why does a surface always have some irregularities in them???
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why does a surface always have some irregularities in them???
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Answer:
Surface irregularities have traditionally been divided into three groups loosely based on scale [49]: (i) roughness, generated by the material removal mechanism such as tool marks; (ii) waviness, produced by imperfect operation of a machine tool and (iii) errors of form, generated by errors of a machine tool,
Explanation:
In mathematics, the irregularity of a complex surface X is the Hodge number {\displaystyle h^{0,1}=\dim H^{1}({\mathcal {O}}_{X})}{\displaystyle h^{0,1}=\dim H^{1}({\mathcal {O}}_{X})}, usually denoted by q.[1] The irregularity of an algebraic surface is sometimes defined to be this Hodge number, and sometimes defined to be the dimension of the Picard variety, which is the same in characteristic 0 but can be smaller in positive characteristic.[2]
The name "irregularity" comes from the fact that for the first surfaces investigated in detail, the smooth complex surfaces in P3, the irregularity happens to vanish. The irregularity then appeared as a new "correction" term measuring the difference {\displaystyle p_{g}-p_{a}}{\displaystyle p_{g}-p_{a}} of the geometric genus and the arithmetic genus of more complicated surfaces. Surfaces are sometimes called regular or irregular depending on whether or not the irregularity vanishes.
For a complex analytic manifold X of general dimension, the Hodge number {\displaystyle h^{0,1}=\dim H^{1}({\mathcal {O}}_{X})}{\displaystyle h^{0,1}=\dim H^{1}({\mathcal {O}}_{X})} is called the irregularity of {\displaystyle X}X, and is denoted by q.
Surface irregularities have traditionally been divided into three groups loosely based on scale [49]: (i) roughness, generated by the material removal mechanism such as tool marks; (ii) waviness, produced by imperfect operation of a machine tool and (iii) errors of form, generated by errors of a machine tool,
Explanation:
In mathematics, the irregularity of a complex surface X is the Hodge number {\displaystyle h^{0,1}=\dim H^{1}({\mathcal {O}}_{X})}{\displaystyle h^{0,1}=\dim H^{1}({\mathcal {O}}_{X})}, usually denoted by q.[1] The irregularity of an algebraic surface is sometimes defined to be this Hodge number, and sometimes defined to be the dimension of the Picard variety, which is the same in characteristic 0 but can be smaller in positive characteristic.[2]
The name "irregularity" comes from the fact that for the first surfaces investigated in detail, the smooth complex surfaces in P3, the irregularity happens to vanish. The irregularity then appeared as a new "correction" term measuring the difference {\displaystyle p_{g}-p_{a}}{\displaystyle p_{g}-p_{a}} of the geometric genus and the arithmetic genus of more complicated surfaces. Surfaces are sometimes called regular or irregular depending on whether or not the irregularity vanishes.
For a complex analytic manifold X of general dimension, the Hodge number {\displaystyle h^{0,1}=\dim H^{1}({\mathcal {O}}_{X})}{\displaystyle h^{0,1}=\dim H^{1}({\mathcal {O}}_{X})} is called the irregularity of {\displaystyle X}X, and is denoted by q.