arXiv research

Rescaling expansiveness proven for k*-expansive vector fields.

We prove a local index theorem of Atiyah-Singer type for Dirac operators on manifolds with a Lie structure at infinity (Lie manifolds for short). With the help of a renormalized supertrace, defined on a suitable class of regularizing operators, the proof of the index theorem relies on a rescaling technique similar in s…

Paper analyzes blowup of regularized Jang solutions and constant expansion surfaces.

We obtain a first order extension of the large deviation estimates in the Gärtner-Ellis theorem. In addition, for a given family of measures, we find a special family of functions having a similar Laplace principle expansion up to order one to that of the original family of measures. The construction of the special fam…

We give an alternate proof of the existence of the asymptotic expansion of the Bergman kernel associated to the $k$-th tensor powers of a positive line bundle $L$ in a $\frac{1}{\sqrt{k}}$-neighborhood of the diagonal using elementary methods. We use the observation that after rescaling the Kähler potential $k\varphi$ …

Study of spacelike discs in Minkowski cones, proving self-similar expansion.

We prove here a general closed-form expansion formula for forward-start options and the forward implied volatility smile in a large class of models, including the Heston stochastic volatility and time-changed exponential Lévy models. This expansion applies to both small and large maturities and is based solely on the p…

Recently Andrews and Bryan [3] discovered a comparison function which allows them to shorten the classical proof of the well-known fact that the curve shortening flow shrinks embedded closed curves in the plane to a round point. Using this comparison function they estimate the length of any chord from below in terms of…

New proof of stability for expanding Kerr-de Sitter spacetimes with smoothness at the boundary.

We prove convergence results for expanding curvature flows in the Euclidean and hyperbolic space. The flow speeds have the form $F^{-p}$, where $p>1$ and $F$ is a positive, strictly monotone and 1-homogeneous curvature function. In particular this class includes the mean curvature $F=H$. We prove that a certain initial…

In this short notes, we discuss monotonicity formulas under various rescaled versions of Ricci flow. The main result is Theorem \ref{theo rescaled}.

The paper calculates spectral torsion for rescaled Dirac operators on manifolds.

New spectral torsion defined for rescaled Dirac operators.

For a Riemannian manifold $M$, we determine some curvature properties of a tangent bundle equipped with the rescaled metric.The main aim of this paper is to give explicit formulae for the rescaled metric on $TM$, and investigate the geodesics on the tangent bundle with respect to the rescaled Sasaki metric.

Study geometric characterization of asymptotic pseudodifferential calculus on spinor bundles.

The paper calculates the noncommutative residue for a rescaled Dirac operator on 6D manifolds.

Formula for heat coefficients of curved conic singularities derived from Riemannian metrics.

Fast algorithm for rescaling vectors with clipping, improving training efficiency.

This work bridges two views of feature learning in neural networks.

A new method to rescale ReLU neural networks based on path-lifting.

Study on the $L^p$ behavior of helix curves' energy density.

This paper tackles non-vacuous generalization bounds in ReLU networks by resolving rescaling invariances.

Localizes Wodzicki residue for logarithm of differential operators.

Study on spectral asymptotics of Toeplitz operators on CR manifolds.

New Lipschitz bound for ReLU networks resists weight rescaling.

We exploit the spinor description of four-dimensional Walker geometry, and conformal rescalings of such, to describe the local geometry of four-dimensional neutral geometries with algebraically degenerate self-dual Weyl curvature and an integrable distribution of alpha-planes (algebraically special real alpha-geometry)…

The paper constructs bundles and recovers Kirillov character formula.

Study proves existence and uniqueness of ancient flows from cones.

This paper extends Lie algebra contractions to infinite-dimensional spaces for better understanding of group limits.

It has been empirically observed that the flatness of minima obtained from training deep networks seems to correlate with better generalization. However, for deep networks with positively homogeneous activations, most measures of sharpness/flatness are not invariant to rescaling of the network parameters, corresponding…

Method determines latent dimensionality in international trade flows.

Improved LLM pre-training performance through better weight and variance control.

Let X and Y be finite-type CW-complexes (X connected, Y simply connected), such that the rational cohomology ring of Y is a k-rescaling of the rational cohomology ring of X. Assume H^*(X,Q) is a Koszul algebra. Then, the homotopy Lie algebra pi_*(Omega Y) tensor Q equals, up to k-rescaling, the graded rational Lie alge…

"Ends of hyperbolic 3-manifolds should support canonical Wick Rotations, so they realize effective interactions of their ending globally hyperbolic spacetimes of constant curvature." We develop a consistent sector of WR-rescaling theory in 3D gravity, that, in particular, concretizes the above guess for many geometrica…

Let $(M,g)$ be an $n-$dimensional Riemannian manifold and $T_{1}^{1}(M)$ be its $(1,1)-$tensor bundle equipped with the rescaled Sasaki type metric $$ which rescale the horizontal part by a nonzero differentiable function $f$. In the present paper, we discuss curvature properties of the Levi-Civita connectio…

A new method to improve deep neural networks using weight rescaling.

The abstract discusses nonuniqueness results for specific Riemannian invariants.

We develop a ``canonical Wick rotation-rescaling theory in 3-dimensional gravity''. This includes: (a) A simultaneous classification that shows how generic maximal globally hyperbolic spacetimes of constant curvature, which admit a complete Cauchy surface (in particular a compact one), as well as complex projective str…

We study the small-time fluctuations for diffusion processes which are conditioned by their initial and final positions, under the assumptions that the diffusivity has a sub-Riemannian structure and that the drift vector field lies in the span of the sub-Riemannian structure. In the case where the endpoints agree and t…

We establish a Lehto--Virtanen-type theorem and a rescaling principle for an isolated essential singularity of a holomorphic curve in a complex space, which are useful for establishing a big Picard-type theorem and a big Brody-type one for holomorphic curves.

We study various covering spectra for complete noncompact length spaces with universal covers (including Riemannian manifolds and the pointed Gromov Hausdorff limits of Riemannian manifolds with lower bounds on their Ricci curvature). We relate the covering spectrum to the (marked) shift spectrum of such a space. We de…

Rescaled ASGD optimizes distributed learning under heterogeneous data.

Study shows uniform decay rate for singular mean curvature flows.

Study on when Bergman metrics of domains are induced by balls.

The study approximates option prices using Hermite polynomials without assuming a specific distribution.

Study of null mean curvature flow on de Sitter lightcone, related to 2d-Ricci flow.

The paper proves a rescaling principle for quasiregular curves and applies it to hyperbolicity.

Uniform diameter bounds for Calabi-Yau fibrations with singular fibers.