Extended logarithm for solvable elements in mapping class groups.
problem Logarithm of Johnson map extension to solvable elements.
method Extension to exponential solvable elements in mapping class groups using solvable Lie groups.
result Solvability of extended logarithm.
In this work, we give a formula for the logarithmic invariant of knots in terms of certain derivatives of the colored Jones invariant. This invariant is related to the logarithmic conformal field theory, and was defined by using the centers in the radical of the restricted quantum group at root of unity. A relation bet…
Localizes Wodzicki residue for logarithm of differential operators.
problem Localizing Wodzicki residue for logarithm of differential operators.
method Localisation formula using rescaled differential operators and spinor bundles.
result Expresses index of Dirac operator in terms of local density involving logarithm.
Solutions near infinity to special Lagrangian equations are asymptotic to quadratic polynomials with logarithmic terms.
problem Solving special Lagrangian equations near infinity with specific conditions.
method Modified Kelvin transforms to characterize remainders in asymptotic expansions.
result Remainders in asymptotic expansions are characterized by a single smooth function in even dimensions and Cn−1,α in odd dimensions. New Poisson bracket connects to logarithmic manifolds.
problem Constructing a new Poisson bracket compatible with existing structures.
method Developed a new local Poisson bracket compatible with Adler-Gelfand-Dickey brackets, leading to a dispersionless limit.
result Leading term defines a logarithmic Dubrovin-Frobenius manifold.
We prove a sharp logarithmic Sobolev inequality which holds for submanifolds in Euclidean space of arbitrary dimension and codimension. Like the Michael-Simon Sobolev inequality, this inequality includes a term involving the mean curvature.
We prove a global residual formula in terms of logarithmic indices for one-dimensional holomorphic foliations, with isolated singularities, and logarithmic along normal crossing divisors. We also give a formula for the total sum of the logarithmic indices if the singular set of the foliation is contained in the invaria…
Lueck expressed the Gromov norm of a knot complement in terms of an infinite series that can be computed from a presentation of the fundamental group of the knot complement. In this note we show that Lueck's formula, applied to torus knots, yields surprising power series expansions for the logarithm function. This gene…
Proves a Baum--Bott formula for foliations by curves with logarithmic terms.
problem Analyzing singularities and smoothness in foliations by curves.
method Logarithmic Baum--Bott residues for foliated triples (X,F,D), relating to Poincaré's Problem and GSV indices. result Logarithmic Baum--Bott residues generalize Aleksandrov logarithmic index for vector fields on hypersurfaces.
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…
We prove that the trace of the logarithmic term of the Toeplitz kernel on a contact manifold is a contact invariant, generalizing K. Hirachi's invariant for the Szego kernel on a CR manifold. When the base manifold is the three-sphere, this vanishes identically.
In the previous article we derived a detailed asymptotic expansion of the heat trace for the Laplace-Beltrami operator on functions on manifolds with conic singularities. In this article we investigate how the terms in the expansion reflect the geometry of the manifold. Since the general expansion contains a logarithmi…
Sharp inequality for submanifolds in curved spaces.
problem Proving a logarithmic Sobolev inequality for submanifolds.
method Analyzes submanifolds in manifolds with nonnegative sectional curvature.
result Sharp inequality including mean curvature term.
Analytic torsion behavior studied for degenerating manifolds with equivariant bundles.
problem Behavior of analytic torsion for degenerating manifolds with equivariant bundles.
method Asymptotic expansion of equivariant analytic torsion, Quillen metrics, L2-metrics, Bott-Chern classes.
result Leading term of analytic torsion has logarithmic singularity, subdominant term has loglog-type singularity.
Study logarithmic flat connections on principal bundles using Lie groupoids.
problem Classify flat connections on principal bundles with logarithmic singularities.
method Use tools from Lie groupoid theory to classify representations and establish van Kampen theorems.
result Obtain a functorial Riemann-Hilbert correspondence for logarithmic connections.
The paper analyzes Q-learning in 2-player Markov games and provides gap-dependent logarithmic regret bounds.
problem Analyzing the cumulative regret of Nash Q-learning in 2-player turn-based stochastic Markov games.
method Proposed gap-dependent logarithmic upper bounds for cumulative regret in episodic tabular setting and discounted game setting.
result The proposed bounds match theoretical lower bounds up to a logarithmic term.
Study quadratic one-forms on logarithmic Higgs bundles on pointed curves.
problem Quadratic one-forms on logarithmic Higgs bundles on pointed curves.
method Use elementary pole cancellation for invariant polynomials.
result Found a logarithmic quadratic one-form.
The paper constructs a Saito basis for a specific class of divisors and applies it to logarithmic Poisson geometry.
problem Investigating a class of non-quasi-homogeneous free divisors and their logarithmic vector fields.
method Explicitly constructing a Saito basis for the module of logarithmic vector fields and applying it to logarithmic Poisson geometry.
result The construction of the Saito basis and the Lie-Rinehart algebra structure on the sheaf of logarithmic 1-forms.
We investigate optimal consumption problems for a Black-Scholes market under uniform restrictions on Value-at-Risk and Expected Shortfall for logarithmic utility functions. We find the solutions in terms of a dynamic strategy in explicit form, which can be compared and interpreted. This paper continues our previous wor…
Paper studies statistical manifolds with logarithmic divergences.
problem Understanding statistical manifolds induced by logarithmic divergences.
method Constructs dual foliation of the statistical manifold.
result Extends dual foliation of a dually flat manifold.
Logarithmic regret achieved in Q-learning with positive gap.
problem Achieving logarithmic cumulative regret in Q-learning with positive sub-optimality gap.
method Optimistic Q-learning with logarithmic regret bound.
result Logarithmic cumulative regret bound proven for optimistic Q-learning.
Logarithmic regret achieved in continuous-time linear-quadratic reinforcement learning.
problem Optimizing control actions in unknown continuous-time systems over a finite time horizon.
method Least-squares algorithm based on continuous-time observations and controls, with perturbation analysis and parameter estimation error analysis.
result Logarithmic regret bound of order O((lnM)(lnlnM)). We study the analytic torsion of the cone over an orientable odd dimensional compact connected Riemannian manifold W. We prove that the logarithm of the analytic torsion of the cone decomposes as the sum of the logarithm of the root of the analytic torsion of the boundary of the cone, plus a topological term, plus a fu…
Sharp upper diameter limit found for Ricci solitons.
problem Bounding the diameter of compact shrinking Ricci solitons.
method Used a sharp logarithmic Sobolev inequality and Vitali-type covering argument.
result Sharp upper diameter bound established in terms of scalar curvature and entropy.
Stabilization technique applied to curve shortening flow in 3D space.
problem Stabilizing curve shortening flow in 3D space.
method Applying stabilization technique developed by T. Zelenyak to curve shortening flow in R3. result Derivation of several new monotonicity formulas for curve shortening flow.
Finite-time queue peaks in stochastic networks have logarithmic scaling after geometric thresholds.
problem Queue peak laws in stochastic networks with geometric thresholds.
method Self-normalization mechanism
result Logarithmic scaling of queue peaks after geometric thresholds.
In this note we consider a heat trace expansion on a manifold with wedge-like singularity. We show that there are two terms in the expansion that contain information about the presence of the singularity, namely the logarithmic term ct−1/2logt and the half power term bt−1/2. We also give a geometric express…
Given a three dimensional pseudo-Einstein CR manifold (M,T1,0M,θ), we study the existence of a contact structure conformal to θ for which the logarithmic Hardy-Littlewood-Sobolev (LHLS) inequality holds. Our approach closely follows \cite{Ok1} in the Riemannian setting. For this purpose, we introduce the notion …
Logarithmic regret achieved in RL with linear function approximation.
problem Achieving logarithmic regret in reinforcement learning with linear function approximation.
method LSVI-UCB for linear MDP assumption, UCRL-VTR for linear mixture MDP assumption.
result Logarithmic regret bounds established for RL with linear function approximation.
We study the logarithmic L(α)-divergence which extrapolates the Bregman divergence and corresponds to solutions to novel optimal transport problems. We show that this logarithmic divergence is equivalent to a conformal transformation of the Bregman divergence, and, via an explicit affine immersion, is equivalent t…
Consider the one-parameter generalizations of the logarithmic and exponential functions which are obtained from the integration of non-symmetrical hyperboles. These generalizations coincide to the one obtained in the context of non-extensive thermostatistics. We show that these functions are suitable to describe and un…
Gradient flows for knot energies ensure long-term existence of knotted loops.
problem Ensuring long-term existence of knotted loops under various energies.
method Banach gradient flows, curves of maximal slope, logarithmic strain control.
result Established long-time existence of gradient flows for knot energies.
This paper tightens the law of the iterated logarithm for empirical KL_inf, applicable to unbounded data.
problem Developing nonasymptotic concentration bounds for empirical KL_inf with optimal constants and rates.
method Presenting a tight law of the iterated logarithm for empirical KL_inf, applicable to unbounded data.
result A tight law of the iterated logarithm for empirical KL_inf, applicable to unbounded data.
Given a compact Kaehler manifold, we consider the complement U of a divisor with normal crossings and a unitary local system V on it. We consider a differential graded Lie algebra (DGLA) of forms with holomorphic logarithmic singularities and vanishing residues. We construct a spectral sequence corresponding to the ant…
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…
Study Bergman kernels on Kähler manifolds, answering Lu-Tian's question.
problem Understanding Bergman kernels on Kähler manifolds and their properties.
method Localization and expansion analysis of Bergman kernels.
result Answered Lu-Tian's question about Bergman kernels having no logarithmic singularity.
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.
Uniform heat kernel and diffusion bridge asymptotics for sub-Riemannian geometry.
problem Analyzing sub-Riemannian heat kernels and their derivatives on incomplete manifolds.
method Localized asymptotic analysis, focusing on minimizing geodesics and the non-abnormal cut locus.
result Uniform bounds and expansions for heat kernels and their derivatives on compacts, including the diffusion bridge measure.
Paper proposes FedQ-Advantage for federated Q-learning with near-optimal regret and low communication cost.
problem Near-optimal federated Q-learning with low communication cost.
method Reference-advantage decomposition for variance reduction, synchronization between agents and server, policy update.
result Achieves almost optimal regret and near-linear regret speedup compared to single-agent learning.
Formula for sections on complex manifolds with non-isolated components.
problem Localization of sections on complex manifolds with non-isolated zero varieties.
method Logarithmic Bott localization formula, current-theoretic formulation.
result Established a formula for sections on compact complex manifolds with non-isolated components.
New algorithm for online portfolio selection with reduced runtime.
problem Maximizing total return in online portfolio selection.
method Minimizes current logarithmic loss regularized by log-determinant of Hessian.
result Achieves regret guarantee similar to Universal Portfolios with reduced runtime.
Researchers compute twisted Reidemeister torsion for hyperbolic 3-manifolds.
problem Computing twisted Reidemeister torsion for hyperbolic 3-manifolds.
method Using Dehn-filling and logarithmic holonomy of meridians.
result Formulas for adjoint twisted Reidemeister torsion in terms of boundary components and edge lengths.
Develops a new option pricing model under G-expectation framework.
problem Modeling uncertainty in financial markets and robust valuation under model uncertainty.
method G-expectation framework, logarithmic transformation, finite difference schemes.
result Unified risk-neutral valuation approach yielding G-Black-Scholes equation.
An efficient algorithm for Riemannian logarithm on Stiefel manifold family.
problem Efficient computation of Riemannian logarithm on Stiefel manifold for various metrics.
method Generalizes a matrix-algebraic approach for the canonical metric to a one-parameter family of metrics.
result Conserves local linear convergence for the family of metrics.
Green functions play an important role in conformal geometry. In this paper, we explain how to compute explicitly the logarithmic singularities of the Green functions of the conformal powers of the Laplacian. These operators include the Yamabe and Paneitz operators, as well as the conformal fractional powers of the Lap…
We give tight concentration bounds for mixtures of martingales that are simultaneously uniform over (a) mixture distributions, in a PAC-Bayes sense; and (b) all finite times. These bounds are proved in terms of the martingale variance, extending classical Bernstein inequalities, and sharpening and simplifying prior wor…
The paper sets bounds on how much regret is unavoidable in adaptive LQR with unknown B-matrix.
problem Understanding the limits of adaptive LQR with unknown B-matrix.
method Local asymptotic minimax regret lower bounds using van Trees' inequality and Bellman error representation.
result Logarithmic regret is impossible if the parametrization induces an uninformative optimal policy.
Study eta invariant remainder on contact manifolds, improving previous results.
problem Eta invariant remainder in metric contact manifolds.
method Analyzes remainder term in semiclassical limit, using volumes of recurrence sets of Reeb flow.
result Improves remainder term for Anosov Reeb flows and certain elliptic flows.