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Exponential inequalities are main tools in machine learning theory. To prove exponential inequalities for non i.i.d random variables allows to extend many learning techniques to these variables. Indeed, much work has been done both on…
We prove the existence of fast traveling pulse solutions in excitable media with non-local coupling. Existence results had been known, until now, in the case of local, diffusive coupling and in the case of a discrete medium, with…
While there has been increasing interest in using neural networks to compute Lyapunov functions, verifying that these functions satisfy the Lyapunov conditions and certifying stability regions remain challenging due to the curse of…
The propagation of stable coherent entities of an electromagnetic field in nonlinear media with parameters varying in space can be described in the framework of iterations of nonlinear integral transformations. It is shown that for a set of…
In this paper, we study nonparametric models allowing for locally stationary regressors and a regression function that changes smoothly over time. These models are a natural extension of time series models with time-varying coefficients. We…
Instabilities in 1D spatially extended systems are studied with the aid of both temporal and spatial Lyapunov exponents. A suitable representation of the spectra allows a compact description of all the possible disturbances in tangent…
This paper provides a first example of constructing Lyapunov functions in a class of piecewise linear systems with limit cycles. The method of construction helps analyze and control complex oscillating systems through novel geometric means.…
We consider a continuous curve of self-adjoint Fredholm extensions of a curve of closed symmetric operators with fixed minimal domain $D_m$ and fixed {\it intermediate} domain $D_W$. Our main example is a family of symmetric generalized…
This work addresses controllability properties for some systems of partial differential equations in which the main feature is the coupling through nonlocal integral terms. In the first part, we study a nonlinear parabolic-elliptic system…
Lyapunov functions play a vital role in the context of control theory for nonlinear dynamical systems. Besides its classical use for stability analysis, Lyapunov functions also arise in iterative schemes for computing optimal feedback laws…
We are interested in nonlocal Eikonal Equations describing the evolution of interfaces moving with a nonlocal, non monotone velocity. For these equations, only the existence of global-in-time weak solutions is available in some particular…
We establish a new functional central limit theorem result for non-invertible measure preserving maps that are not necessarily ergodic, using the Perron-Frobenius operator. We apply the result to asymptotically periodic transformations and…
In this research, we propose a novel technique for visualizing nonstationarity in geostatistics, particularly when confronted with a single realization of data at irregularly spaced locations. Our method hinges on formulating a statistic…
In the setting of several commuting operators on a Hilbert space one defines the notions of invertibility and Fredholmness in terms of the associated Koszul complex. The index problem then consists of computing the Euler characteristic of…
Nonlinearities in finite dimensions can be linearized by projecting them into infinite dimensions. Unfortunately, often the linear operator techniques that one would then use simply fail since the operators cannot be diagonalized. This…
We study linear boundary-value problems for systems of first-order ordinary differential equations with the most general boundary conditions in the normed spaces of continuously differentiable functions on a finite closed interval. The…
The conceptual basis for the nonlocality of accelerated systems is presented. The nonlocal theory of accelerated observers and its consequences are briefly described. Nonlocal field equations are developed for the case of the…
The goal of this paper is to sketch a broader outline of the mathematical structures present in the Nonlinear Maxwell Theory in continuation of work presented in my previous articles. In particular, I display new types of both dynamic and…
Inverse parallel schemes remain indispensable tools for computing the roots of nonlinear systems, yet their dynamical behavior can be unexpectedly rich, ranging from strong contraction to oscillatory or chaotic transients depending on the…
Many complex systems are characterized by intriguing spatio-temporal structures. Their mathematical description relies on the analysis of appropriate correlation functions. Functional integral techniques provide a unifying formalism that…