Sequencings in Semidirect Products via the Polynomial Method

The partial sums of a sequence ${\mathbf x} = x_1, x_2, \ldots, x_k$ of distinct non-identity elements of a group $(G,\cdot)$ are $s_0 = id_G$ and $s_j = \prod_{i=1}^j x_i$ for $0 < j \leq k$. If the partial sums are all different then ${\mathbf x}$ is a linear sequencing and if the partial sums are all different when $|i-j| \leq t$ then ${\mathbf x}$ is a $t$-weak sequencing. We investigate these notions of sequenceability in semidirect products using the polynomial method. We show that every subset of order $k$ of the non-identity elements of the dihedral group of order $2m$ has a linear sequencing when $k \leq 12$ and either $m>3$ is prime or every prime factor of $m$ is larger than $k!$, unless $s_k$ is unavoidably the identity; that every subset of order $k$ of a non-abelian group of order three times a prime has a linear sequencing when $5 < k \leq 10$, unless $s_k$ is unavoidably the identity; and that if the order of a group is $pe$ then all sufficiently large subsets of the non-identity elements are $t$-weakly sequenceable when $p>3$ is prime, $e \leq 3$ and $t \leq 6$.Comment: arXiv admin note: text overlap with arXiv:2203.1665

Two-Terraceable Groups

Given a sequence ${\bf g}: g_0,\ldots, g_{m}$, in a finite group $G$ with $g_0=1_G$, let ${\bf \bar g}: \bar g_0,\ldots, \bar g_{m}$, be the sequence defined by $\bar g_0=1_G$ and $\bar g_i=g_{i-1}^{-1}g_i$ for $1\leq i \leq m$. We say that $G$ is $k$-terraceable, if there exists a sequence ${\bf g}$ in $G$ such that every element of $G$ appears exactly $k$ times in each of ${\bf g}$ and ${\bf \bar g}$. We study the 2-terraceability conjecture which states that every group is 2-terraceable. We give two proofs of $k$-terraceability of abelian groups for all $k\geq 2$. Moreover, we give examples of classes of 2-terraceable non-abelian groups

Biophysical studies of catalytic and starch binding domains of wild-type and mutant glucoamylases from Aspergillus awamori

Aspergillus awamori catalytic and starch binding domains were prepared, purified and characterized. Sedimentation equilibrium experiments and MALDI-TOF spectra reveals that GA470 and SBD contains ~12% and 5% carbohydrate, respectively. Temperature denaturation experiments indicate that catalytic domain unfolds irreversibly whereas that of SBD is reversible with DeltaH of ~410 and 71 kcal/mol, respectively, and T m of 60.9° and 51.3°C, respectively. Analysis of scan rate dependency data suggest that the thermal unfolding of GA470 was partially under kinetic control and do not follow a single two-step model. Comparison of Tm of isolated and intact GA470 and SBD indicates that catalytic domain/linker thermally stabilizes the binding domain by at least 5°C. Solvent accessible surface area calculation along with experimentally determined DeltaH values indicate that glycosylation effect DeltaH as much as 59 kcal/mol for GA470 and 13 kcal/mol for SBD;The three dimensional structures of thermostable glucoamylase mutants, N20C/A27C, N20C/A27C/G137A and N20C/A27C/S30P/G137A were determined at 2.5, 2.4 and 2.3 A resolution by X-ray crystallography. The engineered disulfide bond is right-handed with an average strain energy of 4 kcal/mol and unusual average dihedral angle, c\u272 = 93°. Crystallographic studies show that introduced proline at position 30 adopts a trans conformation, &phis; = -64°, and psi = 131°. We did observe a decrease in the thermal factors of the main chain atoms of proline at site 30 (12 A versus 20 A); however, thermal factors in and around 20--27 loop, G137A and elsewhere in mutant proteins were found to be essentially the same to that of wild-type enzyme. Crystallographic analysis also reveals that replaced Alanine 137 was accommodated within helix-4, supporting the empirical observation that Alanine 137 has high helix forming propensity. Enzymatic thermostability appears to be correlated with the relative accessibility of the substituted residues and a decrease in backbone entropy of unfolding by reducing backbone flexibility. Although, the overall structures of the mutants are very similar compared to wild-type, there are some local conformation differences as indicated from r.m.s. deviations values calculated within 6 A from the mutation site, which supports that the effects of combined mutation in N20C/A27C/S30P/G137A are cumulative

Among graphs, groups, and latin squares

A latin square of order n is an n × n array in which each row and each column contains each of the numbers {1, 2, . . . , n}. A k-plex in a latin square is a collection of entries which intersects each row and column k times and contains k copies of each symbol. This thesis studies the existence of k-plexes and approximations of k-plexes in latin squares, paying particular attention to latin squares which correspond to multiplication tables of groups. The most commonly studied class of k-plex is the 1-plex, better known as a transversal. Although many latin squares do not have transversals, Brualdi conjectured that every latin square has a near transversal—i.e. a collection of entries with distinct symbols which in- tersects all but one row and all but one column. Our first main result confirms Brualdi’s conjecture in the special case of group-based latin squares. Then, using a well-known equivalence between edge-colorings of complete bipartite graphs and latin squares, we introduce Hamilton 2-plexes. We conjecture that every latin square of order n ≥ 5 has a Hamilton 2-plex and provide a range of evidence for this conjecture. In particular, we confirm our conjecture computationally for n ≤ 8 and show that a suitable analogue of Hamilton 2-plexes always occur in n × n arrays with no symbol appearing more than n/√96 times. To study Hamilton 2-plexes in group-based latin squares, we generalize the notion of harmonious groups to what we call H2-harmonious groups. Our second main result classifies all H2-harmonious abelian groups. The last part of the thesis formalizes an idea which first appeared in a paper of Cameron and Wanless: a (k,l)-plex is a collection of entries which intersects each row and column k times and contains at most l copies of each symbol. We demonstrate the existence of (k, 4k)-plexes in all latin squares and (k, k + 1)-plexes in sufficiently large latin squares. We also find analogues of these theorems for Hamilton 2-plexes, including our third main result: every sufficiently large latin square has a Hamilton (2,3)-plex

ERROR CORRECTING CODE USING LATIN SQUARE

ABSTRACT:Digital data stored in computers or transmitted over computer networks are constantly subject to error due to the physical medium in which they are stored or transmitted. Error-correction codes are means of introducing redundancy in the data so that even if part of it is corrupted or completely lost, the original data can be recovered. Error correcting codes are used in modern technology to protect information from errors. Burst error correcting codes are needed in virtually uncountable applications. Such codes will be called complete burst error correcting codes. There are quite a few constructions for complete burst error correcting codes. This paper presents an error correcting code based on the concept and the theory of the Latin Squares, where it employ the characteristics of the orthogonal Latin Squares to correct the errors. That is not complete burst error correcting codes, since it can correct most burst pattern of length i „T n, but not all of them. However, if the number of uncorrectable patterns is sufficiently small, this code can be used in practice as a burst error correcting code

The Gordon game

In 1992, about 30 years after Gordon introduced group sequencings to construct row-complete Latin squares, John Isbell introduced the idea of competitive sequencing, the Gordon Game. Isbell investigated the Gordon Game and found solutions for groups of small order. The purpose of this thesis is to analyze the Gordon Game and develop a brute force method of determining solutions to the game for all groups of order 12 (up to isomorphism) as well as for abelian groups of order less than 20. The method used will be a depth first search program written in MATLAB. Consequently, group representation using matrices will be studied within the thesis --Document