Study classifies solutions to a triharmonic Lane-Emden equation.
problem Classifying solutions to a specific triharmonic Lane-Emden equation.
method Derive monotonicity formula, classify solutions (positive or sign-changing, radial or not).
result New monotonicity formula for triharmonic maps as a byproduct.
We study existence, uniqueness and stability of radial solutions of the Lane-Emden-Fowler equation −Δgu=∣u∣p−1u in a class of Riemannian models (M,g) of dimension n≥3 which includes the classical hyperbolic space Hn as well as manifolds with sectional curvatures unbounded below. Sign properties…
Paper shows how KL exponent is preserved via inf-projection for optimization problems.
problem Estimating the KL exponent for optimization problems.
method Shows KL exponent preservation via inf-projection for optimization problems.
result KL exponent is preserved via inf-projection for several important convex optimization models.
Infinite 2-groups of bounded exponent cannot act faithfully on compact manifolds.
problem Actions of infinite 2-groups of bounded exponent on compact manifolds.
method Analyzing properties of 2-groups of bounded exponent.
result Infinite 2-groups of bounded exponent cannot act faithfully on compact manifolds.
Study shows a specific Carnot group violates a curvature exponent bound.
problem Understanding the curvature exponent in step-two Carnot groups.
method Examined convergence of Lie algebra structure constants.
result Found a Carnot group where curvature exponent bound is violated.
New groups found with critical exponents close to but less than max.
problem Finding discrete isometry groups with critical exponents near maximum.
method Analyzing complex hyperbolic spaces to construct groups.
result Discrete isometry groups with critical exponents arbitrarily close to max but less.
Study the Lyapunov exponent in SL(2,C) families as parameters approach poles.
problem Understanding the asymptotic behavior of Lyapunov exponents in meromorphic families of matrices.
method Analyzing the blow-up of Lyapunov exponent and relating it to non-Archimedean Lyapunov exponent.
result The blow-up of Lyapunov exponent is governed by a quantity interpretable as the non-Archimedean Lyapunov exponent.
New proof for certain groups in higher dimensions.
problem Properties of discrete subgroups in higher dimensions.
method Proving convex-cocompactness for specific groups.
result Finitely generated Kleinian groups with small critical exponent are convex-cocompact.
Extends Nash-Kuiper theorem to higher Hölder exponents.
problem Constructing isometric immersions beyond Borisov's exponent.
method Novel corrugation ansatz, integration by parts, and algebraic decomposition.
result Flexibility of C1,α isometric immersions beyond Borisov's exponent. In this paper, we show how the sampling properties of the Hurst exponent methods of estimation change with the presence of heavy tails. We run extensive Monte Carlo simulations to find out how rescaled range analysis (R/S), multifractal detrended fluctuation analysis (MF-DFA), detrending moving average (DMA) and genera…
Study of deep neural networks using finite-time Lyapunov exponents.
problem Understanding the geometric structures in input space formed by deep neural networks.
method Analogy with dynamical systems, computing finite-time Lyapunov exponents.
result Ridges of large positive exponents divide input space into regions associated with different classes.
Study shows critical exponents for tree-acting groups.
problem Understanding critical exponents of discrete groups on trees.
method Explicit construction of edge-indexed graphs.
result Proven existence of groups with specific critical exponents.
Totally geodesic limit of quasi-Fuchsian groups with converging critical exponent.
problem Understanding convergence of quasi-Fuchsian groups.
method Critical exponent convergence analysis.
result Convergence of quasi-Fuchsian groups to totally geodesic representations.
Study proves boundedness of operators in variable exponent Morrey spaces.
problem Boundedness of operators in global Morrey-type spaces with variable exponents.
method Analysis of Hardy-Littlewood maximal operator and potential type operator in variable exponent Morrey spaces.
result Boundedness of the Hardy-Littlewood maximal operator and potential type operator in global Morrey-type spaces with variable exponents.
Constructs free semigroups with critical exponents close to but less than ambient groups.
problem Creating free semigroups with critical exponents close to but less than ambient groups.
method Constructing finitely generated free subsemigroups with specific properties.
result Free semigroups with critical exponents arbitrarily close to but strictly less than ambient groups.
Proves critical exponent for Θ−positive representations in discrete subgroups.
problem Determining the critical exponent for Θ−positive representations. method Analyzes discrete subgroups Γ⊂PSL(2,R) and their geometric properties. result Equality of critical exponent holds if and only if Γ is a lattice for geometrically finite Γ. Study critical exponent for geodesic currents using quasi-metric spaces.
problem Understanding the critical exponent for geodesic currents.
method Associated a quasi-metric space to geodesic currents and defined a metric for filling currents, studying the critical exponent and its relation to curve intersection growth.
result The critical exponent equals the exponential growth rate of the intersection function for closed curves.
New solutions found using rational exponents in complex-valued geometry.
problem Difficult non-linear problems in differential geometry.
method Generalizing from polynomial solutions to rational exponents.
result Many new solutions found.
Study Lyapunov exponents on curves, relating them to holomorphic bundles and moduli spaces.
problem Bounding Lyapunov exponents for flat bundles over hyperbolic curves.
method Refined lower bounds on Lyapunov exponents, relating them to holomorphic subbundles and developing maps.
result Equality of sum of exponents and asymptotic degree on compact curves, unbounded on maximal Shatz stratum.
New solutions found for complex-valued harmonic morphisms with rational exponents.
problem Finding new solutions to complex-valued harmonic morphisms.
method Generalizing from polynomial solutions to rational exponents.
result Many new solutions to the problem.
Study critical exponents in normal subgroups of higher rank Lie groups.
problem Understanding critical exponents in normal subgroups of higher rank Lie groups.
method Analyzing subgroups and their critical exponents in a higher rank semi-simple Lie group.
result Critical exponents of normal subgroups coincide under certain conditions.
We empirically investigated the relationships between the degree of efficiency and the predictability in financial time-series data. The Hurst exponent was used as the measurement of the degree of efficiency, and the hit rate calculated from the nearest-neighbor prediction method was used for the prediction of the dire…
New bounds on geodesic dimension and curvature exponent in Carnot groups.
problem Characterizing geodesic dimension and curvature exponent in Carnot groups.
method Characterization and lower bound calculation for geodesic dimension and curvature exponent.
result Found an example where curvature exponent is greater than geodesic dimension.
Study approximates top Lyapunov exponents for surface mapping classes.
problem Approximating topological Lyapunov exponents for surface mapping classes.
method Periodic approximation and joint spectral radius extension.
result Top Lyapunov exponents can be approximated by periodic orbits.
Paper analyzes KL exponent for first-order methods and applies it to linear convergence.
problem Analyzing convergence rate of first-order methods for optimization problems.
method Developed calculus rules to deduce KL exponent for new functions and showed its value for specific models.
result Many optimization models have a KL exponent of 1/2, enabling explicit convergence rates for first-order methods.
Study on curvature exponent of sub-Finsler Heisenberg groups, proving N_min ≥ 5.
problem Determining the curvature exponent of sub-Finsler Heisenberg groups.
method Analyzing the measure contraction property and constructing sub-Finsler structures.
result Proved that curvature exponent N_min ≥ 5, with equality if sub-Riemannian.
We study the relationship between the Lyapunov exponents of the geodesic flow of a closed negatively curved manifold and the geometry of the manifold. We show that if each periodic orbit of the geodesic flow has exactly one Lyapunov exponent on the unstable bundle then the manifold has constant negative curvature. We a…
Proves lower bounds for Lyapunov exponents of flat bundles on curves.
problem Proving lower bounds for Lyapunov exponents of flat bundles on curves.
method Generalized from Teichmueller curves to any local system over a curve with non-expanding cusp monodromies.
result Obtained large genus limits of individual Lyapunov exponents in hyperelliptic strata of Abelian differentials.
Paper analyzes error exponent in agnostic PAC learning.
problem Analyzing performance of agnostic PAC learning.
method Using error exponent from Information Theory to analyze PAC learning.
result Improved distribution-dependent error exponent for agnostic learning.
In previous work, the author fully classified orbit closures in genus three with maximally many (four) zero Lyapunov exponents of the Kontsevich-Zorich cocycle. In this paper, we prove that there are no higher dimensional orbit closures in genus three with any zero Lyapunov exponents. Furthermore, if a Teichmüller curv…
Study critical exponents on hyperbolic surfaces with long boundaries using Weil-Petersson measures.
problem Analyzing critical exponents on hyperbolic surfaces with long boundaries.
method Using spine graph construction and comparing normalized Weil-Petersson and Kontsevich measures.
result Asymptotic convergence-in-mean result of normalized Weil-Petersson measures to normalized Kontsevich measures.
Optimizes trading returns using Hurst exponent and Q-learning.
problem Maximizing returns from momentum and mean reversion strategies.
method Classifies assets using Hurst exponent and uses Q-learning to improve trading algorithms.
result Trading with Hurst exponent can achieve higher returns but at higher risk.
In the presence of a layer of metaprobabilities (from uncertainty concerning the parameters), the asymptotic tail exponent corresponds to the lowest possible tail exponent regardless of its probability. The problem explains "Black Swan" effects, i.e., why measurements tend to chronically underestimate tail contribution…
Global Nash-Kuiper theorem extended for compact manifolds with optimal Hölder exponent.
problem Isometric immersions of compact manifolds with optimal Hölder exponent.
method Global extensions of Nash-Kuiper theorem for C1,θ isometric immersions. result Nash-Kuiper non-rigidity prevails up to exponent θ<1/5 for isometric embeddings of convex compact surfaces. We apply the Hurst exponent idea for investigation of DJIA index time-series data. The behavior of the local Hurst exponent prior to drastic changes in financial series signal is analyzed. The optimal length of the time-window over which this exponent can be calculated in order to make some meaningful predictions is di…
mfBm models and forecasts volatility with different Hurst exponents and correlations.
problem Modeling and forecasting volatility with varying Hurst exponents and correlations.
method Multivariate fractional Brownian motion (mfBm) with component-wise Hurst exponents, novel estimation method, time-reversibility test.
result mfBm reduces forecasting errors compared to a one-dimensional model and outperforms HAR model.
We study the Bouchaud-Mézard model on a regular random network. By assuming adiabaticity and independency, and utilizing the generalized central limit theorem and the Tauberian theorem, we derive an equation that determines the exponent of the probability distribution function of the wealth as x→∞. Th…
The common assumption of universal behavior in stock market data can sometimes lead to false conclusions. In statistical physics, the Hurst exponents characterizing long-range correlations are often closely related to universal exponents. We show, that in the case of time series of the traded value, these Hurst exponen…
Study shows critical exponents of normal subgroups behave similarly to Kazhdan distances.
problem Analyzing critical exponents and Kazhdan distances in pinched Hadamard manifolds.
method Analyzing the spectrum of transfer operators associated with subshifts of finite type.
result Critical exponents of normal subgroups have the same coarse behavior as Kazhdan distances.
Explains Lyapunov exponents and their importance in three fields.
problem None explicitly stated, focuses on explaining concepts.
method Expository writing based on talks and research by Artur Avila.
result Lyapunov exponents are crucial in smooth ergodic theory, Teichmüller theory, and spectral theory.
This study examines Bitcoin's market efficiency, liquidity, and multifractality over time.
problem Investigating market efficiency, liquidity, and multifractality of Bitcoin.
method Dynamic analysis of Bitcoin's time series data, including Hurst exponent and multifractal degree.
result Market efficiency improves as liquidity increases, but multifractality remains anti-persistent.
In this paper we propose a new approach to estimation of the tail exponent in financial stock markets. We begin the study with the finite sample behavior of the Hill estimator under α-stable distributions. Using large Monte Carlo simulations, we show that the Hill estimator overestimates the true tail exponent and can …
This paper investigates the scaling dependencies between measures of "activity" and of "size" for companies included in the FTSE 100. The "size" of companies is measured by the total market capitalization. The "activity" is measured with several quantities related to trades (transaction value per trade, transaction val…
Classifies GL(2,R)-invariant subvarieties with zero Lyapunov exponents.
problem Classifying GL(2,R)-invariant subvarieties with specific properties.
method Classification based on homological dimensions and Lyapunov exponents.
result Explicit exceptions list for GL(2,R)-invariant subvarieties with zero Lyapunov exponents.
Python package for estimating Hurst exponent in fBm.
problem Estimating Hurst exponent in fractional Brownian motion.
method Whittle's likelihood method applied to fractional Gaussian noise.
result Implementation achieves state-of-the-art accuracy and speed.
We describe all the situations in which the Kontsevich-Zorich cocycle has zero Lyapunov exponents. Confirming a conjecture of Forni, Matheus, and Zorich, this only occurs when the cocycle satisfies additional geometric constraints. We also describe the real Lie groups which can appear in the monodromy of the Kontsevich…
This study examines evolving networks of P2P lending relationships, revealing scale-free characteristics and the impact of interest rate and term.
problem Understanding the structural characteristics of evolving networks of debtor-creditor relationships in P2P lending.
method Modeling P2P lending networks as evolving networks with addition and deletion of nodes, analyzing attributes and factors affecting the scale-free exponent.
result P2P lending networks are scale-free with significant influence from interest rate and term on the exponent of power-law.
The paper finds free semigroups in dense subgroups of Lie groups with critical exponents arbitrarily close to the subgroup's.
problem Finding free semigroups with critical exponents arbitrarily close to a subgroup's in dense subgroups of Lie groups.
method Analyzing Zariski dense discrete subgroups of Lie groups, showing the existence of free semigroups with critical exponents arbitrarily close to the subgroup's.
result The existence of free semigroups with critical exponents arbitrarily close to the subgroup's in dense subgroups of Lie groups.