The Economic Behavior (Function) of Chinese Militia during the Second World War (Anti-Japanese War)

A militia is an organization that operates like an army but whose members are not professional soldiers. The functions of a militia vary in different periods, particularly during a war, when the militia not only plays a special role in national defense but also plays a unique role in economic production, which has a vital impact on a war scenario. This paper is about the militia in the Shandong Province of China during the Second World War. It explores archival materials to examine the unique economic functions of the militia during this period, and it shows how the militia’s economic role affected the Pacific War

Vertex-based Networks to Accelerate Path Planning Algorithms

Path planning plays a crucial role in various autonomy applications, and RRT* is one of the leading solutions in this field. In this paper, we propose the utilization of vertex-based networks to enhance the sampling process of RRT*, leading to more efficient path planning. Our approach focuses on critical vertices along the optimal paths, which provide essential yet sparser abstractions of the paths. We employ focal loss to address the associated data imbalance issue, and explore different masking configurations to determine practical tradeoffs in system performance. Through experiments conducted on randomly generated floor maps, our solutions demonstrate significant speed improvements, achieving over a 400% enhancement compared to the baseline model.Comment: Accepted to IEEE Workshop on Machine Learning for Signal Processing (MLSP'2023

Operators which are polynomially isometric to a normal operator

Let $\mathcal{H}$ be a complex, separable Hilbert space and $\mathcal{B}(\mathcal{H})$ denote the algebra of all bounded linear operators acting on $\mathcal{H}$. Given a unitarily-invariant norm $\| \cdot \|_u$ on $\mathcal{B}(\mathcal{H})$ and two linear operators $A$ and $B$ in $\mathcal{B}(\mathcal{H})$, we shall say that $A$ and $B$ are \emph{polynomially isometric relative to} $\| \cdot \|_u$ if $\| p(A) \|_u = \| p(B) \|_u$ for all polynomials $p$. In this paper, we examine to what extent an operator $A$ being polynomially isometric to a normal operator $N$ implies that $A$ is itself normal. More explicitly, we first show that if $\| \cdot \|_u$ is any unitarily-invariant norm on $\mathbb{M}_n(\mathbb{C})$, if $A, N \in \mathbb{M}_n(\mathbb{C})$ are polynomially isometric and $N$ is normal, then $A$ is normal. We then extend this result to the infinite-dimensional setting by showing that if $A, N \in \mathcal{B}(\mathcal{H})$ are polynomially isometric relative to the operator norm and $N$ is a normal operator whose spectrum neither disconnects the plane nor has interior, then $A$ is normal, while if the spectrum of $N$ is not of this form, then there always exists a non-normal operator $B$ such that $B$ and $N$ are polynomially isometric. Finally, we show that if $A$ and $N$ are compact operators with $N$ normal, and if $A$ and $N$ are polynomially isometric with respect to the $(c,p)$-norm studied by Chan, Li and Tu, then $A$ is again normal.Comment: submitte

Normal operators with highly incompatible off-diagonal corners

Let $\mathcal{H}$ be a complex, separable Hilbert space, and $\mathcal{B}(\mathcal{H})$ denote the set of all bounded linear operators on $\mathcal{H}$. Given an orthogonal projection $P \in \mathcal{B}(\mathcal{H})$ and an operator $D \in \mathcal{B}(\mathcal{H})$, we may write $D=\begin{bmatrix} D_1& D_2 D_3 & D_4 \end{bmatrix}$ relative to the decomposition $\mathcal{H} = \mathrm{ran}\, P \oplus \mathrm{ran}\, (I-P)$. In this paper we study the question: for which non-negative integers $j, k$ can we find a normal operator $D$ and an orthogonal projection $P$ such that $\mathrm{rank}\, D_2 = j$ and $\mathrm{rank}\, D_3 = k$? Complete results are obtained in the case where $\mathrm{dim}\, \mathcal{H} < \infty$, and partial results are obtained in the infinite-dimensional setting.Comment: submitte

Stable homotopy, 1-dimensional NCCW complexes, and Property (H)

In this paper, we show that the homomorphisms between two unital one-dimensional NCCW complexes with the same KK-class are stably homotopic, i.e., with adding on a common homomorphism (with finite dimensional image), they are homotopic. As a consequence, any one-dimensional NCCW complex has the Property (H).Comment: Add motivation and backgroun

On Specht's Theorem in UHF C⁎-algebras

The final publication is available at Elsevier via https://doi.org/10.1016/j.jfa.2020.108778 © 2020. This manuscript version is made available under the CC-BY-NC-ND 4.0 licenseSpecht's Theorem states that two matrices A and B in Mn(C) are unitarily equivalent if and only if tr(w(A;A )) = tr(w(B;B )) for all words w(x; y) in two non-commuting variables x and y. In this article we examine to what extent this trace condition characterises approximate unitary equivalence in uniformly hyper nite (UHF) C -algebras. In particular, we show that given two elements a; b of the universal UHF algebra Q which generate C -algebras satisfying the UCT, they are approximately unitarily equivalent if and only if (w(a; a )) = (w(b; b )) for all words w(x; y) in two non-commuting variables (where denotes the unique tracial state on Q), while there exist two elements a; b in the UHF-algebra M21 which fail to be approximately unitarily equivalent despite the fact that they satisfy the trace condition. We also examine a consequence of these results for ampliations of matrices.L.W. Marcoux's research is supported in part by NSERC (Canada); Y.H. Zhang's research is supported in part by National Natural Science Foundation of China (Nos.: 12071174, 11671167), Science and Technology Development Project of Jilin Province (No.: 20190103028JH)