An operator $T$ on a separable, infinite dimensional, complex Hilbert space $\mathcal{H}$ is called conjugate normal if $C|T|C = |T^∗|$ for some conjugate linear, isometric involution $C$ on $\mathcal{H}.$ This paper focuses on the invariance of conjugate normality under similarity. Given an operator $T,$ we prove that every operator $A$ similar to $T$ is conjugate normal if and only if there exist complex numbers $λ_1$, $λ_2$ such that $(T−λ_1)(T−λ_2)=0.$

Recently, Yu et al. presented a modified fixed point iterative (MFPI) method for solving large sparse absolute value equation (AVE). In this paper, we consider using accelerated overrelaxation (AOR) splitting to develop the modified fixed point iteration (denoted by MFPI-JS and MFPI-GSS) methods for solving AVE. Furthermore, the convergence analysis of the MFPI-JS and MFPI-GSS methods for AVE are also studied under suitable restrictions on the iteration parameters, and the functional equation between the parameter $\tau$ and matrix $Q.$ Finally, numerical examples show that the MFPI-JS and MFPI-GSS are efficient iteration methods.

This work presents a stochastic Chebyshev-Picard iteration method to efficiently solve nonlinear differential equations with random inputs. If the nonlinear problem involves uncertainty, we need to characterize the uncertainty by using a few random variables. The nonlinear stochastic problems require solving the nonlinear system for a large number of samples in the stochastic space to quantify the statistics of the system of response and explore the uncertainty quantification. The computational cost is very expensive. To overcome the difficulty, a low rank approximation is introduced to the solution of the corresponding nonlinear problem and admits a variable-separation form in terms of stochastic basis functions and deterministic basis functions. No iteration is performed at each enrichment step. These basis functions are model-oriented and involve offline computation. To efficiently identify the stochastic basis functions, we utilize the greedy algorithm to select some optimal samples. Then the modified Chebyshev-Picard iteration method is used to solve the nonlinear system at the selected optimal samples, the solutions of which are used to train the deterministic basis functions. With the deterministic basis functions, we can obtain the corresponding stochastic basis functions by solving linear differential systems. The computation of the stochastic Chebyshev-Picard method decomposes into an offline phase and an online phase. This is very desirable for scientific computation. Several examples are presented to illustrate the efficacy of the proposed method for different nonlinear differential equations.

Kernel learning forward backward stochastic differential equations (FBSDE) filter is an iterative and adaptive meshfree approach to solve the nonlinear filtering problem. It builds from forward backward SDE for Fokker-Planker equation, which defines evolving density for the state variable, and employs kernel density estimation (KDE) to approximate density. This algorithm has shown more superior performance than mainstream particle filter method, in both convergence speed and efficiency of solving high dimension problems. However, this method has only been shown to converge empirically. In this paper, we present a rigorous analysis to demonstrate its local and global convergence, and provide theoretical support for its empirical results.

In [T. Wu et al., arXiv2310.09775, 2023], a general Dabrowski-Sitarz-Zalecki type theorem for odd dimensional manifolds with boundary was proved. In this paper, we give the proof of the another general Dabrowski-Sitarz-Zalecki type theorem for the spectral Einstein functional associated with the Dirac operator on odd dimensional manifolds with boundary.