We consider four-dimensional spacetimes (M,g) which obey the Einstein equations G = T, and admit a global spacelike G1 = R isometry group. By means of dimensional reduction and local analyis on the reduced (2 + 1) spacetime, we obtain a sufficient condition on T which guarantees that (M,g) cannot contain apparent horizons. Given any (3 + 1) spacetime with spacelike translational isometry, the n...

In this chapter, we are interested in the underlying structure of aesthetically pleasing plane figures. By a plane figure, we mean any subset of the plane. To understand the underlying structure, we examine the symmetries of the figure. An isometry of the plane is a distance-preserving transformation of the plane. This means that for any pair of points P and Q, the distance between the images u...

Amatrix is said to possess the Restricted Isometry Property (RIP) if it acts as an approximate isometry when restricted to sparse vectors. Previous work has shown it to be np-hard to determine whether a matrix possess this property, but only in a narrow range of parameters. In this work, we show that it is np-hard to make this determination for any accuracy parameter, even when we restrict ours...

Let F = Fg,n be a surface of genus g with n punctures. We assume 3g − 3 + n > 1 and that (g, n) 6= (1, 2). The purpose of this paper is to prove, for the Weil-Petersson metric on Teichmuller space Tg,n, the analogue of Royden’s famous result [15] that every complex analytic isometry of Tg,0 with respect to the Teichmuller metric is induced by an element of the mapping class group. His proof inv...

When C ⊆ F is a linear code over a finite field F, every linear Hamming isometry of C to itself is the restriction of a linear Hamming isometry of F to itself, i.e., a monomial transformation. This is no longer the case for additive codes over non-prime fields. Every monomial transformation mapping C to itself is an additive Hamming isometry, but there exist additive Hamming isometries that are...

In this paper, we first give a description of a surjective unit-preserving real-linear uniform isometry $ T : A longrightarrow B$,  where $ A $ and $ B $ are complex function spaces on compact Hausdorff spaces $ X $ and $ Y $, respectively, whenever ${rm ER}left (A, Xright ) = {rm Ch}left (A, Xright )$ and ${rm ER}left (B, Yright ) = {rm Ch}left (B, Yright )$. Next, we give a description of $ T...

Recall that the quasi-isometry group QI(X) of a metric space X is the set of equivalence classes of quasi-isometries f : X → X, where two quasiisometries f1, f2 are equivalent iff supx d(f1(x), f2(x)) <∞ (here we consider G as a metric space with a word metric). One approach to this question, which has been the most successful one, is to find an ‘optimal’ space X quasi-isometric to G and show t...

The early work of Mostow, Margulis and Prasad on rigidity of arithmetic lattices has evolved into a broad use of quasi-isometry techniques in group theory and low dimensional topology. The word metric on a finitely generated group makes it into a metric space which is uniquely determined up to the geometric relation called quasi-isometry, despite the fact that the metric depends on the choice o...

We prove the equivalence of the two important facts about finite metric spaces and universal Urysohn metric spaces U, namely theorem A and theorem B below: Theorem A (Approximation): The group of isometry ISO(U) contains everywhere dense locally finite subgroup; Theorem G(Globalization): For each finite metric space F there exists another finite metric space F̄ and isometric imbedding j of F to ...