Šime Ungar
Professor Department of Mathematics Josip Juraj Strossmayer University of Osijek Trg Ljudevita Gaja 6 Osijek, HR31000, Croatia¸

Research Interests
 Geometric and Algebraic Topology
 Shape theory
Degrees
 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.
Publications
Journal Publications
 R. Scitovski, K. Sabo, D. Grahovac, Š. Ungar, Minimal distance index — A new clustering performance metrics, Information Sciences (2023), prihvaćen za objavljivanjeWe 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 wellseparated 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.
 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), 18In 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 COVID19 pandemic in Croatia, but the method can be applied to any new wave of the COVID19 pandemic, as well as to any other possible pandemic. We also supply freely available Mathematica modules to implement the method.
 J. Pečarić, Š. Ungar, On the twopoint Ostrowski inequality, Mathematical Inequalities and Applications 13/2 (2010), 339347We prove the L_{p}version of an inequality similar to the twopoint Ostrowski inequality of Matić and Pečarić.
 M. Matić, Š. Ungar, More on the twopoint Ostrowski inequality, Journal of Mathematical Inequalities 3/3 (2009), 417426We improve the previous results of Pečarić and Ungar on the L_{p}version of an inequality similar to the twopoint Ostrowski inequality of Matić and Pečarić.
 Š. Ungar, The Koch Curve: A Geometric Proof, The American Mathematical Monthly 114/1 (2007), 6065The wellknown 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.
 J. Pečarić, Š. Ungar, On an inequality of Grüss type, Mathematical Communications 11/2 (2006), 137141We prove an inequality of Grüss type for pnorm, which for p=∞ gives an estimate similar to a result of Pachpatte.
 J. Pečarić, Š. Ungar, On an inequality of Ostrowski type, Journal of Inequalities in Pure and Applied Mathematics 7/151 (2006), 15We prove an inequality of Ostrowski type for pnorm, generalizing a result of Dragomir.
 Š. Ungar, Partitions of sets and the Riemann integral, Mathematical Communications 11 (2006), 5561We 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 the Riemann integral is the limit of Darboux sums when the mesh of the partition approaches zero.
 I. Herburt, Š. Ungar, Rigid sets of dimension n1 in R^{n}, Geometriae Dedicata 76 (1999), 331339We 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 (n1)dimensional sets rigid in R^{n}.
 R. Scitovski, Š. Ungar, D. Jukić, Approximating surfaces by moving total least squares method, Applied mathematics and computation 93/23 (1998), 219232We 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.
 Š. Ungar, A remark on the composition of celllike maps, Glasnik Matematički 22 (1987), 459461We give sufficient conditions for the composition of celllike maps between metric spaces to be celllike. In particular the composition X → Y → Z of celllike maps is celllike provided X is finite dimensional.
 Š. Ungar, On a homotopy lifting property for inverse sequences, Berichte der mathematischstatistischen Sektion in der Forschungsgesellschaft Joanneum Graz (1987), 111We define pseudoapproximate 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 pseudoapproximate homotopy lifting property. We show that every shape fibration is a pseudoapproximate fibration and that if p: E → B is a fibration where B is an ANR, then it is also a pseudoapproximate fibration. Further, a pseudoapproximate fibration need not be a shape fibration.
 Š. Ungar, Shape bundles, Topology and its Applications 12 (1981), 8999We define shape bundles as a generalization of shape cellbundles and prove that they are weak shape fibrations. We show that shape cellbundles coincide with celllike maps and therefore with shape bundles in case the fibers are cells or the Hilbert cube.
 Š. Ungar, On local homotopy and homology progroups, Glasnik Matematički 14 (1979), 151158In 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 progroups of topological spaces.
 Š. Ungar, Van Kampen theorem for fundamental progroups, Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques 27 (1979), 171181We prove the analogue of the van Kampen theorem for fundamental progroups for topological spaces.
 Š. Ungar, nConnectedness of inverse systems and applications to shape theory, Glasnik Matematički 13 (1978), 371396Let (X, A, x) be an nconnected inverse system of CWpairs such that the restriction (A, x) is mconnected. We prove that there exists an isomorphic inverse system (Y, B, y) having nconnected terms such that the terms of the restriction (B, y) are mconnected. This result is then applied in proving analogues of Hurewicz and BlakersMassey theorems for homotopy progroups and shape groups.
 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), 275280We prove the analogue of the Freudenthal suspension theorem in shape theory. Our result is in terms of homotopy progroups. In the movable metric case the result can be expressed in terms of shape groups.
 S. Mardešić, Š. Ungar, The relative Hurewicz theorem in shape theory, Glasnik Matematički 9 (1974), 317327The 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 progroups and is valid for arbitrary pairs of connected topological spaces. In the special case of movable pairs of metric compacta, the homotopy and homology progroups can be replaced by their limits, i.e., by shape groups and Čech homology groups.
Refereed Proceedings
 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, 311316
 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, 196201
 Š. Ungar, A remark on shape paths and homotopy progroups, General Topology and its relations to Modern Analysis and Algebra V, Prag, 1981, 642647We define the notion of shape paths for topological spaces and the action they induce on homotopy progroups and shape groups. We also exhibit the relationships between connectedness by shape paths, connectedness by continua of trivial shape, and connectedness by shape 1connected continua.
Others
 Š. Ungar, Slutnja koja je postala teorem, Matematičko fizički list 61/1 (2010), 2023Poincaréova 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.
Books
 R. Scitovski, K. Sabo, F. MartínezÁlvarez, 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.
 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.
 Š. Ungar, Matematička analiza u R^{n}, Golden marketing  Tehnička knjiga, Zagreb, 2005.
 Š. Ungar, Ne baš tako kratak uvod u LaTeX, Odjel za matematiku Sveučilišta J.J. Strossmayera u Osijeku, Osijek, 2002.
 Š. Ungar, Matematička analiza 3, PMFMatematički odjel, Zagreb, 1994.
Projects
Professional Activities
Editorial Boards
Committee Memberships
Refereeing/Reviewing
Service Activities
Teaching
Konzultacije (Office Hours): Prema dogovoru (by appointment)
NASTAVA:
Uvod u algebarsku topologiju — zimski semestar
Integralni račun — utorak 8:15 – 9:45 (D1)
Uvod u teoriju skupova i matematičku logiku — utorak 12:15 – 13:45 (D2)
Metrički prostori — utorak 10:15 – 11:45 i srijeda 8:15 – 9:45 (D22)
Personal
Here goes the private stuff.