arXiv subjects

Alexander Teplyaev

Publications and source records attributed to Alexander Teplyaev.

At least 19 recordsLinked to original sources

arXiv2026

Tangential and normal traces for extension domains with non-Lipschitz boundaries

We generalize the classical vector-valued tangential and normal trace theory on Lipschitz domains to the setting of non-Lipschitz $H^1$-extension domains, which includes domains with fractal boundaries such as the Koch snowflake. We define and study these operators based on the surjectivity of the trace operator for elements of $H^1(\Omega)$ and the Hilbert structure of the associated trace space. The normal trace operator is defined on $\Hdiv$ in $\mathbb{R}^n$. A generalized Stokes formula allows us to introduce the tangential trace operator on $\Hcurl$ and $\Hc^1(\Omega)$ in two and three dimensions. Following the approach of Buffa, Costabel and Sheen (2002) for Lipschitz domains, we define two abstract tangential boundary spaces as images of these trace operators, establish their Hilbert structure, and construct an abstract rotation operator linking them, in place of the geometric rotation based on the normal vector. We also extend Costabel's (1991) coercive bilinear form approach for Maxwell's equations to non-Lipschitz domains, yielding a theory that supports the treatment of the Hodge-Dirac operator and Green's formulas and enables the solution of boundary-value problems for the $\rotv \rotv +1$ operator within the $H^1$-extension domains framework.

math.AP↗

arXiv2026

Spectral properties of the Schreier graphs of the basilica group

We study the spectral properties of Laplacians on the Schreier graphs $\Gamma_n$ of the basilica group, the iterated monodromy group of the polynomial $z^2 - 1$, which is an important example in the theory of self-similar, amenable but not elementarily amenable, automaton groups. Building heavily on results by Brzoska, Jarvis, George, Rogers and Teplyaev about certain subgraphs of the basilica graphs, we develop a new recursive framework for computing the characteristic polynomials of these Laplacians. Our analysis reveals a simple underlying dynamical system and proves approximation results for the Kesten-von Neumann-Serre (KNS) spectral measure.

math.GR↗

arXiv2025

Distributional Convergence of the Empirical Laplacians with Integral Kernels on Domains with Boundaries

Motivated by the problem of understanding theoretical bounds for the performance of the Belkin-Niyogi Laplacian eigencoordinate approach to dimension reduction in machine learning problems, we consider the convergence of random graph Laplacian operators to a Laplacian-type operator on a manifold. For $\{X_j\}$ i.i.d.\ random variables taking values in $\mathbb{R}^d$ and $K$ a kernel with suitable integrability we define random graph Laplacians \begin{equation*} D_{\epsilon,n}f(p)=\frac{1}{n\epsilon^{d+2}}\sum_{j=1}^nK\left(\frac{p-X_j}{\epsilon}\right)(f(X_j)-f(p)) \end{equation*} and study their convergence as $\epsilon=\epsilon_n\to0$ and $n\to\infty$ to a second order elliptic operator of the form \begin{align*} \Delta_K f(p) &= \sum_{i,j=1}^d\frac{\partial f}{\partial x_i}(p)\frac{\partial g}{\partial x_j}(p)\int_{\mathbb{R}^d}K(-t)t_it_jd\lambda(t)\\ &\quad +\frac{g(p)}{2}\sum_{i,j=1}^d\frac{\partial^2f}{\partial x_i\partial x_j}(p)\int_{\mathbb{R}^d}K(-t)t_it_jd\lambda(t). \end{align*} Our results provide conditions that guarantee that $D_{\epsilon_n,n}f(p)-\Delta_Kf(p)$ converges to zero in probability as $n\to\infty$ and can be rescaled by $\sqrt{n\epsilon_n^{d+2}}$ to satisfy a central limit theorem. They generalize the work of Gin\'e--Koltchinskii~\cite{gine2006empirical} and Belkin--Niyogi~\cite{belkin2008towards} to allow manifolds with boundary and a wider choice of kernels $K$, and to prove convergence under weaker smoothness assumptions and a correspondingly more precise choice of conditions on the asymptotics of $\epsilon_n$ as $n\to\infty$.

math.FA↗

arXiv2025

Fractal curvatures and short-time asymptotics of heat content

The aim of our paper is twofold. First, we present new mathematical developments on the analysis of de Gennes' hypothesis on the short-time asymptotics of the heat content for bounded domains with smooth boundary and with fractal boundary. Second, we discuss new findings and concepts related to fractal curvatures for domains with fractal boundary. We conjecture that fractal curvatures and their scaling exponents will emerge in the short-time heat content asymptotics of domains with fractal boundary and the results discussed here are small initial contributions towards a resolution.

math.AP↗

arXiv2024

Convergence of layer potentials and Riemann-Hilbert problem on extension domains

We prove the convergence of layer potential operators for the harmonic transmission problem over a sequence of converging two-sided extension domains. Consequently, the Neumann-Poincar{\'e} operators, Calder{\'o}n projectors, and associated Neumann series converge in this setting. As a result, we generalize the notion of Cauchy integrals and, in a sense, of Hilbert transforms for a class of extension domains. Our approach relies on dyadic approximations of arbitrary open sets, considering convergence in terms of characteristic functions, Hausdorff distance, and compact sets.

math.AP↗

arXiv2024

Dirichlet metric measure spaces: spectrum, irreducibility, and small deviations

We show that for ultracontractive irreducible Dirichlet metric measure spaces, the Dirichlet spectrum is discrete for a restriction to any connected open set without any assumption on regularity of the boundary. The main applications include small deviations for the corresponding Hunt process and large time asymptotics for the generalized heat content. Our examples include Riemannian and sub-Riemannian manifolds, as well as non-smooth and fractal spaces.

math.PR↗

arXiv2024

Convergence, optimization and stability of singular eigenmaps

Eigenmaps are important in analysis, geometry, and machine learning, especially in nonlinear dimension reduction. Approximation of the eigenmaps of a Laplace operator depends crucially on the scaling parameter $\epsilon$. If $\epsilon$ is too small or too large, then the approximation is inaccurate or completely breaks down. However, an analytic expression for the optimal $\epsilon$ is out of reach. In our work, we use some explicitly solvable models and Monte Carlo simulations to find the approximately optimal range of $\epsilon$ that gives, on average, relatively accurate approximation of the eigenmaps. Numerically we can consider several model situations where eigen-coordinates can be computed analytically, including intervals with uniform and weighted measures, squares, tori, spheres, and the Sierpinski gasket. In broader terms, we intend to study eigen-coordinates on weighted Riemannian manifolds, possibly with boundary, and on some metric measure spaces, such as fractals.

math.PR↗

arXiv2024

Layer potential operators for transmission problems on extension domains

We use the well-posedness of transmission problems on classes of two-sided Sobolev extension domains to give variational definitions for (boundary) layer potential operators and Neumann-Poincar{\'e} operators. These classes of domains contain Lipschitz domains, and also domains with fractal boundaries. Although our variational formulation does not involve any measures on the boundary, we recover the classical results in smooth domains by considering the surface measure on the boundary. We discuss properties of these operators and generalize basic results in imaging beyond the Lipschitz case.

math.AP↗

arXiv2022

Besov class via heat semigroup on Dirichlet spaces III: BV functions and sub-Gaussian heat kernel estimates

With a view toward fractal spaces, by using a Korevaar-Schoen space approach, we introduce the class of bounded variation (BV) functions in a general framework of strongly local Dirichlet spaces with a heat kernel satisfying sub-Gaussian estimates. Under a weak Bakry-Émery curvature type condition, which is new in this setting, this BV class is identified with a heat semigroup based Besov class. As a consequence of this identification, properties of BV functions and associated BV measures are studied in detail. In particular, we prove co-area formulas, global $L^1$ Sobolev embeddings and isoperimetric inequalities. It is shown that for nested fractals or their direct products the BV class we define is dense in $L^1$. The examples of the unbounded Vicsek set, unbounded Sierpinski gasket and unbounded Sierpinski carpet are discussed.

math.MG↗

arXiv2022

Quantitative approach to Grover's quantum walk on graphs

In this paper, we study Grover's search algorithm focusing on continuous-time quantum walk on graphs. We propose an alternative optimization approach to Grover's algorithm on graphs that can be summarized as follows: instead of finding specific graph topologies convenient for the related quantum walk, we fix the graph topology and vary the underlying graph Laplacians. As a result, we search for the most appropriate analytical structure on graphs endowed with fixed topologies yielding better search outcomes. We discuss strategies to investigate the optimality of Grover's algorithm and provide an example with an easy tunable graph Laplacian to investigate our ideas.

math-ph↗

arXiv2022

Gaps labeling theorem for the Bubble-diamond self-similar graphs

Motivated by the appearance of fractals in several areas of physics, especially in solid state physics and the physics of aperiodic order, and in other sciences, including the quantum information theory, we present a detailed spectral analysis for a new class of fractal-type diamond graphs, referred to as bubble-diamond graphs, and provide a gap-labeling theorem in the sense of Bellissard for the corresponding probabilistic graph Laplacians using the technique of spectral decimation. Labeling the gaps in the Cantor set by the normalized eigenvalue counting function, also known as the integrated density of states, we describe the gap labels as orbits of a second dynamical system that reflects the branching parameter of the bubble construction and the decimation structure. The spectrum of the natural Laplacian on limit graphs is shown generically to be pure point supported on a Cantor set, though one particular graph has a mixture of pure point and singularly continuous components.

math-ph↗

arXiv2022

Spectral decimation of piecewise centrosymmetric Jacobi operators on graphs

We study the spectral theory of a class of piecewise centrosymmetric Jacobi operators defined on an associated family of substitution graphs. Given a finite centrosymmetric matrix viewed as a weight matrix on a finite directed path graph and a probabilistic Laplacian viewed as a weight matrix on a locally finite strongly connected graph, we construct a new graph and a new operator by edge substitution. Our main result proves that the spectral theory of the piecewise centrosymmetric Jacobi operator can be explicitly related to the spectral theory of the probabilistic Laplacian using certain orthogonal polynomials. Our main tools involve the so-called spectral decimation, known from the analysis on fractals, and the classical Schur complement. We include several examples of self-similar Jacobi matrices that fit into our framework.

math.SP↗

arXiv2021

Spectral decimation of a self-similar version of almost Mathieu-type operators

We introduce self-similar versions of the one-dimensional almost Mathieu operators. Our definition is based on a class of self-similar Laplacians instead of the standard discrete Laplacian, and includes the classical almost Mathieu operators as a particular case. Our main result establishes that the spectra of these self-similar almost Mathieu operators can be completely described by the spectra of the corresponding self-similar Laplacians through the spectral decimation framework used in the context of spectral analysis on fractals. In addition, the self-similar structure of our model provides a natural finite graph approximation model. This approximation is not only helpful in executing the numerical simulation, but is also useful in finding the spectral decimation function via Schur complement computations of given finite-dimensional matrices. The self-similar Laplacians used in our model were considered recently by Chen and Teplyaev who proved the emergence of singularly continuous spectra for specific parameters. We use this result to arrive at similar conclusions in the context of the self-similar almost Mathieu operators. Finally, we derive an explicit formula of the integrated density of states of the self-similar almost Mathieu operators as the weighted pre-images of the balanced invariant measure on a specific Julia set.

math.SP↗

arXiv2021

Boundary value problems on non-Lipschitz uniform domains: Stability, compactness and the existence of optimal shapes

We study boundary value problems for bounded uniform domains in $\mathbb{R}^n$, $n\geq 2$, with non-Lipschitz (and possibly fractal) boundaries. We prove Poincar\'e inequalities with trace terms and uniform constants for uniform $(\varepsilon,\infty)$-domains within bounded common confinements. We then introduce generalized Dirichlet, Robin and Neumann problems for Poisson type equations and prove the Mosco convergence of the associated energy functionals along sequences of suitably converging domains. This implies a stability result for weak solutions, and this also implies the norm convergence of the associated resolvents and the convergence of the corresponding eigenvalues and eigenfunctions. Based on our earlier work, we prove compactness results for parametrized classes of admissible domains, energy functionals and weak solutions. Using these results, we can verify the existence of optimal shapes in these classes.

math.AP↗

arXiv2021

Mixed boundary valued problem for linear and nonlinear wave equations in domains with fractal boundaries

The weak well-posedness, with the mixed boundary conditions, of the strongly damped linear wave equation and of the non linear Westervelt equation is proved in the largest natural class of Sobolev admissible non-smooth domains. In the framework of uniform domains in R^2 or R^3 we also validate the approximation of the solution of the Wester-velt equation on a fractal domain by the solutions on the prefractals using the Mosco convergence of the corresponding variational forms.

math.AP↗

arXiv2021

Non-Lipschitz uniform domain shape optimization in linear acoustics

We introduce new parametrized classes of shape admissible domains in R^n , n $\ge$ 2, and prove that they are compact with respect to the convergence in the sense of characteristic functions, the Hausdorff sense, the sense of compacts and the weak convergence of their boundary volumes. The domains in these classes are bounded ($ε$, $\infty$)-domains with possibly fractal boundaries that can have parts of any non-uniform Hausdorff dimension greater or equal to n -- 1 and less than n. We prove the existence of optimal shapes in such classes for maximum energy dissipation in the framework of linear acous-tics. A by-product of our proof is the result that the class of bounded ($ε$, $\infty$)-domains with fixed $ε$ is stable under Hausdorff convergence. An additional and related result is the Mosco convergence of Robin-type energy functionals on converging domains.

math.AP↗

arXiv2021

Hamiltonian systems, Toda lattices, Solitons, Lax Pairs on weighted Z-graded graphs

We consider discrete one dimensional nonlinear equations and present the procedure of lifting them to Z-graded graphs. We identify conditions which allow one to lift one dimensional solutions to solutions on graphs. In particular, we prove the existence of solitons {for static potentials} on graded fractal graphs. We also show that even for a simple example of a topologically interesting graph the corresponding non-trivial Lax pairs and associated unitary transformations do not lift to a Lax pair on the Z-graded graph.

math-ph↗

arXiv2020

On the existence of optimal shapes in architecture

We consider shape optimization problems for elasticity systems in architecture. A typical question in this context is to identify a structure of maximal stability close to an initially proposed one. We show the existence of such an optimally shaped structure within classes of bounded Lipschitz domains and within wider classes of bounded uniform domains with boundaries that may be fractal. In the first case the optimal shape realizes the infimum of a given energy functional over the class, in the second case it realizes the minimum. As a concrete application we discuss the existence of maximally stable roof structures under snow loads.

math.AP↗