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Let I be a monomial ideal in the polynomial ring S=K [x1,... xn]. We study the Stanley depth of the integral closure Ī of I. We prove that for every integer k ≥ 1, the inequalities (S/Ik) ≤ sdepth (S/Ī) and sdepth(Ik) ≤ sdepth(Ī) hold. We also prove that for every monomial ideal I⊂ S there exist integers k1,k2≥ 1, such that for every s≥ 1, the inequalities sdepth (S/Isk1) ≤ sdepth(S/Ī) and sdepth (Isk2) ≤ sdepth (Ī) hold. In particular, mink{sdepth(S/Ik)} ≤ sdepth(S/Ī) and min̄k {sdepth (Ik)}≤ sdepth(Ī). We conjecture that for every integrally closed monomial ideal I, the inequalities sdepth(S/I)≥ n-l (I) and sdepth (I)≥ n-l (I)+1 hold, where l (I) is the analytic spread of I. Assuming the... 

Let δ be a (d-1)-dimensional simplicial complex and let h(δ)=(h0,h1,...,hd) be its h-vector. A recent result of Murai and Terai guarantees that if δ satisfies Serre's condition (Sr), then (h0,h1,...,hr) is an M-vector and hr+hr+1+...+hd is nonnegative. In this article, we extend the result of Murai and Terai by giving r extra necessary conditions. More precisely, we prove that if δ satisfies Serre's condition (Sr), then iihr+i+1ihr+1+...+i+d-rihd, 0≤i≤r≤d, are all nonnegative  

Let F be a field, char (F) ≠ 2, and S ⊆ GLn (F), where n is a positive integer. In this paper we show that if for every distinct elements x, y ∈ S, x + y is singular, then S is finite. We conjecture that this result is true if one replaces field with a division ring. © 2009 Elsevier Inc. All rights reserved  

Let G be a simple graph of order n and size m which is not a tree. If ℓ; ≤ 3 is a natural number and the length of every cycle of G is divisible by ℓ, then m ≤l/l-2 (n -2), and the equality holds if and only if the following hold: (i) ℓ is odd and G is a cycle of order ℓ or (ii) ℓ is even and G is a generalized 6>-graph with paths of length |l/2 It is shown that for a (0 mod ℓ)-cycle graph, m/n < l/l-1 if ℓ is odd, and for a given e > 0, there exists a (0 mod ℓ)-cycle graph G with m/n > l/l-2 - e. Also m/n > l/l-2 if ℓ is even, and for a given e > 0, there exists a (0 mod ℓ)-cycle graph G with m/n l/l-2-e  

Let K{double-struck} be a field, and let R = K{double-struck}[x1,.., xn] be the polynomial ring over K{double-struck} in n indeterminates x1,.., xn. Let G be a graph with vertex-set {x1,.., xn}, and let J be the cover ideal of G in R. For a given positive integer k, we denote the kth symbolic power and the kth bracket power of J by J(k) and J[k], respectively. In this paper, we give necessary and sufficient conditions for R/Jk, R/J (k), and R/J [k] to be Cohen-Macaulay. We also study the limit behavior of the depths of these rings  

A very well-covered graph is a well-covered graph without isolated vertices such that the height of its edge ideal is half of the number of vertices. In this survey article, we gather together most of the old and new results on the edge and cover ideals of these graphs. © 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG  

"Web 2.0" refers to a second generation of web development and design, that facilitates communication, secure information sharing, interoperability, and collaboration on the World Wide Web. The truth is that Web 2.0 is a difficult term to define, even for web experts. Usually phrases like "the web as platform" and "architecture of participation" are used to describe this term. Examples of Web 2.0 include web-based communities, hosted services, web applications, social-networking sites, video-sharing sites, wikis, blogs, mashups and folksonomies. The Internet is changing medicine and Web 2.0 is the current buzz word in the World Wide Web dictionary. Radiology in the Web 2.0 probably refers to...