Perfect Sets and ff-Ideals

A square-free monomial ideal $I$ is called an {\it $f$-ideal}, if both $\delta_{\mathcal{F}}(I)$ and $\delta_{\mathcal{N}}(I)$ have the same $f$-vector, where $\delta_{\mathcal{F}}(I)$ ($\delta_{\mathcal{N}}(I)$, respectively) is the facet (Stanley-Reisner, respectively) complex related to $I$. In this paper, we introduce and study perfect subsets of $2^{[n]}$ and use them to characterize the $f$-ideals of degree $d$. We give a decomposition of $V(n, 2)$ by taking advantage of a correspondence between graphs and sets of square-free monomials of degree $2$, and then give a formula for counting the number of $f$-ideals of degree $2$, where $V(n, 2)$ is the set of $f$-ideals of degree 2 in $K[x_1,\ldots,x_n]$. We also consider the relation between an $f$-ideal and an unmixed monomial ideal.Comment: 15 page

Determination of multifractal dimensions of complex networks by means of the sandbox algorithm

Complex networks have attracted much attention in diverse areas of science and technology. Multifractal analysis (MFA) is a useful way to systematically describe the spatial heterogeneity of both theoretical and experimental fractal patterns. In this paper, we employ the sandbox (SB) algorithm proposed by T\'{e}l et al. (Physica A, 159 (1989) 155-166), for MFA of complex networks. First we compare the SB algorithm with two existing algorithms of MFA for complex networks: the compact-box-burning (CBB) algorithm proposed by Furuya and Yakubo (Phys. Rev. E, 84 (2011) 036118), and the improved box-counting (BC) algorithm proposed by Li et al. (J. Stat. Mech.: Theor. Exp., 2014 (2014) P02020) by calculating the mass exponents tau(q) of some deterministic model networks. We make a detailed comparison between the numerical and theoretical results of these model networks. The comparison results show that the SB algorithm is the most effective and feasible algorithm to calculate the mass exponents tau(q) and to explore the multifractal behavior of complex networks. Then we apply the SB algorithm to study the multifractal property of some classic model networks, such as scale-free networks, small-world networks, and random networks. Our results show that multifractality exists in scale-free networks, that of small-world networks is not obvious, and it almost does not exist in random networks.Comment: 17 pages, 2 table, 10 figure

A words-of-interest model of sketch representation for image retrieval

In this paper we propose a method for sketch-based image retrieval. Sketch is a magical medium which is capable of conveying semantic messages for user. It’s in accordance with user’s cognitive psychology to retrieve images with sketch. In order to narrow down the semantic gap between the user and the images in database, we preprocess all the images into sketches by the coherent line drawing algorithm. During the process of sketches extraction, saliency maps are used to filter out the redundant background information, while preserve the important semantic information. We use a variant of Words-of-Interest model to retrieve relevant images for the user according to the query. Words-of-Interest (WoI) model is based on Bag-ofvisual Words (BoW) model, which has been proven successfully for information retrieval. Bag-of-Words ignores the spatial relationships among visual words, which are important for sketch representation. Our method takes advantage of the spatial information of the query to select words of interest. Experimental results demonstrate that our sketch-based retrieval method achieves a good tradeoff between retrieval accuracy and semantic representation of users’ query