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The transition density of the diffusion process plays an important role in calculating the dynamic characteristics of the underlying variables as well as the model estimation. In this article, we combine the idea of conditional probability density function with non parametric kernel regression, and introduce the kernel estimation of joint density function and marginal density function, then construct the non parametric kernel estimator of the transition density of second-order diffusion process based on discrete observational samples. In order to obtain the asymptotic properties of the new kernel estimator, we analyze the asymptotic expectation and asymptotic variance of the proposed estimator under some mild conditions. Finally, the consistency and asymptotic normality of the new proposed non parametric estimator of the transition density function are proved. [ABSTRACT FROM AUTHOR]

AbstractIn this article, we study the parameter estimation problem of the continuous-time continuous-state stochastic SIR model in the context of discrete-time observations. Specifically, we propose estimators of the efficient contact rate and recovery rate based on weighted conditional least squares estimators (WCLSEs) and establish the relevant asymptotic distribution of the corresponding estimators under appropriate moment conditions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce three new inertial-like Bregman projection methods with a nonmonotone adaptive step-size for solving quasi-monotone variational inequalities in real Hilbert spaces. Under some suitable conditions, the weak convergence of these methods is proved without the prior knowledge of the Lipschitz constant of the operator and the strong convergence of some proposed methods under a strong quasi-monotonicity assumption of the mapping is also provided. Finally, several numerical experiments and applications in image restoration problems are provided to illustrate the performance of the proposed methods. [ABSTRACT FROM AUTHOR]

Our study in this paper is focused on the split equality fixed-point problem with firmly quasi-non-expansive operators in infinite-dimensional Hilbert spaces. A self-adaptive simultaneous scheme is introduced, and its weak convergence is established under mild and standard assumptions. The new proposed scheme generalizes and extends some related works in the literature, and its simple structure makes it easy for implementation and numerical testing. Primary experiments presented in this paper, in finite- and infinite-dimensional spaces, emphasize their practical advantages over existing results. [ABSTRACT FROM AUTHOR]

In this work, we consider the stochastic Cahn-Hilliard/Allen-Cahn equation with fractional noise, which is fractional in time and white in space. We obtain the existence, uniqueness, and Hölder regularity of the solution. Also, using a weak convergence approach, we prove that the law of solution associated with the above equation with a small perturbation satisfies the large deviation principle in the Hölder norm. [ABSTRACT FROM AUTHOR]

The projected reflected gradient method has been shown to be a simple and elegant method for solving variational inequalities. The method involves one projection onto the feasible set and one evaluation of the cost operator per iteration and has been shown numerically to be more efficient than most available methods for solving variational inequalities. Convergence results for methods with similar elegant structures of projected reflected gradient method are still rare. In this paper, we present weak and linear convergence of a projected reflected gradient method with an inertial extrapolation step and give some applications arising from optimal control problems. We first obtain weak convergence result for the projected reflected gradient method with an inertial extrapolation step for solving variational inequalities under standard assumptions with self-adaptive step sizes. We further obtain a linear convergence rate when the cost operator is strongly monotone and Lipschitz continuous. Finally, we give some numerical applications arising from optimal control. Preliminary results show that our method is effective and efficient when compared to other related state-of-the-art methods in the literature and show the advantage gained by incorporating inertial terms into the projected reflected gradient methods. [ABSTRACT FROM AUTHOR]

We consider the Ginzburg–Landau equation in the perforated domain, with rapidly oscillating coefficients. We derive the homogenized Ginzburg–Landau equation with a 'strange term' (potential) and prove that the trajectory attractors of the given equation tend in a weak sense to the trajectory attractors of the homogenized one. Assuming additional conditions to be satisfied for the coefficients, we provide also a convergence of the global attractor. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an inertial accelerated algorithm for solving the split equality common fixed-point problem of directed operators in real Hilbert space. Our algorithm includes the simultaneous iterative algorithm as special case which has been proposed by Moudafi and Al-Shemas for solving the split equality common fixed-point problem. We establish a weak convergence theorem for the proposed iterative algorithm, which combines the primal-dual method and the inertial technique. In our algorithm, the step sizes are chosen self-adaptively so that the implementation of the algorithm does not need any prior information about bounded linear operator norms. The efficiency of the proposed algorithm is illustrated by some numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two modified Tseng's extragradient algorithms with a new generalized adaptive stepsize for solving monotone variational inequalities (VI) in reflexive Banach spaces. The advantage of our methods is that stepsizes do not require prior knowledge of the Lipschitz constant of the cost mapping. Based on Bregman projection-type methods, we prove weak and strong convergence of the proposed algorithms to a solution of VI. Some numerical experiments to show the efficiency of our methods including a comparison with related methods are provided. [ABSTRACT FROM AUTHOR]

We propose an extension of the Cox-Ross-Rubinstein (CRR) model based on q-binomial (or Kemp) random walks, with application to default with logistic failure rates. This model allows us to consider time-dependent switching probabilities varying according to a trend parameter on a non-self-similar binomial tree. In particular, it includes tilt and stretch parameters that control increment sizes. Option pricing formulas are written using q-binomial coefficients, and we study the convergence of this model to a Black-Scholes type formula in continuous time. A convergence rate of order O(N-1/2) is obtained. [ABSTRACT FROM AUTHOR]

To address the split best proximity point and monotone variational inclusion problems in real Hilbert spaces, we present and investigate projection and viscosity approximation methods. Under a few reasonable assumptions, we prove some weak and strong convergence theorems for the aforementioned methods. The efficiency of the proposed method is demonstrated by some numerical examples. Some well-known recent results in this area have been improved, generalized, and extended as an outcome of this paper. [ABSTRACT FROM AUTHOR]

In this paper, we study the weak convergence and convergence rate in the weak law of large numbers for weighted sums of a class of random variables satisfying the Rosenthal type inequality. The necessary and sufficient conditions for the convergence rates in the weak law of large numbers under some mild conditions are provided. Moreover, the main results that we established are applied to simple linear errors-in-variables regression models and nonparametric regression models based on a class of random errors. Finally, we present some numerical simulations to assess the finite sample performance of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new inertial relaxed CQ algorithm for solving split feasibility problems in Hilbert spaces. The main advantages of the proposed algorithm are that the introduced stepsize is bounded away from zero and relaxation parameter sequences imposed on the stepsize itself are removed. These features help accelerate our method. Also, a weak convergence theorem for our method is established without Lipschitz continuous and firmly-nonexpasive conditions of split feasibility problem related gradient and projection mappings, respectively. As an application, we consider multiple-sets split feasiblity problems. Finally, some preliminary numerical experiments are provided for illustration and comparison. [ABSTRACT FROM AUTHOR]

In this paper, weak convergence of balanced stochastic one-step methods and especially balanced stochastic Runge–Kutta (SRK) methods for Itô multidimensional stochastic differential equations is analyzed. Generalizing a corresponding result obtained by H. Schurz for the standard Euler method, it is shown that under certain conditions, balanced one-step methods preserve the weak convergence properties of their underlying methods. As an application, this allows to prove the weak convergence order of the balanced SRK methods presented in earlier work by A. Rathinasamy, P. Nair and D. Ahmadian. [ABSTRACT FROM AUTHOR]

In this work, we establish two new self-adaptive parallel algorithms to solve the generalized split common fixed point problem which is to find a point which belongs to the intersection of finite family of fixed point sets of demimetric mappings such that its image under a finite number of linear transformations belongs to the intersection of another finite family of fixed point sets of demimetric mappings in the image space. Under suitable assumptions, the weak and strong convergence theorems are analysed. The obtained results generalize and improve the recent results announced by many other authors in the framework of split inverse problem. As a direct consequence of our two main algorithms, we obtain several new algorithms. Preliminary numerical experiments are provided to illustrate the efficiency and implementation of our new methods and also to compare with others. [ABSTRACT FROM AUTHOR]

A nested occupancy scheme in a random environment is a generalization of the classical Karlin infinite balls-in-boxes occupancy scheme in a random environment (with random probabilities). Unlike the Karlin scheme in which the collection of boxes is unique, there is a nested hierarchy of boxes, and the hitting probabilities of boxes are defined in terms of iterated fragmentation of a unit mass. In the present paper, we assume that the random fragmentation law is given by stick-breaking in which case the infinite occupancy scheme defined by the first level boxes is known as the Bernoulli sieve. Assuming that n balls have been thrown, denote by K n (j) the number of occupied boxes in the jth level and call the level j intermediate if j = j n → ∞ and j n = o (log ⁡ n) as n → ∞. We prove a multidimensional central limit theorem for the vector (K n (⌊ j n u 1 ⌋) , ... , K n (⌊ j n u ℓ ⌋) , properly normalized and centred, as n → ∞ , where j n → ∞ and j n = o ((log ⁡ n) 1 / 2). The present paper continues the line of investigation initiated in the article [D. Buraczewski, B. Dovgay, and A. Iksanov, On intermediate levels of nested occupancy scheme in random environment generated by stick-breaking I, Electron. J. Probab. 25(123) (2020), pp. 1–24] in which the occupancy of intermediate levels j n → ∞ , j n = o ((log ⁡ n) 1 / 3) was analysed. [ABSTRACT FROM AUTHOR]

In this article, we propose a mathematical model that describes hydrodynamics and deformation mechanics within a solid tumor which is embedded in or adjacent to a healthy (normal) tissue. The tumor and normal tissue regions are assumed to be deformable and the theory of mixtures is adapted to mass and momentum balance equations for fluid flow and tissue deformation mechanics in each region. The momentum balance equations are coupled via forces that interact between the phases (fluid and solid). Continuity of normal velocities, displacements, and normal stresses along with the Beaver–Joseph–Saffman condition are imposed at the interface between the tumor and tissue regions. The physiological transport parameters (such as hydraulic resistivity or permeability) are assumed to be heterogeneous and deformation dependent which makes the model nonlinear. We establish the existence of a weak solution using Galerkin and weak convergence methods. We show further that the solution is unique and depends continuously on the given data. [ABSTRACT FROM AUTHOR]

The goal of this paper is to study quasi-monotone variational inequality problems in infinite dimensional Hilbert spaces and to prove that the iterative sequence generated by an extragradient algorithm converges weakly to a solution. [ABSTRACT FROM AUTHOR]

In this paper, we consider reflected three-operator splitting methods for monotone inclusion problems in real Hilbert spaces. To do this, we first obtain weak convergence analysis and nonasymptotic O (1 / n) convergence rate of the reflected Krasnosel'skiĭ-Mann iteration for finding a fixed point of nonexpansive mapping in real Hilbert spaces under some seemingly easy to implement conditions on the iterative parameters. We then apply our results to three-operator splitting for the monotone inclusion problem and consequently obtain the corresponding convergence analysis. Furthermore, we derive reflected primal-dual algorithms for highly structured monotone inclusion problems. Some numerical implementations are drawn from splitting methods to support the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of ϕ -nonexpansive sequences and maps in a Banach space E, and prove an ergodic theorem, a fixed point theorem, and some weak and strong convergence theorems for such sequences and maps in E, with rather mild assumptions on the Banach space E, weaker than those assumed by previous authors for nonexpansive maps. These results give a partial answer to the still open problem raised in 1981 by Djafari Rouhani for nonexpansive sequences in a Banach space. [ABSTRACT FROM AUTHOR]

We consider a nonlinear filtering problem of multiscale non-Gaussian signal processes and observation processes with jumps. First, we prove that the dimension for the signal system can be reduced by a homogenized approach. Second, convergence of the corresponding nonlinear filtering to the homogenized filtering is shown by a weak convergence technique. Finally, we give an example to explain our result. [ABSTRACT FROM AUTHOR]

Exact and asymptotic distributional properties are discussed in detail for two mean-squared error consistent point predictors of future-generalized order statistics (GOSs) based on two-parameter exponential distribution. These predictors work even if some observed data were missing. For each point predictor, the asymptotic distribution of the normalized difference between the future GOS and its point predictor is derived, when the scale parameter is known or unknown. It is revealed that the asymptotic distributions of these normalized differences are equal when the scale parameter is known. Two asymptotic prediction intervals of the future GOS are constructed whenever the scale parameter is known or unknown. Furthermore, two tests of outliers are proposed relying on the point predictors. Finally, a simulation study is conducted and a real data set is analyzed for illustrative purposes. [ABSTRACT FROM AUTHOR]

Weak convergence of inertial iterative method for solving variational inequalities is the focus of this paper. The cost function is assumed to be non-Lipschitz and monotone. We propose a projection-type method with inertial terms and give weak convergence analysis under appropriate conditions. Some test results are performed and compared with relevant methods in the literature to show the efficiency and advantages given by our proposed methods. [ABSTRACT FROM AUTHOR]

We perform a numerical analysis of a class of randomly perturbed Hamiltonian systems and Poisson systems. For the considered additive noise perturbation of such systems, we show the long-time behaviour of the energy and quadratic Casimirs for the exact solution. We then propose and analyse a drift-preserving splitting scheme for such problems with the following properties: exact drift preservation of energy and quadratic Casimirs, mean-square order of convergence 1, weak order of convergence 2. These properties are illustrated with numerical experiments. [ABSTRACT FROM AUTHOR]

The class of balanced implicit methods (BIMs) has been introduced by Milstein, Platen and Schurz (1998) for numerical integration of d -dimensional systems of ordinary Itô-type stochastic differential equations (SDEs) driven by independent Wiener processes. They also established the rate r = 0.5 of mean square convergence towards exact solution along equidistant partitions under appropriate assumptions. The other major convergence criterion is that of weak convergence (i.e. the convergence of moments E [ f (X) ]) of them with respect to certain test classes of functions f and remains to be an open question. We fill that gap by proving weak convergence of a subclass of BIMs with rate r = 1.0 with respect to the test class of C 2 -functions and with variable step sizes. This class of stochastic-numerical approximations represents a linear-implicit family of schemes stabilizing p -th moments by a computationally efficient implementation. The novel proof relies on a random version of Clarkson-Beckner inequality to estimate its p -th moments. Moreover, we prove stability and boundedness of a more general class of these methods in any dimension d. [ABSTRACT FROM AUTHOR]

In this work, we study the partial sums of independent and identically distributed random variables with the number of terms following a fractional Poisson (FP) distribution. The FP sum contains the Poisson and geometric summations as particular cases. We show that the weak limit of the FP summation, when properly normalized, is a mixture between the normal and Mittag-Leffler distributions, which we call by Normal-Mittag-Leffler (NML) law. A parameter estimation procedure for the NML distribution is developed and the associated asymptotic distribution is derived. Simulations are run to check the performance of the proposed estimators under finite samples. An empirical illustration on the daily log-returns of the Brazilian stock exchange index (IBOVESPA) shows that the NML distribution captures better the tails than some of its competitors. Related problems such as a mixed Poisson representation for the FP law and the weak convergence for the Conway-Maxwell-Poisson random sum are also addressed. [ABSTRACT FROM AUTHOR]

We give a particle picture interpretation of two recently discovered classes of self-similar stable processes with stationary increments studied by Samorodnitsky et al. and Dombry and Guillotin-Plantard. We study the occupation times of certain Poissonian systems of particles with ±1 charges and i.i.d. heavy-tailed weights, moving independently according to Lévy processes. We also obtain a new class of self-similar stable processes. [ABSTRACT FROM AUTHOR]

A novel iterative algorithm is introduced for solving the split common fixed point problem with demicontractive operators. Weak and strong convergence of the algorithm is discussed. The step-sizes of the algorithm are chosen in such a way that the implementation of the algorithm does not depend on operator norms. The main results proven in this paper extend and improve some results of Yao et al. [Algorithms with strong convergence for the split common solution of the feasibility problem and fixed point problem. Fixed Point Theor Appl. 2014;2014:183], Boikanyo [A strongly convergent algorithm for the split common fixed point problem. Appl Math Comput. 2015;265:844–853] and Cui and Wang [Iterative methods for the split common fixed point problem in Hilbert spaces. Fixed Point Theor Appl. 2014;2014:78]. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two inertial accelerated algorithms for solving the split common fixed-point problem of directed operators in real Hilbert space. The proposed iterative algorithms combine the primal-dual method and the inertial method with the self-adaptive stepsizes such that the implementation of our algorithms does not need any prior information about bounded linear operator norms. Under suitable conditions, the weak and strong convergence results of the algorithms are obtained. Numerical results which involve image restoration problems are reported to show the effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]

Two new iterative algorithms are introduced for solving the split common null point problem (SCNPP). The first algorithm is shown to converge weakly and the second is shown to converge strongly to a solution of SCNPP in a Hilbert space. The algorithms are forward methods (i.e, involving no computation of the resolvent of a monotone operator). Another feature of the algorithms is that the selection of the step-sizes does not need prior knowledge of operator norms. Numerical experiments are presented to illustrate the performance of the algorithms. [ABSTRACT FROM AUTHOR]

A convergence rate of Mann iteration has been studied by many mathematicians. Recently, its rate was improved in a Hilbert space. In this paper, we consider a convergence rate in a geodesic space and generalize the known result. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the convergence analysis of an iterative algorithm with inertial extrapolation step for finding an approximate solution of split monotone inclusion problem in real Hilbert spaces. Weak convergence of the sequence of iterates generated from the proposed method is obtained under some mild assumptions. Some special cases of the general problem are given and we give some numerical implementations to support the theoretical analysis and give justification for the addition of the extrapolation step in the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new iterative algorithm from primal-dual methods for solving the split equality common fixed-point problem of quasi-nonexpansive mappings in real Hilbert space. Our algorithm includes the simultaneous iterative algorithm as special case which has been proposed by Moudafi and Al-Shemas for solving the split equality common fixed-point problem. We use a way of selecting the stepsizes such that the implementation of our algorithm does not need any prior information about bounded linear operator norms. It avoids the difficult task of estimating the operator norms. Under suitable conditions, we get the weak convergence of the proposed algorithm. The performance of the proposed algorithm is also illustrated by preliminary numerical experiments. The results presented in the paper improve and extend some corresponding results. [ABSTRACT FROM AUTHOR]

Nonreversible Markov chain Monte Carlo methods often outperform their reversible counterparts in terms of asymptotic variance of ergodic averages and mixing properties. Lifting the state-space is a generic technique for constructing such samplers. The idea is to think of the random variables we want to generate as position variables and to associate to them direction variables so as to design Markov chains which do not have the diffusive behavior often exhibited by reversible schemes. In this article, we explore the benefits of using such ideas in the context of Bayesian model choice for nested models, a class of models for which the model indicator variable is an ordinal random variable. By lifting this model indicator variable, we obtain nonreversible jump algorithms, a nonreversible version of the popular reversible jump algorithms. This simple algorithmic modification provides samplers which can empirically outperform their reversible counterparts at no extra computational cost. The code to reproduce all experiments is available online1. [ABSTRACT FROM AUTHOR]

In this paper, by using Girsanov's transformation and the property of the corresponding reference stochastic differential equations, we investigate weak existence and uniqueness of solutions and weak convergence of Euler-Maruyama scheme to stochastic functional differential equations with Hölder continuous drift driven by fractional Brownian motion with Hurst index H ∈ (1 / 2 , 1). [ABSTRACT FROM AUTHOR]

This paper proposes two algorithms that are based on a subgradient and an inertial scheme with the explicit iterative method for solving pseudomonotone equilibrium problems. The weak convergence of both algorithms is well-established under standard assumptions on the cost bifunction. The advantage of these algorithms is that they did not require any line search procedure or any knowledge about bifunction Lipschitz-type constants for step-size evaluation. A practical explanation of this is that they use a sequence of step-size that are revised at each iteration based on some previous iteration. For a numerical experiment, we consider a well-known Nash-Cournot equilibrium model of electricity markets and also other test problems to assist the well-established convergence results and be able to see that our proposed algorithms have a competitive advantage over the time of execution and the number of iterations. [ABSTRACT FROM AUTHOR]

In this article, we construct two families of processes, from a unique Lévy process, the finite dimensional distributions of which converge in law towards the finite dimensional distributions of the two independent Gaussian processes. As applications of this result, we obtain families of processes that converge in law towards fractional Brownian motion, sub-fractional Brownian motion and bifractional Brownian motion, respectively. [ABSTRACT FROM AUTHOR]

In this paper, extending the ideas given by (Deheuvels 1979) for the empirical copula, we first assume that we are sampling from the product copula, and we construct C m n the d-sample copula of order m, which is an estimator of C (m) the checkerboard approximation of order m, see (Li et al. 1997), then we study the distribution of the number of observations in each of the boxes generated by the regular partition of order m in [ 0 , 1 ] 2 , and we also give the corresponding moments and correlations. We also prove the weak convergence of the sample process m n ( C m m n − C (m) ) to a centered Gaussian process. In fact, we use the known results of the weak convergence for the empirical process, to prove the weak convergence of the sample process. We first show that the sample copula can be written as a linear functional of the empirical copula, and then we observe that this functional is Hadamard differentiable. Therefore, we can apply the delta method to get the weak convergence of the sample process. Finally, we performed several simulations of the sample process at a given point of [ 0 , 1 ] 2 , to study the sample size needed to get the convergence to a centered Gaussian process. [ABSTRACT FROM AUTHOR]

In this work, we investigate the asymptotic behaviour of the intermediate and central order statistics of a bivariate data by using the Reduced Ordering Principle (R-ordering). When, the sup-norm is used, we reveal the interrelation between the R-ordering principle and Marginal-Ordering Principle (M-ordering). The asymptotic behaviour of the intermediate and central bivariate order statistics based on sup-norms is completely determined. Besides, we highlight the role of the normalizing constants in characterizing the limit types of order statistics and new results concerning the intermediate order statistics are given. [ABSTRACT FROM AUTHOR]

Weak convergence and moment convergence issues are investigated for the New Better than Average Failure Rate (NBAFR) family (introduced by Loh (1984)). We explore the validity of these results in the context of a more general ageing class that we introduce. We prove some new properties of this class and derive its interrelationships with other non-monotonic ageing families. Reliability and moment bounds are obtained and an interesting characterization of exponentiality is proved. Special cases of our results lead to new theorems for the NBAFR class. Finally weak convergence and related issues are established for this class. [ABSTRACT FROM AUTHOR]

In this paper, an improvement of the extragradient algorithm is considered for finding a solution of pseudo-monotone equilibrium problems in real Hilbert spaces. We replace the second minimization problem onto a closed convex set in the extragradient method, with a subgradient projection onto some constructible half-space and propose a new method of getting the step-size. A convergence theorem for the generated sequence is proved and applied for finding a solution of variational inequality problem. [ABSTRACT FROM AUTHOR]

In this paper, we introduce lacunary weak statistical convergence of order α and weak N θ α -convergence for 0 < α ≤ 1 . We prove some properties of these modes of convergence and give some inclusion relations. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the exponential-type inequalities for maximal partial sums of negatively superadditive dependent (NSD) random variables, which extends the corresponding results for independent and negatively associated (NA) random variables. Using these inequalities, we further investigate the weak convergence of the M-estimators in the generalized linear model with NSD errors, which generalize and improve the corresponding results of the independent random errors to that of NSD random errors. [ABSTRACT FROM AUTHOR]

In medical and epidemiological studies, it is often interest to study time-to-event distributions under competing risks that involve two or more failure types. Nonparametric analysis of competing risks is typically focused on the cumulative incidence function or nonparametric quantile function. However, the existing estimators may be very unstable due to their unsmoothness. In this paper, we propose a kernel nonparametric quantile estimator for right-censored competing risks data, which is a smoothed version of Peng and Fine's nonparametric quantile estimator. We establish the Bahadur representation of the proposed estimator. The convergence rate of the remainder term for the proposed estimator is substantially faster than Peng and Fine's quantile estimator. The pointwise confidence intervals and simultaneous confidence bands of the quantile functions are also derived. Simulation studies illustrate the good performance of the proposed estimator. The methodology is demonstrated with two applications of the Supreme Court Judge data and AIDSSI data. [ABSTRACT FROM AUTHOR]

In this note we provide a systematic reasoning to arrive at the reflexivity of the underlying Banach space as a sufficient condition for guaranteeing that any compact operator transforms weak convergence in strong convergence. Our starting point is an adaptation of the proof for the analogue result holding in the case of the weak convergence. Then, along the way, and as a by-product of the analysis, we characterize the existence of what we call the inverse-adjoint operator. [ABSTRACT FROM AUTHOR]

It is shown that in general the popular coefficient alpha estimator for reliability of multi-component measuring instruments converges almost surely to a quantity that is not equal to the population reliability coefficient. This convergence with probability 1 is a stronger statement than convergence in probability (consistency) and convergence in distribution for the alpha estimator, which have been studied in the past. In the special case of congeneric measures with uncorrelated errors and equal loadings on the common true score, the alpha estimator converges almost surely to the population reliability coefficient that equals population alpha, which implies also its consistency as a reliability estimator. When the loadings are unequal but sufficiently high and similar, the alpha estimator converges almost surely to population alpha that is essentially indistinguishable from the population reliability coefficient, which implies alpha's approximate consistency then. For the general case, the results entail that the alpha estimator is not a consistent estimator of reliability. The findings add to the critical literature on coefficient alpha in the general case, as well as to the justification of its use as a dependable measuring instrument reliability estimator in special cases and settings resulting under appropriate restrictive conditions, and are illustrated using a numerical example. [ABSTRACT FROM AUTHOR]

We consider reaction-diffusion systems with random rapidly oscillating coefficient. We do not assume any Lipschitz condition for the nonlinear function in the system, so, the uniqueness theorem for the corresponding initial-value problem may not hold for the considered reaction-diffusion system. Under the assumption that the random function is ergodic and statistically homogeneous in space variables we prove that the trajectory attractors of these systems tend in a weak sense to the trajectory attractors of the homogenized reaction-diffusion systems whose coefficient is the average of the corresponding term of the original systems. [ABSTRACT FROM AUTHOR]

In this paper, we present two penalty-splitting inspired iteration schemes (PFFSA) and (PFFSA) for hierarchical equilibrium problems in Hilbert space. Based on the Opial-Passty lemma, we propose weak ergodic convergence and weak convergence of the iterative sequences generated by the Forward-Forward algorithm (PFFSA) and the Forward-Backward algorithm (PFFSA), which are proved under quite mild conditions: the bifunction of the two level equilibrium problems are supposed pseudomonotone. For strong convergence, we first add a strong monotonicity condition on the objective bifunction. We present after, a strong convergence result of algorithm (PFFSA) by adding a topological assumption, i.e. the objective bifunction is of class . Some examples are given to illustrate our results. The first example deals with pseudomonotone variational inequalities and convex minimization problem in the upper level problem. In the second one, we propose a convex minimization in the lower-level problem, where strong convergence of (PFFSA) to a minimum point is insured under infcompactness condition for objective function. These convergence results are new and generalize some recent results in this field. [ABSTRACT FROM AUTHOR]

In this paper, a hybrid proximal algorithm with inertial effect is introduced to solve a split variational inclusion problem in real Hilbert spaces. Under mild conditions on the parameters, we establish weak convergence results for the proposed algorithm. Unlike the earlier iterative methods, we do not impose any conditions on the sequence generated by the proposed algorithm. Also, we extend our results to find a common solution of a split variational inclusion problem and a fixed-point problem. Finally, some numerical examples are given to discuss the convergence and superiority of the proposed iterative methods. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a contraction algorithm for solving monotone variational inclusion problem. To reach this goal, our main iterative algorithm combine Dong’s projection and contraction algorithm with resolvent operator. Under suitable assumptions, we prove that the sequence generated by our main iterative algorithm converges weakly to the solution of the considered problem. Finally, we give two numerical examples to verify the feasibility of our main algorithm. [ABSTRACT FROM AUTHOR]