This paper deals with the initial-boundary value problem for the system of motion equations of an incompressible viscoelastic medium with Jeffreys constitutive law in an arbitrary domain of two-dimensional or three-dimensional space. The existence of weak solutions of this problem is obtained.

Given a dense set of points lying on or near an embedded submanifold $M_0\subset {ℝ}^n$ of Euclidean space, the manifold fitting problem is to find an embedding $F:M\to {ℝ}^n$ that approximates $M_0$ in the sense of least squares. When the dataset is modeled by a probability distribution, the fitting problem reduces to that of finding an embedding that minimizes $E_d\left[F\right]$, the expected square of the distance from a point in ${ℝ}^n$ to $F\left(M\right)$. It is shown that this approach to the fitting problem is guaranteed to fail because the functional $E_d$ has no local minima. This problem is addressed by adding a small multiple $k$ of the harmonic energy functional to the expected square of the distance. Techniques from the calculus of variations are then used to study this modified functional.

Let $u$ be a given bounded uniformly continuous mild solution of a higher-order abstract functional differential equation of delay or advance type. We give a so-called Massera-type criterion for the existence of a mild solution, which is a “spectral component” of $u$ with spectrum similar to the one of the forcing term $f$. Various spectral criteria for the existence of almost periodic and quasiperiodic mild solutions are given.

We establish the existence of three positive periodic solutions for a class of delay functional differential equations depending on a parameter by the Leggett-Williams fixed point theorem.