Let be a root system with a subset of positive roots, +. We consider edges of the Hasse diagrams of some posets associated with +. For each edge one naturally defines its type, and we study the partition of the set of edges into types. For +, the type is a simple root, and for the posets of ad-nilpotent and Abelian ideals the type is an affine simple root. We give several descriptions of the se...

which is impossible since r is a primitive root of I and the exponent is odd. 1 Proc. Nat. Acad. Sci., 15, 45 (1929). 2 We say that an ideal p belongs to an exponent n when p" is a principle ideal but pd is not a principal ideal for 0 < d < n. Trans. Amer. Math. Soc., 31, 632 (1929). Duke Math. Jour., 5, 419-420 (1939). 6 Trans. Amer. Math. Soc., 31, 632-633 (1929). 6 Dickson, Jour. fair. Math....

Let k k be a field and alttext="upper T"> T encoding="application/x-tex">T an algebraic -t...

We study the full moduli space of vacua 6d worldvolume SCFTs on M5 branes probing an $A$-type singularity, focusing geometric incarnation discrete gauging mechanism which acts as a quotient Higgs branch fibered over tensor branch. combine insights from brane constructions and magnetic quiver techniques, in is implemented through concept decoration introduced [arXiv:2202.01218]. discover charact...

Based on the local Shannon entropy concept in information theory, a new measure of aromaticity is introduced. This index, which describes the probability of electronic charge distribution between atoms in a given ring, is called Shannon aromaticity (SA). Using B3LYP method and different basis sets (6-31G**, 6-31+G** and 6-311++G**), the SA values of some five-membered heterocycles, C(4)H(4)X, a...

A new formulation of a pose refinement technique using “active” models is described. An error term derived from the detection of image derivatives close to an initial object hypothesis is linearised and solved by least squares. The method is particularly well suited to problems involving external geometrical constraints (such as the ground-plane constraint). We show that the method is able to r...