Fine shape theory extends strong shape to noncompact metrizable spaces.
problem Computational complexity in extending strong shape to noncompact spaces.
method Introducing FDR-embeddings and mapping cylinders to extend SSDR-maps to noncompact spaces.
result Fine shape category can be represented as a left fraction localization.
Study shows Dehn twist coefficients are consistent across different actions on surfaces.
problem Consistency of fractional Dehn twist coefficients under various actions.
method Analyzes left orderings of mapping class groups and uses cofinality properties.
result Fractional Dehn twist coefficients are independent of the underlying action for surfaces with genus > 1.
Study asymptotics of Poisson kernel and Green's functions for fractional conformal Laplacian.
problem Asymptotics of Poisson kernel and Green's functions for fractional conformal Laplacian.
method Sharp expansions derived for the Poisson kernel and Green's functions near singularities.
result Sharp expansions of the Green's functions solve the first part of Kim-Musso-Wei's conjecture.
We define symplectic fractional twists, which generalize Dehn twists, and use these in open books to investigate contact structures. The resulting contact structures are invariant under a circle action, and share several similarities with the invariant contact structures that were studied by Lutz and Giroux. We show th…
Characterizes braid types and estimates twist coefficients.
problem Understanding braid types and their properties.
method Birman-Ko-Lee left canonical form of braids.
result Characterization of almost strongly quasipositive braids and estimates of fractional Dehn twist coefficient.
A new framework for pricing the European currency option is developed in the case where the spot exchange rate fellows a time-changed fractional Brownian motion. An analytic formula for pricing European foreign currency option is proposed by a mean self-financing delta-hedging argument in a discrete time setting. The m…
New graph feedback model for bandits with improved regret bounds.
problem Understanding how graph structure affects regret in bandit problems.
method Introduced fractional weak domination number and k-packing independence number to capture upper and lower bounds on regret. Used strong duality theorem to derive upper and lower bounds. result Proved general upper and lower bounds on regret for various graph structures, showing tightness up to a logarithmic factor.
Left orderability for surgeries on the [1,1,2,2,2j] two-bridge knotsmath.GT The paper proves left-orderable surgeries for a specific type of knot.
problem Existence of left-orderable Dehn surgeries on a specific knot.
method Cohomological criterion applied to two-bridge knots.
result Proves left-orderable Dehn surgeries for the [1,1,2,2,2j] two-bridge knots. We study elliptic gradient systems with fractional laplacian operators on the whole space (−Δ)su=∇H(u) in Rn, where u:Rn→Rm, H∈C2,γ(Rm) for γ>max(0,1−2min{si}), $\mathbf s=(s_1,\cdot…
Fractional porous media equations yield q-Gaussian solutions for stock price returns.
problem Modeling stock price returns using fractional porous media equations.
method Analyzed three types of fractional extensions of the porous media equation.
result Local and non-local fractional extensions fit S&P 500 data better than classical models.
Local fractional derivatives affect Riemann curvature tensor to zero.
problem Investigating how local fractional derivatives influence the Riemann curvature tensor.
method Introduced a general local fractional derivative operator and defined a specific Riemannian metric tensor field.
result The Riemann curvature tensor of the new metric is identically zero, indicating local isometry to Euclidean space.
Let Sg be a closed orientable surface of genus g≥2 and C a simple closed nonseparating curve in F. Let tC denote a left handed Dehn twist about C. A \textit{fractional power} of tC of \textit{exponent} $\fraction{\ell}{n}$ is an $h \in \Mod(S_g)$ such that hn=tCℓ. Unlike a root of a $t…
Study examines surfaces with bounded fractional mean curvature, proving control over local parametrization.
problem Understanding surfaces with bounded fractional mean curvature.
method Investigates bounded L^p-norm of fractional mean curvature, proving control over local parametrization.
result Proves control over local parametrization, leading to lower Ahlfors-regularity, weak Michael-Simon type inequality, and stability application.
We establish a relationship between Heegaard Floer homology and the fractional Dehn twist coefficient of surface automorphisms. Specifically, we show that the rank of the Heegaard Floer homology of a 3-manifold bounds the absolute value of the fractional Dehn twist coefficient of the monodromy of any of its open book d…
Study finds non-uniqueness in sphere metrics with constant fractional curvature.
problem Non-uniqueness of metrics with constant positive fractional curvature on spheres.
method Bifurcation techniques applied to non-local equations with critical non-linearity.
result Non-uniqueness results for complete metrics on Sn∖Sk. The study calculates average crosscap numbers for 2-bridge knots.
problem Determining the average crosscap number of 2-bridge knots.
method Using continued fraction expansions and recursion, the study provides exact formulas for average crosscap numbers.
result The study shows that the limit of the average crosscap number of 2-bridge knots approaches zero as the crossing number increases.
New algorithms solve word and conjugacy problems in braid group B3.
problem Word and conjugacy problems in braid group B3.
method Classical interpretation of braid group B3 as central extension of modular group, theory of continued fractions.
result Simple and efficient algorithms to solve word and conjugacy problems in braid group B3.
The paper uses FRFT to fit GTS distribution to asset returns.
problem Modeling asset returns with GTS distribution.
method Fractional Fourier Transform (FRFT) for fitting.
result GTS distribution fits SPY ETF and Bitcoin BTC returns.
The purpose of the present paper is to study the globally and locally φ-T-symmetric (ε)-para Sasakian manifold in dimension 3. The globally φ-T-symmetric 3-dimensional (ε)-para Sasakian manifold is either Einstein manifold or h…
Paper introduces new fractional Dirac operator and Q-curvature.
problem Fractional Dirac operator and Q-curvature in spinors.
method Caffarelli-Silvestre extension, energy inequalities, weighted Sobolev inequality.
result Introduction of conformal fractional Dirac operator and Q-curvature.
Geometry of the tracks left by a bicycle is closely related with the so-called Prytz planimeter and with linear fractional transformations of the complex plane. We describe these relations, along with the history of the problem, and give a proof of a conjecture made by Menzin in 1906.
Improved Thompson Sampling using fractional posteriors achieves better regret bounds.
problem Optimizing regret in stochastic multi-armed bandit problems.
method Using α-posterior distributions, derived frequentist regret bounds. result Instance-dependent and instance-independent regret bounds established.
New framework for ranking distributions using variable fractional parameters.
problem Ordering distributions with varying steepness and local non-concavities.
method Introducing a function γ:Ro[0,1] to replace the fixed parameter in fractional SD. result Enables ranking of a broader range of distributions and incorporates dynamic greediness.
LU-Net improves cardiac segmentation accuracy and robustness.
problem Robustness and accuracy of deep learning cardiac segmentation.
method Multi-task end-to-end network designed to improve cardiac segmentation.
result Outperforms current best deep learning solution, reducing outliers and improving clinical indices.
Introduces q-transpose for q-deformed modular group matrices.
problem Understanding q-deformed rational numbers and their properties. method Introduces q-transpose and applies it to refine q-deformed modular group actions. result New proof and refinement of Leclere and Morier-Genoud's trace palindromicity theorem.
We study in this paper the fractional Yamabe problem first considered by Gonzalez-Qing on the conformal infinity (Mn,[h]) of a Poincaré-Einstein manifold (Xn+1,g+) with either n=2 or n≥3 and (Mn,[h]) is locally flat - namely (M,h) is locally conformally flat. However, as for the classic…
Fractional Laplacian inverse problem solved for connection Laplacians.
problem Determining structures from metric, bundle, and map knowledge.
method Local knowledge of metric, bundle, and map determines global structures.
result Global structures determined from local knowledge of metric, bundle, and map.
Study on error rates for approximating rough volatility models.
problem Simulation of rough volatility models with fractional Brownian motion.
method Analysis of weak error rates for numerical schemes, focusing on fBm and cubic test functions.
result Convergence rates for approximations are (3H+21)∧1 for exact left-point discretization and H+21 for hybrid schemes. It has been a long-standing problem to efficiently learn a halfspace using as few labels as possible in the presence of noise. In this work, we propose an efficient Perceptron-based algorithm for actively learning homogeneous halfspaces under the uniform distribution over the unit sphere. Under the bounded noise condit…
This paper explores how rational numbers on the Stern-Brocot diagram map to lines when terms are extended.
problem Understanding the geometry of rational numbers on the Stern-Brocot diagram.
method Analyzing continued fraction expansions and their geometric implications on the diagram.
result Vertices of the Stern-Brocot diagram corresponding to extended rational numbers lie on two Euclidean lines.
Quantifies fractional isoperimetric inequality with strong control over boundary oscillation.
problem Fractional isoperimetric inequality and its quantitative aspects.
method Regularization process with a new spirit.
result Stability estimates for fractional Cheeger inequality.
The study bounds the number of closed geodesics in a specific orbit closure of surfaces.
problem Counting closed geodesics in a specific orbit closure of surfaces.
method Analyzes triangulations and Teichmüller geodesics to bound the number of closed geodesics.
result Obtains exponential bounds on the number of closed geodesics of length at most R.
Inspired by the role geometric structures play in our understanding of surfaces and three-manifolds, and Berger's observation that a surface of constant sectional curvature is determined up to local isometry by its Laplace spectrum, we explore the extent to which compact locally homogeneous three-manifolds are characte…
Robust testing of sparse signals in corrupted data.
problem Testing the norm of high-dimensional sparse signals in the presence of arbitrary corruption.
method Two observation models: i.i.d. samples from N(θ,Id) and sparse linear regression model. result The robust testing requires significantly more samples than non-robust testing.
Study proves certain algebraic structures are symmetric Frobenius algebras.
problem Understanding algebraic structures in bordered surfaces.
method Analyzing stated skein algebras and their fraction rings.
result Fraction ring of stated skein algebra is a symmetric Frobenius algebra.
New algorithm for robust density estimation in corrupted data.
problem Density estimation in the presence of adversarial corruption.
method Proposes an algorithm for constructing a density estimator within a star-shaped density class, derived minimax bounds for estimation.
result Obtained minimax upper and lower bounds for density estimation under adversarial corruption.
The study provides a criterion for fractional-linear integrals of geodesics on surfaces.
problem Existence and classification of fractional-linear integrals for geodesic flows on Riemannian surfaces.
method Criterion and analysis of moduli space of local integrals.
result The moduli space of such local integrals is either the 2D projective plane or finite points.
Linear ODEs are solved by geodesics in hyperbolic geometry.
problem Solving real linear second order ODEs.
method Defined a Riemannian hyperbolic geometry and showed that solutions to ODEs correspond to geodesics in this geometry.
result Local solutions to ODEs correspond to geodesics in a specific hyperbolic geometry.
In this paper, the fractional order curvature equation (−Δ)γu=(1+εK(x))uN−2γN+2γ in RN is considered. Assuming K(x) has two critical points satisfying certain local conditions, we prove the existence of two-peak solutions.
Study shows compact Sasakian manifolds are locally Heisenberg up to deformation.
problem Characterizing compact Sasakian manifolds.
method Analyzing basic Chern classes and using left invariant Sasakian structures.
result Compact Sasakian manifolds are locally isomorphic to the real Heisenberg group.
The study examines the chaos of fractional Brownian fields as Hurst parameter approaches zero.
problem Understanding the chaos of fractional Brownian fields as their Hurst parameter tends to zero.
method Defining normalizing kernels and using Berestycki's ``good points'' approach to derive the limiting measure of multiplicative chaos.
result The limiting measure of multiplicative chaos converges to a log-correlated Gaussian field as the Hurst parameter approaches zero.
Paper introduces a new optimization method for imbalanced datasets.
problem Overfitting in imbalanced datasets, especially in financial fraud detection.
method Fractional Weyl Integral optimization algorithm.
result Significantly improved performance in financial fraud detection (40% improvement in PR-AUC).
The paper enhances representations to show left-orderability of certain 3-manifold groups.
problem Left-orderability of 3-manifold groups using enhanced representations.
method Recalibration of Calegari and Dunfield's flipping construction for $\mbox{Homeo}_+(S^1)$-representations.
result Branched covers of links are left-orderable, generalizing known results.
We consider four dimensional Lie groups with left-invariant Riemannian metrics. For such groups we classify left-invariant conformal foliations with minimal leaves of codimension two. These foliations produce local complex-valued harmonic morphisms.
Classifies left invariant Kundt structures on 3D Lie groups.
problem Understanding Kundt spacetimes and their properties.
method Analyzes local structure and properties of left invariant Kundt structures.
result Classifies all left invariant Kundt structures on 3D simply connected unimodular Lie groups.
The study characterizes elements of a group quotient and finds a non-locally indicable left-orderable group.
problem Understanding the structure and properties of a specific group quotient.
method Introduced an invariant for quasi-isometries of the positive real line and split it into units.
result Found a quotient of the quasi-isometry group of the positive real line that is left-orderable but not locally indicable.
We study the localization of sets with constant nonlocal mean curvature and prescribed small volume in a bounded open set with smooth boundary, proving that they are {\em sufficiently close} to critical points of a suitable non-local potential. We then consider the fractional perimeter in half-spaces. We prove the exis…
We prove that the geodesic equations of all Sobolev metrics of fractional order one and higher on spaces of diffeomorphisms and, more generally, immersions are locally well posed. This result builds on the recently established real analytic dependence of fractional Laplacians on the underlying Riemannian metric. It ext…