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Start Over Coverage 253-268

1. Jestliže a sestry její: Obraz jedné spojky v korpusu SYN2000

Creator:
Šulc, Michal
Format:
print
Type:
model:internalpart and TEXT
Language:
Czech and English
Description:
In the article a comparison is made between the lexicographical description of "jestliže“ - as presented in Czech dictionaries of the second half of the 20th century - and a linguistic description of "jestliže“ that we can compile ourself on the basis of the 100-milion Czech National Corpus SYN2000. The focus is on collocations and on their linguistic interpretations.
Rights:
http://creativecommons.org/licenses/by-nc-sa/4.0/ and policy:public

2. On Hong’s conjecture for power LCM matrices

Creator:
Cao, Wei
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
gcd-closed set, greatest-type divisor(GTD), maximal gcd-fixed set(MGFS), east common multiple matrix, power LCM matrix, and nonsingularity
Language:
English
Description:
A set $\mathcal{S}=\lbrace x_1,\ldots ,x_n\rbrace $ of $n$ distinct positive integers is said to be gcd-closed if $(x_{i},x_{j})\in \mathcal{S}$ for all $1\le i,j\le n $. Shaofang Hong conjectured in 2002 that for a given positive integer $t$ there is a positive integer $k(t)$ depending only on $t$, such that if $n\le k(t)$, then the power LCM matrix $([x_i,x_j]^t)$ defined on any gcd-closed set $\mathcal{S}=\lbrace x_1,\ldots ,x_n\rbrace $ is nonsingular, but for $n\ge k(t)+1$, there exists a gcd-closed set $\mathcal{S}=\lbrace x_1,\ldots ,x_n\rbrace $ such that the power LCM matrix $([x_i,x_j]^t)$ on $\mathcal{S}$ is singular. In 1996, Hong proved $k(1)=7$ and noted $k(t)\ge 7$ for all $t\ge 2$. This paper develops Hong’s method and provides a new idea to calculate the determinant of the LCM matrix on a gcd-closed set and proves that $k(t)\ge 8$ for all $t\ge 2$. We further prove that $k(t)\ge 9$ iff a special Diophantine equation, which we call the LCM equation, has no $t$-th power solution and conjecture that $k(t)=8$ for all $t\ge 2$, namely, the LCM equation has $t$-th power solution for all $t\ge 2$.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public