# Numerical Modeling of Heat and Mass Transfer during Cryopreservation Using Interval Analysis
## Appendix A Directed Interval Arithmetic  
Let us consider a directed interval \(\overline{a}\), which can be defined as a set \(\mathbf{D}\) of all directed pairs of real numbers defined as \(\overline{a}=[a^{-},\ a^{+}]:=\{\overline{a}\in\ \mathbf{D}|\ a^{-}\,\ a^{+}\in \mathbf{R}\}\), where \(a^{-}\) and \(a^{+}\) denote the beginning and the end of the interval, respectively [62; 63]. The left or the right endpoint of interval \(\overline{a}\) can be denoted as \(a^{s},\;\;s\in\{+,\;-\}\), where \(s\) is a binary variable. This variable can be expressed as a product of two binary variables and is defined as:  
\[ \begin{array}{ccccc}+\;+\;=\;-\;-\;=\;+\\ +\;-\;=\;-\;+\;=\;-\end{array}  \tag{1}\]  
An interval is called proper if \(a^{-}<a^{+}\), improper if \(a^{-}>a^{+}\) and degenerate if \(a^{-}=a^{+}\). The set of all directed interval numbers can be described as \(\mathbf{D}=\mathbf{P}\cup\mathbf{I}\), where \(\mathbf{P}\) denotes a set of all directed proper intervals and \(\mathbf{I}\) denotes a set of all improper intervals. Additionally a subset \(\mathbf{Z}=\mathbf{Z}_{\mathbf{P}}\cup\mathbf{Z}_{\mathbf{I}}\in\mathbf{D}\) should be defined, where \(\mathbf{Z}_{\mathbf{P}}=\{\overline{a}\in\mathbf{P}|\,a^{-}\leq 0\leq a^{+}\}\) and \(\mathbf{Z}_{\mathbf{I}}=\{\overline{a}\in\mathbf{I}|\,a^{+}\leq 0\leq a^{-}\}\). For directed interval numbers, two binary variables are defined. The first of them is the direction variable:  
\[\tau(\overline{a})= \begin{cases}+,\;\;\;if\;\;a^{-}\leq a^{+}\\ -,\;\;\;if\;\;a^{-}>a^{+}\end{cases}  \tag{2}\]  
and the other one is the sign variable:  
\[\sigma(\overline{a})= \begin{cases}+,\;\;\;if\;\;a^{-}>0,\;a^{+}>0\\ -,\;\;\;if\;\;a^{-}<0,\;a^{+}<0\end{cases} ,\qquad\overline{a}\in\mathbf{D} \backslash\mathbf{Z} \tag{3}\]  
In the set of directed interval numbers, the basic mathematical operations for \(\overline{a}=[a^{-},\;\;a^{+}]\) and \(\overline{b}=[b^{-},\;b^{+}]\) belonging to \(\mathbf{D}\) can be defined as follows:  
\[\overline{a}+\overline{b}=\left[a^{-}+b^{-},\;a^{+}+b^{+}\right] \tag{4}\]  
\[\overline{a}-\overline{b}=\left[a^{-}-b^{+},a^{+}-b^{-}\right] \tag{5}\]  
\[\overline{a}\cdot\overline{b}=\left\{ \begin{array}{ll}\left[a^{-\sigma( \overline{b})}\cdot b^{-\sigma(\overline{a})},\;a^{\sigma(\overline{b})}\cdot b ^{\sigma(\overline{a})}\right]&\overline{a},\;\overline{b}\in\mathbf{D} \backslash\mathbf{Z}\\ \left[a^{\sigma(\overline{a})\tau(\overline{b})}\cdot b^{-\sigma(\overline{a })},\;a^{\sigma(\overline{a})\tau(\overline{b})}\cdot b^{\sigma(\overline{a})} \right],&\overline{a}\in\mathbf{D}\backslash\mathbf{Z},\;\overline{b}\in \mathbf{Z}\\ \left[a^{-\sigma(\overline{b})}\cdot b^{\sigma(\overline{b})\tau(\overline{a })},\;a^{\sigma(\overline{b})}\cdot b^{\sigma(\overline{b})\tau(\overline{a})} \right],&\overline{a}\in\mathbf{Z},\;\overline{b}\in\mathbf{D}\backslash \mathbf{Z}\\ \left[\min(a^{-}\cdot b^{+},\;a^{+}\cdot b^{-}),\;\max(a^{-}\cdot b^{-},\;a^{+ }\cdot b^{+})\right],&\overline{a},\;\overline{b}\in\mathbf{Z}_{\mathbf{P}} \\ \left[\max(a^{-}\cdot b^{-},\;a^{+}\cdot b^{+}),\;\min(a^{-}\cdot b^{+},\;a^{+ }\cdot b^{-})\right],&\overline{a},\;\overline{b}\in\mathbf{Z}_{\mathbf{I}} \\ 0,&\left(\overline{a}\in\mathbf{Z}_{\mathbf{P}},\;\overline{b}\in\mathbf{Z}_{ \mathbf{I}}\right)\cup\left(\overline{a}\in\mathbf{Z}_{\mathbf{I}},\;\overline {b}\in\mathbf{Z}_{\mathbf{P}}\right)\end{array} \right. \tag{6}\]  
and  
\[\overline{a}/\overline{b}=\left\{ \begin{array}{ll}\left[a^{-\sigma(\overline {b})}/b^{\sigma(\overline{a})},\;a^{\sigma(\overline{b})}/b^{-\sigma(\overline {a})}\right],&\overline{a},\;\overline{b}\in\mathbf{D}\backslash\mathbf{Z}\\ \left[a^{-\sigma(\overline{b})}/b^{-\sigma(\overline{b})\tau(\overline{a})},\; a^{\sigma(\overline{b})}/b^{-\sigma(\overline{b})\tau(\overline{b})}\right],& \overline{a}\in\mathbf{Z},\;\overline{b}\in\mathbf{D}\backslash\mathbf{Z} \end{array} \right.