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We prove weak convergence of order one for a class of exponential based integrators for SDEs with non-globally Lipschitz drift. Our analysis covers tamed versions of Geometric Brownian Motion (GBM) based methods as well as the standard exponential schemes. The numerical performance of both the GBM and exponential tamed methods through four different multi-level Monte Carlo techniques are compared. We observe that for linear noise the standard exponential tamed method requires severe restrictions on the step size unlike the GBM tamed method. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new method for solving variational inclusion problems in Hilbert spaces. This algorithm uses two inertial terms to speed up the convergence. In order to avoid computing the Lipschitz stepsize, we use an updated stepsize which is not necessary to know the Lipschitz constant of the operator. The weak convergence is established under some mild conditions. We present numerical performance of the proposed algorithm and compare our algorithm with other algorithms in literature. Finally, we deduce our algorithm for solving the convex minimization problem and give an application to the data classification problem of heart failure dataset. [ABSTRACT FROM AUTHOR]

We introduce a novel class of asymptotically α-hemicontractive mappings and demonstrate its relationship with the existing related families of mappings. We establish certain interesting properties of the fixed point set of the new class of mappings. Furthermore, we propose and investigate a new iterative algorithm for solving split common fixed point problem for the new class of mappings. In particular, weak and strong convergence theorems for solving split common fixed point problem for our new class of mappings in Hilbert spaces are proved. Moreover, using our method, we require no prior knowledge of norm of the transfer operator. The results presented in the paper extend and improve the results of Censor and Segal [Censor, Y.; Segal, A. The split common fixed point problem for directed operators. J. Convex Anal. 16 (2009), no. 2, 587-600.], Moudafi [Moudafi, A. The split common fixed-point problem for demicontractive mappings. Inverse Problems 26 (2010), no. 5:055007.; Moudafi, A. A note on the split common fixed-point problem for quasi-nonexpansive operators. Nonlinear Anal. 74 (2011), no. 12, 4083-4087.], Chima and Osilike [Chima, E. E.; Osilike, M. O. Split common fixed point problem for class of asymptotically hemicontractive mappings. J. Nigerian Math. Soc. 38 (2019), no. 3, 363-390.], Fan et al [Fan, Q.; Peng, J.; He, H. Weak and strong convergence theorems for the split common fixed point problem with demicontractive operators. Optimization 70 (2021), no. 5-6, 1409-1423.] and host of other related results in literature. [ABSTRACT FROM AUTHOR]

This work is motivated by a gap in the functional data analysis literature, particularly in the context of neuroimaging, regarding the ability of functional models to robustly accommodate intra‐observation dependence. In response, we propose an M‐estimator based on generalized empirical likelihood for the varying‐coefficient model with a functional response. We develop statistical inference procedures, simultaneous confidence regions, and a global general linear hypothesis test for the model's functional coefficient. Our theoretical results establish the weak convergence of the log‐likelihood ratio process, a nonparametric version of Wilks' theorem for the log‐likelihood ratio, and asymptotic properties of the proposed estimator. Through a simulation study, we show that the proposed confidence sets have close‐to‐nominal coverage probabilities. In a real‐world application to a neuroimaging dataset, we show that mini‐mental state examination score and apolipoprotein E genotype have significant associations with fractional anisotropy, while associations with gender and age are only present at high quantile levels. [ABSTRACT FROM AUTHOR]

This manuscript proposes a generalization of weak convergence and study of fixed point in a real Hilbert space for quasi-nonexpanding maps. In this work, we introduce a class of fixed points theorems for nonexpansive mapping and generalized form of nonexpansive mapping under the Hilbert space. In addition, we obtained under quasi nonexpansive mapping weak convergence with respect to Hilbert space by Mann’s Type. [ABSTRACT FROM AUTHOR]

First-order optimization methods have long been established as effective methods for solving large-scale unconstrained optimization problems. Over time, these methods have been extended to address the task of finding zeros of single-valued monotone maps within the framework of finite-dimensional Hilbert spaces. Notably, Abubakar et al. (Comput Appl Math 39(2):129, 2020) recently introduced a projection method that incorporates a convex combination of two distinct positive spectral coefficients to find the zeros of a single-valued monotone map. Motivated by their work, this paper generalises the Abubakar et al. technique to a more general setting of infinite-dimensional Hilbert spaces. Under mild assumptions, we establish weak convergence of the generated sequence of iterates. Furthermore, the extended method does not involve computing the resolvent of the monotone operator. [ABSTRACT FROM AUTHOR]

This paper investigates numerical solutions of pseudomontone variational inequality problems in real Hilbert spaces. Two new algorithms based on the celebrated extragradient method are introduced. Under suitable conditions, it is proved that the two algorithms achieve weak convergence and linear convergence rates. The results extend and improve upon related works in the literature on the associated extragradient algorithms. [ABSTRACT FROM AUTHOR]

We consider a class of stationary processes exhibiting both long-range dependence and heavy tails. Separate limit theorems for sums and for extremes have been established recently in the literature with novel objects appearing in the limits. In this article, we establish the joint sum-and-max limit theorems for this class of processes. In the finite-variance case, the limit consists of two independent components: a fractional Brownian motion arising from the sum and a long-range dependent random sup measure arising from the maximum. In the infinite-variance case, we obtain in the limit two dependent components: a stable process and a random sup measure whose dependence structure is described through the local time and range of a stable subordinator. For establishing the limit theorem in the latter case, we also develop a joint convergence result for the local time and range of subordinators, which may be of independent interest. [ABSTRACT FROM AUTHOR]

A control reconstruction problem for dynamic deterministic affine-control systems is considered. This problem consists of constructing piecewise constant approximations of an unknown control generating an observed trajectory from discrete inaccurate measurements of this trajectory. It is assumed that the controls are constrained by known nonconvex geometric constraints. In this case, sliding modes may appear. To describe the impact of sliding modes on the dynamics of the system, the theory of generalized controls is used. The notion of normal control is introduced. It is a control that generates an observed trajectory and is defined uniquely. The aim of reconstruction is to find piecewise constant approximations of the normal control that satisfy given nonconvex geometric constraints. The convergence of approximations is understood in the sense of weak convergence in the space superscript 퐿 2 . A solution to the control reconstruction problem is proposed. [ABSTRACT FROM AUTHOR]

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]

In this paper, we introduce generalized difference weak sequence space classes by utilizing the difference operator Δ ı ȷ \Delta^{\jmath}_{\imath} and the de la Vallée–Poussin mean, denoted as [ ( V , λ ) w , Δ ı ȷ ] m [(\mathscr{V},\lambda)_{w},\Delta^{\jmath}_{\imath}]_{m} for m = 0 m=0 , 1, and ∞. Further, we explore some algebraic and topological properties of these spaces, including their nature as linear, normed, Banach, and BK spaces. Additionally, we examine properties such as solidity, symmetry, and monotonicity. Finally, we define and establish some inclusion relations among generalized difference weak statistical convergence, generalized difference weak 휆-statistical convergence, and generalized difference weak [ V , λ ] [\mathscr{V},\lambda] -convergence. [ABSTRACT FROM AUTHOR]

In this article, we introduced the idea of lacunary weak convergence of sequences defined by the class of Orlicz function. Various topological and algebraic properties, such as solidity, symmetry, and monotonicity, have been studied. To validate the failure instances, we gave appropriate examples and detailed discussions. Additionally, we have established some inclusion relationships between these spaces. [ABSTRACT FROM AUTHOR]

We consider a nonlinear, hyperbolic population balance equation that incorporates both aggregation and collisional breakage events simultaneously. Our approach revolves around the development of a novel time-explicit finite volume scheme. Under a suitable time-step stability condition, we prove the convergence of the approximate solution for any non-uniform mesh. A first-order convergence is obtained by a thorough error analysis of the proposed scheme for a suitable choice of kernels. Finally, we compute some numerical test examples to explore the behavior of the solution in steady-state conditions as well as the occurrence of gelation phenomena. [ABSTRACT FROM AUTHOR]

Let (K,d) be a connected compact metric space and p\in (1, \infty). Under the assumption of Kigami [ Conductive homogeneity of compact metric spaces and construction of p-energy , Memoirs of the European Mathematical Society, vol. 5, Europea Mathematical Society (EMS), Berline, 2023, Assumption 2.15] and the conductive p-homogeneity, we show that \mathcal {W}^p(K)\subset C(K) holds if and only if p>\operatorname {dim}_{AR}(K,d), where \mathcal {W}^p(K) is Kigami's (1,p)-Sobolev space and \operatorname {dim}_{AR}(K,d) is the Ahlfors regular dimension. [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]

By the Bagchi theorem, the Riemann hypothesis (all non-trivial zeros lie on the critical line) is equivalent to the self-approximation of the function ζ (s) by shifts ζ (s + i τ) . In this paper, it is determined that the Riemann hypothesis is equivalent to the positivity of density of the set of the above shifts approximating ζ (s) with all but at most countably many accuracies ε > 0 . Also, the analogue of an equivalent in terms of positive density in short intervals is discussed. [ABSTRACT FROM AUTHOR]

We have developed the generalized quasilinearization method (QM) for an initial value problem (IVP) of dynamic equations on time scales by using comparison theorems with a coupled lower solution (LS) and upper solution (US) of the natural type. Under some conditions, we observed that the solutions converged to the unique solution of the problem uniformly and monotonically, and the rate of convergence was investigated. [ABSTRACT FROM AUTHOR]

We propose a finite element discretization for the steady, generalized Navier-Stokes equations for fluids with shear-dependent viscosity, completed with inhomogeneous Dirichlet boundary conditions and an inhomogeneous divergence constraint. We establish (weak) convergence of discrete solutions as well as a priori error estimates for the velocity vector field and the scalar kinematic pressure. Numerical experiments complement the theoretical findings. [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]

In the paper, we prove a joint limit theorem in terms of the weak convergence of probability measures on C 2 defined by means of the Epstein ζ (s ; Q) and Hurwitz ζ (s , α) zeta-functions. The limit measure in the theorem is explicitly given. For this, some restrictions on the matrix Q and the parameter α are required. The theorem obtained extends and generalizes the Bohr-Jessen results characterising the asymptotic behaviour of the Riemann zeta-function. [ABSTRACT FROM AUTHOR]

The metric flow is introduced and extensively studied by Bamler [Compactness theory of the space of super Ricci flows, preprint (2020), arXiv:2008.09298; Structure theory of non-collapsed limits of Ricci flows, preprint (2020), arXiv:2009.03243], especially as an 픽-limit of a sequence of smooth Ricci flows with uniformly bounded Nash entropy, in which case each regular point on the limit is a point of smooth convergence. In this note, we shall consider the 픽-convergence of a sequence of 픽-limit flows, and, like Bamler, show that each regular point on the limit is also a point of smooth convergence. The main result will be applied in another work of the authors [P.-Y. Chan, Z. Ma and Y. Zhang, Dimension reduction for positively curved steady solitons, preprint (2023), arXiv:2310.14020]. [ABSTRACT FROM AUTHOR]

In this study, we suggest a new class of forward-backward (FB) algorithms designed to solve convex minimization problems. Our method incorporates a linesearch technique, eliminating the need to choose Lipschitz assumptions explicitly. Additionally, we apply double inertial extrapolations to enhance the algorithm’s convergence rate. We establish a weak convergence theorem under some mild conditions. Furthermore, we perform numerical tests, and apply the algorithm to image restoration and data classification as a practical application. The experimental results show our approach’s superior performance and effectiveness, surpassing some existing methods in the literature. [ABSTRACT FROM AUTHOR]

A simple, transparent, two-dimensional, nonlinear model of cell reorientation is constructed in this paper. The cells are attached to a substrate by "focal adhesions" that transmit the deformation of the substrate to the "stress fibers" in the cell. When the substrate is subjected to a deformation, say an in-plane bi-axial deformation with stretches λ 1 and λ 2 , the stress fibers deform with it and change their length and orientation. In addition, the focal adhesions can detach from the substrate and reattach to it at new nearby locations, and this process of detachment and reattachment can happen many times. In this scenario the (varying) fiber angle Θ in the reference configuration plays the role of an internal variable. In addition to the elastic energy of the stress fibers, the energy associated with the focal adhesions is accounted for by a wiggly energy ϵ a cos Θ / ϵ , 0 < ϵ ≪ 1 . Each local minimum of this energy corresponds to a particular configuration of the focal adhesions. The small amplitude ϵ a indicates that the energy barrier between two neighboring configurations is relatively small, and the small distance 2 π ϵ between the local minima indicates that a focal adhesion does not have to move very far before it reattaches. The evolution of this system is studied using a gradient flow kinetic law, which is homogenized for ϵ → 0 using results from weak convergence. The results determine (a) a region of the λ 1 , λ 2 -plane in which the (referential) fiber orientation remains stuck at the angle Θ and does not evolve, and (b) the evolution of the orientation when the stretches move out of this region as the fibers seek to minimize energy. [ABSTRACT FROM AUTHOR]

This paper discusses the large deviations for stochastic predator–prey model driven by multiplicative Lévy noise. Using Galerkin approximation, we initially prove the existence and uniqueness of solution. Due to the equivalence between Laplace principle and large deviation principle under a Polish space, the method of weak convergence has been followed in order to establish our results for this coupled system of equations. [ABSTRACT FROM AUTHOR]

Let X be a linear diffusion taking values in (ℓ , r) and consider the standard Euler scheme to compute an approximation to E [ g (X T) 1 { T < ζ } ] for a given function g and a deterministic T , where ζ = inf { t ≥ 0 : X t ∉ (ℓ , r) } . It is well known since Gobet (Stoch. Process. Appl. 87:167–197, 2000) that the presence of killing introduces a loss of accuracy and reduces the weak convergence rate to 1 / N with N being the number of discretisations. We introduce a drift-implicit Euler method to bring the convergence rate back to 1 / N , i.e., the optimal rate in the absence of killing, using the theory of recurrent transformations developed in Çetin (Ann. Appl. Probab. 28:3102–3151, 2018). Although the current setup assumes a one-dimensional setting, multidimensional extension is within reach as soon as a systematic treatment of recurrent transformations is available in higher dimensions. [ABSTRACT FROM AUTHOR]

Keywords: Attractors; homogenization; reaction-diffusion system; nonlinear equations; weak convergence; randomly perforated domains; random porous medium; topological space; compactness.GraphBy Kuanysh A. Bekmaganbetov; Gregory A. Chechkin and Vladimir V. ChepyzhovReported by Author; Author; Author [Extracted from the article]

The empirical likelihood (EL) method possesses desirable qualities such as automatically determining confidence regions and circumventing the need for variance estimation. As an extension, a quantile-based EL (QEL) method is considered, which results in a simpler form. In this paper, we explore the framework of the QEL method. Firstly, we explore the weak convergence of the −2 log empirical likelihood ratio for ROC curves. We also introduce a novel statistic for testing the entire ROC curve and the equality of two distributions. To validate our approach, we conduct simulation studies and analyze real data from hepatitis C patients, comparing our method with existing ones. [ABSTRACT FROM AUTHOR]

In this paper, we consider universality in short intervals for the zeta-function attached to a normalized Hecke-eigen cusp form with respect to the modular group. For this, we apply a conjecture for the mean square in short interval on the critical strip for that zeta-function. The proof of the obtained universality theorem is based on a probabilistic limit theorem in the space of analytic functions. [ABSTRACT FROM AUTHOR]

In this paper, combining the subgradient extragradient method with inertial method, we introduce a new iterative algorithm for solving equilibrium problems in real Hilbert spaces. Moreover, we present a new inertial self-adaptive scheme for solving variational inequalities in real Hilbert spaces, which it is not necessary to know the Lipschitz constant of the mapping. We prove the weak convergence of the generated iterates by presented algorithms. To illustrate the usability of our results and also to show the efficiency of the proposed methods, we present some comparative examples with several existing schemes in the literature. [ABSTRACT FROM AUTHOR]

We consider the problem of detecting distributional changes in a sequence of high dimensional data. Our approach combines two separate statistics stemming from L p norms whose behavior is similar under H 0 but potentially different under H A , leading to a testing procedure that that is flexible against a variety of alternatives. We establish the asymptotic distribution of our proposed test statistics separately in cases of weakly dependent and strongly dependent coordinates as min { N , d } → ∞ , where N denotes sample size and d is the dimension, and establish consistency of testing and estimation procedures in high dimensions under one-change alternative settings. Computational studies in single and multiple change point scenarios demonstrate our method can outperform other nonparametric approaches in the literature for certain alternatives in high dimensions. We illustrate our approach through an application to Twitter data concerning the mentions of U.S. governors. [ABSTRACT FROM AUTHOR]

We study the homogenization problem for a system of stochastic differential equations with local time terms that models a multivariate diffusion in the presence of semipermeable hyperplane interfaces with oblique penetration. We show that this system has a unique weak solution and determine its weak limit as the distances between the interfaces converge to zero. In the limit, the singular local times terms vanish and give rise to an additional regular interface-induced drift. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce a new inertial iterative algorithm involving the linesearch rule for the split variational inclusion in real Hilbert spaces. We use two inertial terms to accelerate the convergence speed of the proposed algorithm. Under suitable conditions, we prove its weak convergence theorem. Finally, in applications, we apply our algorithm to image deblurring and data classification problem in predicting diabetes mellitus. [ABSTRACT FROM AUTHOR]

In this work, we propose inclusion problems based on a novel class of forward-backwardforward algorithms. Our approach incorporates multi-inertial extrapolations and utilizes a self-adaptive technique to eliminate the need for explicitly selecting Lipschitz assumptions to enhance the speed convergence of the algorithm. We establish a weak convergence theorem under suitable assumptions. Furthermore, we conduct numerical tests on image deblurring as a practical application. The experimental results demonstrate that our algorithm surpasses some existing methods in the literature, which shows its superior performance and effectiveness. [ABSTRACT FROM AUTHOR]

This paper presents a two inertial technique with a projection Mann forward-backward splitting algorithm for solving variational inclusion problems that exhibit weak convergence under suitable conditions in Hilbert spaces. Furthermore, we provide a numerical example in infinitely dimensional spaces to support the main result. Finally, we provide an application for data classification using an extreme learning machine. According to data provided by theWorld Health Organization (WHO), stroke is recognized as the predominant contributor to mortality and disability on a global scale. To appraise the efficacy of our algorithm, we procured a dependable dataset for stroke prediction from the Kaggle website. The best algorithm that performed this task is ours compared to other machine learning methods. [ABSTRACT FROM AUTHOR]

Arriaza et al (Metrika 82:99–124, 2019) introduced the right and left shape functions, which enjoy interesting properties in terms of describing the global form of a distribution. This paper proposes and studies nonparametric estimators of those functions. The estimators involve nonparametric estimation of the quantile and density functions. Pointwise and uniform consistency are proved under general regularity assumptions, as well as the limit in law. Simulations are included to study the practical performance of the proposed estimators. The analysis of a real data set illustrates the methodology. [ABSTRACT FROM AUTHOR]

In this work, we propose a novel class of forward‐backward‐forward algorithms for solving monotone inclusion problems. Our approach incorporates a self‐adaptive technique to eliminate the need for explicitly selecting Lipschitz assumptions and utilizes two‐step inertial extrapolations to enhance the convergence rate of the algorithm. We establish a weak convergence theorem under mild assumptions. Furthermore, we conduct numerical tests on image deblurring and data classification as practical applications. The experimental results demonstrate that our algorithm surpasses some existing methods in the literature which shows its superior performance and effectiveness. [ABSTRACT FROM AUTHOR]

We derive a Cramér–von Mises test for testing a class of time dependent coefficients Coditional Heteroscedastic AutoRegressive Non Linear (CHARN) models. The test statistic is based on the log-likelihood ratio process whose weak convergence in a suitable Fréchet space is studied under the null hypothesis and under the sequence of local alternatives considered. This study makes use of the locally asymptotically normal (LAN) result previously established. Using the Karhunen–Loève expansion of the limiting process of the log-likelihood ratio process, the asymptotic null distribution and the power of the test statistic are accurately approximated. These results are applied to change-point analysis. An empirical study is done for evaluating the performance of the methodology proposed. [ABSTRACT FROM AUTHOR]

In this paper, the stochastic fluctuation of buffer stock level at time t is investigated. Therefore, random walk processes X(t) and Y(t) with two specific barriers have been defined to describe the stochastic fluctuation of the product level. Here X(t) = Y(t) -- a and the parameter a specifies half capacity of the buffer stock warehouse. Next, the one-dimensional distribution of the process X(t) has calculated. Moreover, the ergodicity of the process X(t) has been proven and the exact formula for the characteristic function has been found. Then, the weak convergence theorem has been proven for the standardized process W(t) = X(t)/a, as a → ∞. Additionally, exact and asymptotic expressions for the ergodic moments of the processes X(t) and Y (t) are obtained. [ABSTRACT FROM AUTHOR]

This paper aims to investigate the weak convergence of the Rosenbrock semi-implicit method for semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive noise. We are interested in SPDEs where the nonlinear part is stronger than the linear part, also called stochastic reaction dominated transport equations. For such SPDEs, many standard numerical schemes lose their stability properties. Exponential Rosenbrock and Rosenbrock-type methods were proved to be efficient for such SPDEs, but only their strong convergence were recently analyzed. Here, we investigate the weak convergence of the Rosenbrock semi-implicit method. We obtain a weak convergence rate which is twice the rate of the strong convergence. Our error analysis does not rely on Malliavin calculus, but rather only uses the Kolmogorov equation and the smoothing properties of the resolvent operator resulting from the Rosenbrock semi-implicit approximation. [ABSTRACT FROM AUTHOR]

In this paper, we suggest and analyze four inertial algorithms for solving fixed point problems of Bregman quasi-nonexpansive mappings in the framework of reflexive Banach spaces. Our first two algorithms, we propose inertial-like methods based on Mann-type and Halpern-type iterations, and in the others, we propose relaxed inertial-like methods based on Mann-type and Halpern-type iterations. The weak and strong convergence of the algorithms are established under some appropriate conditions on the parameters. As an application, we utilize our main results to find a zero of the sum of Bregman inverse strongly monotone mappings and maximal monotone operators in real reflexive Banach spaces. Also, we provide several numerical experiments to show the convergence behaviour of our algorithms in both finite-dimensional and infinite-dimensional spaces. Finally, we further utilize our algorithms to numerically solve the data classification problems of lung cancer. [ABSTRACT FROM AUTHOR]

This paper derived the homogenization of the heat equation with random convolutional potential. By Tartar's method of oscillating test function, the solution of the heat equation with random convolutional potential was shown to converge in distribution to the solution of the effective equation with determined convolutional potential.

$ U $-statistics represent a fundamental class of statistics arising from modeling quantities of interest defined by multi-subject responses. $ U $-statistics generalize the empirical mean of a random variable $ X $ to sums over every $ m $-tuple of distinct observations of $ X $. Stute [182] introduced a class of so-called conditional $ U $-statistics, which may be viewed as a generalization of the Nadaraya-Watson estimates of a regression function. Stute proved their strong pointwise consistency to: $ r^{(m)}(\varphi, \mathbf{t}): = \mathbb{E}[\varphi(Y_{1}, \ldots, Y_{m})|(X_{1}, \ldots, X_{m}) = \mathbf{t}], \; \mbox{for}\; \mathbf{ t}\in \mathcal{X}^{m}. $ In this paper, we are mainly interested in the study of the $ k $NN conditional $ U $-processes in a functional mixing data framework. More precisely, we investigate the weak convergence of the conditional empirical process indexed by a suitable class of functions and of the $ k $NN conditional $ U $-processes when the explicative variable is functional. We treat the uniform central limit theorem in both cases when the class of functions is bounded or unbounded satisfying some moment conditions. The second main contribution of this study is the establishment of a sharp almost complete Uniform consistency in the Number of Neighbors of the constructed estimator. Such a result allows the number of neighbors to vary within a complete range for which the estimator is consistent. Consequently, it represents an interesting guideline in practice to select the optimal bandwidth in nonparametric functional data analysis. These results are proved under some standard structural conditions on the Vapnik-Chervonenkis classes of functions and some mild conditions on the model. The theoretical results established in this paper are (or will be) key tools for further functional data analysis developments. Potential applications include the set indexed conditional U-statistics, Kendall rank correlation coefficient, the discrimination problems and the time series prediction from a continuous set of past values.

With the increasing interest in the south polar region of the Moon, there is an urgent need for high-resolution mapping products to support future exploration. Bundle block adjustment (BBA) is able to improve the spatial positioning accuracy of images used to generate high-precision mapping products. However, the impacts of weak-convergence geometry and poor illumination conditions in lunar south pole on BBA still need to be solved. This article proposes a large-scale robust BBA method for narrow angle camera (NAC) images in south pole. The NAC images are taken from the Lunar Reconnaissance Orbiter, which are scanner-type images requiring special treatment beyond the classical BBA of framing camera images. To handle the weak-convergence geometry issue of stereo NAC imagery, a topographic constraint using reference digital elevation model is integrated into BBA with an appropriate weighting scheme to estimate local topographic relief. In addition, a two-stage outlier elimination strategy for BBA with absolute-relative thresholding and iteratively reweighting methods is presented to reject outliers caused by the poor illumination conditions in lunar south pole. Thousands of images are used for experimental tests and lunar orbiter laser altimeter digital elevation models (DEMs) with different resolutions are used as reference data. The satisfactory experimental results demonstrate the effectiveness and reliability of our BBA method. The proposed large-scale BBA method reduces the relative positioning errors in block network to within 1 m, and decreases the inconsistency between NAC images caused by the orientation errors to less than 0.5 pixel, which offers a reliable solution for large-scale controlled mapping.

We consider optimal control problems for a system governed by a stochastic differential equation driven by a d-dimensional Brownian motion where both the drift and the diffusion coefficient are controlled. It is well known that without additional convexity conditions the strict control problem does not admit an optimal control. To overcome this difficulty, we consider the relaxed model, in which admissible controls are measure-valued processes and the relaxed state process is governed by a stochastic differential equation driven by a continuous orthogonal martingale measure. This relaxed model admits an optimal control that can be approximated by a sequence of strict controls by the so-called chattering lemma. We establish optimality necessary conditions, in terms of two adjoint processes, extending Peng's maximum principle to relaxed control problems. We show that relaxing the drift and diffusion martingale parts directly as in deterministic control does not lead to a true relaxed model as the obtained controlled dynamics is not continuous in the control variable. [ABSTRACT FROM AUTHOR]

Let P be the set of generalized prime numbers, and ζ P (s) , s = σ + i t , denote the Beurling zeta-function associated with P. In the paper, we consider the approximation of analytic functions by using shifts ζ P (s + i τ) , τ ∈ R . We assume the classical axioms for the number of generalized integers and the mean of the generalized von Mangoldt function, the linear independence of the set { log p : p ∈ P } , and the existence of a bounded mean square for ζ P (s) . Under the above hypotheses, we obtain the universality of the function ζ P (s) . This means that the set of shifts ζ P (s + i τ) approximating a given analytic function defined on a certain strip σ ^ < σ < 1 has a positive lower density. This result opens a new chapter in the theory of Beurling zeta functions. Moreover, it supports the Linnik–Ibragimov conjecture on the universality of Dirichlet series. For the proof, a probabilistic approach is applied. [ABSTRACT FROM AUTHOR]