Paper improves volume comparisons and entropy estimates for integral Ricci curvature.
problem Volume comparisons and entropy estimates for integral Ricci curvature.
method Several volume comparisons and an estimate for volume entropy.
result Improved estimates for volume entropy and algebraic entropy.
We extend the theory of Patterson-Sullivan measure to any regular covering of a compact manifold using the Busemann compactification and derive an integral formula for the volume entropy. As applications we prove some rigidity theorems for the volume entropy.
Paper proves integrability and entropy compactness for Kähler potentials with uniform log-log threshold.
problem Integrability and entropy compactness for Kähler potentials with specific density.
method Skoda-Zeriahi type integrability theorem and log-log threshold detection.
result Positivity of integrability threshold and entropy compactness for uniform log-log threshold.
An example of a real-analytic metric on a compact manifold whose geodesic flow is Liouville integrable by C∞ functions and has positive topological entropy is constructed.
The paper proves a quantitative rigidity result for spaces with specific curvature bounds.
problem Understanding the rigidity of spaces with almost maximal volume entropy.
method Analyzing Riemannian manifolds and RCD-spaces with specific curvature conditions. result Spaces with almost maximal volume entropy are closely related to hyperbolic space forms.
This study uses Tsallis entropy to analyze diversification and integration in Italian stock market companies.
problem Examining the industrial structure and market reactions of cross-shareholding networks.
method Developed Tsallis entropy approach to model diversification and integration using copulas.
result Entropy analysis reveals insights into market polarisation and fairness.
The study shows almost maximal volume entropy rigidity for certain manifolds with integral Ricci curvature.
problem Volume entropy rigidity for manifolds with lower integral Ricci curvature bound.
method Analyzing manifolds with specific integral Ricci curvature bounds, diameter, and volume entropy.
result The universal cover of the manifold is close to a hyperbolic space form under certain conditions.
Ecker's and Huisken's quantities agree for ancient mean curvature flows.
problem Understanding the finiteness of integral quantities for ancient mean curvature flows.
method Comparison of Ecker's and Huisken's integral quantities.
result Finiteness of Ecker's integral quantity implies finiteness of entropy at infinity.
As we have proved in [L], the geodesic flows associated with the flat metrics on T^2 minimize the polynomial entropy. In this paper, we show that, among the geodesic flows that are Bott integrable and dynamically coherent, the geodesic flows associated to flat metrics are local strict minima for the polynomial entropy.…
A relation between the conformal anomaly and the logarithmic term in the entanglement entropy is known to exist for CFT's in even dimensions. In odd dimensions the local anomaly and the logarithmic term in the entropy are absent. As was observed recently, there exists a non-trivial integrated anomaly if an odd-dimensio…
For any toric automorphism with only real eigenvalues a Riemannian metric with an integrable geodesic flow on the suspension of this automorphism is constructed. A qualitative analysis of such a flow on a three-solvmanifold constructed by the authors in math.DG/9905078 is done. This flow is an example of the geodesic f…
PAN improves graph neural networks using path integrals.
problem Efficiency and performance of graph neural networks.
method PAN uses path integrals to generalize graph Laplacian, incorporating all paths between nodes.
result PAN achieves state-of-the-art performance on benchmark tasks.
The entropy-degree theorem applies to Alexandrov spaces with curvature constraints.
problem Geometric obstructions and volume bounds in singular spaces.
method Developed new degree theorem for Alexandrov spaces using integral currents.
result Entropy-volume minimization prevents metric singularities in Gromov-Hausdorff limits.
The entropy of a hypersurface is a geometric invariant that measures complexity and is invariant under rigid motions and dilations. It is given by the supremum over all Gaussian integrals with varying centers and scales. It is monotone under mean curvature flow, thus giving a Lyapunov functional. Therefore, the entropy…
The paper proves lower bounds for Gaussian-weighted curvature integrals of self-shrinkers.
problem Proving lower bounds for Gaussian-weighted \(L^2\)-curvature integrals of self-shrinkers.
method Combining normal coordinate functions with weighted Poincaré inequalities and first-eigenvalue estimates.
result Explicit lower bounds in terms of entropy for closed self-shrinkers, leading to curvature gaps.
Proposes integrating global and local entropy for more reliable LLMs.
problem Uncertainty in large language models (LLMs) leads to unreliable predictions.
method Measures global uncertainty from hidden-state matrices and local uncertainty from tokens, combining them via a multiplicative gate.
result Global-Local Uncertainty (GLU) outperforms unsupervised baselines across multiple models and benchmarks.
Solves a complex geometric problem for symmetric convex bodies.
problem Conditions for a measure to be the dual curvature measure of a symmetric convex body.
method Variational approach using entropy and quermassintegrals, with estimates on entropy and curvature measures.
result Explicit conditions for the measure concentration, leading to a full solution for 1<q<n. 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. …
Unique continuation result for expanding Ricci solitons.
problem Unique continuation of expanding Ricci solitons.
method Optimal relative integral convergence rate, relative entropy.
result Well-defined relative entropy for expanding solitons.
This paper uses copulas and entropies to analyze cross-shareholding configurations in stock markets.
problem Analyzing the concentration and integration of companies in cross-shareholding networks.
method The approach combines copula theory and entropy measures to study the stochastic dependence of diversification and integration.
result The copula approach reveals the dependence structure leading to market polarization or fairness.
A regularization procedure developed in [1] for the integral curvature invariants on manifolds with conical singularities is generalized to the case of squashed cones. In general, the squashed conical singularities do not have rotational O(2) symmetry in a subspace orthogonal to a singular surface Σ so that the surfa…
We introduce a new entropy functional for nonnegative solutions of the heat equation on a manifold with time-dependent Riemannian metric. Under certain integral assumptions, we show that this entropy is non-decreasing, and moreover convex if the metric evolves under super Ricci flow (which includes Ricci flow and fixed…
Let M be a compact Riemannian manifold without boundary and V:M→R a smooth function. Denote by Pt and dμ=eVdx the semigroup and symmetric measure of the second order differential operator L=Δ+∇V⋅∇. For some suitable convex function Φ:I→R define…
Study on flexibility of metric entropy and geodesic flow constraints on surfaces.
problem Flexibility and constraints of metrics and flows on surfaces.
method Analysis of geodesic flow, entropy, and metrics in a fixed conformal class.
result Additional restrictions on topological entropy and new geometric properties.
ED-VAE improves VAEs by explicitly including entropy components in ELBO.
problem Limitations of traditional VAEs with ELBO in generating high-quality samples and interpreting latent spaces.
method Introduces ED-VAE, a re-formulation of ELBO that includes entropy and cross-entropy components.
result Significantly enhances model flexibility and improves interpretability and generative performance.
The holographic description in the presence of gravitational Chern-Simons term is studied. The modified gravitational equations are integrated by using the Fefferman-Graham expansion and the holographic stress-energy tensor is identified. The stress-energy tensor has both conformal anomaly and gravitational or, if re-f…
The problem of determining the joint probability distributions for correlated random variables with pre-specified marginals is considered. When the joint distribution satisfying all the required conditions is not unique, the "most unbiased" choice corresponds to the distribution of maximum entropy. The calculation of t…
Researchers explore gauge freedom in entropies of q-Gaussian measures.
problem Exploring the gauge freedom of entropies in q-Gaussian measures. method Introducing a refined q-logarithmic function to demonstrate gauge freedom. result Different escort expectations can lead to the same entropy but different relative entropies.
Adaptive HMC improves sampling efficiency by optimizing mass matrix.
problem Inefficient HMC performance due to mass matrix choice.
method Gradient-based adaptation of mass matrix to maximize proposal entropy.
result Adaptation method outperforms HMC variants by optimizing mass matrix.
Introduces REVE, a regularization scheme that compresses class conditioned entropy.
problem Improving generalization performance of deep learning models.
method Identifies a variable responsible for final prediction, compresses class conditioned entropy, introduces a variational upper bound, and integrates a tractable loss into training.
result Demonstrates the efficiency of REVE on various neural networks and datasets.
Geometrically constructs twist-field correlation functions in CFT.
problem Understanding entanglement entropy in quantum systems.
method Using Cauchy-Hadamard renormalization of Polyakov anomaly integral on surfaces with conical singularities.
result Provides a purely mathematical interpretation of entanglement entropy results.
A new CoVaR framework integrates expert views using entropy pooling.
problem Risk assessment and spillover effects from diverse expert views.
method Entropy pooling method to integrate expert views and compute general CoVaR.
result General CoVaR shows linear relationships with expectations and differences in expectations, and nonlinear dependencies with variance, quantiles, and correlation.
In the present work we consider the behavior of the geodesic flow on the unit tangent bundle of the 2-torus T2 for an arbitrary Riemannian metric. A natural non-negative quantity which measures the complexity of the geodesic flow is the topological entropy. In particular, positive topological entropy implies chaotic…
Information-theoretic measures such as the entropy, cross-entropy and the Kullback-Leibler divergence between two mixture models is a core primitive in many signal processing tasks. Since the Kullback-Leibler divergence of mixtures provably does not admit a closed-form formula, it is in practice either estimated using …
Study shows flows from double cones remain symmetric, finds non-symmetric example.
problem Understanding flows from double cones under mean curvature flow.
method Analyzes Brakke flows, proves symmetry, constructs non-symmetric examples.
result Non-self-similar flows exist for entropy at most two.
Consider a financial market in which an agent trades with utility-induced restrictions on wealth. For a utility function which satisfies the condition of reasonable asymptotic elasticity at −∞ we prove that the utility-based super-replication price of an unbounded (but sufficiently integrable) contingent claim i…
The paper studies Q-curvature and volume entropy on manifolds, proving polynomial growth polyharmonic function finiteness and rigidity.
problem Investigating Q-curvature and volume entropy on conformally flat manifolds.
method Introducing a new volume entropy, establishing identities, and proving rigidity results.
result Each polynomial growth polyharmonic function on such manifolds is of finite dimension, and the Cohn-Vossen inequality achieves equality under specific conditions.
The article examines entropy-information inequalities for continuous-time Markov chains under curvature-dimension conditions.
problem Proving Li-Yau inequalities and modified logarithmic Sobolev inequalities for reversible Markov chains.
method Introducing the CDΥ(κ,F) condition and deriving entropy-information inequalities. result Derives functional inequalities relating entropy to Fisher information.
Information theory provides principled ways to analyze different inference and learning problems such as hypothesis testing, clustering, dimensionality reduction, classification, among others. However, the use of information theoretic quantities as test statistics, that is, as quantities obtained from empirical data, p…
Ricci flow controls curvature on manifolds with bounds.
problem Controlling curvature on manifolds with given bounds.
method Ricci flow with curvature bounds and entropy controls.
result Global curvature control at positive times for manifolds.
Bayesian models use hyperparameters to indirectly assign priors, and this work shows how these priors can be derived from maximum entropy principles.
problem Understanding the assumptions and dependencies in Bayesian hierarchical models.
method Demonstrates how canonical distributions and maximum entropy principles can be used to derive marginal priors in hierarchical models.
result Marginal priors in hierarchical models derived from maximum entropy principles have different constraints compared to the original priors.
The paper establishes sub-gradient estimates and entropy formulas for quaternionic contact geometry heat equations.
problem Developing sub-gradient estimates and entropy formulas for quaternionic contact geometry.
method Establishing sub-gradient estimates and entropy formulas for the quaternionic contact heat equation.
result Two Perelman-type entropy formulas and sub-gradient estimates for the quaternionic contact heat equation.
Computes entanglement entropy using Chern-Simons theory and symmetric webs.
problem Determining if a product state implies unlinked components.
method Using symmetric webs to compute colored link invariants and write multi-partite entangled states.
result Written down multi-partite entangled states of any given link.
We adapt tools from information theory to analyze how an observer comes to synchronize with the hidden states of a finitary, stationary stochastic process. We show that synchronization is determined by both the process's internal organization and by an observer's model of it. We analyze these components using the conve…
Market entropy analysis reveals horizon dependence in asset prices.
problem Quantifying horizon dependence of asset prices in high-frequency data.
method Cluster entropy approach to quantify price dynamics over different temporal horizons.
result Systematic dependence of cluster entropy and Market Dynamic Index on temporal horizon.
This study uses moving average cluster entropy to analyze financial market dynamics.
problem Understanding long-range dependence in financial markets.
method Moving average cluster entropy approach applied to ARFIMA and FBM processes.
result Long-range positive correlation in financial markets is linked to the cluster entropy behavior.
New DRL algorithm improves sample efficiency and exploration performance.
problem High sample cost in deep reinforcement learning.
method Combines entropy and bootstrap techniques with Tsallis entropy regularization.
result Demonstrates more efficient and effective exploration on Atari games.
Entropy measure assesses market volatility and price heterogeneity.
problem Quantifying short-term market heterogeneity in financial time series.
method Entropy measure based on intersecting a random sequence with its moving average.
result Entropy of volatility series varies by market, while price series is market-invariant.