This paper employs the symmetric GARCH and the asymmetric GJR-GARCH(1,1) and E-GARCH(1,1) models to explain the dynamics of the PLN/EUR and PLN/USD exchange rates in Poland for the periods of January 2015 to July 2022. The result of our study shows that the USD rate is more susceptible to market fluctuations and events than the EUR rate. Additionally, both rates' volatility persists after a market crisis for a while, with the EUR rate taking longer until volatility subsides. Using the Akaike information criterion and Bayesian information criterion, we find the E-GARCH model to be the best model out of all three models considered.

In this paper, we establish general differential summation formulas for integration by parts (IBP), more importantly a powerful tool that promotes exploration and creativity.

In the literature, we can find various methods for generating artistic patterns. One of the methods is the orbit trap method. In this paper, we propose various modifications of a variant of the orbit trap method that generates patterns with wallpaper symmetry. The first modification relies on replacing the Picard iteration (used in the original method) with the S-iteration known from the fixed point theory. Moreover, we extend the parameters in the S-iteration from scalar to vector ones. In the second modification, we replace the Euclidean metric used in the orbit traps with other metrics. Finally, we propose three new orbit traps. The presented examples show that using the proposed method, we are able to obtain a great variety of interesting patterns. Moreover, we show that a proper selection of the orbit traps and the mapping used by the method can lead to patterns that possess a local fractal structure.