Related papers: Weak Operator Continuity for Evolutionary Equation…
Partial differential equations (PDEs) form a central component of scientific computing. Among recent advances in deep learning, evolutionary neural networks have been developed to successively capture the temporal dynamics of time-dependent…
For every closed subset $X$ of a stratifiable [resp. metrizable] space $Y$ we construct a positive linear extension operator $T:R^{X\times X}\to R^{Y\times Y}$ preserving constant functions, bounded functions, continuous functions,…
We investigate microlocal properties of partial differential operators with generalized functions as coefficients. The main result is an extension of a corresponding (microlocalized) distribution theoretic result on operators with smooth…
Topology applied to real world data using persistent homology has started to find applications within machine learning, including deep learning. We present a differentiable topology layer that computes persistent homology based on level set…
The topological derivative represents the sensitivity of a domain-dependent functional with respect to a local perturbation of the domain and is a valuable tool in topology optimization. Motivated by an application from electrical…
We generalize the phenomenon of continuation from complex anal- ysis to locally operator monotone functions. Along the lines of the egde-of- the-wedge theorem, we prove continuations exist dependent only on geometric features of the domain…
Functional evolution equations are used in the modeling of numerous physical processes. In this work, our main tool is perturbation theory of strongly continuous semigroups. The advantage of this technique is that one can provide functional…
This work examines the problem of topology inference over discrete-time nonlinear stochastic networked dynamical systems. The goal is to recover the underlying digraph linking the network agents, from observations of their state-evolution.…
Koopman operator is a composition operator defined for a dynamical system described by nonlinear differential or difference equation. Although the original system is nonlinear and evolves on a finite-dimensional state space, the Koopman…
Persistent topological properties of an image serve as an additional descriptor providing an insight that might not be discovered by traditional neural networks. The existing research in this area focuses primarily on efficiently…
We define various higher-order Markov properties for stochastic processes $(X(t))_{t\in \mathbb{T}}$, indexed by an interval $\mathbb{T} \subseteq \mathbb{R}$ and taking values in a real and separable Hilbert space $U$. We furthermore…
Motivated by problems in machine learning, we study a class of variational problems characterized by nonlocal operators. These operators are characterized by power-type weights, which are singular at a portion of the boundary. We identify a…
We study nonlocal equations from the area of peridynamics on bounded domains. In our companion paper, we discover that, on $\mathbb{R}^n$, the governing operator in peridynamics, which involves a convolution, is a bounded function of the…
We study a system of all-to-all weakly coupled uniformly expanding circle maps in the thermodynamic limit. The state of the system is described by a probability measure and its evolution is given by the action of a nonlinear operator, also…
We establish shape holomorphy results for general weakly- and hyper-singular boundary integral operators arising from second-order partial differential equations in unbounded two-dimensional domains with multiple finite-length open arcs.…
We revisit a well-established model for highly re-entrant semi-conductor manufacturing systems, and analyze it in the setting of states, in- and outfluxes being Borel measures. This is motivated by the lack of optimal solutions in the…
We consider evolution equations generated by quadratic operators admitting a decomposition in creation-annihilation operators without usual ellipticity-type hypotheses; this class includes hypocoercive model operators. We identify the…
The main problem considered in this article is the following: if $\mathbf{F}$, $\mathbf{E}$ are normed spaces of continuous functions over topological spaces $X$ and $Y$ respectively, and $\omega:Y\to\mathbb{C}$ and $\Phi:Y\to X$ are such…
Reciprocal transformations of Hamiltonian operators of hydrodynamic type are investigated. The transformed operators are generally nonlocal, possessing a number of remarkable algebraic and differential-geometric properties. We apply our…
We introduce a new class of quasilinear nonlocal operators and study equations involving these operators. The operators are degenerate elliptic and may have arbitrary growth in the gradient. Included are new nonlocal versions of p-Laplace,…