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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.

Hairtail fish samples were stored at different freezing temperatures of −5 °C, −20 °C, −40 °C and −80 °C. To establish an effective quality evaluation approach for hairtail samples during frozen storage, fractal dimension was used to observe the porous microstructure that resulted from the ice crystal formation in frozen hairtail meat. The results showed that the initial value of fractal dimension of all the samples was 1.968. After frozen storage, the fractal dimension of samples at −5 °C, −20 °C, −40 °C and −80 °C decreased to 1.539, 1.682, 1.856 and 1.896, respectively. Traditional quality indicators such as water activity, color and textural properties (i.e., hardness, springiness and chewiness) also exhibited a similar decreasing trend, and the rate of change decreased with a decrease in storage temperature. The relationships were analyzed, and these traditional quality indicators were correlated with the fractal dimension with determined correlation coefficients within ±0.900. Moreover, based on the fractal dimension model, the first-order kinetic equation of fractal dimension variation during storage was $${{\bf{d}}}_{{\bf{t}}}={\bf{1.968}}\,{{\bf{e}}}^{{\boldsymbol{-}}{\bf{0.928}}{\bf{t}}{{\bf{e}}}^{{\boldsymbol{-}}{\bf{1431.396}}/{\bf{T}}}}$$ d t = 1.968 e − 0.928 t e − 1431.396 / T , which can be used to predict the shelf life of preserved hairtails at different storage temperatures. The results demonstrated fractal dimension was a novel and feasible method to evaluate the quality of hairtails in frozen storage.

In this paper we present a precise description of the connectedness locus of the family of polynomials (z2 + x)2 + y, where x, y are real numbers.

In this paper, we prove that the attractor of C 1,α bi-Lipschitz IFS in ℝ is uniformly perfect if it is not a singleton. Then we construct an example to show that this does not hold for C 1 bi-Lipschitz IFS in ℝn.

Let f i (x) = A i x + b i (1 < i < n) be affine maps of Euclidean space R N with each A i nonsingular and each f i contractive. We prove that the self-affine set K of {f 1 ,..., f n } is uniformly perfect if it is not a singleton.

Let $f:\hat \mathbf C\rightarrow \hat \mathbf C$ be a rational map of degree $n\geq 3$ and with exactly two critical points. Assume that the Julia set $J(f)$ is a proper subcontinuum of $\hat \mathbf C$ and there is no completely invariant Fatou component under the iterates $f^{2}$. It is shown that if there is no buried points in $J(f)$, then the Julia set $J(f)$ is a Lakes of Wada continuum, and hence is either an indecomposable continuum or the union of two indecomposable continua.

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.

For N-shaped mapping g: [0,1] → [0,1] given by g(x) = 3x, if 0 ≤ x ≤ 1/3, g (x) = 2 - 3x, if 1/3 ≤ x ≤ 2/3 and g (x) = 3x - 2 otherwise, there exists a chaotic set of g in the sense of Xiong, which is stronger than Li-Yorke’s, with Hausdorff dimension one everywhere. On the other hand, it is shown that any chaotic set of g is of zero one-dimensional Hausdorff measure.