arXiv subjects

Min-Ruei Lin

Publications and source records attributed to Min-Ruei Lin.

5 recordsLinked to original sources

arXiv2026

Metric Rigidity in Anchored Sobolev Spaces on Intervals

For $1\le p\le\infty$ and $i=1,2$, let $W^{k_i,p}(\Omega_i)$ be the Sobolev space on a bounded open interval $\Omega_i$ with differentiability order $k_i$. We equip $W^{k_i,p}(\Omega_i)$ with an anchored Sobolev norm and the order $\ge_{k_i,p}$ defined by $f^{(j)}(x_i)\ge 0$ for each $j=0,\ldots,k_i-1$ and $f^{(k_i)}\ge 0$ a.e. We show that the positive unit spheres of $W^{k_1,p}(\Omega_1)$ and $W^{k_2,p}(\Omega_2)$ are surjectively isometric if and only if $k_1=k_2$. Every such isometry extends uniquely to a complex-linear isometric order isomorphism, for which we obtain a coordinate representation. The same conclusions hold for surjective phase-isometries. For $1<p<\infty$, they also hold for surjective norm-additive maps.

math.FA↗

arXiv2026

A variant of Tingley's problem on ordered Banach spaces of absolutely continuous functions

For each $1\le p\le\infty$ and $j=1,2$, let $AC^p(\Omega_j)$ denote the Banach space of complex-valued absolutely continuous functions on a closed unit interval $\Omega_j=[x_j,x_j+1]$. We equip $AC^p(\Omega_j)$ with the $p$--norm $\|f\|_{AC,p}$, and the order $\ge_{AC}$ defined by $f(x_j)\ge0$ and $f'\ge 0$ a.e. Set $$S(AC^p(\Omega_j))^+=\{f\in AC^p(\Omega_j):\|f\|_{AC,p}=1,\ f\ge_{AC}0\}.$$ We prove that, for each $1\le p\le\infty$, every surjective isometry $S(AC^p(\Omega_1))^+\to S(AC^p(\Omega_2))^+$ extends uniquely to a complex--linear isometric order isomorphism from $AC^p(\Omega_1)$ onto $AC^p(\Omega_2)$. As an application, we obtain a corresponding extension theorem for surjective phase--isometries.

math.FA↗

arXiv2026

Surjective isometries on the positive parts of the unit spheres of some function spaces

We consider the space $C^1[0, 1]$ of continuously differentiable functions on the closed unit interval $[0, 1]$ and the space $\operatorname{Lip}[0, 1]$ of Lipschitz continuous functions on $[0, 1]$, equipped with the norms \begin{align*} \|f\|_{\sigma, p} = \begin{cases} \sqrt[p]{|f(0)|^p + \|f'\|_\infty^p} & (1 \le p < \infty), \\ \max\{\, |f(0)|, \|f'\|_\infty \,\} & (p = \infty). \end{cases} \end{align*} We show that every surjective isometry on the positive part of the unit sphere extends to a surjective complex-linear isometry on the entire space. As a corollary, every such isometry also extends to an isometric order isomorphism on the real subspaces $C^1_{\mathbb{R}}[0, 1]$ and $\operatorname{Lip}_{\mathbb{R}}[0, 1]$.

math.FA↗

arXiv2026

A variant of Tingley's problem for positive unit spheres of continuous functions that vanish at infinity

Let $S(C_0(X))^+$ and $S(C_0(Y))^+$ denote the positive parts of the unit spheres of $C_0(X)$ and $C_0(Y)$, where $X$ and $Y$ are locally compact Hausdorff spaces. We prove that every surjective isometry from $S(C_0(X))^+$ onto $S(C_0(Y))^+$ is a composition operator induced by a homeomorphism between $X$ and $Y$ . As a consequence, such a map extends to a surjective reallinear isometry from $C_0(X)$ onto $C_0(Y)$. We also characterize surjective phase-isometries on the positive unit sphere.

math.FA↗

arXiv2022

Quaternion-based machine learning on topological quantum systems

Topological phase classifications have been intensively studied via machine-learning techniques where different forms of the training data are proposed in order to maximize the information extracted from the systems of interests. Due to the complexity in quantum physics, advanced mathematical architecture should be considered in designing machines. In this work, we incorporate quaternion algebras into data analysis either in the frame of supervised and unsupervised learning to classify two-dimensional Chern insulators. For the unsupervised-learning aspect, we apply the principal component analysis (PCA) on the quaternion-transformed eigenstates to distinguish topological phases. For the supervised-learning aspect, we construct our machine by adding one quaternion convolutional layer on top of a conventional convolutional neural network. The machine takes quaternion-transformed configurations as inputs and successfully classify all distinct topological phases, even for those states that have different distributuions from those states seen by the machine during the training process. Our work demonstrates the power of quaternion algebras on extracting crucial features from the targeted data and the advantages of quaternion-based neural networks than conventional ones in the tasks of topological phase classifications.

quant-ph↗