Continuous Functions Research Articles

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.

Exponential Sampling Type Kantorovich Max-Product Neural Network Operators

This article deals with the study of approximation behavior of the exponential sampling type Kantorovich max-product neural network operators based on sigmoidal activation function. We study point-wise and uniform convergence theorems in the space of continuous functions for these operators and determine the rate of convergence by means of logarithmic modulus of continuity. We also investigate the approximation of p -th ( 1 ≤ p < ∞ ) Lebesgue integrable functions by the aforementioned operators and study the rate of convergence exploiting Peetre’s K − functional.

Adaptive asymptotic tracking control of constrained nonlinear MIMO systems subject to unknown hysteresis input: A novel network-based strategy

This paper presents a referential adaptive asymptotic tracking control scheme for nonlinear multi-input and multi-output (MIMO) systems with time-varying full-state constraints and unknown hysteresis input. In order to achieve good asymptotic tracking effect, a zero-limit positive continuous function is introduced into the adaptive backstepping design process while thoroughly considering the impact of disturbance-like terms on the tracking effect. Additionally, a new variable is also imported to replace the reciprocal of unknown coefficient of the Bouc–Wen hysteresis model. Then, a new adaptive law about the new variable is added by combining the positive function, which can not only lessen the impact of the unknown hysteresis input on the tracking effect, but also avoid the “singularity” problem. Aiming at the time-varying full-state constraints, a appropriate time-varying log-type barrier Lyapunov function (TLBLF) is constructed to ensure that all the states of the system are restricted within the constraint scope. Finally, a significant adaptive asymptotic tracking scheme is designed based on multi-dimensional Taylor network (MTN) approximation technology. Apart from achieving high-precision asymptotic tracking performance, the proposed scheme ensures the boundedness of all signals of the closed-loop system without violating the full-state constraints. And, three simulations are given to verify the effectiveness of the scheme.

On the classification of function algebras on subvarieties of noncommutative operator balls

We study algebras of bounded noncommutative (nc) functions on unit balls of operator spaces (nc operator balls) and on their subvarieties. Considering the example of the nc unit polydisk we show that these algebras, while having a natural operator algebra structure, might not be the multiplier algebra of any reasonable nc reproducing kernel Hilbert space (RKHS). After examining additional subtleties of the nc RKHS approach, we turn to study the structure and representation theory of these algebras using function theoretic and operator algebraic tools. We show that the underlying nc variety is a complete invariant for the algebra of uniformly continuous nc functions on a homogeneous subvariety, in the sense that two such algebras are completely isometrically isomorphic if and only if the subvarieties are nc biholomorphic. We obtain extension and rigidity results for nc maps between subvarieties of nc operator balls corresponding to injective spaces that imply that a biholomorphism between homogeneous varieties extends to a biholomorphism between the ambient balls, which can be modified to a linear isomorphism. Thus, the algebra of uniformly continuous nc functions on nc operator balls, and even its restriction to certain subvarieties, completely determine the operator space up to completely isometric isomorphism.

Norm-closed and norm-reflecting ideals in rings of continuous functions

Norm-closed and norm-reflecting ideals in rings of continuous functions

Deep Learning Evidence for Global Optimality of Gerver’s Sofa

The moving sofa problem, introduced by Leo Moser in 1966, seeks to determine the maximal area of a 2D shape that can navigate an L-shaped corridor of unit width. Joseph Gerver’s 1992 solution, providing a lower bound of approximately 2.2195, is the best known, though its global optimality remains unproven. This paper leverages neural networks’ approximation power and recent advances in invexity optimization to explore global optimality. We propose two approaches supporting Gerver’s conjecture that his sofa is the unique global maximum. The first approach uses continuous function learning, discarding assumptions about the monotonicity, symmetry, and differentiability of sofa movements. The sofa area is computed as a differentiable function using our “waterfall” algorithm, with the loss function incorporating both differential terms and initial conditions based on physics-informed machine learning. Extensive training with diverse network initialization consistently converges to Gerver’s solution. The second approach applies discrete optimization to the Kallus–Romik upper bound, improving it from 2.37 to 2.3337 for five rotation angles. As the number of angles increases, our model asymptotically converges to Gerver’s area from above, indicating that no larger sofa exists.

The curvature entropy inequalities of convex bodies

Abstract There are many entropy inequalities in geometry, some of them can be seen as the Minkowski inequalities in the form of entropy, which play important roles in convex geometry. In this article, let P P and Q Q be convex bodies, we introduce the mixed curvature entropy M f ( P , Q ) {M}_{f}\left(P,Q) and the curvature entropy E f ( P , Q ) {E}_{f}\left(P,Q) with respect to a continuous function f f . Moreover, we establish the log \log -Minkowski inequalities of the mixed curvature entropy and the curvature entropy for two classes of three-dimensional convex bodies.

On a variant of extremal disconnectedness in locales and spaces

We study a variant of extremal disconnectedness in frames, defined by requiring that any two disjoint open sublocales should be separable by disjoint open sublocales each of which is the interior of some zero-sublocale. In spaces, this notion was recently introduced in [1] in order to study some properties of frames of z-ideals of rings of continuous functions. The point-free approach adopted here brings to the fore some topological results not considered in [1], such as that a dense z-embedded subspace of a Tychonoff space has this property if and only if the space containing it has the property. In [1], this was proved only for the embedding X ⊆ βX. We give a ring-theoretic characterization by defining a property of f-rings in a transparent manner which shows why the property should be defined the way we do. The property is that if two annihilator ideals meet trivially, then the annihilator of each of the annihilator ideals contains a positive element such that the sum of the two elements is a non-zerodivisor.

Noise-resilient designs and analysis for optical neural networks

Abstract All analog signal processing is fundamentally subject to noise, and this is also the case in next generation implementations of Optical Neural Networks (ONNs). Therefore, we propose the first hardware-based approach to mitigate noise in ONNs. A tree-like and an accordion-like design are constructed from a given Neural Network (NN) that one wishes to implement. Both designs have the capability that the resulting ONNs give outputs close to the desired solution. To establish the latter, we analyze the designs mathematically. Specifically, we investigate a probabilistic framework for the tree-like design that establishes the correctness of the design, i.e., for any feed-forward NN with Lipschitz continuous activation functions, an ONN can be constructed that produces output arbitrarily close to the original. ONNs constructed with the tree-like design thus also inherit the universal approximation property of NNs. For the accordion-like design, we restrict the analysis to NNs with linear activation functions and characterize the ONNs’ output distribution using exact formulas. Finally, we report on numerical experiments with LeNet ONNs that give insight into the number of components required in these designs for certain accuracy gains. The results indicate that adding just a few components and/or adding them only in the first (few) layers in the manner of either design can already be expected to increase the accuracy of ONNs considerably. To illustrate the effect we point to a specific simulation of a LeNet implementation, in which adding one copy of the layers components in each layer reduces the Mean Squared Error (MSE) by 59.1% for the tree-like design and by 51.5% for the accordion-like design. In this scenario, the gap in accuracy of prediction between the noiseless NN and the ONNs reduces even more: 93.3% for the tree-like design and 80% for the accordion-like design.