We prove that a self-mapping T on a complete metric space (X,D), satisfying a generalized contractive condition involving its ρ-th iterate Tρ (ρ∈N,ρ≥2) and a uniform coefficient bound L<1, possesses a unique fixed point. The proof establishes that the Picard iteration sequence is Cauchy with the further assumption of continuity of T, and fixed point uniqueness follows directly from the contractive inequality.

In modern reliability and survival analysis, modeling bounded-lifetime data under complex sampling mechanisms remains a challenging yet practically significant problem, particularly in biomedical and engineering applications where experimental time and cost constraints are critical. This study develops a comprehensive inferential framework for the unit-Lindley (ULind) distribution under the improved adaptive progressive Type-II censoring (I-AP-CT2) strategy, a flexible design that ensures bounded test duration while accommodating progressive removal of experimental units. The proposed approach integrates both classical and Bayesian paradigms to enable robust estimation of the model parameter, reliability function, and hazard rate function. Maximum likelihood estimators are derived, and their asymptotic properties are established, with interval estimation constructed via normal approximation and log-transformed techniques. To address analytical intractability, a Bayesian framework via MCMC-based is formulated with a gamma prior, yielding credible and highest posterior density intervals. An extensive Monte Carlo simulation study is conducted to evaluate estimator performance under diverse censoring scenarios, demonstrating that Bayesian procedures consistently outperform their frequentist counterparts in terms of accuracy, stability, and interval efficiency. The practical relevance of the proposed methodology is illustrated through two real-world applications involving kidney dialysis survival data and petroleum engineering reliability data, representing critical domains where accurate modeling of failure behavior directly impacts clinical decision-making and industrial risk management. The findings highlight the flexibility of the ULind model in capturing complex hazard rate shapes and confirm the effectiveness of the I-AP-CT2 mechanism as a realistic and efficient experimental design.

A new approach in constructing a univariate absolutely continuous probability model via power transformation is adopted. The proposed distribution appears to subsume several popular univariate continuous probability models that are already existent in the literature. In addition, the hazard rate function of the proposed distribution appears to possess all possible shapes, increasing, decreasing, and upside-down which is not present in all existing extensions and generalizations of the log-logistic (LL) distribution. Various established estimation strategies under the classical approach such as maximum product spacing, weighted least squares estimation, etc., are adopted to exhibit the flexibility of the proposed probability distribution. The emphasis has been made on the estimation procedure and the new probability model is indeed augmenting the existing literature in many aspects which are highlighted in the simulation and also in the real data application section.

This paper gives expansions for the distribution, density and quantiles of any estimate that is a smooth function of the sample cross-moments of a stationary process. Three versions of these are given, depending on whether an exact, approximate, or asymptotic form is used for the variance or covariance of the estimate. Eight examples are provided, including sample autocovariances and autocorrelations. Their Central Limit Theorems extend those in the literature, such as Bartlett’s formula, by allowing for the effect of the mean and higher order cross-cumulants. Their distribution and quantiles are given to magnitude n−r/2 up to r=3, where n is the sample size.

Addition theorems have been indispensable tools for the reduction of quantum transition amplitudes. They are normally utilized at the start of the process to move the angular dependence within plane waves, Coulomb potentials, and the like, into a sum over spherical harmonics that allows the angular integration to be carried out. These have historically been “two-range” addition theorems, characterized by the two-fold notation r>=Max[r1,r2] and r<=Min[r1,r2] and comprising a single infinite series. More recently, “one-range” addition theorems have been created that have no such piecewise notation, but at the cost of the introduction of another infinite series. We use a very different approach to derive an infinite set of addition theorems for Slater orbitals, hydrogenic and Hylleraas wave functions, and so on, that retain the one-range variable dependence but have, at worst, a finite second series rather than an infinite one. In addition, unlike previous addition theorems, they are applicable to more than one coordinate system. One of these addition theorems may also be used for Yukawa-like functions that may appear late in the reduction of amplitude integrals, and we show its utility for an integral that has stubbornly defied reduction to analytic form for nearly sixty years. Finally, we craft indefinite integrals of 15 half-integer Macdonald functions multiplied by (inverse) powers and negative exponentials containing squares of the integration variable.

This paper introduces the New Bipolar Intuitionistic Fuzzy Metric Space (NBIFM-space)—a mathematical framework that extends intuitionistic and previously proposed bipolar intuitionistic structures by providing a complete three-component formulation based on positive similarity, negative similarity, and indeterminacy. Unlike earlier bipolar intuitionistic models, the NBIFM-space employs normalized metric components and coordinated triangular norms denoted by t-norm/t-conorm interactions, yielding a fully consistent topological and analytic setting. We have developed the basic properties of this structure and have demonstrated its effectiveness in image processing, where the explicit separation of attraction, repulsion, and uncertainty leads to robust edge-preserving filtering. Furthermore, a Banach-type fixed point theorem is established in the full NBIFM framework.

This paper studies a new family of iterative methods for the simultaneous approximation of polynomial zeros with known multiplicities. The methods are obtained by combining the Wang–Zheng iteration function with an arbitrary iteration function. This approach leads to a class of methods referred to as Wang–Zheng-type methods with correction for multiple zeros. A local convergence analysis is developed for a wide class of iteration functions. The analysis describes the conditions under which the proposed methods converge locally. Several known iterative methods are examined as special cases of the general results. In particular, the family constructed by Kyurkchiev and Andreev (1990) is included. For every positive integer N, the N-th method of this family has convergence order 3N+1. The main local convergence theorem extends, complements and improves earlier results by Wang and Wu (1987) and by Kyurkchiev and Andreev (1990).

This paper proposes an impulsive SIRQ model for the analysis of computer network resilience against malware propagation and distributed denial-of-service (DDoS) attacks. The model extends classical epidemic frameworks by combining the continuous-time dynamics of malicious object spreading with discrete control actions corresponding to mass updates, node isolation, and access control policies. A qualitative analysis of the resulting system of impulsive differential equations is performed. The basic reproduction number R0, identified as a threshold parameter characterizing the intensity of attack propagation, and sufficient conditions for the global asymptotic stability of the infection-free state are established. It is shown that, under periodic impulsive control, the infection-free state can be stabilized with respect to the target population coordinates even when R0>1. An exponential decay estimate for the total active threat is derived, guaranteeing the asymptotic extinction of the infected and quarantined node populations. The proposed approach provides quantitative criteria for the effectiveness of impulsive cyber defense strategies and offers a theoretical foundation for the design of adaptive multi-layer protection systems for critical information infrastructures. Practical interpretation of the results illustrates the dependence of the critical impulsive control period on the model parameters and demonstrates the applicability of the approach to cybersecurity strategy design.

The primal-dual hybrid gradient (PDHG) method is widely used for convex–concave saddle-point problems, yet its extrapolated variants are typically asymmetric because only one side is extrapolated. We propose a symmetry-preserving refinement, E-PDHG, which performs dual-side extrapolation followed by an explicit correction step. Under standard step-size conditions, we establish global convergence for all η∈(−1,1) and derive a pointwise (non-ergodic) O(1/t) rate for the last iterate. The method does not improve the asymptotic complexity order of PDHG; instead, it enlarges the practically stable parameter region while retaining the same per-iteration cost. Numerical experiments on image deblurring/inpainting and additional machine learning benchmarks (logistic regression and LASSO) demonstrate improved finite-iteration stability and efficiency.

In this paper, stronger versions of Stein–Weiss inequalities and reverse Stein–Weiss inequalities are established.