The study shows that ergodic measures are not generic on non-positively curved manifolds.
problem Determining the genericity of ergodic measures on non-positively curved Riemannian manifolds.
method Investigates the existence of an open isometric embedding of a product manifold with a factor isometric to S1. result The closure of the set of ergodic measures does not encompass all invariant measures, indicating the failure of genericity.
Unified positive mass theorem and Dirac operator study on weighted manifolds.
problem Establishing a unified positive mass theorem for weighted manifolds and smooth metric measure spaces.
method Analyzing Dirac operators on warped product manifolds and applying results to the positive mass theorem.
result Equivalence of weighted positive mass theorem to usual positive mass theorem.
Study SRB measures for Anosov actions on manifolds.
problem Characterize SRB measures for Anosov actions.
method Use Ruelle-Taylor resonances and properties of Sinai-Ruelle-Bowen measures.
result SRB measures have properties like smooth disintegrations, positive basins, and are unique under certain conditions.
New insights into Markov chain geometry via positive transition measures.
problem Lack of statistical meaning in the space of transition probabilities.
method Constructing an extension of the space of transition probabilities using Amari's theory of positive measures.
result Introduction of a new dually flat structure for the space of positive transition measures.
Monetary risk measures are usually interpreted as the smallest amount of external capital that must be added to a financial position to make it acceptable. We propose a new concept: intrinsic risk measures and argue that this approach provides a direct path from unacceptable positions towards the acceptance set. Intrin…
This paper presents non-parametric estimates of spectral risk measures applied to long and short positions in 5 prominent equity futures contracts. It also compares these to estimates of two popular alternative measures, the Value-at-Risk (VaR) and Expected Shortfall (ES). The spectral risk measures are conditioned on …
Paper solves open question about non-positive kernels by decomposing them into PD kernels.
problem Can non-positive definite kernels be decomposed into the difference of two positive definite kernels?
method Introduced signed measure to transform positive decomposition into measure decomposition, providing a sufficient and necessary condition.
result First random features algorithm for unbiased estimation of non-positive kernels.
We present a general framework for measuring the liquidity risk. The theoretical framework defines a class of risk measures that incorporate the liquidity risk into the standard risk measures. We consider a one-period risk measurement model. The liquidity risk is defined as the risk that a given security or a portfolio…
Set risk measures extend traditional risk measures to handle sets of positions.
problem Handling sets of positions with a single capital requirement.
method Developed an axiomatic framework for set risk measures, dual representation through topology and measures.
result Characterized worst-case set risk measures and provided examples.
New metric to measure liquidity position PNL, delta hedging algorithm for automated market makers.
problem Vulnerability of liquidity positions to price changes in underlying assets.
method Proposes a new metric for measuring PNL, delta hedging algorithm for various AMMs.
result New metric more accurately measures net value change due to price movement.
Derives stability for curvature measure near constant density, proving dual Minkowski problem solutions.
problem Stability of curvature measure near constant density
method Derives stability result for curvature measure, proves existence and uniqueness of solutions to dual Minkowski problem.
result Existence and uniqueness of solutions to dual Minkowski problem for positive indices, stability result for curvature measure.
New kernels defined for various spaces, including measures.
problem Defining kernels on non-standard spaces like measures.
method Integrally strictly positive definite and characteristic kernels on Hilbert, Banach, and metric spaces.
result Explicit classes of kernels on Lp spaces and sets of measures. We propose an iterative scheme for feature-based positioning using a new weighted dissimilarity measure with the goal of reducing the impact of large errors among the measured or modeled features. The weights are computed from the location-dependent standard deviations of the features and stored as part of the referenc…
Curves inscribe rectangles with positive area.
problem Finding angles for inscribing rectangles within Jordan curves.
method Proving existence of a subset of angles with measure at least A/R^2.
result Angles inscribing rectangles have measure at least A/R^2.
The paper finds a geometric way to minimize a convex function to construct isotropic measures.
problem Constructing isotropic measures in convex bodies.
method Minimizing a convex function in a high-dimensional space.
result Geometric interpretation of the minimizer as a derivative.
Extends Dirac operator results to foliations with invariant measures.
problem Positive scalar curvature on foliations with invariant measures.
method Relative measured index theorem for foliations with invariant transverse measures.
result Space of positive scalar curvature metrics has infinitely many path components.
The study examines entropy and pressure at infinity in negatively curved manifolds, linking them to strong positive recurrence.
problem Investigating strong positive recurrence in negatively curved manifolds.
method Defining and comparing entropy and pressure at infinity through different measures.
result Strong positive recurrence potentials admit finite Gibbs measures.
Proposes new deviation measures using Minkowski gauges.
problem Lack of suitable acceptance sets for deviation measures.
method Derives deviation measures through Minkowski gauges of acceptable sets.
result Any positive homogeneous deviation measure can be accommodated in the framework.
Study positive entropy actions by higher-rank lattices, proving rigidity and conjugacy results.
problem Positive entropy actions by higher-rank lattices in Lie groups.
method Analysis of sub-actions, fiber entropy upper semicontinuity, and conjugacy arguments.
result Actions by higher-rank lattices in SL(n,R) are conjugate to affine actions on (infra-)tori. Introduces an asymmetric model for measuring market risk.
problem Existing models are symmetric and do not account for asymmetric risk.
method Develops an asymmetric capital asset pricing model that considers position-dependent market risk.
result Long positions in Apple stock have lower volatility than the market, contrary to the standard model.
The paper develops robust risk measures for uncertain loss positions.
problem Risk assessment for loss positions with uncertain distributions.
method Robust optimized certainty equivalents and generalized quantiles are proposed and analyzed.
result Robust expectiles with specific penalization functions are coherent risk measures.
The paper studies Harnack inequalities on Finsler metric measure spaces.
problem Analyzing Harnack inequalities on Finsler metric measure spaces.
method Using weighted Ricci curvature and distortion conditions, the authors derive an elliptic p-Harnack inequality.
result The paper establishes an elliptic p-Harnack inequality and derives Hölder continuity and gradient estimates for positive harmonic functions.
Paper characterizes star-shaped risk measures and their properties.
problem Characterizing risk measures in the presence of liquidity risk and competitive delegation.
method Characterization of star-shaped risk measures, study of their properties.
result Star-shaped risk measures include all practically used risk measures.
We show that a positive Borel measure of positive finite total mass, on compact Hermitian manifolds, admits a Holder continuous quasi-plurisubharmonic solution to the Monge-Ampere equation if and only if it is dominated locally by Monge-Ampere measures of Holder continuous plurisubharmonic functions.
We show that the Kahler-Ricci flow on an algebraic manifold of positive Kodaira dimension and semi-ample canonical line bundle converges to a unique canonical metric on its canonical model. It is also shown that there exists a canonical measure of analytic Zariski decomposition on an algebraic manifold of positive Koda…
The celebrated KAM Theory says that if one makes a small perturbation of a non-degenerate completely integrable system, we still see a huge measure of invariant tori with quasi-periodic dynamics in the perturbed system. These invariant tori are known as KAM tori. What happens outside KAM tori draws a lot of attention. …
A non-vector-based dissimilarity measure is proposed by combining vector-based distance metrics and set operations. This proposed compound dissimilarity measure (CDM) is applicable to quantify similarity of collections of attribute/feature pairs where not all attributes are present in all collections. This is a typical…
Fingerprinting-based positioning, one of the promising indoor positioning solutions, has been broadly explored owing to the pervasiveness of sensor-rich mobile devices, the prosperity of opportunistically measurable location-relevant signals and the progress of data-driven algorithms. One critical challenge is to contr…
The paper proves a Harnack inequality for heat equations on Finsler metric measure manifolds.
problem Proving a Harnack inequality for positive solutions to heat equations on Finsler metric measure manifolds.
method Volume comparison theorem, weighted Poincaré inequality, local uniform Sobolev inequality, mean value inequalities.
result Derives a Harnack inequality for positive solutions to heat equations.
Paper introduces a new distance measure for Gaussian Mixture Models.
problem Developing a new distance measure for Gaussian Mixture Models.
method Embedding K-component Gaussian Mixture Models into the manifold of symmetric positive definite matrices and calculating a lower bound for the Fisher-Rao metric.
result Demonstrated effectiveness through experiments on standard datasets.
We construct symbolic dynamics on sets of full measure (w.r.t. an ergodic measure of positive entropy) for C1+ε flows on compact smooth three-dimensional manifolds. One consequence is that the geodesic flow on the unit tangent bundle of a compact C∞ surface has at least const ×(ehT/T) simple clos…
Paper extends ranking metrics theory for financial positions.
problem Developing a new class of functionals for evaluating financial positions.
method Axiomatic framework based on monotonicity and cash-quasiconcavity.
result Linking ranking metrics to families of acceptance sets and risk measures.
Paper extends ranking metrics theory for financial positions.
problem Developing a new class of performance evaluation methods.
method Axiomatic framework based on monotonicity and cash-quasiconcavity.
result Linking ranking metrics to families of acceptance sets and risk measures.
It is well known that Expected Shortfall (also called Average Value-at-Risk) is a convex risk measure, i. e. Expected Shortfall of a convex linear combination of arbitrary risk positions is not greater than a convex linear combination with the same weights of Expected Shortfalls of the same risk positions. In this shor…
We address the problem of curvature estimation from sampled compact sets. The main contribution is a stability result: we show that the gaussian, mean or anisotropic curvature measures of the offset of a compact set K with positive μ-reach can be estimated by the same curvature measures of the offset of a compact set…
In this paper, I shall demonstrate that sufficiently high-dimensional closed positively-curved Riemannian manifolds are either diffeomorphic to a spherical space form, or isometric to a locally compact rank one symmetric space. This surprising classification of positively-curved Riemannian manifolds results from combin…
New formulations for comparing metric measure spaces with arbitrary positive measures.
problem Comparing metric measure spaces with arbitrary positive measures.
method Two novel formulations: a divergence and a conic lifting approach.
result Efficiently solvable formulations for comparing metric spaces with arbitrary positive measures.
New risk measures adjust for tail risk inadequacies.
problem Tail risk inadequacy in classical risk measures.
method Developed a family of adjusted risk measures using target risk profiles.
result Analyzed and derived properties of adjusted risk measures.
Proves bounded subsolution theorem for complex Monge-Ampère equation on compact Hermitian manifolds.
problem Complex Monge-Ampère equation with positive Radon measure on compact Hermitian manifolds.
method Proves bounded subsolution theorem.
result Establishes bounded subsolution theorem for complex Monge-Ampère equation.
We prove a positive mass theorem for continuous Riemannian metrics in the Sobolev space Wloc2,n/2(M). We argue that this is the largest class of metrics with scalar curvature a positive a.c. measure for which the positive mass theorem may be proved by our methods.
Upper bound found for systolic geometry on manifolds with positive scalar curvature.
problem Relationship between systolic geometry and positive scalar curvature.
method Spinorial methods combined with geometric measure theory and curvature estimates.
result Upper bound for the two-dimensional stable systole on certain manifolds.
Develops risk measures for markets with constraints and costs.
problem Risk measures in markets with portfolio constraints and transaction costs.
method Embeds portfolio constraints and transaction costs into securities market; provides comprehensive analysis of risk measures properties.
result Establishes dual representations for convex and quasiconvex risk measures.
Let {Tt} be a smooth flow with positive speed and positive topological entropy on a compact smooth three dimensional manifold, and let μ be an ergodic measure of maximal entropy. We show that either {Tt} is Bernoulli, or {Tt} is isomorphic to the product of a Bernoulli flow and a rotational flow. Appli…
We define non-pluripolar products of closed positive currents on a compact Kaehler manifold. We show that a positive non-pluripolar measure can be written in a unique way as the top degree self-intersection (in the non-pluripolar sense) of a closed positive current in given big cohomology class. The solution is shown t…
We prove that manifolds with complicated enough fundamental group admit measure-preserving homeomorphisms which have positive stable fragmentation norm with respect to balls of bounded measure.
The family of admissible positions in a transaction costs model is a random closed set, which is convex in case of proportional transaction costs. However, the convexity fails, e.g. in case of fixed transaction costs or when only a finite number of transfers are possible. The paper presents an approach to measure risks…
Billiard trajectories (broken generalised geodesics) are considered in the exterior of an obstacle K with smooth boundary on an arbitrary Riemannian manifold. We prove a generalisation of the well-known Santalo's formula. As a consequence, it is established that if the set of trapped points has positive measure, then…
Voice Onset Time (VOT), a key measurement of speech for basic research and applied medical studies, is the time between the onset of a stop burst and the onset of voicing. When the voicing onset precedes burst onset the VOT is negative; if voicing onset follows the burst, it is positive. In this work, we present a deep…