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using universal formula for free abelian group and set theory of a group
I’m trying to understand (or learn from) the proof of Theorem 2 here:
It says that if we have a relation of the form $a=c\cdot(x-y)$ with $x,y\in\mathbb{R}$, then for any subgroup $H$ of the additive group of $\mathbb{R}$, the set $H-\{0\}$ has a unique (up to additive inverse) $c$-expansion if and only if $\{x,y\}$ is not a subset of $H$. I understand this to mean that any element of $H-\{0\}$ can be written uniquely as $u_1\cdot x+u_2\cdot y$ for $u_1,u_2\in\mathbb{Z}$.
I also understand that if we use the universal property for the free abelian group on $2$ elements, and the map $f:a\mapsto(a,0)$, then for any subgroup $H$ of $\mathbb{Z}^2$, the set $H-\{0\}$ has a unique (up to additive inverse) $c$-expansion if and only if $\{x,y\}$ is not a subset of $H$. However, I don’t understand the specific properties of this universal map, nor the universal property. In particular, how do we make sure $f$ satisfies the universal property for the free abelian group?
A:
If $A= \langle a \rangle \subseteq \mathbb{Z}^2$ is a subgroup then $\bar{A}= \langle a,0 \rangle$ is a subgroup of $\mathbb{Z}^
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ghostrecon2001crackdownload1. Field of the Invention
The present invention relates to a monolithic ceramic capacitor.
2. Description of the Related Art
As is well known in the art, a capacitor is a kind of electronic component which is widely used to condenser such as signal processing circuits and attenuating circuits of mobile phones and so forth.
In recent years, such capacitors tend to be reduced in size as the trend of the mobile phone becomes to be miniaturized. As a result, the size of the components such as ceramic capacitors tend to be reduced and thus those ceramic capacitors are required to be made thin.
Under the circumstances, a xe2x80x9cmonolithic ceramic capacitorxe2x80x9d is used in which dielectric layers each include a plurality of thinner dielectric layers. The monolithic ceramic capacitor may be made thin by virtue of the thinner dielectric layers.
However, the thinner dielectric layers have a small heat capacity and thus they tend to be decreased in thermal resistance. This will cause a problem that the thin dielectric layers can be damaged by a heat generated by a ceramic element.
In view of the above problems, it is an object of the present invention to provide a monolithic ceramic capacitor which is capable of stably protecting a thin dielectric layer thereof.
According to the present invention, a monolithic ceramic capacitor includes a ceramic element including a plurality of ceramic layers each having a relatively large thickness and a plurality of thinner ceramic layers each having a relatively small thickness. The monolithic ceramic capacitor also includes an internal electrode formed on the plurality of ceramic layers and an external electrode connected to one end of the ceramic element.
According to the present invention, a monolithic ceramic capacitor includes a ceramic element including a plurality of ceramic layers each having a relatively large thickness and a plurality of thinner ceramic layers each having a relatively small thickness. The monolithic ceramic capacitor also includes an internal electrode formed on the plurality of ceramic layers and an external electrode connected to one end of the ceramic element. In this monolithic ceramic capacitor, the ceramic element is made by laminating and firing a ceramic powder and a pressing agent including metal powder to form the ceramic layers and the external electrode is connected to the external electrode of the ceramic element.
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