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Carlos Tomei

Publications and source records attributed to Carlos Tomei.

At least 19 recordsLinked to original sources
arXiv2026

Diagonalization of the Toda flow for arbitrary isospectral symmetric matrices

We construct coordinates on orthogonal conjugacy classes of traceless real symmetric matrices with arbitrary spectrum (the isospectral manifold) that diagonalize the non-periodic Toda vector field. The coordinates, defined on a neighborhood of any diagonal matrix decouple the Toda vector field into a sum of multiples of the Euler vector field in $\mathbb{R}$. The domain of each set of coordinates is dense and their union covers the isospectral manifold. The construction relies on a new matrix factorization which is of independent interest.

math.DG↗
arXiv2025

Computing multiple solutions from knowledge of the critical set

{We explore a simple {\it geometric model} for functions between spaces of the same dimension (in infinite dimensions, we require that Jacobians be Fredholm operators of index zero). The model combines standard results in analysis and topology associated with familiar global and local aspects. Functions are supposed to be proper on bounded sets. The model is valid for a large class of semilinear elliptic differential operators. It also provides a fruitful context for numerical analysis. For a function $F: X \to Y$ between real Banach spaces, continuation methods to solve $F(x) = y$ may improve from considerations about the global geometry of $F$. We consider three classes of examples. First we handle functions from the Euclidean plane to itself, for which the reasoning behind the techniques is visualizable. The second, between spaces of dimension 15, is obtained by discretizing a nonlinear Sturm-Liouville problem for which special right hand sides admit abundant solutions. Finally, we compute the six solutions of a semilinear elliptic equation $-\Delta u - f(u) = g$ studied by Solimini.}

math.AP↗
arXiv2025

Diagonalization of the Toda flow on conjugacy classes of matrices with simple spectrum

We construct coordinates on conjugacy classes of traceless complex matrices with simple spectrum that diagonalize the non-periodic Toda vector field. By this we mean that the coordinates, defined on an open and dense neighborhood of any diagonal matrix in the conjugacy class, decouple the Toda vector field into a sum of multiples of the Euler and rotational vector field in $\mathbb{C}$. Using Lie theoretic methods, we extend this construction from $\mathfrak{sl}_\mathbb{C}$ to arbitrary complex semisimple Lie algebras and to their real forms

math.DG↗
arXiv2025

Efficient implementation of high-order isospectral symplectic Runge-Kutta schemes

Isospectral Runge-Kutta methods are well-suited for the numerical solution of isospectral systems such as the rigid body and the Toda lattice. More recently, these integrators have been applied to geophysical fluid models, where their isospectral property has provided insights into the long-time behavior of such systems. However, higher-order Isospectral Runge-Kutta methods require solving a large number of implicit equations. This makes the implicit midpoint rule the most commonly used due to its relative simplicity and computational efficiency. In this work, we introduce a novel algorithm that simplifies the implementation of general isospectral Runge-Kutta integrators. Our approach leverages block matrix structures to reduce the number of implicit equations per time step to a single one. This equation can be solved efficiently using fixed-point iteration. We present numerical experiments comparing performance and accuracy of higher-order integrators implemented with our algorithm against the implicit midpoint rule. Results show that, for low-dimensional systems, the higher-order integrators yield improved conservation properties with comparable computational cost. For high-dimensional systems, while our algorithm continues to show better conservation properties, its performance is less competitive, though it can be improved through parallelization.

math.NA↗
arXiv2023

Positive eigenvectors and simple nonlinear maps

For linear operators $L, T$ and nonlinear maps $P$, we describe classes of simple maps $F = I - P T$, $F = L - P$ between Banach and Hilbert spaces, for which no point has more than two preimages. The classes encompass known examples (homeomorphisms, global folds) and the weaker, geometric, hypotheses suggest new ones. The operator $L$ may be the Laplacian with various boundary conditions, as in the original Ambrosetti-Prodi theorem, or the operators associated with the quantum harmonic oscillator, the hydrogen atom, a spectral fractional Laplacian, elliptic operators in non-divergent form. The maps $P$ include the Nemitskii map $P(u) = f(u)$ but may be non-local, even non-variational. For self-adjoint operators $L$, we employ familiar results on the nondegeneracy of the ground state. On Banach spaces, we use a variation of the Krein-Rutman theorem.

math.FA↗
arXiv2023

Linearizing Toda and SVD flows on large phase spaces of matrices with real spectrum

We consider different phase spaces for the Toda flows and the less familiar SVD flows. For the Toda flow, we handle symmetric and non-symmetric matrices with real simple eigenvalues, possibly with a given profile. Profiles encode, for example, band matrices and Hessenberg matrices. For the SVD flow, we assume simplicity of the singular values. In all cases, an open cover is constructed, as are corresponding charts to Euclidean space. The charts linearize the flows, converting it into a linear differential system with constant coefficients and diagonal matrix. A variant construction transform the flows into uniform straight line motion. Since limit points belong to the phase space, asymptotic behavior becomes a local issue. The constructions rely only on basic facts of linear algebra, making no use of symplectic geometry.

math.SP↗
arXiv2022

Higher dimensional versions of theorems of Euler and Fuss

We present higher dimensional versions of the classical results of Euler and Fuss, both of which are special cases of the celebrated Poncelet porism. Our results concern polytopes, specifically simplices, parallelotopes and cross polytopes, inscribed in a given ellipsoid and circumscribed to another. The statements and proofs use the language of linear algebra. Without loss, one of the ellipsoids is the unit sphere and the other one is also centered at the origin. Let $A$ be the positive symmetric matrix taking the outer ellipsoid to the inner one. If $trace A = 1$, there exists a bijection between the orthogonal group $O(n)$ and the set of such labeled simplices. Similarly, if $trace A^2 = 1$, there are families of parallelotopes and of cross polytopes, also indexed by $O(n)$.

math.MG↗
arXiv2022

The geometry of spectral interlacing

We provide a detailed description of the maps associated with spectral interlacing, for rank one perturbations and bordering of symmetric and Hermitian matrices. The arguments rely on standard techniques of nonlinear analysis.

math.SP↗
arXiv2021

An atlas adapted to the Toda flow

We introduce an atlas adapted to the Toda flow on the manifold of full flags of any non-compact real semisimple Lie algebra, and on its Hessenberg-type submanifolds. In our local coordinates the Toda flow becomes linear. We use these new coordinates to show that the Toda flow on the manifold of full flags is Morse-Smale, which generalizes the main result of \cite{CSS1} to arbitrary non-compact real semisimple Lie algebras. As a byproduct we describe new features of classical constructions in matrix theory.

math.DG↗
arXiv2020

On the open Toda chain with external forcing

We consider the open Toda chain with external forcing, and in the case when the forcing stretches the system, we derive the longtime behavior of solutions of the chain. Using an observation of Jürgen Moser, we then show that the system is completely integrable, in the sense that the $2N$-dimensional system has $N$ functionally independent Poisson commuting integrals, and also has a Lax-Pair formulation. In addition, we construct action-angle variables for the flow. In the case when the forcing compresses the system, the analysis of the flow remains open.

math.DS↗
arXiv2020

Directed graphs and interferometry

The observed output of an interferometer is the result of interference among the parts of the input light beam traveling along each possible optical path. In complex systems, writing down all these possible optical paths and computing their cumulative effect can become a difficult task. We present an intuitive graph-based method for solving this problem and calculating electric fields within an interferometric setup, classical and quantum. We show how to associate a weighted directed graph to an interferometer and define rules to simplify these associated graphs. Successive application of the rules results in a final graph containing information on the desired field amplitudes. The method is applied to a number of examples in cavity optomechanics and cavity-enhanced interferometers.

quant-ph↗
arXiv2020

Discrete Line Fields on Surfaces

Vector fields and line fields, their counterparts without orientations on tangent lines, are familiar objects in the theory of dynamical systems. Among the techniques used in their study, the Morse--Smale decomposition of a (generic) field plays a fundamental role, relating the geometric structure of phase space to a combinatorial object consisting of critical points and separatrices. Such concepts led Forman to a satisfactory theory of discrete vector fields, in close analogy to the continuous case. In this paper, we introduce discrete line fields. Again, our definition is rich enough to provide the counterparts of the basic results in the theory of continuous line fields: a Euler-Poincaré formula, a Morse--Smale decomposition and a topologically consistent cancellation of critical elements, which allows for topological simplification of the original discrete line field.

cs.CG↗
arXiv2018

Global folds between Banach spaces as perturbations

Global folds between Banach spaces are obtained from a simple geometric construction: a Fredholm operator $T$ of index zero with one dimensional kernel is perturbed by a compatible nonlinear term $P$. The scheme encapsulates most of the known examples and suggests new ones. Concrete examples rely on the positivity of an eigenfunction. For the standard Nemitskii case $P(u) = f(u)$ (but $P$ might be nonlocal, non-variational), $T$ might be the Laplacian with different boundary conditions, as in the Ambrosetti-Prodi theorem, or the Schrödinger operators associated with the quantum harmonic oscillator or the Hydrogen atom, a spectral fractional Laplacian, a (nonsymmetric) Markov operator. For self-adjoint operators, we use results on the nondegeneracy of the ground state. On Banach spaces, a similar role is played by a recent extension by Zhang of the Krein-Rutman theorem.

math.AP↗
arXiv2017

Discrete Gradient Line Fields on Surfaces

A line field on a manifold is a smooth map which assigns a tangent line to all but a finite number of points of the manifold. As such, it can be seen as a generalization of vector fields. They model a number of geometric and physical properties, e.g. the principal curvature directions dynamics on surfaces or the stress flux in elasticity. We propose a discretization of a Morse-Smale line field on surfaces, extending Forman's construction for discrete vector fields. More general critical elements and their indices are defined from local matchings, for which Euler theorem and the characterization of homotopy type in terms of critical cells still hold.

math.GT↗
arXiv2017

Results of Ambrosetti-Prodi type for non-selfadjoint elliptic operators

The well-known Ambrosetti-Prodi theorem considers perturbations of the Dirichlet Laplacian by a nonlinear function whose derivative jumps over the principal eigenvalue of the operator. Various extensions of this landmark result were obtained for self-adjoint operators, in particular by Berger and Podolak, who gave a geometrical description of the solution set. In this text we show that similar theorems are valid for non self-adjoint operators. In particular, we prove that the semilinear operator is a global fold. As a consequence, we obtain what appears to be the first exact multiplicity result for elliptic equations in non-divergence form. We employ techniques based on the maximum principle.

math.AP↗
arXiv2016

Scaling limits of discrete copulas are bridged Brownian sheets

For large $n$, take a random $n \times n$ permutation matrix and its associated discrete copula $X_n$. For $a, b = 0, 1, \ldots, n$, let $y_n(\frac{a}{n},\frac{b}{n}) = \frac{1}{n} ( X_{a,b} - \frac{ab}{n} )$; define $y_n: [0,1]^2 \to R$ by interpolating quadratically on squares of side $\frac{1}{n}$. We prove a Donsker type central limit theorem: $\sqrt{n} y_n$ approaches a bridged Brownian sheet on the unit square.

math.PR↗
arXiv2015

The Toda lattice, old and new

Originally a model for wave propagation on the line, the Toda lattice is a wonderful case study in mechanics and symplectic geometry. In Flaschka's variables, it becomes an evolution given by a Lax pair on the vector space of real, symmetric, tridiagonal matrices. Its very special asymptotic behavior was studied by Moser by introducing norming constants, which play the role of discrete inverse variables in analogy to the solution by inverse scattering of KdV. It is a completely integrable system on the coadjoint orbit of the upper triangular group. Recently, bidiagonal coordinates, which parameterize also non-Jacobi tridiagonal matrices, were used to reduce asymptotic questions to local theory. Larger phase spaces for the Toda lattice lead to the study of isospectral manifolds and different coadjoint orbits. Additionally, the time one map of the associated flow is computed by a familiar algorithm in numerical linear algebra. The text is mostly expositive and self contained, presenting alternative formulations of familiar results and applications to numerical analysis.

math.DS↗