the stochasticity of wind speed. To accurately predict short-term wind speed under uncertainties, this paper proposed a multi-variable stacked LSTMs model (MSLSTM). The proposed method uti…
Using our earlier proposal for Ramond-Ramond fields in an H-flux on loop space, we extend the Hori isomorphism of Bouwknegt-Evslin-Mathai from invariant differential forms, to invariant exotic differential forms such that the momentum and winding numbers are exchanged, filling in a gap in the literature. We also extend…
Applications of shapelet transform to time series classification of earthquake, wind and wave datacs.LG
Shapelet transform improves time series classification for earthquake, wind, and wave events.
problem Autonomous detection of specific events from large time series datasets in civil engineering.
method Shapelet transform for local similarity in time series subsequences, combined with machine learning.
result Shapelet transform yields a new feature representation for time series signals in civil engineering.
We prove that the Garside length a braid is equal to a winding-number type invariant of the curve diagram of the braid.
We conjecture that a non-flat -real-dimensional compact Calabi-Yau manifold, such as a quintic hypersurface with D=6, or a K3 manifold with D=4, has locally length minimizing closed geodesics, and that the number of these with length less than L grows asymptotically as L^{D}. We also outline the physical arguments b…
Improved wind speed forecasts for power generation using machine learning.
problem Improving the accuracy and reliability of wind speed predictions for power generation.
method A novel machine learning approach for calibrating wind speed ensemble forecasts.
result The proposed method improves the calibration and accuracy of probabilistic and point forecasts.
Improving solar wind forecasting using Data Assimilationphysics.space-ph
DA improves solar wind forecasts by updating model boundary conditions.
problem Improving solar wind forecasting accuracy.
method Variational Data Assimilation with solar wind model and in-situ observations.
result DA forecasts are more accurate than non-DA forecasts, especially when STEREO-B's latitude is offset from Earth.
Paper finds exotic diffeomorphisms on specific 4-manifolds.
problem Understanding exotic diffeomorphisms on 4-manifolds with .
method Comparing winding numbers of parameter families.
result 2\mathbb{C}\mathbb{P}^2 \# 10 (-\mathbb{C}\mathbb{P}^2) admits exotic diffeomorphisms.