Publications by Branko Ćurgus

There are five groups of publications below:
submitted, peer reviewed, invited, theses, and published only on this website.

Peer reviewed publications
  1. Indefinite Sturm-Liouville operators in polar form. (with Volodymyr Derkach and Carsten Trunk) Integral Equations and Operator Theory 96 (2024) no. 1, Paper No. 2, 58 pp. doi:10.1007/s00020-023-02746-3.
  2. Operators without eigenvalues in finite-dimensional vector spaces: essential uniqueness of the model. (with Aad Dijksma). Linear Algebra and its Applications. 679 (2023) 86-98. doi:10.1016/j.laa.2023.09.001.
  3. Operators without eigenvalues in finite-dimensional vector spaces. (with Aad Dijksma). Linear Algebra and its Applications. 605 (2020) 63-117. doi:10.1016/j.laa.2020.07.007.
  4. Finite-codimensional compressions of symmetric and self-adjoint linear relations in Krein spaces. (with Tomas Azizov and Aad Dijksma) Integral Equations and Operator Theory 86 (2016) no. 1, 71-95. doi:10.1007/s00020-016-2313-2.
  5. Partially fundamentally reducible operators in Krein spaces. (with Vladimir Derkach) Integral Equations and Operator Theory 82 (2015) no. 4, 469-518. doi:10.1007/s00020-014-2204-3.
  6. Somewhat stochastic matrices. (with Robert I. Jewett) The American Mathematical Monthly 122 (2015) no. 1, 36-42. doi:10.4169/amer.math.monthly.122.01.36.
  7. A squeeze for two common sequences that converge to $e$. The College Mathematics Journal 45 (2014) no. 5, 391-392. doi:10.4169/college.math.j.45.5.391.
  8. Outer median triangles. (with Árpád Bényi). Mathematics Magazine 87 (2014) no. 3, 185-195. doi:10.4169/math.mag.87.3.185.
  9. Ceva's triangle inequalities. (with Árpád Bényi). Mathematical Inequalities & Applications 17 (2014) no. 2, 591-609.
  10. Problem 11749. The $p$-norm on ${\mathbb C}^n$ is a convex function of $p$. The American Mathematical Monthly 121 (2014) no. 1, 83.
  11. A proof of the main theorem on Bezoutians. (with Aad Dijksma). Elemente der Mathematik 69 (2014) no. 1, 33-39. doi:10.4171/EM/243.
  12. The Riesz basis property of an indefinite Sturm-Liouville problem with non-separated boundary conditions. (with Andreas Fleige and Aleksey Kostenko). Integral Equations and Operator Theory 77 (2013) no. 4, 533-557. doi:10.1007/s00020-013-2093-x.
  13. A generalization of Routh's triangle theorem. (with Árpád Bényi). The American Mathematical Monthly 120 (2013) no. 9, 841-846. doi:10.4169/amer.math.monthly.120.09.841.

    The animation that is cited in this paper can be found here.

  14. Triangles and groups via cevians. (with Árpád Bényi). Journal of Geometry 103 (2012) no. 3, 375-408.

    Here you can find more information about this paper including an animation which inspired the paper.

  15. On the reproducing kernel of a Pontryagin space of vector valued polynomials. (with Aad Dijksma) Linear Algebra and its Applications 436 (2012), no. 5, 1312-1343.
  16. Stability of roots of polynomials under linear combinations of derivatives. (with Vania Mascioni). Constructive Approximation 32 (2010) no. 3, 523-541.
  17. Riesz basis of root vectors of indefinite Sturm-Liouville problems with eigenparameter dependent boundary conditions, II. (with Paul Binding). Integral Equations and Operator Theory 63 (2009) no. 4, 473-499.

    Detailed calculations for the opening example in this paper can be found here.

  18. On a convex operator for finite sets. (with Krzysztof Kołodziejczyk). Discrete Applied Mathematics 155 (2007) no. 13, 1774-1792.

    Paper's abstract and a motivating image can be found here.

  19. Root preserving transformations of polynomials. (with Vania Mascioni) Mathematics Magazine 80 (2007) no. 2, 136-138.
  20. Perturbations of roots under linear transformations of polynomials. (with Vania Mascioni) Constructive Approximation 25 (2007) no. 3, 255-277.
  21. An unexpected limit of expected values. (with Robert I. Jewett) Expositiones Mathematicae 25 (2007) no. 1, 1-20.
  22. An exceptional exponential function. The College Mathematics Journal 37 (2006) no. 5, 344-354.
  23. Riesz basis of root vectors of indefinite Sturm-Liouville problems with eigenparameter dependent boundary conditions, I. (with Paul Binding) in Operator Theory and Indefinite Inner Product Spaces, The Proceedings of the Colloquium held on the occasion of the retirement of Heinz Langer, Vienna, Austria. 75-96, Operator Theory: Advances and Applications, Vol. 163, Birkhäuser, 2006.
  24. Roots and polynomials as homeomorphic spaces. (with Vania Mascioni) Expositiones Mathematicae 24 (2006) no. 1, 81-95.
  25. A counterexample in Sturm-Liouville completeness theory. (with Paul Binding) Proceedings of the Royal Society of Edinburgh: Section A: Mathematics 134 (2004) no. 2, 241-248.
  26. A contraction of the Lucas polygon. (with Vania Mascioni) Proceedings of the American Mathematical Society 132 (2004) no. 10, 2973-2981.
  27. Continuous embeddings, completions and complementation in Krein spaces. (with Heinz Langer) Radovi Matematički 12 (2003) no. 1, 37-79.
  28. Standard symmetric operators in Pontryagin spaces: A generalized von Neumann formula and minimality of boundary coefficients. (with Tomas Azizov and Aad Dijksma) Journal of Functional Analysis 198 (2003) no. 2, 361-412.
  29. On the location of critical points of polynomials. (with Vania Mascioni) Proceedings of the American Mathematical Society 131 (2003) no. 1, 253-264.
  30. Form domains and eigenfunction expansions for differential equations with eigenparameter dependent boundary conditions. (with Paul Binding) Canadian Journal of Mathematics 54 (2002) no. 6, 1142-1164.
  31. Discreteness of the spectrum of second order differential operators and associated embedding theorems. (with Thomas T. Read) Journal of Differential Equations, 184 (2002) no. 2, 526-548.
  32. The linearization of boundary eigenvalue problems and reproducing kernel Hilbert spaces. (with Aad Dijksma and Tom Read) Linear Algebra and its Applications 329 (2001) no. 1-3, 97-136.
  33. On singular critical points of positive operators in Krein spaces. (with Aurelian Gheondea and Heinz Langer) Proceedings of the American Mathematical Society 128 (2000) no. 9, 2621-2626.
  34. Boundary value problems in Krein spaces. Dedicated to the memory of Branko Najman. Glasnik Matematički Ser. III 35(55) (2000) no. 1, 45-58.

  35. Positive differential operators in Krein space $L^{2}({\mathbb R}^n)$. (with Branko Najman) Contributions to Operator Theory in Spaces with an Indefinite Metric, The Heinz Langer Anniversary Volume, 113-129, Edited by A. Dijksma, I. Gohberg, M. A. Kaashoek, and R. Mennicken. Operator Theory: Advances and Applications, Vol. 106, Birkhäuser, 1998.

  36. Examples of positive operators in Krein space with $0$ a regular critical point of infinite rank. (with Branko Najman) 51-56, Operator Theory: Advances and Applications, Vol. 102, Birkhäuser, Basel, 1998.
  37. Preservation of the range under perturbations of an operator. (with Branko Najman) Proceedings of the American Mathematical Society 125 (1997) no. 9, 2627-2631.

  38. Positive differential operators in Krein space $L^{2}({\mathbb R})$. (with Branko Najman) Recent developments in operator theory and its applications (Winnipeg, MB, 1994), 95-104, Operator Theory: Advances and Applications, Vol. 87, Birkhäuser, Basel, 1996.

  39. Quadratic eigenvalue problems. (with Branko Najman) Mathematische Nachrichten 174 (1995) 55-64.
  40. Quasi-uniformly positive operators in Krein space. (with Branko Najman) Operator theory and boundary eigenvalue problems (Vienna, 1993), 90-99, Operator Theory: Advances and Applications, Vol. 80, Birkhäuser, Basel, 1995.

  41. The operator $(\mbox{sgn}\,x)\frac{d^{2}}{dx^{2}}$ is similar to a self-adjoint operator in $L^{2}({\mathbb R})$. (with Branko Najman) Proceedings of the American Mathematical Society 123 (1995) no. 4, 1125-1128.

  42. A Krein space approach to elliptic eigenvalue problems with indefinite weights. (with Branko Najman) Differential and Integral Equations 7 (1994) no. 5-6, 1241-1252.
  43. Definitizable extensions of positive symmetric operators in a Krein space. Integral Equations and Operator Theory 12 (1989) no. 5, 615-631.
  44. Characteristic functions of unitary colligations and bounded operators in Krein spaces. (with Aad Dijksma, Heinz Langer, Henk S.V. de Snoo) in The Gohberg Anniversary Collection, Vol. II, 125-152, Edited by H. Dym, S. Goldberg, M. A. Kaashoek, P. Lancaster, Operator Theory: Advances and Applications, Vol. 41, Birkhäuser Verlag, Basel, (1989).
  45. A Krein space approach to symmetric ordinary differential operators with an indefinite weight function. (with Heinz Langer) Journal of Differential Equations 79 (1989) no. 1, 31-61.
  46. On the regularity of the critical point infinity of definitizable operators. Integral Equations and Operator Theory 8 (1985) no. 4, 462-488.
  47. Nonmeasurable sets and pairs of transfinite sequences. (with Harry I. Miller) Akad. Nauka Umjet. Bosne Hercegov. Rad. Odjelj. Prirod. Mat. Nauka LXIX (1982) 39-43.

Invited publications

  1. Numbers in the Sky(viewing Sculpture). Isamu Noguchi and Skyviewing Sculpture. Proceeding of symposia and special lectures. Western Washington University, Bellingham, Wahington, (2004) 75-82.
  2. Spectral properties of self-adjoint ordinary differential operators with an indefinite weight function. (with H. Langer) Proceedings of the 1984 Workshop ``Spectral theory of Sturm-Liouville differential operators,'' ANL-84-73, Argonne National Laboratory, Argonne, Illinois, (1984) 73-80.
Theses

  1. Definitizable operators in Krein spaces. Applications to ordinary self-adjoint differential operators with an indefinite weight function. (Serbo-Croatian) PhD thesis. University of Sarajevo, 1985.
  2. Spectral theorem for definitizable $J$-unitary and $J$-self-adjoint operators. (Serbo-Croatian) Master thesis. University of Sarajevo, 1984.

Papers published only on this website

  1. On the variation of $3\!\times\!3$ stochastic matrices. A computer generated proof of a claim in Example 6.3 in the first version of the paper "Somewhat stochastic matrices". (with Robert I. Jewett)
  2. In this paper, I present proof of the following famous equality \[ \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6} \] from first principles, as much as possible. The problem of proving the preceding equality is called the Basel problem. I was hoping for a shorter proof. But, on the other hand, there are many short proofs on the internet that use "well-known" facts and cite big theorems from different branches of Mathematics. I wanted a proof that will collect everything that is needed in one document.
  3. In this note, I present direct proof that the number $e$ is irrational. The note is an exercise in writing clear, understandable proof.
Last updated: September 19, 2021.