Sharp inequalities and existence results for Liouville equations on spheres.
problem Sharp inequalities and existence results for Liouville equations on spheres.
method Sharp Onofri-type inequality and non-existence of extremals for a Moser-Tudinger functional on the sphere.
result Existence of critical points and sufficient conditions under symmetry or nondegeneracy assumptions.
Classifies graphs with Bonnet-Myers sharp curvature and connects it to Bakry-Émery curvature.
problem Classifying graphs with specific curvature properties.
method Introducing Bonnet-Myers and Lichnerowicz sharpness in Ollivier Ricci curvature, combinatorial reformulation, and relating to Bakry-Émery curvature.
result Classification of graphs including hypercubes, cocktail party graphs, and Johnson graphs J(2n,n). ASAM improves deep neural network generalization by adapting sharpness to scale.
problem Fixed-radius sharpness measure is sensitive to parameter scaling, weakening its connection to generalization.
method Introduces adaptive sharpness, a scale-invariant measure, and proposes ASAM for deep learning.
result ASAM significantly improves model generalization performance across various datasets.
Sharp log-Sobolev inequalities proved for CD(0,N) spaces.
problem Proving log-Sobolev inequalities in noncompact metric measure spaces.
method Sharp isoperimetric inequality, symmetrisation, scaling argument, Hamilton-Jacobi inequality, Sobolev regularity.
result Sharp log-Sobolev inequalities established in CD(0,N) spaces. Sharp bounds on Alexandrov spaces' boundaries with rigidity analysis.
problem Volume bounds on Alexandrov spaces' boundaries.
method Sharp volume bounds and rigidity analysis of Alexandrov spaces.
result New sharp volume bounds and classification of rigidity cases.
Sharp heat equation gradient estimates on compact manifolds.
problem Gradient estimates for positive solutions on weighted manifolds.
method Proving sharp gradient estimates for positive solutions to the weighted heat equation.
result Refined gradient estimates and Liouville theorems for ancient solutions.
Sharpness minimization algorithms don't solely improve generalization.
problem Why do overparameterized neural networks generalize?
method Theoretical and empirical investigation of two-layer ReLU networks.
result Sharpness minimization algorithms do not always lead to better generalization.
Sharp spectral gap estimates on manifolds with integral curvature bounds.
problem Proving spectral gap estimates on manifolds with integral curvature bounds.
method Generalizing previous results to include integral curvature bounds.
result Confirms a conjecture about spectral gap estimates on manifolds with integral curvature bounds.
Sharp bounds on mean curvature and geodesic lengths in convex hypersurfaces.
problem Finding sharp bounds on total mean curvature of convex hypersurfaces.
method Sharp lower bounds for mean width and Birkhoff invariant, characterizing spheres.
result Generalization of Álvarez Paiva's result to convex hypersurfaces.
Sharp chord-arc estimates for curve shortening flow on spheres.
problem Understanding the behavior of curves on spheres under curve shortening flow.
method Proving sharp chord-arc estimates and curvature control.
result Simple spherical curves either contract to points or converge to great circles.
Paper connects Sharpe ratio and Student t-statistic, providing exact distribution and asymptotic behavior.
problem Error-prone Sharpe ratio due to statistical estimation of expected returns and volatilities.
method Derive exact distribution of Sharpe ratio for independent normally distributed returns, extend to AR(1) assumptions.
result Empirical Sharpe ratio is asymptotically optimal and achieves Cramer Rao bound.
Sharp estimate for flow in any dimension.
problem Interior gradient estimate for graphical mean curvature flow.
method Proving sharp interior gradient estimate for area decreasing graphical mean curvature flow in arbitrary codimension.
result Generalized result in arbitrary codimension.
Generalized Blaschke rolling theorem for curved spaces.
problem Extending classical theorem to curved spaces.
method Generalization to Riemannian manifolds with bounded curvature.
result Sharp results in arbitrary dimensions, new even in constant curvature spaces.
SAM improves neural network generalization by penalizing sharpness, clarifying its exact notion and mechanism.
problem Improving deep neural network generalization for various settings.
method Sharpness-Aware Minimization (SAM) technique that penalizes a notion of sharpness of the model.
result SAM regularizes the third notion of sharpness, most likely preferred for practical performance.
We prove that two homogeneous ultra-metric spaces X,Y are coarsely equivalent if and only if Ent♯(X)=Ent♯(Y) where Ent♯(X) is the so-called sharp entropy of X. This classification implies that each homogeneous proper ultra-metric space is coarsely equivalent to the a…
Optimal option portfolios under Sharpe Ratio maximization with skew-elliptical t-distributed returns
problem Optimal option portfolios under Sharpe Ratio maximization
method Formulation for explicit portfolio weights
result Different optimal portfolios for Sharpe Ratio and return-to-Value-at-Risk (VaR) ratio
The Fano ratio offers a more diversified portfolio than the Sharpe ratio in optimization.
problem Optimizing portfolio performance using the Sharpe ratio.
method Introducing and optimizing the Fano ratio as an alternative to the Sharpe ratio.
result Optimizing for the Fano ratio leads to more diversified and less skewed portfolios.
We address the question of determining the eigenvalues λ_n (listed in nondecreasing order, with multiplicities) for which Courant's nodal domain theorem is sharp i.e., for which there exists an associated eigenfunction with n nodal domains (Courant-sharp eigenvalues). Following ideas going back to Pleijel (1956), …
Sharp criterion for Chern-Gauss-Bonnet integral using Q curvature.
problem Quantifying the Chern-Gauss-Bonnet integral using Q curvature.
method New approach involving singular integral estimation.
result Derivation of asymptotic formula for Q curvature equation.
Sharp comparison theorems for 3D manifolds with scalar curvature bound.
problem Understanding the geometry and topology of 3D manifolds with scalar curvature constraints.
method Sharp comparison results for Green's function and spectrum, derived from scalar curvature bounds.
result Sharp upper and lower bounds for the Green's function and spectrum of 3D manifolds.
Deep linear networks minimize sharpness, avoiding large eigenvalues.
problem Understanding optimization dynamics in deep linear networks for regression.
method Analyzing sharpness (largest eigenvalue of Hessian) of minimizers and gradient flow solutions.
result Gradient flow implicitly regularizes towards flat minima, with sharpness bounded by a constant.
Sharp gradient estimate for heat kernels on metric measure spaces.
problem Establishing gradient estimates for heat kernels on metric measure spaces.
method Elliptic local Li-Yau gradient estimate for weak solutions of the heat equation.
result Sharp gradient estimate for the logarithm of heat kernels.
Sharp inscribed radius estimate for curved surfaces.
problem Estimating the inscribed radius of curved surfaces.
method Proving a sharp estimate for inscribed radius under fully nonlinear curvature flows.
result The estimate is asymptotically sharp on cylinders.
Sharp results link DLN gradient flow to basis pursuit optimization and GHA phase transitions.
problem Understanding implicit regularization in Diagonal Linear Networks.
method Sharp convergence bounds and characterization of ℓ1 minimizers. result Gradient flow of DLNs with tiny initialization approximates minimizers of basis pursuit optimization problem.
Sharp boundaries for detecting dense subhypergraphs established.
problem Detecting dense subhypergraphs in random hypergraphs.
method Established sharp detection boundaries for known and unknown edge probabilities.
result Sharp detectable regions differ significantly from graph counterparts.
Estimates true Sharpe ratio of selected assets with various methods.
problem Estimating the true Sharpe ratio of a selected asset with high in-sample ratio.
method Polyhedral lemma, James Stein shrinkage, debiasing, thresholding, empirical Bayes.
result James Stein estimator performs best across various parameter values.
New knot invariant s♯ reveals unexpected phenomena.
problem Understanding the new knot invariant s♯ and its properties. method Computations and new characterizations of s♯ using immersed cobordisms. result Invariant s♯ does not agree with s and is not additive under connected sums. Sharp curvature estimates for mean curvature flow in spheres.
problem Understanding the behavior of surfaces evolving under mean curvature flow in spheres.
method Proving asymptotically sharp curvature pinching estimates and using them to derive derivative and convexity estimates.
result Partial classification of singularity models and new rigidity results for ancient solutions.
Sharp Sobolev inequalities proved on manifolds with non-negative Ricci curvature.
problem Proving sharp Sobolev inequalities on noncompact Riemannian manifolds with non-negative Ricci curvature.
method Using Optimal Mass Transportation with quadratic distance cost.
result Sharp Lp-Sobolev and Lp-logarithmic Sobolev inequalities established for p>1 and p=1. Sharp rigidity theorem for quasilinear Liouville equation on manifolds with nonnegative Ricci curvature.
problem Characterizing solutions to the quasilinear Liouville equation on manifolds with nonnegative Ricci curvature.
method Using a sharp logarithmic lower bound and a sharp upper bound on the total volume of the solution.
result If a solution satisfies a specific logarithmic lower bound, the manifold is isometric to Euclidean space and the solution is a standard bubble solution.
Study energy-minimizing maps in projective spaces, proving sharp bounds.
problem Finding optimal mappings in projective spaces.
method Proving lower bounds and characterizing energy-minimizing maps.
result Sharp lower bounds and characterization of energy-minimizing maps.
Sharp gradient estimates for positive Ricci curvature manifolds.
problem Understanding geometric properties of manifolds with positive Ricci curvature.
method Proving sharp gradient estimates and monotonicity formulae.
result Sharp gradient estimates and monotonicity formulae for positive Ricci curvature manifolds.
We prove a comparison theorem on the first Neumann eigenvalue on Bakry-Emery manifolds. Examples are constructed to illustrate the sharpness of the result. A linear explicit lower bound is also proved. We also discuss the asymptotic sharpness of such a result.
Sharp gradient estimates for heat equation in Riemannian manifolds proved.
problem Proving sharp gradient estimates for heat equation in Riemannian manifolds.
method Analyzing the heat equation ut=Δu+aulogu in complete noncompact Riemannian manifolds. result Gradient estimates are sharp and imply that if u is a positive solution of ut=Δu and logu is of sublinear growth, then u must be constant. We derive a sharp cusp count for finite volume complex hyperbolic surfaces which admit smooth toroidal compactifications. We use this result, and the techniques developed in [DiC12], to study the geometry of cusped complex hyperbolic surfaces and their compactifications.
Paper proposes a robust Sharpe ratio portfolio method.
problem Comparing portfolios with unknown risk-free asset.
method Cross-efficiency evaluation to maximize Sharpe ratio.
result Explicit expression for robust Sharpe ratio portfolio.
GD monotonically decreases GFS sharpness in neural networks and scalar models.
problem Oscillatory behavior of loss in GD training.
method Analysis of GFS sharpness and empirical validation.
result GFS sharpness decreases monotonically during GD training.
Tests Sharpe ratio for skill vs luck in asset management.
problem Accuracy of Sharpe ratio in measuring skill vs luck.
method Statistical tests to assess the significance of Sharpe ratios.
result Tests reveal the statistical significance of Sharpe ratios and their impact of auto-correlation.
Sharp-MAML improves MAML by reducing saddle points in few-shot learning.
problem Challenges in optimizing MAML due to complex loss landscape.
method Sharpness-aware minimization applied to MAML.
result Sharp-MAML and its variant outperform plain MAML on few-shot learning tasks.
Sharp inequality found on three-balls for fourth order Sobolev traces.
problem Fourth order Sobolev trace inequality on three-balls.
method Established through equivalence to a third order Sobolev inequality on two-spheres.
result Sharp fourth order Sobolev trace inequality on three-balls.
We find empirically a characteristic sharp peak-flat trough pattern in a large set of commodity prices. We argue that the sharp peak structure reflects an endogenous inter-market organization, and that peaks may be seen as local ``singularities'' resulting from imitation and herding. These findings impose a novel strin…
Sharp signature bound for positive four-braids derived from three-braids.
problem Optimizing the signature bound for positive four-braids.
method Combining bounds for positive three-braids with Gordon and Litherland's approach to signature via unoriented surfaces and Goeritz forms.
result Improved linear bounds for the signature in terms of the three-genus of closures and first Betti number.
Sharp upper bounds found for solutions of a specific equation on Riemannian manifolds.
problem Finding upper bounds for solutions of a specific equation on Riemannian manifolds.
method Proved sharp upper estimates of weak subsolutions to the Leibenson equation on Riemannian manifolds with non-negative Ricci curvature.
result Improved and proved a conjecture about upper bounds for solutions of the Leibenson equation.
Noise in linear networks minimizes sharpness and leads to shrinkage-thresholding.
problem Minimizing sharpness in diagonal linear networks.
method Stochastic sharpness-aware minimization (SAM) with isotropic noise.
result Noise forces shrinkage-thresholding of true parameters.
Sharp bounds on hyperbolic surfaces with geodesic boundaries.
problem Finding maximal injectivity radii for hyperbolic surfaces with geodesic boundaries.
method Sharp upper bounds derived for all surfaces with any fixed topology, independent of boundary lengths.
result Extends previous results to surfaces with geodesic boundaries.
The paper improves Fréchet bounds and sharpens their applicability.
problem Uncertainty in joint distribution and marginals in multivariate distributions.
method Optimal transport duality, representation of increasing convex functionals, explicit computation of conjugates.
result Improved Fréchet-Hoeffding bounds are pointwise sharp under uncertainty in marginals but not in absence.
Sharp bounds on eigenfunctions of the Ornstein-Uhlenbeck operator.
problem Understanding growth rates of eigenfunctions.
method Proving sharp bounds up to lower order terms.
result Sharp bounds for eigenfunctions growth rates.
Sharp estimates for mean curvature flow of graphs are shown and examples are given to illustrate why these are sharp. The estimates improves earlier (non-sharp) estimates of Klaus Ecker and Gerhard Huisken.