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Extreme Points of the Vandermonde Determinant on the Sphere and Some Limits Involving the Generalized Vandermonde Determinant

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Book cover Algebraic Structures and Applications (SPAS 2017)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 317))

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Abstract

The values of the determinant of Vandermonde matrices with real elements are analyzed both visually and analytically over the unit sphere in various dimensions. For three dimensions some generalized Vandermonde matrices are analyzed visually. The extreme points of the ordinary Vandermonde determinant on finite-dimensional unit spheres are given as the roots of rescaled Hermite polynomials and a recursion relation is provided for the polynomial coefficients. Analytical expressions for these roots are also given for dimension three to seven. A transformation of the optimization problem is provided and some relations between the ordinary and generalized Vandermonde matrices involving limits are discussed.

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References

  1. Lundengård, K., Österberg, J., Silvestrov, S.: Extreme points of the Vandermonde determinant on the sphere and some limits involving the generalized Vandermonde determinant (2013). arXiv:1312.6193 [math.ca]

  2. Lundengård, K., Österberg, J., Silvestrov, S.: Optimization of the determinant of the Vandermonde matrix and related matrices. Methodol. Comput. Appl. Probab. 20, 1417–1428 (2018)

    Article  MathSciNet  Google Scholar 

  3. Muhumuza, A.K., Lundengård, K., Österberg, J., Silvestrov, S., Mango, J.M., Kakuba, G.: Extreme points of the Vandermonde determinant on surfaces implicitly determined by a univariate polynomial. In: Silvestrov, S., Malyarenko, A., Rančić., M. (eds.), Algebraic structures and Applications, Springer Proceedings in Mathematics and Statistics, vol. 317. Springer (2020)

    Google Scholar 

  4. Muhumuza, A. K., Lundengård, K., Österberg, J., Silvestrov, S., Mango, J. M., Kakuba, G.: Optimization of the Wishart joint eigenvalue probability density distribution based on the Vandermonde determinant. In: Silvestrov, S., Malyarenko, A., Rančić., M. (eds.) Algebraic structures and Applications, Springer Proceedings in Mathematics and Statistics, vol. 317. Springer (2020)

    Google Scholar 

  5. Lundengård, K., Rančić, M., Javor, V., Silvestrov, S.: Electrostatic discharge currents representation using the multi-peaked analytically extended function by interpolation on a \(D\)-optimal design. Facta Univ. Ser.: Electron. Energ. 32(1), 25–49 (2019)

    Article  Google Scholar 

  6. Dimitrov, D., Shapiro, B.: Electrostatic problems with a rational constraint and degenerate Lamé operators. Potential Anal. 52(4), 645–659 (2020)

    Google Scholar 

  7. Fröberg, R., Shapiro, B.: Vandermonde varieties and relations among schur polynomials (2013). arXiv:1302.1298 [math.AG]

  8. Fröberg, R., Shapiro, B.: On Vandermonde varieties. Math. Scand. 119(1), 73–91 (2016)

    Article  MathSciNet  Google Scholar 

  9. Kalman, D.: The generalized Vandermonde matrix. Math. Mag. 57, 15–21 (1984)

    Article  MathSciNet  Google Scholar 

  10. Ernst, T., Silvestrov, S.: Shift difference equations, symmetric polynomials and representations of the symmetric group. U. U. D. M. Rep. 1999, 14 (1999)

    Google Scholar 

  11. Ernst, T.: Generalized Vandermonde determinants. U. U. D. M. Rep. 2000, 6 (2000)

    Google Scholar 

  12. Klein, A., Spreij, P.: Some results on Vandermonde matrices with an application to time series analysis. Siam J. Matrix Anal. Appl. 25, 213–223 (2003)

    Article  MathSciNet  Google Scholar 

  13. Szegő, G.: Orthogonal Polynomials. American Mathematics Society (1939)

    Google Scholar 

  14. Serre, D.: Matrices: Theory and Applications. Springer, Berlin (2002)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Division of Applied Mathematics, School of Education, Culture and Communication, Mälardalen University, Box 883, 72123, Västerås, Sweden

    Karl Lundengård, Jonas Österberg & Sergei Silvestrov

Authors
  1. Karl Lundengård
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  2. Jonas Österberg
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  3. Sergei Silvestrov
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Corresponding author

Correspondence to Karl Lundengård .

Editor information

Editors and Affiliations

  1. Division of Applied Mathematics, School of Education, Culture and Communication, Mälardalen University, Västerås, Sweden

    Sergei Silvestrov

  2. Division of Applied Mathematics, School of Education, Culture and Communication, Mälardalen University, Västerås, Sweden

    Anatoliy Malyarenko

  3. Division of Applied Mathematics, School of Education, Culture and Communication, Mälardalen University, Västerås, Sweden

    Milica Rančić

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Lundengård, K., Österberg, J., Silvestrov, S. (2020). Extreme Points of the Vandermonde Determinant on the Sphere and Some Limits Involving the Generalized Vandermonde Determinant. In: Silvestrov, S., Malyarenko, A., Rančić, M. (eds) Algebraic Structures and Applications. SPAS 2017. Springer Proceedings in Mathematics & Statistics, vol 317. Springer, Cham. https://doi.org/10.1007/978-3-030-41850-2_32

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