Existence results for a system of nonlinear operator equations and block operator matrices in locally convex spaces

The purpose of this paper is to prove some fixed point results dealing with a system of nonlinear equations defined in an angelic Hausdorff locally convex space ( X , { | ⋅ | p } p ∈ Λ ) (X,\{\lvert\,{\cdot}\,\rvert_{p}\}_{p\in\Lambda}) having the 𝜏-Krein–Šmulian property, where 𝜏 is a weaker Hausdorff locally convex topology of 𝑋. The method applied in our study is connected with a family Φ Λ τ \Phi_{\Lambda}^{\tau} -MNC of measures of weak noncompactness and with the concept of 𝜏-sequential continuity. As a special case, we discuss the existence of solutions for a 2 × 2 2\times 2 block operator matrix with nonlinear inputs. Furthermore, we give an illustrative example for a system of nonlinear integral equations in the space C ⁢ ( R + ) × C ⁢ ( R + ) C(\mathbb{R}^{+})\times C(\mathbb{R}^{+}) to verify the effectiveness and applicability of our main result.

Algebraic extension of $\mathcal{A}^{*}_{n}$ operator

$T\in L(H_{1}\oplus H_{2})$ is said to be an algebraic extension of a $\mathcal{A}^{*}_{n}$ operator if $$ T = \begin{pmatrix} T_{1} & T_{2} \\O & T_{3} \end{pmatrix} $$ is an operator matrix on $H_{1}\oplus H_{2}$, where $T_{1}$ is a $\mathcal{A}^{*}_{n}$ operator and $T_{3}$ is a algebraic.In this paper, we study basic and spectral properties of an algebraic extension of a $\mathcal{A}^{*}_{n}$ operator. We show that every algebraic extension of a $\mathcal{A}^{*}_{n}$ operator has SVEP, is polaroid and satisfies Weyl's theorem.

New fixed point theorems for countably condensing maps with an application to functional integral inclusions

This paper presents new fixed point theorems for 2 × 2 block operator matrix with countably condensing or countably 𝓓-set-contraction multi-valued inputs. Our theory will then be used to establish some new existence theorems for coupled system of functional differential inclusions in general Banach spaces under weak topology. Our results generalize, improve and complement a number of earlier works.

A New Method to Solve Dual Systems of Fractional Integro-Differential Equations by Legendre Wavelets

The method that will be presented, is numerical solution based on the Legendre wavelets for solving dual systems of fractional integro-differential equations (FIDEs). First of all we make the operational matrix of fractional order integration. The application of this matrix is transforming FIDEs to a system of algebric equations. By this changing, we are able to solve it by a simple solution. In this way, the Legendre wavelets and their operator matrix are the most important keys of our solution. After explaining the method we test on some illustrative examples which numerical solutions of these examples demonstrate the validity and applicability of suggested method.

Randomized Projection Learning Method for Dynamic Mode Decomposition

A data-driven analysis method known as dynamic mode decomposition (DMD) approximates the linear Koopman operator on a projected space. In the spirit of Johnson–Lindenstrauss lemma, we will use a random projection to estimate the DMD modes in a reduced dimensional space. In practical applications, snapshots are in a high-dimensional observable space and the DMD operator matrix is massive. Hence, computing DMD with the full spectrum is expensive, so our main computational goal is to estimate the eigenvalue and eigenvectors of the DMD operator in a projected domain. We generalize the current algorithm to estimate a projected DMD operator. We focus on a powerful and simple random projection algorithm that will reduce the computational and storage costs. While, clearly, a random projection simplifies the algorithmic complexity of a detailed optimal projection, as we will show, the results can generally be excellent, nonetheless, and the quality could be understood through a well-developed theory of random projections. We will demonstrate that modes could be calculated for a low cost by the projected data with sufficient dimension.

On local spectral properties of operator matrices

In this paper, we focus on a $2 \times 2$ 2 × 2 operator matrix $T_{\epsilon _{k}}$ T ϵ k as follows: $$\begin{aligned} T_{\epsilon _{k}}= \begin{pmatrix} A & C \\ \epsilon _{k} D & B\end{pmatrix}, \end{aligned}$$ T ϵ k = ( A C ϵ k D B ) , where $\epsilon _{k}$ ϵ k is a positive sequence such that $\lim_{k\rightarrow \infty }\epsilon _{k}=0$ lim k → ∞ ϵ k = 0 . We first explore how $T_{\epsilon _{k}}$ T ϵ k has several local spectral properties such as the single-valued extension property, the property $(\beta )$ ( β ) , and decomposable. We next study the relationship between some spectra of $T_{\epsilon _{k}}$ T ϵ k and spectra of its diagonal entries, and find some hypotheses by which $T_{\epsilon _{k}}$ T ϵ k satisfies Weyl’s theorem and a-Weyl’s theorem. Finally, we give some conditions that such an operator matrix $T_{\epsilon _{k}}$ T ϵ k has a nontrivial hyperinvariant subspace.

A new inverse modeling approach for emission sources based on the DDM-3D and 3DVAR techniques: an application to air quality forecasts in the Beijing–Tianjin–Hebei region

We develop a new inversion method which is suitable for linear and nonlinear emission source (ES) modeling, based on the three-dimensional decoupled direct (DDM-3D) sensitivity analysis module in the Community Multiscale Air Quality (CMAQ) model and the three-dimensional variational (3DVAR) data assimilation technique. We established the explicit observation operator matrix between the ES and receptor concentrations and the background error covariance (BEC) matrix of the ES, which can reflect the impacts of uncertainties of the ES on assimilation. Then we constructed the inversion model of the ES by combining the sensitivity analysis with 3DVAR techniques. We performed the simulation experiment using the inversion model for a heavy haze case study in the Beijing–Tianjin–Hebei (BTH) region during 27–30 December 2016. Results show that the spatial distribution of sensitivities of SO2 and NOx ESs to their concentrations, as well as the BEC matrix of ES, is reasonable. Using an a posteriori inversed ES, underestimations of SO2 and NO2 during the heavy haze period are remarkably improved, especially for NO2. Spatial distributions of SO2 and NO2 concentrations simulated by the constrained ES were more accurate compared with an a priori ES in the BTH region. The temporal variations in regionally averaged SO2, NO2, and O3 modeled concentrations using an a posteriori inversed ES are consistent with in situ observations at 45 stations over the BTH region, and simulation errors decrease significantly. These results are of great significance for studies on the formation mechanism of heavy haze, the reduction of uncertainties of the ES and its dynamic updating, and the provision of accurate “virtual” emission inventories for air-quality forecasts and decision-making services for optimization control of air pollution.