General Inequality Research Articles

A parametrized approach to generalized fractional integral inequalities: Hermite–Hadamard and Maclaurin variants

This paper introduces a novel parametrized integral identity that forms the basis for deriving a comprehensive class of generalized fractional integral inequalities. Building on recent advancements in fractional calculus, particularly in conformable fractional integrals, our approach offers a unified framework for various known inequalities. The novelty of this work lies in its ability to generate new and more general inequalities, including Hermite–Hadamard-, Maclaurin-, and corrected Maclaurin-type inequalities, by selecting specific parameter values. These results extend the scope of fractional integral inequalities and provide new insights into their structure. To demonstrate the practical applicability and accuracy of the theoretical findings, we present a detailed numerical example along with graphical representations.

Generalized Tensorial Simpson type Inequalities for Convex functions of Selfadjoint Operators in Hilbert Space

Several generalized Simpson tensorial type inequalities for self adjoint operators have been obtained with variation depending on the conditions imposed on the function f.

A gentle introduction to gradient-based optimization and variational inequalities for machine learning

Abstract The rapid progress in machine learning in recent years has been based on a highly productive connection to gradient-based optimization. Further progress hinges in part on a shift in focus from pattern recognition to decision-making and multi-agent problems. In these broader settings, new mathematical challenges emerge that involve equilibria and game theory instead of optima. Gradient-based methods remain essential—given the high dimensionality and large scale of machine-learning problems—but simple gradient descent is no longer the point of departure for algorithm design. We provide a gentle introduction to a broader framework for gradient-based algorithms in machine learning, beginning with saddle points and monotone games, and proceeding to general variational inequalities. While we provide convergence proofs for several of the algorithms that we present, our main focus is that of providing motivation and intuition.

Concentration of a Sparse Bayesian Model With Horseshoe Prior in Estimating High‐Dimensional Precision Matrix

ABSTRACTPrecision matrices are crucial in many fields such as social networks, neuroscience and economics, representing the edge structure of Gaussian graphical models (GGMs), where a zero in an off‐diagonal position of the precision matrix indicates conditional independence between nodes. In high‐dimensional settings where the dimension of the precision matrix exceeds the sample size and the matrix is sparse, methods like graphical Lasso, graphical SCAD and CLIME are popular for estimating GGMs. While frequentist methods are well‐studied, Bayesian approaches for (unstructured) sparse precision matrices are less explored. The graphical horseshoe estimate, applying the global‐local horseshoe prior, shows superior empirical performance, but theoretical work for sparse precision matrix estimations using shrinkage priors is limited. This paper addresses these gaps by providing concentration results for the tempered posterior with the fully specified horseshoe prior in high‐dimensional settings. Moreover, we also provide novel theoretical results for model misspecification, offering a general oracle inequality for the posterior. A concise set of simulations is performed to validate our theoretical findings.

Geometrical criterions of generalized subadditivity

The motivation of our studies is a theorem of J. Matkowski and T. Świątkowski from 1993, which provides convenient and sufficient conditions for the subadditivity of real functions. In the work was constructed a family of sets P⊂R×R with the property that each function whose graph is contained in P is subadditive. The purpose of this article is to generalize this result in several directions. We will discuss a more general inequality:f(∑i=1Nαixi)⩽∑i=1Nβif(xi). Moreover, our family of sets is constructed in such a way that an essentially larger class of solutions are covered. Geometric method used in presented paper may be applied to other functional inequalities like Hlawka's, submultiplicative and subquadratic inequalities as well as to obtaining irregular functions satisfying certain inequalities.

General Trapezoidal-Type Inequalities in Fuzzy Settings

In this study, trapezoidal-type inequalities in fuzzy settings have been investigated. The theory of fuzzy analysis has been discussed in detail. The integration by parts formula of analysis of fuzzy mathematics has been employed to establish an equality. Trapezoidal-type inequality for functions with values in the fuzzy number-valued space is proven by applying the proven equality together with the properties of a metric defined on the set of fuzzy number-valued space and Höler’s inequality. The results proved in this research provide generalizations of the results from earlier existing results in the field of mathematical inequalities. An example is designed by defining a function that has values in fuzzy number-valued space and validated the results numerically using the software Mathematica (latest v. 14.1). The p-levels of the defined fuzzy number-valued mapping have been shown graphically for different values of p∈0,1.

Nonsmooth projection-free optimization with functional constraints

AbstractThis paper presents a subgradient-based algorithm for constrained nonsmooth convex optimization that does not require projections onto the feasible set. While the well-established Frank–Wolfe algorithm and its variants already avoid projections, they are primarily designed for smooth objective functions. In contrast, our proposed algorithm can handle nonsmooth problems with general convex functional inequality constraints. It achieves an $$\epsilon $$ ϵ -suboptimal solution in $$\mathcal {O}(\epsilon ^{-2})$$ O ( ϵ - 2 ) iterations, with each iteration requiring only a single (potentially inexact) Linear Minimization Oracle call and a (possibly inexact) subgradient computation. This performance is consistent with existing lower bounds. Similar performance is observed when deterministic subgradients are replaced with stochastic subgradients. In the special case where there are no functional inequality constraints, our algorithm competes favorably with a recent nonsmooth projection-free method designed for constraint-free problems. Our approach utilizes a simple separation scheme in conjunction with a new Lagrange multiplier update rule.

The equivalent conditions for norm of a Hilbert-type integral operator with a combination kernel and its applications

Introducing adaptation parameters σ,σ1, formal parameters λi(i=1,2,3,4),κ,τ, and type parameters μ,ν, the integration operator is defined as T:Lpp(1−μσˆ)−1(R+)→Lppνσˆ−1(R+), Tf(y)=∫R+eλ1xμyν+κe−λ2xμyνeλ3xμyν+τe−λ4xμyνf(x)dx,y∈R+. Using the weight function method, a general Hilbert-type integral inequality is obtained, thereby proving the boundedness of the operator. The constant factor of the general Hilbert-type inequality is the best possible if and only if the adaptation parameters satisfy σ=σ1. From this, the formula for calculating the operator norm is obtained. In terms of application, some results from the references have been consolidated by discussing the combination of formal parameters and type parameters, and many new operator inequalities of different types and forms have been derived.

Trends in educational inequalities in smoking-attributable mortality and their impact on changes in general mortality inequalities: evidence from England and Wales, Finland, and Italy (Turin)

BackgroundSocioeconomic mortality inequalities are persistent in Europe but have been changing over time. Smoking is a known contributor to inequality levels, but knowledge about its impact on time trends in...

A gradient-based optimization algorithm to solve optimal control problems with conformable fractional-order derivatives

In this paper, we solve numerically a class of fractional optimal control problems in the conformable sense. We first propose a nonlinear fractional optimal control problem subject to general equality and inequality constraints. Then, based on a novel numerical integration scheme, we present a discretization method for the governed fractional-order system as well as the cost and constraint functionals, which yields a discretized approximate problem. We further derive the gradients of the cost and constraint functionals in regard to the discretized controls. On the basis of this result, a numerical optimization approach to solve the approximate problem is developed. Finally, three non-trivial example problems are solved to illustrate the applicability and effectiveness of the developed approach.

Distributed formation containment control for multi-agent systems via dynamic event-triggering communication mechanism

This paper focuses on resolving the distributed formation containment control issue in general multi-agent systems (MASs) possessing multiple leader agents subject to limited communication resources. The unnecessary data transmissions among agents in MASs are reduced by utilizing a unique dynamic event-triggering communication mechanism (DETCM), which can build communication time sequences for each agent automatically. Based on the sampled states of agents at event-triggering instants, a novel distributed formation containment control protocol is proposed. By mathematical analysis, the formation containment control problem is turned into the problem of stability in a reduced-order system. The global formation containment of the MASs is finally achieved through the application of Lyapunov-Krasovskii functionals and general inequalities. Moreover, a valid example is also given to support the validity and effectiveness of the findings.

Synchronization of Intermittently Coupled Neural Networks With Coupling Delay.

In recent years, the synchronization of coupled neural networks (CNNs) has been extensively studied. However, existing results heavily rely on assuming continuous couplings, overlooking the prevalence of intermittent couplings in reality. In this article, we address for the first time the synchronization challenge posed by intermittently CNNs (ICNNs) with coupling delay. To overcome the difficulties arising from intermittent couplings, we put forward a general piecewise delay differential inequality to characterize the dynamics during both coupled intervals and decoupled intervals. Based on the proposed inequality, we establish delay-independent synchronization criteria (DISCs) for ICNNs, enabling them to tackle general coupling delay. Notably, unlike previous studies, the achievement of synchronization in our approach does not rely on external control. Furthermore, for ICNNs that synchronize only under small delays, we formulate non-linear matrix inequality (LMI)-based delay-dependent synchronization criteria (DDSCs) that are computationally efficient and do not require delay differentiability. Finally, we provide illustrative examples to demonstrate our theoretical results.

Sociodemographic Inequalities in COVID-19 Booster Dose Vaccination Coverage: a Retrospective Study of 196 Provinces in Peru.

The COVID-19 vaccination coverage shows variability in booster doses between residency areas or ethnicity. The aim of this study was to evaluate how sociodemographic conditions influence unequal vaccination coverage with booster doses against COVID-19 in Peru. A retrospective, ecological study with an evaluation of 196 provinces in Peru. The sociodemographic conditions were evaluated as sources of inequality (sex, age group, educational level, residence area, and ethnic group). The inequality measure used was the GINI, an index that show the inequal vaccination coverage with third and fourth booster doses against COVID-19 in Peruvians provinces. The index allow determinate a higher inequality when the value is near to 1, and a lower inequality when the value is near to 0. Also, the impact of each sociodemographic condition in the general inequality was evaluate with a decomposition analysis of GINI coefficient into Sk (composition effect), Gk (redistribution effect), Rk (differential effect). In provinces evaluated the mean vaccine coverage for the third and fourth booster doses was 57.00% and 22.19%, respectively at twelve months since the beginning of vaccination campaign. The GINI coefficient was 0.33 and 0.31, for the third and fourth booster doses coverage, respectively. In the decomposition analysis, twelve months after the start of the third and fourth dose vaccination campaign, revealed higher Sk values for people living in rural areas (Sk = 0.94 vs. Sk = 2.39, respectively for third and fourth dose), while higher Gk values for Aymara (Gk = 0.92 vs. Gk = 0.92, respectively), Quechua (Gk = 0.53 vs. Gk = 0.53, respectively), and Afro-Peruvians (Gk = 0.61 vs. Gk = 0.61, respectively). Also, higher negative correlation in Rk values for people with elementary education (Rk=-0.43 vs. Rk=-0.33, respectively), aged between 15 and 19 years (Rk=-0.49 vs. Rk=-0.37, respectively), and Aymara (Rk=-0.51 vs. Rk=-0.66, respectively). The rural residency area, lower education and Quechua, Aymara or Afro-Peruvians ethnicity determinated inequalities in vaccination coverage with booster doses against COVID-19 in Peruvian provinces.

Synthetic approach to Bell's inequalities and a fundamental Bell-type theorem

A fundamental Bell-type inequality is derived by referring solely to laws of logic, tertium non datur and the transitivity axiom, that are used to express the local realism assumption. The general form of the achieved inequality makes it a root of other more specific Bell-type theorems involving additional mathematical formalism and experimental conditions. Its universality is demonstrated explicitly by showing that the most famous theorems provided by Bell, Clauser, Horne, Shimony and Holt (CHSH), Clauser and Horne (CH), Eberhard, d'Espagnat and Maccone are direct corollaries of the general inequality. The method to retrieve Bell's inequalities of different kinds, as being based on a single scheme of reasoning, unifies diversity of proofs presented in the literature. Additionally, some of the known inequalities can be easily achieved in a more general form compared to their original versions and still suitable, or possibly even more convenient, for experimental investigations. By extracting local realism in the form of an explicit logical propositions and making it a cornerstone of a systematic reasoning leading to the universal Bell's inequality, the paper offers a synthetic perspective on Bell-type theorems and their logical basis.

Recent Developments in General Quasi Variational Inequalities

In this paper, we present a number of new and known numerical techniques for solving general quasi variational inequalities, introduced by Noor [34] in 1988, using various techniques including projection, Wiener-Hopf equations, auxiliary principle, dynamical systems coupled with finite difference approach and sensitivity analysis. Convergence analysis of these methods is investigated under suitable conditions. Sensitivity analysis is also investigated. Some special cases are discussed as applications of the main results. Several open problems are suggested for future research.

Updated Ostrowski inequalities over a spherical shell

Here we present general multivariate mixed Ostrowski type inequalities over spherical shells and balls. We cover the radial and not necessarily radial cases. The proofs derive by the use of some estimates coming out of some new trigonometric and hyperbolic Taylor’s formulae ([2]) and reducing the multivariate problem to a univariate one via general polar coordinates.

Low conservative stability criteria for discrete-time Lur’e systems with sector and slope constrained nonlinearities

Lur’e system is an important nonlinear system in engineering, which is composed of a feedback connection of a linear dynamic system and a nonlinear system that satisfies the sector constraint. In this paper, for nominal and uncertain discrete Lur’e systems with time-varying delays, the absolute and robustly absolute stability are studied in detail with the nonlinearity satisfying the sector and slope constraints. Firstly, an augmented Lyapunov–Krasovskii functional (LKF) is designed, in which some augmented vectors are selected to increase the coupling information between the delay intervals and other system state variables. At the same time, some effective integral terms of nonlinear state variables are extended in the LKF according to the characteristics of sector constraint and slope constraint. Secondly, a general lemma of summation inequality for free-weighted matrices is given, which adds some coupling information between vectors and extends the search range for solving linear matrix inequalities (LMIs). Thirdly, by synthesizing the general summation inequality and the newly constructed LKF, different absolute stability criteria are derived respectively, and the absolute stability criteria are extended to robustly absolute stability cases with uncertain parameters. The stability criteria reduce the conservatism of some previously proposed results. Finally, some common numerical examples and simulations are used to verify the results of this paper.

Out-of-Equilibrium Fluctuation-Dissipation Bounds.

We prove a general inequality between the charge current and its fluctuations valid for any weakly interacting coherent electronic conductor and for any stationary out-of-equilibrium condition, thereby going beyond established fluctuation-dissipation relations. The developed fluctuation-dissipation bound saturates at large temperature bias and reveals additional insight for heat engines, since it limits the output power by power fluctuations. It is valid when the thermodynamic uncertainty relations break down due to quantum effects and provides stronger constraints close to thermovoltage.

Novel Multivariate Integral Operators Incorporating Trigonometric Transformations

The creation of new nonlinear multivariate integral operators is motivated by the need for mathematical tools that can handle the complex interdependencies that naturally arise in contemporary applications. From an abstract scientific point of view, it is also necessary to develop new operator theories beyond existing ones to offer original research perspectives. This article contributes to these complementary aspects. We present two nonlinear multivariate integral operators that have the particularity of incorporating trigonometric transformations of the main function. Thanks to their trigonometric nature, they completely stand out from existing operators, offering a new and complete framework. Therefore, we take advantage of advanced mathematical techniques for trigonometric functions to address the challenges they pose. In particular, we show that they have manageable integrals and series expansions, that they are solutions of specific differential and functional equations, and that they are involved in general inequalities of various types (Hölder-type, convex-type, etc.). In the application part, we use some of these properties to propose a wide collection of trigonometric inequalities that are both original and precise. Figures are produced to illustrate them for a direct visual check.

Insecurity and Nigeria’s Socio-Economic Development

Many scholars have identified insecurity in Nigeria as one of the major reasons for the country’s continuous under-development. This paper analyzes the effects of insecurity on the socio-economic development of Nigeria, with the main objective of proffering policy solutions to address insecurity challenges that have almost brought the county to her knees and stunted her development. The paper adopted the conflict theory propounded by Karl Marx as the theoretical framework to interrogate the causes of insecurity and the effects on various aspects of the Nigerian socio-economic system. The study employed qualitative research methods to critically assess the relationship between insecurity and Nigeria’s socio-economic development. It is theoretically based, with the use of secondary data, using in-depth explanatory analysis that produces results with understanding, meanings and views. The study identified severe unemployment, endemic poverty, ethno-religious conflicts, corruption, deprivation, inequalities and small arms and light weapons proliferation as the major causes of the unending insecurity challenge in Nigeria. These have led to population displacement, social dislocation, depression and trauma among the people, declining health situation, worsening school attendance, food insecurity and lack of foreign investment. The study recommended that governments at all levels should implement policies that will ensure serious reductions in unemployment rate, poverty and general inequality, by providing infrastructures and the enabling environment that will encourage entrepreneurs and small-scale industries to thrive, revamp the country’s entire security architecture, strengthen border security, tackle the proliferation of firearms, improve the legal system, among many others.