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Hairtail fish samples were frozen at −3℃, −18℃, −40℃, and −80℃, respectively, followed by different numbers of freeze–thaw cycle treatment. In order to evaluate quality changes associated with microstructure and biochemical changes of hairtail samples, fractal dimension was applied to observe porous microstructure due to the formation of ice crystals in frozen hairtail meat, as well as selected quality parameters were determined. Results showed that fractal dimension presented a decreasing trend with increasing numbers of freeze–thaw cycles from the initial value of 1.9679 to 1.3867, 1.5652, 1.5955, and 1.7445 for −3℃, −18℃, −40℃, and −80℃ freezing temperatures, respectively. Increased numbers of freeze–thaw cycles were accompanied by decrease in water holding capacity, increase in microbiological counts, and a greater degree of protein and lipid oxidation, as evidenced by higher contents of total volatile basic nitrogen and the thiobarbituric acid-reactive substances value. The rates of these parameters decreased with the decrease in freezing temperature. Besides, the relationships were analysed between fractal dimension, and these traditional quality indicators and the determined correlations were within ±0.90. The results demonstrated that fractal dimension proved to be a novel and feasible method to evaluate the quality of hairtails in freezing–thawing processes.

Let $f_i(x)=A_ix+b_i (1\le i\le n)$ be affine maps of Euclidean space $\mathbb{R}^N$ with each $A_i$ nonsingular and each $f_i$ contractive. We prove that the self-affine set $K$ of $\{f_1,\dots, f_n\}$ is uniformly perfect if it is not a singleton. [ABSTRACT FROM AUTHOR]

In this paper we show that the Julia set $J(G)$ of a finitely generated rational semigroup $G$ is connected if the union of the Julia sets of generators is contained in a subcontinuum of $J(G)$. Under a nonseparating condition, we prove that the Julia set of a finitely generated polynomial semigroup is connected if its postcritical set is bounded. [ABSTRACT FROM AUTHOR]