A Tight Bound for Stochastic Submodular Cover

We show that the Adaptive Greedy algorithm of Golovin and Krause achieves an approximation bound of (ln(Q/η)+1) for Stochastic Submodular Cover: here Q is the “goal value” and η is the minimum gap between Q and any attainable utility value Q'<Q. Although this bound was claimed by Golovin and Krause in the original version of their paper, the proof was later shown to be incorrect by Nan & Saligrama. The subsequent corrected proof of Golovin and Krause gives a quadratic bound of (ln(Q/η)+1)2. A bound of 56(ln(Q/η)+1) is implied by work of Im et al. Other bounds for the problem depend on quantities other than Q and η. Our bound restores the original bound claimed by Golovin and Krause, generalizing the well-known (ln m+1) approximation bound on the greedy algorithm for the classical Set Cover problem, where m is the size of the ground set.

Comment on: “Bi-interior ideals of semigroups”

This is about the paper “Bi-interior ideals of semigroups” by M. Murali Krishna Rao in Discuss. Math. Gen. Algebra Appl. 38 (2018) 69–78. According to Theorem 3.11 (also Theorem 3.3(8)) of the paper, the intersection of a bi-interior ideal [Formula: see text] of a semigroup [Formula: see text] and a subsemigroup [Formula: see text] of [Formula: see text] is a bi-interior ideal of [Formula: see text]. Regarding to Theorem 3.6, every bi-interior ideal of a regular semigroup is an ideal of [Formula: see text]. We give an example that the above two results are not true for semigroups. According to the same paper, if [Formula: see text] is a regular semigroup then, for every bi-interior ideal [Formula: see text], every ideal [Formula: see text] and every left ideal [Formula: see text] of [Formula: see text], we have [Formula: see text]. The proof is wrong, we provide the corrected proof. In most of the results of the paper the assumption of unity is not necessary. Care should be taken about the proofs in the paper.

ABC’s of Publishing a Scientific Paper in a Journal for the Novice Researchers

Writing a scientific paper, choosing a journal, submitting/uploading the paper in the journal website, the peer review process, revising the paper based on the reviewer's comments, and galley proofreading after the acceptance of the paper are the essential components of publishing a paper. Publishing is the ultimate goal of all researchers. Writing a scientific paper requires an extensive literature search, collection of reference articles, acquisition of data of research work, analysis of data and discussing the results comparing with other findings published in similar papers. The final version of the paper should be read by all authors and approved before the submission of the manuscript. One has to select the journal and edit the paper as per the author's instructions of that journal before submission. The article will be reviewed by two experts in that field and they will send their comments about the contents of the paper. The comments should be answered point by point, and the revised paper should be sent again to the editor. If required one has to be prepared to do more than one revision of the paper. If the paper is rejected, one should not be disappointed. You can further improve the quality of the paper by including the answers for the deficiencies and send the revised paper to another suitable journal. Finally, when it is accepted, the galley proof of the article should be read carefully and send the corrected proof to the editor in-time. The 'pdf' copy of the published paper should be kept for sending a copy to the people who request a reprint of your article.

Corrigendum: Analytic representation theory of Lie groups: general theory and analytic globalizations of Harish-Chandra modules

We correct the proof of the main result of the paper, Theorem 5.7. Our corrected proof relies on weaker versions of a number of intermediate results from the paper. The original, more general, versions of these statements are not known to be true.