In this paper, we investigate minimal submanifolds in Euclidean space with positive index of relative nullity. Let M m be a complete Riemannian manifold and let f : M m → R n be a minimal isometric immersion with index of relative nullity at least m − 2 at any point. We show that if the Omori-Yau maximum principle for the Laplacian holds on M m , for instance, if the scalar curvature of M m does not decrease to −∞ too fast or if the immersion f is proper, then the submanifold must be a cylinder over a minimal surface.

In this paper we investigate m-dimensional complete minimal submanifolds in Euclidean spheres with index of relative nullity at least m − 2 at any point. These are austere submanifolds in the sense of Harvey and Lawson [19] and were initially studied by Bryant [3]. For any dimension and codimension there is an abundance of non-complete examples fully described by Dajczer and Florit [7] in terms of a class of surfaces, called elliptic, for which the ellipse of curvature of a certain order is a circle at any point. Under the assumption of completeness, it turns out that any submanifold is either totally geodesic or has dimension three. In the latter case there are plenty of examples, even compact ones. Under the mild assumption that the Omori-Yau maximum principle holds on the manifold, a trivial condition in the compact case, we provide a complete local parametric description of the submanifolds in terms of 1-isotropic surfaces in Euclidean space. These are the minimal surfaces for which the standard ellipse of curvature is a circle at any point. For these surfaces, there exists a Weierstrass type representation that generates all simply connected ones.

We investigate complete minimal submanifolds f : M 3 → H n in hyperbolic space with index of relative nullity at least one at any point. The case when the ambient space is either the Euclidean space or the round sphere was already studied in [6] and [7], respectively. If the scalar curvature is bounded from below we conclude that the submanifold has to be either totally geodesic or a generalized cone over a complete minimal surface lying in an equidistant submanifold of H n . of a geodesic sphere or an equidistant hypersurface or a horosphere if c > 0, c < 0 or c = 0, respectively. Let g : L 2 → Q n−ℓ c be an isometric immersion of a two-dimensional Riemannian manifold. The normal bundle of h = i • g : L 2 → H n splits orthogonally aswhere exp denotes the exponential map of H n . Then we denote by M m , m = 2 + ℓ, the open subset of N i Q n−ℓ c where G is an immersion endowed with the metric induced by the map G.Definition: The generalized cone in hyperbolic space over a surface g :1 Preliminaries Let f : M m → H n denote an isometric immersion of an m-dimensional Riemannian manifold M m into hyperbolic space. The relative nullity subspace of f at x ∈ M m is defined as D(x) = {X ∈ T x M : α(X, Y ) = 0 for all Y ∈ T x M} where α : T M × T M → N f M denotes the second fundamental form with values in the normal bundle.

Our main contribution is to show that the behaviour of kernels across multiple layers of a convolutional neural network can be approximated using a logic program. The extracted logic programs yield accuracies that correlate with those of the original model, though with some information loss in particular as approximations of multiple layers are chained together or as lower layers are quantised. We also show that an extracted program can be used as a framework for further understanding the behaviour of CNNs. Specifically, it can be used to identify key kernels worthy of deeper inspection and also identify relationships with other kernels in the form of the logical rules. Finally, we make a preliminary, qualitative assessment of rules we extract from the last convolutional layer and show that kernels identified are symbolic in that they react strongly to sets of similar images that effectively divide output classes into sub-classes with distinct characteristics.

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