English
norskBlar i forfatter "Riener, Cordian"
Viser treff 1-11 av 11
-
Attitudes towards mathematics, achievement, and drop-out intentions among STEM and Non-STEM students in Norway
(Journal article; Tidsskriftartikkel; Peer reviewed, 2023-02-22)High mathematical ability is central in many domains. In the present study, we investigated relations between students’ high school mathematics background and current study program, and various mathematics-related outcomes. We expected that students who had attended higher-level mathematics courses in high school would report more positive attitudes towards mathematics and higher achievement in ... -
Generalized eigenvalue methods for Gaussian quadrature rules
(Journal article; Tidsskriftartikkel; Peer reviewed, 2020)A quadrature rule of a measure <i>µ</i> on the real line represents a conic combination of finitely many evaluations at points, called nodes, that agrees with integration against <i>µ</i> for all polynomials up to some fixed degree. In this paper, we present a bivariate polynomial whose roots parametrize the nodes of minimal quadrature rules for measures on the real line. We give two symmetric ... -
On the equivariant Betti numbers of symmetric definable sets: vanishing, bounds and algorithms
(Journal article; Tidsskriftartikkel; Peer reviewed, 2018-03-02)Let R be a real closed field. We prove that for any fixed <i>d</i>, the equivariant rational cohomology groups of closed symmetric semi-algebraic subsets of R<sup><i>k</i></sup> defined by polynomials of degrees bounded by <i>d</i> vanishes in dimensions <i>d</i> and larger. This vanishing result is tight. Using a new geometric approach we also prove an upper bound of [mathematical formula] on the ... -
On the Isotypic Decomposition of Cohomology Modules of Symmetric Semi-algebraic Sets: Polynomial Bounds on Multiplicities
(Journal article; Tidsskriftartikkel; Peer reviewed, 2018-04-30)We consider symmetric (under the action of products of finite symmetric groups) real algebraic varieties and semi-algebraic sets, as well as symmetric complex varieties in affine and projective spaces, defined by polynomials of degrees bounded by a fixed constant <i>d</i>. We prove that if a Specht module, S<sup>λ</sup>, appears with positive multiplicity in the isotypic decomposition of the ... -
Real Root Finding for Equivariant Semi-algebraic Systems
(Conference object; Konferansebidrag, 2018)Let <i><b>R</b></i> be a real closed field. We consider basic semi-algebraic sets defined by <i>n</i>-variate equations/inequalities of s symmetric polynomials and an equivariant family of polynomials, all of them of degree bounded by 2<i>d</i> < <i>n</i>. Such a semi-algebraic set is invariant by the action of the symmetric group. We show that such a set is either empty or it contains a point ... -
Reflection groups and cones of sums of squares
(Journal article; Tidsskriftartikkel; Peer reviewed, 2023-03-15)We consider cones of real forms which are sums of squares and invariant under a (finite) reflection group. Using the representation theory of these groups we are able to use the symmetry inherent in these cones to give more efficient descriptions. We focus especially on the <i>A</i><sub>n</sub>, <i>B</i><sub>n</sub> and <i>D</i><sub>n</sub> case where we use so-called higher Specht polynomials to ... -
Reflection groups, arrangements, and invariant real varieties
(Journal article; Tidsskriftartikkel; Peer reviewed, 2017-10-18)Let <i>X</i> be a nonempty real variety that is invariant under the action of a reflection group <i>G</i>. We conjecture that if <i>X</i> is defined in terms of the first <i>k</i> basic invariants of <i>G</i> (ordered by degree), then <i>X</i> meets a <i>k</i>-dimensional flat of the associated reflection arrangement. We prove this conjecture for the infinite types, reflection groups of rank at most ... -
Return of the evolute
(Journal article; Tidsskriftartikkel; Peer reviewed, 2023)Below we consider the evolutes of plane real-algebraic curves and discuss some of their complex and real-algebraic properties. In particular, for a given degree d ≥ 2, we provide lower bounds for the following four numerical invariants: 1) the maximal number of times a real line can intersect the evolute of a real-algebraic curve of degree d; 2) the maximal number of real cusps which can occur on ... -
Symmetric ideals, Specht polynomials and solutions to symmetric systems of equations
(Journal article; Tidsskriftartikkel; Peer reviewed, 2021-02-18)An ideal of polynomials is symmetric if it is closed under permutations of variables. We relate general symmetric ideals to the so called Specht ideals generated by all Specht polynomials of a given shape. We show a connection between the leading monomials of polynomials in the ideal and the Specht polynomials contained in the ideal. This provides applications in several contexts. Most notably, this ... -
Symmetric Non-Negative Forms and Sums of Squares
(Journal article; Tidsskriftartikkel; Peer reviewed, 2020-05-21)We study symmetric non-negative forms and their relationship with symmetric sums of squares. For a fixed number of variables <i>n</i> and degree 2<i>d</i>, symmetric non-negative forms and symmetric sums of squares form closed, convex cones in the vector space of <i>n</i>-variate symmetric forms of degree 2<i>d</i>. Using representation theory of the symmetric group we characterize both cones in a ... -
Symmetry Reduction in AM/GM-Based Optimization
(Journal article; Tidsskriftartikkel; Peer reviewed, 2022)The arithmetic mean/geometric mean inequality (AM/GM inequality) facilitates classes of nonnegativity certificates and of relaxation techniques for polynomials and, more generally, for exponential sums. Here, we present a first systematic study of the AM/GM-based techniques in the presence of symmetries under the linear action of a finite group. We prove a symmetry-adapted representation theorem and ...
