ON THE DIVISOR PRODUCTS AND PROPER DIVISOR PRODUCTS SEQUENCES

Let n be a positive integer, Pd(n) denotes the product of all positive divisors of n, qd(n) denotes the product of all proper divisors of n

A NUMBER THEORETIC FUNCTION AND ITS MEAN VALUE PROPERTY

Let p be a prime, n be any positive integer, a(n,p) denotes the power of p in the factorization of n!

Mechanism of Weiwei granules in the treatment of chronic active Helicobacter pylori gastritis with atrophy based on the TLR4/NF-κB/COX-2 inflammatory signaling pathway

Objective. Our paper aimed to elucidate the mechanism of Weiwei granules in the treatment of Helicobacter pylori (Hp)-positive chronic atrophic gastritis (CAG) based on the TLR4/NF-κB/COX-2 inflammatory signaling pathway. Methods. Hp-positive CAG patients were randomized into the control group (treated with quadruple therapy) or the observation group (treated with Weiwei granules based on the control group). The clinical efficacy, Hp clearance rate, and efficacy of traditional Chinese medicine (TCM) symptoms were compared between the two groups after six months of treatment. The scores of various histopathology variables, serum levels of inflammatory factors (interleukin-6 [IL-6], interleukin-8 [IL-8], and tumor necrosis factor-alpha [TNF-α]), gastrin-17 (G-17) and motilin (MTL), pepsinogen (PG) I and PG II, as well as serum levels of gastrointestinal hormone endothelin (ET), epidermal growth factor (EGF), and calcitonin gene-related peptide (CGRP), were compared between the two groups before and after treatment. TLR4, NF-κB, and COX-2 mRNA levels were compared in gastric mucosal tissues before and after treatment in the two groups. Results. After treatment, the clinical efficacy, Hp clearance rate, and efficacy of TCM symptoms of patients in the observation group were higher than those in the control group. After treatment, the scores of various histopathology variables, serum levels of inflammatory factors (IL-6, IL-8, and TNF-α), gastrointestinal hormones (ET and EGF), and the expression levels of TLR4, NF-κB, and COX-2 mRNA in the gastric mucosal tissues were lower and G-17, MTL, CGRP, and PG I levels were higher in the observation group than in the control group. Conclusion. Weiwei granules can effectively improve Hp-positive CAG patients and reduce the expression levels of TLR4, NF-κB, and COX-

Enhancing Recommendation Interpretability with Tags: A Neural Variational Model

Recommender systems are widely used for assisting consumers finding interested products, and providing suitable explanations for recommendation is particularly important for enhancing consumers’ trust and satisfaction with the system. Tags can be used to annotate different types of items, yet their potential for providing interpretability is not well studied previously. Therefore, it is worthy to study how to leverage tags to enhance recommendation systems in terms of both interpretability and accuracy. This paper proposes a novel model that seamlessly fuse topic model and recommendation model, where the topic model can analyze tags to infer understandable topics, and the recommendation model can conduct accurate and interpretable recommendations based on these topics. We develop variational auto-encoding method to take advantage of neural networks to infer model parameters. Experiments on real-world datasets illustrate that the proposed method can not only achieve great recommendation performance, but also provide interpretability for the recommendation results

Strong Optical and UV Intermediate-Width Emission Lines in the Quasar SDSS J232444.80-094600.3: Dust-Free and Intermediate-Density Gas at the Skin of Dusty Torus ?

Emission lines from the broad emission line region (BELR) and the narrow emission line region (NELR) of active galactic nuclei (AGNs) are extensively studied. However, between these two regions emission lines are rarely detected. We present a detailed analysis of a quasar SDSS J232444.80-094600.3 (SDSS J2324$-$0946), which is remarkable for its strong intermediate-width emission lines (IELs) with FWHM $\approx$ 1800 \kmps. The IEL component is presented in different emission lines, including the permitted lines \lya\ $\lambda$1216, \civ\ $\lambda$1549, semiforbidden line \ciii\ $\lambda$1909, and forbidden lines \oiii\ $\lambda\lambda$4959, 5007. With the aid of photo-ionization models, we found that the IELs are produced by gas with a hydrogen density of $n_{\rm H} \sim 10^{6.2}-10^{6.3}~\rm cm^{-3}$, a distance to the central ionizing source of $R \sim 35-50$ pc, a covering factor of CF $\sim$ 6\%, and a dust-to-gas ratio of $\leq 4\%$ times of SMC. We suggest that the strong IELs of this quasar are produced by nearly dust-free and intermediate-density gas located at the skin of the dusty torus. Such strong IELs, served as a useful diagnose, can provide an avenue to study the properties of gas between the BELR and the NELR

Construction of a Cities Evolution Tree, with Applications

China has been experiencing urbanization at an unprecedented rate over the last two decades. This study analyzes the impact of urbanization on land occupation. Cities are clustered by their functions and development stages, which is illustrated by a cluster tree, a dynamic tree that depicts the evolution of cities. The evolution tree in one year is used to predict the state of a city in a future time period. Another application of the evolution tree is to predict urban-type relevant phenomena, such as urban occupation. It is found that comprehensive cities, business cities, and manufacturing cities have higher urban expansion rates than tourist cities, with a few exceptions that focus on both industry and tourism. Meanwhile, the speed and extent of city land growth are dominated by industrialization stages and economic patterns, as well as leap-development. The methodology presented in this study is especially suitable for identifying transition paths of a stochastic process in a complex dataset of 253 cities in China