We show that if T is an isometry (as metric spaces) between the invertible groups of unital Banach algebras, then T is extended to a surjective real-linear isometry up to translation between the two Banach algebras. Furthermore if the underling algebras are closed unital standard operator algebras, (T (eA)) −1 T is extended to a surjective real algebra isomorphism; if T is a surjective isometry...

It is well known that the initialization of weights in deep neural networks can have a dramatic impact on learning speed. For example, ensuring the mean squared singular value of a network’s input-output Jacobian isO(1) is essential for avoiding the exponential vanishing or explosion of gradients. The stronger condition that all singular values of the Jacobian concentrate near 1 is a property k...

Mathieu and Ruddy proved that if be a unital spectral isometry from a unital C*-algebra Aonto a unital type I C*-algebra B whose primitive ideal space is Hausdorff and totallydisconnected, then is Jordan isomorphism. The aim of this note is to show that if be asurjective spectrum preserving additive map, then is a Jordan isomorphism without the extraassumption totally disconnected.

Based on previous results that study the completion of fuzzy metric spaces, we show that every intuitionistic fuzzy quasi-metric space, using the notion of fuzzy metric space in the sense of Kramosil and Michalek to obtain a generalization to the quasi-metric setting, has a bicompletion which is unique up to isometry.

The action of the isometry algebra Uh(sl(2)) on the h–deformed Lobachevsky plane is found. The invariant distance and the invariant 2–point functions are shown to agree precisely with the classical ones. The propagator of the Laplacian is calculated explicitely. It is invariant only after adding a “non–classical” sector to the Hilbert space.

We consider the problem of recovering fusion frame sparse signals from incomplete measurements. These signals are composed of a small number of nonzero blocks taken from a family of subspaces. First, we show that, by using a-priori knowledge of a coherence parameter associated with the angles between the subspaces, one can uniformly recover fusion frame sparse signals with a significantly reduc...

This paper first present a new general completely perturbed compressed sensing (CS) model y=(A+E)(x+u)+e,called noise folding based on general completely perturbed CS system, where y ∈ Rm, u ∈ Rm, u 6= 0, e ∈ Rm, A ∈ Rm×n, m ≪ n, E ∈ Rm×n with incorporating general nonzero perturbation E to sensing matrix A and noise u into signal x simultaneously based on the standard CS model y=Ax+e. Our cons...

In this note, we prove that matrices whose entries are all 0 or 1 cannot achieve good performance with respect to the Restricted Isometry Property (RIP). Most currently known deterministic constructions of matrices satisfying the RIP fall into this category, and hence these constructions suffer inherent limitations. In particular, we show that DeVore’s construction of matrices satisfying the RI...

For $p\geq 2$, the $p$-adic ring $C^*$-algebra $\mathcal {Q}_p$ is universal generated by a unitary $U$ and an isometry $S_p$ such that $S_pU=U^pS_p$ $\sum _{l=0}^{p-1}U^lS_pS_p^*U^{-l}=1$. any $k$ coprime to $p$ we define end