S.S. Chern raised the problem to find necessary and sufficient conditions for a given Riemannian manifold to be realizable on a minimal submanifold of a Euclidean space. The aim of this paper is to provide new necessary conditions. For minimal submanifolds in a Euclidean space we consider the negative of the Ricci tensor as defining a new metric, which is nothing but the third fundamental form,...

The discovery of pseudoparticle solutions to the euclidean SU(2) Yang-Mills theory [1 ] has suggested the possibility that analogous solutions might occur in Einstein's theory of gravitation. The existence of such solutions would have a profound effect on the quantum theory of gravitation [2,3]. Since fire Yang-Mills pseudoparticles possess self-dual field strengths, one likely possibility is t...

Compact metric spaces that admit intrinsic isometries to the Euclidean d-space are considered. Roughly, the main result states that the class of such spaces coincides with the class of inverse limits of Euclidean d-polyhedra. §

Given a scattering metric on the Euclidean space. We consider the Schrödinger equation corresponding to the metric, and study the propagation of singularities for the solution in terms of the homogeneous wavefront set. We also prove that the notion of the homogeneous wavefront set is essentially equivalent to that of the quadratic scattering wavefront set introduced by J. Wunsch [21]. One of th...

The discovery of self-dual instanton solutions in Euclidean Yang-Mills theory [I] has recently stimulated a great deal of interest in self-dual solutions to Einstein’s theory of gravitation. One would expect that the relevant instanton-like metrics would be those whose gravitational fields are self-dual, localized in Euclidean spacetime and free of singularities. In fact, solutions have been fo...

Many natural language processing still depend on the Euclidean distance function between the two feature vectors, but it has severe defects as to feature weightings and feature correlations. In this paper we propose an optimal metric distance function that can be used as an alternative to the Euclidean distance, accommodating the two problems at the same time. This metric is optimal in the sens...

The deterministic border collision normal form describes the bifurcations of a discrete time dynamical system as a fixed point moves across the switching surface with changing parameter. If the position of the switching surface varies randomly, but within some bounded region, we give conditions which imply that the attractor close to the bifurcation point is the attractor of an Iterated Functio...

Invented in the 1960’s, permutation codes have reemerged in recent years as a topic of great interest because of properties making them attractive for certain modern technological applications. In 2011 a decoding method called LP (linear programming) decoding was introduced for a class of permutation codes with a Euclidean distance induced metric. In this paper we comparatively analyze the Eucl...

We give a sufficient condition for a projective metric on a subset of a Euclidean space to admit a bi-Lipschitz embedding into Euclidean space of the same dimension.