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We prove, under certain assumptions, algebraicity of the ratio $L(m, \Pi \times \chi)/L(m, \Pi \times \chi')$, where $\Pi$ is a cuspidal automorphic cohomological unitary representation of $\mathrm{GL}_n(\mathbb{A}_\mathbb{Q})$, and $\chi$, $\chi'$ are finite order Hecke characters such that $\chi_{\infty} = \chi'_{\infty} = \mathrm{sgn}^{r}$, and $m, r$ are specific positive integers which depends only on $\Pi_{\infty}$. The methods in this article are a generalization of those in the work of Mahnkopf [Cohomology of arithmetic groups, parabolic subgroups and the special values of $L$-functions of GL(n), J. Inst. Math. Jussieu, 4 (2005)].

Given a constructible sheaf $F$ on a complex manifold, Kashiwara-Schapira defined the notion of singular support and characteristic cycle of $F$. On the other hand for a Zariski constructible \'{e}tale sheaf $F$ on an algebraic variety $X$, Beilinson defined the notion of singular support of $F$ and Saito defined the notion of characteristic cycle of $F$. In this article we compare these notions and prove that they agree in a suitable sense. In the appendix we discuss extension of the notions of singular support and characteristic cycles developed by Umezaki-Yang-Zhao and Barrett., Comment: 12 pages

In this article, we prove the $t$-exactness of a Brylinski-Radon transformation taking values in sheaves on flag varieties. This implies several weak Lefschetz type results for cohomology. In particular, we obtain de Cataldo-Migliorini's P=Dec(F) and Beilinson's basic lemma, the latter was an important ingredient in their proof of P=Dec(F). Our methods also allow the sharpening of Esnault-Katz's cohomological divisibility theorem and estimates for the Hodge level. Finally, we upgrade P=Dec(F) to an equivalence of functors which is also valid over a base., Comment: Comments are welcome!

We use Morris' theory of Eisenstein series for reductive groups over global function fields in order to extend Harder's computation of Tamagawa numbers to quasi-split groups.

This work studies how we can obtain feature-level ratings of the mobile products from the customer reviews and review votes to influence decision making, both for new customers and manufacturers. Such a rating system gives a more comprehensive picture of the product than what a product-level rating system offers. While product-level ratings are too generic, feature-level ratings are particular; we exactly know what is good or bad about the product. There has always been a need to know which features fall short or are doing well according to the customer's perception. It keeps both the manufacturer and the customer well-informed in the decisions to make in improving the product and buying, respectively. Different customers are interested in different features. Thus, feature-level ratings can make buying decisions personalized. We analyze the customer reviews collected on an online shopping site (Amazon) about various mobile products and the review votes. Explicitly, we carry out a feature-focused sentiment analysis for this purpose. Eventually, our analysis yields ratings to 108 features for 4k+ mobiles sold online. It helps in decision making on how to improve the product (from the manufacturer's perspective) and in making the personalized buying decisions (from the buyer's perspective) a possibility. Our analysis has applications in recommender systems, consumer research, etc., Comment: 10 pages, 7 figures, and accepted by IEEE Transactions on Computational Social Systems

People might not be close-at-hand but they still are - by virtue of the social network. The social network has transformed lives in many ways. People can express their views, opinions and life experiences on various platforms be it Twitter, Facebook or any other medium there is. Such events constitute of reviewing a product or service, conveying views on political banters, predicting share prices or giving feedback on the government policies like Demonetization or GST. These social platforms can be used to investigate the insights of the emotional curve that the general public is generating. This kind of analysis can help make a product better, predict the future prospects and also to implement the public policies in a better way. Such kind of research on sentiment analysis is increasing rapidly. In this research paper, we have performed temporal analysis and spatial analysis on 1,42,508 and 58,613 tweets respectively and these tweets were posted during the post-GST implementation period from July 04, 2017 to July 25, 2017. The tweets were collected using the Twitter streaming API. A well-known lexicon, National Research Council Canada (NRC) emotion Lexicon is used for opinion mining that exhibits a blend of eight basic emotions i.e. joy, trust, anticipation, surprise, fear sadness, anger, disgust and two sentiments i.e. positive and negative for 6,554 words.