In this paper, we introduce a new F-normed space, namely Orlicz–Lorentz spaces equipped with the Mazur–Orlicz F-norm. Some basic properties in Orlicz–Lorentz spaces equipped with the Mazur–Orlicz ...F-norm are given. We find a tool to study the geometry property of Orlicz–Lorentz function spaces, the necessary and sufficient conditions for strict monotonicity, lower local uniform monotonicity, upper local uniform monotonicity in Orlicz–Lorentz spaces endowed with the Mazur–Orlicz F-norm are obtained without any assumptions. The tool also can simplify the proof of the corresponding results of Orlicz–Lorentz spaces equipped with the Luxemburg norm without condition (+).
•Addresses numerical algorithms for pseudo-monotone variational inequalities.•Proves the convergence of Tseng’s FBF method and validates the theoretical results with numerical experiments.•Emphasizes ...the interplay between discrete and continuous time approaches to variational inequalities.
Tseng’s forward–backward–forward algorithm is a valuable alternative for Korpelevich’s extragradient method when solving variational inequalities over a convex and closed set governed by monotone and Lipschitz continuous operators, as it requires in every step only one projection operation. However, it is well-known that Korpelevich’s method converges and can therefore be used also for solving variational inequalities governed by pseudo-monotone and Lipschitz continuous operators. In this paper, we first associate to a pseudo-monotone variational inequality a forward–backward–forward dynamical system and carry out an asymptotic analysis for the generated trajectories. The explicit time discretization of this system results into Tseng’s forward–backward–forward algorithm with relaxation parameters, which we prove to converge also when it is applied to pseudo-monotone variational inequalities. In addition, we show that linear convergence is guaranteed under strong pseudo-monotonicity. Numerical experiments are carried out for pseudo-monotone variational inequalities over polyhedral sets and fractional programming problems.
Elman network is a classical recurrent neural network with an internal delay feedback. In this paper, we propose a recalling-enhanced recurrent neural network (RERNN) which has a selective memory ...property. In addition, an improved conjugate algorithm with generalized Armijo search technique that speeds up the convergence rate is used to train the RERNN model. Further enhancement performance is achieved with adaptive learning coefficients. Finally, we prove weak and strong convergence of the presented algorithm. In other words, as the number of training steps increases, the following has been established for RERNN: (1) the gradient norm of the error function with respect to the weight vectors converges to zero, (2) the weight sequence approaches a fixed optimal point. We have carried out a number of simulations to illustrate and verify the theoretical results that demonstrate the efficiency of the proposed algorithm.
The logrank test is a well-known nonparametric test which is often used to compare the survival distributions of two samples including right-censored observations, it is also known as the ...Mantel-Haenszel test. The
family of tests, generalizes the logrank test by using weights assigned to observations. In this paper, we present a switch monotonicity property for the
family of tests, which was motivated by the need to derive bounds for the test statistic in case of imprecise data observations. This property states that, when all observations from two independent groups are ranked together, the value of the z-test statistic is monotonically increasing after switching a pair of adjacent values from the two groups. Two examples are provided to motivate and illustrate the result presented in this paper.
A system identification-based framework is used to develop monotone fuzzy If-Then rules for formulating monotone zero-order Takagi-Sugeno-Kang (TSK) fuzzy inference systems (FISs) in this paper. ...Convex and normal trapezoidal and triangular fuzzy sets, together with a strong fuzzy partition strategy (either fixed or adaptive), is adopted. By coupling the strong fuzzy partition with a set of complete and monotone fuzzy If-Then rules, a monotone TSK FIS model can be guaranteed. We show that when a clean multiattribute monotone dataset is used, a system identification-based framework does not guarantee the production of monotone fuzzy If-Then rules, which leads to nonmonotone TSK FIS models. This is a new learning phenomenon that needs to be scrutinized when we design data-based monotone TSK FIS models. Two solutions are proposed: 1) a new monotone fuzzy rule relabeling-based method and 2) a constrained derivative-based optimization method. A new modeling framework with an adaptive fuzzy partition is evaluated. The results indicate that TSK FIS models with better accuracy (a lower sum square error) and a good degree of monotonicity (measured with a monotonicity test) are achieved. In short, the main contributions of this study are validation of the new learning phenomenon and introduction of useful methods for developing data-based monotone TSK FIS models.
In the course of a computational experiment on bank customer churn data, we demonstrate the explanatory and predictive capacity of monotonic decision rules. The data exhibit a partially ordinal ...character, as certain attribute value sets describing the clients are ordered and demonstrate a monotonic relationship with churn or non-churn outcomes. The data are structured by the Variable Consistency Dominance-based Rough Set Approach (VC-DRSA) prior to the induction of monotonic decision rules. The supervised learning is conducted using an extended version of VC-DRSA, implemented in RuLeStudio and RuleVisualization programs. The first one is designed to experiment with parameterized rule models, and the second one is used for visualization and a thorough examination of the rule model. The monotonic decision rules give insight into the bank data, characterizing loyal customers and the ones who left the bank. Such an approach is in line with explainable AI, aiming to obtain a transparent decision model, that can be easily understood by decision-makers. We also compare the predictive performance of monotonic rules with some well-known machine learning models.
•Dominance-based Rough Set Approach to customer satisfaction analysis is presented.•Monotonic decision rules are induced from ordinal data.•Explanatory and predicting capacity of monotonic rule model is demonstrated.•Computational experiment with monotonic rules and other learning models is performed.
Many theories in finance imply monotonic patterns in expected returns and other financial variables. The liquidity preference hypothesis predicts higher expected returns for bonds with longer times ...to maturity; the Capital Asset Pricing Model (CAPM) implies higher expected returns for stocks with higher betas; and standard asset pricing models imply that the pricing kernel is declining in market returns. The full set of implications of monotonicity is generally not exploited in empirical work, however. This paper proposes new and simple ways to test for monotonicity in financial variables and compares the proposed tests with extant alternatives such as
t-tests, Bonferroni bounds, and multivariate inequality tests through empirical applications and simulations.
In this paper, we develop a sliding method for the fractional p-Laplacian. We first obtain the key ingredient needed in the sliding method in a bounded domain–the narrow region principle. Then using ...nonlinear equations involving the fractional p-Laplacian in both bounded domains and in the whole space, we illustrate how this new sliding method can be employed to obtain monotonicity of solutions. During these processes, we introduce a new idea–estimating the singular integrals defining the fractional p-Laplacian along a sequence of approximate maximum points.
We believe that the new ideas and methods employed here can be conveniently applied to study a variety of nonlocal problems with more general operators and more general nonlinearities.
We study the strong unique continuation property backwards in time for the nonlocal equation in Rn+1(0.1)(∂t−Δ)su=V(x,t)u,s∈(0,1). Our main result Theorem 1.2 can be thought of as the nonlocal ...counterpart of the result obtained in 30 for the case when s=1. In order to prove Theorem 1.2 we develop the regularity theory of the extension problem for the equation (0.1). With such theory in hands we establish:(i)a basic monotonicity result for an adjusted frequency function which plays a central role in this paper, see Theorem 1.3 below;(ii)an extensive blowup analysis of the so-called Almgren rescalings associated with the extension problem. We feel that our work will also be of interest e.g. in the study of certain basic open questions in free boundary problems, as well as in nonlocal segregation problems.
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