Continuous Functions Research Articles

Accurate and fast quaternion fractional-order Franklin moments for color image analysis

Quaternion type moments (QTM) based on hypergeometric functions and harmonic functions have attracted interest in the imaging field. The Franklin system is a complete orthonormal system consisting of piecewise linear continuous functions with Haar wavelet collocation points. However, due to the lack of efficient computational methods and explicit expressions, research on piecewise polynomials with Haar wavelet collocation points remains underexplored. Furthermore, the impact of magnitude and phase of quaternion on the color image features extracted by QTM has not been comprehensively explored. To address these problems, a fast and accurate computation method for Franklin functions is proposed. Quaternion fractional-order Franklin moments (QFFM) are proposed and using the Minkowski distance as a prior parameter for selecting unit pure quaternions. Experiments are conducted to demonstrate the superior performance of QFFM in image reconstruction and color image classification tasks compared with QTM-based and deep learning-based methods. Moreover, with the help of experiments, it is confirmed the effectiveness of using the Minkowski distance to select magnitude and phase of quaternion.

Closing duality gaps of SDPs completely through perturbation when singularity degree is one

Let ( P , D ) be a primal-dual pair of SDPs with a nonzero finite duality gap. Under such circumstances, P and D are weakly feasible and if we perturb the problem data to recover strong feasibility, the (common) optimal value function v as a function of the perturbation is not well-defined at zero (i.e. at the unperturbed data). Nevertheless, under the assumption that both P and D have singularity degree one, we show that a limiting version v a of v is a well-defined monotone decreasing continuous bijective function with domain [ 0 , π / 2 ] and image [ v ( P ) , v ( D ) ] , where v ( P ) and v ( D ) are the optimal values of P and D , respectively. The domain [ 0 , π / 2 ] corresponds to directions of perturbation defined in a certain manner. Thus, v a ‘completely fills’ the nonzero duality gap under a mild regularity condition. Our result is tight in that there exists an instance with singularity degree two for which v a is not continuous.

N-cadherin facilitates trigeminal sensory neuron outgrowth and target tissue innervation.

During embryonic development, diverse cell types coordinate to form functionally complex tissues. Exemplifying this process, the trigeminal ganglion emerges from the condensation of two distinct precursor cell populations, cranial placodes and neural crest, with neuronal differentiation of the former preceding the latter. While its dual cellular origin has been understood for decades, the molecules orchestrating trigeminal ganglion formation remain relatively obscure. Initial assembly of the trigeminal ganglion is mediated by cell adhesion molecules, including neural cadherin (N-cadherin), which is first expressed by placodal neurons and is required for their proper condensation with other neurons and neural crest cells. Axon outgrowth first occurs from placodal neurons, but as gangliogenesis proceeds, neural crest cells also differentiate into N-cadherin-expressing neurons, and together both extend axons toward target tissues. However, a role for N-cadherin in regulating axon outgrowth and innervation of target tissues by trigeminal neurons has not been explored. To this end, we depleted N-cadherin from chick trigeminal placode cells and observed decreases in trigeminal ganglion size, nerve growth, and target innervation in vivo, phenotypes that could only partially be attributed to increased apoptosis early in gangliogenesis. Accordingly, neurite number and branching of neural crest-derived neurons was decreased in vitro in response to N-cadherin knockdown in placode cells, providing a novel non-cell autonomous explanation for these morphological changes. Inhibiting N-cadherin-mediated adhesion with a function-blocking antibody prevented axon extension in most, but not all, placode-derived trigeminal neurons in vitro, indicating potential unique requirements for N-cadherin in various neuronal subtypes. Collectively, these findings reveal persistent cell autonomous and non-cell autonomous functions for N-cadherin, thus highlighting the critical role of N-cadherin in mediating reciprocal interactions between neural crest and placode neuronal derivatives during trigeminal ganglion development.

Electrical conductivity fluctuations as a tracer to determine time-dependent transport characteristics in hyporheic sediments

Assessing solute transport in riverbed sediments is important for quantifying the effective reactivity of hyporheic sediments and the magnitude of exchange flows between rivers and their river beds. A typical approach of estimating transport in riverbed sediments is by measuring natural tracers such as fluctuations of temperature or electrical conductivity (EC) and fitting models to them that assume time-independent travel time distributions, implying steady-state flow. Here, we use a transport parameterization that is based on the advection–dispersion equation (ADE) with coefficients that continuously vary in time. The ADE is solved numerically and its solution is fitted to measured EC time series using Bayesian parameter inference. A continuous function of model parameters is constructed by smoothly interpolating between point values with different temporal resolution, and Tikhonov regularization is used to avoid spurious parameter fluctuations. The approach is tested using EC time series synchronously measured in river water and hyporheic porewater of two urban rivers in Germany and one urban river in South Australia. For all datasets the goodness of fit was improved by introducing a time-dependent EC offset. Estimated porewater velocities were highly transient in three out of the four datasets with values increasing by up to a factor of six over the course of a day. Flux transients were likely related to both variations of hydraulic gradients along and spatial shifting of flow paths. Comparison to stationary, non-parametric deconvolution indicated that transient flow may induce multimodality in stationary travel time distributions. Given the high temporal dynamics of transport in the streambed sediments of the three investigated urban rivers, we envision that the presented model is also a valuable tool to improve the assessment of reactive transport in riverbed sediments.

On the modulus of continuity of fractional Orlicz-Sobolev functions

AbstractNecessary and sufficient conditions are presented for a fractional Orlicz-Sobolev space on $${\mathbb {R}}^n$$ R n to be continuously embedded into a space of uniformly continuous functions. The optimal modulus of continuity is exhibited whenever these conditions are fulfilled. These results pertain to the supercritical Sobolev regime and complement earlier sharp embeddings into rearrangement-invariant spaces concerning the subcritical setting. Classical embeddings for fractional Sobolev spaces into Hölder spaces are recovered as special instances. Proofs require novel strategies, since customary methods fail to produce optimal conclusions.

Efficiency-Centered Fault Diagnosis of In-Service Induction Motors for Digital Twin Applications: A Case Study on Broken Rotor Bars

The uninterrupted operation of induction motors is crucial for industries, ensuring reliability and continuous functionality. To achieve this, we propose an innovative approach that utilizes an efficiency model-based digital shadow system for in situ failure detection and diagnosis (FDD) in induction motors (IMs). The shadow model accurately estimates IM losses and efficiency across various operational conditions. Our proposed method utilizes efficiency as the primary indicator for fault detection, while losses serve as condition indicators for fault diagnosis based on real-time motor parameters and loss sources. We introduce a bond graph as a fault diagnosis network, linking loss sources, motor parameters, and faults. This interconnected approach is the key aspect of our proposed diagnostic method and aims to be used in fault diagnosis as a general method. A case study of a broken rotor bar is used to validate the proposed method using a dataset of five motors. Among these, one motor operates without failure, while the remaining four exhibit broken rotor faults categorized as 1, 2, 3, and 4. The proposed method achieves 99.99% precision in identifying one to four defective rotor bars in IMs. Comparative analysis demonstrates good performance compared to vibration-based FDD approaches. Moreover, our methodology is computationally efficient and aligned with Industry 4.0 requirements.

Q-CSKDF: A Continuous and Security Key Derivation Function for Quantum Key Distribution

Q-CSKDF: A Continuous and Security Key Derivation Function for Quantum Key Distribution

Application of a specialist nurse–led wireless smart monitoring system in elderly patients with persistent atrial fibrillation

Background: With the increase in the elderly population, the prevalence of persistent atrial fibrillation (AF) among older adults has significantly risen, becoming one of the crucial cardiovascular diseases affecting the health and quality of life of the elderly. Patients with persistent AF require continuous cardiac function monitoring to prevent possible complications such as stroke and heart failure. Traditional monitoring methods often necessitate frequent hospital visits, increasing the economic burden and psychological stress on patients. With advancements in medical technology, wireless smart monitoring systems have emerged as a new remote health care technology, offering a more convenient and effective monitoring option for patients with persistent AF. Specialist nurses play a pivotal role in monitoring and managing patients using these systems. Their professional knowledge and skills are vital for improving monitoring efficiency and patient compliance. Therefore, this study aims to explore the application of wireless smart monitoring systems led by specialist nurses in elderly patients with persistent AF, in hopes of providing a scientific basis for improving the monitoring and management of this patient group. Methods: A total of 88 elderly patients with persistent atrial fibrillation admitted to the Cardiology Ward of a tertiary hospital in Weifang City from October 2021 to April 2023 were selected as research subjects. They were divided into an intervention group and a control group with conventional care, with 44 cases in each group, according to the order of admission. The control group received conventional hospitalization guidance, while the intervention group utilized a wireless smart monitoring system led by specialist nurses. The effects before and after the intervention in both groups were compared. Results: The intervention group patients had better scores in terms of mental stress, sleep quality, and comfort compared with the control group, and the nursing shift change time during hospitalization for the intervention group was shorter than that of the control group (P<0.05). Conclusions: The wireless smart monitoring system led by specialist nurses can improve the mental stress and sleep conditions of elderly patients with persistent atrial fibrillation, increase comfort, and enhance the efficiency of nursing shift changes.

Different types of continuous functions and their applications

Different types of continuous functions and their applications

A Converse Lyapunov-Type Theorem for Control Systems with Regulated Cost

Given a nonlinear control system, a target set, a nonnegative integral cost, and a continuous function W, we say that the system is globally asymptotically controllable to the target withW-regulated cost, whenever, starting from any point z, among the strategies that achieve classical asymptotic controllability we can select one that also keeps the cost less than W(z). In this paper, assuming mild regularity hypotheses on the data, we prove that a necessary and sufficient condition for global asymptotic controllability with regulated cost is the existence of a special, continuous Control Lyapunov Function, called a Minimum Restraint Function. The main novelty is the necessity implication, obtained here for the first time. Nevertheless, the sufficiency condition extends previous results based on semiconcavity of the Minimum Restraint Function, while we require mere continuity.

Robust support function machines for set-valued data classification

Support function machines (SFMs) have been proposed to handle set-valued data, but they are sensitive to outliers and unstable for re-sampling due to the use of the hinge loss function. To address these problems, we propose a robust SFM model with proximity functions. We first define a family of proximity functions that are used to convert set-valued data into continuous functions in a Banach space, and then we use the margin maximization in a Banach space to construct the pinball SFMs (PinSFMs). We study some properties of the proposed model, and it is interesting to observe that the optimal measure of the proposed model has a specific representation under the total variation norm. Using the representation of the optimal measure, we convert an infinite-dimensional optimization problem into a finite-dimensional optimization problem. Unlike SFMs, we employ a sampling strategy to tackle the finite-dimensional optimization problem. We theoretically show that the sparse solution determines the sparsity of the sampling points though the sampling strategy causes uncertainty for the sampling points. In addition, we achieve kernel versions of proximity functions, and the attractive property of this kernelization is that the proposed model is convex even if indefinite kernels are employed. Experiments on a series of data sets are performed to demonstrate that the proposed model is superior to some existing models in the presence of outliers.

Rethinking Propellant-Optimal Powered Descent Guidance

The propellant-optimal powered descent solution requires a bang-bang thrust magnitude profile that switches instantaneously between the upper and lower bounds of the engine thrust. Implementing a bang-bang thrust pattern for guidance can be challenging for the engine and may cause practical and operational difficulties. Motivated by the observation that excellent propellant performance does not appear to be predicated on necessarily having a bang-bang thrust structure, a new powered descent guidance method is proposed in this work. The foundation lies in a propellant-optimal problem in which the thrust magnitude is parameterized by a user-prescribed continuous function of time. In particular, constant and linear functions are considered. The solution determines the optimal thrust direction, time of flight, and design parameters defining the thrust function. An indirect-method-based guidance algorithm is developed to solve this problem rapidly and reliably. Substantial evidence is provided to demonstrate that the solutions proposed in this paper in general yield a propellant consumption very close to that of the optimal bang-bang solution. Given its comparable propellant performance, implementability, and robustness, this paper makes a strong case that the proposed propellant-optimal powered descent guidance approach is preferred to the long-standing bang-bang guidance approach.

Convergence and high order of approximation by Steklov sampling operators

In this paper we introduce a new class of sampling-type operators, named Steklov sampling operators. The idea is to consider a sampling series based on a kernel function that is a discrete approximate identity, and which constitutes a reconstruction process of a given signal f, based on a family of sample values which are Steklov integrals of order r evaluated at the nodes k/w, k∈Z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k \\in {\\mathbb {Z}}$$\\end{document}, w>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$w>0$$\\end{document}. The convergence properties of the introduced sampling operators in continuous functions spaces and in the Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}-setting have been studied. Moreover, the main properties of the Steklov-type functions have been exploited in order to establish results concerning the high order of approximation. Such results have been obtained in a quantitative version thanks to the use of the well-known modulus of smoothness of the approximated functions, and assuming suitable Strang-Fix type conditions, which are very typical assumptions in applications involving Fourier and Harmonic analysis. Concerning the quantitative estimates, we proposed two different approaches; the first one holds in the case of Steklov sampling operators defined with kernels with compact support, its proof is substantially based on the application of the generalized Minkowski inequality, and it is valid with respect to the p-norm, with 1≤p≤+∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1 \\le p \\le +\\infty $$\\end{document}. In the second case, the restriction on the support of the kernel is removed and the corresponding estimates are valid only for 1<p≤+∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1 < p\\le +\\infty $$\\end{document}. Here, the key point of the proof is the application of the well-known Hardy–Littlewood maximal inequality. Finally, a deep comparison between the proposed Steklov sampling series and the already existing sampling-type operators has been given, in order to show the effectiveness of the proposed constructive method of approximation. Examples of kernel functions satisfying the required assumptions have been provided.

Salvaging intraluminal peritoneal dialysis catheter obstruction from blood clot with a targeted thrombolytic agent: an innovation.

Peritoneal dialysis (PD) catheter malfunction commonly leads to the removal of the catheter and eventually to a transfer to hemodialysis. The most common cause is intraluminal obstruction caused by blood and fibrin clots. Recommended interventions include irrigation of the catheter with heparinized saline; if this method fails, thrombolytic agents may be used. Mechanical methods such as intraluminal brushing are also utilized, typically after medical treatment fails. Here, we present a case of a patient who developed an intraluminal blood clot that persisted despite attempts with intraluminal thrombolytic drugs and intraluminal brushing. To salvage the catheter, targeted thrombolysis was performed using an endoscopic retrograde cholangiopancreatography (ERCP) guidewire to reinforce the coiled PD catheter and puncture the clots. Additionally, a Swing Tip cannula was employed for direct injection of the thrombolytic agent. These interventions successfully preserved the catheter, resolving the clot and ensuring continued functionality.

Sutjeska Memorial Park (1958–75): Yugoslav World War II Memorial Complexes and Embodied Strategies for Solidarity

ABSTRACT This article investigates the expansion of World War II memorial sites in socialist Yugoslavia, with a focus on the Sutjeska Memorial Park in Bosnia-Herzegovina. Beginning in the early 1960s, a new typology of memorial complexes emerged, merging domestic tourism and commemoration. These sites operated as forms of social infrastructure that provided ground not solely for the remembrance of the historical events of the War but for enabling solidaristic sentiments required to prevent its resurgence. The Sutjeska Memorial Park exemplifies how artists and architects experimented with new forms to produce embodied affective strategies to engage new generations without personal war memories. Expanded memorial parks attempted to evoke somatic responses to fascist horror while using new tourist amenities to stimulate a sense of significance across diverse generational and cultural communities. Drawing on archival materials and the formal elements of Sutjeska’s monuments, this research examines understudied sites that produced immersive environments intended to open the realm of visual experience to other embodied possibilities and to instil new cultural practices of comradeship and commemoration. This paper illustrates how the design of the Sutjeska Memorial Park as social infrastructure facilitates its continued function today despite Yugoslavia’s dissolution, maintaining much of its original antifascist character.

Remarks on SHD spaces and more divergence properties

The class of SHD spaces was recently introduced in [12]. The first part of this paper focuses on answering most of the questions presented in that article. For instance, we exhibit an example of a non-SHD Tychonoff space X such that F[X], the Pixley-Roy hyperspace of X, βX, the Stone-Čech compactification of X, and Cp(X), the ring of continuous functions over X equipped with the topology of pointwise convergence, are SHD.In the second part of this work we will present some variations of the SHD notion, namely, the WSHD property and the OHD property. Furthermore, we will pay special attention to the relationships between X and F[X] regarding these new concepts.

The asymptotic solution of the elasticity theory problem for a radially inhomogeneous cylinder

The elasticity theory problem for a radially inhomogeneous cylinder of small thickness, whose elastic moduli are arbitrary continuous functions depending on the radius of the cylinder, is considered. It is assumed that the side surface of the cylinder is stress-free, and the boundary conditions that keep the cylinder in equilibrium are given at its seats. Asymptotic solutions are constructed by the asymptotic integration method. It is shown that the asymptotic solution consists of the sum of the penetrating solution, simple boundary effect and boundary layer solutions. The character of the stress-strain state corresponding to the penetrating solution, simple boundary effect and boundary layer solutions is determined. Asymptotic formulas are obtained for displacements and stresses, which allow to calculate the stress-strain state of a cylinder.The problem of torsion of a radially inhomogeneous cylinder is studied, with its lateral surface free from stress and the boundary conditions keeping it in equilibrium at its seats. By applying the asymptotic integration method, it is determined that the asymptotic solution for the torsion problem consists of the sum of the penetrating solution and boundary layer solutions.Numerical analysis is performed and the effect of material inhomogeneity on the stress-strain state of the cylinder is evaluated.

Ulam–Hyers–Rassias Mittag-Leffler stability of ϖ–fractional partial differential equations

This paper offers a comprehensive analysis of solution representations for ϖ-fractional partial differential equations, specifically focusing on the linear case of the Darboux problem. We exhibit a representation of the solutions for the Darboux problem of ϖ-fractional partial differential equations in the linear case in the space of continuous functions. Through the application of the generalized Gronwall inequality, we establish the Ulam–Hyers–Rassias Mittag–Leffler stability in the space of continuous functions. Three numerical examples are presented to show the effectiveness and the applicability of our results.

Asymptotics of block Toeplitz determinants with piecewise continuous symbols

AbstractWe determine the asymptotics of the block Toeplitz determinants as for matrix‐valued piecewise continuous functions with a finitely many jumps under mild additional conditions. In particular, we prove that where , , and are constants that depend on the matrix symbol and are described in our main results. Our approach is based on a new localization theorem for Toeplitz determinants, a new method of computing the Fredholm index of Toeplitz operators with piecewise continuous matrix‐valued symbols, and other operator theoretic methods. As an application of our results, we consider piecewise continuous symbols that arise in the study of entanglement entropy in quantum spin chain models.

Level-2 IFS thermodynamic formalism: Gibbs probabilities in the space of probabilities and the push-forward map

We will denote by M the space of Borel probabilities on the symbolic space Ω = { 1 , 2 ⋯ , m } N . M is equipped Monge–Kantorovich metric. We consider here the push-forward map T : M → M as a dynamical system. The space of Borel probabilities on M is denoted by M . Given a continuous function A : M → R , an a priori probability Π 0 on M , and a certain convolution operation acting on pairs of probabilities on M , we define an associated Level-2 IFS Ruelle operator. We show the existence of an eigenfunction and an eigenprobability Π ^ ∈ M for such an operator. Under a normalization condition for A, we show the existence of some T -invariant probabilities Π ^ ∈ M . We are able to define the variational entropy of such Π ^ and a related maximization pressure problem associated to A. In some particular examples, we show how to get eigenprobabilities solutions on M for the Level-2 Thermodynamic Formalism problem from eigenprobabilities on M for the classical (Level-1) Thermodynamic Formalism; this shows that our approach is a natural generalization of the classic case.