Fakultet primijenjene matematike i informatike

Šime Ungar


Department of Mathematics
Josip Juraj Strossmayer University of Osijek
Trg Ljudevita Gaja 6
Osijek, HR-31000, Croatia¸
phone: +385-31-224-809
fax: +385-31-224-801
email:  sime @ mathos.hr
office:  23/I


Research Interests

Geometric and Algebraic Topology
Shape theory


PhD in mathematics, Department of Mathematics, University of Zagreb , 1977.
MSc in mathematics, Department of Mathematics, University of Zagreb , 1972.
BSc in mathematics, Department of Mathematics, University of Zagreb, 1969.


Journal Publications

  1. R. Scitovski, K. Sabo, D. Grahovac, Š. Ungar, Minimal distance index — A new clustering performance metrics, Information Sciences (2023), prihvaćen za objavljivanje
    We define a new index for measuring clustering performance called the Minimal Distance Index. The index is based on representing clusters by characteristic objects containing the majority of cluster points. It performs well for both spherical and ellipsoidal clusters. This method can recognize all acceptable partitions with well-separated clusters. Among such partitions, our minimal distance index may identify the most appropriate one. The proposed index is compared with other most frequently used indexes in numerous examples with spherical and ellipsoidal clusters. It turned out that our proposed minimal distance index always recognizes the most appropriate partition, whereas the same cannot be said for other indexes found in the literature.
  2. R. Scitovski, K. Sabo, Š. Ungar, A method for forecasting the number of hospitalized and deceased based on the number of newly infected during a pandemic, Scientific Reports - Nature 12/4773 (2022), 1-8
    In this paper we propose a phenomenological model for forecasting the numbers of deaths and of hospitalized persons in a pandemic wave, assuming that these numbers linearly depend, with certain delays τ>0 for deaths and δ>0 for hospitalized, on the number of new cases. We illustrate the application of our method using data from the third wave of the COVID-19 pandemic in Croatia, but the method can be applied to any new wave of the COVID-19 pandemic, as well as to any other possible pandemic. We also supply freely available Mathematica modules to implement the method.
  3. J. Pečarić, Š. Ungar, On the two-point Ostrowski inequality, Mathematical Inequalities and Applications 13/2 (2010), 339-347
    We prove the <i>L<sub>p</sub></i>-version of an inequality similar to the two-point Ostrowski inequality of Matić and Pečarić.
  4. M. Matić, Š. Ungar, More on the two-point Ostrowski inequality, Journal of Mathematical Inequalities 3/3 (2009), 417-426
    We improve the previous results of Pečarić and Ungar on the <i>L<sub>p</sub></i>-version of an inequality similar to the two-point Ostrowski inequality of Matić and Pečarić.
  5. Š. Ungar, The Koch Curve: A Geometric Proof, The American Mathematical Monthly 114/1 (2007), 60-65
    The well-known Koch curve is often used as an example to illustrate a continuous but nowhere differentiable function and as an example of a nonrectifiable curve. Usually only the fact that it is not rectifiable is proved. The proof that it is indeed a curve and that at no point does this curve have a tangent line is omitted. Rarely is even a reference given, and then usually to Koch's original paper from 1906. We give a simple geometric proof that the Koch curve is indeed an arc (i.e., the homeomorphic image of a straight line segment) and that it at no point has a tangent line.
  6. J. Pečarić, Š. Ungar, On an inequality of Grüss type, Mathematical Communications 11/2 (2006), 137-141
    We prove an inequality of Grüss type for <i>p</i>-norm, which for <i>p</i>=&infin; gives an estimate similar to a result of Pachpatte.
  7. J. Pečarić, Š. Ungar, On an inequality of Ostrowski type, Journal of Inequalities in Pure and Applied Mathematics 7/151 (2006), 1-5
    We prove an inequality of Ostrowski type for <i>p</i>-norm, generalizing a result of Dragomir.
  8. Š. Ungar, Partitions of sets and the Riemann integral, Mathematical Communications 11 (2006), 55-61
    We will discuss the definition of Riemann integral using general partitions and give an elementary explication, without resorting to nets, generalized sequences and such, of what is meant by saying that <i>the Riemann integral is the limit of Darboux sums when the mesh of the partition approaches zero</i>.
  9. I. Herburt, Š. Ungar, Rigid sets of dimension n-1 in Rn, Geometriae Dedicata 76 (1999), 331-339
    We give conditions allowing an intrinsic isometry on a dense subset to be extended to an isometry of the whole set. This enables us to find examples of (<i>n</i>-1)-dimensional sets rigid in <b>R</b><sup><i>n</i></sup>.
  10. R. Scitovski, Š. Ungar, D. Jukić, Approximating surfaces by moving total least squares method, Applied mathematics and computation 93/2-3 (1998), 219-232
    We suggest a method for generating a surface approximating the given data (xi, yi, zi) ϵ R^3, i = 1, …. m, assuming that the errors can occur both in the independent variables x and y, as well as in the dependent variable z. Our approach is based on the moving total least squares method, where the local approximants (local planes) are determined in the sense of total least squares. The parameters of the local approximants are obtained by finding the eigenvector, corresponding to the smallest eigenvalue of a certain symmetric matrix. To this end, we develop a procedure based on the inverse power method. The method is tested on several examples.
  11. Š. Ungar, A remark on the composition of cell-like maps, Glasnik Matematički 22 (1987), 459-461
    We give sufficient conditions for the composition of cell-like maps between metric spaces to be cell-like. In particular the composition <i>X &rarr; Y &rarr; Z</i> of cell-like maps is cell-like provided <i>X</i> is finite dimensional.
  12. Š. Ungar, On a homotopy lifting property for inverse sequences, Berichte der mathematisch-statistischen Sektion in der Forschungsgesellschaft Joanneum Graz (1987), 1-11
    We define pseudo-approximate fibration for metric compacta in an attempt to generalize both fibration and shape fibration simultaneously. The definition employs inverse sequences of ANRs along with a notion called the pseudo-approximate homotopy lifting property. We show that every shape fibration is a pseudo-approximate fibration and that if <i>p: E &rarr; B</i> is a fibration where B is an ANR, then it is also a pseudo-approximate fibration. Further, a pseudo-approximate fibration need not be a shape fibration.
  13. Š. Ungar, Shape bundles, Topology and its Applications 12 (1981), 89-99
    We define shape bundles as a generalization of shape cell-bundles and prove that they are weak shape fibrations. We show that shape cell-bundles coincide with cell-like maps and therefore with shape bundles in case the fibers are cells or the Hilbert cube.
  14. Š. Ungar, On local homotopy and homology pro-groups, Glasnik Matematički 14 (1979), 151-158
    In this note we give a proof of Hurewicz theorem for inverse systems of pointed topological spaces and study some properties of local homotopy and homology pro-groups of topological spaces.
  15. Š. Ungar, Van Kampen theorem for fundamental pro-groups, Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques 27 (1979), 171-181
    We prove the analogue of the van Kampen theorem for fundamental pro-groups for topological spaces.
  16. Š. Ungar, n-Connectedness of inverse systems and applications to shape theory, Glasnik Matematički 13 (1978), 371-396
    Let (<i>X, A, x</i>) be an <i>n</i>-connected inverse system of CW-pairs such that the restriction (<i>A, x</i>) is <i>m</i>-connected. We prove that there exists an isomorphic inverse system (<i>Y, B, y</i>) having <i>n</i>-connected terms such that the terms of the restriction (<i>B, y</i>) are <i>m</i>-connected. This result is then applied in proving analogues of Hurewicz and Blakers-Massey theorems for homotopy pro-groups and shape groups.
  17. Š. Ungar, The Freudenthal suspension theorem in shape theory, Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques 24 (1976), 275-280
    We prove the analogue of the Freudenthal suspension theorem in shape theory. Our result is in terms of homotopy pro-groups. In the movable metric case the result can be expressed in terms of shape groups.
  18. S. Mardešić, Š. Ungar, The relative Hurewicz theorem in shape theory, Glasnik Matematički 9 (1974), 317-327
    The purpose of this paper is to establish a Hurewicz theorem in shape theory for pointed pairs of spaces. Our result is expressed in terms of homotopy and homology pro-groups and is valid for arbitrary pairs of connected topological spaces. In the special case of movable pairs of metric compacta, the homotopy and homology pro-groups can be replaced by their limits, i.e., by shape groups and Čech homology groups.

Refereed Proceedings

  1. D. Jukić, R. Scitovski, Š. Ungar, The best total least squares line in R^3, 7th International Conference on Operational Research KOI 1998, Rovinj, 1998, 311-316
  2. R. Scitovski, Š. Ungar, D. Jukić, M. Crnjac, Moving total least squares for parameter identification in mathematical model, Symposium on Operations Research SOR '95, Passau, 1996, 196-201
  3. Š. Ungar, A remark on shape paths and homotopy pro-groups, General Topology and its relations to Modern Analysis and Algebra V, Prag, 1981, 642-647
    We define the notion of shape paths for topological spaces and the action they induce on homotopy pro-groups and shape groups. We also exhibit the relationships between connectedness by shape paths, connectedness by continua of trivial shape, and connectedness by shape 1-connected continua.


  1. Š. Ungar, Slutnja koja je postala teorem, Matematičko fizički list 61/1 (2010), 20-23
    Poincaréova hipoteza, jedna od najpoznatijih matematičkih slutnji, stotinu je godina odolijevala nastojanjima mnogih vrhunskih matematičara, prije svega topologa i diferencijalnih geometričara, da ju dokažu ili opovrgnu. Napokon je, početkom 21. stoljeća, Poincaréova slutnja dokazana.


  1. R. Scitovski, K. Sabo, F. Martínez-Álvarez, Š. Ungar, Cluster Analysis and Applications, Springer, Cham, 2021.
    Clear and precise definitions of basic concepts and notions in clustering, and analysis of their properties. Analysis and implementation of most important methods for searching for optimal partitions. Covers different primitives in clustering, such as points, lines, multiple lines, circles, and ellipses. A new efficient principle of choosing optimal partitions with the most appropriate number of clusters. Detailed description and analysis of several important applications.
  2. Š. Ungar, Ne baš tako kratak Uvod u TEX s naglaskom na pdfLATEX i osvrtom na XƎLATEX, Sveučilište J. J. Strossmayera u Osijeku, Odjel za matematiku, Osijek, 2019.
  3. Š. Ungar, Matematička analiza u Rn, Golden marketing - Tehnička knjiga, Zagreb, 2005.
  4. Š. Ungar, Ne baš tako kratak uvod u LaTeX, Odjel za matematiku Sveučilišta J.J. Strossmayera u Osijeku, Osijek, 2002.
  5. Š. Ungar, Matematička analiza 3, PMF-Matematički odjel, Zagreb, 1994.


Professional Activities

Editorial Boards



Committee Memberships






Service Activities



Konzultacije (Office Hours): Prema dogovoru (by appointment)


Uvod u algebarsku topologiju — zimski semestar

Integralni račun — utorak 8:15 – 9:45    (D-1)

Uvod u teoriju skupova i matematičku logiku — utorak 12:15 – 13:45    (D-2)

Metrički prostori — utorak 10:15 – 11:45 i srijeda 8:15 – 9:45    (D-22)



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