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We investigate the asymptotic properties of the trajectories generated by a second-order dynamical system with Hessian driven damping and a Tikhonov regularization term in connection with the minimization of a smooth convex function in Hilbert spaces. We obtain fast convergence results for the function values along the trajectories. The Tikhonov regularization term enables the derivation of strong convergence results of the trajectory to the minimizer of the objective function of minimum norm.

We propose a complete-search algorithm for solving a class of non-convex, possibly infinite-dimensional, optimization problems to global optimality. We assume that the optimization variables are in a bounded subset of a Hilbert space, and we determine worst-case run-time bounds for the algorithm under certain regularity conditions of the cost functional and the constraint set. Because these run-time bounds are independent of the number of optimization variables and, in particular, are valid for optimization problems with infinitely many optimization variables, we prove that the algorithm converges to an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}ε-suboptimal global solution within finite run-time for any given termination tolerance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon > 0$$\end{document}ε>0. Finally, we illustrate these results for a problem of calculus of variations.

In this paper, we prove the weak convergence of a modified extragradient algorithm for solving a variational inequality problem involving a pseudomonotone operator in an infinite dimensional Hilbert space. Moreover, we establish further the R-linear rate of the convergence of the proposed algorithm with the assumption of error bound. Several numerical experiments are performed to illustrate the convergence behaviour of the new algorithm in comparisons with others. The results obtained in the paper have extended some recent results in the literature.

The present paper is a continuation of our research that was devoted to the theory of the boundary behavior of mappings in the Sobolev classes (mappings with generalized derivatives) on Riemann surfaces. Here we develop the theory of the boundary behavior of the mappings in the class of FLD (mappings with finite length distortion) first introduced for the Euclidean spaces in the article of Martio-Ryazanov-Srebro-Yakubov at 2004 and then included in the known book of these authors at 2009 on the modern mapping theory. As was shown in the recent papers of Kovtonyuk-Petkov-Ryazanov at 2017, such mappings, generally speaking, are not mappings in the Sobolev classes, because their first partial derivatives can be not locally integrable. At the same time, this class is a natural generalization of the well-known significant classes of isometries and quasiisometries. We prove here a series of criteria in terms of dilatations for the continuous and homeomorphic extensions to the boundary of the mappings with finite length distortion between domains on Riemann surfaces by Caratheodory prime ends. The criterion for the continuous extension of the inverse mapping to the boundary is turned out to be the very simple condition on the integrability of the dilatations in the first power. The criteria for the continuous extension of the direct mappings to the boundary have a much more refined nature. One of such criteria is the existence of a majorant for the dilatation in the class of functions with finite mean oscillation, i.e., having a finite mean deviation from its mean value over infinitesimal disks centered at boundary points. As consequences, the corresponding criteria for a homeomorphic extension of mappings with finite length distortion to the closures of domains by Caratheodory prime ends are obtained.

In this paper, we have pointed out that the proof of Theorem 11 in the recent paper (Lafhim in Positivity, 2019. https://doi.org/10.1007/s11117-019-00685-1 ) is erroneous. Using techniques from variational analysis, we propose other proofs to detect necessary optimality conditions in terms of Karush–Kuhn–Tucker multipliers. Our main results are given in terms of the limiting subdifferentials and the limiting normal cones. Completely detailed first order necessary optimality conditions are then given in the smooth setting while using the generalized differentiation calculus of Mordukhovich.

In this paper, we consider fractional ordinary differential equations and the fractional Euler–Poisson–Darboux equation with fractional derivatives in the form of a power of the Bessel differential operator. Using the technique of the Meijer integral transform and its modification, fundamental solutions to these equations are derived in terms of the Fox–Wright function, the Fox H-function, and their particular cases. We also provide some explicit formulas for the solutions to the corresponding initial-value problems in terms of the generalized convolutions introduced in this paper.

The augmented Lagrangian method (ALM) is fundamental in solving convex programming problems with linear constraints. The proximal version of ALM, which regularizes ALM’s subproblem over the primal variable at each iteration by an additional positive-definite quadratic proximal term, has been well studied in the literature. In this paper we show that it is not necessary to employ a positive-definite quadratic proximal term for the proximal ALM and the convergence can be still ensured if the positive definiteness is relaxed to indefiniteness by reducing the proximal parameter. An indefinite proximal version of the ALM is thus proposed for the generic setting of convex programming problems with linear constraints. We show that our relaxation is optimal in the sense that the proximal parameter cannot be further reduced. The consideration of indefinite proximal regularization is particularly meaningful for generating larger step sizes in solving ALM’s primal subproblems. When the model under discussion is separable in the sense that its objective function consists of finitely many additive function components without coupled variables, it is desired to decompose each ALM’s subproblem over the primal variable in Jacobian manner, replacing the original one by a sequence of easier and smaller decomposed subproblems, so that parallel computation can be applied. This full Jacobian splitting version of the ALM is known to be not necessarily convergent, and it has been studied in the literature that its convergence can be ensured if all the decomposed subproblems are further regularized by sufficiently large proximal terms. But how small the proximal parameter could be is still open. The other purpose of this paper is to show the smallest proximal parameter for the full Jacobian splitting version of ALM for solving multi-block separable convex minimization models.

In the paper (Greenblatt in J Funct Anal, https://doi.org/10.1016/j.jfa.2018.05.014 , 2018) the author proved $$L^p$$ Sobolev regularity results for averaging operators over hypersurfaces and connected them to associated Newton polyhedra. In this paper, we use rather different resolution of singularities techniques along with oscillatory integral methods applied to surface measure Fourier transforms to prove $$L^p$$ Sobolev regularity results for a class of averaging operators over surfaces which can be of any codimension.

Sampling in shift-invariant spaces is a realistic model for signals with smooth spectrum. In this paper, we consider phaseless sampling and reconstruction of real-valued signals in a high-dimensional shift-invariant space from their magnitude measurements on the whole Euclidean space and from their phaseless samples taken on a discrete set with finite sampling density. The determination of a signal in a shift-invariant space, up to a sign, by its magnitude measurements on the whole Euclidean space has been shown in the literature to be equivalent to its nonseparability. In this paper, we introduce an undirected graph associated with the signal in a shift-invariant space and use connectivity of the graph to characterize nonseparability of the signal. Under the local complement property assumption on a shift-invariant space, we find a discrete set with finite sampling density such that nonseparable signals in the shift-invariant space can be reconstructed in a stable way from their phaseless samples taken on that set. In this paper, we also propose a reconstruction algorithm which provides an approximation to the original signal when its noisy phaseless samples are available only. Finally, numerical simulations are performed to demonstrate the robustness of the proposed algorithm to reconstruct box spline signals from their noisy phaseless samples.

This paper is concerned with a type of time-symmetric stochastic system, namely the so-called forward&ndash, backward doubly stochastic differential equations (FBDSDEs), in which the forward equations are delayed doubly stochastic differential equations (SDEs) and the backward equations are anticipated backward doubly SDEs. Under some monotonicity assumptions, the existence and uniqueness of measurable solutions to FBDSDEs are obtained. The future development of many processes depends on both their current state and historical state, and these processes can usually be represented by stochastic differential systems with time delay. Therefore, a class of nonzero sum differential game for doubly stochastic systems with time delay is studied in this paper. A necessary condition for the open-loop Nash equilibrium point of the Pontriagin-type maximum principle are established, and a sufficient condition for the Nash equilibrium point is obtained. Furthermore, the above results are applied to the study of nonzero sum differential games for linear quadratic backward doubly stochastic systems with delay. Based on the solution of FBDSDEs, an explicit expression of Nash equilibrium points for such game problems is established.

There are two parts for this paper. In the first part, we extend some results in a recent paper by Du, Guth, Li and Zhang to a more general class of phase functions. The main methods are Bourgain-Demeter's $l^2$ decoupling theorem and induction on scales. In the second part, we prove some positive results for the maximal extension operator for hypersurfaces with positive principal curvatures. The main methods are sharp $L^2$ estimates by Du and Zhang, and the bilinear method by Wolff and Tao., 24 pages, revised version incorporating referees' suggestions

In this paper, we present a smoothing Newton method for solving the monotone weighted complementarity problem (WCP). In each iteration of our method, the iterative direction is achieved by solving a system of linear equations and the iterative step length is achieved by adopting a line search. A feature of the line search criteria used in this paper is that monotone and nonmonotone line search are mixed used. The proposed method is new even when the WCP reduces to the standard complementarity problem. Particularly, the proposed method is proved to possess the global convergence under a weak assumption. The preliminary experimental results show the effectiveness and robustness of the proposed method for solving the concerned WCP.

The purpose of this paper is to introduce the split combination of variational inclusion problem which combines the concept of the modified variational inclusion problem introduced by Khuangsatung and Kangtunyakarn and the split variational inclusion problem introduced by Moudafi. Using a modified Halpern iterative method, we prove the strong convergence theorem for finding a common solution for the hierarchical fixed point problem and the split combination of variational inclusion problem. The result presented in this paper demonstrates the corresponding result for the split zero point problem and the split combination of variation inequality problem. Moreover, we discuss a numerical example for supporting our result and the numerical example shows that our result is not true if some conditions fail.

Many researchers have been interested in the concept of statistical convergence because of the fact that it is stronger than the classical convergence. Also, the concepts of statistical equal convergence and equi-statistical convergence are more general than the statistical uniform convergence. In this paper we define a new type of statistical convergence by using the notions of equi-statistical convergence and statistical equal convergence to prove a Korovkin type theorem. We show that our theorem is a non-trivial extension of some well-known Korovkin type approximation theorems which were demonstrated by earlier authors. After, we present an example in support of our definition and result presented in this paper. Finally, we also compute the rates of statistical equi-equal convergence of sequences of positive linear operators.

In this paper, we study the mean value property for both the harmonic functions and the functions in the domain of the Laplacian on the tetrahedral Sierpinski gasket. This paper is a continuation of the work of Strichartz and the first author (Qiu and Strichartz, J Fourier Anal Appl 19:943–966, 2013)where the same property on p.c.f. self-similar sets with Dihedral-3 symmetry was considered.

This is a continuation of our previous paper [14]. In [14], we introduced the first Aeppli–Chern class on compact complex manifolds, and proved that the (1, 1) curvature form of the Levi–Civita connection represents the first Aeppli–Chern class which is a natural link between Riemannian geometry and complex geometry. In this paper, we study the geometry of compact complex manifolds with Levi–Civita Ricci-flat metrics and classify minimal complex surfaces with Levi–Civita Ricci-flat metrics. More precisely, we show that minimal complex surfaces admitting Levi–Civita Ricci-flat metrics are Kahler Calabi–Yau surfaces and Hopf surfaces.

In the present paper, we introduce a new form of generalized Rayleigh distribution called the Alpha Power generalized Rayleigh (APGR) distribution by following the idea of extension of the distribution families with the Alpha Power transformation. The introduced distribution has the more general form than both the Rayleigh and generalized Rayleigh distributions and provides a better fit than the Rayleigh and generalized Rayleigh distributions for more various forms of the data sets. In the paper, we also obtain explicit forms of some important statistical characteristics of the APGR distribution such as hazard function, survival function, mode, moments, characteristic function, Shannon and Rényi entropies, stress-strength probability, Lorenz and Bonferroni curves and order statistics. The statistical inference problem for the APGR distribution is investigated by using the maximum likelihood and least-square methods. The estimation performances of the obtained estimators are compared based on the bias and mean square error criteria by a conducted Monte-Carlo simulation on small, moderate and large sample sizes. Finally, a real data analysis is given to show how the proposed model works in practice.

This expository paper, based on a Current Events Bulletin talk at the January, 2016 Joint Meetings, introduces the concept of Lyapunov exponents and discusses the role they play in three areas: smooth ergodic theory, Teichm\"uller theory, and the spectral theory of one-frequency Schr\"odinger operators. The inspiration for this paper is the work of 2014 Fields Medalist Artur Avila, and his work in these areas is given special attention., Comment: 27 pages. To appear in the Bulletin of the AMS

In this paper, we propose an iterative algorithm to find the common element of set of solutions of a split equilibrium problem and set of fixed points of an asymptotically nonexpansive mapping in Hilbert spaces. The new method is used to prove the strong convergence for the result of this paper. The result extends the corresponding one in the literature.

This paper studies the approximation of the Hilbert transform f ? = H f of continuous functions f with continuous conjugate f ? based on a finite number of samples. It is known that every sequence { H N f } N ? N which approximates f ? from samples of f diverges (weakly) with respect to the uniform norm. This paper conjectures that all of these approximation sequences even contain no convergent subsequence. A property which is termed strong divergence.The conjecture is supported by two results. First it is proven that the sequence of the sampled conjugate Fejer means diverges strongly. Second, it is shown that for every sample based approximation method { H N } N ? N there are functions f such that ? H N f ? ∞ exceeds any given bound for any given number of consecutive indices N .As an application, the later result is used to investigate a problem associated with a question of Ul'yanov on Fourier series which is related to the possibility to construct adaptive approximation methods to determine the Hilbert transform from sampled data. This paper shows that no such approximation method with a finite search horizon exists.

In this paper a non-convex vector optimization problem among infinite-dimensional spaces is presented. In particular, a generalized Lagrange multiplier rule is formulated as a necessary and sufficient optimality condition for weakly minimal solutions of a constrained vector optimization problem, without requiring that the ordering cone that defines the inequality constraints has non-empty interior. This paper extends the result of Donato (J. Funct. Analysis261 (2011), 2083–2093) to the general setting of vector optimization by introducing a constraint qualification assumption that involves the Fréchet differentiability of the maps and the tangent cone to the image set. Moreover, the constraint qualification is a necessary and sufficient condition for the Lagrange multiplier rule to hold.

The object of the present paper is to introduce a new curva-ture tensor, named generalized quasi-conformal curvature tensor whichbridges conformal curvature tensor, concircular curvature tensor, pro-jective curvature tensor and conharmonic curvature tensor. Flatness andsymmetric properties of generalized quasi-conformal curvature tensor arestudied in the frame of (k,µ)-contact metric manifolds. 1. IntroductionIn 1968, Yano and Sawaki [27] introduced the notion of quasi-conformalcurvature tensor which contains both conformal curvature tensor as well asconcircular curvature tensor, in the context of Riemannian geometry. In tunewith Yano and Sawaki [27], the present paper attempts to introduce a newtensor field, named generalized quasi-conformal curvature tensor. The beautyof generalized quasi-conformal curvature tensor lies in the fact that it has theflavour of Riemann curvature tensor R, conformal curvature tensor C [8] con-harmonic curvature tensor Cˆ [9], concircular curvature tensor E [26, p. 84],projective curvature tensor P [26, p. 84] and m-projective curvature tensor H[15], as particular cases. The generalized quasi-conformal curvature tensor isdefined asW(X,Y)Z =2n−12n+1[(1−b+2na)−{1+2n(a+b)}c]C(X,Y )Z+[1−b+2na]E(X,Y)Z +2 n (b−a) P(X,Y )Z+2 n−12 n+1(1.1) (c −1){1+2 n(a +b)} Cˆ(X,Y)Zfor all X,Y,Z ∈ χ(M), the set of all vector field of the manifold M, where a,b and c are real constants. The above mentioned curvature tensors are defined

This paper proposes two numerical algorithms for solving Euler's elastica-based inpainting model. The minimizing functional is nonsmooth and nonconvex and involves high-order derivatives, that traditional gradient descent based methods converge very slowly. Recent alternating minimization methods show fast convergence when a good choice of parameters is used. The objective of this paper is to introduce efficient algorithms which have simple structures with fewer parameters. These methods are based on operator splitting and alternating direction method of multipliers, and subproblems can be solved efficiently by Fourier transforms and shrinkage operators. For the first method, we relax the normal vector in the curvature term of the Euler's elastica model and exploit two operator splitting techniques to propose a Relaxed Normal Two Split (RN2Split) method. The second method, $\kappa$-weighted Total Variation ($\kappa$TV), solves the Euler's elastica minimization problem as a weighted total variation. We pre...

In this paper, we introduce a system of split variational inequality problems in real Hilbert spaces. Using a projection method, we propose an iterative algorithm for solving this system of split variational inequality problems. Further, we prove that the sequence generated by the iterative algorithm converges strongly to a solution of the system of split variational inequality problems. Furthermore, we discuss some consequences of the main result. The iterative algorithms and results presented in this paper generalize, unify and improve the previously known results of this area.

Adaptive regularization with cubics (ARC) is an algorithm for unconstrained, non-convex optimization. Akin to the popular trust-region method, its iterations can be thought of as approximate, safe-guarded Newton steps. For cost functions with Lipschitz continuous Hessian, ARC has optimal iteration complexity, in the sense that it produces an iterate with gradient smaller than $\varepsilon$ in $O(1/\varepsilon^{1.5})$ iterations. For the same price, it can also guarantee a Hessian with smallest eigenvalue larger than $-\varepsilon^{1/2}$. In this paper, we study a generalization of ARC to optimization on Riemannian manifolds. In particular, we generalize the iteration complexity results to this richer framework. Our central contribution lies in the identification of appropriate manifold-specific assumptions that allow us to secure these complexity guarantees both when using the exponential map and when using a general retraction. A substantial part of the paper is devoted to studying these assumptions---relevant beyond ARC---and providing user-friendly sufficient conditions for them. Numerical experiments are encouraging., Comment: 48 pages, 3 figures

In this paper we study the number of customers in infinite-server queues with a self-exciting (Hawkes) arrival process. Initially we assume that service requirements are exponentially distributed and that the Hawkes arrival process is of a Markovian nature. We obtain a system of differential equations that characterizes the joint distribution of the arrival intensity and the number of customers. Moreover, we provide a recursive procedure that explicitly identifies (transient and stationary) moments. Subsequently, we allow for non-Markovian Hawkes arrival processes and non-exponential service times. By viewing the Hawkes process as a branching process, we find that the probability generating function of the number of customers in the system can be expressed in terms of the solution of a fixed-point equation. We also include various asymptotic results: we derive the tail of the distribution of the number of customers for the case that the intensity jumps of the Hawkes process are heavy-tailed, and we consider a heavy-traffic regime. We conclude the paper by discussing how our results can be used computationally and by verifying the numerical results via simulations., v3: some minor corrections

This paper is centred on the spectral study of a Random Fourier matrix, that is an $n\times n$ matrix $A$ whose $(j, k)$ entries are $\exp(2i\pi m X_jY_k)$, with $X_j$ and $Y_k$ two i.i.d sequences of random variables and $1\leq m\leq n$ is a real number. When they are uniformly distributed on a symmetric interval, this may be seen as a random discretization of the Finite Fourier transform, whose spectrum has been extensively studied in relation with band-limited functions. Our study is two-fold. Firstly, by pushing forward concentration inequalities, we find an accurate comparison in $\ell^2$- norm between the spectrum of $A^*A$ and the one of an integral operator that can be defined in terms of the two probability laws chosen for the rows and the columns. Our study includes the one of stationary Hermitian kernel matrices and can be generalized to non stationary ones, for which the same kind of comparison with an integral operator is possible. Because of possible applications in the data science area, these last matrices have been largely studied in the literature and our results are compared with previous ones. Secondly we concentrate on uniform distributions for the laws of $X_j$'s and $Y_k$'s, for which the integral operator is the well-known Sinc-kernel operator with parameter $m.$ Our previous study allows to translate to random Fourier matrices the knowledge that we have on the spectrum of this operator. We have for them asymptotic results for $m, n$ and $n/m$ tending to $\infty$, as well as non asymptotic bounds in the spirit of recent work on the integral operators. As an application, we give fairly good approximations of the number of degrees of freedom and the capacity of a MIMO wireless communication network approximation model. Finally, we provide the reader with some numerical examples that illustrate the theoretical results of this paper.

This paper focuses on the problem of finite-time stabilization of homogeneous, non-Lipschitz systems with dilations. A key contribution of this paper is the design of a virtual recursive Hölder, non-Lipschitz state feedback which renders the non-Lipschitz systems in the special case dominated by a lower-triangular nonlinear system, finite-time stable. The proof is based on a recursive design algorithm developed recently, to construct the virtual Hölder continuous, finite-time stabilizer as well as a C1 positive definite and proper Lyapunov function that guarantees finite-time stability of the non-Lipschitz non-linear systems.

The aim of this paper is to derive and analyze a variational model for the joint estimation of motion and reconstruction of image sequences, which is based on a time-continuous Eulerian motion model. The model can be set up in terms of the continuity equation or the brightness constancy equation. The analysis in this paper focuses on the latter for robust motion estimation on sequences of twodimensional images. We rigorously prove the existence of a minimizer in a suitable function space setting. Moreover, we discuss the numerical solution of the model based on primal-dual algorithms and investigate several examples. Finally, the benefits of our model compared to existing techniques, such as sequential image reconstruction and motion estimation, are shown., The work of the first author was also supported by the German Science Foundation DFG via EXC 1003 Cells in Motion Cluster of Excellence, M¨unster, Germany

The cyclic projections algorithm is an important method for determining a point in the intersection of a finite number of closed convex sets in a Hilbert space. That is, for determining a solution to the ''convex feasibility'' problem. This is the third paper in a series on a study of the rate of convergence for the cyclic projections algorithm. In the first of these papers, we showed that the rate could be described in terms of the ''angles'' between the convex sets involved. In the second, we showed that these angles often had a more tractable formulation in terms of the ''norm'' of the product of the (nonlinear) metric projections onto related convex sets. In this paper, we show that the rate of convergence of the cyclic projections algorithm is also intimately related to the ''linear regularity property'' of Bauschke and Borwein, the ''normal property'' of Jameson (as well as Bakan, Deutsch, and Li's generalization of Jameson's normal property), the ''strong conical hull intersection property'' of Deutsch, Li, and Ward, and the rate of convergence of iterated parallel projections. Such properties have already been shown to be important in various other contexts as well.

In the first part of this paper, we consider nonlinear extension of frame theory by introducing bi-Lipschitz maps F between Banach spaces. Our linear model of bi-Lipschitz maps is the analysis operator associated with Hilbert frames, p-frames, Banach frames, g-frames and fusion frames. In general Banach space setting, stable algorithms to reconstruct a signal x from its noisy measurement $$F(x)+\epsilon $$ may not exist. In this paper, we establish exponential convergence of two iterative reconstruction algorithms when F is not too far from some bounded below linear operator with bounded pseudo-inverse, and when F is a well-localized map between two Banach spaces with dense Hilbert subspaces. The crucial step to prove the latter conclusion is a novel fixed point theorem for a well-localized map on a Banach space. In the second part of this paper, we consider stable reconstruction of sparse signals in a union $$\mathbf{A}$$ of closed linear subspaces of a Hilbert space $$\mathbf{H}$$ from their nonlinear measurements. We introduce an optimization framework called a sparse approximation triple $$(\mathbf{A}, \mathbf{M}, \mathbf{H})$$ , and show that the minimizer $$\begin{aligned} x^*=\mathrm{argmin}_{\hat{x}\in {\mathbf M}\ \mathrm{with} \ \Vert F(\hat{x})-F(x^0)\Vert \le \epsilon } \Vert \hat{x}\Vert _{\mathbf M} \end{aligned}$$ provides a suboptimal approximation to the original sparse signal $$x^0\in \mathbf{A}$$ when the measurement map F has the sparse Riesz property and the almost linear property on $${\mathbf A}$$ . The above two new properties are shown to be satisfied when F is not far away from a linear measurement operator T having the restricted isometry property.

In this paper, we present a new framework of bi-level unconstrained minimization for development of accelerated methods in Convex Programming. These methods use approximations of the high-order proximal points, which are solutions of some auxiliary parametric optimization problems. For computing these points, we can use different methods, and, in particular, the lower-order schemes. This opens a possibility for the latter methods to overpass traditional limits of the Complexity Theory. As an example, we obtain a new second-order method with the convergence rate $$O\left( k^{-4}\right) $$ O k - 4 , where k is the iteration counter. This rate is better than the maximal possible rate of convergence for this type of methods, as applied to functions with Lipschitz continuous Hessian. We also present new methods with the exact auxiliary search procedure, which have the rate of convergence $$O\left( k^{-(3p+1)/ 2}\right) $$ O k - ( 3 p + 1 ) / 2 , where $$p \ge 1$$ p ≥ 1 is the order of the proximal operator. The auxiliary problem at each iteration of these schemes is convex.

UDC 517.9 This paper deals with the problem of stability of nonlinear differential equations with perturbations. Sufficient conditions for global uniform asymptotic stability in terms of Lyapunov-like functions and integral inequality are obtained. The asymptotic behavior is studied in the sense that the trajectories converge to a small ball centered at the origin. Furthermore, an illustrative example in the plane is given to verify the effectiveness of the theoretical results.

In this paper, inference upon stress-strength reliability is considered for unit-Weibull distributions with a common parameter under the assumption that data are observed using progressive type II censoring. We obtain di_erent estimators of system reliability using classical and Bayesian procedures. Asymptotic interval is constructed based on Fisher information matrix. Besides, boot-p and boot-t intervals are also obtained. We evaluate Bayes estimates using Lindley's technique and Metropolis-Hastings (MH) algorithm. The Bayes credible interval is evaluated using MH method. An unbiased estimator of this parametric function is also obtained under know common parameter case. Numerical simulations are performed to compare estimation methods. Finally, a data set is studied for illustration purposes.

In this paper, we propose a new linesearch iterative scheme for finding a common solution of split equilibrium and fixed point problems without pseudomonotonicity of the bifunction f in a real Hilbert space. When setting the solution of dual equilibrium problem is nonempty, we obtain a strong convergence theorem which is generated by the iterative scheme. Moreover, we also receive a new linesearch iterative scheme for finding a solution of the split equilibrium problem in suitable assumptions, and report some numerical results to illustrate the convergence of the proposed scheme.

In this paper, we investigate the β -stability in q-th moment for neutral impulsive stochastic functional differential equations with Markovian switching (NISFDEwMS). Moreover, β -stability in q-th moment is studied by using the Lyapunov techniques and a new Razumikhin-type theorem to prove our result. Finally, we check the main result by a numerical example.

In this paper, we use the analytic methods and the properties of the classical Gauss sums to study the properties of the error term of the fourth power mean of the generalized cubic Gauss sums and give two recurrence formulae for it.

This is the second paper of the authors in a series concerned with the development of a deterministic model for the transfer matrix of a MIMO system. Starting from the Maxwell equations, we have described in \cite{BCFM} the generic structure of such a deterministic transfer matrix. In the current paper we apply the results of \cite{BCFM} in order to study the (Shannon-Foschini) capacity behavior of a MIMO system as a function of the deterministic spread function of the environment, and the number of transmitting and receiving antennas. The antennas are assumed to fill in a given, fixed volume. Under some generic assumptions, we prove that the capacity grows much more slowly than linearly with the number of antennas. These results reinforce previous heuristic results obtained from statistical models of the transfer matrix, which also predict a sublinear behavior., Comment: 12 pages, to appear in Math. Meth. Appl. Sci

In the present paper we give some criteria for statistical convergence with respect to power series method. The main tool for providing this result is the inverse Fourier transformation. We will also deal with sequences x that admit a factorization $$x=yz$$ in which y is strongly convergent and z is statistically convergent with respect to a power series method.

The purpose of the paper is to find an approximate solution of the two-dimensional nonlinear fuzzy Volterra integral equation, as homotopy analysis method (HAM) is applied. Studied equation is converted to a nonlinear system of Volterra integral equations in a crisp case. Using HAM we find approximate solution of this system and hence obtain an approximation for the fuzzy solution of the nonlinear fuzzy Volterra integral equation. The convergence of the proposed method is proved. An error estimate between the exact and the approximate solution is found. The validity and applicability of the HAM are illustrated by a numerical example.

In this paper, we propose a simple method to generate generalized Gaussian noises using the inverse transform of cumulative distribution. This inverse is expressible by means of the inverse incomplete Gamma function. Since the implementation of Newton's method is rather simple, for approximating inverse incomplete Gamma function, we propose a better and new initial value exploiting the close relationship between the incomplete Gamma function and its piecewise linear interpolant. The numerical results highlight that the proposed method simulates well the univariate and bivariate generalized Gaussian noises.

In this paper, we introduce a new algorithm for solving pseudomonotone variational inequalities with a Lipschitz-type condition in a real Hilbert space. The algorithm is constructed around two algorithms: the subgradient extragradient algorithm and the inertial algorithm. The proposed algorithm uses a new step size rule based on local operator information rather than its Lipschitz constant or any other line search scheme and functions without any knowledge of the Lipschitz constant of an operator. The strong convergence of the algorithm is provided. To determine the computational performance of our algorithm, some numerical results are presented.

This paper considers the nonlinear symmetric conic programming (NSCP) problems. Firstly, a type of strong sufficient optimality condition for NSCP problems in terms of a linear-quadratic term is introduced. Then, a sufficient condition of the nonsingularity of Clarke’s generalized Jacobian of the Karush–Kuhn–Tucker (KKT) system is demonstrated. At last, as an application, this property is used to obtain the local convergence properties of a sequential quadratic programming- (SQP-) type method.

This paper deals with second-order optimality conditions and regularity of Lagrange multipliers for a class of optimal control problems governed by semilinear elliptic equations with mixed pointwise constraints in which controls act both in the domain and on the boundary. We give some criteria under which the optimality conditions are of KKT type and the multipliers are of $$L^p$$ -spaces. Moreover, we show that the multipliers are Lipschitz continuous functions.

National audience; This paper is devoted to the study of the internal stabilization of the Bernoulli-Euler plate equation in a square. The continuous and the space semi-discretizated problems are successively considered and analyzed using a frequency domain approach. For the infinite dimensional problem, we provide a new proof of the exponential stability result, based on a two dimensional Ingham's type result. In the second and main part of the paper, we propose a finite difference space semi-discretization scheme and we prove that this scheme yields a uniform exponential decay rate (with respect to the mesh size).

A linear mapping ϕ from an algebra $${\cal A}$$ into its bimodule $${\cal M}$$ is called a centralizable mapping at G ∈ $${\cal A}$$ if ϕ(AB) = ϕ(A)B = Aϕ(B) for each A and B in $${\cal A}$$ with AB = G. In this paper, we prove that if $${\cal M}$$ is a von Neumann algebra without direct summands of type I1 and type II, $${\cal A}$$ is a *-subalgebra with $${\cal M}$$ ⊆ $${\cal A}$$ ⊆ LS ( $${\cal M}$$ ) and G is a fixed element in $${\cal A}$$ , then every continuous (with respect to the local measure topology t( $${\cal M}$$ )) centralizable mapping at G from $${\cal A}$$ into $${\cal M}$$ is a centralizer.

This paper focuses on the problem of variational inequalities with monotone operators in real Hilbert space. The Tseng algorithm constructed by Thong replaced a high-precision step. Thus, a new Tseng-like gradient method is constructed, and the convergence of the algorithm is proved, and the convergence performance is higher.

Summary Stochastic gradient Markov chain Monte Carlo algorithms have received much attention in Bayesian computing for big data problems, but they are only applicable to a small class of problems for which the parameter space has a fixed dimension and the log-posterior density is differentiable with respect to the parameters. This paper proposes an extended stochastic gradient Markov chain Monte Carlo algorithm which, by introducing appropriate latent variables, can be applied to more general large-scale Bayesian computing problems, such as those involving dimension jumping and missing data. Numerical studies show that the proposed algorithm is highly scalable and much more efficient than traditional Markov chain Monte Carlo algorithms.

The main purpose of this paper is to study higher-order sensitivity analysis in parametric vector optimization problems. Firstly, the higher-order proto-differentiability/the higher-order asymptotic proto-differentiability of the feasible map of a parametric vector optimization problem are considered. Then, we verify that the weak efficient solution map and the weak perturbation map of a parameterized vector optimization problem are higher-order proto-differentiable/higher-order asymptotic proto-differentiable under some suitable qualification conditions.

In this paper, we study the dilation of g-frame generators and the dual g-frame generators for an abstract wavelet system. By the semi-orthogonality of a special unitary system, we prove the dual g-frame generators can be dilated to a pair of dual g-Riesz basis generators for some larger Hilbert space. We first show the existence of the dual g-frame generators with the same structure of a g-frame generator for a unitary system. We then get a sufficient and necessary condition for the existing of the dual g-frame generators for unitary groups on a subspace.