Monotone Sequences Research Articles

Complete monotonicity-preserving numerical methods for time fractional ODEs

The time fractional ODEs are equivalent to convolutional Volterra integral equations with completely monotone kernels. We therefore introduce the concept of complete monotonicity-preserving ($\mathcal{CM}$-preserving) numerical methods for fractional ODEs, in which the discrete convolutional kernels inherit the $\mathcal{CM}$ property as the continuous equations. We prove that $\mathcal{CM}$-preserving schemes are at least $A(\pi/2)$ stable and can preserve the monotonicity of solutions to scalar nonlinear autonomous fractional ODEs, both of which are novel. Significantly, by improving a result of Li and Liu (Quart. Appl. Math., 76(1):189-198, 2018), we show that the $\mathcal{L}$1 scheme is $\mathcal{CM}$-preserving, so that the $\mathcal{L}$1 scheme is at least $A(\pi/2)$ stable, which is an improvement on stability analysis for $\mathcal{L}$1 scheme given in Jin, Lazarov and Zhou (IMA J. Numer. Analy. 36:197-221, 2016). The good signs of the coefficients for such class of schemes ensure the discrete fractional comparison principles, and allow us to establish the convergence in a unified framework when applied to time fractional sub-diffusion equations and fractional ODEs. The main tools in the analysis are a characterization of convolution inverses for completely monotone sequences and a characterization of completely monotone sequences using Pick functions due to Liu and Pego (Trans. Amer. Math. Soc. 368(12):8499-8518, 2016). The results for fractional ODEs are extended to $\mathcal{CM}$-preserving numerical methods for Volterra integral equations with general completely monotone kernels. Numerical examples are presented to illustrate the main theoretical results.

Positivity of Discrete Time-Fractional Operators with Applications to Phase-Field Equations

We present general criteria ensuring the positivity of quadratic forms of convolution type generated by sequences of real numbers. A sharp result is obtained in the case of completely monotone sequences. Applications to widely used approximations of fractional integral and differential operators, including convolution quadrature and L1 formula on uniform temporal meshes, are presented and new inequalities are established. The results are found to be fundamental in the investigation of the numerical stability for time-fractional phase-field models. It is shown through a standard energy stability analysis and without the use of a fractional Grönwall inequality that several numerical schemes satisfy discrete energy dissipation laws.

Kaplan classes of a certain family of functions

We give the complete characterization of members of Kaplan classes of products of power functions with all zeros symmetrically distributed in \(\mathbb{T} := \{z \in\mathbb{C} : |z| = 1\}\) and weakly monotonic sequence of powers. In this way we extend Sheil-Small’s theorem. We apply the obtained result to study univalence of antiderivative of these products of power functions.

Uniform convergence criterion for non-harmonic sine series

We show that for a nonnegative monotonic sequence the condition is sufficient for the series to converge uniformly on any bounded set for , and for any odd it is sufficient for it to converge uniformly on the whole of . Moreover, the latter assertion still holds if we replace by any polynomial in odd powers with rational coefficients. On the other hand, in the case of even it is necessary that for the above series to converge at the point or at . As a consequence, we obtain uniform convergence criteria. Furthermore, the results for natural numbers remain true for sequences in the more general class . Bibliography: 17 titles.

On the best Ulam constant of a first order linear difference equation in Banach spaces

We obtain some results on Ulam stability for the linear difference equation $$x_{n+1}=a_nx_n+b_n,\, n\geq0$$ , in a Banach space X. If there exists $$\lim_{n\to\infty}|a_n|=\lambda$$ , then the equation is Ulam stable if and only if $$\lambda\neq 1$$ . Moreover if $$(|a_n|)_{n\geq 0}$$ is a monotone sequence, then the best Ulam constant of the equation is $$\frac{1}{|\lambda-1|}$$ .

Event-triggered bipartite consensus over cooperation-competition networks under DoS attacks

This paper addresses the bipartite consensus over cooperation-competition networks affected by denial-of-service (DoS) attacks. Consider that a network consists of multiple interactive agents, and the relationship between neighboring agents is cooperative or competitive. Meanwhile, information transmission among the agents is unavailable during the intervals of attacks. In order to save communication resources and exclude the Zeno behavior, an event-triggered scheme depending on the sampled-data information from neighboring agents is proposed, and efficient defense strategies in response to the attacks are put forward. Suppose that the frequency and duration of DoS attacks meet certain requirements, then according to the signed graph theory, the LaSalle’s invariance principle, and the convergence of monotone sequences, the results of bipartite consensus via the event-triggered protocol are provided, which are mainly related to the communication topology of the network, the sampling period, and the threshold parameters in the event-triggered scheme. It is shown that the bipartite consensus is realized even though the DoS attacks take place frequently. Furthermore, this paper discusses the bipartite consensus in the presence of DoS attacks with a random unsuccessful rate. Finally, numerical simulations illustrate the theoretical results.

Auditory local\u2013global temporal processing: evidence for perceptual reorganization with musical expertise

The way the visual system processes different scales of spatial information has been widely studied, highlighting the dominant role of global over local processing. Recent studies addressing how the auditory system deals with local–global temporal information suggest a comparable processing scheme, but little is known about how this organization is modulated by long-term musical training, in particular regarding musical sequences. Here, we investigate how non-musicians and expert musicians detect local and global pitch changes in short hierarchical tone sequences structured across temporally-segregated triplets made of musical intervals (local scale) forming a melodic contour (global scale) varying either in one direction (monotonic) or both (non-monotonic). Our data reveal a clearly distinct organization between both groups. Non-musicians show global advantage (enhanced performance to detect global over local modifications) and global-to-local interference effects (interference of global over local processing) only for monotonic sequences, while musicians exhibit the reversed pattern for non-monotonic sequences. These results suggest that the local–global processing scheme depends on the complexity of the melodic contour, and that long-term musical training induces a prominent perceptual reorganization that reshapes its initial global dominance to favour local information processing. This latter result supports the theory of “analytic” processing acquisition in musicians.

Some results on entropy dimension for non-autonomous systems

In this paper, the preimage branch t-entropy and entropy dimension for nonautonomous systems are studied and some systems with preimage branch t-entropy zero are introduced. Moreover, formulas calculating the s-topological entropy of a sequence of equi-continuous monotone maps on the unit circle are given. Finally, examples to show that the entropy dimension of non-autonomous systems can be attained by any positive number s are constructed.

Monotone Iterative Technique for Nonlinear Periodic Time Fractional Parabolic Problems

In this paper, the existence and uniqueness of the weak solution for a linear parabolic equation with conformable derivative are proved, the existence of weak periodic solutions for conformable fractional parabolic nonlinear differential equation is proved by using a more generalized monotone iterative method combined with the method of upper and lower solutions. We prove the monotone sequence converge to weak periodic minimal and maximal solutions. Moreover, the conformable version of the Lions-Magness and Aubin–Lions lemmas are also proved.

A new Hermite\u2013Hadamard type inequality for coordinate convex function

In the article, we establish a new Hermite–Hadamard type inequality for the coordinate convex function by constructing two monotonic sequences. The given result is the generalization and improvement of some previously obtained results.

Some new dynamic Hardy-type inequalities with kernels involving monotone functions

The main objective of the present paper is an extension of several classical integral Hardy-type inequalities with kernels to a time scales setting. The established inequalities refer to a class of monotone functions and they are characterized by appropriate relations for the accompanying weight functions. As an application, we obtain the corresponding discrete Hardy-type inequalities for monotone sequences, which are essentially new.

Analysis of age and spatially dependent population model: Application to forest growth

In this paper, we consider an age-structured population model with diffusion. We first establish a comparison principle. We then apply the comparison principle to show the existence and uniqueness of solutions by constructing monotone sequences of weak upper and lower solutions. We also use the comparison principle to study the long-time behavior of the solution and present extinction and boundedness results under certain conditions on the parameters in the model. Additionally, as an example of its application, we apply the model to describe the dynamics of a forest population.

Transformation of Fourier Series by Means of General Monotone Sequences

Estimates of the norms and the best approximations of the generalized Liouville–Weyl derivative in the two-dimensional case are obtained.

Computer Simulation and Iterative Algorithm for Approximate Solving of Initial Value Problem for Riemann-Liouville Fractional Delay Differential Equations

The main aim of this paper is to suggest an algorithm for constructing two monotone sequences of mild lower and upper solutions which are convergent to the mild solution of the initial value problem for Riemann-Liouville fractional delay differential equation. The iterative scheme is based on a monotone iterative technique. The suggested scheme is computerized and applied to solve approximately the initial value problem for scalar nonlinear Riemann-Liouville fractional differential equations with a constant delay on a finite interval. The suggested and well-grounded algorithm is applied to a particular problem and the practical usefulness is illustrated.

Quasilinearization technique for solving nonlinear Riemann-Liouville fractional-order problems

In this work, we deal with the quasilinearization technique for a class of nonlinear Riemann-Liouville fractional-order two-point boundary value problems. Using quasilinearization technique, we construct a monotone sequence of approximate solutions which has quadratic convergence to the unique solution of the original problem, and establish the corresponding convergence estimates. Moreover, the performance of the technique is examined through a numerical example, which shows that our regularization method is available and stable.

Some New Weighted Dynamic Inequalities for Monotone Functions Involving Kernels

In the present article, we derive several new weighted dynamic inequalities for monotone functions involving kernels, some of which are the Hardy-type inequalities. The established inequalities are characterized by appropriate relations for the accompanying weight functions. Our results are time scale extensions of several classical weighted inequalities known from the literature. As an application, we obtain the corresponding discrete weighted inequalities for monotone sequences, which are essentially new.

Existence of extremal solutions of fractional Langevin equation involving nonlinear boundary conditions

Fractional Langevin equation describes the evolution of physical phenomena in fluctuating environments for the complex media systems. It is a sequential fractional differential equation with two fractional orders involving a memory kernel, which leads to non-Markovian dynamics and subdiffusion. Here by establishing a general solution of the linear fractional Langevin equations involving initial conditions with the help of well-known Mittag–Leffler functions and using the special properties of these functions, we construct a new comparison result related to linear fractional Langevin equation. Meanwhile, we investigate the existence of extremal solutions for nonlinear boundary value problems with advanced arguments. The method is a constructive method that yields monotone sequences that converge to the extremal solutions. At last an example is presented to illustrate the main results.

A primer on the characterization of the exchangeable Marshall–Olkin copula via monotone sequences

While derivations of the characterization of the $d$-variate exchangeable Marshall–Olkin copula via $d$-monotone sequences relying on basic knowledge in probability theory exist in the literature, they contain a myriad of unnecessary relatively complicated computations. We revisit this issue and provide proofs where all undesired artefacts are removed, thereby exposing the simplicity of the characterization. In particular, we give an insightful analytical derivation of the monotonicity conditions based on the monotonicity properties of the survival probabilities.

The existence of upper and lower solutions to second order random impulsive differential equation with boundary value problem

In this article, we consider the existence of upper and lower solutions to a second-order random impulsive differential equation. We first study the solution form of the corresponding linear impulsive system of the second-order random impulsive differential equation. Based on the form of the solution, we define the resolvent operator. Then, we prove that the fixed point of this operator is the solution to the equation. Finally, we construct the sum of two monotonic iterative sequences and prove that they are convergent. Thus, we conclude that the system has upper and lower solutions.

Demand Forecast of Weapon Equipment Spare Parts Based on Improved Gray-Markov Model

Abstract The demand for spare parts of weapons and equipment is time-varying and random. It is difficult to predict the demand for spare parts. Therefore, on the basis of gray GM(1,1), a state transition probability matrix based on improved state division is used to establish a demand forecast model for weapon equipment and spare parts. The model not only considers the characteristics of the GM(1,1) model’s strong handling of monotonic sequences, but also extracts the characteristics of random fluctuation response of data through the transformation of the state transition probability matrix, avoiding the phenomenon of the worst prediction results when the maximum probability state is not the actual state. It is proved through experiments that the prediction result based on the improved gray-Markov model is superior to the traditional model and the classic gray-Markov prediction model, and the accuracy of the improved model is about 1.46 times higher than that of the gray model.