Extremal length is an important conformal invariant on Riemann surface. It is closely related to the geometry of Teichmuller metric on Teichmuller space. By identifying extremal length functions with energy of harmonic maps from Riemann surfaces to -trees, we study the second variation of extremal length fu…
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48 results for “extremal function”
Develops statistical framework for analyzing functional data extremes.
problem Analyzing extremes of functional data in Hilbert spaces.
method Regular variation in Hilbert spaces, Peaks-Over-Threshold framework, functional PCA.
result Proposes a dimension reduction method for functional extreme observations.
Sharp inequalities and extremizers for J functional on Kähler manifolds.
problem Understanding the large scale asymptotic of the J functional on Kähler metrics.
method Proving sharpness of inequalities and studying extremizing potentials/rays on toric Kähler manifolds, and existence of radial extremizers on general Kähler manifolds.
result Sharpness of inequalities and existence of extremizing potentials/rays on toric Kähler manifolds, and equivalence with plurisupported currents on general Kähler manifolds.
In this paper, we show that the extremal length functions on Teichmüller space are log-plurisubharmonic. As a corollary, we obtain an alternative proof of L.Liu and W.Su's results on the plurisubharmonicity of extremal length functions. We also obtain alternative proofs of S.Krushkal's results that a function defined b…
Uniqueness of weighted extremal metrics on Kähler manifolds proven.
problem Uniqueness of weighted extremal Kähler metrics on compact Kähler manifolds.
method Proof of uniqueness using modified Mabuchi energy and weighted K-semistability.
result Uniqueness of weighted extremal Kähler metrics up to automorphisms.
We give geometrical conditions under which there exist extremal functions for the sharp -Nash inequality.
We study singularities of geodesics flows in two-dimensional generalized Finsler spaces (pseudo-Finsler spaces). Geodesics are defined as extremals of a certain auxiliary functional whose non-isotropic extremals coincide with extremals of the action functional. This allows to consider isotropic lines as (unparametrized…
Extends fractional uncertainty principles with extremizers and stability results.
problem Investigating uncertainty principles in fractional settings.
method Analyzing the fractional Schrödinger equation to find extremal functions and sharp constants.
result Proves stability of extremizers for fractional uncertainty inequalities.
We investigate the properties of a modulus of a foliation on a Riemannian manifold. We give necessary and sufficient conditions for the existence of an extremal function and state some of its properties. We obtain the integral formula which, in a sense, combines the integral over the manifold with integral over the lea…
We consider strictly stationary heavy tailed time series whose finite-dimensional exponent measures are concentrated on axes, and hence their extremal properties cannot be tackled using classical multivariate regular variation that is suitable for time series with extremal dependence. We recover relevant information ab…
Proves regularity of extremal function on compact Kähler manifolds.
problem Regularity of extremal function on compact Kähler manifolds.
method Local property analysis and equivalence of continuity and Hölder continuity.
result Equivalence of classical notions of local -regularity and locally Hölder continuous property.
Improves forecast calibration for extreme events using modified loss functions.
problem Improperly specified models do not issue calibrated forecasts for extreme events.
method Adapting loss functions based on weighted scoring rules and tail miscalibration regularization.
result Calibrated forecasts for extreme wind speeds can be improved by suitable adaptations to the loss function during model training.
New acquisition function for extreme rewards in bandits.
problem Online decision making with extreme payoffs in multi-armed bandits.
method Modeling payoffs as Gaussian processes and using a novel UCB acquisition function.
result Demonstrated benefits across synthetic and real-world benchmarks.
Let $\M$ be a complete, connected noncompact manifold with bounded geometry. Under a condition near infinity, we prove that the Log Sobolev functional (\ref{logfanhan}) has an extremal function decaying exponentially near infinity. We also prove that an extremal function may not exist if the condition is violated. This…
Study on extremizers for Sobolev inequality on curved manifolds.
problem Existence of extremizers for the sharp -Sobolev inequality on Riemannian manifolds with nonnegative curvature.
method Nonsmooth concentration compactness methods and Mosco-convergence results for Cheeger energy.
result Almost extremal functions are close to radial Euclidean bubbles and almost zero globally under nonnegative curvature.
Quantum Extremal Learningquant-ph
Quantum algorithm finds extremal values without direct function access.
problem Finding extremal values of hidden functions without direct access.
method Parametric quantum circuit trained with a trainable quantum feature map.
result Algorithm successfully finds extremal values even with sparse training data.
Researchers identify valid auxiliary functions for extreme value distributions and their max-domains of attraction.
problem Characterize valid auxiliary functions for extreme value distributions and their max-domains of attraction.
method Introduced 'universal' auxiliary functions valid for both VR and vMR representations, identified sets of valid auxiliary functions, and proposed a method for finding appropriate auxiliary functions.
result Characterized valid auxiliary functions for both VR and vMR representations for the entire MDA distribution families.
Non-convex extremal length found in surface metrics.
problem Extremal length functions on surfaces are not always convex.
method Used harmonic maps to -trees and minimal surfaces in .
result Found measured foliations with non-convex extremal length functions.
New loss functions improve extreme classification with missing labels.
problem Large number of infrequent labels and missing labels in XMC.
method Derive unbiased loss functions for XMC, incorporating them into existing algorithms.
result Significant improvement in extreme classification performance (up to 20%) over existing methods.
In this paper, we discuss a Donaldson's version of the modified -energy associated to the Calabi's extremal metrics on toric manifolds and prove the existence of the weak solution for extremal metrics in the sense of convex functions which minimizes the modified -energy.
Three distinct 3D components found in local extremal Kähler metrics.
problem Characterizing extremal Kähler metrics in one dimension.
method Defined local extremal Kähler metrics and analyzed their germs.
result Space of germs of local extremal Kähler metrics in one dimension comprises three distinct components.
Researchers found abnormal extremals on specific Lie groups.
problem Identifying abnormal extremals on four-dimensional Lie groups.
method Using left-invariant sub-Finsler quasimetrics and seminorms on Lie algebra.
result Established a criterion for strict abnormality of extremals.
The first eigenvalue of the Laplacian on a surface can be viewed as a functional on the space of Riemannian metrics of a given area. Critical points of this functional are called extremal metrics. The only known extremal metrics are a round sphere, a standard projective plane, a Clifford torus and an equilateral torus.…
In the practice of point prediction, it is desirable that forecasters receive a directive in the form of a statistical functional, such as the mean or a quantile of the predictive distribution. When evaluating and comparing competing forecasts, it is then critical that the scoring function used for these purposes be co…
New method uses extreme value theory to estimate neural network errors.
problem Quantifying the error of neural networks, especially for large values.
method Applying extreme value theory to approximate the distribution of error.
result Developed a new estimator for the shape parameter of the Pareto distribution.
Let E be the Engel group and D be a rank 2 bracket generating left invariant distribution with a Lorentzian metric, which is a nondegenerate metric of index 1. In this paper, we first prove that timelike normal extremals are locally maximizing. Second, we obtain a parametrization of timelike, spacelike, lightlike norma…
This paper addresses the problem of estimating, in the presence of random censoring as well as competing risks, the extreme value index of the (sub)-distribution function associated to one particular cause, in the heavy-tail case. Asymptotic normality of the proposed estimator (which has the form of an Aalen-Johansen i…
Study moment maps coupled with convex functions to find critical points.
problem Understanding critical points of moment maps coupled with convex functions.
method Develop a theory of moment maps coupled with an Ad_K-invariant convex function f on k*.
result Interpret Kähler-Ricci solitons as a special case of generalized extremal metrics.
Capillarity functionals are parameter invariant functionals defined on classes of two-dimensional parametric surfaces in R3 as the sum of the area integral and a non homogeneous term of suitable form. Here we consider the case of a class of non homogenous terms vanishing at infinity for which the corresponding capillar…
New Kähler metrics generalize Calabi's and relate to Fano manifolds.
problem Existence of extremal Kähler metrics on Fano manifolds.
method Introducing -extremal Kähler metrics and relating them to multiplier Hermitian-Einstein metrics.
result Existence of -extremal Kähler metrics implies existence of multiplier Hermitian-Einstein metrics on Fano manifolds.
Develops deep models to handle nonstationary spatial extremal dependence.
problem Challenges in modeling nonstationary extremal dependence in spatial data.
method Deep compositional spatial models to capture nonstationarity in extremal dependence.
result Efficient estimation of warped space for nonstationary spatial data.
Continuity method proves existence of Mabuchi solitons on Fano manifolds.
problem Existence of Mabuchi solitons on extremal Fano manifolds.
method Continuity method applied to Mabuchi's generalized Kähler-Einstein metrics.
result Analytic proof of Mabuchi soliton existence without minimal model program.
We study Laplace eigenvalues on Kähler manifolds as functionals on the space of Kähler metrics with cohomologous Kähler forms. We introduce a natural notion of a -extremal Kähler metric and obtain necessary and sufficient conditions for it. A particular attention is paid to the -extremal properties of K…
In this paper, we prove the equivalence of the existence of extremal Kahler metrics and the properness of the modified K energy on projective bundles. Moreover, we discuss the relations of the lower boundedness of the K energy, the infimum of the Calabi energy and the extremal polynomials. In particular, we give an exa…
The -th eigenvalue of the Laplace-Beltrami operator on a surface can be considered as a functional on the space of all Riemannian metrics of unit volume on this surface. Surprisingly only few examples of extremal metrics for these functionals are known. In the present paper a new countable family of extremal m…
Explain convexity of K-energy leading to unique metrics.
problem Uniqueness of constant scalar curvature Kahler metrics and extremal metrics.
method Convexity of K-energy along weak geodesics in Kahler potentials.
result Uniqueness of extremal metrics up to automorphisms.
Let (M^n, g) be a closed smooth Riemannian spin manifold and denote by D its Atiyah-Singer-Dirac operator. We study the variation of Riemannian metrics for the zeta function and functional determinant of D^2, and prove finiteness of the Morse index at stationary metrics, and local extremality at such metrics under gene…
We study the Sasaki cone of a CR structure of Sasaki type on a given closed manifold. We introduce an energy functional over the cone, and use its critical points to single out the strongly extremal Reeb vectors fields. Should one such vector field be a member of the extremal set, the scalar curvature of a Sasaki extre…
This paper tackles efficient optimization for nonlinear embeddings in similarity learning.
problem Learning similarity with nonlinear embeddings is challenging due to the large number of pairs.
method Detailed derivations and efficient optimization methods for nonlinear embeddings are developed.
result Efficient optimization methods for nonlinear embeddings are shown to be highly effective.
New method identifies extreme risk propagation in financial networks.
problem Understanding extreme risk in financial networks.
method Max-linear structural equation model, hard-thresholding, Hamming distance.
result Sparse DAG for extreme risk propagation estimated.
We study the Euler-Lagrange equation for several natural functionals defined on a conformal class of almost Hermitian metrics, whose expression involves the Lee form of the metric. We show that the Gauduchon metrics are the unique extremal metrics of the functional corresponding to the norm of the codifferential of…
It will be discussed the statistics of the extreme values in time series characterized by finite-term correlations with non-exponential decay. Precisely, it will be considered the results of numerical analyses concerning the return intervals of extreme values of the fluctuations of resistance and defect-fraction displa…
The paper examines stability of Sobolev inequalities on manifolds with Ricci curvature bounds.
problem Stability of Sobolev inequalities on Riemannian manifolds with Ricci curvature lower bounds.
method Generalized Lions' concentration compactness and rigidity results of Sobolev inequalities on singular spaces.
result Almost extremal functions are close to extremal functions on the round sphere and Euclidean Sobolev inequality.
We develop some pluripotential theoretic techniques for the transversally holomorphic foliation of a Sasakian manifold. We prove the convexity of the K-energy along weak geodesics for Sasakian manifolds. This implies that the K-energy is bounded below if a constant scalar curvature structure exists with those metrics m…
New model predicts financial tail events using RIA-EVT-Copula.
problem Predicting financial tail events for risk management.
method RIA-EVT-Copula framework combining POT, RIA, and copulas.
result Improved accuracy in predicting financial extremes.
From curves to currentsmath.GT
Extends curve functions to geodesic currents with a simple criterion.
problem Continuous extension of curve functions to geodesic currents.
method Simple criterion based on smoothing property.
result Extends known curve functions and introduces new examples.
The paper examines extreme value statistics of high-dimensional sample covariances, with applications in finance and image analysis.
problem Statistical validation of normal conditions in high-dimensional time series data.
method Generalizes the maximal deviation of sample autocovariances to high dimensions and applies Gumbel-type extreme value asymptotics.
result Gumbel-type extreme value asymptotics holds true for high-dimensional sample covariances.
Inference over tails is usually performed by fitting an appropriate limiting distribution over observations that exceed a fixed threshold. However, the choice of such threshold is critical and can affect the inferential results. Extreme value mixture models have been defined to estimate the threshold using the full dat…