A general method for constructing local Lie groups which are not contained in any global Lie group is described. These examples fail to satisfy the global associativity axiom which, by a theorem of Mal’cev, is necessary and sufficient for globalizability. Furthermore, we prove that every local Lie group can be characterized by such a covering construction, thereby generalizing Cartan’s global v...

The scalar initial value problem ut = ρDu+ f(u), is a model for dispersal. Here u represents the density at point x of a compact spatial region Ω ∈ R and time t, and u(·) is a function of t with values in some function space B. D is a bounded linear operator and f(u) is a bistable nonlinearity for the associated ODE ut = f(u). Problems of this type arise in mathematical ecology and materials sc...

Suppose that d ≥ 1 and 0 < β < α < 2. We establish the existence and uniqueness of the fundamental solution q(t, x, y) to a class of (possibly nonsymmetric) non-local operators L = ∆ + S, where Sf(x) := A(d,−β) ∫ R ( f(x+ z)− f(x)−∇f(x) · z1{|z|≤1} ) b(x, z) |z|d+β dz and b(x, z) is a bounded measurable function on R×R with b(x, z) = b(x,−z) for x, z ∈ R. Here A(d,−β) is a normalizing constant ...

To escape from the curse of dimensionality, we claim that one can learn non-local functions, in the sense that the value and shape of the learned function at x must be inferred using examples that may be far from x. With this objective, we present a non-local non-parametric density estimator. It builds upon previously proposed Gaussian mixture models with regularized covariance matrices to take...

On a conformal manifold with boundary, we construct conformally invariant local boundary conditions B for the conformally invariant power of the Laplacian k , with the property that ( k , B) is formally self-adjoint. These boundary problems are used to construct conformally invariant nonlocal operators on the boundary Σ, generalizing the conformal Dirichlet-to-Robin operator, with principal par...

Stapp has recently argued from a version of the Hardy type experiments that quantum mechanics must be non-local, independent of any additional assumptions like realism or hidden variables. I argue that his argument does include an additional assumption about the reality of certain counter-factual statements. In a recent paper [1] H. Stapp has argued that quantum mechanics is non-local. He bases...

A new approach to topological clean-up of 2D meshes is presented. Instead of searching for patterns in a mesh and replacing them as in other methods, the proposed method replaces a region in a mesh only according to the boundary of that region. This simplifies the classification of the different cases, and allows mesh modification over greater regions in the mesh. An algorithm for quadrilateral...

We claim and present arguments to the effect that a large class of manifold learning algorithms that are essentially local and can be framed as kernel learning algorithms will suffer from the curse of dimensionality, at the dimension of the true underlying manifold. This observation suggests to explore non-local manifold learning algorithms which attempt to discover shared structure in the tang...

We define, in a consistent way, non-local pseudo-differential operators acting on a space of analytic functionals. These operators include the fractional derivative case. In this context we show how to solve homogeneous and inhomogeneous equations associated with these operators. We also extend the formalism to d-dimensional space-time solving, in particular, the fractional Wave and Klein-Gordo...

In this paper we study a general class of “quasilinear non-local equations” depending on the gradient which arise from tug-of-war games. We establish a C/C/C regularity theory for these equations (the kind of regularity depending on the assumptions on the kernel), and we construct different non-local approximations of the p-Laplacian.