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 pedagogical introduction to the Slavnov formulation of quantum Yang-Mills theory

Over the last few years, Slavnov has proposed a formulation of quantum Yang-Mills theory in the Coulomb gauge which preserves simultaneously manifest Lorentz invariance and gauge invariance of the ghost field Lagrangian. This paper presents in detail some of the necessary calculations, i.e. those dealing with the functional integral for the S-matrix and its invariance under shifted gauge transformations. The extension of this formalism to quantum gravity in the Prentki gauge deserves consideration.Comment: 16 pages, review pape

A Facile Method to Synthesize 3D Pomegranate-like Polydopamine Microspheres

Nanospheres have found versatile applications in the biomedical field; however, their possible harmful effects on immune and inflammatory systems are also a crucial concern. Inspired by a pomegranate structure, we demonstrated a novel structure for the nanostructured microspheres to overcome the drawbacks of nanospheres without compromising their merits. In this study, 3D pomegranate-like polydopamine microspheres (PDAMS) were synthesized by self-oxidative polymerization of dopamine hydrochloride. Herein, controlling the pH during polymerization led to synthesizing homogeneous agglomerated nano-sized spheres (400–2000 nm) and finally forming tunable and monodisperse micron-sized particles (21 µm) with uniform spherical shape porous microstructure. PDAMS interaction with the potential targets, Bone morphogenetic protein-2 (BMP2), Decorin, and Matrilin-1, was investigated via molecular calculations. Theoretical energy analysis revealed that PDAMS interaction with BMP2, Decorin, and Matrilin-1 is spontaneous, so that a protein layer formation on the PDAMS surface suggests application in bone and cartilage repair. It was also observed that PDAMS presented in-vitro degradation within 4 weeks. Here, disappearance of the UV-VIS spectrum peak at 280 nm is accompanied by the degradation of catechol groups. Pomegranate-like PDAMS support the biomimetic formation of hydroxyapatite-like layers, making them appropriate candidates for hard tissue applications. Herein, the appearance of peaks in XRD spectrum at 31.37, 39.57, 45.21, and 50.13° attributed to hydroxyapatite-like layers formation. All these results demonstrated that self-oxidative polymerization under a controllable pH can be a green and straightforward technique for preparing the pomegranate-like PDAMS and providing an innovative basis for further pre-clinical and clinical investigations

Fillet quality grading of common carp (Cyprinus carpio) by fish size using mathematical equations

Quality grading and nutrition value determination of fish are nowadays necessary for aquatic processing. The present study was conducted to investigate a new, applied and cheap method of fillet quality grading for common carp (Cyprinus carpio) using fish size, based on mathematical equations. For this reason, 61 specimens of cultured market size carp were used and after filleting, their moisture, lipid, protein, energy and ash content were measured. Then, the relationships between proximate composition components of fillet and fillet length were studied. Results showed that there is an inverse linear regression relationship between logarithm of moisture content and logarithm of fish length (P0.05). Regarding the relationship between proximate composition of cultured carp fillets and fillet length machine fish sorting based on fish length and grading of moisture, lipid and energy content is feasible. We also found it feasible to determine type of processing and proximate composition of the prepared common carp