This study presents a C*-algebraic framework for optimizing intrusion mitigation in Internet of Things (IoT) networks by integrating mathematical models for cyberattack propagation with optimization-based security strategies. Theoretical results demonstrate that the spectral radius of the attack operator ρ(A) governs the recovery of IoT networks under attack, where ρ(A) < 1 ensures system recovery, and ρ(A) ≥ 1 leads to persistent or growing attack impact. The framework combines blockchain-based trust, AI-driven intrusion detection systems (IDS), and Zero-Trust Architecture (ZTA) to provide a multi-layered, adaptive defence system. Unlike probabilistic models that simplify attack dynamics, this approach rigorously models threats using bounded linear operators, thereby offering scalability and robustness. Optimization ensures computational efficiency, making the model suitable for resource-constrained IoT environments, with the operator norm and the spectral radius acting as key constraints. Validation on real-world datasets such as CIC-IoT2023, UNSW-NB15, and BoT-IoT revealed that the AI-IDS models achieved near-perfect performance, while the unified model integrating blockchain, IDS, and ZTA showed an accuracy of 51.0

Reliability of a software system is an important attribute of software quality, and its importance has increased significantly due to the growing demand for high-quality software systems. Evaluating reliability during the design phase is essential for improving system performance and ensuring effective planning. At this stage, an important challenge is to identify suitable methods for achieving the desired reliability of the software system. Reliability allocation methods can be applied to assign reliability targets to individual components prior to the actual system design. To address this, a new hierarchical model is proposed that integrates the perspectives of users, software programmers, and software managers. The system is decomposed into multiple levels, including functions, programs, modules, and submodules, to enable systematic analysis. To handle uncertainty and vagueness in human judgments, the fuzzy analytic hierarchy process (FAHP) is employed, incorporating the geometric mean method with trapezoidal fuzzy numbers. The proposed approach determines reliable weights and effectively allocates reliability targets at each level. A comparison with the classical AHP method demonstrates that FAHP provides a more flexible and realistic representation by capturing uncertainty, making it more suitable for reliability allocation in complex software systems.

This paper establishes a linear profile decomposition for bounded sequences in the homogeneous Dirichlet–Sobolev space H ̇_D^1 (Ω), where Ω=R^3\O is the exterior of a smooth, compact obstacle O⊂R^3. Given a bounded sequence {f_n}⊂H ̇_D^1 (Ω), we prove that, after passing to a subsequence, it decomposes as f_n=∑_(j=1)^J▒ϕ_n^j +w_n^J, where the profiles {ϕ_n^j} are asymptotically orthogonal and the remainder w_n^J vanishes in all Strichartz spaces L_t^q L_x^r as J→∞. The decomposition satisfies an exact energy identity ∥∇f_n ∥_(L^2)^2=∑_j^▒∥ ∇ϕ_n^j ∥_(L^2)^2+∥∇w_n^J ∥_(L^2)^2+o(1). Four distinct geometric concentration regimes are identified according to the behaviour of the scale sequence {λ_n^j} relative to the distance d(x_n^j ) to ∂Ω: profiles localised inside Ω; profiles dispersing to spatial infinity (limiting domain R^3); profiles concentrating deep inside Ω away from the boundary; and profiles concentrating near ∂Ω (limiting domain: a half-space). As an application, we prove small-data scattering in H ̇_D^1 (Ω) for the defocusing, energy-subcritical NLS i∂_t u+Δ_Ω u=|u|^(p-1) u with 1<p<5 and Dirichlet boundary condition. We emphasise that the NLS is a Hamiltonian (conservative) system: its energy is conserved, not decaying, and the scattering result follows from Strichartz estimates rather than from any dissipative mechanism. 

Steganography attempts to conceal messages in plain sight while steganalysis seeks to identify them or, more importantly, to extract the embedded data. Low-payload and spatially localized steganographic embedding is increasingly used to evade detection by classical steganalysis methods. While such strategies preserve global image statistics and remain visually imperceptible, they can disrupt natural pixel-level behavior. This work proposes a behavioral steganalysis framework inspired by process mining that detects image steganography by analyzing localized behavioral deviation using regional behavioral contrast and behavioral amplification. Experiments on lossless grayscale PNG images from the USC SIPI database and 10,000 images from the BOWS2 dataset using 1-bit LSB embedding show that the proposed framework reliably identifies steganographic embedding. On the USC SIPI dataset, conventional statistical detectors, including chi-square analysis and the StegExpose tool, showed limited detection capability under the evaluated localized embedding settings. Despite high perceptual quality of stego images (PSNR > 55 dB), significant behavioral deviation is consistently observed within embedded regions. These results demonstrate that the proposed process mining-inspired framework provides an interpretable and complementary direction for image steganalysis, particularly under low-payload and localized embedding scenarios.

Artificial-intelligence workloads increasingly process proprietary and personally identifiable data, yet conventional security controls protect data only at rest and in transit, leaving computation itself exposed. This paper presents CC-Shield, a five-layer confidential-computing architecture that combines hardware trusted execution environments (Intel SGX, AMD SEV-SNP), differentially private federated aggregation, remote attestation, encrypted model lifecycle management, and LSTM-based anomaly detection into a single, formally analysed defence-in-depth stack. We derive a closed-form leakage bound that jointly composes TEE side-channel capacity and differential-privacy noise, prove three attack-resistance theorems covering membership inference, model inversion, and active-adversary integrity, and connect security overhead to system throughput via a queuing-theoretic performance model. On ResNet-50/ImageNet, BERT-base/SST-2, and a clinical MLP on MIMIC-III, CC-Shield with differential privacy (ε=1) reduces membership-inference attack success to 51.8% (statistically indistinguishable from the 50% random-chance baseline at a 95% confidence half-width of approximately 1.0 percentage point over 10,000 attack queries), versus 71.3% for an unprotected baseline, while introducing only 11.9%-13.9% inference latency overhead – more than three orders of magnitude lower than a homomorphic-encryption baseline. A seven-dimension qualitative comparison against five prior frameworks shows CC-Shield is the only approach satisfying data-in-use protection, computation integrity, training- and inference-time protection, quantum resistance, sub-15% latency overhead, and a formal security proof simultaneously.

The one-dimensional non-homogeneous wave equation subject to non-homogeneous boundary and initial conditions presents significant analytical challenges, particularly when external forcing and irregular boundary data are simultaneously present. Classical methods such as separation of variables are restricted to homogeneous settings and fail to accommodate non-homogeneous terms in a unified framework. This study employs the Fourier transform method to derive an exact analytical solution by decomposing the total wave displacement into two components: u(x,t) = V(x,t) + ψ(x), where V(x,t) is the oscillatory component satisfying the homogeneous wave equation and ψ(x) is the spatially adjusted component encoding the influence of non-homogeneous boundary conditions and external forcing. The analytical solution is verified by direct substitution and benchmarked against a second-order explicit finite-difference scheme on a grid of 500 spatial points, yielding a maximum point-wise absolute error below 8×10⁻³, consistent with the second-order truncation error of the numerical scheme. For the representative test case with unit wave speed, unit domain length, constant spatial forcing F(x) = 2, and initial displacement u₀(x) = sin(πx), the steady-state component is recovered exactly as ψ(x) = x − x², and the dominant Fourier coefficient is A₁ ≈ 0.742. A direct point-by-point comparison with published benchmark values further quantifies the sensitivity of wave solutions to boundary condition specification. The proposed framework accommodates both finite and infinite spatial domains and offers a systematic, closed-form alternative to purely numerical approaches for this class of wave propagation problems, with relevance to acoustics, structural dynamics, and materials science.

Curve similarity plays a crucial role in various domains where comparing functional or dynamic shapes is essential, including bioassay analysis, trajectory studies, spectroscopy, medical signal interpretation, and functional genomics. Despite its broad impact, research on curve similarity methods remains fragmented across statistical, computational geometry, and signal processing communities, leading to a lack of unified terminology and systematic comparison. To address this gap, this study adopts a structured literature review methodology, in which relevant studies are identified through a comprehensive search of major academic databases and selected based on predefined inclusion criteria, including peer-reviewed publications focusing on similarity measures for curves and time series. The review systematically examines mathematical and statistical approaches to curve similarity, focusing on their theoretical foundations, statistical properties, and practical applications. The selected methods are categorized into five groups: distance-based, alignment-based, topology-oriented, statistical and hypothesis testing, and learning-based approaches. For each category, key aspects such as mathematical formulation, invariance properties, robustness to sampling variability, and computational characteristics are analyzed. In addition, application domains, method comparisons, and common limitations are discussed, along with available software tools that support curve similarity analysis. By providing a structured and methodologically grounded synthesis, this review assists researchers in selecting appropriate techniques and highlights potential directions for developing more robust and scalable similarity assessment frameworks.

This paper focuses on the methods for determining how uncertainties affect people in everyday life. Since almost nothing is deterministic, anything might become impractical in real life. We will encounter many difficulties if we lack anticipatory ideas. The business organizations are facing uncertainties in demand, supply, cost of raw materials, prices of finished products etc. in everyday life. To address these uncertainties, this study will analyse stochastic Linear Programming Problems (SLPs) aiming to bridge the gap between theoretical concepts and practical applications. This paper specifically explores the effects of assuming stochastic pricing and demand in the Uncapacitated Facility Location Problem (UCFLP), shedding light on the variations in profit. We will develop a stochastic UFLP for a company of Bangladesh known as Unilever Company of Bangladesh and it may use for any company in the world. For this, we will collect and analyse the data from the company in stochastic atmosphere. The model will help the company to be able to aware of how uncertainties of demand, supply and other factors can affect the profit and loss of the company. This approach underscores the importance of considering various factors, including demand patterns, cost-effectiveness, and regional dynamics. In the model, we will consider the data from five different regions of Bangladesh.