Continuous Functions Research Articles

$\delta(\tau_{1},\tau_{2})$-Continuous Functions

This paper introduces a new class of functions called $\delta(\tau_{1},\tau_{2})$-continuous functions. Several characterizations of $\delta(\tau_{1},\tau_{2})$-continuous functions are investigated. The relationships between $\delta(\tau_{1},\tau_{2})$-continuity and the other types of $\delta(\tau_{1},\tau_{2})$-continuity are also discussed.

Pythagorean Fuzzy Soft Somewhat Continuous Functions

In this work, we introduce the concept of Pythagorean fuzzy soft somewhat open sets utilizing the Pythagorean fuzzy soft interior operator, extending its application to Pythagorean fuzzy soft topological spaces. This study aims to enhance decision-making processes in future-assisted economies by addressing the limitations of existing fuzzy set theories. We investigate the distinctive properties of Pythagorean fuzzy soft somewhat open sets as a subclass of Pythagorean fuzzy soft somewhere dense sets. Additionally, we explore Pythagorean fuzzy soft somewhat metamorphism’s within the context of Pythagorean fuzzy soft somewhat continuous functions, offering new insights into their topological invariant. Through detailed analysis and examples, we demonstrate the applicability of these concepts in various scientific and engineering problems. This work provides a comprehensive framework for understanding and utilizing Pythagorean fuzzy soft sets in complex decision-making scenarios. Finally, we compare various relationships across some generalizations of Pythagorean fuzzy soft continuous functions.

Generalized Linear Differential Equation using Hyers - Ulam Stability Approach

In this paper, We demonstrate the Hyers - Ulam stability of linear differential equation of fourth order. We interact with the differential equation\begin{align*}\gamma^{iv} (\omega) + \rho_1 \gamma{'''} (\omega)+ \rho_2 \gamma{''} (\omega) + \rho_3 \gamma' (\omega) + \rho_4 \gamma(\omega) = \chi(\omega),\end{align*}where $\gamma \in c^4 [\alpha,\beta], \chi \in [\alpha,\beta]$. Hyers-Ulam stability concerns the robustness of solutions of functional equations under small perturbations, ensuring that a solution approximately satisfying the equation is close to an exact solution. We extend this concept to fourth-order linear differential equations and continuous functions. Using fixed-point methods and various norms, we establish conditions under which such equations exhibit Hyers-Ulam stability. Several illustrative examples are provided to demonstrate the application of these results in specific cases, contributing to the growing understanding of stability in higher-order differential equations. Our findings have implications in both theoretical research and practical applications in physics and engineering.

Upper and Lower Rarely m-I-Continuous Multifunctions

In 1979, Popa [24] first introduced rarely continuous functions. In this paper, we introduce upper and lower rarely m-continuous multifunctions. Moreover, we extend this concept to a multifunction F : (X, τ, I) → (Y, σ), where (X, τ, I) is an ideal topological space.

Distributed Cooperative Control of Heterogeneous Multiagent Systems for Air‐to‐Ground Formation

ABSTRACTThis paper is concerned with a distributed cooperative formation problem for a heterogeneous multiagent system with a car‐like mobile robot, a quadrotor, and an omnidirectional wheeled mobile robot. In the heterogeneous multiagent system, a three‐layer distributed cooperative formation control strategy is constructed to achieve air‐to‐ground formation, the car‐like mobile robot tracks a virtual leader, the quadrotor, and the omnidirectional wheeled mobile robot track the car‐like mobile robot. Two nonlinear leader‐follower controllers based on exponential virtual control laws are designed to maintain formation for the quadrotor and the omnidirectional wheeled mobile robot. A kinematics controller with continuous functions, an improved adaptive sliding mode controller and a suitable back‐stepping controller are introduced in the heterogeneous multiagent system, respectively. Multiagent communication is one of the key steps to accomplish multiagent time synchronization. Experimental results are given to illustrate the applicability of the proposed distributed cooperative control strategy.

Exchange Formulae for the Stieltjes–Poisson Transform over Weighted Lebesgue Spaces

This paper aims to develop exchange formulae for the Stieltjes–Poisson transform by using Mellin-type convolutions in the context of weighted Lebesgue spaces. A key result is the introduction of bilinear and continuous Mellin-type convolutions, expanding the scope of the analysis to include the space of weighted L1 functions and the space of continuous functions vanishing at infinity.

Representation and inequalities involving continuous linear functionals and fractional derivatives

We investigate how continuous linear functionals can be represented in terms of generic operators and certain kernels (Peano kernels), and we study lower bounds for the operators as a consequence, in the space of square-integrable functions. We apply and develop the theory for the Riemann–Liouville fractional derivative (an inverse of the Riemann–Liouville integral), where inequalities are derived with the Gaussian hypergeometric function. This work is inspired by the recent contributions by Fernandez and Buranay (J Comput Appl Math 441:115705, 2024) and Jornet (Arch Math, 2024).

On Autonomous Superposition Operators in Spaces of μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}–Almost Periodic Functions and Applications to Linear Differential Equations

In this article we focus mainly on the class of almost periodic functions in view of the Lebesgue measure (briefly: μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}-a.p. functions) and on some if its subclasses. We are going to deal with autonomous superposition operators acting in the space of μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}-a.p. functions. We will indicate necessary and sufficient conditions under which the autonomous superposition operator maps the space under consideration into itself as well as conditions under which it is continuous. As a corollary from these results we indicate when the autonomous superposition operator defined on that space is a bijection. Next, we will analyse in detail the situation when the composition of μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}-a.p. function with a continuous function or with a homeomorphism gives a Stepanov almost periodic function. As an application of our results we indicate a subclass of μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}-a.p. functions for which linear differential equations with a non-homogeneous term belonging to this subclass may not have μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}-a.p. solutions.

Two remarks on properties of functions of bounded variation

In terms of variations, a sufficient condition for the uniform convergence of sequences of continuous functions is proved. Using this result, we obtain an addition to the classical Helly theorem on the selection of convergent sequences of functions with uniformly bounded variations in the case when the limit function is continuous. Also, by using an example we show that the condition of continuous differentiability of a function, ensuring the differentiability of its variation with the variable upper limit, is in a certain sense sharp.

Printing Cell Embedded Sacrificial Strategy for Microvasculature using Degradable DNA Biolubricant

Microvasculature is essential for the continued function of cells in tissue and is fundamental in the fields of tissue engineering, organ repair and drug screening. However, the fabrication of microvasculature is still challenging using existing strategies. Here, we developed a general PRINting Cell Embedded Sacrificial Strategy (PRINCESS) and successfully fabricated microvasculatures using degradable DNA biolubricant. This is the first demonstration of direct cell printing to fabricate microvasculature, which eliminates the need for a subsequent cell seeding process and the associated deficiencies. Utilizing the shear‐thinning property of DNA hydrogels as a novel sacrificial, cell‐laden biolubricant, we can print a 70‐μm endothelialized microvasculature, breaking the limit of 100 μm. To our best knowledge, this is the smallest endothelialized microvasculature that has ever been bioprinted so far. In addition, the self‐healing property of DNA hydrogels allows the creation of continuous branched structures. This strategy provides a new platform for constructing complex hierarchical vascular networks and offers new opportunity towards engineering thick tissues. The extremely low volume of sacrificial biolubricant paves the way for DNA hydrogels to be used in practical tissue engineering applications. The high‐resolution bioprinting technique also exhibits great potential for printing lymphatics, retinas and neural networks in the future.

Printing Cell Embedded Sacrificial Strategy for Microvasculature using Degradable DNA Biolubricant.

Microvasculature is essential for the continued function of cells in tissue and is fundamental in the fields of tissue engineering, organ repair and drug screening. However, the fabrication of microvasculature is still challenging using existing strategies. Here, we developed a general PRINting Cell Embedded Sacrificial Strategy (PRINCESS) and successfully fabricated microvasculatures using degradable DNA biolubricant. This is the first demonstration of direct cell printing to fabricate microvasculature, which eliminates the need for a subsequent cell seeding process and the associated deficiencies. Utilizing the shear-thinning property of DNA hydrogels as a novel sacrificial, cell-laden biolubricant, we can print a 70-μm endothelialized microvasculature, breaking the limit of 100 μm. To our best knowledge, this is the smallest endothelialized microvasculature that has ever been bioprinted so far. In addition, the self-healing property of DNA hydrogels allows the creation of continuous branched structures. This strategy provides a new platform for constructing complex hierarchical vascular networks and offers new opportunity towards engineering thick tissues. The extremely low volume of sacrificial biolubricant paves the way for DNA hydrogels to be used in practical tissue engineering applications. The high-resolution bioprinting technique also exhibits great potential for printing lymphatics, retinas and neural networks in the future.

Analysis of the three‐dimensional elasticity theory problem for a radially inhomogeneous cylinder

AbstractIn this paper an axisymmetric problem of elasticity theory for a radially inhomogeneous cylinder of small thickness is studied. It is assumed that homogeneous mixed boundary conditions are specified on lateral surfaces of the cylinder, and boundary conditions that leave the cylinder in equilibrium are specified on the ends of the cylinder. It is considered that elastic moduli are arbitrary continuous functions of a variable along the radius of the cylinder. The formulated boundary value problem is reduced to a spectral problem containing a small parameter characterizing the thinness of the cylinder walls. To determine its solution, the method of asymptotic integration is used, based on three iteration processes. All solutions of the equilibrium equation are constructed, satisfying the specified homogeneous boundary conditions on lateral surfaces. It is obtained that the solution of the problem consists of a penetrating solution and a solution of the nature of the boundary layer. An analysis of stress‐strain states corresponding to different types of solutions is carried out. Based on the constructed asymptotic solutions, a connection is found between the solutions obtained by the equation of elasticity theory and by the applied theory of shells. It is shown that the penetrating solution determines the internal stress‐strain state of a radially inhomogeneous cylinder. The stress state determined by the penetrating solution is equivalent to the principal vector of stress acting in a cross section perpendicular to the axis of the cylinder.The first terms of asymptotic expansions of the penetrating solution in a small parameter coincide with solutions in the applied theory of shells. New classes of solutions are defined as the character of a boundary layer, which are localized at the ends of the cylinder and decrease exponentially with distance from the ends. These solutions are absent in the applied theories of shells. The first terms of the asymptotic expansion of the solution, having the nature of a boundary layer, are equivalent to the Saint‐Venant edge effect in the theory of heterogeneous plates. Asymptotic formulas for displacement and stresses are constructed, allowing one to calculate the three‐dimensional stress‐strain state of a radially heterogeneous cylinder of small thickness with any predetermined accuracy. Based on the obtained asymptotic expansions, one can estimate the applicability areas of applied theories and construct a refined applied theory for radially heterogeneous cylindrical shells.

Decomposition of pseudo-uninorms with continuous underlying functions via ordinal sum

The decomposition of all pseudo-uninorms with continuous underlying functions, defined on the unit interval, via Clifford's ordinal sum is described. It is shown that each such pseudo-uninorm can be decomposed into representable and trivial semigroups, and special semigroups defined on two points, where the corresponding semigroup operation is the projection to one of the coordinates. Linear orders, for which the ordinal sum of such semigroups yields a pseudo-uninorm, are also characterized.

Revolutionizing Textile Manufacturing: Sustainable and Profitable Production by Integrating Industry 4.0, Activity-Based Costing, and the Theory of Constraints

The textile industry, a cornerstone of daily life and a highly globalized sector, faces significant environmental challenges due to its high water and energy consumption and extensive chemical usage. This study proposes a comprehensive green production planning and control model integrating Industry 4.0 concepts, activity-based costing (ABC), and the theory of constraints (TOC). The model utilizes mathematical programming to optimize product mix, maximize profitability, and minimize environmental impact. It leverages real-time sensing technologies and ERP systems to facilitate waste recovery, reduce carbon emissions, and achieve energy savings. Various carbon emission cost models, including continuous and discontinuous tax functions, are explored to balance corporate profitability with environmental sustainability. The findings demonstrate the model’s potential in optimizing resource utilization, reducing the environmental footprint, and enhancing profitability.

Continuous inverse ambiguous functions on Lie groups

Continuous inverse ambiguous functions on Lie groups

Approximation by Schurer Type λ-Bernstein–Bézier Basis Function Enhanced by Shifted Knots Properties

In this article, a novel Schurer form of λ-Bernstein operators augmented by Bézier basis functions is presented by utilizing the features of shifted knots. The shifted knots form of Bernstein operators and the Schurer form of the Bézier basis function are used in this article, then, new operators, the Schurer type λ-Bernstein shifted knots operators are constructed in terms of the Bézier basis function. First, the test functions are calculated and the central moments for these operators are obtained. Then, Korovkin’s type approximation properties are studied by the use of a modulus of continuity of orders one and two. Finally, the convergence theorems for these new operators are obtained by using Peetre’s K-functional and Lipschitz continuous functions. In the end, some direct approximation theorems are also obtained.

Comparing functional countability and exponential separability

A space is functionally countable if every real-valued continuous function has countable image. A stronger property recently defined by Tkachuk is exponential separability. We start by studying these properties in GO spaces, where we extend results by Tkachuk and Wilson, and prove a conjecture by Dow. We also study some subspaces of products that are functionally countable and the influence of the Gδ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G_\\delta $$\\end{document}-topology on exponential separability. Finally, we give some examples of functionally countable spaces that are separable and uncountable.

A step function based recursion method for 0/1 deep neural networks

The deep neural network with step function activation (0/1 DNNs) is a fundamental composite model in deep learning which has high efficiency and robustness to outliers. However, due to the discontinuity and lacking subgradient information of the 0/1 DNNs model, prior researches are largely focused on designing continuous functions to approximate the step activation and developing continuous optimization methods. In this paper, by introducing two sets of network node variables into the 0/1 DNNs and by exploring the composite structure of the resulted model, the 0/1 DNNs is decomposed into a unary optimization model associated with the step function and three derivational optimization subproblems associated with the other variables. For the unary optimization model and two derivational optimization subproblems, we present a closed form solution, and for the third derivational optimization subproblem, we propose an efficient proximal method. Based on this, a globally convergent step function based recursion method for the 0/1 DNNs is developed. The efficiency and performance of the proposed algorithm are validated via theoretical analysis as well as some illustrative numerical examples on classifying MNIST, FashionMNIST and Cifar10 datasets.

Topological Interactions Between Homotopy and Dehn Twist Varieties

The topological Dehn twists have several applications in mathematical sciences as well as in physical sciences. The interplay between homotopy theory and Dehn twists exposes a rich set of properties. This paper generalizes the Dehn twists by proposing the notion of pre-twisted space, orientations of twists and the formation of pointed based space under a homeomorphic continuous function. It is shown that the Dehn twisted homotopy under non-retraction admits a left lifting property (LLP) through the local homeomorphism. The LLP extends the principles of Hurewicz fibration by avoiding pullback. Moreover, this paper illustrates that the Dehn twisted homotopy up to a base point in a based space can be formed by considering retraction. As a result, two disjoint continuous functions become point-wise continuous at the base point under retracted homotopy twists. Interestingly, the oriented Dehn twists of a pre-twisted space under homotopy retraction mutually commute in a contractible space.

On bivariate fractal interpolation for countable data and associated nonlinear fractal operator

Abstract Fractal interpolation has been conventionally treated as a method to construct a univariate continuous function interpolating a given finite data set with the distinguishing property that the graph of the interpolating function is the attractor of a suitable iterated function system. On the one hand, attempts have been made to extend the univariate fractal interpolation from a finite data set to a countably infinite set. On the other hand, fractal interpolation in higher dimensions, particularly the theory of fractal interpolation surfaces (FISs), has received increasing attention for more than a quarter century. This article targets a two-fold extension of the notion of fractal interpolation by providing a general framework to construct FISs for a prescribed set consisting of countably infinite data on a rectangular grid. By using this as a crucial tool, we obtain a parameterized family of bivariate fractal functions simultaneously interpolating and approximating a prescribed bivariate continuous function. Some elementary properties of the associated nonlinear (not necessarily linear) fractal operators are established, thereby allowing the interaction of the notion of fractal interpolation with the theory of nonlinear operators.