Set Of Discontinuity Points Research Articles

On the general form of a linear continuous functional in the space of regulated functions

The author's research continues on the theory of regulated functions (functions having finite one-sided limits at each point) and $\sigma$-continuous functions (bounded functions having no more than a countable set of discontinuity points), as well as on the theory of the *-integral. The representability of a regulated function in the form of a sum of a right-continuous function and a left-continuous function is proved ($rl$-representability of the proper function). It is shown that the general form of a linear continuous functional in the space of regulated functions ($\sigma$-continuous functions) is the *-integral of a regulated ($\sigma$-continuous) function over a function of bounded variation.

On sets of discontinuities of functions continuous on all lines

Answering a question asked by K. C. Ciesielski and T. Glatzer in 2013, we construct a $C^1$-smooth function $f$ on $[0,1]$ and a closed set $M \subset {\rm graph} f$ nowhere dense in ${\rm graph} f$ such that there does not exist any linearly continuous function on ${\mathbb R}^2$ (i.e., function continuous on all lines) which is discontinuous at each point of $M$. We substantially use a recent full characterization of sets of discontinuity points of linearly continuous functions on ${\mathbb R}^n$ proved by T. Banakh and O. Maslyuchenko in 2020. As an easy consequence of our result, we prove that the necessary condition for such sets of discontinuities proved by S. G. Slobodnik in 1976 is not sufficient. We also prove an analogue of this Slobodnik's result in separable Banach spaces.

On Density and Sigma-Porosity in Some Families of Darboux Functions

A cliquish function $f: \mathbb R\to\mathbb R$ is internally cliquish if its set of discontinuity points is nowhere dense. A Baire $1$ function $f: \mathbb R\to\mathbb R$ is internally Baire $1$ if its set of discontinuity points is nowhere dense. In this paper we proved that the set of Darboux internally cliquish (Baire $1$) functions is dense and $\sigma$-strongly porous in the family of Darboux cliquish (Baire $1$) functions.

Dense and σ-Porous Subsets in Some Families of Darboux Functions

G. Ivanova and E. Wagner-Bojakowska shown that the set of Darboux quasi-continuous functions with nowhere dense set of discontinuity points is dense in the metric space of Darboux quasi-continuous functions with the supremum metric. We prove that this set also is σ-strongly porous in such space. We obtain the symmetrical result for the family of strong Świątkowski functions, i.e., that the family of strong Świątkowski functions with nowhere dense set of discontinuity points is dense (thus, “large”) and σ-strongly porous (thus, asymmetrically, “small”) in the family of strong Świątkowski functions.

Dense and σ-porous subsets in some families of cliquish functions

A function f: ℝ → ℝ is internally cliquish (internally quasi-continuous) if it is cliquish (quasi-continuous) and the set of discontinuity points of f is nowhere dense. We prove that the set of internally cliquish (internally quasi-continuous) functions is dense and σ-strongly porous set in the family of cliquish (quasi-continuous) functions. The analogous result is obtained also for the family of functions having the Świątkowski property.

Formation Control and Distributed Goal Assignment for Multi-Agent Non-Holonomic Systems

This paper presents control algorithms for multiple non-holonomic mobile robots moving in formation. Trajectory tracking based on linear feedback control is combined with inter-agent collision avoidance. Artificial potential functions (APF) are used to generate a repulsive component of the control. Stability analysis is based on a Lyapunov-like function. Then the presented method is extended to include a goal exchange algorithm that makes the convergence of the formation much more rapid and, in addition, reduces the number of collision avoidance interactions. The extended method is theoretically justified using a Lyapunov-like function. The controller is discontinuous but the set of discontinuity points is of zero measure. The novelty of the proposed method lies in integration of the closed-loop control for non-holonomic mobile robots with the distributed goal assignment, which is usually regarded in the literature as part of trajectory planning problem. A Lyapunov-like function joins both trajectory tracking and goal assignment analyses. It is shown that distributed goal exchange supports stability of the closed-loop control system. Moreover, robots are equipped with a reactive collision avoidance mechanism, which often does not exist in the known algorithms. The effectiveness of the presented method is illustrated by numerical simulations carried out on the large formation of robots.

A critical anisotropic problem with discontinuous nonlinearities

In this paper we prove existence of a nonnegative solution for nonlinear anisotropic elliptic equations of the type −[∑i=1N∂∂xi(|∂u∂xi|pi−2∂u∂xi)]=f(x,u)+|u|p∗−2uinΩ,u=0on∂Ω,where Ω is a smooth bounded domain in RN with homogeneous Dirichlet boundary conditions. Here f can have an uncountable set of discontinuity points.

Characterization of Uninorms With Continuous Underlying T-norm and T-conorm by Their Set of Discontinuity Points

Uninorms with continuous underlying t-norm and t-conorm are discussed and properties of the set of discontinuity points of such a uninorm are shown. This set is proved to be a subset of the graph of a special symmetric, u-surjective, nonincreasing set-valued function, which gives us a necessary condition for a uninorm to have continuous underlying functions. A sufficient condition for a uninorm to have continuous underlying operations is also given. Several examples are included.

Nonparametric Bayesian Analysis for Masked Data From Hybrid Systems in Accelerated Lifetime Tests

Under constant stress accelerated test, we analyze the masked data from hybrid systems including series–parallel system and parallel–series system. The Bayesian posterior distribution and estimates of components’ subsurvival functions are obtained by assuming a prior of the multivariate Dirichlet process. By establishing a relationship between subsurvival and survival functions of components, the estimates of reliabilities for components and system are derived from the estimates of subsurvival functions. It is not confined to the common restriction that the sets of discontinuity points of the survival functions have to be disjointed. For a complex system, we represent it to the proposed series–parallel system or parallel–series system. Thus, the nonparametric Bayesian approach is also applicable for the complex system with s-independent components. A simulated example is presented to demonstrate the efficiency of the method. Finally, the method is applied to a real data of coal wine monitoring power.

Characterizing set-valued functions of uninorms with continuous underlying t-norm and t-conorm

The uninorms with continuous underlying functions were characterized by their set of discontinuity points in the previous work of the author, using the characterizing set-valued function. In this paper properties of this characterizing set-valued function are studied. It is shown that the type of the monotonicity of such a set-valued function is always changed in idempotent points of the corresponding uninorm. Several additional properties of the characterizing set-valued function of a uninorm with continuous underlying functions are shown and an example is included. Results shown in this paper are used for a complete characterization of uninorms with continuous underlying functions via the ordinal sum construction, which is shown in the consecutive paper.

Characterization of uninorms with continuous underlying t-norm and t-conorm by means of the ordinal sum construction

The uninorms with continuous underlying t-norm and t-conorm are characterized via the ordinal sum construction of Clifford. Using the previous results of the author, where each uninorm with continuous underlying operations was characterized by properties of its set of discontinuity points, it is shown that each such a uninorm can be decomposed into an ordinal sum of a countable number of semigroups related to representable uninorms, continuous Archimedean t-norms, continuous Archimedean t-conorms and internal uninorms (including the min and the max operator).

Operator Lipschitz Functions and Model Spaces

Let H ∞ denote the space of bounded analytic functions on the upper half-plane ℂ+. We prove that each function in the model space H ∞ ∩ Θ $$ \overline{H^{\infty }} $$ is an operator Lipschitz function on ℝ if and only if the inner function Θ is a usual Lipschitz function, i.e., Θ′ ∈ H ∞. Let (OL)′(ℝ) denote the set of all functions f ∈ L ∞ whose antiderivative is operator Lipschitz on the real line ℝ. We prove that H ∞ ∩ Θ $$ \overline{H^{\infty }} $$ ⊂ (OL)′(ℝ) if Θ is a Blaschke product with zeros satisfying the uniform Frostman condition. We also deal with the following questions. When does an inner function Θ belong to (OL)′(ℝ)? When does each divisor of an inner function Θ belong to (OL)′(ℝ)? As an application, we deduce that (OL)′(ℝ) is not a subalgebra of L ∞(ℝ). Another application is related to a description of the sets of discontinuity points for the derivatives of operator Lipschitz functions. We prove that a set ℰ,ℰ ⊂ ℝ, is a set of discontinuity points for the derivative of an operator Lipschitz function if and only if ℰ is an F σ set of first category. A considerable proportion of the results of the paper is based on a sufficient condition for operator Lipschitzness which was obtained by Arazy, Barton, and Friedman. We also give a sufficient condition for operator Lipschitzness which is sharper than the Arazy–Barton–Friedman condition. Bibliography: 27 titles.

Darboux functions with nowhere dense graphs

We study the set of discontinuity points of real-valued Darboux functions with nowhere dense graphs. It is shown to be meager. By generic constructions we show that it can be any meager countable union of perfect sets and any meager countable union of pairwise disjoint closed sets.

A Note on the Set of Discontinuity Points of Multifunctions of two Variables

Abstract In this paper, we consider some problem connected with separate properties of a multifunction of two variables which ensure that the set of its joint discontinuity points is a meager set.

Homogeneous Hilbert boundary-value problem with countable set of discontinuities and logarithmic singularity of index

We consider the Hilbert boundary-value problem for the upper half-plane with a countable set of discontinuity points of coefficient of the boundary condition and with a two-side curling at infinity of a logarithmic order. We obtain formulas for the general solution to the problem.

Nonparametric Bayesian Estimation of Reliabilities in a Class of Coherent Systems

Usually, methods evaluating system reliability require engineers to quantify the reliability of each of the system components. For series and parallel systems, there are limited options to handle the estimation of each component's reliability. This study examines the reliability estimation of complex problems of two classes of coherent systems: series-parallel, and parallel-series. In both of the cases, the component reliabilities may be unknown. We developed estimators for reliability functions at all levels of the system (component and system reliabilities). The main assumption required is that, for all the distributions of the components of a particular system, the sets of discontinuity points have to be disjoint. Nonparametric Bayesian estimators of all sub-distribution and distribution functions are derived, and a Dirichlet multivariate process as a prior distribution is considered for the nonparametric Bayesian estimation of all distributions. For illustration, two simulated numerical examples are presented. The estimators are s-consistent, and one may observe from the examples that they have good performance. Our estimator can accommodate continuous failure distributions, as well as distributions with mass points.

On non-regular g-measures

We prove that g-functions whose set of discontinuity points has strictly negative topological pressure and which satisfy an assumption that is weaker than non-nullness, have at least one stationary g-measure. We also obtain uniqueness by adding conditions on the set of continuity points.

Systems of Sticky Particles Governed by Burgers' Equation

We show the existence of two sticky particles models with the same velocity function ut(x) which is the entropy solution of the inviscid Burgers' equation. One of them is governed by the set of discontinuity points of u0. Thus, the trajectories t↦Xt coincide; however one has different mass distributions ∂xut=du0∘Xt-1 and λ∘Xt-1. Here, λ denotes the Lebesgue measure.

Fracture Surfaces and the Regularity of Inverses for BV Deformations

Motivated by nonlinear elasticity theory, we study deformations that are approximately differentiable, orientation-preserving and one-to-one almost everywhere, and in addition have finite surface energy. This surface energy \({\mathcal{E}}\) was used by the authors in a previous paper, and has connections with the theory of currents. In the present paper we prove that \({\mathcal{E}}\) measures exactly the area of the surface created by the deformation. This is done through a proper definition of created surface, which is related to the set of discontinuity points of the inverse of the deformation. In doing so, we also obtain an SBV regularity result for the inverse.

Reliability Nonparametric Bayesian Estimation in Parallel Systems

Relevant results for (sub-)distribution functions related to parallel systems are discussed. The reverse hazard rate is defined using the product integral. Consequently, the restriction of absolute continuity for the involved distributions can be relaxed. The only restriction is that the sets of discontinuity points of the parallel distributions have to be disjointed. Nonparametric Bayesian estimators of all survival (sub-)distribution functions are derived. Dual to the series systems that use minimum life times as observations, the parallel systems record the maximum life times. Dirichlet multivariate processes forming a class of prior distributions are considered for the nonparametric Bayesian estimation of the component distribution functions, and the system reliability. For illustration, two striking numerical examples are presented.