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Spatially selective nerve stimulation is an active area of research, with the capability to reduce side effects and increase the clinical efficacy of nerve stimulation technologies. Several research groups have demonstrated proof-of-concept devices capable of performing spatially selective stimulation with multi-contact cuff electrodes in vivo; however, optimizing the technique is difficult due to the large possibility space granted by a multi-electrode cuff. Our work attempts to elucidate the most valuable stimulation montages (current ratios between stimulating electrodes) provided by a multi-contact cuff. We characterized the performance of five different montage types when stimulating fibers in different “electrode configurations”, with configurations including up to three rings of electrode contacts, 13 different counts of electrodes per ring, and five electrode arc lengths per electrode count (for 195 unique configurations). Selected montages included several methods from prior art, as well as our own. Among montage types, the most spatially selective stimulation was one we refer to as “X-Adjacent” stimulation, in which three adjacent electrodes are active per ring. Optimized X-adjacent montages achieved an average fiber specificity of 71.9% for single-ring electrode configurations when stimulating fibers located at a depth of two-thirds of the nerve radius, and an average fiber specificity of 77.2% for two-ring configurations. These values were the highest among montages tested, and in combination with our other metrics, led these montages to perform best in the majority of cost functions investigated. This success leads us to recommend X-Adjacent montages to researchers exploring spatially selective stimulation. Full article

In this paper, we study a class of optimization problems with separable constraint structures, characterized by a combination of convex and nonconvex constraints. To handle these two distinct types of constraints, we introduce a partial augmented Lagrangian function by retaining nonconvex constraints while relaxing convex constraints into the objective function. Specifically, we employ the Moreau envelope for the convex term and apply second-order variational geometry to analyze the nonconvex term. For this partial augmented Lagrangian function, we study its saddle points and establish their relationship with KKT conditions. Furthermore, second-order optimality conditions are developed by employing tools such as second-order subdifferentials, asymptotic second-order tangent cones, and second-order tangent sets. Full article

This study investigates how process manufacturing enterprises can optimize semi-finished inventory (SFI) distribution in delayed production models, with particular attention to differences in cost volatility between single- and multi-period planning scenarios. To address this research gap, we develop a mixed-integer programming model that determines optimal customer order decoupling point (CODP)/product differentiation point (PDP) positions and SFI quantities (both generic and dedicated) for each production period, employing particle swarm optimization for solution derivation and validating findings through a comprehensive case study of a steel manufacturer with characteristic long-period production processes. The analysis yields two significant findings: (1) single-period operations demonstrate marked cost sensitivity to service level requirements and delay penalties, necessitating end-stage inventory buffers, and (2) multi-period optimization generates a distinctive cost-smoothing effect through strategic order deferrals and cross-period inventory reuse, resulting in remarkably stable total costs (≤2% variation observed). The study makes seminal theoretical contributions by revealing the convex cost sensitivity of short-term inventory decisions versus the near-flat cost trajectories achievable through multi-period planning, while establishing practical guidelines for process industries through its empirically validated two-period threshold for optimal order deferral and inventory positioning strategies. Full article

The Volume Bragg Grating (VBG) imaging technique provides a novel approach to gaze-type hyperspectral imaging. However, collimation constraints of the incident beam during narrow-band filtering and high-spatial-resolution imaging introduce system complexity, hindering miniaturization and modularization of the optical system. To address these limitations, this paper proposes a convex transmissive VBG structure with tunable design parameters to enhance the field of view (FOV), relax collimation requirements, improve imaging quality, narrow filter spectral bandwidth, and simplify the optical system design. For the precise analysis and optimization of convex VBG performance, we established a physical model for filtered imaging using a convex transmissive VBG with polychromatic extended sources. An evaluation metric termed the “Maximal Splitting Angle (MSA)” was introduced to quantify the dispersion extent of image spots. This approach was employed to investigate the intrinsic correlations between structural parameters (such as the radius of curvature, vector tilt angle, grating period, and thickness) and key system performance indicators (spatial resolution and spectral resolution). The necessity of optimizing these parameters was rigorously demonstrated. Theoretical analysis confirms that convex transmissive VBG achieves superior spatial and spectral resolution over planar VBG under reduced collimation constraints. The experimental results show a 58.5% enhancement in spatial resolution and a 63.6% improvement in spectral bandwidth for the convex transmissive VBG system. Crucially, while planar transmissive VBG suffers from stray fringe interference during wavelength tuning, its convex counterpart remains unaffected. This study proposes a novel device structure, offering new perspectives for optimizing VBG-filtered spectral imaging systems. Full article

Leveraging non-terrestrial networks for edge computing is crucial for the development of 6G, the Internet of Things, and ubiquitous digitalization. In such scenarios, diverse tasks often exhibit continuously distributed attributes, while existing research predominantly relies on qualitative thresholds for task classification, failing to accommodate quantitatively continuous task requirements. To address this issue, this paper models a multi-task scenario with continuously distributed attributes and proposes a three-tier cloud-edge collaborative offloading architecture comprising UAV-based edge nodes, LEO satellites, and ground cloud data centers. We further formulate a system cost minimization problem that integrates UAV network load balancing and satellite energy efficiency. To solve this non-convex, multi-stage optimization problem, a two-layer multi-type-agent deep reinforcement learning (TMDRL) algorithm is developed. This algorithm categorizes agents according to their functional roles in the Markov decision process and jointly optimizes task offloading and resource allocation by integrating DQN and DDPG frameworks. Simulation results demonstrate that the proposed algorithm reduces system cost by 7.82% compared to existing baseline methods. Full article

A novel category of convex functions, termed multiplicatively trigonometric convex functions, are introduced in this paper. We explore their algebraic characteristics and establish connections between such functions and other forms of convex functions. We even show that these functions are symmetric with respect to their components. Furthermore, we prove the Hermite–Hadamard inequality for the mentioned category of functions. In addition, we present new structures of the Hermite–Hadamard inequality within the framework of multiplicative integrals. By broadening these inequalities, the purpose is to reveal some properties and relations that help the advancement of more robust mathematical techniques. Full article

Using Taylor-type expansions, we obtain identity expressions for functions on three intervals and differences for two pairs of Csiszár $\varphi$-divergence. With some more assumptions in these identities, inequalities for functions on three intervals and Csiszár $\varphi$-divergence can be obtained as special cases. They can also deduce the known generalized trapezoid type inequality. Furthermore, we use the identity to obtain a new extension for Levinson inequality; thus, new refinements and reverses for Ky Fan-type inequalities are established, which can be used to compare or estimate the yields in investments. Special cases of Csiszár $\varphi$-divergence are given, and we obtain new inequalities concerning different pairs of Kullback–Leibler distance, Hellinger distance, $\alpha$-order entropy and ${\chi }^2$-distance. Full article

The rapid growth of Industrial Internet of Things (IIoT) terminals has resulted in tasks exhibiting increased concurrency, heterogeneous resource demands, and dynamic priorities, significantly increasing the complexity of task scheduling in edge computing. Cloud–edge–end collaborative computing leverages cross-layer task offloading to alleviate edge node resource contention and improve task scheduling efficiency. However, existing methods generally neglect the joint optimization of task offloading, resource allocation, and priority adaptation, making it difficult to balance task execution and resource utilization under resource-constrained and competitive conditions. To address this, this paper proposes a two-stage dynamic-priority-aware joint task offloading and resource allocation method (DPTORA). In the first stage, an improved Multi-Agent Proximal Policy Optimization (MAPPO) algorithm integrated with a Priority-Gated Attention Module (PGAM) enhances the robustness and accuracy of offloading strategies under dynamic priorities; in the second stage, the resource allocation problem is formulated as a single-objective convex optimization task and solved globally using the Lagrangian dual method. Simulation results show that DPTORA significantly outperforms existing multi-agent reinforcement learning baselines in terms of task latency, energy consumption, and the task completion rate. Full article

This work presents generalizations and extensions of previous results by incorporating weighted integrals and a refined class of second-type $(h,m)$-convex functions. By utilizing classical inequalities, such as those of Hölder and Young and the Power Mean, we establish new Hermite–Hadamard-type inequalities. The findings offer a broader and more flexible analytical framework, enhancing existing results in the literature. Potential applications of the developed inequalities are also explored. Full article

The advent of sixth-generation (6G) wireless networks and cell-free massive multiple-input multiple-output (MIMO) architectures underscores the need for efficient resource allocation to support federated learning (FL) at the network edge. Existing approaches often treat communication, computation, and learning in isolation, overlooking dynamic heterogeneity and fairness, which leads to degraded performance in large-scale deployments. To address this gap, we propose a joint optimization framework that integrates communication–computation co-design, fairness-aware aggregation, and a hybrid strategy combining convex relaxation with deep reinforcement learning. Extensive experiments on benchmark vision datasets and real-world wireless traces demonstrate that the framework achieves up to 23% higher accuracy, 18% lower latency, and 21% energy savings compared with state-of-the-art baselines. These findings advance joint optimization in federated learning (FL) and demonstrate scalability for 6G applications. Full article

The article’s primary focus is on the study of the number of countable non-isomorphic models of linearly ordered theories. The orthogonality of 1-types and their convex closures is employed to analyse a class of theories with a specific type of monotonic non-orthogonality, which includes weakly o-minimal theories. For such theories, a theorem analogous to L. Mayer’s result on the independence of any pairwise independent family of 1-types in o-minimal theories is proven. The article provides conditions for the infinity and maximality of the countable spectrum of weakly o-minimal theories. Full article

The sharp increase in Internet of Things (IoT) terminal numbers imposes significant pressure on energy and wireless spectrum resources. Battery-free IoT technology has become an effective solution to address the high power consumption and cost issues of traditional IoT systems. While leveraging backscatter communication, battery-free IoT faces challenges such as low throughput and poor fairness among wireless links. To tackle these problems, this study proposes a low-power hybrid communication mechanism for terminals. Within this mechanism, a time-frame partitioning method for hybrid communication strategies is designed based on sensing results of licensed spectrum channels. Considering terminal power constraints, quality of service (QoS) requirements of primary communication links, and time resource limitations, a hybrid communication strategy model is established to jointly optimize fairness and maximize throughput. To resolve the non-convexity in the Multi-objective Lexicographical Optimization Problem (MLOP), the Block Coordinate Descent (BCD) method and auxiliary variables are introduced. Simulation results demonstrate that, compared to the baseline scheme, the proposed approach reduces the throughput gap between links from 85.4% to 0.32% when the channel gain differences are small, while the total system throughput decreases by only 8.81%. As the channel gain disparity increases, the baseline scheme exhibits a more pronounced disadvantage in terms of throughput fairness, while the proposed approach still reduces the throughput gap between the best and worst links from 91.02% to 0.684% at the cost of a 9.18% decrease in total system throughput. These results demonstrate that the proposed scheme effectively balances fairness and throughput performance across diverse channel conditions, ensuring relatively equitable quality of service for all users in the IoT network. Full article

We consider first-order impulses for impulsive stochastic differential equations driven by fractional Brownian motion (fBm) with Hurst parameter $H\in ({\displaystyle \frac{1}{2}},1)$ involving a nonlinear $\varphi$-Laplacian operator. The system incorporates both state and derivative impulses at fixed time instants. First, we establish the existence of at least one mild solution under appropriate conditions in terms of nonlinearities, impulses, and diffusion coefficients. We achieve this by applying a nonlinear alternative of the Leray–Schauder fixed-point theorem in a generalized Banach space setting. The topological structure of the solution set is established, showing that the set of all solutions is compact, closed, and convex in the function space considered. Our results extend existing impulsive differential equation frameworks to include fractional stochastic perturbations (via fBm) and general $\varphi$-Laplacian dynamics, which have not been addressed previously in tandem. These contributions provide a new existence framework for impulsive systems with memory and hereditary properties, modeled in stochastic environments with long-range dependence. Full article

The corrosion of welded joints creates widespread issues for the ocean engineering, petrochemical, and nuclear power industries. Geometric discontinuity of the weld reinforcement height plays an important role in weld corrosion, but the mechanism is still unclear. The corrosion behavior of flat and convex SA106B welded joints is investigated at different flow velocities by experiments and simulation. The damage components of the material and geometric discontinuity are quantified. Electrochemical measurements, morphology observations, and flow field simulations are conducted. The results show that the corrosion of the welded joints is influenced by mass transfer and galvanic corrosion. The corrosion of the welded joints is aggravated by geometric discontinuity and increased flow velocity. The damage component introduced by the material of the welded joint decreases with increasing flow velocity, and the maximum value is 91.56% at 0.5 m/s. The damage component introduced by the geometry of the weld reinforcement height increases with increasing flow velocity, reaching up to 45.77% at 6.9 m/s. The corrosion mechanism is also discussed. Full article

Purpose: Transcatheter heart valve replacements (TVR) are typically designed in a closed shape with initial leaflet coaptation. However, recent studies suggest a semi-closed geometry without a predefined coaptation zone, relying on diastolic pressure and clinical oversizing of 10–20 % for closure. This approach may minimize pinwheeling, a phenomenon linked to early valve degeneration. Method: Seven valve geometries were assessed: one closed design (G0) and six semi-closed variations (G1–G6). The semi-closed designs differed in free edge shape (linear, concave, convex) and opening degree, defined as the relative distance from the leaflet to the valve center in the unloaded state. The opening degree was systematically increased across G1–G6, with G6 exhibiting the highest value. 30 mm valves were fabricated using porcine pericardium and self-expanding nitinol stents. Performance was assessed in a pulse duplicator system, evaluating transvalvular pressure gradient (TPG), effective orifice area (EOA), regurgitation fraction (RF) and a novel pinwheeling index (PI) which was validated by finite element simulations. Results: Finite element simulations demonstrated that semi-closed geometries achieve valve closure at a diameter reduction of >5%. In vitro tests confirmed these findings with more homogeneous coaptation and reduced pinwheeling. With increased opening degree the RF reduced significantly (RF_G0 = 18.54 ± 8.05%; RF_G6 = 8.22 ± 1.27%; p < 0.0001), while valve opening remained comparable (p = 0.4519). Conclusions: A semi-closed leaflet geometry enhances valve closure, reducing regurgitation and pinwheeling while preserving effective opening. With clinical oversizing, a higher opening degree improves coaptation and may enhance durability by mitigating structural deterioration, ultimately improving the long-term performance and lifespan of transcatheter valve replacements. Full article