2,285 research outputs found
Self-dual polygons and self-dual curves
We study projectively self-dual polygons and curves in the projective plane.
Our results provide a partial answer to problem No 1994-17 in the book of
Arnold's problems
Addressing Integration Error for Polygonal Finite Elements Through Polynomial Projections: A Patch Test Connection
Polygonal finite elements generally do not pass the patch test as a result of
quadrature error in the evaluation of weak form integrals. In this work, we
examine the consequences of lack of polynomial consistency and show that it can
lead to a deterioration of convergence of the finite element solutions. We
propose a general remedy, inspired by techniques in the recent literature of
mimetic finite differences, for restoring consistency and thereby ensuring the
satisfaction of the patch test and recovering optimal rates of convergence. The
proposed approach, based on polynomial projections of the basis functions,
allows for the use of moderate number of integration points and brings the
computational cost of polygonal finite elements closer to that of the commonly
used linear triangles and bilinear quadrilaterals. Numerical studies of a
two-dimensional scalar diffusion problem accompany the theoretical
considerations
Polygons of the Lorentzian plane and spherical simplexes
It is known that the space of convex polygons in the Euclidean plane with
fixed normals, up to homotheties and translations, endowed with the area form,
is isometric to a hyperbolic polyhedron. In this note we show a class of convex
polygons in the Lorentzian plane such that their moduli space, if the normals
are fixed and endowed with a suitable area, is isometric to a spherical
polyhedron. These polygons have an infinite number of vertices, are space-like,
contained in the future cone of the origin, and setwise invariant under the
action of a linear isometry.Comment: New text, title slightly change
Cluster Toda chains and Nekrasov functions
In this paper the relation between the cluster integrable systems and
-difference equations is extended beyond the Painlev\'e case.
We consider the class of hyperelliptic curves when the Newton polygons
contain only four boundary points. The corresponding cluster integrable Toda
systems are presented, and their discrete automorphisms are identified with
certain reductions of the Hirota difference equation. We also construct
non-autonomous versions of these equations and find that their solutions are
expressed in terms of 5d Nekrasov functions with the Chern-Simons
contributions, while in the autonomous case these equations are solved in terms
of the Riemann theta-functions.Comment: 32 pages, 13 figures, small corrections, references adde
Distance-regular graphs
This is a survey of distance-regular graphs. We present an introduction to
distance-regular graphs for the reader who is unfamiliar with the subject, and
then give an overview of some developments in the area of distance-regular
graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,
Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page
A Two-Level Method for Mimetic Finite Difference Discretizations of Elliptic Problems
We propose and analyze a two-level method for mimetic finite difference
approximations of second order elliptic boundary value problems. We prove that
the two-level algorithm is uniformly convergent, i.e., the number of iterations
needed to achieve convergence is uniformly bounded independently of the
characteristic size of the underling partition. We also show that the resulting
scheme provides a uniform preconditioner with respect to the number of degrees
of freedom. Numerical results that validate the theory are also presented
A virtual element method for the vibration problem of Kirchhoff plates
The aim of this paper is to develop a virtual element method (VEM) for the
vibration problem of thin plates on polygonal meshes. We consider a variational
formulation relying only on the transverse displacement of the plate and
propose an conforming discretization by means of the VEM which is
simple in terms of degrees of freedom and coding aspects. Under standard
assumptions on the computational domain, we establish that the resulting
schemeprovides a correct approximation of the spectrum and prove optimal order
error estimates for the eigenfunctions and a double order for the eigenvalues.
The analysis restricts to simply connected polygonal clamped plates, not
necessarily convex. Finally, we report several numerical experiments
illustrating the behaviour of the proposed scheme and confirming our
theoretical results on different families of meshes. Additional examples of
cases not covered by our theory are also presented
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