I will start with a description of the background related to this thesis. The student initially chose a completely different topic with another supervisor. Later, however, she began looking for a new topic and contacted members of the department, including myself. I believe that the topic I offered did not fully interest her, as she was reportedly more interested in applications. Nevertheless, after some time she returned and enrolled in my topic. Work on the thesis effectively began only around the turn of the year. The original topic focused mainly on a comparison of discrete and continuous calculus, highlighting similarities and differences. These were to be explained not only in general terms but also through suitable examples. I wanted to accommodate her wishes, and therefore modified the topic to some extent by adding selected applications. Unfortunately, this objective was only partially fulfilled. I consider the most significant shortcoming to be the relatively high degree of ignoring my ongoing comments, which were intended to improve the quality of the work. Here I list only some of the deficiencies that should have been corrected (but were not): the note in section 2.2.2; point 2 in sections 2.7.3 and 2.7.4; error in 2.8.1–2; moving the formulas with indefinite sums to p. 18; a large number of typographical errors; inconsistent notation for sequences; failure to supplement the linear model for saving interest; misunderstanding of Sturm’s separation theorem; inconsistent citation of sources; confused description of the eigenvalue method; 4.8.2 is not a “solutions method”; poor text formatting on p. 53; insufficient number of numerical simulations. Furthermore, it is unclear to me where reference [9] came from. Unfortunately, I am not able to assess to what extent AI was used.

On the other hand, it must be said that the student put in some effort, thanks to which at least some passages look quite solid. She also demonstrated the ability to master and understand material that goes beyond the standard curriculum. The main objectives have been achieved, and in view of the above, I recommend the thesis for defense with an overall grade of E.

The text is divided into a theoretical part, which introduces the fundamental concepts, operators, and the theory of linear and nonlinear difference equations, and an application part, which presents the analysis of a two-dimensional nonlinear macroeconomic model.

The submitted master's thesis sets out to systematically develop the foundations of discrete calculus (difference and summation calculus), compare it with classical continuous calculus, analyze selected difference and differential equations, and apply the resulting apparatus to selected discrete models. Having compared the stated aims of the thesis with its actual content, I must conclude that the aims were not fully met. While the thesis formally adheres to the structure of the assignment, the genuine, deeper methodological comparison of the two cases is limited to merely placing subsections side by side, without any deeper analysis or emphasis on the differences in the qualitative behavior of the discrete and continuous cases.

The thesis also contains a large number of typos, mathematical errors, and inaccuracies (I list some of them below). There is also a conflation (interchanging) of the criteria for the discrete and the continuous case, which is unfortunate in a thesis focused precisely on comparing them.

- Example 3.13 (p. 32) — inconsistent initial conditions vs. the stated Fibonacci formula. The example gives the initial conditions in the form F0 = 1, F1 = 1, but the solution is then presented in the form of the explicit Fibonacci formula, which corresponds to the standard indexing F0 = 0, F1 = 1.

- Section 4.5.1 (p. 51) — incorrect condition for finding the equilibrium point for a discrete system. The author states the condition f1(x1, x2) = 0, f2(x1, x2) = 0, which, however, is the procedure for a continuous system. For a discrete system, the equilibrium satisfies the condition f1(x1, x2) = x1, f2(x1, x2) = x2.

- Example 4.10 (p. 50) — application of the stability criterion. The stability criterion for a discrete system ("inside, or outside, the unit circle") is applied to a continuous system, where instead it is the sign of the real part of the eigenvalues that is decisive.

I would also fault the author for the surprisingly small number of actual comparisons between the discrete and the continuous case. An ideal example could have been a comparison of the continuous logistic differential equation and the discrete logistic difference equation — these two cases differ fundamentally in qualitative terms: whereas the solution of the continuous model exhibits smooth convergence to the carrying capacity of the environment, the discrete logistic model behaves differently as the parameter increases, generating period doubling and leading to deterministic chaos. This analogy, for instance, could have been used to illustrate some of the differences between discrete and continuous models.

I appreciate that the author went beyond the aims of the thesis by including an applied economic model, but unfortunately, this part too is not without shortcomings. In the text, the author declares that the model is taken from source [9] (M. Persson, 2009). Despite considerable effort, however, I was unable to locate this source. I believe that the primary source of this model is the cited article [10] (T. Puu and I. Sushko, 2004), to which the author does not, however, refer.

For this model, an analysis of the equilibria is also carried out, in which the author correctly classifies the stable and unstable node, focus, and center. Here, however, it would be appropriate to include numerical simulations (plots) of the phase portraits, which would confirm the stated classification. These would also make clearer the behavior of the solutions in the neighborhood of the equilibria and the economic interpretation of the model.

I recommend the thesis for defense; in view of the shortcomings noted above, I grade it with an E. Topics for thesis defence:

1. Could the author mathematically describe the models of discrete and continuous compound interest? And further, could she compare the two models, ideally using some real-world examples?