arXiv subjects

Chuang Zheng

Publications and source records attributed to Chuang Zheng.

6 recordsLinked to original sources

arXiv2025

Solving fuzzy linear systems in Gaussian PDMF space

We solve the fuzzy linear systems in a fuzzy number space $\mathcal{X}$, namely the Gaussian probability density membership function (Gaussian-PDMF) space. The fuzzy linear systems include two types: the semi-fuzzy linear system (SFLS) and the fully-fuzzy linear system (FFLS). First, we solve the SFLS $A \bm{\tilde{x}}= \bm{\tilde{b}}$, where $A\in \mathbb{R}^{m\times n}$ is a real-valued matrix, $\bm{\tilde{b}}$ is a fuzzy number vector, and $\bm{\tilde{x}}$ is the unknown fuzzy number vector. The elements of both $\bm{\tilde{b}}$ and $\bm{\tilde{x}}$ belong to $\mathcal{X}$. We present the Cramer's rule to calculate the solution with square matrix $A$ and find out that its solution set is a $5(n-R(A))$ dimensional affine space with $A\in \mathbb{R}^{m\times n}$ and $R(A)$ being the rank of $A$. The explicit form of the solution for RREF matrix $A$ is stated to ensure usability for modeling. Secondly, we solve the FFLS $\bm{\tilde{A}}\bm{\tilde{x}}=\bm{\tilde{b}}$, where $\bm{\tilde{A}}$ is a fuzzy matrix with all components in $\mathcal{X}$. We analyze its solution set and present the parametric form of solutions under the fuzzy RREF matrix. We then adapt Gaussian elimination method to fuzzy matrices and systems by restricting it to the unit group of ring $\mathcal{X}$, proving the equivalence of solution sets after elementary row operations. We also establish the connection between FFLS and SFLS by confining elements of $\bm{\tilde{A}}$ to a subset of $\mathcal{X}$ that forms a field. Two numerical examples are given to illustrated our method. All results in this paper are explicit since the Gaussian-PDMF space $\mathcal{X}$, to which the membership function of the fuzzy number belongs, possesses a complete algebraic structure. The proposed framework offers a systematical tool for solving the mathematical models using fuzzy linear systems with uncertainty and fuzziness.

math.GM↗

arXiv2023

Algebraic structure of the Gaussian-PDMF space and applications on fuzzy equations

In this paper, we extend the research presented in [Wang and Zheng, Fuzzy Sets and Systems, p108581, 2023] by establishing the algebraic structure of the Gaussian Probability Density Membership Function (Gaussian-PDMF) space. We consider fixed objective and subjective entities, denoted as $(h,p)$, and provide the explicit form of the membership function. Consequently, every fuzzy number with the membership function in $X_{h,p}(\mathbb{R})$, denoted as $\tilde{x}$, can be uniquely identified by a vector $\langle x; d^-, d^+, μ^-,μ^+\rangle$. Here, $x\in \mathbb{R}$ represents the "leading factor" of the fuzzy number $\tilde{x}$ with a membership degree equal to $1$. The parameters $d^-$ (left side) and $d^+$ (right side) denote the lengths of the compact support, while $μ^-$ (left side) and $μ^+$ (right side) represent the shapes. We introduce five operators: addition, subtraction, multiplication, scalar multiplication, and division. We demonstrate that, based on our definitions, the Gaussian-PDMF space exhibits a well-defined algebraic structure. For instance, $X_{h,p}(\mathbb{R})$ is a vector space over $\mathbb{R}$, featuring a subspace that forms a division ring, allowing for the representation of fuzzy polynomials, among other properties. We provide several examples to illustrate our theoretical results.

math.GM↗

arXiv2023

A class of fuzzy numbers induced by probability density functions and their arithmetic operations

In this paper we are interested in a class of fuzzy numbers which is uniquely identified by their membership functions. The function space, denoted by $X_{h, p}$, will be constructed by combining a class of nonlinear mappings $h$ (subjective perception) and a class of probability density functions (PDF) $p$ (objective entity), respectively. Under our assumptions, we prove that there always exists a class of $h$ to fulfill the observed outcome for a given class of $p$. Especially, we prove that the common triangular number can be interpreted by a function pair $(h, p)$. As an example, we consider a sample function space $X_{h, p}$ where $h$ is the tangent function and $p$ is chosen as the Gaussian kernel with free variable $μ$. By means of the free variable $μ$ (which is also the expectation of $p(x; μ)$), we define the addition, scalar multiplication and subtraction on $X_{h, p}$. We claim that, under our definitions, $X_{h, p}$ has a linear algebra. Some numerical examples are provided to illustrate the proposed approach.

math.GM↗

arXiv2021

Non-homogeneous initial boundary value problems for the biharmonic Schrödinger equation on an interval

In this paper we consider the initial boundary value problem (IBVP) for the nonlinear biharmonic Schrödinger equation posed on a bounded interval $(0,L)$ with non-homogeneous Navier or Dirichlet boundary conditions, respectively. For Navier boundary IBVP, we set up its local well-posedness if the initial data lies in $H^s(0, L)$ with $s\geq 0$ and $s\neq n+1/2, n\in \mathbb{N}$, and the boundary data are selected from the appropriate spaces with optimal regularities, i.e., the $j$-th order data are chosen in $H_{loc}^{(s+3-j)/4}(\mathbb {R}^+)$, for $j=0,2$. For Dirichlet boundary IBVP the corresponding local well-posedness is obtained when $s>10/7$ and $s\neq n+1/2, n\in \mathbb{N}$, and the boundary data are selected from the appropriate spaces with optimal regularities, i.e., the $j$-th order data are chosen in $H_{loc}^{(s+3-j)/4}(\mathbb {R}^+)$, for $j=0,1$.

math.FA↗

arXiv2017

Optimal Impulse Control of a Simple Reparable System in a Nonreflexive Banach Space

We discuss the problem of optimal impulse control representing the preventive maintenance of a simple reparable system. The system model is governed by coupled transport and integro-differential equations in a nonreflexive Banach space. The objective of this paper is to construct nonnegative impulse control inputs at given system running times that minimize the probability of the system in failure mode. To guarantee the nonnegativity of the controlled system, we consider the control inputs to depend on the system state. This essentially leads to a bilinear control problem. We first present a rigorous proof of existence of an optimal controller and then apply the variational inequality to derive the first-order necessary conditions of optimality.

math.OC↗

arXiv2013

Inverse problems for the fourth order Schrödinger equation on a finite domain

In this paper we establish a global Carleman estimate for the fourth order Schrödinger equation posed on a $1-d$ finite domain. The Carleman estimate is used to prove the Lipschitz stability for an inverse problem consisting in retrieving a stationary potential in the Schrödinger equation from boundary measurements.

math.AP↗