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!

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

Empirical Tm modeling in the region of Guangxi

Abstract:This paper presents three strategies for modeling the regional empirical Tm (the weighted mean temperature of the atmosphere) to obtain more accurate determinations in a regional empirical model that is better adapted to the geographical and climatic characteristics of the applied area. The proposed models utilize data from four radiosonde stations in Guangxi, at Nanning, Guilin, Wuzhou and Baise, over an 11 month period (from Jan. to Nov. of 2011). The experimental results demonstrated the following: (1) there is no significant difference between monthly and annual regression results at each site; (2) it is more reasonable and feasible to use the proposed regional Hybrid model for the area far away from the radiosonde site; (3) from the analysis of the possible temperature conditions, the precision of the proposed regional Hybrid model is higher than that of the well-known Bevis formula and of some other existing models and can reach an accuracy within 1mm for the GPS-derived PWV estimates for the applied region