The structure of the coincidence symmetry group of an arbitrary n-dimensional lattice in the n-dimensional Euclidean space is considered by describing a set of generators. Particular attention is given to the coincidence isometry subgroup (the subgroup formed by those coincidence symmetries that are elements of the orthogonal group). Conditions under which the coincidence isometry group can be ...

We show in this paper that every bijective linear isometry between the continuous section spaces of two non-square Banach bundles gives rise to a Banach bundle isomorphism. This is to support our expectation that the geometric structure of the continuous section space of a Banach bundle determines completely its bundle structures. We also describe the structure of an into isometry from a contin...

We give a necessary and sufficient condition for a locally compact group to be isomorphic to a closed cocompact subgroup in the isometry group of a Diestel–Leader graph. As a consequence of this condition, we see that every cocompact lattice in the isometry group of a Diestel–Leader graph admits a transitive, proper action on some other Diestel–Leader graph. We also give some examples of lattic...

If M is a connected n-dimensional Kobayashi-hyperbolic complex manifold, then the group Aut(M) of holomorphic automorphisms of M is a (real) Lie group in the compact-open topology, of dimension d(M) not exceeding n + 2n, with the maximal value occurring only for manifolds holomorphically equivalent to the unit ball B ⊂ C [Ko1], [Ka]. We are interested in describing hyperbolic manifolds with low...

In this paper we give a constructive characterisation of ultraweakly continuous linear functionals on the space of bounded linear operators on a separable Hilbert space. Let H be a separable complex Hilbert space, with orthonormal basis (en)∞n=1, and B(H) the set of bounded linear operators on H . The weak operator norm associated with the orthonormal basis (en) is defined on B(H) by ‖T ‖w ≡ ∞ ...

We argue that the chiral anomaly of N = 1 super Yang-Mills theory admits a dual description as spontaneous symmetry breaking in M theory on G2 holonomy manifolds. We identify an angle of the G2 background dual to the anomalous U(1)R current in field theory. This angle is not an isometry of the metric and we therefore develop a theory of “massive isometry” to describe fluctuations about such ang...

One of the difficulties of training deep neural networks is caused by improper scaling between layers. Scaling issues introduce exploding / gradient problems, and have typically been addressed by careful scale-preserving initialization. We investigate the value of preserving scale, or isometry, beyond the initial weights. We propose two methods of maintaing isometry, one exact and one stochasti...

In previous work, numerical experiments showed that ` minimization with 0 < p < 1 recovers sparse signals from fewer linear measurements than does ` minimization. It was also shown that a weaker restricted isometry property is sufficient to guarantee perfect recovery in the ` case. In this work, we generalize this result to an ` variant of the restricted isometry property, and then determine ho...

Let k be the reporducing kernel for a Hilbert space H(k) of nanlytic functions on Bd, the open unit ball in C, d ≥ 1. k is called a complete NP kernel, if k0 ≡ 1 and if 1 − 1/kλ(z) is positive definite on Bd × Bd. Let D be a separable Hilbert space, and consider H(k) ⊗ D ∼= H(k,D), and think of it as a space of D-valued H(k)-functions. A theorem of McCullough and Trent, [10], partially extends ...

for all a, b ∈ E and 0 ≤ t ≤ 1. Equivalently, f is affine if the map T :E → F , defined by Tx = fx− f(0), is linear. An isometry need not be affine. To see this, let E be the real line R, let F be the plane with the norm ‖x‖ = max(|x1|, |x2|), and let φ:R → R be any function such that |φ(s)−φ(t)| ≤ |s−t| for all s, t ∈ R, for example, φ(t) = |t| or φ(t) = sin t. Setting f(s) = (s, φ(s)) we get ...