60,693 research outputs found
Teichm\"uller spaces of Riemann surfaces with orbifold points of arbitrary order and cluster variables
We generalize a new class of cluster type mutations for which exchange
transformations are given by reciprocal polynomials. In the case of
second-order polynomials of the form these
transformations are related to triangulations of Riemann surfaces of arbitrary
genus with at least one hole/puncture and with an arbitrary number of orbifold
points of arbitrary integer orders . We propose the dual graph description
of the corresponding Teichm\"uller spaces, construct the Poisson algebra of the
Teichm\"uller space coordinates, propose the combinatorial description of the
corresponding geodesic functions and find the mapping class group
transformations.Comment: 20 pages, notations and many essential typos corrected, most
significantly, formulae 2.3, 2.5, proof of Lemmata 2.6 and 4.5. Journal
reference is added (published version contains typos
Finite element differential forms on cubical meshes
We develop a family of finite element spaces of differential forms defined on
cubical meshes in any number of dimensions. The family contains elements of all
polynomial degrees and all form degrees. In two dimensions, these include the
serendipity finite elements and the rectangular BDM elements. In three
dimensions they include a recent generalization of the serendipity spaces, and
new H(curl) and H(div) finite element spaces. Spaces in the family can be
combined to give finite element subcomplexes of the de Rham complex which
satisfy the basic hypotheses of the finite element exterior calculus, and hence
can be used for stable discretization of a variety of problems. The
construction and properties of the spaces are established in a uniform manner
using finite element exterior calculus.Comment: v2: as accepted by Mathematics of Computation after minor revisions;
v3: this version corresponds to the final version for Math. Comp., after
copyediting and galley proof
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