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 ...

Abstract A periodic lattice in Euclidean space is the infinite set of all integer linear combinations basis vectors. Any can be generated by infinitely many different bases. This ambiguity was partially resolved, but standard reductions remain discontinuous under perturbations modelling atomic displacements. paper completes a continuous classification 2-dimensional lattices up to isometry (or c...

If f is an isometry, then every distance r > 0 is conserved by f , and vice versa. We can now raise a question whether each mapping that preserves certain distances is an isometry. Indeed, Aleksandrov [1] had raised a question whether a mapping f : X → X preserving a distance r > 0 is an isometry, which is now known to us as the Aleksandrov problem. Without loss of generality, we may assume r =...

for all x, y ∈ X . A distance r > 0 is said to be preserved (conservative) by a mapping f : X → Y if ‖ f (x)− f (y)‖ = r for all x, y ∈ X with ‖x− y‖ = r. If f is an isometry, then every distance r > 0 is conservative by f , and conversely. We can now raise a question whether each mapping that preserves certain distances is an isometry. Indeed, Aleksandrov [1] had raised a question whether a ma...

for all x, y ∈ X . A distance r > 0 is said to be preserved by a mapping f : X → Y if ‖ f (x)− f (y)‖ = r for all x, y ∈ X whenever ‖x− y‖ = r. If f is an isometry, then every distance r > 0 is preserved by f , and conversely. We can now raise a question whether each mapping that preserves certain distances is an isometry. Indeed, Aleksandrov [1] had raised a question whether a mapping f : X → ...

Abstract We observe that every map between finite-dimensional normed spaces of the same dimension respects fixed semi-inner products must be automatically a linear isometry. Moreover, we construct uniformly smooth renorming Hilbert space $$\ell _2$$ ℓ 2 and continuous i...

it is shown that every  almost linear bijection $h : arightarrow b$ of a unital $c^*$-algebra $a$ onto a unital$c^*$-algebra $b$ is a $c^*$-algebra isomorphism when $h(3^n u y) = h(3^n u) h(y)$ for allunitaries  $u in a$, all $y in a$, and all $nin mathbb z$, andthat almost linear continuous bijection $h : a rightarrow b$ of aunital $c^*$-algebra $a$ of real rank zero onto a unital$c^*$-algebra...

In compressed sensing, it is often desirable to consider signals possessing additional structure beyond sparsity. One such structured signal model – which forms the focus of this paper – is the local sparsity in levels class. This class has recently found applications in problems such as compressive imaging, multi-sensor acquisition systems and sparse regularization in inverse problems. In this...