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

1. K. Sabo, R. Scitovski, Š. Ungar, Z. Tomljanović, A method for searching for a globally optimal k-partition of higher-dimensional datasets, Journal of Global Optimization 89 (2024), 633-653
   Abstract

   The problem with finding a globally optimal k-partition of a set A is a very intricate optimization problem for which in general, except in the case of one-dimensional data, i.e., for data with one feature (A\subset\R), there is no method to solve. Only in the one-dimensional case there exist efficient methods that are based on the fact that the search for a globally optimal partition is equivalent to solving a global optimization problem for a symmetric Lipschitz-continuous function using the global optimization algorithm DIRECT. In the present paper, we propose a method for finding a globally optimal k-partition in the general case (A\subset \R^n, n\geq 1), generalizing an idea for solving the Lipschitz global optimization for symmetric functions. To do this, we propose a method that combines a global optimization algorithm with linear constraints and the k-means algorithm. The first of these two algorithms is used only to find a good initial approximation for the $k$-means algorithm. The method was tested on a number of artificial datasets and on several examples from the UCI Machine Learning Repository, and an application in spectral clustering for linearly non-separable datasets is also demonstrated. Our proposed method proved to be very efficient.
2. R. Scitovski, K. Sabo, D. Grahovac, F. Martínez-Álvarez, Š. Ungar, A partitioning incremental algorithm using adaptive Mahalanobis fuzzy clustering and identifying the most appropriate partition, Pattern Analysis and Applications 28/3 (2024), 1-14
   Abstract

   This paper deals with the problem of determining the most appropriate number of clusters in a fuzzy Mahalanobis partition. First, a new fuzzy Mahalanobis incremental algorithm is constructed to search for an optimal fuzzy Mahalanobis partition with clusters. Among these partitions, selecting the one with the most appropriate number of clusters is based on appropriately modified existing fuzzy indexes. In addition, the Fuzzy Mahalanobis Minimal Distance index is defined as a natural extension of the recently proposed Mahalanobis Minimal Distance index for non-fuzzy clustering. The new fuzzy Mahalanobis incremental algorithm was tested on several artificial data sets and the color image segmentation problems from real-world applications: art images, nature photography images, and medical images. The algorithm includes multiple usage of the global optimization algorithm DIRECT. But unlike previously known fuzzy Mahalanobis indexes, the proposed Fuzzy Mahalanobis Minimal Distance index ensures accurate results even when applied to complex real-world applications. A possible disadvantage could be the need for longer CPU time. Furthermore, besides effective identification of the partition with the most appropriate number of clusters, it is shown how to use the proposed Fuzzy Mahalanobis Minimal Distance index to search for an acceptable partition, which proved particularly useful in the above-mentioned real-world applications.
3. A. Morales-Esteban, R. Scitovski, K. Sabo, D. Grahovac, Š. Ungar, Earthquake analysis of clusters of the most appropriate partition, Journal of Seismology (2024), prihvaćen za objavljivanje
   Abstract

   In our paper, we propose the most appropriate partition to depict the seismogenic zones of an active seismic region.To do so,the earthquake data considered are the location and magnitude. To determine three ellipsoidal layers of shallow, intermediate, and deep earthquakes, we switch from the geoid to a solid ball model and solve an appropriate multiple concentric sphere detection problem. Considering the Iberian Peninsula region, by using the Mahalanobis incremental algorithm with the help of the Mahalanobis area index and Mahalanobis minimal distance index, we first determine the most appropriate partition of earth quake positions, consisting of as compact and mutually separated clusters as possible. The result shows four clusters representing the main seismogenic zones of that area. In each of these clusters, we analyze some important earthquake properties, notably the hypocentral depths—a less researched property. Furthermore, we show how to generate a smooth surface best fitting the hypocenters in the considered area, and since the data contain many outliers, for that purpose we use the moving least absolute deviation method. In addition, for each cluster of the most appropriate partition, we ponder the question of estimating the Gutenberg–Richter’s b-value. To avoid the known drawbacks mentioned in the literature for estimating the b-parameter in the Gutenberg–Richter law, we propose the estimation of parameters a and b by using the least absolute deviation method. We also found that the hypocenters are notably deeper in the southwestern Iberian Peninsula and the Azores-Gibraltar fault zone, where the largest earthquakes take place. Finally, one should emphasizethat the hypocenters study proposed in this research demonstrated that the most hazardous zone encompasses the most deep focuses. The CPU-time required for all calculations has been moderate. The methodology, used in this work, could easily be applied to other seismological areas, for which we list our freely available Mathematica-modules.
4. R. Scitovski, K. Sabo, D. Grahovac, Š. Ungar, Minimal distance index — A new clustering performance metrics, Information Sciences 640/119046 (2023)
   Abstract

   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. Furthermore, among all acceptable partitions, the one with the largest number of clusters, not necessarily the most appropriate ones, has a special significance in image analysis. Namely, following Mahalanobis image segmentation, our index recognizes partitions that might not be the most appropriate ones but are the ones using colors that significantly differ from each other. The minimal distance index recognizes partitions with dominant colors, thus making it possible to select specific details in the image. We apply this approach to some real-world applications such as the plant rows detection problem, painting analysis, and iris detection. This may also be useful for medical image analysis.
5. K. Sabo, R. Scitovski, Š. Ungar, Multiple spheres detection problem—Center based clustering approach, Pattern Recognition Letters 176 (2023), 34-41
   Abstract

   In this paper, we propose an adaptation of the well-known -means algorithm for solving the multiple spheres detection problem when data points are homogeneously scattered around several spheres. We call this adaptation the -closest spheres algorithm. In order to choose good initial spheres, we use a few iterations of the global optimizing algorithm DIRECT , resulting in the high efficiency of the proposed -closest spheres algorithm. We present illustrative examples for the case of non-intersecting and for the case of intersecting spheres. We also show a real-world application in analyzing earthquake depths.

6. 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
   Abstract

   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.
7. J. Pečarić, Š. Ungar, On the two-point Ostrowski inequality, Mathematical Inequalities and Applications 13/2 (2010), 339-347
   Abstract

   We prove the <i>L_p</i>-version of an inequality similar to the two-point Ostrowski inequality of Matić and Pečarić.
8. M. Matić, Š. Ungar, More on the two-point Ostrowski inequality, Journal of Mathematical Inequalities 3/3 (2009), 417-426
   Abstract

   We improve the previous results of Pečarić and Ungar on the <i>L_p</i>-version of an inequality similar to the two-point Ostrowski inequality of Matić and Pečarić.
9. Š. Ungar, The Koch Curve: A Geometric Proof, The American Mathematical Monthly 114/1 (2007), 60-65
   Abstract

   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.
10. J. Pečarić, Š. Ungar, On an inequality of Grüss type, Mathematical Communications 11/2 (2006), 137-141
    Abstract

    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.
11. J. Pečarić, Š. Ungar, On an inequality of Ostrowski type, Journal of Inequalities in Pure and Applied Mathematics 7/151 (2006), 1-5
    Abstract

    We prove an inequality of Ostrowski type for <i>p</i>-norm, generalizing a result of Dragomir.
12. Š. Ungar, Partitions of sets and the Riemann integral, Mathematical Communications 11 (2006), 55-61
    Abstract

    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>.
13. I. Herburt, Š. Ungar, Rigid sets of dimension n-1 in Rⁿ, Geometriae Dedicata 76 (1999), 331-339
    Abstract

    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>^<i>n</i>.
14. R. Scitovski, Š. Ungar, D. Jukić, Approximating surfaces by moving total least squares method, Applied mathematics and computation 93/2-3 (1998), 219-232
    Abstract

    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.
15. Š. Ungar, A remark on the composition of cell-like maps, Glasnik Matematički 22 (1987), 459-461
    Abstract

    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.
16. Š. Ungar, On a homotopy lifting property for inverse sequences, Berichte der mathematisch-statistischen Sektion in der Forschungsgesellschaft Joanneum Graz (1987), 1-11
    Abstract

    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.
17. Š. Ungar, Shape bundles, Topology and its Applications 12 (1981), 89-99
    Abstract

    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.
18. Š. Ungar, On local homotopy and homology pro-groups, Glasnik Matematički 14 (1979), 151-158
    Abstract

    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.
19. 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
    Abstract

    We prove the analogue of the van Kampen theorem for fundamental pro-groups for topological spaces.
20. Š. Ungar, n-Connectedness of inverse systems and applications to shape theory, Glasnik Matematički 13 (1978), 371-396
    Abstract

    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.
21. 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
    Abstract

    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.
22. S. Mardešić, Š. Ungar, The relative Hurewicz theorem in shape theory, Glasnik Matematički 9 (1974), 317-327
    Abstract

    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.

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
   Abstract

   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
   Abstract

   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.
   Abstract

   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.