arXiv research

Solves Tian's stabilization problem for toric Fano manifolds.

Study proves stability of big bang singularity in complex system.

Researchers find explicit solutions to complex Monge-Ampère equation.

Formula for harmonic current dimension on foliated surfaces, extending Brunella's inequality.

Researchers find Kähler-Einstein metrics near isolated log terminal singularities.

Study asymptotic behavior of second Chern forms on degenerating Kähler-Einstein surfaces.

New groups found with critical exponents close to but less than max.

We give sharp $C^{2,α}$ estimates for solutions of some fully nonlinear elliptic and parabolic equations in complex geometry and almost complex geometry, assuming a bound on the Laplacian of the solution. We also prove the analogous results to complex Monge-Ampère equations with conical singularities. As an application…

We prove the existence of Veech groups having a critical exponent strictly greater than any elementary Fuchsian group (i.e. $>\frac{1}{2}$) but strictly smaller than any lattice (i.e. $<1$). More precisely, every affine covering of a primitive L-shaped Veech surface $X$ ramified over the singularity and a non-periodic …

Wavelet analysis reveals limitations in detecting multifractality in signals with isolated singularities.

We study the holonomy cocycle H of a holomorphic foliation \Fc by Riemann surfaces defined on a compact complex projective surface X satisfying the following two conditions: 1) its singularities E are all hyperbolic; 2) there is no holomorphic non-constant map \C\to X such that out of E the image of \C is locally conta…

In this note, we prove the existence of a closed geodesic of positive length on any compact developable orbifold of dimension 3, 5, or 7. The argument uses the stratification of the singular locus, and reduces the problem of existence of a closed geodesic on a compact developable orbifold to the case of even dimensiona…

The miltifractal properties and scaling behaviour of the exchange rate variations of the Iranian rial against the US dollar from a daily perspective is numerically investigated. For this purpose the multifractal detrended fluctuation analysis (MF-DFA) is used. Through multifractal analysis, the scaling exponents, gener…

Study the relationship between canonical polynomials and elliptic sequences for elliptic singularities.

In this article, we solve the strong openness conjecture on the multiplier ideal sheaves for the plurisubharmonic functions posed by Demailly. We prove two conjectures about the growth of the volumes of the sublevel sets of plurisubharmonic functions related to the complex singularity exponents and quasi-plurisubharmon…

Recent results on ergodic theory for Riemann surface laminations and foliations.

The paper proves optimal smoothness for certain Lagrangian graphs with specific Hölder continuity.

P2P lending activities have grown rapidly and have caused the huge and complex networks of debtor-creditor relationships. The aim of this study was to study the underlying structural characteristics of networks formed by debtor-creditor relationships. According attributes of P2P lending, this paper model the networks o…

The paper proves rigidity for complex Kleinian groups.

Study on virtual singular braid groups with algebraic properties and homomorphisms.

Improved loss scaling for stochastic momentum algorithms in high dimensions.

For $n \geq 5$, we consider positive solutions $u$ of the biharmonic equation \[ Δ^2 u = u^\frac{n+4}{n-4} \qquad \text{on}\ \mathbb R^n \setminus \{0\} \] with a non-removable singularity at the origin. We show that $|x|^{\frac{n-4}{2}} u$ is a periodic function of $\ln |x|$ and we classify all periodic functions obta…

We started from computer experiments with simple one-dimensional ergodic dynamical systems called interval exchange transformations. Correlators in these systems decay as a power of time. In the simplest non-trivial case the exponent is equal to 1/3. We found a formula connecting characteristic exponents with explicit …

Study on $Δ$ invariants of plumbed 3-manifolds, comparing with Heegaard-Floer homology.

Optimal batch size minimizes training time for neural networks.

Reverse Hölder inequalities on Fano metrics with applications to geodesics and singularities.

Paper analyzes error exponent in agnostic PAC learning.

We study several new invariants associated to a holomorphic projective structure on a Riemann surface of finite analytic type: the Lyapunov exponent of its holonomy which is of probabilistic/dynamical nature and was introduced in our previous work; the degree which measures the asymptotic covering rate of the developin…

We study qualitative properties for nonnegative solutions to a conformally invariant coupled system of fourth order equations involving critical exponents. For solutions defined in the punctured space, there exist essentially two cases to analyze. If the origin is a removable singularity, we prove that non-singular sol…

A new flow connects manifold invariants with critical exponents.

Consider a flat bundle over a complex curve. We prove a conjecture of Fei Yu that the sum of the top k Lyapunov exponents of the flat bundle is always greater or equal to the degree of any rank k holomorphic subbundle. We generalize the original context from Teichmueller curves to any local system over a curve with non…

New learning rate approach reveals phase transitions in SGD performance.

Paper relaxes symmetry conditions for universal feature selection in noisy data.

Study critical exponents for L^p-cohomology of higher rank Lie groups and manifolds.

Neural network learns low-dimensional polynomials with SGD near information-theoretic limit.

Study of spacelike singularities in spherical spacetimes with scalar matter.

We present an explicit formula for the discrete power function introduced by Bobenko, which is expressed in terms of the hypergeometric τfunctions for the sixth Painlevé equation. The original definition of the discrete power function imposes strict conditions on the domain and the value of the exponent. However, we sh…

The aim of this paper is to prove the existence of weak solutions to the equation $Δu + u^p = 0$ which are positive in a domain $Ω\subset {\Bbb R}^N$, vanish at the boundary, and have prescribed isolated singularities. The exponent $p$ is required to lie in the interval $(N/(N-2), (N+2)/(N-2))$. We also prove the exist…

Keeping a basic tenet of economic theory, rational expectations, we model the nonlinear positive feedback between agents in the stock market as an interplay between nonlinearity and multiplicative noise. The derived hyperbolic stochastic finite-time singularity formula transforms a Gaussian white noise into a rich time…

This work improves online SGD's sample complexity for multi-index models by considering higher-order terms.

We prove a generalization of the fundamental inequality of Guivarc'h relating entropy, drift and critical exponent to Gibbs measures on geometrically finite quotients of CAT(-1) metric spaces. For random walks with finite superexponential moment, we show that the equality is achieved if and only if the Gibbs density is…

Motivated by the equation satisfied by the extremals of certain Hardy-Sobolev type inequalities, we show sharp $L^q$ regularity for finite energy solutions of p-laplace equations involving critical exponents and possible singularity on a sub-space of $\mathbb{R}^n$, which imply asymptotic behavior of the solutions at i…

Stock market comovements are examined using cointegration, Granger causality tests and nonlinear approaches in context of mutual information and correlations. Underlying data sets are affected by non-stationarities and trends, we also apply AMF-DFA and AMF-DXA. We find only 170 pair of Stock markets cointegrated, and a…

Study on sample complexity of policy gradient for stabilizing linear systems under multiplicative noise.

We present new minimax results that concisely capture the relative benefits of source and target labeled data, under covariate-shift. Namely, we show that the benefits of target labels are controlled by a transfer-exponent $γ$ that encodes how singular Q is locally w.r.t. P, and interestingly allows situations where tr…

Regularization is used to find a solution that both fits the data and is sufficiently smooth, and thereby is very effective for designing and refining learning algorithms. But the influence of its exponent remains poorly understood. In particular, it is unclear how the exponent of the reproducing kernel Hilbert space~(…

Study efficient estimation of hidden subspaces in Gaussian Multi-index models.

Gradient flossing stabilizes RNN training by controlling Lyapunov exponents.