Storage stability of encapsulated barberry's anthocyanin and its application in jelly formulation

The barberry (Berberis vulgaris) extract which is a rich source of anthocyanin was used for encapsulation with three different wall materials i.e., combination of gum Arabic and maltodextrin (GA+MD), combination of maltodextrin and gelatin (MD+GE) and maltodextrin (MD) by spray drying process. In this context, the storage stability of encapsulated pigments was investigated under four storage temperatures (4, 25, 35 and 42 °C), four relative humidities (20, 30, 40 and 50%) and light illumination until 90 days. All wall materials largely increased the half-life of the encapsulated pigments during storage compared with non-encapsulated anthocyanins. MD+GA showed the highest encapsulation efficiency, lower degradation rate in all temperatures and was found as the most effective wall material in stabilizing the pigments. The encapsulated pigments were utilized in coloring jelly powder as an alternative of synthetic color. Sensory evaluation were run to identify best encapsulated natural color concentration in jelly powder formulation according to acceptability by consumers. A jelly with added 7% encapsulated color had higher scores than the commercial jelly containing synthetic color for all the sensory attributes evaluated. Physicochemical properties of produced jelly including moisture content, hygroscopicity, acidity, ash content and texture were not significantly different with control sample while, syneresis and solubility of the samples prepared with encapsulated color was significantly reduced. © 2016 Elsevier Ltd

Inclusion Matrices and Chains

Given integers $t$, $k$, and $v$ such that $0\leq t\leq k\leq v$, let $W_{tk}(v)$ be the inclusion matrix of $t$-subsets vs. $k$-subsets of a $v$-set. We modify slightly the concept of standard tableau to study the notion of rank of a finite set of positive integers which was introduced by Frankl. Utilizing this, a decomposition of the poset $2^{[v]}$ into symmetric skipless chains is given. Based on this decomposition, we construct an inclusion matrix, denoted by $W_{\bar{t}k}(v)$, which is row-equivalent to $W_{tk}(v)$. Its Smith normal form is determined. As applications, Wilson's diagonal form of $W_{tk}(v)$ is obtained as well as a new proof of the well known theorem on the necessary and sufficient conditions for existence of integral solutions of the system $W_{tk}\bf{x}=\bf{b}$ due to Wilson. Finally we present anotherinclusion matrix with similar properties to those of $W_{\bar{t}k}(v)$ which is in some way equivalent to $W_{tk}(v)$.Comment: Accepted for publication in Journal of Combinatorial Theory, Series

On the volumes and affine types of trades

A $[t]$-trade is a pair $T=(T_+, T_-)$ of disjoint collections of subsets (blocks) of a $v$-set $V$ such that for every $0\le i\le t$, any $i$-subset of $V$ is included in the same number of blocks of $T_+$ and of $T_-$. It follows that $|T_+| = |T_-|$ and this common value is called the volume of $T$. If we restrict all the blocks to have the same size, we obtain the classical $t$-trades as a special case of $[t]$-trades. It is known that the minimum volume of a nonempty $[t]$-trade is $2^t$. Simple $[t]$-trades (i.e., those with no repeated blocks) correspond to a Boolean function of degree at most $v-t-1$. From the characterization of Kasami--Tokura of such functions with small number of ones, it is known that any simple $[t]$-trade of volume at most $2\cdot2^t$ belongs to one of two affine types, called Type\,(A) and Type\,(B) where Type\,(A) $[t]$-trades are known to exist. By considering the affine rank, we prove that $[t]$-trades of Type\,(B) do not exist. Further, we derive the spectrum of volumes of simple trades up to $2.5\cdot 2^t$, extending the known result for volumes less than $2\cdot 2^t$. We also give a characterization of "small" $[t]$-trades for $t=1,2$. Finally, an algorithm to produce $[t]$-trades for specified $t$, $v$ is given. The result of the implementation of the algorithm for $t\le4$, $v\le7$ is reported.Comment: 30 pages, final version, to appear in Electron. J. Combi