Motion planning problems can be simplified by admissible projections of the configuration space to sequences lower-dimensional quotient-spaces, called sequential simplifications. To exploit simplifications, we present Quotient-space Rapidly-exploring Random Trees (QRRT) algorithm. QRRT takes as input a start and goal configuration, sequence quotient-spaces. The algorithm grows trees on quotient...

For Gorenstein quotient spaces C d =G, a direct generalization of the classical McKay correspondence in dimensions d 4 would primarily demand the existence of projective, crepant desingularizations. Since this turned out to be not always possible, Reid asked about special classes of such quotient spaces which would satisfy the above property. We prove that the underlying spaces of all Gorenstei...

We study stable maps / : M —*U of closed orientable 3-manifolds M into the plane using their quotient spaces, which are defined to be the spaces of the connected components of/-fibres and which are known to be 2-dimensional polyhedra. We show that every 3-manifold admits a stable map whose quotient space is homeomorphic to that of a stable map of 5 \ We also deduce a Morse-type inequality for s...

In the theory of Type Two Effectivity (Weihrauch 2000), computation on non-discrete spaces is performed by type two Turing machines, which compute with infinite words acting as names for elements of spaces. The connection between names and their associated elements is specified by a many-to-one relation called a representation. Such representations induce a topology on the named elements, namel...

We define a relative version of tiling cohomology for the purpose of comparing the topology of tiling dynamical systems when one is a factor of the other. We illustrate this with examples, and outline a method for computing the cohomology of tiling spaces of finite type.

In algebraic geometry one often encounters the problem of taking the quotient of a scheme by a group. Despite the frequent occurrence of such problems, there are few general results about the existence of such quotients. These questions come up again and again in the theory of moduli spaces. When we want to classify some type of algebraic objects, say varieties or vector bundles, the classifica...

Usual statistics are defined, studied and implemented on Euclidean spaces. But what about statistics on other mathematical spaces, like manifolds with additional properties: Lie groups, Quotient spaces, Stratified spaces etc? How can we describe the interaction between statistics and geometry? The structure of Quotient space in particular is widely used to model data, for example every time one...

For Gorenstein quotient spaces C/G, a direct generalization of the classical McKay correspondence in dimensions d ≥ 4 would primarily demand the existence of projective, crepant desingularizations. Since this turned out to be not always possible, Reid asked about special classes of such quotient spaces which would satisfy the above property. We prove that the underlying spaces of all Gorenstein...

New concepts related to approximating a Lipschitz function between Banach spaces by a ne functions are introduced Results which clarify when such approximations are possible are proved and in some cases a complete characterization of the spaces X Y for which any Lipschitz function from X to Y can be so approx imated is obtained This is applied to the study of Lipschitz and uniform quotient mapp...

New concepts related to approximating a Lipschitz function between Banach spaces by affine functions are introduced. Results which clarify when such approximations are possible are proved and in some cases a complete characterization of the spaces X, Y for which any Lipschitz function from X to Y can be so approximated is obtained. This is applied to the study of Lipschitz and uniform quotient ...