Keywords: Elliptic equations Nonlocal boundary value problems Difference schemes Stability a b s t r a c t The well-posedness of the Bitsadze–Samarskii type nonlocal boundary value problem in Hölder spaces with a weight is established. The coercivity inequalities for the solution of the nonlocal boundary value problem for elliptic equations are obtained. The stable second order of accuracy diff...

It is well known that integrable hierarchies in (1+1) dimensions are local while the recur-sion operators that generate them usually contain nonlocal terms. We resolve this apparent discrepancy by providing simple and universal sufficient conditions for a (nonlocal) recursion operator in (1+1) dimensions to generate a hierarchy of local symmetries. These conditions are satisfied by virtually al...

This is the first paper in a series to study vertex algebra-like objects arising from infinite-dimensional quantum groups (quantum affine algebras and Yangians). In this paper we lay the foundation for this study. For any vector space W , we study what we call quasi compatible subsets of Hom(W,W ((x))) and we prove that any maximal quasi compatible subspace has a natural nonlocal (namely noncom...

Hidden symmetries of ordinary differential equations (ODEs) are studied with nonlocal group generators. General forms are given for an exponential nonlocal group generator of an ODE that is reduced from a higher-order ODE, which is expressed in canonical variables and which is invariant under a two-parameter Lie group. The nonlocal group generator identifies a type I hidden symmetry. Type II hi...

We ask the question: is it possible to measure nonlocal properties of a state instantaneously? Relativistic causality restricts how this can be done, but we show here, for the bipartite two dimensional case, that every nonlocal variable of the system can be measured instantaneously. A necessary condition for a general nonlocal measurement on a bipartite system is presented. This essay was writt...

Quantum mechanics permits nonlocality—both nonlocal correlations and nonlocal equations of motion—while respecting relativistic causality. Is quantum mechanics the unique theory that reconciles nonlocality and causality? We consider two models, going beyond quantum mechanics, of nonlocality— “superquantum” correlations, and nonlocal “jamming” of correlations—and derive new results for the jammi...

Two new integrable nonlocal Davey–Stewartson equations are introduced. These equations provide two-spatial dimensional analogues of the integrable, nonlocal nonlinear Schrö-dinger equation introduced in Ablowitz and Musslimani (2013 Phys. Rev. Lett. 110 064105). Furthermore, like the latter equation, they also possess a PT symmetry and, as it is well known, this symmetry is important for the ...

We establish Fredholm properties for a class of nonlocal differential operators. Using mild convergence and localization conditions on the nonlocal terms, we also show how to compute Fredholm indices via a generalized spectral flow, using crossing numbers of generalized spatial eigenvalues. We illustrate possible applications of the results in a nonlinear and a linear setting. We first prove th...

Nonlocal Hamiltonians are used widely in first-principles quantum calculations; the nonlocality stems from eliminating undesired degrees of freedom, e.g., core electrons. To date, attempts to couple nonlocal systems to external electromagnetic (EM) fields have been heuristic or limited to weak or long wavelength fields. Using Feynman path integrals, we derive an exact, closed-form coupling of a...

We construct the covariant nonlocal action for recently suggested long-distance modifications of gravity theory motivated by the cosmological constant and cos-mological acceleration problems. This construction is based on the special non-local form of the Einstein-Hilbert action explicitly revealing the fact that this action within the covariant curvature expansion begins with curvature-squared...