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Register a new userA note on the rings of functions which are discontinuous on some finite sets
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In this paper, we study some properties of the ring $C(X)_F$ of all real valued functions which are continuous except on some finite subsets of $X$. We show that $C(X)_F$ is closed under uniform limit if and only if the set of all non-isolated points of $X$ is finite. We also initiate and investigate the zero divisor graph of the ring $C(X)_F$.
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For any ideal $mathcal{P}$ of closed sets in $X$, let $C_mathcal{P}(X)$ be the family of those functions in $C(X)$ whose support lie on $mathcal{P}$. Further let $C^mathcal{P}_infty(X)$ contain precisely those functions $f$ in $C(X)$ for which for each $epsilon >0, {xin X: lvert f(x)rvertgeq epsilon}$ is a member of $mathcal{P}$. Let $upsilon_{C_{mathcal{P}}}X$ stand for the set of all those points $p$ in $beta X$ at which the stone extension $f^*$ for each $f$ in $C_mathcal{P}(X)$ is real valued. We show that each realcompact space lying between $X$ and $beta X$ is of the form $upsilon_{C_mathcal{P}}X$ if and only if $X$ is pseudocompact. We find out conditions under which an arbitrary product of spaces of the form locally-$mathcal{P}/$ almost locally-$mathcal{P}$, becomes a space of the same form. We further show that $C_mathcal{P}(X)$ is a free ideal ( essential ideal ) of $C(X)$ if and only if $C^mathcal{P}_infty(X)$ is a free ideal ( respectively essential ideal ) of $C^*(X)+C^mathcal{P}_infty(X)$ when and only when $X$ is locally-$mathcal{P}$ ( almost locally-$mathcal{P}$). We address the problem, when does $C_mathcal{P}(X)/C^mathcal{P}_{infty}(X)$ become identical to the socle of the ring $C(X)$. Finally we observe that the ideals of the form $C_mathcal{P}(X)$ of $C(X)$ are no other than the $z^circ$-ideals of $C(X)$.
Hurewicz proved completely metrizable Menger spaces are /sigma-compact. We extend this to Cech-complete Menger spaces and consistently to projective Menger metrizable spaces. On the other hand, it is consistent that there is a co-analytic Menger space that is not /sigma-compact.
We consider sets of positive integers containing no sum of two elements in the set and also no product of two elements. We show that the upper density of such a set is strictly smaller than 1/2 and that this is best possible. Further, we also find the maximal order for the density of such sets that are also periodic modulo some positive integer.
The genus graphs have been studied by many authors, but just a few results concerning in special cases: Planar, Toroidal, Complete, Bipartite and Cartesian Product of Bipartite. We present here a derive general lower bound for the genus of a abelian Cayley graph and construct a family of circulant graphs which reach this bound.
We analyze Mode Coupling discontinuous transition in the limit of vanishing discontinuity, approaching the so called $A_3$ point. In these conditions structural relaxation and fluctuations appear to have universal form independent from the details of the system. The analysis of this limiting case suggests new ways for looking at the Mode Coupling equations in the general case.
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