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 Huseyin Cakalli Mathematics , 2011, Abstract: A sequence $(x_{n})$ of points in a topological group is called $\Delta$-quasi-slowly oscillating if $(\Delta x_{n})$ is quasi-slowly oscillating, and is called quasi-slowly oscillating if $(\Delta x_{n})$ is slowly oscillating. A function $f$ defined on a subset of a topological group is quasi-slowly (respectively, $\Delta$-quasi-slowly) oscillating continuous if it preserves quasi-slowly (respectively, $\Delta$-quasi-slowly) oscillating sequences, i.e. $(f(x_{n}))$ is quasi-slowly (respectively, $\Delta$-quasi-slowly) oscillating whenever $(x_{n})$ is. We study these kinds of continuities, and investigate relations with statistical continuity, lacunary statistical continuity, and some other types of continuities in metrizable topological groups.
 Mathematics , 2013, Abstract: A double sequence $\textbf{x}=\{x_{k,l}\}$ of points in $\textbf{R}$ is slowly oscillating if for any given $\varepsilon>0$, there exist $\alpha=\alpha(\varepsilon)>0$, $\delta=\delta (\varepsilon) >0$, and $N=N(\varepsilon)$ such that $|x_{k,l}-x_{s,t}|<\varepsilon$ whenever $k,l\geq N(\varepsilon)$ and $k\leq s \leq (1+\alpha)k$, $l\leq t \leq (1+\delta)l$. We study continuity type properties of factorable double functions defined on a double subset $A\times A$ of $\textbf{R}^{2}$ into $\textbf{R}$, and obtain interesting results related to uniform continuity, sequential continuity, and a newly introduced type of continuity of factorable double functions defined on a double subset $A\times A$ of $\textbf{R}^{2}$ into $\textbf{R}$.
 Mathematics , 2010, Abstract: Suppose $\alpha$ is an orientation preserving diffeomorphism (shift) of $\mR_+=(0,\infty)$ onto itself with the only fixed points $0$ and $\infty$. We establish sufficient conditions for the Fredholmness of the singular integral operator $(aI-bW_\alpha)P_++(cI-dW_\alpha)P_-$ acting on $L^p(\mR_+)$ with $1  Mathematics , 2010, Abstract: Suppose$\alpha$is an orientation-preserving diffeomorphism (shift) of$\mR_+=(0,\infty)$onto itself with the only fixed points$0$and$\infty$. In \cite{KKLsufficiency} we found sufficient conditions for the Fredholmness of the singular integral operator with shift $(aI-bW_\alpha)P_++(cI-dW_\alpha)P_-$ acting on$L^p(\mR_+)$with$1
 Mathematics , 2014, Abstract: We study Mellin pseudodifferential operators (shortly, Mellin PDO's) with symbols in the algebra $\widetilde{\mathcal{E}}(\mathbb{R}_+,V(\mathbb{R}))$ of slowly oscillating functions of limited smoothness introduced in \cite{K09}. We show that if $\mathfrak{a}\in\widetilde{\mathcal{E}}(\mathbb{R}_+,V(\mathbb{R}))$ does not degenerate on the "boundary" of $\mathbb{R}_+\times\mathbb{R}$ in a certain sense, then the Mellin PDO ${\rm Op}(\mathfrak{a})$ is Fredholm on the space $L^p$ for $p\in(1,\infty)$ and each its regularizer is of the form ${\rm Op}(\mathfrak{b})+K$ where $K$ is a compact operator on $L^p$ and $\mathfrak{b}$ is a certain explicitly constructed function in the same algebra $\widetilde{\mathcal{E}}(\mathbb{R}_+,V(\mathbb{R}))$ such that $\mathfrak{b}=1/\mathfrak{a}$ on the "boundary" of $\mathbb{R}_+\times\mathbb{R}$. This result complements a known Fredholm criterion from \cite{K09} for Mellin PDO's with symbols in the closure of $\widetilde{\mathcal{E}}(\mathbb{R}_+,V(\mathbb{R}))$.