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

A New Proof of a Classical Theorem in Design Theory

AbstractWe present a new proof of the well known theorem on the existence of signed (integral) t-designs due to Wilson and Graver and Jurkat

Latin bitrades derived from groups

A latin bitrade is a pair of partial latin squares which are disjoint, occupy the same set of non-empty cells, and whose corresponding rows and columns contain the same set of entries. Dr\'apal (\cite{Dr9}) showed that a latin bitrade is equivalent to three derangements whose product is the identity and whose cycles pairwise have at most one point in common. By letting a group act on itself by right translation, we show how some latin bitrades may be derived from groups without specifying an independent group action. Properties of latin trades such as homogeneousness, minimality (via thinness) and orthogonality may also be encoded succinctly within the group structure. We apply the construction to some well-known groups, constructing previously unknown latin bitrades. In particular, we show the existence of minimal, $k$-homogeneous latin trades for each odd $k\geq 3$. In some cases these are the smallest known such examples.Comment: 23 page

The study of production Artemia enrichment liquid SELCO1 &SUPER2 SELCO with internal capacities

Artemia as a live food has multiple applications in aquaculture. Artemia contains few unsaturated 3-omega fatty acids particularly eicozapentanoeic acid (EPA) and has no 6-omega fatty acids particularly decozahexanoeic acid (DHA), so Artemia naplious is enriched to improve its food values. The most famous enrichment emulsion are selco and super-selco made by Euro-American INVE Company. This study was performed to make Artemia enrichment emulsions by internal potentials. At first, the final composition of Artemia enrichment emulsion (selco) was determined in Urmia university chemical analysis laboratory. Then, aquatic fishing resources in south of the country such as eye oil of tuna fish, shark liver, cuttlefish and plant oils of sunflower, olive and beef oil were used. Fatty acids profiles were analyzed by Gas Chromatography (GC). The results showed that cuttlefish may produce 13± 3 % wet weight of fatty acids. We made 3 enrichment oils which contained 60± 10 % of plant and animal oils. These suspensions fatty acids were analyzed and the results were compared with control sample. Field tests were performed on 200000 Artemia urmiana naplious by enriching the Artemia with enrichment emulsions and the products were analyzed by GC. The results indicated that the rate of emulsions absorbance in imported and internal samples were 37.47, 25.30, 18.88, 22.14 and 10.32 respectively. In the next stage, enriched Artemia naplious were fed as live food to 500 trout larvae in Ziveh Aquaculture Company as follows: Treatment 1- Control consisted of new feeding larvae fed by concentrated food Treatment 2- new feeding larvae fed by concentrated food plus un-enriched naplious Treatment 3- new feeding larvae fed by concentrated food plus selco oil enriched naplious Treatment 4- new feeding larvae fed by concentrated food plus emulsion-1 oil enriched naplious Treatment 5- new feeding larvae fed by concentrated food plus emulsion-2 oil enriched naplious Treatment 6- new feeding larvae fed by concentrated food plus emulsion-3 oil enriched naplious The results indicated that treatments 1, 2 had significant difference with treatments 3, 4, 5, 6 from survival rate, growth coefficient, obesity coefficient, total length, food conversion coefficient, final weight and protein percent. Abnormalities rates in treatments 1, 2 had significant difference with treatments 3, 4, 5 , 6 in which enriched emulsions were not used, but these indices had no significant difference with commercial samples which shows internal made emulsions can easily be used. The data were analyzed by one-way analyses variance and Duncan test in SPSS and EXCELL softwares. In conclusion, we can make enrichment selco oils in the country by internal potentials which the foreign samples can be replaced by them

Commutative association schemes

Association schemes were originally introduced by Bose and his co-workers in the design of statistical experiments. Since that point of inception, the concept has proved useful in the study of group actions, in algebraic graph theory, in algebraic coding theory, and in areas as far afield as knot theory and numerical integration. This branch of the theory, viewed in this collection of surveys as the "commutative case," has seen significant activity in the last few decades. The goal of the present survey is to discuss the most important new developments in several directions, including Gelfand pairs, cometric association schemes, Delsarte Theory, spin models and the semidefinite programming technique. The narrative follows a thread through this list of topics, this being the contrast between combinatorial symmetry and group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes (based on group actions) and its connection to the Terwilliger algebra (based on combinatorial symmetry). We propose this new role of the Terwilliger algebra in Delsarte Theory as a central topic for future work.Comment: 36 page