We give conditions on a knot on which the Morton-Franks-Williams inequality is not sharp. As applications, we show infinitely many examples of knots where the inequality is not sharp and also prove (by giving examples) that the deficit of the inequality can be arbitrarily large.
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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.
Sharp proof of sub-Riemannian length-minimizing curves being at least
problem Smoothness of sub-Riemannian length-minimizing curves
method Study of a class of sub-Riemannian structures, proving regularity
result Theorem 1.1 in [6] is sharp
We discuss - in what is intended to be a pedagogical fashion - generalized "mean-to-risk" ratios for portfolio optimization. The Sharpe ratio is only one example of such generalized "mean-to-risk" ratios. Another example is what we term the Fano ratio (which, unlike the Sharpe ratio, is independent of the time horizon)…
A simple function shows how neural nets can converge despite high sharpness.
problem Understanding why neural nets converge with high sharpness.
method Constructed a minimal example function and analyzed its training dynamics rigorously.
result Final converging point has sharpness close to .
Sharp Hardy and spectral gap inequalities found on special irreversible Finsler manifolds.
problem Understanding Hardy and spectral gap inequalities on irreversible Finsler manifolds.
method Finslerian extension of the method of Riccati pairs.
result Sharpness of Hardy and spectral gap inequalities on specific Finsler manifolds.
A simple example shows that losing all money is compatible with a very high Sharpe ratio (as computed after losing all money). However, the only way that the Sharpe ratio can be high while losing money is that there is a period in which all or almost all money is lost. This note explores the best achievable Sharpe and …
Going Further: Flatness at the Rescue of Early Stopping for Adversarial Example Transferabilitycs.LG
SAM minimizes loss sharpness, improving adversarial transferability.
problem Improving adversarial transferability of deep neural networks.
method Evaluating surrogate models trained with seven minimizers, focusing on loss sharpness and flat neighborhoods.
result SAM minimizes loss sharpness, leading to better adversarial transferability.
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.
Using Seifert fibered three-manifold examples of Boileau and Zieschang, we demonstrate that the Reshetikhin-Turaev quantum invariants may be used to provide a sharp lower bound on the Heegaard genus which is strictly larger than the rank of the fundamental group.
Sharp bounds on hyperbolic metrics in Ptolemaic spaces are derived.
problem Finding sharp bounds on hyperbolic metrics in Ptolemaic spaces.
method Construction of metrics on open subsets of Ptolemaic spaces.
result Sharp parameter bounds for hyperbolic and strongly hyperbolic metrics are derived.
Sharp inequalities in unit ball with constraints on moments.
problem Establishing Sobolev trace inequalities with constraints.
method Constructing smooth test functions for higher order moments.
result Almost optimal Sobolev trace inequalities for 2nd and 4th orders.
New extension theorem for projective manifolds.
problem Extension of twisted canonical forms on hypersurfaces.
method Established a new extension result for canonical forms on hypersurfaces with simple normal crossings.
result Obtained sharp bounds for the extension.
Researchers found the first and second eigenvalues are Courant-sharp on a Möbius strip.
problem Determining Courant-sharp eigenvalues on a Möbius strip.
method Analyzing the eigenvalues and nodal patterns of the Möbius strip.
result Only the first and second eigenvalues are Courant-sharp on the Möbius strip.
The study analyzes sharpness dynamics in neural networks, revealing mechanisms and conditions.
problem Understanding sharpness in neural network training.
method Fixed point analysis and edge of stability analysis in a simplified 2-layer linear network.
result Reveals mechanisms behind sharpness trends, conditions for edge of stability, and a period-doubling route to chaos.
WARPd method solves inverse problems with approximate sharpness conditions.
problem Reconstruction of signals from undersampled and noisy measurements.
method First-order method based on primal-dual iterations with restart-reweight scheme.
result WARPd achieves stable linear convergence under generic approximate sharpness condition.
Sharp decay estimates for the logarithmic fast diffusion equation and the Ricci flow on surfacesmath.AP
We prove the sharp local L^1 - L^\infty smoothing estimate for the logarithmic fast diffusion equation, or equivalently, for the Ricci flow on surfaces. Our estimate almost instantly implies an improvement of the known L^p - L^\infty estimate for p larger than 1. It also has several applications in geometry, providing …
New examples show inequalities can be sharp even when self-linking and genus differ.
problem Comparing upper bounds for knot self-linking number.
method Examining specific knots to test inequalities.
result Inequalities can be sharp even when self-linking and genus differ.
Sharp Hardy inequalities on Riemannian submanifolds with non-negative curvature.
problem Establishing Hardy inequalities for submanifolds in Riemannian geometry.
method Analyzing distance functions and using Riemannian submanifolds with non-negative curvature.
result Sharp weighted Hardy inequalities valid for compact and non-compact submanifolds, even in compact ambient manifolds.
We give examples showing that Kidwell's inequality for the maximal degree of the Brandt-Lickorish-Millett-Ho polynomial is in general not sharp.
Sharp estimates for mean curvature flow confirm bounded diameter conjecture.
problem Bounding the diameter of mean curvature flow in 3D.
method Quantitative estimates on second fundamental form and singular set.
result Uniform boundedness of intrinsic diameter and sharp estimates on flow properties.
Explains optimal functional inequalities, focusing on Sobolev and fractional Sobolev.
problem Optimal functional inequalities and their stability.
method Compactness theorems, characterization of optimizers, and quantitative stability analysis.
result Characterization and stability of optimizers for Sobolev inequalities and their fractional generalizations.
The VIX is used to enhance quantitative trading strategies.
problem Improving Sharpe ratio and reducing trading risks in quantitative strategies.
method Postprocessing quantitative strategies with VIX signals.
result Increased Sharpe ratio and reduced trading risks.
We present a new methodology of computing incremental contribution for performance ratios for portfolio like Sharpe, Treynor, Calmar or Sterling ratios. Using Euler's homogeneous function theorem, we are able to decompose these performance ratios as a linear combination of individual modified performance ratios. This a…
Bakry-Émery curvature sharpness and curvature flow in finite weighted graphs. II. Implementationmath.CA
Flow preserves curvature sharpness on weighted graphs.
problem Curvature flow on weighted graphs.
method Adapting Bakry-Émery calculus for Markovian preservation and analyzing limits.
result Flow limits to curvature sharp weighted graphs.
Study finds only first and second eigenvalues are Courant-sharp for flat Klein bottle and cylinders.
problem Determining Courant-sharp eigenvalues for compact flat surfaces.
method Analyzing flat Klein bottle and cylinders, proving only first and second eigenvalues are Courant-sharp.
result Only first and second eigenvalues are Courant-sharp for flat Klein bottle and cylinders.
Liouville theorems for harmonic map heat flow along ancient super Ricci flow via reduced geometrymath.DG
Study Liouville theorems for harmonic maps along ancient super Ricci flows.
problem Proving Liouville theorems for harmonic maps under specific geometric conditions.
method Using Perelman's reduced geometric viewpoint, derive Liouville theorems with controlled growth.
result Sharp growth conditions and new Liouville theorems for both non-positively and positively curved target spaces.
We provide the optimal linear bound for the signature of positive four-braids in terms of the three-genus of their closures. As a consequence, we improve previously known linear bounds for the signature in terms of the first Betti number for all positive braid links. We obtain our results by combining bounds for positi…
Sharp inequalities proved for RCD spaces, showing equality conditions.
problem Proving sharp inequalities for RCD spaces and identifying equality conditions.
method Analyzing and spaces to prove inequalities and identify equality conditions.
result Equality conditions for Buser's and Cheeger's inequalities in RCD spaces.
Unwinding spiralsmath.GT
We show that there is no bi-Lipschitz homeomorphism of that maps a spiral with a sub-exponential decay of winding radii to an unwinded arc. This result is sharp as shows an example of a logarithmic spiral.
We prove a sharp pinching estimate for immersed mean convex solutions of mean curvature flow which unifies and improves all previously known pinching estimates, including the umbilic estimate of Huisken, the convexity estimates of Huisken--Sinestrari and the cylindrical estimate of Huisken--Sinestrari. Namely, we show …
New examples show flat singular sets can be arbitrarily complex.
problem Understanding the structure of singular sets in almost-minimizing currents.
method Construction of specific examples of area almost-minimizing currents.
result Flat singular sets can contain any closed empty interior subset of a plane.
Adversarial training makes logistic regression weight loss landscapes sharper.
problem Understanding why adversarial training sharpens the weight loss landscape in logistic regression.
method Theoretical analysis of linear logistic regression model with L2 norm constraints, and experiments on ResNet18.
result Adversarial training sharpens the weight loss landscape in linear logistic regression models.
Study cohomology classes related to -harmonic morphisms and -harmonic maps.
problem Understanding cohomology classes associated with -harmonic morphisms and -harmonic maps.
method Utilizing the -conservation law (2.6) to obtain sharp results.
result Sharp results on cohomology classes related to -harmonic morphisms and -harmonic maps.
We study the profitability of optimal mean reversion trading strategies in the US equity market. Different from regular pair trading practice, we apply maximum likelihood method to construct the optimal static pairs trading portfolio that best fits the Ornstein-Uhlenbeck process, and rigorously estimate the parameters.…
Sphere theorems proved for manifolds with specific curvature conditions.
problem Sphere theorems for Riemannian manifolds with curvature operator constraints.
method Investigation of eigenvalues and curvature operator conditions.
result Proved sphere theorems in dimensions three and four, homological sphere theorem in higher dimensions.
We prove that the quasi-Einstein metrics found by Lü, Page and Pope on -bundles over Fano Kähler-Einstein bases are conformally Kähler and that the Kähler class of the conformal metric is a multiple of the first Chern class. A detailed study of the lowest-dimensional example of such metrics on $\mathbb…
On the under-reaching phenomenon in message-passing neural PDE solvers: revisiting the CFL conditioncs.LG
The paper sets limits for GNNs solving PDEs to avoid under-reaching phenomenon.
problem Under-reaching phenomenon in GNNs solving PDEs.
method Sharp lower bounds for message-passing iterations based on PDE characteristics.
result Proposed lower bounds ensure efficient information propagation in GNNs.
Characterizes diagrams achieving Morton-Franks-Williams inequality for positive knots and links.
problem Understanding when the Morton-Franks-Williams inequality holds for positive knots and links.
method Combinatorial characterisation and generating examples.
result Examples of diagrams achieving crossing number, braid index, and maximal self-linking number.
Sharp bounds for anisotropic p-capacity of Euclidean compact sets derived using flow methods.
problem Sharp bounds for anisotropic p-capacity of Euclidean compact sets.
method Inverse anisotropic mean curvature flow (IAMCF) and anisotropic Hawking mass.
result Upper bounds for anisotropic p-capacity derived using flow methods.
The paper compares three hypoelliptic Laplacians on a specific 5D Cartan group.
problem Defining suitable hypoelliptic Laplacians for sharp estimates on Carnot groups.
method Introducing and comparing three hypoelliptic Laplacians on a specific Carnot group.
result Sharp div-curl type inequalities for the three hypoelliptic Laplacians.
We use a geometric construction to exhibit examples of autonomous Lagrangian systems admitting exactly two homoclinics emanating from a nondegenerate maximum of the potential energy and reaching a regular level of the potential having the same value of the maximum point. Similarly, we show examples of Hamiltonian syste…
We show that recent work of Ni and Wilking yields the result that a noncompact nonflat Ricci shrinker has at most quadratic scalar curvature decay. The examples of noncompact Kähler--Ricci shrinkers by Feldman, Ilmanen, and Knopf exhibit that this result is sharp.
We develop a theory for pricing non-diversifiable mortality risk in an incomplete market. We do this by assuming that the company issuing a mortality-contingent claim requires compensation for this risk in the form of a pre-specified instantaneous Sharpe ratio. We prove that our ensuing valuation formula satisfies a nu…
In this paper we provide examples of maps from almost complex domains into pseudo-Riemannian symmetric targets, which are pluriharmonic and not integrable, i.e. do not admit an associated family. More precisely, for one class of examples the source has a non-integrable complex structure, like for instance a nearly Kaeh…
Sharp gradient bound found for compact manifolds.
problem Finding a sharp gradient bound for compact manifolds.
method Using a specific function α=1 in the Li-Yau gradient bound.
result A sharp gradient bound is found for compact manifolds.
Proves convergence of mean curvature flow on cylinders with unique continuation.
problem Understanding the convergence and uniqueness of mean curvature flow on cylindrical surfaces.
method Proves convergence and provides unique continuation results for mean curvature flow on cylinders.
result Proves that rescaled mean curvature flow on cylinders converging super-exponentially must coincide with the cylinder itself.
We obtain very sharp results about the lack of validity of the Poincare lemma for the tangential Cauchy Riemann equations, acting on tangential forms, tangential to a CR manifold M of general CR dimension n, and general CR codimension k. This generalizes the classical nonsolvability example of H. Lewy. We also discuss …