This work introduces a novel characterization of the free Poisson (FP) law within the Cauchy-Stieltjes kernel (CSK) families framework, based on Boolean additive convolution and measure dilation. This approach highlights the FP law's invariance and structural stability under these operations and provides new insights into the analytic properties of its associated CSK family.

In this paper, we investigate the practical exponential stability of a class of nonlinear systems governed by the tempered Pi-Caputo fractional derivative. A new Lyapunov-based criterion is established to derive sufficient conditions ensuring Pi-practical exponential stability. The obtained result is formulated in a general framework involving suitable growth bounds on the Lyapunov function together with a tempered fractional derivative inequality and a boundedness condition on a weighted integral term. The proposed theorem provides an explicit practical exponential estimate for the system trajectories and extends existing stability results that are available for standard fractional and tempered fractional systems. To demonstrate the applicability of the developed theory, two applications are presented. First, the general criterion is applied to a class of perturbed tempered Pi-fractional systems, for which verifiable sufficient conditions are derived in terms of quadratic Lyapunov functions and perturbation bounds. Second, a state-feedback stabilization result is established for a class of nonlinear tempered fractional control systems, showing that the proposed theorem can be used as an effective tool for closed-loop practical exponential stabilization. Finally, numerical examples are provided to validate the theoretical developments and to illustrate the effectiveness of the proposed approach. An additional test case with eta(3 )> 0 is included to demonstrate the nontrivial range of Theorem 1. Furthermore, a socio-economic tempered fractional cobweb model is incorporated to show how the proposed criterion applies to price-adjustment dynamics with memory and persistent market perturbations.

In this paper, we establish a novel weakly singular integral inequality of Wendroff type. Using a combination of analytical and fractional calculus techniques, we derive sufficient conditions for the validity of this inequality. As a key application, we investigate the existence, uniqueness, and Ulam-Hyers stability of solutions to a class of nonlinear partial fractional differential equations. Our approach not only generalizes previous results but also provides a unified framework for analyzing fractional-order systems with singular kernels. An illustrative example is presented to demonstrate the applicability and effectiveness of the proposed method.

Accuracy in fractional numerical integration is often limited by the regularity of the integrand. This work proposes a flexible error estimation framework for proportional Caputo-hybrid integral operators based on s-convexity. We introduce a parametric Newton–Cotes formula (ν∈[0,1]) that bridges the gap between classical quadrature rules, recovering the fractional Trapezoidal, Midpoint, and Simpson’s methods as specific instances. In order to confirm the correctness of our results, we provide an illustrative example with graphical representations. Furthermore, we provide some additional results using Hölder’s and power mean inequalities and employ a verification strategy based on an Artificial Neural Networks (ANNs) model. The ANN approach allows for high-dimensional parameter space exploration, demonstrating that the proposed inequalities provide robust and precise error estimates.

This study establishes novel sufficient conditions to guarantee the well-posedness and Hyers-Ulam stability of solutions for a class of nonlinear ψ -fractional integro-differential equations subject to functional boundary constraints. By employing fixed-point theorems (Banach’s contraction principle, and Leray-Schauder’s theorem), multivariate Mittag-Leffler functions, and Babenko’s approach, we derive rigorous analytical criteria for the solvability and stability of the proposed equation. Furthermore, we investigate Hyers–Ulam stability to quantify the robustness of solutions under perturbations. Two examples are presented to validate the theoretical findings. This study contributes to the broader understanding of ψ -fractional integro-differential systems by introducing a flexible framework for analyzing nonlinear dynamics under functional constraints.

The objective of this paper is to demonstrate the existence and uniqueness (EU) of solutions to a class of Fractional Integro-Stochastic Differential Equations (FISDEs) by utilizing the fixed-point technique (FPT) and stochastic techniques. Additionally, the paper proves the continuous dependence (CD) of solutions on the initial data. We examine the Hyers–Ulam stability (HUS) of FISDEs by applying Gronwall inequalities. Two theoretical examples are presented to demonstrate our findings.

The estimate of domain mean is a significant issue in sample surveys. However, if the data is missing, it becomes very necessary. In the case of missing data, this paper proposes some direct and synthetic domain mean estimators using simple random sampling. To evaluate the performance of the suggested estimators against existing estimators, the algebraic formula of mean square errors is deduced. Additionally, a thorough, extensive simulation study was conducted utilizing a normally distributed population. Certain applications that contain actual data are also made available. The results of the simulation show the superiority of the suggested direct and synthetic Searls power ratio imputation approaches over the direct and synthetic mean imputation approaches, direct and synthetic ratio imputation approaches, and direct and synthetic power ratio imputation approaches by minimum mean square error and maximum percent relative efficiency. Furthermore, the proposed direct and synthetic imputation approaches are demonstrated using a real data based on the crop production from Agra district, located in the Indian state of Uttar Pradesh.

We propose the neutrosophic Poisson moment exponential distribution (NPMExD) as an extension of the Poisson moment exponential distribution (PMExD) originally developed by Ahsan-ul Haq. We detail the application of neutrosophic logic to the PMExD framework, enhancing its capability to handle uncertainty and indeterminacy. The study explores various statistical and mathematical properties of the NPMExD, including the survival function, moment generating function, hazard rate function, order statistics distribution, cumulative hazard function, index of dispersion, and related measures. Parameter estimation is performed using the maximum likelihood estimation method, followed by a comprehensive simulation study to assess the estimator performance. Finally, the practical efficacy of the proposed distribution is demonstrated through the analysis of two real-world data sets: remission times (in weeks) for 20 leukemia patients and 59 months of actual tax revenue (monthly) data from Egypt. The results indicate that the NPMExD provides a superior fit compared to the neutrosophic discrete Ramos-Louzada distribution for these data sets.

In this paper, we introduce a property of the inverse Semicircle and the free Gamma laws based on the dilation of measures in the context of Cauchy-Stieltjes Kernel (CSK) families. Assume that the CSK family produced by a non-degenerate probability measure lambda on Rwith support limited from above is F+ (lambda ) = {Q(m)(lambda)(dy): m is an element of (m(1)(lambda), m(+)(lambda))}. For alpha is an element of R\0, H-alpha, x bar right arrow alpha x and provide the set of measures H alpha(F+ (lambda)) = {H-alpha (Q(m)(lambda) (dy)) : m is an element of (m(1)(lambda), m(+)(lambda))}. Let us say that alpha > 0. We demonstrate that if H-alpha (F+ (lambda)) is a re-parametrization of F+ (lambda) (i.e., H-alpha (F+(lambda)) = F+(lambda)), then lambda is either the free Gamma type law or the inverse Semicircle type law up to scaling.

This paper establishes the existence and uniqueness (EU) of solutions for a class of pantograph fractional integro-stochastic differential equations (PFISDEs) via the Banach fixed point theorem (BFPT). Furthermore, the Ulam-Hyers stability (UHS) is analyzed using Gronwall's inequality. The applicability of the theoretical findings is demonstrated through two illustrative examples.

This paper present a new characterization of the Wigner's semicircle law. Denote by P (respectively by P-ba) the set of non-degenerate real probabilities (respectively, with one sided support boundary from above). For v is an element of P and a is an element of R, consider the transformation of measure v, denoted T-a (v), defined by F-Ta(nu)(w)=F-nu(w-a)+a, where F-nu(-) is the inverse of the Cauchy-Stieltjes transformation of v. On the other hand, let F+(mu)= { Q(1)(mu)(dx): 1 is an element of(m(0)(mu),m(+)(mu)) be the (CSK) family induced by mu is an element of P-ba with finite first moment. Define a novel family of probabilities T(F+()) = {()() : is an element of ((0),(+))} For a not equal 0, we prove that (F+()) =F+(()),(with() =+), if and only if mu is of the Wigner's type measure up to affinity.

To define useful numbers having values from 0 to 1, as percentages, probabilities, and proportions, the unit probability models are often employed in probability and statistics. In this paper, a new two-parameter model on the unit interval, entitled the unit Mirra distribution (UMD) is suggested which is a modification of the Mirra distribution (MD). Various statistical properties of the proposed UMD as the rth moment, variance, inverse moments, and incomplete moments are presented. Also, the distributions of order statistics, coefficients of variation and skewness, stochastics ordering, and median and mean deviations are provided. The reliability functions such the hazard function, survival function, odds function, reversed hazard rate function, and the mean residual function are provided with supported graphical representation. The Lorenz curve, Gini index, and Bonferroni curve of the UMD with some simulations are included. The distribution parameters are estimated based on the maximum likelihood estimation, ordinary least squares estimation, Anderson-Darling and maximum product of spacing’s estimation. Finally, two different applications of actual data sets that represent the times to infection of kidney dialysis patients in months and the trade share data on unit interval are carried out to evaluate the model's performance as compared to some competitors. It turns out that the suggested UMD can serve as a good alternative for modeling these data sets.

Sustainable construction and demolition waste management relies heavily on the attitudes and actions of its constituents; nevertheless, deep analysis for introducing the best estimator is rarely attained. The main objective of this study is to perform a comparison analysis among different approaches of Structural Equation Modeling (SEM) in Construction and Demolition Waste Management (C&DWM) modeling based on an Extended Theory of Planned Behaviour (Extended TPB). The introduced research model includes twelve latent variables, six independent variables, one mediator, three control variables, and one dependent variable. Maximum likelihood (ML), partial least square (PLS), and Bayesian estimators were considered in this study. The output of SEM with the Bayesian estimator was 85.8%, and among effectiveness of six main variables on C&DWM Behavioral (Depenmalaydent variables), five of them have significant relations. Meanwhile, the variation based on SEM with ML estimator was equal to 78.2%, and four correlations with dependent variable have significant relationship. At the conclusion, the R-square of SEM with the PLS estimator was equivalent to 73.4% and three correlations with the dependent variable had significant relationships. At the same time, the values of the three statistical indices include root mean square error (RMSE), mean absolute percentage error (MPE), and mean absolute error (MSE) with involving Bayesian estimator are lower than both ML and PLS estimators. Therefore, compared to both PLS and ML, the predicted values of the Bayesian estimator are closer to the observed values. The lower values of MPE, RMSE, and MSE and the higher values of R-square will generate better goodness of fit for SEM with a Bayesian estimator. Moreover, the SEM with a Bayesian estimator revealed better data fit than both the PLS and ML estimators. The pattern shows that the relationship between research variables can change with different estimators. Hence, researchers using the SEM technique must carefully consider the primary estimator for their data analysis. The precaution is necessary because higher error means different regression coefficients in the research model.

A notion of generalized (two-parameterized) t-transformation of free convolution, also called (t=(a,b))-deformed free convolution, is introduced for a∈R and b>0. In this article, some results of t-deformed free convolution are given within the theory of Cauchy-Stieltjes Kernel (CSK) families. The variance function is a fundamental concept in CSK families. An expression is provided for the variance function under t-deformed free convolution power. In addition, through the use of the variance function, an approximation is provided for members of the t-deformed free Gaussian CSK family and members of the t-deformed free Poisson CSK family respectively. Furthermore, by involving the free multiplicative convolution, a new limit theorem is provided with respect to t-deformed free convolution.

In free probability, the theory of Cauchy–Stieltjes Kernel (CSK) families has recently been introduced. This theory is about a set of probability measures defined using the Cauchy kernel similarly to natural exponential families in classical probability that are defined by means of the exponential kernel. Within the context of CSK families, this article presents certain features of the Marchenko–Pastur law based on the Fermi convolution and the t-deformed free convolution. The Marchenko–Pastur law holds significant theoretical and practical implications in various fields, particularly in the analysis of random matrices and their applications in statistics, signal processing, and machine learning. In the specific context of CSK families, our study of the Marchenko–Pastur law is summarized as follows: Let K+(μ)={Qmμ(dx);m∈(m0μ,m+μ)} be the CSK family generated by a non-degenerate probability measure μ with support bounded from above. Denote by Qmμ•s the Fermi convolution power of order s>0 of the measure Qmμ. We prove that if Qmμ•s∈K+(μ), then μ is of the Marchenko–Pastur type law. The same result is obtained if we replace the Fermi convolution • with the t-deformed free convolution t.

In this article, we provide some new limiting laws related to the free multiplicative law of large numbers and involving free and Boolean additive convolutions. Some examples of these limiting laws are presented within the framework of non-commutative probability theory.

Several authors are interested in the study of limiting distributions for large symmetrical random matrices. Our approach of studying limiting distributions is different and is related to the new concept of variance functions of Cauchy–Stieltjes Kernel (CSK) families of probabilities. By means of the variance functions machinery, some novel limiting probability measures are provided involving the free multiplicative law of large numbers, using both free and Boolean (additive and multiplicative) convolutions. Several examples of these limiting probability measures are given in the context of free probability.

Let F+(sigma) = { Q sigma m ( dy ) ; m E ( m sigma 1 ,m sigma+)} be the Cauchy-Stieltjes Kernel (CSK) family induced by a (non-degenerate) probability measure sigma possessing a one sided support boundary from above. For tau >= 0, denote by U e tau the tau-deformation of measures introduced in [6, Section 5]. In this article, a property is presented for the CSK families based on the stability under tau-deformation of measures. A CSK family satisfying such stability property is nothing but the CSK family generated by the Semicircle type law, up to affinity.

Let F(νj)={Qmjνj,mj∈(m−νj,m+νj)}, j=1,2, be two Cauchy–Stieltjes Kernel (CSK) families induced by non-degenerate compactly supported probability measures ν1 and ν2. Introduce the set of measures F=F(ν1)⊞F(ν2)={Qm1ν1⊞Qm2ν2,m1∈(m−ν1,m+ν1)andm2∈(m−ν2,m+ν2)}. We show that if F remains a CSK family, (i.e., F=F(μ) where μ is a non-degenerate compactly supported measure), then the measures μ, ν1 and ν2 are of the Marchenko–Pastur type measure up to affinity. A similar conclusion is obtained if we substitute (in the definition of F) the additive free convolution ⊞ by the additive Boolean convolution ⊎. The cases where the additive free convolution ⊞ is replaced (in the definition of F) by the multiplicative free convolution ⊠ or the multiplicative Boolean convolution ⨃ are also studied.

Let K+(μi)={Qsiμi,si∈(m0μi,m+μi)}, i=1,2, be two CSK families generated by the nondegenerate probability measures μ1 and μ2 with support bounded from above. Define the set of measures L=K+(μ1)•K+(μ2)={Qs1μ1•Qs2μ2,s1∈(m0μ1,m+μ1)ands2∈(m0μ2,m+μ2)}, where Qs1μ1•Qs2μ2 denotes the Fermi convolution of Qs1μ1 and Qs2μ2. We prove that if L is still a CSK family (that is, L=K+(σ) for some nondegenerate probability measure ()σ), then the probability measures σ, μ1 and μ2 are of the free Poisson type and follow the free Poisson law up to affinity. The same result, regarding the free Poisson measure, is obtained if we consider the t-deformed free convolution t replacing the Fermi convolution • in the family of measures L.