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This paper addresses the problems of passivity and robust passivity for a class of stochastic switched inertial neural networks (SSINNs) with time‐varying parametric uncertainties. First, the original inertial neural networks can be converted into first‐order differential systems using a suitable variable transformation approach. Moreover, by using a proper Lyapunov–Krasovskii functional (LKF) theory, Jensen's inequality, and the state‐dependent switching (SDS) law approach, several adequate criteria are derived in terms of linear matrix inequalities (LMIs) to ensure passivity and robust passivity analysis of SSINNs with parametric uncertainties and time‐varying delays. It is demonstrated that the developed SDS law may guarantee the passivity requirements of the above‐considered system made up of all unstable subnetworks. Furthermore, the gains are obtained by solving a set of LMIs that can be easily verified by some standard numerical packages. Ultimately, two examples with numerical simulations are provided to demonstrate the efficacy and feasibility of the suggested method. [ABSTRACT FROM AUTHOR]

We aim to extend some recent refinements of the Jensen inequality. First, we establish even more precise estimates for the Jensen inequality. Then, we derive an improvement of the Jensen inequality with regard to h$$ h $$‐convexity. As an application, our main results are applied in deriving refinements of power mean inequalities and the Hölder inequality. Finally, we obtain more accurate estimates for f$$ f $$‐divergences, important in the information theory. [ABSTRACT FROM AUTHOR]

We aim to establish refinements of the Jensen inequality for the classes of completely convex and absolutely convex functions. In the first case the refinement is expressed in terms of the alternating sum of Lidstone polynomials, while in the second case we deal with the sum of the Lidstone polynomials. As an application, more accurate power mean inequalities are derived. In particular, we obtain strengthened versions of arithmetic–geometric mean inequality in a difference and a quotient form. Finally, we also establish more accurate form of the Hölder inequality. [ABSTRACT FROM AUTHOR]

In this paper, we mainly explore the averaging principle of Caputo-type fractional delay stochastic differential equations with Brownian motion. Firstly, the solutions of this considered system are derived with the aid of the Picard iteration technique along with the Laplace transformation and its inverse. Secondly, we obtain the unique result by using the contradiction method. In addition, the averaging principle is discussed by means of the Burkholder-Davis-Gundy inequality, Jensen inequality, Hölder inequality and Grönwall-Bellman inequality under some hypotheses. Finally, an example with numerical simulations is carried out to prove the relevant theories. [ABSTRACT FROM AUTHOR]

In the present paper, utilizing a wide class of fractional integral operators (namely the Raina fractional operator), we develop novel fractional integral inequalities of the Hermite–Hadamard type. With the help of the well-known Riemann–Liouville fractional operators, s-type convex functions are derived using the important results. We also note that some of the conclusions of this study are more reasonable than those found under certain specific conditions, e.g., s = 1 , λ = α ,   σ (0) = 1 , and w = 0 . In conclusion, the methodology described in this article is expected to stimulate further research in this area. [ABSTRACT FROM AUTHOR]

This paper investigates the stability of complex-valued neural networks (CVNNs) with semi-Markovian jump (sMJ) and generally uncertain transition rates. Each transition rate may be totally unknown or its estimate is determined. Firstly, two improved reciprocally convex inequalities (RCIs) and three less conservative integral inequalities are generalized to the complex-valued domain. Secondly, the existence and uniqueness of the addressed networks are proposed by the complex-valued homeomorphism theorem. Thirdly, by constructing a Lyapunov–Krasovskii functional (LKF), delay-dependent robust stability criteria of the CVNNs are obtained by utilizing the improved complex-valued RCIs and integral inequalities. Finally, simulations are presented to demonstrate the effectivenes and practicality of the established method. [ABSTRACT FROM AUTHOR]

Foraging is risk sensitive if choices depend on the variability of returns from the options as well as their mean return. Risk‐sensitive foraging is important in behavioural ecology, psychology and neurophysiology. It has been explained both in terms of mechanisms and in terms of evolutionary advantage. We provide a critical review, evaluating both mechanistic and evolutionary accounts. Some derivations of risk sensitivity from mechanistic models based on psychophysics are not convincing because they depend on an inappropriate use of Jensen's inequality. Attempts have been made to link risk sensitivity to the ecology of a species, but again these are not convincing. The field of risk‐sensitive foraging has provided a focus for theoretical and empirical work and has yielded important insights, but we lack a simple and empirically defendable general account of it in either mechanistic or evolutionary terms. However, empirical analysis of choice sequences under theoretically motivated experimental designs and environmental settings appears a promising avenue for mapping the scope and relative merits of existing theories. Simply put, the devil is in the sequence. [ABSTRACT FROM AUTHOR]

Motivated by a paper of Dragomir, we give new refinements for both discrete and integral Jensen inequalities using the Jensen's gap. As applications, we give refinements of various inequalities verifiable by Jensen's inequality. Topics covered: norms, quasi-arithmetic means, Hölder's inequality and f-divergences in information theory. [ABSTRACT FROM AUTHOR]

In this paper, we explore the stability of a new class of Caputo-type fractional stochastic delay differential systems with Poisson jumps. We prove the Hyers–Ulam stability of the solution by utilizing a version of fixed point theorem, fractional calculus, Cauchy–Schwartz inequality, Jensen inequality, and some stochastic analysis techniques. Finally, an example is provided to illustrate the effectiveness of the results. [ABSTRACT FROM AUTHOR]

In this paper, a theorem is obtained that gives sufficient input-to-state stability conditions for linear systems with control delay and additive disturbances. Stabilizing feedback is considered available in the absence of delay and disturbances. The mathematical tools are the Lyapunov–Krasovskii functional, the Jensen inequality and the double Hadamard inequality. The critical delay is highlighted. [ABSTRACT FROM AUTHOR]

This document discusses the estimation of random mobility models (RMMs) using the expectation-maximization (EM) method. RMMs are important for evaluating mobile wireless networks, particularly in the context of unmanned aerial vehicles (UAVs) as platforms for airborne communication networks. The study focuses on RMMs formulated as random switching systems and develops a generic method based on EM to estimate the parameters in the random variables. The method is shown to outperform existing estimation methods and is applied to real UAV trajectories. The research was supported by the Army Research Office and the National Science Foundation. [Extracted from the article]

In this paper we introduce necessary and sufficient conditions for a real-valued function to be preinvex. Some properties of preinvex functions and new versions of Jensen's integral type inequality in this setting are given. Several examples are also involved. [ABSTRACT FROM AUTHOR]

Some fundamental properties of the Muckenhoupt class A p of weights and the Gehring class G q of weights on time scales and some relations between them will be proved in this paper. To prove the main results, we will apply an approach based on proving some properties of integral operators on time scales with powers and certain mathematical relations connecting the norms of Muckenhoupt and Gehring classes. The results as special cases cover the results for functions following David Cruz-Uribe, C. J. Neugebauer, and A. Popoli, and when the time scale equals the positive integers, the results for sequences are essentially new. [ABSTRACT FROM AUTHOR]

This work studies the convexity of a class of positive definite probability measures with density functions ρ = c λ ∑ n = 0 ∞ g (n) C n λ w λ , that are generated through the interval characteristic sequences. We discuss when such measures supported in an interval are convex and demonstrate how convexity is preserved under multiplication by certain multipliers and under the intertwining product. A key message to be delivered from this work is: any integrable function f on the interval with a polynomial expansion that has relatively fast absolute convergence, can be decomposed into a non-unique pair of positive convex interval probabilities (f + , f −) ∈ E λ ′ × E λ ′ up to normalization, in the sense that f = f + − f − , where E λ ′ is a class of convex functions. Consequently not only the study of an interval distribution is reduced to the study of the properties of class E λ ′ , but also the discontinuous probabilistic Galerkin schemes can be simplified. As concrete examples the decomposition of polynomials and truncated normal distribution are computed and graphically illustrated. Such interval measures lead to localized spherical probability measures, and as a convex expression of probability measure is a Jensen type inequality for spherical random variables. Based on the spherical information entropy models, one dependent on the domain partition and the other not, we propose three minimization formulations for the probabilistic finite element method, including one that mixes the least-squares and entropy models. • A novel way to decompose an interval distribution into two convex positive parts. • Explore the convexity of interval probability measures by interval characteristic sequences. • Preserve convexity through multipliers and intertwining product. • Build least-squares and entropy mixed spherical model. [ABSTRACT FROM AUTHOR]

In this paper we give a necessary and sufficient condition for the discrete Jensen inequality to be satisfied for real (not necessarily nonnegative) weights. The result generalizes and completes the classical Jensen–Steffensen inequality. The validity of the strict inequality is studied. As applications, we first give the form of our result for strongly convex functions, then we study discrete quasi-arithmetic means with real (not necessarily nonnegative) weights, and finally we refine the inequality obtained. [ABSTRACT FROM AUTHOR]

This paper primarily focuses on the chaos synchronisation analysis of neural networks (NNs) under a hybrid controller. Firstly, we design a suitable hybrid controller with saturated impulse control, combined with time-dependent intermittent control. Both controls are low-energy consumption and discrete, aligning well with industrial development needs. Secondly, the saturation function in the chaotic neural network is addressed using the polyhedral representation method and the sector nonlinearity method, respectively. By integrating the Lyapunov stability theory, Jensen's inequality, the mathematical induction method, and the inequality reduction technique, we establish suitable time-dependent Lyapunov generalised equations. This leads to the estimation of the domain of attraction and the derivation of local exponential stability conditions for the error system. The validity of the achieved theoretical criteria is eventually demonstrated through numerical experiment simulations. [ABSTRACT FROM AUTHOR]

Within the recent wave of research advancements, mathematical inequalities and their practical applications play a notably significant role across various domains. In this regard, inequalities offer a captivating arena for scholarly endeavors and investigational pursuits. This research work aims to present new improvements for the integral majorization inequalities using an interesting aproach. Certain previous improvements have been achieved for the Jensen inequality as direct outcomes of the main results. Additionally, estimates for the Csiszár divergence and its cases are provided as applications of the main results. The circumstances under which the principal outcomes offer enhanced estimations for majorization differences are also underscored and emphasized. [ABSTRACT FROM AUTHOR]

The purpose of this work is to introduce fractional cumulative residual inaccuracy (FCRI) information, Jensen-cumulative residual inaccuracy (JCRI), and Jensen-fractional cumulative residual inaccuracy (JFCRI) information measure. Further, we study the FCRI information for some well-known models used in reliability, economics and survival analysis. The associated results reveal some interesting connections between the FCRI information measure and cumulative residual entropy and Gini mean difference measures. Applications to two chaotic discrete-time dynamical systems (Chebyshev and Logistic) are presented to illustrate the behavior of the proposed information measures. FCRI and JFCRI measures allow to determine regions of discrepancy between systems, depending on their respective fractional and chaotic map parameters. [ABSTRACT FROM AUTHOR]

Spectral line shapes provide a window into the local environment coupled to a quantum transition in the condensed phase. In this paper, we build upon a stochastic model to account for non-stationary background processes produced by broad-band pulsed laser stimulation, as distinguished from those for stationary phonon bath. In particular, we consider the contribution of pair-fluctuations arising from the full bosonic many-body Hamiltonian within a mean-field approximation, treating the coupling to the system as a stochastic noise term. Using the Itô transformation, we consider two limiting cases for our model, which lead to a connection between the observed spectral fluctuations and the spectral density of the environment. In the first case, we consider a Brownian environment and show that this produces spectral dynamics that relax to form dressed excitonic states and recover an Anderson–Kubo-like form for the spectral correlations. In the second case, we assume that the spectrum is Anderson–Kubo like and invert to determine the corresponding background. Using the Jensen inequality, we obtain an upper limit for the spectral density for the background. The results presented here provide the technical tools for applying the stochastic model to a broad range of problems. [ABSTRACT FROM AUTHOR]

One of the fundamental inequalities which is used in many inequities is Jensen inequality. In fact, it is a base of some inequality such as the arithmetic mean, harmonic mean inequality also in inequality with respect to entropies and information theory. The purpose of this research paper is to give a new interesting refinement of Jensen inequality for two particular finite sequences by using uniformly convex function. Also, we give some applications of this in information theory. [ABSTRACT FROM AUTHOR]

Intraspecific trait variation is ubiquitous and is likely to influence species coexistence. Despite theoretical progress, empirical work on the effects of intraspecific variation on the dynamics of competing species is rare. This is because of the formidable empirical requirements necessary to link intraspecific variation in species' functional traits with intraspecific variation in the demographic and competitive rates that mediate coexistence. Here we partially overcome these challenges to determine how intraspecific variation in reproductive phenology in a native Californian annual plant species Lasthenia californica affects its ability to coexist with two non‐native species Bromus madritensis and Lactuca serriola that display contrasting phenological patterns. Using data from a field experiment, we empirically parameterize a model of competitive population dynamics, accounting for the effects of intraspecific phenological trait variation on the native species' response to both intra‐ and interspecific competition. We find that intraspecific variation in phenology drives differences in the native species' response to competition. Moreover, simulations of the parameterized model show that this variation improves the competitive performance of the native species. This occurs because of the effects of nonlinear averaging mediated by a nonlinear, concave‐up competition function that is a general feature of competition across a wide range of taxa. While intraspecific variation improves competitive performance, we also find that the magnitude of the benefit is predicted to be insufficient to prevent competitive exclusion against the non‐native species with early phenogy Bromus. Against the second non‐native species with later phenology Lactuca, intraspecific variation is predicted to result in coexistence where competitive exclusion would otherwise occur, but we could not rule out alternative qualitative outcomes for this interaction. [ABSTRACT FROM AUTHOR]

Recently, Rattan and the first author (Ann. Comb. 25 (2021) 697–728) proved a conjectured inequality of Berkovich and Uncu (Ann. Comb. 23 (2019) 263–284) concerning partitions with an impermissible part. In this article, we generalize this inequality upon considering t impermissible parts. We compare these with partitions whose certain parts appear with a frequency which is a perfect t th power. In addition, the partitions that we study here have smallest part greater than or equal to s for some given natural number s. Our inequalities hold after a certain bound, which for given t is a polynomial in s, a major improvement over the previously known bound in the case t = 1 . To prove these inequalities, our methods involve constructing injective maps between the relevant sets of partitions. The construction of these maps crucially involves concepts from analysis and calculus, such as explicit maps used to prove countability of N t , and Jensen's inequality for convex functions, and then merge them with techniques from number theory such as Frobenius numbers, congruence classes, binary numbers and quadratic residues. We also show a connection of our results to colored partitions. Finally, we pose an open problem which seems to be related to power residues and the almost universality of diagonal ternary quadratic forms. [ABSTRACT FROM AUTHOR]

We generalize cyclic Jensen's inequality utilizing the theory of 4-convex function under the effect of introduced Green functions. We formulate result for power means and quasi means. Also we give applications in information theory by giving new estimations of generalized Csiszár divergence, Rényi-divergence, Shannon-entropy, Kullback-Leibler divergence and χ2-divergence. [ABSTRACT FROM AUTHOR]

The Jensen, Jensen–Mercer, and Hermite–Hadamard inequalities have been key in advancing convex inequalities for scalars and operators, with several applications in fields like mathematical physics, operator theory and mathematical inequalities. This paper presents a new argument leading to interesting refinements of these celebrated inequalities. Applications that include entropy and operator inequalities will be presented. [ABSTRACT FROM AUTHOR]

Whole‐ecosystem interactions and feedbacks constrain ecosystem responses to environmental change. The effects of these constraints on responses to climate trends and extreme weather events have been well studied. Here we examine how these constraints respond to changes in day‐to‐day weather variability without changing the long‐term mean weather. Although environmental variability is recognized as a critical factor affecting ecological function, the effects of climate change on day‐to‐day weather variability and the resultant impacts on ecosystem function are still poorly understood. Changes in weather variability can alter the mean rates of individual ecological processes because many processes respond non‐linearly to environmental drivers. We assessed how these individual‐process responses to changes in day‐to‐day weather variability interact with one another at an ecosystem level. We examine responses of arctic tundra to changes in weather variability using stochastic simulations of daily temperature, precipitation, and light to drive a biogeochemical model. Changes in weather variability altered ecosystem carbon, nitrogen, and phosphorus stocks and cycling rates in our model. However, responses of some processes (e.g., respiration) were inconsistent with expectations because ecosystem feedbacks can moderate, or even reverse, direct process responses to weather variability. More weather variability led to greater carbon losses from land to atmosphere; less variability led to higher carbon sequestration on land. The magnitude of modeled ecosystem response to weather variability was comparable to that predicted for the effects of climate mean trends by the end of the century. [ABSTRACT FROM AUTHOR]

The objective of this research is to study in detail the fractional variants of Ostrowski–Mercer-type inequalities, specifically for the first and second order differentiable s-convex mappings of the second sense. To obtain the main outcomes of the paper, we leverage the use of conformable fractional integral operators. We also check the numerical validations of the main results. Our findings are also validated through visual representations. Furthermore, we provide a detailed discussion on applications of the obtained results related to special means, q-digamma mappings, and modified Bessel mappings. [ABSTRACT FROM AUTHOR]

Hyperdiverse tropical forest tree communities illustrate a fundamental problem in ecology: How can many species coexist given relatively few limiting resources? Neutral theory provides a solution by positing that species have equal fitness and hence drift to extinction slowly. However, neutral theory seriously under‐predicts temporal changes in species abundances. This can be remedied by breaking neutrality and adding temporal environmental stochasticity (TES), but the mechanisms mediating the effects of TES on species richness remain unclear. Here, we make progress by analysing a local community model with species competing for a common resource under TES, to derive formulae partitioning species richness according to different mechanisms.By applying our formulae to generic parameter sets for tropical forest tree communities, we found that when the autocorrelation time of TES was short, the dominant mechanism driving species richness was non‐linear averaging of the interspecific competition term over time, which reduced the typical strength of interspecific competition and boosted richness relative to the neutral case. However, greater immigration to the community resulted in more species and hence weaker non‐linear averaging due to the law of large numbers. In contrast, when the autocorrelation time of TES was long, the dominant mechanism driving species richness was strong selection between changes in environmental conditions, which increased the typical strength of interspecific competition and reduced richness relative to the neutral case.By applying our formulae to a specific parameter set for a tropical forest tree community in Panama, we found that TES had minor effects on species richness because (i) the immigration rate was sufficiently large for non‐linear averaging of the interspecific competition term to be weak and (ii) the autocorrelation time was sufficiently short to suppress the effects of selection.Synthesis. We provide a novel mechanistic explanation of how TES affects tree species richness in tropical forests, in particular how TES often has minor effects on richness despite having substantial effects on temporal changes in species abundances. This provides a deeper insight into why a neutral model with added TES can accurately capture static and dynamic aspects of tree community diversity in the tropics. [ABSTRACT FROM AUTHOR]

We prove the result above in the specific case of PCA-spectral cut then HT ht for HT ht (then HT ht ). Lemma 1 Let HT ht be the Karhunen-Loeve expansion of HT ht given at page 334. [Extracted from the article]

In this paper, we introduce an universal definition (fgh-convex) that results in several types of convexity. Particular cases of the fgh-convex are for instance the harmonically convex, geometrically convex, GA-convex, log-convex, and several others. Also, we obtain some useful inequalities such as Jensen, generalization of Jensen Hermite-Hadamard, Mercer inequalities. Moreover, with the use of these inequalities, we obtained bounds for Shannon's entropy and Kapur's entropy. Finally, we found an application of the obtained inequalities in means. [ABSTRACT FROM AUTHOR]

This paper demonstrates through a numerical experiment that utilization of strongly convex functions strengthens the bound presented for the Jensen gap in [4]. Consequently the improved result enables to present improvements in the bounds obtained for the Hölder and Hermite-Hadamard gaps and proposes such improvements in the results obtained for various entropies and divergences in information theory. [ABSTRACT FROM AUTHOR]

New proofs of the classical Fejer inequality and discrete Hermite-Hadamard inequality (HH) are presented and several applications are given, including (HH)-type inequalities for the functions, whose derivatives have inflection points. Morever, some estimates from below and above for the first moments of functions f: [a, b] R about the midpoint c = (a+b)/2 are obtained and the reverse Hardy inequality for convex functions f: (0,8) (0,8) is established. [ABSTRACT FROM AUTHOR]

This document is a comment on a previous article about the estimation of prediction error in linear regression models. It provides a different perspective and focuses on linear models, including nonparametric regression fitting methods. The comment discusses the standard result for the expected value of a quadratic form and examines prediction errors in a standard linear model. It also discusses the use of prediction ellipsoids and significance tests in linear models, emphasizing the importance of estimating the covariance matrix and comparing it to the mean squared error to determine the reasonableness of the models. The document also discusses the issue of overfitting in statistical modeling and its impact on predictions and variance estimates. It explains that overfitting occurs when a model is too complex and fits the current data too closely, resulting in poor predictions for new data. The document also discusses the estimation of heteroscedasticity in statistical models and the challenges of estimating parameters with limited observations. It concludes by mentioning that the estimation of squared residuals can be used as an unbiased estimator of the variance in the homoscedastic case, but not in the heteroscedastic case. Lastly, the document discusses the estimation of low leverage points in a linear model and the estimation of v using least squares. The author concludes that the estimate is reasonable if one accepts the linear model for the variances. [Extracted from the article]

Employing some properties of multivariate Baskakov–Durrmeyer operators and utilizing modified K-functional and a decomposition technique, the authors obtain the direct theorem and weak type inverse theorem in the Orlicz spaces. [ABSTRACT FROM AUTHOR]

Climate change has profound effects on infectious disease dynamics, yet the impacts of increased short-term temperature fluctuations on disease spread remain poorly understood. We empirically tested the theoretical prediction that short-term thermal fluctuations suppress endemic infection prevalence at the pathogen's thermal optimum. This prediction follows from a mechanistic disease transmission model analyzed using stochastic simulations of the model parameterized with thermal performance curves (TPCs) from metabolic scaling theory and using nonlinear averaging, which predicts ecological outcomes consistent with Jensen's inequality (i.e., reduced performance around concave-down portions of a thermal response curve). Experimental observations of replicated epidemics of the microparasite Ordospora colligata in Daphnia magna populations indicate that temperature variability had the opposite effect of our theoretical predictions and instead increase endemic infection prevalence. This positive effect of temperature variability is qualitatively consistent with a published hypothesis that parasites may acclimate more rapidly to fluctuating temperatures than their hosts; however, incorporating hypothetical effects of delayed host acclimation into the mechanistic transmission model did not fully account for the observed pattern. The experimental data indicate that shifts in the distribution of infection burden underlie the positive effect of temperature fluctuations on endemic prevalence. The increase in disease risk associated with climate fluctuations may therefore result from disease processes interacting across scales, particularly within-host dynamics, that are not captured by combining standard transmission models with metabolic scaling theory. Experiments using a Daphnia-microparasite system reveal that disease transmission models that only model transmission dynamics at the among-host scale incorrectly predict the direction of effect of short-term thermal variability on endemic infection prevalence. [ABSTRACT FROM AUTHOR]

The main goal of this work is to investigate the following weakly damped nonlinear suspensionbridge equation utt(x, y, t) + ∆2u(x, y, t) + α(t)g(ut) = 0, and establish explicit and general decay results for the energy of solutions of the problem. Our decay results depend on the functions α and g and obtained without any restriction growth assumption on g at the origin. The multiplier method, the properties of the convex and the dual of the convex functions, Jensen’s inequality and the generalized Young inequality are used to establish the stability results. [ABSTRACT FROM AUTHOR]

In the recent era of research developments, mathematical inequalities and their applications perform a very consequential role in different aspects, and they provide an engaging area for research activities. In this paper, we propose a new approach for the improvement of the classical majorization inequality and its weighted versions in a discrete sense. The proposed improvements give several estimates for the majorization differences. Some earlier improvements of the Jensen and Slater inequalities are deduced as direct consequences of the obtained results. We also discuss the conditions under which the main results give better estimates for the majorization differences. Applications of the acquired results are also presented in information theory. [ABSTRACT FROM AUTHOR]

In this research, we aim to explore generalizations of Hardy-type inequalities using nabla Hölder's inequality, nabla Jensen's inequality, chain rule on nabla calculus and leveraging the properties of convex and submultiplicative functions. Nabla calculus on time scales provides a unified framework that unifies continuous and discrete calculus, making it a powerful tool for studying various mathematical problems on time scales. By utilizing this approach, we seek to extend Hardy-type inequalities beyond their classical continuous or discrete settings to a more general time scale domain. As specific instances of our discoveries, we have the integral inequalities previously established in the existing literature. [ABSTRACT FROM AUTHOR]

In this study, we apply Hölder's inequality, Jensen's inequality, chain rule and the properties of convex functions and submultiplicative functions to develop an innovative category of dynamic Hardy-type inequalities on time scales delta calculus. A time scale, denoted by T , is any closed nonempty subset of R. In time scale calculus, results are unified and extended. As particular cases of our findings (when T = R ), we have the continuous analogues of inequalities established in some the literature. Furthermore, we can find other inequalities in different time scales, such as T = N , which, to the best of the authors' knowledge, is a largely novel conclusion. [ABSTRACT FROM AUTHOR]

In this paper, the problem of robust stability analysis for a type of uncertain stochastic switched inertial neural networks (SSINNs) with time‐varying delay is investigated. First, the original second‐order system is converted into first‐order differential equations using the variable transformation method. Next, some sufficient conditions in terms of linear matrix inequalities (LMIs) are obtained using Lyapunov‐Krasovskii functional (LKF), state‐dependent switching (SDS) method, and Jensen's integral inequality for estimating integral inequalities so that the augmented system is robust, global, and asymptotically stable in the mean square for the uncertain SSINNs with time‐varying delay. It is shown that the stability of the above considered system composed of all unstable subsystems can be achieved by using the SDS law. Finally, two numerical simulations are provided to demonstrate the effectiveness of the proposed SDS law. [ABSTRACT FROM AUTHOR]

The article presents a study which proposed a principled framework to model the organization of higher-order systems, which allows detecting communities in hypergraphs with accuracy exceeding that of state-of-the-art approaches. Topics discussed include optimization procedure; identifiability, interpretation, and theoretical implications; and practical implementation and efficiency.

The aim of this article is to give generalization of Jensen Mercer inequality on delta integral along with applications to Ky Fan inequality and related results. [ABSTRACT FROM AUTHOR]

Summary: This article is concerned with the problem of robust fuzzy fault tolerant control for the nonlinear active suspension system via adaptive hybrid triggered scheme. To describe the system with high nonlinearity, a Takagi‐Sugeno fuzzy approach is applied by weighting a series of linear subsystems. During the design of the controller, the actuator failure is taken into consideration. To ease the network communication pressure, a novel adaptive hybrid triggered scheme is proposed for the suspension system. By applying a Bessel‐Legendre inequality and a Jensen's inequality together, the control algorithm can be derived in the form of linear matrix inequalities with less conservatism. Finally, compared with the existing sampled‐data H∞$$ {H}_{\infty } $$ control and adaptive event triggered H∞$$ {H}_{\infty } $$ control, numerical simulations are performed to demonstrate the effectiveness and superiority of the proposed controller. [ABSTRACT FROM AUTHOR]

In recent years, the theory of convexity has influenced every field of mathematics due to its unique characteristics. Numerous generalizations, extensions, and refinements of convexity have been introduced, and one of them is set-valued convexity. Interval-valued convex mappings are a special type of set-valued maps. These have a close relationship with symmetry analysis. One of the important aspects of the relationship between convex and symmetric analysis is the ability to work on one field and apply its principles to another. In this paper, we introduce a novel class of interval-valued (I.V.) functions called CR - γ -convex functions based on a non-negative mapping γ and center-radius ordering relation. Due to its generic property, a set of new and known forms of convexity can be obtained. First, we derive new generalized discrete and integral forms of Jensen's inequalities using CR - γ -convex I.V. functions. We employ this definition and Riemann-Liouville fractional operators to develop new fractional versions of Hermite-Hadamard's, Hermite-Hadamard-Fejer, and Pachpatte's type integral inequalities. We examine various key properties of this class of functions by considering them as special cases. Finally, we support our findings with interesting examples and graphical representations. [ABSTRACT FROM AUTHOR]

This paper considers a distributed observer design for a class of nonlinear systems with time delays in a network. Specifically, each distributed observer has access to partial output as it communicates with its neighboring observers through consensus protocols defined on the network. The network includes N local observers where adjacent observers transmit and receive information through a delayed telecommunication graph network. The design method is based on Lipschitz condition of nonlinear terms while having time-delay cases in view. Appropriate selection of Lyapunov functions is made and the Jensen inequality applied in delayed nonlinear systems in undirected, strongly connected, and directed networks containing spanning trees. Also, via the design algorithm, the necessary and sufficient conditions for the design of parameters are defined so as to stabilize the error dynamics. Finally, numerical simulations are provided to verify the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This article presents new developments and applications of Jensen’s inequality. We explore various variations of Jensen’s inequality related to modified (h, m)-convex functions. The obtained results extend the applicability of the inequality to a broader class of functions and contexts. In addition to the fact that the results presented in the article provide additional results available in the literature, we also give examples of their application. [ABSTRACT FROM AUTHOR]

Using Jensen's inequality and integration by parts, some tight upper bounds are derived on the Gaussian Q‐function. The tightness of the bounds obtained by Jensen's inequality can be improved by increasing the number of exponential terms, and one of them is invertible. A piece‐wise upper bound is obtained and its application in the analysis of the symbol error probability of various modulation schemes in different channel models is shown. [ABSTRACT FROM AUTHOR]

We propose the concept of Riemann diamond-alpha integrals for time scales interval-valued functions. We first give the definition and some properties of the interval Riemann diamond-alpha integral that are naturally investigated as an extension of interval Riemann nabla and delta integrals. With the help of the interval Riemann diamond-alpha integral, we present interval variants of Jensen inequalities for convex and concave interval-valued functions on an arbitrary time scale. Moreover, diamond-alpha Hölder's and Minkowski's interval inequalities are proved. Also, several numerical examples are provided in order to illustrate our main results. [ABSTRACT FROM AUTHOR]

We study the convergence of a family of numerical integration methods where the numerical integration is formulated as a finite matrix approximation to a multiplication operator. For bounded functions, convergence has already been established using the theory of strong operator convergence. In this article, we consider unbounded functions and domains which pose several difficulties compared to the bounded case. A natural choice of method for this study is the theory of strong resolvent convergence which has previously been mostly applied to study the convergence of approximations of differential operators. The existing theory already includes convergence theorems that can be used as proofs as such for a limited class of functions and extended for a wider class of functions in terms of function growth or discontinuity. The extended results apply to all self-adjoint operators, not just multiplication operators. We also show how Jensen's operator inequality can be used to analyse the convergence of an improper numerical integral of a function bounded by an operator convex function. [ABSTRACT FROM AUTHOR]