Cosine Operator Research Articles

On a cosine operator function framework of approximation processes in Banach space

We introduce the cosine-type approximation processes in abstract Banach space setting. The historical roots of these processes go back to W. W. Rogosinski in 1926. The given new definitions use a cosine operator functions concept. We proved that in presented setting the cosine-type operators possess the order of approximation, which coincide with results known in trigonometric approximation. Moreover, a general method for factorization of certain linear combinations of cosine operator functions is presented. The given method allows to find the order of approximation using the higher order modulus of continuity. Also applications for the different type of approximations are given.

Driving Factor Analysis and Forecasting of CO2 Emissions from Power Output in China Using Scenario Analysis and CSCWOA-ELM Method

With power consumption increasing in China, the CO2 emissions from electricity pose a serious threat to the environment. Therefore, it is of great significance to explore the influencing factors of power CO2 emissions, which is conducive to sustainable economic development. Taking the characteristics of power generation, transmission and consumption into consideration, the grey relational analysis method (GRA) is adopted to select 11 influencing factors, which are further converted into 5 main factors by hierarchical clustering analysis (HCA). According to the possible variation tendency of each factor, 48 development scenarios are set up from 2018–2025, and then an extreme learning machine optimized by whale algorithm based on chaotic sine cosine operator (CSCWOA-ELM) is established to predict the power CO2 emissions respectively. The results show that gross domestic product (GDP) has the greatest impact on the CO2 emissions from power output, of which the average contribution rate is 1.28%. Similarly, power structure and living consumption level also have an enormous influence, with average contribution rates over 0.6%. Eventually, the analysis made in this study can provide valuable policy implications for power CO2 emissions reduction, which can be regarded as a reference for China’s 14th Five-Year development plan in the future.

An Extended Version of Flower Pollination Algorithm

Flower pollination algorithm (FPA) is a novel meta-heuristic algorithm inspired from the pollination of flowers. It has been applied to various fields of research and has proved its worth. But the algorithm suffers from certain limitations which confines its scope to different field of applications. In the present work, four modified versions of FPA have been proposed to improve exploration, searching accuracy, convergence speed and to maintain good balance between intensification and diversification. Sine Cosine operator has been added instead of Levy flight for improving the exploration and searching accuracy. Secondly, an improved probability switching has been used to balance the exploration and exploitation. The last modification uses improved distribution parameter for enhancing the local search phase. The results of proposed versions are tested on nineteen benchmark functions for different population sizes. The best modified version of algorithm has been named as linear–exponential flower pollination algorithm (LEFPA). The results of LEFPA are then compared with firefly algorithm, differential evolution, bat algorithm, bat flower pollination algorithm, Sine Cosine algorithm, particle swarm optimization, genetic algorithm, cuckoo search and FPA. Experimental results show that LEFPA algorithm is better when compared to other algorithms and can be considered as the potential candidate to become state of the art. Further, statistical testing has been done to prove the significance of the proposed approach.

Optimal Control of Second Order Stochastic Evolution Hemivariational Inequalities with Poisson Jumps

The purpose of this article is to study the optimal control problem of second order stochastic evolution hemivariational inequalities with Poisson jumps by virtue of cosine operator theory in the Hilbert space. Initially, the sufficient conditions for existence of mild solution of the proposed system are verified by applying properties of Clarke's subdifferential operator and fixed point theorem in multivalued maps. Further, we formulated and proved the existence results for optimal control of the proposed system with corresponding cost function by using Balder theorem. Finally an example is provided to illustrate the main results.

Hölder-Lebesgue regularity and almost periodicity for semidiscrete equations with a fractional Laplacian

We study the equations\begin{document}$\begin{align}\partial_t u(t, n) = L u(t, n) + f(u(t, n), n); \partial_t u(t, n) = iL u(t, n) + f(u(t, n), n)\end{align}$\end{document}and\begin{document}$\begin{align}\partial_{tt} u(t, n) =Lu(t, n) + f(u(t, n), n), \end{align}$\end{document}where $n∈ \mathbb{Z}$, $t∈ (0, ∞)$, and $L$ is taken to be either the discrete Laplacian operator $Δ_\mathrm{d} f(n)=f(n+1)-2f(n)+f(n-1)$, or its fractional powers $-(-Δ_{\mathrm{d}})^{σ}$, $0<σ<1$. We combine operator theory techniques with the properties of the Bessel functions to develop a theory of analytic semigroups and cosine operators generated by $Δ_\mathrm{d}$ and $-(-Δ_\mathrm{d})^{σ}$. Such theory is then applied to prove existence and uniqueness of almost periodic solutions to the above-mentioned equations. Moreover, we show a new characterization of well-posedness on periodic Hölder spaces for linear heat equations involving discrete and fractional discrete Laplacians. The results obtained are applied to Nagumo and Fisher-KPP models with a discrete Laplacian. Further extensions to the multidimensional setting $\mathbb{Z}^N$ are also accomplished.

Controllability of isotropic viscoelastic bodies of Maxwell–Boltzmann type

In this paper we consider a viscoelastic three dimensional body (of Maxwell–Boltzmann type) controlled on (part of) the boundary. We assume that the material is isotropic and homogeneous. If the body is elastic ( i . e . no dissipation due to past memory), controllability has been studied by several authors. We prove that the viscoelastic body inherits the controllability properties of the corresponding purely elastic system. The proof relays on cosine operator methods combined with moment theory.

Disjoint topological transitivity for cosine operator functions on groups

In this paper, we study finite sequences of operators, generated by the powers of weighted translations on discrete groups, and give sufficient conditions for such sequences to be disjoint topologically transitive and mixing in terms of the group elements and weights. The sequences of operators are cosine operator functions. Moreover, we also obtain necessary conditions for cosine operator functions to be disjoint transitive and mixing.

On approximation processes defined by a cosine operator function

On approximation processes defined by a cosine operator function

Recurrence of Cosine Operator Functions on Groups

AbstractIn this note, we study the recurrence and topologically multiple recurrence of a sequence of operators on Banach spaces. In particular, we give a sufficient and necessary condition for a cosine operator function, induced by a sequence of operators on the Lebesgue space of a locally compact group, to be topologically multiply recurrent.

Approximate controllability of a second-order neutral stochastic differential equation with state dependent delay

In this paper, the existence and uniqueness of mild solution is initially obtained by use of measure of noncompactness and simple growth conditions. Then the conditions for approximate controllability are investigated for the distributed second-order neutral stochastic differential system with respect to the approximate controllability of the corresponding linear system in a Hilbert space. We construct controllability operators by using simple and fundamental assumptions on the system components. We use the lemma, which implies the approximate controllability of the associated linear system. This lemma is also described as a geometrical relation between the range of the operator B and the subspaces Ni⊥, i = 1, 2, 3, associated with sine and cosine operators in L2([0, a], X) and L2([0, a], LQ). Eventually, we show that the reachable set of the stochastic control system lies in the reachable set of its associated linear control system. An example is provided to illustrate the presented theory.

Homogenization of hyperbolic equations

We consider a self-adjoint matrix elliptic operator Ae, e > 0, on L2(Rd;Cn) given by the differential expression b(D)*g(x/e)b(D). The matrix-valued function g(x) is bounded, positive definite, and periodic with respect to some lattice; b(D) is an (m × n)-matrix first order differential operator such that m ≥ n and the symbol b(ξ) has maximal rank. We study the operator cosine cos(τAe1/2), where τ ∈ R. It is shown that, as e → 0, the operator cos(τAe1/2) converges to cos(τ(A0)1/2) in the norm of operators acting from the Sobolev space Hs(Rd;Cn) (with a suitable s) to L2(Rd;Cn). Here A0 is the effective operator with constant coefficients. Sharp-order error estimates are obtained. The question about the sharpness of the result with respect to the type of the operator norm is studied. Similar results are obtained for more general operators. The results are applied to study the behavior of the solution of the Cauchy problem for the hyperbolic equation ∂τ2ue(x, τ) = −Aeue(x, τ).

The research on electric load forecasting based on nonlinear gray bernoulli model optimized by cosine operator and particle swarm optimization

According to the characteristics of the power system with nonlinear operation, this paper analyzed the application limitations of traditional gray model. This study processed original data sequence with Cosine function, which could weaken the influence of outliers, improve the smoothness of the sequence and reduce reduction error. The particle swarm optimization algorithm was employed to optimize the parameter r and the background value structure coefficient α in nonlinear gray bernoulli model with the purpose of searching for the optimal parameters of the model, which could make up for the insufficient caused by given the parameters on experience. Finally, this paper discussed the implementation process of the optimized model. The history load data of Beijing grid was applied to inspect the forecasting effect of the optimized model. The numerical results and error analysis illustrated that the model has a favorable prediction effect and wide applications.

Dynamics of the solutions of the water hammer equations

A water hammer is a pressure wave that occurs, accidentally or intentionally, in a filled liquid pipeline when a tap is suddenly closed, or a pump starts or stops, or when a valve closes or opens. A water hammer wave propagates through pipes reflecting on features and boundaries. This phenomenon is governed by a pair of coupled quasi-linear partial differential equations of first order, that are usually solved using the method of characteristics.In this note we provide a representation of the solution using an operator theoretical approach based on the theory of C0-semigroups and cosine operator functions, when considering this phenomenon on a compressible fluid along an infinite pipe. We provide an integro-differential equation that represents this phenomenon and it only involves the discharge. In addition, the representation of the solution in terms of a specific C0-semigroup lets us show that hypercyclicity and the topologically mixing property can occur when considering this phenomenon on certain weighted spaces of integrable and continuous functions on the real line.

Topological transitivity for cosine operator functions on groups

Let G be a locally compact group and let 1≤p<∞. We give a necessary and sufficient condition for a cosine operator function induced by a weighted translation operator on the Lebesgue space Lp(G) to be topologically transitive. By the same argument, the characterization for mixing cosine operator functions is given. We also determine when the direct sum of a sequence of cosine operator functions is transitive.

Trajectory Controllability of Second Order Nonlinear Integro-Differential System: An Analytical and a Numerical Estimation

In this short article we have discussed trajectory controllability of the second order nonlinear integro-differential system by using theory of monotone operators and tools of cosine operators. We will then describe a general method for computing numerical approximations to control functions. We will illustrate this method with a nonlinear integro-differential equations and a nonlinear hyperbolic equation in one and two spatial dimensions.

Chaos for Cosine Operator Functions Generated by Shifts

Let 1 ≤ p &lt; ∞. We give the sufficient and necessary condition for cosine operator functions, generated by bilateral weighted shifts on ℓp(ℤ), to be chaotic. Moreover, such a cosine operator function is chaotic if, and only if, its weighted shift is chaotic.

Chaos for Cosine Operator Functions on Groups

Let1≤p&lt;∞andGbe a locally compact group. We characterize chaotic cosine operator functions, generated by weighted translations on the Lebesgue spaceLp(G), in terms of the weight condition. In particular, chaotic cosine operator functions and chaotic weighted translations can only occur simultaneously. We also give a necessary and sufficient condition for the direct sum of a sequence of cosine operator functions to be chaotic.

Improved optical orthogonal frequency‐division multiplexing performance using non‐linear signal compression in intensity modulation and direct detection transmission systems incorporating parameter‐relaxed digital‐to‐analogue converters/analogue‐to‐digital converters

Adaptively modulated optical orthogonal frequency-division multiplexing (AMOOFDM) using non-linear signal compression (NSC) (AMOOFDM-NSC) is proposed and numerically explored, for the first time, in intensity modulation and direct detection (IMDD) single-mode fibre transmission systems for cost-sensitive optical access networks. AMOOFDM-NSC utilises a simple cosine operation to compress digital electrical OFDM signals prior to performing digital-to-analogue conversion and optical intensity modulation in the transmitter. Compared with conventional AMOOFDM, AMOOFDM-NSC can not only significantly lower requirements on minimum quantisation bits and sampling rates of digital-to-analogue/analogue-to-digital convertors, but also simultaneously doubles the signal transmission capacity for quantisation bits of < 4. In addition, AMOOFDM-NSC also shows excellent system flexibility and has good performance robustness to variations in system characteristics. AMOOFDM-NSC may have great potential for providing a cost-effective technical solution for practical implementations in cost-sensitive access networks.

Topological mixing for cosine operator functions generated by shifts

Let 1⩽p<∞. We characterize topologically mixing cosine operator functions, generated by unilateral and bilateral weighted shifts on ℓp(N0) and ℓp(Z) respectively. We also give sufficient conditions for such cosine operator functions to be topologically transitive.

Global uniqueness and stability in determining the damping coefficient of an inverse hyperbolic problem with non-homogeneous Dirichlet B.C. through an additional localized Neumann boundary trace

We consider a second-order hyperbolic equation on an open bounded domain Ω in ℝ n for n ≥ 2, with C 2-boundary , Γ0 ∩ Γ1 = ∅, subject to non-homogeneous Dirichlet boundary conditions on the entire boundary Γ. We then study the inverse problem of determining the interior damping coefficient of the equation by means of an additional measurement of the Neumann boundary trace of the solution, in a suitable, explicit sub-portion Γ1 of the boundary Γ, and over a computable time interval T > 0. Under sharp conditions on the complementary part Γ0 = Γ\\Γ1, T > 0, and under sharp regularity requirements on the data, we establish the two canonical results in inverse problems: (i) uniqueness and Lipschitz stability (at the H θ-level, 0 < θ ≤ 1, ). The latter (ii) is the main result of this article. Our proof relies on three main ingredients: (a) sharp Carleman estimates at the H 1 × L 2-level for second-order hyperbolic equations [I. Lasiecka, R. Triggiani, and X. Zhang, Nonconservative wave equations with unobserved Neumann B.C.: Global uniqueness and observability in one shot, Contemp. Math. 268 (2000), pp. 227–325] (b) a correspondingly implied continuous observability inequality at the same energy level Lasiecka et al.; (c) sharp interior and boundary regularity theory for second-order hyperbolic equations with Dirichlet boundary data [I. Lasiecka, J.L. Lions, and R. Triggiani, Non homogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl., 65 (1986), pp. 149–192; I. Lasiecka and R. Triggiani, A cosine operator approach to modelling L 2(0, T; L 2(Ω)) boundary input hyperbolic equations, Appl. Math. Optimiz., 7 (1981), 35–83; I. Lasiecka and R. Triggiani, Regularity of hyperbolic equations under L 2(0, T; L 2(Ω))-Dirichlet boundary terms, Appl. Math. Optimiz., 10 (1983), pp. 243–290]. The proof of the linear uniqueness result (Section 4, Step 5) also takes advantage of a convenient tactical route ‘post-Carleman estimates’ suggested by Isakov in [V. Isakov and M. Yamamoto, Carleman estimate with Neumann B.C. and its application to the observability inequality and inverse hyperbolic problems, Contemp. Math. 268 (2000), pp. 191–225, Theorem 8.2.2, p. 231].