General Time Scale Research Articles

Correlation analysis of degrading systems based on bivariate Wiener processes under imperfect maintenance

AbstractThis article focuses on the correlation between the degradation levels of the two components that form a system. The degradation evolution of each component is modeled using Wiener processes. Both components are dependent and this dependence is described using the trivariate reduction method. To reduce the degradation and extend the system lifetime, preventive maintenance actions are periodically performed. These preventive maintenance actions are imperfect and they are modeled by using an arithmetic reduction of degradation of infinite order model with a determined maintenance efficiency parameter. The evolution of the maintained system is analysed by assessing the expectation and variance of both degradation processes at successive maintenance times. The novelty of this work is the analysis of the Pearson correlation coefficient between the degradation levels of the two components. Different properties of the monotonicity of the Pearson correlation coefficient between the two degradation paths are obtained by considering equal maintenance efficiency and equal general time scales functions for the two Wiener degradation processes associated to each degrading component.

Constructing predictive models for seismic oscillation parameters using covariance functions and Doppler effect phenomena: A case study of InSight mission V2 data

An ability to construct predictive models for identifying seismic oscillation parameters by using the mathematics of covariance functions and Doppler effect phenomena is examined in this work. In the calculations, the Mars seismic oscillations measurement data from InSight Mission V2, observed in the months May, June and July of 2019, was used. To analyze the observation data arrays the Doppler phenomena and the expressions of covariance functions were employed. The seismic oscillations trend's intensity vectors were assessed by least squares method, and the random errors of measurements at the stations were eliminated partially as well. The estimates of the vector's auto-covariance and cross-covariance functions were derived by altering the quantization interval on the general time scale while varying the magnitude of the seismic oscillation vector on the same time scale. To detect the mean values of z —the main parameter of Doppler expression— we developed a formula by involving the derivatives of cross-covariance functions of a single vector and algebraic sum of the relevant vectors.

Exploring Generalized Hardy-Type Inequalities via Nabla Calculus on Time Scales

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.

Exponential Stability of Integro-Differential Volterra Equation on Time Scales

Abstract We study the Volterra integro-differential equation on time scales and provide sufficient conditions for boundness of all solutions of considered equation. Using that result, we present the conditions for exponential stability of considered equation. All the results proved on the general time scale include results for both integral and discrete Volterra equations.

Synchronization of Delayed Complex Networks on Time Scales via Aperiodically Intermittent Control Using Matrix-Based Convex Combination Method.

This article reconsiders synchronization problem of linear complex networks with time-varying delay on time scales. For different types of time scales, aperiodically intermittent control scheme is established by using a matrix-based convex combination method, which has great potential in reducing control consumption and saving communication bandwidth. By employing a common Lyapunov function, aperiodically intermittent controllers are utilized successfully to achieve synchronization of linear delayed complex networks on special time scales onto an isolated node. Next, by constructing a special Lyapunov function with time-varying coefficients, sufficient criteria that consist of two linear matrix inequalities are demonstrated to make linear delayed complex networks on general time scales synchronized onto an isolated system with an exponential convergence rate given in advance. Due to delayed complex networks in this article defined on time scales, the proposed control schemes are applicable to continuous-time networks, their discrete-time forms, and any combination of them. Four numerical examples are offered to highlight the effectiveness and superiority of the proposed aperiodically intermittent control schemes at last.

Timescale standard to discriminate between hyperbolic and exponential discounting and construction of a nonadditive discounting model

Under the presupposition that human time perception is distorted in intertemporal choice, this study constructs a time scale in the framework of axiomatic measurement. First, the conditions (homogeneity of degree one or two) to identify the form of a time scale are proposed so that one can determine whether the hyperbolic or exponential is a more suitable function for modeling people’s discounting. Homogeneity of degree one implies that subjective time delay is measured by a power scale and its discount function becomes a power-exponential type, whereas homogeneity of degree two implies that subjective time delay is measured by a log scale and its discount function becomes a hyperbolic type. Second, a method is shown for constructing a model that can reflect subadditive and superadditive discounting. A non-commutative and non-associative concatenation operation is provided to generate a time string consisting of subdivided durations that incorporates the effect of repeated delay with a subdivided duration. The discounting model is yielded by substituting these strings in an exponential model equipped with a generalized time scale, and the determination of subadditivity or superadditivity depends on whether the discount function value of a product by the non-commutative and non-associative operation is, respectively, smaller or greater than that of a product by an operation of the usual extensive structure.

Some New Generalizations of Reverse Hilbert-Type Inequalities via Supermultiplicative Functions

Our work in this paper is based on the reverse Hölder-type dynamic inequalities illustrated by El-Deeb in 2018 and the reverse Hilbert-type dynamic inequalities illustrated by Rezk in 2021 and 2022. With the help of Specht’s ratio, the concept of supermultiplicative functions, chain rule, and Jensen’s inequality on time scales, we can establish some comprehensive and generalize a number of classical reverse Hilbert-type inequalities to a general time scale space. In time scale calculus, results are unified and extended. At the same time, the theory of time scale calculus is applied to unify discrete and continuous analysis and to combine them in one comprehensive form. This hybrid theory is also widely applied on symmetrical properties which play an essential role in determining the correct methods to solve inequalities. As a special case of our results when the supermultiplicative function represents the identity map, we obtain some results that have been recently published.

On dynamic inequalities in two independent variables on time scales and their applications for boundary value problems

Our work is based on the multiple inequalities illustrated by Boudeliou and Khalaf in 2015. With the help of the Leibniz integral rule on time scales, we generalize a number of those inequalities to a general time scale. Besides that, in order to obtain some new inequalities as special cases, we also extend our inequalities to discrete, quantum, and continuous calculus. These inequalities may be of use in the analysis of some kinds of partial dynamic equations on time scales and their applications in environmental phenomena, physical and engineering sciences described by partial differential equations.

Hirota-Miwa equations on a time space scale

We derive a general time scale version of the Hirota–Miwa equation as a compatibility condition arising from a new version of a three-dimensional Lax system, construct its 3-soliton solutions, and consider various examples. We prove the invariance of the compatibility condition of the two-dimensional Lax system under a gauge transformation. The 3-soliton solution obtained here may be used in modeling in view of its variable graininess and a large number of free parameters.

Some new dynamic Steffensen-type inequalities on a general time scale measure space

<abstract><p>Our work is based on the multiple inequalities illustrated by Josip Pečarić in 2013, 1982 and Srivastava in 2017. With the help of a positive $ \sigma $-finite measure, we generalize a number of those inequalities to a general time scale measure space. Besides that, in order to obtain some new inequalities as special cases, we also extend our inequalities to discrete and continuous calculus.</p></abstract>

Seasonal and interannual variability of salinity in a large West-African lagoon (Nokoué Lagoon, Benin)

Nokoué Lagoon in the South of Benin is a large intermittent coastal water body in West Africa, which supports one of the largest inland fisheries of the region. The seasonal and interannual variability of its salinity was studied, based on 3 years of monthly surveys (Dec 2017–Dec 2020). This dataset allows us to identify fine-scale salinity structures and to better understand the salinization/desalinization processes at seasonal scales. During the rainy season from May to November, under the influence of large freshwater inflows from the rivers on its northern shores, the lagoon desalinizes to a salinity of zero in October–November. During the dry season from December to April, under the effect of the ocean tide, Nokoué lagoon becomes progressively saltier, reaching typical salinities of ∼25 in April. On average, the Nokoué lagoon is saltier in its southwestern part and fresher towards the river's mouths.Vertical salinity stratification is largest in December at the beginning of the main dry season. The lagoon displays a very marked interannual variation with mean surface (bottom, respectively) salinity of 25 (25) in April 2018 and 2020, respectively, against 16 (18) in April 2019. In the absence of river inflow data, a box model shows that the average salinity of the lagoon is very sensitive to small changes in river inflow (or discharge), with observed interannual differences in salinity induced by small variations of 10–15 m3 s−1 in inflow during the dry season. During the salinization phase, the model suggests that ∼30% of the seawater entering the lagoon during flood-tides remains trapped and enhances the Nokoué lagoon salinity. This model also indicates that a complete desalinization of the lagoon occurs for river inflow greater than ∼50–60 m3 s−1. The general mixing time scale of the lagoon is of 30–40 days.

A Variety of Dynamic Steffensen-Type Inequalities on a General Time Scale

This work is motivated by the work of Josip Pečarić in 2013 and 1982 and the work of Srivastava in 2017. By the utilization of the diamond-α dynamic inequalities, which are defined as a linear mixture of the delta and nabla integrals, we present and prove very important generalized results of diamond-α Steffensen-type inequalities on a general time scale. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities.

Is there a general time scale for hydrogen storage with metal hydrides or activated carbon?

The charging and discharging times are some of the very important factors for the design of highly efficient hydrogen storage chambers with metal hydrides or activated carbon. However, experimental research or numerical simulations have to be performed to determine these factors, which could be time consuming and expensive. In this study, the possible existence of a general time scale for hydrogen storage chambers with large amount of sorption heat is discussed. By scale analysis of the energy equation and rapid heat generation and slow cooling assumption, it is found that there likely exists a general time scale for hydrogen charging and discharging. The time scale is determined as the thermal diffusion time scale of the porous storage bed. Its validity is demonstrated by comparing with the available metal hydrides and activated carbon data in previous literature. Furthermore, a minimum bed thickness scale for a specific charging time is also derived according to the developed general time scale. The proposed temporal and spatial scales may serve as a low cost and efficient tool for the design and optimization of hydrogen storage devices with metal hydrides or activated carbon.

More on Hölder’s Inequality and It’s Reverse via the Diamond-Alpha Integral

In this paper, we investigate some new generalizations and refinements for Hölder’s inequality and it’s reverse on time scales through the diamond-α dynamic integral, which is defined as a linear combination of the delta and nabla integrals, which are used in various problems involving symmetry. We develop a number of those symmetric inequalities to a general time scale. Our results as special cases extend some integral dynamic inequalities and Qi’s inequalities achieved on time scales and also include some integral disparities as particular cases when T=R.

Some Dynamic Hilbert-Type Inequalities on Time Scales

Throughout this article, we will demonstrate some new generalizations of dynamic Hilbert type inequalities, which are used in various problems involving symmetry. We develop a number of those symmetric inequalities to a general time scale. From these inequalities, as particular cases, we formulate some integral and discrete inequalities that have been demonstrated in the literature and also extend some of the dynamic inequalities that have been achieved in time scales.

S-Almost Automorphic Solutions for Impulsive Evolution Equations on Time Scales in Shift Operators

In this paper, based on the concept of complete-closed time scales attached with shift direction under non-translational shifts (or S-CCTS for short), as a first attempt, we develop the concepts of S-equipotentially almost automorphic sequences, discontinuous S-almost automorphic functions and weighted piecewise pseudo S-almost automorphic functions. More precisely, some novel results about their basic properties and some related theorems are obtained. Then, we apply the introduced new concepts to investigate the existence of weighted piecewise pseudo S-almost automorphic mild solutions for the impulsive evolution equations on irregular hybrid domains. The obtained results are valid for q-difference partial dynamic equations and can also be extended to other dynamic equations on more general time scales. Finally, some heat dynamic equations on various hybrid domains are provided as applications to illustrate the obtained theory.

Square-mean almost automorphic solution of a stochastic cellular neural network on time scales

We derive some sufficient conditions for the existence of a square-mean almost automorphic solution for a stochastic cellular neural network on time scales by using Krasnoselskii’s fixed point theorem. The Banach contraction principle is also used to show the uniqueness of a solution with some additional restrictions. Exponential stability of the obtained solution is also discussed by taking a suitable Lyapunov functional on time scale. At the end, we give a numerical example to illustrate the effectiveness of the obtained theoretical results. Our results are valid for general time scale, and the cases 𝕋=ℝ or ℤ are particular cases.

New Weighted Opial-Type Inequalities on Time Scales for Convex Functions

Our work is based on the multiple inequalities illustrated in 1967 by E. K. Godunova and V. I. Levin, in 1990 by Hwang and Yang and in 1993 by B. G. Pachpatte. With the help of the dynamic Jensen and Hölder inequality, we generalize a number of those inequalities to a general time scale. In addition to these generalizations, some integral and discrete inequalities will be obtained as special cases of our results.

Convergence Rate of Inertial Proximal Algorithms with General Extrapolation and Proximal Coefficients

In a Hilbert space setting ${\mathcal{H}}$, in order to minimize by fast methods a general convex lower semicontinuous and proper function ${\Phi }: {\mathcal{H}} \rightarrow \mathbb {R} \cup \{+\infty \}$, we analyze the convergence rate of the inertial proximal algorithms. These algorithms involve both extrapolation coefficients (including Nesterov acceleration method) and proximal coefficients in a general form. They can be interpreted as the discrete time version of inertial continuous gradient systems with general damping and time scale coefficients. Based on the proper setting of these parameters, we show the fast convergence of values and the convergence of iterates. In doing so, we provide an overview of this class of algorithms. Our study complements the previous Attouch–Cabot paper (SIOPT, 2018) by introducing into the algorithm time scaling aspects, and sheds new light on the Guler seminal papers on the convergence rate of the accelerated proximal methods for convex optimization.

Dynamic Hilbert-Type Inequalities with Fenchel-Legendre Transform

Our work is based on the multiple inequalities illustrated in 2020 by Hamiaz and Abuelela. With the help of a Fenchel-Legendre transform, which is used in various problems involving symmetry, we generalize a number of those inequalities to a general time scale. Besides that, in order to get new results as special cases, we will extend our results to continuous and discrete calculus.