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...

The EPDiff equation for geodesic flow on the diffeomorphisms admits a remarkable ansatz for its singular solutions, called “diffeons.” Because this solution ansatz is a momentum map, the diffeons evolve according to canonical Hamiltonian equations. We examine diffeon solutions on Einstein spaces that are “mostly” symmetric, i.e., whose quotient by a subgroup of the isometry group is 1-dimension...

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...

Let Q be a generalized quadrangle of order s with a regular point x. The set x⊥ together with all spans which are contained in x⊥ define a projective plane of order s. We introduce a property (Py) for every point y of Q noncollinear with x and prove that this property is equivalent with the regularity of the point y. We will use this to give an elementary proof for the following result: every g...

We study a class of Monte Carlo algorithms for the nonlinear σ-model, based on a Wolff-type embedding of Ising spins into the target manifold M . We argue heuristically that, at least for an asymptotically free model, such an algorithm can have dynamic critical exponent z ≪ 2 only if the embedding is based on an (involutive) isometry of M whose fixed-point manifold has codimension 1. Such an is...

We present a new 1/8 supersymmetric intersecting M-brane solution of D=11 supergravity with two independent rotation parameters. The metric has a non-singular event horizon and the near-horizon geometry is adS3×S×S×E (just as in the non-rotating case). We also present a method of determining the isometry supergroup of supergravity solutions from the Killing spinors and use it to show that for t...

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 ...