Monotone Functions Research Articles

Series expansion, higher-order monotonicity properties and inequalities for the modulus of the GrÖtzsch ring

Abstract For $r\in(0,1)$ , let $\mu \left( r\right) $ be the modulus of the plane Grötzsch ring $\mathbb{B}^2\setminus[0,r]$ , where $\mathbb{B}^2$ is the unit disk. In this paper, we prove that \begin{equation*} \mu \left( r\right) =\ln \frac{4}{r}-\sum_{n=1}^{\infty }\frac{\theta _{n}}{ 2n}r^{2n}, \end{equation*} with $\theta _{n}\in \left( 0,1\right)$ . Employing this series expansion, we obtain several absolutely monotonic and (logarithmically) completely monotonic functions involving $\mu \left( r\right) $ , which yields some new results and extend certain known ones. Moreover, we give an affirmative answer to the conjecture proposed by Alzer and Richards in H. Alzer and K. Richards, On the modulus of the Grötzsch ring, J. Math. Anal. Appl. 432(1): (2015), 134–141, DOI 10.1016/j.jmaa.2015.06.057. As applications, several new sharp bounds and functional inequalities for $\mu \left( r\right) $ are established.

A 3D index for measuring economic resilience with application to modern financial crises

AbstractThis paper proposes a vector index R composed of monotonous functions capturing three theoretical aspects (3D) of economic resilience (engineering, ecological, evolutionary) and performs an analysis of GDP data of 200 countries worldwide for the period 1960–2020 and 15 economic crises (shocks). The empirical analysis, first, reveals that economic resilience appears positively related to gross value added (GVA), employment, education, sectorial structure and diversification, agricultural production, tax revenue and natural resources, whereas it is negatively associated with urbanization degree, primary sector specialization and trade activity. Also, it shows that the global economy shifted to a less resilient performance through time, and it reveals the pros and cons of the continents' performance in economic resilience, highlighting a relative sensitivity of European countries against the available shocks and relatively higher resilience of Asian and African countries. Finally, the analysis of transition probabilities reveals the 2008 economic crisis as the only shock with significant transitions among different states of resilience, providing insights into the insignificant geographical scale of previous shocks in the global economy. Overall, in the complex conceptualization and measurement framework of economic resilience, the proposed 3D index contributes to a multilevel consideration of diverse theoretical aspects of this concept, provides empirical insights into the global economy's resilience and facilitates better management of economic resilience‘s computational complexity.

Long-time behaviors of wave equations stabilized by boundary memory damping and friction damping

In this paper, we study the long-time behaviors of wave equations subject to boundary memory damping and friction damping. Different from assumptions that memory kernel is a nonnegative, monotone function in the previous literatures, we assume that the primitive of the memory kernel is a generalized positive definite kernel (abbreviated to GPDK), which may vary sign or oscillate. The key to the problem lies in establishing the connection between memory damping and energy terms. By combining the properties of the positive definite kernel with classical multiplier methods, and constructing auxiliary systems, we ultimately establish the asymptotic stability, exponential stability and polynomial stability of systems featuring boundary memory damping and friction damping. To illustrate our theoretical results, we provide some numerical simulations.

A Novel Class of Unfolding Models for Binary Preference Data

Abstract We develop a new class of spatial voting models for binary preference data that can accommodate both monotonic and non-monotonic response functions, and are more flexible than alternative “unfolding” models previously introduced in the literature. We then use these models to estimate revealed preferences for legislators in the U.S. House of Representatives and justices on the U.S. Supreme Court. The results from these applications indicate that the new models provide superior complexity-adjusted performance to various alternatives and that the additional flexibility leads to preferences’ estimates that more closely match the perceived ideological positions of legislators and justices.

Trends in satellite-based ocean parameters through integrated time series decomposition and spectral analysis: 1. chlorophyll, sea surface temperature, and sea level anomaly

Abstract This study investigated trends in satellite-based chlorophyll-a (Chl-a; 1998–2022), sea surface temperature (SST; 1982–2022), and sea level anomaly (SLA; 1993–2021) from the European Space Agency’s Climate Change Initiative records, integrating time series decomposition and spectral analysis. Trends in parameters signify prolonged increases, decreases, or no changes over time. These are time series in the same space as original parameters, excluding seasonalities and noise, and can exhibit nonlinearity. Trend rates approximate the pace of change per time unit. We quantified trends using conventional linear-fit and three incrementally advancing methods for time series decomposition: simple moving average (SMA), seasonal-trend decomposition using locally estimated scatterplot smoothing (STL), and multiple STL (MSTL), across the global ocean, the Bay of Bengal, and the Chesapeake Bay. Challenges in decomposition include specifying accurate seasonal periods that are derived here by combining Fourier and Wavelet Transforms. Globally, SST and SLA trend upwards, and Chl-a has no significant change, yet regional variations are notable. We highlight the advantage of extracting multiple periods with MSTL and, more broadly, decomposition’s role in disentangling time-series components (seasonality, trend, noise) without resorting to monotonic functions, thereby preventing overlooking episodic events. Illustrations include extreme events temporarily counteracting background trends, e.g., the 2010–2011 SLA drop due to La Niña-induced rainfall over land. The continuous analysis clarifies the warming hiatus debate, affirming sustained warming. Decadal trend rates per grid cell are also mapped. These are ubiquitously significant for SST and SLA, whereas Chl-a trend rates are globally low but extreme across coasts and boundary currents.

Applications of the Hahn-Banach Theorem, a Solution of the Moment Problem and the Related Approximation

We start by an application the of Krein–Milman theorem to the integral representation of completely monotonic functions. Elements of convex optimization are also mentioned. The paper continues with applications of Hahn–Banach-type theorems and polynomial approximation to obtain recent results on the moment problem on the unbounded closed interval [0,+∞). Necessary and sufficient conditions for the existence and uniqueness of the solution are pointed out. Operator-valued moment problems and a scalar-valued moment problem are solved.

Chemo-elasto-electro free energy of non-uniform system in the diffuse interface context

In the present work, we propose an alternative approach for deriving the free energy formulation of a non-uniform system. Compared with the work of Cahn and Hilliard (1958 J.Chem. Phys.28258-67), our approach provides a more comprehensive explanation for the individual energy contribution in a non-uniform system, including entropy, interaction energy, and internal energy. By employing a fundamental mathematical calculus, we reformulate the local composition within the interface region. Utilizing the reformulated local composition as well as classic thermodynamic principles, we establish formal expressions for entropy, interaction energy, and the internal energy, which are functions of both composition and composition gradients. We obtain a comprehensive free energy expression for a non-uniform system by integrating these energy density formulations. The obtained free energy expression is consistent with the formula type of Cahn and Hilliard and prodives more deeper physical interpretation. Moreover, using the same approach, we derive formulations for elastic energy and electric potential energy in a non-uniform system. However, the proposed approach encounters a limitation in the special case of a non-uniform fluid contacting a solid substrate. Due to the significant difference in the length scales between the solid-fluid and fluid-fluid interfaces, the wall free energy formulation based on the aforementioned concept is unsuitable for this multi-scale system. To address this limitation, we reformulate the wall free energy as a function of the average composition over the solid-fluid interface. Additionally, the previous derivation relies on an artificial treatment of describing the composition variation across the interface by a smooth monotone function, while the true nature of this variation remains unclear. By utilizing the concept of average composition, we circumvent the open question of how the composition varies across the interface region. Our work provides a thorough understanding for the construction of free energy formulations for a non-uniform system in condensed matter physics.

Geometric Characterization of Validity of the Lyapunov Convexity Theorem in the Plane for Two Controls under a Pointwise State Constraint

This paper concerns control BVPs, driven by ODEs x′t=ut, using controls u0· &u1· in L1a,b,R2. We ask these two controls to satisfy a very simple restriction: at points where their first coordinates coincide, also their second coordinates must coincide; which allows one to write (u1−u0)·=v·1,f· for some f·. Given a relaxed non bang-bang solution x¯·∈W1,1a,b,R2, a question relevant to applications was first posed three decades ago by A. Cellina: does there exist a bang-bang solution x^· having lower first-coordinate x^1·≤x¯1·? Being the answer always yes in dimension d=1, hence without f·, as proved by Amar and Cellina, for d=2 the problem is to find out which functions f· “are good”, namely “allow such 1-lower bang-bang solution x^· to exist”. The aim of this paper is to characterize “goodness of f·” geometrically, under “good data”. We do it so well that a simple computational app in a smartphone allows one to easily determine whether an explicitly given f· is good. For example: non-monotonic functions tend to be good; while, on the contrary, strictly monotonic functions are never good.

On submodular prophet inequalities and correlation gap

Prophet inequalities and secretary problems have been extensively studied in recent years due to their elegance, connections to online algorithms, stochastic optimization, and mechanism design problems in game theoretic settings. Rubinstein and Singla [31] developed a notion of combinatorial prophet inequalities in order to generalize the standard prophet inequality setting to combinatorial valuation functions such as submodular and subadditive functions. For non-negative submodular functions they demonstrated a constant factor prophet inequality for matroid constraints. Along the way they showed a variant of the correlation gap for non-negative submodular functions.In this paper we revisit their notion of correlation gap as well as the standard notion of correlation gap and prove much tighter and cleaner bounds. Via these bounds and other insights we obtain substantially improved constant factor combinatorial prophet inequalities for both monotone and non-monotone submodular functions over any constraint that admits an Online Contention Resolution Scheme. In addition to improved bounds we describe efficient polynomial-time algorithms that achieve these bounds.

The Chermak–Delgado measure as a map on posets

The Chermak–Delgado measure of a finite group is a function which assigns to each subgroup a positive integer. In this paper, we give necessary and sufficient conditions for when the Chermak–Delgado measure of a group is actually a map of posets, i.e., a monotone function from the subgroup lattice to the positive integers. We also investigate when the Chermak–Delgado measure, restricted to the centralizers, is increasing.

Uniform Convergence of Fourier Series for Hölder Continuous Functions and Functions of Bounded Variation

This article discusses two significant results about the uniform convergence of Fourier series. One is the uniform convergence of the Fourier series for Hölder-α(α>0) continuous functions, while the other is the uniform convergence of the Fourier series for functions that can be expressed as the sum of finitely many continuous monotonic functions. The two results are particularly useful. Because Hölder continuity condition ensures a specific regularity that facilitates the analysis and provides a robust framework for proving uniform convergence and many practical functions can be decomposed into monotonic components, making the theorem broadly applicable. By providing detailed proofs and applications, this study demonstrates the significant implications of uniform convergence in both theoretical and practical contexts. Moreover, this work paves the way for future research in higher-dimensional Fourier analysis, non-periodic functions, and adaptive Fourier techniques, underscoring the ongoing relevance and versatility of Fourier series in modern mathematics. Several well-known results emerge as corollaries of these findings, highlighting the robustness and applicability of these theorems in Fourier analysis.

Monotone Function Intervals: Theory and Applications

A monotone function interval is the set of monotone functions that lie pointwise between two fixed, -monotone functions. We characterize the set of extreme points of monotone function intervals and apply this to a number of economic settings. First, we leverage the main result to characterize the set of distributions of posterior quantiles that can be induced by a signal, with applications to political economy, Bayesian persuasion, and the psychology of judgment. Second, we combine our characterization with properties of convex optimization problems to unify and generalize seminal results in the literature on security design under adverse selection and moral hazard. (JEL C61, C65, D72, D82, D91, G12)

Using a monotonic density ratio model to increase the power of the goodness-of-fit test for logistic regression models with case-control data.

Logistic regression models are widely used in case-control data analysis, and testing the goodness-of-fit of their parametric model assumption is a fundamental research problem. In this article, we propose to enhance the power of the goodness-of-fit test by exploiting a monotonic density ratio model, in which the ratio of case and control densities is assumed to be a monotone function. We show that such a monotonic density ratio model is naturally induced by the retrospective case-control sampling design under the alternative hypothesis. The pool-adjacent-violator algorithm is adapted to solve for the constrained nonparametric maximum likelihood estimator under the alternative hypothesis. By measuring the discrepancy between this estimator and the semiparametric maximum likelihood estimator under the null hypothesis, we develop a new Kolmogorov-Smirnov-type statistic to test the goodness-of-fit for logistic regression models with case-control data. A bootstrap resampling procedure is suggested to approximate the -value of the proposed test. Simulation results show that the type I error of the proposed test is well controlled and the power improvement is substantial in many cases. Three real data applications are also included for illustration.

Synthesis of Optimal Correction Functions in the Class of Disjunctive Normal Forms

The paper proposes to consider individual heuristics as unreliably operating parts of the information processing system. In a separate case, several different heuristics are adopted to solve the same problem, and the results obtained are adjusted in a certain way. In this case, problems arise that are close in methodology to the problems of synthesizing reliable circuits from unreliable elements or making a collective expert decision. The work solves the problem of constructing an optimal correction function based on control material; classes of functions of k-valued logic under monotonicity restrictions are studied. A theorem on the completeness of the class of monotonic functions of k-valued logic for arbitrary k is proved, and a basis in the given class is proved and constructed. The problem of constructing an optimal corrector in the class of disjunctive normal forms of k-valued functions is solved.

Tropical convexity in location problems

We investigate location problems where the optimal solution is found within the tropical convex hull of the given input points. Our initial focus is on geodesically star-convex sets, using the asymmetric tropical distance. We introduce the concept of tropically quasiconvex functions, which have sub-level sets with this shape, and are closely related to monotonic functions. Our findings demonstrate that location problems using tropically quasiconvex functions as distance measures will result in an optimal solution within the tropical convex hull of the input points. We also extend this result to cases where the input points are replaced with tropically convex sets. Finally, we explore the applications of our research in phylogenetics, highlighting the properties of consensus methods that arise from our class of location problems.

Computation of Choquet integrals: Analytical approach for continuous functions

In the continuous case, analytical computations of the Choquet integral are limited, despite being commonly used in various applications. One can either use the definition, which is computationally demanding and impractical, or apply already existing formulas restricted only to monotone nonnegative functions on a real interval starting at zero. This article aims to present more convenient computational formulas for continuous functions without imposing restrictions on their monotonicity given any real interval. First, a more general approach to monotone functions is provided for both positive and negative functions. Then, reordering techniques are introduced to compute the Choquet integral of an arbitrary continuous function, and with these, a monotone equivalent to every function can be constructed. This equivalent function preserves the final Choquet integral value, implying that only formulas for monotone functions are required. In addition to general fuzzy measures, the article assumes particular cases of distorted Lebesgue measures and distorted probabilities as the most commonly used fuzzy measures.

Hardy Type Inequalities in Classical and Grand Lebesgue Spaces $L_{p)}$, $0<p\le 1$, for Quasi-Monotone Functions

In 2020 Rovshan A. Bandaliev et al. proved the boundedness of Hardy operator for monotone functions in grand Lebesgue spaces $L_{p)} (0,1)$, $0<p\leq1$. In particular, they established similar results for the Hardy operator in classical weighted Lebesgue spaces. Moreover, it is proved that the grand Lebesgue space $L_{p) } (0,1)$ is a quasi-Banach function space. In this work, we are interested in Hardy inequalities applied to quasi-monotonic functions in classical Lebesgue spaces and grand Lebesgue spaces. We establish the boundedness of Hardy operator for quasi-monotone functions in grand Lebesgue spaces $L_{p)}$, $w(0,1),$ $0<p\leq1$. In addition some integral inequalities for the Hardy operator are proved in classical weighted Lebesgue spaces $L_{p,w}(0,1)$, $0<p<1,$ for quasi-monotone functions. All inequalities are proved with sharp constants. Some results of Rovshan A. Bandaliev et al. are deduced as particular cases. Also other estimates are obtained in classical Lebesgue spaces for Hardy's operator and its dual.

Research on Talent Cultivation of Local Product Design Professionals Based on Cooperative Education

The booming growth of creative industries has resulted in a substantial rise in the need for talented individuals in innovative product design. A multitude of institutions and universities possess open-product students majoring in design. Nevertheless, due to the alterations in consumer demand, traditional product design urgently needs reform due to monotonous functions and serious homogenisation. In recent years, the cultural and creative sectors have been actively seeking innovative and intelligent products since they have become a new trend in the consumer market. Moreover, universities' current talent training model for cultivating such needs still needs improvement. Drawing from the current condition of talent development in product design and integrating the pedagogical knowledge gained recently, this paper presents a novel idea of local product design professional talent cultivation founded on the principles of cooperative education.

Nonlinear crossing strategy-based particle swarm optimizations with time-varying acceleration coefficients

In contemporary particle swarm optimization (PSO) algorithms, to efficiently explore global optimum solutions, it is common practice to set the inertia weight to monotonically decrease over time for stability, while allowing the two acceleration coefficients, representing cognitive and social factors, to adopt decreasing or increasing functions over time, including random variations. However, there has been little discussion on a unified design approach for these time-varying acceleration coefficients. This paper presents a unified methodology for designing monotonic decreasing or increasing functions to construct nonlinear time-varying inertia weight and two acceleration coefficients in PSO, along with a control strategy for exploring global optimum solutions. We first construct time-varying coefficients by linearly amplifying well-posed monotonic functions that decrease or increase over normalized time. Here, well-posed functions ensure satisfaction of specified conditions at the initial and terminal points of the search process. However, many of the functions employed thus far only satisfy well-posedness at either the initial or terminal points of the search time, prompting the proposal of a method to adjust them to virtually meet specified initial or terminal points. Furthermore, we propose a crossing strategy where the developed cognitive and social acceleration coefficients intersect within the search time interval, effectively guiding the search process by pre-determining crossing values and times. The performance of our Nonlinear Crossing Strategy-based Particle Swarm Optimization (NCS-PSO) is evaluated using the CEC2014 (Congress on Evolutionary Computation in 2014) benchmark functions. Through comprehensive numerical comparisons and statistical analyses, we demonstrate the superiority of our approach over seven conventional algorithms. Additionally, we validate our approach, particularly in a drone navigation scenario, through an example of optimal 3D path planning. These contributions advance the field of PSO optimization techniques, providing a robust approach to addressing complex optimization problems.

Time-Variant Reliability for Systems with Non-monotonic Limit-State Functions

Most engineering time-variant reliability problems are the result of component degradation and stochastic loading. The resultant failure modes, and their resultant limit-state functions, produce limit-state surfaces with unpredictable temporal trajectories that may exhibit a combination of increasing and decreasing failure probabilities. In many cases the trajectories are monotonic so that failure increases predictably: in other cases, this is not so. In this paper we present the discrete-time set theory derivation for non-monotonic situations wherein the limit-state surface may recede to provide, what only appears to be, ever decreasing failure probability. The presence of both monotonic and non-monotonic limit-state functions can be easily detected by a parametric polar plot of the most-likely failure points in standard normal space. The polar plot reveals the temporal limit-state surfaces that need to be retained to represent the system limit-state surfaces at any time instant. The minimum set herein is called the extreme limit-state surface. The impact of the work is that the cumulative distribution function (cdf) can be provided with a minimum of failure and safe events. This in turn gives rise to several solution options such as the multi-normal integral method or a special Monte Carlo simulation that obviates the tedious marching-out routine. A series system and a parallel system show the efficacy of the theory.