arXiv research

We introduce a scalable measure of curvature for analyzing training dynamics of large language models.

In complex systems like financial market, risk tolerance of individuals is crucial for system resilience.The single-security price limit, designed as risk tolerance to protect investors by avoiding sharp price fluctuation, is blamed for feeding market panic in times of crash.The relationship between the critical market…

Sharp lower bound found for integral varifolds' mean curvature.

Sharp stability in Almgren problem solved in any dimension.

We prove sharp pointwise decay estimates for critical Dirac equations on $\mathbb{R}^n$ with $n\geq 2$. They appear for instance in the study of critical Dirac equations on compact spin manifolds, describing blow-up profiles, and as effective equations in honeycomb structures. For the latter case, we find excited state…

RGRR allocates between QQQ and DIA based on relative states, improving Sharpe and CAGR.

Sharp inequality for compactifying Poincaré-Einstein manifolds.

The paper identifies a new geometric and spectral phenomenon in the critical hyperbolic catenoid family.

Study on curve diffusion flows with scale-critical curvature term.

In multivariate regression, a $K$-dimensional response vector is regressed upon a common set of $p$ covariates, with a matrix $B^*\in\mathbb{R}^{p\times K}$ of regression coefficients. We study the behavior of the multivariate group Lasso, in which block regularization based on the $\ell_1/\ell_2$ norm is used for supp…

Paper sharpens inequality linking curvature and spectrum on manifolds.

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 fractional Sobolev inequalities on closed manifolds identified.

In this article, we investigate the geometry of critical metrics of the volume functional on an $n$-dimensional compact manifold with (possibly disconnected) boundary. We establish sharp estimates to the mean curvature and area of the boundary components of critical metrics of the volume functional on a compact manifol…

We prove a sharp Onofri-type inequality and non-existence of extremals for a Moser-Tudinger functional on the sphere in the presence of potentials having positive order singularities. We also investigate the existence of critical points and give some sufficient conditions under symmetry or nondegeneracy assumptions.

Sharp changes in time series representing market dynamics are studied by means of the self--similar analysis suggested earlier by the authors. These sharp changes are market booms and crashes. Such crises phenomena in markets are analogous to critical phenomena in physics. A simple classification of the market crisis p…

Sharp decay found for solutions of a specific equation in Lie groups.

Study critical metrics on manifolds with boundary using integral and boundary estimates.

We prove some sharp systolic inequalities for compact $3$-manifolds with boundary. They relate the (relative) homological systoles of the manifold to its scalar curvature and mean curvature of the boundary. In the equality case, the universal cover of the manifold is isometric to a cylinder over a disk of nonnegative c…

Solves Dirichlet problem for Lagrangian phase equation with critical and supercritical phase.

Let (W,M,M'), dim W > 5, be a non-trivial h-cobordism (i.e., the Whitehead torsion of (W,V) is non-zero). We prove that every smooth function f: W --> [0,1], f(M)=0, f(M')=1 has at least 2 critical points. This estimate is sharp: W possesses a function as above with precisely two critical points.

Sharpness minimization algorithms don't solely improve generalization.

Overparameterization enhances SAM's effectiveness in minimizing sharpness.

We classify local minimizers of $\intσ_2+\oint H_2$ among all conformally flat metrics in the Euclidean $(n+1)$-ball, $4\leq n\leq 5$, for which the boundary has unit volume, subject to an ellipticity assumption. We also classify local minimizers of the analogous functional in the critical dimension $n+1=4$. If minimiz…

The Hawking energy is nonnegative and rigid on area-constrained surfaces in general relativity.

Estimates for $p$-capacities on symmetric manifolds.

The relative equilibria of a symmetric Hamiltonian dynamical system are the critical points of the so-called augmented Hamiltonian. The underlying geometric structure of the system is used to decompose the critical point equations and construct a collection of implicitly defined functions and reduced equations describi…

Study critical quasilinear equations on Riemannian manifolds with curvature constraints.

In an incomplete market, including liquidly-traded European options in an investment portfolio could potentially improve the expected terminal utility for a risk-averse investor. However, unlike the Sharpe ratio, which provides a concise measure of the relative investment attractiveness of different underlying risky as…

In this paper, we define a relative Morse complex for manifold with boundary using the handlebody decomposition of the manifold. We prove that the homology of the relative Morse complex is isomorphic to the relative singular homology. Furthermore, we construct $A_\infty$-category structure on the relative Morse complex…

When trading incurs proportional costs, leverage can scale an asset's return only up to a maximum multiple, which is sensitive to its volatility and liquidity. In a model with one safe and one risky asset, with constant investment opportunities and proportional costs, we find strategies that maximize long term returns …

This thesis covers different aspects of the p-Laplace operators on Riemannian manifolds. Chapter 2. Potential theoretic aspects: the Khasmkinskii condition. Chapter 3: sharp eigenvalue estimates with Ricci curvature lower bounds. Chapter 4: Critical sets of (2-)harmonic functions.

We investigate fourth order Paneitz equations of critical growth in the case of $n$-dimensional closed conformally flat manifolds, $n \ge 5$. Such equations arise from conformal geometry and are modelized on the Einstein case of the geometric equation describing the effects of conformal changes of metrics on the $Q$-cu…

New method improves deep policy gradient algorithms by learning relative state values.

In this paper, based on the local comparison principle in [12], we study the local behavior of the difference of two spacelike graphs in a neighborhood of a second contact point. Then we apply it to the constant mean curvature equation in 3-dimensional Lorentz-Minkowski space $\mathbb{L}^3$ and get the uniqueness of cr…

The paper proves inequalities for hypersurfaces in weighted manifolds.

Sharp bounds on Fano varieties' heights proven for specific cases.

Study on convergence rates for optimal transport with regularization.

Modeling financial markets with sandpile model to understand price volatility and arbitrage constraints.

For Fano manifolds T. Mabuchi introduced a generalization of the Kähler-Einstein metric, which is characterized as the critical point of the Ricci-Calabi functional. We show that a Fano manifold admits Mabuchi's metric if and only if it is uniformly relatively D-stable.

Sharp estimates for p-capacity on manifolds with Ricci curvature bounds.

Continuous phase transitions identified in Doi-Onsager, noisy transformer, and Hegselmann-Krause models.

Study causal inference under specific sampling methods with monotonicity assumptions.

Off-policy learning exhibits greater instability when compared to on-policy learning in reinforcement learning (RL). The difference in probability distribution between the target policy ($π$) and the behavior policy (b) is a major cause of instability. High variance also originates from distributional mismatch. The var…

The study examines MCMC methods for arbitrary objectives and finds likelihood sharpness impacts performance and regularization.

Sharp constants in curl-Sobolev inequalities on spheres determined.

Willmore flow converges globally for surfaces with rotational symmetry below a specific energy threshold.

Characterizes diagrams achieving Morton-Franks-Williams inequality for positive knots and links.