Two groups have a common model geometry if they act properly and cocompactly by isometries on the same proper geodesic metric space. We consider free products of uniform lattices in isometry rank-1 symmetric spaces prove, within each quasi-isometry class, that residually finite are abstractly commensurable. Our result gives first examples hyperbolic quasi-isometric but do not virtually geometry...

This paper is an investigation of the universal separable metric space up to isometry U discovered by Urysohn. A concrete construction of U as a metric subspace of the space C[0, 1] of functions from [0, 1] to the reals with the supremum metric is given. An answer is given to a question of Sierpiński on isometric embeddings of U in C[0, 1]. It is shown that the closed linear span of an isometri...

We propose a novel local isometry based dimensionality reduction method from the perspective of vector fields, which is called parallel vector field embedding (PFE). We first give a discussion on local isometry and global isometry to show the intrinsic connection between parallel vector fields and isometry. The problem of finding an isometry turns out to be equivalent to finding orthonormal par...

Inspired by significant real-life applications, in particular, sparse phase retrieval and sparse pulsation frequency detection in Asteroseismology, we investigate a general framework for compressed sensing, where the measurements are quasi-linear. We formulate natural generalizations of the well-known Restricted Isometry Property (RIP) towards nonlinear measurements, which allow us to prove bot...

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...

Abstract Let and $H$?> be locally compact groups with fixed two-sided invariant Haar measures. A polyhomomorphism $G\rightarrowtail H$?> is a closed subgroup $R\subset G\times measure, whose marginals on are dominated by the measures . can regarded as multi-valued map sending points to sets equipped ‘uniform’ For two polyhomomorphisms ${G\rightarrowtail H}$?> $H\rightarrowtail K$?...

Non-adaptive joint source network coding of correlated sources is discussed in this paper. By studying the information flow in the network, we propose quantized network coding as an alternative for packet forwarding. This technique has both network coding and distributed source coding advantages, simultaneously. Quantized network coding is a combination of random linear network coding in the (i...

We use the notion of uniform measure equivalence to prove that the Novikov-Shubin invariants resp. the capacities of amenable groups are invariant under quasi-isometry. Further, we comment on the connection to Gaboriau’s theorem on the invariance of L-Betti numbers under orbit equivalence.

The linear code equivalence problem is to decide whether two linear codes over Fq are identical up to a linear isometry of the Hamming space. The support splitting algorithm [23] runs in polynomial time for all but a negligible proportion of all linear codes, and solves the latter problem by recovering the isometry when it is just a permutation of the code support. While for a binary alphabet i...