Search Results (1,022)

This paper studies parameter estimation of Wiener systems with quantized inputs and multiple-valued observations in the presence of replay attacks. The proposed approach effectively addresses the destruction of the data temporal structure caused by replay attacks as well as the information loss induced by multiple-valued observations, thereby achieving consistent parameter estimation of Wiener systems under constrained observation conditions. First, the influence of replay attacks on the consistency of parameter estimation is theoretically analyzed. It is shown that the attack-free estimation algorithm converges to a biased limit under replay attacks, and an explicit expression of this limit is derived. To address this issue, two classes of consistent estimation algorithms are proposed. When the attack strategy is known, a compensation matrix is constructed to correct the compromised data, thereby eliminating the bias introduced by replay attacks and achieving consistent parameter estimation. When the attack strategy is unknown, a data preprocessing mechanism with structured markers is designed to estimate the attack strategy, based on which a compensated parameter estimation method is further developed. Moreover, the statistical properties of the proposed algorithms are analyzed, and the asymptotic normality of the estimators is established together with analytical expressions for the covariance matrices. On this basis, an optimization problem aimed at minimizing the estimation variance is formulated, yielding the optimal design parameters for the consistent estimation algorithms. Simulation results demonstrate that the proposed methods can effectively suppress the influence of replay attacks on system identification and achieve accurate and consistent parameter estimation for Wiener systems under multiple-valued observation. Full article

This paper presents the first comprehensive inferential framework for the unit linear hazard rate distribution under Type-II censoring. The unit linear hazard rate distribution is a flexible two-parameter bounded lifetime model obtained by transforming the classical linear hazard rate distribution. It accommodates a wide range of density and hazard rate shapes, including increasing and bathtub forms, making it suitable for modeling proportions, normalized lifetimes, and bounded reliability data. Its fundamental statistical properties are established through analytical, numerical, and graphical investigations. Parameter estimation under Type-II censoring is developed using maximum likelihood and maximum product spacing methods. Although maximum likelihood estimation enjoys standard asymptotic properties, the maximum product spacing approach offers superior numerical stability and competitive finite-sample performance, particularly under heavy censoring. Bootstrap confidence intervals based on the maximum product spacing estimators are also constructed. In addition, novel point and interval prediction procedures for future censored observations are developed using conditional distributions, predictive likelihood, and a maximum product spacing-based predictive framework. Extensive Monte Carlo simulations demonstrate the effectiveness of the proposed methods, while two real data applications illustrate the flexibility and superior practical performance of the proposed model compared with several competing lifetime distributions. Full article

This paper establishes existence and uniqueness results for a generalized Ćirić–Reich–Rus interpolative contraction involving distinct forward horizons $p,q>1$ and a delay parameter $r>1$. By employing a Matkowski comparison function $\phi$ and introducing a composite condition $\Psi \left(t\right)=\phi \left(p^{\beta }{q}^{\gamma }{t}^{\alpha +\beta +\gamma }\right)<t$, we overcome the geometric expansion induced by multi-step triangle inequalities. The framework guarantees asymptotic regularity and the Cauchy property of Picard sequences, yielding a unique fixed point under mild continuity and right-continuity assumptions. We further demonstrate generalized Ulam–Hyers stability with explicit error bounds under a summability condition on the iterates of $\Psi$; apply the abstract result to fractional integral, Volterra, Hammerstein, and perturbed integral equations; and provide a sharp counterexample illustrating the necessity of the composite hypothesis. The analysis reveals how asymmetric iteration parameters optimize expansion absorption, offering a tunable framework for nonlinear discrete and integral dynamics. Full article

Although the standard model of cosmology ($\mathrm{\Lambda }$CDM) has been successful in explaining a wide range of observational phenomena, it still faces significant limitations, particularly regarding the nature and properties of the dark sector of the universe. These challenges motivate the exploration of alternative approaches that can deepen our understanding of cosmic evolution. This study proposes an extension of the $\mathrm{\Lambda }$CDM model by incorporating irreversible processes through a viscous fluid with out of equilibrium pressure. This fluid interacts dynamically with the dark sector components, offering new possibilities for describing the expansion of the universe. To constrain the proposed scenarios, we perform a Bayesian statistical analysis using the Pantheon+ Type Ia supernova compilation, Cosmic Chronometers, and DESI DR2 Baryon Acoustic Oscillation measurements. The observational results show that all three interaction models successfully reproduce the late-time expansion history while preserving a dark matter barotropic index close to the pressureless limit expected in the standard cosmological scenario. In contrast, the dark energy barotropic index consistently favors positive values, suggesting a mild departure from the cosmological constant description. The incorporation of Cosmic Chronometers and, particularly, BAO measurements substantially improves the constraints on the matter sector, leading to a significant reduction in the uncertainty of the present dark matter density parameter, whereas the bulk-viscosity parameter and the asymptotic ratio remain only weakly constrained. Finally, a Bayesian model comparison based on the Bayes factor shows that the interacting dissipative models remain observationally compatible with the reference $\mathrm{\Lambda }$CDM cosmology. According to the Jeffreys scale, the resulting Bayesian evidence is inconclusive, indicating that current observations neither favor nor disfavor these extensions with respect to the standard cosmological model. Full article

The original Bessel differential equation that describes, among many others cylindrical acoustic or vortical waves, is a particular case of zero degree of the generalised Bessel differential equation that describes coupled acoustic–vortical waves. The solutions of the generalised Bessel differential equation can be obtained for all possible combinations of complex variable, order and degree by three alternative methods: (i) convergent power series of Frobenius–Fuchs type around the regular singularity at the origin; (ii) asymptotic expansions of Thomé normal integral type around the irregular singularity at infinity; and (iii) Laplace transform along suitable paths in the complex plane which are the focus of the present paper. This leads to the generalised Bessel, Neumann and Hankel functions of two kinds, for which are obtained: (i) power series; (ii) asymptotic expansions; and (iii/iv) representations as definite and contour integrals. For the generalised Bessel functions are obtained four integral representations: (i) as two definite integrals along the unit interval with branch-points at both ends; (ii) as integrals along tear-drop loops surrounding one branch-point each; and (iii) as an integral along a Pochhammer double-laced loop around both branch-points. For the modified generalised Neumann function are obtained two integral representations: (i) along the negative real axis joining infinity to the branch-point at origin and (ii) along a Hankel-type semi-infinite loop around the single branch-point. The generalised Hankel functions of two kinds are linear combinations of generalised Bessel and Neumann functions, and are used to describe cylindrical acoustic or vortical waves and their coupling. The acoustic–vortical waves in a cylindrical duct are illustrated as plots of generalised Bessel functions with different orders and degrees. The original Bessel functions describe purely acoustic or vortical cylindrical waves that, for large radius, are asymptotically decaying and oscillating, hence being stable. The generalised Bessel functions describe coupled acoustic–vortical waves that are asymptotically monotonic and unstable. This provides a physical interpretation for the different mathematical properties of the original and modified Bessel functions. Full article

Numerous systems in electrical engineering, biology, and mechanical structures can be modeled using dynamical systems theory. This paper examines the behavior of a 3D dynamical system with six parameters, specifically its damped or periodic oscillations and asymptotic properties as functions of six physical parameters. The system is integrated explicitly through a smooth solution of a third order nonlinear differential equation, yielding exact parametric expressions that describe a heteroclinic orbit. To analyze parameter influence, we apply the Multistage Optimal Homotopy Perturbation Method (MOHPM). Its main advantage is the small number of iterations required, due to the effective choice of auxiliary convergence control functions. The MOHPM solutions agree closely with numerical results, demonstrated qualitatively through figures and quantitatively through tables. Accuracy is further assessed by comparison with the Optimal Homotopy Perturbation Method (OHPM). A qualitative analysis of errors is also provided. Full article

The problem of recovering the relaxation spectrum of the process described by the stretched exponential Kohlrausch–Williams–Watts (KWW) model is considered using the Post–Widder Laplace transform inversion rule. Based on the specific properties of the stretched exponential function, an analytical formula was derived that describes an arbitrarily high-order Post–Widder approximation of the spectrum directly in terms of the relaxation modulus, without any—neither analytical nor numerical—differentiation of the modulus as a product of finite power series of the relaxation times and the relaxation modulus. An alternative recurrence formula defining a sequence of the Post–Widder approximate models of the relaxation spectrum was developed. The positive definiteness, multiple differentiability, and zero asymptotic properties of the spectrum model are demonstrated; its extreme properties are discussed, and a sensitivity analysis with respect to the model parameters is conducted. The developed algorithm ensures fast convergence of the generated model sequence by applying a simple adaptive rule to select subsequent model orders, which relates them to the discrepancy between successive models and results in high-order models in at most a dozen iterations. Detailed numerical studies performed for nine values of the stretching exponent for KWW spectra covering relaxation times from 3 to 41 decades show that the proposed approach allows for generating an excellent approximation of the KWW spectrum by using only the relaxation modulus values in simple algebraic calculations. Full article

With the implementation of the Western Development Strategy and the Belt and Road Initiative, engineering activities on the Loess Plateau have expanded substantially in both scale and depth. Consequently, the disintegration of Lishi loess, which has received relatively limited attention, has become an increasingly important concern in relation to geological hazards and engineering stability. This study investigated the disintegration behavior of remolded Lishi loess specimens with different initial water contents under controlled dry-density conditions and developed a mathematical model to characterize the disintegration process. In addition, three-dimensional particle flow code (PFC3D) simulations were performed to provide a particle-scale mechanical interpretation of the observed behavior. The experimental results showed that the disintegration curves of the specimens exhibited a typical asymmetric S shape at all tested initial water contents. The Gompertz model provided a compact empirical description of these curves, with coefficients of determination ranging from 0.993 to 0.999. In the model, α denotes the upper asymptote of the disintegration curve, λ characterizes the growth-rate behavior, and β represents the characteristic time associated with the inflection point. These parameters should be interpreted as empirical descriptors of the disintegration process rather than direct measures of the microscopic properties of loess. Although the model closely reproduced the experimental curves, further validation using independent datasets is required before it can be applied predictively to other specimens or test conditions. The PFC3D simulations provided an equivalent mesoscopic representation of contact-network weakening, bond breakage, and progressive particle detachment at different initial water contents. The simulated evolution qualitatively reproduced the progression from boundary-particle detachment to the gradual loss of specimen integrity and final particle accumulation. These findings characterize the water-content-dependent disintegration behavior of remolded Lishi loess and provide a preliminary basis for understanding the water sensitivity of disturbed or reworked Lishi loess in engineering applications. Full article

This paper extends the Theory of Functional Connections (TFC) to a new class of admissibility conditions, referred to as divergent constraints, which prescribe an unbounded asymptotic behavior of the solution at one or more specified locations. While the existing TFC literature has established a comprehensive treatment of pointwise, derivative, integral, and general linear-operator constraints, no systematic framework was available for constraints that force the solution to diverge at prescribed points. The main contribution of this work is a rigorous univariate formulation that simultaneously embeds an arbitrary number of regular linear constraints and divergent constraints into a single analytical functional, together with a proof of a decoupling property that allows the regular and divergent switching functions to be computed by two independent families of linear systems. A singularity-aware quadrature is also introduced to evaluate the integrals arising in the assembly of the divergent-switching-function systems to machine precision. The theoretical developments are supported by numerical experiments that verify the prescribed regular and divergent behavior for several representative constraint configurations. Full article

We present a systematic derivation of asymptotic expansions of nonequilibrium thermodynamics for chemical reaction networks (CRNs) based on singular perturbation theory. For a general reversible CRN with fast–slow kinetics, we obtain the first and second laws of thermodynamics for the asymptotic expansion model. We derive composite expansions of the enthalpy, entropy, entropy production rate, and relative entropy. The slow-varying outer parts of these thermodynamic quantities capture the long-time trend, while the fast-varying corrected inner parts decay to zero as the fast time variable tends to infinity. The convergence order of these quantities is determined by the local Lipschitz properties of the respective functions. The enthalpy retains the same convergence order as the kinetic variables, whereas the entropy, relative entropy, and entropy production rate involve logarithmic terms that cause their gradients to diverge when some concentrations approach zero, reducing their theoretical convergence order by one. The general theory is validated on the reversible Michaelis–Menten reaction, for which both leading-order and first-order matched asymptotic expansions are obtained analytically. Numerical simulations confirm the uniform accuracy of the composite thermodynamic approximations and further reveal that the entropy production rate converges with a higher order than theoretically predicted. The results demonstrate that the composite expansion provides a rigorous and physically consistent tool for analyzing energy and entropy balances in multiscale CRNs. Full article

A variable selection method is put forward for multivariate linear regression models which obey normality. This method hinges on minimizing a generalized $C_p$ ($G{C}_p$) criterion which is defined by adding a positive constant value (the product of $\alpha$ and the number of parameters in the mean structure) to the minimum value of the multivariate residual sum of squares. The paper seeks to clarify the sufficient condition for $\alpha$ to simultaneously satisfy asymptotically loss- and mean-efficient properties in an asymptotic framework such that the sample size always goes to ∞, but the dimension of the vector of response variables can be either fixed or infinite. Based on this, we propose an asymptotically loss- and mean-efficient $G{C}_p$ criterion by using $\alpha$, which satisfies the obtained sufficient condition even with high dimensionality of the vector of response variables. Full article

This paper proposes a construction of algebraically stabilizing blocks for quantized and finite-precision neural networks. The approach is based on linear transformations defined by integer-valued matrices satisfying a condition of the form $W^k=I+\mu D$, which specifies algebraically controlled $k$-step behavior and modular periodicity in the integer-valued setting. The proposed property is invariant under conjugation, allowing the stabilizing construction to be transferred across equivalent linear representations. The resulting module can be integrated locally into existing neural architectures without requiring the algebraic structure to be imposed globally. The theoretical results concern algebraic and modular stabilization of the linear block and do not constitute a general guarantee of classical spectral or asymptotic stability in real-valued space. The approach is evaluated experimentally under multi-component harmonic, impulsive, and noisy inputs in both floating-point and INT8-quantized settings. Across several experimental configurations, the proposed block reduces output energy, component-wise variation, selected amplitude-related measures, and finite-precision deviation relative to floating-point reference trajectories. Norm-matched control experiments further suggest that the observed effects are not attributable solely to a reduction in operator magnitude, but may also reflect structural properties of the algebraically constructed operator. The proposed construction is particularly relevant to neural systems operating under limited numerical precision, including FPGA-, ASIC-, and edge-oriented implementations. It provides a structural approach for incorporating formally specified algebraic properties into the design of neural-network architectures. Full article

In this paper, we introduce a new probability distribution defined on the unit interval $(0,1)$, called the unit extended exponential (UEE) distribution. The proposed model is obtained through a transformation of a random variable following the extended exponential distribution, leading to a flexible family able to describe skewed and boundary-concentrated data. The structural properties of this distribution are derived, including its density, cumulative distribution, survival, hazard rate and quantile function, along with the mode and moments. A characterization based on its hazard function is also presented. Parameter estimation is performed via maximum likelihood. Under standard regularity conditions, the asymptotic properties of the estimators are established. A Monte Carlo simulation study is carried out to assess the finite-sample performance of the estimators. Finally, the usefulness of the model is illustrated through two applications to real proportion data, where our proposal is compared with competing models such as the unit-Lindley and Beta distributions using information criteria. The results suggest that the proposed distribution provides a competitive and often superior fit. Full article

Molecular tunnel ionization sets the stage for the subsequent attosecond processes. Therefore, a comprehensive understanding of its dependence on various molecular properties, such as binding energy, wave function shape, dipole moment, and polarizability, is crucial for tailoring attosecond science concepts for scientific and technological applications. While the first three properties have been relatively well understood, the effects of polarizability have not been explored to the same extent in studies of molecular tunnel ionization. To address this gap, in this work we reveal the peak-shifting effect of permanent polarizability in tunnel ionization rates as the ionizing field varies. By analyzing a specially defined rate gradient, we determine the characteristic directions of the shift from the sign of the local slope of the gradient curve. Furthermore, we show how ionization into higher parabolic channels can transform minima of orientation-dependent rates into maxima. These findings demonstrate how the common assumption that the orientation dependence of tunneling rates resembles orbital shape can break down. We benchmark the method employed, the weak-field asymptotic theory (WFAT), and demonstrate good agreement with ab initio results, underscoring the efficiency and accuracy of WFAT in simulating strong-field ionization at low-to-moderate field strengths. Full article

This paper investigates statistical inference for the Shannon entropy of the Transmuted Weibull Distribution under progressively Type-II censored samples. The Transmuted Weibull Distribution is obtained by applying the quadratic rank transmutation map to the cumulative distribution function of the two-parameter Weibull distribution, thereby substantially enhancing its modeling flexibility while preserving the analytical tractability of the baseline distribution. Consequently, it provides greater flexibility for modeling lifetime data exhibiting pronounced skewness and complex hazard rate behaviors. First, a closed-form expression for the Shannon entropy of the Transmuted Weibull Distribution is derived. From a frequentist perspective, the maximum likelihood estimators of the model parameters are obtained numerically using the Newton–Raphson algorithm, and the corresponding maximum likelihood estimator of Shannon entropy is derived through the invariance property of maximum likelihood estimation. To quantify estimation uncertainty, asymptotic confidence intervals are constructed using the Delta method together with the observed Fisher information matrix, while Bootstrap confidence intervals are also developed to improve finite-sample inference. From a Bayesian perspective, posterior inference is conducted using a hybrid Gibbs sampling algorithm within the Markov chain Monte Carlo framework. Bayesian point estimators of Shannon entropy are obtained under the squared error loss function, the absolute error loss function, and the 0–1 loss function, corresponding to the posterior mean, posterior median, and posterior mode, respectively. In addition, highest posterior density credible intervals are constructed for the Shannon entropy. The proposed methods are evaluated through an extensive Monte Carlo simulation study under three representative progressively Type-II censoring schemes. Estimation performance is assessed in terms of bias, mean squared error, interval coverage probability, and average interval length. The simulation results demonstrate that the Bayesian estimators consistently outperform the maximum likelihood estimator, particularly for small sample sizes and heavy censoring, while the highest posterior density credible intervals achieve more accurate coverage probabilities and shorter interval lengths. Finally, the proposed inferential procedures are illustrated using a real dataset consisting of remission times from 128 bladder cancer patients, demonstrating their practical applicability and robustness. Full article