The Schwarz problem for infinite sets of intervals

We solve the Schwarz problem for boundary contours consisting of countable sets of segments with limit point at infinity, including the periodic case. The solution is a result of a reduction to corresponding Riemann boundary-value problems.     

Rotation sets of surface homeomorphisms

We consider the rotation setR of a homeomorphismf, isotopic to the identity, of a closed surface of genusg2. We show if Int(R) is nonempty and contains an element which is realized by an asymptotic measure, then all the rational points of Int(R) are realized by periodic orbits. We raise an example to show that the second condition above is indispensable ifg2. We also show that ifR contains a (g+1)-simplex whose vertices are realizable by periodic orbits, then the topological entropy off is positive.     

Soliton parameters and chaotic sets

In this work we apply a nonperturbative approach to analyze soliton bifurcation in the presence of surface tension, which is a reformulation of standard methods based on the reversibility properties of the system. The hypothesis is nonrestrictive and the results can be extended to a much wider variety of systems. The usual idea of tracking intersections of unstable manifolds with some invariant set is again used, but reversibility plays an important role establishing in a geometrical point of view some kind of symmetry which, in a classical way, is unknown or nonexistent. Using a computer program we determine soliton solutions and also their bifurcations in the space of parameters giving a picture of the chaotic structural distribution to phase and amplitude shift phenomena.     

Congruences with intervals and arbitrary sets

Given a prime p, an integer $$H\in [1,p)$$, and an arbitrary set $${\mathcal {M}} \subseteq {\mathbb {F}} _p^*$$, where $${\mathbb {F}} _p$$ is the finite field with p elements, let $$J(H,{\mathcal {M}} )$$ denote the number of solutions to the congruence $$\begin{aligned} xm\equiv yn~\mathrm{mod}~ p \end{aligned}$$for which $$x,y\in [1,H]$$ and $$m,n\in {\mathcal {M}} $$. In this paper, we bound $$J(H,{\mathcal {M}} )$$ in terms of p, H, and the cardinality of $${\mathcal {M}} $$. In a wide range of parameters, this bound is optimal. We give two applications of this bound: to new estimates of trilinear character sums and to bilinear sums with Kloosterman sums, complementing some recent results of Kowalski et al. (Stratification and averaging for exponential sums: bilinear forms with generalized Kloosterman sums, 2018, arXiv:1802.09849).     

Rotation sets and almost periodic sequences

We study the rotational behaviour on minimal sets of torus homeomorphisms and show that the associated rotation sets can be any type of line segment as well as non-convex and even plane-separating continua. This shows that the restriction which hold for rotation sets on the whole torus are not valid on minimal sets. The proof uses a construction of rotational horseshoes by Kwapisz to transfer the problem to a symbolic level, where the desired rotational behaviour is implemented by means of suitable irregular Toeplitz sequences.     

Rotation sets and monotone periodic orbits for annulus homeomorphisms

Iff is a homeomorphism of the annulus andp/q is a rational in lowest terms that is contained in the rotation set off thenf has a (p, q)-topologically monotone periodic orbit. In addition, iff has ap/q-period orbit that is not topologically monotone then the Farey interval ofp/q is contained in the rotation set off.     

The Hausdorff dimension of chaotic sets for interval self-maps

We obtain two sufficient conditions for an interval self-map to have a chaotic set with positive Hausdorff dimension. Furthermore, we point out that for any interval Lipschitz maps with positive topological entropy there is a chaotic set with positive Hausdorff dimension.     

Monoidal intervals of clones on infinite sets

Let X be an infinite set of cardinality κ. We show that if L is an algebraic and dually algebraic distributive lattice with at most 2^κ completely join irreducibles, then there exists a monoidal interval in the clone lattice on X which is isomorphic to the lattice 1+L obtained by adding a new smallest element to L. In particular, we find that if L is any chain which is an algebraic lattice, and if L does not have more than 2^κ completely join irreducibles, then 1+L appears as a monoidal interval; also, if λ?2^κ, then the power set of λ with an additional smallest element is a monoidal interval. Concerning cardinalities of monoidal intervals these results imply that there are monoidal intervals of all cardinalities not greater than 2^κ, as well as monoidal intervals of cardinality 2^λ, for all λ?2^κ.     

Exact Hausdorff measure and intervals of maximum density for Cantor sets

Consider a linear Cantor set , which is the attractor of a linear iterated function system (i.f.s.) , , on the line satisfying the open set condition (where the open set is an interval). It is known that has Hausdorff dimension given by the equation , and that is finite and positive, where denotes Hausdorff measure of dimension . We give an algorithm for computing exactly as the maximum of a finite set of elementary functions of the parameters of the i.f.s. When (or more generally, if and are commensurable), the algorithm also gives an interval that maximizes the density . The Hausdorff measure is not a continuous function of the i.f.s. parameters. We also show that given the contraction parameters , it is possible to choose the translation parameters in such a way that , so the maximum density is one. Most of the results presented here were discovered through computer experiments, but we give traditional mathematical proofs.

     

Coloring of distance graphs with intervals as distance sets

Let D be a set of positive integers. The distance graph G(Z,D) with distance set D is the graph with vertex set Z in which two vertices x,y are adjacent if and only if |x−y|D. The fractional chromatic number, the chromatic number, and the circular chromatic number of G(Z,D) for various D have been extensively studied recently. In this paper, we investigate the fractional chromatic number, the chromatic number, and the circular chromatic number of the distance graphs with the distance sets of the form D_m,[k,k^′]={1,2,…,m}−{k,k+1,…,k^′}, where m, k, and k^′ are natural numbers with m≥k^′≥k. In particular, we completely determine the chromatic number of G(Z,D_m,[2,k^′]) for arbitrary m, and k^′.     

On intervals and sets of hypermatrices (tensors)

Interval hypermatrices (tensors) are introduced and interval \alpha-hypermatrices are uniformly characterized using a finite set of 'extreme' hypermatrices, where \alpha can be strong P, semi-positive, or positive definite, among many others. It is shown that a symmetric interval is an interval (strictly) copositive-hypermatrix if and only if it is an interval (E) E₀-hypermatrix. It is also shown that an even-order, symmetric interval is an interval positive (semi-) definite-hypermatrix if and only if it is an interval P (P₀)-hypermatrix. Interval hypermatrices are generalized to sets of hyper-matrices, several slice-properties of a set of hypermatrices are introduced and sets of hypermatrices with various slice-properties are uniformly characterized. As a consequence, several slice-properties of a compact, convex set of hyper-matrices are characterized by its extreme points.     

Limit cycles and chaotic invariant sets in autonomous hybrid planar systems

This paper demonstrates that complex dynamical behaviors such as limit cycles and chaotic invariant sets can exist in some simple autonomous hybrid planar systems governed by simple transition laws.     

Border collision bifurcations and chaotic sets in a two-dimensional piecewise linear map

A two-dimensional piecewise linear continuous model is analyzed. It reflects the dynamics occurring in a circuit proposed as chaos generator, in a simplified case. The parameter space is investigated in order to classify completely regions of existence of stable cycles, and regions associated with chaotic behaviors. The border collision bifurcation curves are analytically detected, as well as the degenerate flip bifurcations of k-cycles and the homoclinic bifurcations occurring in cyclic chaotic regions leading to chaos in one-piece.     

Early fault detection of rotating machinery through chaotic vibration feature extraction of experimental data sets

Fault detection of rotating machinery by the complex and non-stationary vibration signals with noise is very difficult, especially at the early stages. Also, many failure mechanisms and various adverse operating conditions in rotating machinery involve significant nonlinear dynamical properties. As a novel method, phase space reconstruction is used to study the effect of faults on the chaotic behavior, for the first time. Strange attractors in reconstructed phase space proof the existence of chaotic behavior. To quantify the chaotic vibration for fault diagnosis, a set of new features are extracted. These features include the largest Lyapunov exponent; approximate entropy and correlation dimension which acquire more fault characteristic information. The variations of these features for different healthy/faulty conditions are very good for fault diagnosis and identification. For the first time, a new chaotic feature space is introduced for fault detection, which is made from chaotic behavior features. In this space, different conditions of rotating machinery are separated very well. To obtain more generalized results, the features are introduced into a neural network to identify different faults in rotating machinery. The effectiveness of the new features based on chaotic vibrations is demonstrated by the experimental data sets. The proposed approach can reliably recognize different fault types and have more accurate results. Also, the performance of the new procedure is robust to the variation of load values and shows good generalization capability for various load values.     

Nonparametric prediction intervals for future record intervals based on order statistics

In this article, outer and inner prediction intervals for future record intervals as well as record spacings are derived based on observed order statistics from the same parent distribution. These intervals are exact and are distribution-free in that they do not depend on the sampling distribution. Three different cases are considered and in each case an exact explicit expression is obtained for the prediction coefficient. Finally, we compare the obtained results with similar intervals based on records, and also present a numerical example in order to illustrate the derived results.     

Confidence estimation for tolerance intervals

The post-data performances of normal tolerance intervals are studied. Under a robust Bayesian predictive scheme, we establish the ordering and bounds of the confidence estimators. It is found that the nominal confidence coefficient tends to be extreme yet coincides with the limiting Bayes estimators in some scenarios. A remark on the choice of beta priors is also given.     

Convexity and level sets for interval-valued fuzzy sets

Fuzzy Optimization and Decision Making - Convexity is a deeply studied concept since it is very useful in many fields of mathematics, like optimization. When we deal with imprecision, the convexity...