arXiv subjects

Mahmood Ettehad

Publications and source records attributed to Mahmood Ettehad.

9 recordsLinked to original sources

arXiv2025

On the spectra of periodic discontinuous quantum graphs

We consider periodic Schrödinger operators on the hexagonal lattice with self-adjoint vertex conditions that allow discontinuity and concentrated mass at the vertices. This model generalizes the periodic Schrödinger operator on the hexagonal lattice with Neumann vertex conditions, known as the graphene Hamiltonian quantum graph. After formulating the corresponding Hamiltonian as a quantum graph and introducing the self adjoint vertex conditions, we provide its dispersion relation, spectrum, eigenvalues, and Dirac points. We also give explicit formulations of our results for the corresponding free operator (zero potential) and show that Borg's theorem is not valid for the Hamiltonians we study, that is, non-degenerate spectral gaps exist for the free operator with our vertex conditions.

math.SP↗

arXiv2022

On the Spectra of Periodic Elastic Beam Lattices: Single-Layer Graph

We present full description of spectra for a Hamiltonian defined on periodic hexagonal elastic lattices. These continua are constructed out of Euler-Bernoulli beams, each governed by a scalar-valued self-adjoint operator, which is also known as the fourth order Schrödinger operator, equipped with a real periodic symmetric potential. In contrast to the second order Schrödinger operator commonly applied in quantum graph literature, here vertex matching conditions encode geometry of the underlying graph by their dependence on angles at which edges are met. We show that for a special equal angle lattice, known as graphene, dispersion relation has a similar structure as reported for the periodic second order Schrödinger operator on hexagonal lattices. This property is then further utilized to prove existence of singular Dirac points. We further discuss reducibility of Fermi surface at uncountably many low-energy levels for this special lattice. Applying perturbation analysis, we extend the developed theory to derive dispersion relation for angle-perturbed Hamiltonian of hexagonal lattices in a geometric neighborhood of graphene. In these graphs, unlike graphene, dispersion relation is not splitted into purely energy and quasimomentum dependent terms, however singular Dirac points exist similar to the graphene case.

math-ph↗

arXiv2022

Hamilton-Jacobi equations on graphs with applications to semi-supervised learning and data depth

Shortest path graph distances are widely used in data science and machine learning, since they can approximate the underlying geodesic distance on the data manifold. However, the shortest path distance is highly sensitive to the addition of corrupted edges in the graph, either through noise or an adversarial perturbation. In this paper we study a family of Hamilton-Jacobi equations on graphs that we call the $p$-eikonal equation. We show that the $p$-eikonal equation with $p=1$ is a provably robust distance-type function on a graph, and the $p\to \infty$ limit recovers shortest path distances. While the $p$-eikonal equation does not correspond to a shortest-path graph distance, we nonetheless show that the continuum limit of the $p$-eikonal equation on a random geometric graph recovers a geodesic density weighted distance in the continuum. We consider applications of the $p$-eikonal equation to data depth and semi-supervised learning, and use the continuum limit to prove asymptotic consistency results for both applications. Finally, we show the results of experiments with data depth and semi-supervised learning on real image datasets, including MNIST, FashionMNIST and CIFAR-10, which show that the $p$-eikonal equation offers significantly better results compared to shortest path distances.

math.AP↗

arXiv2022

On Vertex Conditions In Elastic Beam Frames: Analysis on Compact Graphs

We consider three-dimensional elastic frames constructed out of Euler-Bernoulli beams and describe extension of matching conditions by relaxing the vertex-rigidity assumption and the case in which concentrated mass may exists. This generalization is based on coupling an (elastic) energy functional in terms of field's discontinuities at a vertex along with purely geometric terms derived out of first principles. The corresponding differential operator is shown to be self-adjoint. Although for planar frames with a class of rigid-joints the operator decomposes into a direct sum of two operators, this property only holds for a special class of the proposed model. Application of theoretical results is then discussed in details for compact frames embedded in Euclidean spaces with different dimensions. This includes extension of the established results for rigid-joint case on exploiting the symmetry present in a frame and decomposing the operator by restricting it onto reducing subspaces corresponding to irreducible representations of the symmetry group. Derivation of characteristic equation based on the idea of geometric-free local spectral basis and enforcing geometry of the graph into play by an appropriate choice of the coefficient set will be discussed. Finally, we prove the limit conditions in parameter space which results in decomposing of vector-valued beam Hamiltonian to a direct sum of scalar-valued ones.

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arXiv2021

Three dimensional elastic beam frames: rigid joint conditions in variational and differential formulation

We consider three-dimensional elastic frames constructed out of Euler--Bernoulli beams and describe a simple process of generating joint conditions out of the geometric description of the frame. The corresponding differential operator is shown to be self-adjoint. In the special case of planar frames, the operator decomposes into a direct sum of two operators, one coupling out-of-plane displacement to angular (torsional) displacement and the other coupling in-plane displacement with axial displacement (compression). Detailed analysis of two examples is presented. We actively exploit the symmetry present in the examples and decompose the operator by restricting it onto reducing subspaces corresponding to irreducible representations of the symmetry group. These ``quotient'' operators are shown to capture particular oscillation modes of the frame.

math.AP↗

arXiv2020

Instances of Computational Optimal Recovery: Dealing with Observation Errors

When attempting to recover functions from observational data, one naturally seeks to do so in an optimal manner with respect to some modeling assumption. With a focus put on the worst-case setting, this is the standard goal of Optimal Recovery. The distinctive twists here are the consideration of inaccurate data through some boundedness models and the emphasis on computational realizability. Several scenarios are unraveled through the efficient constructions of optimal recovery maps: local optimality under linearly or semidefinitely describable models, global optimality for the estimation of linear functionals under approximability models, and global near-optimality under approximability models in the space of continuous functions.

math.OC↗

arXiv2019

Approximability Models and Optimal System Identification

This article considers the problem of optimally recovering stable linear time-invariant systems observed via linear measurements made on their transfer functions. A common modeling assumption is replaced here by the related assumption that the transfer functions belong to a model set described by approximation capabilities. Capitalizing on recent optimal-recovery results relative to such approximability models, we construct some optimal algorithms and characterize the optimal performance for the identification and evaluation of transfer functions in the framework of the Hardy Hilbert space and of the disc algebra. In particular, we determine explicitly the optimal recovery performance for frequency measurements taken at equispaced points on an inner circle or on the torus.

math.OC↗

arXiv2019

Optimizing Consistent Merging and Pruning of Subgraphs in Network Tomography

A communication network can be modeled as a directed connected graph with edge weights that characterize performance metrics such as loss and delay. Network tomography aims to infer these edge weights from their pathwise versions measured on a set of intersecting paths between a subset of boundary vertices, and even the underlying graph when this is not known. Recent work has established conditions under which the underlying directed graph can be recovered exactly the pairwise Path Correlation Data, namely, the set of weights of intersection of each pair of directed paths to and from each endpoint. Algorithmically, this enables us to consistently fused tree-based view of the set of network paths to and from each endpoint to reconstruct the underlying network. However, in practice the PCD is not consistently determined by path measurements. Statistical fluctuations give rise to inconsistent inferred weight of edges from measurement based on different endpoints, as do operational constraints on synchronization, and deviations from the underlying packet transmission model. Furthermore, ad hoc solutions to eliminate noise, such as pruning small weight inferred links, are hard to apply in a consistent manner that preserves known end-to-end metric values. This paper takes a unified approach to the problem of inconsistent weight estimation. We formulate two type of inconsistency: \textsl{intrinsic}, when the weight set is internally inconsistent, and \textsl{extrinsic}, when they are inconsistent with a set of known end-to-end path metrics. In both cases we map inconsistent weight to consistent PCD within a least-squares framework. We evaluate the performance of this mapping in composition with tree-based inference algorithms.

math.OC↗

arXiv2018

Graph Reconstruction from Path Correlation Data

A communication network can be modeled as a directed connected graph with edge weights that characterize performance metrics such as loss and delay. Network tomography aims to infer these edge weights from their pathwise versions measured on a set of intersecting paths between a subset of boundary vertices, and even the underlying graph when this is not known. In particular, temporal correlations between path metrics have been used infer composite weights on the subpath formed by the path intersection. We call these subpath weights the Path Correlation Data. In this paper we ask the following question: when can the underlying weighted graph be recovered knowing only the boundary vertices and the Path Correlation Data? We establish necessary and sufficient conditions for a graph to be reconstructible from this information, and describe an algorithm to perform the reconstruction. Subject to our conditions, the result applies to directed graphs with asymmetric edge weights, and accommodates paths arising from asymmetric routing in the underlying communication network. We also describe the relationship between the graph produced by our algorithm and the true graph in the case that our conditions are not satisfied.

math.CO↗