Narrow Results

Let $(X,d)$ be a compact metric space and $F=\{f_1,f_2,\dots ,f_m\}$ be an $m$-tuple of continuous maps from $X$ to itself. In this paper, we investigate the multiple mappings dynamical system $(X,F)$ with Hausdorff metric Li–Yorke chaos, distributional chaos and distributional chaos in a sequence properties from a set-valued view. On the basis of this research, we draw main conclusions as follows: (i) two topological conjugacy dynamical systems to multiple mappings have simultaneously Hausdorff metric Li–Yorke chaos or distributional chaos. (ii) Hausdorff metric Li–Yorke $\delta$-chaos is equivalent to Hausdorff metric distributional $\delta$-chaos in a sequence. (iii) By giving two examples, we show that there is non-mutual implication between that the multiple mappings $F=\{f_1,f_2,\dots ,f_m\}$ is Hausdorff metric Li–Yorke chaos and that each element $f_i$$(i=1,2,\dots ,m)$ in $F$ is Li–Yorke chaos. (iv) For the multiple mappings, weakly mixing implies the Hausdorff metric strongly Li–Yorke chaos and Hausdorff metric distributional chaos in a sequence.

By generating Fourier descriptors based upon the waveform induced by a pattern's geometric projection, a number of classic difficulties with the Fourier-descriptor methodology are mitigated. Not only are the descriptors invariant with respect to scale, translation, and rotation (as is usually the case), they are also continuous in the Hausdorff metric and robust with respect to both point noise and occlusion. An additional advantage is that they can be computed relative to a thresholded image without first finding an edge, thereby avoiding the difficulties typically present in thinning and orientation determination. The present paper discusses the method of projection-generated Fourier descriptors, as well as a study of the sensitivity to point noise. A companion paper will present the morphological properties and the effect of pattern occlusion.

Fourier descriptors based upon the waveform induced by a pattern's projection overcome a number of classic difficulties with Fourier-descriptor methodology. Not only are the descriptors invariant with respect to scale, translation, and rotation (as is usually the case), they are also continuous in the Hausdorff metric and robust with respect to both point noise and occlusion; insensitivity with respect to minimum occlusions is perhaps their most significant advantage. Continuity in the Hausdorff metric allows prediction of the effect on the descriptors when morphologically filtering a pattern. The effect of occlusion is also predictable.

Let $X$ be a local dendrite and let $f:X\to X$ be a monotone map. Denote by $P(f)$, RR$(f)$, UR$(f)$, $R(f)$ the set of periodic (resp., regularly recurrent, uniformly recurrent, recurrent) points and $\mathrm{\Lambda }(f)$ the union of all $\omega$-limit sets of $f$. We show that if $P(f)$ is nonempty, then (i) $\mathrm{\Lambda }(f)=R(f)=\mathrm{\text{UR}}(f)=\mathrm{\text{RR}}(f)=\overline{P(f)}$. (ii) R$(f)=X$ if and only if every cut point is a periodic point. If $P(f)$ is empty, then (iii) $\mathrm{\Lambda }(f)=R(f)=\mathrm{\text{UR}}(f)$. (iv) R$(f)=X$ if and only if $X$ is a circle and $f$ is topologically conjugate to an irrational rotation of the unit circle ${𝕊}^1$. On the other hand, we prove that $f$ has no Li–Yorke pair. Moreover, we show that the family of all $\omega$-limit sets of $f$ is closed with respect to the Hausdorff metric.

We prove that if a recurrent double sequence coincide in a sufficiently large starting square with a double sequence produced by context-free substitutions, then they must coincide everywhere. We apply this result for some examples.

Self-similarity is a common tendency in nature and physics. It is wide spread in geo-physical phenomena like diffusion and iteration. Physically, an object is self-similar if it is invariant under a set of scaling transformation. This paper gives a brief outline of the analytical and set theoretical properties of different types of weak self-similar sets. It is proved that weak sub self-similar sets are closed under finite union. Weak sub self-similar property of the topological boundary of a weak self-similar set is also discussed. The denseness of non-weak super self-similar sets in the set of all non-empty compact subsets of a separable complete metric space is established. It is proved that the power of weak self-similar sets are weak super self-similar in the product metric and weak self-similarity is preserved under isometry. A characterization of weak super self-similar sets using weak sub contractions is also presented. Exact weak sub and super self-similar sets are introduced in this paper and some necessary and sufficient conditions in terms of weak condensation IFS are presented. A condition for a set to be both exact weak super and sub self-similar is obtained and the denseness of exact weak super self similar sets in the set of all weak self-similar sets is discussed.

We consider Pascal’s Triangle $rmodp$ to be the entries of Pascal’s Triangle that are congruent to $rmodp$. Such a representation of Pascal’s Triangle exhibits fractal-like structures. When the Triangle is mapped to a subset of the unit square, we show that such a set is nonempty and exists as a limit of a sequence of coarse approximations. We then show that for any given prime $p$, any such sequence converges to the same set, regardless of the residue(s) considered. As an obvious consequence, this allows us to conclude that the fractal (box-counting) dimension of this nonempty, compact representation of Pascal’s Triangle $rmodp$ is independent of $r$.

Strassen’s theorem circa 1965 gives necessary and sufficient conditions on the existence of a probability measure on two product spaces with given support and two marginals. In the case where each product space is finite, Strassen’s theorem is reduced to a linear programming problem which can be solved using flow theory. A density matrix of bipartite quantum system is a quantum analog of a probability matrix on two finite product spaces. Partial traces of the density matrix are analogs of marginals. The support of the density matrix is its range. The analog of Strassen’s theorem in this case can be stated and solved using semidefinite programming. The aim of this paper is to give analogs of Strassen’s theorem to density trace class operators on a product of two separable Hilbert spaces, where at least one of the Hilbert spaces is infinite-dimensional.

The mapping properties of the time evolution operator E(t) of nonlinear hyperbolic scalar conservation laws is investigated. It is shown that this operator is Lipschitz in the Hausdorff metric in one space dimension whenever the flux is convex and one of the initial conditions satisfies a one-sided Lipschitz condition. The Hausdorff distance between the graphs of the solutions measures the closeness in L^∞ in the regions where the solutions are smooth, as well as the closeness between the locations of shocks. A similar result on Hausdorff stability is proved with respect to a perturbation of the flux function. These results complement the well known L¹ contractivity of the solution operator. They are used in a subsequent paper to prove new smoothness results for solutions to such conservation laws. Negative results are proved in the case of non-convex and genuinely multidimensional fluxes.

The smoothness of the solutions of 1D scalar conservation laws is investigated and it is shown that if the initial value has smoothness of order α in L^q with α > 1 and q = 1/α, this smoothness is preserved at any time t > 0 for the graph of the solution viewed as a function in a suitably rotated coordinate system. The precise notion of smoothness is expressed in terms of a scale of Besov spaces which also characterizes the functions that are approximated at rate N^-α in the uniform norm by piecewise polynomials on N adaptive intervals. An important implication of this result is that a properly designed adaptive strategy should approximate the solution at the same rate N^-α in the Hausdorff distance between the graphs.

I.i.d. random sequence is the simplest but very basic one in stochastic processes, and statistically self-similar set is the simplest but very basic one in random recursive sets in the theory of random fractal. Is there any relation between i.i.d. random sequence and statistically self-similar set? This paper gives a basic theorem which tells us that the random recursive set generated by a collection of i.i.d. statistical contraction operators is always a statistically self-similar set.

For a continuous domain D, some characterization that the convex powerdomain CD is a domain hull of Max(CD) is given in terms of compact subsets of D. And in this case, it is proved that the set of the maximal points Max(CD) of CD with the relative Scott topology is homeomorphic to the set of all Scott compact subsets of Max(CD) with the topology induced by the Hausdorff metric derived from a metric on Max(CD) when Max(CD) is matrizable.

We describe a generalisation of the multiple-instance learning model in which a bag's label is not based on a single instance's proximity to a single target point. Rather, a bag is positive if and only if it contains a collection of instances, each near one of a set of target points. We then adapt a learning-theoretic algorithm for learning in this model and present empirical results on data from robot vision, content-based image retrieval, and protein sequence identification.

A Wasserstein space is a metric space of sufficiently concentrated probability measures over a general metric space. The main goal of this paper is to estimate the largeness of Wasserstein spaces, in a sense to be made precise.

In the first part, we generalize the Hausdorff dimension by defining a family of bi-Lipschitz invariants, called critical parameters, that measure largeness for infinite-dimensional metric spaces. Basic properties of these invariants are given, and they are estimated for a natural set of spaces generalizing the usual Hilbert cube. These invariants are very similar to concepts initiated by Rogers, but our variant is specifically suited to tackle Lipschitz comparison.

In the second part, we estimate the value of these new invariants in the case of some Wasserstein spaces, as well as the dynamical complexity of push-forward maps. The lower bounds rely on several embedding results; for example we provide uniform bi-Lipschitz embeddings of all powers of any space inside its Wasserstein space and we prove that the Wasserstein space of a d-manifold has "power-exponential" critical parameter equal to d. These arguments are very easily adapted to study the space of closed subsets of a compact metric space, partly generalizing results of Boardman, Goodey and McClure.

One of the most common way to generate a fractal is by using an iterated function system (IFS). In this paper, we introduce an (n, m)-IFS, which is a collection of n IFSs and discuss the attractor of this system. Also we prove the continuity theorem and collage theorem for (n, m)-IFS.

If $f_n:X\to X$, $n=1,2,\dots ,$ is a sequence of mappings on a metric space $(X,d)$, a point $x$ in $X$ is called an inverse fixed point of the sequence $(f_n)$, if $inf\{d(x,y):y\in f_n^{-1}(x)\}$ tends to zero as $n$ tends to infinity. This definition is proposed and studied in this paper to obtain a few fixed point results.

Since hyperspaces of complete (separated) uniform spaces are not complete in general, it is highly remarkable that in the more general context of semiuniform convergence spaces even a hyperspace completion exists which preserves several invariants, e.g. precompactness (and thus compactness), connectedness (and uniform connectedness), the property of being a filter space (or a semiuniform space), etc. This completion is used to characterize the subspaces of the compact spaces in the realm of semiuniform convergence spaces axiomatically. The complete hyperspace structure is coarser than the usual uniform hyperspace structure in case uniform spaces are considered.