Jon Eivind Vatne

Applications of Mathematics Doi: https://doi.org/10.21136/AM.2022.0010-22

We prove that eight dihedral angles in a pyramid with an arbitrary quadrilateral base always sum up to a number in the interval (3π, 5π). Moreover, for any number in (3π, 5π) there exists a pyramid whose dihedral angle sum is equal to this number, which means that the lower and upper bounds are tight. Furthermore, the improved (and tight) upper bound 4π is derived for the class of pyramids with parallelogramic bases. This includes pyramids with rectangular bases, often used in finite element mesh generation and analysis.

Numerical Geometry, Grid Generation and Scientific Computing. Proceedings of the 10th International Conference, NUMGRID 2020 / Delaunay 130, Celebrating the 130th Anniversary of Boris Delaunay, Moscow, Russia, November 2020 p. 241-248 Doi: https://doi.org/10.1007/978-3-030-76798-3_15

In this paper we discuss some strategy for red refinements of product elements and show that there are certain structure characteristics (d-sines of angles formed by certain edges in the initial partition) which remain constant during refinement processes. Such a property immediately implies the validity of the so-called maximum angle condition, which is a strongly desired property in interpolation theory and finite element analysis. Our construction also gives a clear refinement scheme preserving shape regularity.

Numerical Mathematics and Advanced Applications ENUMATH 2019 p. 633-640 Doi: https://doi.org/10.1007/978-3-030-55874-1_62

International journal of computational geometry and applications 31(4) p. 183-192 Doi: https://doi.org/10.1142/S0218195922500017

Annali di Matematica Pura ed Applicata p. 1-33 Doi: https://doi.org/10.1007/s10231-020-01013-1

We are interested in differential forms on mixed-dimensional geometries, in the sense of a domain containing sets of d-dimensional manifolds, structured hierarchically so that each d-dimensional manifold is contained in the boundary of one or more d+1-dimensional manifolds. On any given d-dimensional manifold, we then consider differential operators tangent to the manifold as well as discrete differential operators (jumps) normal to the manifold. The combined action of these operators leads to the notion of a semi-discrete differential operator coupling manifolds of different dimensions. We refer to the resulting systems of equations as mixed-dimensional, which have become a popular modeling technique for physical applications including fractured and composite materials. We establish analytical tools in the mixed-dimensional setting, including suitable inner products, differential and codifferential operators, Poincaré lemma, and Poincaré–Friedrichs inequality. The manuscript is concluded by defining the mixed-dimensional minimization problem corresponding to the Hodge Laplacian, and we show that this minimization problem is well-posed.

Differential and difference equations with applications: ICDDEA 2019, Lisbon, Portugal, July 1–5 p. 681-687

Applied Numerical Mathematics 156 p. 174-191 Doi: https://doi.org/10.1016/j.apnum.2020.04.018

In this paper we introduce the semiregularity property for a family of decompositions of a polyhedron into a natural class of prisms. In such a family, prismatic elements are allowed to be very flat or very long compared to their triangular bases, and the edges of quadrilateral faces can be nonparallel. Moreover, the triangular faces of each element are either parallel or skew to each other. To estimate the error of the interpolation operator defined on the finite space whose basis functions are defined on the general prismatic elements, we consider quadratic polynomials as the basis functions for that space which are bilinear on the reference prism. We then prove that under this modification of the semiregularity criterion, the interpolation error is of order O(h) in the H1-norm.

Lecture Notes in Computational Science and Engineering 126 p. 753-760 Doi: https://doi.org/10.1007/978-3-319-96415-7_70

Numerical Geometry, Grid Generation and Scientific Computing Proceedings of the 9th International Conference, NUMGRID 2018 / Voronoi 150, Celebrating the 150th Anniversary of G.F. Voronoi, Moscow, Russia, December 2018 p. 101-108 Doi: https://doi.org/10.1007/978-3-030-23436-2_7

Computers and Mathematics with Applications 80 p. 367-370 Doi: https://doi.org/10.1016/j.camwa.2019.05.020

Journal of Computational and Applied Mathematics 358 p. 29-33 Doi: https://doi.org/10.1016/j.cam.2019.03.003

Applications of Mathematics 63(3) p. 237-257 Doi: https://doi.org/10.21136/AM.2018.0357-17

Journal of Number Theory 172 p. 413-415 Doi: https://doi.org/10.1016/j.jnt.2016.08.015

Applications of Mathematics 62(3) p. 213-223 Doi: https://doi.org/10.21136/AM.2017.0187-16

Annali dell’Università di Ferrara Doi: https://doi.org/10.1007/s11565-011-0139-z

Banach Center Publications 93 p. 19-39 Doi: https://doi.org/10.4064/bc93-0-2

Communications in Algebra 37(11) p. 3861-3873 Doi: https://doi.org/10.1080/00927870802555900

The purpose of this article is to develop tools for producing multiple structures on smooth varieties, based on the theory for curves by Banica and Forster [1, 2]. By recursively extending schemes, we show how all Cohen-Macaulay scheme structures of this kind can be found. Similar results have been obtained by Manolache [12, 15-17], using a different recursive construction. The construction in this article complements Manolache's methods, and for some of the applications we have in mind, our construction gives more flexibility [18, 19]. As an application of the theory, we reformulate Hartshorne's Conjecture on complete intersections in codimension two in terms of multiple schemes of degrees two and three.

Mathematische Nachrichten 281(3) p. 434-441

There are very many non-reduced schemes. In this paper, we consider two examples to back this statement: we give lists of double scheme structures on a twisted cubic, and we construct rank two bundles on projective 3-space with prescribed Chern classes, from double structures on smooth rational curves. (C) 2008 WILEY-VCH Verlag GmbH & Co. KG A, Weinheim.

Journal of Algebra 302(1) p. 116-155

Communications in Algebra 33(9) p. 3121-3

Communications in Algebra 33(9) p. 3121-3136

http://xxx.lanl.gov/abs/math.AG/0209061