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AllAuthorsArticle TitleHeadingsExhibitionsMuseum and Gallery HoldingsAuction RecordsBibliographyOccupationImage CaptionAbbreviation
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Benezit Dictionary of Artists (1)
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Publisher or Printer x
Painting and Drawing x
(1) Georg Philipp Rugendas I
Gode Krämer
Rugendas family
(2) Johann Moritz Rugendas
Roberto Pontual
Lechter, Melchior
German, 19th – 20th century, male.
Born 2 October 1865, in Münster; died 8 October 1937, in Raron (Valais, Switzerland).
Painter (including glass), pastellist, illustrator, draughtsman, decorative designer, graphic designer, writer, publisher.
Melchior Lechter was initially apprenticed to a painter of cartoons for stained-glass windows in Münster, before enrolling at the Hochschule der Künste, Berlin in ...
Okumura Masanobu
Juliann Wolfgram
[Baiō]
(b 1686; d 1764).
Japanese print designer, painter, book illustrator and publisher. Although Masanobu's artistic career spanned six decades, Edo-period (1600–1868) documents reveal little about his life. However, his prolific artistic output and technical innovations make him one of the leading figures of the early history of Japanese woodblock printing and ukiyoe ('pictures of the floating world', see Japan §X 2., (iii)). He began his career in 1701 with a copy of an album of courtesans known as Keisei ehon ('Yoshiwara picture book'; Chicago, IL, A. Inst.) by Torii Kiyonobu I (see Torii family, §1). His earliest sumizurie ('black-and-white pictures') were based on the subject-matter and style of the Torii school and were published in sets of 12 large prints (ōban) or in illustrated books (ehon). Masanobu illustrated no less than 19 novelettes and produced over 30 ehon (see Japan §X 2.). During the formative stage of his career, Masanobu also wrote popular fiction, which led him to develop a pictorial means of conveying literary wit and humour. Through the production of visual parodies of classical themes, known as ...
Gode Krämer and Roberto Pontual
German family of painters, draughtsmen, and engravers. (1) Georg Philipp Rugendas I, an esteemed painter and graphic artist in Augsburg, established a print publishing house there in 1735. His sons Georg Philipp Rugendas II (1701–74), Christian Rugendas (1708–81), and Jeremias Gottlob Rugendas (1710–72) helped with this business, as did their descendants, including Georg Philipp II's grandson, the engraver Johann Lorenz Rugendas II (1775–1826). (2) Johann Moritz Rugendas, the son of Johann Lorenz II, was noted particularly for his drawings and paintings of Brazil and other Latin American countries.
(b Augsburg, Nov 27, 1666; d Augsburg, May 9, 1742).
His father, a watchmaker, trained him in copper engraving, but after a fistula on his right hand forced him to abandon this he was apprenticed (...
Shemyakin, Mikhail
M. N. Sokolov
(Mikhaylovich) [Chemiakin, Mihail]
(b Moscow, May 4, 1943).
Russian painter, graphic designer, sculptor and publisher. One of the most important representatives of the St Petersburg tradition of nonconformist art, he was born to a military family and spent his early years in the German Democratic Republic. His family returned to the USSR in 1957 and until 1961 he studied at the secondary school of art attached to the Il'ya Repin Institute of Painting, Sculpture and Architecture, Leningrad (now St Petersburg). His work combines the World of Art tradition with the surreal grotesque, portraying the world as a colourful carnival, intimidating in its terrifying metamorphoses, but drawing upon a wealth of artistic styles and psychologically striking tones. He was a master of the anarchic, bohemian life, and the poet Andrey Voznesensky (b 1933) described him as the 'black prince of the Russian Underground'. After confrontations with the authorities, notably his participation in a group exhibition by underground artists of the ...
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IATF 2023
Abidjan Exhibition Center, Abidjan, Cote d'Ivoire
The IATF initiative supports the implementation of the African Continental Free Trade Area (AfCFTA). It is organised by Afreximbank in collaboration with the African Union and the AfCFTA Secretariat , marking an important step to sustainably addressing the gap in trade and market information for the successful realisation of the AfCFTA objectives. The Government of the Republic of Côte d'Ivoire will be hosting the 3rd edition of the IATF (IATF2023) in Abidjan from 21 – 27 November 2023, under the theme: Connecting African Markets.
A strong focus will be placed on recognising growth opportunities for Intra-African, cross-border trade and investments. IATF2023 will provide a unique and valuable platform for businesses to access an integrated African market of over 1.3 billion people with a GDP of over US$3.5 trillion created under the African Continental Free Trade Area.
The inaugural edition of IATF, which took place in 2021, attracted over 32,541 visitors and buyers and over 1,501 exhibitors from 69 countries resulting in over $42.1 billion worth of trade and investment deals that were signed at the event. The 3rd edition of IATF is expected to be even bigger and aims to attract over 35,000 visitors, more than 180 hosted buyers from 75 countries and over 1,600 exhibitors. IATF2023 is expected to generate up to US$43 billion in trade and investment deals.
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Despite the lack of detail on any concrete progress, stock markets jumped in response to Trump's tweet. Investors have worried about the president's threats to levy tariffs on an additional US$267 billion in Chinese goods and the damage the move could do to American consumers and companies. The duties would come on top of tariffs already imposed on US$250 billion in Chinese imports.
The tweet comes just five days before Tuesday's midterm elections. Numerous candidates across the nation — particularly those in areas that export agricultural products to China — have pushed for Trump to ease the trade tensions.
"Nothing is set in stone right now" on the potential new tariffs, the National Economic Council director said.
The president has long pushed to crack down on what he calls Chinese trade abuses. He has used tariffs to try to bring Beijing to the negotiating table, describing any economic pain as a temporary necessity to secure a better deal.
The two sides last held talks in late August. Reports last month indicated that a U.S. desire for a plan to address alleged intellectual property theft emerged as a sticking point.


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Home » History of the Lexington Motor Company
History of the Lexington Motor Company
When someone mentions Lexington, Kentucky, a number of different things may pop into your mind. Perhaps you'll remember that it was the home of President Abraham Lincoln. Maybe you'll recognize that it's the location of the University of Kentucky and the iconic Kentucky Wildcats athletic program. The economy may come to mind, including the presence of the Jif Peanut Butter plant.
One overlooked fact may be that Lexington housed one of the most important automobile manufacturers in United States history: the Lexington Motor Car Company. You probably won't find many of the brand's used car lots in Kentucky, but it's at least worthwhile to recognize the historical significance of the company…
The Beginning Years
The entire story really starts in 1899, when mechanic John C. Moore began designing an experimental vehicle based on the technology already available. Taking advantage of the components he'd seen on other makes of cars, Moore was able to produce a new make for a relatively low cost. Moore soon began working in Lexington for Blue Grass Automobile Company, and his work soon caught the eye of some investors.
The company was founded in 1909 by Kentucky race horse promoter Kinzea Stone, and the Lexington Motor Car Company grew faster than expected, as they were forced to move buildings as production increased. Contributing to the increasing interest in the car was the company's showing at the 1909 Glidden Reliability Tour, and the subsequent praise it got for it's sleek design. This was pretty good timing, as a group of Connersville, Indiana businessmen had recognized their community's reliance on buggies and carriages. This seemingly contradicted the booming automobile industry.
The Lexington Motor Car Company was asked to move once again to a larger plant in Connersville, and Moore, the company's chief engineer, immediately worked on ways to improve his company's vehicles. Among the first innovations were the inclusion of multiple exhaust, which reportedly gave a vehicle 30-percent more power on significantly less fuel. The original Lexington cars were typical four-cylinder automobiles, typically built in five, seven, or limousine capacities. That wasn't it for features.
"Rather than use an indifferent lot of parts, as constructed in their own shop, the same having been running but a few months, the company have given prospective customers the benefit of the doubt, and will use only tried and proven parts as made by the country's best specialists in those particular lines," an advertisement for the 1909 Lexington Model A read.
A New Owner Appears…
As the company struggled through financial difficulties in 1913, Lexington Motor Car Company was purchased by E.W. Ansted, who was tasked with creating a six-cylinder Howard. After a couple years of the 'Lexington-Howard' company, the Lexington Motor Car Company once again resumed using it's former name. However, the inclusion of Ansted wasn't a bad thing, as he supplemented the four-cylinder engines with a 'light six' (capable of producing 29 horsepower) or 'supreme six' (capable of 41 horsepower) engine.
These new engines resulted in a popularity boom for Lexington, and the company again moved factories in 1915. Their new location included the iconic smokestack with the Lexington name written on the side. Among the changes in the next five years were the revamping of the car's frame, including a new rigid box cross-section that eliminated any jamming door issues. Emergency brakes were soon added, and hardtop enclosures were included thereafter.
As this time, Lexington cars were generally in the middle of the pack in regards to pricing. The Thoroughbred Six sold for $2,875, which was essentially the priciest car on the market, but the Minute Man Six tourer and convertible sedan sold for $1,185 and $1,350, respectively. Meanwhile, the Enger 40 ($2,000), the Oakland 40 ($1,600) and Colt Runabout ($1,500) all topped the Lexington vehicles, while the Oldsmobile Runabout ($650) and Ford Model T ($440) cost significantly less.
The company further raised their notoriety following the 1920 Pikes Peak hill climb, when two short-wheelbase race cars (accompanied by the Ansted engine) finished first and second in the race. That same year, the company merged with Ansted Engineering Company and Connersville Foundry Corporation in a $10 million merger. The year would also be the peak of the company's production, with over 6,000 vehicles built.
A Wrench in the Gears
Despite the fact that Lexington's parent company, the United State Automotive Corporation, owned ten different factories around the country, the Lexington Motor Car Company began running into issues in the 1920s. General Motors founder William C. Durant asked the company to produce 30,000 Ansted engines for his Durant Six, a vehicle produced by Durant Motors , Inc. Lexington was later sued by General Motors consulting engineer (and Brush Runabout designer) Alanson P. Brush, with the engineer claiming that the Ansted engines infringed on a number of his patents.
By 1923, Lexington's production fell to about 2,000 vehicles. This decline could be attributed to the negative press surrounding the Brush lawsuit, as well as the post-World War I recession that resulted in the failings of many American automobile companies. Between 1926 and 1927, E.L. Cord's Auburn Automobile Company purchased the Lexington Motor Car Company, and Lexington was soon phased out. The factories were later used to make Auburns, Cords, Packard-Darrins and Army Jeeps.
While many car companies in the early 1900s were just trying to keep up with the competition, the Lexington Motor Car Company consistently strived to be innovative. Their six-cylinder engine was ahead of it's time, and the company's performance in various races brought it unmatched notoriety. There's no guessing what could have became of the company if not for a pesky lawsuit and an unavoidable recession. Regardless, the Lexington Motor Company had solidified it's place in automotive history.
The next time you're driving around Lexington, Kentucky, remember the rich automotive history that accompanies the city (in addition to Lexington Motor Company, the Toyota Motor Manufacturing Kentucky plant is in Georgetown, which is within the Lexington CSA.) When you ultimately decide to purchase a car, head into Dan Cummins dealership. You likely won't see anything as old as a Lexington, but it will certainly make you appreciate your new vehicle, nonetheless.


Volkswagen released the current Touareg back in 2010 and since then they really didn't change much about it. Unlike most of their other cars at the time, the car continued to use a rather old architecture. Well, it looks like this is going to change with the future 2019 Volkswagen Touareg. So far they haven't released much information about it. However, we have seen the camouflaged test mule and we have to say, the car definitely looks impressive. For starters, the car will get a new platform, new engines and a brand new look. Over its predecessor, the new model should feel a lot more like a premium car. This should allow VW to finally get a better share of the market for themselves.
The VW will be the car aimed at people who are looking for comfort over performance.
However, unlike these two, the VW will be the car aimed at people who are looking for comfort over performance.
The base model will have a softer suspension setting than the rest. The higher end versions of the car on the other hand will use an air-ride system with different levels of adjustment.
The car will also feature a longer wheelbase which should translate in more space inside the cabin.
It should hit the market sometime in the second quarter of 2018.
Despite the fact VW didn't say a word about it, we managed to see the fully uncovered model not that long ago. Considering the car looked ready for production, it is safe to assume its reveal is not that far.
In fact, it should hit the market sometime in the second quarter of 2018. Its price will likely be nearly identical to the ongoing generation.
While the current model looks like a bigger Golf, this is no longer the case.
One of the most impressive things about the new 2019 Volkswagen Touareg has to be the way it looks. While the current model looks like a bigger Golf, this is no longer the case.
In fact, the new car has a unique design scheme which is not usually the case with VW. In the front the car boasts a large grille, integrated headlights and a relatively aggressive bumper.
From the profile, we can clearly see more muscular lines than before while the wider fenders give the car a meaner look. The rear of the car also seems to boast new tail lights, a roof spoiler as well as a slightly more aggressive rake for the rear window.
Sadly this is one of the biggest unknown factors about the new car. Even so, it is safe to assume VW will take the same route they took with their other new models.
We expect a relatively minimalist design, far better materials than before as well as multiple touch-based interfaces.
These should give the cabin the premium look they are looking for. However, if they won't do it right then these may become an issue in the long term. In terms of space, both the front and the rear passengers will benefit from more than before.
A higher end version of the car, likely sporting a turbocharged 3.0 liter V6, should be an option.
It is safe to assume the base model of the 2019 Volkswagen Touareg will feature a 2.0 liter turbo-four engine. This one should deliver anything between 250 and 300 horsepower and right around 270 lb-ft of torque. A higher end version of the car, likely sporting a turbocharged 3.0 liter V6, should be an option.
This particular model will likely get closer to 350 horsepower and 370 lb-ft of torque. Both engines will come equipped with an 8 speed automatic, likely a regular torque converter unit.
Some also suggested VW may also release a hybrid. The details are still scarce but it looks like this may be a similar unit to the one found in the Cayenne.
A diesel engine is also a real possibility, but only in Europe. Here, the car is also going to feature a 6 speed manual as an option. The diesel should provide anything between 220 and around 280 horsepower, all from 3.0 liters of displacement.


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Dr. Tony Evans is founder and senior pastor of the 10,000-member Oak Cliff Bible Fellowship in Dallas, founder and president of The Urban Alternative, former chaplain of the NFL's Dallas Cowboys, and present chaplain of the NBA's Dallas Mavericks. His radio broadcast, "The Alternative with Dr. Tony Evans", can be heard on more than 1000 US radio outlets daily and in more than 130 countries.
The Urban Alternative is the national ministry of Dr. Tony Evans and is dedicated to restoring hope and transforming lives through the proclamation and application of the Word of God. There is more online at tonyevans.org.


Congratulations on your new baby! You are doing it great as a new mom, but are you doing justice to your well-being. We understand how hard it is to care for a small baby. But at the same time, you need to take care of yourself too. Do you remember the last time you had proper sleep? Probably not, right? It is essential that you take a proper sleep, especially in the first few months after the delivery, not only for your sanity but for your safety as well.
Sleep deprivation can put you at risk for many health problems including diabetes, obesity and heart diseases. Having proper sleep will help you get a higher energy level for taking care of yourself, your family and your little bundle of joy. Listed here are the sleeping tips that might help you get as much sleep as possible.
Any experienced mom will tell you to take a nap when your baby sleeps. This is one of the best ways to beat postpartum sleep deprivation. When baby sleeps don't involve yourself in other chores. Remember, everything can wait, but not the baby.
Don't care about the daily chores, or to make phone calls or clean a messy house. Always keep in mind that nothing is more important than your health.
All the moms are in sync with their baby's crying. Many moms don't sleep due to the fear that they won't be able to attend to their baby when they cry. If you are concerned about it that the nursery is far away from your bedroom, buy a monitor and keep it aside you. Remember, your baby is safe and it's OK if he/she cries for a few minutes before you hear it.
Majority of the new moms faces back pain issues. Ensure that you are getting proper back and spine support while resting. You can optimize your sleep by selecting an orthopedically correct Mattress, with correct firmness and softness that accommodates the various needs of your body after the postpartum period.
Correct firmness and softness of the mattress are highly essential to isolate noise and motion transfer during night trips to the nursery. It also aids in providing proper spine supports and alleviating pains and aches in the lower back, neck, shoulders, hips and other body areas.
If you tend to suffer from postpartum depression, loss of appetite, mood swings, do consult a doctor.
Sleep like your baby does! Happy sleeping!
Why the Change of Season Disrupts your Sleep?


Let me share some stats of Polish VBA & RPA (Robotic Process Automation) job market. I'm using data collected from pracuj.pl portal, by my repoBot – dated January to August 2018. In addition I checked what's on civil service job portal nabory.kprm.gov.pl – same dates.
On a monthly average – there was bit less than 60 vacancies in RPA area and around 430 vacancies with VBA. Here's monthly summary.
It is not a big surprise, that most of RPA vacancies are categorized as IT – Software Development. Most popular RPA vacancy name, again with no surprises, is Robotic Process Automation Developer or RPA Developer. What's more interesting is VBA and majority of offers in Finance / Economy category, most common vacancy name is Analyst or Financial Analyst.
Stop words and unimportant phrases like workplace were removed.
Please leave some feedback if and what you like about this summary and what you'd like to extend or see more here?


Discover Palmetto Park Terrace homes for sale in Boca Raton, FL below! Located in East Boca Raton just south of Palmetto Park Road and east of Interstate 95, Palmetto Park Terrace real estate is a close-knit community of single-family homes that offer close proximity to outstanding amenities and recreation.
Quiet, quaint and quintessential, Palmetto Park Terrace homes for sale offer incredible value in a highly desirable and vibrant community. Typically priced below the $1-million mark, Palmetto Park Terrace properties appeal to a wide range of homebuyer looking for a comfortable and affordable home ideal for the entire family.
Palmetto Park Terrace real estate consists entirely of detached single-story single-family homes along mature tree-lined streets that meander around the lake that is central to the community. Offering fantastic Mediterranean and classic South Florida designs with lush landscaping, Palmetto Park Terrance homes exude pride of ownership and convey a warm and welcoming feeling from the minute you approach the front door.
Home interiors feature fantastic finishes and floorplans including hardwood and ceramic flooring, bright and airy living spaces and large gourmet kitchens complete with marble countertops, custom cabinetry and professional grade stainless steel appliances. Spacious backyards with lush professional landscaping and private swimming pools provide outstanding outdoor entertaining opportunities and provide an oasis to escape the hustle and bustle of the busy day.
Palmetto Park Terrace properties have access to the best recreation and amenities in the Boca Raton area. The community is just a short drive or even a bike ride away from the white sandy beaches and warm waters of South Beach Park to the east. To the west, Sugar Sand Park offers baseball fields, an in-line hockey rink and an active Community Center. World-class shopping, dining and entertainment are just minutes away at both Royal Palm Place shopping area and Mizner Park in Downtown Boca Raton. Not only are Palmetto Park Terrace homes for sale a wise investment in prime Boca Raton real estate but an investment in the Boca Raton lifestyle.
Didn't find what you were looking for in Palmetto Park Terrace real estate? Fill out the form below and let your Boca Raton dream property find you instead!


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Ревни́тели благоче́стия, или боголю́бцы — небольшой круг духовных и светских лиц, объединившийся в конце 1640-х — начале 1650-х годов в Москве вокруг Стефана Вонифатьева, духовника́ царя Алексея Михайловича. «Ревнители» стремились к исправлению церковной и гражданской жизни в России путём утверждения благочестия на основе строгого следования церковным уставам и постановлениям Стоглавого собора 1551 года. В него входили «люди начитанные и искусные в деле проповеди»: настоятель Казанского собора Иван Неронов, протопопы Аввакум и Логгин; Лазарь, Даниила; из светских лиц им сочувствовал Фёдор Ртищев и его сестра Анна, из чёрного духовенства — игумен Павел, будущий епископ Коломенский.

Боголюбцы были реформаторами ещё до реформы Никона, и сам Никон (тогда архимандрит Новоспасского монастыря) им сначала сочувствовал и многому от них научился. Сначала они действовали заодно с патриархом Иосифом, но скоро явились некоторые противоречия с патриархом.

Да и в самом кружке Стефана, благодаря быстрому течению событий, скоро обнаружились несогласие и раздор. Толчок к этому был дан приездом киевских учёных монахов в Москву. Они смотрели свысока на московских начётчиков и грамотеев и в своих церковных взглядах во многом не были согласны с московской стариной. Кружок Стефана был составлен именно из московских начётчиков и староверов, а Стефан вместе с Никоном и Ртищевым явился почитателем и защитником киевских старцев.

Со вступлением Никона на патриарший престол в 1652 году кружок распался. Когда Никон вступил на патриарший престол и начал преобразования, притом властно и резко, обнаружился раскол в собственном смысле этого слова. Прежние соратники стали выступать против Никона.

Примечания

Литература

 Лавров А. С. Новые данные о «ревнителях благочестия»: Документы РГАДА, 1649—1650 гг. // Исторический Архив. 2008. — № 1. — С. 195—211

  отрывок

Религиозно-политические организации
История Русской церкви
Исчезли в 1652 году

redpajama_set_name: RedPajamaWikipedia


The Ajanta Caves, a UNESCO World Heritage Site, are located about 100 kilometers outside of Aurrangabad, India (it takes about 7 hours on a train to get to Aurrangabad from Mumbai) and while the oldest of the caves date back to the 2nd century BCE, they weren't widely known until 1819 when a British tiger hunting party stumbled across them. The caves are a series of 30 rock-cut caves that depict Buddhist religious art. They range from small and very simple to multi-storied ornate spaces. They've been fairly well-preserved, with some of the interior cave paintings still in decent condition.
I stopped by from 'Looky Here'… I love everything about your blog–as I am mostly an 'armchair' traveler, I always enjoy seeing where others travels take them! Love the layout of your blog–crisp and clean, and the photos are awesome, and then there's your writing! Yep, there is a book there!


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Maryland Casinos Net $162 Million In April
Posted on May 12, 2021 - Last Updated on June 15, 2022
by Blair Ames
Maryland followed up a record-breaking March with another impressive haul in April. The state's six casinos generated $162 million in gaming revenue, the third-highest monthly total ever reported.
The April 2021 revenue numbers represent a 12% increase over April 2019 figures, according to the Maryland Lottery. A comparison for 2020 is not available since the state's casinos were closed last April due to the COVID-19 pandemic.
For the casino industry, the past year has presented unique challenges. These include capacity limitations, social distancing guidelines and closures.
Maryland's casinos first closed on March 16, 2020, before reopening with capacity limitations on June 19, 2020. With capacity limits lifted in recent months and citizens looking for ways to spend what is left of their stimulus checks, Marylanders have been frequenting their local casinos. In March, the state's casinos generated $169.2 million in gaming revenue, an all-time monthly record.
MGM National Harbor paces the field
Despite local orders continuing to limit capacity to 50% in April, MGM National Harbor once again led the way in monthly revenue for the state's casinos. MGM's reported April revenue was down slightly from its March report of $66.5 million.
April 2021 Gaming Revenue
MGM National Harbor $62.4 million $60.8 million
Live! Casino & Hotel Maryland $58.2 million $47.8 million
Horseshoe Casino Baltimore $19.7 million $19.7 million
Hollywood Casino Perryville $8.5 million $6.2 million
Ocean Downs Casino $7.7 million $5.9 million
Rocky Gap Casino Resort $5.7 million $4.8 million
Horseshoe Casino Baltimore also remained limited to 50% capacity during the previous month.
The state's remaining four casinos had no capacity limitations. However, some slot machine and table game seats could not be occupied because casinos continue to comply with social-distancing guidelines.
Casino gaming contributions to the state totaled $69.9 million in April 2021, a $9.7 million increase compared to April 2019. Specifically, the Maryland Education Trust Fund received $50.8 million.
Maryland casinos preparing for sports betting launch
It's been a strong start to 2021 for Maryland's casinos, with impressive revenue numbers in both March and April, but looming on the horizon is another avenue for gaming operators to reach the customer: sports betting.
The Maryland General Assembly passed sports betting legislation on April 12 that allows for a multitude of online and retail sportsbook licenses.
The state's casinos have wasted no time in preparing their retail sportsbook offerings. Maryland Live! recently unveiled its new Sports & Social Maryland lounge, which will soon be home to a FanDuel Sportsbook thanks to the partnership between FanDuel and the Cordish Gaming Group.
Meanwhile, MGM National Harbor's planned sportsbook is moving through the permitting process. The MGM Sportsbook will be located near its National Market food court. Three National Market venues are being removed to clear space for the new operation.
The expectation is that casino sportsbooks will be up and running by the start of the 2021 NFL season, although mobile offerings and sports betting at smaller venues may have to wait due to the proposed license selection process.
Considering the number of licenses on the table, Marylanders expect more major sportsbook operators to enter the market.
Blair Ames
View all posts by Blair Ames


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text:
The status and responsibilities of the National Ombudsman are established by the Dutch Constitution (Article 78a). As an institution, the National Ombudsman is one of the five 'High Councils of State' (the others being the Senate, the House of Representatives, the Council of State and the Court of Audit). An Act of Parliament (Wet Nationale ombudsman) was passed in 1981 to define the tasks and authority of the ombudsman.
In essence, the National Ombudsman is empowered to scrutinize the manner in which public sector authorities fulfil their statutory responsibilities. An investigation may be instigated at the Ombudsman's own initiative or further to a complaint from a member of the public. The National Ombudsman is impartial and independent. He is appointed by the House of Representatives for a period of six years.
The National Ombudsman is concerned with virtually every area of public administration, from government ministries and their executive agencies (such as the Tax and Customs Administration), to the bodies which administer social benefits, the police, water boards, provinces and municipalities. Since 10 October 2010, the National Ombudsman also handles complaints relating to administrative agencies on the islands of Bonaire, Sint Eustatius and Saba, which are now 'special municipalities' of the Netherlands in the Caribbean.
The mission of the National Ombudsman is to safeguard the rights of the citizen in all dealings with administrative authorities. The Ombudsman will attempt to help citizens by investigating their complaints. He will advise where appropriate and can refer cases to the relevant public administration. He is alert to potential conflicts and will propose solutions intended to improve relations between the parties. The National Ombudsman helps governmental authorities to improve their own processes and procedures based on the findings of his investigations and published reports.
The strategy of the National Ombudsman is based on four core values:
Alertness: the National Ombudsman is a modern, alert organization with a professional staff who know what citizens may expect from the government. The Ombudsman encourages public sector authorities to respect the rights of the citizen and to improve their own procedures and processes accordingly.
Engagement: the National Ombudsman takes a proactive approach to contact with the general public. All complaints receive careful attention and are treated with respect.
Creativity: the National Ombudsman encourages open discussion of any problems, providing encouragement and assistance to government authorities as they seek creative solutions or improvements.
Accessibility: the National Ombudsman is accessible to all. Communication with the public is straightforward and relies on various channels, including social media.
Anyone who has a complaint about the public administration of the Netherlands can contact the National Ombudsman. The complaint may be submitted in writing (a letter), by phone or online. The Ombudsman may refer the complaint to another competent authority. He may advise the complainant about how to proceed. In all cases, his aim will be to resolve the problem quickly. To that end, he will wish to hear both sides of the story.
There are several ways in which the National Ombudsman can deal with a complaint:
Intervention. The Ombudsman will contact the public administration concerned to discuss the possibility of a rapid resolution. He will encourage the authority to contact the complainant directly by phone, letter or email in order to arrive at an outcome which is acceptable to all concerned.
Mediation. The Ombudsman acts as an independent mediator between the complainant and the public administration. He may invite the parties to meet face to face. He will attempt to improve relations, clarify any points which require clarification, and may suggest ways in which to solve the problem amicably.
Investigation and written report. If the Ombudsman decides to investigate a complaint in detail, he will ask both parties to provide further information. They are under a legal obligation to do so. Each party is given the opportunity to respond to the other's point of view. The Ombudsman will then formulate his findings which are presented in a written report. This type of report is published (online).
A letter. Where the outcome of an investigation is of interest only to the complainant, the Ombudsman may opt to present his findings in the form of a letter. He will also do so if he is unable to arrive at any firm conclusions. Letters are not published.
Qualitative Study
In 2017, at the National Ombudsman's request, an external company, Van de Bunt, carried out a study into the effectiveness of own initiative investigations by the National ombudsman. Read more about it here.


Next generation of horticultural talent discusses the science of healthy roots.
Giovanna quickly fell in love with landscaping and horticulture.
A 2017 Ontario Horticultural Trades Foundation post-secondary scholarship recipient, Catenecci graduated from Humber College in April 2018.
In September 2018, Catenacci will study urban arboriculture at Humber where she will earn an arborist certification.
"It's a one year program that includes tree identification, proper maintenance, disease identification as well as safety when cutting down trees," she explains.
Catenacci hopes to start her own business that specializes in maintenance and sustainability of properties.


text:
\section*{Введение}

В теории массового обслуживания (ТМО) и в смежных задачах очень важно знать числовые характеристики рассматриваемой системы – как в стационарном, так и в достационарном режиме.
В ряде случаев такие характеристики могут быть вычислены, но это возможно для ограниченного количества реальных моделей.
Однако в большинстве случаев возможно вычисление или оценка стационарных значений характеристик исследуемых моделей.
Поведение подавляющего количества систем массового обслуживания (СМО), систем надёжности и сетей массового обслуживания (СеМО) может быть описано с помощью линейчатых марковских процессов, которые во многих случаях являются регенерирующими.
В том случае, когда период регенерации марковского процесса имеет конечное среднее значение, этот процесс эргодичен.
Метод склеивания может применяться для получения строгих оценок сверху для скорости сходимости распределения регенерирующих марковских процессов (РМП) к стационарному распределению (естественно, в тех случаях, когда рассматриваемый РМП эргодичен). Этот метод может применяться к анализу систем массового обслуживания, сетей массового обслуживания и систем надёжности.

К сожалению, в отечественной литературе практически нет информации о методе склеивания (см., например, \cite{1.}, \cite{5.}), хотя отечественные учёные использовали этот метод (\cite{6.} и др.).

\section{Кусочно-линейные Марковские процессы \linebreak(КЛМП)}

Поведение систем массового обслуживания (СМО), сетей массового обслуживания (СеМО) и систем надёжности определяется, как правило, последовательностями случайных величин (сл.в.), представляющих собой времена работы или обслуживания требований (восстанавливаемых элементов или узлов СеМО), интервалы между поступлениями требований в исследуемую систему, случайные времена ожидания обслуживания (в случае "нетерпеливых" требований) и пр.

Такого сорта поведение технических систем принято описывать кусочно-линейными (или линейчатыми) процессами, введёнными в \cite{4.}.

Следуя \cite{4.}, определим кусочно-линейный марковский процесс следующим образом.

Кусочно-линейным Марковским процессом называется случайный процесс $X_t = \left(\nu(t), \vec\upsilon(t)\right)$, определяемый следующим образом.
Пространство состояний процесса $\mathscr{X}$ -- это множество пар $\left(i,\vec \xi_i\right)$, где $i$ -- элемент конечного или счётного пространства, а $\vec\upsilon_i$ -- вектор $=(\upsilon_1, \ldots \upsilon_{|i|})$, где $|i|\geqslant 0$ является ``размером'' {\it базового состояния} $i$; $\upsilon_j\geqslant O$.
Поведение процесса $X_t$ описывается так.

Пусть $X_t = (i,\vec y),$ $\vec y = (y_l, \ldots y_{|i|})$.
Тогда с вероятностью $\lambda_{i}(X_t)\times p_{ij}(X_t) \,\mathrm{d} t$
за время $\,\mathrm{d} t$ произойдёт случайный переход $X_t$ в {\it базовое состояние} $j$.
После перехода в состояние $j$ новое значение компоненты $ \vec\upsilon(t)$ случайно и определяется измеримой по $\vec y$ функцией распределения (ф.р.)
$$
B_{ij}^{(0)}(\vec x| \vec y) =\ensuremath{\mathbf{P}}\{ \vec\upsilon(t + \,\mathrm{d} t) < \vec x | \vec\upsilon(t) = \vec y,\, \nu(t ) = i, \,\nu(t + \,\mathrm{d} t) = j\}
$$

Вероятность того, что за малое время $h$ произойдёт более одного случайного перехода есть $o(h)$.
При этом, если $\nu(t + \,\mathrm{d} t) = \nu(t)$, то $\vec\upsilon(t + \,\mathrm{d} t) =\vec\upsilon(t) + \vec \alpha_i \,\mathrm{d} t$, где $\vec \alpha_i= (\alpha_{il} , \ldots , \alpha_{i|i|})$ -- вектор с неотрицательными компонентами, если на интервале $(t, t + \,\mathrm{d} t)$ не было случайных переходов (см. \cite{4.}).

Для СМО номер $i$ может обозначать количество находящихся в СМО требований, а вектор $\vec\upsilon$ состоит из: времени, прошедшего со времени прихода последнего требования, а также из времён пребывания имеющихся требований с СМО (или стоимость уже проведённых работ по их обслуживанию и проч.).

В СеМО $i$ может нумеровать все возможные состояния СеМО (количество требований в каждом узле и в очередях на этих узлах), а вектор $\vec\upsilon$ может характеризовать состояние (прошедшее время обслуживания или ожидания обслуживания) находящихся на обслуживании на каждом узле требований.

Таких вариантов описания моделей ТМО и смежных задач множество.

Отметим, что в случае, когда $\lambda(X_t)\equiv\lambda(\nu_t)$ и $p_{ij}(X_t)\equiv p(i,j)$, процесс $X_t$ представляет собой цепь Маркова в непрерывном времени (см. \cite{4.}, \cite{7.}).

\subsection{\it Вложенная цепь Маркова}

КЛМП имеет вложенную {\it неоднородную} цепь Маркова -- с вероятностями перехода $p_{ij}(X(t))$ (обычно рассматривается случай, когда $p_{ij}(X(t))\equiv p(i,j)=\ensuremath{\mbox{const }}\!\!$).

Мы предполагаем, что эта Марковская цепь неприводима и положительно возвратна.
При этом $\lambda_{i}(X_t)$ -- это интенсивность окончания времени пребывания в текущем состоянии $\nu(t)=i$.
Как уже говорилось, в случае, если $\lambda_{i}(X_t)\equiv\lambda (\nu(t))$, КЛМП -- это цепь Маркова в непрерывном времени.

Хорошо известно, что поведение цепи Маркова в непрерывном времени описывается уравнениями Колмогорова.

Если среднее время пребывания во всех состояниях цепи Маркова конечно, что соответствующий КЛМП эргодичен(см. \cite{7.}).

\section{Регенерирующие процессы}

Поведение большинства изучаемых СМО, СеМО и систем надёжности описывается регенерирующими процессами.

\begin{mydef}
[Регенерирующего процесса]
Процесс $(X_t,\,t\geqslant 0)$, заданный на вероятностном пространстве $(\Omega,\mathscr{F} , \ensuremath{\mathbf{P}} )$, с измеримым пространством состояний $(\mathscr{X} , \mathscr{B} (\mathscr{X} ))$ называется регенерирующим, если существует возрастающая последовательность $\left\{ \theta_n\right\} $ $(n\in\mathbb{Z}_+)$ Марковских моментов по отношению к фильтрации $\mathscr{F} _{t\leqslant 0}$ таких, что последовательность
$$
\left\{ \Theta_n\right\} =\left\{ B_{t+\theta_{n-1}}-B_{\theta_{n-1}}, \theta_n- \theta_{n-1},t\in[\theta_{n-1},\theta_n)\right\} ,\quad n\in\mathbb{N}
$$
состоит из независимых одинаково распределённых (н.о.р.) случайных элементов, заданных на $(\Omega,\mathscr{F} , \ensuremath{\mathbf{P}} )$.
Если $\theta_0\neq 0$, то регенерирующий процесс $(X_t,\,t\geqslant 0)$ называется процессом с запаздыванием.
Моменты $\left\{ \theta_n\right\} $ называются моментами (точками) регенерации.
{\hfill \ensuremath{\triangleright}}
\end{mydef}
Обозначим $\xi_n\stackrel {\text{\rm def}}{=\!\!\!=} \theta_n- \theta_{n-1}$, и пусть $F(s)=\ensuremath{\mathbf{P}} \left\{ \xi_n\leqslant s\right\} $ -- ф.р. длины периода регенерации; везде далее мы предполагаем, что {\it распределение $F$ нерешётчато} и, более того, для простоты будем считать, почти всюду при $t>0$ существует положительная плотность распределения.

В ТМО моментами регенерации обычно являются моменты начала периода занятости (при экспоненциальном входящем потоке это могут быть и моменты окончания периодов занятости).
В теории надёжности при анализе поведения {\it одного } восстанавливаемого элемента точки регенерации -- это моменты отказов или восстановлений.
При анализе СеМО с экспоненциальными распределениями времени пребывания требований в узлах СеМО и входящих экспоненциальных потоках КЛМП представляет собой цепь Маркова в непрерывном времени; в этом случае в качестве точки регенерации можно выбрать любое {\it возвратное} базовое состояние.

\subsection {\it Наша цель}

\quad Пусть $\mathscr{P}_t$ -- распределение КЛМП $(X_t,\,t\geqslant 0)$ в момент $t$.

\quad Если $\ensuremath{{\mathbb{E}}}\,\xi_n<\infty$, то существует и единственно стационарное инвариантное распределение $\mathscr{P}$ такое, что $\mathscr{P}_t\Longrightarrow \mathscr{P}$.

Наша цель -- вычисление строгой оценки сверху для скорости сходимости $\mathscr{P}_t\Longrightarrow \mathscr{P}$.

Для этого будет использован {\it метод склеивания} Марковских процессов, т.е. исследуемые регенерирующие процессы $(X_t,\,t\geqslant 0)$ должны быть Марковскими.
Если процесс $(X_t,\,t\geqslant 0)$ не является Марковским, то можно расширить пространство его состояний $\mathscr{X}$ таким образом, что на новом (расширенном) пространстве состояний $\overline{\mathscr{X}}$ ``уточнённый'' процесс $(\overline X_t,\,t\geqslant 0)$ окажется марковским.

Например, в состояние регенерирующего процесса $X_t$, $t\in[\theta_{n-1},\theta_n)$ можно включить полную историю этого процесса на интервале $[\theta_{n-1},t]$: процесс $\overline X_t\stackrel {\text{\rm def}}{=\!\!\!=} \left\{ X_s, s\in[\theta_{n-1},t] |t<\theta_n\right\} $ -- марковский и регенерирующий на расширенном пространстве состояний $\overline{ \mathscr{X}} $.

\subsection{\it Что уже известно о регенерирующих процессах (см. \cite{2.})}

\begin{myth}
Если для некоторого $\alpha>0$ выполнено условие
$$
\mathbf{E}\,e^{\alpha \xi}<\infty,
$$
то для любого $\beta<\alpha$ существует постоянная $C(\beta)$ такая, что для всех $A\in\sigma(\mathscr{X})$ и всех $t>0$ верно:
$
|\mathscr{P}_t( A)-\mathscr{P}(A) |<C(\beta)\exp(-\beta t).
$ \QEDB
\end{myth}
Оценки для постоянной $C(\beta)$ неизвестны.

\begin{myth}
Если для некоторого $k>1$ выполнено условие
$$
\qquad\mathbf{E}\,\xi^k<\infty,
$$
то для любого $\kappa\leqslant k-1$ существует постоянная $C(\kappa)$ такая, что\linebreak
$
|\mathscr{P}_t( A)-\mathscr{P}(A) |<C(\kappa)t^{-\kappa}.
$ \QEDB
\end{myth}
Оценки для постоянной $C(k)$ неизвестны.

\subsection{\it Вложенный процесс восстановления}

Рассмотрим регенерирующий процесс $X_t$.
Этот процесс имеет вложенный процесс восстановления $N_t$

\begin{figure}[h]
\center{\includegraphics[width=400pt]{regen+ren}}
\caption{Регенерирующий процесс ${{X}_{t}}$ и вложенный процесс восстановления.}
\label{fig0}
\end{figure}
Распределение процесса $X_\tau$ -- это функция от значения случайной величины $B_\tau$ -- обратного времени восстановления.

Значит, если известны оценки скорости сходимости распределения обратного времени восстановления процесса восстановления $ N_t $, то известны оценки скорости сходимости распределение регенерирующего процесса $ X_t $.

\subsection{\it Процесс восстановления и неравенство Лордена}

Рассмотрим процесс восстановления $N _t\stackrel{\text{\rm def}}{=\!\!\!=}\ensuremath{\displaystyle}\ensuremath{\sum\limits} _{i=1} ^\infty \ensuremath{\mathbf{1}}\left \{ \ensuremath{\sum\limits} _{s=1} ^i \xi _k\leqslant t\right \} $, где
$\left \{\xi _1, \xi _2, ...\right \} $ -- н.о.р. положительные случайные величины (сл.в.) с ф.р. $F(s)$.
$N _t$ -- это считающий процесс, который меняет своё значение в моменты времени $t _k=S _k\stackrel{\text{\rm def}}{=\!\!\!=}\ensuremath{\displaystyle}\ensuremath{\sum\limits} _{j=1} ^s \xi _j$.
Эти моменты $t _k$ называются моментами восстановления.

\begin{figure}[h]
\centering
\begin{picture}(300,45)
\put(0,20){\vector(1,0){300}}
\put(0,18){\line(0,1){4}}
\thicklines
{\qbezier(0,20)(30,40)(60,20)
{\qbezier(60,20)(80,40)(100,20)}
\qbezier(100,20)(150,40)(200,20
\qbezier(200,20)(240,40)(280,20
\put(280,30){\ldots \ldots \ldots}
\put(58,10){$t _1$}
\put(98,10){$t _2$}
\put(198,10){$t _3$}
\put(278,10){$t _4$}
\put(27,40){$\xi _1$}
\put(77,40){$\xi _2\stackrel{\mathscr{D}}{=} \xi _1$}
\put(147,40){$\xi _3\stackrel{\mathscr{D}}{=} \xi _1$}
\put(237,40){$\xi _4\stackrel{\mathscr{D}}{=} \xi _1$}
\put(-5,10){$t _0=0$}
\put(148,10){$t$}
\put(195,15){\line(1,1){10}}
\put(195,25){\line(1,-1){10}}
\color{black}
\qbezier(150,20)(175,0)(200,20)
\qbezier(150,20)(175,1)(200,20)
\qbezier(150,20)(175,2)(200,20)
\put(170,0){$W _t$}
\color{black}
\qbezier(100,20)(125,0)(150,20)
\qbezier(100,20)(125,1)(150,20)
\qbezier(100,20)(125,2)(150,20)
\put(120,0){$B _t$}
\end{picture}
\caption{$B _t$ -- перескок, $W _t$ -- недоскок в момент времени $t$ (см. \cite{4.}, \cite{8.})}
\label{fig1
\end{figure}
На Рис. \ref{fig1} показаны обратное время восстановления или перескок $B _t$ и прямое время восстановления или недоскок $W _t$ в {\bf фиксированный} момент времени $t$:
$B _t\stackrel{\text{\rm def}}{=\!\!\!=} t-S _{N _t}; \quad W _t\stackrel{\text{\rm def}}{=\!\!\!=} S _{N _t+1}-t.$

Напомним, что в наших условиях предельное распределение величины перескока и недоскока при $t\to \infty $ таково: $$\tilde{F}(s)=1-\frac{\int\limits_{0}^{s}{(1-F(}u))\text{d}u}{\int\limits_{0}^{\infty }{(1-F(}u))\text{d}u}=1-\frac{\int\limits_{0}^{s}{(1-F(}u))\text{d}u}{\mathbb{E}\xi }$$ –- см.
\cite{8.}.

\begin{myth}[Г.Лордена, см. \cite{11.}]
Неравенство Лордена даёт оценку сверху математического ожидания (м.о.) перескока:
$$
\ensuremath{{\mathbb{E}}}\, B _t \leqslant \frac{\ensuremath{{\mathbb{E}}}\, \xi ^2}{\ensuremath{{\mathbb{E}}}\, \xi}\stackrel{\text{\rm def}}{=\!\!\!=} \Xi\big(=\mbox{функционал от }F(s)\big).\eqno{\QEDB}
$$
\end{myth}
Это неравенство будет использовано для получения оценки сверху для скорости сходимости распределения перескока к стационарному распределению в метрике полной вариации.

Напомним, что
\begin{multline*}
\ensuremath{\mathbf{P}}\{\mbox{\it имеется хотя бы одно восстановление на интервале }[t, t+\Delta]|{\color{black} B_t}=y\} =\\
= \frac{F(y+\Delta)- F(y)}{1-F(y)}
=\intl_y^{y+\Delta}\lambda(s)\,\mathrm{d} s=\lambda(y)\Delta + o(\Delta),
\end{multline*}
где
\begin{equation}\label{1}
\lambda(t)= \ensuremath{\displaystyle}\frac{F'(t)}{1-F(t)}={\frac{\,\mathrm{d}}{\,\mathrm{d} t}\Big(-\ln(1-F(t)\Big)},
\end{equation}
и $\lambda(t)$ называется {\it интенсивностью процесса восстановления}.

При этом
\begin{equation}\label{2}
{F(s)=1-\exp\left(\intl_0^s(-\lambda(u))\,\mathrm{d} u\right)}, \qquad F'(s)=\lambda(s)\exp\left(\intl_0^s(-\lambda(u))\,\mathrm{d} u\right).
\end{equation}

\begin{myrem}
Обычно предполагается, что распределение $F(s)$ абсолютно \linebreak непрерывно. Однако в некоторых приложениях теории восстановления ф.р. сл.в. $\xi_i$ не абсолютно непрерывны.

Т.е. ф.р. сл.в. $\xi_i$ может иметь скачки (естественно, в прикладных задачах теории вероятностей мы не рассматриваем сингулярные сл.в.).

Положим
$
f(s)=
\begin{cases}
F'(s), & \mbox{если } F'(s) \mbox{ существует; }\\
0, & \mbox{в противном случае}.
\end{cases}
$

В дальнейшем полагаем $\lambda(s)\stackrel{\text{\rm def}}{=\!\!\!=} \ensuremath{\displaystyle}\frac{f(s)}{1-F(s)}-\ensuremath{\sum\limits}_{i}\delta(s-a_i)\ln(F(a_i+0)-F(a_i-0))$, где $\{a_i\}$ -- множество точек разрыва $F(s)$.
Такое определение интенсивности оставляет в силе первую формулу в (\ref{2})  -- это следует из (\ref{1}). \hfill \ensuremath{\triangleright}
\end{myrem}

 Заметим, что процесс $\color{black} B_t$ -- это простейший Марковский КЛМП или {\it линейчатый процесс} (см. \cite{4.}).

\section{Метод склеивания (Марковских процессов) -- см., например, \cite{9.}}

Применение метода склеивания основано на следующем соображении.

{\it Если два однородных независимых Марковских процесса с одними и теми же переходными вероятностями, но с разными начальными условиями, совпадут в некоторый момент $\tau$, то после этого момента $\tau$ их распределения совпадают.
Момент $\tau$ называется временем (моментом) склеивания или склейкой.}

Напомним, что процесс $X_t$ с пространством состояний $\mathscr{X}$ называется Марковским, если для него выполнено {\it Марковское свойство}:
\\
$\forall A_0,A_1,\ldots,A_n\in\sigma(\mathscr{X})$,\; $\forall t_1,t_2,\ldots,t_n, t:$\; $0\leqslant t_i<t_{i+1}<t$ верно равенство:
$$\ensuremath{\mathbf{P}}\{X_t\in A_0|X_{t_1}\in A_1, X_{t_2}\in A_2, \ldots, X_{t_n}\in A_n\}=\ensuremath{\mathbf{P}}\{X_t\in A_0|X_{t_n}\in A_n\},$$
т.е. поведение процесса после момента $t_n$ зависит только от его состояния в момент $t_n$ и не зависит от того, что было раньше.

Пусть теперь два независимых Марковских процесса $X_t$ и $X_t'$ имеют одинаковые переходные вероятности: $\ensuremath{\mathbf{P}}\{X_{t+\theta}\in A|X_t\in B\}\equiv \ensuremath{\mathbf{P}}\{X'_{s+\theta}\in A|X_s'\in B\}$ для всех $\theta>0$ и $t\geqslant 0$, $s\geqslant 0$, при этом $X_0\neq X'_0$.
И пусть известно, что в некоторый момент времени $\tau$ эти процессы совпали: $X_\tau=X'_\tau=\mathfrak{X}$.
Тогда, в соответствии с переходной функцией и Марковским свойством, для всех $t=\tau+s>\tau$ и $A\in \sigma(\mathscr{X})$ выполняется: $\ensuremath{\mathbf{P}}\{X_t\in A\}=\ensuremath{\mathbf{P}}\{X_t\in A|X_\tau=\mathfrak{X}\}=\ensuremath{\mathbf{P}}\{X_t'\in A|X'_\tau=\mathfrak{X}\}=\ensuremath{\mathbf{P}}\{X'_t\in A\}$.

Если известно распределение (или оценка распределения) момента склеивания $\tau$ (эта величина, вообще говоря, зависит от начальных условий процессов $X_t$ и $X_t'$), то выписывается {\it основное неравенство склеивания}:
\begin{multline}\label{3}
|\ensuremath{\mathbf{P}}\{X_t\in S\}-\ensuremath{\mathbf{P}}\{X_t'\in S\}|=
\\ \\
=|\ensuremath{\mathbf{P}}\{X_t\in S\}-\ensuremath{\mathbf{P}}\{X_t'\in S\}|\times (\ensuremath{\mathbf{1}}(\tau> t)+ \ensuremath{\mathbf{1}}(\tau\leqslant t))\leqslant \ensuremath{\mathbf{P}}\{\tau> t\}    \end{multline}
(здесь $\ensuremath{\mathbf{1}}(\cdot)$ обозначает индикатор события).

Процессы $\left(X_t,\,t\geqslant 0\right)$ и $\left(X_t',\,t\geqslant 0\right)$ имеют распределения \linebreak $\mathscr{P}_t(A)\stackrel{\text{\rm def}}{=\!\!\!=} \ensuremath{\mathbf{P}}\left\{X_t\in A\right\}$, и $\mathscr{P}_t'(A)\stackrel{\text{\rm def}}{=\!\!\!=} \ensuremath{\mathbf{P}}\left\{X_t'\in A\right\}$ соответственно.

$
\tau \left(X_0,X_0' \right)\stackrel{\rm{def}}{=\!\!\!=\!\!\!=} \inf \left\{ t>0: \,X_t=X_t' \right\} .
$
Предположим, что для некоторой положительной возрастающей функции $\varphi(t)$ вычислена или оценена сверху величина $\ensuremath{{\mathbb{E}}}\,\varphi\left(\tau \left(X_0,X_0' \right)\right)=C \left(X_0,X_0' \right)<\infty$ .
Подставим эту функцию в последнее выражение (\ref{3}) и применим неравенство Маркова:
\begin{multline*}
\left|\mathscr{P}_t( A)-\mathscr{P}_t'( A) \right|\leqslant \mathbf P\left\{\tau \left(X_0,X_0' \right) >t \right\}= \mathbf P\left\{\varphi\left(\tau \left(X_0,X_0' \right)\right) >\varphi(t) \right\}\leqslant
\\ \\
\leqslant \left|\mathscr{P}_t( A)-\mathscr{P}_t'( A) \right|\leqslant\ensuremath{\displaystyle} \frac{\mathbf{E}\, \varphi \left(\tau \left(X_0,X_0' \right) \right)}{\varphi(t)} .
\end{multline*}
Если известна оценка стационарного распределения $\mathscr{P}$ процесса $X_t$ (и $X_t'$ -- это одно и то же), то последнее неравенство можно проинтегрировать по стационарной мере $\mathscr{P}$, и тогда получаем
$$
\left|\mathscr{P}_t( A)-\mathscr{P}(A) \right|\leqslant (\varphi(t))^{-1} \underbrace{\int\limits_{\mathscr{X}}\varphi \left (\tau \left (X_0,X_0' \right) \right ) \,\mathrm{d}}_{\widehat C\left (X_0\right )}\mathscr{P} \left (X_0'\right)= \frac{\widehat C\left (X_0\right )}{\varphi(t)},
$$
т.е.
$$
\left\|\mathscr{P}_t^{X_0}-\mathscr{P} \right\|_{\mbox{ПВ}}\leqslant \frac{\widehat C\left (X_0\right )}{\varphi(t)}.
$$

Напомним
\begin{mydef}
Расстояние в метрике полной вариации между двумя вероятностными мерами $\mathscr{P}_1$ и $\mathscr{P}_2$ на $\sigma$-алгебре $\mathscr{F}$ подмножеств вероятностного пространства ${ \Omega }$ определяется как

$${\displaystyle \delta (\mathscr{P}_1,\mathscr{P}_2)=\|\mathscr{P}_1-\mathscr{P}_2\|_{\mbox{ПВ}}=\sup _{A\in {\mathscr{F}}}\left|\mathscr{P}_1(A)-\mathscr{P}_2(A)\right|.}
\eqno{\hfill \ensuremath{\triangleright}}$$

\end{mydef}

Метод склеивания впервые был предложен в 1938 г. молодым французским математиком немецкого происхождения В.Доблином (V.Doblin или \linebreak W.Doeblin) для обычных цепей Маркова.
К сожалению, в отечественной учебной литературе практически нет информации об этом методе.
Он упоминается в «Дополнении» в пособии \cite{3.}; см. также \cite{9.}.

Для процессов с непрерывным временем применение метода склеивания ``напрямую'' невозможно, поскольку должны совпасть непрерывные сл.в. -- а вероятность такого совпадения равна нулю.

Т.е. в общем случае для процессов в непрерывном времени $$\ensuremath{\mathbf{P}} \left\{ \tau \left (X_0,X_0' \right )<\infty \right\} <1,$$ и стандартное применение метода склеивания невозможно.
Поэтому применяется процедура конструирования {\it параллельного склеивания}.
\subsection{\it Параллельное склеивание (Successful coupling -- см. \cite{10.}}

Для оценки скорости сходимости друг к другу распределений двух {\it независимых} однородных Марковских процессов $\left(X_t,\,t\geqslant 0\right)$ и $\left(X_t',\,t\geqslant 0\right)$ с одинаковыми переходными функциями конструируется (на специально выбранном вероятностном пространстве) парный случайный процесс the $\mathscr{Z}_t= \left(\left (Z_t ,Z_t ' \right),t\geqslant0 \right)$ такой, что:

1. Для всех $t\geqslant 0$ сл.в. $X_t$ и $Z_t$ имеют одинаковое распределение, также как и пара $X_t'$ и $Z_t'$. Иначе говоря, маргинальные распределения процессов $X_t$ и $Z_t$ (или $X_t'$ и $Z_t'$) совпадают -- но это не значит, что совпадают конечномерные распределения у этих пар процессов.

2. $\mathbf{E}\,\tau \left(Z_0 ,Z_0 ' \right)<\infty\mbox{ где }\tau \left(Z_0 ,Z_0 ' \right)={\tau} (\mathscr Z_0 )\stackrel{\rm{def}}{=\!\!\!=\!\!\!=} \inf \left\{ t\leqslant 0:\,Z_t =Z_t ' \right\}
$ -- момент склеивания процессов $Z_0$ и $Z_0 '$.

3. $Z_t =Z_t '$ для всех $t\geqslant {\tau} \left(Z_0 ,Z_0 ' \right)$.

Парный случайный процесс $\mathscr{Z}_t=\left(\left (Z_t ,Z_t ' \right),t\geqslant0 \right)$, удовлетворяющий условиям 1--3, называется is called {\it параллельным склеиванием}, а момент $\tau$ называется {\it успешной склейкой}.

.

{\it Повторимся, что конечномерные распределения процессов $\left(Z_t^{(i)},\,t\geqslant0\right)$ могут отличаться от конечномерных распределений процессов $\left(X_t^{(i)},\,t\geqslant0\right)$; более того, при конструировании параллельного склеивания процессы $Z_t$ и $Z_t'$, как правило, зависимы.}

Для всех $A\in \mathscr{B}(\mathscr{X})$ основное неравенство склеивания (\ref{3}) переписывается так:
\begin{multline}\label{4}
\left|\mathscr{P}^{X_0 }_t(A) - \mathscr{P}_t^{X_0 '}(A) \right|= \left|\ensuremath{\mathbf{P}} \left\{ X_t \in A \right\} - \ensuremath{\mathbf{P}} \left\{ X_t '\in A \right\} \right|=
\\
\\
= \left|\ensuremath{\mathbf{P}} \left\{ Z_t \in A \right\} - \ensuremath{\mathbf{P}} \left\{ Z_t '\in A \right\} \right| \leqslant
\ensuremath{\mathbf{P}} \left\{ {\tau}\left (Z_0 ,Z_0 '\right )\geqslant t \right\}
\leqslant
\\
\\
\leqslant\frac{\mathbf {E} \,\varphi \left({\tau} \left(Z_0 ,Z_0 ' \right) \right)}{\varphi(t)}\leqslant \frac{C \left(Z_0 ,Z_0 ' \right) }{\varphi(t)},
\end{multline}
где значение $C \left(Z_0 ,Z_0 ' \right)$ вычисляется по начальным условиям процесса $\mathscr{Z}_t$.

Но $Z_0=X_0$ и $Z_0'=X_0'$, и поэтому правая часть неравенства (\ref{4}) зависит только от $(X_0,X_0')$.

Поэтому если $\mathscr{P}_t^{X_0 }\Longrightarrow \mathscr{P} $ для всех начальных состояний $X_0$, можно использовать интегрирование по стационарной мере $\mathscr{P} $ как и выше.

{\it Хотя в ряде случаев это интегрирование вызывает некоторые трудности.}

\section{Конструирование параллельного склеивания для перескоков процесса восстановления}

В этом разделе приводится базовая схема конструирования параллельного склеивания, на основе которой можно строить оценки скорости сходимости распределений различных регенерирующих процессов.

Итак, мы конструируем параллельное склеивание
$$(\mathscr Z_t,\,t\geqslant0)=\left(\left(Z_t , Z_t '\right),\,t\geqslant0\right)$$
для перескоков вложенных процессов восстановления $\left(B_t ,\,t\geqslant0\right)$ и $\left( B_t ',\,t\geqslant0\right)$ для исследуемых регенерирующих процессов $X_t$, $X_t'$ с различными начальными состояниями.
Это значит, что процессы $B_t $, и $B_t '$ имеют разные начальные состояния.

Т.е. соответствующие процессы восстановления начинаются {\it до} момента $t=0$, и в момент $t=0$ значения перескоков вложенных процессов восстановления различны: $B_0 ={b}$, $B_0 '=b'$; что можно обозначить как $B_t =B_t^{b}$, $B_t '=B_t^{b'}$.

Распределения этих процессов в момент $t$ обозначим соответственно $\mathscr{P}^{b}_t$ и $\mathscr{P}^{b'}_t$.

Если мы оценим сл.в. $\tau(b,{b'})=\tau(b,{b'})\stackrel {\text{\rm def}}{=\!\!\!=} \inf\left\{ t>0:\, Z_t = Z_t '\right\}$, то можно получить оценку
$$\left\|\mathscr{P}^{b}_t(A) - \mathscr{P}^{b'}(A) \right\|_{\mbox{ПВ}}\leqslant \ensuremath{\mathbf{P}} \left\{ \tau({b},{b'})>t\right\}\leqslant \frac{\ensuremath{{\mathbb{E}}}\, \varphi( \tau({b},{b'}))}{\varphi(t)}.
$$
\subsection{\it Некоторые сведения}

В ходе конструирования параллельного склеивания надо уметь конструировать случайные величины (на некотором вероятностном пространстве).
\begin{myrem}
Напомним, как конструировать сл.в. по её ф.р. $F(s)$.

Определим обратную функцию к ф.р. $F(s)$ как
$$
F^{-1}(y)\stackrel{\text{\rm def}}{=\!\!\!=}\inf\{x:\,F(x)\geqslant y\}.
$$
Если $\mathscr{U}$ -- равномерно распределённая на $[0,1)$ сл.в., то $\xi\stackrel{\text{\rm def}}{=\!\!\!=} F^{-1}(\mathscr{U})$ имеет ф.р. $F(s)$. \hfill \ensuremath{\triangleright}
\end{myrem}

Ранее говорилось, что  $B_t$ -- это перескок процесса восстановления $N_t$ с ф.р. времени восстановления $F(s)$;
$N_t\stackrel {\text{\rm def}}{=\!\!\!=} \sum\limits_{i=0}^\infty \mathbf{1} (S_n<t)$ где $S_n\stackrel {\text{\rm def}}{=\!\!\!=} \sum\limits_{n=0}^n \xi_i$,
$\mathbf {E} \,\xi_i<\infty$ и $\xi_i$, $i\in \mathbb Z_+$ независимы в совокупности;
$\ensuremath{\mathbf{P}} \left\{ \xi_i\leqslant s\right\} =F(s)$, $i\in\mathbb N$.
$B_t\stackrel {\text{\rm def}}{=\!\!\!=} (t-S_{N_t})$ -- перескок процесса восстановления $N_t$.

Если $B_0=b$ при $t=0$, значит, последнее восстановление процесса $N_t$ было в момент $t=-b$, и остаточное время периода восстановления (недоскок) $W_0$ имеет ф.р.
\begin{multline}\label{5}
\ensuremath{\mathbf{P}}\{W_0\leqslant s|B_0=b\}= \ensuremath{\mathbf{P}}\{\xi_0-b\leqslant s|\xi_0>b\}=\ensuremath{\mathbf{P}} \left\{\xi_0\in(b,b+s]| \xi_0>b\right\} =
\\ \\
=\ensuremath{\displaystyle}\frac{F(s+b)-F(b)}{1-F(b)} \stackrel {\text{\rm def}}{=\!\!\!=}F_b(s) .
\end{multline}

Также напомним, что
\begin{equation}\label{6}
\lim\limits_{t\to \infty}\ensuremath{\mathbf{P}} \left\{ B_t\geqslant s\right\} =\mu^{-1}\ensuremath{\displaystyle}\int\limits_{s}^\infty (1-F(u))\,\mathrm{d}\, u\stackrel {\text{\rm def}}{=\!\!\!=} 1-\widetilde F(s),
\end{equation}
где
$\mu\stackrel {\text{\rm def}}{=\!\!\!=} \mathbf {E} \,\xi_1$ (см. \cite{8.}).

\subsection{\it Первый шаг: конструирование процессов восстановления}

Везде в дальнейшем $\mathscr{U}_i$, $\mathscr{U}_i'$, $\mathscr{U}_i''$, $\mathscr{U}_i'''$,\ldots -- независмые равномерно распределённые на $[0,1)$ сл.в..

1. Конструирование процесса $(B _t,\,t\geqslant 0)$с начальным состоянием $B_0 =b$.
Обозначим $t_1\stackrel{\text{\rm def}}{=\!\!\!=} F_{b}^{-1}(\mathscr{U}_1)$, $t_k\stackrel{\text{\rm def}}{=\!\!\!=} t_{k-1}+F^{-1}(\mathscr{U}_k)$, $k>1$.

\begin{figure}[h]
    \centering
\begin{picture}(350,50)
\put(0,20){\vector(1,0){350}}
\thicklines
{\qbezier(0,33)(40,34)(60,20)
{\qbezier(60,20)(80,40)(100,20)}
\qbezier(100,20)(140,40)(180,20
\qbezier(180,20)(200,40)(220,20
\qbezier(220,20)(250,40)(280,20
\qbezier(280,20)(290,40)(300,20
\qbezier(300,20)(320,31)(330,30
\thinlines
\put(5,37){\small $\xi_0\sim F_b$}
\put(65,37){\small $\xi_1\sim F$}
\put(125,37){\small $\xi_2\sim F$}
\put(185,37){\small $\xi_3\sim F$}
\put(235,37){\small $\xi_4\sim F$}
\put(210,11){\small $t_3$}
\put(58,11){\small $t_0$}
\put(98,11){\small $t_1$}
\put(178,11){\small $t_2$}
\put(278,11){\small $t_4$}
\put(298,11){\small $t_5$}
\put(0,11){\small $0$}
\end{picture}
    \caption{Процесс ${{X}_{t}}$ (соответствующий ${{B}_{t}}$).}
    \label{fig3}
\end{figure}

2. Аналогично для $B_t '$ с начальным состоянием $ B_0 '=b'$ положим $t_1'\stackrel{\text{\rm def}}{=\!\!\!=} F_{b'}^{-1}(\mathscr{U}_1')$, $t_k'\stackrel{\text{\rm def}}{=\!\!\!=} t_{k-1}'+F^{-1}(\mathscr{U}_k')$, $k>1$.

\begin{figure}[h]
    \centering
\begin{picture}(350,50)
\put(0,20){\vector(1,0){350}}
\thicklines
{\qbezier(0,33)(30,34)(40,20)
{\qbezier(40,20)(60,40)(80,20)}
\qbezier(80,20)(120,40)(160,20
\qbezier(160,20)(200,40)(240,20
\qbezier(240,20)(260,40)(280,20
\qbezier(280,20)(320,31)(330,30
\thinlines
\put(0,37){\small $ \xi_0'\sim F_{b'}$}
\put(45,37){\small $\xi_1'\sim F$}
\put(105,37){\small $\xi_2'\sim F$}
\put(185,37){\small $\xi_3'\sim F$}
\put(255,37){\small $\xi_4'\sim F$}
\put(238,11){\small $ t_3'$}
\put(38,11){\small $ t_0'$}
\put(78,11){\small $ t_1'$}
\put(158,11){\small $ t_2'$}
\put(278,11){\small $ t_4'$}
\put(0,11){\small $0$}
\end{picture}
    \caption{Процесс ${{{X}'}_{t}}$ (соответствующий ${{{B}'}_{t}}$).}
    \label{fig4}
\end{figure}

Понятно, что вероятность склеивания (т.е. совпадения двух точек $t_i$ и $t_j$) за конечное время равна нулю.

Более того, построенные процессы независимы, а в параллельном склеивании они будут зависеть.

Для конструирования параллельного склеивания (т.е. модификации предложенной выше конструкции) нам понадобится Основная Лемма Склеивания (см., например, \cite{12.}).

\begin{mylem} [Основная Лемма Склеивания]
Пусть $f_i(s)$ -- плотности распределения (п.р.) сл.в. $\theta_i$ ($i=1,2$).
и $$\intl_{-\infty}^ \infty \min(f_1(s),f_2(s))\,\mathrm{d} s=\varkappa>0.$$

Тогда на некотором вероятностном пространстве существует две сл.в. $\vartheta_i$ такие, что $\vartheta_i\stackrel{\mathscr{D}}{=} \theta_i$, и $\ensuremath{\mathbf{P}}\{\vartheta_1=\vartheta_2\}\geqslant \varkappa$. \hfill \ensuremath{\triangleright}
\end{mylem}
Величина $\ensuremath{\displaystyle}\intl_{-\infty}^ \infty \min(f_1(s),f_2(s))\,\mathrm{d} s=\varkappa$ называется {\it общей частью} распределений $\theta_i$.
\begin{proof}
Пусть
$$\varphi(s)\stackrel {\text{\rm def}}{=\!\!\!=} \min\left(f_1(s), f_2(s)\right); \qquad \ensuremath{\displaystyle}\int\limits_0^\infty \varphi(s)\,\mathrm{d}\, s\stackrel {\text{\rm def}}{=\!\!\!=} \varkappa=\Phi(+\infty)>0,$$
где $\Phi(s)\stackrel {\text{\rm def}}{=\!\!\!=} \ensuremath{\displaystyle}\int\limits_0^s \varphi(u)\,\mathrm{d}\, u$.

Обозначим
$$\Psi(s)\stackrel {\text{\rm def}}{=\!\!\!=} F_1(s)-\Phi(s), \;\;  \Psi_c(s)\stackrel {\text{\rm def}}{=\!\!\!=} F_2 (s)-\Phi(s); \;\; \Psi_1(+\infty)=\Psi_2(+\infty)=1-\varkappa,
$$
где $F_i(s)$ -- ф.р. сл.в. $\theta_i$.

Положим для независимых равномерно распределённых на $[0,1)$ сл.в. $\mathscr{U}$, $\mathscr{U}'$, $\mathscr{U}''$
$$
\xi_1(\mathscr{U},\mathscr{U}',\mathscr{U}'')\stackrel {\text{\rm def}}{=\!\!\!=} \mathbf{1} (\mathscr{U}<\varkappa)\Phi^{-1}(\varkappa \mathscr{U}')+\mathbf{1} (\mathscr{U}\geqslant \varkappa) \Psi^{-1}((1-\varkappa) \mathscr{U}'');$$
$$\xi_2(\mathscr{U},\mathscr{U}',\mathscr{U}'')\stackrel {\text{\rm def}}{=\!\!\!=} \mathbf{1} (\mathscr{U}<\varkappa)\Phi^{-1}(\varkappa \mathscr{U}')+\mathbf{1} (\mathscr{U}\geqslant \varkappa) \Psi_c^{-1}((1-\varkappa) \mathscr{U}'').
$$

Несложно заметить, что
$$\ensuremath{\mathbf{P}}\Big\{\xi_1(\mathscr{U},\mathscr{U}',\mathscr{U}'')\leqslant s\Big\}=F_1(s);\quad \ensuremath{\mathbf{P}}\Big\{\xi_2(\mathscr{U},\mathscr{U}',\mathscr{U}'')\leqslant s\Big\}= F_2 (s);
$$
и $\ensuremath{\mathbf{P}}\left\{\xi_1(\mathscr{U},\mathscr{U}',\mathscr{U}'')=\xi_2(\mathscr{U},\mathscr{U}',\mathscr{U}'')\right\}=\varkappa$.
\end{proof}
\begin{myrem}
Фактически
$$
\xi_i=
\begin{cases}
\widetilde \xi & \mbox{с вероятностью } \varkappa;
\\ \\
\widehat \xi_i & \mbox{с вероятностью } 1-\varkappa,
\end{cases}
$$
где независимые сл.в. $\widetilde \xi$ и $\widehat \xi_i$ имеют ф.р. $\ensuremath{\displaystyle}\frac{\Phi(s)}{\varkappa}$ и $\ensuremath{\displaystyle}\frac{\Psi(s)}{1-\varkappa}$. \hfill \ensuremath{\triangleright}
\end{myrem}

\subsection{\it Второй шаг: модификация конструирования пары $(B_t,B_t '$) в параллельное склеивание}

Вернёмся к нашим {\it независимым} процессам $B_t $ и $B_t '$.
У нас $B_0 =b$, $B_0 '=b'$, это -- перескоки процессов восстановления.

Т.е. времена $t_1$ и $t_1'$ -- остаточные времена периодов регенерации с ф.р. $F_{b}(s)$ и $F_{b'}(s)$ соответственно (см. (\ref{5})).
Первый этап конструирования -- это построение {\it независимых} сл.в. $t_1$ и $t_1'$.

После момента $T_1\stackrel{\text{\rm def}}{=\!\!\!=} \max\left\{t_1 ,t_1 '\right\}$ оба процесса начинают удовлетворять условиям Теоремы Лордена (очевидно, $T_1\leqslant t_1 +t_1 '\stackrel{\text{\rm def}}{=\!\!\!=} \theta_0$).

На рисунке \ref{fig5} $T_1= t_1 '$. После момента $T_1$ к процессу $B_t'$ можно применять неравенство Лордена, т.е. $\ensuremath{{\mathbb{E}}}\,B_{t '_1} \leqslant\Xi$.

\begin{figure}[b]
\centering
\begin{picture}(450,100)
\put(0,20){\vector(1,0){450}}
\put(0,18){\line(0,1){4}}
\thicklines
{\qbezier(0,35)(50,35)(100,20)
{\qbezier(100,20)(140,40)(180,20)}
\qbezier(180,20)(240,40)(300,20
\qbezier(300,20)(340,40)(380,20
\put(400,40){\ldots \ldots \ldots}
\put(95,5){$t _1$}
\put(175,5){$t _2$}
\put(298,5){$t _3$}
\put(378,5){$t _4$}
\put(40,40){\small $\xi _1=t _1$}
\put(137,40){\small $\xi _2$}
\put(220,40){\small $\xi _3$}
\put(330,40){\small $\xi _4$}
\put(-5,5){$t_0=0$}
\put(243,5){$t_2 '$}
\color{black}
\put(0,65){\vector(1,0){450}}
\put(0,63){\line(0,1){4}}
\thicklines
{\qbezier(0,80)(80,80)(150,65)
{\qbezier(150,65)(200,85)(243,65)}
\qbezier(243,65)(269,85)(300,65
\qbezier(300,65)(340,85)(380,65
\put(150,75){$t '_1$}
\put(243,75){$t '_2$}
\put(300,75){$t '_3$}
\put(378,75){$t '_4$}
\put(40,85){\small$ \xi '_1=t '_1$}
\put(200,85){\small$ \xi '_2$}
\put(269,85){\small $\xi '_3$}
\put(330,85){\small$ \xi '_4$}
\put(-5,50){$t_0=0$}
\color{black}
\multiput(150, 5)(0,5){15}{\line(0,1){3}}
\color{black}
\put(150,5){$t '_1$}
\color{black}
\qbezier(180,20)(210,0)(243,20)
\put(120,-1){\small $B_{t '_1} $}
\color{black}
\qbezier(100,20)(125,0)(150,20)
\put(210,-1){\small $B_{t '_2} $}

\color{black}
\multiput(243, 5)(0,5){15}{\line(0,1){3}}
\color{black}
\thinlines
\color{black}
\multiput(300, 5)(0,5){15}{\line(0,1){3}}
\put(305,55){\small\it склейка}
\put(310,50){\vector(-1,-1){10}}
\color{black}
\end{picture}
 \caption{На этом рисунке в точке ${{{t}'}_{2}}$ происходит склеивание.}
    \label{fig5}
\end{figure}

По неравенству Маркова, для любого $\Theta>\Xi\geqslant \ensuremath{{\mathbb{E}}}\,B_{t '_1}'$ верно неравенство $\ensuremath{\displaystyle}\ensuremath{\mathbf{P}}\left\{B_{t '_1} '<\Theta\right\}\geqslant 1-\frac{\Xi}{\Theta}=p_0$ и в случае, если произойдёт событие $B_{t '_1} <\Theta$, можно оценить величину
$$\ensuremath{\displaystyle}\ensuremath{\sup\limits}_{u\in (0,\Theta)}\ensuremath{\displaystyle}\intl_{0}^{\infty} \min(\lambda(s+u),\lambda(s))\,\mathrm{d} s\geqslant \ensuremath{\sup\limits}_{u\in (0,\Theta)}\intl_{0}^{\infty} \min(\varphi(s+u),\varphi(s))\,\mathrm{d} s=\kappa>0,$$
т.к. почти всюду $\varphi(s)>0$ (поскольку предполагается, что плотность распределения сл.в. $\xi_i$ п.в. положительна).

Теперь, используя Основную Лемму Склеивания, можно продолжить остаточное время периода восстановления процесса $B_t$ и следующий период восстановления $B '_t$ таким образом, что с вероятностью $\varkappa\stackrel{\text{\rm def}}{=\!\!\!=} p_0\kappa$, процессы $B _t$ и $B '_t$ совпадут в момент $t_2 '\stackrel{\text{\rm def}}{=\!\!\!=} \theta_1$.

(Заметим, что здесь использовалась некоторая величина $\Theta>\Xi$; выбор этой величины влияет на значение $\varkappa$ -- так что можно искать значение $\Theta$, при котором значение $\varkappa$ максимально.)

Если в момент $t_2 '=\theta_1$ не произошло совпадения процессов (с вероятностью $\leqslant (1-\varkappa)$), то в следующий момент восстановления $B '_t$ мы повторяем описанную выше конструкцию с перескоком $B _t$ процесса восстановления в момент $t_2 '=\theta_2$ таким образом, что процессы $B _t$ и $B '_t$ совпадут с вероятностью $p_0\varkappa$ в момент $t_3 '=\theta_3$, и т.д..
Таким образом, при условии, что в момент $t_n '=\theta_{n-1}$ произойдёт совпадение процессов $B _t$ и $B '_t$,
$$\tau\left(B_0 ,B_0^{(t)}\right)\leqslant T_1+\ensuremath{\displaystyle}\ensuremath{\sum\limits}_{i=2}^n \xi_i '$$
с вероятностью $\varkappa(1-\varkappa)^{n-2}$ (здесь $T_1\leqslant t_1 +t_1 '$).

Иначе говоря, момент склеивания $\tau\left(B_0 ,B_0^{(t)}\right)$ оценивается сверху {\it условной} геометрической суммой периодов восстановления, включая сумму первый (возможно, неполных) периодов восстановления $B_t$ и $B '_t$.

Если существует конечный момент $\ensuremath{{\mathbb{E}}}\,(\xi)^{\ell+1}=C_\ell\left(B_0 ,B_0'\right)$, то можно получить оценку сверху для
$$
\ensuremath{{\mathbb{E}}}\,\left(\tau\left(B_0 ,B_0^{(t)}\right)\right)^\ell=C_\ell\left(B_0 ,B_0 '\right)=C_\ell\left(b,b'\right).
$$
Для этого можно использовать хорошо известное \\
{\bf Неравенство Йенсена}

Для действительнозначной выпуклой (вниз) функции $\varphi$, чисел $ x_{1},x_{2},\ldots ,x_{n}$ из её непрерывной области определения, и положительных величин $a_{i}$, неравенство Йенсена можно сформулировать так:
$$
\varphi \left({\frac {\sum a_{i}x_{i}}{\sum a_{i}}}\right)\leq {\frac {\sum a_{i}\varphi (x_{i})}{\sum a_{i}}}
$$
и, следовательно,
$$
\varphi \left({\frac {\sum x_{i}}{n}}\right)\leq {\frac {\sum \varphi (x_{i})}{n}}.
$$

Если существует $\ensuremath{{\mathbb{E}}}\,\xi^{\ell+1}<\infty$, оценим
\begin{multline*}
\ensuremath{{\mathbb{E}}}\,\left(T_1+\ensuremath{\sum\limits}_{i=1}^{\nu} \xi_i\right)^\ell=\ensuremath{{\mathbb{E}}}\,\left(\frac{(\nu+1)\times\left(T_1+\ensuremath{\sum\limits}_{i=1}^{\nu} \xi_i\right)}{\nu+1}\right)^\ell\leqslant
\ensuremath{{\mathbb{E}}}\,(\nu+1)^\ell\frac{\left(T_1\right)^\ell+\ensuremath{\sum\limits}_{i=1}^\nu \xi_i^\ell}{\nu+1}=
\\
=\ensuremath{{\mathbb{E}}}(\nu+1)^{\ell-1}\times\ensuremath{{\mathbb{E}}}\,T_1^\ell +\ensuremath{\sum\limits}_{i=1}^\infty \left((i+1)\times\ensuremath{\sum\limits}_{j=1}^{i-1}\ensuremath{{\mathbb{E}}}\,(\xi_j^\ell|\mathscr{E}_i)\ensuremath{\mathbf{P}}(\mathscr{E}_i)+\ensuremath{{\mathbb{E}}}\,(\xi_i^\ell|\mathscr{E}_i)\ensuremath{\mathbf{P}}(\mathscr{E}_i) \right),
\end{multline*}
где $\mathscr{E}_i=\bigcup\limits_{j=1}^{i-1}\overline \mathscr{S}_j\cup\mathscr{S}_i$, а $\mathscr{S}_i$ -- событие $\{$момент $\theta_i$ -- это момент склеивания $\tau\}$.

Заметим, что $\ensuremath{\mathbf{P}}(\mathscr{S}_i)= (1-\varkappa)^{i-1}(1-\varkappa)\leqslant (1-\varkappa)^{i-1}$, а также используем очевидное неравенство
\begin{equation}\label{7}
\ensuremath{{\mathbb{E}}}\,(\xi|A)\ensuremath{\mathbf{P}}(A)\leqslant \ensuremath{{\mathbb{E}}}\,(\xi).
\end{equation}

Поэтому $\ensuremath{{\mathbb{E}}}\,(\xi_j^\ell|\mathscr{E}_i)\ensuremath{\mathbf{P}}(\mathscr{E}_i)\leqslant\ensuremath{{\mathbb{E}}}\,(\xi_j^\ell|\mathscr{E}_i)\prod\limits_{j=1}^{i-1}(1-\ensuremath{\mathbf{P}}(\mathscr{S}_j))\leqslant \ensuremath{{\mathbb{E}}}\, \xi_j^\ell (1-\varkappa)^{i-2}$; $\ensuremath{{\mathbb{E}}}\,(\xi_i^\ell|\mathscr{E}_i)\ensuremath{\mathbf{P}}(\mathscr{E}_i)\le \ensuremath{{\mathbb{E}}}\,\xi_i^\ell (1-\varkappa)^{i-1}\le (1-\varkappa)^{i-2}$.

Таким образом, $$\ensuremath{\displaystyle}\ensuremath{{\mathbb{E}}}\,\left(T_1+\ensuremath{\sum\limits}_{i=1}^{\nu} \xi_i\right)^\ell\le \ensuremath{{\mathbb{E}}}(\nu+1)^{\ell-1}\times\ensuremath{{\mathbb{E}}}\,T_1^\ell+\ensuremath{{\mathbb{E}}}\,\xi_i^\ell \ensuremath{\sum\limits}_{i=1}^\infty (i+1)^2(1-\varkappa)^{i-1}=\Upsilon(b,b',\Theta, F(\cdot)),$$
эта величина $\Upsilon(\ell,b,b',\Theta, F(\cdot))$ может быть вычислена или оценена.

Учитывая, что предельное (стационарное) распределение перескока процесса восстановления известно (\ref{5}), и только величина $T_1$ зависит от $b$ и $b'$, интегрирование величины $\Upsilon(\ell,b,b',\Theta, F(\cdot))$ не представляет больших трудностей.
Таким образом, верна
\begin{theorem}
Если конечна величина $\ensuremath{{\mathbb{E}}}\,\xi^{\ell+1}$, то можно вычислить величину $\ensuremath{\displaystyle}\widetilde \Upsilon(\ell,b,b',\Theta, F(\cdot))=\intl_0^\infty \Upsilon(\ell,b,\Theta, F(\cdot)) \,\mathrm{d} \widetilde F(b')$ такую, что
$$
\left\|\mathscr{P}^{b}_t - \widetilde\mathscr{P} \right\|_{\mbox{ПВ}}\leqslant \frac{ \widetilde \Upsilon(\ell,b,\Theta, F(\cdot))}{t^\ell}.
\eqno{\triangleright}$$
\end{theorem}

Оценка экспоненциальных моментов сл.в. $\tau(b,b')$ существенно сложнее, т.к. неравенство (\ref{7}) не даёт возможности вычислить оценку $\ensuremath{{\mathbb{E}}}\exp(\beta \tau)$; здесь для получения оценки можно использовать свойства конкретного распределения $F(s)$ или более тонкие оценки при конструировании параллельного склеивания, на чём мы здесь не будем останавливаться.
Тем не менее, и если $\ensuremath{{\mathbb{E}}}\,\exp(\alpha\xi)$, то можно оценить
$$
\ensuremath{{\mathbb{E}}}\,\exp\left(\beta \times \left(T_1+\ensuremath{\sum\limits}_{i=1}^{\nu} \xi_i\right)\right)
$$
для некоторого $\beta\in(0, \alpha)$, и поэтому верна
\begin{theorem}
Если $\ensuremath{{\mathbb{E}}}\,\exp(\alpha \Xi)<\infty$, то для некоторого $\beta\in (0,\alpha)$ можно вычислить постоянную $K_\beta(b,b')$ такую, что
$\ensuremath{{\mathbb{E}}}\,\exp(\beta \tau(b,b'))\leqslant K_\beta(b,b')$,
и
$$
\left\|\mathscr{P}^{b}_t - \widetilde\mathscr{P} \right\|_{\mbox{ПВ}}\leqslant \widetilde K_\beta(b)\exp(-\beta t),
$$
где $\widetilde K_\beta(b)=\ensuremath{\displaystyle}\intl_0^\infty \exp(K_\beta(b,b')) \,\mathrm{d} \widetilde F(b')$.\hfill \ensuremath{\triangleright}
\end{theorem}

\section{Заключение}

Использование предложенных способов вычисления строгих оценок сверху для регенерирующих процессов может быть использован при анализе поведения различных сложных СМО, СеМО и систем надёжности.
Оценив время $\mathcal T$ достижения приемлемого уровня близости распределения исследуемой модели, исследователь может с помощью имитационного моделирования проанализировать поведение системы до достижения времени $\mathcal T$ и использовать эти данные при прогнозировании поведения аналогичных систем.

Для случая, когда исследуемая модель включает несколько параллельных технологических процессов, может применяться естественное обобщение Основной Леммы Склеивания:
\begin{mylem}[Обобщение Основной Леммы Склеивания]
Пусть $f_i(s)$ -- плотности распределения сл.в. $\theta_i$ ($i=1,\ldots,n$).
и пусть $$\ensuremath{\displaystyle}\intl_{-\infty}^ \infty \min\limits_{i=1,\ldots,n}(f_i(s))\,\mathrm{d} s=\varkappa>0.$$

Тогда на некотором вероятностном пространстве существует $n$ сл.в. $\vartheta_i$ ($i=1,\ldots,n$), таких, что $\vartheta_i\stackrel{\mathscr{D}}{=} \theta_i$ для всех $i=1,\ldots,n$, и $$\ensuremath{\mathbf{P}}\{\vartheta_1=\vartheta_2=\ldots=\vartheta_n\}\geqslant \varkappa. \hfill \eqno{\triangleright}$$
\end{mylem}
Доказательство этой Леммы конструктивно и подобно доказательству Основной Леммы Склеивания  (см., например, \cite{13.}).

Используя Обобщение Основной Леммы склеивания и предложенный в \cite{13.} подход к конструированию многомерных марковски модулированных процессов, можно получать оценки момента склеивания (склейки) сконструированных по предложенной выше схеме ``параллельных'' (т.е. имеющих те же маргинальные распределения) случайных процессов для широкого круга задач
ТМО, СеМО и теории надёжности.

Умение оценивать сверху, а не только указывать тип скорости сходимости (степенной, экспоненциальный) важно для практического использования различных стохастических моделей.
Использование таких оценок позволяет узнать, когда распределение используемой системы станет достаточно близко к предельному (стационарному) распределению.
До этого момента можно оценивать поведение используемой системы с помощью имитационного моделирования, что не всегда бывает экономически оправдано.

Поэтому так важно уметь оценивать {\it гарантированное} время до приближения распределения используемой системы к стационарному режиму.

\section*{Благодарности }
Авторы благодарят анонимного рецензента за ценные замечания, способствовавшие улучшению текста.
Работа частично поддержана РФФИ (проекто № 20-01-00575А)

redpajama_set_name: RedPajamaArXiv


….The majority of men appearing on posters and billboards are celebrities. Although the wide use of cosmetic surgery is making men look increasingly similar, they are often associated not merely with a product, but also with a popular drama, and in some cases, a steamy bed or bathroom scene. That is not something the average worker would ever seek to emulate, nor be able to, as the nation's corporate dress code remains conservative.
See "The bra boys of South Korea" at World News Australia for more, which is mostly about the kkotminam (꽃미남) phenomenon, or here for more on the disproportionate role of celebrity endorsements in the Korean media (source, right: Wonbin Thailand).
…a great short clip instructing women workers newly employed in industrial factories during World War II on how to do their hair to maximize safety. It assumes both ignorance and vanity on the part of women and speaks to the lack of efficiency caused by efforts to remain attractive on the line.
As I pointed out in — yes, again — Part 4, those assumptions about vanity need to be placed in the context of wartime shortages, when attention to beauty and fashion were viewed as extravagant and unpatriotic. But despite that, women's anxieties about both were still explicitly encouraged, preyed upon, and/or encouraged by industry, and actually even by the government itself. The ensuing contradictions, double-standards, hypocrisy, and backlash are very similar to what has been occurring in South Korea since the 2000s with women's rapid entrance into the (part-time) workforce, and make comparisons very useful and compelling.
"Alfred T. Palmer who worked for the Office of War Information (responsible for promoting patriotism, war news management and women recruitment)" whose photos "had to lure young women into the factories by showing women workers as glamorous and even fashionable." (My emphasis; see example on right).
Update: also see Kathryn M. Brown's 2010 MA thesis Patriotic Support: The Girdle Pin-up of World War 2 (it can't be directly linked sorry—type the title into the searchbar) for more on how malleable and adaptable — and, as explained, ultimately hypocritical and contradictory — the language, prevailing standards for, and attitudes towards beauty and fashion proved to be for the needs of government and industry (see Part 3 for modern Korean and earlier US parallels also).
Previous Post Announcements: Two Very Worthy Causes to Support!
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text:
Service to Our Community
The Verde Valley Vettes supports local community activities and charities. For more than a decade we have partnered with Yavapai Big Brothers Big Sisters to provide Christmas gifts for their most needy families and assisted the Mingus High School Hot Rod Club each year with their car shows. We have provided funds for the Cornville Community Association to purchase bicycles for their Easter egg hunt and supported various other local charities.
The club has also supported the National Kidney Foundation through our affiliation with the National Council of Corvette Clubs (NCCC), and are proud to have been the only club in our region to have achieved 100% participation in that alliance every year for its duration through 2015. Looking forward to 2016 and beyond, we continue our strong support of the recently designated new NCCC charity, St. Jude's Children's Reasearch Hospital.
With support from our primary sponsor, Larry Green Chevrolet, we host an annual car show each May and a rally weekend in October to fund our charitable activities. Vsit our Affiliates page to view all of our major sponsors and please support them.


text:
Lloyd Kaufman, Kerry Marlowe, Lauren Lakis, Abbey Anderson, Alyssa Hunter
Evan Marlowe
Kerry Marlowe
Slater Brothers Entertainment
"Horror House" an Entertaining Horror Collective
If you've been paying attention to my horror reviews, then you know that I've been covering an increasingly popular "collective" of largely horror shorts from JABB Pictures going under the title "The Collective." While Horror House is different in scope and intention, it's a similarly entertaining horror collection being distributed by Slater Brothers Enteratinment - The concept is relatively simple. There are five shorts in the collective and all center around the rather horrific stories that have unfolded in one "for sale" house being offered by a rather enthusiastic real estate agent. The real gem of this collective comes in the form of that real estate agent - played by none other than indie horror legend Lloyd Kaufman.
Never Let Go is our first tale, a creepy little story that doesn't really go the way that you're likely expecting, though it is possible for you to catch on before the story itself unfolds. Sarah (Catherine Widell) is your typical suburban housewife/mom, though it appears that the family's under a bit of duress with hubby having recently been kicked out and at least one of the two kids (played by Katelyn and Alyssa Hunter) seemingly having a bit of a challenge adjusting. It's not too long before we catch on that we have a good sister/bad sister scenario going on here, and the bad sister is really getting bad. Director Evan Marlowe does a nice job of building a believable sibling relationship between the two sisters, but also has fun really playing up the sinister suspense. Things aren't what they seem, and screenwriter Kerry Marlowe gives us a nice climax that makes sure we've really been paying attention.
Marlowe herself is in the second short, Be Careful What You Wish For, and it's this piece that may show off the film's highest production values as we come face-to-face with a woman finding out her husband has been killed. Instead of distraught, her reaction is surprisingly timid. Of course, we shouldn't really be surprised about anything here, eh?
In Hot Stuff, Kaden Graves plays a bachelor who could best be described as a player. When he "plays" the wrong woman, however, he finds out the hard way the error of his ways. This story is practically tailor made for the horror genre, and it's probably the short most likely to put a smile on your face - especially if you have a really dark side. Graves keeps you invested in his character, a man whom you're likely to hate but you're also going to be stuck to the screen watching.
As a sexual abuse survivor, I can definitely call Lifelike a "trigger" film. This is a hardcore, honest film that unfolds in a way that's truly dark and intense and unforgettable. Lauren Lakis and Elain Rineheart give wonderful performances and Lifelike, despite freaking the heck out of me, is easily my favorite film from this collection.
The Leapling closes out our collection with a kid-centered monster tale that likely lends itself more easily than the other stories to moving into a full-length scenario. There's very clearly a dynamic story here and it feels a bit incomplete in this format. That said, I enjoyed the cast and the production quality with this film.
Overall, while Horror House occasionally shows its low-budget nature it's also a consistently well done and entertaining horror collective with solid storytelling from Kerry Marlowe and stories that both thread together nicely and stand on their own thanks to Evan Marlowe's solid direction. If you get a chance, check it out.


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When you need money to pay your rent, utility bills or get through a difficult time, there are 2 utility payment and rental assistance programs in the Alberta, Virginia area.
DSS is a division of county government that provides the following services: Adult Medicaid, Aging and Adult services, Children services, Child Support,Families First program, Family and Children Medicaid, Food and Nutritional services, Senior Center, Work First, and Individual and Family support.
Follow State and Federal guidelines for individual programs.
Contact Social Services Ofice to get individual program intake procedures.
Agency provides general assistance with utilities, rent, medicine, clothing and food.
Must apply in person,have picture ID, and SS card for each person in the household. The bill requiring assistance with must have clints name on it and the client must be responsible for the bill.
The greatest obstacle to discovery is not ignorance-it is the illusion of knowledge.
Looking For Services In A City Near Alberta, VA?


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Deque President Preety Kumar to Attend White House 20th Anniversary Celebration of the Americans With Disabilities Act Today
On Monday, July 26, President Obama will hold an event at the White House to commemorate the 20th Anniversary of the Americans with Disabilities Act, which was signed into law by President George W. Bush on this date in 1990.
Preety Kumar, President & Founder of Deque Systems, one of the first web accessibility-focused companies in the United States, has been invited to attend the festivities.
In addition to remarks by President Obama, attendees will enjoy performances by Nathaniel Anthony Ayers, Patti LaBelle and Marlee Matlin, and remarks by Marca Bristo, President of Access Living, White House Senior Advisor Valerie Jarrett and Director of the White House Domestic Policy Council Melody Barnes.
Today's ADA celebration, which begins at 5:30 Eastern time, will be streamed live at http://www.whitehouse.gov/live
By Deque Systems July 26, 2010
Preety Kumar
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