(1) A main theme of this report is the relationship of approximation to learning and the primary role of sampling (inductive inference). We try to emphasize relations of the theory of learning to the mainstream of mathematics. In particular, there are large roles for probability theory, for algorithms such as least squares, and for tools and ideas from linear algebra and linear analysis. An advantage of doing this is that communication is facilitated and the power of core mathematics is more easily brought to bear. We illustrate what we mean by learning theory by giving some instances. (a) The understanding of language acquisition by children or the emergence of languages in early human cultures. (b) In Manufacturing Engineering, the design of a new wave of machines is anticipated which uses sensors to sample properties of objects before, during, and after treatment. The information gathered from these samples is to be analyzed by the machine to decide how to better deal with new input objects (see [43]). (c) Pattern recognition of objects ranging from handwritten letters of the alphabet to pictures of animals, to the human voice. Understanding the laws of learning plays a large role in disciplines such as (Cognitive) Psychology, Animal Behavior, Economic Decision Making, all branches of Engineering, Computer Science, and especially the study of human thought processes (how the brain works). Mathematics has already played a big role towards the goal of giving a universal foundation of studies in these disciplines. We mention as examples the theory of Neural Networks going back to McCulloch and Pitts [25] and Minsky and Papert [27], the PAC learning of Valiant [40], Statistical Learning Theory as developed by Vapnik [42], and the use of reproducing kernels as in [17] among many other mathematical developments. We are heavily indebted to these developments. Recent discussions with a number of mathematicians have also been helpful. In

http://pages.cs.wisc.edu/~brecht/cs838docs/cuckersmale.pdf

On the mathematical foundations of learning

A quantum graph is a graph equipped with a self-adjoint differential or pseudo-differential Hamiltonian. Such graphs have been studied recently in relation to some problems of mathematics, physics and chemistry. The paper has a survey nature and is devoted to the description of some basic notions concerning quantum graphs, including the boundary conditions, self-adjointness, quadratic forms, and relations between quantum and combinatorial graph models.

/pdf/quantum-graphs-i-some-basic-structures-2sf0fhd9ca.pdf

Quantum graphs: I. Some basic structures

A Picone's identity for the p -Laplacian and applications

During the last few years quantum graphs have become a paradigm of quantum chaos with applications from spectral statistics to chaotic scattering and wavefunction statistics. In the first part of this review we give a detailed introduction to the spectral theory of quantum graphs and discuss exact trace formulae for the spectrum and the quantum-to-classical correspondence. The second part of this review is devoted to the spectral statistics of quantum graphs as an application to quantum chaos. In particular, we summarize recent developments on the spectral statistics of generic large quantum graphs based on two approaches: the periodic-orbit approach and the supersymmetry approach. The latter provides a condition and a proof for universal spectral statistics as predicted by random-matrix theory.

https://www.physik.fu-berlin.de/en/einrichtungen/ag/ag-von-oppen/papers/nlin_CD_0605028.pdf

Quantum graphs: Applications to quantum chaos and universal spectral statistics

A brief survey on graph models for wave propagation in thin structures is presented. Such models arise in many areas of mathematics, physics, chemistry and engineering (dynamical systems, nanotechnology, mesoscopic systems, photonic crystals etc). Considerations are limited to spectral problems, although references to works with other studies are provided.

Graph models for waves in thin structures

Double operator integrals are a convenient tool in many problems
arising in the theory of self-adjoint operators, especially in the perturbation
theory. They allow to give a precise meaning to operations with functions
of two ordered operator-valued non-commuting arguments. In a different language,
the theory of double operator integrals turns into the problem of scalarvalued
multipliers for operator-valued kernels of integral operators. The paper gives a short survey of the main ideas, technical tools and
results of the theory. Proofs are given only in the rare occasions, usually they
are replaced by references to the original papers. Various applications are
discussed.

Double operator integrals in a Hilbert space

The spectral theory of operators in a finite-dimensional space first appeared in connection with the description of the frequencies of small vibrations of mechanical systems (see Arnol’d et al. 1985). When the vibrations of a string are considered, there arises a simple eigenvalue problem for a differential operator. In the case of a homogeneous string it suffices to use the classical theory of Fourier series. For an inhomogeneous string it becomes necessary to consider the general Sturm-Liouville problem, which is the eigenvalue problem for a simple one-dimensional differential operator with variable coefficients. Failing to be explicitly soluble, the problem calls for a qualitative and asymptotic study (see Egorov and Shubin 1988a, §9). When considering the vibrations of a membrane or a three-dimensional elastic body, we arrive at the eigenvalue problems for many-dimensional differential operators. Such problems also arise in the theory of shells, hydrodynamics, and other areas of mechanics. One of the richest sources of problems in spectral theory, mostly for Schrodinger operators, is quantum mechanics, in which the eigenvalues of the quantum Hamiltonian, and, more generally, the points of the spectrum of the Hamiltonian, are the possible energy values of the system.

Spectral Theory of Differential Operators

We consider the Dirichlet Laplacian Δ∈ in a family of bounded domains {−a < x < b, 0 < y < eh(x)}. The main assumption is that x = 0 is the only point of global maximum of the positive, continuous function h(x). We find the two-term asymptotics in e → 0 of the eigenvalues and the one-term asymptotics of the corresponding eigenfunctions. The asymptotic formulas obtained involve the eigenvalues and eigenfunctions of an auxiliary ODE on ℝ that depends on the behavior of h(x) as x → 0. The proof is based on a detailed study of the resolvent of the operator Δ∈.

https://math.arizona.edu/~friedlan/papers/narrow1.pdf

On the spectrum of the Dirichlet Laplacian in a narrow strip

The Laplacian on a metric tree is on its edges, with the appropriate compatibility conditions at the vertices. We study the eigenvalue problem on a rooted tree : -\lambda \Delta u = V u \quad \text{on } \Gamma,
 \qquad u(o)=0.
 Here , and . The eigenvalues for such a problem decay no faster than , this last case being typical for one-dimensional problems. We obtain estimates for the eigenvalues in the classes , and their weak analogues . The results for are of different character. The case is studied in more detail. To analyse this case, we use two methods; in both of them the problem is
 reduced to a family of one-dimensional problems. One of the methods is based upon a useful orthogonal decomposition
 of the Dirichlet space on . For a particular class of trees and weights, this method leads to the complete spectral analysis of the problem. We illustrate this in several examples, where we were able to obtain the asymptotics of the eigenvalues. 1991 Mathematics Subject Classification: primary 47E05, 05C05;
 secondary 34L15, 34L20.

Eigenvalue Estimates for the Weighted Laplacian on Metric Trees

A metric tree r is a tree whose edges are viewed as non-degenerate line segments. The Laplacian Δ on such a tree is the operator of second order differentiation on each edge, complemented by the Kirchhoff matching conditions at the vertices. The spectrum of Δ can be quite varied, reflecting the geometry of a tree. We consider a special class of trees, namely the so-called regular metric trees. Any such tree r possesses a rich group of symmetries. As a result, the space L 2 (Γ) decomposes into the orthogonal sum of subspaces reducing the operator Δ. This leads to detailed spectral analysis of Δ. We survey recent results on this subject.

/pdf/on-the-spectrum-of-the-laplacian-on-regular-metric-trees-2tx4ordc0c.pdf

Michael Solomyak

Papers

Double operator integrals in a Hilbert space

Spectral Theory of Differential Operators

On the spectrum of the Dirichlet Laplacian in a narrow strip

Eigenvalue Estimates for the Weighted Laplacian on Metric Trees

On the spectrum of the Laplacian on regular metric trees