In a large saucepan, saute the garlic, pepper, and onion in oil until fragrant, about 5 minutes. Add beef and saute until brown, stirring to break up the meat. Add the remaining ingredients.
Saute, stirring occasionally, about 20 minutes, until thickened. I served mine sprinkled with cheddar on sesame rolls.
Sorry about the unexpected hiatus! I've been busy developing recipes for clients, making dinners that have appeared here before and going out for exciting dinners with friends and for our tenth anniversary. Excuses, excuses! I'm back now and bringing you this new recipe for Sloppy Joe's my mom has deemed Sloppy Johnny's after her childhood interest in Johnny Appleseed because they feature apple butter. I was lucky enough to have been contacted by Musselman's to become an apple butter blogger in 2015. I love apple butter so much, we go to the Apple Butter Festival in Berkeley Springs, WV each year despite it being two hours away. Musselman's Apple Butter is sweet, of course, but it also has a great spicy note to it that really lends itself to savory applications like these Sloppy Johnny's. Apple butter is more than just a spread. Normally I use tomato paste in Sloppy Joe's but the apple butter is thick enough to serve as a substitute. I added some serrano peppers, mustard powder and ground chipotle to kick up the heat and complement the ginger and cinnamon flavor of the apple butter. This made it a Sloppy Joe that was not only suitable for children but for adults as well.
Check out the Musselman's Apple Butter Pinterest account for more ideas.
In a large bowl, mix together the coconut, salt and sugar. Then stir in (or beat in with the paddle attachment if using a mixer) the cream and then the vanilla paste until a firm, sticky, paste-like mass forms. Set aside.
Whisk together the cinnamon and cocoa in a small shallow bowl. Set aside.
Use your hands to form 1 1/2-2 inch long ovals out of the coconut mixture. Roll in the cocoa mixture to coat. Repeat for remaining coconut mixture.
*This recipe uses a lot of cinnamon, if you have a Costco (or similar warehouse store) membership, good quality cinnamon can be purchased there quite cheaply.
Growing up I had a friend who's mom would make homemade candy for every holiday. Tons of fudge for Christmas, homemade candy eggs at Easter and at St. Patrick's Day, Irish Potato Candy. I hadn't thought of them in years but after passing the remnants of the St Patrick's Day parade this weekend, I got to thinking about how odd it was that I see so many references to corned beef and cabbage; whiskey, beer and Irish soda bread this time of year but no real mention of Irish Potato Candy. I came home and googled it and it turns out there is a reason I don't hear much about the candy; it is a regional treat central to Philadelphia! Well, Baltimore is not Philadelphia but we are only 1 1/2 hrs south and have a good sized Irish-American population so it makes sense they'd trickle down here, even if in just in homemade form. Despite their name, they don't have potatoes in them, they are coconut candies rolled in cinnamon to mimic a potato pulled fresh from the ground, still covered in dirt.
This article in Serious Eats gives a good overview of the history of the candy and a behind-the-scenes peek at the Oh Ryan's Irish Potato factory in Linwood, Pa. I made mine to the best of my memory with a bit of a tweak: instead of just cinnamon, I coated them in unsweetened cocoa as well. I like cinnamon, but straight cinnamon is a bit overwhelming. You could also dip them in just cocoa powder, the effect will be the same. I did read about cream cheese versions, but I am sure the ones I had years ago did not use cream cheese so used heavy cream instead. The lack of cream cheese also means these can be stored at room temperature, in a sealed, air-tight container, up to 5 days but I will say they are best consumed the first day or two.
Boil the potatoes in a large pot of water until fork-tender. Add the green beans for the last 2 minutes of cooking. Drain and allow to cool to room temperature. Toss with the radish. Set aside.
Whisk together dressing ingredients. Drizzle over salad and toss to coat.
I was excited when Spanish Black Radishes showed up in my produce boxI was excited when Spanish Black Radishes showed up in my produce box one week. I had heard of them before but I had never had the chance to try them, they are a late winter radish and winter farmers markets are rare here in Baltimore. I was surprised when they showed up and they were huge! One radish (the one I used for this dish) was over half a pound and at least the size of softball. I had heard they were quite pungent so I grated it and made a salad with milder ingredients to balance the flavor. I found they smelled stronger than they tasted; it was sort of like fresh horseradish crossed with a regular radish and with the texture of a rutabaga. I loved the textures in the dish, the crisp beans contrasted nicely with the creamy potatoes and shredded radish. A great side dish to help you transition into spring.
Preheat oven to 350. Line a baking sheet with parchment paper, set aside.
Stir together the dressing ingredients. Spread a thin layer of dressing on the bottom half of each roll. Top with sliced corned beef, a bit of sauerkraut and a slice of Swiss. Place on the baking sheet. Bake for 5 minutes or until the cheese is fully melted. Serve immediately.
*I heated the corned beef for 4 hours on high in the slow cooker.
I'm always excited this time of year because between St Patrick's Day and Passover brisket and pre-corned beef is always on sale pretty cheaply. I like to stock up and freeze some for later in the year when it is a little more difficult to find. This year I kept it simple and made sliders; in previous years I made corned beef and cabbage bao and corned beef stuffed cabbage (among other things, check out my corned beef Pinterest board) but they were very good. Matt isn't the biggest Reuben fan (I agree, putting cheese on corned beef sounds wrong) but even he liked this version. The homemade dressing really makes it; it pulls together all of the pickling flavors and is much fresher tasting than the bottle stuff.
Cook the bacon in a non-stick skillet until crisp. Remove the bacon and drain all but 1 tablespoon of fat from the pan. Add the pimento, garlic and shallot and saute until the shallot is translucent, about 5 minutes. Add the wine or lemon juice and cook 2 minutes or until dissolved. Remove from heat and cool. Stir in the parmesan and bread crumbs.
Meanwhile, steam the clams in 1 cup water, for 3 minutes. Remove the top shells and places the clams on a baking sheet. Spoon filling on each clam, top with bacon bits and then bake 10 minutes. Serve hot.
Growing up in Baltimore and spending most of my days (even vacations) on the East Coast I've eaten a lot of seafood. Baltimore is the land of oysters and crabs but clams are also popular here and in the surrounding states. Clams Casino is a personal favorite that hails from just a bit further up the coast in Rhode Island. It can be found on menus up and down the Atlantic. There are a few variations (I even had a Clams Casino pizza this weekend!) but the basics are the same; clams, bacon, bread crumbs, peppers and shallot or onion. Most recipes call for bell peppers but I like using diced pimento, not only is it perfectly pre-diced, the pimento is peeled so you don't get bits of skin in your final dish.
Anyway, I've been looking for an excuse to make Clams Casino for years so when Lay's brand contacted me about new flavors of potato chips, I knew this was my chance! I loved prawn cocktail and chicken dinner crisps in the UK so I was sure other complex flavors can make a great chip. The classic, regional flavors of Clams Casino would translate just as well to a chip as they did the pizza I had on Staten Island this weekend. Fresh, crisp, briny, oniony with a hint of pepper; what more could a seafood lover want more?
Check out Lay's "Do Us A Flavor" Contest to see official contest rules and submit your own winning chip idea for a chance to win $1 million!
Combine all ingredients in a medium bowl until evenly distributed. Form into 4 equal patties**. Heat a small amount of canola oil in a skillet. Cook 2-3 minutes on each side or until cooked all the way through, adding cheese slices in the last minute of cooking. Serve on buns and top with caramelized onions.
One good thing about snow days is that Matt works from home and has taken over some of dinner duty. He came up with the idea of chicken burgers filled with pretzel crumbs and mustard and I helped him refine it. Chicken burgers can be tricky because ground chicken tends to be damper and stickier than beef so I didn't want to add too much in the way of wet ingredients so we used dry mustard and garlic powder instead of fresh mustard and garlic.
Chicken burgers can be on the bland side but these were anything but bland! The mustard powder and seeds added a lot of flavor without the whole burger tasting just like mustard and the pretzel was a fun alternative to breadcrumbs (and there was no reason to add additional salt!) and added a bit of pretzel to every bite. I think it was the best chicken burger I've ever had! Oh, and I know some of you have said you have a difficult time finding ground chicken, ground turkey would work too!
Preheat the oven to 350. Line 2 cookie sheets with parchment paper. In a medium sized bowl, whisk together the dry ingredients. In a large bowl, cream the butter and sugars until light and fluffy. Add the egg, zest and vanilla paste and combine thoroughly. Gradually add the dry ingredients to the butter mixture and mix until a very thick dough forms. Fold in the chips. Form cookies by dropping 1 teaspoon of dough (I use my on the sheet two inches apart. Flatten slightly then bake until light brown, about 12 minutes. Remove (using a flat turner or spatula) to a wire rack to cool.
One tip: The center of these cookies should look slightly soft (but not shiny) when they are ready to be taken out of the oven. The trick is to take them out when their bottoms are just lightly brown-generally at exactly 12 minutes. They will set up as they cool. If the cookies are large, slide them out on the parchment paper and allow them to cool 1-2 minutes on the parchment on the wire rack before removing the parchment and allowing them to cool directly on the wire rack.
This weather (still snowing and icing in March, really?) has my cooking down. I keep having to scramble together to make dinner and haven't been super inspired. Then I had a sunny respite in Orlando attending the Food Blog Forum where I ate and drank my way through all of the countries in Epcot except Morroco and England. So fun and food-inspiring! Unfortunately, I had to return to Baltimore just in time for an ice storm and now snow. My savory ideas will have to wait. I did have all the ingredients for cookies on hand so I whipped this up this dreary afternoon. I added a bit of sunshine by way of the tangelo zest and since I love citrus with five spice powder, I tossed some in too. I added chocolate chips because, why not? I like non-chocolate cookies but honestly, I always want a bit of chocolate after eating one. Don't you? Try these cookies when you need a bit of a pick-me-up.


Come out to Mannion's Pub on December 29, to celebrate the life and music of Hank Rosenthal. There will be Stevie, Elton, and even Hank Rosenthal tunes played, along with other originals and covers. 8:30 to Midnight! Grab a drink and bring friends. Promises to be a good time!


When a new entrepreneur must delegate do the job in an IT AGAIN workforce bureau or perhaps provider, the actual in general classifies with 1 of 2 different types. However these are: plan (short-term position) in addition to lasting (long-term position). Contract THE ITEM services could take only an hours to fulfill the very job information or it could take them months to accomplish . These kinds of jobs normally middle on many similar jobs such simply because software growth, pest repairs… virtually any task which may satisfy a finish. The main tasks which demand any much more permanent team typically focus on the exact areas needed desk operates, network service, maintenance along with security. If you are on the actual lookout just for several everlasting staffing needs solutions, you'll find several THEM establishments around the globe which you might give typically the work it is important to. These establishments could have staff who sadly are professional in a great many regions, watching for people to work with them . They have a tendency to function around the clock inside order to be able to satisfy the company's necessities. Working with these internet businesses are great alternatives to hiring and paying off your have IT AGAIN team.
When you want for you to stuff a good contract location, you own more options during your convenience. In particular, you can make the most of the services of just about any one about the area or intercontinental IT headhunter groups, who all have a catalog of IT AGAIN workers that may be beneficial to your small business. When you decide on this particular path on your company, you may see a great THAT offerings personnel in order to place roof top payroll regularly, should the main method turned into available. As you must bring in someone with your office, your alternatives are a good tiny bit minimal. With the types of conditions, you'll want to speak to a staffing company that can not get while skilled around employing for doing this and also excellent work situations. Even so, they will however have here are the contacts for individuals that have the exact specified credentials you are looking at to help fulfill the task requirements. Nearly all these headhunter companies refuses to have the actual same principles that happen to be witnessed along with IT staffing requirementws firms. However, they are usually within using the services of a professional obtain your work done after you don't own several choices.
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This video shows images and videos of Pillars Hotel and its amenities to display the sophisticated and elegant environment within the hotel.
Copyright © 2019 Pillars Hotel | Fort Lauderdale, Florida | Luxury Boutique Beach Accommodation. All rights reserved.


Stuart Twin Launch Steam Engine. This Twin Launch Engine was built by my great uncle. He was a WWII airplane mechanic and machinist in Chicago. Im posting four of his completed Stuart Model live steam engines. He had his own machine shop in his basement.
He built models all 25 years of his retirement. This engine was perfectly made in his home. He was a master at what he did.
He ran this on air. There isnt any rust, and it works very well. He kept this on a shelf in his library as a piece of art.
It is painted cream , and is mounted on finished hardwood. He made cutouts on the bottom so you can see the pistons moving. There is a dot of red permanent marker on 4 spots, so you can track how one of the pistons moves the engine (you can see it in the pictures). When my uncle was a teacher, used this as a visual tool to teach his students a lesson about machines.
The marker will come off very easily with a touch from a magic eraser. It is my hope that this engine will go to someone who will enjoy it as much as he did. This one is completed and has been barely used by an adult. The "Twin Launch" series of vertical steam engines is a popular type for use in model steam boats. These engines are a robust resign & if they are well engineered in the first place, will give great results in a model steam boat.
The Stuart Turner "Twin Launch" engines have 1 inch bore cylinders and are also slide valve engines. These Stuart "Twin Launch" engines are very powerful. Included are several publications that my uncle stored with this model a Live Steam magazine that talked about it, and a couple catalogs.
They are all in good, used condition (kept on a shelf in his library). There is a photo of each cover included. Live Steam Magazine: Build Rudy Kouhoupts Fore & Aft Compound Engine , March 1988. Coles Power Models Catalog , 1986 (Picture of Triple Expansion Steam Engine). Look at the pictures and ask any questions you might have. I will do my best. The item "PERFECT LIVE STEAM ENGINE Stuart Twin Launch Model, Vintage Toy, plus booklets" is in sale since Sunday, April 24, 2016.
This item is in the category "Toys & Hobbies\Models & Kits\Tools, Supplies & Engines". The seller is "maxwelshouse1" and is located in Stanwood, Washington.


PhD candidate Maya Marshak writes about a public seminar in Durban.
At the end of January I attended a public seminar on GM Contamination in the Durban Botanical gardens, organised by Biowatch South Africa and the Bio-economy Chair at UCT. Speakers included: two farmers – Thombithini Ndwandwe, co-founder of the Zimele Rural Empowerment Organisation in Mtubatuba and Petros Makhanya from Kwangwanase; Vanessa Black from Biowatch; Ignacio Chapela from the University of California, Berkeley; Rachel Wynberg from UCT, and Angelika Hilbeck from ETH-Zurich.
Whilst sitting in Cape Town and reflecting on last week's seminar the theme of drought and seed feels very relevant to write about. Cape Town is currently experiencing the worst drought in recorded history and water supplies are so low that even with severe water restrictions (25 litres per day per person) taps are predicted to run dry on the 12th of April this year. Cape Town's 4.5 million residents will have to queue for water at 200 water points throughout the city to receive daily rations. For months restrictions have meant that watering gardens, including food gardens, has not been an option. Remarkably however with below average rain for 3 years many plants have managed to survive on the mountain and in gardens. Over the past few weeks I have been noticing tomatoes and rocket shooting up in the cracks in the pavement in our neighbourhood. These plants have been rapidly growing, and putting out seed in the hope that rain will come soon so that some will have the chance of survival. It is amazing to witness the evolutionary resilience of these plant species and how this may be absolutely vital in the future of food.
Angelika Hilbeck's talk at the seminar, titled 'The GMO Push in Africa and the Drought Tolerance Trojan Horse' explored drought resistant GMOs and the many controversies surrounding this in the African context. Angelika explained how while big promises were made (at the onset of GM crops being released over 20 years ago) concerning the development of new traits and how these would solve world hunger for example, in reality very few genetic innovations have been made. In terms of maize only two significant traits have been developed, Bt (where GM plants express Bacillus thuringiensis (Bt) toxins throughout their cells which kill stem borers) and Roundup Ready (which are able to withstand the common glyphosate-based herbicide, Roundup). A further innovation has been to stack these traits so that plants express both Bt and Roundup ready characteristics.
However more recently drought resistance has been the focus of genetic engineers. Monsanto's DroughtGard contains the gene for 'cold shock protein B' (cspB) from Bacillis subtilis bacteria. While this evolved in bacteria to withstand the stress of cold shock, it is also intended to help plants survive in similarly water stressed hot conditions.
Drought has been identified as an increasing reality on the African continent in the face of climate change and in 2008 a public-private partnership known as WEMA (Water Efficient Maize for Africa) was established to focus on developing drought resistant maize varieties for the African context. Initially this involved the development of hybrid maize varieties and as explained by Angelika, was relatively successful in developing hybrids that were more tolerant to water stressed conditions. In 2015 WEMA's track changed when Monsanto became a partner organisation. At this point Monsanto donated the insect resistant trait CRY1AB which was the active trait in MON810. However MON810 was unpopular as insects quickly developed resistance to it. Another addition was the cspB gene first used in MON87460, or DroughtGard maize which was first commercially released in the United States in 2011. In South Africa WEMA intends to make these traits readily available to small-scale farmers who normally can't afford to buy GM varieties through making 'seed products available to African seed companies of all sizes, royalty free, so they can offer these hybrid seeds to smallholder farmers'.
Angelika's talk pointed to the fact that there has not been a lot of evidence to show that this innovation in genetic technology has managed to tackle the very complex issue of water stress in crops. Monsanto themselves state that it can produce 'moderate' yield improvements under 'moderate drought conditions'. It is therefore not conclusive that it is able to perform well under severe drought conditions. As explained in a report by the Union of Concerned Scientists, there are many complexities associated with drought such as that 'droughts vary in their severity and their timing in relation to crop growth. Related factors such as soil quality affect the ability of crops to withstand drought'.
The African Centre for Biodiversity have warned that through WEMA making this drought resistant maize to small-scale farmers they may be undermining the drought tolerance of farmer crops (developed over time and in situ) that are lost when farmers adopt new GM seed in the hope that it will be a silver bullet solution.
It is important that we don't lose the agricultural diversity which may hold the key to attaining resilience in a very uncertain climatic future in favour of technological solutions being put forward as the answers to such a complex problem.


text:
\section{Introduction}

In metals and semiconductors electrons interact among themselves
and with nuclei which are localized, in ordered solids, near lattice
sites.  Even after ignoring the translational degrees of
freedom of the nuclei, we are left with a host of different
many-body problems for interacting electron systems.
In elemental solids, we have $\sim 100$ different
interacting electron problems specified, for example, by the
atomic numbers and lattice constants of the periodic table
on the inside front cover of the text by Ashcroft and
Mermin.\cite{ashcroftmermin}  Problems of current interest
in many-electron physics are more likely to be found in
complicated compounds such as the quaternary cuprate
superconductors, or in artificially fabricated systems such as
metallic magnetic multilayers or semiconductor quantum dots.
The set of possible many-electron problems is endless
and, fortunately for this scientific subfield, the
history of condensed matter physics teaches
that discoveries of unanticipated properties in newly
synthesized materials will also be endless.

Given the large number of many-electron problems,
the hopeless difficulty of each, and
the difference between the largest energy
scales of these problems and the energy scales
of the excitations that determine physical properties
at room temperature and below, a common strategy of
many-body theories is to work with models that are intended
to faithfully describe essential aspects of the observed behavior
of classes of materials without introducing incidental complications.
Perhaps the most commonly studied models are the
Hubbard model, which emphasizes the physics of
strong repulsion between electrons near the same
lattice site, and the electron gas model
which mistreats atomic-like correlations
at short length scales and captures instead the
physics of screening and long-length scale correlations
in metals.  All electron {\it ab-initio} calculations
are generally limited to the consideration of
physical properties and systems for which
self-consistent-field approximations, still
present in practice even in modern
density-functional\cite{densfunc} based
electronic-structure calculations, are adequate.
This situation frequently creates a difficulty in judging
whether discrepancies between many-body theories and
experiment are due to inadequacies of a model, or due
to inadequacies of the approximations used to estimate
the properties of a model.  The case of electrons
in semiconductor quantum wells is unusual in this
respect.  At energies below the subband splitting of the
quantum well, these systems are extremely accurately
modeled by a two-dimensional (2D) version of the
electron-gas model.  In this paper we discuss some recent experimental
and theoretical work on the many-body physics of the
2D electron gas.  We limit our attention
to the one-particle Greens function, which can
be probed experimentally in these systems by 2D-2D tunneling
spectroscopy.\cite{2d2dtunexpt,ashoori,2d2dtun}

The paper is organized as follows.  In section II we
discuss the case of a 2D electron system in
a strong magnetic field with a partially filled
Landau level.  In this limit, the electronic ground
state is not a Fermi liquid and the one-particle Greens function
exhibits strongly non-perturbative behavior in which a large gap surrounding
the Fermi energy is present in the
tunneling density-of-states.
This behavior is characteristic of strongly correlated
electronic states.  In section III we discuss the
case of zero magnetic field, where experimental
results confirm the Fermi-liquid nature of the
electronic state, and tunneling experiments
imply quasiparticle lifetimes in semi-quantitative
agreement with simple random-phase-approximation
estimates.  Section IV offers some
speculations on changes that should be expected
in the zero-field tunneling density of states as
the electron density is lowered and the electronic system
becomes more strongly correlated.  We conclude in Section V
with a brief sumamry.

\section{One-particle Greens Function at Strong Magnetic Fields}

For a non-interacting electron system, the one-particle
Greens function $G_i(\epsilon)$ has a single pole
at $\epsilon = \epsilon_i$, where $\epsilon_i$ is an
eigenvalue of the one-body Hamiltonian.
Since many observables can be expressed entirely in
terms of the one-particle Greens function,\cite{mbtexts}
the way in which it is altered by interactions is a central topic
of many-body theory. In 2D systems tunneling experiments\cite{2d2dtun}
have proved particularly valuable for experimental
studies of the one-particle Greens function.  In a
tunneling-Hamiltonian formalism,\cite{wolf} the tunneling current
between weakly linked electronic systems is given at zero temperature by
\begin{equation}
I = \frac{2 \pi e}{\hbar} \sum_{i_L,i_R}
 |t(i_L,i_R) |^2 \int_{\mu}^{\mu + eV}  d \epsilon
A^{+}_{i_L}(\epsilon) A^{-}_{i_R}(\epsilon - e V).
\label{tuncond}
\end{equation}
where $\mu$ is the Fermi energy,
$L$ and $R$ labels are used to distinguish 2D
systems on opposite sides of the tunneling barrier,
$A_{i}^{+}(\epsilon)$ and $A_{i}^{-}(\epsilon)$ are the
particle and hole contributions to the spectral weight function
for single-particle label $i$ and $V$ is the bias voltage.
The spectral weights are related to the exact one-particle
Greens function as described below.  When interactions are
neglected $A_{i}^{+}(\epsilon) = \delta (\epsilon - \epsilon_i)$
if state $i$ is empty, $A_{i}^{-}(\epsilon) = \delta(\epsilon - \epsilon_i)$
if state $i$ is occupied and the spectral weights are otherwise zero.
In strongly correlated systems, the spectral function can be
qualitatively altered.

In the presence of a perpendicular magnetic field, the
single-particle kinetic-energy spectrum consists of
Landau levels\cite{leshouches} with energy $\epsilon_n =
\hbar \omega_c (n + 1/2)$ and macroscopic degeneracy
$N_{\phi} = A B / \Phi_0$.  Here $\omega_c = e B/ m c$ is
the cyclotron frequency, $A$ is the system area, $B$ is
the magnetic field strength, and $\Phi_0 = hc/e$ is the
magnetic flux quantum.  The one-particle Greens function is defined by
\begin{equation}
G_n(\epsilon) = \int_{- \infty}^{\infty}  \frac{dt}{\hbar} \exp (i \epsilon t / \hbar ) G_n(t)
= -i \int_{-\infty}^{\infty} \frac{dt}{\hbar} \exp (i \epsilon t / \hbar )
 \langle \Psi_0 | T [ c_{n,m}(t) c^{\dagger}_{n,m}(0) ] | \Psi_0 \rangle
\label{gfsf}
\end{equation}
where $c_{n,m}$ and $c_{n,m}^{\dagger}$ are respectively fermion
annihilation and creation operators for one of the degenerate
states within the $n$'th Landau level, and $|\Psi_0\rangle$ is the
ground state of the interacting electron system.\cite{caveatgf}
We restrict our attention here to $n=0$ and drop the
Landau level index on $G_{n}(\epsilon)$.  The Greens function can then
be written in the form
\begin{equation}
G(\epsilon) = \int d \epsilon ' \big[
\frac{A^{+}(\epsilon ')}{\epsilon - \epsilon ' + i \eta}
+ \frac{A^{-}(\epsilon ')}{\epsilon -\epsilon ' - i \eta} \big]
\label{gfsrep}
\end{equation}
where the particle and hole spectral functions $A^+$ and $A^-$ have
the following formal expression in terms of exact eigenstates of the
Hamiltonian:
\begin{eqnarray}
A^+(\epsilon)&=&\sum_{\alpha}|\langle \Psi_{\alpha} (N+1)
|c^{\dagger}_m|\Psi_0(N) \rangle|^2 \nonumber \\
&&  \quad \times \delta(\epsilon-[E_{\alpha}(N+1) - E_0 (N)] ) \nonumber\\
\noalign{\vskip0.2cm}
A^-(\epsilon)&=&\sum_{\alpha}|\langle | \Psi_{\alpha} (N-1) |c_m|\Psi_0(N)
\rangle  |^2 \nonumber \\
&&  \quad \times \delta(\epsilon-[E_0(N) - E_{\alpha}(N-1)])
\label{eq:tdosdef}
\end{eqnarray}
\begin{figure}[ht]
\epsfysize=3.5in
\centerline{\epsfbox{fig1.eps}}
\vspace{6pt}
\caption[]{Low temperature tunneling I-V characteristics for bulk 2D to 2D
tunneling in the fractional Hall regime.  The traces are
at magnetic fields separated by $0.25$ Tesla and for
this sample cover a range of filling factors from
$\nu = 0.48$ to $\nu = 0. 83$.  The higher bias
potential peak is due to Landau level mixing while the lower
bias potential peak is due to tunneling within the lowest
Landau level.  The tunneling current is the convolution of
particle and hole spectral weight functions, at energies
separated by $e V$.  This experiment shows that the tunneling current
is strongly suppressed near zero bias and hence that the
spectral weight functions are suppressed at energies near the
Fermi energy over a broad range of filling factors.  Features
in the tunneling data associated with particular
incompressible states, for example the one that occurs at $\nu = 2/3$, are
weak. The inset shows the onset and peak values of the intra
Landau level tunneling currents.
(After Eisenstein \textit{et al.} in Ref.11.)
}
\label{fig1}
\end{figure}

When interactions are neglected, states in the lowest Landau level
are degenerate and are occupied with probability $\nu$ where $\nu
= N/N_{\phi}$ and $N$ is the number of electrons in the many-particle system.
It follows that $A_{+}(\epsilon) = (1 - \nu)
\delta (\epsilon)$ and $A_{-}(\epsilon)=\nu \delta(\epsilon)$.
When these spectral functions are inserted in Eq.(~\ref{tuncond})
we find that $I \propto \nu (1 - \nu) \delta (eV)$.  If interactions
were unimportant, a sharp peak in the tunneling current would occur
near zero bias.  The experimental results\cite{tundossfe} of Eisenstein
{\it et al.} shown in Fig.[1] evidence quite
the opposite behavior; the tunneling current is immeasurably small
at zero bias and is peaked instead
near a bias voltage of $\sim 5 {\rm mV}$.  Since $I(V)$ depends on the
product of electron and hole spectral functions at energies within
$eV$ of the Fermi energy, it follows that both are extremely small
close to the Fermi energy.  Similar conclusions can be drawn on the
basis of 3D-2D tunneling experiments\cite{ashoori,tundossfa}
by Ashoori and collaborators.
This experimental
result can be understood in qualitative terms
as a consequence of strong correlations
in the ground state of the electron system.  When an additional
electron is added to the ground state
of an $N$ electron system, it is
not strongly correlated with the electrons already present.
The resulting $N+1$ particle state will have a small overlap with the
ground state or any low energy states of the strongly correlated $N+1$ electron
system.  $A_{+}(\epsilon)$ should therefore be peaked at energies well above the
energy difference between the ground states of the $N+1$ and
$N$ particle systems, the chemical potential $\mu$.

Despite the achievement of a number of valuable insights,\cite{sftheory}
we still do not have a completely satisfactory theory for
the spectral weight function in the strong magnetic field limit.
Nevertheless, the connection between the energy at which the tunneling
current is peaked and correlations in the ground state of
the electron system suggested by the preceding words
can be made more precise.\cite{haussmann,renn}  The following sum rules
for the zeroth and first moments of
$A(\epsilon) \equiv A_{-}(\epsilon) + A_{+}(\epsilon) $ \,
follow from Eqs.(~\ref{eq:tdosdef}):
\begin{equation}
\int d \epsilon A(\epsilon) =
\langle \Psi_0 | c_m^{\dagger} c_m + c_m c_m^{\dagger} | \Psi_0 \rangle = 1
\label{zeroth}
\end{equation}
and
\begin{eqnarray}
\int d \epsilon \epsilon A(\epsilon)
&=&\langle \Psi_0 | [H,c_m^{\dagger}] c_m +
c_m [H, c_m^{\dagger}] | \Psi_0 \rangle \nonumber \\
&=& \epsilon_{HF}.
\label{first}
\end{eqnarray}
In Eq.(~\ref{first}),
\begin{equation}
\epsilon_{HF} = \nu \sum_{m'}[ \langle m,m' | V | m, m' \rangle
- \langle m',m | V | m, m' \rangle ]
\label{hfock}
\end{equation}
is the single-particle Hartree-Fock energy for this system.
The last identity
in Eq.(~\ref{first}) follows from translational invariance
which requires that $\langle \Psi_0 | c_m^{\dagger} c_m | \Psi_0 \rangle
= \nu $ for all $m$.  These sum rules should be compared with those
for $A_{-}$:
\begin{equation}
\int d \epsilon A_{-}(\epsilon) = \langle \Psi_0 |
c_m^{\dagger} c_m | \Psi_0 \rangle =  \nu
\label{mzeroth}
\end{equation}
and
\begin{equation}
\int d \epsilon \epsilon A_{-}(\epsilon)
=\langle \Psi_0 | [H,c_m^{\dagger}] c_m | \Psi_0 \rangle  = 2 \epsilon_0(\nu).
\label{mfirst}
\end{equation}
where $\epsilon_0(\nu) = E_0/N_{\phi}$ is the many-particle ground state energy
per lowest-Landau-level orbital.  The last identity is an adaptation, to
the case of Hamiltonians that include only an interaction term,
of the expression for the ground state energy in terms of the
one-particle Greens function, derived using an equation of motion
approach in many-body theory texts.\cite{mbtexts}  Its application here rests
on the observation that
$\langle \Psi_0 | [H,c_m^{\dagger}] c_m | \Psi_0 \rangle$ is independent of $m$.
The integrated weight of the hole part of the spectral weight function
is $\nu$ and its mean energy is therefore
$ \langle \epsilon_{-} \rangle  = 2 \epsilon_0(\nu)/\nu$.
For $A(\epsilon)$, the integrated weight is $1$ and the
mean energy is $\epsilon_{HF}$.   It follows that the mean energy in the particle
part of the spectral weight function is
\begin{equation}
\langle \epsilon_{+} \rangle  = \langle \epsilon_{-} \rangle +
\frac{ \nu \epsilon_{HF} - 2 \epsilon_0(\nu)}{\nu (1 - \nu)}
\label{gap}
\end{equation}
The second term on the right hand side of this equation is a sum rule
estimate of the gap separating peaks in the hole and particle
portions of the spectral weight, $\Delta_{sr}$ .
$\Delta_{sr}$ will give an accurate value for the peak position
whenever the peak is sharp and its position well defined;
a property established for the present case by experiment.
The dependence of $\Delta_{sr}$ on $\nu$ is illustrated in Fig.[2]; when
corrected for finite quantum well width effects these results agree
well with experiment.
\begin{figure}[ht]
\epsfxsize=3.5in
\centerline{\epsfbox{fig2.eps}}
\vspace{6pt}
\caption[]{Sum rule estimate for the gap between
particle and hole contributions to the spectral weight
function of a two-dimensional electron gas at strong
magnetic fields as a function of Landau level
filling factor $\nu$.  The tunneling conductance is peaked when
$eV = \Delta_{sr}$.  These estimates are for the case of
an ideal two-dimensional electron gas and need to be
reduced to account for finite thickness effects when
comparing with experimental systems.  The gap
is expressed in units of $e^2/\epsilon \ell$ where
$\ell = (\hbar c/eB)^{1/2}$ is the magnetic length.
After Mori {\it et al.} in Ref. 14.}
\label{fig2}
\end{figure}

Since in the present case there is no one-body term in the Hamiltonian,
the ground state energy per orbital in the Hartree-Fock approximation
is $\nu \epsilon_{HF}/2$.  The gap is proportional to the amount by
which the ground-state energy lies below the Hartree-Fock approximation
ground-state energy, {\it i.e.}, the gap is proportional to the correlation
energy.  We see that in the strong-magnetic-field case where the kinetic
energy can be completely removed from the problem, the intuitive
notion that there should be a gap in the electronic spectral function
related to ground-state correlations has surprising exactness.
Now we turn to the zero magnetic field limit, where interactions
are in competition with the kinetic energy.

\section{2D Tunneling Spectroscopy of the Fermi Liquid State}

At zero magnetic field, we can label states
in the 2D electron system by two-dimensional wavevectors.  A
unique feature of 2D-2D tunneling between quantum wells in
epitaxial layered semiconductors, is the near perfect translational invariance
perpendicular to the tunneling direction which selects
momentum-conserving tunneling processes.  Accounting for
spin-degeneracy at zero field, the tunneling current is
\begin{equation}
I = \frac{4 \pi e |t|^2 }{\hbar} \sum_{\vec k}
  \int_{\mu}^{\mu + eV}  d \epsilon
A^{+}_{\vec k}(\epsilon) A^{-}_{\vec k}(\epsilon - e V).
\label{2dtuncond}
\end{equation}
For non-interacting 2D electrons, the spectral-weight function
had a delta-function peak in the hole contribution for $|\vec k| < k_F$ and
a delta-function peak in the particle contribution for $|\vec k| > k_F$
where $k_F$ is the Fermi wavevector.
The tunneling current consequently has a delta-function peak at zero
bias.  At zero magnetic field, it is generally expected that the
interacting 2D electron system should be in a
Fermi-liquid state in which a large fraction of the spectral weight lies in
a sharp Lorentzian quasiparticle peak centered at the
quasiparticle energy:
\begin{equation}
A_{\vec k}(\epsilon) = \frac{z_{\vec k}}{\pi}\frac{\Gamma_{\vec k}}{(\epsilon - E_{\vec k})^2 + \Gamma_{\vec k}^2} + A^{inc}_{\vec k}(\epsilon)
\label{flspec}
\end{equation}
Here $\Gamma_{\vec k} = \hbar / 2 \tau_{\vec k}$ where $\tau_{\vec k}$ is the
quasiparticle lifetime, $E_{\vec k}$ is the
quasiparticle energy, $z_{\vec k} < 1 $ is the quasiparticle normalization
factor, and $A^{inc}_{\vec k}$ is the part of the spectral function
non included in the quasiparticle peak.  In a Fermi liquid, the quasiparticle
lifetime diverges in the limit of zero temperature as $k$ approaches
$k_F$.

If we ignore the renormalization of the quasiparticle energy due
to interactions and the decrease of the quasiparticle lifetime
on moving away from the Fermi surface, the contribution to
the integrals in Eq.(~\ref{2dtuncond}) from the quasiparticle portion
of the spectral function can be evaluated analytically with the result
\begin{equation}
G^{qp}(V) = \frac{I(V)}{V} = \frac{z_{k_F}^2 e^2 |t|^2 A \nu_0}{\hbar}
\frac{ 2 \hbar / \tau}{ (e V)^2 + ( \hbar/ \tau)^2}
\label{tuncondfl}
\end{equation}
where $\nu_0$ is the 2D Fermi-gas density of states.  A finite
quasiparticle lifetime turns the $\delta$ function peak in
the tunneling conductance into a Lorentzian.  This is exactly what is
seen in experiment,\cite{murphybzero} providing a very direct
confirmation of the Fermi-liquid nature of the 2D electron-gas
state at zero field.  The zero-bias {\it peak} in the tunneling
conductance at zero field
stands in stark contrast with the extremely small conductance at
small bias found at strong fields and discussed in the previous section.
The 2D tunneling-spectroscopy experiment is a sensitive indicator
of the nature of the many electron state.
This simple result for the tunneling-conductance-spectroscopy line shape
applies when all quasiparticle lifetimes are approximated by their
value at $|\vec k| = k_F$.  Detailed calculations\cite{jungwirth,zheng}
show that the true line shape is not Lorentzian, but that
the voltage at which $G^{qp}$ is reduced to half of
its $V=0$ value is increased by only $20\%$ when the $\vec k$
dependence of the quasiparticle lifetime is taken into account.
Thus it is possible to read off the quasiparticle lifetime at the Fermi energy by
looking at the width of the zero-bias tunneling conductance peak.
As illustrated in Fig.[3], the measured lifetimes are in
reasonable accord with random-phase-approximation theoretical
estimates.   Theoretical and experimental lifetimes can
be brought into closer agreement by incorporating proposals
for improving on the random-phase approximation from the
electron-gas many-body-theory literature.  The improved
agreement supports the
efficacy of exchange-correlation local-field corrections
to the random phase approximation.\cite{locfield}
\begin{figure}[ht]
\epsfxsize=3.5in
\centerline{\epsfbox{fig3.eps}}
\vspace{6pt}
\caption[]{ Half-width at half-maximum for the zero-bias 2D-2D
tunneling spectroscopy peak in the Fermi liquid state of a 2D
electron gas.  This width is inversely proportional to the quasiparticle
lifetime at the Fermi energy.  Theoretical results including
local-field corrections to the RPA (full
line), theoretical results in the RPA (dashed line), and experimental
results (dotted line) reported by Murphy {\it et al.} in Ref. 17.
After Jungwirth {\it et al.} in Ref. 17.}
\label{fig3}
\end{figure}

The sharp peak in the tunneling conductance near zero bias
is due entirely to quasiparticle peaks in the spectral weight
functions.  As long as the 2D electron system is a Fermi liquid,
this peak will be present.  However, as the electron system
becomes more strongly correlated, the quasiparticle normalization
factor $z_{\vec k}$ will become smaller and the incoherent part
of the spectral function will become more dominant.  We now turn our
attention to some speculations on what will happen in this regime.

\section{Strong Correlations at Zero Magnetic Field}

A 2D electron system becomes increasingly correlated at low electron
densities.  The density is usually parameterized in terms of the
electron gas parameter $r_s$ defined by the following equation:
\begin{equation}
n = N/A = (\pi r_s^2 a_0^2 )
\label{2drs}
\end{equation}
where $a_0 = \hbar^2 \epsilon / m^* e^2$  is the host semiconductor
Bohr radius, $m^*$ is the band-structure effective mass of the
electrons, and $\epsilon$ is the host semiconductor
dielectric constant.  When $r_s$ is large, the typical
interaction-energy scale is much larger than the typical kinetic-energy
scale, and the electron system is strongly correlated.  For $r_s$
larger than\cite{wcrefs} about $30$ a phase transition between the
Fermi-liquid electronic state and a Wigner-crystal state with broken
translational symmetry is expected to occur.  Existing tunneling
experiments in 2D electron systems have been performed on samples
with $r_s < 2$, which have moderate correlations adequately described
by the random-phase approximation.  We expect that when experiments
are performed in samples with much lower density, the zero-bias
Fermi-liquid peak in the tunneling conductance will be
embedded in a broader structure with a minimum near zero bias
similar to what has already been observed for strongly correlated
states in partially filled Landau levels.  This expectation follows
from a sum-rule analysis similar to that in Section II.

At zero magnetic field, the one-particle Greens function is
diagonal in a representation of wavevectors and spins,
$\vec k$ and $\sigma$.  Although, unlike the strong field case,
the Greens function is dependent on its single-particle state
labels, it still satisfies similar identities for the zeroth and
first moments of the full spectral function:
\begin{equation}
\int_{- \infty}^{\infty} A_{\vec k, \sigma} (\epsilon) =
\langle \Psi_0 | c_{\vec k,\sigma}^{\dagger} c_{\vec k, \sigma}  +
c_{\vec k \sigma} c_{\vec k \sigma} ^{\dagger} | \Psi_0 \rangle = 1
\label{b0zeroth}
\end{equation}
and
\begin{eqnarray}
\int d \epsilon \epsilon A_{\vec k,\sigma}(\epsilon)  &=&
\langle \Psi_0 | [H,c_{\vec k,\sigma}^{\dagger}] c_{\vec k, \sigma}  +
c_{\vec k \sigma} [H,c_{\vec k \sigma} ^{\dagger}] | \Psi_0 \rangle \nonumber \\
&=& \epsilon_{\vec k} + \frac{N}{A} V(\vec q = 0)
- \frac{1}{A} \sum_{\vec k'} V(\vec k - \vec k') n_{\vec k',\sigma} \nonumber \\
&=& E_{\vec k}
\label{b0first}
\end{eqnarray}
Note that in Eq.(~\ref{b0first}), $n_{\vec k,\sigma} =
\langle \Psi_0 | c_{\vec k,\sigma}^{\dagger} c_{\vec k, \sigma} | \Psi_0 \rangle$
rather than the non-interacting Fermi gas occupation number.  If not for this
distinction, $E_{\vec k}$ would be the Hartree-Fock approximation
to the quasiparticle energy.  Below we compare the total energy of
the electron system with the quantity
\begin{equation}
\tilde E \equiv \frac{1}{2} \sum_{\vec k,\sigma} n_{\vec k,\sigma}
(\epsilon_{\vec k} + E_{\vec k}).
\label{etilde}
\end{equation}
If $n_{\vec k}$ were equal to its non-interacting value,
$\tilde E$ would be the Hartree-Fock approximation to the
ground-state energy.  Actually, $\tilde E$ will be larger than the
Hartree-Fock ground-state energy, since the kinetic energy is increased
and the exchange energy reduced in magnitude
by the partial occupation of states outside the Fermi sea.
The difference between the exact ground-state energy $E$ and
the approximate value obtained in a Hartree-Fock
approximation is generally referred to as the correlation energy
of an interacting electron system.  In the following, we will
appropriate this language in referring to the difference between $E$ and
$\tilde E$.

We employ an exact identity\cite{mbtexts} that follows from the
equation of motion for the one-particle Greens function
and relates the hole part of its spectral weight to the
total energy:
\begin{equation}
E = \frac{1}{2} \sum_{\vec k,\sigma} n_{\vec k}\big[ \epsilon_{\vec k}
+ \langle \epsilon_{\vec k}^{-} \rangle  \big] .
\label{totalenergy}
\end{equation}
Here we have noted that $n_{\vec k}$ is the zeroth moment of
$ A_{\vec k}^{(-)}(\epsilon) $ and defined the average hole
energy $\langle \epsilon_{\vec k}^{-} \rangle $ as the ratio of the first and
zeroth moments.  Comparing this expression with the expression for
$\tilde E$, we find that the correlation energy can be written in the
form:
\begin{equation}
E_{corr} \equiv E - \tilde E = \frac{-1}{2} \sum_{\vec k,\sigma}
n_{\vec k} (1 - n_{\vec k}) (\langle \epsilon_{\vec k}^{+} \rangle -
\langle \epsilon_{\vec k}^{-} \rangle )
\label{correnergy}
\end{equation}
where $\langle  \epsilon_{\vec k}^{+} \rangle $ is the mean energy in the particle
portion of the spectral weight function so that
\begin{equation}
n_{\vec k} \langle  \epsilon_{\vec k}^{-} \rangle  + (1 - n_{\vec k})
\langle  \epsilon_{\vec k} ^{+} \rangle = E_{\vec k}.
\label{eplusbar}
\end{equation}
For a low-density 2D electron gas, correlations are strong.  The correlation
energy is negative and comparable in magnitude to the total
energy per electron.  One way in which this can be consistent with
Eq.(~\ref{correnergy}) is if, for $\sim N$ values of $\vec k$,
the spectral-weight function $A_{\vec k}(\epsilon)$
has a substantial portion of its weight distributed between
{\it non-quasiparticle} peaks located near
$\langle \epsilon_{\vec k}^{-} \rangle $ and
$\langle \epsilon_{\vec k}^{+} \rangle $ and relatively little weight attached to
its dispersive quasiparticle peak.  This is precisely what happens
in the strong field limit where, in accord with the non-Fermi-liquid
character of the ground state, the quasiparticle peak is entirely absent.
We now consider two quite
different examples, in which this scenario is realized.
The generality of the behavior convinces us that it is
ubiquitous in strongly correlated fermion systems.

The one-band Hubbard model is a lattice model in which
electrons interact only if they occupy the same lattice
site.  The one-band version of this model has been
extremely widely studied\cite{elbio} in connection with the strong
correlations that exist in the planar cuprate superconductors,
especially in the underdoped regime.  Electrons in this model become strongly
correlated when the band is near half-filling and
when the on-site interaction $U$ is larger than
the intersite hopping energy $t$.  The one-particle
Greens function is readily evaluated at half-filling
in the narrow-band ($t \to 0$) limit.  The ground state has eigenenergy
$0$ and is  $2^{N}$ fold degenerate.  The zero energy states
are those with one
electron of either spin on each lattice site.  Averaging over
these states gives $n_{\vec k,\sigma} = 1/2$.
Removing an electron from any lattice site in any of the $2^{N}$ states
gives a state with zero energy so that
for each wavevector in the Brillouin zone,
$A_{\vec k,\sigma}^{-}(\epsilon) = (1/2) \delta (\epsilon ) $ and
$\langle \epsilon_{\vec k}^{-} \rangle = 0$.  Adding an electron on any lattice
site produces a state with energy $U$ so that
$A_{\vec k,\sigma}^{+}(\epsilon) = (1/2) \delta (\epsilon -U ) $ and
$\langle \epsilon_{\vec k}^{+} \rangle = U$.  These two peaks in the
spectral function are referred to as the lower and upper Hubbard
bands and they broaden when $t \ne 0$.  The mean-field single-particle
energy, calculated from the mean occupation numbers but including
exchange corrections which eliminate interactions between parallel
spins on the same site, is $E_{\vec k} = U/2$ and, taking note of
the double-counting factor for this interaction energy,
the corresponding total energy is $\tilde E = N U/4$.
In this case, the spectral weight is entirely comprised of
non-dispersive delta-function hole and particle contributions.
This non-Fermi liquid state has no dispersive quasiparticle
pole which sharpens on crossing the Fermi energy.
In precise agreement with Eq.(~\ref{correnergy}),  the two peaks
are split by an amount proportional to the correlation energy
per particle.  For finite $t$, the Hubbard model is generally
expected to have a Fermi liquid ground state away from the half-filled
band condition.  An important issue\cite{elbio} in the many-body theory
of the Hubbard model has been the question of whether the dispersive
quasiparticle peaks
at the Fermi energy are created by moving one of the Hubbard bands to the
Fermi energy or by drawing weight away from the Hubbard bands to
quasiparticle bands at the Fermi energy.  Since the correlation
energy can be strongly dependent on band-filling, the sum rules
discussed here appear to preclude the former possibility and argue
for quasiparticle bands which are energetically separate from
both quasiparticle bands.

Another example that is relevant to the case of a
2D electron gas occurs when the translational symmetry is broken
in the Wigner-crystal ground state.  The broken symmetry
requires a modification of our analysis that we will not
detail here, since the Greens function in no longer diagonal in wavevector.
Nevertheless, the connection between
strong correlations and peaks in the spectral function
that are away from the chemical potential is again present.
The chemical potential in this system is the energy
increase upon adding a particle by slightly decreasing
the lattice constant, or the energy decrease upon removing
a particle by slightly increasing the lattice constant.
The particle portion of the spectral function
has a peak at energies far above the chemical potential
since the particle can be added at an arbitrary position, including
energetically unfavorable positions near one of the crystal lattice sites.
The hole portion of the spectral-weight function is peaked at energies
well below the chemical potential, since the state with
a particle removed from a lattice site will have an energy substantially
larger than that of the adjusted lattice constant state.  The correlation
energy of the Wigner crystal state is proportional to
the separation of these two non-quasiparticle
peaks in the spectral weight function.

The 2D electron system is believed to have
a Fermi-liquid ground state at all values of $r_s$ below
those at which the transition to the Wigner-crystal
state occurs.  For large values of $r_s$, where
a relatively small fraction of the spectral weight
lies in the quasiparticle peak, the
correlation energy is very close to that of the
Wigner-crystal state.  The electronic state is
nearly as strongly correlated as the quantum-fluid states that occur
at strong magnetic fields in partially filled
Landau levels.  Based on the preceeding
examples and the sum rules discussed above, this
suggests that the non-quasiparticle portion
of the spectral function at zero field and low
electron densities will be qualitatively
similar to those at strong fields, leading to
2D-2D tunneling spectroscopy results which have
a small Fermi-liquid peak at the chemical potential
but are otherwise similar to those illustrated
in Fig.[1].  Since $n_{\vec k} \ll 1$ for all
$\vec k$ in this density regime, it follows from
Eq.(~\ref{correnergy}) that the non-Fermi-liquid
peak in the tunneling conductance should occur when
$eV$ is comparable to the correlation energy per particle.
The situation should be quite close to that
expected in Hubbard model systems where strongly dressed
quasiparticle bands are separate from the lower and
upper Hubbard bands.  Tunneling spectroscopy studies of
2D electron gas systems have the potential to provide even
more detailed information on the spectral weight function than
what has been extracted from exhaustive angle-resolved-photoemission
studies in the cuprate superconductor case.

\section{Summary}

Using the spectral-weight function of the one-particle
Greens function as an example, we have attempted to
illustrate the rich many-body physics of the
two-dimensional electron gas system.  Compared to
other strongly correlated fermion systems, the
2DEG has the advantage that it is accurately described
by a simple translationally invariant
model of fermions interacting via long-range forces.
The band structure of the host semiconductor enters only
through an effective mass parameter which is
accurately known.  At strong magnetic fields,
the 2DEG exhibits a rich variety of strong-correlation
physics that has been widely studied in connection with the
fractional quantum Hall effect.  In this regime, 2D
tunneling-spectroscopy experiments have demonstrated that the
spectral weight has a wide and deep gap
around the chemical potential.  The size of this gap can
be related to the ground-state correlation energy of
the system by invoking a first-moment sum rule for the
spectral function.  To date, most experiments on 2D electron
systems in the absence of a field have been performed
in situations where the electrons are moderately correlated.
Experiments show clear Fermi liquid behavior, which is
evidenced in 2D tunneling-spectroscopy experiments by a
{\it peak} in the tunneling conductance at the Fermi energy.
The quasiparticle lifetime can be extracted from these
measurements and is in qualitative agreement with a
simple random-phase approximation for the electronic
self-energy in the Fermi-liquid state.  Systems with lower
electron densities, which are becoming available because of
continuous advances in the epitaxial growth of
semiconductor quantum wells, are expected to be
more strongly correlated.  We propose that, in this regime,
the spectral function will develop peaks away from the
Fermi energy that are analogous to both those which
occur at strong magnetic fields and to the lower and
upper Hubbard-band peaks thta occur in strongly correlated
lattice models.  As new frontiers in materials preparation are
reached, it will be possible to confidently
compare theory and experiment for extremely strongly
correlated Fermi-liquid states.  For the 2D electron gas, tunneling
spectroscopy provides a uniquely powerful probe of a strongly
correlated Fermi liquid.  The open many-body
physics questions in the low-density 2D electron gas have a large overlap
with open issues in the many-body physics of other
systems, and especially with the physics of planar
high-temperature-superconductor materials.\cite{elbio}

\section*{Acknowledgments}

Instructive interactions with Jim Eisenstein, Rudolf Haussmann,
Tomas Jungwirth, Hiro Mori, Sheena Murphy, and Lian Zheng are gratefully
acknowledged.  I am grateful to Ulrich Zuelicke for helpful comments
on a draft of this paper.  This work was supported by
the National Science Foundation under grant DMR-9714055.

\vspace*{-9pt}

\section*{References}

redpajama_set_name: RedPajamaArXiv


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Home / Markets / Stock Markets / Demat accounts up by 41% in October, incremental additions dropping: Report
Demat accounts up by 41% in October, incremental additions dropping: Report
1 min read . Updated: 20 Nov 2022, 12:10 PM IST Livemint
The number of demat accounts rose by 41% till October this year. (PTI)
The analysis by Motilal Oswal Financial Services also stated the primary reason for the declining trend in additions of new demat accounts.
The number of demat accounts rose to 10.4 crores in October, around 41% higher than last year. However, the additions dropped to 18 lakh this October as compared to the same time period last year, according to an analysis by Motilal Oswal Financial Services.
As per PTI reports, incremental additions of demat accounts have been continuously declining since August with 26 lakh in numbers, followed by a further drop to 20 lakh in September, and 18 lakh in October 2022.
In October 2021, the incremental addition in dematerialized (demat) accounts was 36 lakh.
The analysis by Motilal Oswal Financial Services also stated the primary reason for the declining trend in additions of new demat accounts, as per PTI reports.
Roop Bhootra, CEO Investment Services, Anand Rathi Shares and Stock Brokers said, "The primary reason is the market volatility seen in the current calendar year due to global factors and relative under-performance of broader markets compared to frontline indices."
He added that a lesser number of new initial public offerings (IPOs) hitting markets this year compared to 2021 has also contributed to the lower number of Demat accounts being opened in the last few months, PTI reported.
Nitin Aggarwal, Senior Group VP, Research-Banking & Insurance, Institutional Equities, Motilal Oswal Financial Services, said since January, additions have been slowing down as the market volatility increased post the Russia-Ukraine conflict.
It is important to mention that the decline in incremental addition in October month could also be due to the festive season as there were effectively only 18 working days in October 2022 as compared to 22 working days in September.
However, Aggarwal believes that the momentum will come back along with the strength in equity markets in the longer term given the deep under-penetration.
On a month-on-month basis, the number of demat accounts grew by just 2 percent during the month under review from 10.3 crores in September.
(With PTI inputs)
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Home » Air Transport » Red Wings to move to Ulyanovsk
Red Wings to move to Ulyanovsk
Posted on December 9th, 2015 by Russian Aviation Insider in Air Transport, Russia
Red WIngs' Superjet 100 in Ulyanovsk airport (Photo by Red Wings)
Russia's Red Wings airline will change its legal address from Moscow to Ulyanovsk in 2016, the Ulyanovsk Region administration has reported. To support the relocation, the local authorities have committed to creating "a favorable administrative and taxation regime" for the carrier, in exchange for new routes from Ulyanovsk.
The airline is happy with the tax conditions proposed, says Red Wings General Director Evgeny Klyucharev. The carrier has been operating direct flights from Ulyanovsk to Moscow and Simferopol (Crimea), and is planning to add several more destinations, including St. Petersburg and Sochi, next year. Red Wings says other destinations from Uyanovsk may also include the Russian cities of Kaliningrad, Kazan, Grozny, Saratov, Rostov, Mineralnye Vody, and Makhachkala.
Ukyanovsk Region Governor Sergey Morozov says Red Wings operates Tupolev Tu-204 and Sukhoi SSJ100 aircraft; the former is built, and the lattet outfitted, at the Ulyanovsk-based Aviastar-SP facility. The Ulyanovsk airport possesses the infrastructure to maintain the airline's fleet, and a Tu-204 and SSJ100 pilot training program is also locally available.
Ilyushin Finance Co. (IFC), the leasing company preparing to acquire a 74% stake in Red Wings, is also interested in the airline's base change. An IFC subsidiary, which will be the carier's designated owner, has already obtained its legal registration in Ulyanovsk. The region's legislation facilitates leasing activities in terms of taxes, IFC head Alexander Rubtsov explains.
Red Wings is currently based at Moscow's Domodedovo airport. It operates eight Tu-204s, seven of which are on lease from IFC, and also several SSJ100s.
Dniproavia introduces Ukraine's shortest route
Sky Gates starts commercial flights
Russian government has SSJ100 plans for Red Wings
Aeroflot to purchase 100 Russian-made SSJ100 regional aircraft
LCC FlyArystan to increase its fleet to 10 aircraft and the number of its destinations by a factor of five next year


text:
package com.company;

/**
 * Created by frederik on 2/11/15.
 */
public class Employee {
    public String name;
    public int social_number;
    public int age;
    public String department;
    public int salary;
}

redpajama_set_name: RedPajamaGithub


text:
Bertrand Chamayou @ Wigmore Hall, London
by John-Pierre Joyce
published: 25 Oct 2015 in Opera and Classical Reviews
Bertrand Chamayou
(Photo: Marco Borggreve)
French pianist Bertrant Chamayou made a welcome return to the Wigmore Hall with a chronological overview of works by Maurice Ravel.
Whereas Chamayou has previously impressed audiences and critics with lesser-known pieces from the French repertoire, here he stuck to some of Ravel's most celebrated compositions, although familiarity didn't prevent him from making his own mark on these works. Pavane pour une infante défunte (1899), for example, was played as a light-footed courtly dance rather than the usual funereal dirge. Jeux d'eau (1901), too, was humorously playful, reflecting the poetic line (by Henri de Régnier) at the head of Ravel's score about the 'river god laughing at the water that tickles him'.
In some ways the neo-classical restraint of the Sonatine (1903-5) is a surprise after the impressionist sparkle of Jeux d'eau. But dig a little deeper, as Chamayou did, and the similarities become apparent. The opening Moderé and closing Animé rippled with fast moving currents, while the central minuet movement recalled the joyous Romanticism of Chabrier and his contemporaries. In Gaspard de la nuit, Chamayou was less fettered by neo-classical restraint, and invested each of the three movements with more characterisation: Ondine recalling the watery harmonies of Jeux d'eau, the sinister darkness of Le gibet, and the mocking cruelty of Scarbo.
In the second half of his recital, Chamayou presented two substantial and contrasting works: Valses nobles et sentimentales (1911) and Le tombeau de Couperin. In his hands, the harder harmonies of the Valses seemed softened, with more than a nod towards Schubert and Johann Strauss. But the first and seventh waltzes were played with a stridency, even bitterness, as if pointing towards the destructiveness of the orchestral La Valse of 1920.
As a classicist and heir of Rameau and the French Baroque tradition, Ravel didn't allow any overt Romanticism or sentimentality to seep into the finished version of Le tombeau de Couperin (1917). Each movement is dedicated to the memory of friends killed during the First World War, and the suite was completed after the devastating death of Ravel's mother. But it maintains a formal elegance and restraint that belies the painful circumstances of its composition. Chamayou, though, allowed a touch of personal feeling to enter his interpretation, particularly in the wistful Fugue and the quietly anguished Menuet. But there was also plenty of brilliance, particularly in the dizzying Toccata which – justifiably – brought members of the usually settled Wigmore Hall audience to their feet.
Further details of Wigmore Hall concerts can be found at wigmore-hall.org.uk.
Bertrant Chamayou MP3s or CDs
Bertrant Chamayou on Spotify
More on Bertrant Chamayou


GPA: 5.2606. Hometown: Plantation. Accomplishments: National Honor Society, president; Speech and Debate, vice president; Student Government Association, treasurer; Harvard Prize Book Award; QuestBridge National College Match Finalist. Planned College: Florida State University. Planned Major: Psychology.


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text:
\section{Introduction}

Belief Propagation (BP) is an algorithm finding ML solution or marginal probabilities on a graph
without loops, a tree. The algorithm was introduced in  \cite{63Gal} as
an efficient heuristic for decoding of sparse (so called graphical) codes and it
was independently considered in the context of graphical models of artificial
intelligence \cite{88Pea}. Originally the algorithm was primarily thought of as an
iterative procedure. \cite{01YFW,05YFW}, inspired by earlier works of \cite{35Bet}
and  \cite{36Pei} in statistical physics, suggested to use a more fundamental notion
of the Bethe free energy. Extrema of the Bethe free energy represent fixed points of
the iterative BP algorithm on graphs with cycles. Equations describing the
stationary points of the Bethe free energy and the fixed points of the iterative BP
are called Belief Propagation, or Bethe-Peierls, BP equations.

The significance of BP, understood as an algorithm looking for a
minimum of the Bethe free energy, was further elucidated within the
framework coined loop calculus \cite{06CCa,06CCb}. It was shown that algorithm finding an extremum of the
Bethe Free Energy is not just an approximation/heursistics in the loopy case, but it
allows explicit reconstruction of the exact inference in terms of a
series,  where each term corresponds to a loop on the graph.

If the graphical model is dense there are many loops, and thus many
contributions to the loop series. However, not all loops are equal.
Thus, considering models characterized by the same graph but
different factor functions or local weights (exact definitions will
follow) one expects strong sensitivity of an individual loop
contribution (and its significance within the loop series) on the factor
functions. In this context it is of interest to study the following question:
are there graphical models, defined on an arbitrary graph
but with specially tuned factor functions, such that BP provides
exact inference?

Positive answer to this question was given, independently and for two different
models, by  \cite{05KW} and \cite{08BSS}. It was shown in \cite{05KW} that for a
graphical model defined on an arbitrary graph in terms of binary variables with
pairwise sub-modular interaction a properly defined version of BP (linear
programming relaxation underlying the tree-reweighted method of \cite{03WJW})
yields a globally optimal Maximum Likelihood solution. This model is equivalent to
the Ferromagnetic Random Field Ising (FRFI) model popular in statistical physics of
disordered systems, see e.g. \cite{02HR}. Maximum weight matching problem on a bipartite graph
was analyzed in \cite{08BSS} and later in \cite{07SMW}, where it was shown that, in spite of the fact that the
underlying graph has many short cycles, an algorithm of BP type does converge to
correct ML assignment. This consideration was also extended to the problem of weighted b-matchings on an arbitrary graph (which is
yet another problem solvable exactly by BP) in \cite{07BBCZ,07HJ}.
Closely related general results, discussing convexified version of Bethe Free energy  and
an iterative convex-BP scheme converging to respective LP, were reported in \cite{07WYM,07GJ}.

\begin{figure}
\begin{center}
\includegraphics[width=12cm]{scheme}
\end{center}
\caption{Scheme illustrating the sequence of transformations and
relations discussed in the paper. } \label{fig:scheme}
\end{figure}

In this paper we use the Bethe free energy approach of \cite{05YFW} to suggest a
complementary and unifying explanation to these remarkable, and somehow surprising, results of
\cite{05KW,08BSS}. In two subsequent Sections we consider two models, FRFI discussed
in \cite{05KW} and a binary model with Totally Uni-Modular (TUM) constraints
generalizing the weight matching problem considered in \cite{08BSS}. Statistical
weights are defined for both models in terms of a characteristic temperature, $T$.
Our strategy in dealing with both models is illustrated in Fig.~\ref{fig:scheme}. It
consists of the following three steps.
\begin{itemize}
\item
Starting from the original setting we first go anti-clockwise,
getting an Integer Programming (IP) formulation for the ML, $T\to0$,
version of the problem. The most important feature of the two models
is that the LP relaxation of the respective IP, shown as LP-A in the
Figure, is tight/exact. In both cases this reduction from IP to LP
is exact due to the Total-Uni-Modularity (TUM) feature of the
underlying matrix of constraints.

\item Then we return to the
original setting and start moving clockwise (see the Figure), first to the Bethe
free energy formulation of the problem. We call the gedanken algorithm,  finding global
minima of the Bethe free energy, g-BP \footnote{The Bethe free energy is non-convex,
therefore funding the global minimum at a finite temperature is not necessarily
straightforward. Acknowledging importance of the problem,  we will not discuss in this manuscript plausibility and
details of respective iterative algorithm convergent to the global minimum of the Bethe free energy.
We refer interested reader to comprehensive discussion of such iterative
schemes in the general context in \cite{07WYM,07GJ} and for FRFI model and maximum weight matching model in \cite{05KW} and in \cite{08BSS,07SMW} respectively.} In the $T\to 0$ limit the
Bethe free energy turns to respective self-energy (the entropy term multiplied by
temperature is irrelevant) which is a linear functional of beliefs. Thus one gets to
an LP problem here as well, the one shown as LP-B in the Figure. This transformation
from g-BP to LP-B is analogous to similar relation between g-BP and LP-decoding
introduced in the coding theory in \cite{05FWK,03WJ}.

\item Finally, we show that LP-A and LP-B are identical, thus
demonstrating that g-BP in the $T\to 0$ limit outputs the ML solution.
\end{itemize}
Note, that convexity of the Bethe free energy at finite temperature, playing the key role in analysis of \cite{05KW,07WYM,07GJ}, is not a required
part our consideration. Moreover, the Bethe free energy of the binary model with TUM constraints is generally not convex.

\section{Ferromagnetic Random Field Ising model}

Consider an  undirected graph ${\cal G}$,  consisting of $n$
vertexes, ${\cal V}=\{1,\cdots,n\}$ and weighted edges, ${\cal E}$,
with the weight matrix $(J_{ij}|i,j=1,\cdots,n)$ such that whenever
the two vertexes are connected by an edge, i.e. $i\in j$ or $j\in
i$, $J_{ij}> 0$, and $J_{ij}=0$, otherwise. It is also useful to
introduce the directed version of the graph, ${\cal G}_d$, where any
undirected edge of ${\cal G}$ is replaced by two directed edges of
${\cal G}_d$, with the weights $J_{i\to j}=J_{j\to i}=J_{ij}/2$
respectively. The Ferromagnetic Random Field Ising (FRFI) model is
defined by the following statistical weight associated with any
configuration of ${\bm \sigma}=(\sigma_i=\pm 1|i=1,\cdots,n)$ on
${\cal V}$:
\begin{eqnarray}
 && p({\bm\sigma})=Z^{-1}\exp\left(
 \frac{1}{2T}\sum_{(i,j)\in{\cal G}}
 J_{ij}\sigma_i\sigma_j+\frac{1}{T}\sum_i
 h_i\sigma_i\right)\nonumber\\ && =Z^{-1}\exp\left(
 \frac{1}{2T}\sum_{(i\to j)\in{\cal G}_d}
 J_{i\to j}\sigma_i\sigma_j+\frac{1}{T}\sum_i h_i\sigma_i\right), \label{psigma}
\end{eqnarray}
where $h_i$ can be positive or negative; $T$ is the temperature; $Z$
is the partition function, enforcing the probability normalization
condition, $\sum_{\bm\sigma}p({\bm\sigma})=1$; and $(i,j)$/$(i\to
j)$ marks an undirected/directed edge of ${\cal G}$/${\cal G}_d$
connecting the two neighbors $i$ and $j$.

\subsection{From FRFI to the Min-Cut Problem}

Maximum Likelihood (ground state) solution of Eq.~(\ref{psigma})
turns into the problem of quadratic integer programming
\begin{eqnarray}
\left.\min_{\bm\sigma}\left(-\frac{1}{2}\sum_{(i\to j)\in{\cal
G}_d}J_{i\to j}\sigma_i\sigma_j-\sum_{i\in{\cal G}_d}
h_i\sigma_i\right)\right|_{\forall\ i\in{\cal G}_d:\ \
\sigma_i=\pm1}. \label{Quadr1}
\end{eqnarray}
It is well known that any sub-modular energy function (and the
quadratic function in Eq.~(\ref{Quadr1}) is of this type) can be
minimized in polynomial time by reducing the task to the maximum
flow/min-cut problem \cite{02BH,02HR}. In this Subsection we will
reproduce these known results.

To unify linear and quadratic terms in Eq.~(\ref{Quadr1}), one
constructs a new graph, ${\cal G}'_d$, adding two new nodes to ${\cal G}_d$:
source (s) and destination (d), with $\sigma_s=+1$ and $\sigma_d=-1$
respectively. The (s)-node is linked to all the nodes of ${\cal G}_d$ with
$h_i>0$, while any node of ${\cal G}_d$ with $h_i<0$ is linked to (d).
Weights of the newly introduced directed edges of ${\cal G}'_d$ are
\begin{eqnarray}
 J_{s\to i}=
 2h_i, \mbox{ if } h_i>0;
 \quad
 J_{i\to d}= 2|h_i|, \mbox{ if }
 h_i<0.
 \label{sd}
\end{eqnarray}
This results in the following version of Eq.~(\ref{Quadr1})
\begin{eqnarray}
\left.\min_{\bm\sigma}\left(-\frac{1}{2}\sum_{(i\to j)\in{\cal
G}'_d}J_{i\to j}\sigma_i\sigma_j\right)\right|_{\forall\ i\in{\cal
G}_d:\ \ \sigma_i=\pm1; \sigma_s=+1;\sigma_d=-1}. \label{Quadr2}
\end{eqnarray}

Reduction from quadratic integer programming (\ref{Quadr2}) to an
integer linear programming is the next step. This is achieved via
transformation to the edge variables,
\begin{eqnarray}
 \forall (i\to j)\in{\cal G}_d:\quad \eta_{i\to j}=\left\{\begin{array}{cc}
 1,& \sigma_i=1,\sigma_j=-1\\
 0,& \mbox{otherwise}.
\end{array}\right. \label{pij}
\end{eqnarray}
The relations can also be restated
\begin{eqnarray}
&& \forall (i\to j),(j\to i)\in {\cal G}_d:\quad
\sigma_i\sigma_j+\sigma_j\sigma_i=2-4(\eta_{i\to j}+\eta_{j\to i})
\label{sigma-eta}\\
 && \forall (d\to i),(j\to t):\quad
 \sigma_i\sigma_d=1-2\eta_{d\to i},\quad \sigma_s\sigma_j=1-2\eta_{j\to
 s},\label{sigma-eta_sd}
\end{eqnarray}
Therefore,  taking into account that ,  $J_{i\to j}=J_{j\to i}$ for
any $(i\to j),(j\to i)\in{\cal G}_d$, substituting
Eqs.~(\ref{pij},\ref{sigma-eta},\ref{sigma-eta_sd}) into
Eq.~(\ref{Quadr2}) and changing variables from $\sigma_i=\pm 1$ to
$p_i=(1-\sigma_i)/2=0,1$ one arrives at
\begin{equation}
 \hspace{-2cm}-\frac{1}{2}\sum\limits_{(i\to j)\in{\cal G}'_d}J_{i\to j}+
 \left.\min\limits_{\{\eta,p\}}\sum_{(i\to j)\in{\cal
 G}'_d}J_{i\to j} \eta_{i\to j}\right|_{
 \begin{array}{c}
 \forall i\in{\cal G}'_d, p_i=0,1;\\
 \forall (i\to j)\in{\cal G}'_d:\
 \ p_i-p_j+\eta_{i\to j}=0,1;\\
 p_s=0,\quad p_d=1
 \end{array} }.
 \label{min_cutIP_d}
\end{equation}
This expression is nothing but the integer programming formulation
of the famous min-cut problem, calculating the minimum weight cut
splitting all the nodes of the directed graph into two parts such
that the group including the source node has all variables in the
$0$ state while the other group, including the destination node, has
all variables in the $1$ state.

Any $\{\eta,p\}$ configuration which satisfies conditions in
Eq.~(\ref{min_cutIP_d}) requires that either $\eta_{i\to j}=0$ and $\eta_{j\to i}=1$
or $\eta_{i\to j}=1$ and $\eta_{j\to i}=0$ for any pair of directed edges $(i\to
j),(j\to i)\in {\cal G}_d$. This suggests that Eq.~(\ref{min_cutIP_d}) can  be restated in
terms of the undirected graph $G'$, equivalent to the original ${\cal G}$ supplemented by
the source and destination vertexes and edges with the following positive weights
\begin{eqnarray}
 J_{si}=
 2h_i, \mbox{ if } h_i>0;
 \quad
 J_{id}= 2|h_i|, \mbox{ if }
 h_i<0.
 \label{sd_u}
\end{eqnarray}
One derives the following undirected version of
Eq.~(\ref{min_cutIP_d})
\begin{equation}
 \hspace{-2cm}-\frac{1}{2}\sum\limits_{(i,j)\in{\cal G}'}J_{ij}+
 \left.\min\limits_{\{\eta,p\}}\sum_{(i,j)\in{\cal
 G}'}J_{ij} \eta_{i\to j}\right|_{
 \begin{array}{c}
 \forall i\in{\cal G}', p_i=0,1;\\
 \forall (i,j)\in{\cal G}':\
 \ p_i-p_j+\eta_{ij}=0,1;\\
 p_s=0,\quad p_d=1
 \end{array} }.
 \label{min_cutIP}
\end{equation}

The min-cut problem (\ref{min_cutIP}) is solvable in polynomial
time. This means, in particular, that solution of the Integer
Programming Eq.~(\ref{min_cutIP}) and solution of the respective
relaxed LP-A,
\begin{eqnarray}
 -\frac{1}{2}\sum\limits_{(i,j)\in{\cal G}'}J_{ij}+
 \left.\min\limits_{\{\eta,p\}}\sum_{(i,j)\in{\cal
 G}'}J_{ij} \eta_{ij}\right|_{
 \left\{\begin{array}{c}
 \forall i\in{\cal G}', 1\geq p_i\geq 0;\\
 \forall (i,j)\in{\cal G}':\
 p_i-p_j+\eta_{ij}\geq 0;\\
 p_s=0,\quad p_d=1
 \end{array} \right.},
 \label{min_cutLP}
\end{eqnarray}
are identical. The tightness of the relaxation is, e.g., discussed
in \cite{98PS}. (See Ch. 6.1 and specifically Theorems 6.1,6.2 in
\cite{98PS}.) Also, this observation is closely related to the fact
that the matrix of constraints in the max-flow problem,  which is
dual to Eq.~(\ref{min_cutIP}), is Totally Uni-Modular (TUM), i.e.
such that any square minor of the matrix has determinant which is
$0,+1$ or $-1$. (See e.g. Ch. 13.2 of \cite{98PS} for discussion of
the TUM IP/LP problems.)

\subsection{Bethe Free Energy and Belief Propagation for FRFI}

Discussing the FRFI model defined in Eq.~(\ref{psigma}) and
following the general heuristic approach to the graphical models,
suggested in \cite{05YFW}, one introduces beliefs, i.e. estimated
probabilities, for vertexes and edges, $b_i(\sigma_i)$,
$b_{ij}(\sigma_i,\sigma_j)$, related to each other according to
\begin{eqnarray}
\forall i\ \ \&\ \ \forall j\in i:\quad
b_i(\sigma_i)=\sum_{\sigma_j}b_{ij}(\sigma_i,\sigma_j), \label{bb}
\end{eqnarray}
and also satisfying the obvious normalization condition
\begin{eqnarray}
\forall i:\quad \sum_{\sigma_i}b_i(\sigma_i)=1. \label{norm}
\end{eqnarray}

Then the Bethe free energy functional of the beliefs is defined as
\begin{eqnarray}
 {\cal F}=E-T S,\quad E=-\frac{1}{2}\sum_{(i,j)}\sum_{\sigma_i,\sigma_j}
 b_{ij}(\sigma_i,\sigma_j)J_{ij}\sigma_i\sigma_j
 -\sum_i\sum_{\sigma_i}b_i(\sigma_i)h_i\sigma_i,\label{En}\\
 S=\sum_{(i,j)}\sum_{\sigma_i,\sigma_j}b_{ij}(\sigma_i,\sigma_j)
 \ln b_{ij}(\sigma_i,\sigma_j)
 -\sum_i\sum_{\sigma_i}b_i(\sigma_i)\ln b_i(\sigma_i).
 \label{Ent}
\end{eqnarray}

Introducing Lagrangian multipliers associated with the constraints
(\ref{bb},\ref{norm}), one defines the Lagrangian functional
\begin{equation}
 \hspace{-2cm}{\cal L}={\cal F}+\sum_i\sum_{j\in i}
 \sum_{\sigma_i}\eta_{ij}(\sigma_i)\left( b_i(\sigma_i)-\sum_{\sigma_j}b_{ij}(\sigma_i,\sigma_j)\right)
 +\sum_i\lambda_i\left(\sum_{\sigma_i}b_i(\sigma_i)-1\right).
\label{Lagr}
\end{equation}
Looking for the stationary point of the Lagrangian over all the
parameters (the beliefs and the Lagrangian multipliers) will define
the Belief Propagation (BP) equations. Iterative solution of the BP
equations constitutes the celebrated BP algorithm, which is often
used as an efficient heuristic for estimating marginal
probabilities in sparse graphical models.

\subsubsection{Ground State}

In the $T\to 0$ limit the entropy terms in the expression for the
Bethe free energy in Eqs.~(\ref{En}) can be neglected and the task
of finding the absolute minimum  of the Bethe free energy functional
turns into minimization of  the self-energy, $E$ from
Eq.~(\ref{En}), under the set of constraints (\ref{bb},\ref{norm}).
Both the optimization functional and the constraints are linear in
the beliefs, therefore one gets here the following Linear
Programming optimization:
\begin{equation}
\hspace{-2.6cm}\left.\min\limits_{\{b_i;b_{ij}\}} \left(-\!\!\!\!\sum_{(i,j)\in{\cal
G}}\sum_{\sigma_i,\sigma_j}
b_{ij}(\sigma_i,\sigma_j)\frac{J_{ij}}{2}\sigma_i\sigma_j
 \!-\!\sum_{i\in{\cal G}}\sum_{\sigma_i}b_i(\sigma_i)h_i\sigma_i\right)\right|_{
 \begin{array}{c}\forall i,(i,j)\in{\cal G}:\\
b_i(\sigma_i)=\sum_{\sigma_j}b_{ij}(\sigma_i,\sigma_j)\\
\sum_{\sigma_i}b_i(\sigma_i)=1\end{array}}, \label{LP1}
\end{equation}
where it is also assumed that all the beliefs are positive and
smaller than or equal to unity (as we are looking only for
physically sensible solutions of the optimization problem).

Making the transformation from the original graph ${\cal G}$ to its
extended version, ${\cal G}'$, i.e. introducing new edges with
weights defined in Eqs.~(\ref{sd_u}), and requiring that the spin
values of the source/destination are fixed to
 $\pm 1$ respectively, i.e. $b_{s}(+)=b_d(-)=1$ and
 $b_s(-)=b_d(+)=0$, one rewrites Eq.~(\ref{LP1}) as
\begin{equation}
\hspace{-2.6cm}\left.\min\limits_{\{b_i;b_{ij}\}} \left(-\sum_{(i,j)\in{\cal
G}'}\sum_{\sigma_i,\sigma_j}
b_{ij}(\sigma_i,\sigma_j)\frac{J_{ij}}{2}\sigma_i\sigma_j\right)\right|_{
 \begin{array}{c}\forall i,(i,j)\in{\cal G}':\quad
b_i(\sigma_i)=\sum_{\sigma_j}b_{ij}(\sigma_i,\sigma_j)\\
\forall i\in{\cal G}':\quad
\sum_{\sigma_i}b_i(\sigma_i)=1\\
b_s(+)=1\ \ \& \ \ b_d(-)=1
\end{array}}. \label{LP2}
\end{equation}

The structure of the optimization functional in Eq.~(\ref{LP2})
suggests to reduce the number of variables (beliefs), thus keeping
only the edge variables
\begin{eqnarray}
 \mu_{ij}\equiv b_{ij}(+,-)+b_{ij}(-,+)=1-b_{ij}(+,+)-b_{ij}(-,-),\label{muij}
\end{eqnarray}
defined  as the probabilities to observe the edge $(i,j)$ either in
the state $(+,-)$ or in the state $(-,+)$.  Thus,  by construction,
$1\geq \mu_{ij}\geq 0$. The $\mu_{ij}$ variables defined at
different edges are related to each other through local beliefs,
$\pi_i=b_i(-)=1-b_i(+)$, which all satisfy, $0\leq \pi_i\leq 1$.
Taking all these observations into account one rewrites
Eq.~(\ref{LP2}) as
\begin{equation}
\hspace{-2.6cm} -\frac{1}{2}\sum\limits_{(i,j)\in{\cal
G}'}\!\!J_{ij}+\left.\min\limits_{\{\mu,\pi\}}\sum_{(i,j)\in{\cal G}'}J_{ij}
\mu_{ij}\right|_{
 \begin{array}{c}
 \forall i\in{\cal G}',\ \ \forall j\in i:\
 \pi_i-\pi_j+\mu_{ij}\geq 0;\ \ \ 1\geq\mu_{ij}\geq 0\\
 \forall i\in{\cal G}':\quad 1\geq \pi_i\geq 0\\
 \pi_s=0,\quad \pi_d=1
 \end{array} }.
 \label{LP3}
\end{equation}
One finds that, up to an obvious change of variables from $\mu$ to
$\eta$ and from $\pi$ to $p$, the LP-B of Eq.~(\ref{LP3}) is
identical to the LP-A (\ref{min_cutLP}). According to the Theorem
6.1 of \cite{98PS}, solutions of Eq. (\ref{LP3}), or
Eq.~(\ref{min_cutLP}), are integers, $\forall (i,j)\in\Gamma'$
$\mu_{ij},\eta_{ij}=0,1$ and $\forall i\in G'$ $p_i,\pi_i=0,1$.

Summarizing, it was just shown that as $T\to 0$ the BP solution of
the FRFI model, understood as the global minimum of the Bethe Free
energy, is also the ML solution of the model.

\section{Binary model with Totally Uni-Modular Constraints}

Consider $N$ binary variables combined in the vector
${\bm\sigma}=(\sigma_i=0,1|i=1,\cdots,N)$, and associate the
following normalized probability to any possible value of the vector
\begin{eqnarray}
p({\bm\sigma})=Z^{-1}\exp\left(-T^{-1}\sum_i
h_i\sigma_i\right)\prod\limits_\alpha\delta\left(\sum_iJ_{\alpha
i}\sigma_i,m_\alpha\right),\label{ps1}
\end{eqnarray}
where $\delta(x,y)$ is one if $x=y$ and it is zero otherwise;
$\alpha=1,\cdots,M$ ; matrix
$\hat{J}\equiv\left(J_{\alpha
i}=0,1|i=1,\cdots,N;\alpha=1,\cdots,M\right)$ is Totally Uni-Modular
(TUM),  i.e.  determinant of any square minor
of the matrix is $0,1$ or $-1$; the vector ${\bm
m}=(m_\alpha|\alpha=1,\cdots,M)$ is constructed from positive
integers, so that $\forall\alpha$: $m_\alpha\leq
q_\alpha\equiv\sum_i J_{\alpha i}$. The partition function $Z$
is introduced in Eq.~(\ref{ps1}) to guarantee normalization, $\sum_{\bm\sigma}
p({\bm\sigma})=1$.

The model Eq.~(\ref{ps1}) can be viewed as a graphical model defined on the
bipartite graph consisting of ``bits", $\{i\}$, and ``checks", $\{\alpha\}$. Also,
there may be other graphical interpretations. Thus,  for the weighted matching
problem,  e.g. studied in \cite{08BSS}, the binary variables in the formulation of
Eq.~(\ref{ps1}), are associated with edges of the complete bipartite graph. (In this
case of the weighted matching, one can show that the resulting matrix of constraints
is indeed TUM.)

\subsection{Efficient ML solution}

We, first of all, observe that the problem of finding the Maximum
Likelihood of Eq.~(\ref{ps1}) is equivalent to the following Integer
Programming (IP)
\begin{eqnarray}
 \left.\min_{\bm\sigma}\sum_i h_i\sigma_i\right|_{
 \left\{\begin{array}{c}
 \forall i:\ \ \sigma_i=0,1\\
 \forall \alpha:\ \ \sum_i J_{\alpha i}\sigma_i=m_\alpha
 \end{array}\right.}.
 \label{IP1}
\end{eqnarray}

Relaxing the IP to respective LP-A. with $\sigma_i=0,1$ changed to
$s_i=[0;1]$,
\begin{eqnarray}
 \left.\min_{\bm\sigma}\sum_i h_i s_i\right|_{
 \left\{\begin{array}{c}
 \forall i:\ \ 0\leq s_i\leq 1\\
 \forall \alpha:\ \ \sum_i J_{\alpha i} s_i=m_\alpha
 \end{array}\right.}.
 \label{LPA1}
\end{eqnarray}
one finds that the relaxation is tight. In other words,  the
solutions of the IP problem and the LP problem are exactly
equivalent. This is due to the Theorem (see e.g. Theorem 13.1 of
\cite{98PS}) stating that if $\hat{J}$ is TUM and ${\bm m}$ is
integer, then all feasible solutions of the LP problem are integer.

\subsection{Bethe Free Energy \& BP}

Here we discuss the Bethe free energy/Belief Propagation (BP)
approach to the model defined in Eq.~(\ref{ps1}). The Bethe free
energy functional is
\begin{eqnarray}
&& F=E-TS,\quad E=\sum_i h_i b_i(1),\label{FE_TUM}\\ &&
S=\sum_\alpha\sum_{{\bm\sigma}_\alpha}b_\alpha({\bm\sigma}_\alpha)\ln
b_\alpha({\bm\sigma}_\alpha)-\sum_i\left(q_\alpha-1\right)b_i(\sigma_i)\ln
b_i(\sigma_i), \nonumber
\end{eqnarray}
where a vector ${\bm\sigma}_\alpha\equiv(\sigma_i|\forall i\mbox{
s.t. } J_{\alpha i}=1;\sum_i J_{\alpha i}\sigma_i=m_\alpha)$ defines
the set of allowed configurations of variables marked by index $i$
associated and consistent with the given constrained $\alpha$. For
any given $m_\alpha$ the number of such allowed
vectors/configurations of ${\bm\sigma}_\alpha$ is
$C_{m_\alpha}^{q_\alpha}=m_\alpha
!/((m_\alpha-q_\alpha)!q_\alpha!)$. As usual,
$b_\alpha({\bm\sigma}_\alpha)$ and $b_i(\sigma_i)$ are beliefs
(estimations for the respective probabilities) associated with the
variables and the constraints. The two types of beliefs are related
to each other via the following compatibility constraints:
\begin{eqnarray}
\forall i\ \ \&\ \ \forall \alpha\mbox{ s.t. }  J_{\alpha i}=1:\ \
b_i(\sigma_i)=\sum_{{\bm\sigma}_\alpha\setminus\sigma_i}b_\alpha({\bm\sigma}_\alpha),
\label{compL}
\end{eqnarray}
and one should also impose the normalization constraint
\begin{eqnarray}
\forall\ \ i:\ \  \sum_{\sigma_i} b_i(\sigma_i)=1. \label{normL}
\end{eqnarray}

Incorporating the compatibility and normalization constraints in the
form of Lagrangian multipliers into the variational functional one
derives the Lagrangian
\begin{equation}
 \hspace{-2cm}{\cal L}=F+\sum_i \sum_{\alpha\ni i}\mu_{\alpha i}(\sigma_i)
 \left(b_i(\sigma_i)-\sum_{{\bm\sigma}_\alpha\setminus\sigma_i}b_\alpha({\bm\sigma}_\alpha)\right)+
 \sum_i\lambda_i(\sigma_i)\left(\sum_{\sigma_i} b_i(\sigma_i)-1\right).
\label{LagrL}
\end{equation}
Looking for the stationary points of the Lagrangian with respect to
all the beliefs and the Lagrangian multipliers, $\lambda$,$\mu$, one
arrives at the Belief Propagation equations for the problem.

\subsection{$T\to 0$ limit of the Bethe free energy}

In the $T\to 0$ limit the entropy term in Eq.~(\ref{FE_TUM}) can be
dropped and the problem turns into minimization of the LP type
\begin{eqnarray}
 \left.\min_{\{b_i;b_\alpha\}} \sum_i h_i b_i(1)\right|_{
 \left\{\begin{array}{c}
 \forall i:\ \ 0\leq b_i(\sigma_i)\leq 1\\
 \forall i\ \ \&\ \ \forall \alpha\mbox{ s.t. }  J_{\alpha i}=1:\ \
 b_i(\sigma_i)=\sum_{{\bm\sigma}_\alpha\setminus\sigma_i}b_\alpha({\bm\sigma}_\alpha)\\
 \forall\ \ i:\ \  \sum_{\sigma_i} b_i(\sigma_i)=1
 \end{array}\right.}
\label{LPL1}
\end{eqnarray}

It is straightforward to verify that the beliefs associated with
$\alpha$ could be completely removed from Eq.~(\ref{LPL1}), and the
LP problem can be restated solely in terms of the $i$-related
variables, $\beta_i\equiv b_i(1)=1-b_i(0)$.

Let us illustrate this point on example of a single $\alpha$
constraint with $m_\alpha=2$ and $q_\alpha=3$. Then the set of
allowed $\alpha$-beliefs are
\begin{equation}
 b_\alpha(1,1,0),b_\alpha(1,0,1),b_\alpha(0,1,1), \label{b23_1}
\end{equation}
and the respective set of relations (\ref{compL}) between
$\beta_1,\beta_2,\beta_3$ associated with the check $\alpha$ and the
$\alpha$ beliefs are
\begin{equation}
\hspace{-2.6cm}\beta_1=b_\alpha(1,1,0)+b_\alpha(1,0,1),\ \
\beta_2=b_\alpha(1,1,0)+b_\alpha(0,1,1),\ \ \beta_3=b_\alpha(1,0,1)+b_\alpha(0,1,1).
\label{b23_2}
\end{equation}
On the other hand the normalization condition, restated in terms of
the $\alpha$-beliefs (\ref{b23_1}), is
\begin{eqnarray}
b_\alpha(1,1,0)+b_\alpha(1,0,1)+b_\alpha(0,1,1)=1. \label{b23_3}
\end{eqnarray}
Summing Eqs.~(\ref{b23_2}) and accounting for Eq.~(\ref{b23_3}), one
finds
\begin{eqnarray}
\beta_1+\beta_2+\beta_3=2. \label{b23_4}
\end{eqnarray}
In general, one finds that the relation between $\beta$ variables
associated with an $\alpha$-constraint is
\begin{eqnarray}
\sum_i J_{\alpha i}\beta_i=m_\alpha.\label{beta_rel}
\end{eqnarray}

One derives that Eq.~(\ref{LPL1}) reduces to a simpler LP-B problem
stated solely in terms of the $\beta$ variables
\begin{eqnarray}
 \left.\min_{\beta_i} \sum_i h_i \beta_i\right|_{
 \left\{
 \begin{array}{c}
 \forall i:\ \ 0\leq \beta_i\leq 1\\
 \forall \alpha:\ \ \sum_i J_{\alpha i}\beta_i=m_\alpha\end{array}\right.}.
\label{LPL2}
\end{eqnarray}

Furthermore, one observes that, up to re-definition of $\beta_i$ to
$s_i$, Eq.~(\ref{LPL2}) is equivalent to Eq.~(\ref{LP1}). In other
words, we just showed that the $T\to 0$ solution of the BP
equations, understood as the global minimum of the Bethe free
energy, is tight,  i.e. it gives exactly the ML solution of the
binary model (\ref{ps1}).

As a side remark, one notes that it is suggestive to start exploration of the Bethe Free Energy at finite $T$ from the LP solution discussed above. It might be especially useful to initiate BP with the (easy to get) LP solution when the Bethe Free Energy optimization at finite $T$ is non-convex.

\section{Summary and Path Forward}

In this work we discussed easy problems when a zero-temperature BP scheme generates exact ML result.
We argued that this special feature of BP is due to the fact that the related LP optimization is tight (i.e. the LP outputs ML solution as well). Our consideration was  based on the flexibility and convenience provided by the so-called Bethe Free Energy formulation, naturally relating BP and LP. The results were illustrated on two examples, FRFI model and perfect matching model. Also, we briefly discussed a broader class of easy examples related to LP with TUM matrix of constraints.

We conclude briefly mentioning some future challenges which follow from our analysis.
\begin{itemize}

\item It is useful to continue further exploration of other
models of statistical inference with loops allowing computationally
efficient optimal solutions. Thus, it would be interesting to find
examples of ``easy" non-binary problems, also these which allow
efficient and optimal finite temperature evaluation of marginal
probabilities or partition function. In this context, one mentions
exactness of BP marginals at any temperature known to hold for continuous variable Gaussian model on an arbitrary graph
\cite{01WF_Gaussian,06MJW} and also recently established, $T\to 0$, relation between an iterative algorithm
of BP type and Quadratic optimization problem \cite{06MV}.

\item Probably the most intriguing future challenge is to analyze problems
that are not computationally easy, but still close, in some metric,
to easy problems. Thus, the models discussed above, however
considered at finite, not zero, temperature may not allow an
explicit efficient solution. Similarly, perturbation of the FRFI
model with some number of graph local frustrations (e.g. some number
of randomly thrown negative $J_{ij}$ violating the TUM-feature of
the model) sets another ``close to easy" problem of theoretical and
applied interest. As suggested in \cite{05KW}, BP can be utilized as
an efficient heuristic in these ``close to easy" cases.
Note, that in this case finding minima of the Bethe Free energy may be a challenge, and the problem
turns into the quest of devising an efficient algorithms for the optimization of non-convex functions
\cite{02Yui,03YR}.
Here novel BP-convexifixation ideas developed in \cite{03WJW,05KW,07WYM,07GJ} might be helpful.
Notice also, that the loop calculus approach of \cite{06CCa,06CCb} is another useful
tool which may come handy in perturbative and non-perturbative analysis of these ``close to easy" problems.

\item BP is the algorithm of choice for decoding of
error-correction codes stated in terms of sparse graphs
\cite{63Gal}. On the other hand, the above discussion suggests that
for BP to decode optimally, or close to optimally, the graphical
structure should not necessarily be sparse. Therefore, an intriguing
question is: can one design a class of dense codes decoded
optimally (or close to optimally) by an algorithm of BP type?

\end{itemize}

The author acknowledges inspiring discussions with V. Chernyak, M. Vergassola, D.
Shah, B. Shraiman and M. Wainwright. The work was carried out under the auspices of
the National Nuclear Security Administration of the U.S. Department of Energy at Los
Alamos National Laboratory under Contract No. DE-AC52-06NA25396. The author also
acknowledges the Weston Visiting Professorship Program supporting his stay at the
Weizmann Institute, where the work was completed.

\section*{Bibliography}

\bibliographystyle{plain}


Prism (n.) A solid whose bases or ends are any similar, equal, and parallel plane figures, and whose sides are parallelograms.
Prism (n.) A form the planes of which are parallel to the vertical axis. See Form, n., 13.