Nowww this is a story all about how this doll named Magic came into my life and how she became part of my Positive Forward Movement of Spiritual Love.
First, Magic didn't have a name until a couple months ago when my Soul Sangin' Sister Afua, donned her Magic. I'm so grateful because it fits!
This gorgeous doll reminds me of the things I've overcome, the things I am working toward, and that Magic is Normal.
I moved away from Raleigh after my life fell apart. I didn't feel like I had a place here. There was nothing tying me here but pain so I decided it was time to go. I needed a change and a place where I could just be without the influence of others, and found the perfect place.
I settled out in Lil Lexington. A very small town, that became one of my favorite places in North Carolina, in a quiet little apartment not to be found on any GPS. I was in heaven!!
While there I focused on getting my life together. I studied tarot, grew my crystal collection, practiced my own magic, spent time in nature and around good people. I focused on bettering myself and changing my life. I worked on my degree and was laser focused on my goals.
All of which came to be known as Positive Forward Movement. I finally had a name for the process I was going through. I healed, grew, and blossomed beautifully.
While in Lexington I had an amazing mentor, who was like a sister, mother, and friend all rolled up into one. I helped her with her life and business and she helped me with mine. At that time my only business was my life.
The doll now known as Magic entered my life around 2010ish. I found her cleaning out a recently vacant apartment and had to have her. I couldn't throw her away, there was something about that doll that made me happy. I brought her home cleansed and blessed her and she has been with me ever since.
Magic was a decorative piece in Kaylah's room for years, then made her way to Savvi's room, but she had a bigger purpose once I started practicing energy healing techniques.
Although Magic has a deep, personal meaning to me she serves as a proxy for energy healing sessions. She hangs proudly on the wall of my office in The North Raleigh Well Nest. When I do a remote energy healing session, meaning the client isn't able to come into my office, I use Magic in their place.
I invite my clients' energy into the room and ground it in Magic, then I proceed to do my work. Once the session is over my clients' energy returns to them and I walk them through a grounding exercise to help them anchor their energy in. Then, Magic is cleansed and she returns to her space. I don't have to use a proxy, but I choose to because that is how I work best.
So if you enter The North Raleigh Well Nest and see Magic, come in and take a picture with us and know that she is doing her job helping me spread love and light all over.
Previous Post: « What is Spiritual Love?


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In the end, it will be okay because God is "always "with us.
Whether we like it or not, growing up brings change. For many children, this conjures fears that are sometimes real and sometimes imagined. How can kids learn that even when they face new and unfamiliar situations, they don't have to be afraid?
Little Seed and Little Fox are facing changes and brand new circumstances--and they don't like it one bit Through this unlikely friendship, children will discover that no matter how new or fearful their circumstances, God is always with them. The whimsical art by Natalia Moore will put a smile on kids' faces and put their minds at ease. Just as Little Seed and Little Fox learn to trust that the Farmer is good and kind, children will also learn to trust God.
Written by "New York Times" bestselling author Lysa TerKeurst, "It Will Be Okay" will help kids discover that the end, it really will be okay because we have a God who is good and kind and always with us.


text:
"NEW EYES" SHOW
Congregation Shearith Israel 9401 Douglas Avenue
Dallas, TX 75225 972-559-9545 Israeldallascenter@shearith.org
Contact Anat Solomon
Israeldallascenter@shearith.org
https://shearith.org/calendar/new-eyes-show/
$ Cost $ 25.00
NEW EYES is a critically acclaimed one-woman show, starring Israeli-American actress, Yafit Josephson, that made its debut in Los Angeles in 2010 and has been a smash hit playing with rave reviews across the US for over 5 years. NEW EYES tells the life story of Yafit Josephson, a young woman living in Israel who joins the Israeli army as part of her mandatory service to her country. Upon completing her service, she moves to Los Angeles to follow her dream of becoming an actress but finds herself constantly cast as the villain. Just as her native country wrestles with the world's false perceptions, Yafit must decide how far she is willing to go to challenge, change, or accept how others see her, and what she is willing to risk to fulfill her dreams and destiny. NEW EYES explores universal themes of self-esteem, family relationships, identity and cultural differences. Yafit Josephson portrays 18 characters from over 5 different countries as she conveys her riveting and uplifting story, as she sees the world, and it sees her, through NEW EYES.
Sponsor: ISRAEL DALLAS CENTER, Congregation Shearith Israel, Israeli American Council

redpajama_set_name: RedPajamaCommonCrawl


text:
Food is spoiling as lengthy power outages continue after Dorian
CTV Atlantic Published Tuesday, September 10, 2019 11:00PM ADT Last Updated Tuesday, September 10, 2019 11:01PM ADT
With some people on Cape Breton Island having to wait until Thursday to see their power fully restored, small business owners and residents are feeling the effects of Dorian in the pocket book.
That's because no power means spoiled food, and lots of it.
Marie McPherson and her daughter have been without power for more than 50 hours. Today was the first time she was able to have a hot meal, provided at a comfort centre set up inside Sydney's main arena.
"It's been very cold," McPherson said. "No food. We're getting very low on food right now, but I have good company with my daughter."
McPherson says what's worse is that she has a freezer and fridge full of food at home. She hasn't opened them since the power went out, but she's guessing most of it will have to be thrown out.
"Probably about $250 to $300 dollars," McPherson said. "It's a lot of money when you're a single parent on low income."
In Glace Bay, the food bank is in high demand and the shelves are bare.
Michelle Kalbhenn, the Glace Bay Food Bank co-ordinator, said they've been getting more visits.
"Our numbers are going up," she said. "Instead of feeding 50 meals a day, we are up to 65. We had 72 one day."
Volunteer Kimberly McPherson said the power outage has forced many to head to the food bank.
"New clients are coming in because they've had all of their food ruined in their fridges and freezers," she said.
The food bank is also dealing with its own damage. Their newly constructed green house was flattened from the wind. McPherson estimates it will cost nearly $3,000 to replace.
"I walked around the corner and it was just devastating," McPherson said. "It was heartbreaking. There were pieces everywhere."
Thousands are still in the dark on the island. Nova Scotia Power says some communities in Cape Breton have an estimated restoration time on Thursday.
Four comfort centres remained open in the Cape Breton Regional Municipality on Tuesday.
"The comfort centres are in direct relation to the power outages," said CBRM Mayor Cecil Clarke. "So, we will assess as recovery efforts become more clearly identified."
For McPherson, she's hoping the power will come on sooner rather than later.
Two comfort centres remained open Tuesday evening. The Gabarus fire hall was open until 8 p.m., while The North Sydney Fire Hall was open until midnight.
Salvation Army will be helping get food and water to those in need Post-storm starting tomorrow. There will be a number to call that will be live on Wednesday morning at 9 o'clock. (902-562-1717)
With files from CTV Atlantic's Kyle Moore.
NEW: Salvation Army will be helping get food and water to those in need Post-storm starting tomorrow.
There's a # to call, below.
@CTVAtlantic pic.twitter.com/NBCThpFF7y
— Ryan MacDonald (@RyanMacD_CTV) September 11, 2019
Marie McPherson estimates that she'll have to throw out about $250 to $300 worth of food from her fridge and freezer."It's a lot of money when you're a single parent on low income," she said.


In the 2015 edition of the test, all seven of the largest lenders in the country passed, although two struggled.
The seven biggest banks in the UK have all passed the stress test administered by the Bank of England for the second year in a row, to gauge their preparedness in case of a financial crisis.
None of the banks were found in need of a new capital plan under the scenario used in the test – crude oil at $38 a barrel and a slowing global economy, caused by a crash in China and other emerging markets. However, two of the seven, Standard Chartered and the Royal Bank of Scotland (RBS), were told by the central bank to take steps to improve their capital strength in order to be able to survive such a scenario.
The Bank of England noted that the Royal Bank of Scotland was particularly close to failing the test but added that it had used the participating lenders' balance sheet as of the end of 2014. Since then, RBS has done well for itself, shoring up its finances and eventually stabilizing its balance sheet. It was this progress that discouraged the test administrator from excluding the bank from the pass grade.
Standard Chartered, on the other hand, does most of its business in emerging markets, which made it particularly vulnerable to the scenario used – it missed its Tier 1 capital ratio target of 6%. Yet the bank had already taken steps to fix this shortcoming by issuing $5.1 billion worth of new shares to boost its Tier 1 ratio.
In the 2015 edition, the Bank of England tested the banks on the basis of a common equity Tier 1 ratio of 4.5% of risk-weighted assets, plus a 3% additional leverage ratio threshold. The participants included Barclays, HSBC, Lloyds, Nationwide, and Santander UK, RBS, and Standard Chartered.
Good as the results may seem, the Bank of England did not miss the chance to tell the country's biggest lenders to set aside more capital. This advice is linked to a new initiative undertaken by the central bank aiming to strengthen banks' resilience further against cyclical events, hence the name "countercyclical capital buffer." The CCB is one more element of the Basel III framework for comprehensive global banking reform in the wake of the 2008 financial crisis.


text:
Muzzling The Press In Sri Lanka
The following are the words of Lasantha Wickrematunga, a Sri Lankan newspaper editor who for years exposed official corruption and voiced criticism of the Sri Lankan government:
"No other profession calls on its practitioners to lay down their lives for their art save the armed forces and, in Sri Lanka, journalism. In the course of the past few years, the independent media have increasingly come under attack. Electronic and print-media institutions have been burnt, bombed, sealed and coerced. Countless journalists have been harassed, threatened and killed. It has been my honor to belong to all those categories and now especially the last."
Mr. Wickrematunga was assassinated by masked gunmen as he drove to work on the morning of January 8th. The paragraph quoted earlier is from his final column, which was published 3 days after his death.
Media groups say that Sri Lanka is one of the most difficult countries in the world in which to report. According to the 2008 report by Reporters Without Borders, a Paris-based international non-governmental organization that advocates freedom of the press, Sri Lanka was ranked 165th out of 173 countries in the Reporters Without Borders 2008 press freedom index. This was the lowest ranking of any democratic country.
In both the rebel and government controlled areas, a number of journalists have been intimidated, threatened, harassed, kidnapped or killed. The premises of some media outlets, particularly those perceived as critical of the government, have been attacked and destroyed. And although the government of Sri Lanka has condemned such actions and promised to investigate the crimes, little or no investigation has been done and no one has been brought to justice.
"The United States strongly condemns the murder of Lasantha Wickrematunga," Deputy Spokesman of the Department of State Robert Wood said in a written statement. "The murder ... is a shocking blow to independent media in Sri Lanka. This is ... just the latest in a string of incidents against journalists. The United States is deeply concerned that such attacks undermine efforts to build a united and democratic Sri Lanka where the rights of all people are protected. We call on the Government of Sri Lanka to investigate these attacks expeditiously, bring the perpetrators to justice, and take all possible measures to protect freedom of expression for members of the press."


Gated Luxury Condo. Large 3 bedroom 2 bath 2154 square feet. Chefs kitchen with granite counters and designer cabinets. Exceptional detail. 10 foot trey ceilings, arches, pillars, and crown molding. Enjoy the open floor plan. French doors open to a covered screened patio with views of a wide open pond and stately oaks. This unit includes a garage with entrance to the building. Coded security entrance to the building as well as the condo. Exclusive club house with pool, spa, kitchen, fitness center, and entertainment room. This is a great location close to fine restaurants and shopping and minutes to the Atlantic Ocean. Come and take a look at luxury living. 1/12 Gated Luxury Condo. Large 3 bedroom 2 bath 2154 square feet. Chef's kitchen with granite counters and designer cabinets. Exceptional detail: 10 foot trey ceilings arches pillars and crown molding. Enjoy the open floor plan. French doors open to a covered screened patio with views of a wide open pond and stately oaks. this unit includes a garage with entrance to the building. Coded security entrance to the building as well as the condo. Exclusive club house with pool spa kitchen fitness center and entertainment room. this is a great location close to fine restaurants and shopping and minutes to the Atlantic Ocean. come and take a look a luxury living.


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text:
\section{Introduction}

For the duration of this introduction, let $\Gamma$ be a 4-regular graph and let $G=\Gamma - v$ for some $v\in V(\Gamma)$.  The graph $\Gamma$ can be uniquely reconstructed from~$G$; we call $\Gamma$ the \emph{completion} of $G$ and $G$ a \emph{decompletion} of $\Gamma$.  We can think of~$G$ as a Feynman diagram, specifically as a 4-point graph in scalar $\phi^4$ theory.  It is natural, then, to ask about the Feynman integral of $G$.  Simplifying by ignoring all physical parameters and considering only graphs with no subdivergences, it makes sense to define a number known as the \emph{period} of~$G$ \cite{bek, Brbig, Mar, Sphi4} (defined more thoroughly in Section~\ref{relationtofeynman}).  This period is the residue of the Feynman integral under a variety of regularizations and so gives a largely renormalization scheme independent aspect of the Feynman integral.

There are a number of graph theoretic operations which are known to preserve the period.  If $G$ is planar, then $G$ and its planar dual have the same period \cite{bkphi4}; this is a consequence of taking a Fourier transform of the Feynman integral.  If $G$ and another graph $G'$ have isomorphic completions, then $G$ and $G'$ have the same period \cite{bkphi4, Sphi4}.  If $\Gamma$ and $\Gamma'$ relate by the Schnetz twist (see Figure~\ref{twist}) and $G$ and $G'$ are decompletions of $\Gamma$ and $\Gamma'$ respectively, then $G$ and $G'$ have the same period \cite{Sphi4}.  Furthermore, if $\Gamma$ has a 3-vertex cut, then the period of $G$ is a product of the periods of two particular minors \cite{bkphi4, Sphi4}.

In view of this, we are interested in graph theoretic properties or invariants which are preserved by planar duality, completion followed by decompletion, and Schnetz twist.  The $c_2$ invariant is an arithmetic graph invariant defined by counting points on the Kirchhoff polynomial which is conjectured to have these properties (see \cite{BrS}); duality is proven in \cite{Dor} for graphs that meet a specific subgraph condition, and it is established in \cite{BrS} that if decompletion is true then the 3-cut condition follows.  We are not aware of any other nontrivial graph invariants thought to satisfy these properties.

In this paper we introduce the following graph invariant; let $G$ be a graph with $|E(G)| = k(|V(G)|-1)$ for some integer $k$. Construct a signed incidence matrix from $G$ and delete a single arbitrary row. From this, construct a block matrix by stacking the modified incidence matrix $k$ times. Up to sign, we call the permanent of this matrix modulo $k+1$ the \emph{graph permanent}. Extending the concept of completion and decompletion to arbitrary regular graphs (Definition \ref{decompdef}), we prove the following:

\begin{theorem*} Suppose $\Gamma$ and $\Gamma '$ are connected $2k$-regular graphs.
\begin{itemize}
\item Any two decompletions of $\Gamma$ have equal graph permanent (Theorem~\ref{invariance}).
\item If $\Gamma$ and $\Gamma'$ differ by a Schnetz twist, any pair of decompletions of $\Gamma$ and $\Gamma'$ will have equal graph permanents  (Proposition~\ref{schnetz}).
\end{itemize}
Further, let $G$ be a graph such that $|E(G)| = 2(|V(G)|-1)$ and $G^*$ its planar dual.
\begin{itemize}
\item The graph permanents of $G$ and $G^*$ are equal (Proposition~\ref{dual}).
\end{itemize}
Finally, suppose $\Gamma$ is a $4$-regular graph.
\begin{itemize}
\item If $\Gamma$ has a $3$-vertex cut, then the graph permanent of any decompletion of $\Gamma$ is the product of the graph permanents of two particular minors (Corollary~\ref{3cut}).
\end{itemize} \end{theorem*}

\noindent Further we prove a product property when $K$ has a 4-edge cut (Theorem~\ref{4edgecut}) that corresponds to the case of subdivergences in the Feynman graph.

 Our invariant is well defined on a wider class of graphs than decompletions of 4-regular graphs and relates naturally to flow questions on graphs.  Indeed, when this invariant is nonzero, it implies, by way of the Alon-Tarsi polynomial technique (\cite{AlTar}), that the graph in question has a modular $k$ orientation (or equivalently a $\mathbb{Z}_k$ flow using only the values $\pm 1$).  Furthermore, in the key case of 4-regular graphs, the polynomial is closely related to the Feynman integrand.  This will be explained in detail in Section~\ref{flow_sec}

Completion invariance for $2k$-regular graphs can be distilled into a curious identity for graphs (Theorem~\ref{agreementidentity}), and we close the introduction with a  description of this.   Let $G$ be $2k$-regular, and fix an orientation of the edges of $G$ which we call the \emph{reference orientation}.  Now define an arbitrary orientation of $G$ to be \emph{odd} \emph{(even)} if the number of edges for which this orientation disagrees with the reference orientation is odd (even).  Let $s,t$ be distinct vertices of $G$.  Any orientation for which $\mathrm{deg}^+(s) = 2k = \mathrm{deg}^-(t)$ and $\mathrm{deg}^+(v) = k = \mathrm{deg}^-(v)$ for every $v \in V(G) \setminus \{s,t\}$ will be called an $s$-\emph{to}-$t$ orientation.  Let $E_{s, t}$ $(O_{s,  t})$ denote the number of even (odd) $s$-to-$t$ orientations.  With this terminology, we can state this new identity as follows.

\begin{theorem*}
If $G$ is a $2k$-regular graph, and $(s,t)$, $(s',t')$ are pairs of distinct vertices of $G$, then
\[ E_{s , t} - O_{s , t} \equiv    E_{s' , t'} - O_{s' , t'}  \pmod {k+1}.\]
\end{theorem*}

\section{A block matrix construction.}

Throughout this paper, all graphs are assumed to be connected and loop-free. We allow parallel edges.

\begin{definition} Let $G$ be a graph. Arbitrarily apply directions to the edges in $G$, and  let $M^*$ be the incidence matrix associated with this digraph; columns indexed by edges and rows by vertices. Select a vertex $w$ in $V(G)$, and delete the row indexed by $w$ in $M^*$. Call this new matrix $M$. Let $k$ be a positive integer. Define a \emph{$k$-duplicated signed incidence matrix} (herein \emph{$k$DSI matrix}) of $G$ to be the block matrix $$\left. \left[ \begin{array}{c} M\\ \hline M \\ \hline \vdots \\ \hline M \end{array} \right] \right\}\text{$k$ times}.$$ Further, we call $w$ the \emph{special vertex} in the construction of this $k$DSI matrix. \end{definition}

Our interests lie in graphs $G$ that have $|E(G)| = k(|V(G)|-1)$ for some integer $k$, as this results in a square $k$DSI matrix and allows for permanent calculations.

\begin{definition} Let $A=(a_{i,j})$ be an $n$-by-$n$ matrix. The \emph{permanent} of $A$ is $$\text{Perm}(A) = \sum_{\sigma \in S_n} \prod_{i=1}^n a_{i,\sigma(i)},$$ where the sum is over all elements of the symmetric group $S_n$. \end{definition}

If a particular $\sigma \in S_n$ is such that $ \prod_{i=1}^n a_{i,\sigma(i)} \neq 0$, we will say that it \emph{contributes} to the permanent. We may alternately define a contribution from an appropriate selection of non-zero elements in the matrix.

From the definition of the permanent, we see that it is the determinant with signs not taken into account. In fact, the permanent also can be computed using cofactor expansion, similar to the determinant. As $1 \equiv -1 \pmod{2}$, any square matrix $M$ has $\text{Perm}(M) \equiv \det (M) \pmod{2}$, which suggests that the permanent may have some interesting properties modulo integers.

\begin{rem} \label{rowops} From the definition of the permanent, it is clear that we may interchange two rows or columns without affecting the permanent. Further, multiplying a row or column by a constant results in the permanent being multiplied by that constant. \end{rem}

What happens when a multiple of one row is added to another is less clear, and in general not well behaved. However, there is greater control with the $k$DSI matrix modulo $k+1$, which will be examined in Lemma~\ref{redlem} and Corollary~\ref{reduction}.

\begin{lem}\label{comrow} For an $n \times n$ matrix $M$, if there is a set $\{a_1,a_2,...,a_m\}$ such that rows $r_{a_1}, r_{a_2}, ..., r_{a_m}$ are equal, there is a factor of $m!$ in the permanent of $M$. \end{lem}

\begin{proof}We may write \begin{align*} \text{Perm}(M) &= \sum_{\sigma \in S_n} \prod_{i=1}^n a_{i,\sigma(i)} \\ &=m! \sum_{\sigma \in S_n^*} \prod_{i=1}^n a_{i,\sigma(i)}, \end{align*} where $S_n^*$ is the set of elements of the symmetric group such that $\sigma(a_1)<\sigma(a_2)< \cdots < \sigma(a_m)$, and the $m!$ term allows for further permutations of these elements.   \end{proof}

\begin{lem}\label{redlem} Let $M$ be a matrix and $r_i$ and $r_j$ rows of $M$, $r_i \neq r_j$ as vectors. Suppose there are $k$ copies of $r_j$ in $M$.  Let $M'$ be a matrix derived from $M$ by adding a constant integer multiple of $r_j$ to $r_i$. Then $\text{Perm}(M) \equiv \text{Perm}(M') \pmod {k+1}$. \end{lem}

\begin{proof}   Suppose that $$M = (m_{x,y}) = \left[ \begin{array}{c} r_1 \\ r_2 \\ \vdots  \end{array} \right], \text{ } M' = ({m'}_{x,y}) =  \left[ \begin{array}{c} r_1 \\ \vdots \\ r_i +cr_j  \\ \vdots  \end{array} \right].$$ Define $N$ as the matrix $M$ with row $i$ removed. We will use $N_t$ to denote the matrix $N$ with column $t$ removed. By cofactor expansion along the $i^\text{th}$ row, \begin{align*} \text{Perm}\left(M\right) &= \sum_{t=1}^{kn} m_{i,t}  \text{Perm}\left(N_t\right),\\ \text{Perm}\left(M'\right) &= \sum_{t=1}^{kn} {m'}_{i,t} \text{Perm}(N_t) \\ &=  \sum_{t=1}^{kn}(m_{i,t} + cm_{j,t}) \text{Perm}(N_t) \\ &=  \text{Perm}\left(M\right) + c  \text{Perm} \left[ \begin{array}{c} r_1 \\ \vdots \\ r_{i-1} \\ r_j \\ r_{i+1} \\ \vdots  \end{array} \right]  .\end{align*} As this last matrix has $k+1$ copies of row $r_j$, it has permanent congruent to zero modulo $k+1$ by Lemma~\ref{comrow}.\end{proof}

Throughout this paper, we will consider only matrix operations performed simultaneously in all blocks. The following corollary follows immediately from Lemma \ref{redlem}.

\begin{corollary}\label{reduction} Suppose $M$ is a block matrix made of $k$ identical blocks stacked, and $r_i$ and $r_j$ are rows of $M$ in a common block, $i \neq j$. Let $M'$ be a matrix derived from $M$ by adding a constant integer multiple of $r_j$ to $r_i$ in each block. Then $\text{Perm}(M) \equiv \text{Perm}(M') \pmod {k+1}$. \end{corollary}

\begin{prop} \label{specialchoice} The choice of special vertex only affects the overall sign of the permanent modulo $k+1$  in a $k$DSI matrix. If $k$ is odd, changing the special vertex results in a sign change. If $k$ is even, changing special vertex has no effect on the permanent. \end{prop}

\begin{proof} For signed incidence matrix $M^*$, let $r_1, ... , r_n$ be the rows associated to vertices $1,...,n$ in the original graph $G$, and suppose vertex $i$ is the special vertex, $i \in \{1,...,n\}$. Then, $$r_i = -(r_1 + r_2 + \cdots + r_{i-1} + r_{i+1} + \cdots + r_n),$$ a property of the signed incidence matrix. For all blocks in $M$, we may therefore turn row $r_j$, $i \neq j$, into row $r_i$ using the above equation. By Corollary~\ref{reduction}, only the multiplication of a row in each block by $-1$ affects the permanent modulo $k+1$, flipping the overall sign once for each block. This produces the $k$DSI matrix where $j$ was the special vertex; the permanent is unaffected if there is an even number of blocks, and multiplied by $-1$ if there is an odd number of blocks.\end{proof}

\begin{corollary} \label{oddcase} Any $k$DSI matrix from a graph $G$ with $|V(G)|>2$ and odd $k$ has permanent zero modulo $k+1$. \end{corollary}

\begin{proof} Consider $k$DSI matrix $M$ from graph $G$ where $v_1$ is the special vertex, and suppose $v_2,v_3 \in V(G) \setminus v_1$. Let $M_{v \rightarrow w}$ denote the matrix where the special vertex has been changed from $v$ to $w$, as in Proposition \ref{specialchoice}. If $k$ is odd, then, \begin{align*} \text{Perm}(M) &= -\text{Perm}(M_{v_1 \rightarrow v_2})= \text{Perm}\left( (M_{v_1 \rightarrow v_2})_{v_2 \rightarrow v_3} \right) \\ \text{Perm}(M) &= -\text{Perm}(M_{v_1 \rightarrow v_3}).\end{align*}  As $\left( (M_{v_1 \rightarrow v_2})_{v_2 \rightarrow v_3} \right)$ and $M_{v_1 \rightarrow v_3}$ differ only by interchanged rows, $$\text{Perm}\left( (M_{v_1 \rightarrow v_2})_{v_2 \rightarrow v_3} \right) = \text{Perm}(M_{v_1 \rightarrow v_3}),$$ and it follows that $\text{Perm}(M) = 0$.   \end{proof}

To understand why the restriction in the previous corollary that $|V(G)|>2$ is necessary, we require the following theorem.

\begin{wilson} A number $p$ is prime if and only if $(p-1)! \equiv -1\pmod{p}$. For a composite number $n>4$, $(n-1)! \equiv 0 \pmod{n}$. \end{wilson}

\begin{theorem}\label{specialinvariant} Suppose $G$ is a graph such that $|E(G)| = k(|V(G)|-1)$. With a fixed orientation to the edges, the permanent of the $k$DSI matrix of $G$ is invariant under choice of special vertex modulo $k+1$. \end{theorem}

\begin{proof} By Proposition \ref{specialchoice} and Corollary \ref{oddcase}, the permanent is invariant modulo $k+1$ under choice of special vertex if $k$ is even or $k$ is odd and $|V(G)|>2$. As a graph with only a single vertex creates an empty matrix, it remains to be shown that the permanent is invariant if $|V(G)| = 2$ and $k+1$ is even.

 Suppose then that $G$ is a graph with two vertices and $k$ parallel edges. Applying an arbitrary orientation to the edges, the $k$DSI matrix of this graph has entirely nonzero entries, each column either all $1$ or all $-1$. Thus, it trivially has permanent $\pm k!$. As $k+1$ is even, it follows from Wilson's Theorem that $k! \equiv 0 \pmod{k+1}$ if $k+1 > 4$. Hence, we need only consider $k \in \{1,3\}$. As $\pm 1! \equiv 1 \pmod{2}$ and $\pm 3! \equiv 2 \pmod{4}$, the permanent is invariant under any choice made in constructing the $k$DSI matrix, as desired. \end{proof}

\begin{definition} Let $G$ be a graph with $|E(G)| = k(|V(G)|-1)$ for some integer $k$. Let $M$ be a $k$DSI matrix of $G$. The \emph{graph permanent} of $G$ is $\pm \text{Perm}(M) \pmod {k+1}$, which we will by convention take as a residue in $\left[0,\frac{k+1}{2}\right]$. \end{definition}

The fact that we must choose $\pm \text{Perm}(M)$ is the result of variability in the choice of underlying edge orientation. In changing the direction of an edge in the orientation, a column of the matrix is multiplied by negative one, and hence the permanent changes sign.

\section{Graphic Interpretation}

Consider a graph $G$ with $|E(G)| = k(|V(G)|-1)$ and an associated $k$DSI matrix $M$ with special vertex $v \in V(G)$. For each contribution to the permanent, precisely one non-zero value is selected from each row and similarly from each column. Fix such a contribution. Given the block structure that is used to create matrix $M$, we may associate each block with a unique colour. Then, each edge is selected once, and each non-special vertex $k$ times. Assign colour $c$ to an edge if the contribution uses a value in the associated column that is in the $c^\text{th}$ block. For each coloured edge, assign a tag on the edge close to the vertex that uses that edge in $M$. Such a contribution and colouring scheme on $K_4$ is given in Figure~\ref{markedmatrix}.

\begin{figure}[h]
  \centering
      \includegraphics[scale=.95]{markedmatrix}
  \caption{A 2DSI matrix of $K_4$ with special vertex $0$ and a contribution to the permanent.}
\label{markedmatrix}
\end{figure}

Note from this construction that the special vertex cannot receive a tag. All other vertices must receive precisely $k$ tags, one on an edge of each colour. In fact, an arrangement of edge tags and colours on $G$ that assigns each non-special vertex $k$ tags - one on an edge of each colour - and no tags to the special vertex can immediately be turned into a selection of non-zero entries in the $k$DSI matrix.

\begin{rem} \label{tagcolourbij} There is a bijection between these assignments of tags and colours and the contributions to the permanent. \end{rem}

We will use this bijection in a number of proofs as a way of considering these contributions as a property of the graph itself.

\begin{rem} \label{ops} There are two operations on the tags and colours of the graph that produce other contributions to the permanent. The first is, for any non-special vertex, we may permute the colours of the $k$ edges that have a tag at that vertex. There are $k!$ ways to perform this permutation. The second, at its most intuitive, is that we may switch which vertex receives the tag on every edge in a cycle where all edges have the same colour. Since the colours of edges that have tags at a common vertex are all interchangeable, though, this may be restated as reversing the directions of the tags of a cycle $C$ in $G$ such that, in $C$, each vertex receives one tag. \end{rem}

Suppose then that the graph $G$ has $n$ vertices. From the colour permuting operation, each valid configuration of tags produces $(k!)^{n-1}$ valid colourings. As the tags determine the position in the original matrix that is selected, choice of edge colours does not affect the value of the contribution. As such, $(k!)^{n-1}$ is a factor in the permanent, and the colours do not matter.

\begin{prop} \label{prime} For non-prime $k+1$, the permanent of any square $k$DSI matrix associated to a graph $G$ with $|V(G)|>2$ is zero modulo $k+1$. \end{prop}

\begin{proof} By Corollary~\ref{oddcase}, we may assume that $k+1$ is odd. If $k+1$ is composite and odd it must be greater than four, and by Wilson's Theorem $k! \equiv 0 \pmod{k+1}$. As $k!$ is a factor in the permanent, the result follows. \end{proof}

It is interesting to note, then, that the value of the permanent for a $k$DSI matrix is determined completely by the tag assignments.

\begin{prop} For a $k$DSI matrix, we may produce all contributions to the permanent from a single contribution and the two operations stated in Remark~\ref{ops}. \end{prop}

\begin{proof} By Remark~\ref{ops}, it is sufficient to show that all valid tag placements can be obtained. Starting from a fixed orientation, then, consider getting to another by switching which vertex receives the tag on a set of edges. Since each edge must receive $k$ tags, it is immediate that this selection of edges must be a collection of cycles. \end{proof}

\section{Invariance Under Period Preserving Operations}

We now begin to explore the invariance of the graph permanent under the graph operations that are known to preserve the Feynman period, as mentioned in the introduction. Since we are always restricted to graphs $G$ such that $|E(G)| = k(|V(G)|-1)$ for some integer $k$, we split this section into three subsections, corresponding to specific graph classes that are subsets of this larger class; decompleted $2k$-regular graphs, graphs where $|E(G)| = 2(|V(G)| -1)$, and finally the intersection of the previous two classes, decompleted $4$-regular graphs.

\subsection{Decompleted $2k$-regular graphs}

The following definition is generalized from the introduction.

 \begin{definition}\label{decompdef} For a regular graph $\Gamma$ and any $v \in V(\Gamma)$, the graph $G= \Gamma - v$ is a \emph{decompletion} of $\Gamma$, and $\Gamma$ is the unique \emph{completion} of $G$. \end{definition}

\begin{theorem} \label{bijection} Let $\Gamma$ be a $2k$-regular graph. For vertices $v,w \in V(\Gamma)$, let $M$ be the $k$DSI matrix of $\Gamma - v$ with respect to special vertex $w$, and $N$ the matrix from opposite deletion and special vertex. Then, $\text{Perm}(M) = \pm \text{Perm}(N)$. If $k$ is even and an edge orientation based on an Eulerian circuit in $\Gamma$ is used to construct $M$ and $N$, then $\text{Perm}(M) = \text{Perm}(N)$. \end{theorem}

\begin{proof} By Remarks~\ref{tagcolourbij} and \ref{ops}, it suffices to find a bijection between contributions to the two permanents that is consistent in either maintaining or changing signs. Hence, consider an extension of the taggings from the decompleted graphs to $\Gamma$ by assuming the decompleted vertex received the tags of all edges incident to it. By this extension, the special vertex receives no tags, the decompletion vertex receives $2k$, and all other vertices receive $k$ tags. The bijection that arises naturally, then, is to switch which vertex receives a tag on every edge. By construction, this switches between extensions of contributions to $M$ and $N$.

We now consider the signs of the contributions. Consider a fixed orientation in $\Gamma$ from an Eulerian circuit, and use this to define the underlying orientation for $\Gamma-v$ and $\Gamma-w$. Switching the position of a tag on a single edge changes the sign of an entry of the contribution. Hence, each edge common to both $\Gamma-v$ and $\Gamma-w$ causes a sign change. Suppose now that there are $l$ edges  between $v$ and $w$ in $\Gamma$.  If $l$ is even, then an even number of changes are ignored in deleting $v$ or $w$, and hence this causes no additional sign change. If $l$ is odd, then $w$ in $\Gamma-v$ and $v$ in  $\Gamma-w$ will be incident with opposite parity number of tags that disagree with the underlying orientation, causing an additional sign change. In all cases, it was the structure of the graph, not that actual position of the tags, that determined sign changes between the two permanents. Hence, this bijection either preserves the signs of all contributions or changes all signs. Therefore, $\text{Perm}(M) = \pm \text{Perm}(N)$, as desired.

If, in addition, $k$ is even, $|E(\Gamma)| = k|V(\Gamma)|$ is even. Again, supposing that edge $\{u,v\}$ occurs $l$ times, there are $k|V(\Gamma)| - 4k +l$ edges not incident with either $v$ or $w$ in $\Gamma$. No matter the parity of $l$, then, an even number of sign changes made are made in the bijection, and overall sign is preserved. \end{proof}

\begin{theorem}\label{invariance} For a fixed $2k$-regular graph $\Gamma$, the graph permanents of all possible decompletions of $\Gamma$ are equal. \end{theorem}

\begin{proof} By Theorem~\ref{specialinvariant}, the choice of special vertex does not affect the permanent up to sign modulo $k+1$. From Theorem~\ref{bijection} and the fact that changing the direction of an edge in the underlying orientation only affect the overall sign of the permanent, the choice of deleted and special vertex may be interchanged without affecting the permanent up to sign. Thus, any two vertices are interchangeable with any other two as the deleted and special vertex, only potentially changing overall sign modulo $k+1$. As the graph permanent from $k$DSI matrix $M$ is $\pm \text{Perm}(M) \pmod{k+1}$, this completes the proof. \end{proof}

The extension of edge taggings from decompleted graph $G$ to $2k$-regular completion $\Gamma$, as seen in Theorem~\ref{bijection} provides the framework for a more graph theoretic look at the taggings. As stated in the introduction, fix a \emph{reference orientation} on the edges of $\Gamma$. We say that an arbitrary orientation is \emph{odd} (\emph{even}) if it disagrees with the reference orientation on an odd (even) number of edges. For distinct vertices $s,t \in V(\Gamma)$, we call any orientation that has $\mathrm{deg}^+(s) = 2k = \mathrm{deg}^-(t)$ and $\mathrm{deg}^+(v) = k = \mathrm{deg}^-(v)$ for all $v \in V(\Gamma) \setminus \{s,t\}$ a \emph{$s$-to-$t$ orientation}. Let $E_{s,t}$ ($O_{s,t}$) denote the number of even (odd) $s$-to-$t$ orientations. The following theorem, mentioned in the introduction, can now be proved.

\begin{theorem}\label{agreementidentity} Let $\Gamma$ be a $2k$-regular graph and $(s,t)$, $(s',t')$ pairs of distinct vertices. Then, $$E_{s,t} - O_{s,t} \equiv E_{s',t'} - O_{s',t'}\pmod{k+1}.$$ \end{theorem}

\begin{proof} Consider first a contribution to the permanent of a $k$DSI matrix $M$ from the decompletion of $\Gamma$ created by deleting vertex $t$ and making $s$ special. With each element selection, we assign a tag that either agrees or disagrees with the reference orientation. If it agrees, we have selected a $1$ from $M$. Otherwise, we have selected a $-1$. As in Theorem~\ref{bijection}, we extend the tagging of our decompleted graph to $2k$-regular graph to create an $s$-to-$t$ orientation where $s$ is the special vertex and $t$ is the decompleted vertex. Assuming an underlying edge orientation from an Eulerian circuit, a $2k$-regular graph will disagree with an $s$-to-$t$ orientation on precisely $k$ edges incident to $t$. Hence $$\text{Perm}(M) = (-1)^k \left( E_{s,t} - O_{s,t}\right).$$

It follows from Theorem~\ref{bijection} that if $k$ is even, $E_{s,t} - O_{s,t} = E_{t,s} - O_{t,s}$. From this construction and Theorem~\ref{specialinvariant}, $E_{s,t} - O_{s,t} \equiv E_{s',t} - O_{s',t} \pmod{k+1}$ for any $s' \in V(\Gamma) \setminus t$. Hence, using an Eulerian circuit as an underlying orientation, $E_{s,t} - O_{s,t} \equiv E_{s',t'} - O_{s',t'}\pmod{k+1}.$

For odd $k$, it was noted in Corollary~\ref{oddcase} and the proof of Theorem~\ref{specialinvariant} that if $|V(\Gamma)| > 3$, or $|V(\Gamma)| = 3$ and $k>3$, the permanent of any decompletion is equal to zero modulo $k+1$, which completes the proof in this case. Finally, by Wilson's Theorem again, if $|V(\Gamma)| = 3$ and $k \in \{1,3\}$, the permanent of any $k$DSI matrix is invariant under sign.

We complete the proof by noting that in turning an Eulerian circuit into any other orientation, each change in the direction of an edge multiplies both sides of the equation by negative one. \end{proof}

\begin{prop}\label{schnetz} Consider two $2k$-regular graphs that differ by a Schnetz twist, seen in Figure~\ref{twist}. Decompletions of these graphs have equal graph permanents. \end{prop}

\begin{figure}[h]
  \centering
      \includegraphics[scale=0.80]{twist}
  \caption{The Schnetz twist.}
\label{twist}
\end{figure}

\begin{proof} As the graph permanent is invariant under choice of special and decompleted vertex, in each graph choose vertex $v_4$ for deletion and $v_3$ as the special vertex. Using Remark~\ref{tagcolourbij} and as in the proof of Theorem~\ref{bijection}, we may capture all contributions to the permanent of the decompleted graph by treating the deleted vertex as a vertex that has received all tags from incident edges, and the special vertex as a vertex that has received none. This will allow for a natural bijection, to be defined below, between contributions after the twisting operation. As both graphs are assumed to be $2k$-regular, it must be the case that vertices $v_3$ and $v_4$ have an equal number of edges in the left side of the graph, say $d_3$, and an equal number of edges in the right side of the graph, which would then be $2k-d_3$. We may say the same for vertices $v_1$ and $v_2$, and denote these $d_1$ and $2k-d_1$.

Suppose then that the left portion of the graph has $n$ vertices contained properly inside. Then, there are $kn + d_1 + d_3$ edges on this side, and hence an equal number of tags. If vertex $v_1$ receives $m$ tags, then vertex $v_2$ must receive $$(kn + d_1 + d_3) - (kn + d_3 + m) = d_1 - m,$$ accounting for all tags on this side. Similarly, since all vertices receive $k$ tags, $v_1$ must receive $k-m$ tags on the right, and $v_2$ must receive $k-d_1 + m$.

After the twist, then, consider switching the sides of the tags on the right side only. Again, the deleted vertex receives $2k$ tags and the special vertex receives none. Further, vertex $v_1$ receives $m + ((2k-d_1) -(k+m-d_1)) = k$ tags, and similarly $v_2$ receives $(d_1-m)+((2k-d_1) - (k-m)) = k$ tags. Hence, this is a contribution to the permanent. This operation is clearly bijective and at most changes the overall sign of all contributions, and hence the graph permanents must be equal. \end{proof}

\subsection{Graphs $G$ where $|E(G)| = 2(|V(G)| -1)$}

\begin{prop}\label{dual} For a graph $G$ where $|E(G)| = 2(|V(G)| -1)$ and planar dual $G^*$, the graph permanents for $G$ and $G^*$ are equal. \end{prop}

\begin{proof} By Remark~\ref{rowops} and Corollary~\ref{reduction} we may row perform row operations without affecting the graph permanent. After row reduction, we may write the 2DSI matrix $$M_G = \left[ \begin{array}{c|c} I & A \\ \hline I & A \end{array} \right].$$ The permanent of this matrix is $2^{|A|}\text{Perm}(A)$. Dually, we have 2DSI matrix $$M_{G^*} = \left[ \begin{array}{c|c} -A^t & I \\ \hline -A^t & I \end{array} \right],$$ with permanent $(-2)^{|A|} \text{Perm} (A)$.\end{proof}

\begin{lem}\label{pigeon} If a square block matrix has a non-square block that is the only block containing non-zero entries in its particular row and column set, the permanent is zero.  \end{lem}

\begin{proof} This follows immediately from the pigeonhole principle and the definition of the permanent. \end{proof}

\begin{theorem} \label{2vertexcut} Consider the graph $G$ and two minors $G_1$ and $G_2$ seen in Figure~\ref{2cut}. If for $G^* \in \{G,G_1,G_2\}$, $2|V(G^*)| = |E(G^*)|-2$, then the graph permanent of $G$ is equal to the product of the graph permanents of $G_1$ and $G_2$. \end{theorem}

\begin{figure}[h]
  \centering
      \includegraphics[scale=1.20]{2cut}
  \caption{Operation on a two vertex cut.}
\label{2cut}
\end{figure}

\begin{proof} Let $G$ be the original graph, and $G_1$ and $G_2$ the minors as seen in Figure~\ref{2cut}. Maintain a constant edge orientation across all graphs, with the edge $\{v_1,v_2\}$ oriented towards $v_2$. Choosing $v_2$ as the special vertex for all graphs, the respective $2$DSI matrices are
 $$M_G = \left[ \begin{array}{ccc|ccc}
& \bf{\bf{G_1}} & & & \bf{0}& \\
&C& && D& \\
 & \bf{0} & & & \bf{\bf{G_2}} & \\ \hline
& \bf{\bf{G_1}} & & & \bf{0}& \\
&C& &&D& \\
 & \bf{0} & & & \bf{\bf{G_2}} & \\  \end{array} \right], $$

 $$M_{G_1} = \left[ \begin{array}{ccc|c}
& \bf{G_1}& & \bf{0} \\
&C& & 1 \\ \hline
& \bf{\bf{G_1}} & & \bf{0} \\
&C& & 1
 \end{array} \right], $$ and

 $$M_{G_2} = \left[ \begin{array}{c|ccc}
\bf{0} && \bf{\bf{G_2}} &  \\
1 &&D&  \\ \hline
\bf{0} && \bf{\bf{G_2}} &  \\
1&&D&
 \end{array} \right], $$ where $(C|D)$ is the row corresponding to vertex $v_1$.

Using cofactor expansion along the last column, we get $$\text{Perm}(M_{G_1}) =  \text{Perm}\left[ \begin{array}{c} \bf{G_1} \\  \bf{G_1} \\  C \end{array} \right] + \text{Perm}\left[ \begin{array}{c} \bf{G_1} \\  C \\  \bf{G_1} \end{array} \right] =  2  \text{Perm}\left[ \begin{array}{c} \bf{G_1} \\  \bf{G_1} \\  C \end{array} \right] .$$ Similarly, $$ \text{Perm}(M_{G_2}) = 2  \text{Perm} \left[ \begin{array}{c} \bf{G_2} \\ \bf{G_2} \\ D  \end{array}  \right].$$  Again, use $N_r$ to denote matrix $N$ with column $r$ deleted. Letting $C = (c_1,c_2,...)$ and $D=(d_1,d_2,...)$, we use cofactor expansion along the bottom row to get, \begin{align*}  \text{Perm}(M_{G_1}) &= 2 \sum_i c_i  \text{Perm} \left[ \begin{array}{c} \bf{G_1} \\ \bf{G_1} \end{array} \right]_i, \\ \text{Perm}(M_{G_2}) &= 2 \sum_i d_i  \text{Perm} \left[ \begin{array}{c} \bf{G_2} \\ \bf{G_2} \end{array} \right]_i. \end{align*}

We now compute $\text{Perm}(M_G)$ by cofactor expansion along the two rows $(C|D)$. Suppose the block $G_1$ contains $n$ columns and block $G_2$ contains $m$ columns. Blocks will be square only if in the expansion we have deleted one column in the first $n$ columns and one in the last $m$. By Lemma~\ref{pigeon}, then, \begin{align*} \text{Perm}(M_G) &= 2 \sum_{\substack{1 \leq i \leq n \\ n < j \leq n+m}} c_i d_j \text{Perm} \left[ \begin{array}{c|c} \begin{array}{c} \bf{G_1} \\ \bf{G_1} \end{array} & \bf{0}\\ \hline \bf{0} & \begin{array}{c} \bf{G_2} \\ \bf{G_2} \end{array} \end{array} \right]_{i,j} \\ &= 2 \sum_{\substack{ 1\leq i \leq n \\ 1 \leq j \leq m}} c_i d_j \text{Perm} \left[ \begin{array}{c} \bf{G_1} \\ \bf{G_1} \end{array} \right]_i   \text{Perm} \left[ \begin{array}{c} \bf{G_2} \\ \bf{G_2} \end{array} \right]_j \\ & \equiv - \text{Perm}(M_{G_1}) \text{Perm}(M_{G_2}) \pmod{3}.\end{align*} Hence, the graph permanent of $G$ is equal to the product of graph permanents of $G_1$ and $G_2$.
\end{proof}

\subsection{Decompleted $4$-regular graphs}

\begin{corollary}\label{3cut}[to Theorem~\ref{2vertexcut}] With 4-regular graphs $\Gamma$, $\Gamma_1$, and $\Gamma_2$ as in Figure~\ref{3vertexcut}, the graph permanents of any decompletion of $\Gamma$ is equal to the product of graph permanents of decompletions of $\Gamma_1$ and $\Gamma_2$. \end{corollary}

\begin{figure}[h]
  \centering
      \includegraphics[scale=1.10]{3cut}
  \caption{The completed graph with three vertex cut, corresponding to the completion of the graphs in Figure~\ref{2cut}.}
\label{3vertexcut}
\end{figure}

\begin{proof} By Theorem~\ref{invariance}, graph permanent of a decompleted 4-regular graphs is invariant under choice of decompletion vertex. Hence, decompleting each at the vertex labeled with a hollow circle, we produce graphs as in Theorem~\ref{2vertexcut}, and the result follows.  \end{proof}

\begin{theorem}\label{4edgecut} Let graphs $\Gamma$, $\Gamma_1$, and $\Gamma_2$ in Figure~\ref{4edgecutpic} be 4-regular. Construct $2$DSI matrices by deleting the vertices labeled with a hollow circle and making the vertices labeled with a square the special vertex, all distinct from vertices incident with the edges in the 4-edge cut. The permanent of the $2$DSI matrix associated to $\Gamma$ is equal to the product of the permanents associated to $2$DSI matrices of $\Gamma_1$ and $\Gamma_2$.  \end{theorem}

\begin{figure}[h]
  \centering
      \includegraphics[scale=1.30]{4edgecut}
  \caption{The graph permanent reduction for a graph with a 4-edge cut.}
\label{4edgecutpic}
\end{figure}

\begin{proof} Graph $\Gamma$ has associated $2$DSI matrix $$M_{G} = \left[ \begin{array}{c|cc}
I_4 & A & \bf{0} \\
\bf{0} & B & \bf{0} \\
-I_4 & \bf{0} & C \\
\bf{0} & \bf{0} & D \\ \hline
I_4 & A & \bf{0} \\
\bf{0} & B & \bf{0} \\
-I_4 & \bf{0} & C \\
\bf{0} & \bf{0} & D \end{array} \right], $$ where blocks $A$ and $B$ correspond to the left side of the graph, and blocks $C$ and $D$ correspond to the right. Supposing there are $l$ edges in the left subgraph and $k$ edges in the right, then block $B$ consists of $\frac{l-4}{2}$ rows and block $D$ has $\frac{k-8}{2}$ rows.

Consider cofactor expansion along the first four columns. By Lemma~\ref{pigeon} only matrix minors where the deleted row meets $A$ produce a non-zero permanent. Given that each column has two potential values, and since we may interchange rows freely, $$\text{Perm}(M_{G}) = 2^4 \cdot \text{Perm} \left[ \begin{array}{c|c}
A & \bf{0} \\
 B & \bf{0} \\
\bf{0} & C \\
\bf{0} & D \\ \hline
B & \bf{0} \\
\bf{0} & C \\\bf{0} & D \end{array} \right] = 16 \cdot \text{Perm} \left[ \begin{array}{c} B\\mathcal{A}\\mathcal{B} \end{array} \right] \cdot \text{Perm} \left[ \begin{array}{c} C\\D\\mathcal{C}\\D \end{array} \right].$$  Notice that $$\text{Perm}(M_{G_1}) = \text{Perm} \left[ \begin{array}{c|c} I_4 & A \\ \bf{0} & B \\ \hline I_4 & A \\ \bf{0} & B \end{array} \right] = 16 \cdot \text{Perm} \left[ \begin{array}{c} B\\mathcal{A}\\mathcal{B} \end{array} \right] ,$$ and $$\text{Perm}(M_{G_2}) = \text{Perm} \left[ \begin{array}{c}C\\D\\mathcal{C}\\D \end{array} \right].$$ This completes the proof.  \end{proof}

\subsection{Relation to Feynman periods}
\label{relationtofeynman}

Let $\Gamma$ be a 4-regular graph and let $G=\Gamma - v$ for some $v\in V(\Gamma)$.
Once again we are thinking of $G$ as a Feynman diagram, specifically as a 4-point graph in scalar $\phi^4$ theory.  The vertices where $G$ used to have edges connecting to $v$ are the \emph{external edges} of $G$.  If we think of $G$ representing some particle interactions, then the external edges represent the four particles coming into or out of the process.

To each edge $e$ of $G$ associate a variable $a_e$ and define the \emph{period} of $G$ to be $$\int_{a_i \geq 0} \frac{\Omega}{\Psi^2}, $$ where
$$\Omega = \sum_{i=1}^{|E(G)|}(-1)^i \prod_{\substack{j=1 \\ j \neq i}}^{|E(G)|}da_j, \hspace{6mm}
\Psi = \sum_{\substack{T \\\text{spanning} \\ \text{tree of }G}}\prod_{e\not\in E(T)}a_e.$$

Provided $K$ is internally 6-edge-connected the period integral converges \cite{bek}.  If $K$ has an internal 4- or 2-edge-cut then we say $G$ has a \emph{subdivergence}.  If $G$ has no subdivergences then we say it is \emph{primitive}.  In quantum field theory, graphs with subdivergences are more complicated because the subdivergences need to be dealt with first in the renormalization process (in this language see \cite{BrKr}).  The period of a primitive graph is renormalization-scheme independent, but is still an informative part of the full Feynman integral.

Periods of 4-point graphs in $\phi^4$ are preserved by the three operations we have been looking at throughout this paper, namely planar dual, completion followed by decompletion, and Schnetz twist \cite{bkphi4, Sphi4}.  These identities of periods explain all known cases where two primitive 4-point graphs in $\phi^4$ have the same period (see \cite{Sphi4}).  Furthermore if $\Gamma$ has a 3-vertex cut then the period of any decompletion of $\Gamma$ is equal to the product of the periods of graphs $G_1$ and $G_2$ as seen in Figure~\ref{2cut}. Thus, having found a graph invariant which is preserved by the three operations and with the appropriate product property, namely the graph permanent, the following conjecture is natural.

\begin{conjecture} Suppose $G_1$ and $G_2$ are two primitive 4-point graphs in scalar $\phi^4$ theory. If $G_1$ and $G_2$ have equal periods, then they have equal graph permanents. \end{conjecture}

Now consider the case of subdivergences.  Internal 2-edge cuts in $\Gamma$ are not very interesting in this context because they automatically yield small vertex cuts as well.  Suppose $\Gamma$ has an internal 4-edge cut, as in Figure~\ref{4edgecutpic}.  Decompleting to $G$, say $G_2$ is the side which did not have the decompletion vertex.  Then $G_2$ is the subdivergence inside $G$.  The leading term in the renormalized period of $G$ is the product of the periods of $G/G_2$ and $G_2$, which is what is given by the graph permanent.  This further strengthens the suggestion that the graph permanent is measuring something about the period, and gives some initial hints of what kind of thing it could be measuring.

In comparison, there is one two-valued invariant which we have some handle on, namely whether or not the period is of full transcendental weight (see \cite{BrY, Sphi4}).  This invariant has a different behaviour on subdivergences, namely all subdivergences give weight drop regardless of the weight of their pieces.  This shows that the graph permanent is capturing something different about the period.  We can also compare these two invariants on specific graphs.  For example, using names from \cite{Sphi4}, $P_{6,1}$ and $P_{6,4}$ both have graph permanent of $0$ while $P_{6,1}$ is full weight and $P_{6,4}$ has weight drop. Alternately, $P_{6,2}$ and $P_{6,3}$ both have nonzero graph permanent while $P_{6,2}$ is full weight and $P_{6,3}$ has weight drop.  Note that the names here label the completed graphs, or alternately label families of 4-point graphs related by completion.

The $c_2$ invariant is conjecturally another graph invariant with the same symmetries of the period \cite{BrS}.  It does specialize to capture the question of weight drop and, relatedly, is $0$ on graphs with subdivergences.  Thus the graph permanent is again capturing something different about the period.

Unfortunately, it simply is not clear what about the period the graph permanent is measuring.  The graph permanents for the completion-families of graphs are given in Appendix~\ref{calculatedperms}.

A final hint at the existence and potential nature of a connection between the graph permanent and the Feynman period comes from the fact that both of them are closely related to questions of (momentum) flows in graphs.  This is explained in the next section after the required graph theory of flows is presented.

\section{A connection to nowhere-zero flows}
\label{flow_sec}

The graph permanent also arises naturally in the study of nowhere-zero flows on graphs, and in this section we will develop this interesting connection.  To  this end we demonstrate that a graph has a certain orientation (closely related to the study of flows) under the assumption that a spanning subgraph has nonzero graph permanent.

Suppose that $G$ is a graph and (as before) direct the edges of $G$ arbitrarily.  For an abelian group $\mathcal{G}$, a function $\phi : E(G) \rightarrow \mathcal{G}$ has an associated boundary function $\partial \phi : V(G) \rightarrow \mathcal{G}$ given by the following rule (here $\delta^+(v)$ denotes the edges directed away from $v$ and $\delta^-(v)$ the edges directed to $v$);
\[ \partial \phi (v) = \sum_{ e \in \delta^+(v) } \phi(e) - \sum_{ e \in \delta^-(v) } \phi(e). \]
Note that $\sum_{v \in V(G)} \partial \phi (v) = 0$ since each edge $e$ contributes $\phi(e) - \phi(e) = 0$ to the total sum.  We define $\phi$ to be a $\mathcal{G}$-\emph{flow} if $\partial \phi$ is the zero function, and we say that $\phi$ is \emph{nowhere-zero} if $0 \not\in \phi ( E(G) )$, where $\phi(E(G))$ is the \emph{range} of $\phi$.  Note that if $\phi$ is a nowhere-zero flow, we may alter the orientation of $G$ by reversing the direction of some edge $e$ and modify $\phi$ by replacing $\phi(e)$ with its negation, and this results in another nowhere-zero flow.  Therefore, the question of whether our graph has a nowhere-zero flow in a particular group $\mathcal{G}$ will be independent of the chosen orientation.  Accordingly, we will say that an undirected graph has a nowhere-zero flow if some (and thus every) orientation permits such a map.   Nowhere-zero flows have a rich history initiated by Tutte who proved all of the following properties.  (Throughout we let $\mathbb{Z}_k = \mathbb{Z}/ k \mathbb{Z}$).

\begin{theorem}[Tutte \cite{Tu49} \cite{Tu54}] $\mbox{}$
\begin{enumerate}
\item If $G$ and $G^*$ are dual planar graphs, then $G$ has a $k$-colouring if and only if $G^*$ has a nowhere-zero $\mathbb{Z}_k$-flow.

\item A graph $G$ has a nowhere-zero $\mathbb{Z}_k$-flow if and only if it has a $\mathbb{Z}$-flow with range a subset of $\{ \pm 1, \pm 2, \ldots, \pm (k-1) \}$.

\item If $G$ has a nowhere-zero $\mathcal{G}$-flow for a finite abelian group $\mathcal{G}$, then it has a nowhere-zero $\mathcal{G}'$-flow for every abelian group $\mathcal{G}'$ with $|\mathcal{G}'| \ge |\mathcal{G}|$.
\end{enumerate}
\end{theorem}

In addition, Tutte made three famous conjectures which have motivated a tremendous amount of investigation, but all remain unsolved.

\begin{conjecture}[Tutte] $\mbox{}$
\begin{enumerate}
\item (5-Flow) Every graph without a cut-edge has a nowhere-zero $\mathbb{Z}_5$-flow.
\item (4-Flow) Every graph without a cut-edge and without a Petersen minor has a nowhere-zero $\mathbb{Z}_4$-flow.
\item (3-Flow) Every 4-edge-connected graph has a nowhere-zero $\mathbb{Z}_3$-flow.
\end{enumerate}
\end{conjecture}

In his investigations of flows, Jaeger introduced the following interesting concept.  For an (undirected) graph $G$, define an orientation of its edges to be a \emph{modulo} $k$ orientation if every vertex $v$ satisfies $|\delta^+(v)| - |\delta^-(v)| \equiv 0 \pmod{k}$.  If we have a modulo $k$ orientation of a graph, then we can obtain a nowhere-zero $\mathbb{Z}_k$ flow by assigning each edge to have flow value $1$.  On the other hand, if we have found a flow $\phi: E(G) \rightarrow \mathbb{Z}_k$ for which $\phi ( E(G) ) \subseteq \{ -1, 1\}$ then by reversing the edges with flow value $-1$, we obtain a modulo $k$ orientation.  So, in short, a modulo $k$ orientation is equivalent to the existence of a $\mathbb{Z}_k$-flow with range a subset of $\{-1,1\}$.  Jaeger offered the following unifying conjecture which, if true, is known to imply both Tutte's 5-flow and 3-flow conjectures.

\begin{conjecture}[Jaeger \cite{Ja84}]
Every $4k$-edge-connected graph has a modulo $2k+1$ orientation.
\end{conjecture}

A recent flurry of activity started by Thomassen's proof of a weak version of the 3-flow conjecture \cite{Th} has resulted in a proof of a weak version of the above conjecture.  Namely, Lov\'{a}sz, Thomassen, Wu, and Zhang \cite{LoThWuZh} have recently proved that every $(3k-3)$-edge-connected graph has a modulo $k$ orientation.

With an eye toward constructing flows and modulo $k$ orientations, let us now return to incidence matrices.  As before, we shall use $M^*$ to denote the oriented incidence matrix of $G$.  If we regard $M^*$ as a matrix with entries in $\mathcal{G}$, then a flow in $\mathcal{G}$ is precisely a vector in the nullspace of $M^*$.  Since the sum of the rows of $M^*$ is zero, the matrix $M^*$ will have the same nullspace as the matrix $M$ which we obtain from $M^*$ by deleting the row corresponding to an arbitrarily chosen special vertex $w$.  This recasts the problem of finding a nowhere-zero $\mathcal{G}$ flow in $G$ as one of finding a vector in the nullspace of $M$ with no zero entries.  Turning our attention to the special case $\mathcal{G} = \mathbb{Z}_k$, we note that the existence of a modulo $k$ orientation in $G$ is equivalent to the existence of a $\pm 1$ valued vector in the nullspace of $M$.

The main tool we will require for our result is the following ``polynomial method''.

\begin{theorem}[Alon and Tarsi \cite{AlTar}]
\label{polymeth}
Let $\mathbb{F}$ be a field and let $f(x_1, \ldots, x_n)$ be a polynomial in $\mathbb{F}[x_1, \ldots, x_n]$.  Suppose that the coefficient of $x_1^{d_1} x_2^{d_2} \ldots x_n^{d_n}$ is nonzero and that $\deg(f) = d_1 + d_2 + \ldots + d_n$.  Then for every $S_1, \ldots, S_n \subseteq \mathbb{F}$ with $|S_i| > d_i$ for $1 \le i \le n$, there exist $s_i \in S_i$ for $1 \le i \le n$ so that $f(s_1, \ldots, s_n) \neq 0$.
\end{theorem}

Now we shall restrict our attention to the case when $\mathcal{G} = \mathbb{Z}_p$ for a prime $p$ so that our matrix $M$ has its entries in a field.  In this setting, the graph permanent arises naturally in conjunction with the above theorem to produce a certificate for a graph which guarantees the existence of a modulo $p$ orientation of $G$.  Namely, we will prove the following result which appears implicitly in \cite{AlLiMe}.

\begin{theorem}[Alon, Linial, Meshulam]
Let $p$ be prime, let $G$ be an $n$ vertex graph, and let $H$ be a spanning subgraph of $G$ with $|E(H)| = (p-1)(n-1)$.  If a $(p-1)$DSI matrix for $H$ has nonzero permanent modulo $p$, then $G$ has a modulo $p$ orientation.
\end{theorem}

\begin{proof}  Begin by orienting the edges of $G$ arbitrarily.  Define $H' = G - E(H)$ and choose $\phi' : E( H' ) \rightarrow \{ -1, 1 \}$ arbitrarily.  Our goal will be to use the polynomial method to prove that $\phi'$ may be extended to a $\mathbb{Z}_p$ flow of $G$ with range a subset of $\{-1,1\}$.  To do so, define $V' = V \setminus \{w\}$ and for every $v \in V'$ define $A_v = \mathbb{Z}_p \setminus \{\partial \phi'(v) \}$.  Now we shall construct a polynomial $f$ with a variable $x_e$ for every edge $e \in E(H)$ by the following rule:
\[ f = \prod_{v \in V' } \prod_{ a \in A_v } \left( -a + \sum_{e \in \delta_H^+(v)} x_e - \sum_{e \in \delta_H^-(v)} x_e  \right). \]

Let us pause to consider what it would mean for this polynomial to be nonzero on a particular assignment to the variables.  Namely, let $\phi : E(H) \rightarrow \mathbb{Z}_p$ and suppose that evaluating $f$ where each variable $x_e$ is assigned the value $\phi(e)$ gives a nonzero value.  Considering the innermost product in our equation, we see that in order for $f$ to be nonzero when evaluated at $\phi$ it must be that $\partial \phi (v)$ is not equal to $a$ for every $a \in  A_v$.  However, this is precisely equivalent to the statement that $\partial \phi(v) = - \partial \phi'(v)$.  Since this must hold at every $v \in V'$ we have that the function $\phi \cup \phi'$ (i.e. the function which maps each $e \in E(H')$ to $\phi'(e)$ and each $e \in E(H)$ to $\phi(e)$) is a flow.  Indeed, this polynomial evaluated at $\phi$ will result in a nonzero value precisely when $\phi \cup \phi'$ is a flow.

Now  consider the coefficient of $\prod_{e \in E(H)} x_e$ in the expansion of $f$.  Since this term has degree $|E(H)| = (p-1)(n-1)$ we can see that this is the same as the coefficient of the same term in the expansion of the polynomial
\[ \prod_{v \in V' } \left( \sum_{e \in \delta_H^+(v)} x_e - \sum_{e \in \delta_H^-(v)} x_e  \right)^{p-1}. \]
However, this is precisely the permanent of the $(p-1)$DSI matrix of $H$ with special vertex $w$.  So, by assumption, this coefficient is nonzero, and then (since $\deg(f) \le (p-1)(n-1)$) by the Alon-Tarsi theorem, we may choose an assignment to these variables $\phi : E(H) \rightarrow \{ -1, 1 \}$ in such a way that evaluating $f$ on these inputs is nonzero.  As we have seen, this gives us a $\mathbb{Z}_p$-flow $\phi \cup \phi'$ with range a subset of $\{-1,1\}$, or equivalently a modulo $p$ orientation of $G$, as desired.
\end{proof}

Further evidence of the connection between our invariant and the Feynman integral can be found in the polynomial appearing in the previous proof.  Indeed, just as the period of a graph can be computed using a number of different integrals (one of which features integrating over a basis of the cycle space), so the polynomial coefficient in our proof has many essentially equivalent variations.  To give a concrete instance of this, let $G = (V,E)$ be a graph with $|E| = 2 |V| - 2$ and assume that $G$ is equipped with an arbitrary orientation of the edges.  Following the line of the previous proof, we will construct a polynomial and use the polynomial technique to show the existence of a nowhere-zero $\mathbb{Z}_3$-flow on $G$ (under certain additional assumptions).  Though the polynomial we construct will be quite different from that in the proof of the previous theorem, it will turn out that the coefficient of interest will be same (up to sign).

Choose a spanning tree $T$ of $G$ and introduce a variable (from $\mathbb{Z}_3$) denoted $y_e$ for every edge $e \in E \setminus E(T)$.  These variables may be viewed as indexing the $\mathbb{Z}_3$-cycle space of $G$.  For $f \in E(T)$ there is a unique edge-cut in $G$ which contains $f$ but no other edge of $T$, called the fundamental cut of $f$.  Define $C_f^+$ to be the set of edges in this cut (other than $f$) which are oriented the same as $f$ relative to this cut, and $C_f^-$ to be those edges in the cut with opposite orientation to $f$.  Using these, we define the following linear polynomial (with variables $\{ y_e \}_{e \in E \setminus E(T)}$)
\[ g_f = \sum_{e \in C_f^-} y_e - \sum_{e \in C_f^+} y_e. \]
It follows from basic theory that every assignment to all of the $y_e$ variables extends uniquely to a $\mathbb{Z}_3$-flow of $G$.  Furthermore, for this flow, the value on an edge $f \in E(T)$ is given by $g_f$.

Now we will proceed in our attempt to use the polynomial technique to find a nowhere-zero $\mathbb{Z}_3$-flow in $G$.  To do this, define the polynomial
\[ g = \prod_{f \in E(T)} g_f. \]
If the coefficient of the monomial $\prod_{e \in E \setminus E(T)} y_e$ in the expansion of $g$ is nonzero, then by Theorem \ref{polymeth}, there exists an assignment to the variables using only the elements $\{-1,1\}$ so that $g$ evaluates to a nonzero number.  However, this is precisely what is required to have a nowhere-zero flow.  So, in short, we have now found another polynomial coefficient which, if nonzero, implies the existence of a nowhere-zero $\mathbb{Z}_3$-flow in our graph.

Although the polynomial $g$ we have constructed depends on the choice of spanning tree, the coefficient of the term $\prod_{e \in E \setminus E(T)} y_e$ (up to sign) does not depend on this choice.  Furthermore, (up to sign) this coefficient is the same as the coefficient of $\prod_{e \in E(H)} x_e$ in the expansion of $f$ from the previous proof.  To see this, let $M$ be a matrix obtained from the incidence matrix of $G$ by removing a row, and assume (for convenience) that the columns of $M$ associated with edges in $T$ appear before those associated with edges in $E \setminus E(T)$.  It follows from basic theory that the matrix $M$ may be turned by row operations  into a matrix for which the columns associated with edges in $T$ form an identity matrix.  So, by row operations we may transform $M$ into a matrix of the form $\begin{bmatrix} I & A \end{bmatrix}$.  Now, working in the field $\mathbb{Z}_3$ we have
\[ \text{Perm} \begin{bmatrix} M \\ M \end{bmatrix} = \pm \text{Perm} \begin{bmatrix} I & A \\ I & A \end{bmatrix} = \pm \text{Perm} A. \]
By the previous proof, the coefficient of $\prod_{e \in E(H)} x_e$ in the expansion of $f$ is $\text{Perm} \begin{bmatrix} M \\ M \end{bmatrix}$, and by elementary reasoning, the coefficient of $\prod_{ e \in E \setminus E(T) } y_e $ in our polynomial $g$ is equal to $\text{Perm} A$, thus yielding the desired connection.

This polynomial $g$ is closely related to the Feynman integrand in momentum space. As before we are taking the simplest possible case of a Euclidean massless scalar field theory.  To set up the Feynman integrand in momentum space first take a basis of the cycle space and assign a variable to each cycle in the basis; in particular the $y_e$ are appropriate.  These variables represent the momentum flowing around the cycles and we view them as taking values in $\mathbb{R}^4$.  To each edge associate the signed sum of the variables for the cycles running through that edge; this is the momentum flowing through that edge and is $g_f$ for $f\in T$ and is $y_e$ itself for $e \not\in T$.  Let
\[
\tilde{g} = \prod_{f\in T}|g_f|^2\prod_{e\not\in T}|y_e|^2
\]
where the norms are the usual Euclidean norm.
The Feynman integrand is $1/\tilde{g}$ and the integral runs over all values of the $y_e$.  After modifying the integral to use the projective volume measure, $\sum(-1)^i \prod_{\substack{j \neq i}}dy_j$ analogously to subsection~\ref{relationtofeynman}, this calculates the same period (see \cite{Sphi4} for discussion and proof in close to this language).

Returning to the polynomials, another way to look at the construction of $\tilde{g}$ is to assign a momentum variable to each edge but then impose momentum conservation at each vertex, which is just different language for the flow condition.  Furthermore, we see that $\tilde{g}$ and $g$ only differ in two ways.  First the norm squared has replaced the simple appearance of variables which is a natural adjustment to vector valued variables.  Second $\tilde{g}$ has a factor for the edges not in $T$.  If we put an analogous factor in $g$ we would be multiplying $g$ by the product of all the variables.  This has essentially no impact on the use of the polynomial technique as it simply shifts up all degrees.

Ultimately, this close relationship between flow calculations in graph theory and Feynman integrals should not be surprising since the momentum space Feynman integral is the integral over all possible momentum flows through the graph.

\section*{Acknowledgment}

The authors acknowledge the support of NSERC.

redpajama_set_name: RedPajamaArXiv


text:
GJR book review: 'This Town' offers no prescription for reforming America's gilded capital
September 25, 2013 by Chris Burnett Leave a Comment
'Mark Leibovich. This Town: Two Parties and a Funeral – Plus Plenty of Valet Parking – in America's Gilded Capital (2013) New York: Blue Rider Press, a Member of the Penguin Group (USA) Inc. 371 pages
By Chris Burnett
The funerals of Tim Russert, longtime host of "Meet the Press," and Richard Holbrooke, American diplomat and adviser to Democratic presidents, involved two men who would seem on the surface to have had little in common. Holbrooke, the hard-charging special envoy to Afghanistan and Pakistan at the time of his death in 2010, moved in the highest diplomatic circles, never shied away from controversy and made no attempt to keep his ego in check. Russert, who died two years earlier, was the avuncular everyman, known for his fair-minded but tough questioning of American political figures. He clearly had an ego, too, but he kept it under check, preferring to talk and write about his affection for his father, Big Russ.
In Washington journalism and politics, however, the worlds of these two men – and the worlds of countless other journalists and politicians – are shared. Funerals become special events, where the high and mighty (and political climbers from the media and politics) are on hand, as much to be seen as sensitive. Bill and Hillary Clinton, the apex of the social circle of today's Washington, attended both funerals. In the world of insider Washington, funerals, just like April's gala White House Correspondent's Dinner, are not just attended and televised. They're also tweeted about, covered for days and weeks before and after, and become Washington's version of Hollywood on the Potomac. To be at these events, after all, is to be in "The Club" of Washington journalistic and political elite.
Mark Leibovich, chief national correspondent for the New York Times Magazine, and himself most likely considered a member of "The Club," tells this story of journalistic and political Washington during the Obama presidency in the book "This Town." It's the ideal beach read – and if you're into the world of politics and the media of the Beltway sort, it's a real page-turner.
As a Washington reporter in the late 1970s and throughout the 1980s, and as an unreformed political junkie, I loved the book. It described a Washington journalism that still resembles that of the 1980s, but which also has changed. In the 1980s, journalists went with their political friends (generally those they covered or celebrities they were able to land) to the annual White House Correspondent's Dinner, held at the Washington Hilton Hotel. They also angled to get jobs that would get them into "The Club," which then consisted of the big national newspapers, networks and CNN. For those in the second tier (I was a regional reporter for the Columbus (Ohio) Dispatch) members of "The Club" were those you worked next to, not really with. I'd go to the Correspondent's Dinner, but not with the best prom date or invitation to the most glittering before and after parties.
What's changed in the last 20 or so years, according to Leibovich, is the expansion of cable television news starting in the 1990s and the explosion of the Internet. Organizations such as Politico, the political news wire service started in 2007 that distributes its content in print, radio, television – and, most significantly, on the Internet – have taken away much of the agenda-setting role once played by the New York Times and Washington Post. The book is at its best in its profile of veteran Club reporter Mike Allen, whose daily "Playbook" column in Politico is a must-read ticker of the day's news, with a focus on the personalities in the media and politics on Capitol Hill and in the White House. At 8 a.m. on any given day, those on the campaign bus are either reading or have already read "The Playbook," Leibovich says.
Leibovich calls all this a sort of pigpen, with politicians and political "spinmeisters" jumping into journalism (as in Obama pal David Axelrod now serving as a commentator for MSNBC) or setting up lobbying shops, and journalists angling for higher-paying jobs in public and government relations. With the 24-hour news cycle feeding an orgy of the trivial and the trendy, and with insiders reluctant to burn the sources they are likely to run into at tomorrow's party, what happens to serious coverage of government?
This book provides no answer to this question, unfortunately, that's its biggest fault. This book entertains, even amuses, but provides no prescriptions. For those living outside the Washington Beltway, the book – which has no index, lest people quoted seek out reference to their name – has got to be a tad depressing. The fact that all these narcissists are running and reporting on major events involving the country, getting rich or at least living comfortably, while the rest of the nation struggles through an economic recession can get tiring when thrown in your face page after page. Still, I found "This Town" an immensely entertaining read. It describes a political Washington that President Obama promised to reform when he came to office, and it leaves you chuckling at stories but frustrated by what is most likely the impossibility of that task. Still, the book begs the question: Is this the best we can do?
Chris Burnett is professor and chair of the Department of Journalism and Mass Communication at California State University, Long Beach. He reported on Congress and the Supreme Court and the politics of Washington from 1979 to 1989.
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David Butler is a graduate of the Dun Laoghaire Film Institute of Ireland, I.A.D.T. Animation Production Course.
After graduating David worked in Animation in Moving Still Productions on Fionn-A Story from Ireland part of The Tales of the World TV Series for S4C and Channel Four.He later worked as an Inbetween and Clean up Artist on Moving Still and Vinegar Hill Productions Sir Gawain and the Green Knight for Channel Four. The "Best TV Animation" BAFTA award winning Feature film.
He also worked on two Irish Film Board Frameworks project's as Inbetweener and Clean Up artist. From an Evil Cradling for Kavaleer Productions and Rowlandson Rides Again for Moving Still.
Having worked on a few TV projects and award winning short films. He created his own Tv series with comedy writer Paul Woodfull (I Keano, Stew, Val Falvey TD). "Frankie" recieved some development funding from the Irish Film Board but never made it on the Silver screen.
David then tried out a few things like, Nightclub promotions, No budget, live action music videos and band management.
David recently has had a Independantly Produced film "Just Ask Me" currently on Film Festivals circuit and is developing a few ideas with the "Just ask me" team for Irish Film Board funding, and an Historic Documentary with Soilsiu Films for Arts Council funding.
David has also animated some 3D Stop motion Animated viral video content for Animatazz – as seen on RTÉ s"Dragons Den" and The Late Late Toy Show in December 2010.
As an artist David had his first Exhibition of his painting at the Derby Festival Kildare in 2010. David most recently worked on the Graphic Novel styled webcomic for WartGirl for RTÉ Storyland Series 3 for the online Comedy, Drama produced in Newbridge Co. Kildare.
David has recently illustrated a Childrens book "Do Fish Wear Pyjamas? - The Quest for the Great Book of Kildare" which was published in October 2011.
To find out more about the projects I have worked on please view my Resume/ C.V. here.


text:
VA CIO Rob Thomas retiring from government
Sep 20, 2017 | FEDSCOOP
The Department of Veterans Affairs' acting CIO, Rob Thomas, will retire from government service in October.
Thomas took over as CIO and assistant secretary for information and technology in an acting capacity after LaVerne Council stepped down from the position in January at the change in administrations. Before that, Thomas served as deputy CIO at VA.
Scott Blackburn will take over as acting CIO effective this week, Thomas said in an email to VA staff announcing his departure. Secretary David Shulkin appointed Blackburn acting deputy secretary of the VA in February.
"We have made significant progress, and we are continuing to move in the right direction with our ongoing transformation and modernization efforts and by the steadfast leadership of Secretary Shulkin on behalf of our Nation's 23 million Veterans," Thomas wrote. "With this in mind, I am proud of what we have accomplished, and continue to accomplish together, and I hope you are as well. From modernizing both our electronic health record and benefits delivery network to our continuing effort to improve enterprise cybersecurity, VA's scheduling systems and consolidating the financial management system, OI&T is committed to the Secretary's initiatives to deliver better service to our Veterans."
Thomas also spent time as deputy CIO and acting CIO at the Federal Emergency Management Agency, and as a member of the Air Force. He is retiring with 35 years of federal service.
"Please know that it has been an honor, privilege, and pleasure to serve with you at OI&T," he said. "This has been an enriching experience; serving our Nation's Veterans, and equally important, having the opportunity to work side-by-side with outstanding public servants."
Federal News Radio first reported Thomas' departure.
CIO, Department of Veterans Affairs (VA), Rob Thomas, Scott Blackburn


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Decription: A single mother moves into a new house with her daughter. Soon after the young girl has her first baby tooth fall off, she begins to recount that she is having nocturnal visits by a tooth fairy. It seems the house has a sinister history.


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text:
Antonio Basoli, né le  à Castel Guelfo di Bologna, près de Bologne, et mort le  à Bologne, est un peintre, architecte d'intérieur, scénographe et graveur néo-classique italien du  actif principalement à Bologne.

Biographie
Né à Castel Guelfo di Bologna, Antonio Basoli apprend les bases de l'art auprès de son père Lelio Andrea Basoli. Pendant ses études, il a toujours été motivé par une soif insatiable dans l'art classique, dans la littérature, ainsi que par un intérêt des travaux et décorations de Giovanni Battista Piranesi. Il a souvent travaillé et œuvré avec ses frères Luigi et Francesco.

De 1803 à sa mort, il était enseignant d'ornement et de 1804 jusqu'à 1826, professeur d'art à l'Académie des beaux-arts de Bologne. Il a eu pendant ces années-là, une amitié étroite avec deux de ses élèves : Pelagio Palagi et Domenico Corsini. Il a aussi travaillé pendant un moment à l'atelier de Petronio Fancelli.

Œuvres

Entre 1801 et 1803, Basoli a exécuté plusieurs fresques dans le Frioul, notamment à la Casa Muratori de Trieste et à un palais de Spilimbergo.

Il a aussi travaillé comme scénographe et artiste de rideaux et était le décorateur de plusieurs théâtres comme le Teatro Marsigli-Rossi, le Teatro comunale di Bologna, ainsi que le , où il a le plus travaillé. Malheureusement, la plupart de ses travaux n'ont survécu qu'à travers des références dans des livres répertoriant ses œuvres en gravure, en dessin, ou encore à l'aquarelle, dont le Collezione di varie scene teatrali, qui a été écrit en 1821. Il décore, outre les théâtres, des palais municipaux comme le Palazzo Rosselli del Turco, le  et le .

Basoli a aussi reçu des commandes pour effectuer la scénographie dans des théâtres de Rome, en 1815, et à Naples (commande de Gioachino Rossini), qu'il a refusées. En 1818, il voyage à Milan, où il découvre la sala Sanquirico, atelier des plus éminents scénographes de La Scala, desquels il a pris inspiration pour le décor de l'opéra Semiramide de 1820, le décor d'une représentation d'Œdipe roi au Contavalli en 1822 et pour les peintures d'arrière-plan du Teatro dei Cavalieri dell'Unione à Santarcangelo di Romagna la même année. Pendant cette même période, il travaille notamment sur la  et le Palazzo Baciocchi (aujourd'hui le ).

Parmi ses expositions à l'Académie figurent une collection de peintures romantiques aux thèmes exotiques. Par exemple, en 1842, il expose seize peintures à l'huiles  représentant entre autres les quatre Colosses de Memnon en Égypte, le port de Rhodes (vraisemblablement avec son Colosse), la Babylone de Nabuchodonosor II (très certainement avec ses jardins suspendus), la  de Domitien (détruite pendant l'Antiquité), des vues des quatre coins du globe (Europe, Asie, Afrique, Amériques), ainsi que cinq temples incluant celui de Philæ, un temple aux lions gardiens en Chine, celui de Zeus à Olympie et celui d'Artémis à Éphèse. On pouvait aussi y voir une ville du royaume de Pégou (aujourd'hui en Birmanie), une tombe arabe, le Grand hall de Charlemagne et le palais de la sorcière Alcina.

Plus tard dans sa vie, il crée une série de vingt-six vignettes, représentant l'alphabet archéologique en aquarelle, qu'il a surnommé l'Alfabeto pittorico.

Publications

Basoli a publié plusieurs recueils de ses œuvres. En voici quelques-unes :
 Raccolta di prospettive serie, rustiche e di paesaggio (1810) ;
 Guarnizioni diverse di maniera antica (1814) ;
 Porte della città di Bologna (1817) ;
 Esemplare di Elementi d'Ornato che contiene lo studio della pianta d'accanto (1817) ;
 Collezione di varie scene teatrali (1821) ;
 Compartimenti di camere (1827) ;
 Vedute pittoresche della città di Bologna (1833) ;
 Definizioni geometriche (1837) ;
 Raccolta di diversi ornamenti (1838) ;
 Alfabeto pittorico (1839).

Notes et références

Annexes

Bibliographie
  Antonio Basoli, La vita artistica di Antonio Basoli, Minerva Edizioni, Bologne, 2006 .
  S. Sermisoni (éd.), Antonio Basoli. Fantasie e Studi di Cose Egizie (intro. Fabrizio Clerici), Galleria Antonia Jannone Disegni di Architettura, Milan, 1985.
 .
  Antonio Basoli, Collezione di Collezione di Varie Scene Teatrali, Benjamin Blom, New York,  .
  E. Lavagnino, L'arte moderna, Volume I, Turin, 1956.
  , Scenografie di Antonio Basoli, con 4 illustrazioni,  fascicule, février 1924.

Article connexe
 Liste de peintres italiens

Liens externes

 Exposition sur Basoli à la Pinacothèque nationale de Bologne en collaboration avec la municipalité de Castel Guelfo di Bologna et la ville de Bologne.
 Images de plus de 200 des œuvres de Basoli, sur cineca.it.
 Collection d'œuvres de Basoli au Smithsonian Museum.
 L'Alphabet de Basoli

Peintre néo-classique italien
Peintre italien du XVIIIe siècle
Peintre italien du XIXe siècle
Graveur italien du XVIIIe siècle
Graveur italien du XIXe siècle
Architecte d'intérieur
Scénographe italien
Naissance en avril 1774
Naissance à Castel Guelfo di Bologna
Décès en mai 1848
Décès à Bologne
Décès à 74 ans

redpajama_set_name: RedPajamaWikipedia


text:
Assistant Director, Research Facilitation and Outreach (Job ID #158316)
OVCI/ ACER
The University of Illinois at Chicago is growing (ranked 1st in diversity in the Midwest, top 25 public university, top 10 best value and many more). Our research portfolio is rapidly expanding and is around $445M annually. Recently, UIC played a critical role in the COVID19 vaccine and drug trials. The Advanced Cyberinfrastructure for Education and Research (ACER) is responsible for delivering cyberinfrastructure (e.g., HPC, AI, Cloud, Data Lake) and consulting services to over 3000 researchers. We are rapidly growing our team of engineers, computational scientists, and data scientists to deliver on UIC's inspiring 5-year (2021-2025) Research Cyberinfrastructure (RCI) strategic plan.
ACER seeks a qualified Assistant Director, Research Facilitation and Outreach to join our impactful team and advance scientific discovery. If you are looking for an inclusive environment that provides challenges, learning opportunities, and work-life balance, then this opportunity is for you to consider. We also support our staff's pursuit of further education through UIC's significant tuition waiver benefit. ACER also encourages staff exploration of emerging technologies such as deep learning, blockchain, IoT, and quantum computing.
Collaborate to successfully implement UIC's 5-year Research Cyberinfrastructure strategic plan
Lead ACER engineering team of computational scientists, data scientists and research software engineers. Must have the ability to develop staff through coaching and provide stretch opportunities.
Optimize, communicate, and extend RCI services portfolio aligned with the strategic goals of the University and future research institutes.
Recruit and retain talent to grow the size and capabilities of the research facilitation team
Provide insights into latest technological advances in research computing, storage, and networking. Attend and directly contribute to strategic meetings with leadership, faculty, and vendors.
Plan, direct, oversee and optimize service delivery, providing periodic assessments and reports to the campus leadership team and ACER director.
Lead, participate, and present at national conferences, working groups, academic coalitions, etc.
Define and communicate the overall vision and goals of the facilitation team.
Requires PhD in an Engineering or Science discipline and 5+ years of relevant experience of which a minimum of 1 year should be in supervisory or management capacity.
Excellent communication and relationship-building skills. Experience in delivering services to diverse population. Prior experience with research facilitation is highly preferred. Experience in communicating and strategizing with researchers, vendors, engineers etc.
HPC Cluster Management Tools (e.g., Bright, XCAT, Warewulf). HPC Schedulers (e.g., Slurm, Moab/Torque). HPC Compute Architectures (i.e., CPU/GPU architectures). HPC Storage Environments (e.g., Parallel File Systems, NFS, Tape Libraries etc.). HPC Networks (e.g., Infiniband-HDR/EDR, Ethernet, etc.). Familiarity with AI/ML.
Hybrid and flexible work schedules available.


text:
Resolved for 2022: Rebuilding with the Basics
By James C. Howell
As we segue into 2022, we are realizing that some of the things we did well during the pandemic have revealed and maybe even exacerbated some challenges we were coping with long before the pandemic. Once exposed, we see more clearly what our work looks like next year.
Early on, and wisely, we declared that "getting back" were the words that must not be said. We don't plan to get back to church. We ask and dream of what we're being called to going forward. This provided space to name new, more robust ways of being church. And we were lucky enough to have a campaign for a new building that provided a big, forward-looking dream.
But who was moving forward? Who was dreaming with us? We had a solid online presence pre-COVID. Then with the in-person shutdown, we attracted a whole new audience. Worship numbers skyrocketed. People went so far as to join our church from other cities, states, and countries. Our small groups morphed into a mix of local and distant people. We were strutting a little over this technological evangelism. We've resolved to be hybrid moving forward.
Now we understand, a little vaguely, who wasn't tuning in. I'd bump into some of our most active folks, and they were clueless about what we were doing. Some are now finding their way into the building, but we enter 2022 knowing we've lost – a lot. Some who weren't deeply formed in the faith – reminding us we don't do this so well – didn't miss church much. Some discovered what even my staff and I discovered: Sunday mornings for lounging at home or hiking are way cool. And some who felt disenfranchised because of their political ideology or cross feelings about inclusion or immigrant efforts used the long radio silence as a big door to exit without having to explain things.
Of course, people were drifting before COVID, which only hastened the drifting. How do we re-engage? Do our energies go now to our newer people, or to recovering the lost sheep? Both – but do we have the bandwidth? It's about personal contact – which was what we lost since March 2020. One by one, saying We love you, we miss you, you matter. How to do this minus physical presence, how to get them to pay attention: we'll try this and that, and enlist those who stayed – we hope.
And there's this: how many people over the past 20 months have said "I saw you on TV"? Worship as entertainment, passively watching: the problem is as old as Kierkegaard (or older), but pandemic viewing has made this more entrenched, even enjoyable, but unsustainable and not very fruitful. We kept a list of "things we learned during COVID we'll keep," and at the top was the use of children and laity in worship. We'll redouble that in 2022, and keep naming why it matters to show up.
Americans were all about convenience in worship and church life pre-COVID. Unwittingly we fed the beast by giving lots of options and trying to lure people to show up. We are embracing twin ministry needs. Can we seize this moment in time to move the needle from people dipping their toes into church when it suits them toward being all in, for owning not just renting? The other (which may be a prerequisite for the first) popped up for me when a young man whom we didn't know (who began tuning in during COVID) showed up, said I'm brand new to Christianity. How do I go about being a Christian and doing it well? I hope my face didn't turn red when I fumbled for an answer. Hasn't COVID shown us all how malformed, underformed, or unformed our people have been?
And so, programmatically, our programmatic and liturgical themes for early 2022 are pretty basic. Following the lead of our children's curriculum, partly because it's on point, and partly to use this as leverage to re-engage our young families who are our biggest (understandably!) drifters, we'll walk through the life of Jesus, and why he matters. Utterly basic stuff – but with tags to connect with and actually help (we hope) people with things they've shared, again not new things, but intensified by the weird pandemic season. Jesus and relationships. Jesus and home – so in our homes, church as home, offering home for others. Jesus and mental health. Jesus and… Blanks are still being filled in.
Our year-long stewardship theme is one we hope to use as a scaffold on which to hang all these learning endeavors: We Love Our Church. I mentioned our building campaign: we will have the quirky good fortune of bearing and hopefully delighting in the best visual aid ever, as 2022 will witness the demolition of one-third of our whole facility, and then with the cranes and trucks rolling in. How do we rebuild the church? That was how Jesus commissioned St. Francis. Speaking to him from a crucifix in Assisi, Jesus said "Rebuild my church, for as you can see, it is falling into ruin." Ours has been for longer than we care to admit.
And we'll be wearing hard-hats. How often will I toss into a sermon Annie Dillard's sharp wisdom: "It is madness to wear ladies' velvet hats to church. We should all be wearing crash helmets." With the stinging truth that we haven't "the foggiest idea what sort of power we blithely invoke," suspecting rightly that "no one believes a word of it," she wonders if "the sleeping God may wake someday and take offense, or the waking God may draw us out to where we can never return." Tearing down, rebuilding, and waiting on a shocking annihilation and rebirth from the Spirit. I hope this can be our prayer, if we don't drift once more into passivity and timidity.
James C. Howell
Dr. James C. Howell has been senior pastor of Myers Park United Methodist Church since 2003, and has served read more…
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Discussion in 'Minecraft Creations' started by Efcluke94, Oct 21, 2013.
So this is a build which I made months and months ago as a survival games map, Candyland!
SheepDoesn'tPlayMC and MorganDoezn'tPlayMc like this.
WOW, OMG WE MUST HAVE THIS. So Creative!
Nice, now.. whens it added?
Wow! So creative! This deserves to be added in Cubecraft!
Looks pretty nice, i do not like the materials you used for the surrounding walls though, could look better and less confined.
Although I will be remaking this and make it 10 times more candilicious!!!!
Efcluke94's twin sister made this. They never met because they were separated at birth by Herobrine's non-digital counterpart. They both posses great building skills except one's work is very feminen.
What, you are giving me confused faces? My guess could be right. As crazy as it sounds I could be on to an unexplainable work of art's origin.
"I was completely joking about that comment above"
um interesting theory john, but I just have a girly way of building. Wait wut. Anyway here's the upgraded version, of Candyland. It made me sooo hungry building it, I think I put on weight building this.
awesomepingu2005 and ModernRevolution like this.
Is it going to become a kit pvp map?


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