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Start Over Coverage 417-428

1. Some notes on oscillation of two-dimensional system of difference equations

Creator:
Opluštil, Zdeněk
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
two-dimensional system, linear difference equation, and oscillatory solution
Language:
English
Description:
Oscillatory properties of solutions to the system of first-order linear difference equations ∆uk = qkvk ∆vk = −pkuk+1, are studied. It can be regarded as a discrete analogy of the linear Hamiltonian system of differential equations. We establish some new conditions, which provide oscillation of the considered system. Obtained results extend and improve, in certain sense, results presented in Opluštil (2011).
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

2. Strong separativity over exchange rings

Creator:
Chen, Huanyin
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
strong separativity, exchange ring, and regular ring
Language:
English
Description:
An exchange ring $R$ is strongly separative provided that for all finitely generated projective right $R$-modules $A$ and $B$, $A\oplus A\cong A \oplus B\Rightarrow A\cong B$. We prove that an exchange ring $R$ is strongly separative if and only if for any corner $S$ of $R$, $aS+bS=S$ implies that there exist $u,v\in S$ such that $au=bv$ and $Su+Sv=S$ if and only if for any corner $S$ of $R$, $aS+bS=S$ implies that there exists a right invertible matrix $\begin{pmatrix} a&b\\ *&* \end{pmatrix} \in M_2(S)$. The dual assertions are also proved.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public