Deep RL improves space heating control with faster computation and robustness.
problem Suboptimal performance and inability to adapt to dynamic conditions in classical heating control methods.
method Deep reinforcement learning algorithm for optimal control of space heating systems.
result Outperforms rule-based control by 5-10% in simulated environments.
Formula found for heat equation control and backward problems.
problem Exact control of nonhomogeneous backward heat equations.
method Time analyticity and eigenfunctions of the Laplacian.
result Explicit formula for control function in terms of heat kernel.
New heat semigroup characterizes Sobolev and BV spaces in Carnot groups.
problem Lack of explicit representations and symmetry in heat kernels in sub-Riemannian geometry.
method Establishes a new heat semigroup characterisation using integral decoupling property.
result Characterizes Sobolev and BV spaces in Carnot groups.
LGAC enhances heat transfer in turbulent boundary layers using slot jets.
problem Enhancing convective heat transfer in turbulent boundary layers.
method Artificial intelligence-based linear genetic algorithms control (LGAC) with slot jets.
result LGAC optimizes heat transfer and flow asymmetry in turbulent boundary layers.
This paper optimizes a power-to-heat system using reinforcement learning for cost minimization under uncertain conditions.
problem Optimizing a power-to-heat system with fluctuating renewable energy sources.
method Stochastic optimal control, reinforcement learning (Q-learning).
result Reinforcement learning provides an efficient solution to the optimization problem.
Study Liouville theorems for harmonic maps along ancient super Ricci flows.
problem Proving Liouville theorems for harmonic maps under specific geometric conditions.
method Using Perelman's reduced geometric viewpoint, derive Liouville theorems with controlled growth.
result Sharp growth conditions and new Liouville theorems for both non-positively and positively curved target spaces.
Proves spectral inequality and null-controllability for elliptic operators on closed manifolds.
problem Proving spectral inequalities and null-controllability for elliptic pseudo-differential operators.
method Periodization approach in time inspired by global pseudo-differential calculus.
result Established spectral inequality and null-controllability for elliptic operators on closed manifolds.
We consider the heat equation associated with a class of second order hypoelliptic Hörmander operators with constant second order term and linear drift. We describe the possible small time heat kernel expansion on the diagonal giving a geometric characterization of the coefficients in terms of the divergence of the dri…
The paper proves recurrence relations for heat kernels on hyperbolic and spherical spaces.
problem Understanding recurrence relations of heat kernels on different space forms.
method Direct proof and computation of recurrence relations for heat kernels on hyperbolic and spherical spaces.
result Computed diagonal of heat kernels for odd dimensional hyperbolic spaces and heat trace asymptotic expansions for odd dimensional spheres.
We consider the heat equation associated with a class of hypoelliptic operators of Kolmogorov-Fokker-Planck type in dimension two. We explicitly compute the first meaningful coefficient of the small time asymptotic expansion of the heat kernel on the diagonal, and we interpret it in terms of curvature-like invariants o…
Study the heat kernel on quaternionic anti-de Sitter spaces and related spaces.
problem Understanding the heat kernel on quaternionic anti-de Sitter spaces and related spaces.
method Detailed study of the geometry, derivation of the horizontal Laplacian and subelliptic heat kernel formulas, derivation of small time asymptotics.
result Explicit formulas for the horizontal Laplacian and subelliptic heat kernel of the quaternionic anti-de Sitter fibration.
Monotonicity and rigidity of W-entropy proved in singular spaces.
problem Entropy behavior in singular metric measure spaces.
method Space-time Wasserstein control to show monotonicity and rigidity.
result Entropy dissipation rate and rigidity models in singular spaces.
Proves heat kernel superconvexity in hyperbolic space.
problem Heat kernel superconvexity in hyperbolic space.
method Proves conjecture by Bernstein in all dimensions.
result Analog of Huisken's monotonicity formula for mean curvature flow.
Proves metric spaces with Euclidean heat kernel are isometric to Euclidean space.
problem Characterizing metric measure spaces with specific heat kernels.
method Analyzes Dirichlet forms and heat kernels to prove rigidity.
result Metric measure spaces with Euclidean heat kernel are isometric to Euclidean space.
Study on biharmonic map heat flow with monotonicity formula.
problem Properties of biharmonic heat kernel and extrinsic biharmonic map heat flow.
method Derived an entropy type quantity exhibiting monotonicity behaviors.
result Monotonicity formula for extrinsic biharmonic map heat flow.
EBMs trained on discrete data using heat equations on graph structures.
problem Training EBMs on discrete or mixed data.
method Heat equations on graph structures for data perturbation.
result Efficacy demonstrated in various applications.
This paper describes results characterizing the range of the time-t heat operator on various manifolds, including Euclidean spaces, spheres, and hyperbolic spaces. The guiding principle behind these results is this: The functions in the range of the heat operator should be, roughly, those functions having an analytic c…
We consider Laplacians acting on sections of homogeneous vector bundles over symmetric spaces. By using an integral representation of the heat semi-group we find a formal solution for the heat kernel diagonal that gives a generating function for the whole sequence of heat invariants. We show explicitly that the obtaine…
Sharp gradient estimate on hyperbolic spaces derived from heat kernel.
problem Finding sharp Li-Yau type gradient estimates for positive solutions of heat equations.
method Introduced Li-Yau multiplier set and used recurrence relations of heat kernels on hyperbolic spaces.
result Optimal Li-Yau gradient estimate on hyperbolic spaces.
We develop a new method for the calculation of the heat trace asymptotics of the Laplacian on symmetric spaces that is based on a representation of the heat semigroup in form of an average over the Lie group of isometries and obtain a generating function for the whole sequence of all heat invariants.
We study the heat equation on time-dependent metric measure spaces (as well as the dual and the adjoint heat equation) and prove existence, uniqueness and regularity. Of particular interest are properties which characterize the underlying space as a super Ricci flow as previously introduced by the second author. Our ma…
Overview of geometric analysis for manifold learning.
problem Analyzing high-dimensional data via spectral embeddings.
method Heat kernel and eigenfunctions on Riemannian manifolds.
result Uniform control of spectral embeddings on key classes of manifolds.
Sharp gradient estimates found for heat equation on hyperbolic spaces.
problem Finding sharp gradient estimates for heat equations on hyperbolic spaces.
method Introduced a general form of Li-Yau type gradient estimate and used explicit heat kernel expressions.
result Sharp Li-Yau type gradient estimates obtained for heat equation on hyperbolic spaces.
Safe Bayesian optimization reduces HVAC costs by 32%.
problem Optimizing room temperature PID control for energy savings and comfort.
method Safe Contextual Bayesian Optimization.
result 32% reduction in room temperature PID control costs.
Researchers derived heat kernel expansions for non-compact spaces using Witten deformation.
problem Heat kernel expansions on non-compact spaces, especially for Witten Laplacians.
method Introduced parabolic distance and used it to derive asymptotic expansions.
result Derived an asymptotic expansion of trace of heat kernel for small-time t. Study subelliptic heat kernel on octonionic anti-de Sitter space.
problem Heat kernel of octonionic anti-de Sitter space.
method Lift Laplacian of octonionic hyperbolic space and use sub-Laplacian.
result Two integral representations for subelliptic heat kernel.
Heat kernels map RCD spaces to Riemannian manifolds.
problem Mapping RCD spaces to Riemannian manifolds using heat kernels.
method Using heat kernels to map RCD spaces into L2 space and then normalizing to achieve isometric immersions. result Compact RCD spaces with isometrically heat kernel immersions are isometric to unweighted smooth Riemannian manifolds.
The paper studies heat kernel behavior on symmetric spaces.
problem Large-time behavior of heat operator traces on symmetric spaces.
method Uses representation theory and Carmona's proof of Vogan's lambda map.
result Provides an asymptotic formula for heat kernel behavior.
We prove a completely new integral criterion for the existence and completeness of the wave operators W±(−Δh,−Δg,Ig,h) corresponding to the (unique self-adjoint realizations of) the Laplace-Beltrami operators −Δj, j=1,2, that are induced by two quasi-isometric complete Riemannian metrics g and h o…
Study heat content on RCD(K,N) spaces with specific boundary conditions.
problem Analyzing heat content in RCD(K,N) spaces with irregular boundaries.
method Proved first-order asymptotics using measured interior geodesic condition.
result Established first-order heat content asymptotics on RCD(K,N) spaces.
The paper establishes new inequalities for Finsler measure spaces.
problem Developing inequalities for Finsler measure spaces.
method Study of linearized heat semigroup and application of Li-Yau's inequalities.
result Established new Li-Yau's type inequalities for Finsler measure spaces.
Ancient solutions to biharmonic heat equation bounded by polynomial dimensions.
problem Bounding ancient solutions to biharmonic heat equation.
method Using polynomial volume growth and dimensions of biharmonic functions.
result Ancient solutions are bounded by polynomial dimensions.
We consider the heat operator acting on differential forms on spaces with complete and incomplete edge metrics. In the latter case we study the heat operator of the Hodge Laplacian with algebraic boundary conditions at the edge singularity. We establish the mapping properties of the heat operator, recovering and extend…
Heat flow on lens spaces settles into Morse functions with four critical points.
problem Understanding the behavior of heat flow on lens spaces.
method Analyzing the asymptotic spectral expansion of the heat flow.
result Generic heat evolutions on lens spaces \(L(p,q)\) with \(p\geq2\) and \(1\leq q\leq p/2\) tend to settle into Morse functions with exactly four critical points.
Lipschitz regularity proved for harmonic map heat flows into CAT(0) spaces.
problem Proving Lipschitz regularity for harmonic map heat flows into CAT(0) spaces.
method Elliptic approximation method
result Every weak solution of the harmonic map heat flow into CAT(0) spaces is Lipschitz continuous in both space and time.
The paper proves boundedness of a Riesz transform on weighted manifolds.
problem Establishing \(L^p\)-boundedness of the covariant Riesz transform on differential forms.
method Heat-kernel criterion, volume doubling, heat kernel estimates, curvature control, gradient bounds.
result The covariant Riesz transform is \(L^p\)-bounded for \(p>2\) on weighted Riemannian manifolds.
New method predicts heat load in thermal grids using latent variables.
problem Predicting heat load in district energy systems.
method Combines nominal model for outdoor temperature with latent variable model for residual heat load.
result Proposed method achieves better prediction accuracy than artificial neural networks.
We consider Laplacians acting on sections of homogeneous vector bundles over symmetric spaces. By using an integral representation of the heat semi-group we find a formal solution for the heat kernel diagonal that gives a generating function for the whole sequence of heat invariants. We argue that the obtained formal s…
Improved graph-based connectivity estimation using heat modelling.
problem Lack of explicit model-based, dynamic, multivariate, and directed connectivity estimation methods.
method Noise-driven heat modelling on graphs with relaxed assumptions and regularisation.
result Demonstrated ability to capture meaningful spatial structure across real-world datasets.
The aim of this paper is to show that the dynamics of Lp heat semigroups (p>2) on a symmetric space of non-compact type is very different from the dynamics of the Lp heat semigroups if p≤2. To see this, it is shown that certain shifts of the Lp heat semigroups have a chaotic behavior if p>2 and that …
Heat flow fails to preserve concavity in curved spaces.
problem Non-preservation of concavity properties in curved spaces.
method Analysis of Dirichlet heat flow on Riemannian manifolds.
result No concavity properties are preserved unless curvature is zero.
Study subelliptic heat kernel on lifted sphere from octonionic projective space.
problem Analyzing sub-Laplacian on lifted sphere from octonionic projective space.
method Explicit formulas for heat kernel and Green function derived.
result Explicit formulas for heat kernel and Green function.
In this paper, we prove pointwise convergence of heat kernels for mGH-convergent sequences of RCD∗(K,N)-spaces. We obtain as a corollary results on the short-time behavior of the heat kernel in RCD∗(K,N)-spaces. We use then these results to initiate the study of Weyl's law in the RCD setting
We consider the heat kernel (and the zeta function) associated with Laplace type operators acting on a general irreducible rank 1 locally symmetric space X. The set of Minakshisundaram- Pleijel coefficients {A_k(X)}_{k=0}^{\infty} in the short-time asymptotic expansion of the heat kernel is calculated explicitly.
The paper studies harmonic map heat flow stability and decay rates.
problem Analyzing stability and decay rates of harmonic map heat flow solutions.
method Use of homogeneous Besov space B˙p,∞pd(Rd) for small initial data and self-similar decay assumption. result Decay rates for solutions of the harmonic map flow of the form ∥ablau(t)∥L∞(Rd)≤Ct−21 and self-similar decay under stronger initial conditions. Study shows global oscillatory solutions for Yang-Mills heat flow in 4D space.
problem Investigating long-time dynamics of Yang-Mills heat flow with specific initial data.
method Analysis of SO(4)-equivariant Yang-Mills heat flow with SU(2) group in 4D space. result Global solutions can exhibit oscillatory behavior at time infinity.
Entropy study on synthetic spaces with curvature bounds.
problem Entropy functional on synthetic spaces with curvature bounds.
method Rigorous justification of entropy formula, monotonicity, and rigidity properties; heat kernel bounds.
result Bounds for heat equation solutions on synthetic spaces.
Survey on heat kernels and path integrals.
problem Approximating Wiener measure on compact manifolds.
method Review of recent results on approximating Wiener measure.
result Approximation of Wiener measure by measures on spaces of piece-wise geodesics.