Search

Let $G$ be a finite group and $K$ a splitting field of $G$ of characteristic $p>0$. Denote by $KG$ the group algebra of $G$ over $K$ and $Z(KG)$ the center of $KG$. Let $V_G$ be the $K$-subspace of trace-zero elements of $KG$. We give some numerical sufficient and necessary conditions for $V_G$ and $V_G\cap Z(KG)$, respectively, to be Mathieu subspaces of $KG$ in terms of the degrees of irreducible representations of $G$ over $K$. The same numerical conditions also characterize the finite groups $G$ that $KG$ has no nonzero trace-zero idempotents and the finite groups $G$ that $KG$ has no nonzero central trace-zero idempotents, respectively., Comment: Latex 17 pages

In the paper, we prove that the derivation $D=y\partial_x+(a_2(x)y^2+a_1(x)y+a_0(x))\partial_y$ of $K[x,y]$ with $a_2(x),a_1(x),a_0(x)\in K[x]$ is simple iff the following conditions hold: $(1)$ $a_0(x)\in K^*$, $(2)$ $\deg a_1(x)\geq1$ or $\deg a_2(x)\geq1$, $(3)$ there exist no $l\in K^*$ such that $a_2(x)=la_1(x)-l^2a_0(x)$. In addition, we prove that the image of the derivation $D=\partial_x+{\sum_{i=1}^n \gamma_i(x) y_i^{k_i}}{\partial_i}$ is a Mathieu-Zhao space iff $D$ is locally finite. Moreover, we prove that the image of the derivation $D={\sum_{i=1}^n \gamma_i y_i^{k_i}}{\partial_i}$ of $K[y_1,\ldots,y_n]$ is a Mathieu-Zhao space iff $k_i\leq 1$ for all $1\leq i\leq n$, $n\geq 2$., Comment: 18pages

In the paper, we study the relation between the images of polynomial derivations and their simplicity. We prove that the images of simple Shamsuddin derivations are not Mathieu-Zhao spaces. In addition, we also show that the images of some simple derivations in dimension three are not Mathieu-Zhao spaces. Thus, we conjecture that the images of simple derivations in dimension greater than one are not Mathieu-Zhao spaces. We also prove that locally nilpotent derivations are not simple in dimension greater than one., Comment: 12 pages

In our paper, we give a necessary and sufficient conditions for the kernels of the linear maps of finite Abelian group algebras to be Mathieu-Zhao spaces of $K[G]$ if $G$ is a finite Abelian group and $K$ is a split field for $G$. Hence we classify all Mathieu-Zhao spaces of the finite Abelian group algebras if $K$ is a split field for $G$., Comment: 10 pages

Let $K$ be an algebraically closed field of characteristic zero, $\delta$ a nonzero $\mathcal{E}$-derivation of $K[x]$. We first prove that $\operatorname{Im}\delta$ is a Mathieu-Zhao space of $K[x]$ in some cases. Then we prove that LFED Conjecture is true for all $\delta=I-\phi$, where $\phi$ is an affine polynomial homomorphism of $K[x_1,x_2]$. Finally, we prove that LFED Conjecture is true for some $\delta$ of $K[x_1,x_2,x_3]$., Comment: 18pages

Let $K$ be a field of characteristic $p$, $\delta$ a nonzero $\mathcal{E}$-derivation and $D=f(x_1)\partial_1$. We first prove that $\operatorname{Im}D$ is not a Mathieu-Zhao space of $K[x_1]$ if and only if $f(x_1)=x_1^rf_1(x_1^p)$ and $r\neq 1$. Then we prove that $\operatorname{Im}\delta$ is a Mathieu-Zhao space of $K[x_1]$ if and only if $\delta$ is not locally nilpotent. Finally, we classify some nilpotent derivations of $K[x_1]$ and give a sufficient and necessary condition for $D(I)$ to be a Mathieu-Zhao space of $K[x_1]$ for any ideal $I$ of $K[x_1]$., Comment: 17pages

Automated breast ultrasound (ABUS) is a new and promising imaging modality for breast cancer detection and diagnosis, which could provide intuitive 3D information and coronal plane information with great diagnostic value. However, manually screening and diagnosing tumors from ABUS images is very time-consuming and overlooks of abnormalities may happen. In this study, we propose a novel two-stage 3D detection network for locating suspected lesion areas and further classifying lesions as benign or malignant tumors. Specifically, we propose a 3D detection network rather than frequently-used segmentation network to locate lesions in ABUS images, thus our network can make full use of the spatial context information in ABUS images. A novel similarity loss is designed to effectively distinguish lesions from background. Then a classification network is employed to identify the located lesions as benign or malignant. An IoU-balanced classification loss is adopted to improve the correlation between classification and localization task. The efficacy of our network is verified from a collected dataset of 418 patients with 145 benign tumors and 273 malignant tumors. Experiments show our network attains a sensitivity of 97.66% with 1.23 false positives (FPs), and has an area under the curve(AUC) value of 0.8720., Comment: Early Accepted by MICCAI 2020

In the paper, we first classify all polynomial maps of the form $H=(u(x,y,z),v(x,y,z), h(x,y))$ in the case that $JH$ is nilpotent and $\deg_zv\leq 1$. After that, we generalize the structure of $H$ to $H=\big(H_1(x_1,x_2,\ldots,x_n),\allowbreak b_3x_3+\cdots+b_nx_n+H_2^{(0)}(x_2),H_3(x_1,x_2),\ldots,H_n(x_1,x_2)\big)$., Comment: 24 pages

Skeleton-based action recognition has attracted considerable attention in computer vision since skeleton data is more robust to the dynamic circumstance and complicated background than other modalities. Recently, many researchers have used the Graph Convolutional Network (GCN) to model spatial-temporal features of skeleton sequences by an end-to-end optimization. However, conventional GCNs are feedforward networks which are impossible for low-level layers to access semantic information in the high-level layers. In this paper, we propose a novel network, named Feedback Graph Convolutional Network (FGCN). This is the first work that introduces the feedback mechanism into GCNs and action recognition. Compared with conventional GCNs, FGCN has the following advantages: (1) a multi-stage temporal sampling strategy is designed to extract spatial-temporal features for action recognition in a coarse-to-fine progressive process; (2) A dense connections based Feedback Graph Convolutional Block (FGCB) is proposed to introduce feedback connections into the GCNs. It transmits the high-level semantic features to the low-level layers and flows temporal information stage by stage to progressively model global spatial-temporal features for action recognition; (3) The FGCN model provides early predictions. In the early stages, the model receives partial information about actions. Naturally, its predictions are relatively coarse. The coarse predictions are treated as the prior to guide the feature learning of later stages for a accurate prediction. Extensive experiments on the datasets, NTU-RGB+D, NTU-RGB+D120 and Northwestern-UCLA, demonstrate that the proposed FGCN is effective for action recognition. It achieves the state-of-the-art performance on the three datasets., Comment: 18 pages, 5 figures

In the paper, we give an affirmative answer to the conjecture in \cite{13}. We prove that a Shamsuddin derivation $D$ is simple if and only if $\operatorname{Aut}(K[x,y_1,\allowbreak\ldots,y_n])_D=\{id\}$. In addition, we calculate the isotropy groups of the Shamsuddin derivations $d=\partial_x+\sum_{j=1}^r(a(x)y_j+b_j(x))\partial_j$ of $K[x,y_1,\ldots,y_r]$. We also prove that $d$ is a Mathieu-Zhao subspace if and only if $a(x)\in K$., Comment: 11 pages

In the paper, we first study the subgroup of $ K$-automorphisms of $K[x_1,\allowbreak \ldots,x_n]$ which commutes with a simple derivation of $K[x_1,\ldots,x_n]$. We show that the subgroup of $ K$-automorphisms of $K[x_1,\ldots,x_n]$ which commutes with a simple derivation of $K[x_1,\ldots,x_n]$ consists of translations under certain hypothesis. We also prove that the subgroup of $K$-automorphisms of $K[x_1,x_2]$ which commutes with a simple derivation is trivial under the same hypothesis, and the subgroup of $ K$-automorphisms of $K[x_1,\ldots,x_n]$ which commutes with simple Shamsuddin derivations is trivial under the same hypothesis. Then we give an affirmative answer to the conjecture proposed in \cite{13} for $n=2$., Comment: 10 pages

In the paper, we first classify all polynomial maps $H$ of the following form: $H=\big(H_1(x_1,x_2,\ldots,x_n),H_2(x_1,x_2),H_3(x_1,x_2),\ldots,H_n(x_1,x_2)\big)$ with $JH$ nilpotent. After that, we generalize the structure of $H$ to $H=\big(H_1(x_1,x_2,\ldots,x_n),H_2(x_1,x_2),H_3(x_1,x_2,H_1),\ldots,H_n(x_1,x_2,H_1)\big)$., Comment: Major revision with some new results. arXiv admin note: text overlap with arXiv:1710.02802, arXiv:1710.02800

In the paper, we first classify all polynomial maps of the form $H=(u(x,y),v(x,y,z), h(x,y))$ in the case that $JH$ is nilpotent and $(\deg_yu,\deg_yh)\leq 3$, $H(0)=0$. Then we classify all polynomial maps of the form $H=(u(x,y,z),v(x,y,u), h(x,y))$ in the case that $JH$ is nilpotent and $(\deg v(x,y,0),\deg h)\leq 3$, $H(0)=0$. Finally, we classify polynomial maps of the form $H=(u(x,y,z),v(x,y,z), h(x,y))$ in certain conditions., Comment: 28 pages

In this paper, we first prove that $u,v,h$ are linearly dependent over ${\bf K}$ if $JH$ is nilpotent and $H$ has the form: $H=(u(x,y,z),v(u,h),h(x,y))$ with $H(0)=0$ or $H=(u(x,y),v(u,h),h(x,y,z))$ with $H(0)=0$. Then we classify polynomial maps of the form $H=(u(x,y),v(x,y,z), h(x,y))$ in the case that $JH$ is nilpotent and $(\deg_yu,\deg_yh)\leq 2$., Comment: 21 pages

Jedrzejewicz showed that a polynomial map over a field of characteristic zero is invertible, if and only if the corresponding endomorphism maps irreducible polynomials to irreducible polynomials. Furthermore, he showed that a polynomial map over a field of characteristic zero is a Keller map, if and only if the corresponding endomorphism maps irreducible polynomials to square-free polynomials. We show that the latter endomorphism maps other square-free polynomials to square-free polynomials as well. In connection with the above classification of invertible polynomial maps and the Jacobian Conjecture, we study irreducible properties of several types of Keller maps, to each of which the Jacobian Conjecture can be reduced. Herewith, we generalize the result of Bakalarski, that the components of cubic homogeneous Keller maps with a symmetric Jacobian matrix (over C and hence any field of characteristic zero) are irreducible. Furthermore, we show that the Jacobian Conjecture can even be reduced to any of these types with the extra condition that each affinely linear combination of the components of the polynomial map is irreducible. This is somewhat similar to reducing the planar Jacobian Conjecture to the so-called (planar) weak Jacobian Conjecture by Kaliman., Comment: 22 pages

In this paper, we discuss several additional properties a power linear Keller map may have. The Structural Conjecture by Druzkowski in [Dru] asserts that two such properties are equivalent, but we show that one of this properties is stronger than the other. We even show that the property of linear triangularizability is strictly in between. Furthermore, we give some positive results for small dimensions and small Jacobian ranks., Comment: 21 pages, minor changes w.r.t. v2

In this paper, we first show that the Jacobian Conjecture is true for non-homogeneous power linear mappings under some conditions. Secondly, we prove an equivalent statement about the Jacobian Conjecture in dimension $r\geq 1$ and give some partial results for $r=2$. Finally, for a homogeneous power linear Keller map $F=X+H$ of degree $d \ge 2$, we give the inverse polynomial map under the condition that $JH^3=0$. We shall show that ${\operatorname{deg}}(F^{-1})\leq d^k$ if $k \le 2$ and $JH^{k+1}=0$, but also give an example with $d = 2$ and $JH^4=0$ such that ${\operatorname{deg}}(F^{-1})> d^3$., Comment: 15 pages

In this note, we show that, if the Druzkowski mappings $F(X)=X+(AX)^{*3}$, i.e. $F(X)=(x_1+(a_{11}x_1+...+a_{1n}x_n)^3,...,x_n+(a_{n1}x_1+...+a_{nn}x_n)^3)$, satisfies $TrJ((AX)^{*3})=0$, then $rank(A)\leq 1/2(n+\delta)$ where $\delta$ is the number of diagonal elements of A which are equal to zero. Furthermore, we show the Jacobian Conjecture is true for the Druzkowski mappings in dimension $\leq 9$ in the case $\prod_{i=1}^{n}a_{ii}\neq0$., Comment: 5 pages

In this paper, we first show that homogeneous Keller maps are injective on lines through the origin. We subsequently formulate a generalization, which is that under some conditions, a polynomial endomorphism with $r$ homogeneous parts of positive degree does not have $r$ times the same image point on a line through the origin, in case its Jacobian determinant does not vanish anywhere on that line. As a consequence, a Keller map of degree $r$ does not take the same values on $r > 1$ collinear points, provided $r$ is a unit in the base field. Next, we show that for invertible maps $x + H$ of degree $d$, such that $\ker \jac H$ has $n-r$ independent vectors over the base field, in particular for invertible power linear maps $x + (Ax)^{*d}$ with $\rk A = r$, the degree of the inverse of $x + H$ is at most $d^r$., Comment: 11 pages

In this paper, we will first show that, the homogeneous polynomials which satisfy the Jacobian condition are injective on the lines that pass through the origin. Secondly, we will show that $F$ and $G'$ are paired, where $F$ is a Druzkowski map and $G'$ is a cubic homogeneous polynomial which related to $F$. Finally, we will find a more exactly bound for the degree of $F^{-1}$, where $F$ is a invertible map., Comment: 5pages