arXiv research

Edge subdivision affects the Perron eigenvalue of tree Ricci matrices.

Paper finds principal eigenvalue for infinity Laplacian in metric spaces.

Study of endperiodic maps on infinite graphs, proving homotopy and eigenvalue properties.

The symplectic representation of mapping classes is not surjective for certain types of mapping classes.

A pseudo-Anosov surface automorphism $φ$ has associated to it an algebraic unit $λ_φ$ called the dilatation of $φ$. It is known that in many cases $λ_φ$ appears as the spectral radius of a Perron-Frobenius matrix preserving a symplectic form $L$. We investigate what algebraic units could potentially appear as dilatatio…

Existence and uniqueness of discrete Einstein metrics on trees proven.

In this paper, we give sufficient conditions for a Perron number, given as the leading eigenvalue of an aperiodic matrix, to be a pseudo-Anosov dilatation of a compact surface. We give an explicit construction of the surface and the map when the sufficient condition is met.

Using an idea of Doug Lind, we give a lower bound for the Perron-Frobenius degree of a Perron number that is not totally-real. As an application, we prove that there are cubic Perron numbers whose Perron-Frobenius degrees are arbitrary large; a result known to Lind, McMullen and Thurston. A similar result is proved for…

The k-dimensional Dehn (or isoperimetric) function of a group bounds the volume of efficient ball-fillings of k-spheres mapped into k-connected spaces on which the group acts properly and cocompactly; the bound is given as a function of the volume of the sphere. We advance significantly the observed range of behavior f…

Every weak Perron number is realized as a stretch factor of a homeomorphism on a surface.

Stochastic discount factor (SDF) processes in dynamic economies admit a permanent-transitory decomposition in which the permanent component characterizes pricing over long investment horizons. This paper introduces an empirical framework to analyze the permanent-transitory decomposition of SDF processes. Specifically, …

Characterizes bi-Perron numbers with specific Galois conjugates.

Reproducing kernel Hilbert spaces (RKHSs) play an important role in many statistics and machine learning applications ranging from support vector machines to Gaussian processes and kernel embeddings of distributions. Operators acting on such spaces are, for instance, required to embed conditional probability distributi…

This paper extends transfer operator theory to McKean-Vlasov equations.

The paper explores growth rates and Perron numbers in Coxeter systems with low-dimensional Davis complexes.

Deep learning framework for kernel methods using RKHM and Perron-Frobenius operators.

We apply stochastic Perron's method to a singular control problem where an individual targets at a given consumption rate, invests in a risky financial market in which trading is subject to proportional transaction costs, and seeks to minimize her probability of lifetime ruin. Without relying on the dynamic programming…

Solves Monge-Ampère equations on compact Hessian manifolds using the Perron method.

Study complex Monge-Ampère flows on Kähler manifolds using Perron method.

Study of hypersurfaces with specific expansion properties.

In this paper we consider the growth rates of 3-dimensional hyperbolic Coxeter polyhedra some of its dihedral angles are $\fracπ{m}$ for $m\geq{7}$. By combining with the classical result by Parry \cite{Pa} and the main result of \cite{Y}, we prove that the growth rates of 3-dimensional hyperbolic Coxeter groups are Pe…

We define a large class of abstract Coxeter groups, that we call $\infty$--spanned, and for which the word growth rate and the geodesic growth rate appear to be Perron numbers. This class contains a fair amount of Coxeter groups acting on hyperbolic spaces, thus corroborating a conjecture by Kellerhals and Perren. We a…

Paper proves existence and uniqueness of circle patterns on surfaces with assigned geodesic curvatures.

We extend the stochastic Perron method to analyze the framework of stochastic target games, in which one player tries to find a strategy such that the state process almost surely reaches a given target no matter which action is chosen by the other player. Within this framework, our method produces a viscosity sub-solut…

I-BBS identifies latent sub-manifolds from distance matrices, robust to noise.

In this note, we deduce a partial answer to the question in the title. In particular, we show that asymptotically almost all bi-Perron algebraic unit whose characteristic polynomial has degree at most $2n$ do not correspond to dilatations of pseudo-Anosov maps on a closed orientable surface of genus $n$ for $n\geq 10$.…

We prove existence of solutions to boundary value problems and obstacle problems for degenerate-elliptic, linear, second-order partial differential operators with partial Dirichlet boundary conditions using a new version of the Perron method. The elliptic operators considered have a degeneracy along a portion of the do…

In this paper, we adapt stochastic Perron's method to analyze a stochastic target problem with unbounded controls in a jump diffusion set-up. With this method, we construct a viscosity sub-solution and super-solution to the associated Hamiltonian-Jacobi-Bellman (HJB) equations. Under comparison principles, uniqueness o…

The development of a metric for structural data is a long-term problem in pattern recognition and machine learning. In this paper, we develop a general metric for comparing nonlinear dynamical systems that is defined with Perron-Frobenius operators in reproducing kernel Hilbert spaces. Our metric includes the existing …

Using the Perron method, we prove the existence of hypersurfaces of prescribed special Lagrangian curvature with prescribed boundary inside complete Riemannian manifolds of non-positive curvature.

We shall use the classical Perron envelope method to show a general existence theorem to degenerate complex Monge-Ampère type equations on compact Kähler manifolds.

Study of complex Hessian equations using subharmonic functions and geodesics.

Let $f\colon M\to N$ be a continuous map between closed irreducible graph manifolds with infinite fundamental group. Perron and Shalen showed that if $f$ induces a homology equivalence on all finite covers, then $f$ is in fact homotopic to a homeomorphism. Their proof used the statement that every graph manifold is fin…

New methods avoid spectral pollution in transfer operators for accurate analysis.

Investor aims to meet financial goals with deadlines and target amounts, considering stock trading costs.

We construct infinite series of non-simple ideal hyperbolic Coxeter 4-polytopes whose growth rates are Perron numbers. This infinite series is the first example of such a non-compact infinite polytopal series.

We describe a novel technique for solving the Plateau problem for constant curvature hypersurfaces based on recent work of Harvey and Lawson. This is illustrated by an existence theorem for hypersurfaces of constant Gaussian curvature in $\Bbb{R}^{n+1}$.

This paper aims to make a new contribution to the study of lifetime ruin problem by considering investment in two hedge funds with high-watermark fees and drift uncertainty. Due to multi-dimensional performance fees that are charged whenever each fund profit exceeds its historical maximum, the value function is expecte…

Study on self-consuming generative models with diverse human curation, focusing on convergence and stability.

Data-driven methods link graphon limits to random walks and spectral clustering.

IGNN captures long-range graph dependencies using fixed-point equations.

New methods for clustering graphs using spectral analysis.

A framework for goal-based investing with penalties for fund transfers.

Paper introduces RKHM for more explicit variable structures analysis.

We proof existence theorems for the Dirichlet problem for hypersurfaces of constant special Lagrangian curvature in Hadamard manifolds. The first results are obtained using the continuity method and approximation and then refined using two iterations of the Perron method. The a-priori estimates used in the continuity m…

Modernizes Thurston's proof of entropy theorem for traintrack maps.

Building on the line of work [DIRT15a], [DIRT15b], [NS17a], [DT17], [HLS18], [HS18] we continue the study of particle systems with singular interaction through hitting times. In contrast to the previous research, we (i) consider very general driving processes and interaction functions, (ii) allow for inhomogeneous conn…

We prove that non-elementary hyperbolic groups grow exponentially more quickly than their infinite index quasiconvex subgroups. The proof uses the classical tools of automatic structures and Perron-Frobenius theory. We also extend the main result to relatively hyperbolic groups and cubulated groups. These extensions us…