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If X is an affine algebraic variety, then the set of all regular functions on X forms a ring called the coordinate ring of X. For a projective variety, there is an analogous ring called the homogeneous coordinate ring. Those rings are essentially the same things as varieties: they correspond in essentially a unique way.
Let be a group, written multiplicatively, and let be a ring. The group ring of over , which we will denote by [], or simply , is the set of mappings : of finite support (() is nonzero for only finitely many elements ), where the module scalar product of a scalar in and a mapping is defined as the mapping (), and the module group sum of two mappings and is defined as the mapping () + ().
A homocycle or homocyclic ring is a ring in which all atoms are of the same chemical element. [1] A heterocycle or heterocyclic ring is a ring containing atoms of at least two different elements, i.e. a non-homocyclic ring. [2] A carbocycle or carbocyclic ring is a homocyclic ring in which all of the atoms are carbon. [3]
The cubic plane curve (red) defined by the equation y 2 = x 2 (x + 1) is singular at the origin, i.e., the ring k[x, y] / y 2 − x 2 (x + 1), is not a regular ring. The tangent cone (blue) is a union of two lines, which also reflects the singularity. The k-vector space m/m 2 is an algebraic incarnation of the cotangent space.
The 3D framework [Cu 3 (btc) 2 (H 2 O) 3] (btc: benzene-1,3,5-tricarboxylate) used in this study was first reported by Williams et al. [72] The open framework of [Cu 3 (btc) 2 (H 2 O) 3] is built from dimeric cupric tetracarboxylate units (paddle-wheels) with aqua molecules coordinating to the axial positions and btc bridging ligands. The ...
At near-IR wavelengths, the characteristics of the 1.65, 3.1, and 4.53 μm water absorption lines are dependent on the ice temperature and crystal order. [160] The peak strength of the 1.65 μm band as well as the structure of the 3.1 μm band are particularly useful in identifying the crystallinity of water ice. [161] [162]
In naphth[a]azulene, a naphthalene ring is condensed at the 1,2-positions of azulene. In one such system [18] deviation from planarity is found, similar to that of tetrahelicene. Guaiazulene (1,4-dimethyl-7-isopropylazulene) is an alkylated derivative of azulene with an almost identical intensely blue colour. It is commercially available to the ...
Indole is an organic compound with the formula C 6 H 4 CCNH 3. Indole is classified as an aromatic heterocycle. It has a bicyclic structure, consisting of a six-membered benzene ring fused to a five-membered pyrrole ring. Indoles are derivatives of indole where one or more of the hydrogen atoms have been replaced by substituent groups.