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Hearty, full of flavors and quick, this pasta dish is perfect for a weeknight dinner. Sophisticated enough for a date night dinner, too!
This is one of my favorite pasta dishes. Mushrooms provide a rich and earthy flavor that’s brightened by asparagus and lemon. And it’s pasta after all! Generous amount of Parmesan cheese or only a little bit won’t hurt either. It only takes about 30 minutes from start to finish to make it. On a busy weeknight I need all the help I can get.
This pasta dish looks and tastes pretty sophisticated thanks to the main ingredients, asparagus, mushrooms and lemon juice. You would never guess that it is so easy to make it.
This pasta dish is wonderful with any kind of mushrooms but some mushrooms like shiitake or oysters have more flavor than crimini, for example. I tend to have the brown crimini mushrooms in my fridge, so that’s what I often use to make this pasta, but if I plan ahead I will try to buy a few different kinds of mushrooms.
Mushrooms have tons of health benefits. They provide several vitamins and minerals, boost our immune system, are a great source of antioxidants and speed up our metabolism. What is interesting is a fact that brown crimini mushrooms and portobello mushrooms are the same type of a mushroom harvested at different points in their growth.
Umami, or savory taste, is one of the 5 tastes that all humans have. The sensors on our tongues can detect sweet, salty, bitter and sour tastes in addition to the savory one. Umami is what makes a particular dish taste delicious. A very specific amino acid called glutamate and two other chemical compounds found in certain foods create the umami taste. Foods rich in umami include ripe tomatoes, aged cheese like Parmesan cheese, fish and shellfish, mushrooms, soy sauce and cured meat.
So, even though this Lemon Pasta with Mushrooms and Asparagus calls for only few ingredients, they are the “right” kind making it very delicious and sophisticated.
Start the recipe by putting a large pot of water to boil. In the meantime, trim the tough ends off of asparagus. Some chefs recommend snapping off the ends with your hands saying that the really tough part won’t snap. So, if it snaps it is nice and crunchy. I like to use a knife and if there is a resistance where I try to cut I move the knife higher. Next, cut each spear diagonally into 1 inch pieces. Put the cut asparagus into boiling and salted water and blanch for about 1 minute. With a slotted spoon transfer blanched asparagus into a bowl full of very cold water. It will stop cooking and preserve the vibrant green color. Keep the water simmering, you will use it to cook the pasta in it since the water is now full of flavor.
When the asparagus is done start preparing the mushrooms. Wash them quickly without letting them absorb too much water, then dry. Trim the ends and then cut the mushrooms into medium size pieces. If using portobello mushrooms remove the gills first by gently scraping them out using a spoon.
When the mushrooms are prepared, chop the onion and mince the garlic. Heat a large skillet up and add some olive oil. Cook the onion first, then add the garlic. After 1 minute add the mushrooms, salt and pepper. Cook for about 5 minutes then add some vegetable broth and lemon juice. Simmer for 7-10 minutes.
Return the asparagus water to boil (add more if not enough) and add pasta. Make sure the water is salty enough. While the pasta is cooking, return the drained asparagus into the skillet. Add chopped parsley and sunflower seeds. Stir and let it simmer until the pasta is cooked. The pasta I like to use here is called Lemon Pepper Pappardelle from Trader Joe’s. If you can’t get it make sure you use both lemon juice and zest to make this dish full of lemon flavor.
When the pasta is cooked drain it but leave a bit of water. Add the pasta and 4-5 tablespoons of water into the skillet. Stir everything gently. Season with more salt, pepper and lemon juice/zest if needed. Add grated Parmesan cheese, stir and serve.
Serves: 4 Ready in : 30 minutes
- 1 lb (1 bunch) asparagus, tough ends removed, cut diagonally into 1 inch pieces
- 1/4 cup olive oil
- 1 medium size onion, diced
- 3 garlic cloves, minced
- 1/2 lb mushrooms ( crimini, portobellos, shiitakes ), cleaned and quartered
- salt and pepper
- 1/2 cup vegetable broth
- juice of 1 lemon
- if using plain pasta add zest of 1 lemon
- 1/4 cup chopped parsley
- a handful of sunflower seeds
- 2 packages of Trader Joe’s Lemon Pepper Pappardelle Pasta or 1 lb fettuccine pasta
- 1/2 cup Parmesan cheese, divided in half
- Bring a big pot of salted water to boil. In the meantime, trim off tough ends of asparagus and cut the spears diagonally into 1 inch pieces.
- Blanch asparagus in boiling water for 1 minute. Transfer to a bowl full of cold water to prevent discoloration. Preserve the cooking water. Drain asparagus.
- Clean and chop the mushrooms. If using portobello mushrooms remove the gills from underside using a spoon. You want to have about 1 inch long pieces of mushrooms to match the asparagus spears.
- Heat a large skillet up and add the olive oil. Add chopped onion, cook for about 5 minutes, then add garlic. Stir a few times and add chopped mushrooms. Season with salt and pepper and cook for about 5 minutes, until softer.
- Add vegetable broth, lemon juice and zest (if using plain pasta). Simmer for 10 minutes.
- Return the cooking water to boil. Add pasta and cook 7-9 minutes. While pasta is cooking add the drained asparagus into the skillet. Simmer.
- Drain the pasta leaving a bit of water. Add pasta into the skillet and 4-5 tablespoons of water if it’s too dry. Stir and simmer for 3-4 minutes. Adjust the seasonings. Add about 1/4 cup of Parmesan cheese into the skillet.
- Serve. Pass more cheese at the table.
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domain: food


LEO BURNETT WORLDWIDE CEO TOM BERNARDIN HOLDS ANNUAL PLAN MEETING IN LEBANON
Leo Burnett Middle East North Africa (MENA) held its 2007 Annual Plan Meeting in Lebanon despite the tragic events that shook the country two months earlier as a sign of faith in the country and the agency’s unshakable spirit.
The meeting, presided over by Tom Bernardin, CEO of Leo Burnett Worldwide, brought together Farid Chehab, Chairman MENA and Chief Creative Officer CEEMEA (Central & Eastern Europe, Middle East & Africa), Raja Trad, CEO MENA region, with members of the MENA and the European region management teams.
During his two day visit, Mr. Bernardin insisted upon the importance of holding such a meeting in Beirut, which has proven time and again to be a city for the ages and a city that breaks through the clutter by defying all challenges to create world-class advertising. But most importantly, he wanted to demonstrate through his visit his and Leo Burnett Worldwide’s personal commitment to the Beirut agency and its staff.
Taking the floor at a gathering attended by all Leo Burnett and MS&L Beirut staff, Tom Bernardin said: “I am truly amazed by the outstanding creative results delivered by the Beirut office throughout the past two years. Despite the difficult and dangerous working conditions, the Beirut affiliate for Leo Burnett was committed to constantly delivering exceptional advertising, and with its creative results it kept standards as high as ever with inspirational ideas that were on a truly international level.”
Raja Trad, CEO of Leo Burnett MENA region was quoted as saying: “We are sincerely thankful to Tom Bernardin for choosing to hold the annual plan meeting in Beirut. This yearly gathering could have been done anywhere in the region, but despite the risk of having it in Lebanon, Tom insisted on holding it in Beirut as a sign of faith in our country and in the agency’s staff.”
Founded in Chicago in 1935 with eight employees and three clients, today Leo Burnett Worldwide, Inc. operates a global network of over 200 operating units across 83 markets, including 97 full-service advertising agencies and a variety of specialty marketing services including direct, database and interactive marketing, sales promotion and brand consulting. Leo Burnett's mission is to be "the best in the world, bar none, at turning brand buyers into brand believers," and does so utilizing its proprietary Brand Belief System.™ Leo Burnett currently handles many of the world's most valuable brands including McDonald's, Walt Disney, Marlboro, Kellogg’s, Tampax and Nintendo.


This story about the Kona electric SUV from Hyundai was first published by Gas2.
In the US car business today, you either build crossover SUVs and light pickup trucks or you go home. And if you really want to sell electric cars, you target them to fit into those market segments. Hyundai has three SUVs in its model lineup — the Tucson, Santa Fe, and Santa Fe Sport — but none of them goes head to head with the hottest selling compact crossovers — the Chevrolet Trax, Honda HR-V, Toyota CH-R, and Jeep Renegade. The new Hyundai Kona is supposed to fill that gap in the lineup and will be the first electric SUV from the Korean manufacturer.
The standard Kona is of little interest to the rabid electric car fans who frequent Gas2 and CleanTechnica, but it will have a battery electric version that will get the attention of EV fans. At first, the reports out of South Korea were that the electric Kona would have a 40 kWh battery, which most American carbuyers would consider too small for their needs.
But now comes word that Hyundai will offer a larger 64 kWh battery in the Kona. So equipped, it will have 500 kilometers of range. Convert that to miles and adjust for the optimistic European test cycle and you wind up with an electric SUV that should have an EPA range of about 210 miles. Now we’re talking. That puts it close to the range of the Chevy Bolt (238 miles) and the base model Tesla Model 3 (220 miles). Base price of the Kona Electric with the larger battery is projected to be a whisker under $40,000 when it arrives in the US next year.
What is interesting about the Kona is that much of its powertrain comes from LG Chem, the same folks who provide the motor and battery for the Chevy Bolt. In fact, the motor for the Kona is the same 204 horsepower unit found in the Bolt. The battery size of the Kona is just slightly larger than that supplied to General Motors for use in the Bolt.
GM has just made a major announcement that it will have two new electric cars on offer within the next 18 months. As suppliers become more involved in the manufacturing process, some convergence between models should be expected. Will the Kona and one of the new GM electric car models be twin sons of different mothers? Could be.
When it gets here, the Kona Electric will have a full complement of electronic safety features and upgraded charging that will let it take advantage of the 150 kW chargers expected to become part of the charging infrastructure in coming years. It may not be suitable for the Rubicon Trail (few vehicles are, regardless of the macho stickers on their flanks), but it could be just what many shoppers want — an all-electric crossover SUV that looks the part and has more than 200 miles of range. Bring it on!
Source: Side3 News, Norway Hat Tip: Leif Hansen
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Former Tesla Battery Expert Leading Lyten Into New Lithium-Sulfur Battery Era — Podcast:
I don't like paywalls. You don't like paywalls. Who likes paywalls? Here at CleanTechnica, we implemented a limited paywall for a while, but it always felt wrong — and it was always tough to decide what we should put behind there. In theory, your most exclusive and best content goes behind a paywall. But then fewer people read it! We just don't like paywalls, and so we've decided to ditch ours. Unfortunately, the media business is still a tough, cut-throat business with tiny margins. It's a never-ending Olympic challenge to stay above water or even perhaps — gasp — grow. So ...


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From Bangkok to Phuket: Thailand’s Hottest Slot Destinations for Gamblers
Thailand is well-known for its stunning beaches, vibrant culture, and delectable cuisine. However, many people are unaware that the country also offers a thriving gambling scene. With numerous casinos and entertainment venues, Thailand provides an exciting destination for gamblers to try their luck. In this article, we will explore the two most popular slot destinations for gambling enthusiasts: Bangkok and Phuket.
I. Gambling in Thailand:
While gambling is largely illegal in Thailand, there are several legal exceptions that cater to the growing demand for gambling experiences. These exceptions mainly include state-run lotteries and betting on horse racing, as well as underground gambling venues that operate discreetly. However, the government has recognized the potential economic benefits of legalizing and regulating gambling, leading to discussions on the establishment of integrated resorts in selected areas.
II. Bangkok: A Hub of Luxury and Entertainment:
Bangkok, Thailand’s capital city, is not only a cultural and commercial hub but also offers a range of luxurious casinos and entertainment venues for those seeking a thrilling gambling experience. The city boasts extravagant resorts, including the Grand Diamond City Casino and Resort and the NagaWorld Casino. These establishments offer an array of slot machines and table games, as well as world-class entertainment, gourmet dining, and luxurious accommodations.
III. Phuket: A Tropical Paradise with a Gambling Twist:
Phuket, one of Thailand’s most popular tourist destinations, is renowned for its stunning beaches and vibrant nightlife. In recent years, it has also emerged as a hotspot for gambling enthusiasts. While Phuket does not have land-based casinos, it offers several offshore casino cruises that allow visitors to enjoy gambling while cruising along the beautiful Andaman Sea. These cruises feature a variety of slot machines, table games, and live entertainment, creating an unforgettable gambling experience on the crystal-clear waters of Phuket.
IV. Frequently Asked Questions (FAQ):
1. Is gambling legal in Thailand?
Gambling is mostly illegal in Thailand, but there are exceptions such as state-run lotteries and horse racing betting. However, discussions are ongoing regarding the potential legalization and regulation of gambling to boost tourism and the economy.
2. Are there any legal casinos in Bangkok?
Yes, Bangkok has several legal casinos, including the Grand Diamond City Casino and Resort and the NagaWorld Casino. These establishments offer a range of gambling options, entertainment shows, and upscale accommodations.
3. Can I gamble in Phuket?
While Phuket does not have land-based casinos, it offers offshore casino cruises. These cruises provide a unique gambling experience while enjoying the stunning views of the Andaman Sea.
4. Are slot machines available in Thai casinos?
Yes, both Bangkok and Phuket offer a wide selection of slot machines alongside table games. Visitors can enjoy various slot themes and denominations, catering to different preferences and budgets.
5. What other entertainment options are available in these destinations?
Both Bangkok and Phuket offer a plethora of entertainment options beyond gambling. Visitors can explore cultural landmarks, indulge in local cuisine, shop at vibrant markets, and experience thrilling nightlife.
Thailand is emerging as a popular destination for gamblers looking for unique experiences. With legal casinos in Bangkok and innovative offshore casino cruises in Phuket, the country offers exciting slot destinations for gambling enthusiasts. While gambling in Thailand is still predominantly illegal, the government’s discussion on legalizing and regulating the industry further showcases its potential economic benefits. So, whether you fancy the hustle and bustle of Bangkok’s luxury casinos or the tranquil waters of Phuket’s offshore cruises, Thailand has something to offer every gambling enthusiast.

domain: geography


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East Coast Black Age of Comics Convention 2020
May 16, 2020 Postponed
Temple University Student Faculty Center
The East Coast Black Age of Comics Convention is an annual gathering of comic book artists, writers, and their fans who are interested in discussing, buying and selling comic books, sci-fi, action figures and related material by and / or about Black superheroes / super-powered characters / adventures. In addition, the event features panel discussions, a costume contest, self-publishing and graphic arts workshops for aspiring creators, and film screenings of works of veterans and amateurs alike.
East Coast Black Age of Comics Convention 2022 — July 16, 2022
East Coast Black Age of Comics Convention 2018 — May 19, 2018

domain: hobby


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Hangtown, Rhinoceros Theater Festival, at the Lunar Cabaret. "Nobody cares about history anymore!" laments Eva. But even as she utters these words she's surrounded by history, having returned on the occasion of her mother's death to the family farmhouse, once the site of the Last Chance Dance Saloon and Honky-Tonk. Moreover the ghosts of her ancestors come to offer advice, which plays an important part in her future plans as well as those of her sanctimonious brother and Hangtown's resident lawyer, a man who might have been her childhood sweetheart.
Heidi Broadhead's script is constructed on several levels: the characters speak of doing "scenes" or don professorial spectacles to address us from an imaginary downstage podium. And Aunt Juniper carols playful dance-hall ditties from time to time in defiance of her more churchly kin. This many conceits could easily become chaotic, but Ned O'Reilly's dexterous direction quickly dispels the initial genealogical confusion, establishing characters and their eras despite a uniformly youthful cast.
In fact the discussions are so stimulating and the personalities so distinct that by the time Eva finally chooses between Hangtown simplicity and big city glamour, we hardly care. Mere mortals and their petty decisions cannot help but be less entertaining than Broadhead's jolly clan of shades, speaking their minds with the fearless candor and wisdom that are the special province of the dead.

domain: literature


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A look at gun control’s racist history. What better way to maintain dominance over an ethnic group, than to ensure they are disarmed!
Tag Archive: Colion Noir
Anti-gun politicians know exactly what they are doing when they make misleading statements about the ability to buy guns online. In this video, Colion Noir does a fantastic job of explaining the truth about online gun sales. Want to help promote the Second Amendment? Share this post with as many people as possible!!
You know that scary Gun Show Loophole that the anti-gun politicians talk about? Guess what… there is no such thing as a Gun Show Loophole! So, why would the anti-gunners make up a phrase like that? Simple… because many people will believe it. In the following video, Colion Noir does a great job of explaining …

domain: sociology


I would like to mount a bracket onto the head tube of a bike frame to measure the degree of rotation the bike's handlebars have been turned. Ideally, this would be a contactless solution to increase the chance of the bracket fitting correctly.
Two of the options I have researched are:
- A Hall Effect sensor on the fixed head tube and a magnet on the rotating bike fork
- Optical sensor on the fixed head tube and encoded object on the rotating bike fork
Before I delve too deep into creating a Hall Effect prototype, does anyone have a better suggestion than the above options or any tips on how to best design this sensor?
Thanks in advance!
@TelaClavo @RocketMagnet Thankyou both of you guys for the detailed answers. I think due to the requirement of mounting the sensor(s)/magnet on the perimeter of the head tube I am more favoured towards TelaClavo's solution. The sensor would be located in a bracket fitting on the front of the head tube, just above the moving ring between the head tube and fork. The magnets attached around the ring.
The assumption I am making now, is that in order to accommodate head tubes of different diameters, a universal bracket can be manufactured to house the PCB components and sensors to fit flush to the head tube in both the sensor and magnets.


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Autumn lady wearing big scarf
Summer landscape with old barn..
autumn in the city very detailed
Art abstract autumn background
single beech tree at autumn
Halloween background with tree,..
Beautiful Vineyard Landscape with..
original drawing of the wine label..
Active senior drinking water from..
Panoramic view of beautiful mountain..
Beautiful sunset above the sea..
Tree by the lake on a foggy morning
Fog in the morning forest
Portrait mature woman standing..
Passionate love under the tree..
A field of golden ripe barley in..
beautiful black hair woman and..
illustrazione di halloween
Night view on Vltava river whit..
Clouds - Seamless Pattern
Autumn landscape by the pond at..
Sunset at coast of the sea
Calm river and autumn forest at..
Majestic sunset in the mountains..

domain: artistic


text:
Jump to Main Content
Mechanisms of seed ageing under different storage conditions for Vigna radiata (L.) Wilczek: lipid peroxidation, sugar hydrolysis, Maillard reactions and their relationship to glass state transition
- Murthy, U.M.N., Kumar, P.P., Sun, W.Q.
- Journal of experimental botany 2003 v.54 no.384 pp. 1057-1067
- viability, carbohydrate metabolism, seed germination, Maillard reaction, vigor, glucose, lipid peroxidation, hydrolysis, storage
- Two primary biochemical reactions in seed ageing (lipid peroxidation and non-enzymatic protein glycosylation with reducing sugars) have been studied under different seed water contents and storage temperatures, and the role of the glassy state in retarding biochemical deterioration examined. The viability loss of Vigna radiata seeds during storage is associated with Maillard reactions; however, the contribution of primary biochemical reactions varies under different storage conditions. Biochemical deterioration and viability loss are greatly retarded in seeds stored below a high critical temperature (approximately 40 degrees C above glass transition temperature). This high critical temperature corresponds to the cross-over temperature (T(c)) of glass transition where molecular dynamics changes from a solid-like system to a normal liquid system. The data show that seed ageing slows down significantly, even before seed tissue enters into the glassy state.

domain: chemistry


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Featuring the works of new members Tobias Inigo, Saundra Goodman, Vanessa McCaffery, and Harry Littell
Spirit, Dreams, Invention
Solo exhibition from Eva Capobianco
“Roots – Real and Remembered,” a show of sculpture and cross-stitched work by Eva M. Capobianco, will be held at State of the Art Gallery during May. The “real” roots in her work, Ms. Capobianco says, have been found on trails and roads near her home and beyond and they are combined with re-used bits of old furniture, glass and other objects collected over time. This series is a continuation of themes she has explored for years.
The “remembered” roots are memories from her childhood and youth, a new autobiographical series, stitched into both words and images. Not every piece in the show contains a root, she says, but this is the dominant connecting element in both series.
Meredith Ann Fuller will be in the gallery to discuss her award winning debut novel.
Kirkus Reviews has released their list of Best Indie Books of 2017 and Meredith Ann Fuller’s debut novel Quarry has made the list. Quarry portrays the difficulty of coming of age in immigrant families that have suppressed painful histories.
Fuller, who lives in Omaha, Nebraska, stated, “Being included on this prestigious list is quite humbling and very exciting for Quarry. As a writer I want to tell stories that resonate, educate, entertain and allow the reader to be a part of the narrative. I’m thrilled that Quarry is being recognized and accepted in the literary community.”
Resurrecting a lost 19th century tradition, the artist Joan Anderson has designed a
stunningly illustrated novel for adult readers of our time. Published by Mountain
Water Press, Quarry is a gorgeous book to hold and savor. Over forty illustrations
grace the pages, inviting readers to follow Rose’s journey toward understanding her
Painting and sculpture by Jane Dennis and Diana Ozolins.
Dennis’ metaphorical paintings and sculptures explore the psychological and bureaucratic structures that people create to order or disorder their world, while Ozolins’ realistic paintings celebrate the beauty and wonder of our natural world.
“Members’ Show, Part I,” works of art by half of the gallery members: Carol Abitabilo Ast, Patty Brown, Eva Capobianco, Jane Dennis, Ileen Kaplan, Jan Kather, Margy Nelson, Diana Ozolins, Nancy Ridenour, Michael Sampson, Sheryl Sinkow, Jim Spitznagel, Ethel Vrana, and Connie Zehr. On exhibition will be paintings, work in pastel, sculpture, photography, and more. Show dates: January 3 through 28, 2018 with a reception for the artists on Friday, January 5, 5-8pm. Members not showing in January will show in February.


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Here are some information on Foot Anatomy to share with you:
An adult’s foot and ankle contains:
- 26 bones
- 33 joints
- 107 ligaments
- 19 muscles and tendons
- With approx. 250,000 sweat glands
Babies feet have 22 partially developed bones, their bones are still very soft hence can be more easily damaged by abnormal pressures and forces. It is therefore vital to take a real interest in looking after their feet in the early stages of growth, as most problems/ issues they experience in adulthood could stem from possible issues experienced during their childhood (e.g. spinal pains, gait patterns, misalignment of bones etc).
Children’s feet become fully developed around the age of 18 and sometimes we as parents may overlook just how important their feet actually are. It is just as important as their eye sight, hearing abilities including their teeth, especially that they will mostly be walking on these two feet for the rest of their lives. This is why it is extremely important to provide the best foundation for your child’s feet development.
Key features of healthy shoes
Children usually take their first steps between the age of 9 to 12 months, although some children may take a bit longer. Once they start walking, it is best to have them walk around barefoot in safe environments such as their homes as much as possible. However, we strongly recommend protecting their delicate feet as soon as they start venturing independently outside etc.. While it is vital for us to do so with them wearing healthy shoes that not only protect their feet but support their development correctly.
A healthy pair of shoes should:
- Be flexible
- Have thin soles – to ease walking and promote nearly barefoot experiences
- Have wide space and closed edge protection for toes
- Have well supported ankles and toes – fixed at both places to support walking/ running (unlike flip flops)
- Be firm with padded ankle – to support the feet properly, sandals should have an enclosed back
- Be light weight – the lighter the shoe is the easier it is for feet and legs to work as they naturally should
- Have adjustable design – laces/ velcro or buckles allow for easy size adjustments

domain: medical


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Running doesn’t cause arthritis in your knees, but running injuries can raise your arthritis risk. Get tips for protecting your joints when you run.
Tight leg muscles and muscle imbalances can cause patellofemoral pain syndrome (runner’s knee). Stretching and strengthening exercises like wall squats can help.
Osteoarthritis is considered “wear-and-tear arthritis” and is the most common cause of knee pain. An orthopaedic surgeon walks us through how to find relief.
Where is your knee pain, and when does it feel better or worse? Answer these questions to help your doctor determine whether your knee pain is caused by a tendinitis, arthritis, or something else.
Cleveland Clinic is a non-profit academic medical center. Advertising on our site helps support our mission. We do not endorse non-Cleveland Clinic products or services. Policy
Whether it’s a painful shoulder, knee or elbow, tendonitis and bursitis are troublesome conditions that can affect just about anyone. The pain can come gradually, building up over time. Or it can feel sudden and severe. If you feel concerned or unsure about the cause of pain, see your doctor. But in many cases, it’s … Read More

domain: health


Distributed Time-Varying Graph Filtering
Autoregressive Moving Average Graph Filtering
One of the cornerstones of the field of signal processing on graphs are graph filters, direct analogues of classical filters, but intended for signals defined on graphs. This work brings forth new insights on the distributed graph filtering problem. We design a family of autoregressive moving average (ARMA) recursions, which (i) are able to approximate any desired graph frequency response, and (ii) give exact solutions for specific graph signal denoising and interpolation problems. The philosophy, to design the ARMA coefficients independently from the underlying graph, renders the ARMA graph filters suitable in static and, particularly, time-varying settings. The latter occur when the graph signal and/or graph topology are changing over time. We show that in case of a time-varying graph signal our approach extends naturally to a two-dimensional filter, operating concurrently in the graph and regular time domain. We also derive the graph filter behavior, as well as sufficient conditions for filter stability when the graph and signal are time-varying. The analytical and numerical results presented in this paper illustrate that ARMA graph filters are practically appealing for static and time-varying settings, as predicted by theoretical derivations.
Keywords— distributed graph filtering, signal processing on graphs, infinite impulse response graph filters, autoregressive moving average graph filters, time-varying graph signals, time-varying graphs.
Due to their ability to capture the complex relationships present in many high-dimensional datasets, graphs have emerged as a favorite tool for data analysis. Indeed, in recent years we have seen significant efforts to extend classical signal processing methods to the graph setting, where, instead of regular low-dimensional signals (e.g., a temporal or spatial signals), one is interested in graph signals, i.e., signals defined over the nodes of a graph . The introduction of a Fourier-like transform for graph signals brought the tool to analyze these signals not only in the node domain, but also in the graph frequency domain [2, 3, 4]. One of the key tools of graph signal analysis are graph filters. In a direct analogy to classical filters, graph filters process a graph signal by selectively amplifying its graph Fourier coefficients. This renders them ideal for a wide range of tasks, ranging from graph signal smoothing and denoising [5, 6], classification [7, 8, 9] and interpolation , segmentation , wavelet construction , and dictionary learning —among others.
Distributed implementations of filters on graphs emerged as a way of increasing the scalability of computation [6, 14, 15, 16]. In this way, a desired graph filtering operation is performed by only local information exchange between neighbors and there is no need for a node to have access to all the data. Nevertheless, being inspired by finite impulse response (FIR) graph filters, these methods are sensitive to time variations, such as time-varying signals and/or graphs. An alternative approach, namely distributed infinite impulse response (IIR) graph filtering, was recently proposed [17, 18]. Compared to FIR graph filters, IIR filters allow for the computation of a larger family of responses, and give exact rather than approximate solutions to specific denoising  and interpolation  problems. Yet the issue of time variations has so far been unresolved.
In a different context, we introduced IIR filter design (in fact, prior to ) using an autoregressive process called the potential kernel [19, 11]. These graph filters were shown to facilitate information processing tasks in sensor networks, such as smoothing and event region detection, but, due to their ad-hoc design, they only accomplished a limited subset of filtering objectives. In this paper, we build upon our prior work to develop more general autoregressive moving average (ARMA) graph filters of any order, using parallel or periodic concatenations of the potential kernel. The design philosophy of these graph filters allows for the approximation of any desired graph frequency response without knowing the structure of the underlying graph. In this way, we design the filter coefficients independently of the particular graph. This allows the ARMA filters to be universally applicable for any graph structure, and in particular when the graph varies over time, or when the graph structure is unknown to the designer.
Though ARMA graph filters belong to the class of IIR graph filters, they have a distinct design philosophy which bestows them the ability to filter graph signals not only in the graph frequency domain, but also in the regular temporal frequency domain (in case the graph signal is time-varying). Specifically, our design extends naturally to time-varying signals leading to two-dimensional ARMA filters: a filter in the graph domain as well as a filter in the time domain.
Our contributions are twofold:
(i) Distributed graph filters (Sections III and IV). We propose two types of autoregressive moving average (ARMA) recursions, namely the parallel and periodic implementation, which attain a rational graph frequency response. Both methods are implemented distributedly, attain fast convergence, and have message and memory requirements that are linear in the number of graph edges and the approximation order. Using a variant of Shanks’ method, we are able to design graph filters that approximate any desired graph frequency response. In addition, we give exact closed-form solutions for tasks such as Tikhonov and Wiener-based graph signal denoising and graph signal interpolation under smoothness assumptions.
(ii) Time-varying graphs and signals (Section V). We begin by providing a complete temporal characterization of ARMA graph filters w.r.t. time-varying graph signals. Our results show that the proposed recursions naturally extend to two-dimensional filters operating simultaneously in the graph-frequency domain and in the time-frequency domain. We also discuss the ARMA recursion behavior when both the graph topology and graph signal are time-varying. Specifically, we provide sufficient conditions for filter stability, and show that a decomposition basis exists (uniquely determined by the sequence of graph realizations), over which the filters achieve the same frequency response as in the static case.
Notation and terminology.
We indicate a scalar valued variable by normal letters (i.e., or ); a bold lowercase letter will indicate a vector variable and a bold upper case letter a matrix variable. With and we will indicate the eneries of and , respectively. For clarity, if needed we will refer to these entries also as and and to the -th column of as . We indicate by the absolute value of and by and the 2-norm and the spectral norm of the vector and matrix , respectively. To characterize convergence, we adopt the term linear convergence, which asserts that a recursion converges to its stationary value exponentially with time (i.e., linearly in a logarithmic scale) .
Consider an undirected graph of nodes and edges, where indicates the set of nodes and the set of edges. Let be the graph signal defined on the graph nodes, whose -th component represents the value of the signal at the -th node, denoted as .
Graph Fourier transform (GFT).
The GFT transforms a graph signal into the graph frequency domain by projecting it into the basis spanned by the eigenvectors of the graph Laplacian , typically defined as the discrete Laplacian or the normalized Laplacian . Since the Laplacian matrix of an undirected graph is symmetric, its eigenvectors form an orthonormal basis, and the forward and inverse GFTs of and are and , respectively, where the -th column of is indicated as .The corresponding eigenvalues are denoted as and will indicate the graph frequencies. For an extensive review of the properties of the GFT, we refer to [2, 3]. To avoid any restrictions on the generality of our approach, in the following, we present our results for a general representation matrix . We only require that is symmetric and local: for all , whenever and are not neighbors and otherwise. We derive our results for a class of graphs with general Laplacian matrices in some restricted set . We assume that for every the minimum eigenvalue is bounded below by and the maximum eigenvalue is bounded above by . Hence, all considered graphs have a bounded spectral norm, i.e., . For instance, when , we can take and , with related to the maximum degree of any of the graphs. When , we can consider and .
A graph filter is an operator that acts upon a graph signal by amplifying or attenuating its graph Fourier coefficients as
where denotes the usual inner product. Let and be the minimum and maximum eigenvalues of over all possible graphs. The graph frequency response controls how much amplifies the signal component of each graph frequency
We are interested in how we can filter a signal with a graph filter having a user-provided frequency response . Note that this prescribed is a continuous function in the graph frequency and describes the desired response for any graph. This approach brings benefits in those cases when the underlying graph structure is not known to the designer, or in cases the graph changes in time. The corresponding filter coefficients are thus independent of the graph and universally applicable. Using universal filters, we can design a single set of coefficients that instantiate the same graph frequency response over different bases. To illustrate the universality property, consider the application of a universal graph filter to two different graphs and of and nodes with graph frequency sets , and eigenvectors , . The filter will attain the same response over both graphs, but, in each case, supported over a different set of graph frequencies: For , filtering results in , whereas for the filtering operator will be Thus, the universality lies in the correctness to implement on all graphs, which renders it applicable for time-varying graph topologies.
It is well known that we can approximate a universal graph filter in a distributed way using a -th order polynomial of , for instance using Chebychev polynomials . Define FIR as the -th order approximation given by
where the coefficients can be both found by Chebyshev polynomial fitting  or in a least-squares sense, after a (fine) gridding of the frequency range, by minimizing
Observe that, in contrast to traditional graph filters, the order of the considered universal graph filters is not necessarily limited to . By increasing , we can approximate any filter with square integrable frequency response arbitrarily well. On the other hand, a larger FIR order implies a longer convergence time in a distributed setting, since each node requires information from all its -hop neighbors to attain the desired filter response.
To perform the filter distributedly in the network , we assume that each node is imbued with memory, computation, and communication capabilities and is in charge of calculating the local filter output . To do so, the node has to its disposal direct access to , as well as indirect access to the memory of its neighbors. For simplicity of presentation, we pose an additional restriction to the computation model: we will assume that nodes operate in synchronous rounds, each one consisting of a message exchange phase and a local computation phase. In other words, in each round may compute any (polynomial-time computable) function which has as input, variables from its local memory as well as those from the memory of its neighbors in . Since the algorithms examined in this paper are, in effect, dynamical systems, in the following we will adopt the term iteration as being synonymous to rounds. Furthermore, we assume that each iteration lasts exactly one time instant. In this context, the convergence time of an algorithm is measured in terms of the number of iterations the network needs to perform until the output closely reaches the steady-state, i.e., the asymptotic output of the dynamical system.
The computation of FIR is easily performed distributedly. Since , each node can compute the -th term from the values of the -th term in its neighborhood, in an iterative manner. The algorithm performing the FIR graph filter terminates after iterations, and if a more efficient recursive implementation is used , in total, each node exchanges values with its neighbors, meaning that, overall, the network exchanges variables, amounting to a communication cost of . Further, since for this computation each node keeps track of the values of its neighbors at every iteration, the network has a memory complexity of .
However, FIR filters exhibit poor performance when the graph signal or/and graph topology are time-varying, since the intermediate steps of the recursion cannot be computed exactly. This is for two reasons: i) First, the distributed averaging is paused after iterations, and thus the filter output is not a steady state of the iteration , which for gives as above. Accumulated errors in the computation alter the trajectory of the dynamical system, rendering intermediate states and the filter output erroneous. ii) Second, the input signal is only considered during the first iteration. To track a time-varying signal , a new FIR filter should be started at each time step having as input, significantly increasing the computation, message and memory complexities.
To overcome these issues and provide a more solid foundation for graph signal processing, we study ARMA graph filters.
Iii ARMA Graph Filters
where and are the complex numerator and denominator polynomials of order . Note that this structure resembles the frequency response of temporal ARMA filters , in which case , with being the temporal frequency. Though both polynomials are presented here to be of the same order, this is not a limitation: different orders for and are achieved trivially by setting specific constants or to zero.
Iii-a Arma graph filters
Before describing the full fledged ARMA filters, it helps first to consider a 1-st order graph filter. Besides being simpler in its construction, an ARMA lends itself as the basic building block for creating filters with a rational frequency response of any order (cf. Section III-B). We obtain ARMA filters as an extension of the potential kernel . Consider the following 1-st order recursion
for arbitrary and where the coefficients , and are complex numbers (to be specified later on). For this recursion, we can prove our first result.
The frequency response of the ARMA is
with residue and pole given by and , respectively, and with being the spectral radius bound of . Recursion (6) converges to it linearly, irrespective of the initial condition and graph Laplacian .
We can write the recursion (6a) at time in the expanded form as
When and as this recursion approaches the steady state
irrespective of . Based on Sylvester’s matrix theorem, the matrix has the same eigenvectors as and its eigenvalues are equal to . It is also well known that invertible matrices have the same eigenvectors as their inverse and eigenvalues that are the inverse of the eigenvalues of their inverse. Thus,
and the desired frequency response (7) follows by simple algebra. We arrive at (10) by considering a specific realization of , thus the set of eigenvalues is discrete. However, the same result is achieved for every other graph realization matrix with a potentially different set of eigenvalues, still in . Thus, we can write (7) for all . ∎
The stability constraint in (7) can be also understood from a dynamical system perspective. Comparing recursion (6) to a discrete-time state-space equation, it becomes apparent that, when the condition holds, recursion (6) achieves frequency response (7). It is useful to observe that, since , an increment of the stability region can be attained, if we work with a shifted version of the Laplacian and thereby decrease the spectral radius bound . For instance, the following shifted Laplacians can be considered: with and or with and . Due to its benefits, we will use the shifted versions of the Laplacians in the filter design phase and numerical results.111Note that from Sylvester’s matrix theorem, the shifted version of the Laplacians share the same eigenvectors as the original ones.
Recursion (6) leads to a simple distributed implementation of a graph filter with 1-st order rational frequency response: at each iteration , each node updates its value based on its local signal and a weighted combination of the values of its neighbors . Since each node must exchange its value with each of its neighbors, the message/memory complexity at each iteration is of order , which leads to an efficient implementation.
Note that there is an equivalence between the ARMA filter and FIR filters in approximating rational frequency responses. Indeed, the result obtained in (9) from the ARMA in can also be achieved with an FIR filter of order . Further, from (8) we can see that in finite time, i.e., and the ARMA output is equivalent to an FIR filter with coefficients .
This suggests that: the same output of an FIR filter can be obtained form (6) and (ii) the ARMA graph filter can be used to design the FIR coefficients to approximate frequency responses of the form (7). As we will see later on, due to their implementation form (6), the ARMA filters are more robust than FIRs in a time-varying scenario (time-varying graph and/or time-varying graph signal).
Iii-B Arma graph filters
Next, we use ARMA as a building block to derive distributed graph filters with a more complex frequency response. We present two constructions: The first uses a parallel bank of ARMA filters, attaining linear convergence with a communication and memory cost per iteration of . The second uses periodic coefficients in order to reduce the communication costs to , while preserving the linear convergence as the parallel ARMA filters.
Parallel ARMA filters.
A larger variety of filter responses can be obtained by adding the outputs of a parallel ARMA filter bank. Let’s denote with superscript the terms that correspond to the -th ARMA filter for . With this notation in place, the output of the ARMA filter at time instant is
where is arbitrary.
The frequency response of a parallel ARMA is
with the residues and poles , and the spectral radius of . Recursion (11) converges to it linearly, irrespective of the initial conditions and graph Laplacian .
Follows straightforwardly from Theorem 1. ∎
The frequency response of a parallel ARMA is therefore a rational function with numerator and denominator polynomials of order (presented here in a partial fraction form). In addition, since we are simply running ARMA filters in parallel, the communication and memory complexities are times that of the ARMA graph filter. Note also that the same considerations of Remark 1 can be extended to the parallel ARMA filter.
Periodic ARMA filters.
We can decrease the memory requirements of the parallel implementation by letting the filter coefficients periodically vary in time. Our periodic filters take the following form
where is the arbitrary, the output is valid every iterations, and coefficients are periodic with period : , with an integer in .
The frequency response of a periodic ARMA filter is
subject to the stability constraint
with being the spectral radius bound of . Recursion (13) converges to it linearly, irrespective of the initial condition and graph Laplacian .
(The proof is deferred to the appendix.)
By some algebraic manipulation, we can see that the frequency response of periodic ARMA filters is also a rational function of order . We can also observe that the stability criterion of parallel ARMA is more involved than that of the parallel implementation. As now we are dealing with ARMA graph filters interleaved in time, to guarantee their joint stability one does not necessarily have to examine them independently (requiring for instance that, for each , . Instead, it is sufficient that the product is smaller than one. To illustrate this, notice that if , the periodic ARMA can be stable even if some of the ARMA graph filters it is composed of are unstable.
When computing a periodic ARMA distributedly, in each iteration each node stores and exchanges values with its neighbors, yielding a memory complexity of , rather than the of the parallel one (after each iteration, the values are overwritten). On the other hand, since the output of the periodic ARMA is only valid after iterations, the communication complexity is again . The low memory requirements of the periodic ARMA render it suitable for resource constrained devices.
Iv ARMA Filter Design
In this section we focus on selecting the coefficients of our filters. We begin by showing how to approximate any given frequency response with an ARMA filter, using a variant of Shanks’ method . This approach gives us stable filters, ensuring the same selectivity as the universal FIR graph filters. Section IV-B then provides explicit (and exact) filter constructions for two graph signal processing problems which were up-to-now only approximated: Tiknohov and Wiener-based signal denoising and interpolation under smoothness assumptions [6, 23] and .
Iv-a The Filter Design Problem
Given a graph frequency response and filter order , our objective is to find the complex polynomials and of order that minimize
Even if we constrain ourselves to pass-band filters and we consider only the set of for which , it is impossible to design our coefficients based on classical design methods developed for IIR filters (e.g., Butterworth, Chebyshev). The same issue is present also using a rational fitting approach, e.g., Padé and Prony’s method. This is due to the fact that, now, the filters are rational in the variable and the notion of frequency does not stem from nor from . This differentiates the problem from the design of continuous and discrete time filters. Further, the stability constraint of ARMA is different from classical filter design, where the poles of the transfer function must lie within (not outside) the unit circle.
To illustrate this remark, consider the Butterworth-like graph frequency response , where is the desired cut-off frequency. For , one finds that it has two complex conjugate poles at . Thus, the behavior of these filters depends on the cut-off frequency and stability is not always guaranteed. For this particular case, and for a parallel implementation, whenever the filters are not stable.
Similar to Shanks’ method, we approximate the filter coefficients as follows:
Denominator. Determine for by finding a order polynomial approximation of using polynomial regression, and solving the coefficient-wise system of equations .
Numerator. Determine for by solving the least-squares problem of minimizing , w.r.t. .
Once the numerator () and denominator () coefficients of the target rational response are found:
(i) Parallel design. Perform the partial fraction expansion to find the residuals () and poles (). Then, the filter coefficients and can be found by exploiting their relation with and in Theorem 2.
(ii) Periodic design. Identify by computing the roots of the (source) denominator polynomial in (14) and equating them to the roots of the (target) denominator . It is suggested to set and for , which has the effect of putting the two polynomials in similar form. Once coefficients (and ) have been set, we obtain by equating the numerator target and source polynomials.
The method is also suitable for numerator and denominator polynomials of different orders. We advocate however the use of equal orders, because it yields the highest approximation accuracy for a given communication/memory complexity.
The most crucial step is the approximation of the denominator coefficients. By fitting to instead of , we are able to compute coefficients independently of . Increasing often leads to a (slight) increase in accuracy, but at the price of slower convergence and higher sensitivity to numerical errors (such as those caused by packet loss). Especially for sharp functions, such as the ones shown in Fig. 1, a high order polynomial approximation results in very large coefficients, which affect the numerically stability of the filters and push the poles closer to the unit circle. For this reason, in the remainder of this paper we set .
Though the proposed design method does not come with theoretical stability guarantees, it has been found to consistently produce stable filters222This has been observed while working with shifted Laplacians, and especially with the shifted normalized Laplacian .. Fig. 1 illustrates in solid lines the frequency responses of three ARMA filters (), designed to approximate a step function (top) and a window function (bottom). For reproducibility, Table I, which is featured in the Appendix, summarizes the filter coefficients of the parallel implementation for different .
Iv-B Exact and Universal Graph Filter Constructions
We proceed to present exact (and in certain cases explicit) graph filter constructions for particular graph signal denoising and interpolation problems. In contrast to previous work, the proposed filters are universal, that is they are designed without knowledge of the graph structure. Indeed, the filter coefficients are found independently from the eigenvalues of the graph Laplacian. This makes the ARMA filters suitable for any graph, and ideal for cases when the graph structure is unknown or when the complexity of the eigenvalue decomposition becomes prohibitive.
where the first term asks for a denoised signal that is close to , and the second uses the quadratic form of to penalize signals that are not smooth. In (17), admitted choices of are limited to the discrete Laplacian or the normalized Laplacian (without shifting). The positive regularization weight allows us to trade-off between the two terms of the objective function. Being a convex problem, the global minimum is found by setting the gradient to zero, resulting in
It follows that (18) can be approximated with an ARMA with frequency response
with and . From the stability condition of the parallel ARMA, we have stable filters as long as that , which for the particular expression of becomes .
where now for . Again, from the stability of ARMA , or equivalently , we now obtain stable filters as long as
are satisfied. For the shifted normalized Laplacian, and , the stability condition simplifies to
which is always met for the standard choices of (quadratic regularization) and (total variation)333Even though is a free parameter, for K = 1, 2 the value in (23) will be either 0 or -1, due to the expression of .. For these values of , and for different values of the regularized weight , we show in Fig. 2 the normalized error between the output of the ARMA recursion and the solution of the optimization problem (17), as a function of time.
When the statistical properties of the graph signal and noise are available, it is common to opt for a Wiener filter, i.e., the linear filter that minimizes the mean-squared error (MSE)
where as before is the graph signal which has been corrupted with additive noise. It is well known that, when and are zero-mean with covariance and , respectively, the solution of (24) is
given that matrix is non-singular. The above linear system can be solved by a graph filter when matrices and share the eigenvectors of the Laplacian matrix . Denote by the eigenvalue of matrix the which corresponds to the -th eigenvector of , and correspondingly . We then have that
It follows that, when and are rational functions (of ) of order , the Wiener filter corresponds to an ARMA graph filter. Notice that the corresponding filters are still universal, as the ARMA coefficients depend on the rational functions and , but not on the specific eigenvalues of the graph Laplacian. In a different context, similar results have been also observed in semi-supervised learning .
Let us illustrate the above with an example. Suppose that is normally distributed with covariance equal to the pseudoinverse of the Laplacian . This is a popular and well understood model for smooth signals on graphs with strong connections to Gaussian Markov random fields . In addition, let the noise be white with variance . Substituting this into (26), we find
which is identical to the Tikhonov-based denoising for and corresponds to an ARMA with and , which as previously shown has always stable implementation. Note that even though the stability is ensured for the considered case, it does not necessarily hold for every covariance matrix. Indeed, the stability of the filter must be examined in a problem-specific manner.
Graph signal interpolation.
Suppose that only out of the values of a signal are known, and let be the vector which contains the known values and zeros otherwise. Under the assumption of being smooth w.r.t. or , we can estimate the unknowns by the regularized problem
where is the diagonal matrix with if is known and otherwise. Such formulations have been used widely, both in the context of graph signal processing [10, 26] and earlier by the semi-supervised learning community [9, 27]. Similar to (17), this optimization problem is convex and its global minimum is found as
Most commonly, , and can be re-written as
which is an ARMA filter designed for the Laplacian matrix with the -th eigenpair of . For larger values of , the interpolation cannot be computed distributedly using ARMA filters. That is because the corresponding basis matrix cannot be appropriately factorized into a series of local matrices.
At this point we have characterized the filtering and convergence properties of ARMA graph filters for static inputs. But do these properties hold when the graph and signal are a function of time? In the following, we characterize ARMA graph filters with respect to time-variations in the graph signal (cf. Section V-A), as well as in the graph topology (cf. Section V-B).
V-a Joint Graph and Temporal Filters
To understand the impact of graph signal dynamics we broaden the analysis to a two-dimensional domain: the first dimension, as before, captures the graph (based on the graph Fourier transform), whereas the second dimension captures time (based on the Z-transform ). This technique allows us to provide a complete characterization of the ARMA filter subject to time-variations. First, we show that the ARMA filter output remains close to the correct time-varying solution (under sufficient conditions on the input), which implies that our algorithms exhibit a certain robustness to dynamics. Further, we realize that ARMA graph filters operate along both the graph and temporal dimensions. We find that a graph naturally dampens temporal frequencies in a manner that depends on its spectrum. Exploiting this finding, we extend the ARMA designs presented in Section III so as to also allow a measure of control over the temporal frequency response of the filters.
As previously, we start our exposition with the ARMA recursion (6), but now the input graph signal is time dependent (thus, indexed with the subscript )
The dimension of the above recursion can be reduced by restricting the input graph signal to lie in the subspace of an eigenvector with associated eigenvalue , i.e., , where now is a scalar and similarly, we take .444This is a standard way to derive the frequency response of the system. By orthogonality of the basis, the filter only alters the magnitude relative to the eigenvector and not the direction of . Therefore, (31) is equivalent to
where are simply the magnitudes of the vectors lying in the eigenspace of , and we can write and . Taking the Z-transform on both sides, we obtain the joint graph and temporal frequency transfer function
It can be shown that the temporal-impulse response for each graph frequency is
with the impulse function. From (34) we deduce that the region of convergence (ROC) of the filter is , for all and that the filter is causal.
The joint transfer function characterizes completely the behavior of ARMA graph filters for an arbitrary yet time-invariant graph: when , we return to the constant result and stability condition of Theorem 1, while for all other we obtain the standard frequency response as well as the graph frequency one. As one can see, recursion (31) is an ARMA filter in the graph domain as well as in the time domain. Observe also that the poles of obey the fixed relationship . This yields an interesting insight: the temporal frequency response of the filter differs along each graph frequency , meaning that temporal dynamics affecting signals lying in low graph frequency eigenspaces are dampened to a smaller extent.
The joint graph and temporal frequency transfer function of a parallel ARMA is
subject to the stability conditions of Theorem 2.
The joint graph and temporal frequency transfer function of a periodic ARMA is
subject to the stability conditions of Theorem 3.
(The proofs are deferred to the appendix.)
As in the first order case, Theorems 4 and 5 describe completely the behavior of the parallel and periodic implementations. We can see that both filters are ARMA filters in the graph and temporal domain. In particular, the parallel and periodic filters have up to distinct poles abiding respectively to
To provide further insight, Fig. 3 plots the joint graph and temporal frequency response of a parallel and a periodic graph filter of third order, both designed (only in the graph domain) to approximate an ideal low pass response with cut-off frequency . In the figure, the horizontal axis measures the graph frequency with smaller corresponding to lower variation terms. The temporal axis on the other hand measures the normalized temporal frequency such that, for , one obtains the standard graph frequency response.
We make two observations: First, both graph filters ensure almost the same frequency response as for the static case () for low temporal variations . This suggests that these filters are more appropriate for slow temporal variations. Whereas for graph signals lying in eigenspaces with close to all temporal frequencies are damped. This is a phenomenon that transcends the filter implementation (parallel or periodic) and the particular filter coefficients. It is attributed to the shifting of the Laplacian in the design phase and to the multiplicative relation of the response poles. Second, the parallel concatenation of ARMA filters results in a more stable implementation that is more fit to tolerate temporal dynamics than the periodic implementation. As shown in Figure 3, for and , temporal fluctuations with frequencies exceeding cause the periodic filter output to blow up by an order of magnitude, effectively rendering the periodic implementation unusable. The poor stability of the periodic implementation is also seen from (36), where the terms tend to push the poles closer to the unit circle, and it is the price to pay for its small memory requirements. Due to its superior stability properties and convenient form (less coefficients and simpler design), we suggest the parallel implementation for dealing with time-varying graph signals.
V-B Time-Varying Graphs and Signals
Time variations on the graph topology bring new challenges to the graph filtering problem. First, they render approaches that rely on knowledge of the graph spectrum ineffective. Approaches which ensure stability by designing the poles to lie outside the set of the Laplacian eigenvalues of a given graph, may lead to unstable filters in a different graph where some eigenvalues may over-shoot one of the poles. Due to their different design philosophy, the presented ARMA graph filters handle naturally the aforementioned issues. We can, for instance, think that the different graph realizations among time enjoy an upper bound on their maximum eigenvalue . In case this is not possible, or difficult to determine, we can always work with the normalized Laplacian and thus take . In this way, by designing the filter coefficients in order to ensure stability w.r.t. , we automatically impose stability for all different graph realizations. Furthermore, by designing once the filter coefficients for a continuous range of frequencies, the ARMA recursions also preserve the desired frequency response for different graph realizations.
The second major challenge is characterizing the graph filter behavior. Time-varying affine systems are notoriously difficult to analyze when they possess no special structure . To this end, we devise a new methodology for time-varying graph filter analysis. We show that a decomposition basis always exists, over which ARMA graph filters (and as a consequence parallel ARMA filters) have the same frequency response as in the static case. Furthermore, this decomposition basis depends only on the sequence of graph realizations.
In case of a time-varying graph topology, yet with a fixed number of nodes , as well as a time-varying graph signal, the ARMA recursion (6) can be written as
where the time-varying graph is shown by indexing with the subscript . Expanding the recursion we find that, for any sequence of graph realizations with corresponding Laplacians , the output signal is given by
where for , and otherwise.
Since the output depends on the entire sequence of graph realizations, the spectrum of any individual Laplacian is insufficient to derive the graph frequency of the filter. To extend the spectral analysis to the time-varying setting, we define a joint Laplacian matrix that encompasses all the individual shifted graph Laplacians. The intuition behind our approach is to think of a time-varying graph as one large graph that contains all nodes of the graphs , as well as directional edges connecting the nodes at different timesteps. We then interpret the spectrum of the associated Laplacian matrix as the basis for our time-varying graph Fourier transform. This idea is a generalization of the joint graph construction , used to define a Fourier transform for graph signals which change with time (though in the previous work the graph was considered time-invariant). Similar to , we will construct by replicating each node once for each timestep . Denote the -th replication of the -th node as . For each and , will then contain directed edges between and with being a neighbor of in . Therefore, in contrast to previous work, here the edges between nodes and are a function of time. By its construction, captures not only the topology of the different graphs, but also the temporal relation between them: since the exchange of information between two neighbors incurs a delay of one unit of time, at each timestep , a node has access to the values of its neighbors at .
To proceed, define to be the cyclic shift matrix with ones below the diagonal and construct as the permuted block-diagonal matrix
For consistency with the established theory on GFT, when and the graph is time-invariant, we define . Let be the -dimensional canonical unit vector with if and , otherwise. Defining as the vector of dimension which encompasses all input graph signals, we can then write
In those cases when the non-symmetric matrix enjoys an eigendecomposition, we have with (, ) the -th eigenpair. Specifically, is the -th diagonal element of and is the -th column of . The total number of eigenpairs of is . To ease the notation, we will denote as the respective -th column of .


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That same year, Sante magazine gave City Tavern its Restaurant Wine and Spirits Cooking Award, and the TV Food Network declared it the Best Historic Restaurant. And in 2001, the American Academy of Hospitality Sciences awarded Chef Staib and the City Tavern its coveted Five Star Diamond Award for their “commitment to hospitality in every aspect of [the] operation.”
A veteran of the field since 1960, Walter Staib still seems as tireless as ever. He says, “It’s very simple. I drive myself because I enjoy what I’m doing. I’m in quest for the best, whatever it is, whatever it means. City Tavern is still my primary business, but I travel the world looking for exciting concepts in dining.”
Most readers know that on July 4, 1994, Staib revived and reopened the historic tavern where representatives of the Second Continental Congress met to decide the fate of the nation. Through his extensive research into the cuisine and ambience of late 18th century America, Staib has created a unique and authentic dining experience.
Moreover, in his best-selling “City Tavern Cookbook” (Running Press, 2000, $27.50), he has culled 200 years of classic recipes from the nation’s first gourmet restaurant. Staib likes to remind us that Philadelphia, located on the Southern Trade Route, was the largest and most cosmopolitan city in the New World, and had easy access to food we still consider exotic today – mangoes, plantains, carambola, pomegranates, plus curries and hot peppers from African and West Indian sources.
The menu is rife with authentic 18th century favorites enjoyed by Philadelphia’s discerning diners, including starters such as spicy West Indies Pepperpot Soup ($5.25), with pork, beef, greens, Scotch bonnet peppers and taro root; Giant Cornmeal Fried Oysters with Piccapeppa Remoulade ($6.95); Basil Shrimp ($9.25) wrapped in apple-smoked bacon with fresh basil, fresh horseradish and barbecue sauce; and Romaine & Roquefort Salad ($7.25) with fresh bacon bits and Burgundy-Dijon vinaigrette.
Entree highlights include Roasted Duckling ($21.95) glazed with pecan honey, served with peach chutney, herbed barley and vegetables of the season; Martha Washington Style Colonial Turkey Pot Pie ($19.95); Medallions of Venison ($24.95); Braised Rabbit ($19.95) in mushroom-red wine sauce and large egg noodles served in a pewter skillet; Ale-Braised Mallard Duck Sausage ($18.85) simmered in Thomas Jefferson’s Ale, with onions, horseradish mashed potatoes and sweet and sour red cabbage; and Stuffed Shrimp ($24.95) baked with crabmeat, sherry cream sauce, herbed barley and vegetable of the season.
New Chef de Cuisine Craig Peck, who studied and later taught at the world-renowned French Culinary Institute in Manhattan, says he has always loved French peasant cuisine and finds “the hearty, historical ‘comfort food’ of City Tavern exceptional. A chef’s biggest challenge is to be creative within simple dishes, but if you do it perfectly, it’s still special.”
In this he is the perfect complement to Walter Staib, who is high on his chef de cuisine as well as on Pastry Chef Christian Gatti, a former history major who also thrives on the authenticity of the tavern’s sweet treats. Prepared fresh on the premises every day, the desserts include the likes of Pear and Sour Cherry Cobbler, Sweet Potato Cheese Cake, Custard Bread Pudding and Martha Washington Chocolate Mousse Cake.
Speaking of our first president’s devoted wife brings us full circle – to Walter Staib’s upcoming dining adventures. Never one to be lulled into complacency during the usual mid-winter doldrums, the indefatigable restaurateur has planned a series of tantalizing culinary events, beginning with Valentine’s Day Lovers’ Fare on Thursday, Feb. 14 (call for details) and continuing with Presidents’ Week at City Tavern – all to be prepared in three new high-tech kitchens the restaurateur has found time to install recently, when the tavern never skipped its normal pulse.
From Friday, Feb. 15 through Sunday, Feb. 24, the restaurant will offer a special menu featuring our first presidents’ favorite culinary delights. The First Courses will be Smoked Pheasant en Croute ($9.95), smoked pheasant (Jefferson’s favorite food at Monticello) and winter squash wrapped in puff pastry, served on a mirror of Madeira peppercorn sauce, and Crab & Corn Chowder ($4.95), a New England favorite attributed to John Adams.
Entrees include Martha Washington’s Turkey Stew ($18.95) topped with fried oysters and leeks; Veal Olive ($22), tender roulades of veal stuffed with crabmeat and topped with sherry cream sauce, served with herbed barley and seasonal vegetables; and Quails ($21) with cornmeal and pecan stuffing topped with port wine sauce accompanied by seasonal vegetables.
Each dish is served with Thomas Jefferson sweet potato and pecan biscuits. You might also enjoy washing down your meal with City Tavern’s Colonial Ale Sampler ($8), four authentic brews from the colonial era researched and re-created for the tavern by Yards Brewing Company, Philadelphia-on-Schuylkill. They include George Washington’s Porter and Thomas Jefferson’s 1774 Ale (now available for take away), plus Ben Franklin’s Special Ale and Philadelphia Pale Ale.
One week after the Presidents’ Week favorites, City Tavern will offer a special menu featuring recipes inspired by the Philadelphia Flower Show, the largest in the world, running at Pennsylvania Convention Center March 3 – 10. Besides dishes like Greenhouse Tomato stuffed with crabmeat and brie ($8.95) and Grilled Rack of Lamb Hunter Style ($23.95), served with garden vegetables in a tomato herb sauce, the flower-inspired dessert tray will include an assortment of edible flowers as garnish.
Finally, on Monday March 18, starting at 5 p.m., City Tavern will welcome Food Network mega-star David Rosengarten for a special The Book & The Cook event featuring recipes from Rosengarten’s latest cookbook, “Taste.” The dining event will cost $50 per person. Reservations are highly recommended.
So there you have it – just a few weeks in the life of one of the busiest, most innovative and most beloved culinary adventurers the region has ever known. In his spare time, he is working with Christian Gatti on a sequel to his “City Tavern Cookbook,” this one on pastry of the Colonial Era, due for publication in 2003.
He wears his mantle well as Culinary Ambassador to the City of Philadelphia. And like Ulysses, Chef Staib seems destined to wander and search for new concepts in hospitality and dining until his spirit runs dry. But given his impressive credentials and drive, don’t expect that to happen too soon.
138 S. 2nd St. at Walnut
Phone: (215) 413-1443
Fax: (215) 423-3043
Mon. – Sat., 11:30 – 4;
Sunday, 11:30 a.m. – 3 p.m.
Mon. – Thurs., 4 – 9 p.m.;
Fri. & Sat., 4 – 10 p.m.;
Sunday, 3 – 8 p.m.
All major credit cards.
Facilities for handicapped.
Smoking on outside terrace
and veranda. On-street parking
and commercial lots nearby.
Entrees: $17.95 – $25.95.


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Development of fundamental computational methods for high accuracy simulation of high speed flow and aerodynamic noise:
The key to successful high accuracy flow and acoustic simulation is to employ high order numerical methods. They are capable of maintaining low numerical dispersion and dissipation errors without using excessive number of grid points. This drastic departure from traditional CFD methods brings exciting new opportunities as well as difficult challenges. Main investigation topics include high order finite difference methods, dispersion relation preserving method, non-reflecting numerical boundary conditions, spectral element method based on orthogonal polynomials, shock capture schemes for high order methods.
H. Shen and C.K.W. Tam, "Three-Dimensional Numerical Simulation of the Jet Screech Phenomenon", AIAA Journal, v40, n1,pp.33-41, January 2002.
H. Shen and C.K.W. Tam, "The Effects of Jet Temperature and Nozzle Lip Thickness on Screech Tones", AIAA Journal, v38, n5, pp.762-767, February 2000.
H. Shen and C.K.W. Tam, "Numerical Simulation of the Generation of Axisymmetric Mode Jet Screech Tones", AIAA Journal, v36, n10, pp.1801-1807, October 1998.
H. Shen and C.K.W. Tam, "Numerical Simulation of the Jet Screech Phenomenon by Computational Aeroacoustics Method", Advances in DNS/LES, ed. by C. Liu and Z. Liu, pp. 581-588, 1997.
H. Shen, C.K.W. Tam and G. Raman, "Screech Tones of Supersonic Jets from Bevelled Rectangular Nozzles", AIAA Journal, v35, n7, pp.1119-1125, July 1997.
H. Shen and C.K.W. Tam, "Direct Computation of Nonlinear Acoustic Pulses Using High Order Finite Difference Schemes", AIAA Paper 93-4325, October 1993.


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The floatation vest options huge reflective strips for low mild conditions, a cushty seize handle, and a strong D ring for leash attachment. Vibrant colours assist increase your dog’s visibility within the water. The most effective life jacket colours life jacket for dogs are fluorescent yellow, brilliant orange, sizzling pink, and purple. We’ve compiled a list of the highest quality and most popular canine life jackets. Our dachshund Django (@djangothegent) currently wears #6 on the record.
You can’t go mistaken with any of the dog life jackets on this list. All in all, this can be a fantastically designed, premium quality canine life jacket that will present your pet with wonderful comfort and security for years to come back. The higher a dog life vest fits, the more amenable your dog might be to sporting it. It’s additionally important that you get a vest with adjustable straps, thus permitting you to provide your canine the most snug match. A dog life jacket and not using a grab or rescue handle makes it troublesome to assist your dog out of the water. It’s an especially useful feature for instances when you’re on a ship or have to get your canine out quickly. If not for its super-excessive worth, this might need been our total greatest canine life jacket.
Designed for more excessive activities than paddling in the pool, this vest is another that provides a bit of extra buoyancy to keep your canine afloat in sudden conditions. It will also maintain them in an upright place in choppier waters.
The flotation is distributed evenly and keeps the canines horizontal in the water. The use of Velcro additionally prevents the jacket from loosening once it turns into moist. However, some canine owners commented that the Velcro tended to draw shedding hair. The orange Outward Hound is actually quite fashionable with silver reflective piping so that it can be seen easily day or night time.
Veterinarians recommend they’re adjusted for a comfortable fit, but not so cosy that your pup chafes or can’t move round freely. Let’s take a look at the various kinds of canine life preservers provided and how to choose the best dimension on your furry pal. We get lots of questions about the differences between our dog life vests, so we prepared a fast guide under that can assist you get started.
However, should you measure the widest space round their rib cage, you’ll get your dog’s girth properly ninety eight.5% of the time. The first time I measured my Pug, I had no clue what to measure for.
Odds are, your canine will love the exercise and will have the best day of their lifeevery single time. Reviewers recognize the affordability of the Vivaglory Jacket and say it’s straightforward to put on and take off. No matter your finances, preserving your canine secure is vital, and with a lower cost product like Vivaglory, you possibly can nonetheless enjoy the dog days of summer time. „The EzyDog Doggy Life Vest is a superb choice for big breed canines.” My three choices are the Ruffwear (have their walking harnesses for my different canines, have 5 dogs-too many!), the EZ float and the Kurgo surf n turf.
It’s value noting that this dog life jacket has a entrance float that may keep your dog’s head above the water in nearly any condition. This dog life preserver is a security packed performer that won’t break the budget. If you have multiple canine and wish a number of life vests this is the only option. As Amazon’s quantity-one bestseller in canine life jackets, Outward Hound’s model of the basic safety vest simplifies discovering a life jacket that’s excellent in your pup. You must know only your canine’s girth to get the proper measurement, which implies there’s an ideal match for animals of any top, weight, and length.
Chest girth could be measured by wrapping a tape measure across the widest part of your canine’s chest. The bellyband is made out of mesh materials, which helps with water drainage, that means your pup will get dry quicker. The jacket additionally has a leash attachment, which is a nice element. You can shortly put your canine on a leash in case he simply received’t stop chasing birds across the seaside, for example. Sure, some breeds take to water like geese, but others can sink like a stone with out the right introduction to the water world. The popularGranby Splash Life Jacket is great for all-round floatation use.
Its sturdy rescue deal with makes it straightforward to lift your dog out and in of water. However, that isn’t the case with Paws Abroad, which is made of HD Neoprene for sturdy yet snug wear. Its Velcro straps securely fasten around the neck and stomach area. For quick and simple wear, the life vest features fast-launch buckles and adjustable nylon straps.
These premium dog lifejackets wrap fully around the canine’s torso to provide a snug fit. These well-reviewed dog lifejackets have adjustable straps and a neoprene chest band for comfort. There are two helpful seize handles on top and it’s brilliant in colour for simple spotting. If they will be swimming in the deep ocean, rough currents, or unfamiliar waters, then a life vest is a good suggestion. Also, some canines merely aren’t strong swimmers, and they are going to be safer and happier in the backyard pool with a life vest.
It will also improve the percentages of him “barrel rolling,” and ending up the other way up in the water, which is obviously a dangerous state of affairs. It’ll be much simpler to put the life vest on your dog if you assist instill a optimistic attitude about wearing the floatation system from the outset.

domain: pet


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The Georgia High School Association’s women’s state track meet was last weekend in Albany with several members of the Bradwell High Lady Tigers vying for honors. But it was the solid performance of Lady Tiger Quintunya Chapman (seen here running at a local event) who earned a second place in the 400-meter dash with a time of 55.895. The Lady Tigers ended the state meet ranked number 21 with a total of eight meet points.
Lady Tiger takes second at state
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domain: news


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Just a minute walk from Yokosuka Naval Base’s Womble Gate, popular Nepalese-Indian restaurant Gorkha Palace offers a home to American sailors and local Japanese with its tasty fare and casual atmosphere.
Japan takes the cake when it comes to themed dining. From cat cafes to geeky maid bars to izakayas modeled after elementary school classrooms, there’s no limit to the intersection of food and amusement, especially in Tokyo.
With so many things to see and do in Tokyo, it can be overwhelming trying to decide what to do in a day. If you want to skip the crowds and experience something new, why not try your hand and learn about Japanese food at a Tokyo cooking class?

domain: travel


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I visited this restaurant before Doncaster Races for a pre races breakfast and I was so impressed by the standard of the food and service that I came back after the races for my dinner! Had an excellent time, the staff were all very polite, the atmosphere was great and most importantly the food was to die for! As a doncaster local I must say this is one of the best restaurants in the area. 5* for me!
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posted by Michelle .
You initially contributed $200,000 of your own money and in return you received 2 million shares of stock. Since then, you have sold an additional 1 million shares of stock to angel investors. You are now considering raising capital from a venture capital firm. This venture capital firm would invest $5 million and would receive 2 million newly issued shares in return.
The post-money valuation of your firm is closest to:

domain: finance


MOSCOW - A Russian court on Wednesday reduced the sentence of jailed tycoon Mikhail Khodorkovsky's co-defendant in a ruling that could pave the way for the Kremlin foe's own early release.
The surprise decision to cut more than three years from former Menatep bank boss Platon Lebedev's jail term may still be appealed by the prosecution, but it represents a rare victory for Khodorkovsky, one of President Vladimir Putin's most vocal and long-standing opponents.
Khodorkovsky's ruined Yukos oil empire once made him Russia's richest man and generated buzz that he could become president.
The northern Arkhangelsk district court near the prison colony where Khodorkovsky and Lebedev are serving time ruled that Lebedev should be freed in March instead of October 2016 after a recent easing of penalties for business crimes.
The two close colleagues were initially jailed on tax evasion charges in 2004 after waging a fierce political and business campaign against Kremlin interests that included plans to develop a separate Yukos oil pipeline to China.
They were convicted and sentenced for a second time in 2010 just months ahead of their scheduled release in a hearing that drew strong condemnation from the West and put Russia's damaged investment image in further disrepute.
Lebedev's attorney appealed the second conviction, and the court ruled on Wednesday that recent legal changes easing the punishment for white-collar crimes should apply to the Yukos case.
"I am partially granting the defence lawyers' appeal," the Interfax news agency quoted presiding judge Viktor Ivanov as saying.
Khodorkovsky's attorney said the defence team had decided to make Lebedev into a test case to see how the judge would rule on the recent change in legislation.
"We will now be deciding whether to make the same appeal on behalf of Mikhail Khodorkovsky. We wanted to test this first - this was a tactical move. Now we will look at the results and decide," said defence attorney Yury Schmidt. "The judge cannot avoid complying with this law - especially now that he has put it into practice in respect to Lebedev," Schmidt told AFP.
A successful appeal by Khodorkovsky has the potential of invigorating a Russian protest movement that has lost much of its drive since Putin's thumping return to a third term as president in May.
Before his arrest, Khodorkovsky had spent freely on efforts to prop up both the liberal and communist opposition to Putin in parliament.
He has continued to hotly criticise the Kremlin from jail by writing opinion pieces for top Western dailies and promoting a more cohesive political force against Putin's rule.
But the financially-ruined tycoon's impact on contemporary Russian politics remains unclear because the opposition has since rallied around a new generation of leaders who view Khodorkovsky's era of big business with disdain. Schmidt said Khodorkovsky's early release could actually help Putin by easing some of the Western criticism of his first three months back in the Kremlin.
"I have always thought that releasing Khodorkovsky would be the perfect diversion manoeuvre for the Kremlin as it imposes other restrictions and goes after the protesters," the attorney said.


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