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Networks constitute fundamental organizational structures across biological systems, although conventional graph-theoretic analyses capture exclusively pairwise interactions, thereby omitting the intricate higher-order relationships that…
We study the subelliptic heat kernel of the sub-Laplacian on a 2n+1-dimensional anti-de Sitter space H2n+1 which also appears as a model space of a CR Sasakian manifold with constant negative sectional curvature. In particular we obtain an…
The heat kernel expansion on even-dimensional hyperbolic spaces is asymptotic at both short and long times, with interestingly different Borel properties for these short and long time expansions. Resummations in terms of incomplete gamma…
We study the subelliptic heat kernels of the CR three dimensional solvable Lie groups. We first classify all left-invariant sub-Riemannian structures on three dimensional solvable Lie groups and obtain representations of these groups. We…
This is a mini-review of the heat kernel expansion for generalized Laplacians on various noncommutative spaces. Applications to the spectral action principle, renormalization of noncommutative theories and anomalies are also considered.
We calculate the closed analytic form of the solution of heat kernel equation for the anisotropic generalizations of flat Laplacian. We consider a UV as well as UV/IR interpolating generalizations. In all cases, the result can be expressed…
We construct the heat kernel on curvilinear polygonal domains in arbitrary surfaces for Dirichlet, Neumann, and Robin boundary conditions as well as mixed problems, including those of Zaremba type. We compute the short time asymptotic…
We develop new techniques to efficiently evaluate heat kernel coefficients for the Laplacian in the short-time expansion on spheres and hyperboloids with conical singularities. We then apply these techniques to explicitly compute the…
We derive estimates of the derivatives of the heat kernel on noncompact symmetric spaces and on locally symmetric spaces. Applying these estimates we study the $L^{p}$-boundedness of Littlewood-Paley-Stein operators and the Laplacian of the…
By making use of the potentials of the heat conduction equation the integral equations are derived which determine the heat kernel for the Laplace operator $-a^2\Delta$ in the case of compound media. In each of the media the parameter $a^2$…
In this paper, we continue the development of the Cartan neural networks programme, launched with three previous publications, by focusing on some mathematical foundational aspects that we deem necessary for our next steps forward. The…
An approach for solving scattering problems, based on two quantum field theory methods, the heat kernel method and the scattering spectral method, is constructed. This approach converts a method of calculating heat kernels into a method of…
It is known that the couple formed by the two dimensional Brownian motion and its L\'evy area leads to the heat kernel on the Heisenberg group, which is one of the simplest sub-Riemannian space. The associated diffusion operator is…
Let M be a compact Riemannian manifold without boundary and let H be a self-adjoint generalized Laplace operator acting on sections in a bundle over M. We give a path integral formula for the solution to the corresponding heat equation.…
The spherical domains $S^d_\beta$ with conical singularities are a convenient arena for studying the properties of tensor Laplacians on arbitrary manifolds with such a kind of singular points. In this paper the vector Laplacian on…
This paper considers the properties of Dirichlet Spaces of Homogeneous type which consist of band limited functions that are nearly exponential localizations on $\mathbb{R}^k.$ This is a powerful tool in harmonic analysis and it makes…
In the manifold setting, we provide a series of spectral convergence results quantifying how the eigenvectors and eigenvalues of the graph Laplacian converge to the eigenfunctions and eigenvalues of the Laplace-Beltrami operator in the…
Smoothing a signal based on local neighborhoods is a core operation in machine learning and geometry processing. On well-structured domains such as vector spaces and manifolds, the Laplace operator derived from differential geometry offers…
We obtain sharp two-sided heat kernel estimates on spaces with varying dimension, in which two spaces of general dimension are connected at one point. On these spaces, if the dimensions of the two constituent parts are different, the volume…
In this paper, we obtain an explicit formula for the heat kernel on the infinite Cayley graph of the modular group $\operatorname{PSL}_2\mathbb{Z}$, given by the presentation $\langle a,b\mid a^2=1, b^3=1\rangle$. Our approach extends the…