Upper and Lower Bounds of Table Sums

For a group [Formula: see text], we produce upper and lower bounds for the sum of the entries of the Brauer character table of [Formula: see text] and the projective indecomposable character table of [Formula: see text]. When [Formula: see text] is a [Formula: see text]-separable group, we show that the sum of the entries in the table of Isaacs' partial characters is a real number, and we obtain upper and lower bounds for this sum.

Cameron-Liebler $k$-sets in $\mathrm{AG}(n,q)$

We study Cameron-Liebler $k$-sets in the affine geometry, so sets of $k$-spaces in $\mathrm{AG}(n,q)$. This generalizes research on Cameron-Liebler $k$-sets in the projective geometry $\mathrm{PG}(n,q)$. Note that in algebraic combinatorics, Cameron-Liebler $k$-sets of $\mathrm{AG}(n,q)$ correspond to certain equitable bipartitions of the association scheme of $k$-spaces in $\mathrm{AG}(n,q)$, while in the analysis of Boolean functions, they correspond to Boolean degree $1$ functions of $\mathrm{AG}(n,q)$. We define Cameron-Liebler $k$-sets in $\mathrm{AG}(n,q)$ by intersection properties with $k$-spreads and show the equivalence of several definitions. In particular, we investigate the relationship between Cameron-Liebler $k$-sets in $\mathrm{AG}(n,q)$ and $\mathrm{PG}(n,q)$. As a by-product, we calculate the character table of the association scheme of affine lines. Furthermore, we characterize the smallest examples of Cameron-Liebler $k$-sets. This paper focuses on $\mathrm{AG}(n,q)$ for $n > 3$, while the case for Cameron-Liebler line classes in $\mathrm{AG}(3,q)$ was already treated separately.

Accounts for the groups SL(2,U), U = 41 and 43

The circular retail for the groups (2,) where = 41 and 43 compute in this paper from the ordinary character table and the character table (ch.t.) of rational representations (r.rep.) for each group.

Cyclic partition for the groups of PSL(2,41) and PSL(2,43)

The ordinary character table and the character table (cha.ta.) of rational representations (ra.repr.) for projective special linear groups (2,41) and (2,43) find in this work to find the cyclic partition for each group

Integral cayley graphs over semi-dihedral groups

Classifying integral graphs is a hard problem that initiated by Harary and Schwenk in 1974. In this paper, with the help of character table, we treat the corresponding problem for Cayley graphs over the semi-dihedral group SD8n = ?a,b | a4n = b2 = 1; bab = a2n-1?, n ? 2. We present several necessary and sufficient conditions for the integrality of Cayley graphs over SD8n, we also obtain some simple sufficient conditions for the integrality of Cayley graphs over SD8n in terms of the Boolean algebra of hai. In particular, we give the sufficient conditions for the integrality of Cayley graphs over semi-dihedral groups SD2n (n?4) and SD8p for a prime p, from which we determine several infinite classes of integral Cayley graphs over SD2n and SD8p.

A Compression System for Unicode Files Using an Enhanced Lzw Method

Data compression plays a vital and pivotal role in the process of computing as it helps in space reduction occupied by a file as well as to reduce the time taken to access the file. This work relates to a method for compressing and decompressing a UTF-8 encoded stream of data pertaining to Lempel-Ziv-welch (LZW) method. It is worth to use an exclusive-purpose LZW compression scheme as many applications are utilizing Unicode text. The system of the present work comprises a compression module, configured to compress the Unicode data by creating the dictionary entries in Unicode format. This is accomplished with adaptive characteristic data compression tables built upon the data to be compressed reflecting the characteristics of the most recent input data. The decompression module is configured to decompress the compressed file with the help of unique Unicode character table obtained from the compression module and the encoded output. We can have remarkable gain in compression, wherein the knowledge that we gather from the source is used to explore the decompression process.