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Employee Profile

# Jon Eivind Vatne

Associate Professor - Campus Bergen

Department of Economics

## Biography

Jon Eivind Vatne studied mathematics at the University of Bergen, specializing in algebraic geometry. After finishing his doctorate in 2002, he had temporary positions as Post.Doc. and associate professor at the University of Bergen (funded by the Norwegian Research Council) and NTNU. He then held a permanent position at Western Norway University of Applied Sciences until he joined BI full-time from August 2021.

## Publications

Korotov, Sergey & Vatne, Jon Eivind (2023)

### On Dihedral Angle Sums of Prisms and Hexahedra

International journal of computational geometry and applications, 33(3-4), s. 85- 95. Doi: 10.1142/S0218195923500036

Various angle characteristics are used (e.g. in finite element methods or computer graphics) when evaluating the quality of computational meshes which may consist, in the three-dimensional case, of tetrahedra, prisms, hexahedra and pyramids. Thus, it is of interest to derive (preferably tight) bounds for dihedral angle sums, i.e. sums of angles between faces, of such mesh elements. For tetrahedra this task was solved by Gaddum in 1952. For pyramids, this was resolved by Korotov, Lund and Vatne in 2022. In this paper, we compute tight bounds for the remaining two cases, hexahedra and prisms.

Korotov, Sergey; Lund, Lars Fredrik Kirkebø & Vatne, Jon Eivind (2022)

### Tight bounds for the dihedral angle sums of a pyramid

Applications of Mathematics Doi: 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.

Korotov, Sergey; Lund, Lars Fredrik Kirkebø & Vatne, Jon Eivind (2021)

### Improved Maximum Angle Estimate for Longest-Edge Bisection

International journal of computational geometry and applications, 31(4), s. 183- 192. Doi: 10.1142/S0218195922500017

Khademi, Ali; Korotov, Sergey & Vatne, Jon Eivind (2021)

### On mesh regularity conditions for simplicial finite elements

Vermolen, Fred J. & Vuik, Cornelis (red.). Numerical Mathematics and Advanced Applications ENUMATH 2019

Korotov, Sergey & Vatne, Jon Eivind (2021)

### Preserved Structure Constants for Red Refinements of Product Elements

Garanzha, Vladimir; Kamenski, Lennard & Si, Hang (red.). 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

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.

Korotov, Sergey & Vatne, Jon Eivind (2020)

### On regularity of tetrahedral meshes produced by some red-type refinements.

Pinelas, Sandra; Graef, John R, Hilger, Stefan, Kloeden, Peter & Schinas, Christos (red.). Differential and difference equations with applications: ICDDEA 2019, Lisbon, Portugal, July 1–5

Boon, Wietse; Nordbotten, Jan Martin & Vatne, Jon Eivind (2020)

### Functional analysis and exterior calculus on mixed-dimensional geometries

Annali di Matematica Pura ed Applicata, s. 1- 33. Doi: 10.1007/s10231-020-01013-1 - Full text in research archive

Khademi, Ali & Vatne, Jon Eivind (2020)

### Estimation of the interpolation error for semiregular prismatic elements

Applied Numerical Mathematics, 156, s. 174- 191. Doi: 10.1016/j.apnum.2020.04.018 - Full text in research archive

Korotov, Sergey & Vatne, Jon Eivind (2020)

### The minimum angle condition for d-simplices

Computers and Mathematics with Applications, 80, s. 367- 370. Doi: 10.1016/j.camwa.2019.05.020

Khademi, Ali; Korotov, Sergey & Vatne, Jon Eivind (2019)

### On Equivalence of Maximum Angle Conditions for Tetrahedral Finite Element Meshes

Garanzha, Vladimir (red.). 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

Khademi, Ali; Korotov, Sergey & Vatne, Jon Eivind (2019)

### On the generalization of the Synge-Křížek maximum angle condition for d-simplices

Journal of Computational and Applied Mathematics, 358, s. 29- 33. Doi: 10.1016/j.cam.2019.03.003

Vatne, Jon Eivind (2019)

### Spaces of simplicial shapes

Lecture Notes in Computational Science and Engineering, 126, s. 753- 760. Doi: 10.1007/978-3-319-96415-7_70

Khademi, Ali; Korotov, Sergey & Vatne, Jon Eivind (2018)

### On Interpolation Error on Degenerating Prismatic Elements

Applications of Mathematics, 63(3), s. 237- 257. Doi: 10.21136/AM.2018.0357-17

Vatne, Jon Eivind (2017)

### Simplices rarely contain their circumcenter in high dimensions

Applications of Mathematics, 62(3), s. 213- 223. Doi: 10.21136/AM.2017.0187-16

Vatne, Jon Eivind (2017)

### The sequence of middle divisors is unbounded

Journal of Number Theory, 172, s. 413- 415. Doi: 10.1016/j.jnt.2016.08.015

Vatne, Jon Eivind (2012)

### Monomial multiple structures

Annali dell’Università di Ferrara Doi: 10.1007/s11565-011-0139-z

Fløystad, Gunnar & Vatne, Jon Eivind (2011)

### Artin-Schelter regular algebras of dimension five

Banach Center Publications, 93, s. 19- 39. Doi: 10.4064/bc93-0-2

Vatne, Jon Eivind (2009)

### Multiple Structures and Hartshorne's Conjecture

Communications in Algebra, 37(11), s. 3861- 3873. Doi: 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.

Vatne, Jon Eivind (2008)

### Double structures on rational space curves

Mathematische Nachrichten, 281(3), s. 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.

Fløystad, Gunnar & Vatne, Jon Eivind (2006)

### PBW-deformations of N-Koszul algebras

Journal of Algebra, 302(1), s. 116- 155.

Vatne, Jon Eivind (2005)

### (Bi)-Cohen-Macaulay Simplicial Complexes and Their Associated Coherent Sheaves

Communications in Algebra, 33(9), s. 3121- 3.

Fløystad, Gunnar & Vatne, Jon Eivind (2005)

### (Bi)-Cohen-Macaulay simplicial complexes and their associated coherent sheaves

Communications in Algebra, 33(9), s. 3121- 3136.

Vatne, Jon Eivind & Fløystad, Gunnar (2002)

### (Bi-) Cohen-Macaulay simplicial complexes and their associated coherent sheaves

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

Gulbrandsen, Martin G; Kleppe, Johannes, Kro, Tore August & Vatne, Jon Eivind (2015)

### Matematikk for ingeniørfag - oppgaver og fasit

[Textbook]. Gyldendal Norsk Forlag A/S.

Gulbrandsen, Martin G; Kleppe, Johannes, Kro, Tore August & Vatne, Jon Eivind (2013)

### Matematikk for ingeniørfag

Mørken, Knut Martin; Malthe-Sørensen, Anders, Simonsen, Ingve, Hammer, Hugo Lewi, Vatne, Jon Eivind, Nøst, Elisabeth, Løyning, Terje Brinck, Dahl, Lars Oswald & Sasaki, Nina (2011)

### Computing in Science Education. A guide for universities and colleges in Norway

[Report]. Det matematisk-naturvitenskaplige fakultet, Universitetet i Oslo.

Mørken, Knut Martin; Simonsen, Ingve, Malthe-Sørensen, Anders, Hammer, Hugo Lewi, Løyning, Terje Brinck, Vatne, Jon Eivind, Nøst, Elisabeth, Dahl, Lars Oswald & Sasaki, nina (2011)

### Beregninsorientert utdanning: En veileder for universiteter og høgskoler i Norge

[Report]. Universitetet i Oslo, Det matematisk naturvitenskaplige fakultet.

Vatne, Jon Eivind (2005)

### PBW-deformations of N-Koszul algebras and Their A_\infty Ext Algebras

[Academic lecture]. Workshop in noncommutative geometry.

Fløystad, Gunnar & Vatne, Jon Eivind (2004)

### PBW-deformations of N-Koszul algebras

[Report]. Institutt Mittag-Leffler, Report No. 32..

Fløystad, Gunnar & Vatne, Jon Eivind (2003)

### (Bi)-Cohen-Macaulay simplicial complexes and their associated coherent sheaves

[Academic lecture]. Formal Power Series and Algebraic Combinatorics.

Vatne, Jon Eivind & Fløystad, Gunnar (2002)

### (Bi-) Cohen-Macaulay complexes and their associated coherent sheaves

[Academic lecture]. Internatiional conefernce on Algebraic Geometry, Commutative Algebra and Topology.