In this paper, we use a remarkable recent result of Bramson and Kalikow, [BK], to produce an example of a C expanding map of the circle which preserves more than one absolutely continuous probability measure. We first introduce some abbreviations and definitions. An expanding map of the circle is a differentiable map f from the circle to itself such that |Dxf | ≥ C for some fixed constants C > ...

We give digital analogues of classical theorems of topology for continuous functions defined on spheres, for digital simple closed curves. In particular, we show the following: 1) A digital simple closed curve S of more than 4 points is not contractible, i.e., its identity map is not nullhomotopic in S ; 2) Let X and Y be digital simple closed curves, each symmetric with respect to the origin, ...

We consider piecewise monotone interval mappings which are topologically mixing and satisfy the Markov property. It has previously been shown that the invariant densities of the natural approximations converge exponentially fast in uniform pointwise topology to the invariant density of the given map provided its derivative is piecewise Lipshitz continuous. We provide an example of a map which i...

We study all possible weak limits of a minimizing sequence, for p-energy functionals, consisting of continuous maps between Riemannian manifolds subject to a Dirichlet boundary condition or a homotopy condition. We show that if p is not an integer, then any such weak limit is a strong limit and, in particular, a stationary p-harmonic map which is C 1;; continuous away from a closed subset of th...

A method of efficiently implementing a continuous MAP decoding algorithm is presented. A sliding window technique is used to reduce the block size at the expense of extra computations. The smaller block size results in a large reduction in memory storage requirements, reduced delay, and the ability to implement a continuous MAP decoder. A technique for reducing the synchronisation time for a tu...

In this paper, we study a nonconvex continuous relaxation of MAP inference in discrete Markov random fields (MRFs). We show that for arbitrary MRFs, this relaxation is tight, and a discrete stationary point of it can be easily reached by a simple block coordinate descent algorithm. In addition, we study the resolution of this relaxation using popular gradient methods, and further propose a more...

The concept of continuous-time trajectory representation has brought increased accuracy and efficiency to multi-modal sensor fusion in modern SLAM. However, regardless of these advantages, its offline property caused by the requirement of global batch optimization is critically hindering its relevance for real-time and life-long applications. In this paper, we present a dense map-centric SLAM m...

For a smooth harmonic map flow u : M× [0, T ) → N with blow-up as t ↑ T , it has been asked ([6], [5], [7]) whether the weak limit u(T ) : M→ N is continuous. Recently, in [12], we showed that in general it need not be. Meanwhile, the energy function E(u(·)) : [0, T ) → R, being weakly positive, smooth and weakly decreasing, has a continuous extension to [0, T ]. Here we show that if this exten...

Computing the MAP assignment in graphical models is generally intractable. As a result, for discrete graphical models, the MAP problem is often approximated using linear programming relaxations. Much research has focused on characterizing when these LP relaxations are tight, and while they are relatively well-understood in the discrete case, only a few results are known for their continuous ana...

In the present work, we develop a continuous model based on spline functions, in order to determine the T2 Map in Magnetic Resonance Imaging. The T2 parameter corresponds to the time decay parameter of spin relaxation, which cannot be obtained directly from one image, since it has to be calculated by fitting an exponential decaying functions from series of images. The calculation of the T2 map ...