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This paper shows that under very general conditions, there exists a locally stable Nash equilibrium in games of strategic complements (GSC), as well as in the more general case of games with non-decreasing best response correspondences. While it is well known that in such cases a unique equilibrium is globally stable, no equilibrium can be globally stable when multiple equilibria exist. However, the existence of a locally stable equilibrium remains an open question, as we give examples of GSC in which no stable equilibrium exists. One main advantage of our approach is that our results can be derived simply by exploiting the monotonicity properties of the game, and do not require any differentiability assumptions. Results on equilibrium refinement follow as a corollary under slightly stronger assumptions, in the sense that games with two equilibria possess exactly one locally stable equilibrium.

The asymptotic spectrum of graphs, introduced by Zuiddam (arXiv:1807.00169, 2018), is the space of graph parameters that are additive under disjoint union, multiplicative under the strong product, normalized and monotone under homomorphisms between the complements. He used it to obtain a dual characterization of the Shannon capacity of graphs as the minimum of the evaluation function over the asymptotic spectrum and noted that several known upper bounds, including the Lov\'asz number and the fractional Haemers bounds are in fact elements of the asymptotic spectrum (spectral points). We show that every spectral point admits a probabilistic refinement and characterize the functions arising in this way. This reveals that the asymptotic spectrum can be parameterized with a convex set and the evaluation function at every graph is logarithmically convex. One consequence is that for any incomparable pair of spectral points $f$ and $g$ there exists a third one $h$ and a graph $G$ such that $h(G)

In this paper, we are concerned with the existence of positive solutions for nonlinear fourth-order equation with clamped beam boundary conditions $$\begin{aligned} \left\{ \begin{array}{ll} &{}u''''(x)=\lambda a(x)f(u(x))\ \ \ \text {for}\ \ x\in (0,1),\\ &{}u(0)=u(1)=u'(0)=u'(1)=0, \end{array} \right. \end{aligned}$$ where $$\lambda >0$$ is a parameter, $$a\in L^{1}(0,1)$$ may change sign and $$f\in C([0,\infty ),[0,\infty ))$$ . We establish some new results of existence of positive solutions to this problem if the nonlinearity f is monotone on $$[0,\infty )$$ . The proofs of our main results are based upon a monotone iteration technique and the Schauder’s fixed point theorem. Finally, an example is presented to illustrate the application of our main results.

We consider a class of two-dimensional integral equations in ℝ2 with monotone nonlinear terms and prove the existence of a nonnegative bounded solution. We study the asymptotic behavior of this solution with respect to each variable. The result is illustrated by an example.

In this work, we develop a systematic framework for computing the resolvent of the sum of two or more monotone operators which only activates each operator in the sum individually. The key tool in the development of this framework is the notion of the “strengthening” of a set-valued operator, which can be viewed as a type of regularisation that preserves computational tractability. After deriving a number of iterative schemes through this framework, we demonstrate their application to best approximation problems, image denoising and elliptic PDEs. FJAA and RC were partially supported by the Ministry of Science, Innovation and Universities of Spain and the European Regional Development Fund (ERDF) of the European Commission, Grant PGC2018-097960-B-C22. MKT is supported in part by ARC grant DE200100063.

The article deals with the problem of state observer design for a linear time-varying plant. To solve this problem, a number of realistic assumptions are considered, assuming that the model parameters are polynomial functions of time with unknown coefficients. The problem of observer design is solved in the class of identification approaches, which provide transformation of the original mathematical model of the plant to a static linear regression equation, in which, instead of unknown constant parameters, there are state variables of generators that model non-stationary parameters. To recover the unknown functions of the regression model, we use the recently well-established method of dynamic regressor extension and mixing (DREM), which allows to obtain monotone estimates, as well as to accelerate the convergence of estimates to the true values. Despite the fact that the article deals with the problem of state observer design, it is worth noting the possibility of using the proposed approach to solve an independent and actual estimation problem of unknown time-varying parameters.

Motivated by computing medians and means, two proximal splitting methods, backward-backward scheme introduced by Passty [Ergodic convergence to a zero of the sum of monotone operators in Hilbert sp...

We solve the open problem of spectral stability of smooth periodic waves in the Camassa–Holm equation. The key to obtaining this result is that the periodic waves of the Camassa–Holm equation can be characterized by an alternative Hamiltonian structure, different from the standard formulation common to the Korteweg-de Vries equation. The standard formulation has the disadvantage that the period function is not monotone and the quadratic energy form may have two rather than one negative eigenvalues. We prove that the nonstandard formulation has the advantage that the period function is monotone and the quadratic energy form has only one simple negative eigenvalue. We deduce a precise condition for the spectral and orbital stability of the smooth periodic travelling waves and show numerically that this condition is satisfied in the open region where the smooth periodic waves exist.

The main goal of this article is to show that many inequalities that are not valid in operator theory become true if we add a separation condition on the spectra. The applications include showing how monotone functions behave like operator monotone functions and how the Choi-Davis inequality becomes valid for convex functions under this separation condition.

We examine the problem of computing the highest density region (HDR) in a computational context where the user has access to a density function and quantile function for the distribution (e.g., in the statistical language R). We examine several common classes of continuous univariate distributions based on the shape of the density function; this includes monotone densities, quasi-concave and quasi-convex densities, and general multimodal densities. In each case we show how the user can compute the HDR from the quantile and density functions by framing the problem as a nonlinear optimisation problem. We implement these methods in R to obtain general functions to compute HDRs for classes of distributions, and for commonly used families of distributions. We compare our method to existing R packages for computing HDRs and we show that our method performs favourably in terms of both accuracy and average speed.

The present analysis provides insights and analytical findings for a cooperative system formulated with non-linear diffusion. This work discusses, from an analytical perspective, the interaction of two species when both cooperate to explore new territories. Firstly, the analysis focuses on regularity, existence and uniqueness of weak solutions considering slightly positivity in the non-linear operator together with a monotone argument in the independent terms. Afterwards, analytical solution profiles are provided making use of selfsimilar solutions. Non-linear diffusion induces the existence of a propagating front and determines the property of finite speed of propagation in both species when they cooperate to explore new territories.

Relevance and diversity are two desirable properties in data retrieval applications, an important field in data science and machine learning. In this paper, we consider three maximization problems to balance these two factors. The objective function in each problem is the sum of a monotone submodular function f and a supermodular function g, where f and g capture the relevance and diversity of any feasible solution, respectively. In the first problem, we consider a special supermodular diversity function g of a sum-sum format satisfying the relaxed triangle inequality, for which we propose a greedy-type approximation algorithm with an $$\left( 1-1/e,1/(2\alpha )\right) $$ -bifactor approximation ratio, improving the previous $$\left( 1/(2\alpha ),1/(2\alpha )\right) $$ -bifactor approximation ratio. In the second problem, we consider an arbitrary supermodular diversity function g, for which we propose a distorted greedy method to give a $$\min \left\{ 1-k_{f}e^{-1},1-k^{g}e^{-(1-k^{g})}\right\} $$ -approximation algorithm, improving the previous $$k_f^{-1}\left( 1-e^{-k_f(1-k^{g})}\right) $$ -approximation ratio, where $$k_f$$ and $$k^g$$ are the curvatures of the submodular function f and the supermodular funciton g, respectively. In the third problem, we generalize the uniform matroid constraint to the p matroid constraints, for which we present a local search algorithm to improve the previous $$\frac{1-k^g}{(1-k^g)k^f+p}$$ -approximation ratio to $$\min \left\{ \frac{p+1-k_f}{p(p+1)},\left( \frac{1-k^g}{p}+\frac{k^g(1-k^g)^2}{p+(1-k^g)^2}\right) \right\} $$ .

In this paper we present sufficient conditions for convergence of projection and fixed-point algorithms used to compute dynamic user equilibrium with elastic travel demand (E-DUE). The assumption of strongly monotone increasing path delay operators is not needed. In its place, we assume path delay operators are merely weakly monotone increasing, a property assured by Lipschitz continuity, while inverse demand functions are strongly monotone decreasing. Lipschitz continuity of path delay is a very mild regularity condition. As such, nonmonotone delay operators may be weakly monotone increasing and satisfy our convergence criteria, provided inverse demand functions are strongly monotone decreasing. We illustrate convergence for nonmonotone path delays via a numerical example.

In this article, we are going to look at the requirements regarding a monotone function f ∈ ℝ →ℝ ≥0, and regarding the sets of natural numbers ( A i ) i = 1 ∞ ⊆ d m n ( f ) \left( {{A_i}} \right)_{i = 1}^\infty \subseteq dmn\left( f \right) , which requirements are sufficient for the asymptotic ∑ n ∈ A N P ( n ) ≤ N θ f ( n ) ∼ ρ ( 1 / θ ) ∑ n ∈ A N f ( n ) \sum\limits_{\matrix{{n \in {A_N}} \hfill \cr {P\left( n \right) \le {N^\theta }} \hfill \cr } } {f\left( n \right) \sim \rho \left( {1/\theta } \right)\sum\limits_{n \in {A_N}} {f\left( n \right)} } to hold, where N is a positive integer, θ ∈ (0, 1) is a constant, P(n) denotes the largest prime factor of n, and ρ is the Dickman function.

Finitary monads on Pos are characterized as precisely the free-algebra monads of varieties of algebras. These are classes of ordered algebras specified by inequations in context. Analogously, finitary enriched monads on Pos are characterized: here we work with varieties of coherent algebras which means that their operations are monotone.

He (J. Inequal. Appl. 2012:Article ID 162 2012) introduced the proximal point CQ algorithm (PPCQ) for solving the split equilibrium problem (SEP). However, the PPCQ converges weakly to a solution of the SEP and is restricted to monotone bifunctions. In addition, the step-size used in the PPCQ is a fixed constant μ in the interval $(0, \frac{1}{ \| A \|^{2} } )$ ( 0 , 1 ∥ A ∥ 2 ) . This often leads to excessive numerical computation in each iteration, which may affect the applicability of the PPCQ. In order to overcome these intrinsic drawbacks, we propose a robust step-size $\{ \mu _{n} \}_{n=1}^{\infty }$ { μ n } n = 1 ∞ which does not require computation of $\| A \|$ ∥ A ∥ and apply the adaptive step-size rule on $\{ \mu _{n} \}_{n=1}^{\infty }$ { μ n } n = 1 ∞ in such a way that it adjusts itself in accordance with the movement of associated components of the algorithm in each iteration. Then, we introduce a self-adaptive extragradient-CQ algorithm (SECQ) for solving the SEP and prove that our proposed SECQ converges strongly to a solution of the SEP with more general pseudomonotone equilibrium bifunctions. Finally, we present a preliminary numerical test to demonstrate that our SECQ outperforms the PPCQ.

In this manuscript, we study a certain classical second-order fully nonlinear coupled system with generalized nonlinear coupled boundary conditions satisfying the monotone assumptions. Our new results unify the existence criteria of certain linear and nonlinear boundary value problems (BVPs) that have been previously studied on a case-by-case basis; for example, Dirichlet and Neumann are special cases. The common feature is that the solution of each BVPs lies in a sector defined by well-ordered coupled lower and upper solutions. The tools we use are the coupled lower and upper solutions approach along with some results of fixed point theory. By means of the coupled lower and upper solutions approach, the considered BVPs are logically modified to new problems, known as modified BVPs. The solution of the modified BVPs leads to the solution of the original BVPs. In our case, we only require the Nagumo condition to get a priori bound on the derivatives of the solution function. Further, we extend the results presented in (Franco et al. in Extr. Math. 18(2):153–160, 2003; Franco et al. in Appl. Math. Comput. 153:793–802, 2004; Franco and O’Regan in Arch. Inequal. Appl. 1:423–430, 2003; Asif et al. in Bound. Value Probl. 2015:134, 2015). Finally, as an application, we consider the fully nonlinear coupled mass-spring model.

Given a set F of k disjoint monotone orthogonal polygons with a total of m vertices, we present bounds on the number of vertex guards required to guard the free space and the boundaries of the polygons in F when the range of vision of each guard is bounded by 180∘ (the region in front of the guard). When the orthogonal polygons are axis aligned we prove that m 2 + ⌊ k 4 ⌋ + 4 vertex guards are always sufficient. When the orthogonal polygons are arbitrary oriented, we show that m 2 + k + 1 vertex guards are sometimes necessary and conjecture the bound is tight.

Inspired by the classical works on obtaining order from semigroups, recently, many researchers have proposed orders based on associative fuzzy logic connectives. However, the use of these monotone operators succinctly assumes and implies the presence of an (existing) order on the underlying set. In this work, we consider associative operations F on a non-empty set P without recourse to any ordering that may or may not be available on it. Picking the most general of the definitions of order proposed so far, that of Karacal and Kesiciogˇlu, we determine the necessary and sufficient conditions on an associative operation F to obtain a poset on P . Following this we investigate the classes of t-norms, t-conorms, uninorms and nullnorms – which are the typical fuzzy logic operations considered so far to obtain orders – that satisfy these conditions and also do a comparative study of the structures obtained from the different orders proposed so far. Finally, we explore further conditions required on an associative operation F to obtain richer order-theoretic structures.

Given an aggregation function, and two fuzzy subgroups of a group, this study investigates whether the aggregation of them is also a fuzzy subgroup. It is proven that if the cardinal of the group is a prime power, then it is always a fuzzy subgroup. For a general group, it is proven that the existence of a binary relation between the fuzzy subgroups determines if their aggregation is a fuzzy subgroup. Moreover, given two fuzzy subgroups, if the aggregation of them is a fuzzy subgroup for some strictly monotone aggregation function, then it is a fuzzy subgroup for any aggregation function. The present study also examines conditions on the aggregation function that characterise the case where the aggregation of two arbitrary fuzzy subgroups is also a fuzzy subgroup.

The problem of monotone missing data has been broadly studied during the last two decades and has many applications in various fields such as bioinformatics or statistics. Commonly used imputation techniques require multiple iterations through the data before yielding convergence. Moreover, those approaches may introduce extra noises and biases to the subsequent modeling. In this work, we derive exact formulas and propose a novel algorithm to compute the maximum likelihood estimators (MLEs) of a multiple class, monotone missing dataset when all the covariance matrices of all categories are assumed to be equal, namely Efficient Parameter Estimation for Multiple Class Monotone Missing Data (EPEM). We then illustrate an application of our proposed methods in Linear Discriminant Analysis (LDA). As the computation is exact, our EPEM algorithm does not require multiple iterations through the data as other imputation approaches, thus promising to handle much less time-consuming than other methods. This effectiveness was validated by empirical results when EPEM reduced the error rates significantly and required a short computation time compared to several imputation-based approaches. We also release all codes and data of our experiments in a GitHub repository to contribute to the research community related to this problem.

We study the nonlinear newsvendor problem concerning goods of a non-discrete nature, and a class of stochastic dynamic programs with several application areas such as supply chain management and economics. The class is characterized by continuous state and action spaces, either convex or monotone cost functions that are accessed via value oracles, and affine transition functions. We establish that these problems cannot be approximated to any degree of either relative or additive error, regardless of the scheme used. To circumvent these hardness results, we generalize the concept of fully polynomial-time approximation scheme allowing arbitrarily small additive and multiplicative error at the same time, while requiring a polynomial running time in the input size and the error parameters. We develop approximation schemes of this type for the classes of problems mentioned above. In light of our hardness results, such approximation schemes are “best possible”. A computational evaluation shows the promise of this approach.

A non-empty subset S together with an associative function f from S × S×S into the family of all non-empty subsets of S is called a ternary semihypergroup. In this paper, we consider a semihypergroup (S, f) besides a binary relation ≤, where ≤ is a partial order relation on S such that satisfies the monotone condition. This structure is called an ordered ternary semihypergroup. We introduce and investigate the notions of bi-hyperideal and quasi-hyperideal in ordered ternary semihyperroups. In particular, we prove that an ordered ternary semihypergroup is left and right simple if and only if it does not contain proper bi-hyperideals

A projected extrapolated gradient method is designed for solving monotone variational inequality in Hilbert space. Requiring local Lipschitz continuity of the operator, our proposed method improves the value of the extrapolated parameter and admits larger step sizes, which are predicted based a local information of the involved operator and corrected by bounding the distance between each pair of successive iterates. The correction will be implemented when the distance is larger than a given constant and its main cost is to compute a projection onto the feasible set. In particular, when the operator is the gradient of a convex function, the correction step is not necessary. We establish the convergence and ergodic convergence rate in theory under the larger range of parameters. Related numerical experiments illustrate the improvements in efficiency from the larger step sizes.

This paper introduces a new subclass of convex functions called uniformly convex functions [this class is a slight extension of the class introduced earlier by Zalinescu (J Math Anal Appl 95:344–374, 1983; Convex analysis in general vector spaces, World Scientific, Singapore)] and studies some of its properties. It also defines a new subdifferentiability called $$(\phi ,c)$$ -subdifferentiability and investigates some of its properties. As a main result, we prove that a lower semicontinuous function defined on a Banach space is uniformly convex if and only if its Clarke subdifferential is uniformly monotone set-valued map. Then we move to present a result concerning optimization problems for uniformly convex functions. Finally, we apply our theory to conclude the Minty-Browder theorem for a subclass of monotone operators as well as the Lax–Milgram theorem without recourse to use the Riesz–Frechet theorem. Consequently, we establish the existence and uniqueness of weak solution for the p-Laplacian problem.

In this paper, we present some convergence results for various iterative algorithms built from Hardy-Rogers type generalized nonexpansive mappings and monotone operators in Hilbert spaces. We obtain some comparison results for the rates of convergence of these algorithms to the solution of variational inequality problem including Hardy–Rogers type generalized nonexpansive mappings and monotone operators. A numerical example is given to validate these results. We apply the iterative algorithms handled herein to solve convex minimization problem and illustrate this result by providing a non-trivial numerical example. The presented results considerably improve the corresponding results in Ali et al. (Comput Appl Math 39:74, 2020. https://doi.org/10.1007/s40314-020-1101-4 ).

Sample average approximation (SAA) approach for two-stage stochastic variational inequalities (SVIs) with continuous probability distributions, where the second-stage problems have multiple solutions, may not promise convergence assertions as the sample size tends to infinity. In this paper, a regularized SAA approach is proposed to numerically solve a class of two-stage SVIs with continuous probability distributions, where the second-stage problems are monotone and allowed to have multiple solutions. We first give some structural properties. After that, the convergence analysis of the regularized SAA approach for two-stage SVIs is investigated as the regularization parameter tends to zero and the sample size tends to infinity. Finally, we employ the progressive hedging algorithm to report some numerical results.

A new renewable warranty policy is suggested that increases the probability of its success and can decrease the corresponding costs for certain ranges of parameters. It deals with heterogeneous populations of items from two subpopulations (‘weak’ and ‘strong’) and is aimed at elimination and further replacement of weak items after screening at some optimal time. This elimination is performed when degradation described by the corresponding mixed degradation process reaches some optimally predetermined level. Probabilistic and cost analyses of the model are performed and the illustrative example is presented. It is shown that the proposed warranty policy with inspection outperforms the conventional one in a probabilistic sense. Furthermore, the proposed policy becomes economically beneficial especially when the additional penalty cost caused by a sudden failure is large.

In this paper, a derivative-free R M I L conjugate gradient projection method for solving large-scale nonlinear monotone equations with convex constraints is proposed. The proposed method is a modification of an R M I L conjugate gradient method combined with the projection techniques. We establish its global convergence under appropriate conditions. The method is then compared with other existing methods in the literature and the numerical results indicate that the method is efficient. Furthermore, the proposed method is used to recover a sparse signal from an incomplete and contaminated sampling measurements and the results are promising.

Entropy is the most important concept used in information theory and measuring uncertainty. In Choquet calculus, Sugeno (2013) [10] and Torra and Narukawa (2016) [2] studied Choquet integral and derivative with respect to monotone measures on the real line. Then as a very challenging problem, the definition of entropy and relative entropy on monotone measures for infinite sets based on Choquet integral was proposed by Torra (2017) [1] and Agahi (2019) [12] . These results show that based on the submodularity condition on monotone measures, entropy and relative entropy for Choquet integral are non-negative. In this paper, we first introduce the concept of Lin divergence (Lin, 1991, [8] ), including Choquet integral and derivative with respect to monotone measures. Then some fundamental properties of this concept in information theory are given. In special case, we show that we can omit the submodularity condition in previous results on entropy and relative entropy for Choquet integral.

The line sum optimization problem asks for a ( 0 , 1 ) -matrix minimizing the sum of given functions evaluated at its row and column sums. We show that the uniform problem, with identical row functions and identical column functions, and the monotone problem, over matrices with nonincreasing row and column sums, are polynomial time solvable.

Rearranging objects on a planar surface arises in a variety of robotic applications, such as product packaging. Using two arms can improve efficiency but introduces new computational challenges. This article studies the problem structure of object rearrangement using two arms in synchronous, monotone tabletop setups and develops an optimal mixed-integer model. It then describes an efficient and scalable algorithm, which first minimizes the cost of object transfers and then moves between objects. This is motivated by the fact that, asymptotically, object transfers dominate the cost of solutions. Moreover, a lazy strategy minimizes the number of motion planning calls and results in significant speedups. Theoretical arguments support the benefits of using two arms and indicate that synchronous execution, in which the two arms perform together either transfers or moves, introduces only a small overhead. Experiments support these claims and show that the scalable method can quickly compute solutions close to the optimal for the considered setup. Note to Practitioners —Monotone tabletop rearrangement challenges arise in a variety of automation scenarios, including product sorting or packing. Performing this task with two robotic manipulators introduces the overhead of coordinating them in the shared workspace, as well as an increase in the size of the underling search space. The objective of this work is to study the feasibility of such dual-arm solutions, providing both theoretical bounds, as well as a fast, and approximate solution. The approach leverages an effective algorithmic decomposition of the problem so as to take advantage of efficient motion planners and mixed-integer linear programming solvers. The proposed solution has been evaluated in settings that include delta robots as well as seven-degree-of-freedom (DOF) manipulators. Interesting extensions of this work correspond to studying the case of additional arms, nonmonotone, and general manipulation scenarios.

Log-rank conjecture is one of challenging problems in communication complexity. It is known to hold for some special function classes such as XOR functions $\text {f}(\text {x}\oplus \text {y})$ when the outer function f is monotone, symmetric, $\mathsf {AC}^{0}$ , low $\mathbb {F}_{2}$ -degree, linear threshold or read-k. However, for AND functions $\text {f}(\text {x}\wedge \text {y})$ , Log-rank conjecture is only known to hold when the outer function is a monotone function or a function with small alternating numbers. In this paper, we show that Log-rank conjecture holds for AND functions whose outer functions are read-once $\mathbb {F}_{2}$ -degree polynomials, constant- $\mathbb {F}_{2}$ -degree polynomials, and symmetric functions. For read-once and symmetric AND functions, their proof are based on the elementary analysis of Mobius sparsity of the outer functions and designing efficient protocols for computing the corresponding AND functions in terms of the Mobius sparsity of their outer functions. For a constant- $\mathbb {F}_{2}$ -degree AND function $\text {f}(\text {x}\wedge \text {y})$ , we use the concept of the variable rank to design a decision tree for computing f where the tree depth is bounded above by a polynomial in the logarithm of the Mobius sparsity of f. The communication protocol can be obtained easily from the constructed decision tree. Our results deepen the understanding of the log-rank conjecture for AND functions.

Abstract A quasilinear parabolic problem with a time fractional derivative of the Caputo type and mixed boundary conditions is considered. The coefficients of the elliptic operator depend on the gradient of the solution, and this operator is uniformly monotone and Lipschitz-continuous. For this problem, unconditionally stable linear regularized semi-discrete scheme is constructed based on the $$L1$$-approximation of the fractional time derivative. Stability estimates are obtained by the variational method. Accuracy estimates are given provided that the initial data and the solution to the differential problem are sufficiently smooth. The proved result of stability of the semi-discrete scheme is valid for the mesh scheme obtained from the semi-discrete problem using the finite element method in spatial variables.

In the companion letter (Part 1), we have defined and exemplified meta distributions (MDs) as a natural extension of the concepts of the mean and distribution of a random variable. Here we provide an in-depth discussion of the properties and interpretations of MDs. It includes original results on the calculation of MDs in the monotone case and two applications to simple Poisson wireless networks models.

In this article, we present a study on constructing a second order local approximation monotone difference schemes on spatial non-uniform grids for the parabolic equation of convection-diffusion type with a third kind boundary condition without using the basic differential equation at the boundary of the domain. The goal is a combination of the differential inequality, the regularization principle and the assumption of the existence and uniqueness of a smooth solution. In this case, the boundary conditions are directly approximated with the second order on a two-point stencil. The convergence of the proposed algorithms to the solution of the original differential problem with the second order is proved. With the help of difference maximum principle, two-sided estimates of the difference solution are established and an important a priori estimate in a uniform $$C$$ -norm is obtained.

This study deals with relaxing the regressor square integrability condition $$\omega (t)\notin \mathrm {L}_{2} $$ in the DREM procedure so as to ensure the monotone asymptotic convergence of the parametric error in the problem of estimating constant parameters of a linear regression dependence.

Complex symmetric Sylvester matrix equations appear in many applications, such as the numerical solution of the complex Helmholtz equations. In this paper, by designing a global complex symmetric M -Lanczos process we develop a global variant of the conjugate A-orthogonal conjugate residual method (Gl-COCR) for solving the Sylvester matrix equation A X + X B = C with complex symmetric coefficient matrices. To obtain the smooth and monotone convergence behavior, we also propose a smoothed Gl-COCR method, denoted by SGl-COCR. Finally, numerical examples are given to illustrate the performances of our methods.

This paper addresses the problem of verifying discrete-time stochastic systems against omega-regular specifications using finite-state abstractions. Omega-regular properties allow specifying complex behavior and encompass, for example, linear temporal logic. We focus on a class of systems with mixed monotone dynamics. This class has recently been show to be amenable to efficient reachable set computation and models a wide-range of physically relevant systems. In general, finite-state abstractions of continuous state stochastic systems give rise to augmented Markov Chains wherein the probabilities of transition between states are restricted to an interval. We present a procedure to compute a finite-state Interval-valued Markov Chain abstraction of discrete-time, mixed monotone stochastic systems subject to affine disturbances given a rectangular partition of the state-space. Then, we suggest an algorithm for performing verification against omega-regular properties in IMCs. Specifically, we aim to compute bounds on the probability of satisfying the specification of interest from any initial state in the IMC. This is achieved by solving a reachability problem on sets of so-called winning and losing components in the Cartesian product between the IMC and a Rabin automaton representing the specification. Next, the verification of IMCs may yield a set of states whose acceptance status is undecided with respect to the specification, requiring a refinement of the abstraction. We describe a specification-guided approach that compares the best-case and worst-case behaviors of accepting paths in the IMC and targets the appropriate states accordingly. Finally, we show a case study.

Properties of suns in the spaces $$L^1$$ and $$C(Q)$$ are studied. It is shown that every boundedly compact sun in $$L^1$$ is convex and every boundedly weakly compact sun in $$C(Q)$$ is monotone path-connected.

Despite the popularity and practical importance of the fuzzy inference system (FIS), the use of an FIS model as an n -ary aggregation function, which is characterized by both the monotonicity and boundary properties, is yet to be established. This is because research on ensuring that FIS models satisfy the monotonicity property, i.e., monotone FIS, is relatively new, not to mention the additional requirement of satisfying the boundary property. The aim of this article, therefore, is to establish the parametric conditions for the Takagi–Sugeno–Kang (TSK) FIS model to operate as an n -ary aggregation function (hereafter denoted as n -TSK-FIS) via the specifications of fuzzy membership functions and fuzzy rules. An absorption property with fuzzy rules interpretation is outlined, and the use of n -TSK-FIS as a uninorm is explained. Exploiting the established parametric conditions, a framework for which an n -TSK-FIS model can be constructed from data samples is formulated and analyzed, along with a number of remarks. Synthetic data sets and a benchmark example on education assessment are presented and discussed. To be best of the authors’ knowledge, this article serves as the first use of the TSK-FIS model as an n -ary aggregation function.

This paper considers the optimal investment-reinsurance problem under the monotone mean-variance preference. The monotone mean-variance preference is a monotone version of the classical mean-variance preference. First of all, we reformulate the original problem as a zero-sum stochastic differential game. Secondly, the optimal strategy and the optimal value function for the monotone mean-variance problem are derived by the approach of dynamic programming and the Hamilton-Jacobi-Bellman-Isaacs equation. Thirdly, the efficient frontier is obtained and it is proved that the optimal strategy is an efficient strategy. Finally, the continuous-time monotone capital asset pricing model is derived.

Recent research shows that the search for Bayesian estimation of concave production functions is a fruitful area of investigation. In this article, we use a flexible cost function that satisfies gl...

In a Banach space, we study an operator equation with a monotone operator T. The operator is an operator from a Banach space to its conjugate, and T=AC, where A and C are operators of some classes. The considered problem belongs to the class of ill-posed problems. For this reason, an operator regularization method is proposed to solve it. This method is constructed using not the operator T of the original equation, but a more simple operator A, which is B-monotone, B=C−1. The existence of the operator B is assumed. In addition, when constructing the operator regularization method, we use a dual mapping with some gauge function. In this case, the operators of the equation and the right-hand side of the given equation are assumed to be perturbed. The requirements on the geometry of the Banach space and on the agreement conditions for the perturbation levels of the data and of the regularization parameter are established, which provide a strong convergence of the constructed approximations to some solution of the original equation. An example of a problem in Lebesgue space is given for which the proposed method is applicable.

This paper is devoted to the study of the uniqueness and stability of bistable traveling waves for monotone semiflows in an abstract setting. Under appropriate assumptions, we establish the uniqueness and stability of bistable waves for discrete and continuous-time semiflows in a continuous habitat by appealing to a global convergence theorem for monotone semiflows. We also extend such a result to time-periodic semiflows, and apply the general theory to a class of reaction-diffusion-advection systems in a cylinder.

We show that the space of anti-symplectic involutions of a monotone $S^2\times S^2$ whose fixed points set is a Lagrangian sphere is connected. This follows from a stronger result, namely that any two anti-symplectic involutions in that space are Hamiltonian isotopic., 8 pages, 1 figure

In this paper, we prove the boundedness of Hardy operator for monotone functions in grand Lebesgue spaces Lp)(0,1)(0

This paper investigates the exponential stability of traveling wave solutions for nonlinear delayed cellular neural networks. As a continuity of the past work (Wu and Niu, 2016; Yu, et al., 2011) on the existence and uniqueness of the traveling wave solutions, it is very reasonable and interesting to consider the exponential stability of the traveling wave solutions. By the weighted energy method, comparison principle and the first integral mean value theorem, this paper proves that, for all monotone traveling waves with the wave speed $$c < c_1^ \ast < 0$$ or $$c > c_2^ \ast > 0$$ , the solutions converge time-exponentially to the corresponding traveling waves, when the initial perturbations decay at some fields.