text:
Unadilla (village), New York facts for kids
1.1 sq mi (2.8 km2)
1,010 ft (308 m)
1,030/sq mi (403/km2)
Eastern (EST) (UTC-5)
EDT (UTC-4)
Unadilla is a village located in the Town of Unadilla in Otsego County, New York, USA. The population was 1,128 at the 2010 census.
The Village of Unadilla is located in the south part of the town, southwest of Oneonta. The community refers to itself as "The Village Beautiful."
The town was originally the site of an Indian village at the confluence of the Susquehanna and Unadilla Rivers where the Village of Sidney (in Delaware County) is now located. The name Unadilla, which represents one spelling from a number of different interpretations of several Indian dialects, is said to mean "Meeting Place", indicating the place where two rivers meet.
Unadilla was first settled by whites around 1770. The location of the village now called Unadilla was first settled about the same time by a group of Scots-Irish at the junction of the Ouleout Creek and the Susquehanna River. Both villages were destroyed by the Continental Army in October 1778 because they were being used by Loyalists as a base for raiding frontier communities.
Shortly thereafter, Unadilla Village was again settled, and originally called "Wattles Ferry," It grew after it became an important stop for travel west, the Catskill Turnpike crossing the Susquehanna at Wattles Ferry, located in the eastern end of the village. Eventually, the turnpikes were replaced in importance by the Albany and Susquehanna Railroad (later the Delaware and Hudson Railway), while the small community continued to take advantage of this progress and prosper.
Unadilla set itself off from the town by incorporating as a village in 1827, finally incorporated in 1889.
The village itself contains many historical homes, some built before the Civil War in 1860.
Unadilla's Boy Scout Troop #1 was chartered in 1910, and is recognized as the oldest continuously operating troop in the United States.
The Unadilla Village Historic District and Roswell Wright House are listed on the National Register of Historic Places.
Unadilla is located at Lua error in package.lua at line 80: module 'Module:mw' not found. (42.326694, -75.314863).
According to the United States Census Bureau, the village has a total area of 1.1 square miles (2.8 km²). None of the area is covered with water.
The Village of Unadilla is located in Otsego County, New York adjacent to the Susquehanna River by the county line shared with Delaware County. The village is about three miles upstream from the Village of Sidney located in Delaware County. The Village of Unadilla is located in the south part of the Town of Unadilla on New York Route 7 (Main Street) and north of Interstate 88.
1870 875 —
1880 922 5.4%
1910 1,009 −13.9%
Est. 2015 1,088 −3.5%
U.S. Decennial Census
As of the census of 2000, there were 1,127 people, 470 households, and 314 families residing in the village. The population density was 1,045.2 people per square mile (402.9/km²). There were 542 housing units at an average density of 502.7 per square mile (193.8/km²). The racial makeup of the village was 97.07% White, 0.53% Black or African American, 0.18% Native American, 0.09% Asian, 0.09% Pacific Islander, 1.42% from other races, and 0.62% from two or more races. Hispanic or Latino of any race were 1.06% of the population.
There were 470 households out of which 30.2% had children under the age of 18 living with them, 48.7% were married couples living together, 13.2% had a female householder with no husband present, and 33.0% were non-families. 28.1% of all households were made up of individuals and 13.8% had someone living alone who was 65 years of age or older. The average household size was 2.40 and the average family size was 2.91.
In the village, the population was spread out with 24.5% under the age of 18, 9.8% from 18 to 24, 25.5% from 25 to 44, 23.2% from 45 to 64, and 17.0% who were 65 years of age or older. The median age was 40 years. For every 100 females there were 92.0 males. For every 100 females age 18 and over, there were 87.4 males.
The median income for a household in the village was $34,886, and the median income for a family was $39,097. Males had a median income of $29,808 versus $22,500 for females. The per capita income for the village was $19,687. About 11.1% of families and 10.8% of the population were below the poverty line, including 20.7% of those under age 18 and 2.1% of those age 65 or over.
Municipalities and communities of Otsego County, New York, United States
County seat: Cooperstown
Butternuts
Hartwick
Roseboom
Burlington Flats
Fly Creek
Schuyler Lake
South Edmeston
Springfield Center
West Edmeston‡
Unadilla (village), New York Facts for Kids. Kiddle Encyclopedia.


Guan Yu can use any red card as an attack.
Once per turn, Guan Yu can compete with any other player. If Guan Yu wins, the other person cannot use any cards from their handor optional character abilities for the rest of the turn. If Guan Yu loses, he can let the other person recover one health.
"Fighting Saint" is the same power as the original Guan Yu.
When Guan Yu wins the competition with "Absolute Loyalty", the target not only loses access to their hand, they lost all their optional character abilities. This means that the only powers that they will still have are the ones in which they do not have a choice in using.
If Guan Yu loses the competition, he can decide if the other person recovers or not. This means that Guan Yu can compete with a friend to give them more health and does not suffer if he loses the competition to an enemy.


text:
Q: Variant, Recursive Function and Type Inference I'm very new at OCaml but worked all the past two days in order to get a good understanding of how to use it.
I've been doing a lot of thing lately but something is keeping me from moving forward.
I'm trying to implement an evalexpr in OCaml. Pretty easy using this langage you would say : so I thought, and the first I did, using regular ints, worked fine.
But now I'm trying to do so using my OWN type and my own functions to solve operations : and of course it's not as easy as I expected.
type expr =
  | Number of MyInt.myint
  | Sum of (expr * expr)
  | Sub of (expr * expr)
  | Product of (expr * expr)
  | Divide of (expr * expr)
  | Modulo of (expr * expr)

let rec evalexpr expr = function
  | Number n       -> n
  | Sum (a, b)     -> MyInt.add (evalexpr a) (evalexpr b)
  | Sub (a, b)     -> MyInt.sub (evalexpr a) (evalexpr b)
  | Product (a, b) -> MyInt.mul (evalexpr a) (evalexpr b)
  | Divide (a, b)  -> MyInt.div (evalexpr a) (evalexpr b)
  | Modulo (a, b)  -> MyInt.modulo (evalexpr a) (evalexpr b)

This seems ok to me... but the compiler disagrees.
I think it's pretty obvious that the "(evalexpr a)" is of type MyInt.myint, as it is the only final return value the evalexpr function can return : still, the compiler think its type is "expr -> MyInt.myint".
Does it means it doesn't apply the function evalexpr and returns the function itself ?? if so, why the hell would it do that ?
I just can't figure it out. And I can't think of another way to do what I'm trying to achieve here.

A: Remove expr from your evalexpr function.
let rec evalexpr = function
    ...

By having it there, you are declaring that the function takes 2 paramemters.  The first will be bound to the variable expr and the second will be matched in the function body.  Then when you try to recursively call it, you call evalexpr with single parameters yielding functions expecting the second parameter.  Then naturally your functions cannot operate on functions hence the error.

A: Ok, I think it should be like this:
let rec evalexpr = function
  | Number n       -> n
  | Sum (a, b)     -> MyInt.add (evalexpr a) (evalexpr b)
  | Sub (a, b)     -> MyInt.sub (evalexpr a) (evalexpr b)
  | Product (a, b) -> MyInt.mul (evalexpr a) (evalexpr b)
  | Divide (a, b)  -> MyInt.div (evalexpr a) (evalexpr b)
  | Modulo (a, b)  -> MyInt.modulo (evalexpr a) (evalexpr b)

function statement already takes an argument for you so what you actually declared was a function that takes one argument and returns a function

redpajama_set_name: RedPajamaStackExchange


text:
\section{Introduction}

Belief Propagation is a widely used method for inference in graphical models. We
study this algorithm in the context of Gaussian graphical models.  There have
been several studies of Gaussian belief propagation (GaBP)
\cite{Weiss01,Rusmevichientong01,Plarre04} as well as numerous applications
\cite{Moallemi06,Bickson08a,Bickson08b}. The best known sufficient condition for
its convergence is the walk-summable condition \cite{Johnson06,Malioutov06} (see
also \cite{Cseke08,Moallemi09}), which also provides new insights into the
algorithm by interpreting it as computing weighted sums of walks (walk-sums)
within the graph. Our aim in this present paper is to extend this
graphical/combinatorial view of GaBP to include estimation of the determinant
(partition function) of the Gaussian graphical model.  This work is also
inspired by the loop-series correction method for belief propagation
\cite{Chertkov06} that was recently extended to Gaussian graphical models
\cite{Chernyak08}.

Our present study leads to a new perspective on GaBP having close ties to
graphical zeta functions \cite{Stark96}. We find that for walk-summable models
the determinant may be represented as a product over all orbits (cyclic walks)
of the graph.  The estimate of the determinant provided by GaBP only captures
totally backtracking orbits, which can be embedded as orbits in the
computation tree (universal cover) of the graph.  The missing orbits may then be
grouped into equivalence classes corresponding to backtrackless orbits.  The
orbit-product over each such equivalence class may be simply computed from the
solution of GaBP. Also, the product over all backtrackless orbits may be
interpreted as the determinant of a backtrackless adjacency matrix of the graph
with appropriately defined edge weights based on the GaBP solution. Finally, we
propose a simple, efficient method to compute truncated orbit-products including
all orbits up to some specified length and provide an error-bound on the
resulting estimates. In certain classes of graphs (e.g., grids), this leads to
an efficient method with complexity linear in the number of nodes and the
required precision of the determinant estimate.

This paper differs fundamentally from \cite{Chernyak08} in that we rely heavily
on the walk-summable property to develop multiplicative expansions using
infinitely many orbits of the graph, whereas \cite{Chernyak08} develops additive
expansions over a finite number of ``generalized loops'' (which may be
disconnected) using methods of Grassman calculus.  Our present approach leads
naturally to approximation methods (with accuracy guarantees in walk-summable
models) based on truncated orbit-products.

\section{Preliminaries}

\subsection{Walks and Orbits of a Graph}

Let $G$ be a graph on vertices (nodes) $V = \{1,\dots,n\}$ with undirected edges
$\{i,j\} \in G$.  We may also treat each undirected edge $\{i,j\}$ as a
symmetric pair of directed edges $(ij)$ and $(ji)$. A \emph{walk} $w$ is a
sequence of adjacent vertices $(w_0 \dots w_L)$ ($w_t  \in V$ for $t=0,\dots,L$
and $\{w_t,w_{t+1}\} \in G$ for $t = 1,\dots,L-1$) where $|w| \triangleq L$ is
the \emph{length} of the walk. A walk may be equivalently specified as a
sequence of \emph{steps} $w = ((w_0 w_1)(w_1 w_2)\dots(w_{L-1} w_L))$ such that
each
step is a (directed) edge of the graph that ends where the following step
begins.  A walk may visit the same node or cross the same edge multiple times
and may also \emph{backtrack}---that is, it may step back to the preceding
vertex.  A walk is \emph{closed} if it begins and ends at the same node $w_0 =
w_L$.  A closed walk is \emph{primitive} if it is not a multiple of some shorter
walk (e.g., the walk $(1231)$ is primitive but $(1231231)$ is not).  We define
an \emph{orbit} $\ell = [w]$ to be an equivalence class of closed primitive
walks, where two walks are considered equivalent $[w] = [w']$ if one is a
cyclic shift of the other (i.e., if $w_t = w'_{t+s(\mathrm{mod} \, \ell)}$ for
some $s$ and $t=1,\dots,L$).  Hence, there is a one-to-one correspondence
between orbits and (non-terminating) cyclic walks.

The following classification of walks and orbits plays an essential role in our
analysis: A walk (or orbit) is said to be \emph{reducible} if it contains a
\emph{backtracking pair} of consecutive steps $\dots(ij)(ji)\dots$, otherwise
the walk is \emph{irreducible}.  By repeatedly deleting backtracking pairs until
none remain one obtains the (unique) \emph{irreducible core} $\gamma =
\Gamma(w)$ of the walk $w$. For closed walks it may happen that $\gamma =
\emptyset$,  where $\emptyset \triangleq ()$ denotes the trivial (empty) walk.
We then say that the walk is \emph{totally reducible}. We say that a walk is
\emph{non-trivial} if it is not totally reducible. Totally reducible walks have
been called \emph{backtracking} \cite{Malioutov06},  although \emph{totally
backtracking} is perhaps a better description. Irreducible walks have been
called \emph{backtrackless} (or \emph{non-backtracking}) elsewhere in the
literature.

\emph{Notation.} $\mathcal{L}$ denotes the set of all orbits of $G$,
$\Gamma(\mathcal{L})$ denotes the irreducible (backtrackless) orbits, and we
partition $\mathcal{L}$ into (disjoint) equivalence classes $\mathcal{L}_\gamma
\triangleq \{ \ell \in \mathcal{L} | \Gamma(\ell) = \gamma \}$ for $\gamma \in
\Gamma(\mathcal{L})$.   In particular, $\mathcal{L}_\emptyset$ denotes the class
of totally backtracking orbits, which plays a special role in our interpretation
of GaBP.  We will use $\ell$ to denote a generic orbit and reserve $\gamma$ to
denote irreducible orbits.

\emph{Example.} Orbits $[1231]$, $[1231451]$, $[123421561]$ are
backtrackless; $[1234321]$, $[1232421]$, $[1231321]$ are totally backtracking; $[123241]$
is both reducible and non-trivial (neither backtrackless nor totally backtracking).

\subsection{Gaussian Belief Propagation}

A \emph{Gaussian graphical model} is a probability distribution \begin{equation}
p(x) = \mathcal{Z}^{-1} \exp\left\{ -\tfrac{1}{2} x^T J x + h^T x \right\}
\end{equation} of random variables $x \in \mathbb{R}^n$ where $J$ is a sparse,
symmetric, positive-definite matrix.  The fill-pattern of $J$ defines a graph
$G$ with vertices $V = \{1,\dots,n\}$ and edges $(ij)$ for all $J_{ij} \neq 0$.
The partition function is defined by $\mathcal{Z}(h,J) \triangleq \int \exp\{
-\frac{1}{2} x^T J x + h^T x \} dx = \left[ (2\pi)^n \det J^{-1} \right]^{1/2}
e^{\tfrac{1}{2} h^T J^{-1} h}$ so as to normalize the distribution.  Given such
a model, we may compute the mean vector $\mu \triangleq \int p(x) x dx = J^{-1}
h$ and covariance matrix $K \triangleq \int p(x) (x-\mu)(x-\mu)^T dx = J^{-1}$.
This generally requires $\mathcal{O}(n^3)$ computation in dense graphs using
Gaussian elimination.  If $G$ is sparse and only certain elements of $K$ are
required (the diagonal and edge-wise covariances), then the complexity of
Gaussian elimination may be substantially reduced (e.g., $\mathcal{O}(n^{3/2})$
in planar graphs using nested dissection) but still generally has complexity
growing as $\mathcal{O}(w^3)$ in the \emph{tree-width} $w$ of the graph.

\emph{Gaussian belief propagation} (GaBP) is a simple, distributed, iterative
message-passing algorithm to estimate the marginal distribution $p(x_i)$ of each
variable, which is specified by its mean $\mu_i$ and variance $K_{ii}$. GaBP is
parameterized by a set of messages $m_{ij}(x_j) = e^{ \frac{1}{2} \alpha_{ij}
x_j^2 + \beta_{ij} x_j}$ defined on each directed edge $(ij)$ of the graph
($m_{ij}$ is regarded as a message being passed from $i$ to $j$). The GaBP
equations are:
\begin{displaymath}
m_{ij}(x_j) \propto \int \psi_i(x_i) \prod_{k \in \partial i \setminus j}
m_{ki}(x_i)
\psi_{ij}(x_i,x_j) dx_i
\end{displaymath}
where $\psi_i = e^{-\tfrac{1}{2}J_{ii}x_i^2+h_i x_i}$, $\psi_{ij} =
e^{-J_{ij}x_ix_j}$ and $\partial i$ denotes the set of neighbors of $i$ in $G$.
This reduces to the following rules for computing $(\alpha,\beta)$-messages:
\begin{eqnarray}
\alpha_{ij} &=&  J_{ij}^2 (J_{ii} - \alpha_{i \setminus j})^{-1} \nonumber\\
\beta_{ij} &=&  -J_{ij} (J_{ii} - \alpha_{i \setminus j})^{-1} (h_i + \beta_{i \setminus
j}) \label{eq:gabp_eqs}
\end{eqnarray}
where $\alpha_{i \setminus j} \triangleq \sum_{k \in \partial i \setminus j}
\alpha_{ki}$ and $\beta_{i \setminus j} \triangleq \sum_{k \in \partial i \setminus j}
\beta_{ki}$.  These equations are solved by iteratively recomputing each message
from the other messages until convergence. The marginal distribution is then
estimated as $p^\mathrm{bp}(x_i) \propto \psi_i(x_i) \prod_{k \in \partial i}
m_{ki}(x_i)$, which gives
variance estimates $K^\mathrm{bp}_i = (J_{ii}-\sum_{k} \alpha_{ki})^{-1}$
and mean estimates $\mu^\mathrm{bp}_i = K^\mathrm{bp}_i (h_i+\sum_{k}
\beta_{ki})$. In trees, this method is equivalent to Gaussian elimination,
terminates after a finite number of steps and then provides the correct
marginals. In loopy graphs, it may be viewed as performing Gaussian elimination
in the computation tree (universal cover) of the graph
\cite{Plarre04,Malioutov06} (obtained by ``unrolling'' loops) and may therefore
fail to converge due to the
infinite extent of the computation tree. If it does converge, the mean estimates
are still correct but the variances are only approximate. We also obtain an
estimate of the pairwise covariance matrix on edges $\{i,j\} \in G$:
\vspace{-.15cm}
\begin{displaymath}
K^\mathrm{bp}_{(ij)} =
\left(
\begin{array}{cc}
J_{ii}-\alpha_{i \setminus j} & J_{ij} \\
J_{ij} & J_{jj}-\alpha_{j \setminus i}
\end{array}
\right)^{-1}
\end{displaymath}
In this paper, we are concerned with the GaBP estimate of the determinant $Z =
\det K = \det J^{-1}$ (which is closely linked to computation of the partition
function $\mathcal{Z}$).  We obtain an estimate of $Z$ from the GaBP solution
as:
\vspace{-.1cm}
\begin{equation} \label{eq:Z_bp_def}
Z^\mathrm{bp} = \prod_{i \in V} Z_i^\mathrm{bp} \prod_{\{i,j\} \in G} \frac{Z_{ij}^\mathrm{bp}}{Z_i^\mathrm{bp}
Z_j^\mathrm{bp}}
\end{equation}
where $Z_i^\mathrm{bp} = K^\mathrm{bp}_i$ and $Z_{ij}^\mathrm{bp} = \det K^\mathrm{bp}_{(ij)}$.   The motivation
for this form of estimate is that it becomes exact if $G$ is a tree. In loopy
graphs, there may generally be no stable solution to GaBP (or multiple unstable
solutions).  The main objective of this paper is to interpret the estimate
$Z^\mathrm{bp}$ in the context of walk-summable models (described below), for which
there is a well-defined stable solution, and to suggest methods to correct this
estimate. Note that the variance and covariance estimates (and hence the
determinant estimate $Z^\mathrm{bp}$) are independent of $h$ and the $\beta$-messages in
(\ref{eq:gabp_eqs}), they are determined solely by the $\alpha$-messages
determined by $J$.  Since GaBP correctly computes the means in walk-summable
models, we are mainly concerned with how to correct $Z^\mathrm{bp}$ (and hence its
derivatives, which correspond to the GaBP estimates of variances/covariances).

\paragraph{Walk-Sum Interpretation}

Our approach in this paper may be considered as as extension of the walk-sum
interpretation of GaBP \cite{Malioutov06}. Let $J$ be normalized to have
unit-diagonal, such that $J=I-R$ with $R$ having zeros along its diagonal. The
walk-sum idea is based on the series $K = (I-R)^{-1} = \sum_k R^k$, which
converges if $\rho(R)<1$ where $\rho(R)$ denotes the \emph{spectral radius} of
the matrix $R$ (the maximum modulus of the eigenvalues of $R$). This allows us
to interpret $K_{ij}$ as a sum over all walks in the graph $G$ which begin at
node $i$ and end at node $j$ where the weight of a walk is defined as $R^w =
\prod_{(ij) \in w} r_{ij}^{n_{ij}(w)}$ and $n_{ij}(w)$ is a count of how many
times step $(ij)$ occurs in the walk.  We write this \emph{walk-sum} as $K_{ij}
= \sum_{w: i \rightarrow j} R^w$.  However, in order for the walk-sum to be
well-defined, it must converge to the same value regardless of the order in
which we add the walks.  This is equivalent to requiring that it converges
absolutely.  Thus, we say that $R$ is \emph{walk-summable} if $\sum_{w: i
\rightarrow j} |R^w|$ converges for all $i,j \in V$.  This is equivalent to the
spectral condition that $\rho(|R|)<1$ where $|R| \triangleq (|r_{ij}|)$ is the
element-wise
absolute-value matrix of $R$. A number of other equivalent or
sufficient conditions are given in \cite{Malioutov06}.

In walk-summable models it then holds that variances correspond to closed
walk-sums $K_{ii} = \sum_{w: i \rightarrow i} R^w$ and means correspond to a
(reweighted) walk-sum over all walks which end at a specific node $\mu_i =
\sum_{w: * \rightarrow i} h_* R^w$ (here $*$ denotes the arbitrary starting
point of the walk).  Moreover, we may interpret the GaBP message parameters
$(\alpha,\beta)$ as recursively computing walk-sums within the \emph{computation
tree} \cite{Malioutov06}.  This implies that GaBP converges in walk-summable
models and converges to the same ``walk-sum'' solution independent of the order
in which we update messages. This interpretation also shows that GaBP computes
the correct walk-sums for the means but only computes a subset of the closed
walks needed for the variances. Specifically, $K^\mathrm{bp}_i$ only includes totally
backtracking walks at node $i$. This is seen as a walk is totally backtracking
if and only if it can be embedded as a closed walk in the computation tree of
the graph and it is these closed walks of the computation tree which GaBP
captures in its variance estimates.

\section{Orbit-Product Interpretation of Gaussian BP}

\subsection{Determinant $Z$ as Orbit-Product}

Let $Z(R) \triangleq \det(I-R)^{-1}$.  In walk-summable models, we may give this
determinant another graphical interpretation as a product over orbits of a
graph, one closely related to the so-called \emph{zeta function} of a graph
\cite{Stark96}.

\begin{theorem} If $\rho(|R|)<1$ then it holds that $Z(R) =
  \prod_\ell (1-R^\ell)^{-1} \triangleq \prod_\ell Z_\ell$ where the
  product is taken over all orbits of $G$ and $R^\ell = \prod_{(ij)
    \in \ell} r_{ij}^{n_{ij}(\ell)}$ where $n_{ij}(\ell)$ is the
  number of times step $(ij)$ occurs in orbit $\ell$.
\end{theorem}

\emph{Proof.} $\log \det (I-R)^{-1}
 = \mathrm{tr} \log (I-R)^{-1}
 = \mathrm{tr} \sum_k \frac{R^k}{k} \linebreak
 = \sum_{\mathrm{closed}\, w} \frac{R^w}{|w|}
 = \sum_{\mathrm{primitive}\, w} \sum_{m=1}^\infty \frac{(R^w)^m}{m |w|}
 = \sum_{\mathrm{primitive}\, w} \frac{1}{|w|} \log (1-R^w)^{-1}
 = \sum_{\mathrm{orbits}\, \ell} \log (1-R^\ell)^{-1}
 = \log \prod_\ell (1-R^\ell)^{-1}$.
We have used the identity $\log\det
A = \mathrm{tr}\log A$ and the series expansion
$\log (I-A)^{-1} = \sum_{k=1}^\infty \frac{A^k}{k}$.
Each closed walk is expressed as a multiple of a primitive
walk. Every primitive walk $w$ has exactly $|w|$ distinct cyclic shifts. $\diamond$

We emphasize that $\rho(|R|)<1$ is necessary to insure that the the
orbit-products we consider are well-defined. \emph{This condition is assumed
throughout the remainder of the paper.}

\subsection{$Z^\mathrm{bp}$ as Totally Backtracking Orbits}

Totally backtracking walks play an important role in the walk-sum interpretation
of the GaBP variance estimates. We now derive an analogous interpretation of
$Z^\mathrm{bp}$ defined by (\ref{eq:Z_bp_def}):

\begin{theorem} $Z^\mathrm{bp} = \prod_{\ell \in \mathcal{L}_\emptyset} Z_\ell$
  where the product is taken over the set of totally backtracking orbits
  of $G$.
\end{theorem}

Although this result seems intuitive in view of prior work, its
proof is non-trivial involving arguments not used previously.
To prove the theorem, we first summarize some useful lemmas.
Consider a block matrix $A = (A_{11} A_{12}; A_{21} A_{22})$.
The \emph{Schur complement} of block $A_{11}$ is
$A_{22}^* \triangleq A_{22}-A_{21} A_{11}^{-1} A_{21}$. It holds
that $\det A = \det A_{11} \det A_{22}^*$ and $(A_{22}^*)^{-1} = (A^{-1})_{22}$.
Using these well-known identities, it follows:

\begin{lemma} Let $R = (R_{11} R_{12}; R_{21} R_{22})$ and $K = (I-R)^{-1} =
  (K_{11} K_{12}; K_{21} K_{22})$.  Then $\det K_{11} = \frac{Z(R)}{Z(R_{22})}$.
\end{lemma}
For walk-summable models, we then have
\begin{displaymath}
\det K_{11} = \frac{\prod_{\ell \in G} Z_\ell}{\prod_{\ell \in G_2} Z_\ell}
= \prod_{\ell \in G | \ell \,\mathrm{intersects}\, G_1} Z_\ell
\end{displaymath}
where the final orbit-product is taken over all orbits of $G$ which include any
node of subgraph $G_1$ (corresponding to submatrix $R_{11}$). Next, using this
result
and the interpretation of GaBP as inference on the computation tree, we
are led to the following interpretation of the quantities $Z_i^\mathrm{bp}$ and
$Z_{ij}^\mathrm{bp}$ appearing in (\ref{eq:Z_bp_def}).
Let $T_i$ denote the computation tree of the
graph $G$ with one copy of node $i$ marked.  Let $T_{ij}$ denote the computation
tree with one copy of edge $\{i,j\} \in G$ marked. Then,

\begin{lemma} $Z_i^\mathrm{bp} = \prod_{\ell \in T_i | i \in \ell} Z_\ell$
  where the product is over all orbits of $T_i$ that include the
  marked node $i$.  $Z_{ij}^\mathrm{bp} = \prod_{\ell \in T_{ij} | i \in \ell
\,\mathrm{or}\, j \in \ell} Z_\ell$ where the product is
  over all orbits of $T_{ij}$ that include either endpoint of the
  marked edge $\{i,j\}$.
\end{lemma}

\ignore{\emph{Proof.} Let $R_T$ denote the weighted adjacency matrix of the
(infinite) computation tree and $K_T = (I-R_T)^{-1}$ its covariance
matrix.  Let node $i$ be marked in the computation tree. Then $Z_i^\mathrm{bp}
= \det (K_T)_{ii}$. Applying Lemma 2 and Theorem 1, we have:
\begin{displaymath}
Z_i^\mathrm{bp}
= \frac{Z(R_T)}{Z(R_{T \setminus i})}
= \frac{\prod_{\ell \in T_i} Z_\ell}{\prod_{\ell \in T_i | i \not\in \ell}
Z_\ell}
= \prod_{\ell \in T_i | i \in \ell} Z_\ell
\end{displaymath}
A similar argument holds for $Z_{ij}^\mathrm{bp}$:
\begin{displaymath}
Z_{ij}^\mathrm{bp}
= \frac{\prod_{\ell \in T_{ij}} Z_\ell}
{\prod_{\ell \in T_{ij} | i \not\in \ell \wedge j\not\in \ell} Z_\ell}
= \prod_{\ell \in T_{ij} | i \in \ell \vee j \in \ell} Z_\ell \;\;\; \diamond
\end{displaymath}}

\emph{Proof of Theorem 2.} Using Lemma 2 and the correspondence between orbits
of the computation tree and totally backtracking orbits of $G$, we may expand
(\ref{eq:Z_bp_def}) to express  $Z^\mathrm{bp}$ entirely as a product over totally
backtracking orbits $Z^\mathrm{bp} = \prod_{\ell \in \mathcal{L}_\emptyset}
Z_\ell^{N_\ell}$ where $N_\ell$ is the count of how many times $\ell$
appears in the orbit-product---the number of times it appears in the numerator
of (\ref{eq:Z_bp_def}) minus the number of times in appears in the denominator.
It remains to show
that $N_\ell=1$ for each totally backtracking orbit.  This may be seen by
considering the subtree $T_\ell$ of the computation tree $T$ traced out by orbit
$\ell$. Let $v$ and $e$ respectively denote the number of nodes and edges of
$T_\ell$ (hence, $e = v-1$) and let $c$ denote the number of edges of $T$ with
exactly one endpoint in $T_\ell$.  First, we count how many powers of $Z_\ell$
appear in the orbit product $\prod_i Z_i^\mathrm{bp}$.  For each vertex $i \in T_\ell$
we may pick this as the marked node in the computation tree and this shows one
way that $\ell$ can be embedded in $T_i$ so as to include its marked node.
Thus, $v$ gives the total number of multiples of $Z_\ell$ in $\prod_i Z_i^\mathrm{bp}$.
Similarly, we could mark any edge $\{i,j\} \in T$ with one or both endpoints in
$T_\ell$ and this gives one way to embed $\ell$ into $T_{ij}$ so as to
intersect the marked edge.  Thus, the product $\prod_{ij} Z_{ij}^\mathrm{bp}$
contributes $e+c$ powers of $Z_\ell$.  Lastly, the product $\prod_{ij} Z_i^\mathrm{bp}
Z_j^\mathrm{bp}$ contains $2e+c$ powers of $Z_\ell$.  This represents the number of ways
we may pick a \emph{directed} edge $(ij)$ of $T$ such that at least one
endpoint is in $T_\ell$. The total count is then $N_\ell = v + (e+c) - (2e+c) =
v-e = 1$. $\diamond$

Combining Theorems 1 and 2, we obtain the following orbit-product correction
to $Z^\mathrm{bp}$:
\begin{corollary} $Z = Z^\mathrm{bp} \times \prod_{\ell \not\in \mathcal{L}_\emptyset}
Z_\ell$.
\end{corollary}
This formula includes a correction for every missing orbit, that is,
for every non-trivial orbit.  This implies that $Z = Z^\mathrm{bp}$ for
trees since \emph{all} orbits of trees are totally backtracking.

\subsection{$Z^\mathrm{bp}$ Error Bound}

One useful consequence of the orbit-product interpretation of $Z^\mathrm{bp}$ is that it
provides
a simple error bound on GaBP. Let $g$ denote the \emph{girth} of the graph $G$,
defined as the length of the shortest cycle of $G$.  We note that the missing
orbits $\ell \not\in \mathcal{L}_\emptyset$ must all have length greater than or
equal to
$g$.  Then,
\begin{theorem}
$\frac{1}{n} \left|\log \frac{Z^\mathrm{bp}}{Z}\right| \le \frac{\rho(|R|)^g}{g
(1-\rho(|R|))}$.
\end{theorem}
\emph{Proof.}  We derive the chain of inequalities:
$\left|\log \frac{Z}{Z^\mathrm{bp}} \right|
\stackrel{\mathrm{(a)}}{=} \left| \sum_{\ell \not\in \mathcal{L}_\emptyset} \log
Z_\ell \right|
\le \sum_{|\ell| \ge g} |\log Z_\ell|
\stackrel{\mathrm{(b)}}{\le} \sum_{|\ell| \ge g} \log (1-|R|^\ell)^{-1}
\stackrel{\mathrm{(c)}}{=} \mathrm{tr} \sum_{k=g}^\infty \frac{|R|^k}{k}
\le n \sum_{k=g}^\infty \frac{\rho^k}{k}
\le \frac{n \rho^g}{g} \sum_{k=0}^\infty \rho^k
=  \frac{n \rho^g}{g (1-\rho)}$.
(a) Corollary 1.
(b) $|\log Z_\ell|
= \left| \sum_{k=1}^\infty \frac{(R^\ell)^k}{k} \right|
\le \sum_{k=1}^\infty \frac{(|R|^\ell)^k}{k}
= \log (1-|R|^\ell)^{-1}$.
(c) The proof of Theorem 1 shows that $\mathrm{tr} \sum_{k \ge 1} \frac{R^k}{k} =
\sum_{\ell} \log (1-R^\ell)^{-1}$.  Similarly,
$\mathrm{tr} \sum_{k \ge g} \frac{|R|^k}{k}
= \sum_{|\ell| \ge g} \log (1-|R|^\ell)^{-1}$.
$\diamond$

This is consistent with the usual intuition that belief propagation is most
accurate in large girth graphs with weak interactions.

\section{Backtrackless Orbit Correction}

In this section we show that the set of orbits omitted in the GaBP estimate can
be grouped into equivalence classes corresponding to backtrackless
orbits and that the orbit-product over each such equivalence class is simply
computed with the aid of the GaBP solution:

\begin{theorem} $Z = Z^\mathrm{bp} \times \prod_{\gamma \neq \emptyset}  Z'_\gamma$
where the
product is over all \emph{backtrackless orbits} of $G$ and we define
\begin{displaymath}
Z'_\gamma = (1 - \prod_{(ij) \in \gamma} (r'_{ij})^{n_{ij}(\gamma)})^{-1}
\end{displaymath}
where $r'_{ij} \triangleq \frac{r_{ij}}{1 - \alpha_{i \setminus j}}$ and
$\alpha_{i \setminus j} = \sum_{k \in \partial i \setminus j} \alpha_{ki}$ is computed
from the solution of GaBP.
\end{theorem}

In comparison to Corollary 1, here the correction factor is expressed as an
orbit-product over just the backtrackless orbits (whereas Corollary 1 uses a
separate correction for each non-trivial orbit).  However, all orbits are still
correctly accounted for because we modify the edge weights of the graph so as to
include a factor $(1-\alpha_{i \setminus j})^{-1}$ (computed by GaBP) which serves
to ``factor in'' totally-backtracking excursions at each point along the
backtrackless orbit, thereby generating all non-trivial orbits.

The basic idea underlying this construction is depicted in Figure 1.  For each
backtrackless orbit $\gamma$ we define an associated \emph{computation graph}
$G_\gamma$ as follows. First, we start with a single directed cycle based on
$\gamma = [\gamma_1\gamma_2\cdots\gamma_L]$ (any duplicated nodes of the orbit
map to distinct nodes in this directed cycle).  Then, for each node $\gamma_k$
of this graph, we attach a copy of the computation tree $T_{\gamma_k \setminus
\gamma_{k+1}}$, obtained by taking the full computation tree $T_{\gamma_k}$
rooted at node $\gamma_k$ and deleting the branch $(\gamma_k,\gamma_{k+1})$
incident to the root.  This construction is illustrated in Figure 1(a,b) for the
graph $G = K_4$ and orbit $\gamma = [(12)(23)(31)]$.  The cycle has ``one-way''
directed edges whereas each computation tree has ``two-way'' undirected edges.
This is understood to mean that walks are allowed to backtrack within the
computation tree but not within the cycle.  The importance of this graph is
based on the following lemma (the proof is omitted):

\begin{figure}
\centering
(a)\hspace{-.3cm}\input{K4.pstex_t}\\
\vspace{.1cm}
(b)\hspace{-.3cm}\input{loop_graph.pstex_t}\\
\vspace{.1cm}
(c) \input{loop_graph2.pstex_t}\\
\vspace{.1cm}
(d)\hspace{-.1cm}\input{loop_graph3.pstex_t}\\
\vspace{-.2cm}
\caption{Illustration of construction to combine equivalent orbits.  (a) The
graph $G=K_4$. (b) The computation graph $G_\gamma$ for $\gamma =
[(12)(23)(31)]$. (c) Finite graph with self-loops at each node to capture
totally backtracking walks. (d) Equivalent graph with modified edge weights to
capture totally backtracking walks.}
\vspace{-.3cm}
\end{figure}

\begin{lemma} Let $\gamma$ be a backtrackless orbit of $G$.  Then,
  there is a one-to-one correspondence between the class of orbits
  $\mathcal{L}_\gamma$ of $G$ and the non-trivial orbits of $G_\gamma$.
\end{lemma}

\ignore{\emph{Proof.} We aim to show that each orbit $\ell \in
\mathcal{L}_\gamma$ may be embedded in the graph $G_\gamma$ in exactly one way.
We use the equivalence between orbits and infinitely long cyclic walks.  We
trace the cyclic walk $w$ described by $\ell$ in the computation tree (universal
cover) $T$ of $G$. This walk reduces to an infinitely long path $w' = \Gamma(w)$
of the computation tree. This path $w'$ is also the cyclic walk corresponding to
the backtrackless orbit $\gamma = \Gamma(\ell)$. Now, we break up the walk $w$
into totally backtracking segments as follows.  For each edge of the path $w'$
in the computation tree, the walk $w$ crosses this edge some number of times
(once more in the forward direction than in the reverse direction).  We mark the
step of $w$ corresponding to the first time that it crosses this edge.  After
doing this for every edge of $w'$, we will have broken up the walk $w$ into
segments as $w = \dots(\gamma_1\gamma_2)w_2(\gamma_2\gamma_3)w_3
\dots(\gamma_L\gamma_1)w_1\dots$ (where this pattern is repeating).  It is
simple to observe that each walk $w_k$ may be embedded as a closed-walk at the
root node of $T_{\gamma_k \setminus \gamma_{k+1}}$.  Thus, this decomposition of the
cyclic walk $w$ also determines how $\ell$ may be embedded into $G_\gamma$. This
defines a map $\tau$ from $\mathcal{L}_\gamma$ to the non-trivial orbits of
$G_\gamma$.  One can verify that this mapping is bijective.  The inverse map is
defined by $\tau^{-1}(\ell') = [\nu(\ell'_1)\nu(\ell'_2)\dots \nu(\ell'_L]$
where $\nu$ is the map from (distinct) vertices of $G_\gamma$ back to the
corresponding vertices of $G$. $\diamond$}

Next, we demonstrate how to compute all of the orbits within an equivalence
class as a simple determinant calculation based on the backtrackless orbit
$\gamma$ and the GaBP solution.  Let $R'_\gamma$ be defined as the edge-weight
matrix of a simple single-loop graph based on $\gamma$ with edge-weights defined
by
$r'_{\gamma_k,\gamma_{k+1}} = \frac{r_{\gamma_k,\gamma_{k+1}}}{1 -
\alpha_{\gamma_k \setminus \gamma_{k+1}}}$. This construction is illustrated in
Figure 1(d).  Then,

\begin{lemma} $Z'_\gamma = \det (I-R'_\gamma)^{-1}
= \prod_{\ell \in \mathcal{L}_\gamma} Z_{\ell}$.
\end{lemma}
\emph{Proof.} Using Lemma 3, we see that the orbit-product $\prod_{\ell \in
\mathcal{L}_\gamma} Z_\ell$ is equal to the product over all non-trivial orbits
of the graph $G_\gamma$, that is, the product over all orbits in $G_\gamma$
which intersect the subgraph corresponding to $\gamma$. Using Lemma 1, this is
equivalent to computing the determinant of the corresponding submatrix of
$K_{G_\ell} = (I-R_{G_\ell})^{-1}$ where $R_{G_\ell}$ is the edge-weight matrix
of the computation graph.  This is equivalent to first eliminating each
computation tree (by Gaussian elimination/GaBP) attached to each node of
$\gamma$ to obtain a reduced graphical model $I-R_\gamma$ and then computing
$\det (I-R_\gamma)^{-1}$.  Using the GaBP solution, the effect of eliminating
each computation tree is to add a ``self-loop'' (diagonal element) to $R_\gamma$
with edge-weight $\alpha_{\gamma_k \setminus \gamma_{k+1}} = \sum_{v \neq
\gamma_{k+1}} \alpha_{v,\gamma_k}$, obtained by summing the incoming messages to
node $\gamma_k$ from each of its neighbors in the subtree $T_{\gamma_k \setminus
\gamma_{k+1}}$.  This elimination step is illustrated if Figure 1(b,c).  We may
use the orbit-product formula to compute the determinant. However, there are
infinitely many orbits in this graph due to the presence of a self-loop at each
of the remaining nodes.  At each node, an orbit may execute any number of steps
$m$ around this self-loop each with edge-weight $\alpha_{\gamma_k \setminus
\gamma_{k+1}}$. Summing these, we obtain $\sum_{m=0}^\infty \alpha_{\gamma_k
\setminus \gamma_{k+1}}^m = (1-\alpha_{\gamma_k \setminus \gamma_{k+1}})^{-1}$.
Hence, we can delete each self-loop and multiply the following edge's weight by
$(1-\alpha_{\gamma_k \setminus \gamma_{k+1}})^{-1}$ and this preserves the value of
the determinant. This final reduction step is illustrated in Figure 1(c,d).
Then, the orbit-product $\prod_{\ell \in \mathcal{L}_\gamma} Z_\ell$ is equal to
$\det (I-R'_\gamma)^{-1}$ (e.g., based on the graph seen in Figure 1(d)). It is
straight-forward to compute the resulting determinant with respect to the single
(directed) cycle graph with edge-weights $R'_\gamma$.  There is only one orbit
in this graph and hence $\det (I-R'_\gamma)^{-1} = Z'_\gamma \triangleq
(1-(R')^\gamma)^{-1}$ where $(R')^\gamma = \prod_{(ij) \in \gamma}
(r'_{ij})^{n_{ij}(\gamma)}$ and $r'_{ij} = r_{ij} (1-\alpha_{i \setminus j})^{-1}$.
$\diamond$

\emph{Proof of Theorem 4.} Using these results, it is now simple
to show $\frac{Z}{Z^\mathrm{bp}}
= \prod_{\ell \not\in \mathcal{L}_\emptyset} Z_\ell
= \prod_{\gamma \neq \emptyset} \prod_{\ell \in \mathcal{L}_\gamma} Z_\ell
= \prod_{\gamma \neq \emptyset} \det(I-R'_\gamma)^{-1}
= \prod_{\gamma \neq \emptyset} (1-(R')^\gamma)^{-1}$. $\diamond$

\section{Backtrackless Determinant Correction}

Next, we show that the correction factor \begin{displaymath} \frac{Z}{Z^\mathrm{bp}} =
\prod_{\gamma \neq \emptyset} Z'_\gamma = \prod_{\gamma \neq \emptyset}
\det(I-R'_\gamma)^{-1} \end{displaymath} may also be calculated as a
\emph{single determinant} based on the following \emph{backtrackless adjacency
matrix} of the graph.  We define $R' \in \mathbb{R}^{2|G| \times 2|G|}$ as
follows.  Let the rows and columns of $R'$ be indexed by directed edges $(ij)$
of the graph $G$.  Then, the elements of $R'$ are defined \begin{equation}
R'_{(ij),(kl)} = \left\{ \begin{array}{ll} r'_{kl}, & j=k \mbox{ and } i \neq
l\\ 0, & \mbox{otherwise}. \end{array} \right. \end{equation} This construction
is illustrated in Figure 2. Note that the walks generated by taking powers $R'$
correspond to backtrackless walks of the graph $G$.  The weight of an edge
$((ij)(jk))$ in $R'$ is defined as the (modified) edge-weight $r'_{jk}$ of the
endpoint $(jk)$.  The weight of an orbit in $R'$ may then be equivalently
defined as the product of \emph{node} weights $r'_{ij}$ taken over the orbit in
$R'$, which is equal to the weight of the corresponding backtrackless orbit of
$G$ (using the modified edge weights $r'_{ij}$).

\begin{figure}
(a) \hspace{.2cm} \input{grid.pstex_t} \hspace{.3cm} (b)
\hspace{-.2cm}\input{grid2.pstex_t}
\vspace{-.2cm}
\caption{(a) $3 \times 3$ grid $G$. (b) Graph $G'$ representing
the backtrackless adjacency matrix $R'$. Each node $ij$ represents
a directed edge of $G$, directed edges are drawn between nodes $ij$ and $jk$
which are
non-backtracking ($k \neq i$).}
\vspace{-.3cm}
\end{figure}

\begin{theorem} $Z = Z^\mathrm{bp} \times Z'$ where $Z' \triangleq \det(I-R')^{-1}$,
that is,
$\det(I-R)^{-1} = Z^\mathrm{bp} \times \det(I-R')^{-1}$.
\end{theorem}

Before providing the proof, we establish that walk-summability with respect to
$R$ implies walk-summability with respect to $R'$:

\begin{lemma} If $\rho(|R|)<1$ then $\rho(|R'|) \le \rho(|R|)$.  \end{lemma}

\emph{Proof.} Once the $\alpha$-parameters converge, the $\beta$-parameters
follow a linear system $\beta_{k+1} = R' \beta_k + b$ \cite{Moallemi09}. Hence,
the asymptotic convergence rate of GaBP is $\rho(R')$.  Compare this to the
Gauss-Jacobi (GJ) iteration $\mu_{k+1} = \mu_k + (h-J \mu_k) =
\sum_{t=0}^{k+1} R^t h$ ($\mu_0 = 0)$, which has convergence
rate $\rho(R)$.  It is clear that the GaBP iteration captures a superset of
those walks computed by GJ at each iteration (because the depth-$k$ computation
tree includes all $k$-length walks). Hence, for non-negative models ($R \ge 0$
and $h \ge 0)$ it must hold that the error in the GaBP estimate of $\mu$ is less
than or equal to the error of GJ (at every iteration). This implies $\rho(R')
\le \rho(R)$ if $R \ge 0$, from which we conclude $\rho(|R'|) \le \rho(|R|)$ in
walk-summable models.  $\diamond$

\emph{Proof of Theorem 5.}  By construction, there is a one-to-one
weight-preserving correspondence between orbits of $G'$ and
backtrackless orbits of $G$. The result then follows from the orbit-product
representation of $Z'$ over $G'$  (Theorem 1, Lemma 5), which is
equivalent to the backtrackless orbit-product of Theorem 4. $\diamond$

One useful consequence of this result is that the error bound of Theorem 3 can
be improved to $\frac{1}{n} \left| \log \frac{Z^\mathrm{bp}}{Z} \right| = \frac{1}{n}
\left| \log Z' \right| \le \frac{\rho(|R'|)^g}{g(1-\rho(|R'|))}$.

It is impractical to compute the complete correction factor $Z' = \det
(I-R')^{-1}$, as this is not easier than directly computing $Z = \det
(I-R)^{-1}$.  However, because $R'$ is itself walk-summable, we can use this
representation as a starting point for constructing approximate corrections
such as the one considered in the next section.

\section{Block-Resummation Method}

Next, we consider an efficient method to approximate $Z(A)=\det(I-A)^{-1}$ for
walk-summable models $\rho(|A|)<1$.  This method can be used to either directly
approximate $Z(R)$ ($A=R$) or to approximate the GaBP-correction $Z'$ ($A=R')$.

Given a graph $G$ based on vertices $V$, we specify a set of blocks $\mathcal{B}
= (B_k \subset V, k=1,\dots,|\mathcal{B}|)$ chosen such that: (1) Every short
orbit $|\ell| < L$ is covered by some block $B \in \mathcal{B}$, and (2) If
$B,B' \in \mathcal{B}$ then $B \cap B' \in \mathcal{B}$. We also define block
weights $w_B$ as follows:  $w_B=1$ for maximal blocks (not contained by another
block) and $w_B = 1 - \sum_{B' \supsetneq B} w_{B'}$ for
non-maximal blocks (these weights may be negative). This insures that
$\sum_{B' \supseteq B} w_{B'} = 1$ for each $B \in \mathcal{B}$. Then, we define
our estimate
\begin{equation}
Z_{\mathcal{B}} \triangleq \prod_B Z_B^{w_B} \triangleq \prod_B
(\det(I-A_B)^{-1})^{w_B}
\end{equation}
where $A_B$ denotes the $|B| \times |B|$ principle submatrix of $A$
corresponding to $B$.

This approximation method is similar in spirit to approximations used
elsewhere (e.g., Kikuchi approximations to free-energy \cite{Yedidia05}).
However, the new insights offered by the orbit-product view allows us to
give our estimate a precise interpretation in walk-summable Gaussian
models:

\begin{theorem} $Z_{\mathcal{B}} = \prod_{\ell \in \mathcal{L}_{\mathcal{B}}}
Z_\ell$
where $\mathcal{L}_{\mathcal{B}} \triangleq \cup_{B \in \mathcal{B}}
\mathcal{L}_B$ and
$\mathcal{L}_B$ is the set of all orbits covered by $B$.
\end{theorem}
\emph{Proof.} $Z_{\mathcal{B}} = \prod_B \prod_{\ell \in B} Z_\ell^{w_B} =
\prod_{\ell \in \mathcal{L}_{\mathcal{B}}} Z_\ell^{\sum_{B \supset \ell} w_B} =
\prod_{\ell \in \mathcal{L}_{\mathcal{B}}} Z_\ell$ where $\sum_{B \supset \ell}
w_B = 1$ follows from the definition of the block weights. \ignore{Let $B'$
denote the smallest block that covers $\ell$.  Then, $\sum_{B \supset \ell} w_B
= \sum_{B \subseteq B'} w_B = 1$.} $\diamond$

Moreover, we can then bound the error of the estimate.  Noting
that $\mathcal{L}_\mathcal{B}$ includes all short orbits of the
graph, we can derive the following result by a similar proof as
for Theorem 3:

\begin{corollary} $\tfrac{1}{n} \left| \log \frac{Z_{\mathcal{B}}}{Z} \right|
\le
  \frac{\rho(|A|)^L}{L(1-\rho(|A|))}$.
\end{corollary}

Thus, for the class of models with $\rho < 1$, we obtain an approximation scheme
which converges to the correct determinant as the parameter $L$ is made large
with error decaying exponentially in $L$. The estimate $Z_{\mathcal{B}}(R)$
includes all orbits that are covered by some block.  The improved GaBP-based
estimate $Z^\mathrm{bp} Z_{\mathcal{B}}(R')$ includes all orbits $\ell$ such that
$\gamma = \Gamma(\ell)$ is covered by some block.  Thus, the GaBP-based
correction includes many more orbits. We also note that the error-bound using
the GaBP-based estimate is typically smaller as we have shown that $\rho(|R'|)
\le \rho(|R|)$ (if $\rho(|R|)<1$).

\begin{figure}
\centering
\hspace{.16cm}\epsfig{file=fig1.eps,scale=0.2}\hspace{.2cm}
\epsfig{file=fig2.eps,scale=0.2}\\
\vspace{-.1cm}
\hspace{.2cm}(a)\hspace{3.6cm}(b)\\
\epsfig{file=fig3.eps,scale=0.2}\hspace{.2cm}
\epsfig{file=fig4.eps,scale=0.2}\\
\vspace{-.1cm}
\hspace{.2cm}(c)\hspace{3.6cm}(d)\\
\vspace{-.2cm}
\caption{Demonstration of determinant approximation method for $256\times256$
periodic grid with uniform edge weights $r \in (0,.25)$. Plots of (a)
$\rho(|R|)$ and $\rho(|R'|)$ vs $r$; (b) ($\tfrac{1}{n} \log$ of) $Z$, $Z^\mathrm{bp}$
and $Z_\mathcal{B}$ (with $L = 2,4,8,16,32$) vs $r$; (c) $\tfrac{1}{n} \log
(Z^\mathrm{bp} Z'_\mathcal{B})$ vs $r$; and (d) $\tfrac{1}{n} |\log
(Z^{-1}Z_\mathcal{B})|$ and $\tfrac{1}{n}|\log (Z^{-1}Z^\mathrm{bp} Z'_\mathcal{B})|$ vs
$L=2,4,8,16,32$ for $r=.23$. In (b) and (c) the estimates for $L=8,16,32$ are
all nearly exact (and therefore hard to distinguish in the plot) and the errors
are largest near the walk-summable threshold $r=.25$. Estimates (b) are not
based on GaBP and are actually worst than $Z^\mathrm{bp}$ for $L=2$. However, the
GaBP-corrections (c) are strictly better than $Z^\mathrm{bp}$.} \vspace{-.3cm}
\end{figure}

\paragraph{Construction of $\mathcal{B}$ for Grids}

To achieve an error bound $\tfrac{1}{n} |\log\tfrac{Z_{\mathcal{B}}}{Z}| \le
\varepsilon$ we must choose $L \sim \log \varepsilon^{-1}$.  Then, the
computation needed to achieve this precision will depend on both the number of
blocks and the block size needed to cover all orbits up to this length.  In
certain classes of sparse graphs, it should be possible to control the
complexity of the method.  As an example, we demonstrate how to choose blocks
for $2D$ grids. Consider the $\sqrt{n} \times \sqrt{n}$ square grid in which
each vertex is connected to its four nearest neighbors.  We may cover this graph
by $L \times L$ blocks shifted (both vertically and horizontally) in increments
of $\tfrac{L}{2}$ (let $L$ be even).  It can be seen that this set of blocks
covers all loops shorter than $L$. To include all intersections of blocks, we
add $L \times \tfrac{L}{2}$, $\tfrac{L}{2} \times L$ and $\tfrac{L}{2} \times
\tfrac{L}{2}$ blocks.  The block weights are $w_{L \times L} = 1$, $w_{L \times
L/2} = w_{L/2 \times L} = -1$ and $w_{L/2 \times L/2} = 1$.  The complexity of
computing the determinant of an $L \times L$ block is $\mathcal{O}(L^3)$ and the
total number of blocks is $\mathcal{O}(n/L^2)$.  Hence, the total complexity is
$\mathcal{O}(nL) = \mathcal{O}(n \log \varepsilon^{-1})$.  \ignore{Hence, in
grids we achieve both linear complexity in the number of variables and linear
complexity in the precision $\log \varepsilon^{-1}$ (which roughly corresponds
to the number of significant digits of our estimate $\tfrac{1}{n} \log Z$).}

We test our approach numerically on a $256\times256$ square grid (with periodic
boundary conditions).  We set all edge weights to $r$ and test the quality of
approximation using both estimates $Z_\mathcal{B}(R)$ and $Z^\mathrm{bp}
Z_\mathcal{B}(R')$ for $r \in (0,.25)$ ($J=I-R$ becomes indefinite for
larger values of $r$) and block sizes $L = 2,4,8,16,32$.  The results are shown
in Figure 3.  As expected, accuracy rapidly improves with increasing $L$ in both
methods and the GaBP-correction approach is more accurate.

\section{Conclusion and Future Work}

We have demonstrated an orbit-product representation of the determinant (the
partition function of the Gaussian model) and interpreted the estimate obtained
by GaBP as corresponding to totally backtracking orbits.  Furthermore, we have
shown how to correct the GaBP estimate in various ways which involve
incorporating backtrackless orbits (e.g. cycles) of the graph. In particular, we
demonstrated an efficient block-resummation method to compute truncated
orbit-products in sparse graphs (demonstrated for grids). These methods also
extend to address estimation of the matrix inverse (the covariance matrix of the
Gaussian model), which may in turn be used as an efficient preconditioner for
iterative solution of linear systems.  We leave these extensions for a longer
report.

In future work, we plan to extend the method of constructing an efficient set of
blocks to other classes of sparse graphs.   It may also be fruitful to extend
our analysis to generalized belief propagation \cite{Yedidia05} in Gaussian
models. In a related direction, we
intend to explore methods to ``bootstrap'' GaBP using the factorization $Z(R) =
\left(\prod_{k=0}^\infty Z(-R^{2^k}))\right)^{-1}$, which follows from the
formula $(I-R)^{-1} = \prod_k (I+R^{2^k})$. By computing $Z^\mathrm{bp}(-R^{2^k})$ for
small values of $k$ we may capture short backtrackless orbits of the graph.
Another direction is to investigate generalization of the formula $Z = Z^\mathrm{bp} Z'$
to non-walksummable models, perhaps using methods of \cite{Chernyak08}.  A
related idea is to approximate a non-walksummable model by a walk-summable one
and then correct estimates obtained from the walk-summable model to better
approximate the non-walksummable model.

redpajama_set_name: RedPajamaArXiv


text:
Website of Canadian composer Michael Matthews
Sijo (1900)
Duration (min): 10
Instrumentation: Soprano and Percussion
This work is a setting of a poem by Song Sammun (1418-1456). He was one of the scholars commissioned by Korea's famous king Sejong to devise the Korean alphabet. He was also a great loyalist who was put to death by the usurping king Sejo for attempting to replace the ousted boy king Tanjong on the throne. In his book The Bamboo Grove (University of California Press, 1971), Korean scholar Richard Rutt writes: "It (the poem) makes a statement . . which is essentially symbolic, based on the oriental cliché of the pine tree as a symbol of the upright heart. The subject of the poem is not, as the western reader might suppose, a declaration of a belief in immortality, but a protestation of loyalty, a declaration that the poet's steadfastness will be remembered forever as an inspiring example." The Mt. Pongnae or Pongnae-san refered to in the poem has two meanings – the name refers to the the traditional oriental fairyland Pongnae, and it is also the name given to the the famous and spectacular Diamond Mountains of Korea in the summer, because the meaning of Pongnae-san suggests abundant foliage. Rutt writes "The use of "white snow" at the beginning of the last line is, therefore a dramatic change from the category of summer images to that of winter hardship, while the final pine-tree image, though appropriate to winter (as an evergreen), retains the tree imagery implied in the name of Pongnae-san."
The sijo (pronounced "shee-jo" or in the International Phonetic Alphabet, Si dJo) is a traditional Korean lyric of three lines or verses, each line made up of four phrase-groupings with a major pause after each grouping. Extremely elastic in form, the sijo differs from Chinese and Japanese verse forms in that it does not adhere to a strict syllable count. The theme is stated in the first line, developed in the second and an anti-theme or twist is introduced in the third, which rounds off the whole in terms of resolution. The word sijo consists of two Sino-Korean characters meanting "time" or "period" and "rhythm" or "harmony." The term sijo refers not only to a poetic form but also to a type of vocal art music, with the singer accompanied by a changgo or hourglass drum.
This work was written while I was living and teaching music in Korea (1983-85). Rather than being an attempt to write Korean music for western performers, Sijo is more a reflection of my own experiences with Korean music, and with it I have tried to capture a bit of the spirit of Korea's rich musical tradition.
© 1974-2021, Michael Matthews. All rights reserved. Design and development by Underscorefunk Design.

redpajama_set_name: RedPajamaCommonCrawl


It would be easy for Northside to get lost in trying to climb the 7A/6A-Central volleyball standings.
It would be easy for them to be more concerned about Thursday's match against Russellville.
But despite the enormity of the match, the Lady Bears took time out to help a good cause - the Susan G. Komen for the Cure.
"I just think it's important to give back to the community in all aspects of your life no matter if the central point of what you're doing is to give back," Northside senior Merit Jennings said. "Even in volleyball I think we should do something for everyone else."
"It means so much to me because there is cancer in my family so when the whole community comes out to support a cause like this it makes me really proud to be a part of Fort Smith and a Lady Bear," Northside senior Leilani Ocasio said.
Northside donned pink jerseys and took donations in support of breast cancer awareness. Then the Lady Bears won anyway as they topped Russellville 25-20, 26-24, 25-21 to take over second place in the conference standings.
The Lady Bears' program has been on the rise the past few seasons as a strong group of seniors has put a charge into the program.
"We've just had a group of girls come over that have bought into the program," Northside coach Kelsey Orendorff said. "They get along. They have great chemistry. They enjoy the game and they really enjoy each other."
"When we came in as sophomores we just had a passion for the sport and the want to win so it wasn't really us focusing on turning the program around but more just playing what we love," Jennings said.
"We came in every day, worked hard and came together as a team and listened to coach and just did what she said and used our talents to become the best we can be," Northside senior Nicole Knight said.
Northside trails only crosstown rival Southside in the 7A/6A-Central standings.


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El'ad Gips
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Min of a Matrix
Return the minimum value in the given matrix.
Finding perimeter of a rectangle
A rectangle has a length of x centimeters and a width of w centimeters. Find the perimeter.
Sum of the Matrix Elements
Add up all the elements in a NxM matrix where N signifies the number of the rows and M signifies the number of the columns. E...
Add two numbers
Given a and b, return the sum a+b in c.
Find the sum of all the numbers of the input vector
Find the sum of all the numbers of the input vector x. Examples: Input x = [1 2 3 5] Output y is 11 Input x ...
Make the vector [1 2 3 4 5 6 7 8 9 10]
In MATLAB, you create a vector by enclosing the elements in square brackets like so: x = [1 2 3 4] Commas are optional, s...


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Experience the Joy of 'Alive and Kicking'
Alive and Kicking, Lindy Hop, Susan Glatzer, Swing Dance, The Savoy
The film from director and co-writer Susan Glatzer explores the roots of Swing Dancing.
PHOTO: SUSAN GLATZER
I love the old adage that if you smile the whole world smiles with you. While watching the gloriously infectious documentary Alive and Kicking about the international resurgence in Lindy Hop and Swing Dance, I felt that there should be an equally strong and compelling addendum to the smile factor.
Wouldn't it be great if you danced and the whole world danced right along with you, too?
Director and co-writer Susan Glatzer, who partnered with Heidi Zimmerman on the shape and scope of the narrative, works a particular kind of magic, offering up one hell of a call and then daring audiences to not respond. The film explores the roots of Swing Dancing, going all the way back to Harlem, where the Savoy Ballroom was the first dance club that allowed whites and blacks to not only share the same space, but also begin to intermingle as partners. Some may have seen Swing, linked to improvisation much like Jazz (which it was set to), as a safe means of interacting. With its wildly energetic approach, there seemed to be little concern for the intermingling to cross over into legally banned intimacy.
Yet, as seen and heard in anecdotes from a new generation of dancers from all over the globe, Swing Dance might have been one of the most charged forms of dance. Despite lacking the formality of the ballroom showcases or the romance of the Tango, Swing revels in the rubber band snap that comes from throwing a partner out and then snapping them back into a fast and fevered embrace countless times. Constant contact in clenched hands, held firm by a pleading trust, makes momentary strangers feel like born lovers.
Alive and Kicking takes us inside the lives of an inner circle of adherents — some partnered couples (although none of those featured are romantically involved), some solo swingers who latch onto the next available dancer and hang on for dear life — and let's them tell what we come to see as a universal story about their lifestyle. Also, the film takes the time to engage respected elders, like Frankie Manning, whose love of the form and the resulting culture sustained him during years when it seemed as if Swing had all but disappeared.
Each and every one of them heeds the same call to dance with all their might for as long as possible. Hours of instruction and practice lead to events and contests staged all over the globe where the spirit of competition achieves levels generally associated with the ideal of the Olympics. Glory doesn't simply come from winning; instead it is about surrendering to the eternal love of the dance.
All of this sounds like some kind of utopian cult of personality, engendering a cynical reaction to the notion. But, trust me, that will fade away in the face of the athletic and rhythmic moves of the dancers, caught in their religious ecstasy on the verge of breaking free from the planet's gravitational pull. Watching them hints at what it must feel like to speak in tongues or to have an out-of-body experience.
I spoke earlier of the idea that Alive and Kicking sets up a call and response, reminiscent of watching a Fast and Furious movie and then wanting to push the pedal to the metal for a quarter mile at a time on the way home. After the credits roll at the end and the lights come up, you will be overcome by the feverish need to head to the nearest dance floor with a kicking and swinging band so that you can Lindy Hop the night away.
Thanks to Cincinnati World Cinema, local audiences won't have to go far. The 7 p.m. southwestern Ohio premiere of Alive and Kicking at Memorial Hall on Saturday will be followed by a dance demo and lesson by the Cincinnati Lindy Society, live music by the Ron Purdon Quintet and time for participants to put the impromptu lessons to practice during an opening dancing segment. Proceeds from this one-night only event — dubbed Cincinnati Swings — will benefit Cincinnati World Cinema and WMKV Radio.
ALIVE AND KICKING screens 7 p.m. Saturday at Memorial Hall. More info: cincyworldcinema.org.


The late Mr. [James Cusack] Hassall was born at 'Wotton Vale,' Coleraine, Victoria, in 1858. He was educated at the Geelong Grammar School, and was a school mate of Dr. Jackson, of Brisbane. Forty-eight years ago he married Miss Frances Percy Faris. He came to Queensland first in 1882. Later he went back to Victoria and returned again to Queensland in 1889, when he settled on Dandine Station, as manager, during which time he experienced the highest flood ever known on the Condamine. Dandine Station being cut up, he came to Moggill, where he engaged in general farming, but was ruined by the 1893 flood, losing everything with the exception of two ponies and a buggy. Deceased came to Mt. Ubi station in 1893, and two years later was appointed manager, which position he held until the estate was sold in 1925, on the death of the owner, Senator J. T. Walker, of Sydney. During his residence there he experienced the highest flood known at Kenilworth. In the early days he carried his butter on pack horses to Eumundi. Later, at the request of residents, he started the first cream run, and personally drove the first four-wheeled vehicle from Mt. Ubi to Eumundi for the Silverwood Dairy Co. He was one of the first to transport cream by motor lorry from Mt. Ubi to Eumundi. His great grandfather, a clergyman, came to Australia in 1819, and married a daughter of 'Farmer' Marsden, a personal friend of Governor Macquarie.
The funeral, which was largely and representatively attended moved from St. John's Church of England to the Nambour cemetery on Friday last, after service, the last rites at the graveside being conducted by the Rev. G. T. Hobbs.
Maroochy Shire Council; Mr. and Mrs. D. Beattie and family; Mr. and Mrs. F. Rodgers and family; the Chadwick family; Mr. and Mrs. A. W. Thynne and family; Mr. and Mrs. T. Walker, Jack and Don; Mr. A. J. Evans and family; Mr. and Mrs. W. Parsons; Nichols family; Mr. and Mrs. F. Pearce; Mr. and Mrs. J. T.. Lowe and family; Mr. and Mrs. W. Purdon and family, and Mrs. Pearce Senr.; 'Amy,'; the Wilkinson family; Mr. and Mrs. S. Kelly; A. L. McGregor; Mr. and Mrs. E. Pickering and family; Mr and Mrs. Hill; Jacky Nandree; Bridges family; from the children; from 'Per'.
'Hassall, James Cusack (1858–1936)', Obituaries Australia, National Centre of Biography, Australian National University, http://oa.anu.edu.au/obituary/hassall-james-cusack-16214/text28148, accessed 22 April 2019.


I am Reader in Documentation working as part of Research at Camberwell, Chelsea and Wimbledon and Ligatus. I completed my PhD in the RCA/V&A Conservation (run in association with Imperial College). My PhD work was on Computer Applications to Conservation and more specifically Conservation Documentation.
I work with Linked Data technologies, ontologies, knowledge organisation systems to model records from memory institutions. I am a member of the CIDOC-CRM special interest group working on improving the standard and extending it for the field of conservation.
My research focuses on documenting practice with particular interest in conservation and archiving. I am looking into different ways of recording and documenting objects. I work closely with Professor Nicholas Pickwoad on the documentation of binding structures. We have jointly developed a methodology for recording significant features of bindings which can assist in understanding the history of the book. I have developed an XML schema to formalise and validate these records. We led a consortium of partners who worked together to publish a thesaurus of bookbinding terms: the Language of Bindings Thesaurus.
I proposed an archiving methodology called "Creative Archiving" as a way to interpret artists' ideas through the organisation of their own papers. I have researched the work of the artist John Latham through analysis of his archive.
I have a strong interest in the Drupal content management system as a programming platform and a web-application delivery platform.
I am a keen supporter of open source software and open distribution of knowledge.


Ingredients: dibasic calcium phosphate, microcrystalline cellulose, stearic acid, magnesium stearate and modified cellulose gum.
Warnings/Cautions: If you are pregnant, may become pregnant, or breastfeeding, consult your health care professional before using this product. Do not use if either tamper-evident seal is broken or missing. Keep out of the reach of children.


Isobel is turning 16 and has to decide if she wants to work for the boss or leave the Territory, a reimagined American West full of magic. She has grown up in a saloon, where the Devil lets you gamble at the tables, but is willing to make a bargain if desired. And the Devil always keeps his word and honors his bargains. Isobel decides to stay, but instead is sent out on the road on horseback with mentor Gabriel to learn the Territory and to become the Devil's Left Hand, a job that doesn't come with a manual. Isobel and Gabriel encounter talking animals, a traveling magician, shamans, terrifying supernatural winds, and small towns completely empty of people. Young as she is, Isobel needs to take responsibility and find out what is going on in the Territory. A sequel, The Cold Eye, is being published in January, and I'm intrigued to find out what Isobel will encounter next.


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With a spin control, users can click arrow buttons to change incrementally the value within its associated numeric text box. The term spin box refers to the combination of a text box and its associated spin control.
A typical spin box.
Users often prefer spin controls because they can make changes without moving their hands from the mouse. When the spin control is paired with a text box, users can type or paste input directly in the text box, so use of the spin control is optional.
While spin controls are used for numeric input, the input doesn't have to be a pure whole number. The input can be decimal numbers and it can have negative signs, delimiters (such as colons or hyphens), and unit modifiers.
Guidelines related to text boxes and layout are presented in separate articles.
Is this the right control?
To decide, consider these questions:
Is the control used for numeric input? If not, use another control, such as a drop-down list or slider, to select from a fixed set of values. Use scroll bars for scrolling.
Do users think of the value as a relative quantity, not a numeric value? If so, use a slider instead. Use spin boxes only for exact, known numeric values. For example, users think about setting their audio volume to low or medium—not about setting the value to 2 or 5.
Is the control paired with a text box? If not, don't use. Spin controls shouldn't be used alone or with other types of controls besides a text box.
Incorrect:
In this example, a spin control is used to control a dynamic graphic.
Are contiguous value ranges valid? If not, use a drop-down list of valid values instead.
In this example, not all disk drive numbers are valid, so a drop-down list is a better choice.
Is using the spin control practical? Using a spin control is practical for:
Entering a small number, typically under 100.
Making small changes to an existing or default value.
While spin controls can be used for any numeric input, they are inefficient in situations other than these.
Is the spin control helpful? Is the control used in a context where users are likely to be using their mouse? If not, consider a spin control optional.
Are the sibling controls drop-down lists? If there are other drop-down lists, consider using a drop-down list for consistency.
In this example, a spin box could be used, but a drop-down list is used for consistency.
Are touch or pen users a primary target? If so, consider using a drop-down list instead. The arrow buttons in a spin control are too small to be used efficiently with touch or a pen.
If a slider or a spin box is possible, use a spin box if:
Screen space is tight.
A user is likely to prefer using the keyboard.
Use a slider if:
Users will benefit from instant feedback.
Use spin controls whenever they are practical and helpful. See Is this the right control?
Exception: To be consistent with other text boxes on the same user interface (UI), use spin controls even if they aren't always practical.
Correct:
In this example, a spin control is used with the year control for consistency, even though it isn't always practical.
In this example, the spin control is unusable.
Always make a spin control the "buddy" of the text box. Doing so places the spin control inside the text box.
In the correct example, the spin control is placed inside its associated text box.
Disable a spin control when its associated text box is disabled. The spin control is a supplemental input method—never the only input method.
Define the top button to increase the value by one unit and the bottom button to decrease by one unit. Typically, the unit is one, but it should be the smallest common change in value. Ideally, the spin control should cover all valid values, and it should be more convenient than typing in the text.
In this example, clicking a spin control changes the values by .1, which is the smallest common change in value. Using a smaller unit would cover the range of valid values but make the spin controls unusable.
Use the spin control to limit input to valid values. Using a spin control should never result in an incorrect value.
At the end of a range of valid values, restart the range. The spin control metaphor is that the user is spinning a wheel of values, hence this wheel-like behavior.
Exception: Don't restart the range if the resulting value is certain to be incorrect.
In this example, clicking the down arrow button doesn't restart the range (by going to the maximum value) because that value is certain to be incorrect.
Use text instead of special numeric values. Allow users to spin to these special values instead of having to know them and type them in.
In this example, Never is a special value but users can spin to it.
If the value has delimiters, the associated text box should have multiple input focus points. Doing so allows the numeric segments to be manipulated individually.
In this example, the spin control affects the values for hours, minutes, seconds, and A.M./P.M.—whichever has the focus.
If the value has units, use the spin control to change those units as well.
In this example, the spin control can be used to change units.
Apply the text box labeling guidelines to label the associated text box. Spin controls are never labeled directly.
When referring to spin controls:
Don't refer to spin controls in user documentation. Instead, refer to the label of the associated text box.
Refer to spin controls and spin boxes only in programming and other technical documentation.
Example: In the Date box, type or select the part of the date you want to change.


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NHL roundup: Flames down Ducks, extend hot streak
February 23, 2019 ayman022	0 Comments
Andrew Mangiapane scored the game-winning goal late in the third period, and TJ Brodie collected a goal and an assist as the host Calgary Flames won their fourth consecutive game, defeating the Anaheim Ducks 2-1 on Friday night.
Feb 22, 2019; Calgary, Alberta, CAN; Calgary Flames left wing Andrew Mangiapane (88) celebrates with temmates after scoring a goal during the third period against the Anaheim Ducks at Scotiabank Saddledome. Mandatory Credit: Sergei Belski-USA TODAY Sports
Goalie Mike Smith made 25 saves for the Flames, who strengthened their hold on the top spot in the Western Conference.
Mangiapane broke the tie with 3:29 remaining in regulation. He tried to make a pass on a two-on-one rush only to see it blocked, but the puck came back to the speedy rookie, and he promptly buried a short-side offering for his third goal of the season, all in his past seven games.
Brodie and Anaheim's Derek Grant scored in the second period. Ducks goalie Ryan Miller made 26 saves.
Jets 6, Golden Knights 3
Patrik Laine scored twice to snap a 15-game goalless drought as Winnipeg won in Las Vegas.
Nikolaj Ehlers, Kyle Connor, Adam Lowry and Andrew Copp also scored for the Jets, who snapped a three-game losing streak and reclaimed the lead in the NHL's Central Division, moving a point ahead of the Nashville Predators. Winnipeg goaltender Connor Hellebuyck made 29 saves.
Pierre-Edouard Bellemare, Shea Theodore and Jonathan Marchessault scored for the Golden Knights, who are 3-9-1 in their past 13 games and have won just once in their past eight home games (1-6-1). Goalie Marc-Andre Fleury stopped 23 of 28 shots.
Blue Jackets 3, Senators 0
Sergei Bobrovsky made 22 saves to record his fourth shutout of the season as Columbus skated to a win over host Ottawa, hours after acquiring pending unrestricted free agent forward Matt Duchene from its opponent.
Josh Anderson scored his seventh goal in 11 games for the Blue Jackets. Cam Atkinson netted his team-leading 33rd tally and Oliver Bjorkstrand also scored to help Columbus snap a two-game skid. Duchene had three shots on goal against his former team.
Ottawa rookie forward Brady Tkachuk fell on Bobrovsky late in the second period to raise tensions between the teams. The two-time Vezina Trophy recipient barely was tested the rest of the way en route to his 28th career shutout.
Avalanche 5, Blackhawks 3
J.T. Compher had two goals and an assist, including a tiebreaking goal on a breakaway with 5:22 remaining in the third period as Colorado held on for a win at Chicago.
Carl Soderberg also scored twice for the Avalanche, who won their third consecutive game. Gabriel Landeskog added a goal and an assist for Colorado.
The loss spoiled a historic night for Blackhawks forward Patrick Kane, who tallied at least one point for the 20th game in a row. Kane owns the two longest point streaks by a U.S.-born player in NHL history, including his 26-game streak during the 2015-16 season.
Wild 3, Red Wings 2
Devan Dubnyk made 26 saves to lift Minnesota to a win at Detroit, the first time since Jan. 23 that the Wild followed up a win with another victory. Minnesota had lost nine of its previous 11 games, going 2-6-3 in that span.
Luke Kunin had a goal and an assist for the Wild, who also got tallies from Jordan Greenway and Eric Staal.
Tyler Bertuzzi and Anthony Mantha had goals for the Red Wings, with Dylan Larkin assisting on both plays.
Patrik Laine scored twice to snap a 15-game goalless drought as the Winnipeg Jets defeated the Vegas Golden Knights 6-3 Friday night in Las Vegas.
← Syria Kurds evacuate civilians from IS redoubt, hail Trump troop reversal
Brazil prosecutors seek arrest of senior Vale exec after deadly disaster →
Tokyo Games savings at $4.3 billion but more to come: CEO Muto
October 8, 2018 ayman022 0
Tennis: Del Potro suffers fractured kneecap after fall
United run out of patience with Mourinho, Pochettino in the frame


New Openings in downtown Spokane.
Last night I went to a preview of the new Anthropologie that opens today. Oh Spokane, you're getting pretty cool. Fun merchandise and inspiring displays.
The new store next door, &Kloth opens today as well. This is a sister store to Fringe, a local boutique that started on the South Hill and now has a second store in Wandermere.
Both of these stores are located in downtown Spokane on W Main across from Nordstrom. Ohh Nordy's you have some trendy competition.


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Historic Monuments of Ancient Kyoto (Kyoto, Uji and Otsu Cities)
The history of Japan and that of Kyoto are inseparable. From 794 AD when the city was established, Kyoto served as the imperial capital of Japan until mid 19th century. The city was modeled against the capitals of ancient China.
Kyoto was the cultural center of Japan for more than 1000 years and thus its historic center demonstrates evolution of Japanese architecture from wooden structures to stone houses. The religious architecture, art and gardens found within the city form a landscape that has influenced the world for many generations.
The historic monuments of ancient Kyoto include Kyoto, Uji and Otsu Cities. The 3 cities are made up of 17 component parts.
Most of the 198 buildings and 12 gardens you find in this complex were built between the 10th and 17th centuries and form an integral part of the modern Japanese urban culture. All the 17 components that form the city that have been inscribed into the UNESCO World Heritage Sites are religious establishemtns apart from the castle of Nijo-jo. Together, the establishements occupy 1056 hectares of land and have been given a buffer zone of 3579 hectares.
Karmwakeikauchi-jinja (Shinto shrine)
Amomioya-jinja (Shinto shrine)
Kyo-o-gokoku-ji To-ji (Buddhist temple)
Kiyornim-dera (Buddhist temple)
Enryaku-ji (Buddhist temple)
The two large Buddhist temples of Daigo-ji and Ninna-ji from the Heian Period
The Buddhist temple of Byodo-in and the Ujigami-jinja
The Sekisui-in at Kozan-ji
Temples of the Rinzai Zen sect such as Temyu-ji
Zen gardens such as Saiho-ji
The Buddhist temple Rokuon-ji
The list of attraction in the Historic Center of Kyoto cannot be exhausted by in a simple blog. For a city with a history that spans more than 1500 years, you can expect to see a lot of ancient attractions. You are better off visiting it than reading about it.
Most of Japan's attractions are affordable. The Japanese are proud of their culture and attractions. They are also very hospitable and would give you more than your fair dosage of history if you show the slightest interest. I guess this is what makes Japan a great country. Why not exploit their hospitality to learn something useful?
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text:
"Province"
The X-Men battle Fitzroy in Dallas.
Writes: Lobdell
Pencils: Romita Jr.
Inks: Green & Panosian
Vacations & Lettering: Eliopoulos & Babcock
Colors: Buccellato
Edits: Bob Harras
Chiefs: Tom DeFalco
As Storm struggles to contain the debris from the Aerie explosion, Bishop searches the rubble for any sign of Forge, Mystique or Fitzroy. Meanwhile, Xavier worries about Illyana's worsening health. In Dallas, Storm is attacked by a mob, but a local police officer comes to her defense. In the remains of the Aerie, Forge emerges from a protective shield along with an unconscious Mystique, while on the ground, the mob reasons they can probably overpower both Storm and the cop, just as Iceman arrives and helps contain the situation. Above. Fitzroy attacks Bishop & Forge, but his armor is destroyed by Jean, while Colossus leaps into the fray, pummeling him in rage to the point that Colossus has to be pulled off the unconscious villain by his teammates. Back at the mansion, a studying Jubilee is interrupted by the sudden arrival of Kitty Pryde, who came as soon as she heard about Illyana.
Illyana's condition continues to deteriorate, with Xavier at a loss as to how to stop the further degradation of her genetic code. He also dances around calling the cause of her illness the Legacy Virus (though that's what it is).
Having heard of her illness off-panel, Kitty Pryde shows up at the end of the issue, the first time she's appeared in this series since issue #227 (when she watched the X-Men die in Dallas). This kicks off a string of guest appearances for her across a half dozen issues or so of this and Adjectiveless for the course of "Fatal Attractions".
Fitzroy is defeated and, presumably, captured this issue; he will next appear in the "Child's Play" crossover between X-Force and New Warriors, which wraps up the Upstarts plotline.
Bishop mentions that, as Genesis, Forge founds the XSE in the future. He also mentions the Summers Rebellion for the first time, an event from his timeline which will occasionally get name-checked through years before being depicted by Peter David in his return to X-Factor, a story which will also develop an actual character for Fitzroy beyond "sneering psychopath",
Kitty appears here, and in her subsequent non-Excalibur appearances around this time, between issues #70 and #71 of that series.
Bishop peppers his dialogue with future slang inspired by X-Men sound effects in this issue, first using "snikt" as a synonym for "burn out", then "bamf" in place of moving from one place to another.
He also says "by the single eye!" in the same way we might say "by Lucifer's beard!".
Iceman continues to stretch his powers, appearing here in a massive, spiky form that I've always considered the specific inspiration for his second action figure.
Colossus vents his building frustration by savagely pummeling Fitzroy.
This leads to a well-written discussion of morality, in which Bishop says that in executing the criminals Fitzroy brought back in time earlier in the series, he was acting in accordance with the laws and morals of his own time, but said actions aren't as appropriate in the present time.
Jubilee says that she's the worst student to come through Xavier's other than Warlock.
Kitty mentions that she met Jubilee in England, an un-footnoted reference to Excalibur #57-58.
Claremontisms
Jubilee, complaining about Illyana's constant coughing, refers to her by the very Claremontian "'Lanna".
Storm, in the midst of trying to prevent debris from Fitzroy's explosion from raining down on civilians, is accosted by an anti-mutant civilian.
Later, a local police office comes to her defense. in a Claremontian moment of injecting a one-off minor character with a fair amount of gravitas, shades of Sabrina Morrell.
There's ad for the fourth series of Marvel Universe cards in this issue.
Aside from when I first read it off the stands, this issue has always felt overshadowed by the issue which follows it. In part, that's probably because, knowing what's coming, there's a sense of foreboding permeating this issue, from Xavier's disheveled inability to treat Illyana (his stubble the international sign for "haggard and running out of options") to Jubilee's casual irritation towards Illyana's constant coughing (and concern for waking her up), suggesting just how sick she's gotten, to Kitty's arrival to be with her best friend. In part, it's also because the bulk of this issue is a fairly basic action story in which the X-Men rather quickly dispatch Fitzroy, still, to use the modern parlance, a "nothingburger" of a character Fitzroy. As a result, it's Colossus' increasingly unhinged attack on Fitzroy that really stands out, as he releases his growing rage and frustration on the villain. And thus, even there, hindsight once again reminds, that for Colossus, the worst is still yet to come.
Tomorrow, "Fatal Attractions" in X-Factor #92. Friday, Wolverine vs Sauron in Wolverine #71. Next week, X-Men #22.
Collected Editions
Posted by Austin Gorton on August 16, 2017
Labels: "Fatal Attractions", Comic Book reviews, comics, Lobdell/Romita Jr., Uncanny X-Men, X-aminations, X-Men
wwk5d August 17, 2017 at 1:01 AM
"Fitzroy is defeated and, presumably, captured this issue; he will next appear in the "Child's Play" crossover between X-Force and New Warriors, which wraps up the Upstarts plotline."
And what a great plotline it has been so far. Honestly though, I do think the idea of the Upstarts was a good one, it just had a really lousy execution.
"Bishop peppers his dialogue with future slang inspired by X-Men sound effects in this issue, first using "snikt" as a synonym for "burn out", then "bamf" in place of moving from one place to another."
I realize some people like this, but I was always annoyed by it.
"Iceman continues to stretch his powers, appearing here in a massive, spiky form that I've always considered the specific inspiration for his second action figure."
Other than breaking up with Opal, this is pretty much Bobby's main character and plot arc for the next few years. And it isn't even consistent, since he spends the next year or so actually trying to stretch his powers, then after AOA, he's back to whining about not living up to his potential again...
"This leads to a well-written discussion of morality, in which Bishop says that in executing the criminals Fitzroy brought back in time earlier in the series, he was acting in accordance with the laws and morals of his own time, but said actions aren't as appropriate in the present time."
It is one of Lobdell's better written scenes so far. A little reminiscent of Rachel going after Selene. Good thing Colossus calmed down here and wasn't throwing around any ultimatums, or Bishop might have had to snikt him...
"Later, a local police office comes to her defense. in a Claremontian moment of injecting a one-off minor character with a fair amount of gravitas, shades of Sabrina Morrell."
With a Claremontian accent, too!
Teemu August 17, 2017 at 1:26 AM
Now that you mention it, it kind of feels like a semi-conscious rolling-back to more Claremontian UNCANNY after the excesses of the re-Image-nation era.
+1 on annoyance on the onomatopoetic verbiage of the future. Though, if we were to travel back in time to the early 90's and would refer to our communication in the future as "tweets", the subsequent hits on our noses would so be deserved.
Note that Mystique mumbles about "Gray" while she's injured- this is a clue that she's Graydon Creed's mother.
Matt August 17, 2017 at 11:26 AM
I'm with you guys on Bishop's epithets. I'm glad that was done away with. They just sound dopey to me.
I wasn't reading UNCANNY regularly when this one came out, so I never saw Kitty's "we met in England" line. As I also didn't read EXCALIBUR, I just assumed Illyana's funeral was the first time Kitty had seen most of the X-Men in a really long time! (Similar to how I believed issue 300 was Nightcrawler's first reunion with the group since "Mutant Massacre".)
Ian Miller August 17, 2017 at 11:50 AM
It's ironic that this is the era of the X-Men I first discovered when I was 6 or 7, and I loved it a lot. But reading these reviews now after discovering Claremont's run and all those great classic stories, I can't help but feel something is missing from this era, and I'm not finding that joy in them I once had.
You know how Claremont said that when Magneto "died" in X-Men #3 that this was *HIS* Magneto dying, and that any other version after that wasn't the *TRUE* Magneto? I feel the same way about Claremont and the X-Men as a whole - Without him writing it, things just don't feel right. Claremont knew these characters inside and out, and I feel that anyone else writing them is just doing a karaoke version of them. I think Claremont would've done a much better job exploring how the original 5 members interacted with the newer members. And the whole shitting on Colossus thing just doesn't sit right with me, it feels so out of left field.
I'm thinking of checking out X-Men Forever and seeing how Claremont's plans could've turned out. Who knows, it may make me like Lobdell and Nicieza's runs better.
Licínio Miranda August 18, 2017 at 11:40 AM
I can't agree with you regarding how Claremont would handle the original X-Men's interaction to the new ones. We saw it during Inferno and it was awkward. (Arch) Angel had worked with them, was saved by them once, and they treated him as a stranger. Claremont never cared about Iceman.
Knowing Claremont as I do, he'd turn Jean Grey into a main character and make everything revolve around her. He's also take time to show his contempt for Cyclops as well.
wwk5d August 18, 2017 at 12:28 PM
I thought Claremont write Cyclops well. Not sure where his contempt for the character comment comes from.
Yeah, if you actually believe that X-Men: Forever had anything to do with where Claremont would have taken the book, then you might end up appreciating Lobdell and Nicieza far more!
In actuality, X-Men: Forever has nothing to do with the stated premise, of showing what Claremont would have done with the books had he stayed.
Instead, Claremont seems to want to convince people that had he remained on X-Men, he would have written all the important moments in latter-day Marvel history (up to time of X-Men: Forever's publication) first.
The series isn't an awful read, and is better than a great deal of Claremont's post-1991 comic work. Just don't expect that it's going to live up the billing, of being anything remotely like what X-Men would have been with Claremont writing.
cyke68 August 17, 2017 at 1:07 PM
I do wonder if Claremont went in a different direction on Forever because he'd already spoken at length about his post-1991 X-Men plans in interviews. Maybe he no longer had any interest revisiting those ideas or just wanted to throw people a curve ball. The first few issues start out promising enough, then Tony Stark shows up, and... yeah.
Really, I don't know how you enforce a premise like that in the first place. A current day editor wouldn't be expected to came down hard on Chris Claremont for his writing something that he would NOT have been allowed to get away with on the main books 20 years ago, but is totally irrelevant in the "what if?" vacuum of the Forever-verse. It's largely dependent on him editing himself, which he's probably not gonna do. Still, I suppose there is a middle ground, since it didn't feel like the book was even remotely adhering to the spirit of the concept by the end of that first volume.
Ian Miller August 17, 2017 at 2:44 PM
Wow, I just looked up the plot of X-Men Forever on Wikipedia, and it's definitely not what I imagined it'd be. It says that Claremont pretty much changed the premise to it being an alternate reality as opposed to being a continuation of his X-Men run.
I've also read Claremont's return to the core books over the years, and while I enjoyed them, I think they felt really disconnected from the rest of the series. Claremont's original run worked really well because he had years and years of continuity building up to create interesting subplots that could simmer for months or even years. But when he shows up after a major run like Grant Morrison's, it's not the same because too many different hands have been playing in his sandbox.
But yeah, going back to my original post, I'm curious to see if I'd have liked the X-Men around this period better if they were still written by Claremont, and to see whether or not the quality would still be there or if he'd get burned out.
I think Claremont admitted he ended up writing something completely different for X-men: Forever, since he had already pretty much revealed what his original plans were anyway in various interviews (Wolverine being killed and coming back as an assassin of the Hand, the final battle with the Shadow King, Xavier dying and Magneto replacing him, etc). Rather than just doing what we all expected him to, he took a chance and decided to do something different.
Granted, his original plans would have either not included the first 3 issues of Vol. 2, unless he planned or resurrecting Magneto himself in time for the final battle with the Shadow King.
Yeah, that's the other screwy part - the jumping off point is a little arbitrary and disingenuous. It's not like Uncanny X-Men #94-280 and X-Men #1-3 are Claremont's 100% pure, unadulterated vision. His influence wanted significantly at the tail end of the run, and I think it's safe to say vol. 2 never would have existed in that form if he had his druthers.
Of course, he did sort of try working around this with the Nathan Christopher and Storm/X-Tinction Agenda retcons. That just made things feel even more forced, as the events of X-Men #1-3 apparently happened as read, but there's all this weird vague shit preceding them that doesn't add up because it isn't Claremont's preferred status quo.
I posted this once before, I believe on Not Blog X when G. Kendall looked at X-MEN FOREVER, but this seems a good time to bring it up again: I really think Marvel and Chris Claremont fumbled X-MEN FOREVER horribly. It was advertised as Claremont continuing his run, but that's not what it was at all. I feel like a lot of the negative fan reaction could've been averted if they'd tried something like this:
First, in issue #0, Claremont writes his originally intended conclusion to UNCANNY #279 ("The Muir Island Saga). Then from there, whether or not he would choose to disavow X-MEN #1 - 3 (and I think there's a good case to be made for him doing so), each issue of X-MEN FOREVER would map to an issue of UNCANNY -- i.e., FOREVER #1 would be UNCANNY #280. Claremont would then write UNCANNY X-MEN #280 - 300 (XMF 1 - 21) in a pointed effort to tie up all the loose ends from his original run and put a nice bow on it for the longtime fans.
Then from there, Marvel could've launched X-MEN FOREVER volume 2, and he could've gone nuts with burnout, the Consortium, Perfect Storm, and whatever else his heart desired.
I don't know that this would've staved off cancellation any longer than the actual title did, but I think it would've been a more honest way to do the series as advertised, as it would've delivered what the older fans expected going in, while still allowing him to explore all his new ideas going forward.
Jason September 1, 2017 at 4:15 PM
I had had a similar thought, except I think starting with right after X-Men #3 was the right idea, simply because it's cleaner.
But the idea of each issue mapping onto an issue from the time is very similar to my own thoughts. But I had thought that since the series was twice monthly, each issue could have mapped onto one of the two X-series, leapfrogging along. (So, issue 1 maps onto an issue of adjectiveless, issue 2 onto an Uncanny issue, then issue 3 back to adjectiveless, etc.) Which is maybe silly, but it would have let Claremont explore two different things at once by having a blue team and a gold team to play with, having simultaneous adventures.
And it needed a better editor, someone to remind Claremont that, 'Hey, Scalphunter wasn't called "Greycrow" or whatever in 1991, and Magneto's name wasn't "Erik" back then either. Here's where things sat in 1991, here are the storylines that were still outstanding. Why don't we focus on paying off all that stuff first.'
The series could have just been one satisfying payoff after another, but instead it just ignored the danglers that were still in play in 1991 and created like eighty new ones.
Jeff August 17, 2017 at 5:54 PM
These two issues are weird because a bunch of well-regarded characters were jobbed out to Fitzroy to build him up and then he gets taken out fairly easily in two issues in his second appearance. I feel like the creators could have at least tried to give a better send off to the Upstarts plot even if they were going to toss it off as soon as possible.
For some reason, even though the current plot in Adjectiveless actually contradicts his stories, it still feels more like a continuation of Claremont at to me. I can't justify it, but it just feels that way for some reason.
Blam August 27, 2017 at 8:52 AM
I think I'm just as unlikely to exclaim "By Lucifer's beard!" as I am "By the single eye!" but YMMV…
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