In this manuscript, we present the generation of Mandelbrot sets, Julia set fractals, and Biomorphs using a two-step viscosity approximation method applied to the complex function , where and . The two-step viscosity approximation method is employed to establish new escape criteria for Julia sets, Mandelbrot sets, and Biomorphs. Subsequently, the viscosity approximation process is extended by incorporating m-convexity, and the corresponding escape criteria are generalized for these fractals. Further, we introduce the viscosity approximation process with s-convexity and develop the associated escape criteria for the same fractal structures. Additionally, we provide a comparative visualization of Mandelbrot sets, Julia set fractals, and Biomorphs using the standard viscosity approximation process, as well as its m-convex and s-convex variants, applied to the same complex polynomial function. High-resolution fractal images are produced using MATLAB R2024a with 50 iterations and a figure resolution of 800, highlighting the differences in the resulting structures for identical parameter values.

A simple and efficient iterative translation approximation method for simulating stationary non-Gaussian stochastic vector processes

Qualitative models of Alzheimer’s pathology often posit that amyloid accumulation follows a sigmoid curve, indicating that the rate of deposition wanes over time. Longitudinal PET data now allow us to investigate amyloid accumulation trajectories with greater detail and over longer follow-up periods. We combine inferences from simulated amyloid trajectories, empirical PET data from the Alzheimer’s Disease Neuroimaging Initiative (ADNI), and the sampled iterative local approximation algorithm (SILA) to assess whether amyloid accumulation reaches a physiologic ceiling. We find that SILA reliably detects a ceiling, when present, across a range of simulated scenarios that impose a sigmoid shape. When fit to empirical data from ADNI, however, SILA does not appear to indicate the presence of a ceiling. Thus, we conclude that amyloid trajectories may not reach a physiologic ceiling during the stages of Alzheimer’s disease typically observed while patients remain under follow-up in cohort studies. Fits using SILA indicate that illustrative models of biomarker cascades, while useful tools for conceptualizing and interrogating pathologic processes, may not represent the shapes of amyloid trajectories accurately.

In this paper, we study iterative approximation of attractive points for finitely many families of generalized nonexpansive mappings in a uniformly convex Banach space. We introduce a new algorithm combining a viscosity step with an inertial extrapolation. Under suitable control of the inertial and viscosity parameters and standard conditions on the mappings, we prove that the generated sequence is bounded. We then show that every weak cluster point of the sequence is an attractive point common to all mapping families. The main result establishes that the sequence converges weakly to a unique such attractive point. This extends earlier results confined to two mappings by considering finite family. These findings confirm that the proposed viscosity-inertial iteration successfully approximates the common attractive point under the stated hypotheses. Overall, the work broadens convergence theory in Banach spaces by enabling new classes of algorithms for approximating solution points of generalized nonlinear problems.

385 participants from the Alzheimer's Disease Neuroimaging Initiative (ADNI; mean (SD) age = 73.4 (7.3) years) with longitudinal flortaucipir tau PET and 288 participants from the Wisconsin Registry for Alzheimer's Prevention and Wisconsin Alzheimer's Disease Research Center (collectively referred to as WISC; mean (SD) age = 67.4 (6.7) years) with longitudinal MK-6240 tau PET were included in the study. Standard uptake value ratios (SUVRs) in the entorhinal cortex and a meta-temporal ROI were modeled with SILA separately, for each cohort and region. Forward and backward SUVR and T+/- prediction were characterized with ten-fold cross-validation and in-sample validation techniques. Accuracy of estimated T+ onset ages (ETOA) was characterized in T- to T+ converters. Differences in ETOA were tested between APOE-e4 carriers and non-carriers, as well as differences in time T+ between levels of cognitive impairment. SILA was able to accurately estimate retrospective change in tau SUVR in the meta-temporal region regardless of age, sex, APOE-e4 carriage, tau SUVR, and dementia (p >0.05) whereas dementia was associated with model residuals in entorhinal cortex (p ≤0.05; ADNI). In subsets of observed T- to T+ converters, the difference between "observed" and estimated meta-temporal T+ onset age [95% CI] was 0.12 [-0.27, 0.52] years for ADNI and -0.09 [0.93, 0.74] years for WISC. ETOA was significantly earlier, and odds of SILA-estimated T+ status were higher amongst APOE-e4 carriers (p <0.05) and those with dementia (p <0.05). Our results suggest SILA can be used to accurately model longitudinal tau PET trajectories and retrospectively estimate individual T+ onset ages in the meta-temporal region. The accuracy of SILA time estimates in entorhinal cortex worsened amongst those with dementia in ADNI suggesting entorhinal cortex may only be suitable for studying the temporal progression of tau during the preclinical time frame.

Abstract The application of uncertainty quantification (UQ) methods to realistic industrial-scale structures remains a challenging task, due to the typical geometric complexity, the existence of various sources of uncertainties and the high dimension of structural models. These challenges are even bigger when dealing with composite material structures, which are knowingly prone to uncertainties arising from manufacturing processes and environmental influences. In this context, the present paper intends to contribute to increase the maturity level of UQ techniques to composite aeronautic structures, under the combined effects of space-dependent material and environmental fluctuations. Variations in temperature, laminate thickness and fiber volume fraction are jointly considered, being represented as random fields discretized using the Karhunen-Loève Expansion (KLE), while fiber angles are treated as random variables. Aiming at expanding the range of situations possibly found in practice, both Gaussian and non-Gaussian random fields are considered within a methodology combining the Iterative Translation Approximation Method (ITAM) and KLE. A micromechanical model is used to represent temperature- and moisture-dependent material properties, capturing the coupled effects of environmental degradation of material properties and hygrothermally-induced stresses. Monte Carlo Simulation (MCS) is employed to perform uncertainty quantification for buckling loads and vibration natural frequencies of a regional aircraft composite wing structure modeled with a relatively high-dimension finite element model. Additionally, global sensitivity analysis based on Sobol’ indices is conducted to identify the most influential random parameters, where structural responses are approximated using artificial neural network (ANN)-based surrogate models. From the simulation scenarios analyzed, accounting for different values of standard deviations attributed to random variables and correlation lengths assigned to random fields, the statistics of structural responses are assessed. The significant spread of structural responses highlights the importance of incorporating the considered types of uncertainty in analysis and design procedures for achieving robust and reliable aerospace composite structures.

Abstract This paper investigates the approximation of fixed points for quasi-nonexpansive mappings in Banach spaces using the general Picard-Mann (GPM) algorithm. Under mild conditions such as the demiclosedness of $$I - \digamma $$ I - ϝ at zero and the Opial property, we derive weak and strong convergence theorems for the iterative sequences generated by the GPM method. We establish several stability results for the GPM scheme, including summably almost stability property for quasi-contractive mappings. The theoretical findings are applied to the classical relaxation method for solving systems of linear inequalities, demonstrating the practical relevance of our approach. Numerical examples in $$\mathbb {R}^4$$ R 4 and $$\ell ^2$$ ℓ 2 are provided to illustrate the efficiency and convergence behavior of the proposed algorithm. The results presented herein extend and complement existing work in fixed point theory and iterative approximation methods.

This paper presents a computer-aided design (CAD) system for removable partial denture (RPD) frameworks, addressing the challenges of dentition defects. The system takes a digitized dental model obtained via optical scanning as input and generates an RPD framework model ready for 3D printing. Key technologies include spline curve editing and modeling, mesh offsetting, texture image-based modeling, and component models fusion. The system utilizes conformal mapping between the dental model and a disk. This enables spline curve editing to be executed in the parameterized 2D domain, ensuring both accuracy and efficiency. An iterative approximation method with adaptive mesh simplification is introduced to achieve precise mesh offsetting while avoiding self-intersections. Furthermore, texture mapping enables interactive modeling of holes for denture base connectors and 3D branch-like wax patterns for major connectors. An enhanced Boolean algorithm, combined with smoothing and simplifying techniques for intersecting regions, is utilized to ensure seamless and natural integration of various components. Clinical evaluations demonstrate that the system achieves a performance level comparable to advanced commercial CAD systems, having successfully completed over 30,000 clinical designs with high reliability and meeting all required standards.

A novel stochastic resonance assisted deconvolution method and its application on damage identification of wind turbine bearings

This study shows the presence and uniqueness of the optimal proximity point for several classes of generalized cyclic enriched contractions, and offers such fundamental results. We provide convergence results for this contraction. We also provide the conditions in which an iterative method can yield the optimal proximity point. To further illustrate the effectiveness of the ishi technique for generalized cyclic enriched contractions, we present a numerical solutions with comparison table and grapshical analysis, which show that our proposed iterative scheme converges faster than the other schemes. Our results are generalization of many comparable results in literature. In addition, the theoretical framework developed in this study extends classical fixed point and best proximity point results by relaxing standard contraction assumptions. The proposed approach allows a broader class of mappings to be analyzed within a unified setting. The convergence analysis is supported by rigorous proofs, ensuring the reliability of the proposed iterative method. Moreover, the numerical experiments validate the theoretical findings and demonstrate the stability and efficiency of the method under different initial conditions. The comparison with existing iterative schemes highlights the superiority of the proposed algorithm in terms of convergence speed and accuracy. These results indicate that the ishi technique is a powerful and flexible tool for solving proximity point problems arising in nonlinear analysis. Consequently, the findings of this study contribute meaningfully to the existing literature and open new directions for further research in generalized contraction mappings and iterative approximation methods.

Background. According to a number of researchers, fractal antennas, including those for the gigahertz frequency range, have a fundamental multi-frequency, which makes it possible to reduce their number in a compact device for wireless communication designed for various operating bands. At the same time, there are publications in which such an advantage of fractal antennas in comparison with their canonical counterparts is denied. It is important to note that a fractal is different from a fractal. There are fractals obtained by scale-invariant fragmentation of some initial object («Sierpinski gasket», «Koch curve», «Cantor set», etc.). But there are also fractals obtained by scale-invariant branching of the original object («fractal tree», «Julia set», «Newton basins», etc.). There is reason to believe that antennas created on the basis of fractals of fragmentation and branching will have different properties. Aim. Investigation of the characteristics of a fractal patch antenna of the «fractal tree» type, designed for the gigahertz frequency range, in its various iterative approximations. Methods. This goal is achieved by using electrodynamic modeling in the CST Microwave Studio software package by comparing the characteristics of a gigahertz patch antenna obtained for the first three iterations of the fractal tree as a prototype of a fractal antenna. Results. Quasi-fractal patch antennas (the first three iterations) of the «fractal tree» type in the form of an ideally conductive patch on a dielectric substrate with a relative permittivity of 3,55 mm and a thickness of 0,203 mm, on the opposite side of which there is a grounded metal shield, are studied. The frequency dependences of the element S11 of the scattering matrix, the standing wave coefficient of the voltage and the input impedance are constructed. The radiation patterns of the studied antennas in polar and spherical coordinates at the observed operating frequencies are also presented. Conclusion. Step-by-step simulation of a «fractal tree» type patch antenna in the frequency range 0-50 GHz has shown that as the branched type fractal evolves, the number of antenna operating bands can be increased.

BackgroundAmyloid pathology drives tau accumulation, i.e., the key driver of clinical worsening in Alzheimer's disease (AD). Yet, there is considerable heterogeneity in the time of tau onset, as well as in the rates and patterns of tau accumulation, which jointly determine symptom onset and clinical trajectories. The exposure to amyloidosis is predictive of AD progression and may therefore predict tau onset and trajectories. Therefore, we investigated how the age and duration of amyloid onset influence tauopathy onset and accumulation.MethodsWe included 479/390 ADNI/A4 participants with Flortaucipir tau‐PET, and Florbetaben/Florbetapir amyloid‐PET. Using sampled iterative local approximation, we determined subject‐specific estimated onset ages of amyloid‐PET positivity (centiloid>20), and tau‐PET positivity (SUVR>1.3). Using robust linear regression, we investigated the associations between estimated amyloid‐PET and tau‐PET onset ages, the delay between amyloid and tau onset and the effect of amyloid onset on tau‐PET change rates.ResultsYounger estimated age of amyloid onset predicted younger estimated age of tau onset in the temporal meta ROI (ADNI/A4, b=0.6871/0.7148, p <0.001/0.001, Figure 1A). This result pattern was pronounced in tau vulnerable temporo‐parietal regions, while sparing late Braak regions (Figure 1B). However, a younger estimated age of amyloid onset also predicted a longer delay between amyloid and tau onset, indicating that patients with young onset amyloidosis require longer to develop abnormal tau (Figure 2). By combining sliding window analyses across amyloid onset ages and bootstrapping, we identified that younger amyloid onset is linked to faster tau accumulation, with stronger involvement of parieto‐frontal vs. more pronounced temporal lobe tau accumulation in individuals with later amyloid onset (Figure 3).ConclusionsEarlier amyloid onset predicts earlier tau onset and faster more neocortically pronounced tau accumulation. At the same time, younger amyloid onset is linked to a longer delay to tauopathy compared to individuals with older‐age amyloid onset. A longer delay between amyloidosis and tauopathy in patients with earlier onset of amyloidosis may widen the window of opportunity for anti‐amyloid drugs to prevent more aggressive tauopathy in these at risk individuals.

In contrast to the conventional uniform array, nonuniform arrays have demonstrated their potential in equalizing channel eigenvalues and thereby enhancing the overall system efficiency. Focusing on near-field communication (NFC) environments, this study explores the optimization of non-uniform planar array (NUPA) for multi-user multi-input single-output (MU-MISO) systems. Within the context of a generalized Rician channel, accounting for uniform spherical wave (USW) propagation, we initially establish the geometrical relations for the base station (BS) and user equipments. Our objective pivots around maximizing the ergodic sum-rate from a statistical standpoint, for which we develop an approximate mathematical expression, ultimately guiding us towards obtaining a stable array topology configuration. By leveraging the Kronecker mixture product, higher-order Taylor expansion, and iterative convex approximation methods, the challenge of optimizing for array element positions is skillfully addressed in a highly efficient manner. Remarkably, our approach reveals unique characteristics of the array topological arrangement, i.e., a sparse peripheral layout and a dense core, that starkly contrasts the setup for single-user scenarios. Numerical simulations demonstrate substantial performance improvements over conventional layouts, thus providing an efficient array topology design guideline for near-field multi-user systems.

In this paper, we analyze the Picard–S iteration in the context of Banach spaces and establish a range of strong and weak convergence results for mappings satisfying condition (E). These results play an important role in examining the regularity properties of nonlinear dynamical evolution equations.

Novel Explicit Iterative Approximation Schemes for Extended Hierarchical Variational Inequalities Involving Multiple Nonlinear Operators

The randomized progressive iterative approximation (RPIA) is a local and approximate geometric iteration method designed for large-scale data fitting. At each iteration, RPIA updates only the control points indexed by a specific set, leaving the others unchanged. In this work, we introduce a two-dimensional RPIA (D2RPIA) for fitting B-spline curves and surfaces. Unlike RPIA, D2RPIA updates the control points with an adaptive step-size, which is determined by imposing a constraint on the new control points. This adaptive step-size allows D2RPIA to achieve the current optimal result, thereby enhancing the convergence rate compared to RPIA. We prove that D2RPIA converges linearly in the mean square to the least-squares solution. Several numerical studies are presented to validate our theoretical results.

ABSTRACT With the rapid advancement of communication technologies, space‐air‐ground integrated networks (SAGIN) have become a pivotal research frontier in current and future communication domains. To tackle critical challenges in SAGIN scenarios, such as excessive task‐related energy consumption and insufficient communication‐computing resources, this paper proposes a three‐tier edge computing architecture integrating satellites, unmanned aerial vehicle (UAV) swarms, and ground systems. Aiming to minimize the system's weighted energy consumption and latency, we investigate the joint optimization of task allocation, user‐UAV association, UAV deployment, and resource allocation between UAVs and low‐earth orbit (LEO) satellites. Formulated as a non‐convex mixed‐integer nonlinear combinatorial optimization problem, this work integrates the branch‐and‐bound method, multi‐start global optimization, and gray wolf optimization (GWO) to develop a suboptimal solution based on block coordinate descent (BCD), which decouples the original problem into three subproblems for independent solving and iterative approximation of the optimal solution. Experimental results show that the proposed algorithm reduces the total system cost by 7.81%, 11.99%, and 45.69% compared with baseline algorithms with random user‐UAV association, random UAV positioning, and random task assignment, respectively, effectively cutting down overall energy consumption and task latency.

This paper investigates fixed point theory for mean nonexpansive mappings in $p$-uniformly convex metric spaces. It first establishes the existence of fixed points together with a demiclosedness principle in this setting. Building on these foundations, the two-step Karakaya iteration scheme is introduced, and a detailed convergence analysis is provided. In particular, both a $\Delta$-convergence theorem and a strong convergence theorem for mean nonexpansive mappings are proved. To illustrate the applicability of the results, new examples are constructed that clarify the scope of the assumptions. Furthermore, a numerical application to a nonlinear fractional Volterra integral equation within the framework of a $p$-uniformly convex metric space is presented. The existence of a Bochner solution is demonstrated and approximated using the Karakaya iteration scheme, with its numerical performance compared to that of the S-iteration and Thakur schemes.

BackgroundWith continued collection of amyloid positron emission tomography (PET) neuroimaging and new quantitative approaches to analyze longitudinal PET data, the Alzheimer’s disease (AD) research field is now positioned to determine amyloid trajectories empirically. Previous studies proposing a physiologic ceiling generally do not evaluate the direct relationship between amyloid levels and time. However, newer studies using sampled iterative local approximation (SILA), a nonparametric algorithm that estimates trajectories with data reflecting differential scan ages/intervals, have not indicated the presence of an accumulation plateau at high amyloid burden. These findings contradict temporal models of AD development that argue for a physiologic ceiling.MethodWe simulated amyloid trajectories informed by Alzheimer’s Disease Neuroimaging Initiative (ADNI) study data and the Jack model of AD pathogenesis. Empirically informed stochastic parameters included age at first PET scan, number of scans per individual, and inter‐scan intervals. Estimated age of amyloid positivity onset was drawn from a distribution based on prior published literature (Betthauser et al. 2022). Simulations assume interindividual variability in the physiologic ceiling and rates of amyloid accumulation.ResultReimplementing SILA in ADNI shows an apparent lack of a physiologic ceiling for amyloid, consistent with prior literature (Figure 1). Simulations that assume a physiologic ceiling show qualitatively different trajectories from trajectories in ADNI data (Figure 2), and lack an increase in density at high Centiloids characteristic of individuals approaching a ceiling (Figure 3). We are developing an R package for performing realistic amyloid trajectory simulations under various assumptions about trajectory shape and variability.ConclusionsAmyloid trajectories in ADNI aligned based on SILA‐estimated age of amyloid‐positivity onset suggest an apparent lack of a physiologic ceiling for amyloid. This result is in contrast to prior studies which argue for a ceiling based on differences in amyloid by clinical stage and baseline amyloid level. Additionally, generating data based on influential models of temporal AD biomarker evolution does not produce trajectories consistent with those observed in ADNI. Growing longitudinal data availability and new quantitative tools should allow us to formally evaluate prevailing models of AD biomarker evolution.

BackgroundWith continued collection of amyloid positron emission tomography (PET) neuroimaging and new quantitative approaches to analyze longitudinal PET data, the Alzheimer's disease (AD) research field is now positioned to determine amyloid trajectories empirically. Previous studies proposing a physiologic ceiling generally do not evaluate the direct relationship between amyloid levels and time. However, newer studies using sampled iterative local approximation (SILA), a nonparametric algorithm that estimates trajectories with data reflecting differential scan ages/intervals, have not indicated the presence of an accumulation plateau at high amyloid burden. These findings contradict temporal models of AD development that argue for a physiologic ceiling.MethodWe simulated amyloid trajectories informed by Alzheimer's Disease Neuroimaging Initiative (ADNI) study data and the Jack model of AD pathogenesis. Empirically informed stochastic parameters included age at first PET scan, number of scans per individual, and inter‐scan intervals. Estimated age of amyloid positivity onset was drawn from a distribution based on prior published literature (Betthauser et al. 2022). Simulations assume interindividual variability in the physiologic ceiling and rates of amyloid accumulation.ResultReimplementing SILA in ADNI shows an apparent lack of a physiologic ceiling for amyloid, consistent with prior literature (Figure 1). Simulations that assume a physiologic ceiling show qualitatively different trajectories from trajectories in ADNI data (Figure 2), and lack an increase in density at high Centiloids characteristic of individuals approaching a ceiling (Figure 3). We are developing an R package for performing realistic amyloid trajectory simulations under various assumptions about trajectory shape and variability.ConclusionsAmyloid trajectories in ADNI aligned based on SILA‐estimated age of amyloid‐positivity onset suggest an apparent lack of a physiologic ceiling for amyloid. This result is in contrast to prior studies which argue for a ceiling based on differences in amyloid by clinical stage and baseline amyloid level. Additionally, generating data based on influential models of temporal AD biomarker evolution does not produce trajectories consistent with those observed in ADNI. Growing longitudinal data availability and new quantitative tools should allow us to formally evaluate prevailing models of AD biomarker evolution.