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The excellent [git-flow](https://github.com/nvie/gitflow) extension is fully supported, allowing you to run flow commands with sublime commands. To enable `git-flow` integration you must set `show_git_flow_commands` to `true` in `Packages/User/GitSavvy.sublime-settings`.
Most commands attempt to mirror `git-flow` 's interface with the added ability to select a target from local branches/remotes.
## `flow: init`
A required step when you wish to setup a project to use `git-flow`. This will present a series of prompts to configure `git-flow` very much like the interactive shell command does.
## `flow: feature/release/hotfix/support start`
When running this command, you will prompted first for the feature/release/hotfix/support name (without the prefix), and then the branch will be created and checked out.
## `flow: feature/release/hotfix/support finish`
When running this command when an existing feature/release/hotfix/support branch is checked out, you will asked to confirm finish. Otherwise, you will be asked to select the relevant branch. This flow merges the changes from this branch into the "develop" branch (without fast-forwarding, unless branch has only a single commit).
## `flow: feature/release/hotfix publish`
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## `flow: feature/release track`
When running this command you will be prompted to provide a feature name. The command will pull a feature/release from a configured remote and check it out.
## `flow: feature pull`
This will pull a feature from a given remote (not necessarily the configured default remote) and check it out. You will be first prompted to select a remote and then to provide a feature name.
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,369 |
\section{Introduction} \label{s.introduction} We are interested in topics related to one-sided maximal operators on Euclidean spaces. Our focus is on the one-dimensional case and the main operator under study in this paper is the forward one-sided Hardy-Littlewood maximal operator defined by
\[
\mathsf{M}^+f(x)\coloneqq \sup_{h>0}\frac {1}{h} \int_x ^{x+h} |f(t)|dt,\quad x\in\mathbb R,\quad f\in L^1 _{\text{loc}}(\mathbb R).
\]
By a weight $w$ we always mean a non-negative, locally integrable function on the real line. The weights $w$ for which $\mathsf{M}^+:L^p(w)\to L^{p,\infty}(w)$ have been identified and extensively studied. See for example \cites{Reyes, MPT, MOT,MT, ORT, Saw}. Thus it is well known that the appropriate one-sided Muckenhoupt class can be defined by
\[
[w]_{A_p ^+}\coloneqq \sup_{a<b<c}\frac{w(a,b)}{|(a,c)|} \Big( \frac{\sigma(b,c)}{|(a,c)|}\Big) ^{p-1},
\]
where $\sigma\coloneqq w^{-\frac{1}{p-1}}$, and we have that $\mathsf{M}^+:L^p(w)\to L^{p,\infty}(w)$, $1<p<\infty$, if and only if $w\in A_p ^+$. For $p=1$ we also have the endpoint result that $\mathsf{M}^+:L^1(w)\to L^{1,\infty}(w)$ if and only if
\[
[w]_{A_1 ^+}\coloneqq \NOrm \frac{\mathsf{M}^-w}{w}. L^\infty(\mathbb R).<+\infty,
\]
where $\mathsf{M}^-f(x)\coloneqq \sup_{h>0} \frac {1}{h}\int_{x-h} ^x |f(t)|dt$; see for example \cites{MOT,Saw}.
The corresponding one-sided $A_\infty ^+$ class has also been studied and one can define for example
\[
A_\infty ^+ \coloneqq \bigcup_{p>1} A_p ^+;
\]
see \cite{MPT}. The class $A_\infty ^+$ can be defined in many other equivalent ways. Here we adopt the definition through a Fujii-Wilson-type constant which amounts to demanding that
\[
[w]_{A_\infty ^+}\coloneqq \sup_{a<b} \frac{1}{w(a,b)}\int_{(a,b)} \mathsf{M}^-(w\textnormal{\textbf 1}_{(a,b)})<+\infty.
\]
We note here that there is no standard definition for the $A_\infty ^+$ constant of one-sided Muckenhoupt weights $w$. The definition above appears in \cite{MT}*{Definition 1.7} where it is shown that $w\in A_\infty ^+$ if and only if $[w]_{A_\infty ^+}<+\infty$. A similar definition appears in \cite{KS}. We note also that for two-sided weights the constant above was introduced independently by Fujii \cite{Fujii} and Wilson \cites{W1,W2}. In the two-sided case the constant above was shown to be the appropriate quantity in order to prove sharp weighted norm inequalities for singular integral and maximal operators; see for example \cite{HytP}. Similar results for the one-sided maximal operator appear in \cite{MT}.
In this paper we pursue a characterization of $A_\infty ^+$ in terms of \emph{tauberian constants}, in the spirit of \cites{HLP,HP2}. Given a locally integrable function $w\in A_\infty ^+$ and $\alpha\in(0,1)$ we define
\[
\mathsf{C_w ^+}(\alpha)\coloneqq \sup_{0<w(E)<+\infty} \frac{1}{w(E)} w(\{x\in\mathbb R:\,\mathsf{M}^+\textnormal{\textbf 1}_E(x)>\alpha\}).
\]
We call $\mathsf{C_w ^+}(\alpha)$ the \emph{sharp weighted tauberian constant} of $\mathsf{M}^+$. It is obvious that $\mathsf{C_w ^+}(\alpha)<+\infty$ for all $\alpha\in(0,1)$ whenever $\mathsf{M}^+:L^p(w)\to L^{p,\infty}(w)$. In this paper we show that an apparently much weaker converse to this statement holds, namely that $\mathsf{C_w ^+}(\alpha_o)<+\infty$ for a single value $\alpha_o\in(0,1)$ already implies that $M^+:L^p(w)\to L^p(w)$ for sufficiently large $p>1$ and thus that $w\in A_\infty ^+$.
This is the content of our main theorem.
\begin{theorem}\label{t.main} Let $w$ be a non-negative, locally integrable function on the real line. The following are equivalent:
\begin{enumerate}
\item[(i)] We have that $w\in A_\infty ^+$.
\item[(ii)] There exists $\delta>0$ such that $\mathsf{C_w ^+}(\alpha)-1\lesssim (1-\alpha)^\delta$ as $\alpha\to 1^-$.
\item[(iii)] There exists $\alpha_o\in (0,1)$ such that $\mathsf{C_w ^+}(\alpha_o)<+\infty$.
\end{enumerate}
\end{theorem}
The study of asymptotic estimates of the type $\mathsf{C_w ^+}(\alpha)-1\lesssim (1-\alpha)^\delta$ as $\alpha\to 1^-$ has a small history. In \cite{Sol}, Solyanik proved corresponding estimates for the usual Hardy-Littlewood maximal function with respect to axes parallel cubes in $\mathbb R^n$, both in its centered and non-centered version, as well as for the strong maximal function. In \cites{HP1} the first two authors continued these investigations and introduced the term \emph{Solyanik estimate} to indicate the validity of $\lim_{\alpha\to 1^-}\mathsf{C}(\alpha)=1$, whenever $\mathsf{C}(\alpha)$ denotes the sharp tauberian constant with respect to some geometric maximal operator. Finally, in \cite{HP2}, Solyanik estimates are shown to hold in the classical (two-sided) weighted setting and in fact they characterize the usual $A_\infty$ class of Muckenhoupt weights.
Going back to our main result above and assuming that $w\in A_\infty ^+$ we can give a very precise version of the Solyanik estimate (ii).
\begin{corollary}\label{c.main1} Let $w\in A_\infty ^+$ be a one-sided Muckenhoupt weight on the real line. There exists a numerical constant $c>0$ such that
\[
\mathsf{C_w ^+}(\alpha)-1\lesssim (1-\alpha)^{(c[w]_{A_\infty ^+})^{-1}}\quad\text{for all}\quad 1>\alpha>1-e^{-c[w]_{A_\infty ^+}}
\]
with the implicit constant independent of everything. Furthermore this estimate is optimal up to numerical constants; if $w$ is a locally integrable function that satisfies
\[
\mathsf{C_w ^+}(\alpha)-1\lesssim (1-\alpha)^\frac{1}{\beta}\quad\text{whenever}\quad 1>\alpha>1-e^{-\beta}
\]
for some constant $\beta>1$ then $w\in A_\infty ^+$ and $[w]_{A_\infty ^+}\lesssim \beta$.
\end{corollary}
We note in passing that the sharpness direction in Corollary~\ref{c.main1} above relies on exhibiting a sharp reverse H\"older inequality for one-sided Muckenhoupt weights. The validity of the reverse H\"older inequality is actually proved in \cite{MT}. We show in Theorem~\ref{t.ainfty} that the reverse H\"older inequality of \cite{MT} is best possible, which in turn allows to prove the optimality of the Solyanik estimate in the corollary above.
Secondly, it is of some importance to note that our method of proof together with the corollary above allows us to conclude a quantitative embedding of $A_\infty ^+$ into $A_p ^+$.
\begin{corollary}\label{c.main2} Let $w\in A_\infty ^+$ be a one-sided Muckenhoupt weight on the real line and denote by $[w]_{A_\infty ^+}$ its Fujii-Wilson constant. Then there exists a numerical constant $c>0$ such that $w\in A_p ^+$ for all $p>e^{c[w]_{A_\infty ^+}}$ and $[w]_{A_p ^+}\leq \exp(\exp(c[w]_{A_\infty ^+}))$.
\end{corollary}
We close this introductory section with a few comments concerning our motivation for studying these estimates. In \cites{KS,Saari} the third author has considered higher dimensional weighted norm inequalities for one-sided maximal operators. These arise naturally in the study of solutions of appropriate doubly nonlinear partial differential equations. Other approaches to defining and studying higher dimensional one-sided maximal operators appear for example in \cite{FMO},\cite{LerOmb}, and \cite{Ombrosi}. A common feature of all these studies is that, in the higher dimensional case, the connection between the one-sided classes $A_p ^+$ and $A_\infty ^+$ remains elusive. This should be contrasted to the one-dimensional case where we have a full analogue of the two-sided Muckenhoupt theory.
It is customary for most of the one-dimensional papers studying $A_\infty ^+$ in the literature to set everything up with respect to some other measure $g$ and thus study the classes $A_p ^+(g)$. This setup is particularly suited for some symmetry arguments that allow one to show that a weight in $A_\infty ^+$ belongs to $A_p ^+$ for some $p>1$. This approach does not seem to be available in higher dimensions as many general results that hold for arbitrary measures in one-dimension fail in higher dimensions. The current paper avoids this setup and thus proposes another self-contained strategy for proving that $\bigcup_{p>1} A_p^+=A_\infty ^+$. We plan to pursue the higher dimensional analogues in a future work.
\section{Notation} We use the letters $C,c$ to denote generic positive constants which might change even in the same line of text. We write $A\lesssim B$ if $A\leq C B$ and $A\eqsim B$ if $A\lesssim B$ and $B\lesssim A$. Throughout the text $w$ is a nonnegative, locally integrable function on the real line and we write $w(a,b)\coloneqq \int_{(a,b)}w(t)dt$. Finally, given $\beta\in(0,1)$ and an interval $(a,b)\subset \mathbb R$ we say that a function $f:(a,b)\to \mathbb R$ lies in the local H\"older class $C^\beta(a,b)$ if for all compact $K\subset (a,b)$ and all $x,y\in K$ we have that
$|f(x)-f(y)|\lesssim_K |x-y|^\beta$ for all $x,y\in K$.
\section{Preliminaries} In this section we collect classical results about one-sided Muckenhoupt weights that we will need throughout the paper.
The first is a version of the classical rising sun lemma adjusted to our setup and is a minor modification of \cite{Saw}*{Lemma 2.1}.
\begin{lemma}\label{l.rsun} Let $\lambda>0$ and $f\geq 0$ with compact support. Then
\[
\{x\in\mathbb R:\, \mathsf{M}^+f(x)>\lambda\}=\bigcup_j (a_j,b_j)
\]
where the intervals $\{(a_j,b_j)\}_j$ are pairwise disjoint and for every $j$ we have that $\fint _x ^{b_j} f> \lambda $ for all $x\in(a_j,b_j)$. In particular $\fint_{a_j} ^{b_j} f=\lambda$ for all $j$.
\end{lemma}
\begin{proof}Let $E_\lambda\coloneqq \{x\in\mathbb R:\, \mathsf{M}^+f(x)>\lambda\}$. By \cite{Saw}, we have that $E_\lambda=\bigcup_j (a_j,b_j)$ where $\{(a_j,b_j)\}_j$ is a disjoint collection such that $\fint_{a_j}^{b_j}f = \lambda$ for every $j$ and such that $\fint_{x}^{b_j}f \geq \lambda$ whenever $x \in (a_j, b_j)$ for some $j$. Now, for every $x\in (a_j,b_j)$ there exists $r>x$ such that $\fint_x ^r f>\lambda.$ If $r=b_j$ then we are done. If $r>b_j$ then since $\fint_{b_j} ^r f\leq \lambda$ we can conclude again that $\fint_x ^{b_j}f>\lambda$. Finally, if $r<b_j$ we consider the maximal $s\geq r$ such that $\fint_r ^s f\geq\lambda$. Necessarily $s\geq b$ due to the maximality of $s$. From this and the fact that $\fint_{b_j} ^sf\leq \lambda$ we conclude that $\fint_r ^{b_j} f\geq \lambda$. Thus $\int_x ^{b_j} f=\int_x ^r f +\int_r ^{b_j}f >\lambda(b_j-x)$ and we are done.
\end{proof}
Whenever we have a one-sided weight $w\in A_\infty ^+$ we will need to use quantitative estimates of the type
\[
\frac{w(E)}{w(a,c)}\lesssim \big(\frac{|E|}{|(b,c)|}\big)^\delta
\]
whenever $a<b<c$ and $E\subseteq (a,b)$. In fact it is well known that the estimate above provides an equivalent definition for the class $A_\infty ^+$; see for example \cite{MPT}. Another equivalent way to define $A_\infty ^+$ goes through the validity of appropriate reverse H\"older inequalities.
The following theorem summarizes these equivalences. Note that the direct implications (i) and (ii) below are directly taken from \cite{MT}. The optimality of these estimates for $w\in A_\infty ^+$, contained in (iii), appears to be new as the authors in \cite{MT} didn't pursue this direction.
\begin{theorem}\label{t.ainfty} Let $w$ be a nonnegative, locally integrable function on $\mathbb R$. Then the following hold.
\begin{enumerate}
\item[(i)] If $w\in A_\infty ^+$ then for $0<\epsilon\leq \frac{1}{2[w]_{A_\infty ^+}}$ and for all $a<b<c$ we have the one-sided reverse H\"older inequality
\[
|(b,c)|^\epsilon \int_{(a,b)} w^{1+\epsilon}\leq 2 \bigg(\int_{(a,c)} w \bigg)^{1+\epsilon}.
\]
\item[(ii)] If $w\in A_\infty ^+$ then for $0<\epsilon\leq \frac{1}{2[w]_{A_\infty ^+}}$ and for all $a<b<c$ and for all measurable sets $E\subseteq (a,b)$ we have
\[
\frac{w(E)}{w(a,c)}\leq 2 \big(\frac{|E|}{|(b,c)|}\big)^{\frac{\epsilon}{1+\epsilon}}\leq 2 \big(\frac{|E|}{|(b,c)|}\big)^{\frac{1}{3[w]_{A_\infty ^+}}}.
\]
\item[(iii)] Conversely, if the conclusion of (i) or (ii) holds for some $0<\epsilon<1$ then $w\in A_\infty ^+$ and $[w]_{A_\infty ^+}\lesssim \frac{1}{\epsilon}$; thus (i), (ii) are best possible up to numerical constants.
\end{enumerate}
\end{theorem}
\begin{proof} The statement and proof of (i) is \cite{MT}*{Theorem 3.4}. Part (ii) follows trivially by an application of the standard H\"older inequality.
Let us move to the proof of (iii). First we observe that the validity of (ii) for some $0<\epsilon<1$ implies the reverse H\"older inequality
\[
|(b,c)|^\frac{\epsilon}{2}\int_{(a,b)} w^{1+\epsilon/2} \leq 6 \bigg(\int_{(a,c)}w\bigg)^{1+\epsilon/2}
\]
To see this we can assume that $w(a,c)/|(b,c)|=1$. Setting $E_\lambda\coloneqq\{x\in(a,b):\, w(x)>\lambda\}$ we have
\[
\frac{|E_\lambda|}{|(b,c)|}\leq \frac{1}{\lambda}\frac{w(E_\lambda)}{w(a,c)}\leq 2 \frac{1}{\lambda}\frac{|E_\lambda|^\frac{\epsilon}{1+\epsilon}}{|(b,c)|^\frac{\epsilon}{1+\epsilon}}
\]
and thus we have proved the estimate
\[
\frac{|E_\lambda|}{|(b,c)|}\leq 2 ^{1+\epsilon}\frac{1}{\lambda^{1+\epsilon}}.
\]
Using (ii) again this implies $\frac{w(E_\lambda)}{w(a,c)}\leq 2^{1+\epsilon}\frac{1}{\lambda^\epsilon}$. Now
\[\begin{split}
\int_{(a,b)} w^{1+\epsilon/2}&=\int_{(a,b)}w^\frac{\epsilon}{2}w \leq w(a,b)+\frac{\epsilon}{2}\int_1 ^\infty \lambda^{\epsilon/2-1} w(E_\lambda) d \lambda
\\
&\leq w(a,b)+2^{\epsilon} \epsilon\int_1 ^\infty\frac{w(a,c)}{\lambda^{1+\epsilon/2}}d\lambda \leq 5 w(a,c)=5|(b,c)|.
\end{split}
\]
This shows the claimed reverse H\"older inequality.
Now we show that a reverse H\"older inequality with exponent $1+\epsilon$ implies that $[w]_{A_\infty ^+}\lesssim 1/\epsilon$. For this fix $a<b<c$ and set $r\coloneqq 1+\epsilon$. We write $(a,b)=\cup_{j=1} ^\infty I_j$ where $I_j=(x_{j},x_{j+1}]$, and $x_0\coloneqq a$, $x_1\coloneqq \frac{a+b}{2}$ and $x_j\coloneqq \frac{x_{j-1}+b}{2}$.
Assuming that a reverse H\"older inequality, as in (i), holds with exponent $r=1+\epsilon$ we can use the bound $\|\mathsf{M}^-\|_{L^r\to L^r}\lesssim r'$ to estimate
\[
\begin{split}
\int_{(a,b)}\mathsf{M}^-(w\textnormal{\textbf 1}_{(a,b)})&\leq \sum_{j=1} ^\infty \int_{(a,b)} \mathsf{M}^-(w\textnormal{\textbf 1}_{I_j})=\sum_{j=1} ^\infty \int_{\cup_{k\geq j} I_k} \mathsf{M}^-(w\textnormal{\textbf 1}_{I_j})
\\
& \lesssim r' \sum_{j=1} ^\infty \Abs{\bigcup_{k\geq j}I_k}^\frac{1}{r'} \bigg( \int_{I_j}w^r\bigg)^\frac{1}{r}
\\
&\eqsim r'\sum_{j=1} ^\infty |I_j|^\frac{1}{r'} \bigg(\int_{I_j\cup I_{j+1}} w\bigg) |I_{j+1}|^{-\frac{1}{r'}}\lesssim r' w(a,b)
\end{split}
\]
and thus $[w]_{A_\infty ^+}\lesssim r'\eqsim \frac{1}{\epsilon}$ as we wanted.
\end{proof}
\section{Proof of the main theorem}
We divide the proof of the main theorem into two parts. In the first part we show Corollary~\ref{c.main1} which also shows that (i) implies (ii) in Theorem~\ref{t.main}.
\begin{proof}[Proof of Corollary~\ref{c.main1}] Let $E\subset \mathbb R$ be a compact subset of the real line with $0<w(E)<+\infty$ and let $\alpha>0$. We use Lemma~\ref{l.rsun} to write
\[
\{x\in\mathbb R:\, \mathsf{M}^+ \textnormal{\textbf 1}_E(x)>\alpha\}=\bigcup_j (a_j,b_j),
\]
where $\fint_{x}^{b_j} \chi_E > \alpha$ for all $x \in (a_j, b_j)$ and $\fint_{a_j}^{b_j}\chi_E = \alpha$ for each $j$.
Let $(a,b)\in\{(a_j,b_j)\}_j$. By Lemma~\ref{l.rsun} we have that $\fint_{a} ^{b} \textnormal{\textbf 1}_E=\alpha$ and $\fint_x ^b \textnormal{\textbf 1}_E > \alpha $ for all $x\in (a,b)$. Using an idea from \cite{Reyes} we choose an increasing sequence $\{x_k\}_{k=0} ^\infty$ such that $x_0\coloneqq a$, $(a,b)=\cup_{k\geq 1}(x_{k-1},x_k]$, and $\int_{x_{k-1}} ^{x_k} \textnormal{\textbf 1}_E =\int_{x_k} ^b \textnormal{\textbf 1}_E$ for all $k\geq 1$. Then we can estimate for all $k\geq 1$
\[
\begin{split}
w((x_{k-1},x_k]\setminus E) &=\frac{w((x_{k-1},x_k]\setminus E)}{w(x_{k-1},x_{k+1})} w(x_{k-1},x_{k+1})
\\
&\lesssim \bigg(\frac{|(x_{k-1},b)\setminus E|}{|(x_k,x_{k+1})|}\bigg)^\frac{1}{3[w]_{A_\infty ^+}} w(x_{k-1},x_{k+1})
\end{split}
\]
by (ii) of Theorem~\ref{t.ainfty}. By Lemma~\ref {l.rsun} we can conclude that $|(x_{k-1},b)\setminus E|\leq \frac{1-\alpha}{\alpha}\int_{x_{k-1}} ^b \textnormal{\textbf 1}_E$. Remembering the definition of the sequence $\{x_k\}_k$ we can further calculate
\[
\begin{split}
\int_{x_{k-1}} ^b\textnormal{\textbf 1}_E=\int_{x_{k-1}} ^{x_k} \textnormal{\textbf 1}_E + \int_{x_k} ^b \textnormal{\textbf 1}_E = 2\int_{x_k} ^b \textnormal{\textbf 1}_E.
\end{split}
\]
By a single recursion of this formula and another use of the definition of the sequence $\{x_k\}_k$ we thus have
\[
\int_{x_{k-1}} ^b\textnormal{\textbf 1}_E=4\int_{x_{k+1}} ^b \textnormal{\textbf 1}_E =4\int_{x_k} ^{x_{k+1}}\textnormal{\textbf 1}_E\leq 4(x_{k+1}-x_k).
\]
We have proved that
\[
w((x_{k-1},x_{k}]\setminus E)\lesssim (1-\alpha)^{(3[w]_{A_\infty ^+})^{-1}}w(x_{k-1},x_{k+1}).
\]
Summing over $k\geq 1$ we conclude that for every $j$ we have
\[
w((a_j,b_j)\setminus E))\lesssim (1-\alpha)^{(3[w]_{A_\infty ^+})^{-1}} w(a_j,b_j).
\]
Summing over $j$ we get the desired estimate
\[
\mathsf{C_w ^+}(\alpha)-1 \lesssim (1-\alpha)^{(c[w]_{A_\infty ^+})^{-1}}
\]
whenever $\alpha>1-e^{-c[w]_{A_\infty ^+}}$, for some numerical constant $c>1$.
We now proceed to exhibit the optimality of the Solyanik estimate just proved. For this let us assume that we have an estimate $\mathsf{C_w ^+}(\alpha)-1\lesssim(1-\alpha)^\frac{1}{\beta}$ for $\alpha>1-e^{-\beta}$. We will prove that for all $a<b<c$ and measurable $E\subset (a,b)$ we have the estimate
\[
\frac{w(E)}{w(a,c)}\lesssim \bigg(\frac{|E|}{|(b,c)|}\bigg)^\frac{1}{\beta}
\]
By Theorem~\ref{t.ainfty} this will imply that $w\in A_\infty ^+$ and $[w]_{A_\infty ^+}\lesssim \beta $ as claimed.
We now fix real numbers $a<b<c$ and consider a measurable set $E\subseteq (a,b)$. We set $E'\coloneqq (a,c)\setminus E$ and consider two cases.
If $|E|/|(b,c)|\geq e^{-\beta}$ then
\[
\frac{w(E)}{w(a,c)}\leq 1 = e e^{-1}\leq e \bigg(\frac{|E|}{|(b,c)|}\bigg)^\frac{1}{\beta}
\]
and we are done.
In the complementary case we have for all $x\in(a,b)$
\[
\begin{split}
\mathsf{M}^+\textnormal{\textbf 1}_{E'}(x)&\geq \frac{|E'\cap(x,c)|}{|(x,c)|}=\frac{|(b,c)|+|(x,b)|-|(x,b)\cap E|}{|(x,c)|}\geq 1 -\frac{|(x,b)\cap E|}{|(x,c)|}
\\
& \geq 1 -\frac{|E|}{|(b,c)|}>1-e^{-\beta}.
\end{split}
\]
Obviously $\mathsf{M}^+\textnormal{\textbf 1}_{E'}=1$ on $(b,c)$ and thus $(a,c) \subseteq \big\{x\in\mathbb R:\, \mathsf{M}^+\textnormal{\textbf 1}_{E'}(x)\geq 1-|E|/|{b,c}| \big\}$. As $1-|E|/|(b,c)|>1-e^{-\beta}$ we can use the assumed Solyanik estimate to conclude that
\[
w(a,c)\leq \mathsf{C_w ^+}\big(1-|E|/|(b,c)|\big)w(E')=\mathsf{C_w ^+}\big(1-|E|/|(b,c)|\big) (w(a,c)-w(E))
\]
and thus
\[
w(E)\leq \big(\mathsf{C_w ^+}\big(1-|E|/|(b,c)|\big)-1\big)w(a,c)\lesssim \big(\frac{|E|}{|(b,c)|}\big)^\frac{1}{\beta}w(a,c)
\]
as we wanted.
\end{proof}
It is trivial that (ii) implies (iii) in Theorem~\ref{t.main} so we move on to prove that (iii) implies (i). It will clearly suffice to show the following.
\begin{proposition}\label{p.restricted} Suppose that $w$ is a non-negative, locally integrable function that satisfies $\mathsf{C_w ^+}(\alpha_o)<+\infty$ for some $\alpha_o\in(0,1)$. Then $\mathsf{M}^+$ is of restricted weak type $(p,p)$ with respect to $w$ for $p=\log\mathsf{C_w ^+}(\alpha_o)/\log(1/\alpha_o)$, with constant $\mathsf{C_w ^+}(\alpha_o)^\frac{1}{p}$.
\end{proposition}
The proof of Proposition~\ref{p.restricted} relies on the notion of the \emph{Halo extension} of a set $E$, defined as follows. Given $\lambda\in(0,1)$ the Halo extension of $E$ is
\[
\mathcal H_\lambda ^+(E)\coloneqq \{x\in\mathbb R:\, \mathsf{M}^+\textnormal{\textbf 1}_E(x)>\lambda\}.
\]
We also set $\mathcal H^{+,0} _\lambda(E)\coloneqq E$ and for a positive integer $k>1$
\[
\mathcal H^{+,k} _\lambda(E)\coloneqq \mathcal H_\lambda ^+ (\mathcal H_\lambda ^{+,k-1}(E)).
\]
The heart of the matter is the following lemma.
\begin{lemma}\label{l.haloiter} Let $0<\lambda< \alpha<1$ and $E\subset \mathbb R$ be a measurable set with $0<w(E)<+\infty$. Then
\[
\mathcal H ^+ _\lambda(E)\subseteq \mathcal H^{+,N} _\alpha (E),
\]
where
\[
N =\Big\lceil\frac{\log\frac{1}{\lambda}}{\log\frac{1}{\alpha}}\Big\rceil.
\]
Here $\lceil x\rceil$ denotes the smallest integer which is no less than $x$.
\end{lemma}
\begin{proof} Let $(a,b)$ be one of the component intervals of $\mathcal H_\lambda ^+ (E)$ provided by Lemma~\ref{l.rsun}. The same lemma allows us to write $\mathcal H^+ _\alpha(E)$ as the union of disjoint intervals $\cup_{j} (\xi_j,\eta_j)$, where $\fint_{\xi_j}^{\eta_j} \chi_E = \alpha$. Since $\lambda<\alpha<1$ we have that $a,b\notin(\xi_j,\eta_j)$ for any $j$. Indeed, if say $a\in(\xi_j,\eta_j)$ then there exists some $h>0$ such that
\[
\alpha h <\int_a ^{a+h} \textnormal{\textbf 1}_E \leq \lambda h
\]
since $a\notin\mathcal H^+ _\lambda(E)$. This however contradicts the condition $\lambda<\alpha$. Similarly we see that $b\notin(\xi_j,\eta_j)$. Thus $\mathcal H_{\alpha} ^+(E)\cap (a,b)=\cup_{j\in J}(\xi_j,\eta_j)$. For $x\in (a,b)\setminus \mathcal H_\alpha ^+ (E)$ let $J_x\coloneqq \{j\in J:\, \xi_j>x\}$. Then if $x\in (a,b)\setminus \mathcal H_\alpha ^+(E) $ we can calculate
\[
|\mathcal H^+ _\alpha(E)\cap (x,b)|=\sum_{j\in J_x} |(\xi_j,\eta_j)|=\frac{1}{\alpha}\Abs{\bigcup_{j\in J_x}(\xi_j,\eta_j)\cap E}=\frac{1}{\alpha}|E\cap (x,b)|>\frac{\lambda}{\alpha}|(x,b)|,
\]
the last inequality following by Lemma~\ref{l.rsun}. Thus $(a,b)\subseteq \mathcal H^+ _{\frac{\lambda}{\alpha}}(\mathcal H^+ _\alpha(E))$ and accordingly $\mathcal H^+ _\lambda(E)\subseteq \mathcal H^+ _{\frac{\lambda}{\alpha}}(\mathcal H^+ _\alpha(E))$.
Let $K\geq 1$ be the smallest integer such that $\lambda/\alpha^K>\alpha$. Iterating the estimate above we conclude that
\[
\mathcal H_\lambda ^+ (E)\subseteq \mathcal H_{\frac{\lambda}{\alpha^K}} ^+ (\mathcal H_\alpha ^{+,K} (E))\subseteq \mathcal H_\alpha ^{+,K+1}(E).
\]
Note that $K$ satisfies $\lambda/\alpha^{K-1}< \alpha$ and thus $K+1 = \lceil \log(1/\lambda)/\log(1/\alpha) \rceil.$
\end{proof}
We can now give the proof of Proposition~\ref{p.restricted}.
\begin{proof}[Proof of Proposition~\ref{p.restricted}] Let $\lambda\in(0,1)$. Since $\mathsf{C_w ^+}(\alpha_o)<+\infty$ we trivially get that for all $\lambda>\alpha_o$ and every $E\subset \mathbb R$
\[
w(\{x\in\mathbb R:\,\mathsf{M}^+\textnormal{\textbf 1}_E(x)>\lambda\})\leq \mathsf{C_w ^+}(\alpha_o) w(E)\leq \frac{\mathsf{C_w ^+}(\alpha_o)}{\lambda^q}w(E)
\]
for any $q\geq 1$. It thus suffices to consider the case $0<\lambda<\alpha_o<1$. Let $1>\alpha>\alpha_o$. Then for any set $S\subset \mathbb R$ we have
\[
w(\mathcal H_\alpha ^+(S))\leq \mathsf{C_w ^+}(\alpha_o)w(S).
\]
By Lemma~\ref{l.haloiter} applied for $\alpha>\lambda>0$ we can conclude that
\[
w(\mathcal H_\lambda ^+(E))\leq \mathsf{C_w ^+}(\alpha_o) ^N w(E)
\]
with $N = \lceil {\log\frac{1}{\lambda}}/{\log\frac{1}{\alpha}} \rceil \leq {\log\frac{1}{\lambda}}/{\log\frac{1}{\alpha}} +1$. We get
\[
w(\{x\in\mathbb R:\,\mathsf{M}^+\textnormal{\textbf 1}_E(x)>\lambda\})\leq \mathsf{C_w ^+}(\alpha_o) \big(\frac{1}{\lambda}\big)^{\frac{\log\mathsf{C_w ^+}(\alpha_o)}{\log\frac{1}{\alpha}}}w(E)
\]
for all $\alpha>\alpha_o$. Letting $\alpha\to \alpha_o ^+$ we conclude that $\mathsf{M}^+$ is of restricted weak type $(p,p)$ with respect to $w$ for $p=\log\mathsf{C_w ^+}(\alpha_o)/\log(1/\alpha_o)$, with constant $\mathsf{C_w ^+}(\alpha_o)^\frac{1}{p}$.
\end{proof}
We conclude this section with the proof of Corollary~\ref{c.main2}
\begin{proof}[Proof of Corollary~\ref{c.main2}] Assuming that $w\in A_\infty ^+$ with constant $[w]_{A_\infty ^+}$, Theorem~\ref{t.main} implies that $\mathsf{C_w ^+}(\alpha_o)\lesssim 1$ for $\alpha_o=1-e^{-c[w]_{A_\infty ^+}}$, and $c>1$ is a numerical constant. Now Proposition~\ref{p.restricted} implies that $\mathsf{M}^+$ is of restricted weak type $(p,p)$ with respect to $w$ for $p \eqsim (\log(1/\alpha_o))^{-1}\eqsim e^{c[w]_{A_\infty ^+}}$ and thus $w\in A_p ^+$ for $p>e^{c[w]_{A_\infty ^+}}$. By carefully examining the interpolation constants and using the estimate
\[
\|\mathsf{M}^+ \|_{L^p(w)\to L^p(w)}\gtrsim [w]_{A_p ^+} ^\frac{1}{p}
\]
which is contained in \cite{MT}*{Theorem 1.4} we can conclude that $[w]_{A_p ^+}\leq e^{e^{c[w]_{A_\infty ^+}}}$ for some numerical constant $c>0$. See also \cite{HP2}*{p. 21} for the details of this calculation.
\end{proof}
\section{Local H\"older continuity} The methods of this paper easily imply that $\mathsf{C_w ^+}$ is locally H\"older continuous on $(0,1)$ whenever $w\in A_\infty ^+$.
\begin{corollary} There exists a numerical constant $c>1$ such that for all $w\in A_\infty ^+$ we have $\mathsf{C_w ^+} \in C^\frac{1}{ c[w]_{A_\infty ^+} }(0,1)$.
\end{corollary}
\begin{proof} Let $E\subset\mathbb R$ and $0<\lambda<\alpha<1$. By the proof of Lemma~\ref{l.haloiter} we have that
\[
\mathcal H^+ _\lambda(E)\subseteq \mathcal H^+ _\frac{\lambda}{\alpha}(\mathcal H^+ _\alpha(E)).
\]
We conclude that
\[
0<\mathsf{C_w ^+}(\lambda)-\mathsf{C_w ^+}(\lambda/\alpha)\leq\mathsf{C_w ^+}(\lambda/\alpha)(\mathsf{C_w ^+}(\alpha)-1).
\]
Using Corollary~\ref{c.main1} we get that for all $0<x<y<1$ we have
\[
|\mathsf{C_w ^+}(x)-\mathsf{C_w ^+}(y)|\lesssim \mathsf{C_w ^+}(y)\big(\frac{y-x}{x}\big)^\frac{1}{c[w]_{A_\infty ^+}}
\]
for some numerical constant $c>0$. The local H\"older continuity of $\mathsf{C_w ^+}$ follows immediately from the estimate above.
\end{proof}
\section*{Acknowledgment} We are indebted to Francisco Javier Mart{\'{\i}}n-Reyes for enlightening discussions related to the subject of the paper.
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Yale-led study reveals biology of leptin, the hunger hormone
By Ziba Kashef
Leptin's molecular structure (© stock.adobe.com)
In a new study, Yale researchers offer insight into leptin, a hormone that plays a key role in appetite, overeating, and obesity. Their findings advance knowledge about leptin and weight gain, and also suggest a potential strategy for developing future weight-loss treatments, they said.
The study, led by investigators at Yale and Harvard, was published the week of June 17, 2019, in the journal PNAS.
Leptin, which is secreted by fat cells, informs the brain when fuel stored in body fat and in the liver is becoming depleted. It has not been well understood how low leptin concentrations in plasma — the largest component of blood — increase appetite. The researchers studied the biology of leptin in rodents. They also investigated the influence of nerve cells in the brain known as AgRP neurons, which regulate eating behavior.
The researchers discovered that the mechanisms by which reductions in plasma leptin concentrations stimulate food intake are not limited to the brain, as previously thought. In rodents, fasting first activates leptin receptors in the brain, followed by an intermediary step that involves the endocrine system. This system includes the pituitary and adrenal glands, which secrete another hormone, corticosterone, that regulates energy, stress responses, and food intake.
The research team learned that this chain of events is required for leptin to stimulate hunger when food is restricted, or when diabetes is poorly controlled and plasma leptin concentrations drop below a critical threshold, said Gerald Shulman, M.D., the George R. Cowgill Professor of Medicine at Yale School of Medicine, and co-corresponding author of the study.
In further experiments, the researchers also showed that plasma corticosterone activates AgRP neurons, which increases hunger when either leptin or blood-sugar levels are low, Shulman noted. In humans, leptin and blood sugar drop when people diet.
These findings add to scientists' knowledge of leptin, which has been the focus of research on obesity and weight loss since its discovery in the 1990s. The study reveals "the basic biology of leptin, and how the endocrine system is mediating its effect to regulate food intake under conditions of starvation and poorly controlled diabetes," said Shulman.
The research also lends support to a different strategy for developing drugs that treat obesity. "It suggests that AGRP neurons may be an attractive therapeutic target," he said.
Other study authors are Rachel J. Perry, Jon M. Resch, Amelia M. Douglass, Joseph C. Madara, Aviva Rabin-Court, Hakan Kucukdereli, Chen Wu, Joongyu D. Song, and Bradford B. Lowell.
This study was funded by grants from the U.S. Public Health Service.
Leptin hormone spurs body's shift from burning carbs to fat
Lab 'failure' leads to potential treatment for obesity
Ziba Kashef: ziba.kashef@yale.edu, 203-436-9317 | {
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François d'Amboise (* 1550 in Paris; † 1619 in Paris) war ein französischer Schriftsteller. Er war der Sohn von Jean d'Amboise und der Bruder von Adrien d'Amboise, Bischof von Tréguier, und Jacques d'Amboise, Rektor der Universität Paris. Er war Conseiller am Parlement der Bretagne und Generaladvokat am Grand Conseil.
Er studierte am Collège de Navarre Rhetorik und Philosophie. 1568 lehrte er selbst Philosophie am Collège. Im gleichen Jahr schrieb er eine Élégie sur le trépas d'Anne de Montmorency. Im August 1572 begleitete er den Herzog von Anjou und späteren König von Frankreich Heinrich III. zur Königswahl nach Polen. Der Dramatiker Pierre de Larivey, sein Freund, und sein Vetter Louis de Clermont, seigneur de Bussy d'Amboise nahmen ebenfalls an der Reise teil.
1575 wurde er Procureur de la nation (nationaler Staatsanwalt), 1581 Anwalt des Königs an der Chambre du trésor, 1586 avocat général und Grand conseil des Königs. 1589 wurde er von Heinrich III. zu Ritter ernannt.
1596 ernannte ihn König Heinrich IV. zum Maître des requêtes, und 1604 zum Conseiller d'État (Staatsrat).
François d'Amboise war Herr von Vezeul, Bourot, Neuillé-le Lierre, Brouard, Lespinière, La Huardière in der Touraine, Houvoy, Malnoue, Courserin, Plessis-Bourré, Hémery sowie Baron von La Chartre-sur-le-Loir.
Er war befreundet mit Robert Garnier, Pierre Matthieu und Gilles Bourdin, bei dem er die Dramatiker Guillaume-Gabriel Le Breton, Odet de Turnèbe und Pierre de Larivey kennenlernte.
Er schrieb unter anderem eine Verskomödie, Néapolitaines (1584), und einige Gedichte und gab 1616 die Werke von Petrus Abaelardus heraus.
Sein Sohn war Antoine d'Amboise, Herr von Clos Lucé, Oberst im Regiment Amboise und Feldmarschall.
François d'Amboise wurde in der Kirche Saint-Paul-et-Saint-Louis im Marais begraben.
Er ist kein Mitglied des Hauses von Amboise.
Werke
Regrets facétieux et plaisantes harengues funèbres sur la mort de divers animaux pour passer le temps et resveiller les esprits mélencoliques, non moins remplis d'éloquence que d'utilité et gaillardise, Paris, N. Bonfons, 1583.
Néapolitaines, Komödie, 1584
Lettres d'Héloise et d'Abélard, 1616
Literatur
Histoire de la ville et du château de la Chartre-sur-le-Loir par l'abbé L-J Denis - Imp. Aveline - 1965 . P.75 et 76
Einzelnachweise
Autor
Literatur (Französisch)
Franzose
Geboren 1550
Gestorben 1619
Mann | {
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094.0863 11 Supervillain 15-1 Ortiz J L Servis John C.
093.7424 2 The Exception 12-1 Batista J A Gonzalez Oscar M.
093.2774 9 Impossible Mission 20-1 Hernandez. Jr. B J McPeek Kenneth G.
092.6173 7 Calamity Kat 12-1 Juarez N Pino Michael V.
089.1052 12 Imperial Majesty(b+) 30-1 Reyes L Monjes Ruben A. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,320 |
Jared Roberts has plenty of reasons to be angry. It has been nearly a year since his father, Joe, simply vanished after being fired from him job at Stone Industries because management believed he was stealing from the factory. Now, at thirteen years of age, Jared works part-time for the same company, while his mother, Mary, toils there full-time in addition to having taken a part-time job at a store in order to provide for Jared and his three siblings. The family was never wealthy to begin with, residing in the poorest area of town known as "The Sink." Ironically, Stone Industries later caught the real thief and apologized to the Roberts family, but the damage was done — Joe has not been seen or heard from and Mary is struggling to accept that he is probably dead and will never return home. Jared and his thug-like friends are planning to do something terrible to the Stone family.
Meanwhile, Sarah Stone, the only daughter of widower Jonas, owner of Stone Industries, is a privileged and haughty thirteen-year-old living in a mansion and surrounded by servants. The only thing she isn't surrounded by is love. Sarah's mother was killed by a drunk driver a few years ago, robbing Jonas of the love of his life, and his focus has been on keeping busy running his business in order to manage his heartbreak. But in the process, his daughter has become a casualty, as well. Demanding, selfish, and cripplingly lonely, Sarah attends a private school but has no true friends and she secretly longs for her father's attention.
Because of a nasty prank Sarah played on her chauffeur/chaperone, he quit. As the story opens, Sarah's frustration over not being able to go to the mall to shop — one of her few pleasure — amuses her personal maid, who is barely able to tolerate her. But a strange little man named Nicholas is about to enter Sarah's life, as well as Jared's, and change everything.
With Mary's Son ~ A Tale of Christmas, first-time author Darryl Nyznyk borrows elements from other classic stories such as Mary Poppins, Miracle on 34th Street, and It's a Wonderful Life to create a charming, modern story about lost souls who come to understand the true meaning of Christmas. Unlike those other stories, Mary's Son ~ A Tale of Christmas welds the secular and religious messages of the season together. The result is a sweet tale about understanding the reasons we celebrate the birth of the Christ child each year.
When Sarah looks out her bedroom window, she sees a strange little man being admitted onto the property and is instantly mesmerized by him. He insists that his name is simply "Nicholas" because he hasn't used his last name in so many years that he can no longer remember what it is. He claims to be applying for the just-vacated chauffeur/chaperone position even though it hasn't yet been advertised and is capable of some Mary Poppins-esque magic that confounds but beguiles Sarah, as well as the suspicious household staff and her father, Jonas. Thus, the adventure begins.
Whether Mary's Son ~ A Tale of Christmas is a true story is not the real issue; for it is not the facts that matter but rather the message of love that comes from the truth of the birth of Jesus Christ.
In Nyznyk's short, entertaining story, Nicholas enchants everyone who meets him as he sets about accomplishing his goal: to teach Sarah and Jared what Christmas is really all about. But in Nyznyk's hands, Nicholas is much more than just a jolly elf with mysterious powers who believes in the magic of the Christmas season. Rather, he believes that God became flesh in that manger all those years ago. He talks to God, prays for assistance, and reminds folks of why Christmas exists. When his initial efforts to touch Sarah and Jared's hardened hearts fail, he resorts to drastic measures, teleporting them back in time to that most magical and mysterious night the world has ever known so that they can experience first-hand how Christmas came to be and return home to spread good cheer.
Sarah learns that Christmas is about giving rather than receiving gifts, Jared releases his anger, and by the time Christmas arrives, the lives and hearts of all concerned have been transformed by Nicholas. Because Mary's Son ~ A Tale of Christmas is a holiday story, it will come as no surprise that it has a happy ending and a concise message about what really matters — family. It is a delightful, quick read that is appropriate for children and families. In fact, the book is the embodiment of a story that Nyznyk had heard years earlier and told in 1994 to a group of seven-and-eight-year-olds attending his daughter's slumber party, so its re-telling could easily become a tradition for your family.
Click here to read author Darryl Nyznyk's guest post, Remembering Jesus at Christmas!
I read Mary's Son ~ A Tale of Christmas in conjunction with the 2011 Read 'n' Review and Outdo Yourself Reading Challenges.
Author Darryl Nyznyk has generously provided one copy of Mary's Son ~ A Tale of Christmas to be awarded to a lucky Colloquium reader! Submit your entry utilizing the Rafflecopter widget.
Just my hubby and the kids. dinner and a Xmas movie. Hope you had a good one too!
Spent it with family and relatives, ate a dinner with salmon and salad..a lot of salad. Exchanged gifts and caught up with my cousins who were out of state.
Well, we didn't get to celebrate this year. We went instead to take care of my daughter in law who has bacterial meningitis. We had the spirit of christmas and remembered the true meaning but no decorations were up since she had been in the hospital for so long. I know it's not what you wanted to hear!
We always spend all day with extended family at my grandparents' house.
We spent Christmas Eve at my son's house and Christmas Day at my daughter's. There were lots of relatives and we all enjoyed spending time together!
My son was married this past summer, so this was our first Christmas with our daughter-in-law. It was fabulous.
I celebrated Christmas with my brother, sister-in-law, niece, sister, mom, and dad on Christmas Eve. We had turkey for dinner, then opened gifts, and then went to church.
We had our daughter and son-in-law with our two gandkids. Had a great meal, opened presents, and had a lot of fun.
Had a great year….found out on Friday that I am now cancer free after a long four year fight. so had a wonderful year with the kids and their spouses home.
Lots of family and friends, good food, old memories and pictures.
Our Christmas celebration was wonderful–with my best friend, my mom, and my daughter. We have great fun!
Christmas was celebrated at my 82 year old Mother's home. My 2 daughters were able to come home plus all my siblings and their families were there. Wonderful time together. I love family time.
My adult children and their dates were all home for Christmas. Everything went well.
I spent Christmas day at my brother's house. It was fabulous!
Lots of family, food, and presents! | {
"redpajama_set_name": "RedPajamaC4"
} | 7,672 |
\section{Introduction}
A dyadic stationary subdivision operator $S_a$ acts on a sequence $c :
\ZZ \to \RR$ by means of the convolution like and hence stationary
operation
$$
c \mapsto S_a c := \sum_{\alpha \in \ZZ} a ( \cdot - 2\alpha ) \, c(\alpha).
$$
Here $a$, the so-called \emph{mask} of the subdivision operator, is a
finitely supported sequence. There are various ways of generalizing
subdivision operators. For example, one can consider several variables, dilation factors greater than $2$ or even expansive dilation
matrices, or vector- or matrix-valued data which requires the mask
to be a finitely supported matrix-valued sequence, cf.\ \cite{cavaretta91}. A \emph{subdivision
scheme} is an iteration of subdivision operators that may even depend
on the level of iteration, where the $n$th iteration is seen as data
defined on the grid $2^{-n} \ZZ$. Since these grids get finer and
finer, there is the concept of a \emph{limit function} of subdivision
schemes, cf. \cite{cavaretta91}.
\emph{Hermite subdivision} is a special case of subdivision operators
with matrix masks acting on
vector data, where the components of these vectors are interpreted as
consecutive derivatives. Such schemes have been considered and
analyzed first in \cite{dyn95,merrien92}.
The chain rule then enforces a subdivision
process of a mildly level-de\-pendent form that consists of a left and
right multiplication by dyadic diagonal matrices. Also the notion of
\emph{convergence} is special for Hermite subdivision schemes: If the
input data is in $\RR^{d+1}$, the limit function is vector-valued of size $d+1$ and
consists of a $C^d$ function and its derivatives up to order $d$.
It is well--known in subdivision theory \cite{cavaretta91,dyn92} that
the regularity of a limit function implies the preservation of certain
polynomials by the subdivision scheme. For Hermite
subdivision schemes this is usually formulated in terms of the \emph{spectral
condition} and has been related to Taylor polynomials in
\cite{dubuc09}. In
\cite{merrienSauer12:_from_hermit} it is shown that the spectral condition is essentially
equivalent to an operator factorization of the form
\begin{equation}
\label{eq:TSASBT}
T S_{\Ab} = S_{\Bb} T
\end{equation}
where $T$ is the so--called \emph{Taylor operator}. $T$ is a discrete version
of the Taylor formula and relates successive entries of vector-valued data
in accordance with the assumption that they are consecutive
derivatives. Moreover, the contractivity of $S_{\Bb}$ plays an important role in the analysis of convergence, cf. \cite{merrienSauer12:_from_hermit}.
In \cite{MerrienSauer18S} it is conjectured that convergence implies a generalized spectral condition of order at least $d$ to be satisfied. This is in accordance with similar results for scalar subdivision schemes, cf.\ \cite{cavaretta91}. Therefore, if one is interested in Hermite schemes of regularity $n > d$, that is, limit functions consisting of a $C^n$ function and its first $d$ derivatives, the Hermite scheme should satisfy a spectral condition of order at least $n$. Schemes of regularity $n>d$ are considered in e.g.\ \cite{conti14,jeong17,moosmueller18}.
We call this phenomenon \emph{polynomial overreproduction} and it is the main topic of this paper. We describe conditions under which the subdivision operator $S_{\Ab}$ satisfies a spectral
condition of degree \emph{higher} than $d$, providing a generalization of \cite{moosmueller18b}. It turns out that
this property fits well into the existing theory: $S_\Ab$ has
to have a factorization by means of a Taylor operator as in
\eqref{eq:TSASBT} and the \emph{rank one} vector subdivision
scheme $S_\Bb$ has
to be factorizable in the sense defined in
\cite{MicchelliSauer97a,micchelli98}.
There is, however, a peculiarity:
The matrices that appear in the factorizations of rank one schemes
are derived from the spectral condition, but do not depend on the concrete choice of $\Ab$.
The paper is organized as follows. We start by introducing notation and give
detailed definitions of the above properties in
Section~\ref{sec:Notation}; factorizations of subdivision operators
are revised in Section~\ref{sec:factorization}. In
Section~\ref{sec:Stirling} we introduce Stirling numbers and their connection to $p$--Cauchy numbers.
Based on the technical preliminaries of Section~\ref{sec:aux},
the main result of the paper, namely the factorization with respect to
the \emph{augmented Taylor operator}, is given in
Section~\ref{sec:augmented} with a rather short proof.
\section{Notation and subdivision schemes}
\label{sec:Notation}
Throughout this paper, $d$ denotes an integer, and $d\geq 1$. Vectors
in $\RR^{d+1}$ are written as $\cb$, that is, with boldface lowercase
letters, while matrices $\Ab$ are written with boldface uppercase
letters. The standard basis in $\RR^{d+1}$ is denoted by
$\eb_0,\ldots,\eb_d$. The identity matrix of dimension $d+1
$ is denoted by $\Ib_{d+1}$. We also use the Matlab-like notation
$\cb_{k:\ell}$ to extract subvectors.
Furthermore, for a vector $\cb \in \RR^{d+1}$ we introduce the notation
$
\hat{\cb} = \left(
\cb_0,
\ldots,
\cb_{d-1},
0
\right)^T
$
for the canonical embedding of $\cb$ into $\RR^{d+1+k}$, $k \ge 1$.
The space of all polynomials in one variable is written as $\Pi$, while $\Pi_n$ denotes all such polynomials with degree at most $n$.
By $\ell^{d+1}(\ZZ)$ we denote the space of all sequences $\cb: \ZZ \to \RR^{d+1}$, while $\ell^{(d+1)\times (d+1)}(\ZZ)$ is the space of matrix-valued sequences $\Ab: \ZZ \to \RR^{(d+1)\times (d+1)}$. We use the same notation for vectors (matrices) and sequences of vectors (matrices); it will be clear from the context what is meant.
The notation $\ell^{d+1}_{00}(\ZZ)$ and $\ell^{(d+1)\times
(d+1)}_{00}(\ZZ)$ is used to denote sequences with finite support.
To distinguish them from input data for subdivision schemes, we denote
sequences of vector valued parameters by $\cb_n, n \in \NN$, in
accordance with the notation $\eb_0,\dots,\eb_d$ of the unit coordinate vectors.
The
$k-$th entry of an element of such a sequence is accessed by
$\cb_{n,k}, k=0,\ldots,d, n\in \NN$.
The \emph{forward difference operator} $\Delta$ is used both in the context of functions and sequences. If $f$ is a function, then
$(\Delta f)(x) = f(x+1)-f(x), x\in \RR$. For $\cb \in \ell^{d+1}(\ZZ)$ we have $(\Delta \cb)(\alpha)=\cb(\alpha+1)-\cb(\alpha), \alpha \in \ZZ$. Higher order forward difference operators are defined by $\Delta^n=\Delta(\Delta^{n-1})$, $n\geq 1$, with $\Delta^0=\operatorname{id}$.
A \emph{stationary subdivision operator} with \emph{mask} $\Ab \in \ell^{(d+1)\times (d+1)}_{00}(\ZZ)$ is a map $S_{\Ab}:\ell^{d+1}(\ZZ) \to \ell^{d+1}(\ZZ)$ defined by
\begin{equation*}
\left(S_{\Ab}\cb\right)(\alpha)=\sum_{\beta \in \ZZ}\Ab(\alpha-2\beta)\cb(\beta), \quad \alpha \in \ZZ, \quad \cb \in \ell^{d+1}(\ZZ).
\end{equation*}
We consider a vector $\cb \in \RR^{d+1}$ as a constant sequence, so that $S_{\Ab}\cb$ means the application of $S_{\Ab}$ to the constant sequence $\cb(\alpha) = \cb, \alpha \in \ZZ$.
A level-dependent \emph{subdivision scheme} $(S_{\Ab^{[n]}},n\in \NN)$ is the procedure of iteratively constructing vector-valued sequences by
\begin{equation}\label{eq:sds}
\cb^{[n+1]}=S_{\Ab^{[n]}}\cb^{[n]},\quad n\in \NN,
\end{equation}
from initial data $\cb^{[0]} \in \ell^{d+1}(\ZZ)$.
In this paper we consider two cases of such subdivision schemes based
on stationary subdivision operators:
\emph{vector subdivision schemes} which use the same mask in every
iteration level, i.e.\ $\Ab^{[n]}=\Ab, n \in \NN$,
cf. \cite{micchelli98}, and
\emph{Hermite subdivision schemes} which use the mildly level-dependent masks
\begin{equation}\label{eq:stationary_mask}
\Ab^{[n]}=\Db^{-n-1}\,\Ab\,\Db^{n}
\end{equation}
where $\Db=\operatorname{diag}\left(1, 2^{-1},\ldots,2^{-d} \right)$ and $\Ab \in \ell^{(d+1) \times (d+1)}_{00} (\ZZ)$ is fixed. In Hermite subdivision, the data $c^{[n]}$ represents function and consecutive derivative values at $2^{-n}\alpha,\alpha\in \ZZ$, leading to the mask \eqref{eq:stationary_mask} via the chain rule.
For $p \in \Pi$ we define the vector-valued function
\begin{equation}\label{eq:spectral_polynomial}
\vb(p)(x) := \left( p^{(k)}(x) : k = 0,\dots,d \right)^T, \quad x\in \RR.
\end{equation}
We also consider $\vb(p)$ as a sequence in $\ell^{d+1}(\ZZ)$, by evaluating at integers only. The particular meaning of $\vb(p)$ will be clear from the context.
A Hermite subdivision scheme is said to satisfy the \emph{spectral condition of order $n\geq d$} if there exist $p_k \in \Pi_k$, normalized as $p_k (x) =
\frac1{k!} x^k + \cdots$, such that
\begin{equation}
\label{eq:SpecCond}
S_\Ab \vb (p_k) = 2^{-k} \vb (p_k), \qquad k=0,\dots,n.
\end{equation}
The spectral condition for $n=d$ has first been introduced by
\cite{dubuc09}, see also \cite{merrienSauer12:_from_hermit}. The case
$n > d$ is a higher order spectral condition studied in
\cite{conti14}, and we denote it by \emph{polynomial overreproduction}. The recent paper \cite{MerrienSauer18S} introduces
\emph{spectral chains}, which generalize \eqref{eq:SpecCond}. We
briefly discuss spectral chains in Section \ref{sec:augmented}.
While the spectral condition of order $d$ is important for factorization of Hermite subdivision operators \cite{merrienSauer12:_from_hermit}, it has been shown that it is not necessary for convergence \cite{merrienSauer17:_exten_hermit,MerrienSauer18S}.
\section{Factorization of subdivision operators}\label{sec:factorization}
The factorization of subdivision operators is a standard method for
proving convergence of the associated subdivision schemes and
regularity of their limits.
More precisely, convergence of subdivision schemes can often be
characterized by a factorization and \emph{contractivity} of the
factor scheme while regularity of the limit functions is described by
a factorization and \emph{convergence} of the factor scheme. It is
important, however, to emphasize that the nature of the factorization
has to be adapted to the nature of the subdivision scheme. In
particular, although vector and Hermite subdivision schemes both act
by means of matrix masks on vector valued data, the associated
factorizations are of a siginificantly different nature that reflects
the different conceptual nature of the schemes.
In this paper, we are concerned with
factorizations of rank $1$ vector schemes as derived in
\cite{charina05,MicchelliSauer97a,micchelli98,sauer02} and Taylor
factorizations of Hermite schemes
\cite{conti16,merrienSauer12:_from_hermit,MerrienSauer18S}. We now
introduce these concepts.
Following \cite{micchelli98}, for a subdivision operator $S_{\Bb}$, we define
\begin{equation}
\cE_{\Bb} = \left\{\cb \in \RR^{d+1}: S_{\Bb}\cb = \cb \right\},
\end{equation}
which is the eigenspace (of constant sequences) of $S_{\Bb}$ with
respect to the eigenvalue $1$. The dimension $\dim \cE_\Bb$ is called
the \emph{rank} of the subdivision scheme. In this paper we are only
concerned with \emph{rank $1$ schemes}, i.e.\ operators $S_{\Bb}$
satisfying $\dim\, \cE_{\Bb}=1$, cf.\ \cite{MicchelliSauer97a}.
We call a matrix $\Vb = \left(v_0, \ldots, v_d\right)$ with $v_j \in \RR^{d+1}, j=0,\ldots,d$, an $\cE_{\Bb}$-\emph{generator} if $\left\{v_0, \ldots, v_d\right\}$ is a basis of $\RR^{d+1}$ and if $v_d$ spans $\cE_{\Bb}$.
With the operator
\begin{equation}
D :=
\begin{pmatrix}
\Ib_d & \\ & \Delta
\end{pmatrix}
\end{equation}
the following result has been shown, cf. \cite{MicchelliSauer97a,micchelli98}:
\begin{lemma}\label{lem:rank1_factorization}
Let $S_{\Bb}$ be a subdivision operator with $\dim\,\cE_{\Bb}=1$. If $\Vb$ is an $\cE_{\Bb}$-generator, then there exists a subdivision operator $S_{\Cb}$ such that
\begin{equation*}
D\Vb^{-1}\,S_{\Bb}=S_{\Cb}\,D\Vb^{-1}.
\end{equation*}
Furthermore, $\dim\,\cE_{\Cb}=1$.
\end{lemma}
From \cite{merrienSauer12:_from_hermit} recall the
\emph{(incomplete) Taylor operator}
\begin{equation*}
T_d =
\begin{pmatrix}
\Delta & -1 & -\frac12 & \dots & -\frac{1}{d!} \\
& \ddots & \ddots & \ddots & \vdots \\
& & \Delta & -1 & -\frac{1}{2!} \\
& & & \Delta & -1 \\
& & & & 1 \\
\end{pmatrix}
\end{equation*}
and the \emph{complete Taylor operator}
\begin{equation*}
\widetilde T_d =
D \, T_d = \begin{pmatrix}
\Delta & -1 & -\frac12 & \dots & -\frac{1}{d!} \\
& \ddots & \ddots & \ddots & \vdots \\
& & \Delta & -1 & -\frac{1}{2!} \\
& & & \Delta & -1 \\
& & & & \Delta \\
\end{pmatrix}.
\end{equation*}
We also consider the following operator which has been defined and studied in \cite{dubuc09}:
$$
T_d' = \begin{pmatrix}
\Delta & -1 & \dots & -\frac{1}{(d-1)!}&0 \\
& \ddots & \ddots & \vdots & \vdots\\
& & \Delta & -1 & 0 \\
& & & \Delta & 0 \\
& & & & 1 \\
\end{pmatrix}.
$$
We furthermore define $\widetilde{T}_0=\Delta$ and $T_0=T_0'=\operatorname{id}$.
Generalizations of these Taylor operators have been introduced in
\cite{MerrienSauer18S}; we discuss them in Section \ref{sec:augmented}.
It has been shown in \cite[Theorem 4]{merrienSauer12:_from_hermit} that a subdivision operator $S_{\Ab}$ satisfying the
spectral condition of order $d$ \eqref{eq:SpecCond} can be factorized with respect to the Taylor operator: There exists a subdivision operator $S_{\Bb}$ such that
\begin{equation}\label{eq:Taylor_factorization}
T_d S_\Ab = 2^{-d} S_\Bb T_d.
\end{equation}
If $S_{\Ab}$ factorizes as in \eqref{eq:Taylor_factorization}, but stepwise, i.e.\ with respect to operators
\begin{equation*}
\begin{pmatrix}
T_k & \\
& \Ib_{d-k}
\end{pmatrix}, \quad k = 0,\ldots, d,
\end{equation*}
then this
is even a characterization of the spectral condition of order $d$ \eqref{eq:SpecCond},
cf. \cite[Corollary 2.12]{merrienSauer17:_exten_hermit}.
Furthermore, $\cE_{\Bb}$ is spanned by $\eb_d$. Therefore $\Vb = \Ib_{d+1}$ is an $\cE_{\Bb}$-generator and by Lemma \ref{lem:rank1_factorization} there exists a subdivision operator $S_{\Cb}$ such that
\begin{equation*}
D\, S_{\Bb} = S_{\Cb}\,D.
\end{equation*}
The latter implies
\begin{equation*}
\widetilde T_d S_\Ab
= D \, T_d S_\Ab = 2^{-d} D S_\Bb T_d = 2^{-d} S_{\Cb} D T_d
= 2^{-d} S_{\Cb} \widetilde T_d,
\end{equation*}
which is the complete Taylor factorization of \cite[Theorem 4]{merrienSauer12:_from_hermit}:
\begin{equation}\label{eq:level0}
\widetilde T_d S_\Ab = 2^{-d} \, S_{\Cb} \widetilde T_d.
\end{equation}
In this paper we prove a generalization of \eqref{eq:level0} to operators $S_{\Ab}$ which satisfy the spectral condition \eqref{eq:SpecCond} for $n>d$ (Theorem \ref{thm:factorization}). In particular we prove that every such operator factorizes with respect to the \emph{augmented Taylor operator} of order $n$:
\begin{definition}[Augmented Taylor operators] \label{def:augmented}
For $d\geq 1$ and $n\geq d$ we define the \emph{augmented Taylor operator} of order $n$ by
\begin{equation*}
\widetilde T_d^n :=
\begin{pmatrix}
\widetilde{T}_{d-1} & -\displaystyle{\sum_{k=0}^{n-d}} G_k^{d:1} \Delta^{k} \\
& \Delta^{n+1-d} \\
\end{pmatrix}
=
\begin{pmatrix}
\Delta & -1 & -\frac12 & \dots & -\frac{1}{(d-1)!} &
-\displaystyle{\sum_{k=0}^{n-d}} G_k^d\Delta^k\\
& \ddots & \ddots & & \vdots & \vdots\\
& & \ddots & \ddots & \vdots & \vdots\\
& & & \Delta & -1 & -\displaystyle{\sum_{k=0}^{n-d}} G_k^2\Delta^k\\[0.2cm]
& & & & \Delta & -\displaystyle{\sum_{k=0}^{n-d}} G_k^1\Delta^k\\[0.2cm]
& & & & & \Delta^{n+1-d}
\end{pmatrix},
\end{equation*}
where $G_k^{d:1}=\left(G_k^d,G_k^{d-1},\ldots,G_k^1 \right)^T$, and
$G_k^{\ell}, k \geq 0, \ell \geq 1$ are the \emph{coefficients for
repeated integration with forward differences} \cite{salzer47}.
\end{definition}
\begin{remark}
Normalizing the coefficients $G_k^n$ as in
\eqref{eq:p_Cauchy} leads to the \emph{$p$--Cauchy numbers of the
first kind} , see \cite{Rahmani16}. Since $G_k^1$
are known, among others, as \emph{Gregory coefficients},
cf. \cite{blagouchine16}, one could call these numbers
\emph{generalized Gregory coefficients}. We discuss them in more
detail in Section \ref{sec:Stirling}.
\end{remark}
The existence of such a factorization follows from combining
the Taylor factorization \eqref{eq:Taylor_factorization} of \cite{merrienSauer12:_from_hermit} with iterated factorizations for rank $1$ schemes (Lemma \ref{lem:rank1_factorization}) of
\cite{MicchelliSauer97a,micchelli98}. The contribution of this paper is the explicit computation of the augmented Taylor operators via computing $\cE_{\Bb_j}$ for every iteration $j=d,\ldots,n$ of rank $1$ factorizations. In particular, we show that the spectral condition \eqref{eq:SpecCond}, but \emph{not} the choice of spectral polynomials, already determines all $\cE_{\Bb_j}, j=d,\ldots,n$. We thus also extend the results of \cite{moosmueller18b}.
\section{Stirling and $p$--Cauchy numbers}
\label{sec:Stirling}
Following \cite{graham94}, we recall the definition of Stirling numbers.
The \emph{Stirling numbers of the first kind}, denoted by $\stiri{n}{m}$, count the numbers of ways to arrange $n$ elements into $m$ cycles. From the initial conditions
\begin{equation*}
\stiri{0}{0}=1, \quad \stiri{n}{0}=\stiri{0}{n}=0, \quad n\geq 1,
\end{equation*}
they can be computed via the following recurrence relation:
\begin{equation*}
\stiri{n+1}{m}=n\stiri{n}{m}+\stiri{n}{m-1}, \quad m\geq 1.
\end{equation*}
The \emph{signed Stirling numbers of the first kind} are defined by
\begin{equation}\label{def:signed_stir1}
s(n,m) = (-1)^{n-m}\stiri{n}{m}.
\end{equation}
They satisfy the recurrence relation
\begin{equation}\label{signed_stir1_recur}
s(n+1,m)=s(n,m-1)-n\, s(n,m),
\end{equation}
with initial conditions
\begin{equation*}
s(n,n)=1, \quad s(n,m)=0 \quad \text{ if } m<n \text{ or } n<m.
\end{equation*}
The \emph{Stirling numbers of the second kind}, denoted by $\stir{n}{m}$, count the number of ways to split a set of $n$ elements into $m$ non-empty subsets.
They satisfy the following recurrence relation
\begin{equation}\label{stir2_rec}
\stir{n+1}{m} = m \, \stir{n}{m}+\stir{n}{m-1}, \quad m \geq 1.
\end{equation}
with initial conditions
\begin{equation*}
\stir{0}{0}=1, \quad \stir{n}{0}=\stir{0}{n}=0, \quad n \geq 1.
\end{equation*}
The Stirling numbers of the second kind can be computed using Binomial coefficients
\begin{equation*}
\stir{n}{m}=\frac{1}{m!}\sum_{j=0}^m {m \choose j} (-1)^{m-j}j^n.
\end{equation*}
We also need the following relation between the Stirling numbers of the second kind and the Binomial coefficients (see \cite[Eq. 6.15]{graham94}):
\begin{equation}\label{eq:Stir_Bino}
\stir{n+1}{m+1}=\sum_{k=m}^n {n \choose k}\stir{k}{m}.
\end{equation}
Following \cite{salzer47}, we define the \emph{coefficients for repeated integration with forward differences}, $G_n^k$ for $k,n\geq 1$, by
\begin{equation}\label{eq:coeff_int1}
G_n^{1}=\frac{1}{n!}\,\int_{0}^1 x(x-1)\cdots (x-n+1)
dx, \quad n \geq 1,
\end{equation}
and
\begin{equation}\label{eq:coeff_int}
G_n^{k}=\frac{1}{n!}\,\int_{0}^1\int_0^{x_2} \cdots \int_0^{x_k} x(x-1)\cdots (x-n+1)\,
dx dx_k \cdots dx_2, \quad n \geq 1, k\geq 2.
\end{equation}
We also define
\begin{equation}\label{eq:coeff_0}
G^k_0=\frac{1}{k!}, \quad k\geq 1.
\end{equation}
The coefficients $G_n^k$ are connected to the \emph{$p$--Cauchy numbers of the first kind}, $\cC_{n,p}$, defined in \cite{Rahmani16}, via
\begin{equation}\label{eq:p_Cauchy}
\cC_{n,p-1} = n!\,p! \,G_n^p.
\end{equation}
The sequence $G_n^1$ are the \emph{Gregory coefficients}, since \eqref{eq:coeff_int1} is their well-known integral representation, see e.g.\ \cite{merlini06}. The Gregory coefficients are a well-studied sequence in number theory and are also known as the \emph{Cauchy numbers of the first kind}, the \emph{Bernoulli numbers of the second kind} and the \emph{reciprocal logarithmic numbers}, see e.g.\ \cite{blagouchine17,kowalenko10,merlini06}.
In this sense, the coefficients in \eqref{eq:coeff_int} are a generalization of the Gregory coefficients. Another generalization of the Gregory coefficients can be found in \cite[Eq.\ (63)]{blagouchine18}.
In \cite{salzer47}, the following recursion is shown to hold:
\begin{equation}\label{eq:coeff_rec}
G_n^k=\frac{1}{1-k}\left((n-1)G_n^{k-1}+(n+1)G_{n+1}^{k-1}\right), \quad k\geq 2, n\geq 1,
\end{equation}
compare also to the equivalent recursion for $p$--Cauchy numbers in \cite[Theorem 2.5]{Rahmani16}.
Via \eqref{eq:p_Cauchy}, Corollary 2.3 \& Theorem 2.2 of \cite{Rahmani16} imply
\begin{equation}\label{lem:stir2_coeff}
\sum_{r=1}^{j}\stir{j}{r}\,r!\,G_r^k=\frac{1}{(j+1)\cdots(j+k)}, \quad j,k\geq 1
\end{equation}
and
\begin{equation}\label{eq:stir1_coeff}
G_n^k=\frac{1}{n!}\sum_{j=1}^n \frac{s(n,j)}{(j+1)\cdots (j+k)}, \quad j,k\geq 1.
\end{equation}
For $k=1$, \eqref{lem:stir2_coeff} and \eqref{eq:stir1_coeff} are proved in \cite{merlini06}.
\begin{remark}
The case $k=2$ of \eqref{eq:stir1_coeff} can also be found on oeis.org (sequence A002687 resp.\ A002688) under ``formula''.
\end{remark}
\section{Auxiliary results}\label{sec:aux}
We start by proving that the Stirling numbers of the second kind relate forward differences to derivatives:
\begin{lemma}\label{lem:high_forward_diff}
For $p \in \Pi_n, \ell \leq n,\, 1\leq k\leq n-\ell$ we have
\begin{equation*}
\frac{1}{k!}\,\Delta^{k}p^{(\ell)}=\sum_{m=k}^{n-\ell} \frac{1}{m!}\stir{m}{k}p^{(m+\ell)}.
\end{equation*}
\end{lemma}
\begin{pf}
We prove this by induction on $k$. For $k=1$
the Taylor formula gives
\begin{equation*}
\Delta p (x) = p(x+1) - p(x) = \sum_{m=1}^{n} \frac1{m!} p^{(m)}
(x)
\end{equation*}
and for $\ell \leq n$
\begin{equation}\label{eq:Delta}
\Delta p^{(\ell)} (x) = \sum_{m=1}^{n-\ell} \frac{1}{m!} p^{(\ell+m)} (x).
\end{equation}
We assume the statement is true for $k$ and prove it for $k+1$ using \eqref{stir2_rec}, \eqref{eq:Stir_Bino} and \eqref{eq:Delta}:
\begin{align*}
\Delta^{k+1}p^{(\ell)}&=\Delta \, \Delta^k p^{(\ell)}
= \Delta \sum_{m=k}^{n-\ell} \frac{k!}{m!}\stir{m}{k}p^{(m+\ell)}
= \sum_{m=k}^{n-\ell} \frac{k!}{m!}\stir{m}{k}\sum_{s=1}^{n-m-\ell}
\frac{1}{s!}p^{(s+m+\ell)}\\
&=\sum_{m=k}^{n-\ell}\sum_{s=m+1}^{n-\ell} \frac{k!}{s!}{s \choose m}\stir{m}{k} p^{(s+\ell)}
=\sum_{s=k+1}^{n-\ell}\sum_{m=k}^{s-1} \frac{k!}{s!}{s \choose m}\stir{m}{k} p^{(s+\ell)}\\
&=\sum_{s=k+1}^{n-\ell}\frac{(k+1)!}{s!}\stir{s}{k+1}p^{(s+\ell)}.
\end{align*}
This concludes the induction.
\end{pf}
\begin{definition}\label{def:yn}
Define the following vector-valued sequences for $j\geq 0$:
\begin{align*}
\ab_{j}&:= \left(\frac1{(j+d)!}, \frac{1}{(j+d-1)!},\ldots,\frac{1}{(j+1)!}, \frac{1}{j!}\right)^T ,\\
\yb_j&:= \left( G^d_j,\ldots, G^1_j, 0 \right)^T.
\end{align*}
\end{definition}
The following lemma is essential for the main result of this paper,
Theorem \ref{thm:factorization}, since it identifies the sequence
$\yb_j$ as the correct coefficients for factorization.
\begin{lemma}\label{lem:y2}
The sequences $(\yb_j, j\geq 0)$, and $(\ab_j,j\geq 0)$, from Definition \ref{def:yn} satisfy the following property
\begin{align}\label{al:y0}
\yb_0 &= \hat{\ab}_0\\ \label{al:yj}
\sum_{m=1}^{j}\gamma^{j}_{m}\, \yb_{m} &=\hat{\ab}_{j}, \quad
j\geq 1.
\end{align}
where
\begin{equation*
\gamma^j_m:=\frac{m!}{j!}\stir{j}{m}.
\end{equation*}
\end{lemma}
\begin{proof}
Equation \eqref{al:y0} follows from the definition of $G_0^k,k=1,\ldots,d$,
in \eqref{eq:coeff_0}.
For $j\geq 1$ and $k=1,\ldots,d$ equation \eqref{al:yj} is equivalent to
\begin{align*}
\sum_{m=1}^{j}\gamma^{j}_{m}\, \yb_{m,\,k}=\frac{1}{(j+k)!}
\Longleftrightarrow &\, \sum_{m=1}^{j}\stir{j}{m}\,m!\,
\yb_{m,\,k}=\frac{j!}{(j+k)!}=\frac{1}{(j+1)\cdots (j+k)}
\end{align*}
Since $\yb_{m,k}=G_m^k$ for $k=1,\ldots,d$, \eqref{al:yj} is true by \eqref{lem:stir2_coeff}.
For $k=0$, \eqref{al:yj} is correct because both sides equal $0$.
\end{proof}
\begin{remark}\label{rem:Taylor}
Lemma \ref{lem:y2} implies $\widetilde T_d^d= \widetilde T_d$.
\end{remark}
\begin{lemma}\label{lem:augmented_iterated}
For $d\geq 1$ and $j\geq d,$ the augmented Taylor operator satisfies
$$
\widetilde{T}_d^j = D \left( \Ib_d - \yb_{j-d} \eb_d^T \right) \cdots D\left( \Ib_d -
\yb_{0} \eb_d^T \right) T_d',
$$
with $(\yb_j,j\geq 0)$ from Definition \ref{def:yn}.
\end{lemma}
\begin{pf}
Recall from Definition \ref{def:yn} that
\begin{equation*}
\yb_j= \left( G^d_j,\ldots, G^1_j, 0 \right)^T
= \left( G^{d:1}_j,0 \right)^T
\end{equation*}
and from Lemma \ref{lem:y2} that $\yb_0=\hat{\ab}_0$.
Furthermore, note that for any vector $\cb \in \RR^{d+1}$ with $\cb_d=0$ we have
\begin{equation*}
D \left( \Ib_d - \cb \eb_d^T \right)=
\begin{pmatrix}
\Ib_{d-1} & \\
& \Delta
\end{pmatrix}
\begin{pmatrix}
\Ib_{d-1} & -\cb_{0:d-1}\\
& 1
\end{pmatrix}
=
\begin{pmatrix}
\Ib_{d-1} & -\cb_{0:d-1}\\
& \Delta
\end{pmatrix}.
\end{equation*}
We prove the Lemma by induction on $j$. For $j=d$, by Remark \ref{rem:Taylor} we have
\begin{equation*}
\widetilde{T}^d_d=\widetilde{T}_d=
\begin{pmatrix}
\widetilde{T}_{d-1} & -\ab_{0,0:d-1} \\
& \Delta
\end{pmatrix}
=
\begin{pmatrix}
\Ib_{d-1} & -\yb_{0,0:d-1} \\
& \Delta
\end{pmatrix}
\begin{pmatrix}
\widetilde{T}_{d-1} & \\
& 1
\end{pmatrix}
=
D \left( \Ib_d - \yb_0 \eb_d^T \right)
T_d'.
\end{equation*}
Assume that the Lemma is true for $j$, we prove it for $j+1$.
\begin{align*}
\widetilde{T}_d^{j+1}&=
\begin{pmatrix}
\widetilde{T}_{d-1} & -\displaystyle{\sum_{k=0}^{j+1-d}} G_k^{d:1} \Delta^{k} \\
& \Delta^{j+2-d} \\
\end{pmatrix}
=
\begin{pmatrix}
\Ib_{d-1} & -\yb_{j+1-d,0:d-1}\\
& \Delta \\
\end{pmatrix}
\begin{pmatrix}
\widetilde{T}_{d-1} & -\displaystyle{\sum_{k=0}^{j-d}} G_k^{d:1} \Delta^{k} \\
& \Delta^{j+1-d}
\end{pmatrix}\\
& =
D \left( \Ib_d - \yb_{j+1-d} \eb_d^T \right)
D \left( \Ib_d - \yb_{j-d} \eb_d^T \right) \cdots D\left( \Ib_d -
\yb_{0} \eb_d^T \right) T_d',
\end{align*}
which concludes the induction step.
\end{pf}
The next lemma follows from \cite{merrienSauer12:_from_hermit} and Lemma \ref{lem:high_forward_diff}:
\begin{lemma}\label{lem:Taylor_applied}
For $p\in \Pi$ with $\operatorname{deg}(p)=n >d $ we have
\begin{equation*}
\widetilde{T}_d\vb(p)= \sum_{k=1}^{n-d}\ab_{k} p^{(k+d)}.
\end{equation*}
If $n\leq d$ then $ \widetilde{T}_d\vb(p)=0$.
\end{lemma}
We write the polynomial of Lemma \ref{lem:Taylor_applied} in the following
form
\begin{equation*}
\label{eq:PolyRewrite}
\sum_{k=1}^{n-d} \ab_k \, p^{(k+d)} = \eb_d q + \sum_{k=1}^{n-d} \hat \ab_k \,
p^{(k+d)},
\end{equation*}
where
\begin{equation}\label{eq:q}
q = \sum_{k=1}^{n-d} \ab_{k,d}p^{(k+d)}.
\end{equation}
If $\operatorname{deg}(p)=n>d$ then $\operatorname{deg}(q)=n-d-1$.
\begin{lemma}\label{lem:q}
For $n>d$, $0\leq k < n-d$ and the polynomial $q$ from \eqref{eq:q} we have:
\begin{equation*}
\Delta^k q = \sum_{s=k+1}^{n-d} \gamma_{k+1}^s p^{(s+d)},
\end{equation*}
with $\gamma$ defined in Lemma \ref{lem:y2}.
\end{lemma}
\begin{proof}
Note that the result is true for $k=0$. For $k\geq 1$ we use
Definition \ref{def:yn}, Lemma \ref{lem:high_forward_diff},
\eqref{stir2_rec}, and \eqref{eq:Stir_Bino}:
\begin{align*}
\lefteqn{\frac{1}{k!}\Delta^k q
= \frac{1}{k!}
\sum_{\ell=1}^{n-d-k} \ab_{\ell,d} \Delta^k p^{(\ell + d)}
= \sum_{\ell=1}^{n-d-k} \ab_{\ell,d}
\sum_{m=k}^{n-d-\ell} \frac{1}{m!}\stir{m}{k}p^{(m+\ell+d)}}\\
& = \sum_{\ell=1}^{n-d-k}\sum_{m=k}^{n-d-\ell}
\frac{1}{\ell! \, m!}\stir{m}{k}p^{(m+\ell+d)}
= \sum_{\ell=1}^{n-d-k}\sum_{s=k+\ell}^{n-d}
\frac{1}{\ell! \, (s-\ell)!}\stir{s-\ell}{k}p^{(s+d)}\\
& = \sum_{s=k+1}^{n-d}\sum_{\ell=1}^{s-k}
\frac{1}{\ell! \, (s-\ell)!}\stir{s-\ell}{k}p^{(s+d)}
= \sum_{r=1}^{n-d-k}\sum_{\ell=1}^{r}
\frac{1}{\ell! \, (r+k-\ell)!}\stir{r+k-\ell}{k}p^{(r+k+d)}\\
& = \sum_{r=1}^{n-d-k}\sum_{s=k}^{r+k-1}
\frac{1}{ (r+k-s)!s!}\stir{s}{k}p^{(r+k+d)}
= \sum_{r=1}^{n-d-k}\frac{1}{(r+k)!} \sum_{s=k}^{r+k-1}
{r+k \choose s}\stir{s}{k}p^{(r+k+d)}\\
& = \sum_{r=1}^{n-d-k} \frac{1}{(r+k)!}
\left(\stir{r+k+1}{k+1}-\stir{r+k}{k} \right) p^{(r+k+d)}
= \sum_{r=1}^{n-d-k} \frac{(k+1)}{(r+k)!}\stir{r+k}{k+1}p^{(r+k+d)}.
\end{align*}
This implies
\begin{equation*}
\Delta^k q = \sum_{r=1}^{n-d-k} \gamma_{k+1}^{r+k}p^{(r+k+d)}
= \sum_{s=k+1}^{n-d} \gamma_{k+1}^{s}p^{(s+d)}. \qedhere
\end{equation*}
\end{proof}
\begin{lemma}\label{lem:IterLemma2}
For $p \in \Pi, \operatorname{deg}(p)=n$, $n>d$ and $(\cb_k,k\geq 1)$
such that $\cb_{k,d}=0$ for all $k$, we have
\begin{equation*
D \left( \Ib_d - \cb_j \eb^T_d \right) \cdots D \left( \Ib_d - \cb_{1}
\eb^T_d \right) \sum_{k=1}^{n-d} \ab_k \, p^{(k+d)}
=\, \eb_d \Delta^{j} q - \sum_{k=0}^{j-1}\hat{\cb}_{k+1}\Delta^{k}q
+\sum_{k=1}^{n-d} \hat \ab_k \, p^{(k+d)},
\end{equation*}
for some $1\leq j \leq n-d$.
\end{lemma}
\begin{proof}
We prove this lemma by induction on $j$.
First note that the operator $D \left( \Ib - \cb \eb^T_d \right)$ for
any $\cb$ with $\cb_d=0$,
acts as the identity operator on vectors with last component equal to $0$.
Therefore
\begin{align*}
D \left( \Ib - \cb \eb^T_d \right)\sum_{k=1}^{n-d}\ab_{k} p^{(k+d)}&=
D \left( \Ib - \cb \eb^T_d \right)\eb_d q + \sum_{k=1}^{n-d} \hat \ab_k
=
\begin{pmatrix}
\Ib_{d-1} & -\cb_{0:d-1}\\
& \Delta
\end{pmatrix}
\begin{pmatrix}
0 \\
q
\end{pmatrix}
+ \sum_{k=1}^{n-d} \hat \ab_k\\
& =
\begin{pmatrix}
-\cb_{0:d-1}q \\
\Delta q
\end{pmatrix}
+ \sum_{k=1}^{n-d} \hat \ab_k\\
&=
\eb_d\Delta q -\hat{\cb}q + \sum_{k=1}^{n-d} \hat \ab_k.
\end{align*}
This proves the case $j=1$.
Assume that the lemma is true for $j$, we prove it for $j+1$.
\begin{align*}
&D \left( \Ib_d - \cb_{j+1} \eb^T_d \right) \cdots D \left( \Ib_d - \cb_{1}
\eb^T_d \right) \sum_{k=1}^{n-d} \ab_k \, p^{(k+d)} \\
&=D \left( \Ib_d - \cb_{j+1} \eb^T_d \right)\left( \eb_d \Delta^{j} q
- \sum_{k=0}^{j-1}\hat{\cb}_{k+1}\Delta^{k}q
+\sum_{k=1}^{n-d} \hat \ab_k \, p^{(k+d)}\right)\\
&=
\begin{pmatrix}
\Ib_{d-1} & -\cb_{j+1,0:d-1}\\
& \Delta
\end{pmatrix}
\begin{pmatrix}
0 \\
\Delta^j q
\end{pmatrix}
- \sum_{k=0}^{j-1}\hat{\cb}_{k+1}\Delta^{k}q
+\sum_{k=1}^{n-d} \hat \ab_k \, p^{(k+d)}\\
&=
\begin{pmatrix}
-\cb_{j+1,0:d-1}\Delta^{j}q\\
\Delta^{j+1}q
\end{pmatrix}
- \sum_{k=0}^{j-1}\hat{\cb}_{k+1}\Delta^{k}q
+\sum_{k=1}^{n-d} \hat \ab_k \, p^{(k+d)}\\
&= \eb_d \Delta^{j+1}q - \sum_{k=0}^{j}\hat{\cb}_{k+1}\Delta^{k}q
+\sum_{k=1}^{n-d} \hat \ab_k \, p^{(k+d)},
\end{align*}
which concludes the induction step.
\end{proof}
\noindent
Lemma \ref{lem:q} also has the following consequence.
\begin{corollary}\label{cor:IterCor}
With notation as in Lemma \ref{lem:IterLemma2} we have
\begin{equation*
D \left( \Ib_d - \cb_j \eb^T_d \right) \cdots D \left( \Ib_d - \cb_{1}
\eb^T_d \right) \sum_{k=1}^{n-d} \ab_k \, p^{(k+d)}
=\, \eb_d \Delta^{j} q
+\sum_{s=1}^{n-d} \left( \hat \ab_s
-\sum_{k=1}^{\min\{ s,j\}}\gamma_{k}^s \hat{\cb}_k \right) p^{(s+d)}.
\end{equation*}
\end{corollary}
\begin{lemma}\label{Lem:IterConst}
For $p \in \Pi, \operatorname{deg}(p)=n$, $n>d$, normalized such that
$p(x)= \frac{1}{n!}x^n+\ldots,$
and $(\yb_k,k\geq 1)$ from Definition \ref{def:yn}, we have
\begin{equation*
D \left( \Ib_d - \yb_{n-d-1} \eb^T_d \right) \cdots D \left( \Ib_d - \yb_{1}
\eb^T_d \right) \sum_{k=1}^{n-d} \ab_k \, p^{(k+d)}
= \eb_d + \yb_{n-d}.
\end{equation*}
\end{lemma}
\begin{pf}
Lemma \ref{lem:q} implies $\Delta^{n-d-1}q=p^{(n)}=1$, since $p$ is normalized.
Corollary \ref{cor:IterCor} and Lemma \ref{lem:y2} now imply
\begin{align*}
&D \left( \Ib - \yb_{n-d-1} \eb^T_d \right) \cdots D \left( \Ib - \yb_{1}
\eb^T_d \right) \sum_{k=1}^{n-d} \ab_k \, p^{(k+d)}\\
&= \eb_d \Delta^{n-d-1} q
+\sum_{s=1}^{n-d} \left(\hat \ab_s -
\sum_{k=1}^{\min\{s,n-d-1\}} \gamma^{s}_{k}\, \yb_{k} \right) p^{(s+d)}\\
&= \eb_d
+\sum_{s=1}^{n-d-1} \left(\hat \ab_s -\sum_{k=1}^{s}
\gamma^{s}_{k}\, \yb_{k} \right) p^{(s+d)}
+ \left(\hat \ab_{n-d} -\sum_{k=1}^{n-d-1}
\gamma^{n-d}_{k}\, \yb_{k} \right) p^{(n)}\\
&= \eb_d + \yb_{n-d}.
\end{align*}
This concludes the proof.
\end{pf}
\noindent
Finally,
Lemma \ref{lem:augmented_iterated} and Lemma \ref{Lem:IterConst} imply
the following result.
\begin{corollary}\label{cor:augmented_eigenvectors}
With notation as in Lemma \ref{Lem:IterConst} we have
\begin{equation*}
\widetilde{T}_d^{n-1}\vb(p)=\eb_d+\yb_{n-d}.
\end{equation*}
\end{corollary}
\section{Factorization with respect to the augmented Taylor operator}
\label{sec:augmented}
We can now apply the results from the preceding sections to describe
the factorization for Hermite schemes with a spectral condition of possibly
higher order $n \ge d$. It is based on the augmented Taylor operator,
hence on combining Taylor operators with appropriate difference
operators of rank $1$.
\begin{theorem}[Main result]\label{thm:factorization}
If $S_\Ab$ satisfies the spectral condition
\eqref{eq:SpecCond} with $n\geq d$,
then there exist subdivision operators $S_{\Bb_j}, j=d,\ldots,n$, such that we can factorize
\begin{equation}
\label{eq:OverFactorization}
\widetilde{T}_d^j S_\Ab = 2^{-j} \, S_{\Bb_j} \widetilde{T}_d^j,
\end{equation}
with the augmented Taylor operator $\widetilde{T}_d^j$ from Definition \ref{def:augmented}. Furthermore $\dim\,\cE_{\Bb_j}=1,j=d\ldots,n,$ and
the factorization \eqref{eq:OverFactorization} is independent of the concrete spectral polynomials in
\eqref{eq:SpecCond}.
\end{theorem}
\begin{pf}
Denote by $p_k,k=0,\ldots,n$, the spectral polynomials from
\eqref{eq:SpecCond}. Due to their normalization we have $p^{(k)}_k=1$.
We prove this result by induction on $j$.
From Remark \ref{rem:Taylor} we have $\widetilde{T}_d^d=\widetilde{T}_d$ and
the existence of $S_{\Bb_d}$ follows from \cite{merrienSauer12:_from_hermit}, see \eqref{eq:Taylor_factorization}. Also $\dim\,\cE_{\Bb_d}=1$ follows from \cite{merrienSauer12:_from_hermit}.
This
shows the case $j=d$.
We assume that the theorem is true for $j$ and prove it for $j+1$.
Lemma \ref{lem:Taylor_applied} and Corollary \ref{cor:augmented_eigenvectors}
imply
\begin{equation*}
\widetilde{T}_d^j\vb(p_{j+1})=\eb_d + \yb_{j+1-d}
\end{equation*}
The spectral condition implies
\begin{align*}
2^{-j-1}(\eb_d + \yb_{j+1-d})&= 2^{-j-1}\widetilde{T}_d^j \vb(p_{j+1})
=\widetilde{T}_d^j S_\Ab \vb(p_{j+1}) =
2^{-j} \, S_{\Bb_j} \widetilde{T}_d\vb(p_{j+1})\\
&=2^{-j} \, S_{\Bb_j} (\eb_d + \yb_{j+1-d}),
\end{align*}
and thus
\begin{equation*}
2\, S_{\Bb_{j}}\left( \eb_d + \yb_{j+1-d}\right) = \eb_d + \yb_{j+1-d}.
\end{equation*}
Therefore $\eb_d + \yb_{j+1-d}$ lies in $\cE_{2\,\Bb_j}$ and since by assumption
the dimension of this space is $1$, it is spanned by $\eb_d + \yb_{j+1-d}$. Now we use Lemma \ref{lem:rank1_factorization} to factorize further. The Gau{\ss} matrix
\begin{equation}\label{eq:Gauss}
\Ib_d + \yb_{j+1-d} \eb_d^T =
\begin{pmatrix}
1 & & & \yb_{j+1-d,0} \\
& \ddots & & \vdots \\
& & 1 & \yb_{j+1-d,d-1} \\
& & & 1 \\
\end{pmatrix},
\end{equation}
is an $\cE_{2\, \Bb_j}$-generator.
It is easy to check that $\left( \Ib_d + \yb_{j+1-d} \eb_d^T
\right)^{-1} = \Ib_d - \yb_{j+1-d} \eb_d^T$.
Lemma \ref{lem:rank1_factorization} thus implies that there exists a subdivision
operator $S_{\Bb_{j+1}}$ such that
\begin{equation}\label{eq:vector_factor}
2 D\left(\Ib_d - \yb_{j+1-d} \eb_d^T\right) S_{\Bb_j}=
S_{\Bb_{j+1}}D\left(\Ib_d - \yb_{j+1-d} \eb_d^T\right)
\end{equation}
and such that $\dim\,\cE_{\Bb_{j+1}}=1$.
The factorization \eqref{eq:vector_factor} further implies
\begin{align*}
D\left(\Ib_d - \yb_{j+1-d} \eb_d^T\right)\widetilde{T}_d^j S_{\Ab}&=
2^{-j}D\left(\Ib_d - \yb_{j+1-d} \eb_d^T\right)S_{\Bb_j}\widetilde{T}_d^j\\
& =
2^{-j-1}S_{\Bb_{j+1}}D\left(\Ib_d - \yb_{j+1-d} \eb_d^T\right)\widetilde{T}_d^j.
\end{align*}
From Lemma \ref{lem:augmented_iterated} we know that
$D\left(\Ib_d - \yb_{j+1-d} \eb_d^T\right)\widetilde{T}_d^j=\widetilde{T}^{j+1}_d$.
This concludes the induction.
\end{pf}
\begin{remark}\label{rem:greg_operator}
Theorem \ref{thm:factorization} for $d = 1$ and Definition \ref{def:augmented} give
\begin{equation*}
\widetilde{T}_1^j = \begin{pmatrix}
\Delta & - \sum_{k=0}^{n-1}G_k^1 \Delta^k \\
0 & \Delta^j
\end{pmatrix} = \left(\begin{array}{cc} 0 & 1 \\ 1 & 0\end{array}\right) \mathcal{G}^{[j]},
\end{equation*}
where $G^1_k$ are the Gregory coefficients, see Section \ref{sec:Stirling}, and $\mathcal{G}^{[j]}$ is the \emph{Gregory operator} derived in \cite{moosmueller18b}. Therefore, Theorem \ref{thm:factorization} generalizes \cite{moosmueller18b}. Note that the matrix $\left(\begin{array}{cc} 0 & 1 \\ 1 & 0\end{array}\right)$ appears since we use \eqref{eq:Gauss} to transform to $\eb_1$ while \cite{moosmueller18b} uses an equivalent factorization as in Lemma \ref{lem:rank1_factorization} where a transform to $\eb_0$ is needed. The factorization is correct in both cases.
\end{remark}
\begin{remark}
The paper \cite{jeong19} proves factorization and convergence results for level-dependent Hermite subdivision schemes of dimension $d=1$. In particular it considers schemes \eqref{eq:sds}, where the operators $S_{\Ab^{[j]}}, j \in \NN$, are
\emph{not} restricted to the form \eqref{eq:stationary_mask}. From results 5.6 -- 5.8 in \cite{jeong19} we can deduce an interesting connection to the augmented Taylor operator.
Consider a subdivision operator $S_{\Ab^{[j]}}$ of dimension $d=1$ which reproduces $\{1,x,e^{\lambda x}\}$ (this implies that it satisfies the spectral condition \eqref{eq:SpecCond} with the functions $1,x$ and $e^{\lambda x}$). Then there exists a subdivision operator $S_{\Bb^{[j]}}$ such that
\begin{equation*}
R^{[j+1]}S_{\Ab^{[j]}}=2^{-2}\zeta(j)S_{\Bb^{[j]}}R^{[j]},
\end{equation*}
where $R^{[j]}$ is given by
\begin{equation*}
R^{[j]}= \left(\begin{array}{cc} 0 & \delta_j \Delta \\ \Delta & -1-\eta(j)\Delta \end{array} \right),
\end{equation*}
with $\zeta,\eta$ from \cite[Proposition 5.8 (ii)]{jeong19}:
\begin{align*}
\zeta(j) = \frac{2}{e^{\lambda 2^{-j-1}}+1}, \quad
\eta(j) = \frac{e^{\lambda 2^{-j}}-1-\lambda 2^{-j}}{\lambda 2^{-j}(e^{\lambda 2^{-j}}-1)}
\end{align*}
and
\begin{equation*}
(\delta_j \cb)(\alpha) = e^{-\lambda 2^{-j}}\cb(\alpha +1)-\cb(\alpha), \quad \cb \in \ell^{2}(\ZZ).
\end{equation*}
Furthermore, with Definition \ref{def:augmented}, \eqref{eq:coeff_int1} and \eqref{eq:coeff_0}, we obtain
\begin{equation}\label{eq:level_dep_limit}
\lim_{j \to \infty}R^{[j]} = \left(\begin{array}{cc} 0 & \Delta^2 \\ \Delta & -1-2^{-1}\Delta \end{array}\right)=\mathcal{G}^{[2]}=
\left(\begin{array}{cc} 0 & 1 \\ 1 & 0\end{array}\right)
\widetilde{T}_1^2.
\end{equation}
The transformation $\left(\begin{array}{cc} 0 & 1 \\ 1 & 0\end{array}\right)$ and the Gregory operator $\mathcal{G}^{[2]}$ (cf. \cite{moosmueller18b}) appear for the same reason as in Remark \ref{rem:greg_operator}.
Eq.\ \eqref{eq:level_dep_limit} implies that factorizing level-dependent schemes of dimension $d=1$ reproducing $\{1,x,e^{\lambda x}\}$ is connected to factorizing stationary schemes of the same dimension reproducing $\{1,x,x^2\}$ via limits.
The level-dependent factorizations of \cite{jeong19} thus depend on $S_{\Ab}$ satisfying a type of overreproduction, in contrary to the factorizations of \cite{cotronei18}.
Through this overreproduction, the connection to the augmented Taylor operator is not surprising, considering that the cancellation operator for level-dependent Hermite schemes reproducing exponentials of \cite{conti16} converges to the Taylor operator, cf.\ \cite[Corollary 2]{conti16}. This also indicates that a generalization of \cite{jeong19} to $d>1$ and multiple exponentials, has to be an operator which converges to $\widetilde{T}^j_d$.
\end{remark}
A generalization of the spectral condition \eqref{eq:SpecCond} to so-called \emph{spectral chains} is proposed in \cite{MerrienSauer18S}. We mention two special spectral chain for which the augmented Taylor operator can be computed easily. Consider a subdivision operator $S_{\Ab}$ with spectral chain
\begin{equation}\label{eq:new_spec_poly}
\vb(p_k) =
\begin{pmatrix}
\Delta^j p_k : j=0,\dots,d
\end{pmatrix}^T
, \quad k=0,\ldots n.
\end{equation}
This implies that $S_{\Ab}$ satisfies \eqref{eq:SpecCond} with \eqref{eq:new_spec_poly}. In this case $S_{\Ab}$ factorizes with respect to a \emph{complete Taylor operator} of the form
$$
\begin{pmatrix}
\Delta & -1 \\
& \ddots & \ddots \\
& & \Delta & -1 \\
& & & \Delta
\end{pmatrix},
$$
cf.\ \cite{MerrienSauer18S}.
Applying the augmented Taylor construction, analogous to Theorem \ref{thm:factorization}, we obtain that $S_{\Ab}$ factorizes with respect to the operators
$$
\begin{pmatrix}
\Delta & -1 & &\\
& \ddots & \ddots &\\
& & \Delta & -1 \\
& & & \Delta^{j+1-d}
\end{pmatrix}, \quad j=d,\ldots,n.
$$
Note that in this case all vectors $\yb$ are zero.
We also consider the following spectral chain which is connected to B-Splines:
\begin{equation}\label{eq:new_spec_poly2}
\vb(p_k) =
\begin{pmatrix}
\Delta^j p_k(\cdot - j) : j=0,\dots,d
\end{pmatrix}^T
, \quad k=0,\ldots n,
\end{equation}
see \cite{MerrienSauer18S}. In \cite{MerrienSauer18S} it is proved that a subdivision operator $S_{\Ab}$ with spectral chain \eqref{eq:new_spec_poly2} factorizes with respect to the generalized Taylor operator
$$
\begin{pmatrix}
\Delta & -1 & \cdots & -1\\
& \ddots & \ddots & \vdots \\
& & \Delta & -1 \\
& & & \Delta
\end{pmatrix}.
$$
With the augmented Taylor construction we obtain that $S_{\Ab}$ factorizes with respect to
$$
\begin{pmatrix}
\Delta & -1 & \cdots & -1 & -1-\Delta\\
& \ddots & \ddots & \vdots &\vdots \\
& & \Delta & -1 &-1 -\Delta \\
& & & \Delta &-1 -\Delta \\
& & & & \Delta^{j+1-d}
\end{pmatrix}, \quad j=d+1,\ldots,n.
$$
Note that in this case $\yb_0=\left(1.\ldots,1,0\right)^T$ and $\yb_j=\mathbf{0}, j>0$.
\section{Interpretation of the augmented Taylor operator
The coefficients $G_n^k$ appear in the following approximations for integrating functions $f$ (see \cite{phillips72,salzer47}):
\begin{align}\label{integral_coeff}
\int_{x_0}^{x_1}\int_{x_0}^{x_2}\cdots\int^{x_{k}}_{x_0}f(x)dxdx_{k}\cdots dx_2
=(x_1-x_0)^k\sum_{n=0}^m G_n^k\Delta^{n}f(x_0) +R_m^kf(x_1;x_0),
\end{align}
where $R_m^kf(x_0;x_1)$ denotes the remainder term.
Via \eqref{integral_coeff} we derive an interpretation of the augmented Taylor operator $\widetilde T _d^n$ (Theorem \ref{thm:interpret_augm_taylor}).
Let $f\in C^d(\RR)$ and denote by $\cT_n f(x_1;x_0)$ its $n$-th Taylor polynomial, i.e.\
\begin{equation*}
\cT_n f(x_1;x_0)=\sum_{k=0}^n\frac{f^{(k)}(x_0)}{k!}(x_1-x_0)^k, \quad n=0,\ldots, d.
\end{equation*}
In analogy we define
\begin{equation}\label{eq:I}
\cI^k_nf(x_1;x_0):=\sum_{m=0}^nG_m^k\Delta^{m}f(x_0)(x_1-x_0)^k.
\end{equation}
Thus \eqref{integral_coeff} becomes
\begin{equation*}
\int_{x_0}^{x_1}\int_{x_0}^{x_2} \cdots \int_{x_0}^{x_k} f(x)dxdx_k\cdots dx_2=\cI^k_nf(x_1;x_0)+R_n^kf(x_1,x_0).
\end{equation*}
It is easy to see that
\begin{equation*}
\int_{x_0}^{x_1}\int_{x_0}^{x_2} \cdots \int_{x_0}^{x_{d-j}} f^{(d)}(x)dxdx_{d-j}\cdots dx_2
=f^{(j)}(x_1)-\cT_{d-j-1}f^{(j)}(x_1;x_0),
\end{equation*}
for $j=0,\ldots,d-1.$
Thus we get
\begin{equation}\label{eq:IT}
\cI_n^kf^{(d)}(x_1;x_0)=f^{(d-k)}(x_1)-\cT_{k-1}f^{(d-k)}(x_1;x_0)-R_n^kf^{(d)}(x_1,x_0).
\end{equation}
From \cite{merrienSauer12:_from_hermit} we know
\begin{equation*}
\left(\widetilde{T}_d\vb(f)(x)\right)_j=f^{(j)}(x + 1)-\cT_{d-j}f^{(j)}(x+1; x), \quad j=0,\ldots,d-1,
\end{equation*}
i.e.\ the remainder term, when Taylor expanding $f^{(j)}(x+1)$ at $x$ with order $d-j$.
Now consider the augmented Taylor operator in view of \eqref{eq:I} and \eqref{eq:IT}:
\begin{align*}
\left(\widetilde{T}_d^n\vb(f)(x)\right)_j&=
f^{(j)}(x+1)-\cT_{d-j-1}f^{(j)}(x+1;x)-\sum_{k=0}^{n-d}G_k^{d-j}\Delta^kf^{(d)}\\
&=f^{(j)}(x+1)-\cT_{d-j-1}f^{(j)}(x+1;x)-\cI^{d-j}_{n-d}f^{(d)}(x+1;x)\\
&=R^{d-j}_{n-d}f^{(d)}(x+1;x),
\end{align*}
that is, the remainder term, when integrating $f^{(d)}, (d-j)$-times with precision $n-d$.
We summarize this result in the following theorem.
\begin{theorem}\label{thm:interpret_augm_taylor}
Let $f \in C^d(\RR)$. Then
\begin{equation*}
\widetilde{T}_d^n \vb(f)(x) =
\widetilde{T}_d^n
\begin{pmatrix}
f(x) \\
f'(x) \\
\vdots \\
f^{(d)}(x)
\end{pmatrix}
=
\begin{pmatrix}
R^{d}_{n-d}f^{(d)}(x+1;x) \\[1em]
R^{d-1}_{n-d}f^{(d)}(x+1;x) \\[1em]
\vdots \\[1em]
R^{0}_{n-d}f^{(d)}(x+1;x)
\end{pmatrix},
\quad x\in \RR,
\end{equation*}
with the remainder terms $R_d^{d-j}f, j=0,\ldots,d$, given in \eqref{integral_coeff}.
\end{theorem}
\bibliographystyle{plainnat}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]{%
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,029 |
Know North Johnston High Class of 1962 graduates that are NOT on this List? Help us Update the 1962 Class List by adding missing names.
More 1962 alumni from North Johnston HS have posted profiles on Classmates.com®. Click here to register for free at Classmates.com® and view other 1962 alumni.
The North Johnston High class of '62 alumni are listed below. These are former students from North Johnston HS in Kenly, NC who graduated in 1962. Alumni listings below have either been searched for or they registered as members of this directory. | {
"redpajama_set_name": "RedPajamaC4"
} | 474 |
'use strict';
const models = require('./index');
/**
* Describes all the settings to be used when analyzing a video in order to
* detect all the faces present.
*
* @extends models['Preset']
*/
class FaceDetectorPreset extends models['Preset'] {
/**
* Create a FaceDetectorPreset.
* @property {string} [resolution] Specifies the maximum resolution at which
* your video is analyzed. The default behavior is "SourceResolution," which
* will keep the input video at its original resolution when analyzed. Using
* "StandardDefinition" will resize input videos to standard definition while
* preserving the appropriate aspect ratio. It will only resize if the video
* is of higher resolution. For example, a 1920x1080 input would be scaled to
* 640x360 before processing. Switching to "StandardDefinition" will reduce
* the time it takes to process high resolution video. It may also reduce the
* cost of using this component (see
* https://azure.microsoft.com/en-us/pricing/details/media-services/#analytics
* for details). However, faces that end up being too small in the resized
* video may not be detected. Possible values include: 'SourceResolution',
* 'StandardDefinition'
*/
constructor() {
super();
}
/**
* Defines the metadata of FaceDetectorPreset
*
* @returns {object} metadata of FaceDetectorPreset
*
*/
mapper() {
return {
required: false,
serializedName: '#Microsoft.Media.FaceDetectorPreset',
type: {
name: 'Composite',
polymorphicDiscriminator: {
serializedName: '@odata.type',
clientName: 'odatatype'
},
uberParent: 'Preset',
className: 'FaceDetectorPreset',
modelProperties: {
odatatype: {
required: true,
serializedName: '@odata\\.type',
isPolymorphicDiscriminator: true,
type: {
name: 'String'
}
},
resolution: {
required: false,
serializedName: 'resolution',
type: {
name: 'String'
}
}
}
}
};
}
}
module.exports = FaceDetectorPreset;
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,006 |
Refurbished and pre-owned electronics ecommerce ecosystem, Galaxy eSolutions, announces a Token Sale to bridge the market with the technology of the Blockchain. The Token Sale bringing Galaxy's native GES tokens to market will run from 18th February 2018 to 31st March 2018.
Galaxy eSolution is a huge and already well established ecommerce giant reaching nearly 20 Million USD in sales within its first year; the company has a firm grasp on the ever-booming market of refurbished and pre-owned electronics, which is tipped to be worth billions and is predicted to grow further. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,768 |
package jdepend.framework.ui.action;
import java.awt.event.ActionEvent;
import java.io.PrintWriter;
import java.io.StringWriter;
import java.util.HashMap;
import java.util.LinkedHashMap;
import java.util.Map;
import javax.swing.AbstractAction;
import javax.swing.JComponent;
import javax.swing.JOptionPane;
import javax.swing.JScrollPane;
import jdepend.framework.exception.JDependException;
import jdepend.framework.ui.component.JDependFrame;
import jdepend.framework.ui.component.TextViewer;
/**
* 异步执行Action(具有进度条跟踪)
*
* @author wangdg
*
*/
public abstract class AsynAction extends AbstractAction {
private String name;
private JDependFrame frame;
private Map<String, JComponent> result;
private static running running;
public AsynAction(JDependFrame frame, String name) {
super();
this.frame = frame;
this.name = name;
if (running == null) {
running = new running();
}
}
@Override
public void actionPerformed(final ActionEvent e) {
synchronized (running) {
if (running.running) {
JOptionPane.showMessageDialog(frame, "正在有命令运行!", "alert", JOptionPane.INFORMATION_MESSAGE);
return;
} else {
running.running = true;
}
}
result = new LinkedHashMap<String, JComponent>();
try {
this.beforeAnalyse();
this.frame.startProgressMonitor(this.getProcess());
// 在独立的线程中执行分析
new Thread() {
public void run() {
try {
// 执行核心服务
analyse(e);
// 截止进度条
frame.stopProgressMonitor();
stopProgressLater();
// 显示第一批结果
frame.show(result);
// 显示后续结果
new Thread() {
public void run() {
try {
showResultLater();
} catch (JDependException e2) {
e2.printStackTrace();
result = createErrorResult(e2);
synchronized (running) {
running.running = false;
}
// 显示结果
frame.show(result);
}
// 设置执行截止标志
synchronized (running) {
running.running = false;
}
}
}.start();
} catch (Exception e1) {
e1.printStackTrace();
result = createErrorResult(e1);
synchronized (running) {
running.running = false;
}
// 显示结果
frame.show(result);
// 截止进度条
frame.stopProgressMonitor();
}
}
}.start();
} catch (JDependException e2) {
e2.printStackTrace();
result = createErrorResult(e2);
synchronized (running) {
running.running = false;
}
// 显示结果
frame.show(result);
}
}
protected void stopProgressLater() {
frame.showStatusMessage("完成了[" + name + "]分析。");
}
protected void beforeAnalyse() throws JDependException {
}
protected void showResultLater() throws JDependException {
}
protected void addResult(String label, JComponent component) {
result.put(label, component);
}
protected void addResults(Map<String, ? extends JComponent> results) {
result.putAll(results);
}
protected void progress() {
this.frame.progress();
}
protected Map<String, JComponent> createErrorResult(Exception e) {
Map<String, JComponent> r = new HashMap<String, JComponent>();
StringBuilder error = new StringBuilder(100);
error.append("[" + name + "]分析运行失败!\n");
StringWriter sw = new StringWriter();
e.printStackTrace(new PrintWriter(sw));
error.append(sw.toString());
TextViewer errorText = new TextViewer();
errorText.setText(error.toString());
r.put("error", new JScrollPane(errorText));
return r;
}
protected abstract int getProcess() throws JDependException;
protected abstract void analyse(ActionEvent e) throws JDependException;
class running {
boolean running = false;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,935 |
{"url":"https:\/\/math.stackexchange.com\/questions\/2835289\/showing-a-sequence-process-is-a-positive-martingale","text":"# Showing a sequence\/process is a positive martingale\n\nFor $(M_t)_{t\\geq0}$ to be a martingale w.r.t a filtration $\\mathcal{F}_t$, we require\n\n1. $M_t \\in L^1$\n2. $E[M_t \\mid \\mathcal{F}_s] = M_s$, $t\\geq s$\n\nIf it is known that $M_t \\geq 0$, does one have to prove condition 1? Or does it fall out of condition 2 since $$E[M_t] = E[E[M_t \\mid \\mathcal{F}_0]] = E[M_0] < \\infty?$$ My concern is whether the law of iterated expectations holds if we don't know apriori that $M_t \\in L^1$. Relatedly, are there examples of stochastic processes where $M_t \\notin L^1$ but $E[M_t \\mid \\mathcal{F}_s] = M_s$?\n\nIt is part of the definition of the conditional expectation that $$E[M_t 1_A] = E[E[M_t \\mid \\mathcal{F}_0] 1_A]$$ for any $A \\in \\mathcal{F}_0$. By taking $A = \\Omega$ we see that the law of iterated expectations is an immediate consequence of the definition (without assuming that $M_t \\in L^1$).\nAs a result, if $M_0 \\in L^1$ and $(M_t)_{t \\geq 0}$ satisfies condition $2$ then by your reasoning we have that $M_t$ is a martingale.\nIt's also fairly easy to see that there exist processes $M_t \\not \\in L^1$ such that the conditional expectations exist and $E[M_t \\mid \\mathcal{F}_s] = M_s$. For example, let $X$ be a non-integrable, non-negative random variable and for every $t \\geq 0$, let $\\mathcal{F}_t = \\sigma(X)$ and $M_t = X$. Then we obviously have $$E[M_t \\mid \\mathcal{F}_s] = E[X \\mid \\sigma(X)] = X.$$","date":"2019-05-21 11:08:02","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9280694127082825, \"perplexity\": 104.34885627166065}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-22\/segments\/1558232256314.52\/warc\/CC-MAIN-20190521102417-20190521124417-00284.warc.gz\"}"} | null | null |
De molen Sint-Jan staat in het Limburgse Stramproy (gemeente Weert).
Het is een standerdmolen die in 1783 gebouwd is. In 1804 werd de molen verplaatst naar een locatie dichter bij het dorp. De molen was tot 1978 in gebruik. Als eerbetoon aan de laatste molenaar staat er een beeld van hem naast de molen. In 2008/2009 is de molen geheel gerestaureerd.
Molen in Weert
Rijksmonument in Weert
Standerdmolen
Korenmolen
Maalvaardige molen | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,983 |
New two bedroom apartment for sale in a traditional building in Mouraria.
Well-set in Mouraria – one of Lisbon's traditional areas – the property enjoys the proximity to various amenities offered by the neighbourhood. A garden, a theatre, an art exhibition, a park, and a viewpoint to enjoy the sights of Lisbon old town – all can be reached in a matter of minutes. Tram 28 route – one of the most popular tourist attractions – is also close by. Main public transport links of bus, metro and tam are virtually around the corner.
Benefiting from a quality refurbishment design and a convenient location close to Mouraria, this apartment for sale represents an ideal future Lisbon home. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,065 |
← Family of Korryn Gaines, A Maryland Black Woman, Awarded $37 Million In Wrongful Death Lawsuit.
New Study Published In Critical Care Medicine Journal Shows An Alarming Rate Of Medication Errors In Transfers From ICU To Non-ICU.
A new study titled "Evaluation of Medication Errors at the Transition of Care from an ICU to a Non-ICU Location," and published in Critical Care Medicine, shows that almost 50% of patients transferred from the intensive care unit (ICU) to a non-ICU location experience a medication error at the time of transition of care, according to new data. While the majority of these medication errors did, in fact, impact the patients, they tended not to cause significant harm, although the study authors noted that strategies to mitigate these errors should still be implemented. The causes of these errors were multi-factored and varied.
The study examined 985 adult patients from 58 ICUs that were transferred within the same institution to a non-ICU location. The findings revealed that 45.7% of patients experienced a medication error during their transfer. The three most common types of errors were: a continuation of medication, with an ICU-only indication at 28%; an indication with no pharmacotherapy at 19%; and pharmacotherapy with no indication at 12%. The highest odds of an error occurring were associated with anti-infective hematologic agents, and intravenous fluids, electrolytes or diuretics.
In total, 94% of the errors were severity category C, meaning that they reached the patient but did not cause harm. The biggest factors that were related to medication errors occurring were the number of medication orders and the need for renal replacement therapy. According to the authors, factors associated with decreased odds of error occurrence included daily patient care rounds in the ICU and orders being discontinued and rewritten at the time of transfer from the ICU.
If you or a loved one of yours has been injured or died as a result of a medication error, it is critical that you do not hesitate to get in touch with one of the expert Boston medication error accident lawyers here at the Law Offices of Gilbert R. Hoy, Jr. and Affiliates. We can be reached 24/7 by phone at 617-787-3700 or by email at info@gilhoylaw.com. Contact us today for your free and confidential consultation. Your needs are our top priority!
This entry was posted in Boston Accident Injury Lawyer, Boston Medical Malpractice Lawyer, Boston Medication Mistake Lawyer, Boston Negligence Lawyer, Boston Personal Injury Lawyer, Boston Wrongful Death Lawyer. Bookmark the permalink. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,104 |
LagunAir was een Spaanse luchtvaartmaatschappij met thuisbasis in León.
Geschiedenis
LagunAir werd opgericht in 2001 door de Grupo Agelco. In 2003 werd gestart met vluchten, maar op 9 oktober 2008 werden ze weer definitief stilgelegd.
Bestemmingen
LagunAir voert lijnvluchten uit naar: (juli 2007)
Alicante
Barcelona
Ibiza
Jerez de la Frontera
León
Madrid
Málaga
Minorca
Palma de Mallorca
Salamanca
Sevilla
Valencia
Valladolid
Vloot
De vloot van LagunAir bestaat uit: (augustus 2007)
2 Embraer ERJ-145
3 Saab SF340A
Spaanse luchtvaartmaatschappij
Economie in Castilië en León
Voormalig Spaans bedrijf
León | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,199 |
Q: How to extract string after some character I need to extract a known portion from a given string.
Ex:
test_str="http://localhost:8080/course-single.html"
I need to extract "course-single" from above string. What is the most easiest way to do that?
A: Chain a couple of str.split calls:
test = "http://localhost:8080/course-single.html"
test.rsplit('/', 1)[-1].split('.')[0]
# 'course-single'
Details
First, get everything after the last /:
temp = test.rsplit('/', 1)[-1]
temp
# 'course-single.html'
Next, get everything before the .html portion:
temp = temp.split('.')
temp
# ['course-single', 'html']
temp[0]
# 'course-single'
A: Using Python 3, use the urllib module, then trim out the parts you don't want from the resulting "path" you want:
>>> from urllib.parse import urlparse
>>> url_str = "http://localhost:8080/course-single.html"
>>> urlparse(url_str).path.split('.')[0][1:]
'course-single'
Details:
urlparse(url_str) will yield:
ParseResult(
scheme='http', netloc='localhost:8080', path='/course-single.html',
params='', query='', fragment=''
)
So, you extract the path:
urlparse(url_str).path: to get: '/course-single.html'
Then you need to remove the parts you are not interested in, so split on the ., which will give you ['/course-single', 'html']. So, just access the first item ([0]), remove the first slash with [1:] slice and that gives you your final result of course-single.
Multiple slashes
If you have a longer url, and you need that last part, then you can do the following, and it should work for whatever length url you provide:
>>> url_str = "http://localhost:8080/a/b/c/course-single.html"
>>> urlparse(url_str)
ParseResult(scheme='http', netloc='localhost:8080', path='/a/b/c/course-single.html', params='', query='', fragment='')
>>> urlparse(url_str).path.rsplit('/', 1)[-1].split('.')[0]
'course-single'
A: Use regular expressions:
word = re.search('\d/(.*?)\.html', test)
print word.groups()[0]
Output:
course-single
A: You can use this:
test="http://localhost:8080/course-single.html"
str = test[test.find('8080/') + 5: test.find('.html')]
print str
Details:
test.find('8080/') - find place where 8080/ string is located ( + 5 because we only have interested in string comes next in line ).
test.find('.html') - find where .html located.
test[test.find('8080/') + 5: test.find('.html')] - get string between this 2 located strings.
| {
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You are here: Home » COACH PORTFOLIOS » Power Tools » Power Tool: Inclusion vs. Exclusion
Power Tool: Inclusion vs. Exclusion
A Coaching Power Tool Created by Erica Cote
(Transformational Coach, UNITED STATES)
Inclusion is defined as;
The state of being included. dictionary.com
Exclusion is defined as;
An act or instance of excluding. dictionary.com
When we think of inclusion, it can invoke thoughts of children and their interactions with friends and playground pals. The innocence and enjoyment of taking precedence over all other things. A place of fun, exploration, and learning that creates building blocks for future situations and decisions. A time when seemingly small actions can create a large and lasting impact. Inclusion can look like a little boy asking the new girl to play on the slide with him. But exclusion can be the mirror image of such things, where the little girl is not allowed to play in a group game because she is new.
This seemingly small action can create feelings of inferiority, or implant the idea that not allowing some people to join the group is okay as a practice. It can then become a part of how decisions and perceptions move through to adulthood. What if you knew that the little girl was a different race than the little boy? Or was in a wheelchair? Maybe appeared to be of a different financial class than the little boy? Would these things change the way you viewed the actions, or how they could create lasting memories and impact? This is a simple way to describe the concept of inclusion vs exclusion and how it can seem innocuous but really be a much bigger concern. The impact of leaving this area untapped is anything but small, and attention to it is a must.
Sometimes inclusion or exclusion can happen without the plan for its existence, yet ignoring it simply isn't an option anymore. Ignorance of the need for inclusion is no longer an excuse, and there is access to support for creating this in just about any space. The possibility of how you can make inclusion more of a standard, a more equitable baseline is normalized and can support business owners in making shifts for good.
How can inclusion and exclusion be a part of life and business?
Inclusion is the act of providing space for people that look, live, love, and exist differently than some to be and feel included. This includes but is not limited to the physical act of being welcomed into places (physical or virtual), being considered when guidelines or rules are built, and the emotional space extended that makes an individual feel welcome, despite being different in some way. This sets the stage for an environment where everyone feels heard and valued as a part of the collective whole and its purpose.
Exclusion is the opposite of these things but can create extensive damage, whether it is a conscious decision or not. Sometimes there is a blatant exclusion in the way that some individuals or groups are treated, or just not addressed as if this is an oversight (even though this "oversight" only provides plausible deniability). A group or organization that will accept anyone, yet not allow them to be a part of the whole through decision making or other such actions is still practising exclusion, even if passively. The lasting results of passively creating an environment that fosters exclusion are extensive and not supportive of the type of environment most businesses desire.
Just as inclusion can create new opportunities to heal divides and create a connection between people or groups that are different, exclusion can reinforce separation and ways of existing that systematically limit entire groups of people. As the need and desire to create connection, empathy, and equity grows the need for inclusion is hard to ignore. And even harder to ignore is the presence of exclusion. Exclusion because of race, gender, sexual orientation, ideology, or any other reason creates barriers. The existence, and hopefully subsequent dismantling of these barriers has become a charge many are taking up in different and increasingly visible ways today through their businesses.
The awareness of the need for intentional inclusion brings many business owners to a place of integrating their personal beliefs, ethics, and values into their organizations for the greater good. Here people have become more aware of how all of these things intersect to increase inclusion and equity. Enlisting a coach to facilitate the integration of inclusionary efforts is a step that moves an idea into action. Here a coach can support their client through how they are striving for inclusion to be a part of the organization, evaluating the why and how, and moving the client to a place of empowerment for action.
This brings the topic of equity to the table. Equity is the opportunity for all to thrive in life in ways that feel good to them across boundaries and identifiers like race, gender, and religion, equally. This includes things like financial opportunities, job access, pay equality, and educational opportunities. Visibility is another example, where being seen and heard allows marginalized groups to have more of a say in things. Equity is a viable piece that can move someone from a space of exclusion to inclusion. This tool can create an integration in ways that give everyone a seat at the table, as well as a voice in the choices and outcomes.
The perspective of Inclusion in Action: How am I being authentic to this value?
Shannon, a business owner, is very clear on her business being a vessel to support WOC (women of colour). As a white woman, she is clear on the privilege and advantages she has easy access to and chooses to use this platform to create more equity whenever possible. This has increasingly become more of a must and less of a choice as her boundaries around these values become a non-negotiable. Shannon regularly speaks at events and mentors upcoming entrepreneurs while making a concerted effort to operate from a place of inclusion. This includes events and conferences she attends, as well as those that she chooses to lend her voice to. She has been a part of many events, and as her actions on her values increased she's had to withdraw her support due to the lack of inclusion and diversity. Shannon chooses carefully what to continue to support, so as she began to look into the lineup for a conference she had participated in for years she had a rude awakening.
This conference normally had a very diverse lineup but because of some challenges on the end of the organizer, there was a severe lack of diversity. When she discovered this, Shannon felt very conflicted with staying on as a speaker. It felt like such direct conflict with her core values. It didn't feel aligned with her way of operating and supporting others that didn't look like her or providing an opportunity for others to see how they could move to a more inclusive place in their businesses. Yet she also felt like pulling out as a speaker was not professional at such a late point with no replacement available. The desire to be involved in an event that was diverse and inclusive of the type of business she was trying to build was not something she was willing to compromise on. Realizing the deficit, and deciding whether to stay involved in the event out of obligation or bow out from the contradictions with her values and ethics created an extremely difficult choice for her.
Shannon had seen first-hand what can exist when intentionality meets efforts to create an environment of inclusion. Diversity facilitates this goal in a way that she had seen work time and time again. In intended ways, and some desirable yet unexpected ways. Diversity creates an environment for growth mindset expansion that can empower everyone. But a space that is exclusionary of any forms of diversity (age, race, religion, sexual orientation, etc) deliberately or by default, is not one that Shannon wanted to add her voice to. In speaking with her coach and sharing her concerns and predicament, another possible option was presented for consideration.
Shannon's coach inquired with some questions that made Shannon dig a little deeper and discover additional options she could consider.
How have you chosen to support your values in other situations?
What is the worst thing that could happen here?
How do you feel about your current choices?
How can you model the change you want to see?
This brought light to the situation that started to uncover other possibilities. Asking Shannon how she felt about her choices initially made her feel like she only had one choice to assess. Then she got quiet and thought more about it, and she then began to see that she had another choice right in front of her. Shannon thought about ways she had previously been involved in events that were inclusive and diverse. How she watched others she knew and respected make choices to support their values.
This prompted her to consider the one initial choice she thought she had, and then it's mirror image as a second choice. Shannon saw another way she could participate AND maintain her ethical boundaries. By stepping into spaces that felt less inclusive and addressing the disparities there. Here she could speak to how it could be of everyone's benefit in creating a less exclusionary space.
Shannon hadn't even considered this and found it an interesting step she could take to speak up about what she was seeing happening, whether the exclusions were intentional or not. Speaking up about the lack of inclusion and how things could be done differently for the benefit of all became a way she could honour her commitment, as well as her beliefs. The opportunity to use her voice as a conduit for change made the situation seem like more of a gift and no longer a curse.
Choosing to look at the situation as more of an AND instead of an EITHER OR choice made a big difference for Shannon, and as an unexpected benefit. It prompted the conversation around how to make a situation useful, even when it isn't optimal on the surface. All those that watched how she executed her choice felt the inspiration behind her actions. Deciding to participate AND use the platform to speak up for change made a difference in how she felt, and subsequently acted. Seeing exclusion presented an opportunity to present inclusion as an option and a more desirable mindset shift.
The apprehension of the choice created what initially felt like no choice at all for Shannon. Remaining as a part of the event and feeling ethically conflicted, or stepping out and feeling unreliable. The reality of what happened was an opportunity to step up versus step back in the face of difficulty, and to examine the situation objectively to seek additional options. The confidence of feeling as if you have viable choices that are ethically sound for you can change the entire emotion around the event, which Shannon found out first hand.
When have you experienced exclusion?
Where does inclusion show up in your business?
How can you embody your beliefs, ethics, and values?
Filed Under: Power Tools Tagged With: coach united states, erica cote, transformational coach | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,144 |
{"url":"http:\/\/www.physicspages.com\/2015\/11\/16\/number-operator\/","text":"# Number operator\n\nReferences: Mark Srednicki, Quantum Field Theory, (Cambridge University Press, 2007) \u2013 Chapter 1, Problem 1.3.\n\nThe number operator is defined as\n\n$\\displaystyle N\\equiv\\int d^{3}x\\;a^{\\dagger}\\left(\\mathbf{x}\\right)a\\left(\\mathbf{x}\\right) \\ \\ \\ \\ \\ (1)$\n\nApplied to a quantum state, it counts the number of particles in that state:\n\n$\\displaystyle Na^{\\dagger}\\left(\\mathbf{x}_{1}\\right)\\ldots a^{\\dagger}\\left(\\mathbf{x}_{n}\\right)\\left|0\\right\\rangle =na^{\\dagger}\\left(\\mathbf{x}_{1}\\right)\\ldots a^{\\dagger}\\left(\\mathbf{x}_{n}\\right)\\left|0\\right\\rangle \\ \\ \\ \\ \\ (2)$\n\nAnother property of ${N}$ is that it commutes with any other operator that contains an equal number of creation and annihilation operators. To see this, look at the individual commutators as follows (where ${a_{i}\\equiv a\\left(\\mathbf{x}_{i}\\right)}$).\n\n $\\displaystyle \\left[N,a_{i}^{\\dagger}\\right]$ $\\displaystyle =$ $\\displaystyle \\int d^{3}x\\;\\left(a^{\\dagger}\\left(\\mathbf{x}\\right)a\\left(\\mathbf{x}\\right)a_{i}^{\\dagger}-a_{i}^{\\dagger}a^{\\dagger}\\left(\\mathbf{x}\\right)a\\left(\\mathbf{x}\\right)\\right)\\ \\ \\ \\ \\ (3)$ $\\displaystyle$ $\\displaystyle =$ $\\displaystyle \\int d^{3}x\\;\\left[a^{\\dagger}\\left(\\mathbf{x}\\right)\\left(\\delta\\left(\\mathbf{x}-\\mathbf{x}_{i}\\right)+a_{i}^{\\dagger}a\\left(\\mathbf{x}\\right)\\right)-a_{i}^{\\dagger}a^{\\dagger}\\left(\\mathbf{x}\\right)a\\left(\\mathbf{x}\\right)\\right]\\ \\ \\ \\ \\ (4)$ $\\displaystyle$ $\\displaystyle =$ $\\displaystyle \\int d^{3}x\\;a^{\\dagger}\\left(\\mathbf{x}\\right)\\delta\\left(\\mathbf{x}-\\mathbf{x}_{i}\\right)\\ \\ \\ \\ \\ (5)$ $\\displaystyle$ $\\displaystyle =$ $\\displaystyle a_{i}^{\\dagger}\\ \\ \\ \\ \\ (6)$ $\\displaystyle \\left[N,a_{i}\\right]$ $\\displaystyle =$ $\\displaystyle \\int d^{3}x\\;\\left(a^{\\dagger}\\left(\\mathbf{x}\\right)a\\left(\\mathbf{x}\\right)a_{i}-a_{i}a^{\\dagger}\\left(\\mathbf{x}\\right)a\\left(\\mathbf{x}\\right)\\right)\\ \\ \\ \\ \\ (7)$ $\\displaystyle$ $\\displaystyle =$ $\\displaystyle \\int d^{3}x\\;\\left[a^{\\dagger}\\left(\\mathbf{x}\\right)a\\left(\\mathbf{x}\\right)a_{i}-\\left(\\delta\\left(\\mathbf{x}-\\mathbf{x}_{i}\\right)+a^{\\dagger}\\left(\\mathbf{x}\\right)a_{i}\\right)a\\left(\\mathbf{x}\\right)\\right]\\ \\ \\ \\ \\ (8)$ $\\displaystyle$ $\\displaystyle =$ $\\displaystyle -\\int d^{3}x\\;a\\left(\\mathbf{x}\\right)\\delta\\left(\\mathbf{x}-\\mathbf{x}_{i}\\right)\\ \\ \\ \\ \\ (9)$ $\\displaystyle$ $\\displaystyle =$ $\\displaystyle -a_{i} \\ \\ \\ \\ \\ (10)$\n\nHere we\u2019ve used the commutation relations\n\n $\\displaystyle \\left[a\\left(\\mathbf{x}\\right),a\\left(\\mathbf{x}^{\\prime}\\right)\\right]$ $\\displaystyle =$ $\\displaystyle 0\\ \\ \\ \\ \\ (11)$ $\\displaystyle \\left[a^{\\dagger}\\left(\\mathbf{x}\\right),a^{\\dagger}\\left(\\mathbf{x}^{\\prime}\\right)\\right]$ $\\displaystyle =$ $\\displaystyle 0\\ \\ \\ \\ \\ (12)$ $\\displaystyle \\left[a\\left(\\mathbf{x}\\right),a^{\\dagger}\\left(\\mathbf{x}^{\\prime}\\right)\\right]$ $\\displaystyle =$ $\\displaystyle \\delta^{3}\\left(\\mathbf{x}-\\mathbf{x}^{\\prime}\\right) \\ \\ \\ \\ \\ (13)$\n\nNow suppose we have an operator ${X}$ which contains ${n}$ creation operators ${a_{i}^{\\dagger}}$, ${i=1,\\ldots,n}$ and ${m}$ annihiliation operators ${a_{j}}$, ${j=1,\\ldots,m}$:\n\n$\\displaystyle X=a_{i1}^{\\dagger}\\ldots a_{in}^{\\dagger}a_{j1}\\ldots a_{jm} \\ \\ \\ \\ \\ (14)$\n\nThen\n\n $\\displaystyle \\left[N,X\\right]$ $\\displaystyle =$ $\\displaystyle Na_{i1}^{\\dagger}\\ldots a_{in}^{\\dagger}a_{j1}\\ldots a_{jm}-a_{i1}^{\\dagger}\\ldots a_{in}^{\\dagger}a_{j1}\\ldots a_{jm}N\\ \\ \\ \\ \\ (15)$ $\\displaystyle$ $\\displaystyle =$ $\\displaystyle \\left(a_{i1}^{\\dagger}N+a_{i1}^{\\dagger}\\right)a_{i2}^{\\dagger}\\ldots a_{in}^{\\dagger}a_{j1}\\ldots a_{jm}-a_{i1}^{\\dagger}\\ldots a_{in}^{\\dagger}a_{j1}\\ldots a_{jm}N\\ \\ \\ \\ \\ (16)$ $\\displaystyle$ $\\displaystyle =$ $\\displaystyle X+a_{i1}^{\\dagger}\\left[N,a_{i2}^{\\dagger}\\ldots a_{in}^{\\dagger}a_{j1}\\ldots a_{jm}\\right] \\ \\ \\ \\ \\ (17)$\n\nWe can see that the commutator in the last line can be worked out recursively until we\u2019ve processed all the creation operators up to ${a_{in}^{\\dagger}}$, giving\n\n$\\displaystyle \\left[N,X\\right]=nX+a_{i1}^{\\dagger}\\ldots a_{in}^{\\dagger}\\left[N,a_{j1}\\ldots a_{jm}\\right] \\ \\ \\ \\ \\ (18)$\n\nThe last commutator gives us\n\n $\\displaystyle \\left[N,a_{j1}\\ldots a_{jm}\\right]$ $\\displaystyle =$ $\\displaystyle Na_{j1}\\ldots a_{jm}-a_{j1}\\ldots a_{jm}N\\ \\ \\ \\ \\ (19)$ $\\displaystyle$ $\\displaystyle =$ $\\displaystyle \\left(a_{j1}N-a_{j1}\\right)a_{j2}\\ldots a_{jm}-a_{j1}\\ldots a_{jm}N\\ \\ \\ \\ \\ (20)$ $\\displaystyle$ $\\displaystyle =$ $\\displaystyle -a_{j1}\\ldots a_{jm}+a_{j1}\\left[N,a_{j2}\\ldots a_{jm}\\right]\\ \\ \\ \\ \\ (21)$ $\\displaystyle$ $\\displaystyle =$ $\\displaystyle -m\\left(a_{j1}\\ldots a_{jm}\\right) \\ \\ \\ \\ \\ (22)$\n\nTherefore\n\n $\\displaystyle a_{i1}^{\\dagger}\\ldots a_{in}^{\\dagger}\\left[N,a_{j1}\\ldots a_{jm}\\right]$ $\\displaystyle =$ $\\displaystyle -m\\left(a_{i1}^{\\dagger}\\ldots a_{in}^{\\dagger}a_{j1}\\ldots a_{jm}\\right)\\ \\ \\ \\ \\ (23)$ $\\displaystyle$ $\\displaystyle =$ $\\displaystyle -mX\\ \\ \\ \\ \\ (24)$ $\\displaystyle \\left[N,X\\right]$ $\\displaystyle =$ $\\displaystyle \\left(n-m\\right)X \\ \\ \\ \\ \\ (25)$\n\nSo if ${n=m}$ (the numbers of creation and annihiliation operators are equal), the operator ${X}$ commutes with ${N}$. In particular, the hamiltonian we met last time satisfies this criterion, so ${\\left[N,H\\right]=0}$ and this hamiltonian conserves particle numbers.","date":"2017-05-25 20:00:29","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 81, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9378124475479126, \"perplexity\": 82.67122551995037}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-22\/segments\/1495463608416.96\/warc\/CC-MAIN-20170525195400-20170525215400-00201.warc.gz\"}"} | null | null |
import java.util.ArrayList;
import java.util.Arrays;
import java.util.Scanner;
import java.util.stream.Collectors;
import java.util.stream.IntStream;
public class p06_Stuck_Zipper {
static ArrayList<Integer> firstList;
static ArrayList<Integer> secondList;
public static void main(String[] args) {
Scanner scanner = new Scanner(System.in);
firstList = IntStream.of(Arrays.stream(scanner.nextLine().split(" "))
.mapToInt(Integer::parseInt).toArray())
.boxed().collect(Collectors.toCollection(ArrayList::new));
secondList = IntStream.of(Arrays.stream(scanner.nextLine().split(" "))
.mapToInt(Integer::parseInt).toArray())
.boxed().collect(Collectors.toCollection(ArrayList::new));
int minimumDigits = 0;
boolean minDigitsFound = false;
// Finding smallest amount of digits in both of the lists
while (!minDigitsFound) {
int minFirst = findMinFirst();
int minSecond = findMinSecond();
if (minFirst < minSecond) {
minimumDigits = minFirst;
minDigitsFound = true;
} else {
minimumDigits = minSecond;
minDigitsFound = true;
}
}
removeElements(minimumDigits);
// Zipping both lists
int position = 0;
if (firstList.size() > 0 || secondList.size() > 0) {
for (int i = 0; i < firstList.size(); i++) {
secondList.add(position + 1, firstList.get(i));
position += 2;
if (position >= secondList.size()) {
position--;
}
}
}
System.out.println(secondList.stream().map(Object::toString).collect(Collectors.joining(" ")));
}
private static int findMinFirst() {
int minDigits = Integer.MAX_VALUE;
for (int i = 0; i < firstList.size(); i++) {
int currentNum = Math.abs(firstList.get(i));
String numToStr = "" + currentNum;
if (numToStr.length() < minDigits) {
minDigits = numToStr.length();
}
}
return minDigits;
}
private static int findMinSecond() {
int minDigits = Integer.MAX_VALUE;
for (int i = 0; i < secondList.size(); i++) {
int currentNum = Math.abs(secondList.get(i));
String numToStr = "" + currentNum;
if (numToStr.length() < minDigits) {
minDigits = numToStr.length();
}
}
return minDigits;
}
private static void removeElements(int minDigits) {
int currentNum = 0;
String numToStr = "";
for (int i = 0; i < firstList.size(); i++) {
currentNum = Math.abs(firstList.get(i));
numToStr = "" + currentNum;
if (minDigits < numToStr.length()) {
firstList.remove(i);
i--;
}
}
currentNum = 0;
numToStr = "";
for (int i = 0; i < secondList.size(); i++) {
currentNum = Math.abs(secondList.get(i));
numToStr = "" + currentNum;
if (minDigits < numToStr.length()) {
secondList.remove(i);
i--;
}
}
}
}
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The Best Self Help Books For Men of 2019 – Fatherly
Want to have better routines? Form a less toxic relationship with your phone? Make your mind a bit less busy? The best self-help books of 2019, written by psychologists, legal scholars, philosophers and more, offer unique perspectives on these topics and more. Each is smart, helpful, and written in a refreshing bluntness that makes them feel like smart things said over coffee by a kindly mentor who's not afraid to confront you with deep questions or crack the whip. We're certain there's one on here that will help bring a small, or maybe even big, change to your life.
Are you up to the task of going without devices for a little while? You should be.
Here, Cal Newport, a computer scientist at Georgetown and author of Deep Work, makes a case for a 'digital detox.' He argues that smartphones, apps, and screen time have greatly diminished our quality of life, not just because we're looking at screens and engaging in a non-physical social world, but largely because of what he refers to as a 'fragmenting' effect — that the 10 seconds it takes for you to look at your phone greatly diminishes the quality of any in-person experience you may be having at that time. His solution? A 30-day plan where followers pare down all non-essential technology, and after those 30 days are over, begin using tech again with intention. It's something from which all of us can benefit, if we can muster the courage to do so.
For anyone who needs a little mindfulness and a lot more chill in their life.
#Chill is the book for the guy who knows that they can't really change overnight. For those who bring the office home with them at night and over the weekends, and would rather spend that time relaxing with family and friends, this book gives a bit of a guidebook to engage with a monthly program to "stop the cycle of over-work." Bryan Robinson largely employs the use of mindfulness and meditation practices that help the over-worked take a deep breath and remain present.
A therapist on therapist look at the importance of understanding our internal monologues.
In writing about her own need to see a therapist, Gottlieb, a sought-after psychotherapist with decades of experience who pens The Atlantic's "Dear Therapist" column, breaks the wall of the therapeutic class (Yes, they judge) and examines, as both a therapist and someone on the other side of the couch, the stories we all tell ourselves.
A book that helps everyone get their act together
Wendy Wood is a psychologist and the Provost Professor of Psychology and Business at the University of Southern California. In this book, she explores the nature of habit formation in her with a refreshing frankness. Forming habits doesn't have to be difficult. Neither does doing away with a bad habit. That's not to say it's as easy as doing nothing, but you certainly can't create good habits by engaging in self-denial and limiting yourself. It's all about changing perspective. This book will help you do just that.
To share or not to share? That's the question this book interrogates.
"Sharenting," posting photos and stories of moments from a child's early life to social media is a rite of modern parenting. While this can help make the unsteady and isolating new world of mother and fatherhood bearable, it deserves a deeper look. Should so much information be out there? Should a child have say about what moments are or aren't posted? Is it right to have the world aware of every step of a child's life before they're even born? In Sharenthood, Plunkett, a Faculty Associate at the Berkman Klein Center for Internet and Society at Harvard University, presents, with humor, insight, and a laudable broad-mindedness, a look at all concerns, both hypothetical and glaringly real, that parents should consider.
A case for cultivating stillness for the sake of yourself and your family.
In his new book Stillness is the Key, philosopher and author Ryan Holiday, whose other books include The Obstacle is the Way and Ego is the Enemy draws from the lessons of history to show how leaders and thinkers — from Marcus Aurelius to JFK — drew on stillness to benefit themselves and the world at times of crisis. He builds the case for all of us to set up systems, interrogate our own reactions, and think: What can I do to be the best person for myself and my family? It's a tough question, certainly. But one this book is more than armed to answer.
Marching orders we could all use more often
An uncomfortable truth: We can all be dicks. At one time or another, we all have dickish tendencies. We're grumpy, tired, stressed, pissed off and, try we might, those emotions manifest as pure dick-itude. Another uncomfortable truth: A lot of us like to think that it can't be us, that we must be feeling isolated and misunderstood because we were at the mercy of other dicks. Wrong. Borg makes this clear and lays out a game plan to interrogate our behavior and be more mindful of the choices we make.
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,459 |
\section{Introduction: Runaway OB Stars} \label{sec:intro}
Field OB stars constitute a significant subset of the massive star
population in star-forming galaxies. Given the power-law cluster mass
distribution, \citet{oey2004} showed that field OB stars typically
comprise 20 -- 30\% of massive stars. However, the field additionally
includes significant numbers of high-velocity, runaway stars
ejected from clusters \citep{blaauw1961,hoogerwerf2000}. The classic work by
\citet{blaauw1961} found that about 20\% of early B stars and 3\% of O
stars are runaways, and \citet{Moffat1998} find a runaway
fraction of O and Wolf-Rayet stars of 14\% from {\sl
HIPPARCOS} space velocities. However, infrared work by
\cite{deWit2005} suggests that over 90\% of O stars are runaways,
and some studies suggest that {\it all} truly
isolated field massive stars are runaways
\citep[e.g.,][]{Pflamm-Altenburg2010,
gvaramadze2011b}. On the other hand, a variety of observational
evidence suggests that field objects formed in relative isolation are
also a major, if not dominant, component of the field massive star
population \citep[e.g.,][]{Lamb2016,Oey2013}.
Two principal mechanisms are responsible for generating runaway
stars. One is dynamical ejection from gravitationally unstable
configurations \citep{Poveda1967, Leonard1988};
another is the acceleration of a star when its binary
companion explodes as a supernova \citep[SN;][]{blaauw1961}.
These are dominated, for higher velocity runaways, by explosions
that generate a recoil ``kick'' to the companion, rather than simple
``slingshot'' acceleration \citep[e.g.,][]{Renzo2018}.
The relative importance of the dynamical vs SN mechanisms is poorly
known. For the latter, a minority of runaways should retain
their neutron star companions, while SNe disrupt most of these binaries
\citep[e.g.,][]{Brandt1995,Renzo2018}. This is supported by searches
for runaways with neutron star companions \citep[e.g.,][]{Philp1996, Sayer1996}.
The dynamical ejection mechanism takes place
primarily via binary-binary interactions
\citep{Poveda1967, Leonard1988}.
This is the only process that can yield binary runaways consisting of
two non-compact stars, in addition to single runaways.
From 42 runaways in the Galactic field O star sample,
the frequency of non-compact, multiple runaways is at least $\sim15$\% of
O star runaways, based on detections of double-lined spectroscopic binaries (SB2) and
astrometric binaries \citep{Mason2009}. If such a value is confirmed,
dynamical ejection may well dominate the massive runaway population.
However, dynamical ejection may require unusual mass ratios and orbital
parameters \citep{Leonard1990}. For example, for a binary-binary
ejection model, \citet{Clarke1992} require nearly all O stars
to have close binary companions with mass ratios $>0.25$, to achieve
a runaway fraction $>10$\%. Although most O stars indeed have
close OB companions \citep{Sana2012}, there is a
significant contribution from lower-mass close companions as well \citep{Moe2015}.
Hence, the statistical properties and fundamental parameters of the OB
runaway population offer critical diagnostics of the ejection
mechanisms, and their statistics
also depend strongly on cluster properties and dynamical evolution
\citep[e.g.,][]{Poveda1967, Hills1980}.
Evaluating the frequency and properties of runaways is therefore vital to
understanding the nature of both the field population and clusters
\citep[e.g.,][]{Clarke1992, PortegiesZwart2000}.
However, testing such predictions has been limited to date by
the substantial uncertainties for runaway statistics and
inhomogeneous data in the Milky Way.
\begin{figure*}
\epsscale{1.1}
\plotone{figure1a.pdf}
\plotone{figure1b.pdf}
\caption{
Vector map of 315 {\sl GAIA} DR2 PM residuals for RIOTS4 field OB stars
superposed on H$\alpha$\ image from \citet{Smith2005}.
Panel ($a$, top) shows only corrections for geometric perspective and SMC
systemic velocity. Panel ($b$, bottom) additionally includes a separate
peculiar PM and RV correction for Wing stars. The adopted boundary between the
so-called Wing and Bar regions is shown by the dashed line, with stars removed from the
PM samples indicated in green. The vectors for the Wing peculiar
PM correction and SMC systemic PM are shown, as is the adopted
kinematic center. Colors show available RVs for 216 stars. Symbols
indicate RV source, with triangles and squares showing
stars with systemic RV measurements from multi-epoch monitoring;
T1, T2 and T3 refer to data from tables in \citet[][see text, Section 2]{Lamb2016}.
The direction toward the Magellanic Bridge is indicated.
\label{f_vectors}}
\end{figure*}
Here, we examine the kinematics of
runaway OB stars in an extragalactic environment: the Small
Magellanic Cloud (SMC), where statistical completeness is easily
evaluated. {\sl GAIA} DR2 \citep{lindegren2018} now offers an outstanding
data set of proper motions for the SMC, which
is located at high Galactic latitude and low extinction, owing to its
low metallicity. Our study is based on the Runaways and Isolated
O-Type Star Spectroscopic Survey of the SMC (RIOTS4)
which is yielding a detailed, quantitative characterization of this
field OB population \citep[e.g.,][]{Lamb2016}.
This sample also offers an opportunity to look at
large-scale stellar kinematics of the SMC's young population.
\section{RIOTS4 Proper Motions from {\sl GAIA}}
The RIOTS4 field star sample is defined from \citet{oey2004}, who
used the $UBVR$ photometric survey of the SMC by \citet{Massey2002} to
identify OB-star candidates based on having reddening-free parameter
$Q_{UBR}\leq -0.84$ and $B\leq 15.21$. These serve as uniform
selection criteria for stars earlier than spectral type $\sim$B0.5.
Field and cluster stars were defined using the friends-of-friends
algorithm of Battinelli (1991), adopting a clustering length of 28 pc,
which yields 374 field stars, or 28\% of all SMC OB stars identified by
\citet{oey2004}. An additional 23 O stars in the RIOTS4 sample were
identified using UV photometric criteria on data from the {\sl Ultraviolet Imaging
Telescope}, yielding a total of 397 stars. The two subsamples are
given in Tables 1 and 2, respectively, of \citet[][T1, T2]{Lamb2016}.
Spectroscopic observations of these RIOTS4 stars were obtained at Magellan
using the IMACS and MIKE spectrographs \citep{Lamb2016}, with
on-going, multi-year monitoring in the Wing region using the M2FS multi-fiber
spectrograph. The latter data yield systemic radial velocities (RVs) for
detected binary systems; a smaller region in the SMC Bar was similarly
monitored with IMACS, and systemic RVs reported in Table~3 of
\citet[][T3]{Lamb2016}.
We identify the RIOTS4 stars in the {\sl GAIA} DR2
catalog by specifying a position match within $3\arcsec$ and
magnitude match $|G-V|<0.3$,
yielding 328 matches.
We further vet the sample by including only
stars having both RA and Dec proper motion (PM) errors $<1\sigma$ from
the median error,
eliminating 12 stars. We also delete one more star that has
RA or Dec errors $>3.5 \sigma$ after the initial clip.
The final RA and Dec standard
deviations are 55 ${\rm\,km\,s^{-1}}$ and 37 ${\rm\,km\,s^{-1}}$, respectively.
Our stars are generally in the range $12.5 < G < 15.5$.
Figure~\ref{f_vectors}$a$ depicts the PMs for these 315
RIOTS4 stars, adopting a mean SMC distance modulus of 18.99 \citep{cioni2000}
(Table~\ref{t_master}).
The PMs are residuals relative to the SMC
systemic PM of $(\mu_\alpha,\mu_\delta)=(0.754,-1.252)$ mas yr$^{-1}$
from \citet{Piatek2008}, which is the published value
that minimizes residuals in the SMC Bar. The shown vectors are also
corrected for geometric perspective using the model of
\citet{vandermarel2002}, but adopting a center of motion at the midpoint
between the {\sc H\thinspace i}\ \citep{stanimirovic2004} and stellar
\citep{ripepi2017} kinematic centers. Figure~\ref{f_vectors} reveals
a pattern of motion consistent with that of \citet{zivick2018}.
We also apply geometric corrections to the RVs and subtract
the median SMC systemic RV of 152 ${\rm\,km\,s^{-1}}$ for the Bar.
It is apparent in Figure~\ref{f_vectors} that the SMC Wing shows a median systemic PM of
$(\mu_\alpha, \mu_\delta) = (0.207\pm 0.025, -0.060\pm 0.016)\ \rm mas\ yr^{-1}$,
corresponding to velocities
$(v_\alpha, v_\delta)=(62\pm7, -18\pm5)\ {\rm\,km\,s^{-1}}$ and
a total transverse velocity $v_\perp=64\pm 8 {\rm\,km\,s^{-1}}$. This Wing peculiar
motion is obtained after deleting 11 stars within 0.5 degree of the
boundary shown in Figure~\ref{f_vectors}. Panel $b$
shows the PMs with the 68 Wing stars corrected for this additional
peculiar motion. This effect is robust to the choice of systemic PM and
kinematic center.
In applying geometric corrections to the RVs, we find that the RV
offset reported by \citet{Lamb2016}, who did not correct for
perspective, is due primarily to this effect. The Wing median RV offset
from our data is now $+4.5\pm 5.0\ {\rm\,km\,s^{-1}}$.
\begin{deluxetable*}{cccccccccccccc}
\tablecaption{Kinematic Data for RIOTS4 Field OB
Stars\tablenotemark{a} \label{t_master}}
\tablewidth{0pt}
\tablehead{
\colhead{ID\tablenotemark{b}} & \colhead{Subgroup\tablenotemark{c}} &
\colhead{$v_\perp$\tablenotemark{d}} &
\colhead{$v_{\rm loc,\perp}$\tablenotemark{d}} & \colhead{Quality\tablenotemark{e}} &
\colhead{RV\tablenotemark{d}} &
\colhead{$v_{\rm RA}$\tablenotemark{f}} &
\colhead{err\tablenotemark{g}} & \colhead{$v_{\rm Dec}$\tablenotemark{f}} &
\colhead{err\tablenotemark{g}} & \colhead{$v_{\rm loc,\rm RA}$\tablenotemark{f}} &
\colhead {err\tablenotemark{g}} &
\colhead{$v_{\rm loc,\rm Dec}$\tablenotemark{f}} & \colhead{err\tablenotemark{g}}\\
\colhead{- } & \colhead{- } & \colhead{${\rm\,km\,s^{-1}}$} &
\colhead{${\rm\,km\,s^{-1}}$} & \colhead{-} &
\colhead{${\rm\,km\,s^{-1}}$} & \colhead{${\rm\,km\,s^{-1}}$} &
\colhead{${\rm\,km\,s^{-1}}$} & \colhead{${\rm\,km\,s^{-1}}$} &
\colhead{${\rm\,km\,s^{-1}}$} & \colhead{${\rm\,km\,s^{-1}}$} &
\colhead{${\rm\,km\,s^{-1}}$} & \colhead{${\rm\,km\,s^{-1}}$} &
\colhead{${\rm\,km\,s^{-1}}$}}
\startdata
107 & -,-,-,B,- & 42 & 20 & -,0,0 & ... & 186 & 17 & -347 & 12 & 167 & 5 & -349 & 7 \\
1037 & -,-,-,B,- & 145 & 99 & -,0,0 & -43 & 85 & 24 & -300 & 18 & 166 & 4 & -356 & 4 \\
1600 & -,E,-,B,- & 22 & 43 & -,0,0 & -58 & 188 & 22 & -401 & 21 & 187 & 5 & -358 & 4 \\
1631 & -,-,-,B,- & 87 & 51 & -,0,0 & -33 & 190 & 17 & -298 & 15 & 175 & 2 & -347 & 7 \\
1830 & -,-,-,B,- & 75 & 32 & -,0,0 & ... & 143 & 20 & -341 & 17 & 174 & 4 & -349 & 6 \\
2034 & -,-,-,B,- & 47 & 21 & -,0,0 & ... & 186 & 21 & -339 & 15 & 174 & 2 & -357 & 5 \\
2093 & -,-,-,B,- & 105 & 63 & -,0,0 & ... & 122 & 20 & -318 & 18 & 178 & 5 & -347 & 8 \\
3224 & -,-,-,B,- & 39 & 9 & -,0,0 & -51 & 169 & 15 & -368 & 15 & 169 & 3 & -359 & 4 \\
3459 & -,-,-,B,- & 39 & 7 & -,0,0 & 50 & 183 & 21 & -353 & 17 & 179 & 5 & -359 & 3 \\
3815 & -,-,-,B,- & 181 & 147 & -,0,1 & ... & 182 & 21 & -204 & 20 & 181 & 4 & -351 & 3 \\
\enddata
\tablenotetext{a}{Table~\ref{t_master} is available in its entirety on-line.}
\tablenotetext{b}{From \citet{Massey2002}}
\tablenotetext{c}{`E', `S', `X' indicate EB, SB2 and HMXB,
respectively; `B', `W', `D' indicate Bar, Wing, and boundary stars,
respectively; `m' indicates object in multi-epoch spectroscopic sample.}
\tablenotetext{d}{
Final residual velocity relative to SMC and Wing systemic motion.
The $v_{\rm loc,\perp}$ values are computed relative to local velocity fields
(see text). RV errors are typically $10\ {\rm\,km\,s^{-1}}$.}
\tablenotetext{e}{`a' indicates object meets $\sim 10$\% asymmetry
criterion in RA vs Dec; the second and third values give the number
of stars within 1$\arcsec$ and 1.5$\arcsec$, respectively.}
\tablenotetext{f}{Transverse velocity computed from proper
motion, without geometric or systemic velocity corrections.
\tablenotetext{g}{Measurement errors, not including systematic errors
(see text).}}
\end{deluxetable*}
Table~\ref{t_master} lists the total residual transverse velocities
$v_\perp$ and RV, along with non-residual PMs for our field OB stars.
We also give locally determined transverse velocities (see \S 3.2).
Stars in the Wing region (Figure~\ref{f_vectors}) are indicated, and
their residual values are corrected for
the Wing peculiar motions in PM and RV. All PM values are based
on the original SMC geometric correction described above; our models
show that a specific correction for the Wing would modify the
velocities by at most 3.5 ${\rm\,km\,s^{-1}}$, whereas systematic errors on the
geometric correction are on the order of 30 ${\rm\,km\,s^{-1}}$.
The FWHM of the Cepheid distance distribution yields a variation of 15\%
\citep[e.g.,][]{ripepi2017}, with extremes up to 50\%,
given the SMC's end-on orientation to the line of sight. These
imply distance uncertainties that propagate directly to our transverse
velocities.
\section{SMC field OB kinematics}
\subsection{Proper motion of the SMC Wing}
While it is necessary to correct for the Wing's systemic motion to
identify runaway stars,
our data also clearly reveal that {\it the Wing and Bar are
kinematically distinct components,} with a 3-D offset of
$64\pm10\ {\rm\,km\,s^{-1}}$.
The Wing has been identified as the southeast component of the
SMC, and extends $\gtrsim 2^\circ$ beyond our observed data set,
merging with the Magellanic Bridge linking the SMC to the Large
Magellanic Cloud (LMC).
Previous work \citep[e.g.,][]{Brueck1978, Dobbie2014} shows
an older coexisting Wing stellar population having RVs similar to those of
young stars. This suggests that our sample is a good tracer of the bulk motion of
this region, but follow-up examination of {\sl GAIA} PMs for red giant
stars is needed to understand differentials between the old and young populations.
The dynamical state of the Wing
provides a vital kinematic discriminant for dynamical models of the internal
structure of the SMC, the recent encounter history of the Magellanic
Clouds and formation of the Magellanic Bridge
\citep[e.g.,][]{Besla2012,zivick2018}.
In particular, the Wing kinematics seen here are consistent with
transverse motion along the Bridge towards the LMC, instead of
perpendicular to the Bridge. This
vividly confirms models for a recent,
direct collision between the Clouds
100 -- 200 Myr ago,
for which gas velocities are
expected to be aligned with the Bridge. In contrast, motions
perpendicular to the Bridge are theoretically expected in a tidal
stripping scenario of an SMC that did not collide with the LMC,
allowing it to retain ordered rotation. The absence of perpendicular
motion is consistent with the results of \citet{zivick2018}, who find
little evidence of rotation in the SMC,
which also supports the direct collision model.
\subsection{Field OB kinematics and runaway stars}
\begin{figure*}
\epsscale{1.2}
\plotone{figure2.pdf}
\caption{
Comparison of residual RVs with ($a$) RA and ($b$) Dec velocities. Triangles and squares
show objects with systemic RVs obtained through multi-epoch monitoring.
The remainder of the sample are single-epoch RV measurements.
Subsample labels are as in Figure~\ref{f_vectors}.
\label{f_pmrv}}
\end{figure*}
\begin{figure*}
\epsscale{1.2}
\plotone{figure3.pdf}
\caption{
Distribution of ($a$) $v_\perp$ and ($b$) $v_{\rm loc,\perp}$.
EB, SB2, and HMXB populations are shown as
indicated. The vertical lines correspond to 1-$\sigma$ and
2-$\sigma$ velocities from the medians, using values in
Table~\ref{t_binaries}. The lower insets show the $y$-axis
zoomed for clarity.
\label{f_histkine}}
\end{figure*}
As noted in \S 1, non-compact binaries that are runaway systems must
result from the dynamical ejection mechanism.
In addition to SB2s, non-compact binaries also can be
identified as eclipsing binaries (EBs), which are given
in the OGLE-III EB catalog for the SMC
\citep{Pawlak2016}. Of the 315 stars, only 295 are covered by the
OGLE-III survey, since the easternmost Wing region is excluded.
Furthermore, it is also possible to identify
binaries with a compact remnant as high-mass X-ray binaries (HMXBs),
which are compiled by \citet{Haberl2016}. Runaway HMXBs result from the
SN acceleration mechanism. Stars found to be SB2, EB, and/or
HMXB are indicated in Table~\ref{t_master}.
In Table~\ref{t_binaries}, we list the numbers of stars in each
population, along with the mean and median $v_\perp$ and
standard deviations $\sigma$ for each binary population.
The median {\sl GAIA} errors in
$v_\alpha,\ v_\delta,$ and $v_\perp$
for the sample are 22, 17, and 28 ${\rm\,km\,s^{-1}}$, with
standard deviations on the errors of 5, 3, and 6 ${\rm\,km\,s^{-1}}$,
respectively; thus the errors are more than a factor of 2 below the
respective observed $\sigma_\alpha,\ \sigma_\delta,$ and $\sigma_{\perp}$.
Therefore, these standard deviations reflect actual velocity dispersions
convolved with the substantial errors.
Figure~\ref{f_pmrv} compares PM and RV for the 207
stars that also have RVs reported by \citet{Lamb2016}, omitting the stars
near the Wing-Bridge boundary. Stars
shown by triangles and squares are, respectively, systemic RVs estimated from our
multi-epoch monitoring surveys of the Bar
(Table~3 of \citet{Lamb2016})
and Wing region (on-going). We caution that the single-epoch
observations often include significant binary motions.
Figure~\ref{f_pmrv} confirms that the measured PMs and RVs are comparable.
It is apparent that the PMs show a larger spread in RA than Dec, which
is due to the asymmetric {\sl GAIA} errors \citep[e.g.,][]{lindegren2018}.
Thus, velocities, especially large ones, for any given star may not be
real, but the kinematics of subsamples may be compared. We have also
inspected the OGLE-III \citep{udalski2008} images
available for 304 of our target stars to evaluate PSF asymmetry and
crowding, which can degrade the astrometry; stars that may be thus affected
are flagged in Table~\ref{t_master}. Measured $v_\perp$ have no
apparent dependence on PSF asymmetry when flagging those with $\gtrsim 10$\%
variation in RA vs Dec. But whereas $\sim$10\% of all targets
have neighbors within 1.5$\arcsec$, targets having $v_\perp >
1\sigma_\perp$ are much more likely (24\%, 10 out of 41) to have such
close neighbors. Thus we caution that stars
with $v_\perp\gtrsim 200\ {\rm\,km\,s^{-1}}$ are likely dominated by spurious values
\citep[e.g.,][]{Platais2018}.
Figure~\ref{f_histkine}$a$ shows $v_\perp$
distributions for the 304 stars, which exclude those
near the Wing-Bar boundary. We also show the
contribution of each binary population. The peak
of the $v_\perp$ distribution
occurs at the value corresponding to the median error of 28 ${\rm\,km\,s^{-1}}$,
reflecting the {\sl GAIA} {\bf detection} limit at $G<15$.
The median $v_\perp$ of 44 ${\rm\,km\,s^{-1}}$ is therefore significant, occurring in
the runaway velocity regime, since the typical velocity dispersion in OB
associations is $\sim5\ {\rm\,km\,s^{-1}}$ \citep[e.g.,][]{Melnik2017}.
Since half the sample has $v_\perp$ greater than the median value,
which in turn is larger than typical bound velocity
dispersions, this therefore implies that well over half the sample
corresponds to unbound runaways, since many unbound stars also occur
at velocities below the median \citep[e.g.,][]{Renzo2018}.
The population of ``unclassified'' stars
in Table~\ref{t_binaries} simply refers to
the remainder of the sample excluded by the other categories, and
therefore includes any unidentified binaries. Thus, the EB, SB2, and HMXB
populations are lower limits on the true numbers of non-compact and
compact binary systems. Table~\ref{t_binaries} identifies 22
non-compact systems and 15 compact.
We caution that the field includes a likely
substantial population of non-runaway stars that formed in situ
\citep[e.g.,][]{Oey2013}; analysis of the binary
frequencies will be presented in a future work.
Figure~\ref{f_histkine} and Table~\ref{t_binaries} show that the
kinematics of the EB and SB2 populations are generally consistent with
those of the total population, showing similar transverse velocity
dispersions. In contrast, the spreads
for the HMXBs are much lower, with the non-compact binaries
having values about 50\% larger than for the HMXBs. In fact,
none of the HMXBs have $v_\perp >1\sigma$ from the median of the total sample
(Figure~\ref{f_histkine}$a$). We caution that one SB2 with velocity
$> 1\sigma$, star 76253, has another star within $1.5\arcsec$ to the
north, which may affect the {\sl GAIA} astrometry (Table~\ref{t_master}).
The above kinematics are derived from only two assumed systemic
components, Wing and Bar, as described in \S 2. Since there may be additional,
higher-order systemic motions, we also examine the PMs of our
sample stars relative to their local velocity fields. We use
the {\sl GAIA} PMs of stars from the \citet{Massey2002} catalog of OB
stars within a 5$\arcmin$ (90 pc) radius of the target
star to obtain the mean local velocity of the young
population. We fitted the local PM distributions in RA and Dec
with gaussians having $\sigma=45\ {\rm\,km\,s^{-1}}$ and 55 ${\rm\,km\,s^{-1}}$,
respectively; these are the mean values for the Bar.
The local transverse velocities $v_{\rm loc}$ obtained in this
way are given in Table~\ref{t_master}.
Figure~\ref{f_histkine}$b$ and Table~\ref{t_binaries} show the
resulting residual PM kinematics.
We see a similar pattern as before, with the HMXBs again showing smaller
standard deviations $\sigma_{\rm loc}$ when measured relative to the
local fields.
\begin{deluxetable*}{lccccr}
\tablecaption{Kinematics of Binary SMC Field OB Stars \label{t_binaries}}
\tablewidth{0pt}
\tablehead{
\colhead{\ } &
\colhead{Unclassified\tablenotemark{a}} &
\colhead{EB} &
\colhead{SB2} &
\colhead{HMXB} &
\colhead{Total}
}
\startdata
Number & 267 & 16\tablenotemark{b} & 10\tablenotemark{b} & 15 & 304 \\
$\sigma_{\perp}/{\rm\,km\,s^{-1}}$\tablenotemark{c} & 43 & 42 & 45 & 23 & 42 \\
$\sigma_{\alpha}/{\rm\,km\,s^{-1}}$ & 56 & 52 & 47 & 32 & 55 \\
$\sigma_{\delta}/{\rm\,km\,s^{-1}}$ & 37 & 33 & 42 & 26 & 37 \\
median($v_{\perp}/{\rm\,km\,s^{-1}}$) & 44 & 39 & 38 & 38 & 43 \\
mean($v_\alpha/{\rm\,km\,s^{-1}}$) & --3 & 16 & 16 & 1 & --1 \\
mean($v_\delta/{\rm\,km\,s^{-1}}$) & 7 & 8 & --25 & 9 & 6 \\
\hline
$\sigma_{\rm loc,\perp}/{\rm\,km\,s^{-1}}$\tablenotemark{c} & 42 & 51 & 34 & 21 & 41 \\
$\sigma_{\rm loc,\alpha}/{\rm\,km\,s^{-1}}$ & 53 & 50 & 42 & 31 & 52 \\
$\sigma_{\rm loc,\delta}/{\rm\,km\,s^{-1}}$ & 35 & 33 & 37 & 23 & 35 \\
median($v_{\rm loc,\perp}/{\rm\,km\,s^{-1}}$) & 39 & 29 & 50 & 31 & 39 \\
mean($v_{\rm loc,\alpha}/{\rm\,km\,s^{-1}}$) & 10 & 11 & 33 & 17 & 10 \\
mean($v_{\rm loc,\delta}/{\rm\,km\,s^{-1}}$) & 3 & 2 & --29 & --4 & 2 \\
\enddata
\tablenotetext{a}{Includes unidentified binaries.}
\tablenotetext{b}{There are 4 stars identified as both EB and SB2.}
\tablenotetext{c}{Values for $\sigma_\perp$ and $\sigma_{\rm
loc,\perp}$ are standard deviations from the median.}
\end{deluxetable*}
We caution that K-S tests show that the difference between the
$v_\perp$ distributions of the binary populations is not statistically
significant. However, our sample likely contains a
substantial, perhaps even dominant, contribution from non-runaway, field
stars that formed {\it in situ} \citep{Lamb2016,Oey2013}, which
will significantly dilute the non-compact binary population in our
sample at the lowest velocities. The fact that our HMXBs have smaller
velocity dispersions than non-compact binaries is consistent with the
expectation that bound compact binaries represent systems with less
energetic SN kicks that failed to unbind the components. Moreover,
dynamical ejections from dense clusters can accelerate runaways to
higher velocities than the SN mechanism, since cluster acceleration
can leverage the gravitational energy from multiple stars.
Models by, e.g., \citet{Brandt1995}
and \citet{Renzo2018}
show that HMXBs have runaway
velocities $< 100\ {\rm\,km\,s^{-1}}$, and typically half that value, depending on
the assumed kick velocities and pre-SN orbital parameters. In
contrast, \citet{perets2012} find that dynamically ejected runaways
from clusters having masses on the order of $10^4\ \rm\,M_\odot$ can reach
$200\ {\rm\,km\,s^{-1}}$, including significant fractions of binaries.
Despite contamination from non-runaways systems, the non-compact
binaries show velocity distributions
that are not only larger than for the HMXBs, but also similar
to that for unclassified field OB stars (Table~\ref{t_binaries}).
Since the latter include single-star runaways from both mechanisms,
this suggests that dynamically ejected
objects dominate over {\it in situ} field stars in the SMC. Furthermore,
\citet{Renzo2018} predict that $\sim 14$\% of post-SN binaries
fail to disrupt, of which some fraction are observed as HMXBs.
They also expect $\sim 3$\% of post-SN binaries to generate single
runaways faster than 30 ${\rm\,km\,s^{-1}}$. These estimates have large
uncertainties, so we might expect roughly similar
numbers of these two groups. However, there are only 15 HMXBs, whereas
roughly half (134) of unclassified stars are fast runaways
(median $v_\perp = 44\ {\rm\,km\,s^{-1}}$). While these numbers are subject to
various biases, the large disparity does suggest that
dynamical ejections likely dominate.
We will examine additional properties, including frequencies, masses,
and rotation of these field OB runaways in future work.
\acknowledgements
We thank Roeland van der Marel for the use of his geometric correction
code, and Michal K. Szymanski for help using OGLE data in preparatory
work for this study. MSO appreciates helpful discussions and generous
hospitality from Kaitlin Kratter and the University of Arizona in
hosting an extended visit supporting this collaboration.
We thank the anonymous referee for helpful comments and suggestions.
This work was funded by the National Science Foundation, grant AST-1514838 to MSO and
the University of Michigan. MM acknowledges support from
NASA's Einstein Postdoctoral Fellowship program PF5-160139 and NASA
ATP grant 17-ATP17-0070; NK is supported by NSF CAREER award AST-1455260.
\vspace{5mm}
\facilities{GAIA, Magellan, astropy}
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package org.chocosolver.solver.explanations;
import org.chocosolver.solver.ICause;
import org.chocosolver.solver.Solver;
import org.chocosolver.solver.constraints.Propagator;
import org.chocosolver.solver.explanations.antidom.AntiDomain;
import org.chocosolver.solver.explanations.store.BufferedEventStore;
import org.chocosolver.solver.search.loop.monitors.IMonitorDownBranch;
import org.chocosolver.solver.search.strategy.decision.Decision;
import org.chocosolver.solver.variables.BoolVar;
import org.chocosolver.solver.variables.IntVar;
import org.chocosolver.solver.variables.Variable;
import org.chocosolver.solver.variables.events.IntEventType;
import org.chocosolver.solver.variables.events.PropagatorEventType;
/**
* UNSAFE
* Created by cprudhom on 18/11/14.
* Project: choco.
*/
public class ThreadExplanationEngine extends ExplanationEngine implements IMonitorDownBranch {
private final BufferedEventStore eventStore;
/**
* Builds an ExplanationEngine
*
* @param slv associated solver's environment
*/
public ThreadExplanationEngine(Solver slv, BufferedEventStore eventStore) {
super(slv);
this.eventStore = eventStore;
slv.plugMonitor(this);
}
/**
* This is the main reason why we create this class.
* Record operations to execute for explicit call to explanation.
*
* @param var an integer variable
* @param val a value
* @param cause a cause
*/
@Override
public void removeValue(IntVar var, int val, ICause cause) {
eventStore.push(var, cause, IntEventType.REMOVE, val, 0, 0);
}
/**
* This is the main reason why we create this class.
* Record operations to execute for explicit call to explanation.
*
* @param intVar an integer variable
* @param value a value
* @param cause a cause
* @value old previous LB
*/
@Override
public void updateLowerBound(IntVar intVar, int old, int value, ICause cause) {
eventStore.push(intVar, cause, IntEventType.INCLOW, old, value, 0);
}
/**
* This is the main reason why we create this class.
* Record operations to execute for explicit call to explanation.
*
* @param var an integer variable
* @param value a value
* @param cause a cause
* @value old previous LB
*/
@Override
public void updateUpperBound(IntVar var, int old, int value, ICause cause) {
eventStore.push(var, cause, IntEventType.DECUPP, old, value, 0);
}
/**
* This is the main reason why we create this class.
* Record operations to execute for explicit call to explanation.
*
* @param var an integer variable
* @param val a value
* @param cause a cause
* @param oldLB previous lb
* @param oldUB previous ub
*/
@Override
public void instantiateTo(IntVar var, int val, ICause cause, int oldLB, int oldUB) {
eventStore.push(var, cause, IntEventType.INSTANTIATE, val, oldLB, oldUB);
}
/**
* This is the main reason why we create this class.
* Record operations to execute for explicit call to explanation.
*
* @param var the reified variable
* @param propagator the propagator to awake.
*/
@Override
public void activePropagator(BoolVar var, Propagator propagator) {
eventStore.push(var, propagator, PropagatorEventType.FULL_PROPAGATION, 0, 0, 0);
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
@Override
public void request() {
eventStore.hurryUp();
}
@Override
public AntiDomain getRemovedValues(IntVar v) {
return eventStore.getRemovedValues(v);
}
@Override
public ValueRemoval getValueRemoval(IntVar var, int val) {
throw new UnsupportedOperationException();
}
@Override
public Deduction explain(IntVar var, int val) {
return eventStore.explain(var, val);
}
@Override
public Deduction explain(Deduction deduction) {
throw new UnsupportedOperationException();
}
@Override
public Explanation flatten(IntVar var, int val) {
throw new UnsupportedOperationException();
}
@Override
public Explanation flatten(Explanation expl) {
return eventStore.flatten(expl);
}
@Override
public Explanation flatten(Deduction deduction) {
throw new UnsupportedOperationException();
}
@Override
public Explanation retrieve(IntVar var, int val) {
throw new UnsupportedOperationException();
}
@Override
public BranchingDecision getDecision(Decision decision, boolean isLeft) {
return eventStore.getDecision(decision, isLeft);
}
@Override
public PropagatorActivation getPropagatorActivation(Propagator propagator) {
throw new UnsupportedOperationException();
}
@Override
public void store(Deduction deduction, Explanation explanation) {
eventStore.store(deduction, explanation);
}
@Override
public void removeLeftDecisionFrom(Decision decision, Variable var) {
eventStore.removeLeftDecisionFrom(decision, var);
}
@Override
public void beforeDownLeftBranch() {
eventStore.hurryUp();
}
@Override
public void afterDownLeftBranch() {
}
@Override
public void beforeDownRightBranch() {
eventStore.hurryUp();
}
@Override
public void afterDownRightBranch() {
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,686 |
Q: I have a split() input of different types(int and str), when I try to perform mathematical operations I get an error message I have a split() function of different types of user input values. When I try to perform mathematical operations on the int types, I get an error saying "unsupported operand type(s) for -: 'int' and 'str'"
I've tried placing int before the mathematical operations but I still get some sort of the same error message
integer = int(input("N> "))
f_or_b,change_min,hours,current_min = (input("INPUT> ").split())
if (f_or_b == 'F'):
if (int(change_min + current_min) > 60):
m = int(60 - (change_min + current_min))
print(m)
I want to be able to input something like: F 20 23 50
and receive an output of 10 (since 20 + 50 > 60, 60 - sum = 10)
A: change_min and current_min will be strings. So, change_min + current_min will do string concatenation and not mathematical addition.
Try to understand the below code:
a, b = input().split()
print(a + b)
print(int(a) + int(b))
Input:
1 2
Output:
12
3
Your code should be something like:
if f_or_b == 'F':
if (int(change_min) + int(current_min)) > 60:
m = 60 - (int(change_min) + int(current_min))
print(m)
Or you can convert:
change_min = int(change_min)
current_min = int(current_min)
And then use them as integers like change_min + current_min.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,944 |
Drop the top triangle one level. The Triforce is one big triangle on the outside. The way you stacked it here makes it look odd. All the outside edges have to be even.
I need help to make it look more... Well. Right. Any tips/help?
So if you made it larger, like 20-30 blocks wide (whatever makes the math work), you could probably fit the top triangle inbetween the other two like you should. The top triangle may need to be like 2 blocks smaller, but in a larger triangle, it'll be harder to notice. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,656 |
Larus cachinnans 2CY 415P October 22 2009, Barneveld, the Netherlands. Picture: PieterGeert Gelderblom.
Bird ringed as pullus on May 22 2008 at Jankowice Babice Malopolskie, Poland.
Note tiny mirror on 2nd gen P10. Hybrid option not fully eliminated. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,880 |
Q: Maximum sum of non adjacent elements in a 2D array I'm trying to look for an algorithm that will help me find the maximum sum of non adjacent elements in a 2D array.
For 1D arrays, I've found helpful solutions from:
1) http://www.geeksforgeeks.org/maximum-sum-such-that-no-two-elements-are-adjacent/
2) https://www.youtube.com/watch?v=UtGtF6nc35g
For example in a 1D array: {3, 2, 6, 2, 10}
I will get a maximum sum of 19, because 3, 6 and 10 are non adjacent.
However, I'm unable to find one that can help me with a 2D array. How can i find the maximum sum of integers in this array without horizontal or vertical adjacent elements? Diagonally adjacent elements are allowed.
For example:
[3, 2, 6, 2, 10]
[1, 5, 2, 5, 1]
[5, 1, 7, 2, 9]
[3, 9, 1, 8, 2]
Is there an existing algorithm to solve this problem? Or would it be another method to solve this problem if I used another data structure instead of a 2D array?
A: Problem can be stated as maximum-weight independent set or mixed integer linear programming problem.
To convert it to maximum-weight independent set, create graph where vertices are elements and edges are adjacency.
I am not expert on integer programming. There is similar (even more complicated) question where sascha demonstrated how to transform problem to MIP, and power of solvers.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,138 |
"""
Module contains current pmagpy version number.
Version number is displayed by GUIs
and used by setuptools to assign number to pmagpy/pmagpy-cli.
"""
"pmagpy-3.9.1-5"
version = 'pmagpy-3.9.1-5'
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,262 |
Glassy - Sona Family Ft. Hard Kaur
If you've spoken to me over the last few days, you'll know how much I love this song. I've never really paid attention to Hard Kaur before this track and it's pretty coincidental that two lady MCs hit my playlist at relatively the same time. I've been listening to this so much that I'm now singing it in its entirety even when it's not playing. Lovely.
Dil Nai Lagda - Feroz Khan & Vinni
Another track from Aman Hayer's Groundshaker album currently doing the rounds (and I think it's in the medley playing on television too). As good as Tharti Hildi, but in a different way, the mix of vocals in this duet work really well.
Tags: asian, music, review
Food: Tayyabs
Where The Hell Is The Isle of Man?
Champions of Europe
Karaoke!
Film: Star Wars Episode 3: Revenge Of The Sith
Operation Muslim Vote
Book: A Brief History of Saudi Arabia, James Wynbr...
Film: Ong-Bak
Film: The Jacket
Life Isn't All Ha Ha Hee Hee
Lifesaver, Plus
Well, The Results Are In...
Film: xXx2: The Next Level
I Just Refuse To Lie Down
Film: The Hitchhiker's Guide To The Galaxy | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,175 |
namespace Azure.Management.Compute.Models
{
/// <summary> Contains the data disk images information. </summary>
public partial class DataDiskImage
{
/// <summary> Initializes a new instance of DataDiskImage. </summary>
public DataDiskImage()
{
}
/// <summary> Initializes a new instance of DataDiskImage. </summary>
/// <param name="lun"> Specifies the logical unit number of the data disk. This value is used to identify data disks within the VM and therefore must be unique for each data disk attached to a VM. </param>
internal DataDiskImage(int? lun)
{
Lun = lun;
}
/// <summary> Specifies the logical unit number of the data disk. This value is used to identify data disks within the VM and therefore must be unique for each data disk attached to a VM. </summary>
public int? Lun { get; }
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,259 |
{"url":"https:\/\/www.hcm.uni-bonn.de\/events\/eventpages\/2013\/hausdorff-kolloquium-2013\/","text":"# Hausdorff-Kolloquium 2013\n\nDate: April 24, 2013 - July 10, 2013\n\nVenue: Mathematik-Zentrum, Lipschitz Lecture Hall, Endenicher Allee 60, Bonn\n\n## Wednesday, April 24\n\n 15:15 Jean Bertoin (Z\u00fcrich): Giant and almost giant clusters for percolation on large trees. 16:45 J\u00fcrg Kramer (Humboldt, Berlin): Mathematical concepts from school to current research: an example.\n\n## Wednesday, June 12\n\n 15:15 G\u00e9rard Laumon (Paris Sud, Orsay): On the counting of Hitchin bundles 16:45 Laure Saint-Raymond (ENS, Paris): The irreversibility in gas dynamics, a matter of probability\n\n## Wednesday, July 10\n\n 15:15 Martin Hairer (Warwick): Dynamics near criticality 16:45 Nalini Anantharaman (Paris Sud, Orsay): Quantum Ergodicity on Large Regular Graphs\n\n## Abstracts:\n\n##### Nalini Anantharaman: Quantum Ergodicity on Large Regular Graphs\n\n\"Quantum ergodicity\" usually deals with the study of eigenfunctions of the Laplacian on Riemannian manifolds, in the high-frequency asymptotics. The rough idea is that, under certain geometric assumptions (like negative curvature), the eigenfunctions should become spatially uniformly distributed, in the high-frequency limit. I will review the many conjectures in the subject, some of which have been turned into theorems recently. Physicists like Uzy Smilansky or Jon Keating have suggested looking for similar questions and results on large (finite) discrete graphs. Take a large graph $G=(V, E)$ and an eigenfunction $\\psi$ of the discrete Laplacian -- normalized in $L^2(V)$. What can we say about the probability measure$|\\psi(x)|^2$ (x\\in V)? Is it close to uniform, or can it, on the contrary, be concentrated in small sets? I will talk about ongoing work with Etienne Le Masson, in the case of large regular graphs.\n\nTop\n##### Jean Bertoin: Giant and almost giant clusters for percolation on large trees\n\nMotivated by a celebrated work of Erd\u00f6s and R\u00e9nyi, we consider Bernoulli bond percolation on a tree with size $n \\gg 1$, where the parameter of percolation depends on the size of the tree.\nOur purpose is to investigate the asymptotic behavior of the sizes of the largest clusters for appropriate regimes.\nWe shall first provide a simple characterization of tree families and percolation regimes which yield giant clusters, answering a question raised by David Croydon.\nIn the second part, we will review briefly recent results concerning two natural families of random trees with logarithmic heights, namely recursive trees and scale-free trees.\nWe shall see that the next largest clusters are almost giant, in the\nsense that their sizes are of order $\\frac{n}{\\ln{n}}$, and obtain precise limit theorems in terms of certain Poisson random measures. A common feature in the analysis of percolation for these models is that, even though one addresses a static problem, it is useful to consider dynamical versions in which edges are removed, respectively vertices are inserted, one after the other in certain order as time passes.\n\nTop\n##### Martin Hairer: Dynamics near criticality\n\nHeuristically, one can give arguments why the fluctuations of classical models of statistical mechanics near criticality are typically expected to be described by nonlinear stochastic PDEs. Unfortunately, in most examples of interest, these equations seem to make no sense whatsoever due to the appearance of infinities or of terms that are simply ill-posed. I will give an overview of a new theory of \"regularity structures\" that allows to treat such equations in a unified way, which in turn leads to a number of natural conjectures. One interesting byproduct of the theory is a new (and rigorous) interpretation of \"renormalisation group techniques\" in this context. At the technical level, the main novel idea involves a complete rethinking of the notion of \"Taylor expansion\" at a point for a function or even a distribution. The resulting structure is useful for encoding \"recipes\" allowing to multiply distributions that could not normally be multiplied. This provides a robust analytical framework to encode renormalisation procedures.\n\nTop\n##### J\u00fcrg Kramer: Mathematical concepts from school to current research: an example\n\nIn our talk we plan to illustrate by means of the well known example of the irrationality of the square root of 2 (which is usually discussed in grades 8 or 9 in schools to motivate the real numbers) how this topic provides the foundations of far reaching concepts in number theory beginning with the (classical) study of rational points on algebraic curves and leading to current developments in Arakelov geometry to study cycles on higher dimensional algebraic varieties.\n\nTop\n##### G\u00e9rard Laumon : On the counting of Hitchin bundles\n\nOn a curve over a finite field, the total mass of vector bundles (of a given rank and of a given degree) is finite. A beautiful formula established by Siegel, expresses this mass in terms of values of the zeta function of the curve.\n\nOn the opposite, the total mass of the Hitchin bundles (also called Higgs bundles or Hitchin pairs) on the curve is infinite and we have no analog of the Siegel formula. Nevertheless there are only finitely many semistable Hitchin bundles and it makes sense to try to find a formula for their total mass.\n\nWith Pierre-Henri Chaudouard, we apply to this question some techniques developed by Langlands and Arthur for the trace formula. Our main motivations are the conjecture of Hausel and Rodriguez-Villegas, and the understanding of the nilpotent part of the Arthur-Selberg race formula.\n\nIn the talk I will present the method that we use and some results that we have obtained.\n\nTop\n##### Laure Saint-Raymond: The irreversibility in gas dynamics, a matter of probability\n\nThe goal of this lecture is to present a derivation of the Boltzmann equation starting from the hamiltonian dynamics of particles in the Boltzmann-Grad limit, i.e. when the number of particles $N\\to\\infty$ and their size $\\eps \\to 0$ with $N\\eps^2 = 1$. We will especially discuss the origin of irreversibility and the phenomenon of relaxation towards equilibrium, which are apparently paradoxical properties of the limiting dynamics.\n\nTop","date":"2020-07-02 22:52:47","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 8, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7025349736213684, \"perplexity\": 741.1144036338693}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-29\/segments\/1593655880243.25\/warc\/CC-MAIN-20200702205206-20200702235206-00228.warc.gz\"}"} | null | null |
{"url":"https:\/\/www.transtutors.com\/questions\/question1-question-2-please-explain-why-question1-have-no-loss-while-question-2-have-2567555.htm","text":"# Question1 Question 2 Please explain why Question1 have no LOSS, While Question 2 have LOSS, why...\n\nQuestion1\n\nQuestion 2\n\nPlease explain why Question1 have no LOSS, While Question 2 have LOSS, why and how to determine?\n\nDecker Company has five products in its inventory. Information about the December 31, 2016, inventory follows Unit Unit Selling Product Quantity Cost Price 1,000 $10$16 800 15 18 600 200 600 14 13 The cost to sell for each product consists of a 15 percent sales commission. Required 1. Determine the book value of inventory at December 31, 2016, assuming the lower of cost and net realizable value rule is applied to individual products. Product Inventory Cost NRV (units) A (1,000) 10.000 13600 10,000 12,000 12,240 12,000 B (800) C (600) 1,800 4,080 1,800 1,020 1,020 D (200) 1,400 6,630 8,400 E (600) 6,630 33,600 37,570 a\\$ 31,450 2. Determine the book value of inventory at December 31, 2016, assuming the lower of cost and net realizable value rule is applied to the entire inventory. Also, assuming that Decker reports an inventory write-down as a line item in the income statement, determine the amount of the loss. Inventory book value 33,600 Loss from inventory write- down\n\n## Plagiarism Checker\n\nSubmit your documents and get free Plagiarism report\n\nFree Plagiarism Checker","date":"2021-09-21 08:03:56","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.18205082416534424, \"perplexity\": 3694.265756355882}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-39\/segments\/1631780057199.49\/warc\/CC-MAIN-20210921070944-20210921100944-00658.warc.gz\"}"} | null | null |
package com.example.facebookfriendviewer.tasks;
import com.example.facebookfriendviewer.BaseActivity;
import com.example.facebookfriendviewer.objects.Album;
import com.example.facebookfriendviewer.objects.Photo;
import com.example.facebookfriendviewer.utils.Utils;
import com.facebook.Request;
import com.facebook.Response;
import com.facebook.Session;
import com.facebook.model.GraphObject;
import java.util.List;
public class FacebookPhotosLoadTask extends AbstractLoadTask {
private static final String PHOTOS = "/photos";
private Album album;
public FacebookPhotosLoadTask(BaseActivity photosActivity, Album album,
LoadCompleteListener loadCompleteListener) {
super(photosActivity, loadCompleteListener);
this.album = album;
}
@Override
public Void call() {
Response response = Request
.newGraphPathRequest(Session.getActiveSession(),
album.getAlbumId() + PHOTOS, null).executeAndWait();
List<GraphObject> list = Utils.listGraphObjectFromResponse(response);
if (list != null && list.size() > 0) {
dbService.deletePhotos(album.getId());
Photo photo;
for (GraphObject obj : list) {
photo = Utils.convertToPhoto(obj);
photo.setAlbumId(album.getId());
dbService.savePhoto(photo);
}
}
return null;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,388 |
Help your older pet climb up to a chair or bed. Each step has a 6" rise and folds up for easy storage. Patterns drawn FULL SIZE. Measures 18"H. Order Bold & Spacer kit and one piece of 1/4" walnut for overlays separately. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,512 |
{"url":"https:\/\/www.lpsm.paris\/mathdoc\/preprints\/bally.Thu_May_30_14_58_34_CEST_2002.html","text":"Universit\u00e9 Paris 6Pierre et Marie Curie Universit\u00e9 Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilit\u00e9s et Mod\u00e8les Al\u00e9atoires''\n\n### First order schemes in the numerical quantization method\n\nAuteur(s):\n\nCode(s) de Classification MSC:\n\n\u2022 60G40 Stopping times; optimal stopping problems; gambling theory, See also {62L15, 90D60}\n\u2022 65C05 Monte Carlo methods\n\u2022 65C20 Models, numerical methods\n\u2022 65N50 Mesh generation and refinement\n\u2022 90A09 Finance, portfolios, investment\n\nR\u00e9sum\u00e9: The numerical quantization method is a grid method which relies on the approximation of the solution of a nonlinear problem by piecewise constant functions. Its purpose is to compute a large number of conditional expectations along the path of the associated diffusion process. We give here an improvement of this method by describing a first order scheme based on piecewise {\\em linear} approximations. Main ingredients are correction terms in the transition probability weights. We emphasize the fact that in the case of optimal quantization, many of these correcting terms vanish. We think that this is a strong argument to use it. The problem of pricing and hedging American options is investigated and {\\em a priori} estimates of the errors are proposed.\n\nMots Cl\u00e9s: Numerical quantization ; american options ; Malliavin calculus\n\nDate: 2002-05-30\n\nPr\u00e9publication num\u00e9ro: PMA-735\n\nPdf file : PMA-735.pdf","date":"2018-07-19 21:10:24","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7605789303779602, \"perplexity\": 1683.2986919416828}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-30\/segments\/1531676591296.46\/warc\/CC-MAIN-20180719203515-20180719223515-00055.warc.gz\"}"} | null | null |
\section{Introduction}
An important property of the solar magnetic
fields in sunspots and active regions is their
helical nature. Quantitative studies of this
effect have been carried out for a long time,
e.g., Seehafer (1990), Pevtsov \& Canfield
(1994), Bao \& Zhang (1998), Hagino \& Sakurai
(2004). \cite{Zhang_ea2010}(thereafter Z10) presented a systematic
study of
current helicity density and
twist of photospheric magnetic fields in solar
active regions. The obtained butterfly diagram 2D
(latitude-time) plot covers the two consequent
solar cycles. The quantities which are presented
in Z10 are the radial part of current helicity
and twist (force-free factor) averaged over a
statistically significant sample of active
regions. Vector magnetographic data have been
collected for more than 6,000 individual
magnetograms over the period of almost two
sunspots cycles 1988-2005, and they demonstrate
significant cyclic dynamics of these quantities.
At the same time they give relatively reliable
magnitudes of these helical quantities which
would be attractive to obtain by some ad-hoc
analysis of the very basic process of sunspot
formation and compare it with the observational
values.
The magnetic helicity plays an importance role in
hydromagnetic dynamo theory as it is an inviscid
invariant which is conserved in turbulent
convective motion for very large magnetic
Reynolds numbers. It can be used as a constraint
for the theoretical modelling. Current helicity
and twist are observational tracers of magnetic
helicity. Furthermore, as the observational
results have been complimented by constructions
of dynamo models based on helicity conservation
argument (e.g. Kleeorin et al. 2003; Zhang et al.
2012, and references therein).
In the present paper we estimate the tilt (the angle
between a line connecting the leading and following sunspots
and the solar equator), current helicity and twist of magnetic fields in solar active regions.
\section{The Model}
Let us consider a simple bipolar active
region with the distance $L$ between the opposite
polarities. We use a model of a turbulent convective cell of the size of
super-granulation with a depth $L/2 \approx 10^9{\rm cm}$.
This scale is of the order of the density stratification scale in the convective zone $L/2 \approx
H_\rho=-\displaystyle{\left[{d\over
dr}{\log{\rho_0(r)}}\right]^{-1}}$, where
$\rho_0$ is the fluid density at the depth $H=10^9{\rm cm}$.
This scale we associate with the depth of the sunspot formation region.
In a classical Rayleigh-B\'{e}nard convective roll, horizontal and vertical scales are the same.
A superposition of the three convective rolls forms a hexagonal structure (see, e.g., \cite{Chandra1961}, Chap.~16, p.~48, Fig.~7a). This implies that the ratio of horizonal to vertical sizes in the hexagonal structure is about 2.
In the Sun and stars the convection is fully turbulent and is different from classical Rayleigh-B\'{e}nard convection. In particular, the convection rolls are formed with their horizontal scales greater than the vertical scales approximately by factor 2. Similar phenomena are observed in Earth's atmosphere and other natural turbulent convection systems.
Large-scale structures like the convection rolls can be isolated from turbulent eddies using scale separation ideas (Elperin et al 2002; Bukai et al. 2009) being applied to solar active regions.
This consideration is in agreement with observations of super-granulation convection in the Sun.
Therefore, the total horizontal extent of this active region is $2L$.
Let us assume that the active region scales are small
compared to the solar radius, and put it at
heliographic latitude $\phi$ being counted from
the solar equator. In terms of observable active
regions, we estimate this scale by order of
magnitude as $L \sim 20-50 \,{\rm Mm} =
(2-5)\cdot10^{9}\,{\rm cm}$.
We consider the momentum equation for ${\bf u}$
applying the anelastic approximation at the boundary between the solar convective
zone and the photosphere
\begin{equation}
\frac{\partial {\bf u}}{\partial t} = -\grad \left(
\frac{p_{tot}}{\rho_0}\right)
-{\bf g} S
+ \mathcal{{\bf F}}_{mag}
+ \mathcal{{\bf F}}_{hd}
+ \mathcal{{\bf F}}_{visc}
+ \mathcal{{\bf F}}_{cor} ,
\label{vandak}
\end{equation}
which includes the effects of the total pressure, hydrodynamic and magnetic buoyancy,
nonlinear local hydrodynamic Coriolis force, viscous forces and global Coriolis force.
The formulation of equation (\ref{vandak})originates from the book \cite{vand76}, also see the recent work of \cite{Kleeorin2020}.
The detailed forms of these quantities are decribed in Appendix~A.
Let us estimate sunspot twisting time $\tau_D$ as the ratio
$L/v_a$, where Alfven speed is
$v_a=B_{eq}/ \sqrt{4\pi
\rho_0}$. At the upper part of the convective zone typical equipartition value of
the mean magnetic field is $B_{eq} \sim 300\, {\rm G}$, and
density of the solar plasma $\rho_0$
according to estimates of Spruit (1974),
is of the order of
$4.5 \cdot 10^{-7} {\rm g}\,{\rm cm}^{-3}$.
Thus, Alfven speed is of the order of
$v_a \sim 1.2 \cdot 10^5 {\rm cm}\,{\rm s}^{-1}$,
and therefore the time scale
$\tau_D \sim (2-4) \cdot 10^4{\rm s} \approx 6-12$~hours.
The duration of the flux tube emerging from the bottom of the convective cell
is of the order of $\tau_F=L/2u_r$.
We use the anelastic approximation for convection in the solar convection zone ${\rm div} (\rho \bec u) = 0$. The density stratification is in the radial direction, so we have ${\rm div} \bec u =
-u_r\displaystyle{\frac{d}{dr}} \log \rho
= \displaystyle{\frac{u_r}{H_\rho}}
\approx \displaystyle{\frac{1}{\tau_F}}$, where $H_\rho$ is density stratification scale.
Estimate for the radial component of vorticity $\omega_r$ for the motion of bipolar sunspots
is obtained in Appendix~A:
\begin{equation}
\omega_r
\approx
- \Omega_\odot \displaystyle{\frac{\tau_D}{\tau_F}}
\left(
4\, \sin{\phi}
+
\frac{1}{\xi}
{\cos{\phi}
\right)
\,,
\end{equation}
where instead of co-latitude $\theta$ we use the latitude $\phi=\pi/2-\theta$,
then $\tau_F=H_\rho/u_r$
and $\xi$ is defined below.
The latitudinal derivative of the vertical convective velocity can be estimated as
$\displaystyle{\frac{1}{r}\frac{\partial u_r}{\partial \theta}}
\approx
\displaystyle{\frac{u_r}{\xi L}}$, where $\xi$ being dependent on the structure of active regions varies in sign, and in absolute value is around $1-2$ due to the hexagonal structure of convection.
The sign of $\xi$ can be considered random for super-granulation convection in the Sun.
If we consider roll-like convection, it is close to unity while for hexagonal convective cells comprising of three rolls, it is close to 2 (see Elperin et al 2002; Bukai et al. 2009).
\subsection{Estimate for Tilt of Sunspots}
Given the time of evolution of the active region
during the sunspot formation (which contributes to
the tilt angle of the opposite polarities), is
comparable with the flux tube emerging time
$\tau_F$, so that we can estimate the tilt as
\begin{equation}
\delta \approx w_r \tau_F
= \displaystyle{-\frac{2\pi\tau_D}{T_{\odot}}}
\left(4\sin{\phi}
+\displaystyle{\frac{\cos{\phi}}{\xi}}
\right) ,
\label{tilt}
\end{equation}
where the solar siderial rotation period ${T_{\odot}}\approx 25\,$d
approximately corresponding to Carrington rotation.
The coefficient in the front of the brackets in this formula is of order $0.25-0.5$.
Since the value of $\sin{\phi}$ for low
latitudes where sunspots mainly occur is comparable with the latter term within the brackets, that is of order $1/8-1/4$, we can expect the tilt angles of individual active regions to vary a lot and even change sign. This estimate also implies the less variability in tilt angles the higher latitude, which may need to be verified with observations.
Given the sign of $\xi$ randomly fluctuates and its value varies with super-granulation, for the averaged tilt $\langle \delta \rangle $ we may obtain the range $(0.25-0.5) \sin{\phi}$, which gives the order
up to $15^\circ$ for middle latitudes, and so it fits perfectly well with the observational results of Howard (1991).
The observational magnitude of tilt is indeed
increasing with departure from the solar equator
almost linearly with latitude, or like
$\sin{\phi}$, and for middle-range latitudes it is
in average $5-15^\circ$, see e.g. Stenflo \& Kosovichev (2012);
\cite{TlatovIllarionov2013,StenfloKosovichev2012} and references therein.
\subsection{Estimates for Current Helicity and Twist from Solar Observations}
We estimate the current helicity in terms of
the magnetic energy and the vertical variation scale of
magnetic field. This scale is of the order
of several density stratification
scales $H_\rho$.
The flux tubes are mainly formed by large-scale MHD instability
(e.g., the magnetic buoyancy instability
\cite{1955ApJ...121..491Parker_sunspot}
and the negative effective magnetic pressure instability \cite{Kleeorin1990,Brandenburg2016})
from the dynamo generated large-scale magnetic field.
These magnetic flux tubes are
unstable when the mean field strength exceeds a critical
value. The rise of magnetic flux tubes to the surface
due to magnetic buoyancy can cause formation of sunspots and active regions.
As the horizontal fields $B_x$ and $B_y$ are getting
tilted with the active region formation process,
they form horizontal gradients which are
observable in high resolution vector
magnetographic measurements and used for
computation of electric currents, and,
subsequently, current helicity density. These
titled distortions can be estimated as
\begin{equation}
\Delta B_y \approx
\delta
\displaystyle{
\frac{\partial B_x}{\partial x}} x
\qquad
{\rm and}
\qquad
\Delta B_x \approx
-\delta
\displaystyle{
\frac{\partial B_y}{\partial y}} y .
\end{equation}
Using solenoidality condition ${\rm div} \bec B = 0$,
we determine the vertical component of the $\curl$ as
\begin{equation}
(\curl B)_z =
\displaystyle{
\frac{\partial \Delta B_y}{\partial x}}
-
\displaystyle{
\frac{\partial \Delta B_x}{\partial y}}
\,=\,
\delta
\displaystyle{
\left(
\frac{\partial B_x}{\partial x}
+\frac{\partial B_y}{\partial y}
\right)
}
=-\delta
\displaystyle{
\frac{\partial B_z}{\partial z}
}
\,.
\end{equation}
Now we can relate the vertical part of the current helicity with twist as
\begin{equation}
H_c=B_z(\curl B)_z =
-\delta
B_z\displaystyle{
\frac{\partial B_z}{\partial z}
}=
\delta\,
\displaystyle{
\frac{B_z^2}{H_B}
}
\,,
\label{heli2}
\end{equation}
where $H_B$ is the vertical magnetic field variation scale, which is of the order of
the sunspot size ($\sim 10-20$ Mm $= 1-2\cdot10^9$ cm).
Correspondingly, the estimate of twist $\Upsilon$ for typical tilt angles of order $\delta \sim 0.1-0.2$ ($5-10^\circ$) reads
\begin{equation}
\Upsilon=
\delta/H_B
\sim 10^{-10}\,{\rm cm}^{-1} = 10^{-8}\,{\rm m}^{-1}
\,,
\label{twist}
\end{equation}
which matches quite well the order of magnitude of the observational results, e.g. \cite{Zhang_ea2002}; Zhang el al. (2010).
Notice that the vertical magnetic field energy proportional to ${B_z^2}$ is of the order of the equipartition magnetic field.
For estimation of current helicity we take into account that the typical value of the magnetic field varies from a few G in the quiet Sun to kG in sunspots. Keeping in mind, however, that for comparison with vector magnetographic observations we must focus mainly on penumbral parts of active regions, the reasonable estimate for the field would be hundreds of gauss. Setting $B_z \sim 300$~G we obtain the estimate for $H_c \sim 10^{-3} {\rm G}^2 {\rm m}^{-1}$, in agreement with observations of Bao \& Zhang (1998); Zhang el al. (2010).
These naive add-hoc estimates of the tilt angles as well as current helicity and twist in solar active regions based on local considerations are in accordance with the observational ranges of current helicity and twist.
\section{Estimates for Current Helicity and Twist in Dynamo Models}
After we have estimated the tilt angles as well as current helicity and twist using local effects for rising flux tubes in solar active regions, we are going to estimate these values from the axially symmetric spherical shell dynamo model by Zhang et al. (2012), see also references therein.
Let us use Equation (5) of Zhang et al. (2012) for the mean current helicity
of the active region,
\begin{eqnarray}
H_c^{\rm AR}=
\langle {\bf B}^{\rm AR} {\bf
\cdot} \bec{\rm curl} \,{\bf B}^{\rm AR} \rangle
\approx
- {1 \over L^2_{\rm ar}} {\bec{ A}} {\bf \cdot} {\bec{ B}}
= - {B_{*}^2 R_\odot \over
L^2_{\rm ar}} \tilde{A} \tilde{B},
\label{D1}
\end{eqnarray}
where $A=R_\sun \tilde A B_{*}$,
$B=\tilde B B_{*}$; $B_{*}=10 \rho_0^{1/2} \eta_T/R_\sun$ is the characteristic magnetic field produced by the dynamo mechanism and $\eta_T$ is the turbulent magnetic diffusion coefficient
(see Zhang et al. (2012); Kleeorin et al. (1995)), and
$R_\sun \approx 7\times10^{10}\,{\rm cm}$ is the solar radius.
The twist of magnetic fields of an active region can be estimated as
\begin{eqnarray}
\Upsilon \equiv {\langle {\bec B}^{\rm AR} {\bec
\cdot} \bec{\curl} \,{\bec B}^{\rm AR} \rangle
\over \langle ({\bec B}^{\rm AR})^2 \rangle}
\approx - {{\bec{ A}} {\bec \cdot} {\bec{ B}}
\over L^2_{\rm AR} {\bec{ B}}^2} \approx
-{{\tilde{A}} \over {\tilde{B}}}
{{R_\odot} \over{ L^2_{\rm aAR}}},
\label{D2}
\end{eqnarray}
where we assumed that
$\langle ({\bf B}^{\rm AR})^2 \rangle \approx {\bec{ B}}^2$.
Note that the product and ratio of $\tilde A$ and $\tilde B$ vary within the solar cycle but does not vary from one (odd/even) cycle to another (even/odd). The sign of ${\bec{A}} \cdot {\bec{B}}$ is mainly negative (~3/4 of the period), so the sign of $\Upsilon >0$, i.e. opposite to the one produced by Coriolis force.
According to observations the horizontal size of the active region is about the size of a super-granule,
i.e., $L_{\rm AR} \sim (2-5)\cdot H_\rho$,
where $H_\rho \sim 10^9$~cm.
So, we estimate $L_{AR} \sim (2-5)\cdot 10^{9}\,{\rm cm}
= 20-50\,{\rm Mm} $.
Using Eq.~(\ref{D2}), we estimate the twist of the magnetic field in the active region as
\begin{eqnarray}
\Upsilon \approx -
{{\tilde{A}} \over {\tilde{B}}}
{{R_\odot} \over
{{L_{\rm AR}}^2}} \sim
- (0.3-1) \cdot 10^{-10}
{\rm cm}^{-1},
\label{D3}
\end{eqnarray}
where typically $\tilde{A}/\tilde{B} \sim
6\times 10^{-3}$
(see Appendix~B and also the figures of \cite{Zhang et al. (2012)}).
This value is of order $10^{-2}$ which is typical for the most kinematic dynamo models of $\alpha\Omega$ type.
Now, let us take the value of $\tilde{B}$ of
order $0.2-0.5$ (as it is on in the bottom of the
convective zone). If we adopt the value of
$B_{eq}\sim 500-1000 \,{\rm G}$ as somewhere in the
sunspot umbra near or just beneath the photosphere,
we estimate the magnitude for the current helicity as
\begin{eqnarray}
H_c^{\rm ar} = \Upsilon {\bec{ B}}^2 {\tilde{B}}^2
\sim - 10^{-5}
{\rm G}^{2} {\rm cm}^{-1}
= - 10^{-3} {\rm G}^{2} {\rm m}^{-1}
\,.
\label{D4}
\end{eqnarray}
This value is comparable with the observational
results of Zhang et al. (2010) giving the order
of $(1-2)\cdot10^{-5}\,{\rm G}^2\,{\rm cm}^{-1}$
for current helicity, and
$(1-2)\cdot10^{-10}\,{\rm cm}^{-1}$ for the
twist.
We can apply the other method of estimation of ratio $A/B$ using the numerical simulations of the 2D mean-field dynamo model in a spherical shell, e.g., by Zhang et al. (2012). For the variety of depths and latitudes, this ratio is of order $10^{-3}-10^{-2}$, most typically $(2-7)\cdot10^{-3}$ (private communications with late David Moss
\footnote{deceased in 2020}).
The estimate~({\ref{D3}}) for twist yields the same order of magnitude $ -(10^{-10}- 10^{-11}){\rm cm}^{-1}$ as estimated above.
For small-size active regions, $L \sim (0.5-1)\cdot H_\rho$, the contributions to current helicity from both
effects caused by the Coriolis force and the dynamo mechanism are getting comparable.
\section{Discussion}
In this paper we estimated the effects of two mechanisms of formation of tilt, twist and current helicity in solar active regions.
One mechanism is related to the action of Coriolis force on rising active regions, so for that
the original magnetic field is not assumed twisted at the initial stage of rising magnetic flux tube.
The second mechanism is related to production of magnetic helicity by the dynamo process on the mean magnetic field. The study by Zhang et al. (2012) shows that the rising magnetic fields may already be helical at the very beginning stage because the magnetic helicity is produced together with the mean magnetic field generation.
The magnetic helicity of the mean field has the opposite sign to the current helicity of the active region.
The second mechanism of the origin of tilt, twist and current helicity is apparently independent of the first one.
These two mechanism cause opposite sign contributions to the tilt, twist and current helicity in the most but not all phases of the solar cycle.
The overall sign of tilt, twist and current helicity may depend on the particular ratio of the two contributions at the given time and location.
This may explain certain irregularities of the tilt from the classic hemispheric rule (e.g. \cite{TlatovIllarionov2013,Pevtsov_ea_rev2014}).
Having estimated the tilt, current helicity and twist from local considerations in active regions we notice that even the sign of them may vary from one active region to another.
On the other hand,
the averaging of the tilt, current helicity and twist
over all active regions does not depend on the phase of solar cycle and the sign of magnetic field (Hale's polarity law), so from one cycle to another cycle the sign of these quantities holds.
This is in good agreement with Joy's law for tilt as well as Hemispheric Sign Rule for current helicity and twist. Given the considered effects do not vary much with the phase of the solar cycle, we may have the same trend everywhere. However, as the quantities which are produced by the solar dynamo may vary with the phase of the cycle, we can impute this fact to systematic change in sign and reversal of the Hemispheric Sign Rule for current helicity and twist noticed by Bao et al. (2000) and obtained in detail for some phases of the solar cycle and some latitudes by Z10.
The results of our estimations show that the
value of helicity which can be formed by flux
tube arising in the convective zone during the
sunspot formation can well be comparable
with amount of helicity generated by the entire
dynamo process. This makes important simultaneous
studies of the origin of helicity in the Sun by
both mechanisms
(see, \cite{Kleeorin2020}).
Our vision of the phenomena of helicity and tilt in solar active regions is significantly different from one reported by Longcope et al. (1998), see also Fisher, Fan et al. (1998), as the contribution to helicity and tilt due to Coriolis force is linearly proportional to the value of tilt while in those papers it appears quadratic (with the sign oppisite to the sign of tilt) in the value of tilt. In our vision, we operate not with helicity of magnetic tubes but with helicity of physical magnetic field in active regions. This naive consideration looks to us more relevant to observations. We impute the variability of sign of overall twist and helicity by the evolution of the dynamo contribution with time-latitudinal evolution in the solar cycle on the one hand, and the relatively stable contribution by Coriolis force, on the other.
However, these two mechanisms look apparently independent only from the first instance. The reason is that the current helicity is intimately related to magnetic helicity (see, e.g. \cite{BergerField1984}; \cite{Pevtsov_ea_rev2014} and references therein) and in absence of helicity flux the latter is a local integral in a non-dissipative MHD flow. In the scales which are much greater than turbulent scale of basic granulation and less than the typical size of an active region, we expect that the current helicity of the active region is determined by the magnetic helicity produced by the mean-field dynamo mechanism, see formula (\ref{D1}).
In the presence of helicity fluxes some fraction of the dynamo-born helicity along with some (let us assume the same fraction) of helicity due to Coriolis force are removed from the
photosphere and injected into the solar corona. Let us denote the fraction of this helicity injection as $\epsilon$. Then combining formulae (\ref{tilt}), (\ref{D1}), and (\ref{D4}), we can derive the following expression for the remaining total current helicity of an active region
\begin{eqnarray}
H_c^{\rm ar TOT} =
-\, \epsilon
\displaystyle{\frac{2\pi\tau_D}{T_{\odot}}}
\left(
4\,\sin{\phi}
+
\frac{1}{\xi}
{\cos{\phi}
\right)
\frac{B_z^2}{H_B}
-\, (1-\epsilon)
{1 \over L^2_{\rm ar}} {\bec{ A}} {\bf \cdot} {\bec{ B}} \,.
\label{D5}
\end{eqnarray}
Hereby the ejection of helicity from an active region into the corona is
\begin{eqnarray}
H_c^{\rm ar FL} =
\epsilon
\displaystyle{\frac{2\pi\tau_D}{T_{\odot}}}
\left(
4\,\sin{\phi}
+
\frac{1}{\xi}
{\cos{\phi}
\right)
\frac{B_z^2}{H_B}
-\, \epsilon
{1 \over L^2_{\rm ar}} {\bec{ A}} {\bf \cdot} {\bec{ B}} \,.
\label{D6}
\end{eqnarray}
Note that the sum of the total remaining and the flux parts of helicities in formulae (\ref{D4}) and (\ref{D5}) is equal to the amount of helicity produced by dynamo as in formula (\ref{D1}). The particular value of $\epsilon$ in formula (\ref{D5}) is not known and it can be estimated from comparison of theoretical dynamo models and the observational data for tilt, vertical magnetic field, twist and current helicity in solar active regions. The main contribution in the total helicity for large active regions is probably due to Coriolis force while for smaller one from dynamo generation mechanism. The distinction between the two can be determined by the latitude and phase of the solar cycle. This question requires further investigation with the use of calibrated dynamo models and available observational data. Please also note that the first part (contribution from large active regions) is scaled by $\tau_D \sim L$, so the relatively to the second part it is scaled as $\sim L^3$.
If for equation (\ref{D5}) we divide both sides by $B_z^2$ then we can treat this formula as expression for overall effective tilt. It contains both constant and oscillatory with the cycle parts.
\vspace{0.1cm}
\noindent
{\bf Acknowledgements.}
Stimulating discussions with participants
of the NORDITA program on ''Solar Helicities in Theory and Observations:
Implications for Space Weather and Dynamo Theory"
(March 2019) are acknowledged. NK, KK, IR would like to acknowledge the hospitality of NORDITA for this and earlier occasions where this work has been outlined and developed. Private communications with late David Moss are acknowledged.
The work of KK concerning the tilt and twist computations was supported by the grant from the Russian Science Foundation
(RNF 18-12-00131) at the Crimean Astrophysical Observatory.
DS would like to acknowledge support from RFBR grant 18-02-00085.
The observational estimates of helicity and twist are obtained due to collaboration of IZMIRAN team with National Astronomical Observatories of China, Key Laboratory for Solar Astivity, Chinese Academy of Sciences, supported in part by CAS PIFI visiting program and RFBR of Russia - NNSF of China joint grant 19-52-53045 GFEN.a.
HZ would also like to acknowledge support by grants from the National Natural Science Foundation of China (NSFC 11673033, 11427803, 11427901) and by Huairou Solar Observing Station, Chinese Academy of Sciences.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,948 |
The principal depends on staff to change/implement/follow-through on making the change a reality while he provides limited investment of time and resources in leading the actual implementation and change process.
The school becomes flooded with "add-on solutions or changes" that have never been implemented, a waste of money, and at the cost of a healthy relationship with staff.
Staff become resistant to change because the "solutions" provided appear to contradict each other, don't seem to be genuine, don't address their needs as learners nor as educators, or simply aren't a priority for them in comparison to pervasive problems they are facing with students, with staff in the work environment, or with systematic problems that have not been addressed.
The change is embodied by a single staff member, who becomes the expert everyone else refers to when there is any question about what is going on related to the change. That teacher is self-taught about the change and cannot refer to the principal for support, as the principal too directs questions to that teacher.
The change initially seemed to be a priority, but in reality descended down the list of priorities as more new changes were quickly adopted.
After reading this article, what beliefs do you think underscore fixed mindsets? Can you think of some changes you've led in your school? Did you ask similar questions? How do you think asking questions could have affected the change for better or worse? Have you experienced the difference in school change with a principal with a fixed mindset vs. a growth mindset? What did you learn? | {
"redpajama_set_name": "RedPajamaC4"
} | 7,881 |
Electrical wiring set up is very important to work in the houses and commercial buildings. It should be finished with adequate care and protection. Whether you lived in an old house or new one, the electrical installation should be updated and correctly installed. Wrong wiring is very hazardous and risky but it could be worse if not noticed on time or taken adequate care. Usually, the electrical wiring is inspected at the time of renovation but they should be checked after every 10 years. You can inspect the signs of wrong wiring by yourself too. In the event that you find any fault or wrong wiring call an electrical contractor in Singapore instantly. After many years, you may see signs that you need to change certain areas of wiring. So, what are these possible signs?
These are the signs which are very helpful to you to find out when you need an electrical wiring upgrade.
Shocking and jolting switches and outlets are the sign that you need to upgrade them or change them with new one. The causes of shocking switches are the cable in the circuit shorting out to the conduit enfolding the wires.
Ever notice any buzzing sounds coming out of the electrical connections? It could be a very risky sign. The buzzing noise that comes out of the electrical wires may be due to the loose connections or outlets, and wobble wires.
If you continuously change and replace your electricity fuses, it may be a symptom of wrong electrical wiring. It is the clear sign that the electricity is not going through a right path. Call an electrical contractor in Singapore to inspect the situation.
People generally may not be aware that certain appliance does not work in Singapore's household. For example, inserting a high power consuming appliance that may be bought from oversea in a single port simultaneously with other appliances may result in overloading, thus overheating the socket. So always look out for what is plugged in and how many is plugged into the outlet so as to ensure that the port is not overheated.
These are a few signs that would be very helpful to find out when to go for an electrical wiring upgrade. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,889 |
import java.lang.reflect.*;
public class runme {
static {
try {
System.loadLibrary("example");
} catch (UnsatisfiedLinkError e) {
System.err.println("Native code library failed to load. See the chapter on Dynamic Linking Problems in the SWIG Java documentation for help.\n" + e);
System.exit(1);
}
}
public static void main(String argv[])
{
System.out.println("ICONST = " + example.ICONST + " (should be 42)");
System.out.println("FCONST = " + example.FCONST + " (should be 2.1828)");
System.out.println("CCONST = " + example.CCONST + " (should be 'x')");
System.out.println("CCONST2 = " + example.CCONST2 + " (this should be on a new line)");
System.out.println("SCONST = " + example.SCONST + " (should be 'Hello World')");
System.out.println("SCONST2 = " + example.SCONST2 + " (should be '\"Hello World\"')");
System.out.println("EXPR = " + example.EXPR + " (should be 48.5484)");
System.out.println("iconst = " + example.iconst + " (should be 37)");
System.out.println("fconst = " + example.fconst + " (should be 3.14)");
// Use reflection to check if these variables are defined:
try
{
System.out.println("EXTERN = " + example.class.getField("EXTERN") + " (Arg! This shouldn't print anything)");
}
catch (NoSuchFieldException e)
{
System.out.println("EXTERN isn't defined (good)");
}
try
{
System.out.println("FOO = " + example.class.getField("FOO") + " (Arg! This shouldn't print anything)");
}
catch (NoSuchFieldException e)
{
System.out.println("FOO isn't defined (good)");
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,165 |
{"url":"http:\/\/www.physicsforums.com\/showthread.php?t=211194","text":"## Geometic Series that sums to circle?\n\nDoes anyone know if there is a way to divide up the area of a circle using similar polygons, with a common ratio? I was just curious if there is a way, or if it has been proven impossible.\n\nFor example, I tried inscribing a square inside a circle and making an infinite series of triangles with the remaining area, but the triangles do not have a common ratio.\n Recognitions: Gold Member Homework Help Science Advisor IF we can write the area of the unit disk as the limit of a geometric series, then, with a0 being the area of the largest sub-figure, and k the constant ratio, then we would necessarily have the following equation: $$\\frac{a_{0}}{1-k}=\\pi$$ But, since the ratio between rational numbers itself must be rational, it follows that either a0, k, or both must be irrational numbers. And that sort of deflates the attractiveness of the procedure, don't you agree?\n This probably doesn't make a difference but the area of circle is equal to pi*r^2 where r is the radius of the circle. But I don't know why that would deflate the attractiveness of the procedure. Why couldn't the ratio be the sqrt(2) or something?\n\nRecognitions:\nGold Member\nHomework Help","date":"2013-05-23 00:56:15","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8296464085578918, \"perplexity\": 220.29476667258822}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2013-20\/segments\/1368702718570\/warc\/CC-MAIN-20130516111158-00030-ip-10-60-113-184.ec2.internal.warc.gz\"}"} | null | null |
Note: much of this information comes from the website, RoboticPoolCleanersCompared.com. Check it out for more in-depth reviews and video trials.
The Nautilus only weighs 18.75 pounds. This robotic pool cleaner has many great features and is simple to use. This is one of the robotic pool cleaners that has two ways of cleaning. It not only uses suction to absorb debris it also has scrubbers attached. In this way, it vacuums and polishes the floor of the pool.
Even though this device is small, the filtration is very powerful. Its rate of filtration can get up to 4,233 gallons per hour. This is one of the best robotic pool cleaners that can reach lengths of fifty feet.
The power cable is sixty feet in length and has a unique swivelling mechanism. This stops the cable from twisting which could impede the cleaning process. It also features a program mapping system. It can scan the shape of the pool to determine the best method of cleaning it.
The Nautilus has a filter cartridge, not counterproductive bags. The cartridge is simple to take out, clean, and put back in. This is one of the robotic pool cleaners that will clean for three hours before shutting off. This is great because once the system is done cleaning it will not run up the electric bill.
The TigerShark is the best robotic pool cleaners available on the market. It too uses filter cartridges instead of bags. Owners of this robotic cleaner can take them out and clean them with a garden hose. In online reviews, first-time owners are typically shocked by how efficient the motor is. fuel given by a small twenty-four Volt motor and the entire apparatus weighs only twenty-one point five pounds.
This is one of the longest extended warranties offers when buying a robotic pool cleaner. This robotic pool cleaner will not only clean from the bottom it will clean the bottom and sides too. It also does an excellent job of cleaning steps. The material of the pool also will not affect how well it cleans. It can be gunite, vinyl, or fibreglass and still do a fantastic job of removing debris.
At the push of a button, the Pool Rover starts its twenty-four-volt motor to start cleaning. It uses vacuum suction as well as jet drive propulsion to perform its functions. A forty foot were floating cable also comes with the Pool Rover Junior. The Rover Junior can clean five thousand four hundred square feet per hour.
It can filter four thousand two hundred gallons of water a minute. Even though it only weighs ten and a half pounds it can clean almost any pool in 30 minutes. It will run for two full hours before shutting off. Its non-marring wheels never leave streaks behind or cause damage to the pool. The forty foot cable means it can reach long distances. This swivel cord is certain not to get tangled up in the pool.
This is one of the most tech-savvy robotic pool cleaners of all time. It has a built-in computer that processes the shape and size of the pool. The ActivMotion Sensor determines the most efficient cleaning method. Even though other robotic pool cleaners can do a similar job of cleaning, the Polaris is super fast.
Owners can watch dirt drop down and get sucked up by this cleaner. It can actually reach a filtration rate of an astonishing seventy gallons per minute. The filter canister holds up to one hundred sixty ounces of unwanted materials. It can clean a pool with a maximum length of fifty feet.
It weighs sixteen pounds and comes with a two-year warranty. What's interesting about the Polaris is its range of motion. It can flip over or fall down then regain its usual stance. Those who buy it love that the Polaris comes with a transport caddy as well.
The Water Tech Blue Diamond is one of the most lightweight robotic pool cleaners to make the list. Even though it is tiny, it features a filtration rate of four thousand two hundred fifty-five gallons per hour. It can clean up to five thousand nine hundred square feet per hour. Surprisingly it only takes twelve Volts of power. It can clean almost every pool in an hour.
It can clean almost every pool in an hour. Its PVA brushes are soft, durable, and absorbent. A special detector will keep it from getting blocked by steps and obstructions. The swivel cord mechanism also keeps it from becoming tangled up. Every model gets tested at seventeen-inch underwater depth to ensure the quality of its cleaning power.
These are five of the most highly ranked robotic pool cleaners on the market.
Since these products are never one-size-fits-all, it's important to read reviews before purchasing one.
What is right for one pool owner may not be the perfect choice for the next.
Shoppers will want to consider the length of cord they'll need, its efficiency, and of course the price. | {
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President Donald Trump gave a rather perfunctory speech during the National Prayer Breakfast this morning, basically paring down Tuesday's State of the Union address.
There was one unwelcome addition: Trump touted his administration's move last month to sanction taxpayer-funded discrimination by allowing South Carolina foster care agencies to reject prospective parents or volunteers they deem to be the "wrong" religion.
"My administration is working to ensure that faith-based adoption agencies are able to help vulnerable children find their forever homes while following their deeply held beliefs," Trump said at the breakfast – ignoring that if publicly funded foster care agencies are allowed to turn away qualified parents because of their religion, it will take even longer for children in the system to find loving, stable homes.
Unbelievably, in the very next breath, Trump spoke of his administration's efforts to combat anti-Semitism – even though the South Carolina foster care agency that triggered his administration's new policy was discriminating against Jewish parents and volunteers.
Miracle Hill Ministries, a large agency in South Carolina that contracts with the state to provide foster care and other social services, has a policy of turning away prospective parents and volunteers who don't have the same evangelical Christian beliefs – meaning they reject Jews, Catholics, nonbelievers and other religious minorities.
Lydia is not alone; other people, including Beth Lesser, also a Jewish woman, have come forward to explain how Miracle Hill wouldn't let them volunteer or foster children in need because of their religious beliefs.
Rather than stopping Miracle Hill from discriminating with taxpayer dollars and denying children the chance for a loving and stable home with qualified parents, South Carolina Gov. Henry McMaster (R) asked the Trump administration to give Miracle Hill and other South Carolina foster care providers a religious exemption so they can discriminate against prospective parents without risk of losing federal funding. Trump's Department of Health and Human Services granted that waiver on Jan. 23.
While the waiver only applies to South Carolina for now, it sets a dangerous nationwide precedent of elevating the beliefs of a government-funded agency over the best interests of the children in their care. Texas also has requested a waiver from HSS, and several states have passed laws sanctioning similar discrimination.
Because Trump and his Religious Right allies continue to weaponize religious freedom, Americans United is urging Congress to pass the Do No Harm Act (DNHA). The act would restore our fundamental principle of religious freedom to its original intent – acting as a shield that protects religious expression while clarifying that it may not be used as a sword to harm others. | {
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<?xml version="1.0" encoding="utf-8"?>
<!DOCTYPE TS>
<TS version="2.1" language="cs">
<context>
<name>BMMDialog</name>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/BMMDialog.ui" line="14"/>
<source>Manage Bookmarks</source>
<translation>Spravovat záložky</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/BMMDialog.ui" line="35"/>
<source>Name</source>
<translation>Název</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/BMMDialog.ui" line="40"/>
<source>Path</source>
<translation>Cesta</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/BMMDialog.ui" line="52"/>
<source>Remove Bookmark</source>
<translation>Odstranit záložku</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/BMMDialog.ui" line="65"/>
<source>Rename BookMark</source>
<translation>Přejmenovat záložku</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/BMMDialog.ui" line="91"/>
<source>Finished</source>
<translation>Hotovo</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/BMMDialog.cpp" line="58"/>
<source>Rename Bookmark</source>
<translation>Přejmenovat záložku</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/BMMDialog.cpp" line="58"/>
<source>Name:</source>
<translation>Název:</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/BMMDialog.cpp" line="64"/>
<source>Invalid Name</source>
<translation>Neplatný název</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/BMMDialog.cpp" line="64"/>
<source>This bookmark name already exists. Please choose another.</source>
<translation>Záložka s tímto názvem již existuje. Zvolte jiný název.</translation>
</message>
</context>
<context>
<name>DirWidget</name>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.ui" line="20"/>
<source>Form</source>
<translation>Formulář</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.ui" line="246"/>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.ui" line="249"/>
<source>Open item</source>
<translation>Otevřít položku</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.ui" line="271"/>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.ui" line="274"/>
<source>Open item (select application)</source>
<translation>Otevřít položku (vybrat aplikaci)</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.ui" line="300"/>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.ui" line="303"/>
<source>Add item to personal favorites</source>
<translation>Přidat položku do oblíbených</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.ui" line="322"/>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.ui" line="325"/>
<source>Rename item</source>
<translation>Přejmenovat položku</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.ui" line="351"/>
<source>Cut items</source>
<translation>Vyjmout položku</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.ui" line="354"/>
<source>Cut items (add to the clipboard)</source>
<translation>Vyjmout položku (přidat do schránky)</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.ui" line="373"/>
<source>Copy items</source>
<translation>Kopírovat položky</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.ui" line="376"/>
<source>Copy items to the clipboard</source>
<translation>Kopírovat položky do schránky</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.ui" line="395"/>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.ui" line="398"/>
<source>Paste items from clipboard</source>
<translation>Vložit položky ze schránky</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.ui" line="424"/>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.ui" line="427"/>
<source>Delete Items</source>
<translation>Odstranit položky</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.ui" line="507"/>
<source>Stop loading the directory</source>
<translation>Zastavit načítání složky</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.ui" line="90"/>
<source>Add selected images to slideshow</source>
<translation>Přidat vybrané obrázky do prezentace</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.ui" line="77"/>
<source>Create a new directory</source>
<translation>Vytvořit novou složku</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.ui" line="80"/>
<source>New Dir</source>
<translation>Nová složka</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.ui" line="64"/>
<source>Create a new file</source>
<translation>Vytvořit nový soubor</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.ui" line="67"/>
<source>New File</source>
<translation>Nový soubor</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.ui" line="93"/>
<source>Slideshow</source>
<translation>Prezentace</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.ui" line="109"/>
<source>Enqueue selection in multimedia player</source>
<translation>Zařadit výběr do přehrávače</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.ui" line="112"/>
<source>Play</source>
<translation>Přehrát</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.ui" line="465"/>
<source>Back</source>
<translation>Zpět</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.ui" line="468"/>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.ui" line="471"/>
<source>Go back to previous directory</source>
<translation>Zpět do předchozí složky</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.ui" line="479"/>
<source>Up</source>
<translation>Nahoru</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.ui" line="482"/>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.ui" line="485"/>
<source>Go to parent directory</source>
<translation>Přejít do nadřazené složky</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.ui" line="493"/>
<source>Home</source>
<translation>Domů</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.ui" line="496"/>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.ui" line="499"/>
<source>Go to home directory</source>
<translation>Přejít do domovské složky</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.ui" line="518"/>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.ui" line="521"/>
<source>Close this browser</source>
<translation>Zavřít tento prohlížeč</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.cpp" line="136"/>
<source>Name</source>
<translation>Název</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.cpp" line="140"/>
<source>Size</source>
<translation>Velikost</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.cpp" line="143"/>
<source>Type</source>
<translation>Typ</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.cpp" line="147"/>
<source>Date Modified</source>
<translation>Datum změny</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.cpp" line="150"/>
<source>Date Created</source>
<translation>Datum vytvoření</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.cpp" line="207"/>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.cpp" line="484"/>
<source>(Limited Access) </source>
<translation>(Omezený přístup) </translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.cpp" line="469"/>
<source>Capacity: %1</source>
<translation>Kapacita: %1</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.cpp" line="474"/>
<source>Files: %1 (%2)</source>
<translation>Soubory: %1 (%2)</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.cpp" line="479"/>
<source>Dirs: %1</source>
<translation>Složky: %1</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.cpp" line="781"/>
<source>New Document</source>
<translation>Nový dokument</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.cpp" line="781"/>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.cpp" line="802"/>
<source>Name:</source>
<translation>Název:</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.cpp" line="792"/>
<source>Error Creating Document</source>
<translation>Chyba při vytváření dokumentu</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.cpp" line="792"/>
<source>The document could not be created. Please ensure that you have the proper permissions.</source>
<translation>Dokument se nepodařilo vytvořit. Ujistěte se, prosím, že máte náležitá oprávnění.</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.cpp" line="802"/>
<source>New Directory</source>
<translation>Nová složka</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.cpp" line="812"/>
<source>Invalid Name</source>
<translation>Neplatný název</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.cpp" line="812"/>
<source>A file or directory with that name already exists! Please pick a different name.</source>
<translation>Soubor nebo složka se stejným jménem již existuje! Použijte prosím jiné jméno.</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.cpp" line="816"/>
<source>Error Creating Directory</source>
<translation>Chyba při vytváření složky</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.cpp" line="816"/>
<source>The directory could not be created. Please ensure that you have the proper permissions to modify the current directory.</source>
<translation>Dokument se nepodařilo vytvořit. Ujistěte se, prosím, že máte náležitá oprávnění ke změně nynějšího adresáře.</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.cpp" line="839"/>
<source>Current</source>
<translation>Nynější</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.cpp" line="949"/>
<source>File Checksums:</source>
<translation>Kontrolní součty souborů:</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.cpp" line="966"/>
<source>Missing Utility</source>
<translation>Chybějící nástroj</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.cpp" line="966"/>
<source>The "lumina-fileinfo" utility could not be found on the system. Please install it first.</source>
<translation>Nástroj "lumina-fileinfo" nebyl nalezen.Prosím nainstalujte ho první.</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.cpp" line="984"/>
<source>Open</source>
<translation>Otevřít</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.cpp" line="985"/>
<source>Open With...</source>
<translation>Otevřít s…</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.cpp" line="987"/>
<source>Rename...</source>
<translation>Přejmenovat...</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.cpp" line="988"/>
<source>View Checksums...</source>
<translation>Zobrazit kontrolní součty...</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.cpp" line="999"/>
<source>File Properties...</source>
<translation>Vlastnosti souboru...</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.cpp" line="992"/>
<source>Cut Selection</source>
<translation>Vyjmout výběr</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.cpp" line="993"/>
<source>Copy Selection</source>
<translation>Kopírovat výběr</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.cpp" line="994"/>
<source>Paste</source>
<translation>Vložit</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.cpp" line="996"/>
<source>Delete Selection</source>
<translation>Smazat výběr</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/DirWidget.cpp" line="1002"/>
<source>Open Terminal here</source>
<translation>Otevřít terminál zde</translation>
</message>
</context>
<context>
<name>FODialog</name>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/FODialog.ui" line="14"/>
<source>Performing File Operations</source>
<translation>Provádění operací se soubory</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/FODialog.ui" line="39"/>
<source>%v/%m</source>
<translation>%v/%m</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/FODialog.ui" line="74"/>
<source>Stop</source>
<translation>Zastavit</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/FODialog.cpp" line="19"/>
<source>Calculating</source>
<translation>Počítá se</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/FODialog.cpp" line="131"/>
<source>Overwrite Files?</source>
<translation>Přepsat soubory?</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/FODialog.cpp" line="131"/>
<source>Do you want to overwrite the existing files?</source>
<translation>Chcete přepsat stávající soubory?</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/FODialog.cpp" line="131"/>
<source>Note: It will just add a number to the filename otherwise.</source>
<translation>Poznámka: Jinak přidá pouze číslo k názvu souboru.</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/FODialog.cpp" line="148"/>
<source>Removing: %1</source>
<translation>Odstraňování: %1</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/FODialog.cpp" line="149"/>
<source>Copying: %1 to %2</source>
<translation>Kopírování: %1 do %2</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/FODialog.cpp" line="150"/>
<source>Restoring: %1 as %2</source>
<translation>Obnovení: %1 jako %2</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/FODialog.cpp" line="151"/>
<source>Moving: %1 to %2</source>
<translation>Přesunování: %1 do %2</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/FODialog.cpp" line="164"/>
<source>Could not remove these files:</source>
<translation>Nelze smazat tyto soubory:</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/FODialog.cpp" line="165"/>
<source>Could not copy these files:</source>
<translation>Nelze kopírovat tyto soubory:</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/FODialog.cpp" line="166"/>
<source>Could not restore these files:</source>
<translation>Nelze obnovit tyto soubory:</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/FODialog.cpp" line="167"/>
<source>Could not move these files:</source>
<translation>Nelze přesunout tyto soubory:</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/FODialog.cpp" line="169"/>
<source>File Errors</source>
<translation>Chyby souboru</translation>
</message>
</context>
<context>
<name>FOWorker</name>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/FODialog.cpp" line="320"/>
<source>Invalid Move</source>
<translation>Neplatný přesun</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/FODialog.cpp" line="320"/>
<source>It is not possible to move a directory into itself. Please make a copy of the directory instead.
Old Location: %1
New Location: %2</source>
<translation>Není možné přesunout složku do sebe sama. Raději vytvořte kopii této složky.
Původní umístění: %1
Nové umístění: %2</translation>
</message>
</context>
<context>
<name>MainUI</name>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="14"/>
<source>Insight</source>
<translation>Náhled</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.cpp" line="95"/>
<source>Shift+Left</source>
<translation>Shift+šipka vlevo</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.cpp" line="96"/>
<source>Shift+Right</source>
<translation>Shift+šipka vpravo</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="114"/>
<source>File</source>
<translation>Soubor</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="126"/>
<source>View</source>
<translation>Zobrazit</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="130"/>
<source>View Mode</source>
<translation>Režim zobrazení</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="136"/>
<source>Group Mode</source>
<translation>Režim seskupení</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="154"/>
<source>Bookmarks</source>
<translation>Záložky</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="162"/>
<source>External Devices</source>
<translation>Vnější zařízení</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="169"/>
<source>Edit</source>
<translation>Upravit</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="188"/>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="191"/>
<source>New Browser</source>
<translation>Nový prohlížeč</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="267"/>
<source>Search Directory...</source>
<translation>Prohledat složku...</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="300"/>
<source>Increase Icon Size</source>
<translation>Zvětšit velikost ikony</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="305"/>
<source>Decrease Icon Size</source>
<translation>Zmenšit velikost ikony</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="310"/>
<source>Larger Icons</source>
<translation>Větší ikony</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="313"/>
<source>Ctrl++</source>
<translation>Ctrl++</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="321"/>
<source>Smaller Icons</source>
<translation>Menší ikony</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="324"/>
<source>Ctrl+-</source>
<translation>Ctrl+-</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="332"/>
<source>New Window</source>
<translation>Nové okno</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="335"/>
<source>Ctrl+N</source>
<translation>Ctrl+N</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="343"/>
<source>Add Bookmark</source>
<translation>Přidat záložku</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="346"/>
<source>Ctrl+D</source>
<translation>Ctrl+D</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="398"/>
<source>Delete Selection</source>
<translation>Smazat výběr</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="401"/>
<source>Del</source>
<translation>Del</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="409"/>
<source>Refresh</source>
<translation>Obnovit</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="420"/>
<source>Close Browser</source>
<translation>Zavřít prohlížeč</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="194"/>
<source>Ctrl+T</source>
<translation>Ctrl+T</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="202"/>
<source>Exit</source>
<translation>Ukončit</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="205"/>
<source>Ctrl+Q</source>
<translation>Ctrl+Q</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="213"/>
<source>&Preferences</source>
<translation>&Nastavení</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="224"/>
<source>Show Hidden Files</source>
<translation>Zobrazit skryté soubory</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="229"/>
<source>Scan for Devices</source>
<translation>Hledat zařízení</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="237"/>
<source>Manage Bookmarks</source>
<translation>Spravovat záložky</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="251"/>
<source>Show Action Buttons</source>
<translation>Ukázat tlačítka činností</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="262"/>
<source>Load Thumbnails</source>
<translation>Nahrát náhledy</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="270"/>
<source>Ctrl+F</source>
<translation>Ctrl+F</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.cpp" line="66"/>
<source>Detailed List</source>
<translation>Podrobný seznam</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.cpp" line="67"/>
<source>Basic List</source>
<translation>Základní seznam</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="423"/>
<source>Ctrl+W</source>
<translation>Ctrl+W</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.cpp" line="68"/>
<source>Prefer Tabs</source>
<translation>Upřednostňovat karty</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.cpp" line="69"/>
<source>Prefer Columns</source>
<translation>Upřednostňovat sloupce</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="412"/>
<source>F5</source>
<translation>F5</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="379"/>
<source>Ctrl+C</source>
<translation>Ctrl+C</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="354"/>
<source>Rename...</source>
<translation>Přejmenovat...</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="357"/>
<source>F2</source>
<translation>F2</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="365"/>
<source>Cut Selection</source>
<translation>Vyjmout výběr</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="376"/>
<source>Copy Selection</source>
<translation>Kopírovat výběr</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="387"/>
<source>Paste</source>
<translation>Vložit</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="390"/>
<source>Ctrl+V</source>
<translation>Ctrl+V</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.ui" line="368"/>
<source>Ctrl+X</source>
<translation>Ctrl+X</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.cpp" line="196"/>
<source>Invalid Directories</source>
<translation>Neplatné adresáře</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.cpp" line="196"/>
<source>The following directories are invalid and could not be opened:</source>
<translation>Následující adresáře jsou neplatné a nepodařilo se je otevřít:</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.cpp" line="335"/>
<source>Root</source>
<translation>Kořen</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.cpp" line="341"/>
<source>%1 (Type: %2)</source>
<translation>%1 (typ: %2)</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.cpp" line="345"/>
<source>Filesystem: %1</source>
<translation>Souborový systém: %1</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.cpp" line="566"/>
<source>Browser</source>
<translation>Prohlížeč</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.cpp" line="593"/>
<source>New Bookmark</source>
<translation>Nová záložka</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.cpp" line="593"/>
<source>Name:</source>
<translation>Název:</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.cpp" line="598"/>
<source>Invalid Name</source>
<translation>Neplatné jméno</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.cpp" line="598"/>
<source>This bookmark name already exists. Please choose another.</source>
<translation>Záložka s tímto názvem již existuje. Zvolte jiný název.</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.cpp" line="705"/>
<source>Multimedia</source>
<translation>Multimédia</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.cpp" line="723"/>
<source>Slideshow</source>
<translation>Prezentace</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.cpp" line="900"/>
<source>Items to be removed:</source>
<translation>Soubory k odstranění:</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.cpp" line="422"/>
<source>Verify Quit</source>
<translation>Ověřit ukončení</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.cpp" line="97"/>
<source>Ctrl+H</source>
<translation>Ctrl+H</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.cpp" line="98"/>
<source>Ctrl+L</source>
<translation type="unfinished"></translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.cpp" line="422"/>
<source>You have multiple tabs open. Are you sure you want to quit?</source>
<translation>Máte otevřeno několik karet. Opravdu si přejete skončit?</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.cpp" line="899"/>
<source>Verify Removal</source>
<translation>Ověřit odstranění</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.cpp" line="899"/>
<source>WARNING: This will permanently delete the file(s) from the system!</source>
<translation>Varování: Tímto bude soubor(y) trvale odstraněn ze systému!</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.cpp" line="899"/>
<source>Are you sure you want to continue?</source>
<translation>Opravdu chcete pokračovat?</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.cpp" line="860"/>
<source>Rename File</source>
<translation>Přejmenovat soubor</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.cpp" line="860"/>
<source>New Name:</source>
<translation>Nový název:</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.cpp" line="874"/>
<source>Overwrite File?</source>
<translation>Přepsat soubor?</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/MainUI.cpp" line="874"/>
<source>An existing file with the same name will be replaced. Are you sure you want to proceed?</source>
<translation>Stávající soubor se stejným názvem bude nahrazen. Opravdu chcete pokračovat?</translation>
</message>
</context>
<context>
<name>MultimediaWidget</name>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/MultimediaWidget.ui" line="14"/>
<source>Form</source>
<translation>Formulář</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/MultimediaWidget.ui" line="28"/>
<source>Go To Next</source>
<translation>Přejít na následující</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/MultimediaWidget.ui" line="107"/>
<source>(No Running Video)</source>
<translation>(Žádný běžící obrazový záznam)</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/MultimediaWidget.cpp" line="124"/>
<source>Playing:</source>
<translation>Přehrávání:</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/MultimediaWidget.cpp" line="130"/>
<source>Stopped</source>
<translation>Zastaveno</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/MultimediaWidget.cpp" line="157"/>
<source>Error Playing File: %1</source>
<translation>Chyba při přehrávání souboru %1</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/MultimediaWidget.cpp" line="168"/>
<source>Finished</source>
<translation>Hotovo</translation>
</message>
</context>
<context>
<name>SlideshowWidget</name>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/SlideshowWidget.ui" line="14"/>
<source>Form</source>
<translation>Formulář</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/SlideshowWidget.ui" line="36"/>
<source>Delete this image file</source>
<translation>Smazat tento obrázkový soubor</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/SlideshowWidget.ui" line="56"/>
<source>Rotate this image file counter-clockwise</source>
<translation>Otočit tento obrázek proti směru hodinových ručiček</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/SlideshowWidget.ui" line="69"/>
<source>Rotate this image file clockwise</source>
<translation>Otočit tento obrázek po směru hodinových ručiček</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/SlideshowWidget.ui" line="89"/>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/SlideshowWidget.ui" line="92"/>
<source>Zoom in</source>
<translation>Přiblížit</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/SlideshowWidget.ui" line="105"/>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/SlideshowWidget.ui" line="108"/>
<source>Zoom out</source>
<translation>Oddálit</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/SlideshowWidget.ui" line="212"/>
<source>Go to Beginning</source>
<translation>Přejít na začátek</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/SlideshowWidget.ui" line="215"/>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/SlideshowWidget.ui" line="231"/>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/SlideshowWidget.ui" line="304"/>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/SlideshowWidget.ui" line="320"/>
<source>...</source>
<translation>...</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/SlideshowWidget.ui" line="218"/>
<source>Shift+Left</source>
<translation>Shift+šipka vlevo</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/SlideshowWidget.ui" line="228"/>
<source>Go to Previous</source>
<translation>Přejít na předchozí</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/SlideshowWidget.ui" line="234"/>
<source>Left</source>
<translation>Vlevo</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/SlideshowWidget.ui" line="267"/>
<source>File Name</source>
<translation>Název souboru</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/SlideshowWidget.ui" line="301"/>
<source>Go to Next</source>
<translation>Přejít na následující</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/SlideshowWidget.ui" line="307"/>
<source>Right</source>
<translation>Vpravo</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/SlideshowWidget.ui" line="317"/>
<source>Go to End</source>
<translation>Přejít na konec</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/SlideshowWidget.ui" line="323"/>
<source>Shift+Right</source>
<translation>Shift+šipka vpravo</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/SlideshowWidget.cpp" line="125"/>
<source>Verify Removal</source>
<translation>Potvrdit odstranění</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/SlideshowWidget.cpp" line="125"/>
<source>WARNING: This will permanently delete the file from the system!</source>
<translation>VAROVÁNÍ: Toto trvale odstraní soubory z vašeho systému!</translation>
</message>
<message>
<location filename="../../lumina-git/src-qt5/desktop-utils/lumina-fm/widgets/SlideshowWidget.cpp" line="125"/>
<source>Are you sure you want to continue?</source>
<translation>Opravdu chcete pokračovat?</translation>
</message>
</context>
</TS>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,532 |
Q: Performing setRGB on BufferedImage changes pixel to black instead of color ** Important update, see below! **
I am creating a program that changes the pixels of a BufferedImage to a certain color when that pixel fulfills a set of conditions in Java. However, when I write the image to disk, the pixels that should be colored are instead black.
First I define the color, using RGB codes:
Color purple = new Color(82, 0, 99);
int PURPLE = purple.getRGB();
Then I read the image I want to alter from a File into a BufferedImage called "blank":
BufferedImage blank = ImageIO.read(new File("some path"));
Now, loop through the pixels, and when a pixel at location (x, y) matches a criteria, change its color to purple:
blank.setRGB(x, y, PURPLE);
Now, write "blank" to the disk.
File output = new File("some other path");
ImageIO.write(blankIn, "png", output); // try-catch blocks intentionally left out
The resulting file should be "blank" with some purple pixels, but the pixels in question are instead black. I know for a fact that the issue is with setRGB and NOT any import or export functions, because "blank" itself is a color image, and gets written to file as such. I read around and saw a lot of posts recommending that I use Graphics2D and to avoid setRGB, but with no discussion of pixel-by-pixel color changing.
I also tried direct bit manipulation, like this:
blank.setRGB(x, y, ((82 << 16) + (0 << 8) + 99));
I'm probably doing that wrong, but if I put it in correctly it wouldn't matter, because the pixels are getting set to transparent when I do this (regardless of what the numbers say, which is very strange, to say the least).
** When I try this:
blank.setRGB(x, y, Color.RED.getRGB());
My output file is grayscale, so that means setRGB is, in fact, modifying my picture in grayscale. I think this is actually a rather simple issue, but the solution eludes me.
A: Based on the insights in https://stackoverflow.com/a/21981173 that you found yourself ... (a few minutes after posting the question) ... it seems that it should be sufficient to simply convert the image into ARGB directly after it was loaded:
public static BufferedImage convertToARGB(BufferedImage image)
{
BufferedImage newImage = new BufferedImage(
image.getWidth(), image.getHeight(),
BufferedImage.TYPE_INT_ARGB);
Graphics2D g = newImage.createGraphics();
g.drawImage(image, 0, 0, null);
g.dispose();
return newImage;
}
A: The original image that was imported into Java was actually grayscale, so when Java read it into the BufferedImage, it simply imported it as a grayscale BufferedImage. By adding a very small but imperceptible colored dot in the corner of my image, I was able to get Java to output a correctly colored image.
Unfortunately, this is only a half solution, because I do not know how to fix this programmatically.
SOLUTION:
Convert the BufferedImage from grayscale to ARGB with this snippet:
BufferedImage blank2 = blank;
// Create temporary copy of blank
blank = new BufferedImage(blank.getWidth(), blank.getHeight(), BufferedImage.TYPE_INT_ARGB);
// Recreate blank as an ARGB BufferedImage
ColorConvertOp convert = new ColorConvertOp(null);
// Now create a ColorConvertOp object
convert.filter(blank2, blank);
// Convert blank2 to the blank color scheme and hold it in blank
You want to add this right after blank = ImageIO.read(new File("some path")).
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,355 |
IIT Bombay Racing is a Formula Student team from India based at Indian Institute of Technology Bombay since 2007. The team consists of 70 members who aim to conceive, design and fabricate a formula style racecar through innovation to compete in international Formula Student events.
IIT Bombay Racing made its debut at Formula Student Michigan, 2008 with an entry car Vayu as only Indian team participating in the event. In the following year, the team participated in Formula Student UK, 2009 at Silverstone Circuit with entry car Agni and was 2nd best Asian entry at Formula Student UK. The team also achieved highest score among Indian participating teams along with a 2nd rank in Cost Event.
In the year 2012, IIT Bombay Racing developed Prithvi 3.0 securing 1st in low weight sustainability and won the best car award by peers at the Baja SAE India event held at Pithampur. The same year also witnessed the launch of EVo 1.0, India's first student designed electric racing vehicle.
Since 2012, the team has launched 7 electric vehicles with giant leaps in design and performance. Recently IIT Bombay Racing's 7th electric Racecar EvoK participated in Formula Student UK, 2019.
Team Philosophy
IIT Bombay Racing team has been a unique team being a leading Indian team developing electric race car. The team is driven by a common vision of 'Revolutionizing electric mobility in India focusing on sustainable technologies and innovations'. Bringing electric car technologies in India has been goal of the team for last 5 years. The team has been working besides competition to increase awareness about electric vehicle technologies in Indian society. It has aided many Indian FSAE teams to develop own electric vehicle. The team will further continue to prove power of electric vehicles.
Racing Seasons
Vayu
Vayu was the first car fabricated by IIT Bombay Racing. Vayu participated at FSAE Michigan in 2008. It was the only Indian team participating at the competition. The team won perseverance award and best new entrant award in the competition.
Technical Specifications of Vayu
Agni
Agni participated at FSAE UK in 2009 at Silverstone Circuit, UK. It was 2nd best Asian entry at FSUK competition. The team achieved highest score among participating Indian teams along with 2nd rank in Cost Event and 9th rank in Business Presentation event.
Technical Specifications of Agni
Prithvi 2.0
Prithvi 2.0 was IIT Bombay Racing's second entry in Baja SAE India competition 2009 at Pithampur. It won Raftar award of 1 Lakhs INR for being lightest and fastest vehicle in the competition.
Prithvi 3.0
Prithvi 3.0 was IIT Bombay Racing's third entry in Baja SAE India competition 2012 at Pithampur. Prithvi 3.0 received 1st prize in low weight sustainability as well as received best car award by peers.
EVo 1.0
IIT Bombay Racing launched EVo 1.0 in the same year 2012 as Prithvi 3.0. Being India's first electric race car, it was a breakthrough in Indian Racing history.
Technical Specifications of EVo 1.0
EVo 2.0
EVo 2.0, as a 2nd electric race car developed by IIT Bombay Racing, participated in FSAE UK 2013. EVo 2.0 was first Indian race car to implement electric differential successfully. The team achieved an overall rank of 54th out of 120 teams being top among Indian teams. EVo 2.0 was 1st Indian car to complete all safety checks and participate in dynamic events in competition.
Technical Specifications of EVo 2.0
EVo 3.0
EVo 3.0, as a 3rd electric race car developed by IIT Bombay Racing, participated in FSAE UK 2014. EVo 3.0 was first Indian FS car to complete all static and dynamic events. Also, 1st Indian FS car to qualify for endurance event. EVo 3.0 achieved a rank of 15th among 33 participating electric cars in the event. The team also achieved Formula Student Award of 3000 GBP which is given to only 2 non-UK teams annually.
Technical Specifications of EVo 3.0
EVo 4.0
EVo 4.0, as a 4th consecutive electric race car developed by IIT Bombay Racing, participated in FSAE UK 2015. IIT Bombay Racing fabricated Aluminum honeycomb monocoque chassis which was a feat done first time by an Indian team. The team achieved Formula Student Award of 3000 GBP which is given to only 2 non-UK teams annually.
Technical Specifications of EVo 4.0
Orca
Orca, as a 5th electric race car developed by IIT Bombay Racing, participated in FSAE UK 2016. IIT Bombay Racing fabricated Carbon Fiber bodyworks in-house which was done first time by a student team at IIT Bombay. The team achieved Formula Student Award of 3000 GBP for 3rd time consecutively which is given to only 2 non-UK teams annually. In the event, team completed all static events, and participated in endurance event.
Technical Specifications of Orca
References
External links
http://indianexpress.com/article/education/iit-b-india-fastest-electric-racing-car-orca-2825209/
https://www.youtube.com/watch?v=WMRcTdSIw0w
http://www.insightiitb.org/2016/full-throttle/
http://www.insightiitb.org/2015/iitb-racings-performance-at-formula-student-15/
http://www.autocarpro.in/news-national/iit-bombay-develops-fifth-electric-racecar-compete-formula-student-uk-20025
http://www.dailyo.in/technology/iit-bombay-racing-formula-electric-car-student-tech-fests/story/1/11449.html
http://www.thehindu.com/news/cities/mumbai/iit-bombay-team-unveils-electric-racing-car/article7216205.ece
Formula Student | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 586 |
Q: Magento: Update store logo programmatically I've noticed that Magento keeps the URL to the logo it uses for each store you setup in the core_config_data table.
If I run this SQL:
SELECT *
FROM core_config_data`
WHERE path = 'design/header/logo_src'
I get a list of stores and their associated logo. I also get a config_id and a scope_id.
I'd like to be able to update these logo's programmatically behind the scenes but I can't figure out how to relate this table's data back to a store name.
config_id and store_id got to somehow link back to another table that sets up the relationship. Magento's EAV model, ugh :)
Any ideas?
A: Magento provides an API to set and retrieve configuration values. I wrote an article about fetching data out of the config. You'll want to review it if any of the terminology below confuses you.
As for setting the variables, it's not quite as simple as saying "I want to set the design/header/logo_src" variable to "this value". You also need to specify which website you're saving the values for, and which store you're saving the values for.
It's important to be careful here. I haven't looked deeply at the system to see if Magento is enforcing the "show for website, show for site" granularity of the config values at the back-end/resource level, or only at the UI level. My instincts say it's the later, which mean you might end up saving a value at the website/store level that wasn't intended to be savable at that level, and that could lead to the dreaded Undefined Behavior.
In the 1.4 community edition you can see where Magento is saving the config values at the following location
File: app/code/core/Mage/Adminhtml/controllers/System/ConfigController.php
//around line 126
Mage::getModel('adminhtml/config_data')
->setSection($section)
->setWebsite($website)
->setStore($store)
->setGroups($groups)
->save();
Caveats out of the way, the code for you to update just the logo for the top level store/website config would be.
//create a groups array that has the value we want at the rigth location
$groups_value = array();
$groups_value['header']['fields']['logo_src']['value'] = 'images/logo.gif';
Mage::getModel('adminhtml/config_data')
->setSection('design')
->setWebsite(null)
->setStore(null)
->setGroups($groups_value)
->save();
The line
Mage::getModel('adminhtml/config_data')
instantiates a Magento model.
The 'design' in the setSection method as well as the 'header' and 'logo_src' string correspond to the config URI
design/header/logo_src
I'm pretty sure the nulls passed to setWebsite and setStore tell the object you're saving this value at the top level. You could also pass in store and website IDs. (I'm a little shaky on that, but when you save the logo normally that's what the system is doing)
The 'fields' and 'values' are hard coded. The intent of the object is to save multiple config values at once from all the groups in a section.
Checkout the saveAction method in the above controller if you're curious.
A: The most easiest and shortest method is:
Mage::getConfig()->saveConfig('design/header/logo_src', $logo);
A: Thanks for the in-depth reply. I truly appreciate it.
I figured out that I can just insert a new record like this:
INSERT INTO core_config_data (`scope`, `scope_id`, `path`, `value`)
VALUES ('websites', $websiteId, 'design/header/logo_src', $logo);
I tested it and it appears to add a different logo for the store in the configuration - design page, but when i reference the logo by <?php echo $this->getLogoSrc() ?> in my header.phtml it is still pulling the default logo from the top level configuration.
I think that using your post as a guideline I can start debugging using Mage::log() and see where it takes me.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,357 |
Acknowledgments
A lot goes into writing a book. This one started with my parents, David and Anita Kingdon, and their emphasis on the value of education and learning, a value I myself have pursued throughout my life. I am grateful for their constant support, as well as that of my brothers, Robbie and Andy Kingdon, and my best friend and surrogate brother, Randal Blandon. Finally, this book wouldn't have been written without the love, support, and constructive suggestions of my husband, Chris Backe; I had wanted to write this for years, but he was the one who finally inspired me to get it started, and kept me supplied with enough encouragement to stick with it once I'd started.
Of course, I also have to thank Tuttle Publishing for their work in publishing this book, not to mention the many other books they've published that have been of use to me in my own language studies. Regarding this book in particular, I am grateful to Eric Oey for getting the ball rolling, to my editors Sandra Korinchak and Nancy Goh, to Mike Page for marketing, to Terri Jadick for general assistance throughout the process, and to all others who worked behind the scenes to make the book a reality. My Korean editors Sanghyun Ahn and Hannah Han were instrumental in making sure all the Korean in the book was correct, and I also have to thank my former Korean tutor Kyehyun Park and all my professors at Yonsei for their instruction.
Finally, I'd like to thank Gordon and Ruth Kingdon, Bill and Joan Brown, Murray and Christine Brown, Beth Lloyd, Euen Moore, Bruce McAuley, Becky Strople, Dave Cook, Jan Barkhouse, Mark Currie, Jon Watts, Carolyn Watts, Lisa Wadden, Ed Charlton, Ian MacK-enzie, Stacey van Dyk, Rachel Davison, Dan and Elizabeth Rolfe, Alex Kennedy, Steve Fehr, Paul Benjamin, Amy Bowler, Bob Adamson, Luke Roberts, Azrael Jeffrey, Debbie Kim, Chelsey Mathews, Aaron Seymour, Bill Carver, Emma Chaitongkao, Holly Sroymalai, Ken May, Matthijs van Rooyen, Christian Frech, Park Chansoon, Lee Kayoung, Peter Daley, Jessica Suchan, JD DeLemont, and everyone else who has supported me along the way.
A comprehensive reference
for learners at every level
LAURA KINGDON
TUTTLE Publishing
Tokyo | Rutland, Vermont | Singapore
The Tuttle Story
"Books to Span the East and West"
Many people are surprised to learn that the world's leading publisher of books on Asia had humble beginnings in the tiny American state of Vermont. The company's founder, Charles E. Tuttle, belonged to a New England family steeped in publishing.
Immediately after WWII, Tuttle served in Tokyo under General Douglas MacArthur and was tasked with reviving the Japanese publishing industry. He later founded the Charles E. Tuttle Publishing Company, which thrives today as one of the world's leading independent publishers.
Though a westerner, Tuttle was hugely instrumental in bringing a knowledge of Japan and Asia to a world hungry for information about the East. By the time of his death in 1993, Tuttle had published over 6,000 books on Asian culture, history and art—a legacy honored by the Japanese emperor with the "Order of the Sacred Treasure," the highest tribute Japan can bestow upon a non-Japanese.
With a backlist of 1,500 titles, Tuttle Publishing is more active today than at any time in its past—still inspired by Charles Tuttle's core mission to publish fine books to span the East and West and provide a greater understanding of each.
#
Published by Tuttle Publishing, an imprint of Periplus Editions (HK) Ltd.
www.tuttlepublishing.com
Copyright © 2015 Laura Kingdon
All rights reserved. No part of this publication may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without prior written permission from the publisher.
Library of Congress Control Number: 2015936092
ISBN 978-0-8048-4431-4; ISBN 978-1-4629-1775-4 (ebook)
Distributed by
North America, Latin America & Europe
Tuttle Publishing
364 Innovation Drive
North Clarendon
VT 05759-9436 U.S.A.
Tel: 1 (802) 773-8930
Fax: 1 (802) 773-6993
info@tuttlepublishing.com
www.tuttlepublishing.com
Japan
Tuttle Publishing
Yaekari Building, 3rd Floor
5-4-12 Osaki, Shinagawa-ku
Tokyo 141 0032
Tel: (81) 3 5437-0171
Fax: (81) 3 5437-0755
sales@tuttle.co.jp
www.tuttle.co.jp
Asia Pacific
Berkeley Books Pte. Ltd.
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Singapore 534167
Tel: (65) 6280-1330
Fax: (65) 6280-6290
inquiries@periplus.com.sg
www.periplus.com
First edition
18 17 16 15
10 9 8 7 6 5 4 3 2 1
1504MP
Printed in Singapore
TUTTLE PUBLISHING® is a registered trademark of Tuttle Publishing, a division of Periplus Editions (HK) Ltd.
#
Contents
How to Use This Book
The Rating System (or, How Important Is This Grammar Point to Me?)
Grammatical Terms Used in This book
[PART ONE
THE BASICS OF KOREAN](../Text/Part1.xhtml#heading_id_2)
Pronouns and Contractions
How to Talk to People Without Being Rude
How to Add Expressions to Verbs
How to Handle Verbs, Part 1: Changing Their Form
How to Handle Verbs, Part 2: Irregular Verbs
Basic Particles
Helping Verbs
Passives and Causatives
[PART TWO
GRAMMAR POINTS](../Text/Part2-01.xhtml#heading_id_2)
Agreement
Disagreement/Negatives
Asking Questions
And: Basic Forms
And (So On, and So On)
Linking Sentences: And
And: Putting Things Together
Linking Sentences: Or
Linking Sentences—Cause and Effect: The Big Three
Linking Sentences—Cause and Effect: More Ways to Say "So"
Linking Sentences—Cause and Effect: Through, Because Of, For the Sake Of
Cause and Effect: Origins
Cause and Effect, or Lack Thereof
Cause and Effect: Multiple Causes
Linking Sentences: Basic Comparisons
Linking Sentences—Contrast: Although/But
Linking Sentences—Contrast: Not A, But B
Linking Sentences—Comparisons: As
Linking Sentences—Comparisons: Equals (or Not)
Thinking: Making Guesses
Thinking: Knowing
Thinking: Making Decisions
Emotions: Surprise
Emotions: Regret
Emotions: Thanks
Emotions: Desires
Emotions: Personal Experiences
Obligations, Orders and Permission
Warnings
Plans
Accidents
Progression: Review of Past and Future Tenses
Progression: While
Progression: Continuing States
Progression: Almost, But Not Quite
Progression: Putting Things in Temporal Order
If: If
If: Possibility and The Lack Thereof
Talking about What Others Have Said: Review of Reported Speech
Talking about What Others Have Said: Combinations
Talking about What Others Have Said: Spreading Rumors
Talking about What Others Have Said: According to
Giving Explanations
Location
Limits and Excess
Habits
Faking it
[PART THREE
FREQUENTLY SEEN WORD PARTS](../Text/Part3.xhtml#heading_id_2)
Talking about People
Prefixes
Appendix 1: How to Sound Like a Native
Adverbs of Time
Adverbs of Frequency
Adverbs of Degree
Giving Your Opinion and Sounding Smart
How to Deal with 그렇다
Appendix 2: Useful Korean Language Resources
Index
#
How to Use This Book
When I first started studying Korean, I found there were any number of books, blogs, podcasts, classes, etc., available for beginners, and I rejoiced. However, as my studies progressed, the quantity of appropriate materials dropped sharply and while there were still many textbooks, very few of them explained the material in a useful way: for instance, though they would happily explain that 어서 eoseo and 으니까 eunikka both roughly meant "so," there was little guidance as to when it was appropriate to use each form. These books also tended to be poorly organized, with ㄴ다고 해서 dago haeseo in one chapter, ㄴ다고 하니까 dago hanikka several chapters later and ㄴ다고 하는데 n dago haneunde a few chapters again after, even though these are all very similar expressions.
In this book, I have attempted to synthesize all the information gained from my studies into a format that's useful and convenient for others to follow. I am not planning to teach you these expressions for the first time but rather to provide a useful guide to help you make sense of what you've already partially learned and to help you use these expressions more fluently, much like a toolbox to help you form correct and natural-sounding Korean sentences rather than ending every single sentence with the same old 아/어/여요 a/eo/yeoyo.
To this end, I have mainly focused on verbs and adverbs. Constructing Korean sentences is really all about what you do with the verbs, and a whole lot of implications are packed into different verb endings. There's a little about pronouns because contractions are used all the time (just like in English) but rarely taught, and then some information about adverbs at the end because a good adverb can really spice up a sentence (and because I myself find them confusing at times and so I suspect others might as well).
I really haven't focused on vocabulary since you can probably use a dictionary just as well as I can. There are some good vocabulary books listed in the Appendix if you want a little more direction.
There are only a few different ways to attach verbs and endings in Korean and I have summarized these on page , which you can refer back to anytime. Each expression comes with its own set of "hooks." For example, every expression starting with "ㄴ/는" attaches to verbs the same way: ㄴ to verbs ending in vowels and 는 to verbs ending in consonants, regardless of tense or any other factor. This seems daunting at first, but with practice it will become second nature.
Because this book is intended to take you from beginning to advanced Korean, I recognize that my readers will be at somewhat different levels and will know or not know different things, so I've summarized what I consider to be the basic points of Korean grammar in "The Basics of Korean" for you to check if you need a little help. I haven't spent much time on them because it's not terribly difficult to find good-quality materials on basic Korean grammar. If you find you're not quite ready for this book yet, I've listed some useful beginning Korean books in the Appendix.
I have also tried to keep the language in my examples fairly simple. When possible, they all end in the 아/어/여요 (polite) form because those are the most useful and I wanted to keep everything consistent for the sake of clarity. Unless I say otherwise, you can use any of these expressions with any level of politeness.
#
The Rating System
(or, How Important Is This Grammar Point to Me?)
All the information in the section "The Basics of Korean" is material you should know before reading the rest of this book. Material in the appendixes is stuff I think might be useful for you to know, but you don't have to study it if you'd rather not. Beyond that, I've rated everything in the main part of the book according to a star system, which works like this:
★★★★★ | Critical; with only the five-star expressions, you can communicate almost everything you need to say, and without them you'll have trouble understanding any more than the most basic of sentences.
---|---
★★★★ | Very helpful; you should learn this if you want to communicate well in Korean
★★★ | Useful, fairly common, and will help you sound more natural and fluent
★★ | Not that important; grammar only the TOPIK (Test of Proficiency in Korean) people care about
★ | Don't worry about this unless there's some reason you need to know it
Expressions are also rated according to whether or not they're used more in speaking or in writing.
| The expression is:
---|---
Speaking > Writing | used more often in speaking than in writing
Speaking < Writing | used more often in writing than in speaking
Speaking = Writing | used as often in writing as in speaking
Speaking | used only in speaking
Writing | used only in writing
#
Grammatical Terms Used in This Book
I've tried to keep grammatical terms to a minimum here so as to avoid confusion and because, honestly, you don't really need to know what a desiderative auxiliary verb is in order to speak Korean. Here are the absolute bare-minimum terms you should know.
Term | Korean term | Meaning | Examples
---|---|---|---
Noun | 명사 myeongsa | A word that names an object, person or place | 책 chaek (book)
사람 saram (person)
동대문 (Dongdaemun—place)
민수 (Minsu—male's name)
Action verb | 동사 dongsa | A word that tells you what someone or something does | 가다 gada (to go)
하다 hada (to do)
먹다 meokda (to eat)
살다 salda (to live)
Descriptive verbA | 형용사 hyeong yongsa | A word that describes what someone or something is | 아름답다 areumdapda (to be beautiful)
빨갛다 ppalgata (to be red)
중요하다 jungyohada (to be important)
덥다 deopda (to be hot)
Adverb | 부사 busa | A word that describes how something is done | 조용히 joyonghi (quietly)
빨리 ppal-li (quickly)
그래서 geuraeseo (therefore)
Pronoun | 대명사 dae myeongsa | A shorthand for a noun | 나/저 (I) na/jeo
너/당신 neo/dangsin (you)
우리 uri (we)
Directional verb | | A word that describes the action of going or coming | 가다 gada (to go)
오다 oda (to come)
돌아가다 doragada (to go back)
떠나다 tteonada (to leave)
Past Tense | 과거 gwageo | | 했다 haetda (did)
먹었다 meogeotda (ate)
더웠다 deowotda (was hot)
Present Tense | 현재 hyeonjae | | 한다 handa (do)
하고 있다 hago itda (doing)
아름다운 areumdaun (beautiful)
Future Tense | 미래 mirae | | 할 것이다 hal geosida (will do)
할 거야 hal geoya (will do—
반말)
하겠다 hagetda (will do)
PassiveB | 피동사 pidongsa | A verb that describes what is done to something else | 쓰이다 sseu-i-da (to be written/used)
보이다 bo-i-da (to be seen)
먹히다 meokida (to be eaten)
CausativeB | 사동 sadong | A verb that describes the action of making something happen | 안기다 angida (to hug)
먹이다 meogida (to feed)
씌다다 ssu-i uda (to put something
[a hat/glasses] on someone else)
Statement | 서술문 seosulmun | A sentence ending in a period that neither orders nor suggests anything to anyone else | 이렇게 해도 돼요.Ireoke haedo dwaeyo.It can be done like this.
Question | 의문문 uimunmun | A sentence that asks for information and ends with a question mark | 어떻게 하면 돼요?Eotteoke hamyeon dwaeyo? How should it be done?
Command | 명령문 myeong-ryeongmun | An order telling someone else to do something | 이렇게 해 보세요. Ireoke hae boseyo.Try to do it like this.
Suggestion | 청유문 cheongyumun | A suggestion to someone else that they do something | 이렇게 할까요?Ireoke halkkayo?Shall we do it this way?
A In Korean, adjectives are also considered verbs. In their dictionary form they translate to "to be." For example, 아름답다 means "to be beautiful" and if you want to use it to describe someone, you have to conjugate it appropriately.
B For more on passives and causatives, and the mysteries of their creation, see page .
#
1 The Basics of Korean
##
Pronouns and Contractions
Let's start by reviewing the basics here. As you should know by this point, you change your pronouns depending on how much respect you want to give the person you're talking to.
> English | Lower | Higher
> ---|---|---
> I | 나 na | 저 jeo
> You | 너 neo | 당신 dangsin
> We | 우리 uri | 저희 jeohui
> This | 이것 igeot | 이것
> That | 그것 geugeot | 그것
> That (over there) | 저것 jeogeot | 저것
너 is used all the time in 반말 (informal language), but if you want to be respectful, it's far more common to refer to someone you're talking to in the third person: 선생님 seonsaengnim (teacher), 계현씨 gyehyeonssi (polite way to address a person named 계현) or even 민정 엄마 minjeong eomma (Minjeong's mother) or something similar. Although 당신 is technically the polite way to say "you," it's almost never used in spoken Korean and if it is, it often means a fight is about to break out. So be very careful with 당신.
As you should also know, subjects and objects take endings: 은/는, 이/가, 을/를 eun/neun, i/ga, eul/reul are the ones we'll deal with here. Go to page if you need to learn about any of these endings.
Here's a table to show you how to contract each word.
Original word | 저 jeo | 나 na | 우리 uri | 저희 jeohui | 너 neo | 이(그/저)것 igeoseun
---|---|---|---|---|---|---
+는 neun | 저는 jeoneun | 나는 naneun | 우리는 urineun | 저희는 jeohuineun | 너는 neoneun | 이것은 igeoseun
Contraction | 전 jeon | 난 nan | 우린 urin | 저흰 jeohuin | 넌 neon | 이건 igeon
+이/가 i/ga | 제가 jega | 내가 naega | 우리가 uriga | 저희가 jeohuiga | 네가 nega | 이것이 igeosi
Contraction | | | | | | 이게 ige
+을/를 eul/reul | 저를 jeoreul | 나를 nareul | 우리를 urireul | 저희를 jeohuireul | 너를 neoreul | 이것을 igeoseul
Contraction | 절 jeol | 날 nal | 우릴 uril | 저흴 jeohuil | 널 neol | 이걸 igeol
+의 ui | 저의 jeoui | 나의 naui | 우리의 uriui | 저희의 jeohuiui | 너의 neoui | 이것의 igeosui
Contraction | 제 je | 내 nae | | | 네 ne |
저걸로 주세요. Jeogeollo juseyo. Give me that.
이건 어때요? Igeon eottaeyo? How about this?
More Contractions:
이렇다/그렇다/저렇다/어떻다 ireota/geureota/jeoreota/eotteota
As you may know, these mean "to be a certain way." 이렇다 means "to be this way," 그렇다 means "to be that way" and 저렇다 means "to be that way over there" and isn't really used that much. These are very versatile expressions that you'll see. and they are used in all kinds of ways. 어떻다 by itself doesn't translate well, but you'll see it all the time as 어떻게 (how).
The most common way you'll probably see them conjugated is by adding the ending 게, which turns a verb into an adverb. We don't have these adverbs in English, at least not as single words, but if "thisly," "thatly," and "that over therely" were words, they'd be translated this way.
어떻게
eotteoke
How?
이렇게
ireoke
Like this.
그렇게
geureoke
Like that.
저렇게
jeoreoke
Like that over there.
Let's try making some sentences:
어떻게 하면 돼요?
Eotteoke hamyeon dwaeyo?
What's a good way to do it? (How can I do this?)
그렇게 하면 안 돼요.
Geureoke hamyeon an dwaeyo.
You can't do it like that.
이렇게 어려운 책을 읽지 못 해요.
ireoke eoryeoun chaegeul ikji mot taeyo.
I can't read such a difficult book.
And here's how to abbreviate 이렇다, 그렇다, 저렇다, and 어떻다 using the tense markers ㄴ and ㄹ. (See page for more information on tense markers).
Original word | 이렇다 ireota | 그렇다 geureota | 저렇다 jeoreota | 어떻다 eotteota
---|---|---|---|---
+ㄴ | 이렇 + ㄴ | 그렇 + ㄴ | 저렇 + ㄴ | 어떻 + ㄴ
Contraction | 이런 ireon | 그런 geureon | 저런 jeoreon | 어떤 eotteon
+ㄹ | 이렇 + ㄹ | 그렇 + ㄹ | 저렇 + ㄹ |
Contraction | 이럴 ireol | 그럴 geureol | 저럴 jeoreol |
이런 헤어 스타일은 어떠세요?
Ireon heeo seu-ta-i-reun eotteoseyo?
What do you think about this hairstyle?
어떤 헤어 스타일을 좋아하세요?
Eotteon heeo seu-ta-i-reul joahaseyo?
What kind of style do you like?
저런 헤어 스타일을 좋아해요.
Jeoreon heeo seu-ta-i-reul joahaeyo.
I like that kind of hairstyle.
More ways to use 이렇다, 저렇다 and especially 그렇다 are on page . For now, let's look at 어떻다 and its unique contractions.
When you add 어떻게 plus 하다, you can keep on using 어떻게 하다 or you can contract the whole thing to 어쩌다. This contraction happens in many commonly used expressions.
■ 어쩔 수 없다 Eojjeol su eopda
A: 저기 버스가 가네요! Jeogi beoseuga ganeyo! The bus is leaving!
B: 어쩔 수 없죠, 뭐. 다음 버스 타요. Eojjeol su eopjo, mwo. Daeum beoseu tayo. Oh well, it can't be helped. Let's take the next one.
■ 어쩌면 eojjeomyeon Maybe
내일 어쩌면 비가 올 지도 몰라요. Naeil eojjeomyeon biga ol jido mollayo. Maybe it'll rain tomorrow.
■ 어쩐지 eojjeonji Somehow
A: 저 감기 걸린 것 같아요. Jeo gamgi geollin geot gatayo. I think I caught a cold.
B: 어쩐지, 얼굴이 안좋아보이더라고요. Eojjeonji, eolguri an-jo-a-bo-i-deo-ra-go-yo. Somehow, your face doesn't look that good.
(Here "somehow" means "in some vague way." And while it's strange to tell people in English that their face doesn't look so good, it's done in Korean all the time.)
##
How to Talk to People Without Being Rude
반말 and 존댓말 and All Their Permutations
There are seven levels of speech in Korean. There used to be many more, but thankfully they've been greatly simplified. Even more thankfully, most of the seven aren't that commonly used. 반말 banmal is the lowest form, used toward children or people very close to you, and it actually covers five of the seven levels, including all the ones you don't need to worry about. The next two levels are both called 존댓말 jondaenmal. One is what I'll refer to as "informal polite" and is by far the most common and useful level. It's used toward strangers, people you don't know well, or people older than you. The other is super-polite and is used when speaking to people higher in rank or status or when addressing crowds. Just as an example, I use 반말 to my students and informal polite to my co-teachers. I just about never use super-polite, but I hear it all the time in subway announcements. Finally, you don't really use any of the levels in writing, so I'll show you how to end sentences when you're writing, too.
As you may also know, Korean grammar much depends on what kind of sentence you're forming. These can be classified as follows:
> 1. Statements with action verbs
>
> 2. Statements with descriptive verbs
>
> 3. Statements with nouns
>
> 4. Commands
>
> 5. Questions
>
> 6. Suggestions
Here's how to end each kind of sentence in each level of speech. Note as well that there are many ways to end sentences—this whole book's worth, in fact. The ones listed here are just the basics. Let's start with the three very common ways of speaking.
Sentence type | Tense | 반말 (해체 haeche) (casual) | 존댓말 (해요체 haeyoche) (informal polite) | 존댓말 (하십시오체 hasipsi-o-che) (super polite) | 문어체 muneoche (writing)
---|---|---|---|---|---
Action verb statements ending in vowels | Past | 했어 haeseo | 했어요 haeseoyo | 했습니다 haetseupnida | 했다 haetda
Present | 해 hae | 해요 haeyo | 합니다 hapnida | 한다 handa
Future | 할 거야 hal geoya | 할 거예요 hal geoyeyo | 하겠습니다 hagetseumnida | 할 것이다 hal geosida
Action verb statements ending in consonants | Past | 먹었어 meogeoseo | 먹었어요 meogeoseoyo | 먹었습니다 meogeotseupnida | 먹었다 meogeotda
Present | 먹어 meogeo | 먹어요 meogeoyo | 먹습니다 meokseubnida | 먹는다 meokneunda
Future | 먹을 거야 meogeul geoya | 먹을 거예요 meogeul geoyeyo | 먹을 겁니다 meogeul geopnida | 먹을 것이다 meogeul geosida
Descriptive verb statements ending in vowels | Past | 작았어 jagasseo | 작았어요 jagasseoyo | 작았습니다 jagatseumnida | 작았다 jagatda
Present | 작아 jaga | 작아요 jagayo | 작습니다 jakseumnida | 작다 jakda
Future | 작을 거야 jageul geoya | 작을 거예요 jageul geoyeyo | 작을 겁니다 jageul geopnida | 작을 것이다 jageul geosida
Descriptive verb statements ending in consonants | Past | 예뻤어 yeppeoseo | 예뻤어요 yeppeoseoyo | 예뻤습니다 yeppeotseumnida | 예뻤다 yeppeotda
Present | 예뻐 yeppeo | 예뻐요 yeppeoyo | 예쁩니다 yeppeupnida | 예쁘다 yeppeuda
Future | 예쁠 거야 yeppeul geoya | 예쁠 거예요 yeppeul geoyeyo | 예쁠 겁니다 yeppeul geopnida | 예쁠 것이다 yeppeul geosida
Noun statements ending in vowels | Past | 남자였어 namjayeosseo | 남자였어요 namjayeosseoyo | 남자였습니다 namjayeotseumnida | 남자였다 namjayeotda
Present | 남자야 namjaya | 남자예요 namjayeyo | 남자입니다 namjaipnida | 남자이다 namja-i-da
Future | 남자일 거야 namja-il geoya | 남자일 거예요 namja-il geoyeyo | 남자일 겁니다 namja-il geopnida | 남자일 것이다 namja-il geosida
Noun statements ending in consonants | Past | 물이었어 murieosseo | 물이었어요 murieosseoyo | 물이었습니다 murieotseumnida | 물이었다 murieotda
Present | 물이야 muliya | 물이에요 mul-i-eyo | 물입니다 mulipnida | 물이다 mulida
Future | 물일 거야 mulil geoya | 물일 거예요 mulil geoyeyo | 물일 겁니다 mulil geopnida | 물일 것이다 mulil geosida
Commands ending in vowels | Present | 해 hae | 하세요 haseyo | 하십시오 hasipsio | 해라E haela
Commands ending in consonants | Present | 먹어 meogeo | 잡으세요C jabeuseyo | 잡으십시오C jabeusipsio | 잡아라E jabala
Questions ending in vowels | Past | 했어? haesseo? | 했어요? haesseoyo? | 했습니까? haetseumnikka? | 했나?A haenna?
Present | 해? hae? | 해요? haeyo? | 합니까? hapnikka? | 하나?A hana?
Future | 할 거야? hal geoya? | 할 거예요? hal geoyeyo? | 할 겁니까? hal geopnikka? | 하겠나?A hagenna?
Questions ending in consonants | Past | 먹었어? meogeosseo? | 먹었어요? meogeosseoyo? | 먹었습니까? meogeotseumnikka? | 먹었나?A meogeonna?
Present | 먹어? meogeo? | 먹어요? meogeoyo? | 먹습니까? meokseupnikka? | 먹나?A meokna?
Future | 먹을 거야?A meogeul geoya? | 먹을 거예요? meogeul geoyeyo? | 먹을 겁니까? meogeul geopnikka? | 먹겠나?A meokgenna?
Suggestions ending in vowels | Present | 하자 haja | 할래요 hallaeyo/ 할까요 halkkayo | 합시다 hapsida | 하자 haja
Suggestions ending in consonants | Present | 먹자 meokja | 먹을래요 meogeullaeyo/먹을 까요 meogeulkkayo | 먹읍시다 meogeupsida | 먹자 meokja
Now let's take a look at some of the more uncommon forms. You'll most likely never hear these in real life. 하오체 and 하게체 are used only by older people (and on warning signs, in the case of [으]시오), while 해라체 is used by people talking either to themselves or to very young children. You may note that it's very similar to the written style (문어체) listed above: in fact, in most respects, they're the same. You have to show great respect to your audience while addressing a crowd verbally, which is almost always done in the super-polite 하십시오체 form, but none at all while addressing them in writing.
Sentence type | Tense | 하오체 haeche | 하게체 haeyoche | 해라체 muneoche
---|---|---|---|---
Action verb statements ending in vowels | Past | 했소 haetso | 했네 haenne | 했다 haetda
Present | 하오 hao | 하네 hane | 한다 handa
Future | 할 거요 hal geoyo | 할 거네 hal geone | 하겠다/할 것이다/할 거다 hagetda/hal geotsida/hal geoda
Action verb statements ending in consonants | Past | 먹었소 meogeotso | 먹었네 meogeonne | 먹었다 meogeotda
Present | 먹소 meokso | 먹네 meokne | 먹다 meokda
Future | 먹을 거요 meogeul geoyo | 먹을 거네 meogeul geone | 먹겠다/먹을 것이다/먹을 거다 meokgetda/meogeul geosida/meogeul geoda
Descriptive verb statements ending in vowels | Past | 작았소 jagatso | 작았네 jaganne | 작았다 jagatda
Present | 작소 jakso | 작네 jakne | 작다 jakda
Future | 작을 거요 jageul geoyo | 작을 거네 jageul geone | 작겠다/작을 것이다/작을 거다 jakgetda/jageul geosida/jageul geoda
Descriptive verb statements ending in consonants | Past | 예뻤소 yeppeotso | 예뻤네 yeppeonne | 예뻤다 yeppeotda
Present | 예쁘오 yeppeuo | 예쁘네 yeppeune | 예쁘다 yeppeuda
Future | 예쁠 거요 yeppeul geoyo | 예쁠 거네 yeppeul geone | 예쁘겠다/예쁠 것이다/예쁠 거다 yeppeugetda/yeppeul geosida/yeppeul geoda
Noun statements ending in vowels | Past | 남자였소 namjayeotso | 남자였네 namjayeonne | 남자였다 namjayeotda
Present | 남자요 namjayo | 남자이네 namja-i-ne | 남자다/남자이다B namjada/namja-i-da
Future | 남자일 거요 namja-il geoyo | 남자일 거네 namja-il geone | 남자일 갓이다/남자일 거다 namja-il gasida/namja-il geoda
Noun statements ending in consonants | Past | 물이었소 mulieotso | 물이었네 mulieonne | 물이었다 mulieotda
Present | 물이오 mulio | 물이네 muline | 물이다 mulida
Future | 물일 거요 mulil geoyo | 물일 거네 mulil geone | 물일 것이다/물일 거다 mulil geosida/mulil geoda
Commands ending in vowels | Present | 하시오 hasio | 하게 hage | 해라E haela
Commands ending in consonants | Present | 잡으시오C jabeusio | 먹게 meokge | 먹어라E meogeola
Questions ending in vowels | Past | 했소? haetso? | 했나?A haetna? | 했니? haetni?
Present | 하오? hao? | 하나?A hana? | 하니? hani?
Future | 할 거요? hal geoyo? | 할 건가? hal geonga? | 할 것이니?/할 거니?D hal geosini?/hal geoni?
Questions ending in consonants | Past | 먹었소? meogeotso? | 먹었나?A meogeonna? | 먹었니? meogeonni?
Present | 먹소? meokso? | 먹나?A meokna? | 먹니? meokni?
Future | 먹을 거요? meogeul geoyo? | 먹을 건가? meogeul geonga? | 먹을 것이니?/할 거니?D meogeul geosini?/hal geoni?
Suggestions ending in vowels | Present | 합시다 hapsida | 하세 hase | 하자 haja
Suggestions ending in consonants | Present | 먹읍시다 meogeupsida | 잡세C jabse | 먹자 meokja
A These are actually conjugated using the 나/(으)ㄴ가 question forms, which you can find on page . Action verbs such as the ones above are conjugated with 나 and descriptive verbs take (으)ㄴ가. So you would say 하나? but 예쁜가?, and 먹나? but 작은가?
B 남자다 and 남자이다 are both acceptable.
C This example was changed from 먹다 to 잡다 because it's impolite to say 먹으세요 and 먹으십시오: you have to change them to 드세요 and 드십시오.
D 거니 is a contraction of 것이니 and is much more commonly used.
E These are conjugated with 어/아/여라, so they actually depend on the vowel in the first syllable rather than whether it ends in a vowel or a consonant.
##
How to Add Expressions to Verbs
This looks confusing at first, but with some practice it will become second nature to you. The best way to learn it is to learn how different expressions can connect to different verbs, and the good news here is that there are actually only a few ways to do this. First, find the stem of the verb you want to conjugate. That part is easy: just drop the 다, and there's your verb stem. Thus, the stem of 하다 is 하, the stem of 먹다 is 먹, the stem of 모으다 moeuda is 모으, and so on. You will never add a Korean grammar expression to a verb including the 다 ending—always use the stem.
Irregular verbs can be tricky, even for advanced students. However, even they are not totally lawless and will always interact with the same kinds of expressions in the same way. Go to the end of the irregular verbs section to find out how to combine them with each type of ending.
Expressions that Don't Change Verbs
Many expressions, particularly those beginning with ㅈ or ㄱ, can simply be added to verbs on their own without any special adaptation. In these cases you won't see anything in parentheses before the expression. Let's take a look at three: 거든, 잖아요, and 지만.
| | 거든 geodeun | 지만 jiman | 잖아요 janhayo
---|---|---|---|---
Action verbs ending in a vowel | 하다 hada | 하거든 hageodeun | 하지만 hajiman | 하잖아요 hajanhayo
Action verbs ending in a consonant | 먹다 meokda | 먹거든 meokgeodeun | 먹지만 meokjiman | 먹잖아요 meokjanhayo
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 yeppeuda | 예쁘거든 yeppeugeodeun | 예쁘지만 yeppeujiman | 예쁘잖아요 yeppeujanhayo
Descriptive verbs (adjectives) ending in a consonant | 작다 jakda | 작거든 jakgeodeun | 작지만 jakjiman | 작잖아요 jakjanhayo
Expressions with 으 or 이 Prefixes
You'll see many expressions in this book that have either 으 or 이 before them in parentheses. In these cases, if your verb stem or noun ends in a consonant, add that 으 or 이 first. Some examples are (으)니까, (으)나 and (이)라서. Normally 으 is used with expressions that are added to verbs and 이 with expressions that are added to nouns.
| | (으)니까 (eu)nikka | (으)나 (eu)na | (이)라서 (i)laseo
---|---|---|---|---
Action verbs ending in a vowel | 하다 hada | 하니까 hanikka | 하나 hana |
Action verbs ending in a consonant | 먹다 meokda | 먹으니까 meogeunikka | 먹으나 meogeuna |
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 yeppeuda | 예쁘니까 yeppeunikka | 예쁘나 yeppeuna |
Descriptive verbs (adjectives) ending in a consonant | 작다 jakda | 작으니까 jageunikka | 작으나 jageuna |
Nouns ending in a vowel | 남자 namja | 남자이니까 namja-i-nikka | 남자이나 namja-i-na | 남자라서 namjalaseo
Nouns ending in a consonant | 물 mul | 물이니까 mulinikka | 물이나 mulina | 물이라서 mulilaseo
Expressions with 아/어/여
In these cases, you have to check the last syllable of your verb stem to know how to conjugate the verb.
If that last syllable contains an 아 or an 오, you should add 아. (That includes syllables with 애, 얘, 야 or 요, though I've never seen any verb stems ending in the latter three.)
If it contains any other vowel (어, 우, 으 or 이) then add 어. Again, that includes 여, 유, 에 and 예.
If the verb is 하다, add 여. This makes it 해 plus the rest of the expression; the only time you'll see 하여 is in formal situations, usually in writing. Even in this case, it's not all that commonly used except in the past tense: 하였다.
Let's take a look at 아/어/여서 and 았/었/였이다.
Verb stems ending in vowels can be a somewhat special; see page 26 on how to add 아/어/여 to vowels.
| | 아/어/여서 a/eo/yeoseo | 았/었/였이다 at/eot/yeosida
---|---|---|---
Verbs with 아 or 오 | 작다 jakda | 작아서 jagaseo | 작았다 jagatda
Regular verbs with 어, 우, 으 or 이 (see below for exceptions) | 먹다 meogda | 먹어서 meogeoseo | 먹었다 meogeotda
하다 | 하다 hada | 해서 haeseo | 했다 haetda
(으)ㄴ/는, and ㄴ/는
Here's where it gets interesting. First, check the title of the expression carefully to see which of the above sets you should use.
First, we have (으)ㄴ/는. When you see this, you have to go a step further and differentiate between active verbs and descriptive verbs. Descriptive verbs are like adjectives: 예쁘다 yeppeuda, 작다 jakda, 크다 keuda, 조용하다 joyonghada, 중요하다 jungyohada, and so on, while active verbs are verbs that describe actions: 먹다 meokda, 가다 gada, 걷다 geodda, 띄다 ttuida, and so on.
With (으)ㄴ/는 expressions, active verbs always take 는. Descriptive verbs are conjugated with ㄴ if they end in a vowel and 은 if they end in a consonant. Expressions that are conjugated in this way include (으)ㄴ/는데 and (으)ㄴ/는 탓에.
| | (으)ㄴ/는데 (eu)n/neunde | (으)ㄴ/는 탓에 (eu)n/neun tase
---|---|---|---
Action verbs ending in a vowel | 하다 | 하는데 haneunde | 하는 탓에 haneun tase
Action verbs ending in a consonant | 먹다 | 먹는데 meokneunde | 먹는 탓에 meokneun tase
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁜데 yeppeunde | 예쁜 탓에 yeppeun tase
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작은데 jageunde | 작은 탓에 jageun tase
Finally, ㄴ/는 is added to verbs which use indirect speech particles; see page on how to conjugate these. Basically, descriptive verbs take 다 plus whatever else you're using in your expression (다고 하다 dago hada, 다기보다 dagiboda, 다면 damyeon) while active verbs take either ㄴ or 는. ㄴ is added to action verb stems ending in vowels while 는 goes after action verb stems ending in consonants.
##
How to Handle Verbs, Part 1: Changing Their Form
Changing Verbs to Nouns: (으)ㅁ, 기, (으)ㄴ/는 것
■ (으)ㅁ
This can be added to any kind of verb or even to nouns with 이 in case you need to change a noun into a verb and then back into a noun. It's often used in writing and less so in speaking. You're likely to see it on warning signs and other formal notices. There are a few common nouns which always use ㅁ: 꿈 kkum ("dream" from 꾸다 kkuda), 잠 jam ("sleep" from 자다 jada), and 얼음 eoleum ("ice" from 얼다 eolda) come to mind.
While 기 tends to have more to do with activities and appearances, (으)ㅁ is an introvert; it's more concerned with thoughts and ideas.
ㅁ follows vowels and 음 follows consonants. 음 can follow the past tense, but not the future tense.
하다 hada (to do) > 함 ham (doing)
슬프다 seulpeuda (to be sad) > 슬픔 seulpeum (sadness)
살다 salda (to live) > 삶 salm (life) (irregular)
믿다 mitda (to believe) > 믿음 mideum (belief)
■ 기
기 gi also turns verbs into nouns and can be added to any kind of verb. It's often used in speaking and there are quite a number of grammar points which demand 기 if you want to use a verb with it. Like (으)ㅁ, some words just like to be used with 기. I'm sure you're all trying to improve your Korean 듣기 deudgi, 쓰기 sseugi, 말하기 malhagi and 읽기 ilggi.
(으)ㅁ is the "quiet, contemplative" way to turn verbs into nouns while 기 is the "extrovert." 기 is normally used more for activities and actions as well as being more common in spoken Korean. It's also used more in proverbs and slogans.
하다 hada (to do) > 하기 hagi (doing)
찾다 chajda (to find) > 찾기 chajgi (finding)
보다 boda (to see) > 보기 bogi (seeing)
크다 keuda (to be big) > 크기 keugi (size)
■ (으)ㄴ/는 것
는 것 neun geot is a very easygoing expression. You can use it just about anywhere to turn a verb into a noun. 것 means "thing" and 는 is the present tense marker, but don't think of 는 것 that way; use 는 with any tense and 것 whether you're talking about something concrete or not. Compared to 기 and (으)ㅁ, 는 것 can be used in a greater variety of ways without the same kind of nuances.
하다 hada (to do) > 하는 것 haneun geot (doing)
찾다 chajda (to find) > 찾는 것 chajneun geot (finding)
살다 salda (to live) > 사는 것 saneun geot (living) (irregular)
믿다 mitda (to believe) > 믿는 것 mitneun geot (believing)
This 것 will often be followed by the subject marker 이 (assuming, of course, that it is the subject of your sentence). In that case it can be shortened to 게. This is used more often in conversation while 것이 is more commonly seen in writing.
Here's a table that shows how to conjugate 는 것. See page for more on 던 deon.
| Example | Past | Present | Future
---|---|---|---|---
Action verbs ending in vowels | 하다 | 했던 것 haetdeon geot | 하는 것 haneun geot | 할 것 hal geot
Action verbs ending in consonants | 찾다 | 찾던 것 chajdeon geot | 찾는 것 chajneun geot | 찾을 것 chajeul geot
Descriptive verbs ending in vowels | 예쁘다 | 예뻤던 것 yeppeotdeon geot | 예쁜 것 yeppeun geot | 예쁜 것 yeppeun geot
Descriptive verbs ending in consonants | 작다 | 작았던 것 jagatdeon geot | 작은 것 jageun geot | 작은 것 jageun geot
Changing Action Verbs to Descriptive Verbs: (으)ㄴ/는/(으)ㄹ
(으)ㄴ/는/(으)ㄹ are tense markers and are covered in greater detail on page . For now, let's look at how to use them to change an action verb into an adjective. English isn't very efficient about this; if you want to talk about a person visiting your house, you have to say "the person who came," "the person who is coming," or "the person who will come." In Korean, it's much simpler.
ㄴ and 은 are past tense markers. ㄴ goes after verbs ending in vowels and 은 after verbs ending in consonants.
는 is the present tense marker and follows any verb.
ㄹ and 을 are future tense markers. ㄹ follows vowels and 을 follows consonants.
Going back to the visitor we had:
온 사람 on saram the person who came
오는 사람 oneun saram the person who is coming
올 사람 ol saram the person who will come
Who, of course, needs to eat from time to time:
먹은 음식 meogeun eumsig the food that was eaten
먹는 음식 meogneun eumsig the food that is being eaten
먹을 음식 meogeul eumsig the food that will be eaten
By adding (으)ㄴ/는/(으)ㄹ, you can use any verb to describe any noun. Please note that your choice of (으)ㄴ/는/(으)ㄹ depends on the time the action happened relative to the sentence, not relative to right now.
올 사람은 김종국이라고 했어요.
Ol sarameun Kim-jong-guk-i-lago haesseoyo.
The person who was going to come was called Kim Jongkook.
온 사람은 김종국이라고 했어요.
On sarameun Kim-jong-guk-i-lago haesseoyo.
The person who had come was called Kim Jongkook.
오는 사람은 김종국이라고 했어요.
Oneun sarameun Kim-jong-guk-i-lago haesseoyo.
The person who was coming was called Kim Jongkook.
Descriptive verbs usually take only (으)ㄴ or maybe sometimes (으)ㄹ. This (으)ㄴ is built into many of them: you've probably talked about things that were 큰 or 작은 or 아름다운. These are all based on verbs (크다, 작다 and 아름답다) that were conjugated using (으)ㄴ.
예쁜 아이 yeppeun a-i a beautiful child
멋있는 남자 meo-sinneun namja a handsome man
아름다운 여자 areumdaun yeoja a beautiful woman
Changing Verbs to Adverbs: 히, 이, 게
■ 게
This is the most common way to form an adverb. You just take the 다 off your verb and add 게.
조용하다 joyonghada (to be quiet) > 조용하게 joyonghage (quietly)
즐겁다 jeulgeobda (to be pleasant) > 즐겁게 jeulgeobge (pleasantly)
슬프다 seulpeuda (to be sad) > 슬프게 seulpeuge (sadly)
In particular, any verb not ending in 하다 should be changed to an adverb using 게. 하다 verbs can always be changed this way as well if you'd like, but in many cases it's more natural to use 히. See below.
■ 히
You can also form an adverb by taking a verb ending in 하다 and changing that 하다 to 히.
적당하다 jeogdanghada (to be suitable) > 적당히 jeogdanghi (suitably)
무사하다 musahada (to be safe) > 무사히 musahi (safely)
편하다 pyeonhada (to be comfortable) > 편히 pyeonhi (comfortably)
부지런하다 bujileonhada (to be diligent) > 부지런히 bujileonhi (diligently)
■ 이
This is the rarest of the adverbial forms, and I've seen it only a few times:
~없이 eobsi without
~빠듯이 ppadeusi barely, narrowly
밖이 bakki outside (as an adverb)
깊이 gipi deeply
굳이 guji firmly, stubbornly
깨끗이 kkaekkeusi cleanly
This isn't a comprehensive list, but it does cover most of the more common 이 adverbs.
Changing Descriptive Verbs to Action Verbs: 아/어/여지다, 게 되다
To change an adjective (descriptive verb) into an action verb, you need to add "become." So "big" becomes "to become big." "Beautiful" becomes "to become beautiful." There are two ways to add this "become," and which one you use depends on whether you're focusing on the situation changing or the fact that the situation has already changed. 아/어/여지다 a/eo/yeojida means the focus is on the change itself while 게 되다 ge doeda means the focus is on the finished product. In situations where you're using 아/어/여지다 in the past tense (아/어/여졌다), although the change has been completed, the focus is on the situation changing rather than the final result. This is usually not a very big difference.
게 되다 can be added to any verb. 아/어/여지다 follows the same rules as 아/어/여: 아 follows verbs with 아 or 오 as their last vowel, 어 follows verbs with 어, 우, 으 or 이 as their last vowel, and 여 follows 하다.
예쁘다 yeppeuda (to be beautiful) > 예뻐지다 yeppeojida (to become beautiful)
작다 jakda (to be small) > 작아지다 jagajida (to become small)
조용하다 joyonghada (to be quiet) > 조용해지다 joyonghaejida or 조용하게 되다 joyonghage doeda (to become quiet)
Changing Nouns to Verbs: 하다, 이다
A large number, I'd even say most, nouns can be changed to verbs simply by the addition of 하다 hada. This is especially true of words derived from 한자 hanja. You'll notice Korean has many two-syllable nouns to which you can add 하다 and get a four-syllable word; this is usually two 한자 characters plus 하다. This is also done with many words derived from English.
So if you ever need to change a noun to a verb in a hurry and don't have a grammar reference guide handy, try 하다. It's usually a good guess.
운전 unjeon (driving) > 운전하다 unjeonhada (to drive)
공부 gongbu (studying) > 공부하다 gongbuhada (to study)
지각 jigak (tardiness) > 지각하다 jigakhada (to be late)
인쇄/프린트 inswae/peulinteu (printing) > 인쇄하다/프린트 하다 inswaehada/peulinteu hada (to print)
게임 geim (game) > 게임 하다 geim hada (to play a game)
블로그 beullogeu (blog) > 블로그 하다 beullogeu hada (to blog)
The second way to change a noun is to add 이다, which means "it is." This is necessary for many grammar patterns that will accept only verbs. If you want to sneak a noun in, you can quite often get away with it by adding 이다 to the noun.
학생 haksaeng (student) > 학생이다 haksaeng-i-da (it's/he's/she's a student)
남자 namja (man) > 남자이다 namja-i-da (he's a man)
물 mul (water) > 물이다 murida (it's water)
책 chaek (book) > 책이다 chaegida (it's a book)
Changing Nouns to Descriptive Verbs: 적
Finally, here's how to turn a noun into an adjective. It works with the two-syllable 한자 hanja nouns we talked about in the last section. Add 적 jeok to the end and the result is an adjective. However, these adjectives can't be conjugated like regular Korean adjectives (descriptive verbs); for that, you need to add 이다 ida to the end of them as if they were nouns.
개인 gaein (individual) > 개인적 gaeinjeok (private) > 개인적이다 gaeinjeogida
과학 gwahak (science) > 과학적 gwahakjeok (scientific) > 과학적이다 gwahakjeogida
효율 (efficiency) hyoyul > 효율적 hyoyuljeok (efficient) > 효율적이다 hyoyuljeogida
이렇게 개인적인 질문을 하지 마세요. Ileoke gaeinjeogin jilmuneul haji maseyo. Please don't ask such personal questions.
이것은 더 효율적인 연료인데요. Igeoseun deo hyoyuljeogin yeonryoindeyo. This is a more efficient fuel.
##
How to Handle Verbs, Part 2: Irregular Verbs
Vowels
When a verb stem ends in a vowel and the expression you're adding to the end begins with a vowel (어/아 or 었/았 expressions), you have to combine the two vowels. This isn't difficult as long as you learn how each set combines. In all other cases, ㄴ/는/ㄹ, expressions that have one form for vowels and another for consonants, and expressions that don't change no matter what they follow, you don't have to worry about verbs ending in vowels at all. They're very easy to deal with most of the time.
ㅏ, ㅗ, ㅑ and ㅐ have 아 added to them:
가다 gada > 가 > 가 (아 plus 아 = no change)
사다 sada > 사 > 사
오다 oda > 오 > 와요 wayo (오 plus 아 becomes 와)
보다 boda > 보 > 봐요 bwayo
ㅐ and 야 are based on ㅏ and so are conjugated like 아 verbs.
내다 naeda > 내 > 내요 naeyo
Verbs ending in ㅓ, ㅜ, ㅡ, ㅣ, ㅕ or ㅔ take 어 and this is added as follows:
서다 seoda > 서 > 서요 seoyo (어 plus 어 = no change)
켜다 kyeoda > 켜 > 켜요 kyeoyo (여 plus 어 = no change)
치다 chida > 치 > 쳐요 chyeoyo (이 plus 어 = 여)
지다 jida > 지 > 져요 jyeoyo
마시다 masida > 마시 masi > 마셔요 masyeoyo
으 is a meek shy little vowel and prefers to give way whenever it encounters a stronger vowel. This means that it completely disappears from the word, like this:
크다 keuda > 크 > 커요 keoyo
끄다 kkeuda > 끄 > 꺼요 kkeoyo
우 has 어 added to it. Some words can be written with the 우 and 어 together or in separate syllables; this is mostly a matter of custom.
태우다 taeuda > 태우 > 태워요 taewoyo
세우다 seuda > 세우 > 세워요 sewoyo
주다 juda > 주 > 줘요 jwoyo
나누다 nanuda > 나누 > 나눠요 nanwoyo
두다 duda > 두 > 두어요 dueoyo or 둬요 dwoyo
Verbs ending in 에 absorb their 어 like so:
세다 seda > 세 > 세요 seyo
ㄹ irregular Verbs
Verbs ending in ㄹ and followed by expressions starting with ㅂ, ㅅ, or ㄴ (like ㅂ니다, 세요 or 는) lose the ㄹ altogether.
> 알다 alda > 알 al > 아세요, aseyo 압니다, 아는, 아시다시피, 아니까, abnida, aneun, asidasipi, anikka...
>
> 알아요, 알아, 알면, 알고, 알지만, arayo, ara, almyeon, algo, aljiman...
When verb stems ending in 알 are changed into nouns using the ending ㅁ (see page ), it just gets added into the same syllable, like so:
알다 alda > 알 > 앎 alm
살다 salda > 살 > 삶 salm
This doesn't happen with verb stems ending in any other consonant.
Tenses with ㄴ/는/ㄹ look strange with this particular irregularity; see the table below.
Verb | Past Tense | Present Tense | Future Tense
---|---|---|---
알다 alda | 안 an | 아는 aneun | 알 al
팔다 palda | 판 pan | 파는 paneun | 팔 pal
울다 ulda | 운 un | 우는 uneun | 울 ul
르 irregular Verbs
Verb stems ending in 르 leu are totally regular except when they have to deal with an ending starting with 아/어/여: for instance, 아/어/여요 or 아/어/여서. In these cases, the ㄹ in the verb stem needs an extra ㄹ at the end of the previous syllable with these verbs.
모르다 moreuda > 모르 moreu > 몰ㄹ mol-l > 몰라요 mollayo
고르다 goreuda > 고르 goreu > 골ㄹ gol-l > 골라요 gollayo
바르다 bareuda > 바르 bareu > 발ㄹ bal-l > 발라요 ballayo
빠르다 ppareuda > 빠르 ppareu > 빨ㄹ ppal-l > 빨라요 ppallayo
ㅎ irregular Verbs
These include just about all the color words in Korean, plus 이렇다/그렇다/저렇다 ireota/geureota/jeoreota and 어떻다 eotteota. Other verbs ending in ㅎ are conjugated normally.
놓다 nota > 놓 > 놓아요 noayo
넣다 neota > 넣 > 넣어요 neoeoyo
Color words typically drop their ㅎ and have 이요 added instead of 아요.
빨갛다 ppalgata > 빨갛 > 빨가 ppalga > 빨간, 빨개요 ppalgan, ppalgaeyo
까맣다 kkamata > 까맣 > 까마 kkama > 까만, 까매요 kkaman, kkamaeyo
하얗다 hayata > 하얗 > 하야 haya > 하얀, 하얘요 hayan, hayaeyo
이렇다 ireota/그렇다 geureota/저렇다 jeoreota and 어떻다 eotteota also function like these words.
이렇다 > 이렇 > 이러 ireo > 이런 ireon, 이래요 iraeyo
그렇다 > 그렇 > 그러 geureo > 그런 geureon, 그래요 geuraeyo
저렇다 > 저렇 > 저러 jeoreo > 저런 jeoreon, 저래요 jeoraeyo
어떻다 > 어떻 > 어떠 eotteo > 어떤 eotteon, 어때요 eottaeyo
They're strange enough that, particularly in the case of 그렇다, which is used all over the place, you're better off simply memorizing their permutations. See page for a detailed list of the various ways in which 그렇다 can be conjugated.
Irregular Verbs: Stems Ending with ㅂ, ㅅ or ㄷ
Basically, each of these irregular verbs has two forms: in any instance in which an expression that begins (or can begin) with a vowel is added, the verb will change, while in any instance in which an expression that can begin with only a consonant is added, the verb will be conjugated the regular way. The table below shows how each one changes:
Kind of irregular verb (example) | Verb stem before vowels | Verb stem before consonants
---|---|---
ㅂ (돕다) dopda | 도우 dou (ㅂ changes to 우) | 돕 dob
ㅅ (짓다) jitda | 지 ji (ㅅ disappears) | 짓 jit
ㄷ (걷다) geotda | 걸 geol (ㄷ changes to ㄹ) | 걷 geot
And here are some examples of the verbs in action:
Kind of verb | Example | \+ consonant (지만) | \+ consonant (는) | \+ consonant (고) | \+ vowel (아/어/여서) | \+ vowel ([으]ㄹ) | \+ vowel ([으]면)
---|---|---|---|---|---|---|---
ㅂ irregular | 돕다 dopda | 돕지만 dopjiman | 돕는 dopneun | 돕고 dopgo | 도와서 dowaseo | 도울 doul | 도우면 doumeyon
| 밉다 mipda | 밉지만 mipjiman | 밉는 mipneun | 밉고 mipgo | 미워서 miwoseo | 미울 miul | 미우면 miumyeon
| 굽다 gupda | 굽지만 gupjiman | 굽는 gupneun | 굽고 gupgo | 구워서 guwoseo | 구울 guul | 구우면 gu-u-myeon
| 눕다 nupda | 눕지만 nupjiman | 눕는 nupneun | 눕고 nupgo | 누워서 nuwoseo | 누울 nuul | 누우면 nu-u-myeon
ㅅ irregular | 짓다 jitda | 짓지만 jitjiman | 짓는 jitneun | 짓고 jitgo | 지어서 jieoseo | 지을 jieul | 지으면 jieumyeon
| 잇다 itda | 잇지만 itjiman | 잇는 itneun | 잇고 itgo | 이어서 ieoseo | 이을 ieul | 이으면 ieumyeon
| 붓다 butda | 붓지만 butjiman | 붓는 butneun | 붓고 butgo | 부어서 bueoseo | 부을 bueul | 부으면 bueumyeon
| 낫다 natda | 낫지만 natjiman | 낫는 nanneun | 낫고 natgo | 나아서 na-a-seo | 나을 naeul | 나으면 naeumyeon
ㄷ irregular | 걷다 geotda | 걷지만 geotjiman | 걷는 geotneun | 걷고 geotgo | 걸어서 georeoseo | 걸을 georeul | 걸으면 georeumyeon
| 싣다 sitda | 싣지만 sitjiman | 싣는 sitneun | 싣고 sitgo | 실어서 sireoseo | 실을 sireul | 실으면 sireumyeon
| 묻다 mutda | mutjiman | 묻는 mutneun | 묻고 mutgo | 물어서 mureoseo | 물을 mureul | 물으면 mureumyeon
| 듣다 deutda | 듣지만 deutjiman | 듣는 deutneun | 듣고 deutgo | 들어서 deureoseo | 들을 deureul | 들으면 deureumyeon
It's important to remember that unlike ㄹ, 르, ㅎ, or vowel-based irregularities, which apply to all verbs with that particular spelling pattern, verb stems ending in ㅅ, ㄷ, and ㅂ can be regular or irregular. You just have to memorize which ones are which. Here are a few common ones to get you started:
Regular ㅂ verbs: | 입다 ipda, 잡다 japda, 집다 jipda, 씹다 ssipda, 줍다 jupda, 짧다 jjalda, 넓다 neolda, 좁다 jopda
---|---
Irregular ㅂ verbs: | 덥다 deopda, 쉽다 swipda, 맵다 maepda, 눕다 nupda, 굽다 gupda, 돕다 dopda, 밉다 mipda, 귀엽다 gwiyeopda, 아름답다 areumdapda, 곱다 gopda, 춥다 chupda
Regular ㅅ verbs: | 씻다 ssitda, 빗다 bitda, 벗다 beotda
Irregular ㅅ verbs: | 붓다 butda, 젓다 jeotda, 짓다 jitda, 잇다 itda, 낫다 natda, 긋 다 geutda
Regular ㄷ verbs: | 받다 batda, 쏟다 ssotda, 닫다 datda, 믿다 mitda
Irregular ㄷ verbs: | 듣다 deutda, 걷다 geotda, 싣다 sitda, 묻다 mutda
##
Basic Particles
You should have some familiarity with most of the following particles before using this book. Otherwise you'll have a hard time understanding the example sentences. There are plenty of good beginning books out there that cover this material; see the list of Useful Korean Language Resources in Appendix 2. As reference, here are the basic particles with their general meanings:
> 이 | i | subject particle for words ending in consonants
> ---|---|---
> 가 | ga | subject particle for words ending in vowels
> 은 | eun | topic particle for words ending in consonants
> 는 | neun | topic particle for words ending in vowels
> 을 | eul | object particle for words ending in consonants
> 를 | reul | object particle for words ending in vowels
> 에서 | eseo | from, at
> 에 | e | to, at, concerning
> 더러 | deoreo | to/from a person (very informal)
> 에게 | ege | to a person (informal)
> 에게서 | egeseo | from a person (informal)
> 한테 | hante | to/from a person (somewhat polite)
> 께 | kke | to a person (super-polite)
> 께서 | kkeseo | from a person (super-polite)
> 로 | ro | through, to for words ending in vowels
> 으로 | euro | through, to for words ending in consonants
> 부터 | buteo | from
> 까지 | kkaji | to, until
> 의 | ui | 's (possessive particle)
> 들 | deul | s (pluralizing particle)
> 씩 | ssik | each, at a time
> 마다 | mada | each, every
> 시 | si | added to verbs when talking about people higher in status than you
> 님 | nim | Mr./Mrs./Ms./Miss (very polite)
> 씨 | ssi | Mr./Mrs./Ms./Miss (somewhat polite)
> 만 | man | only
> 도 | do | also, too
All these particles are simply added to the ends of nouns when you want to use them.
##
Helping Verbs
These verbs follow 아/어/여 verb endings. When added to a verb plus one of those three endings, they mean "to take the result of that verb and do something with it." Here are the more common "helping" verbs:
> ~가다 gada/오다 oda | to go/to come
> ---|---
> ~두다 duda/놓다 nota | to keep/to put aside
> ~넣다 neota | to put
> ~대다 daeda | to repeat
> ~가지다 gajida | to carry (in this context, the result of ~ was carried over to something else)
> ~버리다 beorida | to throw out (in this context, to throw out or to finish/be done with)
> ~보다 boda | to watch (in this context, to try)
> ~주다 juda | to give
> ~내다 naeda | to pay (in this context, it implies accomplishment: ~ was something you did with some effort)
있다 can also follow verbs with 아/어/여, but its use is a little more specialized; see page .
As an example, let's try adding these words to 하다 and see the result:
> 가다/오다 > 해 가다/해 오다
>
> gada/oda > hae gada/hae oda
>
> | to do and go/come with
> ---|---
> 두다/놓다 > 해 두다/해 놓다
>
> duda/nota > hae duda/hae nota
>
> | to keep/to put aside
> 넣다 > 해 넣다
>
> neota > hae neota
>
> | to put
> 대다 > 해 대다
>
> daeda > hae daeda
>
> | to do repeatedly
> 가지다 > 해 가지다
>
> gajida > hae gajida
>
> | to do and take
> 버리다 > 해 버리다
>
> beorida > hae beorida
>
> | to do and throw out/get rid of/be done with
> 보다 > 해 보다
>
> boda > hae boda
>
> | to try to do
> 주다 > 해 주다
>
> juda > hae juda
>
> | to do and give/to do something for someone else
> 내다 > 해 내다
>
> naeda > hae naeda
>
> | to accomplish/to finish doing
And let's see one more: 만들다 mandeulda, to make.
> 가다/오다 > 만들어 가다/만들어 오다
>
> mandeureo gada/mandeureo oda
>
> | to make and go/come with
> ---|---
> 두다/놓다 > 만들어 두다/만들어 놓다
>
> mandeureo duda/mandeureo nota
>
> | to make and keep/put aside
> 넣다 > 만들어 넣다
>
> mandeureo neota
>
> | to make and put
> 대다 > 만들어 대다
>
> mandeureo daeda
>
> | to repeatedly make
> 가지다 > 만들어 가지다
>
> mandeureo gajida
>
> | to make and take
> 버리다 > 만들어 버리다
>
> mandeureo beorida
>
> | to make and throw out/get rid of/ be done with
> 보다 > 만들어 보다
>
> mandeureo boda
>
> | to try to make
> 주다 > 만들어 주다
>
> mandeureo juda
>
> | to make and give/to make for someone else
> 내다 > 만들어 내다
>
> mandeureo naeda
>
> | to finish making (something very difficult)
##
Passives and Causatives
Passives
In a passive-voice sentence, make the object of the sentence the subject and don't worry too much about who did the action. For instance, let's take the sentence: "He did his homework." In the passive voice, this would become "His homework was done." Alternatively, let's try "The mother hugged the child." In the passive, it changes to "The child was hugged by the mother."
Korean verbs are changed to the passive voice by adding 이, 히, 리 or 기. Which syllable you add usually depends on the letter the verb ends with; however, there are many exceptions to this rule and you simply have to memorize most of the words.
Having said that, let's discuss the general rules first and common exceptions later.
■ Verbs ending in vowels or ㅎ normally have 이 added.
보다 boda (to see) > 보이다 bo-ida (to be showing/seen)
쌓다 ssata (to pile) > 쌓이다 ssa-ida (to be piled up)
놓다 nota (to put) > 놓이다 no-ida (to be put)
바꾸다 bakkuda (to change) > 바뀌다 bakkwida (to be changed)
■ Verbs ending in ㅂ, ㄷ or ㄱ take 히.
잡다 japda (to catch) > 잡히다 japida (to be caught)
읽다 ilgda (to read) > 읽히다 ilkhida (to be read)
먹다 meokda (to eat) > 먹히다 meokida (to be eaten)
닫다 datda (to close) > 닫히다 dachida (to be closed)
■ Verbs ending in ㄹ almost always take 리.
걸다 geolda (to hang) > 걸리다 geollida (to be hung)
열다 yeolda (to open) > 열리다 yeollida (to be opened)
듣다 deutda (to listen) > 들리다 deullida (to be heard)
■ Verbs ending in ㄴ, ㅁ, ㅅ or ㅊ take 기.
안다 anda (to hug) > 안기다 angida (to be hugged)
끊다 kkeunta (to stop) > 끊기다 kkeunkida (to be stopped)
쫓다 jjotda (to chase) > 쫓기다 jjotgida (to be chased)
담다 damda (to put something in) > 담기다 damgida (to be put in)
When an active verb ends in 하다, you can change it to passive simply by changing 하다 to 되다.
사용하다 sayonghada (to use) > 사용되다 sayongdoeda (to be used)
활용하다 hwallyonghada (to use) > 활용되다 hwallyongdoeda (to be used)
이동하다 idonghada (to move) > 이동되다 idongdoeda (to be moved)
You can also change verbs by adding either 아/어/여지다 or 게 되다; see page . For quick reference, here are some examples:
예쁘다 yeppeuda (to be beautiful) > 예뻐지다 yeppeojida or 예쁘게 되다 yeppeuge doeda (to become beautiful)
작다 jakda (to be small) > 작아지다 jagajida or 작게 되다 jakge doeda (to become small)
조용하다 joyonghada (to be quiet) > 조용해지다 joyonghaejida or 조용하게 되다 joyonghage doeda (to become quiet)
Causatives
If you're an active rather than a passive type and found all that talk of having things happen to you to be unpleasant, you'll enjoy this next part a bit more. This is how to make causative sentences in Korean. In the passive voice, "The child was dressed by the mother"; in the causative voice, "The mother dressed the child."
Just to make things interesting, Korean uses almost exactly the same particles for this as it does for passives. They are 이, 히, 리, and 기. For causatives you also sometimes need 우 and 추.
Just to make it even more fun, the particles sometimes follow the same rules as passives and sometimes not. There are many more exceptions with causative verbs as well.
Again, let's start with the general rules and then move on to exceptions.
■ 이 follows verbs ending in vowels, ㄱ, and ㅎ.
보다 boda (to see) > 보이다 bo-ida (to show)
먹다 meokda (to eat) > 먹이다 meogida (to feed)
끓다 kkeulta (to be boiling) > 끓이다 kkeulida (to boil [something])
죽다 jukda (to die) > 죽이다 jugida (to kill)
■ 히 follows ㅂ.
입다 ipda (to wear) > 입히다 iphida (to clothe or to cover)
눕다 nupda (to lie down) > 눕히다 nuphida (to lay [something/someone] down)
■ 리, again, follows ㄹ.
알다 alda (to know) > 알리다 allida (to let something be known)
살다 salda (to live) > 살리다 salida (to save [someone's life])
울다 ulda (to cry) > 울리다 ullida (to make [someone] cry)
■ 기 follows ㅅ and ㅁ.
숨다 sumda (to hide [oneself]) > 숨기다 sumgida (to hide [something])
남다 namda (to remain) > 남기다 namgida (to leave)
웃다 utda (to smile) > 웃기다 utgida (to make [someone] smile or laugh)
벗다 beotda (to take off [someone's clothes]) > 벗기다 beotgida (to take off [someone's clothes])
■ 우 isn't actually 우, but rather 이우. It follows verbs ending in vowels.
타다 tada (to ride/to burn) > 태우다 taeuda (to give a ride/to burn something)
자다 jada (to sleep) > 재우다 jaeuda (to put to sleep)
서다 seoda (to stand) > 세우다 seuda (to park)
깨다 kkaeda (to wake up) > 깨우다 kkaeuda (to wake up [someone])
■ 추 follows ㅈ.
낮다 natda (to be low) > 낮추다 natchuda (to lower)
늦다 neutda (to be late) > 늦추다 neutchuda (to delay)
맞다 matda (to be right) > 맞추다 matchuda (to fit, to make right)
Here are a few common exceptions:
읽다 ilkda (to read) > 읽히다 ilkhida (to be read)
앉다 anda (to sit) > 앉히다 anchida (to make someone sit)
맡다 matda (to assume, to be in charge) > 맡기다 matgida (to entrust)
Like its partner 게 되다 ge doeda, you can use 게 하다 ge hada to make someone do something or to make something become a certain way. It's the active version of 게 되다 ge doeda. See page for how to use 게 하다 ge hada.
Certain other causatives are based on the verb 나다 nada. When changed to causative form, this becomes 내다 naeda. That's true for all variations of 나다 and 내다.
나다 nada (to happen) > 내다 naeda (to make/pay)
끝나다 kkeutnada (to be finished) > 끝내다 kkeutnaeda (to finish)
힘나다 himnada (to gain strength) > 힘내다 himnaeda (to strengthen oneself)
알아내다 aranaeda (to discover)
그 일이 벌써 끝났어요. geu iri beolsseo kkeutnasseoyo. The work is already finished. (passive)
그 일을 벌써 끝냈어요. Geu ireul beolsseo kkeutnaesseoyo. I've already finished the work. (causative)
Note that the subject of the sentence changes: for 나다 verbs, the subject will always be whatever was done or whatever happened; for 내다 verbs, the subject should be the person that did the action.
해내다 haenaeda to accomplish (see page )
해결해 내다 haegyeolhae naeda to find a solution
#
2 Grammar Points
##
Agreement
First of all, most of the time you won't need any special grammar lessons to agree with people in Korean. 응, 네, 예, 그래(요), 당연하죠, 그렇군요/그렇구나, eung, ne, ye, geu-rae(yo), dang-yeon-ha-jyo, geu-reo-kun-nyo/geu-reo-ku-na and 맞아(요) ma-ja(yo) are all very common ways of saying "Yeah, uh huh, you're right, absolutely, I agree."
However, should you want to agree in a slightly more spectacular fashion, see below:
민수가 오늘도 늦었어요?
Min-su-ga o-neul-do neu-jeo-seo-yo?
Was Minsu late today as well?
늦고 말고요. 민수는 일 때문에 정시에 오지 못 해요.
Neut-go mal-go-yo. Min-su-neun il ttae-mun-e jeong-si-e o-ji mot hae-yo.
Of course he was late. Because of his job, he can't come on time.
Do people ever ask you questions with obvious answers? If so, you'll want to know how to use this expression. It is used to express strong agreement with ~, which is really obvious and beyond doubt. This can be used when replying to a request ("Of course I'll do it!") or when confirming something you think is so obvious that it really doesn't need confirmation.
HOW IT'S FORMED
고 말다 go malda is its own expression (see page ) and it means that something that was expected finally happened. 고요 goyo is an expression used at the end of a sentence to add information. Neither of these expressions has much relevance to 고 말고요 go malgoyo, which always goes at the end of a sentence.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 ha-da | 하고 말고요 ha-go mal-go-yo
했고 말고요 haet-go mal-go-yo
| 하고 말고요 ha-go mal-go-yo | 하고 말고요 ha-go mal-go-yo
할거고 말고요 hal-geo-go mal-go-yo
Action verbs ending in a consonant | 먹다 meok-da | 먹고 말고요 meok-go mal-go-yo
먹었고 말고요 meo-geot-go mal-go-yo
| 먹고 말고요 meok-go mal-go-yo | 먹고 말고요 meok-go mal-go-yo
먹을거고 말고요 meo-geul-geo-go mal-go-yo
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 ye-ppeu-da | 예쁘고 말고요 ye-ppeu-go mal-go-yo
예뻤고 말고요 ye-ppeot-go mal-go-yo
| 예쁘고 말고요 ye-ppeu-go mal-goyo | 예쁘고 말고요 ye-ppeu-go mal-go-yo
Descriptive verbs (adjectives) ending in a consonant | 작다 jak-da | 작고 말고요 jak-go mal-go-yo
작았고 말고요 ja-gat-go mal-go-yo
| 작고 말고요 jak-go mal-go-yo | 작고 말고요 jak-go mal-go-yo
Nouns ending in a vowel | 남자 nam-ja | 남자이고 말고요 nam-ja-i-go mal-go-yo
남자였고 말고요 nam-ja-yeot-go
| 남자이고 말고요 nam-ja-i-go mal-goyo | 남자이고 말고요 nam-ja-i-go mal-go-yo
남자일거고 말고요 nam-ja-il-geo-go mal-go-yo
Nouns ending in a consonant | 물 mul | 물이고 말고요 mul-i-go mal-go-yo
물이었고 말고요 mul-i-eot-go mal-go-yo
| 물이고 말고요 mul-i-go mal-go-yo | 물이고 말고요 mul-i-go mal-go-yo
물일거고 말고요 mu-lil-geo-go mal-go-yo
TAKE NOTE
This expression is used only at the end of very short sentences. If you want to explain yourself further, start a new sentence and do so.
While you can use go 고 말고요 mal-go-yo with nouns, it's not very common; people generally prefer one of the expressions listed at the beginning of this section.
고 말았다 Go ma-rat-da (see page ) is not at all the same as 고 말고요. 고 말았다 means that something happened accidentally and/or unfortunately. Likewise, 고 말겠다 go mal-get-da (see page ) is used to talk about definite plans—it has nothing to do with 고 말고요.
EXAMPLE SENTENCES
일을 벌써 끝냈어요?
I-leul beol-sseo kkeun-nae-seo-yo?
Have you already finished your work?
끝냈고 말고요. 다섯시간 전에 그 일을 시작했어요.
Kkeun-naet-go mal-go-yo. Da-seot si-gan jeon-e geu i-leul si-jak-hae-seo-yo.
Of course I've finished. I started five hours ago.
__________
미스 코리아가 예쁜가요?
Miseu koriaga yeppeungayo?
Is Miss Korea beautiful?
예쁘고 말고요. 모델이잖아요.
Yeppeugo malgoyo. Moderijanayo.
Of course she's beautiful! She's a model.
##
Disagreement/Negatives
Why can't we all just get along? Sometimes you need to object to or disagree with something someone else has said or to state that something is untrue. This section will show you how to do that.
First, however, let's review basic negatives. The simplest way to negate something in Korean is to simply add 지 않다 ji anta to the end of the verb. This works with any kind of verb and after either a vowel or a consonant.
> 먹다 meokda | to eat | 먹지 않다 meokji anta | to not eat
> ---|---|---|---
> 하다 hada | to do | 하지 않다 haji anta | to not do
> 작다 jakda | to be small | 작지 않다 jakji anta | to not be small
> 예쁘다 yeppeuda | to be pretty | 예쁘지 않다 yeppeuji anta | to not be pretty
Another option is to put 안 an in front of the verb. This negates the verb. It's not really used much in formal situations or in writing. In common, everyday speech, either 안 or 지 않다 ji anta is usually fine.
> 먹다 meokda | to eat | 안 먹다 an meokda | to not eat
> ---|---|---|---
> 하다 hada | to do | 안 하다 an hada | to not do
> 작다 jakda | to be small | 안 작다 an jakda | to not be small
> 예쁘다 yeppeuda | to be pretty | 안 예쁘다 an yeppeuda | to not be pretty
If what you're talking about has to do with capability, 지 못하다 should be used. This is covered on page in the section on Possibility.
Another expression which can also belong in this section is (으)ㄴ/는/(으)ㄹ걸요. It's often used for guesses as well, so it appears in that section on page . When used as a negative, it's quite similar to 기는요.
What's with All These 말s?
You'll see a few expressions here and there that look kind of like this:
A(expression) 말 mal (expression) 하다 hada
In these cases the 말 is negating A, so the expression will have something to do with "A or not". The table below summarizes and briefly explains these expressions; I hope it clears up any confusion.
Expression | Page | Meaning | Example
---|---|---|---
~(으)ㄹ락 말락 하다 (eu) r-rak mallak hada | | ~ almost, but not quite, happened/is happening | 그 사고에서 죽을락 말락 했어요. Geu sagoeseo jugeullak mallak haesseoyo. I was hovering on the verge of death after that accident.
~다가 말다가 하다 daga maldaga hada | | Doing ~ on and off | 운동을 하다가 말다가 하면 근육에 무리가갈 거예요. Undongeul hadaga maldaga hamyeon geunnyuge muriga gal geoyeyo. If you keep exercising on and off like that, it will be too stressful for your muscles.
~(으)ㄹ지 말지 하다 (eu)r-ji malji hada | | I can't decide whether to ~ or not. | 운동을 할지 말지 결정 못했어요. Undongeul halji malji gyeoljeong motaesseoyo. I couldn't decide whether or not to exercise.
~(으)ㄹ까 말까 하다 (eu)r-kka malkka hada | | I can't decide whether to ~ or not. | 운동을 할까 말까 해요. undongeul halkka malkka haeyo. I can't decide whether to exercise or not.
~는 둥 마는 둥 하다 neun dung maneun dung hada | | ~ almost didn't happen, but ultimately did | 비가 오는 둥 마는 둥 했어요. Biga oneun dung maneun dung haesseoyo. It rained just a little.
A 거나 말거나 geona malgeona B | | B, whether or not A. | 그녀가 예쁘거나 말거나, 저는그녀를사랑해요. Geunyeoga yeppeugeona malgeona jeoneun geunyeoreul saranghaeyo. Whether or not she's beautiful, I love her.
A: 오늘 늦었어요? Oneul neujeosseoyo? Were you late today?
B: 늦기는요. 정시에 왔어요. Neutgineunnyo. Jeongsie wasseoyo. Late? Not at all. I was on time.
This is a nice way to deny something, especially a compliment. It's often used when you want to be modest such as when you say 안녕 하세요 annyeong haseyo to a Korean and are met with heaps of praise for your wonderful Korean abilities.
HOW IT'S CONJUGATED
| | Past | Present
---|---|---|---
Action verbs ending in a vowel | 하다 | 하기는요 hagineunnyo 했기는요 haetgineunnyo | 하기는요 hagineunnyo
Action verbs ending in a consonant | 먹다 | 먹기는요 meokgineunnyo 먹었기는요 meogeotgineunnyo | 먹기는요 meokgineunnyo
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁘기는요 yeppeugineunnyo 예뻤기는요 yeppeotgineunnyo | 예쁘기는요 yeppeugineunnyo
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작기는요 jakgineunnyo 작았기는요 jagatgineunnyo | 작기는요 jakgineunnyo
Nouns ending in a vowel | 남자 | 남자였기는요 namjayeotgineunnyo | 남자기는요 namjagineunnyo
Nouns ending in a consonant | 물 | 물이었기는요 murieotgineunnyo | 물이기는요 murigineunnyo
TAKE NOTE
As this expression is used to deny something, it's used only when you're responding to something someone has already said. It's normally used in a sentence by itself along with the part that you're not accepting. Then, if you need to explain, start a new sentence.
Keep in mind that since it's used to deny something, you need to add it to the opposite of whatever you think is the truth.
기는요 and (으)ㄴ/는/(으)ㄹ걸요 (see page ) are both used for similar purposes. They're both gentle ways to contradict what was said. The main difference between them is that when you use 기는요 you have to state the thing you're contradicting, whereas with 걸요 you say the opposite. So if someone says you speak Korean well and you don't think you do (or you know you do, but you want to be modest), you can say 잘 하기는요 jal hagineunnyo.
EXAMPLE SENTENCES
A: 한국은 10월이면 더워요? Hangukeun shiworinyeon deowoyo? Is Korea hot in October?
B: 덥기는요. 10월에 시원해요. Deopgineunnyo. Shiwore siwonhaeyo. Hot? Not at all. It's cool in October.
__________
A: 한국어를 아주 잘 하시네요! Hangukeoreul aju jal hasineyo! You speak Korean very well!
B: 잘 하기는요. Jal hagineunnyo. I really don't.
A: 오늘 늦었어요? Oneul neujeosseoyo? Were you late today?
B: 늦기는커녕 8시에 왔어요. Neutgineunkeonyeong yeodeolsie waseoyo. Not only was I not late, but I was here at eight.
A isn't actually the case. This is usually followed by B, an explanation of what exactly the case is. The expressions above are used in a sentence about something negative, and B is often something easier to accomplish than A but still impossible to achieve.
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Action verbs ending in a vowel | 하다 | 하기는커녕 hagineunkeonyeong
Action verbs ending in a consonant | 먹다 | 먹기는커녕 meokgineunkeonyeong
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁘기는 커녕 yeppeugineun keonyeong
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작기는커녕 jakgineunkeonyeong
Nouns ending in a vowel | 남자 | 남자는커녕 namjaneunkeonyeong
Nouns ending in a consonant | 물 | 물은커녕 mureunkeonyeong
TAKE NOTE
It's interchangeable with 은/는 말할 것 없다 eun/neun malhal geot eopda.
When speaking, 커녕 is usually dropped. For example, if you want to say 잘 하기는커녕, jal hagineun keonyeong the sentence simply becomes 잘 하기는.
EXAMPLE SENTENCES
A: 한국은 10월이면 더워요? Hangukeun shiworimyeon deowoyo? Is Korea hot in October?
B: 덥기는커녕 시원해요. Deopgineunkeonyeong siwonhaeyo. Not at all. In fact, it's cool.
__________
A: 한국어를 아주 잘 하시네요! Hangukeoreul aju jal hasineyo! You speak Korean very well!
B: 잘 하기는커녕 기본적인 표현 밖에 몰라요. Jal hagineunkeonyeong gibonjeogin pyohyeon bakke mollayo. Not only do I not speak it well, but I don't know anything except basic expressions.
왜 과속해요? 일찍 가면 뭘 해요? 안전이 제일 중요하지요.
Wae gwasokaeyo? Iljjik gamyeon mwol haeyo? Anjeoni jeil jungyohajiyo.
Why are you speeding? What's the point of getting there early? Safety is more important.
Ever had someone be amazed by something that really wasn't a big deal? Ever want to bring them down a notch or two? Here's how you can do that. This expression means "What's the point of ~?" or "So what if ~?" and is usually followed by an explanation of what you think is more important. For instance, in the example sentence above, you use the expression 면 뭔 해요 to ask "What's the point of speeding?" and then follow it with an explanation of what you think is more important—safety.
HOW IT'S FORMED
면 means "if" (see page ); 뭘 is 무엇 mueot contracted with the object marker 를.
HOW IT'S CONJUGATED
This expression is attached to verbs just like the (으)면 with which it starts.
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했으면 뭘 해요? haesseumyeon mwol haeyo? | 하면 뭘 해요? hamyeon mwol haeyo? | 하면 뭘 해요?
Action verbs ending in a consonant | 먹다 | 먹었으면 뭘 해요? meogeoseumyeon mwol haeyo? | 먹으면 뭘 해요? meogeumyeon mwol haeyo? | 먹으면 뭘 해요?
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤으면 뭘 해요? yeppeoseumyeon mwol haeyo? | 예쁘면 뭘 해요? yeppeumyeon mwol haeyo? | 예쁘면 뭘 해요?
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았으면 뭘 해요? jagaseumyeon mwol haeyo? | 작으면 뭘 해요? jageumyeon mwol haeyo? | 작으면 뭘 해요?
Nouns ending in a vowel | 남자 | 남자였으면 뭘 해요? namjayeosseumyeon mwol haeyo? | 남자이면 뭘 해요? namja-i-myeon mwol haeyo? | 남자이면 뭘 해요?
Nouns ending in a consonant | 물 | 물이었으면 뭘 해요? murieoseumyeon mwol haeyo? | 물이면 뭘 해요? murimyeon mwol haeyo? | 물이면 뭘 해요?
TAKE NOTE
The explanation for why ~ doesn't matter often follows ~(으)면 뭘 해요? (eu)myeon mwol hae-yo? in a second sentence. It often ends with either 아/어/여야지요 ("you must~"; see page ) or ㄴ/는데요; see page 69).
EXAMPLE SENTENCES
돈이 많이 모으면 뭘 해요? 건강이 제일 중요한데요.
Doni mani moeumyeon mwol haeyo? Geongangi jeil jungyohandeyo.
What's the point of saving a lot of money? Health is the most important thing.
건강에 대해서 계속 걱정하면 뭘 해요? 한번 검진 받았으니까 걱정할 필요가 없어요.
Geongange daehaeseo gyesok geokjeonghamyeon mwol haeyo? Hanbeon geomjin badasseunikka geokjeonghal pillyoga eobseoyo.
What's the point of constantly worrying about your health? If you'd just get a checkup, you won't need to worry.
##
Asking Questions
어제 늦었나요? Eoje neujeonnayo? Were you late yesterday?
Let's face it: life in Korea can be pretty confusing sometimes, and you'll probably have a few questions about things. You can (and Koreans, more often that not, do) ask questions using standard verb forms like 해요? or 가세요?, but what fun is that? Here's how to sound better when you're actually clueless.
HOW IT'S CONJUGATED
If your question ends in an action verb like 가다 or 먹다 or with a past tense form, you can end it with 나요. If it ends in a descriptive verb like 예쁘다 or 똑똑하다 ttokttokhada, end with (으)ㄴ가요.
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했나요? haennayo? | 하나요? hanayo? | 할 건가요? hal geongayo?
Action verbs ending in a consonant | 먹다 | 먹었나요? meogeonnayo? | 먹나요? meoknayo? | 먹을 건가요? meogeul geongayo?
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤나요? yeppeonnayo? | 예쁜가요? yeppeungayo? | 예쁠 건가요? yeppeul geongayo?
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았나요? jagannayo? | 작은가요? jageungayo? | 작을 건가요? jageul geongayo?
TAKE NOTE
ㄴ가 and 나 reappear later on in expressions such as ㄴ가/나 봐 (see page ), so they're really useful to learn. They also appear at the end of 뭐 mwo, 언제 eonje, 누구 nugu, 무엇 mueot, and 어디 eodi.
In writing, you'll often see these endings as 나? and (으)ㄴ가? such as in a newspaper interview where a reporter is asking questions:
> 뭔가 mwonga | Whatever it is/something | |
> ---|---|---|---
> 언젠가 eonjenga | Whenever it is/sometime | 언제나 eonjena | any time
> 누군가 nugunga | Whoever it is/someone | 누구나 nuguna | anyone
> 무언가 mueonga | Whatever it is/something | 무엇이나 mueosina | anything
> 어딘가 eodinga | Wherever it is/somewhere | 어디나 eodina | anywhere
EXAMPLE SENTENCES
여자친구가 예쁜가요?
Yeojachinguga yeppeungayo?
Is your girlfriend beautiful?
아버지가 선생님인가요?
Abeojiga seonsaengnimingayo?
Is your father a teacher?
어디로 가야 하나요?
Eodiro gaya hanayo?
Where must I go?
쓰레기 봉투가 있나요?
Sseuregi bongtuga innayo?
Do you have trash bags?
어제 숙제를 했나요?
Eoje sukjereul haennayo?
Did you do your homework yesterday?
##
And: Basic Forms
These are all added to the ends of nouns and are fairly easy to use.
Somewhat counterintuitively, 와 goes with words ending in vowels and 과 goes with words ending in consonants. They both simply mean "and" and can also be used to mean "together with," depending on the context. When used to mean "together with," they are often paired with either 같이 gachi or 함께 hamkke, but you can leave these out, unless you really need to clarify the context.
매운 음식과 짠 음식을 좋아해요.
Maeun eumsikgwa jjan eumsigeul joahaeyo.
I like spicy foods and salty foods.
홍대와 이태원에 술집이 많아요.
Hongdaewa Itaewone suljibi manayo.
There are many bars in Hongdae and Itaewon.
친구와 같이 영화 봤어요.
Chinguwa gachi yeonghwa bwasseoyo.
I saw a movie with a friend.
These have exactly the same meaning as 와/과; in other words, "and" or "together with." They are somewhat more colloquial, but otherwise the same. 이랑 is used after vowels and 랑 after consonants.
매운 음식이랑 짠 음식을 좋아해요.
Maeun eumsigirang jjan eumsigeul joahaeyo.
I like spicy foods and salty foods.
홍대랑 이태원에 술집이 많아요.
Hongdaerang Itaewone suljibi manayo.
There are many bars in Hongdae and Itaewon.
친구랑 같이 영화 봤어요.
Chingurang gachi yeonghwa bwasseoyo.
I saw a movie with a friend.
하고 is used the same way as 와/과 and 랑/이랑.
매운 음식하고 짠 음식을 좋아해요.
Maeun eumsikhago jjan eumsigeul joahaeyo.
I like spicy foods and salty foods.
홍대하고 이태원에 술집이 많아요.
Hongdaehago Itaewone suljibi manayo.
There are many bars in Hongdae and Itaewon.
친구하고 같이 영화 봤어요.
Chinguhago gachi yeonghwa bwasseoyo.
I saw a movie with a friend.
This is just another short, simple way to say "and." You can use (이)며 as many times as you'd like in a sentence; just put it after each noun you want to list, including the last one. Since it follows nouns, there's no need to worry about any kind of conjugation. I've changed the third example since 며 can't be used to mean "together with," but can be used for any kind of list. 며 follows nouns ending in vowels and 이며 follows nouns ending in consonants.
매운 음식이며 짠 음식이며 좋아해요.
Maeun eumsigimyeo jjan eumsigimyeo joahaeyo.
I like spicy foods and salty foods.
홍대며 이태원이며 갔어요.
Hongdaemyeo Itaewonimyeo gasseoyo.
I went to Hongdae and Itaewon.
민수며 가영이며 같이 영화 봤어요.
Minsumyeo Kayeongimyeo gachi yeonghwa bwasseoyo.
I saw a movie with Minsu and Kayoung.
This is not exactly a form of "and," but I'm including it in this section anyway because it combines things in order to make a group and it's very useful to know. It means "a group of ~" or "among ~" and is attached to the end of nouns.
친구끼리 chingukkiri a group of friends, among friends
우리끼리 urikkiri our group, among ourselves
학생끼리 haksaengkkiri a group of students, among students
우리끼리 먼저 갈게요. Urikkiri meonjeo galgeyo.
Our group will go first.
아이들끼리 놀고 있어요. A-i-deulkkiri nolgo isseoyo.
The children are playing amongst themselves.
##
And (So On, and So On...)
Sometimes two is not enough and you want to go on and on—how is this done in Korean?
This is much like "etc." in English. It follows as many examples as you provide (at least two, but you can go on and on if you'd like and if you have patient listeners). After 등, you should go on to explain what A and B are examples of. 등 gets particles added to it just like any other nouns; if you'd normally use a particle after the noun, use it after 등 instead.
종이, 연필 등을 시험에 가지고 오세요.
Jongi, yeonpil deungeul siheome gajigo oseyo.
Bring paper, a pencil, etc. to the test.
그 슈퍼에서 물, 맥주, 사이다 등 음료를 팔아요.
Geu syupeoeseo mul, maekju, sa-ida deung eumnyoreul parayo.
That shop sells drinks like water, beer, cider, etc.
이 책을 한국어, 일본어 등의 언어로 읽을 수 있어요.
I chaegeul hangukeo, ilboneo deungui eoneoro ilgeul su isseoyo.
You can read this book in Korean, Japanese, etc.
This one is more like "like, for example." You list your examples before it (A) and put the category they belong to (B) afterward.
종이, 연필 같은 학습준비물을 시험에 가지고 오세요.
Jongi, yeonpil gateun hakseupjunbimureul siheome gajigo oseyo.
Bring study materials like paper, a pencil, etc. to the test.
그 슈퍼에서 물, 맥주, 사이다 같은 음료를 팔아요.
Geu syupeoeseo mul, maekju, sa-ida gateun eumnyoreul parayo.
That shop sells drinks like water, beer, cider, etc.
이 책을 한국어, 일본어 같은 언어로 읽을 수 있어요.
I chaegeul hangukeo, ilboneo gateun eoneoro ilgeul su isseoyo.
You can read this book in languages like Korean, Japanese, etc.
This one is a little more complicated. It can be thought of as a slash. If only one example is listed, it becomes more like "and so on." (으)ㄹ 겸 can be used after verbs or 겸 directly after nouns. There is also an expression, (으)ㄹ 겸 해서, which is used to focus on one cause out of many. This is covered on page .
아침 겸 점심을 먹었어요.
Achim gyeom jeomsimeul meogeosseoyo.
I ate brunch. (breakfast/lunch)
그는 기자 겸 사진사이에요.
Geuneun gija gyeom sajinsaieyo.
He is a journalist/photographer.
쇼핑도 하고 영화도 볼 겸, 명동에 갔어요.
Syopingdo hago yeonghwado bol gyeom, Myeongdonge gasseoyo.
I went to Myeongdong to go shopping and see a movie.
This is a way of listing examples. The 을/를 are object markers; see page . The category you're giving examples of shouldn't be too general: for instance, "action movies" is fine, but just "movies" is too broad a category.
카리비안의 해적을 비롯한 액션 영화를 봤어요.
Karibianui haejeogeul birotan aeksyeon yeonghwareul bwasseoyo.
I watched some action movies, including Pirates of the Caribbean.
비빔밥을 비롯한 한국 음식을 만들어 봤어요.
Bibimbapeul birotan hanguk eumsigeul mandeureo bwasseoyo.
I tried making some Korean food, including bibimbap.
보드카, 위스키, 소주를 비롯한 독한 술을 마셨어요.
Bodeuka, wiseuki, sojureul birotan dokhan sureul masyeosseoyo.
I drank strong alcohol, including beer, whiskey and soju.
##
Linking Sentences: And
Focus: 고
고 by itself is one way of saying "and." You can also put it together with other expressions to get plenty of interesting and unexpected combinations. Here's a quick list if you're confused:
Expression | Page | Meaning | Sample sentence
---|---|---|---
A고B | 51 | A and B | 민수가 정시에 오고 가영이 늦게 왔어요. Minsuga jeongsie ogo Kayeongi neutge wasseoyo. Minsu came on time and Kayoung was late.
~고요 | 52 | ~, too | 민수가 늦어요. 음식도 가지고 오지 않고요. Minsuga neujeoyo. Eumsikdo gajigo oji ankoyo. Minsu's late. He also didn't bring any food.
~고 있다 | | Is ~ing | 그 아이가 놀고 있어요 Geu aiga nolgo isseoyo. The child is playing.
A고도B | | Although A, B | 서두르고도 늦었어요. Seodureugodo neujeosseoyo. Although I hurried, I was late.
A고는B | – | Because of A, B (unexpectedly) happened | 그 책을 읽고는 성격이 달라졌어요. Geu chaegeul ilgoneun seonggyeogi dallajyeosseoyo. Since reading that book, his personality changed completely.
A고서B | | A, then B | 지하철을 타고서 늦었어요. Jihacheoreul tagoseo neujeosseoyo. I took the subway and then I was late.
A고서는B | | While A, B (can't happen) | 신발을 신고서는 집에 들어올 수 없어요. Sinbareul singoseoneun jibe deureool su eobseoyo. You can't come in the house while wearing your shoes.
A고서야B | | A, then B | 우리가 지하철을 타고서야 정시에 올 수 있었어요. Uriga jihacheoreul tagoseoya jeongsie ol su isseosseoyo. We took the subway, and then we were able to arrive on time (at least partly because of taking the subway).
A고 나서B | | A and then B | 저녁을 먹고 나서 담배를 피웠어요. Jeonyeogeul meokgo naseo dambaereul piwosseoyo. I ate dinner and then had a cigarette.
A고자B | | I plan to A, so B | 늦지 않게 오고자 집에서 일찍 출발했어요. Neutji anke ogoja jibeseo iljjik chulbalhaesseoyo. I left the house early so as not to be late.
A고 보니 (까)B | | I did A and then discovered that B | 지하철에 타고 보니 택시보다 더 느렸어요. Jihacheore tago boni taeksiboda deo neuryeosseoyo. I took the subway, but found it was slower than a taxi.
A고 해서B | | A, so B | 열쇠를 일어버리고 해서 늦게 왔어요. Yeolsoereul ireobeorigo haeseo neutge wasseoyo. I lost my keys (among other things), so I arrived late.
~고 말고(요) go malgo(yo) | 37 | Of course, ~ | 늦고 말고요. 민수는 일 때문에 정시에 오지 못 해요. Neutgo malgoyo. Minsuneun il ttaemune jeongsie oji mot haeyo. Of course he was late. Because of his job, he can't come on time.
~고 말겠다 go malgetda | | I (definitely) plan to ~ | 이번에는 꼭 정시에 오고 말겠어요. Ibeoneneun kkok jeongsie ogo malgesseoyo. This time I'm definitely going to come on time.
~고 말았다 go maratda | | (Unfortunately), ~ happened. | 정시에 온다는 것이 결국 늦게 오고 말았어요. Jeongsie ondaneun geosi gyeolguk neutge ogo marasseoyo. I meant to come on time, but (unfortunately) I ended up being late.
~고는/곤 하다 goneun/gon hada | | Habitually do ~ | 민수가 대학교 때 자주 수업에 늦게 오곤 했어요. Minsuga daehakgyo ttae jaju sueobe neutge ogon haesseoyo. During university, Minsu was habitually late for class.
##
And: Putting Things Together
The Basics:
민수가 정시에 오고 가영이 늦게 왔어요.
Minsuga jeongsie ogo Kayeongi neutge wasseoyo.
Minsu came on time and Kayoung was late.
고 is the workhorse when it comes to combining things in Korean. However, it's a bit of a prima donna, and you can't just stick it in any sentence and expect it to do its job. See the "Take note" section below on how to properly use 고.
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Action verbs ending in a vowel | 하다 hada | 하고 hago
Action verbs ending in a consonant | 먹다 meokda | 먹고 meokgo
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 yeppeuda | 예쁘고 yeppeugo
Descriptive verbs (adjectives) ending in a consonant | 작다 jakda | 작고 jakgo
Nouns ending in a vowel | 남자 namja | 남자이고 namjaigo
Nouns ending in a consonant | 물 mul | 물이고 muligo
TAKE NOTE
First of all, the subjects of both clauses, A and B, must be the same you're using one of these three verbs to end A: 가다, 오다 or 일어나다 ireonada. In these cases you should use 어서 instead of 고. You can use 가다, 오다 or 일어나다 with 고 if the subjects of both clauses are different from each other.
EXAMPLE SENTENCES
영화를 보고 쇼핑을 하러 강남에 갔어요.
Yeonghwareul bogo syopingeul hareo Gangname gasseoyo.
I went to Gangnam to see a movie and go shopping.
그 수업이 재미없고 어려워요.
Geu sueobi jaemieopgo eoryeowoyo.
That class is boring and difficult.
내일 한국어 공부하고 책을 읽을 거예요.
Naeil hangukeo gongbuhago chaegeul ilgeul geoyeyo.
Tomorrow I'm going to study Korean and read a book.
민수가 가고 가영이 왔어요.
Minsuga gago Kayeongi wasseoyo.
Minsu left and Kayoung came.
Note that in the last example I used 가다 but changed the subject. If Minsu left and then went to see a movie, you would say 민수가 가서 영화를 봤어요 Minsuga gaseo yeonghwareul bwasseoyo.
In all the variations of 고 (고요, 고도, 고서, etc.), the same rules apply: same subjects and no 가다, 오다, or 일어나다 unless the subject of the sentence is different.
민수가 항상 늦어요. 음식도 가지고 오지 않고요.
Minsuga hangsang neujeoyo. Eumsikdo gajigo oji ankoyo.
Minsu is always late. He also never brings any food.
Sometimes you say something and then remember something else you wanted to say. In this case, you can add a second sentence using 고요.
HOW IT'S FORMED
고 plus the polite ending 요. Of course, if you want to use this in 반말, you can; just drop the 요.
HOW IT'S CONJUGATED
It's added to the verb at the end of a second sentence, the one where you mention your afterthought.
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했고요 | 하고요 | 할 거고요
Action verbs ending in a consonant | 먹다 | 먹었고요 | 먹고요 | 먹을 거고요
TAKE NOTE
You don't have to bother conjugating this expression in the past or future tenses if you don't want to; Koreans often don't. See the last example below.
If you have an object in the second clause (like "Korean" and "a book" in the example below) you should add 도 (too) at the end of it instead of 을/를. It sounds more natural that way.
It may or may not be strictly correct, but I've often heard Koreans use this expression even if the afterthought comes long afterward or is implied. For instance, a teacher will tell her students to "그림 그리세요 geurim geuriseyo" (draw a picture) and then five minutes later, "색칠도 하고요 saekchil do hagoyo" (also, color it).
EXAMPLE SENTENCES
그 수업이 재미없어요. 어렵고요.
Geu sueobi jaemieobseoyo. Eoryeopgoyo.
That class is boring. It's also difficult.
내일 한국어 공부할 거예요. 책도 읽을 거고요.
Naeil hanguleo gongbuhal geoyeyo. Chaekdo ilgeul geogoyo.
내일 한국어 공부할 거예요. 책도 읽고요.
Naeil hangukeo gongbuhal geoyeyo. Chaekdo ilkgoyo.
Tomorrow I'm going to study Korean. I'm also going to read a book.
민수가 늦은 데다가 음식도 가지고 오지 않았어요.
Minsuga neujeun dedaga eumsikdo gajigo oji anasseoyo.
Minsu was late. He also didn't bring any food.
These expressions are for those who like to cook. They can be used to add one thing to another thing. 에다(가) can be used with nouns to mean "adding one thing to another thing" or "putting something in a specific location."
HOW IT'S FORMED
다가 is a very versatile expression which is explained in detail on page . The 에 and (으)ㄴ/는 데 before it simply mark the thing to which something else is being added: there's no other meaning behind it. The 가 part of 다가 is optional.
HOW IT'S CONJUGATED
The difference between the two noun forms is that while 물인 데다가 murin dedaga means "It's water, and on top of that...," 물에다가 muredaga means to add something to the water.
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 한 데다가 han dedaga | 하는 데다가 haneun dedaga | 하는 데다가
Action verbs ending in a consonant | 먹다 | 먹은 데다가 meogeun dedaga | 먹는 데다가 meokneun dedaga | 먹는 데다가
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁜 데다가 yeppeun dedaga | 예쁜 데다가 | 예쁜 데다가
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작은 데다가 jageun dedaga | 작은 데다가 | 작은 데다가
Nouns ending in a vowel | 남자 | 남자인 데다가 namjain dedaga 남자에다가 namja-e-daga | 남자인 데다가 남자에다가 | 남자인 데다가 남자에다가
Nouns ending in a consonant | 물 | 물인 데다가 murin dedaga 물에다가 muredaga | 물인 데다가 물에다가 | 물인 데다가 물에다가
TAKE NOTE
The subjects of both clauses should be the same.
The expressions here and (으)ㄹ 뿐만 아니라/더러 (eu)r ppunman anira/deoreo have similar meanings, but they are used in different situations. See page 55 for when the 뿐 expressions can and can't be used.
EXAMPLE SENTENCES
■ A ㄴ/은/는 데다(가) n/eun/neun deda(ga) B
맛있어 보이는 데다가 냄새도 좋아요.
Masisseo boineun dedaga naemsaedo joayo.
It not only looks delicious, but it smells good too.
그는 축구를 잘 하는 데다가 농구도 잘 해요.
Geuneun chukgureul jal haneun dedaga nonggudo jal haeyo.
He is not only good at soccer, but also at basketball.
■ A 에다(가) eda(ga) B
열에다가 스물을 더하면 서른이에요.
Yeoredaga seumureul deohamyeon seoreun-i-e-yo.
Ten plus twenty is thirty.
그것을 트럭에다가 실었어요.
Geugeoseul teureogedaga sireo-sseoyo.
He put it in the truck.
큰 그릇에다 밀가루와 물을 넣으세요.
Keun geureuseda milgaruwa mureul neo-eu-seyo.
Put flour and water into a big bowl.
민수가 늦었을 뿐만 아니라 음식도 가지고 오지 않았어요.
Minsuga neujeosseul ppunman anira eumsikdo gajigo oji anasseoyo.
민수가 늦었을 뿐더러 음식도 가지고 오지 않았어요.
Minsuga neujeosseul ppundeoreo eumsikdo gajigo oji anasseoyo.
Minsu was late. He also didn't bring any food.
(으)ㄹ 뿐만 아니라 is a very common expression which is quite simple to use, and it essentially means "not just A, but also B". (으)ㄹ 뿐더러 is less common than ㄹ 뿐만 아니라 and not really used these days except by older people, but you may as well learn them together since they're similarly formed and have exactly the same meaning.
HOW IT'S FORMED
We're going to discuss 뿐 later on, on page . It's an expression used to limit something: to say that there was only one instance of it. 만 is another limiting expression which means "only." 아니라 is an expression on its own (see page ) which negates what's before it and asserts what's after it.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했을 뿐만 아니라 haesseul ppunman anira
했을 뿐더러 haesseul ppundeoreo
| 할 뿐만 아니라 hal ppunman anira
할 뿐더러 hal ppundeoreo
| 할 뿐만 아니라
할 뿐더러
Action verbs ending in a consonant | 먹다 | 먹었을 뿐만 아니라 meogeosseul ppunman anira
먹었을 뿐더러 meogeosseul ppundeoreo
| 먹을 뿐만 아니라 meogeul ppunman anira
먹을 뿐더러 meogeul ppundeoreo
| 먹을 뿐만 아니라
먹을 뿐더러
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤을 뿐만 아니라 yeppeosseul ppunman anira
예뻤을 뿐더러 yeppeosseul ppundeoreo
| 예쁠 뿐만 아니라 yeppeul ppunman anira
예쁠 뿐더러 yeppeul ppundeoreo
| 예쁠 뿐만 아니라
예쁠 뿐더러
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았을 뿐만 아니라 jagasseul ppunman anira
작았을 뿐더러 jagasseul ppundeoreo
| 작을 뿐만 아니라 jageul ppunman anira
작을 뿐더러 jageul ppundeoreo
| 작을 뿐만 아니라
작을 뿐더러
Nouns ending in a vowel | 남자 | 남자 뿐만 아니라 namja ppunman anira | 남자 뿐만 아니라 | 남자 뿐만 아니라
Nouns ending in a consonant | 물 | 물 뿐만 아니라 mul ppunman anira | 물 뿐만 아니라 | 물 뿐만 아니라
TAKE NOTE
(으)ㄹ 뿐더러 (eu)r ppundeoreo is not used after nouns.
Since these expressions are used for adding two things together, you can't have a positive statement in A and a negative one in B. Your expressions must either be both positive or both negative. In fact, 뿐더러 can't be used with positive statements at all.
그 애는 예쁠 뿐만 아니라 공부도 잘해요.
Geu aeneun yeppeul ppunman anira gongbudo jalhaeyo. (acceptable)
That child is not only pretty, but she also studies well.
그 애는 공부를 못할 뿐만 아니라 놀기도 못해요.
Geu aeneun gongbureul mothal ppunman anira nolgido mothaeyo. (acceptable)
Not only does that child not study well, but he can't even play.
그 애는 예쁠 뿐더러 착해요.
Geu aeneun yeppeul ppundeoreo chakhaeyo. (ungrammatical)
That child is both pretty and kind.
그 애는 공부를 잘할 뿐더러 놀기도 잘해요.
Geu aeneun gongbureul jalhal ppundeoreo nolgido jalhaeyo. (ungrammatical)
That child not only studies well, but also plays well.
그 애는 공부를 못 할 뿐더러 예쁘지도 않아요.
Geu aeneun gongbureul mothal ppundeoreo yeppeujido anayo. (acceptable)
Not only does that child not study well, she's not even pretty.
There is commonly a 도 after the noun in clause B, and that 도 means "too." See the examples below.
EXAMPLE SENTENCES
맛있어 보일 뿐만 아니라 냄새도 좋아요.
Mas-isseo bo-il ppunman anira naemsaedo joayo.
맛있어 보일 뿐더러 냄새도 좋아요.
Mas-isseo bo-il ppundeoreo naemsaedo joayo.
It not only looks delicious, but it smells good too.
그는 축구 뿐만 아니라 농구도 잘 해요.
Geuneun chukgu ppunman anira nonggudo jal haeyo.
축구를 잘 할 뿐더러 농구도 잘 해요.
Chukgureul jal hal ppundeoreo nonggudo jal haeyo.
He is not only good at soccer, but also at basketball.
민수가 물은 물론 음식도 가지고 오지 않았어요.
Minsuga mureun mullon eumsikdo gajigo oji anasseoyo.
Minsu didn't bring any water or food.
물론 means "of course," but not here. Taken as a whole, this phrase is yet another way to express "not just A, but also B." In other words, it's very much like (으)ㄹ 뿐만 아니라 (eu)r ppunman anira, but much less common.
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Nouns ending in a vowel | 남자 | 남자는 물론 namjaneun mullon
Nouns ending in a consonant | 물 | 물은 물론 mureun mullon
TAKE NOTE
Don't confuse 물은 with 물론 ("of course"). 물론 by itself is more commonly found at the beginning or sometimes at the end of sentences, and is an adverb that stands on its own. You can tell them apart by the (으)ㄴ/는 before 물론 in this expression and by context.
This is interchangeable with 뿐만 아니라 ppunman anira and 뿐더러 ppundeoreo (see page 55), but it's fairly uncommon.
EXAMPLE SENTENCES
맛있어 보이는 것은 물론 냄새도 좋아요.
Mas-isseo bo-i-neun geoseun mullon naemsaedo joayo.
It not only looks delicious, but smells good too.
그는 축구는 물론 농구도 잘 해요.
Guneun chukguneun mullon nonggudo jal haeyo.
He is not only good at soccer, but also at basketball.
그는 축구 말고 농구도 잘 해요.
Geuneun chukgu malgo nonggudo jal haeyo.
그는 축구 외에 농구 잘 해요.
Geuneun chukgu oe-e nonggu jal haeyo.
He is not only good at soccer, but also at basketball.
I'm putting these two together because they're quite similar. A말고 B and A외에B mean "not just A, but B."
HOW IT'S FORMED
말다 is a verb that means "to not be," and appears in many other expressions such as 할까 말까 (to do it or not to do it). 외 means "outside" and so 외에 refers to the something outside another thing. An equivalent English expression would be, "other than" or "aside from that." The person in the example sentences above is good at soccer (and other than/aside from that) basketball. 도 is used here as an intensifier; see above for how it alters the expression.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 한 것(이) 말고(도) han geos(i) malgo(do)
한 것(이) 외에(도) han geos(i) oe-e(do)
| 하는 것(이) 말고(도) haneun geos(i) malgo(do)
하는 것(이) 외에(도) haneun geos(i) oe-e(do)
| 하는 것(이) 말고(도) 할 것(이) 말고(도) hal geos(i) malgo(do)
하는 것(이) 외에(도) 할 것(이) 외에(도) hal geos(i) oe-e(do)
Action verbs ending in a consonant | 먹다 | 먹은 것(이) 말고(도) meogeun geos(i) malgo(do)
먹은 것(이) 외에(도) meogeun geos(i) oe-e(do)
| 먹는 것(이) 말고(도) meokneun geos(i) malgo(do)
먹는 것(이) 외에(도) meokneun geos(i) oe-e(do)
| 먹는 것(이) 말고(도) 먹을 것(이) 말고(도) meogeul geos(i) malgo(do)
먹는 것(이) 외에(도) 먹을 것(이) 외에(도) meogeul geos(i) oe-e(do)
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁜 것(이) 말고(도) yeppeun geos(i) malgo(do)
예쁜 것(이) 외에(도) yeppeun geos(i) oe-e(do)
| 예쁜 것(이) 말고(도) 예쁜 것(이) 외에(도) | 예쁜 것(이) 말고(도) 예쁜 것(이) 외에(도)
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작은 것(이) 말고(도) jageun geos(i) malgo(do)
작은 것(이) 외에(도) jageun geos(i) oe-e(do)
| 작은 것(이) 말고(도) 작은 것(이) 외에(도) | 작은 것(이) 말고(도) 작은 것(이) 외에(도)
Nouns ending in a vowel | 남자 | 남자 말고(도) namja malgo(do)
남자 외에(도) namja oe-e(do)
| 남자 말고(도) 남자 외에(도) | 남자 말고(도) 남자 외에(도)
Nouns ending in a consonant | 물 | 물 말고(도) mul malgo(do)
물 외에(도) mul oe-e(do)
| 물 말고(도) 물 외에(도) | 물 말고(도) 물 외에(도)
TAKE NOTE
If you put 도 on the end, the expression changes a little. In this case your B clause should be something very general and broader than your A clause.
EXAMPLE SENTENCES
■ A말고/외에 B
민수가 똑똑하는 것이 말고 잘 생긴 것이에요.
Minsuga ttokttokhaneun geosi malgo jal saenggin geosieyo.
민수가 똑똑하는 것이 외에 잘 생긴 것이에요.
Minsuga ttokttokhaneun geosi oe-e jal saenggin geosieyo.
Minsu is not just smart, but he is also handsome.
맛있어 보인 것이 말고 냄새도 좋은 것이에요.
Mas-isseo boin geosi malgo naemsaedo joeun geosieyo.
맛있어 보인 것이 외에 냄새도 좋은 것이에요.
Mas-isseo boin geosi oe-e naemsaedo joeun geosieyo.
It not only looks delicious, but it smells good too.
■ A말고도/외에도 B
민수가 똑똑하는 것이 말고도 좋은 점이 많아요.
Minsuga ttokttokhaneun geosi malgodo joeun jeomi manayo.
민수가 똑똑하는 것이 외에도 좋은 점이 많아요.
Minsuga ttokttokhaneun geosi oe-e-do joeun jeomi manayo.
Minsu is not just smart, but he has many good qualities too.
맛있어 보인 것이 말고도 음식이 많아요.
Mas-isseo boin geosi malgodo eumsigi manayo.
맛있어 보인 것이 외에도 음식이 많아요.
Mas-isseo boin geosi oe-e-do eumsigi manayo.
It not only looks delicious, but there's also lots of it.
그는 축구 말고도 스포츠를 다 잘 해요.
Geuneun chukgu malgodo seupo-cheureul da jal haeyo.
그는 축구 외에도 스포츠를 다 잘 해요.
Geuneun chukgu oe-e-do seupo-cheureul da jal haeyo.
He is not only good at soccer, but also at all sports.
##
Linking Sentences: Or
This is the most easy, simple, basic way to say "or" in Korean. It goes between two nouns and means "A or B."
커피나 차
keopina cha
coffee or tea
지하철이나 택시
jihacheolina taeksi
subway or taxi
야구나 축구를 할까요?
Yaguna chukgureul halkkayo?
Shall we play baseball or soccer?
가영이나 혜진을 만났어요?
Kayoungina Hyejineul mannasseoyo?
Did you meet Kayoung or Hyejin?
"스타 워즈"나 "스타 트렉" 을 보고 싶어요.
Seuta Wojeuna Seuta Teurekeul bogo sipeoyo.
I want to watch Star Wars or Star Trek.
If you just use (이)나 by itself after a noun without another noun, it means "or something." You can also use (이)나마 (i)nama, which has the same meaning and implies that whatever is being offered isn't the best choice, but it'll do.
커피나 마실래요?
Keopina masillaeyo?
Shall we drink some coffee or something?
책이나 읽고 싶어요.
Chaegina ilggo sipeoyo.
I want to read a book or something.
책이 없으니까 민수 책이나마 보세요.
Chaegi eobseunikka Minsu chaeginama boseyo.
Since you have no book, you can look at Minsu's.
김 선생님을 찾지 못하면 이메일이나마 쓰세요.
Kim seonsaengnimeul chatji mothamyeon imeilinama sseuseyo.
If you can't find Mr. Kim, you can send an email or something.
Here are some commonly used expressions with (이)나:
누구나
nuguna
anyone
언제나
eonjena
any time
어디서나
eodiseona
anywhere (used only with 가다, 오다 and other verbs of motion)
어디에서나
eodi-e-seona
anywhere
무엇이나
mueosina
anything
아무거나
amugeona
anything
누구나 갈 수 있어요.
Nuguna gal su isseoyo.
Anyone can go.
언제나 할 수 있어요.
Eonjena hal su isseoyo.
We can do it any time.
어디서나 하면 돼요.
Eodiseona hamyeon dwaeyo.
We can do it anywhere.
어디에서나 하면 돼요.
Eodi-e-seona hamyeon dwaeyo.
We can do it anywhere.
아무거나 주세요.
Amugeona juseyo.
Just give me anything.
This is the most basic way to say "or" when you're talking about verbs. Put it after each of two verbs to mean "A or B." You don't need any past tense markers on your verbs; the verb at the end of your sentence tells people all they need to know about when it happened. What you're really doing here is changing a verb into a noun using (으)ㄴ/는 것, adding 이나 (see page 60), and then shrinking the whole expression to 거나.
수영하거나 자전거를 타거나
suyeonghageona jajeongeoreul tageona
go swimming or go biking
영화를 보거나 쇼핑을 하거나
yeonghwareul bogeona syopingeul hageona
see a movie or go shopping
커피를 마시거나 식사를 하거나 할까요?
Keopireul masigeona siksareul hageona halkkayo?
Shall we drink coffee or eat dinner?
지금 은행에 가거나 내일 인터넷 뱅킹을 하거나 해야 해요.
Jigeum eunhaenge gageona naeil inteonet baengkingeul hageona haeya haeyo.
I have to either go to the bank now or log on to online banking tomorrow.
You can use A거나 말거나 geona malgeona B to say that whether or not you do A, B will happen.
서두르거나 말거나 늦을 거예요.
Seodureugeona malgeona neujeul geoyeyo.
Whether you hurry or not, you'll be late.
아이들이 듣거나 말거나 선생님은 계속 말했어요.
A-i-deuri deutgeona malgeona seonsaengnimeun gyesok malhaesseoyo.
Whether or not the children were listening, the teacher kept talking.
그녀가 예쁘거나 말거나, 저는 그녀를사랑해요.
Geunyeoga yeppeugeona malgeona, jeoneun geunyeoreulsaranghaeyo.
Whether or not she's beautiful, I love her.
This can best be translated as "or something" as in "Do you want to get some coffee or something?" It implies that "something" is not necessarily the first choice, but it will do.
커피라도 먹을래요?
Keopirado meogeullaeyo?
Do you want to get some coffee or something? (I don't really care if we get coffee or something else.)
이번 주말에 부산이라도 갈까요?
Ibeon jumare Busanirado galkkayo?
Shall we go to Busan or something this weekend? (I'm not that excited about Busan, but I'd like to go somewhere.)
이거라도 주세요.
Igeorado juseyo.
Just give me this one. (It's not my first choice, but it'll do.)
It can also be used after question words like 누구, 언제, nugu, eonje, or 어디 eodi or after 아무 amu to mean "any."
누구라도
nugurado
anyone
아무라도
amurado
anyone
언제라도
eonjerado
any time
어디라도
eodirado
anywhere (used only with 가다, 오다 and other verbs of motion)
어디에서라도
eodieseorado
anywhere
무엇이라도
mueosirado
whatever
아무거라도
amugeorado
anything
누구라도 갈 수 있어요.
Nugurado gal su isseoyo.
Anyone can go.
언제라도 할 수 있어요.
Eonjerado hal su isseoyo.
We can do it any time.
어디에서라도 하면 돼요.
Eodi-e-seorado hamyeon dwaeyo.
We can do it anywhere.
아무라도 주세요.
Amurado juseyo
Just give me anything.
무엇이라도 주세요.
Mueosirado juseyo.
Just give me whatever.
This goes after nouns or verbs and means that either A or B is fine. It's the same as 거나. When used after verbs, it is always in the form A든지 B든지 and can be used after 았/었/였 if your sentence is in the past tense. After nouns, you should use (이)든지: 남자든지 여자든지 namjadeunji yeojadeunji or 물이든지 불이든지 murideunji burideunji.
You can also shorten the entire expression to A든 B든 or use A든지 안 A든지, A든지 말든지, or A든지 못 A든지.
커피를 마시든지 식사를 하든지 할까요?
Keopireul masideunji siksareul hadeunji halkkayo?
커피를 마시든 식사를 하든 할까요?
Keopireul masideun siksareul hadeun halkkayo?
Shall we drink coffee or eat dinner?
커피든지 차든지 마실래요?
Keopideunji chadeunji masillaeyo?
Let's drink some coffee or tea.
비싸든지 싸든지 사야 돼요.
Bissadeunji ssadeunji saya dwaeyo.
싸든지 말든지 사야 돼요.
Ssadeunji maldeunji saya dwaeyo.
I have to buy it no matter how much it costs. (whether it's cheap or expensive)
네가 가든지 못 가든지, 내 친구는 나와 있을 거예요.
Nega gadeunji mot gadeunji, nae chinguneun nawa isseul geoyeyo.
Whether or not you can go, my friend will be with me.
Just like 거나 and 라도, 든지 can be added to the end of question words or 아무 to form words that mean "any."
누구든지
nugudeunji
anyone
언제든지
eonjedeunji
any time
어디든지
eodideunji
anywhere (used only with 가다, 오다 and other verbs of motion)
어디에서든지
eodieseodeunji
anywhere
무엇이든지
mueosideunji
anything
누구든지 갈 수 있어요.
Nugudeunji gal su isseoyo.
Anyone can go.
언제든지 할 수 있어요.
Eonjedeunji hal su isseoyo.
We can do it any time.
어디에서든지 하면 돼요.
Eodieseodeunji hamyeon dwaeyo.
We can do it anywhere.
무엇이든지 주세요.
Mueosideunji juseyo.
Just give me whatever.
This is almost exactly the same as 거나 above, but it's used in situations where A and B are either examples or are about important issues, or are examples used while talking about an important issue. So you can use it to talk about how to solve the city's traffic problem, for example, but it would be strange to use it to talk about your weekend plans. You could, however, use it to give examples of things you normally do on the weekend.
Add an indirect speech particle (see page ) to 거나, and repeat the particle plus 거나 after every example.
수영한다거나 자전거를 탄다거나
suyeonghandageona jajeongeoreul tandageona
go swimming or go biking
영화라거나 쇼핑이라거나
yeonghwarageona syopingirageona
a movie or shopping
소설이란다거나 만화란다거나 책을 다른 종류를 다 좋아해요.
Soseorirandageona manhwarandageona chaegeul dareun jongnyureul da joahaeyo.
I like all kinds of books like novels or comic books.
패스트 푸드를 끊는다거나 운동을 더 열심히 한다거나 하면 콜레스케롤 수치를 낮출 수 있어요.
Paeseuteu pudeureul kkeunneundageona undongeul deo yeolsimhi handageona hamyeon kolleseuterol suchi reul natchul su isseoyo.
If you quit eating fast food or if you exercise more, you can lower your cholesterol.
You may recognize 라, 는다, ㄴ다, and 다 as indirect speech endings. In this case they aren't used to talk about something someone else said. However, they still attach to words the same way: (이)라든가 after nouns, 는/ㄴ다든가 after action verbs, and 다든가 after descriptive verbs. (See page for more on indirect speech endings.)
라든가 and its companions mean "or" and are used to talk about two choices. In the case of (이)라든가, B is normally followed by an explanation of how exactly A and B are similar: "비빔밥이라든가 냉면이라든가 한국 음식 같은 Bibimbapiradeunga naengmyeoniradeunga hanguk eumsig gateun" ("Korean foods like bibimbap or naengmyeon") 는다든가, ㄴ다든가 and 다든가, on the other hand, are usually followed by 하다, which can then be conjugated however you'd like.
While (이)라든가 is normally used only after A, 는다든가/ㄴ다든가/다든가 follows both A and B.
비빔밥이라든가 불고기 같은 대표적인 한국 음식을 먹어 보세요.
Bibimbapiradeunga bulgogi gateun daepyojeogin hanguk eumsigeul meogeo boseyo.
Please try a representative (famous) Korean food like bibimbap or bulgogi.
커피라든가 차 같은 따뜻한 것을 마시고 싶어요.
Keopiradeunga cha gateun ttatteuthan geoseul masigo sipeoyo.
I want to drink something warm like coffee or tea.
스트레스가 쌓일 때 산책이라든가 수영이라든가 운동을 해요.
Seuteuleseuga ssahil ttae sanchaegiradeunga suyeongiradeunga undongeul haeyo.
When I'm stressed, I like to do some exercise like walking or swimming.
스트레스가 쌓일 때 산책을 한다든가 음악을 듣는다든가 해요.
Seuteuleseuga ssahil ttae sanchaegeul handadeunga eumageul deunneundadeunga haeyo.
When I'm stressed, I like to take a walk or listen to music.
방학 때 여행을 간다든가 친구와 논다든가 하고 싶어요.
Banghak ttae yeohaengeul gandadeunga chinguwa nondadeunga hago sipeoyo.
During vacation I want to travel or hang out with friends.
This is a shorter version of 거나, but it's used to mean "whether you do A or B." A and B should be opposites and 건 should follow each of them.
지하철을 타건 택시를 타건 늦을 거예요.
Jihacheoreul tageon taeksireul tageon neujeul geoyeyo.
Whether you take the subway or a taxi, you'll be late.
한국어를 잘 하건 못 하건 최선을 다 해야 돼요.
Hangukeoreul jal hageon mot hageon choeseoneul da haeya dwaeyo.
Whether you're good at Korean or not, you must do your best.
돈이 있건 없건 더 어려운 사람을 도와 주었으면 좋겠어요.
Doni itgeon eobsgeon deo eoryeoun sarameul dowa jueosseumyeon jokesseoyo.
Whether you have money or not, you should help those who need it.
##
Linking Sentences—Cause and Effect: The Big Three
There are three extremely common expressions that all essentially mean "so," and they are used in a few ways. The big three are ㄴ/는데, 아/어/여서 n/neunde, a/eo/yeoseo, and (으)니까 (eu)nikka. You'll see them used in three ways. One is in the middle of a sentence, one is at the end, and the other is roughly at the beginning tagging along with 그렇다 geureota. Let's start by talking about each expression and when to use it.
차가 막혀서 늦었어요.
Chaga makyeoseo neujeosseoyo.
There was a lot of traffic, so I was late.
This expression essentially means "so" and is extremely important. Not only is it used all the time, but it also forms the basis for many other expressions you'll want to learn.
HOW IT'S FORMED
아/어/여서 is an expression all on its own; (이)라서 is a combination of the indirect speech particle (이)라 and 아서.
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Action or descriptive verbs with 오 or 아 | 작다 | 작아서 jagaseo
Action or descriptive verbs with 어, 우, 으 or 이 | 먹다 | 먹어서 meogeoseo
하다 | 하다 | 해서 haeseo
Nouns ending in a vowel | 남자 | 남자라서 namjaraseo
Nouns ending in a consonant | 물 | 물이라서 muriraseo
TAKE NOTE
This can be used only in statements (no questions, suggestions, or commands). In other words, if your B clause is a question, suggestion or command, you should use a different expression. (으)니(까) is a good one; see page 67.
It's also used to replace 고 (see page 51) in expressions where 고 follows a verb that means "to come" or "to go" (오다, 가다, 들어오다, etc...) See the example below about shopping and Dongdaemun.
EXAMPLE SENTENCES
■ 아/어/여서
아침을 안 먹어서 지금 배고파요.
Achimeul an meogeoseo jigeum baegopayo.
I didn't eat breakfast, so I'm hungry.
공부를 열심히 해서 시험에 100점을 받았어요.
Gongbureul yeolsimhi haeseo siheome baekjeomeul badasseoyo.
I studied hard, so I got 100 on the test.
동대문에 가서 쇼핑을 할 거예요.
Dongdaemune gaseo syopingeul hal geoyeyo.
I'm going to Dongdaemun, so I can go shopping.
■ (이)라서
저는 외국인이라서 한국어를 잘 못 해요.
Jeoneun oeguginiraseo hangukeoreul jal mot haeyo.
I'm a foreigner, so I don't speak Korean well.
오늘 생일이라서 케이크 먹을 거예요.
Oneul saengiriraseo keikeu meogeul geoyeyo.
Today's my birthday, so I'm going to eat cake.
차가 람보르기니라서 아주 비싸요.
Chaga Ramboreuginiraseo aju bissayo.
The car is a Lamborghini, so it's very expensive.
차가 막혔으니까 늦었어요.
Chaga makyeosseunikka neujeosseoyo.
Traffic was heavy, so I was late.
This is another common and versatile way to say "so."
HOW IT'S FORMED
You will see this expression written two ways: 니 and 니까. They mean exactly the same thing, but 니 is only used formally or in writing.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했으니까 haesseunikka | 하니까 hanikka | 하니까 할 거니까 hal geonikka
Action verbs ending in a consonant | 먹다 | 먹었으니까 meogeosseunikka | 먹으니까 meogeunikka | 먹으니까 먹을 거니까 meogeul geonikka
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤으니까 yeppeosseunikka | 예쁘니까 yeppeunikka | 예쁘니까 예쁠 거니까 yeppeul geonikka
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았으니까 jagasseunikka | 작으니까 jageunikka | 작으니까 작을 거니까 jageul geonikka
Nouns ending in a vowel | 남자 | 남자였으니까 namjayeosseunikka | 남자이니까 namjainikka | 남자이니까 남자일 거니까 namjail geonikka
Nouns ending in a consonant | 물 | 물이었으니까 murieosseunikka | 물이니까 mulinikka | 물이니까 물 일거니까 mul ilgeonikka
TAKE NOTE
니(까) can be used in any kind of sentence and in any tense, though it isn't usually used when you're talking about yourself. If you feel narcissistic at the moment, you'd be better off using 어서.
Be aware that there is another expression, 니, which is used in 반말 at the end of questions: 점심을 먹었니? That's different. This 니 is in the middle of a sentence.
Also, 습니까 is not the same 니까 used here. 습니까 is the super-polite ending for a question (see page ). That 니까 always goes after 습, while this one never does.
있다 and 없다 can be conjugated either as 있니(까) inni(kka) and 없니(까) eobsni(kka) or 있으니(까) isseuni(kka) and 없으니(까) eobseuni(kka). Koreans do it both ways, so just use whichever you think sounds better.
EXAMPLE SENTENCES
어머니의 생신이니까 파티에 갈 수 없어요.
Eomeoniui saengsininikka patie gal su eobseoyo.
It's my mother's birthday, so I can't go to the party.
이 신발은 너무 작으니까 큰 것으로 바꿔 주세요.
I sinbareun neomu jageunikka keun geoseuro bakkwo juseyo.
These shoes are too small, so could you please change it to a bigger size?
이 회의가 아주 중요한데 민수가 늦었어요.
I hoeuiga aju jungyohande Minsuga neujeosseoyo.
This meeting is very important, but Minsu was late.
(으)ㄴ/는데 functions much like an English semicolon. Clause A explains the background for clause B, which can be a statement, question, suggestion or command: this expression works with any kind of sentence. It can also be used to contrast two situations. It doesn't always mean "so," but it is often used that way and functions similarly to the expressions above, so I've put it in this section.
HOW IT'S CONJUGATED
ㄴ/은데 n/eunde is used after descriptive verbs and 는데 neunde after action verbs or past tense forms. If you want to use this expression after a noun, add 이 to the end of the noun and then conjugate it as if it were a descriptive verb.
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했는데 haenneunde | 하는데 haneunde | 할 건데 hal geonde
Action verbs ending in a consonant | 먹다 | 먹었는데 meogeonneunde | 먹는데 meokneunde | 먹을 건데 meogeul geonde
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤는데 yeppeonneunde | 예쁜데 yeppeunde | 예쁠 건데 yeppeul geonde
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았는데 jaganneunde | 작은데 jageunde | 작을 건데 jageul geonde
Nouns ending in a vowel | 남자 | 남자였는데 namjayeonneunde | 남자인데 namjainde | 남자일 건데 namjail geonde
Nouns ending in a consonant | 물 | 물이었는데 murieonneunde | 물인데 murinde | 물일 건데 muril geonde
TAKE NOTE
There is another expression, (으)ㄴ/는 데, which is totally different and has to do with intentions. It's covered on page . You can tell it apart from (으)ㄴ/는데 by the space in the middle, but sometimes this space is omitted, so look for context.
EXAMPLE SENTENCES
어머니의 생신인데 무슨 선물이 좋을까요?
Eomeoniui saengsininde museun seonmuri joeulkkayo?
It's my mother's birthday; what kind of present would be good?
자고 있었는데 갑자기 전화가 왔어요.
Jago isseonneunde gapjagi jeonhwaga wasseoyo.
I was sleeping when the phone suddenly rang.
Ending sentences with 어서, and 니까 (and 거든!) Speaking
These mean essentially the same thing when used at the end of a sentence. You can use them to explain the reason for an action. It's much like answering a question in English with "Because." They're more or less all interchangeable, but 서요 is quite uncommon while you'll hear the others all the time. 거든 is also usually used for commonplace situations about daily life, while the other two can be (but are not always) somewhat more formal.
거든 itself, in the middle of a sentence, has a slightly different meaning than 아/어/여서 and 니까; see pages 66, 67 and . Here however, they are the same.
A: 왜 비빔밥을 또 시켰어요? Wae bibimbapbeul tto sikyeosseoyo? Why did you order bibimbap again?
B: 맛있으니까요. Masisseunikkayo. Because it's delicious.
__________
A: 왜 비빔밥을 또 시켰어요? Why did you order bibimbap again?
B: 맛있거든요. Masissgeodeunyo. Because it's delicious.
__________
A: 왜 비빔밥을 또 시켰어요? Why did you order bibimbap again?
B: 맛있어서요. Masisseoseoyo. Because it's delicious.
새로운 코트를 샀어요. 요즘 날씨가 추워지니까요.
Saeloun koteureul sasseoyo. Yojeum nalssiga chuwojinikkayo.
I bought a new coat. It's because the weather is getting colder nowadays.
새로운 코트를 샀어요. 요즘 날씨가 추워지거든요.
Yojeum nalssiga chuwojigeodeunyo.
I bought a new coat. It's because the weather is getting colder nowadays.
새로운 코트를 샀어요. 요즘 날씨가 추워져서요.
Yojeum nalssiga chuwojyeoseoyo.
I bought a new coat. It's because the weather is getting colder nowadays.
오늘 피곤해요. 어젯밤에 일이 너무 많아서 별로 못 잤으니까요.
Oneul pigonhaeyo. Eojetbame iri neomu manaseo byeollo mot jasseunikkayo.
Today I'm tired. It's because last night I had too much work, so I hardly slept.
오늘 피곤해요. 어젯밤에 일이 너무 많아서 별로 못 잤거든요.
Oneul pigonhaeyo. Eojetbame iri neomu manaseo byeollo mot jatgeodeunyo.
Today I'm tired. It's because last night I had too much work, so I hardly slept.
Another important note about 거든: it has a second use which means "of course" and is used to explain something really obvious to people. When you use it in this way the intonation is different (and it's almost always in 반말). If you're using 거든 in the first way, to explain a reason, the intonation should go down at the end as if stating a fact. In the second way, it goes up and the emphasis tends to be on 거든 rather than on the rest of the sentence. It shouldn't be too hard to tell which 거든 Koreans are using: apart from the context, the intonation they use in the second is exactly how it sounds—someone who's been wrongly accused and is annoyed about it.
숙제를 했거든.
Sukjereul haetgeodeun.
Of course I did my homework!
아침을 먹었거든.
Achimeul meogeotgeodeun.
Of course I ate breakfast!
남자친구가 있거든.
Namjachinguga itgeodeun.
Of course I have a boyfriend!
Ending sentences with (으)ㄴ/는데요 (eu)n/neundeyo Speaking
You can think of this ending as an ellipsis (...). It implies there's more to the story than what's just been said, as if you started a sentence that was going to have a second part, but then stopped before you got there. After (으)ㄴ/는데고 you can start a new sentence to explain yourself or if the second part of the story is mutually understood, just stop at 데요. I have heard (으)ㄴ/는데요 used as a more polite version of the second 거든 above.
Please also note that 는데, 는 데, and 는대 are all different. Isn't grammar fun?! I'll cover 는 데 on page and 는대 along with the other indirect speech forms on page .
A: 소개팅 할래요? Sogaeting hallaeyo? Want to try 소개팅? (a kind of arranged dating)
B: 아니요. 나 남자친구 있는데요. Aniyo. Na namjachingu inneundeyo. No. I already have a boyfriend. (so I don't need another date)
__________
A: 숙제 했어요? 보여 주세요. Sukje haesseoyo? Boyeo juseyo. Did you do your homework? Show me.
B: 숙제 했는데요... 집에 두고 왔어요. Sukje haenneundeyo...jibe dugo wasseoyo. I did it, but I left it at home.
A: 너, 오늘도 숙제 안 했지? Neo, oneuldo sukje an haetji? You, you also didn't do your homework today, right?
B: 오늘은 숙제 했는데요. Oneuleun sukje haenneundeyo. I did my homework today. (so why are you accusing me of not having done it?)
Starting sentences with 는데, 어서, and 니까 Speaking = Writing
You're in the middle of a lengthy explanation of the whys and hows of what happened. Naturally, sometimes you'll want to pause for breath (or to give your listeners a chance to catch their breath). So how can you do that while explaning things in Korean? 그렇다 geureota is here to help.
I'll discuss all the forms of 그렇다 further on page along with other ways of putting an argument together, but let's put it in here also since it's used very commonly with the verb forms we're learning. In this section, we will finish one statement and then explain further with:
그래서
geuraeseo
and, so, then, thereupon
그러니까
geureonikka
so, therefore, for that reason
그런데
geureonde
now, by the way, but, and (depending on the context)
Let's see some examples.
아침을 안 먹었어요. 그래서 배고파요.
Achimeul an meogeosseoyo. Geuraeseo baegopayo.
I didn't eat breakfast, so I'm hungry.
공부를 열심히 했어요. 그래서 시험에 100점을 받았어요.
Gongbureul yeolsimhi haesseoyo. Geuraeseo siheome baekjeomeul badasseoyo.
I studied hard, so I got 100 on the test.
비가 오고 있어요. 그러니까 우산을 가지고 오세요.
Biga ogo isseoyo. Geureonikka usaneul gajigo oseyo.
It's raining, so please bring an umbrella.
어머니 생신이에요. 그러니까 파티에 갈 수 없어요.
Eomeoni saengsinieyo. Geureonikka patie gal su eobseoyo.
It's my mother's birthday, so I can't go to the party.
이 신발을 너무 작아요. 그러니까 큰 것으로 바꾸어 주세요.
I sinbareul neomu jagayo. Geureonikka keun geoseuro bakkueo juseyo.
These shoes are too small, so could you change it to a bigger size?
친구가 많아요. 그런데 슬퍼요.
Chinguga manayo. Geureonde seulpeoyo.
I have many friends. But I'm sad.
피곤해요. 그런데 파티에 가고 싶어요.
Pigonhaeyo. Geureonde patie gago sipeoyo.
I'm tired. But I want to go to the party.
그는 한국사람 이에요. 그런데 한국음식을 못 먹어요.
Geuneun hanguksaram ieyo. Geureonde hangukeumsigeul mot meogeoyo.
He is a Korean. But he cannot eat Korean food.
##
Linking Sentences—Cause and Effect: More Ways to Say "So"
There are a couple of other useful expressions you can use to say that one thing resulted in another.
민수가 늦게 오기에 잠시 인터넷 하고 있어요.
Minsuga neutge ogie jamsi inteonet hago isseoyo.
Since Minsu is coming late, I'm using the internet for a while.
기에 and 길래 mean exactly the same thing, but 기에 is the written form of this expression and 길래 is the spoken or colloquial form. They mean "Since A, then B." A is a situation that you have observed. As a result of seeing and judging A, you have done, are doing, or are recommending B. You can also use this expression to ask the reason or background for a decision.
The second, similar meaning is a little more difficult to explain in English, but I think you can probably figure it out. It's pretty simple and is similar to 때문에 ttaemune ("because of"), which I'll cover on page .
When you add 는, the meaning of the expression changes a little. If you add the 는 for emphasis, your second clause should be something negative: something is B to A. If I were to translate "Too Legit to Quit" into Korean, it would be done using 기에는.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했기에(는) haetgie(neun)
했길래 haetgillae
| 하기에(는) hagie(neun)
하길래 hagillae
| 하기에(는) 하길래
Action verbs ending in a consonant | 먹다 | 먹었기에(는) meogeotgie(neun)
먹었길래 meogeotgillae
| 먹기에(는) meokgie(neun)
먹길래 meokgillae
| 먹기에(는) 먹길래
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤기에(는) yeppeotgie(neun)
예뻤길래 yeppeotgillae
| 예쁘기에(는) yeppeugie(neun)
예쁘길래 yeppeugillae
| 예쁘기에(는) 예쁘길래
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았기에(는) jagatgie(neun)
작았길래 jagatgillae
| 작기에(는) jaggie(neun)
작길래 jaggillae
| 작기에(는) 작길래
Nouns ending in a vowel | 남자 | 남자이었기에(는) namjaieotgie(neun)
남자이었길래 namjaieotgillae
| 남자기에(는) namjagie(neun)
남자길래 namjagillae
| 남자기에(는) 남자길래
Nouns ending in a consonant | 물 | 물이었기에(는) murieotgie(neun)
물이었길래 murieotgillae
| 물이기에(는) murigie(neun)
물이길래 murigillae
| 물이기에(는) 물이길래
TAKE NOTE
기에 and 길래 can be used only with statements—not commands, questions, or suggestions. The subject of the first clause shouldn't be the speaker, and the subjects of both clauses should be different. Also, with 기에 and 길래 (but not 기에는), the first clause must be some new knowledge that was discovered by the speaker.
EXAMPLE SENTENCES
차가 많이 막히길래, 늦게 출발했어요.
Chaga mani makigillae, neutge chulbalhaesseoyo.
차가 많이 막히기에, 늦게 출발했어요.
Chaga mani makigie, neutge chulbalhaesseoyo.
Since traffic was heavy, I left late.
맛있길래 다시 시켰어요.
Masitgillae dasi sikyeosseoyo.
맛있기에 다시 시켰어요.
Masitgie dasi sikyeosseoyo.
Since it was delicious, I ordered it again.
청바지가 싸길래 샀어요.
Cheongbajiga ssagillae sasseoyo.
청바지가 싸기에 샀어요.
Cheongbajiga ssagie sasseoyo.
Since the jeans were cheap, I bought them.
Here are some examples with 기에는. Note that the B clause is always a negative thing.
이 책을 학생이 읽기에는 너무 어려워요.
I chaegeul haksaengi ilkgieneun neomu eoryeowoyo.
This book is too difficult for students to read.
라지 피자를 혼자 먹기에는 너무 많아요.
Lagi pijareul honja meokgieneun neomu manayo.
A large pizza is too much to eat alone.
그 차를 공무원이 사기에는 너무 비싸요.
Geu chareul gongmuwoni sagieneun neomu bissayo.
That car is too expensive for a public servant to buy.
열쇠를 찾느라고 집에서 늦게 출발했어요.
Yeolsoereul chatneulago jibeseo neutge chulbalhaesseoyo.
I left the house late because I was looking for my keys.
Be careful about using 느라고 neurago because there are rules about where and when it can be used. It is the prima donna of the Korean grammar world, and is extraordinarily picky about where it can be used properly. That said, if you learn how to use it, you can legitimately feel really cool and smart.
You can also use this expression to mean "in order to." In this case, you did clause B on purpose in order to bring about clause A. In this case B doesn't have to be negative (although it often is).
HOW IT'S FORMED
This expression is normally written and said as 느라고, but 느라 neura is also possible.
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Action verbs ending in a vowel | 하다 | 하느라고 haneurago
Action verbs ending in a consonant | 먹다 | 먹느라고 meokneurago
TAKE NOTE
This is somewhat tricky because it can be used only under very specific conditions. First of all, just like 고, the subjects of clause A and clause B must be identical. The first clause needs to end in an action verb: something is being done or has been done. It must be an action—not a condition like catching a cold, even though most of the time that 걸리다 geollida is considered to be an action verb—and it must have been done by an agent, not an inanimate object. That something takes or took a long time. As a result of this lengthy action in clause A, clause B unfortunately happened or became impossible. Clause A is stated in the present, not the past, and clause B must be something that has already happened or is happening now: this sentence structure can't be used to predict the future, however certain you may be about it.
> To sum up:
>
> A: An action, not a condition, done by an agent, taking place over a long time.
>
> B: Same subject as A, something bad, not in the future.
>
> Here are some examples:
EXAMPLE SENTENCES
■ "B because of A"
컴퓨터 게임을 하느라고 늦게 잤어요.
Keompyuteo geimeul haneurago neutge jasseoyo.
I didn't get to sleep until late because I was playing a computer game.
그 일을 하느라고 바빴어요.
Geu ireul haneurago bappasseoyo.
I was busy because I was doing that work.
■ "B in order to A"
고향에 갔다 오느라고 돈을 많이 썼어요.
Gohyange gatda oneurago doneul mani sseosseoyo.
I spent a lot of money in order to visit my home.
부산에 가느라고 숙제를 안 했어요.
Busane ganeurago sukjereul an haesseoyo.
I didn't do my homework because I went to Busan.
친구를 만나느라고 천안까지 갔어요.
Chingureul mannaneurago Cheonankkaji gasseoyo.
I went to Cheonan in order to meet my friend.
늦었으니만큼 제값을 다 안 내도 돼요.
Neujeosseunimankeum jegabseul da an naedo dwaeyo.
Since you were late, you don't have to pay the full price.
This would translate easily as "inasmuch as" in formal English.
HOW IT'S FORMED
This is a combination of (으)니(까) ("so"; see page 67) and 만큼 ("as much as"; see page ).
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했으니만큼 haesseunimankeum | 하니만큼 hanimankeum | 하겠으니만큼 hagesseunimankeum
Action verbs ending in a consonant | 먹다 | 먹었으니만큼 meogeosseunimankeum | 먹으니만큼 meogeunimankeum | 먹겠으니만큼 yeppeugesseunimankeum
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤으니만큼 yeppeosseunimankeum | 예쁘니만큼 yeppeunimankeum | 예쁘겠으니만큼 yeppeugesseunimankeum
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았으니만큼 jagasseunimankeum | 작으니만큼 jageunimankeum | 작겠으니만큼 jakgesseunimankeum
Nouns ending in a vowel | 남자 | 남자였으니만큼 namjayeosseunimankeum | 남자이니만큼 namjainimankeum | 남자이겠으니만큼 namjaigesseunimankeum
Nouns ending in a consonant | 물 | 물이었으니만큼 murieosseunimankeum | 물이니만큼 murinimankeum | 물이겠으니만큼 murigesseunimankeum
TAKE NOTE
만큼 all by itself is used for comparisons: it means "as much as"; (으)니만큼 isn't really the same expression.
The subject of the sentence normally isn't the speaker.
EXAMPLE SENTENCES
공부를 열심히 했으니만큼 시험을 잘 볼 거예요.
Gongbureul yeolsimhi haesseunimankeum siheomeul jal bol geoyeyo.
Since you studied hard, I'm sure you'll do well on the test.
오늘 휴일이니만큼 일 하지 마세요.
Oneul hyuirinimankeum il haji maseyo.
Since today's a holiday, please don't work.
날씨가 추우니만큼 등산 가지 말아요.
Nalssiga chuunimankeum deungsan gaji marayo.
Since it's cold, let's not go hiking.
##
Linking Sentences—Cause and Effect: Through, Because Of, For the Sake Of
There are actually thirteen different grammar forms to cover here, and that's not even including 기에 which you can review on page if you'd like. One of its uses also fits here, but because it's not too difficult and also fits in the previous section, there's no need to go over it again. The nice thing about all of these patterns is that you can use them any time—in the past, present or future—to talk about anyone or anything. There aren't many special rules about usage.
교통 체증 때문에 늦게 왔어요.
Gyotong chejeung ttaemune neutge wasseoyo.
I was late because of traffic.
This is probably the most common of all these patterns. It goes after a noun (or a verb, if you put 기 at the end of the verb) and means "Because of A, B." It can, but often doesn't, have negative connotations.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했기 때문에 haetgi ttaemune | 하기 때문에 hagi ttaemune | 하기 때문에
Action verbs ending in a consonant | 먹다 | 먹었기 때문에 meogeotgi ttaemune | 먹기 때문에 meokgi ttaemune | 먹기 때문에
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤기 때문에 yeppeotgi ttaemune | 예쁘기 때문에 yeppeugi ttaemune | 예쁘기 때문에
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았기 때문에 jagatgi ttaemune | 작기 때문에 jakgi ttaemune | 작기 때문에
Nouns ending in a vowel | 남자 | 남자 때문에 namja ttaemune | 남자 때문에 | 남자 때문에
Nouns ending in a consonant | 물 | 물 때문에 mul ttaemune | 물 때문에 | 물 때문에
TAKE NOTE
A remark about the nouns above: 남자 때문에 means "because of the man," while "남자이기 때문에" means "because he is a man." See the second example below.
If you're blaming someone, 때문에 is a good expression to use, but 는 바람에, 는 통에, neun barame, neun tong e, or (으)ㄴ/는 탓에 (eu) n/neun tase (see page 79) are even more negative. If you want to say that something good happened because of something else, you should use (으)ㄴ/는 덕분에 (eu)n/neun deokbune (see page 80).
EXAMPLE SENTENCES
담배 때문에 건강이 나빠졌어요.
Dambae ttaemune geongangi nappajyeosseoyo.
My health has worsened because of cigarettes.
그는 장애인이기 때문에 잘 못 걸어요.
Geuneun jangaeinigi ttaemune jal mot georeoyo.
He can't walk well because he's handicapped.
(If I'd just said 장애인 때문에 jangaein ttaemune, it would mean "because of the handicapped person.")
차가 막히는 통에 늦게 왔어요.
Chaga makineun tonge neutge wasseoyo.
Because of traffic, I was late.
차가 막히는 바람에 늦게 왔어요.
Chaga makineun barame neutge wasseoyo.
Because of traffic, I was late.
차가 막히는 탓에 늦게 왔어요.
Because of traffic, I was late.
Chaga makineun tase neutge wasseoyo.
Again, "because of A, B," but these expressions can be used only when B is a negative result of A.
HOW IT'S FORMED
탓 tat literally means "fault" or "blame," while 통 tong just means "through."
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 하는 바람에 haneun barame 하는 통에 haneun tonge 한 탓에 han tase | 하는 바람에 하는 통에 하는 탓에 haneun tase | 하는 바람에 하는 통에 하는 탓에
Action verbs ending in a consonant | 먹다 | 먹는 바람에 meokneun barame 먹는 통에 meokneun tonge 먹은 탓에 meokeun tase | 먹는 바람에 먹는 통에 먹는 탓에 | 먹는 바람에 먹는 통에 먹는 탓에
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤던 탓에 yeppeotdeon tase | 예쁜 탓에 yeppeun tase | 예쁜 탓에
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작었던 탓에 jageotdeon tase | 작은 탓에 jageun tase | 작은 탓에
Nouns ending in a vowel | 남자 | 남자였던 탓에 namjayeotdeon tase | 남자인 탓에 namjain tase | 남자인 탓에
Nouns ending in a consonant | 물 | 물이었던 탓에 murieotdeon tase | 물인 탓에 murin tase | 물인 탓에
TAKE NOTE
None of these expressions can be used with questions, suggestions, or commands; they all describe events that took place in the past, so you can use them only with statements that refer to something that's done. Only 탓에 can be used with descriptive verbs or nouns; the other two are for actions only.
There are some small differences among the three. 바람에 barame is used when the reason was unexpected and can be used for positive but unexpected results. 통에 tonge is used only when the event was something chaotic and drawn-out such as a war or a disaster.
Please be aware that 바람 by itself is a noun that means "wind"; I don't know if that has anything to do with the origin of this expression or not, but the 바람 in this sentence isn't "wind" and "wind" by itself isn't necessarily the kind of thing that warrants the use of 바람에. There's also 바랍니다 barapnida, which you will hear all the time on the subway and which is a form of 바라다 barada that means "to want." These are not what we're looking at here.
EXAMPLE SENTENCES
전쟁 통에 그녀가 고아가 되었어요.
Jeonjaeng tonge geunyeoga goaga doeeosseoyo.
Because of the war, she became an orphan.
너무 많이 걸은 통에 다리가 아파요.
Neomu mani georeun tonge dariga apayo.
Because I walked too much, my leg hurts.
사고가 나는 바람에 그녀가 고아가 되었어요.
Aagoga naneun barame geunyeoga goaga doeeosseoyo.
Because of an accident, she was orphaned.
다리를 다치는 바람에 못 걸어요.
Darireul dachineun barame mot georeoyo.
Because I hurt my leg, I can't walk.
전쟁 탓에 그녀가 고아가 되었어요.
Jeonjaeng tase geunyeoga goaga doeeosseoyo.
Because of the war, she became an orphan.
너무 많이 걸은 탓에 다리가 아파요.
Neomu mani georeun tase dariga apayo.
Because I walked too much, my leg hurts.
Notice that I had to change my examples for 바람에 because the causes in the original examples were not unexpected, so I couldn't use 바람에 with them.
택시를 탄 덕분에 늦지 않았어요.
Taeksireul tan deokbune neutji anasseoyo.
Because I took a taxi, I wasn't late.
Do you like Thanksgiving? Who doesn't? Unfortunately, I don't have any turkey for you, but I can show you how to give thanks in Korean. This expression means "Thanks to A, B." It's used only when A was a good thing with a positive result, B.
HOW IT'S CONJUGATED
| | Past | Present
---|---|---|---
Action verbs ending in a vowel | 하다 | 한 덕분에 han deokbune | 하는 덕분에 haneun deokbune
Action verbs ending in a consonant | 먹다 | 먹은 덕분에 meogeun deokbune | 먹는 덕분에 meogneun deokbune
Nouns ending in a vowel | 남자 | 남자 덕분에 namja deokbune | 남자 덕분에
Nouns ending in a consonant | 물 | 물 덕분에 mul deokbune | 물 덕분에
TAKE NOTE
This is used only when you're thankful for something, so don't use it with negative expressions. Unless, of course, you enjoy misery.
EXAMPLE SENTENCES
열심히 일한 덕분에 성공하게 되었어요.
Yeolsimhi ilhan deokbune seonggonghage doeeosseoyo.
Thanks to your hard work, it was a success.
민수 덕분에 그 일을 일찍 끝냈어요.
Minsu deokbune geu ileul iljjik kkeutnaesseoyo.
Thanks to Minsu, we finished the work early.
명절 덕분에 마침내 쉴 수 있어요.
Myeongjeol deokbune machimnae swil su isseoyo.
Thanks to the holiday, I can finally rest.
교사의 요청에 의해서 일찍 왔어요.
Gyosaui yocheonge uihaeseo iljjik wasseoyo.
I came early by the teacher's request.
This expression can be used in a couple of different situations; one would be said as "by" in English (as in the example above) and the other is more like "according to." It means that something happened because of/in accordance with something else. This is useful if you want to say that something was done in accordance with a certain law or regulation or that something was decided by someone/something.
HOW IT'S FORMED
에 (ending preposition) plus 의하다 (roughly meaning "based on") plus (optionally) 여서.
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Nouns ending in a vowel | 남자 | 남자에 의해(서) namjae uihae(seo)
Nouns ending in a consonant | 물 | 물에 의해(서) mule uihae(seo)
TAKE NOTE
에 의하여 is exactly the same expression as 에 의해, but is more formal (as 하여 hayeo is normally used in more formal situations than 해 hae). 에 의해서 means exactly the same thing and can be used interchangeably with 에 의하여 or 에 의해. Despite the resemblance, this expression is not interchangeable with 에 의하면 e uihamyeon.
EXAMPLE SENTENCES
투표에 의하여 대통령이 재선되었어요.
Tupyoe uihayeo daetongryeongi jaeseondoe-eoss-eoyo.
The president was reelected by ballot.
필요에 의해 결정되었어요.
Pilyoe uihae gyeoljeongdoe-eoss-eoyo.
The decision was made according to need.
습관에 의해서 그렇게 했어요.
Seupgwane uihaeseo geureoke haesseoyo.
I did it from force of habit.
이로써 주문이 완료 됐어요.
Irosseo jumuni wallyo dwaesseoyo.
With this, the order is complete.
This pattern means that A is a means to achieve B. It's the type of expression that would work really well on propaganda posters, although it's by no means limited to that kind of use; you can use it to talk about relieving stress (through whatever means you use) or about coercing someone into doing what you want. Also, there are a few common expressions which you can see in the "Take note" section below.
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Action verbs ending in a vowel | 하다 | 함으로써 hameurosseo
Action verbs ending in a consonant | 먹다 | 먹음으로써 meogeumeurosseo
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁨으로서 yeppeumeuroseo
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작음으로써 jageumeurosseo
Nouns ending in a vowel | 남자 | 남자로써 namjarosseo
Nouns ending in a consonant | 힘 him* | 힘으로써 himeurosseo
* Be aware that I changed my example for nouns ending in a consonant; this is because 물 and 로써 start with the same letter and so become 물로써. 힘 (strength) is a more accurate example of how 으로써 is normally used.
TAKE NOTE
Don't confuse (으)로서 (eu)roseo and (으)로써 (eu)rosseo. While they have a few things in common, ~(으)로서 means "as ~" rather than "through ~."
Here are some commonly used expressions that feature (으)로써...
이로써
irosseo
Through this/like this
예로써
yerosseo
By way of example (followed by the example)
사실로써
sasillosseo
Truthfully/with facts
권력으로써
gwollyeogeurosseo
Through power
노력으로써
noryeoguerosseo
Through effort
All are followed by what happened through or because of them, as you can see in the examples below.
EXAMPLE SENTENCES
한 예로써 작년에는 쌀값이 50%로 올랐어요.
Han yerosseo jagnyeoneneun ssalgabsi osippeosenteulo ollasseoyo.
For example, last year the price of rice rose by 50%.
노력으로써 어떤 일도 할 수 있어요.
Noryeogeurosseo eotteon ildo hal su isseoyo.
Through effort, we can do anything.
운동을 함으로써 20킬로를 뺄 수 있었어요.
Undongeul hameurosseo ishipkilloreul ppael su isseosseoyo.
Through exercise I could lose 20 kilos.
인터넷을 통해서 그 사실을 알았어요.
Inteoneseul tonghaeseo geu sasireul arasseoyo.
I learned it through the Internet.
This is just another way to say "through" a person or a thing. You can use it if you want to explain how you know something, how you learned something, how you decided on something, etc.
HOW IT'S FORMED
을/를 are the object marker endings and 통하다 is a verb which means "to go through." 서, from 여서 that means "so," is optional.
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Nouns ending in a vowel | 남자 | 남자를 통해(서) namjareul tonghae(seo)
Nouns ending in a consonant | 물 | 물을 통해(서) mureul tonghae(seo)
TAKE NOTE
는 통에 neun tonge is a different expression that means "because of" in negative situations—don't get it confused with the 을/를 통해 that we are studying here.
The expression is also used more literally in sentences like:
운하를 통해서 한강으로 갈 수 있어요.
Unhareul tonghaeseo Hangangeuro gal su isseoyo.
You can go to the Han River through the canal.
EXAMPLE SENTENCES
누구를 통해 그 사실을 아셨어요?
Nugureul tonghae geu sasireul asyeosseoyo?
From whom did you find out that information?
민수를 통해서 그 사실을 알았어요.
Minsureul tonghaeseo geu sasireul arasseoyo.
I learned it from Minsu.
This means that B was caused by A.
HOW IT'S FORMED
It comes from the verb "인하다 inhada," which literally means "to be caused by" or "to arise from." The 로 part is our good friend which means "through" (see page ).
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Nouns ending in a vowel | 남자 | 남자로 인해 namjaro inhae
Nouns ending in a consonant | 밥 | 밥으로 인해 babeuro inhae
EXAMPLE SENTENCES
교통 체증으로 인해 늦게 왔어요.
Gyotong chejeungeuro inhae neutge wasseoyo.
I was late because of traffic.
담배로 인해 건강이 나빠졌어요.
Dambaero inhae geongangi nappajyeosseoyo.
My health has worsened because of cigarettes.
그는 장애로 인해 잘 못 걸어요.
Geuneun jangaero inhae jal mot georeoyo.
He can't walk well because he's handicapped.
차가 막힌 까닭에 늦었어요.
Chaga makhin kkadalge neujeosseoyo.
He was late because traffic was heavy.
까닭 means reason, so this is another way to say that A is the reason for B: "Since A, then B." You can also use 까닭 by itself as a noun or with 으로 to give or ask a reason for something.
HOW IT'S CONJUGATED
| | Past | Present
---|---|---|---
Action verbs ending in a vowel | 하다 | 한 까닭에 han kkadalge | 하는 까닭에 haneun kkadalge
Action verbs ending in a consonant | 먹다 | 먹은 까닭에 meogeun kkadalge | 먹는 까닭에 meokneun kkadalge
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁜 까닭에 yeppeun kkadalge | 예쁜 까닭에
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작은 까닭에 jageun kkadalge | 작은 까닭에
TAKE NOTE
까닭 doesn't have any kind of emotion attached to it: it can be used with either positive or negative sentences. When it's used positively, you can use (으)ㄴ/는 덕분에 (see page 80). If it's used negatively, you can use the 바람에, 통에, or 탓에 expressions (see page 79). 기 때문에 (see page 78) is interchangeable with (으)ㄴ/는 까닭에, but much more common and can also be used with nouns.
까닭 is normally used to answer a question. For instance, in the soccer example below, your conversation partner might have asked, "왜 그렇게 기뻐요? Wae geureoke gippeoyo?" ("Why are you so happy?") Then you could answer using 까닭에. If the question isn't asked, pick a different expression: 때문에, 덕분에, 바람에, 탓에, or 통에 (see above).
EXAMPLE SENTENCES
Here are some examples of 까닭 used by itself to mean "reason."
무슨 까닭으로 왔어요?
Museun kkadalgeuro wasseoyo?
For what reason (why) did you come here?
까닭 없이 왔어요.
Kkadak eobsi wasseoyo.
I just came for no reason.
And here it is as a linking expression.
우리 축구 팀이 이긴 까닭에 기뻤어요.
Uri chukgu timi igin kkadalge gippeosseoyo.
Since our soccer team won, we were very happy.
라면만 먹은 까닭에 건강이 나빠졌어요.
Ramyeonman meogeun kkadalge geongangi nappajyeosseoyo.
Since I ate only ramen, my health became worse.
맥주를 좋아하는 까닭에 맥주를 직접 만들어요.
Maekjureul johahaneun kkadalge maekjureul jikjeob mandeureoyo.
Because I like beer, I make it myself.
너는 스톰트루퍼치고는 좀 키가 작지 않은가요?
Neoneun seutomteurupeochigoneun jom kiga jakji aneungayo?
Aren't you a little short for a stormtrooper?
This pattern is used to talk about things that aren't normal for what they are such as sentences like "Aren't you a little short for a stormtrooper?" or "He's very mature for his age." It can also be used to say that all cases of a certain noun are the same: for example, "What Korean doesn't eat kimchi?"
HOW IT'S FORMED
셈 means "conjecture" and 치다 is its associated verb; together, they mean "to make a guess."
HOW IT'S CONJUGATED
Verbs take 셈치고는 preceded by the appropriate tense marker. Nouns just take 치고는.
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 한 셈치고는 han semchigoneun | 하는 셈치고는 haneun semchigoneun | 할 셈치고는 hal semchigoneun
Action verbs ending in a consonant | 먹다 | 먹은 셈치고는 meogeun semchigoneun | 먹는 셈치고는 meokneun semchigoneun | 먹을 셈치고는 meogeul semchigoneun
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁜 셈치고는 yeppeun semchigoneun | 예쁜 셈치고는 | 예쁜 셈치고는
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작은 셈치고는 jageun semchigoneun | 작은 셈치고는 | 작은 셈치고는
Nouns ending in a vowel | 남자 | 남자치고는 namjachigoneun | 남자치고(는) | 남자치고(는)
Nouns ending in a consonant | 물 | 물치고는 mulchigoneun | 물치고(는) | 물치고(는)
TAKE NOTE
A셈치고 B looks somewhat similar, but it isn't the same as 셈치고는; see page .
If 는 isn't enough for you and you really want to emphasize the oddness of what you're saying, you can use a double negative. See the last example below.
B has to be something you wouldn't expect from A such as a child is more mature than expected, a stormtrooper is short, or a Korean doesn't eat kimchi.
EXAMPLE SENTENCES
그는 아이치고는 아주 어른스러워요.
Geuneun aichigoneun aju eoreunseureowoyo.
He's very mature for a child.
한국 사람치고는 김치 안 먹는 사람이 없어요.
Hanguk saramchigoneun gimchi an meokneun sarami eobseoyo.
What Korean doesn't eat kimchi?
그 아이가 작은 셈치고는 농구를 잘 해요.
Geu aiga jageun semchigoneun nonggureul jal haeyo.
For a child so small, he plays basketball well.
공부를 그렇게 열심히 한 셈치고는 시험을 잘 못 봤어요.
Gongbureul geureoke yeolsimhi han semchigoneun siheomeul jal mot bwasseoyo.
He didn't do well on the test considering how hard he studied.
정시에 오는 것은 교통 상황에 달려 있어요.
Jeongsie oneun geoseun gyotong sanghwange dallyeo isseoyo.
Getting there on time depends on the traffic situation.
This expression is useful when things aren' t as straightforward and when you need to explain that something depends on something else. It can also mean "because of" when ~ was the main cause, the thing on which the first part of your sentence was dependent.
HOW IT'S FORMED
달리다 dallida means "to be hung" and is combined here with 어 있다 eo itda (to remain in a certain state), so this expression literally means that something has "been hung" on ~.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 한 것에 달려 있다 han geose dallyeo itda 하기에 달려 있다 hagie dallyeo itda | 하는 것에 달려 있다 haneun geose dallyeo itda 하기에 달려 있다 | 할 것에 달려 있다 hal geose dallyeo itda 하기에 달려 있다
Action verbs ending in a consonant | 먹다 | 먹은 것에 달려 있다 meogeun geose dallyeo itda 먹기에 달려 있다 meokgie dallyeo itda | 먹는 것에 달려 있다 meokneun geose dallyeo itda 먹기에 달려 있다 | 먹을 것에 달려 있다 meogeul geose dallyeo itda 먹기에 달려 있다
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁜 것에 달려 있다 yeppeun geose dallyeo itda | 예쁜 것에 달려 있다 | 예쁜 것에 달려 있다
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작은 것에 달려 있다 jageun geose dallyeo itda | 작은 것에 달려 있다 | 작은 것에 달려 있다
Nouns ending in a vowel | 남자 | 남자에 달려 있다 namjae dallyeo itda | 남자에 달려 있다 | 남자에 달려 있다
Nouns ending in a consonant | 교육 | 교육에 달려 있다 gyoyuge dallyeo itda | 교육에 달려 있다 | 교육에 달려 있다
TAKE NOTE
When following a question word (누구, 무엇, 어디, nugu, mueos, eodi, etc.), you should use the form (느)냐에 달려 있다. (neu) nyae dallyeo itda. (ㄴ)냐 is covered on page .
EXAMPLE SENTENCES
행복은 자기 자신이 어떻게 생각하느냐에 달려 있어요.
Haengbogeun jagi jasini eotteoke saenggakhaneunyae dallyeo isseoyo.
Happiness depends on how one thinks about oneself.
그 체인점의 인기가 음식이 싼 것에 달려 있어요.
Geu cheinjeomui ingiga eumsigi ssan geose dallyeo isseoyo.
That chain is popular because of its cheap food.
아인슈타인의 발견은 그의 지능에 달려 있었어요.
Ainsyutainui balgyeoneun geuui jineunge dallyeo isseosseoyo.
Einstein's discoveries were because of his intelligence.
지각을 하기로는 민수가 최고이죠.
Jigageul hagironeun Minsuga choegoijyo.
When it comes to being late, Minsu is later than anyone (Minsu is the best at being late).
This is a great way to give a compliment, whether straightforward or backhanded. It's used to talk about someone or something being the best or worst at something.
HOW IT'S FORMED
This is simply the 기 that you use to turn a verb into a noun, followed by the 로 which means "through." 는 is optional and can be added for emphasis.
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Action verbs ending in a vowel | 하다 | 하기로(는) hagiro(neun)
Action verbs ending in a consonant | 먹다 | 먹기로(는) meokgilo(neun)
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁘기로(는) yeppeugiro(neun)
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작기로(는) jaggiro(neun)
TAKE NOTE
B should include a sentence indicating superiority. Here are a few examples:
최고이지요, 최고 아닐까요?, 최고예요
choegoijiyo, choego anilkkayo?, choegoyeyo
따를 없어요, 없죠 ttaleul eobseoyo, eobsjyo
제일이에요, 제일이죠 jeilieyo, jeilijyo
기로 하다 giro hada is a different expression used to express determination; see page . There the 기로 is always followed by 하다 while here it will always be followed by a sentence explaining who or what is the best.
EXAMPLE SENTENCES
술을 마시기로는 제 아버지가 제일이에요.
Sureul masigironeun je abeojiga jeirieyo.
When it comes to drinking alcohol, my father can handle more than anyone (my father is the best at handling alcohol).
예쁘기로는 가영을 따를 사람이 없어요.
Yeppeugironeun Kayoungeul ttareul sarami eobseoyo.
When it comes to beauty, no one is more beautiful than Kayoung.
##
Cause and Effect: Origins
우리 숫자는 원래 아라비아에서 비롯됐어요.
Uri sutjaneun wonlae arabiaeseo birotdwaesseoyo.
우리 숫자는 원래 아라비아에 뿌리를 두고 있어요.
Uri sutjaneun wonlae arabiae ppurireul dugo isseoyo.
Our numerals originally came from Arabia.
Which came first: the chicken or the egg? That question may never be answered, but at least after reading this section you'll know how to discuss it along with where other things came from. These expressions are used used to say that A started from or came from B.
HOW IT'S FORMED
비롯되다 birotdoeda literally means "to originate from" and you ought to know 에서 by now. That's all there is to this one.
뿌리 ppuri means roots, like those of a tree. 두다 duda is a verb which means "to keep" and can be used to help many other verbs get their meanings across (see page ). As for 에 well, you know 에, and if you don't, please refer back to the section on "Basic Particles" on page . Put it all together and you get "A has its roots in B."
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Nouns ending in a vowel | 경제 | 경제에서 비롯됐다 gyeongjee-seo birotdwaetda
경제에 뿌리를 두었다 gyeongje-e ppurireul dueotda
| 경제에서 비롯되다 gyeongje-e-seo birot deoda
경제에 뿌리를 두다 gyeongje-e ppurireul duda
| 경제에서 비롯되다 경제에 뿌리를 두다
Nouns ending in a consonant | 교육 | 교육에서 비롯됐다 gyoyugeseo birotdwaetda
교육에 뿌리를 두었다 gyoyuge ppurireul dueotda
| 교육에서 비롯되다 gyoyugeseo birot doeda
교육에 뿌리를 두다 gyoyuge ppurireul duda
| 교육에서 비롯되다 교육에 뿌리를 두다
TAKE NOTE
뿌리를 두다 ppurireul duda should always be conjugated in the present progressive: for example, 뿌리를 두고 있어요 ppurireul dugo isseoyo.
EXAMPLE SENTENCES
아이의 행동은 부모의 행동에서 비롯돼요.
Aiui haengdongeun bumoui haengdongeseo birotdwaeyo.
아이의 행동은 부모의 행동에 뿌리를 두고 있어요.
Aiui haengdongeun bumoui haengdonge ppurireul dugo isseoyo.
A child's behavior originates from its parents' behavior.
그 전쟁은 욕심에서 비롯되었어요.
Geu jeonjaengeun yoksimeseo birotdoe-eoss-eoyo.
그 전쟁은 욕심에 뿌리를 두고 있었어요.
Geu jeonjaengeun yoksime ppurireul dugo isseosseoyo.
The war arose out of greed.
##
Cause and Effect, or Lack Thereof
Here's one for your nihilists out there. This section will teach you how to say that something is futile or had no effect. Another expression that's similar to the two below is A거나 말거나 B, which is covered in the section on 거나 on page .
택시를 타나 마나 늦을 거예요.
Taeksireul tana mana neujeul geoyeyo.
Whether you take a taxi or not, you'll be late.
나 마나 means that whether you do A or not (A나 마나) the result, B, is the same.
HOW IT'S FORMED
There are a number of expressions in Korean using this particular 말, and you can explore them in more detail in the "Making decisions" section on page []. The 마 in this expression is that same 말 but abbreviated a little, and the 나 is (으)나, which normally means "but" (see page ). A verb followed by an expression with 마 means "to ~ or not": for example, 할까 말까 halkka malkka or 가지 말지 gaji malji. With adjectives, you can use the opposite of your A verb instead of 마: for example, 좋나 싫나 johna silhna (whether you like it or not) or 크나 작으나 keuna jageuna (whether it's big or small).
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했으나 마나 haesseuna mana | 하나 마나 hana mana | 하나 마나
Action verbs ending in a consonant | 먹다 | 먹었으나 마나 meogeosseuna mana | 먹으나 마나 meogeuna mana | 먹으나 마나
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤으나 못생겼으나 yeppeosseuna motsaenggyeosseuna | 예쁘나 못 생기나 yeppeuna mot saenggina | 예쁘나 못 생기나
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았으나 컸으나 jagasseuna keosseuna | 작으나 크나 jageuna keuna | 작으나 크나
TAKE NOTE
This is quite similar to 아/어/여 봤자 a/eo/yeo bwatja (see the bottom of this page). The slight difference in meaning is that with 나 마나 the result is definitely fixed, whereas with 봤자/봐야, you won't get what you want regardless of whether or not you do A.
EXAMPLE SENTENCES
약을 먹으나 마나 감기가 떨어지지 않을 거예요.
Yageul meogeuna mana gamgiga tteoleojiji anheul geoyeyo.
Whether you take medicine or not, your cold won't go away.
그건 말하나마다.
Geugeon malhanamada.
It goes without saying. (Something is a fact, whether or not it is mentioned in words.)
급하게 가 봤자 무슨 소용이겠어요? 벌써 늦어요.
Geubhage ga bwatja museun soyongigesseoyo? Beolsseo neujeoyo.
급하게 가 봐야 무슨 소용이겠어요? 벌써 늦어요.
Geubhage ga bwaya museun soyongigesseoyo? Beolsseo neujeoyo.
What's the point of hurrying? We're already late.
This is an expression you can use while wringing your hands and/or sitting on your bed dressed in black listening to The Cure. These two expressions mean that A is pointless or not worth doing.
HOW IT'S FORMED
The first expression combines 아/어/여 보다 ("to try to do something"), the past tense marker, and 자, which can mean "so," "and then," or quite a few other things and which we'll check out on page . Alternately, you can use 아/어/여 봐야, which means the same thing.
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Action verbs with 오 or 아 | 잡다 japda | 잡아 봤자 jaba bwatja
잡아 봐야 jaba bwaya
Action verbs with 어, 우, 으 or 이 | 먹다 | 먹어 봤자 meogeo bwatja
먹어 봐야 meogeo bwaya
하다 | 하다 | 해 봤자 hae bwatja
해 봐야 hae bwaya
TAKE NOTE
For your B clause, you can choose an expression which means "useless." 소용 없어요 so~yong eobseoyo is pretty common, but I'll show you a couple of others in the examples below. Just pick an expression from below, or if you're really advanced and/or a free thinker, invent your own. Either way, B can mean only one thing and that is "it's useless." If your B expression means anything else, this is not the grammar point you're looking for.
You can also skip the B clause altogether, end your sentence with 봤자예요 bwatjayeyo, and leave it to your listeners to deduce the futility of A based on the fact that you used 봤자. However, this doesn't work with 봐야.
Some expressions you can use for B are 아무 의미가 없어요 amu uimiga eobseoyo (there's no point), 시간낭비다 si gannangbida (it's a waste of time), 아무 소용 없을 것 같아요 amu soyong eobseul geot gatayo (it seems like it'll be useless), or 소용없는 짓이다 soyongeobsneun jisida (it tends to be useless).
This expression and (으)나 마나 are similar, but 봤자/봐야 means that whether you do A or not, you won't get the desired result, whereas 나 마나 means that whatever you do, the results won't change.
It's also possible, though less common, to use 았/었/였자 with verbs other than 보다. In this case it means "even if A, B."
열심히 공부해 봤자 시험에 떨어질 거예요.
Yeolsimhi gongbuhae bwatja siheome tteoleojil geoyeyo.
Even if I study hard, I will fail the test.
EXAMPLE SENTENCES
그에게 말을 해 봤자 소용 없어요.
Geuege mareul hae bwatja soyong eobseoyo.
그에게 말을 해 봐야 소용 없어요.
Geuege mareul hae bwaya soyong eobseoyo.
It's useless to try to talk to him.
이렇게 해 봤자 별 수 없어요.
Ireoke hae bwatja byeol su eobseoyo.
이렇게 해 봐야 별 수 없어요.
Ireoke hae bwaya byeol su eobseoyo.
There's no point in doing it like this.
##
Cause and Effect: Multiple Causes
열쇄를 잃어버리고 해서 늦게 왔어요.
Yeolswaereul ilheobeorigo haeseo neutge wasseoyo.
I lost my keys (among other things), so I arrived late.
"But," you say, "things aren't always so simple. Sometimes life gets complicated." It sure does, and that's why you need this expression. It says that A is one cause among many for B, probably the most important cause or you'd hardly be singling it out, right? You can use this when all the other non-A causes aren't really that important or if you just want to emphasize A for some reason.
HOW IT'S FORMED
It's a combination of 고 하다 (see page ) and 아/어/여서.
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Action verbs ending in a vowel | 하다 | 하고 해서 hago haeseo
Action verbs ending in a consonant | 먹다 | 먹고 해서 meokgo haeso
TAKE NOTE
고 can be used only with action verbs and only when the subjects of both clauses A and B are the same.
EXAMPLE SENTENCES
쇼핑을 하고 해서 동대문에 갔어요.
Syopingeul hago haeseo Dongdaemune gasseoyo.
I went to Dongdaemun to go shopping (among other things).
부모님한테 돈을 받고 해서 새로운 코트를 샀어요.
Bumonimhante doneul batgo haeseo saeroun koteureul sasseoyo.
I got some money from my parents (among other things), so I bought a new coat.
쇼핑도 하고 저녁도 먹고 해서 돈을 많이 썼어요.
Syopingdo hago jeonyeokdo meokgo haeseo doneul mani sseosseoyo.
I went shopping and ate dinner, so I spent a lot of money.
광고를 보지 않을 겸 해서 영화에 조금 늦게 왔어요.
Gwanggoreul boji anheul gyeom haeseo yeonghwae jogeum neutge wasseoyo.
I arrived a little late to the movies in order (to skip) the ads (among other things).
HOW IT'S FORMED
(으)ㄹ 겸 is basically a slash and allows you to say you did two or more things. You can brush up on this in the "and" section on page . 서 is from 아/어/여서 (see page ) and means "so."
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Action verbs ending in a vowel | 하다 | 할 겸 해서 hal gyeom haeseo
Action verbs ending in a consonant | 먹다 | 먹을 겸 해서 meogeul gyeom haeseo
TAKE NOTE
You can use this with multiple examples in your A clause or just one. See the example sentences.
A should be the reason for B.
EXAMPLE SENTENCES
부모님한테 돈을 받을 겸 해서 부모님 집에 놀러 갔어요.
Bumonimhante doneul badeul gyeom haeseo bumonim jibe nolleo gasseoyo.
Since I got money from my parents, I went to hang out at their house.
쇼핑을 할 겸 해서 동대문에 갔어요.
Syopingeul hal gyeom haeseo Dongdaemune gasseoyo.
I went to Dongdaemun to go shopping (among other things).
쇼핑도 하고 저녁도 먹을 겸 해서 시장에 갔어요.
Syopingdo hago jeonyeokdo meogeul gyeom haeseo sijange gasseoyo.
I went to the market to go shopping and eat dinner.
##
Linking Sentences: Basic Comparisons
민수가 가영이보다 더 늦었어요. Minsuga Kayoungiboda deo neujeosseoyo.
민수가 가영이보다 늦었어요. Minsuga Kayoungiboda neujeosseoyo.
Minsu was later than Kayoung.
This is the easiest and most basic form of comparison in Korean. All it means is that A is more C (bigger, faster, longer, shorter, easier, prettier, etc.) than B. So if you're all about keeping up with the Joneses and/or you like to one-up other people, you should learn this expression.
HOW IT'S CONJUGATED
보다 directly follows the noun that forms the basis for comparison: the lesser of the two nouns. A is the noun that is better than it. If A is implied, you can leave it out in the actual sentence: for example, 어제보다 기분이 좋아요 eojeboda gibuni joayo means "I feel better (now) than yesterday" and will be understood without you needing to mention that you feel better now and since you're speaking in the present tense, people can reasonably assume you mean now. After B, C should explain exactly what you're comparing. See the examples below for clarification:
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 한 것보다 han geotboda | 하기보다 hagiboda 하는 것보다 haneun geotboda | 할 것보다 hal geotboda
Action verbs ending in a consonant | 먹다 | 먹은 것보다 meogeun geotboda | 먹기보다 meokgiboda 먹는 것보다 meokneun geotboda | 먹을 것보다 meokeul goetboda
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁜 것보다 yeppeun geotboda | 예쁜 것보다 | 예쁜 것보다
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작은 것보다 jageun geotboda | 작은 것보다 | 작은 것보다
Nouns ending in a vowel | 남자 | 남자보다 namjaboda | 남자보다 | 남자보다
Nouns ending in a consonant | 물 | 물보다 mulboda | 물보다 | 물보다
RESTRICTIONS
Don't forget the subject marker after A. B often gets no marker: it can be directly followed by 보다.
This has nothing to do with the 보다 which means "to see." This 보다 is a particle which is always directly attached to a noun; the other 보다 is a verb which does all the things that verbs normally do in Korean.
You can put 는 after the 보다 if you want to add emphasis; see the last example below.
C often includes 더, which means "more."
EXAMPLE SENTENCES
공부하기가 쉬는 것보다 더 힘들어요.
Gongbuhagiga swineun geotboda deo himdeureoyo.
공부하기가 쉬는 것보다 힘들어요.
Gongbuhagiga swineun geotboda himdeureoyo.
Studying is more difficult than resting.
내가 너보다 더 똑똑해.
Naega neoboda deo ttokttokhae.
내가 너보다 똑똑해.
Naega neoboda ttokttokhae.
I'm smarter than you.
한국어가 불어보다 미국인에게 배우기 더 어려워요.
Hangukeoga bureoboda miguginege baeugi deo eoryeowoyo.
한국어가 불어보다 미국인에게 배우기 어려워요.
Hangukeoga bureoboda miguginege baeugi eoryeowoyo.
It's harder for Americans to learn Korean than to learn French.
나는 예쁜 여자보다 귀여운 여자를 더 좋아해요.
Naneun yeppeun yeojaboda gwiyeoun yeojareul deo johahaeyo.
I prefer cute women to beautiful women.
그녀는 예쁘기보다는 귀여워요.
Geunyeoneun yeppeugibodaneun gwiyeowoyo.
She is cute rather than pretty.
가영에 비하면 민수가 더 늦었어요.
Kayounge bihamyeon Minsuga deo neujeosseoyo.
가영에 비해서 민수가 더 늦었어요.
Kayounge bihaeseo Minsuga deo neujeosseoyo.
Minsu was later than Kayoung.
These are other ways to make comparisons. They're a little more official than the other ways stated here, so you can make yourself sound smarter or more knowledgeable by using them. You'll often hear them on the news or read them in newspapers in sentences like "Unemployment is up this year compared to last year."
HOW IT'S FORMED
에 is the same particle you use to talk about going places, and here it simply indicates the basis for a comparison. In a way, with these expressions you are explaining the distance between A and B. 비하다 means "to compare" and can be combined here with 여서 (so) or 면 (if).
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Action verbs ending in a vowel | 하다 | 하기에 비해서 hagie bihaeseo
하기에 비하면 hagie bihamyeon
Action verbs ending in a consonant | 먹다 | 먹기에 비해서 meokgie bihaeseo
먹기에 비하면 meokgie bihamyeon
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁘기에 비해서 yeppeugie bihaeseo
예쁘기에 비하면 yeppeugie bihamyeon
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작기에 비해서 jakgie bihaeseo
작기에 비하면 jakgie bihamyeon
Nouns ending in a vowel | 남자 | 남자에 비해서 namjae bihaeseo
남자에 비하면 namjae bihamyeon
Nouns ending in a consonant | 물 | 물에 비해서 mure bihaeseo
물에 비하면 mure bihamyeon
TAKE NOTE
These two expressions are not exactly identical, but about 90% of the time you can use either one without changing the meaning or ruining the sentence's grammar, and the rest of the time the differences are so small that you don't really need to worry about them. Because 비하면 is made with 면 (see page ), it means "if you compare," whereas 비해서 means "when you compare."
While you can skip using 더 after 보다, you can't after 비하면/비해서. They are used for a wider variety of comparisons than 보다 which is always used to indicate that B is more of something than A. 비해서 and 비하면 are used merely to compare two situations: whether there's more, less, or the same amount, you can still use either one.
EXAMPLE SENTENCES
미국 경제에 비하면 중국 경제가 더 강해요.
Miguk gyeongjee bihamyeon jungguk gyeongjega deo ganghaeyo.
미국 경제에 비해서 중국 경제가 더 강해요.
Miguk gyeongjee bihaeseo jungguk gyeongjega deo ganghaeyo.
The Chinese economy is stronger than the American economy.
5년 전에 비하면 월급은 거의 오르지 않았어요.
O-nyeon jeone bihamyeon wolgeubeun geoui oreuji anasseoyo.
5년 전에 비해서 월급은 거의 오르지 않았어요.
O-nyeon jeone bihaeseo wolgeubeun geoui oreuji anasseoyo.
My salary has hardly increased in the past five years. (Compared to five years ago, these days my salary has hardly increased).
6개월 전만 해도 물가가 이렇게 비싸지 않았어요.
Yuk gaewol jeonman haedo mulgaga ireoke bissaji anasseoyo.
Just six months ago things weren't this expensive.
There are many big problems in this world: global warming, the economy, rising fuel prices, poverty, war, etc. Sometimes you want to talk to people about these problems, but they just don't get it, so instead of talking about the big picture you try to reduce it to something people can grasp by using an example that's small and/or close to home. This is where 만 해도 comes in. It means "if we consider A alone."
Similarly, it can be used to talk about the way things used to be. In this case, it's used with expressions like "10년 전만 해도 sipnyeon jeonman haedo" or "한달 전만 해도 handal jeonman haedo." The first translates to "just ten years ago," and the latter to "just a month ago." You can then talk about how things have changed: the area has gentrified, the price has risen drastically, your friend has slimmed down, and so on.
HOW IT'S FORMED
만, in this case, is a marker used with numbers to emphasize their meaning. Then 하다 ("to do") is combined with 여도 ("although"; see page 104) to get "even if we only do (consider) A."
HOW IT'S CONJUGATED
A is a noun indicating either the time period that you're considering or what you want to specifically look at.
| | All tenses
---|---|---
Nouns ending in a vowel | 남자 | 남자만 해도 namjaman haedo
Nouns ending in a consonant | 물 | 물만 해도 mul mulman haedo
TAKE NOTE
You can't use this expression in the future tense.
It often follows an expression of time with 전 such as in the second example below to mean "just ~ ago." B is then a statement of what was true at that time (but presumably no longer is).
EXAMPLE SENTENCES
휘발유 값만 해도 500원이나 올랐어요.
Hwibalyu gabsman haedo obaekwonina ollasseoyo.
The price of gas alone has risen by 500 won.
요즘 취업이 아주 어려워요. 우리 가족만 해도 제 아버지와 제 삼촌이 실직 상태예요.
Yojeum chwieobi aju eoryeowoyo. Uri gajokman haedo je abeojiwa je samchoni siljik sangtaeyeyo.
These days it's really difficult to get a job. In my family alone my father and uncle are out of work.
요즘 취업이 아주 어려워요. 2년 전만 해도 지금보다 훨씬 좋았어요.
Yojeum chwieobi aju eoryeowoyo. Inyeon jeonman haedo jigeumboda hwolssin joasseoyo.
These days it's really difficult to get a job. Just two years ago, it was much better than now.
제 수업에 정시에 오는 사람들은 있는가 하면 늦게 오는 사람들도 있어요.
Je sueobe jeongsie oneun saramdeureun inneunga hamyeon neutge oneun saramdeuldo isseoyo.
In my class, there are some people who come on time and some who are late.
If you live in Korea, you may often find yourself having to explain that stereotypes aren't true and that different people do different things. That's where you need this expression. It's used to explain that there are cases of both A and B.
HOW IT'S CONJUGATED
This expression is mainly used with two types of sentences. One is A가 있는가 하면 ga inneunga hamyeon B도 있다 do itda (There is A, but there is also B). The other is 어떤 eotteon (사람들은 saramdeureun, 때 ttae, 날 nal, 57) A(으)ㄴ가 하면 어떤 (eu) n-ga hamyeon eotteon (사람들은, 때, 날,57) B.
In the first type of sentence, A and B are nouns. B should be followed by 도 (too). In the second type of sentence, 는가 하면 neunga hamyeon can be used after action verbs while ㄴ가 하면 is used after descriptive verbs ending in vowels and 은가 하면 eunga hamyeon is used after descriptive verbs ending in consonants. You can put 이 after a noun and then conjugate it as a descriptive verb.
A is the first kind of case; B is the second. For example, A might be "students who study hard" while B is "students who don't study at all." You can then use (으)ㄴ/는가 하면 to explain that there are both students who study hard and students who don't study at all. Let's see an example of the first type of sentence:
공부를 열심히 하는 학생들이 있는가 하면 공부를 안 하는 학생들도 있어요.
Gongbureul yeolsimhi haneun haksaengdeuri inneunga hamyeon gongbureul an haneun haksaengdeuldo isseoyo.
And an example of the second type:
어떤 학생들은 공부를 열심히 하는가 하면 어떤 학생들은 공부를 안 해요.
Eotteon haksaengdeureun gongbureul yeolsimhi haneunga hamyeon eotteon haksaengdeureun gongbureul an haeyo.
This expression can be used in the past, present, or future tenses, but these tenses are expressed at the end of the sentence rather than with this expression.
| | A이/가 있는가 하면 B도 있다 | 어떤... A(으)ㄴ/는가 하면 어떤... B...
---|---|---|---
Action verbs ending in a vowel | 하다 | | 하는가 하면 haneunga hamyeon
Action verbs ending in a consonant | 먹다 | | 먹는가 하면 meokneunga hamyeon
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | | 예쁜가 하면 yeppeunga hamyeon
Descriptive verbs (adjectives) ending in a consonant | 작다 | | 작은가 하면 jageunga hamyeon
Nouns ending in a vowel | 남자 | 남자가 있는가 하면 namjaga inneunga hamyeon |
Nouns ending in a consonant | 물 | 물이 있는가 하면 muri inneunga hamyeon |
TAKE NOTE
Be careful about which type of sentence you're using with this, and don't get tangled up in the grammar.
There are a few similar expressions. ~기도 하다 gido hada (see page ) indicates that sometimeŝ also occurs. In this case, ~ is not as important as whatever usually occurs: it's just tagging along. ~(으)ㄹ 수도 있다 (eu)r sudo itda (see page ) is another expression similar to 기도 하다. Again, it means that ~ can happen, but the other thing that happens is more important. (으)ㄴ/는가 하면, however, doesn't tell your listener what's more important or more common: it simply states that both cases A and B occur.
EXAMPLE SENTENCES
우리 학교에는 좋은 선생님들이 있는가 하면 그냥 그런 선생님들도 있어요.
Uri hakgyoeneun joeun seonsaengnimdeuri inneunga hamyeon geunyang geureon seonsaengnimdeuldo isseoyo.
At our school there are some really good teachers and some average teachers.
어떤 사람들은 똑똑한가 하면 또 어떤 사람들은 어리석어요.
Eotteon saramdeureun ttokttokanga hamyeon tto eotteon saramdeureun eoriseogeoyo.
Some people are smart while some people are foolish.
오늘 수업이 많은데, 오늘따라 회의가 있어서 정말 바빴어요.
Oneul sueobi manheunde, oneulttara hoeuiga isseoseo jeongmal bappasseoyo.
Today I had a lot of classes, and on top of that (unlike most days) I had a meeting, so I was really busy.
Have you ever had one of those days where everything goes wrong? When all the things that normally work suddenly don't? When your normally perfect hair refuses to settle down for the important date or you forgot your documents for the big meeting? 따라 is an expression you can use to talk about those days. It means that something happened on the day in question that was different from normal and is often used in expressions such as "Today it was freezing cold, and on top of that the heater at work suddenly broke."
HOW IT'S CONJUGATED
따라 is used directly after either 오늘 oneul or 그날 geunal: 오늘따라 oneulttala, 그날따라 geunalttala. It can be used in the past or present tense. B describes what was different on that day. If you're using it to say "and on top of that," then you can explain the first situation, conclude that with (으)ㄴ/는데 (eu) n/neunde, add 오늘/그날따라 oneul/geunalttala, and then say what was different about that day. See the examples below.
TAKE NOTE
There are a couple of 따라s in Korean. ~에 따라 e ttala means "according to ~" (see page ), and ~를/을 따라 leul/eul ttala means "following ~." The 따라 in question here always follows only 오늘 or 그날 and is never preceded by 에 or 을/를 or any other word except 오늘 or 그날. ~에 따라 and ~을/를 따라 are made by conjugating the verb 따르다, which means "to follow" and isn't really applicable to 오늘따라 and 그날따라.
You can use 마침 machim instead of ~따라; it means the same thing, but can be used for sentences with times other than just this day or that day.
EXAMPLE SENTENCES
어제 많이 추웠는데, 그날 따라 히터가 고장나서 훨씬 더 추웠어요.
Eoje mani chuwonneunde, geunal ttala hiteoga gojangnaseo hwolssin deo chuwosseoyo.
어제 많이 추웠는데, 마침 히터도 고장나서 훨씬 더 추웠어요.
Eoje mani chuwonneunde, machim hiteodo gojangnaseo hwolssin deo chuwosseoyo.
Yesterday was very cold, and on top of that (unlike most days) the heater broke, so I was even more cold.
올해 물가도 많이 올랐는데, 마침 의료비도 많이 나와서 고생했어요.
Olhae mulgado mani ollanneunde, machim uiryobido mani nawaseo gosaenghaesseoyo.
This year prices have risen quite a lot also, and on top of that (unlike other years), I had to pay a lot for medical treatment, so it's been difficult for me. (따라 can't be used in this sentence.)
##
Linking Sentences—Contrast: Although/But
Another expression that can be used to mean "although" is 다가도 dagado. I felt it belonged in the temporal order section rather than here because it's used only while talking about a progression of events.
그녀는 급하게 서두르다가도 항상 다른 생각을 하다가 늦어요.
Geunyeoneun geuphage seodureudagado hangsang dareun saenggageul hadaga neujeoyo.
Even though she hurries, she always thinks about something else and ends up being late.
Likewise, 면서도 myeonseodo combines 면서 and 도 to mean "even while." Since it's about two simultaneous actions, I put it in the "while" section.
서둘렀으면서도 늦었어요.
Seodulleosseumyeonseodo neujeosseoyo.
Even while I hurried, I was late.
다가도 dagado is on page and 면서도 is on page if you feel either of the two is the better fit for your sentence. Otherwise, here are some more applicable expressions to express contrast.
택시를 타도 늦어요.
Taeksileul tado neujeoyo.
Even if I take a taxi, I'll be late.
When it comes to expressing contrast, the big player in Korean grammar is 도. It also has a few other uses, but with a little practice you'll find it easy enough to tell when it's used for contrast and when it's something else. With the former it will appear at the end of other expressions like the ones shown in this section: 아/어/여도, ㄴ/는데도, 면서도, 고도, 다가도, etc. For now, let's look at the most basic form: 아/어/여도. It simply means "although."
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했어도 haesseodo | 해도 haedo | 해도
Action verbs ending in a consonant | 먹다 | 먹었어도 meogeosseodo | 먹어도 meogeodo | 먹어도
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤어도 yeppeosseodo | 예뻐도 yeppeodo | 예뻐도
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았어도 jagassado | 작아도 jagado | 작아도
TAKE NOTE
This expression is similar to 더라도, but it can be used more broadly. 더라도 can be used only hypothetically when the action in A hasn't yet occurred. 아/어/여도 can be used hypothetically in the past tense, like so:
택시를 탔어도 늦었어요.
Taeksireul tasseodo neujeosseoyo.
Even if I had taken a taxi, I would have been late.
EXAMPLE SENTENCES
시간이 없어도 숙제를 해야 해요.
Sigani eobseodo sukjereul haeya haeyo.
Even if you have no time, you must do your homework.
운동을 해도 살이 안 빠져요.
Undongeul haedo sari an ppajyeoyo.
Even though I exercise, I'm not losing weight.
서둘렀지만 늦었어요.
Seodulleotjiman neujeosseoyo.
I hurried, but I was late.
This is by far the most commonly used of the "but" expressions and, along with the versatile 도, the only one you need to worry about if you're simply studying informally. In these sentences A and B have about the same weight: neither one is more important.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했지만 haetjiman | 하지만 hajiman | 하지만
Action verbs ending in a consonant | 먹다 | 먹었지만 meogeotjiman | 먹지만 meogjiman | 먹지만
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤지만 yeppeotjiman | 예쁘지만 yeppeujiman | 예쁘지만
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았지만 jagatjiman | 작지만 jagjiman | 작지만
Nouns ending in a vowel | 남자 | 남자였지만 namjayeotjiman | 남자이지만 namjaijiman | 남자이지만
Nouns ending in a consonant | 물 | 물이었지만 murieotjiman | 물이지만 murijiman | 물이지만
TAKE NOTE
There is another, rather less common, expression "ㄴ지 만" (notice the space between 지 and 만) which is used to tell the period of time that has lapsed since an event. For example, "한국에 온지 만 5년 됐어요. Hanguke onji man onyeon dwaesseoyo." (It's been five years since I came to Korea.) We'll cover this in the section on Progression and putting things in order on page . For now, just remember to watch for the space and the context.
EXAMPLE SENTENCES
그녀 예쁘지만 성격이 까다로워요.
Geunyeo yeppeujiman seonggyeogi kkadarowoyo.
She is beautiful but has a difficult personality.
운동을 했지만 살이 빠지지 않았어요.
Undongeul haetjiman sari ppajiji anasseoyo.
I exercised, but I didn't lose weight.
그 가수가 노래를 잘 하지만 기타를 칠 수 없어요.
Geu gasuga noraereul jal hajiman gitareul chil su eobseoyo.
That singer can sing well but can't play the guitar.
택시를 타더라도 늦을 거예요.
Taeksireul tadeorado neujeul geoyeyo.
Even if I take a taxi, I'll be late.
This is much like 아/어/여도, but it's stronger." The result (B) in a 더라도 expression is even more contrary than that in an 아/어/여도 expression.
HOW IT'S FORMED
더라 is covered on page and is normally used to indicate something you experienced personally; however, that doesn't really matter in this expression. 도 is a particle which means "although."
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했더라도 haetdeorado | 하더라도 hadeorado | 하더라도
Action verbs ending in a consonant | 먹다 | 먹었더라도 meogeotdeorado | 먹더라도 meokdeorado | 먹더라도
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤더라도 yeppeotdeorado | 예쁘더라도 yeppeudeorado | 예쁘더라도
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았더라도 jagatdeorado | 작더라도 jagdeorado | 작더라도
Nouns ending in a vowel | 남자 | 남자였더라도 namjayeotdeorado | 남자이더라도 namja-ideorado | 남자이더라 도
Nouns ending in a consonant | 물 | 물이었더라도 murieotdeorado | 물이더라도 murideorado | 물이더라도
TAKE NOTE
This expression is much like 아/어/여도, but unlike 아/어/여도, it can be used only when the action in A hasn't yet taken place. It's stronger and more hypothetical than 아/어/여도.
EXAMPLE SENTENCES
시간이 없더라도 숙제를 해야 해요.
Sigani eopdeorado sukjereul haeya haeyo.
Even if you have no time, you must do your homework.
운동을 열심히 하더라도 살이 빠지지 않을 거예요.
Undongeul yeolsimhi hadeorado sari ppajiji anheul geoyeyo.
Even if I exercise a lot, I won't lose weight.
더 일찍 왔더라도 그 사람이 죽었을 거예요.
Deo iljjig watdeolado geu sarami jugeosseul geoyeyo.
Even if you had arrived earlier, the man would still have died.
택시를 타는데도 늦을 거예요.
Taeksileul taneundedo neujeul geoyeyo.
Even if I take a taxi, I'll be late.
Just like the above two expressions, this one means, "even though."
HOW IT'S FORMED
(으)ㄴ/는데 is used to give the background for a situation; see page . 도 is the particle used to indicate contrast.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했는데도 haenneundedo | 하는데도 haneundedo | 하는데도
Action verbs ending in a consonant | 먹다 | 먹었는데도 meogeonneundedo | 먹는데도 meokneundedo | 먹는데도
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤는데도 yeppeonneundedo | 예쁜데도 yeppeundedo | 예쁜데도
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았는데도 jaganneundedo | 작는데도 jakneundedo | 작는데도
Nouns ending in a vowel | 남자 | 남자였는데도 namjayeonneundedo | 남자인데도 namjaindedo | 남자인데도
Nouns ending in a consonant | 물 | 물이었는데도 murieonneundedo | 물인데도 murindedo | 물인데도
TAKE NOTE
B can't include a suggestion, a proposition, or the future tense. It often begins with 불구하고 bul guhago, which is a word that often goes along with contrast expressions and emphasizes the contrast a little more.
EXAMPLE SENTENCES
시간이 없는데도 숙제를 해야 해요.
Sigani eobsneundedo sukjereul haeya haeyo.
Even if you have no time, you must do your homework.
운동을 했는데도 살이 빠지지 않았어요.
Undongeul haenneundedo sari ppajiji anasseoyo.
Even though I exercised, I didn't lose weight.
서두르고도 늦었어요.
Seodureugodo neujeosseoyo.
Although I hurried, I was late.
This translates as "although A, B." In other words, it has more to do with 도 (which sets up a contrast) than 고.
HOW IT'S FORMED
고 (and) plus 도 (indicating a contrast).
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Action verbs ending in a vowel | 하다 | 하고도 hagodo
Action verbs ending in a consonant | 먹다 | 먹고도 meokgodo
TAKE NOTE
고도 can be used only with action verbs and only when the subjects of both clause A and clause B are the same.
EXAMPLE SENTENCES
그 사람 돈을 잘 벌고도 자주 돈을 빌리고 싶어해요.
Geu saram doneul jal beolgodo jaju doneul billigo sipeohaeyo.
Although he makes a lot of money, he often wants to borrow more.
밥을 먹고도 배가 고파서 라면을 또 먹었어요.
Babeul meokgodo baega gopaseo ramyeoneul tto meogeosseoyo.
Although I ate a meal (or rice), I was hungry, so I ate ramen again.
이 근처가 조용한 반면에 재미가없어요.
I geuncheoga joyonghan banmyeone jaemiga eobseoyo.
이 근처가 조용한데 반해 재미가 없어요.
I geuncheoga joyonghande banhae jaemiga eobpseoyo.
This area is quiet but boring.
Here are two more expressions you can use to contrast A and B. They're just about exactly the same as the English expression "on the other hand."
HOW IT'S FORMED
반면 literally means "opposite side," and 에 is optional. I think it sounds more natural to use it, but you don't have to if you don't want to. 반하다 banhada means "to oppose" and is conjugated here after (으)ㄴ/는데.
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Action verbs ending in a vowel | 하다 | 하는 반면(에) haneun banmyeon(e)
하는데 반해 haneunde banhae
Action verbs ending in a consonant | 먹다 | 먹은 반면(에) meogeun banmyeon(e)
먹는데 반해 meokneunde banhae
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁜 반면(에) yeppeun banmyeon(e)
예쁜데 반해 yeppeunde banhae
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작은 반면(에) jageun banmyeon(e)
작은데 반해 jageunde banhae
TAKE NOTE
The subjects of A and B should be the same.
This expression is quite similar to 지 (see page 112), 지만 jiman (see page 105) and A기는/긴 하는데/한데 gineun/gin haneunde/hande B (see page 113). Unlike the latter, 반면에 simply contrasts two facts and doesn't, on its own, imply anything about which of the two facts is more important. It has the same meaning as 지 or 지만 but sounds somewhat more official, so it's good for newspapers, reports, official discussions, and things of that nature.
You can use A(으)ㄴ/는데 반해 B interchangeably with A 반면에 B.
EXAMPLE SENTENCES
새로운 대통령이 인기가 많은 반면에 정치 경험이 아직 많지 않아요.
Saeroun daetonglyeongi ingiga manheun banmyeone jeongchi gyeongheomi ajik manchi anayo.
새로운 대통령이 인기가 많은데 반해 정치 경험이 아직 많지 않아요.
Saeroun daetonglyeongi ingiga manheunde banhae jeongchi gyeongheomi ajik manchi anayo.
The new president is very popular, but on the other hand, he has little political experience.
민수가 스포츠를 잘 하는 반면에 음악을 못 해요.
Minsuga seupocheureul jal haneun banyeone eumageul mot haeyo.
민수가 스포츠를 잘 하는데 반해 음악을 못 해요.
Minsuga seupocheureul jal haneunde banhae eumageul mot haeyo.
Minsu is good at sports, but on the other hand, he's not very good at music.
급히 왔으나 늦었어요.
Geupi wasseuna neujeosseoyo.
I hurried, but I was late.
For the writers among you, here's another way to say "but." This one has the same meaning as 지만 but is mainly used in writing.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했으나 haesseuna | 하나 hana | 하나
Action verbs ending in a consonant | 먹다 | 먹었으나 meogeosseuna | 먹으나 meogeuna | 먹으나
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤으나 yeppeosseuna | 예쁘나 yeppeuna | 예쁘나
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았으나 jagasseuna | 작으나 jageuna | 작으나
TAKE NOTE
(으)나 is mostly used in writing; 지만 can be used anywhere. Otherwise they're interchangeable.
(이)나 (i)na means "or" and is used to link two nouns. (으)나 is used after verbs, so check whether the word before 나 is a noun or a verb. As always, it's also important to check the context in which the expression is used—(이)나 and (으)나 are not used in the same contexts.
나? can be a 반말 question particle. It's the 나 of 나요, which is on page . It will always appear at the end of a 반말 question. 나 of (으)나 will always be followed by a B clause.
(으)나 마나 is an expression which uses 나 but is worth investigating on its own; see page .
EXAMPLE SENTENCES
그녀는 예쁘나 성격이 까다로워요.
Geunyeoneun yeppeuna seonggyeogi kkadarowoyo.
She is beautiful but has a difficult personality.
그 가수가 노래를 잘 하나 기타를 칠 수 없어요.
Geu gasuga noraereul jal hana gitareul chil su eobseoyo.
That singer can sing well but can't play the guitar.
교통에도 불구하고 늦지 않았어요.
Gyotongedo bulguhago neutji anasseoyo.
In spite of traffic, I arrived on time.
This is the same expression as 아/어/여도 and means "in spite of/even though," but it is attached to nouns.
HOW IT'S FORMED
불구하다 is optional in this expression.
HOW IT'S CONJUGATED
Note that there are two expressions listed below for each noun. You can use this expression either with a noun alone (such as in the example above, "in spite of traffic"), or you can use it with 이 to mean "in spite of being": for instance, 남자임에도 불구하고 means "in spite of being a man."
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했음에도 haesseumedo (불구하고) | 함에도 hamedo (불구하고) | 할 것임에도 hal geosimedo (불구하고)
Action verbs ending in a consonant | 먹다 | 먹었음에도 meogeosseumedo (불구하고) | 먹에도 meogedo (불구하고) | 먹을 것임에도 meogeul geosimedo (불구하고)
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤음에도 yeppeosseumedo (불구하고) | 예쁨에도 yeppeumedo (불구하고) | 예쁠 것임에도 yeppeul geosimedo (불구하고)
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았음에도 jagasseumedo (불구하고) | 작음에도 jageumedo (불구하고) | 작을 것임에도 jageul geosimedo (불구하고)
Nouns ending in a vowel | 남자 | 남자에도 namjaedo (불구하고) 남자였음에도 namjayeosseumedo (불구하고) | 남자에도 (불구하고) 남자임에도 namjaimedo (불구하고) | 남자에도 (불구하고) 남자일 것임에도 namjail geosimedo (불구하고)
Nouns ending in a consonant | 물 | 물에도 muredo (불구하고) 물이었음에도 murieosseumedo (불구하고) | 물에도 (불구하고) 물임에도 murimedo (불구하고) | 물에도 (불구하고) 물일 것임에도 muril geosimedo (불구하고)
TAKE NOTE
If you want to say "in spite of (verb or adjective)," you should use 아/어/여도 (see page 104) instead of 에도.
This expression is often used with verbs as well. In these cases you conjugate the verb with (으)ㅁ, like the second and third examples below.
This expression is more often used with major problems of the sort that appear on the news. For personal problems or other relatively minor issues, you can use (으)ㄴ/는데 and (으)ㄴ/는데도 불구하고 for more serious personal problems.
EXAMPLE SENTENCES
학교 폭력을 반대하는 캠패인을 함에도 불구하고 학교폭력 사건의 건 수가 계속 증가하고 있어요.
Hakgyo poglyeogeul bandaehaneun kaempaeineul hamedo bulguhago haggyopoglyeog sageonui geon suga gyesog jeunggahago isseoyo.
In spite of the campaign against school violence, the number of incidences continues to rise.
물가가 높음에도 불구하고 사람들이 계속 새로운 물건을 사요.
Mulgaga nopeumedo bulguhago saramdeuri gyesok saeroun mulgeoneul sayo.
Although prices are high, people keep buying new things.
서둘렀지 늦었어요.
Seodulleotji neujeosseoyo.
I hurried, but I was late.
This is really just a shortened version of 지만 (see page 105).
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했지 haetji | 하지 haji | 하지
Action verbs ending in a consonant | 먹다 | 먹었지 meogeotji | 먹지 meokji | 먹지
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤지 yeppeotji | 예쁘지 yeppeuji | 예쁘지
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았지 jagatji | 작지 jakji | 작지
Nouns ending in a vowel | 남자 | 남자였지 namjayeotji | 남자이지 namjaiji | 남자이지
Nouns ending in a consonant | 물 | 물이었지 murieotji | 물이지 muriji | 물이지
TAKE NOTE
There are many different 지s: ~지 않다 ji anta (isn't~), ~ㄴ 지 (since~), ~지 말다 ji malda (not~), etc. You can tell them apart from the context, but if you're confused, check the expression closely: 지 (since) always has ㄴ, 는 or ㄹ before it, and 지 않다 and 지 말다 are expressions all by themselves: you'll never see these 지s without some conjugation of 않다 or 말다 after them, nor will you see this 지 with any version of 않다 or 말다 right afterward.
EXAMPLE SENTENCES
그녀 예쁘지 성격이 까다로워요.
Geunyeo yeppeuji seonggyeogi kkadarowoyo.
She is beautiful but has a difficult personality.
운동을 했지 살을 빠지 지 않았어요.
Undongeul haetji sareul ppajiji anasseoyo.
I exercised, but I didn't lose weight.
그 가수가 노래를 잘 하지 기타를 칠 수 없어요.
Geu gasuga noraereul jal haji gitareul chil su eobseoyo.
That singer can sing well but can't play the guitar.
집에 일찍 출발하기는 했는데 교통이 막혀서 늦었어요.
Jibe iljjik chulbalhagineun haenneunde gyotongi maghyeoseo neujeosseoyo.
I leave my house early, but traffic was heavy, so I was late.
If you've read much of this book, you'll have noticed that I told you in a significant number of expressions that 는 was optional and added emphasis. Well, here it's decidedly not optional because the whole point of this expression is to emphasize ~. You can use this expression to say things like "I did my homework, but..." or "The food was good, but..." It's the part of the Korean language that replaces all these italicized words.
HOW IT'S FORMED
기 turns a verb into a noun and 는 adds emphasis.
HOW IT'S CONJUGATED
하다 normally ends with (으)ㄴ/는데 or 지만 and then a justification (the part of the sentence that comes after the "but"). Instead of using 하다, you can also just repeat the verb you used in A. Both ways are shown below. 기는/긴 하다 is used with descriptive verbs such as 예쁘다 or 똑똑하다, or with nouns with 이, while 하는데 is used with action verbs like 하다 or 가다. This is because of how (으)ㄴ/는데 works; see page .
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 하기는 했다 hagineun haetda | 하기는 하다 hagineun hada | 하기는 할 것이다 hagineun hal geosida
Action verbs ending in a consonant | 먹다 | 먹기는 했다 meokgineun haetda
먹기는 먹었다 meokgineun meogeotda
| 먹기는 하다 meokgineun hada
먹기는 먹다 meokgineun meokda
| 먹기는 할 것이다 meokgineun hal geosida
먹기는 먹을 것이다 meokgineun meogeul geosida
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁘기는 했다 yeppeugineun haetda
예쁘기는 예뻤다 yeppeugineun yeppeotda
| 예쁘기는 하다 yeppeugineun hada
예쁘기는 예쁘다 yeppeugineun yeppeuda
| 예쁘기는 할 것이다 yeppeugineun hal geosida
예쁘기는 예쁠 것이다 yeppeugineun yeppeul geosida
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작기는 했다 jakgineun haetda
작기는 작았다 jakgineun jagatda
| 작기는 하다 jakgineun hada
작기는 작다 jakgineun jakda
| 작기는 할 것이다 jakgineun hal geosida
작기는 작을 것이다 jakgineun jageul geosida
TAKE NOTE
기는 can also be shortened to 긴.
This expression is similar to 지만. The main difference is that with 지만 both parts of the sentence, A and B, have equal weight, while with 기는 or 긴 하는데 or 한데, B is more important. You present A to show that there's another side to the situation, but ultimately you find B more convincing.
EXAMPLE SENTENCES
그 여자가 예쁘기는 예쁘지만 성격이 나빠요.
Geu yeojaga yeppeugineun yeppeujiman seonggyeogi nappayo.
That woman is beautiful, but she doesn't have a good character.
먹기는 먹는데 이 음식 맛이 없어서 많이 먹기는 어려워요.
Meokgineun meokneunde i eumsig masi eobseoseo mani meokgineun eoryeowoyo.
I'm eating it, but the food doesn't taste good, so it's hard to eat a lot.
서두렀긴 했지만 늦었어요.
Seodureotgin haetjiman neujeosseoyo.
Even though I hurried, I was late.
여행을 좋아하긴 하는데 돈이 없어서 자주 안 가요.
Yeohaengeul joahagin haneunde doni eobseoseo jaju an gayo.
I love traveling, but (more importantly) I have no money so I don't go often.
도시 생활이 편리하기는 한데 시끄러워요.
Dosi saenghwari pyeonlihagineun hande sikkeureowoyo.
Life in the city is convenient, but (more importantly) noisy.
택시를 타네 급행열차를 타네 해도 결국 늦을 거예요.
Taeksireul tane geuphaengyeolchareul tane haedo gyeolguk neujeul geoyeyo.
Even if he takes a taxi or a fast train, he will be late.
In this expression, you are essentially saying that A and B don't matter and C is true regardless. For instance, if you just adore living in the country, you might say to someone who's not convinced, "Even if it's remote and there's nothing much to do, I still like living here." Or maybe you are that person who isn't convinced, in which case you could use this expression to tell your friend from the countryside, "Even if it's peaceful and the scenery is great, I still couldn't live here."
Another use of this expression translates more accurately as "whether or not A, C." In this case A and B are opposites, and C is true regardless of whether A or B is true in the case in question.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했네 haenne | 하네 hane | 할 거네 hal geone
Action verbs ending in a consonant | 먹다 | 먹었네 meogeonne | 먹네 meokne | 먹을 거네 meogeul geone
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤네 yeppeonne | 예쁘네 yeppeune | 예쁠 거네 yeppeul geone
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았네 jaganne | 작네 jakne | 작을 거네 jageul geone
Nouns ending in a vowel | 남자 | 남자였네 namjayeonne | 남자네 namjane | 남자일 거네 namjail geone
Nouns ending in a consonant | 물 | 물이었네 murieonne | 물이네 muline | 물일 거네 muril geone
TAKE NOTE
This has nothing to do with the 네s that mean "yes" and "you."
A and B must be either direct opposites, or things that are either both positive or both negative. In the former case, C will be something that's true regardless of whether A is true or not; in the latter, C is true in spite of both A and B. See the examples below for clarification.
EXAMPLE SENTENCES
물가가 비싸네 사람이 많네 해도 도시에 살고 싶어요.
Mulgaga bissane sarami manne haedo dosie salgo sipeoyo.
Even if it's expensive and there are a lot of people, I want to live in the city.
돈이 많네 없네 해도 어려운 사람을 도와 줘야 돼요.
Doni manne eomne haedo eoryeoun sarameul dowa jwoya dwaeyo.
Whether you have a lot of money or not, you must help those who are having trouble.
보드카네 위스키네 해도, 역시 소주가 최고예요.
Bodeukane wiseukine haedo, yeoksi sojuga choegoyeyo.
Even if people are saying that vodka is good and whisky is good, soju is still the best.
아무리 바쁘기로서니 그렇게 늦게 와서야 되겠어요?
Amuri bappeugiroseoni geureoke neutge waseoya doegesseoyo?
Even though you were busy, do you really think it's all right to be that late?
This is for when your friend tells you something that leaves you aghast. You just can't believe he did such a thing.
HOW IT'S FORMED
기 turns verbs into nouns (see page ), 로서 means "through" (see page 130) and 니 is short for 니까, which means "so" (see page ).
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했기로서니 haetgiroseoni | 하기로서니 hagiroseoni | 하기로서니
Action verbs ending in a consonant | 먹다 | 먹었기로서니 meogeotgiroseoni | 먹기로서니 meokgiroseoni | 먹기로서니
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤기로서니 yeppeotgiroseoni | 예쁘기로서니 yeppeugiroseoni | 예쁘기로서니
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았기로서니 jagatgiroseoni | 작기로서니 jakgiroseoni | 작기로서니
Nouns ending in a vowel | 남자 | 남자였기로서니 namjayeotgiroseoni | 남자이기로서 니 namja-i-giroseoni | 남자이기로서 니
Nouns ending in a consonant | 물 | 물이었기로서니 murieotgiroseoni | 물이기로서니 murigiroseoni | 물이기로서니
TAKE NOTE
A should be in the second or third person; this expression is used to express shock at what someone else has done, not talk about what you've done. A often includes 아무리 amuri to express deeper shock.
B should be a question or expression of disbelief. Here are some examples:
~(으)ㄴ/는단 말이야? (see page )
(eu)n/neundan mariya?
Are you saying...?
~(으)ㄹ 수(야) 없지요 (see page )
~(eu)r su(ya) eopjiyo
You know you can't...
~아/어/여서야 되겠냐? (see page )
~a/eo/yeoseoya doegennya?
Do you really think that's all right?
EXAMPLE SENTENCES
그 여자가 아무리 예쁜 여자이기로서니 그렇게 처다볼 수는 없지요.
Geu yeojaga amuri yeppeun yeoja-i-giroseoni geureoke cheodabol suneun eopjiyo.
No matter how beautiful a woman is, you can't stare at her like that.
당신이 아무리 일을 열심히 했기로서니 마음대로 회사에 안 나간단 말이에요?
dangsini amuri ireul lyeolsimhi haetgiroseoni maeumdaero hoesae an nagandan mari eyo?
Even though you worked hard, you can't just skip work whenever you want to.
##
Linking Sentences—Contrast: Not A, But B
Do you enjoy telling people they're wrong? Here's how you can do that in Korean.
내가 아니라 민수 잘못 때문에 늦게 왔어요.
Naega anira Minsu jalmot ttaemune neutge waseoyo.
내가 아니고 민수 잘못 때문에 늦게 왔어요.
Naega anigo Minsu jalmot ttaemune neutge waseoyo.
It was not my fault, but Minsu's, that we were late.
These expressions are based on 아니다 (the dictionary form of good old 아니요) and are used to negate A and assert B: in other words, not A, but B.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 한 것이 아니라 han geosi anira
한 것이 아니고 han geosi anigo
| 하는 것이 아니라 haneun geosi anira
하는 것이 아니고 haneun geosi anigo
| 할 것이 아니라 hal geosi anira
할 것이 아니고 hal geosi anigo
Action verbs ending in a consonant | 먹다 | 먹은 것이 아니라 meogeun geosi anira
먹은 것이 아니고 meogeun geosi anigo
| 먹는 것이 아니라 meogneun geosi anira
먹는 것이 아니고 meogneun geosi anigo
| 먹을 것이 아니라 meogeul geosi anira
먹을 것이 아니고 meogeul geosi anigo
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁜 것이 아니라 yeppeun geosi anira
예쁜 것이 아니고 yeppeun geosi anigo
| 예쁜 것이 아니라 yeppeun geosi anira
예쁜 것이 아니고 yeppeun geosi anigo
| 예쁜 것이 아니라 yeppeun geosi anira
예쁜 것이 아니고 yeppeun geosi anigo
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작은 것이 아니라 jageun geosi anira
작은 것이 아니고 jageun geosi anigo
| 작은 것이 아니라 작은 것이 아니고 | 작은 것이 아니라 작은 것이 아니고
Nouns ending in a vowel | 남자 | 남자(가) 아니라 namja(ga) anira
남자(가) 아니고 namja(ga) anigo
| 남자(가) 아니라 남자(가) 아니고 | 남자(가) 아니라 남자(가) 아니고
Nouns ending in a consonant | 물 | 물(이) 아니라 mul(i) anira
물(이) 아니고 mul(i) anigo
| 물(이) 아니라 물(이) 아니고 | 물(이) 아니라 물(이) 아니고
TAKE NOTE
As you may recall (or can learn on page ), 고 can be used only in sentences where both A and B have the same subject, and that's true of 아니고 as well. However, you probably don't need to worry about it too much since A and B usually have the same subject in these kinds of sentences anyway.
EXAMPLE SENTENCES
그는 의사가 아니라 변호사이에요.
Geuneun uisaga anira byeonhosa-i-eyo.
그는 의사가 아니고 변호사이에요.
Geuneun uisaga anigo byeonhosa-i-eyo.
He is not a doctor, but a lawyer.
비빔밥이 아니라 김밥을 주문했어요.
Bibimbapi anira gimbapeul jumunhaesseoyo.
비빔밥이 아니고 김밥을 주문했어요.
Bibimbapi anigo gimbapeul jumunhaesseoyo.
비빔밥을 주문한 것이 아니라 김밥을 주문했어요.
Bibimbapeul jumunhan geosi anira gimbapeul jumunhaesseoyo.
비빔밥을 주문한 것이 아니고 김밥을 주문했어요.
Bibimbapeul jumunhan geosi anigo gimbapeul jumunhaesseoyo.
비빔밥을 주문한 게 아니라 김밥을 주문했어요.
Bibimbapeul jumunhan ge anira gimbapeul jumunhaesseoyo.
비빔밥을 주문한 게 아니고 김밥을 주문했어요.
Bibimbapeul jumunhan ge anigo gimbapeul jumunhaesseoyo.
I didn't order bibimbap; I ordered kimbap.
(Yes, all six of these sentences mean the same thing. Keep in mind that 게, the abbreviated form of 것이, is conversational Korean while 것이 is a little more formal.)
시간에 맞게 온 대신에 늦었어요.
Sigane matge on daesine neujeosseoyo.
Instead of arriving on time, he was late.
This has two uses. The first one is to say "B instead of A," and the other is used is to make trades. In both cases, the same grammatical rules apply. Let's take a look:
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 한 대신에 han daesine | 하는 대신에 haneun daesine | 하는 대신에
Action verbs ending in a consonant | 먹다 | 먹은 대신에 meogeun daesine | 먹는 대신에 meogneun daesine | 먹는 대신에
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁜 대신에 yeppeun daesine | 예쁜 대신에 | 예쁜 대신에
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작은 대신에 jageun daesine | 작은 대신에 | 작은 대신에
Nouns ending in a vowel | 남자 | 남자(가) 대신에 namja(ga) daesine | 남자(가) 대신에 | 남자(가) 대신에
Nouns ending in a consonant | 물 | 물(이) 대신에 mul(i) daesine | 물(이) 대신에 | 물(이) 대신에
EXAMPLE SENTENCES
Here's 대신에 used to mean "instead of":
비빔밥 대신에 김밥을 먹었어요.
Bibimbap daesine gimbapeul meogeosseoyo.
Instead of bibimbap, I ate kimbap.
기타 치는 대신에 노래를 부르겠어요.
Gita chineun daesine noraereul bureugesseoyo.
Instead of playing the guitar, I intend to sing a song.
And here are some examples of using 대신에 to make trades:
내가 친구 숙제를 도와주는 대신에 친구가 나한테 저녁을 살게요.
Naega chingu sukjereul dowajuneun daesine chinguga nahante jeonyeogeul salgeyo.
In exchange for helping my friend with his homework, he'll buy me dinner.
제가 친구 저녁을 사 주는 대신에 친구가 저에게 점심을 산다고 했어요.
Jega chingu jeonyeogeul sa juneun daesine chinguga jeoege jeomsimeul sandago haesseoyo.
In exchange for buying my friend dinner, he'll buy me lunch.
부모님이 민수에게 용돈을 주는 대신에 민수가 부모님 집청소를 해야 돼요.
Bumonimi Minsuege yongdoneul juneun daesine Minsuga bumonim jipcheongsoreul haeya dwaeyo.
In exchange for his allowance, Minsu must clean his parents' house.
늦을 거예요? 아니면 시간에 맞출 거예요?
Neujeul geoyeyo? Animyeon sigane matchul geoyeyo?
Will you be late? Or will you arrive on time?
Either A or B is possible, but if it's not A then it's B. Think of English expressions like "Are you going to have steak, or are you trying something new today?" Since you obviously don't know if it's A or B (or you wouldn't be using this pattern), this expression is normally used for posing questions or making guesses.
HOW IT'S FORMED
Here's 아니다 (not) again. Add 면, which means "if" (see page ), and you get "If not A, then B."
HOW IT'S CONJUGATED
Verbs can be used with 아니면 if they're first changed into nouns using 것이; however, this is not usually done, and Koreans will normally find another way around this kind of sentence. See, for example, the last two sentences below. It's awkward to say 늦는 것이 아니면 neujneun geos i animyeon and much simpler to divide one sentence into two.
| | All tenses
---|---|---
Nouns ending in a vowel | 남자 | 남자(가) 아니면 namja(ga) animyeon
Nouns ending in a consonant | 물 | 물(이) 아니면 mul(i) animyeon
TAKE NOTE
The difference between 아니면 animyeon and 아니라/아니고 anira/anigo is that in the latter, you think A is definitely not true; in the former, you think it could be true, but if not, then B is definitely true instead. So in 아니라/아니고 you are clearly rejecting A, while in 아니면 you aren't.
Finally, you can use the expression 아니면 좋겠다 animyeon joketda to state "I hope." See page for details on 좋겠다 joketda.
EXAMPLE SENTENCES
그는 의사일걸요? 아니면 변호사일걸요?
Geuneun uisa-ilgeollyo? Animyeon byeonhosa-ilgeollyo?
Is he a doctor? (If not, then) a lawyer?
(ㄹ/을걸요 is an ending used mostly for making rough guesses; see page ).
고기 먹었어요? 아니면 회 먹었어요?
Gogi meogeosseoyo? Animyeon hoe meogeosseoyo?
Did you eat meat? or did you eat sashimi?
그는 의사가 아니면 변호사일걸요.
Geuneun uisaga animyeon byeonhosa-ilgeollyo.
He's a doctor or (if not, then) a lawyer.
스테이크가 아니면 다른 것을 먹을까요?
Seuteikeuga animyeon dareun geoseul meogeulkkayo?
Shall we eat steak, or something else?
늦게 올 거예요? 아니면 시간에 맞춰 올 거예요?
Neutge ol geoyeyo? Animyeon sigane matchwo ol geoyeyo?
Will you be late, or will you arrive on time?
기타를 칠 거예요? 아니면 노래를 부를 거예요?
Gitareul chil geoyeyo? Animyeon noraereul bureul geoyeyo?
Are you going to play the guitar? Or sing a song?
그는 작가라기보다 사진사예요.
Geuneun jakgaragiboda sajinsayeyo.
He's more of a photographer than a writer.
This means "more of B than of A" or "B is more true than A."
HOW IT'S FORMED
First, take an indirect speech particle and change it into a noun using 기. Then add 보다 (see page 96) which is used to compare two things. 는 is optional and can be added for emphasis if you'd like.
HOW IT'S CONJUGATED
| | Past | Present
---|---|---|---
Action verbs ending in a vowel | 하다 | 했다기보다(는) haetdagiboda(neun) | 한다기보다(는) handagiboda(neun)
Action verbs ending in a consonant | 먹다 | 먹었다기보다(는) meogeotdagiboda(neun) | 먹는다기보다(는) meokneundagiboda(neun)
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤다기보다(는) yeppeotdagiboda(neun) | 예쁘다기보다(는) yeppeudagiboda(neun)
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았다기보다(는) jagatdagiboda(neun) | 작다기보다(는) jakdagiboda(neun)
Nouns ending in a vowel | 남자 | 남자라기보다(는) namjaragiboda(neun) | 남자라기보다(는)
Nouns ending in a consonant | 물 | 물이라기보다(는) muriragiboda(neun) | 물이라기보다(는)
EXAMPLE SENTENCES
떡볶이가 식사라기보다 간식이에요.
Ddeokbokkiga siksaragiboda gansigieyo.
Ddeokbokki is more of a snack than a meal.
김밥은 맛있다기보다는 싸기 때문에 자주 먹어요.
Kimbapeun masitdagibodaneun ssagi ttaemune jaju meogeoyo.
I often eat kimbap because it's cheap, not because I like it.
뛴다기보다는 빨리 걸어 가고 있어요.
Ttwindagibodaneun ppalli georeo gago isseoyo.
I'm not running; I'm walking quickly.
늦게 오느니 그냥 오지 마세요.
Neutge oneuni geunyang oji maseyo.
Rather than coming late, just don't come at all.
This expression is perfect for when you need to choose between two options but aren't terribly excited about either. Let's say you hate seafood, but are stuck overnight in a seaside town where the only open restaurant serves either 매운탕 maeuntang (spicy fish soup) or 회 hoe (raw fish). That's when you need 느니. A is the option you're rejecting and B is what you're choosing to do instead.
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Action verbs ending in a vowel | 하다 | 하느니 haneuni
Action verbs ending in a consonant | 먹다 | 먹느니 meogneuni
TAKE NOTE
차라리 charari, which means "rather," often follows 느니.
B normally ends with a future tense: either (으)ㄹ 것이다 (eu)r geosida or 겠다 getda.
EXAMPLE SENTENCES
퇴직하느니 계속 일하겠어요.
Toejikaneuni gyesog ilhagesseoyo.
I'd rather keep working than retire.
매운탕을 먹느니 회를 먹을 거예요.
Maeuntangeul meogneuni hoereul meogeul geoyeyo.
I'd rather eat raw fish than spicy fish soup.
어젯밤과 달리, 오늘 집에 일찍 돌아왔어요.
Eojetbamgwa dalli oneul jibe iljjik dolawasseoyo.
Unlike last night, today I got home early.
This comes from 다르다 dareuda or "to be different." It means "Unlike A, B." 달리 itself can be used in sentences as an adjective and is quite often used in sentences indicating that there is no alternative but A.
HOW IT'S FORMED
와/과 means "and"; see page . 는 is optional and can be used to add emphasis. 달리 comes from 다르다, which means "to be different."
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Nouns ending in a vowel | 남자 | 남자와 달리 namjawa dalli
Nouns ending in a consonant | 물 | 물과 달리 mulgwa dalli
TAKE NOTE
You can also use 달리 by itself to begin a sentence contrasting with the previous sentence or to contrast two clauses. The latter is a nice way to make you sound more fluent.
EXAMPLE SENTENCES
달리 친구가 없어요.
Dalli chinguga eobseoyo.
I don't have any other friends.
달리 할 방법이 없어요.
Dalli hal bangbeobi eobseoyo.
There's no other way to do it.
어렸을 때는 지금과는 달리 축구를 자주 했어요.
Eoryeosseul ttaeneun jigeumgwaneun dalli chukgureul jaju haesseoyo.
When I was a child, I often played soccer, but not any more.
어렸을 때와 달리 이제 자주 안 해요.
Eoryeosseul ttaewa dalli ije jaju an haeyo.
Unlike when I was a child, I rarely do it now.
도시와는 달리, 시골에는 차가 많지 않아요.
Dosiwaneun dalli, sigoreneun chaga manchi anayo.
Unlike in the city, there aren't many cars in the country.
어제와는 달리 오늘은 집에 일찍 들어왔어요.
Eojewaneun dalli oneureun jibe iljjik deureowasseoyo.
Unlike yesterday, I got home early.
##
Linking Sentences—Comparisons: As
돈을 물처럼 썼어요.
Doneul mulcheoreom sseosseoyo.
He spent money like water.
Very easy and very common, this expression simply means that B resembles A in some way. Grammatically, it turns a noun into an adverbial phrase. It can usually be replaced by 같이 if you prefer to do it that way. There are many common expressions with 처럼: like 좀처럼 jomcheoreom (like the first time), 전처럼 jeoncheoreom (like before) or 평소처럼 pyeongsocheoreom (as usual).
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 한 것처럼 han geotcheoreom | 하는 것처럼 haneun geotcheoreom | 할 것처럼 hal geotcheoreom
Action verbs ending in a consonant | 먹다 | 먹은 것처럼 meogeun geotcheoreom | 먹는 것처럼 meogneun geotcheoreom | 먹을 것처럼 meogeul geotcheoreom
Nouns ending in a vowel | 남자 | 남자처럼 namjacheoreom | 남자처럼 | 남자처럼
Nouns ending in a consonant | 물 | 물처럼 mulcheoreom | 물처럼 | 물처럼
TAKE NOTE
It's common to put 마치 machi before A. 마치 ~처럼 means "just like A."
EXAMPLE SENTENCES
그녀는 남자 처럼 행동해요.
Geunyeoneun namja cheoreom haengdonghaeyo.
She acts like a man.
우리는 옛날 처럼 말을 탔어요.
Urineun yennal cheoreom mareul tasseoyo.
We rode horses just like in the olden days.
그 열차가 너무 천천히 가서 마치 걷는 것 처럼 오래 걸려요.
Geu yeolchaga neomu cheoncheonhi gaseo machi geonneun geot cheoreom orae geollyeoyo.
That train is so slow that riding it takes as long as walking.
남자친구를 처음 만났을 때 마치 하늘을 날 것 처럼 기뻤어요.
Namjachingureul cheoeum mannaseul ttae machi haneureul nal geot cheoreom gippeoseoyo.
The first time I met my boyfriend, I was so happy I felt as if I were floating on air.
민수씨는 가영씨만큼 늦었어요.
Minsussineun Kayoungssimankeum neujeosseoyo.
Minsu was as late as Kayoung.
This is another good one for scientific types. It's used for comparing two things and declaring them to be about equal, and is generally used to express the degree of something.
HOW IT'S FORMED
만큼 is an expression by itself. You can add 도 after 만큼 to express an extreme degree of something.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 한 만큼 han mankeum | 하는 만큼 haneun mankeum | 할 만큼 hal mankeum
Action verbs ending in a consonant | 먹다 | 먹은 만큼 meogeun mankeum | 먹는 만큼 meogneun mankeum | 먹을 만큼 meogeul mankeum
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁜 만큼 yeppeun mankeum | 예쁜 만큼 | 예쁜 만큼
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작은 만큼 jageun mankeum | 작은 만큼 | 작은 만큼
Nouns ending in a vowel | 남자 | 남자만큼 namjamankeum | 남자만큼 | 남자만큼
Nouns ending in a consonant | 물 | 물만큼 mulmankeum | 물만큼 | 물만큼
TAKE NOTE
(으)니만큼 is a rarely used, related expression; see page . It means "inasmuch as" (in speech or in writing) or "since" (mostly in writing; not usually used in speech).
만큼 can follow immediately after a noun to say that something is as (adjective) as another (noun). When used with verbs, be sure to add the tense markers (으)ㄴ/는/(으)ㄹ.
만큼 can be interchangeable with 도록 dorok (see page ) under certain rare circumstances in which 도록 is used to express a limit.
너를 죽을 만큼 사랑한다.
Neoreul jugeul mankeum saranghanda.
너를 죽도록 사랑한다.
Neoreul jugdorok saranghanda.
I'll love you until death.
날씨가 추우니만큼 등산에 가지 말래요.
Nalssiga chuunimankeum deungsane gaji mallaeyo.
Since it's cold, let's not go hiking.
EXAMPLE SENTENCES
그 엄마가 자기 아이만큼 작아요.
Geu eommaga jagi aimankeum jagayo.
That mother is as small as her child.
그 엄마가 자기 아이만큼도 작아요.
Geu eommaga jagi aimankeumdo jagayo.
That mother is as small as her child. (emphasizes how unusually tiny this mom's size is)
음식을 먹을 만큼 가지고 가세요.
Eumsigeul meogeul mankeum gajigo gaseyo.
Take as much food as you're going to eat.
그 맥주가 물만큼 맛이 없어요.
Geu maekjuga mulmankeum masi eobseoyo.
That beer is as tasteless as water.
차가 정말 막혔어요. 두 시간 동안안 움직일 정도였어요.
Chaga jeongmal makhyeosseoyo. Du sigan dongan an umjigil jeongdoyeosseoyo.
Traffic was so congested that I was stopped for two hours.
Do you like to exaggerate? Do you want to make your stories sound more exciting than they really are? The expressions above all mean the same thing and are used to say "B, to the extent that A." They are used in Korean equivalents to English expressions like: "It's raining so hard I can't see. I'm so tired I can't even think."
HOW IT'S FORMED
정도 by itself is a noun that means "degree" and you'll see it used that way as well.
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Action verbs ending in a vowel | 하다 | 할 정도 hal jeongdo
Action verbs ending in a consonant | 먹다 | 먹을 정도 meogeul jeongdo
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁠 정도 yeppeul jeongdo
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작을 정도 jageul jeongdo
TAKE NOTE
You can also talk about 이정도 (this much) or 그 정도 (that much) if you want.
그 정도 돈을 쓰면 안 돼요.
Geu jeongdo doneul sseumyeon an dwaeyo.
You can't spend that much money.
EXAMPLE SENTENCES
아무 것도 안 보일 정도로 비가 많이 와요.
Amu geotdo an boil jeongdoro biga mani wayo.
It's raining so hard I can't see anything.
남자친구와 헤어져서 죽을 정도로 슬퍼요.
Namjachinguwa heeojyeoseo jugeul jeongdoro seulpeoyo.
Since I broke up with my boyfriend, I've been so sad I could die.
늦었지만 정시에 온 듯이 행동했어요.
Neujeotjiman jeongsie on deusi haengdonghaesseoyo.
Although I came late, I acted as if I'd been there on time.
These expressions compare B to A and are often used in similes. If you feel like getting poetic and talking, for example, about "tears flowing like rain," you can use these expressions. They're very similar with just a few differences, so I've grouped them together.
HOW IT'S FORMED
This comes from the verb 듯하다, which means "to be like." You take away 하다 to form a noun, 듯, and then change the verb into an adverb using the 이 ending (covered on page ), which turns A into an adverbial expression talking about how things are done. The 이 ending is optional.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 하듯(이) hadeus(i)
한 듯(이) han deus(i)
| 하듯(이)
하는 듯(이) haneun deus(i)
| 하듯(이)
할 듯(이) hal deus(i)
Action verbs ending in a consonant | 먹다 | 먹듯(이) meokdeus(i)
먹은 듯(이) meogeun deus(i)
| 먹듯(이)
먹는 듯(이) meogneun deus(i)
| 먹듯(이)
먹을 듯(이) meogeul deus(i)
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁘듯(이) yeppeudeus(i) | 예쁘듯(이) | 예쁘듯(이)
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작듯(이) jakdeus(i) | 작듯(이) | 작듯(이)
TAKE NOTE
When used to talk about events in the future, (으)ㄴ/는/(으)ㄹ 듯(이) can be used only with action verbs. 듯(이) all by itself can't be used after nouns; 인 듯(이) can, but almost never is, so you'd be better off using 처럼 (see page 125) if you want to compare something to a noun.
는 듯이 can also be added to the indirect speech particles (ㄴ/는)다/(이/으)라/자 (see page ) to mean "as if to say" or "acting as if."
말싸움 후에 제가 아무 일도 없다는 듯이 했어요.
Mmalssaum hue jega amu ildo eobsdaneun deusi haesseoyo.
After the argument, I acted as if nothing had happened.
머리를 좋아한다는 듯이 여자친구의 머리 스타일이 예쁘다고 칭찬 했어요.
Meorireul joahandaneun deusi yeojachinguui meori seutairi yeppeudago chingchanhaesseoyo.
He praised his girlfriend's hairstyle as if he liked it. (The implication here being that he didn't.)
EXAMPLE SENTENCES
학생들이 쥐 죽은 듯이 조용했어요.
Haksaengdeuri jwi jugeun deusi joyonghaesseoyo.
학생들이 쥐 죽을 듯이 조용했어요.
Haksaengdeuri jwi jugeul deusi joyonghaesseoyo.
The students were (as) silent (as a dead mouse).
그는 그녀를 미친 듯이 사랑했어요.
Geuneun geunyeoreul michin deusi saranghaesseoyo.
그는 그녀를 미칠 듯이 사랑했어요.
Geuneun geunyeoreul michil deusi saranghaesseoyo.
He loved her like crazy. (He loved her as if he were crazy.)
선생님으로서 학생들에게 참 엄격해요.
Seonsaengnimeuroseo haksaengdeurege cham eomgyeokhaeyo.
As a teacher, he is very strict.
This one is for those who like to pull rank. (으)로서 means "as" or "in the capacity of." A is what the person or object in question is, and B is what results from them being A. It's another one that's easy to use but hard to explain. It's the expression used in the Korean equivalent of English sentences like: "As a doctor, I advise you to stop smoking. As an artist, he is a total failure."
HOW IT'S FORMED
(으)로 (eu)ro means "through" and 서 comes from 아/어/여서 (see page ) and is followed by a result. In other words, through your capacity as A, you have the authority to say B.
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Nouns ending in a vowel | 남자 | 남자로서 namjaroseo
Nouns ending in a consonant | 선생님 | 선생님으로서 seonsaengnimeuroseo
TAKE NOTE
Don't confuse it with (으)로써 (eu)rosseo, which has a totally different meaning (see page ).
EXAMPLE SENTENCES
대통령으로서 중요한 결정을 자주 해요.
Daetonglyeongeuroseo jungyohan gyeoljeongeul jaju haeyo.
As President, he often makes important decisions.
그 허브가 약으로서 효력이 없어요.
Geu heobeuga yageuroseo hyolyeogi eobseoyo.
That herb is ineffective as a medicine.
늦을까 봐 경주하다시피 해서 빨리 갔어요.
Neujeulkka bwa gyeongjuhadasipi haeseo ppalli gasseoyo.
I was worried I'd be late, so I raced there (went quickly, as if I were racing).
Do you live a busy life full of endless work and Korean language studies? Are you heavily into dieting or physical fitness? Alternatively, do you enjoy exaggerating the extent of your suffering? This expression is used when doing something as if you were nearly doing something else such as "studying as if your life depended on it" or "eating nearly nothing."
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 하다시피 했다 hadasipi haessda | 하다시피 하다 hadasipi hada | 하다시피 할 것이다 hadasipi hal geosida
Action verbs ending in a consonant | 먹다 | 먹다시피 했다 meokdasipi haessda | 먹다시피 하다 meokdasipi hada | 먹다시피 할 것이다 meokdasipi hal geosida
TAKE NOTE
다시피 all by itself is a different expression which means "as you." Don't get them confused. 다시피 is covered on page . It's also far more common, so if you hear anyone talking about 다시피 anything, it's most likely that other 다시피.
EXAMPLE SENTENCES
민수가 우리 집에서 살다시피 해요.
Minsuga uri jibeseo saldasipi haeyo.
Minsu practically lives at our house.
그는 싸움 후에 죽다시피 했어요.
Geuneun ssaum hue jukdasipi haesseoyo.
After the fight, he was almost dead.
##
Linking Sentences—Comparisons: Equals (or Not)
그 컴퓨터가 여전히 빨라서 새것이나 다름없어요.
Geu keompyuteoga yeojeonhi ppallaseo saegeosina dareumeobseoyo.
That computer is still fast, so it's just like a new one.
This literally means "there is no difference between A and B" and can be used when you want to say that one thing is acting just like another one.
HOW IT'S FORMED
(이)나 means "or something" (see page ) and 다름 comes from 다르다, "to be different." 없다 is, of course, "there isn't."
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Nouns ending in a vowel | 남자 | 남자나 다름없었다 namjana dareumeobseossda | 남자나 다름없다 namjana dareumeobsda | 남자나 다름없을 것이아 namjana dareumeobseul geosia
Nouns ending in a consonant | 물 | 물이나 다름없었다 murina dareumeobseossda | 물이나 다름없다 murina dareumeobsda | 물이나 다름없을 것이다murina dareumeobseul geosida
TAKE NOTE
B is usually a general category and nothing specific. It would be ungrammatical to say "There is no difference between 민수 and 가영," but you can say "가영 is just like a sister to me."
민수가 가영이나 다름없어요. (ungrammatical)
Minsuga Kayoungina dareumeobseoyo.
There is no difference between Minsu and Kayoung.
가영이 예쁜 것은 언니나 다름없어요.
Kayoungi yeppeun geoseun eonnina dareumeobseoyo.
Kayoung's beauty is just like that of her elder sister.
EXAMPLE SENTENCES
이모가 나를 잘 챙겨줘서 어머니나 다름없어요.
Imoga nareul jal chaenggyeojwoseo eomeonina dareumeobseoyo.
My aunt took good care of me, so she was just like a mother to me.
그는 한국어를 잘 하고 매운 음식을 잘 먹고 절도 할 줄 알아서 한국 사람이나 다름없어요.
Geuneun hangukeoreul jal hago maeun eumsigeul jal meokgo jeoldo hal jul araseo hanguk sarami na dareumeobseoyo.
He speaks Korean well, eats spicy food and also bows, so he is just like a Korean.
그 컴퓨터가 새로 산 것 못지않게 빨라요.
Geu keompyuteoga saero san geot motjianke ppallayo.
That computer is as good as a new one.
Remember how your teachers always told you not to use a double negative? In Korean, you can! You're finally free! This expression means that "A is roughly as good as B." It can also be used with question words like 무엇, 누구, or 어디, in which case it means "better than anything/anyone/anywhere."
HOW IT'S FORMED
못 is usually used to mean "can't." No doubt you're familiar with expressions such as "한국말을 잘 못 해요 hangukmareul jal mot haeyo." 지 않다 ji anta is another familiar expression that means "isn't." In this particular expression, the 지 and 않다 are usually put together with no space in between. Finally, 게 turns words into adverbs (쉽게 swipge, 빠르게 ppaleuge, etc.) The negatives basically cancel each other out and leave us an adverbial phrase that means "A is as good as B."
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Nouns ending in a vowel | 남자 | 남자 못지않게 namja motjianke
Nouns ending in a consonant | 물 | 물 못지않게 mul motjianke
TAKE NOTE
못지 않게 and 다름 없다 dareum eobsda are quite similar. 못지 않다 motji anta means "as good as" or "as (adjective) as."
한국 음식은 프랑스 음식 못지않게 맛있어요.
Hanguk eumsigeun peuranseu eumsig motjianke masisseoyo.
Korean food is as delicious as French food.
다름 없다 dareum eobsda means "There is no difference." So you can't say 한국 음식은 프랑스 음식과 다름없이 맛있어요 Hanguk eumsigeun peuranseu eumsikgwa dareumeobsi masisseoyo because they are both delicious, but they are different. A better sentence would be:
경기도 음식과 서울 음식은 다름없어요.
Gyeonggi-do eumsikgwa Seoul eumsigeun dareumeobseoyo.
Gyeonggi-do food is not different from Seoul food.
In that case the two items being compared are quite similar, so it's appropriate to use 다름없다. The sentence could also be written with 못지 않게.
You can either explain how A and B are similar in a clause after 못지않게 or just end the sentence right there (못지않아요) if you prefer to be cryptic.
EXAMPLE SENTENCES
이모가 어머니 못지않게 나를 잘 챙겼어요.
Imoga eomeoni motjianke nareul jal chaenggyeosseoyo.
My aunt took good care of me, so she was as good as a mother to me.
그는 한국어를 잘 하고 매운 음식을 잘 먹고 절도 해서 한국 사람 못지않아요.
Geuneun Hangukeoreul jal hago maeun eumsigeul jal meokgo jeoldo haeseo hanguk saram motjianayo.
He speaks Korean well, eats spicy food and also bows, so he is just like a Korean.
민수가 누구 못지않게 축구를 잘 해요.
Minsuga nugu motjianke chukgureul jal haeyo.
Minsu plays soccer better than anyone.
일찍 오는 것이 늦게 오는 것만 못해요.
Iljjig oneun geosi neutge oneun geotman mothaeyo.
Being early is not as good as coming late.
Here's an expression for people who like to criticize. It means "A isn't as good as B."
HOW IT'S FORMED
만 means "only" and 못하다 means "can't." To translate very roughly, A can't keep up with B.
HOW IT'S CONJUGATED
만 못하다 is normally used in the present tense.
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | A하는 것이 haneun geosi B하는 것만 못했다 haneun geotman mothaetda | A하는 것이 B하는 것만 못하다 | A하는 것이 B하는 것만 못하겠다
Action verbs ending in a consonant | 먹다 | A먹는 것이 meokneun geosi B먹는 것만 못했다 meokneun geotman mothaetda | A먹는 것이 B먹는 것만 못하다 | A먹는 것이 B먹는 것만 못하겠다
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | A 예쁜 것이 yeppeun geosi B예쁜 것만 못했다 yeppeun geotman mothaetda | A 예쁜 것이 B예쁜 것만 못하다 | A 예쁜 것이 B예쁜 것만 못할 것이다
Descriptive verbs (adjectives) ending in a consonant | 작다 | A 작은 것이 jageun geosi B 작은 것만 못했다 jageun geotman mothaetda | A 작은 것이 B 작은 것만 못하다 | A 작은 것이 B 작은 것만 못할 것이다
Nouns ending in a vowel | 남자 | 그 남자가 저 남자만 못했다 geu namjaga jeo namjaman mothaetda | 그 남자가 저 남자만 못하다 geu namjaga jeo namjaman mothada | 그 남자가 저 남자만 못하겠다 geu namjaga jeo namjaman mothagetda
Nouns ending in a consonant | 물 | 그 물이 저 물만 못했다 geu muri jeo mulman mothaetda | 그 물이 저 물만 못하다 geu muri jeo mulman motada | 그 물이 저 물만 못하겠다 geu muri jeo mulman motagetda
TAKE NOTE
Keep an eye out for 못하다; A이/가 B만하다 (see farther down this page) means "A is the size of B."
EXAMPLE SENTENCES
쓰기는 민수가 가영만 못해요.
Sseugineun Minsuga Kayoungman motaeyo.
Kayoung is better than Minsu at writing.
90년대의 컴퓨터가 현대의 컴퓨터만 못해요.
Gusipnyeondae-ui keompyuteoga hyeondae-ui keompyuteoman motaeyo.
Computers in the nineties weren't as good as modern ones.
제 고양이가 개만해요.
Je goyangiga gaemanhaeyo.
My cat is as big as a dog.
This is used to compare the size of two objects. It's used in many idiomatic expressions which you can use if you like to exaggerate.
HOW IT'S CONJUGATED
All you have to do here is to substitute the two nouns you're comparing for A and B, choose the appropriate subject marker, and then conjugate 하다 whichever way you want.
| | Past | Present | Future
---|---|---|---|---
Nouns ending in a vowel | 크기 | 크기만했다 keugimanhaetda | 크기만하다 keugimanhada | 크기만할 것이다 keugimanhal geosida
Nouns ending in a consonant | 집 | 집만했다 jipmanhaetda | 집만하다 jipmanhada | 집만할 것이다 jipmanhal geosida
TAKE NOTE
There's another expression, (으)ㄹ 만하다 (see page ) which is used to say that something is worth doing. That one goes after verbs. This one goes after nouns.
Yet another expression with 만 (it's a very common particle) is ~(기)만 하다 (see page ), which means "to only do ~"). First of all, it has a space between 만 and 하다, while the expression we're discussing here doesn't. Secondly, it follows verbs: the only time you'll see it after a noun is when that noun is usually followed by 하다. So that 만 하다 could follow 운동 (운동하다) undong (undonghada) or 공부 (공부하다) gongbu (gongbuhada), but not just any noun.
EXAMPLE SENTENCES
Because this is quite often used in idiomatic expressions, I'm going to focus on a few of these.
방이 손바닥만해요.
Bangi sonbadakmanhaeyo.
The room is tiny. (Literally: the size of the palm of one's hand)
방이 운동장만해요.
Bangi undongjangmanhaeyo.
The room is huge. (Literally: the size of a playing field)
월급이 쥐꼬리만해요.
Wolgeubi jwikkorimanhaeyo.
My salary is small. (Literally: the size of a mouse's tail)
눈이 단춧구멍만해요.
Nuni danchutgumeongmanhaeyo.
His eyes are tiny. (Literally: the size of buttons)
얼굴이 주먹만해요.
Eolguri jumeokmanhaeyo.
His face is tiny. (Literally: the size of a fist)
목소리가 모기 소리만해요.
Moksoriga mogi sorimanhaeyo.
His voice is tiny. (Literally: like the sound of a mosquito)
You can use it in your own expressions, too, if you want. Check the example at the beginning of this section.
##
Thinking: Making Guesses
One thing you may notice if you spend much time around Koreans is that they don't like to directly express their opinion. This is reflected in this section which, as you may have noticed, is the biggest in the entire book. Compare it to the Knowing section, which only has a few expressions. In other words, if you want to be able to speak without sounding like a know-it-all, this is a good section to study.
There are a few expressions which belong more appropriately in the "plans" section, but which can also be used for making guesses. All the expressions with 터이다 teo-ida (테다 teda, 테니까 tenikka, 텐데 tende, 테면 temyeon, 테고 tego and 테지만 tejiman) can be used for making guesses about someone's intentions when the subject of the sentence is in the third person. These expressions can be found from pages 218–225. 셈치고 semchigo (see page ) is another expression that can be used to make plans or guesses. 셈치고 has to do with doing something based on another assumption (if we assume that A, then B) and so it's in the section on "if."
All the expressions below are used to guess or to say that something "looks like it may do something."
민수가 늦을 것 같아요.
Minsuga neujeul geot gatayo.
It seems like Minsu will be late.
This expression is very similar to 듯하다 deutada and functions in much the same way: to comment on a situation you are observing or have observed.
HOW IT'S FORMED
The tense markers (으)ㄴ/는/(으)ㄹ; plus 것, which transforms a verb into a noun; plus 같다, which means "to be the same as." Now, I realize it may not seem to make much sense to transform a noun into a verb and then back again as I've done below, but that's just how the expression goes. 남자 같다 namja gatda can also be used, but it's a little different and appears more often in the middle of a sentence like "남자 같은 여자들 namja gateun yeojadeul" ("women who are like men"). I discussed it on page . (으)ㄹ 것 같다 ends a clause and you can then end the sentence altogether or move on to the next clause.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 한 것 같다 han geot gatda | 하는 것 같다 haneun geot gatda | 할 것 같다 hal geot gatda
Action verbs ending in a consonant | 먹다 | 먹은 것 같다 meogeun geot gatda | 먹는 것 같다 meokneun geot gatda | 먹을 것 같다 meogeul geot gatda
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁜 것 같다 yeppeun geot gatda | 예쁜 것 같다 | 예쁠 것 같다 yeppeul geot gatda
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작은 것 같다 jageun geot gatda | 작은 것 같다 | 작을 것 같다 jageul geot gatda
Nouns ending in a vowel | 남자 | 남자인 것 같다 namja-in geot gatda | 남자인 것 같다 | 남자일 것 같다 namja-il geot gatda
Nouns ending in a consonant | 물 | 물인 것 같다 murin geot gatda | 물인 것 같다 | 물일 것 같다 muril geot gatda
TAKE NOTE
같다 has many uses outside this expression, but they all have similar meanings, so you shouldn't get too confused. 같이 means "together" while 같은, as discussed above and on page , can be used to either compare two items (usually nouns) or to list examples from a given category.
It can usually be replaced by 나 보다, (으)ㄴ/는/(으)ㄹ 듯하다 deuthada or 는 모양이다 neun moyangida without any change in meaning.
While it can be conjugated with any tense marker, the expression, like most other guessing expressions, is most often used with the future tense marker (으)ㄹ.
EXAMPLE SENTENCES
그 여자가 한국사람일 것 같아요.
Geu yeojaga hanguksaramil geot gatayo.
That woman seems like a Korean.
눈이 올 것 같아요.
Nuni ol geot gatayo.
It seems like it will snow.
Here are some examples of the other 같다s to help you see the differences in usage:
같이 갑시다.
Gachi gapsida.
Let's go together.
남자 친구와 같이 영화를 봤어요.
Namja chinguwa gachi yeonghwareul bwasseoyo.
I saw a movie (together) with my boyfriend.
맥주 같은 주류를 너무 마시지 마세요.
Maekju gateun juryureul neomu masiji maseyo.
Don't drink too many alcoholic drinks like beer.
그 어른 같은 아이는 항상 심각한 표정을 하고 있어요.
Geu eoreun gateun a-i-neun hangsang simgakan pyojeongeul hago isseoyo.
That child who seems like an adult always has a very serious face.
민수가 늦나 봐요.
Minsuga neunna bwayo.
It looks like Minsu is late.
This is one of the most common of the expressions used for guessing, and you'll hear Koreans use it all the time. It's used for making assumptions about situations. When you add 려 in front of it, you're going even deeper: not only making an assumption based on what you see, but on what you think someone is planning. If you ever find yourself battling an evil Korean genius, you'll probably want to know this expression.
HOW IT'S FORMED
Remember our question forms 나 and (으)ㄴ가? If not, check page . They mean "Is it?" or "Does it?" They are used here again to begin this expression and are then followed by 보다 ("to see"). It means there is a little hesitation about what's being observed. (으)려 is short for (으)려고 하다 (see page ), which has to do with plans.
HOW IT'S CONJUGATED
나 follows action verbs and past tense forms. (으)ㄴ가 follows adjectives in the present or future tense.
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했나 보다 haenna boda | 하나 보다 hana boda | 하려나 보다 haryeona boda
Action verbs ending in a consonant | 먹다 | 먹었나 보다 meogeotna boda | 먹나 보다 meogna boda | 먹으려나 보다 meogeuryeona boda
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤나 보다 yeppeotna boda | 예쁜가 보다 yeppeunga boda | 예쁜가 보다
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았나 보다 jagatna boda | 작은가 보다 jageunga boda | 작은가 보다
Nouns ending in a vowel | 남자 | 남자였나 보다 namjayeotna boda | 남자인가 보다 namja-inga boda | 남자인가 보다
Nouns ending in a consonant | 물 | 물이었나 보다 murieotna boda | 물인가 보다 muringa boda | 물인가 보다
TAKE NOTE
It's interchangeable with (으)ㄴ/는/(으)ㄹ 것 같다 (eu)n/neun/(eu)r geot gatda, (으)ㄴ/는/(으)ㄹ모양이다 (eu)n/neun/(eu)r moyangida, and (으)ㄴ/는/(으)ㄹ듯하다 (eu)n/neun/(eu)r deuthada (see pages 137, 145 and 153). Like these, it is used to make a guess based on a situation that is being observed.
으려나 보다 euryeona boda is used to talk about other people's plans, not your own; you probably won't need to make guesses about your own plans. If your own plans are tentative, you can use (으)ㄹ까 보다/하다/싶다 (see page ) to express that.
(으)려나 보다 is also used to talk specifically about plans, and not anything else. Thus, it indicates less certainty than either 나 보다 by itself or (으)ㄹ 것 같다.
(으)려나 all by itself at the end of a sentence is a 반말 way to wonder about something: for example, 나중에 비가 오려나? means "I wonder if it will rain later?"
You can add a 긴 before 나/(으)ㄴ가 보다 and then repeat the verb; this isn't very common, but you can do it to add emphasis. It works like this:
아이들이 숙제를 잘 못하는 걸 보니 이 단어를 아직 이해하기가 어렵긴 어려운가 봐요.
A-i-deuri sukjereul jal motaneun geol boni i daneoreul ajig ihaehagiga eoryeopgin eoryeo-unga bwayo.
The children didn't do well on their homework, so it seems the vocabulary is still difficult for them.
마이크씨가 한국어 말하기 능력이 좋아지는 걸 보니 공부를 열심히 하긴 했나 봐요.
Maikeussiga hangukeo malhagi neunglyeogi johajineun geol boni gongbureul yeolsimhi hagin haetna bwayo.
Because Mike's Korean speaking ability has improved, I guess he's been studying hard.
This 긴 ~ 나/(으)ㄴ가 보다 expression usually comes after an observation followed by (으)ㄴ/는 걸 보니, which is an abbreviation of (으)ㄴ/는 것을 보니까. ([으]ㄴ/는 것 turns the verb into a noun, 보다 shows that you observed it, and 니까 means "so.")
EXAMPLE SENTENCES
그 여자가 한국사람인가 봐요.
Geu yeojaga hanguksaraminga bwayo.
That woman seems like a Korean.
눈이 왔나 봐요.
Nuni watna bwayo.
It looks like it snowed.
민수 손이 작은가 봐요.
Minsu soni jageunga bwayo.
Minsu's hands look small.
민수가 다른 약속도 있어서 늦게 오려나 봐요.
Minsuga dareun yaksokdo isseoseo neutge oryeona bwayo.
Minsu has another engagement, so I think he's planning to come late.
가영이 자주 여행 웹사이트에 방문하는데 여행을 가려나 봐요.
Kayoungi jaju yeohaeng websaiteue bangmunhaneunde yeohaengeul garyeona bwayo.
Kayoung often looks at travel websites. I think she's planning to go on a trip.
최근 여기에 공사가 많은데 새로운 가게를 내려나 봐요.
Choegeun yeogie gongsaga manheunde saeroun gagereul naeryeona bwayo.
There's a lot of construction here lately. Maybe they're planning to open a new store.
택시를 탔으면 늦지 않았을 거예요.
Taeksireul tasseumyeon neutji anasseul geoyeyo.
택시를 탔으면 늦지 않았을 거야.
Taeksireul tasseumyeon neutji anasseul geoya.
If you'd taken a taxi, you wouldn't have been late.
This expression is used when you're making a guess about something that's already been done, and it translates directly as "will have." For example: "She'll have arrived at the airport by now" or "You must have been famished!" It can also be used hypothetically, similar to English "would have" in sentences such as "If my father were still alive, he would have been happy to meet his grandson" or "If you had taken a taxi, you wouldn't have been late."
HOW IT'S FORMED
The past tense 았/었/였 is placed before the future tense 을 것이다. It's the other way around in English, but the meaning is the same: will have been, will have done, will have gone, etc.)
HOW IT'S CONJUGATED
| | Past
---|---|---
Action verbs with 오 or 아 | 가다 | 갔을 거예요 gasseul geoyeyo
갔을 거야 gasseul geoya
Action verbs with 어, 우, 으 or 이 | 먹다 | 먹었을 거예요 meogeosseul geoyeyo
먹었을 거야 meogeosseul geoya
하다 | 하다 | 했을 거예요 haesseul geoyeyo
했을 거야 haesseul geoya
Descriptive verbs (adjectives) with 오 or 아 | 예쁘다 | 예뻤을 거예요 yeppeosseul geoyeyo
예뻤을 거야 yeppeosseul geoya
Descriptive verbs (adjectives) with 어, 우, 으 or 이 | 작다 | 작았을 거예요 jagasseul geoyeyo
작았을 거야 jagasseul geoya
Nouns ending in a vowel | 남자 | 남자였을 거예요 namjayeosseul geoyeyo
남자였을 거야 namjayeosseul geoya
Nouns ending in a consonant | 물 | 물이었을 거예요 murieosseul geoyeyo
물이었을 거야 murieosseul geoya
TAKE NOTE
This expression goes really well with 았/었/였다면 (see page ).
Remember that 것이다 conjugates irregularly; in 존댓말 it becomes 거예요, and in 반말 it becomes 거야. Both forms are shown in the table above.
EXAMPLE SENTENCES
민수가 유명한 식당에서 저녁을 먹었어요. 진짜 맛있었을 거예요.
Minsuga yumyeonghan sikdangeseo jeonyeogeul meogeosseoyo.
Jinjja masisseosseul geoyeyo.
민수가 유명한 식당에서 저녁을 먹었어. 진짜 맛있었을 거야.
Minsuga yumyeonghan sikdangeseo jeonyeogeul meogeosseo.
Jinjja masisseosseul geoya.
Minsu ate dinner at a very famous restaurant. It must have been good.
백마 탄 왕자를 오래 기다리지 않았으면 벌써 결혼했을 거예요.
Baekma tan wangjareul orae gidariji anasseumyeon beolsseo gyeolhonhaesseul geoyeyo.
백마 탄 왕자를 오래 기다리지 않았으면 벌써 결혼했을 거야.
Baekma tan wangjareul orae gidariji anasseumyeon beolsseo gyeolhonhaesseul geoya.
If you hadn't waited so long for Prince Charming, you'd be married by now.
그 치마 입을 때 가영이 정말 예뻐 보여요.
Geu chima ibeul ttae Kayoungi jeongmal yeppeo boyeoyo.
Kayoung looks really beautiful in (while wearing) that skirt.
This is a simple, common way to say that something looks a certain way.
HOW IT'S FORMED
보이다 means "to appear."
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Descriptive verbs (adjectives) with 어, 우, 이 or 으 | 넓다 neolpda | 넓어 보이다 neolbeo bo-ida
Descriptive verbs (adjectives) with 오 or 아 | 작다 | 작아 보이다 jaga bo-ida
TAKE NOTE
Be sure to use 어/아 보이다; 아/어/여 보다 means "to try" (see page ).
EXAMPLE SENTENCES
가구를 치우니 이 방이 넓어 보였어요.
Gagureul chiuni i bangi neolbeo boyeosseoyo.
When we took out the furniture, this room looked really big.
그 책은 초등학생 읽기로는 너무 어려워 보여요.
Geu chaegeun chodeunghaksaeng ilkgironeun neomu eoryeowo boyeoyo.
This book looks too difficult for elementary school students.
A: 민수가 왜 아직 안 와요? Minsuga wae ajig an wayo? Why isn't Minsu here yet?
B: 글쎄요. 교통이 막히는지 늦어요. Geulsseyo. Gyotongi makineunji neujeoyo. Who knows? Maybe traffic is heavy, so he's late.
This is a good expression for people who can't stand not knowing or who like to start rumors. It's used to hypothesize that maybe A was the cause of B where B is a result that you can see for yourself. 아/어/여서 그런지 is an extended form which is interchangeable with (으)ㄴ/는지 by itself. At the end of a sentence, you can use 아/어/여서 그를 것이다 instead.
HOW IT'S FORMED
아/어/여서 그런지 is 아/어/여서 plus 그렇다 ("to be like that") plus ㄴ지. The ㄴ지 part makes this expression into a guess: "maybe." Put it all together and you get "A, so maybe it's like that", where A is the reason for whatever question or statement your listener just made. In 그럴 것이다, you're adding the future tense ㄹ 것이다 to 그렇다.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했는지 haetneunji
해서 그런지 haeseo geureonji
해서 그럴 거예요 haeseo geureol geoyeyo
| 하는지 haneunji
해서 그런지 해서 그럴 거예요
| 하는지 해서 그런지 해서 그럴 거예요
Action verbs ending in a consonant | 먹다 | 먹었는지 meogeotneunji
먹어서 그런지 meogeoseo geureonji
먹어서 그럴 거예요 meogeoseo geureol geoyeyo
| 먹는지 meogneunji
먹어서 그런지 먹어서 그럴 거예요
| 먹는지 먹어서 그런지 먹어서 그럴 거예요
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤는지 yeppeotneunji
예뻐서 그런지 yeppeoseo geureonji
예뻐서 그럴 거예요 yeppeoseo geureol geoyeyo
| 예쁜지 yeppeunji
예뻐서 그런지 예뻐서 그럴 거예요
| 예쁜지 예뻐서 그런지 예뻐서 그럴 거예요
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았는지 jagatneunji
작아서 그런지 jagaseo geureonji
작아서 그럴 거예요 jagaseo geureol geoyeyo
| 작는지 jakneunji
작아서 그런지 작아서 그럴 거예요
| 작는지 작아서 그런지 작아서 그럴 거예요
Nouns ending in a vowel | 남자 | 남자였는지 namjayeotneunji
남자라서 그런지 namjaraseo geureonji
남자라서 그럴 거예요 namjaraseo geureol geoyeyo
| 남자인지 namjainji
남자라서 그런지 남자라서 그럴 거예요
| 남자인지 남자라서 그런지 남자라서 그럴 거예요
Nouns ending in a consonant | 물 | 물이었는지 murieotneunji
물이라서 그런지 muriraseo geureonji
물이라서 그럴 거예요 muriraseo geureol geoyeyo
| 물인지 murinji
물이라서 그런지 물이라서 그럴 거예요
| 물인지 물이라서 그런지 물이라서 그럴 거예요
TAKE NOTE
아/어/여서 그런지 is normally used while speaking and can't be used for commands, suggestions, or questions.
EXAMPLE SENTENCES
A: 민수가 왜 아직 안 와요? Minsuga wae ajig an wayo? Why isn't Minsu here yet?
B: 글쎄요. 교통이 막혀서 그런지 늦어요. Geulsseyo. Gyotongi makyeoseo geureonji neujeoyo. Who knows? Maybe traffic is heavy, so he's late.
__________
A: 민수가 왜 아직 안 와요? Minsuga wae ajig an wayo? Why isn't Minsu here yet?
B: 글쎄요. 교통이 막혀서 그럴 거예요. Geulsseyo. Gyotongi makyeoseo geureol geoyeyo. Who knows? Maybe traffic is heavy.
__________
A: 오늘 지하철에서 사람들이 별로 없네요. Oneul jihacheoreseo saramdeuri byeollo eobsneyo.
B: 오늘 명절이라서 사람들이 고향에 갔는지 지하철이 한산해요. Oneul myeongjeoriraseo saramdeuri gohyange ganneunji jihacheori hansanhaeyo.
A: 오늘 지하철에서 사람들이 별로 없네요. Oneul jihacheoreseo saramdeuri byeollo eobsneyo.
B: 오늘 명절이라서 사람들이 고향에 가서 그런지 지하철이 한산해요. Oneul myeongjeoriraseo saramdeuri gohyange gatneunji jihacheori hansanhaeyo.
A: Today there's almost no one on the subway!
B: Since today's a national holiday, maybe everyone went home, so the subway is quiet.
__________
A: 가영씨가 오늘 안 왔어요. 무슨 일 있어요? Kayoungssiga oneul an wasseoyo. Museun il isseoyo?
B: 가영씨가 아픈지 집에서 쉬고 있어요. Kayoungssiga apeunji jibeseo swigo isseo.
A: 가영씨가 오늘 안 왔어요. 무슨 일 있어요? Kayoungssiga oneul an wasseoyo. Museun il isseoyo?
B: 가영씨가 아파서 그런지 집에서 쉬고 있어요. Kayoungssiga apaseo geureonji jibeseo swigo isseoyo.
A: Kayoung didn't come today. Did something happen? B: Maybe she's sick and stayed at home.
민수가 늦을 모양이에요.
Minsuga neujeul moyang-i-eyo.
It seems like Minsu will be late.
This is another expression just like (으)ㄴ/는/(으)ㄹ 것 같다, 나 보다 and (으)ㄴ/는/(으)ㄹ 듯하다. It is used for making guesses about a situation you've observed.
HOW IT'S FORMED
Tense markers are followed by 모양, which is a noun that means "shape." So something is, was, or will be "in the shape of."
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 한 모양이다 han moyangida | 하는 모양이다 haneun moyangida | 할 모양이다 hal moyangida
Action verbs ending in a consonant | 먹다 | 먹은 모양이다 meogeun moyangida | 먹는 모양이다 meokneun moyangida | 먹을 모양이다 meogeul moyangida
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁜 모양이다 yeppeun moyangida | 예쁜 모양이다 | 예쁜 모양이다
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작은 모양이다 jageun moyangida | 작은 모양이다 | 작은 모양이다
Nouns ending in a vowel | 남자 | 남자인 모양이다 namjain moyangida | 남자인 모양이다 | 남자인 모양이다
Nouns ending in a consonant | 물 | 물인 모양이다 murin moyangida | 물인 모양이다 | 물인 모양이다
TAKE NOTE
It's interchangeable with (으)ㄴ/는/(으)ㄹ 것 같다, 나/(으)ㄴ가/려나 보다 and (으)ㄴ/는/(으)ㄹ 듯하다 (see pages 137, 138 and 153 respectively). Like these, it is used to make a guess based on a situation that is being observed.
EXAMPLE SENTENCES
그 여자가 한국사람인 모양이에요.
Geu yeojaga hanguksaramin moyang-i-eyo.
That woman seems Korean.
눈이 올 모양이에요.
Nuni ol moyangieyo.
It looks like it will snow.
아직 배 고파요? 점심을 적게 먹은 모양이에요.
Ajik bae gopayo? Jeomsimeul jeokge meogeun moyang-i-eyo.
Are you still hungry? It seemed that you ate only a little for lunch.
그 남자가 요트도 가지고 있어요? 돈이 많은 모양이에요.
Geu namjaga yoteudo gajigo isseoyo? Doni manheun moyangieyo.
Does he have a yacht? He seems rich.
늦을까 봐 집에 일찍 출발했어요.
Neujeulkka bwa jibe iljjik chulbalhaesseoyo.
I was worried I'd be late, so I left the house early.
Are you a perpetually anxious person? Do you like to overprepare for everything so that you'll be ready for absolutely any situation that comes up? If so, this is an expression for you. This is used to say that you were worried about A, so you did B.
HOW IT'S FORMED
(으)ㄹ까 is an expression that means "shall" and is commonly used in (으)ㄹ까요 (see page ) and (으)ㄹ까 말까 (see page ). 보다 as used here means "to make a guess." 서 is optional and doesn't change the meaning of the expression.
HOW IT'S CONJUGATED
| | Past | Present
---|---|---|---
Action verbs ending in a vowel | 하다 | 했을까 봐(서) haesseulkka bwa(seo) | 할까 봐(서) halkka bwa(seo)
Action verbs ending in a consonant | 먹다 | 먹었을까 봐(서) meogeosseulkka bwa(seo) | 먹을까 봐(서) meogeulkka bwa(seo)
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤을까 봐(서) yeppeosseulkka bwa(seo) | 예쁠까 봐(서) yeppeulkka bwa(seo)
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았을까 봐(서) jagasseulkka bwa(seo) | 작을까 봐(서) jagasseulkka bwa(seo)
Nouns ending in a vowel | 남자 | 남자였을까 봐(서) namjayeosseulkka bwa(seo) | 남자일까 봐(서) namjailkka bwa(seo)
Nouns ending in a consonant | 물 | 물이었을까 봐(서) murieosseulkka bwa(seo) | 물일까 봐(서) murilkka bwa(seo)
TAKE NOTE
This expression can be used only when the action in B is something you're already doing or have done. You can't use this expression if you're talking about something you're only planning to do.
(으)ㄹ까 봐요 at the end of a sentence is not the same expression. It's used to talk about a plan that's not yet certain (see page ).
(으)ㄹ까 봐 is interchangeable with (으)ㄹ까 싶어 and (으)ㄹ지도 몰라 (see page 160) although the latter is used only for things about which one is uncertain and not specifically for worries like (으)ㄹ까 봐.
EXAMPLE SENTENCES
비가 올까 봐 우산을 가지고 왔어요.
Biga olkka bwa usaneul gajigo wasseoyo.
I was worried it would rain, so I brought an umbrella.
맛이 없을까 봐서 요리 할 때 음식 맛을 계속 봤어요.
Masi eobseulkka bwaseo yori hal ttae eumsik maseul gyesok bwasseoyo.
I was worried the food wouldn't taste good, so I kept tasting it while I was cooking.
교육에서는 가장 중요한 것이 부모님의 지지라고 봐요.
Gyoyugeseoneun gajang jungyohan geosi bumonimui jijirago bwayo.
교육에서는 가장 중요한 것은 부모님의 지지라고 생각해요.
Gyoyugeseoneun gajang jungyohan geosi bumonimui jijirago saenggakhaeyo.
교육에서는 가장 중요한 것은 부모님의 지지라고 믿어요.
Gyoyugeseoneun gajang jungyohan geosi bumonimui jijirago mideoyo.
I think the most important thing in education is the parents' support.
If you're ever debating in Korean and need to know how to express your opinion while sounding knowledgeable, here it is. Here are some ways to give your opinion about important topics like the economy, politics, global warming, or world peace.
HOW IT'S FORMED
Indirect speech particles 라고, 다고 and the rest (see page ) are followed by 보다 (to see), 생각하다 (to think) or 믿다 (to believe), which means that "this is how you see or what you think of the situation."
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했다고 보다 haetdago boda
했다고 생각하다 haetdago saenggakhada
했다고 믿다 haetdago mitda
| 한다고 보다 handago boda
한다고 생각하다 handago saenggakhada
한다고 믿다 handago mitda
| 할 거라고 보다 hal georago boda
할 거라고 생각하다 hal georago saenggakhada
할 거라고 믿다 hal georago mitda
Action verbs ending in a consonant | 먹다 | 먹었다고 보다 meogeotdago boda
먹었다고 생각하다 meogeotdago saenggakada
먹었다고 믿다 meogeotdago mitda
| 먹는다고 보다 meongneundago boda
먹는다고 생각하다 meongneundago saenggakada
먹는다고 믿다 meongneundago mitda
| 먹을 거라고 보다 meogeul georago boda
먹을 거라고 생각하다 meogeul georago saenggakada
먹을 거라고 믿다 meogeul georago mitda
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤다고 보다 yeppeotdago boda
예뻤다고 생각하다 yeppeotdago saenggakada
예뻤다고 믿다 yeppeotdago mitda
| 예쁘다고 보다 yeppeudago boda
예쁘다고 생각하다 yeppeudago saenggakada
예쁘다고 믿다 yeppeudago mitda
| 예쁠 거라고 보다 yeppeul georago boda
예쁠 거라고 생각하다 yeppeul georago saenggakada
예쁠 거라고 믿다 yeppeul georago mitda
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았다고 보다 jagatdago boda
작았다고 생각하다 jagatdago saenggakada
작았다고 믿다 jagatdago mitda
| 작다고 보다 jakdago boda
작다고 생각하다 jakdago saenggakada
작다고 믿다 jakdago mitda
| 작을 거라고 보다 jageul georago boda
작을 거라고 생각하다 jageul georago saenggakada
작을 거라고 믿다 jageul georago mitda
Nouns ending in a vowel | 남자 | 남자라고 보다 namjarago boda
남자라고 생각하다 namjarago saenggakada
남자라고 믿다 namjarago mitda
| 남자라고 보다 남자라고 생각하다 남자라고 믿다 | 남자라고 보다 남자라고 생각하다 남자라고 믿다
Nouns ending in a consonant | 물 | 물이라고 보다 murirago boda
물이라고 생각하다 murirago saenggakada
물이라고 믿다 murirago mitda
| 물이라고 보다 물이라고 생각하다 물이라고 믿다 | 물이라고 보다 물이라고 생각하다 물이라고 믿다
TAKE NOTE
(이)라고/(ㄴ/는)다고 보다 is for important topics and used mainly in important discussions. Don't use it to talk about what you thought of that dress 혜진 Hyejin was wearing yesterday. (이)라고/(ㄴ/는)다고 생각하다 would be better for that.
These are all used to state your opinion rather than an objectively verifiable fact. For a fact, you can use a more decisive ending.
A word about 생각하다 and 믿다: the former can be used for both positive and negative things while 믿다 isn't normally used for negatives. So in the first example below, you can use either 생각하다 or 믿다, but you wouldn't say the opposite using 믿다.
EXAMPLE SENTENCES
올해는 경제가 좋아질 거라고 봐요.
Olhaeneun gyeongjega joajil georago bwayo
올해에 경제가 좋아질 거라고 생각해요.
olhaee gyeongjega joajil georago saenggakaeyo.
올해에 경제가 좋아질 거라고 믿어요.
Olhaee gyeongjega joajil georago mideoyo.
I think the economy will improve next year.
올해는 경제가 나빠질 거라고 봐요.
Olhaeneun gyeongjega nappajil georago bwayo.
올해에 경제가 나빠질 거라고 생각해요.
Olhaee gyeongjega nappajil georago saenggakaeyo.
I think the economy will improve next year.
(Here you shouldn't use 믿어요 since you're talking about a negative situation).
요즘 경제가 좋아진다고 봐요.
Yojeum gyeongjega joajindago bwayo.
요즘 경제가 좋아진다고 생각해요.
Yojeum gyeongjega joajindago saenggakaeyo.
요즘 경제가 좋아진다고 믿어요.
Yojeum gyeongjega joajindago mideoyo.
I think the economy is improving these days.
지난해에 경제가 좋아졌다고 봐요.
Jinanhaee gyeongjega joajyeotdago bwayo.
지난해에 경제가 좋아졌다고 생각해요.
Jinanhaee gyeongjega joajyeotdago saenggakaeyo.
올해에 경제가 좋아질 거라고 믿어요.
Olhaee gyeongjega joajil georago mideoyo.
I think the economy improved last year.
나중에 커피를 마셨으면 좋겠다고 생각해요.
Najunge keopireul masyeoseumyeon joketdago saenggakaeyo.
I think it would be nice to have some coffee later on.
(can be said with 생각하다, but not 보다 because it's hardly a major world issue)
민수는 회의가 있어서 아마 늦을걸요.
Minsuneun hoeuiga iseoseo ama neujeulgeollyo.
Minsu has a meeting, so maybe he'll be late.
This is for those who hate committing themselves to anything, as well as those who like to speculate on what's happening, whether or not they really know anything about it. If you'd like to speculate and hate commitment, you can use this expression to make a guess about a situation, but it's specifically for guesses that you're not too confident about: something that you think might be true, but you're not really too sure of.
(으)ㄴ/는/(으)ㄹ걸(요) can also be used to deny things, like so:
A: 오늘 늦었어요? Oneul neujeoseoyo? Were you late today?
B: 늦지 않은걸요. 정시에 왔어요. Neutji aneungeollyo. Jeongsie waseoyo. Late? Not at all. I was here on time.
This is used to gently contradict something someone else has said. It's another expression like 기는요 that comes across as being very modest when used to contradict praise or refuse a compliment that someone has given.
HOW IT'S CONJUGATED
In 반말, this expression is simply (으)ㄹ걸.
Note that you conjugate this expression differently when making guesses about the past than you do when denying things in the past. See below.
| | Past (denial) | Past (guess) | Present (denial) | Present/future (guess)
---|---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했는걸요 haenneungeollyo | 했을걸요 haeseulgeollyo | 하는걸요 haneungeollyo | 할걸요 halgeollyo
Action verbs ending in a consonant | 먹다 | 먹었는걸요 meogeonneun-geollyo | 먹었을걸요 meogeoseul-geollyo | 먹는걸요 meongneun-geollyo | 먹을걸요 meogeulgeollyo
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤는걸요 yeppeonneun-geollyo | 예뻤을걸요 yeppeoseul-geollyo | 예쁜걸요 yeppeun-geollyo | 예쁠걸요 yeppeulgeollyo
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았는걸요 jaganneungeollyo | 작았을걸요 jagaseulgeollyo | 작은걸요 jageungeollyo | 작을걸요 jageulgeollyo
Nouns ending in a vowel | 남자 | 남자였는걸요 namjayeonneungeollyo | 남자였을걸요 namjayeoseul-geollyo | 남자인걸요 namjaingeollyo | 남자일걸요 namjailgeollyo
Nouns ending in a consonant | 물 | 물이었는걸요 murieonneun-geollyo | 물이었을걸요 murieoseulgeollyo | 물인걸요 muringeollyo | 물일걸요 murilgeollyo
TAKE NOTE
When using this expression to make guesses, always use it in the future tense with (으)ㄹ걸(요). If you're denying something that happened in the past or present, you can use (으)ㄴ/는걸(요).
Even in 존댓말, this expression is not really used toward people higher in the hierarchy than you are, so be careful.
This is used only for rough guesses; you have some reason to assume whatever it is you're saying (for example, your assumption is based an something you've observed or something someone has told you), use ~(으)ㄴ/는/(으)ㄹ 것 같다 (see page 137), ~(으)ㄴ가/나 보다 (see page 138), ~(으)ㄴ/는/(으)ㄹ 모양이다 (see page 145), or ~(으)ㄴ/는/(으)ㄹ 듯하다 (see page 153).
기는요 (see page ) and (으)ㄴ/는/(으)ㄹ걸요 are both used for similar purposes: they're both gentle ways to contradict people. The main difference between them is that when you use 기는요 you have to state what it is that you're contradicting, whereas with 걸요 you simply say the opposite of what was said. So if someone says you speak Korean well and you don't think you do, you can say 잘 하기는요 jal hagineunnyo or 잘 못하는걸요 jal motaneungeollyo.
Finally, another useful expression with (으)ㄹ 걸 is ~(으)ㄹ 걸 그랬어요; it's worthy of its own section on see page because it's used only to express regret: "I should have gone" or "I shouldn't have eaten all that cake."
EXAMPLE SENTENCES
■ Guesses
민수가 회의도 있어서 아마 늦을걸요.
Minsuga hoeuido iseoseo ama neujeulgeollyo.
Minsu also has a meeting, so maybe he'll be late.
(You have no solid reason to assume this; if, for example, Minsu called and told you he might be late, you'd say 늦을 것 같아요 neujeul geon gachiyo or 늦나 봐요 neunna bwayo.)
A: 가영이 왜 안 왔어요? Kayoungi wae an waseoyo? Why didn't Kayoung come?
B: 감기를 걸렸을걸요. Gamgireul geollyeoseulgeollyo. Maybe she caught a cold. (You don't know and are just speculating.)
__________
A: 지금 종로에서 교통이 어때요? Jigeum jongnoeseo gyotongi eottaeyo? How's traffic on Jongno right now?
B: 복잡할걸요. Bokjaphalgeollyo. I'd guess it's congested. (You haven't been down Jongno lately or heard any traffic reports.)
■ Denial
A: 한국어를 아주 잘 하시네요! Hangukeoreul aju jal hasineyo! You speak Korean very well!
B: 잘 못하는걸요. Jal motaneungeollyo. I really don't.
__________
A: 한국에서 10월이 더운가요? Hangukeseo shiwori deoungayo? Is Korea hot in October?
B: 10월은 시원할걸요. Shiworeun siwonhalgeollyo. I think October is cool.
지금 10시인데 가영이 공항에 벌써 도착했을걸요
Jigeum yeolsiinde Kayoungi gonghange beolsseo dochakaeseulgeollyo.
It's ten o'clock now, so I think Kayoung will have already arrived at the airport.
무슨 일 있었어요? 회의에 늦었게요.
Museun il iseoseoyo? hoeuie neujeotgeyo.
Did something happen? (I'm guessing so because) you're late to the meeting.
This is an ending you can use to make guesses. A sentence with 게요 normally follows a question. This question is your guess about your listener and the follow-up sentence with 게요 explains why you guessed that. For example: "Do you have a new boyfriend? Because you're sure smiling a lot today."
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했게요 haetgeyo | 하게요 hageyo | 하게요 hageyo
Action verbs ending in a consonant | 먹다 | 먹었게요 meogeotgeyo | 먹게요 meokgeyo | 먹게요 meokgeyo
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤게요 yeppeotgeyo | 예쁘게요 yeppeugeyo | 예쁘게요
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았게요 jagatgeyo | 작게요 jakgeyo | 작게요
Nouns ending in a vowel | 남자 | 남자였게요 namjayeotgeyo | 남자게요 namjageyo | 남자게요
Nouns ending in a consonant | 물 | 물이었게요 murieotgeyo | 물이게요 murigeyo | 물이게요
TAKE NOTE
This is a rarely used, informal expression used primarily in speaking. You'll likely never see it in writing or hear it in formal situations. You're also more likely to hear it in 반말 than in 존댓말.
It's not really the same as 게 in the middle of a sentence; see page .
EXAMPLE SENTENCES
밖에 추워? 손이 정말 차갑게.
Bakke chuwo? Soni jeongmal chagapge.
Is it cold out? (I'm guessing so because) your hands are really cold.
새로운 남자친구 생겼어? 하루 종일 웃고 있었게.
Saeroun namjachingu saenggyeoseo? Haru jongil utgo iseotge.
Did you get a new boyfriend? (I'm guessingso because) you've been smiling all day.
민수가 늦을 듯해요.
Minsuga neujeul deutaeyo.
It looks like Minsu will be late.
This has two similar meanings. One is used to say that something "looks like" something else. The other is used as a prediction of sorts to say that something is "on the verge of" something. Use this expression to speculate on a situation you're observing.
HOW IT'S FORMED
듯하다 is a verb which means "to look like." When combined with the future marker (으)ㄹ, it means "looks like it's about to."
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 한 듯하다 han deutada 한 듯싶다 han deutsipda | 하는 듯하다 haneun deutada 하는 듯싶다 haneun deutsipda | 할 듯하다 hal deutada 할 듯싶다 hal deutsipda
Action verbs ending in a consonant | 먹다 | 먹은 듯하다 meogeun deutada 먹은 듯싶다 meogeun deutsipda | 먹는 듯하다 meongneun deutada 먹는 듯싶다 meongneun deutsipda | 먹을 듯하다 meogeul deutada 먹을 듯싶다 meogeul deutsipda
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁜 듯하다 yeppeun deutada 예쁜 듯싶다 yeppeun deutsipda | 예쁜 듯하다 yeppeun deutada 예쁜 듯싶다 yeppeun deutsipda | 예쁠 듯하다 yeppeul deutada 예쁠 듯싶다 yeppeul deutsipda
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작은 듯하다 jageun deutada 작은 듯싶다 jageun deutsipda | 작은 듯하다 jageun deutada 작은 듯싶다 jageun deutsipda | 작을 듯하다 jageul deutada 작을 듯싶다 jageul deutsipda
Nouns ending in a vowel | 남자 | 남자인 듯하다 namjain deutada 남자인 듯싶다 namjain deutsipda | 남자인 듯하다 namjain deutada 남자인 듯싶다 namjain deutsipda | 남자일 듯하다 namjail deutada 남자일 듯싶다 namjail deutsipda
Nouns ending in a consonant | 물 | 물인 듯하다 murin deutada 물인 듯싶다 murin deutsipda | 물인 듯하다 murin deutada 물인 듯싶다 murin deutsipda | 물일 듯하다 muril deutada 물일 듯싶다 muril deutsipda
TAKE NOTE
It is interchangeable with 나 보다 (see page 138), (으)ㄴ/는/(으)ㄹ 것 같다 (see page 137), and (으)ㄴ/는/(으)ㄹ 모양이다 (eu)n/neun/(eu)r moyangida (see page 145). The only important thing to consider here is that 듯싶다 sentences can't be based on a snap judgement: they are the result of having observed a situation for some time and then making a guess about it.
듯하다 is interchangeable in meaning with 듯싶다 but is used only in writing.
EXAMPLE SENTENCES
그 여자가 한국인인 듯해요.
Geu yeojaga hangukinin deutaeyo.
That woman seems Korean.
눈이 올 듯해요.
Nuni ol deutaeyo.
It looks like it will snow.
민수가 늦을 듯싶어요.
Minsuga neujeul deutsipeoyo.
It looks like Minsu will be late.
그 여자가 한국사람인 듯싶어요.
Geu yeojaga hanguksaramin deutsipeoyo.
That woman seems Korean.
눈이 올 듯싶어요.
Nuni ol deutsipeoyo.
It looks like it will snow.
아침을 아주 늦게 먹었어요. 아점을 먹은 셈이에요.
Achimeul aju neutge meogeoseoyo. Ajeomeul meogeun semieyo.
I ate a very late breakfast. It was sort of a brunch.
Not everything is black and white. If you're one who likes to see the world in shades of grey, this is a good expression for you. It means that something is "kind of but not exactly like something else," or is "almost the same as something else." When used with the future tense, (으)ㄹ 셈이다, it refers to a plan.
HOW IT'S FORMED
셈 is a noun that means "calculation" or "conjecture" by itself. In other words, in this expression, ~ is "calculated" or "conjectured" to be something.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 한 셈이다 han semida | 하는 셈이다 haneun semida | 할 셈이다 hal semida
Action verbs ending in a consonant | 먹다 | 먹은 셈이다 meogeun semida | 먹는 셈이다 meokneun semida | 먹을 셈이다 meogeul semida
TAKE NOTE
Don't confuse 셈 with 셈치; they're two totally different expressions. 셈치 is about a hypothetical situation and is covered on page .
EXAMPLE SENTENCES
중고차를 사서 많이 고쳤어요. 그래서 돈을 많이 썼어요. 새차를 산 셈이에요.
Junggochareul saseo mani gochyeosseoyo. Geuraeseo doneul mani sseosseoyo. Saechareul san semieyo.
I bought a used car and spent a lot of money for repairs. I sort of bought a new car.
(I spent the same amount of money as if I had bought a new car)
제가 갈비 양념을 준비했어요. 가영씨가 요리한 갈비는 제가 만든 셈이에요.
Jega galbi yangnyeomeul junbihaesseoyo. Kayoungssiga yorihan galbineun jega mandeun semieyo.
I prepared the seasoning of the galbi. I made my galbi sort of like Kayoung's.
그가 자신의 생일 파티에도 늦으리라고는 상상도 못했어요.
Geuga jasinui saengil patiedo neujeuriragoneun sangsangdo motaesseoyo.
I never would have thought he'd even be late to his own birthday party.
This is a relatively uncommon expression used to talk about what you believe, think, or never would have thought of. This is a good for expressing outrage.
HOW IT'S FORMED
It's very similar to all the reported speech expressions, and indeed, you can just as well use 다고 생각하다 dago saenggakada, 라고 믿다 rago mitda, or any other verb that has to do with believing/thinking; 생각하다 saenggakada and 믿다 mitda are the most common, but if you have another favorite, feel free to use it instead. Bear in mind that 으리라고 eurirago and 으리라는 euriraneun are a little stronger than other reported speech expressions..
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했으리라고 haesseurirago 했으리라는 haesseuriraneun | 하리라고 harirago 하리라는 hariraneun | 하리라고 하리라는
Action verbs ending in a consonant | 먹다 | 먹었으리라고 meogeosseurirago 먹었으리라는 meogeosseuriraneun | 먹으리라고 meogeurirago 먹으리라는 meogeuriraneun | 먹으리라 고 먹으리라 는
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤으리라고 yeppeoseurirago 예뻤으리라는 yeppeosseuriraneun | 예쁘리라고 yeppeurirago 예쁘리라는 yeppeuriraneun | 예쁘리라 고 예쁘리라 는
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았으리라고 jagasseurirago 작았으리라는 jagasseuriraneun | 작으리라고 jageurirago 작으리라는 jageuriraneun | 작으리라 고 작으리라 는
Nouns ending in a vowel | 남자 | 남자였으리라고 namjayeosseurirago 남자였으리라는 namjayeosseuriraneun | 남자리라고 namjarirago 남자리라는 namjariraneun | 남자리라 고 남자리라 는
Nouns ending in a consonant | 물 | 물이었으리라고 murieosseurirago 물이었으리라는 murieosseuriraneun | 물으리라고 mureurirago 물으리라는 mureuriraneun | 물으리라 고 물으리라 는
TAKE NOTE
A is what you believe, think, or never would have expected, and B is a verb like 생각하다 or 믿다. You can also change B into a negative expression such as "상상도 못 했어요 sangsangdo mot haesseoyo" ("I never could have imagined") or "생각도 못했어요 saenggakdo motaesseoyo" ("I never would have thought"). In this case (으)리라고 is usually followed by 는 and the 고 can be dropped if you'd like. So you can use (으)리라고, (으)리라고는, or (으)리라는.
EXAMPLE SENTENCES
열심히 공부했으니까 시험을 잘 보리라고 생각해요.
Yeolsimhi gongbuhaesseunikka siheomeul jal borirago saenggakaeyo.
Since you studied so hard, I think you'll do well on the test.
열심히 공부했으니까 시험을 잘 보리라는 생각해요.
Yeolsimhi gongbuhaesseunikka siheomeul jal boriraneun saenggakaeyo.
Since you studied so hard, I think you'll do well on the test.
그녀가 그렇게 과속을 계속 하면 교통 사고가 나리라고 생각해요.
Geunyeoga geureoke gwasogeul gyesok hamyeon gyotong sagoga narirago saenggakaeyo.
그녀가 그렇게 과속을 계속 하면 교통 사고가 나리라는 생각해요.
Geunyeoga geureoke gwasogeul gyesok hamyeon gyotong sagoga nariraneun saenggakaeyo.
If she keeps driving that fast, I'm sure she'll have an accident.
늦으리라고는 상상도 못했어요.
Neujeuriragoneun sangsangdo motaesseoyo.
I never would have imagined he'd be late.
그녀가 그렇게 예쁘리라고는 생각도 못했어요.
Geunyeoga geureoke yeppeuriragoneun saenggakdo motaesseoyo.
I never would have thought she'd be this pretty.
그녀가 아직 살아있으리라고는 꿈도 못 꾸었어요.
Geunyeoga ajik sara-isseuriragoneun kkumdo mot kkueosseoyo.
I couldn't have dreamed she'd still be alive.
##
Thinking: Knowing
어제 민수가 늦었잖아요.
Eoje Minsuga neujeotjanayo.
As you know, Minsu was late yesterday.
Are you surrounded by incompetent people? Do you frequently need to remind people of things they either already know or really should know? English does this with tone of voice, but Koreans have a specific expression for it, and that expression is 잖아(요). It can be used to chide, but also to remind people of a fact. If you've ever wondered why Koreans often say "as you know" even when you clearly don't know whatever it is they're saying, it's because they're directly translating from a Korean sentence with 잖아요.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했잖아요 haetjanayo | 하잖아요 hajanayo | 하잖아요
Action verbs ending in a consonant | 먹다 | 먹었잖아요 meogeotjanayo | 먹잖아요 meokjanayo | 먹잖아요
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤잖아요 yeppeotjanayo | 예쁘잖아요 yeppeujanayo | 예쁘잖아요
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았잖아요 jagatjanayo | 작잖아요 jakjanayo | 작잖아요
Nouns ending in a vowel | 남자 | 남자였잖아요 namjayeotjanayo | 남자잖아요 namjajanayo | 남자잖아요
Nouns ending in a consonant | 물 | 물이었잖아요 murieotjanayo | 물이잖아요 murijanayo | 물이잖아요
EXAMPLE SENTENCES
A: 가영은 오늘 왜 일 안 해요? Kayoungeun oneul wae il an haeyo? Why isn't Kayoung working?
B: 가영이 머리가 아프잖아요. Kayoungi meoriga apeujanayo. She has a headache, as you know.
얘들아! 교실에서 뛰면 안 되잖아.
Yaedeura! Gyosireseo ttwimyeon an doejana.
Children! You know you're not allowed to run in the classroom.
요리를 할 줄 알아요.
yorireul hal jul arayo.
I know how to cook.
요리를 할 줄 몰라요.
yorireul hal jul mollayo.
I don't know how to cook.
This is used to talk about either knowing (알다) or not knowing (모르다) how to do something.
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Action verbs ending in a vowel | 하다 | 할 줄 알다 hal jul alda 할 줄 모르다 hal jul moreuda
Action verbs ending in a consonant | 먹다 | 먹을 줄 알다 meogeul jul alda 먹을 줄 모르다 meogeul jul moreuda
TAKE NOTE
If you use 알았다 instead of 알다, the meaning changes completely. (으)ㄴ/는/(으)ㄹ 줄 알았다 is an expression that means "you thought you knew something but later found out you were wrong." It's covered on the next page. Similarly, (으)ㄴ/는/(으)ㄹ 줄 몰랐다 (also on page 159) means you didn't know something that you know now.
EXAMPLE SENTENCES
한국어를 쓸 줄 알아요.
Hangukeoreul sseul jul arayo.
I know how to write Korean.
한국어를 쓸 줄 몰라요.
Hangukeoreul sseul jul mollayo.
I don't know how to write Korean.
스키를 탈 줄 알아요.
Seukireul tal jul arayo.
I know how to ski.
스키를 탈 줄 몰라요.
Seukireul tal jul mollayo.
I don't know how to ski.
민수가 회의에 늦게 온 줄 알았는데 민수는 저보다 일찍 왔어요.
Minsuga hoeuie neutge on jul aranneunde Minsuneun jeoboda iljjig wasseoyo.
I thought Minsu was late to the meeting, but he came earlier than me.
Every so often, you need to admit your mistakes. Here's how to say you were wrong.
HOW IT'S FORMED
Unlike the other 줄 알다/모르다, this one is in the past tense. Bear this in mind; it's very important because it means the situation has since changed and you no longer know, or don't know what you thought you once knew or didn't know. In other words, either you found out you were wrong (줄 알았다) or you learned something you previously didn't know (줄 몰랐다).
HOW IT'S CONJUGATED
If you're talking about the future, you're talking about a guess you made that was wrong. (I thought it was going to rain / I didn't know it was going to rain).
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 한 줄 알았다 han jul aratda
한 줄 몰랐다 han jul mollatda
| 하는 줄 알았다 haneun jul aratda
하는 줄 몰랐다 haneun jul mollatda
| 할 줄 알았다 hal jul aratda
할 줄 몰랐다 hal jul mollatda
Action verbs ending in a consonant | 먹다 | 먹은 줄 알았다 meogeun jul aratda
먹은 줄 몰랐다 meogeun jul mollatda
| 먹는 줄 알았다 meongneun jul aratda
먹는 줄 몰랐다 meongneun jul mollatda
| 먹을 줄 알았다 meogeul jul aratda
먹을 줄 몰랐다 meogeul jul mollatda
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁜 줄 알았다 yeppeun jul aratda
예쁜 줄 몰랐다 yeppeun jul mollatda
| 예쁜 줄 알았다 yeppeun jul aratda
예쁜 줄 몰랐다 yeppeun jul mollatda
| 예쁠 줄 알았다 yeppeul jul aratda
예쁠 줄 몰랐다 yeppeul jul mollatda
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작은 줄 알았다 Jageun jul aratda
작은 줄 몰랐다 jageun jul mollatda
| 작은 줄 알았다 jageun jul aratda
작은 줄 몰랐다 jageun jul mollatda
| 작을 줄 알았다 jageul jul aratda
작을 줄 몰랐다 jageul jul mollatda
Nouns ending in a vowel | 남자 | 남자인 줄 알았다 namjain jul aratda
남자인 줄 몰랐다 namjain jul mollatda
| 남자인 줄 알았다 namjain jul aratda
남자인 줄 몰랐다 namjain jul mollatda
| 남자일 줄 알았다 namjail jul aratda
남자일 줄 몰랐다 namjail jul mollatda
Nouns ending in a consonant | 물 | 물인 줄 알았다 murin jul aratda
물인 줄 몰랐다 murin jul mollatda
| 물인 줄 알았다 murin jul aratda
물인 줄 몰랐다 murin jul mollatda
| 물일 줄 알았다 muril jul aratda
물일 줄 몰랐다 muril jul mollatda
TAKE NOTE
Make sure to put 알다 or 모르다 in the past tense; otherwise you'll be using a different expression, (으)ㄹ 줄 알다/모르다 (see page 158) and people do tend to get somewhat confused when you go around saying stuff like "I know how to fail a test!" (시험에 떨어질 줄 알아요 siheome tteoreojil jul arayo) rather than "I thought I had failed the test (but was wrong)" (시험에 떨어질 줄 알았어요 siheome tteoreojil jul arasseoyo). You can use these expressions without putting them in the past tense if you're later continuing your sentence in a way that makes it clear what you meant (see the last example below).
EXAMPLE SENTENCES
시험에 떨어질 줄 알았어요.
Siheome tteoreojil jul allasseoyo
I thought I had failed the test (but I was wrong).
오늘이 휴일인 줄 모르고 학교에 그냥 갔어요.
Oneuri hyuirin jul moreugo hakgyoe geunyang gasseoyo.
I didn't know it was a holiday, so I went to school anyway.
민수가 얼마나 늦을지 아세요?
Minsuga eolmana neujeulji aseyo?
Do you know how late Minsu will be?
민수가 얼마나 늦을지 알아요.
Minsuga eolmana neujeulji arayo.
I know how late Minsu will be.
민수가 늦을지 몰라요.
Minsuga neujeulji mollayo.
I think Minsu may be late.
No one likes to admit they don't know something, but sometimes you have to. On such an occasion (which, I'm sure, will be rare for you) here's how to say you're not sure about something.
HOW IT'S FORMED
(으)ㄴ/는/(으)ㄹ지 is part of a number of expressions having to do with something happening or not. 도, again, is just used as an intensifier here. It goes only with (으)ㄹ지 모르다 and in this case it is optional. 알다 and 모르다 are, of course, the verbs having to do with knowing: 알다 is "to know" and 모르다 is "to not know."
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했는지 알다 haenneunji alda
했는지 모르다 haenneunji moreuda
했을지(도) 모르다 haeseulji(do) moreuda
| 하는지 알다 haneunji alda
하는지 모르다 haneunji moreuda
할지(도) 모르다 halji(do) moreuda
| 할지 알다 halji alda
할지 모르다 halji moreuda
할지(도) 모르다 halji(do) moreuda
Action verbs ending in a consonant | 먹다 | 먹었는지 알다 meogeonneunji alda
먹었는지 모르다 meogeonneunji moreuda
먹었을지(도) 모르다 meogeoseulji(do) moreuda
| 먹는지 알다 meongneunji alda
먹는지 모르다 meongneunji moreuda
먹을지(도) 모르다 meogeulji(do) moreuda
| 먹을지 알다 meogeulji alda
먹을지 모르다 meogeulji moreuda
먹을지(도) 모르다 meogeulji(do) moreuda
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤는지 알다 yeppeonneunji alda
예뻤는지 모르다 yeppeonneunji moreuda
| 예쁜지 알다 yeppeunji alda
예쁜지 모르다 yeppeunji moreuda
| 예쁜지 알다 yeppeunji alda
예쁜지 모르다 yeppeunji moreuda
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았는지 알다 jaganneunji alda
작았는지 모르다 jaganneunji moreuda
| 작은지 알다 jageunji alda
작은지 모르다 jageunji moreuda
| 작은지 알다 jageunji alda
작은지 모르다 jageunji moreuda
Nouns ending in a vowel | 남자 | 남자였는지 알다 namjayeonneunji alda
남자였는지 모르다 namjayeonneunji moreuda
남자였을지(도) 모르다 namjayeoseulji(do) moreuda
| 남자인지 알다 namjainji alda
남자인지 모르다 namjainji moreuda
남자일지(도) 모르다 namjailji(do) moreuda
| 남자인지 알다 남자인지 모르다 남자일지(도) 모르다
Nouns ending in a consonant | 물 | 물이었는지 알다 murieonneunji alda
물이었는지 모르다 murieonneunji moreuda
물이었을지(도) 모르다 murieoseulji(do) moreuda
| 물인지 알다 murinji alda
물인지 모르다 murinji moreuda
물일지(도) 모르다 murilji(do) moreuda
| 물인지 알다 물인지 모르다 물일지(도) 모르다
TAKE NOTE
I've actually combined two different but similar grammar patterns here. (으)ㄹ지(도) 모르다 is used to make guesses. As such, even if you want to make a guess about something that's already finished, you should keep the (으)ㄹ and just put 았/었/였 before it. ㄹ지 (도) 모르다 means "I don't know if" and is more of a guess that means "you think ~ might be true," but aren't really sure.
With ㄴ/는/ㄹ지 알다/모르다, use the tense markers ㄴ/는/ㄹ as usual to form sentences like "I don't know if it (past tense verb)," "I don't know if it's (present tense verb)," or "I don't know if it'll be (verb)."
Because both uses of ㄹ지 모르다 are about a future you can't possibly know yet, there ends up being very little difference between the two ㄹ지 모르다s. With the former you actually have no idea, whereas with the latter you're making a very tentative guess. See the last examples below (about tomorrow's snow and Minsu's tardiness) for more clarification.
EXAMPLE SENTENCES
약국은 어디 있는지 아세요?
Yakgugeun eodi inneunji aseyo?
Do you know where the pharmacy is?
약국은 어디 있는지 알아요.
Yakgugeun eodi inneunji arayo.
I know where the pharmacy is.
약국은 어디 있는지 몰라요.
Yakgugeun eodi inneunji mollayo.
I don't know where the pharmacy is.
가영이 이 책을 벌써 읽었는지 아세요?
Kayoungi i chaegeul beolsseo ilgeonneunji aseyo?
Do you know if Kayoung has already read this book?
가영이 이 책을 벌써 읽었는지 알아요.
Kayoungi i chaegeul beolsseo ilgeonneunji arayo.
I know if Kayoung has already read this book.
가영이 이 책을 벌써 읽었는지 몰라요.
Kayoungi i chaegeul beolsseo ilgeonneunji mollayo.
I don't know if Kayoung has read this book yet.
(And yes, the middle sentences above are just as strange in Korean as they are in English. Presumably you're going on to explain whether or not Minsu will be late, where the pharmacy is, or if Kayoung has read the book in question. However, that would be an entirely different grammatical point, so I'm not going to pursue this here.)
우체국은 일요일에 문을 여는지 안 여는지 몰라요.
Uchegugeun ilyoire muneul yeoneunji an yeoneunji mollayo.
I don't know if the post office is open or not on Sunday.
내일 눈이 올지도 몰라요.
Naeil nuni oljido mollayo.
I don't know if it'll snow tomorrow. (I think it might.)
내일 눈이 올지 안 올지 몰라요.
Naeil nuni olji an olji mollayo.
I don't know if it'll snow tomorrow. (I have no idea.)
민수가 늦을지 안 늦을지 몰라요.
Minsuga neujeulji an neujeulji mollayo.
I don't know if Minsu will be late. (I have no idea.)
민수가 늦을지도 몰라요.
Minsuga neujeuljido mollayo.
I don't know if Minsu will be late. (I think he might be.)
보다시피 민수가 또 늦어요.
Bodasipi Minsuga tto neujeoyo.
As you can see, Minsu is late again.
Here's an easy one that you can use for presentations, if you have to do presentations in Korean. It means "as you" and can be attached only to a few verbs, all of which have to do with gaining information.
HOW IT'S CONJUGATED
다시피 is often seen with 보다, 알다, 듣다, and 배우다. Thus we get the following constructions:
알다시피, 아시다시피
aldasipi, asidasipi
as you know
보다시피, 보시다시피
bodasipi, bosidasipi
as you can see
들다시피, 들으시다시피
deuldasipi, deureusidasipi
as you heard
배웠다시피
baewotdasipi
as you learned
TAKE NOTE
This expression can be used only with verbs having to do with knowing or learning, similar to the ones above.
Remember that when speaking to people higher in rank or status, you need to add 시 to your verbs in order to be polite and respectful. That's why you so often hear verbs ending with "시다시피."
다시피 하다 dasipi hada is a different expression and is covered on page .
EXAMPLE SENTENCES
아시다시피 회의를 다음 월요일에 할 예정이에요.
Asidasipi hoeuireul daeum wollyoire hal yejeongieyo.
As you know, the meeting will be held next Monday.
오늘 배웠다시피 한국어 동사는 다 규칙적으로 활용시킬 수 없어요.
Oneul baewotdasipi hangukeo dongsaneun da gyuchikjeogeuro hwallyongsikil su eopseoyo.
As you learned today, Korean verbs can't all be conjugated regularly.
민수는 항상 늦어요. 오늘도 늦을 게 틀림없어요.
Minsuneun hangsang neujeoyo oneuldo neujeulge teullimeobseoyo.
Minsu is always late. I'm quite sure he'll be late today as well.
This expression straddles the border between thinking and knowing. It's used to make a guess about something which you feel to be very certain.
HOW IT'S FORMED
Tense markers (으)ㄴ/는/(으)ㄹ plus 것이 (which can be shortened to 게). 틀리다 means "to be wrong" and is a word which you'll probably hear all too often in the course of your Korean studies. Such is education. 틀림 is the noun form of that verb. 없다, of course, means "there isn't." In other words, you really think that your guess (followed by the appropriate tense marker and nounified by 것이) isn't wrong. Put it all together and you've got "There's no doubt that ~" or "I'm almost certain that ~."
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 한 것이 틀림없다 han geosi teullimeobsda 한 게 틀림없다 han ge teullimeobsda | 하는 것이 틀림없다 haneun geosi teullimeobsda 하는 게 틀림없다 haneun ge teullimeobsda | 할 것이 틀림없다 hal geosi teullimeobsda 할 게 틀림없다 hal ge teullimeobsda
Action verbs ending in a consonant | 먹다 | 먹은 것이 틀림없다 meogeun geosi teullimeobsda 먹은 게 틀림없다 meogeun ge teullimeobsda | 먹는 것이 틀림없다 meogneun geosi teullimeobsda 먹는 게 틀림없다 meogneun ge teullimeobsda | 먹을 것이 틀림없다 meogeul geosi teullimeobsda 먹을 게 틀림없다 meogeul ge teullimeobsda
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁜 것이 틀림없다 yeppeun geosi teullimeobsda 예쁜 게 틀림없다 yeppeun ge teullimeobsda | 예쁜 것이 틀림없다 예쁜 게 틀림없다 | 예쁜 것이 틀림없다 예쁜 게 틀림없다
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작은 것이 틀림없다 jageun geosi teullimeobsda 작은 게 틀림없다 jageun ge teullimeobsda | 작은 것이 틀림없다 작은 게 틀림없다 | 작은 것이 틀림없다 작은 게 틀림없다
Nouns ending in a vowel | 남자 | 남자인 것이 틀림없다 namjain geosi teullimeobsda 남자인 게 틀림 없다 namjain ge teullim eobsda | 남자인 것이 틀림없다 남자인 게 틀림 없다 | 남자인 것이 틀림없다 남자인 게 틀림 없다
Nouns ending in a consonant | 물 | 물인 것이 틀림없다 murin geosi teullimeobsda 물인 게 틀림없다 murin ge teullimeobsda | 물인 것이 틀림없다 물인 게 틀림없다 | 물인 것이 틀림없다 물인 게 틀림없다
TAKE NOTE
If you're not all that certain about making your guess, there are plenty of expressions in the "Guess" section to help you out. ~(으)ㄴ가/나 보다 (see page 138) or (으)ㄹ 것 같다 (see page 137) would be good places to start.
EXAMPLE SENTENCES
그 남자는 약을 먹었어요. 아픈 게 틀림없어요.
Geu namjaneun yageul meogeosseoyo. Apeun ge teullimeobseoyo.
I saw him take some medicine. I'm pretty sure he's sick.
민수가 가영을 항상 쳐다봐요. 사랑하는 것이 틀림없어요.
Minsuga Kayoungeul hangsang chyeodabwayo. Saranghaneun geosi teullimeobseoyo.
Minsu is always staring at Kayoung. I'm certain he's in love.
##
Thinking: Making Decisions
늦지 않게 일찍 준비했어요.
Neutji anke iljjik junbihaesseoyo.
I got ready early so as not to be late.
What is life without purpose? Here's how you can express a purpose or intention in Korean. This expression means you do B so that A can happen.
HOW IT'S FORMED
게 by itself is the particle used to make adverbs, which in this case transforms A into an adverbial clause (purpose clause). 게끔 and 게시리 mean the same but are more emphatic, and 게시리 is more literary in style.
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Action verbs ending in a vowel | 하다 | 하게 hage
하게끔 hagekkeum
하게시리 hagesiri
Action verbs ending in a consonant | 먹다 | 먹게 meokge
먹게끔 meokgekkeum
먹게시리 meokgesiri
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁘게 yeppeuge
예쁘게끔 yeppeugekkeum
예쁘게시리 yeppeugesiri
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작게 jakge
작게끔 jakgekkeum
작게시리 jakgesiri
TAKE NOTE
It's also possible to end your sentence with 게 (followed, of course, by the appropriate level of politeness) if the action taken doesn't really need to be stated. Likewise, you can use 게 하다 to mean "to make something a certain way." Again, the action taken to get the thing that way is implied and/or not important enough to the sentence to be mentioned.
This 게 and the 게 which is added to adjectives to turn them into adverbs are the same, but they are used differently. You can tell them apart from context, in most cases, though sometimes the meanings end up being very nearly the same anyway – for example, the teacher speaking loudly in the second example. In other cases, such as 조용하게 침실에 갔어요 ("I went quietly to the bedroom"), it's clear that you didn't go to the bedroom for the purpose of being quiet.
There is another 게, which is an abbreviation of 것이 (see page ). This may seem confusing at first. The easiest way to tell them apart is to remember that the 게 in question here, which makes words into adverbs,, is always added directly to the end of a verb stem with no space in between while the abbreviation of 것이 is always on its own with a space between it and the word before it. Aside from that, you can tell by context and by what parts of speech are around the two 게s. 는 게, (으)ㄴ 게 or (으)ㄹ 게 will always equal 는 것이, (으)ㄴ 것이, and (으)ㄹ 것이, and will never be the adverbial 게 we're discussing here.
게 and 도록 (see next page) can be used interchangeably as long as you're using 게 to turn a phrase into an adverbial phrase. If it's just changing a single word into an adverb, it can't be changed to 도록.
EXAMPLE SENTENCES
늦지 않게끔 일찍 준비했어요.
Neutji ankekkeum iljjik junbihaesseoyo.
늦지 않게시리 일찍 준비했어요.
Neutji ankesiri iljjik junbihaesseoyo.
I got ready early so as not to be late. (more emphatic than 게 alone)
학생들이 잘 들을 수 있게 큰 목소리로 말할 거예요.
Haksaengdeuri jal deureul su itge keun moksoriro malhal geoyeyo.
I'll speak loudly so the students can hear.
후회하지 않게 최선을 다하세요.
Huhoehaji anke choeseoneul dahaseyo.
Please do your best, so you won't have regrets later.
후회하지 않게끔 최선을 다하세요.
Huhoehaji ankekkeum choeseoneul dahaseyo.
후회하지 않게시리 최선을 다하세요.
Huhoehaji ankesiri choeseoneul dahaseyo.
Please do your best, so you won't have regrets later. (more emphatic)
늦지 않도록 일찍 출발하세요.
Neutji antorok iljjik chulbalhaseyo.
Please leave early so as not to be late.
도록 is another very common expression used to express a purpose. It means to do B so that A can happen.
*Read the paragraph below if you want to find out more about the grammatical explanation behind this; otherwise, go to the next section.*
Remember how 게 turns verbs into adverbs? Well, 도록 does the same thing, although its uses are limited to phrases. Its primary meaning is to turn phrases into adverbial phrases indicating purpose, but it also has a secondary meaning, which is to state a limit to some kind of action. This limit can be based on time, degree, or manner. This simply means turning phrases into different kinds of adverbial phrases.
If you're still confused, take a look at the example sentences below to see how 도록 can be used to turn phrases into adverbial phrases indicating manner, time, degree, or purpose.
HOW IT'S FORMED
도록 is an expression that stands by itself, but you can also use ~도록 하다 which translates to "be sure to~" or "do your best to ~."
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Action verbs ending in a vowel | 하다 | 하도록 hadorok
Action verbs ending in a consonant | 먹다 | 먹도록 meokdorok
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁘도록 yeppeudorok
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작도록 jakdorok
TAKE NOTE
도록 and 게 (see page 166) can be used interchangeably most of the time. The difference is that 게 has a definite goal, whereas 도록 is concerned with the conditions around that goal.
늦지 않게 일찍 준비했어요.
Neutji anke iljjik junbihaesseoyo.
늦지 않도록 일찍 준비했어요.
Neutji antoroj iljjik junbihaesseoyo.
Both expressions mean "I got ready early so as not to be late," but in the first example with 게, the point is to not be late. In the second, the point is your own actions – the way you arrange the situation (maybe getting up early, getting your outfit ready the night before, or whatever actions you have done) such that you're not late. It's a very slight difference that you will almost never need to worry about.
도록 can also be used interchangeably with 기 위해(서) (see next page) as long as the subjects of clause A and clause B are the same.
When used to give a limit, 도록 is interchangeable with 만큼 (see page ). Also, there are a number of idiomatic expressions that go with 도록.
눈이 빠지도록 기다리다
nuni ppajidorok gidarida
to wait until your eyes fall out
눈이 퉁퉁 붓도록 울다
nuni tungtung butdorok ulda
to cry until your eyes are swollen
죽도록 사랑하다
jukdorok saranghada
to love until death
귀가 닳도록 듣다
gwiga daltorok deutda
to listen until your ears are worn out
입이 닳도록 말하다
ibi daltorok malhada
to talk until your mouth is worn out
손이 발이 되도록 빌다
soni bari doedork bilda
to beg until your hands become (as tough as) feet
코가 삐뚤어지도록 술을 마시다
koga ppittureojidorok sureul masida
to drink until your nose is crooked
EXAMPLE SENTENCES
감기에 걸리지 않도록 조심하세요.
Gamgie geolliji antorok josimhaseyo.
Be careful not to catch a cold. (Be careful so that you don't catch a cold.)
운전 할 때 사고가 나지 않도록 하세요.
Unjeon hal ttae sagoga naji antorok haseyo.
Be sure not to have an accident while driving.
월급을 받을 수 있도록 계약을 맺으세요.
Wolgeubeul badeul su itdorok gyeyageul maejeuseyo.
Sign the contract in order to get your salary.
늦지 않기 위해서 일찍 준비했어요.
Neutji anki wihaeseo iljjik junbihaesseoyo.
I got ready early so as not to be late.
If you go out drinking with Koreans, you'll be familiar with this one, whether you realize it or not. It's the same expression as the "위하여! wihayeo!" Koreans use when making a toast. 위해(서) means "for." When we toast in English, we say, for example, "To Chris!" or "To a Happy New Year!" In Korean, they just say "To!" following whatever the speaker proposed a toast to. This is that 위하여.
HOW IT'S FORMED
위하다 wihada actually means "to take care of" and is conjugated like a 하다 verb; hence the addition of 여 in 위하여 and the shorter 위해. 서 comes from 아/어/여서 and is optional. This verb can also be conjugated as 위한 wihan if you want to use it as an adjective (see the examples below). As 위하여, 위해, or 위해서 it changes a phrase into an adverbial phrase.
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Action verbs ending in a vowel | 하다 | 하기 위해(서) hagi wihae(seo)
Action verbs ending in a consonant | 먹다 | 먹기 위해(서) meokgi wihae(seo)
Nouns ending in a vowel | 남자 | 남자를 위해(서) namjareul wihae(seo)
Nouns ending in a consonant | 물 | 물을 위해(서) mureul wihae(seo)
TAKE NOTE
위해(서) can be used only when the subjects of A and B are the same. It is interchangeable with 도록 except that 도록 can be used a little more broadly (see page 167).
위하여 is the exact same expression as 위해 or 위해서, but it is more commonly used in formal situations or in writing.
When 위해(서) is added to nouns, the nouns get the object markers 을/를.
EXAMPLE SENTENCES
■ Examples with (기) 위해(서)
용돈을 받기 위해서 부모님을 도와줬어요.
Yongdoneul batgi wihaeseo bumonimeul dowajwosseoyo.
I helped my parents in order to get an allowance.
여자 친구를 위해서 꽃을 샀어요.
Yeoja chingureul wihaeseo kkocheul sasseoyo.
I bought flowers for my girlfriend.
■ Examples with 위한
여자친구를 위한 꽃이에요.
Yeojachingureul wihan kkochieyo.
They are flowers for my girlfriend.
초보를 위한 문법책을 샀어요.
Choboreul wihan munbeopchaegeul sasseoyo.
I bought a grammar book for beginners.
정시에 도착하는 데 정시에 출발하는 것이 필수적이에요.
Jeongsie dochakaneun de jeongsie chulbalhaneun geosi pilsujeogieyo.
In order to arrive on time, you need to leave on time.
Do you like to give advice whether you're asked for it or not? This expression is used to make a statement about something being good or being bad for something else. The entire sentence ends up to mean "B is good or bad (effective or ineffective) for A."
HOW IT'S FORMED
데 can stand by itself but is more properly followed by 에, and you can add 는 for more emphasis.
HOW IT'S CONJUGATED
| | Past | Present
---|---|---|---
Action verbs ending in a vowel | 하다 | 한 데(에)(는) han de(e)(neun) | 하는 데(에)(는) haneun de(e)(neun)
Action verbs ending in a consonant | 먹다 | 먹은 데(에)(는) meogeun de(e)(neun) | 먹는 데(에)(는) meongneun de(e)(neun)
TAKE NOTE
The expression is similar to 기 위해(서), although somewhat more limited in use because it can only be finished in certain ways. The whole expression needs to end with a phrase that indicates good or bad: 좋다/나쁘다 jota/nappeuda, 도움이 되다/안 되다 doumi doeda/an doeda, 종요하다/중요하지 않다 jongyohada/jungyohaji anda, and 필요하다/필요 없다 pillyohada/pillyo eopda are all possible. Statements indicating effectiveness (효과적이다, 효과가 있다 hyogwa jeogida, hyogwaga itda) are often used.
A(으)ㄴ/는 데 B is another expression which is used to talk about the place where an activity is done or an event takes place. Check for context: this 는 데 will always be followed by a sentence indicating effectiveness or quality, or lack thereof, while the other one will always have to do with places.
EXAMPLE SENTENCES
다이어트를 하는 데 건강하게 먹는 것이 효과적이에요.
Daieoteureul haneun de geonganghage meokneun geosi hyogwajeogieyo.
Eating healthy is important for dieting.
아이들이 학교에 가는 데 큰 차가 있으면 좋아요.
Aideuri hakgyoe ganeun de keun chaga isseumyeon joayo.
It is good to have a big car for children to ride to school.
##
Emotions: Surprise
The most commonly used expression to indicate surprise in Korean is ~네요. I've included 군요 below because, although it isn't always used to express surprise, it does serve that function quite often. You can use 라/다/(으)ㄴ다/는다니요 to express surprise at something that someone said. This expression is covered on page because I felt it more appropriately fits in the section on responding to what other people said.
진짜 늦었네요!
You're really late!
There are many things you'll find surprising while living in Korea. Even if you're not living in Korea, there are many aspects you'll find surprising about the Korean language. How can you express this in Korean? 네요 is here and can be thought of as an exclamation point.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했네요 haenneyo | 하네요 haneyo | 하겠네요 hagenneyo
Action verbs ending in a consonant | 먹다 | 먹었네요 meogeonneyo | 먹네요 meokneyo | 먹겠네요 meokgenneyo
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤네요 yeppeonneyo | 예쁘네요 yeppeuneyo | 예쁘겠네요 yeppeugenneyo
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았네요 jaganneyo | 작네요 jakneyo | 작겠네요 jakgenneyo
Nouns ending in a vowel | 남자 | 남자였네요 namjayeonneyo | 남자이네요 namja-i-neyo | 남자이겠네요 namja-i-genneyo
Nouns ending in a consonant | 물 | 물이었네요 murieonneyo | 물이네요 murineyo | 물이겠네요 murigenneyo
TAKE NOTE
Sentences ending in 네요 are usually quite short. There's no rule about this; it's just how the expression is commonly used. 네요 always goes at the end of a sentence.
Sentences ending in 네(요) are also very commonly used as responses to someone talking. You'll notice in talking with Koreans that they'll keep talking over you with these kinds of responses; this is considered polite because it shows they're actively listening to you.
EXAMPLE SENTENCES
한국어를 잘 하시네요!
Hangukeoreul jal hasineyo!
You speak Korean really well!
맛있네요!
Masinneyo!
It's really delicious!
진짜 늦었군요.
Jinjja neujeotgunyo.
(I've just discovered that) you're late.
진짜 늦었구나.
Jinjja neujeotguna.
Hopefully, with this book and your other Korean studies, you're learning many new things. Sometimes you'll probably want to talk about something new and interesting that you've just discovered. So how can we do that? 군요 and 구나 are verb endings used at the end of a sentence where you talk about something you've just realized or just learned. They often, but not always, convey a feeling of surprise; you can tell by someone's intonation whether they're surprised or simply stating a fact.
HOW IT'S CONJUGATED
| | Past | Present
---|---|---|---
Action verbs ending in a vowel | 하다 | 했군요 haetgunyo
했구나 haetguna
| 하는군요 haneungunyo
하는구나 haneunguna
Action verbs ending in a consonant | 먹다 | 먹었군요 meogeotgunyo
먹었구나 meogeotguna
| 먹는군요 meokneungunyo
먹는구나 meokneunguna
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤군요 yeppeotgunyo
예뻤구나 yeppeotguna
| 예쁘군요 yeppeugunyo
예쁘구나 yeppeuguna
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았군요 jagatgunyo
작았구나 jagatguna
| 작군요 jakgunyo
작구나 jakguna
Nouns ending in a vowel | 남자 | 남자였군요 namjayeotgunyo
남자였구나 namjayeotguna
| 남자이군요 namja-i-gunyo
남자이구나 namja-i-guna
Nouns ending in a consonant | 물 | 물이었군요 murieotgunyo
물이었구나 murieotguna
| 물이군요 murigunyo
물이구나 muriguna
TAKE NOTE
This is one of the rare cases where you can't make the 반말 simply by dropping 요. It's 구나, not 군.
The difference between this and 네요 is that while 네요 indicates only surprise (regardless of when you learned whatever you just stated), 군요/구나 sometimes indicates surprise and sometimes merely indicates that you learned something new.
구나 and 군요 always go at the end of a sentence.
EXAMPLE SENTENCES
한국어를 잘 하시는군요.
Hangukeoreul jal hasineungunyo.
(I've just noticed that) you speak Korean really well.
한국어를 잘 하시는구나.
Hangukeoreul jal hasineunguna.
맛있군요!
Masitgunyo!
(I've just realized that) it's really delicious!
맛있구나!
Masitguna!
그렇군요!
Geureokunyo!
Oh, I see!*
그렇구나!
Geureokuna!
*I highly recommend that you learn this one here and now. It's very useful and even if you never plan to use this expression yourself, you will hear it all the time, so it's important that you know what it means.
##
Emotions: Regret
또 늦었어요. 택시를 탈걸 그랬어요.
Tto neujeosseoyo. Taeksireul talgeol geuraesseoyo.
또 늦었어. 택시를 탈걸.
Tto neujeoseo. Taeksireul talgeol.
I'm late again. I should have taken a taxi.
Life doesn't always go exactly the way you want it to. Sometimes you make mistakes. Sometimes you realize you should have done things differently. Here's one for all you forlorn lovers out there. (으)ㄹ걸 그랬다 means "I should/shouldn't have" and implies regret.
HOW IT'S CONJUGATED
The table below shows 존댓말 and 반말 forms for (으)ㄹ 걸 그랬다 (I should have), 안 ~(으)ㄹ걸 그랬다 (I shouldn't have) and 지 말걸 그랬다 (I shouldn't have).
| | Past
---|---|---
Action verbs ending in a vowel | 하다 | 할걸 그랬다 halgeol geuraetda
할걸 halgeol
안 할걸 그랬다 an halgeol geuraetda
안 할걸 an halgeol
하지 말걸 그랬다 haji malgeol geuraetda
하지 말걸 haji malgeol
Action verbs ending in a consonant | 먹다 | 먹을걸 그랬다 meogeulgeol geuraetda
먹을걸 meogeulgeol
안 먹을걸 그랬다 an meogeulgeol geuraetda
안 먹을걸 an meogeulgeol
먹지 말걸 그랬다 meokji malgeol geuraetda
먹지 말걸 meokji malgeol
TAKE NOTE
There is another expression, (으)ㄹ걸(요), which is used to make a guess (see page ). This second (으)ㄹ걸 is used only in informal speech and has a different intonation: rising instead of falling. When using 존댓말 you'll have no trouble telling them apart because this one ends in 그랬어요 and that one ends with 요 right after 걸; however, listen carefully for the difference in 반말.
This expression can be used only in the past tense and it's only for your own regrets: if you want to say someone else should have done something, you can use 았/었/였으면 좋겠다/좋았을 텐데요.
(으)ㄹ걸 그랬다 is similar to 았/었/였어야 했는데 (see next page). 었어야 했는데 is used to express regret for something that was not done or not accomplished while with (으)ㄹ걸 그랬다 you believe you should have (or shouldn't have) done something differently. This is not a very big difference, so don't worry about it too much.
EXAMPLE SENTENCES
The first sentence in each example is formal; the second is the very informal 반말.
시험에 떨어졌어요. 더 열심히 공부할걸 그랬어요.
Siheome tteoreojyeosseoyo. Deo yeolsimhi gongbuhalgeol geuraesseoyo.
시험에 떨어졌어. 더 열심히 공부할걸.
Siheome tteoreojyeosseo. Deo yeolsimhi gongbuhalgeol.
I failed the test. I should have studied harder.
배가 아파요. 그렇게 많이 먹지 말걸 그랬어요.
Baega apayo. Geureoke mani meokji malgeol geuraesseoyo.
배가 아파요. 그렇게 많이 안 먹을걸 그랬어요.
Baega apayo. Geureoke mani an meogeulgeol geuraesseoyo.
배가 아파. 그렇게 많이 먹지 말걸.
Baega apa. Geureoke mani meokji malgeol.
My stomach hurts. I shouldn't have eaten so much.
또 늦었어요. 택시를 탔어야 했는데.
Tto neujeosseoyo. Taeksireul tasseoya haenneunde.
I'm late again. I should have taken a taxi.
This is another way to express regret. It's quite similar to (으)ㄹ걸 그랬다.
HOW IT'S FORMED
았/었/였 past tense marker plus 어야, which implies an obligation or required condition. Then 하다 is put into the past tense and combined with 는데, which uses A as a background for B (see page ). Put it all together and you get "I should have..., so..."
HOW IT'S CONJUGATED
| | Past
---|---|---
Action verbs ending in a vowel | 하다 | 했어야 했는데 haesseoya haenneunde
Action verbs ending in a consonant | 먹다 | 먹었어야 했는데 meogeosseoya haenneunde
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤어야 했는데 yeppeosseoya haenneunde
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았어야 했는데 jagasseoya haenneunde
Nouns ending in a vowel | 남자 | 남자였어야 했는데 namjayeosseoya haenneunde
Nouns ending in a consonant | 물 | 물이었어야 했는데 murieosseoya haenneunde
TAKE NOTE
This expression can be used only in the first person to express regret for something you yourself have done.
B is either a statement of regret (후회가 돼요 huhwega dwaeyo, 죄송해요 joesonghaeyo, ...) or a statement of what you did instead of A. You can also just trail off after 했는데 just as you can after any other 는데. Because the expression is used to show regret, it works only in the past tense.
It's very similar to (으)ㄹ걸 그랬다, but the difference is that 었어야 했는데 is used to express regret for something that was not done or not accomplished, while with (으)ㄹ걸 그랬다 you believe you should have (or shouldn't have) done something differently. It's not a big difference, as you can see if you compare the sentences below with those on the previous page; I didn't even have to change any of the examples to make both expressions work.
EXAMPLE SENTENCES
시험에 떨어졌어요. 더 열심히 공부했어야 했는데 후회가 돼요.
Siheome tteoreojyeosseoyo. Deo yeolsimi gongbuhaesseoya haenneunde huhwega dwaeyo.
I failed the test. I should have studied harder.
배가 아파요. 그렇게 많이 먹지 않았어야 했는데 음식이 맛있어서 계속 먹었어요.
Baega apayo. Geureoke mani meokji anasseoya haenneunde eumsigi masisseoseo gyesok meogeosseoyo.
My stomach hurts. I shouldn't have eaten so much, but the food was so delicious that I kept on eating.
##
Emotions: Thanks
A very common way to say "thanks to ~" in Korean is ~덕분에 deokbune, which was covered on page in the section on cause and effect (through/because of). 망정 mangjeong can also be used in a couple of different ways to express gratitude.
일찍 출발했기에 망정이지 안 그랬으면 늦을 뻔했어요.
Iljjik chulbalhaeggie mangjeongiji an geuraesseumyeon neujeul ppeonhaeseoyo.
일찍 출발했으니까 망정이지 안 그랬으면 늦을 뻔했어요.
Iljjik chulbalhaesseunikka mangjeongiji an geuraesseumyeon neujeul ppeonhaeseoyo.
We're lucky we left early or we would have been late.
Just as you sometimes wish things had turned out differently, other times you're grateful they worked out the way they did. ~기에 or (으)니(까) plus 망정이다 means "thanks to ~." The two are interchangeable.
HOW IT'S FORMED
기에 roughly means "since" (see page ). 망정 by itself is a noun which can mean thanks to (as seen here) or "I'd rather..." (as seen in the expression (으)ㄹ 망정 on page 182). Think of it as a way of expressing a preference: in 기에 망정이다 you prefer what actually happened to what could have happened, and in A(으)ㄹ 망정 B you prefer A to B, which is what could have happened.
(으)니(까) means "so" and is covered on page . The 까 is optional.
HOW IT'S CONJUGATED
They are conjugated the same way as 기에 and (으)니(까) by themselves, so if you remember how to do these, then you're set. If not, check the table below.
| | Past
---|---|---
Action verbs ending in a vowel | 하다 | 했기에 망정이다 haetgie mangjeongida
했으니(까) 망정이다 haesseuni(kka) mangjeongida
Action verbs ending in a consonant | 먹다 | 먹었기에 망정이다 meogeotgie mangjeongida
먹었으니(까) 망정이다 meogeosseuni(kka) mangjeongida
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤기에 망정이다 yeppeotgie mangjeongida
예뻤으니(까) 망정이다 yeppeosseuni(kka) mangjeongida
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았기에 망정이다 jagatgie mangjeongida
작았으니(까) 망정이다 Jagasseuni(kka) mangjeongida
Nouns ending in a vowel | 남자 | 남자였기에 망정이다 namjayeotgie mangjeongida
남자였으니(까) 망정이다 namjayeosseuni(kka) mangjeongida
Nouns ending in a consonant | 물 | 물이었기에 망정이다 murieotgie mangjeongida
물이었으니(까) 망정이다 murieosseuni(kka) mangjeongida
TAKE NOTE
These expressions are always used for replying to someone else's or question don't just toss them out of nowhere into a conversation.
These expressions and (으)ㄹ 망정 are somewhat similar, but not quite the same. This one means it was fortunate that ~ happened while the other one expresses a preference. You can tell them apart by looking at what comes before 망정: in this case it will always be 기에 or 으니(까), while with the other expression the verb before 망정 will always end in ㄹ.
망정이다 is often changed to 망정이지 and then followed by a B clause including a phrase like 안 그랬으면 or 아니면 ("If it hadn't been that way, then ..."). The B clause often finishes with (으)ㄹ 뻔했다 ("I almost"), which you can find on page . 았/었/였을 것이다 (see page ) can also be used.
EXAMPLE SENTENCES
돈이 있었기에 망정이지 안 그랬으면 먹을 수 없을 뻔했어요.
Doni isseotgie mangjeongiji an geuraesseumyeon meogeul su eopsseul ppeonhaesseoyo.
돈이 있었으니 망정이지 안 그랬으면 먹을 수 없을 뻔했어요.
Doni isseosseuni mangjeongiji an geuraesseumyeon meogeul su eopsseul ppeonhaesseoyo.
Fortunately, I had money or I wouldn't have been able to eat.
돈이 있었기에 망정이지요.
Doni isseotgie mangjeongijiyo.
Fortunately, I had money.
##
Emotions: Desires
지금 가고 싶어요.
Jigeum gago sipeoyo.
I want to go now.
This is the most common and basic way to talk about what you want or to ask what other people want.
HOW IT'S CONJUGATED
If you want to use this expression with an adjective such as "I want to be beautiful," then you have to turn the adjective into a verb using 아/어/여지다. For nouns, you can use 되다: for example, 남자가 되고 싶어요 Namjaga doego sipeoyo.
| | Past | Present
---|---|---|---
Action verbs ending in a vowel | 하다 | 하고 싶었다 hago sipeotda | 하고 싶다 hago sipda
Action verbs ending in a consonant | 먹다 | 먹고 싶었다 meokgo sipeotda | 먹고 싶다 meokgo sipda
TAKE NOTE
If you want to talk about what someone else wants, use 고 싶어하다 instead of 고 싶다.
가고 싶어요.
Gago sipeoyo.
I want to go.
민수는 가고 싶어해요.
Minsuneun gago sipeohaeyo.
Minsu wants to go.
EXAMPLE SENTENCES
저녁을 먹고 싶어요?
Jeonyeogeul meokgo sipeoyo?
Do you want to eat dinner?
어제 자고 싶었어요.
Eoje jago sipeosseoyo.
Yesterday I wanted to sleep.
정시에 왔으면 좋겠어요.
Jeongsie wasseumyeon jokesseoyo.
정시에 왔으면 해요.
Jeongsie wasseumyeon haeyo.
정시에 왔으면 싶어요.
Jeongsie wasseumyeon sipeoyo.
It would have been better if you'd arrived on time.
These are good expressions for daydreamers or people trying to give not-so-subtle hints. The three expressions above all mean exactly the same thing. 았/었/였으면 좋겠다 is by far the most common and is, in fact, probably one of the first expressions you learned while studying beginning Korean. If you didn't, now's your chance!
HOW IT'S FORMED
The past tense marker 았/었/였 is followed by 으면, which means "if." The second part of the expression is 좋다 (to be good) in the future tense with 겠. In other words, "if it had been that way, it would be good." 하다 is plain old 하다 and there's no special reason why it should serve the function it does in this expression, but it does. 싶다 is most commonly seen in 고 싶다 (see the previous page) and expresses a desire. It serves the same function here with 었으면.
HOW IT'S CONJUGATED
| | Past
---|---|---
Action verbs ending in a vowel | 하다 | 했으면 좋겠다 haesseumyeon joketda
했으면 하다 haesseumyeon hada
했으면 싶다 haesseumyeon sipda
Action verbs ending in a consonant | 먹다 | 먹었으면 좋겠다 meogeosseumyeon joketda
먹었으면 하다 meogeosseumyeon hada
먹었으면 싶다 meogeosseumyeon sipda
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤으면 좋겠다 yeppeosseumyeon joketda
예뻤으면 하다 yeppeosseumyeon hada
예뻤으면 싶다 yeppeosseumyeon sipda
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았으면 좋겠다 jagasseumyeon joketda
작았으면 하다 jagasseumyeon hada
작았으면 싶다 jagasseumyeon sipda
Nouns ending in a vowel | 남자 | 남자였으면 좋겠다 namjayeosseumyeon joketda
남자였으면 하다 namjayeosseumyeon hada
남자였으면 싶다 namjayeosseumyeon sipda
Nouns ending in a consonant | 물 | 물이었으면 좋겠다 murieosseumyeon joketda
물이었으면 하다 murieosseumyeon hada
물이었으면 싶다 murieosseumyeon sipda
TAKE NOTE
좋겠다 is, again, the most common way to end this expression, and 았/었/였으면 is by far the most common way to start it; however, if you'd like some variety, you can also use 는다면 좋겠다 or 으면 좋겠다. They all mean the same. 는다면 or 으면 are covered on pages and respectively.)
These expressions are normally used at the ends of sentences, though they don't have to be.
EXAMPLE SENTENCES
값이 더 쌌으면 좋겠어요.
Gapsi deo ssasseumyeon jokesseoyo.
값이 더 쌌으면 해요.
Gapsi deo ssasseumyeon haeyo.
값이 더 쌌으면 싶어요.
Gapsi deo ssasseumyeon sipeoyo.
I wish the price were lower.
이 물이 와인이었으면 좋겠어요.
I muri wainieosseumyeon jokesseoyo.
이 물이 와인이었으면 해요.
I muri wainieosseumyeon haeyo.
이 물이 와인이었으면 싶어요.
I muri wainieosseumyeon sipeoyo.
I wish this water were wine.
최선을 다해 주시기 바랍니다.
Choeseoneul dahae jusigi baramnida.
Please do your best.
If you ever take the subway in Korea, you'll hear this all the time. It's an extremely formal way to express a desire and is usually used in sentences such as "We hope you'll have a pleasant journey."
HOW IT'S FORMED
기 makes a verb into a noun. 바라다 means "to want."
HOW IT'S CONJUGATED
Most importantly, because it's very formal, it's almost always conjugated aŝ기 바랍니다 and the respectful 시 is usually added to the verb before it. It doesn't have to be, but that's how you'll almost always hear it used. It can also be conjugated as 바랄게요 if you don't need to be quite that formal.
| | Present
---|---|---
Action verbs ending in a vowel | 하다 | 하기 바라다 hagi barada
Action verbs ending in a consonant | 먹다 | 먹기 바라다 meokgi barada
EXAMPLE SENTENCES
이해해 주시기 바랍니다.
Ihaehae jusigi baramnida.
Please understand.
안전한 여행이 되시기 바랍니다.
Anjeonhan yeohaengi doesigi baramnida.
안전한 여행이 되기 바랄게요.
Anjeonhan yeohaengi doegi baralggeyo.
(less formal) We wish you a safe journey.
늦게 갈 바에야 차라리 가지 않겠어요.
Neutge gal baeya charari gaji ankesseoyo.
늦게 갈 바에는 차라리 가지 않겠어요.
Neutge gal baeneun charari gaji ankesseoyo.
I'd rather not go at all than be late.
Not all things are created equal, and often you're going to have to make a choice between two (or more) things. Here's how to do that. A(으)ㄹ 바에야 B and A(으)ㄹ 바에는 B are used to express a preference for B over A: B may or may not necessarily be your first choice, but it's better than the alternative A.
HOW IT'S FORMED
(으)ㄹ is the future tense marker, which in this case is used to indicate that the continuation of A is hypothetical; it hasn't happened yet. 바 is a noun that can mean a few things, but in this case it means "situation" or "circumstances." See page for more on 바. 에 is used grammatically to set up A as something to which something else is being compared. 야 is from 아/어/여야 and 는 and is used for emphasis.
HOW IT'S CONJUGATED
| | Present/Future
---|---|---
Action verbs ending in a vowel | 하다 | 할 바에야 hal baeya
할 바에는 hal baeneun
Action verbs ending in a consonant | 먹다 | 먹을 바에야 meogeul baeya
먹을 바에는 meogeul baeneun
TAKE NOTE
B often begins with 차라리 charari ("rather") or a similar expression and can end with the present tense or with 겠다. It doesn't have any past tense ever.
You can replace (으)ㄹ 바에야/바에는 with 기 대신에 gi daesine, which means "instead of," but doesn't, by itself, indicate a preference; see page . It's also interchangeable with 느니, which is covered on page .
EXAMPLE SENTENCES
포기할 바에야 차라리 죽겠어요.
Pogihal baeya charari jukgesseoyo.
포기할 바에는 차라리 죽겠어요.
Pogihal baeneun charari jukgesseoyo.
I'd rather die than give up.
구걸할 바에야 그냥 굶어 죽을 거예요.
Gugeolhal baeya geunyang gulmeo jugeul ggeoyeyo.
구걸할 바에는 그냥 굶어 죽을 거예요.
Gugeolhal baeneun geunyang gulmeo jugeul ggeoyeyo.
I'd rather starve than have to beg.
가지 않을 망정 늦게 가지 않겠어요.
Gaji aneul mangjeong neutge gaji ankesseoyo.
I'd rather not go at all than be late. (I won't be late. I'd rather not go.)
Remember (으)ㄹ 바에야 and 바에는? If not, see the previous page. This is a similar expression, only this time instead of preferring B, you prefer A.
HOW IT'S FORMED
(으)ㄹ is the future tense marker used here because A is something that may happen in the future or may continue happening. 망정 is a noun which doesn't really have a clear English equivalent. It's used to talk about preferences here in (으)ㄹ 망정 or also in 기에 망정이다.
HOW IT'S CONJUGATED
| | Future
---|---|---
Action verbs ending in a vowel | 하다 | 할 망정 hal mangjeong
Action verbs ending in a consonant | 먹다 | 먹을 망정 meogeul mangjeong
TAKE NOTE
Don't confuse this expression with 기에 망정이지, which means that something was lucky (see page 176). In the expression in question here, the verb is always conjugated with (으)ㄹ.
B always ends with the future tense—either 겠다 or (으)ㄹ 것이다—and explains what you will do. See the examples below.
You can replace (으)ㄹ 망정 with (으)ㄴ/는 대신에, which means "instead of," but doesn't, by itself, indicate a preference; see page . In this case you have to change the order of A and B.
포기하는 대신에 죽었을 거예요.
Pogihaneun daesine jugeosseul geoyeyo.
I'd die rather than give up.
EXAMPLE SENTENCES
죽을 망정 포기하지 않을 거예요.
Jugeul mangjeong pogihaji aneul geoyeyo.
I'll die rather than give up.
굶을 망정 구걸하지 않겠어요.
Gulmeul mangjeong gugeolhaji ankesseoyo.
I'll starve rather than have to beg.
마음 같아선 정시에 오고 싶어요.
Maeum gataseon jeongsie ogo sipeoyo.
If it were up to me, I'd be on time (but I can't).
같아선 has three different meanings, which can make it rather confusing. Fortunately, it works only with a relatively small number of words in the A clause, so all you have to do is look at A to understand which of the meanings the sentence refers to.
First, it can follow 마음, 생각, 성질, 욕심, maeum, saenggak, seongji, yoksim, or 기분 gibun. You may recognize these as words having to do with what's on your mind. Any of them can be followed by 같아선 to mean "if it were up to me" B must include 고 싶다. An expression with 같아선 implies that what you want to do is impossible.
마음 같아선 그 남자에게 선물을 사주고 싶어요.
Maeum gataseon geu namjaege seonmureul sajugo sipeoyo.
I want to buy that man a present (but I can't).
생각 같아선 회사를 그만두고 싶지만 할 수 없어요.
Saenggak gataseon hoesareul geumandugo sipjiman hal su eopsseoyo.
I want to quit my job (but I can't).
성질 같아선 예의 없는 아줌마에게 화를 내고 싶어요.
Seongjil gataseon ye-ui eobsneun ajummaege hwareul naego sipeoyo.
If it were up to me, I'd scold that rude ajumma.
The second meaning goes with expressions of time like 요즘 yojeum, 옛날 yennal, or 예전 yejeon, and it simply means "If it were like that time." So 옛날 같아선 yennal gataseon means "if it were like old times." This is often used when talking about things that are now possible but used to be impossible.
옛날 같아선 선생님께 말대꾸하는 건 생각조차 못했어요.
Yennal gataseon seonsaengnimkke maldaekkuhaneun geon saenggakjocha motaesseoyo.
In the old days, you wouldn't even dream of arguing with your teacher.
요즘 같아선 살 맛이 없어요.
Yojeum gataseon sal masi eopsseoyo.
These days I don't even feel like living.
The third meaning works only with two expressions: 올 겨울 ol gyeoul (this winter) and 올 여름 ol yeoreum (this summer). The only sentence it's used in is this:
올 겨울 같아선 정말 못 살겠어요.
Ol gyeoul gataseon jeongmal mot salgesseoyo.
This winter is unbearable.
The implication is that it's unbearable because of the cold (or heat, if you want to talk about 올 여름).
HOW IT'S FORMED
같다 means "to be similar"; here it's conjugated with the 아서 verb ending, which means "so." 는 is added to this for emphasis and then 같아서는 gataseoneun is contracted to 같아선.
HOW IT'S CONJUGATED
There's no need for a conjugation table here. 같아선 follows only nouns and doesn't change regardless of the noun.
TAKE NOTE
마음 and 생각; 마음 is normally used for good things and 생각 for bad. So if you want to punch someone, use 생각 같아선; if you want to hug them, use 마음 같아선.
##
Emotions: Personal Experiences
Focus — 더 deo
Korean has a special particle that is used when talking about things that you personally observed; 더 is that particle. It's not the same as the 더 that means "more"; that 더 always appears as a word all by itself, while the particle 더 appears as part of a grammatical expression. There are a number of these expressions which are all covered in their own individual sections, but for the purpose of easy comparison, I've also listed them all here.
Now for the rules in using 더. First of all, it's easy to conjugate; it doesn't ask you to do anything more with your verbs than to stick it at the end of verbs, and it prefers not to be paired with 았/었/였 or 겠 except in a small number of very specific situations.
Secondly, the only time you can use 더 with first-person expressions is when you're talking about how you felt. You can't use it to talk about what you did yourself except with 았/었/였더니 (see page 185). You can always use 더 to talk about what other people did (second or third person), but you can't use it to talk about how they felt; this is something you can't consider yourself to have personally witnessed in Korean unless maybe you're in some kind of strange sci-fi universe where you can live inside another person's head.
Now let's compare some 더 expressions:
Expression | Page | Explanation | Example
---|---|---|---
A던데 deonde B | 188 | "I noticed that A, so/but/and B." | 민수가 어제 늦었던데 아마 오늘도 늦을걸요. Minsuga eoje neujeotdeonde ama oneuldo neujeulgeolyo. (I noticed that) Minsu was late yesterday, so maybe he'll be late today as well.
~던데(요) ~deonde(yo) | 188 | "I noticed that ~." | 민수가 어제 늦었던데요. Minsuga eoje neujeotdeondeyo. (I noticed that) Minsu was late yesterday.
A더니(만) deoni(man) B | 190 | "I noticed that A, so/but/and B." | 어제는 민수가 늦더니 오늘은 정시에 왔어요. Eojeneun minsuga neutdeoni oneureun jeongsie wasseoyo. Yesterday Minsu was late, but today he arrived on time.
A았/었/였이더니B A at/eot/yeotdeoni B | 190 | "I noticed that A, so/but/and B." | 집에서 늦게 출발했더니 늦게 도착했어요. Jibeseo neutge chulbalhaetdeoni neutge dochakhaesseoyo. I left the house late, so I arrived late.
A더라 deora B | 192 | "I noticed that ~!" | 민수가 어제 정말 늦었더라. Minsuga eoje jeongmal neujeotdeora. (I saw that) Minsu was really late yesterday!
~더라고(요) ~deorago(yo) | 195 | "I noticed that ~." | 민수가 어제 정말 늦었더라고요. Minsuga eoje jeongmal neujeotdeoragoyo. (I saw that) Minsu was really late yesterday.
~더구나/더군요 ~deoguna/deogunnyo | 196 | "I noticed that ~!" | 민수가 어제 정말 늦었더구나. Minsuga eoje jeongmal neujeotdeoguna. (I saw that) Minsu was really late yesterday
~던가요 ~deongayo | 195 | "Did you ~?" | 회의에 늦었던가요? Hoe-ui-e neujeotdeongayo? Were you late to the meeting?
You'll notice that most of them are quite similar in meaning. Pay attention to the exclamation point at the end of some translations: it indicates surprise or wonder whereas the others are just straightforward statements about things you noticed. 던데 and 더니 are 더 combined with ㄴ데 and then 니, so if you know how to use ㄴ데 and 니, you can probably figure out how to use 던데 and 더니. They're used to make a statement about something you saw and then followed up with whatever you think follows from that first statement. They're expressions that are flexible to use.
There are a few other expressions which use 더, but with none of its implications: it's just a part of the expression and can be used whether or not you're talking about something you personally saw. These expressions are 던, 더라도, and 았/었/였더라면. They don't belong here and will not be covered in this section, but you can look them up on pages , , and , respectively.
빨리 와 보세요.
Ppalli wa boseyo.
Please try to come quickly.
Are you into experiments? If not, there's always 아/어/여 보지 않아요 a/eo/yeo boji anayo. Otherwise 아/어/여 보다 means "to try."
HOW IT'S FORMED
아/어/여 is the verb ending which means that the verb carries over to the next verb, and 보다 means "to see." In other words, "to do and see" or "to try."
HOW IT'S CONJUGATED
If you want to use a noun with this, you'll have to add a verb after it. For example, "try some kimchi" would be 김치를 먹어 보세요 Kimchireul meogeo boseyo.
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in 아 or 오 | 잡다 | 잡아 봤다 jaba bwatda | 잡아 보다 jaba boda | 잡아 볼 것이다 jaba bol geosida
Action verbs ending in 어, 우, 으 or 이 | 먹다 | 먹어 봤다 meogeo bwatda | 먹어 보다 meogeo boda | 먹어 볼 것이다 meogeo bol geosida
하다 | 하다 | 해 봤다 hae bwatda | 해 보다 hae boda | 해 볼 것이다 hae bol geosida
TAKE NOTE
The meaning of this expression changes a little depending on tense: in the past tense you're talking about a previous experience, and 아/어/여 보다 has a similar meaning to (으)ㄴ 적이 있다 (eu)n jeogi itda (see page 187). In the present it's usually a recommendation, like "try some kimchi" above. In the future you can use it to talk about something you plan to do.
There are many words which already have 아/어/여 보다 built into them. Some examples:
물어보다
mureoboda
to ask
알아보다
araboda
to recognize
One useful conjugation of 아/어/여 보다 is 아/어/여 보니. It's a combination of 아/어/여 보다 and 니(까) (see page ), and it means "to try and then discover..."
그는 만나 보니 아주 친절한 사람이에요.
Geuneun manna boni aju chinjeolhan saramieyo.
When I met him, I discovered that he was really kind.
김치를 먹어 보니 맛이 없었어요.
Kimchireul meogeo boni masi eopseosseoyo.
When I tried kimchi, I found it wasn't very tasty.
It's similar to 고 보니 (see page ) and 다 보니 (see page ), but the difference is in timing. A고 보니 B means you did something, finished doing it, and then discovered B. A아/어/여 보니 B means that you tried it: whether you finished or not, or how many times you tried doesn't really enter into the expression. A 다 보니 B means you did something repeatedly and then noticed B.
김치를 먹어 보니 맛이 없었어요.
Kimchireul meogeo boni masi eopseosseoyo.
When I tried kimchi, it wasn't tasty.
김치를 먹고 보니 배가 아팠어요.
Kimchireul meokgo boni baega apasseoyo.
After I ate kimchi, my stomach hurt.
김치를 먹다 보니 맛있어요.
Kimchireul meokda boni masisseoyo.
After repeatedly eating kimchi, I started to find it tasty.
EXAMPLE SENTENCES
김치를 먹어 봤어요?
Kimchireul meogeo bwasseoyo?
Have you tried kimchi?
김치를 먹어 볼 거예요.
Kimchireul meogeo bol geoyeyo.
I'm going to try kimchi.
스키는 재미있으니까 한번 해 보세요.
Seukineun jaemi-isseunikka hanbeon hae boseyo.
Skiing is fun. You should try it!
수업에 늦은 적이 없어요.
Sueobe neujeun jeogi eopsseoyo.
I've never been late to class.
Here's an expression that's perfect for braggarts or drinking games. You can use it to talk about what you have or have never done.
HOW IT'S FORMED
First, add the past tense marker (으)ㄴ to your verb. 적 (or, alternatively, 일) means "experience" and can take the subject marker 이. Finally, 있다/없다 (there is/there isn't). Before all this it's also very common to add 아/어/여 보다 ("try"; see page 185), so then the entire expression becomes 아/어/여 본 적(이) 있다/없다.
HOW IT'S CONJUGATED
| | Past
---|---|---
Action verbs ending in a vowel with 오 or 아 | 가다 gada | 간 적(이) 있다 gan jeok(i) itda
간 적(이) 없다 gan jeok(i) eopda
가 본 적(이) 있다 ga bon jeok(i) itda
가 본 적(이) 없다 ga bon jeok(i) eopda
Action verbs ending in a consonant with 오 or 아 | 잡다 japda | 잡은 적(이) 있다 jabeun jeok(i) itda
잡은 적(이) 없다 jabeun jeok(i) eopda
잡아 본 적(이) 있다 jaba bon jeok(i) itda
잡아 본 적(이) 없다 jaba bon jeok(i) eopda
Action verbs ending in a vowel with 어, 우, 으 or 이 | 서다 seoda | 선 적(이) 있다 seon jeok(i) itda
선 적(이) 없다 seon jeok(i) eopda
서 본 적(이) 있다 seo bon jeok(i) itda
서 본 적(이) 없다 seo bon jeok(i) eopda
Action verbs ending in a consonant with 어, 우, 으 or 이 | 먹다 meokda | 먹은 적(이) 있다 meogeun jeok(i) itda
먹은 적(이) 없다 meogeun jeok(i) eopda
먹어 본 적(이) 있다 meogeo bon jeok(i) itda
먹어 본 적(이) 없다 meogeo bon jeok(i) eopda
하다 | 하다 hada | 한 적(이) 있다 han jeok(i) itda
한 적(이) 없다 han jeok(i) eopda
해 본 적(이) 있다 hae bon jeok(i) itda
해 본 적(이) 없다 hae bon jeok(i) eopda
TAKE NOTE
This expression is quite often used to talk about where you've been. See the examples.
EXAMPLE SENTENCES
수업에 늦은 적이 있어요. Sueobe neujeun jeogi isseoyo. I've been late to class.
수업에 늦어 본 적이 있어요. Sueobe neujeo bon jeogi isseoyo.
미국에 간 적이 없어요. Miguge gan jeogi eopsseoyo. I've never been to the US.
미국에 가 본 적이 없어요. Miguge ga bon jeogi eopsseoyo.
미국에 간 적이 있어요. Miguge gan jeogi isseoyo. I've been to the US.
미국에 가 본 적이 있어요. Miguge ga bon jeogi isseoyo.
스키를 탄 적 있어요. Seukireul tan jeok isseoyo. I've been skiing.
스키를 타 본 적 있어요. Seukireul ta bon jeok isseoyo.
스키를 탄 적 없어요. Seukireul tan jeok eopsseoyo. I've never been skiing.
스키를 타 본 적 없어요. Seukireul ta bon jeok eopsseoyo.
민수가 어제 늦었던데 아마 오늘도 늦을걸요.
Minsuga eoje neujeotdeonde ama oneuldo neujeulgeolyo.
(I noticed that) Minsu was late yesterday, so maybe he'll be late today too.
Who doesn't love a good story? And who doesn't love telling stories about what they've seen? Here's an expression you can use to do that in Korean. 던데 and 던데요 are expressions used to talk about what the speaker has personally seen or experienced.
HOW IT'S FORMED
The particle 더 is used to indicate that you're talking about something you personally saw or experienced (but not something you yourself did). It's pretty common in Korean , and there's a whole section on it on page 184 if you'd like to know more about how to use it. In this case, it's combined with ㄴ데(요), which is covered on page and has a wide variety of meanings: most often "so" or "but." Put them together and you've got "I noticed that A, so/but B."
HOW IT'S CONJUGATED
더 is very kind to you when it comes to conjugation; it will happily follow any letter without needing any special consideration. A is what you saw and B is what you're saying based on that. You can also just use 던데요 at the end of a sentence to simply mention something you noticed.
As 더 implies recollection, you don't need to put A in the past tense most of the time. If, however, the event you saw is completely finished, you can use 았/었/였이던데 to indicate that. For example, you wouldn't use 었던데 to talk about seeing someone beautiful (unless she has since turned ugly), but you would use it to talk about how it rained earlier (and has since stopped). Although I've put 예뻤던데, 작았던데, 남자였던데 and 물이었던데 in the table below so you can see how to form, they are not the expressions you'll normally use when talking about seeing something that was beautiful or small, or seeing a man or water: 작던데, 예쁘던데, 남자이던데 and 물이던데 will usually be the forms you want. 예뻤던데, 작았던데, 남자였던데 and 물이었던데 would be used if you were talking about someone who was beautiful but isn't any longer, something that was small but has grown, someone who was once a man but now is not, and something that was once water but is now something else.
그녀는 예쁘던데요.
Geunyeoneun yeppeudeondeyo.
I saw her, and she was very pretty.
비가 왔던데 지금은 맑아요.
Biga watdeonde jigeumeun malgayo.
It was raining earlier, but it has since cleared up.
겠던데 getdeonde is also a viable expression, though not commonly used. It's used to make a guess about what's going to happen based on what you have observed. In the example below, maybe you saw that the sky was getting cloudy.
비가 오겠던데 우산을 가지고 오세요.
Biga ogetdeonde usaneul gajigo oseyo.
I think it's going to rain, so bring your umbrella.
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했던데 haetdeonde | 하던데 hadeonde | 하겠던데 hagetdeonde
Action verbs ending in a consonant | 먹다 | 먹었던데 meogeotdeonde | 먹던데 meokdeonde | 먹겠던데 meokgetdeonde
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤던데 yeppeotdeonde | 예쁘던데 yeppeudeonde | 예쁘겠던데 yeppeugetdeonde
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았던데 jagatdeonde | 작던데 jakdeonde | 작겠던데 jakgetdeonde
Nouns ending in a vowel | 남자 | 남자였던데 namjayeotdeonde | 남자이던데 namja-i-deonde | 남자이겠던데 namja-i-getdeonde
Nouns ending in a consonant | 물 | 물이었던데 murieotdeonde | 물이던데 murideonde | 물이겠던데 murigetdeonde
TAKE NOTE
Be careful about past and future tense markers: see the conjugation section for an explanation.
던데 can't be used when talking about something you personally did—only something you saw. The only exception to this is if you're talking about how you felt, in which case it's fine to use 던데(요).
The difference between 던데 and 더니 is the same as the difference between ㄴ데 and 니: see page for more on ㄴ데 and page for more on 니(까).
EXAMPLE SENTENCES
가영의 남자친구가 정말 멋있던데요.
Kayoungui namjachinguga jeongmal meositdeondeyo.
(I noticed that) Kayoung's boyfriend is really cool.
민수가 어제 정말 늦었던데요.
Minsuga eoje jeongmal neujeotdeondeyo.
(I noticed that) Minsu was really late yesterday.
이틀동안 비가 오던데 내일도 비가 올지 모르겠어요.
Iteul dongan biga odeonde naeildo biga olji moreugesseoyo.
(I see that) It's been raining for two days, so I think it might rain tomorrow as well.
어제는 민수가 늦더니 오늘은 정시에 왔어요.
Eojeneun Minsuga neutdeoni oneureun jeongsie wasseoyo.
Yesterday Minsu was late, but today he arrived on time.
Just like 던데, these are expressions in which you use 더 to talk about something you noticed and its result, or how it has since changed. 더니 and 었더니 are not exactly the same expression, but they're very similar in meaning, so I've put them together here.
HOW IT'S FORMED
더 is a particle added to verbs to indicate that the situation in question is something you personally observed; see page 184 for more on 더 and all its uses. 니 is the short form of 니까; they are exactly the same, but 니까 is never used in its full form in either of the 더니 expressions.
HOW IT'S CONJUGATED
| | Past | Present/Future
---|---|---|---
Action verbs ending in a vowel | 하다 | 했더니 haetdeoni
하더니 hadeoni
| 하더니
Action verbs ending in a consonant | 먹다 | 먹었더니 meogeotdeoni
먹더니 meokdeoni
| 먹더니
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁘더니 yeppeudeoni | 예쁘더니
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작더니 jakdeoni | 작더니
Nouns ending in a vowel | 남자 | 남자이더니 namja-i-deoni | 남자이더니
Nouns ending in a consonant | 물 | 물이더니 murideoni | 물이더니
TAKE NOTE
The difference between 더니 and 았/었/였더니 is that 더니, like the rest of the 더 expressions, is normally used only to talk about things you saw—not things you did yourself. That means it can be used only when the subject of the sentence is someone else. 았/었/였더니 is the expression you can use if you want to talk about things you yourself did.
With both expressions, the subjects of A and B must be the same.
There is another expression, 다니, which sounds somewhat similar but has a totally different meaning. It's used to express surprise at something you heard and is covered on page .
The difference between 던데 and 더니 is essentially the same as the difference between (으)ㄴ/는데 (see page ) and (으)니(까) (see page ).
EXAMPLE SENTENCES
■ 았/었/였더니 at/eot/yeotdeoni
집에서 늦게 출발했더니 늦게 도착했어요.
Jibeseo neutge chulbalhaetdeoni neutge dochakhaesseoyo.
I left the house late, so I arrived late.
한국어 소설을 읽었더니 머리가 아팠어요.
Hangugeo soseoreul ilgeotdeoni meoriga apasseoyo.
I read a Korean novel, but it gave me a headache.
말다툼을 했던 사람한테 전화했더니 그녀가 전화를 안 받았어요.
Maldatumeul haetdeon saramhante jeonhwahaetdeoni geunyeoga jeonhwareul an badasseoyo.
I tried to call the person I'd argued with, but she didn't answer.
■ 더니 deoni
가영씨, 열심히 공부하더니 시험을 잘 봤어요?
Kayoungssi, yeolssimhi gongbuhadeoni siheomeul jal bwasseoyo?
Kayoung, I know (I saw that) you studied hard. Did you do well on the test?
민수씨가 날마다 담배를 피우더니 건강이 나빠졌어요.
Minsussiga nalmada dambaereul piudeoni geongangi nappajyeosseoyo.
(I've noticed that) Minsu smokes every day, so his health has worsened.
민수가 어제 정말 늦었더라.
Minsuga eoje jeongmal neujeotdeora.
(I saw that) Minsu was really late yesterday!
더라 is another way of making a statement about something you noticed. It's used only in 반말 and is used for talking about things you noticed which surprised you.
HOW IT'S CONJUGATED
더라 goes at the very end of a sentence.
| | Past | Present
---|---|---|---
Action verbs ending in a vowel | 하다 | 했더라 haetdeora | 하더라 hadeora
Action verbs ending in a consonant | 먹다 | 먹었더라 meogeotdeora | 먹더라 meokdeora
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤더라 yeppeotdeora | 예쁘더라 yeppeudeora
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았더라 jagatdeora | 작더라 jakdeora
Nouns ending in a vowel | 남자 | 남자였더라 namjayeotdeora | 남자더라 namjadeora
Nouns ending in a consonant | 물 | 물이었더라 murieotdeora | 물이더라 murideora
TAKE NOTE
This expression can't be used for sentences about the speaker himself—only sentences with second- or third-person subjects. The only exception to this is when it's used to talk about the speaker's emotions, in which case you can use 더라. You can't use it to talk about other people's emotions. You also can't use it in the future tense.
It can also be used to talk only about things you just learned.
더라고요 deoragoyo, covered a bit further down this page, is a very similar expression. 더라 is used only in 반말: there is no 더라요 or 더랍니다. If you want to use a similar expression in 존댓말, you should use 던데요 deondeyo (see page 188).
Another very slight difference is that using 더라 will make you sound more excited than 더라고요. 더라고요 also emphasizes things a little more than 더라 does, although this is, again, not a big difference. Another similar expression is 더군(요) deogun(yo) (see page 196), which is equivalent to 더라 but can be used in 반말 or 존댓말.
EXAMPLE SENTENCES
그는 아직 학생이더라.
Geuneun ajik haksaengideora.
(I saw that) he is still a student!
가영이 벌써 집에 가더라.
Gayeongi beolsseo jibe gadeora.
(I saw that) Kayoung already went home!
민수가 어제 정말 늦었더라고요.
Minsuga eoje jeongmal neujeotdeoragoyo.
(I saw that) Minsu was really late yesterday.
This is almost the same expression as 더라 (covered on the previous page), but it sounds a little calmer and more objective: less like, "! Minsu was really late!" and more like "So I see Minsu was quite late..."
HOW IT'S CONJUGATED
If you're talking about something that's already completely finished, you can use 았/었/였 to indicate that.
오늘 아침에 비가 오더라고요.
Oneul achime biga odeoragoyo.
This morning it rained. (The rain may or may not have stopped by now.)
오늘 아침에 비가 왔더라고요.
Oneul achime biga watdeoragoyo.
This morning it rained. (It's no longer raining.)
| | Past | Present
---|---|---|---
Action verbs ending in a vowel | 하다 | 했더라고요 haetdeoragoyo | 하더라고요 hadeoragoyo
Action verbs ending in a consonant | 먹다 | 먹었더라고요 meogeotdeoragoyo | 먹더라고요 meokdeoragoyo
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤더라고요 yeppeotdeoragoyo | 예쁘더라고요 yeppeudeoragoyo
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았더라고요 jagatdeoragoyo | 작더라고요 jakdeoragoyo
Nouns ending in a vowel | 남자 | 남자였더라고요 namjayeotdeoragoyo | 남자더라고요 namjadeoragoyo
Nouns ending in a consonant | 물 | 물이었더라고요 murieotdeoragoyo | 물이더라고요 murideoragoyo
TAKE NOTE
Since this is almost exactly the same expression as 더라, it has all the same restrictions. Here they are again for your edification.
This expression can't be used for sentences about the speaker himself: only sentences with second- or third-person subjects. The only exception to this is when it's used to talk about the speaker's emotions, in which case you can use 더라고요. You can't use it to talk about other people's emotions.
It can also be used to talk only about things you just learned, and not things you personally saw. So in the examples above about rain, you didn't actually see it rain, but maybe you noticed the ground was wet when you went outside.
더라, covered on page 192, is a very similar expression, but is used only in 반말. 더라고요 can be used in either 반말 or 존댓말 just like most other grammar points; simply add or drop the 요.
Another very slight difference is that 더라 will make you sound more excited than 더라고요. 더라고요 also emphasizes things a little more than 더라 does, although this is, again, not a big difference. Another similar expression is 더군 (요) (see page 196), which is equivalent to 더라 but can be used in 반말 or 존댓말.
EXAMPLE SENTENCES
그는 아직 학생이더라고요.
Geuneun ajik haksaengideoragoyo.
(I saw that) he is still a student.
가영이 벌써 집에 갔더라고요.
Kayoungi beolsseo jibe gatdeoragoyo.
(I saw that) Kayoung already went home.
대학교 때 민수가 보통 늦었던 것 같아요.
Daehakgyo ttae Minsuga botong neujeotdeon geot gatayo.
I think I remember that in our university days, Minsu was usually late.
This expression is perfect for people with poor long-term memories. It's used when you think of something that used to be a certain way, but aren't a hundred percent sure. Alternatively, you can use it to make educated guesses about the past.
HOW IT'S FORMED
First, take the past-tense marker 았/었/였 and then add yet another past-tense marker 던 (see page ) to talk about things that happened in the past, often things that you experienced yourself. Then 것 turns the entire phrase into a noun. Finally, add 같다, which means "to be like."
HOW IT'S CONJUGATED
| | Past
---|---|---
Action verbs with 아 or 오 | 잡다 | 잡았던 것 같다 Jabatdeon geot gatda
Action verbs with 어, 우, 으 or 이 | 먹다 | 먹었던 것 같다 meogeotdeon geot gatda
Action verbs ending in a vowel | 하다 | 했던 것 같다 haetdeon geot gatda
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤던 것 같다 yeppeotdeon geot gatda
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았던 것 같다 jagatdeon geot gatda
Nouns ending in a vowel | 남자 | 남자였던 것 같다 namjayeotdeon geot gatda
Nouns ending in a consonant | 물 | 물이었던 것 같다 murieotdeon geot gatda
EXAMPLE SENTENCES
옛날에는 사람들이 말을 탔던 것 같아요.
Yennareneun saramdeuri mareul tatdeon geot gatayo.
It seems like people rode horses in the old days.
어렸을 때 축구를 자주 했던 것 같아요.
Eoryeosseul ttae chukgureul jaju haetdeon geot gatayo.
I think I used to play soccer a lot when I was a child.
회의에 늦었던가요?
Hoe-ui-e neujeotdeongayo?
Were you late to the meeting?
You may remember the question form ~(으)ㄴ가요? from the section on questions. If not, go to page . This is essentially that same question form, but here it's added to 더, so you're asking your listener to reminisce about their own personal experience.
HOW IT'S FORMED
더, the personal experience particle (see page 184) plus ㄴ가요?, which is used to ask questions.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했던가요? haetdeongayo? | 하던가요? hadeongayo? | 하겠던가요? hagetdeongayo?
Action verbs ending in a consonant | 먹다 | 먹었던가요? meogeotdeongayo? | 먹던가요? meokdeongayo? | 먹겠던가요? meokgetdeongayo?
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤던가요? yeppeotdeongayo? | 예쁘던가요? yeppeudeongayo? | 예쁘겠던가요? yeppeugetdeongayo?
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았던가요? jagatdeongayo? | 작던가요? jakdeongayo? | 작겠던가요? jakgetdeongayo?
Nouns ending in a vowel | 남자 | 남자였던가요? namjayeotdeongayo? | 남자이던가요? namja-i-deongayo? | 남자이던가요? namja-i-deongayo?
Nouns ending in a consonant | 물 | 물이었던가요? murieotdeongayo? | 물이던가요? murideongayo? | 물이던가요? murideongayo?
TAKE NOTE
You can, and Koreans generally do, simply use the regular question forms or a statement form with rising intonation. They just don't have quite the same nuances as 던가요.
회의에 늦었어요?
Hoe-ui-e neujeosseoyo?
회의에 늦었나요?
Hoe-ui-e neujeonnayo?
Were you late to the meeting?
EXAMPLE SENTENCES
민수의 새로운 여자 친구가 예쁘던가요?
Minsu-ui saeroun yeoja chinguga yeppeudeongayo?
Is Minsu's new girlfriend beautiful?
파티 준비가 다 되었던가요?
Pati junbiga da doe-eotdeongayo?
Do you think everything was ready for the party?
파티 준비가 다 되겠던가요?
Pati junbiga da doegetdeongayo?
Do you think everything will be ready for the party?
(Note: In all cases above, the speaker expects that the listener knows the answer: the listener has met Minsu's new girlfriend and is involved in planning the party.)
민수가 어제 정말 늦었더구나.
Minsuga eoje jeongmal neujeotdeoguna.
(I saw that) Minsu was really late yesterday
This is just another, but much less common, way to say 더라 (which is commonly used by Koreans, grammar guides, AND the TOPIK. This is covered in depth on page 192).
HOW IT'S FORMED
The "personal observation" particle 더 plus 군, which is used to comment on things you learned for the first time. See page for more on 군.
HOW IT'S CONJUGATED
Like 군요 itself, the 반말 form isn't conjugated in the regular manner--in this case it's 더구나.
| | Past | Present
---|---|---|---
Action verbs ending in a vowel | 하다 | 했더군요 haetdeogunnyo
했더구나 haetdeoguna
| 하더군요 hadeogunnyo
하더구나 hadeoguna
Action verbs ending in a consonant | 먹다 | 먹었더군요 meogeotdeogunnyo
먹었더구나 meogeotdeoguna
| 먹더군요 meokdeogunnyo
먹더구나 meokdeoguna
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤더군요 yeppeotdeogunnyo
예뻤더구나 yeppeotdeoguna
| 예쁘더군요 yeppeudeogunnyo
예쁘더구나 yeppeudeoguna
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았더군요 jagatdeogunnyo
작았더구나 jagatdeoguna
| 작더군요 jakdeogunnyo
작더구나 jakdeoguna
Nouns ending in a vowel | 남자 | 남자였더군요 namjayeotdeogunnyo
남자였더구나 namjayeotdeoguna
| 남자더군요 namjadeogunnyo
남자더구나 namjadeoguna
Nouns ending in a consonant | 물 | 물이었더군요 murieotdeogunnyo
물이었더구나 murieotdeoguna
| 물이더군요 murideogunnyo
물이더구나 murideoguna
TAKE NOTE
더군요 and 더라 are identical in meaning.
The usual rules about dealing with 더 apply (see page 184): No first-person sentences unless you're talking about your emotions, and only second- or third-person sentences unless you're talking about other people's emotions; sentences about other people's emotions are not allowed. Also, the event must be something that you personally observed.
더군요 can be used only to talk about things you just learned.
EXAMPLE SENTENCES
민수가 어제 정말 늦었더군요. (존댓말)
Minsuga eoje jeongmal neujeotdeogunnyo.
민수가 어제 정말 늦었더구나. (반말)
Minsuga eoje jeongmal neujeotdeoguna.
(I saw that) Minsu was really late yesterday.
그는 아직 학생이더군요.
Geuneun ajik haksaengideogunnyo.
(I saw that) he is still a student.
가영이 벌써 집에 갔더군요.
Kayoungi beolsseo jibe gatdeogunnyo.
(I saw that) Kayoung already went home.
##
Obligations, Orders and Permission
In addition to the grammar points covered here, you can also use 지(요) to offer to do something or to suggest that someone else do it. This is covered along with the rest of 지요's meanings on page .
늦지 않으려면 집에서 일찍 출발해야 해요.
Neutji aneuryeomyeon jibeseo iljjik chulbalhaeya haeyo.
늦지 않으려면 집에서 일찍 출발해야 돼요.
Neutji aneuryeomyeon jibeseo iljjik chulbalhaeya dwaeyo.
You'd better leave the house early so as not to be late.
아/어/여야 하다 and 되다 are by far the most common ways to express obligation in Korean. 아/어/여야 되다 is a little more colloquial while 아/어/여야 하다 is used slightly more often in writing, but you can use either expression whenever you want to without sounding strange. Personally, I prefer 해야 되다 because 해야 해요 just sounds awkward to me, but that's a matter of opinion; Koreans use both.
HOW IT'S FORMED
아/어/여야 is used to express obligation or necessity (see page 202). 하다 and 되다 are, well, 하다 and 되다. Normally they are different words, but in this expression they both mean exactly the same thing.
HOW IT'S CONJUGATED
These expressions work only with action verbs, so if you want to say something such as "You must be beautiful," you'll need to turn 예쁘다 into an action verb by adding 아/어/여지다 to the end of it; see page .
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 해야 했다 haeya haetda
해야 됐다 haeya dwaetda
| 해야 하다 haeya hada
해야 되다 haeya doeda
| 해야 하겠다 haeya hagetda
해야 되겠다 haeya doegetda
Action verbs ending in a consonant | 먹다 | 먹어야 했다 meogeoya haetda
먹어야 됐다 meogeoya dwaetda
| 먹어야 하다 meogeoya hada
먹어야 되다 meogeoya doeda
| 먹어야 하겠다 meogeoya hagetda
먹어야 되겠다 meogeoya doegetda
TAKE NOTE
The difference between these expressions and 아/어/여야지요 is that the latter has somewhat more of a chiding tone to it and is used to remind someone they have to do something they should already know about. 아/어/여 하다/되다 doesn't have this connotation.
To express prohibition, it's far more natural to use ~(으)면 안 되다 ~(eu)myeon an doeda (see page 200) rather than ~지 않아야 해요 ~ji anaya haeyo, although the latter is not ungrammatical.
EXAMPLE SENTENCES
숙제를 해야 해요.
Sukjereul haeya haeyo.
숙제를 해야 돼요.
Sukjereul haeya dwaeyo.
You must do your homework.
내일 제가 일본에 가야 해요.
Naeil jega Ilbone gaya haeyo.
내일 제가 일본에 가야 돼요.
Naeil jega Ilbone gaya dwaeyo.
Tomorrow I must go to Japan.
늦지 마세요.
Neutji maseyo.
Don't be late.
If you work in the Korean school system, you'll already be very familiar with these two expressions. They're ways to tell people not to do something. If you often find yourself needing to say things such as "Don't run!" or "Don't shout!" or "Don't eat your glue!"—, you'll want to know this expression.
HOW IT'S FORMED
지 plus 말다, which is a verb that means "to stop." You can add 고 if you want to continue the sentence by telling the other person what they should do instead of A; if you're not already familiar with 고 and how it works, it basically means "and" and is explained in detail on page .
HOW IT'S CONJUGATED
You can then conjugate 말다 malda (마 in 반말, 마세요 maseyo in informal polite or 마십시오 masipsio in super-polite language) to end the sentence, or you can use 지 말고 ji malgo and then continue the sentence, in which case, A is what the person shouldn't do and B is what they should do instead.
| | Present
---|---|---
Action verbs ending in a vowel | 하다 |
하지 마 haji ma
하지 마세요 haji maseyo
하지 맙시다 haji mapsida
하지 말고 haji malgo
Action verbs ending in a consonant | 먹다 |
먹지 마 meokji ma
먹지 마세요 meokji maseyo
먹지 맙시다 meokji mapsida
먹지 말고 meokji malgo
TAKE NOTE
No past or future tenses are used with this expression. When you use 지 말고, the subjects of A and B must be the same. 지 말고 can also be used to end a sentence in 반말; just end with 지 말고 instead of 지 마. This may or may not be technically correct, but it's often done.
EXAMPLE SENTENCES
사탕을 너무 많이 먹지 마세요.
Satangeul neomu mani meokji maseyo.
Don't eat too much candy.
싸우지 마십시오. Ssauji masipsio. (존댓말)
싸우지 마. Ssauji ma. (반말)
싸우지 말고. Ssauji malgo. (반말)
Don't fight.
And here are a few for you teachers out there.
뛰지 마. Ttwiji ma. Don't run.
떠들지 마. Tteodeulji ma. Don't be noisy.
싸우지 마. Ssauji ma. Don't fight.
놀지 마. Nolji ma. Don't play.
밀지 마. Milji ma. Don't push.
한국어로 말하지 마. Hangugeoro malhaji ma. Don't speak Korean.
연필을 잡지 마. Yeonpireul japji ma. Don't hold your pencil.
먹지 마. Meokji ma. Don't eat.
만지지 마. Manjiji ma. Don't touch.
늦으면 안 돼요.
Neujeumyeon an dwaeyo.
It's not okay to be late.
Here's another good expression for teachers. (으)면 되다 and 아/어/여도 되다 are ways to say that it's okay to do something. (으)면 되다 normally means that it's okay to do something for some purpose: for example, it's okay to eat that yogurt even though it's slightly after the best-before date. ~아/어/여도 되다 means that you're allowed to do ~. (으)면 안 되다 means it's not okay.
HOW IT'S FORMED
(으)면 means "if" (see page ). 아/어/여도 means "even if" and is covered on page . 되다 is a versatile verb with many meanings, one of which is "is okay." In other words, "if/even if you do ~, it's okay." If you throw in 안 (not) then you get "if you do ~, it's not okay."
HOW IT'S CONJUGATED
| | Present
---|---|---
Action verbs ending in a vowel | 하다 | 하면 (안) 되다 hamyeon (an) doeda
해도 되다 haedo doeda
Action verbs ending in a consonant | 먹다 | 먹으면 (안) 되다 meogeumyeon (an) doeda
먹어도 되다 meogeodo doeda
TAKE NOTE
(으)면 되다 and 아/어/여도 되다 are equivalent expressions.
~지 말다 and ~(으)면 안 되다 are similar in meaning, but 지 말다 is used to directly tell a person not to do something while (으)면 안 되다 is a general statement about what's okay or not okay. So if you want to tell a running child not to do so, you can say 뛰지 마! If you want to say it's not okay to run somewhere, regardless of whether or not anyone is actually running at that moment, you can say 뛰면 안 돼. Of course, if someone is already running and you say 뛰면 안 돼, you may as well be saying 뛰지 마.
Likewise, ~(으)면 되다 and ~아/어/여도 되다 mean that it's okay to do something, regardless of whether or not anyone should do it or wants to do it. It can be used to give someone permission to do something, like in the examples below. If you want to say someone should do something, you can use 아/어/여야지 A/eo/yeoyaji (see the next page) or simply 지 (see page ).
EXAMPLE SENTENCES
여기에서 놀면 안 돼요.
Yeogieseo nolmyeon an dwaeyo.
You're not allowed to play here.
이 방에서 뛰어도 돼요.
I bangeseo ttwieodo dwaeyo.
It's okay to run in this room.
지금 집에 가도 돼요.
Jigeum jibe gado dwaeyo.
지금 집에 가면 돼요.
Jigeum jibe gamyeon dwaeyo.
It's okay to go home now.
머리가 아파요?
Meoriga apayo?
이 약을 먹으면 돼요.
I yageul meogeumyeon dwaeyo.
Do you have headache? If you take this medicine, you'll be okay.
이 약을 먹어도 돼요.
I yageul meogeodo dwaeyo.
You're allowed to take this medicine.
이 약을 먹으면 안 돼요.
I yageul meogeumyeon an dwaeyo.
You're not allowed to take this medicine.
일찍 집에서 출발해야 늦지 않을 거예요.
Iljjik jibeseo chulbalhaeya neutji aneul geoyeyo.
You must leave the house early in order not to be late.
These expressions are for anyone who likes obeying rules, or better yet, making them. It's used to say that A is necessary for B to happen and is a good translation of sentences like "You must study hard in order to do well on the test ," or "You must have a student card in order to use the library."
HOW IT'S FORMED
아/어/여야 stands on its own. You can add 지 or 만 to add emphasis, but neither one is necessary. 지 can be added during informal speech while 만 is more commonly used in formal speech or in writing.
(이)라 is a particle commonly used after nouns when you plan to do something else with these nouns. For example, A(이)라는 is something that is called A. A라고 하다 means that A is said. Add 아야 to 라 and you get 라야. Again, 만 can be added to emphasize that you really, really must be A in order to B.
HOW IT'S CONJUGATED
(이)라야(만) means exactly the same thing as 아/어/여야(만), but it only follows nouns.
If you want to tell someone they must A in order to B, then you can use 지 않아야 with descriptive verbs, 지 말아야 with action verbs, or 이/가 아니어야 with nouns.
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했어야 haesseoya
했어야지 haesseoyaji
했어야만 haesseoyaman
하지 말았어야 haji marasseoya
하지 말았어야지 haji marasseoyaji
하지 말았어야만 haji marasseoyaman
| 해야 haeya
해야지 haeyaji
해야만 haeyaman
하지 말아야 haji maraya
하지 말아야지 haji marayaji
하지 말아야만 haji marayaman
| 해야
해야지
해야만
하지 말어야 하지
말어야지 하지
말아야만
Action verbs ending in a consonant | 먹다 | 먹었어야 meogeosseoya
먹었어야지 meogeosseoyaji
먹었어야만 meogeosseoyaman
먹지 말았어야 meokji marasseoya
먹지 말았어야지 meokji marasseoyaji
먹지 말았어야만 meokji marasseoyaman
| 먹어야 meogeoya
먹어야지 meogeoyaji
먹어야만 meogeoyaman
먹지 말아야 meokji maraya
먹지 말아야지 meokji marayaji
먹지 말아야만 meokji marayaman
| 먹어야
먹어야지
먹어야만
먹지 말아야 먹지
말아야지 먹지
말아야만
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻐야
예뻐야지
예뻐야만
예쁘지 않아야
예쁘지 않아야지
예쁘지 않아야만 yeppeuji anayaman
| 예뻐야 yeppeoya
예뻐야지 yeppeoyaji
예뻐야만 yeppeoyaman
예쁘지 않아야 yeppeuji anaya
예쁘지 않아야지 yeppeuji anayaji
예쁘지 않아야 yeppeuji anayaman
| 예뻐야
예뻐야지
예뻐야만
예쁘지
않아야 예쁘지
않아야지 예쁘지
않아야
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작아야 jagaya
작아야지 jagayaji
작아야만 jagayaman
작지 않아야 jakji anaya
작지 않아야지 jakji anayaji
작지 않아야만 jakji anayaman
| 작아야
작아야지
작아야만
작지 않아야
작지 말아야지
작지 말아야만
| 작아야
작아야지
작아야만
하지 말아야 하지
말아야지 작지
말아야만
Nouns ending in a vowel | 남자 | 남자여야
남자여야지
남자여야만
남자가 아니어야
남자가 아니어야지
남자가 아니어야만
남자이라야
남자이라야만
| 남자여야 namjayeoya
남자여야지 namjayeoyaji
남자여야만 namjayeoyaman
남자가 아니어야 namjaga anieoya
남자가 아니어야지 namjaga anieoyaji
남자가 아니어야만 namjaga anieoyaman
남자이라야 namjairaya
남자이라야만 namjairayaman
| 남자여야
남자여야지
남자여야만
남자가아니어야
남자가아니어야지
남자가 아니어야만
남자이라야
남자이라야만
Nouns ending in a consonant | 물 | 물이어야
물이어야지
물이어야만
물이 아니어야
물이 아니어야지
물이 아니어야만
물이라야
물이라야만
| 물이어야 murieoya
물이어야지 murieoyaji
물이어야만 murieoyaman
물이 아니어야 muri anieoya
물이 아니어야지 muri anieoyaji
물이 아니어야만 muri anieoyaman
물이라야 muriraya
물이라야만 murirayaman
| 물이어야
물이어야지
물이어야만
물이 아니어야
물이 아니어야지
물이 아니어야만
물이라야
물이라야만
TAKE NOTE
B often ends with expressions like (으)ㄹ 수 있다 (see page ) or (으)ㄹ 거예요 (see page ). B must also be a statement rather than a question, suggestion, or command.
If you end a sentence with 아/어/여야지(요), it means you're either telling someone what to do or reminding yourself that you must do something. In the latter case, just use 아/어/여지; it's not common to talk about yourself using 존댓말. The 지 here is an ending that means "isn't it?" and it has a similar connotation in 아/어/여야지(요): when you use this expression, you're normally reminding someone of something they should already know rather than telling them anything new. 아/어/여야겠다 (see page 207) is interchangeable with 아/어/여야지 when talking to yourself.
아/어/여야(지) in the middle of a sentence and 아/어/여야지(요) are grouped together here because they're almost the same expression, but remember that at the end of a sentence this expression is used to tell someone they must do something or to remind yourself that you must do something. In the middle, it refers to a necessary condition in order to do B: it doesn't mean the other person is obligated to do A, but they must do A if they want to do B. If they don't want to do B, they won't really care about your 아/어/여야 sentence in the first place.
EXAMPLE SENTENCES
■ A아/어/여야 a/eo/yeoya B
75프로를 넘어야 시험에 합격할 수 있어요.
Chilsibo-peuroreul neomeoya siheome hapgyeokhal su isseoyo.
You have to get more than 75% in order to pass the test.
신발을 벗어야 집에 들어올 수 있어요
Sinbareul beoseoya jibe deureool su isseoyo.
You have to take off your shoes in order to go in the house.
■ A(이)라야 (i)raya B
남자라야 이 찜질방에 갈 수 있어요.
Namjaraya i jjimjilbange gal su isseoyo.
You must be a man in order to go into this jjimjilbang (sauna).
제일 좋은 학생이라야 그 장학금을 받을 수 있어요.
Jeil joeun haksaengiraya geu janghakgeumeul badeul su isseoyo.
You must be the best student in order to get that scholarship.
■ ~아/어/여야지(요) ~a/eo/yeoyaji(yo)
늦지 않으려면 일찍 집에서 출발해야지요.
Neutji aneuryeomyeon iljjik jibeseo chulbalhaeyajiyo.
You know you'd better leave the house early so as not to be late.
12시 이후에는 먹지 말아야지요.
Yeoldusi ihueneun meokji marayajiyo.
You know you can't eat after 12:00.
남자친구가 나를 행복하게 했어요.
Namjachinguga nareul haengbokhage haesseoyo.
My boyfriend made me very happy.
You can't always do what you want. Sometimes people are going to make you do certain things, and you're going to have to explain those things: that's where this expression comes in. It means "to make someone ~," or "to let someone ~," depending on the situation, so it can be used to say either "My teacher made me stay after school" or "My teacher let me go home early." People will understand your meaning from the context.
HOW IT'S FORMED
게 is the particle that changes verbs into adverbs. In this case it changes ~ into an adverbial phrase. 하다, of course, just means "to do." In this case it implies intent: that someone made that adverbial phrase be that way.
HOW IT'S CONJUGATED
Here's where things get a little complicated. First of all, this can be used with action verbs or descriptive verbs. Descriptive verbs are the easy part, so we'll do them first.
Let's say Y and Z are the two people involved in the situation. Y is the person who makes Z a certain way. So your sentence describing this will look like Y 이/가 Z를/을 ~게 하다. See below for examples.
Now, with action verbs, you have to construct your sentence differently depending on whether your verb is transitive or intransitive. If you already know what that's all about, feel free to skip the next paragraph.
Intransitive verbs are verbs that don't take a direct object. Some common examples are go, sleep, arrive and die. Transitive verbs, on the other hand, do take a direct object. Some examples are "eat the pizza," "play the guitar,"and "carry a book." The direct objects are the things affected by the verb: the pizza, guitar, and book. You'll recognize them as the parts of the sentence that take the 를/을 suffix. These objects can't follow intransitive verbs: you don't "go the house" or "sleep the bed."
Now, when you use 게 하다 with an intransitive verb, the sentence looks the same as it does when you use a descriptive verb: Y이/가 Z를/을 ~게 하다.
With a transitive verb, you need to use 를/을 for the direct object of the verb and thus you can't use it for the person you're making do the action). So in this case, the person gets the suffix 에게 and the direct object keeps 를/을. The sentence structure then becomes Y이/가 Z에게 ~게 하다. Here's an example sentence that may help you understand:
엄마가 민수에게 자주 TV를 보게 해요.
Eommaga Minsuege jaju tibireul boge haeyo.
Minsu's mom often lets him watch TV.
As you can see, the direct object of the verb (TV) gets the 를 ending while Minsu (who would have had that 를 ending for himself if the TV hadn't been in the sentence) gets 에게 instead to show him as the person who is being made (or, in this case, allowed) to do the action (watch TV).
As for tenses, you conjugate 하다 to put the sentence in the past or the future tense.
You can use 못 ~게 하다 to express prohibition.
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 하게 했다 hage haetda
못 하게 했다 mot hage haetda
| 하게 하다 hage hada
못 하게 하다 mot hage hada
| 하게 하겠다 hage hagetda
못 하게 하겠다 mot hage hagetda
Action verbs ending in a consonant | 먹다 | 먹게 했다 meokge haetda
못 먹게 했다 mot meokge haetda
| 먹게 하다 meokge hada
못 먹게 하다 mot meokge hada
| 먹게 하겠다 meokge hagetda
못 먹게 하겠다 mot meokge hagetda
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁘게 했다 yeppeuge haetda | 예쁘게 하다 yeppeuge hada | 예쁘게 하겠다 yeppeuge hagetda
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작게 했다 jakge haetda | 작게 하다 jakge hada | 작게 하겠다 jakge hagetda
TAKE NOTE
Be careful about constructing your sentences correctly with all the different types of verbs: see above on how to do that.
The difference between 게 하다 and 게 되다 is that 하다 (to do) is an active verb whereas 되다 (to become) is passive. Thus, 게 하다 means to make something a certain way (active) whereas 게 되다 means to become a certain way (passive). 게 되다 says nothing about how the thing became that way. You can learn more about it, along with other passive verbs, on page .
게 하다 and 도록 하다 dorok hada are interchangeable in certain situations: 게 하다 can always be changed into 도록 하다, but the reverse isn't true; see page . 게 만들다 ge mandeulda is another similar, although much less common, expression.
게 하다 is a causative expression. Causative verbs are covered in more detail on page . The difference between 게 하다 and causative verbs made with 이, 히, etc., is that causative verbs using 이, 히 and so on imply a physical connection similar to a mother dressing her child (입히다) ipida. 게 하다 doesn't imply any such connection: in this case the mother told her child to put on his clothes (입게 하다) ipge hada.
EXAMPLE SENTENCES
■ Descriptive verbs
명상이 제 마음을 깨끗하게 했어요.
Myeongsangi je maeumeul kkaekkeutage haesseoyo.
Meditation cleansed my spirit. (made my spirit clean)
■ Intransitive action verbs
선생님이 우리를 공부하게 하셨어요.
Seonsaengnimi urireul gongbuhage hasyeosseoyo.
The teacher made us study.
의사가 나를 못 뛰게 하셨어요.
Uisaga nareul mot ttwige ha-syeoss-eoyo.
The doctor told me not to run.
■ Transitive action verbs
엄마가 민수에게 사탕을 못 먹게 해요.
Eommaga Minsuege satangeul mot meokge haeyo.
Minsu's mom won't let him eat candy.
엄마가 민수에게 자주 TV를 보게 해요.
Eommaga Minsuege jaju tibireul boge haeyo.
Minsu's mom often lets him watch TV.
회의에 정시에 도착하기 위해서 집에서 일찍 출발해야겠다.
Hoe-ui-e jeongsie dochakhagi wihaeseo jibeseo iljjik chulbalhaeyagetda.
In order to get to the meeting on time, I'll have to leave my house early.
Here's an expression for busy, forgetful, or busy and forgetful people. 아/어/여야겠다 is used to remind yourself about something you have to do.
HOW IT'S FORMED
아/어/여야 is used to express obligation; see page 202. 겠 is the future tense marker and is covered on page .
HOW IT'S CONJUGATED
Because this expression is used only when talking to oneself, it's always at the end of a sentence and always in 반말, just as it appears above: 아/어/여야겠다.
| | Future
---|---|---
Action verbs with 오 or 아 | 잡다 | 잡아야겠다 jabayagetda
Action verbs with 어, 우, 으 or 이 | 먹다 | 먹어야겠다 meogeoyagetda
Action verbs ending in a vowel | 하다 | 해야겠다 haeyagetda
TAKE NOTE
This expression and 아/어/여야지 can be used interchangeably when talking about something you need to do.
EXAMPLE SENTENCES
가영의 파티에 꼭 가야겠다.
Kayoungui patie kkok gayagetda.
I'll definitely have to go to Kayoung's party.
오늘 더운데 물을 더 마셔야겠다.
Oneul deo-un-de mureul deo masyeoyagetda.
It's hot today. I'll have to drink more water.
민수야, 학교에 빨리 가렴! 늦을 거야.
Minsuya, hakgyoe ppalli garyeom! Neujeul geoya.
민수야, 학교에 빨리 가려무나! 늦을 거야.
Minsuya, hakgyoe ppalli garyeomuna! Neujeul geoya.
Minsu, please hurry up! You're going to be late for school.
This is a way to kindly grant a favor or suggest someone do something, most often used by mothers talking to their children. Please note that it is 반말 and isn't used outside of 반말. So use it toward a child, but not toward someone older unless you want to be rude.
HOW IT'S CONJUGATED
It always goes at the end of a sentence.
| | Present
---|---|---
Action verbs ending in a vowel | 하다 | 하렴 haryeom
하려무나 haryeomuna
Action verbs ending in a consonant | 먹다 | 먹으렴 meogeuryeom
먹으려무나 meogeuryeomuna
TAKE NOTE
This has a similar meaning to 아/어/여라 a/eo/yeora, the simplest way of telling someone to do something in one form of 반말 (see page ). Again, it's only 반말; 아/어/여 주세요 a/eo/yeo juseyo (see page ) is a good way to politely request a favor from someone who is older than you or whom you don't know.
EXAMPLE SENTENCES
그렇게 싸우지 마렴.
Geureoke ssauji maryeom.
그렇게 싸우지 마려무나.
Geureoke ssauji maryeomuna.
Please don't fight like that.
그래. 네 마음대로 하렴.
Geurae. Ne maeumdaero haryeom.
그래. 네 마음대로 하려무나.
Geurae. Ne maeumdaero haryeomuna.
Fine. Go ahead and do what you want.
##
Warnings
계속 이야기 하다 보면 늦을 거예요.
Gyesok iyagi hada bomyeon neujeul geoyeyo.
If you keep talking, you'll be late.
Remember when your mom used to tell you, "If you keep on making that face, your face will stay that way!" or "If you don't eat your vegetables, you won't get tall!" Yeah, me too. I'm still short. For all you moms and teachers out there, this expression's for you. It's either a threat or a promise depending on the content and is used only to talk about the future because it wouldn't be conditional otherwise.
HOW IT'S FORMED
다가 plus 보다 (to watch/try) plus 면 (if). The 가 part of 다가 is optional. Just like the rest of the 다가 expressions, this describes an action that lasts for some time. Just like everything else with 면, it's conditional.
HOW IT'S CONJUGATED
| | Future
---|---|---
Action verbs ending in a vowel | 하다 | 하다가 보면 hadaga bomyeon
Action verbs ending in a consonant | 먹다 | 먹다가 보면 meokdaga bomyeon
TAKE NOTE
Like the other 다가 expressions, make sure the subjects of both clauses are the same and that you use the expression only with action verbs. Again, it's used only in the future. (으)ㄹ 거예요, 겠 and (으)ㄹ 수 있다 are all common endings for the B clause.
살다 보면 (with or without the space) is a commonly used expression meaning "as you live" or "in life."
살다 보면 좋은 일도 생기고 나쁜 일도 생기지요.
Salda bomyeon joeun ildo saenggigo nappeun ildo saenggijiyo.
살다보면 좋은 일도 나쁜 일도 생기지요.
Saldabomyeon joeun ildo nappeun ildo saenggijiyo.
Good things and bad things happen in life.
EXAMPLE SENTENCES
계속 공부하다가 보면 유창해질 거예요.
Gyesok gongbuhadaga bomyeon yuchanghaejil geoyeyo.
If you keep studying, you'll become fluent.
돈을 그렇게 많이 쓰다 보면 돈이 다 없어질 거예요.
Doneul geureoke mani sseuda bomyeon doni da eopsseojil geoyeyo.
If you keep spending that much money, you won't have any left.
이렇게 천천히 준비하다가는 늦을 거예요.
Ireoke cheoncheonhi junbihadaganeun neujeul geoyeyo.
If you take your time getting ready, you'll be late.
While 다가 by itself is the little child who won't let you get anything done, 다가는 is the kind of verb moms love. It's used to foretell a bad result (B) if someone does a specific action (A).
HOW IT'S FORMED
다가 plus 는. There is no special interaction between the two.
HOW IT'S CONJUGATED
| | Future
---|---|---
Action verbs ending in a vowel | 하다 | 하다가는 hadaganeun
Action verbs ending in a consonant | 먹다 | 먹다가는 meokdaganeun
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁘다가는 yeppeudaganeun
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작다가는 jakdaganeun
TAKE NOTE
Because this is used to predict negative results, it cannot be used in a positive sentence. And because it's a type of sentence that predicts, it can be used only in the future. It is usually combined with expressions that indicate guesses, so you should end your B clause with something like ~(으)ㄹ 거예요 (eu)r geoyeyo, ~(으)ㄹ 텐데요 (eu)r tendeyo, or 겠어요 gesseoyo. You can also use A 았/었/였다가는 B for hypothetical sentences. It means "If A were to happen, B would happen."
EXAMPLE SENTENCES
날씨가 이렇게 너무 덥다가는 큰일 날 텐데요.
Nalssiga ireoke neomu deopdaganeun keunil nal tendeyo.
If the weather is too hot like this there'll be many problems.
지금 태풍이 왔다가는 많은 사람들이 죽을 거예요.
Jigeum taepungi watdaganeun maneun saramdeuri jugeul geoyeyo.
If there were to be a typhoon now, many people would die.
##
Plans
If you've ever lived in Korea, you'll know that plans tend to be less like "plans" and more like "vague ideas." They change at any time up to, including, and after the event in question has already started, and you're wise not to take them too seriously. However, you'll probably still want to discuss them, and so here are some ways to do that.
First of all, two ways that aren't covered in this section are 아/어/여야지 a/eo/yeoyaji and 아/어/여야겠다 a/eo/yeoyagetda. These expressions fit better under "obligations." They're both used to remind yourself about something you have to do. If you need to do that, you can find these expressions on pages and , respectively. Another expression that's useful for planning is ~(으)ㄹ 셈이다 (eu)r semida, which simply means that you plan to ~. Because it has a few more uses having to do with guesses, it more properly belongs in the Guess section.
You can also simply use the future expressions (으)ㄹ 것이다 or 겠다. These are covered in the Making Guesses section on past and future tenses on page .
늦을까 걱정이에요. 지금 출발할까요?
Neujeulkka geokjeongieyo. Jigeum chulbalhalkkayo?
I'm worried we'll be late. Shall we leave now?
This can be most easily translated as "shall" or "let's." If you propose something using this ending, you're giving the other person much say in the final result. Further uses include asking what other people think about something and making guesses about an event in the future. You can also use it to ask yourself questions, in which case you should probably leave off the 요 unless you want to get strange looks.
HOW IT'S CONJUGATED
| | Present | Future
---|---|---|---
Action verbs ending in a vowel | 하다 | 할까요? halkkayo? | 할까요?
Action verbs ending in a consonant | 먹다 | 먹을까요? meogeulkkayo? | 먹을까요?
TAKE NOTE
(으)ㄹ까요 is one of the gentlest ways to suggest something. (으)ㄹ래요 (eu)llaeyo is another mild one which means "let's." It's the next expression on this page. If you'd like to be a little more forceful, you can use (으)ㄹ게요 (eu)lgeyo (see page 214), which means you're going to do something if it's okay with the other person. To be even more resolute, try (으)ㄹ 거예요 (eu)l geoyeyo or 겠다 getda (see page ), and if you're really determined to do ~ no matter what, ~기로 하다 gi ro hada conveys that determination; see page 225.
EXAMPLE SENTENCES
비빔밥을 먹을까요?
Bibimbapeul meogeulkkayo?
Shall we eat some bibimbap?
내일 쇼핑을 할까요?
Naeil syopingeul halkkayo?
Let's go shopping tomorrow.
지하철을 탈까요?
Jihacheoreul talkkayo?
Shall we take the subway?
늦을 것 같아요. 지금 출발할래요?
Neujeul geot gatayo. Jigeum chulbalhallaeyo?
I think we are going to be late. Shall we leave now?
This is somewhat similar to ㄹ까요. You can use it to propose something or to express your own intentions.
HOW IT'S CONJUGATED
| | Present | Future
---|---|---|---
Action verbs ending in a vowel | 하다 | 할래요 hallaeyo | 할래요
Action verbs ending in a consonant | 먹다 | 먹을래요 meogeullaeyo | 먹을래요
TAKE NOTE
It can be used only in the first person (singular or plural).
(으)ㄹ래요 can be used to mean "let's" or to tell someone your own intentions. In the latter case, it's not at all forceful but more like (으)ㄹ게요, which suggests that the other person has the ability to veto or change your plans. (으)ㄹ게요 can also be used to answer a question asked with (으)ㄹ래요, like so:
A: 밥 먹을래요? Bap meogeullaeyo? Shall we eat?
B: 그럴게요. Geureolgeyo. Yeah, sure.
(으)ㄹ까요 (see the previous page) is one of the gentlest ways to suggest something. For the rest of this paragraph see Take Note on page 212. (으)ㄹ래요 is another gentle one which means "let's." If you'd like to be a little more forceful, you can use (으)ㄹ게요, which means you're going to do something if it's all right with the other person. To be even more resolute, try (으)ㄹ 거예요 or 겠다, and if you're really determined to do ~ no matter what, ~기로 하다 conveys that determination.
EXAMPLE SENTENCES
비빔밥을 먹을래요?
Bibimbapeul meogeullaeyo?
Let's eat some bibimbap.
내일 쇼핑을 할래요?
Naeil syopingeul hallaeyo?
Let's go shopping tomorrow.
지하철을 탈래요?
Jihacheoreul tallaeyo?
Let's take the subway.
비빔밥을 먹을래요.
Bibimbapeul meogeullaeyo.
I'm going to eat some bibimbap.
내일 쇼핑을 할래요.
Naeil syopingeul hallaeyo.
I'm going shopping tomorrow.
지하철을 탈래요.
Jihacheoreul tallaeyo.
I'm going to take the subway.
Whether a sentence with (으)ㄹ래요 is a statement or a question depends on your intonation, as above. You can also add pronouns if you want to further clarify your meaning.
저는 비빔밥을 먹을래요.
Jeoneun bibimbapeul meogeullaeyo.
I'm going to eat bibimbap.
우리 비빔밥을 먹을래요?
Uri bibimbapeul meogeullaeyo?
Let's eat some bibimbap.
늦을 것 같아요. 지금 출발할게요.
Neujeul geot gatayo. Jigeum chulbalhalgeyo.
I think I am going to be late. I'm going to leave now.
This is very similar to ㄹ 거예요, but it carries the connotation of "Is it all right with you if I?" It's used in spoken Korean and can also be used when you're changing your plans as a result of something someone else said.
HOW IT'S CONJUGATED
| | Future
---|---|---
Action verbs ending in a vowel | 하다 | 할게요 halgeyo
Action verbs ending in a consonant | 먹다 | 먹을게요 meogeulgeyo
TAKE NOTE
You should be aware that ㄹ 거예요 (see page ) and ㄹ게요, while sounding and acting similarly, do have different connotations and you don't want to use ㄹ 거예요 when you mean ㄹ게요 because it's rather more forceful. If you really do mean to say "I'm going to eat bibimbap whether you like it or not!" then yes, by all means, go for ㄹ 거예요. Otherwise you might want to be gentle and use ㄹ게요.
(으)ㄹ까요 (see page 212) is one of the gentlest ways to suggest something. (For the rest, see Take Note on page 212.) (으)ㄹ래요 is another mild one which means "let's." If you'd like to be a little more forceful, you can use (으)ㄹ게요, which means you're going to do something if it's all right with the other person. To be even more resolute, try (으)ㄹ 거예요 or 겠다, and if you're really determined to do ~ no matter what, ~기로 하다 conveys that determination.
EXAMPLE SENTENCES
비빔밥 먹을게요.
Bibimbap meogeulgeyo.
I'm going to eat some bibimbap (if it's all right with you).
내일 쇼핑을 할게요.
Naeil syopingeul halgeyo.
I'm going shopping tomorrow (if it's all right with you).
지하철을 탈게요.
Jihacheoreul talgeyo.
I'm going to take the subway (if it's all right with you).
늦지 않기 위해서 집에서 일찍 출발하려고 해요.
Neutji anki wihaeseo jibeseo iljjik chulbalharyeogo haeyo.
I plan to leave the house early so as not to be late.
This is the most common and basic way to talk about plans.
HOW IT'S FORMED
려 in a grammatical expression almost always indicates that a person is talking about plans. In this case it's combined with 고, which means "and." Finally, 하다 means "to do; " in this case it implies an intention.
HOW IT'S CONJUGATED
If you want to use this expression with adjectives, you need to first change them into verbs; for example, if you plan to become pretty, you can't say 예쁘려고 해요 yeppeuryeogo haeyo. 예뻐지려고 해요 yeppeojiryeogo haeyo is the correct expression. See page to learn how to change adjectives (descriptive verbs) into action verbs.
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 하려고 했다 haryeogo haetda | 하려고 하다 haryeogo hada | 하려고 하다
Action verbs ending in a consonant | 먹다 | 먹으려고 했다 meogeuryeogo haetda | 먹으려고 하다 meogeuryeogo hada | 먹으려고 하다
TAKE NOTE
You can use 들다 instead of 하다, though this is rather less common. It implies a little more stubbornness and a little less uncertainty; in these cases the 려 usually refers to a plan that's already been carried out.
친구가 제말을 안 들으려고 드니 화가 났어요.
Chinguga jemareul an deureuryeogo deuni hwaga nasseoyo.
My friend didn't listen to what I was saying, so I got angry.
(으)ㄹ까 하다 is a similar expression to (으)려고 하다, but if you use (으)려고 하다 you're talking about a plan that's more certain than the one stated using (으)ㄹ까 하다. If you want to talk about a plan that's absolutely definite, use one of the future tense forms: (으)ㄹ 것이다 or 겠다.
(으)ㄹ 테 and (으)려고 하다 (see page 218) have the same meaning and are interchangeable when talking about the future or your own plans, like so:
비가 올 테니까 우산을 준비하세요.
Biga ol tenikka usaneul junbihaseyo.
비가 오려고 하니까 우산을 준비하세요.
Biga oryeogo hanikka usaneul junbihaseyo.
It's going to rain, so take an umbrella.
저는 공부할 테니까 엄마는 주무세요.
Jeoneun gongbuhal tenikka eommaneun jumuseyo.
저는 공부하려고 하니까 엄마는 주무세요.
Jeoneun gongbuharyeogo hanikka eommaneun jumuseyo.
I'm going to study; go to sleep, mom.
However, (으)려고 하다 can't be used when making a guess about someone else's plans.
피곤할 테니까 잠깐 쉬세요.
Pigonhal tenikka jamkkan swiseyo.
You must be tired; rest for a while.
피곤 하려고 하니까 잠깐 쉬세요.
Pigon haryeogo hanikka jamkkan swiseyo.
(ungrammatical)
EXAMPLE SENTENCES
우리는 먼저 영화를 보고 나중에 식사를 하려고 해요.
Urineun meonjeo yeonghwareul bogo najunge siksareul haryeogo haeyo.
First we plan to see a movie and then we plan to have dinner.
한국어를 열심히 공부하려고 해요.
Hangugeoreul yeolsimhi gongbuharyeogo haeyo.
I plan to study Korean hard.
There are any number of possible combinations of 려 (by itself or with 고 하다) and other expressions. Basically, any time you see 려 used in an expression, you can safely assume that expression has to do with guessing or the future (remember, they're not that far apart in meaning in Korean). Here's a table showing some possible combinations:
Expression | Meaning
---|---
~(으)려고 하다 ~(eu)ryeogo hada | Plan to ~
A(으)려고 해도 B A (eu)ryeogo haedo B | Even though I want to A, B
A (으)려고 해서 B A (eu)ryeogo haeseo B | I plan to A, so B
A(으)려면 B A (eu)ryeomyeon B | If you plan to A, then B
A(으)려거든 B A (eu)ryeogeodeun B | If you plan to A, then B
~(으)려나 보다 ~(eu)ryeona boda | I think he's planning to ~
A(으)려다가 B A (eu)ryeodaga B | I planned to A, but was interrupted by B
~(으)려던 참이다 ~(eu)ryeodeon chamida | Just about to ~
Combine (으)려고 하다 with 여도 yeodo and you get A(으)려고 해도 B, which means "Even though I want to A, B."
숙제를 하려고 해도 잘 몰라서 못 해요.
Sukjereul haryeogo haedo jal mollaseo mot taeyo.
Even though I want to do my homework, I don't know how, so I can't.
(으)려고 하다 plus 여서 equals A(으)려고 해서 B, which means "I plan to A, so B."
케이크를 만들려고 해서 밀가루를 샀어요.
Keikeureul mandeullyeogo haeseo milgarureul sasseoyo.
I planned to make a cake, so I bought flour.
Next up is (으)려면. Here you're combining 려 (the planning particle) with 면 (if; see page ) to get "If you plan to A, then B." This one is somewhat more common than the other 려 expressions.
B often ends with an expression indicating obligation: some variant of 아/어/여야 (see page ). 아/어/여 하다 or 되다 (see page ) are very commonly used. You can also use a suggestion or command form (see page for some examples).
늦지 않으려면 택시를 타야 돼요.
Neutji aneuryeomyeon taeksireul taya dwaeyo.
If you don't want to be late, you'd better take a taxi.
그 연극을 보려면 표를 빨리 사야 해요.
Geu yeongeugeul boryeomyeon pyoreul ppalli saya haeyo.
If you want to see that performance, you should buy tickets quickly.
사장님을 만나려면 먼저 비서한테 이메일을 보내 주세요.
Sajangnimeul mannaryeomyeon meonjeo biseohante imeireul bonae juseyo.
If you want to meet the director, please email his secretary first.
If you combine (으)려고 하다 with 다가 and shorten it, you get (으)려다가. 다가 (see page ) in its original form implies an interruption, and that's how it works here as well: you were planning to A, but then you got interrupted by B.
등산을 하려다가 비가 와서 집에 있었어요.
Deungsaneul haryeodaga biga waseo jibe isseosseoyo.
I planned to go hiking, but then it rained, so I stayed home.
(으)려고 하다 plus 거든 equals (으)려고 하거든 which is then shortened to (으)려거든. 거든 is similar in meaning to 면 (if); see page . A(으)려거든 B, likewise, is equivalent to (으)려면. It means "If you're planning to A, then B."
늦지 않으려거든 택시를 타야 돼요.
Neutji aneuryeogeodeun taeksireul taya dwaeyo.
If you don't want to be late, you'd better take a taxi.
Finally, ~(으)려던 참이다 is worthy of its own section and is examined in more detail on page ; briefly, it's a way of saying you were just about to do something.
전 가려던 참이었어요.
Jeon garyeodeon chamieosseoyo.
I was just about to go.
~(으)려나 보다 ~(eu)ryeona boda is really more of a variation on ~나 보다, so it's in that section, which you can find on page .
비빔밥을 먹으러 식당에 갔어요.
Bibimbapeul meogeureo sikdange gasseoyo.
I went to the restaurant to eat bibimbap.
This expression is used when you're planning to go or come somewhere to do something.
HOW IT'S CONJUGATED
You can use 가다 or 오다 depending on whether you're coming or going; make sure you use the verb going in the right direction. 다니다, another motion verb, can also be used with this expression.
TAKE NOTE
(으)러 can be followed only by 가다, 다니다, or 오다. If you want to plan something using any other verb, you should use 려고 하다 (see page 215). Variants of 가다 or 오다 (들어가다, 나가다, 들어오다, etc.) can be used with (으)러.
EXAMPLE SENTENCES
비빔밥을 먹으러 식당에 왔어요.
Bibimbapeul meogeureo sikdange wasseoyo.
I came to the restaurant to eat bibimbap.
공부를 하러 왔어요.
Gongbureul hareo wasseoyo.
I came to study.
공부를 하러 갑시다.
Gongbureul hareo gapsida.
Let's go study.
벌써 8시인데 정시에 올 테야?
Beolsseo yeodeolsi-inde jeongsie ol teya?
It's already eight. Are you planning to come on time?
This is the easiest, simplest way to tell someone you're planning to do something. It's a good one for you evil geniuses out there. It also forms the basis of a few other very useful expressions: (으)ㄹ 테니까(요), (으)ㄹ 텐데(요), (으)ㄹ 테면, (으)ㄹ 테고, and (으)ㄹ 테지만. In other words, even if you're not an evil genius, this is a good expression to learn. It means "I plan to ~" or "Do you plan to ~?" Alternatively, it can be used to make guesses about what other people are planning. These guesses are made with confidence: if you use this expression to make a guess, you really think you're right. (으)ㄹ 테니까 and (으)ㄹ 텐데(요) are by far the most common expressions with 테다: all the others are not used often, but they're easy enough to learn if you want to know them. Here's a summary of the expressions that begin with (으)ㄹ 테다:
Expression | Page | Meaning | Example sentence
---|---|---|---
(으)ㄹ 터이다/테다 (eu)r teoida/teda | – | Plan to ~ | 나중에 파티에 갈 테야. Najunge patie gal teya. Later I'm planning to go to the party.
A(으)ㄹ 테니까 B A (eu)r tenikka B | 221 | I plan to A, so B | 민수가 늦을 테니까 지금 시작합시다. Minsuga neujeul tenikka jigeum sijakhapsida.I think Minsu is going to be late, so let's start.
A(으)ㄹ 텐데 B A (eu)r tende B | 222 | I plan to A; B | 민수가 늦을 텐데 지금 시작합시다. Minsuga neujeul tende jigeum sijakhapsida.I think Minsu is going to be late. Let's start.
A(으)ㄹ 테면 B A (eu)r temyeon B | 223 | If you plan to A, then B | 어떤 얘기든 할 테면 그냥 하세요. Eotteon yaegideun hal temyeon geunyang haseyo. Whatever you want to say, just say it.
A(으)ㄹ 테고 B A (eu)r tego B | 223 | I plan to A, and then B | 버스로 가면 시간이 걸릴 테고 힘들 거예요. Beoseuro gamyeon sigani geollil tego himdeul geoyeyo. If we go by bus, it'll take time and it might be difficult.
A(으)ㄹ 테지만 B A (eu)r tejiman B | 224 | I plan to A, but B | 잘 할 수 있을 거라고 생각해서 공부를 별로 하지 않았을 테지만 시험에 떨어졌어요. Jal hal su isseul georago saenggakaeseo gongbureul byeollo haji anasseul tejiman siheome tteoreojyeosseoyo.I thought I would do well, so I barely studied, but I failed the test.
HOW IT'S FORMED
터 is actually a noun, though it's never used outside of this expression. It refers to either a plan or a guess. (으)ㄹ is the future tense marker; you can, in theory, use the other tense markers (으)ㄴ/는, but only when you're talking about a plan in the past. You will most often see and hear this expression used with (으)ㄹ.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 한 터이다 han teoida
한 테다 han teda
| 할 터이다 hal teoida
할 테다 hal teda
| 할 터이다 할 테다
Action verbs ending in a consonant | 먹다 | 먹은 터이다 meogeun teoida
먹은 테다 meogeun teda
| 먹을 터이다 meogeul teoida
먹을 테다 meogeul teda
| 먹을 터이다 먹을 테다
TAKE NOTE
(으)ㄹ 테다 by itself is colloquial and is used only in 반말. It can be used in first-person sentences to explain your own plans or in second-person sentences to ask about another person's plans, but it cannot be used in third-person sentences. In this case you should use 겠다 or (으)ㄹ 것이다 (see page ).
(으)ㄹ 텐데 and (으)ㄹ 테니까 are very similar expressions. The difference between the two is the same as the difference between ㄴ데 and 니까, both covered in the Cause and Effect section on page if you'd like to review. They're quite often, but not always, interchangeable, although the nuances are different; ㄴ데 is used to explain the background for a situation (in this case, a request or command) and 니까 is used to give a reason. Also, (으)ㄹ 테니까 refers to a slightly firmer plan or expectation.
피곤할 텐데 잠깐 쉬세요.
Pigonhal tende jamkkan swiseyo.
You may be tired. Please rest.
피곤할 테니까 잠깐 쉬세요.
Pigonhal tenikka jamkkan swiseyo.
You must be tired, so please rest.
(으)ㄹ 테 and (으)려고 하다 (see page 215) have the same meaning and are interchangeable when talking about the future or your own plans, like so:
비가 올 테니까 우산을 준비하세요.
Biga ol tenikka usaneul junbihaseyo.
비가 오려고 하니까 우산을 준비하세요.
Biga oryeogo hanikka usaneul junbihaseyo.
It's going to rain, so take an umbrella.
저는 공부할 테니까 엄마는 주무세요.
Jeoneun gongbuhal tenikka eommaneun jumuseyo.
저는 공부하려고 하니까 엄마는 주무세요.
Jeoneun gongbuharyeogo hanikka eommaneun jumuseyo.
I'm going to study; go to sleep, mom.
However, (으)려고 하다 can't be used when making a guess about someone else's plans.
피곤할 테니까 잠깐 쉬세요.
Pigonhal tenikka jamkkan swiseyo.
You must be tired; rest for a while.
피곤 하려고 하니까 잠깐 쉬세요.
Pigon haryeogo hanikka jamkkan swiseyo.
(ungrammatical)
Here are a few example sentences with (으)ㄹ 테다:
나중에 파티에 갈 테야.
Najunge patie gal teya.
Later I'm planning to go to the party.
어떤 영화를 볼 테야?
Eotteon yeonghwareul bol teya?
What movie are you planning to see?
And here are a few more expressions based on (으)ㄹ 테다.
A(으)ㄹ 테니까 (eu)l tenikka B, ~(으)ㄹ 테니까요 (eu)l tenikkayo
What's more fun than bossing other people around? Here's an expression you can use to make suggestions or give commands based on your thoughts and plans. It means "I think or plan to A, so you should B."
HOW IT'S FORMED
(으)ㄹ 테다 is combined here with 니까(요) which means "so"; see page for more. 니까 can be abbreviated to 니, but that's not usually done with this expression.
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했을 테니까(요) haesseul tenikka(yo) | 할 테니까(요) hal tenikka(yo) | 할 테니까(요)
Action verbs ending in a consonant | 먹다 | 먹었을 테니까(요) meogeosseul tenikka(yo) | 먹을 테니까(요) meogeul tenikka(yo) | 먹을 테니까(요)
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤을 테니까(요) yeppeosseul tenikka(yo) | 예쁠 테니까(요) yeppeul tenikka(yo) | 예쁠 테니까(요)
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았을 테니까(요) jagasseul tenikka(yo) | 작을 테니까(요) jageul tenikka(yo) | 작을 테니까(요)
Nouns ending in a vowel | 남자 | 남자였을 테니까(요) namjayeosseul tenikka(yo) | 남자일 테니까(요) namjail tenikka(yo) | 남자일 테니까(요)
Nouns ending in a consonant | 물 | 물이었을 테니까(요) murieosseul tenikka(yo) | 물일 테니까(요) muril tenikka(yo) | 물일 테니까(요)
If you're talking about a plan, A can only be about yourself. If you're talking about what you expect to happen, it can be about a second or third person.
EXAMPLE SENTENCES
나중에 프레젠테이션을 할 테니까 회의실을 좀 준비해 주세요.
Najunge peu-re-jen-te-isyeoneul hal tenikka hoe-ui-sireul jom junbihae juseyo.
I'm doing a presentation later, so please get the meeting room ready.
걱정하지 마세요. 준비를 금방 끝낼 테니까요.
Geokjeonghaji maseyo. Junbireul geumbang kkeunnael tenikkayo.
Don't worry. I plan to finish the preparations soon.
가영의 항공이 벌써 도착했을 테니까 빨리 공항에 갑시다.
Kayoungui hanggongi beolsseo dochakhaesseul tenikka ppalli gonghange gapsida.
I think Kayoung's flight will have already arrived, so let's go to the airport quickly.
민수가 늦을 테니까 지금 시작합시다.
Minsuga neujeul tenikka jigeum sijakhapsida.
I think Minsu is going to be late, so let's start.
A(으)ㄹ 텐데 (eu)r tende B, ~(으)ㄹ 텐데요 (eu)r tendeyo
This expression is a combination of (으)ㄹ 테다 and ㄴ데 (see page ) which is used to explain the background for something. It means "I plan or expect to A; B."
HOW IT'S CONJUGATED
You can reverse the sentence structure and make the suggestion or give the command first and then follow it with the background explanation; in this case you can end your sentence with (으)ㄹ 텐데요.
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했을 텐데 haesseul tende | 할 텐데 hal tende | 할 텐데
Action verbs ending in a consonant | 먹다 | 먹었을 텐데 meogeosseul tende | 먹을 텐데 meogeul tende | 먹을 텐데
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤을 텐데 yeppeosseul tende | 예쁠 텐데 yeppeul tende | 예쁠 텐데
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았을 텐데 jagasseul tende | 작을 텐데 jageul tende | 작을 텐데
Nouns ending in a vowel | 남자 | 남자였을 텐데 namjayeosseul tende | 남자일 텐데 namja-il tende | 남자일 텐데
Nouns ending in a consonant | 물 | 물이었을 텐데 murieosseul tende | 물일 텐데 muril tende | 물일 텐데
Like the other expressions with 테다, this one can be used with sentences in the first person to talk about plans or with sentences which make guesses about a second or third person.
EXAMPLE SENTENCES
나중에 프레젠테이션을 할 텐데 회의실을 좀 준비해 주세요.
Najunge peu-re-jen-te-i-syeoneul hal tende hoe-ui-sireul jom junbihae juseyo.
I'm doing a presentation later. Please get the meeting room ready.
가영의 항공이 벌써 도착했을 텐데 빨리 공항에 갑시다.
Kayoungui hanggongi beolsseo dochakhaesseul tende ppalli gonghange gapsida.
빨리 공항에 갑시다. 가영의 항공이 벌써 도착했을 텐데요.
Ppalli gonghange gapsida. Kayoungui hanggongi beolsseo dochakhaesseul tendeyo.
I think Kayoung's flight will have already arrived. Let's go to the airport quickly.
민수가 늦을 텐데 지금 시작합시다.
Minsuga neujeul tende jigeum sijakhapsida.
I think Minsu is going to be late. Let's start.
A(으)ㄹ 테면 (eu)l temyeon B
(으)ㄹ 테다 is combined with 면, which means "if" and is on page .
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Action verbs ending in a vowel | 하다 | 할 테면 hal temyeon
Action verbs ending in a consonant | 먹다 | 먹을 테면 meogeul temyeon
TAKE NOTE
A can only be a second-person expression. B can only be a command. The verb in B should be the same as the verb in A; see the examples below.
Other expressions with 테다 are (으)ㄹ 텐데 and (으)ㄹ 테니까. They are different from (으)ㄹ 테면 in that they are used to talk about things the speaker is planning or expecting while (으)ㄹ 테면 is only about the listener's plans. Also, 니까 means "so" and ㄴ데 is used to provide background information for whatever comes after it, while 면 means "if."
EXAMPLE SENTENCES
어떤 얘기든 할 테면 그냥 하세요.
Eotteon yaegideun hal temyeon geunyang haseyo.
Whatever you want to say, just say it.
떠날 테면 후회없이 떠나세요.
Tteonal temyeon hu-hoe-eopssi tteonaseyo.
If you want to leave, then leave without regrets.
늦게 올 테면 저녁을 먹고 오세요.
Neutge ol temyeon jeonyeogeul meokgo oseyo.
If you plan to come late, please come after you eat dinner.
A(으)ㄹ 테고 (eu)l tego B
Combine (으)ㄹ 테다 with 고 ("and"; page ) to get this expression, which means "I plan or expect to do A, and then B."
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했을 테고 haesseul tego | 할 테고 hal tego | 할 테고
Action verbs ending in a consonant | 먹다 | 먹었을 테고 meogeosseul tego | 먹을 테고 meogeul tego | 먹을 테고
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤을 테고 yeppeosseul tego | 예쁠 테고 yeppeul tego | 예쁠 테고
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았을 테고 jagasseul tego | 작을 테고 jageul tego | 작을 테고
Nouns ending in a vowel | 남자 | 남자였을 테고 namjayeosseul tego | 남자일 테고 namjail tego | 남자일 테고
Nouns ending in a consonant | 물 | 물이었을 테고 murieosseul tego | 물일 테고 muril tego | 물일 테고
버스로 가면 시간이 걸릴 테고 힘들 거예요.
Beoseuro gamyeon sigani geollil tego himdeul geoyeyo.
If we go by bus, it'll take time and it might be difficult.
10시까지 직장에 도착할 테고 그러면 12시까지 식당에 도착할 수 있을 거예요.
Yeolsi-kkaji jikjange dochakhal tego geureomyeon yeoldusikkaji sikdange dochakhal su isseul geoyeyo.
I plan to arrive at work by 10:00, and if so, I should be able to get to the restaurant by 12:00.
A(으)ㄹ 테지만 (eu)l tejiman
Finally, you can combine (으)ㄹ 테다 with 지만 to get this expression, which means "I plan/expect to do A, but B" (see page ).
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했을 테지만 Haesseul tejiman | 할 테지만 hal tejiman | 할 테지만
Action verbs ending in a consonant | 먹다 | 먹었을 테지만 meogeosseul tejiman | 먹을 테지만 meogeul tejiman | 먹을 테지만
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤을 테지만 yeppeosseul tejiman | 예쁠 테지만 yeppeul tejiman | 예쁠 테지만
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았을 테지만 jagasseul tejiman | 작을 테지만 jageul tejiman | 작을 테지만
Nouns ending in a vowel | 남자 | 남자였을 테지만 namjayeosseul tejiman | 남자일 테지만 namjail tejiman | 남자일 테지만
Nouns ending in a consonant | 물 | 물이었을 테지만 murieosseul tejiman | 물일 테지만 muril tejiman | 물일 테지만
잘할 수 있을거라고 생각해서 공부를 별로 하지 않았을 테지만 시험에 떨어졌어요.
Jalhal su isseulgeorago saengakhaeseo gongbureul byeollo haji anasseul tejiman siheome tteoreojyeosseoyo.
I thought I could do well, so I barely studied, but I failed the test.
정시에 올 수 있을 거라고 생각해서 서두르지 않았을 테지만 늦게 왔어요.
Jeongsie ol su isseul georago saenggakhaeseo seodureuji anasseul tejiman neutge wasseoyo.
I thought I could get there on time, so I didn't hurry, but I was late.
오늘 회의가 있어서 파티에 늦게 갈 예정인데요.
Oneul hoe-ui-ga isseoseo patie neutge gal yejeongindeyo.
Today there's a meeting, so I plan to go late to the party.
Here's a simple, useful way to talk about your future plans.
HOW IT'S FORMED
(으)ㄹ is the future tense marker. 예정 is a noun that literally means "schedule."
HOW IT'S CONJUGATED
| | Future
---|---|---
Action verbs ending in a vowel | 하다 | 할 예정이다 hal yejeongida
Action verbs ending in a consonant | 먹다 | 먹을 예정이다 meogeul yejeongida
TAKE NOTE
This is basically interchangeable with (으)ㄹ 거예요 (eu)l geoyeyo or the (으)ㄹ 테 (eu)l te expressions; it's just another way to talk about plans.
EXAMPLE SENTENCES
오늘 회의가 있을 예정이어서 파티에 늦게 갈 거예요.
Oneul hoe-ui-ga isseul yejeongieoseo patie neutge gal geoyeyo.
Today we plan to have a meeting, so I'll be late to the party.
내일 새로운 식당에서먹을 예정인데 같이 먹을래요?
Naeil saeroun sikdangeseo meogeul yejeonginde gachi meogeullaeyo?
Tomorrow we're planning to eat at the new restaurant; want to join us (eat together)?
내일 일찍 일어나기로 했어요.
Naeil iljjik ireonagiro haesseoyo.
I decided to get up early tomorrow.
This expression is quite strong and implies determination and decision. Often it's like a vow you've made or the result of a lengthy decision-making process.
HOW IT'S FORMED
기 (the particle that turns verbs into nouns) plus 로 ("through") plus 하다. However, these are only sort of applicable to this expression, and don't really provide much insight.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 하기로 했다 hagiro haetda | 하기로 하다 hagiro hada | 하기로 하겠다 hagiro hagetda
Action verbs ending in a consonant | 먹다 | 먹기로 했다 meokgiro haetda | 먹기로 하다 meokgiro hada | 먹기로 하겠다 meokgiro hagetda
TAKE NOTE
This is used only for expressing your own decisions, so it can be used only in the first person. It is normally used in the past tense to talk about a decision you've already made.
Instead of 하다 at the end, you can also use 기로 계획이다 giro gyehoegida ("It's my plan to"), 기로 약속하다 giro yaksokhada ("I promise to") or 기로 결심하다 giro gyeolsimhada ("It's my decision to").
(으)ㄹ까요 (see page 212) is one of the gentlest ways to suggest something. (으)ㄹ래요 is another mild one which means "let's." It's on page 212. If you'd like to be a little more forceful, you can use (으)ㄹ게요 (see page 214), which means you're going to do something if it's all right with the other person. To be even more resolute, try (으)ㄹ 거예요 or 겠다 (see page ), and if you're really determined to do ~ no matter what, ~기로 하다 conveys that determination (see page 225).
EXAMPLE SENTENCES
살을 빼고 싶어서 다이어트를 하기로 했어요.
Sareul ppaego sipeoseo da-i-eo-teureul hagiro haesseoyo.
I wanted to lose weight, so I decided to go on a diet.
돈이 없어서 직업을 찾기로 했어요.
Doni eopsseoseo jigeobeul chatgiro haesseoyo.
I had no money, so I decided to get a job.
다른 약속도 있어서 가영의 파티에 조금 늦게 갈까 해요.
Dareun yaksokdo isseoseo Kayoungui patie jogeum neutge galkka haeyo.
다른 약속도 있어서 가영의 파티에 조금 늦게 갈까 봐요.
Dareun yaksokdo isseoseo Kayoungui patie jogeum neutge galkka bwayo.
다른 약속도 있어서 가영의 파티에 조금 늦게 갈까 싶어요.
Dareun yaksokdo isseoseo Kayoungui patie jogeum neutge galkka sipeoyo.
I have another engagement, so I might come a little late to Kayoung's party.
The three expressions above are all equivalent. (으)ㄹ까 하다 is probably the most common, but the others can also be used. They are all ways to express a tentative plan.
HOW IT'S FORMED
(으)ㄹ까 means "shall" (see page 212) and can be followed here by 하다 (to do), 보다 (to try) or 싶다 (which doesn't really have an English translation, but has to do with wanting and planning).
HOW IT'S CONJUGATED
If you want to talk about what you're planning not to do, you can use 안 ~(으)ㄹ까 하다/보다/싶다 or ~지 말까 하다/보다/싶다.
| | Past | Present/Future
---|---|---|---
Action verbs ending in a vowel | 하다 | 할까 했다 halkka haetda
할까 봤다 halkka bwatda
할까 싶었다 halkka sipeotda
안 할까 했다 an halkka haetda
안 할까 봤다 an halkka bwatda
안 할까 싶었다 an halkka sipeotda
하지 말까 했다 haji malkka haetda
하지 말까 봤다 haji malkka bwatda
하지 말까 싶었다 haji malkka sipeotda
| 할까 하다 halkka hada
할까 보다 halkka boda
할까 싶다 halkka sipda
안 할까 하다 an halkka hada
안 할까 보다 an halkka boda
안 할까 싶다 an halkka sipda
하지 말까 하다 haji malkka hada
하지 말까 보다 haji malkka boda
하지 말까 싶다 haji malkka sipda
Action verbs ending in a consonant | 먹다 |
먹을까 했다 meogeulkka haetda
먹을까 봤다 meogeulkka bwatda
먹을까 싶었다 meogeulkka sipeotda
안 먹을까 했다 an meogeulkka haetda
안 먹을까 봤다 an meogeulkka bwatda
안 먹을까 싶었다 an meogeulkka sipeotda
먹지 말까 했다 meokji malkka haetda
먹지 말까 봤다 meokji malkka bwatda
먹지 말까 싶었다 meokji malkka sipeotda | 먹을까 하다 meogeulkka hada
먹을까 보다 meogeulkka boda
먹을까 싶다 meogeulkka sipda
안 먹을까 하다 an meogeulkka hada
안 먹을까 보다 an meogeulkka boda
안 먹을까 싶다 an meogeulkka sipda
먹지 말까 하다 meokji malkka hada
먹지 말까 보다 meokji malkka boda
먹지 말까 싶다 meokji malkka sipda
TAKE NOTE
These expressions can be used only with statements—not commands, questions, or propositions. If you'd like to ask a question, give a command or propose something based on what you're planning, you can conjugate 하다/보다/싶다 accordingly with 는데 or 니까 and then continue with whatever you wanted to ask, propose, or command. See example 3.
No future tenses allowed, and no adjectives or nouns either. This is a very picky expression. 보다 in is exceptionally picky and also cannot be used in the past tense or in the middle of a sentence.
(으)ㄹ까 하다/보다/싶다 describes a tentative plan. A more certain plan can be described with (으)려고 하다 (see page 215) or any of its associated expressions, and a very certain plan can be described using (으)ㄹ 것이다 or 겠다 (page ).
You can also use (으)ㄹ까 말까 to talk about something that you can't decide whether to do or not; see page 231 for more on this.
A(으)ㄹ까 봐(서) B and A(으)ㄹ까 싶(어서) B go in the middle of a sentence, but they are not the same expressions as the ones we're discussing here. A(으)ㄹ까 봐(서) B and A (으)ㄹ까 싶(어서) B are used when you're worried about something happening (see page ).
EXAMPLE SENTENCES
작년에 일본에서 여행을 할까 했어요.
Jangnyeone ilboneseo yeohaengeul halkka haesseoyo.
작년에 일본에서 여행을 할까 싶었어요.
Jangnyeone ilboneseo yeohaengeul halkka sipeosseoyo.
Last year I was thinking of traveling around Japan (but I didn't).
(Note: (으)ㄹ까 보다 sounds strange in the past tense, so choose one of the other two expressions instead.)
내일 새로운 식당에서 저녁을 먹을까 하는데 같이 갈까요?
Naeil saeroun sikdangeseo jeonyeogeul meogeulkka haneunde gachi galkkayo?
내일 새로운 식당에서 저녁을 먹을까 싶은데 같이 갈까요?
Naeil saeroun sikdangeseo jeonyeogeul meogeulkka sipeunde gachi galkkayo?
내일 새로운 식당에서 저녁을 먹을까 봐요. 같이 갈까요?
Naeil saeroun sikdangeseo jeonyeogeul meogeulkka bwayo. Gachi galkkayo?
I'm thinking about eating dinner at that new restaurant tomorrow. Do you want to go together?
Again, (으)ㄹ까 보다 is picky and shouldn't be used in the middle of a sentence.
오늘 파티에 늦게 올 건가요?
Oneul patie neutge ol geongayo?
Are you planning to come late to today's party?
This is a question form used to talk about plans. I would suggest using it often in Korea because plans there do change often: be sure to reconfirm, reconfirm, reconfirm.
HOW IT'S FORMED
This is really two different grammar points combined: (으)ㄹ 것이다 (the future tense, as seen on page ) and (으)ㄴ가요 (a question form covered on page ). Add them together, shorten it a little and you get (으)ㄹ 건가요?
HOW IT'S CONJUGATED
(으)ㄹ 건가요 is always used at the end of a sentence and never follows any other tense markers.
| | Future
---|---|---
Action verbs ending in a vowel | 하다 | 할 건가요 hal geongayo
Action verbs ending in a consonant | 먹다 | 먹을 건가요 meogeul geongayo
TAKE NOTE
This is used only to confirm plans or to ask others' opinions.
EXAMPLE SENTENCES
누가 이 일을 할 건가요?
Nuga i ireul hal geongayo?
Who's going to do this job?
언제 시험을 볼 건가요?
Eonje siheomeul bol geongayo?
When are we going to take the test?
늦지 않게 오고자 집에서 일찍 출발했어요.
Neutji anke ogoja jibeseo iljjik chulbalhaesseoyo.
I left the house early so as not to be late.
If you ever have to do a job interview in Korean, this is a good one to know. It's a relatively formal expression used for describing plans or intentions, similar to a more official (으)려고. It could be translated as "in order to," "since," or "to."
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Action verbs ending in a vowel | 하다 | 하고자 hagoja
Action verbs ending in a consonant | 먹다 | 먹고자 meokgoja
TAKE NOTE
고 can be used only with action verbs and only when the subjects of both clause A and clause B are the same.
It can also be used at the end of a sentence describing your plans (see the third example below), in which case it is followed by 하다. When used this way, it can't be used with negative particles like 안 or 못.
As stated in the intro, 고자 has the same meaning as (으)려고 but is more formal. It can also be replaced with 기 위해(서) (see page ).
EXAMPLE SENTENCES
영어를 가르치고자 한국에 왔어요.
Yeongeoreul gareuchigoja hanguge wasseoyo.
I came to Korea to teach English.
그 회사에 취직하고자 이력서를 냈어요.
Geu hoesae chwijikhagoja iryeokseoreul naesseoyo.
I sent my resume to that company to get a job.
운동을 한다 한다 하는 게 게으름을 피우다가 안 했어요.
Undongeul handa handa haneun ge geeureumeul piudaga an haesseoyo.
I tried and tried to exercise, but I kept on being lazy and never did it.
We've covered many expressions involving plans and determination. That's all well and good, but plans don't always succeed. Sometimes, no matter how much you want to do something, other things keep getting in the way. This is an expression you can use when that happens.
HOW IT'S FORMED
게 ge is an abbreviation of 것이 geosi and not the adverb form 게.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하 다 | 한다 한다 했는 게 Handa handa haenneun ge | 한다 한다 하는 게 handa handa haneun ge | 한다 한다 하는 게
Action verbs ending in a consonant | 먹 다 | 먹는다 먹는다 했는 게 meokneunda meokneunda haenneun ge | 먹는다 먹는다 하는 게 meokneunda meokneunda haneun ge | 먹는다 먹는다 하는 게
TAKE NOTE
Be sure to repeat A twice.
If A is a verb with a direct object, like 운동을 하다 undongeul hada, you need to repeat only the verb, not the whole expression, the second time, like so: 운동을 한다 한다 하는 게 undongeul handa handa haneun ge.
EXAMPLE SENTENCES
그 공연에 간다 간다 했는 게 계속 잊어버려서 결국 못 갔어요.
Geu gongyeone ganda ganda haenneun ge gyesok ijeobeoryeoseo gyeolguk mot gasseoyo.
I really meant to go to that show, but I kept forgetting, so in the end I wasn't able to go.
순대를 먹는다 먹는다 했는 게 너무 역겨워서 못 먹었어요.
Sundaereul meokneunda meokneunda haenneun ge neomu yeokgyeowoseo mot meogeosseoyo.
I tried and tried to eat soondae, but it was too disgusting, so I couldn't eat it.
운동을 할까 말까 해요.
Undongeul halkka malkka haeyo.
I can't decide whether to exercise or not.
If you're indecisive, here's an expression for you. This is used to say that you still haven't decided "whether to ~ or not."
HOW IT'S FORMED
(으)ㄹ까 means "shall"; see page 212. 말다 contradicts the verb before it, and 하다 is 하다, as always.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 할까 말까 했다 halkka malkka haetda | 할까 말까 하다 halkka malkka hada | 할까 말까 할 것이다 halkka malkka hal geosida
Action verbs ending in a consonant | 먹다 | 먹을까 말까 했다 meogeulkka malkka haetda | 먹을까 말까 하다 meogeulkka malkka hada | 먹을까 말까 할 것이다 meogeulkka malkka hal geosida
TAKE NOTE
(으)ㄹ까 말까 하다 is normally used at the end of short sentences saying nothing but that you couldn't decide whether to ~ or not.
(으)ㄹ락 말락 하다 (eu)rrak mallak hada has a similar form, but while (으)ㄹ까 말까 implies indecision, (으)ㄹ락 말락 means that something was about to (but didn't) do something: that it was on the verge of raining (but didn't) or that someone was about to fall over (but didn't).
When using (으)ㄹ까 말까 to answer a question, it's fairly common to begin with 글쎄 which roughly means "Hmm" or, "I don't know."
You don't have to use 하다 after ㄹ까 말까. The example sentence above could also be written like this:
운동을 할까 말까 결정 못했어요.
Undongeul halkka malkka gyeoljeong motaesseoyo.
I couldn't decide whether or not to exercise.
In this case, (으)ㄹ까 말까 is interchangeable with (으)ㄹ지 말지 (eu)lji malji.
운동을 할지 말지 결정 못했어요.
Undongeul halji malji gyeoljeong motaesseoyo.
I couldn't decide whether or not to exercise.
EXAMPLE SENTENCES
A: 파티에 갈 거예요? Patie gal geoyeyo? Are you going to the party?
B: 글쎄요. 갈까 말까 해요. Geulsseyo. Galkka malkka haeyo. I don't know. I can't decide whether to go or not.
오늘 배가 아파서 점심을 먹을까 말까 해요.
Oneul baega apaseo jeomsimeul meogeulkka malkka haeyo.
My stomach hurts today, so I can't decide whether to eat lunch or not.
이번에는 꼭 정시에 오고 말겠어요.
Ibeoneneun kkok jeongsie ogo malgesseoyo.
This time I'm definitely going to come on time.
This expression is for those who are determined. 고 말았다 means something happened by accident, but in contrast, 고 말겠다 or 고 말 테니까 means something is very firmly planned.
HOW IT'S FORMED
고 말다 plus 겠다. 겠다 is the future tense marker (see page ) and 고 말다 is normally used to talk about something unplanned and unfortunate; in this case, with 겠, it takes the opposite meaning. (으)ㄹ 테니까 (an expression used to talk about plans and covered in detail on page 221) can also follow 고 말다 and together they have the same meaning as 고 말겠다.
HOW IT'S CONJUGATED
| | Future
---|---|---
Action verbs ending in a vowel | 하다 | 하고 말겠다 hago malgetda
하고 말 테니까 hago mal tenikka
Action verbs ending in a consonant | 먹다 | 먹고 말겠다 meokgo malgetda
먹고 말 테니까 meokgo mal tenikka
TAKE NOTE
There are three expressions with 고 말다, and they're all quite different, so it's important to be careful which one you use. 고 말고(요) is used at the end of very short sentences to agree strongly with something another person has said. It's covered on page . 고 말았다, at the bottom of this page, means that something happened by accident. It will always be in the past tense. 고 말겠다 and 고 말 테니까 are always seen in these forms, so if the 고 말다 expression you see or hear has any conjugation of 겠다 or 테니까, it means someone is determined to do something.
If you're not all that determined, you can use (으)려고 하다 (see page 215) and if you're really not determined at all but just have a notion, (으)ㄹ까 하다/보다/싶다 (see page 226) is what you'll want to use. Meanwhile, if what you're planning is absolutely guaranteed to happen, use a future tense expression: 겠다 or (으)ㄹ 것이다 (see page ).
Words like 꼭 kkok or 반드시 bandeusi are often used with 고 말겠다 or 고 말 테니까.
EXAMPLE SENTENCES
이 면접을 꼭 잘 보고 말겠어요.
I myeonjeobeul kkok jal bogo malgesseoyo.
I'm determined to do well in this interview.
내일 그 일을 반드시 끝내고 말 테니까 당신이 맡은 부분도 좀 끝내 주세요.
Naeil geu ireul bandeusi kkeunnaego mal tenikka dangsini mateun bubundo jom kkeunnae juseyo.
Tomorrow I'm definitely going to finish that job, so please finish your part as well.
##
Accidents
정시에 온다는 것이 결국 늦게 오고 말았어요.
Jeongsie ondaneun geosi gyeolguk neutge ogo marasseoyo.
I meant to come on time, but (unfortunately) I ended up being late.
One thing you'll notice if you have lived in Korea for a while is that plans are nice and all, but things quite often don't happen according to plans. Naturally, there's an expression for that. 고 말았다 means that something happened accidentally and unfortunately, and conveys feelings of regret.
HOW IT'S FORMED
Our old friend 고 plus 말다 in the past tense, though neither particularly relates to this expression.If you're really sorry for the way things turned out, you can add 아/어/여 버리다 (which literally means "something is thrown away") in front of the expression to get 아/어/여 버리고 말았다. This strengthens the expression by implying that not only did ~ happen and is now over with, but it's "thrown away"—it can't even be recovered, and there is no way to go back.
HOW IT'S CONJUGATED
| | Past
---|---|---
Action verbs ending in a vowel | 하다 | 하고 말았다 hago maratda
Action verbs ending in a consonant | 먹다 | 먹고 말았다 meokgo maratda
TAKE NOTE
The other 고 말다 expressions are 고 말고(요) and 고 말겠다. The former means "of course" and is on page , and the latter means "I'm determined to ~" and is on page . Pay attention to the tense of any 고 말다 expression: the one in question here is only ever seen in the past tense[말았다], while the others are used only in the present (고 말고[요]) and future (고 말겠다) tenses.
A good expression to use before 고 말았다 is (느)ㄴ다는 것이 (see the next expression), which means that you meant to do A but B ended up happening.
EXAMPLE SENTENCES
잠깐 쉰다는 것이 6시간 동안 자고 말았어요.
Jamkkan swindaneun geosi yeoseotsigan dongan jago marasseoyo.
I planned to rest for just a little while, but then I ended up falling asleep for six hours.
요리를 하고 싶었지만, 가스가 없어서 배달을 시키고 말았어요.
Yorireul hago sipeotjiman, gaseuga eopsseoseo baedareul sikigo marasseoyo.
I wanted to cook, but I didn't have gas and ended up ordering delivery food.
정시에 온다는 것이 결국 늦게 오고 말았어요.
Jeongsie ondaneun geosi gyeolguk neutge ogo marasseoyo.
I meant to come on time, but (unfortunately) I ended up being late.
This expression and 고 말았다 (see page ) go hand in hand; 고 말았다 deals with things that happened accidentally, and A(느)ㄴ다는 것이 B means you meant to A but B ended up happening by accident. If you're prone to making mistakes, you'll want to know this expression well.
HOW IT'S FORMED
(느)ㄴ다 is the indirect speech particle that follows action verbs. It's often used in Korean grammar even when no indirect speech is actually involved, as is the case here. It's conjugated with 는 것이, which turns the verb in A into a noun. 것이 can be shortened to 게.
HOW IT'S CONJUGATED
| | Past
---|---|---
Action verbs ending in a vowel | 하다 | 한다는 것이 handaneun geosi
한다는 게 handaneun ge
Action verbs ending in a consonant | 먹다 | 먹는다는 것이 meokneundaneun geosi
먹는다는 게 meokneundaneun ge
TAKE NOTE
게 is the short form of 것이, not the 게 that turns verbs into adjectives. 게 and 것이 are completely interchangeable.
For your B clause, you should use an expression indicating regret. 아/어/여 버리다 (see page ) and 고 말았다 (see page ) are commonly used.
EXAMPLE SENTENCES
잠깐 쉰다는 것이 6시간 동안 자 버렸어요.
Jamkkan swindaneun geosi yeoseotsigan dongan ja beoryeosseoyo.
I planned to rest for just a little while, but then I ended up falling asleep for six hours.
요리를 한다는 게 음식을 잊어버려서 타 버리고 말았어요.
Yorireul handaneun ge eumsigeul ijeobeoryeoseo ta beorigo marasseoyo.
I wanted to cook, but I forgot about the food and it ended up burned.
##
Progression: Review of Past and Future Tenses
Past (았/었/였다 at/eot/yeotda)
To change a present-tense Korean verb into a past-tense Korean verb, add 았/었/였 to the end. 았 is added to verbs with 오 or 아 as their last vowel:
> 잡다 japda catch | 잡았다 jabatda caught
> ---|---
> 놀다 nolda play | 놀았다 noratda played
> 오다 oda come | 왔다 watda came
And 었다 is added to verbs with any other vowel (어, 우, 이, 으) eo, u, i, eu as their last vowel.
> 먹다 meokda eat | 먹었다 meogeotda ate
> ---|---
> 예쁘다 yeppeuda to be beautiful | 예뻤다 yeppeotda was beautiful
> 숨다 sumda hide | 숨었다 sumeotda hid
> 잡히다 japida to be caught | 잡혔다 japyeotda was caught
였다 is only added to 하다
> 하다 hada do | 했다 did
> ---|---
았다 is also added to verbs ending in 아, but it is shortened to 갔다 to save those few seconds. Likewise, 사다 becomes 샀다. Verbs ending in 오 like 오다 get 았다 added right to that last syllable, so it becomes 왔다. 쏘다 (to shoot) becomes 쐈다. In cases like this where it's awkward to try to say 쐈다, you can just use 쏘았다. However, 왔다 is never 오았다.
> 가다 gada go | 갔다 gotda went
> ---|---
> 사다 sada buy | 샀다 satda bought
> 오다 oda come | 왔다 watda came
> 쏘다 ssoda shoot 쐈 | 다 sswatda or 쏘았다 ssoatda shot
Similar rules apply for verbs ending in 어, 우 or 이. 어 absorbs the 어 in 었다, 우 becomes 웠다, and 이 becomes 였다.
> 서다 seoda stand | 섰다 seotda stood
> ---|---
> 피우다 piuda to smoke | 피웠다 piwotda smoked
> 두다 duda keep | 뒀다 dwotda or 두었다 dueotda kept
> 피다 pida to bloom | 폈다 pyeotda bloomed
Verbs ending in 으 normally lose the 으 because it's a weak vowel sound and easily pushed around by stronger vowel sounds. If the 으 is the only vowel, then you get rid of it and use 었다.
> 쓰다 sseuda to use | 썼다 sseotda used
> ---|---
There are a very small number of cases where 으 is the ending of the second syllable in a multi-syllable word. In these cases, it disappears and you use 았다 or 었다 depending on the vowel in the first syllable.
> 모으다 moeuda to collect | 모았다 moatda collected
> ---|---
Past past (았/었/였었다 at/eot/yeosseotda)
So the past tense tells us that something is finished, but if you really want to emphasize just how finished an event is, you can double the past tense. 았/었/였었다 can be added to a verb to show that it was true in the past, but that it's long over and completely done with now. See page 235 for basic rules on conjugating the past tense; all this expression does is add one more 었.
작년에 부산에 갔었어요.
Jangnyeone Busane gasseoseoyo.
I went to Busan last year. (but came back)
어렸을 때 미국에서 살았었어요.
Eoryeosseul ttae migugeseo sarasseosseoyo.
When I was a child, I lived in the US. (but I no longer do)
이 회사에 취직하기 전에 다른 회사에 면접을 봤었어요.
I hoesae chwijikhagi jeone dareun hoesae myeonjeobeul bwasseosseoyo.
I had an interview with another company before I got a job at this company.
나는 영어를 잠깐 배웠었어요.
Naneun yeongeoreul jamkkan baewosseosseoyo.
I learned English for a short time. (but I no longer do)
You can also, as a joke, just keep on repeating 었었었었 as long as you'd like during your sentence; this further emphasizes the fact that whatever you're talking about is really, definitely, absolutely, positively no longer true.
Past — adjectives ([으]ㄴ, 던, 았/었/였던 [eu]n, deon, at/eot/yeotdeon)
던 is used when you want to turn a verb into an adjective describing something that was done in the past. For example, 하던 일 means "the work that was done," or 보던 사람 means "the person I saw." It is used only for things that happened repeatedly in the past. These can be things that are still done or things that are no longer done. If you'd like to indicate for certain that something is no longer done, you can use (으)ㄴ. If you'd like to talk about something that was done only once in the past, you can use 았/었/였이던 or (으)ㄴ.
항상 늦던 사람이 계속 미안하다고 했어요.
Hangsang neutdeon sarami gyesok mianhadago haesseoyo.
That man who always used to be always late kept on apologizing.
늦었던 사람이 계속 미안하다고 했어요.
Neujeotdeon sarami gyesok mianhadago haesseoyo.
That man who was late (that one time) kept on apologizing.
늦은 사람이 계속 미안하다고 했어요.
Neujeun sarami gyesok mianhadago haesseoyo.
That man who was late kept on apologizing.
내가 마시던 커피가 맛있었어요.
Naega masideon keopiga masisseosseoyo.
That coffee that I used to drink was delicious.
내가 마셨던 커피가 맛있었어요.
Naega masyeotdeon keopiga masisseosseoyo.
That coffee I drank (that one time) was delicious.
내가 마신 커피가 맛있었어요.
Naega masin keopiga masisseosseoyo.
That coffee I drank (but am no longer drinking) was delicious.
더 by itself is a particle having to do with personal recollections (see page ). Thus, 던 and 았/었/였이던 should be used only when talking about things you've personally experienced. Normally 더 isn't used to talk about things you've done yourself, but 던 is often used that way. While 던 is like 더 with the past tense particle ㄴ on the end, it doesn't play by the same rules as 더.
4급에서 공부하던 문법이 참 어려웠어요.
Sageubeseo gongbuhadeon munbeobi cham eoryeowosseoyo
The grammar we studied in level four was really difficult.
지난주 먹었던 케이크가 맛있었어요.
Jinanju meogeotdeon keikeuga masisseosseoyo.
The cake I ate last week was delicious.
어렸을 때 자주 가던 식당이 요즘 없어졌어요.
Eoryeosseul ttae jaju gadeon sikdangi yojeum eopsseojyeosseoyo.
The restaurant I often used to go to a a child is now gone.
Past, present and future — (으)ㄴ, 는, (으)ㄹ (eu)n, neun, (eu)r
은 is the past tense marker for verbs ending in consonants, ㄴ is the past tense marker for verbs ending in vowels, 는 is the present tense marker for anything, 을 is the future tense marker for verbs ending in consonants, and ㄹ is the future tense marker for verbs ending in vowels. Adjectives (descriptive verbs) are almost always conjugated using (으)ㄴ.
These markers are a short, convenient way of turning verbs into adjectives.
> 온 사람 | on saram | the person who came
> ---|---|---
> 오는 사람 | oneun saram | the person who is coming
> 올 사람 | ol saram | the person who will come
> 먹은 음식 | meogeun eumsik | the food that was eaten
> 먹는 음식 | meokneun eumsik | the food that is being eaten
> 먹을 음식 | meogeul eumsik | the food that will be eaten
> 예쁜 여자 | yeppeun yeoja | beautiful woman
> 작은 남자 | jageun namja | small man
In addition to being used that way, the above markers are also part of many grammatical expressions. In this book and every other Korean textbook I've seen, they are considered part of the expression and you will learn them along with the rest of the expression. Some expressions ([으]ㄹ 수 있다/없다, [으]ㄹ 것이다, many of the expressions used to make guesses or plans, etc.) can be used only in certain tenses while others ([으]ㄴ/는/[으]ㄹ 듯이, [으]ㄴ/는/[으]ㄹ 때, etc.) can be used in any tense. You'll have to choose your tense marker according to what you're trying to convey.
One big difference between (으)ㄴ and 던 is that with (으)ㄴ, the action is always completely finished and over with. (This is not true of descriptive verbs: a 예쁜 여자 is still 예쁜 at the time the sentence was formed.) You could say 제가 하던 일 jega hadeon il if you were talking about something you were working on earlier (which may or may not have been finished), but you can't call it 한 일 han il unless it's all done.
It is extremely easy to get confused, particularly with ㄴ and ㄹ, as to whether the ㄴ or ㄹ you see is a past or future tense marker, or just the end of a word. This is particularly true with words like 한 of 한 일 above, where 한 has many other meanings besides 하다 plus ㄴ. There's really no easy way to distinguish the words in this case; it's really just a matter of practice and familiarity.
Note as well that the past, present, or future tense marker with (으)ㄴ/는/(으)ㄹ compares the time of that action to what's happening in the rest of your sentence. So if clause A happened before clause B, use (으)ㄴ regardless of what tense your sentence is in relative to the present.
늦게 온 사람이 계속 미안하다고 했어요.
Neutge on sarami gyesok mianhadago haesseoyo.
That man who was late kept on apologizing.
늦게 온 사람이 계속 미안하다고 해요.
Neutge on sarami gyesok mianhadago haeyo.
That man who was late keeps on apologizing.
If A and B happen at the same time, use 는 regardless of B's tense.
늦게 오는 사람이 계속 미안하다고 했어요.
Neutge oneun sarami gyesok mianhadago haesseoyo.
That man who was late kept on apologizing.
늦은 사람이 계속 미안하다고 해요.
Neujeun sarami gyesok mianhadago haeyo.
That man who was late keeps on apologizing.
The future — (으)ㄹ 것이다, 겠다 (eu)l geosida, getda
There are two expressions most commonly used to talk about the future in Korean. 겠다 is simple to use but a bit more limited, while (으)ㄹ 것이다 is a willing, cheerful participant in almost any phrase. Let's look at (으)ㄹ 것이다 first.
(으)ㄹ 것이다 can be used for just about any statement about the future, no matter who's doing it or why. You can add it to any kind of verb. 을 것이다 is added to verbs ending in consonants and ㄹ 것이다 is added to verbs ending in vowels. For nouns, it's better to use 되다 (to become) and then conjugate that with ㄹ 것이다. See the third example sentence below.
This is conjugated as (으)ㄹ 거예요 in 존댓말 and (으)ㄹ 거야 in 반말. If you need to be super-polite, just conjugate (으)ㄹ 것이다 correctly: (으)ㄹ 것입니다. However, this is somewhat less common. The table below shows the basic form of (으)ㄹ 것이다 plus the polite and 반말 forms.
| | Future
---|---|---
Action verbs ending in a vowel | 하다 | 할 것이다 hal geosida
할 거예요 hal geoyeyo
할 거야 hal geoya
Action verbs ending in a consonant | 먹다 | 먹을 것이다 meogeul geosida
먹을 거예요 meogeul geoyeyo
먹을 거야 meogeul geoya
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁠 것이다 yeppeul geosida
예쁠 거예요 yeppeul geoyeyo
예쁠 거야 yeppeul geoya
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작을 것이다 jageul geosida
작을 거예요 jageul geoyeyo
작을 거야 jageul geoya
Nouns ending in a vowel | 남자 | 남자일 것이다 namjail geosida
남자일 거예요 namjail geoyeyo
남자일 거야 namjail geoya
Nouns ending in a consonant | 물 | 물일 것이다 muril geosida
물일 거예요 muril geoyeyo
물일 거야 muril geoya
And here are some examples:
파티에 늦게 올 거예요. Patie neutge ol geoyeyo.
파티에 늦게 올 거야. Patie neutge ol geoya.
I'll come to the party late.
그 치마를 입으면 예쁠 거예요. Geu chimareul ibeumyeon yeppeul geoyeyo.
그 치마를 입으면 예쁠 거야. Geu chimareul ibeumyeon yeppeul geoya.
If you wear that skirt, you'll be beautiful.
가영이는 의사가 될 거예요. Kayoung-i-neun uisaga doel geoyeyo.
가영이는 의사가 될 거야. Kayoung-i-neun uisaga doel geoya.
Kayoung is going to be a doctor.
(으)ㄹ 거예요 and (으)ㄹ게요 (see page ) are very similar, and it's easy to confuse one for the other. The difference is that (으)ㄹ게요 depends more on your listener. With (으)ㄹ 거예요 you're simply stating what you think will happen in the future, whereas (으)ㄹ게요 has more of a connotation of "I'm going to do ~ (if it's okay with you)." (으)ㄹ게요 can also be used when you're making/changing your plans as a result of what the other person has said. Either way, it always has something to do with that other person's reaction. (으)ㄹ 거예요, on the other hand, has nothing at all to do with what the other person says.
As for 겠다, it's more commonly used with very formal Korean (습니다, 입니다, etc.) You can use it to talk about your own or someone else's intentions for the future, especially if you or they are quite determined to carry out these intentions. You can't normally use it with sentences about the future that are not describing someone's intentions; in these cases you should use (으)ㄹ 것이다.
Additionally, 겠다 is often used to make guesses, much like 나/(으)ㄴ가 보다 (see page ) or (으)ㄹ 것 같다 (see page ). You'll often hear Koreans say things like "맛있겠다!" upon seeing a plate of food or "아프겠다! Apeugetda" upon seeing a friend wearing a cast on their arm. This doesn't mean they're anticipating future pain for the person; it's just a way of making a guess. 맛있겠다 masitgetda means "Looks good!" and 아프겠다 means "Looks painful!"
Finally, there are a few idiomatic expressions with 겠다. The most common are 모르겠다 moreugetda and 알겠다 algetda ("I don't know" and "I know," or more accurately, "I understand"). This has nothing to do with the future or with guessing; it's just it's said.
As for conjugation, simply add 겠 to any action or descriptive verb regardless of its ending. Just like (으)ㄹ 것이다, nouns are normally followed by 되다 and then 겠다 (so, 되겠다).
| | Future
---|---|---
Action verbs ending in a vowel | 하다 | 하겠다 hagetda
Action verbs ending in a consonant | 먹다 | 먹겠다 meokgetda
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁘겠다 yeppeugetda
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작겠다 jakgetda
Here are some examples.
늦겠어요.
Neutgesseoyo.
You're going to be late.
어디로 가시겠어요?
Eodiro gasigesseoyo?
Where would you like to go?
괜찮겠어요?
Gwaen-chan-kesseoyo?
Will you be okay?
##
Progression: While
숙제 하는 동안에 음악을 들었어요.
Sukje haneun dong-ane eumageul deureosseoyo.
While I was doing my homework, I listened to music.
동안에 is another expression that means "while." It is used only while talking about time.
HOW IT'S FORMED
동안 means "while" and can be used on its own. 에 is a preposition of location which, in this case, serves to emphasize 동안. (으)ㄴ and 는 are tense markers which go between verbs and 동안(에).
HOW IT'S CONJUGATED
동안에 can follow either an action verb or a noun indicating a period of time (일 년 il nyeon, 한 달 han dal, 아침 achim, etc) or an action or activity, like a class (수업 sueop) or lunch (점심 jeomsim) that takes a period of time. If it follows a verb, put 는 before that verb regardless of which tense it's in. The assumption is that A was in the present tense at the time B happened. If your verb has to do with coming and going (가다 gada, 오다 oda, 떠나다 tteonada, etc.) and it happened in the past, then you can use (으)ㄴ 동안에. B is something that happened while A was ongoing.
| | All tenses
---|---|---
Action verbs ending in a vowel | 하다 | 하는 동안(에) haneun dongan(e)
Action verbs ending in a consonant | 먹다 | 먹는 동안(에) meokneun dongan(e)
가다, 오다, 떠나다 and other verbs describing coming and going | 가다 | 가는 동안(에) ganeun dongan(e)
Nouns ending in a vowel | 휴가 | 휴가 동안(에) hyuga dongan(e)
Nouns ending in a consonant | 아침 | 아침 동안(에) achim dongan(e)
TAKE NOTE
사이에 is a similar expression which can be used to talk about other things besides just time; see page 251 for other ways to use 사이에. The difference between the two is that 동안에 can be used whether or not the subjects of A and B are the same, while 사이에 can be used only if the subjects are different.
(으)ㄴ/는 동안(에) is completely interchangeable with (으)ㄹ 때; see the next page for more on (으)ㄹ 때.
(도)중에 (page 244) has a similar meaning but can be used only with action verbs.
EXAMPLE SENTENCES
민수가 공부하는 동안에 가영이 TV를 보고 있어요.
Minsuga gongbuhaneun dongane Kayoungi tibireul bogo isseoyo
While Minsu studies, Kayoung is watching TV.
영국에 가 있는 동안에 어머니가 죽었어요.
Yeonguke ga inneun dong-ane eomeoniga jugeosseoyo.
While I was in England, my mother died.
대학교 때 민수가 자주 늦었어요.
Daehakgyo ttae Minsuga jaju neujeosseoyo.
During university Minsu was often late.
때 is an important word to know if you ever want to talk about time in Korean. It's used in many expressions that mean "while," "during," "when," "at times," and the like.
HOW IT'S FORMED
(으)ㄹ, though it appears to be the future tense marker, doesn't really mean anything like that in this expression and the expression remains (으)ㄹ 때(에는) no matter what time you're talking about. 때 is a noun which simply means "time." 에 is a preposition of location that is quite frequently seen in expressions involving time; it roughly means "at," as in "at that time." 는 is the topic marker (see page ) and serves to emphasize the period of time. Both 에 and 는 are optional, so you can use either or both.
HOW IT'S CONJUGATED
A must be a period of time or something that you were doing for a period of time: note that the nouns below had to be changed to things that take some time.
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했을 때 haesseul ttae | 할 때 hal ttae | 할 때
Action verbs ending in a consonant | 먹다 | 먹었을 때 meogeosseul ttae | 먹을 때 meogeul ttae | 먹을 때
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤을 때 yeppeosseul ttae | 예쁠 때 yeppeul ttae | 예쁠 때
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았을 때 jagasseul ttae | 작을 때 jageul ttae | 작을 때
Nouns ending in a vowel | 휴가 | 휴가 때 hyuga ttae | 휴가 때 | 휴가 때
Nouns ending in a consonant | 아침 | 아침 때 achim ttae | 아침 때 | 아침 때
TAKE NOTE
(으)ㄴ/는 동안(에) is completely interchangeable with (으)ㄹ 때; see page 241 for more on (으)ㄴ/는 동안(에). Similar expressions are 사이에 saie and (도)중에 (do)junge (see pages 251 and 244, respectively), but these follow action verbs rather than nouns indicating periods of time. 사이에 can be used to talk about locations, while 동안에 and 중에 cannot. It can also be used for expressions like "among friends" and for actions occurring at the same time. 사이에 is covered on page 251. (으)ㄹ 때, like 동안에, is used to talk only about time.
Some commonly used expressions with 때 include:
어렸을 때/어릴 때 eoryeosseul ttae/eoril ttae during childhood
한창 때 hanchang ttae in one's prime
아무 때 amu ttae at any time
어느 때/어떤 때/어떨 때 eoneu ttae/eotteon ttae/eotteol ttae sometimes
그 때 geu ttae at that time
이 때 i ttae at this time
그맘때 geumamttae around that time
날 때부터 nal ttaebuteo from birth
EXAMPLE SENTENCES
A: 언제 그 일을 끝내야 돼요? Eonje geu ireul kkeunnaeya dwaeyo?
When do I have to finish this job?
B: 아무 때나 끝내도 돼요. Amu ttaena kkeunnaedo dwaeyo.
You can finish it whenever you like.
여기를 걸을 때 조심하세요.
Yeogireul georeul ttae josimhaseyo.
Be careful while walking here.
숙제 하는 중에 음악을 들어요.
Sukje haneun jung-e eumageul deureoyo.
While doing my homework, I listen to music.
중에 and all its variations have two meanings. You can use it to talk about two actions that happen at the same time, or you can use it to present alternatives: Among all the boys in the class, which one(s) would you want to date? Or, between Korean and Western food, which one do you feel like eating for lunch today?
HOW IT'S FORMED
중 jung by itself means "middle" and is very commonly used in words like 중학교 junghakgyo (middle school) and 집중 jipjung ("concentration"; literally "collect at the center"). 에 e and 에서 eseo are prepositions of location which, in this case, emphasize 중. 에 is used in any sentence involving 중 while 에서 is used only when talking about choices; 남자 중에서 namjajungeseo is fine, but 가는 중에서 isn't. 도 is optional and only be added only when 중 means "on the way"; 가는 중에 ganeun junge can become 가는 도중에 ganeun dojunge, but 남자 중에 can't become 남자 도중에.
HOW IT'S CONJUGATED
는 (도)중에 can be used only with action verbs. When used with action verbs, it means "while." 중에 by itself can also follow nouns indicating some period of time: 일 년, 한 달, 수업, 아침, etc.
| | All tenses
---|---|---
Action verbs ending in a vowel | 하다 | 하는 중에 haneun junge
하는 도중에 haneun dojunge
Action verbs ending in a consonant | 먹다 | 먹는 중에 meokneun junge
먹는 도중에 meokneun dojunge
Nouns ending in a vowel | 휴가 | 휴가 중에 hyuga junge
Nouns ending in a consonant | 학생 | 학생 중에 haksaeng junge
중에서 can be used only with nouns and in this case it means "between" or "among," and the next part of the sentence offers a choice.
| | Present
---|---|---
Nouns ending in a vowel | 남자 | 남자 중에 namja junge
남자 중에서 namja jungeseo
Nouns ending in a consonant | 책 | 책 중에 chaek junge
책 중에서 chaek jungeseo
TAKE NOTE
Be careful about all these forms of 중에. Check the tables above to see which one is appropriate when.
The subjects of both clauses should be the same.
Similar expressions are ([으]ㄴ/는) 동안에, ([으]ㄴ/는) 사이에, and (으)ㄹ 때. 동안에 (see page 241) is used only to talk about time and co-occurring actions. 사이에 can be used to talk about locations, while 동안에 and 중에 cannot. It can also be used for expressions like "among friends" and for co-occurring actions. 사이에 is covered on page 251. (으)ㄹ 때 (see page 243), like 동안에, is used only to talk about time. Thus, when talking about two actions that occur at the same time, 중에, 도중에, 동안에, 사이에 and (으)ㄹ 때 are all interchangeable; when talking about making a choice, they are not.
다가 (see page ) describes an interruption in action and is quite often interchangeable with 중에: for example, "My friend called while I was in the shower."
You can also use 는 중이다 at the end of a sentence to describe being in the middle of something. Now, when the aforementioned friend calls you, you'll know how to explain that you were in the middle of taking a shower when he called. 중에 있다 can be used here as well.
EXAMPLE SENTENCES
학교에 가는 중에 사고를 봤어요.
Hakgyoe ganeun junge sagoreul bwasseoyo.
While I was going to school, I saw an accident.
수업 중에 핸드폰을 쓰면 안 돼요.
Sueop junge haendeuponeul sseumyeon an dwaeyo.
During class you can't use your cell phone.
지금 샤워 하는 중이에요.
Jigeum syawo haneun jungieyo
I'm in the middle of taking a shower right now.
지금 샤워 하는 중에 있어요.
Jigeum syawo haneun junge isseoyo.
I'm in the middle of taking a shower right now.
저 남자들 중에서 어떤 남자가 제일 잘 생겼다고 생각하세요?
Jeo namjadeul jungeseo eotteon namjaga je-il jal saenggyeotdago saenggakaseyo?
Which of those men do you think is the most handsome?
책 중에서 하나만 골라 읽어 보세요.
Chaek jungeseo hanaman golla ilgeo boseyo.
Please choose and read just one of the books.
늦을까 봐 운전하면서 계속 시계를 봤어요.
Neujeulkka bwa unjeonhamyeonseo gyesok sigyereul bwasseoyo.
I was afraid I'd be late, so I kept checking my watch as I drove.
This is a very basic way to talk about A and B happening at the same time.
HOW IT'S FORMED
It's an expression on its own and has nothing to do with (으)면 or 아/어/여서.
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Action verbs ending in a vowel | 하다 | 하면서 hamyeonseo
Action verbs ending in a consonant | 먹다 | 먹으면서 meogeumyeonseo
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁘면서 yeppeumyeonseo
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작으면서 jageumyeonseo
Nouns ending in a vowel | 남자 | 남자이면서 namjaimyeonseo
Nouns ending in a consonant | 물 | 물이면서 murimyeonseo
TAKE NOTE
Again, this is completely different from 면; see page for more on 면.
Since A and B happen at the same time, there's no need to conjugate the verb before (으)면서: the verb at the end of the B clause expresses the tense of both clauses. So a verb with (으)면서 is always conjugated as if it were in the present tense.
(으)면서도 (see page 253) is a similar expression, but can be used only when A and B contrast: when B isn't something that would normally be expected to co-occur with A. 면서 can be used instead of 면서도 in all cases, but the reverse is not true.
EXAMPLE SENTENCES
빨래를 하면서 음악을 들어요.
Ppallaereul hamyeonseo eumageul deureoyo.
I listen to music while doing laundry.
아이가 먹으면서 TV를 보고 있어요.
Aiga meogeumyeonseo tibireul bogo isseoyo.
The child is watching TV while eating.
밥을 먹으면서 맥주도 마셨어요.
Babeul meogeumyeonseo maekjudo masyeosseoyo.
While I was eating, I also drank beer.
멀리 가면 갈수록 피곤해졌어요.
Meolli gamyeon galsurok pigonhaejyeosseoyo.
멀리 갈수록 피곤해졌어요.
Meolli galsurok pigonhaejyeosseoyo.
The farther I went, the more tired I became.
This expression simply means that the more A happens, the more B happens, or as A progresses, B progresses as well.
HOW IT'S FORMED
You can use (으)ㄹ수록 on its own or you can repeat the verb in A and follow the first A with (으)면 and the second with (으)ㄹ수록. It's really up to you. The meaning is the same either way.
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Action verbs ending in a vowel | 하다 | (하면) 할수록 (hamyeon) halsurok
Action verbs ending in a consonant | 먹다 | (먹으면) 먹을수록 (meogeumyeon) meogeulsurok
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | (예쁘면) 예쁠수록 (yeppeumyeon) yeppeulsurok
Descriptive verbs (adjectives) ending in a consonant | 작다 | (작으면) 작을수록 (jageumyeon) jageulsurok
Nouns ending in a vowel | 남자 | (남자면) 남자일수록 (namjamyeon) namjailsurok
Nouns ending in a consonant | 물 | (물이면) 물일수록 (murimyeon) murilsurok
TAKE NOTE
B is a sentence describing what happens the more A, so it often ends with an expression meaning "to become": 아/어/여지다 or 게 되다 (see page 251). However, this is not strictly necessary; people will understand from your use of (으)ㄹ수록 (eu)l-surok that the situation is changing.
수록하다/되다 surokada/doeda is a verb having to do with containment; it has nothing to do with this 수록, which always follows a verb stem ending in ㄹ.
갈수록 galsurok is an expression which can literally mean "as you go," but can also mean "as time passes."
EXAMPLE SENTENCES
좋으면 좋을수록 잘 써져요.
Joeumyeon joeulsurok jal sseojyeoyo.
좋을수록 잘 써져요.
Joeulsurok jal sseojyeoyo.
The better it is, the more it's used.
공부하면 공부할수록 듣기 능력이 더 좋아져요.
Gongbuhamyeon gongbuhalsurok deutgi neung-nyeog-i deo joa-jyeo-yo.
공부할수록 듣기 능력이 더 좋아져요.
Gongbuhalsurok deutgi neung-nyeog-i deo joa-jyeo-yo.
The more I study, the better my listening ability gets.
학교에 가는 길에 친구를 만나서 늦었어요.
Hakgyoe ganeun gire chingureul mannaseo neujeosseoyo.
While on the way to school, I met a friend, so I was late.
This is another expression like (도)중에, which means that "something happened while you were in the middle of something else." However, 는 길에 is more particular to use than 도중에.
HOW IT'S FORMED
는 is the present tense marker. 길 literally means "road" and 에 is a preposition meaning "to" or "on." This expression literally means "while on the road."
HOW IT'S CONJUGATED
는 길에 is rarely used with any verbs in A other than those indicating movement. Here are a few examples of verbs with which you can use 는 길에:
가다 gada to go
오다 oda to come
돌아가다, 들어오다 doragada, deureooda, etc. to return, to come in, etc.
떠나다 tteonada to leave
다니다 danida to go around
퇴근하다/출근하다 toegeunhada/chulgeunhada to leave/go to work
Take your A verb and add 는 길에. Although 는 is the present tense marker, it is used with 길에 regardless of when the incident happened: A was still in the present tense when B happened, so it gets the present tense marker.
I can't find any directional verbs that end in a consonant, but if you should need to conjugate any, they're conjugated the same way: just add 는 길에.
| | Past | Present
---|---|---|---
Directional verbs | 가다 | 가는 길에 ganeun gire | 가는 길에
TAKE NOTE
Be careful to use this expression only with verbs of direction.
Two other similar expressions are (으)ㄴ/는 김에 (eu)n/neun gime (on page 250) and 는 (도)중에 (on page 244). The difference between 는 길에 and 는 김에 is that the former can be used for things that happened by accident, while 는 김에 is used only for events or activities that were planned. 는 도중에 can be used with any verb, whether it has to do with coming and going, or not. So if you wanted to talk about what happened while you were working, you could say 일하는 중에 ilhaneun junge or 일하는 도중에 ilhaneun dojunge, but not 일하는 길에 ilhaneun gire. But if something terribly interesting happened while you were out going somewhere and you wanted to tell someone all about it, you could say 가는 중에 ganeun junge, 가는 도중에 ganeun dojunge, or 가는 길에 ganeun gire.
Finally, in certain sentences, 다가 and 는 길에 are interchangeable. A다가 B (see page ) means that B happened during an action and interrupted A. Sentences in which the verb in A is a directional verb can use either 다가 or 는 길에.
You can use 는 길이다 to end a sentence saying "you are or were on your way somewhere."
EXAMPLE SENTENCES
집에 오는 길에 슈퍼에 갔어요.
Jibe oneun gire syupeoe gasseoyo.
While coming home, I went to the supermarket.
홍콩으로 여행을 떠나는 길에 이 선물을 홍콩에 있는 제 친구에게 전해주세요.
Hongkong-euro yeo-haeng-eul tteonaneun gire i seonmureul Hongkonge inneun je chinguege jeon-hae-juseyo
While you are on your trip to Hong Kong, please give this present to my friend in Hong Kong.
미국에서 돌아오는 길에 공항 면세점에서 기념품을 사 주세요.
Migugeseo dor-a-o-neun gire gonghang myeonsejeomeseo gin-yeom-pum-eul sa juseyo.
While you are coming back from USA, please buy a souvenir at the airport duty-free shop.
A: 어디 가는 길이에요? Eodi ganeun girieyo? Where are you going?
B: 은행에 가는 길이에요. Eunhaenge ganeun girieyo. I'm on my way to the bank.
온 김에 같이 저녁 먹을래요.
On gime gachi jeonyeok meogeullaeyo.
Since you're here, let's have dinner together.
This expression is good for people who are lazy and/or good planners. It's used to talk about doing B incidentally or on your way to or from A. You can use it to make requests of people, as in the title of this section, or to talk about errands you combined: "While you're A, would you mind B as well?" or "While on my way to A, I also took care of B" are the kind of sentences you'd translate into (으)ㄴ/는 김에.
HOW IT'S FORMED
The past and present tense markers (으)ㄴ/는 are combined with 김에, which doesn't mean anything by itself.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 한 김에 han gime | 하는 김에 haneun gime | 하는 김에
Action verbs ending in a consonant | 먹다 | 먹은 김에 meogeun gime | 먹는 김에 meokneun gime | 먹는 김에
TAKE NOTE
A is the main objective, the thing you originally set out to do, and B is what you're doing incidentally or asking someone to do.
A somewhat similar expression is (으)ㄴ/는 길에 (see page 248). The difference between the two is that while in both cases A is the main objective, with (으)ㄴ/는 김에 B is always something planned, while with (으)ㄴ/는 길에 B can also have happened accidentally. So you'd use 김에 to talk about stopping by the grocery store on your way to the bank while 길에 is what you'd use to describe accidentally running into your friend on the street on the way to the grocery store or the bank. Also, 길에 can be used only with verbs indicating direction (like 오다 or 가다) while 김에 can be used with any action verb.
은행에 가는 김에 슈퍼에 들렀어요.
Eunhaenge ganeun gime syupeoe deulleosseoyo.
While on my way to the bank, I stopped by the supermarket.
은행에 가는 길에 친구를 만났어요.
Eunhaenge ganeun gire chingureul mannasseoyo.
While on my way to the bank, I (accidentally) ran into my friend.
EXAMPLE SENTENCES
파티에 가는 김에 멋있는 옷도 준비했어요.
Patie ganeun gime meosinneun otdo junbihaesseoyo.
Because I was going to the party, I prepared some stylish clothes.
우체국에 가는 김에 그 옆에 있는 은행도 들렀어요.
Ucheguge ganeun gime geu yeope inneun eunhaengdo deulleosseoyo.
On my way to the post office, I dropped by the bank.
여자친구 선물을 사는 김에 어머니 선물도 같이 샀어요.
Yeoja-chin-gu seonmureul saneun gime eomeoni seonmuldo gachi sasseoyo.
While buying a gift for my girlfriend, I also bought a gift for my mother.
A commonly used expression with 김에 is 말(이) 나온 김에 mal(i) naon gime ~, which means "While we're on the subject of ~."
A: 다음 회의에 그 문제에 대해서 이야기를 더 해야 돼요. Daeum hoe-ui-e geu mun-je-e daehaeseo iyagireul deo haeya dwaeyo. We need to talk about that problem more at our next meeting.
B: 말이 나온 김에 다음 회의 날짜를 정할까요? Mari naon gime daeum hoe-ui naljjareul jeonghalkkayo? While we're on the subject, shall we set the date for our next meeting?
저녁을 먹는 사이에 친구가 들렸어요.
Jeonyeogeul meokneun saie chinguga deul-lyeosseoyo.
While I was eating dinner, my friend dropped by.
You can use 사이에 to talk about people, places, or activities. It can mean either "while" (when talking about activities) or "between" (when talking about people or places).
HOW IT'S FORMED
사이 means between and 에 is the preposition of location. When 사이에 is used to talk about activities: either the past tense marker (으)ㄴ or the present tense marker 는 can be used.
HOW IT'S CONJUGATED
Since we're essentially talking about two different meanings that happen to share an expression, let's divide this into two parts. (으)ㄴ/는 사이에 (the top table, shown below) can be used only with action verbs; it means "while." A and B are two simultaneous activities. Almost all verbs with 사이에 take the present tense marker 는 no matter when they happened. The only exceptions are verbs having to do with coming and going: 가다, 오다, 떠나다, etc. When used in past tense sentences, these verbs take the past tense marker (으)ㄴ.
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 하는 사이에 haneun saie | 하는 사이에 | 하는 사이에
Action verbs ending in a consonant | 먹다 | 먹는 사이에 meokneun saie | 먹는 사이에 | 먹는 사이에
가다, 오다, 떠나다, and other verbs indicating going or coming | 가다 | 간 사이에 gan saie | 가는 사이에 | 가는 사이에
Plain old 사이에 (below) follows nouns. When talking about places, it means "between" (between the post office and the bank, between the hospital and the school, etc.) and goes after the two places it links.
When talking about people it means "between" or "among" (between friends, among ourselves, etc.) and goes after the people it links. This can be two people (너와 나 사이에서 neowa na saieseo) or a group of people (친구 사이에서 chingu saieseo).
| | All tenses
---|---|---
Nouns ending in a vowel | 남자 | 남자 사이에 namja saie
Nouns ending in a consonant | 물 | 물 사이에 mul saie
TAKE NOTE
When using 사이에 to talk about activities (B while A), the subjects of clause A and clause B must be different. If the subjects are the same, you should use 동안에, which is covered on page 241.
사이에서 means a metaphorical space between people while 사이에 means a physical one such as the space between two buildings. See the examples.
Similar expressions are ([으]ㄴ/는) 동안에, ([으]ㄴ/는) (는) (도)중에 (see page 244), and (으)ㄹ 때 (see page 243). 동안에 is used to talk only about time and co-occurring actions. (으)ㄹ 때, like 동안에, is used to talk only about time. Thus, when talking about two actions that occur at the same time, 중에, 도중에, 동안에, 사이에 and (으)ㄹ 때 are all interchangeable; when talking about making a choice, they are not.
EXAMPLE SENTENCES
민수가 공부하는 사이에 가영이 TV를 보고 있어요.
Minsuga gongbuhaneun saie Kayoungi tibireul bogo isseoyo.
While Minsu studies, Kayoung is watching TV.
영국에 간 사이에 어머니가 죽었어요.
Yeong-gug-e gan saie eomeoniga jugeosseoyo.
While I was in England, my mother died.
우리 사이에 비밀이 어디 있어?
Uri saie bimiri eodi isseo?
There are no secrets between us.
외국인들 사이에서 그 식당이 인기 있어요.
Oe-gug-in-deul sa-i-eseo geu sikdangi ingi isseoyo.
That restaurant is popular among foreigners.
지하철 역이 학교와 공원 사이에 있어요.
Jihacheol yeogi hakgyowa gongwon saie isseoyo.
The subway stop is between the school and the park.
서둘렀으면서도 늦었어요.
Seo-dul-leoss-eu-myeon-seo-do neujeosseoyo.
Even though I hurried, I was late.
Here's another way to say "while," but this one adds an element of contrast: you wouldn't expect B to happen while A is happening, but somehow it is.
HOW IT'S FORMED
(으)면서 means "while" (see page 246). 도 is a particle used to indicate contrast and is covered in detail on page .
HOW IT'S CONJUGATED
You can use (으)ㄹ 거면서도 for hypothetical events or events in the future.
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했으면서도 haesseumyeonseodo | 하면서도 hamyeonseodo | 할 거면서도 hal geomyeonseodo
Action verbs ending in a consonant | 먹다 | 먹었으면서도 meogeosseumyeonseodo | 먹으면서도 meogeumyeonseodo | 먹을 거면서도 meogeul geomyeonseodo
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤으면서도 yeppeosseumyeonseodo | 예쁘면서도 yeppeumyeonseodo | 예쁠 거면서도 yeppeul geomyeonseodo
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았으면서도 jagasseumyeonseodo | 작으면서도 jageumyeonseodo | 작을 거면서도 jageul geomyeonseodo
Nouns ending in a vowel | 남자 | 남자였으면서도 namjayeosseumyeonseodo | 남자이면서도 namjaimyeonseodo | 남자일 거면서도 namjail geomyeonseo
Nouns ending in a consonant | 물 | 물이었으면서도 murieosseumyeonseodo | 물이면서도 murimyeonseodo | 물일 거면서도 muril geomyeonseodo
TAKE NOTE
This expression can be changed to (으)면서, but (으)면서 can't always be changed to (으)면서도. (으)면서도 can also be replaced by 지만 (see page ) although it loses some of the nuance. You can't use (으)면서도 in the past tense if the subject is the first person ("I" or "we").
서둘렀으면서도 늦었어요.
Seodulleosseumyeonseodo neujeosseoyo.
Even though I hurried, I was late.
서둘렀지만 늦었어요.
Seodulleotjiman neujeosseoyo.
I hurried, but I was late.
EXAMPLE SENTENCES
남자친구에게 화가 나면서도 그를 사랑해요.
Namjachinguege hwaga namyeonseodo geureul saranghaeyo.
Even when I'm angry at my boyfriend, I still love him.
이 옷은 예쁘면서도 잘 안 팔려요.
I oseun yeppeumyeonseodo jal an pallyeoyo.
Even though these clothes are beautiful, they don't sell well.
늦을까 봐 시계를 봐 가면서 운전했어요.
Neujeulkka bwa sigyereul bwa gamyeonseo unjeonhaesseoyo.
I was worried I'd be late, so I kept looking at my watch while driving.
Like (으)면서, this expression is used to talk about doing two things at the same time. However, in this case B is the main thing you're doing and A is what you're doing occasionally while focusing on B.
HOW IT'S FORMED
아/어/여 가다 means to do something while going. 면서 means "while" and is covered on page 246.
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Action verbs ending in a vowel | 하다 | 해 가면서 hae gamyeonseo
Action verbs ending in a consonant | 먹다 | 먹어 가면서 meogeo gamyeonseo
TAKE NOTE
Remember that B is the main thing; you're saying "I drank coffee while studying" (커피를 마셔 가면서 공부했어요 Keopireul masyeo gamyeonseo gongbuhaesseoyo) rather than "I studied while drinking coffee" (공부해 가면서 커피를 마셨어요 Gongbuhae gamyeonseo keopireul masyeosseoyo), assuming you mean you raised a mug to your lips once in a while in between studying rather than steadily downing pot after pot.
EXAMPLE SENTENCES
담배를 피워 가면서 경치를 봤어요.
Dambaereul piwo gamyeonseo gyeongchireul bwasseoyo.
I looked at the scenery while smoking a cigarette.
이 책을 봐 가면서 새로운 사실을 알게 되었어요.
I chaegeul bwa gamyeonseo saeroun sa-si-reul alge doe-eosseoyo.
While reading this book, I learned many new facts.
##
Progression: Continuing States
제가 새로운 한국어 단어를 공책에 쓰고 있었어요.
Jega saeroun hangugeo dan-eo-reul gongchaege sseugo isseosseoyo.
I was writing the new Korean vocabulary in my notebook.
새로운 한국어 단어가 공책에 써 있었어요.
Saeroun hangugeo dan-eo-ga gongchaege sseo isseosseoyo.
The new Korean vocabulary was written in my notebook.
고 and 아/어/여 있다 are the Korean present progressive. I've put them together because they're really two sides of the same coin: one is used for continued actions while one is used for continued states.
HOW IT'S FORMED
First, take either 고 (see page ) or 아/어. Both have a variety of uses, most of which aren't really relevant to understanding the expressions above. One thing to remember about 아/어 is that it normally is used for either continued states or verbs that have some relevance to the next clause. See the section on helping verbs (page ) for more on this. Finally, 있다 means "there is," which implies continuity; if "there is" a chair in the classroom, the chair has usually been there for some time, whether that time is long or short. That's how 있다 is used in 고 있다 and 아/어 있다: it means that the action or state of ~ exists for some time.
HOW IT'S CONJUGATED
고 있다 is simply added to the end of an action verb. This verb should specifically describe an action that is happening at the time rather than a state that is continued: e. g., 먹다 meokda, 운동하다 undonghada, 피아노를 치다 pianoreul chida, 놀다 nolda. 아/어/여 있다 is added to the end of an action verb, but an action verb describing a state that continues uninterrupted for some time: examples are 서다 seoda (to stand), 앉다 anda (to sit), 넣다 neota (to put), or 눕다 nupda (to lie down). 아 있다 follows verbs with 오 or 아 as their last vowel and 어 있다 follows verbs with 어, 우, 으 or 이 as their last vowel. You can conjugate 있다 in either the past or future.
| | Past | Present | Future
---|---|---|---|---
Verbs describing an action and ending in a vowel | 하다 | 하고 있었다 hago isseotda | 하고 있다 hago itda | 하고 있을 것이다 hago isseul geosida
Verbs describing an action and ending in a consonant | 먹다 | 먹고 있었다 meokgo isseotda | 먹고 있다 meokgo itda | 먹고 있을 것이다 meokgo isseul geosida
Verbs describing a state and with their last vowel as 오 or 아 | 앉다 | 앉아 있었다 anja isseotda | 앉아 있다 anja itda | 앉아 있을 것이다 anja isseul geosida
Verbs describing a state and with their last vowel as 으, 이, 어 or 우 | 넣다 | 넣어 있었다 neoeo isseotda | 넣어 있다 neoeo itda | 넣어 있을 것이다 neoeo isseul geosida
TAKE NOTE
The present progressive is less commonly used in Korean than it is in English. Where we would say "He's swimming," Koreans are more likely to say "그는 수영해요 Geuneun suyeonghaeyo" rather than "그는 수영하고 있어요 Geuneun suyeonghago isseoyo," even though both are correct.
Remember that if the action still isn't finished at the time, you should use 고 있다 whereas if there is no ongoing action but rather a continued state, go with 어/아 있다.
EXAMPLE SENTENCES
아이들이 서 있어요.
Aideuri seo isseoyo.
The children are standing.
아이들이 일어서고 있어요.
Aideuri ireoseogo isseoyo.
The children are (in the middle of) standing (up).
가영이 몸이 아파서 지금 누워 있어요.
Kayoungi momi apaseo jigeum nuwo isseoyo.
Kayoung's in pain, so she's lying down right now.
책상에서 꽃이 놓여 있었어요.
Chaeksange kkochi noyeo isseosseoyo.
The flower was put on the desk.
민수가 책을 읽고 있어요.
Minsuga chaegeul ilgo isseoyo.
Minsu is reading a book.
그 아이가 놀고 있어요.
Geu aiga nolgo isseoyo.
The child is playing.
어머니가 미트 로프를 만들어 두었어요.
Eomeoniga miteu ropeureul mandeureo dueosseoyo.
My mother made (and saved) a meatloaf.
어머니가 미트 로프를 만들어 놓았어요.
Eomeoniga miteu ropeureul mandeureo noasseoyo.
My mother made (and put aside) a meatloaf.
There are some things you just don't want to get rid of. Here's an expression you can use to talk about that meatloaf that's been in your fridge forever.
HOW IT'S FORMED
아/어/여 is a verb ending which usually implies that the result of ~ carries over somehow to the next word. 두다 is a verb which literally means to "put," "keep," "leave behind," "set down," various other similar connotations. 놓다 is a similar verb which means "to put."
책상에 꽃을 두었어요.
Chaeksange kkocheul dueosseoyo.
Someone left a flower on the desk.
책상에 꽃을 놓았어요.
Chaeksange kkocheul noasseoyo.
Someone put a flower on the desk.
HOW IT'S CONJUGATED
This expression doesn't go with adjectives or nouns; if you want to "stay beautiful" you would use an expression like 아름답게 지내다 Areumdapge jinaeda, and if you want to keep that aforementioned meatloaf, you need a verb to go along with it and stay with 두다 duda: 미트 로프를 만들어 두다 Miteu ropeureul mandeureo duda would work.
| | Past | Present | Future
---|---|---|---|---
Action verbs with 아 or 오 | 잡다 | 잡아 두었다 jaba dueotda
잡아 놓았다 jaba noatda
| 잡아 두다 jaba duda
잡아 놓다 jaba nota
| 잡아 둘 것이다 jaba dul geosida
잡아 놓을 것이다 jaba noeul geosida
Action verbs with 어, 우, 으 or 이 | 먹다 | 먹어 두었다 meogeo dueotda
먹어 놓았다 meogeo noatda
| 먹어 두다 meogeo duda
먹어 놓다 meogeo nota
| 먹어 둘 것이다 meogeo dul geosida
먹어 놓을 것이다 meogeo noeul geosida
하다 | 하다 | 해 두었다 hae dueotda
해 놓았다 hae noatda
| 해 두다 hae duda
해 놓다 hae nota
| 해 둘 것이다 hae dul geosida
해 놓을 것이다 hae noeul geosida
TAKE NOTE
While 아/어 있다 and 고 있다 have some similarities to 아/어/여 두다/놓다, 두다 specifically implies that you are saving or keeping something, and 놓다 means you are putting it somewhere. 아/어 있다 and 고 있다 by themselves don't imply anything about what's going to happen after the sentence is finished.
아/어/여 놓다 and 아/어/여 두다 are normally interchangeable: use the one that better fits whatever you mean to say. 두다 is refers to "keeping" and 놓다 refers to putting somewhere.
두다 duda has many meanings beyond this expression; it's beyond the scope of this book to go into that in detail, but they're worth learning. For example, 두다 is the verb used to play board games like chess or 바둑 baduk: you can say 바둑을 두다 badugeul duda (to play 바둑) or 바둑 한 판 둘래요? baduk han pan dullaeyo (Would you like to play a game of 바둑?)
EXAMPLE SENTENCES
창문을 열어 두세요.
Changmuneul yeoreo duseyo.
Please open the window (and keep it open).
창문을 열어 놓으세요.
Changmuneul yeoreo noeuseyo.
Please open the window).
새로운 단어를 공책에 써 두었어요.
Saeroun daneoreul gongchaege sseo du-eosseoyo.
I wrote the new vocabulary words down in my notebook (and kept it).
새로운 단어를 공책에 써 놓았어요.
Saeroun daneoreul gongchaege sseo noasseoyo.
I wrote the new vocabulary words down in my notebook.
늦을까 봐 양복을 입은 채 민수의 생일 파티에 갔어요.
Neujeulkka bwa yangbogeul ibeun chae Minsuui saengil patie gasseoyo.
I was worried I'd be late, so I went to Minsu's party while still wearing my suit.
채(로) is a good expression if you ever need to talk about people who are drunk, lazy, or both. It means to do one thing while in the state of A such as falling asleep with your clothes on, or the door unlocked, or the TV on.
HOW IT'S FORMED
The (으)ㄴ in this expression is the past tense marker; whatever A was happens before B, so the past tense marker must be used regardless of which tense your sentence is actually in. 채 by itself doesn't mean anything other than to remain in a certain state. 로 (through) is optional and adds a little emphasis to the fact that the situation was maintained.
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Action verbs ending in a vowel | 하다 | 한 채(로) han chae(ro)
Action verbs ending in a consonant | 먹다 | 먹은 채(로) meogeun chae(ro)
TAKE NOTE
A and B should always have the same subject.
가다 and 오다 are never used before (으)ㄴ 채로. You can use 아서 (see page ) instead.
A고서는 B (see page ) is similar to (으)ㄴ 채로, but 고서는 is used only in cases where being in the state of A is a necessary condition for doing B. 채로 has no such implications. So 고서는 could be replaced by 채로, but not always vice versa. And while the sentences below are interchangeable:
신발을 신고서는 방을 들어올 수 없어요.
Sinbareul singoseoneun bangeul deureool su eopsseoyo.
신발을 신은 채 방을 들어올 수 없어요.
Sinbareul sineun chae bangeul deureool su eopsseoyo.
A sentence like 창문을 연 채 피아노를 쳤어요 Changmuneul yeon chae pianoreul chyeosseoyo ("I played piano with the window open.") could not be made with 고서는 since your playing of the piano is possible whether or not the window is open.
아/어/여 놓다/두다 (covered on page 257) is often used before (으)ㄴ 채(로). You can think of 채(로) as an extension of these two expressions: 아/어/여 놓다/두다 means that something was done and the result was kept or put aside, while 채(로) keeps going with that thought and tells you what was done with that result.
창문을 열어 두었어요.
Changmuneul yeoreo dueosseoyo.
창문을 열어 놓았어요.
Changmuneul yeoreo noasseoyo.
Both mean "I left the window open."
창문을 연 채 피아노를 쳤어요.
Changmuneul yeon chae pianoreul chyeosseoyo.
창문을 열어 둔 채 피아노를 쳤어요.
Changmuneul yeoreo dun chae pianoreul chyeosseoyo.
창문을 열어 둔 채로 피아노를 쳤어요.
Changmuneul yeoreo dun chaero pianoreul chyeosseoyo.
창문을 열어 놓은 채 피아노를 쳤어요.
Changmuneul yeoreo noeun chae pianoreul chyeosseoyo.
창문을 열어 놓은 채로 피아노를 쳤어요.
Changmuneul yeoreo noeun chaero pianoreul chyeosseoyo.
All mean "I played piano with the window open": taking the open window that the first two sentences told you about and continuing on to tell you what was done while that window was open.
EXAMPLE SENTENCES
늦을까 봐 양복을 입은 채 민수의 생일 파티에 갔어요.
Neujeulkka bwa yangbogeul ibeun chae minsuui saengil patie gasseoyo.
늦을까 봐 양복을 입은 채로 민수의 생일 파티에 갔어요.
Neujeulkka bwa yangbogeul ibeun chaero minsuui saengil patie gasseoyo.
I was worried I'd be late, so I hurried to Minsu's party while still wearing my suit.
아침을 먹지 않은 채 병원에 와 주세요.
Achimeul meokji aneun chae byeongwone wa juseyo.
아침을 먹지 않은 채로 병원에 와 주세요.
Achimeul meokji aneun chaero byeongwone wa juseyo.
Please come to the hospital without eating breakfast.
##
Progression: Almost, But Not Quite...
회의가 너무 오래 걸려서 파티에 늦을 뻔했어요.
Hoeuiga neomu orae geollyeoseo patie neujeul ppeonhaesseoyo.
The meeting took a long time, so I was almost late for the party.
This expression is perfect for people who like to exaggerate. It's frequently used in sentences such as "I almost died!" Of course, you can also use it for more mundane things like almost being late, almost missing your train, and so on.
HOW IT'S CONJUGATED
| | Past
---|---|---
Action verbs ending in a vowel | 하다 | 할 뻔했다 hal ppeonhaetda
Action verbs ending in a consonant | 먹다 | 먹을 뻔했다 meogeul ppeonhaetda
TAKE NOTE
There is another 뻔하다 which means "to be clear." The one in this expression will always follow an action verb and (으)ㄹ; the other one will not.
(으)려던 참이다 (eu)ryeodeon chamida (see the next section) is also used to talk about things almost happening, but it's used only with things that you intended to do. (으)ㄹ 뻔하다 is normally used when talking about things that were accidental and can be used to talk about yourself or anyone else. It's normally used for bad things, but doesn't have to be; if you want to say 복권에 당첨될 뻔했어요 Bokgwone dangcheomdoel ppeonhaesseoyo (I almost won the lottery) or talk about similar blessings that you just barely missed out on, that's fine.
The difference between 뻔하다 and (으)ㄹ락 말락 하다 (eu)rrak mallak hada (see page 265) is that 뻔하다 means that something could have happened, but didn't. You could have died in that car accident, but were spared. (으)ㄹ락 말락 하다 is more like wavering between doing A and not doing A, but ultimately not doing it. Also, (으)ㄹ락 말락 is normally used only in sentences where what almost happened is a result of the subject's will.
그 사고에서 죽을 뻔했어요.
Geu sagoeseo jugeul ppeonhaesseoyo.
I could have died in that accident. (Maybe nothing happened to you at all in the end, or maybe you were seriously hurt but didn't die.)
그 사고에서 죽을락 말락 했어요.
Geu sagoeseo jugeullak mallak haesseoyo.
I was hovering on the verge of death after that accident. (You were seriously injured and close to death, but ultimately survived.)
(으)ㄹ 뻔하다 is quite often used with words like 거의 geoui (almost) or 아마 ama (maybe). It also likes to hang out with hypothetical sentences such as "If I had..." Some common examples are 았/었/였다면 at/eot/yeotdamyeon (see page 289) and 았/었/였더라면 at/eot/yeotdeoramyeon (see page 286).
EXAMPLE SENTENCES
눈 때문에 길이 참 미끄러워서 넘어질 뻔했어요.
Nun ttaemune giri cham mikkeureowoseo neomeojil ppeonhaesseoyo.
Because of the snow, the road is very slippery, so I almost fell.
오늘 아침에 지하철에서 사고가 났어요. 오늘 지하철을 탔으면 죽을 뻔했어요.
Oneul achime jihacheoreseo sagoga nasseoyo. Oneul jihacheoreul tasseumyeon jugeul ppeonhaesseoyo.
This morning, there was an accident on the subway. If I'd taken the subway today I could have died.
현준: 엄마, 학교에 가려던 참이었어요.
Hyeonjun: Eomma, hakgyoe garyeodeon chamieosseoyo.
Hyeonjun: Mom, I was just about to go to school.
Do you have a worried mother, an overbearing boss, or a nagging significant other? If so, you should learn this expression. It means you were just about to do something. Quite often it's said with a bit of an injured tone because you were about to go do the very thing someone just accused you of not doing.
HOW IT'S FORMED
(으)려 comes from (으)려고 하다 (eu)ryeogo hada, which is on page and is used to talk about a plan. 던 is a past-tense marker, which in this case indicates that you had already made that plan. 참 has a few meanings; in this expression it means "on the point of" (I don't like this) and is followed by 이다, which means "you were."
HOW IT'S CONJUGATED
| | Past | Present
---|---|---|---
Action verbs ending in a vowel | 하다 | 하려던 참이었다 haryeodeon chamieotda | 하려던 참이었다 하려던 참이다 haryeodeon chamida
Action verbs ending in a consonant | 먹다 | 먹으려던 참이었다 meogeuryeodeon chamieotda | 먹으려던 참이었다 먹으려던 참이다 meogeuryeodeon chamida
TAKE NOTE
This expression is used only in the first person and can't be used on its own to suggest or to command someone to do something. You can, however, say that you were about to do something and then use that as the basis for suggesting something to another person, perhaps by ending your A clause with ㄴ데 (A려던 참인데 ryeodeon chaminde B). This is often done when, for example, someone drops by your house just as you were about to eat and you invite them to join you.
If you use 참이었다 you are implying you had been planning the action for some time, while 참이다 means you were about to do it right that very second.
Sentences with (으)려던 참이다 very often use 안 그래도 an geuraedo or 그렇지 않아도 geureochi anado (even if you hadn't said that) or 마침 machim (as in "I was just about to").
Koreans use 참이다 and 참이었다 interchangeably, so just pick whichever one you like better for your sentence.
EXAMPLE SENTENCES
엄마: 현준아, 늦을 거야! 왜 학교에 아직 안 갔어?
Eomma: Hyeonjuna, neujeul geoya! Wae hakgyoe ajik an gasseo?
현준: 엄마, 학교에 가려던 참이었어요.
Hyeonjun: Eomma, hakgyoe garyeodeon chamieosseoyo.
엄마: 현준아, 늦을 거야! 왜 학교에 아직 안 갔어?
Eomma: hyeonjuna, neujeul geoya! Wae hakgyoe ajik an gasseo?
현준: 엄마, 학교에 가려던 참이에요.
Hyeonjun: Eomma, hakgyoe garyeodeon chamieyo.
Mom: Hyeonjun, you're going to be late! Why haven't you gone to school yet? Hyeonjun: Mom, I was just about to go to school.
(Children in Korea often, though not by any means always, use 존댓말 when talking to their parents.)
A: 식사를 준비 하겠어요?
Ga: Siksareul junbi hagesseoyo?
Are you going to get dinner ready?
B: 안 그래도 준비하려던 참이에요.
Na: An geuraedo junbiharyeodeon chamieyo.
안 그래도 준비하려던 참이었어요.
Na: An geuraedo junbiharyeodeon chamieosseoyo
Even if you hadn't said that, I was just about to do it.
식사를 먹으려던 참인데 같이 먹을래요?
Siksareul meogeuryeodeon chaminde gachi meogeullaeyo?
식사를 먹으려던 참이었는데 같이 먹을래요?
Siksareul meogeuryeodeon chamieonneunde gachi meogeullaeyo?
I was just about to eat; would you like to join me?
짜증이 나서 죽을 지경이에요.
Jjajeungi naseo jugeul jigyeongieyo.
I'm so annoyed! (so annoyed I could die)
Here's another expression that you can use to exaggerate. This one means "to be on the verge of ~."
HOW IT'S FORMED
(으)ㄹ is a future tense marker and 지경 is a noun which literally means "boundary."
HOW IT'S CONJUGATED
| | Present
---|---|---
Action verbs ending in a vowel | 하다 | 할 지경이다 hal jigyeongida
Action verbs ending in a consonant | 먹다 | 먹을 지경이다 meogeul jigyeongida
TAKE NOTE
It's also possible to form a sentence using one of the other tense markers ([으]ㄴ/는) before 지경. In this case you are talking about a situation that actually happened or is happening, whereas (으)ㄹ 지경이다 is hypothetical.
그는 배고파서 죽은 지경이에요.
Geuneun baegopaseo jugeun jigyeongieyo.
He's starving to death. (He's so hungry he's dying.)
그는 배고파서 죽을 지경이에요.
Geuneun baegopaseo jugeul jigyeongieyo.
He's so hungry he's on the verge of dying.
(으)ㄹ 지경이다 and 아/어/여서 죽겠다 (see page ) are interchangeable, but the latter is used quite often while the former is used only rarely.
지경 is used in many idiomatic expressions, of which the most common is 죽을 지경이다 (to be on the verge of dying). This can be used whether you actually are on the verge of dying or whether you're just milking the situation for sympathy.
지경이다 isn't used in the past tense.
EXAMPLE SENTENCES
오늘 너무 더워서 숨이 막힐 지경이에요.
Oneul neomu deowoseo sumi makil jigyeongieyo.
Today it's so hot I can't breathe. (It is so hot I'm on the verge of losing my breath – i.e., I almost cannot breathe).
그녀가 남자친구 때문에 울 지경이어서 갑자기 방에서 나갔어요.
Geunyeoga namjachingu ttaemune ul jigyeongieoseo gapjagi bangeseo nagasseoyo.
Because of her boyfriend, she suddenly ran out of the room on the verge of tears.
지금 비가 올락 말락 해요.
Jigeum biga ollak mallak haeyo.
Right now it's almost raining.
This is for people who are indecisive. It means that either something was wavering between A and B (A[으]ㄹ락 B[으]ㄹ락 하다) or was wavering between doing A or not doing A (A[으]ㄹ락 말락 하다). It's also good for those days when the weather just can't seem to make up its mind.
HOW IT'S FORMED
(으)ㄹ락202(으)ㄹ락 is the basis of the expression. It's like a grammatical seesaw, with the sentence teetering between the verb before the first (으)ㄹ락 and the verb before the second (으)ㄹ락. These verbs will be either opposites (A and B) or A and 말다 (which just means the opposite of whatever was before it). Then 하다 is added at the end to tie it all together.
HOW IT'S CONJUGATED
| | Past | Present
---|---|---|---
Action verbs ending in a vowel | 하다 | 할락 말락 했다 hallak mallak haetda | 할락 말락 하다 hallak mallak hada
Action verbs ending in a consonant | 먹다 | 먹을락 말락 했다 meogeullak mallak haetda | 먹을락 말락 하다 meogeullak mallak hada
Action verbs ending in ㄹ | 들다 | 들락 말락 했다 deullak mallak haetda | 들락 말락 하다 deullak mallak hada
TAKE NOTE
The difference between this expression and A둥 B둥 하다 A dung B dung hada is that the ㄹ락s mean that something almost happened, but didn't. The 둥s mean something almost didn't happen, but did.
비가 올락 말락 했어요.
Biga ollak mallak haesseoyo.
It almost rained, but didn't.
비가 오는 둥 마는 둥 했어요.
Biga oneun dung maneun dung haesseoyo.
It rained just a little.
If the situation that almost happened is accidental, it's far more common to say (으)ㄹ 뻔했다 (eu)l ppeonhaetda (see page 261). (으)ㄹ락 말락 is normally used in situations where what almost happened is a result of the subject's will. So the first example sentence is a "good" example of this grammar point, and the second, while technically correct, isn't quite natural.
EXAMPLE SENTENCES
갈락 말락 했지만 마지막 순간에 가영이 오지 않는다고 해서 저도 안 갔어요.
Gallak mallak haetjiman majimak sungane Kayoungi oji anneundago haeseo jeodo an gasseoyo.
I almost went, but at the last second Kayoung said she wasn't going, so I didn't go either.
그 사고에서 저는 죽을락 말락 했어요.
Geu sagoeseo jeoneun jugeullak mallak haesseoyo.
I almost died in that accident.
In the sentence above, the grammar is correct, but it's more natural to say 죽을 뻔했어요. Jugeul ppeon haesseoyo.
늦어서 식사를 먹는 둥 마는 둥 하고 나갔어요.
Neujeoseo siksareul meokneun dung maneun dung hago nagasseoyo.
I was late, so I barely ate and immediately left the house.
This can be used in two ways. One is when something almost, but didn't quite happen such as when you get to work and are hurried off to a meeting when you've only just turned on your computer, or when there are just a few flakes of snow coming down instead of steady snow. The other way is when someone is making poor excuses.
HOW IT'S FORMED
는 and (으)ㄹ are present and future tense markers. (으)ㄴ isn't used with this expression. 둥 doesn't really mean anything by itself.
HOW IT'S CONJUGATED
The first meaning (just barely A) works with action verbs. In this case A is what was just barely done and B is either 말다 or the opposite of A.
The second meaning (excuses) is used with descriptive or active verbs. Add (ㄴ/는)다는 둥 after each of A and B, which are two excuses that the person is making.
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 하는 둥 haneun dung (first meaning)
한다는 둥 handaneun dung (second meaning)
| 하는 둥 (first meaning)
한다는 둥 (second meaning)
| 할 둥 hal dung (first meaning)
한다는 둥 (second meaning)
Action verbs ending in a consonant | 먹다 | 먹는 둥 meokneun dung (first meaning)
먹는다는 둥 meokneundaneun dung (second meaning)
| 먹는 둥 (first meaning)
먹는다는 둥 (second meaning)
| 먹을 둥 meogeul dung (first meaning)
먹는다는 둥 (second meaning)
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁜 둥 yeppeun dung | 예쁜 둥 | 예쁜 둥
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작은 둥 jageun dung | 작은 둥 | 작은 둥
TAKE NOTE
The structure of this expression is rather similar to A(으)ㄹ락 B(으)ㄹ락 하다 (see page 265). The difference between the two is that the ㄹ락s mean something almost happened, but didn't. The 둥s mean something almost didn't happen, but did.
비가 올락 말락 했어요.
Biga ollak mallak haesseoyo.
It almost rained, but didn't.
비가 오는 둥 마는 둥 했어요.
Biga oneun dung maneun dung haesseoyo.
It rained just a little.
Also, 둥 is normally used in sentences with negative connotations.
Don't get 둥 confused with 등; 등 means "etc." and is covered on page .
And definitely don't get 둥 confused with 똥.
EXAMPLE SENTENCES
■ First meaning (just barely A):
아이들이 제 말을 듣는 둥 마는 둥 했어요.
Aideuri je mareul deunneun dung maneun dung haesseoyo.
The children barely listened to what I was saying.
내 개가 아파요. 지금 죽을 둥 말 둥 해요.
Nae gaega apayo. Jigeum jugeul dung mal dung haeyo.
My dog is sick. He is about to die.
■ Second meaning (making excuses):
그는 바쁘다는 둥 시간이 없다는 둥 핑계를 대면서 일을 거의 안 해요.
Geuneun bappeudaneun dung sigani eopdaneun dung pinggyereul daemyeonseo ireul geoui an haeyo.
He's always saying he's busy or doesn't have time, and so he hardly works.
##
Progression: Putting Things in Temporal Order
Focus: 다가 daga
다가 is the basis for many very commonly used expressions. They can be somewhat tricky to use and you can accidentally change the meaning of your sentences quite a bit with just a few minor mistakes, so you need to handle them with some care.
When you see 다가 in an expression, it essentially means "and then." All the expressions with 다가 have to do in some way with progression: A happens and then B happens. Of course, the nuances vary depending on which particular 다가 expression you use. 다가 all by itself means that A was interrupted by B; more ways of using it are shown in the chart.
Expression | Page | Meaning | Example
---|---|---|---
A다가 daga B | 269 | "A is interrupted by B" | 학교에 가다가 교통 사고가 나서 늦었어요. Hakgyoe gadaga gyotong sagoga naseo neujeosseoyo. As I was going to school, there was a traffic accident, so I was late.
A아/어/여다가 a/eo/yeodaga B | 270 | "A is done, and then B is done on the basis of A" | 늦을까 봐 선물을 빨리 찾아다가 파티에 갔어요. Neujeulkka bwa seonmureul ppalli chajadaga patie gasseoyo. I was afraid I'd be late, so I quickly bought a present and went to the party.
A았/었/였다가 at/eot/yeotdaga B | 271 | "A is finished and then B is done" | 회의에 도착했다가 늦어서 죄송하다고 했어요. Hoeuie dochakaetdaga neujeoseo joesonghadago haesseoyo. After I got to the meeting, I apologized for being late.
A려다가 ryeodaga B | | "I planned to A, but then B happened" | 등산을 하려다가 비가 와서 집에 있었어요. Deungsaneul haryeodaga biga waseo jibe isseosseoyo. I planned to go hiking, but then it rained, so I stayed home.
A(으)ㄴ/는 데다(가) (eu)n/neun deda(ga) B | | "B is added to A" | 민수가 늦은 데다가 음식도 가지고 오지 않았어요. Minsuga neujeun dedaga eumsikdo gajigo oji anasseoyo. Minsu was late. He also didn't bring any food.
A에다(가) eda(ga) B | | "B is added to A" | 열에다가 스물을 더하면 서른이에요. Yeoredaga seumureul deohamyeon seoreunieyo. Ten plus twenty is thirty.
A다가는 daganeun B | | "Unfortunately, if you do A, B might happen" | 이렇게 천천히 준비하다가는 늦을 거예요. Ireoke cheoncheonhi junbihadaganeun neujeul geoyeyo. If you take your time getting ready, you'll be late.
A다가도 dagado B | 279 | "A is (suddenly/dramatically) interrupted by B" | 그녀는는 급하게 서두르다가도 항상 정신이 산만해져서 늦어요. Geunyeoneun geupage seodureudagado hangsang jeongsini sanmanhaejyeoseo neujeoyo. Even though she hurries, she always gets distracted and ends up being late.
A다(가) 보니(까) da(ga) boni (kka) B | | "After doing A for a long time, I noticed B" | 친구 하고 길게 이야기를 하다가 보니까 수업에 늦게 됐어요. Chingu hago gilge iyagireul hadaga bonikka sueobe neutge dwaesseoyo. I was talking with my friend for a long time and suddenly noticed I was late for class.
A다(가) 보면 da(ga) bomyeon B | | "If you keep on doing A, B will happen" | 계속 이야기 하다 보면 늦을 거예요. Gyesok iyagi hada bomyeon neujeul geoyeyo. If you keep talking, you'll be late.
~다가 말다가 하다 daga maldaga hada | | "Doing ~ on and off " | 그렇게 준비를 하다가 말다가 하면 늦을 거예요. Geureoke junbireul hadaga maldaga hamyeon neujeul geoyeyo. If you keep on getting ready and then stopping like that, you'll be late.
학교에 가다가 교통 사고가 나서 늦었어요.
Hakgyoe gadaga gyotong sagoga naseo neujeosseoyo.
As I was going to school, there was a traffic accident, so I was late.
This is the expression you use when you were in the middle of something which was subsequently interrupted by something else. It's a good expression for those with children or younger siblings.
HOW IT'S CONJUGATED
Be careful not to put the verb in clause A into the past tense (for example, 했다가 haetdaga) because that changes the meaning (see page 271).
| | All tenses
---|---|---
Action verbs ending in a vowel | 하다 | 하다가 hadaga
Action verbs ending in a consonant | 먹다 | 먹다가 meokdaga
TAKE NOTE
Since this expression is used in talking about actions, it is normally used only with action verbs. The subjects of clause A and clause B should be the same.
EXAMPLE SENTENCES
공부하다가 갑자기 잤어요.
Gongbuhadaga gapjagi jasseoyo.
While studying, I suddenly fell asleep.
지하철을 타다가 거지에게 돈을 주었어요.
Jihacheoreul tadaga geojiege doneul jueosseoyo.
While taking the subway, I gave money to a beggar.
늦을까 봐 선물을 빨리 찾아다가 파티에 갔어요.
Neujeulkka bwa seonmureul ppalli chajadaga patie gasseoyo.
I was afraid I'd be late, so I quickly bought a present and went to the party.
Yeah, I had a hard time expressing this succinctly. Basically, in contrast to plain old 다가 which implies that A was interrupted, 아/어/여다가 means that A was done and the result was carried over and used in B. Think of making a snack and bringing it to a party. That's the situation where you would need 아/어/여다가.
HOW IT'S FORMED
Our old friends 아/어/여 are back to play with our new friend 다가. You may have noticed that when a verb takes 아/어/여 along with it, it usually means that verb is there in the sentence to stay: it will stick around and affect clause B. In other words, it means that whatever happened in clause A has a lasting effect. Think of all the helping verbs on page or 아/어 있다. The 가 in 다가 is optional and is often omitted in longer 다가-type expressions: use it depending on which you think sounds better.
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Action verbs with 아/오 | 찾다 | 찾아다가 chajadaga
Action verbs with 어/우/으/이 | 먹다 | 먹다가 meokdaga
하다 | 하다 | 하다가 hadaga
TAKE NOTE
Since this expression is used in talking about actions, it is normally used only with action verbs. The subjects of clause A and clause B should be the same.
EXAMPLE SENTENCES
저 개를 잡아다가 보신탕을 만드세요.
Jeo gaereul jabadaga bosintangeul mandeuseyo.
Please catch the dog, and make dog meat stew out of it.
돈을 가져와다가 김밥을 샀어요.
Doneul gajyeowadaga kimbapeul sasseoyo.
I brought some money and bought kimbap.
김밥을 사다가 먹었어요.
Kimbapeul sadaga meogeosseoyo.
I bought kimbap and ate it.
(In this case the expression with 아다가 looks the same as the expression with just 다가, so check the context carefully to determine which one is being used: 다가 by itself wouldn't make sense in this expression unless you started eating in the middle of buying kimbap, leaving your purchase unfinished).
회의에 도착했다가 늦어서 죄송하다고 했어요.
Hoeuie dochakaetdaga neujeoseo joesonghadago haesseoyo.
After I got to the meeting, I apologized for being late.
In order to use this expression, A must be completely finished. With 다가 by itself, A is interrupted, but 았/었/였이다가 is used only when two different actions follow each other without overlapping. If that's not entirely clear, a few examples (see below) should help.
HOW IT'S FORMED
The past tense marker 았/었/였 tells you A is finished and then 다가 comes along to arrange everything neatly and tells you when B, the next thing, happened. The 가 part of 다가 is optional.
HOW IT'S CONJUGATED
Because this expression is used only when A is completely finished, it is always in the past tense; if you're trying to write 하다가 or 해다가, you're using a different grammatical point (see pages 269 and 270).
| | All tenses
---|---|---
Action verbs with 아/오 | 찾다 | 찾았다가 chajatdaga
Action verbs with 어/우/으/이 | 먹다 | 먹었다가 meogeotdaga
Action verbs ending in a vowel | 하다 | 했다가 haetdaga
TAKE NOTE
Since this expression is used in talking about actions, it is normally used only with action verbs. The subjects of clause A and clause B should be the same. Although A is in the past tense form, you can still use the expression when talking about the present or future: just make sure your B verb uses the appropriate tense. A will have been finished by the time B happens, by definition.
EXAMPLE SENTENCES
Here I've compared the same sentences written with 다가 and with 았/었/였이다가 so that you can see the difference in meaning.
파티에 가다가 친구에게 전화했어요.
Patie gadaga chinguege jeonhwahaesseoyo.
While going to the party, I (stopped and) called my friend.
파티에 갔다가 친구에게 전화했어요.
Patie gatdaga chinguege jeonhwahaesseoyo.
After I went to the party, I called my friend.
TV를 보다가 잤어요.
Tibireul bodaga jasseoyo.
While I was watching TV, I fell asleep.
TV를 봤다가 어머니한테 혼났어요.
Tibireul bwatdaga eomeonihante honnasseoyo.
I watched TV and then my mother scolded me.
저녁을 먹고 나서 담배를 피웠어요.
Jeonyeogeul meokgo naseo dambaereul piwosseoyo.
I ate dinner and then had a cigarette.
This basically just means "and then." It's a subset of 고서, but you can use 고 나서 only when the action in the first sentence is completely finished. All these sentences could also be made with 고서.
HOW IT'S FORMED
고 plus 나다 (to happen) plus 아/어/여서 (which means "then" here).
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Action verbs ending in a vowel | 하다 | 하고 나서 hago naseo
Action verbs ending in a consonant | 먹다 | 먹고 나서 meokgo naseo
TAKE NOTE
고 나서 can be used only with action verbs and only when the subjects of both clause A and clause B are the same.
EXAMPLE SENTENCES
영화를 보고 나서 집에 갔어요.
Yeonghwareul bogo naseo jibe gasseoyo.
I watched a movie and then went home.
공부를 하고 나서 잤어요.
Gongbureul hago naseo jasseoyo.
I studied and then went to sleep.
늦을까 봐 옷을 갈아입자마자 친구 집에 갔어요.
Neujeulkka bwa oseul garaipjamaja chingu jibe gasseoyo.
늦을까 봐 옷을 갈아입는 대로 친구 집에 갔어요.
Neujeulkka bwa oseul garaimneun daero chingu jibe gasseoyo.
I was afraid I'd be late, so I changed my clothes and then immediately went to my friend's house.
옷을 입자 전화가 왔어요.
Oseul ipja jeonhwaga wasseoyo.
As soon as I put on my clothes, the phone rang.
When life is happening quickly, these four expressions are here to help. They all mean that B happened right after A. However, each one has its own peculiarities. This section will compare the four expressions above and explain when you should or shouldn't use each one.
HOW IT'S FORMED
기 turns verbs into nouns; 무섭게 museopge is the well-known 무섭다 museopda (to be scary) but rather is unique to this expression. You can add the subject marker 가 after the verb with 기.
There's nothing special to note here about 자마자, 는 대로, or 자.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 하자마자
하자
하기(가) 무섭게
| 하자마자 hajamaja
하는 대로 haneun daero
하자 haja
하기(가) 무섭게 hagi(ga) museopge
| 하자마자
하는 대로
하기(가) 무섭게
Action verbs ending in a consonant | 먹다 | 먹자마자
먹자
먹기(가) 무섭게
| 먹자마자 meokjamaja
먹는 대로 meokneun daero
먹자 meokja
먹기(가) 무섭게 meokgi(ga) museopge
| 먹자마자
먹는 대로
먹기(가) 무섭게
TAKE NOTE
자마자 can't be used when A is negative (a sentence with 안, 못, 않, etc.).
A는 대로 B, A기 무섭게 B, A자마자 B and A자 B can all be used interchangeably in some cases, but not in others. Let's compare.
는 대로 can normally be used interchangeably with 자마자 unless the sentence is in the past tense. 는 대로 can't be used in the past tense. 는 대로 can be used interchangeably with 자 in the present tense only. 는 대로 can't be used in the past tense and 자 can't be used in the future tense, so the two can only cross paths in the present. 는 대로 can always be changed to 기 무섭게.
기 무섭게 is completely interchangeable with all the other expressions here, but is far less common; in fact, I'm not sure I've ever heard or seen it used in the real Korean world. Nonetheless, if you want to impress people, are cramming for the TOPIK test, or are a completionist, it can be good to know.
자 by itself can be used interchangeably with 자마자 in statements, but not questions, commands, or imperatives. It also can't be used in the future tense. It can imply more time between A and B than 자마자 does.
There are several other ways of saying one thing happened after another: 고서 (see page 280), 고 나서 (see page 272), and some variations of 다가 (see page 268) – but none of them necessarily mean "immediately after." If it's important that your readers/listeners understand that one thing happened immediately after the other, you should use 자마자, 자, 는 대로, or 기 무섭게.
There is another expression, (으)ㄴ/는 대로, which looks exactly the same as the 는 대로 here, but is actually a variation of 는 대로 (see page ). There's no really obvious way to tell them apart, so you'll mainly have to rely on context – the other one means "according to," so it will mainly appear in sentences such as "선생님 말하는 대로 Seonsaengnim malhaneun daero" ("According to what the teacher said"). It also sometimes takes (으)ㄴ instead of 는 or directly follows nouns such as 마음 maeum.
There are two more 자 expressions. One is the 반말 propositive 하자! haja ("Let's do it!") while the other, 자고 하다 jago hada, is the indirect speech propositive. The first will always appear at the end of a sentence while the 자 we're discussing here never will. The second will normally be followed by something else; 자고 jago, 자면 jamyeon, or 자는데 janeunde are examples. The 자 in this section, however, always likes to be the last syllable of whatever it's attached to, so it will never be followed by 고, 면, or anything else except the next clause in the sentence.
EXAMPLE SENTENCES
늦을까 봐 옷을 갈아입자마자 친구 집에 갔어요.
Neujeulkka bwa oseul garaipjamaja chingu jibe gasseoyo.
I was afraid I'd be late, so I changed my clothes and then immediately went to my friend's house.
(This sentence can also be written with 기 무섭게, but not 는 대로.)
늦을까 봐 옷을 갈아입기 무섭게 친구 집에 갔어요.
Neujeulkka bwa oseul garaipgi museopge chingu jibe gasseoyo.
늦을까 봐 옷을 갈아입자 친구 집에 갔어요.
Neujeulkka bwa oseul garaipja chingu jibe gasseoyo.
__________
집에 돌아오자마자 저에게 전화해 주세요.
Jibe doraojamaja jeoege jeonhwahae juseyo.
Please call me as soon as you get home.
(This sentence can be written with 는 대로 and 기 무섭게, but not 자.)
집에 돌아오는 대로 저에게 전화해 주세요.
Jibe doraoneun daero jeoege jeonhwahae juseyo.
집에 돌아오기 무섭게 저에게 전화해 주세요.
Jibe doraogi museopge jeoege jeonhwahae juseyo.
__________
저녁을 먹자마자 갈 거예요.
Jeonyeogeul meokjamaja gal geoyeyo.
I'll come as soon as I eat dinner.
(This sentence can be written with 기 무섭게 and 는 대로, but not 자.)
저녁을 먹는 대로 갈 거예요. Jeonyeogeul meokneun daero gal geoyeyo.
저녁을 먹기 무섭게 갈 거예요. Jeonyeogeul meokgi museopge gal geoyeyo.
__________
매일 일을 끝내는 대로 저녁을 먹어요.
Maeil ireul kkeunnaeneun daero jeonyeogeul meogeoyo.
I eat dinner as soon as I finish work every day.
(This sentence works with 자, 자마자 and 기 무섭게)
매일 일을 끝내자 저녁을 먹어요.
Maeil ireul kkeunnaeja jeonyeogeul meogeoyo.
매일 일을 끝내자마자 저녁을 먹어요.
Maeil ireul kkeunnaejamaja jeonyeogeul meogeoyo.
매일 일을 끝내기 무섭게 저녁을 먹어요.
Maeil ireul kkeunnaegi museopge jeonyeogeul meogeoyo.
집에서 출발한 지 2 시간 만에 친구 집에 도착했어요.
Jibeseo chulbalhan ji dusigan mane chingu jibe dochakaesseoyo.
Two hours after I left my house, I arrived at my friend's house.
한국에 온 지 3년이 지났어요.
Hanguge on ji samnyeoni jinasseoyo.
It's been three years (three years have passed) since I came to Korea.
(으)ㄴ 지 forms the basis for two more expressions which we'll cover here: A(으)ㄴ 지 B이/가 넘다/되다/지나다, and A(으)ㄴ 지 B 만에 C. They're both ways to talk about how much time has passed since A.
HOW IT'S FORMED
(으)ㄴ 지 in this expression means "since."
There are many 만s in Korean, and the one in A(으)ㄴ 지 B 만에 is the one that means "full": in this case, "fully." So "fully B (a period of time) since A, C happened..."
넘다 means "to go over," 되다 means "to become," and 지나다 means "to pass"; in this case they're all interchangeable and mean that B (again, a period of time) has passed since A.
HOW IT'S CONJUGATED
In both expressions, B is a period of time, like 5년 onyeon, 두 달 du dal, 일주일 iljuil. It can also be a non-specific expression of time: 얼마 eolma or 오래 orae, for example. When making a sentence with 만에, you can leave out A if you don't need it, so the expression then becomes B만에 C B mane C: "After B, C happened."
C is an action that took place B (some period of time) since A.
| | All tenses
---|---|---
Action verbs ending in a vowel | 하다 | 한 지 han ji
Action verbs ending in a consonant | 먹다 | 먹은 지 meogeun ji
TAKE NOTE
Watch out for all the other grammar points that use 지! Koreans seem to be quite fond of that syllable. Most of the other expressions with 지 use it before a negative: 지 못하다 ji motada, 지 않다 ji anta, and so on. There is also (으)ㄴ 지 알다/모르다 (eu)n ji alda/moreuda (see page ) in which you will ever see only 지 followed by some conjugation of 알다 or 모르다. If you see 지 and then an expression of time, it's the (으)ㄴ 지 that means "since."
만 is another syllable that you'll see all over the place in different contexts. The 만 in this expression is always followed by 에. Any 만 that's not followed by 에 is another 만.
You can use (으)ㄴ 지 without 만에 or 넘다/되다/지나다; these are just two easy ways to talk about how much time has passed.
EXAMPLE SENTENCES
A: 집에서 출발한 지 얼마나 됐어요?
Ga: Jibeseo chulbalhan ji eolmana dwaesseoyo?
How long has it been since you left your house?
B: 집에서 출발한 지 벌써 2 시간이 넘었어요. 늦을 것 같아요.
Na: Jibeseo chulbalhan ji beolsseo dusigani neomeosseoyo. Neujeul geot gatayo.
I left my house two hours ago (two hours have passed since I left my house). It seems like I'll be late.
__________
한국에 온 지 3년이 됐어요.
Hanguge on ji samnyeoni dwaesseoyo
한국에 온 지 3년이 넘었어요.
Hanguge on ji samnyeoni neomeosseoyo.
한국에 온 지 3년이 지났어요.
Hanguge on ji samnyeoni jinasseoyo.
It's been three years (three years have passed) since I came to Korea.
한국에 온 지 3 년 만에 다시 고향에 돌아갔어요.
Hanguge on ji samnyeon mane dasi gohyange doragasseoyo.
Three years after I came to Korea, I went back to my home.
3년 만에 한국에 왔어요.
Samnyeon mane hanguge wasseoyo.
I came to Korea three years ago.
한국에 온 지 1주일 만에 한글을 배웠어요.
Hanguge on ji iljuil mane hangeureul baewosseoyo.
I learned the Korean alphabet within a week of coming to Korea.
약을 먹은 지 세 시간이 되었는데 아직 머리가 아파요.
Yageul meogeun ji se sigani doeeonneunde ajik meoriga apayo.
It has been three hours since I took the medicine, and I still have a headache.
수업 전에 저녁을 먹었어요.
Sueop jeone jeonyeogeul meogeosseoyo.
I ate dinner before class.
수업 후에 저녁을 먹었어요.
Sueop hue jeonyeogeul meogeosseoyo.
I ate dinner after class.
수업 뒤에 저녁을 먹었어요.
sueop dwie jeonyeogeul meogeosseoyo.
I ate dinner after class.
The expressions above have to do with time and putting B and A in their temporal places. Let's look at each one:
A 전에 A jeone: before A
A 후에 A hue: after A
A 뒤에 A dwie: after A
HOW THEY'RE MADE
전, 뒤 and 후 are all changed into prepositions by the addition of 에.
HOW IT'S CONJUGATED
With all these expressions, A must include a measure of time—two years, five months, 얼마, 오래, or something that takes time or occurs at a point in time to which you are referring: 수업 Sueop, 결혼식 gyeolhonsik, 입학 ipak, etc. You can use it after an action verb if you conjugate the verb using 기 (for 전에) or (으)ㄴ (for 후에 and 뒤에).
| | All tenses
---|---|---
Action verbs ending in a vowel | 하다 | 하기 전에 hagi jeone
한 후에 han hue
한 뒤에 han dwie
Action verbs ending in a consonant | 먹다 | 먹기 전에 meokgi jeone
먹은 후에 meogeun hue
먹은 뒤에 meogeun dwie
Nouns ending in a vowel | 사고 | 사고 전에 sago jeone
사고 후에 sago hue
사고 뒤에 sago dwie
Nouns ending in a consonant | 수업 | 수업 전에 Sueop jeone
수업 후에 sueop hue
수업 뒤에 sueop dwie
TAKE NOTE
All the above expressions have a variety of uses other than the one in question here: consult a dictionary or vocabulary guide for more information. Most commonly, all are used to put actions or events in temporal order and tell what happened before, after, or at the end of the sequence of events.
EXAMPLE SENTENCES
5년 전에 대학교를 졸업했어요.
Onyeon jeone daehakgyoreul joreopaesseoyo.
Five years before, I graduated from college.
5년 후에 대학교를 졸업하겠어요.
Onyeon hue daehakgyoreul joreopagesseoyo.
I will graduate from college after five years.
5년 뒤에 대학교를 졸업하겠어요.
Onyeon dwie daehakgyoreul joreopagesseoyo.
I will graduate from college after five years.
30분 전에 시험이 끝났어요.
Samsipbun jeone siheomi kkeunnasseoyo.
The test was over 30 minutes ago.
30분 후에 시험이 끝날 거예요.
Samsipbun hue siheomi kkeunnal geoyeyo.
After 30 minutes, the test will be over.
30분 뒤에 시험이 끝날 거예요.
Samsipbun dwie siheomi kkeunnal geoyeyo.
After 30 minutes, the test will be over.
아침 먹기 전에 샤워를 했어요.
Achim meokgi jeone syaworeul haesseoyo.
Before eating breakfast, I took a shower.
아침 먹은 후에 샤워를 했어요.
Achim meogeun hue syaworeul haesseoyo.
After eating breakfast, I took a shower.
아침 먹은 뒤에 샤워를 했어요.
Achim meogeun dwie syaworeul haesseoyo.
After eating breakfast, I took a shower.
샤워를 하기 전에 아침을 먹을 거예요.
Syaworeul hagi jeone achimeul meogeul geoyeyo.
Before taking a shower, I'll eat breakfast.
샤워를 한 후에 아침을 먹을 거예요.
Syaworeul han hue achimeul meogeul geoyeyo.
After taking a shower, I'll eat breakfast.
샤워를 한 뒤에 아침을 먹을 거예요.
Syaworeul han dwie achimeul meogeul geoyeyo.
After taking a shower, I'll eat breakfast.
그녀는 급하게 서두르다가도 항상 정신이 산만해져서 늦어요.
Geunyeoneun geupage seodureudagado hangsang jeongsini sanmanhaejyeoseo neujeoyo.
Even though she hurries, she always gets distracted and ends up being late.
This is much like 다가, but this expression implies a sudden and dramatic change in a situation.
HOW IT'S FORMED
다가 plus 도. 도 is used mostly in contrasting sentences (see page ) and also has the effect of adding emphasis. 다가 by itself means only that the situation changed, but when you add 도 you're saying it changed.
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Action verbs ending in a vowel | 하다 | 하다가도 hadagado
Action verbs ending in a consonant | 먹다 | 먹다가도 meokdagado
TAKE NOTE
The subjects of both clauses should be the same.
EXAMPLE SENTENCES
슬프다가도 음악을 들으면 기분이 좋아져요.
Seulpeudagado eumageul deureumyeon gibuni joajyeoyo.
Even when I'm sad, I feel better if I listen to music.
자다가도 전화벨이 울리면 받아요.
Jadagado jeonhwaberi ullimyeon badayo.
Even if he's asleep, he'll answer the phone if it rings.
지하철을 타고서 늦었어요.
Jihacheoreul tagoseo neujeosseoyo.
I took the subway and then I was late.
This shows that B happened after A, and sometimes at least partly because of A.
HOW IT'S FORMED
고 plus 서 (as in 아/어/여서), a combination of "and" and "so."
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Action verbs ending in a vowel | 하다 | 하고서 hagoseo
Action verbs ending in a consonant | 먹다 | 먹고서 meokgoseo
TAKE NOTE
고서 can be used only with action verbs and only when the subjects of both clause A and clause B are the same.
It's very similar to 고 나서 go naseo (see page 272) but can be used in a broader range of sentences.
EXAMPLE SENTENCES
저녁을 빨리 먹고서 영화를 봤어요.
Jeonyeogeul ppalli meokgoseo yeonghwareul bwasseoyo.
I ate dinner quickly and watched a movie.
저녁을 먹고서 담배를 피웠어요.
Jeonyeogeul meokgoseo dambaereul piwosseoyo.
I ate dinner and then had a cigarette.
우리가 지하철을 타고서야 정시에 올 수 있었어요.
Uriga jihacheoreul tagoseoya jeongsie ol su isseosseoyo.
We took the subway and we were on time (at least partly because of taking the subway).
This is much like 고서 but requires that clause A be a condition of clause B. In other words, there's more of a causal relationship between A and B than there is with 고서. A happened before and helped bring about B. In other words, once you've done A, you can finally do B.
HOW IT'S FORMED
고 plus 서 (아/어/여서; so) plus 야 (아/어/여야 is an expression which means A must happen in order for B to happen). Put them all together and you get the meaning of this expression. In 고서야, unlike 아/어/여야, A is not a necessary condition for B, but it does help cause it.
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Action verbs ending in a vowel | 하다 | 하고서야 hagoseoya
Action verbs ending in a consonant | 먹다 | 먹고서야 meokgoseoya
TAKE NOTE
고서야 can be used only with action verbs and only when the subjects of both clause A and clause B are the same.
B should include an expression of possibility such as (으)ㄹ 수 있다 (eu)r su itda (see page 294).
EXAMPLE SENTENCES
공부를 열심히 안 하고서야 시험에 어떻게 합격할 수 있어요?
Gongbureul yeolsimhi an hagoseoya siheome eotteoke hapgyeokal su isseoyo?
If you don't study hard, how will you pass the test?
숙제를 하고서야 수업에 갈 수 있었어요.
Sukjereul hagoseoya sueobe gal su isseosseoyo.
I did my homework and then went to class (partly because I was finally ready).
지하철을 타고 보니 택시보다 더 느렸어요.
Jihacheoreul tago boni taeksiboda deo neuryeosseoyo.
I took the subway, but found it was slower than a taxi.
This expression is used to talk about something you noticed when you tried to do something else. A is something that was done, and B is what you observed after it was done.
HOW IT'S FORMED
고 plus 보다 (in this case, to notice or discover) plus 니(까) (so). The 까 part of 니까 is optional.
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Action verbs ending in a vowel | 하다 | 하고 보니까 hago bonikka
Action verbs ending in a consonant | 먹다 | 먹고 보니까 meokgo bonikka
TAKE NOTE
고 보니(까) is quite similar to another useful expression, 다(가) 보니(까) (see page ). The difference between the two is that while the action in the A clause was repeated and is ongoing in 다가 보니까, it must be finished in order to be used with 고 보니까. So 아버지 되다가 보니까 abeoji doedaga bonikka ("I became a father and then noticed...") is an acceptable construction while 아버지 되고 보니까 abeoji doego bonikka is not because being a father is something that is ongoing. Likewise, you can say 사고가 나고 보니까 sagoga nago bonikka ("There was an accident and then I noticed") while 사고가 나다가 보니까 sagoga nadaga bonikka is ungrammatical because an accident happening is only a single, brief event.
고 보니까 can be used only with action verbs and only when the subjects of both clause A and clause B are the same.
EXAMPLE SENTENCES
이사할 집이 편리할 거라고 생각했는데 이사 하고 보니까 시끄럽기만 했어요.
Isahal jibi pyeollihal georago saenggakaenneunde isa hago bonikka sikkeureopgiman haesseoyo.
I thought the house I was moving to would be more conveniently located, but I found it was just noisy.
친구인 줄 알았는데 인사를 하고 보니 모르는 사람이었어요.
Chinguin jul aranneunde insareul hago boni moreuneun saramieosseoyo.
I thought he was my friend, but when I greeted him I found out he was someone I didn't know.
##
If: If
늦으면 전화해 주세요.
Neujeumyeon jeonhwahae juseyo.
If you're late, please call me.
면 is the workhorse of the Korean conditional world, so learn it well. It simply means "if" and is not at all difficult to use. You'll see it all throughout this book in various other expressions. Here are a few:
Expression | Page | Meaning | Example sentence
---|---|---|---
A(으)면 B A (eu)myeon B | – | If A, then B | 늦으면 전화해 주세요. Neujeumyeon jeonhwahae juseyo. If you're late, please call me.
A(이)라/(ㄴ/는)다/자 면 B A (i)ra/(n/neun)da/ jamyeon B | 286 | If A, then B | 늦었다면 남자친구를 만나지 않았을 거예요. Neujeotdamyeon namjachingureul mannaji anasseul geoyeyo. If I'd been late, I wouldn't have met my boyfriend.
A았/었/였더라면 B A at/eot/yeotdeoramyeon B | 289 | If only A, then B | 택시를 타지 않았더라면 늦었을 거예요. Taeksireul taji anatdeoramyeon neujeosseul geoyeyo. If I hadn't taken a taxi, I would have been late.
A 같으면 B A gateumyeon B | 293 | If it were like A, then B | 왜 이렇게 늦었어? 너가 내 아이 같으면 야단쳤을 거야. Wae ireoke neujeosseo? Neoga nae ai gateumyeon, yadanchyeosseul geoya. Why were you so late? If you were my child, I'd scold you.
A(이)라/(ㄴ/는)다/(으)라/자고 하면 B A (i)ra/(n/neun)da/(eu)ra/jago hamyeon B | – | If you say A, then B | 민수가 빨리 가자고 하면 서둘러요. Minsuga ppalli gajago hamyeon seodulleoyo. If Minsu suggested we go quickly, then let's hurry.
A에 의하면 B A e uihamyeon B | | According to A, B | 일기 예보에 의하면 내일 비가 올 거예요. Ilgi yeboe uihamyeon naeil biga ol geoyeyo. According to the weather forecast, it's going to rain tomorrow.
A 아니면 B A animyeon B | | If not A, then B | 늦을 거예요? 아니면 시간에 맞출 거예요? Neujeul geoyeyo? Animyeon sigane matchul geoyeyo? Will you be late? Or will you arrive on time?
A에 비하면 B A e bihamyeon B | | B, compared to A | 가영에 비하면 민수가 더 늦었어요. Kayounge bihamyeon Minsuga deo neujeosseoyo. Minsu was later than Kayoung.
A(으)ㄴ/는가 하면 B A (eu)n/neunga hamyeon B | | There's A, but there's also B | 제 수업에 정시에 오는 사람들이 있는가 하면 늦게 오는 사람들도 있어요. Je sueobe jeongsie oneun saramdeuri inneunga hamyeon neutge oneun saramdeuldo isseoyo. In my class, there are some people who come on time and some who are late.
A(으)ㄹ 테면 B A (eu)r temyeon B | | If you plan/expe ct to do A, then B | 어떤 얘기든 할 테면 그냥 하세요. Eotteon yaegideun hal temyeon geunyang haseyo. Whatever you want to say, just say it.
A(으)면 A(으)ㄹ수록 B A (eu)myeon A (eu)rsurok B | | The more A, B | 멀리 가면 갈수록 피곤해졌어요. Meolli gamyeon galsurok pigon-hae-jyeosseoyo. The farther I went, the more tired I became.
A다(가) 보면 B A da(ga) bomyeon B | | If you do A, then B | 계속 공부하다가 보면 유창해질 거예요. Gyesok gongbuhadaga bomyeon yuchanghaejil geoyeyo. If you keep studying, you'll become fluent.
~(으)면 되다/안 되다 ~(eu)myeon doeda/an doeda | | ~ is okay/not okay | 늦으면 안 돼요. Neujeumyeon an dwaeyo. It's not okay to be late.
~았/었/였으면 좋겠다/싶다/하다 ~at/eot/yeosseumyeon joketda/sipda/hada | | It would be nice if ~ | 정시에 왔으면 좋겠어요. Jeongsie wasseumyeon jokesseoyo. It would have been better if you'd arrived on time.
~(으)면 뭘 해요? ~(eu)myeon mwol haeyo? | | What's the point of ~? | 왜 과속해요? 일찍 가면 뭘 해요? 안전이 제일 중요하지요. Wae gwasokaeyo? Iljjik gamyeon mwol haeyo? Anjeoni jeil jungyohajiyo. Why are you speeding? What's the point of getting there early? Safety is more important.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했으면 haesseumyeon | 하면 hamyeon | 하면
Action verbs ending in a consonant | 먹다 | 먹었으면 meogeosseumyeon | 먹으면 meogeumyeon | 먹으면
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤으면 yeppeosseumyeon | 예쁘면 yeppeumyeon | 예쁘면
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았으면 jagasseumyeon | 작으면 jageumyeon | 작으면
TAKE NOTE
When using this expression in the past tense (했으면, 먹었으면), you can add 야 for emphasis. So 했으면 becomes 했으면야 haesseumyeonnya, 먹었으면 becomes 먹었으면야 meogeosseumyeonnya, etc. This means exactly the same thing as 했으면 and 먹었으면 by themselves, but emphasizes your regret or gratitude that you didn't A. See the examples below. In this case you normally end B with 았/었/였겠다 at/eot/yeotgetda, 았/었/였을 거예요 ass/eoss/yeosseul geoyeyo or 았/었/였을 텐데(요) ass/eoss/yeosseul tende(yo).
EXAMPLE SENTENCES
비가 오면 꽃이 펴요.
Biga omyeon kkochi pyeoyo.
If it rains, the flowers bloom.
시간이 있었으면 왔을 거예요.
Sigani iss-eoss-eumyeon wasseul geoyeyo.
If I'd had time I would have come.
사업을 했으면 지금쯤 부자가 됐을 거예요.
Saeobeul haesseumyeon jigeumjjeum bujaga dwaesseul geoyeyo.
If I'd started a business, I'd be rich by now.
사업을 했으면야 지금 부자가 됐을 거예요.
Saeobeul haess-eu-myeonnya jigeum bujaga dwaesseul geoyeyo.
If only I'd started a business, I'd be rich by now.
It can also be used with 그렇다 geureota to form 그러면 geureomyeon, which means "if so."
A: 늦을 텐데... Neujeul tende... I think we're going to be late
B: 그러면 빨리 가세요. Geureomyeon ppalli gaseyo. Then hurry up.
이 회의가 이렇게 중요하거든 늦지 마세요.
I hoeuiga ireoke jungyohageodeun neutji maseyo.
If this meeting is so important, then please don't be late.
거든 is similar to 면 myeon but can be used only when your A clause is the basis for a suggestion about your B clause.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했거든 haetgeodeun | 하거든 hageodeun | 하거든
Action verbs ending in a consonant | 먹다 | 먹었거든 meogeotgeodeun | 먹거든 meokgeodeun | 먹거든
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤거든 yeppeotgeodeun | 예쁘거든 yeppeugeodeun | 예쁘거든
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았거든 jagatgeodeun | 작거든 jakgeodeun | 작거든
Nouns ending in a vowel | 남자 | 남자였거든 namjayeotgeodeun | 남자이거든 namjaigeodeun | 남자이거 든
Nouns ending in a consonant | 물 | 물이었거든 murieotgeodeun | 물이거든 murigeodeun | 물이거든
TAKE NOTE
This one can be used only with suggestions, commands, or questions—no statements. Since it's used for things that will happen in the future such as suggestions, it can't be used with sentences in the past tense. The A clause can be in the past tense if that action is already finished, but B can't be.
거든 can also be used at the end of a sentence; in that case it has a somewhat different meaning; in fact, two different meanings depending on your intonation! See page for how to use 거든 at the end of a sentence.
EXAMPLE SENTENCES
어머니 생신이거든, 어서 빨리 가서 선물을 사 드리세요.
Eomeoni saengsinigeodeun, eoseo ppalli gaseo seonmureul sa deuriseyo.
If it's your mother's birthday, then please hurry up and buy a present for her.
이 신발이 너무 작거든 큰 것으로 바꿔 줄게요.
I sinbari neomu jakgeodeun keun geoseuro bakkwo julgeyo.
If these shoes are too small, I'll give you a bigger size.
늦었다면 남자친구를 만나지 않았을 거예요.
Neujeotdamyeon namjachingureul mannaji anasseul geoyeyo.
If I'd been late, I wouldn't have met my boyfriend.
Are you a daydreamer? Dislike reality? Here's an expression for you. This is much like 면, but can be used only to talk about either hypothetical situations contrary to what actually happened or situations in the future that haven't yet happened. In other words, reality has nothing to do with this one. If you imagine 면 and all the expressions that can be made with it as a giant circle, 라/다/ㄴ다/는다/자면 is a smaller circle within that big circle. All the sentences that use this expression could also be made with 면 and would mean the same thing, but the reverse is not true.
HOW IT'S FORMED
It uses the indirect speech particles 라/다/ㄴ다/는다/자, but has nothing to do with indirect speech; the part being "quoted" is just the situation that is being imagined or predicted. This is your A clause. B is what you think would happen if A were or had been true.
HOW IT'S CONJUGATED
자면 is used only after action verbs and is never used to talk about the past tense.
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했다면 haetdamyeon | 한다면 handamyeon
하자면 hajamyeon
| 할 거라면 hal georamyeon
하자면
Action verbs ending in a consonant | 먹다 | 먹었다면 meogeotdamyeon | 먹는다면 meokneundamyeon
먹자면 meokjamyeon
| 먹을 거라면 meogeul georamyeon
먹자면
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤다면 yeppeotdamyeon | 예쁘다면 yeppeudamyeon | 예쁠 거라면 yeppeulgeoramyeon
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았다면 jagatdamyeon | 작다면 jakdamyeon | 작을 거라면 jageul georamyeon
Nouns ending in a vowel | 남자 | 남자였다면 namjayeotdamyeon | 남자라면 namjaramyeon | 남자라면
Nouns ending in a consonant | 물 | 물이었다면 murieotdamyeon | 물이라면 muriramyeon | 물이라면
TAKE NOTE
Keep reality out of it! Use this only for things that aren't true or that you're imagining or predicting. It should go with expressions that are also hypothetical such as ㄹ 것이다 r geosi da, ㄹ 텐데 r tende or 겠어요 gesseoyo.
Just like 았/었/였으면 at/eot/yeosseumyeon (see page 285), you can also add 야 after this expression (한다면야 handamyeonnya, 했다면야 haetdamyeonnya, 먹는다면야 meokneunda-myeonnya, 예쁘다면야 yeppeudamyeonnya, etc.), which emphasizes A and so conveys your feelings of regret or gratitude even more strongly. Compare the examples below.
자면 jamyeon is for hypothetical situations in the future and translates as "If you want to" (이)라면 (i)ramyeon and ([으]ㄴ/는)다면 ((eu)n/neun)damyeon can be used in hypothetical situations in any tense.
Since this is essentially used to make recommendations, B normally (but not always) ends with an expression indicating duty or preference; 아/어/여야 하다/되다 a/eo/yeoya hada/doeda (see page ) and 는 것이 좋다 neun geosi jota are common.
EXAMPLE SENTENCES
늦었다면 남자친구를 만나지 않았을 거예요.
Neujeotdamyeon namjachingureul mannaji anasseul geoyeyo.
If I'd been late, I wouldn't have met my boyfriend.
늦었다면야 남자친구를 만나지 않았을 거예요.
Neujeotdamyeonnya namjachingureul mannaji anasseul geoyeyo.
If I'd been late (and I'm really glad I wasn't), I wouldn't have met my boyfriend.
제가 부자라면 부모님께 큰 집을 사 드릴 거예요.
Jega bujaramyeon bumonimkke keun jibeul sa deuril geoyeyo.
If I were rich, I'd buy my parents a big house.
제가 부자라면야 부모님께 큰 집을 사 드릴 거예요.
Jega bujaramyeonnya bumonimkke keun jibeul sa deuril geoyeyo.
If only I were rich, I'd buy my parents a big house.
비가 왔다면 꽃이 필 거예요.
Biga watdamyeon kkochi pil geoyeyo.
비가 왔으면 꽃이 필 거예요.
Biga wasseumyeon kkochi pil geoyeyo.
If it had rained, the flowers would have bloomed.
In the last example above, both 다면 damyeon and 으면 eumyeon mean the same thing because the expression is in the past and it's now impossible for it to rain. As you can see, 다면 in this case implies some regret on the speaker's part. If a negative consequence was avoided, it can also convey a feeling of thanks. Below you can see how 면 and 다면 differ when used in the present tense.
비가 온다면 꽃이 필 거예요.
Biga ondamyeon kkochi pil geoyeyo.
If it rains, the flowers will bloom. (Hypothetical; it's not raining now.)
비가 오면 꽃이 필 거예요.
Biga omyeon kkochi pil geoyeyo.
If it rains, the flowers will bloom. (It may or may not be raining already.)
And a few examples with 자면:
정시에 도착하자면 집에서 일찍 출발해야 돼요.
Jeongsie dochakajamyeon jibeseo iljjik chulbalhaeya dwaeyo.
If you want to arrive on time, you should leave the house early.
돈을 많이 모으자면 투자를 하면 좋겠어요.
Doneul mani moeujamyeon tujareul hamyeon jokesseoyo.
If you want to save a lot of money, you should invest.
영어를 잘 배우자면 해외 여행을 가야 해요.
Yeongeoreul jal baeujamyeon haeoe yeohaengeul gaya haeyo.
If you want to learn English well, you must go overseas.
택시를 타지 않았더라면 늦었을 거예요.
Taeksireul taji anatdeoramyeon neujeosseul geoyeyo.
If I hadn't taken a taxi, I would have been late.
This is a good expression to know if you're someone who always forgets things or doesn't plan ahead. It's even more hypothetical than 라/다/ㄴ다/는다면 ra/da/nda/neundamyeon and can be used only in hypothetical situations in the past tense, usually (but certainly not always) ones that had negative results in reality: "If only you had saved your money"; "If only you had asked that girl out."
HOW IT'S FORMED
The past tense marker 았/었/였 at/eot/yeot, plus 더라 deora, which is used for things that you personally experienced or heard, plus the conditional marker 면.
HOW IT'S CONJUGATED
| | Past
---|---|---
Action verbs ending in a vowel | 하다 | 했더라면 haetdeoramyeon
Action verbs ending in a consonant | 먹다 | 먹었더라면 meogeotdeoramyeon
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤더라면 yeppeotdeoramyeon
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았더라면 jagatdeoramyeon
Nouns ending in a vowel | 남자 | 남자였더라면 namjayeotdeoramyeon
Nouns ending in a consonant | 물 | 물이었더라면 murieotdeoramyeon
TAKE NOTE
Again, use this only when imagining the opposite of what really happened.
B often ends with (으)ㄹ 거예요 (eu)r geoyeyo, 겠어요 gesseoyo, or (으)ㄹ 텐데요 (eu)r tendeyo.
EXAMPLE SENTENCES
집에서 더 일찍 출발했더라면 늦지 않았을 텐데요.
Jibeseo deo iljjik chulbalhaetdeoramyeon neutji anasseul tendeyo.
If I'd left the house earlier, I wouldn't have been late.
병원에 가지 않았더라면 죽을 뻔했어요.
Byeongwone gaji anatdeoramyeon jugeul ppeonhaesseoyo.
If he hadn't gone to the hospital, he would have died.
See page 261 for an explanation of (으)ㄹ 뻔했다 (eu)r ppeonhaetda. It goes well with 았/었/였더라면 at/eot/yeotdeoramyeon.
저 같은 경우에는 그 보너스 받을 수 없어요.
Jeo gateun gyeongueneun geu boneoseu badeul su eopseoyo.
In cases like mine, I can't receive a bonus.
Here's a good expression to know if you're ever involved in disaster planning or if you just want to know how to talk about what could happen in different situations.
HOW IT'S FORMED
First, take the past, present, or future tense markers (으)ㄴ/는/(으)ㄹ. Then add 경우, which is a noun that means "case" or "occasion." 에는 simply means that whatever happens in B happens in the case of or occasion A. 는 may be dropped.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 한 경우에(는) han gyeongue(neun) | 하는 경우에(는) haneun gyeongue(neun) | 할 경우에(는) hal gyeongue(neun)
Action verbs ending in a consonant | 먹다 | 먹은 경우에(는) meogeun gyeongue(neun) | 먹는 경우에(는) meokneun gyeongue(neun) | 먹을 경우에(는) meogeul gyeongue(neun)
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁜 경우에(는) yeppeun gyeongue(neun) | 예쁜 경우에(는) yeppeun gyeongue(neun) | 예쁠 경우에(는) yeppeul gyeongue(neun)
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작은 경우에(는) jageun gyeongue(neun) | 작은 경우에(는) | 작을 졍우에(는)
Nouns ending in a vowel | 남자 | 남자의 경우에(는) namjaui gyeongue(neun) | 남자의 경우에(는) | 남자의 경우에(는)
Nouns ending in a consonant | 물 | 물의 경우에(는) murui gyeongue(neun) | 물의 경우에(는) | 물의 경우에(는)
TAKE NOTE
This pattern can be changed to 면, but not vice versa.
There are a few commonly used expressions with 경우에는:
저 같은 경우에는
jeo gateun gyeongueneun
In cases like mine
내 경우에는
nae gyeongueneun
In my case
이 경우에는
i gyeongueneun
In this case
그런 경우에는
geureon gyeongueneun
In that case
어떠한 경우에도
eotteohan gyeonguedo
In any case
어떤 경우에는
eotteon gyeongueneun
In some cases
최악의 경우에는
choeagui gyeongueneun
At worst (in the worst case)
모든 경우에는
modeun gyeongueneun
In all cases
EXAMPLE SENTENCES
눈이 많이 오는 경우에는 길이 미끄러워져요.
Nuni mani oneun gyeongueneun giri mikkeureowojyeoyo.
If (in cases where) it snows a lot, the road will become slippery.
급한 일이 생길 경우에는 일찍 퇴근하지 마세요.
Geupan iri saenggil gyeongueneun iljjik toegeunhaji maseyo.
If anything urgent comes up, don't leave work early.
남자의 경우에는 오른쪽에 있는 화장실을 이용해 주세요.
Namjaui gyeongueneun oreunjjoge inneun hwajangsireul iyonghae juseyo.
Men should use the bathroom on the right side.
우리는 아주 늦을 거예요. 그러니까 우리는 없는 셈 치고 먼저 파티를 시작하세요.
Urineun aju neujeul geoyeyo. Geureonikka urineun eopneun sem chigo meonjeo patireul sijakaseyo.
We will be very late, so please assume we're not there and go ahead and start the party.
In this expression, you do B based on the assumption you make in A: you do B as if A were true.
HOW IT'S FORMED
셈 is a noun that means "supposition" and 치다 is the verb used to make that supposition. This expression is just the two words joined together plus whatever else you need to add to make the expression work.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 한 셈치고 han semchigo | 하는 셈치고 haneun semchigo | 할 셈치고 hal semchigo
Action verbs ending in a consonant | 먹다 | 먹은 셈치고 meogeun semchigo | 먹는 셈치고 meokneun semchigo | 먹을 셈치고 meogeul semchigo
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁜 셈치고 yeppeun semchigo | 예쁜 셈치고 | 예쁠 셈치고 yeppeul semchigo
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작은 셈치고 jageun semchigo | 작은 셈치고 | 작을 셈치고 jageul semchigo
Nouns ending in a vowel | 남자 | 남자인 셈치고 namjain semchigo | 남자인 셈치고 | 남자일 셈치고 namjail semchigo
Nouns ending in a consonant | 물 | 물인 셈치고 murin semchigo | 물인 셈치고 | 물일 셈치고 muril semchigo
TAKE NOTE
Don't confuse this with 셈이다 semida (see page ); they're similar, but 셈이다 by itself means "seems like" whereas 셈치다 is used to consider a hypothetical situation: if we assume that A, then B follows from that.
셈치고는 is a different expression which means "for an A" or "considering that A." It's on page . Although it looks very much like 셈치고, the meaning is rather different.
셈치고 is probably the most common use of this expression, but 셈치다 can be conjugated with 고, 면, or whatever else you need, or it can just go at the end of a sentence.
EXAMPLE SENTENCES
속는 셈치고 믿어보세요.
Songneun semchigo mideoboseyo.
Take a chance and believe me this one time. (Even assuming you might be fooled again, just try to believe me this time.)
그 사람이 있는 셈치고 한 번 얘기해보세요.
Geu sarami inneun semchigo han beon yaegihaeboseyo.
Let's talk as if that person were here.
왜 이렇게 늦었어? 너가 내 아이 같으면 야단쳤을 거야.
Wae ireoke neujeosseo? neoga nae ai gateumyeon yadanchyeosseul geoya.
Why were you so late? If you were my child, I'd scold you.
Like 았/었/였더라면 at/eot/yeotdeoramyeon, 같으면 is used to imagine a situation contrary to reality: "If it were like ten years ago" (10년 전 같으면 sipnyeon jeon gateum~yeon) or "If that were my child" (내 아이 같으면).
HOW IT'S FORMED
같다 means "to be like" and is commonly seen in expressions like 같은 or 같이. In this expression it is combined with 면, which means "if." (see page 283).
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Nouns ending in a vowel | 남자 | 남자 같았으면 namja gatasseumyeon | 남자 같으면 namja gateumyeon | 남자 같으면
Nouns ending in a consonant | 물 | 물 같았으면 mul gatasseumyeon | 물 같으면 mul gateumyeon | 물 같으면
TAKE NOTE
If you want to talk about something that might be possible, you should use (이)라면 instead of 같으면. (이)라면 is on page 286. If you want to talk about a verb instead of a noun, use 았/었/였더라면 (see page 289).
EXAMPLE SENTENCES
옛날 같았으면 이렇게 빨리 갈 수 없어요.
Yennal gatasseumyeon ireoke ppalli gal su eopseoyo.
If it were like the old days, we wouldn't be able to go this fast.
한국 같으면 쉽게 할 수 있을 거예요.
Hangug gateumyeon swipge hal su isseul geoyeyo.
If this were like Korea, we could do it easily.
##
If: Possibility and The Lack Thereof
한국어를 잘 할 수 없어요.
Hangugeoreul jal hal su eopseoyo.
I can't speak Korean well.
한국어를 잘 할 수 있어요.
Hangugeoreul jal hal su isseoyo.
I can speak Korean well.
This is one of the first expressions you'll learn while studying Korean. It's particularly helpful at the very beginning when you need to tell people you can't speak Korean. Later on you can mix things up a bit by getting into expressions like (으)ㄹ 수도 있다 or 는 수 있다/없다. It's pretty versatile and definitely worth knowing even if you never plan to take your Korean past the beginner level. ~(으)ㄹ 수 있다 means "can ~" while ~(ㅇ)ㄹ 수 없다 means "can't ~."
HOW IT'S FORMED
The future tense marker (으)ㄹ (see page ) is followed by 수, which has quite a number of meanings; in this case it just means "possibility." Then 있다 (there is) or 없다 (there isn't) is added to show that there is or isn't the possibility of doing ~. Because 수 is a noun, you can add the subject marker 가 to it if you'd like; this doesn't really change the meaning, but it does put a little more emphasis on the fact that ~ is really possible or impossible. 는 is occasionally added to 수 as well, depending on the sentence, and also has the effect of emphasizing the possibility or impossibility of the statement, like so:
한국말을 잘 할 수는 있어요.
Hangukmareul jal hal suneun isseoyo.
Here you can speak Korean well, but can't do anything else: for instance, your writing may be poor. So you're comparing your ability to speak Korean well against something else which may or may not be mentioned.
한국말을 잘 할 수가 있어요.
Hangukmareul jal hal suga isseoyo.
This sentence means it's possible for you to speak Korean well.
HOW IT'S CONJUGATED
Descriptive verbs (adjectives) and nouns can't be conjugated with this expression, but you can turn them into action verbs (see page for how to do that) and then use them with (으)ㄹ 수 있다/없다 if: for example, you want to say that a woman can become beautiful.
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 할 수 있었다 hal su iseotda
할 수 없었다 hal su eopseotda
| 할 수 있다 hal su itda
할 수 없다 hal su eopda
| 할 수 있겠다 hal su itgetda
할 수 없겠다 hal su eopgetda
Action verbs ending in a consonant | 먹다 | 먹을 수 있었다 meogeul su iseotda
먹을 수 없었다 meogeul su eopseotda
| 먹을 수 있다 meogeul su itda
먹을 수 없다 meogeul su eopda
| 먹을 수 있겠다 meogeul su itgetda
먹을 수 없겠다 meogeul su eopgetda
TAKE NOTE
Another expression involving knowing and not knowing is ~(으)ㄹ 줄 알다/모르다 ~(eu)r jul alda/moreuda. That one, however, means "to know how to do ~." ~(으)ㄹ 수 있다/없다 means "to be able to ~." In some cases, like 한국어를 할 수 있다/할 줄 알다 Hangugeoreul hal su itda/hal jul alda, you can use both; in others you can't.
There is also 못, which literally means "can't." This is a shorter way than (으)ㄹ 수 없다 to say you can't do something. You put it directly before the verb you can't do. It's interchangeable with (으)ㄹ 수 없다.
한국어를 잘 못해요.
Hangugeoreul jal motaeyo.
I can't speak Korean well.
알레르기 때문에 땅콩을 못 먹어요.
Allereugi ttaemune ttangkongeul mon meogeoyo.
Because of allergies, I can't eat peanuts.
There are a number of complicated expressions based on (으)ㄹ 수 있다/없다. They're useful to know but mostly not essential, so I'm putting them all in here rather than giving them their own sections because they're all essentially variations of (으)ㄹ 수 있다/없다.
■ ~는 수가 있다 ~neun suga itda
This expression uses the present tense marker 는 instead of the future tense marker (으)ㄹ before 수 있다. While (으)ㄹ 수 있다 only implies potential, 는 수가 있다 means that ~ is unusual but can and sometimes does happen. However, this event is not quite as possible as (으)ㄹ 수도 있다 (eu)r sudo itda.
보통 지하철 열차가 정시에 오지만 무슨 특별한 문제가 있으면 늦는 수가 있어요.
Botong jihacheol yeolchaga jeongsie ojiman museun teukbyeolhan munjega isseumyeon neunneun suga isseoyo.
Usually the subway trains are on time, but if there's a problem, they can also be late.
믿는 도끼에 발등 찍히는 수가 있어요.
Minneun dokkie baldeung jjikineun suga isseoyo.
The ax that you trust can also cut you. (a Korean proverb)
■ ~(으)ㄹ 수도 있다 ~(eu)r sudo itda
This is quite similar to ~기도 하다 ~gido hada, which you can find on page . ~기도 하다 is used to say that ~ also happens sometimes. Likewise, ~(으)ㄹ 수도 있다 means that ~ is also possible.
저녁에 비가 올 것 같아요. 눈이 올 수도 있어요.
Jeonyeoge biga ol geot gatayo. Nuni ol sudo isseoyo.
It seems like it'll rain this evening. It could also snow.
이 거리는 보통 안전하지만 사고가 날 수도 있어요.
I georineun botong anjeonhajiman sagoga nal sudo isseoyo.
This street is usually safe, but sometimes accidents do happen.
■ ~(으)ㄹ 수가 있어야지(요) ~(eu)r suga isseoyaji(yo)
This one is a little confusing. Take (으)ㄹ 수 있다 and add 가, which adds emphasis, and then 어야, which means "must" (see page ), and then 지(요) which means "isn't it?" (see page ). Instead of getting a really forceful confirmation of the possibility of ~, you get.~ is impossible. So this one means the opposite of what it looks like it should mean. A good English translation would be "How could (I/you/it/) possibly ~?"
차가 너무 밀려서 정시에 올 수가 있어야지요.
Chaga neomu millyeoseo jeongsie ol suga isseoyajiyo.
Traffic was too heavy, so I couldn't possibly arrive on time.
돈이 없어서 여행을 갈 수가 있어야지요.
Doni eopseoseo yeohaengeul gal suga isseoyajiyo.
I had no money, so how could I possibly travel?
■ ~(으)ㄹ 수 밖에 없다 ~(eu)r su bakke eopda
In this expression, you take (으)ㄹ 수 없다 (can't) and insert a 밖에 in the middle. The 밖에 means "outside of," making the whole expression "can't do anything except ~." So you had no choice but to ~.
그를 죽일 수 밖에 없었어요.
Geureul jugil su bakke eopseosseoyo.
I had no choice but to kill him.
할 수 밖에 없었어요.
hal su bakke eopseosseoyo.
There was no choice.
■ ~는/(으)ㄹ 수 밖에 별 도리 없다 ~neun/(eu)r su bakke byeol dori eopda
Here we take (으)ㄹ 수 밖에 없다 (see above) and extend it a little more to add 별도리. 도리 means "way," "method," or "means," so this is just a slightly more elaborate way of saying "There was no choice but to."
갈 수 밖에 별 도리가 없어요.
Gal su bakke byeol doriga eopseoyo.
I have no choice but to go.
그를 죽일 수 밖에 도리 없었어요.
Geureul jugil su bakke dori eopseosseoyo.
I had no choice but to kill him.
EXAMPLE SENTENCES
수영 할 수 있어요?
Suyeong hal su isseoyo?
Can you swim?
수영 할 수 없어요.
Suyeong hal su eopseoyo.
I can't swim.
젓가락을 쓸 수 있어요.
Jeotgarageul sseul su isseoyo.
I can use chopsticks.
매운 음식을 먹을 수 있어요.
Maeun eumsigeul meogeul su isseoyo.
I can eat spicy food.
정시에 오지 못했어요.
Jeongsie oji motaesseoyo.
I couldn't arrive on time.
Here's a very easy way to say you can't do something. This is also used to say you're not good at something: whether you can't sing or dance or speak Korean well, you can use 지 못하다 to explain this to people.
HOW IT'S FORMED
지 is a particle often used before negative expressions. 못 means "can't," and 하다 means "to do."
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 하지 못했다 haji motaetda | 하지 못하다 haji motada | 하지 못할 것이다 haji motal geosida
Action verbs ending in a consonant | 먹다 | 먹지 못했다 meokji motaetda | 먹지 못하다 meokji motada | 먹지 못할 것이다 meokji motal geosida
TAKE NOTE
This expression and (으)ㄹ 수 없다 (see page 294) are almost identical except that this one can also be used to say you're poor at doing something whereas (으)ㄹ 수 없다 generally means you can't do it at all.
EXAMPLE SENTENCES
노래를 하지 못해요.
Noraereul haji motaeyo.
I can't sing (well).
우리 반에 영어 못하는 아이들이 많아요.
Uri bane yeongeo motaneun aideuri manayo.
In our class there are many children who aren't good in English.
알레르기 때문에 생선을 먹지 못 해요.
Allereugi ttaemune saengseoneul meokji mot haeyo.
Because of allergies, I can't eat fish.
정시에 오기는 쉬워요.
Jeongsie ogineun swiwoyo.
정시에 오기는 십상이에요.
Jeongsie ogineun sipsangieyo.
정시에 오기는 어려워요.
Jeongsie ogineun eoryeowoyo.
Here are two ways to say something is easy and one to say it's difficult.
HOW IT'S FORMED
기 changes verbs to nouns. If you're familiar with 쉽다 and 어렵다, you'll know they usually do follow nouns. 십상이다 also means "easy" and is a good one for you TOPIK-takers to know.
HOW IT'S CONJUGATED
Add 기 after action verbs or add the verbs themselves directly after nouns. You can also conjugate verbs using 는 것(이), though this is somewhat less commonly done. Conjugate 쉽다/십상이다/어렵다 in any tense or any form you need.
| | All tenses
---|---|---
Action verbs ending in a vowel | 하다 | 하기 쉽다 hagi swipda
하기 십상이다 hagi sipsangida
하기 어렵다 hagi eoryeopda
Action verbs ending in a consonant | 먹다 | 먹기 쉽다 meokgi swipda
먹기 십상이다 meokgi sipsangida
먹기 어렵다 meokgi eoryeopda
Nouns ending in a vowel | 조사 | 조사(가) 쉽다 josa(ga) swipda
조사(가) 십상이다 josa(ga) sipsangida
조사(가) 어렵다 josa(ga) eoryeopda
Nouns ending in a consonant | 시험 | 시험(이) 쉽다 siheom(i) swipda
시험(이) 십상이다 siheom(i) sipsangida
시험(이) 어렵다 siheom(i) eoryeopda
TAKE NOTE
쉽다 and 십상이다 both mean "easy," but aren't quite interchangeable; while 쉽다 simply means "easy," 십상이다 means "something is easy to find or see," or that there's a good chance of it. 어렵다, of course, is the opposite of both and means "difficult."
Watch out for 쉽다. It's an ㅂ-irregular verb, which means it changes to 쉬우 plus whatever you're adding. Don't get it confused with 쉬다, which means "to rest" and changes to 쉬 before being conjugated.
십상이다 isn't usually used with negative sentences. You should also be sure to mention a place when you use it. See the examples below.
EXAMPLE SENTENCES
한국어 듣기가 쉬워요.
Hangugeo deutgiga swiwoyo.
Listening to Korean is easy.
한국어 듣기가 어려워요.
Hangugeo deutgiga eoryeowoyo.
Listening to Korean is difficult.
코리아타운에서는 한국어를 듣기 십상이에요.
Koriatauneseoneun hangugeoreul deutgi sipsangieyo.
You can easily hear Korean in Korea Town.
__________
금메달을 따기는 쉽지 않아요.
Geummedareul ttagineun swipji anayo.
Getting a gold medal isn't easy.
금메달을 따기는 어렵지 않아요.
Geummedareul ttagineun eoryeopji anayo.
Getting a gold medal isn't difficult.
올림픽에서 금메달을 따기는 십상이에요.
Ollimpigeseo geummedareul ttagineun sipsangieyo.
There's a good chance of getting a gold medal.
정시에 도착했을 리가 없어요.
Jeongsie dochakaesseul riga eopseoyo.
There's no way I could have arrived on time.
This is a good way to either emphatically deny something ([으]ㄹ 리 없다) or to ask if you have any chance at all ([으]ㄹ 리 있다/없다?). It's used to talk about whether or not the possibility of ~ exists.
HOW IT'S FORMED
리 literally means "possibility," so this expression is saying there is (있다) or isn't (없다) the possibility of ~. 가 is optional and serves to emphasize the impossibility of whatever is being discussed.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했을 리(가) 있다 haesseul ri(ga) itda
했을 리(가) 없다 haesseul ri(ga) eopda
| 할 리(가) 있다 hal ri(ga) itda
할 리(가) 없다 hal ri(ga) eopda
| 할 리(가) 있다 hal ri(ga) itda
할 리(가) 없다 hal ri(ga) eopda
Action verbs ending in a consonant | 먹다 | 먹었을 리(가) 있다 meogeosseul ri(ga) itda
먹었을 리(가) 없다 meogeosseul ri(ga) eopda
| 먹을 리(가) 있다 meogeul ri(ga) itda
먹을 리(가) 없다 meogeul ri(ga) eopda
| 먹을 리(가) 있다 meogeul ri(ga) itda
먹을 리(가) 없다 meogeul ri(ga) eopda
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤을 리(가) 있다 yeppeosseul ri(ga) itda
예뻤을 리(가) 없다 yeppeosseul ri(ga) eopda
| 예쁠 리(가) 있다yeppeul ri(ga) itda
예쁠 리(가) 없다 yeppeul ri(ga) eopda
| 예쁠 리(가) 있다 yeppeul ri(ga) itda
예쁠 리(가) 없다 yeppeul ri(ga) eopda
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았을 리(가) 있다 jagasseul ri(ga) itda
작았을 리(가) 없다 jagasseul ri(ga) eopda
| 작을 리(가) 있다 jageul ri(ga) itda
작을 리(가) 없다 jageul ri(ga) eopda
| 작을 리(가) 있다 jageul ri(ga) itda
작을 리(가) 없다 jageul ri(ga) eopda
Nouns ending in a vowel | 남자 | 남자였을 리(가) 있다 namjayeosseul ri(ga) itda
남자였을 리(가) 없다 namjayeosseul ri(ga) eopda
| 남자일 리(가) 있다 namjail ri(ga) itda
남자일 리(가) 없다 namjail ri(ga) eopda
| 남자일 리(가) 있다 namjail ri(ga) itda
남자일 리(가) 없다 namjail ri(ga) eopda
Nouns ending in a consonant | 물 | 물이었을 리(가) 있다 murieosseul ri(ga) itda
물이었을 리(가) 없다 murieosseul ri(ga) eopda
| 물일 리(가) 있다 muril ri(ga) itda
물일 리(가) 없다 muril ri(ga) eopda
| 물일 리(가) 있다 muril ri(ga) itda
물일 리(가) 없다 muril ri(ga) eopda
TAKE NOTE
(으)ㄹ 리 있어요 (eu)r ri isseoyo isn't used as a statement, only as a question. The form "~(으)ㄹ 리가 있겠어요?" "~(eu)r riga itgesseoyo?" is frequently used for this. If you want to say there's a possibility of something, use (으)ㄹ 수 있다 or one of its forms instead (see page 294). When used as a question, 리 있어요? (or 있겠어요?) is a bit rhetorical: it often means "How could there be any possibility of ~?", which is exactly the same meaning as 리 없어요. Consider the context to determine what someone means when they ask 리 있어요?
EXAMPLE SENTENCES
정시에 도착했을 리가 없어요.
Jeongsie dochakaesseul riga eopseoyo.
There's no way I could have arrived on time.
정시에 도착했을 리가 있어요?
Jeongsie dochakaesseul riga isseoyo?
How could I possibly have arrived on time? or Was there any chance you could have arrived on time?
당신과 데이트를 할 리가 없어요.
Dangsingwa deiteureul hal riga eopseoyo.
There's no way I would ever date you.
그녀가 너와 데이트를 할 리가 있어?
Geunyeoga neowa deiteureul hal riga isseo?
Is there any chance she could ever date you? or How could she possibly ever date you?
그는 진짜 여자일 리가 있겠어요?
Geuneun jinjja yeojail riga itgesseoyo?
Could that really be a woman? or How could that possibly be a woman?
그는 진짜 여자일 리 없어요.
Geuneun jinjja yeojail ri eopseoyo.
There's no way that could be a woman.
고기를 먹지 않고서는 푸딩을 받을 수 없어요.
Gogireul meokji ankoseoneun pudingeul badeul su eopseoyo.
고기를 먹지 않아서는 푸딩을 받을 수 없어요.
Gogireul meokji anaseoneun pudingeul badeul su eopseoyo.
If you don't eat your meat, you can't have any pudding.
Remember 고서? It's on page 280. 고서는 is the intersection of 고서 and (으)면 (see page 283): it's similar to both but more specific than either. It's a good one for moms or teachers: if you ever need to say, "If you don't eat your meat, you can't have any pudding!" in Korean, this is the expression you want.
HOW IT'S FORMED
고 plus 서 plus 는, which is an intensifier here. Alternately, 아/어/여 plus 서 plus 는.
HOW IT'S CONJUGATED
| | Present
---|---|---
Action verbs with 오 or 아 | 잡다 | 잡아서는 jabaseoneun
잡고서는 japgoseoneun
Action verbs with 어, 우, 이 or 으 | 먹다 | 먹어서는 meogeoseoneun
먹고서는 meokgoseoneun
Action verbs ending in a vowel 하다 | 하다 | 해서는 haeseoneun
하고서는 hagoseoneun
Descriptive verbs with 오 or 아 | 짧다 | 짧아서는 jjalbaseoneun
Descriptive verbs with 어, 우, 이 or 으 | 넓다 | 넓어서는 neolbeoseoneun
TAKE NOTE
고 can be used only with action verbs and only when the subjects of both clause A and clause B are the same.
아/어/여서는 is very similar and can be used in most of the same places. However, 고서는 means "if," "while" or "after," and 아/어/여서는 can mean "if" or "after" but not "while."
고서는/아/어/여서는 can be changed to (으)ㄴ 채로 (see page ), though the reverse is not always true.
신발을 신은 채로는 집에 들어올 수 없어요.
Sinbareul sineun chaeroneun jibe deureool su eopseoyo.
신발을 신고서는 집에 들어올 수 없어요.
Sinbareul singoseoneun jibe deureool su eopseoyo.
You can't come in the house while wearing your shoes.
B always ends with something negative: (으)ㄹ 수 없다 (eu)l su eopda is very common, but 지 않다 ji anta or 지 못 하다 ji mot hada or other similar expressions are also possible.
EXAMPLE SENTENCES
신발을 신고서는 집에 들어올 수 없어요.
Sinbareul singoseoneun jibe deureool su eopseoyo.
신발을 신어서는 집에 들어올 수 없어요.
Sinbareul sineoseoneun jibe deureool su eopseoyo.
You can't come in the house while wearing your shoes.
티셔츠를 입고서는 그 클럽에 갈 수 없어요.
Tisyeocheureul ipgoseoneun geu keulleobe gal su eopseoyo.
티셔츠를 입어서는 그 클럽에 갈 수 없어요.
Tisyeocheureul ibeoseoneun geu keulleobe gal su eopseoyo.
You can't go to that club wearing a T-shirt.
(In both the above examples, 고서는 sounds much better).
아이들이 그렇게 작아서는 이 놀이터에서 놀면 안 돼요.
Aideuri geureoke jagaseoneun i noriteoeseo nolmyeon an dwaeyo.
Children that small aren't allowed to play on this playground.
(As this sentence uses an adjective to end A, it can't be written with 고서는 goseoneun).
시간을 이렇게 많이 낭비해서야 어떻게 정시에 오겠어요?
Siganeul ireoke mani nangbihaeseoya eotteoke jeongsie ogesseoyo?
If you waste this much time, how are you going to get there on time?
What this one essentially means is that doing A makes B difficult or impossible. If you had a Korean mom, she might have used this expression to tell you that if you didn't do your homework you'd get a bad grade, or to ask you how you think you're going to get into college if you never studied.
HOW IT'S FORMED
A아/어/여서 B A a/eo/yeoseo B is a causative expression; if A, so then B. In 아/어/여서야 as well, there will be a causative relationship (usually negative) between A and B. 야 has to do with obligation and is featured in such expressions as 아/어/여야 하다/되다 a/eo/yeoya hada/doeda (see page ) or all by itself as 아/어/여야 a/eo/yeoya (see page ).
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Action verbs with 오 or 아 | 잡다 | 잡아서야 jabaseoya
Action verbs with 어, 우, 으 or 이 | 먹다 | 먹어서야 meogeoseoya
Action verbs ending in a vowel 하다 | 하다 | 해서야 haeseoya
Descriptive verbs (adjectives) with 오 or 아 | 작다 | 작아서야 jagaseoya
Descriptive verbs (adjectives) with 어, 우, 으 or 이 | 넓다 | 넓어서야 neolbeoseoya
TAKE NOTE
This expression is quite often used in questions such as "How can you have any pudding if you can't eat your meat?" In these cases, B often ends with a question such as 어떻게 ~겠어요? eotteoke ~gesseoyo? or 어디 ~겠어요? eodi ~gesseoyo? (In the latter case, 어디 does mean "where," but how: it's just a part of the expression). (으)ㄹ 수 없어요 (eu)r su eopseoyo is also a common ending for sentences with 아/어/여서야.
고서는 goseoneun and 아/어/여서야 a/eo/yeoseoya are very similar and can be used in most of the same places (see page 301 for more on this.) However, 고서는 means "if," "while," or "after," and 아/어/여서는 can mean "if" or "after" but not "while."
EXAMPLE SENTENCES
고기를 먹지 않아서야 어떻게 푸딩을 받을 수 있겠어요?
Gogireul meokji anaseoya eotteoke pudingeul badeul su itgesseoyo?
How can you have any pudding if you don't eat your meat?
이렇게 늦게 자서야 오늘 밤에 잘 수 없을 거예요.
Ireoke neutge jaseoya oneul bame jal su eopseul geoyeyo.
If you sleep this late, you won't be able to sleep tonight.
##
Talking about What Others Have Said: Review of Reported Speech
This is something you absolutely need to know if you're planning to go any further than beginning Korean. Reported speech particles are used not just for indirect speech, but in an array of expressions, especially in situations where you are talking hypothetically or about an assumption you are or someone else is making. You can also use them when talking about things "people say" or "they say" ("They say it's going to be hot tomorrow") even if no one around at the time has said anything of the kind.
Reported speech particles look very intimidating at first and it took me a long time to get used to them. Korean grammar is normally relatively gentle compared to many other languages, but this particular case is a bit of an exception. With practice, all this will come naturally.
Let's take a look at the particles themselves first:
Particle | Example | When it's used | Example
---|---|---|---
라 ra | 남자라 namjara | Nouns ending in vowels | 벌써 8시라고 했어요. Beolsseo yeodeolsirago haesseoyo. He said it's already 8:00.
이라 ira | 물이라 murira | Nouns ending in consonants | 지각이라고 했어요. Jigagirago haesseoyo He called it tardiness.
ㄴ다 nda | 한다 handa | Action verbs ending in vowels | 늦게 온다고 했어요. Neutge ondago haesseoyo. He said we came late.
는다 neunda | 먹는다 meokneunda | Action verbs ending in consonants | 늦는다고 했어요. Neunneundago haesseoyo. He said we were late.
다 da | 예쁘다 yeppeuda | Descriptive verbs (adjectives), past tense verbs with 았/었/였, 있다/없다 | 시간이 벌써 늦다고 했어요. Sigani beolsseo neutdago haesseoyo. He said it was already late.
자 ja | 하자 haja | Suggestions | 친구가 정시에 오자고 했어요. Chinguga jeongsie ojago haesseoyo. My friend suggested we come on time.
냐 nya | 예쁘냐 yeppeunya | Questions using descriptive verbs or nouns ending in vowels to someone of lower status than you | 벌써 8시냐고 했어요. Beolsseo yeodeolsinyago haesseoyo. He asked if it was already 8:00.
이냐 inya | 물이냐 murinya | Questions using nouns ending in consonants to someone of lower status than you | 지각이냐고 했어요. Jigaginyago haesseoyo. He asked if it was tardiness.
으냐 eunya | 작으냐 jageunya | Questions using descriptive verbs ending in consonants to someone of lower status than you | 시간이 늦냐고 했어요. Sigani neunnyago haesseoyo. He asked if it was late.
느냐 neunya | 하느냐 haneunya 먹었느냐 meogeonneunya | Questions using active and/or past tense verbs to someone of lower status than you | 우리가 늦느냐고 했어요. Uriga neunneunyago haesseoyo. He asked if we were late.
라 ra | 하라 hara | Commands ending in vowels | 선생님이 정시에 오라고 했어요. Seonsaengnimi jeongsie orago haesseoyo. The teacher told us to come on time.
으라 eura | 먹으라 meogeura | Commands ending in consonants | 선생님이 우리에게 늦게 오라고 했어요. Seonsaengnimi uriege neutge orago haesseoyo. The teacher told us to come late.
어 달라다 eo dallada | 먹어 달라다 meogeo dallada | Questions/commands to someone of higher status than you: 어, 우, 이 or 으 | 선생님께 늦으면 돼 달라고 했어요. Seonsaengnimkke neujeumyeon dwae dallago haesseoyo. I asked the teacher if it was okay to be late.
아 달라다 a dallada | 잡아 달라다 jaba dallada | Questions/commands to someone of higher status than you: 아 or 오 | 선생님께 정시에 와 달라고 했어요. Seonsaengnimkke jeongsie wa dallago haesseoyo. I asked the teacher to please come on time.
여 달라다 yeo dallada | 해 달라다 hae dallada | Questions/commands to someone of higher status than you: 하다 | 선생님께 좀 지각해 달라고 했어요. Seonsaengnimkke jom jigakae dallago haesseoyo. I asked the teacher to please come a little late.
For now, we're going to look at three kinds of expressions with these particles.
Adding 고 하다 to any of them, as in the examples above, is the basic way to say that someone else said something.
Because Koreans generally don't bother with pronouns, I've left them out here. Thus any of the translations above that don't mention a specific person could also mean something else. The first example, for instance, could mean that "he," "I," "you," "she," "they" or anyone called it tardiness.
늦다 is both an action verb and an adjective in Korean, as it is in English: to "be late" as in "to arrive late" is an action while to "be late" as in "it's late" is an adjective. That's why I can conjugate 늦다 using both the action verb and adjective forms. If this is confusing, don't worry; you'll have plenty of opportunities to see the different forms in action.
In the past tense, you can conjugate 하다 to 했어요. In the future tense, conjugate (으)ㄹ 것이다 (better known as (으)ㄹ 거예요 (eu)r geoyeyo or (으)ㄹ 거야) (eu)r geoya to (으)ㄹ 거라고 해요 (eu)r georago haeyo (or (으)ㄹ 거냐고 (eu)r geonyago or (으)ㄹ 거자고 (eu)r geojago).
내일 갈 거라고 해요.
Naeil gal georago haeyo.
He says he'll go tomorrow.
어제 갔다고 해요.
Eoje gatdago haeyo.
He says he went yesterday.
Another way to use the expression above is to drop the 하다 and just continue the sentence after 고. This is often done when quoting a proverb. Here are a couple of examples.
고생 끝에 낙이 온다고 나중에 좋은 일이 꼭 생길 거예요.
Gosaeng kkeute nagi ondago najunge joeun iri kkok saenggil geoyeyo.
Just like the proverb "At the end of sorrow, there's joy," something good is sure to happen later.
(고생 끝에 낙이 온다 Gosaeng kkeute nagi onda.: A Korean proverb that means "At the end of sorrow, there's joy.")
윗물이 맑아야 아랫물이 맑다고 윗사람이 질서를 잘 지키면 아랫사람도 잘 지킬 거예요.
Winmuri malgaya araenmuri makdago witsarami jilseoreul jal jikimyeon araetsaramdo jal jikil geoyeyo.
According to the proverb, "If the upper waters are clear, so are the lower waters": if people of higher status follow the rules, so will people of lower status.
(윗물이 맑아야 아랫물이 맑다 Winmuri malgaya araenmuri makda.: A Korean proverb that means "If the upper waters are clear, so are the lower waters.")
The particles plus 고 하다 can be shortened at the end of a sentence, as follows:
Particle | Short form | When it's used | Example
---|---|---|---
라고 해요 rago haeyo | 래요 raeyo | Nouns ending in vowels | 벌써 8시래요. Beolsseo yeodeolsiraeyo. He said it's already 8:00.
이라고 해요 irago haeyo | 이래요 iraeyo | Nouns ending in consonants | 지각이래요. Jigagiraeyo. He called it tardiness.
ㄴ다고 해요 ndago haeyo | ㄴ대요 ndaeyo | Action verbs ending in vowels | 늦게 온대요. Neutge ondaeyo. He said we came late.
는다고 해요 neundago haeyo | 는대요 neundaeyo | Action verbs ending in consonants | 늦는대요. Neunneundaeyo. He said we were late.
다고 해요 dago haeyo | 대요 daeyo | Descriptive verbs (adjectives), past tense verbs with 았/었/였, 있다/없다 | 시간이 벌써 늦었대요. Sigani beolsseo neujeotdaeyo. He said it was already late.
자고 해요 jago haeyo | 재요 jaeyo | Suggestions | 친구가 정시에 오재요. Chinguga jeongsie ojaeyo. My friend suggested we come on time.
냐고 해요 nyago haeyo | 냬요 nyaeyo | Questions using descriptive verbs or nouns ending in vowels to someone of lower status than you | 벌써 8시냬요. Beolsseo yeodeolsinyaeyo.He asked if it was already 8:00.
이냐고 해요 inyago haeyo | 이냬요 inyaeyo | Questions using nouns ending in consonants to someone of lower status than you | 지각이냬요. jigaginyaeyo He asked if it was tardiness.
으냐고 해요 eunyago haeyo | 으냬요 eunyaeyo | Questions using descriptive verbs ending in consonants to someone of lower status than you | 시간이 늦으냬요. sigani neujeunyaeyo He asked if it was late.
느냐고 해요 neunyago haeyo | 늦냬요 neunnyaeyo | Questions using active and/or past tense verbs to someone of lower status than you | 우리가 늦느냬요. Uriga neunneunyaeyo.He asked if we were late.
라고 해요 rago haeyo | 래요 raeyo | Commands ending in vowels | 선생님이 정시에 오래요. Seonsaengnimi jeongsie oraeyo. The teacher told us to come on time.
으라고 해요 eurago haeyo | 으래요 euraeyo | Commands ending in consonants | 선생님이 우리에게 늦으래요. Seonsaengnimi uriege neujeuraeyo. The teacher told us to come late.
어 달라다고 해요 eo dalladago haeyo | 어 달래요 eo dallaeyo | Questions to someone of higher status than you: 어, 우, 이 or 으 | 지호가 선생님께 늦으면 돼 달래요. Jihoga seonsaengnimkke neujeumyeon dwae dallaeyo. Jiho asked the teacher if it was okay to be late.
아 달라다 a dallada | 아 달래요 a dallaeyo | Questions to someone of higher status than you: 아 or 오 | 지호가 선생님께 정시에 와 달래요. Jihoga seonsaengnimkke jeongsie wa dallaeyo. Jiho asked the teacher to please come on time.
여 달라다 yeo dallada | 여 달래요 yeo dallaeyo | Questions to someone of higher status than you: 하다 | 지호가 선생님께 좀 지각해 달래요. Jihoga seonsaengnimkke jom jigakae dallaeyo. Jiho asked the teacher to please come a little late.
Note that 래요 raeyo, 대요 daeyo, 재요 jaeyo, and the rest remain the same regardless of tense. Just like in English, it will normally be in the past tense unless you're talking about something someone always says.
Finally, you can conjugate the expressions using 는 if you want to construct a sentence such as "That crazy little thing called love" or "The thing the teacher said."
Particle | With 는 | When it's used | Example
---|---|---|---
라고 해요 rago haeyo | 라는 raneun | Nouns ending in vowels | 수라는 남자 Suraneun namja a boy called Sue
이라고 해요 irago haeyo | 이라는 iraneun | Nouns ending in consonants | 사랑이라는 것 sarangiraneun geot a thing called love
ㄴ다고 해요 ndago haeyo | ㄴ다는 ndaneun | Action verbs ending in vowels | 간다는 남자 gandaneun namja a man who is said to be going
는다고 해요 neundago haeyo | 는다는 neundaneun | Action verbs ending in consonants | 먹는다는 음식 meokneundaneun eumsik a food that is said to be eaten
다고 해요 dago haeyo | 다는 daneun | Descriptive verbs (adjectives), past tense verbs with 았/었/였, 있다/없다 | 좋다는 결과 jotaneun gyeolgwa a result that is called good
자고 해요 jago haeyo | 자는 janeun | Suggestions | 친구가 하자는 운동 chinguga hajaneun undong an exercise my friend suggested
I've left out the rest of the table because you'll just about never hear 냐는 nyaneun, 라는 (as an order) or 달라는. If you really need to know, you can figure out the pattern from the examples above.
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Talking about What Others Have Said: Combinations
Beyond the basic 라고 하다 rago hada, 라는 raneun, etc., you can also combine indirect speech with a number of other grammatical patterns to make your sentences more complex and more precise. These are often contractions of (다, 자, 라, etc.) 고 하다 plus whatever you're adding to it: for instance, 라고 하다 plus 기에 becomes 라기에 ragie, 는다고 하다 neundago hada plus 기에 becomes 는다기에 neundagie, etc. In cases where this is done, I've written the contracted expressions; in cases where it's not normally done, I haven't. Even if an expression is normally contracted, you can still use it in its original form. 라고 하기에 rago hagie isn't wrong. However, if you go around doing this on a regular basis, you might well end up sounding as awkward as you would if you never used contractions in English.
If you're unsure about the differences between the different expressions (e.g. 던데 deonde vs. 는데 neunde, 어서 eoseo vs. 니까 nikka) then you can look them up in their respective sections – 던데 is on page , 는데 on page , 어서 on page and 니까 on page . They normally don't change meaning when added to indirect speech particles.
Finally, a few of the expressions below work or are only commonly used only with 라고 and 다고 expressions. In these cases I've cut off the 자고 jago/냐고 nyago/달라고 dallago part of the table.
Here's a closer look at some of the patterns.
Particle | Plus | Equals | Meaning | Example
---|---|---|---|---
(이/으)라고 하다 (i/eu)rago hada | 여요 yeoyo (see page ) | ~(이/으)라고 요 (i/eu)ragoyo | You said ~? | 그것이 진짜라고요? Geugeosi jinjjaragoyo? You said that was true?
(ㄴ/는)다고 하다 (n/neun)dago hada | 여요 (see page ) | ~(ㄴ/는)다고 요 (n/neun)dagoyo | You said ~? | 민수가 늦는다고요? Minsuga neunneundagoyo? You said Minsu is late?
자고 하다 jago hada | 여요 (see page ) | ~자고요 ~jagoyo | You suggested ~? | 민수가 빨리 가자고요? Minsuga ppalli gajagoyo? Minsu suggested we go quickly?
(이/으/느)냐고 하다 (i/eu/neu) nyago hada | 여요 (see page ) | ~(이/으/느) 냐고요 (i/eu/neu) nyagoyo | You asked ~? | 민수가 언제 오냐고요? Minsuga eonje onyagoyo? Minsu asked when to come?.
(아/어/여) 달라고 하다 (a/eo/yeo) dallago hada | 여요 (see page ) | ~아/어/여 달라고요 a/eo/yeo dallagoyo | You asked ~? | 선생님께 6시에 와 달라고요? Seonsaengnimkke yeoseotsie wa dallagoyo? You asked the teacher to come at 6?
When using this in 반말, drop 요 and just use the 라고, 다고, 자고 etc. part of the question.
Particle | Plus | Equals | Meaning | Example
---|---|---|---|---
(이/으)라고 하다 (i/eu)rago hada | 여서 yeoseo (see page ) | A(이/으)라고 해서 B A (i/eu)rago haeseo B | I heard that/some one said A, so B | 그것이 진짜라고 해서 믿었어요. Geugeosi jinjjarago haeseo mideosseoyo. I heard that was true, so I believed it.
(ㄴ/는)다고 하다 (n/neun)dago hada | 여서 (see page ) | A(ㄴ/는)다고 해서 B A (n/neun)dago haeseo B | I heard that/some one said A, so B | 민수가 늦는다고 해서 나도 늦게 왔어요. Minsuga neunneundago haeseo nado neutge wasseoyo. They said Minsu is late, so I came late too.
자고 하다 jago hada | 여서 (see page ) | A자고 해서 B A jago haeseo B | I heard that/some one suggested A, so B | 민수가 빨리 가자고 해서 서둘렀어요. Minsuga ppalli gajago haeseo seodulleosseoyo. Minsu said to go quickly, so I hurried.
(이/으/느)냐 고 하다 (i/eu/neu)nyago hada | 여서 (see page ) | A(이/으/느)냐 고 해서 B A (i/eu/neu)nyago haeseo B | I heard that/some one asked A, so B | 민수가 언제 오느냐고 해서 6시에 오라고 했어요. Minsuga eonje oneunyago haeseo yeoseotsie orago haesseoyo. Minsu asked when to come, so I told him to come at 6.
(아/어/여) 달라고 하다 (a/eo/yeo)dallago hada | 여서 (see page ) | A아/어/여 달라고 해서 B A a/eo/yeo dallago haeseo B | I heard that/ someone asked A, so B | 선생님께 6시에 와 달라고 해서 나도 빨리 준비했어요. Seonsaengnimkke yeoseotsie wa dallago haeseo nado ppalli junbihaesseoyo. I asked the teacher to come at 6, so I had to get ready quickly too.
(이/으)라고 하다 (i/eu)rago hada | 니(까) ni(kka) (see page ) | A(이/으)라 니(까) B A (i/eu)rani(kka) B | I heard that/someo ne said A, so B | 그것이 진짜라니까 이것은 아마 가짜일 거예요. Geugeosi jinjjaranikka igeoseun ama gajjail geoyeyo. I heard that was true (or real), so I think this is fake.
(ㄴ/는)다고 하다 (n/neun)dago hada | 니(까) (see page ) | A(ㄴ/는)다 니(까) B A (n/neun)dani(kka) B | I heard that/someo ne said A, so B | 민수가 늦는다니, 민수한테 전화해 주세요. Minsuga neunneundani, Minsuhante jeonhwahae juseyo. They said Minsu is late, so please call him.
자고 하다 jago hada | 니(까) (see page ) | A자니(까) B A jani(kka) B | I heard that/someo ne suggested A, so B | 민수가 빨리 가자니까 빨리 서두르세요. Minsuga ppalli gajanikka ppalli seodureuseyo. Minsu suggested to go quickly, so please hurry up.
(이/으/느) 냐고 하다 (i/eu/neu)nyag o hada | 니(까) (see page ) | A(이/으/느) 냐니(까) B A (i/eu/neu)nyani(kka) B | I heard that/someo ne asked A, so B | 민수한테 언제 오냐니까 안 온다고 했어요. Minsuhante eonje onyanikka an ondago haesseoyo. I asked Minsu when he was coming, and he said he would not come.
(아/어/여) 달라고 하다 (a/eo/yeo)dalla go hada | 니(까) (see page ) | A아/어/여 달라니(까) B A a/eo/yeo dallani(kka) B | I heard that/someo ne asked A, so B | 선생님이 6시에 와 달라니까 빨리 준비하세요. Seonsaengnimi yeoseotsie wa dallanikka ppalli junbihaseyo. The teacher asked you to come at 6, so please get ready quickly.
(ㄴ/는)다니 (n/neun)dani is a particularly common expression. Don't confuse it with 더니 deoni (see page ), which has to do with personal experience and is totally different.
You can also use ~(이)라니까(요) ~(i)ranikka(yo) and ~(ㄴ/는)다니까(요) ~(n/neun)danikka(yo) at the end of a sentence; in this case, use it when you want to repeat yourself in order to emphasize what you already said. It's the Korean equivalent of "I said, ~!"
Particle | Plus | Equals | Meaning | Example
---|---|---|---|---
(이/으)라고 하다 (i/eu)rago hada | 는데 neunde (see page ) | A(이/으)라고 하는데 B A (i/eu)rago haneunde B | I heard that/so meone said A; B | 그것이 진짜라고 하는데 믿었어요. Geugeosi jinjjarago haneunde mideosseoyo. I heard that was true, so I believed it.
(ㄴ/는)다고 하다 (n/neun)dago hada | 는데 (see page ) | A(ㄴ/는)다고 하는데/한데 B A (n/neun)dago haneunde/hande B | I heard that/ someone said A; B | 민수가 늦는다고 하는데 전화해 주세요. Minsuga neunneundago haneunde jeonhwahaejuseyo. They said Minsu is late; please call him.
자고 하다 jago hada | 는데 neunde (see page ) | A자고 하는데B A jago haneunde B | I heard that/so meone suggest ed A; B | 민수가 빨리 가자고 하는데 서둘러요. Minsuga ppalli gajago haneunde seodulleoyo. Minsu suggested we go quickly; let's hurry.
(이/으/느)냐 고 하다 (i/eu/neu)nyago hada | 는데 (see page ) | A(이/으/느)냐고 하는데 B A (i/eu/neu)nyago haneunde B | I heard that/so meone asked A; B | 민수가 언제 오냐고 하는데 저도 몰라서 아직 대답 할 수 없어요. Minsuga eonje onyago haneunde jeodo mollaseo ajik daedapal su eopseoyo. Minsu asked when to come; I don't know either, so I can't answer him yet.
(아/어/여) 달라고 하다 (a/eo/yeo)dallag o hada | 는데 (see page ) | A아/어/여 달라고 하는데 B A a/eo/yeo dallago haneunde B | I heard that/so meone asked A; B | 선생님께서 6시에 와 달라고 하는데 빨리 준비할래요. Seonsaengnimkkeseo yeoseotsie wa dallago haneunde ppalli junbihallaeyo. The teacher asked us to come at 6; let's get ready quickly.
Particle | Plus | Equals | Meaning | Example
---|---|---|---|---
(이/으)라고 하다 (i/eu)rago hada | 여도 yeodo (see page ) | A(이/으)라고 해도 B A (i/eu)rago haedo B | B even though someone says A | 그것이 진짜라고 해도 정시에 와야 돼요. Geugeosi jinjjarago haedo jeongsie waya dwaeyo. Even if (you say) that's true, you still have to come on time.
(ㄴ/는)다고 하다 (n/neun)dago hada | 여도 (see page ) | A(ㄴ/는)다고 해도 B A (n/neun)dago haedo B | B even though someone says A | 바쁘다고 해도 정시에 와야 돼요. Bappeudago haedo jeongsie waya dwaeyo. Even if (you say) you're busy, you still have to come on time.
자고 하다 jago hada | 여도 (see page ) | A자고 해도 B A jago haedo B | B even though someone suggests A | 민수가 빨리 가자고 해도 그 일을 끝내 주세요. Minsuga ppalli gajago haedo geu ireul kkeunnaejuseyo. Even if Minsu suggested you leave quickly, please finish that job.
(이/으/느) 냐고 하다 (i/eu/neu)nyag o hada | 여도 (see page ) | A(이/으/느)냐 고 해도 B A (i/eu/neu)nyago haedo B | B even though someone asks A | 민수가 선물이 무엇이냐고 해도 알려 주지 마세요. Minsuga seonmuri mueosinyago haedo allyeojuji maseyo. Even if Minsu asks what the present is, please don't tell him.
(아/어/여) 달라고 하다 (a/eo/yeo)dallago hada | 여도 (see page ) | A아/어/여 달라고 해도 B A a/eo/yeo dallago haedo B | B even though someone asks A | 선생님께서 6시에 와 달라고 해도 우리는 7시 까지 가요. Seonsaengnimkkeseo yeoseotsie wa dallago haedo urineun ilgopsi kkaji gayo. Even if teacher asked us to come at 6, let's go there by 7.
(이/으)라고 하다 (i/eu)rago hada | 던데 deonde (see page ) | A(이)라던데 B A (i)radeonde B | I heard that someone said A; B | 그것이 진짜라던데 진짜예요? Geugeosi jinjjaradeonde jinjjayeyo? I heard that was true; is it?
(ㄴ/는)다고 하다 (n/neun)dago hada | 던데 (see page ) | A(ㄴ/는)다던 데 B A (n/neun)dadeonde B | I heard that someone said A; B | 민수가 늦는다고 하던데 전화해 주세요. Minsuga neunneundago hadeonde jeonhwahaejuseyo. They said Minsu is late; please call him.
The expressions above are normally used in two situations. A reports a fact that you heard. B can either be a recommendation or suggestion based on that fact (as in the example for 라던데 radeonde) or a request for confirmation (진짜예요? Jinjjayeyo? 사실이에요? Sasirieyo? 정말이에요? Jeongmarieyo?) as in the example for 다던데 dadeonde.
Particle | Plus | Equals | Meaning | Example
---|---|---|---|---
(이/으)라고 하다 (i/eu)rago hada | 면 myeon (see page ) | A(이/으)라고 하면 B A (i/eu)rago hamyeon B | If (someone says) A, then B | 그것이 진짜라고 하면 저도 믿을 거예요. Geugeosi jinjjarago hamyeon jeodo mideul geoyeyo. If you say that's true, then I'll believe it.
(ㄴ/는)다고 하다 (n/neun)dago hada | 면 (see page ) | A(ㄴ/는)다고 하면 B A (n/neun)dago hamyeon B | If (someone says) A, then B | 민수가 늦었다고 하면 민수가 벌금을 내야 돼요. Minsuga neujeotdago hamyeon Minsuga beolgeumeul naeya dwaeyo. If they say Minsu was late, he'll have to pay a fine.
자고 하다 jago hada | 면 (see page ) | A자고 하면 B A jago hamyeon B | If (someone suggests) A, then B | 민수가 빨리 가자고 하면 서둘러요. Minsuga ppalli gajago hamyeon seodulleoyo. If Minsu suggested we go quickly, then let's hurry.
(이/으/느)냐 고 하다 (i/eu/neu)nyago hada | 면 (see page ) | A(이/으/느) 냐고 하면 B A (i/eu/neu)nyago hamyeon B | If (someone asks) A, then B | 민수가 언제 와야 하냐고 하면 그냥 오지 말라고 하세요. Minsuga eonje waya hanyago hamyeon geunyang oji mallago haseyo. If Minsu asks when he needs to come, please tell him not to come.
(아/어/여) 달라고 하다 (a/eo/yeo)dallago hada | 면 (see page ) | A아/어/여 달라고 하면 B A a/eo/yeodallago hamyeon B | If (someone asks) A, then B | 선생님께서 6시에 와 달라고 하면 우리 빨리 서둘러요. Seonsaengnimkkeseo yeoseotsie wa dallago hamyeon uri ppalli seodulleoyo. If the teacher asks us to come at 6, then let's get ready quickly.
(이/으)라고 하다 (i/eu)rago hada | ㄹ 수 있다 r su itda (see page ) | ~(이)라고 할 수 있다 ~(i)rago hal su itda | It can be said that ~ | 그것은 진짜라고 할 수 있어요. Geugeoseun jinjjarago hal su isseoyo. That can be said to be true.
(ㄴ/는)다고 하다 (n/neun)dago hada | ㄹ 수 있다 (see page ) | ~(ㄴ/는)다고 할 수 있다 ~(n/neun)dago hal su itda | It can be said that ~ | 그것은 너무 어렵다고 할 수 있어요. Geugeoseun neomu eoryeopdago hal su isseoyo. That can be said to be too difficult.
You can also use 할 수 없다 hal su eopda to express that something can't be said or 할 수도 있다 hal sudo itda to say that something else can be said.
그것이 진짜라고 할 수 없어요.
Geugeosi jinjjarago hal su eopseoyo.
That can't be said to be true.
A: 그것이 진짜라고 할 수 있어요. Geugeosi jinjjarago hal su isseoyo.
That can be said to be true.
B: 이것이 진짜라고 할 수도 있어요. Igeosi jinjjarago hal sudo isseoyo.
This can also be said to be true.
Particle | Plus | Equals | Meaning | Example
---|---|---|---|---
(이/으)라 고 하다 (i/eu)rago hada | 기에 gie (see page ) | A(이/으)라 기에 B A (i/eu)ragie B | Since I heard that A, (I did) B | 친구가 그것은 진짜라기에 믿었어요. Chinguga geugeoseun jinjjaragie mideosseoyo. Since my friend said that was true, I believed it.
(ㄴ/는)다 고 하다 (n/neun)dago hada | 기에 (see page ) | A(ㄴ/는)다 기에 B A (n/neun)dagie B | Since I heard that A, (I did) B | 민수가 늦는다기에 민수를 기다렸어요. Minsuga neunneundagie minsureul gidaryeosseoyo. Since I heard Minsu was late, I waited for him.
자고 하다 jago hada | 기에 (see page ) | A자기에 B A jagie B | Since someone suggested A, (I did) B | 민수가 빨리 가자기에 서둘렀어요. Minsuga ppalli gajagie seodulleosseoyo. Since Minsu said to go quickly, I hurried.
(이/으/느) 냐고 하다 (i/eu/neu)nyag o hada | 기에 (see page ) | A(이/으/ 느)냐기에 B A (i/eu/neu)nyag ie B | Since someone asked A, (I did) B | 민수가 언제 오냐기에 가영한테 전화해서 물어봤어요. Minsuga eonje onyagie Kayounghante jeonhwahaeseo mureobwasseoyo. Since Minsu asked when to come, I called Kayoung and asked her.
(아/어/여) 달라고 하다 (a/eo/yeo)dallago hada | 기에 (see page ) | A아/어/여 달라기에 B A a/eo/yeodallagie B | Since someone asked A, (I did) B | 선생님께서 6시에 와 달라기에 빨리 준비 해야 돼요. Seonsaengnimkkeseo yeoseotsie wa dallagie ppalli junbi haeya dwaeyo. Since I asked the teacher to come at 6, we need to get ready quickly.
(이/으)라고 하다 (i/eu)rago hada | 더라 deora (see page ) | ~(이/으)라더라 고요 ~(i/eu)radeoragoyo | I heard that ~ | 그것은 진짜라더라고요. Geugeoseun jinjjaradeoragoyo. I heard that was true.
(ㄴ/는)다고 하다 (n/neun)dago hada | 더라 (see page ) | ~(ㄴ/는)다더라 고요 ~(n/neun)dadeoragoyo | I heard that ~ | 민수가 늦었다더라고요. Minsuga neujeotdadeoragoyo. I heard Minsu was late.
자고 하다 jago hada | 더라 deora (see page ) | ~자더라고요 ~jadeoragoyo | I heard that ~ | 민수가 빨리 가자더라고요. Minsuga ppalli gajadeoragoyo. I heard that Minsu suggested we go quickly.
(이/으/느)냐 고 하다 (i/eu/neu)nyago hada | 더라 (see page ) | ~(이/으/느)냐더 라고요 ~(i/eu/neu)nyadeorago yo | I heard that ~ | 민수가 언제 오냐더라고요. Minsuga eonje onyadeoragoyo. I heard that Minsu asked when to come.
(아/어/여) 달라고 하다 (a/eo/yeo)dallago hada | 더라 (see page ) | ~아/어/여 달라더라고요 ~a/eo/yeo dalladeoragoyo | I heard that ~ | 선생님께서 6시에 와 달라더라고요. Seon-saeng-nim-kkeseo yeoseotsie wa dal-ladeoragoyo. I heard that the teacher asked you to come at six.
Note that the difference between 라더라고요 radeoragoyo (라더라 radeora in 반말) and the regular 라고 하다 rago hada is that when you add 더라고요 deoragoyo, you're specifically talking about your own personal experiences: a rumor or something that you heard. If you personally saw Minsu come in late, you'd use 늦었더라고요 neujeotdeoragoyo, but if Kayoung told you that Minsu came in late, you could say 늦었다더라고요 neujeotdadeoragoyo: you heard that Minsu came late even though you didn't personally see him enter.
Particle | Plus | Equals | Meaning | Example
---|---|---|---|---
(이/으)라고 하다 (i/eu)rago hada | 지요 jiyo (see page 315) | ~(이/으)라지 요 (i/eu)rajiyo | I heard ~; is it true? | 그것이 진짜라지요? Geugeosi jinjjarajiyo? I heard that was true; it is, isn't it?
(ㄴ/는)다고 하다 (n/neun)dago hada | 지요 (see page 315) | ~(ㄴ/는)다지 요 (n/neun)dajiyo | I heard ~; is it true? | 민수가 늦었다지요? Minsuga neujeotdajiyo? I heard Minsu was late; that's true, right?
자고 하다 jago hada | 지요 (see page 315) | ~자고 하지요 jago hajiyo | I heard ~; is it true? | 민수가 빨리 가자고 했지요? Minsuga ppalli gajago haetjiyo? Minsu suggested we go quickly, didn't he?
(이/으/느)냐 고 하다 (i/eu/neu)nyago hada | 지요 (see page 315) | ~(이/으/느)냐 고 하지요 (i/eu/neu)nyago hajiyo | I heard ~; is it true? | 민수가 언제 오냐고 했지요? Minsuga eonje onyago haetjiyo? Minsu asked when to come, didn't he?
(아/어/여) 달라고 하다 (a/eo/yeo)dallago hada | 지요 (see page 315) | ~아/어/여 달라고 하지요 ~a/eo/yeo dallago hajiyo | I heard ~; is it true? | 선생님께서 6시에 와 달라고 했지요? Seonsaengnimkkeseo yeoseotsie wa dallago haetjiyo? The teacher asked us to come at 6, didn't he?
Particle | Plus | Equals | Meaning | Example
---|---|---|---|---
(이/으)라고 하다 (i/eu)rago hada | 들 deul (see page ) | ~(이)라고들 하다 ~(i)ragodeul hada | They say that ~ | 그것이 진짜라고들 해요. Geugeosi jinjjaragodeul haeyo. They say that's true.
(ㄴ/는)다고 하다 (n/neun)dago hada | 들 (see page ) | ~(ㄴ/는)다고 들 하다 ~(n/neun)dagodeul hada | They say that ~ | 야채가 건강에 좋다고들 해요. Yachaega geongange jotagodeul haeyo. They say vegetables are good for your health.
##
Talking about What Others Have Said: Spreading Rumors
Let's say you heard something from someone and want to confirm it. Here are some ways to do so.
어제 늦었지요?
Eoje neujeotjiyo?
You were late yesterday, weren't you?
This ending has many uses. One of the most commonly used is to confirm information you expect the other person is also aware of, that is to check your facts. I liken it to the stereotypical Canadian use of "eh?" While speaking, it can be, and usually is, contracted to "죠."
You can also use it to offer to do something. When used in this way, it's not very forceful, much like when you offer to do something in English.
Finally, it can be used to make suggestions. When using it toward people of higher status than you, it translates as "Would you like to ~." If you use 지 alone toward people of lower status than you, it's a little more forceful and suggests they should really do ~. You'll hear this used all the time if you work in an elementary school or kindergarten. Below are examples of both uses:
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했지요? haetjiyo? | 하지요? hajiyo? | 하겠지요? hagetjiyo?
Action verbs ending in a consonant | 먹다 | 먹었지요? meogeotjiyo? | 먹지요? meokjiyo? | 먹겠지요? meokgetjiyo?
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤지요? yeppeotjiyo? | 예쁘지요? yeppeujiyo? | 예쁘겠지요? yeppeugetjiyo?
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았지요? jagatjiyo? | 작지요? jakjiyo? | 작겠지요? jakgetjiyo?
Nouns ending in a vowel | 남자 | 남자였지요? namjayeotjiyo? | 남자지요? namjajiyo? | 남자겠지요? namjagetjiyo?
Nouns ending in a consonant | 물 | 물이었지요? murieotjiyo? | 물이지요? murijiyo? | 물이겠지요? murigetjiyo?
TAKE NOTE
지 ji and 지요 jiyo have many uses, and it's important not to get them confused. This 지요 always goes at the end of a sentence, never before anything else, which distinguishes it from 지 말다 ji malda, 지 않다 ji anta and other negatives with 지.
As for telling 지요 (I'll) apart from 지요 (you should) and 지요 (isn't it?), that's mainly a matter of context.
EXAMPLE SENTENCES
오늘이 생일이지요?
Oneuri saengirijiyo?
Today's your birthday, isn't it?
오늘이 생일이죠?
Oneuri saengirijyo?
Today's your birthday, isn't it?
The two sentences above are the same, but the second form with 죠 is more common in spoken Korean.
축구를 좋아하죠?
Chukgureul joahajyo?
You like soccer, don't you?
감기 걸렸죠?
Gamgi geollyeotjyo?
You caught a cold, didn't you?
늦겠어요. 지금 출발하시지요.
Neutgesseoyo. Jigeum chulbalhasijiyo.
You'll be late. Why don't you go now?
Here's 지요, used to offer to do things:
제가 저녁을 사지요.
Jega jeonyeogeul sajiyo.
I'll buy dinner.
내일 보지요. Naeil bojiyo. I'll see you tomorrow.
제가 먼저 가지요.
Jega meonjeo gajiyo.
I'll go first.
And here it is to suggest things to people:
같이 가시지요.
Gachi gasijiyo.
Would you like to go together?
같이 가지.
Gachi gaji.
We really should go together.
비가 오는데 우산을 가지고 오시지요.
Biga oneunde usaneul gajigo osijiyo.
You should bring an umbrella since it's raining.
이 떡을 먹어 보시지요.
I tteogeul meogeo bosijiyo.
Would you like to try this rice cake?
얘들아! 떠들지 말지.
Yaedeura! Tteodeulji malji.
Children! You shouldn't chatter.
숙제를 꼼꼼하게 하지.
Sukjereul kkomkkomhage haji.
You should have done your homework carefully.
어제 늦었다면서요?
Eoje neujeotda-myeon-seo-yo?
I heard you were late yesterday. Is it true?
This is really an indirect speech combination (similar to the rest of pages 304 to 315), but it goes in this section because it's not versatile like the others and its meaning can't be deduced from just looking at it. It's really an expression unto itself. It's used to check your facts and confirm whether something you heard is true or fictitious.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했다면서요 haetdamyeonseoyo | 한다면서요 handamyeonseoyo | 하겠다면서요 hagetdamyeonseoyo
Action verbs ending in a consonant | 먹다 | 먹었다면서요 meogeotdamyeonseoyo | 먹는다면서요 meokneundamyeonseoyo | 먹겠다면서요 meokgetdamyeonseoyo
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤다면서요 yeppeotdamyeonseoyo | 예쁘다면서요 yeppeudamyeonseoyo | 예쁘겠다면서요 yeppeugetdamyeonseoyo
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았다면서요 jagatdamyeonseoyo | 작다면서요 jakdamyeonseoyo | 작겠다면서요 jakgetdamyeonseoyo
Nouns ending in a vowel | 남자 | 남자라면서요 namjaramyeonseoyo | 남자라면서요 | 남자라면서요
Nouns ending in a consonant | 물 | 물이라면서요 muriramyeonseoyo | 물이라면서요 | 물이라면서요
EXAMPLE SENTENCES
오늘이 생일이라면서요?
Oneuri saengiriramyeonseoyo?
I heard it's your birthday. Is it true?
축구를 좋아한다면서요?
Chukgureul joahandamyeonseoyo?
I heard you like soccer. Is it true?
감기가 걸렸다면서요?
Gamgiga geollyeotdamyeonseoyo?
I heard you caught a cold. Is it true?
어제 늦었다는 말이에요?
Eoje neujeotdaneun marieyo?
Are you saying you were late yesterday?
This ending is used to confirm, refute, or emphasize what someone said.
HOW IT'S FORMED
Indirect speech particles are used since you're talking about something you heard. They're conjugated with 는 (see page ) which turns that part of the sentence into an adjective which describes 말, or "word." 이다 just means "it is."
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했다는 말이다 haetdaneun marida | 한다는 말이다 handaneun marida | 하겠다는 말이다 hagetdaneun marida
Action verbs ending in a consonant | 먹다 | 먹었다는 말이다 meogeotdaneun marida | 먹는다는 말이다 meokneundaneun marida | 먹겠다는 말이다 meokgetdaneun marida
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤다는 말이다 yeppeotdaneun marida | 예쁘다는 말이다 yeppeudaneun marida | 예쁘겠다는 말이다 yeppeugetdaneun marida
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았다는 말이다 jagatdaneun marida | 작다는 말이다 jakdaneun marida | 작겠다는 말이다 jakgetdaneun marida
Nouns ending in a vowel | 남자 | 남자라는 말이다 namjaraneun marida | 남자라는 말이다 | 남자라는 말이다
Nouns ending in a consonant | 물 | 물이라는 말이다 muriraneun marida | 물이라는 말이다 | 물이라는 말이다
EXAMPLE SENTENCES
오늘이 생일이라는 말이에요?
Oneuri saengiriraneun marieyo?
Are you saying it's your birthday today?
축구를 좋아한다는 말이에요? 축구 잘 못한다고 하던데요.
Chukgureul joahandaneun marieyo? Chukgu jal motandago hadeondeyo.
Are you saying you like soccer? I heard you're not very good at it.
감기가 걸렸다는 말이에요? 거짓말인것 같은데요.
Gamgiga geollyeotdaneun marieyo? Geojin-marin-geot gateundeyo.
Are you saying you caught a cold? That doesn't seem true.
어제 늦었다니요?
Eoje neujeotdaniyo?
What's this about being late yesterday?
This expression indicates surprise at something you just heard. If someone asks you about that class you're supposed to be teaching when you hadn't heard anything about any class, this is the expression for you. It's yet another way of repeating something that was said, but one that adds a touch of surprise or denial. Use this when someone is telling you something they've heard which you don't think is true; it's also useful in situations where you've been wrongly accused.
HOW IT'S CONJUGATED
A wonderful thing about this expression is that you don't have to worry as much about remembering which of the indirect speech verb endings to use with it because you just use 다니요 with all verbs, whether they're active or descriptive.
| | Past | Present
---|---|---|---
Action verbs ending in a vowel | 하다 | 했다니요 haetdaniyo | 하다니요 hadaniyo
Action verbs ending in a consonant | 먹다 | 먹었다니요 meogeotdaniyo | 먹다니요 meokdaniyo
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤다니요 yeppeotdaniyo | 예쁘다니요 yeppeudaniyo
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았다니요 jagatdaniyo | 작다니요 jakdaniyo
Nouns ending in a vowel | 남자 | 남자라니요 namjaraniyo | 남자라니요
Nouns ending in a consonant | 물 | 물이라니요 muriraniyo | 물이라니요
EXAMPLE SENTENCES
오늘이 생일이라니요?
Oneuri saengiriraniyo?
Birthday? What's this about a birthday?
축구를 좋아하다니요?
Chukgureul joahadaniyo?
What do you mean I like soccer?
감기 걸렸다니요?
Gamgi geollyeotdaniyo?
Cold? What cold?
##
Talking about What Others Have Said: According to
늦게 오더라도 교통법규대로 운전하세요.
Neutge odeorado gyotongbeopgyudaero unjeonhaseyo.
Even if you're late, please obey the traffic rules.
Here's another way to say "according to."
HOW IT'S CONJUGATED
A can be a noun, in which case it's directly followed by 대로 with no space in between. It can also be a present tense action verb conjugated with 는 and followed by a space and then 대로. Descriptive verbs can't be used with 대로, though you can change them to action verbs using 아/어/여지다; see page for how to do that.
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 한 대로 han daero | 하는 대로 haneun daero | 할 대로 hal daero
Action verbs ending in a consonant | 먹다 | 먹은 대로 meogeun daero | 먹은 대로 | 먹을 대로 meogeul daero
Nouns ending in a vowel | 예보 | 예보대로 yebo daero | 예보대로 | 예보대로
Nouns ending in a consonant | 마음 | 마음대로 maeumdaero | 마음대로 | 마음대로
TAKE NOTE
대로 daero can't be used with negative expressions. It can also never be used with adjectives except for 고 싶다 go sipda, 좋다 jota and 편하다 pyeonhada.
A는 대로 B also means that B immediately follows A; see page . There's really no way to tell the two apart except for context, so be careful.
A few commonly used words with 대로:
마음대로
maeumdaero
as you wish
예상대로
yesangdaero
as expected
시키는 대로
sikineun daero
as ordered
얘기한 대로
yaegihan daero
as said
되는 대로
doeneun daero
as it happens (on its own, with no interference)
제멋대로
jemeotdaero
as one likes
EXAMPLE SENTENCES
마음대로 했어요.
Maeumdaero haesseoyo.
I did as I wished.
좋을 대로 하세요.
Joeul daero haseyo.
Please do whatever is best.
그 책대로 연습했어요.
Geu chaekdaero yeonseupaesseoyo.
I practiced according to what that book recommended.
민수에 따르면 가영이가 늦을 거예요.
Minsue ttareumyeon Kayoungiga neujeul geoyeyo.
According to Minsu, Kayoung will be late.
It's good to have opinions (see page ), but it's even better to be able to back up your opinions with facts. Here's a good way to do that in Korean. It's very formal, which is great for those occasions when you want to sound good.
HOW IT'S FORMED
따르다 means "to follow" and can be conjugated as either 따라 or 따르면 to say "according to." Literally translated, it's "following A" or "if we follow A."
HOW IT'S CONJUGATED
A is a noun indicating your source: the newspaper, your mom, an expert in the subject, a drunk guy in a bar... B is what that person said.
| | All tenses
---|---|---
Nouns ending in a vowel | 남자 | 남자에 따라 namjae ttara
남자에 따르면 namjae ttareumyeon
Nouns ending in a consonant | 신문 | 신문에 따라 sinmune ttara
신문에 따르면 sinmune ttareumyeon
TAKE NOTE
If you're using the expression to quote someone (other than the person giving you an order), then you should use 따르면 rather than 따라. 따라 and 따르면 can be used interchangeably when the meaning is "according to A" and A is a rule or an order.
엄마에 따르면 야채를 많이 먹어야 돼요.
Eommae ttareumyeon yachaereul mani meogeoya dwaeyo.
According to my mother, you have to eat a lot of vegetables.
엄마의 말에 따라 야채를 많이 먹었어요.
Eommaui mare ttara yachaereul mani meogeosseoyo.
Because my mother ordered me (according to my mother's orders), I ate a lot of vegetables.
법에 따라 처리하세요.
Beobe ttara cheorihaseyo.
Please deal with it lawfully (according to the law).
따르면 can normally be traded for 의하면 uihamyeon (see the next page), also interchangeably, except that 의하면 is normally not used if the person you're quoting is just anyone. If you're talking about a newspaper, an expert, a book, or something that carries some authority, you can use 의하면; however, if you're quoting your father or the aforementioned drunk guy at the bar, better go with 따르면.
일기 예보에 따르면 내일 비가 올 거예요.
Ilgi yeboe ttareumyeon naeil biga ol geoyeyo.
According to the weather forecast, it's going to rain tomorrow.
일기 예보에 의하면 내일 비가 올 거예요.
Ilgi yeboe uihamyeon naeil biga ol geoyeyo.
According to the weather forecast, it's going to rain tomorrow.
B will normally include an indirect speech particle (see page 304) if you're quoting a person.
따라 (and its parent 따르다) have a number of uses aside from "according to." For instance, you can talk about doing something "according to or depending on" A such as deciding whether or not to travel depending on your finances.
When 따르다 is used to mean "follow"; it normally follows an object with 을/를. When it means "according to," it follows 에.
엄마를 따랐어요.
Eommareul ttarasseoyo.
I followed my mother.
엄마에 따르면 야채를 많이 먹어야 돼요.
Eommae ttareumyeon yachaereul mani meogeoya dwaeyo.
According to my mother, you have to eat a lot of vegetables.
이 길을 따라가면 학교가 곧 보일 거예요.
I gireul ttaragamyeon hakgyoga got boil geoyeyo.
If you follow this street, you'll soon see the school.
이 길에 따르면 학교가 곧 보일 거예요.
I gire ttareumyeon hakgyoga got boil geoyeyo.
According to this street, you'll soon see the school. (nonsensical; applies only to a kind of fairy tale)
EXAMPLE SENTENCES
일기 예보에 따르면 내일 비가 올 거예요.
Ilgi yeboe ttareumyeon naeil biga ol geoyeyo.
According to the weather forecast, it's going to rain tomorrow.
대답은 상황에 따라 달라요.
Daedabeun sanghwange ttara dallayo.
The answer is different, depending on the situation.
뉴스에 의하면 금년 장마가 늦게 올 거예요.
Nyuseue uihamyeon geumnyeon jangmaga neutge ol geoyeyo.
According to the news, the rainy season will be late this year.
Here's another way to say "according to."
HOW IT'S FORMED
의하다 means "according to" and is combined with 면 to get 의하면.
HOW IT'S CONJUGATED
A is your source: a newspaper, an expert, the weather report, and the like. B is what that source said.
| | All tenses
---|---|---
Nouns ending in a vowel | 전문가 jeonmunga | 전문가에 의하면 jeonmungae uihamyeon
Nouns ending in a consonant | 신문 sinmun | 신문에 의하면 sinmune uihamyeon
TAKE NOTE
If you're talking about a newspaper, an expert, a book, or something that carries some authority, you can use 의하면; however, if you're quoting your father or some drunk guy at the bar, better go with 따라 or 따르면 (see page 322). For authoritative sources, 의하면, 따라 and 따르면 are all acceptable.
EXAMPLE SENTENCES
일기 예보에 의하면 내일 비가 올 거예요.
Ilgi yeboe uihamyeon naeil biga ol geoyeyo.
According to the weather forecast, it's going to rain tomorrow.
경제 전문가에 의하면 경제가 좋아질 거예요.
Gyeongje jeonmungae uihamyeon gyeongjega joajil geoyeyo.
According to an economist, the economy is going to improve.
뉴스에 나온 바에 의하면 금년 장마가 늦게 올 거예요.
Nyuseue naon bae uihamyeon geumnyeon jangmaga neutge ol geoyeyo.
According to the news, the rainy season will be late this year.
바 is somewhat different. It's what's called a dependent noun, which means it never appears on its own but only in the company of some friends, usually a statement that was said or shown earlier. 바 is mostly interchangeable with "것" and it normally follows (으)ㄴ (in the past) or 는 (in the present), and after that you can take it two ways. If you follow 바 with 에따르면 or 에 의하면, it means "according to A, B." You can also talk about 바가 있다/없다, and in this case it refers to an experience: something you did or didn't do. In this case it's interchangeable with (으)ㄴ 적 있다 (see page ).
HOW IT'S CONJUGATED
| | Past | Present
---|---|---|---
Action verbs ending in a vowel | 하다 | 한 바 han ba | 하는 바 haneun ba
Action verbs ending in a consonant | 먹다 | 먹은 바 meogeun ba | 먹는 바 meokneun ba
TAKE NOTE
Although 바 can be changed to 것, the reverse isn't always true. When it's used to mean "according to," 바 is used only in cases where something was discovered in some way. You'll see it used quite often to discuss survey results (조사한 바에 따르면 josahan bae ttareumyeon...)
EXAMPLE SENTENCES
일기 예보한 바에 의하면 내일 비가 올 거예요.
Ilgi yebohan bae uihamyeon naeil biga ol geoyeyo.
According to the weather forecast, it's going to rain tomorrow.
경제 전문가가 말한 바에 의하면 경제가 좋아질 거예요.
Gyeongje jeonmungaga malhan bae uihamyeon gyeongjega joajil geoyeyo.
According to an economist, the economy is going to improve.
김치를 먹은 바가 있어요.
Kimchireul meogeun baga isseoyo.
I've eaten kimchi.
스키를 탄 바가 없어요.
Seukireul ta baga eopseoyo.
I've never been skiing.
가영이 듣기에 어제 민수가 또 늦었대요.
Kayoungi deutgie eoje Minsuga tto neujeotdaeyo.
From what Kayoung heard, yesterday Minsu was late again.
기에 is most commonly used to mean "since"; see page . However, it also has this slightly less useful function, which is to talk about something based on what you or someone else can see, hear, etc.
HOW IT'S CONJUGATED
A has to be a verb that's about gaining knowledge. 보다 boda, 생각하다 saenggakada, 알다 alda, and 듣다 deutda are common, but there are others as well. B is what is known from A.
| | All tenses
---|---|---
Knowledge verbs ending in a vowel | 보다 | 보기에 bogie
Knowledge verbs ending in a consonant | 알다 | 알기에 algie
TAKE NOTE
Check the context carefully in order not to get this 기에 confused with the one that means "since." If in doubt, it's probably the other one.
You can also use 기에 gie, 기에는 gieneun, or 기로는 gironeun; the latter two just add emphasis.
EXAMPLE SENTENCES
제가 보기에 그 실수는 민수의 잘못이에요.
Jega bogie geu silsuneun Minsuui jalmosieyo.
From what I can see, this mistake was Minsu's fault.
제가 알기에는 그 실수를 민수의 잘못이에요.
Jega algieneun geu silsureul Minsuui jalmosieyo.
As far as I know, this mistake was Minsu's fault.
민수의 표정으로 봐서는 화가 많이 난 것 같아요. 다시 늦는가 봐요.
Minsuui pyojeongeuro bwaseoneun hwaga mani nan geot gatayo. Dasi neunneunga bwayo.
Looking at Minsu's expression, he seems to be very angry. It looks like he's late again.
This one is good for people who have to do research, presentations, and especially presentations based on their research. You can use it to explain the basis for a judgement, something that you can see through something else – like a fact that's obvious from a graph, or something you can guess from looking at a person's face.
HOW IT'S FORMED
(으)로 means "through" (see page ). 보다 means "to look at" and it's conjugated here with 아서, which means "so" (see page ). 는 emphasizes what you're saying.
HOW IT'S CONJUGATED
A should be a noun on its own or an action verb conjugated with 는 것. A is your source and B is what you've determined or are guessing based on A.
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 한 것으로 봐서는 han geoseuro bwaseoneun | 하는 것으로 봐서는 haneun geoseuro bwaseoneun | 할 것으로 봐서는 hal geoseuro bwaseoneun
Action verbs ending in a consonant | 먹다 | 먹은 것으로 봐서는 meogeun geoseuro bwaseoneun | 먹는 것으로 봐서는 meokneun geoseuro bwaseoneun | 먹을 것으로 봐서는 meogeul geoseuro bwaseoneun
Nouns ending in a vowel | 외모 | 외모로 봐서는 oemoro bwaseoneun | 외모로 봐서는 | 외모로 봐서는
Nouns ending in a consonant | 모양 | 모양으로 봐서는 moyangeuro bwaseoneun | 모양으로 봐서는 | 모양으로 봐서는
TAKE NOTE
B is very often a guess, so it sounds natural to end it with one of the expressions for guessing: (으)ㄹ 것 같다 is a good one, or you can look through pages to to find more.
A is normally not something too specific. Rather than saying "I can see from his surprised expression that..." it would be more normal to simply say "I can see from his expression that..." Likewise, you wouldn't say "I can see from the 95% you got on this test that..." but rather "I can see from your test score that..."
EXAMPLE SENTENCES
시험 결과로 봐서는 듣기 실력이 많이 좋아졌어요.
Siheom gyeolgwaro bwaseoneun deutgi sillyeogi mani joajyeosseoyo.
Looking at your test results, I can see that your listening skills have really improved.
외모로 봐서는 미국인인 것 같아요.
Oemoro bwaseoneun miguginin geot gatayo.
From that person's appearance I'd guess he's an American.
##
Giving Explanations
경제 문제에 대해서 알고 싶어요.
Gyeongje munjee daehaeseo algo sipeoyo.
I want to know more about the economic problem.
All of the above are ways to express talking about something.
HOW IT'S FORMED
대하다 daehada means "to be about or related to" and all the expressions above are variations of that. You can add ㄴ, 여, 여서, or 여서는. They all mean the same thing and can be used interchangeably except for 대한; see the "Take note" section for more information.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 한 것에 대해 han geose daehae
한 것에 대하여 han geose daehayeo
한 것에 대해서 han geose daehaeseo
한 것에 대한 han geose daehan
| 하는 것에 대해 haneun geose daehae
하는 것에 대하여 haneun geose daehayeo
하는 것에 대해서 haneun geose daehaeseo
하는 것에 대한 haneun geose daehan
| 할 것에 대해 hal geose daehae
할 것에 대하여 hal geose daehayeo
할 것에 대해서 hal geose daehaeseo
할 것에 대한 hal geose daehan
Action verbs ending in a consonant | 먹다 | 먹은 것에 대해 meogeun geose daehae
먹은 것에 대하여 meogeun geose daehayeo
먹은 것에 대해서 meogeun geose daehaeseo
먹은 것에 대한 meogeun geose daehan
| 먹는 것에 대해 meokneun geose daehae
먹는 것에 대하여 meokneun geose daehayeo
먹는 것에 대해서 meokneun geose daehaeseo
먹는 것에 대한 meokneun geose daehan
| 먹을 것에 대해 meogeul geose daehae
먹을 것에 대하여 meogeul geose daehayeo
먹을 것에 대해서 meogeul geose daehaeseo
먹을 것에 대한 meogeul geose daehan
Nouns ending in a vowel | 남자 | 남자에 대해 namjae daehae
남자에 대하여 namjae daehayeo
남자에 대해서 namjae daehaeseo
남자에 대한 namjae daehan
| 남자에 대해
남자에 대하여
남자에 대해서
남자에 대한
| 남자에 대해
남자에 대하여
남자에 대해서
남자에 대한
Nouns ending in a consonant | 물 | 물에 대해 mure daehae
물에 대하여 mure daehayeo
물에 대해서 mure daehaeseo
물에 대한 mure daehan
| 물에 대해
물에 대하여
물에 대해서
물에 대한
| 물에 대해
물에 대하여
물에 대해서
물에 대한
TAKE NOTE
에 대한 is an exception. It turns your first clause into an adjective clause, which should then modify a noun. To put this in simpler terms: 대한 must be followed by a noun. "A story about Cinderella" becomes "신데렐라에 대한 이야기 sinderellae daehan iyagi." "A discussion about politics" becomes "정치에 대한 토론 jeongchie daehan toron."
All the other 대하다 expressions don't work like this and can simply be followed by whatever you need to add to them. Keep in mind that since 는 emphasizes the meaning, it sounds far more natural to use it while explaining a contrast; otherwise the emphasis isn't really necessary. See the examples below.
EXAMPLE SENTENCES
정치에 대한 토론을 들었어요.
Jeongchie daehan toroneul deureosseoyo.
정치에 대하여 토론을 들었어요.
Jeongchie daehayeo toroneul deureosseoyo.
정치에 대해 토론을 들었어요.
Jeongchie daehae toroneul deureosseoyo.
정치에 대해서 토론을 들었어요.
Jeongchie daehaeseo toroneul deureosseoyo.
I listened to a discussion about politics.
정치에 대한 토론을 못 들었어요. 그렇지만 경제에 대해서는 토론을 들었어요.
Jeongchie daehan toroneul mot deureosseoyo. Geureochiman gyeongjee daehaeseoneun toroneul deureosseoyo.
I couldn't understand the discussion about politics, but I understood the discussion about economics.
경제 문제에 대하여 알고 싶어요.
Gyeongje munjee daehayeo algo sipeoyo.
경제 문제에 대해 알고 싶어요.
Gyeongje munjee daehae algo sipeoyo.
경제 문제에 대해서 알고 싶어요.
Gyeongje munjee daehaeseo algo sipeoyo.
I want to know more about the economic problem.
(This can't be written with 에 대한 because there's no noun following 대한).
정치에 대해서 관심이 없어요. 그렇지만 경제 문제에 대해서는 알고 싶어요.
Jeongchie daehaeseo gwansimi eopseoyo. geureochiman gyeongje munjee daehaeseoneun algo sipeoyo.
I don't care about politics, but I want to know more about the economic problem.
옛날 옛날에는 백설공주라는 여자가 살았답니다.
Yennal yennareneun baek-seol-gong-juraneun yeojaga saratdamnida.
A long time ago, there lived a girl named Snow White.
This is a relatively formal and somewhat poetic ending that's used in three situations:
1) Introductions—of people or of facts you don't think your listener knows
2) Telling stories, as in in the example above
3) Talking about someone you're proud of
It's used to tell someone things that you know but don't think they do, and so it is often used by tour guides, storytellers, and proud moms. It's not often used toward someone you're close to unless that someone is a group of children to whom you're telling a story.
HOW IT'S FORMED
It's a contraction of the indirect speech forms (이)라고 합니다 (i)rago hamnida and (ㄴ/는)다고 합니다 (n/neun)dago hamnida.
HOW IT'S CONJUGATED
This ending is always used at the end of a sentence.
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했답니다 haetdamnida | 한답니다 handamnida | 할 거랍니다 hal georamnida
Action verbs ending in a consonant | 먹다 | 먹었답니다 meogeotdamnida | 먹는답니다 meokneundamnida | 먹을 거랍니다 meogeul georamnida
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤답니다 yeppeotdamnida | 예쁘답니다 yeppeudamnida | 예쁠 거랍니다 yeppeul georamnida
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았답니다 jagatdamnida | 작답니다 | 작을 거랍니다
Nouns ending in a vowel | 남자 | 남자랍니다 namjaramnida | 남자랍니다 | 남자랍니다
Nouns ending in a consonant | 물 | 물이랍니다 muriramnida | 물이랍니다 | 물이랍니다
TAKE NOTE
I found it somewhat challenging to learn when to use this ending and sound natural with it. With enough exposure to the Korean language, it becomes much easier and being able to use it well in the right situations will make you sound quite fluent.
EXAMPLE SENTENCES
요즘 맞벌이 하는 부부들이 많답니다.
Yojeum matbeori haneun bubudeuri mantamnida.
These days there are many dual-income couples, you know.
우리 첫손자가 민수랍니다.
Uri cheotsonjaga Minsuramnida.
My first grandchild is named Minsu.
우리 딸 가영이 다음 달에 대학교를 졸업한답니다.
Uri ttal Kayoungi daeum dare daehakgyoreul joreopandamnida.
My daughter Kayoung is going to graduate university next month, you know.
지각이란 늦게 오는 것인데요.
Jigagiran neutge oneun geosindeyo.
Tardiness means "coming late."
한국에서 일찍 퇴근하기란 하늘의 별 따기이에요.
Hangugeseo iljjik toegeunhagiran haneurui byeol ttagiieyo.
In Korea, leaving work early is as difficult as picking a star from the sky.
These two expressions have similar forms and similar meanings, but they are used in totally different ways. However, they're not that difficult to understand. (이)란 is used to define words just like in a dictionary. If you want to talk about what love is or what it is to be a leader, use (이)란. 기란 is used to talk about things that are difficult?).
HOW IT'S FORMED
기 turns verbs into nouns. (이)란/기란 are contractions of (이)라는 것/기라는 것.
HOW IT'S CONJUGATED
(이)란 follows only nouns. 기란 follows only action verbs.
| | All tenses
---|---|---
Action verbs ending in a vowel | 하다 | 하기란 hagiran
Action verbs ending in a consonant | 먹다 | 먹기란 meokgiran
| | All tenses
---|---|---
Nouns ending in a vowel | 남자 | 남자란 namjaran
Nouns ending in a consonant | 물 | 물이란 muriran
TAKE NOTE
When using 기란, B must be a sentence which explains that A is difficult. Here are some examples:
쉬운 일이 아니다
swiun iri anida
isn't easy
힘든 일이다
himdeun irida
is difficult
여간 어렵지 않다
yeogan eoryeopji anta
is difficult
하늘의 별 따기이다
haneurui byeol ttagiida
is like picking a star from the sky (a Korean proverb)
EXAMPLE SENTENCES
■ 기란 giran
새로운 언어를 배우기란 쉬운 일이 아니에요.
Saeroun eoneoreul baeugiran swiun iri anieyo.
Learning a new language isn't easy.
아침에 일찍 일어나기란 여간 어렵지 않아요.
Achime iljjik ireonagiran yeogan eoryeopji anayo.
Waking up early is really difficult.
■ 이란 iran
사랑이란 희생인데요.
Sarangiran huisaengindeyo.
Love is sacrifice.
희생이란 무엇인가요?
Huisaengiran mueosingayo?
What is sacrifice?
그 콘서트에 밴드가 다 좋아서 정시에 맞춰 갈 만해요.
Geu konseoteue baendeuga da joaseo jeongsie matchwo gal manhaeyo.
All the bands at that concert are good, so it's worth arriving on time.
This is a way to say that something is worth doing (or seeing or eating or reading or listening to, etc.). If you use this expression, you're recommending ~ to a listener but are probably not wildly excited about it. It's worth the time, but not more than that.
HOW IT'S FORMED
만하다 in this case means "to be worth."
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Action verbs ending in a vowel | 하다 | 할 만하다 hal manhada
Action verbs ending in a consonant | 먹다 | 먹을 만하다 meogeul manhada
TAKE NOTE
만하다 by itself without the (으)ㄹ in front is used to compare the size of two things, mostly in idiomatic expressions. 만하다 will always follow nouns, never verbs.
기만 하다 giman hada means to do one thing and one thing only. It will always follow 기, never (으)ㄹ.
There are also a number of expressions with 만 man: 만에 mane, 만 못하다 man motada, 만 해도 man haedo, and just plain 만. You can look them up in their appropriate sections on pages , , and respectively if you like. They mostly involve 만, which means "only" and aren't followed by 하다.
EXAMPLE SENTENCES
그 새로 나온 책은 읽을 만해요.
Geu saero naon chaegeun ilgeul manhaeyo.
That new book is worth reading.
여행을 하고 싶다니 제주도가 가 볼 만해요.
Yeohaengeul hago sipdani jejudoga ga bol manhaeyo.
Since you said you wanted to travel, I think Jeju do is worth seeing.
##
Location
점심을 먹은 데에서 만날래요?
Jeomsimeul meogeun deeseo mannallaeyo?
Let's meet at the place we had lunch.
데 means place and is used in conjunction with A, which explains what exactly that place is and (maybe) why the listener should care. This expression sets up A as "the place where B happens."
HOW IT'S FORMED
The tense markers (으)ㄴ/는/(으)ㄹ are there to help you explain A, while 데, as stated above, is simply a noun that means "place."
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 한 데 han de | 하는 데 haneun de | 할 데 hal de
Action verbs ending in a consonant | 먹다 | 먹은 데 meogeun de | 먹는 데 meokneun de | 먹을 데 meogeul de
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁜 데 yeppeun de | 예쁜 데 | 예쁠 데 yeppeul de
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작은 데 jageun de | 작은 데 | 작을 데 jageul de
TAKE NOTE
A can't be 이, 그, or 저. In this case you can use 곳 got instead of 데.
Don't confuse 데 and 때. 때 ttae means "time" and is commonly used in the expression "(으)ㄹ 때," which means "while." 데 means "place."
A(으)ㄴ/는 데 B is another expression which is used to talk about effectiveness (see page ). It's never used with (으)ㄹ. Otherwise, you can tell the two apart by context: the other 는 데 will always be followed by a sentence indicating effectiveness (효과적이다 hyogwajeogida, 필수적이에요 pilsujeonieyo, etc.) or lack thereof, while this one will always have to do with places.
EXAMPLE SENTENCES
큰 TV를 놓을 데가 없어요.
Keun tibireul noeul dega eopseoyo.
I have no place to put a big TV.
운동하는 데가 아주 시끄러워요.
Undonghaneun dega aju sikkeureowoyo.
The place I exercise in is very noisy.
##
Limits and Excess
어제 숙취가 심했어요. 하루 종일 누워있기만 했어요.
Eoje sukchwiga simhaesseoyo. Haru jongil nuwoitgiman haesseoyo.
I had a bad hangover yesterday. I did nothing but lie down.
This expression is for addicts and people who have to talk about them. It means to do a certain thing, and only that thing, repeatedly.
HOW IT'S FORMED
기 turns a verb into a noun. 만 means "only" and is added to nouns.
HOW IT'S CONJUGATED
It works with action verbs plus 기. If you want to use a noun which has been turned into a verb by the addition of 하다 (examples are 운동하다 undonghada, 야구하다 yaguhada, 운전하다 unjeonhada), then you can drop the 하다 and follow the noun by itself with 만하다. It's not necessary to say 운동하기만 했다 undonghagiman haetda, for example.
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 하기만 했다 hagiman haetda | 하기만 하다 hagiman hada | 하기만 할 것이다 hagiman hal geosida
Action verbs ending in a consonant | 먹다 | 먹기만 했다 meokgiman haetda | 먹기만 하다 meokgiman hada | 먹기만 할 것이다 meokgiman hal geosida
Nouns ending in a vowel | 야구 | 야구만 했다 yaguman haetda | 야구만 하다 yaguman hada | 야구만 할 것이다 yaguman hal geosida
Nouns ending in a consonant | 게임 | 게임만 했다 geimman haetda | 게임만 하다 geimman hada | 게임만 할 것이다 geimman hal geosida
TAKE NOTE
기만 하다 and (으)ㄹ 뿐이다 (eu)l ppunida (on next page) are similar, but 기만 하다 specifically means to do one thing repeatedly while (으)ㄹ 뿐이다 can be used in a much greater variety of situations.
만 하다 has two other uses. One is to talk about something "worth doing." That's covered on page 333 and can be differentiated because it always follows a verb conjugated with (으)ㄹ. The 만 하다 we're talking about here follows only either verbs with 기 or nouns. The other use of 만 하다 is to compare sizes (see page ). That one's a bit trickier to distinguish, but one key is that it's normally used to talk about objects, while this 만 하다 is used to talk mostly about people and what they're (only) doing. Other than that, check the context.
EXAMPLE SENTENCES
민수가 게임만 해요.
Minsuga geimman haeyo.
Minsu does nothing but play games.
아주 피곤해서 내일 자기만 할 거예요.
Aju pigonhaeseo naeil jagiman hal geoyeyo.
I'm very tired, so tomorrow I'm only going to sleep.
민수가 또 늦어요. 짜증나서 죽겠어요.
Minsuga tto neujeoyo. Jjajeungnaseo jukgesseoyo.
Minsu's late again. I'm so annoyed I could just die.
Drama queens, rejoice! Here's the perfect Korean expression if you're prone to exaggerate. It literally translates as "I'm so ~ I could die" and means that ~ is really bad: the cold, the boredom, the hunger, or whatever minor inconvenience is bothering you that day.
HOW IT'S FORMED
아/어/여서 means "so" (see page ). 죽겠다 is 죽다 (to die) conjugated in the future tense to suggest that while you're not dead yet, you might be soon.
HOW IT'S CONJUGATED.
| | All tenses
---|---|---
Action or descriptive verbs with 아 or 오 | 잡다 | 잡아서 죽겠다 jabaseo jukgetda
Action or descriptive verbs with 어, 우, 으 or 이 | 먹다 | 먹어서 죽겠다 meogeoseo jukgetda
하다 | 피곤하다 | 피곤해서 죽겠다 pigonhaeseo jukgetda
TAKE NOTE
While 아/어/여서 죽겠다 is normally used only with bad things, you can use it for good things as well: "That baby is so cute I could just die!" can be translated almost exactly into Korean.
EXAMPLE SENTENCES
오늘 추워서 죽겠어요.
Oneul chuwoseo jukgesseoyo.
It's cold today!
배가 고파서 죽겠어요.
Baega gopaseo jukgesseoyo.
I'm hungry!
너무 많이 먹어서 죽겠어요.
Neomu mani meogeoseo jukgesseoyo.
I ate too much!
제가 벌써 회의에 늦어서 걱정할 뿐이에요.
Jega beolsseo hoeuie neujeoseo geokjeonghal ppunieyo.
I'm already late for the meeting, so all I'm doing is worrying.
This expression, useful for very modest people or those who like to downplay a fact, means that the situation is or was limited to "~ and only ~." It be used for sentences such as "We're just friends," "All I did was sleep," or "Nonsense, I was only doing what any other person would have done."
HOW IT'S FORMED
뿐 is a noun which means "only." And yes, that's an adverb in English.
HOW IT'S CONJUGATED
| | Past | Present
---|---|---|---
Action verbs ending in a vowel | 하다 | 했을 뿐이다 haeseul ppunida | 할 뿐이다 hal ppunida
Action verbs ending in a consonant | 먹다 | 먹었을 뿐이다 meogeosseul ppunida | 먹을 뿐이다 meogeul ppunida
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤을 뿐이다 yeppeoseul ppunida | 예쁠 뿐이다 yeppeul ppunida
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았을 뿐이다 jagasseul ppunida | 작을 뿐이다 jageul ppunida
Nouns ending in a vowel | 남자 | 남자였을 뿐이다 namjayeosseul ppunida
남자 뿐이다 namja ppunida
| 남자일 뿐이다 namjail ppunida
남자 뿐이다
Nouns ending in a consonant | 물 | 물이었을 뿐이다 murieosseul ppunida
물 뿐이다 mul ppunida
| 물일 뿐이다 muril ppunida
물 뿐이다
TAKE NOTE
A commonly used expression based on 뿐 is A([으]ㄹ) 뿐만 아니라 B ([eu]l) ppunman anira B, which means "not just A, but also B" (see page ).
(으)ㄹ 뿐이다 and 기만 하다 (see page 335) are quite similar, but 기만 하다 can be used when only one action was done repeatedly. (으)ㄹ 뿐이다 can be used much more widely.
You can add 만 or 기만 to A if you want to make your expression even more emphatic.
나는 잠 잤을 뿐이에요.
Naneun jam jasseul ppunieyo.
I did nothing but sleep.
나는 잠만 잤을 뿐이에요.
Naneun jamman jasseul ppunieyo.
No, really, ALL I did was sleep.
뿐이다 and 따름이다 ttareumida (see page 346) are interchangeable except that you can use 뿐이다 directly after nouns, while 따름이다 follows only verbs (or nouns with 이).
EXAMPLE SENTENCES
그녀는 개성이 없어요. 예쁠 뿐이에요.
Geunyeoneun gaeseongi eopseoyo. Yeppeul ppunieyo.
She doesn't have any personality. She's just beautiful (and there's nothing more to her than that).
저 사람은 제 남자친구 아닌데요. 친구일 뿐이에요.
Jeo sarameun je namjachingu anindeyo. Chinguil ppunieyo.
That person isn't my boyfriend. We're just friends.
확실하지 않아요. 소문으로 들었을 뿐이에요.
Hwaksilhaji anayo. Somuneuro deureosseul ppunieyo.
I'm not sure. I just heard a rumor.
늦게 퇴근하지 않는 한 비행기를 놓치지 않을 거예요.
Neutge toegeunhaji anneun han bihaenggireul nochiji aneul geoyeyo.
As long as you don't leave work late, you won't miss your flight.
You can use this expression to say that as long as A happens, B will happen.
HOW IT'S FORMED
한 is a noun which literally means "limit," and it can be used on its own as that noun. In this expression 는 is added to an action verb to turn A into an adjective clause modifying the noun 한.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 한 한 han han | 하는 한 haneun han | 할 한 hal han
Action verbs ending in a consonant | 먹다 | 먹은 한 meogeun han | 먹는 한 meokneun han | 먹을 한 meogeul han
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁜 한 yeppeun han | 예쁜 한 | 예쁜 한
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작은 한 jageun han | 작은 한 | 작은 한
Nouns ending in a vowel | 남자 | 남자인 한 namjain han | 남자인 한 | 남자인 한
Nouns ending in a consonant | 물 | 물인 한 murin han | 물인 한 | 물인 한
TAKE NOTE
아/어/여야, as covered on page , is another expression where A is required for B. In this case A must happen in order for B to happen; with 한, A is not always a necessity, but B will happen as long as A is going on.
EXAMPLE SENTENCES
건강하게 먹는 한 다이어트에 성공할 수 있어요.
Geonganghage meokneun han daieoteue seonggonghal su isseoyo.
As long as you eat healthy, you can succeed in your diet.
다른 수트 케이스가 작은 한 비행기 안으로 가지고 들어갈 수 있을 거예요.
Dareun suteu keiseuga jageun han bihaenggi aneuro gajigo deureogal su isseul geoyeyo.
You will be able to bring a suitcase into an airplane as long as it's a small one.
무료 입장이 12시 이후에 한해서 가능해요.
Muryo ipjangi yeoldusi ihue hanhaeseo ganeunghaeyo.
Free entry is available only after twelve.
Here's a good one for tour guides or amusement park owners. It's a way to say that B is limited or only available to A: For example, free entry is limited to senior citizens only or an offer is limited to today.
HOW IT'S FORMED
한 is a noun that means "limit." Here it's made into a verb using 하다 and then conjugated using 여서 (see page ) to mean "something is limited to A, so B."
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Nouns ending in a vowel | 남자 | 남자에 한해서 namjae hanhaeseo
Nouns ending in a consonant | 물 | 물에 한해서 mure hanhaeseo
TAKE NOTE
This is similar to 는 한 in some ways, but 는 한 follows verbs while 한하다 follows nouns.
EXAMPLE SENTENCES
이 표는 한 사람 한 표에 한해요.
I pyoneun han saram han pyoe hanhaeyo.
This ticket is good only for one person.
이 제공은 한국 시민에 한해서 외국인들은 받을 수 없어요.
I jegongeun hanguk simine hanhaeseo oegugindeureun badeul su eopseoyo.
This offer is limited to Korean citizens, so foreigners can't take advantage of it.
민수가 어찌나 늦었는지 청소부조차 퇴근했어요.
Minsuga eojjina neujeonneunji cheongsobujocha toegeunhaesseoyo.
민수가 어찌나 늦었는지 청소부까지 퇴근했어요.
Minsuga eojjina neujeonneunji cheongsobukkaji toegeunhaesseoyo.
민수가 어찌나 늦었는지 청소부마저 퇴근했어요.
Minsuga eojjina neujeonneunji cheongsobumajeo toegeunhaesseoyo.
민수가 어찌나 늦었는지 청소부도 퇴근했어요.
Minsuga eojjina neujeonneunji cheongsobudo toegeunhaesseoyo.
Minsu was so late that even the cleaning lady had gone home.
These expressions are good to use if you should find yourself in an extreme situation such as final exams.
도 is absolutely essential and Koreans use it all the time, so be sure to learn it well.
조차 is a very negative word which can't be translated exactly, but which appears in sentences like "not even a genius could pass this test," "not even his own mother could love him," or "it's freezing cold, the heater is broken and I even forgot to wear a sweater today."
까지 should be one of the first particles you need to learn in Korean; it means "to," as you probably know well by now. That meaning carries over to this use of 까지, which expresses that something goes really far: as far as ~, which is a little extreme for the situation.
마저 is an expression that only the TOPIK makers seem to care about. If you've found yourself in this section because you are taking the TOPIK, don't worry. 마저 is easy to learn.
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Action verbs ending in a vowel | 하다 | 할 수조차 없다 hal sujocha eopda
Action verbs ending in a consonant | 먹다 | 먹을 수조차 없다 meogeul sujocha eopda
Nouns ending in a vowel | 남자 | 남자조차 namjajocha
남자까지 namjakkaji
남자마저 namjamajeo
남자도 namjado
Nouns ending in a consonant | 물 | 물조차 muljocha
물까지 mulkkaji
물마저 mulmajeo
물도 muldo
TAKE NOTE
조차 is choosy and cannot just be used anywhere. First of all, it's always used in negative sentences: B will normally (but not always) include a word like 못, 아, or 지 않다. If your sentence is already about something bad happening, you may not need these expressions. You can use 조차 with a verb by putting the verb into an expression with (으)ㄹ 수조차 없다 (see page ). Descriptive verbs, however, can't be used with any of the particles; either rearrange your sentence or rearrange your descriptive verb by adding 어/아지다 to it (see page shows you how to do that).
Secondly, 조차, 마저, 까지 (only in negative uses of 까지) and 도 are normally interchangeable, though 조차 makes things out to be slightly worse than any of the other three. If you're using 조차, the situation is really bad, at least in your own eyes, if no one else's. 까지 can be used for positive as well as negative situations, while 마저 and 조차 can't.
Finally, a couple of expressions that go well with these are 얼마나/어찌나 A(으)ㄴ/는지 B eolmana/eojjina A(eu)n/ neunji B (see page 343) and ~은/는 커녕 ~eun/neun keonyeong (see page ). Look at the examples to learn how to use these together with 조차, 까지, and 마저.
EXAMPLE SENTENCES
어제 얼마나 더운지 에어컨조차 소용이 없었어요.
Eoje eolmana deounji eeokeonjocha soyongi eopseosseoyo.
어제 어찌나 더운지 에어컨조차 소용이 없었어요.
Eoje eojjina deounji eeokeonjocha soyongi eopseosseoyo.
어제 얼마나 더운지 에어컨까지 소용이 없었어요.
Eoje eolmana deounji eeokeonkkaji soyongi eopseosseoyo.
어제 어찌나 더운지 에어컨까지 소용이 없었어요.
Eoje eojjina deounji eeokeonkkaji soyongi eopseosseoyo.
어제 얼마나 더운지 에어컨마저 소용이 없었어요.
Eoje eolmana deounji eeokeonmajeo soyongi eopseosseoyo.
어제 어찌나 더운지 에어컨마저 소용이 없었어요.
Eoje eojjina deounji eeokeonmajeo soyongi eopseosseoyo.
어제 얼마나 더운지 에어컨도 소용이 없었어요.
Eoje eolmana deounji eeokeondo soyongi eopseosseoyo.
어제 어찌나 더운지 에어컨도 소용이 없었어요.
Eoje eojjina deounji eeokeondo soyongi eopseosseoyo.
Yesterday it was so hot not even the air conditioning could help.
__________
피자 한판은 커녕 한 조각조차 먹을 수 없어요.
Pija hanpaneun keonyeong han jogakjocha meogeul su eopseoyo.
피자 한판은 커녕 한 조각까지 먹을 수 없어요.
Pija hanpaneun keonyeong han jogakkkaji meogeul su eopseoyo.
피자 한판은 커녕 한 조각마저 먹을 수 없어요.
Pija hanpaneun keonyeong han jogangmajeo meogeul su eopseoyo.
피자 한판은 커녕 한 조각도 먹을 수 없어요.
Pija hanpaneun keonyeong han jogakdo meogeul su eopseoyo.
I can't even eat one slice, let alone an entire pizza.
__________
가영이 테니스뿐만 아니라 수영까지 잘 해요.
Kayoungi teniseuppunman anira suyeongkkaji jal haeyo.
가영이 테니스뿐만 아니라 수영도 잘 해요.
Kayoungi teniseuppunman anira suyeongdo jal haeyo.
Kayoung is not only good at tennis, but swimming as well.
(Can't be used with 마저 or 조차 since they can't be used for positive expressions.)
Here's another expression you can use to talk about extremes. This one means that something was "so A that B."
HOW IT'S FORMED
어찌나 means "so." 얼마나 means "how much" or "how many." (으)ㄴ/는지 is an expression which turns a phrase into a into a noun clause.
HOW IT'S CONJUGATED
Nouns are a little complicated; you can use them with these expressions, but only if you add an adjective before them and an 이 after them. So 어찌나 남자인지 doesn't work, but 어찌나 좋은 남자인지 does.
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 어찌나 했는지 eojjina haenneunji
얼마나 했는지 eolmana haenneunji
| 어찌나 하는지 eojjina haneunji
얼마나 하는지 eolmana haneunji
| 어찌나 할지 eojjina halji
얼마나 할지 eolmana halji
Action verbs ending in a consonant | 먹다 | 어찌나 먹었는지 eojjina meogeonneunji
얼마나 먹었는지 eolmana meogeonneunji
| 어찌나 먹는지 eojjina meokneunji
얼마나 먹는지 eolmana meokneunji
| 어찌나 먹을지 eojjina meogeulji
얼마나 먹을지 eolmana meogeulji
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 어찌나 예뻤는지 eojjina yeppeonneunji
얼마나 예뻤는지 eolmana yeppeonneunji
| 어찌나 예쁜지 eojjina yeppeunji
얼마나 예쁜지 eolmana yeppeunji
| 어찌나 예쁠지 eojjina yeppeulji
얼마나 예쁠지 eolmana yeppeulji
Descriptive verbs (adjectives) ending in a consonant | 작다 | 어찌나 작았는지 eojjina jaganneunji
얼마나 작았는지 eolmana jaganneunji
| 어찌나 작은지 eojjina jageunji
얼마나 작은지 eolmana jageunji
| 어찌나 작을지 eojjina jageulji
얼마나 작을지 eolmana jageulji
Nouns ending in a vowel | 남자 | 어찌나 좋은 남자였는지 eojjina joeun namjayeonneunji
얼마나 좋은 남자였는지 eolmana joeun namjayeonneunji
| 어찌나 좋은 남자인지 eojjina joeun namjainji
얼마나 좋은 남자인지 eolmana joeun namjainji
| 어찌나 좋은 남자일지 eojjina joeun namjailji
얼마나 좋은 남자일지 eolmana joeun namjailji
Nouns ending in a consonant | 물 | 어찌나 좋은 물이었는지 eojjina joeun murieonneunji
얼마나 좋은 물이었는지 eolmana joeun murieonneunji
| 어찌나 좋은 물인지 eojjina joeun murinji
얼마나 좋은 물인지 eolmana joeun murinji
| 어찌나 좋은 물일지 eojjina joeun murilji
얼마나 좋은 물일지 eolmana joeun murilji
TAKE NOTE
얼마나 and 어찌나 are interchangeable in the expressions above.
Action verbs are often used together with adverbs. So rather than 얼마나 먹었는지, you'd say:
얼마나 많이 먹었는지
eolmana mani meogeonneunji
I ate so much that...
얼마나 자주 먹었는지
eolmana jaju meogeonneunji
I ate so often that...
얼마나 잘 먹었는지
eolmana jal meogeonneunji
I ate so well that...
or something similar.
얼마나 A(으)ㄴ/는지 is featured in another rather common expression: 얼마나 A(으)ㄴ/는지 모르다/모르겠다. Eolmana A (eu)n/neunji moreuda/moreugetda. It means "I don't even know how A it is/was," which implies that something was A.
오늘 날씨가 얼마나 추운지 모르겠어요.
Oneul nalssiga eolmana chuunji moreugesseoyo.
오늘 날씨가 얼마나 추운지 몰라요.
Oneul nalssiga eolmana chuunji mollayo.
It's freezing cold out today.
내일 날씨가 얼마나 추울지 모르겠어요.
Naeil nalssiga eolmana chuulji moreugesseoyo.
I don't know how cold it will be tomorrow.
EXAMPLE SENTENCES
어제 어찌나 많이 먹었는지 나중에 배가 아팠어요.
Eoje eojjina mani meogeonneunji najunge baega apasseoyo.
어제 얼마나 많이 먹었는지 나중에 배가 아팠어요.
Eoje eolmana mani meogeonneunji najunge baega apasseoyo.
Yesterday I ate so much that I got a stomachache.
민수가 어찌나 늦었는지 연극이 벌써 끝났어요.
Minsuga eojjina neujeonneunji yeongeugi beolsseo kkeunnasseoyo.
민수가 얼마나 늦었는지 연극이 벌써 끝났어요.
Minsuga eolmana neujeonneunji yeongeugi beolsseo kkeunnasseoyo.
Minsu was so late that the show had already finished.
가영이 어찌나 좋은 학생인지 항상 시험에서 만점을 받아요.
Kayoungi eojjina joeun haksaenginji hangsang siheomeseo manjeomeul badayo.
가영이 얼마나 좋은 학생인지 항상 시험에서 만점을 받아요.
Kayoungi eolmana joeun haksaenginji hangsang siheomeseo manjeomeul badayo.
Kayoung is such a good student that she always gets a 100% on tests.
늦으면 이번에야말로 전화할 거예요.
Neujeumyeon ibeoneyamallo jeonhwahal geoyeyo.
This time I'll definitely call you if I'm late.
(이)야말로 is used to say that one particular thing, among many, is just right. For instance, if your friend has only a day in New York and wants to know that one thing that cannot be missed, you could use (이)야말로 to recommend the Statue of Liberty. You could also use (이)야말로 if you've finally met the man or woman of your dreams, are adopting a pet from an animal shelter, are buying shoes, or are in any situation where one thing is definitely much better than all the rest. You can also use it to say "definitely" or "without fail," as in "This time I'll definitely do it" or "That's definitely the right one."
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Nouns ending in a vowel | 남자 | 남자야말로 namjayamallo
Nouns ending in a consonant | 물 | 물이야말로 muriyamallo
TAKE NOTE
This is often used together with 뭐니 뭐니 해도 mwoni mwoni haedo ("above all else" or "regardless of anything else").
EXAMPLE SENTENCES
한국 음식 중에 김치야말로 대표적인 한국 음식이에요.
Hanguk eumsik junge kimchiyamallo daepyojeogin hanguk eumsigieyo.
뭐니 뭐니 해도 한국 음식 중에 김치야말로 대표적인 한국 음식이에요.
Mwoni mwoni haedo hanguk eumsik junge kimchiyamallo daepyojeogin hanguk eumsigieyo.
Kimchi is the representative food of Korea. (Among Korean food, kimchi is representative of Korean food.)
이 역할은 가영이야말로 잘 할 거예요.
I yeokareun Kayoungiyamallo jal hal geoyeyo.
Kayoung is perfect for this role.
제가 벌써 회의에 늦어서 걱정할 따름이에요.
Jega beolsseo hoeuie neujeoseo geokjeonghal ttareumieyo.
I'm already late for the meeting, so all I'm doing is worrying.
Here's another useful expression for either putting people down or minimizing your own accomplishments. It means "nothing but ~."
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 했을 따름이다 haesseul ttareumida | 할 따름이다 hal ttareumida | 할 따름이다
Action verbs ending in a consonant | 먹다 | 먹었을 따름이다 meogeosseul ttareumida | 먹을 따름이다 meogeul ttareumida | 먹을 따름이다
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예뻤을 따름이다 yeppeosseul ttareumida | 예쁠 따름이다 yeppeul ttareumida | 예쁠 따름이다
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작았을 따름이다 jagasseul ttareumida | 작을 따름이다 jageul ttareumida | 작을 따름이다
Nouns ending in a vowel | 남자 | 남자였을 따름이다 namjayeosseul ttareumida | 남자일 따름이다 namjail ttareumida | 남자일 따름이다
Nouns ending in a consonant | 물 | 물이었을 따름이다 murieosseul ttareumida | 물일 따름이다 muril ttareumida | 물일 따름이다
TAKE NOTE
따름이다 and 뿐이다 (see page 337) are interchangeable except that you can use 뿐이다 directly after nouns, while 따름이다 follows only nouns with 이 added to turn them into verbs.
EXAMPLE SENTENCES
그녀는 개성이 없어요. 예쁠 따름이에요.
Geunyeoneun gaeseongi eopseoyo. Yeppeul ttareumieyo.
She doesn't have any personality. She's just beautiful (and there's nothing more to her than that).
저 사람은 제 남자친구 아닌데요. 친구일 따름이에요.
Jeo sarameun je namjachingu anindeyo. Chinguil ttareumieyo.
That person isn't my boyfriend. We're just friends.
확실하지 않아요. 소문으로 들었을 따름이에요.
Hwaksilhaji anayo. Somuneuro deureosseul ttareumieyo.
I'm not sure. I just heard a rumor.
생각만으로는 정시에 올 수 없어요. 서둘러야겠죠.
Saenggangmaneuroneun jeongsie ol su eopseoyo. Seodulleoyagetjyo.
You won't get there on time just by thinking about it. You have to hurry.
Here's an expression for all you managers. This is a way to say that something on its own just isn't enough for whatever you're trying to accomplish.
HOW IT'S FORMED
만 means "only" (see page ). 으로 means "through" (see page ), and 는 is optional: it simply adds emphasis: "through only A."
HOW IT'S CONJUGATED
Action verbs can be changed to nouns using 는 것.
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 한 것만으로는 han geonmaneuroneun | 하는 것만으로는 haneun geonmaneuroneun | 할 것만으로는 hal geonmaneuroneun
Action verbs ending in a consonant | 먹다 | 먹은 것만으로는 meogeun geonmaneuroneun | 먹는 것만으로는 meokneun geonmaneuroneun | 먹을 것만으로는 meogeul geonmaneuroneun
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁜 것만으로는 yeppeun geonmaneuroneun | 예쁜 것만으로는 | 예쁜 것만으로는
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작은 것만으로는 jageun geonmaneuroneun | 작은 것만으로는 | 작은 것만으로는
Nouns ending in a vowel | 남자 | 남자만으로는 namjamaneuroneun | 남자만으로는 | 남자만으로는
Nouns ending in a consonant | 물 | 물만으로는 mulmaneuroneun | 물만으로는 | 물만으로는
TAKE NOTE
B should be an expression like 부족하다 bujokada or 하기가 힘들다 hagiga himdeulda which indicates that A isn't enough by itself.
만으로는 and 만 by itself are quite similar, but 만으로는 is used a little more specifically to say that something isn't enough by itself.
EXAMPLE SENTENCES
한국어 잘 배우기 위해서 문법 공부만으로 부족해요. 한국 사람과 대화를 해야 돼요.
Hangugeo jal baeugi wihaeseo munbeop gongbumaneuro bujokaeyo. Hanguk saramgwa daehwareul hae ya dwaeyo.
Studying grammar alone isn't enough to learn Korean well. You also need to have conversations with Koreans.
노력만으로는 그 일을 끝내기가 힘들어요. 도움이 필요해요.
Noryeongmaneuroneun geu ireul kkeunnaegiga himdeureoyo. Doumi piryohaeyo.
It'll be difficult to finish that job just by trying. You need help.
A: 저는 살을 빼고 싶으니까 운동을 해야겠어요.
Jeoneun sareul ppaego sipeunikka undongeul haeyagesseoyo.
I want to lose weight, so I'll have to exercise.
B: 운동도 운동이지만 패스트 푸드부터 먹지 마세요.
Undongdo undongijiman paeseuteu pudeubuteo meokji maseyo.
Exercise is all fine and well, but you should start by not eating fast food.
This is for people who need to give advice to friends who are headed off in the wrong direction. Let's say your friend wants to lose weight and says they're going to start exercising. This is a good plan, but you think they'd be better off if they'd just stop eating fast food three meals a day. That's the kind of situation in which you'd use this expression.
HOW IT'S FORMED
기 turns verbs into nouns. 도 means "too." 이 turns nouns into verbs, while 지만 means "but."
HOW IT'S CONJUGATED
Add 기 to verbs to turn them into nouns before putting them before 도.
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 하기도 했지만 hagido haetjiman | 하기도 하지만 hagido hajiman | 하기도 할 거지만 hagido hal geojiman
Action verbs ending in a consonant | 먹다 | 먹기도 먹었지만 meokgido meogeotjiman | 먹기도 먹지만 meokgido meokjiman | 먹기도 먹을 거지만 meokgido meogeul geojiman
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁘기도 예쁘지만 yeppeugido yeppeujiman | 예쁘기도 예쁘지만 | 예쁘기도 예쁘지만
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작기도 작지만 jakgido jakjiman | 작기도 작지만 | 작기도 작지만
Nouns ending in a vowel | 남자 | 남자도 남자이지만 namjado namjaijiman | 남자도 남자이지만 | 남자도 남자이지만
Nouns ending in a consonant | 물 | 물도 물이지만 muldo murijiman | 물도 물이지만 | 물도 물이지만
EXAMPLE SENTENCES
A: 가영이 정말 예쁘지요?
Kayoungi jeongmal yeppeujiyo?
Kayoung is really pretty, isn't she?
B: 예쁘기도 예쁘지만 성격이 나빠서 가영을 싫어요.
Yeppeugido yeppeujiman seonggyeogi nappaseo Kayoungi sireoyo.
Beauty is all fine and well, but she has a bad personality so I don't like her.
__________
A: 한국어를 잘 하고 싶어서 요즘 문법을 열심히 배워요.
Hangugeoreul jal hago sipeoseo yojeum munbeobeul yeolsimhi baewoyo.
I want to speak Korean well, so these days I'm learning grammar.
B: 문법도 문법이지만 한국 사람과 대화를 했으면 좋겠어요.
Munbeopdo munbeobijiman hanguk saramgwa daehwareul haesseumyeon jokesseoyo.
Learning grammar is all fine and well, but you also have to talk to Koreans.
##
Habits
민수가 대학교 때 자주 수업에 늦게 오곤 했어요.
Minsuga daehakgyo ttae jaju sueobe neutge ogon haesseoyo.
During university, Minsu was habitually late for class.
곤 하다, a good grammatical pattern for those who like to live their lives according to a consistent schedule, means "to do something repeatedly or as a habit." It's used in sentences such as "I always have toast and jam for breakfast" or "I often go to that bar after work."
HOW IT'S CONJUGATED
| | Past | Present
---|---|---|---
Action verbs ending in a vowel | 하다 | 하곤 했다 hagon haetda
하고는 했다 hagoneun haetda
| 하곤 하다 hagon hada
하고는 하다 hagoneun hada
Action verbs ending in a consonant | 먹다 | 먹곤 했다 meokgon haetda
먹고는 했다 meokgoneun haetda
| 먹곤 하다 meokgon hada
먹고는 하다 meokgoneun hada
TAKE NOTE
곤 하다 and 고는 하다 are interchangeable, but 곤 하다 is somewhat more common.
곤/고는 하다 is used to describe a habit rather than something that always happens without fail. So if you want to talk about eating toast and jam for breakfast most mornings, you could use 곤 하다, but it would be strange to use it to describe your alarm clock going off every single day at 7:00.
곤 하다 and 기 일쑤이다 gi ilssuida (see page 356) are interchangeable when the sentence is negative. 기 일쑤이다 isn't usually used with positive sentences.
아/어/여 대다 a/eo/yeo daeda (see page 357) means to do something repeatedly, but not necessarily as a habit. If your ex-boyfriend or ex-girlfriend calls you all day long the day after you've broken up with him or her, you can use 아/어/여 대다 to describe the action, but not 곤 하다 because the calling is done repeatedly but is not habitual. If that same ex-boyfriend or ex-girlfriend calls you every day for months on end, then you could use 곤 하다.
EXAMPLE SENTENCES
남자친구와 헤어진 후에 그는 몇일 동안 계속 저에게 전화하곤 했어요.
Namjachinguwa he-eo-jin hue geuneun myeochil dongan gyesok jeoege jeonhwahagon haesseoyo.
After I broke up with my boyfriend, he kept on calling me for several days.
퇴근한 후에 맥주 두세잔 마시곤 해요.
Toegeunhan hue maekju dusejan masigon haeyo.
After work I usually drink a few glasses (two or three glasses) of beer.
민수가 또 늦었어요. 그는 자주 늦는 편이지요?
Minsuga tto neujeosseoyo. Geuneun jaju neunneun pyeonijiyo?
Minsu's late again. He often tends to be late, doesn't he?
This can be used to talk about someone or something which tends to be a certain way.
HOW IT'S FORMED
편 means "side," so this expression literally means "on the side of ~."
HOW IT'S CONJUGATED
| | Past | Present
---|---|---|---
Action verbs ending in a vowel | 하다 | 하는 편이었다 haneun pyeonieotda | 하는 편이다 haneun pyeonida
Action verbs ending in a consonant | 먹다 | 먹는 편이었다 meokneun pyeonieotda | 먹는 편이다 meokneun pyeonida
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁜 편이었다 yeppeun pyeonieotda | 예쁜 편이다 yeppeun pyeonida
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작은 편이었다 jageun pyeonieotda | 작은 편이다 jageun pyeonida
TAKE NOTE
편이다 can never be used in the future tense.
It's not used to talk about objectively verifiable facts: for example, "He tends to be a man." You could, however, say that "He tends to be manly."
그는 남자인 편이에요.
Geuneun namjain pyeonieyo.
He tends to be a man. (ungrammatical)
그는 남자다운 편이에요.
Geuneun namjadaun pyeonieyo.
He tends to be manly. (just fine)
Action verbs are normally used in conjunction with adverbs like 자주 jaju, 가끔 gakkeum, and so on.
A similar expression is ~(으)ㄴ/는 축에 들다 (eu)n/neun chuge deulda (see page 359). There's a very slight difference in meaning between the two. Think of 편이다 as placing the subject on a scale between 0 and 100. Let's say 0 is 제일 작은 jeil jageun and 100 is 제일 큰 jeil keun. If the subject rates over 50 on that scale, 큰 편이다 keun pyeonida. With 축에 들다 chuge deulda, rather than a scale, you're dividing people into groups and saying which group they belong in: the big group or the small group. It's more often used when comparing yourself to family or colleagues, rather than the entire human population. 편이다 compares you to everyone. Another difference is that 편이다 is rather more formal. 축에 들다 shouldn't be used toward elders.
EXAMPLE SENTENCES
저는 라면을 좋아해요. 라면을 자주 먹는 편이에요.
Jeoneun ramyeoneul joahaeyo. Ramyeoneul jaju meokneun pyeonieyo.
I like ramen. I tend to eat ramen a lot.
저 학생은 작은 편이라서 다른 학생보다 더 작은 책상이 필요해요.
Jeo haksaengeun jageun pyeoniraseo dareun haksaengboda deo jageun chaeksangi piryohaeyo.
That student is small (for his grade), so he needs a smaller desk than the other students.
수업에 정시에 오기도 하고 늦기도 해요.
Sueobe jeongsie ogido hago neutgido haeyo.
I sometimes come to class on time and sometimes I come late.
This is a way to talk about things that are known to occur from time to time.
HOW IT'S FORMED
기 turns a verb into a noun. 도 in this case means "too," and 하다 means "to occur" here.
HOW IT'S CONJUGATED
It follows action verbs. 하다 can be conjugated to express the past tense or the future tense.
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 하기도 했다 hagido haetda | 하기도 하다 hagido hada | 하기도 할 것이다 hagido hal geosida
Action verbs ending in a consonant | 먹다 | 먹기도 했다 meokgido haetda | 먹기도 하다 meokgido hada | 먹기도 할 것이다 meokgido hal geosida
TAKE NOTE
This expression and ~(으)ㄹ 수도 있다 ~(eu)l sudo itda are similar, but ~기도 하다 means that ~ actually does happen while ~(으)ㄹ 수도 있다 means that ~ has the potential to happen, whether or not it actually does.
기도하다 or 기도 하다 by itself means "to pray" or "to attempt." 기도 하다 directly following another verb means "~ sometimes happens." Don't get them confused.
You can repeat 기도 하다 in the middle of your sentence and again at the end to say that sometimes you do one thing and sometimes you do another. See the examples.
EXAMPLE SENTENCES
어제 주로 비가 왔어요. 그래도 눈이 오기도 했어요.
Eoje juro biga wasseoyo. Geuraedo nuni ogido haesseoyo.
Yesterday it mainly rained, but occasionally it snowed.
점심에 밥을 먹곤 하지만 라면을 먹기도 해요.
Jeomsime babeul meokgon hajiman ramyeoneul meokgido haeyo.
I usually eat rice for lunch, but sometimes I eat ramen.
어떤 날은 잘 자기도 하지만 어떤 날은 잘 못 자요.
Eotteon nareun jal jagido hajiman eotteon nareun jal mot jayo.
Some days I sleep well, but some days I don't.
친구 하고 길게 이야기를 하다가 보니까 수업에 늦게 됐어요.
Chingu hago gilge iyagireul hadaga bonikka sueobe neutge dwaesseoyo.
I was talking with my friend for a long time and suddenly noticed I was late for class.
You know how sometimes you get carried away doing something for a long time and suddenly you notice something has changed? For example you've been studying Korean for a long time and then suddenly you realize you're able to understand most of what's going on around you? That's what this expression is for.
HOW IT'S FORMED
다가 plus 보다 plus (으)니까. 보다, as you know, means to watch or see; in this expression it means to discover or to notice. 니까, as we saw on page , means "so." Put it all together and you (sort of) get, "After A, I noticed that B." 가 and 까 are both optional, and so A다가 보니까 B, A다 보니까 B, A다가 보니 B, and A다 보니 are all variations of this expression. You can use them interchangeably.
HOW IT'S CONJUGATED
| | All tenses
---|---|---
Action verbs ending in a vowel | 하다 | 하다가 보니까 hadaga bonikka
Action verbs ending in a consonant | 먹다 | 먹다가 보니까 meokdaga bonikka
TAKE NOTE
A is an action that is repeated and/or lasts for some time. If you studied Korean just once and then noticed some improvement, you can't use this expression (but you can use 고 보니 instead.; see page ). If you've been studying for a while, this expression is perfect. It can't be used in the future or with imaginary situations.
As it's an expression used to talk about something that changed, the B clause usually ends with an expression indicating change such aŝ어/아지다 eo/ajida or ~게 되다 ge doeda.
A다가 보면 B is related; it means that if you keep doing A for a long time, B will happen. It's in the section on warnings and can be found on page .
돈을 그렇게 많이 쓰다 보면 돈이 다 없어질 거예요.
Doneul geureoke mani sseuda bomyeon doni da eopseojil geoyeyo.
If you keep spending that much money, you won't have any left.
EXAMPLE SENTENCES
한국어 공부하다 보니 쓰기 능력이 훨씬 더 좋아졌어요.
Hangugeo gongbuhada boni sseugi neungnyeogi hwolssin deo joajyeosseoyo.
After studying Korean (for some time), my writing ability greatly improved.
몇 시간 동안 축구를 하다 보니까 다리가 아파졌어요.
Myeot sigan dongan chukgureul hada bonikka dariga apajyeosseoyo.
After playing soccer for several hours, he hurt his leg.
빨리 세운 계획은 실패하는 법이다.
Ppalli seun gyehoegeun silpaehaneun beobida.
빨리 세운 계획은 실패하기 마련이다.
Ppalli seun gyehoegeun silpaehagi maryeonida.
빨리 세운 계획은 실패하게 마련이다.
Ppalli seun gyehoegeun silpaehage maryeonida.
A plan that's made quickly is bound to fail.
No, not natural like organic oatmeal. Natural like people losing weight if they eat that same organic oatmeal, or gaining weight if they eat only junk food. (으)ㄴ/는 법이다 and 기/게 마련이다 are two interchangeable ways to say that something occurs naturally or is only to be expected.
HOW IT'S FORMED
(으)ㄴ/는 are present tense markers for action verbs and adjectives. 법 means "law," so this expression means that it's "a kind of law that ~."
기 is a way to change a verb into a noun. 게 is a short form of 것이, which is another way to make a noun. 마련이다 is another way to say something is natural.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 하는 법이었다 haneun beobieotda
하기 마련이었다 hagi maryeonieotda
하게 마련이었다 hage maryeonieotda
| 하는 법이다 haneun beobida
하기 마련이다 hagi maryeonida
하게 마련이다 hage maryeonida
| 하는 법일 것이다 haneun beobil geosida
하기 마련일 것이다 hagi maryeonil geosida
하게 마련일 것이다 hage maryeonil geosida
Action verbs ending in a consonant | 먹다 | 먹는 법이었다 meokneun beobieotda
먹기 마련이었다 meokgi maryeonieotda
먹게 마련이었다 meokge maryeonieotda
| 먹는 법이다 meokneun beobida
먹기 마련이다 meokgi maryeonida
먹게 마련이다 meokge maryeonida
| 먹는 법일 것이다 meokneun beobil geosida
먹기 마련일 것이다 meokgi maryeonil geosida
먹게 마련일 것이다 meokge maryeonil geosida
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁜 법이었다 yeppeun beobieotda
예쁘기 마련이었다 yeppeugi maryeonieotda
예쁘게 마련이었다 yeppeuge maryeonieotda
| 예쁜 법이다 yeppeun beobida
예쁘기 마련이다 yeppeugi maryeonida
예쁘게 마련이다 yeppeuge maryeonida
| 예쁜 법일 것이다 yeppeun beobil geosida
예쁘기 마련일 것이다 yeppeugi maryeonil geosida
예쁘게 마련일 것이다 yeppeuge maryeonil geosida
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작은 법이었다 jageun beobieotda
작기 마련이었다 jakgi maryeonieotda
작게 마련이었다 jakge maryeonieotda
| 작은 법이다 jageun beobida
작기 마련이다 jakgi maryeonida
작게 마련이다 jakge maryeonida
| 작은 법일 것이다 jageun beobil geosida
작기 마련일 것이다 jakgi maryeonil geosida
작게 마련일 것이다 jakge maryeonil geosida
Nouns ending in a vowel | 남자 | 남자인 법이었다 namjain beobieotda
남자이기 마련이었다 namjaigi maryeonieotda
남자이게 마련이었다 namjaige maryeonieotda
| 남자인 법이다 namjain beobida
남자이기 마련이다 namjaigi maryeonida
남자이게 마련이다 namjaige maryeonida
| 남자인 법일 것이다 namjain beobil geosida
남자이기 마련일 것이다 namjaigi maryeonil geosida
남자이게 마련일 것이다 namjaige maryeonil geosida
Nouns ending in a consonant | 물 | 물인 법이었다 murin beobieotda
물이기 마련이었다 murigi maryeonieotda
물이게 마련이었다 murige maryeonieotda
| 물인 법이다 murin beobida
물이기 마련이다 murigi maryeonida
물이게 마련이다 murige maryeonida
| 물인 법일 것이다 murin beobil geosida
물이기 마련일 것이다 murigi maryeonil geosida
물이게 마련일 것이다 murige maryeonil geosida
TAKE NOTE
마련하다 means to prepare or arrange; 마련이다 means "to be natural." Be careful with these two.
The expression ~(으)ㄹ 법하다 is similar, but in this case the use of the future (으)ㄹ denotes a general rule which is then contradicted by the situation at hand. It means, "Normally A, but in this case B."
프로 야구 선수들이 야구를 잘 할 법한데 오늘 실수만 해요.
Peuro yagu seonsudeuri yagureul jal hal beopande oneul silsuman haeyo.
Normally professional baseball players are pretty good, but today they're doing nothing but making mistakes.
EXAMPLE SENTENCES
과로하면 피곤한 법이다. Gwarohamyeon pigonhan beobida.
과로하면 피곤하기 마련이다. Gwarohamyeon pigonhagi maryeonida.
과로하면 피곤하게 마련이다. Gwarohamyeon pigonhage maryeonida.
If you work too much, of course you'll be tired.
건강하게 먹으면 살이 빠지는 법이다.
Geonganghage meogeumyeon sari ppajineun beobida.
건강하게 먹으면 살이 빠지기 마련이다.
Geonganghage meogeumyeon sari ppajigi maryeonida.
건강하게 먹으면 살이 빠지게 마련이다.
Geonganghage meogeumyeon sari ppajige maryeonida.
If you eat healthily, you're bound to lose weight.
민수가 대학교 때 자주 수업에 늦게 오기 일쑤였다.
Minsuga daehakgyo ttae jaju sueobe neutge ogi ilssuyeotda.
During university, Minsu was habitually late for class.
This is another, slightly less common way to talk about doing something habitually.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 하기(가) 일쑤였다 hagi(ga) ilssuyeotda | 하기(가) 일쑤이다 | 하기(가) 일쑤일 것이다 hagi(ga) ilssuilgeotsida
Action verbs ending in a consonant | 먹다 | 먹기(가) 일쑤였다 meokgi(ga) ilssuyeotda | 먹기(가) 일쑤이다 | 먹기(가) 일쑤일 것이다 meokgi(ga) ilssuilgeotsida
TAKE NOTE
곤 하다 (see page 349) and 기 일쑤이다 are interchangeable when the sentence is negative. 기 일쑤이다 isn't usually used with positive sentences.
EXAMPLE SENTENCES
남자친구와 헤어진 후에 그는 며칠 동안 계속 저에게 전화하기 일쑤였다.
Namjachinguwa heeojin hue geuneun myeochil dongan gyesok jeoege jeonhwahagi ilssuyeotda.
After I broke up with my boyfriend, he kept on calling me for several days.
가영이는 거짓말하기 일쑤이다.
Kayoungineun geojinmalhagi ilssuida.
Kayoung always lies.
개가 계속 짖어 대서 짜증 났어요.
Gaega gyesok jijeo daeseo jjajeung nasseoyo.
The dog kept barking, so I was annoyed.
This expression is good for people with roommates or those who have to deal with babies crying or dogs barking all night long, or any other situation where someone or something continues to do something annoying.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs with 오 or 아 | 잡다 | 잡아 댔다 jaba daetda | 잡아 대다 jaba daeda | 잡아 댈 것이다 jaba dael geosida
Action verbs with 어, 우, 으 or 이 | 먹다 | 먹어 댔다 meogeo daetda | 먹어 대다 meogeo daeda | 먹어 댈 것이다 meogeo dael geosida
하다 | 하다 | 해 댔다 hae daetda | 해 대다 hae daeda | 해 댈 것이다 hae dael geosida
TAKE NOTE
대다 has a number of meanings outside of this expression: "to put," "to touch," "to provide," "to pay," "to park," "to make an excuse." In these cases, 대다 follows either 을/를 or 에. To distinguish this expression from the others, keep an eye out for the 아/어/여 before 대다 and the context.
아/어/여 대다 is normally used in sentences with negative connotations.
곤 하다 means to do something habitually. If your ex-boyfriend or ex-girlfriend calls you all day long the day after you've broken up with him or her, you can use 아/어/여 대다 to describe the action, but not 곤 하다 because the calling is repeatedly done but is not habitual. If the same ex-boyfriend or ex-girlfriend calls you every day for months on end, then you could use 곤 하다.
EXAMPLE SENTENCES
여자친구가 계속 잔소리를 해 대는데 헤어질까요?
Yeojachinguga gyesok jansorireul hae daeneunde heeojilkkayo?
My girlfriend always nags me; should I break up with her?
선거 때문에 하루 종일 선거 운동원들이 스피커로 소리를 질러 대요.
Seongeo ttaemune haru jongil seongeo undongwondeuri seupikeoro sorireul jilleo daeyo.
Because of the election, campaigners are yelling through speakers all day long.
민수가 늦는 것이 보통이에요.
Minsuga neunneun geosi botongieyo.
민수가 늦는 게 보통이에요.
Minsuga neunneun ge botongieyo.
It's normal for Minsu to be late.
Some things are just not that exciting. Here's how you can talk about something that's normal or usual.
HOW IT'S FORMED
는 것 changes a verb into a noun. You can use 는 게 instead; it's just a contraction of 는 것이. 보통 means "normal" or "usual." So ~ is normal, commonplace, or to be expected.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 하는 것이 보통이었다 haneun geosi botongieotda
하는 게 보통이었다 haneun ge botongieotda
| 하는 것이 보통이다 haneun geosi botongida
하는 게 보통이다 haneun ge botongida
| 하는 것이 보통이 것이다 haneun geosi botongil geosida
하는 게 보통일 것이다 haneun ge botongil geosida
Action verbs ending in a consonant | 먹다 | 먹는 것이 보통이었다 meokneun geosi botongieotda
먹는 게 보통이었다 meokneun ge botongieotda
| 먹는 것이 보통이다 meokneun geosi botongida
먹는 게 보통이다 meokneun ge botongida
| 먹는 것이 보통일 것이다 meokneun geosi botongil geosida
먹는 게 보통일 것이다 meokneun ge botongil geosida
TAKE NOTE
보통 all by itself can mean normal (as in this expression) but also mediocre or average—if you review a date by saying 그 사람은 보통이에요 Geu sarameun botongieyo, it means the person was normal in the sense that they were mediocre, nothing special, and you probably aren't too excited about the possibility of going out with them again.
EXAMPLE SENTENCES
이렇게 높은 점수를 얻는 것이 보통이 아니에요.
Ireoke nopeun jeomsureul eonneun geosi botongi anieyo.
이렇게 높은 점수를 얻는 게 보통이 아니에요.
Ireoke nopeun jeomsureul eonneun ge botongi anieyo.
It's not often that someone gets this high a test score.
저기 온도가 -30도 이하가 되는 것이 보통이에요.
Jeogi ondoga maineoseu samsipdo ihaga doeneun geosi botongieyo.
저기 온도가 -30도 이하가 되는 게 보통이에요.
Jeogi ondoga maineoseu samsipdo ihaga doeneun ge botongieyo.
The temperature there is often lower than -30 degrees.
민수가 또 늦었어요. 그는 학생 중에서 자주 늦는 축에 들지요?
Minsuga tto neujeosseoyo. Geuneun haksaeng jungeseo jaju neunneun chuge deuljiyo?
Minsu's late again. He tends to be later than most of the other students, doesn't he?
This expression, similar to (으)ㄴ/는 편이다 (eu)n/neun pyeonida, means "tends to."
HOW IT'S FORMED
축 means "axis," like on a graph. 들다 means "to belong to."
HOW IT'S CONJUGATED
| | Past | Present
---|---|---|---
Action verbs ending in a vowel | 하다 | 하는 축에 들었다 haneun chuge deureotda | 하는 축에 들다 haneun chuge deulda
Action verbs ending in a consonant | 먹다 | 먹는 축에 들었다 meokneun chuge deureotda | 먹는 축에 들다 meokneun chuge deulda
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁜 축에 들었다 yeppeun chuge deureotda | 예쁜 축에 들다 yeppeun chuge deulda
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작은 축에 들었다 Jageun chuge deureotda | 작은 축에 들다 jageun chuge deulda
TAKE NOTE
This expression shouldn't be used in the future tense.
When using 축에 들다 with action verbs, always add an adverb such as 자주 jaju or 드물게 deumulge.
Just like 편이다, 축에 들다 isn't used to talk about objectively verifiable facts: for instance, "he tends to be a man." You could, however, say that he tends to be manly.
그는 남자인 축에 들어요.
Geuneun namjain chuge deureoyo.
He tends to be a man. (ungrammatical)
그는 남자다운 축에 들어요.
Geuneun namjadaun chuge deureoyo.
He tends to be manly. (just fine)
A similar expression is ~(으)ㄴ/는 편이다 (see page 350). There's a very slight difference in meaning between the two. Think of 편이다 as placing the subject on a scale between 0 and 100. Let's say 0 is 제일 작은 and 100 is 제일 큰. If the subject rates over 50 on that scale, 큰 편이다. With 축에 들다, rather than a scale, you're dividing people into groups and saying which group they belong in: the big group or the small group. It's more often used when comparing yourself to family or colleagues, rather than the entire human population. 편이다 compares you to everyone. Another difference is that 편이다 is rather more formal. 축에 들다 shouldn't be used toward elders.
If you want to use 축에 들다 to compare yourself to others, you can use an expression like 제 가족 중에서 je gajok jungeseo or 제 친구 중에서 je chingu jungeseo. See the examples below.
EXAMPLE SENTENCES
저는 라면을 매일 먹어요. 제 가족 중에서 라면을 자주 먹는 축에 들어요.
Jeoneun ramyeoneul maeil meogeoyo. Je gajok jungeseo ramyeoneul jaju meokneun chuge deureoyo.
I eat ramen every day. Compared to the rest of my family, I eat ramen a lot.
저 학생은 다른 학생보다 작은 축에 들어서 더 작은 책상이 필요해요.
Jeo haksaengeun dareun haksaengboda jageun chuge deureoseo deo jageun chaeksangi piryohaeyo.
That student is small compared to the others, so he needs a smaller desk.
그렇게 준비를 하다가 말다가 하면 늦을 거예요.
Geureoke junbireul hadaga maldaga hamyeon neujeul geoyeyo.
If you keep on getting ready and then stopping like that, you'll be late.
Are you someone who cannot commit to something, such as having trouble keeping New Year's resolutions? If so, this expression's for you.
HOW IT'S FORMED
다가 plus 말다. 말다 basically means "to not be that way" and is used much in expressions indicating indecision: for example, 하거나 말거나 hageona malgeona (see page ) or 하는 둥 마는 둥 haneun dung maneun dung (see page ). So the literal meaning of this expression is "to do ~ and then not do ~ and then." It's normally used for something that is done intermittently. 가 is optional.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 하다가 말다가 했다 hadaga maldaga haetda | 하다가 말다가 하다 hadaga maldaga hada | 하다가 말다가 할 것이다 hadaga maldaga hal geosida
Action verbs ending in a consonant | 먹다 | 먹다가 말다가 했다 meokdaga maldaga haetda | 먹다가 말다가 하다 meokdaga maldaga hada | 먹다가 말다가 할 것이다 meokdaga maldaga hal geosida
TAKE NOTE
The subjects of clause A and clause B should be the same.
EXAMPLE SENTENCES
운동을 하다가 말다가 하면 근육에 무리가 갈 거예요.
Undongeul hadaga maldaga hamyeon geunyuge muriga gal geoyeyo.
If you keep exercising on and off like that, it will be too stressful for your muscles.
공부하다가 말다가 해서 시험 준비를 잘 못했어요.
Gongbuhadaga maldaga haeseo siheom junbireul jal motaesseoyo.
I studied on and off, so I wasn't well-prepared for the test.
##
Faking it
정말 늦게 가고 있어서 다른 약속이 있는 척할 거예요.
Jeongmal neutge gago isseoseo dareun yaksogi inneun cheokal geoyeyo.
정말 늦게 가고 있어서 다른 약속이 있는 체할 거예요.
Jeongmal neutge gago isseoseo dareun yaksogi inneun chehal geoyeyo.
I'm really late, so I'm going to pretend I have another appointment.
These two expressions go together because they're totally interchangeable. They're good for con men and women out there because they're both used to talk about pretending to do or be something.
HOW IT'S CONJUGATED
| | Past | Present | Future
---|---|---|---|---
Action verbs ending in a vowel | 하다 | 한 척했다 han cheokaetda
한 체했다 han chehaetda
| 하는 척하다 haneun cheokada
하는 체하다 haneun chehada
| 하는 척 할 것이다 haneun cheok hal geosida
하는 체 할 것이다 haneun che hal geosida
Action verbs ending in a consonant | 먹다 | 먹은 척했다 meogeun cheokaetda
먹은 체했다 meogeun chehaetda
| 먹는 척하다 meokneun cheokada
먹는 체하다 meokneun chehada
| 먹는 척 할 것이다 meokneun cheok hal geosida
먹는 체 할 것이다 meokneun che hal geosida
Descriptive verbs (adjectives) ending in a vowel | 예쁘다 | 예쁜 척했다 yeppeun cheokaetda
예쁜 체했다 yeppeun chehaetda
| 예쁜 척하다 yeppeun cheokada
예쁜 체하다 yeppeun chehada
| 예쁜 척 할 것이다 yeppeun cheok hal geosida
예쁜 체 할 것이다 yeppeun che hal geosida
Descriptive verbs (adjectives) ending in a consonant | 작다 | 작은 척했다 jageun cheokaetda
작은 체했다 jageun chehaetda
| 작은 척하다 jageun cheokada
작은 체하다 jageun chehada
| 작은 척 할 것이다 jageun cheok hal geosida
작은 체 할 것이다 jageun che hal geosida
Nouns ending in a vowel | 남자 | 남자인 척했다 namjain cheokaetda
남자인 체했다 namjain chehaetda
| 남자인 척하다 namjain cheokada
남자인 체하다 namjain chehada
| 남자인 척 할 것이다 namjain cheok hal geosida
남자인 체 할 것이다 namjain che hal geosida
Nouns ending in a consonant | 학생 | 학생인 척했다 haksaengin cheokaetda
학생인 체했다 haksaengin chehaetda
| 학생인 척하다 haksaengin cheokada
학생인 체하다 haksaengin chehada
| 학생인 척 할 것이다 haksaengin cheok hal geosida
학생인 체 할 것이다 haksaengin che hal geosida
TAKE NOTE
A(으)ㄴ/는 채 B means to be in a certain state. Aside from the different spelling, you can tell it apart from 체 here because 채 isn't followed by 하다 but rather by a sentence explaining what was done while in the state of A. 체 as seen in this section will always be followed by 하다.
알다 and 모르다 are very commonly used with 척하다.
아는 척하다
aneun cheokada
to pretend to know
모르는 척하다
moreuneun cheokada
to pretend not to know
EXAMPLE SENTENCES
민수가 싫으니까 거리에서 봤을 때 모르는 척했어요.
Minsuga sireunikka georieseo bwasseul ttae moreuneun cheokaesseoyo.
민수가 싫으니까 거리에서 봤을 때 모르는 체했어요.
Minsuga sireunikka georieseo bwasseul ttae moreuneun chehaesseoyo.
민수가 싫으니까 거리에서 봤을 때 모른 척했어요.
Minsuga sireunikka georieseo bwasseul ttae moreun cheokaesseoyo.
민수가 싫으니까 거리에서 봤을 때 모른 체했어요.
Minsuga sireunikka georieseo bwasseul ttae moreun chehaesseoyo.
I don't like Minsu, so when I saw him in the street I pretended not to recognize him.
할인 받고 싶어서 학생인 척했어요.
Harin batgo sipeoseo haksaengin cheokaesseoyo.
할인 받고 싶어서 학생인 체했어요.
Harin batgo sipeoseo haksaengin chehaesseoyo.
I wanted to get a discount, so I pretended I was a student.
#
3 Frequently Seen Word Parts
~답다 dapda "typical of ~"
답다 is used to talk about qualities like manliness, womanliness, etc.: qualities that are typical of a certain noun. The most well-known example is 아름답다 (to be beautiful), but you can also add 답다 to any other noun you want.
Add 답다 to the end of the noun which describes the category. The rest of your sentence should describe how ~ is typical or not typical of that category.
> 남자 | 남자답다 namjadapda
> ---|---
> 아름 | 아름답다 areumdapda
Note that 답다 can't be used to describe something that isn't already ~. For instance, you can't call a man 여자답다 yeojadapda. If you do want to talk about a girly man, try 스럽다 seureopda (see the next expression on this page). You can call a man 남자답다 if you think he has all the qualities that make him a perfect specimen of a man. Alternatively, you can say he's 남자답지 않다 namjadapji anta if he's not at all manly.
> 가영이 자주 쇼핑을 가요. 정말 여자다워요.
>
> Kayoungi jaju syopingeul gayo. Jeongmal yeojadawoyo.
>
> Kayoung goes shopping a lot. She's really a typical girl.
>
> 제 남자친구는 늘 예쁜 것을 사주고 문을 열어줘요. 신사답지요?
>
> Je namjachinguneun neul yeppeun geoseul sajugo muneul yeoreojwoyo. Sinsadapjiyo?
>
> My boyfriend always buys me pretty things and opens doors for me. He's really gentlemanly, don't you think?
>
> 이번 방학 때 아름다운 곳에 가고 싶어요
>
> Ibeon banghak ttae areumdaun gose gago sipeoyo.
>
> This vacation, I want to go somewhere beautiful.
~스럽다 seureopda "appears to be ~"
스럽다 is all about appearances. It means that something appears to be a certain way. You'll see it used all the time as part of certain words in 자연스럽다, 어른스럽다, etc..
> 자연 | 자연스럽다 jayeonseureopda | to be natural
> ---|---|---
> 어른 | 어른스럽다 eoreunseureopda | to be mature
> 걱정 | 걱정스럽다 geokjeongseureopda | to be worried
> 사랑 | 사랑스럽다 sarangseureopda | to be loving
> 자랑 | 자랑스럽다 jarangseureopda | to be proud
Unlike 답다, you can't use 스럽다 to describe something that actually ~. For example, you can't use 어른스럽다 to describe an actual adult, but you can use it to describe a child. If you want to say an adult is especially adult-like, use 답다 instead (see the previous page).
> 민수 때문에 걱정스러워요. 요즘 늦은 밤에 집에 돌아오곤 해요.
>
> Minsu ttaemune geokjeongseureowoyo. Yojeum neujeun bame jibe doraogon haeyo.
>
> I'm really worried about Minsu. These days he usually comes home late at night.
>
> 우리 딸이 1등상을 받았어요. 얼마나 자랑스러워요.
>
> Uri ttari ildeungsangeul badasseoyo. Eolmana jarangseureowoyo.
>
> My daughter got first prize. I'm so proud!
>
> 그는 여성스러운 남자일 뿐이에요.
>
> Geuneun yeoseongseureoun namjail ppunieyo.
>
> He's nothing but a girly man.
~롭다 ropda "to be ~"
롭다 is similar to 스럽다 (see just above), but it's used only after certain nouns ending with vowels.
> 자유 | 자유롭다 jayuropda | to be free
> ---|---|---
> 새 | 새롭다 saeropda | to be new
> 향기 | 향기롭다 hyanggiropda | to be fragrant
> 해 | 해롭다 haeropda | to be harmful
> 명예 | 명예롭다 myeongyeropda | to be honorable
>
> 자유롭게 살고 싶어요.
>
> Jayuropge salgo sipeoyo.
>
> I want to live freely.
>
> 어제 새로운 가방을 샀어요.
>
> Eoje saeroun gabangeul sasseoyo.
>
> Yesterday I bought a new bag.
>
> 그 예쁜 꽃이 놀랍게도 향기롭지 않아요.
>
> Geu yeppeun kkochi nollapgedo hyanggiropji anayo.
>
> Surprisingly, that beautiful flower doesn't smell good.
~거리다 georida "does ~ repeatedly"
There are a number of verbs which end in 거리다. In these cases, 거리다 means "to do repeatedly." You'll see it used with many motion verbs and onomatopoeic verbs.
> 깜빡거리다 kkamppakgeorida | to flicker
> ---|---
> 덜렁거리다 deolleonggeorida | to jingle
> 중얼거리다 jungeolgeorida | to mutter
> 비실거리다 bisilgeorida | to stagger
> 씩씩거리다 ssikssikgeorida | to gasp, pant
> 낄낄거리다 kkilkkilgeorida | to giggle
>
> 등산할 때 씩씩거렸어요.
>
> Deungsanhal ttae ssikssikgeoryeosseoyo.
>
> While mountain-climbing, I was panting.
>
> 선생님이 보지 않을 때 아이들이 다 중얼거리고 있어요.
>
> Seonsaengnimi boji aneul ttae aideuri da jungeolgeorigo isseoyo.
>
> While the teacher isn't looking, the children are whispering.
>
> 촛불이 깜빡거리고 있어요.
>
> Chotburi kkamppakgeorigo isseoyo.
>
> The candle is flickering.
##
Talking about People
You'll have noticed that Koreans use different words that all essentially mean "person." There are a couple of reasons for this, and they have to do with both the Korean language's hanja roots and with the great importance the language places on people's ranks.
■ 인, 자, 사, 원, 가, 부, 관 in, ja, sa, won, ga, bu, gwan
These words are all derived from hanja.
인 (人) is used all over the place, and it simply means "person." It's never used by itself and only rarely as a job title.
> 외국인 oegugin | foreigner
> ---|---
> 한국인 hangugin | Korean (person)
> 부인 buin | wife
> 군인 gunin | soldier
자 (者) also simply means person. It's sometimes used as a job title.
> 기술자 gisulja | engineer, technician
> ---|---
> 저자 jeoja | writer
> 피해자 pihaeja | victim
> 기자 gija | reporter
사 (師) means "master" or "teacher." It's used for professions that require some level of mastery.
> 교사 gyosa | teacher
> ---|---
> 사진사 sajinsa | photographer
> 의사 uisa | doctor
> 간호사 ganhosa | nurse
> 기사 gisa | driver
원 (員) means "member" and is used for situations where the person is a member of some organization.
> 회사원 hoesawon | office worker
> ---|---
> 사무원 samuwon | office worker, clerk
> 직원 jigwon | employee
> 은행원 eunhaengwon | bank teller
가 (家) has quite a few different meanings such as "house" (it's also the 가 in 가족) and "shop," but when it's used to talk about people it also implies some level of expertise or specialization.
> 화가 hwaga | painter
> ---|---
> 작가 jakga | writer
> 전문가 jeonmunga | expert
부 (夫) is also the male 부 in 부부 (couple). It means "man," but when applied to jobs it's used for either a man or a woman. Jobs ending in 부 tend to be of the blue-collar kind.
> 어부 eobu | fisherman
> ---|---
> 농부 nongbu | farmer
> 광부 gwangbu | miner
> 청소부 cheongsobu | cleaner
> 배달부 baedalbu | deliveryman
관 (官) is also used to talk about government buildings. It's used to describe people in government positions or official capacities.
> 경찰관 gyeong-chal-gwan | police officer
> ---|---
> 소방관 sobanggwan | firefighter
■ 꾼, 쟁이, 뱅이, 꾸러기, 광 kkun, jaengi, baengi, kkureogi, gwang
These words have nothing to do with hanja. They're pure, 100% Korean. With the exception of 꾼, they're not usually used to describe job titles, but just people who do a certain thing a lot. There's not much difference between them, but they're paired with different words. You just have to remember which ending follows which words.
> 일꾼 ilkkun | laborer
> ---|---
> 나무꾼 namukkun | lumberjack
> 노름꾼 noreumkkun | gambler
> 사냥꾼 sanyangkkun | hunter
> 누리꾼 nurikkun | habitual Internet user (Netizen)
> 점쟁이 jeomjaengi | fortuneteller
> ---|---
> 멋쟁이 meotjaengi | fashionable person
> 무식쟁이 musikjaengi | ignorant person
> 겁쟁이 geopjaengi | coward
> 욕심쟁이 yoksimjaengi | greedy person
> 게으름뱅이 geeureumbaengi | lazy person
> ---|---
> 가난뱅이 gananbaengi | poor person
> 주정뱅이 jujeongbaengi | drunk
> 비렁뱅이 bireongbaengi | beggar, hobo
> 장난꾸러기 jangnankkureogi | someone who always plays
> ---|---
> 잠꾸러기 jamkkureogi | someone who always sleeps
> 말썽꾸러기 malsseongkkureogi | someone who always makes trouble
> 심술꾸러기 simsulkkureogi | a cynic, grumpy person
> 스포츠광 seupocheugwang | sports fan
> ---|---
> 낚시광 naksigwang | fishing maniac
> 살인광 saringwang | homicidal maniac
> 영화광 yeonghwa-gwang | movie buff
■ 사람, 명, 분, 인분 saram, myeong, bun, inbun
These are all just different words that mean "person." 사람 is general and can be used in almost any situation. 명 and 분 are counting markers for people, so you'll talk about 한명, 세명, 네명 hanmyeong, semyeong, nemyeong, and so on. 분 is very formal while 명 is much less so. You'd use 명 to talk about how many people are in your own party while the waiter would use the more polite 분 to talk about that same party. 인분 actually means "servings" (or "excrement") and maybe the two aren't that far apart in certain restaurants.... Thus, you use it when you order food: steak for three people would be 스테이크 삼 인분 주세요. Seuteikeu sam inbun juseyo.
##
Prefixes
Prefix | Meaning | Example Words
---|---|---
외 oe | the only one | 외톨이 oetori (loner), 외아들 oeadeul (only son), 외딸 oettal (only daughter)
되 doe | again | 되찾다 doechatda (to recover), 되묻다 doemutda (to ask again), 되돌아가다 doedoragada (to go back again)
날 nal | raw, uncooked | 날생선 nalsaengseon (raw fish), 날고기 nalgogi (raw meat), 날계란 nalgyeran (raw egg)
풋 put | unripe | 풋사랑 putsarang (puppy love), 풋사과 putsagwa (unripe apple), 풋고추 putgochu (unripe pepper)
군 gun | unnecessary, useless | 군침 gunchim (drooling, e.g. at the smell of food), 군살 gunsal (extra weight), 군말 gunmal (useless chatter)
헛 heot | unnecessary, useless | 헛기침 heotgichim (fake cough), 헛걸음 heotgeoreum (fool's errand), 헛소문 heotsomun (baseless rumor)
덧 deot | in addition, plus | 덧붙이다 deotbuchida (to append), 덧칠 deotchil (to paint over), 덧셈 deotsem (addition)
맨 maen* | bare | 맨발 maenbal (barefoot), 맨얼굴 maeneolgul (without makeup), 맨손 maenson (empty-handed)
* This isn't the same as the 맨 that means "most" (맨 위에 maen wie, 맨 뒤에 maen dwie, etc.). This one is placed directly in front of the word it modifies.
#
Appendix 1:
How to Sound Like a Native
##
Adverbs of Time
Most adverbs are made using 게 or 히; see page to find out how that works. However, there are a significant number of very useful adverbs that don't follow these patterns. They're summarized below.
> 일찍 iljjik | early
> ---|---
> 일단 ildan | once, first of all
> 비로소 biroso | at last, for the first time
> 돌연 dolyeon | suddenly
> ---|---
> 갑자기 gapjagi | suddenly
> 별안간 byeolangan | suddenly
> 느닷없이 neudateopsi | suddenly
> 뚝 dduk | (something was done) suddenly
> ---|---
> 어느덧 eoneudeot | before we knew it
> ---|---
> 어느새 eoneuse | before we knew it
> 냉큼 naengkeum | immediately
> ---|---
> 당장 dangjang | immediately
> 얼른 eolleun | quickly, promptly, eagerly
> ---|---
> 빨리 ppali | quickly, promptly, eagerly
> 급히 geupi | quickly, promptly, eagerly
> 즉시 jeuksi | quickly, promptly, eagerly, immediately
> ---|---
> 곧 got | quickly, promptly, eagerly, immediately
> 금세 geumse | instantly, at once
> ---|---
> 금방 geumbang | instantly, at once
> 바로 baro | at once, immediately, directly, straight, right, exactly
> ---|---
> 새로 sero | newly
> ---|---
> 이미 imi | already
> 벌써 beolsseo | already
> 간밤 ganbam | last night
> ---|---
> 어제 eoje | yesterday
> 오늘 oneul | today
> 그저께 geujeokke | the day before yesterday
> 내일 naeil | tomorrow
> 모래 more | the day after tomorrow
> 당일 dangil | the same day
> 동시에 dongsie | simultaneously
> ---|---
> 한꺼번에 hankkeobeone | simultaneously, all at once
> 확 hwak | (to do something) completely, all at once
> 뜬금없이 tteungeumeopsi | all of a sudden
> ---|---
> 마지막 majimak | the last
> 드디어 deudieo | finally
> ---|---
> 마침내 machimne | finally
> 결국 gyeolguk | eventually
> ---|---
> 계속 gyeosok | continuously
> ---|---
> 당분간 dangbungan | for the time being
> 한참 hancham | for some time
> 슬슬 seulseul | gradually, taking one's time
> ---|---
> 여태 yeote | still, so far, yet
> ---|---
> 아직 ajik | still, so far, yet
> 여전히 yeojeonhi | still, as ever
> ---|---
> 아까 akka | a while ago
> ---|---
> 막 mak | just now
> ---|---
> 방금 banggeum | just now
> 이제 ije | now
> ---|---
> 지금 jigeum | now
> ---|---
##
Adverbs of Frequency
> 평소(에) pyeongso(e) | usually, ordinarily, always
> ---|---
> 늘 neul | always
> 항상 hangsang | always
> 언제나 eonjena | always
> 언제쯤 eonjejjeum | always
> 계속 gyesok | continuously
> ---|---
> 끊임없이 ggeunimeopsi | continuously
> 자꾸 jakku | often, incessantly, persistently
> ---|---
> 빈번히 binbeoni | frequently
> 자주 jaju | often
> 수시로 susiro | frequently, often
> 가끔 gakkeum | sometimes
> ---|---
> 간혹 ganhok | sometimes, occasionally
> 이따금 ittageum | sometimes
> 종종 jongjong | sometimes, often
> 띄엄띄엄 ttuieomttuieom | occasionally
> 드문드문 deumundeumun | sometimes, occasionally
> 틈틈이 teumteumi | at spare moments
> ---|---
> 드물게 deumulge | rarely
> ---|---
> 한때 hantte | once
> ---|---
> 결코 gyeolko | never, by no means
> ---|---
> 절대로 jeoldero | never, by no means
##
Adverbs of Degree
■ None
> 전혀 jeonhyeo | not at all
> ---|---
> 절대로 jeoldero | never, absolutely not
> 도무지 domuji | entirely, never (with negatives)
> 통 tong | entirely, never (with negatives)
■ Not much
> 오직 ojik | only
> ---|---
> 적어도 jeokeodo | at least
> 그리 geuri | not much
> 그다지 geudaji | not much
> 좀처럼 jomcheoreom | rarely, not easily (usually with negative expressions)
> 살며시 salmyeosi | furtively, secretly
> 빠듯이 ppadeutsi | barely, narrowly
> 번쩍 beonjjeok | lightly
> 겨우 gyeou | hardly
> 덜 deol | not much, less than
> 여간 yeogan | rarely, not much (used with the opposite of whatever you're really trying to say)
■ Some
> 적당히 jeokdanghi | somewhat
> ---|---
■ A lot
> 더 deo | more
> ---|---
> 흔히 heunhi | commonly
> 꽤 ggue | fairly, much
> 푹 ggwae | (to rest) deeply, soundly
> 특히 teuki | especially
> 무척 mucheok | very
> 어찌나 eojjina | so
> 훨씬 hweolssin | greatly, by far, much
> 맨 maen | first, most, very
> 유난히 yunanhi | especially
> 하도 hado | too
> 깊이 gipi | deeply
> 워낙 wonak | originally, very, by nature
> 원래 wonre | originally, very, by nature
> 무려 muryeo | as much as
■ Almost entirely
> 거의 geoui | almost
> ---|---
> 대충 daechung | roughly, in general
■ Entirely
> 완전히 wanjeonhi | completely
> ---|---
> 수없이 sueopsi | innumerably
> 온통 ontong | all, entirely
> 가득 gadeuk | full
> 잔뜩 jantteuk | full
> 정성껏 jeonseongkkeot | with all one's heart
> 실컷 silkeot | to one's heart's content, as much as possible
> 마음껏 maeumkkeot | to one's heart's content
##
Giving Your Opinion and Sounding Smart
Should you ever find yourself in a serious debate or having to write a serious essay in Korean, here are a few expressions you can use to state your opinion.
First of all, ~(이)라고/(ㄴ/는)다고 보다 and 생각하다 are expressions you can use to end a sentence that states what you think. They are on page .
Secondly, many variants of 그렇다 are commonly used when starting sentences (see the next page).
> ~말할 나위도 없다 malhal nawido eopda it goes without saying that ~
>
> A뒤에는 B이/가 있다 A dwuieneun B i/ga itda B is behind A
>
> A(으)로 하여금 B게 하다 A (eu)ro hayeogeum B ge hada make A do B
>
> A이/가 B~에 영향을 주다/미치다 A i/ga B ye yeonghyangeul juda/michida A influences B
>
> A(으)ㄹ 겸 해서 B A (eu)r gyeom haeseo B A is one of many possible causes for B
>
> A(으)ㄴ/는 반면에 B A (eu) n/neun banyeone B B, as opposed to A
>
> ~에 맞서(서) e matseo (seo) in opposition to ~
>
> A은/는 B에 비해서/비하면 A eun/neun B e bihaeseo/bihamyeon A, compared to B
>
> 다른 어느 ~보다 dareun eoneu ~ boda more than any other ~
>
> ~(으)로 여기다 (eu)ro yeogida consider/deem as ~
>
> ~게/것이 분명하다 ge/gutsi bunmyeonghada it is obvious that ~
>
> A면서 B(이)가 C게 되다 A myeonseo B(i)ga C ge doeda as A happens, B becomes C
>
> ~(이)라/(ㄴ/는)다니까요 (i)ra (n/neun) danikkayo re-emphasizes ~ (something the speaker already said) forcefully
>
> A의 입장에서는 B A eui ipjangeseoneun B B, from A's point of view
>
> ~에 의해서/의하면 e euihaeseo/euihamyun according to ~
>
> ~에서 보는 바와 같이 eseo boneun bawa gachi as you can see in ~
>
> A(이)란/(ㄴ/는)단 말이 나왔으니까 B A(i)ran/(n/neun)dan mali nawasseunikka B having mentioned A, B
>
> 다음과 같은 daeumgwa gateun like the following
>
> 위에서 (앞에서) 언급했듯이 wieseo (apeseo) eongeuphaetdeusi as I mentioned before/above
>
> 앞서 제시했던 것과 같이 apseo jesihaetdeon geotgwa gachi as I mentioned earlier
>
> 이상에서 알 수 있는 것과 같이 isangeseo al su itneun geotgwa gachi as we can learn from the above
>
> ~(이)라/(ㄴ/는)다고 할 수 있다 (i)ra/(n/neun) dago hal su itda you can say that ~
>
> 적게는 A에서 많게는 B까지 jeokgeneun A eseo mankeneun B kkaji from at least A to a maximum of B
>
> 제가 보기에는 jega bogieneun From what I've seen
>
> 제 생각으로는/제 생각에는 je saenggakeuroneun/je saenggakeneun I think
>
> 제가 말씀드리고 싶은 것은 jega malsseumdeurigo sipeun geoseun What I want to say is that
>
> 저는 이렇게 생각합니다 jeoneun ireoke saenggakhapnida This is what I think.
>
> 제 말의 의미는 je malui uimineun What I mean is that
>
> 저는 ~씨와 생각이 같습니다 jeoneun ~ssiwa saenggaki gatseubnida I agree with ~
##
How to Deal with 그렇다
그렇다, which means "to be like that," can be used to mean many things that may not be immediately obvious. It's quite often used to take a break in the middle of a long sentence. Just as you might drone on in English, pause for a second and then start again with "However" or "Therefore," you can use 그렇다 for similar purposes in Korean. It can be conjugated just like any other descriptive verb. Here are some useful words with 그렇다.
> 그래(요) geure(yo) Yes/I agree/Uh-huh
>
> 그렇구나/그렇군요 geureotkuna/geureotkunyo Oh, really!/Oh, I see!
>
> 그렇지(요) geureotchi(yo) Yes/Absolutely/Uh-huh/That's true, isn't it!
>
> 그렇고말고(요) geureotkomalgo(yo) Of course!/Absolutely!/Definitely!/Obviously!
> 그래서 geureseo Therefore/so
>
> 그러니까 geureonikka Therefore/so
>
> 그러므로 geureomeuro Therefore
>
> 그러하다 geureohada So, like that
> 그러면 geureomyeon If so, then
>
> 그렇다고 geureotago If you say so, then
> 그렇지만 geureochiman But
>
> 그러나 geureona But
>
> 그래도 geuredo In spite of that
> 그렇지 않아도 geureochi anado Even if you hadn't said/done that
#
Appendix 2:
Useful Korean Language Resources
■ Beginning Level
If you're not quite ready for this book yet or you want to review basic grammar, there are a few resources that I found invaluable. The Talk to Me in Korean website does a fantastic job of explaining different expressions, how to form them, and how to use them. It also has much information about real-life Korean language, including vocabulary, useful hanja characters, and stories in Korean. The website is http://www.talktomeinkorean.com/
Another good website is Sogang's free online class. The website is http://korean.sogang.ac.kr and it has various activities and lessons for beginning levels. Activities are quite good in helping you review, especially if you're studying on your own.
Stephen Revere's book Survival Korean is another great beginning level book. He's very good at explaining things in ways that make sense, and he makes the concepts easy to remember. Revere, Stephen. Survival Korean. Nexus, 2005.
Finally, if you are a little more advanced 가나다, Wild Korean by 안상현, is a very good book to help you speak Korean in various everyday situations. It has a list of useful vocabulary for each section along with grammar points and expressions to help you. If you want to learn how to speak Korean when you go to the hospital, to the supermarket, to the bank, and so on, this is the book for you. Ahn Sanghyun. Wild Korean, BookShelf Publishing
■ Intermediate Level
Obviously I'm going to recommend this book...
■ Advanced Level
If this book not enough, there are a two comprehensive reference guides I frequently referred to while writing this grammar reference book.
한국어 문법 사전 (translated roughly as Korean Grammar Dictionary [by Hangukeo Munbeop Sajeon]) covers just about anything you'd ever need. It's all in Korean, so you need to know your grammatical terms if you want to look anything up. But it's all there, and the author does a very good job of explaining when and how each expression can be used.
Finally, if you're very serious about learning grammar, A Reference Grammar of Korean is exactly what it claims to be. However, it's a thick book, is difficult to understand, and uses a strange Romanization system instead of actual Korean characters. It's really intended more for linguists than for students of Korean, but it is about the most comprehensive resource available and there is much additional material on hanja, North and South Korean dialects, and many other topics. Martin, Samuel E. A Reference Grammar of Korean, Tuttle Publishing, 2006.
■ Textbook Series
The best series I've seen to date is Korean Grammar in Use. It goes from beginning through intermediate levels at the time of writing this book. It organizes everything by purpose, just like I did, and compares different expressions so you know when to say what. Korean Grammar in Use is published by 다락원.
#
Index
Using the Index: An Introduction
The index lists grammatical points alphabetically using the Korean alphabet. In cases where an expression changes forms depending on whether it follows a consonant or vowel, the vowel form is used. For example, let's say you want to look up 먹을 거예요 meogeul geo~yeyo. You can tell that the grammatical point 을 거예요 eur geoyeyo follows the verb 먹다, and you can guess that 을 probably wouldn't follow a vowel and that the vowel form of this expression is probably ㄹ 거예요. And you're right!
Tips
Parentheses indicate that a part of an expression is not always used. When you see (으)ㄹ 거예요, this means that sometimes the expression is used as 을 거예요 eur geoyeyo and sometimes as ㄹ 거예요 r geoyeyo.
이 and 으 are the usual vowels used when an expression changes forms to follow a consonant, so if you want to look up something starting with one of those, try starting with the letter right after the 이 or 으 (e.g. for (이)라고 (i)rago, look up 라고).
In cases where expressions change at the beginning in other ways (e.g. the many that begin with (으)ㄴ, 는 or (으)ㄹ), I've listed all the forms separately so you can easily find them without having to guess which variations are applicable to the particular expression you want to learn about.
In some cases, the ends of expressions may look different depending on the level of politeness or dictionary form versus the form used in speech. For example, the (으)ㄹ 거예요 above is actually (으)ㄹ 것이다 (eu)r geosida in its dictionary form. In cases like that which can be confusing, I've tried to list both forms; however, if you don't see what you're looking for but see something similar, try looking it up to see if it's what you're after. With a couple of unique exceptions, all levels of politeness of expressions are NOT listed here – you will not find (으)ㄹ 거야 (eu)r geoya or (으)ㄹ 겁니다 (eu)r geomnida, respectively the familiar and super-polite forms of (으)ㄹ 거예요). This is simply because it would make the index really long and because changes in level of politeness are usually consistent and done at the end of the sentence anyway, so as long as you can find the beginning of the expression, you can usually make an educated guess about whether or not it matches what you want to find.
Combinations of expressions are not usually listed or covered in this book, simply because there are too many and it would make the book very long and repetitive. So if, for example, you want to find something like 라고 해서 rago haeseo, and you don't see it in the index but you do see 라고 하다, look that up first and then look up the second part 여서. (In fact, reported speech combinations such as that one have their own section on page ; other combinations, however, do not.)
Finally, if you need a refresher, here's the order of the South Korean alphabet:
> ㄱ ㄲ ㄴ ㄷ ㄸ ㄹ ㅁ ㅂ ㅃ ㅅ ㅆ ㅇ ㅈ ㅉ ㅊ ㅋ ㅌ ㅍ ㅎ
>
> ㅏ ㅐ ㅑ ㅒ ㅓ ㅔ ㅕ ㅖ ㅗ ㅘ ㅙ ㅚ ㅛ ㅜ ㅝ ㅞ ㅟ ㅠ ㅡ ㅢ ㅣ
Syllables are very important in alphabetical order. For example, the syllable 게 ge comes before 겠 gess alphabetically, so an expression like 게 하다 ge hada comes before 겠다 gessda even though ㅎ follows ㅆ in the alphabet.
Note: Page numbers correspond to the print edition.
가 30
가 만 못하다 134
가 아니고 118
가 아니라 118
가 아니면 121, 284
가 에 뿌리를 두다 90
가 에서 비롯되다 90
가다 31
가지다 31
같아선 183
같으면 284, 293
같은 48
거나 61
거나 말거나 40, 61
거든(요) 70, 285
건 건 65
걸 그랬어요 173
게 되다 25
게 마련이다 354
게 하다 205
게(요) 152, 165
게끔 165
게시리 165
겠다 239
겠던데 189
고 49, 51
고 나서 50, 272
고 말 테니까 232
고 말겠다 50, 232
고 말고요 37, 50
고 말았다 50, 233
고 보니(까) 50, 282
고 싶다 178
고 있다 50, 255
고 해서 50, 94
고는 50
고는 하다 51, 349
고도 50, 108
고서 50, 280
고서는 50, 301
고서야 50, 281
고자 50, 229
곤 하다 51, 349
과 45
과(는) 달리 124
구나 172
군요 172
기 22
기 마련이다 354
기 바라다 180
기 쉽다 298
기 십상이다 298
기 어렵다 298
기 일쑤이다 356
기(가) 무섭게 273
기는 하는데 113
기는 한데 113
기는요 40
기는커녕 41
기도 하다 352
기란 331
기로 하다 225
기로(는) 89
기로서니 116
기에 73, 325
기에 망정이다 176
길래 73
까지 31, 340
께 31
께서 31
끼리 47
(으)ㄴ 23, 237, 238
(으)ㄴ 것 22
(으)ㄴ 것 같다 137
(으)ㄴ 것 처럼 125
(으)ㄴ 경우에(는) 290
(으)ㄴ 김에 250
(으)ㄴ 까닭에 85
ㄴ 답니다 330
(으)ㄴ 대로 320
(으)ㄴ 대신에 119
(으)ㄴ 덕분에 80
(으)ㄴ 데 334
(으)ㄴ 듯(이) 128
(으)ㄴ 듯싶다 153
(으)ㄴ 듯하다 153
(으)ㄴ 만큼 126
(으)ㄴ 모양이다 145
(으)ㄴ 문론 57
(으)ㄴ 바 324
(으)ㄴ 바람에 79
(으)ㄴ 반면(에) 109
(으)ㄴ 반해 109
(으)ㄴ 법이다 354
(으)ㄴ 셈이다 154
(으)ㄴ 셈치고 292
(으)ㄴ 셈치고는 86
(으)ㄴ 적(이) 없다 187
(으)ㄴ 적(이) 있다 187
(으)ㄴ 줄 몰랐다 159
(으)ㄴ 줄 알았다 159
(으)ㄴ 지 276
(으)ㄴ 지 이/가 넘다 276
(으)ㄴ 지 이/가 지나다 276
(으)ㄴ 채(로) 259
(으)ㄴ 탓에 79
(으)ㄴ 통에 79
(으)ㄴ 한 338
(으)ㄴ가 보다 138
(으)ㄴ가 하면 100, 284
(으)ㄴ가요 44
(으)ㄴ걸(요) 149
(으)ㄴ게 틀림없다 164
ㄴ다 ㄴ다 하는 게 230
ㄴ다거나 64
ㄴ다고 믿다 147
ㄴ다고 보다 147
ㄴ다고 생각하다 147
ㄴ다고 하다 305
ㄴ다고 하면 284
ㄴ다기보다(는) 122
ㄴ다는 것이 234
ㄴ다는 말이다 318
ㄴ다든가 64
ㄴ다면 283, 286
(으)ㄴ데(요) 69
(으)ㄴ데다(가) 53
(으)ㄴ데다(가) 268
ㄴ데도 107
(으)ㄴ지 만에 276
(으)ㄴ지 이/가 되다 276
(으)ㄴ지(도) 모르다 160
(으)ㄴ지(도) 알다 160
(으)ㄴ척하다 362
(으)ㄴ축에 들다 359
(으)ㄴ편이다 350
(으)나 110
(이)나 60
(이)나 다름없다 131
(으)나 마나 91
나 보다 138
나요 44
내다 32
(이/으)냐고 하다 305
넣다 31
네 네 해도 115
네요 171
놓다 31
느냐고 하다 305
느니 123
느라(고) 75
는 23, 30, 238
는 것 22
는 것 같다 137
는 것 보통이다 358
는 것 처럼 125
는 게 틀림없다 164
는 경우에(는) 290
는 길에 248
는 김에 250
는 까닭에 85
는 답니다 330
는 대로 273, 320
는 대신에 119
는 덕분에 80
는 데 170, 334
는 둥 는 둥 266
는 둥 마는 둥 하다 40
는 듯(이) 128
는 듯싶다 153
는 듯하다 153
는 만큼 126
는 모양이다 145
는 문론 57
는 바 324
는 바람에 79
는 반면(에) 109
는 반해 109
는 법이다 354
는 셈이다 154
는 셈치고 292
는 수 밖에 (별) 도리 없다 296
는 수가 있다 295
는 줄 몰랐다 159
는 줄 알았다 159
는 척하다 362
는 축에 들다 359
는 탓에 79
는 통에 79
는 편이다 350
는 한 338
는가 하면 100, 284
는걸(요) 149
는구나 172
는군요 172
는다 는다 하는 게 230
는다거나 64
는다고 믿다 147
는다고 보다 147
는다고 생각하다 147
는다고 하다 305
는다고 하면 284
는다기보다(는) 122
는다는 것이 234
는다는 말이다 318
는다든가 64
는다면 283, 286
는데(요) 69
는데다(가) 53
는데다(가) 268
는데도 107
는지(도) 모르다 160
는지(도) 알다 160
는커녕 41
(으)니 만큼 76
(으)니까 망정이다 176
(으)니까(요) 67
님 31
다(가) 보니(까) 269, 353
다(가) 보면 209, 269, 284
다가 268, 269
다가 말다가 하다 40, 269, 360
다가는 210, 268
다가도 269, 279
다고 하다 305
다고 하면 284
다기보다(는) 122
다는 말이다 318
다니요 319
다든가 64
다면 283, 286
다면서요 317
다시피 163
다시피 하다 130
답니다 330
대다 31
더구나 185, 196
더군요 185, 196
더니(만) 185, 190
더라 185, 192
더라고요 185, 193
더라도 106
더러 31
던 237
던가요 185, 195
던데 185, 188
도 31, 340
(기)도 (이)지만 348
(도)중에(서) 244
도록 167
동안에 241
두다 31
뒤에 277
든(지) 63
들 31
등 48
따라 102
(기) 때문에 78
(으)ㄹ 23, 238
(으)ㄹ 거예요 239
(으)ㄹ 건가요 228
(으)ㄹ 걸 그랬어요 173
(으)ㄹ 것 같다 137
(으)ㄹ 것이다 239
(으)ㄹ 게 틀림없다 164
(으)ㄹ 겸 49
(으)ㄹ 겸 해서 95
(으)ㄹ 경우에(는) 290
(으)ㄹ 대로 320
(으)ㄹ 데 334
(으)ㄹ 둥 (으)ㄹ 둥 266
(으)ㄹ 듯(이) 128
(으)ㄹ 듯싶다 153
(으)ㄹ 듯하다 153
(으)ㄹ 따름이다 346
(으)ㄹ 때(에는) 243
(으)ㄹ 리(가) 없다 299
(으)ㄹ 리(가) 있다 299
(으)ㄹ 만큼 126
(으)ㄹ 만하다 333
(으)ㄹ 망정 182
(으)ㄹ 모양이다 145
(으)ㄹ 바에는 181
(으)ㄹ 봐에야 181
(으)ㄹ 뻔하다 261
(으)ㄹ 뿐더러 55
(으)ㄹ 뿐만 아니라 55
(으)ㄹ 뿐이다 337
(으)ㄹ 셈이다 154
(으)ㄹ 셈치(고) 291
(으)ㄹ 수 밖에 (별) 도리 없다 296
(으)ㄹ 수 밖에 없다 296
(으)ㄹ 수 없다 294
(으)ㄹ 수 있다 294
(으)ㄹ 수가 있어야지(요) 296
(으)ㄹ 수도 있다 295
(으)ㄹ 예정 225
(으)ㄹ 정도(로) 127
(으)ㄹ 줄 모르다 158, 159
(으)ㄹ 줄 몰았다 159
(으)ㄹ 줄 알다 158, 159
(으)ㄹ 줄 알았다 159
(으)ㄹ 지 말지 하다 40
(으)ㄹ 지경이다 127
(으)ㄹ 지경이다 264
(으)ㄹ 테고 223
(으)ㄹ 테니까 221
(으)ㄹ 테다 218
(으)ㄹ 테면 223, 284
(으)ㄹ 테지만 224
(으)ㄹ 텐데 222
(으)ㄹ걸(요) 149
(으)ㄹ게요 214
(으)ㄹ까 말까 하다 40, 231
(으)ㄹ까 보다 226
(으)ㄹ까 봐 146
(으)ㄹ까 싶다 226
(으)ㄹ까 하다 226
(으)ㄹ까요 212
(으)ㄹ락 (으)ㄹ락 하다 265
(으)ㄹ락 말락 하다 40, 265
(으)ㄹ래요 212
(으)ㄹ지(도) 모르다 160
(으)ㄹ지(도) 알다 160
(이/으)라거나 64
(이)라고 믿다 147
(이)라고 보다 147
(이)라고 생각하다 147
(으)라고 하다 305
(이)라고 하다 305
(이)라고 하면 284
(이)라기보다(는) 122
(이)라는 말이다 318
(이)라니요 319
(이)라도 62
(이)라든가 64
(이)라면 283, 286
(이)라면서요 317
(이)라야 202
랍니다 330
(이)랑 46
(으)러 218
(으)려(고) 215
(으)려거든 217
(으)려고 해도 216
(으)려고 해서 216
려나 보다 138, 216
려다가 217, 268
(으)려던 참이다 217, 262
(으)려면 217
(으)려무나 208
(으)렴 208
(으)로 31
(으)로 봐서는 326
(으)로 인해 84
(으)로서 130
(으)로써 82
를 31
를 비롯한 49
를 통해(서) 83
(으)리라고 155
(으)리라는 155
(으)ㅁ 22
(으)ㅁ에도 (불구하고) 111
마다 31
마저 340
만 31
(기)만 하다 335
만 해도 99
만으로는 347
만하다 135
말고(도) 58
(이)며 47
(으)면 283
(으)면 되다 200, 284
(으)면 (으)ㄹ수록 247, 284
(으)면 뭘 해요? 43, 284
(으)면 안 되다 200, 284
(으)면서 246
(으)면서도 253
못 지 않게 132
버리다 31
보다 96
부터 31
((으)ㄹ) 뿐이다 337
사이에 251
시 31
씨 31
씩 31
아 가면서 254
아 놓다 257
아 달로고 하다 305
아 대다 357
아 두다 257
아 보니 186
아 보다 185
아 보이다 142
아 봐야 92
아 봤자 92
아 있다 255
아다가 268, 270
아도 104
아도 되다 200
아서 그런지 142
아서 죽겠다 336
아서(요) 66
아서는 301
아서야 303
아야 되다 198
아야 하다 198
아야(지/만) 202
아야겠다 207
아지다 25
았다 235
았다가 268, 271
았다면 283, 286
았더니 185
았더니 185
았더라면 283, 289
았던 237
았던 것 같다 194
았어야 했는데 175
았었다 237
았으면 싶다 178, 284
았으면 좋겠다 178, 284
았으면 하다 178, 284
았을 것이다 140
야말로 345
어 가면서 254
어 놓다 257
어 달라고 하다 305
어 대다 357
어 두다 257
어 보니 186
어 보다 185
어 보이다 142
어 봐야 92
어 봤자 92
어 있다 255
어다가 268, 270
어도 104
어도 되다 200
어서 그런지 142
어서 죽겠다 336
어서(요) 66
어서는 301
어서야 303
어야 되다 198
어야 하다 198
어야(지/만) 202
어야겠다 207
어지다 25
어찌나 (으)ㄴ/는지 343
얼마나 (으)ㄴ/는지 343
었다 235
었다가 268, 271
었다면 283, 286
었더니 185
었더라면 283, 289
었던 237
었던 것 같다 194
었어야 했는데 175
었었다 237
었으면 싶다 178, 284
었으면 좋겠다 178, 284
었으면 하다 178, 284
었을 것이다 140
에 31
에 달려 있다 88
에 대하여 328
에 대한 328
에 대해서 328
에 대해서(는) 328
에 따라 322
에 따르면 322
에 비하면 97, 284
에 비해서 97
에 의하면 284, 324
에 의해서 81
에게 31
에다(가) 53, 268
에서 31
여 가면서 254
여 놓다 257
여 달라고 하다 305
여 대다 357
여 두다 257
여 보니 186
여 보다 185
여 봐야 92
여 봤자 92
여다가 268, 270
여도 104
여도 되다 200
여서 그런지 142
여서 죽겠다 336
여서(요) 66
여서는 301
여서야 303
여야 되다 198
여야 하다 198
여야(지/만) 202
여야겠다 207
여지다 25
였다 235
였다가 268, 271
였다면 283, 286
였더니 185
였더라면 283, 289
였던 237
였던 것 같다 194
였어야 했는데 175
였었다 237
였으면 싶다 178, 284
였으면 좋겠다 178, 284
였으면 하다 178, 284
였을 것이다 140
오다 31
와 45
와(는) 달리 124
외에(도) 58
위해(서) 169
은 30
은커녕 41
을 31
을 비롯한 49
을 통해(서) 83
의 31
의 경우에(는) 290
이 30
이 만 못하다 134
이 아니고 118
이 아니라 118
이 아니면 121
이 에서 비롯되다 90
이란 331
이에 뿌리를 두다 90
자 273
자거나 64
자고 하다 305
자고 하면 284
자마자 273
자면 283, 286
잖아요 157
적 26
전에 277
조차 340
주다 32
지 112
지 말다 199
지 못하다 297
지만 105
지요 315
체하다 362
치고(는) 86
하고 46
한테 31
한해서 339
후에 277
| {
"redpajama_set_name": "RedPajamaBook"
} | 4,568 |
Noel was held out of the previous game with a left quadriceps contusion but is likely to play against the Raptors on Wednesday.
Morris is dealing with a neck issue but will likely be available for Saturday's matchup against the Warriors.
Morris was held out of the previous contest with soreness in his neck and will not play against the Pacers on Thursday.
Morris is sidelined with a neck injury and it is unknown if he will make his team debut Friday against the Jazz.
Schroder has sat out the past two games due to the birth of his child but is expected back Friday against the Jazz.
Grant has missed the previous two games with a right ankle sprain. It is expected he will be available against the Jazz on Friday.
Abrines was absent from the lineup during the last game due to a personal matter and he will not play against the Rockets on Saturday.
Nader sat out the previous game due to illness but is expected to be available against the Magic on Tuesday.
Ferguson was absent from the lineup during the last game due to mid-back spasms and it is unclear if he will be available against the Heat on Friday.
Adams was held out of the previous game with a right ankle sprain and it is uncertain whether he will be in Friday's lineup against the Heat.
Patterson left the last game with an injury to his nose and it is uncertain if he will play Tuesday against the Trail Blazers.
Schroder left Thursday's game due to a quadriceps injury but is likely to return.
Noel suffered a concussion but has passed the league-mandated protocol and is expected to be available against the Lakers on Thursday.
George is nursing a quad injury that kept him out last game but is likely to play Sunday against the Mavericks.
Abrines has been away from the team due to a personal matter and his status for Sunday's contest against the Bucks is in question.
Felton has been suspended one game by the NBA for his role in an altercation but is likely to be reinstated in time for Saturday's matchup against the Jazz.
Schroder has been issued a one-game suspension for his role in an altercation but is expected to be available against the Jazz on Saturday.
Diallo is healing from a left ankle sprain but is likely to return to the lineup against the Nets on Wednesday.
Grant is nursing a right ankle injury but is likely to take part in Wednesday's matchup against the Warriors. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,146 |
Canoe Creek Indian Reserve No. 2, known officially as Canoe Creek 2, is an Indian reserve in British Columbia, Canada, governed by the Canoe Creek/Dog Creek Indian Band, located six miles east of the mouth of Canoe Creek into the Fraser River.
See also
List of Indian reserves in British Columbia
Canoe Creek (disambiguation)
References
Indian reserves in British Columbia
Geography of the Cariboo
Secwepemc | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,009 |
\section{Introduction}
Josephson junctions between spin-singlet superconductors with ferromagnetic
tunneling barriers have recently
attracted much interest.\cite{RevModPhys.77.1321,RevModPhys.76.411,RevModPhys.77.935} Since spin-singlet
superconductivity is strongly suppressed by the exchange splitting of
the Fermi surface in the ferromagnet, such junctions represent a unique
way of probing the interplay of two competing states. A
startling consequence is the so-called $0$-$\pi$ transition: By
varying the width of the ferromagnetic tunneling barrier or changing the
magnitude of the exchange splitting, the current vs. phase relationship
reverses sign relative to the non-magnetic
case.\cite{JETP.74.178,Tanaka1997357,PhysRevLett.98.107002,PhysRevLett.90.137003,PhysRevLett.102.227005,PhysRevB.75.094514,PhysRevLett.86.2427,PhysRevB.43.10124,PhysRevB.68.014501,PhysRevB.62.11812,PhysRevB.83.144520,NatPhys.4.138,PhysRevB.77.214506,Nature.439.825,PhysRevLett.97.247001,PhysRevLett.92.057005}
Roughly speaking, the pair-breaking effect of the exchange fields causes a spatial
oscillation of the superconducting pairing correlations within the
ferromagnetic tunneling barrier, which can therefore connect the two
superconductors with either a $0$ or $\pi$ phase shift. This phenomenon has been
the subject of intense
theoretical\cite{RevModPhys.77.1321,RevModPhys.76.411,RevModPhys.77.935,JETP.74.178,Tanaka1997357,PhysRevLett.98.107002,PhysRevLett.90.137003,PhysRevLett.102.227005,PhysRevB.75.094514,PhysRevB.43.10124,PhysRevB.68.014501,PhysRevB.62.11812,NatPhys.4.138,PhysRevB.83.144520}
and experimental\cite{PhysRevLett.86.2427,PhysRevB.77.214506,Nature.439.825,PhysRevLett.97.247001,PhysRevLett.92.057005} interest and is by now very well
understood.
At least from a theoretical point of view, it is therefore surprising
that the physics of triplet superconductor (TSC)--ferromagnet (FM)
bilayers~\cite{PhysRevB.67.174501,PhysRevB.75.134510,PhysRevB.70.214524,PhysRevB.80.224520,PhysRevB.83.060508,PhysRevB.83.180504}
and junctions has received relatively little
attention.\cite{PhysRevB.75.134508,PhysRevB.75.094514,PhysRevLett.96.047009,PhysRevB.70.214524,PhysRevB.77.104504,JPSP,PhysRevLett.103.147001,springerlink:10.1007/s10909-007-9446-2,Rahnavard}
Due to the intimate relationship between triplet pairing and ferromagnetism, one
would anticipate very different behavior compared to the singlet
case.
Recently, several papers have pointed out that the orientation
of the magnetization relative to the TSC vector order
parameter is a key parameter of such systems,
leading to a variety of exotic effects: A universal $0$-$\pi$ transition
dependent upon
the orientation of the magnetic moment, an intimate connection between
spin and charge Josephson effects, and also phase-independent
Josephson spin
currents.\cite{PhysRevB.77.104504,JPSP,PhysRevLett.103.147001,PhysRevB.80.224520,Rahnavard,PhysRevB.83.180504}
Although none of these predictions have yet been
verified experimentally, the recent creation of superconducting thin
films of Sr$_2$RuO$_4$ is a significant step towards the fabrication of
TSC-FM heterostructures.\cite{AppliedPhysics}
A deeper understanding of the physics of these systems is therefore
desirable.
Most investigations of TSC junctions were performed assuming an atomically thin
barrier modeled by a $\delta$-function separating the superconducting
regions. The effect of a finite barrier
width have only been studied by Rahnavard \emph{et al.}\cite{Rahnavard} for
the case of a $(p_z+ip_y)$-wave gap where the authors obtained results that are
only partially consistent with the previous perturbative
calculations~\cite{PhysRevLett.103.147001} for a $\delta$-function
barrier. This raises the interesting possibility that additional
mechanisms for the Josephson currents appear due to the finite barrier
width. It is therefore the aim of this work to systematically investigate the
relationship between the results in the $\delta$-function limit and for a more
realistic barrier model. In our calculations we utilize a
non-perturbative method to determine the Josephson current by solving the
Bogoliubov-de Gennes (BdG) equation. We compare the results with the
predictions of perturbation theory.~\cite{PhysRevLett.103.147001}
\section{Theory}
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{Fig1_junction.eps}
\caption{Schematic diagram of the junction.}\label{junction}
\end{center}
\end{figure}
We consider a Josephson junction consisting of an FM layer of width
$L$ sandwiched between two TSCs, shown schematically
in Fig.\ \ref{junction}. The BdG equation for quasiparticle states with
energy $E$ reads
\begin{align}\label{BDG}
\left(
\begin{array}{cc}
\hat{\mathcal{H}}_0(\mathbf{r}) & \hat{\Delta}(\mathbf{r})\\
\hat{\Delta}^\dagger(\mathbf{r}) & -\hat{\mathcal{H}}^T_0(\mathbf{r})
\end{array}\right)\Psi(r)=E\Psi(r).
\end{align}
The noninteracting Hamiltonian is
\begin{align}
\hat{\mathcal{H}}_0(\mathbf{r})=\left[-\frac{\hbar^2}{2m}\nabla_\mathbf{r}^2-\mu\right]
\mathbb{1}-\Theta(z)\Theta(L-z)g\mu_B\hat{\bm{\sigma}}\cdot\mathbf{M} ,
\end{align}
where for simplicity we assume two-dimensional circular Fermi surfaces lying
in the $y$-$z$ plane, as well as equal effective masses and chemical potentials
in all three regions of the junction. The magnetic moment $\mathbf{M}$ of the FM
lies in the $x$-$y$ plane and forms an angle $\alpha$ with the $x$ axis so that
\begin{align}
\mathbf{M}=M\, (\cos\alpha,\sin\alpha,0).
\end{align}
This formulation is completely general since the TSCs are invariant under
spin rotations
around the $x$ axis.
The gap matrix in equation (\ref{BDG}) is
\begin{align}
\hat{\Delta}(\mathbf{r})=\hat{\Delta}_L\Theta(-z)+\hat{\Delta}_R\Theta(z-L),
\end{align}
where $\hat{\Delta}_\nu$ ($\nu=L,R$) is related to the $\mathbf{d}$ vector by
$\hat{\Delta}_\nu=i\hat{\sigma}_y\,(\hat{\bm{\sigma}}\cdot\mathbf{d}_\nu)$.
We assume the TSCs to be in an equal-spin-pairing unitary state.
In particular, we take
\begin{eqnarray}
\mathbf{d}_L(\mathbf{k}) & = & \Delta(\mathbf{k})\mathbf{e}_x, \\
\mathbf{d}_R(\mathbf{k}) & = & \Delta(\mathbf{k})e^{i\phi}\mathbf{e}_x,
\end{eqnarray}
where $\Delta(\mathbf{k})$ denotes the orbital pairing state, so that the two
superconductors differ only in the overall phase $\phi$ of the order parameter.
In this work we will be concerned with the three orbital pairing states
\begin{align}
\Delta(\mathbf{k})=
\left\{\begin{array}{cl}
\Delta(T)k_y/k_F,& p_y \text{-wave}, \\
\Delta(T)k_z/k_F,& p_z \text{-wave}, \\
\Delta(T)(k_z+ik_y)/k_F,& p_z+ip_y \text{-wave}, \end{array}\right.
\end{align}
where $\Delta(T)\cong\Delta_0\sqrt{(T_C-T)/T_C}$ is the weak-coupling
temperature-dependent gap magnitude and $k_F$ is the Fermi momentum of the
normal state.
The BdG equation is solved separately in the different regions under the
assumption of low-energy excitations, i.e., $\mu\gg E,$ $|\Delta(\mathbf{k})|$,
so that the electron and hole wavevectors can be regarded as approximately
equal in magnitude, $k_e \approx k_h \approx k_F$, the so-called Andreev
approximation. The dephasing of the
electron and hole wavefunctions in the finite FM layer constrains the validity
of this approximation to $(k_e - k_h)L \approx 2EL/\hbar v_{F} \ll 1$, where
$v_F\approx 10^6\, \mathrm{ms}^{-1}$ is the Fermi velocity
and $E \leq \max\{|\Delta_{\bf k}|\} \approx 0.1\,\mathrm{meV}$ (we assume
$T_C\approx 1\,\mathrm{K}$). This
implies that we restrict ourselves to thin layers with $Lk_F\lesssim100$.
In the TSC
we obtain plane-wave solutions
$\psi_{\mathbf{k},e(h),\sigma}=\phi_{\mathbf{k},e(h),\sigma}e^{i\mathbf{k}\cdot{\bf r}}$ for
electron-like (hole-like) quasiparticles with wavevector $\mathbf{k}$ and spin
$\sigma$. The spinors are
\begin{align}
\phi_{\mathbf{k},e,\uparrow}=(s_{\uparrow\mathbf{k}} u_{\mathbf{k}},0,v_{\mathbf{k}},0)^T,\\
\phi_{\mathbf{k},h,\uparrow}=(s_{\uparrow\mathbf{k}} v_{\mathbf{k}},0,u_{\mathbf{k}},0)^T,\\
\phi_{\mathbf{k},e,\downarrow}=(0,s_{\downarrow\mathbf{k}} u_{\mathbf{k}},0,v_{\mathbf{k}})^T,\\
\phi_{\mathbf{k},h,\downarrow}=(0,s_{\downarrow\mathbf{k}} v_{\mathbf{k}},0,u_{\mathbf{k}})^T,
\end{align}
where $u_{\mathbf{k}}=\sqrt{(E+\Omega_\mathbf{k})/2E}$,
$v_{\mathbf{k}}=\sqrt{(E-\Omega_\mathbf{k})/2E}$,
$\Omega_\mathbf{k}=\sqrt{E^2-|\Delta(\mathbf{k})|^2}$,
$s_{\sigma\mathbf{k}}=-\sigma\Delta(\mathbf{k})/|\Delta(\mathbf{k})|$, and
$|\mathbf{k}|=\sqrt{2m\mu/\hbar^2}=k_F$. In the FM region we obtain plane-wave
solutions $\psi^{FM}_{\mathbf{k}^s,e(h),s}=\phi^{FM}_{e(h),s}e^{i\mathbf{k}^s\cdot\mathbf{r}}$
for electron-like (hole-like) quasiparticles with wavevector $\mathbf{k}^s$. The
spinors of spin $s=\pm$ electrons and holes are
\begin{align}
\phi^{FM}_{e,s}&=(s e^{-i\alpha}/\sqrt{2},1/\sqrt{2},0,0)^T,\\
\phi^{FM}_{h,s}&=(0,0,s e^{i\alpha}/\sqrt{2},1/\sqrt{2})^T.
\end{align}
Due to the exchange splitting in the FM, we have majority-spin and minority-spin
Fermi surfaces with radii
\begin{align}
|\mathbf{k}^s|=k_F^s=\sqrt{\frac{2m}{\hbar^2}(\mu+sg\mu_BM)}=k_F\sqrt{1+s\lambda} ,
\end{align}
where $\lambda$ denotes the ratio of the exchange splitting to the chemical
potential.
\begin{figure*}
\begin{center}
\includegraphics[width=4.5in]{Fig2_scatterbig.eps}
\caption{(Color online) Schematic representation of the various wave
vectors,
amplitudes, and coefficients for a spin-$\sigma$ electron-like quasiparticle
incident from the left TSC, with wavefunction
$\Psi_{e,\sigma}(\mathbf{r})$. The arrows denote the directions of travel of
electron-like quasiparticles (solid blue) and hole-like quasiparticles
(dashed red).\label{scatter}}
\end{center}
\end{figure*}
We use these piecewise solutions to construct the wavefunction ansatz for a
spin-$\sigma$ electron-like quasiparticle incident from the left TSC with
wavevector $\mathbf{k}=(k_y,k_z)=k_F(\sin\theta,\cos\theta)$, where $\theta$ is the
injection angle relative to the $z$ axis:
\begin{align}\label{elinj}
&\Psi_{e,\sigma}(\mathbf{r})=\notag\\ &\Theta(-z)\Big\{\psi^L_{\mathbf{k},e,\sigma}+\sum_{\sigma^\prime=\uparrow\downarrow}\big[a^e_{\sigma\sigma^\prime}\psi_{\mathbf{k},h,\sigma^\prime}^L+b^e_{\sigma\sigma^\prime}\psi^L_{-\hat{\mathbf{k}},e,\sigma^\prime}\big]\Big\}\notag\\
&+\Theta(z)\Theta(L-z)\Big\{\sum_{s=+-}\big[c^e_{\sigma{s}}\psi^{FM}_{\mathbf{k}^s,e,s}+d^e_{\sigma{s}}\psi^{FM}_{-\widetilde{\mathbf{k}}^s,h,s}\big.\notag\\&+\big.e^e_{\sigma{s}}\psi^{FM}_{-\widetilde{\mathbf{k}}^s,e,s}+f^e_{\sigma{s}}\psi^{FM}_{\mathbf{k}^s,h,s}\big]\Big\}\notag\\
&+\Theta(z-L)\Big\{\sum_{\sigma^\prime=\uparrow\downarrow}[g^e_{\sigma\sigma^\prime}\psi^R_{\mathbf{k},e,\sigma^\prime}+h^e_{\sigma\sigma^\prime}\psi^R_{-\hat{\mathbf{k}},h,\sigma^\prime}]\Big\}.
\end{align}
A schematic representation of the wavevectors and transmission and reflection amplitudes appearing in the wavefunction $\Psi_{e,\sigma}(\mathbf{r})$ is
shown in Fig.\ \ref{scatter}. For $z<0$ this ansatz describes Andreev
reflection of hole-like quasiparticles
with wavevector $\mathbf{k}$ and specular reflection of electron-like quasiparticles
with wavevector $-\hat{\mathbf{k}}=(k_y,-k_z)$. These processes are weighted by
their probability amplitudes $a^e_{\sigma\sigma^\prime}$ and
$b^e_{\sigma\sigma^\prime}$, respectively. In the FM (i.e., for $0<z<L$), the
wavevectors of the transmitted and reflected quasiparticles are spin-dependent.
For wavevectors of spin-$s$ right-moving (left-moving) electrons (holes)
we write $\mathbf{k}^s=(k_y,k_z^s)$, while spin-$s$ left-moving (right-moving)
electrons
(holes) have wavevectors $-\widetilde{\mathbf{k}}^s=(k_y,-k_z^s)$. The associated
probability amplitudes for electrons (holes) are $c^e_{\sigma s}$ and
$e^e_{\sigma s}$ ($d^e_{\sigma s}$ and $f^e_{\sigma s}$). For $z>L$ we have
transmitted electron-like and hole-like quasiparticles with wavevectors $\mathbf{k}$
and $-\hat{\mathbf{k}}$, and probability amplitudes $g^e_{\sigma\sigma^\prime}$ and
$h^e_{\sigma\sigma^\prime}$, respectively. The ansatz for an
incident hole-like quasiparticle $\Psi_{h,\sigma}(\mathbf{r})$ is analogous.
The probability amplitudes for this case are distinguished by the superscript
$h$,
e.g., $a^h_{\sigma\sigma^\prime}$, $b^h_{\sigma\sigma^\prime}$ etc. Due to the
translational invariance along the $y$ axis, the wavevector
component parallel to the interface is conserved during scattering, which gives
\begin{align}
k_F\sin\theta = k_F^s\sin\theta^s ,
\end{align}
where $\theta^s$ is the transmission angle for quasiparticles from the spin-$s$
Fermi surfaces. Making use of this relation, $k_z^s$ can be expressed as
\begin{align}
k_z^s=k_F^s\cos\theta^s=k_F\sqrt{\cos^2\theta+s\lambda}.
\end{align}
For injection angles
\begin{align}\label{critangle}
\theta>\theta_\text{crit}=\arccos\sqrt{\lambda} ,
\end{align}
$k_z^s$ is imaginary for minority-spin quasiparticles, i.e., the wave decays
exponentially into the FM region.
The probability amplitudes for the various processes are calculated from
the continuity of the wavefunction and its derivative at the interfaces,
\begin{align}\label{boundarystart}
\Psi_{e(h),\sigma}(\mathbf{r})|_{z=0^-}&=\Psi_{e(h),\sigma}(\mathbf{r})|_{z=0^+},\\
\Psi_{e(h),\sigma}(\mathbf{r})|_{z=L+0^-}&=\Psi_{e(h),\sigma}(\mathbf{r})|_{z=L+0^+},\\
\frac{\partial}{\partial
z}\Psi_{e(h),\sigma}(\mathbf{r})\Big|_{z=0^-}&=\frac{\partial}{\partial
z}\Psi_{e(h),\sigma}(\mathbf{r})\Big|_{z=0^+},\\
\frac{\partial}{\partial
z}\Psi_{e(h),\sigma}(\mathbf{r})\Big|_{z=L+0^-}&=\frac{\partial}{\partial
z}\Psi_{e(h),\sigma}(\mathbf{r})\Big|_{z=L+0^+}\label{boundaryend}.
\end{align}
The results for a $\delta$-barrier can be obtained by taking the limits
$L\rightarrow0$, $\lambda\rightarrow \infty$ while keeping the product
$L\lambda$ constant.
The charge current $I_C$ and spin current $I_S^\mu$ ($\mu=x,y,z$) perpendicular to
the junction interface are obtained using the Furusaki-Tsukada
formulas.\cite{Furusaki,Kasiwaya_Tanaka,Asano}
These are written in terms of the Andreev reflection coefficients
$a^e_{\sigma\sigma^\prime}$ and $a^h_{\sigma\sigma^\prime}$, where the energy
argument is analytically continued to Matsubara frequencies [$E\rightarrow
i\omega_n = i\, (2n-1)\pi/\beta$],
\begin{align}\label{current_pwave}
I_{C}=&\frac{e}{2\hbar\beta}\int dk_{y}\sum_{\omega_n}\sum_{\sigma} \notag\\
&\times\left\{
\frac{|\Delta_\mathbf{k}|}{\Omega_{n,\mathbf{k}}}
a^e_{\sigma\sigma}(\mathbf{k},\omega_n)-\frac{|\Delta_{-\hat{\mathbf{k}}}|}{\Omega_{n,-\hat{\mathbf{k}}}} a^h_{\sigma\sigma}(-\hat{\mathbf{k}},\omega_n) \right\},\\
{I_{S}^y}=&\frac{i}{4\beta}\int
dk_{y}\sum_{\omega_n}\sum_{\sigma}\sigma \notag \\& \times\left\{
\frac{|\Delta_\mathbf{k}|}{\Omega_{n,\mathbf{k}}}
a^e_{\sigma\overline{\sigma}}(\mathbf{k},\omega_n) -\frac{|\Delta_{-\hat{\mathbf{k}}}|}{\Omega_{n,-\hat{\mathbf{k}}}} a^h_{\sigma\overline{\sigma}}(-\hat{\mathbf{k}},\omega_n) \right\},\\
I_{S}^z=&-\frac{1}{4\beta}\int
dk_{y}\sum_{\omega_n}\sum_{\sigma}\sigma \notag\\
&\times\left\{
\frac{|\Delta_\mathbf{k}|}{\Omega_{n,\mathbf{k}}}
a^e_{\sigma\sigma}(\mathbf{k},\omega_n) -\frac{|\Delta_{-\hat{\mathbf{k}}}|}{\Omega_{n,-\hat{\mathbf{k}}}} a^h_{\sigma\sigma}(-\hat{\mathbf{k}},\omega_n) \right\},
\end{align}
where $\Omega_{n,\mathbf{k}}=\sqrt{\omega_n^2+|\Delta_\mathbf{k}|^2}$,
$\overline{\sigma}=-\sigma$ and $\int
dk_{y}$ denotes the integration over all momenta parallel to the
interface.~\cite{meaculpa} The $x$ component of the spin current vanishes
since the Cooper-pair spin is always perpendicular to the $\mathbf{d}$-vector.
\begin{figure*}
\begin{center}\includegraphics[clip,width=2\columnwidth]{Fig3_allstripe_L.eps}\end{center}
\caption{Phase diagram of the junction showing $0$- and $\pi$-states as a
function of the angle
$\alpha$ and barrier width $L$ for the three considered gap symmetries.
In all panels we set $T=0.4\,T_C$ and $\lambda=0.3$.}\label{opimaps1}
\begin{center}
\includegraphics[clip,width=2\columnwidth]{Fig4_allstripe_lamb.eps}
\end{center}
\caption{Phase diagram of the junction showing $0$- and $\pi$-states as a function of the angle
$\alpha$ and exchange splitting $\lambda$ for the three considered gap
symmetries. In all panels we set $T=0.4\,T_C$ and
$L=20k_F^{-1}$.}\label{opimaps2}
\begin{center}
\includegraphics[clip,width=2\columnwidth]{Fig5_allstripe_lamb_L.eps}
\end{center}
\caption{Phase diagram of the junction showing $0$- and $\pi$-states as a function of the
exchange splitting $\lambda$ and the barrier width $L$ for the three considered
gap symmetries. In all panels we set $T=0.4\,T_C$ and $\alpha=0$.}\label{opimaps3}
\end{figure*}
\section{Results}
We have numerically solved Eqs.\ (\ref{boundarystart})--(\ref{boundaryend}) for
the Andreev reflection coefficients at $T=0.4\,T_C$. At least $400$ Matsubara
frequencies have been used in all calculations.
\subsection{Charge current}
\begin{figure*}[t]
\begin{center}
\includegraphics[clip,scale=0.3]{Fig6a_py_crit_L_final.eps}
\includegraphics[clip,scale=0.3]{Fig6b_pz_crit_L_final.eps}
\includegraphics[clip,scale=0.3]{Fig6c_pzipy_crit_L_final.eps}
\end{center}
\caption{Critical charge current as a function of the barrier width $L$. In
all panels we set $T=0.4\,T_C$ and $\lambda=0.3$.
}\label{critcurrent1}
\begin{center}
\includegraphics[scale=0.3]{Fig7a_py_crit_lamb_final.eps}
\includegraphics[scale=0.3]{Fig7b_pz_crit_lamb_final.eps}
\includegraphics[scale=0.3]{Fig7c_pzipy_crit_lamb_final.eps}
\end{center}
\caption{Critical charge current as a function of the exchange splitting $\lambda$.
In all panels we set $T=0.4\,T_C$ and
$L=20\,k_F^{-1}$.}\label{critcurrent2}
\begin{center}
\includegraphics[scale=0.3]{Fig8a_py_trans_final.eps}
\includegraphics[scale=0.3]{Fig8b_pz_trans_final.eps}
\includegraphics[scale=0.3]{Fig8c_pzipy_trans_final.eps}
\end{center}
\caption{Critical charge current (black solid lines, left axis)
and integrated transmissivities $T_\text{int}$ (red dashed lines, right
axis) as functions of the barrier width $L$. In all panels
we set $\alpha=0$, $T=0.4\,T_C$, and
$\lambda=0.3$.}\label{critcurrentweight}
\end{figure*}
We are mainly interested in the occurrence of $0$-$\pi$ transitions, which
involve a change of the phase difference $\phi$ of the global free-energy
minimum
between $\phi=0$ and $\phi=\pi$. Points in parameter space can be identified as
corresponding to a $0$-state or a
$\pi$-state by examining the current-vs.-phase relationship. Specifically, we
have the thermodynamic relation\cite{Tinkham}
\begin{align}
I_C(\phi)=\frac{2e}{\hbar}\frac{\partial F}{\partial \phi}
\end{align}
so that $I_C=0$ implies an extremum of the free energy $F$. In our system,
minima only occur at $\phi=0$ or $\pi$. We therefore numerically calculate the
derivative of $I_C(\phi)$ at these two points to find out which one is a
minimum. If both are minima we integrate $I_C(\phi)$ to obtain the
free-energy difference $\Delta F$ between $\phi=0$ and $\pi$,
\begin{align}
\Delta F = \frac{\hbar}{2e}\int_0^\pi d \phi\, I_C(\phi) ,
\end{align}
which allows us to identify the global free-energy minimum.
Using this method, we construct phase diagrams of the junction for cuts
through the $(\alpha,\lambda,L)$ parameter space, which are presented in
Figs.~\ref{opimaps1}--\ref{opimaps3}.
We find that $0$-$\pi$ transitions are possible for all considered gap
symmetries by increasing the magnitude of the magnetization, rotating the
magnetization within the $x$-$y$ plane, or by increasing the barrier width
$L$. For
$\lambda<1$ in the $(\alpha,\lambda)$ maps and in the $(\alpha,L)$ maps, we find
multiple dome-shaped regions where the junction is in a $\pi$-state. These
$\pi$-state domes are only found for $\alpha<\pi/4$ and their appearance is
shifted to lower values of $\alpha$ by increasing $L$ or $\lambda$. This
effect is strongest for the $p_y$-wave case, while the $\pi$ states are much
more robust for $p_z$-wave and $(p_z+ip_y)$-wave pairing.
In the $(\alpha,\lambda)$ maps the $\pi$-states are strongly suppressed
for $\lambda>1$.
For $\alpha=0$, the distribution of $0$- and $\pi$-states in
Fig.\ \ref{opimaps3} shows a stripe-like pattern, which does not strongly
depend on the pairing symmetry of the TSC but rather upon the characteristics
of the barrier. In particular, the sharp bend of the stripes at $\lambda=1$ in
all panels is a signature of the
onset of half-metalicity since $|\mathbf{k}_F^-|$ vanishes at
$\lambda=1$. Interestingly, these phase diagrams are
very similar to those obtained for a junction involving
\emph{singlet} superconductors on both sides of the
barrier.\cite{PhysRevB.83.144520} The parallels between these different junctions originate from the universal pair-breaking effect of the FM-layer. Note that for the TSC case the FM layer is
purely pair-breaking only for $\alpha=0$, and pair-breaking is completely suppressed upon
increasing $\alpha$ to $\alpha=\pi/2$. In contrast, for a
singlet-superconductor junction the FM layer is purely pair-breaking for all $\alpha$.
A $0$-$\pi$ transition caused by varying the orientation of
the magnetic moment is therefore an unambiguous experimental signature of
triplet superconductivity.
The direct observation of $\pi$-junctions is not trivial, as it requires
the detection of half vortices at the barrier.~\cite{PhysRevLett.92.057005,PhysRevB.77.214506} A more practical experimental
signature is provided by the critical charge current
\begin{align}
I_\text{crit}=\text{max}\left\{|I_C(\phi)|\right\},\;\;\phi\in[0,\pi]\, .
\end{align}
As the junction is tuned through a $0$-$\pi$ transition, the sign change of
the dominant $\sin\phi$ term in the charge current $I_C(\phi)$ causes the
appearance of sharp, cusp-like minima where $I_\text{crit}$ nearly
vanishes.~\cite{JETP.74.178} If $I_C(\phi)$ were exactly proportional to
$\sin\phi$, this sign
change would exactly coincide both with $I_\text{crit}=0$ and with a $0$-$\pi$
transition; the presence of weak higher harmonics in $I_C(\phi)$ complicates
the analysis, but the minima in $I_\text{crit}$ nevertheless remain a good
proxy for the $0$-$\pi$ transition.
We plot $I_\text{crit}$ as a function of $L$ and $\lambda$ for $\alpha=0$,
$\pi/4$, and $\pi/2$ in Figs.\ \ref{critcurrent1} and \ref{critcurrent2}. The
sharp minima in the critical current characteristic of the $0$-$\pi$
transition are only found at $\alpha=0$, and are in good agreement with the
phase boundaries in Figs.\ \ref{opimaps1} and \ref{opimaps2}.
Upon increasing $\alpha$, the sharp minima disappear by
$\alpha=\pi/4$ and are replaced by broad minima
in $I_{\text{crit}}$, which coincide with every second maximum of the $\alpha=0$
critical current. These minima do not correspond to sign changes of
the $\sin\phi$ term. At $\alpha=\pi/2$, most structure in $I_\text{crit}$ has
vanished.
The charge
current found by perturbation theory for a $\delta$-barrier
junction is,\cite{PhysRevLett.103.147001}
\begin{align}\label{pertubation_current}
I_C^\text{pert}(\phi)\propto\left(T_\mathrm{sp}-T_\mathrm{sf}\cos 2\alpha
\right) \,\sin\phi ,
\end{align}
where $T_\mathrm{sp}$ and $T_\mathrm{sf}$ denote the positive tunneling matrix
elements for spin-preserving tunneling and spin-flip tunneling, respectively.
As one can see, a $\pi$-state can only be realized if
$T_\mathrm{sp}<T_\mathrm{sf}$ and $\alpha<\pi/4$, with the latter condition
clearly consistent with the phase diagrams and critical current plots.
It is interesting to see if Eq.\ (\ref{pertubation_current}) is also
valid in the finite-barrier case if we regard $T_\mathrm{sp}$
and $T_\mathrm{sf}$ as functions of $L$ and $\lambda$. The matrix
elements $T_\mathrm{sp}$ and $T_\mathrm{sf}$ are expected to be
roughly proportional to the corresponding
transmissivities.\cite{PhysRevLett.103.147001} Note that the transmissivities
are
independent of $\alpha$ for both the $\delta$-function and finite width
barriers. From Eq.\ (\ref{pertubation_current}) we see that the spin-flip
transmissivity $\mathcal{T}_\mathrm{sf}$ should be larger than the
spin-preserving transmissivity $\mathcal{T}_\mathrm{sp}$ in a $\pi$-state,
whereas the opposite should be
true in a $0$-state. We have obtained $\mathcal{T}_\mathrm{sf}$
and $\mathcal{T}_\mathrm{sp}$ from the scattering matrix of the FM barrier,
which does not change below $T_C$.\cite{PhysRevB.62.11812,EshInTanRev} As the
Josephson current
is expected to be proportional to the square of the TSC gap magnitude, we
introduce an angular weighting of the transmissivities to account for the
direction dependence of the gap. We therefore define the integrated
transmissivity
\begin{align}\label{quantityweight}
T_\text{int} = \frac{\int_{-\pi/2}^{\pi/2}d\theta\,
{\cal{T}}_\mathrm{sf}\widetilde{\Delta}_\theta^2}
{\int_{-\pi/2}^{\pi/2}d\theta\,({\cal{T}}_\mathrm{sp}+{\cal{T}}_\mathrm{sf})
\, \widetilde{\Delta}_\theta^2}
\end{align}
with
\begin{align}
\widetilde{\Delta}_\theta=
\left\{\begin{array}{cl}
\sin\theta & \quad\text{for } p_y \text{-symmetry}, \\
\cos\theta & \quad\text{for } p_z \text{-symmetry}, \\
1 & \quad\text{for } p_z+ip_y \text{-symmetry}. \end{array}\right.
\end{align}
Spin-flip
tunneling is dominant for $T_\text{int}>0.5$, whereas for
$T_\text{int}<0.5$
spin-preserving tunneling is most important. As seen in Fig.\
\ref{critcurrentweight}, the barrier widths for which $T_\text{int}=0.5$
are in very good agreement with the sharp minima of
the critical current at $\alpha=0$ and thus with the 0-$\pi$ transitions. This
shows that Eq.\ (\ref{pertubation_current}) gives a good description of the
relevant physics if one includes the $L$- and $\lambda$-dependence of the
transmissivities.
In Fig.~\ref{critcurrentweight} we also see that the convergence of
$T_{\text{int}}$ towards $0.5$ with increasing $L$ is much faster for
$p_y$-symmetry compared to the other cases. Due to the $p_y$-wave form
factor, shallow incident angles dominate the integral
in~Eq.~(\ref{quantityweight}), which implies averaging over quasiparticle
trajectories corresponding to a large range of effective barrier
widths. Since the probability of spin-flip transmission oscillates with the
effective barrier width, increasing $L$ rapidly leads to equal integrated
probabilities of spin-flip and spin-preserving transmission.
In contrast, $T_{\text{int}}$ decreases more slowly for $p_z$-symmetry
since the weighting favors normal incidence, and thus the average is over a
smaller range of effective barrier widths. The angle-independent weighting for
the $(p_z+ip_y)$-wave case lies between these two
extremes. From Eq.~(\ref{pertubation_current}) we hence deduce that the $\pi$
state in the $p_y$-wave junction is much less robust to increasing
$L$ than for the other cases, consistent with Fig.\ \ref{opimaps1}.
Since increasing $\lambda$ decreases the length scale of the oscillation
between spin-flip and spin-preserving transmission, this argument also
explains the strong suppression of the $\pi$ states in the $p_y$-wave junction
at moderate to strong polarizations in the ($\alpha$, $\lambda$) maps
(Fig.\ \ref{opimaps2}).
The loss of every second peak in the $\alpha=0$ critical current upon
increasing $\alpha$ to
$\pi/4$ can be
understood from Eq.~(\ref{pertubation_current}) as resulting from a vanishing
spin-flip contribution. As evidenced by the plots of $T_\text{int}$
in~Fig.\ (\ref{critcurrentweight}) the relative strength
of $\cal{T}_\mathrm{sf}$ and $\cal{T}_\mathrm{sp}$ oscillates with $L$ and
$\lambda$. By removing the spin-flip contribution at
$\alpha=\pi/4$, the variation of the spin-preserving transmissivity on these
quantities is evidenced in the critical current. Further support for
this interpretation comes from the observation that every second maximum in
the $\alpha=0$ critical current also corresponds to a maximum in
$T_{\text{int}}$, i.e., strong spin-flip tunneling and weak spin-preserving
tunneling.
\begin{figure}[t]
\begin{center}
\includegraphics[width=\columnwidth]{Fig9_spinmecha_both.eps}
\end{center}
\caption{Spin-current components (a) $I_S^y$, which is due to spin filtering,
and (b) $I_S^z$, due to spin flip, as functions of the phase difference $\phi$.
In both panels we set $L=10\,k_F^{-1}$, $\lambda=0.3$, and $T=0.4\,T_C$.
}
\label{spindemo}
\end{figure}
\subsection{Spin current}
\begin{figure*}[t]
\begin{center}
\includegraphics[clip,width=2\columnwidth]{Fig10_stripe_all_lamb.eps}
\end{center}
\caption{$\phi$-independent contributions to be spin current as functions of
the barrier width $L$ for all considered gap symmetries and various exchange
splittings $\lambda$. In all panels we set
$\alpha=0.25\,\pi$ and $T=0.4\,T_C$.}\label{new_reflectivities}
\end{figure*}
\begin{figure*}[t]
\begin{center}
\includegraphics[clip,width=2\columnwidth]{Fig11_allstripe_weighted.eps}
\end{center}
\caption{$\phi$-independent part of the $z$ component of the spin
current (black solid line, left axis) and integrate spin-flip reflectivity
$R_\text{int}$ (red dashed line, right axis) as functions of the
barrier width $L$ for all considered gap symmetries. For $p_z$-wave
the current is multiplied
by $-1$ for better comparability. We set $T=0.4\,T_C$ and $\lambda=0.5$ in all
panels.}\label{weighted_reflectivities}
\end{figure*}
We find that the polarization of the spin current always lies in the
$y$-$z$
plane. The $y$ and $z$ components are plotted in Fig.\ \ref{spindemo}
for the $p_y$-wave case; the results are qualitatively identical for the
other orbital symmetries. We show
below that these two polarizations are due to different mechanisms, as proposed
by Brydon \textit{et al.}\cite{JPSP}
The $y$ component of the spin current shows the $\alpha$ and $\phi$ dependence
expected for a spin-filtering mechanism.\cite{PhysRevB.77.104504,JPSP} It occurs
when the magnetic moment of the barrier has a component perpendicular to both
$\mathbf{d}$-vectors of the TSCs. The spin-$\uparrow$ and spin-$\downarrow$
states with respect to this direction are not mixed by scattering at the barrier
but do experience different barrier
transmissivities, leading to spin filtering of the
current. For a $\delta$-function barrier, the filtering effect requires a
non-magnetic scattering potential at the interface. This is not the case for a finite
magnetic barrier due to the spin splitting of the Fermi surface. Specifically,
minority-spin quasiparticle propagation is suppressed in the FM region above
the critical injection angle $\theta_{\text{crit}}$ defined in Eq.\ (\ref{critangle}),
which is equivalent to a lowered transmissivity of the minority spins compared
to the majority spins, giving rise to spin filtering.
The $z$ component of the spin current shows the $\alpha$ and
$\phi$ dependence expected for a spin-flipping
mechanism.\cite{PhysRevLett.103.147001,JPSP} This occurs due to the
spin-dependent phase shift acquired by a triplet Cooper pair when it
undergoes a
spin-flip during the tunneling across the barrier.
This mechanism requires a component of the magnetic moment parallel to the
$\mathbf{d}$-vectors of both TSCs. In the perturbation theory for a
$\delta$-barrier junction the $z$
component of the spin current satisfies \cite{PhysRevLett.103.147001}
\begin{align}\label{pertubation_spincurrent}
I_S^{z,\text{pert}} \propto ( T_\mathrm{sf}\cos\phi + \gamma R_\mathrm{sf} )\,
\sin 2\alpha \, ,
\end{align}
where $T_\mathrm{sf}$ and $R_\mathrm{sf}$ are tunneling matrix elements of
spin-flip tunneling and spin-flip reflection and $\gamma$ is an
orbital-dependent
factor connected to the phase shift experienced by specularly reflected
quasiparticles. For the $p_y$ junction, there is no phase shift upon reflection
and it is found that $\gamma=1$. In a $p_z$ junction, reflected Cooper pairs
experience a $\pi$ phase shift leading to a factor of $\gamma=-1$ and in the
more complicated case of a $p_z+ip_y$ junction the phase shift of the Cooper
pairs depends on their incident angle as $\pi-2\arctan(k_y/k_z)$ so that
$-1<\gamma<0$.
We obtain the $\phi$-independent contribution $I_{S,0}^z$ by performing
a Fourier transformation
of $I_S^z(\phi)$. Figure \ref{new_reflectivities} shows
that $I_{S,0}^z$ oscillates with increasing
barrier width $L$ for all considered gap symmetries. These oscillations are
due to interference between quasiparticles reflected off the two interfaces.
The sign difference between $I_{S,0}^z$ for the
$p_y$-wave and $p_z$-wave
cases agrees with previous
results;\cite{PhysRevLett.103.147001,PhysRevB.80.224520,PhysRevB.83.180504} in
particular, we find that the
sign of the $\gamma$ factor in
Eq.\ (\ref{pertubation_spincurrent}) for the finite width barrier is the same
as for the $\delta$-function barrier.
The $(p_z+ip_y)$-wave case shown in
Fig.\ \ref{new_reflectivities} is intermediate between these other cases, and
we find that $\gamma$ can
be positive or negative depending on
$\lambda$ and $L$, in contrast to the $\delta$-function results for which the
sign was found to be always negative.\cite{PhysRevLett.103.147001}
Specifically, in Fig.\ \ref{new_reflectivities} we see that for
$(p_z+ip_y)$-wave symmetry $I_{S,0}^z$ is always positive at $\lambda=0.3$,
at $\lambda=0.5$ it is negative for some values of $L$, and it is
mostly negative at $\lambda=0.8$.
In analogy to the procedure for the charge current, we demonstrate the
appropriateness of Eq.\ (\ref{pertubation_spincurrent}) to describe the
relevant physics, by showing that the oscillations of $I^{z}_{S,0}$ are due to
oscillations of the spin-flip reflectivity
$\mathcal{R}_\mathrm{sf}$.
The parameter $R_\mathrm{sf}$ in Eq.\ (\ref{pertubation_spincurrent})
is expected to be roughly proportional to the reflectivity
$\mathcal{R}_\mathrm{sf}$.\cite{PhysRevLett.103.147001}
Since increasing the magnetization of the barrier makes it more likely for
particles to be spin-flip reflected, i.e., $\mathcal{R}_\mathrm{sf}$ should be
enhanced, we see that $I^{z}_{S,0}$ increases with
$\lambda$
for all gap symmetries,
see Fig.\ \ref{new_reflectivities}. We have also compared $I^{z}_{S,0}$ to the
weighted and $\theta$-integrated spin-flip reflectivity
\begin{align}
R_\text{int} = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}d\theta\,
\mathcal{R}_\mathrm{sf}(\theta)\widetilde{\Delta}^2_{\theta}
\end{align}
for a finite-barrier normal metal-FM-normal metal junction. The weighting is
carried out in the same way as for the charge current except for the
$p_z+ip_y$-wave
case, where it is necessary to account for the $\theta$-dependent phase shift
of quasiparticles reflected at the interface. Thus we weight the
normal metal-FM-normal metal reflectivity by
$\widetilde{\Delta}_\theta^2=\sin^2 \theta-\cos^2\theta$.
Fig.\ \ref{weighted_reflectivities} shows that the variation of $I_{S,0}^z$ is in
very good agreement with $R_\text{int}$,
especially for smaller values of $L$ and $\lambda$. We also find that
the sign change in $I_{S,0}^z$ for the $(p_z+ip_y)$-wave case reflects an
important change in the nature of the spin-flip
reflection:~\cite{PhysRevB.80.224520,PhysRevB.83.180504} the positive
$I_{S,0}^z$ at small $\lambda$ indicates
$p_y$-like behaviour, where tunneling of quasiparticles with high incident
angles $\theta\approx\pi/2$ is dominant. At higher values of $\lambda$,
however, the transmission of
quasiparticles with lower incident angles $\theta\approx 0$ is favored, leading
to a $p_z$-like behavior and hence negative $I_{S,0}^z$. We hence conclude
that when the $L$ and $\lambda$ dependence of the transmissivity and
reflectivity are taken into account, Eq.\ (\ref{pertubation_spincurrent})
qualitatively captures the observed behavior of the spin current.
\section{Summary and Discussion}
In this paper we have studied the charge and spin currents in a
TSC-FM-TSC junction.
For the charge current, we have found that transitions from a $0$- to a
$\pi$-state are induced by rotating the magnetization of the FM barrier in the
plane
parallel to the \textbf{d}-vectors of both TSCs or by varying its
magnitude. These effects have already been predicted for a
$\delta$-function barrier.\cite{PhysRevLett.103.147001}
In addition, we have found that changing the barrier
width can also cause a $0$-$\pi$ transition. In all cases, $0$-$\pi$ transitions
are accompanied by sharp minima of the critical current, which can thus serve
as diagnostics. For the case of $(p_z+ip_y)$-wave symmetry, our results for the
critical charge current agree with Rahnavard \emph{et al.}\cite{Rahnavard}\
However, we have gone beyond their work by considering two additional
gap symmetries, by obtaining $0$-$\pi$ phase maps, and by extending the
range of the exchange splitting. Furthermore, we have explained the
origin of the charge current in terms of the variation of the
spin-flip and spin-preserving transmissivities:
for all considered gap symmetries and barrier parameters, the latter dominates
over the former in a $0$-state, whereas the opposite is true for a
$\pi$-state. This result is consistent with lowest-order perturbation theory.
The spin currents in the finite-barrier case can be classified in the same
manner as for the $\delta$-function barrier, i.e., we find spin currents due to
spin filtering and to spin flipping. Unlike for the $\delta$-function
barrier, however, the spin-filtering effect can occur in the absence
of a finite charge scattering potential because of the spin splitting of
the Fermi surface in the FM barrier.
For the spin current in the $(p_z+ip_y)$-wave case we find very different
polarizations and
phase dependences compared to Rahnavard \emph{et
al.}\cite{Rahnavard}\ Specifically, the authors of
Ref. \onlinecite{Rahnavard} found a spin current
polarization parallel to the
$\mathbf{d}$-vector, which is
inconsistent with the fact that the spin of the Cooper
pairs is always perpendicular to the $\mathbf{d}$-vector. The origin of this
discrepancy might be that,
in contrast to our work, in Ref. \onlinecite{Rahnavard} the spin current was
evaluated in the FM region, where the spin current is not a conserved quantity
and is position dependent.
Overall, our results show that approximating the barrier as a
$\delta$-function
gives a qualitatively correct account of the Josephson currents in a TSC-FM-TSC
junction. This demonstrates the robustness and, since a metallic ferromagnetic
barrier is realizable in a Josephson junction, the experimental relevance of
the previously predicted
phenomena.\cite{PhysRevLett.103.147001,Asano,JPSP} In particular, the
sign reversal of the current upon varying the orientation of the
magnetic moment is found to be a robust test of triplet
superconductivity. The
inclusion of a finite barrier width is nevertheless crucial for a
quantitatively realistic description, as the barrier width can by
itself control 0-$\pi$ transitions of the junction.
\acknowledgments
We have profited from discussions with G. Annunziata and T. Yokoyama.
| {
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{"url":"http:\/\/blog.csdn.net\/haisheng95\/article\/details\/7389197","text":"# Rank List\n\n419\u4eba\u9605\u8bfb \u8bc4\u8bba(0)\n\u8fd9\u662f\u6211\u7b2c\u4e8c\u6b21\u505acodeforces\u7684\u6bd4\u8d5b\uff0c\u6210\u7ee9\u6ca1\u7b2c\u4e00\u6b21\u597d\uff0c\u4e3b\u8981\u539f\u56e0\u8fd8\u662f\u90a3\u4e2a\u4e07\u5e74\u4e0d\u53d8\u7684\u539f\u56e0\u2014\u2014\u770b\u4e0d\u61c2\u9898\u610f\u3002\u62a5\u544a\u4e00\u4e0b\u65f6\u95f42012\u5e743\u300124 1:12\u3002\u5148\u4e0d\u5520\u53e8\uff0c\u5f88\u6709\u5fc5\u8981\u611f\u8c22\u6211\u7684\u4e0a\u94fa\u5144\u5f1fzhifa,\u4ed6\u7684\u82f1\u8bed\u6c34\u5e73\u5f88ok\uff0c\u5c31\u8ba9\u4ed6\u5e2e\u6211\u770b\u9898\uff0c\u6211\u592a\u5fc3\u6025\u4e86\uff0c\u5728\u6ca1\u6709\u4e86\u89e3\u9898\u610f\u7684\u60c5\u51b5\u4e0b\u5c31\u6572\u4ee3\u7801\uff0c\u7ed3\u679c\u5462\uff1f\u7ed3\u679c\u60b2\u5267\u4e86\uff01\u5c31\u53c8\u4e00\u6b21\u8bfb\u9898\uff0c\u7ec8\u4e8e\u660e\u767d\u4e86\u9898\u610f\uff0c\u518d\u6b21\u611f\u8c22\u4ed6\uff01\uff08\u7ae5\u978b\u4eec\uff0c\u8fd8\u662f\u5b66\u82f1\u8bed\u5427\uff01\u8c01\u8ba9\u7956\u56fd\u4e0d\u5f3a\u5927\uff0c\u5565\u65f6\u6bd4\u8d5b\u90fd\u6709\u4e2d\u82f1\u5bf9\u7167\u65f6\uff0c\u54c8\u54c8\u3002\u53c8\u5e7b\u60f3\u4e86.....)\n\u8bfb\u9898\u6ce8\u610f\u4ee5\u4e0b\u51e0\u70b9\uff1a\n\u5982\u679c\u89e3\u9898\u6570\u76ee\u548c\u6240\u7528\u65f6\u95f4\u90fd\u76f8\u7b49\u5219\u4ed6\u4eec\u5c31\u5171\u540c\u6392\u540d\u5728\uff08y+1,y+2,..y+x),y\u662f\u6bd4\u4ed6\u4eec\u6210\u7ee9\u597d\u7684\uff0cx\u662f\u540c\u540d\u6b21\u7684\u4eba\u6570\n\u6211\u7684\u601d\u8def\u5148\u6392\u5e8f\uff0c\u518d\u6bd4\u8f83\uff0c\u8fd9\u4e2a\u65b9\u6cd5\u5f88\u7b28\u62d9\uff0c\u5148\u8d34\u4e0a\uff0c\u6539\u5929\u770b\u770b\u5927\u795e\u7684\nA. Rank List\ntime limit per test\n2 seconds\nmemory limit per test\n256 megabytes\ninput\nstandard input\noutput\nstandard output\n\nAnother programming contest is over. You got hold of the contest's final results table. The table has the following data. For each team we are shown two numbers: the number of problems and the total penalty time. However, for no team we are shown its final place.\n\nYou know the rules of comparing the results of two given teams very well. Let's say that teama solvedpa problems with total penalty timeta and teamb solvedpb problems with total penalty timetb. Teama gets a higher place than team b in the end, if it either solved more problems on the contest, or solved the same number of problems but in less total time. In other words, teama gets a higher place than team b in the final results' table if eitherpa\u2009>\u2009pb, orpa\u2009=\u2009pb andta\u2009<\u2009tb.\n\nIt is considered that the teams that solve the same number of problems with the same penalty time share all corresponding places. More formally, let's say there is a group ofx teams that solved the same number of problems with the same penalty time. Let's also say thaty teams performed better than the teams from this group. In this case all teams from the group share placesy\u2009+\u20091,y\u2009+\u20092,...,y\u2009+\u2009x. The teams that performed worse than the teams from this group, get their places in the results table starting from they\u2009+\u2009x\u2009+\u20091-th place.\n\nYour task is to count what number of teams from the given list shared the k-th place.\n\n<P< p><P< p>\nInput\n<P< p>\n\nThe first line contains two integers n andk (1\u2009\u2264\u2009k\u2009\u2264\u2009n\u2009\u2264\u200950). Thenn lines contain the description of the teams: thei-th line contains two integerspi andti (1\u2009\u2264\u2009pi,\u2009ti\u2009\u2264\u200950) \u2014 the number of solved problems and the total penalty time of the i-th team, correspondingly. All numbers in the lines are separated by spaces.\n\n<P< p><P< p>\nOutput\n<P< p>\n\nIn the only line print the sought number of teams that got the k-th place in the final results' table.\n\n<P< p><P< p>\nSample test(s)\n<P< p><P< p>\nInput\n7 2\n4 10\n4 10\n4 10\n3 20\n2 1\n2 1\n1 10\n\nOutput\n3\n\nInput\n5 4\n3 1\n3 1\n5 3\n3 1\n3 1\n\nOutput\n4\n\n<P< p><P< p>\nNote\n<P< p>\n\nThe final results' table for the first sample is:\n\n\u2022 1-3 places \u2014 4 solved problems, the penalty time equals 10\n\u2022 4 place \u2014 3 solved problems, the penalty time equals 20\n\u2022 5-6 places \u2014 2 solved problems, the penalty time equals 1\n\u2022 7 place \u2014 1 solved problem, the penalty time equals 10\n\nThe table shows that the second place is shared by the teams that solved 4 problems with penalty time 10. There are 3 such teams.\n\nThe final table for the second sample is:\n\n\u2022 1 place \u2014 5 solved problems, the penalty time equals 3\n\u2022 2-5 places \u2014 3 solved problems, the penalty time equals 1\n\nThe table shows that the fourth place is shared by the teams that solved 3 problems with penalty time 1. 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Q: How to reserve free space in Home dir for display to prevent GUI freezing and app crashing? As the title says. Since switching to Linux (and actually probably the main thing that prevented me from switching right away many years ago), the system has crashed upwards of 100 times, with data loss each time (usually lost work browsing pages, but sometimes sensitive and important mounted encryption volume file loss).
How does Windows accomplish not running out of free space for it's display? Linux has better uptime stability RAM-wise and for the root system, but these disk storage crashes for the user are a problem. There is a low space notification, but it can be too late or unseen.
Here are a couple subquestions and observations that could be helpful to find the best solution:
*
*Is it Xorg that crashes the user's applications because of "No space
left on device"
*Could a similar scheme to the 5% system reserved space be used?
Reserved space for root on a filesystem - why? ..aparently uses tune2fs and has a set user option
*For Kubuntu, after clearing space and switching back to tty1, there
is an error regarding 'cannot find xrunner..' and the session
terminates. I checked before that from tty2 and Chromium, etc had
crashed. Then when clicking on the error message it crashes the user
profile again back out to the SDDM greeter.
*I've used AND: Auto Nice Daemon to tame the RAM and/or swap freezes, could something like ionice be used similarly:
How to prevent dd from freezing my system?
*Maybe limiting any single process to filling 90% of the free space, stopping it and warning when it does, seems like a logical solution.
If so, why is this not already done? It could be disabled if an
advanced user really wants that extra 10% of space and risk freezing
the User GUI. I realize Linux can run on tiny disks, but this is Ubuntu, trying to be a real Desktop.
A: Here is one partial solution. It only stops files from filling the disk space not reserve space for critical user GUI functions.
The outline is to use iotop output to identify high IO usage or a threshold of cumulative IO write, say 50 or 100mbs, to send a warning, and then to stop the process once it has written 90% of the disk space. And send a warning then and when it reaches 50% or 1GB or other thresholds.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,914 |
# One in Every Crowd
## Stories
## by
## Ivan E. Coyote
# Table of Contents
* Introduction
* One: Kid I Was
* No Bikini
* Walks Like
* Just Reward
* It Doesn't Hurt
* Sticks and Stones
* Bad Luck and Big Feet
* Three Left Turns
* My Hero
* Two: Family I Have
* Objects in the Mirror
* The Curse?
* By Any Other Name
* To Whom It May Concern:
* Single Malt
* Thicker than Water
* Maiden Heart
* All about Herman
* Just a Love Story
* The Rest of Us
* Three: That Boy
* Red Sock Circle Dance
* I Like to Wear Dresses
* A Week Straight
* The Future of Francis
* Four: Kids I Met
* Saturdays and Cowboy Hats
* Schooled
* Teach the Children Well
* This Summer at Gay Camp
* Straight Teens Talk Queer
* My Name Is Sam
* Nobody Ever
* As Good As We Can Make It
* Letter fr Grammar Wizard
* Five: Folks I Felt It Necessary to School in Some Way or Another, With Varying Degrees of Success
* Judging a Book
* Take That
* Bully This
* Imagine a Pair of Boots
* The Bathroom Chronicles
* Dear Lady in the Women's Washroom
* Truth Story
* Six: Wisdom I Found, Learned, or Was Given
* My Dad Told Me
* Spare Change
* My Kind of Guy
* I Will
* About the Author
ONE IN EVERY CROWD
Copyright © 2012 by Ivan E. Coyote
All rights reserved. No part of this book may be reproduced or used in any form by any means—graphic, electronic, or mechanical—without the prior written permission of the publisher, except by a reviewer, who may use brief excerpts in a review, or in the case of photocopying in Canada, a license from Access Copyright.
ARSENAL PULP PRESS
Suite 101, 211 East Georgia St.
Vancouver, BC
Canada V6A 1Z6
_arsenalpulp.com_
The publisher gratefully acknowledges the support of the Canada Council for the Arts and the British Columbia Arts Council for its publishing program, and the Government of Canada (through the Canada Book Fund) and the Government of British Columbia (through the Book Publishing Tax Credit Program) for its publishing activities.
Cover design by Elisha Lim
Printed and bound in Canada
Library and Archives Canada Cataloguing in Publication
Coyote, Ivan E. (Ivan Elizabeth), 1969-
One in every crowd [electronic resource] / Ivan E. Coyote.
Short stories.
Electronic monograph issued in multiple formats.
Also issued in print format.
ISBN 978-1-55152-460-3 (PDF).--ISBN 978-1-55152-460-3 (EPUB)
I. Title.
PS8555.O99O525 2012 jC813'.6 C2012-901142-8
* * *
This ebook was produced with <http://pressbooks.com>.
# Introduction
Dear Kid I Was:
Hey there. It's me. I mean you. It is you/me, writing to me/you, from the future. We are almost forty-three, and I sure do wish there was a way for me to get this message from future me to past you, but so far, humankind hasn't invented anything like that yet, not that I know of, anyway, so all I can do is write this letter and put it in the front of this book, and just maybe it might help out some other poor kid who feels all alone, just like you and I did, way back when. I am hoping this letter might let them know that they are not the only one. Dear Every Kid Who Picks Up This Book: You are not the only one.
We graduated in 1987, you and I, who can believe it, but that was twenty-five years ago. I would tell you right now that the last quarter of a century has flown by, but I know you, and back then you/I wouldn't have believed me. Why? Because I am writing to the me I was in 1985, and you/we are fifteen years old, and smack dab in the middle of grade ten and graduation feels like five lifetimes from now. School sucks today, in fact, life sucks, too, and you are miserable. It is February, and last fall that thing went down with that dude you were dating (here in the future, what he did that night to us would technically be called date rape, by the way), and the two of you broke up, but he was the popular guy, and you were kind of dorky plus also brand new to that school and he was two years older and about to graduate and in almost every club or team or whatever, and now him and his whole group of jocks and pretty boys and their girlfriends don't invite you to any parties, but you know what? You are better off without them. Better off without him, for sure. But you already knew that. You're no dummy, past me. And you only get better, little buddy. Trust me on this one.
So, while I have your ear, I want to tell you a couple of really important things. First, school is more important than you even thought it was. And I don't mean this in a boring I know it all and I am here to tell you kind of adult way, I am talking in the just between you and me as equals kind of a way. Educating yourself right now is your ticket to options, my friend. And I know you. You like options. So trust me when I tell you finishing school is important. Even more important than your mom thinks it is, and you know how she feels about a higher education. Well, she is right. I am here (or is it there) to tell you that every single thing you will ever manage to learn, every skill, every course, every bit of school or college or university you attend is going to help you on your way to becoming exactly who we always dreamed of being.
And you dreamed big. We dreamed of being a writer and a musician. You knew you wanted to be a writer since grade seven, remember, that assignment for what's-his- face's class, the teacher with the chalk marks all around his front pockets, and sometimes even the crotch of his brown cords (why did that man own only one good pair of pants for school?), anyway, remember what's-his-name's English class, and how he made everybody write a short story? Remember how good it felt? Well, one day you will get to do this for a living. A good living, now, after all those years of hard work. In fact, I am sitting in my office in a small home that my wife and I own, and I am writing a story right now, to you, from the future. The view from here is beautiful. But we have to get through high school first, so hang in there, little buddy, you are almost there. I am not going to tell you it gets better, even though it sure did, because really, how is that going to help you get through grade ten?
What you need right now is information. You need someone to tell you it is okay to be you. And hope. You need hope, and someone to say hey, I know exactly what you are going through. So hey, here I am, and I know exactly what you are going through. And you are, as our dear departed gran once told us when I was you and you were very little, exactly who God meant you to be. And I am here with you. I get it, I do.
I went through it too, remember? I am you. How could I not know what you are going through right now?
Where was I? Oh yeah. Important things I need you to know. Or wish I had known then. Stuff I wish someone would have told me when I was you.
You know how you always loved Dolly Parton? Even though she was really your mom's favourite first, and that is supposedly kind of uncool, right, to have the same favourite records as your mother, but who cares? It is Dolly. And one day you will go on to learn that it turns out she really is very cool, and even better, she is a survivor. Dolly Parton came from poverty, like, real poverty, and recorded her first songs when she was thirteen years old. She worked and worked and was smart, and used her brains and her voice _and_ her body and her big breasts, and she built something for herself. And for her family, too. She always took care of her family. We have that in common, you and me and Dolly. Anyway, Dolly has a quote I love, and I want you to hear it now, and know it. It goes:
"Find out who you are and do it on purpose."
So, how, you might ask me, do I find out who I am? Good question, my young friend. One of life's big ones, in fact. Some of the greatest thinkers in the history of humankind have put their noggins in gear to try to answer this one for us all.
I don't really have the perfect answer, but I will tell you one thing I figured out. Or maybe I read it, or saw someone else doing it and copied them, but here it is:
One surefire way to figure out who you are is to never listen to anyone else tell you who or what you can be. Never let someone else decide for you what you are capable of being. Remember when that one music teacher told you that girls don't make good drummers? Well, turns out he was wrong. I am here to tell you that he was very wrong about that. In many other ways he turned out to be a great saxophone teacher, but he was profoundly wrong about the drumming thing. Right now, future you is in an all-butch choir, and you are singing and drumming and strumming your bursting heart right out, and it is a beautiful thing. I only wish that we hadn't listened to the guy who told us when we were thirteen that girls couldn't play drums, because then I would have started playing them thirty years ago.
Remember when everyone in school called boys who liked drama class or played the flute or who wanted to be cheerleaders fags? You know that guy Corey from home economics class, and Michael, and that blond kid, his name started with a D, I forget his name? Remember how that blond kid did that dance number for a talent show that one time in grade nine, and he could really dance, remember him, his name started with a D, and you don't know this yet because it hasn't happened, and you might wanna sit down because I know you, you are sensitive and this next little bit of news I am bringing you from the future is going to tear your tender heart right out, but I need you to know that he will go on to shoot himself in his father's basement with a rifle one winter a couple years after we graduate. And Corey, he will asphyxiate himself in Yellowknife in a car inside a garage in the dark of a long northern winter and see, I don't think either of these boys had to die, and that is why I am going to ask you a favour. Oh, Michael will make it, by the way. In fact, Michael will turn out to be a real nice guy who works with your sister and grows orchids in his spare time. But those other two guys, well, it didn't have to go down the way it did with them.
Past me, I know you have not accepted this about yourself yet, but I am here to tell you that you are queer, and not only is this fact about us nothing to be ashamed of, it will go on to be one of the things that teaches future you/me so much about the world, and about what is really important. Things like love and truth and honesty and compassion and respecting others who are different than you and friendship and community and chosen family.
So I am going to need you to do me a favour, like I said. I need you to go and find those kids they are picking on even worse than they are picking on you right now. I need you to be kind to them. Even when this means taking a risk with what is left of your own coolness at school. I am asking you to be brave and stick up for those kids they are calling a fag or a dyke, I want you to stand up for and beside those kids that get pushed around and left out and picked over and picked on and made to feel less than worthy somehow. Have their back, and in turn, they will have yours. And together, you will be better than the bullies. Braver. Taller. Truer. More righteous. Someone you can look back on one day and be really proud of.
Because trust me, no one ever gets to be forty-three and thinks about their life and what it all means, and wishes that they had stood aside more often and allowed more injustice to happen to more already struggling kids while they were back in high school. That is not going to happen. But what will happen is that you are going to grow up and look back one day and be sorry for your silences, and regret the times you stood by and let someone be cruel to someone else, or even worse, joined in on it all yourself. And I know what you are thinking. You are thinking that I don't get it, that I have forgotten how hard high school really is in your here and now, because it is my way back when now. But I have not forgotten. I know for a fact that standing up and doing the right thing when more popular or prettier or richer kids are doing the wrong thing, or worse, is some of the hardest shit you will ever have to do in our entire life. But it is so worth it. You don't know this yet, but one day we will travel all over the world talking to scared and lonely and bullied and brave and smart and talented queer youth. One day you and I will literally save a kid's life, maybe more than one, even maybe, if we keep it up. But I need you to take that first step for me earlier than I was brave enough to do when I was your age. You feel me? In a way, you are kind of my second chance at doing it better, kid, and I am counting on you. I believe you can do this.
I can't do it all over, but I can pass on some of what I have learned to you. So, to sum it all up, here it is again: Make art. Write stories. Don't pay any attention to the haters. Most of them will grow up to be adult haters, too, and they will not leave anything behind but a bad taste in your mouth as their legacy. You are capable of great things, and beautiful things, but you need to be strong. Please, don't ever let anyone tell you that you can't sing, or dance, or do math or play the drums or hockey or be a full-time working writer. You can and will one day grow up to be a writer who plays the drums while dancing and singing and counting, all at the same time.
School sucks sometimes. This is the plain truth of it. Parts of school, just like parts of society, are meant to train you to conform, to make you afraid to be anything else but just like everybody else. But look around. Kindness and compassion will reward you with so much more than your fear or apathy will. Trust your heart, it is a lot smarter than you know. Find out what moves you, and be it. Be brave. Be fearless. Be fabulous. Make me proud of us. I know you can do it. I am living proof that you can be anything you dream and work and fight to be. I wish someone had given me these words when I was still you, but they did not. So I am giving them to you now. What you do with them is, of course, completely up to you.
With much love and affection,
Ivan
February 9, 2012
P.S. Do us both a big favour, would you? Don't ever start smoking. I hate to admit it twice in the same letter, but our mother was right about that one, too. It _is_ a filthy habit. We quit four years ago today, and we have never looked back.
# One: Kid I Was
# No Bikini
I HAD A SEX CHANGE ONCE, WHEN I WAS six years old.
The Lions pool where I grew up smelled like every other swimming pool everywhere. That's the thing about pools. Same smell. Doesn't matter where you are.
It was summer swimming lessons, it was a little red badge with white trim we were all after: beginners, age five to seven. My mom had bought me a bikini.
It was one of those little girl bikinis, a two-piece, I guess you would call it. The top part fit like a tight cut-off t-shirt, red with blue squares on it, the bottoms were longer than panties but shorter than shorts, blue with red squares. I had tried it on the night before when my mom got home from work and found that if I raised both my arms completely above my head too quickly, the top would slide up over my flat chest and people could see my... you-know-whats.
_You'll have to watch out for that_ , my mother had stated, her concern making lines in her forehead, _maybe I should have got the one-piece, but all they had was yellow and pink left. You don't like yellow either, do you?_
Pink was out of the question. We had already established this.
So the blue and red two-piece it was going to have to be. I was an accomplished tomboy by this time, so I was used to hating my clothes.
It was so easy, the first time, that it didn't even feel like a crime. I just didn't wear the top part. There were lots of little boys still getting changed with their mothers, and nobody noticed me slipping out of my brown cords and striped t-shirt, and padding, bare-chested, out to the poolside alone.
Our swimming instructor was broad-shouldered and walked with her toes pointing out. She was a human bullhorn, bellowing all instructions to us and punctuating each sentence with sharp blasts on a silver whistle which hung about her bulging neck on a leather bootlace.
"Alright, beginners, everyone line up at the shallow end, boys here, girls here, come on come on come on, boys on the left, girls on the right."
It was that simple, and it only got easier after that.
I wore my trunks under my pants and changed in the boys' room after that first day. The short form of the birth name my parents bestowed me with was androgynous enough to allow my charade to proceed through the entire six weeks of swimming lessons, six weeks of boyhood, six weeks of bliss.
It was easier not to be afraid of things, like diving boards and cannonballs and backstrokes, when nobody expected you to be afraid.
It was easier to jump into the deep end when you didn't have to worry about your top sliding up over your ears. I didn't have to be ashamed of my naked nipples, because I had not covered them up in the first place.
The water running over my shoulders and back felt simple, and natural, and good.
Six weeks lasts a long time when you are six years old, so in the beginning I guess I thought the summer would never really end, that grade two was still an age away. I guess I thought that swimming lessons would continue far enough into the future that I didn't need to worry about report card day.
Or maybe I didn't think at all.
_"He_ is not afraid of water over his head?" my mom read aloud in the car on the way home. My dad was driving, eyes straight ahead on the road. "He can tread water without a flotation device?" Her eyes were narrow, and hard, and kept trying to catch mine in the rearview mirror. "Your _son_ has successfully completed _his_ beginner's and intermediate badges and is ready for _his_ level one?"
I stared at the toes of my sneakers and said nothing.
"Now excuse me, young lady, but would you like to explain to me just exactly what you have done here? How many people you have lied to? Have you been parading about all summer half-naked?"
How could I explain to her that it wasn't what I had done, but what I didn't do? That I hadn't lied, because no one had asked? And that I had never, not once, felt naked?
"I can't believe you. You can't be trusted with a two-piece."
I said nothing all the way home. There was nothing to say. She was right. I couldn't be trusted with a two-piece. Not then, and not now.
# Walks Like
THE FABRIC OF THIS MEMORY IS FADED, its edges frayed by time.
The young girl who lived it is now just a ghost inside of me. I can remember only her bones; the skin and flesh of her are brought to me in the stories of others. Mothers, uncles, and aunts remind me of the kind of child I was then.
There was the smell of Christmas everywhere, I do remember that, pine trees and wood smoke and rum cake. The women smelled of gift perfume, the men of new sweaters.
Everywhere were voices, maybe a dozen different conversations woven together in the rise and fall of talk and laughter that is the backdrop of all my mind's snapshots of my family then, a huge room full of people connected to me by their blood.
I was sitting almost too close to the fire. Iced window panes separated us from the bitter white of winter outside. Everyone I'd ever known was still alive.
I was about four years old.
Both of my grandmothers sat in overstuffed chairs next to the fireplace, talking, a trace of Cockney, and a hint of an Irish lilt, respectively.
I sat on the thick rug between them, rolling a red metal fire truck up and down my white-stockinged legs, making motor, gear-changing, braking noises. Listening.
"You should have seen the fuss this morning, getting her into that dress. I tell you, Pat, I'd've never stood for it from any of my girls. You'd've thought I was boiling her in oil, the way she was carrying on. She wanted to wear those filthy brown corduroy pants again, imagine that, and she knows we're going to mass tonight." My mother's mother clicked her tongue and sent a stern glance in my general direction.
"That was what all my boys were like, Flo. Really, if you could have seen me the day that portrait on the wall there was taken, I swear I didn't have a nerve left for them to get on. Like pulling teeth, you know it was, to dress those four."
"Well, you'd expect it from the boys, you know, it's only natural. But her, I don't understand it. Her mother always liked to dress up, and never a speck of dirt could you find on my Norah... look here, come here you." She curled an arthritic finger at me.
I stood up reluctantly and dragged my feet across the carpet toward her, hoping for a good spark.
"Look, see what I mean? Look at her knees, how does she do it? It's only been a couple of hours, and there's only snow on the ground out there. I couldn't find any dirt right now if I went out looking. Here, let me fix up that zipper..."
My small fingers shot up to intercept her, and a rather large bolt of static electricity flashed between us. She pulled her gnarled hand back for a moment, and then brought it down on the back of mine.
"What a nasty thing to do to your poor old gran! Aren't you ashamed of yourself? Now, run and fetch us both a rum ball, and for the love of Mary, don't get it all over the front of you."
I looked over to my other grandmother, at the shadow of an evil smile which pulled at the corners of her mouth. She winked at me, and motioned for me to be off.
"See what I mean about her, Pat? I'm worried sick she'll turn out to be an old maid. What happens when she starts school? Look, now... she even walks like a little boy..."
"You're far too hard on her, Flo," came the voice of the mother of my father from behind me, laced with just a hint of annoyance. "She will be just fine. She just walks like that. That's just how she walks."
# Just Reward
SHE WAS NEVER THAT GOOD AT FRISBEE, but it wasn't about that for me. Her summer brown legs bent with a grace I could never possess, and her straight black hair swung unbraided, always a strand or two across her face, in her mouth.
Her palms were lighter than the backs of her hands, and often she would lay them in the place her hips would be one day, plant both feet in the dust, and throw her head back when she laughed.
She was doing just this the day we found the money. Her Frisbee throws were unpredictable and wobbly, and this one had arced sideways into the juniper bushes that lined the parking lot next to the parched park we were playing in.
Nothing is as dry as July dust in the land of the midnight sun, so I almost missed the brown leather wallet laying in the dirt.
"Valerie, come look here. Look at this."
"I don't want to look at bugs. Come on, throw it here."
She saw the look on my face and went silent, looked down into my hands.
A rectangle of worn calfskin with a brass bill clip inside, pinning down a wad of American bills. I stuffed it into the waistband of my shorts and we ran down to the edge of the river, under the cover of willows.
Eleven one hundred dollar bills, two twenties, four ones.
"One thousand, one hundred and forty-four dollars." Valerie was perched on the balls of her feet, her teeth shining white behind chapped lips. "We have to take it to the police station," she whispered.
"The police? Are you crazy? We could buy practically anything with this."
She shook her head, a wrinkle creasing her forehead. "Our parents would take it away anyhow. The police." She said this like there was no other option.
"We could hide it for a while then, in the fort. We could save for our educations." I appealed to her practical side.
"If we take it to the police, and they can't find whose money it is, then we can keep it. We could be heroes." She raised her eyebrows and rubbed her palms on her shorts for emphasis. "Rich heroes."
It was settled then. I never once thought to argue that it was I who had found the money. I had no name for what I felt for her; we were nine years old and I would have done anything she wanted.
"You fucking did what?" My father was chewing his pork chop with his mouth open.
My mom slapped his arm, right above where his shirtsleeve was rolled up to. "You did the right thing. I'm really proud of you girls, and so is Valerie's mom."
My father looked at me like he couldn't figure out just where he had gone wrong.
The policeman shook his head as he filled out the form. "Well, he was probably an American." This guy was sure to make detective. "No ID, huh?" He narrowed his eyes at us, beads of sweat on his forehead.
We shook our heads simultaneously.
"Beginning of summer, probably on his way up north. To Alaska," he explained, as though there was a multitude of destinations for tourists to choose from. "There's a chance he'll check in on his way back down. No one claims this in six weeks, say, then you two are in the money."
* * *
We spent that money over and over in our heads for the rest of the summer. Valerie wanted a camera, and an easel and paint set. "No cheap stuff. The kind of brushes with horse hair in them."
I wanted a BMX with chrome pedals, and a microscope. "Maybe a chemistry set, too. And walkie talkies. One for me, one for you. We could talk on them late at night. And a rowboat."
"Cowboy boots," she added, swinging in the hammock, a piece of straw between her front teeth. "Red cowboy boots."
It was the ninth of August. We had seven days left.
* * *
The next morning, the phone rang at exactly eight o'clock. I was eating puffed wheat and listening to "Seasons in the Sun" on the radio that sat between the toaster and the plant on the lemon yellow counter next to the window. My mom was filling the kettle, and held the phone between ear and shoulder, motioning silently at me to turn the music down.
"She's right here. I see. Okay, I'll tell her. Thank you, officer." She uncurled the phone cord with her forefinger and hung it up. "Someone claimed the wallet. He's downtown, he wants to give you two a reward. I'll drop you both off on my way to work."
We sat side by each in the back seat of my mom's Tercel, silent and lead-bellied under our seat belts. Valerie smelled like Irish Spring soap and toothpaste. I had forgotten to even brush my hair.
He looked like a caricature of a tourist come magically to life. The buttons of his polyester print shirt strained to hold his belly inside his khaki shorts. He even had waxy hairs sticking out of his ears. He shook our hands, his moist palms unnaturally soft. "Here's my little heroes," he wheezed. He patted us both on our heads, mussing our hair and smiling at the cop behind the counter. "Let's head across the street and get you girls your _re_ -ward."
He stood perspiring in the service window of the Dairy Queen. "What's your favourite flavour of milkshake?"
"She likes strawberry, chocolate for me," I piped up. Talking to strangers was my job. Explaining why we had done what we did to parents was her territory, but strangers were my area of expertise.
"Too early for milkshakes," she whispered to me, as he pulled out his billfold and handed over the four singles. I shushed her. Surely this was just the first phase of our reward.
But ten minutes later we sat alone at the bus stop, the change from our milkshakes stuck to my palm, for bus fare. He had told us what good girls we were and hopped into his motor home. His wife had waved over her knitting at us from the passenger seat. The TravelEase edged back onto the road.
"I hate South Carolina. Never going there." Valerie spit in the dust and tied up her shoe.
My dad was still at home when we got back, strange at this time of day. He was smoking an Export 'A', drinking tea, and reading _Shogun_. We tried to head straight into my room, but he looked up and cleared his throat.
"Whoawhoawhoa. Where're you two going?"
Valerie picked idly at a scratch on her thigh; I stood on one leg, then the other, waiting for the inevitable.
"Didja get your _re_ -ward?" He split the word in two, like someone from South Carolina would.
I nodded almost without moving my head. Valerie shrugged.
"Welllll...?" His one eyebrow raised, his hands perched like spiders on the wooden table.
"We got milkshakes," Valerie said softly.
My dad turned his right ear to us, played with a make believe hearing aid.
"He bought us both milkshakes," I blurted out, the sweetness of chocolate already halfway back up my throat.
"Small or large?" he crowed, slamming both palms flat, slopping tea onto his paperback.
"Large ones." The bottom of Valerie's jaw stuck further out defiantly, her brown palms returning to her hips.
My dad laughed from deep in his belly at us both, and reached for his smokes. "Well, I hope it went down good, because that was the most expensive fuckin' milkshake you're ever gonna drink."
Twenty years later I realized we had, in fact, spent that money on our educations.
# It Doesn't Hurt
My cousin claims he invented the game, but I swear it was me. You need what they call a rat-tail comb, one of those plastic ones you can buy at the drug store; they come in bags of ten. They have a comb part, and then a skinny plastic handle, which, I suppose, is where the name comes from.
You take the comb and heat it up over an element on the stove so you can bend a curve into it, like a hockey stick. Then you get a ping pong ball, or one of those plastic golf balls with the holes, and there you are. Comb ball, we called it. Let the game begin.
The game was invented to be played in a long, narrow hallway, so a mobile home is the perfect stadium. You close all the bathroom and bedroom doors, and each opponent gets on their knees at either end of the hall. Kind of like a soccer goalie, only shorter. Whoever has the ball goes first.
You hold the handle of the comb in one hand and bend the comb back with the other, and let go. The ping pong ball rebounds off the walls and floor at speeds approaching the sound barrier, and the other guy tries to block the ball with any part of his completely unprotected body.
A ping pong ball striking naked skin at the speed of sound is bound to hurt. So there were the obvious injuries: circular welts about the face, neck, and arms were common. There were other hazards, too: carpet burn, bruised elbows and knees. Once, my sister leapt up to block a shot and smashed her head on a door handle and just about bit the tip of her tongue off. My cousin sprained his wrist trying to flip himself back onto his feet for a rebound.
My aunt stepped in, in an attempt to reduce the casualties. She tried to ban comb ball altogether, but was met with such a united front of dismay and pouting that she was forced to compromise. We were only allowed to play until someone cried. And we had to scrub off all the little white marks the ping pong balls left on the wood panelling.
We were only allowed to play until someone cried. Of course, this added a masochistic element to the game we all enjoyed. I would take a stinging shot to the lower lip and kneel motionless in the hallway, breathing deep through clenched teeth. Everyone would stop, searching my face for any sign of moisture, which would signal the end of the game. "Doesn't hurt," I would whisper bravely. "It doesn't hurt. It doesn't hurt. Let's go. My shot." Everyone would let out their breath and continue.
Whoever cried ended the game. Whoever cried sucked. My aunt would march in and grab our combs, and send us outside to play. "It's a beautiful day out there. Quit killing each other in my hallway and go get some exercise."
Playing outside was okay, but there was nothing like a rousing, bloody match of comb ball. We would compare scars afterwards, like soldiers. "Took the skin right off, bled all over the rug too," we would brag, our striped shirts pulled up over our elbows. "And not one tear. Kept right on going."
My cousin Christopher ended it all the day he broke his thumb. This required a trip to emergency, and a splint. He forgot to try not to cry, and the combs were confiscated for good. For a while we were impressed with his sling, and his need for painkillers, but then reality set in. No more comb ball. Christopher was a wimp, and prone to accidents. Remember, he got that concussion that one time and they took the tire swing away? We all mourned the loss of the greatest game that ever came to the trailer park.
We came up with a version of cops and robbers that satisfied our bloodlust for a while. It involved riding around on our bikes and wailing on each other with broken-off car antennas, but it wasn't the same. Crying while playing outside was a different story, because you got to go back into the house. The stakes weren't as high. There was nothing to lose.
I worry today, about my friends' kids. Nothing hurts when you play Nintendo, not even when you die. What are we teaching our children? I still utilize the skills I learned playing comb ball. Just the other day, I fell off the back of a five-ton truck helping my friend move. I leapt up immediately, exclaiming, "It doesn't hurt. It doesn't hurt." And a couple of weeks later, it didn't. Just like the good old days.
# Sticks and Stones
IT SEEMED LIKE A FINE IDEA AT THE TIME. Of course, now I look back and count my ten fingers and toes, my two legs and arms that still function properly, shake my head that sits on top of the neck I have never broken, and thank my guardian angel that I still possess these blessings. But it seemed like a fine idea at the time.
My father is a welder, and his shop was located in the middle of a large and potholed industrial section just off the Alaska Highway on the edge of town. It came complete with snarling guard dogs and broken-down bulldozers, and even had its very own forgotten car and truck graveyard. If you looked up from the dusty ground and buckets of used oil, out behind colourless mechanics' shops and the skeletons of scaffolding, you could see the whole valley stretched out, the Yukon River sparkling blue and snaking through the painted postcard mountains. If you looked up, which I rarely did. There was too much to do.
There were any number of stupid and dangerous activities to pass the day with, untold numbers of rusty edges to tear your skin and clothes on, a myriad of heavy metal objects to fall off of or get pinned beneath. I don't remember whose idea the tires were. They were not just any tires; they had once pounded dust under earth movers, or dump trucks. They were monsters, and they were everywhere. It took the whole pack of mechanics' kids and welders' daughters and crane operators' sons to move them; getting them up and onto their sides was a feat of team effort and determination, aided by crowbars we pinched from the backs of our dads' pick-ups when no one was looking. Rolling them to the edge of the power line without being noticed involved lookouts and quick action. We knew they would stop us if they found out; we didn't need to ask. The covert element of the operation only added to the thrill of it all.
Only two of us could fit in at a time, which was okay, because we had all summer and plenty of tires. Three or four kids would hold the tire steady, teetering on the edge of the cliff at the top of the power line, and two would climb inside. Kind of like gerbils on one of those exercise wheels, except you would face each other, arms and legs pushing out into the inside of the tire to hold yourself in. Gravity pretty much took care of the rest.
It was better than any roller coaster, not that any of us had been on one. It was the random element of the tire's path that did it. There was just no way to know what that tire was going to bump into or off of, and the only thing more fun than the roll down was when the tire started to come to a stop at the bottom, and did that roll-on-its-side, flip-flop dance at the bottom of the hill, kind of like a coin does when you flip it and miss and it lands on the linoleum. Only this was a huge dump truck tire with two dirty kids inside, laughing hysterically, laughing until tears ran and our sides hurt the next day. Only one of us ever puked: the heavy duty mechanic's oldest daughter lost her lunch all over her brother one day, and so we never let her ride after that, just sent her into her dad's shop to distract him while we rolled tires past his big bay doors out front.
I think it was the smell that finally gave us away. My mom kept asking me what the hell had I been up to that day while my dad was at work. There is something unmistakably foul about the smell of the inside of a tire, a cross between pond water and cat pee, I would venture, and my mom couldn't quite pin it down, but she got suspicious.
It was a bright August morning, the day it all ended, and we had a beauty of a big tire all loaded up and ready for take-off when we heard a noise inside our heads, a skull-piercing shriek that stopped our blood. We all froze in our tracks. My mom appeared from out of nowhere and it dawned on me that the noise was originating from her mouth, the words becoming slowly recognizable as she bee-lined toward us, her face all veins bulging red, and the whites of her eyes all you could see: " _What the fuck are you stop right now stop that stop it stop_..." and so forth.
There was really no explaining our way out of this one. What else could we possibly have had in mind? More damning, of course, was the pile of tires already situated at the bottom of the power line; we couldn't even argue that we were just thinking about climbing inside one and rolling it down the hill, but were just about to prudently change our minds and go help our fathers sort bolts and sweep up.
An ad-hoc committee of irate parents was called immediately, and our dads did what any fathers would have done when catching their child about to engage in activities which could only result in grievous bodily harm: they spanked us all senseless. Nothing like pain to remind you of how much you could have been hurt. It was, after all, the seventies. I was also given plenty of time to mull over my decisions for the next two weeks: I was grounded, and spent the rest of the summer inside at home, watching the Seventh Day Adventist kids safely ride their bikes on the road. What could you do? Like I said, it seemed like a fine idea at the time.
# Bad Luck and Big Feet
WE COULDN'T BELIEVE OUR LUCK THAT DAY as we walked into the Sally Ann and saw all those roller skates. Story was, they had just shut down an old rink, and about two hundred pairs of kids' rental roller skates had been donated to the thrift store. The cool kind, too, where the translucent yellow wheels were bolted right to the soles of sturdy-looking running shoes that tied up with wide white shoelaces, not the dorky old-fashioned kind that you had to strap on over your own shoes and tighten up with a key that always went missing after a couple of days. They were all on sale for a buck fifty a pair, because there were so many of them they were clogging up the entire sporting goods aisle, overflowing from the shelves and piled on top of a moth-bitten street hockey net. Gran announced that she was going to buy all of us our very own roller skates, since they were a bargain, plus we had behaved ourselves all day so far. She told us we could go ahead and help ourselves to a pair that fit, but not to bother the poor lady behind the counter with it, she had enough on her plate as it was. We all raced forward, digging about for the best-looking pair in our size, almost climbing over one another, like you see nowadays when they have half-price stuff on sale at the Wal-Mart or somewheres. The only thing better than getting new stuff was not having to share it with anyone. A few springs back Christopher got a skateboard for his birthday, and we all had to take turns on it. Chris wasn't exactly famous for sharing, and he knuckle-punched his brother for hogging it too long. My little sister ran into the house to tell on Chris and woke up my Uncle Kevin who ended up locking the skateboard in the trunk of his car, just so he could get some peace and quiet since he was working the night shift. Four kids, and a hundred pairs of roller skates to pick from. Mine were blue, with three white racing stripes down each side. They were practically brand new, the wheels were just a tiny bit scuffed. The laces even had the little plastic tips on them still, like shiny back-to-school gym shoes do for the first half of September. Carrie wanted the yellow ones, but they were too big so she settled on red, and Danny's were a faded blue, at one time the same colour as mine, but now a little grey after being handled by a thousand different dirty fingers. Now all we had to do was find a pair that fit Christopher. The thing about my cousin Christopher was that he was extremely prone to misfortune. My gran once even crossed herself and said that Chris must have slept in the morning that the good Lord was handing out the luck, because nobody else had ever choked on the only quarter hidden in his own birthday money cake. It was like he was born with a little dark ball of wrong turns and close calls and mishaps that hovered in a tangled cloud above his head, I swear, you could almost see its shadow on the sidewalk beside him some days. When bad luck decided that something needed to happen to someone, it usually picked Chris, since he was never more than an arm's length away from a catastrophe. Spilled milk, bee stings, and deep slivers followed him around like stray dogs. He couldn't help it; it wasn't his fault, he was just born like that. When he first started kindergarten his teacher thought he might be slow, and they almost put Chris in the special class, until my aunt had to go down to the school and give that teacher a piece of her mind. Just because a kid didn't talk much didn't mean he couldn't count backwards from one hundred and already know his alphabet by heart. Not to mention how many slow kids could take a radio apart and put it back together when they were barely five years old and if that teacher had his head screwed on right he would know it. It turned out that Christopher was not part retarded (that's what they called it back then), he was just mostly deaf, but by the time the doctor put the tubes in his ears, the bullies and bigger kids had picked up the scent of blood in the water and closed in on my cousin. He was quiet, and clumsy, and something about the way he blinked his eyes and chewed his bottom lip showed his fear and evoked cruelty in others. He was all awkward elbows and not quite right angles, and his feet and hands looked like they belonged on a much bigger body, like he had recently borrowed them from a boy twice his size. One hundred pairs of discount roller skates, yet somehow none that Christopher could squeeze his feet all the way into. His bottom lip swelled into a quivering pout, and his eyes filled up with tears. I knew it was a sin to hate your own cousin just because he was born with bad luck and big feet, but I couldn't help it, and a sour ball of guilty spit got caught in my throat and refused to be swallowed. "No fair if everyone gets roller skates but me." His voice sounded small, and broken. He picked at a scratch above his knee and rubbed one sock foot with the other. The rest of us stared down at the worn-out floor tiles and pondered this awful truth. Gran's form of justice was swift, and thrifty. After enduring a brief bout of tears from all four of us, followed by a sobering sermon that included such topics as the mouths of gift horses, shoeless children from other countries, counting your blessings, and living on a fixed income, we were given two options. We could go home without any roller skates at all, or Gran would buy Carrie, Dan, and me each a used pair, and then we would take the bus to the mall and find a brand new pair that fit Christopher. Of course, we chose option #2. At the sporting goods store, the only ones we could find were the old fashioned kind that you strapped on over your shoes and tightened with a key. Gran told the salesman that it was highway robbery, what he was charging her for them, and he gave her ten percent off, even though there was usually no senior's discount on sports equipment. It was obvious that she had almost had enough of the whole business, so Chris didn't dare complain that his skates weren't as cool as ours were, and we didn't risk even a sideways gloat. The biggest patch of pavement in our neighbourhood was the parking lot beside the baseball diamond, about two blocks up the hill from our house. We wobbled and rolled up the road as soon as we got home, laughing and leaning on each other for balance. We stopped at the top of the hill for a minute to catch our breath. The street sloped down in a lazy curve and met the steeper road that led into the parking lot. Carrie went first, her bum sticking out and her feet spread too far apart, a squeal of glee trailing behind her as she picked up speed. Dan bent his knees and cannon-balled down the hill, almost overshooting the turn-off, and I followed just far enough behind to avoid rear-ending him. My eyes were fixed on the road, on the lookout for patches of dandelions that had pushed their way up through cracks in the chip-seal, and bits of gravel big enough to catch a wheel on. It dawned on me that none of us had spent much time practicing how we were going to stop before starting down the hill, and as the oldest, this was just the kind of detail I should have thought about. Carrie hit the scruffy lawn and fell forward, her arms and legs splayed in all directions like a starfish. She curled into a ball, clutching the crotch of her shorts since she was prone to peeing herself when she got too excited. Dan managed to grab a signpost with one hand and spun to a stop, and I safely bounced off the tired chainlink fence that sagged around the outside of the ball diamond. Christopher hesitated at the top of the hill. The afternoon sun burned like an egg yolk in the blue behind him, and the air rippled in blurry waves wherever the sky touched the pavement. The toes of his sneakers stuck way out over the front wheels of his skates, and his naked knees were glued together. It crossed my mind that maybe letting the younger kids learn how to roller skate in bare legs was not such a good idea, but it was too late. Chris careened towards us, his arms whirling in giant circles, backpedaling on his heels in a slow-motion slapstick of panic. He fell on his butt and skidded to a stop not even halfway down the hill. There was a breathless second of silence, and then his jaw dropped and an animal sound came out of his open mouth. I had never heard anyone screech like that, it was worse than when Danny burnt his leg on the exhaust pipe of his dad's motorbike, even louder than the time I fell out of the tree and broke my wrist and got eleven stitches in my head. It was more of a siren than a scream, and he didn't stop. My blood stood still in my veins for a moment, then I leapt forward, forgetting I had wheels attached to my feet. I broke my fall with the heels of both hands, tearing identical patches of roadrash into my palms. I ripped both of his roller skates off, and ran my stinging hands up and down his arms and legs, searching his body for blood or broken bones. I pulled him to his feet by one elbow and a belt loop, but he sunk back into a crouch when I let go of his arm, still wailing and spewing tears and snot. I couldn't see anything wrong with him, he wasn't bleeding or holding his ankle or wrist, but he was screaming like he was being skinned alive. When I knelt down beside him to wrap his arm around my shoulders, a foul smell filled my nostrils. He had crapped his pants, and the evidence had escaped his underwear and was smeared down the back of his right leg. I had to take him home, which meant I had to take everyone home, since it was 1981 and it wasn't safe to leave little kids alone in a park in a big city, especially when there was a psycho on the loose and anything could happen. My sister wouldn't touch Christopher because he had poo on him, so I made her carry the roller skates while Dan and I dragged him home in our sock feet. He was made of lead and rubber, and by the time we burst through the screen door on the back porch my knees were wobbling and my breath was burning in the back of my throat. "Gran," I gasped, "Chris wiped out on the hill and I can't see anything wrong with him but he pooped himself and won't shut up or tell me where it is hurting." Chris had stopped screeching, but his chest still heaved in great long sobs and his face was streaked with dirt and tears. He wiped his upper lip with a snotty wrist, and leaned against the laundry room door, which wasn't closed all the way. It swung open with a squeak and Chris stumbled sideways and fell to one knee on the linoleum in front of the washing machine, leaving a brown streak of poop across the white ceramic door of Gran's brand new clothes dryer. "You dirty little bird," she squawked, hauling Chris back onto his feet by one wrist and dragging him toward the bathroom. "Knock it off with all this nonsense and get into the tub or I'll give you something to cry about." She launched him through the bathroom door with one swat across the sagging bum of his shorts. He let out a scream so loud that even Gran clapped her hands over her ears, and the dog bolted down the hall, skidded across the kitchen, and hurled herself under the table, knocking the butter dish to the floor. Not even my little sister could fake that kind of agony, and Gran lifted his limp body into the bathtub and pulled down his pants with a shaky hand. His bum was snow white, and between his poo-smeared butt cheeks bulged a giant purple egg. It was a horrible hybrid of a blood-blister and a bruise, and it throbbed and pulsed like a swollen creature from a science fiction movie. Even the nurse at the walk-in clinic in the mall claimed she had never seen anything quite like it, and said that she bet it hurt something terrible. Then she patted the sweaty curls on top of Chris's head with a manicured hand, and gave him an envelope that had twelve real painkillers in it, not just orange-flavoured baby aspirin, and a frozen bag full of what looked like blue jello. Gran called us a cab from the pay phone outside of the 7-Eleven, even though the mall was only ten blocks or so from our house and we had no luggage or groceries. We were all acting super nice to Chris, since we felt bad for thinking he was being a big baby because it we had mistakenly thought he had barely wiped out at all because we couldn't see any blood or wounds, but how were we supposed to know he had a giant purple lump in his shorts that hurt so much it made him shit himself? My sister even gave him her winning Oh Henry! wrapper she had kept folded up in her pink plastic wallet for weeks. She had been waiting for the right time to trade it in for the free Pepsi she had coming, saving it for a rainy day, she said, which really meant a day when the rest of us were broke. Carrie was like that, she used to hoard a good chunk of her Halloween stash every year too, just so she could haul it out a month later and eat it with dramatic relish in front of the other kids without sharing. But she gave her free pop to Chris, and Gran let him sit in the front seat of the cab alone, and she made him a jam sandwich right before dinner, just because. Chris was good about it all, he didn't lord his injury over us able-bodied kids like some would, he didn't make us get him stuff or hog the couch or fake a limp so he wouldn't have to help with the dishes or go move the sprinkler. Gran watched him wince as he slowly settled himself onto a chair at dinnertime. He sat lopsided, perched painfully atop the blue ice pack, and laced his fingers together so we could say grace. Gran thanked God for the meal we were about to eat, and made the sign of the cross. Then she picked up her fork and shook it at the four of us, clucking her tongue like she did when she was about to say how something was a crying shame. "Look at him, poor little wretch. It's a crying shame none of yous had enough sense to put on long pants before you went out fooling around on them things, just a crying shame. Count yourself lucky that nobody cracked their skull wide open, thank Gawd." She looked up at the ceiling tiles, and crossed herself again. We sat with our hands in our laps and our heads bowed, just in case this particular crying shame needed any further explanation or perhaps an extra prayer. I mulled it all over in my head, wondering how long pants could possibly prevent head injuries, and whether or not Chris would still have got poop all over the dryer door if he had crapped in his jeans instead, and would any of this have happened at all if he hadn't been born with giant feet that only fit into the crappy roller skates? And didn't that mean the whole thing was in fact an act of God? I pressed my lips together to keep myself quiet. It was best not to ask too many questions, especially ones about the Good Lord Above. I was old enough to know it was a sin to blame God when bad things happened, even things that could only be his fault, like floods or earthquakes or innocent children from good homes who died too young or babies who got born with a rare disease or a weak heart. It was blasphemy to question his will or his wisdom or the way he went about his business, to even suggest that God might have thought twice before burdening a boy with feet that did nothing for his self-esteem which was a big part of why he had to repeat grade two and go see the special ed teacher for extra help with his math and spelling instead of going to gym class with all the other kids and that is why he sucked at almost every sport except wrestling and sprinting and long-distance running which he had plenty of practice at from getting chased home or beaten up. The next day Gran took all four pairs of roller skates back to the Salvation Army and traded them in for a bat, three baseball gloves, and a grass-stained softball, even though Chris sucked at baseball too. The only fair thing to do was to give all of us something none of us wanted, and disappointing all of us equally was the only way to keep everyone happy. I knew it wasn't her fault that we had gotten on her last nerve and she had to wash her hands of it all and teach us a valuable lesson so we would think twice next time before risking our necks when she had enough to worry about as it was. I couldn't stay mad at Gran. I blamed God for all of it.
# Three Left Turns
THE AIR SHIMMERED AND TWISTED where it met the earth. The road beneath the tires of my bike was a ribbon of dust, hard-packed and hot, a backroad race-track, and I was gaining on him.
His BMX was kicking up a cloud of pretend motorcycle smoke. I smiled and pedalled through it, teeth grinding grit and lungs burning, because the stakes were so high.
If I won, I was faster, until next time, than my Uncle Jimmy. And if he lost, he was slower, until next time, than a girl.
Is the little brother of the woman who married your father's brother related to you? I called him my Uncle Jimmy, regardless, and he was my hero.
He was four years older and almost a foot taller than me, and I don't think I ever did beat him in a bicycle race, but the threat was always there.
Just allowing a girl into the race in the first place raises the possibility that one might be beaten by a girl, so the whole situation was risky to begin with. We all knew this, and I probably wouldn't have been allowed to tag along as much as I did had I been older, or taller, or a slightly faster pedaller.
Girls complicate everything, you see, even a girl like me, who wasn't like most; you can't just pee anywhere in front of them, for instance, or let them see your bum under any circumstance, or your tears.
There were other considerations, too, precautions to be taken, rules to be observed when girls were around, some that I wasn't even privy to, because I was, after all, a girl myself.
It was the summer I turned six years old, and I was only beginning to see what trouble girls really were.
But I, it was allowed by most, _was_ different, and could be trusted by Jimmy and his friends with certain classified knowledge. I was a good goalie and had my own jackknife, and could, on rare occasions, come in quite handy.
Like that day. That day I had a reason to tag along. I had been given a job to do, a job vital to the mission.
The mission was to kiss the twins. For Jimmy and his skinny friend Grant to kiss the twins.
The twins were eleven, and blonde, and from outside. Being from outside was a catch-all term used by people from the Yukon to describe people who were not from the Yukon, as in:
_Well, you know how she's from outside and all, and always thought she was better than the rest of us_ , or, _I couldn't_ _get the part and had to send it outside to get fixed, cost me a mint_ , or, _well, he went outside that one winter and came back with his ear pierced, and I've wondered about him ever since_.
The twins were only there for the summer. Their dad was there to oversee the reopening of the copper mine. They wore matching everything, and also had a little sister, who was seven.
That's where I came in.
The plan was a simple man's plan, in essence. As we worked out the details, we all stood straddling our bikes in a circle at the end of Black Street where the power line cut up the side of the clay cliffs.
We were all going to pedal over to where the twins and their little sister lived. We had already hidden the supplies in the alley behind their house. The supplies consisted of a small piece of plywood and a short piece of four-by-four fence post.
We would take the plywood and prop one end of it up with the four-by-four (Jimmy and I had two uncles who were carpenters, and he would himself go on to become a plumber) and build a jump for our bikes. Then we would ride and jump off it, right in front of the twins' house, which was conveniently located right across from the park (good cover). This would enchant the unsuspecting kissees-to-be (and most likely their little sister), drawing them out from their house and into the street, where they would be easier to kiss.
We would then gallantly offer the girls a ride on the handlebars of our bikes, having just proven our proficiency with bike trick skills by landing any number of cool jumps. The girls would get on our handlebars, and Jimmy and Grant would ride left down the alley with the twins, and I would take a right with their little sister and keep her occupied while they carried out the rest of the mission. The kiss-the-twins mission.
The only person more likely to tell on us than the girls, after all, was their little sister, and I had it covered. Keep her occupied. Don't tell her the plan. Don't wipe out and rip the knees out of her tights. Drive her around the block a couple of times, and drop her off. Grant and Jimmy would take care of the rest.
We thought we had pretty much everything covered. We even had secondary strategies; if the jump didn't work right away, we could always make it higher, and if that didn't work, I could bravely lie on the ground right in front of it, and they could jump over me.
It was a good plan, and it worked.
What we hadn't foreseen was, I guess, unforeseeable to us at the time. The girl factor, that is.
How could we have known that the twins' little sister would think that I was a boy?
And how had the girls already found out that Jimmy and Grant wanted to kiss them?
And what was I supposed to do if this girl, who was one year older than I was, slid off my handlebars as soon as we rounded the corner into the alley, planted both of her buckle-up shoes in the dust and both her hands on her hips, wanting me to kiss her like my uncle was kissing her older sister?
It hadn't crossed our minds, but that is exactly what she did (and I can't remember her name to this day, and so can't make one up, because this is a true story): the twins' little sister wanted me to kiss her, and I'm sure I must've wanted to oblige her, if only for the sake of the mission. Because that is the first most secret, sacred tomboy rule: never chicken out of the mission.
There was only one problem. The girl problem. She didn't know I was one.
It wasn't that I had deliberately misled her, it just hadn't really come up yet.
And since me kissing anyone was never part of the plan as I knew it, I had not given much thought to the girl factor. But this girl had a plan of her own.
There she was, all puckered up and expectant-like, and it seemed to me I had a full-blown situation on my six-year-old hands.
A mistake had been made, somewhere, by someone. But what was it?
I had a number of options at that point, I guess.
I could have put my left hand on the back of her yellow dress, my right hand over her smaller left one, and given her a long, slow...
No, I would have dropped my bike.
I could have leaned awkwardly over my handlebars and given her a short, sloppy one, and just hoped for the best, hoped that there wasn't something about kissing a girl the boys couldn't tell me, any slip that might reveal my true identity.
I might even have gotten away with it. Who knows? I would have liked for this story to have ended that way.
But it didn't. And because this is a true story, I would like to tell you what really went down with me and the twins' little sister in an alley by the clay cliffs the summer I turned six.
But I don't remember.
What I do recall was that unexplainable complications had arisen because we did not take the girl factor into consideration, rendering this mission impossible for me to carry out.
According to Grant and Jimmy, the little sister started to cry when the dust had cleared and she found herself alone, in an alley, in this weird little town where her dad made her come for the summer, and the twins had to take her home.
And when all three left, two weeks later, unkissed, Grant and Jimmy still considered me a major security risk.
But I don't remember my retreat.
My Aunt Norah was seventeen, and babysitting us that day. She said I came flying up the driveway, dumped my bike on her lawn, streaked past her into the living room, and threw myself on the couch, sobbing incoherently.
I would like to think that at this point she patted my head, or hugged me, or something, to calm me down, but we weren't really that kind of a family. It's not like I was bleeding or anything.
She said that when I finally calmed down enough for her to ask me what was wrong, all I could say was three words, over and over.
I don't know. I don't know. I don't know.
Girls. We can be so complicated.
# My Hero
_Webster's New School_ _and Office Dictionary_ defines a hero as a man of distinguished courage, moral or physical; or the chief character in a play, novel, or poem.
* * *
Her name was Cathy Bulahouski, and she was, among other things, my Uncle John's girlfriend. She had other titles, too—my family is fond of nicknames and in-jokes—she was also referred to as the girl with the large glands, and later, when she left John and he had to pay her for half of the house they had built together, she became and was remembered by the men as "lump sum." The women just smiled, and always called her Cathy.
Cathy Bulahouski, the Polish cowgirl from Calgary. I've wanted to tell this story for years, but never have, because I couldn't think of a better name for a Polish cowgirl from Calgary than Cathy Bulahouski.
I remember sitting in her and John's half-built kitchen, the smell of sawdust all around us, watching her brush her hair. Her hair was light brown and not quite straight, and she usually wore it in a tight braid that hung like a whip between her shoulder blades. When she shook the braid out at night, her hair cascaded in shining ripples right down her back to just past the dips behind her knees.
She would get John to brush it out for her, she sitting at one end of a plain wooden table, he standing behind her on their unpainted plywood floor. I would be mesmerized, watching her stretch her head back, showing the tendons in her neck. Brushing hair seemed like a girl-type activity to me, but John would stroke her hair first with the brush, then smooth it with his other hand, like a pro. My father rarely touched my mother in front of me, and I couldn't take my eyes off this commonplace intimacy passing between them.
* * *
The summer I turned eleven, Cathy was working as a short-order cook at a lodge next to some hot springs. She was also the horse lady. She hired me to help her run a little trail ride operation for the tourists. My duties included feeding, brushing, and saddling up the eight or so horses we had. And the shoveling of shit. I wasn't paid any cash money, but I got to eat for free in the diner, and I got to ride Little Chief, half-Appaloosa and half-Shetland pony, silver grey with a spotted ass. Cathy and I were co-workers and conspirators. Every time we got an obnoxious American guy she would wink at me over his shoulder while he drawled on about his riding days in Texas or Montana, and I would saddle up Steamboat for him, a giant jet black stallion who was famous both for his frightening bursts of uncontrollable galloping and for trying to rub his rider off by scraping his sides up against the spindly lodge-pole pines the trails were lined with.
I always rode behind my aunt; Little Chief was trained to follow her horse. Hers would plod along at tourist speed in front of me, and I would try to make my legs copy the way hers moved, the seamless satin groove her hips fell into with every swing of the horse's step.
Sometimes we would ride alone and she would whistle and kick the insides of her boots in and race ahead of me. Little Chief would pick up the pace a bit like the foot soldier he was, and my heart would begin to pound. Cathy would ride for a while and then whirl her horse around, her long braid swinging, then hanging down her front as she rode back towards me.
One day we were lazily loping alongside the little road that led back to the hot springs when one of Cathy's admirers came up from behind in a pick-up. He honked hello as he and his road-dust drove by, which spooked Little Chief and he bucked me off.
On impact, tears and snot and all the air in my lungs were expelled. I lay on the hard-packed dirt and dry grass for a minute, bawling when I could catch little pieces of my breath.
"Get on," Cathy said, hard-lipped as she rode up beside me. "Get back up on that horse right now. Do it now or you'll be too afraid later."
She was tough like that.
* * *
One Christmas Eve shortly after she and John had finished drywalling, our family all had turkey dinner out at their place. We were each allowed to open one present, and my mom had suggested I bring the one shaped just like a brand new toboggan from under our tree at home.
The coolest thing about Cathy was how she would gear up in a snowsuit in thirty below in the blue-black sky of a Yukon night (which begins at about two in the afternoon around solstice) and go play outside with the rest of the kids. Not in a grown-up, sit-on-the-porch-and-smoke-cigarettes-and-watch kind of way, but in a dirty-kneed, get-roadrash-kid kind of a way. Right after I ripped the last of the wrapping paper off my gleaming red sled, she was searching through the sea of snowboots by the door for her black Sorels and pulling her jacket off the hook behind the door.
"Let's go up the hill behind the house and give it a try. Not much of a trail in winter, but we'll make one."
I suited up right behind her, followed by my sister and a stream of cousins with mittens on strings.
It was so cold outside that the air burned arrows into the backs of our throats and frost collected on our eyelashes above where our scarves ended, which would melt if you closed your eyes for too long and freeze on your cheeks.
We packed as many of us onto the sled as we could so everyone could have a go. We rode and climbed, rode and climbed until our toes began to burn. "Once more, everybody goes one more time, then we should go in," Cathy breathed through her scarf. She pushed her butt to the very back of the sled, and motioned for my little sister to shove in front of her, between her legs. I jumped in the front, and my little sister's snowsuit whistled up against mine as she wrapped her legs around me. The toboggan's most alluring feature was the two metal emergency brakes on either side, with the black plastic handles molded to fit the shape of your hand. I tried to grab both and steer, but my sister also wanted to hold onto one, and started to whine. "You can each have one," Cathy ordered. "You steer one way and you can steer on the other side, okay. you guys. Don't fight about it, everyone else is waiting for their turn. Let's go."
About halfway down we flew off a bump. My little sister hauled on her brake and we screeched off the path and smashed into a tree. Cathy's leg hit first and I heard a snap.
By the time we rolled her onto the sled and pulled her back to the house, her face was glowing blue white and her teeth were chattering. John came running out with a flashlight. He gingerly pulled up the leg of her snow pant, and dropped it again, his face changing from Christmas rum red to moonlight white.
"Jesus Christ. Get the kids in the house and pull the truck around. And get her a blanket, she's going to emergency. Pat, are you okay to drive?"
A weird silence took over the house after they left. I fell asleep in my clothes on the spare bed and barely woke up when my dad carried me out to the truck hours later.
"Did Cathy live?" I whispered into his ear. He smelled like scotch and shampoo and his new sweater.
"She's fine. She's asleep. She's got a cast and a bottle of painkillers. You can call her in the morning."
I told my friend Valerie all about it the next day, bragging like it was me. "I tried to save us, but my sister is too little to steer. Cathy's bone was poking right out of her leg, and she never cried once. There was even blood. Now she has a little rubber thing on the bottom of her cast so she can walk a bit, plus she has crutches."
"My dad cut the tip of his finger off with a saw once. They sewed it back on," she reminded me.
"Yeah, but you weren't even born yet. Besides, a leg is way bigger than a finger. Hurts more."
* * *
Later that winter the wolves got hungry because the government sold too many moose-hunting licenses, and dogs and cats started to disappear. Cathy phoned me one Sunday morning and told me that they had found what was left of Little Chief at the bottom of the mountain the day before.
"I didn't tell you yesterday because your mom told me you had a hockey game. I know you're sad, but horses and wolves are animals, and they follow different rules than we do. He had a good horse life, and now the wolves will make it until spring. You were too big to ride him any more anyway, and your little heart will get better in time. Wolves are wolves and men are just people."
She was tough like that.
When she left John, she was tough too. She took her horse and one duffle bag, and most of his savings to cover her half of the house. She never even cried. Or that's how I saw her leaving in my mind. Dry-eyed in her pick-up, with the radio on and a cloud of dirt-road dust from the Yukon straight back to Alberta.
Bet she never looked back, I thought.
My uncle started to date one of the other cooks from the lodge. She had a university degree so everyone called her the professor.
My little sister grew up and moved to Calgary. She, like myself, inherited our family loyalty, and looked Cathy up.
My mom and I went to visit my sister at Christmas, and she told me Cathy would love it if I could make it out to visit her in Bragg Creek. But we got snowed in, so I called her the day before we left.
Cathy's voice sounded the same as I remembered, except more tired. "I can't believe you're almost thirty years old. I remember you as just a little girl with that white hair and filthy hands, little chicken legs. You had such big eyes. My God, you were cute. Just let me grab my smokes."
I could hear dogs barking and a man cursing at them to shut the hell up. A television droned in the background.
She sounded out of breath when she got back on the phone. "Here I am. Hard dragging myself around since my accident. Did Carrie tell you about my legs?"
She had broken both of her femurs straight through a couple of years ago, and had pins in her knees. She still had to walk on two canes and couldn't work any more.
"I lost the trailer," she explained. "Couldn't get worker's comp because it happened on a weekend, and the unemployment ran out a year ago. Had to move in with Edward and lie about being common-law even to get welfare."
"What are you saying about me?" I heard the man's voice again in the background. "Who you talking to anyway?"
"My niece. Turn down the TV for chrissakes."
"You don't have any nieces. You don't even have any brothers or sisters."
I presumed this was Edward. He obviously didn't understand family loyalty the way we did. Blood and marriage were only part of it.
"'Member that time you broke your ankle on the sled? You didn't even cry. You were my hero back then, you know?"
"Well, I cried when I broke my legs this time. I'm still crying. Some fucking hero I am now, huh?"
I heard the empty in her voice and didn't know what to say. So I told her a story.
"Your old shed is still out behind John's, you know. Nobody ever goes in there. A couple of years ago John said I should go out and see if your leather tools were still out there. Might as well, since maybe I would use them, and so I did. It was like a time machine in there. Everything was still hanging where you left it.
"I took down your old bullwhip. It didn't really want to uncurl, but I played with it a little and it warmed up a bit. I took it outside into the corral and screwed around with it. On about the hundredth try or so, I got it to crack. I got so excited by how it jumps in your hand when you get the roll of the arm right that I hauled off and really let one rip. The end of the whip came whistling past my head, and just the tip of it clipped the back of my ear on the way by, and it dropped me right into the dirt. I was afraid to peel my hand from the side of my head to see if there was still an ear there. Hurt like fuck.
"But I thought of you and made myself try a bunch more times until I got it to crack again, you know, so I wouldn't be too afraid next time. Like you would have done."
She was quiet for a while on the other end of the line. "You still have some imagination, kid. Always did. You gotta come visit me sometime. I'd love to see what you look like all grown up. I don't get into Calgary much any more, only when Edward feels like driving, which is never. You'd have to come out here. Carrie could give you directions."
I've been back to Southern Alberta twice since then, but never made it to Bragg Creek to see Cathy Bulahouski, the Polish cowgirl from Calgary. She can't ride any more, she told me, and I couldn't bear to ask her if she had cut off her hair.
# Two: Family I Have
# Objects in the Mirror
LAST MONTH I SPENT TEN DAYS AT HOME IN THE YUKON, doing research for a new project. I went through as many family photos as I could lay my hands on: sorting through the magic red bag of memorabilia my Aunt Roberta keeps in her basement and sifting through the gigantic mishmash of memories crammed into a box in my mother's guestroom closet. My Grandma Pat won the organization award; hers were some of the only photos actually placed in albums, and each album had a glossary of subjects and decades listed on the inside cover in her bold, confident script. I found a citation for drunk driving from the seventies for one of my uncles, not totally out of character for him, but it was issued at ten o'clock in the morning, which was impressive. I unfolded a stiff and stern letter written by the principal of my father's high school, which would later be my high school, explaining to his parents just why he was going to have to repeat grade ten. It wasn't for lack of intelligence, he made sure to point out. I found a lot of pictures of me as a kid. Way more than I remember anyone taking at the time.
There is the one of me with my dad and my Uncle Rob, who are on either end of a broomstick loaded down with lake trout; I am crouching underneath the fish between the two men, blood spattered up to my elbows, proudly holding up a string of grayling. Me in a campground somewhere up north, exploding out of the willows, carrying a giant log of firewood on my back. Me on the first day of grade one, in a line-up with all the other little girls on the block; all the neighbour girls and my little sister are in sparkly new dresses, their chubby knees scrubbed and squishing out of the tops of sparkling white knee socks. I, on the other hand, am wearing blue corduroys, black rubber boots with red-brown toes, and my Davy Crockett fringed buckskin jacket. Me, in my grade two class photo, front-toothless in a plaid shirt, pearly snaps done right up to my chin, sporting an Andy Gibbish shag do. Me smiling in full hockey gear, lined up with all of my teammates, the only girl in the boy's league.
None of this was surprising to me; I appear to be the same kid I remember being. What I couldn't believe, in retrospect, is that anyone in my family could have actually been surprised when I came out of the closet at eighteen. The evidence was everywhere, right from the start; how could anyone have missed it?
I decided to investigate.
I called up my Aunt Roberta first, because it was almost eight o'clock in the evening, and she goes to bed early. I asked her if she ever suspected that I was gay when I was little, if she ever wondered about the hockey and the buckskin jackets?
I heard the kitchen chair complain about being dragged across the linoleum, and she sat down.
"I know this sounds silly, but I always thought you were just who you were. An amazing little strong personality. Thought you got it from your dad."
I asked her if Gran had ever said anything to her about me and the gay.
"Gran's gone to bed already, but I do remember her saying to me that you were exactly right. All you kids turned out to be exactly who God meant you to be. I mean, you can call her in the morning if you want to, but I know that's what she'll say."
My grandma Pat was good for an awesome quote, as usual.
"I never labeled you as anything. You were just boyish, and you did boyish things. Keep in mind that we didn't think like that back then, you see. Any knowledge of homosexuality I might have had would have gone back to Victorian times. All those novels. You probably skirted under my radar, because you weren't wearing hoop skirts and high button boots."
My mom swore she had no clue whatsoever. "My mind never went there. I just let you be what you wanted to be. Not very helpful, I guess. I'm sorry."
My Aunt Cathy echoed my mom. "I just thought you were a little brat because you refused to wear a dress to our wedding."
My Aunt Norah thought my sister and I were simply polar opposites, that was all. "Carrie was the prissy little girl, and you... weren't. You were just your own little people. When you were in your teens I remember thinking... knowing somehow that you weren't happy, that you seemed tense inside your own skin. I knew there was something going on with you, but I didn't know what it was. We didn't have to have a label for everything back then."
My Uncle John was cooking an omelet in the background when I talked to him. "Sorry, kiddo, but I can't identify the moment we realized you had gone to the dark side. We were just glad you weren't stupid. There's no cure for stupid. There was that one time, you were only six or so, when you gave me supreme shit for not attending to my fishing rod, but I don't think that had much to do with your sexuality."
My Uncle Rob was pensive, thinking over his response a bit before speaking. "Well... you can see why we wouldn't have thought much about it. There's lots of hetero butch chicks out there, to be honest. Especially up here."
"On the other hand," he continued, "maybe a guy should have twigged due to your aversion to wearing a dress, but who cares, anyway? I've always said, it's your soap and your dick, and you can wash it as fast as you want."
So it appears that for all those years, in all those photographs of that little tomboy, there was only one member of my family wondering about me.
And that was me.
# The Curse?
I CALLED MY COUSIN UP THE OTHER DAY, and partway through our usual gab he informed me that Layla, his stepdaughter, had some very exciting news.
"Can I tell Ivan, or do you want to tell her yourself?" he asked her from his end of the living room couch.
I heard her almost teenage voice in the background, saying it was fine; he could go ahead and tell me.
"Layla got her period this morning." He sounded proud, like she had won the science fair, or got straight As, something along those lines.
I was unsure how I should respond, but they both sounded happy on their end of the phone, so I asked to speak to Layla directly.
"Congratulations," I told her. "It sounds like some sort of a celebration is in order. You have anything in mind?"
I couldn't help but think back to my big day. My mom was out of town at the time, and when I called my Dad upstairs to ask him what I should do, he panicked on the other side of the bathroom door, emptied out my mom's drawer in the other bathroom, whacked his toe on a doorjamb, swore profusely, and then tossed me a box of anal suppositories, mistaking them, I believe, for some sort of feminine hygiene product. Neither of us were proud of me, and we never mentioned the subject again.
"Can you take me to see _The Corpse Bride_?" Layla asked me. "And can we have popcorn?"
"That is an excellent plan," I told her. "I'll pick you up at 6:30. It's a date."
On my way over, I pondered whether or not I should discuss the merits of menstruation with my young friend. It was obvious that this was a brave new world, and that Layla was being brought up to believe that her period was not a dirty female secret like it was when I was twelve years old, and this was a good thing. But I wondered if it would be strange for her to chat about it all with her butch relatives, or if my silence on the matter would be noted.
"So, you got your period, huh?" I asked her as she did up her seatbelt.
She nodded casually.
"Cool," I said, feeling like a gigantic dork. "Way to go."
And that was the end of that.
A couple of days later, I got mine. I plodded through the slush on the sidewalks to the corner store for a box of tampons. I went up both aisles twice, and couldn't find them.
Finally the guy behind the counter asked me if I needed any help finding something.
Normally, I would just shake my head and grab a can of soup, so I didn't have to say the word tampon to the guy behind the counter at the corner store, but this was a brave new world. I needed to get with the times, and cast off my shame and embarrassment, for the sake of young girls everywhere.
"Uh, yeah, I'm looking for tampons," I said.
"For what?" There were two other guys waiting to buy their cigarettes, and they both looked at me.
"Tampons."
He shook his head again, and cupped one hand around his ear, signaling that he couldn't hear me, I needed to speak up. I considered my options. I could scream out in a crowded corner store that I needed a box of tampons, or I could run for the door.
I chose the door.
The next corner store had an ample supply, and I let out a huge breath I hadn't realized I had been holding in. I took the box up to the counter along with a couple of other items I didn't really need, for cover.
I don't know why I am uncomfortable saying the word tampon out loud, or acknowledging the fact that I, like almost all estrogen-based organisms my age, get my period. Maybe it is residual Catholicism; maybe it is because most corner store guys think I am a young man on a supply run for his girlfriend or mother. Or maybe I just don't like to talk tampons with strangers.
"What brand is the best?" The guy behind this counter held up my tampons for the entire world to take notice of.
"I beg your pardon?" I was hoping I hadn't heard him properly, that this was not happening to me.
"There are so many brands to choose from, and different sizes, too. I never know what I should order, so I ask my lady customers, which one is the best?"
There was another guy behind me in line now, holding a box of Kraft dinner and a loaf of white bread. He raised an eyebrow.
I felt a sudden rivulet of sweat in my armpits. Running for yet another door at this juncture would send the message that tampons are, indeed, a shameful topic. This thoughtful merchant had come to me for help in serving the needs of women throughout the entire neighbourhood, and it would behoove me to behave accordingly.
I took a deep breath and spoke in a calm, confident tone. "Well, I would say that it is definitely a matter of personal choice, similar to choosing the right condom for the job. A variety of sizes would obviously be a good thing, as there are many sizes of... vaginas out there."
He nodded and leaned forward, interested.
"And as for brand, I always prefer the ones without an applicator for, you know, environmental reasons, but again, I can only speak for my own... I can only speak for myself. I guess as wide a variety as you can carry would be my answer."
He thanked me and rang in my purchases. "Will you be needing a bag today?"
I nodded, and stuffed my tampons in, out of sight for the walk home. "You gonna watch the hockey game tonight?"
He shook his head. "I don't follow the hockey. Myself, I like cricket."
I shrugged. The guy behind me shook his head and stepped up to the counter as I headed for the door.
"Cricket, hey?" He was still shaking his head. "Well, each to their own."
# By Any Other Name
I LEARNED MOST OF WHAT I KNOW ABOUT BEING A MAN from my Uncle Rob. Uncle Rob has never let the fact that I was declared female at birth get in the way of our male bonding, and I've always loved him best for it.
Uncle Rob taught me how to fish, drive a standard, light a match off of my front tooth, and open a beer with a Bic lighter. He taught me how to make a fist, turn into a skid, light a fire, and shoot a gun. He passed on to me everything he has ever managed to learn about women, and all the Zippo tricks he has ever been shown. He taught me how to tell a story, and how to hold my liquor. All the important stuff. Some of the family reckon I look more like my Uncle Rob than I do my own father, and everyone agrees I look just like my dad.
Uncle Rob and Aunt Cathy flew to Vancouver last week, because Rob had an appointment with a fancy eye doctor. Whitehorse General Hospital is equipped to handle your basic medical tests and common ailments, but anything involving a specialist or an expensive machine requires a trip to the big city. Rob called me from the hotel and told me to round up the stray cousins and bring the girlfriend; he was taking us all out for dinner. Cousin Darryl's brand new baby had somehow turned into a seven-year-old girl, and I hadn't seen my cousin Garth since Grandma Pat came to town for her knee replacement three years ago.
I rarely bring a date along to family functions, because more than two or three of us in one room can be hazardous, especially if you are shy, offend easily, are clean and sober, or don't eat meat. The way my family demonstrates our love and affection for each other has occasionally been mistaken for verbal abuse by outsiders, so I usually don't take the risk.
But I knew she could hold her own; she is smart and strong and can take a joke. She loves fishing and hates hippies. There was common ground, and she might just fit right in. Besides, I figured, how could she love me and not like my Uncle Rob? He was the man who taught me everything I knew, and I look just like him.
The appetizers arrived in the middle of a raucous debate about flatulence and love: was unabashed farting in front of the fairer sex an expression of intimacy, or the sign of the death of romance? Was pulling the covers over her head actually a form of foreplay? Was our whole family actually lactose intolerant, or did we just not chew our food enough?
My sweetheart was unfazed, and retained her appetite. Maybe she really was the perfect girl for me.
By the time our entrees arrived, the talk had turned to embarrassing stories from when I was a kid, how I had panic attacks when forced into a dress for weddings, and how I finally gave in and wore a satin gown with dyed-to-match pumps to my high school graduation, just like the normal girls did.
"She looked so pretty," said Aunt Cathy solemnly, like she was giving my eulogy.
"I looked like a drag queen."
Darryl shook his head. "I can't imagine cousin Ivan in a dress."
"I can't imagine calling her Ivan." Cathy stabbed a bit of broccoli with her fork. "She'll never be Ivan to me. That's just, like, your writing name, right? Nobody actually calls you Ivan in person, do they?"
Cathy asks me this, even though the entire table had been calling me Ivan all night. I stopped using my birth name over a decade ago, but Cathy likes to pretend she doesn't know this because it makes her uncomfortable. I love her enough to allow her this tiny corner of cozy denial, and my continued silence on the matter helps to hold up my half of her little charade.
I have lots of people who call me Ivan. I only have the one Aunt Cathy. She has never understood why I changed my name, or why I vote NDP. I've never understood why she collects Santa Claus dolls, or how she can smoke menthols. It doesn't mean we love each other any the less for it.
"I've always called Ivan Ivan," states cousin Darryl, God bless him. No wonder everyone thinks he's gay.
"Are we allowed to have dessert?" squeaks second cousin Rachael.
"Anybody want to try a prawn? Going, going, gone." Rob speaks around a mouthful of his dinner.
"Don't chew and talk at the same time, Robert. You'll set a bad example. There are children present." Cathy half-feigns disgust and backhands her husband in the upper arm, right where his shirtsleeve stopped and his tanline started. This signaled the official change of subject.
"Set a bad example for little Rachael?" Rob smirks, rubbing his arm where she had whacked him one. "It's already too late for Rachael, too late for all of them. I saw it on the Learning Channel. A child's personality is fully formed by the time they turn three. We might as well relax and let it all hang loose. The kid is already who she's gonna be, all we can do now is love her. It's out of our hands."
Rob leans across the table to pinch one of my fries. "Did Garth tell you him and Allison are getting hitched in Fiji? Cath and I are going. You and your lovely lady friend should come too. I'll rent us a boat and we can go fishing. The wedding is still over a year away, so start saving up. Maybe even Darryl will have a girlfriend by then, and we'll all go. A family that fishes together stays together, isn't that what they say? And you two girls would love Fiji. It's the perfect place for you, really: it's beautiful there, and the policemen wear skirts."
# To Whom It May Concern:
I DON'T WANT TO SOUND LIKE SOMEONE'S GRANDMOTHER or anything here, but really, would it be so hard to pick up a phone and call? You don't even have to call me, just call anyone, your brother, your dad, any of us, just to let us know that you are alive. We all talk, you see, hoping that one of us has seen you, or heard word, or even heard a rumour.
I'm not even the worrying kind, you know me, I get really busy too and forget to keep in touch and miss my cousin's birthday or whatever, just like everyone else, and I'm definitely not usually the type to get on anyone's case for stuff like this. It's just that the last time I saw you, you had lost about thirty-five pounds and the crystal meth was starting to turn your back teeth black, and the newspapers and the streets are full of stories about irreversible brain damage and psych wards brimming with lost souls stricken by this addiction, and, well, I worry. It's not like you're backpacking in Europe and just forgot to send a postcard. I don't care about broken promises or the money you owe anyone. I do care that your brother and your dad spent another Christmas wondering where you were, and that they are running out of reasons you haven't seen your niece and nephew. I can't help but care about that, but even that I would let slide.
Some guy asked me for change outside of the bank today. He looked skinny and drawn and nervous, just like you did the last time I ran into you on the Drive, and for some unexplainable reason I felt like punching him. Instead I took a deep breath and asked him when was the last time he called his mother?
The self-help books and the twelve-step doctrines would probably feed me some line right now about how no one can really help you until you are ready to help yourself and to not to allow myself to feel hurt that I haven't heard from you in almost a year, that it is your addiction governing your behaviour right now and not you. But I call bullshit on that. We have known each other since we were kids, I would and have done anything to help you, and I deserve better than this.
This not knowing. Remember when I dragged you off the street and let you sleep it off for days and fed you and helped you track down the bits and pieces of your life so you could start putting them back together? Back then you said you were done with it all, you were ready, you wanted to change your life, and you needed my help.
I told you that night on the back porch I would do whatever it took, anything in my power to see you through this time, but that I had one condition. My one condition wasn't even that you stay clean, because I know what a demon the meth is, and I didn't want you tossing me out with the clean and sober bathwater if you backslid. My one condition was that you didn't lie to me anymore, that if you used I wanted to hear about it from you. No more bullshit.
Maybe that is why you haven't called, maybe the truth was something you thought I wouldn't want to hear, or something you weren't prepared to say out loud.
I asked after you at your favourite old coffee shop the other day. The owner's grandson, the cute one, he surprised me by saying yes, he had seen you, and that you were looking great, that you had cleaned up and were living in the suburbs somewhere, and working construction.
I let out the long breath with your name on it that I had been holding for almost a year, and went straight home to call your brother. I was so glad to have word that you were alive and well that it took me a couple of days to get around to wondering why you hadn't gotten in touch with anyone.
The guy who first said 'No news is good news' obviously never had a best friend fighting the ice.
And the guy who coined the phrase "fair-weather friend" never met either of us. I once told you I knew that if ever I found myself in your shoes, I had every faith you would be there for me, and you hugged me in place of a yes.
I think of you whenever I swim in a lake, whenever I pass a rusty pick-up truck on the highway, whenever I see the northern lights or a blue-eyed dog. I miss you whenever I hit my thumb with a hammer, ride my bike, or walk past a lawn that needs mowing.
I'm not writing this to judge you, or to make you feel guilty. I'm writing this to let you know that whenever you are ready, I will be here. I refuse to give up on you. The fire that burned my house down spared the garage, so I still have most of the tools you stored at my place. A couple of times I had to laugh out loud at the same time as I was cursing your name, as I've moved around a lot since my house burned down, and I must really love you, because I can't think of anyone else I would move an entire set of free weights five times for, myself included.
I will pick up that phone whether you are still using or not, and I will listen to you whether your news is rosy or rainy. I want you to know that I meant what I said on the back porch that night, no matter what. No bullshit. A lot of things have changed for both of us since then, but not my home phone number.
Oh yeah, and my grandmother says to say hello.
# Single Malt
MY DAD USED TO BE EASY TO SHOP FOR. Every Christmas and birthday, for as long as I can remember, I have got him a bottle of single malt scotch. What brand I chose changed yearly; it depended largely on my economic status come shopping time, but that was all okay by him. I knew in my leaner years that he could just pawn the cheaper stuff off on his visitors. I've caught him guiltlessly pouring an Oban for himself, back turned, while simultaneously serving his houseguests Johnnie Walker Red. I once caught him trying to pull this stunt on me, red-faced with a bottle of Canadian Club in his hand, as if he thought I wouldn't know the difference.
I reminded him I had been trained by a professional.
A couple of Christmases ago, my Uncle Rob trapped me coming out of the washroom to have words with me about this. "Why buy booze for a guy who drinks too much?" he asked me with rum and eggnog on his breath. "Why not get him socks or something, like everybody else does now?"
Rob had a point, to be sure, and it wasn't like I hadn't thought about it all. But my dad already owns every tool known to mankind, never wears ties, hates sports of any stripe, and only wears work shirts. The contents of his closet reveal a repeating pattern: GWG boot cut jeans, thirty three-inch waist, thirty three-inch leg. White Stanfield T-shirts, size medium. Blue BVDs, also medium. Tan work boots, size nine men's. Grey and white work socks, the kind with the red stripe. He reckons if you own all the same socks, you don't have to throw both away when you get a hole in one. Easier to sort that way, too. A couple of summers ago he got himself a pair of sandals, and the whole family almost fell over in collective shock. Buying him clothes as a gift would be like going out for supplies.
My dad throws stuff away when he knows he won't use it, even gifts. After a couple of Boxing Day heartbreaks when taking out his garbage, I settled myself into buying him something I was sure he would love, something I knew would never go to waste. Scotch it was.
Last spring my dad called me out of the blue, which should have been my first clue that something big was up. The second alarm bell went off when he asked me how my girlfriend was. Sure, he didn't know her name, but that was as much my fault as it was his: I had stopped telling him years ago. But still, he asked.
Then the bottom fell out of all things predictable. My father interrupted me, stopped himself, and went on to say the following: "I'm sorry, I interrupted you. What was that you were saying?"
I immediately called my grandmother to find out if he was okay. "Is my father dying of cancer or something? He's acting very weird. First of all, he called me up just to talk. Then, he apologized for interrupting me. Is everything alright?"
"Of course," she said, laughing her little laugh, letting air out through her nose like she does. "The new Don takes a little getting used to. Yesterday, he called to let me know he was going to be late for lunch. Very unusual, indeed. Usually he just wouldn't show up and then avoid me until he thought I'd forgotten. But everything is different since he quit drinking."
My mind reeled through the rest of our conversation. The details rolled around my head and only stuck later, because I had yet to fathom the first line. I heard her say he just got up and poured it all down the sink, that it had been over a month now, but I was still stuck on imagining my father without a drink in his hand and wondering what his voice might sound like, thin-tongued and without a tinkling soundscape of ice cubes behind it?
When I visited in July, he was clear-eyed and full of wisdom; simple, yet sublime. "I realized," he said to me, swirling black tea with one sugar and tinned milk, "that it looks like I'm probably going to have the same wife, the same job, the same house, and work on this same little piece of ground for the rest of my life." He paused to light a smoke with steady hands. "And that the only thing I could change was my attitude."
We drove around town, going to pick up sheets of aluminium, welding rods, one-inch square tubing, and two-inch fine thread bolts, talking the whole time. It was just like when I was a kid, the only difference being that my dad now would allow the proud-biceped kid who worked in the warehouse load the really heavy stuff into the back of his pick-up. "Better his spine than mine," he whispered to me out of the corner of his mouth by way of explanation. His hair was a streak of silver, so startling, I could never lose him in the aisles. He seemed shorter, somehow, than I remembered him.
He lit a smoke when we got back into the truck, letting it dangle from one corner of his bottom lip as he backed the truck up with one hand, his right arm draped over the back of the seat between us. He has always been able to do almost anything with a cigarette in his mouth like this. Somehow, the smoke never gets in his eyes. "You know what?" he asks, eyes on the road. "I used to drive downtown, just like this... before."
"Before" is the term my dad uses. He will not say "when I was drinking." He does not use the word alcoholic. Everything is just before, or after. You have to just let him talk. He won't answer direct questions. I don't push him, I'm just so glad it's after now.
He continued, because I said nothing. "I used to drive around, and everything seemed like it was broken, or abandoned, or it needed a paint job. Nobody smiled. I really felt like it was hell right here on earth, some days." He cleared his throat. "But now, I come down here, like today with you, and all I see is new construction, heavy equipment, girls in tight shirts... and a lot of chrome." He looked at me out of the corner of his eye, to register my comprehension. "You know, all the things I like."
After I got back, I was afraid to phone him all summer, afraid to hear that his voice had slowed and dropped, afraid he would let the phone ring and ring and I would know. Scared that things had changed, or worse, that they had gone back to being how I thought they always would be.
I called him last week; I had to finally, to see what he wanted for Christmas. He sounded fine, but the stone in my stomach didn't dissolve until I heard him laugh. He didn't laugh that much, before.
"Don't get me anything. I'm good. Don't you worry about me. Or, how 'bout you get me something we both know I'll like. How 'bout you get me a carton of smokes?"
# Thicker than Water
EVERYBODY ALWAYS SAYS I LOOK JUST LIKE HIM. Every once in a while, my grandmother hauls out the second oldest photo album from her closet and opens it on the kitchen table, next to the cut crystal bowl of sugar cubes and the matching cup that holds the little silver teaspoons. She slides the teapot aside to make room and squints over her bifocals. If I have brought a friend with me, this is the part where she makes them try to pick out which face in the faded black-and-white photos belongs to my father. My dad has three brothers. They are wearing matching plaid shirts, or bathing suits, or cub scout uniforms, or hand-me-down pajamas and holsters for their cap guns. In the background there is a Christmas tree, or a lopsided front porch, or a wall tent, or a brass statue of a war hero from the summer the old man took them to Winnipeg to see the army base and learn some respect for the soldiers who fought and died so the rest of us could sit around in our underwear and read comic books and not eat the peas or the broccoli he worked all day to pay good money for. It is always easy to find my dad's face in the photographs. I look just like him, but without the ears. My grandmother named him Don, after his father, she tells my friend. This is the part where if it is raining or her knees are bad she will confess that she never really loved the old bastard, that he was never half the man his sons turned out to be.
More and more, I find little bits of my father in me. Not just around the eyes or in the shape of my jaw, but how I can't stand to have less than half a tank of gas in my car, because you never know. How I hate cheap tools and dull knives and loose screws. How I own twenty pairs of the exact same underwear. How I can't stop looking for something until I find it, even when I'm late, even if I don't need it until the day after tomorrow. I have to know where it is. My smokes are always in my left pocket, lighter in the right. I can't sleep if the dishes aren't done, can't read only half of a book, and I never turn off the radio until the song is over. I like a little bit of egg, potato, and bacon in every bite of my breakfast. It is a finely tuned ratio, constantly being weighed and adjusted throughout the meal. Nothing worse than winding up with only hashbrowns in the end. Always let your engine warm up before you drive anywhere and cool down a bit before you turn it off. You can double the life of a motor if you treat it right. Driving fast burns more gas and is hard on your brake pads. Besides, you just spend more time waiting for the light to turn green. Don't go grocery shopping on an empty stomach. All of these things I learned from my father. Most of the time I do them without thinking of him, but every once in a while I remember; these are inherited habits. Other fathers might have saved their bacon until last, or ran out of gas, or hired someone else to build their house. Other fathers might have worn dress shoes to work instead of steel-toed boots. A different kind of dad might not have taught me how to weld. A man with sons might not have let his daughter drive the forklift.
Who would I be if he had been someone else?
A couple of months ago, I had a gig in Calgary. An all-queer spoken word show at a sports bar downtown, right in the middle of the hockey playoffs. Strange, but true. I was wearing a dark blue shirt with thin stripes, and a sky blue tie that subtly highlighted the secondary tones of my shirt. The waitress liked my stories and kept slipping me free scotch on the rocks after the show, and I had about four stiff drinks in me when this huge guy in a Flames jersey grabbed me by the necktie and pulled my nose right into his chest hair.
"Your tie is all messed up. Where'd you learn that? Nobody ever taught you how to do a proper double Windsor? You're a disgrace. Come here, lemme show you."
I tried to explain that I had been drinking, and was thus unable to engage in activities that required concentration or hand-eye co-ordination, plus it was dark and my tie was fine anyway, but he pulled my substandard knot loose and laid a drunken death grip on my right shoulder.
"I'm in the Mafia. The Mafia knows how to tie a tie. You going to argue how to tie a tie with the Mafia, or you going to shut up and watch me do this right?"
I mentioned that I had read somewhere that the real Mafia never admits that there is a real Mafia, and that Calgary wasn't known for being a hotbed of organized crime, and that the odds were neither of us would remember any of this in the morning anyway, but he insisted.
I ended up getting a nonconsensual thirty-minute lesson in proper manly attire from a guy with one leg of his track pants accidentally tucked into his white sweat sock. He started with the double Windsor knot demonstration and went on to sum up the billfold versus money clip conundrum for me. He was pontificating on the merits of French cuffs when his buddy interrupted to announce they were all leaving to go catch the peelers.
I woke early the next morning, dry-mouthed and blurry. I pulled a clean shirt and a different tie out of my suitcase and was amazed when my fingers remembered what tying a perfect double Windsor knot felt like. I don't remember who taught me the wrong way to tie a tie, but I know for sure it wasn't my dad. He never wears neckties. He taught me how to tie a boat to a dock, and a fishhook to a line. How to tie double bows in your bootlaces so they never come undone halfway down a ladder or get caught up in a conveyor belt or a lawnmower blade and end up costing you a toe. My father is a wise man. He taught me all the important knots. The double Windsor I learned from a wise guy.
# Maiden Heart
I PUT THIS STORY TOGETHER OVER THE last ten or twelve years, and it is still full of holes. It is a true story in the same way that an old vase that is broken into pieces in the sink and glued back together holds water. Maybe, maybe not. But it is no less beautiful to look at. This is what I think I know.
On October 31, 1997, my father turned fifty years old. He and my mother had split up about two years earlier, and as far as any of us could tell, he was attempting to drink himself into an early grave.
As always, he never let all that booze get in the way of a solid day's work, so he was half in the bag and all the way inside a tanker he was welding on when the phone rang in the little office at the back of his shop. He hung his torch up on the ladder and climbed out to get the phone. It had been ringing all day. Big family and a big birthday. Big pain in the ass.
"Happy Birthday, Don." The woman's voice was husky, with a bit of a smoker's rattle.
"Um, thanks." He didn't know who she was, but he felt as though he should. Something in her voice told him he should recognize her. Not the sound of her voice, more like the way she weighted down her words, like they meant something.
They chatted a short while; how was he doing, did he feel old, that kind of thing. She cleared her throat, paused for a second.
"You don't even know who this is, do you?"
"Keep talking," he insisted, sitting up a little in his greasy rolling chair. "I'll figure it out any minute."
"I should hang up on you right now."
"No no no, don't. Don't hang up. I will never sleep again from wondering. Just give me a hint."
"You forgot our promise."
My father took a sharp breath, dropped the pencil he had been fiddling with onto his desk.
"Patsy?"
She didn't speak, but wasn't silent on the other end of the phone line; a small, animal-like noise escaped her throat by accident, and thirty years hung in the space between them for a long second.
He repeated her name, more sure this time. "Patsy Joseph?"
She nodded, but he couldn't hear her nod, so she swallowed and spoke. "Uh-huh. It's me. And you forgot our promise."
My dad tells me this story in his boat, in 2003. It is August. We are in the middle of Marsh Lake, trolling one of his sweet spots for lake trout.
Patsy Joseph was his very first real girlfriend, he tells me, and she was two years older than him. They had promised each other when she was seventeen and he was fifteen that they would call each other on their fiftieth birthdays, no matter where they were. He had forgotten hers, almost two years earlier. She hadn't. Hadn't forgotten him at all.
They started talking on the phone quite a bit, and soon it was every day. She had left Whitehorse when her father moved to Hope, outside of Vancouver. The two childhood sweethearts never wrote or talked on the phone, he was mostly working in the bush back then, and they lost track of each other. When Patsy came back in the summer of 1969 to look for him, she heard from one of her girlfriends that he had gotten one of the Daws girls pregnant, and that he was married, was building a house up in Porter Creek somewhere. Catholic girl, what else could he do but the right thing?
Patsy was devastated, and left town with a truck driver who told her she had pretty eyes. Ended up in Dawson Creek. Good a place as any. Got a job in an auto parts place, on account of how my dad always made her help him fix up his '53 Mercury Comet convertible and so she knew a little about cars. More than most women did back then anyway. So, she still worked at the auto parts place, yeah, thirty years later, and she still lived with the truck driver, only he didn't drive truck anymore, he was on disability because of his back and maybe he couldn't drive the long hauls like he used to, but he could still beat on her so they weren't really together, these last few years, she lived in the upstairs suite of their house and he lived on the ground floor, and she wanted to sell that house and be rid of him for good, but they couldn't, not with this market, and so there she was. She told my dad she still loved him, always had, that she still had his old letters and birthday cards and some photographs. Kept them hidden from the truck driver all these years. Jealous and mean, you know the type.
My dad and I share a weakness for a lady who needs help. It feeds something big and empty in us to arrive on the scene with a truck or jumper cables or a generator or wide open kind of dumb heart; we like to think it sort of makes up for always saying the wrong thing just when the song ends and the room goes quiet. My dad told his childhood sweetheart that he had not seen in over thirty years not to say a word to her ex-trucker, just to pack up her car when the guy was asleep, take only the stuff she really needed, and drive to work like it was any other day, and he would meet her there. He would take care of the rest. He would take care of everything. He would take care of her. And did she have snow tires?
The next bit of this story I heard much later, not in my dad's boat, but in his 1981 Ford F-150 pickup truck, driving in a full-on blizzard on our way back from spending the night in the little house in Atlin that he was building for when he and Pat finally retired. They had been married for about ten years. The windshield wipers thump-thumped in the quiet but merciless storm; snow devils swirled on the black ice in front of the two stab marks our headlights made into all that oblivion. Nobody but us crazy enough to be out on that road in this weather. Used to be when I was a kid I was never scared when my dad was driving, no matter how big the waves or black the ice. Now I am older. I light his smokes for him so he doesn't have to take his hands off the wheel for long. No streetlights here, just dark and snow and cold all around us, not even a light on in a cabin, not out here, not until we hit the main highway. His face is lit up only a little from the dashboard lights, and the cherry on his cigarette dangles in the dark when he talks.
He tells me how he drove almost all the way through the night, when he went south to go get her, and walked into the auto parts place in Dawson Creek in the early afternoon. She was behind the counter wearing an angora sweater, kind of light blue he thinks it was, and he tried not to let his face show it, but he couldn't believe how old she looked. She said it was time for her coffee break; did he want to come up to the lunchroom with her then? She wouldn't meet his eyes with hers, wouldn't look at him right on at all, kept hiding her face behind her bangs, which were still blonde, but shot all through with silver now. She told him much later she couldn't rest her eyes right on his face at all that day, not even for a second, because she couldn't believe how old he looked. Couldn't look right at the years in his eyes and stamped all over his face. So she stared out the window into the parking lot of the auto parts place even though there was nothing much to look at out there and she had seen it all a million times anyways, but it was better than turning around and seeing your beautiful memory grown old and wrinkled and grey and with a bit of a gut now. And my dad, he has never been any good at knowing the right thing to say, so he tells a joke. And she smiles at the joke, because he's funny, he really is, the old man is, and she turns to look at him just a little and then she laughs.
Light me another smoke, he tells me, and so I do, trying not to get any smoke in me at all because I quit for over a year now, which makes him not trust me in a way that neither of us can put our finger on.
"Anyway," he says, "that's when I saw her. The girl I fell in love with when I was a kid. She laughed, and all those years just fell away somehow, and suddenly it was just her and I standing there, together. So she quit her job and followed me home back to Whitehorse in her Lincoln Continental and I married her up at Nolan's reindeer farm with only an eight-fingered farm hand as a witness. Filthy old Danny Nolan is a justice of the peace, can you believe that? Didn't invite any of the family. Not even my brothers. John still hasn't forgiven me."
My father's eyes are shining with tears he will not allow to slip out and down his cheeks. He opens his window a crack and the wind sucks his cigarette butt out of his square-nailed fingertips and disappears with it. It was true. I had seen it for myself, the previous summer. I had seen Pat's face crack when she laughed and reassemble for a split second into a much younger memory of itself. Almost pretty. I can't even remember what the joke was, but I remember that face, remember wondering where it had come from, and where she hid it the rest of the time.
We finally make it to Whitehorse; the last fifty clicks into town we just trailed behind the snowplow, the storm swirling behind us and filling in our tracks as soon as we were gone. Pat has the coffee on, and all six of their dogs explode in a fit of barking when we stomp through the front door and strip off our wet parkas and heavy boots. Pat is pissed off, she is not saying anything but you can see it in her face. We shouldn't have been driving in that weather, and we both know it, so we say nothing.
I sit down at the little kitchen table and pour a dollop of evaporated milk into my coffee, add a sugar cube from the Roger's box on the counter. My dad is stoking the fire. The television is on but turned down so you almost can't hear it. She won't let him smoke in the house anymore, and it smells like the cinnamon-scented candle burning on the coffee table. There are several pictures that hang on the wall next to the bathroom, above the washer and dryer, right next to her Elvis clock. Pictures of the dogs when they were puppies, stuff like that. There is one of her and my dad, one she kept secret from her ex-trucker somehow, all of those awful years. She has had it blown up and framed. Black and white, my dad and her, back in the mid-sixties, he with his hair slicked back and his smoke pack tucked into the rolled up sleeve of his white crew neck T-shirt, his jeans with wide cuffs and his lips curled in a smile around his cigarette. She has a kind of beehive hairdo, and his arm is around her waist. They are standing in front of an old wall tent, and the chrome on the grill of the Mercury Comet winks in the sunlight beside them, and the soft shape of the mountain next to the Fox Lake campground rolls in the far background. The photo looks like something out of an old ad from _Life_ magazine or something. This photo hangs right next to another one, this one in colour, a shot of the two of them again, her with her new perm and him with his silver shock of hair sticking up all over. He is wearing sandals and a clean work shirt with the sleeves pulled up over the welding scars on his forearms, and they are standing next to the Lincoln Continental, which is parked beside the motor home he traded a guy for some welding a couple of summers ago. In the background is the same mountain; they have returned to the very same campground site, it looks like, right there on the gravel beach of Fox Lake. But there was a forest fire there a couple of years ago, and so the trees left standing on the familiar shape of the mountain are crooked little blackened matchsticks, the fireweed curling up between them and taking over. My dad has a gut and his wife is squinting into the sunlight, her glasses catching a glint so you can't really see her eyes behind them. But none of this matters, really, because it is forty years later, and they are both smiling.
# All about Herman
MY GRANDMOTHER HAS KEPT A JOURNAL FOR MOST OF HER LIFE. All ninety years of it. She loves to write, she tells me on the phone from the Yukon. I can picture her, all the way from Vancouver, it is January, so she has the propane fireplace on in the living room and she is sitting with her legs tucked up beside her on the couch like she does. She is wearing a dress with a floral pattern and the rug needs a good vacuum, which she would do if she could still see the dirt, but she can't. There is the smell of drip coffee and bread dough set aside under a clean tea towel to rise. Newspapers and magazines cover the coffee table, and she has a fresh cup of black tea with cream and sugar in it on the side table, next to a plate empty save for a scattering of toast crumbs. She has lived in this little house on Elm Street in Whitehorse since 1967. It is the only house belonging to anyone in my giant family that has been there all of my life. Everyone else has sold and moved up, to make room for more kids, and later, less room for fewer kids. Only this house remains, as unchanged as playground concrete in all of our memories. I can't imagine my grandmother anywhere else but in this house, and I refuse to think about anyone else ever living here when she is gone.
"Did you get my envelope?" she asks me, as always speaking far too loud into the receiver, as though she doesn't quite trust in the technology. "I sent you a copy of all of my latest scribblings."
She has been going back through her old journals, editing them and typing them up. Her vignettes, as she calls them. She has been sending me envelopes, sometimes containing carefully folded, ten-page-long stories handwritten in her sloping but still solid script, sometimes typewritten in all capital letters, with capital Xs crossing out mistakes, and corrections made in blue pen in the margins. Most are untitled, with just that day's date in the upper right hand corner. I read and re-read them; they are full of old stories, confessions, and advice. Lately her musings have grown somehow more poignant, more emotional, full of regrets.
"What I bitterly regret are the things I didn't say, and the questions I didn't ask," she writes. "I have dreams now, and I dream of the past. I am not old. I'm not an old lady. I am young, vibrant, full of life. I'm like that in all my dreams. So I enjoy my dreams."
Her last letter was four pages long, typewritten. She has titled this one, called it "All about Herman." I don't remember Herman, as a child I knew of him only through escaped bits of stories whispered here and there, nobody talked about Herman much. He died Christmas Day in 1970; I never knew why. I knew next to nothing of the story of Herman until just last year, when my grandmother writes:
"It all started with the morning of March the 9th, 2008. It was his birthday on March the 9th, 1930. He has been dead now for thirty-eight years, and on this morning, I am thinking about him. I remember him, and a week or so later, I can't get him out of my mind."
Herman had been an engineer for the Department of Public Works, and my grandmother was a secretary. She was married; her husband and three of my uncles had recently left the Yukon and travelled ahead of her to New Zealand, where she was to join them in a year, when they had found work and set up a place for them all to live. My dad didn't want to leave the Yukon; he was already working, driving a caterpillar in the bush on a road building crew. My grandmother was to stay behind and save all the money from her government job. At least that was the plan. But that is only sort of what happened.
It turned out Patricia liked being alone. This was unforeseen.
And then the big, rugged engineer began to court her. At first she turned him down. Finally, she agreed to go for a drink with him one night. They began an illicit affair. He took her on trips. He liked classical music. He was well-read. He was in love. And she was in trouble.
Time did what it does, and the day came for her to travel to New Zealand and be reunited with her husband and sons. Herman travelled with her to Vancouver, and put her on a steamship. When she arrived at the little cabin she was to share with two other women for the journey, it was full of flowers Herman had sent her. Her cabin mates thought she was crazy to leave a man like that behind.
"I haven't any words to describe my disappointment when I arrived in Auckland. The boat docked, and there he was. This husband of twenty-odd years that I was committed to spend the rest of my life with. He was there. He takes me home to a rented house, full of furniture bought on the hire-purchase, which I am supposed to get a job right away and pay off. He doesn't say I'm so glad you're here, welcome, I hope I can make you happy. He doesn't say any of those things. He just spreads the newspaper out on the table to look for jobs. For me. I do get a job. I am hired as office manager, switchboard operator, and tea lady. One of the mistakes I made was I used day-old milk; I also bought lemons when there was a lemon tree out in the yard, where I could have picked a lemon. Well, I had picked a lemon, twenty years ago.
"This was not a new life—just more of the same dismal, unhappy existence. Don, the man I had married, was not my friend. I began to dislike him, and that dislike eventually turned into hate. I had brought all of this onto myself. All right, I had allowed myself to have feelings for another man. How was I going to deal with that? Well, he dealt with it."
She eventually left Don and New Zealand, and returned to the Yukon alone. Her youngest son, John, would follow her in a year.
She tells me this part of the story forty years later, at her kitchen table, the part about how she pulled her car over to the side of the road in Cache Creek, at the crossroads, and pondered all those road signs for a long minute. Should she go back to the prairies, and her mother, or was it north she wanted? She claims she wasn't thinking of Herman so much in that moment. She tells me she thought it was over, that they had ended it. But she continued north, so I don't know if I believe her. I don't think it was me she was lying to. I'm no shrink, but I know enough to know when a woman most needs to believe her own lies first.
I get the story from her in snapshots, short bursts, late-night kitchen table talk when the lips are loose with the whiskey. I knew she returned to the north, it was why we were all still here. She tells me part of the story in 2004: she breathes out in one long sentence that my grandfather broke her nose in New Zealand. Just a detail, an aside in another story about something else. She doesn't rest on the memory, and I will myself not to react, so she won't lose her train of thought. She does that now, more and more. Yesterday on the phone she confesses that she never wrote to me much about what happened to her in New Zealand because she hates to remember it, wants her sons to hold a different past in their heads. A different father. My grandfather, and what he did.
"In a fit of pent-up bottled rage, he attacked me. I can't imagine his hatred, and anger that he would smash me in the face over and over again with his fist. The blood was spattered all over the wallpaper. He wanted to mess up my face, so that I wouldn't be attractive to another man. The kids were there, they knew what was going on. They saw it. They had to clean up the mess. Years later I asked Rob, I said what did you do about all that blood on the wall? He said, we cleaned it up."
Pat returned to Whitehorse, alone, and got herself a job. She stayed with friends of the family and made no attempt to contact Herman. But they ran into each other on Main Street.
"He must have thought he'd seen a ghost," she writes. "We didn't speak very much, but he didn't go away. He came back. He came to see me. This time this was serious. We resumed what we had started. I thought our relationship was private. I thought nobody knew. I thought it was a secret. I thought we were kinda sneaking. It was not private. Everybody knew. The whole town knew. I didn't have to make a secret of it anymore. I was acknowledged as his partner, and I started divorce proceedings.
"I realize now the seriousness of his drinking problem. Like he was two people. Dr Jekyll and Mr Hyde. The Dr Jekyll was a good natured, amiable, agreeable, softhearted, generous, loving... what else can I say? But the Mr Hyde could be terrifying. He could charge at me like an enraged bull, and he was bigger, he was twice the size of me. He wasn't fat, he was just meaty. I probably should have been afraid of him, but I wasn't. Because I knew he wouldn't hurt me. The last thing in the world he would do would be to hurt me.
"He built that house, and I know he built it for me, I know he did. We tried to live in it, but it just didn't work. There was just too much. It was battle stations all the time. I know it was the drinking. He spent a lot of time in bars. The Capital Hotel. I was not allowed to go there, and I never went there with him.
"He talked of getting married, but this bothered me. I couldn't see that. But he told me that if I married him, he would give me a sapphire ring that would flash blue like my eyes did when I was mad at him. If that's a proposal, then I guess that's what it was.
"But it ended on Christmas Day, 1970. He collapsed in my house. Right there. Right there on the floor. A big, vital, alive man came crashing to the floor. I called the ambulance. In those days you didn't go with the ambulance, that wasn't done, you were in the way if you did. So I just hid in my second bedroom, I couldn't bear even seeing them taking him away. I didn't visit him until the next day. I went in there and I discovered that he had tried to walk out of the hospital. He had torn out his tubes and whatever they attach to you and tried to walk out. I thought this was probably a good sign that maybe he was going to be all right. Even when somebody said to me how's Herman doing? I said I think he's out of the woods. I said that. He was anything but out of the woods.
"That night I got the call about three o'clock in the morning that he had died. He was forty. Forty years old. The same age as you?
"But had he lived, he would be eighty-one today. He'd probably be as mean as sin. In a way, I am glad he never lived to see me grow old. I'm glad in a way. Because he wouldn't have been very nice about it. He would have been cruel. All in all, we were together about five years. I was as happy as I've ever been in my life.
"Which brings me back to March, 2008. I feel his presence. I don't believe in spirits. I can't imagine him going to heaven. He just wouldn't fit in. And the thought of me having to spend eternity with him in heaven? I'd rather not. We would just fight.
"You should only marry for two reasons. Only two reasons. Love or money. I know what real love is now. And what I had for Herman, there was nothing like it before him, nor has there been since. Passion helps. I mean it helps. It's the glue that holds the love together. Well, all right, sex. Let's face it. My love affair with Herman was passionate. Even when we fought, it was passionate. I think it can actually outlive death, and even time. In retrospect, I believe this. Now, I am ninety. Like sweet ninety and never been kissed? I still feel the same way about him that I did forty years ago. Believe it or not, that is the truth. He told me he liked to hold my little hand. Somehow, I'd like to think he still does."
# Just a Love Story
A COUPLE OF YEARS AGO I WAS CRAMMED into a Honda Civic hatchback with four poets, squinting through the furious wiper blades to find the right exit off of the Number One Highway into Surrey. We were on our way to a suburban high school for a gig.
The slam poet in the back seat with the relentless bad breath squeezed his face into the front seat. "It's Valentine's Day tomorrow. I think we should all do love poems."
There was an exuberant round of agreement from everyone but me. I cracked the passenger window just a little, and an icy spray of February rain hit my cheek. I took a deep breath and rolled the window back up. I was the only storyteller in the car. I am used to this. Used to being lumped in with the poets. This doesn't bother me. I have even stopped telling people I have never written a poem in my entire life. Storyteller, poet, close enough, I guess, for most people. Even though they are not the same thing at all.
"I can't read a love story in a high school in Surrey," I blurt out, feeling a bit like a parent who just busted in on a pillow fight.
"Why not?" the slam poet heavy-breathed from the back seat, his eyebrow raised in a question mark.
I was also the only queer person in the car. I am used to this. This almost never bothers me. Gay person, straight person, what is the difference anymore, right? Aren't we over all that?
Truth is, I have been over it for decades now. Most of us mostly are. But not in a high school. And not here in Surrey, British Columbia. Surrey, where they banned the _Harry Potter_ books from school libraries for encouraging witchcraft. They also banned _Heather Has Two Mommies_ and _One Dad, Two Dads, Brown Dad, Blue Dads_ for promoting anti-family values.
"Because," I say, letting out a long breath, "it is scary enough to be a homo in a high school in Surrey in the first place."
His face shows no sign of recognition, of understanding, of camaraderie, and I suddenly feel in-my-bones tired.
I take another heavy breath. "For you, a love poem is just that. A love poem. And I am glad for you, I truly am. But for me to read a love poem in a high school in the bible belt is a political statement, whether I mean it to be or not, someone will think I am recruiting, armpits will grow moist with tension, I will be pushing the homosexual agenda on unsuspecting adolescents, I will be disrespecting someone's interpretation of the words of their God, you know, the whole tired routine."
"So what?" pipes up the anarchist beat poet who had been slumped in the backseat beside the slam poet. "We've got your back, Coyote, fuck them all, rock the boat. Surrey needs it."
"What if I just want to tell a love story?" I asked. Only the thump of the windshield wipers responded.
I met her the first time eight years ago, in the hospitality room of the Granville Island Hotel, during the Vancouver International Writers' Festival. She was wearing tall red boots and her wool jacket and handbag matched. Silver and black ringlets surrounded her dimples and sparkling smart eyes. Some people you can see how brilliant they are from a distance, like there are little invisible sparks coming out of their brain while it is working, creating static electric charges in the air above their heads. She was electric spark smart, and all I remember is I could make her laugh. Every time she laughed, my heart pounded possibility. When I saw her from across the room, she kind of shone. Like God Himself was pointing her out to me with a glowing finger. I left with too many plastic glasses of free wine in my belly, and without her phone number in my pocket.
I ran into her on the Drive a couple of days later, just like I knew I would.
It was one of those early spring days in Vancouver, where all of a sudden the grey of the previous week gives way and suddenly it is raining cherry blossoms everywhere, a crushed and scented carpet of them underfoot. We were talking about music. Somehow the band Nirvana came up, I can't remember why, I like them all right, maybe they reminded me of some other band I liked better, I can't remember, but she told me that the album _Nevermind_ was her favorite all-time record when she was in grade seven. I quickly did some silent math in my head. How could the sexiest, smartest, silver-hairedest woman I had ever met be too young for me to go out with?
"Grade seven?" I blurted out. "How can you be twenty-three? How did I get to be... if I had met you in 1991 when _Nevermind_ first came out, you would have been..." I shuddered.
"Twelve years old." She laughed again. Like this didn't matter at all. "It's the grey hair, right? That fooled you? I started going grey when I was sixteen. Runs in the family."
My shoulders seemed too heavy to hold up all of a sudden. I told her I was too old for her. She told me that age doesn't matter. I told her the only people who think there is no such thing as too old for you are usually too young to know any better. She told me that she had just come out of the closet, that she wanted an older lover. She told me I was being ageist. I told her I used to think people were just being ageist too, when I was her age. She told me I was being ageist. I told her I know. Then I let out a long sigh. Did what I had to do. Told her that I was a dirty rotten rotter, that I had been around the block a million times, that I had slept with more women than... that I had slept with a fair number of women in my long and lucky life of loving, and that she should pick someone special, that this was her second chance at having a first time, and most people never get a second first-time chance at anything, that she was lucky, and not to waste that chance on a pussy crook like me. Go, I told her, and fall in love with a nice woman. Fall in crazy stupid dumb-struck love and move in together and figure yourself out, don't get a cat, though, and then fall out of love, suffer through a hopefully short but nevertheless nether-region-numbing bout of lesbian bed death, and break up. Lather, rinse, and repeat. I told her that if she still wanted me five years from now, to come and find me. I told her that if she still wanted me then, that I would be honoured. Told her I had to go, before I changed my mind.
I would see her around from time to time. Usually at poetry readings. Started going to a lot of poetry readings. Started dressing up to go to poetry readings. Started ironing my shirts to go to poetry readings.
Five years later I am in my car, waiting to turn left off of Commercial Drive onto First Avenue, on my way to the Home Depot. My girlfriend and I have recently broken up. We still live together, which could have been awkward, but luckily she was often in Portland with her new lover, who made more money than me, had a really hot truck, and a brand new Harley. So of course I was doing what any self-respecting butch does in this kind of situation: I was throwing myself heart-first into a complicated home improvement endeavour.
This next part seems like magic, but it is true. Some would say this is evidence that magic is for real. I was listening to classic rock and Fleetwood Mac was singing about don't stop thinking about tomorrow, and so I was thinking about tomorrow, about how maybe this breakup was for the best anyway, right, because look, I was finally going to get the new floor down in my office, and wasn't I now free to do what I wanted with whomever I wanted, plus, hadn't it been five years now, so couldn't I take that silver fox out on a date now? Thirty-eight and twenty-eight wasn't so bad, right?
And that's when I saw her. Standing on the corner with a coffee in her hand. Her hair now more silver than black, somehow even more beautiful. She waved when she saw me. I unlocked the passenger side door and she jumped in.
"Where you going?" she smiled, showing her one crooked tooth.
"Home Depot," I told her.
"I love Home Depot," she said, and winked.
We didn't get out of bed for three days. She did a lot of yoga, it turned out. I vowed to quit smoking, so I could keep up with her. Eventually, I did. Quit smoking, that is.
Last month we went home to the Yukon. My family loves her, especially my mother. I think she is actually the daughter my mother always wanted. She is so smart and dresses so fine and almost has her PhD and it almost makes up for my mom having me and my even blacker sheep sister as her real children.
I drove her out to one of my favorite places in the world, the Carcross Desert. White sand and mountains and so much sky all over the sky. Some dirt bikers had accidentally burned a huge heart shape into the sand with their back tires. We stood together in the centre of that accidental heart, and it seemed like the perfect spot to put that big old diamond ring on her finger.
My family is beside themselves. At dinner, my cousin Dan insists that I tell his sister the whole story of how we met. It's so romantic, he says. It is just such a love story.
# The Rest of Us
I GOT THE CALL ON A SUNDAY NIGHT. My gran was in the hospital, and the doctor had advised the family that it was time. Time to call everybody home.
I arrived bleary-eyed at the Whitehorse airport the next day. My mom and Aunt Nora were both there to meet me and my cousin Robert and his girlfriend. They looked so tired and worried; the skeleton was showing behind their faces, their eyes red-rimmed and puffy. They took us directly to the hospital, our suitcases stowed away in the trunk of the car.
I knew my gran wasn't going to look good, and I thought I had steeled myself for the worst. Still, my heart stopped and dropped when I laid my eyes on the tiny shape of her, the outline of her hips and legs barely visible under the green sheets and blanket. Impossibly frail and little. Almost gone already, it seemed. I had promised myself I would be strong for my mom, that I wasn't going to cry. So much for that.
"Talk to her," my Uncle Dave said, waving two fingers at Robert and me. "The nurses say she can still hear us."
And so we did. All afternoon we sat and talked. To her, to each other. Remember her bad cooking? Baloney roast? Boiled hamburger? Lemon hard cake, cousin Dan had dubbed her attempt at meringue. How she loved us all, no matter who we were, no matter what we did. I volunteered for night shift, and sat next to the laboured breathing shape of her with my two uncles, whispering stories through the dark to each other, into her ear, slipping our warm hands under the covers to grasp her limp, cold ones.
By early the next afternoon all of us were there. Five of her children, eight grandchildren, plus partners. I began to worry that we were pissing the nursing staff off a little, them trying to work around us, asking us to leave the room so they could change her sheets. Ten or fifteen of us at a time, filing like exhausted soldiers out into the hallway to stand around, teary-eyed and sometimes bickering. I asked one of the nurses if we were driving anyone nuts yet, wasn't it hard trying to do her job with the whole lot of us underfoot? She shook her head and said no, that the First Nations people had taught the nursing staff what an extended family could really look like, and that it is often easier when the family is there to help keep an eye on a patient. She said that what was really hard was when someone was dying without anyone there at all. This choked me up a little, and she shoved a no-name box of Kleenex across the counter at me with a latex-gloved hand. She had said it out loud. The doctor was kind, and had talked around it. Don't get your hopes up, she had said. We are keeping her comfortable, the doctor said. The doctor didn't lie, but it was the nurse who actually said the words. My grandmother was dying.
Florence Amelia Mary Lawless Daws passed away a little after eleven a.m. on May 13, surrounded by seventeen members of her family. Our hands made a circle, all touching her tiny body as her chest rose and fell, and then stopped. I hesitate to say her death was beautiful, because it means I have to miss her now, but it was.
My family asked me to write and read her eulogy. Blessing from the family, the Catholics now call it. I call it what it is. Of course I said yes, I would be honoured, and I was.
I wrote about the values the tiny little Cockney/Irish/Roma woman had lived and died by, and raised us all up to believe in. Love your family, work hard, save your money, have faith, and be grateful for what you have. I worked really hard on the eulogy. I wanted to do justice to her memory, to honour everything she was. There were over four hundred people at the service, and not a dry eye among them when I was finished.
Up at the graveyard, after the internment, I hugged strangers and shook hands. Suddenly I found myself surrounded by Catholic priests. They were being uncommonly nice to me, the queer granddaughter in the shirt and tie. Maybe they make special allowances in the case of a death in the family, I thought. Or maybe they were still hoping to save my soul. The bishop hugged me, and then held both of my hands in his too-soft ones.
"Excellent job, young man. Your grandmother would have been very proud of you today, son. Strong work, young fellow."
My mother heard him too. I saw her freeze. Waiting.
"Thank you, Father," I said. That was why he seemed to like me so much. He didn't know who I really was.
The bishop caught up with me again at the reception, back at the funeral home. We were both leaned over the cheese platters, when he addressed me a second time.
"Once again, I must say, you are a gifted orator. A natural, even. Have you ever considered the priesthood?"
This time it was my Aunt Nora within direct earshot, and she stopped in mid-bite, half a baby carrot removed from her mouth and dropped on a small paper plate. Her eyes met mine, and she tried not to wince.
I took a deep breath. Thought about my beloved gran, about how much she loved the Church, and respected the bishop. He seemed like a nice enough guy.
I'm not going to lie and say that one hundred wise-ass quips didn't run through my head and gather on my tongue. They did. But what counts is what I actually said.
"No, Father, I have to admit, I have never considered the priesthood. But thank you again for the compliment."
The bishop nodded, and everyone around us relaxed and resumed eating and talking.
I like to think my gran would have been real proud of me.
# Three: That Boy
# Red Sock Circle Dance
_August, I974 † Whitehorse, Yukon_
FIVE YEARS OLD AT THE QUANLIN MALL, Saturday shopping, and I was holding open the swing door for my mom and the cart. I remember I had half a cinnamon candy stick in my mouth and a red baseball hat with the plastic thing in the back pushed through a hole that was smaller than the smallest hole in the strap, a hole I had to make myself with the tip of a heated bobby pin.
So the rest of the strap stuck oddly out from one side of the back of my head, but I didn't care, because it was my Snap-On-Tools hat that my dad had given me, just handed it right over to me when the guy at the tool place gave it to him, he was buying rivets or concrete pins or something, and the hat said Northern Explosives too, in black block letters in an arch over the hole in the back part, and come to think of it, what I wouldn't do now for that hat.
So enough about the hat, this American tourist sees me holding the door open, and of course he assumes it's for him, so he won't bump his cameras together pushing past his belly to open it for himself, and he steps through the door, right in front of my mom and her groceries.
He thanks me down his nose in heavy Texan "Thank you, son," and sucks more fresh Yukon air through his teeth. He is about to speak to me again, to meet the people, to engage in a little local colour, in the form of a polite little boy, and perhaps, via a patronizing conversation with him, get to meet his lovely young mother, too, who also had my little sister in tow, perpetual snot on her upper lip, even in summer like this.
My mom interrupts this quaint northern moment, pushing the puffed wheat, two percent, and pork chop-laden cart briskly through the door. "She is not your son," she shoots out the side of her mouth and the door slams shut behind the surprised Texan. I can't see him anymore, there is just myself reflected in the dusty glass, and the back of my mom smaller in the background, as she pushed the cart and dragged my little sister to my dad's Chevy, where he was smoking behind the wheel.
We could hate the tourists a lot more back then, before the mines all shut down.
The pavement was so hot in the parking lot that the bottoms of my sneakers stuck to the tar that patched the cracks on the way back to my Dad's truck.
* * *
_April, 1992 † Vancouver, B.C_.
The van was packed when the call came.
"Is this the girl named Ivan?"
How much can you really guess about a stranger's voice on the phone, but I listened to the soft, smiling lilt of hers rise and fall as she explained that she had been at a going away party for me the night before, a surprise going away party that my friends threw for me because I was driving up to the Yukon today to work for six months. Except the surprise part of the plan had worked just a bit too well, because what nobody besides myself knew was that I was teaching twelve inmates at the Burnaby Correctional Centre for Women how to make leather belts all night, and this was the first I had heard about my own party, and it was over. Quite the surprise it was.
"Great party," she explained, and the sound of her laugh made me think of leprechauns. "Anyway, I was going to take the bus up to Whitehorse today, and well, how do you feel about some company? I cooked a whole ton of pasta salad for the bus."
Now, no amount of gas money and pasta salad can pay for four days on the Alaska Highway with someone who is starting to get on your nerves, because after Prince George you really are in the middle of nowhere, but I liked her voice. I said I'd pick her up in an hour at her sister's place on my way out of town.
Of course, driving over, the doubting began. Just me and the open road home—and a perfect stranger. What if she doesn't smoke, or wants to talk about co-dependency or something like that for two thousand miles? She'll be so glad she's not stuck on a Greyhound that she won't actually say anything; she'll just silently roll down her window in a disapproving fashion and say things like, "I should give you my therapist's number. She specializes in addiction issues."
But I picked her up, she bungee-corded her beat-up mountain bike to the roof, loaded in her pasta salad, lit a smoke, and smiled with an elf mouth that matched her leprechaun laugh as she surveyed my van and said:
"So if she breaks down, I guess I'll just double you the rest of the way on my bike."
Three nights later, in a campground somewhere just outside of Fort Nelson, she slipped her tongue into my ear and her right hand into my Levi's and whispered, "I've wanted to do this since we left Kitsilano."
Six months later, I drove back to Vancouver to go to electrical school, and she stayed. She had met a sweet-faced French-Canadian boy who I thought looked like Leif Garrett, and she was, unbeknownst to all of us at the time, pregnant with their first son.
"You gonna write me, Chris?" I asked her as we loaded the last of my stuff back into my van.
"Probably not, but I'll think about you whenever I eat pasta salad, and if that's not love, then I've never been in it."
This is the closest thing to a commitment you will ever get from a leprechaun, and I knew this at the time.
* * *
_November, 1998 † Whitehorse, Yukon_
It is a balmy November day at Chris's cabin, about three below zero and still no snow. The grass is frost-frozen, sparkling under a sun that shines, not cold, but heatlessly, if there is such a word.
Chris wants to get the kids together and dressed and go into town, about a half-hour drive in a four-by-four. You could still make the road right now in a car, but not after a good snowfall.
I haven't seen Francis, her middle son, since he was a babe in arms. He is now three, and his red brown curls and round face were the first thing I saw at six this morning, when I was still sandpaper-mouthed. He pulled the covers off my face and pronounced in a matter-of-fact falsetto: "I'm not sure who you are, but could you help me out?" His one hand still held the end of the sleeping bag up, and his other hand held a strip of toilet paper, which trailed across the cabin floor and into the cold storage room where I assumed he'd just performed his morning's first production.
Because Francis performs everything. He has just pranced out of his and his brother's bedroom, in a pair of emerald and blue-striped tights, red wool socks, and what looks like part of a sleeve from his dad's old orange sweater stretched up and over his chest, like a tube top.
"Dat dah da dahhh..." sliding in his socks on the bare floor, his smile flits and then disappears, and he comes to a full halt in front of Chris.
"Francis. Warmer clothes. It's minus three."
His shoulders drop like sandbags, and he stomps, his censored artist head down, back to wardrobe, to change. Thirty seconds later, sliding socks and all, he is back out for act two, but with a purple hippie scarf he is whirling around his neck and twirling... his red socks making circles and figure eights, he knows no fear of slivers...
"A sweater. For chrissakes, Francis, don't you want to go into town with Ivan?"
Again with the shoulders, and eventually he is forced to compromise his ensemble altogether and submit to a sweater, and a toque as well. I know how he feels—nobody wears a toque and a tube top at the same time, and then to have to cover it all with a sweater?
"What do you think of my three-year-old drag queen, Ivan?" Chris asks me like she is showing me a brand new old car she just bought with her own money. She thinks that he will be my favourite because he is... well, just like me, and I always thought it would be Emile, because he was the first, and because I was inside of her when he was in her belly and when she came I felt him kick and knew the magic of him then. And then there was Galen, too, and my mom said Chris told her in the truck one day that it was too late for an abortion with him, and that Chris cried when the midwife handed her her third boy, that makes four boys now and her, alone in the cabin, and she knew Galen was going to be the last of it.
But Chris never told me any of this, she just told my mom, and now Galen sits, too, under his crown of cotton ball hair and watches me eat an egg and toast. He is one-and-a-half and drinks cranberry tea from a mug with the rest of us. The kids picked the cranberries themselves.
Galen looks like a little old man shrunk right down, like an owl. There is no baby in his face, and my mom says he will be the most special because Chris almost didn't have him, so he is more of a gift that way. But all Chris tells me is that she has been breast-feeding for five years now, and I couldn't see her in the dark last night when we touched, but her hands felt older.
She smells of wood smoke, and I smell of hair products, and every time I see her the boys are bigger and there is somehow less of her and I meet her sons again, three secrets of her unfolding into their own in a tiny cabin forty miles from anything.
No wonder Chris couldn't wait for me and Francis to meet again. Now that he's walking and talking, and putting on shows. Now that we can relate as equals, he and I. Sure, he's only three, but age has never mattered to a true queen, and it takes one to know one.
Say what you will of nature and nurture and the children of both, scientists and sociologists and endocrinologists and psychologists and psychiatrists and therapists and plastic surgeons can all have their theories, but none of them can explain to me this:
How did Francis get to be Francis in all his Francisness? He doesn't watch TV. He listens to CBC. Francis doesn't know that boys don't wear tube tops. No one has told him this. He just has to wear a sweater too, if it's winter. The magic of this is not lost on me.
He doesn't get it from his father, who doesn't eat anything he doesn't grow, or pick, or preferably shoot, skin, and dress himself with, and his older brother is a five-year-old water-packing, bicepped bushman in his own right, and Galen is only a year and a half.
All four boys seem well aware that Chris is the only female in the house; she owns the only two breasts, the only one without what they have.
Yet Francis, three years old, triumphs like a crocus in a crack in a cliff; how does a lonesome queen even know he exists in a cabin in a frozen field in the Yukon with apparently not another soul around, with an ounce of fashion sense, or even the most minute grasp of the immense and innate drama of it all for miles?
No one but Francis. Until mom drags Uncle Ivan home for a night or two.
This is why I must be there for him, for all those moments, for those drag queen equivalents of baptism, first communion, confirmation, priest, and sainthood, and so on.
The first time he finds the right outfit, the one that really fits, I will hold up the mirror for him and say, "You go, girl." If he wants his ears pierced, he can count on me. [The first time he necks in a closet with the captain of the basketball team, I will be his confessor.] The first time someone calls him a faggot, and he slowly comes to realize that they don't think a faggot is a good thing to be at all, the first time he feels that fear, I want to be there. I will tell him of the time he was three and first did the red sock circle dance in the orange tube top ensemble. I will tell him then that he was born a special kind of creature, one that God never meant for everyone to understand, but that I understand. I will tell him that I will always love that little flower of him, that perfect unknowing differentness that blossomed and danced in a frozen field in spite of everything.
Because drag queens always dance in spite of everything. It's part of the job description.
How can I look at him and not feel relief? He is living proof that I was just born this way. I don't remember my version of the red sock circle dance, but ten to one someone told me to close my legs because you could see my panties when I danced like that, and how do you spell _unladylike_?
But things will be different for Francis, he who will start kindergarten in the year 2000.
Chris and I load the boys into the truck and head into town. I am on a mission: I am taking Francis to meet more of his people.
My friend Cody, the legendary creature with painted nails and black ringlets that reach halfway down his back. It is rumoured that he is a hermaphrodite, that he possesses extra plumbing, perhaps special powers. I have never asked him, because it is none of my business, and Cody has never inquired about the bulge in my own pants. He is a creature of immense grace and beauty, and that is all I need to know.
I take Francis into the cafe where Cody works, to introduce them to each other with all the pomp and circumstance required when in the presence of royalty.
"Cody, I'd like you to meet my godson, Francis. Francis, this is Cody."
But Francis doesn't acknowledge Cody, or his ringlets, or his fingernails at all. Something else more pressing has caught his attention. He reaches his small hand up to caress the fabric of Cody's silver velvet shirt, tight and shimmering over his slender torso. Francis smiles in wonder to himself and his mother places her hand on my shoulder, and laughs like a leprechaun.
"That's my boy," she says, and for a second I am unsure whether she is referring to Francis, Cody, or myself, but it doesn't matter, because we are all where we belong. Home.
# I Like to Wear Dresses
I HADN'T BEEN TO THE YUKON FOR OVER A YEAR, and had been absent from the fold the last three Christmases. I could hardly wait: I love how rush hour in Whitehorse is seven cars long, and how nobody even thinks about washing their vehicles until the end of May.
I think my body was actually designed to function in minus sixteen degrees Celsius, in the clear, blue cold. I like when the air just starts to sting the backs of your hands, the inside of your nostrils, and the back of your mouth. I love to skate on lakes. It was only December, but I needed a fix to shake the grey edge of Vancouver off my shoulders.
I got a chance to go up for the Longest Night Storytelling Festival and a free plane ticket, so I jumped on it.
I hadn't seen my friend Chris's boys since September 2001, and they were all a foot taller now. During intermission, I snuck the three of them backstage. Galen was five and wide-eyed, standing dwarfed in front of the timpani drums. Emile was nonchalant at eight. "I know that," was how he responded, coolly, to each of my careful explanations of rigging, and scrims, and backlights.
And then there was Francis. Seven now and topped with a crown of red-brown curls, he was most impressed with my solo dressing room and the remnants of the smoke machine's fog backstage from the rock star's set just before the intermission. Francis has recently taken up the ukulele, his mother tells me.
I noticed Francis was wearing just jeans and a T-shirt, even though the show is more than enough reason to dress up. Usually, he never passes up a chance to break out one of his velvet skirts or long-flowing ladies' blouses. My stomach dropped for him. Chris, his mom and one of my fondest loves, told me a few months ago that it has started already. They have started calling him a faggot at school. We knew it was going to happen. I guess we were just hoping it would happen, well, later. He is allowing it to fold up the little flower inside of him. Now he mostly keeps his dresses in the closet and wears them only in the safety and freedom of his own home.
Chris tells me later when the kids are in bed that Francis initially had on his long copper velour lace-up blouse, bell bottoms, and pumps when he heard tonight was going to be Uncle Ivan's big show and they were going to the Arts Centre. When he swooped down the stairs to look for his little mittens on strings, Emile reminded him that Sebastian (from school) was going to be there, too. Francis went back to his room and changed into jeans without a word.
I took him alone (after quite a bit of bickering with his brothers about us needing special time together) to see _The Lord of the Rings: The Two Towers_. I for one am scared shitless of the Dark Riders or Ring Wraiths or whatever, and thought maybe it was too scary for a seven-year-old, but he reminded me politely that I had said he could pick. So he, my big old Cheshire cat-grinning dyke buddy Brenda, and I set off for a little queer quality time together, as per the request of his mother.
Francis wasted little time. He spent three dollars on those plastic eggs with rings and miniature tea cups in them, bought popcorn with his own money, and started asking questions, the first of which were brought on by me going to the bathroom.
Francis had leaned across my empty seat to enquire of Brenda just which washroom I used when out at the movies.
Brenda told Francis that to the best of her knowledge, I utilized the ungendered wheelchair-accessible facilities whenever possible, so as to avoid confusing anyone in the men's room or scaring anyone in the ladies'.
Francis then asked Brenda if she knew for sure if I was a boy or a girl. Francis had asked me this himself on several occasions in the past, and each time I explained myself to him as best I could. I'm not sure if he forgets when I go away, or if he just needs to process it all again as a three-, then five-, and now seven-year-old might. Brenda told Francis that she figured that I was technically a girl, but that I had a whole lot of boy in me as well.
I returned to my seat, and Brenda brought me up to speed on their conversation. Francis's eyes were lit up in recognition and he grabbed my wrist. "I'm just like you, but the reverse." He nodded repeatedly and sat up on his heels in his seat. "I'm a boy, but I have a little girl in me too." He lowered his voice and looked left, then right, and continued. "I like to wear dresses," he whispered in his most conspiratorial voice.
My heart felt like it was going to climb out of my mouth for the love of him at that moment, and I hugged him over the armrest between us. He was warm and sinewy and smelled just like his brothers, but he isn't. I don't love them any the less for it; it's just that I love him more.
"I know you like to wear dresses, Francis," I said. "I've known you since you were a baby, remember?"
"Since I was inside of my mom? Since Emile was?"
I told him I knew his mom since before she even met his dad, and he shook his head in amazement, like he couldn't fathom a time that long ago.
"Is that why you like to kiss her on the mouth so much all the time?" he asked loudly, in the not-so-innocent way of babes. I shushed him because the movie was starting.
Turns out that _The Two Towers_ was too scary for both Francis and me, and at one point he grabbed my hand and bravely whispered, "If this is scaring you too much, I wouldn't mind if you wanted to leave early."
But we stuck it out, and then the three of us drove up Grey Mountain and looked at the tiny, snow-silenced metropolis below us. All the way up the mountain Brenda and I told Francis about our people: those of us who are boys with girls inside, and girls with boys inside, and all of the beautiful in-between and shape shifters that are his ancestors. We told him that since before even his older brother was in his mom's belly, there were people like us.
Brenda told Francis that she was like me too, a girl with a whole lotta man in her, just it was harder to tell with her on account of her gynormous breasts.
"Yes, they are big," he responded almost with reverence at her frame, which for years now has been nicknamed by her friends as Tyrannosaurus Rack. We told Francis that his people have forever been artists and mystics and healers and leaders and librarians.
We talked a lot about bullies and their ways. Francis blew me away, as seven-year-olds are known to do with relatives who don't see them everyday as their brilliance unfolds, by explaining that he reckoned that his bully was mean cuz he'd failed grade two twice already, and his mother drank alcohol when he was in her tummy.
I wondered, as Francis's fairy godfather should, when is too soon to warn my young friend about gay bashers, and how exactly I would go about explaining to a northern boy-girl a thing as incomprehensible as what happened to Aaron Webster, who was found naked in Vancouver's busiest downtown park, beaten to death by a crew of teenagers armed with baseball bats and pool cues. Would I leave out the details, and not mention how the police couldn't find any witnesses brave enough to come forward?
I cried at the sight of his face, so determined, and sure, and self-aware of his difference. So entirely void of shame. I cried with relief in the knowledge that my very existence in his life might make it easier for him to make it all the way through grade three. I cried for the hope he makes me feel, now that I'm not the only cross-dresser born in the Yukon in the family, that I will never be alone again. My own seven-year-old loneliness forged my promise to him to see that things will, indeed, be different for us as a team.
Guess what I got Francis for Christmas? Earrings, both dangly and sparkly ones, and fancy French cologne, the same stuff I wear. It all fits perfectly into the jewellery box he got from his older brother.
# A Week Straight
I PICKED HIM UP AT THE AIRPORT LAST WEEK. What struck me first was how ugly his hat was. A fleece baseball cap. I blamed his mother, signed the Unaccompanied Minor form they made him wear on a red elastic string around his neck, crammed the offending hat into his overstuffed yellow backpack, and we left. No checked baggage. I love a kid who travels light.
Often when Francis and I hook up, some time has passed since we last saw each other. This time it had only been three months, but when you're seven-and-a-half, three months can hold a decade of things to catch up on. I studied him out of the corner of my eye while driving over the bridge. He was taller, and his legs were beginning to take up more of him than they looked like they should. Stick legs folded into oversized green rubber boots with laces. Very practical footwear. I always appreciated that trait in his mother too. Warm, lined, navy blue rain jacket.
Just a normal little boy, right?
He touches everything, runs his hands over things, opens the glove box, wide-eyed, staring, and pointing at accordion buses. His knees bounce, his head turns, and his fingers tap. Then I see it.
His pinky fingernails are very long. I'm pretty sure that even Whitehorse Elementary, a notoriously tough place to endure grade two, does not yet have a cocaine problem, even though it is right next to the Quanlin Mall, right in the heart of our throbbing downtown Yukon metropolis.
No, I'm pretty sure Francis has long pinky fingernails because somehow, even though there's a guy in grade two who should by age and weight be in grade four who calls him Francis-pees-his-pantses, Francis has managed to keep a hold on something of the smaller boy he once was: the fairy child bedecked in the sunflower-print dress, before public school, divorce, and reality set in, and someone started calling him queer.
He left his dresses at home in a box under his bed, even to come to Vancouver to see me, but he did bring his blue crushed velvet hotpants and velour copper-coloured top. I breathed a sigh of relief that night when he came out of his room dressed for dinner.
I realized at the Value Village the next day how much I had invested in this little boy, how much my heart counted on him making it through school whole. How much I hoped elementary school wouldn't kick the difference right out of him.
We were going to be pirates for the Fool's Parade, and fortunately, I was already quite prepared. A short stop at the home of the girl up the street (not to be confused with the girl next door) produced a virtual pirate's booty of baubles, sashes, and bandanas.
We were searching through the girl's pants in the Village when it happened.
A pair of black, crushed velvet pants with gold lamé parrots embossed around the bell-bottoms. My eyes lit up and I ripped them off the hanger. The perfect pirate pants. Francis ran his hands over them, and I watched his face go from sparkling to something else altogether. A small cloud crossed inside his brown eyes, a picture played behind it in his head, and he shook it out. He made a face and dropped the pants. "A pirate wouldn't wear those," he said with fierce commitment.
"Dude, are you joking? They have parrots on them." I began to argue with this seven-year-old for a minute, and then stopped myself. I was doing what my mother had done. I remembered the summer I turned eleven, and a yellow and grey dress for my Aunt Norah's wedding had me paralyzed in a dressing room. The shopkeeper stood next to my mother pleading, "Come on out, honey, it's okay, I'm sure you look just lovely. It's a beautiful dress, Pat, and one she can wear anywhere."
I had felt panic that day: both at the thought of looking beautiful, and at the very concept of owning a dress that was both "formal enough for a wedding, but not too dressy for school."
I made a promise to myself to always let the boy dress himself how he wanted, even if it was boring and didn't match at all, and bought him the black cargo pants that he thought a pirate should wear.
He did love the tiny little black patent leather dress shoes we found, almost as much as I did, and I took some comfort in that.
We were standing in the line-up the first time it happened. Women talk to you when you have a child with you, and this woman had been watching Francis try on plastic pearls and clip-on earrings as I waited to pay for our booty. She had been checking me out too, and when I caught her, she gave me the old, "Isn't he just a darling" face.
"Now, are we picking out some jewellery for Mom?" she piped up, in that voice used by women who don't have any children anymore.
Francis froze, his shoulders squared, and he returned the pearls to the metal hook they had been hanging on. He looked guilty, maybe, or sad.
"Oh dear, I do hope I haven't said the wrong thing." She reached for my arm and stroked it, and left her hand on the inside of my elbow.
Francis looked like a small child busted doing something he knows everybody thinks is wrong or weird, and my heart broke for him.
The woman thought he was the confused child of a broken home, and thus I was the grieving divorced single father of one, and her heart broke for us.
"Everything is okay with Mom," I said, letting her off the hook. I smiled, which was easy to do when I imagined Chris in her wool army pants and felt jacket, reeking of wood smoke and wearing rubber boots, Francis's string of plastic pearls, and a clip-on hoop earring.
She breathed a sigh of relief. I was just a nice guy taking my boy out shopping for Mom. In the middle of the day. Maybe I was even one of those new-fangled stay-at-home dads.
Me and the kid, we kind of look like each other too.
I began to revel in my new disguise, my new cloak and mirror. A child: proof of my heterosexuality, even if I was a little swishy myself, and apparently it was rubbing off on the kid. At least I was fertile.
I realized this must suck for straighter-looking moms or dads trying to seek a little action, but it was some novelty to me. No one, no matter what gender they mistake me for, ever mistakes me as straight. I might even have a chance to come out of the closet, for the first time ever, I thought with a kind of glee.
I dropped Francis off at the airport yesterday; he was wearing his sensible shoes. Me and four other Spring Break single fathers milled around the security gate, seeing off our respective unaccompanied minors. We called out last-minute reminders to not eat any dairy, and tell your mom to call me, and don't drink any pop even if they give it to you, and tie up your boots. Francis didn't look back as he let the pretty stewardess take his hand.
The guy with the tight pants and John Deere belt buckle's little boy started to bawl, and his dad teared up himself and waved through the glass, yelling, "Daddy loves you," unabashedly. He turned to me and someone's grandparents and said in a choked voice that pulled at the corners of my eyes, "Now, that's harder than a guy would think, huh? Won't see him till September. I'm a merchant marine."
I nodded like I understood, because he thought I did. I felt secretly proud that Francis didn't cry one bit, that in fact my kid was the toughest one of them all.
# The Future of Francis
THE FIRST TIME I WROTE ABOUT MY little friend Francis, the little boy who liked to wear dresses, he was three years old. The middle son of one of my most beloved friends, he was the fearless fairy child who provided me with living, pirouetting proof that gender outlaws are just born like that, even in cabins in the bush with no running water or satellite television. He confirmed my theory that some of us come out of the factory without a box or with parts that don't match the directions that tell our parents how we are supposed to be assembled. Watching Francis grow up taught me that what makes him and me different was not bred into us by the absence of a father figure or a domineering mother, or being exposed to too many show tunes or power tools at an impressionable stage in our development. We are not hormonal accidents, evolutional mistakes, or created by a God who would later disown us. Most of us learn at a very early age to keep our secret to ourselves, to try to squeeze into clothes that feel like they belong on someone else's body, and hope that the mean kids at school don't look at us long enough to find something they feel they need to pound out of us. But Francis had a mother who let him wear what he wanted, and Francis had evidence that he was not alone, because Francis had me.
He is eleven now, and I got to hang out with him and his brothers last January, up in Dawson City. He doesn't wear dresses anymore, and I didn't see much of his younger self in the gangly boy body he is growing into. He is a tough guy now, too cool to hug me when his friends are around, full of wisecracks and small-town street smarts. He can ride a unicycle, juggle, and do head spins. He listens to hip-hop and is not afraid to get in a fist fight. He calls other kids faggot, just like his friends do, but only when his mother can't hear him.
I can't help but wonder if the politics of public school have pushed him to conform, or if he has just outgrown his cross-dressing phase and become as butch a son as any father could hope for. I try to imagine what it would be like for him to be the only boy in a dress on a playground full of kids whose parents are trappers and hunters. To be labelled queer in a town of 1,700 people and more than its fair share of souls who survived residential schools, families with four generations of inherited memories of same-sex touches that left scars and shame and secrets. I don't blame him for hiding his difference here, for fighting to fit in.
I walk past his school one day on my way to buy groceries, and watch him kick a frozen soccer ball around in the snow with his buddies. He sees me and stands still for a second, breathing silver clouds of steam into the cold. When he was little, he used to fling himself out his front door when I came to visit and jump on me before I was all the way out of my truck. He would wrap his whole body around my neck and hips and whisper wet secrets and slobber kisses into my ear. Now, he barely returns my wave before he turns and disappears into a sea of snowsuits and scarf-covered faces. I find myself searching the crowd for a boy I barely recognize, a Francis who has outgrown my memory of him. I miss the Francis he used to be, the boy-girl who confessed to me when he was five years old that I was his favourite uncle because we were the same kind of different. Now, I can't tell him apart from all the other boys wearing blue parkas.
I realize later I am doing to Francis exactly what I wish the whole world would stop doing to our children: wanting him to be something he is not, instead of just allowing him to be exactly what he is. I don't want Francis to spend his lunch break being tormented and beaten up. I remember growing my hair in junior high and wanting everyone to like me, and I will never forget the blond boy from school who walked like a girl, and that time in grade eight someone slammed his face in a locker door and gave him a concussion because he wanted to try out for the cheerleading team. By grade ten, he had learned to eat his lunch alone in an empty classroom and wear his gym shorts under his jeans instead of braving the boys' change room, but everybody acted like they were his best friend after he shot himself in the head with his stepfather's hunting rifle during spring break the year we all graduated. They hung his school photo up in the hallway, and all the kids pinned paper flowers and rest in peace notes to the wall around his picture, but nobody wrote that they were sorry for calling him faggot or sticking gum in his hair or making fun of how he threw a ball.
I made a silent promise to Francis the day I left Dawson City to always love what he is right now as much as I loved who he was back then. Whether he grows up to become a textbook heterosexual he-man or one day rediscovers his early love for ladies' garments, I will always be his favourite uncle, no matter what he's wearing.
# Four: Kids I Met
# Saturdays and Cowboy Hats
EVERY SATURDAY MORNING ALL SUMMER LONG, the parking lot across the street from me is transformed. Friday night, it's full of sports cars and sparsely moustached, beer-guzzling boys with cell phones and car stereos that shake the glass in my front windows, but come Saturday morning at eight, it's a farmer's market. There is the fey fella selling homemade dog biscuits, the family-run fireweed honey corporation, the lesbian cheese makers from Saltspring Island, a grumpy potter, and a sunburnt man selling bundles of organic mustard greens and butter lettuce. You can buy cherries and maple syrup, visit the latte wagon, and get gardening advice. You can sign petitions and join a jam-making group that donates to the food bank. There are face painters and banjo players. People wear sandals and the dogs rarely get into fights, because everyone is too busy saying hello and showing off their new bedding plants. Yard sales spring up spontaneously on street corners.
All of this appeals to the increasingly not-so-latent hippie in me. I mean, I still like to wear shoes in the city and I wholeheartedly believe in the frequent washing of one's clothing, but there is still something of the small-towner in me—I like to know my neighbours, I like to meet the guy who picked the cherries I'm about to eat.
I usually throw on a pair of jeans and take the dogs with me. We always complete a loop around the lake before we hit the market, to avoid any unsightly squatting in the middle of the town square.
I saw them getting out of a late model minivan, a young, slender mother and her maybe six-year-old kid. She was in a wind-blown dress that wrapped around her legs, the kid in blue cords with frayed cuffs, a red and yellow striped T-shirt, and now colourless canvas sneakers. The mom had a canvas shopping bag over her shoulder and the kid had a comic book rolled up and pushed into the back pocket of his cords.
"Mom, lookit the little dog, he's sooo wee..." The little boy bent down to pet my Pomeranian, and his mom stood up straight and slammed the door of the minivan shut.
"Olivia, you have to ask the man if the dog is friendly before you touch it. Maybe it doesn't like little girls."
I looked at the kid again, and she stared back up at me. Her hair was straw yellow, and cut short. She had one hand on her hip, her elbow resting on the comic in her back pocket. The knees of her cords were worn and grass-stained. One shoelace was hanging untied, flattened, and muddied. The only things about her that matched her name were two tiny stud earrings, dark blue and sparkling, out of place with her tomboy face.
I wondered if Olivia got her ears pierced to make Olivia happy, or her mom. Maybe her grandma took her to the salon in a last-ditch feminine attempt to make up for the striped t-shirts and dirty knees.
"She's not a mister, Ma." Olivia spoke matter-of-factly, rolling her eyes back like kids do when their parents say dumb things. "So can I pet your dog, or what?"
I nodded, struck as dumb as her mother. I couldn't make my mouth work, and there were tears in my eyes. I wanted to show Olivia my new fishing rod; I wanted to build her a tree fort with a rope ladder. I wanted to make her a belt with interchangeable brass buckles and teach her how to perfect her wrist shot. I wanted to play street hockey with a tennis ball, and get headaches from eating our Slurpees too fast.
I wanted to pass her a note written in pencil on a piece torn from a brown paper bag that said: You are not the only one. And one day everything will be fine, I promise you that. Oh, and learn a trade you can fall back on.
Olivia's mom stood next to me on the sidewalk. "She really loves little dogs. She's always begging to get one, but we live in a one-bedroom apartment."
Goliath was flat on his back now, all four legs in the air, working the cute angle. Olivia was scratching his belly with both hands.
"Come on, honey, we have to shop. You've got karate at noon. Say goodbye."
Olivia jumped up, wiping her hands on her faded red and yellow shirt. She looked me up and down. Her eyes rested unabashedly on my dusty workboots, then my jeans, my Snap-On Tools belt buckle, the wallet in my back pocket, my black t-shirt, naked earlobes, and freshly shorn hair. She chewed her gum slowly on one side of her mouth and hooked her thumb through an empty belt loop.
"Thanks fer lettin' me pet him. He's real cute, huh? What's his name?"
"Goliath." I could still barely talk, I was still afraid the tears were going to spill over my bottom lids. I wanted her to remember me as being tall and dry-eyed, just in case I was the first one of her people she had met so far.
"It was really nice to meet you, Olivia." I extended my hand, and she shook it, her face deadly serious. Her mother nodded a polite goodbye. Olivia just kept shaking my hand.
"One more thing..." she said, squinting up at me, the sun bright over my shoulder, "I need to know, where'd ya get that cowboy hat?"
# Schooled
YESTERDAY I SPENT THE DAY IN A HIGH school in Burnaby, telling stories to the grade tens. I was surprised how nervous I was. I tell stories all over the place, often to people who in real life are much more intimidating than a couple hundred fifteen-year-old strangers should be, but right from the time the alarm went off it was there, the big ball of nervous. It hung there in my gut, between my ribs and my belly, all waxy and electric.
The face in the mirror looked pale and rumpled. "That's just perfect," I told my reflection. "A zit. Right in the middle of your chin. On the first day of school."
It's something about the hallways that does it to me, the way sounds are amplified by the polished tiles and painted lockers, all sharp edges and canned echoes. Just the sound of a high school makes me fifteen again.
It didn't help much that all five of us poets and storytellers had to wait in the office for the English teacher to come and escort us to the auditorium. Lined up with our asses slouched in the plastic chairs outside the principal's office, in between the photocopier and the water cooler, the rest of them joked and told anecdotes. I was the quiet one for once, trying to breathe around the inflatable lump in my throat and wondering why my toes were sweating so profusely.
The teacher that had organized the reading was cool; the kind of teacher who would think that poetry in high school was a good thing. Her classroom was the one with the beaded curtain, and the kids who were wrestling or kicking each other in the ass in the hallways didn't straighten up or act like pretend angels when she came around the corner. She explained to us over her shoulder as we walked that the crowd for the lunch-hour show might be a little smaller than they had expected, because today the student council was auctioning off elves in the gymnasium, plus a representative from the community college was answering questions and handing out pamphlets outside the library. We had competition, she told us, but assured us we would have a good house for the afternoon sessions, when attendance was mandatory.
She took us into a place she called the dance room, which meant it looked like a small gym with mirrors lining the walls. She apologized for the fact that we were required to remove our shoes, because they marked up the floor. For some reason this made me uncomfortable. I was about to tell queer stories to a bunch of teenagers, and I wanted my shoes. My sock feet left little sweaty tracks behind if I stood in one place for too long. Two of the other poets were wearing odd socks, and this made me blush. We were here to prove that being a spoken word performer was a viable career option, and I felt that not owning a pair of socks that matched might undermine our position. Then I reminded myself that they had both just come off of a long tour, and I should be glad they were wearing any socks at all. The kids all had to take their shoes off too, which they did in an orderly fashion as they filed into the room. Quite a few of them had on odd socks as well. I changed my position on the matter immediately, thinking maybe it would be something we could bond with them over. Odd socks didn't mean you were poor. Odd socks meant you were a non-conformist.
It turned out that the kids were great. They listened and laughed in all the right places, and asked really smart questions. One kid asked us what the meaning of life was, saying that he had read somewhere that if you asked enough people, one of them might just have an answer. Then he asked me what my favourite Led Zeppelin album was. I told him _Led Zeppelin IV_ , and he nodded, like I had passed his invisible test.
Somewhere between classes I relaxed a bit and started to have fun. Sure, there were a couple of kids slouched along one side of the classroom at the back of the room who already could grow sideburns and snickered and rolled their eyes the whole time, but for the most part they were interested, and engaged. I kept telling myself that I wasn't there to change the mind of the beefy guy in the back with the almost full goatee. I was there for the kid I couldn't see yet, the kid who was seeing me for the first time. The kid who walked the edges of the hallways, one hand trailing the lockers and the walls, hoping they won't be waiting for him at the bus stop today. The kid who hides his _Muscle and Fitness_ magazines behind a ceiling tile in his closet, when his brothers can read them openly because they are not like him. For the girl who doesn't know yet but her parents do. That was who I was there for.
The cool teacher escorted us through the woodworking shop in between classes to a patchy corner of lawn you couldn't see from any windows in the school, so we could have a smoke. The shop was almost empty, because the bell hadn't rung yet. There was a skinny boy with glasses screwing two bits of wood together with a cordless drill. He nodded at the cool teacher as we shuffled past.
"Hello, Vanessa," the cool teacher nodded back at the kid, and I did a double take. The teacher winked at me, and I smiled. All day, I had been searching for signs that things were different than they were when I was in school, that things were getting easier for queer kids, that we really had come a long way, baby. I had overlooked the most obvious sign. Of course things were changing. I was here, wasn't I?
# Teach the Children Well
EVERY TIME I DO A STORYTELLING GIG at a public school, I swear to myself that I will never do it again. I promise myself that this is the last time, that the next time they ask me I will remember that I have decided to avoid attempting to entertain large groups of teenagers for health reasons, that breathing gymnasium air makes me dangerously dehydrated, that hallways lined with lockers can cause painful grade eight flashbacks. High schools remind me of high school, I can't help it. I graduated twenty years ago, but all it takes is the sound of the first period buzzer going off or the smell of floor wax and it is 1985 all over again, when I am skinny and self-conscious. I hate my legs, my flat hair, my flat chest, my chipped front tooth. I am scared of change rooms and crowded cafeterias. I am scared of myself, of the secret heart inside me that doesn't beat like it is supposed to and makes me different. I don't know I'm queer yet, but I know what happens to kids who don't fit in.
Every time I walk through the front doors of another high school, I remember what it was like to hide, to pretend, to practice not being different. I watch the kids, noticing the ones who avoid my eyes instead of staring. I am not here to change the minds of the many. I am here for the kids who think they are alone. The skinny boy with the long eyelashes who knew he was a fag even before they started calling him one in gym class. The Catholic girl who confesses only to her journal and prays that God will make it go away. The oldest daughter of a former beauty queen whose mother makes her see a shrink once a week ever since she got busted French kissing a girl named Marie on the couch in the rec room when they were supposed to be working on a three-dimensional model of a molecule. These are the kids I want to be seen by, the kids I want to stand in front of, unashamed and unafraid. I don't say I'm queer, because I don't need to. I wear cowboy shirts and big black boots and tell stories. I tell them that my writing pays all my bills, that I love my job, that they can be artists too, not just lawyers and dentists and assistant managers.
A couple of months ago, I got an email from an English teacher asking me to come and perform in a high school in Surrey, the conservative town situated southeast of Vancouver. Surrey, with a school board prone to banning books with titles like _Heather_ _Has Two Mommies_. Did I want to risk a gig in Surrey? Absolutely not. I was halfway through writing a polite letter saying that I was busy that day, when I stopped to consider what school must be like there for young homos. How could I turn my back on the queer kids who needed me most? How often was a gay storyteller even allowed inside a high school in Surrey? I said yes, and immediately started to stress out about it. I arranged to bring my friend the punk-rock cello player with me, for moral and musical support.
A couple of weeks before the gig, I got another email from the English teacher. He explained that one of the other teachers had done some research on me and had raised concerns about "inappropriate sexual content" in my work, and would I mind sending copies of all the stories I was planning to read so that the staff could make sure I wouldn't say anything that might offend anyone? There would be a couple of Mormon kids in the audience, he added, and the school wanted to avoid any trouble.
I took a deep breath, smoked two cigarettes, and called him on the phone. I liked him, and I knew he meant well. I told him that the reason I do gigs in high schools is to show the kids that being an artist is a viable career option, to inspire them to believe that writing or painting or playing an instrument is just as important as algebra or volleyball. I told him that I would never do or say anything that would jeopardize the chance to bring other artists into his school, and that I was there to encourage creativity, not homosexuality. I told him that I wouldn't say anything too gay, but that I looked queer and if looking queer was also against school rules then I could recommend another talented storyteller who also might offend the Mormon kids because he is from the Dogrib First Nation and believes in magic and different gods, but at least he was heterosexual.
So the cello player and I did two one-hour sets in a Surrey school last week. I told wholesome stories, and she swore once in one of her songs, but none of the teachers batted an eye, they were so relieved that we didn't bring up how obviously queer we both were. The principal gave us each a mug and a matching pen, and a thank-you card with a cheque inside.
That night we both received MySpace messages from the girl with the purple brush-cut who sat in the front row during the afternoon set. She was smiling in her picture, her cheek pressed up against her girlfriend, who had orange hair and a nose ring. She was just writing to tell us how much she loved our show; that it was the best thing her school had ever seen. I clicked on her profile. It said she was sixteen, a lesbian, and an aspiring writer.
# This Summer at Gay Camp
HE SHONE LIKE A BRAND NEW DIME, that first time. "I want you to meet my son," she had told me. "I want him to meet more gay people. School has been hard on him these last couple of years."
I was in Fort Smith, Northwest Territories, on tour with a mismatched set of other storytellers. It was the first week of June, and the roof of the earth was gearing up for summer solstice. The midnight sun stretched the light so far and long that dusk was bent over backward enough to bump into the next day. The sun cooked the dirt into dust that got into everything, grinding between back teeth and turning my new black boots grey. We were a seven-hour drive by mostly gravel road to Yellowknife. A hell of a place to try to hide yourself. A hell of a place to have to repeat grade ten.
His mother was a solid, smiling Métis woman with a laugh you could hear from the other side of the lake. Her son stepped out of the car and onto the weary pavement of the parking lot outside of the only motel in town, which boasted a restaurant that served both Chinese and Italian cuisine, and I use the term loosely.
He was wearing brand new sneakers, so white they caught the sunlight and bounced it right back, bleaching the backs of my eyelids when I closed them. His tracksuit was also white, both pieces, and so was the singlet he had on underneath. All of his clothes were crisp and pristine, with a fresh-out-of-the-wrapper look that stood out stark and sudden against the frayed and aging backdrop of this little northern town.
He was sapling thin, with cover girl cheekbones and feather duster lashes. Easily one of the prettiest boys I had ever seen, all long fingers and fey hips and wrists. I could imagine him standing in a line-up on Davie Street in Vancouver, waiting to get into a club that would be pounding a dull bassline from inside, surrounded by his twinkie buddies in designer jeans and two-hundred-dollar t-shirts. That such a creature still breathed in a high school in Fort Smith, Northwest Territories, seemed somehow unfathomable to me.
A mud-coloured pick-up pulled up beside us, its tires popping bits of loose gravel sideways. Our hiking guide jumped down from the driver's seat, wearing sturdy boots and a grey beard. He led us on a meandering route past the old graveyard and down a well-worn path through the pines, wide shards of sunlight showing the dust and dandelion seeds floating in the air that smelled so much like home to me. I kept stealing looks at my friend's fairy boy son, him in his immaculate threads and me in my now dirty new Fleuvog boots and vintage leather coat. I loved him at first sight, flying his flaming flag so fiercely, here, so far from a pride parade or leather bar or Mac counter. All of fifteen years old and fearless already.
Later, I pulled his mother aside and told her about a camp in Edmonton for gay youth where I was going to be artist-in-residence in a couple of weeks. It was probably too late for this year, I told her, but what the hell, send in an application, because you never know.
The last week in July, he sashayed through the door of the education centre in Edmonton. Sixty-five queer youth for four days. I wondered if he had ever been around more than one or two queer people at the same time before. I wondered if he felt as overwhelmed as I did. A place where faggot wasn't a bad word anymore. A place where he could be one of many. A place where he could just be.
I got to work, teaching creative writing classes every morning and cajoling my group of youth to choreograph an a cappella synchronized dance number to "I Will Survive." He was in my group, and I spent the better part of four days trying not to hug him too much in front of everybody else.
On Saturday night there was a talent show. One of the local kids organized a fashion show, and he modeled a gold lamé gown complete with fake breasts and walked the runway in heels like he was born in them. I felt like the homosexual version of a hockey dad whose son has just scored in overtime.
I watched him stand taller and smile bigger and swish wider every day. And then, of course, the inevitable came around.
Sunday night. There was a lot of crying, the kind of tears that could only be conjured up by a bunch of queer kids about to return to High River and Moose Jaw and some little town just north of Edmonton. Alone.
I couldn't even look him in the eyes the last time I hugged him. I couldn't tell him what I was thinking. I hoped that the new pride he held in his shoulders wasn't going to be pounded out of him in gym class, or while he tried to learn trigonometry. I felt sad, but mostly I felt rage.
Rage that we are beginning the second decade of the twenty-first century in what is supposed to be one of the most liberal and progressive countries in the world and still we haven't made our schools safe for kids like him. That something as vital to his future as his education happens in a culture of fear and under the threat of violence.
I reminded myself to be thankful that at least he has what a lot of queer kids don't have: an amazing family behind him. I got an e-mail from his mom yesterday. She thanked me for getting him into camp, saying that he really needed this support, and that he seemed so much more confident and wiser since he came home.
The four days of relative safety and acceptance from his peers really did him some good. Now we just have to get to work on the other 361 days of the year. He still has grade ten to get through. Again.
# Straight Teens Talk Queer
RECENTLY I HAD THE PLEASURE OF BEING A teen mentor for a group of nine youths at the Vancouver Public Library's annual book camp. My kids were almost frighteningly smart, and savvy, and hilarious, and of course, well-read.
I decided I was going to put all that intelligence and potential and Internet virtuosity to work and get them to write my column for me this month. We set out to write a piece about homophobia from the point of view of a group of predominantly heterosexual youths. As they were a rather studious lot, we started off by not only defining homophobia for the reader, but by including a historical overview of how definitions of the word homophobia might have changed over the years. Turns out that in 1958, there was no such word as homophobia listed in _The Comprehensive Word Guide;_ all the kids could find was a definition of homosexuality listed under "certain specific sexual aberrations, perversions, abnormal practices, etc." alongside thirty-nine other practices which included bestiality, auto-fellatio, cunnilingus, and coprolagnia, which none of us had ever heard of, but we looked it up. Look it up. I dare you.
We all found it notable that a mere fifty years later, _Webster's_ defined homophobia as "the fear of or contempt for lesbians and gay men, or behavior based on such a feeling."
We then came up with a list of questions, and everybody took them home for homework. This was followed the next day by a rather raucous and ridiculously funny discussion resulting in all of us being resoundingly shushed twice, because we were, after all, in a library. Here is a list of the questions and a sampling of their answers.
_Do you think that homophobia still exists in our society?_
Sarah, age sixteen: It may not be as harsh as it was in the past, but it is still there. People in the gay community are not always beaten for being who they are but they are definitely not always welcomed by all the people around them.
Wednesday, seventeen: Being a high school student myself I can safely say yes, it does. I do believe that acceptance is a lot more common than it was twenty, or even ten years ago. Things are definitely looking up. I see straight boys with their arms around each other as a sign of affection, I see boys wearing pink and not getting called the F word. I see girls holding hands and no one is writing accusatory labels on their lockers.
_Why do you think homophobia still exists?_
Megan, sixteen: I blame religion, or, more accurately, religious fanatics.
Sarah: Not all cultures suppressed it for thousands of years. In Greece they used to wrestle naked. That's how the Olympics got started.
Olivia, fifteen: People prefer the ordinary.
Annalise, fifteen: Some people are closed-minded and not accepting of what is different and strange to them.
Kylee, seventeen: It's all Adam and Eve stuff. People are afraid that if they allow it to happen God will be angry and bring damnation or something down upon them.
Wednesday: I'm not sure that there is only one thing or person to blame, unless you can blame the entire human race and call it a night. But that won't bring back the numerous suicides, and it won't make things any better.
Julian, fifteen: Some bigotry is rooted deeper than just in ignorance, but hopefully those people will eventually succumb to the inevitable and keep their mouths shut.
_Do you want to end homophobia, if indeed you feel it still exists? Why?_
Sarah: Of course I want it to end.
Neil, seventeen: Why should straight people care? Why do white people care that we are mean to black people? It's a moral issue and we have accepted that it is not okay to discriminate... period.
_Does homophobia impact your life in any way, or anyone who you know or care about?_
Sarah: One of my best friends felt so afraid of what would happen to him in my town that he felt the need to move. I haven't seen him in over two years.
Lisa, sixteen: I've grown up in a family that says they find nothing wrong with it, but have some serious issues, and I feel embarrassed. I meet these truly interesting and inspiring people, and it hurts to learn that they have been treated wrongly, especially when I hear the slander coming from the mouths of people I respect and trust. What if, somewhere down the line, I realize that I'm not heterosexual? I won't have a problem with it, but what of my friends and family? Will they be supportive or turn their backs?
_Give an example of ways we could change things._
Sarah: My school tries to stop people from using the term gay in a derogatory fashion by making the student who uses the word write a 5,000-word essay on why the use of that word could be offensive. But I don't think this works because it is hardly ever done or checked up on.
Julian: The fact that Gay/Straight Alliance groups can exist is a sign of the times. Fifty years ago, such groups would have been counterproductive: instead of a safe place, these groups would have been bull's-eyes.
Annalise: Set an example of not being homophobic, and not making homophobic remarks, and hope that others take on that acceptance too.
Megan: My school has a program on sexual orientation; they mix it in with sex ed and suicide awareness. The leaders asked us what we would do if we found out one of our friends were gay. If you were okay, you went to one side of the room; if you weren't, you went to the other side of the room. Only one person stayed on the not okay side.
So. There you have it. I think there is only one right thing to do with our society. We have to turn it over to these people. Which is great, because eventually this is going to happen anyway, whether the rest of us are ready for it or not.
# My Name Is Sam
I WAS SMOKING A CIGARETTE WITH THE PERFORMANCE poet outside the theatre. She smokes like a movie star, making sweeping semicircles with her forearms and revealing glamorous cheekbones with every inhale. When she exhales, a perfectly lipsticked stream of silver escapes her mouth between bits of story. I could watch her smoke until the sun showed up. I'm a Player's Light regular peasant; she's a Benson and Hedges Ultra Light King Size Menthol diva.
We were interrupted by a squeal that belonged to a permed and tinted blonde in a beige pantsuit and dyed-to-match pumps. She sniffed her way through our smoking circle to kiss the poet on both cheeks and hug her without really touching.
"Oh my God," the blonde exclaimed, "I thought that was you. You look fabulous. Haven't changed a bit. It's been a long time. When did we graduate? Nineteen seventy..."
The poet blanched, and interrupted her. "Ivan, this is..."
"Diane. I'm Diane. We went to high school together. Oh, I could tell you some stories."
The poet cleared her throat and took a long drag from her cigarette. "Well, actually, Diane, you graduated a few years ahead of me."
Diane looked confused. I smiled. The performance poet has been lying to me about her age for several years now, and for me to do the math at this juncture would be ungentlemanly. To know her age in people years would be tantamount to seeing the bride in her dress before the ceremony. She is beautiful years old according to the diva calendar, and that is all I've ever needed to know.
Diane changes the subject. "Well, I married Richard of course, we have one son, twenty-three, and one daughter, twenty-one. They're both at the University of Alberta, doing well, and I'm directing _Fiddler on the Roof_ this summer, in the park right across the street. You should come by one night. We're having a gas. The kids are just great. And you, are you still writing poetry?"
"Always." The poet exhales, blinking.
"How interesting. We should do lunch one day, I'd love to hear all about it. Call me. I should be off, though, to round up the kids. It was nice to meet you."
And she was gone, leaving only a hint of Oscar de la Renta in the air.
"She's much older than me," the poet whispered over the sound of Diane's pumps retreating.
"Quite obviously so." I grind my cigarette under the heel of my Daytons. "Let's head in. I'm on in half an hour."
Right at the end of my set, I heard a small kerfuffle in the balcony. It was over quickly, and I thought no more of it.
Post-show, we resumed our spot in the smokers circle, several hours and two beers later. There were five or six of us now, talking poetry, gossip, and business.
A teenage boy paced around our circle a couple of times, took one huge breath, strode up and stood beside me. He seemed nervous, his hands stuffed deep into the pockets of his too-big-for-him black blazer. He waited for a pause in the conversation, and then placed a long-fingered hand on my forearm.
"Sorry to interrupt you," he stammered. "But I have to thank you for your stories tonight. You just changed my life. My life is changed now. I really needed to hear what you just said. I'm a huge fan of spoken word and poetry."
I tuned out everyone else except the boy. This was one of those moments, I could tell, one of those moments you conjure up when you're trying to sleep on the cat pee-scented couch in a chilly basement room on tour somewhere in Manitoba, to remind yourself why you choose to do this for a living. I extended my hand to him.
"My name is Sam. I've been reading Ferlinghetti and Rilke for years, and I'm a huge fan of Sheri-D..." He shook my hand with baby-soft palms. His bangs hung over his caterpillar lashes and brown eyes. He had a peace sign and a Sex Pistols button on his lapel. The knees of his jeans were peeled back to reveal doorknob kneecaps. His dress shoes were spit-shined. I loved him.
"This is Sheri-D right here, I'll introduce you... she doesn't bite, well, not strangers, anyway."
I tapped on the performance poet's elbow. "Sheri-D, I'd like you to meet Sam. He loves poetry."
Sam swallowed, overwhelmed. "Wow, pleased to meet you, all of you, the show was, well, it blew my mind, and I'd do it all over again, it was worth it all, even though I got into trouble."
Sheri-D furrowed her brow and looked sideways at Sam. "You got into trouble for coming to a poetry reading?"
"Well, I skipped out of our meeting after my show. I'm in the play across the street, in the park. I'm the boyfriend of the milkman's daughter."
Suddenly Diane and her pumps and perm were upon us again. "There you are Sam, good, I wanted to talk to you. I want you to know, I'm not angry with you, just disappointed. You can't take off like that without telling anyone where you are going. We were all concerned for your safety. This is downtown Calgary, and I am responsible for all of you. We had to call the police, and security."
The whole picture became apparent to both Sheri-D and I at the same time, and we simultaneously clutched our aching chests with our right hands. Sheri-D spoke first.
"Sam is in trouble for skipping his notes to come and see Ivan tell stories?"
I thought about all the things I ever got busted for when I was fifteen. Poetry readings were not among them. My heart opened and swallowed Sam up.
Diane nodded. "We had to have security remove him from the theatre. They serve alcohol in there. We were looking all over for him. He's been suspended from the play for two nights."
"I'll leave you two tickets at the door for tomorrow night then." Sheri-D smiled at Sam. Diane fixed an acid stare on Sheri-D. "Well, he might as well, since he's not working," Sheri-D shrugged.
I nodded. The boy needed poetry, that much was obvious.
"It is time to get you home, Sam." Diane grabbed the sleeve of his jacket and steered him towards her mini-van.
Sam called back over his shoulder to us as he was led away by one arm. "I'd do it all again. I loved it. They call me Art Fag at school." The sliding door shut, and he was gone.
"What a bitch," Sheri-D breathed sideways at me. "No wonder she looks so much older than I do."
"Decades," I agreed, and lit her next cigarette for her.
# Nobody Ever
IT WAS RAINING THE DAY I MET HER. The kind of rain that hits the pavement and puddles so hard it bounces back at the sky, backward and defiant. It was the kind of evening best spent inside, but there she was, standing soggy on the sidewalk, waiting to talk to me.
As soon as I emerged from the back door of the theatre, she speed-walked in a straight line towards me. Her name was Ruby, she told me, and she was from a small town, about three hours' drive from here. She was almost twelve years old and she wanted to be a firefighter when she grew up, or maybe a marine biologist. Her mom had driven her here, so she could see me perform at the Capitol Theater. It had said on my website that I was going to be reading in Olympia, Washington, and since it was a Saturday and there was no school she had made her mom drive her all this way for my show, but then it turned out that since they were selling alcohol in the theatre she wasn't allowed inside, not until she turned twenty-one, anyways, which was like, ten years away, practically.
She took a deep breath, and continued. She had seen me at the folk festival in Vancouver last summer, and I had read a story about a tomboy I had met at the farmers' market, did I remember the one?
I nodded, yes, I did.
She shifted her weight from one sneakered foot to the other and back again, like she needed to pee, and flipped her head back to shake her shaggy bangs out of her eyes. She blurted out her words like machine gun bullets, like she had been rehearsing them for a while, her mouth pursed in a determined little raisin.
When she first heard that story, well, she was just amazed, she told me. She had begged her mom to buy her all of my books right there on the spot, but her mom only had enough money for one. She had to wait until it was her birthday, which was October by the way, until she could get my next book, and then she got one more from her aunt at Christmas, but when was I going to put out a new one? She liked them all, nearly the same amount, except for _Loose End,_ which of course was her favourite because it had the story "Saturdays and Cowboy Hats" in it, which was the very first story of mine she ever found out about, when she heard me at the park in Vancouver last summer but she had already told me that part.
By this time I was ready to scoop Ruby up in my arms and hug her, but I didn't, because her mom was waiting in the car parked two feet away from where we were standing and I thought it might seem weird.
Ruby stepped sideways, farther under the awning over the door of the theatre. She pulled a love-worn copy of my book out from her rain jacket, and held it out to me.
"Could you sign it for me? To Ruby, Love from Ivan? You could say, To my biggest fan, Ruby, too, if you felt like it. Whatever you want."
I wrote "To Ruby, my biggest fan, Love from your biggest fan, Ivan," and passed it back to her. She tucked it under her armpit for safekeeping. Her fingernails were bitten right down to the quick, just like mine used to be.
"Thanks. I really love your books a lot. Especially the one about the tomboy, cuz, well, the little girl in that story, she reminds me of me." She paused for a second, met my eyes with hers, and held them there. "And nobody ever reminds me of me."
I stepped back out into the rain, hoping that it would look like raindrops sliding down my cheeks, not big hot tears. I pulled one of my CDs out of my bag and passed it to her.
"Here you go, this should hold you until the new book is out."
The last time I saw Ruby, she was waving backwards at me from the passenger seat of a beat-up station wagon. Her mom honked the horn twice goodbye as they turned and disappeared around the corner.
A while ago I was reading at a fundraising dinner in Ottawa, and I met a woman named Hilary. Hilary was in her fifties I would say, wearing black boots and old jeans. She used to own her own house painting company, but she was retired now. I liked how she shook my hand too hard, how the skin of her palms was still callused, how she spooned too much sugar into her coffee. I liked how she ate her salad with her dinner fork and didn't care. Her hair was just getting long enough to brush the collar of her dress shirt and hang over the tops of her ears. This probably bothered her, and she probably had an appointment to get it cut early next week, before it got totally out of hand.
After the gig was over, she helped me pack the rest of my books out to my truck. We talked about everything and nothing: what it used to be like working on a job site twenty years ago, how it is better now but not by much, what a difference a good pair of snow tires can make, how the old back just ain't what it used to be, stuff like that.
The snow was falling in fat lazy flakes. The parking lot was empty, except for two trucks, one hers, the other mine. Finally, she shook my hand hard one last time and then pulled me into a hug.
"Make sure you keep in touch," she told me. "It was great to meet you. You remind me of me when I was a kid."
# As Good As We Can Make It
I HAVE BEEN A ROAD DOG LATELY. Festivals, theatres, conferences, planes, boats, rental cars, road and road and then some more road. And schools. I have been doing a lot of high school gigs too. Sometimes I wonder if maybe I get the opportunity to see the insides and the guts of more high schools, and shake hands with more students, and stumble through more uncomfortable introductions to more principals, and cover more territory, span more provinces and borders and districts and countries, even, than almost anyone working in our education system today. Sure, it's quick and I am there and then gone in a little under two hours, mostly, but still. You get a sense of a place, a taste of it, anyway, and more and more I am sampling the smorgasbord of our school system, and where it best serves our students, and where it is still falling short. I do a one-hour show, designed to fit in between bells, forty-five minutes or so of storytelling, followed by a ten- to fifteen-minute question-and-answer and hopefully discussion period. I don't say the word queer or gay or lesbian during this show, nor do I talk about sexuality at all. I just tell stories. Stories about me, my little sister, and my two little cousins, Dan and Christopher. Christopher was an awkward, clumsy kid who was mercilessly teased and picked on all throughout school, right from the beginning. I tell stories about the four of us, stuff we used to do when we were young, stupid broke-ass bored small-town kid stuff. I tell the story about how Christopher had gigantic feet for his age, size thirteen by the time he was eight years old, and about how we all got second-hand roller skates this one summer, all of us except Christopher, who could not cram his gigantic feet into the cool roller skates, so we had to buy him those crappy old-fashioned kind that you had to buckle up over your own shoes, and anyway long story short, he wipes out and craps his pants. Of course, all of us love a good poop-your-pants story, right? It's a classic, I believe, the great leveler. We all pooped our pants when we were babies, and then accidentally here and there throughout our lives, and of course every single one of us is gonna shit ourselves again at some point on our way out of this world, unless it happens very quickly and we never see it coming, so in this way pooping yourself is one of those things that makes us all human, together. Needless to say this story goes over well with the kids, and I achieve my primary objective, which is to get them all to identify somehow with my clumsy and unlucky little cousin, to invest in him somehow, to care about him, to sympathize. Hopefully we laugh together. Then I sit back and wait for the question, which almost always comes. Almost every show some kid puts up their hand and asks me where is Christopher now? Where is Christopher now? I tell them that I know what they want me to tell them. I tell them I really wish I could tell them what I know they want to hear. I say how much I wish I could tell them that my little clumsy cousin Christopher grew into his gigantic feet and eventually became a tall and handsome man, who would one day marry a tall handsome woman and they had two tall handsome children and now he lives happily in a suburb somewhere and works at his successful and fulfilling job in the IT industry, and that they have a little brown dog and a white picket fence, but I can't. I can't tell them that because Christopher died on Christmas Eve in his twenty-first year of a self-inflicted gunshot wound to the head, and that is why I come into high schools. That I want them all to know that someone cares about them, and that they have a right to access their public education without threat of physical, emotional, or spiritual violence. Then we talk about bullying, and what we can all do to work towards building a safe and respectful learning environment for each and every one of them. Recently I was in St. John's, Newfoundland for a storytelling festival. This festival has historically asked all of their performers do a few school gigs while they are in town. But a couple of weeks ago the festival director called me up to inform me that six St. John's schools had turned down my show, even though the festival was going to pay my fee and it would have cost these schools nothing. She was a little embarrassed about the whole situation. Said this had never happened before. Said the principals were concerned that I might upset certain parents, that perhaps I was... not appropriate, somehow, for a high school environment. So. I ended up doing only two school shows in St. John's, the only two schools that would have me, which were the Catholic school, ironically, and an alternative school for kids who had dropped out of the public school system altogether, many of whom had been bullied right out of an education, and/or battled learning disabilities or other challenges. Both shows were amazing, full of good discussions and intelligent questions. It was a great way to spend a Thursday. But that very same Thursday night in St. John's, the unthinkable happened. One of the students from a school that had turned down my anti-bullying show took his own life. I don't know him, never had the chance to meet him. I don't know if he was gay, or even if he was bullied, and now I will never know. But obviously something was going on for him. There is no way to know if a one-hour storytelling show and discussion might have changed this terrible outcome for this boy, and his family, and his friends and fellow students, who will all carry his death now for the rest of their lives. How do I know this? Because I carry my cousin's, it is right here with me now. I don't know that my show would have changed anything. I don't know that. But what really haunts me is that I don't know that it wouldn't have helped him, either. I send my compassion and love out to his family and classmates. What will it take for school administrations to realize that providing a safe school environment for all is more important than catering to the bigotries of the few? I want to share part of an article that my friend Matt Pearson, an Ottawa writer and journalist, published, called "The Arithmetic of Shame." Matthew writes: "You may have read or seen on the news recently that a teenage boy in Ottawa took his own life after struggling for some time with depression and the challenges of being the only openly gay student at his suburban high school. I covered the tragic story as a reporter for a daily newspaper and have remained troubled by it for days afterward. "I did not know this child. But what I do know, at least in part, was the depth of despair he too often felt. It mirrored what I felt more than 15 years ago as a confused and pimply teen growing up in Woodstock, Ontario. "At my Catholic elementary school, I was called names on the playground years before understanding the full and hurtful meaning of them. I was made to feel different—and not in a good way—because I preferred drama, hung out mostly with girls and didn't like rough-and-tumble competitive sports. I soon became isolated. I developed a deep sense that even if I didn't quite know what those words meant, it must have been something pretty awful, judging by the way some boys I had known for years spit them at me. "I started bringing candy and bubble gum to school to give out freely on the playground at recess. Later, I became one of those chauffeur teens, always glad to give someone a ride somewhere, even if it was way out of my way. "This is the arithmetic of shame. The subconscious calculations I made in hopes people could find in me enough good things to compensate for the one unspeakably ugly part. "I switched to the public school system for high school and hoped my troubles were behind me, as most of my classmates continued on to the Catholic high school. "But it only got worse. I know now, years later, that high school sucks for just about everyone, but back then, I thought it was my own private hell. The verbal harassment was almost unbearable and came from people—often boys, but sometimes girls, too—who I'd never even seen before. How could they know something about me that I was only beginning to understand? "I was never physically beaten up, which is a good thing because far too many young people are victims of assault. But while it may seem masochistic, I often wished I was because then perhaps my teachers and fellow students would see for themselves the black eyes and bruises of hate. "There were hallways, nooks and crannies in that school where I never dared to go, especially on my own. And this was a school where my father—a kind man who has loved and supported me unconditionally since the day I came out a dozen years ago—was principal. I know it still pains him that he could not protect me."
Last year I had a gig in a Vancouver area high school. It was for the Dare to Stand Out conference, a gathering of LGBTQ students and their queer or ally teachers. Even though I was in a high school for an explicitly homo positive event, I still had the familiar heart-pound-mouth-dry-watch-my-back feeling descend upon me as I entered through the double glass doors and past the school office, and followed the rainbow signs that led me into the gymnasium. I was an adult, and I was about to speak to a bunch of queer kids and teachers. I had officially been out of school for longer than I was ever in school, but still, my body's memory took over, and took me back. Back to my own sixteen-year-old self. How do I know that I still need to work to make schools safer for all kids? Because I am still afraid of entering a high school, to this day, even now, even for something like this. My nerves quickly disappeared when I walked into the gym and saw the sixteen-year-old pretty boy with the Mohawk and the eyeliner who was setting up the mikes. Not to mention the awesome kick-ass young woman behind the soundboard, whose nametag read Darth Vader, and her sidekick, dubbed Stormtrooper, of course. Our next generation. I love them all, just on principle, and feel fiercely, almost irrationally protective of them. I want everything to be so much different for them than it was for us. I want them to be able to be unapologetically out and safe in their schools, and I want them to feel nothing but memories of joy and triumph should they ever return to a high school for any reason twenty some years from now. I know, what a dreamer, right? But why not? Why not imagine building a safe, respectful environment for all kids to be educated in right now? Why expect anything less, and why settle? Because we had to? That is simply not good enough. The fact that so many of us, queer or fat or nerdy or smart or slow or brown or from somewhere that is not here, still can't imagine school without the accompanying torment or hassle or trauma is a sign to me of just how much work we still need to do in our schools for all kids, not just the queer ones. This is the gist of what I said to all those young and beautifully out-already faces that day: 1. Always remember that working to make your school safer for queer students, or bisexual students, or gender non-conforming students is not a selfish act. Creating a safe school for yourself will only lead to a safer school for everyone, and everyone deserves a safe place to learn in. Not feeling safe at school can seriously affect your ability to access your own education, which can impact your life for the rest of your life. When you work to make your school better for you, you are doing your school, and everyone in it, and everyone who will ever be in it in the future, a gigantic favour. Never forget that. 2. You deserve so much more than to just be tolerated. You deserve to be loved for exactly who and what you are right now. This is, of course, a double-edged sword. This also means you must return the favour. Learn about racism and sexism and ableism too. You unfortunately are probably already well aware of how much homophobia can hurt, inside and out. Learning more about how different kinds of oppression work and where they intersect will help you build better bridges with others and create a safe and respectful school culture for everyone. Bullies are almost always outnumbered by the bullied. We just need to organize. 3. Remember that not everyone is able to come out to everyone all of the time. Some of us cannot come out to our parents yet, or our employers, or our teammates, or even our friends. It is okay to know who you are and keep it private if your own safety requires it. This does not mean you are any less queer or radical or cool than the guy with the purple hair and the rainbow stockings. It just means that he has different circumstances than you do. 4. It does get better. Especially when you make it better. There are lots of us out there who care a whole lot about you, whether it feels like it sometimes or not. I am one of them, and I will never stop coming into high schools to meet kids just like you, until I stop feeling scared every freaking time I walk through those front doors, I will keep working to make all schools safer for all of us. I promise you that. And in the meantime, when I get home I will watch It Gets Better videos, not because they are any kind of a real solution, but because they make me, a comfortably out-for-almost-twenty-five-years adult, feel a whole lot better, so I can get up the next day and get to work, actually making it better. I really love the It Gets Better series—I know we need these stories, that they can be accessed from any public computer, and that kids need them. But now I want to hear from some from straight kids, adults, teachers—vowing to grow up, step up and make it better. I want to hear stories about straight kids who have moved from fear to humanity and stood up to become allies, I want adult former haters to tell conversion stories. I want teachers who are finally so ashamed of pretending this isn't a literacy issue to challenge other teachers by making their classrooms a place where ignorance and fear are met with information and compassion. I am sick of moving people to tears with stories of casualties from the warfare we let our children wage on each other. I am sick of young dead boys becoming icons of public compassion, and inspiring Rick Mercer rants we can share with each other on Facebook, while at the same time we continue to allow our principals and school administrators to cater to the conservative and religious right and pretend that our kids don't all pay the price for their apathy and cowardice. Bullies grow up—their behaviour gets modified and sometimes their language gets slicked over with education—and they become the political, financial, and social arbiters of life as we know it. I bet you any money that Prime Minister Stephen Harper was a bully in school, and don't we all wish now that someone had nipped him in the bud before it was too late for Canada. It is time for us to write ourselves some new stories, people. So, let me tell you one. Last year I was invited to a high school on the west side of Vancouver. The rich part. It was an arts-enhanced school, and I had been asked to come in specifically because one of the students there was transitioning, in grade nine, and they were throwing him a birthday party. Not because it was his actual birthday, but because he was being reborn. The school threw him a party, and bought him a cake. They informed the entire student body of his new name, his pronoun preference, and that he would now be using the boy's washroom. Amazing, right? I cried throughout most of that day. Hope, relief, and redemption, palpable, caught in my throat, pounding in my heart. Just because I got to see one good story, for a change. But what a change, indeed. What did the school spend to make that kid's life so much better? The cost of one birthday cake, that is what they spent. And what did they save? Maybe his life, or maybe the lives of countless other kids who took heart and hope that maybe school and life doesn't suck as hard as it did yesterday. Maybe school and life were worth sticking around for after all. Ask me now, how important it is for queer teachers and school staff to come out of the closet? How important it is for queer athletes and rock stars and radio hosts and storytellers? How much did you need a role model when you were a kid? And did you really, truly have one? A couple of weeks ago a young butch friend of mine asked me if I would help her out with her art school homework. She said she was doing a photo project, taking pictures of older butches. You know, like, documenting her elders. Sure, I thought to myself, I know several older butches who I could hook her up with. A couple of them have moved to the sunshine coast like they do, but I could certainly track them down, no problem. It slowly dawned on me that she was referring to me. I was the older butch she wanted to document. At first this realization made me laugh, and then it made my right knee ache like it does. I am forty-three. She is twenty-one. I can't help but do the math. I had been out of the closet for three years already by the time she was born. I was navigating my way through the gender binary blues when she was learning to do up the Velcro straps on her first-day-of-school shoes. She has probably never dialed a rotary phone. More and more at my shows, young butches and barely whiskered trans guys have been coming up and telling me that my books and stories helped them get through high school, or even junior high. They thank me for being a role model. This makes me feel simultaneously honoured and terrified. It makes my heart sing to know that they had what I didn't even know I needed when I was a kid: someone they could imagine growing up to be like. It makes my heart pound to know that this means I now have to somehow be worthy of this kind of respect. How can I possibly be a role model, when I feel like I am just now starting to fit into my own skin? When I am still stretching and bending the space around me to make room for myself? How could I possibly give advice away when I just got my hands on it? I find it is way easier to imagine whispering any wisdom I may have gleaned from the last four decades into the ear of a younger me. If I could magically tell my younger self something I know now that I wish I had known then, what would that be? First of all, I would tell myself not to be too proud to ask for advice. Remember, you don't have to take advice just because someone has given it. Of course, my twenty-one-year-old self may not have taken the time to listen to present day me, but I will continue, regardless. Dear younger self: floss your teeth. It turns out you will eventually be a working artist, just like you always dreamed. A man named Stephen Harper will one day rule this land, and he will care nothing for artists, or queers, or even health care. You need teeth, and you alone will be financially responsible for them. Floss is cheaper than even your commie pinko east end lesbian dentist will be. Quit smoking. Please see above. I am not going to say this again. Okay, I am. Quit smoking right now. Your mother is worried that no one will like you, or hire you, or even love you, if you look "like that." She is wrong. This next bit is really important: she does not mean to intentionally do you harm, or cause you to fear who and what you truly are. She worries because she fears what the world might do to you, and because she doesn't know any successful tattooed butch storytellers with biceps and a brush cut. Yet. But one day she will, and she is going to love the hell out of future you. Trust me on this one. Do not cave into the pressure from mainstream society to fit in. You do not, and will not ever fit in. One day you will realize you don't even want to anymore, and that your difference is inherently tied to your beauty, and your bravery, and your giant, mystical, invisible brass balls. You will love these balls, and they will swing majestically between your ears, inside the head you will hold up proudly. Do not cave into the pressure from the queer community to fit in, either. Make your own decisions, and trust your own heart. Being butch is not just a bus stop on the highway to transitioning. You will learn to love your butch self. If you do ever decide to go on testosterone, build yourself into a good man. The last thing the world needs is another misogynist prick. Be the man your father accidentally taught you to be, even if it was only because you didn't have a brother to help him out in the shop. Make and keep long-term friendships. You will need them, and they will need you. This is one of the most important things you will ever do in your entire life. Whenever possible, be polite. In the long run, your good manners will serve you better than even your most righteous rage. Find a tailor, and be good to them. Get your pants hemmed properly, and learn what it feels like to have your clothes really fit your body. People come in all shapes, clothes do not. This is a wrong that can be easily righted. The world is going to try to squeeze you into many things that do not fit you, but your clothes need not be one of them. Seek out a mentor. Listen to what they have to say, and then follow your own path. Keep a journal, because one day, someone is going to look up to you, and even ask you for advice, and you are going to wish you had taken better notes. So go. Find yourself a mentor, and be a role model. Be a leader. Be the change we need to see. Don't wait for it to get better, make it better. Write us some better stories. Because somewhere, there is a kid out there who really needs your strength, and your courage. Someone out there needs you to be every bit of your brave, beautiful, fabulous, talented self.
# Letter fr Grammar Wizard
From: Dark Princess
To: Ivan Coyote
Hello you probly get a lot of emails, so ill try to stay a bit to the point i guess... btw im the girl (Jamie) from the alternative school... kinky switch omni sexual, with ptsd,panic attacks, bullyed from grade 4-12lvl5ish lol didnt really show up after sertian point., etc,.... anywho...sence i was too nervous and over exausted to really speak during your time here on my oppyions on what i think should an dshould not be done about bullying I figured id email you. for one in the murphy centre there is a zero tolerence on bullying and i do personaly find it a safe conforting enviorment over all in terms of free expression etc. How ever.... i do believe in regular school systems they should be taught as a course senstivity training, basic psychology( aka how your actions effect people), how to comunicate your problems, and that it is truely ok to talk to a councolar about your issues.I do not believe everysingle bully is the way they are cause of issues i do believe that some sensitvity training , knowlage, and a good set of communicational skills might help someone who was prone to being a bully... help in terms of lessining the way they do it etc... With that said i do not think it will work for all but some it might... sence that would be a big change a easyer to pass off way to explain it would be not only to show the person how it would effect someone , tell stories like you do, do, but mabye teach people that no one is allone, not the one doing the bullying nore the one being bullyed. As easy as it would be for me to hold hate toward thoes who did bully me the only feelings i allow my self is a will to try to over come everything... This is not perosnaly easy for me i do not think Growing a Thick skin is something people should be Fourced into doing at all i think it is something which should happen with maturity instead.I was bullyed physicaly, verbaly, mostly verbaly i was also psychologialy abused interms of the face my friends would be bullied till they told my secrets and stoped talkling to me anywho back to the view enough personal stories less you want to hear about it all if so feel free to ask i hav no problem telling you . I just find it a bit easyer for me if it alk about it. Back to ideas on how to stop it.I think understanding the person bullied( do not like to refer to them as a victum even tho some are i personaly do not like to view my self as one only because it would mean i had no control and in my eyes quitting school/ending it in someway was some fourm of control.... )understanding them by seeing there prespective, what might be wrong thats makeing kids want to hurt them, if you can help sooth them in anyway what could you do, better councleing and tolerance /senstivity training for school staff members and even Students !!!!!!. Understanding the bully what makes someone bully , what kinds of ways can you help prevent bullies, is there anything you can do to help make a bully less likely to hurt anyone, school punishments like fourced tolerance class work, or senstivity classess, or even videos with tests on how to behave around people im sure after ENOUGH times doing it they will learn or get sick of it , of course also sending them to a guidance councolar before the principal might help as well. Sorry this might be a bit overwalminin in terms of condenced info please email me back with insigh t questions etc.. or even conversation etc......
# Five: Folks I Felt It Necessary to School in Some Way or Another, With Varying Degrees of Success
# Judging a Book
THERE'S AN OLD CLICHÉ, SOMETHING ABOUT how you can choose your friends but you can't choose your family. I travel a lot, and I'd like to add a line, or at least a footnote, about how you also can't choose who you sit next to on an airplane ride, especially if you're flying in economy class.
I am a collector of stories, and a connoisseur of characters, so for the most part I love the random way that travelling strangers enter and exit each other's lives. I relish the chance to spend a few hours listening to the life story of a little old lady who usually only talks to her cat or the postman, or the girl that her family hired to come and clean the house once a week, ever since her daughter got too busy with the twins and the promotion. I notice how thin her skin seems, stretched like tracing paper over the blue veins that map the backs of her hands. How they shake just a little when she holds up a photo for me to see, how she spills a little bit of sugar when she pours it from the tiny packet and has to hold her paper cup with both hands. I savour all these details, and save them as souvenirs. Some people take pictures or buy postcards to remember where they have been. I collect people, and conversations.
One time I spent three hours waiting for the fog to lift in San Francisco with a guy who told me that he spends so much time on the road he never fully unpacks his suitcase, and that he has missed nine of his son's twelve birthday parties. He was a salesman who had single-handedly cornered the North American market for snow globes. His chest swelled proudly when he passed me his business card and announced that if I ever bought a quality snow globe anywhere on the continent, chances were it was one of his. Not the cheap ones, mind you, but the good kind, where the snow floats around for a while before it falls and collects on the bottom.
When he found out I was a writer, he told me he had spent the last ten years working on a novel, mostly at night in hotel rooms, and that when he finally retired, he was going to take a screenwriting class.
"Maybe I'm a writer too," he told me. "You never know. Stranger things have happened."
I told him I thought everyone had at least 1,000 great stories to tell, but we have been taught to believe that only heroes or serial killers or rich people or crime scene investigators live lives worth writing down. He rubbed his bald spot with one hand for a bit, like he was thinking about something he forgot to do, and took a deep breath.
That's when he blurted out that he hated his job, but the only thing he'd ever been better than everyone else at was selling snow globes, and that his wife hadn't touched him in three years, ever since he put on forty pounds after his back surgery, and he was pretty much convinced that she was banging his son's soccer coach and how the worst part was that he didn't even care anymore, but he didn't want to leave her because she would get the house, and he loved that house, and his dog, who had lived to be almost fifteen years old, was buried in the backyard right next to the apple tree, and what if his wife sold the house and bought a condo when the kids moved out so she wouldn't have to mow the lawn, and maybe a dead dog was a terrible reason to stay married to someone who won't look at you without a shirt on, but he was hardly ever home anyways, except for long weekends like this, and if the weather didn't get better he wouldn't make it home at all. Then he apologized and said he didn't know why he was telling me all this, that he hadn't even talked to his best friend about any of it, on account of how they worked for the same company, and getting too personal might put a strain on their business relationship. I hugged a perfect stranger that night because I knew his wife wouldn't, and I think of him now whenever I see snow that falls slowly.
Today I sat next to a man who immediately informed me that he was on his way to Europe to work with the Christian Embassy, spreading the good word of the Lord. Before the plane was off the ground, he asked me if I had a girlfriend. I took this line of inquiry to mean that he thought I was a clean-cut young man, and therefore possessed a soul worth saving. I told him the truth; I did have a girlfriend, and no, we were not married yet, and yes, we were indeed living together, and yes, I was aware that we were living in sin. I smiled inside at just how much sin he didn't realize we were actually living in, and pondered telling him I was not as nice, young, or male as he appeared to think I was. Then I realized how fun it was to listen to a fundamentalist Christian lecture me on how God wanted me to marry my girlfriend, how the family unit in this country was depending on me, and how not fun it might immediately become if he were to find out he was brushing thighs with a full-blown sodomite disguised as a harmless wayward Catholic boy in a crisp shirt and a tie. I knew there was as much chance of me changing his mind about anything as there was that he would ever lead me back to the path of righteousness, so I told him he was right, and that I was going to propose to my girlfriend as soon as I had enough money saved up to buy her a decent conflict-free diamond ring. He took this to mean that he had helped me see the light, and continued the Lord's work all the way to Toronto. When the plane finally landed, he shook my hand and told me that I seemed like a good person, and that if I were ever in Guelph, I should look up his son, who had strayed from God's path a little and had pierced his eyebrow and was pursuing an arts degree.
"I'd like him to meet some friends with ambition. People who realize that appearances matter. I pray that he grows up to be just like you."
"I hope God answers that prayer," I told him. "I really do."
# Take That
I HAD A FORTY-FIVE MINUTE LAYOVER IN OTTAWA, on my way to Halifax. I was halfway through my complimentary newspaper when I heard them arrive at the gate. Forty teenage girls and their thirty-something male chaperone were getting on the plane with me. The chaperone was one of those cool teachers, we all had one at some point, the kind who teaches gym or band and sports a ponytail and over-manicured facial hair. He's the kind of guy who buffs his nails and lets the kids call him by his first name, which is usually Steve or Rick or Darryl. Maybe he smokes a little pot on the weekends, too, and plays a little acoustic guitar. He wears designer jeans and tight t-shirts that show off his well-muscled forearms. The girls all harbour not-so-secret crushes on him, because, you know, he like totally understands them, plus he's handsome. The guys have more mixed emotions, a combination of wanting to be him and wanting to kill him, or at least one day beat him in an arm-wrestle. He calls everybody "buddy" or "kiddo."
Steve or Rick or Darryl clapped his hands together to get the girls' attention. "Okay, ladies, let's line up, and have your ID ready. Don't leave your garbage behind; let's make a positive impression here, okay? Make sure you have your buddy with you."
I buried my face in my newspaper. I've never been overly fond of teenage girls, especially in packs, even when I was attempting to be one. They're a mean, judgmental lot, I find, and they still intimidate me. They whisper, they gawk, and they snicker. It takes me back, I can't help it.
F.H. Collins High School in the early eighties was ruled by her highnesses Wendy, Tracy, Sandra, Jeanie, and Kerri-Anne. It was a time of big hair, small sweaters, and tight jeans. All five girls possessed all of these prerequisites. Their affections were highly sought after, and fleeting. They liked me once for half a day when they found out I was vaguely related by marriage to Jimmy Baker, my dad's brother's wife's little brother, because he was cute, and had his own car. But I soon fell from their favour over my inability to grow breasts or like Depeche Mode.
At least they just pretended I didn't exist after that. It could have been a lot worse. Ask "Pizza-Face'" Andrea Mullen or "Big-Fat" Alice Byers just how bad it could get.
Not much has changed since then. The jeans cover even less skin, and the hair is a lot smaller, but the Wendys, Tracys, Sandras, Jeanies, and Kerri-Annes still rule the school, and I was getting on a plane with forty of them.
Everything was cool until I had to get up and go pee. The girls were all sitting at the back of the plane. The beverage cart was parked in the aisle two or three rows from the bathrooms. I was going to have to wait in line and be scrutinized by forty teenage girls.
My early teenage girl trauma was later complicated and compounded by the fact that I am often mistaken by them for a young man of appropriate cruising age. From eighteen rows away, I must have looked cute enough to check out. She had long, straight, copper-coloured hair and perfect skin and teeth. I disliked her immediately, just on principle. She fixed her blue eyes on me and elbowed her friend in the ribs. I pretended to be fascinated by my thumbnail and hated myself for caring about what I knew was going to happen next. Ten rows away, all three girls held their heads together and giggled, still trying to catch my eye.
But five rows away, the redhead peeled her lashes back from her eyes and sat up straight. She dropped her magazine and gripped her armrests in horror. Her mouth gaped open. She stared shamelessly at me, and then leaned over and covered her face in her hands. Her girlfriends leaned in to see what was the matter. She whispered something to them and they plastered their mouths with their palms. The redhead made pretend gagging motions.
I was right beside them now, and could hear them.
"That is the grossest thing I've seen all year. Oh my God, what is it? Does it have boobs? You look. No, I'm not looking, I feel totally sick. You check, Colleen, you were the one who thought he was cute. Was not. Were too. Oh my God, I can't tell what it is."
My face and ears were on fire. Did they think I couldn't hear them? I calmly put my right hand on the seat back in front of them and leaned into their row. I placed one word in front of the other, in an orderly fashion.
"Why don't you just ask _it_ what it is? Maybe _it_ is a human being with ears, and feelings. Why don't you just ask _it_ what _it_ is? Maybe _it_ can talk, too, and maybe _it_ will tell you. Go ahead, ask _it_. Because _it_ is standing right here."
They just stared straight ahead, wordless. They pretended I wasn't there, like I didn't exist.
I splashed cold water on my face in the tiny bathroom. I thought about finding Steve or Rick or Darryl and telling him that his girls had failed to leave a positive impression here. Then I decided against it. I didn't feel like explaining myself, or receiving a forced, toe-kicking teenage apology.
The girls were still whispering mercilessly as I walked past them. They fell silent when they saw me. My hands were shaking. I hoped they couldn't see that.
I'd calmed myself down by the time the plane landed. I don't like standing up and waiting in the aisle while some guy way up in first class drags his laptop down from the overhead compartment and puts his jacket on in slow motion, all the while holding up the entire disembarking process, so I usually stay in my seat reading until almost everyone is off the plane.
I caught a flash of red hair in my periphery. I swear I didn't think about it. It wasn't planned. I sent no conscious signal to my leg to move, but just as Colleen passed my seat, my foot shot out and tripped her, all on its own. She fell perfectly, knocking over the two girls in front of her as well. The girl behind her tripped over the resulting tangle, and all four of them went down.
The blonde in the striped bell-bottoms leaped up first. "Jesus, Colleen, watch where you're going. I could have chipped a tooth. I just got my braces off."
They righted themselves and left the plane without looking back. I don't think even Colleen knew what I had done.
I sat in my aisle seat, shaking my head at myself. Good thing the two nice old ladies who had been seated next to me had already gotten off the plane, or they would have thought I had cruelly tripped an innocent sixteen-year-old girl with absolutely no provocation.
I tried to feel guilty. I tried to feel ashamed of my behaviour. I was an adult, I told myself, and I should have known better.
But I couldn't. I thought of Andrea Mullen, who is a lawyer now. I thought of Alice Byers, who overdosed on sleeping pills in her third year of university.
I smiled to myself. Take that, Wendy, Tracy, Sandra, Jeanie, and Kerri-Anne. Truth is, I never liked you guys anyways.
# Bully This
I RETURNED HOME FROM DOING my anti-bullying storytelling show in a couple of small towns just in time to open the newspaper and read about right-wing Christian radio hosts and rogue school board members targeting programs like Out in Schools, and "pro-family" organizations taking out full-page ads full of hate and fear-mongering in national publications. I had just performed for 1,500 kids in two days in four schools, and as always, I walked away feeling inspired and full of hope for the world. These kids are smarter and savvier than those on the right seem willing or able to give them credit for, and they are certainly wiser and worldlier than I remember being when I was fifteen. That's the funny thing: that the evangelicals are so convinced that their kids will grow up heterosexual as long as they never cross paths with a living, breathing homosexual. It is like they actually believe that if they can somehow just keep us homos out of schools, or at least keep us in the closet, and keep our lives and our language out of the curriculum, that all of their children will magically grow up to be straight. What they forget is that no matter what kind of self-hatred and misinformed poison they whisper into their kids' ears, an estimated ten percent of them will grow up to be some sort of queer, and that the real question is whether they will somehow find the strength to survive and thrive and live truthfully despite what they were taught to believe about themselves. So in some ways, every time I swallow the lump in my throat and step through those streaky glass doors at the front of every high school I enter, I am there for those kids the most. Because I know that despite how scary high school can be for some kids, for others high school is the only place they might have any hope of acceptance and support, because they are not going to find it at home. The thing is, I don't even say the word queer while I am there. I just tell stories. Stories about growing up with my cousins and little sister, stories about my Gran. Stories about Wendy, Tracy, Sandra, Jeanie, and Kerri-Anne, the mean girls back at my own high school. My show is designed to get the kids talking about bullies and teen suicide and how the way we treat each other impacts the kind of people we are, and the kind of adults we might become. I don't need to say the word queer, because it is not about being queer. It is about each and every one of them feeling safe enough to access their education, and about respecting difference. Because I remember who got picked on in school. The fat kids, the dumb kids, the slow kids, the fast kids, the poor kids, the boys who threw like girls, the kids who weren't white, the quiet kids, the religious kids. That's right, Christian right: your kids. The ones who weren't allowed to go on dates, go to dances, wear makeup or the right clothes, watch the right television shows, or listen to the right music. When I was in school, the most risqué show on TV was _Dallas_ and the dangerous band was Judas Priest; today, maybe it is more like _True Blood_ and Gaga, but the song remains the same. I got this email when I got home, from one of the teachers: "We have had a two-year leadership focus on inclusiveness and anti-bullying and your presentation supported this so beautifully... getting to that part of the audience that may not always be listening or be open to receiving a message. This week we had three different groups come up to our admin to report an incident where a vulnerable grade 10 boy was being harassed by older boys in the lounge. Our principal called all six of the boys up for a visit. They were banned from the lounge for a week and the public shaming was a lovely thing. Two of the boys called in were not harassing the boy, but they didn't say anything. They apologized for not speaking up when they knew they should have and could have stopped the ugly affair. Anyway, we think that your performance may have been fresh on our students' minds and something very good came out of a potentially very bad situation. So thank you again." I also want to share part of a letter I got from a student after a show I did in a high school last spring: "Heeeyyy. So you came to my school today. After I got in the car with my older brother and told him all about you, and he goes, Britney, you are one of those girls. I yelled at him and then gave him the silent treatment the whole way back, but when I wasn't talking to him I was thinking yah, I am one. I'm a Kerri-Anne or a Wendy or that volleyball team. But I don't want to be. So I just wanted to say thanks. Because even though I have all this respect for you, I don't always give that respect to other people, and I know I am not going to change this second cause it's been a part of me since I can remember, but I'm going to be conscious of it all the time now. I know you are making an impact I just wanted to be the extra email that helps motivate you to never quit." So, Britney, I promise you, I will never quit. And evangelicals, you might want to think again about stopping folks like me from doing this kind of work in schools. Because chances are pretty good that it might just be your kid who is going to need us to be there for them the most.
# Imagine a Pair of Boots
IMAGINE A PAIR OF BOOTS. A sturdy, well-made, kind of nondescript pair of boots. They are functional enough, but kind of plain. Imagine that you live in a country where every citizen is issued this one pair of boots at birth, and that there are no other footwear options permitted by law. If you grow out of or wear through the soles of these government-issued boots, you may trade them in for a new pair, always identical to your old ones. Imagine that everyone you know wears these very same boots without question or complaint.
Now imagine that your right foot is two sizes bigger than your left one. That no matter what you do, one boot will chafe and the other will slip, and both will cause blisters. When you mention your discomfort you are told that odd-sized pairs of boots are forbidden, because they cause confusion and excess paperwork. It is explained to you that this footwear system works perfectly for everyone else, and reminded that there are people in other countries who have no boots at all. You are beat up in grade three because none of the other kids have ever seen feet like yours. The teacher tells you that you should probably just learn to keep your boots on. Your parents blame each other. You end up wearing an extra sock on your small foot to compensate, and never go to swimming pools. Your feet sweat profusely in the summer and you always undress in the dark. You hate your feet but need them to walk and stand up on. You hate your boots even more. You dream of things that look like sandals and moccasins, but you have no words for them. You learn things will be easier for you if you just never talk about your feet. One time on the bus, you spot a guy with the exact same limp as you, but you pretend not to see him. He watches you limp off at your bus stop and then looks the other way. You can't stop thinking about the man with the limp for weeks. You are nineteen years old and until that day on the bus you thought you were the only person in the country who couldn't fit into their boots.
I have always felt this way about gender pronouns, that 'she' pinches a little and 'he' slips off me too easily. I'm often asked by well-intentioned people which pronoun I prefer, and I always say the same thing: that I don't really have a preference, that neither pronoun really fits, but thank you for asking, all the same. Then I tell them they can call it like they see it, or mix it up a little if they wish. Or, they can try to avoid using he or she altogether. I suggest this even though I am fully aware of the fact it is almost impossible to talk about anything other than yourself or inanimate objects without using a gender-specific pronoun. It is especially hard at gigs, when the poor host has to get up and introduce me to the audience. No matter which pronoun the host goes with, there is always someone cringing in the crowd, convinced that an uncomfortable mistake has just been made. I know it would be easier if I just picked a pronoun and stuck with it, but that would be a compromise made for the comfort of everyone else but me. A decision that would inevitably leave me with a blister, or even a nasty rash.
Perfect strangers have been asking me if I am a boy or a girl as far back as I can remember. Not all of them are polite about it. Some are just curious, others ask me like they have every right to know, as if my ambiguity is a personal insult to their otherwise completely understandable reality. Few of them seem to realize they have just interrupted my day to demand I give someone I don't know personal information they don't really need to sell me a movie ticket or a newspaper. I have learned the hard way to just answer the question politely, so they don't think I'm rude. In my braver days, when someone asked if I was a boy or a girl, I would say something flip and witty, like 'yes' or 'no' or 'makes you wonder, doesn't it?,' but I found that this type of tactic greatly increased the chances I would get the living shit kicked out of me, so I eventually knocked it off. Then I went through a phase where I would answer calmly, and then casually ask them something equally as personal, such as did they have chest hair or were they satisfied with the size of their penis or were those their real breasts, just so they would see how it felt, but this proved just as ineffective.
A couple of months ago, as I was smoking outside the Anza Club after a gig, this young guy marched up and interrupted the person I was talking with to ask me if I was a man or a woman. I told him I was a primarily estrogen-based organism, and then I asked him the exact same question. He took two steps back and dropped his jaw.
"I'm a man." He seemed visibly shaken by the thought of any other option.
"And were you just born male?" I continued, winking at my companion.
"Well, yeah, of course I was."
"How interesting." I lit another smoke.
"Hard to tell these days," my friend chimed in.
The guy walked off, looking confused and kind of vulnerable.
"He's gone home to grow a moustache," my buddy said, then laughed and shook his head.
I thought about it all later, how the guy's ego had crumpled right in front of us, just because a stranger had questioned his masculinity. How scared he was of not being a real man, how easy it had been to take him down. It dawned on me that if you've never had a blister, then you'll never have a callus, either. And if your soles are too soft, then you are fucked if you ever lose your boots.
# The Bathroom Chronicles
LATELY, I FIND MYSELF ON THE ROAD A LOT. Sleeping in beds unfamiliar with the shape of me, feeling along strange walls to find the light switch in the dark, waking up to wonder at a ceiling I've never seen before in the daylight of a different town. Wearing the same pair of pants for a week and running my fingers over a calling card in my pocket when I miss my girlfriend. Airports and a highway and little tiny soaps and MapQuest and gas stations. Always gas stations. Because no matter where you are, or how much time you have until you have to be somewhere else, you're going to need gas, and someone always has to pee.
For me, the best gas station bathroom scenario is the single stall version with the sturdy locking door with a sign on it that says men-slash-women and you don't have to ask for the key first. These are the bathrooms most conducive to a stress-free urination experience for me, for a number of reasons. First of all, you don't need to ask for the key. The key for the gas station bathroom is usually somewhat wet for some reason, which I find unsanitary and disturbing, and is invariably tied or chained to a filthy germ-harbouring item which is hard to pocket or lose, such as a piece of hockey stick, a giant spoon, or a tire iron. You have to ask for the key from the either bored or harried and always underpaid guy behind the counter, and if there are two keys, one for the men's and another for the women's, then the cashier has either no time if there's a line-up, or lots of time if things are slow, to decide for himself which key he should give you. Keep in mind that he is probably feeling unfulfilled about the fact that he is ten times more likely to be robbed at gunpoint than he is to get a raise anytime in the near future, and that deciding which washroom he thinks I should be using is the most arbitrary power he's been afforded by this job since he caught that twelve-year-old shoplifting condoms and decided not to call the cops because at least the kid was stealing responsibly.
So this is the guy who gets to decide where I get to pee. I have learned that asking for the key to a specific washroom will only increase the odds that he will notice that the washroom I wish to enter doesn't match the hair or voice or footwear of the person he sees in front of him. Maybe he couldn't care less which bathroom I use, maybe his favourite sister is a dyke. But maybe his religion tells him I am damned, maybe him and his buddies almost killed a guy once for wearing a pink shirt, just in case he was a queer, just for fun. Maybe he dreamt of kissing his best friend all the way through grade eight but never did, and he hates me because I remind him of how scared he is of his own insides. I cannot know his mind. I am in a strange town, and something about me doesn't fit. It is best if I let him decide, and don't draw attention, or alert anyone in the line-up behind me to his conundrum.
Maybe you think I'm just paranoid, that I'm a drama queen, or that I exaggerate to make a point. I would say good for you, that your gender or skin colour or economic status have allowed you to feel safe enough that you still think the rest of us are making this stuff up. You probably don't even realize how lucky you are to be able to not believe me when I tell you that every time I have to pee in a public bathroom, I also take a risk that someone will take issue with me being somewhere they believe to be the wrong room, depending on who they mistook me to be, based solely on that first quick glance.
I can pray for a wheelchair-accessible stall, or one of the ungendered kind with a baby-changing station in it, and then hope that no one is waiting there when I slip out, able-bodied and childless. I can cross my fingers that the ladies' room is empty, or bolt quietly for the closest empty stall if it is not. Unfortunately, women and children have many good reasons to fear what they think is a man in their washroom. I have learned to be more forgiving of their concern, and try not to take any hostility too personally. They only want the same thing I'm looking for: a safe place to pull down their pants and pee.
I can hold my nose and use the men's room, and if I'm lucky there will be a seat on the toilet and the guy who comes in to use the urinal will not be the type who hates slightly effeminate men, or the type who likes them a little too much. In men's rooms, I squat and pee quickly, simultaneously relieved and terrified when I am alone.
Over the years I have learned a few techniques, like not drinking pop in movie theatres and holding my pee for probably unhealthy lengths of time. I do my best to be polite and non-confrontational, even when confronted or questioned rudely. One of my favourite methods is to enter the women's room with a preferably ladylike companion who has been previously instructed to ask me if I have a tampon in my purse. I answer her in the most demure and feminine tone I can muster that I left my purse in the car, or that I'm down to my last pantyliner, and dash for the first open stall.
Just recently, I accidentally improvised the perfect line to deliver to the nice but confused lady that I often meet on my way out of the gas station bathroom. She was standing with her hand on the half-open door, looking first at me and then again at the sign that said "Women" on it. She was in her later sixties, and I felt bad that I had startled her, or maybe made her feel even for a moment that she was lost, or in the men's room, where she might not be safe. That I had scared an old woman with a full bladder. Again.
"It's okay," I smiled and said calmly. "It's just me."
# Dear Lady in the Women's Washroom
I CAN ONLY SURMISE FROM OUR RECENT INTERACTION that I startled you in the women's washroom at the mall today. I guess I don't look much like what you seem to think a female washroom user should. This is not the first time this has happened to me; in fact, this was not the first time this has happened to me this week. Forgive me if I was not as patient with you as you seemed to feel I should have been, but I would like to point out that your high pitched squeal startled me, and I needed to urinate very badly. Perhaps I was not as gracious as I could have been.
To ensure that the next time this happens to you, or me, things go more smoothly for everyone involved, I have jotted down a couple of notes for your reference.
Not everyone fits easily into one of the two options provided on your standard public washroom doors. In my world, gender is a spectrum, not a binary. Just because an individual does not present as what you feel a woman should look like, does not mean that they do not belong there. Public washrooms are just that: public. This means that you do not get to decide whom you share them with. I would like to remind you that everyone, regardless of their gender identity or presentation, needs to pee.
For some of us, public washrooms are stressful places. We generally avoid them whenever possible. Please, rest assured that if I have chosen to enter a public washroom in spite of my long and arduous history with them, that I have taken the time to carefully note which door I am about to walk into, and that I am confident I have chosen the lesser of two evils. I am, in fact, hyper aware of which bathroom I am in. It is not necessary for you to stare at me, pointedly refer to the graphic on the door, or discuss my decision loudly with your companions. Gawking, elbowing your friend, and repeatedly clearing your throat are also not helpful. Trust me, I will be in and out as quick as is humanly possible.
The next time this happens to you, I would like you to think twice before screaming. I would like you to imagine what it feels like to be me. Imagine being screeched at by a perfect stranger. Now imagine being screeched at when you really need to pee, or your tampon gave out twenty minutes ago. Sucks, doesn't it?
I want you to know that I understand wanting to feel safe from men while using the bathroom in a public place. This is, in fact, the primary reason I don't just use the other bathroom. That, and I have a very delicate sense of smell, and don't like returning filthy toilet seats to the down position.
I also would like you to know that trans and gender queer people suffer from many more bladder infections, urinary tract issues, and general pee-related health problems than the general population. I humbly ask you to consider why this might be the case.
I would also like you to know that I have had the great pleasure of spending time with a seven-year-old and an eight-year-old tomboy lately. Both young girls have experienced serious bullying at school and day camp over their gender presentation, especially in and around the question of gendered bathrooms. They have both come home from school in tears, and one of them even quit science camp because of it. Hearing that these two sweet, kind, amazing children have both already experienced "the bathroom problem" that I so often face myself not only broke my heart, it enraged me. I feel that this type of bullying has impeded their ability to access a public education, and impacted their desire to participate in valuable activities outside of school as well. I would like you to consider how this might affect their self-esteem, their grades, and their sense of self-worth. I remind you that they are just little kids. They are only in elementary school, and it has started already. Not such a little thing after all, is it?
I ask you to forgive me my impatience with you at the mall today. But how could I possibly not think of my two little friends, and feel anything but rage?
See, when you scream at me without thinking in the women's washroom, you are implicating yourself in a rigid, two-party gender system that tells others that it is okay to discriminate against people like me. Even little children who are like me. This is the very same attitude that results in queer youth suicides, and high school murderers being acquitted because the dead boy asked for it by wearing a skirt and makeup. It is this same attitude that turns its head when trans women are shot at by off duty police officers, and denied services at women's shelters. It is this kind of sentiment that says it is okay to deny us housing, or a job, or the right to adopt children or dance on a freaking reality television show. If you think I am making any of this up, then I encourage you to open up your newspaper and have another look.
I would like to remind you that this very same two-party gender system is enforced on me and others like me everyday, policed by people just like you. It starts very young, and sometimes is subtle, as small as a second look on the way out of a bathroom stall, but sometimes it is deafening, and painful, and violent, even murderous.
So, the next time you meet up with someone like me in the 'ladies room,' please think twice before screaming. I am not there by accident. In fact, I spent a lot more time looking at the sign on the door than you ever have.
# Truth Story
A COUPLE OF YEARS AGO I WAS BACKSTAGE AT A little music festival with my friend and guitar player, Richard. It was a breezy blue-skied July day, drawing quite a decent crowd for a small town. I pulled back the velvet curtain a crack to have a sneak peek at our audience. The entire first row was a beefy, bleeding, tattooed wall of biker-looking types. I swallowed and pulled the curtain back.
"Rico..." I whispered. "I think we're gonna have to change up our set a little. I think maybe we need to drop the Francis story and do the fishing story instead."
The Francis story was a tale about a little boy who liked to wear dresses. I thought maybe a less faggy, more fishing-oriented piece might go over a little better with this crowd.
Richard took a deep breath and gave me his I-am-about-to-tell-you-something-for-your-own-good look.
"First of all," he began, "the truck is parked right backstage. Second, artists are always allowed to talk about stuff that other people would get punched out for bringing up, remember? It's part of the deal."
I nodded, because this was true. Richard inhaled again, obviously not finished yet.
"But most important of all is, don't be a chicken, Coyote. Have some balls. What, you only going to tell that story to people who don't need to hear it?"
"You bastard." I smiled at him.
He shrugged. He knew me. Knew what to say to activate my stubborn streak.
The biggest and most bad-assed-looking of the bikers stood there in the front row, his veiny forearms crossed over his black t-shirt, for the first ten minutes of my set. He even laughed here and there, the skin around his eyes crinkling into well-worn crow's feet every time he smiled. I started to relax a little, and when I started the first couple of lines of the Francis story, Richard tipped his head in my direction in approval and played like an angel beside me.
Halfway through the story, I watched the gigantic man in the front row start to unpeel himself right in front of me. First he uncrossed his arms and let them fall to his sides. Then he bit his lower lip, and his handlebar moustache began to quiver a little. By the end, he was crying giant man-sized tears, unabashedly letting them roll down his dusty cheeks and disappear into his beard. He almost got me choked up too, just watching him. I was used to the drag queens losing it in the last couple of paragraphs of the Francis story, but this was something else altogether.
After, when Richard and I were loading gear into the back of his pick-up, I looked up and he was standing next to the table that held the cheese trays and the juice cooler, waiting to talk to me.
He rushed toward me and picked me right up off the ground in a cigar-scented hug. When he let me back down to the ground, he still held both of my hands in his baseball glove-sized hands, squeezing them until it almost hurt.
"I just had to thank you. Just had to tell you how much that story you told meant to me." He pulled me up close to him, and lowered his voice a couple of decibels. "My baby brother James died from AIDS, ten years ago tomorrow. My only brother. I loved him like crazy when we were kids, but my dad... well... let's just say the old man wasn't very flexible in his beliefs about certain things. He never understood Jamie, right from the get-go, and Christ, he was hard on the kid. Beat the living shit out of him one time when he caught him wearing my sister Donna's lipstick. Finally kicked him out when Jamie was fifteen. Nobody knew, back then, and by the time we did, it was too late. I never stuck up for him, never said a word, and to this day I have never forgiven myself for it. My baby brother, out on the street. How else was he going to get by? He was only a kid."
He looked me right in the eyes. By this time, both of us were crying.
"He was the sweetest fucking kid in the world. Your little friend in that story reminded me of James. There were five of us kids, but he was always my mom's favourite. The old man blamed her, said she babied him, but we all knew he was just born like that. That was just who he always was." He cleared his throat and wiped his eyes on the hair on the back of his hands. Looked a bit sheepish all of a sudden. "Anyways, just wanted to thank you for that. Good stuff."
Then he shook my hand and was gone. I've never forgotten him, and I imagine him standing behind me whenever I find myself scared of the next story I am about to tell, or afraid of the people I'm about to tell it to.
Last week I walked into a classroom at the college in Powell River, to tell stories to a bunch of Adult Education students. Working-class town, working-class guys all lined up in the back row. I found myself wishing with my whole heart I had not chosen to wear a paisley dress shirt that morning. What was I thinking?
Then I took a deep breath and told them a story. I started with the one about my dad. The one where I had almost given up wishing he would quit drinking, but then one day he did. Afterward, this guy with biceps the size of my thighs came up and thanked me. He had sleeve tattoos and could barely squeeze his muscles into his white Stanfield crewneck.
"I really liked the one about your dad," he explained. "I could totally relate to him. I used to be a welder, too."
# Six: Wisdom I Found, Learned, or Was Given
# My Dad Told Me
IT WAS A FRIDAY AFTERNOON, SUNNY AND LAZY. I ran into my friend Sir coming out of her apartment building, and we went for a coffee. She grabbed a table outside on the deck in the warm sun and I went inside for two Americanos.
I squeezed past the lady in the hippie dress and sat beside Sir and her cowboy hat, across from two biker types and their overfilled ashtray. Sir passed me a piece of the newspaper.
"Business section?" I asked her. "What am I gonna do with this? Check my stocks?" She passed me the New Homes, smiling. "Smart ass," I said. "At least give me the Lower Mainland bit. Don't make me roll up Fashion and pummel you with it."
She passes me the front page. A true friend, indeed.
"It's not the same as inside," the bigger of the two bikers laments. "Inside there is a code, you know, a way of being that makes sense... then when you get out..."
"It's an adjustment," his buddy nods. "Took me over a year to be able to sleep past six a.m. Ate pork chops every Tuesday for a while, until I got used to Tuesday isn't pork chop night for the whole planet. You've only been out a coupla weeks. It gets better. When's your kid gonna be here?"
"Ten after."
The second guy stands, extends his hand. Slaps the other guy on the shoulder. They half-hug, awkward. "So I'll make a move, leave you to it. Take care, buddy. Same time, next week?"
My face is hidden behind pictures of Iraqi prisoners. I can't face the news; instead I am eavesdropping on a rare bonding moment between these two men. I sneak a peek at Sir. She is watching the second man disappear around the corner; his wallet is wearing through the denim of his right back pocket, the chain swinging, smokes, cell phone, and truck keys in hand. The sound of his boots on pavement fades with him. She smiles at me. We were both witness.
A tall, pimply boy gets off the bus and crosses the street. He squints into the sun, holds up a knuckly hand across his eyes. He jumps over the guardrail and slumps into the empty chair. He is all right angles and straight lines. His feet seem impossibly big in brand new white runners. One shin is road-rashed and picked.
The biker leans across the table to hug him, the kid moves to meet him and knocks over a half-empty bottle of apple juice. His father catches it before it hits the table.
"Sorry, Dad."
His dad smiles and surveys the boy. "You look great. I think you're finally taller than your father."
"By three-quarters of an inch." The boy raises his eyebrows and grins.
"Your mom?" Dad is staring at his fingernails.
"She's good. You staying at Uncle John's?"
"For a while. I'm looking at a place this weekend. There's a skatepark a block away. I'm getting a pull-out couch for you."
They talk like this for a while. I'm smoking and getting involved in the sorry state of the planet, enough so I'm almost not eavesdropping anymore, until I hear the man ask his kid if he's having any luck with the ladies.
The kid swallows, his oversize Adam's apple plunging in discomfort. He shakes his head. "There was that one chick from Kelowna, remember? She was staying at her Grandma's? That was a while ago..."
"That was last summer, little dude. School's almost out again."
"Yeah, well, I'm not like you. Girls don't like me too much, mostly. I don't have the magic touch like you."
"It's not a magic touch. You want to know my secret? My fail-proof method?"
The kid leans forward. Behind my newspaper, I find I have leaned forward too. Sir has cocked her head, too. We are all waiting.
"Let me just grab myself another coffee, and I'll tell you all about it. Hold that thought. You still drinking iced tea?"
The kid nods. His dad gets up and goes inside. All three of us sit back, impatient. I watch him make his way back to our table. Average height. Over-size biceps. Bleeding tattoos. Not an ugly man by any stretch, but, as my aunts would say, nothing to write home about. He resumes his seat, lights an Export 'A', and stirs his coffee with a hand that makes the spoon look like it came from an Easy-Bake Oven Set.
"Where were we?"
"You were going to tell me how to meet chicks."
"Right. I'll tell you the one thing that women cannot resist in a man. The one thing that will always keep them coming back for more."
_For the love of Christ, spit it out already, man_ , I'm thinking. _We all need to know here._
"Listen to them."
The kid sits up straight with a sideways glance.
"I mean, really listen. Ask her about how her day went. Be interested. Don't just act that way, I mean really be interested in her. What she has to say, what she thinks about things."
"And then...?"
"That's it, son. That's all. You'd be surprised how many guys never figure that one out, but that's it. My big secret. Really listen to her, and then if you're lucky, when you come home from work, there will be a good woman there. Cooking for her every once in a while never hurt a guy in the long run, either."
The kid looks at his father. I look at Sir. Sir looks at the biker, then she meets my eyes. Again, we were both witness.
The biker drains his coffee. "C'mon, kiddo, I'll buy you a slice."
The two of them stand up and walk together down the block—noisy black Dayton boots and silent white runners, respectively.
Sir is shaking her head, smiling. "That was just about the sweetest thing I ever heard. Did you get all that?" she asks me.
I nod reverently.
For the first time, the lady in the hippie dress lowers her paperback and speaks up, her eyes moist and bright blue. "Now if only someone would have told my husband that, I might still own half of that cabin on Salt Spring Island."
# Spare Change
CORNER OF PENDER AND ABBOTT, JUST BEFORE MIDNIGHT. Whose bright idea was it to build a multiplex theatre and high-end mall here? Remember when it was a parking lot? I think I liked it better as a pit filled with water. I knew a guy who was arrested once for canoeing in the flooded hole that gaped where this mall now is.
I light a smoke into cupped palms. Orange glows bright, then dimmer.
Streetlights leak shifting spilled paint reflections off shining sidewalks and pavement. It is chilly tonight, like this evening belongs in a whole different month than the rest of this week.
I smoke with one hand and run fingertips over the ridged edges of quarters in my pocket with the other.
There's a woman, she's just rounded the corner off Carrall onto Pender Street, she's walking towards me. Her dress has two straps; one has fallen to her elbow and remains there, the other clings to a prominent collarbone. I watch her only because there is nobody else on the street to look at. She shuffles along the sidewalk, fists blossoming into five narrow fingers and then closing again. Repeat. Eyes down, back and forth, she searches the sidewalk and gutter. She scoops up a flat cigarette butt and places it into the shapeless front pocket of her dress. A small baggie is picked up, opened, sniffed, licked, and dropped again. She runs a yellow tongue over peeling lips, passes a sleeveless wrist under her nose. Repeat.
I look down as she starts to get close to me. I can hear the sound of her flip-flops sucking and slapping against the wet pavement. The sound stops in front of me. I don't look up. Both hands are in my pockets. My half-smoked cigarette is crushed and soggy, an inch away from the toe of my boot. _What a waste_ , I think. Too late to fix it.
"Spare some change, young fella?" Her voice is deeper than her small frame seems capable of.
I shake my head.
She lifts one lip a little, in my direction. "I know you've got change in your pockets. I can hear it. Heard it all the way up the street."
"You asked me if I could spare some change, not if I had any."
She raises her eyebrows. They have been plucked and then painted back on, but she raises them nonetheless. "We got a wise guy, huh?" She flips then flops back two steps and surveys me closer. "You go to college? Because that, my friend, is lawyer talk."
I shake my head. "I'm a writer. I tell stories."
She snorts. "Same diff. Makin' shit up. Twisting the facts so they end up on your side. I'll ask you again, counsellor. Can I have some of the change I can hear in your pocket?"
"It's not change. It's my car keys." I jingle them for evidence. _Exhibit A._
"Other pocket. Nice try. What, are you afraid I'll go spend your hard-earned money on drugs?"
I half-shrug, half-nod. "What if I get you something to eat?" I motion over my shoulder to the McDonald's behind me, which is getting ready to close up.
She snorts again. "That garbage? Now, that stuff will kill you."
We both laugh. I pull my other hand out of my pocket. Two loonies, a twoonie, three quarters. I hand it over. A nicotine-stained hand shoots out and collects. The change disappears before I can squeeze out a second thought. She doesn't thank me.
"You're welcome," I say.
"What? You want me to thank you now? I took your money to make you feel better about having more of it than me. I just did you a favour, if you think about it. Don't you feel like a better person now, helping out an old woman? I'm the mother of four children. I have three grandchildren. I'm almost sixty-five years old."
"You don't look a day over eighty," I quip.
"Why, thank you."
We laugh again, she coughs.
"Where are your kids, then?"
"My kids? Where are my kids? You mean why don't my kids swoop down and rescue their poor old mother from the mean streets of the Downtown Eastside?"
"Well, yeah I guess that's pretty much what I mean."
"And argue over whose turn it is to keep me in their basement suite? All the free cable I can watch? I tried that. There's one catch. There's always the one catch."
"What's that?"
"I'm never allowed to bring my heroin."
I nod, because there seems to be nothing to say.
"Shit happens, kiddo. Sometimes life gets in the way of all your plans. I'm too old to live under someone else's roof, someone else's laws. Had enough of that when I was married to the bastard, may he rest in peace." She makes the sign of the cross in the air with her right hand and falls silent for a bit.
I nod again and reach into my pocket for my smokes. I offer her one, light both.
She inhales deeply, stares at the red end of her cigarette. "It's the simple things. Tell you what—how 'bout you spare me a couple more of these for later?"
I look down into my pack. There are two left.
"I'd offer to buy them from you, but you'd probably just go spend the money on more cigarettes." She smiles, raises one eyebrow, and then winks at me.
I hand her the rest of the package. Up close, she smells like rose water.
"There now." The pack disappears. She pats my forearm. "Doesn't that feel better?"
# My Kind of Guy
I AM GOOD AT FINDING MY KIND OF place for breakfast. Especially in small towns. This place had all the right elements: it was embedded in the middle of a mini mall, in between a second-hand furniture store and a laundromat. Lots of new pick-ups parked outside. All you can eat Chinese food buffet on Sunday nights. All-day breakfast for five bucks. Neon open sign flashing in the window. Vinyl booths and chrome-edged tables that have been there since the fifties. I pulled up a stool at the counter, and the owner passed me a newspaper and slopped coffee into my cup without asking.
The old guy sat down right next to me a minute or so later, I had seen him and his hand-carved cane coming up the sidewalk when I was parking. GWG jeans, a white Stanfield V-neck t-shirt under a faded red and blue plaid jacket, work boots with stainless steel starting to show at the toes where the leather was worn through. Clean-shaven. Export 'A' cigarette pack peeking out of his breast pocket. I know this kind of man. He has worked hard every day of his life. Paid his bills. Buried his wife. He keeps his garage spotless, draws outlines of hammers in black felt pen on the pegboard above his workbench, repairs the lawnmower of the single lady next door, even though he doesn't like her noisy kids. My father will be this kind of man one day, sooner than I would like to admit.
The owner smiled hello at the old guy. "Soup of the day and pie with ice cream after?"
The old-timer nodded, and then spun his stool around to address the two older ladies tucked into the first booth by the door. "Bea. Helen. Enjoying the sunshine?"
They smiled, exchanged niceties, and then he turned back to me, squinting at the headlines in the open newspaper in front of me. "No good news in there, I read it this morning."
We get to talking. He asks me what I am doing in town, as it is painfully obvious to all of us that I am not from there. I tell him I am a writer, in town to teach some creative writing classes at the high school.
"Ah, an educated man then?" He narrows his eyes at me, and then smiles, as if to let me know he will not hold this against me, even though he should.
I shrug. We move on and talk about other things. As far as I can tell, he continues to think I am a young man. I can tell by his comfortable body language, how he slaps me in the upper arm with the back of a gnarled hand when I crack a joke, the kinds of questions he asks me. The details about his own life he reveals.
Some people would say that I am being dishonest, that I am lying, to not stop him mid-sentence and inform him, even though he has not asked me, that according to what he has been taught to believe about these things, I am female. The people who believe that I am being deceitful have never lived in a skin like mine. I answer his questions with the truth. I mind my pronouns, sure, but I do not lie. Ever. Why? Because I like this old man, and so far, he likes me. Even if I am an educated man.
He tells me that his wife has been dead for ten years. That he is about to turn eighty-one years old. That he hates golf, and doesn't watch hockey. I ask him how many grandchildren he has. He has to think for a minute, moving his fingers in front of his face to count them. Ten he says. All of them turned out pretty okay, except for the one grandson, the druggie, who is sponging off his only daughter, can't keep a job.
I ask him what kind of drugs his grandson is on, and talk a little about my friend, the one I haven't seen in years, and her battles with the meth.
"Does she look hard now?" he asks me, and I think about this for a minute. "You know, older than her years? The drugs, they hit the ladies in the face harder than they do the fellas." He shakes his head, sadly. "Can make it hard to come back from." He holds up one finger, to make a point. "The hard stuff, I'm talking about here. Not the pot. I'll even take a bit of pot myself, now and then, for the arthritis, you know," he winks at me, "but I don't seem to get the same kick off the stuff I used to get. Maybe I'm toking it all wrong, who knows? Anyway, point is, I always stayed off the hard stuff, and now here I am, outliving everyone."
His pie and ice cream comes, and his coffee cup is refilled. We are both quiet for a minute while he eats.
"Don't get me wrong." He clears his throat, pushes his plate away. "I'm no angel. I like my beer, for one thing. But if I was to give you any kind of decent advice, here is what I would say: stay off the hard stuff. By that I mean the hard liquor, the hard drugs, and especially the hard women." He laughs at his own joke then, slaps me on the back with a leathery paw.
I tell him it was great meeting him, and we shake hands. I am thankful for the weightlifting, and the calluses it gives me. As I push the glass door open to leave, he picks up his coffee cup and slides into the booth with his two lady friends. "Just having a little chat with a young fella from the city," he explains. "A writer, he tells me. Just telling him about my secret to sticking around long enough to get to be an old bastard like me."
# I Will
WE PICK UP THE RINGS NEXT WEEK. We drop off the cheque for the florist tomorrow. My custom suit is hanging in my closet, and her dress is nearly finished. It is really happening. We are getting married.
I have learned a lot about what other people think about marriage over the last few months. Next to birth and death, I think it is one of the most ritualized things we do as humans, and people have strong feelings about it. They have ideas. I quickly learned that whenever one of my friends confessed that they were surprised I was getting married, it was because they thought my marriage would mirror what their idea of marriage looked like. Which it often does not. My sweetheart and I have worked really hard to build the kind of relationship that we could live happily in, and this rarely involved tracing the blueprints of others.
This does not mean I am not open to hearing advice about the topic. In fact, last week I called around the family, as I do, and asked them for any words of marital wisdom. My grandmother is ninety-two, and she had a miserable marriage, followed by a passionate love affair, so I was interested in what she had to say, having lived through the extremes. She told me to "foster the ability to really talk to each other. You don't want to know all of his secrets, but honour the ones that he does tell you. And respect each other. Respect is almost a bigger word for love."
Respect turned out to factor big in my family. My Uncle Rob told me to "make sure to marry your best friend. Respect her. You can love all kinds of stuff, you can love ice cream, you can love your new shoes, love is the most misused word in the world. When you respect something, you take care of it. Respect her, and take care of her. Be her best friend. And remember, everybody fucks up. Especially you. You come from a bloodline that is prone to selfishness and narcissism, so keep that in mind. Everybody screws up, but it is probably mostly going to be you."
My Aunt Cathy took the phone away from him to add: "Learn to nod your head when they talk about the boat or the motorhome or whatever. Be kind to each other, don't fight over stupid things. When Rob and I first got married he told me he would make all the big decisions and I would make all the little ones. So far we haven't had any big decisions."
Then she passed him back the phone. I asked Rob how long they had been married for. "Near forty years if you count the time we were living in sin, and the time we've been married."
That would make them both experts, in my book.
My parents were married for twenty-six years, and my mom has been with her beau now for eight. She told me to "be honest with your feelings but always kind delivering the message. And you have to have fun. Laugh together a lot. And let him win once in a while."
Her partner Chuck chimed in from the living room, in the background: "Always make sure to have the last word. They should be 'Yes dear.'"
They were both still laughing about this when they hung up the phone.
My cousin Dan and his wife Sarah have been married for thirteen years. I was interested in their opinions, being from the same generation, and similar radical lefty feministy artistic bent that we come from. Their words echoed those of the previous generations, almost exactly. Communication, always talking about problems before they really become an issue.
"Actively pursuing interests together," Dan says. "Show an interest in her interests. If I hadn't started learning about roller derby, who knows where we'd be by now. I get her to chase me on my bike on her skates. It's a good time for everyone. Even bystanders."
I called my Dad's wife Pat on the phone. She was my Dad's childhood sweetheart, and then her family moved away and they lost touch for thirty years. She stayed in love with him that whole time, and they reconnected on his fiftieth birthday, when she tracked him down and called him. They have been married now for fourteen years.
Pat had a one-word answer for me. "Trust," she said, without hesitation. I asked her how she stayed in love with my Dad for three decades, without even talking to him or seeing him. "I don't know. I can't answer that. I do know that the first time I saw him after all those years, it all just came rushing back into me. Where I kept it all that time, I cannot say."
My father's advice was very practical. He told me to "stay busy. She will get tired of vacuuming around you eventually."
When pressed, not a single member of my family thought that a queer marriage would need a different set of values than their straight ones. "A relationship is a relationship is a relationship," my grandmother informed me. "Whether you've got a piece of paper from the government or not. It is your marriage. You get to make the rules."
I found it interesting that none of my family even brought up things like co-habitation, or enforced monogamy, or rigid gender roles, or "settling down," which were all assumptions made by the predominately queer friends who expressed their shock over our upcoming wedding.
I do lament that I don't have three generations of queer married couples on hand to look to for marriage advice, as it hasn't been legal for long enough to afford us that. Maybe thirty years from now, we'll have a lot more to say to each other about queer marriage than "I never thought it would happen to you."
### About the Author
#### Ivan E. Coyote
* <http://twitter.com/ivancoyote>
* <http://www.facebook.com/ivanecoyote>
* <http://www.ivanecoyote.com/>
Ivan E. Coyote is a writer, performer, and author of six previous books published by Arsenal Pulp Press, the most recent of which is _Missed Her_. Ivan is also the co-editor (with Zena Sharman) of the anthology _Persistence: All Ways Butch and Femme_. One of her stories appears in the anthology _It Gets Better_ , edited by Dan Savage and Terry Miller.
| {
"redpajama_set_name": "RedPajamaBook"
} | 8,223 |
Read some of our Frequently Asked Questions regarding coffee roasting and coffee in general below. For more information and tips view our coffee roasting blog, or contact us.
What Machine Should I Use to Roast at Home?
At Coffee Roasters, we have a selection of home coffee roasters to make roasting at home simple for everyone from beginners to professionals. Our home coffee roasters are a must have for any coffee lover.
It is said that coffee was first found and consumed in the 9th century (800 AD) by the Ethiopians. However, the earliest credible evidence of knowledge of the coffee tree appears in the middle of the fifteenth century, in Yemen.
What is "Fair Trade" Coffee?
Fair trade coffee, or equal exchange coffee, refers to when coffee is traded by bypassing the coffee trader, which therefore provides the coffee producer (and buyer), with higher profit margins.
What size roaster do I need?
It depends on the required production output and frequency you want to cover per week initially and into the future. Another factor to consider is the footprint of the equipment verse the space you plan to locate it in. An easy way to work the batch size is to consider the average hourly output. You can usually base this on approximately 3 batches per hour back to back roasting.
So for example with a 5kg batch roaster, you may be able to achieve 15kg of green coffee or app 12kg of roasted coffee (weight loss is app 15%-20% from green to roasted) per hour. You should also consider how often you want to be roasting to complete your weekly production schedule. It is important to consider the time required for other business practices such as administration, sales and marketing and allocate time accordingly.
Can I run a roaster off a gas bottle?
Yes but you must conform to Australian standards for the installation of the gas bottle and gas lines. This must be done by a certified gas fitter and you will need proper sign off and certification. There are certain requirements relating to the placement of the gas bottle that need to be considered. In Australia you are not allowed to operate from a gas bottle that is located inside a premise. Contact your local gas fitter or energy authority when considering LPG or Natural Gas installation.
How much coffee can I roast in a day?
You can roast as much as the batch capacity of the equipment can handle within a certain period. Refer to "What size roaster do I need?" in relation to working out hourly production capacities of equipment.
What is the difference between an electric and gas roaster?
The difference usually refers to the heating source for roasting the coffee. Either that being of gas heating or electric element heating. Both heating methods have advantages and disadvantages but in our experience, you can achieve comparable results with both methods.
Electric heating has its limitation beyond 3kg batch roasters due to the amount of electricity required to produce enough heat making them not cost effective.
Gas heating is more responsive than electric heating but electric heating has an advantage of producing more subtle radiant heat that has a non-aggressive drying effect to the coffee and can produce good results especially if combined with a perforated drum type.
There are different burner types for gas heating that can produce varying levels of conduction and convection in the roasting process. For a comprehensive understanding of roasting thermodynamics, our Coffee Roasting Fundamentals course covers this comprehensively.
Why do I need a sample roaster?
Sample roasters typically are designed for commercial QA analysis of coffee by commercial green coffee brokers and coffee roasting companies. They usually are designed to roast anywhere from 100-200g per batch which is typically the sample size that green coffee brokers will supply to coffee roasting companies for testing.
Trying to roast samples this small in larger commercial roasters is very difficult so it can be an advantage to have a sample roaster to assist in the initial and ongoing testing of green coffee. This also reduces the waste from the testing process whereas if this was to be done in the larger commercial roasters the amount required to roast effectively would be much larger.
What is a roast control system?
A roast control system is an electronic interface that allows the roaster to control the roasting process. This is designed to make the process of roasting coffee easier and more repeatable and usually involves software that maps the process on a computer.
Coffee Roasters Australia has their own in-house developed coffee roasting control platform called Coffee Sweet. This system is available as either a pure datalogging system or a full automation control system where the roaster can be controlled from a computer.
Do I need to test the moisture levels in my coffee?
This is the best practice as the moisture levels in coffee have a direct effect on the roasting process. Green coffee is quite porous and can lose moisture in dry environments as well as absorb moisture in humid environments. The fluctuation in moisture content of stored green coffee has been proven to have a negative effect on acidity levels.
Ideally, a roasting operation should be checking the moisture content in coffee to confirm bought coffee is within an acceptable level of 9-12%. Anything outside these parameters should be rejected. It is also necessary to record moisture levels of the coffee, as it is being stored to ascertain any issues with the storage environment, fluctuations will have an inverse effect on the quality of the coffee over time.
A moisture analyser is also a good quality took for a coffee roaster to use to check before roasting to consider whether the drying effect in the roaster will be faster or slower than the target recipe. The roaster can then make appropriate adjustment to load, gas and fan settings to account for this change. For example, less moisture means the coffee will dry faster than before and the inverse for more moisture.
Do I need an afterburner?
An afterburner is a very effective emission solution that is connected post the coffee roasters chaff cyclone and oxidises the smoke and odour from the resting process by processing at a temperature of usually 500-600 deg.
An emission solution for odour and smoke from the roasting process may be a requirement from your local council or Environmental Protection Agency (EPA). It is best to consult with your local council whether you will require an emission solution for your operation.
Coffee Roasters Australia is a leading manufacturer and supplier of afterburners in Australia and also overseas. We have supplied afterburners to a variety of coffee roasting brands and types and can custom make solutions to suit most applications.
What does a destoner do?
A coffee destoner is designed to separate foreign matter from roasted coffee. Foreign matter such as sticks, stones, metal and other unwanted materials are classified as a defect in coffee and usually introduced into the coffee at the harvesting and processing level. Some of this foreign matter can be detrimental to grinding equipment and can prove costly if grinder blades need to be replaced from damage caused by a stone or piece of metal.
Typically a destoner will sit at the end of the cooling tray of a coffee roaster and will work like a vacuum cleaner when dumping the cooled coffee by sucking up the roasted coffee at a rate slow enough, more dense and different sized objects will remain in a waste tray.
Coffee Roasters Australia is a manufacturer of high-quality coffee destoners with the flexibility to custom design for various applications. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,491 |
struct SplitData
{
GenTree* root; // root stmt of tree being processed
BasicBlock* block;
Rationalizer* thisPhase;
};
// return op that is the store equivalent of the given load opcode
genTreeOps storeForm(genTreeOps loadForm)
{
switch (loadForm)
{
case GT_LCL_VAR:
return GT_STORE_LCL_VAR;
case GT_LCL_FLD:
return GT_STORE_LCL_FLD;
case GT_REG_VAR:
noway_assert(!"reg vars only supported in classic backend\n");
unreached();
default:
noway_assert(!"not a data load opcode\n");
unreached();
}
}
// return op that is the addr equivalent of the given load opcode
genTreeOps addrForm(genTreeOps loadForm)
{
switch (loadForm)
{
case GT_LCL_VAR:
return GT_LCL_VAR_ADDR;
case GT_LCL_FLD:
return GT_LCL_FLD_ADDR;
default:
noway_assert(!"not a data load opcode\n");
unreached();
}
}
// return op that is the load equivalent of the given addr opcode
genTreeOps loadForm(genTreeOps addrForm)
{
switch (addrForm)
{
case GT_LCL_VAR_ADDR:
return GT_LCL_VAR;
case GT_LCL_FLD_ADDR:
return GT_LCL_FLD;
default:
noway_assert(!"not a local address opcode\n");
unreached();
}
}
// copy the flags determined by mask from src to dst
void copyFlags(GenTree* dst, GenTree* src, unsigned mask)
{
dst->gtFlags &= ~mask;
dst->gtFlags |= (src->gtFlags & mask);
}
// Rewrite a SIMD indirection as GT_IND(GT_LEA(obj.op1)), or as a simple
// lclVar if possible.
//
// Arguments:
// use - A use reference for a block node
// keepBlk - True if this should remain a block node if it is not a lclVar
//
// Return Value:
// None.
//
// TODO-1stClassStructs: These should be eliminated earlier, once we can handle
// lclVars in all the places that used to have GT_OBJ.
//
void Rationalizer::RewriteSIMDOperand(LIR::Use& use, bool keepBlk)
{
#ifdef FEATURE_SIMD
// No lowering is needed for non-SIMD nodes, so early out if featureSIMD is not enabled.
if (!comp->featureSIMD)
{
return;
}
GenTree* tree = use.Def();
if (!tree->OperIsIndir())
{
return;
}
var_types simdType = tree->TypeGet();
if (!varTypeIsSIMD(simdType))
{
return;
}
// If the operand of is a GT_ADDR(GT_LCL_VAR) and LclVar is known to be of simdType,
// replace obj by GT_LCL_VAR.
GenTree* addr = tree->AsIndir()->Addr();
if (addr->OperIsLocalAddr() && comp->isAddrOfSIMDType(addr))
{
BlockRange().Remove(tree);
addr->SetOper(loadForm(addr->OperGet()));
addr->gtType = simdType;
use.ReplaceWith(comp, addr);
}
else if (!keepBlk)
{
tree->SetOper(GT_IND);
tree->gtType = simdType;
}
#endif // FEATURE_SIMD
}
// RewriteNodeAsCall : Replace the given tree node by a GT_CALL.
//
// Arguments:
// ppTree - A pointer-to-a-pointer for the tree node
// fgWalkData - A pointer to tree walk data providing the context
// callHnd - The method handle of the call to be generated
// entryPoint - The method entrypoint of the call to be generated
// args - The argument list of the call to be generated
//
// Return Value:
// None.
//
void Rationalizer::RewriteNodeAsCall(GenTree** use,
Compiler::fgWalkData* data,
CORINFO_METHOD_HANDLE callHnd,
#ifdef FEATURE_READYTORUN_COMPILER
CORINFO_CONST_LOOKUP entryPoint,
#endif
GenTreeArgList* args)
{
GenTreePtr tree = *use;
Compiler* comp = data->compiler;
SplitData* tmpState = (SplitData*)data->pCallbackData;
GenTreePtr root = tmpState->root;
GenTreePtr treeFirstNode = comp->fgGetFirstNode(tree);
GenTreePtr treeLastNode = tree;
GenTreePtr treePrevNode = treeFirstNode->gtPrev;
GenTreePtr treeNextNode = treeLastNode->gtNext;
// Create the call node
GenTreeCall* call = comp->gtNewCallNode(CT_USER_FUNC, callHnd, tree->gtType, args);
call = comp->fgMorphArgs(call);
// Determine if this call has changed any codegen requirements.
comp->fgCheckArgCnt();
#ifdef FEATURE_READYTORUN_COMPILER
call->gtCall.setEntryPoint(entryPoint);
#endif
// Replace "tree" with "call"
if (data->parentStack->Height() > 1)
{
data->parentStack->Index(1)->ReplaceOperand(use, call);
}
else
{
// If there's no parent, the tree being replaced is the root of the
// statement (and no special handling is necessary).
*use = call;
}
// Rebuild the evaluation order.
comp->gtSetStmtInfo(root);
// Rebuild the execution order.
comp->fgSetTreeSeq(call, treePrevNode);
// Restore linear-order Prev and Next for "call".
if (treePrevNode)
{
treeFirstNode = comp->fgGetFirstNode(call);
treeFirstNode->gtPrev = treePrevNode;
treePrevNode->gtNext = treeFirstNode;
}
else
{
// Update the linear oder start of "root" if treeFirstNode
// appears to have replaced the original first node.
assert(treeFirstNode == root->gtStmt.gtStmtList);
root->gtStmt.gtStmtList = comp->fgGetFirstNode(call);
}
if (treeNextNode)
{
treeLastNode = call;
treeLastNode->gtNext = treeNextNode;
treeNextNode->gtPrev = treeLastNode;
}
// Propagate flags of "call" to its parents.
// 0 is current node, so start at 1
for (int i = 1; i < data->parentStack->Height(); i++)
{
GenTree* node = data->parentStack->Index(i);
node->gtFlags |= GTF_CALL;
node->gtFlags |= call->gtFlags & GTF_ALL_EFFECT;
}
// Since "tree" is replaced with "call", pop "tree" node (i.e the current node)
// and replace it with "call" on parent stack.
assert(data->parentStack->Top() == tree);
(void)data->parentStack->Pop();
data->parentStack->Push(call);
}
// RewriteIntrinsicAsUserCall : Rewrite an intrinsic operator as a GT_CALL to the original method.
//
// Arguments:
// ppTree - A pointer-to-a-pointer for the intrinsic node
// fgWalkData - A pointer to tree walk data providing the context
//
// Return Value:
// None.
//
// Some intrinsics, such as operation Sqrt, are rewritten back to calls, and some are not.
// The ones that are not being rewritten here must be handled in Codegen.
// Conceptually, the lower is the right place to do the rewrite. Keeping it in rationalization is
// mainly for throughput issue.
void Rationalizer::RewriteIntrinsicAsUserCall(GenTree** use, Compiler::fgWalkData* data)
{
GenTreeIntrinsic* intrinsic = (*use)->AsIntrinsic();
Compiler* comp = data->compiler;
GenTreeArgList* args;
if (intrinsic->gtOp.gtOp2 == nullptr)
{
args = comp->gtNewArgList(intrinsic->gtGetOp1());
}
else
{
args = comp->gtNewArgList(intrinsic->gtGetOp1(), intrinsic->gtGetOp2());
}
RewriteNodeAsCall(use, data, intrinsic->gtMethodHandle,
#ifdef FEATURE_READYTORUN_COMPILER
intrinsic->gtEntryPoint,
#endif
args);
}
// FixupIfSIMDLocal: Fixup the type of a lclVar tree, as needed, if it is a SIMD type vector.
//
// Arguments:
// comp - the Compiler object.
// tree - the GenTreeLclVarCommon tree to be fixed up.
//
// Return Value:
// None.
//
// TODO-1stClassStructs: This is now only here to preserve existing behavior. It is actually not
// desirable to change the lclFld nodes back to TYP_SIMD (it will cause them to be loaded
// into a vector register, and then moved to an int register).
void Rationalizer::FixupIfSIMDLocal(GenTreeLclVarCommon* node)
{
#ifdef FEATURE_SIMD
if (!comp->featureSIMD)
{
return;
}
LclVarDsc* varDsc = &(comp->lvaTable[node->gtLclNum]);
// Don't mark byref of SIMD vector as a SIMD type.
// Note that struct args though marked as lvIsSIMD=true,
// the tree node representing such an arg should not be
// marked as a SIMD type, since it is a byref of a SIMD type.
if (!varTypeIsSIMD(varDsc))
{
return;
}
switch (node->OperGet())
{
default:
// Nothing to do for most tree nodes.
break;
case GT_LCL_FLD:
// We may see a lclFld used for pointer-sized structs that have been morphed, in which
// case we can change it to GT_LCL_VAR.
// However, we may also see a lclFld with FieldSeqStore::NotAField() for structs that can't
// be analyzed, e.g. those with overlapping fields such as the IL implementation of Vector<T>.
if ((node->AsLclFld()->gtFieldSeq == FieldSeqStore::NotAField()) && (node->AsLclFld()->gtLclOffs == 0) &&
(node->gtType == TYP_I_IMPL) && (varDsc->lvExactSize == TARGET_POINTER_SIZE))
{
node->SetOper(GT_LCL_VAR);
node->gtFlags &= ~(GTF_VAR_USEASG);
}
else
{
// If we access a field of a SIMD lclVar via GT_LCL_FLD, it cannot have been
// independently promoted.
assert(comp->lvaGetPromotionType(varDsc) != Compiler::PROMOTION_TYPE_INDEPENDENT);
return;
}
break;
case GT_STORE_LCL_FLD:
assert(node->gtType == TYP_I_IMPL);
node->SetOper(GT_STORE_LCL_VAR);
node->gtFlags &= ~(GTF_VAR_USEASG);
break;
}
unsigned simdSize = (unsigned int)roundUp(varDsc->lvExactSize, TARGET_POINTER_SIZE);
node->gtType = comp->getSIMDTypeForSize(simdSize);
#endif // FEATURE_SIMD
}
#ifdef DEBUG
void Rationalizer::ValidateStatement(GenTree* tree, BasicBlock* block)
{
assert(tree->gtOper == GT_STMT);
DBEXEC(TRUE, JitTls::GetCompiler()->fgDebugCheckNodeLinks(block, tree));
}
// sanity checks that apply to all kinds of IR
void Rationalizer::SanityCheck()
{
// TODO: assert(!IsLIR());
BasicBlock* block;
foreach_block(comp, block)
{
for (GenTree* statement = block->bbTreeList; statement != nullptr; statement = statement->gtNext)
{
ValidateStatement(statement, block);
for (GenTree* tree = statement->gtStmt.gtStmtList; tree; tree = tree->gtNext)
{
// QMARK nodes should have been removed before this phase.
assert(tree->OperGet() != GT_QMARK);
if (tree->OperGet() == GT_ASG)
{
if (tree->gtGetOp1()->OperGet() == GT_LCL_VAR)
{
assert(tree->gtGetOp1()->gtFlags & GTF_VAR_DEF);
}
else if (tree->gtGetOp2()->OperGet() == GT_LCL_VAR)
{
assert(!(tree->gtGetOp2()->gtFlags & GTF_VAR_DEF));
}
}
}
}
}
}
void Rationalizer::SanityCheckRational()
{
// TODO-Cleanup : check that the tree is rational here
// then do normal checks
SanityCheck();
}
#endif // DEBUG
static void RewriteAssignmentIntoStoreLclCore(GenTreeOp* assignment,
GenTree* location,
GenTree* value,
genTreeOps locationOp)
{
assert(assignment != nullptr);
assert(assignment->OperGet() == GT_ASG);
assert(location != nullptr);
assert(value != nullptr);
genTreeOps storeOp = storeForm(locationOp);
#ifdef DEBUG
JITDUMP("rewriting asg(%s, X) to %s(X)\n", GenTree::NodeName(locationOp), GenTree::NodeName(storeOp));
#endif // DEBUG
assignment->SetOper(storeOp);
GenTreeLclVarCommon* store = assignment->AsLclVarCommon();
GenTreeLclVarCommon* var = location->AsLclVarCommon();
store->SetLclNum(var->gtLclNum);
store->SetSsaNum(var->gtSsaNum);
if (locationOp == GT_LCL_FLD)
{
store->gtLclFld.gtLclOffs = var->gtLclFld.gtLclOffs;
store->gtLclFld.gtFieldSeq = var->gtLclFld.gtFieldSeq;
}
copyFlags(store, var, GTF_LIVENESS_MASK);
store->gtFlags &= ~GTF_REVERSE_OPS;
store->gtType = var->TypeGet();
store->gtOp1 = value;
DISPNODE(store);
JITDUMP("\n");
}
void Rationalizer::RewriteAssignmentIntoStoreLcl(GenTreeOp* assignment)
{
assert(assignment != nullptr);
assert(assignment->OperGet() == GT_ASG);
GenTree* location = assignment->gtGetOp1();
GenTree* value = assignment->gtGetOp2();
RewriteAssignmentIntoStoreLclCore(assignment, location, value, location->OperGet());
}
void Rationalizer::RewriteAssignment(LIR::Use& use)
{
assert(use.IsInitialized());
GenTreeOp* assignment = use.Def()->AsOp();
assert(assignment->OperGet() == GT_ASG);
GenTree* location = assignment->gtGetOp1();
GenTree* value = assignment->gtGetOp2();
genTreeOps locationOp = location->OperGet();
#ifdef FEATURE_SIMD
if (varTypeIsSIMD(location) && assignment->OperIsInitBlkOp())
{
if (location->OperGet() == GT_LCL_VAR)
{
var_types simdType = location->TypeGet();
GenTree* initVal = assignment->gtOp.gtOp2;
var_types baseType = comp->getBaseTypeOfSIMDLocal(location);
if (baseType != TYP_UNKNOWN)
{
GenTreeSIMD* simdTree = new (comp, GT_SIMD)
GenTreeSIMD(simdType, initVal, SIMDIntrinsicInit, baseType, genTypeSize(simdType));
assignment->gtOp.gtOp2 = simdTree;
value = simdTree;
initVal->gtNext = simdTree;
simdTree->gtPrev = initVal;
simdTree->gtNext = location;
location->gtPrev = simdTree;
}
}
else
{
assert(location->OperIsBlk());
}
}
#endif // FEATURE_SIMD
switch (locationOp)
{
case GT_LCL_VAR:
case GT_LCL_FLD:
case GT_REG_VAR:
case GT_PHI_ARG:
RewriteAssignmentIntoStoreLclCore(assignment, location, value, locationOp);
BlockRange().Remove(location);
break;
case GT_IND:
{
GenTreeStoreInd* store =
new (comp, GT_STOREIND) GenTreeStoreInd(location->TypeGet(), location->gtGetOp1(), value);
copyFlags(store, assignment, GTF_ALL_EFFECT);
copyFlags(store, location, GTF_IND_FLAGS);
if (assignment->IsReverseOp())
{
store->gtFlags |= GTF_REVERSE_OPS;
}
// TODO: JIT dump
// Remove the GT_IND node and replace the assignment node with the store
BlockRange().Remove(location);
BlockRange().InsertBefore(assignment, store);
use.ReplaceWith(comp, store);
BlockRange().Remove(assignment);
}
break;
case GT_CLS_VAR:
{
location->SetOper(GT_CLS_VAR_ADDR);
location->gtType = TYP_BYREF;
assignment->SetOper(GT_STOREIND);
// TODO: JIT dump
}
break;
case GT_BLK:
case GT_OBJ:
case GT_DYN_BLK:
{
assert(varTypeIsStruct(location));
GenTreeBlk* storeBlk = location->AsBlk();
genTreeOps storeOper;
switch (location->gtOper)
{
case GT_BLK:
storeOper = GT_STORE_BLK;
break;
case GT_OBJ:
storeOper = GT_STORE_OBJ;
break;
case GT_DYN_BLK:
storeOper = GT_STORE_DYN_BLK;
break;
default:
unreached();
}
JITDUMP("Rewriting GT_ASG(%s(X), Y) to %s(X,Y):\n", GenTree::NodeName(location->gtOper),
GenTree::NodeName(storeOper));
storeBlk->SetOperRaw(storeOper);
storeBlk->gtFlags &= ~GTF_DONT_CSE;
storeBlk->gtFlags |= (assignment->gtFlags & (GTF_ALL_EFFECT | GTF_REVERSE_OPS | GTF_BLK_VOLATILE |
GTF_BLK_UNALIGNED | GTF_BLK_INIT | GTF_DONT_CSE));
storeBlk->gtBlk.Data() = value;
// Replace the assignment node with the store
use.ReplaceWith(comp, storeBlk);
BlockRange().Remove(assignment);
DISPTREERANGE(BlockRange(), use.Def());
JITDUMP("\n");
}
break;
default:
unreached();
break;
}
}
void Rationalizer::RewriteAddress(LIR::Use& use)
{
assert(use.IsInitialized());
GenTreeUnOp* address = use.Def()->AsUnOp();
assert(address->OperGet() == GT_ADDR);
GenTree* location = address->gtGetOp1();
genTreeOps locationOp = location->OperGet();
if (location->IsLocal())
{
// We are changing the child from GT_LCL_VAR TO GT_LCL_VAR_ADDR.
// Therefore gtType of the child needs to be changed to a TYP_BYREF
#ifdef DEBUG
if (locationOp == GT_LCL_VAR)
{
JITDUMP("Rewriting GT_ADDR(GT_LCL_VAR) to GT_LCL_VAR_ADDR:\n");
}
else
{
assert(locationOp == GT_LCL_FLD);
JITDUMP("Rewriting GT_ADDR(GT_LCL_FLD) to GT_LCL_FLD_ADDR:\n");
}
#endif // DEBUG
location->SetOper(addrForm(locationOp));
location->gtType = TYP_BYREF;
copyFlags(location, address, GTF_ALL_EFFECT);
use.ReplaceWith(comp, location);
BlockRange().Remove(address);
}
else if (locationOp == GT_CLS_VAR)
{
location->SetOper(GT_CLS_VAR_ADDR);
location->gtType = TYP_BYREF;
copyFlags(location, address, GTF_ALL_EFFECT);
use.ReplaceWith(comp, location);
BlockRange().Remove(address);
JITDUMP("Rewriting GT_ADDR(GT_CLS_VAR) to GT_CLS_VAR_ADDR:\n");
}
else if (location->OperIsIndir())
{
use.ReplaceWith(comp, location->gtGetOp1());
BlockRange().Remove(location);
BlockRange().Remove(address);
JITDUMP("Rewriting GT_ADDR(GT_IND(X)) to X:\n");
}
DISPTREERANGE(BlockRange(), use.Def());
JITDUMP("\n");
}
Compiler::fgWalkResult Rationalizer::RewriteNode(GenTree** useEdge, ArrayStack<GenTree*>& parentStack)
{
assert(useEdge != nullptr);
GenTree* node = *useEdge;
assert(node != nullptr);
#ifdef DEBUG
const bool isLateArg = (node->gtFlags & GTF_LATE_ARG) != 0;
#endif
// First, remove any preceeding list nodes, which are not otherwise visited by the tree walk.
//
// NOTE: GT_FIELD_LIST head nodes, and GT_LIST nodes used by phi nodes will in fact be visited.
for (GenTree* prev = node->gtPrev; prev != nullptr && prev->OperIsAnyList() && !(prev->OperIsFieldListHead());
prev = node->gtPrev)
{
BlockRange().Remove(prev);
}
// In addition, remove the current node if it is a GT_LIST node that is not an aggregate.
if (node->OperIsAnyList())
{
GenTreeArgList* list = node->AsArgList();
if (!list->OperIsFieldListHead())
{
BlockRange().Remove(list);
}
return Compiler::WALK_CONTINUE;
}
LIR::Use use;
if (parentStack.Height() < 2)
{
use = LIR::Use::GetDummyUse(BlockRange(), *useEdge);
}
else
{
use = LIR::Use(BlockRange(), useEdge, parentStack.Index(1));
}
assert(node == use.Def());
switch (node->OperGet())
{
case GT_ASG:
RewriteAssignment(use);
break;
case GT_BOX:
// GT_BOX at this level just passes through so get rid of it
use.ReplaceWith(comp, node->gtGetOp1());
BlockRange().Remove(node);
break;
case GT_ADDR:
RewriteAddress(use);
break;
case GT_IND:
// Clear the `GTF_IND_ASG_LHS` flag, which overlaps with `GTF_IND_REQ_ADDR_IN_REG`.
node->gtFlags &= ~GTF_IND_ASG_LHS;
break;
case GT_NOP:
// fgMorph sometimes inserts NOP nodes between defs and uses
// supposedly 'to prevent constant folding'. In this case, remove the
// NOP.
if (node->gtGetOp1() != nullptr)
{
use.ReplaceWith(comp, node->gtGetOp1());
BlockRange().Remove(node);
}
break;
case GT_COMMA:
{
GenTree* op1 = node->gtGetOp1();
if ((op1->gtFlags & GTF_ALL_EFFECT) == 0)
{
// The LHS has no side effects. Remove it.
bool isClosed = false;
unsigned sideEffects = 0;
LIR::ReadOnlyRange lhsRange = BlockRange().GetTreeRange(op1, &isClosed, &sideEffects);
// None of the transforms performed herein violate tree order, so these
// should always be true.
assert(isClosed);
assert((sideEffects & GTF_ALL_EFFECT) == 0);
BlockRange().Delete(comp, m_block, std::move(lhsRange));
}
GenTree* replacement = node->gtGetOp2();
if (!use.IsDummyUse())
{
use.ReplaceWith(comp, replacement);
}
else
{
// This is a top-level comma. If the RHS has no side effects we can remove
// it as well.
if ((replacement->gtFlags & GTF_ALL_EFFECT) == 0)
{
bool isClosed = false;
unsigned sideEffects = 0;
LIR::ReadOnlyRange rhsRange = BlockRange().GetTreeRange(replacement, &isClosed, &sideEffects);
// None of the transforms performed herein violate tree order, so these
// should always be true.
assert(isClosed);
assert((sideEffects & GTF_ALL_EFFECT) == 0);
BlockRange().Delete(comp, m_block, std::move(rhsRange));
}
}
BlockRange().Remove(node);
}
break;
case GT_ARGPLACE:
// Remove argplace and list nodes from the execution order.
//
// TODO: remove phi args and phi nodes as well?
BlockRange().Remove(node);
break;
#ifdef _TARGET_XARCH_
case GT_CLS_VAR:
{
// Class vars that are the target of an assignment will get rewritten into
// GT_STOREIND(GT_CLS_VAR_ADDR, val) by RewriteAssignment. This check is
// not strictly necessary--the GT_IND(GT_CLS_VAR_ADDR) pattern that would
// otherwise be generated would also be picked up by RewriteAssignment--but
// skipping the rewrite here saves an allocation and a bit of extra work.
const bool isLHSOfAssignment = (use.User()->OperGet() == GT_ASG) && (use.User()->gtGetOp1() == node);
if (!isLHSOfAssignment)
{
GenTree* ind = comp->gtNewOperNode(GT_IND, node->TypeGet(), node);
node->SetOper(GT_CLS_VAR_ADDR);
node->gtType = TYP_BYREF;
BlockRange().InsertAfter(node, ind);
use.ReplaceWith(comp, ind);
// TODO: JIT dump
}
}
break;
#endif // _TARGET_XARCH_
case GT_INTRINSIC:
// Non-target intrinsics should have already been rewritten back into user calls.
assert(Compiler::IsTargetIntrinsic(node->gtIntrinsic.gtIntrinsicId));
break;
#ifdef FEATURE_SIMD
case GT_BLK:
case GT_OBJ:
{
// TODO-1stClassStructs: These should have been transformed to GT_INDs, but in order
// to preserve existing behavior, we will keep this as a block node if this is the
// lhs of a block assignment, and either:
// - It is a "generic" TYP_STRUCT assignment, OR
// - It is an initblk, OR
// - Neither the lhs or rhs are known to be of SIMD type.
GenTree* parent = use.User();
bool keepBlk = false;
if ((parent->OperGet() == GT_ASG) && (node == parent->gtGetOp1()))
{
if ((node->TypeGet() == TYP_STRUCT) || parent->OperIsInitBlkOp())
{
keepBlk = true;
}
else if (!comp->isAddrOfSIMDType(node->AsBlk()->Addr()))
{
GenTree* dataSrc = parent->gtGetOp2();
if (!dataSrc->IsLocal() && (dataSrc->OperGet() != GT_SIMD))
{
noway_assert(dataSrc->OperIsIndir());
keepBlk = !comp->isAddrOfSIMDType(dataSrc->AsIndir()->Addr());
}
}
}
RewriteSIMDOperand(use, keepBlk);
}
break;
case GT_LCL_FLD:
case GT_STORE_LCL_FLD:
// TODO-1stClassStructs: Eliminate this.
FixupIfSIMDLocal(node->AsLclVarCommon());
break;
case GT_SIMD:
{
noway_assert(comp->featureSIMD);
GenTreeSIMD* simdNode = node->AsSIMD();
unsigned simdSize = simdNode->gtSIMDSize;
var_types simdType = comp->getSIMDTypeForSize(simdSize);
// TODO-1stClassStructs: This should be handled more generally for enregistered or promoted
// structs that are passed or returned in a different register type than their enregistered
// type(s).
if (simdNode->gtType == TYP_I_IMPL && simdNode->gtSIMDSize == TARGET_POINTER_SIZE)
{
// This happens when it is consumed by a GT_RET_EXPR.
// It can only be a Vector2f or Vector2i.
assert(genTypeSize(simdNode->gtSIMDBaseType) == 4);
simdNode->gtType = TYP_SIMD8;
}
// Certain SIMD trees require rationalizing.
if (simdNode->gtSIMD.gtSIMDIntrinsicID == SIMDIntrinsicInitArray)
{
// Rewrite this as an explicit load.
JITDUMP("Rewriting GT_SIMD array init as an explicit load:\n");
unsigned int baseTypeSize = genTypeSize(simdNode->gtSIMDBaseType);
GenTree* address = new (comp, GT_LEA) GenTreeAddrMode(TYP_BYREF, simdNode->gtOp1, simdNode->gtOp2,
baseTypeSize, offsetof(CORINFO_Array, u1Elems));
GenTree* ind = comp->gtNewOperNode(GT_IND, simdType, address);
BlockRange().InsertBefore(simdNode, address, ind);
use.ReplaceWith(comp, ind);
BlockRange().Remove(simdNode);
DISPTREERANGE(BlockRange(), use.Def());
JITDUMP("\n");
}
else
{
// This code depends on the fact that NONE of the SIMD intrinsics take vector operands
// of a different width. If that assumption changes, we will EITHER have to make these type
// transformations during importation, and plumb the types all the way through the JIT,
// OR add a lot of special handling here.
GenTree* op1 = simdNode->gtGetOp1();
if (op1 != nullptr && op1->gtType == TYP_STRUCT)
{
op1->gtType = simdType;
}
GenTree* op2 = simdNode->gtGetOp2();
if (op2 != nullptr && op2->gtType == TYP_STRUCT)
{
op2->gtType = simdType;
}
}
}
break;
#endif // FEATURE_SIMD
default:
// JCC nodes should not be present in HIR.
assert(node->OperGet() != GT_JCC);
break;
}
// Do some extra processing on top-level nodes to remove unused local reads.
if (use.IsDummyUse() && node->OperIsLocalRead())
{
assert((node->gtFlags & GTF_ALL_EFFECT) == 0);
comp->lvaDecRefCnts(node);
BlockRange().Remove(node);
}
assert(isLateArg == ((use.Def()->gtFlags & GTF_LATE_ARG) != 0));
return Compiler::WALK_CONTINUE;
}
void Rationalizer::DoPhase()
{
DBEXEC(TRUE, SanityCheck());
comp->compCurBB = nullptr;
comp->fgOrder = Compiler::FGOrderLinear;
BasicBlock* firstBlock = comp->fgFirstBB;
for (BasicBlock* block = comp->fgFirstBB; block != nullptr; block = block->bbNext)
{
comp->compCurBB = block;
m_block = block;
// Establish the first and last nodes for the block. This is necessary in order for the LIR
// utilities that hang off the BasicBlock type to work correctly.
GenTreeStmt* firstStatement = block->firstStmt();
if (firstStatement == nullptr)
{
// No statements in this block; skip it.
block->MakeLIR(nullptr, nullptr);
continue;
}
GenTreeStmt* lastStatement = block->lastStmt();
// Rewrite intrinsics that are not supported by the target back into user calls.
// This needs to be done before the transition to LIR because it relies on the use
// of fgMorphArgs, which is designed to operate on HIR. Once this is done for a
// particular statement, link that statement's nodes into the current basic block.
//
// This walk also clears the GTF_VAR_USEDEF bit on locals, which is not necessary
// in the backend.
GenTree* lastNodeInPreviousStatement = nullptr;
for (GenTreeStmt* statement = firstStatement; statement != nullptr; statement = statement->getNextStmt())
{
assert(statement->gtStmtList != nullptr);
assert(statement->gtStmtList->gtPrev == nullptr);
assert(statement->gtStmtExpr != nullptr);
assert(statement->gtStmtExpr->gtNext == nullptr);
SplitData splitData;
splitData.root = statement;
splitData.block = block;
splitData.thisPhase = this;
comp->fgWalkTreePost(&statement->gtStmtExpr,
[](GenTree** use, Compiler::fgWalkData* walkData) -> Compiler::fgWalkResult {
GenTree* node = *use;
if (node->OperGet() == GT_INTRINSIC &&
Compiler::IsIntrinsicImplementedByUserCall(node->gtIntrinsic.gtIntrinsicId))
{
RewriteIntrinsicAsUserCall(use, walkData);
}
else if (node->OperIsLocal())
{
node->gtFlags &= ~GTF_VAR_USEDEF;
}
return Compiler::WALK_CONTINUE;
},
&splitData, true);
GenTree* firstNodeInStatement = statement->gtStmtList;
if (lastNodeInPreviousStatement != nullptr)
{
lastNodeInPreviousStatement->gtNext = firstNodeInStatement;
}
firstNodeInStatement->gtPrev = lastNodeInPreviousStatement;
lastNodeInPreviousStatement = statement->gtStmtExpr;
}
block->MakeLIR(firstStatement->gtStmtList, lastStatement->gtStmtExpr);
// Rewrite HIR nodes into LIR nodes.
for (GenTreeStmt *statement = firstStatement, *nextStatement; statement != nullptr; statement = nextStatement)
{
nextStatement = statement->getNextStmt();
// If this statement has correct offset information, change it into an IL offset
// node and insert it into the LIR.
if (statement->gtStmtILoffsx != BAD_IL_OFFSET)
{
assert(!statement->IsPhiDefnStmt());
statement->SetOper(GT_IL_OFFSET);
statement->gtNext = nullptr;
statement->gtPrev = nullptr;
BlockRange().InsertBefore(statement->gtStmtList, statement);
}
m_statement = statement;
comp->fgWalkTreePost(&statement->gtStmtExpr,
[](GenTree** use, Compiler::fgWalkData* walkData) -> Compiler::fgWalkResult {
return reinterpret_cast<Rationalizer*>(walkData->pCallbackData)
->RewriteNode(use, *walkData->parentStack);
},
this, true);
}
assert(BlockRange().CheckLIR(comp));
}
comp->compRationalIRForm = true;
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,101 |
Q: Delphi 7 keeps using old outdated form Changed, updated, form is not used even though uses and project settings seem fine, old form files removed from disk.
Is this a bug in the IDE? I may just delete the form and copy it into another unit with a new name.
A: If it's using an old form it has to be getting it from somewhere--it doesn't appear out of thin air. Two scenarios come to mind:
1) It's somewhere where you don't realize. Search your system for files by that name.
2) Unless you do a build Delphi compiles based on timestamps. If the clock was wrong when it was compiled before the .dcu can have a more recent time and thus it gets skipped in compiling. I've hit this more than once with timezones.
A: A good way to find it is to first move the project to a different new folder and try to compile it. This should produce and error that will help you to find the culprit. If this does not work then it is settings like paths etc in your libraries that are at fault.
Also make sure that you deleted all ".dcu" files in the project before re-compiling.
A: No, it is not a bug in the IDE.
You are referencing that form in some setting in your project or environment, which you didn't find yet and which takes precedence to options you already tweaked.
Where do you need to go to resolve your problem? Well, that's difficult to say without looking at your development environment and your project settings.
A: I've had this happen before. It is always something referenced that I wasn't aware of.
You can do a grep for something from the form and see where it shows up.
A: Thanks for the input. The first one I tried, moving the files, mm2010, showed it was my code that was at fault.
A: Although the form/unit is not included in the project file (dpr), it is still referenced by some other unit. So the compiler links the res into the application. Look for the unit name you want to remove in other units' uses clauses.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,419 |
<?php
namespace PFCD\TourismBundle;
class Constants
{
const ADMIN = 'admin';
const BACK = 'back';
const FRONT = 'front';
const FORM_CREATE = 'create';
const FORM_UPDATE = 'update';
const ORGANIZATION = 'organization';
const ACTIVITY = 'activity';
const ARTICLE = 'article';
const ENQUIRY_FULL = 'unregistered';
const ENQUIRY_MINI = 'registered';
const MONDAY = 1;
const TUESDAY = 2;
const WEDNESDAY = 3;
const THURSDAY = 4;
const FRIDAY = 5;
const SATURDAY = 6;
const SUNDAY = 7;
}
?>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,966 |
The 2020 World Junior Ice Hockey Championship Division I consisted of two tiered groups of six teams each: the second-tier Division I A and the third-tier Division I B. For each tier's tournament, the team which placed first was promoted to the next highest division, while the team which placed last was relegated to a lower division.
To be eligible as a junior player in these tournaments, a player couldn't be born earlier than 2000.
Division I A
The Division I A tournament was played in Minsk, Belarus, from 9 to 15 December 2019.
Participants
Match officials
Four referees and 7 linesmen were selected for the tournament.
Referees
Michał Baca
Andrea Moschen
Alexey Roshchyn
Marcus Wannerstedt
Linesmen
Pavel Badyl
Nicola Basso
Thomas Caillot
Dario Fuchs
Lukáš Kacej
Mikita Paliakou
Toivo Tilku
Standings
Results
All times are local (UTC+3).
Statistics
Top 10 scorers
GP = Games played; G = Goals; A = Assists; Pts = Points; +/− = Plus-minus; PIM = Penalties In Minutes
Source: IIHF.com
Goaltending leaders
(minimum 40% team's total ice time)
TOI = Time On Ice (minutes:seconds); GA = Goals against; GAA = Goals against average; Sv% = Save percentage; SO = Shutouts
Source: IIHF.com
Best Players Selected by the Directorate
Goaltender: Jānis Voris
Defenceman: Ilya Solovyov
Forward: Benjamin Baumgartner
Source: IIHF.com
Division I B
The Division I B tournament was played in Kyiv, Ukraine, from 12 to 18 December 2019.
Participants
Match officials
4 referees and 7 linesmen were selected for the tournament.
Referees
Miha Bulovec
Richard Magnusson
Andrew Miller
Rasmus Toppel
Linesmen
Roman Kaderli
Ilya Khohlov
Ilia Kisil
Artem Korepanov
Andreas Weise Krøyer
Patrick Laguzov
Nazar Slezov
Standings
Results
All times are local (UTC+2).
Statistics
Top 10 scorers
GP = Games played; G = Goals; A = Assists; Pts = Points; +/− = Plus-minus; PIM = Penalties In Minutes
Source: IIHF.com
Goaltending leaders
(minimum 40% team's total ice time)
TOI = Time On Ice (minutes:seconds); GA = Goals against; GAA = Goals against average; Sv% = Save percentage; SO = Shutouts
Source: IIHF.com
Best Players Selected by the Directorate
Goaltender: Artur Ohandzhanyan
Defenceman: Dávid Pokornyi
Forward: Pierrick Dubé
Source: IIHF.com
References
External links
Division I A
Division I B
I
World Junior Ice Hockey Championships – Division I
International ice hockey competitions hosted by Belarus
International ice hockey competitions hosted by Ukraine
2019–20 in Belarusian ice hockey
2019–20 in Ukrainian ice hockey
Sports competitions in Minsk
Sports competitions in Kyiv
IIHF | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,508 |
NMCB-3 Seabees Complete Palawan School House
Story by CM2 Justin Ott, Naval Mobile Construction Battalion 3 Public Affairs
PALAWAN, Philippines – Seabees assigned to Naval Mobile Construction Battalion (NMCB) 3, Construction Civic Action Detail (CCAD) Palawan and members of the Municipality of Aborlan community celebrated the completion of a two-room school house for the Magsaysay School with a ribbon-cutting ceremony Feb. 6.
Seabees assigned to Naval Mobile Construction Battalion 3, Construction Civic Action Detail Palawan, along with their Philippine navy counterparts and members of the Aborlan community, bow their heads for the blessing of a two-room school house during a ribbon cutting ceremony to celebrate its completion, Feb. 7, 2019. (Photo by CM2 Justin Ott)
NMCB-3 Seabees partnered with Philippine Navy Seabees assigned to the second NMCB to build the school house in order to provide some relief to the over crowded classrooms that the students currently attend. The classrooms were built to accommodate the typical class size of 40 students with enough windows and high flow fans to ensure that the temperature stays comfortable for the students throughout the school day.
The principal, Roque F. Badenas, expressed his appreciation of the quality of construction the Seabees provided.
"The building has been built sturdy enough to withstand the storms prevalent on the island long enough to serve the grandchildren of its first students," said Badenas.
Constructing the school house was more challenging than similar projects due to the initial state of the land it was to be built on. Once the land was cleared for building the foundational pad, it was discovered that the ground was swampland with ground water present from ground-level and penetrated four-feet down. With the combined expertise of NMCB-3 Seabees and their Philippine counterparts, the group was able to work hand-in-hand to overcome the challenge and construct the school house to serve generations to come.
"Despite several setbacks, the crew worked hard; and with their dedicated resolve they were able to overcome the obstacles and were able to complete the school on time and under budget," said Lt. j.g. Curtis Weis, officer in charge of NMCB-3, CCAD Palawan.
Lt. j.g. Curtis Weis, officer in charge of Naval Mobile Construction Battalion 3, Construction Civic Action Detail Palawan, delivers remarks during a ribbon cutting ceremony to celebrate the completion of a two-room school house, Feb. 7, 2019. (Photo by CM2 Justin Ott)
CCADs are small detachments of personnel that are placed in areas of need throughout the NMCB's area of operations. The CCAD's mission is to execute engineering civic assistance projects, conduct formal training with the host nation and perform community relations event to help enhance shared capabilities and improve the country's social welfare.
Being part of a CCAD is a unique work environment that appeals to many Seabees for the ability to work with allied nations, give back to communities, and learn more out of rate skills.
"The small number of Seabees assigned to a CCAD creates a camaraderie amongst everyone involved so that when there were challenges with the construction of the school house it brought the team closer together, and therefore the challenges were able to be overcome," said Steelworker Constructionman Apprentice Kalub Walker. "It also creates an environment where everyone becomes more well rounded by gaining a basic knowledge of the other Seabees' rates."
CCAD Palawan Seabees have broken ground on an identical schoolhouse 1.5 hours southwest of the first schoolhouse location. This location will provide numerous challenges to the CCAD due to the travel time it takes to get to the site and its lack of amenities and infrastructure around the site, but it will serve as an opportunity for the Seabees to further demonstrate their skills and interoperability with their counterparts.
Builder 1st Class Zachary Crouse, assigned to Naval Mobile Construction Battalion 3, Construction Civic Action Detail Palawan, unveils a Seabee statue during a ribbon cutting ceremony to celebrate the completion of a two-room school house, Feb. 7, 2019. (Photo by CM2 Justin Ott)
NMCB-3 is forward deployed throughout the Indo-Pacific region and United States ready to support major combat operations, theater security, humanitarian assistance and disaster relief operations. Seabees provide general engineering and civil support to Navy, Marine Corps and joint operational forces globally.
For more information about Seabees and NMCB-3, visit http://seabeemagazine.navylive.dodlive.mil or https://www.facebook.com/NMCB3/
Tags nmcb 3 Palawan zoconus
Previous This Week in Seabee History: February 24 – March 2
Next Seabees Proud to Serve People of Tinian as DSCA Support Comes to Close
NMCB 133 Completes 48- hour Mount-out Exercise By Air with a 24-hour Follow-on Mount-out by Sea
Seabees and Marines Power III MEF and MCIPAC
Story by Cpl. Karis Mattingly, Marine Corps Installations Pacific Day in and day out U.S. … | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 850 |
Operation Prism has opened a lot of eyes and a lot of phone records. The worldwide leak that NSA trolls through people's phone calls and internet usage like an ex on Facebook has put a lot of people on edge. Now data mining has been happening for years, to help people buy everything from cellphones to nuclear reactors (can't barbeque without it!), but the revelation that the government is using data mining to find terrorists as well is just a step too far. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,012 |
Ultimatecore
====
The UltimateCore base command.
Usage: /ultimatecore gendocs|reload|setuppermissions|clearcache|resetuser|error|modules|<br>
Aliases: ultimatecore, uc<br>
Module: [core](../modules/core.md)<br>
Basic permission: uc.core.ultimatecore.base<br>
Basic role: admin<br>
Permissions: <br>
* **uc.core.ultimatecore.base** - Recommended role: admin<br>Allows you to use the /ultimatecore command.
* **uc.core.ultimatecore.clearcache.base** - Recommended role: owner<br>Allows you to use the /ultimatecore clearcache command.
* **uc.core.ultimatecore.resetuser.base** - Recommended role: owner<br>Allows you to use the /ultimatecore resetuser command.
* **uc.core.ultimatecore.modules.base** - Recommended role: admin<br>Allows you to use the /ultimatecore modules command.
* **uc.core.ultimatecore.gendocs.base** - Recommended role: nobody<br>Allows you to use the /ultimatecore gendocs command.
* **uc.core.ultimatecore.error.base** - Recommended role: nobody<br>Allows you to use the /ultimatecore error command.
* **uc.core.ultimatecore.reload.base** - Recommended role: admin<br>Allows you to use the /ultimatecore reload command.
* **uc.core.ultimatecore.setuppermissions.base** - Recommended role: owner<br>Allows you to use the /ultimatecore setuppermissions command.
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,880 |
Guillaume de Montmorency, seigneur de Thoré dit simplement Thoré, est un militaire français des guerres de Religion, cinquième fils du connétable Anne de Montmorency et Madeleine de Savoie, né vers 1546-1547, mort en 1594.
Biographie
Durant les premières guerres de Religion, il se bat dans l'armée royale contre les protestants. En 1572-1574, il est colonel général de la cavalerie légère.
Après le massacre de la Saint-Barthélemy, il fait partie des groupes d'opposants au régime qu'on appelle les malcontents. Il participe activement aux complots et à la suite de l'échec des opérations il s'enfuit avec son frère Méru (1574). Les deux frères trouvent refuge en Allemagne où ils tentent de lever des troupes. Lors de la cinquième guerre de Religion, ils commandent les mercenaires allemands mais sont battus par l'armée royale à Dormans.
Lors de la dernière guerre de Religion, il prend et défend Senlis contre la Ligue en .
Famille
Il épouse en premières noces Léonore d'Humières (fille de Jean et petite-fille de d'Humières) en 1561 qui meurt en 1563, peu après l'exécution de Poltrot de Méré. Il se remarie le avec Anne de Lalaing, décédée en , dont il a :
Madeleine de Montmorency (1582-1615), mariée le avec Henri de Luxembourg, duc de Piney, d'où :
Marguerite de Luxembourg ( – ), duchesse de Piney. Elle est mariée le , en premières noces, avec Léon d'Albert (1582-1630 ; frère cadet de Charles, duc de Luynes). Elle est mariée en secondes noces avec de Clermont-Tonnerre (1640-1674), comte de Clermont-Tonnerre, fils de Charles-Henri de Clermont-Tonnerre. De ces deux unions sont issus :
Henri Léon d'Albert, (1630-1697), duc de Piney.
Madeleine de Clermont-Tonnerre, ( – ), duchesse de Piney, mariée le avec François-Henri de Montmorency (1628-1695), maréchal de France, dont elle a :
Charles François Frédéric de Montmorency-Luxembourg (1662-1726), duc de Piney,
Pierre Henry Thibaud de Montmorency-Luxembourg (1663-1700), abbé de Saint-Michel et d'Orcamps,
Paul Sigismond de Montmorency-Luxembourg (1664-1731), duc de Châtillon,
Angélique Cunégonde de Montmorency (1666-1736),
Christian Louis de Montmorency-Luxembourg (1675-1746), prince de Tingry.
Marie-Liesse de Luxembourg (1611-1660), mariée à Henri de Lévis (1596-1651), de Ventadour, vice-roi de Nouvelle-France (1625-1627).
Ascendance
Sources
Société internationale pour l'étude des femmes de l'Ancien Régime. Hilarion de Coste. Lire en ligne.
Notes
Thoré
Guillaume de Montmorency-Thoré
Colonel général (France) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 673 |
\section{Introduction}
\label{sec:intro}
A common practice in the development of Cyber-Physical Systems (CPS) is to specify CPS behaviors using executable and dynamic models ~\cite{liebel2018model,chowdhury2018automatically,mathworks}. These models support engineers in a number of activities, most notably in automated code generation and early testing and simulation of CPS. Recent technological advancements in the areas of robotics and autonomous systems have led to increasingly more complex CPS whose models are often characterized as \emph{compute-intensive}~\cite{chaturvedi2009modeling,arrieta2016search,Arrieta:2018:MBT:3205455.3205490,sagardui2017multiplex,gonzalez2018enabling}.
Compute-Intensive CPS models (CI-CPS) require a lot of computational power to execute~\cite{arrieta2019pareto} since they include complex computations such as dynamic, non-linear and non-algebraic mathematics, and further, they have to be executed for long durations in order to thoroughly exercise interactions between the CPS and its environment. For example, non-trivial simulations of an industrial model of a satellite system, capturing the satellite behavior for $24h$, takes on average around $84$ minutes (\char`\~1.5 hours)~\cite{Luxspace}.\footnote{Machine \textsc{M1}: $12$-core Intel Core $i7$ $3.20$GHz $32$GB of RAM.} The sheer amount of time required for just a single execution of CI-CPS models significantly impedes testing and verification of these models since many testing and verification strategies require to execute the Model Under Test (MUT) for hundreds or thousands of test inputs.
Approaches to verification and testing of CPS models can be largely classified into \emph{exhaustive verification}, and \emph{white-box} and \emph{black-box testing}. Exhaustive verification approaches often translate CPS models
into the input language of model checkers or Satisfiability Modulo Theories (SMT) solvers.
CPS models, however, may contain constructs that cannot be easily encoded into the SMT solver input languages.
For example, CPS models specified in the Simulink language~\cite{mathworks} allow importing arbitrary C code via S-Function blocks or include other plugins (e.g., the Deep Learning Toolbox~\cite{deep}). In addition, CPS models typically capture continuous dynamic and hybrid
systems~\cite{alur:15}. Translating such modeling constructs into low-level logic-based languages is complex, has to be handled on a case-by case basis and may lead to loss of precision which may or may not be
acceptable depending on the application domain. Furthermore, it is well-known that model checking such systems is in general undecidable~\cite{henzinger1998s,alur1995algorithmic,6064535}.
White-box testing uses the internal structure of the model under test to specifically choose inputs that exercise different paths through the model.
Most white-box testing techniques aim to generate a set of test cases that satisfy some structural coverage criteria (e.g.,~\cite{10.1007/978-3-540-71493-4_27,dang2009coverage}).
To achieve their intended coverage goals, they may rely either on SMT-solvers (e.g.,~\cite{kong2015dreach,gao2013dreal}) or on randomized search algorithms (e.g.,~\cite{Nejati:2019:TCS:3340872.3340874,matinnejad2013automated,dreossi2015efficient}). But irrespective of their underlying technique, coverage-guided testing approaches are not meant to demonstrate that CPS models satisfy their requirements.
More recently, falsification-based testing techniques have been proposed as a way to test CPS models with respect to their requirements~\cite{yaghoubi2017local,7963007,abbas2014functional,Nghiem:2010:MTF:1755952.1755983}. These techniques are black-box and aim to find test inputs violating system requirements. They are guided by (quantitative) fitness functions
that can estimate how far a candidate test is from violating some system requirement. Candidate tests are sampled from the search input space using randomized or meta-heuristic search strategies (e.g.,~\cite{8453180,matinnejad2013automated,Matinnejad:2016:ATS:2884781.2884797}). To compute fitness functions, the model under test is executed for each candidate test input. The fitness values then determine whether the goal of testing is achieved (i.e., a requirement violation is found) or further test candidates should be selected. In the latter case, the fitness values may guide selection of new test candidates. Falsification-based testing has shown to be effective in revealing requirements violations in complex CPS models that cannot be
handled by alternative verification methods. However, serious scalability issues arise when testing CI-CPS models since simulating such models for every candidate test may take such a large amount time to the extent that testing becomes impractical.
In this paper, in order to enable efficient and effective testing of CI-CPS models, we propose a technique that combines falsification-based testing with an approximation-refinement loop. Our technique, shown in Figure~\ref{fig:ourapproach}, is referred to as AppRoxImation-based TEst generatiOn (\ourtool). As shown in the figure, provided with a CI-CPS model under test (MUT), we automatically create an approximation of the MUT that closely mimics its behavior but is significantly cheaper to execute. We refer to the approximation model as \emph{surrogate} model, and generate it using System Identification (SI) (e.g.,~\cite{soderstrom1989system,bittanti2019model}) which is a methodology for building mathematical models of dynamic systems using measurements of the system's inputs and outputs~\cite{soderstrom1989system}. Specifically, we use some pairs of inputs and outputs from the MUT to build an initial surrogate model. We then apply falsification testing to the surrogate model instead of the MUT until we find a test revealing some requirement violation for the surrogate model. The identified failure, however, might be spurious. Hence, we
check the test on the MUT. If the test is spurious, we use the output of the test to retrain, using SI, our surrogate model into a new model that more closely mimics the behavior of the MUT, and continue with testing the retrained surrogate model. If the test is not spurious, we have found a requirement violation by running the MUT very few times.
\ourtool\ is inspired, at a high-level, by the counter-example guided abstraction-refinement (CEGAR) loop~\cite{kurshan1994computer,clarke1994model,clarke2000counterexample} proposed to increase scalability of formal verification techniques.
In CEGAR, boolean abstract models are generated and refined based on counter-examples produced by model checking, while in \ourtool, numerical approximation of CPS models are learned and retrained using test inputs and outputs generated by model testing.
Our contributions are as follows:
\begin{figure}
\includegraphics[width=\columnwidth]{approachsmall.pdf}
\vspace{-0.7cm}
\caption{\ourtool : AppRoxImation-based TEst generatiOn.}
\label{fig:ourapproach}
\vspace{-0.3cm}
\end{figure}
\noindent $\bullet$ \emph{We
developed \ourtool , an approximation-refinement testing technique, to identify requirements violations for CI-CPS models.} \ourtool\ combines falsification-based testing with surrogate models built using System Identification (SI). We have implemented \ourtool\ as a Matlab/Simulink standalone application, relying on the existing state-of-the-art System Identification toolbox of Matlab as well as S-Taliro~\cite{staliro}, a state-of-the-art, open source falsification-based framework for Simulink models.
\noindent $\bullet$ \emph{We compared \ourtool\ and S-Taliro to assess the effectiveness and efficiency of our proposed approximation-refinement testing loop.} Our experiments, performed on five publicly-available Simulink models from the literature, show that, on average, \ourtool\ finds $23.9\%$ more requirements violations than S-Taliro and finds the violations in $31.3\%$ less time than the time S-Taliro needs to find them.
\noindent $\bullet$ \emph{We evaluated usefulness and applicability of ARIsTEO in revealing requirements violations in large and industrial CI-CPS models from the satellite domain.}
We analyzed three different requirements over two different versions of a CI-CPS model provided by our industrial partner. \ourtool\ successfully detected violations in each of these versions and for all the requirements, requiring four hour, on average, to find each violation. In contrast, S-Taliro was not able to find any violation on neither of the model versions and after running for four hours.
\textbf{Structure.}
Section~\ref{sec:running} presents our running example, formulates the problem and describes our assumptions.
Section~\ref{sec:ourtool} describes \ourtool , which is then evaluated in Section~\ref{sec:evaluation}.
Section~\ref{sec:related} presents the related work. Section~\ref{sec:conclusion} concludes the paper.
\section{CPS Models and Falsification-Based Testing}
\label{sec:running}
In this section, we describe how test inputs are generated for black-box testing of CPS models. We then introduce the baseline falsification-based testing framework we use in this paper to test CPS models against their requirements.
\textbf{Black-box testing of CPS models.} We consider CPS models under test (MUT) specified in Simulink since it is a prevalent language used in CPS development~\cite{liebel2018model,Dajsuren}.
Our approach is not tied to the Simulink language, and can be applied to other executable languages requiring inputs and generating outputs that are signals over time (e.g., hybrid systems~\cite{grossman1993hybrid}). Such languages are common for CPS as engineers need to describe models capturing interactions of a system with its physical environment~\cite{alur:15}. We use \texttt{SatEx}, a model of a satellite, as a running example. \texttt{SatEx}\ is a case study from our industrial partner, LuxSpace~\cite{Luxspace}, in the satellite domain.
Let \emph{time domain} $T=[0,b]$ be a non-singular bounded interval of $\mathbb{R}$. A \emph{signal} is a function $f: T \rightarrow \mathbb{R}$. We indicate individual signals using lower case letters, and \emph{sets} of signals using upper case letters. Let $\mathcal{M}$ be a MUT. We write $Y=\mathcal{M}(U)$ to indicate that the model $\mathcal{M}$ takes a set of signals $U=\{u_1, u_2 \ldots u_m\}$ as input and produces a set of signals $Y=\{y_1, y_2 \ldots y_n\}$ as output.
Each $u_i$ corresponds to one model input signal, and each $y_i$ corresponds to one model output signal. We use the notation $u_i(t)$ and $y_i(t)$ to, respectively, indicate the values of the input signal $u_i$ and the output signal $y_i$ at time $t$. For example, the \texttt{SatEx}\ model has four input signals indicating the temperatures perceived by the Magnetometer, Gyro, Reaction wheel and Magnetorquer components, and one output signal representing the orientation (a.k.a attitude) of the satellite.
To execute a Simulink MUT $\mathcal{M}$, the simulation engine receives signal inputs defined over a time domain and computes signal outputs at successive time steps over the same time domain used for the inputs. A test input for $\mathcal{M}$ is, therefore, a set of signal functions assigned to the input signals $\{u_1, u_2 \ldots u_m\}$ of $\mathcal{M}$. To generate signal functions, we have to generate values over the time interval $T=[0,b]$. This, however, cannot be done in a purely random fashion, since input signals are expected to conform to some specific shape to ensure dynamic properties pertaining to their semantic. For examples, input signals may be constant, piecewise constant, linear, piecewise linear, sinusoidal, etc. To address this issue, we parameterize each input signal $u_i$ by an interpolation function, a value range $R$ and a number $n$ of control points (with $n>2$). To generate a signal function for $u_i$, we then randomly select $n$ control points $u_i(t_1)$ to $u_i(t_{n})$ within $\mathbb{R}$ such that $t_1=0$, $t_n=b$ and $t_2$ to $t_{n-1}$ are from $T$ such that $t_1 < t_2 < \ldots < t_{n-1} < t_{n}$.
The values of $t_2 < t_3 < \ldots < t_{n-1}$ can be either randomly chosen or they can be fixed with equal differences between each subsequent pairs, i.e, $(t_{i+1}-t_i) = (t_{i}-t_{i-1}) $. The interpolation function is then used to connect the $n$ control points $u_i(t_1)$ to $u_i(t_n)$.
\ourtool\ currently supports several interpolation functions, such as
piecewise constant,
linear and piecewise cubic interpolation.
For each input $u_i$ of $\mathcal{M}$, we define a triple $\langle \mathit{int}_i, R_i, n_i \rangle$, where $\mathit{int}_i$ is an interpolation function, $R_i$ is the range of signal values and $n_i$ is the number of control points. We refer to the set of all such triples for all inputs $u_1$ to $u_m$ of $\mathcal{M}$ as an \emph{input profile} of $\mathcal{M}$ and denote it by $\texttt{IP}$. Provided with an input profile for a MUT $\mathcal{M}$, we can randomly generate test inputs for $\mathcal{M}$ as sets of signal functions for every input $u_1$ to $u_m$. For example, the input profile for \texttt{SatEx}\ provided by LuxSpace\ is reported in Table~\ref{table:rq3}, where $[-20,50]$, $[-15,50]$, $[-20,50]$, $[-20,50]$ are real value domains.
\textbf{Baseline falsification-based testing.}
The goal
is to produce a test input $U$ that, when executed on the MUT $\mathcal{M}$, reveals a violation of some requirement of $\mathcal{M}$. Algorithm~\ref{algo:fal} represents a high-level overview of falsification-based testing. It is a black-box testing process and includes three main components: (1)~a test input generation component (\Call{Generate}{} in Algorithm~\ref{algo:fal}), (2)~a test objective determining whether, or not, a requirement violation is identified (\Call{TObj}{} in Algorithm~\ref{algo:fal}), and (3)~a search strategy to traverse the search input space and select candidate tests (\Call{Search}{} in Algorithm~\ref{algo:fal}).
\begin{table}[t]
\caption{Input Profile for the \texttt{SatEx}\ case study.
}
\vspace{-0.3cm}
\label{table:rq3}
\scalebox{.8}{\begin{tabular}{c c c c c }
\toprule
& \bf Magnetometer & \bf Gyro & \bf Reaction wheel & \bf Magnetorquer \\
\midrule
$\mathit{int(n)}$ & \texttt{pchip(16)} & \texttt{pchip(16)} & \texttt{pchip(16)} & \texttt{pchip(16)} \\
\midrule
$R$ & [-20,50] & [-15,50] & [-20,50] & [-20,50] \\
\bottomrule
\end{tabular}}
\vspace{-0.3cm}
\end{table}
We describe \Call{Generate}{}, \Call{Search}{} and \Call{TObj}{}.
The input to the algorithm is a MUT $\mathcal{M}$ together with its input profile $\texttt{IP}$ and the maximum number $\texttt{MAX}$ of executions of MUT that can be performed within an allotted test budget time. Note that we choose the maximum number of executions as a loop terminating condition, but an equivalent terminating condition can be defined in term of maximum execution time.
\emph{Initial test Generation} (\Call{Generate}{}). It produces a (candidate) test input $U$ for $\mathcal{M}$ by randomly selecting control points within the
ranges
and applying the interpolation functions as specified in $\texttt{IP}$.
\emph{Iterative search} (\Call{Search}{}). It selects a new (candidate) test input $U$ from the search input space of $\mathcal{M}$. It uses the input profile $\texttt{IP}$ to generate new test inputs. The existing candidate test input $U$ may or may not be used in the selection of the new test input. In particular, $\Call{Search}{\mathcal{M}, \texttt{IP},U}$ can be implemented using different randomized or meta-heuristic search algorithms~\cite{Nghiem:2010:MTF:1755952.1755983,matinnejad2015search,FSE2019Comp}. These algorithms can be purely \emph{explorative} and generate the new test input randomly without considering the existing test input $U$ (e.g., Monte-Carlo search~\cite{Nghiem:2010:MTF:1755952.1755983}), or they may be purely \emph{exploitative} and generate the new test input by slightly modifying $U$ (e.g., Hill Climbing~\cite{matinnejad2015search,FSE2019Comp}). Alternatively, the search algorithm may combine both explorative and exploitative heuristics (e.g., Hill Climbing with random restarts~\cite{luke2013essentials}).
\algrenewcommand\algorithmicindent{0.4em}
\begin{algorithm}[t]
\caption{Baseline Falsification-based Testing.}
\label{algo:fal}
\begin{algorithmic}[1]
\Function{Falsification-Test}{$\mathcal{M}$, $\texttt{IP}$, $\texttt{MAX}$}
\Repeat
\If{$U$ is null}
\State $U$ = \Call{Generate}{$\mathcal{M}$, $\texttt{IP}$}; \Comment{Generate a candidate test input}
\Else
\State $U$ = \Call{Search}{$\mathcal{M}$, $\texttt{IP}$, $U$}; \Comment{Generate next candidate test input}
\EndIf
\State $Y = \mathcal{M}(U)$; \Comment{Execute $\mathcal{M}$ for $U$}
\If{ $\Call{TObj}{U,Y} \leq 0$} \Comment{Check if $U$ reveals a violation}
\State \textbf{return} $U$;
\EndIf
\Until{the number of executions of $\mathcal{M}$ reaches $\texttt{MAX}$}
\State \Return NULL; \label{ln:noinputfals}
\EndFunction
\end{algorithmic}
\end{algorithm}
\emph{Test objective} (\Call{TObj}{}). It maps every test input $U$ and its corresponding output $Y$, i.e., $Y = \mathcal{M}(U)$, into a test objective value $\Call{TObj}{U, Y}$ in the set $\mathbb{R}$ of real numbers. Note that computing test objective values requires simulating $\mathcal{M}$ for each candidate test input. We assume for each requirement of $\mathcal{M}$, we have a test objective $\Call{TObj}{}$ that satisfies the following conditions:
\begin{enumerate}[parsep=0pt,itemindent=0cm,leftmargin=3\parindent]
\item[$\Call{TObj}{}1$] If $\Call{TObj}{U, \mathcal{M}(U)}<0$, the requirement is violated;
\item[$\Call{TObj}{}2$] If $\Call{TObj}{U, \mathcal{M}(U)} \geq 0$, the requirement is satisfied;
\item[$\Call{TObj}{}3$] The more positive the test objective value, the farther the system from violating its requirement;
the more negative, the farther the system from satisfying its requirement.
\end{enumerate}
These conditions ensure that we can infer using the value of $\Call{TObj}{}$ whether a test cases passes or fails, and further, $\Call{TObj}{}$ serves as a distance function, estimating how far a test is from violating model requirements, and hence, it can be used to guide generation of test cases.
The robustness semantics of STL is an example of a semantics that satisfies those conditions~\cite{fainekos2009robustness}.
An example requirement for \texttt{SatEx}\ is:
\begin{enumerate}[leftmargin=3\parindent]
\item[SatReq] \emph{``the difference among the satellite attitude and the target attitude should not exceed 2 degrees"}.
\end{enumerate}
This requirement can be expressed in many languages including formal logics that predicate on signals, such as Signal Temporal Logics (STL)~\cite{maler2004monitoring} and Restricted Signals First-Order Logic (RFOL)~\cite{Socrates}.
For example, this requirement can be expressed in STL as
\begin{align}
\LTLg_{[0, \numprint{24}h]} \left( error<2 \right) \nonumber
\end{align}
where $error$ is the difference among the satellite attitude and the target attitude, $\LTLg$ is the ``globally" STL temporal operator which is parametrized with the interval $[0, \numprint{24}h]$, i.e., the property $error<2$ should hold for the entire simulation time ($\numprint{24}h$).
We define a test objective $\Call{TObj}{}$ for this requirement as
\begin{align}
\Call{TObj}{U, M(U)}=\underset{t \in [0, \numprint{24}h]}{min} \left( error(t)-2 \right) \nonumber
\end{align}
This is consistent with the robustness semantics of STL~\cite{fainekos2009robustness}.
This value ensures the conditions $\Call{TObj}{}1$, $\Call{TObj}{}2$ and $\Call{TObj}{}3$ since if the property is violated, i.e., there exists a time instant $t$ such that $error(t)-2<0$,
a negative value is returned.
In the opposite case, the property is satisfied and $\Call{TObj}{U, \mathcal{M}(U)}$ returns a non negative value.
Furthermore, the more positive the test objective value, the farther the system from violating its requirement; and the more negative, the farther the system from satisfying its requirement.
\vspace*{.2cm}
In our work, we use the S-Taliro tool~\cite{staliro} which implements the falsification-based testing shown in Algorithm~\ref{alg:algorithmaristeo}. S-Taliro is a well-developed, open source research tool for falsification based-testing and has been recently classified as ready for industrial deployment~\cite{7741019}. It has been applied to several realistic and industrial systems~\cite{10.1007/978-3-319-77935-5_30} and based on a recent survey on the topic~\cite{7741019} is the most mature tool for falsification of CPSs.
Further, S-Taliro supports a range of standard search algorithms such as Simulated Annealing, Monte Carlo~\cite{Nghiem:2010:MTF:1755952.1755983}, and gradient descent methods~\cite{abbas2014functional}.
Test objectives
can be defined manually. Alternatively, assuming that the requirements are specified in logic languages, test objectives satisfying the three conditions we described earlier can be generated automatically. In particular, we have identified two existing tools that generate quantitative test objectives from requirements encoded in logic-based languages: Taliro~\cite{fainekos2008user} and Socrates~\cite{Socrates}. In this paper, we use Taliro since it is integrated into
S-Taliro.
To do so, we specified our requirements into Signal Temporal logic (STL)~\cite{maler2004monitoring} and used Taliro to automatically convert them into quantitative test objectives capturing degrees of satisfaction and refutation conforming to our conditions $\Call{TObj}{}1$-$\Call{TObj}{}3$ on test objectives.
\section{\ourtool}
\label{sec:ourtool}
Algorithm~\ref{alg:algorithmaristeo} shows the approximation-refinement loop of \ourtool.
The algorithm relies on the following inputs: a CI-CPS model $\mathcal{M}$ (i.e., the model under test---MUT), the input profile $\texttt{IP}$ of MUT, and the maximum number of iterations $\texttt{MAX\_REF}$ that can be executed by \ourtool . In the first iteration, an initial surrogate model $\hat{\mathcal{M}}$ is computed such that it approximates the MUT behavior (Line~\ref{ln:abstraction}). Note that $\hat{\mathcal{M}}$ is built such that it has the same input profile as $\mathcal{M}$, i.e., $\hat{\mathcal{M}}$ and $\mathcal{M}$ have exactly the same inputs and outputs.
At every iteration, the algorithm applies falsification-based testing to the surrogate model $\hat{\mathcal{M}}$ in order to find a test input $U$ violating the requirement captured by the test objective $\Call{TObj}{}$ (Line~\ref{ln:falsification}). The number $\texttt{MAX}$ of iterations of falsification-based testing for $\hat{\mathcal{M}}$ is an internal parameter of \ourtool, and in general, can be set to a high value since executing $\hat{\mathcal{M}}$ is not expensive.
Once $U$ is found, the algorithm checks whether $U$ leads to a violation when it is checked on the MUT (Line~\ref{ln:conformanceCheck}). Recall from Section~\ref{sec:running} that test objectives $\Call{TObj}{}$ are defined such that a negative value indicates a requirement violation. If so, $U$ is returned as a
failure-revealing test for $\mathcal{M}$ (Line~\ref{ln:return}). Otherwise, $U$ is spurious and in the next iteration it is used to refine the surrogate model $\hat{\mathcal{M}}$ (Line~\ref{ln:refinement}).
If no failure-revealing test
for $\mathcal{M}$ is found after $\texttt{MAX\_REF}$ iterations the algorithm stops and a null value is returned.
\begin{algorithm}[t]
\caption{The \ourtool\ Main Loop.}
\label{alg:algorithmaristeo}
\begin{algorithmic}[1]
\Function{\ourtool }{$\mathcal{M}, \texttt{IP}, \texttt{MAX\_REF}$}
\Repeat
\If{$\hat{\mathcal{M}}$ is null}
\State $\hat{\mathcal{M}}$=\Call{Approximate}{$\mathcal{M}$}; \Comment{Generate a surrogate model} \label{ln:abstraction}
\Else
\State $\hat{\mathcal{M}}$=\Call{Refine}{$\hat{\mathcal{M}}$,$U$,$\mathcal{M}$}; \Comment{Refine the surrogate model} \label{ln:refinement}
\EndIf
\State $U$=\Call{Falsification-Test}{$\hat{\mathcal{M}}, \texttt{IP}, \texttt{MAX}$}; \label{ln:falsification}
\If{\Call{TObj}{$U$, $\mathcal{M}$($U$)} $\leq 0$} \Comment{Test $U$ finds a real violation} \label{ln:conformanceCheck}
\State \Return{$U$}; \label{ln:return}
\EndIf
\Until{the number of executions of $\mathcal{M}$ reaches $\texttt{MAX\_REF}$}
\State \Return NULL; \label{ln:noinput}
\EndFunction
\end{algorithmic}
\end{algorithm}
The falsification-based testing procedure is
described in Section~\ref{sec:running} (Algorithm~\ref{algo:fal}). In
Section~\ref{sec:abstractionAndRefinement}, we describe the \textsc{Approximate} method (line~\ref{ln:abstraction}), and in Section~\ref{sec:refinement}, we describe
the \textsc{Refine} method (line~\ref{ln:refinement}).
\begin{table*}[t]
\caption{Model structure and parameter choices for developing surrogate models.}
\vspace{-0.3cm}
\label{tab:modelstructures}
\scalebox{.8}{
\begin{tabular}{p{0.01\textwidth} p{0.17\textwidth} p{0.87\textwidth}
p{0.1\textwidth}}
\toprule
\multirow{35}{*}{\rotatebox[origin=c]{90}{\emph{Linear}}} & \textbf{Model Structure} & \textbf{Equation} & \textbf{Model Type} \\
\cmidrule{2-4}
& \texttt{arx($na,nb,nk$)}
&
$y(t)=a_1 \cdot y(t-1)+\ldots+a_{na} \cdot y(t-na)+b_1 \cdot u(t-nk)+\ldots+b_{nb} \cdot u(t-nb-nk+1)+e(t)$
&
Discrete\\
\cmidrule{2-4}
& \multicolumn{3}{p{1.2\textwidth}}{\textbf{Description}}\\
\cmidrule{2-4}
& \multicolumn{3}{p{1.2\textwidth}}{
The output $y$ depends on previous input values, i.e., $u(t-nk),$\ldots$,u(t-nb-nk+1)$, and on values assumed by the output $y$ in previous steps, i.e., $y(t-1),$\ldots $,y(t-na)$. $na$ and $nb$ are the number of past output and input values to be used in predicting the next output. $nk$ is the delay (number of samples) from the input to the output.
}
\\
\cmidrule[1pt]{2-4}
& \textbf{Model Structure} & \textbf{Equation} & \textbf{Model Type} \\
\cmidrule{2-4}
& \texttt{armax($na,nb,nk,nc$)}
&
$y(t)=a_1 \cdot y(t-1)+\ldots+a_{na} \cdot y(t-na)+b_1 \cdot u(t-nk)+\ldots+b_{nb} \cdot u(t-nb-nk+1)+c_1 \cdot e(t-1)+ \ldots +c_{nc} \cdot e(t-nc)+e(t)$
&
Discrete
\\
\cmidrule{2-4}
& \multicolumn{3}{p{1.2\textwidth}}{\textbf{Description}}\\
\cmidrule{2-4}
& \multicolumn{3}{p{1.2\textwidth}}{
Extends the \texttt{arx} model by considering how the values $e(t-1),$\ldots$,e(t-nc)$ of the noise $e$ at time $t$, $t-1$, $\ldots$, $t-nc$ influence the value $y(t)$ of the output $y$.
}\\
\cmidrule[1pt]{2-4}
& \textbf{Model Structure} & \textbf{Equation} & \textbf{Model Type} \\
\cmidrule{2-4}
& \texttt{bj($nb,nc,nf,nd,nk$)}
&
$y(t)= \frac{B(z)}{F(z)} \cdot u(t)+ \frac{C(z)}{D(z)} \cdot e(t)$
&
Discrete
\\
\cmidrule{2-4}
& \multicolumn{3}{p{1.2\textwidth}}{\textbf{Description}}\\
\cmidrule{2-4}
& \multicolumn{3}{p{1.2\textwidth}}{
Box-Jenkins models allow a more general noise description than \texttt{armax} models.
The output $y$ depends on a finite number of previous input $u$ and output $y$ values.
The values $n_b$, $n_c$, $n_d$, $n_f$, $n_k$ indicate the parameters of the matrix $B$, $C$, $D$, $F$ and the value of the input delay.
}\\
\cmidrule[1pt]{2-4}
& \textbf{Model Structure} & \textbf{Equation} & \textbf{Model Type} \\
\cmidrule{2-4}
& \texttt{tf($np,nz$)}
&
$y(t)= \frac{b_0+b_1 \cdot s+b_2 \cdot s^2 +\ldots + b_n \cdot s^{n_z}}{1+f_1 \cdot s + f_2 \cdot s^2+ \ldots + f_m \cdot s^{n_p}} \cdot u(t)+e(t)$
&
Continuous\\
\cmidrule{2-4}
& \multicolumn{3}{p{1.2\textwidth}}{\textbf{Description}}\\
\cmidrule{2-4}
& \multicolumn{3}{p{1.2\textwidth}}{
Represents a transfer function model.
The values $n_p$, $n_z$ indicate the number of poles and zeros of the transfer function.
} \\
\cmidrule[1pt]{2-4}
& \textbf{Model Structure} & \textbf{Equation} & \textbf{Model Type} \\
\cmidrule{2-4}
& \texttt{ss($n$)}
&
$x(0) = x0$ \newline
$\dot{x}(t) = Fx(t) + Gu(t) + Kw(t)$ \newline
$y(t) = Hx(t) + Du(t) + w(t)$
&
Continuous
\\
\cmidrule{2-4}
& \multicolumn{3}{p{1.2\textwidth}}{\textbf{Description}}\\
\cmidrule{2-4}
& \multicolumn{3}{p{1.2\textwidth}}{
Uses state variables to describe a system by a set of first-order differential or difference equations.
$n$ is an integer indicating the size of the matrix $F$, $G$, $K$, $H$ and $D$.
}\\
\toprule
\toprule
\multirow{16}{*}{\rotatebox[origin=c]{90}{\emph{Non Linear}}} & \textbf{Model Structure} & \textbf{Equation} & \textbf{Model Type} \\
\cmidrule{2-4}
& \texttt{nlarx($f,na,nb,nk$)}
&
$y(t) = f(y(t - 1), ..., y(t - na), u(t - nk), ..., u(t -nk -nb + 1))$
& Discrete \\
\cmidrule{2-4}
& \multicolumn{3}{p{1.2\textwidth}}{\textbf{Description}}\\
\cmidrule{2-4}
& \multicolumn{3}{p{1.2\textwidth}}{
Uses a non linear function $f$ to describe the input/output relation.
Wavelet, sigmoid networks or neural networks in the Deep Learning Matlab Toolbox~\cite{deep} can be used to compute the function $f$.
$na$ and $nb$ are the number of past output
and input
values used to predict the next output value.
$nk$ is the delay from the input to the output.} \\
\cmidrule[1pt]{2-4}
& \textbf{Model Structure} & \textbf{Equation} & \textbf{Model Type} \\
\cmidrule{2-4}
& \texttt{hw($f,h,na,nb,nk$)}
&
$w(t)= f(u(t))$ \newline
$x(t)=(B(z)/F(z))\cdot w(t)$\newline
$y(t)= h(x(t))$ &
Continuous \\
\cmidrule{2-4}
& \multicolumn{3}{p{1.2\textwidth}}{\textbf{Description}}\\
\cmidrule{2-4}
& \multicolumn{3}{p{1.2\textwidth}}{
Hammerstein-Wiener models describe dynamic systems two nonlinear blocks in series with a linear block. Specifically, $f$ and $h$ are non linear functions,
$B(z)$, $F(z)$, $na$, $nb$, $nk$ are defined as for \texttt{bj} models.
Different nonlinearity estimators can be used to learn $f$ and $h$ similarly to the \texttt{nlarx} case.
} \\
\bottomrule
\end{tabular}}
\end{table*}
\subsection{Approximation}
\label{sec:abstractionAndRefinement}
Given an MUT $\mathcal{M}$, the goal of the approximation
is to produce a surrogate model $\hat{\mathcal{M}}$ such that:
\begin{enumerate*}[label=(\roman*)]
\item[\textbf{(C1)}] $\mathcal{M}$ and $\hat{\mathcal{M}}$ have the same interface, i.e., the same inputs and outputs;
\item[\textbf{(C2)}] provided with the same input values, they generate similar output values; and
\item[\textbf{(C3)}] $\hat{\mathcal{M}}$ is less expensive to execute than $\mathcal{M}$
\end{enumerate*}.
We rely on System Identification (SI) techniques to produce surrogate models~\cite{soderstrom1989system} since their purpose is to automatically build mathematical models of dynamical systems from data when it is difficult to build the models analytically, or when engineers want to build models from data obtained based on measurements of the actual hardware.
Note that the more complex SI structures (i.e., non-linear \texttt{nlarx} and \texttt{hw}) rely on machine learning and neural network algorithms~\cite{ljung2008system}.
To build $\hat{\mathcal{M}}$ using SI, we need some input and output data from the MUT $\mathcal{M}$. Since $\mathcal{M}$ is expensive to execute, to build the initial surrogate model $\hat{\mathcal{M}}$ (line~\ref{ln:refinement}), we run $\mathcal{M}$ for one input $U$ only. Note that an input $U$ of $\mathcal{M}$ is a set $\{u_1, \ldots, u_m\}$ of signal functions over $T=[0,b]$. So, each $u_i$ is a sequence $u_i(0), u_i(\delta), u_i(2\cdot \delta) \ldots u_i(l\cdot \delta)$ where $b = l\cdot \delta$ and $\delta$ is the sampling rate applied to the time domain $[0,b]$. Similarly, the output $Y = \mathcal{M}(U)$ is a set $\{y_1, \ldots, y_n\}$ of signal functions where each $y_j$ is a sequence $y_j(0), y_j(\delta), y_j(2\cdot \delta) \ldots y_j(l\cdot \delta)$ obtained based on the same sampling rate and the same time domain as those used for the input. We refer to the data used to build $\hat{\mathcal{M}}$ as \emph{traning data} and denote it by $\mathcal{D}$. Specifically,
$\mathcal{D}=\langle U, Y \rangle$.
For CI-CPS, the size $l$ of $\mathcal{D}$
tends to be large since we typically execute such models for a long time duration (large $b$) and use a small sampling rate (small $\delta$) for them. For example, we typically run \texttt{SatEx}\ for $b = \numprint{86400}$s ($24$h) and use the sampling rate $\delta = 0.0312$s. Hence, a single execution of \texttt{SatEx}\ generates a training data set $\mathcal{D}$ with size $l=\numprint{2769200}$.
Such training data size is sufficient for SI to build reasonably accurate surrogate models.
We use the System Identification Toolbox~\cite{ljung2008system} of Matlab to generate surrogate models. In order to effectively use SI, we need to anticipate the expected \emph{structure} and \emph{parameters} of surrogate models, a.k.a \emph{configuration}. Table~\ref{tab:modelstructures} shows some standard model structures and parameters supported by SI.
Specifically, selecting the model structure is about deciding which mathematical equation among those shown in Table~\ref{tab:modelstructures} is more likely to fit to our training data and is better able to capture the dynamics of the model $\mathcal{M}$. As shown in Table~\ref{tab:modelstructures}, equations specifying the model structure have some parameters that need to be specified so that we can apply SI techniques. For example, for \texttt{arx($na,nb,nk$)}, the values of the parameters $na$, $nb$ and $nk$ are the model parameters.
Table~\ref{tab:modelstructures} provides a short description for each model structure.
We note that some of the equations in the table are simplified and refer to the case in which the MUT has a single input signal
and a single output signal.
The equations, however, can be generalized to models with
multiple input and output signals. Briefly, model structures can be linear or non-linear in terms of the relation between the inputs and outputs, or they can be continuous and discrete in terms of their underlying training data. Specifically, the training data generated from MUT can be either discrete (i.e., sampled at a fixed rate) or continuous (i.e., sampled at a variable rate). Provided with discrete training data, we can select either continuous or discrete model structures, while for continuous training data, we can select continuous model structures only. As discussed earlier, our training data $\mathcal{D}$ is discrete since it is sampled at the fix sampling rate of $\delta$. Hence, we can choose both types of model structures to generate surrogate models.
In our work we
support training data sampled at a fixed sampling rate to build and refine the surrogate models.
Data sampled at a variable time rate can be then handled by exploiting the resampling procedure of Matlab~\cite{Resampling}.
The users of \ourtool\
need to choose upfront the configuration to be used by the SI, i.e., the model structure and the values of its parameters.
This choice depends on domain specific knowledge that the engineers possess for the model under analysis.
The values of the parameters selected by the user should be chosen such that the resulting surrogate model (i) has the same interface as the MUT to ensure \textbf{C1} and (ii) has a simpler structure than the MUT to ensure \textbf{C3}.
The System Identification Toolbox provides some generic guidance for selecting the parameters ensuring these two criteria~\cite{modelStructure}.
In this work we performed an empirical evaluation over a set of benchmark models to determine the configuration to be used in our experiments (Section~\ref{sec:modelstructure}).
Once a configuration
is selected, SI uses the training data to learn values for the coefficients of the equation from Table~\ref{tab:modelstructures} that corresponds to the selected structure and paramters. For example, after selecting \texttt{arx($na,nb,nk$)} and assigning values to $na$, $nb$ and $nk$, SI generates a surrogate model by learning values for the coefficients: $a_1,\ldots a_{na}$ and $b_1,\ldots b_{nb}$.
Similar to standard machine learning algorithms, SI's objective is to compute the model coefficients by minimizing the difference (error) between the outputs of $\mathcal{M}$ and $\hat{\mathcal{M}}$ for the training data~\cite{soderstrom1989system}. SI uses different standard notions of errors depending on the model structure selected. In our work, we compute the Mean Squared Error (MSE)~\cite{soderstrom1989system} between the outputs of $\mathcal{M}$ and $\hat{\mathcal{M}}$.
SI
learns a surrogate model $\hat{\mathcal{M}}$ by minimizing MSE over the training data $\mathcal{D}$ and hence, ensuring \textbf{C2}. The learning algorithm selected by SI depends on the chosen model structure, on the purpose of the identification process, i.e., whether the identified model will be used for prediction or simulation, and on whether the system is continuous or discrete.
\subsection{Refinement}
\label{sec:refinement}
The refinement step rebuilds the surrogate model $\hat{\mathcal{M}}$ when the test input $U$ obtained by falsification-based testing of the surrogate model is
spurious for MUT (i.e., it does not reveal any failure according to the test objective). Note that $\hat{\mathcal{M}}$ may not be sufficiently accurate to predict the behavior of the MUT. Hence, it is likely that we need to improve its accuracy and we do so by reusing the data obtained when checking a candidate test input $U$ on MUT (line~\ref{ln:conformanceCheck} of Algorithm~\ref{alg:algorithmaristeo}).
Let $U=\{u_1, \ldots, u_m\}$ and $Y=\{y_1, \ldots, y_n\}$ be the spurious test inputs and its output, respectively.
Similar to the data used to build the initial $\hat{\mathcal{M}}$ by the approximate step (line~\ref{ln:conformanceCheck} of Algorithm~\ref{alg:algorithmaristeo}),
the data $\mathcal{D}'=\langle U,Y \rangle$ used to rebuild $\hat{\mathcal{M}}$ is also discretized based on the same sampling rate $\delta$.
To refine the surrogate model, we do not
change
the considered configuration,
but we combine the new
$\mathcal{D}'$
and existing training data $\mathcal{D}$, and refine $\hat{\mathcal{M}}$ using these data.
Alternative policies can be chosen to refine the surrogate model.
For example, the refinement activity may also change the configuration of \ourtool .
This is
a rather drastic change in the surrogate model.
When engineers have a
clear understanding of the underlying model, they may be able to define a systematic methodology on how to move from less complex structures (e.g., linear) to more complex ones (e.g., non-linear). Without proper domain knowledge, such modification may
be too disruptive.
In this paper, our refinement strategy is focused on incrementing the training data and rebuilding the surrogate model without changing the
configuration.
\section{Evaluation}
\label{sec:evaluation}
In this section, we empirically evaluate \ourtool\ by answering the following research questions:
\noindent $\bullet$ \textbf{Configuration - RQ1.} \emph{Which are the optimal (most effective and efficient) SI configurations for \ourtool ? Which of the optimal configurations can be used in the rest of our experiments?}
We investigate the performance of \ourtool\ for different SI configurations (model structures and parameters listed in Table~\ref{tab:modelstructures}) to identify the optimal ones, i.e., those that offer the best trade-offs between effectiveness (revealing the most requirements violations) and efficiency (revealing the violations in less time). We then select one configuration among the optimal ones and use that configuration for the rest of our experiments.
\noindent $\bullet$ \textbf{Effectiveness - RQ2.} \emph{How \emph{effective} is \ourtool\ in generating tests that reveal requirements violations?} We use \ourtool\ with the optimal configuration identified in {\bf RQ1} and evaluate its effectiveness (i.e., its ability in detecting requirements violations) by comparing it with falsification-based testing without surrogate models. We use S-Taliro discussed in Section~\ref{sec:running} for the baseline of comparison.
\noindent $\bullet$ \textbf{Efficiency - RQ3.} \emph{How \emph{efficient} is \ourtool\ in generating tests revealing requirements violations?} We use \ourtool\ with the optimal configuration identified in {\bf RQ1} and evaluate its efficiency (i.e., the time it takes to find violations) by comparing it with falsification-based testing without surrogate models (i.e., S-Taliro).
A key challenge regarding the empirical evaluation of \ourtool\ is that, both \ourtool\ and S-Taliro rely on randomized algorithms. Hence, we have to repeat our experiments numerous times for different models and requirements so that the results can analysed in a sound and systematic way using statistical tests~\cite{Hitchhiker}. This is necessary to answer RQ1-RQ3 that involve selecting an optimal configuration and comparing \ourtool\ with the baseline S-Taliro. Performing these experiments on CI-CPS models is, however, extremely expensive, to the point that the experiments become infeasible. A ballpark figure for the execution time of the experiments required to answer RQ1-RQ3 is around 50 years if the experiments are performed on our CI-CPS model case study (\texttt{SatEx}). Therefore, instead of using CI-CPS models, we use non-CI-CPS models to address RQ1-RQ3. The implications of this decision on the results are assessed and mitigated in Sections~\ref{sec:modelstructure} and~\ref{subsec:rq2-rq3} where we discuss these three research questions in detail.
In addition, to be able to still assess the performance of \ourtool\ on CI-CPS models, we consider an additional research question described below:
\begin{table}[t]
\caption{Non-CI-CPS subject models.
\textbf{ID}: model identifier;
\textbf{\#B}: number of blocks of the Simulink model;
\textbf{\#I}: number of inputs of the Simulink model;
\textbf{int(n)}: input interpolation functions and number of control points;
\textbf{R}: input ranges; and
\textbf{T}: time domain.
}
\vspace{-0.3cm}
\label{table:benchmarkmodels}
\scalebox{.8}{\begin{tabular}{ c c c c c c }
\toprule
\bf ID & \bf \#B & \bf \#I & \bf int(n) & \bf R & \bf T \\
\toprule
RHB(1) & 28 & 1 & \texttt{pchip(4)} & $[-2, 5]^*$ & 24 \\
\midrule
RHB(2) & 31 & 2 &\makecell{\texttt{pchip(4)},\texttt{const(1)}} & \makecell{$[-2, 5]$, $[0.8, 1.2]$}& 24 \\
\midrule
AT& 63 & 1 &\makecell{\texttt{pconst(7)}} & $[0,100]$ & 30 \\
\midrule
AFC & 302 & 2 &\makecell{\texttt{const(1)},\texttt{pulse(10)}} &
\makecell{$ [900,1100] $, $[0,61.1]$} & 50\\
\midrule
IGC & 70 & 10 &
\makecell{\texttt{const(1)},\texttt{const(1)},\\
\texttt{const(1)},\texttt{const(1)},\\
\texttt{const(1)},\texttt{const(1)},\\
\texttt{const(1)},\texttt{const(1)},\\
\texttt{const(1)},\texttt{const(1)}}
&
\makecell{$[40,40]$,$[30,30]$,\\
$[200,200]$,$[40,40]$,\\
$[150,250]$,$[0,80]$,\\
$[20,50]$,$[100,300]$,\\
$[20,70]$,$[-0.3,0.3]$} & 400 \\
\bottomrule
\end{tabular}}
\end{table}
\noindent $\bullet$ \textbf{Usefulness - RQ4.} \emph{How applicable and useful is \ourtool\ in generating tests revealing requirements violations for industrial CI-CPS models?} We apply \ourtool\ with the optimal configuration identified in {\bf RQ1} to our CI-CPS model case study from the satellite industry (\texttt{SatEx}) and evaluate its effectiveness and efficiency. The focus here is to obtain representative results in terms of effectiveness and efficiency based on an industry CI-CPS model. Note that we still apply S-Taliro to \texttt{SatEx}\ to be able to compare it with \ourtool\ for an industry CI-CPS model. This comparison, however, is not meant to be subject to statistical analysis due to the large execution time of \texttt{SatEx}, and is only meant to complement RQ3 with a fully realistic though extremely time consuming study.
\textbf{The subject models.} We used five publicly available non-CI-CPS models (i.e., RHB(1), RHB(2), AT, AFC, IGC)
that have been previously used in the literature on falsification-based testing of CPS models~\cite{fehnker2004benchmarks,dang2004verification,zhao2003generating,jin2014powertrain,sankaranarayanan2012simulating,ernst2019arch}.
The models represent realistic and representative models of CPS systems from different domains.
RHB(1) and RHB(2)~\cite{fehnker2004benchmarks} are from the IoT and smart home domain.
AFC~\cite{jin2014powertrain} is from the automotive domain and has been originally developed by Toyota.
AT~\cite{zhao2003generating} is another model from the automotive domain.
IGC~\cite{sankaranarayanan2012simulating} is from the health care domain.
AT and AFC have also been recently considered as a part of the reference benchmarks in the ARCH competition~\cite{ernst2019arch} -- an
international competition among verification and testing tools for continuous and hybrid systems~\cite{cpsweek}.
In Table~\ref{table:benchmarkmodels}, we report the number of blocks and inputs, the input profiles, input ranges and simulation times for the five non-CI-CPS models.
These models have been manually developed and may violate their requirements due to human error. Some of the violations have been identified by the existing testing tools and are reported in the literature~\cite{fehnker2004benchmarks,jin2014powertrain,zhao2003generating,sankaranarayanan2012simulating,ernst2019arch}.
As for the CI-CPS model to address RQ4, we use the \texttt{SatEx}\ case study that we introduced as a running example in Sections~\ref{sec:running} and \ref{sec:ourtool}.
\texttt{SatEx}\ contains $2192$ blocks and has to be simulated for $24$h for each test case to sufficiently exercise the system dynamics and interactions with the environment. Like the models in Table~\ref{table:benchmarkmodels}, \texttt{SatEx}\ is manually developed by engineers and is likely to be faulty. Its inputs and input profiles are shown in Table~\ref{table:rq3}.
\textbf{Implementation and Data Availability.} We implemented \ourtool\ as a Matlab application and as an add-on of S-Taliro.
Our (sanitized) models, data and tool are available online~\cite{appedix} and are also submitted alongside the paper.
\subsection{RQ1 - Configuration}
\label{sec:modelstructure}
Recall that \ourtool\ requires to be provided with a configuration to build surrogate models.
The universe of the possible configurations is infinite as the model structures in Table~\ref{tab:modelstructures} can be parametrized in an infinite number of ways by associating different values to their parameters.
RQ1 identifies the optimal configurations that yield the best tradeoff between effectiveness and efficiency for \ourtool\ among a reasonably large set of
alternative representative configurations.
It then selects one among the optimal configurations.
We do not evaluate configurations by measuring their prediction accuracy (i.e., by measuring their prediction error when applied to a set of test data as is common practice in assessing prediction models in the machine learning area~\cite{bishop2006pattern})
because our focus is not to have the most accurate configuration but the one that is able to have the most effective impact on \ourtool's approximation-refinement loop by quickly finding requirements violations.
However, it is likely that there exists a relationship between the two.
\textbf{Experiment design.}
We consider five different configurations obtained by five different sets of parameter values for each model structure in Table~\ref{tab:modelstructures}. We denote the five configurations related to each model structure $S$ by $S_1$ to $S_5$. For example, the configurations related to the model structure \texttt{ss} are denoted by \texttt{ss}$_1$ to \texttt{ss}$_5$. The specific parameter value sets for the 35 configurations based on the seven model structures in Table~\ref{tab:modelstructures} are available online~\cite{appedix}.
To answer RQ1, we apply \ourtool\ to the five non-CI-CPS models using each configuration among the 35 possible ones. That is, we execute \ourtool\ for 175 times. We further rerun each application of \ourtool\ for 100 times to account for the randomness in both falsification-based testing and the approximation-refinement loop of \ourtool~\cite{8453180}.
We set the value of \texttt{MAX\_REF}, i.e, the number of iterations of the \ourtool's main loop, to 10 (see Algorithm~\ref{alg:algorithmaristeo}) and the value of \texttt{MAX}, i.e, the number times each iteration of \ourtool\ executes falsification-based testing (see Algorithm~\ref{algo:fal}), to $100$ for RHB(1), RHB(2) and AFC, and to $1000$ for AT and IGC. These values were used
in the original experiments that apply falsification-based testing to these models~\cite{benchmarkconfigurations}. Running all the 17,500 experiments required
\numprint{4315567}\ hours ($\approx$ \numprint{99}\ days).\footnote{
We used the high performance computing cluster at [location redacted]
with $100$ Dell PowerEdge C6320 and a total of 2800 cores with 12.8 TB RAM.
The parallelization reduced the experiments time to approximately $15$ days.}
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{h1.pdf}
\vspace{-0.7cm}
\caption{Effectiveness and efficiency of different configurations across our non-CI-CPS subject models.}
\label{fig:RQ1}
\end{figure}
Due the sheer size of the experiments required to answer RQ1, we used our non-CI-CPS subject models.
While these models are smaller than typical CI-CPS models,
the complexity of their structure (how Simulink blocks are used and connected) is similar to the one of \texttt{SatEx}.
Specifically, the structural complexity index~\cite{plkaska2009quality,olszewska2011simulink}, which provides an estimation of the complexity of the structure of a Simulink model,
is $1.8$, $1.6$, $1.2$, $1.1$, $2.1$ for the RHB(1), RHB(2), AT, AFC and IGC benchmarks, respectively, and $1.5$ for the \texttt{SatEx}\ case study.
We conjecture that given these similarities, the efficiency and effectiveness comparisons of the configurations performed on non-CI-CPS models would likely remain the same should the comparisons be performed on CI-CPS models. However, due to computational time restrictions, we are not able to check this conjecture. Finally, we note that even if we select a sub-optimal configuration, it will be a disadvantage for \ourtool. So, the results for RQ2-RQ4 are likely to improve if we find a way to identify a better configuration for \ourtool\ using CI-CPS models.
\textbf{Results.}
The scatter plot in Figure~\ref{fig:RQ1} shows the results of our experiments.
The x-axis
indicates our \emph{efficiency metric} which is defined as \emph{the number of iterations that \ourtool\ requires to reveal a requirement violation} in a model for a given configuration.
As described in the experiment design, the maximum number of iterations is 10. Given a configuration for \ourtool, the fewer iterations required to reveal a violation, the more efficient that configuration is.
The y-axis
indicates our \emph{effectiveness metric} which is defined as \emph{the number of \ourtool\ runs (out of 100) that can reveal a violation} in a model.
For effectiveness
we are interested to know how often we are able to reveal a requirement violation.
The higher the number of runs detecting violations, the more effective that configuration is. The ideal configuration is the one that
finds requirements violations in 100\% of the runs in just one iteration as indicated by the origin
of the plot in Figure~\ref{fig:RQ1} with coordinates $(1, 100)$.
For each configuration, there is one point in the plot in Figure~\ref{fig:RQ1} whose coordinates, respectively, indicate the average efficiency and effectiveness of that configuration for the non-CI-CPS subject models. As shown in the figure,
bj$_1$ and ss$_2$ are on the Pareto frontier~\cite{pareto} and dominate other configurations in terms of efficiency and effectiveness. That is, any configuration other than bj$_1$ and ss$_2$ is strictly dominated in terms of both efficiency and effectiveness by either bj$_1$ or ss$_2$.
But bj$_1$ does not dominate ss$_2$, and neither does ss$_2$. Specifically, bj$_1$ is more efficient but less effective than ss$_2$, and ss$_2$ is less efficient but more effective than bj$_1$. For our experiments, we select bj$_1$ as the optimal configuration
since efficiency is paramount when dealing with CI-CPS models. In terms of effectiveness, bj$_1$ is only slightly less effective than ss$_2$ (46.4\% versus 52.4\%).
\resq{The answer to \textbf{RQ1} is that, among all the 35 configurations we compared, the bj$_1$ and the ss$_2$ configurations are the optimal configurations offering the best trade-off between efficiency (i.e., time required to reveal requirements violations) and effectiveness (i.e., number of violations revealed) for \ourtool. We select bj$_1$ as we prioritize efficiency.}
\subsection{RQ2 and RQ3 - Effectiveness and Efficiency}
\label{subsec:rq2-rq3}
For RQ2 and RQ3, we compare \ourtool\ (Algorithm~\ref{alg:algorithmaristeo}) with S-Taliro (Algorithm~\ref{algo:fal}). As discussed earlier, due to the large size of the experiments, we use non-CI-CPS models, \emph{but we want to obtain results that are representative for the CI-CPS case}. For such comparisons, we need to execute both tools for an equivalent amount of time and then compare their effectiveness and efficiency.
This is a non trivial problem, because
\begin{itemize}
\item That equivalent amount of time cannot simply translate into identical \emph{execution times}. Non-CI-CPS models, by definition, are very quick to execute. Hence, the benefits of performing the falsification on the surrogate model, as done by \ourtool , would not be visible if we compared the two tools
based on the execution times of non-CI-CPS models. Therefore, comparisons would be in favour of S-Taliro if we fix the execution times of the two tools for non-CI-CPS models.
\item Neither can we can run the two tools for the same \emph{number of iterations}, as commonly done in this domain~\cite{ernst2019arch}, because one iteration of \ourtool\ takes more time than one iteration of S-Taliro. Recall that \ourtool, in addition to performing falsification, builds and refines surrogate models in each iteration. Thus, by fixing the number of iterations for the two tools, comparisons would be in favour of \ourtool.
\end{itemize}
To answer RQ2 and RQ3 without favouring neither of the tools, we propose the following:
Suppose that we could perform {\bf RQ2} and {\bf RQ3} on a CI-CPS model, and that we execute \ourtool\ and S-Taliro on this model for the same time limit $TL$. Let $IA$ and $IB$ be the number of iterations of \ourtool\ and S-Taliro within $TL$, respectively. Recall that one iteration of \ourtool\ typically takes more time than one iteration of the baseline ($IA < IB$). If we know the values of $IA$ and $IB$, we can execute \ourtool\ for $IA$ times and S-Taliro for $IB$ times on non-CI-CPS models and use the results to compare the tools as if they were executing on CI-CPS models.
To run our experiment,
we need to know the relation between $IA$ and $IB$. We approximate this relation empirically using our \texttt{SatEx}\ CI-CPS model. We execute \ourtool\ for $10$ iterations and we set the number of falsification iterations in each iteration of \ourtool\ to $100$ as suggested by the literature on CPS falsification testing~\cite{staliro,benchmarkconfigurations} (i.e., \texttt{MAX\_REF} = 10 and \texttt{MAX} = $100$ in Algorithm~\ref{alg:algorithmaristeo}). We repeated these runs of \ourtool\ for five times. The first iteration of \ourtool\ took, on average, \numprint{16902}s, and the subsequent iterations of \ourtool\ took, on average, \numprint{9865}s. Note that the first iteration of \ourtool\ is always more expensive than the subsequent iterations since \ourtool\ builds surrogate models in the first iteration. Similarly, we executed S-Taliro for 10 iterations on \texttt{SatEx}, and repeated this run for five times. Each iteration of S-Taliro took, on average, \numprint{8336}s on \texttt{SatEx}.
This preliminary experiment took approximately 20 days.
We then solve the two equations below to approximate the relation between $IA$ and $IB$:
\begin{align}
& TL = \numprint{9865} \times (IA-1) +\numprint{16902} &&\label{eq1}\\[-3pt]
& TL = \numprint{8336} \times IB&& \label{eq2}
\end{align}
The above yields $IB = 1.2 \times IA + 0.8$. Though we obtained this relation between $IA$ and $IB$ based on one CI-CPS case study, \texttt{SatEx}\ is a large and industrial system representative of the CPS domain. Further, for CI-CPS models that are more compute-intensive than \texttt{SatEx}, executing the models takes even more time compared to the approximation and refinement time, and hence, the relation above could be further improved in favour of \ourtool.
\begin{figure*}[ht]
\includegraphics[width=\textwidth]{RQ3.pdf}
\vspace{-0.6cm}
\caption{Comparing the efficiency of \ourtool\ and S-Taliro.
The box plots show the execution time (computed using equations~\ref{eq1} and~\ref{eq2}) of \ourtool\ and S-Taliro (in hours) for our non-CI-CPS subject models (labels on the left of the figure)
and over different iterations (labels on the top of the figure). Diamonds depict the average.}
\label{fig:efficiency}
\end{figure*}
\begin{table}[t]
\caption{The effectiveness results.
Percentages of cases in which \ourtool\ (IA$_i$ labelled columns) and S-Taliro (IB$_i$ labelled columns) were able to detect requirements violations for different iteration pairs (IA$_i$ and IB$_i$) and benchmarks.}
\vspace{-0.3cm}
\label{table:rq1results}
\scalebox{.68}{
\begin{tabular}{c | c c | c c | c c | c c | c c | c c }
\toprule
& IA$_1$ & IB$_1$ & IA$_2$ & IB$_2$ & IA$_3$ & IB$_3$ & IA$_4$ & IB$_4$ & IA$_5$ & IB$_5$ & IA$_6$ & IB$_6$ \\
\midrule
RHB(1) & 0\% & 5\% & 2\% & 2\% & 8\% & 8\% & 7\% & 7\% & 11\% & 6\% & 9\% & 8\% \\
RHB(2) & 5\% & 2\% & 6\% & 8\% & 4\% & 9\% & 5\% & 10\% & 13\% & 10\% & 7\% & 10\% \\
AT & 85\% & 7\% & 92\% & 7\% & 93\% & 7\% & 99\% & 4\% & 100\% & 8\% & 100\% & 13\% \\
AFC & 100\% & 77\% & 100\% & 73\% & 100\% & 88\% & 100\% & 86\% & 100\% & 92\% & 100\% & 95\% \\
IGC & 33\% & 4\% & 31\% & 6\% & 34\% & 9\% & 37\% & 15\% & 40\% & 18\% & 13\% & 21\% \\
\bottomrule
\end{tabular}}
\vspace{-0.3cm}
\end{table}
\textbf{Experiment design.}
To answer \textbf{RQ2} and \textbf{RQ3}, we applied \ourtool\ with the configuration identified by \textbf{RQ1} (bj$_1$) and S-Taliro to the five non-CI-CPS models in Table~\ref{table:benchmarkmodels}. We executed \ourtool\ and S-Taliro for the following pairs of iterations: $\langle IA_1=5 , IB_1 =7 \rangle$, $\langle IA_2=7, IB_2 =9 \rangle$, $\langle IA_3=9, IB_3 =12 \rangle$, $\langle IA_4=11, IB_4 =14 \rangle$, $\langle IA_5=13, IB_5 =16 \rangle$, and $\langle IA_6=15, IB_6 =19 \rangle$. Note that every pair approximately satisfies $IB_i = 1.2 \times IA_i + 0.8$. We repeated each run 100 times to account for their randomness.
For \textbf{RQ2}, we compute the \emph{effectiveness metric} as in {\bf RQ1}: the number of runs revealing requirements violations (out of 100) for each tool.
For \textbf{RQ3}, we assess efficiency by computing the \emph{efficiency metric} as in {\bf RQ1}: the number of iterations that each tool requires to reveal a requirement violation. However, as discussed above, the number of iterations of
\ourtool\ and S-Taliro are not comparable. Hence, for \textbf{RQ3}, we report efficiency in terms of the estimated time that each tool needs to perform those iterations on CI-CPS models computed using equations~\ref{eq1} and~\ref{eq2}.
\textbf{Results-RQ2.}
Table~\ref{table:rq1results} shows the effectiveness values for \ourtool\ and S-Taliro for the five iteration pairs discussed in the
experiment design. For the AT, AFC and IGC models, the average effectiveness of \ourtool\ is significantly higher
than that of S-Taliro ($75.4\%$ versus $35.0\%$ on average across benchmarks), while for RHB(1) and RHB(2), \ourtool\ and S-Taliro
reveal almost the same number of violations ($6.4\%$ versus $7.0\%$ on average across benchmarks).
The former difference in proportion is statistically significant as confirmed by a two-sample z-test~\cite{mcdonald2009handbook} with the level of significance ($\alpha$) set to 0.05.
RHB(1) and RHB(2) have more outputs than the other benchmarks and they have
shorter simulation times (see Table~\ref{table:benchmarkmodels}). This is an increased challenge for building accurate surrogate models. In practice, CI-CPS models can have a large number of outputs but they usually involve long simulation times.
\resq{The answer to \textbf{RQ2} is that \ourtool\ is significantly more effective than S-Taliro for three benchmark models while, for the other two models, they reveal almost the same number of violations.
On average, over the five models, \ourtool\ detects $23.9\%$ more requirements violations than S-Taliro (min=-8\%, max=95\%).
}
\textbf{Results-RQ3.}
The execution times (computed using equations~\ref{eq1} and~\ref{eq2}) of \ourtool\ and S-Taliro for our non-CI-CPS subject models
and the iteration pairs $\langle IA_i, IB_i \rangle$ are shown in Figure~\ref{fig:efficiency}.
The box plots in the same row are related to the same benchmark model, while the box plots in the same column are related to the same iteration pair. Recall that we described the iteration pairs $\langle IA_i, IB_i \rangle$ considered for our experiments earlier in the experiment design subsection.
As expected,
the average execution times of the two tools increases with their number of iterations.
To statistically compare the results, we used the Wilcoxon rank sum test~\cite{mcdonald2009handbook} with the level
of significance ($\alpha$) set to $0.05$. The results show that
\ourtool\ is significantly more efficient than S-Taliro for the AT and IGC models (Figure~\ref{fig:efficiency} -- rows 3,5). The efficiency improvement that
\ourtool\ brings about over S-Taliro for AT and IGC across different iterations ranges from $14.4\%$ ($2.2$h) to $73.1\%$ ($31.2$h). Note that, for AT and IGC, \ourtool\ is significantly more effective than S-Taliro (see Table~\ref{table:rq1results}). This shows that, many runs of \ourtool\ for AT and IGC can reveal a requirement violation and stop before reaching the maximum ten iterations, hence yielding better efficiency results of \ourtool\ compared to the other model.
For the RHB(1) and RHB(2) models (Figure~\ref{fig:efficiency} -- rows 1,2), \ourtool\ and S-Taliro yield comparable efficiency results.
The effectiveness results in Table~\ref{table:rq1results} confirm that, for RHB(1) and RHB(2), both \ourtool\ and S-Taliro have to execute for ten iterations most of the times as they cannot reveal violations (low effectiveness). Hence, the efficiency results are worse for RHB(1) and RHB(2) than for the other models. Further, as we run the tools for more iterations, the efficiency results slightly increases as indicated by the increase in the number of outliers. For the AFC model (Figure~\ref{fig:efficiency} -- row 4), \ourtool\ is slightly more efficient than S-Taliro. For AFC, S-Taliro is relatively effective in finding violations, and hence, is efficient. But, its average execution time is slightly worse than that of ARIsTEO. Comparing the interquartile ranges of the box plots shows that ARIsTEO is generally more efficient that S-Taliro. However, a Wilcoxon test does not reject the null hypothesis (p-value = 0.06).
The average execution time of \ourtool\ and S-Taliro across the different models is, respectively, approximately $19$h and $25$h.
Though there is significant variation across the different models, \ourtool\ is, on average, $31.3\%$ more efficient than S-Taliro.
\resq{The answer to \textbf{RQ3} is that \ourtool\ is on average $31.3\%$ (min=$-1.6\%$, max=$85.2\%$) more efficient than S-Taliro.
}
\subsection{RQ4 - Practical Usefulness}
We assess the usefulness of \ourtool\ in revealing requirements violations of a representative industrial CI-CPS model.
\textbf{Experiment design.} We received three different requirements from our industry partner~\cite{Luxspace}. One is the SatReq requirement presented in Section~\ref{sec:running}, and the two others (SatReq1 and SatReq2) are strengthened versions of SatReq that, if violated, indicate increasingly critical violations. We also received the input profile IP (Section~\ref{sec:running}) and a more restricted input profile IP$^\prime$,
representing realistic input subranges associated with more critical violations. For each combination of the requirement (SatReq, SatReq1 and SatReq2) and the input profiles IP and IP$^\prime$, we checked whether \ourtool\ was able to detect any requirement violation, and further, we recorded the time needed by \ourtool\ to detect a violation. In addition, for the two most critical requirements (SatReq1 and SatReq2) and the input profiles IP and IP$^\prime$, we checked whether S-Taliro is able to detect any violation within the time limit required by \ourtool\ to successfully reveal violations for SatReq1 and SatReq2. Running this experiment took approximately four days and both tools were run twice for each requirement and input profile combination.
\textbf{Results}. \ourtool\ found a violation for every requirement and input profile combination in our study in just one iteration, requiring approximately four hours of execution time. Given that simulating the model under test takes approximately an hour and a half, detecting errors in four hours is highly efficient as it corresponds to roughly two model simulations. In comparison, S-Taliro failed to find any violations for SatReq1 and SatReq2 after running the tool for four hours based on the input profiles IP and IP$^\prime$.
\vspace{0.1cm}
\resq{The answer to \textbf{RQ4} is that
\ourtool\ efficiently detected requirements violations -- in practical time -- that S-Taliro could not find, for
three different requirements and two input profiles on an industrial CI-CPS model.
}
\section{Related Work}
\label{sec:related}
Formal verification techniques such as model checking aim to exhaustively check correctness of behavioural/functional models (e.g.,~\cite{fan2017d,henzinger1997hytech}), but they often face scalability issues for complex CPS models.
The CEGAR framework has been proposed to help model checking scale to such models (e.g.,~\cite{alur2003counter,ratschan2007safety,alur2006predicate,clarke2003verification,clarke2003abstraction,
ratschan2007safety,segelken2007abstraction,HARE,
dierks2007automatic,jha2007reachability,prabhakar2015hybrid,10.1007/978-3-319-65765-3_7,sorea2004lazy,zhang2017abstraction,Nellen2016}).
As discussed in Section~\ref{sec:intro}, the approximation-refinement loop of \ourtool, at a general level, is inspired by CEGAR.
Two CEGAR-based model checking approaches have been proposed for hybrid systems capturing CPS models: (a)~abstracting hybrid system models into discrete finite state machines without dynamics~\cite{alur2006predicate,clarke2003verification,clarke2003abstraction,
ratschan2007safety,segelken2007abstraction,sorea2004lazy}
and (b)~abstracting hybrid systems into hybrid systems with simpler dynamics~\cite{HARE,
dierks2007automatic,jha2007reachability,prabhakar2015hybrid,10.1007/978-3-319-65765-3_7}.
These two lines of work, although supported by various automated tools (e.g.,~\cite{ratschan2007safety,Henzinger1997,frehse2008phaver,frehse2011spaceex,chen2013flow}),
are difficult to be used in practice due to implicit and restrictive assumptions that they make on the structure of the hybrid systems under analysis.
Further, due to their limited scalability, they are inadequate for testing CI-CPS models.
For example, Ratschan~\cite{ratschan2007safety} proposes an approach that took more than 10h to verify the RHB benchmark (a non-CI-CPS model also used in this paper). In contrast, our technique tests models instead of exhaustively verifying them. Being black-box, our approach is agnostic to the modeling language used for MUT, and hence, is applicable to Simulink models irrespective of their internal complexities. Further, as shown in our evaluation, our approach can effectively and efficiently test industrial CI-CPS models.
There has been earlier work to combine CEGAR with testing instead of model checking (e.g.,~\cite{ball2005abstraction,935473,Zutshi:2014:MSC:2656045.2656061,Zutshi:2015:FSP:2728606.2728648,Kroening2010,10.1007/978-3-540-70545-1_38,Kroening2010,10.1007/978-3-319-24953-7_35,7741019,10.1007/978-3-319-24953-7_35}). However,
based on a recent survey on the topic~\cite{7741019}, \ourtool\ is the first approach that combines the ideas behind CEGAR with the system identification framework to develop an effective and efficient testing framework for CI-CPS models. Non-CEGAR based model testing approaches for CPS have been presented in the literature~\cite{dreossi2015efficient,Nghiem:2010:MTF:1755952.1755983,7963007,yaghoubi2017local,dreossi2015efficient,Arrieta2017,plaku2007hybrid,sankaranarayanan2012falsification,Bartocci2018} and are supported by tools~\cite{7741019,ernst2019arch,staliro,donze2010breach,akazaki2018falsification,zhang2018two}. Among these, we considered S-Taliro as a baseline for the reasons reported in Section~\ref{sec:ourtool}.
Zhang et al.~\cite{zhang2018two} reduce the number of simulations of the MUT by iteratively evaluating different inputs
for short simulation times and by generating at each iteration the next input based on the final state of the simulation.
This approach assumes that the inputs are piecewise constants
and does not support complex input profiles such as those used in our evaluation for testing our industry CI-CPS model. To reduce the simulation time of CI-CPS models, we can manually simplify the models while preserving the behaviour needed to
test the requirements of interest~\cite{alur:15,popinchalk2012improving}.
However, such manual simplifications are error-prone and reduce maintainability~\cite{arrieta2019pareto}. Further, finding an optimal balance between accuracy and execution time is a complex task~\cite{Siddesh:2015:CSC:2898951}.
\vspace{-0.2cm}
\section{Conclusions}
\label{sec:conclusion}
We presented \ourtool, a technique that combines testing with an approximation-refinement loop to detect requirements violations in CI-CPS models.
We implemented \ourtool\ as a Matlab/Simulink application and compared its effectiveness and efficiency with the one of S-Taliro, a state-of-the-art testing framework for Simulink models.
ARIsTEO finds $23.9\%$ more violations than S-Taliro and finds those violations in $31.3\%$ less time than S-Taliro.
We evaluated
the practical usefulness of ARIsTEO on two versions of an industrial CI-CPS model to check three different requirements.
ARIsTEO successfully triggered requirements violations in every case and required four hours on average for each violation, while S-Taliro failed to find any violations within four-hours.
\begin{acks}
This work has received funding from the
\grantsponsor{erc}{European Research Council}{https://erc.europa.eu/} under the European Union's Horizon 2020
research and innovation programme (grant agreement
No~\grantnum{erc}{694277}), from QRA Corp, and from the University of Luxembourg (grant ``ReACP").\\
We thank our partners LuxSpace and QRA Corp for their support.
\end{acks}
\balance
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Biografia
Degland nacque a Armentières, e visse a Lilla per la maggior parte della sua vita, dove fu il capo del Hôpital Saint-Sauveur, e dove anche morì. Partecipò alla fondazione del Museo di Storia Naturale di Lille, nella quale ci sono anche le sue collezioni zoologiche originali. Pubblicò un catalogo dei coleotteri nel 1821, e un catalogo in due volumi degli uccelli della Francia e dell'Europa nel 1849. Con Zéphirin Gerbe, fu co-autore di Ornithologie européenne, ou, Catalogue descriptif, analytique et raisonné des oiseaux observés en Europe (seconda edizione, 1867).
Melanitta deglandi prese il nome da lui.
Note | {
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The Houston Huskies were a minor league ice hockey team based in Houston, Texas. Formerly known as the Houston Skippers, they were a member of the United States Hockey League, and were active from 1947 to 1949. They were affiliated with the Buffalo Bisons.
The Huskies continued with the same team as the Skippers, and also continued playing in Sam Houston Coliseum. In 1948 former Montreal Canadiens player Toe Blake was named head coach. The Huskies went on to have a great season, eventually winning the 1948 USHL championship.
In June 1949 the Huskies were dropped from the USHL. The Huskies, along with teams from Dallas and Fort Worth attempted to form a Texas Hockey League, but were unable to get firm commitments for dates. Because of this, the Sam Houston Coliseum canceled their contract with the team. Without a place to play, and no other city willing to host them, the Huskies were forced to cancel their 1949-50 season. The team ceased operations in late 1949.
Bob Blake played one season with the Huskies and led the team to a championship trophy. He was inducted into the United States Hockey Hall of Fame in 1985.
References
External links
Hidden History of Houston Hockey
Houston Huskies All-Time player list
Ice hockey teams in Houston
Defunct ice hockey teams in Texas
1947 establishments in Texas
1949 disestablishments in Texas
Ice hockey clubs established in 1947
Ice hockey clubs disestablished in 1949 | {
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Dit is een lijst van burgemeesters van de voormalige Nederlandse gemeente Boskoop in de provincie Zuid-Holland die per 1 januari 2014 is opgegaan in de gemeente Alphen aan den Rijn.
Zie ook
Lijst van burgemeesters van Alphen aan den Rijn
Boskoop | {
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Q: Scaling of probability for multivariate normal. Say I have a multivariate normal given as:
$$f({\bf x})=f({\bf x}_1,{\bf x}_2)=\frac{1}{(2\pi)^{n/2}|\Sigma|^{1/2}}\exp\left[-\frac{1}{2}Q({\bf x}_1,{\bf x}_2)\right] $$
where $Q$ is defined as
$\displaystyle Q({\bf x}_1,{\bf x}_2)$ $\textstyle =$ $\displaystyle ({\bf x}-\mu)^T\Sigma^{-1}({\bf x}-\mu)$ and where ${\bf x}_1$ and ${\bf x}_2$ are two subvectors of respective dimensions $p$ and $q$ with $p+q=n$. Note that $\Sigma=\Sigma^T$, and $\Sigma_{21}=\Sigma_{21}^T$ and
we can write
$$ f({\bf x})=f({\bf x}_1,{\bf x}_2)= \frac{1}{(2\pi)^{p/2} |\Sigma_{11}|^{1/2}}\exp\left[-\frac{1}{2}(x_1-\mu_1)^T\Sigma_{11}^{-1}(x_1-\mu_1)\right] \frac{1}{(2\pi)^{p/2} |A|^{1/2}}\exp\left[-\frac{1}{2}(x_2-\mu_2)^TA^{-1}(x_2-\mu_2)\right] = N({\bf x}_1,\mu_1,\Sigma_{11})\;N({\bf x}_2,b,A)$$
where the b is given as
$$b\stackrel{\triangle}{=}\mu_2+\Sigma_{12}^T\Sigma_{11}^{-1}({\bf x}_1-\mu_1) $$
and the A as
$$A\stackrel{\triangle}{=}\Sigma_{22}-\Sigma_{12}^T\Sigma_{11}^{-1}\Sigma_{12} $$
Now say I define $(x_1 ^{\alpha}, x_2^{\alpha})$ as $F( (x_1,x_2) \leq (x_1^{\alpha}, x_2^{\alpha}) ) > \alpha $. where F is the cumulative normal.
Given a $b$ is there a scaling relation of to figure out what cumulative $ N( x_2^{\alpha},b,A)$ will be in terms of F and/or possibly some other factors.
Essentially I have split the multivariate normal into two parts. And if I know that cumulative probability of the whole part, I am trying to figure out how much of that comes from part one and how much comes from part two.
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Большой Ошмесвай () — деревня в Увинском районе Удмуртии России. Входит в Чеканское сельское поселение.
География
Находится в центре республики, в северной части района, в пределах Центрально-Удмуртской низменности, на реке Ошмесвай, в 31 км к северо-востоку от посёлка Ува и в 48 км к северо-западу от Ижевска.
Единственной улицей деревни является улица Митенская.
Население
Инфраструктура
Личное подсобное хозяйство.
Транспорт
Проходит дорога местного значения.
Примечания
Населённые пункты Увинского района | {
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{"url":"http:\/\/tex.stackexchange.com\/questions\/111868\/pandoc-how-can-i-get-numbered-latex-equations-to-show-up-in-both-pdf-and-html-o\/112045","text":"# Pandoc: How can I get numbered LaTeX equations to show up in both PDF and HTML output\n\nTo make a numbered equation in Pandoc, I defer to using pure LaTeX:\n\n$$a^2 + b^2 = c^2$$\n\n\nIn PDF form, it gives a nice numbered equation, however, this equation doesn't show up at all in the HTML output because the equation is within the \\begin environment and is ignored. I would like to keep the numbering in the PDF, but would like the equation to show up in the HTML, with or without the numbering.\n\n-\n\n## 2 Answers\n\nIn pandoc you can do:\n\n(@foo) $a^2 + b^2 = c^2$\n\nAs (@foo) says, ...\n\n(@bar) $e = x + y$\n\n\nThis will give you running equation numbers in both PDF and HTML. However, the equations will not be centered or display-formatted. (You can use for display math, but then the numbers probably won't be lined up right -- though maybe this can be fixed with CSS.) - Doing it this way, the equation ends up being placed on the left and slightly above the equation itself. Is there a way to have the equation number on the right just like in the regular latex format? \u2013 user1027169 May 3 '13 at 20:11 No, there's no way to get the number on the right. You could probably fix alignment issues with display math using CSS. \u2013 John MacFarlane May 4 '13 at 2:06 I meant in the PDF output, the equation number gets moved to the left. As you mentioned, the HTML shouldn't be an issue with some CSS. \u2013 user1027169 May 4 '13 at 16:25 @JohnMacFarlane just wondering if this (displaymath, right-aligned eqs) is something that might be addressed in later pandoc releases, or if explicit LaTeX environment remains the way to go for pdf output? \u2013 cboettig Jun 23 '14 at 18:20 Mathjax supports \\ref and \\eqref in plain HTML: cdn.mathjax.org\/mathjax\/latest\/test\/sample-eqrefs.html --- can Pandoc users get access to this functionality? (I'm converting multimarkdown to html.) \u2013 Ahmed Fasih Jun 29 '14 at 4:26 An extension exists on github: pandoc-crossref install it with: cabal update cabal install pandoc-crossref or in archlinux using ArchHaskell pacman -Sy pandoc-crossref you can use it by doing: math $${#eq:label} [@eq:label] and compiling with pandoc file.md --filter pandoc-crossref -o file.pdf for mor information see the documentation Alternatively, as described in https:\/\/github.com\/jgm\/pandoc\/issues\/1938#issuecomment-74011358 you can use --mathjax for HTML rendering if your equations are in math.txt $$ a^2 + b^2 = c^2 \n\n\ncreate a file header\n\n<script type=\"text\/x-mathjax-config\">\nMathJax.Hub.Config({ TeX: { equationNumbers: {autoNumber: \"all\"} } });\n<\/script>\n\n\nand compile with\n\npandoc math.txt -t html -s -o test.html --mathjax=https:\/\/cdn.mathjax.org\/mathjax\/latest\/MathJax.js?config=TeX-AMS-MML_HTMLorMML -H header\n\n-\nWelcome to TeX.SX! You can have a look at our starter guide to familiarize yourself further with our format. \u2013\u00a0 Symbol 1 Apr 1 at 9:27","date":"2015-05-28 06:39:09","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 1, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.808532178401947, \"perplexity\": 2263.0682536622485}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-22\/segments\/1432207929256.27\/warc\/CC-MAIN-20150521113209-00343-ip-10-180-206-219.ec2.internal.warc.gz\"}"} | null | null |
layout: post
microblog: true
date: 2009-04-26 03:00 +0300
guid: http://desparoz.micro.blog/2009/04/26/t1619800171.html
---
Qld Coroner recommends review of post dive practices for deep dives. [www.blueflipperdiving.com/news.asp](http://www.blueflipperdiving.com/news.asp?item=3198)
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I have been going to do a write up on this for a while and having digested all the documents from the Coalitions policy I think now is a good time to do it.
The Coalition has now presented a full policy on how they will handle the NBN if they win the election later this year. This involves moving from a full fibre to the premises (FTTP) rollout, to a Fibre to the node Rollout (FTTN) and using the existing hybrid-fibre Coax networks owned by Optus and Telstra as well as the current wireless and satellite options being rolled out by NBN Co.
While there are other points could continue to compare from these policies much of the remainder of the coalition's policy documents deal with a number of assumptions in order to make the Labour NBN seem to be costing more, going slower, or otherwise generally worse then the documents being presented by NBN Co, I could do similar things to the Coalitions policy however I think this would be unproductive.
Over the last 6 months I have been Participating in the SCA as a heavy fighter, recreating pre 17th century knightly combat. One faction of this I have noticed is a tendency for people to focus on the sport aspects of Heavy fighting rather then the recreation aspects, with many people opting for minimal armor or close to it within our rule set. I find my self rather opposed to these noting that the SCA is dedicated to recreating all aspects of life in pre 17th century.
When asking people why they chose to wear minimal armor you will receive many responses ranging from "It's to heavy", to "I don't get hit there anyway", my personal opinion is that neither of these, or the plethora of similar reasons, truly make sense in the context of what we are attempting to achieve. By removing the weight and movement restrictions that wearing armor provides you are no longer attempting to participate in chivalric combat but in a sport that is only loosely based upon combat from this time period, as part of how and why things where done is due to the armor that was being worn by combatants.
On the other side of this argument is people who wear plenty of armor, however use modern substitutes (like zoombang, or plastic)for parts of there armor, with these I find myself being more sympathetic to there reasoning as much of this is based on improving the durability of there armor as they have found (and I have experienced) that some pieces of armor simply does not last, or gets badly damaged from being regularly hit, and replacing our equipment is not a cheap exercise, however there are a number of people who are going for these modern alternatives for reasons such as those above I find difficult to deal with this, particularly noting that a similar effect can be created with period materials.
So with my rant over with what I don't like to see I would like to see more people wearing more armor, perhaps not to the amount that I am wearing, but to a standard where they are wearing some form of period armor on most valid target target areas what they may be hit.
Well I'm up to 6 hours played on Skyrim (according to steam) so I'm going to give it a full review (with out story line spoilers).
Upon loading the game up I was quite impressed, visually its quite easy on the eyes even on medium settings with a distinctly elder scrolls feel to it, and this is achievable on most common PC hardware. Now my PC is no gaming beast but its still above the average hardware that post people are using (again according to the steam statistics) so I'm going to take my experience as being fairly average of what most people would see. The shadows in Skyrim are one of the least impressive aspects of its graphics with them tending to get very jagged at times and detracting from otherwise wonderful settings, however all in all most of the time you would overlook this as your busy doing something (like killing the guy who just jumped out of his grave to kill you!) making it less of an issue that you would first expect. Overall I would place them graphics at ab 8/10.
The UI's in the PC version seem to have been taken directly from the console version and for the first few hours I was fighting them to get what I wanted done and learning where everything was hidden in the 2 separate menus you have available (one for your stuff your character uses, items spells etc, and the other got more game like things like saving the game and viewing active quests). this was rather disorientating particularly as navigating them was clearly designed to be done with a controller. in addition to this some of the default key binding made functions hard to find (or use when you needed them like the dragon shouts that are bound to 'Z' by default. In the end i gave up on the keyboard and mouse and plugged in a Xbox controller and things became much easier to use, this are of the game could have defiantly used some more work in the PC version, however my experience with the controller says they got it right for the console versions. combat also became easier with the controller as all your combat related buttons where not under each other (like the shouts I already mentioned.) making the game much easier to play.
The leveling mechanics in the game make it very fast to level up with the perks giving you options to make a specialist that is viable regardless of your play style (yes even the sneaky archer can kill a dragon, if your careful) however adding perks using the KB and mouse was again hard, and much faster and more intuitive using the controller. but as your not doing this to regularly its not that bad. all up i give the UI and control system a 5/10 using the Keyboard or 9/10 using the controller.
The storyline of the main quest is exceptional (so far) and much better if you don't go and read any spoilers before you play (and that's why I won't put any here) with some nice twists and the odd side quest that actually helps you out, to the point of having done one of the side quests allowed me to practically skip an entire main quest as I had already achieved the objective. while this was a nice bonus it would have been nice for the objectives to be separate, with so much of the story i have not begun to look at I won't comment to much on this section, but quests seem to be relatively compact so far, making it more inviting to do the odd side quest, even if it is just to take some herbs across town.
With so much to do in the game I feel that the price tag on this game is more then justified, with a good mix of content, good graphics and game mechanics this title has earned its place as a AAA title this year, however the shocking menu system for the PC version leaves something to be desired. the game also requires you to link it to a Steam account making it difficult for some people who may wish to play the game with out access to broadband internet, as there was a day 1 patch that I am defiantly not a fan of (Surely buying the game on launch day I should be playing 1.0 not 1.1.21.0?) as this screams that the game could have been better released in a month or 2.
Overall I would rate this a 7/10 game on PC, having an Xbox controller for your PC makes the experience much better, however the keyboard and mouse experience needs some more work to make this a good introduction for players new to the elder scrolls series. if you happen to own a controller this is easily a 9/10 game.
It's been some time since I last played an elder scrolls game, however with skim I have decided to give other a go.
After some shopping around I got my copy, for the PC, and headed home to load it up, and during he install it automatically linked it to my steam account, so no need to keep the cd key on hand thankfully. Steam then went to work getting the day 1 patch for me and updating my game to 1.1.21.0 taking about 10 minutes to download and install. While this was happening I had a quick look at the map that came with my copy of the game, this was a nice high quality printing that would not look astray above your pc so you can see where you are going with out having to refer to the in game map.
With the install now finished I started up the game and it detected the video settings and set my system to use the high settings (and using my native resolution that was a nice surprise.) so I left everything as it suggested and dived head first in to the game. as usual for an elder scrolls game, you start with a scripted sequence with you as a prisoner being taken to an outpost of some kind. where you are introduced to some of the main factions (well I presume they are) and some of the story line to come. After some explanation of the story line you are thrown in to for the main quest line (that I won't spoil for you here) you are placed in to a small tutorial style section of the game that explains some of the mechanics for fighting and gives you some basic equipment before leading you to your first village in the game so continue with either the main quest line or the large number of side quests in the game.
One thing I noticed that could get annoying quickly was when killing the last of a a group of enemy's your character went in to a cut scene style kill move, while h=these are quite nice there is a limited number of them so they tend to repeat, and while doing one of the side quests I noticed I went in to one while I was still being shot at! so I'm a little undecided about that feature, however it is quite nice when you sneak up on someone to see a little shot of you assassinating them to reward your efforts.
On the graphics side of things, its a very good looking game, even with out the settings maxed out. I'm using a gt20 in my pc with a dual monitor set up, placing me below the recommended specs however the game still put me on the high setting, just one below the max, with out having checked any numbers the game was a little jerky so I turned the AA and AF down a little to get it nice and smooth, but even still the quality did not suffer significantly, so with that said if you meet the recommended specs for the PC version you can expect this game to run very well and get the most out of it.
well over the last few months if have been slowly working on setting up a workshop in the garage, at it has finally ha the last vital piece installed. The forge, now this may not seem to be a normal piece to add to a home workshop i find it quite useful and fun to boot.
how this came about is an interesting story that started with some innocent Google searching, and i came across some websites about blacksmithing, now this has always been something that has interested me, however i never thought that i would be doing this at home. however with some help of a friend who had a gas forge and some inspirational websites showing just what could be achieved with very few tools i decided to set out and build my own.
first up was a list of tools I HAD to have to be able to do anything, that was a hammer, something to make it hot, and something to hit it on.
after much searching i came across http://www.tharwavalleyforge.com/ where there was listings for all sorts of tools that would complete my workshop, and a few emails and a couple of months later i was the proud owner of a 30kg anvil and a pair of blacksmiths cross pein hammers. this solved 2 of my 3 must have items, a hammer and something to hit it on, but still left me with no way to make it hot.
hence began the long search for a forge, after a few false starts looking for gas forges and either not being able to get them to sydney, or them being way out of my price range i posted a wanted a on the work classifieds, only to be offered a free solid fuel forge, that was found to be in perfect working order and only about 15 minuets from home,so naturally that was a taker.
well after quite some time of not writing much code I decided to dig out the old code for this site and see what If I could use it for some projects I had in mind.
As it turns out while the code isn't spectacular in the form I had left it it was still a good idea for the most part, so I decided to set about a rewrite/re factor of the code to bring it up to php5.3 standards from its old 4.x standard and make it in to more of a OO framework that i can drop into other projects with out as much need to hack it to pieces to make it work.
with the choice to host this code came the choice of license to put it up under, and after looking through several licenses i ended up going with the modified (3-clause) BSD license.
hopefully this will form a nice, simple, and fast framework that can be used for a range of php software both open and closed source.
Well I have been a bit lasy again putting things up here so I thought its about time I uploaded some more photos.
So I have finaly got around to uploading some photos I took at the 20 twenty big bash last weekend.
I came across an interesting concept the other day of a developer of an open source project who was developing the project "behind the scenes" with out any feedback to the community, now this stuck me as odd while first reading there comments about what they were doing.
What had happened in this case was the developer wasn't happy with being pushed in any particular direction on there project so they had stopped openly advertising what they were doing on the project, with out stopping other people from using there code, this is where i came up with the concept of Open Source Vs Open development.
When most people think of open source the link these two concepts of not only having the source code available but also being directly involved in the development cycle of the project through bug reports, submitting code patches etc, and most project this is how they are run, however in this case it was no longer occurring as the developer had no wish to implement new features and was more focused on maintaining it and fixing bugs at there own pace.
my question to you is have you come across a case of open source but not open development? do you think there is a place where this makes sense for a project to be developed with this methodology? or should all open source projects also employ open development?
IPV6 address space – are we allocating it thoughtfully.
Well recently I got IPv6 enabled on the VPS that is currently running my mail server, now this I thought would be a great opportunity to start to move into the new generation of IP on my hosting by getting this server enabled. when this was assigned I was interested to note my humble VPS with just 128mb of ran was not allocated a single IP address but a whole subnet that I could use. Now knowing that IPv6 has some 340 trillion trillion trillion unique addresses I wasn't so surprised by getting more then one address, but my surprise came when I found out how much I was given.
My VPS was assigned 2607:f128:42:d1::2/64 as its IPv6 allocation, now that may not seem to imposing at first glance but a small amount of research and you will see that this allocation contains more IP address space then the ENTIRE ipv4 address space, this makes me wonder why this is the case, and why the MINIMUM allocation any of the regional address providers will give out to an organization is a /32 block, this seems wasteful and unnecessary.
Now I understand that IPv6 has been designed so there are far more addresses then we will ever use but is that a reason to waste them like this so early in its adoption. as was the case with ipv4 where early in its life cycle large blocks were handed out to the point where we now see that things like NAT are required to keep even a portion of the address space available for use.
Would it not be more prudent to give a single server a much smaller allocation of ip addresses (perhaps just 1 even?) rather then taking such a huge chunk out of the address pool we have seen the effect this has had on IPV4 with some organizations being given entire /8 blocks of address space early in its life cycle (although some have since been returned for reallocation).
Now this may never become an issue with the huge address pool available to us through IPv6 but I believe that this should at least be considered before this becomes so wide spread that every IPS and hosting provider in the world is using a /32 and they are running out.
Well having just finished this book I figure its time for a review, this is the thirteenth book in the series and has been completed by Brandon Sanderson after the death of Robert Jorden in 2007, and while many fans believed that the series would never be the same being finished by another author, my self included, I assure you you wont be disappointed by this book.
Now I will attempt to keep spoilers out of this review for those that have not read the book yet but I can't guarantee I wont give any. If you haven't read all of the proceeding books the please be patient as this will contain spoilers for them, you have been warned. | {
"redpajama_set_name": "RedPajamaC4"
} | 29 |
Störtebekerweg ist der Name eines regionalen Wanderwegs des Wiehengebirgsverbandes Weser-Ems. Der Weg ist 185 Kilometer lang und führt von Leer an der Ems nach Wilhelmshaven am Jadebusen.
Streckenführung
Der Störtebekerweg beginnt in der ostfriesischen Stadt Leer am Bahnhof und führt durch Emden bis an die Mündung der Ems. Die weitere Wegführung an der Nordsee entlang auf, vor oder hinter dem Deich ist nicht markiert, da hier ein Verlaufen nicht möglich ist. Am Strand vom Badeort Schillig führt die Strecke südwärts in Richtung Wilhelmshaven, wo die Strecke im Alten Hafen endet.
Die Strecke ist flach und in acht bis zehn Tagesetappen gut zu bewältigen. Markiert ist der Weg mit einem weißen "S" auf schwarzem Hintergrund.
Weblinks
Offizielle Seite des Wiehengebirgsverbandes Weser-Ems, abgerufen am 25. Januar 2021
GeoLife – Störtebekerweg, abgerufen am 25. Januar 2021
Wanderweg in Niedersachsen
Verkehr (Landkreis Leer)
Verkehr (Wilhelmshaven)
Klaus Störtebeker als Namensgeber | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,125 |
using System.Collections.Generic;
using Newtonsoft.Json;
namespace VaultSharp.V1.SecretsEngines.Database
{
/// <summary>
/// Role definition.
/// </summary>
public class Role
{
/// <summary>
/// The name of the database connection to use for this role.
/// </summary>
[JsonProperty("db_name")]
public DatabaseProviderType DatabaseProviderType { get; set; }
/// <summary>
/// Specifies the TTL for the leases associated with this role.
/// Accepts time suffixed strings ("1h") or an integer number of seconds.
/// Defaults to system/engine default TTL time.
/// </summary>
[JsonProperty("default_ttl")]
public string DefaultTimeToLive { get; set; }
/// <summary>
/// Specifies the maximum TTL for the leases associated with this role.
/// Accepts time suffixed strings ("1h") or an integer number of seconds.
/// Defaults to system/mount default TTL time;
/// this value is allowed to be less than the mount max TTL
/// (or, if not set, the system max TTL), but it is not allowed to be longer.
/// </summary>
[JsonProperty("max_ttl")]
public string MaximumTimeToLive { get; set; }
/// <summary>
/// Specifies the database statements executed to create and configure a user.
/// </summary>
[JsonProperty("creation_statements")]
public List<string> CreationStatements { get; set; }
/// <summary>
/// Specifies the database statements to be executed to revoke a user.
/// </summary>
[JsonProperty("revocation_statements")]
public List<string> RevocationStatements { get; set; }
/// <summary>
/// Specifies the database statements to be executed rollback a create operation in the event of an error.
/// Not every plugin type will support this functionality.
/// </summary>
[JsonProperty("rollback_statements")]
public List<string> RollbackStatements { get; set; }
/// <summary>
/// Specifies the database statements to be executed to renew a user.
/// Not every plugin type will support this functionality.
/// </summary>
[JsonProperty("renew_statements")]
public List<string> RenewStatements { get; set; }
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 2,310 |
Polish journalist taken hostage in Syria
Polish photojournalist Marcin Suder has been kidnapped by Islamist militants in north west Syria, say opposition activists. Suder, who works for the Studio Melon photo agency based in Warsaw, was
25 lipca 2013 0
Joint US-Polish military training to continue
US budget cuts will not affect the joint American-Polish military training program the US's highest-ranking general insisted on Wednesday. General Martin E. Dempsey, Chairman of the Joint Chiefs of Staff, made
Poland sends condolences after dozens die in Spain train crash
Poland's prime minister has expressed his "deepest sympathy" with the Spanish people after at least 78 died in the train crash in north west Spain on Wednesday. "On behalf of
Pole elected as world credit union chief
Poland's Grzegorz Bierecki has been elected chairman of the World Council of Credit Union at its conference in Ottawa, Canada. Members of the World Council include regional and national credit
Security lapse led to president egg attack?
Poland's Ministry of the Interior has indicated that security officers made procedural errors allowing President Bronislaw Komorowski to be struck with an egg in Ukraine, Sunday. "According to preliminary findings,
Israel summons Polish ambassador over kosher ban
Poland's ambassador to Israel has been summoned to Israel's Foreign Ministry in Jerusalem to explain the vote in parliament, which upheld the ban on kosher and halal slaughter. Ministry
Hitch-hikers killed 'lying down' across road
Two Polish hitch-hikers were killed on Wednesday night when a van ran over the pair while they were reportedly lying down in the middle of a road in Seelow, east
Four out of ten Poles attend Sunday Mass
Forty percent of Polish Roman Catholics attend Sunday mass, according to Poland's Statistical Institute of the Catholic Church. The annual count was carried out in the autumn of last year
President Komorowski: "it was a Polish Wimbledon"
Poland's president has awarded Agnieszka Radwanska, Jerzy Janowicz and Lukasz Kubot state honours for their success at the Wimbledon Lawn Tennis Championships. "It was a Polish Wimbledon," Bronislaw Komorowski said
9 lipca 2013 0
Janowicz ranking soars after Wimbledon success
Jerzy Janowicz has advanced to 17th place in the ATP World Tour rankings after reaching the semi-finals in Wimbledon last week. However, the 22-year-old, who was previously World No. 24,
Poland sees drop in illegal immigrants
A report indicates that the number of foreign workers living illegally in Poland dropped by 23 percent in 2012. Between 2010, Poland's National Labour Inspectorate conducted approximately 6300 checks on firms
Poland, India, Brazil snub Snowden asylum application
Poland's foreign ministry says the fax sent by representatives of CIA whistleblower Edward Snowden asking for asylum does not conform to legal procedure and cannot be processed. Foreign Ministry spokesman
TV station fined after Polish flag-in-dog-mess stunt
Poland's Supreme Court upheld an appeal court verdict on Tuesday against a TV station that broadcast a miniature Polish flag being placed in a dog's excrement. Judge Maciej Pacuda insisted
Janowicz triumphs in Wimbledon all-Poland quarter final
Jerzy Janowicz became the first Pole ever to reach the men's semi finals at Wimbledon when he beat fellow countryman Lukasz Kubot in an emotional last-eight match on Wednesday. This
US deports comatose illegal Polish immigrant
A 69-year-old Polish man was deported from the US while still in a coma, after medics discovered that he was an illegal immigrant. Wladyslaw Haniszewski had apparently been living in
27 czerwca 2013 3
Warsaw's second airport open from 2 July
Poland's Civil Aviation Office (ULC) has given permission for Warsaw Modlin Airport to resume flights, after six months out of business due to runway repairs. Marta Chylinska, a spokesperson for
'Anonymous' hackers behind nationwide bomb hoax?
Several of the emails sent to 22 institutions across Poland, Tuesday, claiming there were bombs at the Attorney General's office and other institutions were signed "anon.pl". An Internal Security
LOT to apply for second bailout package
Cash-strapped LOT airlines has applied for a second loan amounting to 100 million euros from the state. The European Commission is currently examining the first 400 million zloty (100 million
Man who torched himself outside PM's Office dies
A man who set himself on fire in the vicinity of the Prime Minister's Office a fortnight ago has died. A spokesman for the Military Medical Institute in Warsaw, where the | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,914 |
{"url":"https:\/\/math.stackexchange.com\/questions\/631649\/nets-and-convergence-why-directed-indices\/631852","text":"# Nets and Convergence: Why directed indices?\n\n1.) Why do we require nets to be indexed by directed sets (apart from it simply works compared to filters and topology). Is there a reason w.r.t. to the notion of convergence?\n\nSo far there are those hints:\n\n\u2022 Notion of a good Direction\n\u2022 Uniqueness of Limit\n\n2.) Moreover, why does it make sense to say that the following net converges even though it grows arbitrarily large on one side: $$\\Lambda:=\\{1,2\\}\\times\\mathbb{N}:(i,n)\\leq(i',n') :\\Leftrightarrow i\\leq i',n\\leq n'\\\\ x_{(1,n)} :=n, x_{(2,n)} :=0$$ Thanks in advance. Cheers, Alex\n\n\u2022 Note that the relation you describe isn't transitive. Probably you mean $(i,n)\\le(i',n')\\iff i\\le i'\\wedge n\\le n'$, which makes this set the product of the directed sets $\\{0<1\\}$ and $\\Bbb N$. \u2013\u00a0Stefan Hamcke Jan 8 '14 at 19:31\n\u2022 Your example net is \"eventually in\" every neighborhood of $0$, which is what we define as convergence. It does not mean \"all but finitely many in\", however. Your example illustrates the difference. \u2013\u00a0GEdgar Jan 8 '14 at 20:44\n\nNets emerges from the work of Moore and Smith to understand integrals, i.e., integrals are the great example of nets. For localizing it, imagine you want to calculate\n\n$$\\int_a^bf(x)\\,dx$$\n\nfor a sufficient good function $f$. Now, we consider all pairs of $(\\{X_i\\}_{i<n},\\{\\xi_i\\}_{i<n})$, where $\\{X_i\\}_{i<n}$ is a partition in measurable set of $[a,b]$, i.e., a family of sets such that $$\\bigcup_{i<n}X_i=[a,b]$$ and such that for each $i$, we can calc the length of $X_i$, $l(X_i)$ and where $\\{\\xi_i\\}_{i<n}$ a family of points such that for all $i$, $\\xi_i\\in X_i$; and for each of these pairs $\\mathcal{X}=(\\{X_i\\}_{i<n},\\{\\xi_i\\}_{i<n})$ we consider the sum $$I(f,\\mathcal{X})=\\sum_{i<n} f(\\xi_i)l(X_i)$$ which is an approximation of our desired integral. The problem is that we need now a way to understand how these sums converge to the desire integral and this can only be done via directed sets and nets. In this way directed sets arise in a natural manner as the only possible way to order these partitions.\n\nIn this particular case, the usual order is that $$\\mathcal{X}=(\\{X_i\\}_{i<n},\\{\\xi_i\\}_{i<n})\\geq \\mathcal{X'}=(\\{X_i'\\}_{i<n'},\\{\\xi_i'\\}_{i<n'})$$ whenever for all $i<n$ there is some $j_i<n'$ such that $X_i\\subseteq X_{j_i}'$, i.e., the first partition is a refinement of the second one. And here the usual axioms of directed sets are translated in the following way:\n\n1. Every partition is a refinement of itself.\n2. If $\\mathcal{X}$ is a refinement of $\\mathcal{X}'$ and $\\mathcal{X}'$ a refinement of $\\mathcal{X}''$, then $\\mathcal{X}$ is a refinement of $\\mathcal{X}''$.\n3. Given any two partitions $\\mathcal{X}$ and $\\mathcal{Y}$, there is always a partition $\\mathcal{Z}_{\\mathcal{X},\\mathcal{Y}}$ which refines both $\\mathcal{X}$ and $\\mathcal{Y}$.\u00b4\n\nIf you haven't, take the dfinition of net and check that $\\{I(f,\\mathcal{X})\\}_\\mathcal{X}$ converges effectively to the integral of $f$.\n\nIn this sense we can speak that the previous partial sums converge to the integral we want to calculate because there is a sense about what means all partitions that refines the one you take. Furthermore, if we take any partition we can refine it to be a refinement of the given one, which means that we know what means we don't miss any\n\nSo, why we need directed sets? Because we need to have a sense of direction to advance so as in the case of the partitions, we have to have a clear notions of what means advancing and this is not only possible with the third axioms. If you omit it, then it is not clear that convergence has the meaning of the net approaching to the point as it would happen if it were not possible for each pair of partitions to build a partition that refines both of them. This is the essence of the direction, giving the sense that the net advance towards to its limit points.\n\nFor your example, even though $$\\lim_{n\\to\\infty} x_{(1,n)}=\\infty\\text{,}$$ it has sense to say that the net converges because when you are advancing whit one fixed, even if you \"arrived\" to $(1,\\infty)$, you have not advance enough because there are points far beyond to advance. This means that by fixing one, you are not advancing all you can to the given direction because there are pints that you will never pass beyond in this way.\n\nNote: If any inquiry, please tell me.\n\n\u2022 The Notion of Advance sounds reasonable.<br> A plain relation gives the Notion of Direction: $n\\geq m$ Now, what we want next is it to be transitive, namely, if sth. holds for all $n\\geq M_0$ and there is a $n\\geq M_0$ for some $m\\geq n$ then it should hold for this $m$ as well. Reflexivity is not so important since one can switch back and forth between $\\geq$ and $>$ without troubles. Antisymmetry resp. Asymmetry I'm not so sure yet. Last, will we always be able to advance all members so that $m\\geq N_0$ and luckily, yes we can.<br> Thank you Tonelli-Cueto, thank you very much! =D \u2013\u00a0C-Star-W-Star Jan 8 '14 at 22:34\n\u2022 @Alex Antysymmetry is not in my definition. For example, in Willard's General Topology you don't have that axiom for directed sets. \u2013\u00a0Josu\u00e9 Tonelli-Cueto Jan 8 '14 at 22:46\n\nTo define a notion of converge of a net $\u03a6$ you need some sort of order on the indexing set, and it should at least be reflexive and transitive. You can think of a net as many \"lines\" $\u03b1 \u2264 \u03b1_1 \u2264\u03b1_2\\le...$ and $\u03b2 \u2264 \u03b2_1 \u2264 \u03b2_2 \u2264 ...$. The third condition ensures that every two lines eventually meet. If a net were defined merely as a function from an arbitrary set $S$, then this $S$ would become irrelevant due to having no structure at all and we could just look at its image $\u03a6(S)$. A limit of $\u03a6$ could then at best be defined as a point $x$ where infinitely many points of $\u03a6(S)$ are gathering. This, however, would not be any different from $x$ being a limit point of the set $\u03a6(S)$ in the space $X$. In other words, there would be no difference between a net in $X$ and just an ordinary subset of $X$.\n\nAnother reason for directed sets is that we get a sort of equivalence between nets and filters. Suppose $\\Phi$ is a net over the directed set $I$. You can get a filter \"Filter$(\u03a6)$\" by taking the tails $T_j=\\{\u03a6_i\\mid i\\ge j\\}$ as a filter base. The condition that the intersection of $T_j$ and $T_k$ contains another $T_l$ comes just from the property that for any $j,k$ there is another $l\\ge k,j$.\nOn the other hand, if you have a filter $\\cal F$, then define the directed set $$I_\\cal F=(\\{(x,F)\\mid x\\in F\\in\\mathcal F\\},\\ (x,F)\\le (y,G)\\iff G\\subseteq F)$$ and let Net$(\\cal F)_{(x,F)}=x$. Note that the tails of Net$(\\cal F)$ are precisely the $F\\in\\cal F$. This implies that $\\text{Filter(Net($\\cal F$))}=\\cal F$. With a reasonable notion of \"subnet\" we can show that $\\text{Net(Filter($\u03a6$))}$ and $\u03a6$ are subnets of each other. This means that going from a net to a filter and back (or from a filter to net and back) yields the original object.\n\n\u2022 Thanks for the hint to subsets ;-) The explanation of the interrelation between nets filters and topology was not asked. \u2013\u00a0C-Star-W-Star Jan 8 '14 at 20:21\n\u2022 But I'm not talkin of a Subset without havin a partial ordering! \u2013\u00a0C-Star-W-Star Jan 8 '14 at 20:29\n\u2022 @Alex: Sorry. I think I got distracted by the answer by Alex ... wait, isn't that the same user as you? Did not notice that until now. Well, in that answer you mentioned the possibility that a net is indexed by an arbitrary set. Maybe you should have made more clear that you mean directed sets in contrast to preordered sets. \u2013\u00a0Stefan Hamcke Jan 8 '14 at 20:36\n\u2022 Oh yes, what I meant was preordered index sets. Thanks @Stefan Hamcke =) \u2013\u00a0C-Star-W-Star Jan 8 '14 at 20:40\n\n1) For convergence you obviously need some sense of direction. Sequences give a simple of direction (towards $\\infty$) but it turns out this is not good enough for dealing with arbitrary topological spaces. So people generalized sequences to nets by using arbitrary directed sets. There was no a priori reason this would work for arbitrary spaces either but it turned out it does work. Filters are equally good and are equivalent to nets.\n\n2) The subnet $y$, given by restricting to the top row, is cofinal in $x$ and it obviously converges to $0$, so $x \\to 0$ as well. Informally, the net eventually goes up. More formally, for every $(a,n) \\in x$, there is $(b, m) \\in y$ such that $(b, m) \\geq (a,n)$. Just take $(b, m) := (2, n)$. If you reverted the vertical arrows, the resulting net would go bottom and wouldn't converge.\n\n\u2022 1.) Yes, I know that it simply works by generalizing sequences and yes, I know nets are sufficient and necessary to describe a topology. But that doesn't answer my question. Is there a reason w.r.t. convergence?<br> Some sort of direction seems to be a good starting point.<br> So one can talk about convergence in which one chooses either to say sth holds true eventually upwards or downwards. 2.) Yes, I know that it converges, as I wrote so. But that was not my question either. Why does it make sense not does it converge. \u2013\u00a0C-Star-W-Star Jan 8 '14 at 20:11\n\u2022 @Alex: 1) I'm not sure what kind of answer you expect. Convergence is a phenomenon of getting near to something. So having sense of a direction is useful. I'm not sure if you can have more intuition than this since bad things happen in pathological spaces. 2) I've explained why it converges both formally and informally. What more do you need? For nets, the concept of direction is more complex than for sequences. Here, the net goes up and right and so for convergence issues only top matters and the net is constant there. So it's eventually close to $0$. I'm not sure what more do you want. \u2013\u00a0Marek Jan 8 '14 at 21:17\n\nCollecting everything, finally we have:\n\n\u2022 Notion of \"Direction\": Plain Order Relation\n\u2022 Notion of \"Eventually All\": Transitive Order\n\u2022 Notion of \"Being Expandable\": Directed Order\n\nIn detail, this means that:\n\n\u2022 First of all, we must be able to talk of a direction which gives rise to the notion of convergence and thus distinguishes sequences from sober subsets. That is, for all $U_x$ there is some $\\Lambda_0$ s.t. for all $\\lambda\\geq \\Lambda_0$ it holds $x_\\lambda \\in U_x$.\n\u2022 Next, we'd like to be sure that whenever sth. eventually holds it won't drop out anymore so we can say we're only getting closer. That is, if $\\lambda_0\\geq \\Lambda_0$ then for every succesor $\\lambda\\geq \\lambda_0$ it holds as well $\\lambda\\geq \\Lambda_0$.\n\u2022 Last, we want to be able to expand everything so that it holds as well. That is, for every $\\lambda$ there is some $\\lambda_0\\geq\\lambda$ s.t. as well $\\lambda_0\\geq\\Lambda_0$.","date":"2019-10-18 16:11:23","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9043228030204773, \"perplexity\": 254.61383551310843}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-43\/segments\/1570986684226.55\/warc\/CC-MAIN-20191018154409-20191018181909-00335.warc.gz\"}"} | null | null |
\section{Introduction}\label{intro}
In finite dimension, studying the behaviour of the orbits of a vector field (or of diffeomorphism) nearby a fixed point is a fundamental and classical problem. The very first natural step into this understanding is to compare the dynamical system with its linearization at the fixed point. This is done by trying to transform the dynamical system into its linear part by a change of coordinates. There are formal obstructions to do so, called resonances. Hence, in general, one can merely expect the dynamical system to be transformed into a {\it normal form}, that is supposed to capture effect the very nonlinearities, through a formal change of coordinates. It was understood by the end of the 19th century that if the convex hull of the eigenvalues of the linear part does not contain the origin (one says then that the linear part is in the "Poincaré domain"), and if an higher order analytic perturbation is formally conjugate to the linear part, then it is also analytically so. When the linear part does not satisfy this property, then one has so-called "small divisors" that may forbid the transformation to be analytic. It was a major step forward made by C.L. Siegel \cite{siegel}, followed by H. R\"ussmann \cite{russmann-ihes}(for diffeomorphisms) and by A.D. Brjuno \cite{bruno} (for vector fields) who devised a sufficient "small divisors condition" ensuring the analycity of a linearizing transformation as soon as there exists a formal one.
Linearizing (resp. Normalizing) problems for diffeomorphisms were devised by J. P\"oschel \cite{Posch} and for commuting families by the second author \cite{stolo-diffeos} (resp. \cite{Stolo-ihes}).
By the end of the 70's, it became clear to few people that some PDE's problems could be translated into an infinite dimensional dynamical systems to which one would have tried to apply methods of finite dimension. In particular, we mention the work by E. Zehnder \cite{zehnder-infinite} and V. Nikolenko \cite{Nik} who gave results similar to finite dimensional ones. It happens that the "small divisors condition" they required are too strong and are rarely satisfied. Furthermore, in general, the notion of formal normal form and formal change of variables should be clarified (for instance if one defines formal polynomials and formal power series it is not in general true that this space has a Poisson algebra structure). Neverteless, in some very peculiar situation, this problem can be handled\cite{stolo-bambusi}.
\\
Starting from the mid 80', there has been a lot of interest {in studying long time behavior of solutions of PDEs. For those PDEs which can be considerered as Hamiltonian (infinite dimensional) dynamical systems related to a symplectic sturucture, one natural way to proceed is to prove the existence of finite dimensional invariant tori in the phase space. This usually implies the existence of quasi-periodic solutions, which are defined for all time. Lot of progresses has been done on the problem of extending KAM theory to PDEs. This circle of problems are very related, though distinct, to the ones solved in this article. Indeed, here one considers a dynamical system close to an elliptic fixed point with the purpose of conjugating it to its most simple {\it normal form}~: its linear part at the fixed point. On the other hand, in KAM theory, one looks for the existence of a { \it finite dimensional invariant flat torus} on which the dynamics is the linear translation by a diophantine frequency. There is by now a wide literature dealing the subject related to semilinear PDEs, starting from \cite{Kuk,Po,KP, W,CW}, (for instance, see \cite{EK,GYX, PPBumi, BKM18,Y} for more recent treatments). It has been early understood that these results might be seen through elaborated versions of "Nash-Moser" theorem see for instance \cite{Bo98,BB3,BCP,CM}.
We finally mention \cite{FGPr, BBHM,BM,FG} for the case of fully-nonlinear PDEs. See also \cite{BMP:almost,CY} and references therein for infinite-dimensional tori. }
Birkhoff normal form (BNF) methods have been used in order to prove long time existence results and control of Sobolev norms for many classes of evolution PDEs close to an elliptic fixed point. Loosely speaking the point is to canonically transform $H$ into a Hamiltonian Normal form which depends only on the actions plus a remainder term whose the Taylor polynomial, at the origin is of degree $\tN+1 $. If one achieves this then initial data which are $\delta$-small (with respect to the norm on the phase space) stay small (in the same norm) for times of order $\delta^{-\suca}$. A more precise formulation is given in the {\it Strategy section }below. Of course in the infinite dimensional setting this stability time depends strongly on the choice of the phase space as well as on the nature of the non-linear terms. A further problem is that in general it is not obvious that one can perform even one step of this procedure, indeed the generating function of the desired change of variables is a {\it formal} polynomial which in infinite dimension is not necessarily analytic. This is a particularly difficult problem in the case of PDEs with derivatives in the nonlinearity.
\\
Let us briefly describe some of the literature. Regarding applications to PDEs (and particularly the NLS) the first results were given in \cite{Bourgain:1996} by Bourgain, who proved that for any $\suca$ there exists $p=p(\suca)$ such that small initial data
in the $H^{p'+p}$ norm stay small in the $H^{p'}$ norm, for times of order $\delta^{-\suca}$.
Afterwards, Bambusi in \cite{Bambu:1999b} proved that superanalytic initial data stay small in analytic norm for subexponentially long times.
Following the strategy proposed in \cite{Bambusi:2003} for the Klein-Gordon equation Bambusi and Gr\'ebert in \cite{Bambusi-Grebert:2003} first considered NLS equations on $\T^d$ and then, in \cite{Bambusi-Grebert:2006}, proved
polynomial bounds for a class of {\it tame-modulus} PDEs.
Similar results were also proved for the Klein Gordon equation on tori and Zoll manifolds in \cite{DS0},\cite{DS},\cite{BDGS}. Successively Faou and Gr\'ebert in \cite{Faou-Grebert:2013} considered the case of analytic initial data and proved subexponential bounds on the stability time
for classes of NLS equations in $\T^d$. In \cite{BMP18} the first author with Biasco and Massetti studied an abstract Birkhoff normal form on {\it sequence spaces} proving subexponential stability times for Gevrey regular initial data. A similar result was proved in \cite{Cong}.
An interesting feature of the last three papers is that instead on relying on tameness properties they use the fact that the equations they study have some symmetries, namely they are gauge and translation invariant (actually in \cite{BMP18} the translation invariance condition is weakened).
\\
All the preceding results regard semilinear PDEs.
Regarding equations with derivatives in the nonlinearity, the first results were in \cite{Yuan-Zhang} for the semilinear case. Then we mention \cite{Delort-2009,DS15} for the Klein-Gordon equation, \cite{Berti-Delort} for the water waves and \cite{FI} for the reversible NLS equation. Recently, Feola and Iandoli, \cite{FI2} prove polynomial lower bounds for the stability times of Hamiltonian NLS equations with two derivatives in the nonlinearity. In the context of infinite chains with a finite range coupling, similar considerations can be done and we mention \cite{BenFroGior}.
\subsection{Statements}
We study Hamiltonians on infinite dimensional sequence spaces, which are higher order ({\it $M$-regular}) analytic perturbations of quadratic Hamiltonians nearby an elliptic fixed point (i.e a zero) and satisfying the {\it Momentum conservation} property, namely they are formally translation invariant, see Definition \ref{mconserv}.
\\
We first show that the space $\cF$ of formal Hamiltonians in infinite variables $u=\pa{u_j}_{j\in \Z}$ satisfying this {\it Momentum conservation property} is well defined and closed w.r.t Poisson brackets, then we define a scaling degree (which is the homogeneity degree minus two, see Definition \ref{scialla}, so that the degree of the Poisson bracket of two functions is the sum of the respective degrees) so that $\cF$ has a natural filtered Lie algebra structure. Thus $\cF$ is decomposed in homogeneous components $\cF^d$ and we define $\cF^{\ge d}:= \widehat{\oplus}_{h\ge d}\cF^{h}$.
\\
Given a rationally independent $\omega\in \R^\Z$, namely such that all non-trivial finite rational combinations of $\omega$ are non zero, we consider the affine space $D_\omega +\cF^{\ge 1}$ of formal Hamiltonians of the form
\begin{equation}
\label{lequazione}
H= D_\omega + P \,,\quad D_\omega= \sum_{j\in \Z} \omega_j |u_j|^2\,,\qquad P= O(u^3)\,,
\end{equation}
and acting on this space we define the group of formal symplectic (i.e canonical) transformations $e^{\set{\cF^{\ge 1},\cdot}}$.
Finally we define the space of normal forms as those formal Hamiltonians which Poisson commute with $D_\omega$. We prove the following
\begin{thm*}
All Hamiltonians $H$ as above are formally symplectically conjugated to normal form. Moreover the normal form Hamiltonian associated to $H$ is unique.
\end{thm*}
Having properly developed the formal framework, we consider the question of formal vs. analytic linearization in the infinite dimensional setting on the phase space of Gevrey regular functions.
\\
In order to keep technical difficulties to a minimum, we work on
Nonlinear Schr\"odinger like Hamiltonians of the form
with the standard symplectic structure on $\ell_2= \ell_2(\Z,\C)$. As phase space we consider the sequences of {\it Gevrey} regularity, namely
we consider the weighted space
\begin{equation}
\label{gevrey}
\th_{s,p,\theta} :=\set{ u\in \ell^2(\Z,\C): \quad |u|^2_s:=\sum_{j\in\Z}\jap{j}^{2p}e^{2s\jap{j}^\theta}|u_j|^2<\infty }
\end{equation}
where $\jap{j}:=\max(|j|,1)$, $s>0$, $p\geq \frac12$ and $0<\theta< 1$. Then, given $r>0$, we consider the space of {\it M-regular Hamiltonians} $P\in \cH_r(\th_{s,p,\theta})$, such that
the Cauchy majorant of the map $u\to X_P(u)$ is analytic from the ball $B_r (\th_{s,p,\theta})$, centered at the origin and of radius $r$ into $\th_{s,p,\theta}$.
\\
Now we consider a Hamiltonian as in \eqref{lequazione}, with the additional condition that $P\in \cH_{r_0}(\th_{s_0,p,\theta})$ and the
frequency $\omega$ is ``Diophantine" in the following sense introduced by Bourgain \cite{Bourgain:2005}.
We set
\begin{equation}
\Omega_{}:=\set{\omega=\pa{\omega_j}_{j\in \Z}\in \R^\Z,\quad \sup_j|\omega_j-j^2| < 1/2 }
\end{equation}
\begin{defn} Given $\gamma>0$ , we denote by $\dg$ the set of \sl{Diophantine} frequencies
\begin{equation}\label{diofantinoBIS}
\dg:=\set{\omega\in \Omega_{}\,:\; |\omega\cdot \ell|> \gamma \prod_{n\in \Z}\frac{1}{(1+|\ell_n|^{2} \jap{n}^{2})}\,,\quad \forall \ell\in \Z^\Z_f\setminus \{0\}}.
\end{equation}
\end{defn}
The map $\pa{\omega_j}_{j\in\Z}\to \pa{j^2-\omega_j}_{j\in\Z}$ identifies $\Omega$ with $[-1/2,1/2]^\Z$. Hence we endow $\Omega$ with the product topology and with the corresponding probability measure. With respect to such measure Diophantine frequencies are typical, namely $\Omega\setminus\tD_\g$ has measure {proportionally bounded by $\g$ (see \cite{BMP18}[Lemma 4.1])}.
\\
Then we prove~:
\begin{thm*}
{\it If $H$ is formally conjugated to $D_\omega$, then there exists $r_1<r_0$, $s_1> s_0$ and a close to identity analytic symplectic change of variables
$
\Psi: B_{r_1}(\th_{s_1,p,\theta}) \to \th_{s_1,p,\theta}
$
such that $H\circ \Psi = \sum_{j\in \Z} \omega_j |u_j|^2$.}
\end{thm*}
\subsection{Strategy}
In order to describe our strategy consider a finite dimensional Hamiltonian system with a non-degenerate elliptic fixed point, which in the standard complex symplectic coordinates
$u_j= \frac{1}{\sqrt2}(q_j+ \im p_j)$
is described by the Hamiltonian
\begin{equation}
\label{finito}
H= \sum_{j=1}^n \omega_j |u_j|^2 + O(u^3)\,,\quad \mbox{where $\omega_j\in \R$ are the {\it linear frequencies}}.
\end{equation}
Here if the frequencies $\omega$ are rationally independent, then one can perform the so-called Birkhoff normal form procedure: for $\suca\geq 1$
Hamiltonian \eqref{finito} is transformed into
\begin{equation}
\label{finitobirk}
\sum_{j=1}^n \omega_j |u_j|^2 +Z +R\,
\,,
\end{equation}
where $Z$ depends only on the actions $(|u_i|^2)_{i=1}^n$ while $ R= O(|u|^{\suca +3})$ has a zero of order at least $\suca + 3$ in $\abs{u}$.
At each step, the generating function of the change of variables is a polynomial, so it is analytic and generates a flow in a sufficiently small ball $B_\delta$ around the origin.
It is well known that this procedure generically diverges in $\suca$, but assuming that $\omega$ is appropriately non resonant, say diophantine\footnote{A vector $\omega\in\R^n$ is called diophantine when it is badly approximated by rationals, i.e. it satisfies, for some $\gamma,\tau>0$, $\abs{k\cdot \omega} \ge \gamma \abs{k}^{-\tau},\quad \forall k\in\Z^n\setminus \set{0}\,$.} one can control $R$ and hence find $\suca=\suca(\delta)$ which minimizes the size of the remainder $R$. It can be shown that it is bounded by an exponentially flat function of $\delta$, of order related to $\tau$ (for a general treatment, see instance, \cite{lombardi-nf,stolo-lombardi}). This phenomenon is also related to Nekhoroshev kind of result \cite{Posch-Nekho,BGG85,Nekho,Nid,BCG}.
\\
If $H$ in \eqref{finito} is "formally linearizable", namely there exists a formal symplectic change of variables which conjugates $H$ to $\sum_{j=1}^n \omega_j |u_j|^2$, and $\omega$ is Diophantine, then at each step of the procedure described above, \bu{we find} $Z=0$ and one can prove convergence.
In order to apply this general scheme in the infinite dimensional setting we first discuss the BNF procedure at the level of formal power series. Here the fundamental difference w.r.t. the finite dimensional case is that even polynomials can be just formal power series, so it is not a priori obvious that the space
of formal power series is well defined and has a Poisson algebra structure (which coincides with the usual one on finite dimensional subspaces).
As a simple example consider the formal power series $ H= \sum_j u_j$, then
\[
\{H,\bar H\}= \sum_i\sum_j \{u_j\,,\bar u_i\}=\infty\,.
\]
We show that for translation invariant formal Hamiltonians the Poisson brackets are well defined (see also \cite{FGP} ), and that formal Hamiltonians are a filtered Lie algebra with respect to a {\it scaling degree}. Then we define a group of formal symplectic changes of variables, and prove our BNF result. In order to define our changes of variables and prove the group structure we strongly rely on the properties of the {\it scaling degree} as well as on the Baker Campbell Hausdorf formula.
\\
Then we restrict to functions on the sequence space $\th_{s,p,\theta}$, introduce the space of regular Hamiltonians and state the main relevant properties. All properties were proved in \cite{BMP18} in the more restrictive case of Gauge invariant Hamiltonians, so we follow the same strategy; for completeness we give all the proofs in the appendix.
One we have all the basic properties needed to perform Birkhoff Normal Form, proving that formal linearizability implies analytic linearizability becomes a relatively straightforward induction.
\section{ Formal Birkhoff Normal on sequence spaces}
As usual given a vector $k\in \Z^\Z$, $|k|:=\sum_{j\in\Z}|k_j|$. We denote $\N^\Z_f$ to be the set of finitely supported sequences of non negative integers, similarly for $\Z^\Z_f$. If $j\in\mathbb{Z}$ then $\be_j\in \Z^\Z_f$ denotes the vector the $j$-coordinate of which is $1$, while the others are zero.
\begin{defn}[Formal power series]\label{Hr}
We consider the space $\cF$ of formal power series expansions
in $u\in \C^\Z$:
$$
H(u) = \sum_{\substack{\bal,\bbt\in\N^\Z_f} }H_{\bal,\bbt}u^\bal \bar u^\bbt\,,
\qquad u\in \C^\Z,\quad
u^\bal:=\prod_{j\in\Z}u_j^{\bal_j} \quad |v|: = \sum_i|v_i|
$$
with the following properties:
\begin{enumerate
\item $H_{0,0}= 0$, $H_{\be_0,0}= H_{0,\be_0}=0$
\item Reality condition:
\begin{equation}\label{real}
H_{\bal,\bbt}= \overline{ H}_{\bbt,\bal}\,;
\end{equation}
\item Momentum conservation:
\begin{equation}
H_{\bal,\bbt}= 0 \quad\mbox{if}\;\, \pi(\bal,\bbt):= \sum_{j\in\Z} j(\bal_j-\bbt_j)\ne 0 \label{mconserv}
\end{equation
\end{enumerate}
\begin{rmk}
The condition (3) means that the formal Hamiltonian is invariant w.r.t. the symmetry $u_j\to e^{ \im j\tau} u_j$, $\tau\in\mathbb{R}$.
\end{rmk}
We shall denote
\[
\cM:=\{(\bal,\bbt) \in \N^\Z_f:\qquad \pi(\bal,\bbt)=0\}
\]
so that $H\in \cF$ can be written as
\[
\sum_{(\bal,\bbt)\in \cM}H_{\bal,\bbt}u^\bal \bar u^\bbt
\]
Finally we define
\begin{equation}\label{nocciolina}
\cK:=\left\{Z\in \cF\, : \, Z(u) = \!\!\sum_{\substack{\bal\in \N^\Z_f}} Z_{\bal,\bal}|u|^{2\bal}\right\}\,,\qquad \cR:=\left\{R\in \cF\, : \, R(u) = \!\!\!\!\!\!\!\!\sum_{\bal,\bbt\in \cM:\, \bal\ne \bbt}\!\!\!\!\!\!\!\! R_{\bal,\bbt}u^\bal\bar u^\bbt\right\}
\end{equation}
and we can decompose $\cF= \cK\oplus \cR$ as each element of $\cF$ can uniquely be expressed in term of monomials the coefficients of which is either zero or not zero.
\end{defn}
\begin{defn}[ scaling degree]\label{scialla}
For $d\in \N$, we denote by $\cF^d\subset \cF$ the vector space of homogeneous formal polynomials of degree $\td+2$, and define
\[
\cF^{\le d}= \oplus_{h\le d} \cF^{h}\,,\quad \cF^{>d}:=\widehat\oplus_{h>d} \cF^{h} \,,\quad \cF^{\ge d}:= \cF^{>d}\oplus \cF^{d}\,,\cF =\cF^{\le d}\oplus\cF^{> d},\dots
\]
We define the projections associated to these direct sum decompositions
\[
\Pi^{(\td)} H= \sum_{|\al|+|\bt|=\td+2}H_{\al,\bt} u^\al\bar u^\bt \,,\quad \Pi^{(> \td)} H= \sum_{|\al|+|\bt|> \td+2}H_{\al,\bt} u^\al\bar u^\bt\,,\dots
\]
Elements of $\cF^{\geq d}$ (resp. $\cF^{> d}$) are said to be {\it of scaling order} $\geq d+2$ (resp. $>d+2$). In the sequel, for simplicity, we shall just say that an element of $f\in \cF^{\geq d}$ is of "order d" and we shall say that $f$ is "exactly of order d" if it has a non vanishing component $\Pi^{(\td))}f$ in $\cF^d$.
Finally we define
\[
\Pi^\cK H = \sum_{\al}H_{\al,\al} |u|^{2\al}\,,\quad \Pi^\cR H = \sum_{\al\ne \bt}H_{\al,\bt} u^\al\bar u^\bt.
\]
We denote by $\cK^d:= \cF^d \cap \cK$ and similarly for $\cR$ and $\ge \td ,\le \td$.
Note that $\cF= \widehat \oplus_d \cF^d$.
\end{defn}
\begin{rmk}
Of course, since we are in infinite dimension, even if the $\cF^d$ are homogeneous they are only formal polynomials. However if we restrict to monomials $u^\bal\bar{u}^\bbt$ with $|\bal_j|+|\bbt_j| =0$ for all $j>N$ we are working on the usual space of polynomials on which we have the standard symplectic structure $\im \sum_{j\le N} d u_j\wedge d \bar u_j$. We now show that such structure extends to $\cF$.
\end{rmk}
\begin{prop}\label{degree decompositionbis}
The following Formula \eqref{well} is well defined and endows $\cF$ with a Poisson algebra structure which is a filtered Lie algebra w.r.t. the $\cF^{\geq d}$'s.
\begin{equation}\label{well}
\set{F,G} := \im\sum_{ (\bal^{(i)},\bbt^{(i))}\in\cM } \bcoeffu{F}\bcoeffd{G} \sum_j \pa{\baluno_j \bbtdue_j - \bbtuno_j \baldue_j }u^{\baluno + \baldue -\be_j}\bar{u}^{\bbtuno + \bbtdue - \be_j}
\end{equation}
\end{prop}
Before proving our assertion we need a technical lemma.
Let $\be_j\in \N^\Z_f$ be the $j$th vector of the standard basis.
\begin{lemma}\label{finmo} 1)\quad Given $\alpha\in \N^\Z_f$ there is only a finite number of pairs $ \alpha^{(1)},\alpha^{(2)}\in \N^\Z_f$ with $\alpha=\alpha^{(1)}+\alpha^{(2)}$.
2)\quad
Given $ (\bal,\bbt)\in \cM$ there is only a finite number of pairs $ (\baluno,\bbtuno),\ (\baldue,\bbtdue)\in \cM$ and indices $j\in\Z$ such that:
\begin{enumerate}
\item[\it i)] $ (\bal,\bbt)= (\baluno,\bbtuno)+(\baldue,\bbtdue)-(\be_j,\be_j)$
\item[\it ii)] one has $\bal_j^{(1)}\bbt_j^{(2)}+ \bal_j^{(2)}\bbt_j^{(1)}\ne 0$
\end{enumerate}
\end{lemma}
\begin{proof} 1) is clear since for all $j$ one has $0\le (\alpha_1)_j\leq \alpha_j $.\smallskip
\\
2)\quad By item 1) we may divide $(\bal,\bbt)= (a^{(1)}, b^{(1)})+ (a^{(2)}, b^{(2)})$ in a finite number of ways. Then
the pairs $(\baluno,\bbtuno),(\baldue,\bbtdue)$ can only have one of the following forms (up to exchanging the indices)
A) $\quad (\baluno,\bbtuno)= (a^{(1)}, b^{(1)})+ (\be_j,\be_j)\,,\quad (\baldue,\bbtdue) = (a^{(2)}, b^{(2)})$
B) $\quad (\baluno,\bbtuno)= (a^{(1)}, b^{(1)})+ (\be_j,0)\,,\quad (\baldue,\bbtdue) = (a^{(2)}, b^{(2)})+(0,\be_j)$,
\\
for some index $j\in \Z$.
\\
If we are in case $A)$ then by condition $ii)$ we have $j\in $Supp$(a^{(2)}+b^{(2)})$, which restricts to a finite number of possible $j's$. Otherwise in case $B)$
by momentum conservation e have $j= -\pi( a^{(1)}, b^{(1)})={\pi} (a^{(2)}, b^{(2)})$ and again $j$ is restricted to a finite number of possible choices.
\end{proof}
\begin{proof}[Proof of Proposition \ref{degree decompositionbis}]
The fact that the Poisson bracket is well defined follows immediately from the previous Lemma.
Indeed by construction
\[
\{F,G\}= \sum_{\bal,\bbt} P_{\bal,\bbt} \buu \in \cF
\]
where $P_{\bal,\bbt}=0$ if $\pi(\bal,\bbt)\ne 0$ and otherwise
\begin{equation}
\label{begon}
P_{\bal,\bbt}= {\im}\sum_j \sum_{\substack {\bal^{(i)},\bbt^{(i)}\in\N^\Z_f:\; \pi(\bal^{(i)},\bbt^{(i)})=0\\ \bal = \baluno+\baldue - \be_j\,, \bbt = \bbtuno+\bbtdue - \be_j }}
\bcoeffu{F}\bcoeffd{G} \pa{\baluno_j \bbtdue_j - \bbtuno_j \baldue_j }\,.
\end{equation}
Then item 2 of the previous Lemma implies that $P_{\al,\bt}$ above is given by a finite sum.
\\
The fact that it endows $\cF$ with a Poisson algebra structure follows from the fact that the infinitely many identities defining such a structure involve only a finite number of elements $u_i,\bar u_i$ and then we are in the canonical Poisson algebra.
\\
The filtered Lie algebra property comes from the fact that in \eqref{begon} we get $|\alpha|+|\beta| = |\baluno|+ |\baldue|+ |\bbtuno|+ |\bbtdue| -2$, this shows that if $F\in \cF^{\ge \td_1}$, and $G\in \cF^{\ge \td_2}$ then
\begin{equation}
\label{a+b}|\alpha|+|\beta| \ge \td_1+2+\td_2+2-2 = \td_1+\td_2+2.
\end{equation} so
$\{F,G\}\in \cF^{\ge \td_1+\td_2}$.
\end{proof}
\begin{rmk}\label{gogna}
Let $H_i\in \cF^{\ge \td_i}$ be a sequence of formal Hamiltonians with $\td_{i+1}\ge \td_i$ for all $i\ge 1$. Then the series
\[
H= \sum_{i=1}^\infty H_i \in \cF^{\ge \td_1}
\]
is well defined since for any $\td\ge \td_0$ the projection \[\Pi^{(\td)} H = \Pi^{(\td)} \sum_{i: \td_i \le \td} H_i \] is a finite sum.
\end{rmk}
We say that a linear operator $L:\cF\to \cF$ is of order (or increase the order by) $\td$ if for all $h$
\[
L:\cF^{ \ge h} \to \cF^{ \ge h+\td}\,.
\]
\begin{lemma}\label{lemniscata}
let $L_n$ be a sequence of linear operators on $\cF$ and let $\td_n$ be the order of $L_n$. If the sequence $\td_n$ increases to infinity then
\[
L:= \sum_{n=1}^\infty L_n \,,\qquad T =\prod_{n=1}^\infty (\id + L_n)-\id
\]
are linear operators on $\cF$ of order $\td_1$.
\end{lemma}
\begin{proof}
For the first statement, for all $\td\in \N$ let $N(\td)$ be the largest $N$ such that $\td_N\le \td$. By construction
$\Pi^{(\le \td)} L_n K=0$ for all $n> N(\td)$ and for any $K\in \cF$.
Then for all $K\in \cF$ and $N>N(\td)$ one has
\[
\Pi^{(\le \td)} \sum_{n=1}^N L_n K = \Pi^{(\le \td)} \sum_{n=1}^{N(\td)} L_n K\,,
\] and the claim follows.
\\
Regarding the second statement we proceed similarly
\[
\prod _{n=1}^N (\id + L_n ) = \prod _{n=1}^{N-1} (\id + L_n ) +L_N \prod _{n=1}^{N-1} (\id + L_n )\,,
\] hence, {for all $\td\geq 0$ and all $N> N(\td)$}
\[
\Pi^{(\le \td)}\prod _{n=1}^N (\id + L_n ) = \Pi^{(\le \td)}\prod _{n=1}^{N(\td)} (\id + L_n )\,.
\]
\end{proof} As a direct consequence we have the following.
\begin{cor}
Given $G\in \cF^{\ge \td}$, with $\td\ge 1$ we define
\begin{equation}
\label{flusso}
\ad_G:= \{G,\cdot \} \,,\quad \Phi_G:= \exp(\{G,\cdot\})= \sum_{k\ge 0} \frac{\ad_G^k}{k!}\,,
\end{equation}
then $ \ad_G$ and $\Phi_G-\id$ are operators of order $\td$, namely
\[
\ad_G,\Phi_G -\id: \cF^{\ge h} \to \cF^{\ge h+\td}\,.
\]
Similarly for any sequence $b_k$ one has that
\[
\sum_{k\ge \tn } b_k{\ad_G^k}: \cF^{\ge h} \to \cF^{\ge h+\td\tn }\,.
\]
\end{cor}
\begin{defn}
Given $G\in \cF^{\ge 1}$ we call the operator $\Phi_G$ defined in \eqref{flusso} a formal symplectic change of variables on $\cF$.
\end{defn}
The following Lemma ensures the group structure of the formal symplectic changes of variables
\begin{lemma}[Baker-Campbell-Haussdorf]\label{BCH}
Given $F\in \cF^{\ge \td_1}$ and $G\in \cF^{\geq \td_2}$, with $\td_i\ge 1$, then there exists $K\in \cF^{\ge 1}$, such that
\[
e^{\{G,\}}e^{\{F,\}}= e^{\{ K,\}} \,,\qquad K-F-G \in \cF^{ \ge \td_1+\td_2}
\]
\end{lemma}
\begin{proof}
By the Baker-Campbell-Hausdorff formula (\cite{serre-Lie1}[p.29]) one has
\[
K:=
\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n} \sum_{r_i+s_i>0} \frac{[G^{r_1}F^{s_1}\dots G^{r_n}F^{s_n}]}
\left(\sum_{i=1}^n(r_i+s_i)\right) \prod_{i=1}^n r_i! s_i!}
\]
where
\begin{equation}
\label{hoho}
[G^{r_1}F^{s_1}\dots G^{r_n}F^{s_n}]:= \begin{cases}
\ad_G^{r_1}\ad_F^{s_1}\dots \ad_G^{r_n}F \quad \mbox{ if} \; s_n=1
\\
\ad_G^{r_1}\ad_F^{s_1}\dots \ad_F^{s_{n-1}} G \quad \mbox{ if}\; s_n=0\,,\; \mbox {and} \; r_n=1
\\
0 \quad \quad \quad \quad \mbox {otherwise}
\end{cases}
\end{equation}
Recalling that $F\in \cF^{\ge \td_1}$ and $G\in \cF^{\geq \td_2}$, each term $\ad_G^{r_1}\ad_F^{s_1}\dots \ad_G^{r_n}F$ (resp. $\ad_G^{r_1}\ad_F^{s_1}\dots \ad_F^{s_{n-1}}G$)
is of order $\left(\sum_{i=1}^{n}r_i\right)\td_2+(\sum_{i=1}^{n}s_i)\td_1\geq n \min(\td_1,\td_2)$.
Hence setting $N(\td)$ to be the largest $N$ such that $ N \min(\td_1,\td_2) \le \td$n
\[
\Pi^{\le \td} K= \Pi^{\le \td} \sum_{n=1}^{N(\td)} \frac{(-1)^{n-1}}{n} \sum_{r_i+s_i>0} \frac{[G^{r_1}F^{s_1}\dots G^{r_n}F^{s_n}]}
\left(\sum_{i=1}^n(r_i+s_i)\right) \prod_{i=1}^n r_i! s_i!}
\]
\\
Moreover if $n\ge 2 $ then the Hamiltonian in \eqref{hoho} is of order $\ge \td_1+\td_2$, so $K-F-G \in \cF^{ \ge \td_1+\td_2}$.
\end{proof}
\begin{lemma} \label{compo} Given a sequence of generating functions $G_i\in \cF^{{\geq} \td_i}$ with $\td_{i+1} > \td_i\ge 1$ then there exists $\cG\in \cF^{\ge d_1}$ such that the composition
\[
\prod_i e^{\{G_i,\}} = e^{\{\cG,\}}
\]
\end{lemma}
\begin{proof}
By Lemma \ref{lemniscata} with $L_n = e^{\{G_n,\}} -\id$ we know that $\prod_i e^{\{G_i,\}}$ is a well defined operator of $\cF$.
Using Lemma \ref{BCH} we can define $F_k\in \cF^{\ge1}$ iteratively so that
\[
e^{\{F_k,\cdot\}}= e^{\{G_k,\cdot\}} e^{\{F_{k-1},\cdot\}}
\]
since $e^{\{G_k,\cdot\}} -\id$ is of order $\td_k$ there exists $N(\td)$ such that if $k>N(\td)$ then
\[
\Pi^{(\le \td )} F_k = \Pi^{(\le \td )} F_{N(\td)}
\]
Then $\cG= \lim_{k\to \infty} F_k$ is well defined.
\end{proof}
For any vector $\omega\in \R^\Z$ such that
\[\omega\cdot\ell \ne 0\,,\quad \forall \ell\in \Z^\Z_f\setminus\{0\}\,,
\]
we define the {\it non-resonant} quadratic Hamiltonian
\[
D_\omega:= \sum_j \omega_j |u_j|^2\,.
\]
\begin{lemma}\label{uff}
The operator $\ad_{D_\omega}$ is invertible on $\cR^{(\td)}$ for all $\td$.
\end{lemma}
\begin{proof}
Given $F\in \cR^{(\td)}$, we have
\begin{equation*}
\set{D_{\omega},G} = \im\sum_{\substack {\bal^{(2)},\bbt^{(2)}\in\N^\Z\,, \\
|\bal^{(2)}|+|\bbt^{(2)}|<\infty\,,\, \pi(\bal^{(2)},\bbt^{(2)})=0}} \bcoeffd{G} \left(\sum_j \omega_j\pa{\bbtdue_j - \baldue_j }\right)u^{\baldue}\bar{u}^{\bbtdue}=F
\end{equation*}
Hence, we have $G:=\ad_{D_\omega}^{-1}(F)$ with for all $\bal^{(2)},\bbt^{(2)}\in\N^\Z$, $\bal^{(2)}\neq\bbt^{(2)}$ with $|\bal^{(2)}|+|\bbt^{(2)}|<\infty$ and $\pi(\bal^{(2)},\bbt^{(2)})=0$,
\[
\bcoeffd{G}:=\bcoeffd{F} \left(\sum_j {\im}\omega_j\pa{\bbtdue_j - \baldue_j }\right)^{-1},\quad G_{\alpha^2,\alpha^2}=0.
\]
\end{proof}
\begin{prop}[Birkhoff Normal Form] \label{giova}Given any formal Hamiltonian of the form
\begin{equation}
\label{Birkhof}
H= D_\omega + Z + R
\end{equation}
where $Z\in \cK^{\ge 2}$ and $R\in \cF^{\ge \td}$ with $\td\ge 1$, then
\begin{enumerate}
\item {\bf Formal Normal Form:} there exists $S\in \cF^{\ge \td}$ such that
\[
e^{\{S,\cdot\}} H = D_\omega + \wtZ\,,\quad \wtZ- Z \in \cK^{\ge \td}\,.
\]
\item {\bf Uniqueness:} if $G\in \cF^{\ge 1}$ is such that $e^{\{G,\cdot\}} H \in \cK$ then $e^{\{G,\cdot\}} H= e^{\{S,\cdot\}} H$. Hence to each $H$ as above we can associate a unique $Z_H\in \cK^{\ge 2}$ such that $e^{\{S,\cdot\}} H= D_\omega+ Z_H$.
\end{enumerate}
\end{prop}
\begin{proof}
For item $(1)$ Let us first consider the case $\td\ge 2$. we start with a Hamiltonian $H_0\in \cF$ of the form $D_\omega+Z_0+ P_0$ with $P_0\in \cF^{\ge d}$ and we iteratively construct a sequence of generating functions
$S_i \in \cR^{\ge 2i+d}$ and Hamiltonians $H_i$ by setting
\[
\{D_\omega,S_i\} =\Pi^{\cR} H_{i}\,,\quad H_{i+1}= e^{\{S_i,\cdot\} }H_{i}\,.
\]
We now show inductively that for each $i$
\[
\Pi^{(< 2i +d)} \Pi^{\cR} H_i =0 \,, \quad S_i \in \cR^{\ge 2i+ d}
\]
so in other words
\[
H_i = D_\omega+ Z_i + P_i\,,\quad Z_i \in \cK\cap \cF^{\le 2i +d-1} \,,\quad P_i\in \cF^{\ge 2i+d} \,.
\]
For $i=0$ we just set $Z_0= \Pi^{< d} Z$ and $P_0= R+ \Pi^{\ge d} Z$.
By induction we assume that $P_i \in \cF^{\ge 2i+d}$. Then by Lemma \ref{uff}, $S_i \in \cF^{\ge 2i +d}$.
\begin{align*}
e^{\{S_i,\cdot\} }H_{i} &=D_\omega+ Z_i + P_i + \{S_i, D_\omega\} + \sum_{h=2}^\infty \frac{\ad_{S_i}^{h-1}}{h!} \{S_i,D_\omega \}
+ \sum_{k=1}^\infty \frac{\ad_{S_i}^k}{k!}(Z_i + P_i) \\
&= D_\omega+ Z_i + \Pi^\cK P_i - \sum_{k=1}^\infty \frac{\ad_{S_i}^{k}}{(k+1)!}\Pi^{\cR} P_{i} + \sum_{k=1}^\infty \frac{\ad_{S_i}^k}{k!}(Z_i + P_i)\,.
\end{align*}
So we may set
\[
Z_{i+1}:= Z_i
\Pi^{(< 2i +d+2)}\Pi^\cK P_i \,,\quad P_{i+1}= e^{\{S_i,\cdot\} }H_{i}- D_\omega - Z_{i+1}
\]
and verify that $P_{i+1}\in \cF^{\ge 2i+d+2}$ by applying Proposition \ref{degree decompositionbis} and noticing that, since $4i +2\td \ge 2i +\td+2$, the term of lowest degree is $\{S_i,Z\}$.
\\
Then
we set
\[
\widetilde Z= \lim_{i\to \infty} Z_i = Z_0+ \sum_{i=0}^\infty \Pi^{\le 2i +d+1}\Pi^\cK P_i = Z_0+ \Pi^\cK \sum_{i=0}^\infty \Pi^{(< 2i +d+2)} \Pi^{(\ge 2i +d) }P_i\,,
\] which is well defined by Remark \ref{gogna}. Finally by Lemma \ref{compo} we can define $S\in \cF^{\ge \td}$ so that
\[
e^{\{S,\cdot\}}= \prod_{i=0}^\infty e^{\{S_i,\cdot\}}\,.
\]
If $\td=1$ we perform a preliminary step in order to increase the degree by one and then we start the procedure explained above.
We start with $H = D_\omega+ P$, with $P:= R+ Z$ . As before we fix $S\in \cR^{\ge 1}$ so that
$\{D_\omega ,S\}=\Pi^\cR H$ we set
\begin{align*}
H_0:= e^{\{S,\cdot\} }H = D_\omega+ \Pi^\cK P - \sum_{k=1}^\infty \frac{\ad_{S}^{k}}{(k+1)!}\Pi^{\cR} P + \sum_{k=1}^\infty \frac{\ad_{S}^k}{k!}P\,.
\end{align*}
then fixing $Z_0:= \Pi^{\le 2} \Pi^\cK P$ and $P_0:= H_0-D_\omega-Z_0$ we are in the setting of the previous case.
Regarding item $(2)$ we remark that If $e^{\{S_1,\cdot\}}$ transforms a normal form $D_{\om}+K_1$ into a normal form $D_{\om}+K_2$, then
$$
e^{\{S_1,\cdot\}}(D_{\omega}+K_1)= D_{\omega}+K_1+\sum_{h=1}^\infty \frac{\ad_{S_1}^{h-1}}{h!} \{S_1,D_\omega+K_1 \}=D_{\omega}+K_2.
$$
Since $\cK=\cK^{\ge 2}$ and $S\in \cH^{\ge 1}$, comparing homogeneous terms of degree $1$ we get $\{S_1,D_\omega\}=0$ so we should have $S_1^{(1)}\in\cK$ which can only be possible if $S_1^{(1)}=0$.
Comparing homogeneous terms of degree $2$, we obtain $K_1^{(2)}-K_2^{(2)}+\{S_1^{(2)}, D_{\om}\}=0$. Recalling that $\{S_1^{(2)}, D_{\om}\}\in \cR$ we have $K_1^{(2)}-K_2^{(2)}\in \cK\cap\cR$ is zero and $S_1^{(2)}\in\cK$. Assuming that $K_1^{(j)}=K_2^{(j)}\in \cK$ and $S_1^{(j)}\in\cK$ for $2\leq j\leq m$. Then we have
\begin{eqnarray*}
K_1^{(m+1)}-K_2^{(m+1)}&+&\{S_1^{(m+1)}, D_{\om}\}+\sum_{h=2}^\infty \frac{1}{h!}\left(\sum_{j_1+\cdots+j_l=m+1}\{S_1^{(j_1)},\{S_1^{(j_2)},\cdots \{S_1^{(j_h)},D_\omega\}\}\}\right.\\
&+&\left.\sum_{j_1+\cdots+j_h+j_{h+1}=m+1}\{S_1^{(j_1)},\{S_1^{(j_2)},\cdots \{S_1^{(j_h)},K_1^{(j_{h+1})} \}\}\}\right)=0
\end{eqnarray*}
By induction and since $D_{\om}$ is non resonant, then both sums above are zero. Hence, we the same reasoning as above, we obtain $K_1^{(m+1)}=K_2^{(m+1)}\in \cK$ and $S_1^{(m+1)}\in\cK$. The result follows from \rp{compo}.
\end{proof}
\begin{cor}\label{ofelia}
For any $H$ as in (\ref{Birkhof}), if for $G\in \cF^{\ge 1}$ one has $e^{\{G,\cdot\}} H= D_\omega+ Z +R$ with $R\in \cF^{\ge d_1}$ then $Z- Z_H\in \cK^{\ge d_1}$.
\end{cor}
\begin{proof}
By Proposition \ref{giova} $(1)$ there exists $S\in \cF^{\ge d_1}$ which normalizes $D_\omega+ Z +R$ to $D_\omega+ \widetilde Z$ with $\widetilde Z-Z\in \cF^{\ge d_1}$. By Lemma \ref{compo} there exists $G_1\in \cF$ such that $e^{\{G_1,\cdot\}}= e^{\{S,\cdot\}}e^{\{G,\cdot\}}$. Since $G_1$ puts $H$ in normal form, by Proposition \ref{giova} $(2)$, $\widetilde Z= Z_H$ and the result follows.
\end{proof}
\begin{defn}
We say that $H$ is formally linearizable if $Z_H=0$.
\end{defn}
\begin{cor}\label{keypoint}
If $H$ is formally linearizable and there exists a formal symplectic change of variables with $e^{\{S,\cdot\}} H= D_\omega+ Z +R$ with $R\in \cF^{\ge d}$ and $Z \in \cK^{< d}$ (this last condition does not imply any loss of generality) then $Z=0$.
\end{cor}
\begin{proof}
This follows directly from Corollary \ref{ofelia}.
\end{proof}
If we know a priori that $H$ is formally linearizable then we get a faster growth of the degree of $P_i$.
\begin{lemma} \label{Birlinform}.
If $H_0\in \cF$ of the form $D_\omega+ P_0$ with $P_0\in \cF^{\ge 1}$ is formally linearizable then the sequence of generating functions
\[
\{D_\omega,S_i\} =\Pi^{\cR} H_{i}\,,\quad H_{i+1}= e^{\{S_i,\cdot\} }H_{i}\,.
\]
satisfies
\[
H_i = D_\omega + P_i \,,\quad P_i\in \cF^{\ge 2^i} \,.
\]
\end{lemma}
\begin{proof}
By induction we assume that $P_i \in \cF^{\ge 2^i}$. Then by construction $S_i \in \cF^{\ge 2^i}$.
\begin{align*}
e^{\{S_i,\cdot\} }H_{i} &=D_\omega + P_i + \{S_i, D_\omega\} + \sum_{h=2}^\infty \frac{\ad_{S_i}^{h-1}}{h!} \{S_i,D_\omega \}
+ \sum_{k=1}^\infty \frac{\ad_S^k}{k!} P_i \\
&= D_\omega+ \Pi^\cK P_i - \sum_{k=1}^\infty \frac{\ad_{S_i}^{k}}{(k+1)!}\Pi^{\cR} P_{i} + \sum_{k=1}^\infty \frac{\ad_{S_i}^k}{k!} P_i\,\\
&=: D_\omega+ \Pi ^{<2^{i+1}} \Pi^\cK P_i + P_{i+1}\,.
\end{align*}
By Proposition \ref{degree decompositionbis} the two series in the formula above are in $\cF^{2^{i+1}}$ so to prove our claim we only need to show $\Pi^\cK \Pi ^{<2^{i+1}} P_i =0$. This is a consequence of Corollary \ref{keypoint}.
\end{proof}
\section{Regular Hamiltonians }
We now revisit the formal Birkhoff normal form in the case of analytic Hamiltonians. We start by introducing an appropriate functional setting.
\subsection{Spaces of Hamiltonians}
Let us consider the weighted space
\[
\th_s= \th_{s,p,\theta} :=\set{ u\in \ell^2(\Z,\C): \quad |u|^2_s:=\sum_{j\in\Z}\jap{j}^{2p}e^{2s\jap{j}^\theta}|u_j|^2<\infty }
\]
where $\jap{j}:=\max(|j|,1)$, $p\geq \frac12$ and $0<\theta\leq1$. The spaces $\th_{s,p,\theta}$ are contained in $\ell^2(\C)$, so we endow them with the standard symplectic structure coming from the Hermitian product on $\ell^2(\C)$.
\\
We identify $\ell^2(\C)$ with $\ell^2(\R)\times \ell^2(\R)$ through $u_j= \pa{x_j+ i y_j}/\sqrt{2}$ and induce on $\ell^2(\C)$ the structure of a real symplectic Hilbert space\footnote{We recall that given a complex Hilbert space $H$ with a Hermitian product $(\cdot,\cdot)$, its realification is a real symplectic Hilbert space with scalar product and symplectic form given by
\[
\langle u,v\rangle = 2{\rm Re}(u,v)\,,\quad \omega(u,v)= 2{\rm Im}(u,v)\,.
\] } by setting, for any $(u^{(1)}, u^{(2)}) \in \ell^2(\C)\times \ell^2(\C)$,
\[
\langle u^{(1)},u^{(2)}\rangle = \sum_j \pa{x_j^{(1)}x_j^{(2)}+ y_j^{(1)}y_j^{(2)}} \,,\quad \omega(u^{(1)},u^{(2)})= \sum_j \pa{y_j^{(1)}x_j^{(2)}- x_j^{(1)}y_j^{(2)}},
\]
which are the standard scalar product and symplectic form $\Omega= \sum_j dy_j\wedge d x_j$. \\
Given $H\in \cF$, we define its majorant
as
\begin{equation}\label{betta}
\und H(u) = \sum_{\substack{\bal,\bbt\in\N^\Z\,, \\
|\bal|+|\bbt|<\infty} }|H_{\bal,\bbt}|u^\bal \bar u^\bbt\,.
\end{equation}
\begin{defn}[M-regular Hamiltonians]\label{Hreg}
For $ r>0$, let
$\cH_{r,s}$
be the subspace of $\cF$ of formal power series $H$ such that $\und{H}$ is pointwise absolutely convergent on
$B_r(\th_{s}) $, the ball of radius $r$ centered at the origin of $\th_s$,
and
$$
|H|_{ B_{r}(\th_{s})}
\equiv
\|H\|_{r,s}
:=
r^{-1} \pa{\sup_{\abs{u}_{\th_{s}}\leq r}
\abs{{X}_{{\underline H}}}_{\th_{s}} } < \infty\,.
$$
\end{defn}
\noindent
{\sl
Note that in $\cF$ one has $H(0)=0$ so this is actually a norm.
}
\smallskip
We shall show in the next subsection that $H\in \cH_{r,s}$ guarantees that the Hamiltonian flow of $H$ exists at least locally and generates a symplectic transformation on $\th_s$, i.e. $\th_s$ is an invariant subspace for the dynamics.
\begin{thm}[Main]
Consider a Hamiltonian of the form
\[
\sum_{j\in \Z} \omega_j |u_j|^2 + P_0\,,\quad P_0\in \cH_{\tr,s_0} \cap \cF^{\ge 1}
\]
where $\omega\in \dg$. Assume that there exists $G\in \cF^{\ge 1}$ such that
\[
e^{\{G,.\}} H = \sum_{j\in \Z} \omega_j |u_j|^2\,,
\]
then there exists $r_1<\tr$, $s_1> s_0$ and a close to identity change of variables $\Psi$
\[
\Psi: B_{r_1}(\th_{s_1}) \to \th_{s_1}
\]
such that $H\circ \Psi = \sum_{j\in \Z} \omega_j |u_j|^2$.
\end{thm}
\subsection{Poisson structure and homological equation}
The following Lemmata are proved in \cite{BMP18} under the extra assumption of mass conservation, we discuss the proof in our slightly more general setting in the apendix.
\begin{lemma}\label{gasteropode}
If $H\in \cH_{r,s}\cap \cF^{\ge \td}$, then for all $\rs\le r$ one has
\begin{equation*}
\norm{H}^{\wc}_{\rs,s} \le \pa{\frac{\rs}{r}}^{\td} \norm{H}^{\wc}_{r,s}\,.
\end{equation*}
\end{lemma}
\begin{lemma}\label{genoveffa}
If $H\in \cH_{r,s}$, then for all $s_1\ge s$ one has
\begin{equation*}
\norm{H}^{\wc}_{r,s_1} \le \norm{H}^{\wc}_{r,s}\,.
\end{equation*}
\end{lemma}
\begin{lemma}[Poisson brakets and Hamiltonian flow]\label{ham flow}
Let $0<\rho< r $, and $F,G\in\cH_{r+\rho,\eta}(\th_{s})$, then
\begin{equation}\label{commXHK}
\norm{\{F,G\}}_{r,s}
\le
4\pa{1+\frac{r}{\rho}}
\norm{F}_{r+\rho,s}
\norm{G}_{r+\rho,s}\,.
\end{equation}
For $S\in\cH_{r+\rho,\eta}(\th_{s})$ with
\begin{equation}\label{stima generatrice}
\norm{S}_{r+\rho,s} \leq\delta:= \frac{\rho}{8 e\pa{r+\rho}}.
\end{equation}
Then the time $1$-Hamiltonian flow
$\Psi^1_S: B_r(\th_{s})\to
B_{r + \rho}(\th_{s})$ is well defined, analytic, symplectic with
\begin{equation}
\label{pollon}
\sup_{u\in B_r(\th_{s})} \norm{\Psi^1_S(u)-u}_{\th_{s}}
\le
(r+\rho) \norm{S}_{r+\rho,s}
\leq
\frac{\rho}{8 e}.
\end{equation}
For any $H\in \cH_{r+\rho,s}$
we have that
$H\circ\Psi^1_S= e^{\set{S,\cdot}} H\in\cH_{r,s}$ and
\begin{align}
\label{tizio}
\norm{\es H}_{r,s} & \le 2 \norm{H}_{r+\rho,s}\,,
\\
\label{caio}
\norm{\pa{\es - \id}H}_{r,s}
&\le \delta^{-1}
\norm{S}_{r+\rho,s}
\norm{H}_{r+\rho,s}\,,
\\
\label{sempronio}
\norm{\pa{\es - \id - \set{S,\cdot}}H}_{r,s} &\le
\frac12 \delta^{-2}
\norm{S}_{r+\rho,s}^2
\norm{H}_{r+\rho,s} \end{align}
More generally for any $h\in\N$ and any sequence $(c_k)_{k\in\N}$ with $| c_k|\leq 1/k!$, we have
\begin{equation}\label{brubeck}
\norm{\sum_{k\geq h} c_k \ad^k_S\pa{H}}_{r,s} \le
2 \|H\|_{r+\rho,s} \big(\|S\|_{r+\rho,s}/2\delta\big)^h
\,,
\end{equation}
where $\ad_S\pa{\cdot}:= \set{S,\cdot}$.
\end{lemma}
\begin{lemma}\label{estim-cohom}
Fix $s\geq 0$ and $\s>0$ and $\omega\in \dg$. For any $R\in\cH_{r,s}^{d}$ with $d\ge 1$ and such that $\Pi_{\cK} R=0$, the Homological equation $L_\omega S = R$ has a unique solution $S=L_\omega^{-1}R\in\cH_{r,s+\s}^d$ such that $\Pi_{\cK} S=0$ and moreover
\begin{equation}\label{cavolfiore gevrey}
\norm{L_\omega ^{-1} R}_{r,s+\s}\le \g^{-1} e^{\Cuno\s^{-\frac{3}{\theta}}}\norm{R}_{r,s}
\end{equation}
\end{lemma}
\subsection{Poof of the main Theorem}
The theorem follows by the following holomorphic version of Lemma \ref{Birlinform}.
If $H_0\in \cF\cap \cH_{\tr,s_0}$ of the form $D_\omega+ P_0$ with $P_0\in \cF^{\ge 1}$ is formally linearizable.
Fix $0< r_0<\tr$ and $s_0>0$ so that
\[
\e_0:=\g^{-1} \|P_0\|_{r_0,s_0} \le \g^{-1}\frac{r_0}{\tr}\|P_0\|_{\tr,s_0}
\]
is appropriately small.
More precisely, fix $\tC = 1+ \pi^2/6$ and assume
\begin{equation}
\label{piccolo1}
\e_0^{-1} \ge \tK\sup_n e^{\cC_2(s_0) n^{\frac6\theta}} n^2 \max(e^{n-\chi^n}, e^{-(2-\chi)\chi^n}) .
\end{equation}
where $\tK$ is an appropriately large absolute constant while $\cC_2(s_0)= \cC_1 {\tC}^{\frac{3}{\theta}}{s_0}^{-\frac{3}{\theta}}$.
\\
Let
\[
r_i = r_{i-1}- \rho_{i-1}\,,\quad s_{i} = s_{i-1} + \s_{i-1} \,,\quad \td_i = 2^{i}\,, \rho_i =\frac{ r_0}{2\tC \jap{i}^2}\,,\quad \s_i = \frac{ s_0}{\tC \jap{i}^2}
\]
so that $r_i \to r_0/2$ and $s_i\to 2s_0$.
Fix $1<\chi<2$ such that\footnote{for example if $\chi= 15/14$ the sup on the left hand side is smaller than $-0,2$.}
\begin{equation}
\label{chi}
\sup_{n\ge 0}2^{n+1}\ln(1- \frac{1}{2\tC n^2} )+\chi^n(\chi-1) \le -0.1 \,
\end{equation}
\begin{lemma}\label{Birlinholo}
The sequence of generating functions and Hamiltonians of Lemma \ref{Birlinform}.
\[
\{D_\omega,S_i\} =\Pi^{\cR} H_{i}\,,\quad H_{i}= e^{\{S_{i-1},\cdot\} }H_{i-1}\,.
\]
satisfies
\[
H_i = D_\omega + P_i \,,\quad P_i\in \cF^{\ge \td_i} \cap \cH_{r_i,s_i}\,.
\]
with the bounds
\[
\|S_{i-1}\|_{r_{i-1},s_i} \le \g^{-1}e^{\cC_1 \s_{i-1}^{-\frac3\theta}}\|P_{i-1}\|_{r_{i-1},s_{i-1}} \,,\quad \| P_i\|_{r_i,s_i} \le \|P_0\|_{r_0,s_0} e^{-\chi^i} \,.
\]
Moreover each $S_{i-1}$ defines a symplextic analytic change of variables $\Psi_{i-1}: B_{r_{i}}(\th_s) \to B_{r_{i-1}}(\th_s)$ for all $s\ge s_i$ satisfying
\begin{equation}
\label{super}
\sup_{|u|_s\leq r_i}|\Psi_i(u) -u|_s \le 2^{-i} r_0
\end{equation}
Finally setting
\[
\Phi_i= \Psi_1\circ\Psi_2\circ\dots \Psi_i
\]
we have that
$\Phi_i\to \Phi_\infty$ where $\Phi_\infty$ is an invertible symplectic map $B_{r_0/2}(\th_{2s_0}) \to B_{r_0}(\th_{2s_0})$
such that
\[
H_0\circ \Phi_\infty = D_\omega
\]
\end{lemma}
\begin{proof}
By induction. Let us denote $\g^{-1}\|P_0\|_{r_0,s_0}:= \e_0$. Fix $k\ge 0$ and assume that for all $i\le k$ the Lemma holds.
By definition
\[
S_k = \ad_{D_\omega}^{-1}\Pi_\cR P_k\,.
\]
\\
For all $s\ge s_k+\s_k\equiv s_{k+1}$, by Lemma \ref{estim-cohom} and \eqref{piccolo1}
\[
\|S_k\|_{r_k,s} \le \|S_k\|_{r_k,s_{k+1}} \le \g^{-1} e^{\cC_1 \s_{k}^{-\frac3\theta}}\|P_{k}\|_{r_{k},s_{k}}\le \e_0 e^{\cC_2(s_0) k^{\frac6\theta}} e^{-\chi^k}\le \frac{1}{16 e 2\tC k^2} \le \frac{\rho_k}{8 e r_k}
\]
so, by Lemma \ref{ham flow} the time one flow $\Psi^1_{S_k}: B_{r_{k+1}}(\th_s)\to B_{r_k}(\th_s)$ is well defined analytic, symplectic
and, by \eqref{pollon} satisfies
\begin{equation}
\sup_{u\in B_{r_{k+1}}(\th_{s})} \abs{\Phi^1_{S_k}(u)-u}_{\th_{s}}
\le
r_k \norm{S_k}_{r_k,s} \le C \e_0 r_0 k^{-2}e^{\cC_2(s_0) k^{\frac6\theta}} e^{-\chi^k}\stackrel{\eqref{piccolo1}}{\le} 2^{-k}r_0\,.\label{pollonk}
\end{equation}
Recalling that
\begin{align*}
H_{k+1}:=e^{\{S_k,\cdot\} }H_{k} &=D_\omega + P_k + \{S_k, D_\omega\} + \sum_{h=2}^\infty \frac{\ad_{S_k}^{h-1}}{h!} \{S_k,D_\omega \}
+ \sum_{h=1}^\infty \frac{\ad_S^h}{h!} P_k \\
&= D_\omega+ \Pi^\cK P_k - \sum_{h=1}^\infty \frac{\ad_{S_k}^{h}}{(h+1)!}\Pi^{\cR} P_{k} + \sum_{h=1}^\infty \frac{\ad_{S_k}^h}{h!} P_k\,\\
&=: D_\omega+ \Pi ^{<2^{k+1}} \Pi^\cK P_k + P_{k+1}\,.
\end{align*}
and that in Lemma \ref{Birlinform} we have proved that $\Pi ^{<2^{k+1}}\Pi^\cK P_k=0$, we get
\[
P_{k+1}= \Pi ^{\ge 2^{k+1}}\Pi^\cK P_k - \sum_{h=1}^\infty \frac{\ad_{S_k}^{h}}{(h+1)!}\Pi^{\cR} P_{k} + \sum_{h=1}^\infty \frac{\ad_{S_k}^h}{h!} P_k
\]
Now
\begin{align*}
\|\Pi ^{\ge 2^{k+1}}\Pi^\cK P_k\|_{r_{k+1},s_{k+1}} \le \pa{\frac{ r_{k+1}}{r_k}}^{\td_{k+1}}\|P_k\|_{r_k,s_k} &\le \e_0(1- \frac{1}{2\tC k^2})^{2^{k+1}} e^{-\chi^k}
\\
\|\sum_{h=1}^\infty \frac{\ad_{S_k}^{h}}{(h+1)!}\Pi^{\cR} P_{k} + \sum_{h=1}^\infty \frac{\ad_{S_k}^h}{h!} P_k\|_{r_{k+1},s_{k+1}}
&\le \frac{16 e r_k}{ \rho_k} \|P_k\|_{r_k,s_k} \|S_k\|_{r_{k+1},s_{k+1}} \\ & \le C \e^2_0 e^{\cC_2(s_0) k^{\frac6\theta}} e^{-2\chi^k} k^2
\end{align*}
The bound on $P_{k+1}$ follows from \eqref{piccolo1} and \eqref{chi} which imply
\[
\e_0(1- \frac{1}{2\tC k^2})^{2^{k+1}} e^{-\chi^k} + C \e^2_0 e^{\cC_2(s_0) k^{\frac6\theta}} e^{-2\chi^k} k^2 \le \e_0 e^{-\chi^{k+1}}
\]
In order to prove the convergence we remark that all the $\Psi_i$ map $ B_{r_{i}}(\th_{2s_0}) \to B_{r_{i-1}}(\th_{2s_0})$, consequently $\Phi_i$ maps
$ B_{r_{i}}(\th_{2s_0}) \to B_{r_0}(\th_{2s_0})$ and, by \eqref{super}, it is a Cauchy sequence.
\end{proof}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,696 |
Q: Docker & Graylog I've been installing the last version of Docker (1.8).
This new version is great because you can set a particular log-driver to send the log to Graylog2 (for example).
Here's the blog post from docker : Docker Blog announcing 1.8
Here's the graylog team blog to make the magic happen : Graylog blog
My problem is that it doesn't work :(
Here is the error message I get when I lauch this command :
docker run –log-driver=gelf –log-opt gelf-address=udp://127.0.0.1:12201 busybox echo Hello Graylog
Error :
Unable to find image '–log-driver=gelf:latest' locally repository name component must match "[a-z0-9]+(?:[._-][a-z0-9]+)*"
Here's an screenshot :
Any idea ?
EDIT
Following the advices of smart people : It looks like the dash was not the good one.
Here's a paste of the correct command :
docker run --log-driver=gelf --log-opt gelf-address=udp://127.0.0.1:12201 busybox echo Hello Graylog
A: You need to write two dashes instead a single one: ... --log-driver=gelf --log-opt ...
EDIT: In that blogpost it is written:
The Docker plugins mechanism is now available in the new Docker experimental channel.
You need to install the experimental docker version like:
curl -sSL https://experimental.docker.com/ | sh
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,840 |
{"url":"https:\/\/www.tutorialspoint.com\/swift-program-to-check-if-two-arrays-are-equal-or-not","text":"# Swift Program to Check if Two Arrays Are Equal or Not\n\nIn this article, we will learn how to write a swift program to check whether two arrays are equal. To check if two arrays are equal to not either, we are going to use the following two methods\n\n\u2022 using == operator\n\n\u2022 elementsEqual(_:) method.\n\nBoth method and operator return true if both the given array has the same elements. Otherwise, they will return false. For example, arr1 = [2, 3, 4, 5] and arr2 = [2, 3, 4, 5] both arrays are equal so both the method and operator will return true. arr1 = [2, 3, 4, 5] and arr2 = [1, 3, 3, 5] both arrays are not equal so both the method and operator will return false.\n\n## Method 1: Using == Operator\n\nIn the following example, we use the == operator to compare two arrays. If both arrays contain the same elements in the same order and of the same type, then this operator will return true. Otherwise, it will return false.\n\n### Syntax\n\nFollowing is the syntax of == operator\n\nArray1 == Array2\n\n\n### Algorithm\n\n\u2022 Step 1\u00a0\u2212 Create two arrays.\n\n\u2022 Step 2\u00a0\u2212 Check if both the array contain the same elements or not using either the == operator or elementsEqual(_:) method.\n\n\/\/ Using == operator\nvar output1 = (Array1 == Array2)\n\n\/\/ Using elementsEqual() method\nvar output = Array1.elementsEqual(Array2)\n\n\u2022 Step 3\u00a0\u2212 Print the output\n\n### Example\n\nIn the following example, we check if the given arrays are equal or not using the == operator.\n\nimport Foundation\nimport Glibc\n\n\/\/ Creating arrays\nvar Array1 = [74, 99, 9, 38, 78, 132]\nvar Array2 = [74, 99, 9, 38, 78, 132]\nvar Array3 = [78, 99, 91, 38, 708, 32]\n\n\/\/ Checking if the arrays are equal or not\n\/\/ Using == operator\nvar output1 = (Array1 == Array2)\nprint(\"Is Array 1 is equal to Array 2?\", output1)\n\nvar output2 = (Array1 == Array3)\nprint(\"Is Array 1 is equal to Array 3?\", output2)\n\n### Output\n\nIs Array 1 is equal to Array 2? true\nIs Array 1 is equal to Array 3? false\n\n\nHere in the above code, we create three arrays with different variations in their values. Now we compare arrays with each other using the == operator. It will return true if the array present on the left-hand side is equal to the array present on the right-hand side of this operator. Otherwise, return false.\n\n## Method 2: Using elementsEqual(_:) Method\n\nThe elementsEqual(_:) method will return true when both arrays contain the same elements in the same order. Otherwise, this method will return false. If the elements are the same but the order is different, then also this method will return false. Out of two given arrays at least one must be the finite array.\n\n### Syntax\n\nFollowing is the syntax of elementsEqual(_:) method \u2212\n\nfunc elementsEqual<anotherSequence>(_other:Another Sequence)->Bool\n\n\nHere, other parameters represent the sequence to which we want to compare.\n\n### Algorithm\n\n\u2022 Step 1\u00a0\u2212 Create two arrays.\n\n\u2022 Step 2\u00a0\u2212 Check if both the array contain the same elements or not using the elementsEqual(_:) method.\n\nvar output = Array1.elementsEqual(Array2)\n\n\u2022 Step 3\u00a0\u2212 Print the output\n\n### Example\n\nimport Foundation\nimport Glibc\n\n\/\/ Creating arrays\nvar Array1 = [\"Bus\", \"Car\", \"bicycle\", \"bike\"]\nvar Array2 = [\"Bus\", \"Car\", \"bicycle\", \"bike\"]\nvar Array3 = [\"Mango\", \"Apple\", \"Banana\", \"Orange\"]\nvar Array4 = [\"Car\", \"Bus\", \"bike\", \"bicycle\"]\n\n\/\/ Checking if the arrays are equal or not\n\/\/ Using elementsEqual() method\nvar output1 = Array1.elementsEqual(Array2)\nprint(\"Is Array 1 is equal to Array 2?\", output1)\n\nvar output2 = Array1.elementsEqual(Array3)\nprint(\"Is Array 1 is equal to Array 3?\", output2)\n\nvar output3 = Array1.elementsEqual(Array4)\nprint(\"Is Array 1 is equal to Array 4?\", output3)\n\n\nIn the following example, we use the elementsEqual(_:) method to check if two arrays are equal or not.\n\n### Output\n\nIs Array 1 is equal to Array 2? true\nIs Array 1 is equal to Array 3? false\nIs Array 1 is equal to Array 4? false\n\n\nHere in the above code, we create four arrays with different variations in their values. Now we compare arrays with each other using elementsEqual(_:) method. It will return true if the Array on which this method is applied is equal to the array present inside the method(as a parameter) are equal. Otherwise, return false.\n\n## Conclusion\n\nSo, this is how we can check if two arrays are equal or not either using == operator or using elementsEqual() method. They both return the same result so you can use them according to your choice. Also, they compare two arrays at a time.","date":"2023-03-23 22:12:46","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.19667106866836548, \"perplexity\": 2177.306404198852}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-14\/segments\/1679296945183.40\/warc\/CC-MAIN-20230323194025-20230323224025-00631.warc.gz\"}"} | null | null |
\section{Introduction}
The nature of the planetary system around GJ 581 has been discussed in the literature by several authors and the preferred orbital properties and number of planets in the system have been revised whenever new data has become available. While such accumulation of knowledge is the very essence of science, contradicting results, especially those based on robust statistical arguments, have seen the system around GJ 581 become a planetary system with four established planets in its orbits and another two hypothetical ones that have been shown to exist in some studies but have been shown detections of false positives in others.
Astronomers first started counting the planets orbiting GJ 581 in 2005 when \citet{bonfils2005} reported they had found a Neptune-mass planet ($m_{p} \sin i = 16.6 $M$_{\oplus}$) with an orbital period of roughly 5.4 days. This discovery, made using the High Accuracy Radial Velocity Planet Searcher (HARPS) spectrograph \citep{mayor2003} mounted on the 3.6 m telescope at the European Southern Observatory (ESO) at La Silla, Chile, was soon accompanied by discoveries of two additional super-Earth-mass planets made by the same team in 2007 \citep{udry2007}. These two additional planets were reported to have minimum masses of 5.0 and 7.7 M$_{\oplus}$ and orbital periods of 12.9 and 84 days. \citet{udry2007} argued that the outer planet (GJ 581 d) was on a slightly eccentric orbit ($e = $0.20$\pm$0.10) but also noted that its orbit could be consistent with a circular one as was the case for the other two planets.
In 2009, \citet{mayor2009} reported that additional HARPS velocities revealed the existence of a fourth planet orbiting GJ 581 with and orbital period of 3.1 days. \citet{mayor2009} detected this planetary companion from 119 velocity measurements and claimed it to have a low minimum mass of 1.9 M$_{\oplus}$. They also revised the orbital period of GJ 581 d to 66.4 days and stated that the period of 84 days obtained by \citet{udry2007} was in fact a yearly alias of the 66.4-day signal. This is a textbook example of problems aliasing can cause to detections of periodic signals in radial velocities and other time series.
Counting the planets orbiting GJ 581 got complicated in 2010 and 2011 when \citet{vogt2010} claimed that a combined dataset with the 119 HARPS measurements of \citet{mayor2009} and another set of 122 High Resolution Echelle Spectrograph \citep[HIRES;][]{vogt1994} velocities from the 10 m Keck-II telescope at Hawaii actually contained the periodic signatures of -- not only the four planets observed by \citet{mayor2009} -- but also two other signals corresponding to low-mass planets with minimum masses of 7.0 and 3.1 M$_{\oplus}$ and orbital periods of 433 and 36.6 days, respectively. The immediate conclusion of \citet{vogt2010} was that the latter might be a good candidate for a habitable planet with its Earth-like mass and its orbit in the middle of the stellar habitable zone of GJ 581. However, the analyses of \citet{tuomi2011} and \citet{gregory2011} soon revealed that the existence of this candidate habitable world was not supported by the data and that the evidence in favour of the outer planet with an orbital period of 433 days was at most suggestive. \citet{gregory2011} and \citet{tuomi2011} concluded that there were five and four planets, respectively, in the system depending on the subjective choices of the probabilistic detection threshold and the prior densities of the parameters of the statistical models. Yet, the existence of the four planets detected by \citet{mayor2009} was easily verified using the Bayesian statistical tests of \citet{tuomi2011} and \citet{gregory2011}.
Other authors have also re-analysed the HARPS and/or HIRES data of \citet{mayor2009} and \citet{vogt2010} and received evidence that supports the existence of the hypothetical habitable zone planet GJ 581 g \citep[e.g.][]{anglada2011,dossantos2012}. However, it has been clear from the start, as also stated by almost every author discussing the system and estimating the number of planets in it, that only additional data can firmly verify or falsify the existence of the two proposed companions GJ 581 f and g.
Such data was made available by \citet{forveille2011} and increased the amount of HARPS data of GJ 581 by another 121 velocities. This increase made the total number of HARPS precision velocities 240. However, despite attempts of recovering the proposed planets f and g from this data using standard Lomb-Scargle periodograms \citep{lomb1976,scargle1982} and $\chi^{2}$ minimisations, \citet{forveille2011} concluded that these two planets could not exist orbiting the star with the masses and orbital properties reported by \citep{vogt2010}. Recently, a re-analysis of these HARPS velocities backed up by dynamical arguments was found to support the existence of a habitable-zone super-Earth with an orbital period of 32 days and a minimum mass of 2.2 M$_{\oplus}$ \citep{vogt2012} but another study taking into account the noise correlations did not find evidence in favour of the existence of this signal in the same data and also casted doubt on the significance of the signal corresponding to GJ 581 d \citep{baluev2012}. Therefore, considering the different estimates for the number of planets (parameter $k$) favoured by the data in the very recent history suggests that this debate is far from over.
If of planetary origin, any periodic variations in radial velocities should be present in data obtained using different telescope-instrument combinations. The availability of such independent confirmation is essential to exclude spurious signals that can be caused by instrument instability and insufficient noise modelling of the features in the measurement noise of the instrument.
Using the combined set of HARPS and HIRES velocities of GJ 581, we perform a global search for low-amplitude signals in an attempt to see whether the conclusions of \citet{forveille2011}, stating that $k=4$, hold for results obtained using Bayesian analysis techniques \citep[e.g.][]{tuomi2011,tuomi2012,tuomi2011b,tuomi2012b}. We also re-analyse the 240 HARPS velocities of \citet{forveille2011} to (1) enable comparison of our results with those of \citet{bonfils2005}, \citet{udry2007}, \citet{mayor2009}, \citet{forveille2011}, \citet{vogt2012}, and \citet{baluev2011}; to (2) quantify the sensitivity of the Bayesian detection methods w.r.t. the frequentist periodogram tools used throughout the literature; and to (3) see whether the apparent sensitivity of the Bayesian detection thresholds is prone to false positives in practice. Our goal is to receive a reliable estimate for $k$ in the case of GJ 581 radial velocities and to determine the significances and properties of possible additional signals if $k > 4$. In our Bayesian analyses, we use the ``standard'' white noise models assuming independent and identically distributed measurements with Gaussian white noise but we also test models with red noise features, i.e. we model the data using a moving average (MA) model that accounts for the noise correlations in the data in a time-scale of few dozen days \citep[e.g.][]{baluev2012,tuomi2012c,tuomi2012d}.
\section{Statistical analysis}\label{sec:statistics}
\subsection{Bayesian statistics}\label{sec:bayes}
When assessing the optimal value of $k$, i.e. the number of signals statistically significantly present in a time series, we take advantage of Bayesian model selection that enables the simultaneous comparison of any number of models -- in this particular case, models with different $k$. We denote these models as $\mathcal{M}_{k}$, for all $k=0, ..., p$, and calculate the corresponding probabilities of each model in a standard way as
\begin{equation}\label{eq:model_probability}
P(\mathcal{M}_{k} | m) = \frac{P(m | \mathcal{M}_{k}) P(\mathcal{M}_{k})}{\sum_{i=p}^{p} P(m | \mathcal{M}_{i}) P(\mathcal{M}_{i})} ,
\end{equation}
where $m$ denotes the measurements; $P(m | \mathcal{M}_{i})$ is the Bayesian evidence, or probability of obtaining the data if the model $\mathcal{M}_{i}$ indeed was a ``correct'' model; and $P(\mathcal{M}_{i})$ is the prior probability of the model $\mathcal{M}_{i}$ that quantifies the subjective level of confidence we have on the validity of this model \emph{a priori} to analysing the measurements. As can be seen in Eq. (\ref{eq:model_probability}), the probabilities are scaled in such a way that the sum of the individual probabilities of models $\mathcal{M}_{k}$, for $k=0, ..., p$, is unity indicating that any posterior probabilities obtained using this equation are dependent on the selected set of candidate models and only reflect the \emph{relative} probabilities of the models in this set.
Finding the optimal value of $k$ based on model selections using Eq. (\ref{eq:model_probability}), however, is not as such a sufficient way of counting the periodic signals in a given data set. While a given $k$ might receive the greatest posterior probability, the claim that there really are $k$ signals in the data should not be made lightly. This is especially the case if the probability of a model with some lower value of $k$ receives estimates of similar magnitude. For this reason, we require that the probability of a model $\mathcal{M}_{k+1}$ has to be at least 150 times that of a model $\mathcal{M}_{k}$ to conclude that there is evidence in favour of the existence of $k+1$ Keplerian signals in the data. This threshold corresponds to an interpretation of ``strong evidence'' given by \citet{kass1995} to make decisions based on data. While completely subjective, we adopt this threshold but do not consider signals to exist in the data unless the other two criteria of \citet{tuomi2012} are also satisfied, i.e. that the estimates of the velocity amplitude parameter $K$ are statistically significantly greater than zero and that the period $P$ is constrained from above and below. If all these criteria are satisfied, we state that the existence of a signal is supported by data.
While the prior probabilities in Eq. (\ref{eq:model_probability}) are only simple numbers, the Bayesian evidences are in fact complicated integrals over the whole parameter space of model parameters that depend on the value of the product of a likelihood function and a prior probability density of the model parameters, i.e. the formulation of the statistical model and the physical interpretation of its parameters given the available data. For a given model, this integral, sometimes called the marginal integral, is of essence in any model comparison problems in the Bayesian context and the ability to compare models in a probabilistic way depends on the ability to estimate this integral. We write this integral as
\begin{equation}\label{eq:integral}
P(m | \mathcal{M}_{k}) = \int_{\theta_{k} \in \Omega_{k}} l(m | \theta_{k}, \mathcal{M}_{k}) \pi(\theta_{k} | \mathcal{M}_{k}) d \theta_{k} ,
\end{equation}
where the parameters of the $k$th model, $\theta_{k}$, in the parameter space $\Omega_{k}$, are interpreted using the likelihood function $l$ and their prior probability density $\pi$ that describe the mathematical details as well as the physical interpretation of the statistical model $\mathcal{M}_{k}$. This is also the reason we write the likelihood and the prior as conditioned on the model.
We estimated the integral in Eq. (\ref{eq:integral}) using the truncated posterior mixture (TPM) estimate of \citet{tuomi2012b}. This estimation requires the availability of a statistically representative sample from the parameter posterior density and we drew such samples using the adaptive Metropolis algorithm \citep{haario2001} that is simply an adaptive version of the famous Metropolis-Hastings Markov chain Monte Carlo (MCMC) algorithm \citep{metropolis1953,hastings1970}. This sampling technique has been used succesfully in e.g. \citet{tuomi2012}.
Apart from the chosen set of candidate models (i.e. the definition of their likelihood functions), the only other subjective issue in Bayesian statistics is the prior information quantified using prior probability densities of the model parameters and prior probabilities of the different models. Standard (frequentist) analyses of radial velocity data actually incorporate prior information as well, which makes them somewhat Bayesian in reality, but do not easily enable testing different prior opinions in practice. For instance, when searching for a solution using typical $\chi^{2}$ minimisations, the amount of excess noise in the measurements, or parameter $\sigma_{J}$, is commonly fixed to an \emph{a priori} selected value (e.g. $\sigma_{0}$), which corresponds to setting the prior probability of this parameter equal to a delta-function as $\pi(\sigma_{J}) = \delta(\sigma_{J} - \sigma_{0})$. This prior assumes that only a value of $\sigma_{J} = \sigma_{0}$ is a possible one and ignores the rather likely possibility that this parameter might not be equal to $\sigma_{0}$. If this prior choice was made in the Bayesian context, any scientifically oriented reader would immediately dismiss this choice as too limiting and would not consider the obtained results trustworthy. However, such \emph{prior limitations}, i.e. setting prior densities to zero in chosen subsets of the parameter space, are commonly used in statistical analyses but should be used with care and only when the physical interpretation of the parameters justifies the limitations.
Similar ``hidden'' prior assumptions are made throughout the literature announcing detections of exoplanets around nearby stars. For instance, finding a solution to some data by assuming that planetary eccentricities are equal to zero represents a similar prior choice. It could be argued that in ``dynamically packed'' systems only eccentricities close to zero are viable by enabling the long-term stabilities of these systems, but the possibility that some non-zero orbital eccentricities could also provide stable systems cannot be ignored because enabling eccentricities as free parameters effectively increases the number of free parameter in the model by $2k$ compared to a model with fixed eccentricities and this can have considerable effects on the significances of the obtained solutions. As fully Bayesian, these hypotheses could be easily tested by treating the model with eccentricities fixed to zero as another model in the set of candidate models. In practice, this can be seen as a comparison of different prior densities of the model parameters.
The argument of hidden prior information can in fact be taken even further. Any non-Bayesian analyses correspond to Bayesian ones with the assumption that the prior probability densities of model parameters are actually ``flat'', i.e. uniform distributions in the parameter space. This is a typical uninformative choice for a prior density and is applied widely in the statistical literature to various parameter estimation problems. However, it still \emph{is} a subjective choice and it \emph{does} have an effect on the obtained results. For instance, a typical non-linear transformation of the parameter vector from $\theta$ to $\theta'$ reveals the consequences of this prior choice. If the prior for parameter $\theta$ is a uniform density, that for $\theta'$ has to be chosen by using the Jacobian of the corresponding transformation to receive the same results. If the prior of $\theta'$ is chosen uniform as well, the obtained analysis results would not be the same for these two parameterisations but would depend heavily on the chosen transformation, which emphasises the fact that prior densities are important entities that cannot be ignored when performing statistical analyses. This in turn implies that the only logically consistent framework of statistics is the Bayesian one. While these issues are discussed widely in the statistical literature, we refer to the excellent book \emph{Statistical Decision Theory and Bayesian Analysis} by J. O. Berger \citep{berger1980}.
Because priors are subjective, we present our results only based on the priors we choose. In practice, we adopt the same prior probability densities of model parameters that were used in \citet{tuomi2012}. We also use the same prior model probabilities as \citet{tuomi2012} to take into account the fact that we believe finding $k+1$ planets in any given system is less likely than finding $k$ of them. Because posterior samplings can be used to obtain estimates of the parameter probability densities that enables the computation of any point- and uncertainty estimates, we report all the results using the maximum \emph{a posteriori} (MAP) estimates and the corresponding 99\% Bayesian credibility sets \citep[BCSs; as defined in e.g.][]{tuomi2009b}.
\subsection{Periodograms}\label{sec:periodograms}
Because it is useful to calculate the periodograms of time series, especially when searching for periodic signals, we discuss the resulting power spectra as well for the GJ 581 velocities. Identifying the strongest powers in the periodogram, such as the famous Lomb-Scargle periodogram \citep{lomb1976,scargle1982}, helps choosing the initial states of posterior samplings by pointing towards the regions in the period space where periodic signals are the most likely to be found. The ability to use information from periodogram analyses can help decreasing the burn-in period, i.e. the length of the Markov chain before it has converged to the global solution, of MCMC samplings considerably. However, because periodogram analyses are commonly used to assess the number $k$ in practice, we first discuss the caveats of such assessments.
As noted by \citet{tuomi2012}, periodograms and related analysis methods rely on the assumption that the model residuals have been calculated correctly. For instance, when searching for a $k+1$th signal in a noisy time series, the residuals are commonly calculated by assuming that there are in fact only $k$ periodicities in the data. This obvious contradiction of the assumption and the statistical test performed to search for another signal is not only made throughout the literature when searching for planets using the radial velocity technique, but its effects on the detectability of periodic signals have not received as much attention in the same literature as they should. A clear example is the case of HD 10180 that was reported to host six, possibly seven, planets orbiting it by \citet{lovis2011}. They used power spectra in their attempt of assessing the number $k$ for this star and fell a victim of the above contradiction of the statistical test and the assumption this test is based on. Contrary, the Bayesian analysis of the same exact data performed by \citet{tuomi2012} concluded that the preferred estimate for $k$ is actually as high as nine. This example only demonstrates the weakness of the periodogram-based methods in assessing $k$ but can also be seen as an obvious consequence of the assumption: if it is assumed that there are $k$ signals in a data set, the statistical tests performed based on this assumption will certainly be biased in favour of this assumption being true and prevent the detection of a $k+1$th signal unless it is a strong one and clearly present in the data.
In an attempt to overcome the above shortcomings of periodogram analyses, \citet{anglada2012b} proposed a generalised version of the periodogram defined by \citet{cumming2004} for the purpose of detecting multiple periodicities in time series. This \emph{recursive periodogram} \citep{anglada2012b} can be used to obtain the power spectrum for model residuals by simultaneously adjusting the parameters of the previously detected $k$ signals for each test frequency of the $k+1$th one. Clearly this method is not prone to such a severe contradiction as the traditional periodograms but it is still not free of its own conseptual problems such as the fact that one has to move sequentially by increasing $k$ one step at a time when searching for several periodicities. While certainly an improvement to the classical periodograms, the consequences of hidden prior information cannot be addressed using this modified periodogram either.
We report the results of standard Lomb-Scargle periodogram analyses together with those from the Bayesian analyses. We also report the resulting false alarm probabilities (FAPs) of the signals we detect in the GJ 581 velocities obtained using the HARPS spectrograph.
\section{Sequential analysis of HARPS velocities}
The HARPS velocities provide an interesting opportunity to compare the sensitivity of the Bayesian analysis methods to the traditional periodogram ones used throughout the literature. For this reason, we construct a timeline for GJ 581 where we analyse the available HARPS data after each yearly observing period. We also analyse the data sets of \citet{bonfils2005}, \citet{udry2007}, and \citet{mayor2009} and compare the best estimate of $k$ with an estimate that \emph{could} have been received had Bayesian tools been applied to these velocities since 2005.
We show the number of HARPS velocities after each observing period and at the time when the three studies describing detections of the four planets orbiting GJ 581 and reporting the data were published \citep[these three studies are:][]{bonfils2005,udry2007,mayor1995}. The numbers of measurements and data baselines of these subsets of HARPS data are shown in Table \ref{tab:timeline}. In the following subsections we describe briefly the results of our analyses of the subsets of HARPS data in Table \ref{tab:timeline}. We note, that we assign numbers to the signals we detect by calling the shortest periodicity as signal 1, the second shortest as signal 2 and so forth. We also refer to the signals corresponding to the four planet candidates by using the letters b, c, d, and e as in \citet{mayor2009}.
\begin{table}
\center
\caption{Number of signals detected from the subsets of HARPS data as a function of time as reported in the literature ($k_{1}$), using Lomb-Scargle periodograms ($k_{2}$), and using Bayesian tools ($k_{3}$). The corresponding data baselines ($T_{\rm obs}$) and numbers of data ($N$) are also shown. The subsets correspond to the published data sets and subsets of \citet{forveille2011} data set after each full observing period (FOP).\label{tab:timeline}}
\begin{tabular}{lccccc}
\hline \hline
Data subset & $N$ & $T_{\rm obs}$ [days] & $k_{1}$ & $k_{2}$ & $k_{3}$ \\
\hline
\citet{bonfils2005} & 20 & 440 & 1 & 1 & 1 \\
First FOP & 25 & 457 & -- & 2 & 2 \\
Second FOP & 43 & 827 & -- & 2 & 2 \\
\citet{udry2007} & 50 & 1050 & 3 & 3 & 3 \\
Third FOP & 76 & 1197 & -- & 4 & 4 \\
\citet{mayor2009} & 119 & 1570 & 4 & 4 & 4 \\
Fifth FOP & 131 & 1904 & -- & 4 & 4 \\
Sixth FOP & 197 & 2312 & -- & 4 & 5 \\
\citet{forveille2011} & 240 & 2543 & 4 & 4 & 5 \\
\hline \hline
\end{tabular}
\end{table}
\subsection{Bonfils et al. data}
When analysing the \citet{bonfils2005} data set, our posterior samplings could rapidly verify that it contains the clear planetary signature of GJ 581 b. This signal was seen easily with posterior samplings (with $P(k=1) \approx 5.9 \times 10^{10} P(k=0)$) and periodograms (FAP $< 0.0001$) and \citet{bonfils2005} were also very confident about its existence based on their analyses that we assume\footnote{In fact, \citet{bonfils2005} do not mention the statistical methods they used to obtain their solution.} relied on periodograms and $\chi^{2}$ minimisations.
Our samplings of a two-Keplerian model also spotted hints of a second periodic signal between roughly 10-16 days. The two-Keplerian model received a significantly greater posterior probability (by a factor of 640) than the one-Keplerian one, though the samplings could not constrain the period of the second signal very accurately. Also, parameter $K$ of this second signal did not differ statistically from zero, which forced us to conclude that there is only one signal significantly present in the \citet{bonfils2005} data set. Yet, this signal exists with little doubt because taking it into account increases the model probability by a factor of 1.1$\times 10^{11}$ compared to the model with $k=0$ despite the low number of measurements (see Table \ref{tab:timeline}).
\subsection{First full observing period}
The end of the first full observing period provided only five more spectra and corresponding velocities of GJ 581 than the data of \citep{bonfils2005}. However, this changed the situation with respect to the second signal considerably. The two-Keplerian model became 4.6$\times 10^{4}$ times more probable than the one-Keplerian one, and parameter $K$ of the second signal was significantly greater than zero with $K_{2} =$ 3.6 [1.5, 5.7] ms$^{-1}$. Despite the fact that the period of this signal had a density with several modes between roughly 12 and 16 days, it was well constrained from above and below and thus satisfied all the detection criteria. The second periodic signal was also detected by the periodograms and the corresponding FAP of this signal was 0.0075.
The probability densities of the velocity amplitude and period of the second signal are plotted in Fig. \ref{fig:densities_case_b}. The multimodality of the period is clear from the top panel in Fig. \ref{fig:densities_case_b} but the period is still constrained from above and below and its 99\% BCS is [11.98, 16.27] days. We note that the MAP estimate of the period is 12.9 days.
\begin{figure}
\center
\includegraphics[angle=-90,width=0.24\textwidth]{rvdist02_rv_GJ581b_dist_P2.ps}
\includegraphics[angle=-90,width=0.24\textwidth]{rvdist02_rv_GJ581b_dist_K2.ps}
\caption{Distributions of the velocity amplitude and period of the second signal (corresponding to GJ 581 c) as obtained from the partial HARPS data after the first full observing period. The solid curve shows a Gaussian function with the same mean and variance.}\label{fig:densities_case_b}
\end{figure}
We sampled a three-Keplerian model as well, but unsurprisingly a third periodic signal did not get constrained by the available 25 velocities.
\subsection{Second full observing period}
At the end of the second full observing period, the amount of velocities had increased to 43, and we could easily sample models with $k=0, ..., 3$. For this data set, the three-Keplerian model implied that there are at least three periodicities in the data but the period of the third signal turned out to have a density with three modes. These modes were found at periods of 42.3, 67, and 84 days. We could not set constraints to the period, and could not be sure whether our Markov chains had indeed converged to this multimodal density -- but assuming they had, the three-Keplerian model would have had a probability of 620 times that of the two-Keplerian one whose parameter space was well sampled and provided two clear periodicities at 5.37 and 12.9 days.
When we sampled the parameter space of a four-Keplerian model, however, we could identify a solution for a fourth period at 3.1 days. This solution, while only suggestive because the period was not constrained from above or below but was only apparent as a global maximum in the otherwise ``flat'' probability density, actually helped constraining the third periodic signal from above and below according to our detection criteria. While this subset of HARPS data could not be used to claim a detection of four periodicities, the results suggested that only a modest increase in the number of available data would be likely to provide detections of four signals. Yet, according to our detection criteria, we could spot only two signals confidently in this data set, namely, those with periods of 5.37 and 12.9 days. These two signals were also detected by the periodogram analyses (Table \ref{tab:timeline}).
\subsection{Udry et al. data}
\citet{udry2007} reported that the \citet{bonfils2005} data already actually suggested there might be an additional companion orbiting GJ 581 with an orbital period of 13 days with a modest significance. With additional 30 data points and improved barycentric corrections, their periodogram analyses and $\chi^2$ minimisations pointed towards two new periodic signals, corresponding to two new planets orbiting GJ 581 with orbital periods of 13 and 84 days.
These two signals were indeed significantly present in the \citet{udry2007} data according to our Bayesian analyses. However, we could go further than that and identify a fourth periodicity at 3.14 days corresponding to the fourth planet candidate reported by \citet{mayor2009}. We show the probability densities of velocity amplitude and period of the fourth signal (Fig. \ref{fig:densities_case_d}) to demonstrate that the fourth periodicity was detected as a clear probability maximum but it did not quite satisfy the detection criteria, namely, parameter $K$ of the 3.14 day signal was not significantly greater than zero by having a tail that extended all the way to zero in the amplitude space (Fig. \ref{fig:densities_case_d}, top panel). However, the signals reported by \citet{udry2007} were clearly detected in the data.
\begin{figure}
\center
\includegraphics[angle=-90,width=0.24\textwidth]{rvdist04_rv_GJ581d_dist_P1.ps}
\includegraphics[angle=-90,width=0.24\textwidth]{rvdist04_rv_GJ581d_dist_K1.ps}
\caption{As in Fig. \ref{fig:densities_case_b} but for the velocity amplitude and period of the signal with the shortest periodicity (corresponding to GJ 581 e) from the \citet{udry2007} data.}\label{fig:densities_case_d}
\end{figure}
The periodogram analyses revealed the existence of three signals in this data set as well. The third most significant signal was found at 82.6 days with a FAP of 0.001 but the interpretation of the origin of this signal corresponding to the reported period of 84 days was less clear than claimed by \citet{udry2007} because the sixth stronges power in the window function was suspiciously close to this period at 87 days making it possible that this periodicity could be due to sampling and not to a genuine signal.
\subsection{Third full observing period}
After the third full observing period with 76 radial velocities, the 3.14 day signal that was close to satisfying the detection criteria in the \citet{udry2007} data set was well constrained and satisfied all the detection criteria. With these velocities, the period of the outer planet candidate received a global maximum at 67 days instead of the periodicity at 84 days in the \citet{udry2007} data set. Essentially, we obtained the same solution that was reported by \citet{mayor2009} after four full observing periods and 43 more velocities.
To demonstrate that the solution indeed satisfied the detection criteria, we show the posterior densities of the period and velocity amplitude of the ``weakest'' 3.14 day signal in Fig. \ref{fig:densities_case_e} (top panels) together with the phase-folded signal (bottom panel). Because the corresponding four-Keplerian solution also had a posterior probability that was 5.4$\times 10^{7}$ times greater than that of the three-Keplerian model, we could conclude that the 76 epochs of HARPS data obtained after three full observing periods were already clearly in favour of a four-Keplerian model. We note that despite the eccentricity prior that penalises high eccentricities, the eccentricity of the outer companion had a MAP estimate of 0.30, yet, its uncertainties were considerable and it was found to be statistically indistinguishable from zero with a 99\% BCS of [0, 0.58]. The four signals were also obtained by the periodogram analyses and the FAP of the fourth signal was 0.001, which indicates that this signal was significantly present in the data.
\begin{figure}
\center
\includegraphics[angle=-90,width=0.24\textwidth]{rvdist04_rv_GJ581e_dist_P1.ps}
\includegraphics[angle=-90,width=0.24\textwidth]{rvdist04_rv_GJ581e_dist_K1.ps}
\includegraphics[angle=-90,width=0.32\textwidth]{rvdist04_scresidc_rv_GJ581e_1.ps}
\caption{Probability densities of parameters $P$ and $K$ of GJ 581 e as in Fig. \ref{fig:densities_case_b} and the phase-folded MAP signal (bottom panel) with the other three signals subtracted.}\label{fig:densities_case_e}
\end{figure}
We sampled the five-Keplerian solution and detected two probability maxima for the fifth signal at 25 and 42 days. However, neither of these was found to correspond to a solution that satisfied the detection criteria of another signal -- the corresponding periods did not get well constrained and the velocity amplitudes received values were not found distinguishable from zero.
\subsection{Mayor et al. data}
\citet{mayor2009} reported the detection of the fourth and innermost planet candidate orbiting GJ 581 with an orbital period of 3.14 days after having observed 119 spectra of the star. This new data set included velocities based on all the existing spectra after improved spectral reductions and corrections accounting for e.g. a changing internal pressure of the Th-Ar calibration lamps caused by aging. \citet{mayor2009} stated that they discovered a fourth periodic signal in the velocities using ``non-linear minimisations with genetic algorithms'' and periodogram analysis.\footnote{\citet{mayor2009} did not actually specify which measure of model goodness they minimised, which makes it impossible to replicate their results. However, the term ``minimisations'' that they used suggests that the chosen measure was $\chi^{2}$ statistics which corresponds to a delta-function prior for the jitter.}
We could obtain the same four-Keplerian solution easily using the \citet{mayor2009} velocities. However, we also tested whether we could spot a fifth signal in this set of 119 radial velocities. Indeed, sampling the parameter space of the five-Keplerian model identified three probability maxima for a fifth periodicity in the data. These maxima were found at periods of 17, 34, and $\sim$400 days but the corresponding Markov chains did not actually converge to a posterior density in the sense that their means and variances did not show converge as a function of the chain length. While this might be a result of insufficient chain length, we drew samples of up to few $10^{8}$ members from the posterior density, which should be a sufficient length given the acceptance rate of roughly 10\% throughout the samplings \citep{tuomi2012}. Also, for the local maxima at 17 and 34 days, the velocity amplitude parameter was not found statistically different from zero, which violates the detection criteria. Similarly, the global maximum of the fifth periodicity at $\sim$400 days (likely corresponding to the periodicity reported by \citet{vogt2010} at 433 days) was found to have an amplitude consistent with zero despite the fact that its MAP estimate was found to be as high as 1.7 ms$^{-1}$. The signal corresponding to this global maximum was also found to have an extremely poor phase-coverage due to annual gaps in the HARPS velocities.
\subsection{Fifth full observing period}
The 131 velocities obtained after the fifth full observing period did not change the solution much with respect to the one obtained using the data set of \citet{mayor2009}. The four signals reported by \citet{mayor2009} were very clearly present in the data and we moved on to sample the parameter space of a five-Keplerian model. This time, we spotted four different probability maxima for the period of the fifth signal at 32, 42, 190, and 420 days. However, none of these maxima corresponded to solutions satisfying the detection criteria by having either posterior probabilities below the detection threshold of by having velocity amplitudes statistically indistinguishable from zero. Therefore, we could not claim that the data supported the existence of more than four signals according to our detection criteria.
Curiously, the 420 day probability maximum corresponds to the signal reported by \citet{vogt2010} whereas the 32 day one corresponds to the signal reported by \citet{vogt2012}. These two probability maxima, despite the fact that they do not satisfy the detection criteria, are therefore existing features of the parameter density of the five-Keplerian model. Based on these 131 velocities, however, it is impossible to state whether these probability maxima correspond to actual signals in the data or false positives in the analyses of \citet{vogt2010} and \citet{vogt2012}.
\subsection{Sixth full observing period}
After the sixth full observing period, the total amount of HARPS velocities for GJ 581 had increased to 197. These data further improved the constrains of the orbits of the four planets detected by \citet{mayor2009} and increased their significances w.r.t. the 131 velocities that had been available a year earlier after the fifth full observing period. When sampling the five-Keplerian model, we could identify an interesting solution for the fifth signal.
According to our samplings, there were actually two probability maxima in the five-Keplerian model corresponding to 170 and 190 day periodicities for the fifth signal. These maxima, by being rather close to one another in the period space, enabled drawing a sample from the posterior density and our Markov chains were found to ``bounce'' between the two maxima yielding a sample from this bimodal solution. This solution satisfied all the detection criteria by having a well constrained period, velocity amplitude that was significantly greater than zero, and a model probability that was 4.7$\times 10^{3}$ times greater than the probability of the four-Keplerian model, which indicates that the favoured value of $k$ is 5 for this data set. While the bimodality is a rather unsatisfactory feature of a probability density -- especially when the goal is to quantify the properties of signals corresponding to candidate planets -- it is by no means surprising considering that we are searching for periodic signals in the first place. Similar bimodal densities have been observed before when dealing with radial velocities \citep[e.g.][]{tuomi2012}. We show the posterior density of the period of the fifth signal in Fig. \ref{fig:6fop_bimodal}. We could not find any other significant periodicities in this data. Also, we could not detect the fifth signal using the periodogram analyses of the data.
\begin{figure}
\center
\includegraphics[angle=-90,width=0.32\textwidth]{rvdist05_rv_GJ581h_dist_P5.ps}
\caption{Probability density of the fifth periodicity in the HARPS data after the sixth full observing period.}\label{fig:6fop_bimodal}
\end{figure}
\subsection{Forveille et al. data}\label{sec:forveille_data}
\citet{forveille2011} reported that they could spot four and only four periodic signals in their set of 240 HARPS velocities. This was also a simple task for our periodogram analyses as well as for the Bayesian posterior samplings and we continued to sample the parameter space of the five-Keplerian model. These samplings indicated that the 190 day periodicity observed in the data after the sixth observing period gained greater significance while the 170 day period ceased to correspond to a probability maximum. The probability of the five-Keplerian model was found to be 5.7$\times 10^{4}$ times more probable than the four-Keplerian one, which shows that the significance of the fifth periodic signal at 190 days received roughly ten times greater significance than it did for the data after the sixth full observing period. This periodicity was also well constrained and satisfied all our detection criteria. The periodogram analyses could not be used to detect a fifth periodogram in the data confidently.
However, as also reported by \citet{vogt2012}, we also spotted another probability maximum corresponding to a periodicity at 32 days. Treating the 32 and 190 day solutions as separate models, we calculated their respective probabilities. These probabilities were found to be 7\% and 93\%, respectively, which indicates that the global solution of the five-Keplerian model corresponds to a fifth periodicity at 190 days. Because of the existence of the probability maximum at 32 days, we performed global samplings of a six-Keplerian model but failed to find a signal satisfying the detection criteria. We did, however, spot a probability maximum for the period of the sixth signal at 32 days, but it could not be constrained from above or below, which violates one of our detection criteria. We show the phase-folded fifth signal in Fig. \ref{fig:forv_fifth} (bottom panel). While the phase-folded signal is not visually very impressive, we also plotted the distributions estimating the period and amplitude of this signal in Fig. \ref{fig:forv_fifth} (top panels) which indicate that this signal complies with our detection criteria.
\begin{figure}
\center
\includegraphics[angle=-90,width=0.24\textwidth]{rvdist05_rv_GJ581i_dist_P5.ps}
\includegraphics[angle=-90,width=0.24\textwidth]{rvdist05_rv_GJ581i_dist_K5.ps}
\includegraphics[angle=-90,width=0.32\textwidth]{rvdist05_scresidc_rv_GJ581i_5.ps}
\caption{As in Fig. \ref{fig:densities_case_e} but for the fifth signal in the \citet{forveille2011} data.}\label{fig:forv_fifth}
\end{figure}
\section{Analysis of combined radial velocities}
In this section we perform a robust analysis of the combined HARPS and HIRES velocities of GJ 581. The baseline of the available 240 HARPS velocities of \citet{forveille2011} is 2543 days. We plotted the four-Keplerian model residuals of the HARPS velocities in Fig. \ref{fig:residuals_4} (top panel). Clearly, these velocities show the remarkable stability (with an RMS of only 1.98 ms$^{-1}$) of the HARPS spectrograph and indicate that, in addition to obvious yearly gaps, the HARPS data can be expected to provide accurate constraints to possible additional periodic signals in the data.
\begin{figure}
\center
\includegraphics[angle=-90,width=0.45\textwidth]{rv_HARPSdist04_residc_rv_GJ581e.ps}
\includegraphics[angle=-90,width=0.45\textwidth]{rv_HIRESdist04_residc_rv_GJ581e.ps}
\caption{HARPS (top) and HIRES (bottom) residuals of the four-Keplerian model.}\label{fig:residuals_4}
\end{figure}
The residuals of the four-Keplerian model of the HIRES precision velocities of \citet{vogt2010} are also shown in Fig. \ref{fig:residuals_4} (bottom panel). These velocities have an RMS of 2.82 ms$^{-1}$, which implies an impressive precision, but falls short of the precision of the HARPS velocities for GJ 581. The baseline of the HIRES data is 4000 days, which is almost twice that of the HARPS one. Therefore, combining the HIRES and HARPS velocities would certainly provide greater accuracy than either one of them alone by having a greater overall precision and longer baseline. Also, the differences in the time-distribution of these two sets can certainly help to distinguish genuine Doppler periodicities and their aliases from one another and from spurious signals generated by noise and insufficient modelling by providing a better sampling than the individual sets.
\subsection{White noise model}
As we already analysed the HARPS velocities individually in Section \ref{sec:forveille_data}, and because the HIRES velocities have been analysed individually in e.g. \citet{gregory2011}, where it was concluded that they contain two significant periodicities (the two strongest signals at 5.4 and 12.9 days), we move on directly to analyse the combined data set with the pure white noise model.
Using the Gaussian white noise model, the four signals reported by \citet{mayor2009} were indeed found clearly in the combined HARPS and HIRES velocities and our Markov chains converged to the corresponding four-Keplerian solution with ease. All these four signals were well constrained and satisfied our detection criteria.
When sampling the parameter space of the five-Keplerian model, we searched for signals in the period space between 12.9 and 66.7 days and the space beyond 66.7 days up to a cutoff period of $T_{\rm max} = 10T_{\rm obs}$, where $T_{\rm obs}$ is the data baseline, separately and treated them as different models. Effectively, this corresponds to comparing two different prior models, i.e. in this case prior limitations (see Section \ref{sec:bayes}), with period ranges as $P \in [12.9, 66.7]$ and $P \in [66.7, T_{\rm max}]$, where $P$ and $T_{\rm max}$ have the unit of days, with each other. The cutoff period exceeds the data baseline, but we chose such a long upper limit for the parameter space because signals with periods exceeding the data baseline can be detected in radial velocity data \citep{tuomi2009} and this cutoff choice also enables the detection of signals that are only present in the data as trends with no or little curvature. We chose the two other cutoffs of the period space because we expect \emph{a priori} the inner system of GJ 581 to be ``dynamically packed'' in the sense that because of the existence of planets with orbital periods of 3.1, 5.4, and 12.9 days, it is likely that there are no stable orbits for additional planets with orbital periods less than 12.9 days. This leaves two subspaces of the period space where stable orbits could exist, namely, the ones we define above, i.e. orbits between GJ 581 c and d and orbits beyond GJ 581 d.
Because of the division of the period space into two parts, we could draw samples from both subspaces relatively easily. Samplings of the parameter space of a five-Keplerian model revealed a clear probability maximum for the period of the fifth signal at 43.7 days. We plotted the convergence of the period and amplitude parameters of the fifth signal to this solution in Fig. \ref{fig:white_fifth_convergence} and the posterior probability densities of these two parameters in Fig. \ref{fig:white_fifth_densities}. This signal was significantly present in the data and the five-Keplerian model was found to have a posterior probability of 460 times that of the four-Keplerian one, which exceeds the detection threshold of 150. We note that around the 2000th member of the Markov chains in Fig. \ref{fig:white_fifth_convergence} (bottom panel), the variance of the proposal density has been decreased heavily to increase the convergence rate because the adaptation of the proposal density to the posterior is very low when the chain discovers a narrow probability maximum in the parameter space, as happens in this particular case because the width of the posterior density of the period parameter is much narrower than the corresponding parameter space (see Fig. \ref{fig:white_fifth_convergence}, top panel).
\begin{figure}
\center
\includegraphics[angle=-90,width=0.45\textwidth]{rvdist05_rv_GJ581f_multiw_P4.ps}
\includegraphics[angle=-90,width=0.45\textwidth]{rvdist05_rv_GJ581f_multiw_K4.ps}
\caption{Convergence of three Markov chains to a signal with a period of 43.7 days and an amplitude of 0.8 ms$^{-1}$ with randomly selected initial states (arrows) in the period space between 12.9 and 66.7 days given the combined HARPS and HIRES data set and a pure white noise model.}\label{fig:white_fifth_convergence}
\end{figure}
\begin{figure}
\center
\includegraphics[angle=-90,width=0.24\textwidth]{rvdist05_rv_GJ581f_dist_P4.ps}
\includegraphics[angle=-90,width=0.24\textwidth]{rvdist05_rv_GJ581f_dist_K4.ps}
\caption{Posterior probability densities of the period and amplitude of the fifth most significant (fourth shortest periodicity) signal at a period of 43.7 days given the combined HARPS and HIRES data set and a pure white noise model.}\label{fig:white_fifth_densities}
\end{figure}
Samplings of both five- and six-Keplerian models revealed an additional probability maximum in the period space between 160-200 days with two maxima at roughly 170 and 190 days, but the posterior samplings did not constrain the amplitude of this signal from below, which violates the detection criteria. Therefore, using the white noise model leads to the conclusion that there are five periodic signals in the combined HARPS and HIRES data. The fact that the 190 day signal observed in the HARPS data alone did not satisfy the detection criteria is unfortunate. While it did appear as a probability maximum given the combined data as well, it could not be constrained as it was for the HARPS data alone. This suggests that this signal is not a genuine Doppler signal of planetary origin but caused by either pure chance in the HARPS velocities or by some (pseudo) periodic source of noise or bias in the HARPS data not accounted for by the pure white noise model.
On the other hand, the nature of the 43.7 day signal in the combined data set is unknown and we cannot rule out its origin as a genuine Doppler signature of planetary origin. It is always possible that a signal cannot be detected independently in any of the available datasets but only in the combined set because of the better phase-coverage, longer baseline, and a greater number of measurements in the combined set. Yet, it could also be caused by systematic differences between the different data sets, such as small instrumental biases that give rise to weak signals in the combined data set when not accounted for. However, we do not find any indication of such biases and in fact the HARPS and HIRES data sets cannot be shown to contradict with one another given the five-Keplerian model based on the Bayesian model inadequacy criterion of \citet{tuomi2011b}. Further, it might be a coincidence that the fifth signal appears at a period that, if of planetary origin, implies a 3:2 commensurability of the corresponding planet with GJ 581 d that has an orbital period of 66.7 days. Such an orbital configuration would likely enable the stability of the system in long term but, while this is compatible with a hypothesis that the 43.7 day signal is caused by a fifth planet candidate in the system, it does not imply the existence of five planets around GJ 581 unless the existence of the fifth signal is supported by future data. Whether this explanation for the 43.7 day signal is valid or not also remains to be investigated with better noise models.
We note that the excess noise levels (estimates of parameter $\sigma_{J}$) of the HARPS and HIRES velocities were 1.50 [1.15, 1.86] and 2.20 [1.63, 2.90] ms$^{-1}$, respectively, given the best model with $k=5$.
\subsection{Red noise model}
We tested the dependence of the observed signals on the selected noise model and the conclusions of \citet{baluev2012} by taking into account red features in the radial velocity noise using a moving average model (MA). This model was constructed as in \citet{tuomi2012c} and \citet{tuomi2012d} but had only one component, making the model a first order MA model with Gaussian white excess noise. Mathematically, this noise model can be written as
\begin{equation}\label{eq:moving_average}
m_{i} = r_{k}(t_{i}) + \gamma + \epsilon_{i} + \phi \big[ m_{i-1} - r_{k}(t_{i-1}) - \gamma \big] \exp \big[ \alpha (t_{i-1} - t_{i}) \big],
\end{equation}
where each measurement $m_{i}$ at epoch $t_{i}$ is described using the superposition of $k$ Keplerian signals ($r_{r}$); a reference velocity ($\gamma$); Gaussian random variable corresponding to the white noise in the data ($\epsilon_{i}$) with zero mean and variance $\sigma_{i}^{2} + \sigma_{J}^{2}$, where $\sigma_{i}$ is the estimated instrument uncertainty from the spectral reduction pipeline and $\sigma_{J}$ is an unknown excess noise component sometimes referred to as ``stellar jitter''; parameter $\phi$ is a parameter describing the MA magnitude and the term in brackets corresponds to an exponential decrease of this magnitude as a function of time difference of the $i$th and $i-1$th measurements in accordance with the formulation of \citet{baluev2012}. Parameter $\alpha$ describes the timescale of the correlations in the measurement noise.
As observed by \citet{baluev2012}, the HIRES data alone actually supports the existence of the planets GJ 581 b, c, and e despite the fact that only b and c could be detected using a pure white noise model and Bayesian analyses of the data \citep{gregory2011}. Using the MA model described above, the samplings of the parameter space of the three-Keplerian model showed beyond reasonable doubt that these three signals are present in the 122 HIRES velocities of \citet{vogt2010} by increasing the posterior probability of the model by a factor of 8.2$\times 10^{5}$ greater than that of the two-Keplerian model that corresponded to the two signals at 5.4 and 12.9 days. The third signal with a period of 3.1 days was constrained well and found consistent with the corresponding signal obtained from the HARPS velocities. We show the posterior densities of the period and velocity amplitude of GJ 581 e in Fig. \ref{fig:densities_HIRES} to demonstrate the significance of this detection.
\begin{figure}
\center
\includegraphics[angle=-90,width=0.24\textwidth]{rvdist03_rv_GJ581a_dist_Pe.ps}
\includegraphics[angle=-90,width=0.24\textwidth]{rvdist03_rv_GJ581a_dist_Ke.ps}
\caption{As in Fig. \ref{fig:densities_case_b} but for the velocity amplitude and period of GJ 581 e as obtained from the 122 HIRES velocities of \citet{vogt2010} data using the MA model.}\label{fig:densities_HIRES}
\end{figure}
We could not spot a fourth significant periodic signal in the HIRES data using the posterior samplings and the MA model. However, we performed samplings anyway and spotted three probability maxima for the period of the fourth signal at roughly 70, 300, and 500 days. However, we could not constrain any solutions for the four-Keplerian model and essentially ended up drawing samples from the prior of the period, though the chains were attracted to these three maxima throughout the samplings.
Using the MA model also decreased the estimated amount of white noise in the data. While this white noise was found to be 2.20 [1.63, 2.90] ms$^{-1}$ as estimated with the parameter $\sigma_{J}$ in the pure white noise model, it decreased to 1.81 [1.37, 2.72] ms$^{-1}$ for the MA model.
The HARPS velocities, however, did appear to favour a model with five periodic signals. We spotted a 190 day periodicity in the HARPS velocities using the MA model of Eq. (\ref{eq:moving_average}). Compared to the four-Keplerian model, the five-Keplerian one had 540 times greater posterior probability and the period and amplitude of the signal at 190 days were well constrained in accordance of our detection criteria. Therefore, the HARPS velocities, as was the case for the analyses with the pure white noise model, supports the existence of five independent sources of periodicity corresponding to the four planets reported in \citet{mayor2009} and one previously unknown signal at roughly 190 day period. We plotted the densities of parameters $P$ and $K$ of this signal in Fig. \ref{fig:MA_fifth_signal} (top panels) together with the corresponding phase-folded signal with the four planetary signatures of GJ 581 b, c, d, and e, and the MA component of the noise removed (Fig. \ref{fig:MA_fifth_signal}, bottom panel). The white noise parameter $\sigma_{J}$ decreased from 1.50 [1.15, 1.86] to 1.19 [0.98, 1.64] ms$^{-1}$ when adding the MA component to the statistical model. This indicates, as was found for the HIRES data, that there are significant correlations in the noise of the HARPS data as well and they have to be accounted for when modelling the velocities.
\begin{figure}
\center
\includegraphics[angle=-90,width=0.24\textwidth]{rvdist05_rv_GJ581i_dist_Pf.ps}
\includegraphics[angle=-90,width=0.24\textwidth]{rvdist05_rv_GJ581i_dist_Kf.ps}
\includegraphics[angle=-90,width=0.32\textwidth]{rvdist05_scresidc_rv_GJ581i_5b.ps}
\caption{As in Fig. \ref{fig:densities_case_e} but for the fifth signal in the \citet{forveille2011} data given the MA model of Eq. (\ref{eq:moving_average}).}\label{fig:MA_fifth_signal}
\end{figure}
When analysing the combined HARPS and HIRES data set with the MA model of Eq. (\ref{eq:moving_average}), we did not have any trouble recovering the signals of the four planets orbiting the star. However, as was the case with the white noise model, we failed to identify the existence of the 190 day signal as a significant one when sampling the period space above 67 days with a five-Keplerian model. We did observe a probability maximum at 190 days in the period space but could not constrain the corresponding signal from above and below in the period space and below in the amplitude space in violation of the detection criteria. Also, we did not find a signal at a period of 43.7 days, which indicates that the corresponding periodicity detected using a pure Gaussian white noise model was in fact most likely a spurious signal generated by insufficient noise modelling. This conclusion is supported by the fact that for $k=4$ the MA model was found to have a Bayesian evidence of -797.0 on the logarithmic scale, whereas the corresponding Bayesian evidence of the white noise model was -813.1. Further, the log-Bayesian evidence of the white noise model with $k=5$ was only -806.3, which implies that the MA model with $k=4$ is the best description of the data out of the set of models tested. We show the parameter MAP estimates and their 99\% BCSs of this best model in Table \ref{tab:parameters}.
\begin{table*}
\center
\caption{Four-Keplerian solution of the HARPS and HIRES velocities of GJ 581 together with the parameters quantifying the noise properties. MAP estimates of the parameters and their 99\% BCSs.\label{tab:parameters}}
\begin{tabular}{lcccc}
\hline \hline
Parameter & e & b & c & d \\
\hline
$P$ [days] & 3.14912 [3.14865, 3.14963] & 5.36863 [5.36840, 5.36884] & 12.9169 [12.9104, 12.9235] & 66.63 [66.31, 66.92] \\
$e$ & 0.16 [0, 0.36] & 0.03 [0, 0.05] & 0.03 [0, 0.20] & 0.17 [0, 0.51] \\
$K$ [ms$^{-1}$] & 1.72 [1.36, 2.11] & 12.63 [12.24, 13.05] & 3.21 [2.64, 3.73] & 1.87 [1.12, 2.47] \\
$\omega$ [rad] & 0.6 [0, 2$\pi$] & 0.9 [0, 2$\pi$] & 1.3 [0, 2$\pi$] & 6.2 [0, 2$\pi$] \\
$M_{0}$ [rad] & 5.8 [0, 2$\pi$] & 5.9 [0, 2$\pi$] & 1.9 [0, 2$\pi$] & 0.6 [0, 2$\pi$] \\
$a$ [AU] & 0.0285 [0.0266, 0.0302] & 0.0407 [0.0380, 0.0431] & 0.0731 [0.0684, 0.0773] & 0.218 [0.204, 0.231] \\
$m_{p} \sin i$ [M$_{\oplus}$] & 1.8 [1.3, 2.3] & 15.9 [13.7, 17.9] & 5.4 [4.2, 6.5] & 5.2 [3.3, 7.5] \\
\hline
Parameter & HARPS & HIRES \\
\hline
$\gamma$ [ms$^{-1}$] & 0.11 [-0.46, 0.61] & 0.14 [-0.87, 1.22] \\
$\sigma_{J}$ [ms$^{-1}$] & 1.23 [1.06, 1.72] & 1.62 [1.15, 2.32] \\
$\alpha$ [day$^{-1}$] & 0.057 [0, 0.446] & 0.120 [0, 0.863] \\
$\phi$ & 0.49 [0.26, 0.95] & 0.91 [0.45, 1] \\
\hline \hline
\end{tabular}
\end{table*}
We note that samplings of a six-Keplerian model did not show any evidence in favour of additional periodic signals in the combined data set.
\section{Discussion}
Estimating the number of planetary companions orbiting GJ 581, or equivalently, estimating the parameter $k$ favoured by the data for this star, has been attempted by several authors over the past few years \citep[e.g.][]{bonfils2005,udry2007,mayor2009,vogt2010,tuomi2011,gregory2011,vogt2012,baluev2012}. Improvements in statistical analysis techniques have proven efficient means of improving this estimate, i.e. especially showing that some signals have actually been false positives, but the most significant developments have relied on the availability of new high-precision data. We have assessed the number of significant periodic signals in the combined HARPS and HIRES data of GJ 581 and conclude that there is evidence in favour of four signals, corresponding to the four planets orbiting the star observed by \citet{mayor2009}.
By re-analysing sequentially the HARPS data after each observing period, i.e. each full year, we have shown that the Bayesian statistical techniques were at least as sensitive in detecting periodic signals in the data as the more commonly applied periodogram-based analysis tools (Table \ref{tab:timeline}). Moreover, we did not find any significant periodicities in these HARPS data subsets that did not show as significant signals when the amount of data was increased. This implies that the Bayesian detection criteria \citep{tuomi2012} are not sensitive to false positives arising from noise by chance in practice and any periodic signals that satisfy these criteria are likely genuine signals present in the data.
However, insufficient modelling and possible biases in any single data set can produce false positives that might be interpreted as planetary signals and could contaminate the statistics of exoplanet properties. For instance, the HARPS data was found to favour five periodicities after the sixth full observing period and 197 velocities. This fifth signal at a period of 190 days did not disappear after the inclusion of another 43 velocity measurements and was clearly present in the data published by \citet{forveille2011}. However, another independent data set obtained using the HIRES did not support the existence of this signal. This was evident when the signal could not be detected significantly in the combined HARPS and HIRES data set. Therefore, we conclude that the 190 day periodicity in the HARPS data set, while being significantly present, was likely not of planetary origin but a spurious periodicity caused by some source of (systematic) noise that had variations resembling periodic behaviour. Alternatively, it is possible that the noise model does not describe the HIRES and/or HARPS data well enough, which might result in the inability to detect this signal in the combined set. However, we do not see any evidence of such inadequate modelling as the two data sets are consistent according to the Bayesian model inadequacy criterion of \citet{tuomi2011b}.
The combined HARPS and HIRES data set, however, did contain a fifth periodic signal. This time the weak fifth signal occurred at a period of 43.7 days. This result was obtained by modelling the two velocity sets using Gaussian white noise models that are almost a standard way of describing radial velocities in practice. As was shown recently by \citet{baluev2012}, the radial velocity noise of GJ 581 is not white but contains significant correlations (red noise) that, if not accounted for, could mimic periodic behaviour. We found this to be the case and observed that the statistical model containing a first order moving average component improved the model significantly and enabled us to describe the combined data the best with only four periodic signals. Therefore, the 43.7 day signal observed in the combined data set using the white noise model is the most likely a spurious signal caused by insufficient modelling and consequent misinterpretation of the data and is not a genuine Doppler signal of planetary origin. This does not, however, imply beyond reasonable doubt that the 43.7 day periodicity is not a genuine signal because it is possible that the noise model is still not optimal\footnote{Actually this is not only possible but rather certain because the origin, nature, time-evolution, and other features of radial velocity noise are very poorly understood at the moment.} and that a better model could enable the recovery of the signal.
These results show that not only can insufficient modelling lead to detections of false positives, but relying on a single data set can also lead to detections of signals that are not supported by other independent measurements. False positives can be caused by these two sources in practice and care is needed to assess whether any given weak signal in any single data set is caused by either of these two effects or not. However, our results suggest that while the Bayesian detection criteria are not particularly sensitive to false positives, they can fall a victim of these two sources of false positives.
Finally, based on our results, it looks like there are four and only four planets orbiting GJ 581. This means that the 32 day signal observed by \citet{vogt2012} was a false positive, though the evidence in favour of this signal in their analyses was moderate at most anyway. Also, the periodogram analyses of \citet{baluev2012} obtained a decreased significance for the companion GJ 581 d because of the inconsistensies in the periodogram analyses described in Section \ref{sec:periodograms} and in e.g. \citet{tuomi2012}. However, the HARPS velocities of \citet{forveille2011} obtained from the spectra using the cross-correlation function method are not optimal \citep{pepe2002} but their accuracy can be significantly improved when using more sophisticated spectral reduction methods, such as the TERRA algorithm \citep{anglada2012,anglada2012b}. Improving the noise modelling can also lead to more accurate descriptions of the velocities and therefore enable the detections of one or some of the signals that were only obtained from the HARPS data or with a particular noise model at 190 and 44 day periods. A better understanding of the noise inherent in measuring RVs for M-dwarf stars is essential because current \citep{barnes2012} and future \citep{ramsey2008,mahadevan2009} planet search surveys aim to focus their attention on the inactive and slowly rotating subset of these stars \citep[see][]{jenkins2009}. For these reasons, and because GJ 581 is still a target of e.g. the HARPS and HIRES exoplanet surveys (according to \citet{vogt2012} there is already a considerable amount of additional HIRES data), we do not expect that the debate over the number of $k$ for GJ 581 is over. This is especially interesting because the period-space between GJ 581 c and d is known to contain dynamically stable orbits for low-mass planets and happens to correspond to the liquid water habitable zone of the star.
\begin{acknowledgements}
M. Tuomi is supported by RoPACS (Rocky Planets Around Cool Stars), a Marie Curie Initial Training Network funded by the European Commission's Seventh Framework Programme. J. S. Jenkins acknowledges funding by Fondecyt through grant 3110004 and partial support from Centro de Astrof\'isica FONDAP 15010003, the GEMINI-CONICYT FUND and from the Comit\'e Mixto ESO-GOBIERNO DE CHILE.
\end{acknowledgements}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,392 |
Q: Proof of "ordinal if and only if is Mostowski collapse" I tried to prove the following, can you tell me please if my proof is correct? Thank you.
Claim: A set $\alpha$ is an ordinal iff $\alpha$ is the Mostowski collapse of a strict well-order $\langle X, \prec \rangle$. The Mostowski collapse of a strict well-order $\langle X, \prec \rangle$ is a unique transitive set $\alpha$ together with a collapsing function $F: X \to \alpha$ such that $F$ is a bijection and $\forall x,x' \in X ( x \prec x' \leftrightarrow F(x) \in F(x'))$, that is, $F$ is an order isomorphism.
Proof: $\implies$ Let $\alpha$ be an ordinal. Then by definition, $\alpha$ is transitive and strictly well-ordered with respect to $\in$. Hence $\mathrm{id}: \alpha \to \alpha$ is a collapsing function and every ordinal is its own Mostowski collapse.
$\Longleftarrow$: Let $\alpha$ be the Mostowski collapse of a strict well-order $\langle X, \prec \rangle$. Then $\alpha$ is transitive. It remains to be verified that $\alpha$ is strictly well-ordered with respect to $\in$. Since we have $\forall x,x' \in X ( x \prec x' \leftrightarrow F(x) \in F(x'))$, $F$ is a bijection and $X$ is a strict total order, it follows that $\alpha$ is a strict total order. Assume $\alpha$ was not well-founded and let $\beta \subset \alpha$ be a set containing an infinite descending chain. Then $F^{-1}\beta$ would be a subset of $X$ containing an infinite descending chain hence contradicting well-foundedness of $\prec$.
A: It looks okay, but there is a shorter argument in the $\Longleftarrow$ direction: $\alpha$ is isomorphic to $\langle X,\prec\rangle$. Therefore $\alpha$ is transitive and well-ordered by $\in$.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,875 |
This art print is created from my original watercolor titled "A Lucky Flight Over the Vineyard."
Many of my creations originally were given little mini stories, the descriptive "story" for this print reads: "This good fortune pug and his Maneki Neko lucky mascot are floating happily over the vineyard. Well laden with grapes and charmed by this duo, these vines are sure to produce a gold medal vintage." The title of the piece and the description appear on the back of the accompanying note card. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,712 |
How to make an EASY paper butterfly heart canvas or wall art, complete with four free butterfly patterns for you to use!
Valentine's Day is fast approaching and I thought it would be lovely to create a pretty heart …. from PAPER BUTTERFLIES! I have a particular fondness for butterflies — one of my first projects was a pop-up butterfly card, and later I made this paper butterfly birdhouse. So I think it's time to make a paper butterfly canvas for my walll!
There's just something about butterflies I love. They spend most of their time being ordinary caterpillars, crawling about, chained to the ground like most of us. Then something amazing happens. They undergo the most startling transformation from a creepy crawly into a glorious explosion of flying color, spreading their wings and soaring high. It's really something special, and underscores the miraculous power of CHANGE for all of us.
So rather than have my butterflies form a complete heart, I put some littler paper butterflies on the edges to create wings so my heart can SOAR! And that's one of the best things about this simple and fun project, because there's no specific way you need to arrange your butterflies. Other than the heart shape, I put these butterflies on in random directions and distances apart. There are other ways to arrange these butterflies, of course, and I'll show you some ideas in this tutorial.
The real beauty of this paper butterfly heart is you don't actually need a lot to make it. All you really need are about 4-5 sheets of paper — I'm using several sheets of paper from this Seafoam Green pack I got at Michaels. I cut out my butterflies — there are about 250–300 of them — using my Cricut cutting machine, but you could also use scissors of course (though it might take a while). There are actually four different kinds of butterflies in this heart — blue morphos, emerald swallowtails, monarchs, and scarlet peacocks. All of the butterfly designs are free here on my blog for you to use!
You'll also need a canvas, board, or sign to attach your butterflies to. I'm using a 16″ x 20″ stretched canvas — I've not painted or prepared it in anyway. I simply took it out of its packaging and went to town!
And finally you'll need a way to attach your butterflies to your canvas. You all had several different ideas on how you might attach them, so I experimented with three different methods — glue dots, raised zots, and glue. I'll show you which method turned out to be both the fastest and most reliable in this tutorial!
This step-by-step video tutorial shows you how to create your own paper butterfly heart canvas or wall art!
I'm excited to show you how to create a paper butterfly heart canvas! This post contains some affiliate links for your convenience (which means if you make a purchase after clicking a link I will earn a small commission but it won't cost you a penny more)! Read my full disclosure policy.
Download my free butterfly files from my free resource library (you can get a password for it by signing up at the bottom of this page). I recommend you cut these butterflies with a cutting machine like a Cricut. Upload the file to your design software. If you're not sure how to upload an SVG cut file to Cricut Design Space, watch this helpful video training series I made. If you're on an iPhone or iPad, here's how to download and upload SVG files to the Cricut Design Space app.
Watch my step-by-step video for directions!
Step 5: Glue your paper butterflies on however you wish!
Watch my step-by-step video to see how I made both of these alternate butterfly canvas designs!
Download my four butterfly cut files by subscribing (free) below to get the password to my resource library.
As it turned out, the best way to apply the paper butterflies was with tacky glue. The glue dots and zots were more time consuming to apply AND they weren't as stable on the canvas. You can use the tacky glue as I have, but really any glue that dries clear should work just fine.
Some folks like to put the paper butterfly canvas heart in a shadowbox with glass over it to keep it dust-free. If you do that, you can also put vinyl on the glass with a name or a sweet message. Or you could just add some text to the corner of your canvas.
If you make butterfly canvas wall art using this tutorial, please share a photo in our amazing Cricut Facebook group, email it to me at [email protected], or tag me on social media with #makershowandtell.
Want to remember this? Save the Paper Butterfly Canvas Wall Art Tutorial to Your Favorite DIY Pinterest Board!
This is simply GORGEOUS, Jennifer!!! You did a MAGNIFICENT job designing this!! In this one BEAUTIFUL canvas, you embodied the whole process of change in a butterfly!!! ABSOLUTELY FANTA-BULOUS (fantastically fabulous)!!!!!!! | {
"redpajama_set_name": "RedPajamaC4"
} | 5,277 |
Nuttallanthus subandinus är en grobladsväxtart som först beskrevs av Friedrich Ludwig Diels, och fick sitt nu gällande namn av David A. Sutton. Nuttallanthus subandinus ingår i släktet indiansporrar, och familjen grobladsväxter. Inga underarter finns listade i Catalogue of Life.
Källor
Indiansporrar
subandinus | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,444 |
{"url":"https:\/\/ch.gateoverflow.in\/310\/gate-chemical-2014-question-48","text":"In a steady incompressible flow, the velocity distribution is given by $\\bar{V}=3x\\hat{i}-Py\\hat{j}+5z\\hat{k}$, where, $V$ is in $m\/s$ and $x,y$ and $z$ are in $m$. In order to satisfy the mass conservation, the value of the constant $P$ (in $s^{-1}$) is _____________","date":"2022-10-02 10:04:45","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8741778135299683, \"perplexity\": 56.52490987744788}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-40\/segments\/1664030337307.81\/warc\/CC-MAIN-20221002083954-20221002113954-00631.warc.gz\"}"} | null | null |
package net.malisis.core.renderer.element.face;
import static net.minecraft.util.EnumFacing.*;
import net.malisis.core.renderer.element.Face;
import net.malisis.core.renderer.element.vertex.BottomNorthWest;
import net.malisis.core.renderer.element.vertex.BottomSouthWest;
import net.malisis.core.renderer.element.vertex.TopNorthWest;
import net.malisis.core.renderer.element.vertex.TopSouthWest;
/**
* @author Ordinastie
*
*/
public class WestFace extends Face
{
public WestFace()
{
super(new TopNorthWest(), new BottomNorthWest(), new BottomSouthWest(), new TopSouthWest());
params.direction.set(WEST);
params.textureSide.set(WEST);
params.colorFactor.set(0.6F);
params.aoMatrix.set(calculateAoMatrix(WEST));
setStandardUV();
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,505 |
\section{Introduction}
\label{sec:intro}
Since its inception \cite{HMC2006,LL2007}, mean field game theory has undergone a phenomenal growth and found applications in diverse areas
\cite{BTB2016,CFS2015,CS2017,DAS2016,HJN2019,LW2011,LZ2019,LT2015,LRS2019,MCH2013,
SMN2018,SGU2016,WH2019,YMMS2012}.
The theory is inspired by ideas in statistical physics and overcomes the dimensionality difficulty in competitive decision problems involving a large population of agents. The reader is referred to \cite{BFY2013,CHM2017,C2013,CD2018} for an overview of basic theory and applications.
While mean field games have been developed with very different modelling frameworks, linear quadratic (LQ) mean field games are of particular importance and have been extensively studied due to their elegant closed-form solutions \cite{BSYY2016, HWW2016, HCM2007,SMN2018}. Huang, Caines and Malham\'e \cite{HCM2007} adopt infinite horizon discounted costs and use the infinite population limit model to design decentralized strategies for the actual model with a large but finite population.
Li and Zhang \cite{LZ2008} study decentralized strategies with ergodic costs.
Wang and Zhang \cite{WZ2012} introduce Markov jumps in the system dynamics and costs.
Bardi and Priuli \cite{BP2014} study LQ $N$-person games and their mean field limit with ergodic costs. Huang, Wang and Wu \cite{HWW2016} adopt backward stochastic differential equations for modelling state processes. Moon and Basar \cite{MB2017} consider risk sensitive costs and address robustness.
Huang and Huang \cite{HH2017} consider linear diffusion dynamics
including model uncertainty treated as an adversarial player.
Tchuendom \cite{T2018} shows nonuniqueness can arise, but interestingly, uniqueness can be restored by the presence of common noise.
LQ mean field games have an extension by including a major player \cite{H2010,NH2012}.
This modelling framework is introduced by Huang \cite{H2010}.
Bensoussan et al \cite{BCLY2017} consider Stackelberg equilibria under state and control delays. Caines and Kizikale \cite{CK2017} consider partial information and filtering based strategies for an LQ model with a major player.
In this paper we study a class of LQ mean field games
with common noise and indefinite weight matrices (simply called weights below) in the cost functional. We adopt the so-called asymptotic solvability framework in \cite{HZ2020}. Starting with feedback perfect state information, this approach aims to determine feedback Nash strategies under such centralized information and next study how the solutions behave when the number of players increases.
It uses a rescaling method to derive a set of Riccati ordinary differential equations (ODEs), which characterizes a necessary and sufficient condition for asymptotic solvability \cite{HZ2020}. This method can be extended to LQ mean field games with a major player \cite{MH2020}. Recently, Huang and Yang \cite{HY2020b} extend this asymptotic solvability notion to mean field social optimization, where the agents cooperatively optimize a social cost. That work further develops a method of asymptotic analysis to obtain tight estimates of optimality loss
when decentralized strategies are implemented. For our current model, the test of asymptotic solvability reduces to checking two Riccati ODEs in a low dimensional space, which, as a result of the controlled individual and common noises, have higher nonlinearity than those Riccati
equations in \cite{HZ2020}.
In the analysis of mean field games, a crucial step is to examine how the strategies obtained in the mean field limit model perform when
implemented in the actual model with a large but finite population.
This can be addressed by establishing the so-called $\epsilon$-Nash equilibrium property, where $\epsilon\to 0$ as $N\to \infty$.
For LQ models \cite{BSYY2016,HWW2016,HCM2007,WZ2012} as well as some nonlinear cases \cite{NC2013}, one can obtain an $O(1/\sqrt{N})$-Nash equilibrium when all players are symmetric. This typically results from cost estimates by the Cauchy-Schwarz inequality.
To our best knowledge on the existing literature, probably only Basna, Hilbert and Kolokoltsov \cite{BHK2014} have obtained an $O(1/N)$-Nash equilibrium result in a finite state mean field game. We will establish an $O(1/N)$-Nash equilibrium for the decentralized strategies obtained from the LQ mean field limit model; our approach is different from that in \cite{BHK2014} which relies on perturbation estimates of generators of continuous-time controlled Markov chains. We will
directly treat the best response control problem of the unilateral agent in a high dimensional space and then employ the rescaling method to obtain accurate information about its performance improvement.
We will develop extensive asymptotic error estimates by building upon techniques in the companion paper \cite{HY2020b} on social optimization.
In a convergence problem of mean field games with common noise, Cardaliaguet et al \cite{CDLL2015} prove that the value functions of $N$ players converge in an average sense to the solution of the master equation, and the averaged error disappears by rate $1/N$ as $N\to \infty$. But their error bound is different from the $O(1/N)$-Nash equilibrium notion.
It will be helpful to briefly explain the route that we will follow in the analysis.
For the LQ Nash game with indefinite weights, we apply dynamic programming to derive a set of large-scale Riccati equations, which is used to
formulate the asymptotic solvability problem of the $N$-player game. In order to get useful information from the large Riccati equations, we exploit their symmetries to achieve dimension reduction and next use a rescaling technique to derive two key Riccati equations, which completely characterize asymptotic solvability. By taking the mean field limit of the solution of the $N$-player game, we construct a set of decentralized strategies, which are then applied to the $N$-player model.
We further obtain explicit formulas for the per agent cost for three scenarios: i) the $N$ players apply the Nash equilibrium strategies $(\hat u_1, \cdots, \hat u_N)$; ii) the $N$ players apply decentralized strategies $(\check u_1, \cdots, \check u_N)$ obtained from the mean field limit model; iii) the player in question takes its best response while the other $N-1$ players apply these decentralized strategies.
When $N\to \infty$, the three cases have the same limit for the per agent cost. The comparison of the costs in scenarios ii) and iii) establishes the $O(1/N)$-Nash equilibrium property. A comparison of the per agent costs
for the mean field game and the mean field social optimum enables us to quantify the efficiency loss of the mean field game with respect to the social optimum; see the comparison in the companion paper \cite{HY2020b}.
\subsection{Organization of the paper}
Section \ref{sec:LQNG} introduces the $N$-player LQ Nash game with indefinite weight matrices in the cost functionals. The set of feedback Nash equilibrium strategies is characterized using a system of Riccati ODEs in Section
\ref{sec:RiccatiEqns}.
The asymptotic solvability problem is studied in Section \ref{sec:AS} and a necessary and sufficient condition is derived. Section \ref{sec:DecentralStrats} constructs a set of decentralized strategies for
the $N$-player game, and
Section \ref{sec:O(1/N)Nash} proves an $O(1/N)$-Nash equilibrium theorem.
A numerical example is presented in Section \ref{sec:num}.
Section \ref{sec:conclusion} concludes the paper.
\subsection{Notation}
Let ${\mathcal S}^n$ be the set of $n\times n$ real symmetric matrices. We denote the quadratic form $[x]_M^2 = x^T M x$ for $M\in {\mathcal S}^n$ and $x\in \mathbb{R}^n$.
We use $I$ to denote an identity matrix of compatible dimensions, and sometimes write $I_k$ to indicate the $k\times k$ identity matrix.
We use $0$ to denote either the scalar zero or a zero vector/matrix of compatible dimensions.
We denote by $|F|$ the Euclidean norm of a vector or matrix $F$, by $\mathbf{1}_{k\times l}$ a $k\times l$ matrix with all entries equal to $1$, by $\otimes$ the Kronecker product, and by the column vectors $\{e^k_1, \cdots, e^k_k\}$ the canonical basis of $\mathbb{R}^k$.
For a function $f(t,x)$, we may write partial derivatives $\partial f/\partial t$ as $\partial_t f$;
$\partial f/\partial x$ as $\partial_x f$; and
$\partial^2 f/\partial x^2$ as $\partial_x^2 f$.
\section{The LQ Nash game}
\label{sec:LQNG}
Consider a system of $N$ players (or called agents) denoted by $\mathcal{A}_i$, $1\leq i \leq N$. The state process $X_i(t)$ satisfies the following stochastic differential equation (SDE)
\begin{align}
\label{Xi}
d X_i(t) & = (A X_i(t) + Bu_i(t) + G X^{(N)}(t)) dt
+ (B_1 u_i(t) + D) d W_i(t) \\
& \quad + ( B_0 u^{(N)}(t) + D_0 ) d W_0(t) , \notag
\end{align}
where we have the state $X_i(t) \in \mathbb{R}^n$, the control
$u_i(t)\in \mathbb{R}^{n_1}$, the mean field state $X^{(N)} \coloneqq(1/N)\sum_{i=1}^N X_i$ and the control mean field $u^{(N)} \coloneqq (1/N)\sum_{i=1}^N u_i$.
The initial states $\{X_i(0): 1 \leq i \leq N\}$ are independent with $\mathbb{E}|X_i(0)|^2<\infty$. The individual noise processes $\{W_i: 1\leq i \leq N\}$ are $1$-dimensional independent standard Brownian motions, which are also independent of $\{X_i(0): 1\leq i \leq N\}$.
The common noise $W_0$ is a $1$-dimensional standard Brownian motion independent of $\{W_i: 1 \leq i \leq N\}$ and $\{X_i(0): 1 \leq i \leq N\}$.
In contrast to \cite{HZ2018a,HZ2020}, each individual noise is affected by that player's control, and the model contains a common noise affected by the control mean field.
The individual cost functional (simply called cost) of $\cal{A}_i$, $1\leq i \leq N$, is given by
\begin{align}
J_i(u_{1}, \cdots, u_N) = & \mathbb{E} \left[ \int_0^T \Big( [X_i(t) - \Gamma X^{(N)}(t)]_Q^2 + [u_i(t)]_{R}^2 \Big) \ dt \right.
\label{Ji}\\
& \quad \left. + [X_i(T) - \Gamma_f X^{(N)}(T)]_{Q_f}^2 \right] ,
\notag
\end{align}
where we denote $[x]_M^2 = x^T M x$ for $M\in\mathcal{S}^n$ and $x\in \mathbb{R}^n$.
The constant matrices $A$, $B$, $B_0$ $B_1$, $D$, $D_0$, $G$, $\Gamma$, $Q$, $R$, $\Gamma_f$, $Q_f$ above have compatible dimensions, and $Q$, $Q_f$, $R$ are symmetric, possibly indefinite, matrices.
Define
\begin{align}
& X(t) = \begin{bmatrix}X_1(t) \\ \vdots \\ X_N(t) \end{bmatrix} \in
\mathbb{R}^{Nn},\quad u_{-i}=(u_1, \cdots, u_{i-1}, u_{i+1}, \cdots, u_N), \notag \\
& \mathbf{A}
= \diag[A, \cdots, A] + \mathbf{1}_{N \times N} \otimes \tfrac{G}{N} \in \mathbb{R}^{Nn \times Nn} , \notag \\
& \mathbf{B}_0 = \mathbf{1}_{N\times 1} \otimes \tfrac{B_0}{N} \in
\mathbb{R}^{Nn \times n_1} , \quad
\mathbf{D}_0 = \mathbf{1}_{N\times 1} \otimes D_0 \in \mathbb{R}^{Nn \times 1} , \notag \\
& \widehat{\mathbf{B}}_k = e^N_k \otimes B \in \mathbb{R}^{Nn \times n_1} , \quad
\mathbf{B}_k = e^N_k \otimes B_1 \in \mathbb{R}^{Nn\times n_1} , \nonumber\\
& \mathbf{D}_k = e^N_k \otimes D \in \mathbb{R}^{Nn \times 1} , \quad 1 \leq k \leq N . \notag
\end{align}
Then $X =(X_1^T, \cdots, X_N^T)^T$ has the following dynamics
\begin{align} \label{X}
d X(t) = & \Big( \mathbf{A} X(t) + \sum_{i=1}^N \widehat{\mathbf{B}}_i u_i(t)\Big) dt + \sum_{i=1}^N ( \mathbf{B}_i u_i(t) + \mathbf{D}_i ) d W_i
\\ & + \Big( \mathbf{B}_0 \sum_{i=1}^N u_i(t) + \mathbf{D}_0 \Big) d W_0 . \notag
\end{align}
We denote
\begin{align}
& \mathbf{K}_i
= [0, \cdots, 0, I_n, 0, \cdots, 0] - (1/N) [\Gamma, \Gamma, \cdots,
\Gamma]\in \mathbb{R}^{n\times Nn} , \notag \\
& \mathbf{K}_{i f} = [0, \cdots, 0, I_n, 0, \cdots, 0] - (1/N) [\Gamma_f , \Gamma_f, \cdots, \Gamma_f ] , \notag \\
& \mathbf{Q}_i = \mathbf{K}_i^T Q \mathbf{K}_i, \quad
\mathbf{Q}_{if} = \mathbf{K}_{i f}^T Q_f \mathbf{K}_{i f} . \notag
\end{align}
The individual cost \eqref{Ji} can be written as
\begin{align}
J_i(u_{i}, u_{-i}) & = \mathbb{E} \left[ \int_0^T \Big( [X(t)]_{\mathbf{Q}_i}^2 + [u_i(t)]_{R}^2 \Big) \ dt
+ [X(T) ]_{\mathbf{Q}_{i f}}^2 \right] .
\label{Ji2}
\end{align}
We begin by solving the LQ Nash game
under closed-loop perfect state (CLPS) information,
where the full state vector $X(t)$ is observed by each player.
The players seek a set of Nash equilibrium strategies $(\hat u_1, \cdots, \hat u_N)$.
For notational simplicity, Sections \ref{sec:RiccatiEqns}--\ref{sec:O(1/N)Nash}
will treat a simplified model \eqref{Xi}--\eqref{Ji} with $D= D_0 = 0$.
The extension to the general case will be discussed in Section \ref{sec:O(1/N)Nash}.
\section{Riccati equations and feedback Nash strategies}
\label{sec:RiccatiEqns}
Based on \eqref{Ji2}, we may naturally define the cost $J(t,\mathbf{x}, u_1, \cdots, u_N)$ where the running cost is integrated on $[t,T]$ instead of $[0,T]$ with initial state $X(t) = \mathbf{x} = (x_1^T, \cdots, x_N^T)^T$.
Let $V_i(t, \mathbf{x})$ denote the value function of player $\mathcal{A}_i$.
The Hamilton--Jacobi--Bellman (HJB) equations of the $N$ players associated with \eqref{X}--\eqref{Ji2} (taking $D_0 = D = 0$) are
\begin{align}
\label{HJBV}
& - \frac{\partial V_i}{\partial t}
= \frac{\partial^T V_i}{\partial \mathbf{x}} \Big( \mathbf{A} \mathbf{x}
+ \sum_{k =1}^N \widehat{\mathbf{B}}_k \hat u_k \Big)
+ \frac{1}{2} \Big( \sum_{k=1}^N \hat u_k \Big)^T \mathbf{B}_0^T \frac{\partial^2 V_i}{\partial \mathbf{x}^2 } \mathbf{B}_0
\Big( \sum_{k=1}^N \hat u_k \Big) \\
&\qquad\qquad + \frac{1}{2} \sum_{k=1}^N ( \mathbf{B}_k \hat u_k )^T \frac{\partial^2 V_i}{\partial \mathbf{x}^2} ( \mathbf{B}_k \hat u_k )
+ \mathbf{x}^T \mathbf{Q}_i \mathbf{x}
+ \hat u_i^T R \hat u_i , \notag \\
& V_i( T, \mathbf{x} ) = \mathbf{x}^T \mathbf{Q}_{if} \mathbf{x} , \quad 1\leq i \leq N . \notag
\end{align}
Each $\hat u_i$ is the minimizer in the HJB equation of $V_i(t, \mathbf{x})$
as specified below. Taking $(u_1, \cdots, u_N)$ in place of
$(\hat u_1, \cdots, \hat u_N)$, we write the right hand side of \eqref{HJBV} in the form:
$${\mathcal H}(\mathbf{x}, \partial_\mathbf{x} V_i, \partial_\mathbf{x}^2 V_i, u_i, u_{-i} ).$$ Then we require
\begin{align}\label{Hminimizer}
\hat u_i= \arg\min_{u_i} {\mathcal H} (\mathbf{x}, \partial_\mathbf{x} V_i, \partial_\mathbf{x}^2 V_i, u_i, \hat u_{-i} ), \quad \forall i.
\end{align}
We will calculate $\hat u_i$ under the following conditions: for all $(t, \mathbf{x}) \in [0, T] \times \mathbb{R}^{Nn}$,
\begin{align}
\label{Ri>0}
& R + \frac{1}{2} \mathbf{B}_i^T \frac{\partial^2 V_i(t, \mathbf{x})}{\partial \mathbf{x}^2} \mathbf{B}_i >0 ,
\\
& I + \frac{1}{2} \sum_{k=1}^N \Big( R + \frac{1}{2} \mathbf{B}_k^T
\frac{\partial^2 V_k(t, \mathbf{x})}{\partial \mathbf{x}^2} \mathbf{B}_k \Big)^{-1} \mathbf{B}_0^T \frac{\partial^2 V_k(t, \mathbf{x})}{\partial \mathbf{x}^2} \mathbf{B}_0 \ \text{is invertible} ,
\label{inv}\\
& R + \frac{1}{2} \mathbf{B}_i^T \frac{\partial^2 V_i(t, \mathbf{x})}{\partial \mathbf{x}^2} \mathbf{B}_i
+ \frac{1}{2} \mathbf{B}_0^T \frac{\partial^2 V_i(t, \mathbf{x})}{\partial \mathbf{x}^2} \mathbf{B}_0 >0 .
\label{R0i>0}
\end{align}
By \eqref{Hminimizer}, we derive
\begin{align}
0 = \widehat{\mathbf{B}}_i^T \frac{\partial V_i}{\partial \mathbf{x}}
+ \mathbf{B}_0^T \frac{\partial^2 V_i}{\partial \mathbf{x}^2} \mathbf{B}_0 \sum_{i\ne k=1}^N \hat u_k + \mathbf{B}_0^T \frac{\partial^2 V_i}{\partial \mathbf{x}^2} \mathbf{B}_0 \hat u_i + \mathbf{B}_i^T \frac{\partial^2 V_i}{\partial \mathbf{x}^2}
\mathbf{B}_i \hat u_i + 2 R \hat u_i , \label{NEuhat}
\end{align}
which implies that
\begin{align}
\hat u_i
= - \frac{1}{2} \Big( R + \frac{1}{2}\mathbf{B}_i^T \frac{\partial^2 V_i}{\partial \mathbf{x}^2}\mathbf{B}_i \Big)^{-1}
\Big( \widehat{\mathbf{B}}_i^T \frac{\partial V_i}{\partial \mathbf{x}}
+ \mathbf{B}_0^T \frac{\partial^2 V_i}{\partial \mathbf{x}^2} \mathbf{B}_0 \sum_{k=1}^N \hat u_k \Big) . \label{uisumu}
\end{align}
Adding up the $N$ equations in \eqref{uisumu} leads to
\begin{align}
\sum_{i=1}^N \hat u_i
= - \frac{1}{2} \sum_{i=1}^N \Big( R + \frac{1}{2}\mathbf{B}_i^T \frac{\partial^2 V_i}{\partial \mathbf{x}^2}\mathbf{B}_i \Big)^{-1}
\Big( \widehat{\mathbf{B}}_i^T \frac{\partial V_i}{\partial \mathbf{x}}
+ \mathbf{B}_0^T \frac{\partial^2 V_i}{\partial \mathbf{x}^2} \mathbf{B}_0 \sum_{k=1}^N \hat u_k \Big) , \notag
\end{align}
which
under condition \eqref{inv} yields
\begin{align}
\sum_{k = 1}^N \hat u_k
= & - \frac{1}{2} \Big[ I + \frac{1}{2} \sum_{k=1}^N ( R + \frac{1}{2} \mathbf{B}_k^T \frac{\partial^2 V_k}{\partial \mathbf{x}^2} \mathbf{B}_k )^{-1} \mathbf{B}_0^T \frac{\partial^2 V_k}{\partial \mathbf{x}^2} \mathbf{B}_0 \Big]^{-1} \cdot \notag \\
&
\quad \sum_{k=1}^N \Big( R + \frac{1}{2} \mathbf{B}_k^T \frac{\partial^2 V_k}{\partial \mathbf{x}^2} \mathbf{B}_k \Big)^{-1} \widehat{\mathbf{B}}_k^T \frac{\partial V_k}{\partial \mathbf{x}}
\notag \\
= : &\ \mathbf{M} . \label{sumu}
\end{align}
Combining \eqref{uisumu} and \eqref{sumu} gives that
\begin{align}
\hat u_i = - \frac{1}{2} \Big( R + \frac{1}{2}\mathbf{B}_i^T \frac{\partial^2 V_i}{\partial \mathbf{x}^2}\mathbf{B}_i \Big)^{-1}
\Big( \widehat{\mathbf{B}}_i^T \frac{\partial V_i}{\partial \mathbf{x}}
+ \mathbf{B}_0^T \frac{\partial^2 V_i}{\partial \mathbf{x}^2} \mathbf{B}_0 \mathbf{M} \Big) .
\label{uieqm}
\end{align}
We substitute \eqref{uieqm} into the right hand side of
\eqref{HJBV} to obtain
\begin{align}
\label{VPDE}
- \frac{\partial V_i}{\partial t} = & \frac{1}{4} \Big[ \mathbf{B}_0^T \frac{\partial^2 V_i}{\partial \mathbf{x}^2} \mathbf{B}_0 \mathbf{M}
- \widehat{\mathbf{B}}_i^T \frac{\partial V_i}{\partial \mathbf{x}} \Big]^T \Big( R + \frac{1}{2}\mathbf{B}_i^T \frac{\partial^2 V_i}{\partial \mathbf{x}^2}\mathbf{B}_i \Big)^{-1}
\Big[ \mathbf{B}_0^T \frac{\partial^2 V_i}{\partial \mathbf{x}^2} \mathbf{B}_0 \mathbf{M} + \widehat{\mathbf{B}}_i^T \frac{\partial V_i}{\partial \mathbf{x}} \Big] \\
& - \frac{1}{2} \frac{\partial^T V_i}{\partial \mathbf{x}} \sum_{i\ne k = 1 }^N
\widehat{\mathbf{B}}_k \Big( R + \frac{1}{2}\mathbf{B}_k^T \frac{\partial^2 V_k}{\partial \mathbf{x}^2}\mathbf{B}_k \Big)^{-1}
\Big[ \mathbf{B}_0^T \frac{\partial^2 V_k }{\partial \mathbf{x}^2} \mathbf{B}_0 \mathbf{M} + \widehat{\mathbf{B}}_k^T \frac{\partial V_k }{\partial \mathbf{x}} \Big] \notag \\
& + \frac{1}{8} \sum_{i\ne k=1}^N
\Big[ \mathbf{B}_0^T \frac{\partial^2 V_k }{\partial \mathbf{x}^2} \mathbf{B}_0 \mathbf{M} + \widehat{\mathbf{B}}_k^T \frac{\partial V_k }{\partial \mathbf{x}} \Big]^T
\Big( R + \frac{1}{2}\mathbf{B}_k^T \frac{\partial^2 V_k}{\partial \mathbf{x}^2}\mathbf{B}_k \Big)^{-1}
\cdot \notag \\
& \qquad\qquad \mathbf{B}_k^T \frac{\partial^2 V_i}{\partial \mathbf{x}^2} \mathbf{B}_k
\Big( R + \frac{1}{2}\mathbf{B}_k^T \frac{\partial^2 V_k}{\partial \mathbf{x}^2}\mathbf{B}_k \Big)^{-1}
\Big[ \mathbf{B}_0^T \frac{\partial^2 V_k }{\partial \mathbf{x}^2} \mathbf{B}_0 \mathbf{M} + \widehat{\mathbf{B}}_k^T \frac{\partial V_k }{\partial \mathbf{x}} \Big] \notag \\
& + \frac{1}{2} \mathbf{M}^T \mathbf{B}_0^T \frac{\partial^2 V_i}{\partial \mathbf{x}^2} \mathbf{B}_0 \mathbf{M}
+ \mathbf{x}^T \mathbf{Q}_i \mathbf{x} + \frac{\partial^T V_i}{\partial \mathbf{x}} \mathbf{A} \mathbf{x} ,
\notag \\
V_i( T, \mathbf{x} ) = & \mathbf{x}^T \mathbf{Q}_{i f} \mathbf{x} , \quad 1\leq i \leq N , \notag
\end{align}
subject to the conditions \eqref{Ri>0}, \eqref{inv} and \eqref{R0i>0}.
We are interested in a solution of the form
\begin{align}
V_i(t, \mathbf{x}) = \mathbf{x}^T \mathbf{P}_i(t) \mathbf{x} , \quad
1\leq i \leq N ,
\label{Vansatz}
\end{align}
where $\mathbf{P}_i(t)$ is a symmetric matrix function of $t\in [0, T]$ and is differentiable in $t$.
Substituting \eqref{Vansatz} into \eqref{VPDE}, we obtain the ODE system for $\mathbf{P}_i$, $1\leq i \leq N$:
\begin{align}
\begin{cases}
- \dot{\mathbf{P}}_i = \mathbf{P}_i \mathbf{A} + \mathbf{A}^T \mathbf{P}_i + \mathbf{Q}_i
- \mathbf{P}_i \widehat{\mathbf{B}}_i
(R + \mathbf{B}_i^T \mathbf{P}_i \mathbf{B}_i )^{-1}
\widehat{\mathbf{B}}_i^T \mathbf{P}_i
\\
\qquad\quad
+ \mathbf{M}_{0}^T \mathbf{B}_0^T \mathbf{P}_i \mathbf{B}_0
(R + \mathbf{B}_i^T \mathbf{P}_i \mathbf{B}_i )^{-1}
\mathbf{B}_0^T \mathbf{P}_i \mathbf{B}_0 \mathbf{M}_{0} \\
\qquad\quad - \mathbf{P}_i \sum_{i\ne k=1}^N
\widehat{\mathbf{B}}_k
( R + \mathbf{B}_k^T \mathbf{P}_k \mathbf{B}_k )^{-1} ( \mathbf{B}_0^T \mathbf{P}_k \mathbf{B}_0 \mathbf{M}_{0} + \widehat{\mathbf{B}}_k^T \mathbf{P}_k ) \\
\qquad \quad - \sum_{i\ne k=1}^N
( \mathbf{B}_0^T \mathbf{P}_k \mathbf{B}_0 \mathbf{M}_{0}
+ \widehat{\mathbf{B}}_k^T \mathbf{P}_k )^T
( R + \mathbf{B}_k^T \mathbf{P}_k \mathbf{B}_k )^{-1}
\widehat{\mathbf{B}}_k^T \mathbf{P}_i
\\
\qquad\quad + \sum_{i\ne k=1}^N ( \mathbf{B}_0^T \mathbf{P}_k \mathbf{B}_0 \mathbf{M}_{0} + \widehat{\mathbf{B}}_k^T \mathbf{P}_k )^T ( R + \mathbf{B}_k^T \mathbf{P}_k \mathbf{B}_k )^{-1}
\mathbf{B}_k^T \mathbf{P}_i \mathbf{B}_k \cdot \\
\qquad\quad \ \
( R + \mathbf{B}_k^T \mathbf{P}_k \mathbf{B}_k )^{-1}
( \mathbf{B}_0^T \mathbf{P}_k \mathbf{B}_0 \mathbf{M}_{0} + \widehat{\mathbf{B}}_k^T \mathbf{P}_k )
+ \mathbf{M}_{0}^T \mathbf{B}_0^T \mathbf{P}_i \mathbf{B}_0 \mathbf{M}_{0}
, \\
\mathbf{P}_i(T) = \mathbf{Q}_{if} ,
\end{cases}
\label{ODEP}
\end{align}
subject to
\begin{align}
\begin{cases}
({\rm i}) \quad \ \ R + \mathbf{B}_i^T \mathbf{P}_i(t) \mathbf{B}_i >0 ,
\forall t \in [0, T] , \\
({\rm ii}) \quad \ R + \mathbf{B}_i^T \mathbf{P}_i(t) \mathbf{B}_i + \mathbf{B}_0^T \mathbf{P}_i(t) \mathbf{B}_0 >0 , \ \forall t\in[0, T] , \\
({\rm iii}) \quad I + \sum_{k=1}^N \left( R + \mathbf{B}_k^T \mathbf{P}_k(t)
\mathbf{B}_k \right)^{-1} \mathbf{B}_0^T \mathbf{P}_k(t) \mathbf{B}_0 \ \text{is invertible} , \ \forall t \in [0, T] ,
\end{cases} \label{Pcon}
\end{align}
where
\begin{align}
\mathbf{M}_{0} := - \Big[ I + \sum_{k=1}^N \left( R + \mathbf{B}_k^T \mathbf{P}_k \mathbf{B}_k
\right)^{-1} \mathbf{B}_0^T \mathbf{P}_k \mathbf{B}_0 \Big]^{-1}
\sum_{k=1}^N \left( R + \mathbf{B}_k^T \mathbf{P}_k \mathbf{B}_k \right)^{-1}
\widehat{\mathbf{B}}_k^T \mathbf{P}_k . \notag
\end{align}
In further analysis, if we just say $(\mathbf{P}_1, \cdots, \mathbf{P}_N )$ is a solution of \eqref{ODEP}, that means \eqref{Pcon} is in effect unless otherwise indicated. Condition \eqref{Pcon}--(ii) is not used in the vector field of the Riccati equation, but will play a role in the best response control problem later.
\begin{remark}
\label{rmk:uniqueP}
If the ODE system \eqref{ODEP} admits a solution $(\mathbf{P}_1, \cdots, \mathbf{P}_N )$ on $[0, T]$, then it is the unique solution since the vector field of the ODE system has a local Lipschitz property along the solution trajectory satisfying \eqref{Pcon}--\emph{(i)} and \emph{(iii)}.
\end{remark}
The following theorem gives a sufficient condition for the existence of feedback Nash strategies in terms of the Riccati equations \eqref{ODEP}. These strategies are called centralized due to the use of full state information by each player.
\begin{theorem}
\label{thm: FBNE}
If \eqref{ODEP} has a solution $(\mathbf{P}_1, \cdots, \mathbf{P}_N )$ on $[0, T]$, then the Nash game \eqref{X}--\eqref{Ji2} has a set of feedback Nash strategies $(\hat u_1, \cdots, \hat u_N)$ given by
\begin{align}
\label{uiP}
\hat u_i(t) = - [ R + \mathbf{B}_i^T \mathbf{P}_i(t)
\mathbf{B}_i ]^{-1}
[ \mathbf{B}_0^T \mathbf{P}_i(t) \mathbf{B}_0 \mathbf{M}_{0}(t) + \widehat{\mathbf{B}}_i^T \mathbf{P}_i(t) ] X(t) ,\ 1\leq i \leq N .
\end{align}
\end{theorem}
\begin{proof}
See Appendix~\ref{appendix:pflmPsubmat}.
\end{proof}
The best response control problem in the proof of Theorem \ref{thm: FBNE} amounts to LQ optimal control with indefinite weights in the cost. The HJB equation \eqref{HJBV} is only used for constructing \eqref{ODEP}. The rigorous proof of $\hat u_1$ as a best response strategy on $[t, T]$ given $(t, \mathbf{x}, \hat u_{-1})$ has been solely based on the Riccati equation system \eqref{ODEP} itself.
\section{Asymptotic solvability}
\label{sec:AS}
We start with a representation of the matrix $\mathbf{P}_i$ if the ODE system \eqref{ODEP} has a solution.
Write the $Nn \times Nn$ identity matrix $I_{Nn}$ as
$I_{Nn} = \diag[I_n, I_n, \cdots, I_n]$.
Let $J_{ij}$ denote the matrix obtained by exchanging the $i$th and $j$th rows of submatrices in $I_{Nn}$.
\begin{lemma}
\label{lm:Psubmat}
Suppose \eqref{ODEP} has a solution $(\mathbf{P}_1, \cdots,
\mathbf{P}_N)$ on $[0, T]$.
Then $\mathbf{P}_i$, $1\leq i \leq N$, have the representation
\begin{align}
\label{Psubmat}
\mathbf{P}_1 = \begin{bmatrix}
\Pi_1^{N} & \Pi_2^{N} & \Pi_2^{N} & \cdots & \Pi_2^{N} \\
\Pi_2^{NT} & \Pi_3^N & \Pi_4^N & \cdots & \Pi_4^N \\
\Pi_2^{NT} & \Pi_4^N & \Pi_3^N & \cdots & \Pi_4^N \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
\Pi_2^{NT} & \Pi_4^N & \Pi_4^N & \cdots & \Pi_3^N \\
\end{bmatrix} , \quad
\mathbf{P}_i = J_{1i}^T \mathbf{P}_1 J_{1i} , \quad \forall
2\leq i \leq N,
\end{align}
where $\Pi_1^N(t)$, $\Pi_3^N(t)$, $\Pi_4^N(t) \in \mathcal{S}^{n}$,
and $\Pi_2^N(t)\in \mathbb{R}^{n\times n}$.
\end{lemma}
\begin{proof}
See Appendix~\ref{appendix:pflmPsubmat}.
\end{proof}
Following the route in \cite{HZ2020}, we introduce the notion of asymptotic solvability of the Nash game \eqref{Xi}--\eqref{Ji} (with $D = D_0 = 0$).
\begin{definition}
\label{def:AS}
The Nash game \eqref{Xi}--\eqref{Ji} is asymptotically solvable if there exist $N_0>0$ and $c_0>0$ such that the ODE system \eqref{ODEP}--\eqref{Pcon} has a solution $(\mathbf{P}_1, \cdots, \mathbf{P}_N)$ on $[0, T]$ for all $N\geq N_0$, and that
\begin{align}
& \sup_{N\geq N_0} \sup_{0\leq t \leq T} ( |\Pi_1^N| + N|\Pi_2^N| + N^2|\Pi_3^N|
+ N^2|\Pi_4^N| ) < \infty , \label{AS2} \\
& R + \mathbf{B}_i^T \mathbf{P}_i \mathbf{B}_i \ge c_0 I , \
\ \forall t\in[0, T] ,\ \forall N\geq N_0,\label{AS3} \\
& I + \sum_{k=1}^N \left( R + \mathbf{B}_k^T \mathbf{P}_k \mathbf{B}_k \right)^{-1} \mathbf{B}_0^T \mathbf{P}_k \mathbf{B}_0 \ \text{is invertible},\ \forall t \in [0, T] , \ \forall N \geq N_0 .
\label{AS4}
\end{align}
\end{definition}
\begin{remark}
The conditions \eqref{AS2}--\eqref{AS3} imply that
\begin{align}
R + \mathbf{B}_i^T \mathbf{P}_i(t) \mathbf{B}_i + \mathbf{B}_0^T \mathbf{P}_i(t) \mathbf{B}_0 \ge (c_0/2) I, \quad \forall t,\notag
\end{align}
as long as a sufficiently large $N_0$ is chosen.
\end{remark}
Define the mapping
$\mathcal{R}_1: \mathcal{S}^n \to \mathcal{S}^n$ by
\begin{align}
\mathcal{R}_1(Z) = R + B_1^T Z B_1 , \quad \mbox{for} \ Z \in \mathcal{S}^n . \notag
\end{align}
For $\Lambda_k\in \mathcal{S}^n$, $k=1,3,4$, and $\Lambda_2\in\mathbb{R}^{n\times n} $, we define the mappings:
\begin{align}
& \Psi_1(\Lambda_1) = \Lambda_1 B H B^T \Lambda_1 - \Lambda_1 A - A^T \Lambda_1 - Q , \notag \\
& \Psi_2(\Lambda_1, \Lambda_2) =
- \Lambda_1 G - \Lambda_2 (A+G) - A^T \Lambda_2
+ \Lambda_1 B H B^T \Lambda_2 + \Lambda_2 B H B^T \Lambda_1 \notag \\
& \qquad\qquad\qquad
+ \Lambda_2 B H B^T \Lambda_2
+ Q \Gamma , \notag \\
& \Psi_3 (\Lambda_1, \Lambda_2, \Lambda_3, \Lambda_4)
=
- \Lambda_3 A - A^T \Lambda_3- ( \Lambda_2^{T} + \Lambda_4 ) G
- G^T(\Lambda_2 + \Lambda_4 ) \notag \\
& \hspace{3.3cm} - (\Lambda_1 + \Lambda_2^T ) B H B_0^T(\Lambda_1 + \Lambda_2 + \Lambda_2^T + \Lambda_4 ) B_0 H B^T (\Lambda_1 + \Lambda_2 ) \notag \\
&\hspace{3.3cm} + \Lambda_3 B H B^T \Lambda_1
+ \Lambda_1 B H B^T \Lambda_3
+ \Lambda_4 B H B^T \Lambda_2 \notag \\
& \hspace{3.3cm}
+ \Lambda_2^T B H B^T ( \Lambda_2 + \Lambda_4)
- \Lambda_1 B H B_1^T \Lambda_3 B_1 H B^T \Lambda_1
- \Gamma^T Q \Gamma , \notag \\
& \Psi_4 (\Lambda_1, \Lambda_2, \Lambda_4)
=- \Lambda_4 A - A^T \Lambda_4- ( \Lambda_2^T + \Lambda_4 ) G
- G^T ( \Lambda_2 + \Lambda_4 )
- \Gamma^T Q \Gamma \notag \\
& \hspace{2.7cm}
- (\Lambda_1 + \Lambda_2^T ) B H B_0^T
( \Lambda_1 + \Lambda_2 + \Lambda_2^T + \Lambda_4 ) B_0 H B^T ( \Lambda_1 + \Lambda_2 ) \notag \\
& \hspace{2.7cm}
+ \Lambda_4 B H B^T ( \Lambda_1 + \Lambda_2 )
+ (\Lambda_1 + \Lambda_2^T ) B H B^T \Lambda_4
+ \Lambda_2^T B H B^T \Lambda_2 , \notag
\end{align}
where we denote $H = ( \mathcal{R}_1(\Lambda_1) )^{-1}$ provided that the inverse matrix exists. It is clear that $\Psi_k$, $k=1,3,4$, are
$\mathcal{S}^n$-valued.
We introduce the following ODE system
\begin{align}
& \begin{cases}
\dot \Lambda_1 = \Psi_1(\Lambda_1) , \\
\Lambda_1 (T) = Q_f , \quad
\mathcal{R}_1(\Lambda_1(t) ) >0, \quad \forall t\in[0, T] ,
\end{cases} \label{ODELam1} \\
& \dot\Lambda_2 = \Psi_2(\Lambda_1, \Lambda_2) , \qquad
\Lambda_2 (T) = - Q_f \Gamma_f , \label{ODELam2} \\
& \dot \Lambda_3 = \Psi_3(\Lambda_1, \Lambda_2, \Lambda_3, \Lambda_4) , \qquad
\Lambda_3 (T) = \Gamma_f^T Q_f \Gamma_f ,
\label{ODELam3} \\
& \dot \Lambda_4 =
\Psi_4(\Lambda_1, \Lambda_2, \Lambda_4 ) , \qquad
\Lambda_4 (T) = \Gamma_f^T Q_f \Gamma_f . \label{ODELam4}
\end{align}
\begin{remark}
\label{rmk:solLam}
Note that \eqref{ODELam1} is the Riccati equation associated with an optimal control problem with controlled diffusion.
If \eqref{ODELam1}--\eqref{ODELam2} admits a solution $(\Lambda_1, \Lambda_2)$, substituting $(\Lambda_1, \Lambda_2)$ into
\eqref{ODELam3}--\eqref{ODELam4} gives a first order linear ODE system of
$(\Lambda_3, \Lambda_4)$, which then admits a unique solution on $[0, T]$.
\end{remark}
\begin{remark}
If $B_1 = 0$ and \eqref{ODELam1}--\eqref{ODELam2} has a solution on $[0,T]$, from \eqref{ODELam3}--\eqref{ODELam4} we obtain a first order linear homogeneous ODE of $\Lambda_3 - \Lambda_4$ with zero terminal condition $\Lambda_3(T) - \Lambda_4(T) = 0$, which implies that $\Lambda_3 - \Lambda_4 = 0$ on $[0, T]$. Such a representation by three submatrices is similar to \cite[Theorem 3]{HZ2020}.
\end{remark}
The following theorem characterizes asymptotic solvability of the Nash game \eqref{Xi}--\eqref{Ji} in terms of the low-dimensional ODE system \eqref{ODELam1}--\eqref{ODELam2}. The proof is postponed near the end of this section.
\begin{theorem}
\label{thm:NSAS}
The Nash game \eqref{Xi}--\eqref{Ji} has asymptotic solvability
if and only if \eqref{ODELam1}--\eqref{ODELam2} has a solution $(\Lambda_1, \Lambda_2)$ on $[0, T]$.
\end{theorem}
Following the rescaling method in \cite{HY2020b,HZ2020,MH2020},
we define
\begin{align}
\Lambda_1^N(t)= \Pi_1^N(t), \
\Lambda_2^N(t) = N\Pi_2^N(t) , \
\Lambda_3^N(t) = N^2 \Pi_3^N(t) , \
\Lambda_4^N(t)=N^2\Pi_4^N(t) . \label{PLam}
\end{align}
We introduce the following ODE system for $(\Lambda_1^N, \cdots, \Lambda_4^N)$:
\begin{align}
& \begin{cases}
\dot \Lambda_1^N = \Psi_1(\Lambda_1^N) + g_1^N , \\
\Lambda_1^{N} (T) = (I - \Gamma_f^T/N ) Q_f (I - \Gamma_f/N ) , \quad
\mathcal{R}_1(\Lambda_1^N(t)) >0, \quad \forall t\in[0, T] ,
\end{cases} \label{ODELam1N} \\
& \begin{cases}
\dot\Lambda_2^N = \Psi_2(\Lambda_1^N, \Lambda_2^N)
+ g_2^N , \\
\Lambda^N_2 (T) = - (I - \Gamma_f^T/N ) Q_f \Gamma_f ,
\end{cases} \label{ODELam2N} \\
&\begin{cases}
\dot \Lambda^N_3 = \Psi_3(\Lambda_1^N, \Lambda_2^N, \Lambda_3^N, \Lambda_4^N) + g_3^N , \\
\Lambda_3^N (T) = \Gamma_f^T Q_f \Gamma_f ,
\end{cases}\label{ODELam3N} \\
& \begin{cases}
\dot \Lambda_4^N =
\Psi_4(\Lambda_1^N, \Lambda_2^N, \Lambda_4^N )
+ g_4^N , \\
\Lambda_4^N (T) = \Gamma_f^T Q_f \Gamma_f ,
\end{cases} \label{ODELam4N}
\end{align}
where $g_k^N$, $1\leq k \leq 4$, are perturbation terms. We have
\begin{align}
g_1^N =& - [ \Lambda_2^N B K^N B^T S_{12}^N
+ S_{12}^{NT} B K^{NT} B^T \Lambda_2^{NT}
] (N-1)/N^3 \notag \\
&
+ [ \Lambda_2^N B H^N B^T \Lambda_2^N + \Lambda_2^{NT} B H^N B^T \Lambda_2^{NT} ] (N-1)/N^2
\notag \\
& - [ S_{12}^{NT} B K^{NT}/N - \Lambda_2^{NT} B H^N ] B_1^T \Lambda_3^N B_1 [ K^N B^T S_{12}^N/N - H^N B^T \Lambda_2^N ]
(N-1)/N^4 \notag \\
& - [ \Lambda_1^N G + G^T \Lambda_1^N ]/N
- [ \Lambda_2^N G + G^T \Lambda_2^{NT} ] (N-1)/N^2
\notag \\
& - S_{12}^{NT} B F^N B^T S_{12}^N /N^2 - ( \Gamma^T Q \Gamma/N - \Gamma^T Q - Q \Gamma )/N \notag
\end{align}
and
\begin{align}
& H^N = (R+ B_1^T \Lambda_1^N B_1 )^{-1} , \notag \\
&S^N = \Lambda_1^N + ( \Lambda_2^N + \Lambda_2^{NT} )(N-1)/N
+ [ \Lambda_3^N
+ \Lambda_4^N(N-2)] (N-1)/N^2 , \notag \\
&S_{12}^N = \Lambda_1^N + \Lambda_2^N (N-1)/N , \notag \\
&S_{34}^N = \Lambda_3^N/N^2 + \Lambda_4^N (N-2)/N^2 , \notag\\
& K^N = H^N B_0^T S^N B_0 (I + H^N B_0^T S^N B_0/N )^{-1} H^N , \notag \\
& F^N = H^N (I + B_0^T S^N B_0 H^N /N )^{-1}
( B_0^T S^N B_0 + B_0^T S^N B_0 H^N B_0^T S^N B_0 / N^2 )
\cdot \notag \\
&\qquad\quad (I + H^N B_0^T S^N B_0/N )^{-1} H^N . \notag
\end{align}
The other terms $g_k^N$, $k=2,3,4$, are listed in Appendix \ref{appendix:gdN}. They depend on $S_{34}^N$ above.
The mappings $g_k^N$, $1\le k\le 4$, are defined for
$\Lambda^N_k\in \mathcal{S}^n$, $k=1,3,4$, and $\Lambda^N_2\in\mathbb{R}^{n\times n} $. If \eqref{ODELam1N}--\eqref{ODELam4N} has a solution on $[0,T]$, then $\Lambda^N_k(t)$ is $\mathcal{S}^n$-valued for $k=1, 3, 4$.
The ODE system \eqref{ODELam1N}--\eqref{ODELam4N} is essentially derived from \eqref{ODEP} by use of the new variables \eqref{PLam}. However, \eqref{ODELam1N}--\eqref{ODELam4N} can stand alone without being immediately related to \eqref{ODEP}. If $(\Lambda_1^N, \cdots, \Lambda_4^N)$ is a solution, the inverse
$ (I + H^N B_0^T S^N B_0/N )^{-1}$ necessarily exists for all $t\in[0, T]$; such a solution is unique.
For
$\Lambda^N_k\in \mathcal{S}^n$, $k=1,3,4$, and $\Lambda^N_2\in\mathbb{R}^{n\times n} $, define the mappings
\begin{align}
\xi(\Lambda_1^N, \Lambda_2^N, \Lambda_3^N, \Lambda^N_4)=& \mathcal{R}_1 ( \Lambda_1^N ) + B_0^T [\Lambda_1^N + ( \Lambda_2^N + \Lambda_2^{NT} ) (N-1)/N \label{zetaL14} \\
&\hskip 0cm + \Lambda_3^N(N-1)/N^2
+ \Lambda_4^N (N-1)(N-2)/N^2 ] B_0 /N^2, \nonumber\\
\xi_0(\Lambda_1^N, \Lambda_2^N, \Lambda_3^N, \Lambda^N_4)= & I + \left( \mathcal{R}_1 ( \Lambda_1^N ) \right)^{-1}
B_0^T [\Lambda_1^N + ( \Lambda_2^N + \Lambda_2^{NT} ) (N-1)/N \label{xi0Lam14} \\
&+ \Lambda_3^N(N-1)/N^2 + \Lambda_4^N (N-1)(N-2)/N^2 ] B_0 /N. \notag
\end{align}
It is easy to show that
\begin{align}\label{IHxi0}
I + H^N B_0^T S^N B_0/N = \xi_0(\Lambda_1^N, \Lambda_2^N, \Lambda_3^N, \Lambda^N_4) .
\end{align}
\begin{lemma}
\label{lm:solPLam}
\emph{(i)} Suppose \eqref{ODEP}--\eqref{Pcon} has a solution $(\mathbf{P}_1, \cdots, \mathbf{P}_N)$ on $[0, T]$, and let
$(\Lambda_1^N, \Lambda_2^N, \Lambda_3^N, \Lambda^N_4)$ be defined using \eqref{Psubmat} and \eqref{PLam}. Then $(\Lambda_1^N, \Lambda_2^N,\Lambda_3^N, \Lambda^N_4 )$ satisfies
\eqref{ODELam1N}--\eqref{ODELam4N}.
\emph{(ii)} Conversely, if \eqref{ODELam1N}--\eqref{ODELam4N} admits a solution $(\Lambda_1^N, \Lambda_2^N,\Lambda_3^N, \Lambda^N_4 )$ on $[0, T]$, and such a solution further satisfies
\begin{align}
\xi(\Lambda_1^N(t), \Lambda_2^N(t), \Lambda_3^N(t), \Lambda^N_4(t)) >0 \nonumber
\end{align}
for all $t\in [0,T]$, then \eqref{ODEP}--\eqref{Pcon} has a solution $(\mathbf{P}_1, \cdots, \mathbf{P}_N)$ on $[0, T]$. Moreover, $\mathbf{P}_i$ may be determined in terms of the above $(\Lambda_1^N, \Lambda_2^N,\Lambda_3^N, \Lambda^N_4)$ using \eqref{Psubmat} .
\end{lemma}
\begin{proof}
(i) By \eqref{Psubmat} and \eqref{PLam}, we have
\begin{align}
R + \mathbf{B}_i^T \mathbf{P}_i \mathbf{B}_i
= \mathcal{R}_1 (\Lambda_1^N ) .
\label{R1(P)=R1(Lam)}
\end{align}
By condition \eqref{Pcon}--(i), $R + \mathbf{B}_i^T \mathbf{P}_i \mathbf{B}_i>0$. Therefore, $ \mathcal{R}_1( \Lambda_1^N(t) )>0$ on $[0, T]$.
It can be shown that
\begin{align}
\label{invP=invLam}
& I + \sum_{k=1}^N \left( R + \mathbf{B}_k^T \mathbf{P}_k(t)
\mathbf{B}_k \right)^{-1} \mathbf{B}_0^T \mathbf{P}_k(t) \mathbf{B}_0
=\xi_0(\Lambda_1^N(t), \Lambda_2^N(t), \Lambda_3^N(t), \Lambda^N_4(t)).
\end{align}
We substitute \eqref{Psubmat} into \eqref{ODEP} and change to the variables $\Lambda_k^N$, $1\le k\le 4$, to verify the equalities \eqref{ODELam1N}--\eqref{ODELam4N}, for which the inverse $(I + H^N B_0^T S^N B_0/N )^{-1}$ exists by condition \eqref{Pcon}--(iii), \eqref{IHxi0} and \eqref{invP=invLam}.
(ii) If \eqref{ODELam1N}--\eqref{ODELam4N} admits a solution $(\Lambda_1^N, \Lambda_2^N,\Lambda_3^N, \Lambda^N_4 )$ on $[0, T]$, let $\mathbf{P}_i$ be defined by \eqref{Psubmat} and \eqref{PLam}.
By $\mathcal{R}_1(\Lambda_1^N)>0$ in \eqref{ODELam1N}, we have
$R+\mathbf{B}_i^T \mathbf{P}_i \mathbf{B}_i >0$.
We can verify
\begin{align}
& R + \mathbf{B}_i^T \mathbf{P}_i \mathbf{B}_i + \mathbf{B}_0^T \mathbf{P}_i \mathbf{B}_0= \xi(\Lambda_1^N, \Lambda_2^N, \Lambda_3^N, \Lambda^N_4) \label{R2(P)=R2(Lam)}
\end{align}
so that $R + \mathbf{B}_i^T \mathbf{P}_i \mathbf{B}_i + \mathbf{B}_0^T \mathbf{P}_i \mathbf{B}_0 >0$ for all $t\in [0,T]$.
Note that
$(I + H^N B_0^T S^N B_0/N )^{-1}$ in \eqref{ODELam1N}--\eqref{ODELam4N}
exists for all $t\in [0,T]$. Recalling \eqref{xi0Lam14}--\eqref{IHxi0} and \eqref{invP=invLam}, we see that
\begin{align}
I + \sum_{k=1}^N \left( R + \mathbf{B}_k^T \mathbf{P}_k(t)
\mathbf{B}_k \right)^{-1} \mathbf{B}_0^T \mathbf{P}_k(t) \mathbf{B}_0 \notag
\end{align}
is invertible for all $t\in [0,T]$.
Now it is straightforward to verify that $(\mathbf{P}_1, \cdots, \mathbf{P}_N)$ defined above solves \eqref{ODEP} subject to \eqref{Pcon}.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:NSAS}]
(i) -- Necessity. Suppose the game \eqref{Xi}--\eqref{Ji} has asymptotic solvability, where $N_0$ and $c_0>0$ have been selected in \eqref{AS2}--\eqref{AS4}.
By Lemma \ref{lm:solPLam}--(i), for all $N\ge N_0$, \eqref{ODELam1N}--\eqref{ODELam4N} has a solution $(\Lambda_1^N, \Lambda_2^N,\Lambda_3^N, \Lambda^N_4 )$ on $[0,T]$, and by \eqref{AS2}--\eqref{AS3} and \eqref{PLam}, we have
\begin{align}
& \sup_{N\geq N_0} \sup_{0\leq t \leq T} ( |\Lambda_1^N| + |\Lambda_2^N| + |\Lambda_3^N|
+ |\Lambda_4^N| ) < \infty , \label{LamB2} \\
& \mathcal{R}_1 ( \Lambda_1^N(t) )\ge c_0 I , \quad \forall t\in [0, T] , \quad \forall N\geq N_0 .
\label{LamB3}
\end{align}
We write \eqref{ODELam1N} in the integral form
\begin{align}
& \Lambda_1^N(t) = \Lambda_1^N(T) - \int_t^T [\Psi_1(\Lambda_1^N) + g_1^N] d \tau , \notag
\end{align}
and do the same for $\Lambda_2^N$, $\Lambda_3^N$ and $\Lambda_4^N$. By
\eqref{LamB2}--\eqref{LamB3} we obtain $\sup_{0\le t\le T, k\le 4}|g_k^N| = O(1/N)$.
Then the functions $\{(\Lambda_1^N(\cdot), \Lambda_2^N(\cdot)),\Lambda_3^N(\cdot)), \Lambda_4^N(\cdot)) \}_{N\geq N_0}$ are uniformly bounded and equicontinuous on $[0, T]$.
By Arzel\`{a}--Ascoli theorem~\cite{Y1980}, there exists a subsequence
$\{ ( \Lambda_1^{N_j}(\cdot) , \Lambda_2^{N_j}(\cdot),\Lambda_3^{N_j}(\cdot) ,\Lambda_4^{N_j}(\cdot) ) \}_{j\geq 1}$ that converges to
$(\Lambda_1^\ast, \Lambda_2^\ast, \Lambda_3^\ast , \Lambda_4^\ast )$ uniformly on $[0, T]$ as $j\to\infty$.
It is easy to see that $(\Lambda_1^\ast, \Lambda_2^\ast, \Lambda_3^\ast, \Lambda_4^\ast )$ solves the system \eqref{ODELam1}--\eqref{ODELam4} and $\mathcal{R}_1 ( \Lambda_1^\ast(t) ) \geq c_0 I$ for all $t\in [0,T]$.
(ii) --
Sufficiency.
Suppose \eqref{ODELam1}--\eqref{ODELam2} has a solution so that we can obtain $(\Lambda_1, \Lambda_2, \Lambda_3, \Lambda_4)$ from
\eqref{ODELam1}--\eqref{ODELam4}.
We proceed to check the solution of \eqref{ODELam1N}--\eqref{ODELam4N}, which now stands alone without using \eqref{ODEP}.
Following the method in the sufficiency proof of Theorem 3.1 in \cite{HY2020b}, we specify a thin ``tube", surrounding the solution trajectory $(\Lambda_1, \Lambda_2, \Lambda_3, \Lambda_4)$, $t\in [0,T]$, of this form:
\begin{align}
\label{tubeC}
{\cal C}=& \{(t, Z_1, Z_2, Z_3 ,Z_4)\in [0,T]\times \mathcal{S}^n\times
\mathbb{R}^{n\times n}\times \mathcal{S}^n\times \mathcal{S}^n:\\
&\qquad \mbox{$\sum_{k\le 4}$}|Z_k-\Lambda_k(t)|<\delta_0 \}, \nonumber
\end{align}
where $\delta_0>0$ is a sufficiently small but fixed constant, and next show that
for all sufficiently large $N$, the solution of \eqref{ODELam1N}--\eqref{ODELam4N} starting from the terminal condition will always remain in this tube. This establishes the global existence of solutions on $[0,T]$,
and the detailed steps are exactly the same as in \cite{HY2020b}. Specifically,
it can be shown that there exist $\hat N_0$ and $c_0>0$ such that we have the following: (a) \eqref{ODELam1N}--\eqref{ODELam4N}
has a solution $(\Lambda_1^N, \Lambda_2^N,\Lambda_3^N, \Lambda_4^N) $ remaining in the tube \eqref{tubeC} on $[0,T]$ for all $N\ge \hat N_0$; (b)
\begin{align}
& \sup_{N\geq \hat N_0} \sup_{0\leq t \leq T} ( |\Lambda_1^N| + |\Lambda_2^N| + |\Lambda_3^N|
+ |\Lambda_4^N| ) < \infty , \label{LamB2s} \\
& \mathcal{R}_1 ( \Lambda_1^N(t) ) \ge c_0 I, \quad \forall t\in [0, T] , \quad \forall N\geq \hat N_0 ;
\label{LamB3s}
\end{align}
(c) for $\xi_0(\cdot)$ defined in \eqref{xi0Lam14}, $\xi_0(\Lambda_1^N(t), \Lambda_2^N(t), \Lambda_3^N(t), \Lambda^N_4(t)) $ is invertible for all $ N\ge \hat N_0$,
so that the term $(I + H^N B_0^T S^N B_0/N )^{-1}$ in \eqref{ODELam1N}--\eqref{ODELam4N}
is well defined.
If $\hat N_1>\hat N_0$ is sufficiently large, by \eqref{LamB3s} we can ensure that
\begin{align}\xi(\Lambda_1^N(t), \Lambda_2^N(t), \Lambda_3^N(t), \Lambda^N_4(t))
>(c_0/2) I, \quad \forall t\in [0,T], \ \forall N\ge \hat N_1, \label{R1c02I}
\end{align}
where $\xi(\cdot)$ is defined in \eqref{zetaL14}.
By \eqref{R1c02I}, we apply Lemma \ref{lm:solPLam}--(ii) to obtain $(\mathbf{P}_1, \cdots, \mathbf{P}_N)$ for \eqref{ODEP} whenever $N\ge \hat N_1$. By \eqref{R1c02I} and \eqref{invP=invLam}, we see that \eqref{AS4} holds for all $N\ge \hat N_1$.
Subsequently, asymptotic solvability holds.
\end{proof}
\begin{corollary}
\label{cor:NSAS:LamN-Lam}
If \eqref{ODELam1}--\eqref{ODELam2} has a solution $(\Lambda_1, \Lambda_2)$ on $[0, T]$, then there exists $\hat N_0>0$ such that for each $N\geq \hat N_0$, \eqref{ODELam1N}--\eqref{ODELam4N} has a solution
$(\Lambda_1^N, \Lambda_2^N, \Lambda_3^N, \Lambda_4^N)$ on
$[0, T]$ and moreover,
$ \sup_{t\in[0, T], k\le 4}
|\Lambda_k^N(t) - \Lambda_k(t)| =O(1/N). $
\end{corollary}
\begin{proof}
Since \eqref{ODELam1}--\eqref{ODELam4} has a solution on $[0, T]$, we take a sufficiently thin tube as in \eqref{tubeC}.
Then by the sufficiency proof of Theorem~\ref{thm:NSAS}, there exists $\hat N_0>0$ such that for each $N\geq \hat N_0$, \eqref{ODELam1N}--\eqref{ODELam4N} has a solution
$(\Lambda_1^N, \Lambda_2^N, \Lambda_3^N, \Lambda_4^N)$ on
$[0, T]$, which is always within the tube.
The desired result then follows from Gr\"{o}nwall's lemma. See similar estimates in \cite[Corollary 3.1]{HY2020b}.
\end{proof}
\section{Decentralized strategies}
\label{sec:DecentralStrats}
By Theorem~\ref{thm:NSAS}, the Nash game \eqref{Xi}--\eqref{Ji} has asymptotic solvability if and only if \eqref{ODELam1}--\eqref{ODELam2} admits a solution $(\Lambda_1, \Lambda_2)$ on $[0, T]$.
We introduce the following assumptions:
\begin{assumption}
\label{assm:solLam}
The ODE system~\eqref{ODELam1}--\eqref{ODELam2} has a solution $(\Lambda_1, \Lambda_2)$ on $[0, T]$.
\end{assumption}
For $X_i(0)$, denote the covariance matrix
$\Sigma_0^i= \mathbb{E}\{ [X_i(0)-\mathbb{E}X_i(0)] [X_i(0)-\mathbb{E}X_i(0)]^T\}$.
\begin{assumption}
\label{assm:initialX}
The initial states $\{X_i(0), i\geq 0\}$ are independent. There exist $\mu_0\in\mathbb{R}^n$ and a constant $C_\Sigma$, both independent of $N$, such that $\mathbb{E}X_i(0) = \mu_0$ and
$|\Sigma_0^i| \leq C_\Sigma$ for all $ i$.
\end{assumption}
Under Assumption \ref{assm:solLam}, the sufficiency proof of Theorem \ref{thm:NSAS} shows that there exists $\hat N_1$ such that
\eqref{ODELam1N}--\eqref{ODELam4N} has a solution $(\Lambda_1^N, \Lambda_2^N, \Lambda_3^N, \Lambda_4^N)$ for all $N\ge \hat N_1$. By Lemma~\ref{lm:solPLam}--(ii), we determine $\mathbf{P}$ in~\eqref{ODEP} by using \eqref{Psubmat} and~\eqref{PLam}, and obtain the Nash equilibrium strategies
\eqref{uiP}, which are displayed below:
\begin{align}
\hat u_i(t) = - [ R + B_1^T \Lambda_1^N(t) B_1 ]^{-1}
[ \mathbf{B}_0^T \mathbf{P}_i(t) \mathbf{B}_0 \mathbf{M}_{0}(t) + \widehat{\mathbf{B}}_i^T \mathbf{P}_i(t) ] X(t) ,\ 1\leq i \leq N .
\notag
\end{align}
Throughout this section we assume $N\ge \hat N_1$.
Before further analysis we introduce some notation:
\begin{align}
& \Theta(t) = ( \mathcal{R}_1(\Lambda_1(t)) )^{-1} B^T \Lambda_1(t) ,
\quad \Theta_1(t) = ( \mathcal{R}_1(\Lambda_1(t)) )^{-1} B^T \Lambda_2(t) , \notag \\
& \widehat{\Theta}(t) = I_N \otimes \Theta(t) ,
\quad \widehat{\Theta}_1 = \mathbf{1}_{N\times 1} \otimes \Theta_1 , \notag \\
& \mathbf{e}_i = ( e_i^N \otimes I_n )^T = (0, \cdots, 0, I_n, 0, \cdots, 0 ) \in \mathbb{R}^{n \times Nn} , \notag \\
& \widehat{\mathbf{B}} = (\widehat{\mathbf{B}}_1, \cdots, \widehat{\mathbf{B}}_N ) \in \mathbb{R}^{Nn\times N n_1}, \quad
\mathbf{I} = (I_n, \cdots, I_n) \in \mathbb{R}^{n\times Nn} . \notag
\end{align}
By using the closed-loop dynamics under $(\hat u_1, \cdots, \hat u_N)$, we consider the SDE of $X^{(N)}$ and let $N\to \infty$.
This gives
the mean field limit state $\overline{X}$ as follows:
\begin{align}
\label{dbarX}
& d \overline{X} = ( A + G - B( \Theta + \Theta_1 ) ) \overline{X} dt - B_0 (\Theta + \Theta_1 ) \overline{X} d W_0 , \quad
t \geq 0 ,
\end{align}
where $\overline{X}(0) = \mu_0 $.
We denote the set of decentralized feedback strategies
\begin{align}
\check{u}_i (t)
= - \Theta(t) \mathbf{e}_i X(t) - \Theta_1(t) \overline{X}(t) , \quad
1 \leq i \leq N .
\label{checkui}
\end{align}
The state dynamics under the decentralized strategies \eqref{checkui} follows
\begin{align}
d X(t) = & ( \mathbf{A} X
- \widehat{\mathbf{B}}( \widehat\Theta X + \widehat\Theta_1 \overline{X} ) ) dt
- \sum_{i=1}^N \mathbf{B}_i (\Theta X_i + \Theta_1 \overline{X}) d W_i \notag \\
& - \mathbf{B}_0 \sum_{i=1}^N (\Theta X_i + \Theta_1 \overline{X} ) d W_0 , \quad t\geq 0 , \notag
\end{align}
where the initial state $X(0)=(X_1^T(0), \cdots, X_N^T(0))^T$ is the same as in \eqref{X}.
Below we evaluate the cost with more general initial conditions.
When all the $N$ players take the decentralized strategies \eqref{checkui}, the cost of player $\mathcal{A}_i$ with initial condition $(X(t), \overline{X}(t))=(\mathbf{x}, \bar{x})$ is denoted by $\check{V}_i(t, \mathbf{x}, \bar{x})$, $t\in[0, T]$, $\mathbf{x} \in \mathbb{R}^{Nn}$, $\bar{x} \in \mathbb{R}^n$.
The Feynman--Kac formula~\cite[Sec. 1.3, 3.5]{P2009} gives the following equation that $\check{V}_i$ satisfies:
\begin{align}
\label{HJBcheckV}
\begin{cases}
- \frac{\partial \check{V}_i}{\partial t}
= \frac{\partial^T \check{V}_i}{\partial \mathbf{x}} ( \mathbf{A} \mathbf{x}
- \widehat{\mathbf{B}}( \widehat\Theta \mathbf{x} + \widehat\Theta_1 \bar{x} ) )
+ \frac{\partial^T \check{V}_i}{\partial \bar{x}} (A+G - B( \Theta + \Theta_1 ) ) \bar{x} \\
\hspace{1.3cm}
+ (1/2) ( \Theta \mathbf{I} \mathbf{x} + \mathbf{I} \widehat\Theta_1 \bar{x} )^T
\mathbf{B}_0^T \frac{\partial^2 \check{V}_i}{\partial \mathbf{x}^2 }
\mathbf{B}_0 ( \Theta \mathbf{I} \mathbf{x} + \mathbf{I} \widehat\Theta_1 \bar{x} ) \\
\hspace{1.3cm} + (1/2) ( ( \Theta + \Theta_1 ) \bar{x} )^T B_0^T\frac{\partial^2 \check{V}_i}{\partial \bar{x}^2} B_0
( \Theta + \Theta_1 ) \bar{x} \\
\hspace{1.3cm} + (1/2) \sum_{k=1}^N ( \Theta \mathbf{e}_k \mathbf{x} + \Theta_1 \bar{x} )^T \mathbf{B}_k^T \frac{\partial^2 \check{V}_i}{\partial \mathbf{x}^2}
\mathbf{B}_k (\Theta \mathbf{e}_k \mathbf{x} + \Theta_1 \bar{x} ) \\
\hspace{1.3cm}
+ ( \Theta \mathbf{I} \mathbf{x} + \mathbf{I} \widehat\Theta_1 \bar{x} )^T
\mathbf{B}_0^T \frac{\partial^2 \check{V}_i }{ \partial \mathbf{x} \partial \bar{x}} B_0 ( \Theta + \Theta_1 ) \bar{x} \\
\hspace{1.3cm} + ( \Theta \mathbf{e}_i \mathbf{x} + \Theta_1 \bar{x} )^T R
( \Theta \mathbf{e}_i \mathbf{x} + \Theta_1 \bar{x} )
+ \mathbf{x}^T \mathbf{Q}_i \mathbf{x} , \\
\check{V}_i ( T, \mathbf{x} ) = \mathbf{x}^T \mathbf{Q}_{if} \mathbf{x} .
\end{cases}
\end{align}
Assume $\check{V}_i(t, \mathbf{x}, \overline{x})$ takes the following form
\begin{align}
\label{checkVansatz}
\check{V}_i(t, \mathbf{x} , \overline{x}) = \mathbf{x}^T
\check{\mathbf{P}}^i_1(t) \mathbf{x} + 2 \mathbf{x}^T
\check{\mathbf{P}}^i_{12} (t) \overline{x} + \overline{x}^T
\check{\mathbf{P}}_2^i (t) \overline{x} , \quad 1\leq i \leq N .
\end{align}
Substituting \eqref{checkVansatz} into \eqref{HJBcheckV}
gives the following system of ODEs for
$\check{\mathbf{P}}_1^i$, $\check{\mathbf{P}}_{12}^i$
and $\check{\mathbf{P}}_2^i$:
\begin{align}
& \begin{cases}
- \frac{d}{dt}\check{\mathbf{P}}_1^i = \check{\mathbf{P}}_1^i
(\mathbf{A} - \widehat{\mathbf{B}} \widehat\Theta )
+ ( \mathbf{A} - \widehat{\mathbf{B}} \widehat\Theta )^T \check{\mathbf{P}}_1^i
+ \mathbf{I}^T \Theta^T \mathbf{B}_0^T \check{\mathbf{P}}^i_1 \mathbf{B}_0 \Theta \mathbf{I} \\
\hspace{1.55cm}
+ \sum_{k=1}^N \mathbf{e}_k^T \Theta^T \mathbf{B}_k^T
\check{\mathbf{P}}_1^i \mathbf{B}_k \Theta \mathbf{e}_k
+ \mathbf{e}_i^T \Theta^T R \Theta \mathbf{e}_i + \mathbf{Q}_i , \\
\check{\mathbf{P}}_1^i (T) = \mathbf{Q}_{if} ,
\end{cases} \label{ODEcheckP1} \\
& \begin{cases}
- \frac{d}{dt}\check{\mathbf{P}}_{12}^i = - \check{\mathbf{P}}_1^i \widehat{\mathbf{B}} \widehat\Theta_1 + (\mathbf{A} - \widehat{\mathbf{B}} \widehat\Theta )^T \check{\mathbf{P}}_{12}^i
+ \check{\mathbf{P}}_{12}^i (A+G - B(\Theta + \Theta_1 ) ) \\
\hspace{1.7cm}
+ \mathbf{I}^T \Theta^T \mathbf{B}_0^T \check{\mathbf{P}}_1^i
\mathbf{B}_0 \mathbf{I}\widehat\Theta_1
+ \sum_{k=1}^N \mathbf{e}_k^T \Theta^T \mathbf{B}_k^T
\check{\mathbf{P}}_1^i \mathbf{B}_k \Theta_1 \\
\hspace{1.7cm}
+ \mathbf{I}^T \Theta^T \mathbf{B}_0^T \check{\mathbf{P}}_{12}^i B_0 ( \Theta + \Theta_1 )
+ \mathbf{e}_i^T \Theta^T R \Theta_1 , \\
\check{\mathbf{P}}_{12}^i(T) = 0 ,
\end{cases} \label{ODEcheckP12} \\
&\begin{cases}
- \frac{d}{dt} \check{\mathbf{P}}_2^i = - \check{\mathbf{P}}_{12}^{iT} \widehat{\mathbf{B}} \widehat\Theta_1 - \widehat\Theta_1^T \widehat{\mathbf{B}}^T \check{\mathbf{P}}_{12}^i
+ \check{\mathbf{P}}_2^i (A+G - B(\Theta + \Theta_1)) \\
\hspace{1.55cm}
+ (A+G - B(\Theta + \Theta_1))^T \check{\mathbf{P}}_2^i
+ ( \mathbf{B}_0 \mathbf{I} \widehat\Theta_1 )^T \check{\mathbf{P}}_1^i
\mathbf{B}_0 \mathbf{I}\widehat\Theta_1 \\
\hspace{1.55cm}
+ \sum_{k=1}^N (\mathbf{B}_k \Theta_1 )^T \check{\mathbf{P}}_1^i \mathbf{B}_k \Theta_1
+ (\Theta + \Theta_1 )^T B_0^T \check{\mathbf{P}}_2^i B_0 (\Theta + \Theta_1 ) \\
\hspace{1.55cm}
+ (\mathbf{B}_0 \mathbf{I} \widehat{\Theta}_1 )^T \check{\mathbf{P}}_{12}^i
B_0 (\Theta + \Theta_1 )
+ ( B_0(\Theta + \Theta_1 ) )^T \check{\mathbf{P}}_{12}^{iT}
\mathbf{B}_0 \mathbf{I} \widehat{\Theta}_1 \\
\hspace{1.55cm}
+ \Theta_1^T R \Theta_1 , \\
\check{\mathbf{P}}^i_2(T) = 0 .
\end{cases} \label{ODEcheckP2}
\end{align}
\begin{remark}
\label{rmk:checkPsol}
\eqref{ODEcheckP1}--\eqref{ODEcheckP2} is a first order linear ODE system and admits a unique solution.
\end{remark}
We have the following submatrix partition of the matrices
$\check{\mathbf{P}}_1^i$, $\check{\mathbf{P}}_{12}^i$, $1\leq i\leq N$.
\begin{lemma}
\label{lm:checkPsubmat}
For \eqref{ODEcheckP1} and \eqref{ODEcheckP12}, the solution
$(\check{\mathbf{P}}^i_{1}, \check{\mathbf{P}}^i_{12})$,
$1\leq i \leq N$, has the representation
\begin{align}
\label{checkP1submat}
& \check{\mathbf{P}}^1_1
= \begin{bmatrix}
\check\Pi_1^N & \check\Pi_2^N & \check\Pi_2^N & \cdots & \check\Pi_2^N \\
\check\Pi_2^{NT} & \check\Pi_3^N & \check\Pi_4^N & \cdots & \check\Pi_4^N \\
\check\Pi_2^{NT} & \check\Pi_4^N & \check\Pi_3^N & \cdots & \check\Pi_4^N \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
\check\Pi_2^{NT} & \check\Pi_4^N & \check\Pi_4^N & \cdots & \check\Pi_3^N \\
\end{bmatrix} ,
\qquad
\check{\mathbf{P}}_1^i = J_{1i}^T \check{\mathbf{P}}_1^1 J_{1i},
\quad \forall 2\leq i \leq N , \\
& \check{\mathbf{P}}_{12}^1 = \begin{bmatrix} \check{\Pi}_{11}^{NT} , \ \check{\Pi}_{12}^{NT} ,\ \cdots , \ \check{\Pi}_{12}^{NT} \end{bmatrix}^T , \quad
\check{\mathbf{P}}_{12}^i = J_{1i}^T \check{\mathbf{P}}_{12}^1 ,
\quad \forall 2\leq i \leq N ,
\label{checkP12submat}
\end{align}
where $\check{\Pi}_1^N(t)$, $\check{\Pi}_3^N(t)$, $\check{\Pi}_4^N(t)\in \mathcal{S}^n$, and
$\check{\Pi}_2^N(t)$, $\check{\Pi}_{11}^N(t)$, $\check{\Pi}_{12}^N(t)\in \mathbb{R}^{n\times n}$.
\end{lemma}
\begin{proof}
The proof is similar to that of Lemma~\ref{lm:Psubmat} or \cite[Theorem 3]{HZ2020} and is omitted.
\end{proof}
We define the new variables:
\begin{align}
\begin{cases}
\check\Lambda_1^N = \check\Pi_1^N , \hspace{0.4cm}
\check\Lambda_2^N =N\check\Pi_2^N , \hspace{0.42cm}
\check\Lambda_3^N = N^2\check\Pi_3^N , \hspace{0.4cm}
\check\Lambda_4^N =N^2\check\Pi_4^N , \\
\check\Lambda_{11}^N =\check\Pi_{11}^N ,
\quad
\check\Lambda_{12}^N = N\check\Pi_{12}^N ,
\quad
\check\Lambda_{22}^N = \check{\mathbf{P}}_2^i.
\end{cases} \label{checkPLam}
\end{align}
Substituting~\eqref{checkP1submat}--\eqref{checkP12submat} into \eqref{HJBcheckV}
and next converting into the new variables in \eqref{checkPLam}, we derive
\begin{align}
& \begin{cases}
- \frac{d}{dt}\check{\Lambda}_1^N = \check\Lambda_1^N ( A - B \Theta )
+ ( A - B \Theta )^T \check\Lambda_1^N
+ \Theta^T ( R + B_1^T \check\Lambda_1^N B_1 ) \Theta \\
\hspace{1.5cm} + Q + \check{g}_1^N , \\
\check\Lambda_1^N(T) = (I - \Gamma_f^T/N) Q_f (I - \Gamma_f/N) ,
\end{cases} \label{ODEcheckLam1} \\
& \begin{cases}
- \frac{d}{dt}\check\Lambda_2^N = \check\Lambda_1^N G
+ \check\Lambda_2^N ( A + G - B \Theta )
+ ( A - B \Theta )^T \check\Lambda_2^N - Q \Gamma
+ \check{g}_2^N , \\
\check\Lambda_2^N (T) = - (I - \Gamma_f^T/N) Q_f \Gamma_f ,
\end{cases} \label{ODEcheckLam2}
\end{align}
\begin{align}
& \begin{cases}
- \frac{d}{dt}\check\Lambda_3^N =
\Theta^T [ B_1^T \check\Lambda_3^N B_1 + B_0^T (\check\Lambda_1^N + \check\Lambda_2^N
+ \check\Lambda_2^{NT} + \check\Lambda_4^N ) B_0 ] \Theta \\
\hspace{1.5cm}
+ ( \check\Lambda_2^{NT}
+ \check\Lambda_4^N ) G
+ G^T ( \check\Lambda_2^N + \check\Lambda_4^N )
+ \check\Lambda_3^N (A - B \Theta ) \\
\hspace{1.5cm}
+ ( A - B \Theta )^T \check\Lambda_3^N
+ \Gamma^T Q \Gamma + \check{g}_3^N , \\
\check\Lambda_3^N (T) = \Gamma_f^T Q_f \Gamma_f ,
\end{cases} \label{ODEcheckLam3}
\end{align}
\begin{align}
& \begin{cases}
- \frac{d}{dt}\check\Lambda_4^N = \check\Lambda_2^{NT} G
+ G^T \check\Lambda_2^N + \check\Lambda_4^N (A + G - B \Theta ) + (A + G - B \Theta )^T \check\Lambda_4^N \\
\hspace{1.65cm}
+ \Theta^T B_0^T (\check\Lambda_1^N + \check\Lambda_2^N
+ \check\Lambda_2^{NT} + \check\Lambda_4^N ) B_0 \Theta
+ \Gamma^T Q \Gamma + \check{g}_4^N , \\
\check\Lambda_4^N(T) = \Gamma_f^T Q_f \Gamma_f ,
\end{cases} \label{ODEcheckLam4}
\end{align}
\begin{align}
& \begin{cases}
- \frac{d}{dt}\check\Lambda_{11}^N =
\check{\Lambda}_{11}^N (A+G - B(\Theta + \Theta_1 ))
+ ( A - B \Theta )^T \check{\Lambda}_{11}^N \\
\hspace{1.6cm}
- ( \check\Lambda_1^N
+ \check\Lambda_2^N ) B \Theta_1
+ \Theta^T ( R + B_1^T \check\Lambda_1^N B_1 ) \Theta_1
+ \check{g}_{11}^N , \\
\check\Lambda_{11}^N(T) = 0 ,
\end{cases} \label{ODEcheckLam11}
\end{align}
\begin{align}
& \begin{cases}
- \frac{d}{dt}\check\Lambda_{12}^N
= - ( \check\Lambda_2^{NT} + \check\Lambda_4^N ) B \Theta_1
+ ( A + G - B \Theta )^T \check\Lambda_{12}^N
+ G^T \check\Lambda_{11}^N \\
\hspace{1.65cm}
+ \check\Lambda_{12}^N (A+G - B (\Theta + \Theta_1 ) ) \\
\hspace{1.65cm}
+ \Theta^T B_0^T ( \check\Lambda_1^N
+ \check\Lambda_2^N + \check\Lambda_2^{ NT}
+ \check\Lambda_4^N ) B_0 \Theta_1 \\
\hspace{1.65cm}
+ \Theta^T B_0^T (\check\Lambda_{11}^N
+ \check\Lambda_{12}^N ) B_0 (\Theta + \Theta_1 )
+ \check{g}_{12}^N , \\
\check\Lambda_{12}^N(T) = 0 ,
\end{cases} \label{ODEcheckLam12}
\end{align}
\begin{align}
& \begin{cases}
- \frac{d}{dt}\check\Lambda_{22}^N =
- ( \check{\Lambda}_{11}^N + \check{\Lambda}_{12}^N
+ \check\Lambda_{22}^N )^T B \Theta_1
- \Theta_1^T B^T ( \check\Lambda_{11}^N
+ \check\Lambda_{12}^N + \check\Lambda_{22}^N ) \\
\hspace{1.65cm}
+ \check\Lambda_{22}^N (A+G - B \Theta )
+ (A+G - B \Theta )^T \check\Lambda_{22}^N \\
\hspace{1.65cm}
+ \Theta^T B_0^T \check\Lambda_{22}^N B_0 \Theta
+ \Theta^T B_0^T ( \check\Lambda_{11}^N
+ \check\Lambda_{12}^N
+ \check\Lambda_{22}^N )^T B_0 \Theta_1 \\
\hspace{1.65cm}
+ \Theta_1^T B_0^T ( \check\Lambda_{11}^N
+ \check\Lambda_{12}^N + \check\Lambda_{22}^N ) B_0 \Theta
+ \Theta_1^T ( R + B_1^T \check\Lambda_1^N B_1 ) \Theta_1 \\
\hspace{1.65cm}
+ \Theta_1^T B_0^T (\check\Lambda_1^N + \check\Lambda_2^N
+ \check\Lambda_2^{NT} + \check\Lambda_4^N \\
\hspace{1.65cm}
+ \check\Lambda_{11}^N + \check\Lambda_{12}^N
+ \check\Lambda_{11}^{NT} + \check\Lambda_{12}^{NT}
+ \check\Lambda_{22}^N ) B_0 \Theta_1
+ \check{g}_{22}^N ,\\
\check\Lambda_{22}^N(T) = 0 .
\end{cases} \label{ODEcheckLam22}
\end{align}
The perturbation terms $\check{g}^N_1,\cdots, \check{g}^N_{22}$ are
listed in Appendix \ref{appendix:gdN}.
\begin{remark}
\label{rmk:solcheckLam}
Under Assumption~\ref{assm:solLam}, the system~\eqref{ODEcheckLam1}--\eqref{ODEcheckLam22} is a first order linear ODE system and admits a unique solution $(\check\Lambda_1^N, \cdots, \check\Lambda_{22}^N)$ on $[0, T]$.
\end{remark}
\begin{remark}
\label{rmk:bdcheckLam}
Let $\psi^N$ stand for any of the functions $\check\Lambda_1^N$, $\check\Lambda_2^N$, $\check\Lambda_3^N$, $\check\Lambda_4^N$,
$\check\Lambda_{11}^N$, $\check\Lambda_{12}^N$ and $\check\Lambda_{22}^N$. Due to the bounded coefficients in the ODE system
\eqref{ODEcheckLam1}--\eqref{ODEcheckLam22}, $\sup_{N\geq \hat{N}_1, 0 \leq t \leq T} |\psi^N| \leq C$ for some fixed constant $C$.
\end{remark}
\begin{remark}
\label{rmk:checkgO(1/N)}
Let $h^N$ stand for any of the functions $\check{g}_1^N$, $\check{g}_2^N$, $\check{g}_3^N$, $\check{g}_4^N$, $\check{g}_{11}^N$, $\check{g}_{12}^N$ and $\check{g}_{22}^N$. Then $\sup_{t\in [0, T]} |h^N(t)|=O(1/N)$.
\end{remark}
Let $(\check\Lambda_1^N, \cdots, \check\Lambda_{22}^N)$ be obtained from \eqref{ODEcheckLam1}--\eqref{ODEcheckLam22}.
By substituting \eqref{checkP1submat} into \eqref{checkVansatz}, which is further expressed in terms of $(\check\Lambda_1^N, \cdots, \check\Lambda_{22}^N)$ via \eqref{checkPLam}, we obtain an explicit representation of a player's cost when all the players take the set of decentralized strategies
$(\check u_1, \cdots, \check u_N)$ in \eqref{checkui}.
The cost of player $\mathcal{A}_i$ is
\begin{align}
J_i(\check{u}_i, \check{u}_{-i})
=& \mathbb{E}\Big[ \check{V}_i(0, X(0), \overline{X}(0))
\Big] \notag \\
= & \mathbb{E} \Big[ X^T(0) \check{\mathbf{P}}_1^i(0) X(0)
+ 2 X^T(0) \check{\mathbf{P}}_{12}^i(0) \overline{X}(0)
+ \overline{X}^T(0) \check{\mathbf{P}}_2^i(0) \overline{X}(0) \Big] . \label{checkJiP}
\end{align}
Denote $\mathcal{N}_{-i}=\{1, \cdots, N\}\setminus\{i\}$.
Under Assumption \ref{assm:initialX}, the first term on the right hand side of \eqref{checkJiP} is
\begin{align}
& \mathbb{E} \Big[ X^T(0) \check{\mathbf{P}}_1^i(0) X(0) \Big] \notag \\
= & \mathbb{E}\Big[ X_i^T(0) \check\Lambda_1^N (0) X_i(0)
+ (2/N) \sum_{j\in \mathcal{N}_{-i} } X_i^T(0) \check\Lambda_2^N(0) X_j(0)
\notag \\
& + \frac{1}{N^2}\sum_{j\in \mathcal{N}_{-i} } X_j^T(0) \check\Lambda_3^N(0) X_j(0)
+ \frac{1}{N^2}\sum_{{ j, k \in \mathcal{N}_{-i}, j \neq k } } X_j^T(0) \check\Lambda_4^N(0) X_k(0) \Big] \notag \\
= & \Tr[ \check\Lambda_1^N(0) \Sigma_0^i ]
+ (1/N^2) \sum_{j\in \mathcal{N}_{-i} } \Tr [ \check\Lambda_3^N(0) \Sigma_0^j] \notag \\
& + \mu_0^T [ \check\Lambda_1^N(0) + \check\Lambda_2^N(0) + \check\Lambda_2^{NT}(0) + \check\Lambda_4^N(0) ] \mu_0 \notag \\
& + \mu_0^T [ - (\check\Lambda_2^N(0) + \check\Lambda_2^{NT}(0)) /N + \check\Lambda_3^N(0) (N-1)/N^2 \notag \\ &\qquad
+ \check\Lambda_4^N(0) (2-3N)/N ] \mu_0.
\label{XcheckP1}
\end{align}
The second term on the right hand side of \eqref{checkJiP} can be written as
\begin{align}
& 2 \mathbb{E} \Big[ X^T(0) \check{\mathbf{P}}_{12}^i(0) \overline{X}(0) \Big] \label{XcheckP12} \\
=& \mu_0^T [ \check\Lambda_{11}^{N}(0)
+ \check\Lambda_{11}^{NT}(0)
+ \check\Lambda_{12}^{N}(0) + \check\Lambda_{12}^{NT}(0) ] \mu_0
- \mu_0^T [ \check\Lambda_{12}^N(0)
+ \check\Lambda_{12}^{NT}(0) ] \mu_0 /N . \notag
\end{align}
The third term on the right hand side of \eqref{checkJiP} can be written as
\begin{align}
\mathbb{E} [ \overline{X}^T(0) \check{\mathbf{P}}_2^i(0) \overline{X}(0) ]
= \mu_0^T \check\Lambda_{22}^N(0) \mu_0 . \label{XcheckP2}
\end{align}
Denote
\begin{align}
& \check{Y}^N := \check\Lambda_1^N + \check\Lambda_2^N
+ \check\Lambda_2^{NT} + \check\Lambda_4^N
+ \check\Lambda_{11}^N
+ \check\Lambda_{11}^{NT} + \check\Lambda_{12}^N
+ \check\Lambda_{12}^{NT}
+ \check\Lambda_{22}^N .
\label{checkYN}
\end{align}
Substituting \eqref{XcheckP1}, \eqref{XcheckP12} and \eqref{XcheckP2} into \eqref{checkJiP} gives
\begin{align}
J_i(\check{u}_i, \check{u}_{-i})
=& \mu_0^T \check{Y}^N(0) \mu_0
+ \Tr[ \check\Lambda_1^N(0) \Sigma_0^i ]
+ (1/N^2) \sum_{j\in \mathcal{N}_{-i} } \Tr [ \check\Lambda_3^N(0) \Sigma_0^j] \notag \\
& + \mu_0^T \{ - (\check\Lambda_2^N(0) + \check\Lambda_2^{NT}(0))/N + \check\Lambda_3^N(0) (N-1)/N^2 \notag \\
& + \check\Lambda_4^N(0) (2-3N)/N^2
- ( \check\Lambda_{12}^N(0) + \check\Lambda_{12}^{NT}(0) ) /N
\} \mu_0 . \label{checkJiLam}
\end{align}
\section{Decentralized $O(1/N)$-Nash equilibrium strategies}
\label{sec:O(1/N)Nash}
In this section we show that the set of decentralized strategies in
\eqref{checkui} has an $O(1/N)$-Nash equilibrium property. More precisely, when the game \eqref{Xi}--\eqref{Ji} is asymptotically solvable and all other players take the decentralized strategies \eqref{checkui}, the extra benefit that a player obtains by unilaterally deviating from the strategy
\eqref{checkui} is at most $O(1/N)$.
\begin{theorem}
\label{thm:O(1/N)Nash}
Under Assumptions \ref{assm:solLam} and \ref{assm:initialX},
the set of decentralized strategies $(\check{u}_1, \cdots, \check{u}_N)$ given by \eqref{checkui} is an $O(1/N)$-Nash equilibrium of the Nash game \eqref{Xi}--\eqref{Ji}, i.e.,
\begin{align}
J_i(\check{u}_i, \check{u}_{-i}) \leq J_i(u_i, \check{u}_{-i}) + O(1/N), \quad \forall 1\le i\le N,\label{epsNash}
\end{align}
where $u_i$ is any admissible control under CLPS information such that the closed-loop system under $(u_i,\hat u_{-i})$ has a well defined solution. \end{theorem}
We will prove Theorem~\ref{thm:O(1/N)Nash} after some technical preparations.
Without loss of generality, we prove \eqref{epsNash} for player $\mathcal{A}_1$.
Suppose that players $\mathcal{A}_i$, $2\leq i \leq N$, use decentralized strategies given by \eqref{checkui}.
Player $\mathcal{A}_1$ seeks its best response strategy $u_1^b$
with respect to $\check{u}_{-1}$ so that $J_1(u_1^b, \check{u}_{-1})= \inf_{u_1} J_1(u_1,\hat u_{-1}) $, where $J_1$ is defined by \eqref{Ji}. This leads to the optimal control problem with dynamics
\begin{align}
d X(t) = & \Big[ \mathbf{A} X + \widehat{\mathbf{B}}_1 u_1
- \widehat{\mathbf{B}}_{-1} ( \widehat\Theta X
+ \widehat\Theta_1 \overline{X} ) \Big] dt + \mathbf{B}_1 u_1 d W_1 \notag \\
& - \sum_{i=2}^N \mathbf{B}_i ( \Theta \mathbf{e}_i X + \Theta_1 \overline{X} ) d W_i
+ \mathbf{B}_0 \Big( u_1 - \sum_{i=2}^N (\Theta \mathbf{e}_i X + \Theta_1 \overline{X} ) \Big) d W_0 , \notag
\end{align}
where we denote $\widehat{\mathbf{B}}_{-1} = (0, \widehat{\mathbf{B}}_2, \cdots , \widehat{\mathbf{B}}_N)$ and
the mean field limit state $\overline{X}$ follows the dynamics
\eqref{dbarX}. The best response $u_1^b$ is to be determined.
We employ a dynamic programming approach to solve player $\mathcal{A}_1$'s optimal control problem.
Let $V_1^b(t, \mathbf{x}, \bar{x})$ be the value function of $\mathcal{A}_1$ with initial state $(X(t), \overline{X}(t))=(\mathbf{x}, \bar{x})$, associated with the cost $J_1(u_1, \check u_{-1})$.
Now $V_1^b(t, \mathbf{x}, \bar{x})$ is formally solved from the following dynamic programming equation:
\begin{align}
\label{HJBV1d}
& - \frac{\partial V_1^b}{\partial t}
= \min_{u_1 \in \mathbb{R}^{n_1}} \left[ \frac{\partial^T V_1^b}{\partial \mathbf{x}}
( \mathbf{A} \mathbf{x} + \widehat{\mathbf{B}}_1 u_1
- \widehat{\mathbf{B}}_{-1} ( \widehat\Theta \mathbf{x}
+ \widehat\Theta_1 \bar{x} ) ) \right. \\
&\qquad\qquad + \frac{\partial^T V_1^b}{\partial \bar{x}} (A+G - B(\Theta + \Theta_1 ) ) \bar{x} \notag \\
&\qquad\qquad \left. + \frac{1}{2} \Big( u_1 - \sum_{i=2}^N (\Theta \mathbf{e}_i \mathbf{x} + \Theta_1 \bar{x} ) \Big) ^T \mathbf{B}_0^T \frac{\partial^2 V_1^b}{\partial \mathbf{x}^2 } \mathbf{B}_0 ( u_1 - \sum_{i=2}^N (\Theta \mathbf{e}_i \mathbf{x} + \Theta_1 \bar{x} ) ) \right. \notag \\
& \qquad\qquad \left. + \frac{1}{2} ( \mathbf{B}_1 u_1 )^T \frac{\partial^2 V_1^b}{\partial \mathbf{x}^2} \mathbf{B}_1 u_1
+ \frac{1}{2} \sum_{i=2}^N ( \mathbf{B}_i (\Theta \mathbf{e}_i \mathbf{x} + \Theta_1 \bar{x} ) )^T \frac{\partial^2 V_1^b}{\partial \mathbf{x}^2} \mathbf{B}_i (\Theta \mathbf{e}_i \mathbf{x} + \Theta_1 \bar{x} ) \right. \notag \\
&\qquad\qquad \left.
+ \frac{1}{2} ( B_0 ( \Theta + \Theta_1 ) \bar{x} )^T \frac{\partial^2 V_1^b}{\partial \bar{x}^2} B_0 ( \Theta + \Theta_1 ) \bar{x}
+ \mathbf{x}^T \mathbf{Q}_1 \mathbf{x} + u_1^{T} R u_1 \right. \notag \\
& \qquad\qquad \left.
- \Big( u_1 - \sum_{i=2}^N (\Theta \mathbf{e}_i \mathbf{x} + \Theta_1 \bar{x} ) \Big)^T \mathbf{B}_0^T \frac{\partial^2 V_1^b}{ \partial \mathbf{x} \partial \bar{x}} B_0 (\Theta + \Theta_1 ) \bar{x} \right] ,
\notag \\
& V_1^b ( T, \mathbf{x} , \bar{x} ) = \mathbf{x}^T \mathbf{Q}_{1f} \mathbf{x} , \quad t \in [0, T], \ \mathbf{x} \in \mathbb{R}^{Nn}, \
\bar{x} \in \mathbb{R}^n . \notag
\end{align}
The first order condition with respect to $u_1$ gives
\begin{align}
u_1^b = &
- \frac{1}{2}\Big( R + \frac{1}{2} \mathbf{B}_1^T \frac{\partial^2 V_1^b}{\partial \mathbf{x}^2} \mathbf{B}_1 + \frac{1}{2}\mathbf{B}_0^T \frac{\partial^2 V_1^b}{\partial \mathbf{x}^2} \mathbf{B}_0 \Big)^{-1} \cdot \label{u1dV} \\
& \Big[ \widehat{\mathbf{B}}_1^T \frac{\partial V_1^b}{\partial \mathbf{x}}
- \mathbf{B}_0^T \frac{\partial^2 V_1^b}{\partial \mathbf{x}^2} \mathbf{B}_0
\sum_{i=2}^N ( \Theta \mathbf{e}_i \mathbf{x} + \Theta_1 \bar{x} )
- \mathbf{B}_0^T \frac{\partial^2 V_1^b}{\partial \mathbf{x} \partial \bar{x}} B_0 ( \Theta + \Theta_1 ) \bar{x} \Big] . \notag
\end{align}
We substitute \eqref{u1dV} into \eqref{HJBV1d} to obtain
\begin{align}
\label{HJBV1d2}
-\frac{\partial V_1^b}{\partial t} =&
- \Big[ \widehat{\mathbf{B}}_1^T \frac{\partial V_1^b}{\partial \mathbf{x}} - \mathbf{B}_0^T \frac{\partial^2 V_1^b}{\partial \mathbf{x}^2} \mathbf{B}_0
\sum_{i=2}^N (\Theta \mathbf{e}_i \mathbf{x} + \Theta_1 \bar{x} ) - \mathbf{B}_0^T \frac{\partial^2 V_1^b}{\partial \mathbf{x} \partial \bar{x}} B_0(\Theta + \Theta_1) \bar{x} \Big]^T \cdot \\
&\quad \frac{1}{4}\Big( R + \frac{1}{2} \mathbf{B}_1^T \frac{\partial^2 V_1^b}{\partial \mathbf{x}^2} \mathbf{B}_1 + \frac{1}{2} \mathbf{B}_0^T \frac{\partial^2 V_1^b}{\partial \mathbf{x}^2} \mathbf{B}_0 \Big)^{-1} \cdot \notag \\
&\quad \Big[ \widehat{\mathbf{B}}_1^T \frac{\partial V_1^b}{\partial \mathbf{x}}
- \mathbf{B}_0^T \frac{\partial^2 V_1^b}{\partial \mathbf{x}^2} \mathbf{B}_0 \sum_{i=2}^N
(\Theta \mathbf{e}_i \mathbf{x} + \Theta_1 \bar{x} )
- \mathbf{B}_0^T \frac{\partial^2 V_1^b}{\partial \mathbf{x} \partial \bar{x}}
B_0(\Theta + \Theta_1 ) \bar{x} \Big] \notag \\
& + \frac{\partial^T V_1^b }{\partial \mathbf{x}} (\mathbf{A} \mathbf{x} - \sum_{i=2}^N \widehat{\mathbf{B}}_i (\Theta \mathbf{e}_i \mathbf{x} + \Theta_1 \bar{x} ) )
+ \frac{\partial^T V_1^b}{\partial \bar{x}} (A+G - B ( \Theta + \Theta_1 ) ) \bar{x} \notag \\
& + \frac{1}{2} \Big( \sum_{i=2}^N (\Theta \mathbf{e}_i \mathbf{x} + \Theta_1 \bar{x} ) \Big)^T \mathbf{B}_0^T \frac{\partial^2 V_1^b}{\partial \mathbf{x}^2} \mathbf{B}_0 \sum_{i=2}^N
(\Theta \mathbf{e}_i \mathbf{x} + \Theta_1 \bar{x} ) \notag \\
& + \frac{1}{2} \sum_{i=2}^N ( \Theta \mathbf{e}_i \mathbf{x} + \Theta_1 \bar{x} )^T \mathbf{B}_i^T \frac{\partial^2 V_1^b }{\partial \mathbf{x}^2} \mathbf{B}_i
( \Theta \mathbf{e}_i \mathbf{x} + \Theta_1 \bar{x} ) + \mathbf{x}^T \mathbf{Q}_1 \mathbf{x} \notag \\
& + \frac{1}{2} ( B_0 ( \Theta + \Theta_1 ) \bar{x} )^T
\frac{\partial^2 V_1^b}{\partial \bar{x}^2} B_0 (\Theta + \Theta_1 ) \bar{x}
\notag \\
& + \Big( \mathbf{B}_0 \sum_{i=2}^N ( \Theta \mathbf{e}_i \mathbf{x} + \Theta_1 \bar{x} ) \Big)^T
\frac{\partial^2 V_1^b}{ \partial \mathbf{x} \partial \bar{x}} B_0 ( \Theta + \Theta_1 ) \bar{x} , \notag \\
V_1^b ( T, \mathbf{x} , \bar{x} ) = & \mathbf{x}^T \mathbf{Q}_{1 f} \mathbf{x} . \notag
\end{align}
Assume $V_1^b$ takes the form
\begin{align}
V_1^b(t, \mathbf{x} , \bar{x}) = \mathbf{x}^T \mathbf{P}_1^b(t) \mathbf{x} + 2 \mathbf{x}^T
\mathbf{P}_{12}^b(t) \bar{x} + \bar{x}^T \mathbf{P}_2^b(t) \bar{x} .
\label{V1dansatz}
\end{align}
We denote $\mathbf{I}_{-1} = (0, I_n, \cdots, I_n) \in \mathbb{R}^{n \times N n}$, and substitute \eqref{V1dansatz} into \eqref{HJBV1d2} to obtain ODEs for $\mathbf{P}_1^b$, $\mathbf{P}_{12}^b$ and
$\mathbf{P}_2^b$:
\begin{align}
\label{ODEP1d}
\begin{cases}
- \dot{\mathbf{P}}_1^b
= - ( \widehat{\mathbf{B}}_1^T \mathbf{P}_1^b - \mathbf{B}_0^T \mathbf{P}_1^b \mathbf{B}_0 \Theta \mathbf{I}_{-1} )^T
( R + \mathbf{B}_1^T \mathbf{P}_1^b \mathbf{B}_1 + \mathbf{B}_0^T \mathbf{P}_1^b \mathbf{B}_0 )^{-1} \cdot \\
\qquad\qquad
( \widehat{\mathbf{B}}_1^T \mathbf{P}_1^b - \mathbf{B}_0^T \mathbf{P}_1^b \mathbf{B}_0 \Theta \mathbf{I}_{-1} )
+ \mathbf{P}_1^b ( \mathbf{A} - \widehat{\mathbf{B}}_{-1} \widehat\Theta ) \\
\hspace{1.2cm}
+ ( \mathbf{A} - \widehat{\mathbf{B}}_{-1} \widehat\Theta )^T
\mathbf{P}_1^b
+ ( \mathbf{B}_0 \Theta \mathbf{I}_{-1} )^T \mathbf{P}_1^b \mathbf{B}_0 \Theta \mathbf{I}_{-1} \\
\hspace{1.2cm}
+ \sum_{k=2}^N ( \mathbf{B}_k \Theta \mathbf{e}_k )^T
\mathbf{P}_1^b ( \mathbf{B}_k \Theta \mathbf{e}_k )
+ \mathbf{Q}_1 , \\
\mathbf{P}_1^b(T) = \mathbf{Q}_1, \quad
R + \mathbf{B}_1^T \mathbf{P}_1^b(t) \mathbf{B}_1 + \mathbf{B}_0^T \mathbf{P}_1^b(t) \mathbf{B}_0 >0 , \ \forall t \in [0, T] ,
\end{cases}
\end{align}
\begin{align}
\label{ODEP12d}
\begin{cases}
- \dot{\mathbf{P}}_{12}^b =
(\mathbf{A} - \widehat{\mathbf{B}}_{-1} \widehat\Theta )^T
\mathbf{P}_{12}^b + \mathbf{P}_{12}^b (A+G - B(\Theta + \Theta_1 ) ) \\
\hspace{1.25cm}
+ (\mathbf{B}_0 \Theta \mathbf{I}_{-1} )^T \mathbf{P}_1^b \mathbf{B}_0 \mathbf{I}_{-1}\widehat\Theta_1
- \mathbf{P}_1^b \widehat{\mathbf{B}}_{-1} \widehat\Theta_1
\\
\hspace{1.25cm}
+ \sum_{k=2}^N ( \mathbf{B}_k \Theta \mathbf{e}_k )^T \mathbf{P}_1^b \mathbf{B}_k \Theta_1
+ (\mathbf{B}_0 \Theta \mathbf{I}_{-1} )^T \mathbf{P}_{12}^b B_0 (\Theta + \Theta_1 ) \\
\hspace{1.25cm}
- ( \widehat{\mathbf{B}}_1^T \mathbf{P}_1^b - \mathbf{B}_0^T \mathbf{P}_1^b \mathbf{B}_0 \Theta \mathbf{I}_{-1} )^T (R + \mathbf{B}_1^T \mathbf{P}_1^b \mathbf{B}_1 + \mathbf{B}_0^T \mathbf{P}_1^b \mathbf{B}_0 )^{-1} \cdot \\
\qquad\qquad\quad
[ \widehat{\mathbf{B}}_1^T \mathbf{P}_{12}^b - \mathbf{B}_0^T \mathbf{P}_1^b \mathbf{B}_0 \mathbf{I}_{-1} \widehat\Theta_1
- \mathbf{B}_0^T \mathbf{P}_{12}^b B_0( \Theta + \Theta_1 ) ] , \\
\mathbf{P}_{12}^b(T) = 0 ,
\end{cases}
\end{align}
\begin{align}
\label{ODEP2d}
\begin{cases}
- \dot{\mathbf{P}}_2^b = - [ \widehat{\mathbf{B}}_1^T \mathbf{P}_{12}^b - \mathbf{B}_0^T \mathbf{P}_1^b \mathbf{B}_0 \mathbf{I}_{-1}\widehat\Theta_1 - \mathbf{B}_0^T \mathbf{P}_{12}^b B_0 ( \Theta + \Theta_1 ) ]^T \cdot \\
\qquad\qquad
(R + \mathbf{B}_1^T \mathbf{P}_1^b \mathbf{B}_1 + \mathbf{B}_0^T
\mathbf{P}_1^b \mathbf{B}_0 )^{-1} \cdot \\
\qquad\qquad
[ \widehat{\mathbf{B}}_1^T \mathbf{P}_{12}^b - \mathbf{B}_0^T \mathbf{P}_1^b \mathbf{B}_0 \mathbf{I}_{-1} \widehat\Theta_1 - \mathbf{B}_0^T
\mathbf{P}_{12}^b B_0 ( \Theta + \Theta_1 ) ] \\
\hspace{1.2cm}
- \mathbf{P}_{12}^{bT} \widehat{\mathbf{B}}_{-1} \widehat\Theta_1
- \widehat\Theta_1^T \widehat{\mathbf{B}}_{-1}^T \mathbf{P}_{12}^b
+ (\mathbf{B}_0 \mathbf{I}_{-1} \widehat\Theta_1 )^T \mathbf{P}_1^b \mathbf{B}_0 \mathbf{I}_{-1} \widehat\Theta_1 \\
\hspace{1.2cm}
+ \mathbf{P}_2^b (A + G - B(\Theta + \Theta_1 ) ) + (A + G - B ( \Theta + \Theta_1 ) )^T \mathbf{P}_2^b \\
\hspace{1.2cm}
+ \sum_{k=2}^N ( \mathbf{B}_k \Theta_1 )^T \mathbf{P}_1^b (\mathbf{B}_k \Theta_1 )
+ (\Theta + \Theta_1 )^T B_0^T \mathbf{P}_2^b B_0 (\Theta + \Theta_1 ) \\
\hspace{1.2cm}
+ ( \mathbf{B}_0 \mathbf{I}_{-1} \widehat\Theta_1 )^T \mathbf{P}_{12}^b B_0 ( \Theta + \Theta_1 )
+ (\Theta + \Theta_1 )^T B_0^T \mathbf{P}_{12}^{bT} \mathbf{B}_0 \mathbf{I}_{-1} \widehat\Theta_1 , \\
\mathbf{P}_2^b(T) = 0 .
\end{cases}
\end{align}
\begin{proposition}\label{prop:br}
Suppose that Assumption~\ref{assm:solLam} holds and that \eqref{ODEP1d} has a solution $\mathbf{P}_1^b$ on $[0, T]$.
Then we may uniquely solve \eqref{ODEP12d}--\eqref{ODEP2d}, and the best response strategy for $\mathcal{A}_1$ is
\begin{align}
\label{u1dP}
u_1^b(t) = &
- \Big( R + \mathbf{B}_1^T \mathbf{P}_1^b \mathbf{B}_1
+ \mathbf{B}_0^T \mathbf{P}_1^b \mathbf{B}_0 \Big)^{-1}
\Big[ \widehat{\mathbf{B}}_1^T ( \mathbf{P}_1^b X(t) + \mathbf{P}_{12}^b \overline{X}(t) ) \\
&\quad - \mathbf{B}_0^T \mathbf{P}_1^b \mathbf{B}_0
\sum_{i=2}^N ( \Theta \mathbf{e}_i X(t) + \Theta_1 \overline{X}(t) )
- \mathbf{B}_0^T \mathbf{P}_{12}^b B_0
( \Theta + \Theta_1 ) \overline{X}(t) \Big] . \notag
\end{align}
\end{proposition}
\begin{proof}
If \eqref{ODEP1d} admits a (unique) solution $\mathbf{P}_1^b$ on $[0, T]$, then we can substitute $\mathbf{P}_1^b$ into
\eqref{ODEP12d} and solve a first order linear ODE for a unique
$\mathbf{P}_{12}^b$. Given $(\mathbf{P}_1^b, \mathbf{P}_{12}^b)$, $\mathbf{P}_2^b$ is again solved from a linear ODE.
Note that the LQ optimal control problem of player $\mathcal{A}_1$ has its Riccati equation given by \eqref{ODEP1d}--\eqref{ODEP2d}.
It then follows from \cite[Theorem 6.6.1]{YZ1999} that player $\mathcal{A}_1$'s optimal control problem is solvable with the optimal control given by \eqref{u1dP}.
\end{proof}
We will later show that for all sufficiently large $N$, \eqref{ODEP1d} indeed has a solution on $[0, T]$ (see Lemma \ref{lm:exissolP1P2d}). The next lemma is
parallel to Lemma~\ref{lm:Psubmat}.
\begin{lemma}
\label{lm:Pdsubmat}
Suppose \eqref{ODEP1d} has a solution $\mathbf{P}_1^b$ on $[0,T]$.
Then for \eqref{ODEP1d} and \eqref{ODEP12d}, $\mathbf{P}_1^b$ and $\mathbf{P}_{12}^b$ have the representations
\begin{align}
& \mathbf{P}_1^b
= \begin{bmatrix}
\Pi_1^{b N} & \Pi_2^{b N} & \Pi_2^{b N} & \cdots & \Pi_2^{b N} \\
( \Pi_2^{b N})^T & \Pi_3^{b N} & \Pi_4^{b N} & \cdots & \Pi_4^{b N} \\
( \Pi_2^{b N} )^T & \Pi_4^{b N} & \Pi_3^{b N} & \cdots & \Pi_4^{b N} \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
( \Pi_2^{b N} )^T & \Pi_4^{b N} & \Pi_4^{b N} & \cdots & \Pi_3^{b N} \\
\end{bmatrix} , \label{P1dsubmat}\\
& \mathbf{P}_{12}^b = \begin{bmatrix} (\Pi_{11}^{b N})^T , \
( \Pi_{12}^{b N} )^T , \ \cdots , \ (\Pi_{12}^{b N})^T \end{bmatrix}^T ,
\label{P12dsubmat}
\end{align}
where $\Pi^{bN}_1(t)$, $\Pi^{bN}_3(t)$, $\Pi^{bN}_4(t) \in \mathcal{S}^{ n}$, and $\Pi^{bN}_2(t)$, $\Pi^{bN}_{11}(t)$, $\Pi^{bN}_{12}(t) \in \mathbb{R}^{n\times n}$.
\end{lemma}
\begin{proof}
The proof is similar to that of Lemma~\ref{lm:Psubmat}, and is thus omitted here.
\end{proof}
We define new variables:
\begin{align}
\begin{cases}
\Lambda_1^{b N} = \Pi_1^{bN} , \quad
\Lambda_2^{b N} = N \Pi_2^{bN} , \quad
\Lambda_3^{b N} = N^2 \Pi_3^{bN} , \quad
\Lambda_4^{b N} = N^2 \Pi_4^{bN} , \\
\Lambda_{11}^{b N} = \Pi_{11}^{bN} , \quad
\Lambda_{12}^{bN} = N \Pi_{12}^{bN} , \quad
\Lambda_{22}^{b N} =\mathbf{P}_2^b ,
\end{cases} \label{PdLam}
\end{align}
and suppose \eqref{ODEP1d} has a solution $\mathbf{P}_1^b$ on $[0, T]$.
We substitute \eqref{P1dsubmat} and \eqref{P12dsubmat} into
\eqref{ODEP1d}--\eqref{ODEP2d} and take a change of variables by \eqref{PdLam} to obtain (under the additional condition that $R+B_1^T \Lambda_1^{b N}(t) B_1 > 0$) the following ODEs:
\begin{align}
& \begin{cases}
\dot{\Lambda}_1^{b N} = \Lambda_1^{b N} B
( \mathcal{R}_1 ( \Lambda_1^{b N} ) )^{-1} B^T \Lambda_1^{b N}
- \Lambda_1^{b N } A - A^T \Lambda_1^{b N } \\
\hspace{1.1cm}
- Q + g_1^{bN} , \\
\Lambda_1^{b N}(T) = (I - \Gamma_f^T/N) Q_f (I - \Gamma_f/N) , \\
R+B_1^T \Lambda_1^{b N}(t) B_1 > 0 , \quad \forall t\in[0, T] ,
\end{cases} \label{ODELam1dN} \\
& \begin{cases}
\dot\Lambda_2^{b N} = \Lambda_1^{b N} B
( \mathcal{R}_1 ( \Lambda_1^{b N} ) )^{-1} B^T\Lambda_2^{b N}
- ( \Lambda_1^{b N} + \Lambda_2^{b N} ) G \\
\hspace{1.1cm}
- A^T \Lambda_2^{b N} - \Lambda_2^{b N} ( A - B \Theta )
+ Q \Gamma + g_2^{bN} , \\
\Lambda_2^{bN}(T) = - (I - \Gamma_f^T/N) Q_f \Gamma_f ,
\end{cases} \label{ODELam2dN}
\end{align}
\begin{align}
& \begin{cases}
\label{ODELam3dN}
\dot\Lambda_3^{bN} = (\Lambda_2^{bN})^T
B ( \mathcal{R}_1( \Lambda_1^{bN} ) )^{-1} B^T \Lambda_2^{b N}
- ( \Lambda_2^{bN} + \Lambda_4^{bN} )^T G \\
\hspace{1.1cm}
- G^T (\Lambda_2^{bN} + \Lambda_4^{bN} )
- \Lambda_3^{b N} (A- B \Theta)
- ( A - B \Theta )^T \Lambda_3^{b N} \\
\hspace{1.1cm}
- \Theta^T B_0^T ( \Lambda_1^{b N} + \Lambda_2^{b N}
+ (\Lambda_2^{b N})^T + \Lambda_4^{bN} ) B_0 \Theta \\
\hspace{1.1cm}
- \Theta^T B_1^T \Lambda_3^{b N} B_1 \Theta
- \Gamma^T Q \Gamma + g_3^{bN} , \\
\Lambda_3^{b N}(T) = \Gamma_f^T Q_f \Gamma_f ,
\end{cases}
\end{align}
\begin{align}
& \begin{cases}
\dot\Lambda_4^{b N} = (\Lambda_2^{b N})^T
B ( \mathcal{R}_1( \Lambda_1^{bN} ) )^{-1} B^T \Lambda_2^{b N}
- (\Lambda_2^{b N})^T G - G^T \Lambda_2^{b N} \\
\hspace{1.1cm}
- \Lambda_4^{b N} (A+G - B \Theta )
- (A+G - B \Theta )^T \Lambda_4^{b N} \\
\hspace{1.1cm}
- \Theta^T B_0^T (\Lambda_1^{b N} + \Lambda_2^{b N}
+ (\Lambda_2^{b N})^T + \Lambda_4^{b N} ) B_0 \Theta \\
\hspace{1.1cm}
- \Gamma^T Q \Gamma + g_4^{bN} , \\
\Lambda_4^{b N} (T) = \Gamma_f^T Q_f \Gamma_f ,
\end{cases} \label{ODELam4dN} \\
& \begin{cases}
\dot\Lambda_{11}^{b N} = \Lambda_1^{b N} B
( \mathcal{R}_1 (\Lambda_1^{bN} ) )^{-1} B^T \Lambda_{11}^{b N}
+ \Lambda_2^{b N} B \Theta_1 - A^T \Lambda_{11}^{bN} \\
\hspace{1.1cm}
- \Lambda_{11}^{bN} (A+G - B( \Theta + \Theta_1 ) ) + g_{11}^{bN} ,\\
\Lambda_{11}^{b N} (T) = 0 ,
\end{cases} \label{ODELam11dN}
\end{align}
\begin{align}
& \begin{cases}
\dot\Lambda_{12}^{b N} = (\Lambda_2^{b N})^T B
( \mathcal{R}_1 (\Lambda_1^{b N} ) )^{-1} B^T \Lambda_{11}^{b N}
- G^T ( \Lambda_{11}^{b N} + \Lambda_{12}^{b N} ) \\
\hspace{1.1cm}
- ( A^T - \Theta^T B^T ) \Lambda_{12}^{b N}
- \Lambda_{12}^{b N} (A+G - B(\Theta + \Theta_1 ) ) \\
\hspace{1.1cm}
- \Theta^T B_0^T
(\Lambda_1^{b N} + \Lambda_2^{b N} + (\Lambda_2^{bN})^T + \Lambda_4^{b N} ) B_0 \Theta_1 \\
\hspace{1.1cm}
- \Theta^T B_0^T ( \Lambda_{11}^{b N} + \Lambda_{12}^{b N} ) B_0 (\Theta + \Theta_1 ) + \Lambda_4^{b N} B \Theta_1
+ g_{12}^{bN} , \\
\Lambda_{12}^{b N}(T) = 0 ,
\end{cases}\label{ODELam12dN}
\end{align}
\begin{align}
& \begin{cases}
\dot{\Lambda}_{22}^{b N}
= (\Lambda_{11}^{b N})^T B
( \mathcal{R}_1( \Lambda_1^{b N} ) )^{-1} B^T \Lambda_{11}^{bN}
\\
\hspace{1.1cm}
- \Lambda_{22}^{b N} (A+G - B ( \Theta + \Theta_1 ) )
+ (\Lambda_{12}^{b N})^T B \Theta_1 \\
\hspace{1.1cm}
- (A+G - B( \Theta + \Theta_1 ) )^T \Lambda_{22}^{b N}
+ \Theta_1^T B^T \Lambda_{12}^{b N} \\
\hspace{1.1cm}
- \Theta_1^T B_0^T ( \Lambda_1^{b N} + \Lambda_2^{b N}
+ (\Lambda_2^{bN})^T + \Lambda_4^{b N} ) B_0 \Theta_1 \\
\hspace{1.1cm}
- ( \Theta + \Theta_1 )^T B_0^T \Lambda_{22}^{b N} B_0
( \Theta + \Theta_1 ) \\
\hspace{1.1cm}
- \Theta_1^T B_0^T ( \Lambda_{11}^{b N} + \Lambda_{12}^{b N} ) B_0 (\Theta + \Theta_1 ) \\
\hspace{1.1cm}
- (\Theta + \Theta_1 )^T B_0^T
( \Lambda_{11}^{b N} + \Lambda_{12}^{b N} )^T B_0 \Theta_1
+ g_{22}^{bN} , \\
\Lambda_{22}^{b N}(T) = 0 .
\end{cases} \label{ODELam22dN}
\end{align}
The terms $g^{bN}_k$, $1\leq k \leq 4$, $g^{bN}_{11}$, $g^{bN}_{12}$ and $g^{bN}_{22}$, as functions of $(N, \Lambda_1^{bN}, \Lambda_2^{bN},\cdots , \Lambda_{22}^{bN})$, are given in Appendix~\ref{appendix:gdN}.
Let $(\Lambda_1^b, \Lambda_2^b,\cdots, \Lambda_{22}^b )$ be determined by the ODE system
\eqref{ODELam1d}--\eqref{ODELam22d} in Appendix ~\ref{appendix:PfsolLamdN}.
\begin{lemma} \label{lm:LambLam124}
Under Assumption \ref{assm:solLam}
we have
\begin{align}
&\Lambda_1^b(t)=\Lambda_1(t), \label{LbL1} \\
&\Lambda_2^b(t)+\Lambda_{11}^b(t)= \Lambda_2(t), \label{LbL2}\\
&\zeta^b(t)= \Lambda_{2}(t)+ \Lambda_{2}^{T}(t)+ \Lambda_{4}(t)\label{LbL3}
\end{align}
for all $t\in [0,T]$,
where
$$
\zeta^b(t):= (\Lambda_{2}^b+ \Lambda_{2}^{bT}+ \Lambda_{4}^b+ \Lambda_{11}^b+ \Lambda_{11}^{bT}+\Lambda_{12}^b+ \Lambda_{12}^{bT}+ \Lambda_{22}^b)(t).
$$
\end{lemma}
\begin{proof}
\eqref{LbL1} is already stated in the proof of Lemma \ref{lemma:Lam1d}.
By considering the ODE of $\Lambda_2^b + \Lambda_{11}^b - \Lambda_2$
and next applying Gr\"{o}wnwall's lemma, we establish
$\sup_{0\le t\le T}|\Lambda_2^b(t) + \Lambda_{11}^b(t) - \Lambda_2(t)|=0$, which implies \eqref{LbL2}.
Define $\zeta(t)=\Lambda_{2}(t)+ \Lambda_{2}^{T}(t)+ \Lambda_{4}(t) $. By use of \eqref{ODELam2}, \eqref{ODELam4}, and
\eqref{ODELam2d}--\eqref{ODELam22d} we write the ODEs:
\begin{align*}
&\dot{\zeta} (t) = \Phi( \Lambda_2, \Lambda_4),\\
&\dot{\zeta}^b(t)= \Phi^b(\Lambda_2^b, \Lambda_4^b, \Lambda^b_{11}, \Lambda_{12}^b, \Lambda_{22}^b),
\end{align*}
where the two vector fields are not fully displayed but can be easily determined.
Note that $\Lambda_1(t)$ and $(\Lambda_1(t), \Lambda_2(t))$ appear in $\Phi$ and $\Phi^b$, respectively, and are treated as known functions of time. Letting $H=(\mathcal{R}_1(\Lambda_1))^{-1}$, we have
\begin{align}
\Phi-\Phi^b =& (\zeta^b-\zeta)(A+G) +(A+G)^T (\zeta^b-\zeta)\\
&+ (\Theta+\Theta_1)^T {B}_0^T (\zeta^b-\zeta){B}_0(\Theta+\Theta_1)\nonumber\\
&- \Lambda_1 BHB^T (\zeta^b-\zeta) -(\zeta^b-\zeta) BHB^T \Lambda_1\nonumber\\
&- (\zeta^b-\zeta) BHB^T \Lambda_2 + \Delta_\Phi, \nonumber
\end{align}
where we have used \eqref{LbL1}--\eqref{LbL2} to
derive the last line to get
\begin{align}
\Delta_\Phi = & \Lambda_2^T BHB^T \Lambda_2^T + \Lambda_2^T BHB^T \Lambda_4 \nonumber\\ &
- \Theta_1^T B^T (\Lambda^{bT}_2 +\Lambda^{bT}_{11} + \Lambda^{bT}_{12} +\Lambda^b_4 +\Lambda^{bT}_{22} +\Lambda^b_{12}). \nonumber
\end{align}
By use of the definition of $\zeta$ and \eqref{LbL2}, we obtain
$\Delta_\Phi= \Lambda_2^T BHB^T (\zeta-\zeta^b)$.
By the ODE of $\zeta - \zeta^b$ and Gr\"{o}nwall's lemma, we obtain
$\sup_{t\in [0,T]}|\zeta(t) - \zeta^b(t)|=0$.
\end{proof}
Although the system \eqref{ODELam1dN}--\eqref{ODELam22dN}
has been constructed based on \eqref{ODEP1d}--\eqref{ODEP2d}, it can stand alone for its existence analysis without using the latter.
\begin{lemma}
\label{lm:solLamdN}
Under Assumption~\ref{assm:solLam}, there exists $N_1>0$ such that for all $N\geq N_1$,
\eqref{ODELam1dN}--\eqref{ODELam22dN} admits a solution $(\Lambda_1^{bN}, \cdots, \Lambda_{22}^{bN})$ on $[0, T]$ satisfying
\begin{align}\label{R1B0e0}
(\mathcal{R}_1(\Lambda_1^{bN} ) + B_0^T S^{bN} B_0/N^2)(t)>\epsilon_0 I , \quad \forall t\in [0,T],
\end{align}
for some small constant $\epsilon_0>0$. In addition, $\sup_{t\in[0, T]} |\Lambda_\iota^{bN} - \Lambda_\iota^b|=O(1/N)$ for $\iota=1, 2, \cdots, 22$, where $\Lambda_1^b$, $\cdots$, $\Lambda_{22}^b$ are given in Appendix~\ref{appendix:PfsolLamdN}.
\end{lemma}
\begin{proof}
We view \eqref{ODELam1dN}--\eqref{ODELam22dN} as a slightly perturbed version of \eqref{ODELam1d}--\eqref{ODELam22d}. By the same thin tube method as in the sufficiency proof of Theorem~\ref{thm:NSAS},
we establish the existence and uniqueness of a solution of
\eqref{ODELam1dN}--\eqref{ODELam22dN} for all sufficiently large $N$.
We may ensure \eqref{R1B0e0} due to $\mathcal{R}_1(\Lambda_1^b) >0$ for all $t\in [0,T]$ and a continuity argument.
The error bound of $O(1/N)$ is obtained by applying Gr\"{o}nwall's lemma as in Corollary \ref{cor:NSAS:LamN-Lam}. \end{proof}
\begin{remark}
\label{rmk:bdLamdN}
Let $\psi^N$ stand for any of the functions $\Lambda_1^{bN}$,
$\Lambda_2^{bN}$, $\Lambda_3^{bN}$, $\Lambda_4^{bN}$,
$\Lambda_{11}^{bN}$, $\Lambda_{12}^{bN}$ and
$\Lambda_{22}^{bN}$.
Then $\sup_{N\geq N_1, 0 \leq t \leq T} |\psi^N| \leq C$ for some fixed constant $C$.
\end{remark}
\begin{remark}
\label{rmk:gdNO(1/N)}
Let $h^N$ stand for any of the functions $g_1^{bN}$, $g_2^{bN}$,
$g_3^{bN}$, $g_4^{bN}$, $g_{11}^{bN}$, $g_{12}^{bN}$ and
$g_{22}^{bN}$. Then $\sup_{t\in [0, T]} |h^N(t)|=O(1/N)$.
\end{remark}
\begin{lemma} \label{lm:exissolP1P2d}
Under Assumption \ref{assm:solLam},
the ODE system \eqref{ODEP1d}--\eqref{ODEP2d} has a solution on $[0,T]$ for all $N\ge N_1$, where $N_1$ is specified in Lemma \ref{lm:solLamdN}. \end{lemma}
\begin{proof}
After obtaining $(\Lambda_1^{bN}, \cdots, \Lambda_{22}^{bN})$ by Lemma \ref{lm:solLamdN}, we define $\mathbf{P}_1^b$ using \eqref{P1dsubmat} and \eqref{PdLam}.
Then we can directly verify that $\mathbf{P}_1^b$ satisfies \eqref{ODEP1d}, where $ R + \mathbf{B}_1^T \mathbf{P}_1^b(t) \mathbf{B}_1 + \mathbf{B}_0^T \mathbf{P}_1^b(t) \mathbf{B}_0 >0 $ holds for all $t\in [0,T]$ since this matrix is equal to the term $ \mathcal{R}_1(\Lambda_1^{bN} ) + B_0^T S^{bN} B_0/N^2 $ appearing in \eqref{ODELam1dN}. Note that \eqref{R1B0e0} holds.
Then we further uniquely solve \eqref{ODEP12d}--\eqref{ODEP2d}.
\end{proof}
Combining Lemma
\ref{lm:exissolP1P2d} with Proposition \ref{prop:br} and Lemma \ref{lm:Pdsubmat}, we have the following facts. Under Assumption \ref{assm:solLam}, for all sufficiently large $N$, the best response control problem for player $\mathcal{A}_1$ has a solution. Next, the value function of the best response control problem can be specified using
\eqref{ODELam1dN}--\eqref{ODELam22dN}, which has a well defined solution.
\begin{lemma}
\label{lm:Lam1dN-Lam1}
$\sup_{t\in[0, T]}|\Lambda_1^{bN}(t) - \Lambda_1(t)| = O(1/N)$.
\end{lemma}
\begin{proof} The lemma follows from Lemma \ref{lm:solLamdN} and \eqref{LbL1}.
\end{proof}
\begin{lemma}
\label{lm:checkLam1-Lam1}
$\sup_{t\in[0, T]} | \check\Lambda^N_1(t) - \Lambda_1(t)| = O(1/N)$.
\end{lemma}
\begin{proof}
Taking the difference of \eqref{ODEcheckLam1} and \eqref{ODELam1} gives
\begin{align}
\begin{cases}
\frac{d}{dt} (\check\Lambda_1^N - \Lambda_1)
= - \Theta^T B_1^T (\check\Lambda_1^N - \Lambda_1) B_1\Theta
- (\check\Lambda_1^N - \Lambda_1) (A-B\Theta) \\
\hspace{2.4cm}
- (A-B\Theta)^T (\check\Lambda_1^N - \Lambda_1) - \check{g}_1^N , \\
\check\Lambda_1^N(T) - \Lambda_1(T) = (I - \Gamma_f^T/N) Q_f (I - \Gamma_f/N) - Q_f .
\end{cases} \notag
\end{align}
By Remark \ref{rmk:checkgO(1/N)}, $\sup_{t\in [0, T]} |\check{g}_1^N(t)|=O(1/N)$. The desired result follows from Gr\"{o}nwall's lemma.
\end{proof}
\begin{lemma}
\label{lm:Lam211dN-Lam2}
$\sup_{t\in[0, T]} |\Lambda_2^{bN}(t) + \Lambda_{11}^{bN}(t) - \Lambda_2(t) | = O(1/N)$.
\end{lemma}
\begin{proof} The lemma follows from Lemma \ref{lm:solLamdN} and \eqref{LbL2}.
\end{proof}
\begin{lemma}
\label{lm:checkYN-YdN}
Let $\check{Y}^N$ be defined by \eqref{checkYN}, and denote
\begin{align}
& Y^{b N} := \Lambda_1^{b N} + \Lambda_2^{b N}
+ (\Lambda_2^{b N})^T + \Lambda_4^{b N}
+ \Lambda_{11}^{b N} + ( \Lambda_{11}^{b N} )^T
+ \Lambda_{12}^{b N} + (\Lambda_{12}^{bN})^T
+ \Lambda_{22}^{b N} . \notag
\end{align}
Then $\sup_{t\in[0, T]} |\check{Y}^N(t) - Y^{bN}(t)|=O(1/N)$.
\end{lemma}
\begin{proof}
Combining the ODEs \eqref{ODEcheckLam1}--\eqref{ODEcheckLam22} and
\eqref{ODELam1dN}--\eqref{ODELam22dN},
we obtain the following ODE of $\check{Y}^N - Y^{bN}$:
\begin{align*}
&\frac{d}{dt} ( \check{Y}^N - Y^{b N} )\\
&= ( Y^{b N} - \check{Y}^N ) (A+G - B ( \Theta + \Theta_1 ) )
+ (A+G - B(\Theta + \Theta_1 ) )^T ( Y^{b N} - \check{Y}^N ) \notag \\
&
\quad+ (\Theta + \Theta_1 )^T
[ B_1^T (\Lambda_1^{b N} - \Lambda_1 ) B_1
+ B_0^T ( Y^{b N} - \check{Y}^N ) B_0 ] ( \Theta + \Theta_1 )
\notag \\
& \quad- ( \Lambda_1^{bN} + \Lambda_2^{bN} + \Lambda_{11}^{bN} - \Lambda_1 - \Lambda_2 )^T B ( \mathcal{R}_1( \Lambda_1 ) )^{-1} B^T
( \Lambda_1^{bN} + \Lambda_2^{bN} + \Lambda_{11}^{bN} - \Lambda_1 - \Lambda_2 ) \notag \\
&\quad - (\Lambda_1 + \Lambda_2 )^T B ( \mathcal{R}_1 ( \Lambda_1 ) )^{-1} B_1^T (\Lambda_1^{bN} - \Lambda_1 ) B_1
( \mathcal{R}_1( \Lambda_1 ) )^{-1} B^T (\Lambda_1 + \Lambda_2 ) \notag \\
&\quad - (\Lambda_1^{bN} + \Lambda_2^{bN} + \Lambda_{11}^{bN} )^T B
( \mathcal{R}_1 ( \Lambda_1 ) )^{-1} B_1^T (\Lambda_1 - \Lambda_1^{bN} ) B_1 \cdot \\
&\qquad ( \mathcal{R}_1 (\Lambda_1^{bN} ) )^{-1} B^T
(\Lambda_1^{bN} + \Lambda_2^{bN} + \Lambda_{11}^{bN} )
+ \rho^N , \\
&\check{Y}^N(T) - Y^{b N}(T) = 0 ,
\end{align*}
where
\begin{align}
\rho^N := & -(\check{g}_1^N + \check{g}_2^N + \check{g}_2^{NT}
+ \check{g}_4^N + \check{g}_{11}^N + \check{g}_{11}^{NT}
+ \check{g}_{12}^N + \check{g}_{12}^{NT} + \check{g}_{22}^N)
\notag \\
& - [g_1^{bN} + g_2^{bN} + (g_2^{bN})^T + g_4^{bN} + g_{11}^{bN} + (g_{11}^{bN})^T + g_{12}^{bN} + (g_{12}^{bN})^T + g_{22}^{bN}] .
\notag
\end{align}
The coefficients of the term $\check{Y}^N - Y^{bN}$ are bounded.
By Lemmas \ref{lm:Lam1dN-Lam1} and \ref{lm:Lam211dN-Lam2},
we have $\sup_{t\in[0, T]}|\Lambda_1^{bN}(t) - \Lambda_1(t)| = O(1/N)$
and
$\sup_{t\in[0, T]}| \Lambda_1^{bN} + \Lambda_2^{bN} + \Lambda_{11}^{bN} - \Lambda_1 - \Lambda_2 | = O(1/N)$.
By Remarks \ref{rmk:checkgO(1/N)} and \ref{rmk:gdNO(1/N)}, we have that
$\sup_{t\in[0, T]} |\rho^N| = O(1/N)$.
The lemma is then proven by applying Gr\"{o}nwall's lemma to the integral form of the ODE of $\check{Y}^N - Y^{bN}$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:O(1/N)Nash}]
When all other players $\mathcal{A}_i$, $2\leq i \leq N$, take the decentralized strategies $\check{u}_{-1} = (\check{u}_2, \cdots, \check{u}_N)$, we compare the cost of player $\mathcal{A}_1$ under $u_1^b$ with the cost under $\check{u}_1$.
The cost $J_1(u_1^b, \check{u}_{-1})$ of $\mathcal{A}_1$ is
\begin{align}
J_1(u_1^b, \check{u}_{-1})
=& \mathbb{E}\Big[ V_1^b(0, X(0), \overline{X}(0)) \Big] \notag \\
= & \mathbb{E} \Big[ X^T(0) \mathbf{P}_1^b(0) X(0)
+ 2 X^T(0) \mathbf{P}_{12}^b(0) \overline{X}(0)
+ \overline{X}^T(0) \mathbf{P}_2^b(0) \overline{X}(0) \Big] ,
\notag \\
=& \mu_0^T Y^{bN}(0) \mu_0
+ \Tr[ \Lambda_1^{bN}(0) \Sigma_0^1 ]
+ (1/N^2) \sum_{ i = 2}^N \Tr [ \Lambda_3^{bN}(0) \Sigma_0^i] \label{J1Pd} \\
& + \mu_0^T \{ - (\Lambda_2^{bN}(0) + (\Lambda_2^{bN}(0))^T) /N + \Lambda_3^{bN}(0) (N-1)/N^2 \notag \\
& + \Lambda_4^{bN}(0) (2-3N)/N^2
- ( \Lambda_{12}^{bN}(0) + (\Lambda_{12}^{bN}(0))^T ) /N
\} \mu_0 . \notag
\end{align}
The cost $J_1(\check{u}_1, \check{u}_{-1})$ can be obtained from \eqref{checkJiP}.
Then we have
\begin{align}\label{diff:Jd-Jcheck}
J_1(u_1^b, \check{u}_{-1}) - J_1(\check{u}_1, \check{u}_{-1})
=& \mu_0^T [ Y^{bN}(0) - \check{Y}^N(0) ] \mu_0
+ \Tr[ (\Lambda_1^{bN}(0) - \check\Lambda_1^N(0)) \Sigma_0^1 ]
\\
& + (1/N^2) \sum_{i=2}^N \Tr [(\Lambda_3^{bN}(0) - \check\Lambda_3^N(0)) \Sigma_0^i] + O(1/N), \notag
\end{align}
where we obtain the estimate $O(1/N)$ using Remarks \ref{rmk:bdcheckLam} and \ref{rmk:bdLamdN}.
By Lemma \ref{lm:checkYN-YdN}, we have
\begin{align}
|\mu_0^T [ Y^{bN}(0) - \check{Y}^N(0) ] \mu_0| = O(1/N).
\label{estmu0Y}
\end{align}
From Lemmas \ref{lm:Lam1dN-Lam1} and \ref{lm:checkLam1-Lam1}, we have $\sup_{t\in[0, T]} |\Lambda_1^{bN}(0) - \check\Lambda_1^N(0)| = O(1/N)$ and thus
\begin{align}
|\Tr[ (\Lambda_1^{bN}(0) - \check\Lambda_1^N(0)) \Sigma_0^1 ] | =O(1/N) . \label{estLam1Sigma}
\end{align}
By Assumption \ref{assm:initialX} and Remarks \ref{rmk:bdcheckLam} and \ref{rmk:bdLamdN}, we have
\begin{align}
& (1/N^2)\Big| \sum_{i=2}^N \Tr [(\Lambda_3^{bN}(0) - \check\Lambda_3^N(0)) \Sigma_0^i]\Big| = O(1/N) . \label{estLam3Sigma}
\end{align}
It follows from \eqref{diff:Jd-Jcheck} and \eqref{estmu0Y}, \eqref{estLam1Sigma} and \eqref{estLam3Sigma} that
\begin{align}
0\le J_1(\check{u}_1, \check{u}_{-1})-J_1(u_1^b, \check{u}_{-1}) = O(1/N).
\label{J1eNEN}
\end{align}
Note that the term $O(1/N) $ in \eqref{J1eNEN} does not depend on which player is selected to apply its best response.
This completes the proof.
\end{proof}
Let $u_i^b$ denote the best response strategy of ${\mathcal A}_i$ when all other players apply their strategies $\check u_{-i}$.
\begin{theorem}\label{thm:J1J2J3}
Under Assumptions \ref{assm:solLam} and \ref{assm:initialX}, we have
\begin{align}
&\max_{1\le i\le N}|J_i(u_i^b, {\check u}_{-i})- J_i(\hat u_i, \hat u_{-i})|
=O(1/N), \label{Jibhat} \\
&\max_{1\le i\le N} |J_i(\check{u}_i, {\check u}_{-i})- J_i(\hat u_i, \hat u_{-i})|=O(1/N). \label{Jichat}
\end{align}
\end{theorem}
\begin{proof}
By using the value function of the $N$-player Nash game, we have
\begin{align}
J_i(\hat u_i, \hat u_{-i}) =
& \mu_0^T [ \Lambda_1^N(0) + \Lambda_2^N(0) + \Lambda_2^{NT}(0) + \Lambda_4^N(0) ] \mu_0
+ \Tr[ \Lambda_1^N(0) \Sigma_0^i ] \label{Jihathat0} \\
& + (1/N^2) \sum_{ i\neq j = 1}^N \Tr [ \Lambda_3^N(0) \Sigma_0^j]
+ \mu_0^T \{ - (\Lambda_2^N(0) + \Lambda_2^{NT}(0)) /N \notag \\
& + \Lambda_3^N(0) (N-1)/N^2 + \Lambda_4^N(0) (2-3N)/N^2 \} \mu_0
\notag \\
=& \mu_0^T[\Lambda_1^N(0) +\Lambda_2^N(0)+\Lambda_2^{NT}(0)+\Lambda_4^N(0)]
\mu_0 \notag \\
& + {\rm Tr}[\Lambda_1^N(0)\Sigma_0^i] +O(1/N)\notag \\
=& \mu_0^T[\Lambda_1(0) +\Lambda_2(0)+\Lambda_2^{T}(0)+\Lambda_4(0)]
\mu_0 \label{JiLam14mu} \\
&
+ {\rm Tr}[\Lambda_1(0)\Sigma_0^i] +O(1/N),\nonumber
\end{align}
where the last equality follows from Corollary \ref{cor:NSAS:LamN-Lam}.
Similarly, we use \eqref{J1Pd} and Lemma \ref{lm:solLamdN} to obtain
\begin{align}
J_i(u_i^b, \check u_{-i})=& \mu_0^TY^{bN}(0)
\mu_0 + {\rm Tr}[\Lambda_1^{bN}(0)\Sigma_0^i] +O(1/N) \nonumber \\
=& \mu_0^T[\Lambda_1(0)+ \zeta^b(0)]
\mu_0 +{\rm Tr}[\Lambda_1(0)\Sigma_0^i] +O(1/N). \label{JibLamzeta}
\end{align}
The term $O(1/N)$ in all estimates obtained above does not depend on $i$.
By \eqref{JiLam14mu}--\eqref{JibLamzeta} and Lemma \ref{lm:LambLam124}, we obtain \eqref{Jibhat}, which combined with Theorem \ref{thm:O(1/N)Nash} yields \eqref{Jichat}.
\end{proof}
\subsection{The general model}
Now we consider a general LQ model where
$D$ and $D_0$ in \eqref{Xi} may be nonzero and where the cost \eqref{Ji} is modified by using the running cost
$ [X_i(t) - \Gamma X^{(N)}(t)-\eta]_Q^2 + [u_i(t)]_{R}^2$ and
the terminal cost
$ [X_i(T) - \Gamma_f X^{(N)}(T)-\eta_f]_{Q_f}^2$ for $\eta, \eta_f\in \mathbb{R}^n$. Then all the previous analysis in Sections \ref{sec:RiccatiEqns}--\ref{sec:O(1/N)Nash} may be easily adapted to this general model.
The value function in \eqref{Vansatz} is now replaced by the form
$$
V^G(t,\mathbf{x})= \mathbf{x}^T \mathbf{P}_i(t) \mathbf{x} + 2 \mathbf{x}^T\mathbf{S}_i^{G}(t) + \mathbf{r}_i^G (t), \quad
1\leq i \leq N .
$$
The same ODE system \eqref{ODEP}--\eqref{Pcon} is used for $(\mathbf{P}_1, \cdots, \mathbf{P}_N)$. If $(\mathbf{P}_1, \cdots, \mathbf{P}_N)$ is given on $[0,T]$, then $ (\mathbf{S}_1^{G}, \cdots, \mathbf{S}_N^{G})$ is uniquely solved from a linear ODE system. Finally, given $(\mathbf{P}_i, \mathbf{S}^G_i)$, $1\le i\le N$, on $[0,T]$, each $\mathbf{r}^G_i$ is again solved from a linear ODE. For this general model,
Definition \ref{def:AS} about asymptotic solvability remains valid, and Theorem \ref{thm:NSAS} still holds. The asymptotic analysis can be extended to treat $\{\mathbf{S}_i^{G}, \mathbf{r}_i^{G}, 1\le i\le N \}$. We can accordingly determine the Nash equilibrium strategies $\hat u_i^G$, $1\le i\le N$, the decentralized strategies $\check u_i^G$, $1\le i\le N$, and the best response strategy $u_i^{Gb}$ given $ \check u_{-i}^G$, which are further used to establish Theorems
\ref{thm:O(1/N)Nash} and \ref{thm:J1J2J3}.
We summarize the following result:
\begin{corollary}
Under Assumptions \ref{assm:solLam} and \ref{assm:initialX}, Theorems \ref{thm:O(1/N)Nash} and \ref{thm:J1J2J3} still hold for the general model with parameters $(D,D_0, \eta, \eta_f)$.
\end{corollary}
\section{Numerical example}
\label{sec:num}
We present a numerical example to illustrate asymptotic solvability and individual costs.
The parameter values are
$A =-1$, $B = 1$, $B_0=-2$, $B_1=4$, $G = 1$, $R=-1$, $Q =8$,
$\Gamma =0.8$,
$Q_f = 8$, $\Gamma_f = 0.8$, and $T=2$.
We take the initial conditions $X_i(0)=1$ for all $i\geq 1$, and so $\overline{X}(0)=1$.
When \eqref{ODELam1}--\eqref{ODELam2} admits a solution $(\Lambda_1, \Lambda_2)$ on $[0, T]$, we use MATLAB ODE solver \texttt{ode45} to solve \eqref{ODELam1}--\eqref{ODELam4} to obtain the solution
$(\Lambda_1, \Lambda_2, \Lambda_3, \Lambda_4)$.
At $t=0$, we obtain $ \Lambda_1(0)=3.9435$, $\Lambda_2(0)=-2.3751$, $\Lambda_3(0)=1.8351$ and $\Lambda_4(0)=1.7786$.
Fig. \ref{fig:Lam12J} (left panel) shows that
\eqref{ODELam1}--\eqref{ODELam2} admits a solution $(\Lambda_1, \Lambda_2)$ on $[0, T]$ so that the Nash game \eqref{Xi}--\eqref{Ji}
has asymptotic solvability. By the initial conditions and
\eqref{JiLam14mu}, under Nash strategies the asymptotic per agent
cost is $\lim_{N\to\infty}J_i(\hat u_i, \hat u_{-i})
= \Lambda_1(0)+2\Lambda_2(0)+\Lambda_4(0)=0.9719$, which is indicated by the dashed horizonal line in Fig. \ref{fig:Lam12J} (right panel).
Fig. \ref{fig:Lam12J} (right panel) shows that as $N$ increases,
the cost $J_i(\check u_i, \check u_{-i})$ of player $\mathcal{A}_i$ under the set of decentralized strategies approaches $\lim_{N\to \infty}J_i(\hat{u}_i, \hat{u}_{-i})$, as asserted by Theorem \ref{thm:J1J2J3}.
\begin{figure}[h]
\begin{centering}
\begin{tabular}{cc}
\hspace{-0.2cm}
\begin{minipage}{0.57\textwidth}
\includegraphics[width=1.2\textwidth, height=5cm]{Lam1Lam2}
\end{minipage} &
\hspace{-1cm}
\begin{minipage}{0.57\textwidth}
\includegraphics[width=1.2\textwidth, height=5cm]{Jhatchecklim}
\end{minipage}
\end{tabular}
\begin{minipage}{0.97\textwidth}
\caption{
Left panel: $(\Lambda_1, \Lambda_2)$
admits a solution on $[0, T]$ with $T=2$.
Right panel: The cost of player $\mathcal{A}_i$ under the set of decentralized strategies $(\check{u}_i, \check{u}_{-i})$ converges to a limit as $N\to \infty$.}
\label{fig:Lam12J}
\end{minipage}
\end{centering}
\end{figure}
\section{Conclusion}
\label{sec:conclusion}
This paper studies an asymptotic solvability problem for LQ mean field games with controlled diffusions and indefinite cost weights.
By a rescaling approach
we derive a necessary and sufficient condition for asymptotic solvability.
We further establish an $O(1/N)$-Nash equilibrium property for the obtained decentralized strategies.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,715 |
Q: ATtiny85 Yikes Error While Burning Bootloader I'm using an Elegoo uno to burn the ArduinoISP example code on to an ATtiny85 SMD. I wired the board according to the "#define USE_OLD_STYLE_WIRING" using pins 13/12/11/10. The wiring diagram for the circuit board is here. I first upload the sketch as "Board -> Arduino Uno" and it uploads successfully. I then click burn bootloader, using the following settings here. I get the error listed below after trying to upload to the ATtiny85. Any ideas?
https://gist.github.com/gmanc2/8e0b354d75c80d648f526b03fd6f8cf1
code: https://gist.github.com/gmanc2/683803105034c2c1ac1393d1f6b26cef
I have run the burn code only with the 1mhz internal crystal and 8mhz internal crystal selected. I if it is the case I accidentally selected an external crystal source, I don't have any 1-8 mhz external crystals. This error happened on my second attempt to burn to the board. Are there any other options besides buying a crystal? I'm fairly certain my wiring is correct. I have an oscilloscope is there any way I can verify the clock speed?
Thanks for any assistance!
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,485 |
\section{Introduction}
\label{intro}
Much of the software industry is characterised by a clear separation of concerns, encouraging users not to be attentive to how a system functions. However with safety critical applications or those that must adhere to tight legal requirements, further assurances are needed. Rigorous software engineering, through formal development or comprehensive testing, has been
very successful at supporting complex system development in many exacting fields. However we are now moving into an era where software is generated from examples.
The concept of the \textit{black box} describes a system that can only be viewed with reference to its observed inputs and resulting outputs. Black box models can be validated using well-established software testing methods based on such observations \cite{Beizer:1995:BTT:202699}. This validates the intended behaviour of a black box model by providing us an \textit{interpretation} of the mechanics of the model. Why the observed behaviour occurs is opaque and any \textit{explanation} for its behaviour remains uncertain.
AI and Machine Learning (ML) models are often black boxes. While there are some ML models that exhibit simple relationships that can be straightforwardly explained (e.g. logistic regression) or that inherently display logical structure (e.g. decision trees), complex models that consist of artificial neural networks (ANNs), deep neural networks (DNNs) and random forests are non-intuitive and are not structured with human-readable logic. This makes us unable to posit the question to a model, \textit{``Why did you do that?''}, to seek explanations of the decisions that ANN-based systems make \cite{Gunning2018}.
To provide explanations of black box models is an intersection of technical and human challenges:
\begin{itemize}
\item How can we generate an explainable model of a black box?
\item How can we ensure these explainable models are human-readable?
\item How do we use these explanations of black boxes?
\end{itemize}
\textbf{Contributions:} In this paper we present a novel method, Rule Induction of CNP Explanations (RICE), that combines sensitivity analysis and logic program synthesis for generating explanation models of black boxes. The explanation models are not intended to be executable, but rather to be human-readable in their explanation of the executable model. Previous work on explaining predictions of ML classifiers, such as the LIME algorithm \cite{Ribeiro:2016:WIT:2939672.2939778}, which uses a local linear approximation of a model's behaviour to highlight what a model has picked out in the input data to make its classification. Such work focuses on explaining classifiers, while our method focuses on explaining the rules and logic learned in a black box.
The paper is structured as follows. In Section~\ref{related} we position our work within the field of black box interpretability. In Section~\ref{explanation} we provide an overview of the RICE methodology. In Section~\ref{demonstration} we go through a simple traffic light use case, showing how the explainable model is produced using RICE. In Section~\ref{discussion} we reflect on our methodology and in Section~\ref{conc} we summarise and list some areas of future work.
\section{Related work}
\label{related}
Black box models, and in particular ML models, now exist in many critical domains such as medicine, finance, and even in military applications. There is a clear problem faced by the industrialization of using such models, where the human understanding of how they behave and work is difficult. Model \textit{interpretability} sets out to produce metrics about models, for example, scoring the performance of predictions against ground truth, and to build trust in models by providing examples of when they perform correctly or otherwise \cite{DBLP:journals/cacm/Lipton18}. When applied to black box models, for example in supervised learning models, such measures are focused on validating the learning of associations between factors. This of course does not guarantee that the model has learned causal relationships. Associations and causality describe the behaviour of models, however, they do not reveal an \textit{explanation} for why an output was generated nor an explanation for how the black box works.
Interpretable models already exist. Of note, there are a number of classification models that are readily interpretable, for example decision trees, rule-based systems, and nearest-neighbour methods, each with varying levels of usability \cite{kar38534}. Decision trees generate a tree-like structure representing a series of tests on different features in a training dataset where leaf nodes represent various labeled classifications. Rules-based systems explicitly map an input to an action through some explicitly defined series of logical assertions (\textit{if-then} rules). Nearest-neighbour algorithms qualify a classification based on the values of attributes in the immediate neighbourhood of some input.
In \cite{Guidotti:2018:SME:3271482.3236009} the authors produced an ``Open the black box taxonomy'' to better understand different approaches to explaining black box models. At the top level there are two options: To formulate a black box explanation, and design a transparent box. The latter has been discussed in some depth in \cite{DBLP:journals/corr/abs-1811-10154} where its author argues that explanations are often unreliable and misleading, and therefore black box models should not be used for critical systems. She cites several examples where lack of transparency and accountability of predictive models have had severe consequences. Counter-arguments to interpretable ML models include black boxes holding significant value and intellectual property for the model authors, and that interpretable models can be more costly to construct.
They further define the \textit{black box explanation problem} that requires an explanation model that is ``globally interpretable''; that is to be able to mimic the behaviour of the black box \textit{and} should be understandable by a human.
Approaches to generating human-readable explanation models include the Automatic Statistician, a system that discovers plausible models from data and automatically presents its findings as figures and as natural language \cite{Ghahramani:2015:Nature:26017444,Lloyd:2014:ACN:2893873.2894066}, LIME (Local Interpretable Model-agnostic Explanations) \cite{Ribeiro:2016:WIT:2939672.2939778}, a method for isolating parts of a given input that contribute most to a classification, and QII (Quantitative Input Influence) \cite{7546525}, a technique for calculating influences of individual inputs or groups of inputs, each provides explanations as summaries of local causal phenomena. A recent survey lists various methods in explainable AI, and notes that due to rise in autonomous systems and complex models, there is even more need for interpretable models \cite{biran2017explanation}.
\section{Explanations through Inductive Synthesis}
\label{explanation}
With RICE, we propose a method that combines sensitivity analysis and inductive logic programming. In contrast to LIME and QII, our approach seeks to generate globally interpretable model explanations for rule-based black box by synthesizing human-readable logic programs.
\begin{algorithm}
\caption{Rule Induced CNP Explanation (RICE)}\label{methodAlgorithm}
\begin{algorithmic}[1]
\Procedure{RICE}{training data}
\State $\textit{model} \gets \text{train}(\text{training data})$
\State $observables \gets $\textit{probe}(model)
\State $explanation \gets $\textit{CNPInduce}(observables)
\EndProcedure
\end{algorithmic}
\end{algorithm}
We start with a ML artefact after the learning step is accomplished. Sensitivity analysis consists of a series of methods developed to inspect the input/output relation of a function-like structure and is used in relation to ANNs to identify which input variables are relevant to the model, and which are not. This in turn is used to optimize the learning by reducing the unused dimensions in the training data. \cite{Zurada1994SensitivityAnalysis}. We use monothetic analysis, that is, systematically trying different values for one of the inputs while keeping the values for other inputs constant \cite{ten_broeke2016}. This is undertaken for each possible assignment of inputs to the model, effectively calculating every partial differential of the model's function. The data points where the partial differential $m$ is greater than an arbitrary $e$ are noted as \textit{critical examples}. Our method relies on the assumption that these extracted critical examples are a good estimate of the model's function. We call this stage of the method \textit{probing}.
For synthesizing logic programs we rely on earlier work on inductive program synthesis, CNPInduce (Induction of CNP) \cite{Pacaci2017CNP,Pacaci2017Representation}. The approach is a meta-interpretative form of inductive logic programming, where the CNPInduce synthesizer is written in Prolog. A meta-interpreter is a higher-level program that executes a program in a language and produces its input/output relation. In RICE, the meta-interpreter of a specially designed target language (CNP) is reversed, and this reversed meta-interpreter is executed with some known examples from its input/output relation thus producing all programs that would produce an input/output relation containing those examples \cite{HamfeltJFN94InductiveMetalogic,HamfeltJFN99InductiveSynthesis}. There are other rule-extraction methods that are applicable but they are mostly domain-specific, and require a user strategy to investigate the model. The strength of CNPInduce is its domain-agnostic technique and its ability to synthesize recursive programs \cite{Pacaci2017CNP,Pacaci2017Representation}. The ability to synthesize recursive programs makes it a good candidate to tackle models that deal with vector and matrix data such as audio and video signals. Even though inductive synthesis is a form of ML itself, it cannot perform well with noisy, high volume data; but it can produce a human-readable output. When coupled with a technique such as ANNs, which can deal with noisy input data, the combination gives a novel technique that has the efficiency of ANNs and the human-readable output of program synthesis.
To combine the two techniques, we take the critical examples extracted from the black box model and export them in a format that can be input to CNPInduce, the synthesizer to generate CNP programs. By using critical examples the synthesizer generates a program in the CNP language which satisfies these observables, which constitutes our explanation model. We call this stage of the method simply \textit{synthesis}. Algorithm \ref{methodAlgorithm} illustrates this sequence of stages to generate explanations using RICE. Since the synthesized program and the ML artefact are semantically correlated through the critical examples, one can interpret the synthesized program instead of the ML artefact. This allows the inspection and validation of the ML model through the synthesized program. In RICE, this program is expressed in a language specifically developed for this purpose, called Combilog with Named Projection or CNP \cite{Pacaci2017Representation}. Its human-readability is improved compared to other forms of variable-free relational programming languages. Moreover, since it is a pure relational language the programs in this language can be automatically translated to more familiar languages, such as first-order logic, definite clauses \cite{Hamfelt1998combinatoryFormOfPure}, or even structured English \cite{Fuchs1996AttemptoControlledEnglish}. We call this stage of the method \textit{validation}. In the next section we demonstrate the RICE method through an example.
\section{A demonstration of model validation}
\label{demonstration}
To demonstrate our method we devised an experiment based on a simplification of a self-driving vehicle's decision system. We assume a case where through its sensors the vehicle's systems have identified the status of a traffic light (red, amber, green) and the distance to it. The vehicle needs to decide to accelerate or to brake continuously. Existing studies using ANNs on traffic lights focus on detection of the state of the traffic light under complex circumstances, and recent work demonstrate this is achievable in real-time \cite{BehrendtNovak2017Bosch}. An action stage which would naturally follow detection is usually left out of the model. For demonstration purposes of RICE we design a model that mimics the action stage. This allows us to focus on the core of the method instead of issues regarding optimization and scale. Let us assume this decision is left to an ANN, the model is trained with data from actual driving sessions, and it seems to function normally. In order to quantify the decision system's reliability, one may want to know precisely how it reacts to specific conditions. In the case of a trained model this is near-impossible since the model consists of binary data. Let us take it from here and show how our method can help us approach validation of this model. The implementation can be accessed in our GitHub repository\footnote{\url{https://github.com/UppsalaIM/rice/releases/tag/iccs19}}. We present this demonstration in multiple stages:
\begin{enumerate}
\item Training of the model
\item Extract examples from the model (probing)
\item Synthesize a human-readable program from examples (synthesis)
\item Inspect the synthesized program as an indirect representation of source artefact (validation).
\end{enumerate}
\subsection{Training of the model}
In our feature vector we encode each light as being on or off with 0 or 1, where position 0 corresponds to the red light, position 1 to the amber light and position 2 to the green light. Our final feature representing the distance of the car from the traffic lights is encoded as a floating point decimal between 0 and 1 representing the range of distances from 0m to 100m, and is placed at position 3 of the feature vector. Our sample states are therefore:
\begin{verbatim}
[1, 0, 0, 0.25] # red and the car is 25m from the lights
[0, 1, 0, 1.0] # amber and the car is 100m from the lights
[0, 0, 1, 0.0] # green and the car is 0m from the lights
\end{verbatim}
A classification model is implemented and trained using TensorFlow\footnote{\url{https://tensorflow.org}}. The network has four layers, the input layer with 4 nodes, two hidden layers with 11 nodes each, and an output layer with 1 node. In the output layer, the value of the single output node represents the action; 1 for \textit{go}, and 0 for \textit{stop}.
Training data was procedurally generated in order to be able to embed algorithmic rules in the training data. While calculating the training labels a 2\% noise was introduced. This is important to show that the method works with noisy data, as inductive synthesis methods by themselves would not be able to work with noisy examples.
\label{subs_generate_and_train}
Training data is generated with random values for the state of the traffic light and the distance, within the valid space of input vectors. The labels are calculated according to the following scheme:
\begin{itemize}
\item If the light is \textbf{red}, and the distance to it is less than 60m, then stop; otherwise go. For example, if input is $[1,0,0,0.5]$, output is $0$; and if the input is $[1,0,0,0.9]$, output is $1$.
\item If the light is \textbf{amber}, and the distance to it is between 10m and 80m, then stop; otherwise go. The minimum distance of 10m is for avoiding stopping when it is too close. For example, if input is $[0,1,0,0.2]$, output is $0$.
\item If the light is \textbf{green}, go. For example, if input is $[0,0,1,0.1]$, output is $1$.
\end{itemize}
In the training set there were 45,000 samples, and in the testing set there were 5,000. The data was generated as 50,000 samples and then split into training versus test data. During 10 training runs with 10 epochs each, accuracy up to 99.7\% was measured, and it was consistently above 96\%.
A model with 99\% accuracy was saved to a HDF5 file which contains the model structure and the weights. In the following stage \textit{probing}, we load this model from a file in order to show that the training data is separate from the next stage.
\subsection{Probing: Extracting examples from a trained model}
The trained model from the previous stage is loaded from a file using TensorFlow with the probing stage implemented in Python. It is evaluated for each combination of the light state, for each possible distance (with a 1/100 granularity). The \verb|distance_sweep| function below displays how the input \textit{sweep} is generated, and the following two lines display how the model is evaluated for the given inputs. Here only the sweep for the red light state is displayed, where the outputs from the model are stored in the variable \verb|red_o|.
\begin{verbatim}
def distance_sweep(lights_state):
return np.array(np.array([lights_state + [x] for
x in np.linspace(0, 1, STEPS+1)]))
red_sweep = distance_sweep([1, 0, 0])
red_o = [np.rint(x[0]) for x in model.predict(red_sweep)]
\end{verbatim}
Once these input/output sweeps are generated for every possible light state, the ones where the partial differential $m$ is equal to 1 (modulo rounding). This gives all the critical points in the sweep where the output changes from 0 to 1, or 1 to 0. These are the points where the change in input results a dramatic change in the output. These input/output pairs are printed out by the probing stage in the input format of CNPInduce. Along with the input/output pairs, a distinct set of all the constants involved in the input/output pairs are extracted as well, and a synthesis job file is generated, as exemplified below:
\begin{verbatim}
jobValence([rd:in, am:in, gr:in, dist:in, go:out]).
jobConstant(0.0).
jobConstant(0.08).
jobConstant(0.09).
jobConstant(0.59).
jobConstant(0.6).
jobConstant(0.78).
jobConstant(0.79).
jobConstant(1.0).
jobObservable([rd:1.00, am:0.00, gr:0.00, dist:0.00, go:0.00], true).
jobObservable([rd:1.00, am:0.00, gr:0.00, dist:0.59, go:0.00], true).
jobObservable([rd:1.00, am:0.00, gr:0.00, dist:0.60, go:1.00], true).
jobObservable([rd:1.00, am:0.00, gr:0.00, dist:1.00, go:1.00], true).
jobObservable([rd:0.00, am:1.00, gr:0.00, dist:0.00, go:1.00], true).
jobObservable([rd:0.00, am:1.00, gr:0.00, dist:0.08, go:1.00], true).
jobObservable([rd:0.00, am:1.00, gr:0.00, dist:0.09, go:0.00], true).
jobObservable([rd:0.00, am:1.00, gr:0.00, dist:0.78, go:0.00], true).
jobObservable([rd:0.00, am:1.00, gr:0.00, dist:0.79, go:1.00], true).
jobObservable([rd:0.00, am:1.00, gr:0.00, dist:1.00, go:1.00], true).
jobObservable([rd:0.00, am:0.00, gr:1.00, dist:0.00, go:1.00], true).
jobObservable([rd:0.00, am:0.00, gr:1.00, dist:1.00, go:1.00], true).
\end{verbatim}
Each \verb|jobObservable| line in this synthesis file gives an input/output pair along with a Boolean flag \verb|true| that indicates if the model \textit{does} produce this output. Examples where the model \textit{does not} produce an output can alternatively be included for the synthesis to be able to eliminate some of the possible programs.
\subsection{Synthesis: Synthesizing a human-readable program}
In the synthesis stage, the synthesis job file prepared by the previous stage is run through CNPInduce by loading it through the \verb|jobFromLocalFile| command. This command loads the constants and examples (observables) from the file, and initiates a synthesis job with these. In this case, the number of domains in the model goes beyond the synthesizer's current efficiency limits. Therefore we manipulate the synthesis job file to only include the observables where the state of the light is red, which reduces the number of domains to 3, as shown:
\begin{verbatim}
jobObservable([rd:1.00, dist:0.00, go:0.00], true).
jobObservable([rd:1.00, dist:0.59, go:0.00], true).
jobObservable([rd:1.00, dist:0.60, go:1.00], true).
jobObservable([rd:1.00, dist:1.00, go:1.00], true).
\end{verbatim}
When the synthesizer is initiated with these observables, the first program it suggests is the following CNP program:
\begin{verbatim}
ande(const(rd,1.0),proj(iif(ltValue(a,0.6),0.0,1.0),[a->dist,o->go]))
\end{verbatim}
This reveals that the synthesizer found a program that involves all of the given arguments (rd, dist, go). CNPInduce also guarantees that the programs it produces are terminating programs. This means that as long as the inputs are constant values the programs generated will terminate and assign an output value. When a program is suggested, the synthesizer gives the option to stop or to look for other programs. When instructed to look for other programs, the synthesizer continues to find others that are correct, but longer and more complex. These induced logic programs suggested by CNPInduce form our explanation models.
In the next section, let us discuss how CNP programs can be interpreted as explanation models to validate the black box model being studied.
\subsection{Validation: Interpreting the program}
CNP is a pure relational language and therefore it may not be straightforward to those who are not familiar to such languages. In definite clause form it can be translated to:
\begin{align*}
& \text{model}(\text{Rd}, \text{Dist}, Go) \leftarrow \text{Rd}=1.0 \wedge \text{Rd} < 0.6 \wedge \text{Go}=1.0 \\
& \text{model}(\text{Rd}, \text{Dist}, Go) \leftarrow \text{Rd}=1.0 \wedge \neg(\text{Rd} < 0.6) \wedge \text{Go}=0.0
\end{align*}
Or in English it can be written as:
\begin{displayquote}
\emph{If the red light is 1.0, when the Dist is less than 0.6 assign Go to 1.0, otherwise assign Go to 0.0.}
\end{displayquote}
A visualization of the input/output states of the model can be seen in Figure \ref{fig_lights_plot}. The red line shows a shift at $Distance=0.6$, which is in line with the CNP program. Visualizations like these are useful but not always applicable due to higher dimensions or discrete data. In these cases a program is much more helpful to make sense of.
\begin{figure}
\includegraphics[width=\textwidth]{light_plots.png}
\caption{Sensitivity analysis on the trained model, visualized.}
\label{fig_lights_plot}
\end{figure}
Once an explanation model expressed in CNP is available for a given black box model, it can be automatically translated to logical rules or to natural language such as structured English. Once in this form, further automated assertions can be made on the explanation model's CNP program. For example, we can use automated testing and validation on the CNP explanations to reason about regulatory and legal compliance, where law may be represented logical axioms \cite{Prakken2015LawAL}.
\section{Discussion and Future Work}
\label{discussion}
While we have demonstrated that RICE can be used to generate explanation models in this paper, we have identified a number of challenges that need to be addressed to improve its applicability, usability and performance.
A weakness of our method lies in indirection. The probing stage extracts examples and these examples are used for program synthesis, which introduces a level of indirection between the synthesized program (the explanation model) and the actual model (the black box being studied). One way we suggest approaching this is to exploit the fact that both models can be evaluated for any input. By generating randomized samples for inputs and comparing the difference of the explanation model's output to those of the black box model, a difference value can be calculated that would converge to 0 if they are semantically identical.
An opportunity that was discovered during the development of probing phase was that it was just as easy to identify erroneous behaviour in irregular system states (such as red light and green light being on at the same time), and specifically how the errors occur. As the probing stage indifferently extracts all critical data points, it identifies irregular states as well, which are reflected in the synthesized program. This is an opportunity that would be useful for identifying incorrect behaviour trained into ML models that might only use positive examples. Another point that needs improvement is the granularity of probing. In the demonstration presented in this paper, 1/100 granularity was used. Ideally the probing can be much more efficient using techniques such as logarithmic search and dynamic granularity. Using heuristics to make this stage more efficient is an acceptable approach since the final program can always be confirmed with randomized input/output pairs against the actual model.
An advantage of our generic probing stage is that the only manual requirement it needs is the valid value ranges of input/output variables, and a name for each so the synthesized program can refer to these names. The probing stage can be written in a completely generic fashion and since the synthesis stage is also domain-agnostic, the whole method can be considered generic.
We found that the success of synthesis is highly sensitive to the accuracy of the model itself. In the example, the model was measured to have over 99\% accuracy. With models under 90\% accuracy we did not find the synthesis successful in identifying a program.
Scaling of the RICE method is a significant issue. Currently, the CNPInduce program synthesizer is implemented as a Prolog program, which cannot be parallelized trivially. But the simplicity of the search algorithm behind CNPInduce \cite{Pacaci2017Representation} may allow for a MapReduce implementation that is more amenable to parallelization \cite{Chu2007MapReduceForML}.
Finally, while the usability of CNP has been studied \cite{Pacaci2017Representation}, its use as a target language for explanation models needs to be tested. Studies have shown that model interpretability (transparent models vs. explained black box models) and the styles of algorithmic explanations have varying degrees to which they build trust in ML models and there is currently no `best' approach emerging \cite{DBLP:journals/corr/abs-1802-07810,Binns:2018:RHP:3173574.3173951}.
\section{Conclusions}
\label{conc}
In this paper we have demonstrated a novel method of assigning meaning to opaque software artifacts whose specification is unknown. Our methodology called RICE, does not try to interfere, steer or model the black box directly. Instead we have adopted a three stage post-processing approach in which we use the artefact as an opaque prototype on which to extract meaning. Our first stage, called \textit{probing}, inspects the generated artefact using sensitivity analysis to create a set of valid input/output pairs. From these pairs we have shown how to generate a logic-based explanation model, using \textit{synthesis} as a second stage, that subsequently allows the artefact to be interpreted. This is achieved through a final stage, called \textit{validation}, that produces a readable version.
This paper presents a proof of concept and much work remains to successfully scale the technique up to larger and more complex examples. Our interests lie not in the popular areas of machine learning, such as image recognition and consumer marketing, but in applications that require legal interpretation and certification.
Finally while the interoperability problem is usually presented as a concern for systems based on machine learning, the scope is far greater. Legacy systems for which there is no longer a readable source code, or agent based systems that have been finely calibrated are two further examples that would benefit from the ability to reverse-engineer an explainable interpretation that our methodology permits.
\bibliographystyle{acm}
| {
"redpajama_set_name": "RedPajamaArXiv"
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People learn best when their bodies and minds are engaged in discovery.
The following activities and games demonstrate what we know about evolution and how we know that evolution occurs.
This icon indicates links to activities found on this site (some are found only here). Unmarked links go off site.
DNA Pop-Beads This hands-on activity demonstrates DNA structure and replication using "Pop-Beads" available from science education supply companies.
DNA Beads DNA is the language of biological evolution. This simple hands-on activity provides a hands on way to learn about DNA's structure, replication, transcription and translation using beads on pip cleaners.
Tinkering With Evolution – This hands-on activity lets students "tinker" with DNA to observe duplication and modification as important sources of genetic variation. Velcro simulates genes. Tinkering with these Velcro "genes" demonstrates point mutations, duplications and DNA mobility.
Isolating DNA from Strawberries Using household ingredients, you can extract DNA from a strawberry.
Mighty Mutation Maker– The Canadian Museum of Nature has a fun online game that teaches lets you make mutations in DNA and see how these affect the amino acids in a protein.
Build DNA ModelGenetic Science Learning Center has a great, web-based activity that lets you build a virtual DNA molecule.
Gel Electrophoresis – This excellent, web-based interactive shows how DNA is analyzed using gel electrophoresis. From the Genetic Science Learning Center.
Geee! In Genome – Contains lots of interactive games that teach important concepts in genomics.
Candy Cells: Exploring Symbiosis in Cell Evolution This easy and delicious activity demonstrates how simple cells merged to form more complex cell with greater powers than either alone.
Web of Life Use yarn to demonstrate the interdependence of all life.
Skulls Lab– by Martin Nickels of Evolution & the Nature of Science Institutes.
The Genographic Project– This web site lets you have your own DNA tested, so that you may be part of this project that shows the how all people alive today descended from a group of African Ancestors around 60,000 years ago. Sponsored by National Geographic.
Family Tree DNA – Send in your DNA for testing to determine your evolution within your family tree.
Bird Beak Adaptations – Hands on investigation of how shape and structure of beaks affects the type of food that birds are able to eat.
Evolution of the Blue Grabber– This simulation uses mythical creatures to demonstrate the principles of evolution by natural selection.
Peppered Moth Web 'Interactive active Simulation– Fun and informative simulation of the evolution of dark and light peppered moths.
Candy Dish Selection by Carol Tang – Students select from different types of candies from a bowl and have an opportunity to think about what traits brought about the "survival" of some candies.
Natural Selection Scarf Tag Will the long tails or the short tails survive?
Population Growth Game – Demonstration of exponential population growth.
Deep Time – activities that explore vast scales of time.
Epic of Evolution Beads – students learn about major evolutionary events as they string together beads representing each event.
Tiles Through Time– (coming soon). Students create and then place tiles along path to represent major events in the Epic of Evolution.
Walk Through Time– a mile long walk represents a 5 million years of the Epic of Evolution. This web based activity has pictures that illustrate major events on this walk.
TimeLine Project Cash-register tape and pictures model timeline of the universe. By Thomas Atkins.
Dating Techniques – How do we know how old something is?
Thirteen Ways to Tell Time Backwards– from the Evolution & the Nature of Science Institutes web site.
Date A Rock – This activity uses beans and corn to simulate the use of radioisotope half-lives to provide a reliable measure for age-dating rocks. Age-dating rocks involves counting atoms and comparing the counts. Download worksheet .
Cephied Variables – Using variable stars to determine the age of the stars.
Drake Equation Calculator– The SETI institute has an online calculator to help you guesstimate the number of civilizations in our galaxy likely to contact us.
Extreme Life Card Game – If life exists elsewhere in the universe, it most likely will be similar to microbes on Earth that can live in extreme environments.
Fossils in Strata – Digging fossils from simulated sedimentary layers.
Snack Tectonics – Create a tasty model that illustrates plate tectonic motions. By Windows to the Universe. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,794 |
I'm Not in Love: "I'm Not in Love" is a song by English group 10cc, written by band members Eric Stewart and Graham Gouldman. It is known for its innovative and distinctive backing track, composed mostly of the band's multitracked vocals. Released in the UK in May 1975 as the second single from the band's third album The Original Soundtrack, it became the second of the group's three number-one singles in the UK between 1973 and 1978, topping the UK singles chart for two weeks. The song was also the band's breakthrough hit worldwide, reaching number one in Ireland and Canada and number two on the Billboard Hot 100 in the US, as well as reaching the top 10 in Australia, New Zealand and several European countries. Angel Dusted: Co-starring Patrick Cassidy, a young Helen Hunt and Jean Stapleton's real-life son Putch in a terrific lead performance, Angel Dusted features an early score by award-winning composer James Horner. Based on the book by Ursula Etons, the film was adapted by Darlene Craviotto, who also plays the boy's psychiatrist.
クレス・オルデンバーグ(Claes Oldenburg, 1929年1月28日 - )はスウェーデン生まれのアメリカ合衆国の彫刻家。日常のありふれた物を超巨大に複製したパブリックアート・インスタレーションで知られる。他にも、日常の同様のソフト・スカルプチュアを制作している。フリー百科事典『ウィキペディア(Wikipedia)』より。 使用曲:Pop Up! - Hot! | {
"redpajama_set_name": "RedPajamaC4"
} | 7,976 |
{"url":"https:\/\/stats.stackexchange.com\/questions\/300776\/oob-score-vs-test-set-accuray-random-forest","text":"# OOB Score vs test set accuray Random Forest\n\nI'm new to machine learning and trying to train a Random Forest with time series data. I used a time series split to generate my training set and test set. In addition, because I'm working with time series data, in order to verify the robustness of the model, I am doing a walk forward test starting with 50% of the data.\n\nSo my first walk forward step is training the first 50% of the data and use the next 10% as my test set. and then for the next step, I used the first 60% of the data and use the next 10% of my test set. Repeat the steps until the end.\n\nThe oob scores are always around 63%. but the test set accuracy are all over the places(not very stable) it ranges between .48 to .63 for different steps. Is it because the RF is overfitted? Am I missing some important features ? I have 8 features and roughly 30K data points in total. I'm using 2000 trees, max features .5, min_samples_leaf 10 and max_depth 10. Is it true that oob score might not be the optimal metric to reflect the general accuracy with time series data?\n\nThanks guys\n\n\u2022 It is relatively hard to overfit with a random forest. Max depth is a little low, min samples is a little high, and max features candidated is crazy low. Could you move that up to the late 80's? What does the convergence look like that drives 2000 trees? For many cases 200 is overkill and 80 to 100 will do. Eight features is not a lot of features for 30k samples. You could augment with a hundred first lags (or leads depending on ordering) to get a better model. I don't see you talking about how 3 measurements ago informs next prediction. Jul 19, 2019 at 18:09\n\u2022 @EngrStudent 2000 trees might be overkill in the sense of wasted time\/computation, but adding too many trees doesn't cause overfitting under RF.\n\u2013\u00a0Ceph\nJul 19, 2019 at 20:31\n\u2022 There should be purpose in decisions. Are the classes balanced? I hear you talking about time series, but I don't see how you are handling lags. Can you talk about the shape of your data? If I had 30k rows and 8 features, I would trade it in for 29k rows and 8000 features. That is data more \"fit\" for a random forest. I would use Boruta to chew that 8k features to 40 features that would do some real good. Jul 25, 2019 at 0:30\n\nThe OOB score is not \"looking forward\" to the next 10 %. For each observation it takes all the trees where it was not used for training this tree to predict this observation. So it uses also observations for training that come after this observation in a time series.\n\nIt can be used as alternative to normal cross-validation.\n\n\u2022 This answer doesn't seem to shed much light on why the OOB score is not serving as a more accurate estimate of the test error -- which seems to me to be OP's question.\n\u2013\u00a0Ceph\nJul 19, 2019 at 20:30\n\u2022 This answer does really involve the problem. Because it is used to test other trees, the OOB error is a \"validation\", which is part of the training\/tuning process and it is not a \"test\" which is a set of data that is pristine: never once been touched by the learning process. Only \"test\" is going to show real-world performance; this is why the test set is retained in the untouched state. Jul 23, 2019 at 14:49\n\nQuestion\nIs it true that oob score might not be the optimal metric to reflect the general accuracy with time series data? Is (my error) because the RF is overfitted?\n\nGoal:\nThe solution from training MUST \"work well\" on the validation set from a different time series.\n\nThere are two Issues:\n\n1. We could just be overfitting\n2. We could be very bad at predicting \"out of domain data\" (e.g., our validation set from a different time series)\n3. Bit of BOTH\n\nThesis:\n\nThe OOB accuracy captures only one of the issues (overfitting). The OOB prediction from each row of data could be in the same time series as in the training (atleast there is a high chance for every row).\n\nHaving said that, it is still a useful metric to check out which of the two issues you are having. This allows us to see whether the model is overfitting, without needing a separate validation set.\n\nInterpretation Example\n\nIn this lecture they show a training error of 17%, OOB error of 21% and a validation error of 23%.\n\nThe training error is LESS THAN OOB error indicating overfitting. Because the OOB error is \"much less\" than the validation error, we can gather that there is \"something else\" in addition to overfitting (i.e., perhaps something related to time).\n\nConclusion:\n\nYes it doesn't seem optimal as it is \"similar\" to cross validation where the time series issue would not be captured. However it is a clever way of determining if you are overfitting.\n\ntraining error < OOB --> overfitting\nOOB != validation --> domain shift\n\nP.S.\n\n1. Regarding your question on why the test accuracy is all over the place, perhaps can you provide the data? Maybe it is a kaggle competition?\n2. I borrowed this heavily from Jeremy Howard (founder of fastai)from his course on Deeplearning.","date":"2022-07-01 11:34:25","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.45748573541641235, \"perplexity\": 959.2466078854661}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-27\/segments\/1656103940327.51\/warc\/CC-MAIN-20220701095156-20220701125156-00250.warc.gz\"}"} | null | null |
{"url":"http:\/\/the-biologist-is-in.blogspot.com\/2014\/12\/mathematical-recreations-ramanujans.html","text":"## Tuesday, December 30, 2014\n\n### Mathematical Recreations : Ramanujan's Nested Radical\n\nSrinivasa Ramanujan posed the following puzzle to the Journal of Indian Mathematical Society in April 1911.\n\nWhat is the value of the following?\n$$\\sqrt{1+2\\sqrt{1+3\\sqrt{1+4\\sqrt{\\cdots}}}}$$\n\nHe waited over six months, and when nobody replied he gave his solution. The result he provided was 3.\n\nThe algebraic solution provided by Ramunujan:\n\nConsider the identity\n$$(x+n)^2 = n^2 + 2nx + x^2 = n^2 + x[(x+n)+n]$$\nTake square roots to get\n$$[x+n] = \\sqrt{n^2+x[(x+n)+n]}$$\nNow inside the brackets you have (something + n), so you can\nsubstitute in the same equation with (x+n) replacing (x) to get\n$$x+n = \\sqrt{n^2+x\\sqrt{n^2+(x+n)[(x+2n)+n]}}$$\nand repeat the process to get\n$$x+n = \\sqrt{n^2 + x\\sqrt{n^2+(x+n)\\sqrt{n^2+(x+2n)\\sqrt{\\cdots}}}}$$\nIf you now set n = 1, x = 2 you get\n$$3 = \\sqrt{1+2\\sqrt{1+3\\sqrt{1+4\\sqrt{\\cdots}}}}$$\n\nAn alternate way of calculating the equation reveals some interesting behavior.\n\n$$3 = \\sqrt{1+2\\sqrt{1+3\\sqrt{1+4\\sqrt{\\cdots}}}}$$\n$$3^2 = 1+2\\sqrt{1+3\\sqrt{1+4\\sqrt{\\cdots}}}$$\n$$3^2-1= 2\\sqrt{1+3\\sqrt{1+4\\sqrt{\\cdots}}}$$\n$$\\frac{3^2-1}{2}= \\sqrt{1+3\\sqrt{1+4\\sqrt{\\cdots}}}$$\n$$4 = \\sqrt{1+3\\sqrt{1+4\\sqrt{1+5\\sqrt{\\cdots}}}}$$\n$$\\vdots$$\n$$n = \\sqrt{1+(n-1)\\sqrt{1+(n)\\sqrt{1+(n+1)\\sqrt{\\cdots}}}}$$\n\nThe value of each sequential nested radical forms a series...\n\n$$3, 4, 5, 6, \\cdots, \\infty$$\n\n...which is perfectly valid. If 3 was not a valid answer to the equation, then this calculation would have revealed a contradiction. Proof by contradiction is one of my favorite methods of proving.\n\nSo, what happens if we start with a different starting assumption?\n\n$$4 = \\sqrt{1+2\\sqrt{1+3\\sqrt{1+4\\sqrt{\\cdots}}}}$$\n$$\\frac{3\\cdot5}{2} = \\sqrt{1+3\\sqrt{1+4\\sqrt{1+5\\sqrt{\\cdots}}}}$$\n$$\\frac{13\\cdot17}{2^{2}3} = \\sqrt{1+4\\sqrt{1+5\\sqrt{1+6\\sqrt{\\cdots}}}}$$\n$$\\frac{11\\cdot19\\cdot223}{2^{6}3^{2}} = \\sqrt{1+5\\sqrt{1+6\\sqrt{1+7\\sqrt{\\cdots}}}}$$\n$$\\frac{17\\cdot27902701}{2^{12}3^{4}} = \\sqrt{1+6\\sqrt{1+7\\sqrt{1+8\\sqrt{\\cdots}}}}$$\n$$\\vdots$$\n\nThe value of each sequential radical forms another series.\n\n$$4, 7.5, 18.41\\bar6, 84.543402\\bar7, 1429.71739065, \\cdots, \\infty$$\n\nIt is harder to predict subsequent entries of the series than for the original series, but no less valid. I've put some effort into constructing an algebraic function that also produces this series, but have set aside the challenge for now.\n\nSo, let's try another starting assumption.\n\n$$2 = \\sqrt{1+2\\sqrt{1+3\\sqrt{1+4\\sqrt{\\cdots}}}}$$\n\nThe value of each sequential radical forms a third series...\n\n$$2, 1.5, 0.41\\bar6, -0.206597\\bar2, -0.191463517554, \\cdots, -\\infty$$\n\n...which is not valid under the convention that a radical can't have a negative value.\n\nGeneralizing this computation method, Ramanujan's Nested Radical has an infinite number of valid answers...\n\n$$3 \\ge \\sqrt{1+2\\sqrt{1+3\\sqrt{1+4\\sqrt{\\cdots}}}}$$\n\nEvery starting value greater than 3 results in a series of radical values that increase at more than 1 per step and increase to positive infinity. Every starting value less than 3 results in a series of radical values that increases at less than 1 per step and eventually turns negative. I don't yet know how to prove this general statement, but the pattern holds. The system behaves very simply when the starting value is 3, more than 3, or less than 3.\n\nRamanujan's solution to his puzzle is correct, but it is also incomplete.\n\nThe algebraic method is unable to find the full solution to the equation. Algebra doesn't effectively deal with infinites, so it shouldn't entirely be a surprise that an algebraic intuition might make some faulty (or at least incomplete) inferences when infinity is involved. This is interesting because it hints at the limits of algebra as a way of knowing truth. The G\u00f6del Incompleteness Theorems discuss something similar to this and are worth reading more about.\n\nHow does this connect with biology?\n\nOne detail of my training to be a scientist is learning to look for untested assumptions in arguments that are presented to me. This was never explicitly stated in any of my graduate coursework, but it has been important to do well in the academic environment I experienced in getting my PhD. When I find such an assumption, I see it as a puzzle to be explored. Am I seeing something real? What inferences is the presenter drawing from the assumption? Why is what I'm seeing not already addressed?\n\nWhat assumptions have you made today?\n\nReferences:\n1. Ramanujan: www.thefamouspeople.com\/profiles\/srinivasa-ramanujan-503.php\n2. Algebraic solve: mathforum.org\/library\/drmath\/view\/52674.html\n3. Numerical solve: www.johndcook.com\/blog\/2013\/09\/13\/ramanujans-nested-radical\/\n4. Proof by contradiction: en.wikipedia.org\/wiki\/Proof_by_contradiction\n5. Incompleteness theorems: en.wikipedia.org\/wiki\/G%C3%B6del%27s_incompleteness_theorems","date":"2017-02-26 23:32:46","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6919801235198975, \"perplexity\": 770.9832882174834}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-09\/segments\/1487501172156.69\/warc\/CC-MAIN-20170219104612-00316-ip-10-171-10-108.ec2.internal.warc.gz\"}"} | null | null |
Bel Canto Choir Vilnius, after its last year's intensive Open to the World concert series, invites listeners to ACOUSTICO, the choir's first performance of the year. 8PM on Friday, May 24, 2019, Bel Canto Choir Vilnius will present an intimate spring concert, mostly featuring music by contemporary composers from neighboring Latvia.
ACOUSTICO will also star some of the choir's good friends – guitarist Saulius Lipčius, one of the country's best young cellists Augustas Gocentas, children's percussion ensemble from the Vilnius Methodist Art Centre (director Saulius Auglys), and pianist Raminta Gocentienė.
The highlight of the concert will be the Lithuanian premiere of mass for choir and classical guitar by Latvian composer and guitarist Kaspars Zemītis. This will be performed by the choir together with the composer's good friend Saulius Lipčius.
Mesa by Kaspars Zemītis is a unique and modern blend of the traditional Latin mass genre together with the sounds of the classical guitar. The contemporary re-imagining of the mass is especially calming, tender, and thought-provoking. The composition features a range of solo passages on the classical guitar, which elegantly intertwine with individual vocal parts of the choir.
The finale – Agnus Dei and Amen – stands out with its especially subtle and tender soprano solo, whose fluttering, wordless duet creates a musical contrast with the slower, more careful choral sounds that follow. This leads to the final section of the mass, Amen, an uplifting ending with the words 'hallelujah' and 'amen' repeated and eventually fading as the composition reaches its peaceful and calm finale.
During their end-of-spring concert, Bel Canto Choir Vilnius will also perform pieces by well-known contemporary composers Eric Whitacre and Ēriks Ešenvalds, somewhat of a staple of the choir's recent concerts. The programme also features pieces by pop musicians and composers, such as Carly Simon, Dolly Parton, Josiah Kelley Atwood arranged for choir.
Bel Canto Choir Vilnius' Artistic director and conductor Dr Egidijus Kaveckas, second conductor and choirmaster Milda Zapolskaitė, choirmaster and concertmaster Raminta Gocentienė, vocal coach Liudas Norvaišas. Producer Audrius Valatkevičius. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,336 |
\section{Introduction}
The study of entanglement properties in quantum many-body systems
has attracted considerable attention in recent years (see \cite{Eisert2010} for a review).
The measures of entanglement provide a promising tool for understanding universal
properties of quantum systems, in particular, at the vicinity of quantum phase transition.
Typically, as a simple estimation, the entanglement is measured between two fictitiously
partitioned subsystems $\A$ and $\B$ in the ground state $\vert \Psi_{0}\rangle $ of the whole
system. A quantitative measure of this bipartition entanglement is the von-Neumann entropy of the reduced density matrix
$\rho_\A=\tr_{\B}\rho$, defined as
\begin{equation}
\mathcal{E}_{\A:\B}=-\tr_{\A}\rho_{\A}\ln \rho_{\A},
\end{equation}
where $\rho=\vert \Psi_0 \rangle\langle \Psi_0 \vert$ is the density matrix of the entire system.
This entanglement estimator has been widely studied in several quantum systems
\cite{Holzhey1994,Vidal2003,Korepin2004,Gioev2006,Li2008,Calabrese2008,Alba2012}.
Although the quantity appears extensive, it is typically found to be
proportional to the area of the hyper-surface separating the two
subsystems, particularly when the system is in a gaped phase. This is the
celebrated area law \cite{Srednicki1993,Wolf2008,Eisert2010}. What is more
interesting is that,
when the system is critical (or gap-less) there is correction to this area law. Moreover, the correction has universal properties.
For a one dimensional quantum system which exhibits a conformal symmetry, it was shown that the entanglement estimator $\mathcal{E}_{\A:\B}$
diverges logarithmically with the increasing system length $L$ \cite{Holzhey1994,Calabrese2004}. Moreover, the divergence obeys the following scaling form:
\begin{equation}
\label{eq:area_div}
\mathcal{E}_{\A:\B}=\frac{c}{3}\ln \left[ L \psi\left( \frac{\ell}{L}
\right)\right]+\textrm{constant},
\end{equation}
where $\ell$ is the size of the subsystem $\A$, and $\psi$ is the scaling
function. The additive constant in \eref{eq:area_div} is non-universal in the sense that it depends on the
microscopic details of the model. Remarkably, the constant $c$ turns out to be
universal. For periodic systems, $c$ is equal to the central charge of the underlying conformal field
theory whereas for open systems it is equal to half of the central charge \cite{Holzhey1994,Vidal2003,Calabrese2004}.
For critical quantum systems, the central charge characterizes the long-distance physics of the system. For example, the central charge of the quantum Ising system is equal to $1/2$.
Our understanding is less complete for higher-dimensional systems.
The area law has been generally proven in the gaped phases of a
systems with short-range interactions \cite{Wolf2008}. At criticality there are mixed examples: in some systems, such as free bosons, the area law
is found to be satisfied \cite{Plenio2005,Cramer2006,Calabrese2012}, whereas in
other systems, such as free fermions, there are logarithmic corrections to the area law \cite{Song2011,Barthel2006,Wolf2006,Gioev2006,Swingle2010}.
There are also other bipartite estimators of entanglement such as the R\'enyi entropy
\cite{Calabrese2010}, the mutual information \cite{Um2012,Alcaraz2013}, the
quantum discord \cite{Ollivier2001}, the logarithmic negativity \cite{zyczkowski1998volume,eisert1999comparison}, which
exhibit similar universal behavior \cite{Korepin2004,Sarandy2009}.
Naively, all these estimators measure the information shared between the degrees of
freedom in the two subsystems.
A natural question to ask is whether in classical systems information estimators, similar
to the entanglement entropy, exhibit an area law. Moreover, it is interesting to study whether such estimators
exhibit logarithmic corrections to the area law at criticality, in a manner that characterizes the universality class of the classical system.
The first question has been addressed by Wolf \textit{et al.} \cite{Wolf2008}, who
studied an estimator known as the mutual information.
It quantifies the amount of information acquired about the configuration of
one subsystem by measuring the state of the other. They have shown that the mutual information
of any classical system with a finite correlation length obeys an area-law.
The second question, about criticality, has recently been addressed by Alcaraz and Rittenberg \cite{Alcaraz2010}, who
studied the scaling of the mutual information as well as other information
estimators for several classical stochastic models. They have shown that at
criticality the estimators exhibit logarithmic corrections to the area-law,
with a scaling similar to \eref{eq:area_div}, observed in quantum systems.
The coefficient $c$ was found to depend on the model and the estimator studied.
However, for a specific model and estimator, $c$ was found to be independent of the parameters of the model
and thus remains constant along the critical line. This universal behavior suggests that as in the quantum case, the
coefficients $c$ could be useful in characterizing the universal properties of classical many-body systems.
Unfortunately, there is almost no other example where the scaling properties of the
shared-information estimators have been studied in classical many-body systems. In some cases,
the mutual information has been proposed as a means to detect phase transition in classical
spin models, where it was shown numerically to exhibit non-analytic behaviour
\cite{Iaconis2013,Lau2013}. However, an analytical computation of the mutual information
estimator is often very difficult.
In this paper, we address the issue by studying shared-information estimators
in classical spin-chains with mean-field interactions of a general form.
Due to the long-range nature of their interactions such models exhibit non-trivial phase diagrams.
At the same time they are simple enough for detailed analytic calculations.
Our goal is twofold: first, to study different estimators and compare their behavior
across the phase diagram, particularly along the critical line. Second, to study how the scaling behavior changes from
one thermodynamic ensemble to the other, in particular from the canonical to the
micro-canonical ensemble. We study two shared-information estimators: the mutual information $(\mathcal{I}_{\A:\B})$ and separation entropy
$(S_{\A:\B})$.
By carrying out an explicit analytical calculation, we find that within the canonical
ensemble and away from criticality, both the estimators remain finite as the system length tends to infinity.
This is not obvious a priori for systems with long-range interactions. At criticality, we find a different scenario. The
mutual information exhibits a logarithmic divergence similar to
\eref{eq:area_div} with $c=1/4$, with $\ell$ and $L-\ell$ denoting the number of spins in the two partitions. On the other hand, the separation
entropy has a $\sqrt{L}$-divergence in addition to the $\ln L$ term.
In both estimators, the coefficient of the logarithmic term does not depend on the
microscopic details of the model, as in the quantum case. On the hand, the scaling function $\psi$ does depend on the details of model.
We also demonstrate that
for both estimators the coefficient of $\ln L$ remains the same even in the presence of additional
short-range interactions.
This suggests that the value of this coefficient is characteristic of the mean-field universality class.
Unlike in the entanglement entropy used for quantum systems, the coefficient does not depend on the number of states each spin takes.
It is important to note that when considering only mean-field interactions the notion of geometry is lost, and thus the
area-law is not well defined. The fact that the estimators remain finite in the thermodynamic limit, can be considered as equivalent to the area-law of one dimensional systems.
The spin-spin correlation in mean-field models,
$c(r)=\langle\sigma_{i}\sigma_{i+r}\rangle-\langle\sigma_{i}\rangle\langle\sigma_{i+r}\rangle$
does not depend on $r$ because all spins interact with all the other spins. The critical point in these models is characterized by a change in the scaling of $c(r)$ from $c(r) \sim 1/L$ away from criticality to $c(r) \sim 1/\sqrt{L}$ at criticality. This is reflected in the divergence of the information estimators at the critical point.
In the micro-canonical ensemble, where the energy is strictly conserved, the
estimators exhibit a very different behavior. In our analysis we find that
the fixed-energy constraint imposes additional correlations between the local
degrees of freedom of the subsystems, which result in an additional $(1/2)\ln L$ terms
in both the estimators. As a result, even away from criticality we find a logarithmic divergence similar to \eref{eq:area_div}. At criticality,
the mutual information scales as $(3/4)\ln L$ while the separation entropy
scales as $(1/4) \ln L$. Notably, the leading $\sqrt{L}$ scaling
seen in separation entropy in the canonical ensemble is absent in the
micro-canonical ensemble. In our detailed analysis we show that this term is
associated to the fluctuations of the total energy which are absent in the micro-canonical ensemble.
To test our analysis we compare with numerical results of two particular
realizations of the general Hamiltonian, which we studied analytically: one is the mean-field
variant of the Blume-Emery-Griffiths model \cite{Blume1971,Barr2001} and the second is the
Nagel-Kardar model \cite{Nagle1970,Bonner1971,Kardar1983,Mukamel2005}. The former is a
$3$-state spin model with pure mean-fields interactions.
The latter is an Ising model with additional mean-field interaction. Both models have been
studied extensively in the past, serving as prototypical models of the long-range
interacting systems. Results from our numerical analysis of these two models is found to be in good agreement with
our analytical results. Another instance of our generic Hamiltonian is the
Curie-Weiss model. This has been studied analytically by Wilms \textit{et al.}, who computed the mutual information within the canonical ensemble \cite{Wilms2012}. In our analysis we recover their results.
The layout of this paper is as follows. \Sref{sec:background} provides the
background for our theoretical analysis, introducing the information estimators
and a brief description of the models considered.
The main results of our study are summarized in \sref{sec:main}. A
detailed analysis of the information estimators are then presented for the
generic model in \sref{sec:generic_model}. Concluding remarks are given in \sref{sec:conc}.
\section{Background}
\label{sec:background}
\subsection{Estimators of shared information in classical spin chains}
\label{sec:estimators}
The idea to measure mutual information between two random variables was first
introduced by Shannon in the context of the theory of communication \cite{Shannon}.
In recent years this approach has been extended to systems with many degrees of freedom. While a measure
of the information among all variables in a system (multipartite information) is hard to compute, we can learn much from measuring the
mutual information between two macroscopic parts of the system (bipartite information). It is possible to define
more than one estimator of shared information in bipartite systems, as demonstrated in \cite{Alcaraz2010}.
They all measure in different ways the mutual dependence between two compartments of a system and quantify
the amount of uncertainty about one subsystem when knowing only the state of the other.
In this paper, we study two such bipartite information estimators, namely, the separation entropy estimator ($S_{\A:\B}$), the mutual-information estimator
($\mathcal{I}_{\A:\B}$).
We define the estimators for a classical spin chain of size $L$,
where every site is occupied by a spin variable $\sigma_{i}$
that takes $p$ discrete values, $\sigma_i=1,\dots,p$. We consider a spatial bipartition of the
system into two parts, $\A$ and $\B$, of size $\ell$ and $(L-\ell)$,
respectively, such that sites $\{1,...,\ell\}$ belong to subsystem
$\A$ and the remainder to subsystem $\B$. A configuration of the system is denoted
by $({\sig}^{\A},{\sig}^{\B})$, where ${\sig}^{\A}\equiv\{\sigma_{1},\dots,\sigma_{\ell}\}$
and ${\sig}^{\B}\equiv\{\sigma_{\ell+1},\dots,\sigma_{L}\}$
are the spin configurations of the two subsystems. We denote the equilibrium probability of a configuration by $P\left({\sig}^{\A},{\sig}^{\B}\right)$.
In order to define the information estimators one has to consider also the marginal probability distribution of each subsystem,
obtained by summing over the configuration of the other subsystem, yielding
\begin{equation}
P_\M^\A({\sig}^{\A})=\sum_{{\sig}^{\B}}P({\sig}^{\A},{\sig}^{\B}),\qquad
\textrm{and} \qquad P_\M^\B({\sig}^{\B})=\sum_{{\sig}^{\A}}P({\sig}^{\A},{\sig}^{\B}).
\end{equation}
In addition, we consider the probability distributions of the two subsystems when they
are physically decoupled, denoted by $P(\sig^{\A})$ and $P(\sig^{\B})$.
The decoupling is obtained by turning off all the interactions between spins belonging to different subsystems.
Note that in the decoupled state, the distribution of the composite system is given by a product form,
$P(\sig^{\A},\sig^{\B})=P(\sig^{\A})P(\sig^{\B})$,
yielding $P^{A}_\M(\sig^{A})=P(\sig^{A})$ and $P^{B}_\M(\sig^{B})=P(\sig^{B})$.
The two information estimators are defined in terms of the above distribution functions as follows:
\begin{enumerate}
\item the mutual information:
\begin{equation}
\label{eq:I_def}
\fl \mathcal{I}_{\A:\B} =
\sum_{{\sig}^{\A}}\sum_{{\sig}^{\B}}P({\sig}^{\A},{\sig}^{\B})\ln\left[\frac{P({\sig}^{\A},{\sig}^{\B})}{P_\M^\A({\sig}^{\A})P_\M^\B({\sig}^{\B})}\right],
\end{equation}
\item and the separation entropy:
\begin{equation}
\label{eq:S_def}
S_{\A:\B} = \sum_{{\sig}^{\A}}P({\sig}^{\A})\ln
P({\sig}^{\A})+\sum_{{\sig}^{\B}}P({\sig}^{\B})\ln P({\sig}^{\B}) -
\sum_{{\sig}^{\A},{\sig}^{\B}}P({\sig}^{\A},{\sig}^{\B})\ln
P({\sig}^{\A},{\sig}^{\B}).
\end{equation}
\end{enumerate}
These estimators can be written in a more compact form using the Shannon entropy
$H[P(\sig)]=-\sum_{\sig}P(\sig)\ln P(\sig)$, as
\begin{eqnarray}
\label{eq:Idef2}
I_{\A:\B}=H[P_\M^\A(\sig^\A)]+H[P_\M^\B(\sig^\B)]-H[P(\sig^{\A},\sig^\B)], \\\nonumber
\label{eq:Sdef2}
S_{\A:\B}=H[P(\sig^{\A},\sig^\B)]-H[P(\sig^\A)]-H[P(\sig^\B)].
\end{eqnarray}
Of the two, the mutual information has been studied more extensively in the context of quantum systems \cite{Eisert2010}.
\subsection{Mean-field models}
\label{sec:models_intro}
It is in general quite difficult to compute the above information estimators for a
classical many-body systems in two or higher dimensions, and one has to resort to numerical methods
\cite{Iaconis2013}. The analytical calculations are simpler in
one-dimension, but the absence of phase transitions in short-range interacting
systems makes the computation redundant. For this reason, we consider
models with mean-field interactions which are known to exhibit rich phase
diagrams even in one dimension, providing non-trivial examples for studying
shared information.
In order to identify generic properties we consider a classical spin model with both mean-field and short-range
interactions of a general type. The model is defined on a one-dimensional lattice of length $L$. In the case where there is only mean-field
interactions, the notion geometry is lost and $L$ denotes simply the number of spins in the systems.
Every site is occupied by a $(p+1)$-state spin variable. We choose $p+1$ states rather than simply $p$ in order
to simplify the notation in the detailed calculation. We consider a general form of the mean-field interaction among the spins, defined
in terms of $Q_{k}$ variables with $k=1,\dots,p+1$, which counts the number of spins in
the $k^{\mathrm{th}}$ state and defined as
\begin{equation}
\label{eq:app_Qj}
Q_k(\sig)=\sum_{i=1}^{L} \delta_{l_k,\sigma_i}.
\end{equation}
Here $\delta$ denotes the Kronecker delta, $l_k$ is the value of $\sigma_i$ in the $k^{\mathrm{th}}$ state and $\sig$ denotes a spin
configuration of the entire system. We define the Hamiltonian as
\begin{equation}
\label{eq:Ham_generic}
\HH(\sig,L ) = L ~\epsilon\Big(\frac{{\bf Q}(\sig)}{L}\Big)+\sum_{i,j} \phi_{i,j}(\sig),
\end{equation}
where ${\bf Q}=\{Q_1,\dots,Q_{p}\}$, and $\epsilon$ is an arbitrary function
which accounts for the mean-field interaction. The function $\epsilon$ depends only on $p$ parameters, rather than $p+1$, because the sum of the $Q$'s is always
$L$, i.e. $Q_{p+1}=L-\sum_{i=1}^p Q_i$.
The second interaction term in \eref{eq:Ham_generic}, $\phi_{i,j}$,
represents a general short-range interaction potential among the spins, which
vanishes when $\vert i-j\vert$ is larger than some finite distance, $R$, which does not scale with
the system length \footnote{ For simplicity we focus here on the case of interval boundary conditions in the definition of $\phi_{i,j}$. The generalization of the derivation below to other boundary conditions is straightforward, and
their effect is found only in the constant term in the scaling form in \eref{eq:area_div}.}.
Note that, because the ${\bf Q}(\sig)/L$ is intensive, the Hamiltonian remains extensive
in spite of the infinite-range interaction.
The above form of the Hamiltonian describes a large class of mean-filed models.
The two specific instances of the model which have been studied extensively in
the past are the mean-field Blume, Emery and Griffiths (BEG) model \cite{Blume1971,Barr2001} and the Nagel-Kardar (NK) model \cite{Nagle1970,Bonner1971,Kardar1983,Mukamel2005}.
Despite being one-dimensional, both models display a rich phase diagram. In the
following two subsections we present a brief description of the phase diagram that will be relevant for
our analysis.
\subsubsection{The mean-field BEG model:}
This is a three state spin model with variable
$\sigma_{i}=\left\{ -1,1,0 \right\}$ and a Hamiltonian
\begin{equation}
\label{eq:beg_ham}
\HH(\sig,L)=\Delta\sum_{i=1}^{L}\sigma_{i}^{2}-\frac{J}{2L}\left(\sum_{i=1}^{L}\sigma_{i}\right)^{2}.
\end{equation}
The parameter $\Delta$ is the on-site field strength and $J$ is the strength of the infinite range interaction between all the spins.
This is a special case of the general Hamiltonian in
\eref{eq:Ham_generic} with $p=2$ and $\epsilon(q_{1},q_2)=\Delta (q_1+q_2) -
J (q_2-q_{1})^2/2$ and $\phi_{i,j}(\sig)=0$.
The BEG model has been used in the past as a prototypical model of
long-range interacting systems, particularly in the study of ensemble inequivalence,
whereby a model exhibits different phase diagrams within two different ensembles \cite{Barr2001,Leyvraz2002,Ellis2004}. In both the micro-canonical and the canonical ensembles
the BEG model undergoes a phase transition between a paramagnetic (disordered) phase where
the average magnetization $ m = L^{-1} \langle \sum_{i}\sigma_i \rangle=0$, to
a ferromagnetic (ordered) phase, where $m\neq0$. The phase diagram in the two ensembles is shown
in \fref{fig:BEG}a. The temperature in the micro-canonical ensemble is defined by
the thermodynamic relation $T^{-1}=k_{B}\beta =\partial S/\partial E$ with
$S$ and $E$ being the entropy and the energy, respectively, and $k_B$ denoting the Boltzmann constant.
The two thermodynamic phases are separated by the following critical line:
\begin{equation}
\label{eq:critical line BEG}
\beta J=\frac{1}{2}\exp\left(\beta\Delta\right)+1,
\end{equation}
which meets a first order transition line at a tricritical point.
The inequivalence of the two ensembles can be seen in the position of the tricritical point and in the first order transition line.
In the canonical ensemble the first order transition is denoted by a thick solid line, where the average magnetization in the system,
$m$, changes discontinuously. In the micro-canonical ensemble the first order transition is denoted by two stability lines, which encompass a region
where the ordered and disordered phases are both either stable or meta-stable.
This inequivalence is a common feature in the long-range interacting systems \cite{Dauxois2009,Dauxois2010}.
\begin{comment}
An important property of the BEG Hamiltonian is that all the sites are
equivalent and hence the notion of locality and the dimension is lost. As a consequence,
the reduced correlation $c(r)=\langle\sigma_{i}\sigma_{i+r}\rangle-\langle\sigma_{i}\rangle\langle\sigma_{i+r}\rangle$
does not change with the distance $r$, and scales as $c(r) \sim 1/L$. At criticality
the scaling changes to $c(r) \sim 1/\sqrt{L}$, and is reflected in the behavior
of information estimators in \sref{sec:main}.
\end{comment}
\begin{figure}
\begin{center}
\includegraphics[scale=0.74]{fig1}
\caption{Phase diagram of (a) the BEG model and (b) the NK model in the
canonical and the micro-canonical ensemble. For both models, the critical line
(thin, solid) separating the ordered and disordered phases, is identical
in the two ensembles. The line terminates at different tricritical points in each ensemble, denoted by a star ($\star$) in the canonical ensemble
and by a square ($\blacksquare$) in the micro-canonical ensemble. Below the
tricritical points the phases are separated by first order transition line,
denoted by thick solid line in the canonical ensemble and by dashed lines in the
micro-canonical ensemble. In the latter the intermediate region between the
dashed lines does not have a well defined temperature. In the BEG model, the two tricritical points are found to be
very close to each other, and thus appear to coincide in (a).
\label{fig:BEG}}
\end{center}
\end{figure}
\subsubsection{The NK model:}
This is a variant of the nearest-neighbor Ising spin chain with an additional
mean-field interaction term. The spin variables are $\sigma_{i}=\{-1,1\}$ and the Hamiltonian is given by
\begin{equation}
\label{eq:NK model_energy_first}
\HH(\sig,L)=-\frac{K}{2}\sum_{i=1}^{L-1}\left(\sigma_{i}\sigma_{i+1}-1\right)-\frac{J}{2L}\left(\sum_{i=1}^{L}\sigma_{i}\right)^{2}.
\end{equation}
The parameters $K$ and $J$ denote the short-range and long-range interaction strengths, respectively.
Similarly to the BEG model, the mean-field interaction strength is rescaled with
the system length $L$ to keep the energy extensive. This is a special case of the
general Hamiltonian \eref{eq:Ham_generic} with $p=1$, $\epsilon(q_1)= -\frac{J}{2}(2q_1-1)^2$ and $\phi_{i,j}(\sig) = - \frac{K}{2} (\sigma_{i}\sigma_{j}-1) \delta_{j-i,1}$.
The model has been studied within both the micro-canonical \cite{Nagle1970,Bonner1971,Kardar1983} and then canonical \cite{Mukamel2005}
ensembles. The phase diagrams corresponding to the two ensembles are shown in \fref{fig:BEG}b.
Similarly to the BEG model, the NK model exhibits a disordered phase with
vanishing average magnetization, and an ordered phase where the magnetization is
non-zero. At small values of the ratio $K/J$ the two phases are separated by a
second order transition line which in both the ensembles is given by
\begin{equation}
\beta J =\exp(-\beta K).
\end{equation}
As $K/J$ increases the second order transition line turns into a first order
line at a tricritical point, which is different for the two ensembles.
This ensemble inequivalence is qualitatively similar
to the one observed in the BEG model, as evident by the similarities between \fref{fig:BEG}a and \fref{fig:BEG}b.
\section{The main results}
\label{sec:main}
In this section, we summarize the main results of our study of the information
estimators in the general model defined in
\eref{eq:Ham_generic}. A detailed derivation of these results is given in \sref{sec:generic_model}.
We consider the fictitious partitioning of the system into two subsystems, $\A$ and $\B$ of size $\ell$ and $L-\ell$, respectively, within two limits:
one where the sizes of both subsystems scales linearly with $L$, i.e. $1 \ll \ell \sim L$, and the other where
$\ell$ is large but does not scale with $L$, i.e. $1\ll \ell \ll L$. Both of these limits
have been considered in the past studies of entanglement in quantum systems. One would expect that the scaling
behavior of entanglement estimators in the second limit can be obtained by taking $\ell/L\to 0$ in the results obtained from the first limit ($1\ll \ell \ll L$).
To our surprise, we
find that for the mutual information estimator at criticality this is not true. This difference between the two limits can be understood by a careful analysis, presented in \sref{sec:beg_finite}, which we also verify using a numerical computation.
The results below are presented first in the $\ell \sim L$ limit for the
canonical and microcanonical ensemble in
\sref{sec:sum_can} and \sref{sec:sum_mc}, respectively. The differences found in the $1\ll \ell \ll L$ limit are summerized in \sref{sec:sum_finite_size}.
\subsection{Canonical ensemble:}
\label{sec:sum_can}
{\it Away from the critical line}, both the
information estimators follow the area law, \textit{i.e.}, they remain finite as $L\to \infty$.
To leading order in $L$ we obtain that
\begin{eqnarray}
\label{eq:sum_I_beg_can_away}
I_{\A:\B} &=&\frac{1}{2}\ln\left[ g(\alpha)g(1-\alpha)\right] + \mathcal{O}(1), \\
S_{\A:\B}&=& O(1),\nonumber
\end{eqnarray}
where $\alpha \equiv \ell / L$ is the fractional volume of the subsystem
$\A$ and $g$ is a scaling function.
The symbol $\mathcal{O}(1)$ denotes terms that do not increase with either $L$ or
$\ell$, and do not depend of $\alpha$. These terms involve the microscopic details of the model.
In general, we find that the scaling function $g(\alpha)$ has a non-universal form that
depends on the details in the Hamiltonian \eref{eq:Ham_generic}. For a
$(p+1)$-state spin chain, it is a polynomial of degree $p$ defined as
\begin{equation}
\label{eq:sum_g}
g(\alpha)= b_{p} \alpha^{p} + b_{p-1} \alpha^{p-1}+\ldots + b_2 \alpha^2 + \alpha+b_0,
\end{equation}
where $b_i$ depend on the details of the model. Note that, the coefficient of the
linear term is $1$.
Along the critical line, both the estimators diverge with $L$ and resemble the
scaling seen in the entanglement estimators in quantum systems. The mutual
information is given {\it at criticality} by
\begin{eqnarray}
\label{eq:sum_I_beg_can}
I_{\A:\B}=\frac{1}{4}\ln \left[L
g(\alpha)^{2}g(1-\alpha)^{2}\right] + \mathcal{O}\left( 1 \right).
\end{eqnarray}
At criticality $g(0)$ vanishes and thus $b_0=0$.
In the example of the NK model, where $p=1$, this leads to a simple form of the scaling
function $g(\alpha)=\alpha$. For the BEG model, although $p=2$, the scaling
function is also give by $g(\alpha)=\alpha$. This is because the $\alpha^{2}$ term is excluded due to
a particular symmetry of the Hamiltonian, discussed in \sref{sec:gen_can}.
For the separation entropy {\it at criticality} the leading divergence with $L$ is
$\sqrt{L}$ with a negative sub-leading logarithmic term. The overall scaling form is thus given by
\begin{eqnarray}
\label{eq:sum_S_beg_can}
\mathcal{S}_{\A:\B}=\gamma L^{1/2} \left[(1-\alpha)^{1/2}+\alpha^{1/2}-1\right]
-\frac{1}{4}\ln\left[ L \alpha(1-\alpha)\right]+\mathcal{O}(1),
\end{eqnarray}
where $\gamma$ is a non-universal coefficient whose explicit form is
derived in \sref{sec:beg_degeneracy}.
It is a strictly positive quantity resulting a positive separation entropy
$S_{\A:\B}$. We have verified this numerically for the BEG model as shown in the
\fref{fig:gamma}.
This is consistent with the fact that the entropy of the composite
system is higher than the combined entropy of the isolated subsystems.
Unlike the coefficient of the $\sqrt{L}$ term, the coefficient of the sub-leading
$\ln L$ term is universal and remains constant along the critical line.
\begin{figure}
\begin{center}
\includegraphics[scale=0.65]{fig2}
\caption{\label{fig:gamma}
The coefficient $\gamma$ of the leading order term in the separation entropy at criticality in the canonical BEG model as a function of $\Delta/J$.
The coefficient diverges at the tricritical point, $\Delta/J \simeq 0.462$, where the critical line turns into a first order transition line. }
\end{center}
\end{figure}
\subsection{Micro-canonical ensemble:}
\label{sec:sum_mc}
In the micro-canonical ensemble the total energy of the system is strictly fixed. This global constraint is expected to induce
correlations between microscopic fluctuation in the two subsystems, and one
would expect the area law to break down. This is exactly what we find in
our analysis where both the information estimators have an additional
$\frac{1}{2}\ln L$ term everywhere in the parameter space, even away from criticality.
{\it Away from criticality} we find the following leading $L$ dependence of the shared-information estimators:
\begin{eqnarray}
I_{\A:\B}
&=&\frac{1}{2}\ln\left[ L g(\alpha)g(1-\alpha)\right] +
\mathcal{O}(1),\label{eq:I_mc_away} \\
S_{\A:\B}&=&\frac{1}{2}\ln\left[ L \alpha(1-\alpha) \right]
+O(1),\label{eq:sum_mc_away}
\end{eqnarray}
whereas {\it at criticality} it changes into
\begin{eqnarray}
I_{\A:\B}
&=&\frac{3}{4}\ln \left[ L (g(\alpha)g(1-\alpha))^{2/3}\right]+
\mathcal{O}(1), \label{eq:Iab micro critical} \\
S_{\A:\B}&=&\frac{1}{4}\ln\left[ L \alpha(1-\alpha) \right]
+\mathcal{O}(1).\label{eq:S_micro}
\end{eqnarray}
The scaling function $g(\alpha)$ has the same form as in \eref{eq:sum_g}, with
the coefficients $b_i$ depending on the microscopic details of the model.
For the BEG model, we find that the scaling function has a simple form
$g(\alpha)=\alpha+b_{0}$, where the constant $b_{0}$ vanishes along the critical
line. A plot of the scaling function $g(\alpha)g(1-\alpha)$ for the
BEG model is found in \fref{fig:scaling_I} for representative points in the parameter-space.
Another notable feature is the absence of the leading $\sqrt{L}$ term in
\eref{eq:S_micro} as compared to the form of the separation entropy in the canonical ensemble, given in
\eref{eq:sum_S_beg_can}. This $\sqrt{L}$ divergence in the canonical ensemble results from finite-size corrections to the total energy, as discussed below \eref{eq:gamma1}. In the micro-canonical ensemble where the total energy is strictly fixed this term vanishes.
It is important to stress that the additional $\frac{1}{2}\ln L$ terms observed in the micro-canonical ensemble are due to the fixed energy constraint.
In general, such $\ln L$ terms are related to long-range correlations which in our case can result from either the explicit long-range interactions
or from the total energy constraint. The source of the $\frac{1}{2}\ln L$ term can be verified by setting
the mean-field interaction term to zero in our model, resulting in a model with only short-range interactions.
Following the derivation presented below one obtains a similar $\frac{1}{2}\ln L$ difference between the canonical and microcanonical calculations,
which implies that this difference is indeed due to the total energy constraint.
\begin{figure}
\begin{center}
\includegraphics[scale=0.65]{fig3}
\caption{\label{fig:scaling_I}
The scaling function of the leading order term in the mutual information in the micro-canonical BEG model, $g(\alpha)g(1-\alpha)=(b_0+\alpha)(b_0+(1-\alpha))$.
The function is plotted for $\Delta/J=0.35$ and for different values of the
energy corresponding to the homogeneous, critical and ordered phases.
For convenience we provide the corresponding values of the temperature defined
in the micro-canonical ensemble using the relation $T=(ds/d\epsilon)^{-1}$.
In the homogeneous phase $k_BT/J=0.6$ and $b_0\simeq 0.137$, in the critical phase $k_BT_c/J\simeq 0.497$ and $b_0=0$ and in the ordered phase $k_B T/J=0.49$ and
$b_0\simeq 0.21$.}
\end{center}
\end{figure}
\subsection{Small $\ell/L$ scaling:\label{sec:sum_finite_size}}
As mentioned above, in most cases studied here the scaling behavior of the information estimators in the limit $1\ll \ell \ll L$ can be obtained by
taking $\alpha\equiv \ell/L \to 0$ in the results presented above. The only exception is the behaviour of the mutual information estimator at criticality, for which the $\ell/L\to0$ and $L\to \infty$ limits do not commute. Before we discuss this case, let us summarize the results where this procedure does work.
In the canonical ensemble and {\it away from criticality}, since the scaling function
$g(\alpha)$ in \eref{eq:sum_g} generally obeys $g(0)\neq0$, the information estimators
\eref{eq:sum_I_beg_can_away} are finite when considering $\alpha\rightarrow 0$, i.e.
\begin{eqnarray}
I_{\A:\B}=\mathcal{O}(1),
\qquad S_{\A:\B}=\mathcal{O}(1).
\end{eqnarray}
The same reasoning works in the non-critical micro-canonical ensemble, where the
scaling forms in \eref{eq:I_mc_away}-\eref{eq:sum_mc_away} yield in the $\alpha\rightarrow 0$ limit
\begin{eqnarray}
I_{\A:\B} = &\frac{1}{2}\ln L + \mathcal{O}(1),\qquad
S_{\A:\B}= &\frac{1}{2}\ln \ell +O(1).
\end{eqnarray}
{\it At criticality} in the canonical ensemble taking $\alpha\rightarrow 0$ limit in \eref{eq:sum_S_beg_can} yields
\begin{equation}
S_{\A:\B}= \gamma \ell^{1/2} -\frac{1}{4}\ln \ell +\mathcal{O}(1),
\end{equation}
whereas in the micro-canonical ensemble taking the same limit in \eref{eq:S_micro} leads to
\begin{equation}
S_{\A:\B}=\frac{1}{4}\ln \ell +\mathcal{O}(1).
\end{equation}
Taking the same $\alpha\rightarrow 0$ limit in the expression for mutual
information \eref{eq:sum_I_beg_can} at criticality leads to a negative, diverging result
since $g(0)=0$. This cannot be the right result as the mutual
information is a positive quantity. A detailed microscopic derivation in the
limit $1\ll \ell \ll L$, given in \sref{sec:beg_finite}, yields the correct result whereby
\begin{equation}
I_{\A:\B}=\mathcal{O}(1),
\end{equation}
in the canonical ensemble {\it at criticality}. A similar analysis within the micro-canonical ensemble
shows that the mutual information diverges {\it at criticality} with the system size $L$, as
\begin{equation}
I_{\A:\B} =\frac{1}{2}\ln L + \mathcal{O}(1).
\end{equation}
Similarly to the $\ell \sim L$ limit, this divergence can be attributed to the fixed total energy constraint.
\section{Explicit calculation of the information estimators}
\label{sec:generic_model}
In this section we compute the scaling form of the shared-information for large $L$ using the saddle point method. This calculation is relatively straightforward for pure mean-field models.
As shown below, the inclusion of the short-range interactions
in the generic Hamiltonian (\ref{eq:Ham_generic}) does not alter the derivation significantly and affects only the sub-leading terms in $L$ of the information estimators. The calculation
is first carried out in detail within the canonical ensemble, in \sref{sec:gen_can}, where it is relatively simple. For the microcanonical case, we present a sketch of the calculation in \sref{sec:gen_micro}. Additional issues which include ground-state degeneracy, small $\ell$ scaling and higher order critical points are discussed in section \ref{sec:beg_degeneracy}, \ref{sec:beg_finite} and \ref{sec:higher_order}, respectively.
\subsection{Canonical ensemble}
\label{sec:gen_can}
We consider first the generic model within the canonical ensemble, where the probability of a micro-state of the whole system, $\sig$, is given by
\begin{equation}
P(\sig)= Z^{-1}e^{-\beta\HH(\sig,L)}.
\end{equation}
The partition function, $Z$, is defined as
\begin{equation}
\label{eq:Z_can_delta}
Z=\sum_{\sig}e^{-\beta\HH(\sig)}=\sum_{\sig}e^{-L\beta\epsilon\big({\bf Q}(\sig)/L\big)-\beta\sum_{i,j}\phi_{i,j}(\sig)}.
\end{equation}
The first step in computing the shared-information estimators is the computation of the partition function.
\subsubsection{Partition function:}
The computation of $Z$ can be carried out using a standard technique, employed repeatedly in this paper, by which
we replace the $e^{-L\beta\epsilon\big({\bf Q}(\sig)/L\big)}$ with an integral over a continuous variable ${\bf q}$, yielding
\begin{equation}
e^{-L\beta\epsilon\big({\bf Q}(\sig)/L\big)} = \int d{\bf q}e^{-L\beta\epsilon({\bf q})} \prod_{j=1}^{p}\delta\big( Q_{j}(\sig)/L- q_j\big),
\end{equation}
where $\delta$ denotes the Dirac delta function. For the partition function, this procedure yields
\begin{equation}
Z=L^p \int d{\bf q} e^{-L\beta\epsilon({\bf q})}\sum_{\sig}e^{-\beta\sum_{i,j}\phi_{i,j}(\sig)}\prod_{j=1}^{p}\delta\big( Q_{j}(\sig)-L q_j\big).
\end{equation}
The delta function can be replaced by a set of integrals over a $p$-vector-field ${\bf h}$, yielding
\begin{equation}
\label{eq:Z_can_0}
Z= L^p\int d{\bf q} d {\bf h} e^{-L\beta[\epsilon({\bf q})+\mathbf{h}\cdot\mathbf{q}]}\sum_ {\sig}e^{-\beta\sum\limits_{i,j}\phi_{i,j}(\sig)+\beta\mathbf{h}\cdot\mathbf{Q}(\sig)}.
\end{equation}
The sum over $\sig$ in \eref{eq:Z_can_0} is in fact the partition sum of
a system with a short-range interaction term,
$\phi_{i,j}(\sig)$, and a field conjugate to $Q_{j}(\sig)$, denoted by ${\bf h}$, whose Hamiltonian is thus given by
\begin{equation}
\label{eq:Ham_short}
\HH'(\sig)=\sum_{i,j}\phi_{i,j}(\sig)-{\bf h}\cdot {\bf Q}(\sig)
\end{equation}
In the following we demonstrate that the partition sum of this Hamiltonian is given to leading order in $L$ by $e^{L\ln\lambda_1+\mathcal{O}(1)}$,
where $\lambda_1$ is the largest eigenvalue of the transfer matrix corresponding to $\HH'$, and $\mathcal{O}(1)$ denotes terms that do not scale with $L$.
The crucial point is that there are no $\ln L$ terms in the exponent.
To this end, the partition sum is written in terms of transfer matrices as
\begin{equation}
\label{eq:Z_can_1}
\sum_ {\sig}e^{-\beta\sum_{i,j}\phi_{i,j}(\sig)+\beta\mathbf{h}\cdot\mathbf{Q}(\sig)}=\langle 1 | T_{\beta,\mathbf{h}}^{L}|1\rangle
\end{equation}
where $T_{\beta,\mathbf{h}}$ is the transfer matrix corresponding to the $\HH'$ and $\langle 1 |, |1 \rangle$ are the left and right identity vectors.
An important property of $T_{\beta,\mathbf{h}}$ is that its dimension does not scale with $L$.
In the case $\phi_{i,j}$ describes only nearest neighbours interactions, denoted by $\phi_{i,j}(\sig)=K_{\sigma_i, \sigma_{j}}\delta_{j,i+1}$, the transfer matrix is of dimension $(p+1)\times (p+1)$ and it is given by
\begin{equation}
\big(T_{\beta,\mathbf{h}}\big)_{q,r}=e^{-\beta (K_{q,r}-h_q)}.
\end{equation}
For more general interaction range, $R$, the transfer matrix is constructed similarly, but by taking into account the state of the $R$ nearest neighbours. The dimension of the matrix is therefore at most $(p+1)^R\times (p+1)^R$. The fact that $T_{\beta,\mathbf{h}}$ is of finite dimension implies by the Perron-Frobenius theorem that its largest and its second largest eigenvalues
differ by a gap which is independent of $L$. Denoting the eigenvalues by $\lambda_k$ and the corresponding eigenvectors by $|v_k\rangle$, we obtain that
\begin{eqnarray}
\label{eq:Z_can_2}
\langle 1|T_{\beta,\mathbf{h}}^{L}|1\rangle=\lambda_1^L\big( \langle 1|v_1\rangle\big)^2 \big[1+\mathcal{O}(|\lambda_2|^L/\lambda_1^L)\big].
\end{eqnarray}
Inserting the leading order term in \eref{eq:Z_can_2} into the partition sum in \eref{eq:Z_can_0}, one obtains an integral which can be evaluated
in the $L\to \infty$ limit using the saddle point approximation (SPA) \footnote{
The equivalence of statistical ensembles in short-range interacting systems implies that $\lambda_1$ is a convex function of ${\bf h}$. This allows us to perform
the Laplace transform \eref{eq:Z_can_delta} and the corresponding inverse transform to obtain the correct leading order contribution to the original sum. }.
We denote the result of the approximation of the integral over ${\bf h}$ by
\begin{equation}
\label{eq:Z_can_3}
\int d{\bf h} e^{-L\beta\mathbf{h}\cdot\mathbf{q}}\left\langle 1| T_{\beta,\mathbf{h}}^{L}|1\right\rangle = \omega(\mathbf{q}) e^{-L\beta f_\phi(\beta,{\bf q})}\big[1+\mathcal{O}(L^{-1})\big] ,
\end{equation}
where $f_\phi(\beta,{\bf q})$ denotes the Landau free energy density of ${\bf q}$ in a system with only short-range interactions, given by $\phi_{i,j}(\sig)$.
The function $\omega(\mathbf{q})$ accounts for the $\mathcal{O}(1)$ pre-factor of the leading order term in $\langle 1| T_{\beta,\mathbf{h}}^{L}|1\rangle$ and additional
pre-factors that result from the saddle point approximation. A specific example of $f_\phi(\beta,{\bf q})$ and $\omega({\bf q})$ can be obtained
for pure mean-field models, where $\phi_{i,j}(\sig)=0$. In this case one obtains from combinatorial considerations that
\begin{eqnarray}
f_\phi(\beta,{\bf q}) &=& \frac{1}{\beta}\sum_{j=1}^p q_j \ln q_j - \frac{1}{\beta}(1-\sum_{j=1}^p q_j) \ln (1-\sum_{j=1}^p q_j) \equiv - \frac{1}{\beta} s_0({\bf q}) ,\\
\omega({\bf q}) &=& \Big[ \pi^p (\prod_{j=1}^p q_j) (1-\sum_{j=1}^p q_j)\Big]^{-1/2} \equiv \omega_0({\bf q}).
\end{eqnarray}
Here $s_0({\bf q})$ is the entropy of a noninteracting spin system for a given value of ${\bf q}$.
Inserting \eref{eq:Z_can_3} into the partition function in \eref{eq:Z_can_1} yields finally
\begin{equation}
\label{eq:app_Z2}
Z=L^{p/2}\int d{\bf q} \omega(\mathbf{q}) e^{-L\beta f_\HH(\beta,{\bf q})}\big[1+\mathcal{O}(L^{-1})\big],
\end{equation}
where $f_\HH(\beta,{\bf q})=\epsilon({\bf q})+f_\phi(\beta,{\bf q})$ is the Landau free energy density of the complete system, which includes
the short-range and the long-range interaction terms.
This integral in \eref{eq:app_Z2} can be further evaluated using the SPA, which we choose to separate into two steps. In the first step we approximate the integrals over
$q_2,\dots,q_p$, by expanding the exponent to quadratic order in these variables. The next step is to approximate the remaining one-dimensional
integral, by expanding the exponent to order $q_1^2$ away from criticality and to order $q_1^4$ at criticality.
In cases where $p=1$, such as in the NK model, the first step is skipped.
The first step of the SPA yields,
\begin{equation}
\label{eq:app_Z_first_step}
Z = L^{1/2} \int dq_1 \tilde{\omega}(q_1) e^{-L \beta f_\HH(\beta,\tilde{{\bf q}})}\big[1+\mathcal{O}(L^{-1})\big]
\end{equation}
%
where $\tilde{\bf q}(q_1)=\left(q_1,{\tilde q}_2(q_1),{\tilde q}_3(q_1),\dots, {\tilde q}_p(q_1)\right)$ is the solution of the set of equations $\partial f_\HH/ \partial q_i =0$
for $i=2,3,\dots,p$.
The function $\tilde{\omega}(q_1)$ accounts for the contribution from $\omega (\tilde{{\bf q}})$ and the coefficients that results from the SPA,
\begin{equation}
\label{eq:omega_tilde}
\tilde{\omega}(q_1)\equiv \omega (\tilde{{\bf q}}(q_1)) \pi^{(p-1)/2} \Big[ \det \Big( \frac{\beta \partial^2 f_\HH}{\partial q_i \partial q_j} \Big\vert_{\tilde{{\bf q}}(q_1)} \Big)\Big]^{-1/2},
\end{equation}
for $i,j=2,3,\dots,p$.
{\it Away from the critical line}, the SPA over of the integral over $q_1$ in \eref{eq:app_Z_first_step} yields
\begin{eqnarray}
\label{eq:app_Z_away}
Z \simeq \tilde{\omega}(q^\star_{1}) e^{-L \beta f_\HH(\beta,{\bf q}^\star)} \chi_{0,2}\left( \beta \frac{d^2 f_\HH(\tilde{{\bf q}}(q_1))}{d q_1^2} \Big\vert_{q^\star_{1}} \right),
\end{eqnarray}
where ${\bf q}^\star = (q^\star_{1},q^\star_{2},\dots,q^\star_{p})$ is the global minimum of $f_\HH$ and $\chi_{2,0}(a)=\sqrt{\pi/a}$ denotes the coefficient
which results from the Gaussian integral.
Since such integrals are performed frequently in the rest of the paper, it is convenient to define
the following notation:
\begin{equation}
\label{eq:general_gausian_integral}
\fl\qquad
\int_{-\infty}^{\infty}dx x^{r}e^{-L a
x^{s}}=\frac{1}{L^{\frac{r+1}{s}}}\left[\frac{2}{s\times
a^{\frac{r+1}{s}}}\Gamma(\frac{r+1}{s})\right] \equiv \frac{1}{L^{\frac{r+1}{s}}}\chi_{r,s}(a),
\end{equation}
where $a>0$ and $r,s$ are positive even integers. In this section and in \sref{sec:gen_micro} we assume that the Landau free energy, $f_\HH(\beta,{\bf q})$, has a single global minimum. The effect
of degenerate minima is discussed in \sref{sec:beg_degeneracy}, where the degeneracy is shown to affect only the $\mathcal{O}(1)$ term of the information estimators.
On the critical line, we assume without loss of generality that the determinant in \eref{eq:omega_tilde} does not vanish.
This implies that the order parameter of the transition is a combination of the $q$'s that necessarily involves $q_1$. On the other hand, the argument
of $\chi_{0,2}$ in \eref{eq:app_Z_away} does vanish at criticality. As a result, the SPA of the integral in \eref{eq:app_Z_first_step} has to be carried out by expanding the
exponent in \eref{eq:app_Z_first_step} to order $q_1^4$, yielding {\it at criticality} the following scaling form:
\begin{eqnarray}
\label{eq:app_Z_at}
Z \simeq L^{1/4} \tilde{\omega}(q^\star_{1}) e^{-L\beta f_\HH(\beta,{\bf q}^\star)} \chi_{0,4}\left( \beta \frac{d^4f_\HH(\beta,\tilde{{\bf q}}(q_1))}{d q_1^4} \Big\vert_{q^\star_{1}}\right).
\end{eqnarray}
The resulting expressions for $Z$ will be used below in the derivation of $S_{\A:\B}$ and $\mathcal{I}_{\A:\B}$.
\subsubsection{Separation entropy:}
The Shannon separation entropy, $S_{\A:\B}$, can be derived directly from the expression Shannon entropy of the whole system, given by
\begin{equation}
\label{eq:S3}
S=-\sum_{\sig} P(\sig) \ln P(\sig) = \ln Z+Z^{-1}\sum_{\sig}e^{-\beta\HH(\sig)}\beta\HH(\sig).
\end{equation}
Following a derivation similar to that of $Z$, the entropy can be expressed in terms of the transfer matrix of a short-range interacting system, whose Hamiltonian is $\HH'$, as
\begin{equation}
S=\ln Z+Z^{-1}L^p\int d{\bf q} d{\bf h} e^{-L\beta[\epsilon({\bf q})+\mathbf{h}\cdot\mathbf{q}]}\Big[L\beta\epsilon({\bf q})\langle 1| T_{\beta,\mathbf{h}}^{L}|1\rangle -\frac{\partial}{\partial\beta}\langle 1| T_{\beta,\mathbf{h}}^{L}|1\rangle\Big].
\end{equation}
Similarly to the derivation of \eref{eq:app_Z2}, in the $L\to\infty$ limit one may consider only the leading order term of $\langle 1| T_{\beta,\mathbf{h}}^{L}|1\rangle$
and evaluate the integral over ${\bf h}$ of using the SPA, yielding
\begin{equation}
\label{eq:S_int}
S=\ln Z+Z^{-1}L^{p/2}\int d{\bf q}\omega(\mathbf{q})e^{-L\beta f_\HH(\beta,\mathbf{q})}L[\beta\epsilon(\mathbf{q})+\beta\phi(\beta,\mathbf{q})]+\mathcal{O}(1),
\end{equation}
where $\phi(\beta,\mathbf{q})\equiv\frac{\partial}{\partial \beta} [\beta f_\phi(\beta,\mathbf{q})]$ is the average energy of the short-range interacting
system, when constrained on a specific value of the coarse variables, ${\bf Q}(\sig)=L\mathbf{q}$.
One can define in a similar manner the average entropy of this system, $s({\bf q})\equiv-\beta f_\phi(\beta,\mathbf{q})+\beta\phi(\beta,\mathbf{q})$, which will be used below.
As in the case of $Z$, \eref{eq:S_int} can be evaluated using a two-step saddle point approximation of the integral over ${\bf q}$. In the first step, the SPA of the integrals over $q_2,\dots,q_p$ yields
\begin{equation}
\label{eq:app_S3}
S \simeq -\ln Z - L \beta\frac{ \int d q_1 \tilde{\omega}(q_1) [ \epsilon(\tilde{{\bf q}}(q_1))+\phi(\beta,\tilde{{\bf q}}(q_1))] e^{-L \beta f_{\HH}(\beta,\tilde{{\bf q}}(q_1))} }
{\int d q_1 \tilde{\omega}(q_1)e^{-L \beta f_{\HH}(\beta,\tilde{{\bf q}}(q_1))} }.
\end{equation}
The SPA of the integral over $q_1$ is done by expanding the exponent to order $q_1^2$ {\it away from criticality} yielding
\begin{equation}
\label{eq:app_S_away}
S = L s({\bf q}^\star) + \mathcal{O}(1).
\end{equation}
On the other hand, {\it at criticality} the exponent need to be expanded to order $q_1^4$, leading to the following scaling form:
\begin{equation}
\label{eq:app_S_at}
S = L s({\bf q}^\star) + \gamma L^{1/2} - \frac{1}{4} \ln L + \mathcal{O}(1).
\end{equation}
Note that in deriving \eref{eq:app_S_away} and \eref{eq:app_S_at} $Z$ has been replaced by its expression in \eref{eq:app_Z_away} and \eref{eq:app_Z_at}, respectively.
The coefficient $\gamma$, given by
\begin{equation}
\label{eq:gamma1}
\gamma = \beta \frac{\chi_{2,4}(\beta\frac{d^4 f_{\HH}}{d q^4_1}|_{q_1^\star})}{\chi_{0,4} (\beta\frac{d^4 f_{\HH}}{d q^4_1}|_{q_1^\star}) \omega({\bf q}^\star)} \frac{d}{dq_1^2} \big\{ \tilde{\omega}(q_1)[\epsilon( \tilde{\bf q}(q_1))+\phi(\beta, \tilde{\bf q}(q_1)) - \epsilon({\bf q}^\star)-\phi(\beta,{\bf q}^\star) ] \big\} \Big|_{q_1^\star},
\end{equation}
depends in general on the parameters of the model. Its form suggests that it is related to the finite-size corrections to the mean energy.
This can be clearly understood by noting that the source of the $\gamma L^{1/2}$ term is the second term in the RHS of \eref{eq:S3}, which corresponds to the average energy in the systems, i.e. $Z^{-1}\sum_{\sig}e^{-\beta\HH(\sig)}\beta\HH(\sig) = \langle \HH \rangle = [\epsilon({\bf q}^\star)+\phi(\beta,{\bf q}^\star)] L + \gamma L^{1/2} + O(\log L)$.
This coefficient is studied in more detail in the BEG model in \fref{fig:gamma}, where it is found shown to be
strictly positive.
This coefficient was found to diverge at the tricritical point, where
$\frac{d^2 f_{\HH}}{d q^2_1}|_{q_1^\star}=\frac{d^4 f_{\HH}}{d q^4_1}|_{q_1^\star} =0$, indicating that the exponent in \eref{eq:app_S3} has to be expanded to order $q_1^6$ in
order to obtain the correct scaling of $S$. This divergence is evident in \fref{fig:gamma}. The behaviour of $S$ at tricritical points is discussed in \sref{sec:higher_order}.
As mentioned above, the separation entropy measures the difference between the Shannon entropy
of the whole system and that of the two subsystems when they are physically decoupled.
The two decoupled subsystems $\A$ and $\B$, are assumed to obey the Gibbs-Boltzmann distribution
with respect to the Hamiltonian, $\HH(\sig,\ell )$ and $\HH(\sig,L-\ell )$, respectively, where $\HH$ is defined in \eref{eq:Ham_generic}. This implies
that after the separation the interaction strength in the Hamiltonian of each subsystem
has to be rescaled with the size of the each subsystem. This rescaling ensures that the separated subsystems would have the same
values of ${\bf q}^\star$ as those of the composite system.
Since the decoupled subsystems maintain the form of the Hamiltonian of the whole system, their Shannon entropies are given by the above expression with the size
$L$ replaced by $\ell$ for subsystem $\A$ and by $L-\ell$ for subsystem $\B$.
As a result the extensive terms in the separation entropy in \eref{eq:S_def} cancel and we obtain that {\it away from criticality} to leading order
\begin{eqnarray}
\label{eq:can_S_away}
S_{\A:\B}=\mathcal{O}\left( 1 \right).
\end{eqnarray}
The cancelation of the $\mathcal{O}(L)$ terms in $S_{\A:\B}$ suggests that the rescaling of the Hamiltonians, described above, is a physically sensible way to define the separation process.
{\it At criticality}, the extensive terms still cancel but the $\sqrt{L}$ and $\log L$ terms do not, yielding
\begin{eqnarray}
\label{eq:can_S_at}
S_{\A:\B} = \gamma L^{1/2}\left[\sqrt{\alpha}+\sqrt{1-\alpha}- 1
\right]-\frac{1}{4}\ln\left[ L \alpha \left( 1-\alpha \right)
\right]+\mathcal{O}\left( 1 \right).
\end{eqnarray}
The results in \eref{eq:can_S_away} and \eref{eq:can_S_at} are verified numerically in \fref{fig:can_SAB} for the BEG and NK models.
\begin{figure}
\begin{center}
\includegraphics[scale=0.6]{fig4}
\caption{
Log-log plot of the Shannon separation estimator in the canonical ensemble as a function of the system-size for $\ell/L=1/2$.
Figure (a) shows the results for the BEG model at criticality ($\Delta/J\approx 0.379$, $k_B T/J\approx 0.473$) and away from criticality ($\Delta/J\approx 0.441$, $k_B T/J\approx 0.552$), denoted by $\star$ and $\bullet$ respectively.
Similarly, figure (b) shows the results for the NK model at criticality ($K/J\approx -0.268$, $k_B T/J\approx 0.670$) and away from criticality ($K/J\approx -0.287$, $k_B T/J\approx 0.718$).
At criticality, the leading order term in $S_{\A:\B}$ is expected to scale as $\sqrt{L}$, depicted by the dashed lines, whereas away from criticality $S_{\A:\B}$ is expected to converge to a constant.
\label{fig:can_SAB}}
\end{center}
\end{figure}
\subsubsection{Mutual information:}
The mutual information estimator is based on the marginal probability distribution of the bipartition, which for subsystem $\A$ is defined as
$P_{\M}^{\A}({\sig}^{\A})=\sum_{{\sig}^{\B}}P({\sig}^{\A},{\sig}^{\B})$. The derivation below is done mainly for subsystem $\A$. The results for subsystem $\B$
can be obtained by replacing $\A \to \B$ and $\ell \to L-\ell$ in the expressions below. The marginal distribution of $\A$ can be written as
\begin{equation}
\label{eq:PMA_three_term}
P_{\M}^{\A}({\sig}^{\A})= \frac{1}{Z}e^{-\beta\sum_{i,j\in\A}\phi_{i,j}(\sig^\A)} Z^{\B}(\mathbf{Q}({\sig}^{\A})/L,\sig^\A)
\end{equation}
where $Z^{\B}(\mathbf{q}^{\A},\sig^\A)$ is the partition function of subsystem $\B$, defined as
\begin{equation}
Z^{\B}(\mathbf{q}^{\A}, \sig^\A) \equiv \sum_{{\sig}^{\B}}e^{-L\beta \epsilon(\mathbf{q}^\A+\mathbf{Q}({\sig}^{\B})/L)-\beta\sum_{i\in \B,j \in \B}\phi_{i,j}(\sig)-2\beta\sum_{i\in \A,j \in \B}\phi_{i,j}(\sig)}.
\end{equation}
Note that the term $\sum_{i\in \A, j\in \B}\phi_{i,j}(\sig)$ in the exponentials corresponds to the short-range interactions on the boundary between the two subsystems,
and thus involves a number of terms that does not scale with $L$. This fact will be used below to neglect its contribution.
In the $L\to\infty$ limit the leading order term in $Z^\B$ can be simplified using the same technique employed in the computation of $Z$ above \eref{eq:app_Z2}, yielding
\begin{eqnarray}
\label{eq:ZBa}
\fl \qquad \qquad Z^{\B}(\mathbf{q}^{\A},\sig^\A)&=&
(L-\ell)^{p}\int d{\bf q}^{\B} d{\bf h}^{\B}
e^{-L\beta[\epsilon(\mathbf{q}^{\A},\mathbf{q}^{\B})+{\bf h}^\B \cdot \mathbf{q}^{\B}] }\big\langle \sig^{\A}_{\mathrm{b}}|T_{\beta,\mathbf{h}^{\B}}^{L-\ell}|1\big\rangle \\
\fl \qquad \qquad &=&(L-\ell)^{p/2}\int d{\bf q}^{\B}e^{-(L-\ell)\beta f_{\HH,1-\alpha}(\beta,{\bf q}^\A,{\bf q}^\B)}\omega(\mathbf{q}^{\B},\sig^\A)
\big[1+\mathcal{O}(L^{-1})\big], \nonumber
\end{eqnarray}
where
\begin{equation}
\label{eq:app_f_def}
f_{\HH,x}(\beta,{\bf q}^\A,{\bf q}^\B) \equiv \frac{1}{\beta x}\epsilon( (1-x) \mathbf{q}^{\A}+ x \mathbf{q}^{\B})+f_\phi(\beta,\mathbf{q}^{\B}).
\end{equation}
The function $f_{\HH,x}(\beta,{\bf q}^\A,{\bf q}^\B)$ can be regarded as the Landau free energy corresponding of a single subsystem of size $xL$, given the values of the coarse variables in the complementary subsystem, denoted by ${\bf q}^\A$. The vector $\big\langle \sig^{\A}_{\mathrm{b}}|$ in \eref{eq:ZBa} denotes configuration of the {\it boundary spins} in
subsystem $\A$ which interact with subsystem $\B$.
This boundary condition affects only the $\mathcal{O}(1)$ coefficient of the leading order term in $L$, denoted by $\omega(\mathbf{q}^{\B},\sig^\A)$.
In general, evaluating the integral in \eref{eq:ZBa} involves a complicated expression for the saddle point of the integrand as a function of ${\bf q}^\A$.
However as will be shown below, for the purpose of computing the $\ln L$ terms in the mutual information it is sufficient to evaluate $Z^\B$ only for ${\bf q}^\A={\bf q}^\star$,
where ${\bf q}^\star$ is the saddle point of Landau free energy of the full system, $f_{\HH}(\beta,{\bf q})$.
For the case that ${\bf q}^\A={\bf q}^\star$, it can be easily shown that the saddle point of $f_{\HH,1-\alpha}(\beta,{\bf q}^\star,{\bf q}^\B)$
is found at ${\bf q}^\B={\bf q}^\star$ as well.
In contrast to the calculation the partition function, the quadratic terms in $q_j^\B$ of the exponential in \eref{eq:ZBa} do not vanish at criticality (since $f_{\HH,x}(\beta,{\bf q}^\star,{\bf q})\neq
f_\HH(\beta,{\bf q})$ for $x<1$). This implies that the SPA of the integral in \eref{eq:ZBa} yields the same results both {\it at criticality} and {\it away from criticality}, given by
\begin{equation}
\label{eq:app_PMA}
Z^\B ( {\bf q}^\star,\sig^\A) = \omega({\bf q}^\star,\sig^\A) \pi^{p/2} \left[\det( A_{1-\alpha})\right]^{-1/2} e^{-L(1-\alpha)\beta f_{\HH,1-\alpha}(\beta,{\bf q}^\star,{\bf q}^\star)}\big[1+\mathcal{O}(L^{-1})\big],
\end{equation}
where the Hessian matrix $A_{1-\alpha}$ is given for a general subsystem-size by
\begin{equation}
\label{eq:app_A_def}
(A_{x})_{i,j}= \beta \frac{\partial ^2 f_{\HH,x} }{\partial q_i^\B \partial q_j^\B} \Big\vert_{{\bf q}^\B={\bf q}^\star}.
\end{equation}
The Shannon entropy of the marginal probability distribution, denoted here by $S_{\M}^{\A}\equiv-\sum_{{\sig}^{\A}}P_{\M}^{\A}({\sig}^{\A})\ln P_{\M}^{\A}({\sig}^{\A})$, is the key ingredient in the mutual information
estimator, which can also be written as $\mathcal{I}_{\A:\B}=S_{\M}^{\A}+S_{\M}^{\B}-S$. Using \eref{eq:PMA_three_term} the marginal entropy can be expressed as
\begin{equation}
\label{eq:SMA_gen}
S_{\M}^{\A} = \ln Z+\big\langle \beta \sum_{i,j\in \A} \phi_{i,j}(\sig) \big\rangle_\HH +
\big\langle \ln Z^{\B}(\mathbf{q}({\sig}^{\A}),{\sig}^{\A}) \big\rangle_\HH + \mathcal{O}(1)
\end{equation}
where $\langle f(\sig) \rangle_\HH\equiv Z^{-1}\sum_{\sig}e^{-\beta\HH(\sig)}f(\sig)$ for a general function $f(\sig)$ and $\mathbf{q}(\sig)\equiv\mathbf{Q}(\sig)/l(\sig)$ with $l(\sig)$ denoting the number of spins in $\sig$. Using the above expression for $S$ and $S_{\M}^{\A}$ in \eref{eq:S_int} and \eref{eq:SMA_gen}, and the expression for $S_\M^\B$, obtained in a similar way as \eref{eq:SMA_gen}, the mutual information can be written as
\begin{eqnarray}
\label{eq:I_genA}
\fl\qquad I_{\A:\B}=\ln Z+\big\langle2\beta\sum_{i\in\A,j\in\B}\phi_{i,j}(\sig)\big\rangle_{\HH}-\big\langle\ln[Z^{\A}(\mathbf{q}({\sig}^{\B}),\sig^\B)Z^{\B}(\mathbf{q}({\sig}^{\A}),{\sig}^{\A})]\big\rangle_{\HH} \nonumber\\
\qquad\qquad\qquad\qquad-L\big\langle\beta\epsilon(\mathbf{q}(\sig))\big\rangle_{\HH}+\mathcal{O}(1).
\end{eqnarray}
The second term in the RHS of the above equation corresponds to the average of the short-range interaction term over the boundary of the bipartition. In one-dimension, the number of terms
in this sum does not increase with $L$, and it therefore contributes only to the $\mathcal{O}(1)$ term in $I_{\A:\B}$. The third and forth terms are evaluated below using the SPA.
In order to compute the term $\big\langle\ln[Z^{\A}(\mathbf{q}({\sig}^{\B}),\sig^\B)Z^{\B}(\mathbf{q}({\sig}^{\A}),{\sig}^{\A})]\big\rangle_{\HH}$,
it is useful to consider the ensemble average of a general function of the $q$ variables in each of the
two subsystems, denoted by $\big\langle g({\bf q}({\sig}^{\A}),{\bf q}({\sig}^{\B})) \big\rangle_\HH$. Using the technique used above in the derivation of $Z$ \eref{eq:app_Z2}, the
average can be written as
\begin{eqnarray}
\fl \quad \big\langle g({\bf q}({\sig}^{\A}),{\bf q}({\sig}^{\B})) \big\rangle_\HH = Z^{-1}\ell^{p}(L-\ell)^{p}\int d{\bf q}^{\A}d{\bf q}^{\B} \int d{\bf h}^{\A}d{\bf h}^{\B} \\
\fl \qquad \quad \times g(\mathbf{q}^{\A},\mathbf{q}^{\B})e^{-L[\beta\epsilon(\alpha\mathbf{q}^{\A}+(1-\alpha)\mathbf{q}^{\B})+\alpha\mathbf{h}^{\A}\cdot\mathbf{q}^{\A}+(1-\alpha)\mathbf{h}^{\B}\cdot\mathbf{q}^{\B}]}\big\langle 1|T_{\beta,\mathbf{h}^{\A}}^{\ell}T_{\beta,\mathbf{h}^{\B}}^{L-\ell}|1\big\rangle\big[1+\mathcal{O}(L^{-1})\big]. \nonumber
\end{eqnarray}
Evaluating using the SPA the integrals over ${\bf h}^\A$ and ${\bf h}^\B$ of the leading order term in $L$ in $\big\langle g({\bf q}({\sig}^{\A}),{\bf q}({\sig}^{\B})) \big\rangle_\HH$ yields
\begin{equation}
\label{eq:avg_qa_qb}
\big\langle g({\bf q}({\sig}^{\A}),{\bf q}({\sig}^{\B})) \big\rangle_\HH \simeq Z^{-1} \ell^{\frac{p}{2}}(L-\ell)^{\frac{p}{2}}\int d{\bf q}^{\A} d{\bf q}^{\B}g(\mathbf{q}^{\A},\mathbf{q}^{\B}) \omega(\mathbf{q}^{\A},\mathbf{q}^{\B}) e^{-L\beta f_{\HH}(\beta,\mathbf{q}^{\A},\mathbf{q}^{\B})},
\end{equation}
where
\begin{equation}
f_{\HH}(\beta,{\bf q}^\A,{\bf q}^\B) = \epsilon(\alpha \mathbf{q}^{\A}+(1-\alpha) \mathbf{q}^{\B})+\alpha f_\phi(\beta,\mathbf{q}^{\A})+(1-\alpha)f_\phi(\beta,\mathbf{q}^{\B}),
\end{equation}
is the Landau free energy of $({\bf q}^\A,{\bf q}^\B)$. The function
$\omega(\mathbf{q}^{\A},\mathbf{q}^{\B})$ accounts for the $\mathcal{O}(1)$ pre-factor of the leading order term in $\big\langle g({\bf q}({\sig}^{\A}),{\bf q}({\sig}^{\B})) \big\rangle_\HH$ and additional pre-factors that result from the SPA. For pure mean-field systems, where $\phi_{i,j}(\sig)=0$, it can be easily shown that $\omega(\mathbf{q}^{\A},\mathbf{q}^{\B})=\omega_0(\mathbf{q}^{\A})\omega_0(\mathbf{q}^{\B})$.
Using \eref{eq:avg_qa_qb} to evaluate the term $\big\langle\ln[Z^{\A}(\mathbf{q}({\sig}^{\B}),\sig^{\B})Z^{\B}(\mathbf{q}({\sig}^{\A}),\sig^{\A})]\big\rangle_{\HH}$ in \eref{eq:I_genA} yields different results depending whether the system is critical or not. Away from criticality, the SPA of the integrals over ${\bf q}^\A$ and ${\bf q}^\B$ can be performed by expanding the
exponential in \eref{eq:avg_qa_qb} to quadratic order in these variables. In this case since $\ln[Z^\A({\bf q}^\B,\sig^{\B}) Z^\B({\bf q}^\A,\sig^{\A})]$ is a slowly varying function in comparison to the exponential, its
leading order contribution involves only $\ln[Z^\A({\bf q}^\star,\sig^{\B}) Z^\B({\bf q}^\star,\sig^{\A})]$.
When inserting the result into \eref{eq:I_genA} the $\mathcal{O}(L)$ term cancels with that of $\big\langle\epsilon(\mathbf{q}(\sig))\big\rangle_{\HH}$, obtained in \eref{eq:app_S3}, yielding {\it away from criticality} the following result:
\begin{eqnarray}
\label{eq:app_I_away}
\mathcal{I}_{\A:\B}= \frac{1}{2}\ln\left[ g(\alpha)g(1-\alpha)\right] + \mathcal{O}(1),
\end{eqnarray}
where $g(x)$ is in general a non-generic scaling function.
In order to derive $g$ it is useful to note that the only $\alpha$-dependent contribution to \eref{eq:app_I_away} comes from $(\det A_{1-\alpha})^{-1/2}$ term in \eref{eq:app_PMA}, which yields $g(x)=\det A_{1-x}$.
According to \eref{eq:app_f_def} and \eref{eq:app_A_def}, each element in $A(x)$ is a linear polynomial of $x$, whose coefficients depend in general
on the parameters of the model. The determinant, $\det A_x$, is thus a polynomial of the form $a_p x^p + a_{p-1} x^{p-1}+\ldots + a_1x + a_0$.
However, since the scaling function is determined up to a constant, it can be written in terms of the rescaled parameters $b_i=a_i/a_1$ as
\begin{equation}
\label{eq:gx_can}
g(x)= \det A_{1-x} = b_p x^p + b_{p-1} x^{p-1}+\ldots + b_2 x^2 + x +b_0.
\end{equation}
This implies that $g(x)$ depends in fact only on $p$ parameters.
In general, one expects the leading term in $\mathcal{I}_{\A:\B}$ resulting from the SPA to scale as $\sqrt{L}$ at criticality. This is because $\mathcal{I}_{\A:\B}$
involves the term $L\big\langle\epsilon(\mathbf{q}(\sig))\big\rangle_{\HH}$ which leads to a divergence of this kind in the case of
$\mathcal{S}_{\A:\B}$. In \ref{app:vanish} it shown, however, that the $\sqrt{L}$ term that comes from the energy
cancels exactly with the one that come from $\big\langle\ln[Z^{\A}(\mathbf{q}({\sig}^{\B}))Z^{\B}(\mathbf{q}({\sig}^{\A}))]\big\rangle_{\HH}$.
The remaining leading order term in \eref{eq:I_genA} comes from the $\ln Z$. Inserting the form of $Z$ in (\ref{eq:app_Z_at}) yields {\it at criticality}
\begin{eqnarray}
\label{eq:app_I_at}
\mathcal{I}_{\A:\B}=\frac{1}{4}\ln L+ \frac{1}{2}\ln\left[ g(\alpha)g(1-\alpha)\right]+
\mathcal{O}\left( 1 \right).
\end{eqnarray}
The results in \eref{eq:app_I_away} and \eref{eq:app_I_at} are verified numerically in \fref{fig:can_IAB} for the BEG and the NK models.
At criticality, since $f_{\HH,1}$ is the landau free energy of the whole system, we find that $\det A_1=0$. This implies in turn that $b_0=0$ in \eref{eq:gx_can},
and that $g(x)$ involves only on $p-1$ non-generic parameters. As a result, for $p=1$, such as in the NK model, $g$ has a generic form, $g(x)=x$.
It is interesting to note that $b_p$ is proportional to the determinant of the Hessian (discriminant) of $\epsilon({\bf q})$, $b_p = a_1^{-1} \det ( \partial ^2 \epsilon / \partial q_i\partial q_j \vert_{{\bf q}^\star} )$.
In the BEG model, where $p=2$, one would expect to obtain the a non-generic scaling function of the form $g(x)=b_2 x^2+ x $.
However, the fact that the discriminant of the energy vanishes at criticality leads to a generic scaling function, $g(x)=x$.
\begin{figure}
\begin{center}
\includegraphics[scale=0.6]{fig5}
\caption{
\label{fig:can_IAB}
Numerical evaluation of the mutual information estimator in the canonical ensemble for $\ell/L=1/2$, plotted as a function of the system-size, which is
given in a logarithmic scale. Figure (a) and (b) shows the results for the BEG model and NK model, respectively.
The parameters used in the computation are identical to those described in the caption of \fref{fig:can_SAB}.
At criticality, the leading order term in $I_{\A:\B}$ is expected to scale as $\frac{1}{4}\log L$, denoted by the dashed lines, whereas away from criticality $I_{\A:\B}$ is expected to converge to a constant.}
\end{center}
\end{figure}
\subsection{Micro-canonical ensemble}
\label{sec:gen_micro}
In this section we study the behavior of the information estimators in the generic model, defined in \eref{eq:Ham_generic}, within the micro-canonical ensemble, where the total energy is fixed.
The main results of this calculation are summarized in \sref{sec:sum_mc}.
The computation is done by following the lines of derivation presented in the previous section, while omitting several of the steps for the sake of brevity.
\subsubsection{Degeneracy:}
In the micro-canonical ensemble the probability distribution is uniform over all spin configurations with a certain energy, $E$. Mathematically this can be written as
\begin{equation}
P\left(\sig\right)=\Omega^{-1} \int_{E-\delta /2}^{E+\delta/2}dE'\delta(E'- \HH( \sig,L)),
\end{equation}
where $\delta\sim \mathcal{O}(1)$ is a finite parameter and $\Omega=L\int_{E-\delta/2}^{E+\delta/2} dE'\sum_{\sig}\delta(E'-\HH(\sig))$ is the number of micro-states with
energy between $E-\delta/2$ and $E+\delta/2$.
Similarly to the derivation of $Z$ in the previous section, it is useful to express the degeneracy, $\Omega$, in terms of an integral over continuous variable as
\begin{equation}
\label{eq:OmegaE_00}
\Omega=L^{p}\int dE'\int d{\bf q} \sum_{\sig} \delta\big[E'-\sum_{i,j}\phi_{i,j}(\sig)-L\epsilon(\mathbf{q})\big]\prod_{j=1}^{p}\delta( Q_{j}(\sig)-L q_j).
\end{equation}
As before, the delta function can be replaced by an integral over the fields ${\bf h}$, yielding
\begin{equation}
\label{eq:OmegaE_0}
\Omega=L^{p}\int d{\bf q} d{\bf h} dE' d\beta e^{\beta E'-L\beta\epsilon(\mathbf{q})}\langle 1 | T_{\beta,\mathbf{h}}^{L} |1\rangle,
\end{equation}
where $\langle 1 | T_{\beta,\mathbf{h}}^{L} |1\rangle$ denotes as in \eref{eq:Z_can_1} the partition function of a short-range interacting systems whose Hamilton
is given in \eref{eq:Ham_short}.
Because the integrand in \eref{eq:OmegaE_0} does not vary significantly in the interval $E'\in [E-\delta/2,E+\delta/2]$, the integral over $E'$ can be replace by the value of the integrand at $E'=E$. This would result in errors that
scale as $e^{-\beta\delta}$ which can be written as $\mathcal{O}(1)$.
The integral over $\beta$, on the other hand, has to be evaluated using the SPA, yielding
\begin{equation}
\label{eq:OmegaE_A}
\Omega=L^{p/2-1/2}\int d{\bf q} e^{-L y_\HH(E/L,{\bf q})}\omega(\mathbf{q})[\mathcal{O}(1)+\mathcal{O}(L^{-1})],
\end{equation}
where $y_\HH(\varepsilon,{\bf q})\equiv \beta^\star(\varepsilon,{\bf q}) [ \varepsilon - f_\HH (\beta^\star(\varepsilon,{\bf q}),\mathbf{q})]$ can be regarded as the Landau free energy of the micro-canonical system and $f_\HH$ is defined below \eref{eq:app_Z2}. Here $\beta^\star(\varepsilon,{\bf q})$ is the saddle point of the integral over $\beta$, defined via the equation
\begin{equation}
\phi(\beta^\star,{\bf q})=\varepsilon-\epsilon (\mathbf{q}).
\end{equation}
At this inverse temperature the average energy in the short-range interacting system is equal to the difference between the overall energy and the mean-field energy.
In the case of a pure mean-field system, where $\phi_{i,j}(\sig)=0$, the integral over $\beta$ can be replaced by a delta function, yielding
\begin{equation}
\label{eq:OmegaE_B}
\Omega=L^{p/2-1}\int d{\bf q} \delta(\epsilon (\mathbf{q})-E/L) \omega(\mathbf{q})[\mathcal{O}(1)+\mathcal{O}(L^{-1})].
\end{equation}
As expected, in the absence of additional short-range interactions, the mean-field energy is strictly fixed, $\epsilon (\mathbf{q})=E/L$.
The derivations of the information estimators for $\phi_{i,j}(\sig)\neq0$ and for $\phi_{i,j}(\sig)=0$ are slightly
different, as indicated by the difference between \eref{eq:OmegaE_A} and \eref{eq:OmegaE_B}. In both cases, however, one finds the same leading order scaling of $\mathcal{S}_{\A:\B}$ and $\mathcal{I}_{\A:\B}$. For brevity, we present only the analysis of the more general case where $\phi_{i,j}(\sig)\neq0$.
As in the case of the analysis of $Z$ in the previous section, the integral in \eref{eq:OmegaE_A} can be evaluated using the SPA, which is performed in
two steps. In the first step we approximate the integrals over
$q_2,\dots,q_p$, by expanding the exponent to quadratic order in these variables. The next step is to approximate the remaining one-dimensional
integral, by expanding the exponent to order $q_1^2$ away from criticality and to order $q_1^4$ at criticality.
The first step of the SPA yields,
\begin{equation}
\label{eq:Omega_first_step}
\Omega = \int dq_1 \tilde{\omega}(q_1) e^{ -L y_\HH(E/L,\tilde{{\bf q}})}\big[\mathcal{O}(1)+\mathcal{O}(L^{-1})\big]
\end{equation}
%
where in this section $\tilde{\bf q}=\left(q_1,{\tilde q}_2(q_1),{\tilde q}_3(q_1),\dots, {\tilde q}_p(q_1)\right)$ is the solution of the set of equations
$\partial y_\HH/ \partial q_i =0$ for $i=2,3,\dots,p$.
The function $\tilde{\omega}(q_1)$ accounts for the contribution from $\omega (\tilde{{\bf q}})$ and the coefficients that results from the SPA and thus
\begin{equation}
\label{eq:app_omega_tilde}
\tilde{\omega}(q_1)\equiv \omega (\tilde{{\bf q}}(q_1)) \pi^{(p-1)/2} \Big[ \det \Big( \frac{\beta \partial^2 y_\HH(E/L,{\bf q})}{\partial q_i \partial q_j} \Big\vert_{\tilde{{\bf q}}(q_1)} \Big)\Big]^{-1/2},
\end{equation}
for $i,j=2,3,\dots,p$.
{\it Away from criticality} the second step of the SPA yields
\begin{eqnarray}
\label{eq:gen_Omega_away}
\Omega \simeq e^{L s_\phi(\frac{E}{L},{\bf q}^\star)} L^{-1/2}\tilde{\omega}(q_1^\star) \chi_{0,2} \Big( \frac{d^2 y_\HH (\frac{E}{L},\tilde{{\bf q}}(q_1))}{d q_1^2} \Big\vert_{q^\star_{1}} \Big),
\end{eqnarray}
where $s_\phi(\varepsilon,{\bf q}) = \beta\big[ f_\phi (\beta^\star(\varepsilon,{\bf q}),\mathbf{q}) - \phi(\beta^\star(\varepsilon,{\bf q}),\mathbf{q})\big]$ is the entropy
of the short-range interacting system, described by $\HH'$ in \eref{eq:Ham_short}, for $\beta=\beta^\star(\varepsilon,{\bf q})$ and with ${\bf h}$ set such that $\langle{\bf Q}(\sig)\rangle_{\HH'}/L={\bf q}$. The point ${\bf q}={\bf q}^\star$ is the global minimum of $y_\HH(\frac{E}{L},{\bf q})$. Since for ${\bf q}={\bf q}^\star$, the energy terms in $y_\HH$ cancel, yielding $y_\HH(\varepsilon,{\bf q}^\star)= -s_\phi(\varepsilon,{\bf q}^\star)$, one can use $s_\phi$ in the exponent
in \eref{eq:gen_Omega_away}.
As in the derivation of $Z$ in \eref{eq:app_Z_at}, {\it at criticality} the SPA of the integral in \eref{eq:OmegaE_A}
yields a different polynomial-dependence in $L$, given by
\begin{eqnarray}
\label{eq:gen_Omega_at}
\Omega \simeq e^{L s_\phi(\frac{E}{L},{\bf q}^\star)} L^{-1/4}\tilde{\omega}(q_1^\star) \chi_{0,4} \Big( \frac{d^4 y_\HH (\frac{E}{L},\tilde{{\bf q}}(q_1))}{d q_1^4} \Big\vert_{q^\star_{1}} \Big).
\end{eqnarray}
\subsubsection{Separation entropy:} In order to compute the separation entropy, one has to compute first the Shannon entropy of the whole system, which in
the micro-canonical ensemble is given simply by $S= \ln \Omega$.
Here we consider the separation process discussed in the case of the canonical ensemble, whereby the Hamiltonians of the separated subsystems are given by \eref{eq:Ham_generic} with
$L$ replaced by the corresponding length of each subsystem. This assures that the
average values of ${\bf q}$ of the separated subsystems are identical to those of the composite system.
Since the decoupled subsystems maintain the form of the Hamiltonian of the whole system, their entropies are given by $S=\ln \Omega$ where $\Omega$ is given by \eref{eq:gen_Omega_away} and
\eref{eq:gen_Omega_at} with
$L$ replaced by $\ell$ for subsystem $\A$ and by $L-\ell$ for subsystem $\B$.
As a result the extensive terms in the separation entropy cancel and the remaining leading order terms are given {\it away from criticality} by
\begin{eqnarray}
\label{eq:BEG_micro_Omega_away}
S_{\A:\B}=\frac{1}{2}\ln\left[ L \alpha \left( 1-\alpha \right)
\right]+\mathcal{O}\left( 1 \right),
\end{eqnarray}
and {\it at criticality} they are equal to
\begin{eqnarray}
\label{eq:BEG_micro_Omega_at}
S_{\A:\B}=\frac{1}{4}\ln\left[ L \alpha \left( 1-\alpha \right)
\right]+\mathcal{O}\left( 1 \right).
\end{eqnarray}
This scaling form of $S_{\A:\B}$ is identical to that obtained in the canonical ensemble, up to an addition of a $\frac{1}{2}\ln L$ term both at criticality
and away from criticality. This term is due to the fixed energy constraint, $L \epsilon({\bf Q}(\sig)/L)+\sum_{i,j} \phi_{i,j}(\sig) = E$, which introduces additional correlations between the spin variables.
Similarly to the canonical case, these results can be verified numerically, as shown in \fref{fig:mc_SAB}.
Here, however, the constant term in $S_{\A:\B}$ was found to oscillate with some finite scale. A convincing fit thus required sampling a large number of system sizes.
In order to avoid the arbitrariness in value of the parameter $\delta$, the integral over $E'$ in \eref{eq:OmegaE_00} was performed numerically over $E'\in (-\infty,E]$ instead of
$E'\in[E-\delta/2,E+\delta/2]$. These two definitions of the micro-canonical ensemble can be shown in our case to yield the same scaling form of $S_{\A:\B}$ as well as of $I_{\A:\B}$, computed
below.
\begin{figure}
\begin{center}
\includegraphics[scale=0.6]{fig6}
\caption{
\label{fig:mc_SAB} Numerical evaluation of the Shannon separation estimator in the micro-canonical ensemble for $\ell/L=1/2$, plotted as a function of the system-size, given in a logarithmic scale.
Figure (a) shows the results for the BEG model at criticality ($\Delta/J\approx 0.347$, $E/L\Delta= 0.5$) and away from criticality ($\Delta/J\approx 0.530$, $E/L\Delta= 0.5$), denoted by $\star$ and $\bullet$ respectively.
Similarly, figure (b) shows the results for the NK model at criticality ($K/J\approx -0.333$, $E/L K = 0.65$) and away from criticality ($K/J\approx -0.5$, $E/L K = 0.65$).
At criticality and away from criticality, the leading order term in $S_{\A:\B}$ is expected to scale as $\frac{1}{4}\log{L}$ and $\frac{1}{2}\log{L}$, respectively. These $\log L$ terms are denoted by the straight dashed lines.}
\end{center}
\end{figure}
\subsubsection{Mutual information}
The mutual information is computed from the marginal probability distribution. In the micro-canonical ensemble the latter is given by
a sum over microstates with a energy between $E-\delta/2$ and $E+\delta/2$, which can be written as
\begin{eqnarray}
\label{eq:gen_pm_micro}
P_{\M}^{\A}({\sig}^{\A})&=&\Omega^{-1}\int_{E-\delta/2}^{E+\delta/2} dE'\sum_{\sig^{\B}}\delta(E'-\HH({\sig}^{\A},{\sig}^{\B})) \\ &=&\Omega^{-1}Z^{\B}\Big(\varepsilon,\mathbf{q}({\sig}^{\A}),\sig^{\A}\Big)\mathcal{O}(1), \nonumber
\end{eqnarray}
where $\varepsilon=\frac{E}{L}-\frac{1}{L}\sum_{i,j\in\A}\phi_{i,j}(\sig^\A)$ and the partition function over subsystem $\B$ is defined in this section as
\begin{equation}
Z^{\B}(\varepsilon,\mathbf{q}^{\A},\sig^{\A})\equiv \int d\beta e^{L\beta \varepsilon }\sum_{{\sig}^{\B}}e^{-L\beta\epsilon(\mathbf{q}({\sig}^{\A}),\mathbf{q}({\sig}^{\B}))-\beta\sum_{i,j\in\B}\phi_{i,j}(\sig)}e^{-2\beta\sum_{i\in\A,j\in\B}\phi_{i,j}(\sig)}.
\end{equation}
The $\mathcal{O}(1)$ term in \eref{eq:gen_pm_micro} comes from the approximation of the integral over $E'$ by a delta function at $E'=E$.
As in the derivation of $Z^\B$ in the canonical ensemble in \eref{eq:ZBa}, one can introduce an integral over the $q$ variables of subsystem
$\B$ and replace the resulting delta function by an integral over ${\bf h}$, yielding
\begin{equation}
Z^{\B}(\varepsilon,\mathbf{q}^{\A},\sig^{\A})=L^{p}\int d{\bf q}^{\B}d{\bf h}^{\B}d\beta e^{\beta L\epsilon-L\beta\epsilon(\mathbf{q}^{\A},
\mathbf{q}^{\B})}\big\langle \sig^{\A}_{\mathrm{b}}|T_{\beta,\mathbf{h}^{\B}}^{L-\ell}|1\big\rangle.
\end{equation}
Considering only the leading order contribution in $L$ to $\big\langle \sig^{\A}_{\mathrm{b}}|T_{\beta,\mathbf{h}^{\B}}^{L-\ell}|1\big\rangle$ and evaluating the integral
over ${\bf h}^\B$ and $\beta$ using the SPA yields
\begin{equation}
\label{eq:gen_micro_zB}
Z^{\B}(\varepsilon,\mathbf{q}^{\A},\sig^{\A})=L^{(p-1)/2}\int d{\bf q}^{\B}e^{-L y_{\HH,1-\alpha}(\varepsilon,\mathbf{q}^{\A},\mathbf{q}^{\B}) }\omega(\mathbf{q}^{\B},\sig^{\A})
\big[1+\mathcal{O}(L^{-1})\big],
\end{equation}
where
\begin{equation}
y_{\HH,x}(\varepsilon,\mathbf{q}^{\A},\mathbf{q}^{\B}) = \beta^{\star}_x[\varepsilon-x f_{\HH,x}(\beta^{\star}_x,\mathbf{q}^{\A},\mathbf{q}^{\B})].
\end{equation}
Here $\beta^\star_x$ is the saddle point of the integral over $\beta$, defined via the equation $x \phi(\beta^\star_x,{\bf q}^\B)=\varepsilon-\epsilon (\mathbf{q}^\A,\mathbf{q}^\B)$.
The function $y_{\HH,x}(\epsilon,{\bf q}^\A,{\bf q}^\B)$ can be regarded as the Landau free energy of a subsystem of size $xL$, given the values of the $q$ variables in the complementary subsystem, denoted by ${\bf q}^\A$.
Using \eref{eq:gen_pm_micro} the mutual information estimator can be written as
\begin{equation}
\label{eq:micro_IAB}
\mathcal{I}_{\A:\B}= \ln \Omega -\big\langle \ln \big[Z^{\B}(\frac{E}{L}-\phi^\A,\mathbf{q}^{\A},\sig^{\A}) Z^{\A}(\frac{E}{L}-\phi^\B,\mathbf{q}^{\B},\sig^{\B})\big]\big\rangle_{\HH} + \mathcal{O}(1),
\end{equation}
where $\phi^k\equiv\frac{1}{L}\sum_{i,j\in k} \phi_{i,j}(\sig)$ for $k=\A,\B$. Following the same reasoning described in \ref{app:vanish} it can be shown that the second term in RHS above does not yield a $\sqrt{L}$-divergence at criticality and that its leading order contribution the same as that of
$\big\langle \ln \big[Z^{\B}(\frac{E}{L}-\alpha\phi(\beta^\star,{\bf q}^\star),{\bf q}^\star,\sig^{\A}) Z^{\A}(\frac{E}{L}-(1-\alpha)\phi(\beta^\star,{\bf q}^\star),{\bf q}^\star,\sig^{\B})\big]\big\rangle_{\HH}$. For subsystem $\B$ one can compute $Z^{\B}(\frac{E}{L}-\alpha\phi(\beta^\star,{\bf q}^\star),{\bf q}^\star,\sig^{\A})$
by evaluating the integral in \eref{eq:gen_micro_zB} using the SPA which yields
\begin{equation}
\label{eq:gen_micro_zBstar}
Z^{\B}(\frac{E}{L}-\alpha \phi(\beta^\star,{\bf q}^\star),{\bf q}^\star,\sig^{\A}) =
L^{-1/2}\omega({\bf q}^\star,\sig^\A) \pi^{p/2} e^{L\alpha s_\phi(\frac{E}{L},{\bf q}^\star)}\big[1+\mathcal{O}(L^{-1})\big],
\end{equation}
where in this section the Hessian matrix is defined as
\begin{equation}
\label{eq:app_A_def1}
(A_x)_{i,j}= \beta \frac{\partial ^2 y_{\HH,x} (E/L-\phi(\beta^\star,\mathbf{q}^{\star}),\mathbf{q}^{\star}, \mathbf{q}^\B) }{\partial q_i^\B \partial q_j^\B} \Big\vert_{{\bf q}^\B={\bf q}^\star}.
\end{equation}
The same expression for $Z^\A$ is obtained by replacing $\alpha \to 1 - \alpha$ and exchanging $\A$ and $\B$ in \eref{eq:gen_micro_zBstar}.
Inserting \eref{eq:gen_micro_zBstar} and the expression for $\Omega$ in (\ref{eq:BEG_micro_Omega_away}) and (\ref{eq:BEG_micro_Omega_at}) into \eref{eq:micro_IAB}
yields {\it away from criticality} the following scaling form:
\begin{eqnarray}
\label{eq:micro_I_away}
\mathcal{I}_{\A:\B}= \frac{1}{2}\ln L + \frac{1}{2} \ln [g(\alpha)g(1-\alpha)] + \mathcal{O}(1),
\end{eqnarray}
whereas {\it at criticality} the mutual information is given by
\begin{eqnarray}
\label{eq:micro_I_at}
\mathcal{I}_{\A:\B}=\frac{3}{4}\ln L + \frac{1}{2} \ln [g(\alpha)g(1-\alpha)]+\mathcal{O}\left( 1 \right).
\end{eqnarray}
Here $g(x)=\det A_{x}$ with $A_x$ defined in \eref{eq:app_A_def1}. Similarly to the canonical case, these results can be verified numerically, as shown in \fref{fig:mc_IAB}.
Here, however, the mutual information appears to converge more slowly with $L$ than in the canonical ensemble.
As in the canonical ensemble, one can show that $g(x)$ is a polynomial of degree $p$, of the form given in \eref{eq:gx_can},
and that at criticality $b_0=0$. This implies that $g(x)$ depends on $p-1$ parameters at criticality and on $p$ parameters away from criticality.
For $p=1$ such as in the NK model, $g$ therefore has a generic form at criticality, $g(x)=x$.
In the case of pure mean-field models, the fixed energy constraint reduces the dimension of $A(x)$ to $(p-1)\times(p-1)$, which implies that in this case
$g(x)$ is a polynomial of degree $p-1$. At criticality, $g(x)$ therefore depends on $p-2$ parameters in pure mean-field model. This implies that the BEG model, which is a pure mean-field model with $p=2$, also exhibits at criticality a generic scaling function of the form $g(x)=x$. The scaling function, $g(\alpha)g(1-\alpha)$, of the BEG model is plotted for several temperatures in \fref{fig:scaling_I}.
\begin{figure}
\begin{center}
\includegraphics[scale=0.6]{fig7}
\caption{
\label{fig:mc_IAB} Numerical evaluation of the mutual information estimator in the canonical ensemble for $\ell/L=1/2$, plotted as a function of the system-size. Figure (a) and (b) shows the results for the BEG model and NK model, respectively.
The parameters used in the computation are identical to those described in the caption of \fref{fig:mc_SAB}.
At criticality and away from criticality, the leading order term in $I_{\A:\B}$ is expected to scale as $\frac{3}{4}\log{L}$ and $\frac{1}{2}\log{L}$, respectively. These $\log L$ terms are denoted by the straight dashed lines.}
\end{center}
\end{figure}
\subsection{Ground-state degeneracy and first order transitions}
\label{sec:beg_degeneracy}
In all the above analysis the Landau free energies, $f_\HH(\beta,{\bf q})$ in the canonical ensemble and $y_\HH(E/L,{\bf q})$ in the microcanonical ensemble,
were assumed to have a single ground state, denoted by ${\bf q}^\star$.
However, specific symmetry properties may lead to degenerate ground states. This is true in the BEG model and the NK model, where the spin flip symmetry, $\sig \to -\sig$,
yields two degenerate ordered states with opposite magnetization. Even models without symmetry exhibit degenerate ground state at first order phase transition points.
From a dynamical point, in a mean-field system the tunneling time between these ground-states grows exponentially with $L$.
The effect of having multiple ground states on the information estimators is therefore
relevant for finite systems that are measured over a very long period of time. Nevertheless, this degeneracy can be taken into account in the above derivation, as demonstrated in this section. The results show that the degeneracy affects only the $\mathcal{O}(1)$ terms of the
information estimators. Specifically, this implies that $\mathcal{S}_{\A:\B}$ and $\mathcal{I}_{\A:\B}$ remain finite at first order transitions points.
We demonstrate the effect of the ground-state degeneracy within the canonical ensemble,
where we denote the global minima of $f_\HH(\beta,{\bf q})$ by ${\bf q}^{\star \,(j)}$ with $j=1,\ldots,K$ and $K$ being the number of degenerate states.
The first calculation in \sref{sec:gen_can} to be affected by this degeneracy is the evaluation the integral in $Z$,
\begin{equation}
Z=L^{p/2}\int d{\bf q} \omega(\mathbf{q}) e^{-L\beta f_\HH(\beta,{\bf q})}\big[1+\mathcal{O}(L^{-1})\big],
\end{equation}
given in \eref{eq:app_Z2} and rewritten here for convenience. In the case of degeneracy the SPA of the integral over ${\bf q}$ yields,
\begin{equation}
Z = \sum_{j=1}^K \tilde{\omega}(q^{\star\,(j)}_{1}) e^{-L \beta f_\HH(\beta,{\bf q}^{\star\,(j)})} \chi_{0,2}\left( \beta \frac{d^2 f_\HH(\tilde{{\bf q}}(q_1))}{d q_1^2} \Big\vert_{q^{\star\,(j)}_{1}} \right)\big[1+\mathcal{O}(L^{-1})\big],
\end{equation}
which is in fact a sum over the degenerate ground states of the result obtained in the case of a single ground state (\ref{eq:app_Z_away}).
It is easy to show that a similar sum appears in the expression for $S$ and thus also in the expression
for $S_{\A:\B}$. This eventually yields $S_{\A:\B}=\mathcal{O}(1)$ away from criticality and on the first order transition line.
At criticality the sum of $S_{\A:\B}$ over the different minima would yield
the same form as in the case of a single ground state, only with a different coefficient in front of the $\sqrt{L}$ term, given by
\begin{eqnarray}
\label{eq:gamma details}
\gamma &=& \sum_{j=1}^K \beta \frac{\chi_{2,4}(\beta\frac{d^4 f_{\HH}}{d q^4_1}|_{q_1^{\star\,(j)}})}{\chi_{0,4} (\beta\frac{d^4 f_{\HH}}{d q^4_1}|_{q_1^{\star\,(j)}}) \omega({\bf q}^{\star,(j)})} \\
&& \times \frac{d}{dq_1^2} \big\{ \tilde{\omega}(q_1)[\epsilon( \tilde{\bf q}(q_1))+\phi(\beta, \tilde{\bf q}(q_1)) - \epsilon({\bf q}^{\star,(j)})-\phi(\beta,{\bf q}^{\star,(j)}) ] \big\} \Big|_{q_1^{\star\,(j)}}.\nonumber
\end{eqnarray}
It is important to note, however, that in simple models one does not expect to find degenerate ground state at criticality. This would happen in models that have several symmetries that are broken at different critical points.
Using the same reasoning in the calculation of the mutual information, the term
$\big\langle\ln[Z^{\A}(\mathbf{q}({\sig}^{\B}),\sig^\B)Z^{\B}(\mathbf{q}({\sig}^{\A}),{\sig}^{\A})]\big\rangle_{\HH}-L\big\langle\epsilon(\mathbf{q}(\sig))\big\rangle_{\HH}$
in \eref{eq:I_genA} can be written as a sum over the degenerate ground states. This sum involves the value of $\ln Z^{\B}(\mathbf{q}^{\star,(j)},{\sig}^{\A})$ and
$\ln Z^{\A}(\mathbf{q}^{\star(j)},{\sig}^{\B})$. The evaluation of latter expressions using the SPA does not yield a sum over the degenerate ground-states. This can be understood by writing
$Z^{\B}(\mathbf{q}^{\star\,(j)},\sig^\A)$ explicitly as
\begin{equation}
\label{eq:ZB_B}
Z^{\B}(\mathbf{q}^{\star\,(j)},\sig^\A)=(L-\ell)^{p/2}\int d{\bf q}^{\B}e^{-(L-\ell)\beta f_{\HH,1-\alpha}(\beta,\mathbf{q}^{\star\,(j)},{\bf q}^\B)}\omega(\mathbf{q}^{\B},\sig^\A)
\big[1+\mathcal{O}(L^{-1})\big],
\end{equation}
where $f_{\HH,1-\alpha}(\beta,\mathbf{q}^{\star\,(j)},{\bf q}^\B) \equiv \frac{1}{1-\alpha}\epsilon( \alpha \mathbf{q}^{\star\,(j)}+ (1-\alpha) \mathbf{q}^{\B})+f_\phi(\beta,\mathbf{q}^{\B})$. Using the fact that $\mathbf{q}^{\star\,(j)}$ are minima of $f_\HH(\mathbf{q})=\epsilon(\mathbf{q})+f_\phi(\mathbf{q})$
it is easy to see that only $\mathbf{q}^{\star\,(j)}$ minimizes $f_{\HH,1-\alpha}(\beta,\mathbf{q}^{\star\,(j)},{\bf q}^\B)$.
Physically this implies that the value of ${\bf q}$ in subsystem $\A$ has broken the symmetry of
the free energy in $\B$. As a result the SPA of \eref{eq:ZB_B} involves only $\mathbf{q}^{\star\,(j)}$. Using a similar argument for $Z^{\A}(\mathbf{q}^{\star,(j)},{\sig}^{\B})$ one can show that
that $I_{\A:\B}$ is also given by the same scaling forms obtained in the previous sections.
\subsection{Small $\ell/L$ scaling}
\label{sec:beg_finite}
The scaling forms of the information estimators, obtained in \sref{sec:gen_can} and \sref{sec:gen_micro}, have been derived for the case where each subsystem, $\A$ and $\B$, comprises
a finite fraction of the entire system, i.e. $\ell\sim L$.
It is interesting to study how the scaling of the information estimators changes in the case where one of the subsystems, chosen here to be $\A$, is much smaller than the other but is still very large (allowing
SPA to be employed). We denote this limit as $1\ll \ell \ll L$.
As discussed above, the separation entropy is given by $S_{\A:\B}=S(L)-S(\ell)-S(L-\ell)$, where $S(x)$ denotes the entropy of a system of size $x$.
The expression of $S(x)$, computed in sections \ref{sec:gen_can} and \ref{sec:gen_micro} for the generic model, remains valid for any $x\gg1$. Inserting the expressions of $S(x)$ into
the $S_{\A:\B}$ is equivalent to taking the $\alpha \to 0$ limit in the expressions for $S_{\A:\B}$ obtained above.
The results of this calculation are summarized in \sref{sec:sum_finite_size} and not repeated here for the sake of brevity.
This approach cannot be applied in the case of the mutual information, where the $L\to\infty$ and $\alpha\to0$ limit do not commute.
In the canonical ensemble $\mathcal{I}_{\A:\B}$ was found to be given by
$\mathcal{I}_{\A:\B}=\frac{1}{4}\ln L+ \frac{1}{2}\ln[ g(\alpha)g(1-\alpha)]+\mathcal{O}( 1 )$, where $g(x)$ is a non-generic polynomial.
At criticality, the constant term in $g(x)$ vanishes and hence $g(\alpha)= b_1 \alpha+O(\alpha^2)$ for $\alpha\ll1$.
This yields $\mathcal{I}_{\A:\B} \sim \frac{1}{4}\ln\frac{\ell^2}{L}$, which diverges to $-\infty$ with $L$ for $\ell \ll L^{1/2}$. This is in contrast with the intuition that the mutual information should be positive and diverge only with $\ell$ for $\ell \ll L$.
The term $\ln\frac{\ell^2}{L}$ suggests that there exists an intermediate scale, $\ell\sim L^{1/2}$, where the derivation in sections \ref{sec:gen_can} and \ref{sec:gen_micro} ceases to be correct.
In order to show this explicitly and obtain the correct scaling of $I_{\A:\B}$ we consider limit where
\begin{equation}
\ell= a L^\zeta, \qquad 0<\zeta<1.
\end{equation}
We begin the analysis of $I_{\A:\B}$ from \eref{eq:I_genA}, which, by using the result of \ref{app:vanish}, can be written as
\begin{equation}
\label{eq:I_genA1}
I_{\A:\B}=\ln Z-\big\langle\ln[Z^{\A}(\mathbf{q}^\star,\sig^\B)Z^{\B}(\mathbf{q}^\star,{\sig}^{\A})]\big\rangle_{\HH}-L\big\langle\beta\epsilon(\mathbf{q}(\sig))\big\rangle_{\HH}+\mathcal{O}(1).
\end{equation}
We analyze first the expression for $Z^{\A}$. Starting from \eref{eq:app_Z2}, it can be evaluated the SPA with the exponentials expanded to quadratic order, yielding
\begin{eqnarray}
Z^{\A}(\mathbf{q}^{\star},\sig^\B) &=& \ell^{p/2}\int d{\bf q}^{\A}e^{-\ell\beta f_{\HH,\alpha}(\beta,{\bf q}^{\star},{\bf q}^\A)}\omega(\mathbf{q}^{\A},\sig^\B)\big[1+\mathcal{O}(L^{-1})\big]
\\ &=& \ell^{p/2}\int d{\bf q}^{\A}e^{-L [ \epsilon ( {\bf q}^\star )+\mathcal{O}(\alpha)]}\omega(\mathbf{q}^{\B},\sig^\A) = e^{-L\epsilon( {\bf q}^\star)} \times \mathcal{O}(1). \nonumber
\end{eqnarray}
The leading order term of $\langle\ln Z^\A(\mathbf{q}^\star,\sig^\B)\rangle_{\HH}$ is therefore does not involve $\zeta$ and cancels with the leading order term of $L\langle\beta\epsilon(\mathbf{q}(\sig))\rangle_{\HH}$ in \eref{eq:I_genA1} for all $0<\zeta<1$.
The key quantity that depends on $\zeta$ is $Z^{\B}$. Starting from \eref{eq:app_Z2} it can be analyzed similarly to $Z$, by first integrating over the variables $q_2,\dots,q_p$ using the SPA, yielding
\begin{eqnarray}
\fl \quad & Z^{\B}(\mathbf{q}^{\star},\sig^\A) =(L-\ell)^{p/2}\int d{\bf q}^{\B}e^{-(L-\ell)\beta f_{\HH,1-\alpha}(\beta,{\bf q}^{\star},{\bf q}^\B)}\omega(\mathbf{q}^{\B},\sig^\A)
\big[1+\mathcal{O}(L^{-1})\big] \\
\fl &=L^{1/2} e^{-L\beta f_\HH (\beta,{\bf q}^\star)} \int dq_{1}^{\B}e^{-L(\det A_{1-\alpha})(q_{1}^{\B})^{2}-L\beta f_{\HH,1-\alpha}^{(4)}(q_{1}^{\B})^{4}+O[(q_{1}^{\B})^{6}]}\tilde{\omega}(q_{1}^{\B},\sig^{\A})\big[1+\mathcal{O}(\ell/L)\big]. \nonumber
\end{eqnarray}
Away from criticality, where $\det A_{1-\alpha}=g(\alpha)=b_0+O(\alpha)$ this integral can be evaluated using a SPA which involve only the first term in the exponential,
yielding $Z^\B \sim O(1)\times e^{-L\beta f_\HH (\beta,{\bf q}^\star)}$, which is similar to the leading order term in $Z$. As a result all the
extensive terms in \eref{eq:I_genA1} cancel, yielding {\it away from criticality} the following expression:
\begin{eqnarray}
I_{\A:\B}=\mathcal{O}(1) \qquad 0<\zeta\leq 1.
\end{eqnarray}
{\it At criticality}, where $\det A_{1-\alpha}=g(\alpha)=b_1 \alpha+O(\alpha^2)$, one obtains the following expressions for $Z^{\B}$:
\begin{equation}
\label{eq:ZB_finite}
Z^{\B}(\mathbf{q}^{\star},\sig^{\A}) = L^{1/2} e^{-L\beta f_\HH (\beta,{\bf q}^\star)}\int dq_{1}^{\B}e^{-\ell b_{1} (q_{1}^{\B})^{2}-L(\beta f_{\HH,1-\alpha}^{(4)})(q_{1}^{\B})^{4}+O[(q_{1}^{\B})^{6}]}\tilde{\omega}(q_{1}^{\B},\sig^{\A}) .
\end{equation}
The SPA of the above integral should be evaluated differently depending on the value of $\zeta$. With the appropriate change of variables the leading order term is given by
\begin{equation}
\label{eq:ZB_finite1}
Z^{\B}(\mathbf{q}^{\star},\sig^{\A})\approx e^{-L\beta f_\HH (\beta,{\bf q}^\star)} \tilde{\omega}(q_{1}^\star,\sig^{\A}) \left\{ \begin{array}{ccc}
L^{(1-\zeta)/2}\int dye^{-y^{2}b_{1}} & \quad & \frac{1}{2}<\zeta\leq 1\\
L^{1/4}\int dze^{-z^{2}a \, b_{1}-z^{4}\beta f_{\HH,1-\alpha}^{(4)}} & \quad & \zeta=\frac{1}{2}\\
L^{1/4}\int dze^{-z^{4}\beta f_{\HH,1-\alpha}^{(4)}} & \quad & 0<\zeta<\frac{1}{2}
\end{array}\right. ,
\end{equation}
where $y\equiv \ell^{1/2}q_1^\B$ and $z\equiv L^{1/4}q_1^\B$. This expression should be compared with that of the partition function $Z=L^{1/4} e^{-L\beta f_\HH (\beta,{\bf q}^\star)} \times \mathcal{O}(1)$.
For $\zeta > 1/2$, the partition function of subsystem $\B$ is smaller that of the whole system, $Z^\B\ll Z$, whereas for $\zeta \leq 1/2$ they have the same leading order terms.
The effect of subsystem $\A$ on $\B$ becomes negligible in the latter limit. Inserting \eref{eq:ZB_finite1} into \eref{eq:I_genA1} yields in the canonical ensemble the following result {\it at criticality}:
\begin{eqnarray}
I_{\A:\B}=\left\{ \begin{array}{ccc}
(\frac{\zeta}{2}-\frac{1}{4})\ln L+\mathcal{O}(1) & \quad & \frac{1}{2}<\zeta\leq 1\\
\mathcal{O}(1) & \quad & 0<\zeta\leq\frac{1}{2}
\end{array}\right. .
\end{eqnarray}
The two limits of $\ell\ll L^{1/2}$ and $L^{1/2} \ll \ell \lesssim L$ can be connected smoothly by
considering $\ell = a L^{1/2}$. The $\mathcal{O}(1)$ part of the mutual information contains in this case a term of the form $\ln [\int dze^{-z^{2}a \, b_{1}-z^{4}\beta f_{\HH,1-\alpha}^{(4)}}]$.
As expected, this term diverges to $-\infty$ for $a\to\infty$ and converges to the constant term obtained for $\zeta<1/2$ in the $a\to 0$ limit.
The discussion above can be shown to apply also to the micro-canonical ensemble, where obtains the following scaling {\it at criticality}:
\begin{eqnarray}
I_{\A:\B}=\left\{ \begin{array}{ccc}
( \zeta-\frac{1}{4} ) \ln L+\mathcal{O}(1) & \quad & \ \frac{1}{2}<\zeta\leq 1\\
\frac{\zeta}{2}\ln L+\mathcal{O}(1) & \quad & 0<\zeta\leq\frac{1}{2}
\end{array}\right. .
\end{eqnarray}
As argued in \sref{sec:gen_micro}, the additional $\frac{\zeta}{2}\ln L$ term, which can be written as $\frac{1}{2}\ln \ell$, is due to the fixed energy constraint.
\subsection{Higher order critical points}
\label{sec:higher_order}
Both the BEG model and the NK model exhibit tricritical points, where the second and first order transition lines meet, as shown in \fref{fig:BEG}.
These point are defined by the vanishing of the 4$^{\mathrm{th}}$ order term in the Landau expansion. In the canonical ensemble, we denote the expansion of the Landau free energy
as $ f_\HH(\beta,\tilde{{\bf q}})=f_{\HH,2} q_1^2 + f_{\HH,4} q_1^4+f_{\HH,6} q_1^6+\ldots$
\footnote{We assumed here that the model is invariant under $q_1\to-q_1$ as in the case when $q_1$ is the overall magnetization in the BEG and NK model.
This implies that all the odd coefficients in the Landau expansion vanish. Other cases can be treated by following the lines of derivation presented here.}.
In terms of this expansion the tricritical point is defined by the equation $f_{\HH,2}=f_{\HH,4}=0$.
As a result, the saddle point approximation performed throughout \sref{sec:generic_model} has to be computed based on the 6$^{\mathrm{th}}$ order terms. For brevity,
we state only the results of this calculation, which is done by following the same lines of derivation presented above.
We define an $r$-order critical point as such where the Landau expansion is given by $f_\HH(\beta,\tilde{{\bf q}})=f_{\HH,2r} q_1^{2r}+\mathcal{O}(q_1^{2r+2})$. For such a general critical point we
obtain the following results in the canonical ensemble:
\begin{eqnarray}
\fl \qquad S_{\A:\B} & =& \gamma_r L^{1-\frac{1}{r}}\left[\alpha^{1-\frac{1}{r}}+(1-\alpha)^{1-\frac{1}{r}}- 1
\right]-\frac{1}{2r}\ln\left[ L \alpha \left( 1-\alpha \right)
\right]+\mathcal{O}\left( 1 \right), \\
\fl \qquad \mathcal{I}_{\A:\B}&=&\frac{1}{2r}\ln L+ \frac{1}{2}\ln\left[ g(\alpha)g(1-\alpha)\right]+
\mathcal{O}\left( 1 \right),
\end{eqnarray}
where $\gamma_r$ denotes a non-universal constant.
At the tricritical point ($r=3$), the separation entropy thus diverges as $L^{2/3}$ as compared to $L^{1/2}$ at the critical point ($r=2$). Similarly, the mutual information
diverges as $\frac{1}{3}\log L$ as compared to $\frac{1}{4}\log L$ at the critical point. This conforms with our intuition that the fluctuation in the order parameter
of the transition are stronger at higher order critical points.
A similar calculation a $r$-order critical point in the micro-canonical ensemble yields
\begin{eqnarray}
\fl \qquad S_{\A:\B} &=& \frac{1}{r}\ln\left[ L \alpha \left( 1-\alpha \right)
\right]+\mathcal{O}\left( 1 \right), \\
\fl \qquad \mathcal{I}_{\A:\B}&=&(\frac{1}{2}+\frac{1}{2r})\ln L+ \frac{1}{2}\ln\left[ g(\alpha)g(1-\alpha)\right]+
\mathcal{O}\left( 1 \right).
\end{eqnarray}
As in the critical case, we obtain an additional logarithmic divergence due to the global energy constraint and no
polynomial divergence in $L$ due to the absence of energy fluctuations.
\section{Conclusions}
\label{sec:conc}
We computed the scaling behavior of two bipartite information estimators, namely the
mutual information and the separation entropy, in a $(p+1)$-state
classical spin chain. The Hamiltonian we considered involves a
mean-field and a short-range interaction term, both of a general form, thus
encompassing a large class of infinite-range interacting models. Two particular examples are the BEG model and NK model.
Models of this type are particularly interesting because they exhibit a rich
phase diagram while being analytically tractable.
Because mean-field models often exhibit ensemble inequivalence, we chose to study the
information estimators both in the canonical and in the micro-canonical ensemble. We first studied the
limit where the system is fictitiously divided into two macroscopic subsystems, with total number of spins $\ell$ and $L-\ell$, respectively.
In the canonical ensemble and away from the critical line , both the estimators remain finite in the
thermodynamic limit. However, at criticality, this ceases to be true. The
mutual information diverges as $(1/4)\ln L$ where the pre-factor $1/4$ appears to
be a characteristic of the mean-field interaction. The coefficient is universal
in the sense that it does not depend on the microscopic details of the model, and thus remains constant along the critical line.
This is not true for the scaling function of the mutual information, which depends on the details of the model.
The separation entropy at criticality exhibits a
$\sqrt{L}$ divergence, whose coefficient in general is non-universal. However, the sub-leading term diverges logarithmically with $L$ and exhibits a universal scaling form $(1/4)\ln
\left[L \alpha (1-\alpha)\right]$, where $\alpha \equiv \ell/L$.
It is important to note that unlike in the entanglement entropy used for quantum systems, the coefficient of the $\ln L$ term in the two estimators does not depend on the number of states each spin takes.
In the micro-canonical ensemble we find a different scaling behaviour.
Both of the estimators exhibit $\ln L$ corrections to the area law even away from the critical line. This
divergence is argued to be due to the fixed total energy constraint in
the microcanonical ensemble, which leads to non-trivial correlation among all spins.
This difference between the canonical and the micro-canonical calculations is not related to the ensemble inequivalence observed in
long-range interacting systems. The same difference is observed in spin-chains without interactions or with only short-range interactions. This can be easily seen by dropping the mean-field interactions term from our analysis.
The universal scaling of the shared-information estimators opens a new direction
in analyzing critical phenomena in classical systems. They could be useful in detecting phase transitions and in identifying
universality classes. In this paper we find universal scaling behavior of the bipartite information
estimators in a one-dimensional mean-field spin
chain. It would be interesting to investigate how the scaling behavior changes in
higher-dimensions, and in continuous models. A few promising non-trivial models to
analyze are the classical two-dimensional Ising model and ice-type models, where
many beautiful exact results are available in the literature.
Another interesting direction would be to study the information estimator in classical
non-equilibrium spin-chains, where equilibrium concepts such as free energy cannot be used to analyze critical phenomena.
Non-equilibrium systems are particularly relevant
for our study, because similar to the mean-field model they generally exhibit
long-range correlations \cite{Spohn1983,Sadhu2014}, ensemble inequivalence
\cite{Cohen2012} and phase transitions in one-dimension \cite{David2000}. It
would be interesting to know how the scaling behavior of the information estimators in these systems compare with the results of our present study.
\ack We thank F.C. Alcaraz, H. Hinrichsen, T. Mori and D. Mukamel for helpful discussions. The support of the Israel Science
Foundation (ISF) is gratefully acknowledged.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,173 |
Sport Football Arsenal stays top of EPL after late goals
6:13am, Jan 2, 2014 Updated: 12:55pm, Jan 2
Arsenal stays top of EPL after late goals
Arsenal have stayed top of the Premier League after scoring two late goals in a 2-0 win against Cardiff while other title contenders also started the year with victories.
Nicklas Bendtner and Theo Walcott scored Arsenal's goals at the Emirates to down a resilient Cardiff team which played in front of expected new manager Ole Gunnar Solskjaer and frustrated Arsenal for 88 minutes in the driving rain.
"We have a special spirit in the side and you know we'll fight until the last minute in the game," Arsenal manager Arsene Wenger said on Wednesday.
Manchester City remained second, one point behind the Gunners, after a 3-2 win at Swansea, while Chelsea's 3-0 victory at Southampton in awful weather conditions ensured Jose Mourinho's team stayed in third place, two points behind the leaders.
After two consecutive losses, Liverpool climbed back into the top four after Luis Suarez scored his 20th goal of the season in a 2-0 win over Hull.
Arsenal had to wait until two minutes from time before Bendtner gave the Gunners reasons to celebrate the start of a new year when he pounced from a rebound.
The striker picked up an injury while scoring and was substituted before Walcott chipped the ball over goalkeeper David Marshall.
"It doesn't look too good, but it's too early to know. Let's hope it's good news tomorrow morning," Wenger said about Bendtner's injury.
In other games, Everton fell to fifth when it was held to a 1-1 draw by Stoke, Aston Villa beat last-place Sunderland 1-0, Fulham earned a comeback 2-1 win over West Ham to climb out of the relegation zone, Crystal Palace drew 1-1 with Norwich and West Brom beat Newcastle 1-0. Manchester United hosted Tottenham in a late game.
Phoenix stun Wanderers with 3-1 victory | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,517 |
Best College Basketball Parlay Picks of the Weekend (Jan 28-29)
Written by Kenny Bell on January 26, 2017
NCAA Basketball Betting News
There are plenty of interesting interconference matchups in November and December in college basketball, but the SEC and Big 12 are doing something unique this Saturday as ESPN, ESPN2 and ESPNU will present the 10 games of the fourth annual SEC/Big 12 Challenge. The highlight game of that is No. 2 Kansas and No. 4 Kentucky, which I will preview here at MyBookie individually. But here are some other games Saturday I would parlay together in the NCAAB lines, including two from the Challenge.
Arizona is up to No. 7 in the @CBSSports Top 25 (and one). RANKINGS:https://t.co/CStwVC87Az
— CBS Sports CBB (@CBSSportsCBB) January 22, 2017
No. 5 Baylor at Ole Miss
It's possible the Bears, from the Big 12, could return to the No. 1 spot in the polls on Monday if they win here and following losses by No. 1 Villanova, No. 2 Kansas and No. 4 Kentucky over the weekend. The Bears might also need No. 3 Gonzaga to lose this weekend. Baylor has the nation's No. 1 RPI and has 8 top-50 RPI wins, 2nd-most nationally behind FSU (9). Baylor has faced a second-half deficit and come back to win in 9 games this season. Baylor's defense this season has been its best of the Scott Drew era — BU ranks 3rd nationally in adjusted defensive efficiency, and Drew's previous best defense was ranked No. 23 nationally in 2014-15. Baylor is 1 of 3 teams ranked in the nation's top-25 in offensive and defensive field goal percentage. The Bears (23rd O, 7th D) join Gonzaga (5th O, 5th D) and Virginia (11th O, 10th D). Ole Miss ranks second in the SEC league pulling down 40.3 rpg and have been outrebounded only four times this year. Ole Miss ranks sixth in the SEC grabbing 12.5 offensive rpg this season. Senior F Sebastian Saiz has 13 double-doubles this season, which leads the SEC and ranks fifth nationally. Saiz leads the SEC and ranks sixth in the nation pulling down 11.5 rebounds a game and is the only player in the SEC this season averaging a double-double. Go Baylor here.
Kansas State at Tennessee
Could be a major letdown game for the SEC's Vols after they upset No. 4 Kentucky on Tuesday, 82-80. Robert Hubbs III scored 25 points and Tennessee surprised Kentucky to snap the Wildcats' seven-game winning streak. UT beat Kentucky in Knoxville for the second straight season and earned its first victory over a top-five team since an 83-76 triumph over No. 3 Pittsburgh on Dec. 11, 2010. The win also marked the first over a ranked opponent this season for the Volunteers after losing their previous five against ranked foes. The Vols scored the third most points of any of UK's opponents this season while shooting 47 percent from the field (30-for- 64) and holding the Wildcats to their third-lowest point total of the year. Kansas State of the Big 12 should be an NCAA Tournament team but comes off a 70-65 loss to Iowa State. Down 46-26 just 44 seconds into the second half after senior Matt Thomas connected on his seventh 3-pointer, the Wildcats slowly chipped away at the deficit. The team used a 20-3 run over a more than an eight-minute span to take a 58-57 edge. But then ISU regained the lead and wouldn't look back. I like the Wildcats here.
Georgetown at No. 11 Butler
It's a Big East game from Indianapolis. Rodney Pryor leads the Hoyas, and is fourth in the Big East, with 18.0 points per game. He is tied for second on the team with 4.9 rebounds band has hit a team-best 49 3-point field goals. He is third in the league in 3-point field goals made, averaging 2.5 per game. Through the first 20 games of the season, the Hoyas had played five games against nationally-ranked teams – beating then-No. 13 Oregon (65-61), falling to then-No. 16 Wisconsin (73-57), to No. 17 Xavier (81-76), to No. 18 Butler in overtime and at No. 22 Xavier – and another that was receiving votes, Maryland (L, 76-75). Butler goes for the season sweep here thanks to that earlier 85-76 overtime win in D.C. on Jan. 7. Bulldogs freshman Kamar Baldwin scored a career-high 16 points and helped hold GU's Pryor scoreless. Georgetown led by three points entering the final minute of regulation and had a chance to avoid overtime, but L.J. Peak's turnaround jumper at the buzzer clanked off the rim. The No. 11 ranking represents the Bulldogs' highest ranking since they were No. 9 on Dec. 28, 2015. Their highest rank ever was No. 8 on Feb. 19, 2008. Butler will roll here.
2020-21 College Basketball Week 9 Power Rankings Update
Which under-the-radar team can challenge #1 Gonzaga?
Indiana vs #4 Iowa | 2021 NCAA Basketball Expert Analysis
Davidson Wildcats vs. Fordham Rams: 1/20/2021 Betting Preview, Predictions, Computer Picks, Odds and Trends
East Tennessee State Buccaneers vs. Western Carolina Catamounts: 1/20/2021 Betting Trends, Predictions, Computer Picks, Odds, Line and Preview
Mercer Bears vs. Citadel Bulldogs: 1/20/2021 Betting Trends, Predictions, Computer Picks, Odds, Line and Preview
Saint Bonaventure Bonnies vs. VCU Rams: 1/20/2021 Betting Preview, Predictions, Computer Picks, Odds and Trends
Rhode Island Rams vs. Duquesne Dukes 1/20/2021 Betting Preview: Predictions, Computer Picks, Odds, Line, Spread and Trends
North Carolina Tar Heels vs. Wake Forest Demon Deacons: 1/20/2021 Matchup Betting Preview, Predictions, Computer Picks, Odds and Trends
Loyola Chicago Ramblers vs. Valparaiso Crusaders: 1/20/2021 Betting Trends, Predictions, Computer Picks, Odds, Line and Preview
Georgia Tech Yellow Jackets vs. Clemson Tigers: 1/20/2021 Predictions, Computer Picks, Odds, Line, Spread and Betting Trends
Creighton Bluejays vs. Providence Friars: 1/20/2021 Matchup Betting Preview, Predictions, Computer Picks, Odds and Trends
Kentucky Wildcats vs. Georgia Bulldogs: 1/20/2021 Betting Preview, Predictions, Computer Picks, Odds and Trends
George Mason Patriots vs. Saint Joseph's (PA) Hawks: 1/20/2021 Betting Preview, Predictions, Computer Picks, Odds and Trends
Furman Paladins vs. VMI Keydets: 1/20/2021 Betting Trends, Predictions, Computer Picks, Odds, Line and Preview | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,337 |
(function($) {
/**
* Class: $.jqplot.PieRenderer
* Plugin renderer to draw a pie chart.
* x values, if present, will be used as slice labels.
* y values give slice size.
*
* To use this renderer, you need to include the
* pie renderer plugin, for example:
*
* > <script type="text/javascript" src="plugins/jqplot.pieRenderer.js"></script>
*
* Properties described here are passed into the $.jqplot function
* as options on the series renderer. For example:
*
* > plot2 = $.jqplot('chart2', [s1, s2], {
* > seriesDefaults: {
* > renderer:$.jqplot.PieRenderer,
* > rendererOptions:{
* > sliceMargin: 2,
* > startAngle: -90
* > }
* > }
* > });
*
* A pie plot will trigger events on the plot target
* according to user interaction. All events return the event object,
* the series index, the point (slice) index, and the point data for
* the appropriate slice.
*
* 'jqplotDataMouseOver' - triggered when user mouseing over a slice.
* 'jqplotDataHighlight' - triggered the first time user mouses over a slice,
* if highlighting is enabled.
* 'jqplotDataUnhighlight' - triggered when a user moves the mouse out of
* a highlighted slice.
* 'jqplotDataClick' - triggered when the user clicks on a slice.
* 'jqplotDataRightClick' - tiggered when the user right clicks on a slice if
* the "captureRightClick" option is set to true on the plot.
*/
$.jqplot.PieRenderer = function(){
$.jqplot.LineRenderer.call(this);
};
$.jqplot.PieRenderer.prototype = new $.jqplot.LineRenderer();
$.jqplot.PieRenderer.prototype.constructor = $.jqplot.PieRenderer;
// called with scope of a series
$.jqplot.PieRenderer.prototype.init = function(options, plot) {
// Group: Properties
//
// prop: diameter
// Outer diameter of the pie, auto computed by default
this.diameter = null;
// prop: padding
// padding between the pie and plot edges, legend, etc.
this.padding = 20;
// prop: sliceMargin
// angular spacing between pie slices in degrees.
this.sliceMargin = 0;
// prop: fill
// true or false, whether to fil the slices.
this.fill = true;
// prop: shadowOffset
// offset of the shadow from the slice and offset of
// each succesive stroke of the shadow from the last.
this.shadowOffset = 2;
// prop: shadowAlpha
// transparency of the shadow (0 = transparent, 1 = opaque)
this.shadowAlpha = 0.07;
// prop: shadowDepth
// number of strokes to apply to the shadow,
// each stroke offset shadowOffset from the last.
this.shadowDepth = 5;
// prop: highlightMouseOver
// True to highlight slice when moused over.
// This must be false to enable highlightMouseDown to highlight when clicking on a slice.
this.highlightMouseOver = true;
// prop: highlightMouseDown
// True to highlight when a mouse button is pressed over a slice.
// This will be disabled if highlightMouseOver is true.
this.highlightMouseDown = false;
// prop: highlightColors
// an array of colors to use when highlighting a slice.
this.highlightColors = [];
// prop: dataLabels
// Either 'label', 'value', 'percent' or an array of labels to place on the pie slices.
// Defaults to percentage of each pie slice.
this.dataLabels = 'percent';
// prop: showDataLabels
// true to show data labels on slices.
this.showDataLabels = false;
// prop: dataLabelFormatString
// Format string for data labels. If none, '%s' is used for "label" and for arrays, '%d' for value and '%d%%' for percentage.
this.dataLabelFormatString = null;
// prop: dataLabelThreshold
// Threshhold in percentage (0-100) of pie area, below which no label will be displayed.
// This applies to all label types, not just to percentage labels.
this.dataLabelThreshold = 3;
// prop: dataLabelPositionFactor
// A Multiplier (0-1) of the pie radius which controls position of label on slice.
// Increasing will slide label toward edge of pie, decreasing will slide label toward center of pie.
this.dataLabelPositionFactor = 0.52;
// prop: dataLabelNudge
// Number of pixels to slide the label away from (+) or toward (-) the center of the pie.
this.dataLabelNudge = 2;
// prop: dataLabelCenterOn
// True to center the data label at its position.
// False to set the inside facing edge of the label at its position.
this.dataLabelCenterOn = true;
// prop showDataCategory
// true to show the str category of a slice.
this.showDataCategoryOnHover == true;
// prop: startAngle
// Angle to start drawing pie in degrees.
// According to orientation of canvas coordinate system:
// 0 = on the positive x axis
// -90 = on the positive y axis.
// 90 = on the negaive y axis.
// 180 or - 180 = on the negative x axis.
this.startAngle = 0;
this.tickRenderer = $.jqplot.PieTickRenderer;
// Used as check for conditions where pie shouldn't be drawn.
this._drawData = true;
this._type = 'pie';
// if user has passed in highlightMouseDown option and not set highlightMouseOver, disable highlightMouseOver
if (options.highlightMouseDown && options.highlightMouseOver == null) {
options.highlightMouseOver = false;
}
$.extend(true, this, options);
if (this.sliceMargin < 0) {
this.sliceMargin = 0;
}
this._diameter = null;
this._radius = null;
// array of [start,end] angles arrays, one for each slice. In radians.
this._sliceAngles = [];
// index of the currenty highlighted point, if any
this._highlightedPoint = null;
this.highlightLabelSliceRenderer = new $.jqplot.PieHighlighterLabelSliceRenderer(this);
// labels of slices.
this.labelsSlices = [];
// set highlight colors if none provided
if (this.highlightColors.length == 0) {
for (var i=0; i<this.seriesColors.length; i++){
var rgba = $.jqplot.getColorComponents(this.seriesColors[i]);
var newrgb = [rgba[0], rgba[1], rgba[2]];
var sum = newrgb[0] + newrgb[1] + newrgb[2];
for (var j=0; j<3; j++) {
// when darkening, lowest color component can be is 60.
newrgb[j] = (sum > 570) ? newrgb[j] * 0.8 : newrgb[j] + 0.3 * (255 - newrgb[j]);
newrgb[j] = parseInt(newrgb[j], 10);
}
this.highlightColors.push('rgb('+newrgb[0]+','+newrgb[1]+','+newrgb[2]+')');
}
}
this.highlightColorGenerator = new $.jqplot.ColorGenerator(this.highlightColors);
plot.postParseOptionsHooks.addOnce(postParseOptions);
plot.postInitHooks.addOnce(postInit);
plot.eventListenerHooks.addOnce('jqplotMouseMove', handleMove);
plot.eventListenerHooks.addOnce('jqplotMouseDown', handleMouseDown);
plot.eventListenerHooks.addOnce('jqplotMouseUp', handleMouseUp);
plot.eventListenerHooks.addOnce('jqplotClick', handleClick);
plot.eventListenerHooks.addOnce('jqplotRightClick', handleRightClick);
plot.postDrawHooks.addOnce(postPlotDraw);
};
$.jqplot.PieRenderer.prototype.setGridData = function(plot) {
// set gridData property. This will hold angle in radians of each data point.
var stack = [];
var td = [];
var sa = this.startAngle/180*Math.PI;
var tot = 0;
// don't know if we have any valid data yet, so set plot to not draw.
this._drawData = false;
for (var i=0; i<this.data.length; i++){
if (this.data[i][1] != 0) {
// we have data, O.K. to draw.
this._drawData = true;
}
stack.push(this.data[i][1]);
td.push([this.data[i][0]]);
if (i>0) {
stack[i] += stack[i-1];
}
tot += this.data[i][1];
}
var fact = Math.PI*2/stack[stack.length - 1];
for (var i=0; i<stack.length; i++) {
td[i][1] = stack[i] * fact;
td[i][2] = this.data[i][1]/tot;
}
this.gridData = td;
};
$.jqplot.PieRenderer.prototype.makeGridData = function(data, plot) {
var stack = [];
var td = [];
var tot = 0;
var sa = this.startAngle/180*Math.PI;
// don't know if we have any valid data yet, so set plot to not draw.
this._drawData = false;
for (var i=0; i<data.length; i++){
if (this.data[i][1] != 0) {
// we have data, O.K. to draw.
this._drawData = true;
}
stack.push(data[i][1]);
td.push([data[i][0]]);
if (i>0) {
stack[i] += stack[i-1];
}
tot += data[i][1];
}
var fact = Math.PI*2/stack[stack.length - 1];
for (var i=0; i<stack.length; i++) {
td[i][1] = stack[i] * fact;
td[i][2] = data[i][1]/tot;
}
return td;
};
function calcRadiusAdjustment(ang) {
return Math.sin((ang - (ang-Math.PI) / 8 / Math.PI )/2.0);
}
function calcRPrime(ang1, ang2, sliceMargin, fill, lineWidth) {
var rprime = 0;
var ang = ang2 - ang1;
var absang = Math.abs(ang);
var sm = sliceMargin;
if (fill == false) {
sm += lineWidth;
}
if (sm > 0 && absang > 0.01 && absang < 6.282) {
rprime = parseFloat(sm) / 2.0 / calcRadiusAdjustment(ang);
}
return rprime;
}
$.jqplot.PieRenderer.prototype.drawSlice = function (ctx, ang1, ang2, color, isShadow) {
if (this._drawData) {
var r = this._radius;
var fill = this.fill;
var lineWidth = this.lineWidth;
var sm = this.sliceMargin;
if (this.fill == false) {
sm += this.lineWidth;
}
ctx.save();
ctx.translate(this._center[0], this._center[1]);
var rprime = calcRPrime(ang1, ang2, this.sliceMargin, this.fill, this.lineWidth);
var transx = rprime * Math.cos((ang1 + ang2) / 2.0);
var transy = rprime * Math.sin((ang1 + ang2) / 2.0);
if ((ang2 - ang1) <= Math.PI) {
r -= rprime;
}
else {
r += rprime;
}
ctx.translate(transx, transy);
if (isShadow) {
for (var i=0, l=this.shadowDepth; i<l; i++) {
ctx.save();
ctx.translate(this.shadowOffset*Math.cos(this.shadowAngle/180*Math.PI), this.shadowOffset*Math.sin(this.shadowAngle/180*Math.PI));
doDraw(r);
}
for (var i=0, l=this.shadowDepth; i<l; i++) {
ctx.restore();
}
}
else {
doDraw(r);
}
ctx.restore();
}
function doDraw (rad) {
// Fix for IE and Chrome that can't seem to draw circles correctly.
// ang2 should always be <= 2 pi since that is the way the data is converted.
// 2Pi = 6.2831853, Pi = 3.1415927
if (ang2 > 6.282 + this.startAngle) {
ang2 = 6.282 + this.startAngle;
if (ang1 > ang2) {
ang1 = 6.281 + this.startAngle;
}
}
// Fix for IE, where it can't seem to handle 0 degree angles. Also avoids
// ugly line on unfilled pies.
if (ang1 >= ang2) {
return;
}
ctx.beginPath();
ctx.fillStyle = color;
ctx.strokeStyle = color;
ctx.lineWidth = lineWidth;
ctx.arc(0, 0, rad, ang1, ang2, false);
ctx.lineTo(0,0);
ctx.closePath();
if (fill) {
ctx.fill();
}
else {
ctx.stroke();
}
}
};
$.jqplot.PieHighlighterLabelSliceRenderer = function(serie){
this.serie = serie;
};
$.jqplot.PieHighlighterLabelSliceRenderer.prototype.highlight = function(pid){
if(this.serie.labelsSlices[pid]){
this.serie.labelsSlices[pid].removeClass("hidden");
}
};
$.jqplot.PieHighlighterLabelSliceRenderer.prototype.unhighlight = function(pid){
for (var i = this.serie.labelsSlices.length - 1; i >= 0; i--) {
if(this.serie.labelsSlices){
this.serie.labelsSlices[i].addClass("hidden");
}
};
};
$.jqplot.PieHighlighterLabelSliceRenderer.prototype.move = function(pid, x, y){
if (this.serie.labelsSlices[pid]){
var pointLblPosition = this.serie.labelsSlices[pid].position();
var tooltip = this.serie.labelsSlices[pid].find(".tooltip");
var topPos = y - pointLblPosition.top - (this.serie.labelsSlices[pid].height()*2 + tooltip.height());
var leftPos = x - pointLblPosition.left - this.serie.labelsSlices[pid].width() - tooltip.width() / 2;
tooltip.css({left: leftPos, top: topPos});
}
};
// called with scope of series
$.jqplot.PieRenderer.prototype.draw = function (ctx, gd, options, plot) {
this.labelsSlices= [];
var i;
var opts = (options != undefined) ? options : {};
// offset and direction of offset due to legend placement
var offx = 0;
var offy = 0;
var trans = 1;
var colorGenerator = new $.jqplot.ColorGenerator(this.seriesColors);
if (options.legendInfo && options.legendInfo.placement == 'insideGrid') {
var li = options.legendInfo;
switch (li.location) {
case 'nw':
offx = li.width + li.xoffset;
break;
case 'w':
offx = li.width + li.xoffset;
break;
case 'sw':
offx = li.width + li.xoffset;
break;
case 'ne':
offx = li.width + li.xoffset;
trans = -1;
break;
case 'e':
offx = li.width + li.xoffset;
trans = -1;
break;
case 'se':
offx = li.width + li.xoffset;
trans = -1;
break;
case 'n':
offy = li.height + li.yoffset;
break;
case 's':
offy = li.height + li.yoffset;
trans = -1;
break;
default:
break;
}
}
var shadow = (opts.shadow != undefined) ? opts.shadow : this.shadow;
var fill = (opts.fill != undefined) ? opts.fill : this.fill;
var cw = ctx.canvas.width;
var ch = ctx.canvas.height;
var w = cw - offx - 2 * this.padding;
var h = ch - offy - 2 * this.padding;
var mindim = Math.min(w,h);
var d = mindim;
// Fixes issue #272. Thanks hugwijst!
// reset slice angles array.
this._sliceAngles = [];
var sm = this.sliceMargin;
if (this.fill == false) {
sm += this.lineWidth;
}
var rprime;
var maxrprime = 0;
var ang, ang1, ang2, shadowColor;
var sa = this.startAngle / 180 * Math.PI;
// have to pre-draw shadows, so loop throgh here and calculate some values also.
for (var i=0, l=gd.length; i<l; i++) {
ang1 = (i == 0) ? sa : gd[i-1][1] + sa;
ang2 = gd[i][1] + sa;
this._sliceAngles.push([ang1, ang2]);
rprime = calcRPrime(ang1, ang2, this.sliceMargin, this.fill, this.lineWidth);
if (Math.abs(ang2-ang1) > Math.PI) {
maxrprime = Math.max(rprime, maxrprime);
}
}
if (this.diameter != null && this.diameter > 0) {
this._diameter = this.diameter - 2*maxrprime;
}
else {
this._diameter = d - 2*maxrprime;
}
// Need to check for undersized pie. This can happen if
// plot area too small and legend is too big.
if (this._diameter < 6) {
$.jqplot.log('Diameter of pie too small, not rendering.');
return;
}
var r = this._radius = this._diameter/2;
this._center = [(cw - trans * offx)/2 + trans * offx + maxrprime * Math.cos(sa), (ch - trans*offy)/2 + trans * offy + maxrprime * Math.sin(sa)];
if (this.shadow) {
for (var i=0, l=gd.length; i<l; i++) {
shadowColor = 'rgba(0,0,0,'+this.shadowAlpha+')';
this.renderer.drawSlice.call (this, ctx, this._sliceAngles[i][0], this._sliceAngles[i][1], shadowColor, true);
}
}
for (var i=0; i<gd.length; i++) {
var sliceColor = colorGenerator.next();
if(this.labelColor == "complementary"){
var labelColor = '#'+rgbToHex(moveColor(hexToRGB(sliceColor),180));
}
else if (opts.labelColor !=""){
var labelColor = this.labelColor;
}
else {
var labelColor = "#FFFFFF";
}
this.renderer.drawSlice.call (this, ctx, this._sliceAngles[i][0], this._sliceAngles[i][1], sliceColor, false);
if (this.showDataLabels && gd[i][2]*100 >= this.dataLabelThreshold) {
var fstr, avgang = (this._sliceAngles[i][0] + this._sliceAngles[i][1])/2, label;
if (this.dataLabels == 'label') {
fstr = this.dataLabelFormatString || '%s';
label = $.jqplot.sprintf(fstr, gd[i][0]);
}
else if (this.dataLabels == 'value') {
fstr = this.dataLabelFormatString || '%d';
label = $.jqplot.sprintf(fstr, this.data[i][1]);
}
else if (this.dataLabels == 'percent') {
fstr = this.dataLabelFormatString || '%d%%';
label = $.jqplot.sprintf(fstr, gd[i][2]*100);
}
else if (this.dataLabels.constructor == Array) {
fstr = this.dataLabelFormatString || '%s';
label = $.jqplot.sprintf(fstr, this.dataLabels[i]);
}
var fact = (this._radius ) * this.dataLabelPositionFactor + this.sliceMargin + this.dataLabelNudge;
var x = this._center[0] + Math.cos(avgang) * fact + this.canvas._offsets.left;
var y = this._center[1] + Math.sin(avgang) * fact + this.canvas._offsets.top;
var labelelem;
if (this.showDataCategoryOnHover){
var fstrTooltip = this.dataLabelFormatString || '%s';
labelTooltip = $.jqplot.sprintf(fstrTooltip, gd[i][0]);
var tooltip = '<span class="tooltip" style="position:absolute;color:'+labelColor+';">' + labelTooltip + '</span>';
var txt = '<span class="jqplot-pie-series jqplot-data-label" style="position:absolute;color:'+labelColor+';">' + label + tooltip + + '</span>'
var labelelem = $(txt).insertBefore(plot.eventCanvas._elem);
this.labelsSlices.push(labelelem);
labelelem.addClass('jqplot-pie-show-hover hidden');
}else{
labelelem = $('<div class="jqplot-pie-series jqplot-data-label" style="position:absolute;color:'+labelColor+';">' + label + '</div>').insertBefore(plot.eventCanvas._elem);
}
if (this.dataLabelCenterOn) {
x -= labelelem.width()/2;
y -= labelelem.height()/2;
}
else {
x -= labelelem.width() * Math.sin(avgang/2);
y -= labelelem.height()/2;
}
x = Math.round(x);
y = Math.round(y);
labelelem.css({left: x, top: y});
}else if(this.showDataCategoryOnHover && gd[i][2]*100 >= this.dataLabelThreshold){
var fstr, avgang = (this._sliceAngles[i][0] + this._sliceAngles[i][1])/2, label;
if (this.dataLabels == 'label') {
fstr = this.dataLabelFormatString || '%s';
label = $.jqplot.sprintf(fstr, gd[i][0]);
}
else if (this.dataLabels == 'value') {
fstr = this.dataLabelFormatString || '%d';
label = $.jqplot.sprintf(fstr, this.data[i][1]);
}
else if (this.dataLabels == 'percent') {
fstr = this.dataLabelFormatString || '%d%%';
label = $.jqplot.sprintf(fstr, gd[i][2]*100);
}
else if (this.dataLabels.constructor == Array) {
fstr = this.dataLabelFormatString || '%s';
label = $.jqplot.sprintf(fstr, this.dataLabels[i]);
}
var fact = (this._radius ) * this.dataLabelPositionFactor + this.sliceMargin + this.dataLabelNudge;
var x = this._center[0] + Math.cos(avgang) * fact + this.canvas._offsets.left;
var y = this._center[1] + Math.sin(avgang) * fact + this.canvas._offsets.top;
var labelelem;
var fstrTooltip = this.dataLabelFormatString || '%s';
labelTooltip = $.jqplot.sprintf(fstrTooltip, gd[i][0]);
var tooltip = '<span class="tooltip" style="position:absolute;color:'+labelColor+';">' + labelTooltip + '</span>';
var txt = '<span class="jqplot-pie-series jqplot-data-label" style="position:absolute;color:'+labelColor+';">' + label + tooltip + + '</span>'
var labelelem = $(txt).insertBefore(plot.eventCanvas._elem);
this.labelsSlices.push(labelelem);
labelelem.addClass('jqplot-pie-show-hover hidden');
}
}
};
$.jqplot.PieAxisRenderer = function() {
$.jqplot.LinearAxisRenderer.call(this);
};
$.jqplot.PieAxisRenderer.prototype = new $.jqplot.LinearAxisRenderer();
$.jqplot.PieAxisRenderer.prototype.constructor = $.jqplot.PieAxisRenderer;
// There are no traditional axes on a pie chart. We just need to provide
// dummy objects with properties so the plot will render.
// called with scope of axis object.
$.jqplot.PieAxisRenderer.prototype.init = function(options){
//
this.tickRenderer = $.jqplot.PieTickRenderer;
$.extend(true, this, options);
// I don't think I'm going to need _dataBounds here.
// have to go Axis scaling in a way to fit chart onto plot area
// and provide u2p and p2u functionality for mouse cursor, etc.
// for convienence set _dataBounds to 0 and 100 and
// set min/max to 0 and 100.
this._dataBounds = {min:0, max:100};
this.min = 0;
this.max = 100;
this.showTicks = false;
this.ticks = [];
this.showMark = false;
this.show = false;
};
$.jqplot.PieLegendRenderer = function(){
$.jqplot.TableLegendRenderer.call(this);
};
$.jqplot.PieLegendRenderer.prototype = new $.jqplot.TableLegendRenderer();
$.jqplot.PieLegendRenderer.prototype.constructor = $.jqplot.PieLegendRenderer;
/**
* Class: $.jqplot.PieLegendRenderer
* Legend Renderer specific to pie plots. Set by default
* when user creates a pie plot.
*/
$.jqplot.PieLegendRenderer.prototype.init = function(options) {
// Group: Properties
//
// prop: numberRows
// Maximum number of rows in the legend. 0 or null for unlimited.
this.numberRows = null;
// prop: numberColumns
// Maximum number of columns in the legend. 0 or null for unlimited.
this.numberColumns = null;
$.extend(true, this, options);
};
// called with context of legend
$.jqplot.PieLegendRenderer.prototype.draw = function() {
var legend = this;
if (this.show) {
var series = this._series;
this._elem = $(document.createElement('table'));
this._elem.addClass('jqplot-table-legend');
var ss = {position:'absolute'};
if (this.background) {
ss['background'] = this.background;
}
if (this.border) {
ss['border'] = this.border;
}
if (this.fontSize) {
ss['fontSize'] = this.fontSize;
}
if (this.fontFamily) {
ss['fontFamily'] = this.fontFamily;
}
if (this.textColor) {
ss['textColor'] = this.textColor;
}
if (this.marginTop != null) {
ss['marginTop'] = this.marginTop;
}
if (this.marginBottom != null) {
ss['marginBottom'] = this.marginBottom;
}
if (this.marginLeft != null) {
ss['marginLeft'] = this.marginLeft;
}
if (this.marginRight != null) {
ss['marginRight'] = this.marginRight;
}
this._elem.css(ss);
// Pie charts legends don't go by number of series, but by number of data points
// in the series. Refactor things here for that.
var pad = false,
reverse = false,
nr,
nc;
var s = series[0];
var colorGenerator = new $.jqplot.ColorGenerator(s.seriesColors);
//console.log(s.seriesColors);
if (s.show) {
var pd = s.data;
if (this.numberRows) {
nr = this.numberRows;
if (!this.numberColumns){
nc = Math.ceil(pd.length/nr);
}
else{
nc = this.numberColumns;
}
}
else if (this.numberColumns) {
nc = this.numberColumns;
nr = Math.ceil(pd.length/this.numberColumns);
}
else {
nr = pd.length;
nc = 1;
}
var i, j;
var tr, td1, td2;
var lt, rs, color;
var idx = 0;
var div0, div1;
for (i=0; i<nr; i++) {
tr = $(document.createElement('tr'));
tr.addClass('jqplot-table-legend');
if (reverse){
tr.prependTo(this._elem);
}
else{
tr.appendTo(this._elem);
}
for (j=0; j<nc; j++) {
if (idx < pd.length){
lt = this.labels[idx] || pd[idx][0].toString();
color = colorGenerator.next();
if (!reverse){
if (i>0){
pad = true;
}
else{
pad = false;
}
}
else{
if (i == nr -1){
pad = false;
}
else{
pad = true;
}
}
rs = (pad) ? this.rowSpacing : '0';
td1 = $(document.createElement('td'));
td1.addClass('jqplot-table-legend jqplot-table-legend-swatch');
td1.css({textAlign: 'center', paddingTop: rs});
div0 = $(document.createElement('div'));
div0.addClass('jqplot-table-legend-swatch-outline');
div1 = $(document.createElement('div'));
div1.addClass('jqplot-table-legend-swatch');
div1.css({backgroundColor: color, borderColor: color});
td1.append(div0.append(div1));
td2 = $(document.createElement('td'));
td2.addClass('jqplot-table-legend jqplot-table-legend-label');
td2.css('paddingTop', rs);
if (this.escapeHtml){
td2.text(lt);
}
else {
td2.html(lt);
}
if (reverse) {
td2.prependTo(tr);
td1.prependTo(tr);
}
else {
td1.appendTo(tr);
td2.appendTo(tr);
}
pad = true;
}
idx++;
}
}
}
}
return this._elem;
};
$.jqplot.PieRenderer.prototype.handleMove = function(ev, gridpos, datapos, neighbor, plot) {
if (neighbor) {
var ins = [neighbor.seriesIndex, neighbor.pointIndex, neighbor.data];
plot.target.trigger('jqplotDataMouseOver', ins);
if (plot.series[ins[0]].highlightMouseOver && !(ins[0] == plot.plugins.pieRenderer.highlightedSeriesIndex && ins[1] == plot.series[ins[0]]._highlightedPoint)) {
plot.target.trigger('jqplotDataHighlight', ins);
highlight (plot, ins[0], ins[1]);
}
}
else if (neighbor == null) {
unhighlight (plot);
}
};
// this.eventCanvas._elem.bind($.jqplot.eventListenerHooks[i][0], {plot:this}, $.jqplot.eventListenerHooks[i][1]);
// setup default renderers for axes and legend so user doesn't have to
// called with scope of plot
function preInit(target, data, options) {
options = options || {};
options.axesDefaults = options.axesDefaults || {};
options.legend = options.legend || {};
options.seriesDefaults = options.seriesDefaults || {};
// only set these if there is a pie series
var setopts = false;
if (options.seriesDefaults.renderer == $.jqplot.PieRenderer) {
setopts = true;
}
else if (options.series) {
for (var i=0; i < options.series.length; i++) {
if (options.series[i].renderer == $.jqplot.PieRenderer) {
setopts = true;
}
}
}
if (setopts) {
options.axesDefaults.renderer = $.jqplot.PieAxisRenderer;
options.legend.renderer = $.jqplot.PieLegendRenderer;
options.legend.preDraw = true;
options.seriesDefaults.pointLabels = {show: false};
}
}
function postInit(target, data, options) {
for (var i=0; i<this.series.length; i++) {
if (this.series[i].renderer.constructor == $.jqplot.PieRenderer) {
// don't allow mouseover and mousedown at same time.
if (this.series[i].highlightMouseOver) {
this.series[i].highlightMouseDown = false;
}
}
}
}
// called with scope of plot
function postParseOptions(options) {
for (var i=0; i<this.series.length; i++) {
this.series[i].seriesColors = this.seriesColors;
this.series[i].colorGenerator = $.jqplot.colorGenerator;
}
}
function highlight (plot, sidx, pidx) {
var s = plot.series[sidx];
var canvas = plot.plugins.pieRenderer.highlightCanvas;
canvas._ctx.clearRect(0,0,canvas._ctx.canvas.width, canvas._ctx.canvas.height);
s._highlightedPoint = pidx;
plot.plugins.pieRenderer.highlightedSeriesIndex = sidx;
// need to regenerate highlight color (for theme change color fixes).
// set highlight colors if none provided
if (s.highlightColors.length == 0) {
for (var i=0; i<s.seriesColors.length; i++){
var rgba = $.jqplot.getColorComponents(this.seriesColors[i]);
var newrgb = [rgba[0], rgba[1], rgba[2]];
var sum = newrgb[0] + newrgb[1] + newrgb[2];
for (var j=0; j<3; j++) {
// when darkening, lowest color component can be is 60.
newrgb[j] = (sum > 570) ? newrgb[j] * 0.8 : newrgb[j] + 0.3 * (255 - newrgb[j]);
newrgb[j] = parseInt(newrgb[j], 10);
}
s.highlightColors.push('rgb('+newrgb[0]+','+newrgb[1]+','+newrgb[2]+')');
}
}
s.highlightColorGenerator = new $.jqplot.ColorGenerator(s.highlightColors);
s.renderer.drawSlice.call(s, canvas._ctx, s._sliceAngles[pidx][0], s._sliceAngles[pidx][1], s.highlightColorGenerator.get(pidx), false);
s.highlightLabelSliceRenderer.highlight(pidx);
}
function unhighlight (plot) {
var canvas = plot.plugins.pieRenderer.highlightCanvas;
canvas._ctx.clearRect(0,0, canvas._ctx.canvas.width, canvas._ctx.canvas.height);
for (var i=0; i<plot.series.length; i++) {
plot.series[i]._highlightedPoint = null;
}
plot.plugins.pieRenderer.highlightedSeriesIndex = null;
plot.target.trigger('jqplotDataUnhighlight');
for (var i = plot.series.length - 1; i >= 0; i--) {
plot.series[i].highlightLabelSliceRenderer.unhighlight();
};
}
function handleMove(ev, gridpos, datapos, neighbor, plot) {
if (neighbor) {
var ins = [neighbor.seriesIndex, neighbor.pointIndex, neighbor.data];
var evt1 = jQuery.Event('jqplotDataMouseOver');
evt1.pageX = ev.pageX;
evt1.pageY = ev.pageY;
plot.target.trigger(evt1, ins);
if (plot.series[ins[0]].highlightMouseOver && !(ins[0] == plot.plugins.pieRenderer.highlightedSeriesIndex && ins[1] == plot.series[ins[0]]._highlightedPoint)) {
var evt = jQuery.Event('jqplotDataHighlight');
evt.which = ev.which;
evt.pageX = ev.pageX;
evt.pageY = ev.pageY;
plot.target.trigger(evt, ins);
for (var i = plot.series.length - 1; i >= 0; i--) {
plot.series[i].highlightLabelSliceRenderer.unhighlight();
};
highlight (plot, ins[0], ins[1]);
}else{
plot.series[ins[0]].highlightLabelSliceRenderer.move(ins[1], evt1.pageX, evt1.pageY);
}
}
else if (neighbor == null) {
unhighlight (plot);
}
}
function handleMouseDown(ev, gridpos, datapos, neighbor, plot) {
if (neighbor) {
var ins = [neighbor.seriesIndex, neighbor.pointIndex, neighbor.data];
if (plot.series[ins[0]].highlightMouseDown && !(ins[0] == plot.plugins.pieRenderer.highlightedSeriesIndex && ins[1] == plot.series[ins[0]]._highlightedPoint)) {
var evt = jQuery.Event('jqplotDataHighlight');
evt.which = ev.which;
evt.pageX = ev.pageX;
evt.pageY = ev.pageY;
plot.target.trigger(evt, ins);
highlight (plot, ins[0], ins[1]);
}
}
else if (neighbor == null) {
unhighlight (plot);
}
}
function handleMouseUp(ev, gridpos, datapos, neighbor, plot) {
var idx = plot.plugins.pieRenderer.highlightedSeriesIndex;
if (idx != null && plot.series[idx].highlightMouseDown) {
unhighlight(plot);
}
}
function handleClick(ev, gridpos, datapos, neighbor, plot) {
if (neighbor) {
var ins = [neighbor.seriesIndex, neighbor.pointIndex, neighbor.data];
var evt = jQuery.Event('jqplotDataClick');
evt.which = ev.which;
evt.pageX = ev.pageX;
evt.pageY = ev.pageY;
plot.target.trigger(evt, ins);
}
}
function handleRightClick(ev, gridpos, datapos, neighbor, plot) {
if (neighbor) {
var ins = [neighbor.seriesIndex, neighbor.pointIndex, neighbor.data];
var idx = plot.plugins.pieRenderer.highlightedSeriesIndex;
if (idx != null && plot.series[idx].highlightMouseDown) {
unhighlight(plot);
}
var evt = jQuery.Event('jqplotDataRightClick');
evt.which = ev.which;
evt.pageX = ev.pageX;
evt.pageY = ev.pageY;
plot.target.trigger(evt, ins);
}
}
// called within context of plot
// create a canvas which we can draw on.
// insert it before the eventCanvas, so eventCanvas will still capture events.
function postPlotDraw() {
this.highlightLabelSliceRenderer = new $.jqplot.PieHighlighterLabelSliceRenderer(this);
// Memory Leaks patch
if (this.plugins.pieRenderer && this.plugins.pieRenderer.highlightCanvas) {
this.plugins.pieRenderer.highlightCanvas.resetCanvas();
this.plugins.pieRenderer.highlightCanvas = null;
}
this.plugins.pieRenderer = {highlightedSeriesIndex:null};
this.plugins.pieRenderer.highlightCanvas = new $.jqplot.GenericCanvas();
// do we have any data labels? if so, put highlight canvas before those
var labels = $(this.targetId+' .jqplot-data-label');
if (labels.length) {
$(labels[0]).before(this.plugins.pieRenderer.highlightCanvas.createElement(this._gridPadding, 'jqplot-pieRenderer-highlight-canvas', this._plotDimensions, this));
}
// else put highlight canvas before event canvas.
else {
this.eventCanvas._elem.before(this.plugins.pieRenderer.highlightCanvas.createElement(this._gridPadding, 'jqplot-pieRenderer-highlight-canvas', this._plotDimensions, this));
}
var hctx = this.plugins.pieRenderer.highlightCanvas.setContext();
//this.eventCanvas._elem.bind('mouseleave', {plot:this}, function (ev) { unhighlight(ev.data.plot); });
}
$.jqplot.preInitHooks.push(preInit);
$.jqplot.PieTickRenderer = function() {
$.jqplot.AxisTickRenderer.call(this);
};
$.jqplot.PieTickRenderer.prototype = new $.jqplot.AxisTickRenderer();
$.jqplot.PieTickRenderer.prototype.constructor = $.jqplot.PieTickRenderer;
})(jQuery);
function getHue (R,G,B) {
return (180/Math.PI*Math.atan2( Math.sqrt(3)*(G-B) , 2*R-G-B ));
}
function RGBToHSL(color) {
var R = ( color[0] / 255 ); //RGB from 0 to 255
var G = ( color[1] / 255 );
var B = ( color[2] / 255 );
var Min = Math.min(R,G,B); //Min. value of RGB
var Max = Math.max(R,G,B); //Max. value of RGB
var del_Max = Max - Min; //Delta RGB value
var L = ( Max + Min ) / 2;
if ( del_Max == 0 ){
H = 0;
S = 0;
}
else {
if ( L < 0.5 ) S = del_Max / ( Max + Min );
else S = del_Max / ( 2 - Max - Min );
var del_R = ( ( ( Max - R ) / 6 ) + ( del_Max / 2 ) ) / del_Max;
var del_G = ( ( ( Max - G ) / 6 ) + ( del_Max / 2 ) ) / del_Max;
var del_B = ( ( ( Max - B ) / 6 ) + ( del_Max / 2 ) ) / del_Max;
if ( R == Max ) var H = del_B - del_G;
else if ( G == Max ) var H = ( 1 / 3 ) + del_R - del_B;
else if ( B == Max ) var H = ( 2 / 3 ) + del_G - del_R;
if ( H < 0 ) H += 1;
if ( H > 1 ) H -= 1;
}
return ([H,S,L]);
}
function HSLtoRGB(hsl) {
var H = hsl[0];
var S = hsl[1];
var L = hsl[2];
if ( S == 0 ){
var R = L * 255;
var G = L * 255;
var B = L * 255;
}
else{
if ( L < 0.5 ) var var_2 = L * ( 1 + S );
else var var_2 = ( L + S ) - ( S * L );
var var_1 = 2 * L - var_2;
var R = Math.floor(255 * Hue_2_RGB( var_1, var_2, H + ( 1 / 3 ) ));
var G = Math.floor(255 * Hue_2_RGB( var_1, var_2, H ));
var B = Math.floor(255 * Hue_2_RGB( var_1, var_2, H - ( 1 / 3 ) ));
}
return ([R,G,B]);
}
function Hue_2_RGB( v1, v2, vH )
{
if ( vH < 0 ) vH += 1;
if ( vH > 1 ) vH -= 1;
if ( ( 6 * vH ) < 1 ) {
return ( v1 + ( v2 - v1 ) * 6 * vH );
}
else if ( ( 2 * vH ) < 1 ) {
return ( v2 );
}
else if ( ( 3 * vH ) < 2 ) {
return ( v1 + ( v2 - v1 ) * ( ( 2 / 3 ) - vH ) * 6 );
}
else return ( v1 );
}
function hexToRGB(h) {
if (h.charAt(0)=="#") h = h.substring(1,7);
return [parseInt(h.substring(0,2),16),parseInt(h.substring(2,4),16),parseInt(h.substring(4,6),16)];
}
function moveColor(color,degrees){
var hsl = RGBToHSL(color);
hsl[0] = (((hsl[0]*360)+degrees)%360)/360;
return HSLtoRGB(hsl);
}
function rgbToHex(color) {return toHex(color[0])+toHex(color[1])+toHex(color[2])}
function toHex(n) {
n = parseInt(n,10);
if (isNaN(n)) return "00";
n = Math.max(0,Math.min(n,255));
return "0123456789ABCDEF".charAt((n-n%16)/16)
+ "0123456789ABCDEF".charAt(n%16);
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 108 |
{"url":"https:\/\/www.gradesaver.com\/textbooks\/math\/algebra\/algebra-1-common-core-15th-edition\/chapter-1-foundations-for-algebra-1-5-adding-and-subtracting-real-numbers-practice-and-problem-solving-exercises-page-34\/47","text":"## Algebra 1: Common Core (15th Edition)\n\n$-4-(-5)$ =$1$\nThe answer is $1$ not $-1$ because when subtracting an integer, the double negatives becomes positive. $-4-(-5)$ becomes $-4+5$ which is equal to $1$.","date":"2019-01-21 18:11:11","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4567374289035797, \"perplexity\": 1101.8647408107918}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-04\/segments\/1547583804001.73\/warc\/CC-MAIN-20190121172846-20190121194846-00257.warc.gz\"}"} | null | null |
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