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 → ...

We consider the elementary operator L, acting on the Hilbert-Schmidt Class C2(H), given by L(T ) = ATB, with A and B bounded operators on a separable Hilbert space H. In this paper we establish results relating isometric properties of L with those of the defining symbols A and B. We also show that if A is a strict n−isometry on a Hilbert space H then {I, A∗A, (A∗)2A2, . . . , (A∗)n−1An−1} is a ...

It is shown that for many finite dimensional normed vector spaces V over C, a linear projection P : V → V will have nice structure if P + λ(I − P ) is an isometry for some complex unit not equal to one. From these results, one can readily determine the structure of bicircular projections, i.e., those linear projections P such that P + μ(I − P ) is a an isometry for every complex unit μ. The key...

We address the problem of correspondence between 3D isometric shapes. We present an automatic method that finds the optimal correspondence between two given (nearly) isometric shapes by minimizing the amount of deviation from isometry. We optimize the isometry error in two steps. In the first step, the 3D points uniformly sampled from the shape surfaces are transformed into spectral domain base...

Compressed sensing is a technique for finding sparse solutions to underdetermined linear systems. This technique relies on properties of the sensing matrix such as the restricted isometry property. Sensing matrices that satisfy this property with optimal parameters are mainly obtained via probabilistic arguments. Given any matrix, deciding whether it satisfies the restricted isometry property i...

Given polar spaces (V, β) and (V, Q) where V is a vector space over a field K, β a reflexive sesquilinear form and Q a quadratic form, we have associated classical isometry groups. Given a subfield F of K and an F -linear function L : K → F we can define new spaces (V, Lβ) and (V, LQ) which are polar spaces over F . The construction so described gives an embedding of the isometry groups of (V, ...

In this paper we analyze the local and global boundary rigidity problem for general Riemannian manifolds with $(M,g)$. We show that distance function, i.e., $d_g|_{\partial M \times \partial M}$, known near a point $p\in M$ at which $\partial is strictly convex, determines $g$ in suitable neighborhood of $p$ $M$, up to natural diffeomorphism invariance problem. also consider closely related len...