text:
\section{Introduction}\label{sec1}

All our graphs are simple and undirected, and we allow uncountable graphs. We start by stating basic definitions. For graphs $F$, $G$ and $H$, we write $F \to (G,H)$ if every red-blue coloring of the edges of $F$ produces a red copy of $G$ or a blue copy of $H$. A red-blue coloring of $F$ is \emph{$(G,H)$-good} if it produces neither a red copy of $G$ nor a blue copy of $H$. If $F \to (G,H)$ and every subgraph $F'$ of $F$ is such that $F' \not\to (G,H)$, then $F$ is \emph{$(G,H)$-minimal}. The collection of all $(G,H)$-minimal graphs is denoted by $\mathcal{R}(G,H)$. A pair $(G,H)$ \emph{admits a Ramsey-minimal graph} if $\mathcal{R}(G,H)$ is nonempty.

If $G$ and $H$ are both finite, then a $(G,H)$-minimal graph exists. Indeed, we can delete finitely many vertices and/or edges of $K_{r(G,H)}$ until it is $(G,H)$-minimal. This observation does not necessarily hold when at least one of $G$ and $H$ is infinite, even though there exists a graph $F$ such that $F \to (G,H)$ in the countable case by the Infinite Ramsey Theorem \cite{ramsey1930} and in general by the Erd\H{o}s--Rado Theorem \cite{erdos1956}. In fact, a pair of countably infinite graphs almost never admits a Ramsey-minimal graph---see Proposition \ref{prop3}. In 2021, Barrett and the second author \cite{barrett2021} introduced a characterization problem for pairs of graphs according to whether or not they admit a minimal graph.

\begin{mainproblem}[\cite{barrett2021}]
Determine which pairs $(G,H)$ admit a Ramsey-minimal graph and which ones do not.
\end{mainproblem}

The primary motivation for posing the main problem is the classic problem of determining whether there are finitely or infinitely many $(G,H)$-minimal graphs. This problem was first introduced in 1976 \cite{burr1976,nesetril1976}, and it was studied for finite graphs in general by Ne{\v{s}}et{\v{r}}il and R{\"o}dl \cite{nesetril1978a,nesetril1978b} and for various classes of graphs by Burr et al.\ \cite{burr1978a,burr1980,burr1981a,burr1982,burr1985}. A result by Burr et al.\ on Ramsey-minimal graphs for finite star forests is relevant to our discussion.

\begin{theorem}[\cite{burr1981b}]\label{burr-et-al}
The pair of star forests $(\bigcup_{i = 1}^s S_{n_i}, \bigcup_{j = 1}^t S_{m_j})$ admits infinitely many Ramsey-minimal graphs for $n_1 \ge \dots \ge n_s \ge 2$ and $m_1 \ge \dots \ge m_t \ge 2$ when $s \ge 2$ or $t \ge 2$.
\end{theorem}

The formulation of the main problem is also motivated by the more recent work of Stein \cite{stein2010,stein2011a,stein2011b} on extremal infinite graph theory. It is a subfield of extremal graph theory that developed after the notion of end degrees was introduced a few years prior \cite{bruhn2007,stein2007}.

Barrett and the second author mainly studied the main problem for pairs $(G,H)$ in general. The following is one of the main results presented in their paper.

\begin{theorem}[\cite{barrett2021}]\label{bar-vit}
Let $G$ and $H$ be graphs, and suppose that $\mathcal{F}$ is a (possibly infinite) collection of graphs such that:
\begin{enumerate}[\normalfont 1.]
    \item For all $F \in \mathcal{F}$, we have $F \to (G,H)$.
    \item For every graph $\Gamma$ with $\Gamma \to (G,H)$, there exists an $F \in \mathcal{F}$ that is contained in $\Gamma$.
\end{enumerate}
The following statements hold:
\begin{enumerate}[\normalfont(i)]
    \item If $F$ is a $(G,H)$-minimal graph, then $F \in \mathcal{F}$ and $F$ is non--self-embeddable.
    \item Suppose that any two different graphs $F_1,F_2 \in \mathcal{F}$ do not contain each other. A graph $F$ is $(G,H)$-minimal if and only if $F \in \mathcal{F}$ and $F$ is non--self-embeddable.
\end{enumerate}
\end{theorem}

This paper instead focuses on pairs $(G,H)$ involving a \emph{star forest}---a union of stars. Our first main result shows that any pair of star forests such that at least one of them involves an infinite-star component admits no Ramsey-minimal graph.

\begin{theorem}\label{mainthm1}
Let $G$ and $H$ be star forests. If at least one of $G$ and $H$ contains an infinite-star component, then no $(G,H)$-minimal graph exists.
\end{theorem}

This theorem is in contrast to Theorem \ref{burr-et-al}, which states that there are infinitely many $(G,H)$-minimal graphs when $G$ and $H$ are disconnected finite star forests with no single-edge components. Loosely speaking, the existence of infinitely many finite minimal graphs hence does not give an indication that a corresponding infinite minimal graph exists.

\begin{figure}
    \centering
    \begin{tikzpicture}[x=11mm,y=11mm]
        \draw[thick] (-1,0) -- (3,0);
        \draw[thick] (4,0) -- (8,0);
        \draw[thick] (1,0) -- (1,1);
        \draw[thick] (6,0) -- (6,1);
        \draw[thick,dashed] (3,0) -- (4,0);
        \draw[dashed] (-1,0) -- (-2,0);
        \draw[dashed] (8,0) -- (9,0);

        \draw[fill=black] (-1,0) circle (2pt);
        \draw[fill=black] (0,0) circle (2pt);
        \draw[fill=black] (1,0) circle (2pt);
        \draw[fill=black] (2,0) circle (2pt);
        \draw[fill=black] (3,0) circle (2pt);
        \draw[fill=black] (4,0) circle (2pt);
        \draw[fill=black] (5,0) circle (2pt);
        \draw[fill=black] (6,0) circle (2pt);
        \draw[fill=black] (7,0) circle (2pt);
        \draw[fill=black] (8,0) circle (2pt);
        \draw[fill=black] (1,1) circle (2pt);
        \draw[fill=black] (6,1) circle (2pt);

        \node at (1,-0.3) {$0$};
        \node at (2,-0.3) {$1$};
        \node at (3,-0.3) {$2$};
        \node at (4,-0.3) {$d-2$};
        \node at (5,-0.3) {$d-1$};
        \node at (6,-0.3) {$d$};
    \end{tikzpicture}

    \caption{The graph $F_d$.}
    \label{fig1}
\end{figure}

Similarly, the existence of only finitely many finite minimal graphs does not imply that there are only finitely many corresponding infinite minimal graphs. For $n \in \mathbb{N}$---where $\mathbb{N}$ is the set of positive integers---we denote the $n$-edge star by $S_n$. It is known from \cite{burr1978b} that there are only finitely many $(nS_1,H)$-minimal graphs for $n \in \mathbb{N}$ and $H$ a finite graph. On the other hand, if $\mathbb{Z}$ is the \emph{double ray}---the two-way infinite path---then $(2S_1,\mathbb{Z} \cup S_3)$ admits infinitely many minimal graphs. Indeed, we have $2F_d \in \mathcal{R}(2S_1,\mathbb{Z} \cup S_3)$ for every $d \ge 3$, where $F_d$ is the graph illustrated in Figure \ref{fig1}.

A graph is \emph{leafless} if it contains no vertex of degree one, and it is \emph{non--self-embeddable} if it is not isomorphic to any proper subgraph of itself. Following \cite[p.\ 79]{harary1969}, we denote the \emph{subdivision graph} of $G$ by $S(G)$, which is a graph obtained from $G$ by performing a subdivision on each one of its edges. For example, if $P_n$ denotes the $n$-vertex path, then $S(P_n)=P_{2n-1}$ for $n \in \mathbb{N}$.

For our second main result, we construct a Ramsey-minimal graph for a finite star forest versus the subdivision graph of a connected, leafless, non--self-embeddable graph. In 2020, subdivision graphs were used by Wijaya et al.\ \cite{wijaya2020} to construct new $(nS_1,P_4)$-minimal graphs.

\begin{theorem}\label{mainthm2}
Let $G$ be a connected, leafless, non--self-embeddable graph. For any finite star forest $H$, there exists a $(S(G),H)$-minimal graph.
\end{theorem}

For future investigation, it would be interesting to consider whether every pair of non--self-embeddable graphs admits a minimal graph. If true, this would generalize the observation that a pair of finite graphs always admits a minimal graph, since finite graphs are non--self-embeddable.

\begin{question}\label{ques1}
Is it true that every pair $(G,H)$ of non--self-embeddable graphs admits a Ramsey-minimal graph?
\end{question}

We give an outline of this paper. Section \ref{sec2} discusses self-embeddable graphs and their relevance to the study of Ramsey-minimal graphs. In Section \ref{sec3}, we briefly discuss the Ramsey-minimal properties of $(G,H)$ when $H$ is a union of graphs. Finally, our two main theorems are proved in Sections \ref{sec4} and \ref{sec5}.

\section{Self-embeddable graphs}\label{sec2}

We first provide several preliminary definitions. A \emph{graph homomorphism} $\varphi\colon G \to H$ is a map from $V(G)$ to $V(H)$ such that $\varphi(u)\varphi(v) \in E(H)$ whenever $uv \in E(G)$. A graph homomorphism is an \emph{embedding} if it is an injective map of vertices. Following \cite{bonato2003,bonato2006}, we write $G \le H$ if $G$ embeds into $H$; that is, there exists an embedding $\varphi\colon G \to H$. Unlike in \cite{bonato2003,bonato2006}, however, we do not require that the graph image of $\varphi$ is an induced subgraph of $H$.

A graph $G$ is \emph{self-embeddable} if $G \cong G'$ for some proper subgraph $G'$ of $G$, and the corresponding isomorphism $\varphi\colon G \to G'$ is its \emph{self-embedding}. Examples of self-embeddable graphs include the \emph{ray} $\mathbb{N}$---the one-way infinite path---and a complete graph on infinitely many vertices. On the other hand, finite graphs and the double ray $\mathbb{Z}$ are non--self-embeddable.

Proposition \ref{prop1} provides a necessary and sufficient condition for a graph to be self-embeddable in terms of its components. This proposition is quite similar to \cite[Theorem 2.5]{shekarriz2015} for \emph{self-contained graphs}, the ``induced'' version of self-embeddable graphs.

\begin{proposition}\label{prop1}
A graph $G$ is self-embeddable if and only if at least one of the following statements holds:
\begin{enumerate}[\normalfont (i)]
    \item There exists a self-embeddable component of $G$.
    \item There exists a sequence of distinct components $(C_i)_{i \in \mathbb{N}}$ of $G$ such that $C_1 \le C_2 \le \cdots$.
\end{enumerate}
\end{proposition}

\begin{proof}
The backward direction can be easily proved by defining a suitable self-embedding of $G$ for each of the two cases; it remains to show the forward direction.

Suppose that $G$ has a self-embedding $\varphi$ that embeds $G$ into $G-p$, where $p$ is either a vertex or an edge of $G$, and $G$ contains no self-embeddable component. Let $C_0$ be the component of $G$ containing $p$. We write $v \simeq w$ if the vertices $v$ and $w$ belong to the same component, and we denote $\varphi^k$ as the $k$-fold composition of $\varphi$.

We claim that if $u \in V(C_0)$, then for $0 \le i<j$, we have $\varphi^i(u) \not\simeq \varphi^j(u)$. We use induction on $i$. Let $i=0$, and suppose to the contrary that $u$ and $\varphi^j(u)$, where $j>0$, both belong to $C_0$. If $v \simeq u$, we then have
\[\varphi^j(v) \simeq \varphi^j(u) \simeq u,\]
so $\varphi^j(v) \in V(C_0)$ for every $v \in V(C_0)$. Also, since $\varphi$ embeds $G$ into $G-p$, the map $\varphi^j$ also embeds $G$ into $G-p$. Hence $\varphi^j$ carries $C_0$ into $C_0-p$, which contradicts the non--self-embeddability of $C_0$. Now suppose $i \ge 1$, and suppose to the contrary that $\varphi^i(u)$ and $\varphi^j(u)$, where $j>i$, both belong to the same component $C$. If $v \simeq \varphi^i(u)$, we then have
\[\varphi^{j-i}(v) \simeq \varphi^j(u) \simeq \varphi^i(u),\]
so $\varphi^{j-i}(v) \in V(C)$ for every $v \in V(C)$. We now prove that $\varphi^{j-i}$ carries $C$ into $C-\varphi^i(u)$; this would contradict the non--self-embeddability of $C$. Suppose that $\varphi^{j-i}(v)=\varphi^i(u)$ for some vertex $v$. We have $\varphi^{j-i-1}(v)=\varphi^{i-1}(u)$ by injectivity. By the induction hypothesis, we also have $\varphi^{i-1}(u) \not\simeq \varphi^{j-1}(u)$, so $\varphi^{j-i-1}(v) \not\simeq \varphi^{j-1}(u)$. It follows that $v \not\simeq \varphi^i(u)$, and since $\varphi^i(u) \in V(C)$, we infer that $v \notin V(C)$. Therefore, $\varphi^i(u)$ cannot be the image of a vertex of $C$ under $\varphi^{j-i}$, as desired.

Let $u \in V(C_0)$. Define a sequence $(C_i)_{i \in \mathbb{N}}$ such that $C_i$ is the component containing $\varphi^i(u)$. This sequence consists of pairwise distinct components by the previous claim. It is clear that $\varphi$ carries $C_i$ to $C_{i+1}$, so $C_i \le C_{i+1}$ for $i \in \mathbb{N}$, and we are done.
\end{proof}

Proposition \ref{prop1} implies, as an example, that the union of finite paths is self-embeddable, but the union of finite cycles with different lengths is not. Also, we obtain the following corollary.

\begin{corollary}\label{cor1}
A star forest is self-embeddable if and only if it is infinite.
\end{corollary}

For nonempty graphs $G$, a stronger property than self-embeddability is the property that $G \le G-e$ for all $e \in E(G)$. The ray and an infinite complete graph, for example, enjoy this stronger property. On the other hand, the disjoint union $\mathbb{N} \cup \mathbb{Z}$ is self-embeddable, but does not embed into $\mathbb{N} \cup (\mathbb{Z}-e)$, where $e$ is any edge of $\mathbb{Z}$. Thus $\mathbb{N} \cup \mathbb{Z}$ does not possess this stronger property.

\begin{proposition}\label{prop2}
If $G$ is a nonempty graph such that $G \le G-e$ for all $e \in E(G)$, then no $(G,H)$-minimal graph exists for any graph $H$.
\end{proposition}

\begin{proof}
We will prove that for every graph $F$ such that $F \to (G,H)$, we have $F-e \to (G,H)$ for some $e \in E(F)$. This would show that $(G,H)$ admits no minimal graph.

Let $F$ be a graph, and let $e$ be any one of its edges. Set $F'=F-e$. Suppose that $F' \not\to (G,H)$---there exists a $(G,H)$-good coloring $c'$ of $F'$. We show that $F \not\to (G,H)$. Define a coloring $c$ on $F$ such that $c\restriction_{E(F')}=c'$ and $e$ is colored red. By this definition, no blue copy of $H$ is produced in $F$. We claim that $c$ does not produce a red copy of $G$ either. Suppose to the contrary that a red copy of $G$, say $\widehat{G}$, is produced in $F$. Since $\widehat{G} \le \widehat{G}-e$, we can choose a red copy of $G$ in $F$ that does not contain $e$; that is, there exists a red copy of $G$ in $F'$. This contradicts the $(G,H)$-goodness of $c'$. As a consequence, $c$ is a $(G,H)$-good coloring of $F$, and thus $F \not\to (G,H)$.
\end{proof}

We note that Proposition \ref{prop2} does not hold for self-embeddable graphs $G$ in general---see Example \ref{ex1}.

If $R$ is the Rado graph, then $R-e$ is also the Rado graph for every $e \in E(R)$ via \cite[Proposition 2(b)]{cameron2013}. As a result, the Rado graph satisfies the hypothesis of Proposition \ref{prop2}. Consequently, by \cite{erdos1963}, the following holds.

\begin{proposition}\label{prop3}
For $H$ a fixed graph, almost all countably infinite graphs $G$ produce a pair $(G,H)$ which admits no Ramsey-minimal graph.
\end{proposition}

\section{Graph unions}\label{sec3}

Before we focus on star forests proper, we provide a quick background on graph unions in general. Consider graphs $G$, $H_1$, and $H_2$; let $F_i \in \mathcal{R}(G,H_i)$ for $i \in \{1,2\}$. Possible candidates for a $(G,H_1 \cup H_2)$-minimal graph include $F_1$, $F_2$, and $F_1 \cup F_2$.

Although $F_1 \cup F_2 \to (G,H_1 \cup H_2)$, it not necessarily true that $F_1 \cup F_2 \in \mathcal{R}(G,H_1 \cup H_2)$. Indeed, let us take $H_1=H_2=S_1$. For $G$ connected, we have $2G \in \mathcal{R}(G,2S_1)$ provided that $G \in \mathcal{R}(G,S_1)$. This was discussed in \cite{barrett2021} but also follows from Proposition \ref{prop4}. On the other hand, if $G$ is disconnected, we have $3\mathbb{Z} \in \mathcal{R}(2\mathbb{Z},2S_1)$---not $4\mathbb{Z}$---even though $2\mathbb{Z} \in \mathcal{R}(2\mathbb{Z},S_1)$.

\begin{proposition}\label{prop4}
Let $G$ and $H$ be nontrivial, connected graphs, and let $n \in \mathbb{N}$. If $F_i \in \mathcal{R}(G,H)$ for $1 \le i \le n$, then
\[\bigcup_{i=1}^n F_i \in \mathcal{R}(G,nH).\]
Consequently, the existence of a $(G,nH)$-minimal graph is assured provided that a $(G,H)$-minimal graph exists.
\end{proposition}

\begin{proof}
The arrowing part is obvious, so we only show the minimality of $\bigcup_{i=1}^n F_i$. It is clear that $F_i \not\to (G,2H)$, since otherwise we would have $F_i \notin \mathcal{R}(G,H)$. Let $e$ be an edge of $F_k$ for some $1 \le k \le n$. Color $F_k-e$ by a $(G,H)$-good coloring and $F_i$, for $i \neq k$, by a $(G,2H)$-good coloring. This coloring on $\left(\bigcup_{i=1}^n F_i\right)-e$ is easily shown to be $(G,nH)$-good from the connectivity of $G$ and $H$. Since $e$ is arbitrary, the proposition is proved.
\end{proof}

In contrast to Proposition \ref{prop4}, the following proposition considers $F_i$ as a candidate for being in $\mathcal{R}(G,H_1 \cup H_2)$. A sufficient condition is provided for a $(G,H_1)$-minimal graph to be $(G,H_1 \cup H_2)$-minimal.

\begin{proposition}\label{prop5}
Let $G$, $H_1$, and $H_2$ be graphs, and let $F \in \mathcal{R}(G,H_1)$. If $F-V(\widehat{H}_1) \to (G,H_2)$ for every $\widehat{H}_1$ a copy of $H_1$ in $F$, then $F \in \mathcal{R}(G,H_1 \cup H_2)$.
\end{proposition}

\begin{proof}
We first prove that $F \to (G,H_1 \cup H_2)$. Suppose $c$ is a coloring on $F$ that produces no red copy of $G$. It follows from $F \to (G,H_1)$ that $c$ produces a blue copy of $H_1$, say $\widehat{H}_1$, in $F$. Let $F'=F-V(\widehat{H}_1)$. Since $F' \to (G,H_2)$ and $F'$ contains no red copy of $G$, there exists a blue copy of $H_2$, say $\widehat{H}_2$, in $F'$. We observe that $\widehat{H}_1$ and $\widehat{H}_2$ are disjoint, so $c$ produces a blue copy of $H_1 \cup H_2$. Hence $F \to (G,H_1 \cup H_2)$. Its minimality follows immediately from the $(G,H_1)$-minimality of $F$.
\end{proof}

\begin{example}\label{ex1}
Let
\begin{align*}
&G=2S_1, \\
&H_1=\mathbb{Z}, \\
&H_2=\mathbb{N},\text{ and} \\
&F=2\mathbb{Z}.
\end{align*}
The graph $2\mathbb{Z}$ is $(2S_1,\mathbb{Z})$-minimal, and $\mathbb{Z} \to (2S_1,\mathbb{N})$, so we can conclude by Proposition \ref{prop5} that $2\mathbb{Z} \in \mathcal{R}(2S_1,\mathbb{Z} \cup \mathbb{N})$. This serves as an example of a pair $(G,H)$ involving a self-embeddable graph that admits a minimal graph. We note, however, that no $(S_1,\mathbb{Z} \cup \mathbb{N})$-minimal graph exists since $\mathbb{Z} \cup \mathbb{N}$ is self-embeddable. Thus it is possible that a $(2G,H)$-minimal graph exists even though no $(G,H)$-minimal graph exists.
\end{example}

\section{Proof of Theorem \ref{mainthm1}}\label{sec4}
We fix star forests $G$ and $H$ such that at least one of them contains a star component on infinitely many vertices. We prove in this section that $(G,H)$ admits no Ramsey-minimal graph.

Suppose that $F \to (G,H)$. Since one of $G$ and $H$ contains a vertex of infinite degree, there exists a vertex $v$ of infinite degree in $F$. We choose an arbitrary edge $e$ at $v$. We prove that $F' \to (G,H)$, where $F'=F-e$. Toward a contradiction, suppose that $F'$ admits a $(G,H)$-good coloring $c'$. Since $\deg(v)$ is infinite, there are two possible cases: $v$ is incident to infinitely many red edges or infinitely many blue edges under the coloring $c'$.

Suppose that $v$ is incident to infinitely many red edges. Define a coloring $c$ on $F$ such that $c\restriction_{E(F')}=c'$ and $e$ is colored red. This coloring produces no blue copy of $H$, so by $F \to (G,H)$ it produces a red copy of $G$, say $\widehat{G}$, in $F$. There exists a star component $S$ of $\widehat{G}$ that contains $e$ since otherwise, $\widehat{G} \subseteq F'$, which contradicts the $(G,H)$-goodness of $c'$.

If $S$ is infinite, then $F'$ clearly contains a red copy of $G$ by removing $e$ from $\widehat{G}$. On the other hand, let us suppose that $S$ has $n$ vertices. We can pick a red star $S'$ on $n$ vertices that is centered on $v$ but does not contain $e$, since $v$ is incident to infinitely many red edges. The graph $F'$ can then be shown to contain a red copy of $G$ by exchanging $S$ from $\widehat{G}$ for $S'$. In both cases, we obtain a contradiction.

The case when $v$ is incident to infinitely many blue edges can be handled similarly, so our proof of Theorem \ref{mainthm1} is complete.

\section{Subdivision graphs vs.\ star forests}\label{sec5}

\subsection{Bipartite graphs}\label{subsec5.1}

Recall that a graph is \emph{bipartite} if its vertex set can be partitioned into two \emph{parts} such that each part is an independent set. Let $K$ be a bipartite graph with bipartition $\{A,B\}$ such that $\deg(u) \neq \infty$ for all $u \in A$. Before we work on subdivision graphs $S(G)$, we construct for $n \in \mathbb{N}$, a graph $\Gamma(K,A,n)$ such that $\Gamma(K,A,n) \to (K,S_n)$.

\begin{figure}
    \centering
    \begin{tikzpicture}[x=11mm,y=11mm]
        \draw[thick] (0,0) -- (6,0);
        \draw[dashed] (0,0) -- (-1,0);
        \draw[dashed] (6,0) -- (7,0);

        \draw[fill=white] (0,0) circle (2pt);
        \draw[fill=black] (1,0) circle (2pt);
        \draw[fill=white] (2,0) circle (2pt);
        \draw[fill=black] (3,0) circle (2pt);
        \draw[fill=white] (4,0) circle (2pt);
        \draw[fill=black] (5,0) circle (2pt);
        \draw[fill=white] (6,0) circle (2pt);

        \node at (0,-0.3) {$-1$};
        \node at (1,-0.3) {$0$};
        \node at (2,-0.3) {$1$};
        \node at (3,-0.3) {$2$};
        \node at (4,-0.3) {$3$};
        \node at (5,-0.3) {$4$};
        \node at (6,-0.3) {$5$};

        \node at (3,-0.9) {(a) $\mathbb{Z}$};

        \draw[thick] (0,-3) -- (6,-3);
        \draw[thick] (0,-3) to [bend left=15] (1,-2.6) to [bend left=15] (2,-3) to [bend left=15] (3,-2.6) to [bend left=15] (4,-3) to [bend left=15] (5,-2.6) to [bend left=15] (6,-3);
        \draw[thick] (0,-3) to [bend right=15] (1,-3.4) to [bend right=15] (2,-3) to [bend right=15] (3,-3.4) to [bend right=15] (4,-3) to [bend right=15] (5,-3.4) to [bend right=15] (6,-3);
        \draw[thick] (0,-3) to [bend left=15] (1,-2.2) to [bend left=15] (2,-3) to [bend left=15] (3,-2.2) to [bend left=15] (4,-3) to [bend left=15] (5,-2.2) to [bend left=15] (6,-3);
        \draw[thick] (0,-3) to [bend right=15] (1,-3.8) to [bend right=15] (2,-3) to [bend right=15] (3,-3.8) to [bend right=15] (4,-3) to [bend right=15] (5,-3.8) to [bend right=15] (6,-3);
        \draw[dashed] (0,-3) -- (-1,-3);
        \draw[dashed] (6,-3) -- (7,-3);
        \draw[dashed] (-1,-2.6) to [bend left=15] (0,-3) to [bend left=15] (-1,-3.4);
        \draw[dashed] (-1,-2.2) to [bend left=15] (0,-3) to [bend left=15] (-1,-3.8);
        \draw[dashed] (7,-3.4) to [bend left=15] (6,-3) to [bend left=15] (7,-2.6);
        \draw[dashed] (7,-3.8) to [bend left=15] (6,-3) to [bend left=15] (7,-2.2);
        \draw[thick,dashed] (1,-3) ellipse (13pt and 34pt);
        \draw[thick,dashed] (3,-3) ellipse (13pt and 34pt);
        \draw[thick,dashed] (5,-3) ellipse (13pt and 34pt);

        \draw[fill=black] (1,-2.6) circle (2pt);
        \draw[fill=black] (3,-2.6) circle (2pt);
        \draw[fill=black] (5,-2.6) circle (2pt);
        \draw[fill=black] (1,-3.4) circle (2pt);
        \draw[fill=black] (3,-3.4) circle (2pt);
        \draw[fill=black] (5,-3.4) circle (2pt);
        \draw[fill=black] (1,-2.2) circle (2pt);
        \draw[fill=black] (3,-2.2) circle (2pt);
        \draw[fill=black] (5,-2.2) circle (2pt);
        \draw[fill=black] (1,-3.8) circle (2pt);
        \draw[fill=black] (3,-3.8) circle (2pt);
        \draw[fill=black] (5,-3.8) circle (2pt);
        \draw[fill=white] (0,-3) circle (2pt);
        \draw[fill=black] (1,-3) circle (2pt);
        \draw[fill=white] (2,-3) circle (2pt);
        \draw[fill=black] (3,-3) circle (2pt);
        \draw[fill=white] (4,-3) circle (2pt);
        \draw[fill=black] (5,-3) circle (2pt);
        \draw[fill=white] (6,-3) circle (2pt);

        \node at (1,-4.4) {$A_0$};
        \node at (3,-4.4) {$A_2$};
        \node at (5,-4.4) {$A_4$};
        \node at (3,-5.1) {(b) $\Gamma(\mathbb{Z},A=\text{the set of even vertices},3)$};

    \end{tikzpicture}

    \caption{The construction of $\Gamma(K,A,n)$ for $K=\mathbb{Z}$ and $n=3$.}
    \label{fig2}
\end{figure}

We define $\Gamma(K,A,n)$ by adding additional vertices and edges to $K$. For every $u \in A$, we add vertices $u_1,...,u_{m(n-1)}$---each not already in $V(K)$---to $K$, where $m=\deg(u)$. We then insert an edge between each $u_i$ and a vertex $v$ of $K$ if $uv$ exists in $K$. We denote the resulting graph by $\Gamma(K,A,n)$. Also, for each $u \in A$, we define $A_u$ as the set $\{u,u_1,...,u_{m(n-1)}\}$. As a result, $\Gamma(K,A,n)$ admits a bipartition $\{\bigcup_{u \in A} A_u,B\}$. Figure \ref{fig2} shows the result of this construction when $K=\mathbb{Z}$ and $n=3$.

There is a natural projection $\pi\colon \Gamma(K,A,n) \to K$ that is also a homomorphism. It is defined as
\begin{equation}\label{eq1}\pi(v)=\begin{cases}
u, & v \in A_u \text{ for some } u \in A, \\ v, & v \in B.
\end{cases}\end{equation}

\begin{proposition}\label{prop6}
Let $K$ be a bipartite graph with bipartition $\{A,B\}$ such that $\deg(u) \neq \infty$ for all $u \in A$. For $n \in \mathbb{N}$, we have $\Gamma(K,A,n) \to (K,S_n)$. Consequently for $k \in \mathbb{N}$,
\[\bigcup_{i=1}^k \Gamma(K,A,n_i) \to \left(K,\bigcup_{i=1}^k S_{n_i}\right),\]
where $n_1,\dots,n_k \in \mathbb{N}$.
\end{proposition}

\begin{proof}
Suppose that $c$ is a coloring on $\Gamma(K,A,n)$ that produces no blue copy of $S_n$. We prove that $c$ produces a red copy of $K$.

We claim that for all $u \in A$, there exists $v_u \in A_u$ such that $v_u$ is incident to red edges only. By construction, the vertices in $A_u$ share the same neighborhood $N$ of $m$ vertices, and $|A_u|=m(n-1)+1$. If every vertex in $A_u$ is incident to at least one blue edge, then the vertices in $N$ in total are incident to at least $m(n-1)+1$ blue edges. Since $|N|=m$, there exists a vertex in $N$ that is incident to at least $n$ blue edges by the Pigeonhole Principle. This is impossible since $\Gamma(K,A,n)$ does not contain a blue copy of $S_n$. Therefore, $A_u$ must contain a vertex that is incident to only red edges.

Hence we can define an embedding $\varphi\colon K \to \Gamma(K,A,n)$ as
\[\varphi(u)=\begin{cases}
v_u, & u \in A, \\ u, & u \in B.
\end{cases}\]
The graph image of $\varphi$ is a red copy of $K$ in $\Gamma(K,A,n)$, as desired.
\end{proof}

The graph $\Gamma(K,A,n)$ is not necessarily $(K,S_n)$-minimal in general. For example, let us take $K=S_k$ and $A$ as the set of leaf vertices of $S_k$. We have $\Gamma(S_k,A,n)=S_{kn}$, which is not $(S_k,S_n)$-minimal for $k,n \ge 2$ since $S_{k+n-1} \in \mathcal{R}(S_k,S_n)$. However, we potentially have $\Gamma(K,A,n) \in \mathcal{R}(K,S_n)$ when $K=S(G)$ for some graph $G$ as stated in Theorem \ref{mainthm2}.

\subsection{Proof of Theorem \ref{mainthm2}}\label{subsec5.2}

Fix a connected, leafless, non--self-embeddable graph $G$. Building upon Subsection \ref{subsec5.1}, we prove that for $n_1,\dots,n_k \in \mathbb{N}$, we have
\begin{equation}\label{eq2}\bigcup_{i=1}^k \Gamma(S(G),A,n_i) \in \mathcal{R}\left(S(G),\bigcup_{i=1}^k S_{n_i}\right),\end{equation}
where $A$ is taken as the set of vertices of $S(G)$ that subdivide the edges of $G$. We note that $\deg(u)=2$ for all $u \in A$. First, we show that the three properties of $G$ transfer to $S(G)$, and that $S(G)$ is \emph{$C_4$-free}---it contains no $4$-cycles. The following lemma can be verified using elementary means.

\begin{lemma}\label{lem1}
Let $G$ and $H$ be connected, bipartite graph with bipartition $\{A,B\}$ and $\{C,D\}$, respectively. For any isomorphism $\varphi\colon G \to H$, either $\varphi(A)=C$ and $\varphi(B)=D$, or $\varphi(A)=D$ and $\varphi(B)=C$.
\end{lemma}

\begin{proposition}\label{prop7}
If $G$ is a connected, leafless, non--self-embeddable graph, then $S(G)$ is also a connected, leafless, non--self-embeddable graph. In addition, $S(G)$ is $C_4$-free.
\end{proposition}

\begin{proof}
The first two properties obviously transfer, and $S(G)$ is $C_4$-free since $G$ contains no multiple edges. We now prove that $G$ is self-embeddable given that $S(G)$ is self-embeddable.

Suppose that $\varphi$ is a self-embedding of $S(G)$. Let $A$ be the set of vertices of $S(G)$ that subdivide the edges of $G$, and let $B=V(G)$. Since $S(G)$ is connected and bipartite with bipartition $\{A,B\}$, there are by Lemma \ref{lem1} two cases to consider.

\textbf{Case 1:} $\varphi(A) \subseteq A$ and $\varphi(B) \subseteq B$. We claim that $\varphi$, restricted to $V(G)$, gives rise to a self-embedding $\widehat{\varphi}$ of $G$. It is straightforward to show that $\widehat{\varphi}$ is an embedding, so we only prove that there is an edge of $G$ not in the image of $\widehat{\varphi}$. Suppose that $uv$, where $u \in A$ and $v \in B$, is an edge of $S(G)$ not in the image of $\varphi$, and suppose that $u$ subdivides an edge $vw$ of $G$.

We prove that $vw$ is not in the image of $\widehat{\varphi}$. Suppose toward a contradiction that $\widehat{\varphi}(a)=v$ and $\widehat{\varphi}(b)=w$ for two adjacent vertices $a,b \in V(G)$. Let $c$ be the vertex that subdivides $ab$. It is apparent that $\{\varphi(c),v\}$ and $\{\varphi(c),w\}$ are edges of $S(G)$. Also, we cannot have $\varphi(c)=u$ since $uv$ is not in the image of $\varphi$. But then the vertices in the set $\{v,u,w,\varphi(c)\}$ induce a $4$-cycle on $S(G)$, which contradicts the fact that $S(G)$ is $C_4$-free.

\textbf{Case 2:} $\varphi(A) \subseteq B$ and $\varphi(B) \subseteq A$. The map $\varphi^2$ is a self-embedding of $S(G)$ that carries $A$ into $A$, and $B$ into $B$. So by appealing to Case 1, we can obtain a self-embedding of $G$.
\end{proof}

Armed with Proposition \ref{prop7}, we are ready to prove Theorem \ref{mainthm2}. But first, let us provide a straightforward application of the membership statement of (\ref{eq2}) that we will prove later.

\begin{example}\label{ex2}
Choose $G=\mathbb{Z}$ and $H=S_3$. Since $\mathbb{Z}$ is connected, leafless, and non--self-embeddable, and $S(\mathbb{Z})=\mathbb{Z}$, the graph of Figure \ref{fig2}(b) is $(\mathbb{Z},S_3)$-minimal by (\ref{eq2}).
\end{example}

\begin{proof}[Proof of Theorem \ref{mainthm2}]
First, suppose
\[H=\bigcup_{i=1}^k S_{n_i}, \text{ where } 1 \le n_1 \le \dots \le n_k.\]
Let $A$ be the set of vertices of $S(G)$ that subdivide the edges of $G$ so that $\deg(u)=2$ for all $u \in A$. Define $\Gamma_i=\Gamma(S(G),A,n_i)$, and let $\Gamma=\bigcup_{i=1}^k \Gamma_i$. Denote the corresponding set to $A_u$ that belongs to $\Gamma_i$ by $A_{u,i}$. We have $|A_{u,i}|=2n_i-1$. If $B_i=V(\Gamma_i) \backslash \bigcup_{u \in A} A_{u,i}$, then $\Gamma_i$ admits a bipartition $\{\bigcup_{u \in A} A_{u,i},B_i\}$.

We prove for each $e \in E(\Gamma)$ that there is a $(S(G),H)$-good coloring of $\Gamma-e$. This, along with Proposition \ref{prop6}, would show that $\Gamma \in \mathcal{R}(S(G),H)$.

\begin{lemma}\label{lem2}
For every $e \in E(\Gamma)$, there exists a coloring $c$ on $\Gamma-e$ such that both of the following statements hold:
\begin{enumerate}[\normalfont(i)]
    \item The coloring $c$ produces no blue copy of $H$.
    \item There exists $u \in A$ such that for $1 \le i \le k$, every vertex in $A_{u,i}$ is incident to exactly one red edge.
\end{enumerate}
\end{lemma}

\begin{proof}
Suppose that $e$ is an edge of some $\Gamma_j$, where $1 \le j \le k$, and that $e$ is at a vertex $v \in A_{u,j}$ for some $u \in A$. We color each edge in every $\Gamma_i$, minus the edge $e$ when $i=j$, by the following rules:

\textbf{Case 1:} $i<j$. Recall that $|A_{u,i}|=2n_i-1$ and that all vertices in $A_{u,i}$ share the same neighborhood $\{a,b\}$. Arbitrarily partition $A_{u,i}$ into sets $S$ and $T$ such that $|S|=n_i$ and $|T|=n_i-1$. Color all the edges in $E(S,a) \cup E(T,b)$ blue, where $E(S,a)$ denotes the set of all edges between the vertex set $S$ and the vertex $a$; this produces two blue stars of sizes $n_i$ and $n_i-1$, respectively. Color the rest of $\Gamma_i$ red.

\textbf{Case 2:} $i=j$. As before, let $a$ and $b$ be the vertices adjacent to each vertex in $A_{u,j}$. Partition $A_{u,j} \backslash v$ into sets $S$ and $T$ both of size $n_j-1$. Similarly to Case 1, we color all the edges in $E(S,a) \cup E(T,b)$ blue. This produces two blue stars of size $n_j-1$. Color the rest of $\Gamma_j$ red.

\textbf{Case 3:} $i>j$. Let $a$ be a vertex adjacent to each vertex in $A_{u,i}$. Color $E(A_{u,i},a)$ blue; this produces a blue star of size $2n_i-1$. As previously, we color the rest of $\Gamma_i$ red.

Denote the preceding coloring scheme by $c$. It is obvious from the preceding construction of $c$ that (ii) holds for our $u \in A$, so it remains to prove that (i) holds.

Let $j'$ be the least positive integer such that $n_{j'}=n_j$. Observe that we only produce blue stars of size at least $n_j$ in Case 3 and, if $j'<j$, in Case 1 also. Every $\Gamma_i$ such that $j'\le i \le k$ and $i \neq j$ contributes exactly one blue star of size at least $n_j$, so exactly $k-j'$ such blue stars are produced in $\Gamma-e$ in total. But $H$ contains $k-j'+1$ stars of size at least $n_j$, so no blue copy of $H$ can be produced in $\Gamma-e$ by the coloring $c$.
\end{proof}

We take the coloring $c$ of Lemma \ref{lem2}. To prove that $c$ is $(S(G),H)$-good, we need to show that $c$ does not produce a red copy of $S(G)$ in $\Gamma-e$.

Suppose to the contrary that there exists an embedding $\xi\colon S(G) \to \Gamma_i$ such that its graph image is a red copy of $S(G)$. Set $\varphi=\pi \circ \xi$, where $\pi\colon \Gamma_i \to S(G)$ is a projection that sends each vertex in $A_{u,i}$ to $u$ and is defined similarly to Eq.\ (\ref{eq1}). We prove that $\varphi$ is a self-embedding of $S(G)$, which would contradict the non--self-embeddability of $S(G)$. For illustration, we provide the following commutative diagram of graph homomorphisms:
\[\begin{tikzcd}[row sep=9mm,column sep=8mm]
    S(G) \arrow[hook,r,"\xi"] \arrow[dr,"\varphi"'] & \Gamma_i \arrow[two heads,d,"\pi"] \\
    & S(G)
\end{tikzcd}\]

Suppose that $\varphi(a)=b$ for some vertices $a$ and $b$ of $S(G)$. If $b \in A$, then the vertex $\xi(a)$ belongs in $A_{b,i}$. Recall that $\deg(a) \ge 2$ since $S(G)$ is leafless. Since the graph image of $\xi$ is red, $\xi(a)$ needs to be incident to at least two red edges as a result. We infer that $b \neq u$, where $u \in A$ is taken from Lemma \ref{lem2}(ii). This shows that the vertex $u$ of Lemma \ref{lem2}(ii) is not in the image of $\varphi$.

Since $S(G)$ is $C_4$-free and $\xi$ is an embedding, there cannot be a $C_4$ in the graph image of $\xi$. We now prove that $\varphi$ is injective. Let $a$ and $b$ be distinct vertices of $S(G)$. Since $a$ and $b$ have degree at least two, the vertices $\xi(a)$ and $\xi(b)$ also have degree at least two. As a result, $\xi(a)$ and $\xi(b)$ cannot both belong in $A_{u,i}$ for some $u \in A$, since that would create a $C_4$ in the graph image of $\xi$. Therefore, $\varphi$ is injective. This completes the proof that $\varphi$ is a self-embedding and finishes our proof of Theorem \ref{mainthm2}.
\end{proof}

\bibliographystyle{elsarticle-harv}

redpajama_set_name: RedPajamaArXiv


Assisted Parking is available on Sunday mornings from 8 a.m. to 12:30 p.m. at the Mulberry Street entrance. Pull up to the assisted parking stand in the Mulberry lot and personnel will park your car and have it ready for you after services. There are also reserved spots for first-time visitors in the Mulberry lot.
See 360 degree views of the Sanctuary, Chapel and available meeting spaces. Choose a room from the menu.


Yesterday's signals were not triggered, as there was no bearish price action when the resistance level at $3,936 was reached.
Trades must be entered between 8am New York time Wednesday and 5pm Tokyo time Thursday.
Go long at a bullish price action reversal on the H1 time frame following the next touch of $3,936, $3,921, or $3,876.
I wrote yesterday that a look at the daily chart showed a slightly bearish picture, so short trades from rejections of resistance levels were likely to be the best bet – I was ready to take a bearish bias if $3,876 broke down convincingly or if the price retraced to $3,936 and strongly rejected that level.
In fact, the price turned around with the support at $3,876 continuing to hold, and recent hours have seen a relatively strong upwards movement which has probably flipped the former resistance levels at $3,936 and below to become new higher support. There are initial signs that the price may be topping out near $4,000 where is there is key resistance close by. If the price can close above $4,008 for a couple of hours, that would be a bullish sign. There is nothing of high importance due today concerning the USD.


text:
Il teatro Bellini è un teatro lirico di Acireale. Fu costruito dall'ingegnere catanese Carmelo Sciuto Patti ed inaugurato nel 1870. Nel 1952 un devastante incendio doloso ne distrusse l'interno. Si attende, ancora dopo oltre mezzo secolo, il suo restauro e la sua riapertura.

Note

Altri progetti

Teatri della città metropolitana di Catania
Architetture di Acireale

redpajama_set_name: RedPajamaWikipedia


Families of children with special needs have many health questions. Each person's circumstances are unique, so please contact us if we can help. We are parents ourselves of children with special needs.
1. I spend a lot of time overseeing my child's care. Is there any help available?
Overseeing your child's care can be time consuming and, at times overwhelming. Our Care Notebook can help you organize and keep track of information related to your child. Your child's primary care office or Medical Home may have a Care Coordinator who can help you access resources and information, as well as support you in keeping your family and child well. Many practices have adopted the core concepts of patient-and family-centered care, dignity and respect, information sharing, participation, and collaboration.
2. We are over income for Vermont Medicaid (Dr. Dynasaur). Are there other Medicaid options for my child who has a disability?
The Disabled Children's Home Care (DCHC), also known as Katie Beckett Medicaid, is another way to access Medicaid. This program is for children under the age of 19. Parent's income and resources are not counted toward eligibility, only the child's. The determination depends on the degree of the individual child's functional limitations and his or her need for specialized care and services. Many families who have private insurance also apply for Medicaid. Contact Green Mountain Care or Vermont Family Network to request an application packet. It's important to complete the application packet correctly and completely. Contact us if you would like assistance with, or tips on completing the application forms. Refer to our Six Ways to Access Medicaid booklet for more information.
3. My child sees many pediatric specialists. When will we need to switch to adult providers?
Adult specialists are better able to care for adult bodies; referrals and expertise are better suited to our young adults in the adult healthcare system. Specialists vary as to when they will guide your child to transition to an adult practice. Ask your child's doctors or talk with your Care Coordinator if you have one. Got Transition has an abundance of information including a Health Care Transition Resources guide.
4. How can I help plan for my child's financial future?
A Special Needs Trust (SNT) can be set up with the help of an attorney. The funds in the SNT are not counted as income for benefits eligibility. There are different types of SNTs.
ABLE accounts, which stand for Achieving a Better Life Experience, will allow individuals with disabilities to save up to a maximum of $100,000 for qualifying expenses without it affecting their benefits. Vermont's ABLE program is called STABLE. For information and to sign up, visit the State's website. If you would like more information regarding ABLE accounts, please call us and ask to speak with a Family Support Consultant.
5. What should I be thinking about as my child enters their teens?
For families with children with disabilities, making plans for the future is an ongoing process from the moment of diagnosis. Our Holistic Transition Guide for Families of Children with Disabilities was created by parents, for parents, family members, and caregivers of children with disabilities. Topics covered in the Guide include: housing options, independent living skills, recreation, friendships, safety, transportation, education, health, and employment/income.
6. Is there any help for my child who struggles with behavior and anxiety?
Many children and adolescents have mental health challenges that affect their daily functioning. Parenting a child who is experiencing a mental health challenge can be a confusing and difficult process. Please call us for help and resources.
7. Our child with a disability needs assistance with developing personal care and daily living skills. Is there help available?
Children's Personal Care Services is a Medicaid benefit for children, under the age of 21, diagnosed with a verifiable long-term disability or health condition, which impacts age-appropriate development of activities of daily living (ADLs). A verifiable long-term disability is defined as a health condition or disability that is expected to last one year or more. The federal government defines ADLs as dressing, bathing, grooming, toileting, eating or feeding, and mobility and transfers. This program is intended to provide supplemental assistance to the primary caregiver as individualized one-to-one support for the identified child related to their ADLs. Eligibility is determined by an Assessment of Need evaluation, which is completed by a health professional with the family. To apply for Children's Personal Care Services, contact the Vermont Department of Health or your local Designated Agency. Finding and hiring a child's Personal Care Attendant can be challenging. Review our Fact Sheet for helpful tips.
8. Who can I talk to who understands what I'm going through?
We provide at no cost to you Parent Matches. Talking on the phone with a parent who has "been there" can offer a source of empathetic listening, hope, encouragement, and problem solving. Vermont Family Network connects families of children with special needs with other families for caring and support. Parents should never feel alone! Also, many parents find it helpful to attend a support group where they can meet and talk with other parents.
9. What support is available for my child's siblings?
make new friends, learn new games, do crafts, share a meal, and have a lot of fun.
Sibshops is a program of the Sibling Support Project, which is dedicated to the life-long and ever-changing concerns of millions of brothers and sisters of people with special health, developmental, and mental health concerns. The Sibling Support Project also connects siblings and offers online support.
10. I'm exhausted and need a break. Where can I get help?
For parents of children with special needs, the physical, emotional, and financial demands can be overwhelming. We periodically receive funding to provide parents with one-time respite. COMFORT (College of Medicine Friends Offer Respite Time) is a program run by medical students and undergrads at the University of Vermont through which student volunteers spend a few hours each week with a child with special needs. If you or family members of any age (child to elder) need to hire someone to provide support in your home, check out Rewarding Work or Care.com.
11. Will my child receive services as an adult?
Vermont's Agency of Human Services, Department of Disabilities, Aging and Independent Living, offers a variety of programs and services as does the Department of Mental Health.
The transition from high school is an exciting time for youth and their families. See our Transition to Adulthood Toolkit for a checklist of what to consider along with many helpful resources.
12. My child needs specialized care out of state. Are there resources to help with travel expenses like housing?
Traveling out-of-state for specialized care can be stressful and financially difficult. Housing, food, the cost of gas, and medical expenses are just a few things that you will need to consider when traveling away from your home. A medical social worker through Children with Special Health Needs may be able to help with resources such as housing or completing a Medicaid application if you and your family are eligible. Talking to another parent who has had a similar experience through our Parent Match program may also be helpful. They may have suggestions or tips on how to get food vouchers when at the hospital or fun things to do when you have some down time. Please call us for help.


text:
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Clean Power Professionals Group
This special interest group is for professionals to connect and discuss all types of carbon-free power alternatives, including nuclear, renewable, tidal and more.
Electric Vehicle Installation: New Business Models Emerge with the Infrastructure Law
Diane Cherry 8,086
Principal Diane Cherry Consulting, LLC
Diane Cherry is a woman owned small business providing clean energy consulting services for local government, clean energy companies, non-profits and educational institutions. Her projects and...
78 items added with 20,301 views
Aug 30, 2022Aug 30, 2022 8:30 pm GMT
The Bipartisan Infrastructure Law seeks to develop a national electric vehicle (EV) charging network of 500,000 chargers by 2030. It includes up to $7.5 billion in dedicated funding to increase EV charging accessibility, with money distributed based on the state's federal highway funding formula. There is also a $2.5 billion discretionary grant program and a $5 billion National Electric Vehicle Infrastructure Formula Program (NEVI). These are applied along Alternative Fuel Corridors (AFC), a U.S. Department of Transportation (U.S. DOT) designated network of plug-in EV charging (and other fuels) infrastructure along national/state highway systems.
Each state's Department of Transportation is tasked with making sure that its funding is disseminated thoughtfully, and is required to submit a plan for proposed spending to the U.S. DOT by August 1st.
Opportunities for Charging Businesses
The current EV charging market has a few large providers. Currently, California-based EV charging infrastructure company ChargePoint operates the largest EV charging network in the country with 30,000 stations and over 47,000 individual charging ports. Tesla, the second largest, has fewer than 6,000 station locations and about 25,000 charging ports. While the majority of Tesla's charging ports are DC fast chargers, most ChargePoint locations are Level 2. Level 2 chargers operate between 208-240 volts and the power output provides 18-28 miles per hour; a full charge takes about 8 hours. On the other hand, DC fast charging has a maximum output of 350 volts and can charge an EV battery up to 80 percent between 20-40 minutes.
With the rise of EV charging in the next couple of years, there will be many ways to deploy options for ownership, financing, and serving customers (see examples in the Figure below). Until recently, companies used one of only two business models. Under the first, an EV charging provider sells electricity from public charging stations it owns and operates. Under the second, the provider doesn't get paid for the electricity, but instead collects service fees for installing, operating, and maintaining charging stations at locations like shopping malls and parking garages. But now, there are so many other options because of the large federal EV investment and one of those is highlighted below.
Service Aggregator as a One New EV Business Model
Consumer vehicles are parked most of the time and a parked EV can connect to the power grid with opportunities for charge coordination and energy management. A connected EV offers flexibility on how and when energy is delivered.
To that end, increased numbers of parked EVs may facilitate a business model for service aggregators. An aggregator is a group in a power system such as energy storage providers, rooftop PV solar, or EVs that act as one entity when they engage in the power system or sell services. It is possible to aggregate on several levels: (1) the first is on site where EV charging infrastructure is installed and connected; (2) the second is to aggregate locally such as in a neighborhood or an industrial site that spans several miles; and (3) the last is serving a company that operates as a virtual power plant (VPP), aggregating different distributed energy resources to provide grid services. In this way, an aggregator acts as a middle man between the system operator and the EVs. It can be seen as a large generator source that provides ancillary services such as frequency and regulating reserve. Currently the large EV charging companies like ChargePoint can serve as an aggregator or an entirely new company may enter the market and provide those aggregation services.
The rapid growth of clean energy technologies like solar also presents an opportunity for EV chargers to integrate with existing or new clean energy infrastructure. Tying solar, storage, and EV charging together in a microgrid that can be islanded from the larger electric grid is another technological solution that is likely to be seen. It will make a facility resilient and self-sustaining in the event of power failures or forced outages.
Predictive analytics is another business opportunity with the EV infrastructure growth. An EV car can generate terabytes (1 trillion bytes) of data. Within that data are answers to questions about the patterns and the potential for monetizing connected car data is massive. It remains to be seen which business model is the most successful but a year from now, we should expect to see new players in the EV charging market.
electric-transportation
infrastructure-investment-and-jobs-act
Diane Cherry
Thank Diane for the Post!
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Regulatory Analyst IV
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Dover, Deleware

redpajama_set_name: RedPajamaCommonCrawl


You are not pleased with Omegle apps on your mobile? So you can still try some Omegle alternatives if you are using Windows Phone. There are good and reliable mobile chat apps at Microsoft store.
WeChat doesn't providing a stranger chat service like Omegle but it has better options like Nearby Chat and Shake. You will able to meet people who are living (or standing) around of you with nearby chat feature. You can also meet people from different countries with Shake feature. How does Shake feature work? You will need to enter Shake section of WeChat and shake your phone. You will meet with some random who is shaking his/her phone in the same time with you. It's also another kind of random chat and it's quite fun. Sometimes both features doesn't give 100 percent performance. If you love to talk to Asian people, this application will be quite interesting for you since there are many Chinese people who are using Shake feature on the application. You can also meet people from other countries too. We Chat is also a good messenger app for your device. You can add your contacts and talk them with video calling and audio calling!


text:
O Kum-ran  (born 18 September 1981,) is a North Korean women's international footballer who plays as a midfielder. She is a member of the North Korea women's national football team. She was part of the team at the 2003 FIFA Women's World Cup.

International goals

References

1981 births
Living people
North Korean women's footballers
North Korea women's international footballers
Place of birth missing (living people)
2003 FIFA Women's World Cup players
Women's association football midfielders
Footballers at the 2002 Asian Games
Asian Games gold medalists for North Korea
Asian Games medalists in football
Medalists at the 2002 Asian Games


Major price adjustment.....This is a very unique property. Home has 3BR, living room, den and screened-in porch. There is a 2 car detached carport. There is also a 3 car detached garage with a half bath, storage area and office area. There are 3 garage doors, one of them is electric. Lots of shelving and storage throughout the garage. The garage is 27'x70' with 2 steel beams running length of garage. There is a second floor to the garage, it is 27'x45'. It is divided into five rooms. There is a possibly that it could be finished into an apartment. There are steps from outside leading up to this area, and steps inside the garage. This property probably has one of the largest lots in the area. It is about 3/4 of an acre. Garage has water and heat. The seller's husband ran a business out of the detached garage. Many different possibilities for this property. The home needs some TLC. This property is being sold "as is". Don't wait to show this property - Might want to think about your investors also. Square footage taken from appraisal.
203 Brunswick Avenue is located in Greenville, SC 29609. This 1000 square foot home has 3 bedrooms and 1 bathrooms.
I was searching for a Property and found this listing (MLS® #1380320). Please send me more information regarding 203 Brunswick Avenue, Greenville, SC, 29609. Thank you!
I'd like to request a showing of 203 Brunswick Avenue, Greenville, SC, 29609 (MLS® #1380320). Thank you!


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Watch and discuss How to Survive a Plague at the Klamath County Library
Host Klamath County Library
Location Klamath County Library/126th S. 3rd
Time 02:00 pm - 04:00 pm
Saturday, April 21st 2pm
Downtown Klamath County Library
The downtown Klamath County Library and the Klamath Regional Health Equity Coalition are hosting a screening and discussion of the documentary How to Survive a Plague on Saturday, April 21st at 2 pm.
The film screening is part of the Klamath Regional Health Equity Coalition's Diversity Through the Arts program, a series of discussions about films and books that describe culture clashes and the benefits of embracing diversity.
How to Survive a Plague covers the early years of the AIDS epidemic, and how advocacy groups like ACT UP (AIDS Coalition to Unleash Power) and TAG (Treatment Action Group) rose up to agitate for more robust and affordable medical treatment for people with AIDS. The groups shut down Wall Street and the offices of the Federal Drug Administration, and organized "die-ins" and other protests around the country in their fight to get the U.S. government to take the spread of the disease more seriously.
After the screening, why not read the film's accompanying book, How to Survive a Plague by David France? You can pick up a copy at the downtown Klamath County Library or download one through our ebook partner site, Library2Go.
How to Survive a Plague is unrated and runs for 2 hours.
For more information, please call 541-882-8894 or stop by the Information and Reference desk.


We all have been there. Looking at a DSLR camera like it was some spaceship. Like it was some larger than life object that only the best in the business will possess. For us, normal cameras will never ever make the cut. Even though smartphone cameras have come quite a long way on the quality and clarity front, it still doesn't feel right for us enthusiasts who are waiting for that 'my own professional camera' moment.
Social media lovers will bask in this new found popularity to post pictures with the best of quality. It all started when that rich kid in the group brought his/hers 'professional camera' to click some farewell pictures.
Times have changed now. Drastically. Cameras of all sizes have increased the amount of quality they deliver and DSLRs are no exception. Manufacturers have keenly noted this boom in popularity and have done their best to cash in at this boom.
Looking for DSLR? We are here to help you out in the best way possible. The market today is flooded with options which can be quite overwhelming for the normal buyer. Need not worry because this article's sole aim is to make sure you bring home the best DSLR camera for beginners that suits all your parameters. We have condensed some attributes that you need to look out for so that you make no mistake in purchasing your treasured tool to show the world through your perspective.
We have tried our best to not get too much into the technical side and keep things as simple as possible. For beginners, not all the settings will be easy to grasp in the first try. This is a skill that has to be perfectly honed with practice.
Without further ado, let us get into the topic in question.
This is one technical term all people interested in photography would have come across. Because this is the main advertising factor for all manufacturers when it comes to cameras in general or the cameras in smartphones. The general myth is that every megapixel counts. It isn't really the case. You really won't be able to see the difference when it is less than 5 between two cameras.
Megapixels matter the most to photographers who like to print out their photos for sale. Landscape photographers have the most affiliation towards megapixel count as the more the megapixels are, the better the image quality and size translated when printed.
Pixel is the fundamental unit that makes a picture. A megapixel is roughly a million pixels. These are small dots that comprise a picture. A jump from one megapixel to two will be felt hard but a jump from 6 to 7 megapixels will be barely visible as the percentage of increase is not all that much.
Most entry-level and mid-range DSLRs pack at least 18 megapixels and that is more than enough for a beginner to have their lessons on. As your experience and need rise, bigger cameras are the way to go.
This is the measure that you should actually be worried about. Even though every camera in the beginner level makes use of a cropped sensor, this is something you need to know for further upgrades.
There are two frame sizes that matter when it comes to DSLRs. The full frame sensor and the cropped sensor. The full frame sensor has the same measurements of a standard 35mm film format. If you know cameras, you will know that the older cameras that used film kept this as a standard measurement.
Full frame cameras are found only in more expensive and advanced models as they are tougher to package in and way better in terms of quality reproduction.
The main difference between a full frame and a cropped sensor is the size. Any sensor smaller than a full frame can safely be categorized as a cropped sensor. But the most famous and the most sued size is the APS-C cropped sensor.
A full frame will have better low light capabilities as it is bigger and can hold in more light. But cropped sensors make sure that the camera is small and compact. When it comes to long-range photography with telephoto zoom lenses, the cropped nature of the sensor makes sure that the zooming is more profound. This is because no lens using a cropped sensor has the same focal length as prescribed on it.
This is due to the multiplier effect. Nikon, for instance, has a multiplier quotient of 1.5x. This means that a 50mm lens is actually a 75mm lens due to this multiplication.
Nowadays APS-C has grown a long way and you don't have to worry much about quality as it is just a few steps below the very best.
The reason the DSLR is actually popular is due to its property to be compatible with different lenses. Usually, most lenses are compatible with both full frame and cropped sensors. Your choice of lenses totally depends on the type of photography you aspire to pursue.
For example, wildlife photography is tricky. You need to be quite far from the subject as they might get startled and disrupt the whole scene if you try to get closer. This is where a long zoom ranged telephoto lens comes into play. So that you can maintain your distance and take a clear shot.
When you want to take portraits, the best is to go with a prime lens. A prime lens is one with a fixed focal length. These lenses usually have fast apertures so that the photo is crisp and well exposed with a shallow depth of field.
Image stabilization is a newfound treasure for photography as the involuntary vibrations and shakes are considerably reduced. Since these cameras are pretty cheap, image stabilization is usually built into the lenses rather than the camera itself. These lenses are usually costlier.
The best tip ever is to get a feel of the camera. Most of the cameras in this price point might appear to be completely unchanged even across different brands to the naked eye. But only when you get to use them, the feel of the camera is predictable. The weight, the grip and the size of the camera matter way more than the features as your comfort and preferred ergonomics play an important role in your performance with the camera.
There are numerous brands in the market and all of them seem to offer the best in that particular price range. Here is a condensed list that we think qualifies as the best DSLRs for beginners available on the market today.
Nikon is a manufacturer with immense expertise in the photography field. They have been around for more than a century bringing out class-leading innovation. In India, they have discovered some great potential in the beginner DSLR market and have started rolling out models that suit the budgets and the feature seeking mentality of our people.
The D3400 is a camera specifically designed for beginners. This camera has been marketed as an entry-level DSLR. This camera boasts some pretty awesome specs that once upon a time were found only in the more expensive ones. This camera comes with a healthy 24 megapixels.
There is a built-in guide mode that will help out the budding user in figuring out certain basics of a professional camera. That is a feature that is missing in the rival models. There are special effects to spruce up the photography experience.
The ISO range is 100-25600. This is a pretty impressive range for an entry-level camera. Low Light capabilities are immense.
The shutter speed range is 30 seconds to 1/4000th of a second. This is in par with the best in the business.
This camera is capable of taking Full HD 1080p in 60 frames per second.
The D3400 weighs around 450 grams and the dimensions are 4.9 x 3.9 x 3 in.
Excellent battery life. Other rival models can only dream of such range.
Bluetooth connectivity makes sharing photos way easier.
No touchscreen. Rival models offer touch screens for more ease of use.
Autofocusing is pretty average when it comes to moving objects.
Nikon's beginner/semi-advanced lineup is in our opinion the most promising and most VFM oriented of all the lineups out there. The D5600 camera has been designed with a carbon fibre monocoque and that means it is extremely light. This camera then seems like it has been built with not many compromises to show.
The D5600 comes with a 24-megapixel shooter and a vari-angle LCD monitor that is also a touchscreen. This makes zooming into pictures and scanning through pictures a joy to behold.
The ISO range is a hefty 100-25600 and that means low light performance is going to be excellent. It comes packed with an EXPEED 4 processor which makes speed and performance a mundane practice.
The Shutter speed range is a good 30 seconds to 1/4000th of a second which means you won't be missing many special moments.
This camera is capable of shooting at 5 frames per second and has 39 autofocus points to make sure that locking onto a target never goes out of sync.
The D5600 weighs around 465 grams and the dimensions are 4.9 x 3.9 x 2.8 in.
A plethora of connectivity options like Wi-Fi, Bluetooth and make this one tough product to beat.
The touch screen makes operation a whole lot easier.
Excellent battery life makes sure that you take more photos than rivals.
No GPS feature for geotagging photos.
Pentax is brand which is practically unheard of in India at least. It is a brand name used by the company Ricoh situated in Japan. This is an optics company. This is a company with rich history and numerous years of expertise.
The Pentax K50 is honestly a mixed bag of tricks. It can many things most cameras cannot in this price range but still falls short on some basic stuff.
The K50 is a 16-megapixel shooter with a 3 inch TFT LCD monitor at the rear.
The shutter speed range is the highest here of any camera. The range is a whopping 30 seconds to 1/6000th of a second. This is on par with cameras that cost 2-3 times more.
The practical ISO range is also the highest of all the camera here. The low light capabilities are immense and there is also a mode dedicated to astrophotography.
This camera has an inbuilt Sensor-shift anti-shake system and also has inbuilt weather protection to withstand tough weather.
This camera is capable of shooting at 6 frames per second and Full HD video at 30 frames per second to deliver some crisp and vibrant images.
This camera has an 11 point autofocus system but it is often regarded as a slow system.
This camera weighs around 650 grams and the dimensions are 5.07 x 3.77 x 2.7in.
This is the only weather protected camera on this entire list. That is a great property to have.
There is a stabilization system that has been inbuilt into the camera. So no worries about using lenses not having stabilization.
Low light capabilities are the best in class.
This is a heavy camera due to the weather protection and needs some getting used to.
Battery life needs some serious improvement.
Canon is definitely upping its game on the Indian shores with the 200D. This camera is one of the recent launches from Canon and this has been making some noise in this category. The features all have been designed to rival the likes of Nikon's D5300 and D5500.
The 200D is a 24-megapixel camera with a touch capacitive LCD screen in the rear which is also articulated.
The ISO range is pretty much the best in the business for low light photography with a healthy range of 100-25600.
The shutter speed range is a standard 30 seconds to 1/4000th of a second.
This camera is capable of shooting up to 5 frames per second. That means missing out on sudden and beautiful events will probably never happen.
This camera is capable of capturing Full HD videos at 60 frames per second and that is way more than enough for a nice and crisp playback.
This camera weighs around 453 grams and the dimensions are 4.8 x 3.6 x 2.7in.
Great Battery life makes sure you take more photos than rival models.
ISO range is healthy and Low Light photos are great.
A very light camera to behold.
Options like WiFI, Bluetooth and NFC make connectivity great.
There is no GPS option for geotagging your photos.
Autofocusing capabilities are not up to the mark for a camera that has had its launch very recently.
The 1300D is one important camera for Canon. This is because it replaces the 1200D which was the best selling entry level DSLR in India. This was achieved due to a very low price with the balance of average specs and performance which made it accessible to the masses who were just stepping into the DSLR game.
This camera is not something that would set the spec sheet ablaze. This is a camera built down to a budget so that it can be accessible to beginners who are in a dilemma or in financial constraints. There are a few tweaks from the same spec sheet of the 1200D to make the 1300D.
This camera comes out of the box with an 18-megapixel sensor. The rear LCD is 3 inches in size. This camera is capable of handling 3 shots per second and that is pretty much adequate for most scenarios.
The ISO range is a mediocre 100-6400. It can be boosted but the grain creeps in too much.
The shutter speed range is a quick 30 second to 1/4000th of a second. That is impressive for such a small camera.
There is a 9 point autofocus system for tracking objects in motion and it is best to say that it is an average system.
The 1300D is capable of shooting videos in Full HD at 30 frames per second.
The 1300D weighs around 485 grams and the dimensions are 5.08 x 3.99 x 3.06 in.
The cheapest way to get into the DSLR family.
Built in WIFI and NFC makes it easy to share photos to other devices.
Low light capabilities are pretty much limited as the ISO range is pretty low.
Canon has been recently launching some new models with the sole aim of competing directly with Nikon. The 800D on paper looks like it is going head to head with the king of the segment, the Nikon D5600.
The Canon 800D is a 24-megapixel shooter with a 3 inch LCD touchscreen at the rear for ease of use. This means that it is on tie with the Nikon here.
The ISO range is a good 100-25600 which means low light capabilities are on par with the best.
The Shutter speed range is a standard 30 seconds to 1/4000th of a second. This is way more than enough for extensive use.
The camera has an impressive 45 focus point system which makes it superior to the Nikon in terms of autofocusing on objects in motion.
This camera is capable of shooting at 6 frames per second which makes it one shot better than the Nikon. This is an impressive number.
The 800D is capable of shooting full HD 1080p videos at 60 frames per second. Video playback in bigger screens is a treat.
The 800D weighs around 532 grams and the dimensions are 5.1 x 3.9 x 3in.
Stabilization is built into the camera and so all lenses can be used without worry.
Autofocusing is pretty much the best in class.
Battery life definitely needs some improvement.
It is pretty heavy and needs some getting used to.
This might seem to be a pleasant departure for you from the usual big giants like Nikon and Canon. Sony has had troubled times in India when it comes to DSLRs. This is Sony's attempt to get it all worked out again. This is a DSLR with no actual moving mirror as in traditional DSLRs. There is an electronic viewfinder rather than a manual one.
The Alpha 68 is a camera that has 24 megapixels up its sleeve. It has a 2.7 inch LCD screen that is articulated.
The ISO range is a whopping 100-25600 and this is possible thanks to an awesome processor the BIONZ X.
The shutter speed range is a good 30 seconds to 1/4000th of second.
This camera is capable of shooting at 8 frames per second and that is a lot. This is not much behind the elite and more expensive cameras.
This camera has a whopping 79 point Autofocus system which means this is one great camera when it comes to capturing fast moving objects.
This camera is capable of recording Full HD video at 60 frames per second. This means that videos are to the best of their quality even in big screen playback.
This camera weighs around 600 grams and the dimensions are 5.6 x 4.1 x 3.2in.
The continuous shooting range for this camera is pretty impressive considering its price range.
The image stabilization is built into the camera and not the lens. This means every picture you take with every lens is properly stabilized.
Low light capabilities are great considering the practical ISO range.
There is a secondary display for displaying the important attributes which make things way easier.
It is pretty heavy when compared with rivals.
No GPS function included for geotagging photos.
There aren't many lenses for this camera on the market.
Nikon is a brand that needs no introduction to people who know photography. The camera in question is in our opinion the best beginner camera that money can buy. This prestigious position for this camera has been given after extensive research and after going through thousands of user reviews. We feel this is the camera that balances the budget and feature attributes in the best way possible. This camera has a lot to give and doesn't ask much from your wallet in return.
The D5300 comes out of the box with 24 megapixels along with a rear LCD screen. This screen can articulate which makes shooting in difficult angles a breeze. This makes the camera very versatile.
The ISO range of this camera is a manageable 100-12800 and can be boosted too. Low light photos are pretty good with minimal noise/grain.
The shutter speed is a healthy 30 seconds to 1/4000th of second and this is a range that is not behind by much to the cameras that cost 5 times.
This camera is capable of shooting at 5 frames per second which means it can take 5 photos per second. That is a good number for a camera of this price. This camera has 39 autofocus points which mean autofocusing is great even with moving objects.
The D5300 weighs around 530 grams and the dimensions are 4.9 x 3.9 x 3 in.
When it comes to VFM and a boatload of features, the D5300 is the one to go for in the entry-level DSLR.
Has almost all of the features that its bigger brother the D5600 has for a significantly lesser price.
Has GPS function for geotagging photos and also WiFI for smartphone control and easy transfer of photos.
Availability of Lens bundle kits makes this one smart buy.
Touch screen should have been included as many rivals have that.
Even though this camera is not really better than the camera it intended to rival, this is the only alternative. This camera was launched to aim squarely at the Nikon D5500. This camera is the only way you can go if you want a change of brand.
As told before, the 750D was introduced to directly rival the D5500. The features offered are pretty much on par with the D5500 in most ways. This camera comes with a 24-megapixel shooter and 3 inch LCD vari-angle touch screen for more intuitive usage and fiddling of settings and playback.
The ISO range is an adequate 100-12800 which means that low light performance is quite good. The more powerful DIGIC 6 processor is employed to make things smooth and fast.
The shutter speed range is a quick 1/4000th of a second to 30 seconds which means long exposure and quick shots are equally balanced.
This camera is capable of shooting at 5 frames per second which is translated as 5 photos per second. That means this is not much behind the more elite cameras that cost way more.
This camera is capable of shooting in full HD 1080p resolution at 30 frames per second which makes sure that your videos are taken with crisp and sharp details.
The 750D weighs approximately 555 grams and the dimensions are 5.1 x 3.9 x 3.06in.
This camera has smartphone control which enables the user to control the shutter with their smartphone.
NFC makes sharing photos to a smartphone lightning quick.
The inclusion of touch screen makes it much easier to use.
Battery life definitely needs a bump.
It is considerably heavier than its rivals.
This camera is the latest addition to the entry-level DSLR lineup for Canon in India. Canon has been the king when it comes to shifting entry level cameras as everyone seems to get a Canon as their first camera. The 1200D which was Canon's most successful model in India was a best seller in Amazon from the day it was launched. This is because of the very compelling pricing that Canon had adopted. Now they had to update the model as the competition had grown fierce. The update was through the Canon 1500D.
This is a camera that many choose to be their first DSLR. The 1500D comes with a 24-megapixel shooter and a 3 inch LCD screen at the rear.
The ISO range is a mediocre 100-6400 which can be boosted but a lot of grain does start to creep in when boosted.
The shutter speed range is an impressive 30 seconds to 1/4000th of a second. This is an impressive range when the price is considered.
This camera is capable of shooting approx. 3 shots per second and that is pretty much adequate.
This camera has a humble 9 point autofocus system for locking on moving objects.
The 1500D weighs around 475 grams and the dimensions are 5.08 x 3.99 x 3.06in.
Probably one of the cheapest ways to get into the DSLR game now.
The battery life is pretty good for an entry-level camera.
Low light capabilities are not this camera's biggest strength.
Autofocus points are less in number and that hinders autofocusing abilities when the object is in motion.
Choose the Best DSLR Camera for Beginners!
You reached the end of this article. That's great! Hope this article was helpful to you in some way. All the products that have been included in this list have been extensively researched keeping in mind the pros and cons along with an overall Value For Money perspective. Because at the end of the day, it is your hard earned money that is being invested here.
The best DSLR for you will show itself up the day you decide to try out every option available in your price range. You are the best judge of your wants and needs. We are sure that none of these products will disappoint you when it comes to shooting good photos. We hope you have a great time with your trusty camera with many more awesome memories to come. Good luck!


home > home loans > home equity loans > which home equity lender is right for you?
Which Home Equity Lender Is Right for You?
Choosing a home equity lender is as important as choosing a mortgage lender. Most home equity loans are long term sources of financing, and many are issued in revolving terms that allow them to stay active for years. Further, you are placing your home as collateral on a home equity loan just as you did with your mortgage. If you default, your home can be foreclosed on even if your mortgage is active. For these reasons, it is important to approach sourcing a home equity loan on the same terms you approached sourcing your mortgage lender.
For many borrowers, using the same lender for a mortgage and a home equity loan is very convenient. The lender has already run your credit and approved you for a very large debt. If the lender gave you good terms on your mortgage, the lender may extend similar good terms on your home equity line. However, some mortgage lenders will be wary of extending another loan because they will be aggregating their own risk. The lender approved the limits of your mortgage knowing they could recover your home to pay off the loan if you defaulted. By adding a home equity loan, the lender may be over-extending limits and risking the possibility they will not recover in a default.
Dedicated home equity companies often offer the fastest solutions for your financing. You may find they contact you the moment you purchase your new home. These lenders are willing to extend very high limits in most cases because they have less to risk than a mortgage lender does in the same situation. For home equity lenders, they are extending a loan worth only a small portion of the value of your home. If you default, they have the option of purchasing your mortgage and forcing your home into foreclosure to recover. Choose a dedicated home equity lender if you want financing fast and are looking for high limits.
Credit cards can be secured with home equity and used like a home equity line of credit. For this option, you may not need to seek an actual home equity lender. This is a good option if you would like a more permanent financing source for minor home projects but not big repairs. The limits may be lower, but the options are potentially more flexible.
A final and often ignored option for home owners is seeking a government rehabilitation loan. There are a number of loans guaranteed by the Federal Housing Administration that allow you to rehabilitate, improve or otherwise alter your home at a low rate. These are excellent options if you are in an older home that needs a lot of work or if you are customizing your home for a disability. Safety improvements, like updating drainage or a septic system, may also be affordable through FHA secured loans. You will need to get a private loan, but the FHA can guarantee the loan under one of its rehabilitation program to decrease your expense.


My January Ipsy Glam Bag arrived today ... the theme is carpe p.m.
Subscriptions are $10US a month or $110US for a year. Shipping is free in the U.S. and $4.95US per month to Canada. It works out to $226.36CN for the year ($18.86CN a month).


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AANZFTA Overview Opportunities
Agreement Establishing The AANZFTA Understanding On Article 1 On Chapter 2 of AANZFTA Reviews of AANZFTA
ECONOMIC COOPERATION
AECSP Overview AECSP At A Glance Reviews of AECSP AECSP Supported Projects
SECTORAL PORTAL
Trade in Goods - Rules of Origin - Tariff and Non-Tariff Measures - Customs Procedures and Trade Facilitation - Sanitary and Phytosanitary Measures - Standards, Technical Regulations and Conformity Assessment Procedures Trade in Services - Financial Services - Telecommunication Services Investment Intellectual Property Competition and Consumer Protection E-Commerce Government Procurement
Business Engagement Strategy and Tools Statements and Reports Publications Other Official Documents Useful Links
Media Releases Calendar of Events AANZFTA In The News
Annex on Telecommunications
Scope and Coverage
This Annex applies to measures by a Party affecting trade in public telecommunications transport networks and services.
Notwithstanding Paragraph 1, this Annex shall not apply to measures by a Party affecting the distribution of broadcasting and audio-visual services, as defined in each Party's domestic legal framework.
Nothing in this Annex shall be construed to require a Party to allow the supply of public telecommunications transport networks or services in relation to which it has not made specific commitments under this Chapter.
For the purposes of this Annex:
co-location (physical) means access to space in order to install, maintain or repair equipment at premises owned or controlled and used by a major supplier to supply public telecommunications transport services;
cost-oriented means based on cost, and may include a reasonable profit, and may involve different cost methodologies for different facilities or services;
essential facilities means facilities of a public telecommunications transport network or service that:
are exclusively or predominantly provided by a single or limited number of suppliers; and
cannot feasibly be economically or technically substituted in order to provide a service;
facilities-based suppliers means suppliers of public telecommunications transport networks or services that:
are licensed carriers in Australia;
are classified as Access Seekers in accordance with the Telecommunications Act 2001 as amended from time to time in New Zealand;
are the Infrastructure Provider for the Telecommunication Industry (InTi) licensees in Brunei Darussalam;
are licensed as network facility provider and licensed network services provider under domestic law in Cambodia;
are licensed as telecommunication network provider in Indonesia;
are authorised to establish an enterprise to provide telecommunications service under the Telecommunications Act of 2001 in Lao PDR;
are licensed as Network Facilities Provider and Network Services Provider in Malaysia;
are telecommunications operators licensed as network facility provider and/or network service provider; and operators authorised by the Ministry of Communications, Posts and Telegraphs to provide facility based services in Myanmar;
are licensed public telecommunications entities as defined in the Public Telecommunications Policy Act of the Philippines;
are facilities-based operators in Singapore;
are duly licensed under domestic law as facilities-based supplier in Thailand; and
are facilities-based operators duly licensed in Viet Nam;
interconnection means linking with suppliers providing public telecommunications transport networks or services in order to allow the users of one supplier to communicate with users of another supplier and to access services provided by another supplier;
leased circuits[1] means telecommunications facilities between two or more designated points that are set aside for the dedicated use of, or availability to, a particular user;
major supplier means a supplier which has the ability to materially affect the terms of participation, having regard to price and supply, in the relevant market for the supply of public telecommunications transport networks or services, or parts thereof, as a result of:
control over essential facilities; or
use of its position in the market;
non-discriminatory means treatment no less favourable than that accorded to any other user of like public telecommunications transport networks or services in like circumstances;
public telecommunications transport network means the public telecommunications infrastructure which permits telecommunications between and among defined network termination points;
public telecommunications transport service means any telecommunications transport service required, explicitly or in effect, by a Party to be offered to the public generally. Such services may include, inter alia, telegraph, telephone and data transmission typically involving the real-time transmission of customer-supplied information between two or more points without any end-to-end change in the form or content of the customer's information;
telecommunications means the transmission and reception of signals by any electromagnetic means;
telecommunications regulatory body means any body or bodies in the territory of a Party which is or are responsible, under the Party's domestic legal framework, for the regulation of telecommunications; and
user means service consumers and service suppliers.
Transitional Arrangements
Noting each Party's different stage of development, and noting each Party's commitments under GATS, a Party may delay the application of Article 4 (Competitive Safeguards), Article 6 (Interconnection), Article 7 (Co-location), Article 8 (Leased Circuits Services) and Article 9.2 (Resolution of Disputes) in accordance with the timetable set out in this Annex's Appendix on Transitional Arrangements.
Competitive Safeguards
Subject to Article 3 (Transitional Arrangements), each Party shall prevent suppliers of public telecommunications transport networks or services who, alone or together, are major suppliers in its territory, from engaging in or continuing anti-competitive practices.
The anti-competitive practices referred to in this Article shall include:
engaging in anti-competitive cross-subsidisation;
using information obtained from competitors with anti-competitive results; and
not making available to other suppliers of telecommunications transport networks or services, in a timely fashion, technical information about essential facilities or commercially relevant information, which is necessary for such suppliers to provide public telecommunications transport networks or services.
Each Party shall ensure that, where a licence is required, all measures relating to the licensing of suppliers of public telecommunications transport networks or services in its territory are published or, where publication is not practicable, otherwise made publicly available, including:
circumstances in which a licence is required;
licence application procedures;
criteria used to assess licence applications;
standard terms and conditions applicable to licences;
the period of time normally required to reach a decision concerning a licence application;
the cost of and/or fees for applying for and/or obtaining a licence; and
the period of validity of a licence.
Each Party shall ensure that the reasons for the denial of a licence are made known to an applicant upon request.
Interconnection[2]
Subject to Article 3 (Transitional Arrangements), each Party shall ensure that major suppliers in its territory provide interconnection to suppliers of public telecommunications transport networks or services of other Parties at any technically feasible point in the major supplier's network. Such interconnection shall be:
provided in a timely fashion, on terms and conditions (including technical standards and specifications), and at cost-oriented rates, that are reasonable (having regard to economic feasibility), non-discriminatory and transparent;
sufficiently unbundled, such that the supplier of public telecommunications transport networks or services seeking interconnection need not pay for network components or facilities that it does not require for the service to be provided;
of a quality no less favourable than that provided for the major supplier's own like services, or for like services of non-affiliated service suppliers, or for its subsidiaries or other affiliates; and
provided upon request, at points in addition to the network termination points offered to the majority of users, subject to charges that reflect the cost of construction of necessary additional facilities.
Each Party shall ensure that the terms, conditions and rates (including technical standards and specifications) for interconnection between major suppliers in its territory and suppliers of public telecommunications transport networks or services of other Parties are able to be established (at least):
through commercial negotiation; or
by reference to a set of standard terms, conditions and rates that the major supplier offers generally to other suppliers of public telecommunications transport networks or services, and that are approved or set out by a telecommunications regulatory body.
Each Party shall ensure that the procedures for interconnection with major suppliers in its territory are published or otherwise made publicly available.
Subject to Article 3 (Transitional Arrangements), each Party shall ensure that major suppliers in its territory:
provide to suppliers of public telecommunications transport networks or services of other Parties that are facilities-based suppliers in the territory of that Party, physical co-location of equipment necessary for interconnection; and
in situations where physical co-location referred to in Subparagraph (a) is not practical for technical reasons or because of space limitations, co-operate with suppliers of public telecommunications transport networks or services of other Parties that are facilities-based suppliers in the territory of that Party, to find and implement a practical and commercially viable alternative solution.[3]
Each Party shall ensure that major suppliers in its territory provide the physical co-location or practical and commercially viable alternative solution referred to in Paragraph 1 in a timely fashion and on terms and conditions (including technical standards and specifications), and at rates, that are reasonable (having regard to economic feasibility), non-discriminatory and transparent.
Each Party may determine, in accordance with its domestic laws and regulations, the locations at which it requires major suppliers in its territory to provide the physical co-location or the practical and commercially viable alternative solutions referred to in Paragraph 1.
Leased Circuits Services
Subject to Article 3 (Transitional Arrangements), each Party shall, unless it is not technically feasible, ensure that major suppliers in its territory make leased circuits services (that are public telecommunications transport services) available to suppliers of public telecommunications transport networks or services of other Parties in a timely fashion and on terms and conditions (including technical standards and specifications), and at rates, that are reasonable (having regard to economic feasibility), non-discriminatory and transparent.
Resolution of Disputes
Each Party shall ensure that a supplier of public telecommunications transport networks or services of another Party who requests interconnection with a major supplier that is authorised to supply public telecommunications transport networks or services in the Party's territory has recourse to a telecommunications regulatory body to resolve disputes in relation to such interconnection, including in relation to terms, conditions or rates:
within a reasonable period of time, according to a procedure that has been published or otherwise made publicly available; and
at the request of the affected supplier of public telecommunications transport networks or services of the other Party.
Subject to Article 3 (Transitional Arrangements), each Party shall ensure that a supplier of public telecommunications transport networks or services of another Party who requests co-location with or leased circuits services from a major supplier that is authorised to supply public telecommunications transport networks or services in the Party's territory has recourse to a telecommunications regulatory body or a competition regulatory body to address issues in relation to such co-location or leased circuits services, including in relation to terms, conditions or rates:
Each Party shall ensure that its telecommunications regulatory body or bodies provide, upon request by a supplier of public telecommunications transport networks or services of another Party, a written explanation of any decision by a telecommunications regulatory body that affects the supplier of public telecommunications transport networks or services of the other Party, unless such explanation is otherwise publicly available.
Each Party shall endeavour to make information that the Party is required to publish or make publicly available pursuant to this Annex available on the internet.
Telecommunications Regulatory Body
Each Party shall establish or maintain, as part of its domestic legal framework, a telecommunications regulatory body.
Each Party shall ensure that every telecommunications regulatory body that it establishes or maintains is separate from, and not accountable to, any supplier of public telecommunications transport networks or services.
Each Party shall ensure that the functions and responsibilities of the telecommunications regulatory body or bodies, which shall include enforcement of the commitments set out in Article 6 (Interconnection), and all of its decision-making powers, shall be set out in the Party's domestic laws or regulations.
Each Party shall ensure that the decisions of, and the procedures used by, its telecommunications regulatory body or bodies are impartial with respect to all interested persons.
Each Party shall ensure that any supplier of public telecommunications transport networks or services of another Party that is aggrieved, or whose interests are adversely affected by a determination or decision of a telecommunications regulatory body of that Party, may obtain review of the determination or decision by an administrative, arbitral or judicial tribunal or authority or according to administrative, arbitral or judicial procedures. Where such procedures are not independent of the telecommunications regulatory body, the Party shall ensure that the procedures in fact provide for an objective and impartial review.
Universal Service
Each Party has the right to define the kind of universal service obligation it wishes to maintain. Such obligations, including any cross subsidisation policy set out under each Party's domestic laws, shall not be regarded as anti-competitive per se, provided they are administered in a transparent, non-discriminatory and competitively neutral manner and are not more burdensome than necessary for the kind of universal service defined by the Party.
Allocation and Use of Scarce Resources[4]
Each Party shall administer its procedures for the allocation and use of scarce resources, including frequencies and numbers, in an objective, timely, transparent and non-discriminatory manner.
Each Party shall publish or otherwise make publicly available the current state of allocated frequency bands.[5]
Parties are not required to publish identification of frequencies allocated for specific government uses, or to otherwise make them publicly available.
1 In the case of Thailand, "leased circuits" means telecommunications facilities between two designated points that are set aside for the dedicated use of, or availability to, a particular user.
2 For the sake of clarity, nothing in this Article shall be construed to require Thailand or Viet Nam to allow cross-border supply of public telecommunications transport networks or services in relation to which it has not made specific commitments under this Chapter.
3 Such solutions may include:
permitting facilities-based suppliers to locate equipment in a nearby building and to connect such equipment to the major supplier's network;
conditioning additional equipment space or virtual co-location;
optimising the use of existing space; and
finding adjacent space.
4 Decisions on the allocation and assignment of spectrum and frequency management are not measures that are per se inconsistent with Article 4 (Market Access) of Chapter 8 (Trade in Services). Accordingly, each Party retains the ability to exercise its spectrum and frequency management policies, which may affect the number of service suppliers, provided that this is done in a manner that is consistent with this Chapter. Each Party also retains the right to allocate frequency bands taking into account existing and future needs.
5 Parties are not required to publish information about the allocation of individual frequencies to specific licencees.
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Forums › General Discussion › Hardware & Software Discussion › Post your PC pictures!
This topic contains 97 replies, has 29 voices, and was last updated by aalio 3 months, 1 week ago.
Got a new case for my PC, so took some pictures. It's so quiet, first thing you hear in my apartment when you come in now is my fridge, and not my stupid PC.
Reply To: Post your PC pictures!
This reply was modified 5 years, 10 months ago by TunaTrunks.
Nice desk, Tuna! , similar to mine, damn i cant find my cameras usb cable.
I work from home most of the time so have my own office. Work laptop and PC on a kvm, NAS and HP microserver.
What kind of Microserver is that, the N40L or?
Looking into those aswell. Seems cheap.
HP were doing a £100 cashback last year, so with that it was only £150.
And clean desk is a productive desk!
Not really a picture of my PC but here's my rather festive desk.
2early for christmas decorations, Linc!
I am planning to buy a new chair and I really like yours. Can you share some info of it (price, where did you buy it, is it custom)?
I'll upload my config today.