Lin introduced the partition function $\text {PDO}_t(n)$, which counts the total number of tagged parts over all the partitions of n with designated summands in which all parts are odd. Lin also proved some congruences modulo 3 and 9 for $\text {PDO}_t(n)$, and conjectured certain congruences modulo $3^{k+2}$ for $k\geq 0$. He proved the conjecture for $k=0$ and $k=1$ [‘The number of tagged parts over the partitions with designated summands’, J. Number Theory 184 (2018), 216–234]. We prove the conjecture for $k=2$. We also study the lacunarity of $\text {PDO}_t(n)$ modulo arbitrary powers of 2 and 3. Using nilpotency of Hecke operators, we prove that there exists an infinite family of congruences modulo any power of 2 satisfied by $\text {PDO}_t(n)$.

Biomimetic hierarchical surface structures that exhibit features having multiple length scales have been used in many technological and engineering applications. Their surface topographies are most commonly analyzed using scanning electron microscopy (SEM), which only allows for qualitative visual assessments. Here we introduce fractal and lacunarity analyses as a method of characterizing the SEM images of hierarchical surface structures in a quantitative manner. Taking femtosecond laser-irradiated metals as an example, our results illustrate that, while the fractal dimension is a poor descriptor of surface complexity, lacunarity analysis can successfully quantify the spatial texture of an SEM image; this, in turn, provides a convenient means of reporting changes in surface topography with respect to changes in processing parameters. Furthermore, lacunarity plots are shown to be sensitive to the different length scales present within a hierarchical structure due to the reversal of lacunarity trends at specific magnifications where new features become resolvable. Finally, we have established a consistent method of detecting pattern sizes in an image from the oscillation of lacunarity plots. Therefore, we promote the adoption of lacunarity analysis as a powerful tool for quantitative characterization of, but not limited to, multi-scale hierarchical surface topographies.

Recent studies have shown that ghrelin increases pancreatic exocrine secretion. However, the potential effects of ghrelin on the morphology of exocrine pancreas (EP) remain unknown. In this work, using fractal analysis, we demonstrate that centrally administered ghrelin increases structural complexity and tissue disorder in rat EP. The study was carried out on a total of 40 male Wistar rats divided into four groups (n = 10): ghrelin-treated animals (average age, 1.5 months), ghrelin-treated animals (8.5 months), and controls (1.5 and 8.5 months). The pancreas tissue sections were stained with hematoxylin/eosin and visualized by light microscopy. For each animal, the average values of tissue fractal dimension, lacunarity, as well as parameters of co-occurrence matrix texture, were determined using tissue digital micrographs. The results indicate that ghrelin administration increases EP fractal dimension and textural entropy, and decreases lacunarity, regardless of the age. To our knowledge, this is the first study to investigate the effects of ghrelin on the morphological properties of pancreatic tissue, and also the first to apply fractal and textural analysis methods in quantification of EP tissue architecture.