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Peter Galasso
Art League of Long Island's latest exhibit, Stan Brodsky and Friends, set to remember abstract master
by Melissa Arnold - April 12, 2019
Stan Brodsky in his studio. Photo by Peter Scheer
By Melissa Arnold
For Stan Brodsky, painting was so much more than just a skill or even a career. It was a language, a love affair, a truly sensual experience. The artist shared those feelings openly with students over the course of a renowned teaching career that spanned more than 50 years.
Several months ago, the Art League of Long Island in Dix Hills began to prepare Stan Brodsky and Friends, a springtime exhibit celebrating Brodsky's work along with nearly 30 of his dearest friends, many of whom were former students and mentees.
'Woman in a Car,' oil/acrylic on canvas by Doug Reina
On March 30, just two weeks before the exhibit's scheduled opening, Stan Brodsky passed away at the age of 94. He had continued to work and teach until the final weeks of his life, just as he wanted it. Brodsky's students noted that the World War II veteran tried to retire a few years ago, but he couldn't stand being away from doing what he loved.
The Art League is moving forward with the show as planned, with the exhibit running from April 13 to 28. A reception on April 14 at 3:30 p.m. will allow the artists and those who loved Brodsky to honor his life and legacy.
Participating artists include Ennid Berger, Susan Bird, Susan Canin, Denise DiGiovanna, Simon Fenster, Stuart Friedman, Peter Galasso, Lenore Ann Hanson, Ginger Balizer-Hendler, Caroline Isacsson, Vincent Joseph, Deborah Katz, Marceil Kazickas, Denise Kramer, Barbara Miller, Catherine Morris, Pamela Long Nolan, Dianne Parker, Alicia R. Peterson, Doug Reina, Fran Roberts, Susan M. Rostan, Ellen Hallie Schiff, Laura Powers-Swiggett, Janice Sztabnik, Lois Walker and Hiroko Yoshida.
Stan has touched so many lives, inspiring them to pursue their passions," said Susan Peragallo, coordinator and curator of the Art League's Jeanie Tengelsen Gallery. "The exhibit will be a chance for everyone to celebrate him — the 27 artists in the show are only a small segment of those who were influenced by him over the years."
A master abstract expressionist, Brodsky studied photojournalism and fine art before receiving a doctorate in art education from Columbia University in 1959. Originally from Greenwich Village, he moved to Huntington in 1965. Most of his teaching years were spent at Long Island University's C.W. Post Campus in Brookville, and a collection of his notes and sketches from 1951 to 2004 can be found at the Smithsonian Institution.
'Superficial Information,' oil on canvas by Marceil Kazickas
Brodsky's relationship with the Art League began in the late '90s when he became an instructor. The classes were small in the beginning, with just five students enrolled in 1994, but grew rapidly, and eventually people had to be turned away from lack of space. "It's not so much that he was popular, but he was inspiring and generous in his critiques, and people really responded to that," Peragallo said.
Peter Galasso of Setauket remembers that Brodsky could often be found in the same way over the years as students arrived for class — sitting at his desk, usually eating an egg sandwich, always poring over an art history text.
"He had a contagious passion, and was constantly reading and continuing to study," said Galasso, who began studies under Brodsky 20 years ago, eventually becoming a friend and traveling companion. "He was always looking to travel somewhere new or different. He wanted to be inspired by the local color of a place."
Susan Rostan of Woodbury remembers entering Brodsky's classroom for the first time while pursuing a master's in fine art. Brodsky arranged the students in a circle and asked each one to introduce themselves. When it was her turn, Rostan simply told him, "I've heard I'm either going to love you or hate you, but I'm cautiously optimistic."
'She Wears Her Heart on Her Sleeve …,' mixed media by Susan Canin
Many years later, Rostan was sitting in a different class of Brodsky's, this one at the Art League. But she was stunned by the striking realization that nothing had changed: He still wore the same striped sweaters and paint-splattered jeans. She painted a full-length portrait of him that day that will appear in the exhibit.
"He taught us as much about ourselves as he did about painting," said Rostan, who is now working on a biography of Brodsky. "He was an unusual teacher in that he approached his students as equals and opened himself up to be vulnerable and form friendships with them, which allowed him to encourage them particularly well."
Brodsky's friendship and deep encouragement were beloved by so many of his students, said Doug Reina of Setauket. In fact, some of them continued to take his classes for decades just to spend more time with him.
"Stan had this ability to make you feel special. He was genuinely curious about you, and that means a lot," Reina said. "In the old days before taking his classes, I would look at a scene and just try to copy it. But through him I learned to paint in a way that also expresses how I feel about the subject and the sensuousness of the paint itself. Stan painted with his own language and created something truly unique for the world."
Stan Brodsky and Friends will be on view at the Art League of Long Island's Jeanie Tengelsen Gallery, 107 E. Deer Park Road, Dix Hills. Admission is free. For more information, call 631-462-5400 or visit www.artleagueli.net
Local artist coalition to host third annual Open Studio Tour
by Heidi Sutton - October 12, 2018
Visit artist Doug Reina in his Setauket studio during the tour.
By Heidi Sutton
Back by popular demand, the North Shore Artist Coalition will host its 2018 Open Studio Tour this weekend, Oct. 13 and 14, from 11 a.m. to 5 p.m. The free event will showcase the studios of 15 award-winning artists in Setauket, Stony Brook, Port Jefferson and St. James.
Visit ceramic artist Hugh McElroy during the tour.
The coalition, whose founding members include Pam J. Brown, Jim Molloy, Doug Reina, Mary Jane van Zeijts and Nancy Bueti-Randall, started this tour three years ago with the goal of bringing more awareness to professional artists that are living in the Three Village area.
"We felt that by coming together and pooling our talents and ideas that we could have some kind of creative impact in the community and the studio tour was one of those ideas," said Reina in a recent interview. "It's nice to do this with like-minded people."
While Molloy will be unable to participate this year, the group has invited artists Al Candia, Peter Galasso, Sungsook Hong Setton, Christian Stuyvesant White, Hugh J. McElroy, Marlene Weinstein, Christine Mannone Carolan, Cindy Crowell, Leslie M. Cross, and mother/daughter duoFlo and Karen Kemp to join them for the weekend event. "It's good that they're on board. They're good artists and I know they're excited to be part of this," said Reina.
Reached by phone, Brown said visitors to the event "can expect to see the works of an eclectic mix of professional artists who are illustrators, photographers, sculptors and painters." Most importantly, she said, the tour will offer an intimate look into their art studio.
Sunsook Setton will give a tour of her studio during the event
That, said Reina, is what makes this event so unique. "Honestly, how often do you get to see the inner workings of an artist's creative process?" he asked. "Usually you see the paintings hanging up [in a gallery] but you don't really get a chance to see where the artwork gets created." The Setauket artist added that those that "are at all interested in the technical part of art or getting into art or becoming a little bit more serious about your art" would benefit from this tour.
Finished works as well as works in progress will be on view and several artists will be giving demonstrations.
In the two previous tours, each artist welcomed 80 to 100 visitors to their studio and Brown is excited to see what the future holds.
"People go to Gallery North, there's the Reboli Center, the Setauket Artists, Neil Watson at The Long Island Museum is doing unbelievable things, we now have the Brick Studio, we have The Atelier at Flowerfield and then you have the Mills Pond Gallery. That's a lot of art organizations — there's a lot happening — so I think it's really great for local artists to be connected as much as possible and build our community and try to build awareness for people outside of our community," she explained.
"We would love in the future to have all these local organizations on board so this becomes a big cultural attraction, an art destination for people who are looking to get away for the weekend," Brown continued. "It is my hope that this event continues to grow."
The Artist Open Studio Tour map and addresses may be found at https://www.facebook.com/NorthShoreArtistCoalition. Admission is free and refreshments will be served at some of the studios. For further information, please call 631-834-9036.
North Shore artists take part in studio tour
by Heidi Sutton - November 15, 2017
Hugh McElroy of Port Jefferson with some of the masks he created
Hugh McElroy of Port Jefferson speaks about his art process
Sungsook Setton in her Setauket studio
Christian White in his St. James Studio
Jim Molloy of Miller Place in his home studio
Nanci Bueti-Randall of Stony Brook in her St. James studio
Nanci Bueti Randall of Stony Brook in her St. James studio
Peter Galasso of Setauket talks about his art process
A COLORFUL PALATE The North Shore Artists Coalition held its second annual Artists Open Studio Tour on Nov. 11 and 12. Eleven award-winning artists welcomed visitors into their studios to see where the magic happens including, clockwise from top left, Peter Gallasso of Setauket, Hugh McElroy of Port Jefferson, Christian White of St. James, Nanci Bueti-Randall of Stony Brook (in her St. James studio), Sungsook Setton of Setauket, and Jim Molloy of Miller Place. Other participating artists included Doug Reina, Mary Jane van Zeijts, Marlene Weinstein, Pam Brown and Kelynn Z. Alder. Guests were able to view artwork for sale, ask questions and enjoy refreshments.
Photos by Heidi Sutton
North Shore Artist Coalition to host second annual Open Studio Tour
by Heidi Sutton - November 8, 2017
Abstract artist Peter Galasso's Setauket studio will be one of the stops on the tour. Photo from Peter Galasso
The North Shore is teeming with talented artists and local venues like the Mills Pond House Gallery, the Reboli Center for Art and History, The Long Island Museum, Gallery North, the Port Jefferson Village Center and libraries are more than eager to show off their artwork. But what if you could hit the rewind button and observe the artist working on the piece right before your eyes?
That rare opportunity will arise this weekend as the North Shore Artist Coalition hosts its second annual Artist Open Studio Tour.
Doug Reina will be working on a painting during the tour. Photo from Doug Reina
The self-guided event will offer an intimate glimpse into the working studios of 11 award-winning artists living in Miller Place, St. James, Setauket, Port Jefferson and Stony Brook, giving visitors a personal opportunity to meet and talk with artists about their work and the creative process.
The coalition, whose core artist group is Nancy Bueti-Randall, Mary Jane van Zeijts, Jim Molloy, Doug Reina and Pam Brown, formed last year to contribute to the community through exhibitions, open studio tours and educational programs. "Fundamentally we believe the arts improve our lives and enrich our communities culturally, socially and economically. As well our mission is to promote and increase regional awareness of professional artists working in a wide range of styles and studio practices," said Brown in a recent email.
Artists Peter Galasso, Kelynn Z. Adler, Sungsook Setton, Christian White, Hugh McElroy and Marlene Weinstein were invited to participate in this year's tour.
Artist Pam Brown's studio is one of the stops on the tour. Photo from Pam Brown
"Each year we invite artists from the Three Village and surrounding areas [to join us]," Brown explained. "Our goal is to invite more artists each year as the Studio Tour grows." In addition to the Artist Studio Tour, the group aims to have yearly pop-up exhibitions. This past June, Mary Jane van Zeijts hosted an exhibition at her Setauket studio titled Five @ 268 Art on Main that featured the core group. Upcoming projects include a curated exhibition of the Artists in the Studio Tour. "Currently we are looking for exhibition spaces in our local area," said Brown.
At each studio, which can be in a backyard barn, garage, house, storefront or outdoor space, visitors will be able to talk freely with the artists and ask questions about their approach and individual styles to making art. "They can visit as long as they like," said Brown. "Additionally some of the artists will be doing demonstrations and talking directing about their processes and the materials they use."
Visitors can also expect to see a variety of artwork on display including original paintings, sculptures, ceramics, pastels, photographs, prints and textiles. Additionally, artwork will be for sale.
Brown is hoping to attract even more visitors to this year's event. "Last year's Studio Tour was very successful and well-attended, we received positive feedback from local art organizations and other cultural venues, plus we had incredible support from fellow artists, friends and our community at large," she said.
"It is our hope that people walk away with a deeper meaning and understanding of art, the Artist, and the important role that art plays in our community."
The North Shore Artist Coalition's Artist's Open Studio Tour will be held on Saturday, Nov. 11 and Sunday, Nov. 12 from 11 a.m. to 5 p.m. The event will be held rain or shine and admission is free. Refreshments will be served at several studios. For a list of the 11 locations, visit www.facebook.com/NorthShoreArtistCoalition. For more information, call 631-834-9036.
Abstract art takes center stage in new show at the Art League of Long Island
by Ellen Barcel - April 5, 2017
'Blue in Green' by Peter Galasso
By Ellen Barcel
"You rarely see a show of all abstract art," said artist Peter Galasso, of the Art League of Long Island's new show, Long Island Abstraction: 2 Generations, on view at the league's Jeanie Tengelsen Gallery now through April 15.
'Accents Red' by Frank Wimberley
Four artists, Stan Brodsky, Laura Powers-Swiggett, Frank Wimberley and Galasso have filled the gallery with approximately 50 of their abstract works. What else unites these four artists? They are all award-winning artists with strong ties to Long Island. Two, Galasso and Powers-Swiggett were influenced by their mentor, Brodsky. The fourth, Wimberley was added to the exhibit by Galasso.
"I've been showing for 25 years. I met Frank [Wimberley] about 10 years ago at a gallery show. I admired his work," said Galassao in a recent interview. The two became friends and Galasso suggested his work for an exhibit held at the Art League about two years ago. When the concept for the current exhibit was broached, "I told him of my idea of two generations of abstract artists from Long Island …" The idea was very specific. "He could see how this would work."
Where did the two generations come from? Both Brodsky and Wimberley are in their 90s, the senior members of the foursome. Powers-Swiggett and Galasso, the younger members, were both students of Brodsky. Brodsky was not only a mentor to these two, but many, many others as professor of art at C.W. Post College for over 30 years. In his artist's statement, Brodsky noted, "I've been an exhibiting artist in New York City for more than 50 years — and my passion for painting is a strong now as ever."
'Descending Light 2' by Stan Brodsky
Galasso described Wimberley's work saying, "I admire his work — movement and color. He uses a lot of acrylic medium, a very thick mixture. It moves spontaneously across the canvas."
Susan Peragallo, gallery coordinator, said that abstract art is nonrepresentational and "about expressing an idea or emotion using color, line and form." But what inspires each of these four artists? In his artist's statement Brodsky noted, "I have traveled extensively absorbing the colors and textures of new landscapes," and Powers-Swiggett's paintings are landscape-based abstractions exploring spatial and color relationships. Galasso's works have been described as "an exploration of feeling, memory and a unique vision …"
Abstract art can be very freeing for both the artist and the viewer. The realist must represent the scene accurately, but the abstract artist uses a scene as inspiration. Said Wimberley in his artist statement, "The abstract painter can commence his drawing or canvas generally with only a preconceived notion, reflection or emotion … he has far less guarantees than perhaps the realist painter or photographer that the finished expression with extended from calculated reason or logic. This for me provides the excitement of taking the theme or feeling from the very first stroke, and following it to its own particular conclusion. It is very much like creating the controlled accident."
'Wawapek' by Laura Powers-Swiggett
While each of the four artists decided which of their works were to be shown, it was Peragallo who decided which paintings would be hung together, making them, "flow together. That was my job. It was sort of like putting a puzzle together. You want the works to speak to each other but one shouldn't overpower the other. They should gradually draw the viewer into the show." "It's a wonderful show, really beautiful," said Peragallo. "People who don't normally like abstract art come in and say 'Wow.' It's a happy show, so colorful and uplifting," she added.
Long Island Abstraction: 2 Generations will be on view at The Jeanie Tengelsen Gallery of the Art League of Long Island, 107 East Deer Park Road, Dix Hills through April 15. The gallery is open Monday through Thursday from 9 a.m. to 9 p.m., Friday from 9 a.m. to 4 p.m. and weekends from 11 a.m. to 4 p.m. Admission is free. For further information, call 631-462-5400 or go to www.artleagueli.org.


The integration of discrete mathematics into the secondary school curriculum (grades 7-12) is an important consideration because the mathematical area is dynamic and interesting, providing students the development of mathematical thinking. Also, it provides for teachers the opportunity to develop innovative mathematics instruction. Since the publication of the document Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989), it has been difficult to determine how many schools have integrated discrete mathematics as a separate or as an integrated course in the school mathematics curriculum. Moreover, the mathematics education research community has, for the most part, not focused on teachersâ perceptions about teaching and learning discrete mathematics as an area of investigation. Because of the lack of research in this area, the researcher investigated preservice secondary mathematics teachers' perceptions about discrete mathematics and their reactions to the integration of discrete mathematis into the school curriculum. The researcher purposely selected four preservice secondary teachers who were enrolled in a mathematics course in the fall of 2005. Various data sources were used to get a deep understanding of each participant, including selected coursework, an online survey, and interviews. Results indicated that these preservice teachers perceive discrete mathematics as meaningful to students, as it emphasizes processes such as problem solving and mathematical thinking, and it provides opportunities to use innovative instruction. Because of this, the preservice teachers believe that discrete mathematics should be integrated in the school mathematics curriculum. In addition, several factors that affect the integration of discrete mathematics in the school were identified. These factors are the state curriculum and testing, the historical emphasis of algebra and calculus in the school curriculum, the National Council of Teachers of Mathematics (1989, 2000) Standards documents' views of discrete mathematics, teachers' views of discrete mathematics, the lack of knowledge of discrete mathematics, and the lack of materials and guidelines for teaching discrete mathematics.


On a remote site in regional Queensland, the challenge is to transfer contaminated soil with solids and rocks up to 10-12mm in size. The customer was originally using vacuum trucks but this become an ongoing costly expense. They required the pump to be single phase.
We originally looked at using a peristaltic hose pump but they would have needed a very large pump to handle the solid size, which did not fit the customer's budget. The customer was offered a stainless steel pump by an alternative supplier, but this would have been too costly to maintain and repair. We selected a SlurryPro centrifugal slurry pump with hard metal liners due to the heavy duty design and ease of maintainance, which also enabled budget savings for the customer.
Watch this space - we will report on the outcomes gained by the customer once the pump is installed and operational. Note that the customer and site location will remain confidential.
SlurryPro is a slurry pump manufacturer with presence in Australia, UK and Argentina.
By working closely with mining and other industries around the world, SlurryPro has developed wear materials that outlast conventional pump designs, impacting significantly on the bottom line for plant operators.
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Climate Adaptation Program Manager
Lisa works with New Hampshire communities to build awareness of and resilience to the impacts of a changing climate. About half of her time is dedicated to supporting climate adaptation in the coastal watershed, working on projects with the NH Coastal Adaptation Workgroup and conducting programming with coastal communities. She is also working to develop new programming for climate adaptation work in the rest of the state, including participating in the Upper Valley Adaptation Workgroup, and she represents Cooperative Extension on the USDA Northeast Climate Hub. Recent climate adaptation programming has focused on outreach to conservation commissions about how they can address climate change impacts on natural resources, a Climate in the Classroom program with several junior high classes in the seacoast, public engagement in municipal planning related to climate, and more.
Prior to joining Extension in January 2017, Lisa worked on projects with Great Bay National Estuarine Research Reserve, the Great Bay 2020 partners, the National Estuarine Research Reserve System Science Collaborative, and Infrastructure and Climate Network (ICNet). She completed both her bachelor's and master's degrees at the University of New Hampshire. She is a graduate of the internship-based TIDES Masters program (Training for the Integration of Decision-making and Ecosystem Science), during which she worked with the Hudson River National Estuarine Research Reserve in New York, earned a Master of Science in Integrated Coastal Ecosystem Science, Policy and Management, and completed the UNH certificate program in Geospatial Studies.
M.S., Coastal Ecosystem Management, University of New Hampshire, 2015
Graduate Certificate in Geospatial Science, University of New Hampshire, 2015
B.S., Environmental Conservation Studies, University of New Hampshire, 2012
The Coastal Society
American Society of Adaptation Professionals
lisa.wise@unh.edu
Lisa is Affiliated with
Aquaculture & Marine Fisheries
What Conservation Commissions Can Do to Protect Natural Resources in a Changing Climate [fact sheet]
5 Ways Communities can Protect Wildlife in a Changing Climate [factsheet]
What NH Atlantic Coast Municipalities Need to Know about Preparing for Storm Surge, Sea-level Rise, and Extreme Precipitation [factsheet]
What NH Great Bay Municipalities Need to Know about Preparing for Storm Surge, Sea-level Rise, and Extreme Precipitation [factsheet]

redpajama_set_name: RedPajamaCommonCrawl


Occams razor states that given a choice between differing explanations one should always choose the simplest. Of course this begs the question as to what one could mean by simplest.
I'm interested in situations in which the opposite principle has been found correct. That the more complex explanation wins out.
A possible example could be: There is life only on this planet and it is carbon-based as opposed to there are many kinds of life on many planets and they are not neccessarily carbon-based.
Given that choice the simplest seems to be the first, whereas I think the second, the more complex explanation is likely to be correct.
That's a pretty poor restatement of Occam's razor. What he actually said was Numquam ponenda est pluralitas sine necessitate, or "Plurality ought never be posited without necessity."
In other words, one should not introduce an entity into an explanation which is not necessary.
In order for that to be an example of Occam's razor, we'd need two competing explanations for the same phenomenon. That's not the case here.
Not the answer you're looking for? Browse other questions tagged philosophy-of-science occams-razor or ask your own question.


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Pollard, Bravo omission a self-inflicted blow, says Lara
Emrit healthy to face Jamaica
18th February 2015 Comments Off on WI regroup ahead of Pakistan clash Views: 1353 News
WI regroup ahead of Pakistan clash
The West Indies team, seeking to regroup following their opening round thrashing at the hands of minnows Ireland in the ICC Cricket World Cup, arrived in Christchurch yesterday for their second round clash against Pakistan which will be played on Saturday, at Hagley Oval.
Captain Jason Holder's side, jolted by an excruciating four-wicket loss to Ireland at Nelson, will be seeking to revive their campaign when they take on the Pakistanis. West Indies have been criticised in many quarters for Monday's defeat to Ireland and must now win matches to keep alive their chances of securing a place in the quarter-finals phase.
The Caribbean team collapsed to 87 for five before Lendl Simmons (102) and Darren Sammy (89) guided them to a total of 304 for seven against Ireland.
"The West Indies, I do not understand how they could ever think that just over 300 runs that they would just automatically win," insisted former fast bowler Michael Holding, a member of the West Indies side that won the 1979 World Cup. "They cannot take things for granted. You should be able to think for yourself, 'OK we got a reasonable score but we still have a fight on our hands and we still have to go out there and perform".
West Indies will be resting today and will only resume training tomorrow at the Hagley Oval in Christchurch. They will also train on Friday ahead of the crucial clash against Pakistan, who lost to the defending World Cup champions India by 76 runs in their opening encounter at the Adelaide Oval, in Australia.
Taken from Trinidad Express


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Solar Powered Roofing: Pros and Cons
by chari | Mar 2, 2022 | Roof Types | 0 comments
Solar roofing has become more and more popular as homeowners seek to reduce utility costs and limit their carbon footprints. Is this roofing option right for you? Read on to learn the advantages and disadvantages associated with solar roofing tiles.
What Are Solar Roofs?
Solar tiles are made of thin-film solar cells consisting of copper indium gallium selenide. Light and flexible, the cell material is a great semiconductor. A single solar tile can produce from 50 to 114 watts or power. To get the most power, however, it's best to install the tiles along the true south.
With solar tiles, each individual unit is installed and wired individually. This can be very tedious, and mistakes can be costly. This is why it's very important to hire a professional with extensive expertise. It's also important to know the pros and cons of this sort of roofing system so you can make an intelligent decision for your unique needs.
Solar Roofing Pros
Solar roofing systems require a heavy up-front investment, but this comes with some pretty compelling benefits, including:
They are energy efficient.
This is the greatest advantage of solar tiled roofs and the most appealing selling point for potential buyers. Solar roofs can help lower your utility bill by producing clean, reliable energy. Solar shingles produce about 100 to 110 Watts per meter squared, while tiles produce about 65 to 70 watts per meter squared.
They are environmentally friendly.
Solar roofs are much more eco-friendly compared to fossil-fueled power sources. While it is true that the manufacturing process does rely on fossil fuels; in the long-run, solar shingles and solar roofing tiles are much more environmentally friendly because they draw power from the sun while producing no waste.
They are durable.
Solar roofs are made out of durable glass that's resistant to impact. In fact, solar tiles typically exceed the highest Class 4 impact resistance. Solar roof shingles and tiles are also made to meet the highest standards for fire resistance (Class A fire rating).
Solar tiles are infrangible which means you can walk on them without causing breakage. Rated for wind speeds up to 130 mph, solar tiles have an estimated life expectancy of around 25 to 40 years.
They are attractive.
Unlike bulky solar panels, solar tiles are sleek and discrete. They also blend in with the roof and give it a clean contemporary look. Available in a variety of different forms and shapes, solar tiles can be made to look similar to various roofing materials, including shakes, clay and slate tiles.
Solar Roofing Cons
While they offer a number of attractive traits, solar roofing systems do have some drawbacks, including:
They aren't cheap.
While prices can vary by company, solar tiles can cost you anywhere from $1,100 to $2,600 per square. That said, you can recoup a good portion of these costs over time thanks to lower utility bills.
They require expert installers.
Solar tiles require specialists trained in both roofing and electrical work. It also takes longer to install compared to traditional roofs. To avoid issues, it's important to work with a reputable local contractor with extensive experience installing solar tile roofing.
Repairs can be complicated.
Repairing solar tiles can be a complex process requiring special ordering and technical expertise. Again, it's best to work with an expert to reduce the risk of problems.
Your ROI can take a while.
Solar roof shingles and tiles have a high upfront cost. And while you can recoup much of this expense from energy savings; it can take at least 15 years to get your return on investment. That equates to about half the life expectancy of a solar roof system.
They require rewiring.
Solar tiled roofs demand new wiring to install battery storage and connect your home to the roof. You must also get a new roof when you install the solar tiles because you typically can't install them over an existing roof system. The old roof system must be completely removed before installing the solar tiles, which is different from many other kinds of roof systems, such as SPF, TPO, asphalt shingles, etc. which allow you to cover the existing roof system.
Whether you need a traditional roof, energy-efficient shingles or stylized roofing design, A to Z is here to help. For decades, the professionals at A to Z Roofing have served countless satisfied customers throughout the Denver metro and the Front Range. For nearly a quarter of a century, we have been the go-to source for expert commercial and residential roofing, repair and restoration for homeowners and businesses throughout Colorado. During this time, we've provided comprehensive service using only top-quality products. Contact our attentive team to learn how we can restore, repair or replace your damaged or aging roof and get a free estimate!


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Ballads ist ein Kompilationsalbum des britischen Rockmusikers Eric Clapton, das dreizehn Rockballaden enthält. Es erschien am 23. Dezember 2003 ausschließlich für den japanischen Markt beim Label Warner Brothers Japan und war nur drei Monate lang erhältlich.

Rezeption
Kritiker Stephen Thomas Erlewine von der Musikwebsite Allmusic schrieb, dass in die Zusammenstellung des Albums der psychedelische Titel Badge nicht hineinpasse. Das Album sei aber für Gelegenheits-Fans von Clapton, die nach schöner Stimmungsmusik suchten, lohnend. Er vergab insgesamt 2,5 von 5 möglichen Bewertungseinheiten für das Album. Ballads erreichte Platz 7 der japanischen Albumcharts und blieb 18 Wochen dort. Musikjournalist Ayano Miyahara der Yomiuri Shimbun bezeichnete das Album als exzellent und lobte sowohl Claptons Gesang als auch sein Gitarrenspiel. Zum Titel Wonderful Tonight schrieb er: "In diesem Lied kommt sowohl Claptons leidenschaftliche maskuline als auch seine sentimentale Seite zum Ausdruck." Weiter fand Miyahara, dass Liedtext und Melodie perfekt harmonieren. Kritiker der Website artistdirect.com vergaben 2,5 von 5 möglichen Punkten. Mister Udo von Udo Artists Japan bezeichnete das Werk als "wundervolle Erfahrung". todomusica.org vergab 5 von 5 Sternen für Ballads.

Vermarktung
Vor allem in den japanischen Städten Tokio, Yokohama, Osaka und Nagoya wurden intensive Werbemaßnahmen getroffen, um eine möglichst hohe Zahl an verkauften Tonträgern in nur 98 Tagen zu erreichen. Das Album wurde im japanischen Fernsehen vorgestellt und zahlreiche Werbeplakate an öffentlichen Plätzen sowie an Bussen und Zügen angebracht. Die Werbekampagne startete Mitte Dezember 2003 und wurde Ende März des folgenden Jahres beendet.

Titelliste

Verkaufszahlen

Weblinks

Einzelnachweise

Album (Bluesrock)
Album (Rock)
Kompilation (Musik)
Album 2003
Eric-Clapton-Album

redpajama_set_name: RedPajamaWikipedia


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At Mayfaire, Wilmington Scientists To Open The STEM Labs For Kids
By Cece Nunn, posted Mar 1, 2018
The STEM Labs is expected to open in May at 820 Town Center Drive in Mayfaire Town Center. (Photo by Cece Nunn)
Two Wilmington scientists have synthesized a business that will offer students in kindergarten through 12th grade the opportunity to experience science, technology, engineering and math (STEM) through a variety of programs.
Tina and Brad Catalone, a married couple with backgrounds in the pharmaceutical and biotechnology industries, are opening The STEM Labs at Mayfaire Town Center, in new space at 820 Town Center Drive.
The STEM Labs enrichment programs will include after-school care, summer camps, workshops, a leadership academy for middle- and high-schoolers, tutoring, academic advising, preschool labs, birthday parties and parents' night out events. The space will be upfit with miniature laboratories.
To boil it down: things like robots, slime-making, 3-D printers, Legos and drones are part of the curriculum.
The Catalones plan to open their new venture at the Mayfaire location in May, with future plans for growth elsewhere.
"We want to start it here in Wilmington and have a really solid beginning, but we know that there is so much demand right now in trying to educate in the STEM professions, and we know that there's a huge opportunity here," Tina Catalone said. "So we'd really like to grow, and grow pretty quickly in terms of reaching as many students as we can and to implement this new idea."
Tina Catalone has a doctorate in cell and molecular biology from Penn State College of Medicine, and Brad Catalone has a doctorate in microbiology and immunology from the same school. He was vice president of laboratory operations at Alcami Corp., and Tina Catalone was a project manager in laboratory and development services at the same company, a drug development and manufacturing firm that has moved its headquarters from Wilmington to Durham. The Catalones both left Alcami in September of last year.
The couple has four children, ages 14, 11, 8 and 6, who have served as inspirations for The STEM Labs.
"We both are researchers and scientists. I met my husband in grad school. We worked together and published together early on. I went on the path of being a research scientist – I did a lot of vaccine development work and pharmaceutical biomedical research," Tina Catalone said.
Brad Catalone, meanwhile, also earned an MBA from Penn State University.
"He started going into the business operations side, so really the two of us coming together for this initiative is a great team because we complement each other so well. I also went down the track of education as a college professor for a number of years as well and still have an affiliation at UNCW," Tina Catalone said.
She said she's working with the University of North Carolina Wilmington with a workforce development grant that aims to give students real-life experiences and create internships.
"I just feel like that is one of those missing links out there. And that is really what makes all the difference and that comes back to why we are building The STEM Labs for a community partnership to try to take that kind of concept and bring it down to another level for the students," said Tina Catalone, who is president of The STEM Labs, while her husband is vice president. Brad Catalone is also chief science officer at TSO3, a sterilization processes firm in Myrtle Beach, South Carolina.
They'll have a roster of 20 part-time employees, many of whom have experience as STEM professionals or executives.
"We're going to have a diverse group of really talented people who believe so strongly in the mission of giving back to these students and really helping them to grow in ways that they may not have exposure to right now," Tina Catalone said. "We want to supplement what they've already been getting in the education system; we want to offer something a little bit above and beyond and different and bring some gadgets in they may not have exposure to otherwise."
One of the sessions they're planning will bring in an aviculturist (a person who keeps and breeds birds) to talk about zoology.
"She'll be talking about what a day in the life of a zookeeper is really like at one of the largest zoos in the country," Tina Catalone said.
She said the aim of The STEM Labs is to help students absorb knowledge in a hands-on way. Tina Catalone said, "The more you're exposed to it and the more opportunities you have to work with it [STEM] and different perspectives you take, it all comes together in the end."
Story On The Map
Finalists announced for commercial real estate awards
Cape Fear Industrial Complex sells for more than $8M
Developers plan modern industrial building in Pender
Big Sky Design moving to Oleander Drive
Tractor Supply Co. store coming to Leland, bringing jobs
Redevelopment project could add to Castle Street activity
Put Happy In The New Year
Diane Durance - UNCW Center for Innovation and Entrepreneurship
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text:
Home Android Samsung teases the February 20 launch of its foldable phone
Samsung teases the February 20 launch of its foldable phone
Samsung drops a new teaser of a foldable phone on its Twitter handle. Well, the teaser reads a caption that says "The future of mobile will unfold on February 20, 2019." The company didn't reveal the phone, but the text reads "The Future Unfolds."
The future of mobile will unfold on February 20, 2019. #SamsungEvent pic.twitter.com/MHvwrt7Rf4
— Samsung Mobile (@SamsungMobile) February 11, 2019
Samsung's Foldable phone will name after Galaxy Fold. Looking at this teaser, we can acknowledge that the company will bring this phone alongside the Galaxy S10 series smartphones, on February 20. We have already seen the glimpse of the phone in 2018, and some people also got a chance to look at the phone at CES 2019.
We've also seen the phone in a promo video from Samsung. The idea behind the Samsung Foldable smartphone looks pleasing. The two screens on the phone will measure 4.58-inch and 7.3-inch respectively. Even more, the outer cover of the phone will bear an aspect ratio of 21:9 and a resolution of 1960 x 480 pixels. The side with a larger screen will have an aspect ratio of 4.2:3 along with a resolution of 2152 x 1536 pixels.
The phone is not far away from its official launch, and we are hoping to see the best. Anyways, Do you like the Galaxy Fold concept? Do let us know in the comments section below.
Previous articleMeizu Note 9 appears on TENAA – Shows from all sides!
Next articleSamsung Galaxy S10+ banner leaks – Confirms the Design!
Nokia 3.1 Plus and Nokia 7.1 pre-orders now open in Russia
Anshul Nischal - October 16, 2018


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text:
\section{Introduction}

Congruence is one of the most important properties of a behavioural semantics. The reason is that the fundamental issue in process algebra - providing sound and complete axiomatisations for collections of process operators - requires that these operators are compositional. Only then we can use equational logic priciples and provide sound axioms.

There is a large amount of research to find ways of ensuring the congruence property. The basic methodology is to impose restrictions on operator definitions; there is a notion of a rule format for transition system specifications which provide operational semantics for process algebras. If a process operator is defined with rules that fit within a format, then the semantics in question is a congruence with respect to this operator. Examples include the panth format for bisimulation semantics \cite{Ver95} and formats designed specifically for several decorated trace semantics \cite{BlFoGl04}. The focus here is on \textit{semantics}; rule formats are most often defined with one particular process semantics in mind. Interestingly, in \cite{BlFoGl04}, the modal characterization of a process semantics is taken as starting point to derive the syntactic constraints of the congruence format for this semantics. A modal characterization of a semantics is a sublanguage of Hennessy-Milner logic such that two processes are semantically equivalent if and only if they satisfy exactly the same formulas in the modal characterization of the semantics. For almost all process semantics in van Glabbeek's spectrum \cite{Gla01} there is a corresponding modal characterization.

In this paper, we attempt to look at the compositionality issue from an \textit{operator's} point of view. For a number of basic process operators, we determine conditions that a process semantics should satisfy in order to be congruence with respect to such an operator. To be more precise, given a process operator, we develop syntactic constraints on modal characterizations; if the modal characterization of a process semantics satisfies these constraints, then the process operator is guaranteed to be compositional with respect to this semantics. So instead of going from a process semantics to a class of transition system specifications for which that semantics is a congruence, we go from the transition rules of a process operator to a class of process semantics for which this operator is compositional. This approach gives us an orthogonal view on compositionality, and provides further insight into connections between process algebra and modal logic.

\section{Preliminaries}

We work in the usual setting of labelled transition systems (LTSs), which consist of a set $S$ of states $p$ (also called processes), a set ${\it Act}$
of actions $a$, and a set of transitions $p \stackrel{a}{\rightarrow} p'$.

\subsection{Hennessy-Milner logic}
\label{sec:hml}

Hennessy-Milner logic (${\it HML}$) \cite{HeMi85} is a modal logic for specifying properties of states in an LTS.
There exist different versions of ${\it HML}$ \cite{HeMi85,Gla01,BlRiVe01}. The choice of syntax is important here,
even if two logics have the same expressivity; compositionality requirements established for some version of ${\it HML}$
(e.g.\ with diamond, conjunction and negation only) may become insufficient when we add other operators
(e.g.\ box), because these extra operators may require syntactic requirements of their own.
Our point of departure is the infinitary ${\it HML}$ variant without box and disjunction. The ${\it HML}$ syntax is therefore as follows:
\begin{center}
$
\varphi ~~::=~~ {\sf T} ~\mid~ \bigwedge_{i\in I}\varphi_i ~\mid~ \diam{a}\varphi ~\mid~ \neg\varphi
$
\end{center}
where $I$ is an arbitrary index set, and $a$ ranges over the set {\it Act} of actions.
Furthermore, we use ${\sf F}$ as an abbreviation for $\neg{\sf T}$. We introduce some additional notations, based on the standard notion of context. A context, notation $C[]$, is a ${\it HML}$ formula with one occurrence of $[]$. A \textit{multicontext $C[]_{i \in I}$} is a ${\it HML}$ formula containing one or more $[]$ symbols, indexed by the elements from $I$. For a (multi)context $C[\varphi_i]_{i \in I}$, a formula is obtained by replacing the $[]_i$ symbols with formulas $\varphi_i$.
Finally, we introduce an \textit{$n$-level context}, which means that the context symbol has $n$ diamond operators above it. It is defined inductively as follows:
 \begin{itemize}
 \item $[]$ is a ${\bf 0}$-level context;
 \item if $C_{n}[]$ is an $n$-level context, then $\neg C_{n}[]$ and $C_{n}[] \wedge \bigwedge_{i \in I} \varphi_i$ are $n$-level contexts;
 \item if $C_{n}[]$ is an $n$-level context, then $\diam{a}C_{n}[]$ is an ($n+1$)-level context.
 \end{itemize}
An example of a ${\bf 0}$-level context is $\diam{a}\diam{b} {\sf T} \wedge \neg []$, while $\diam{a} (\diam{a}\diam{b} {\sf T} \wedge [])$ is a $1$-level context.

A sublanguage ${\cal O}$ of ${\it HML}$ gives rise to a process equivalence by identifying those processes which satisfy exactly the same formulas from ${\cal O}$:
\begin{center}
$p \sim_{\hmo} q \,\stackrel{\rm def}\Leftrightarrow\, \forall_{\varphi \in {\cal O}}: (p \models \varphi \Leftrightarrow q \models \varphi)$.
\end{center}
We call ${\cal O}$ a  modal characterization of $\sim_{\cal O}$. Below, examples of modal characterizations of standard process equivalences from the literature are given (see \cite{Gla01}):
\begin{itemize}
	\item \textit{trace observations}:\\
	 ${\cal O}_{T} \hspace{5.0 pt} \varphi ::= {\sf T} \hspace{3.0 pt}|\hspace{3.0 pt} \diam{a} \varphi '~(\varphi' \in {\cal O}_{T})$
	\item \textit{completed trace obervations}:\\
	 ${\cal O}_{CT} \hspace{5.0 pt} \varphi ::= {\sf T} \hspace{3.0 pt}|\hspace{3.0 pt} \diam{a} \varphi '~(\varphi' \in {\cal O}_{CT})\hspace{3.0 pt}|\hspace{3.0 pt} \bigwedge_{a \in {\it Act}} \neg \diam{a} {\sf T}$
	\item \textit{failures observations}:\\
	 ${\cal O}_{F} \hspace{5.0 pt} \varphi ::= {\sf T} \hspace{3.0 pt}|\hspace{3.0 pt} \diam{a} \varphi '~(\varphi' \in {\cal O}_{F})\hspace{3.0 pt}|\hspace{3.0 pt} \bigwedge_{i \in I} \neg \diam{a_i} {\sf T}$
  \item \textit{readiness observations}:\\
	${\cal O}_{R} \hspace{5.0 pt} \varphi ::= {\sf T} \hspace{3.0 pt}|\hspace{3.0 pt} \diam{a} \varphi '~(\varphi' \in {\cal O}_{R})\hspace{3.0 pt}|\hspace{3.0 pt} \bigwedge_{i \in I} \neg \diam{a_i} {\sf T} \wedge \bigwedge_{j \in J} \diam{b_j} {\sf T}$
	\item \textit{failure trace observations}:\\
	${\cal O}_{FT} \hspace{5.0 pt} \varphi ::= {\sf T} \hspace{3.0 pt}|\hspace{3.0 pt} \diam{a} \varphi '~(\varphi' \in {\cal O}_{FT})\hspace{3.0 pt}|\hspace{3.0 pt} \bigwedge_{i \in I} \neg \diam{a_i} {\sf T} \wedge \varphi '~(\varphi ' \in {\cal O}_{FT})$
	\item \textit{ready trace observations}:\\
	${\cal O}_{RT} \hspace{5.0 pt} \varphi ::= {\sf T} \hspace{3.0 pt}|\hspace{3.0 pt} \diam{a} \varphi '~(\varphi' \in {\cal O}_{RT})\hspace{3.0 pt}|\hspace{3.0 pt} \bigwedge_{i \in I} \neg \diam{a_i} {\sf T} \wedge \bigwedge_{j \in J} \diam{b_j} {\sf T} \wedge \varphi '~(\varphi ' \in {\cal O}_{RT})$
	\item \textit{simulation observations}:\\
	${\cal O}_{1S} \hspace{5.0 pt} \varphi ::= {\sf T} \hspace{3.0 pt}|\hspace{3.0 pt} \diam{a} \varphi '~(\varphi' \in {\cal O}_{1S}) \hspace{3.0 pt}|\hspace{3.0 pt} \bigwedge_{i \in I} \varphi_{i}~(\varphi_{i} \in {\cal O}_{1S})$
	\item \textit{ready simulation observations}:\\
	${\cal O}_{RS} \hspace{5.0 pt} \varphi ::= {\sf T} \hspace{3.0 pt}|\hspace{3.0 pt} \diam{a} \varphi '~(\varphi' \in {\cal O}_{RS}) \hspace{3.0 pt}|\hspace{3.0 pt} \neg \diam{a} {\sf T} \hspace{3.0 pt}|\hspace{3.0 pt} \bigwedge_{i \in I} \varphi_{i}~(\varphi_{i} \in {\cal O}_{RS})$
	\item \textit{$n$-nested simulation observations for $n \geq 2$}:\\
	${\cal O}_{nS} \hspace{5.0 pt} \varphi ::= {\sf T} \hspace{3.0 pt}|\hspace{3.0 pt} \diam{a} \varphi '~(\varphi' \in {\cal O}_{nS}) \hspace{3.0 pt}|\hspace{3.0 pt} \bigwedge_{i \in I} \varphi_{i}~(\varphi_{i} \in {\cal O}_{nS}) \hspace{3.0 pt}|\hspace{3.0 pt} \neg \varphi '~(\varphi' \in {\cal O}_{(n-1)S})$
	\item \textit{ bisimulation observations}:\\
	${\cal O}_{B} \hspace{5.0 pt} \varphi ::= {\sf T} \hspace{3.0 pt}|\hspace{3.0 pt} \diam{a} \varphi '~(\varphi' \in {\cal O}_{B}) \hspace{3.0 pt}|\hspace{3.0 pt} \bigwedge_{i \in I} \varphi_{i}~(\varphi_{i} \in {\cal O}_{B})\hspace{3.0 pt}|\hspace{3.0 pt} \neg \varphi '~(\varphi ' \in {\cal O}_{B}) $
	\end{itemize}

\vspace{2mm}
We write $\varphi\equiv\varphi'$ if $p\models\varphi\Leftrightarrow p\models\varphi'$ for any process $p$ in any LTS.
Given an ${\cal O}\subseteq{\it HML}$, we write ${\cal O}^\equiv$ for the set of HML formulas $\varphi$ for which there
exists a $\varphi'\in{\cal O}$ with $\varphi\equiv\varphi'$.

\subsection{BCCSP}

Any LTS isomorphic with a finite tree can be described with the following process algebra BCCSP, consisting of three operators:

\begin{itemize}
\item a nullary process ${\bf 0}$ which does not have any behaviour;
\item action prefix $a.()$ for $a\in{\it Act}$: a unary operator which represents execution of a single action followed by the process given as the argument, defined by the transition rule
\begin{center}
\[ \frac{}{ax \stackrel{a}{\rightarrow} x} \]
\end{center}
\item alternative composition ($+$), a nondeterministic choice between two processes, defined by the transition rules
\begin{center}
\[ \frac{x \stackrel{a}{\rightarrow} x'}{x + y \stackrel{a}{\rightarrow} x'}
~~~~~~~~~~ \frac{y \stackrel{a}{\rightarrow} y'}{x + y \stackrel{a}{\rightarrow} y'}\]
\end{center}
\end{itemize}

In this paper we focus on several process operators from the literature, and try to establish which syntactic properties a modal language ${\cal O} \subseteq {\it HML}$ should satisfy to guarantee that the induced equivalence is a congruence with respect to the given operator. That is, given a process operator $f$, we will search for a syntactic condition $C$ such that if ${\cal O}$ satisfies $C$, then $f$ is compositional with respect to $\sim_{\hmo}$.

\section{Basic operators}

\subsection {Alternative composition}

We start with alternative composition, which expresses a nondeterministic choice between two processes. We want to find a general property of a modal language that would guarantee congruence of the induced equivalence with respect to alternative composition. Our first observation is that the behaviour of an alternative composition $p_1+p_2$ \textit{after performing the first step} is completely determined by the behaviour of one of the components. For example, $p_1+p_2 \models \diam{a}\varphi$ if and only if either $p_1 \models \diam{a} \varphi$ or $p_2 \models \diam{a} \varphi$. The only potential problem can occur when there is a formula with a conjunction at level 0 (i.e., not in the scope of an action prefix). For instance, consider $ {\cal O} = \{ \diam{a} {\sf T} \wedge \diam{b} {\sf T} \} $.  We have $a{\bf 0} \sim_{\cal O} {\bf 0}$ and $b{\bf 0} \sim_{\cal O} {\bf 0}$, but $a{\bf 0} + b{\bf 0} \not\sim_{\cal O} {\bf 0} + {\bf 0}$. As it turns out, it suffices to simply close the language on sub-conjunctions at level 0.

\begin{theorem}
Let ${\cal O} \subseteq {\it HML}$. If for any 0-level context $C_{0}[]$ and $\varphi_i\in {\it HML}$ for $i\in I$,
\begin{center}
(AC)~~~~~$C_{0}[\bigwedge_{i \in I} \varphi_i]\in{\cal O}$ implies that $\forall_{i \in I}: (\varphi_i \in {\cal O}^\equiv)$,
\end{center}
then $\sim_{\cal O}$ is a congruence with respect to alternative composition ($+$).
\end{theorem}
\begin{proof} Assume a modal language ${\cal O}$ with the AC property. Let $p_1 \sim_{\cal O} q_1$ and $p_2 \sim_{\cal O} q_2$. We show that for any $\varphi\in {\cal O}$:
\begin{center}
$p_1 + p_2 \models \varphi ~\Rightarrow~ q_1 + q_2 \models \varphi$
\end{center}
(the converse implication "$\Leftarrow$" is symmetric).  We apply induction on the structure of $\varphi$. The base case (${\sf T}$) is trivial. We proceed with the inductive step. Assume that $p_1 + p_2 \models \varphi$. We have to consider the following cases:

\begin{itemize}
\item $\varphi = \diam{a} \psi$: then either $p_1 \models \diam{a} \psi$ or $p_2 \models \diam{a} \psi$. From the equivalence of components we have either $q_1 \models \diam{a} \psi$ or $q_2 \models \diam{a} \psi$, which yields $q_1 + q_2 \models \diam{a} \psi$.

\item $\varphi = \bigwedge_{i \in I} \varphi_i$: we have $p_1 + p_2 \models \bigwedge_{i \in I}\, \varphi_i ~\Leftrightarrow~ \forall_{i \in I}: p_1 + p_2 \models \varphi_i ~\Leftrightarrow~ \forall_{i \in I}: q_1 + q_2 \models \varphi_i$ (AC + inductive hypothesis) $ \Leftrightarrow~ q_1 + q_2 \models \bigwedge_{i \in I} \varphi_i$.

\item $\varphi = \neg \psi$: let $\psi'$ be the outermost subformula of $\varphi$ which does not begin with a "$\neg$" symbol (so $\varphi = (\neg)^{n} \psi'$). Then $\varphi$ is logically equivalent to either $\psi'$ or $\neg \psi'$. The case $\psi' = {\sf T}$ is trivial. Also, the case where $\varphi \equiv \psi'$ can be handled analogously as the first two cases. We thus have to consider two possibilities:

\begin{itemize}
\item $\varphi \equiv \neg \diam{a} \varphi'$: we have $p_1 + p_2 \models \neg \diam{a} \varphi' ~\Leftrightarrow~ p_1 \models \neg \diam{a} \varphi' \wedge p_2 \models \neg \diam{a} \varphi' ~\Leftrightarrow~ q_1 \models \neg \diam{a} \varphi' \wedge q_2 \models \neg \diam{a} \varphi'$ (equivalence of components) $\Leftrightarrow~q_1 + q_2 \models \neg \diam{a} \varphi'$.

\item $\varphi \equiv \neg \bigwedge_{i \in I} \varphi_i$: we have $p_1 + p_2 \models \neg \bigwedge_{i \in I}\, \varphi_i ~\Leftrightarrow~ \exists_{i \in I}: p_1 + p_2 \models \neg \varphi_i ~\Leftrightarrow~ \exists_{i \in I}: q_1 + q_2 \models \neg \varphi_i$ (AC + inductive hypothesis) $ \Leftrightarrow~ q_1 + q_2 \models \neg \bigwedge_{i \in I} \varphi_i  $.
\end{itemize}

\end{itemize}

\end{proof}

For example, consider ${\cal O} = \{  \diam{a}(\diam{a}{\sf T} \wedge \diam{b}{\sf T})  \wedge \neg \diam{b}{\sf T}, \diam{a}(\diam{a}{\sf T} \wedge \diam{b}{\sf T}), \neg \diam{b}{\sf T} \}$. The language ${\cal O}$ satisfies the AC requirement, and so the corresponding equivalence $\sim_{\cal O}$ is a congruence with respect to $+$.

\vspace{2mm}

Almost all modal characterizations of standard process semantics from Section \ref{sec:hml} fulfill AC. The only exception is the modal characterization of completed trace equivalence,
although we can provide an alternative characterization that meets the AC requirement:
\begin{center}
${\cal O}^\ast_{CT} \hspace{5.0 pt} \varphi ::= \varphi'~(\varphi' \in {\cal O}_{CT})\hspace{3.0 pt}|\hspace{3.0 pt} \neg \diam{a} {\sf T}$
\end{center}
The characterization ${\cal O}^\ast_{CT}$ is the same as ${\cal O}_{CT}$, except that it includes formulas $\neg \diam{a} {\sf T}$. Clearly this does not change the corresponding semantics.

\subsection{Action prefix}

In the case of action prefix, it is easy to obtain a sufficient congruence requirement; the crucial observatioin is that $a.p \models \diam{a} \varphi$ if and only if $p \models \varphi$, so we need to make sure that for each formula $C_0[\diam{a}\varphi] \in {\cal O}$, the subformula $\varphi$ also belongs to the language ${\cal O}$. If this is not the case, an equivalence might not be a congruence. For instance, if ${\cal O} = \{ \diam{a}\diam{a}{\sf T} \}$, then $a.{\bf 0} \sim_{\cal O} {\bf 0}$, but $a.a.{\bf 0} \not\sim_{\cal O} a.{\bf 0}$.

\begin{theorem} Let ${\cal O} \subseteq {\it HML}$ and fix $a \in {\it Act}$. If for any 0-level context $C_0[]$ and $\varphi\in {\it HML}$,
\begin{center}
(AP)~~~~~$C_0[\diam{a} \varphi] \in {\cal O}$ implies that $\varphi \in {\cal O}^\equiv$,
\end{center}
then $\sim_{\cal O}$ is a congruence with respect to the action prefix operator $a.()$.
 \end{theorem}
\begin{proof}
Let $p \sim_{\cal O} q$. We need to show that for any $\varphi \in {\cal O}$, $a.p \models \varphi \Leftrightarrow a.q \models \varphi$.

Take any $\varphi \in {\cal O}$. Let $\varphi = C[\diam{a_i} \varphi_i]_{i \in I}$ such that the multicontext $C[]_{i \in I}$ does not contain any action prefix symbols. That is, the $\diam{a_i} \varphi_i$ for $i\in I$ are all action prefix subformulas of $\varphi$ that appear at level zero. Since $C[]_{i\in I}$ is built from only ${\sf T}$, conjunction and negation, whether a process satisfies $\varphi$ is completely determined by the satisfiability of $\diam{a_i} \varphi_i$ for $i \in I$ by this process. In other words, if $\forall_{i \in I}: (p_1 \models \diam{a_i} \varphi_i \Leftrightarrow p_2 \models \diam{a_i} \varphi_i)$, then $p_1 \models \varphi \Leftrightarrow p_2\models \varphi$. Coming back to our setting with $a.p$ and $a.q$, take an arbitrary $i \in I$. We have:\\
$a.p \models \diam{a_i} \varphi_i$\\
$\Leftrightarrow (a_i = a) \wedge p \models \varphi_i$\\
$\Leftrightarrow (a_i = a) \wedge q \models \varphi_i$ (AP + $p\sim_{\cal O} q$)\\
$\Leftrightarrow a.q \models \diam{a_i} \varphi_i$.\\
The choice of $i$ was arbitrary, hence the earlier remark yields: $a.p \models \varphi \Leftrightarrow a.q \models \varphi$.
\end{proof}
The AP condition is satisfied by all modal characterizations from Section \ref{sec:hml}.

\section{Restriction operators: projection and encapsulation}

We now consider projection and encapsulation operators. The $n$th projection of a process $p$, for $n\geq 0$, mimicks the behaviour of $p$ up to level $n$:
\begin{center}
\[ \frac{x \stackrel{a}{\rightarrow}x'}{\pi_{n+1}(x)\stackrel{a}{\rightarrow}\pi_{n}(x')} \]
\end{center}
Applying encapsulation with parameter $B \subseteq {\it Act}$ removes all transitions whose labels are in $B$ from the process:
\begin{center}
\[ \frac{x \stackrel{a}{\rightarrow}x'~~(a \not \in B)}{\partial_{B}(x)\stackrel{a}{\rightarrow}\partial_{B}(x')} \]
\end{center}
More generally, we consider unary restriction operators $f$ such that given a process $p$, the process $f(p)$ can be viewed as a subgraph of $p$. Below we will give a precise description of which restriction operators are covered. For the projection operator $\pi_n$ as well as for the encapsulation operator $\partial_B$, given any ${\it HML}$ formula we can deduce in advance which of its subformulas $\diam{a} \varphi$ will always yield false, regardless of the process $\pi_n(p)$ or $\partial_B(p)$ for which the ${\it HML}$ formula is evaluated. In case of a process $\pi_n(p)$, any subformula $\diam{a} \varphi$ that appears at level $n$ can be replaced by ${\sf F}$. And in case of a process $\partial_B(p)$, any subformula $\diam{b} \varphi$ with $b \in B$ can be replaced by ${\sf F}$.

We cannot reason in this way about any restriction operator. For example, consider the priority operator $\theta$, which assumes a partial order $<$ on the set of actions and allows us to execute an action only if no action with higher priority is executable at the same time:
\begin{center}
\[ \frac{x \stackrel{a}{\rightarrow} x' ~~~~~~ \forall b\in{\it Act}\,(a<b\,\Rightarrow\, x \not \stackrel{b}{\rightarrow})} {\theta(x) \stackrel{a}{\rightarrow} \theta(x')} \]
\end{center}
Suppose that $a > b$ and there is a process $p$ of which we only know that it satisfies $\diam{b} {\sf T}$. This knowledge is not sufficient to determine whether $\theta(p) \models \diam{b} {\sf T}$.

Let $f$ be a unary operator such as $\pi_n$ or $\partial_B$. We would like to define for each formula $\varphi \in {\it HML}$ a corresponding formula ${\it cut}_f(\varphi)$ in which every subformula $\diam{a} \varphi'$ which is known in advance to be unsatisfiable when evaluating any process $f(p)$ is replaced by ${\sf F}$. Actually this means that either we can replace a larger subformula by ${\sf T}$, or the entire formula becomes ${\sf F}$. Namely, we can replace the first innermost negation symbol (closest to the introduced ${\sf F}$) and the following subformula by ${\sf T}$; if the ${\sf F}$ symbol does not appear within the scope of a negation symbol, then the whole formula yields ${\sf F}$. If a language ${\cal O}\subseteq{\it HML}$ is closed under ${\it cut}_f$, then it induces a congruence with respect to $f$. The whole idea is made formal below.

\begin{lemma}
\label{lem:cut1} Let $f$ be a unary process operator. Suppose there exists a function  ${\it cut}_f: {\it HML} \longrightarrow {\it HML}$ such that for any process $p$ and $\varphi\in{\it HML}$,
\begin{center}
\begin{tabular}{l l}
(CUT)~~ & $f(p) \models \varphi~\Leftrightarrow~p \models {\it cut}_f(\varphi)$\\
\end{tabular}
\end{center}
Then for any language ${\cal O}$ satisfying
\begin{center}
$\varphi \in {\cal O}~\Rightarrow~({\it cut}_f(\varphi) \in {\cal O}^\equiv~\lor~{\it cut}_f(\varphi)\equiv {\sf F})$
\end{center}
the corresponding equivalence $\sim_{\cal O}$ is a congruence with respect to $f$.
\end{lemma}
\begin{proof} Suppose ${\cal O} \subseteq {\it HML}$ and $p \sim_{\cal O} q$. We have
$f(p) \models \varphi ~\Leftrightarrow~ p \models {\it cut}_f(\varphi)$ (CUT) $\Leftrightarrow q \models {\it cut}_f(\varphi)$ (either because ${\it cut}_f(\varphi) \in {\cal O}^\equiv$ and $p \sim_{\cal O} q$, or because ${\it cut}_f(\varphi)\equiv {\sf F}$) $\Leftrightarrow f(q) \models \varphi$ (CUT).
\end{proof}
The next lemma gives an explicit condition for a modal language to induce a congruence in case ${\it cut}_f$ formulas are obtained from the original ones by turning certain subformulas $\diam{a} \varphi$ into ${\sf F}$.

\begin{lemma}
\label{lem:cut2}
Assume $f$ and ${\it cut}_f$ are as in Lem.~\ref{lem:cut1}, and satisfy CUT. Suppose that for each $\varphi \in {\it HML}$ there exists a multicontext $C[]_{ i \in I}$ such that
$\varphi = C[\diam{a_i} \varphi_i]_{i \in I}$ and ${\it cut}_f(\varphi) \equiv C[{\sf F}]_{i \in I}$.
Then for each language ${\cal O} \subseteq {\it HML}$ that satisfies for any context $C'[]$ and $\varphi \in {\it HML}$,
\begin{center}
\begin{tabular}{l l}
(RES) & $C'[\neg \varphi] \in {\cal O}$ implies $C'[{\sf T}] \in {\cal O}^\equiv$,
\end{tabular}
\end{center}
the corresponding equivalence $\sim_{\cal O}$ is a congruence with respect to $f$.
\end{lemma}
\begin{proof} By Lem.\ \ref{lem:cut1} it suffices to prove that for all $\varphi \in {\cal O}$ either ${\it cut}_f(\varphi) \in {\cal O}$ or ${\it cut}_f(\varphi) \equiv {\sf F}$. Take any $\varphi \in {\cal O}$ such that ${\it cut}_f(\varphi) \not \equiv {\sf F}$. By assumption, ${\it cut}_f(\varphi) \equiv C[{\sf F}]_{i \in I}$ for some multicontext $C[]_{i \in I}$. Since ${\it cut}_f(\varphi) \not \equiv {\sf F}$, clearly each occurrence of ${\sf F}$ in this formula must be within the scope of a negation symbol.
Hence ${\it cut}_f(\varphi) \equiv C'[\neg D^i[{\sf F}]]_{i \in I}$, where we can choose contexts $D^i[]$ for $i\in I$ such that in each $D^i[]$, $[]$ is not within the scope of a negation.
Then $C'[\neg D^i[{\sf F}]]_{i \in I} \equiv C'[{\sf T}]_{i \in I}$. Since ${\cal O}$ satisfies RES, $C'[{\sf T}]_{i \in I} \in {\cal O}^\equiv$. Hence ${\it cut}_f(\varphi) \in {\cal O}^\equiv$.
\end{proof}
We have provided a compositionality framework for a general class of restriction operators. What remains is to provide ${\it cut}_f$ functions for the projection and encapsulation operators.

\begin{lemma}
\label{lem:cut3}
The functions ${\it cut}_f$ defined below are proper cutting functions (i.e., they satisfy condition CUT of Lem.~\ref{lem:cut1}).\vspace{4mm}\\
a) For the projection operators $\pi_n$ with $n\geq 0$:\vspace{2mm}\\
\begin{tabular}{lll}
${\it cut}_n({\sf T}) = {\sf T}$ &
${\it cut}_n(\bigwedge_{i \in I} \varphi_i) = \bigwedge_{i \in I} {\it cut}_n(\varphi_i)$ &
${\it cut}_n(\neg \varphi) = \neg {\it cut}_n(\varphi)$ \vspace{2mm} \\
${\it cut}_0(\diam{a} \varphi) = {\sf F}$ & ${\it cut}_{n+1}(\diam{a}\varphi) = \diam{a}{\it cut}_n(\varphi)$
\end{tabular}
\\ \\ \\
b) For the encapsulation operators $\partial_B$ with $B \subseteq {\it Act}$:\vspace{2mm}\\
\begin{tabular}{lll}
${\it cut}_B({\sf T}) = {\sf T}$ &
${\it cut}_B(\bigwedge_{i \in I} \varphi_i) = \bigwedge_{i \in I} {\it cut}_B(\varphi_i)$ &
${\it cut}_B(\neg \varphi) = \neg {\it cut}_B(\varphi)$ \vspace{2mm} \\
${\it cut}_B(\diam{a} \varphi) = {\sf F}$ if $a \in B$ &
${\it cut}_B(\diam{a}\varphi) = \diam{a}{\it cut}_B(\varphi)$ if $a \not \in B$
\end{tabular}
\end{lemma}
\begin{proof}
a) We prove CUT by induction on the structure of $\varphi$.
\begin{itemize}
\item $\varphi = {\sf T}$:

$\pi_n(p) \models {\sf T}$  and $p\models {\it cut}_n({\sf T})={\sf T}$.

\item $\varphi = \diam{a} \psi$:

We distinguish the cases $n=0$ and $n > 0$.
Clearly $\pi_0(p) \not\models \diam{a} \psi$ and $p\not\models {\it cut}_0(\diam{a} \psi)={\sf F}$.

If $ n > 0$, then $\pi_{n}(p) \models \diam{a} \psi
\Leftrightarrow \exists p': p \stackrel{a}{\rightarrow} p' \wedge \pi_{n-1}(p') \models \psi$ (transition rule for $\pi_n$)
$\Leftrightarrow \exists p': p \stackrel{a}{\rightarrow} p' ~\wedge~ p' \models {\it cut}_{n-1}(\psi)$ (structural induction)
$\Leftrightarrow p \models \diam{a}{\it cut}_{n-1}(\psi)
\Leftrightarrow p \models {\it cut}_{n}(\diam{a} \psi)$ (definition of ${\it cut}_n$).

\item $\varphi = \bigwedge_{i \in I} \psi_i$:

$\pi_n(p) \models \bigwedge_{i \in I} \psi_i
\Leftrightarrow \forall_{i \in I}: \pi_n(p) \models \psi_i
\Leftrightarrow \forall_{i \in I}: p \models {\it cut}_n(\psi_i)$ (structural induction)
$\Leftrightarrow p \models {\it cut}_n(\bigwedge_{i \in I} \psi_i)$ (definition of ${\it cut}_n$).

\item $\varphi = \neg \psi$:

$\pi_n(p) \models \neg \psi
\Leftrightarrow \pi_n(p) \not \models \psi
\Leftrightarrow p \not \models {\it cut}_n(\psi)$ (structural induction)
$\Leftrightarrow p \models \neg {\it cut}_n(\psi)$ $\Leftrightarrow p \models {\it cut}_n(\neg \psi)$ (definition of ${\it cut}_n$).
\end{itemize}

b) Again we use structural induction on $\varphi$.
\begin{itemize}
\item $\varphi = {\sf T}$:

$\partial_B(p) \models {\sf T}$  and $p \models {\it cut}_B({\sf T})={\sf T}$.

\item $\varphi = \diam{a} \psi$:

Suppose first that $a  \in B$. Then $\partial_B(p) \not \models \diam{a} \psi$ (transition rule for $\partial_B$)
and $p \not \models {\it cut}_B(\diam{a} \psi)={\sf F}$ (definition of ${\it cut}_B$).

Suppose now that $a \not \in B$.
Then $\partial_B(p) \models \diam{a} \psi$
$\Leftrightarrow \exists p': p \stackrel{a}{\rightarrow} p' \wedge \partial_B(p') \models \psi$ (transition rule for $\partial_B$)
$\Leftrightarrow \exists p': p \stackrel{a}{\rightarrow} p' \wedge p' \models {\it cut}_B(\psi)$ (structural induction)
$\Leftrightarrow p \models \diam{a}{\it cut}_B(\psi) \Leftrightarrow p \models {\it cut}_B(\diam{a}\psi)$ (definition of ${\it cut}_B$).

\item $\varphi = \bigwedge_{i \in I} \psi_i$:

$\partial_B(p) \models \bigwedge_{i \in I} \psi_i
\Leftrightarrow \forall_{i \in I}: \partial_B(p) \models \psi_i
\Leftrightarrow \forall_{i \in I}: p \models {\it cut}_B(\psi_i)$ (structural induction)
$\Leftrightarrow p \models {\it cut}_B(\bigwedge_{i \in I} \psi_i)$ (definition of ${\it cut}_B$).

\item $\varphi = \neg \psi$:

$\partial_B(p) \models \neg \psi$ ~
$ \Leftrightarrow \partial_B(p) \not \models \psi
\Leftrightarrow p \not \models {\it cut}_B(\psi)$ (structural induction)
$\Leftrightarrow p \models \neg {\it cut}_B(\psi)$ $\Leftrightarrow p \models {\it cut}_B(\neg \psi)$ (definition of ${\it cut}_B$).
\end{itemize}

\end{proof}
\begin{theorem}
For any language ${\cal O} \subseteq {\it HML}$ satisfying RES, the corresponding equivalence $\sim_{\cal O}$ is a congruence with respect to the projection operators $\pi_n$ (for $n\geq 0$) and the encapsulation operators $\partial_B$ (for $ B \subseteq {\it Act}$).
\end{theorem}

\begin{proof}
By Lem.\ \ref{lem:cut3}, the functions ${\it cut}_n$ and ${\it cut}_B$ satisfy CUT.
Observe that the ${\it cut}_n$ and ${\it cut}_B$ functions defined in the Lem.~\ref{lem:cut3} only replace certain subformulas $\diam{a} \psi$ of the original formula with ${\sf F}$. So they meet the requirements of Lem.~\ref{lem:cut2}. Congruence is thus an immediate consequence of Lem.\ \ref{lem:cut2}.
\end{proof}

To demonstrate that the RES requirement is essential, consider the following counterexamples.
\begin{itemize}
\item
For projection, take ${\cal O} = \{ \neg \diam{a} \neg \diam{a} {\sf T} \}$. We have $aa{\bf 0} \sim_{\cal O} {\bf 0}$, but $\pi_1(aa{\bf 0}) \not\sim_{\cal O} \pi_1({\bf 0})$.
\item
 For encapsulation, take ${\cal O} = \{ \diam{a} \neg \diam{b} {\sf T}\}$. We have $ab{\bf 0} \sim_{\cal O} {\bf 0}$, but $\partial_{\{b\}}(ab{\bf 0}) \not \sim_{\cal O} \partial_{\{b\}}({\bf 0})$.
\end{itemize}

The RES requirement is satisfied by every characterization from Section \ref{sec:hml}, except for completed trace observations. Completed trace equivalence is a congruence with respect to projection operators, but not encapsulation. Take for instance the completed trace equivalent processes $a(b{\bf 0}+c{\bf 0})$ and $ab{\bf 0}+ac{\bf 0}$. We have $\partial_{\{b\}}(a(b{\bf 0}+c{\bf 0})) \,\sim_{CT}\, ac{\bf 0} \,\not \sim_{CT}\, a{\bf 0}+ac{\bf 0} \,\sim_{CT}\, \partial_{\{b\}}(ab{\bf 0}+ac{\bf 0})$.

\section{Parallel composition ($||$)}

We now consider the parallel composition operator (without communication). That is, $p ||q$ behaves as $\lmerge{p}{q} + \lmerge{q}{p}$ where the left-merge operator is defined by
\begin{center}
\[ \frac{x \stackrel{a}{\rightarrow} x'}{\lmerge{x}{y} \stackrel{a}{\rightarrow} x' || y} \]
\end{center}

Let us restrict for a moment to only trace formulas (meaning that conjunctions are disregarded). The following example shows that the requirement AP and even being closed under substrings is not sufficient (by a substring of $w$ we mean a subsequence constisting of elements appearing \textit{consecutively} in $w$).
Take ${\cal O} = \{ \diam{a}{\sf T},\diam{b}{\sf T},\diam{a}\diam{b}{\sf T},\diam{a}\diam{b}\diam{a}{\sf T},\diam{b}\diam{a}{\sf T} \}$. This language not only satisfies AP, but is also closed under prefixes and substrings (but not arbitrary subsequences). However, we have $aa{\bf 0} \sim_{\cal O} a{\bf 0}$, but $aa{\bf 0} || b{\bf 0} \models \diam{a}\diam{b}\diam{a}{\sf T}$ while $a{\bf 0}||b{\bf 0}$ does not satisfy this formula.

This example suggests that if a trace $\sigma$ belongs to ${\cal O}$, then all \textit{subsequences} of $\sigma$ must belong to the language as well. This is not unexpected; the behaviour of parallel composition consists of all possible interleavings of the component processes, and all of these interleavings should be described in the modal characterization.

It is also necessary to close the language on subconjunctions. Indeed, take ${\cal O} = \{ \diam{a}{\sf T} \wedge \diam{b} {\sf T} \}$, a language which does not meet this condition. We have $a{\bf 0} \sim_{\cal O} b{\bf 0}$, but $a{\bf 0}||b{\bf 0}\models\diam{a}{\sf T} \wedge \diam{b} {\sf T}$ while $b{\bf 0}||b{\bf 0}$ does not satisfy this formula.

In case of general ${\it HML}$ formulas, we first define a generalization of a subsequence for an arbitrary formula $\varphi \in {\it HML}$ by specifying a set of subformulas with possible replacement from a lower level. We thus define $Sub(\varphi)$ as the smallest set of ${\it HML}$ formulas satisfying:
\begin{itemize}
\item
$\varphi \in Sub(\varphi)$;
\item
$\varphi' \in Sub(\varphi) \Rightarrow \{ D[\psi] ~|~ \varphi' = D[C[\psi]] \} \subseteq Sub(\varphi)$.
\end{itemize}

We now define a tool to infer satisfaction of modal formulas by a parallel composition $p||q$ from the formulas satisfied by the component processes $p$ and $q$. This is accomplished by the function $Par$, which given $A\subseteq {\it HML}$ and $B\subseteq {\it HML}$, returns the collection of formulas that are certainly satisfied by a parallel composition of two processes satisfying $A$ and $B$ respectively. One can view $Par(A,B)$ as parallel composition operator on collections of modal formulas.

Formally, $Par: {\cal P}({\it HML}) \times {\cal P}({\it HML}) \longrightarrow {\cal P}({\it HML})$ is defined with induction on the structure of formulas.
\begin{itemize}
\item ${\sf T} \in Par(A,B)$
\item $\diam{a} \varphi \in Par(A,B) $
$~\stackrel{\rm def}\Leftrightarrow~$ $( \exists \diam{a} \varphi_A \in A: \varphi \in Par(\varphi_A,B))$
$\vee ( \exists \diam{a} \varphi_B \in B: \varphi \in Par(A,\varphi_B))$
\item $ \bigwedge_{i \in I} \varphi_i \in Par(A,B)$
$~\stackrel{\rm def}\Leftrightarrow~ \forall_{i \in I}: \varphi_i \in Par(A,B)$
\item $\neg \varphi \in Par(A,B)$
$ ~\stackrel{\rm def}\Leftrightarrow~ \forall C,D \subseteq Sub(\varphi): \varphi \in Par(C,D) ~ (\exists \psi_C \in C: \neg \psi_C \in A) \vee (\exists \psi_D \in D: \neg \psi_D \in B)$
\end{itemize}

By abuse of notation, we let $A\subseteq{\it HML}$ also denote the formula $\bigwedge_{\varphi\in A}\varphi$.

\begin{lemma} Let $\varphi \in {\it HML}$.
\label{lem:par}
\begin{center}
$p || q \models \varphi \Leftrightarrow \exists A,B \subseteq Sub(\varphi): (p \models A \wedge q \models B \wedge \varphi \in Par(A,B))$.
\end{center}
\end{lemma}

\begin{proof}
 We use induction on the structure of formulas. The base case ($\varphi = {\sf T}$) is immediate. We proceed with the inductive step:

$"\Rightarrow"$: Assume that $p || q \models \varphi$. We prove that $\exists A,B \subseteq Sub(\varphi): (p \models A \wedge q \models B \wedge \varphi \in Par(A,B))$.

\begin{itemize}
\item $\varphi = \diam{a} \psi$: Without loss of generality, suppose $\lmerge{p}{q} \models \diam{a} \psi$ (the case $\lmerge{q}{p} \models \diam{a} \psi$ is symmetric), so $p \stackrel{a}{\rightarrow} p' \wedge p' || q \models \psi$. From the inductive hypothesis we know that there are $A',B' \subseteq Sub(\psi)$ such that $p' \models A' \wedge q \models B' \wedge \psi \in Par(A',B')$. We take $A = \diam{a}(\bigwedge_{\varphi\in A'}\varphi)$ and $B =B'$.

\item $\varphi = \bigwedge_{i\in I} \varphi_i$: By the inductive hypothesis, for each $i\in I$ there are $A_i,B_i\subseteq Sub(\varphi_i)$ such that $p\models A_i \wedge q\models B_i \wedge \varphi_i\in Par(A_i,B_i)$. We can take $A = \bigcup_{i \in I} A_i$ and $B = \bigcup_{i \in I} B_i$.

\item $\varphi = \neg \psi$: We have:\\
$p ||q \models \neg \psi $ $\Leftrightarrow$ $ \neg (p ||q \models \psi)$\\
$\Leftrightarrow \neg (\exists C,D \subseteq Sub(\psi): (p \models C \wedge q \models D \wedge \psi \in Par(C,D)))$ (inductive hypothesis)\\
$\Leftrightarrow \forall C,D \subseteq Sub(\psi): \psi \in Par(C,D)
(\exists \psi_C \in C: p \not \models \psi_C) \vee (\exists \psi_D \in D: q \not \models \psi_D)$.\\
We define:\\
$ A_p(\psi) = \bigcup_{C,D \subseteq Sub(\psi): \psi \in Par(C,D) } \{ \neg \psi_C ~|~ p \not \models \psi_C \wedge \psi_C \in C \} $\\
$ B_q(\psi) = \bigcup_{C,D \subseteq Sub(\psi): \psi \in Par(C,D) } \{ \neg \psi_D ~|~ q \not \models \psi_D \wedge \psi_D \in D \} $\\
These are the $A$ and $B$ we are looking for.
\end{itemize}

$"\Leftarrow"$: Suppose that $\exists A,B \subseteq Sub(\varphi): (p \models A, q \models B \wedge \varphi \in Par(A,B))$. We prove that $p||q \models \varphi$.
\begin{itemize}

\item $\varphi = \diam{a} \psi$: From $\diam{a} \psi \in Par(A,B)$ we have $( \exists \diam{a} \psi_A \in A: \psi \in Par(\psi_A,B))$
$\vee ( \exists \diam{a} \psi_B \in B: \psi \in Par(A,\psi_B))$. Without loss of generality suppose that $(\exists \diam{a} \psi_A \in A: \psi \in Par(\psi_A,B))$. Then $p \stackrel{a}{\rightarrow} p': p' \models \psi_A$. From $\psi \in Par(\psi_A,B)$ and the inductive hypothesis we have $p' || q \models \psi$. Since $p || q \stackrel{a}{\rightarrow} p' || q$, we finally obtain $p||q \models \diam{a} \psi$.

\item $\varphi = \bigwedge_{i\in I} \varphi_i$: According to the definition of $Par$ we have $\forall_{i \in I}: \varphi_i \in Par(A,B)$. The inductive hypothesis yields $\forall_{i \in I}: p||q \models \varphi_i$, and hence $p||q \models \bigwedge_{i\in I} \varphi_i$.

\item $\varphi = \neg \psi$:  We have $\forall C,D \subseteq Sub(\psi): \psi \in Par(C,D) ~ (\exists \psi_C \in C: \neg \psi_C \in A) \vee (\exists \psi_D \in D: \neg \psi_D \in B)$. Suppose, towards a contradiction, that $p||q \models \psi$. Then according to the inductive hypothesis there exist $C,D$ such that $p \models C$, $q \models D$ and $\psi \in Par(C,D)$. But from the earlier remark, we have either $\in C: \neg \psi_C \in A$ or $\psi_D \in D: \neg \psi_D \in B$. This contradicts the fact that $p \models A,C$ and $q \models B,D$.

\end{itemize}
\end{proof}

\begin{theorem} For any language ${\cal O} \subseteq {\it HML}$ satisfying
\begin{center}
(PAR)~~~~~$\varphi \in {\cal O} \Rightarrow Sub(\varphi) \subseteq {\cal O}^\equiv$,
\end{center}
 $\sim_{\cal O}$ is a congruence with respect to parallel composition $||$.
\end{theorem}
\begin{proof}
Suppose $p_1 \sim_{\cal O} q_1$ and $p_2 \sim_{\cal O} q_2$. Suppose that $p_1 || p_2 \models \varphi \in {\cal O}$. According to Lem.~\ref{lem:par}, there exist $ A,B \subseteq Sub(\varphi)$ such that $p_1 \models A, p_2 \models B$ and $\varphi \in Par(A,B)$. Since $\varphi \in {\cal O}$, by condition PAR, $A,B\subseteq{\cal O}^\equiv$. Since $p_1 \sim_{\cal O} q_1$ and $p_2 \sim_{\cal O} q_2$, it follows that $q_1 \models A$ and $q_2 \models B$. According to Lem.~\ref{lem:par} this implies $q_1 || q_2 \models \varphi$.

\end{proof}

As an example, if we want to define a modal language that would be a congruence with respect to parallel composition, which includes behaviour described by a formula $\diam{a}(\neg \diam{b} {\sf T} \wedge \diam{c} \diam{d}{\sf T})$, we should include the following formulas in the characterization (we omit irrelevant formulas like $\diam{a}\neg {\sf T}$):
$\diam{a}{\sf T}$,
$\diam{a}\neg \diam{b} {\sf T}$,
$\diam{a}\diam{c} {\sf T}$,
$\diam{a}\diam{c} \diam{d} {\sf T}$,
$\diam{a}\diam{d} {\sf T}$,
$\diam{a}(\neg \diam{b} {\sf T} \wedge \diam{c}{\sf T})$
$\diam{a}(\neg \diam{b} {\sf T} \wedge \diam{d}{\sf T})$.

All basic equivalences except for completed trace have modal characterizations that satisfy the condition PAR. We note that parallel composition is compositional with respect to completed trace equivalence.

\section{Conclusions and future work}

We have presented, for a number of process operators from the literature, general conditions that guarantee congruence of process equivalences defined by means of a modal characterization. To the best of our knowledge it is the first such attempt.

Our conditions are sufficient, but by no means necessary. We believe that it is difficult (if not impossible) to provide a syntactic restriction on a modal language that would characterize the class of congruences for a given operator (strictly speaking, languages that induce congruences). We aimed at clear and comprehensible rather than slightly relaxed but more complicated conditions.

As the next step, we would like to investigate other process operators (e.g.\ sequential composition, renaming, merge with communication), consider the setting of weak semantics and different modal languages. In the last case, if we consider e.g.\ ${\it HML}$ with recursion or the $\mu$-calculus, we may attempt to combine our work with existing results on characteristic formulas \cite{AcInSa09}. In that setting, instead of modal language properties, we could focus on compositionality of single formulas.

redpajama_set_name: RedPajamaArXiv


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"You didn't give me much of a choice," she retorted back.
"He's not my father," Jonathan said.
"I have my passport." She kept her voice even, and kept her gaze off of him.
"You're the one who's meant to understand the symbol your father left." He pulled out a tablet and began tapping on it, then offered the tablet to her. There, scanned in, was a copy of the symbol on one of the letters.
She hesitated. "When I was growing up, I was very into Ancient Roman studies. One of the superstitions the Romans had were curse tablets. They believed that if you wanted to curse someone, you wrote the curse on a tablet and then hid it in a place that no one could find. When I was nine, I wrote a curse on my Etch A Sketch and buried it in the backyard of my childhood home. And because I wanted to find it later, I carved that symbol onto the tree." She pointed at the symbol on the paper.


Finch, Pruyn & Company was formed in 1865 when Jeremiah and Daniel Finch, together with Samuel Pruyn, purchased the Glens Falls Company. Shortly thereafter they bought the Wing Mill, on which site they are still located. In 2007 the company was renamed Finch Paper LLC.
They started out with various lines of business including lumber, and it was only in 1905 that they started making paper. In the early years they made newsprint and hanging paper, the basis for wallpapers. Only in the 1950s did book-paper-making get going. Finch Opaque, the sheet best known to the publishing community, was only introduced in 1963, around the time when the company installed an odor-free pulping process and moved from coal power to oil. The mill is quite close to the middle of town — three or four blocks — so you can imagine the highly scented life in the town back then.
As is usually the case the mill is built next to a constant source of water, the Hudson River. In the early days this water would provide power. The falls are just upstream from the Finch Mill. In this old postcard view the mill is just to the left at the northern end of the bridge across the river. A mill lade leads off from the upper river and flows into the mill site. Not, I'm sure, that it had any influence on the founders decision-making, but the literary-minded may be interested to know that just under that bridge one can find the cave in which Hawkeye and his companions hid in James Fenimore Cooper's Last of the Mohicans. Part of the mill can be seen in the background of this photo of the cave.
Doubtless in the olden days the lade would also serve as a delivery method for their raw material. Logs, as we all sort of know, used not to be hauled around on trucks: they were thrown into the river and dramatically floated downstream to civilization. The last river drive carrying logs from the Adirondack Mountains down the Hudson River took place in 1950. This video, from Finch, shows a drive from the thirties. A more exciting job than driving a truck for sure.


Literature References: The term chamomile refers to the dried flowerheads of either of two distinct plants of the family Asteraceae (Compositae): German chamomile, Chamomilla recutita (L.) Rauschert, also known as Matricaria recutita L. or M. chamomilla, and Roman chamomile, Chamaemelum nobile (L.) All. (formerly Anthemis nobilis L.). Both chamomiles have been used in traditional medicines as antispasmodic and anti-inflammatory agents. Evaluation of hydroalcoholic extracts for antispasmodic activity: H. B. Forster et al., Planta Med. 40, 309 (1980); for anti-inflammatory activity: A. Tubaro et al., ibid. 50, 359 (1984). Stability of extracts: R. Carle et al., ibid. 55, 540 (1989). HPLC determn of flavones and coumarins in extracts: P. Pietta et al., J. Chromatogr. 404, 279 (1987). Reviews: A. Y. Leung, Encyclopedia of Common Natural Ingredients (Wiley-Interscience, New York, 1980) pp 110-112; C. Mann, E. J. Staba in Herbs, Spices, and Medicinal Plants: Recent Advances in Botany, Horticulture, and Pharmacology vol. 1 (Oryx Press, 1986) pp 235-280.
Literature References: Habit. Europe, northern and western Asia; naturalized in North America. Cultivated in Hungary, Romania, Germany. Constit. 0.24-1.9% volatile oil; flavonoids including apigenin, luteolin, quercetin; coumarins such as herniarin, umbelliferone; proazulenes (matricin, matricarin, etc.); phenolic carboxylic acids (anisic, vanillic, syringic, caffeic); polysaccharide mucilage.
Literature References: Extracted from flower heads by steam distillation. Constit. (-)-a-bisabolol (up to 50%), bisabolol oxides, chamazulene, farnesene. Review of constituents: B. M. Lawrence, Perfum. Flavor. 12, 35-52 (1987).
Properties: Deep ink-blue oil. d 0.91-0.95. Ester value: <40. Acid value: 5 to 50. Sol in propylene glycol. Insol in glycerin, mineral oil.
Literature References: Habit. Southern and western Europe; naturalized in North America. Cultivated in Great Britain, Belgium, France, U.S. Constit. up to 1.75% volatile oil; up to 0.6% sesquiterpene lactones of the germacranolide type (mainly nobilin, 3-epinobilin); flavonoids (apigenin, luteolin, apiin, etc.); phenolic carboxylic acids (caffeic, ferulic); coumarins; and thiophene derivatives.
Literature References: Extracted from flowerheads by steam distillation. Constit. esters of angelic and tiglic acids (~85%), chamazulene, a-pinene, farnesol.
Properties: Blue oil. n20 1.44-1.45. [a]20 -1 to +4°. Ester value: 250 to 310. Acid value <15. Sol in mineral oil and most fixed oils; sol with haziness in propylene glycol. Practically insol in glycerin.
Use: Flowers in herbal teas. Extracts and oils in perfumery; as flavor in foods and alcoholic beverages; hair dye.
Therap-Cat: Carminative; vulnerary. Extracts and oils as topical anti-inflammatory.


black and white Trevira Velvet.
Limited Edition of only 1090 pieces. Size 35cm. Gold plated Button in Ear.
Comes in a limited edition box with certificate of authenticity.
This entry was posted in Uncategorized on February 11, 2019 by Dave.


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India Today has reported India's triumphs and its despair, over the past 25 years. It was during the Kargil conflict of 1999 that we realised simple reporting wasn't enough. Whenever stories of disasters were published, offers for help from readers came flooding in, but there was never an opportunity to properly tap this wellspring of goodwill. On August 1999, the India Today group set up CARE TODAY, a registered society. It's mandate is simple: financial contributions from readers of India Today to address problems, report back on the progress through the pages of the magazine, and now through this web site.
India is suffering one of the worst droughts in living memory. Cattle are dying in the thousands. Millions of hungry, thirsty Indians, made penniless by the drought are leaving their homes in a desperate attempt to survive. We must all help as best we can. CARE TODAY, a trust run by the India Today Group, announces the launch of Fight the Drought, a fund for the people affected by the drought. The fund will be put together by contributions from the India Today group, readers and web users and then administered by the trust, as on two previous, proven occassions: the Kargil conflict and the Orissa supercyclone.
Words are often inadequate to describe the tragedy unfolding across the great plains of India. Here is an selection of images--some may be disturbing--that graphically tell you the story of the drought.
Using total donations of Rs. 29 lakh we supported the following projects. Seven drought proofing projects in Rajasthan Four drought proofing projects in Gujarat.


A South Carolina assistant principal resigned his position this week after a controversy broke over some of his Facebook posts, with hundreds of people signing a petition asking for his termination. The man, Charles Fowler, posted a picture to his Facebook page of a six-year-old girl shopping in a Wal-Mart with the caption "Honey Boo-Boo in Wal-Mart." The girl is a kindergartener in the same school district as Fowler's employer, Walhalla High School.
Posting online can often feel like it doesn't have consequences; obviously, the man wouldn't have said what he wrote on his Facebook page to the girl's face. But, despite this feeling of anonymity, poorly-worded social media posts can be exceedingly harmful to a person's career, particularly someone with a public profession.
If you're going to make off-color jokes on your Facebook page, it's a good idea to check your privacy settings and make sure only the people you want seeing your content can find it. Of course, it's smart to keep potentially risky posts off the Internet all together if you can help it, particularly the ones that have any chance at all of coming back to bite you.