Researchers create a method to join hyperboloidal data sets without violating the shear-free condition.
problem Creating consistent initial data sets for simulations of spacetime.
method Developed a new gluing procedure that maintains the shear-free condition using special Hölder spaces and elliptic operators.
result Successfully constructed hyperboloidal initial data sets that preserve the shear-free condition.
Study umbilical properties of spacelike 2D submanifolds in semi-Riemannian geometry.
problem Characterize umbilical properties of spacelike 2D submanifolds.
method Introduce total shear tensor and shear operators; analyze relationships; consider novel umbilical notions; prove necessary and sufficient conditions for umbilical submanifolds.
result Unique umbilical direction exists unless submanifold is totally umbilical.
Geodesic patterns, shears, and Anosov representations of the modular group.
problem Understanding representations of the modular group into Isom(X).
method Analyzing geodesic patterns, shears, and foliations.
result The Barbot component is homeomorphic to R^2 x [0,∞), with interior and boundary properties.
Study of circle homeomorphisms with square summable diamond shears.
problem Characterizing circle homeomorphisms with specific summability properties.
method Analysis of homeomorphisms in modular coordinates and comparison to Weil-Petersson class.
result Sharp results comparing new class to Weil-Petersson class and Hölder classes.
Invariants of braids found using shear coordinates in hyperbolic geometry.
problem Finding invariants of braids.
method Using shear coordinates in hyperbolic geometry.
result Developed a method for calculating braids invariants.
New model reduces bias in cosmic shear measurements.
problem Bias in cosmic shear measurements due to non-well-defined ellipticity.
method Hybrid physical and deep learning Hierarchical Bayesian Model.
result Unbiased estimate of shear on realistic galaxies.
Bounding shears in ideal triangulations on hyperbolic surfaces.
problem Bounding shears in ideal triangulations on hyperbolic surfaces.
method Showing an ideal triangulation with bounded shear parameters on hyperbolic surfaces.
result An upper bound on shear parameters depends logarithmically on the surface's topology.
Study fluid spacetimes, proving shear-free implies vanishing expansion or vorticity.
problem Understanding shear and vorticity in perfect-fluid spacetimes.
method Analyzing perfect-fluid spacetimes using Weyl tensor and divergence.
result Proves shear-free implies vanishing expansion or vorticity for perfect fluids.
Shear construction builds solvable Lie algebras from \(\mathbb{R}^n\).
problem Building new solvable Lie algebras from \(\mathbb{R}^n\).
method Using vector bundles with flat connections, shears are defined to construct any solvable Lie algebra from \(\mathbb{R}^n\).
result Any solvable Lie algebra can be obtained by a succession of shears starting from almost Abelian Lie algebras.
Thurston introduced shear deformations (cataclysms) on geodesic laminations - deformations including left and right displacements along geodesics. For hyperbolic surfaces with cusps, we consider shear deformations on disjoint unions of ideal geodesics. The length of a balanced weighted sum of ideal geodesics is defined…
Study counts orbits of mapping class group in shearing coordinates.
problem Counting orbits of mapping class group in shearing coordinates.
method Uses shearing coordinates and asymptotics of Teichmüller space.
result Asymptotic behavior of mapping class group orbits in shearing coordinates.
We parametrize the space Z of Zygmund vector fields on the unit circle in terms of infinitesimal shear functions on the Farey tesselation. Then we express the Hilbert transform and the Fourier coefficients of the Zygmund vector fields in terms of the above parametrization by infinitesimal shear functions. F…
Defines conditions for umbilical submanifolds in arbitrary dimensions.
problem Conditions for umbilical submanifolds in arbitrary dimensions.
method Using the total shear tensor and defining shear and umbilical spaces.
result The sum of dimensions of shear and umbilical spaces equals the co-dimension.
Bayesian Monte-Carlo method assesses uncertainty in shear stress entropy models.
problem Uncertainty in evaluating shear stress entropy models remains an open question.
method Bayesian Monte-Carlo (BMC) uncertainty method to evaluate four entropy models.
result FOCB statistic index determines certainty of entropy models in shear stress estimation.
The paper analyzes thin-shell limits for viscous operators on Riemannian hypersurfaces.
problem Analyzing boundary conditions and thin-shell limits for viscous operators on arbitrary smooth hypersurfaces.
method Decomposing the ambient Bochner Laplacian into intrinsic and radial pieces, proving results for stress-free and Hodge boundary conditions.
result Universal thin-shell limits for viscous operators on arbitrary smooth hypersurfaces, including stress-free and Hodge boundary conditions.
Study predicts shear stress in compound channels using data mining and machine learning.
problem Predicting shear stress distribution in symmetric compound channels.
method Conducted experiments to measure shear stress. Used data mining and machine learning models (RF, M5P, RC, KStar, AR) to predict.
result Random Forest (RF) model showed highest accuracy with R2=0.9.
Twisted SL2C local systems on surfaces of finite type appear often in geometry and physics. Most of them arise geometrically as local systems of charts for pleated hyperbolic structures. Bonahon and Thurston's "shear-bend coordinates" parameterize these local systems of charts. On a surface …
Link between Teichmüller and anti de Sitter geometry via length functions.
problem Understanding the geometry of Teichmüller space and anti de Sitter manifolds.
method Establishing a connection between Teichmüller space and anti de Sitter geometry through length functions.
result New purely anti de Sitter proofs of Teichmüller theory results.
The twist construction is a geometric model of T-duality that includes constructions of nilmanifolds from tori. This paper shows how one-dimensional foliations on manifolds may be used in a shear construction, which in algebraic form builds certain solvable Lie groups from Abelian ones. We discuss other examples of geo…
New geometrization of gravitational wave phase space.
problem Understanding the geometry of null-infinity in asymptotically flat space-times.
method Proposes a new geometrization using tractor calculus adapted to degenerate conformal metrics.
result Gravitational waves correspond to a class of tractor connections called 'null-normal'.
We give parameterizations of homeomorphisms, quasisymmetric maps and symmetric maps of the unit circle in terms of shear coordinates for the Farey tesselation.
A new proof shows how to characterize maps using simple geometry.
problem Characterizing quasisymmetric maps on the unit circle.
method Elementary proof using normal family argument and hyperbolic geometry.
result Characterizes quasisymmetric maps via shear coordinates on the Farey tesselation.
Shearing deformations in Hitchin representations are computed for a symplectic form.
problem Computing symplectic form pairings for Hitchin representations.
method Shearing deformations of Hitchin representations.
result Pairings of shearing deformations computed for the Atiyah-Bott-Goldman symplectic form.
Sparse regression models CMs from oscillatory shear data efficiently.
problem Discovering parsimonious constitutive models from oscillatory shear experiments.
method Sparse regression with tensor basis functions, l1 regularization, and greedy two-stage algorithm.
result Inferred CMs extrapolate well beyond training data and flow conditions.
Physics-informed neural networks simulate solute dispersion in shear flows, validating complex transport mechanisms.
problem Simulating complex solute dispersion in asymmetric reactive environments.
method Physics-informed neural networks (PINNs) embedded with governing equations and boundary conditions.
result PINNs accurately predict solute dispersion, validating transport diagnostics.
Study circular foliations and shear-radius coordinates on hyperbolic cone surfaces.
problem Characterize Teichmüller spaces of hyperbolic cone surfaces.
method Construct circular foliations and shear-radius coordinates on Teichmüller spaces of hyperbolic cone surfaces.
result Shear-radius coordinates provide global coordinates on Teichmüller spaces and converge to specific metrics.
The paper extends optimal transport for linear separability of sheared distributions in supervised learning.
problem Learning on the space of probability measures using shifts and scalings.
method Embedding probability measures into L2 spaces using optimal transport, then applying regular machine learning techniques. result Sheared distributions can be linearly separated under certain conditions, with bounds on transformations.
Classifies two-step solvable Lie groups with SKT structures.
problem Classifying Lie groups with SKT structures.
method Shear construction and analysis of SKT shear data on Abelian Lie algebras.
result Large part of the classification for two-step solvable SKT algebras of dimension six.
Shear moves connect square-tiled surfaces in quadratic differentials.
problem Connecting square-tiled surfaces via specific moves.
method Shear moves corresponding to diagonal flips preserving square-tiled properties.
result Connected components of reconfiguration problem are in bijection with moduli space of quadratic differentials.
The paper studies properties of triangle and shearing invariants in PSL(n,R) and connects them to a slice of Hitchin components.
problem Understanding invariants of PSL(n,R)-Fuchsian representations and their relationship to Hitchin components.
method Examined triangle and shearing invariants, used Bonahon-Dreyer parameterization.
result The Fuchsian locus of Hitchin components corresponds to a slice.
Machine learning improves cosmic shear measurements by compensating for feature noise.
problem Accurately measuring cosmic shear from galaxy images in the presence of various nuisance effects.
method Supervised machine learning with artificial neural networks trained on simulated data.
result Demonstrated competitive low shear biases in Euclid-like images.
Study quasisymmetric maps on hyperbolic plane boundaries.
problem Identify quasisymmetric maps corresponding to specific lambda lengths and flip distances.
method Analyze maps on Farey triangulation, relate to shearing coordinates and flip distance.
result Identify quasisymmetric maps corresponding to pinched lambda lengths and flip distances.
Enhanced Teichmüller space for surfaces with decorations and enhancements.
problem Parameterizing and understanding Teichmüller spaces with enhancements and decorations.
method Introduced a new variation of Teichmüller space, constructed parameterization, and introduced lamination space.
result Compatibility of shear coordinates and λ-length coordinates in the new deformation space.
Abstract: Poisson bracket on shear coordinates relates to Fenchel-Nielsen bracket on gluing parameters.
problem Relationship between Fenchel-Nielsen coordinates and shear coordinates on Riemann surfaces.
method Explicitly showed the Poisson bracket on shear coordinates induces the Fenchel-Nielsen bracket on gluing parameters.
result Poisson bracket on shear coordinates relates to Fenchel-Nielsen bracket on gluing parameters.
We start by describing how ideal triangulations on a surface degenerate under pinching of a multicurve. We use this process to construct a homomorphism from the Ptolemy groupoid of a surface to that of a pinched surface which is natural with respect to the action of the mapping class group. We then apply this construct…
We show that any polyhomogeneous asymptotically hyperbolic constant-mean-curvature solution to the vacuum Einstein constraint equations can be approximated, arbitrarily closely in Hölder norms determined by the physical metric, by shear-free smoothly conformally compact vacuum initial data.
We prove that shear-free perfect fluid solutions of Einstein's field equations must be either expansion-free or non-rotating (as conjectured by Treciokas and Ellis) for all linear equations of state p=wρ except for six values of w.
Geometric model of unbounded sl3 laminations with tropical coordinates.
problem Modeling unbounded laminations in cluster varieties.
method Introducing tropical cluster coordinates and geometric gluing procedures.
result Established a geometric gluing procedure for unbounded sl3 laminations.
We show that the length function of a measured geodesic lamination is convex in Thurston's shear coordinates over Teichmüller space and strictly convex for generic laminations. We give some consequences of this result in the context of Thurston's asymmetric metric on Teichmüller space.
A simple property of Weyl tensor in shear-free, vorticity-free, acceleration-free velocity fields.
problem Proving a property of the Weyl tensor in specific velocity fields.
method Analyzing the Weyl tensor's divergence and contraction properties in shear-free, vorticity-free, acceleration-free velocity fields.
result The covariant divergence of the Weyl tensor is zero if the contraction of the Weyl tensor with the velocity is zero, and vice versa.
The paper models financial order books using geometric shears and directional liquidity.
problem Understanding the geometry and dynamics of financial order books.
method Structural framework modeling liquidity as emergent observables, geometric shears, and directional imbalances.
result The geometry of financial order books can be described by a rigid drift and geometric shear, leading to a gamma-like profile of projected liquidity.
New method connects veering triangulations to dynamic pairs.
problem Understanding veering triangulations and their properties.
method Shearing decomposition of veering triangulations.
result Canonically associated dynamic pairs of branched surfaces.
We consider the Einstein-Maxwell-fluid constraint equations, and make use of the conformal method to construct and parametrize constant-mean-curvature hyperboloidal initial data sets that satisfy the shear-free condition. This condition is known to be necessary in order that a spacetime development admit a regular conf…
We describe the relationship between complex-valued harmonic morphisms from Minkowski 4-space} and the shear-free ray congruences of mathematical physics. Then we show how a horizontally conformal submersion on a domain of Euclidean 3-space gives the boundary values at infinity of a complex-valued harmonic morphism on …
Symplectic structures derived on Teichmüller spaces with holes and bordered cusps.
problem Deriving symplectic structures on Teichmüller spaces with holes and bordered cusps.
method Using fat-graph description, Thurston shear coordinates, and Penner's λ-lengths, a symplectic structure Ω_WP is derived.
result The derived symplectic structure Ω_WP is similar to the Kontsevich symplectic structure for ψ-classes.
Extends earthquake and horocycle flows to new measures.
problem Ergodic theory of earthquake flow on measured laminations.
method Generalizes shear coordinates to arbitrary measured laminations.
result Classifies ergodic measures for P action on bundle of quadratic differentials.
Let S be a closed oriented surface of genus at least 2, and denote by T(S) its Teichm{ü}ller space. For any isotopy class of closed curves γ, we compute the first three derivatives of the length function ℓ_γ:T(S)→R_+ in the shearing coordinates associated to a maxim…
Equivalences between conformal foliations on Euclidean 3-space, Hermitian structures on Euclidean 4-space, shear-free ray congruences on Minkowski 4-space, and holomorphic foliations on complex 4-space are explained geometrically and twistorially; these are used to show that 1) any real-analytic complex-valued …