text:
Mien shiang ( meaning face () reading ()) is a physiognomic and fortune-telling practice in Chinese culture and traditional Chinese medicine which purports to determine aspects of person's character, personality, and (future) health by analyzing their face according to the five phases ("wu xing"). The five phases (namely wood, fire, earth, metal, and water) are metaphors devised by ancient Chinese philosophers to explain the relationship, interaction, and ongoing change of everything in the universe. In recent times the art of face reading has become more popular and schools that teach  more wide spread.

Historical records
There are early written records of . In Book of Rites, it writes, "Those who look up are arrogant; Those who look down are worried; Those who look sideways are sly". After Gou Jian () of Yue () settled the State of Wu, Fan Li () (536–488 BCE) knew that Gou Jian had a "long neck and sharp upper lips", which meant they could face hardship together but not enjoy peace together. Fan Li later fled with Xi Shi () to the State of Qi and changed their names to lead a new life. Another historical example in which Mien Shiang was used was Wang Anshi () of Song Dynasty. Wang's eyes resembled those of ox, and his head shaped like those of tigers. It was said that if Wang could stare straight at the sun, he had a face with good fortune ().

Mien shiang terminology
The words below are a description of  terminology, such as "three parts of one's face" () which includes the upper part (), middle part (), and lower part ().

 Yin-Yang ()
 Three parts of one's face ()are the upper part, middle part, and lower part.
 Upper part ()is area between the forehead between the hairline and eyebrows. This section represents the fortune between ages 10 to 20, which is one's youth.
 Middle part () is the area between the eyebrows and the bottom of the nose. This section represents the fortune between ages 20 to 40, which is one's adult years.
 Lower part () is the area between the bottom of the nose to the chin. This section represents the fortune after ages 40, which is one's older adult years.
 Five organs on one's face ()includes one's ears, eyebrows, eyes, nose, and mouth.
 Face Shape ()includes ten major kinds: You (), Jia (), Shen (), Tian (), Tong (), Wang (), Yuan (), Mu (), Yong (), and Feng ().
 Twelve Houses ()includes the 12 basic parts on one's face that represent different aspects of life. They are the life house (), wealth house (), sibling house (), marriage house (), children house (), health house (), travelling house (), assistant house (), career house (), property house (), fortune & emotion house (), and parents house ().
 Wrinkle (), scar (), mole (), speckle ().
 Nei Shiang ()is the internal aspect of the body including shoulders, waist, back, chest, abdomen, umbilicus, forearms, limbs, neck, breasts (for women), and others.
 Gu Shiang ()is bone reading, including the skull and the bones in the body.
 Dong Shiang ()is the individual's movement, including walking (), sitting and standing (), eating (), lying (), crying (), and laughing ().
 Sheng Shiang ()is the sounds that one makes.

See also
 Metoposcopy
 Physiognomy
 Phrenology

References

Chinese culture
Divination
Physiognomy
Taoist practices
Traditional Chinese medicine


Plate stands have sturdy acrylic bases which are machine-riveted to a heavy vinyl coated steel wire back brace. All are completely adjustable!
Plate stands for displaying serving trays, plates, china, artwork, framed photos, and collectibles.


This alpaca blog is intended to give a light hearted view of the day to day activities of an alpaca breeder, as well as providing an opportunity to share experiences and knowledge gained along the way. Beck Brow Alpacas of Cumbria is an alpaca farm situated within the beautiful Eden Valley in the North of England. Our aim is to breed the finest quality alpacas available...this is our journey.
Yesterday we had a very early start (4.15 to be precise) as we were off to Westmorland County Show. We only live an hour away, but as organisers of the alpaca show we have to be early on site. Thankfully, all the exhibitors kindly heeded my advice to arrive in plenty of time, as the alpaca marquee was down a rather muddy lane with no turning point; reversing skills were definitely tested!
We took seven alpacas in total (six of our own and one belonging to Robyn and Craig) and all were placed between 1st and 3rd. The standard was high this year, with a number of the alpacas entered already having proven themselves as rosette winners at the British Alpaca Futurity. We only took white and fawns so were in some of the most competitive classes of the day. Beck Snooty Boots came a very credible second behind the eventual fawn female champion, with judge Cathy Lloyd commenting that there was really nothing between the top two, and that she would be very happy if both of them were in her own herd.
Beck Brow Rumour Had It also came second behind the white female champion, as is always the case she got praised for her prettiness, as well as her fleece characteristics. Beck Brow Quintos and Beck Brow Magellan came second and third in the intermediate white male class (remember Quintos...first fleece like a Suri...beat Magellan!!!). Now, this is where the pattern ends; the boys didn't come second to the eventually white male champion. No, the white male champion was an adult...the white male champion was...Beck Brow Explorer!
I had a lovely day, it is such a friendly show and it was great to catch up with so many breeders. An added bonus was Craig getting the sales man of the year award, for selling us out of all the adult knitwear that we had taken.
It was a day of reflected glory; firstly for owning Explorer and secondly for my choice of judge. Cathy Lloyd did a fantastic job, which was commented on by so many of the exhibitors at the end of the show. It all ran very smoothly, and everyone knew exactly why they had been placed where they had. I think that we all came away having gained valuable knowledge about the strengths and weaknesses of our alpacas.
Beck Brow Explorer before taking the championship...he was having none of it when it came to posing with his sash on!
Beck Brow Explorer bowing to the President of the Westmorland County Show....Actually I think he might be saying "if you pat me on the head I'll have ya"...he is in fact supposed to be getting his photo taken receiving the trophy but he wasn't playing ball...it was raining though and he doesn't do rain!
Roma is still having nothing to do with the adorable Foxy Sox...thanks go to Amanda, and to Mum and Dad, who shared bottle feeding duties yesterday. Hopefully next year Sox will be joining us at Westmorland as part of our show team!
P.S I missed at Beck Brow Synergy...2nd placed beige intermediate male!
Wow! What a star he is. Well done and many congratulations.
Well done, great results, especially old Lardy pants!
Explorer is looking fine in that top photo - congratulations on a great day all around! I do so love alpacas.
As I said yesterday well done on your results and organisation of the show, having done it I know the effort that goes into organising a show.


text:
ETHEREUMFeatured
Ethereum (ETH): The community has nothing to fear from the Merge?
Despite the Merge being dangled by Vitalik Buterin and his gang, investors' concerns have not been allayed. What will happen to their Ethereum holdings throughout and after The Merge? Will ETH 2 tokens exist? People are currently plagued by a plethora of questions. On the other side of the river, though, there are also assured investors. They increase the amount of ETH placed in the blockchain deposit contract as a sign of their commitment to Ethereum.
Ethereum 2.0, increasing ETH blockage
Times Tabloid came to the conclusion that the amount of ETH deposited in the Ethereum deposit contract is fluctuating by using data from the Oklink platform. In fact, there are currently 13.3 million ETH equivalents.
Since August 1, there have been an average of 36,000 coins staked per week, according to the placement history of ETH 2.0. Adding up to 153,000 ETH in just 23 days.
However, keep in mind that ordinary investors are only permitted to sell, transport, and trade the stacked ethereum coins they own on the blockchain. Suspension of withdrawals until further notice.
The best course of action for Coinbase is to halt deposits and withdrawals before and during the Merge. to prevent too many investors from being concerned about what will happen to their ETH holdings.
Only a few days remain till the Merge.
Is the Merge's date already known? More precisely. It has been reported in numerous media outlets that it will happen on September 16.
Everything appears to be in place for the Merge's preparation. We only recently discovered that Ethereum developers have invited users of Ethereum 2.0 Teku to upgrade. This is a phase in the Bellatrix upgrade integration process, which necessitates downloading Teku v.22.81.
After The Merge, there won't be any more space for Proof-of-Work, that much is certain. Chandler Guo and other children will not enjoy this circumstance. Ethereum 2.0, according to Vitalik Buterin, would be devoid of selfishness.
The greatest significant benefit of fusion, however, is energy savings, which are said to be 99.95% higher.
Finally, after so much anticipation, the Ethereum Merge is within reach. An occurrence that generated a lot of discussion and served as the basis for several theories. If investors express concerns about this transition, that is very normal. However, they would better wait for the adjustments patiently because Vitalik Buterin is not shy about providing explanations, and the crew working on the blockchain is quite engaged.
The Merge: Towards the Final Transition
Ethereum (ETH): SEBA Bank launches ETH staking before The Merge!

redpajama_set_name: RedPajamaCommonCrawl


text:
MAICHONG Minnesota
Water is important to every living thing in the world and by preventing from getting dirty, the government and people should help each other to maintain clean water.
Dear president,
Water is the main source to all living thing in the world. Without water, many things will start to die. Human tend to use water without a second thought, but there are places in the world that don't have much access to water and it's hard to survive. This is why, the new President, please help make our water cleaner in the world. While all the water around us is clean, it will be convenient because water is essential to humans, keep sea animal fresh, and appropriate for irrigation.
All organisms need water in order to live, mainly included human whose body needed it. Water is very important and essential to human because according to freedrinkingwater.com, the human body is two thirds made up of water and 95% of water made up of the brain, blood is 82%, and lungs is 90%. If the water is dirtied, it can be contaminated and cause real deadly illness. As apply to the United Nation from thehealthyfish.com; about 1.8 million people died every year due to dirty water. The water is either contaminated with diarrhea diseases or illness such as cholera. By just having clean water is good enough since it gave oxygen into the atmosphere as well.
Sea animals also need clean water as well because fish will be consuming by humans and give more benefit to human. Dirtying water is such a big threat to fish especially if people throw cans, wrapper, and plastic bag into water. According to whoi.edu, some turtle has mistaken a plastic bag for a jellyfish just because people did not clean the water. Keeping water clean can keep the fish fresh and healthy after consuming them, and it will not make an illness to us. This is why big companies who sells lake grown always perform consistent water testing and making sure that they have the best kind of water for the fishes. There has been progress but some part of the world still need help, and I hope our new president will help to reach out to those places to be as healthy as other places. As related to whoi.edu, in 1988, The Marine Plastic Pollution Research and Control Act (MARPOL) was pass out on Dec. 31st. It was an act to restrict people from dumping plastic into the sea, lake, river, or ocean. Once again according to whoi.edu, people has been dumping nearly 7 million tons of unnecessary using into the water. This is bad, and it will kill the creatures in the sea.
Good clean sources of water can also help irrigation. For farmers many water was used yearly to provide the organic goods for humans. As present in scotland.lovefoodhatewaste.com, 800 gram of breads is already used up to 1,608 liters of water. Cheese needed about 3,178 liters and rice is 2,497 liters of water. There have been more humans in the world than it is in the past and if our water is in poor condition, there's no way that the crops the farmer made will be enough to sell in supermarkets. Most water used for irrigation are from underground that were bumped up into the surface but it came from the same route from the ocean, sea, or rivers. According to the United States Geological Survey (USGS), about 330 million acres are used for agriculture, and apply to fao.org, the whole world uses agriculture except Europe and North America. Dirty water will not only rise the crops badly but can also bring illness to the body as well.
Water is the main sources keeping living thing alive. It is for us to use directly or to be used for our foods that we will eventually eat too. The water alone has help the world a lot, and this is why we should protect the water, we seem to take it for granted and never really thought through how important water is. Even there are more serious matter in this world, the new president should take a look at our water and help improve because there are many people dying every day that is due to poor water. We do not want to let people die because we want happiness and healthiness to all the people we love.
Maichong Lee
water water contamination water pollution water problems
More from this group More from this site More from Minnesota More about these issues
St. Paul Central High School
Approaching Analysis Hours 4 and 5
10 Quest 4th and 5th hour students
Oliver Y. Minnesota
Academic Profiling
Students of color are unevenly distributed throughout education
Xue X. Minnesota
Are guns really the problem?
Chai Y. Minnesota
Abolishing Worldwide Poverty
Poverty occurs throughout the whole world. No one should have to go through "Poverty".
Bee y. Minnesota
Abortion should be banned.
Eli M. Minnesota
Global warming is happening now
Amy B. Minnesota
Undocumented Immigrants
Undocumented immigrants should have an easier time becoming citizens.
Jack Minnesota
Toxic-Masculinity
Toxic-Masculinity is one of the worst attitudes plaguing our country and here's why.
Abby Minnesota
Equal Pay For Equal Play
Female athletes should be payed equally as men.
Hue X. Minnesota
Free College Tuition
Free College Tuition for everyone will make this world a better place.
Emma F. Minnesota
Abortion needs to be accessible to everyone
The decision to end a pregnancy is a basic human right that should be available anyone wishing to have an abortion; it needs to be a choice.
Jenny L. Minnesota
Immigrants Enrich The U.S
This letter is about immigrants shouldn't be taken for granted because they will actually help us or the United States increase our digits in econo...
Shai P. Minnesota
Why Guns Should Not be Ban
In this text I am writing to the next president why they shouldn't ban guns in the U.S.
Kiin N. Minnesota
Women around the world deserve equality.
Isabelle Lutter Minnesota
The True Definition of 'Feminism'
Feminism is fighting for the equality of the sexes. In today's society, it means many different things for different people. We need to bring this ...
Luke M. Minnesota
Mosquitos are the real enemy
We should kill all mosquitos on the planet because they spread diseases, they have no large benefits or impact on the world, and because they are h...
Melinda C. Minnesota
Higher Minimum Wage for Women
Raising minimum wage can prevent lots of things.
PZ T. Minnesota
Statistics for LGBTQ hate crimes and comparisons to heterosexual people.
Marissa Providence Minnesota
How people are discriminated for speaking their native language and how it impacts not only the individual but the rest of society.
Izaak D. Minnesota
Anti-Gun Laws Won't Stop Killings
Anti-gun laws have not effect on killings because criminals will find other means of getting guns or if no guns are available they will find other ...
Riley H. Minnesota
Anyone who abuses animals should get in trouble with the law.
Abbey Minnesota
GMO Food Labeling
GMO food labels should be mandatory because American consumers have a right to know what they are putting into their bodies.
Matthew Minnesota
Gun control ismsomething that needs to be addressed
Lacey Minnesota
Raises in Minimum Wage
The minimum wage must be raised to help the economy and lifestyles of Americans.
Giannalisa Minnesota
Our future president should be concerned with keeping everyone in line and out of danger.
Clara N. Minnesota
Women should not be abused and seen as not confident and weak, women are just as strong as men.
Anna S. Minnesota
Animal abuse has been going on for a long time and its increasing over the years. Animals deserve a life too, so this needs to be stopped.
Tyler M. Minnesota
Issues at the border!
We need to focus more on the real issues that come with immigration
All letters from Minnesota →
Jennifer G. Wisconsin
Clean Water in America
Water pollution can have a severe effect on our earth and it's time for this generation to take some sort of action to prevent further damage.
Max N. New Jersey
This letter is about water pollution. Water pollution is a big deal. The fish could eat the garbage and die. We need to help.
Jack W. Idaho
Pollution around the world is at its worst and it needs to be fixed.
Callie T. Ohio
Cleaning Up The World
Parents and kids clean up their yard why not clean a little farther? Pool people clean pools why not the ocean? Everyone breaths air why not make s...
Anisah A. Louisiana
Fracking and its effects on the environment.
The concern of fracking has surfaced recently, and it is a enormous issue.
Noelle Illinois
Solving The Freshwater Crisis
The world is losing water, and it is your job as president to help the world conserve and care for its waters.
Julianna V.
Polluted Waters
Polluted water affects many including sea creatures, humans, and other land animals. It's an important issue that must be dealt with immediately.
Chase Oklahoma
I understand that you are currently receiving hundreds of letters with the people's' various points and view. But that won't stop me from trying to...


You go to the online and do a bit of research before you purchase. With just a little bit of research and just a little little bit of flair you will have your portable DVD player listings ranking excessive and promoting out in no time. 2. Any cell phone spying app for iPhone, iPad or iPod, that will get installed to the gadget you need to monitor, will require that you simply jailbreak that system. When you've got a computer at residence that has a 56k dial up modem put in then all you need is a few sneaky software program to start out recording all calls made to and from your phone. However, in case you don't have any probability to get entry to the goal telephone, it might become a problem for you in order for you to observe Snapchat. The information logs shall be presented in a neat on-line dashboard which you'll be able to entry by logging into your MSpy account by way of the MSpy official web site.
5. Surely, positively not simply Message knowledge type, you can even access the other iOS knowledge, like contacts, images, call logs, notes and more, and assist you put them where you want. As a mystery shopper you might be sort of like a spy. Seems to be like you have got a knack for recipe hubs too! Oh, how times have changed. Some have formed an inability to focus by too much technology utilization. Ah, I should have scrolled to the very backside of web page! These features ought to all the time be available on any respectable cellphone spy application. Phone spying app is most reliable and trusted the app. The explanation TrackMyFone didn't make it to our prime three of spying apps is that while this program is a complete monitoring resolution with a wide range of features, it can not rightfully compete with the leaders of the market. The laborious question to answer remains whether or not or not your device is being tracked or hacked in any means, merely because of how they're designed to be fully invisible, undetectable and very stealthy while in operation.
Receiving RSS feeds is also a relatively straightforward activity to accomplish with a large variety of RSS feed readers being available. We assure that you may ask for a refund inside 2 weeks after the purchase. If you are not linked to the Internet, you can still view the archives! 4. Instantly Hear Reside Calls, Learn Textual content Messages, View Pictures, See Activity Logs and way more! When expertise adjustments, factors of view change. The truth is, it could actually take immense effort to get past a breakup. https://dailyweblog.com/spy-text-messages/ are you'll wish to follow fingerprinting before the celebration, so you understand how a lot ink to use an the best way to roll the fingers to get a pleasant, clear print. Can https://allstarnewspaper.com/spy-text-messages/ receive messages? Today, text messages are in all probability probably the most revealing thing, which may tell you 1,000,000 things about a person. Relaxation assured our developers work non-stop to maintain FlexiSPY up to date, so you possibly can monitor the conversations necessary to you.
It can work wherever in the world! I talk with my ex fairly commonly, and we call each other a lot. I have to admit that my grandkids win most of the time on this recreation. After making a variety, you should make the payment by an internet gateway. Thanks, Invoice. I am sure that's true. Watch movies similar to "What Must you Count on When Meeting A few Divorce" and other videos in our Video Gallery. 2. I signed up for Built-in Messaging. It is smart, does it? ABC Discover Me Similar to Highway Alphabet. Scroll through to find "Cydia Installer." Faucet on it. Should you store for a card reader, be sure that it reads SIM cards.


Note: As promised, I took your comments from Cause Marketing Challenge: The Boston Common and offered our help to the City of Boston. Based on my outreach to my contacts at City Hall, they advised I send our suggestions to Toni Pollak, Boston Parks Commissioner. Here's a copy of the email I sent her this morning.
My name is Joe Waters and I'm a local cause marketer who writes the blog Selfishgiving.com. I read recently that the City is considering opening the Boston Common to company sponsorships to pay for repairs and improvements. I got to thinking about how cause marketing might play a supporting role.
I'm unsure how familiar you are with cause marketing. There are other definitions, but I define it as a partnership between a nonprofit and a for-profit for mutual profit. The "profit" for the cause is generally funds and greater public awareness. The business "profits" from enhanced favorability in the public eye and perhaps increased sales.
You can read more about cause marketing here.
After I read about the Common I wrote a post inviting my readers to brainstorm on ways cause marketing could support the Common. I got a great response on my blog and Twitter, and wanted to share with the Mayor and you what my readers suggested.
Yyou can read their comments here.
First, based on the conversation I came up with a realistic outline of what cause marketing could truly accomplish.
Second, there was a general consensus that we needed to leverage the Common and the businesses around it for fundraising. The Common has been the site of so many fundraisers, why not one for the Common itself?
Third, we talked about commemorative items stamped with the Boston Common image that could be sold in area stores with a percentage going back to support the Common.
Fourth, what about a Boston Common affinity credit card? I noticed in the article I read that neighborhood groups support the Common. Perhaps they could kick-off a credit card that would donate 1% back to the Common. Can a charity license plate be far behind?
Fifth, a couple of readers pointed to other cities that are doing city park sponsorships well, including Chicago's Millennium Park. They should be a model for our efforts. As should the National Parks Foundation. Including the Boston Common in a larger group of city parks across the country would make it a more attractive partner for national cause marketing.
Finally, thanks to its great foot traffic and tourists, cause-based businesses have great potential on the Common. Everything from photo booths to food vendors to souvenir hawkers selling licensed Boston Common gear could support restoring the area.
Ms. Pollak, I hope you'll take a moment to read my post and the suggestions my readers and I offered. I'm happy to volunteer my time to meet with you or anyone from the City to discuss our recommendations and to help the Boston Common partner with local and national companies.
I'll follow up with you shortly, but in the meantime don't hesitate to contact me first if you wish to further discuss these ideas.


In this fast changing environment it's important to have robust risk management processes in place. ISO/TC 262 'Risk Management' develops international standards in the field of risk management to support organizations in all their activities including making decisions to manage and minimize the effects of accidents, disasters and faults in technical systems as well as response and recovery from major disruptive risks. The technical committee aims to create a logical and consistent family of standards that provides guidance based on market needs.
Apart from these standards the technical committees is also continuously reviewing new areas where standards can support organizations.
Find out more about these standards in the Projects section of this site.
ISO/TC 262 has 55 Participating Countries and 18 Observing Counties as well as numerous liaisons.
BSI in the United Kingdom is responsible for the secretariat.


text:
About Gifu
"Gifu Future Center (Environment Seminar)" Opened for Wanouchi Town Residents
Gifu University hosted "Gifu Future Center (Environment Seminar) for the residents of Wanouchi Town, Gifu Prefecture on Sunday, December 11, 2022 (attended by 25 people: of them, 11 Wanouchi citizens, 14 Gifu University students).
"Future Center" is defined as a "place for dialogues for citizens" and people gather to discuss complex, difficult socioeconomic issues of today from the perspectives of "creating better future" and "producing new values for future."
Gifu University hosted this environment seminar as part of a commissioned project by Wanouchi Town called, "Zero-Carbon City Promotion and Environmental Human Resource Development Project in Wanouchi Town." On Friday, December 9, Gifu University also hosted the same seminar for the students of Wanouchi Junior High School.
On December 11, participants learned the current status of environment conservation activities now undertaken at Wanouchi Town. Young citizens and university students who are expected to play a greater role for the better future of their hometowns, exchanged their thoughts on the theme, "Declaration on zero-carbon: think about the future of Wanouchi Town." They discussed what they can do for their hometowns, and shared their future visions for the community. They were also encouraged to consider themselves as an integral member of society to bring "Zero-Carbon Declaration" to fruition.
First, two lectures were given for the participants. First, a lecture by Gifu University titled, "How to build a "Zero-Carbon City" in Wanouchi Town: reform rather than patience for the better future of our society." Listeners got some insight into what will be necessary to realize a "Zero-Carbon City." Second, a lecture by Wanouchi Town titled, "Towards "Zero-Carbon City: missions and goals of Wanouchi Town" which introduced Wanouchi Town's "Zero-Carbon City Declaration" in March 2022, and the town's commitment to comply with the Declaration.
These lectures were followed by multiple group works by participants. They discussed what individuals can do in their lives for the Declaration, and drew one big illustration to envision "Zero-Carbon Wanouchi Town" in the near future. In the following presentations, a number of proposals were made by referring to the illustrations. These proposals include how to reduce food loss, promote reusing, advance renewable energy technologies, and use of ammonia for electricity generation.
Lastly, Mr. KINO Takayuki, Mayor of Wanouchi Town wrapped up the seminar by summarizing the presentations and mentioning Wanouchi's future programs as his closing remarks.
We are expecting that each one of the participant can figure out and carry out what he/she can do for the environment, and realize their future visions for Wanouchi Town.
Scene of the lecture
Speech by Wanouchi Town Mayor
Detailed explanation of icons
Original sites
File links
About Gifu University
Recruitment of faculty members
1-1 Yanagido, Gifu City 501-1193, JAPAN Phone; +81-(0)58-230-1111


text:
Tabata is de naam van een type intervaltraining waarbij de sporter zelf de intensiteit van de training kan bepalen. De gehele work-out duurt 4 minuten en bestaat uit acht intervallen.

Dr. Izumi Tabata is een Japanse onderzoeker die een methode zocht om in een korte tijd toch een complete work-out te kunnen hebben. Met de 20-10-interval zorgde hij voor een training die in minder dan 4 minuten zorgde voor zowel een aerobe als anaerobe work-out. De bedoeling is dat men 20 seconden een bepaalde oefening redelijk intensief doet. Daarna neemt men 10 seconden rust. Dan is men weer 20 seconden intensief bezig. Dit wordt 8 keer herhaald.

De intensiteit kan zelf bepaald worden, maar men dient ervoor te zorgen dat deze steeds gelijk is.

Voorbeeld
Men kan het bijvoorbeeld verpakken in een looptraining.

20 seconden joggen + 10 seconden rust.
20 seconden joggen + 10 seconden rust.
20 seconden joggen + 10 seconden rust.
20 seconden joggen + 10 seconden rust.
20 seconden joggen + 10 seconden rust.
20 seconden joggen + 10 seconden rust.
20 seconden joggen + 10 seconden rust.
20 seconden joggen + 10 seconden rust.

Men kan ook voor andere oefeningen kiezen zoals sprints, squats, burpees en push-ups

Zie ook
 Coopertest

Referenties

Fitness

redpajama_set_name: RedPajamaWikipedia


Help Santa show those elves the true meaning of the word "strike".
Interested in the Full Version?: This download is the free trial version of Super Elf Bowling, if you enjoy the program and would like to get the full unlimited version, please return to 5 Star and click the green Get Full Version button beside the program's description.


Sooner or later, you'll want desired that you may record calls onto your Windows PC to store that dialog with formal members or family conversation with a mentor. It really is among the recorder applications that allow users to record calls in addition. Let us take a glance at the effortless process of preparing your computer to record Skype calls. It has really a few advantages which happen to be fun, for the point moreover this TalkHelper of Skype discussion. It's just as it is and might never complicate your life when you've got it for your call recording. It's really a enjoyable and exciting mode that you may talk about how your screen looks like. The single limiting factor in this case may be the space on your own personal computer. We are likely to produce you the Skype recorder for the windows PC. You see the reason why we branded it a Skype cataloging add-on for Windows.
These editing comes with transition effects that could turn your recordings. The above discussed apps show a brief description on ways to record your Skype calls using the applications's without hassle. If you liked this article and you also would like to acquire more info regarding Read Webpage kindly visit our website. This tool is convenient because within a call, you're able to fool around with the pause and play with , better stop the telephone number. A survey in to the market showed that the required upgrades making them work better together with Skype are not performed by a large proportion of those call programs today. All these are amazing features that Skype has been able to offer. The difference between these two is that infinite recording can not be supported by the you, whereas the one that is paid is unquestionably unlimited, allowing you to have the terrific features which are not seen from the free one. Third party developers have developed software that are able to do that for you and TalkHelper appears to be one.
Pamela call recorder enables you to record Skype calls and is totally free to down load. TalkHelper is a Skype add on that may make it possible for users to capture their video calls, providing top quality of sound and video recording. Considering that that the extensive research of the various relevant elements that you should consider when seeking out for recording your own Skype conversations on Windows, a software, we believe you will be better armed to create judgements of preference. Well, the debut of the Skype on Mac is currently feasible for users to record their conversations which can be both in sound and video format. Step 4 After making all necessary settings, then click"REC" button start recording Skype video/audio. this article usually means that should you chance to be having a version which is not supported by the other Skype call recorders, the talk helper will probably come in handy personally. Additionally, it records into Skype, which means which in fact means it will capture all the frames of this video telephone of the recording window you are currently using to capture calls. Create trendy mood notes – Having a recorder that enables you to incorporate memos or short notes is a cool item, this indicates you can include your recorded telephone and any wording and get it memorable.
With this method, talk assistant will begin instantly you start Windows which can allow it to be simpler for one to record Skype calls. Furthermore, saving your documented Skype calls is achievable only when your computer camera can record high quality pictures! Saving your recorded Skype calls is achievable when your computer camera could record premium quality videos! Once you connect Skype and talk helper, you will find a way to mechanically list any outgoing or incoming calls. This program opens automatically the moment OS is booted and it'll begin its own job of recording both the Skype conversations and audio without restrictions attached. It's a simple interface that will enable you Document each of side, Record other side only and to record Skype conversations in every angles such as Record sound only. Maintaining your Skype calls is achievable only if your computer camera may capture superb high quality pictures!


Pictured above, the recent Broadway show, "Newsies," about the Newsboys Strike of 1899 in New York City.
Excuse us for living, but some of us grew up in the heyday of the American newspaper industry in the 1940's! And the growing population helped circulation numbers increase until the 1970's. Those circulation levels remained stable until the 1990's when the numbers began to decline. The toll of radio, television, and more recently the internet were too much for the printed word. The downward trend of circulation numbers and advertisement revenues have been undoubtedly devastating to the print media and its publishers, not only newspapers but news magazines as well. However, maybe there is a "Catch-22," a self-fulfilling prophecy, in this dilemma. Just maybe this downslide can be offset by the newspapers and news magazines not giving up. They must rediscover their mission of providing diversified information to educate the public-at-large.
Circulation of daily newspapers: 62 million, 1990, down to 43 million, 2010 (Sunday circulation about the same).
Advertising revenues of newspapers: $46 million, 2003, down to $24 million, 2011.
Number of daily newspapers: 1,611, 1990. reduced to 1,387, 2009.
Circulation of news magazines: down -1%, 2008, -2.2%, 2009, -1.5%, 2010.
Advertising revenues of news magazines: down -12%, 2008, -25%, 2009, & "stopped the bleeding" flat at -.1%, 2010.
Hand-in-hand with this steady decline in the print circulation and the accompanying drop in advertising revenues has been the transition to and increase in digital journalism. This internet news and analysis not only includes websites and their blogs such as huffingtonpost.com, but also newspaper and news magazine on-line versions/editions while they continue their hard-copy publications as well.
But the salutary point here is that the newspapers and news magazines themselves do not make the situation/the trend any better by their actions. In fact, often their hardcopy editions and on-line versions resemble one-another, each trying to look like the other! The "Catch-22," the self-fulfilling prophecy compounding the problem, goes something like this: circulation decreases, resulting in lower advertising revenues, causing a downsizing of the number of pages in the periodical. Then since the publication is smaller and offers less information, the readership declines, revenues continue to drop, the number of pages shrink, and the cycle down-spirals.
Newsweek, a striking example of this collateral damage, saw its demise hastened. Then Editor Jon Meacham attempted in May 2009 to make Newsweek read more like an intellectual journal. After 77 years, 1933-2010, it was sold for $1.00 & assumption of $50 million in liabilities by the Washington Post Company in 2010. Newsweek ceased as a print publication December 31, 2012 & put out its first digital only issue January 4, 2013, merging with thedailybeast.com.
Now just maybe this downslide could be slowed, stopped, if not reversed by newspapers and news magazines hanging tough. Don't "cut and run," but rediscover yourself and how it all began. Publications used to be the source of information which broadened the horizons of the reader, i.e., educated its readership. Newspapers and news magazines used to be mini-versions of The New York Times on Sundays with its various sections from news to the arts, books, business, sports and the like. – – Kind of a daily or weekly Cultural Literacy, a book by E. D. Hirsch, 1988, in which the author describes and provides the basic knowledge, background information, that enables a person to function in our contemporary American society.
Instead, what we have is dumbed-down editions which assume the reader gets his or her news and information elsewhere. And guess what? The self-fulfilling prophecy is they do! People go to their favorite TV cable program or website that tells them exactly what they wish to hear and know in their niche of information.
Never mind that people just might learn something, gain something, by being exposed to a broad spectrum of information that they would peruse on their way to that topic they are looking for. Indeed, they might stop and read something from another field beside their own and learn something, perhaps, about art, history, literature, etc. It may even give them a new idea that they can use in their respective fields of work.
Excuse us for living, but that's what the newspaper, the news magazine, used to do for us. The mission was to provide diversified information to educate us. Only a citizenry armed with sufficient information on a wide range of topics can take a look at our American past, deal with the present, and address the problems of our common future.
Comments: Please, besides the fact that the below sources are mostly on-line and that this is an on-line blog and not a newspaper column!!!
Sources: Wikipedia.org, multimedia.journalism.berkely.edu, stateofthemedia.org, Cultural Literacy: What Every American Needs To Know, 1988, by E. D. Hirsch, Jr.
Call me old fashioned, but I continue to receive the local newspaper every day all week. Local news, local sports and such are easily readable on paper, and faster to skip topics, plus more local ads to see. I can see how national news is immediately available on satellite radio and on TV cable service. Why would us elderlies spend more time on the PC screen reading news, when we (meeself) already spend so much time with email, activity coordinations, and even purchasing power and online banking.
I read everything on the internet. I am so happy that we are no longer destroying trees to print news papers. The internet gives me access to the NY Times, Washington Post,, LA Times, Chicago Tribune, and every .magazine printed. There are blogger and all kinds of media that is available. I find that the news business has become so partisan that to get the real story you need to read a balanced variety to get the real news.
I love the internet–I love research–I love the immediate answers to questions that would take days to research.
But my eyes get tired reading from the computer.
It is a cold, heavy object and very proper. (the phones are too small and not very restful).
The newspaper is fluid, messy and warm with many surprises. I take sections and travel with them. I am a slow reader so it works for me.
I forgot to check the comment box……..And have a great day.
I agree that the newspapers are dumbed down; take our Orlando sentinel for instance. I love a good personal interest story, but it seems that the Sentinel has outdone itself with these stories. Not enough news. Even when my Michigan cousin comes to visit he says our newspaper is not very "newsy".
I do remember "before the internet" how I loved to sit down after putting the kids to bed and would read the paper from cover to cover. Now I get on the 'puter' or read a book….
So that is my story and I am sticking to it…lol….
Like a lot of the others who have left comments, I much prefer reading from print instead of on screen. I print most stories online that I feel are important and want to absorb and then read at my convenience. I can read print a lot faster with better comprehension and retainment than the same article on screen. Plus I would really miss the daily page of comics.
First, the newsprint itself has become so thin and flimsy that if you open the paper and then lay it on a table, the corners curl up so badly that you have to hold down those portions of the page you want to read.
Second, they only cover the preps (high school) activities of schools in the Boulder valley. I have to go to the Denver Post to find out what happened in the Denver area and other parts of the state.
a) I am absolutely dumbfounded about the small number of people who understand the difference between "your" (as in your cat) and you're (as in you're [you are] going to school today). There is even a small difference in the pronunciations, with your having a bit more of an "o" sound, and you're having a bit of an "e" sound.
b) Another is usage of less and fewer – there is less water in the glass after I drank some, and there are fewer pieces of toast on the platter since I've taken two slices. Simply – few(er) has to do with a small number of countable items, where less has to do change in measurement such as volume.
c) Bring and take. Bring the ball to me. Now take it to Harry. People just don't seem to use "bring" anymore.
d) Good and fine – the nun does good work and I feel fine. When someone asks, "How do you feel?", the answer should not be, "I'm good" but "I'm fine." The work that I do is good or bad, but not fine.
There are others but these examples show what I was trying to point out – that todays newspaper are not edited as closely as they used to be. Part of the blame for this is with the internet and texting as a way of life (I'm not there yet). With the effort being about getting the jist of the message sent as quickly as possible, and with the reliance on "auto-correct", people just don't care about usage. Therefore the newspapers, and weekly periodicals, should be more vigilant so that proper usage is not lost.
Thank you for your comment & kind words of compliment. I am thrilled that the content of my article on the decline of newspapers was substantive enough to be of use to your research. It would have been better for you if my sources listed at the end were footnoted to the information above. I hope the sources provide you with further leads for your research & writing. Best wishes for success in your work. Feel free to further communicate with me at fontana14@optonline.net in an e-mail.
Those numbers include the U-T San Diego, which, sadly, has not been a newspaper since Doug Manchester bought it in December 2011. He has turned it into his personal toy with whole sections of the paper (it still is a paper, just not a newspaper in the traditional sense of the word) devoted to God and Jesus Christ, happenings with his family (marriages, deaths, births, vacations, sightings about town………gag), diatribes against the sick, the poor, the homeless…….. I finally had to cancel my subscription in August 2012. I gave him a chance but I really don't need daily information about him and his family.


text:
019 Trails-Jesse Chisholm-img006
Published August 23, 2020 at 1170 × 730 in Short Stories of Harvey County, Kansas – A Series of Pictorial Histories – Unit 3B of 8
(I) Eastern, Abilene and Chisholm Trail System of 1867-1889 – A Cattle Trail. Within today's Harvey County, a prominent cattle trail came from the south. It skirted the west edge of today's city of Sedgwick. The trail paralleled both sides of Sand Creek and went through today's Newton. Going north from there, the route was in close proximity to today's Hwy K-15 as it continued to Abilene, KS. In the 1860s and 1870s, the drovers and others involved in the cattle business referred to this route as "The Trail," the "Abilene Trail," the "Kansas Trail," the "Northern Trail" or the "Texas Trail" when they spoke about trailing cattle through central Kansas. Formally mapped as the Eastern Trail, Abilene Trail and Chisholm's Trail, this trail system was a linking of several existing trails. Using the old Shawnee Trail (Eastern in Texas) from south Texas to Waco, it went up the western fork, the Shawnee – Arbuckle Trail, to go through Ft. Worth and across the Red River to the (south) Canadian River in Indian Territory. It utilized an old freight route, Chisholm's Wagon Road, between Fort Arbuckle, the N. Canadian River and the Cimarron River. The Black Beaver/Col. Emory Trail was used from there to Wichita. The Wichita to Abilene leg paralleled much of the six year old Black Beaver/Emory Trail. Joseph McCoy directed Timothy Hersey and a crew to construct mounds of dirt to guide drovers from Wichita to Abilene. By following the common path used by the expeditions of Cpt. Marcy/Black Beaver in 1849, Maj. Hamilton W. Merrill in 1855 (Fort Belknap, Texas to the mouth of the Little Arkansas River), Col. Emory in 1861 and Jesse Chisholm's wagon route, the first herd to Abilene plus all the subsequent herds followed an existing trail to get to Hersey's 1867 route. Jesse Chisholm (1805 or 1806—1868). Photo credit: https://www.kansasmemory.org/item/25070. 22 Mar 2020.


DBRS on Dec. 22 confirmed all ratings of Bank of Hawaii Corp. and unit Bank of Hawaii, reflecting the company's "strong and defensible banking franchise" particularly within Hawaii, and its "solid earnings generation capacity and strong balance sheet fundamentals."
Confirmed were the company's issuer and senior debt rating of A (low), subordinated debt rating of BBB (high) and short-term instruments rating of R-1 (low). The bank's A deposits and senior debt rating, A (low) subordinated debt rating and R-1 (low) short-term instruments rating were also confirmed.
The trend for all ratings is still stable.


Editor's Introduction: For over a decade I worked with Canadian Power company clients, oil and gas associations and governments to promote the wise use of energy in all its forms. These assignments took myself, my two sons and other staff from Long Beach, CA in the Southwest to Quebec City in the Northeast. Along the way we received the US Industrial Film Festival's Creative Excellence Award twice for our efforts. It was all a grand education, and we're still learning.
News Release: BC Hydro and Calpine Canada Power Holdings Ltd. (Calpine) are proposing to develop a$300 million, 265 MW (295 MW with duct firing) high efficiency combined-cycle, natural gas-fired electricity generation facility. The preferred site is in the Port Alberni Industrial Park at the end of Tebo Avenue. The proposed project will utilize state-of-the-art turbine and emission control technology, and be one of the cleanest facilities in North America. Calpine, based in Calgary, Alberta, is a subsidiary of Calpine Corporation of San Jose, California, one of the leading independent power producers in the US. Calpine is currently developing the Calgary Energy Centre, a 250 MW combined-cycle, natural gas-fired facility near Calgary.
Why is this project being developed?
This project is being developed to meet the anticipated demand for power on Vancouver Island in the winter of 2003/2004. That demand is projected to grow at a rate of about 35 MW per annum. It will also help to replace electricity that is currently being delivered to Vancouver Island via submarine cables, some of which are deteriorating and will be decommissioned in stages between now and 2007.
BC Hydro is shopping for buyers for three of its operations.
October 2001: BC Hydro, a Crown agency has reported that it's looking for joint partnerships with the private sector in its customer and fleet services divisions, and Westech, Hydro's information technology branch. More to come.


"Take a floating teashop-ship, staffed by would-be pirates, a wicked professor, and a parrot named Toothpick....." Another Mahy pirate tale!
We try to keep Moomins in stock and at the time of this writing have two others, less expensive. Use the Search.
Miss Rivers and Miss Bridges.
During their London holiday, the two girls transform themselves into the matronly Suffragettes, Miss Rivers and Miss Bridges. Full of hilarious adventures, as Atlanta's clever plots and Pansy's sweetness see them through. NOTE: This Book has been sold.
The Shy Stegosaurus of Cricket Creek.
Paji ran away from his elephant carving job at his uncle's Big Cheap Elephant Factory. He and his dreamy bullock friend Push-ba ran to an ancient dagoba and there Paji made beautiful carvings, each different. A charming story in fluid three colour lithos.


text:
Q: Verb tense after the word tommorow Would be grateful for any advice please.
I am attempting to write a novel (not for publication but for personal satisfaction only).
I am now running into grammar problems I have never even thought about before!
Here is the sentence that is causing my current confusion!

"Tomorrow, when they made their vows in front of family and friends, both old and new, it would all be part of an ultimate grand design."

I am confused about the word "made" being correct or not. As "tomorrow" is in the future maybe it should be "make" ?
Any help would be much appreciated.
Many thanks

A: As @BobaFit said in a comment, "The question is, where is the speaker relative to the events?".
Context always matters, and to have an easy to read and understandable style, one must maintain a consistent point of view.

As he sat in the waiting room, he thought about the significant events they had shared.
He remembered how nervous they had felt on that plane flight, not because of any fear of flying, but because of why they were flying.
Tomorrow, when they made their vows in front of family and friends, both old and new, it would all be part of an ultimate grand design.
Then, at the wedding itself they had … .

That paragraph contains your sentence as you wrote it, and it fits well.
It shows their point of view as seen from his point of view.
But, in other contexts, from other points of view, it would be wrong and confusing.

redpajama_set_name: RedPajamaStackExchange


I am a professional visual artist, founder and team leader at Faces Up Uganda, an NGO that empowers young people (10-25 years) in mentorship through creative arts to instil the 5Cs; Creativity, Critical thinking, Confidence, Consciousness and Contentment to reach their full potential. Joining the YALI Regional Leadership Center East Africa was something I had anticipated for a while. I had heard about the great work YALI RLC EA was doing in empowering young leaders all over Africa and being a civic leader optimistic about the future of Africa, it was an absolute delight to be there. I was impressed and at the same time positively challenged when I learned about the great work other young leaders were doing in their respective countries, which created a yearning for more of such stories. There was a great vibe in between activities at the Center; whether during or after sessions with a variety of cultural music tunes playing in the background in the process exhibiting the rich diversity Africa is blessed with. The Design Thinking sessions were my favorite, probably because the work I do is inspired by creativity, human-centred and seeks to address the challenges in the education system in Africa from simple, appropriate and practical angles. Seven Habits of Highly Effective People stood out for me too and I apply the principles every day.
YALI RLC EA imparted in me the spirit of networking and teamwork to support other people and so far, I have managed to collaborate with others in my cohort. I have worked with Eleanor Odhiambo Sharon (Cohort 2) to initiate an arts activity at Kenyatta National Hospital in the cancer children's ward. I drafted her art activity work plans which she uses as a guideline to execute art activities. She has successfully conducted one and I was really happy to see the smiles from the children when she sent me photos. I have written another activity plan, which she will be executing soon and can't wait to hear of the results.
Another great connection was with Emmanuel Achapah (Cohort 27) with whom we have mutual interest in creative work and humanitarian work. . He offered to redesign Faces Up website and printing of message cards designed by the kids that we support.
I have also done work for alumni; Shain Kazibwe (Uganda), John Froggy Ajingdit (Sudan), Karangi Brian (Tanzania), and Zachuas Ogonji (Kenya) to design logos and posters for their organisations in support of the great work they are already doing. I also often meet alumna, Dorothy Nabakooza (Uganda) to brainstorm ideas for her initiative that seeks to address bullying amongst young people in schools and other communities. I have also engaged many more in different capacities like through Faces Talks, one of our projects where we host various speakers on our Twitter page where they tell their impact stories to the World. It is always uplifting when fellow alumni get in touch with me and ask, "How are the children? How is your project going?"
After the YALI RLC EA program, I was shortlisted for an award competition called "What's your solution" by Sawa World International Uganda. Although we didn't win, it was another great platform for us to create awareness about our project and learn what other people are doing in a bid to transform Africa. I was also invited as a mentor into a mentorship program called Sauti Ya Sasa which seeks to build capacity for young people to improve their civic engagement in Uganda. It is implemented by Action For Development which is funded by GIZ and co-founded by European Union in Uganda.
Juliana Warija, 11 years old: Being a young refugee girl from South Sudan has not stopped her from following her dream of becoming a great footballer and runway model. We identified her interest of being a football player and model within three months during our engagement with her. We supported her by enrolling her in to a football academy where she now trains and we are always on ground to cheer her up and also transport her back to where she lives. World Bank got interested in her and donated three footballs to support her dream, and also featured her story on their website so that her story can inspire other refugees to follow their dreams. Daily Monitor and New Vision, major newspapers in Uganda also wrote about her.
Kato Katongole Osman, 12 years old: We first engaged Kato in 2016. He aspires to be an international hip-hop dancer and trainer who looks up to Abrahams Tekya a professional dance instructor and Reneus Bbosa, an award winning dancer. Our excitement here is that we have managed to link him up to both of them, whom he now trains with and gets supported by. In July this year, we helped him record a documentary with Deutsche Welle TV a media company from Germany in a documentary called #77 Percent which was well received. He has gone ahead to perform at various platforms including the the Break Fast Jam annual dance competition on November 2018.
For more information about Faces Up, check their website facesup.org.


Want to live longer, stay healthy and be happy? By teaching you how to change old patterns and find healing, we can help you to a happier healthier life promoting well-being.
Our programs run from our facilities in the Southern Highlands, allowing you to focus on your own life without distraction and learn valuable life skills and strategies for healing and peace.
Practical strategies and techniques that help you live well with mental anguish, cancer and other illnesses and post-trauma stress.
Quest for Life has assisted over 120,000 people with grief, depression, anxiety, PTSD, cancer, chronic pain and other illnesses.
Help someone to blossom. Often, those most in need of our life changing programs are the least able to afford them. Your donation makes our programs accessible to more people.
"We call them life's Ds. Whether it's a diagnosis, divorce, death, depression or disaster everyone faces a significant challenge at some point.
Quest for Life can make a profound and positive difference in your life. We take a whole-person, recovery orientated approach which provides evidence-based practical strategies and tools to help you to heal the past, build resilience for the future and live in the present.
You will be nurtured, nourished and restored with the confidence and skills to embrace the world again."
Would you like Petrea to speak at your event?
Petrea King is at the forefront of wellness education and is a frequent lecturer at medical and other conferences. Petrea sees crisis as a catalyst for personal growth and understanding and as an opportunity for healing and peace. She is a valued speaker, lecturer and facilitator of workshops.


text:
\section{Introduction}
Reachability is an important property that describes the behavior of control systems.
In a classic reachability problem, one
specifies a target set in the state space and then aims to find a set of initial states of the trajectories that can reach the target set
within a given time horizon \cite{a1,a2,a2.5}. Such a set is referred to as the reachable set.
Such problems are of great interest in engineering, for example, in those involving feasibility \cite{a3} or safety \cite{a4},
where feasible or safe system states sometimes refer to those states that can reach the target set in a given duration.

So far, most researches on reachability are based on deterministic and exact modeling of the system,
and they discuss a "yes" or "no" problem,
i.e. whether it is possible to reach the target set in a given time starting from a certain set.
Various methods have been proposed to deal with such problems, including the ellipsoidal method \cite{a2.5,a5,a6},
the polyhedral method \cite{i1,a7}, and the level set method \cite{a1,a2,i2}.
The level set method requires less form for dynamical systems and can be used to solve nonlinear problems.
Moreover, several mature toolboxes have been developed based on the level set method \cite{a8,a9,a10}, therefore,
the level set method has become the most widely used method and is
applied in a large number of fields such as flight control systems \cite{a13,a14},
ground traffic management systems \cite{i3,a12} and air traffic management systems \cite{a11,a15}.
In recent years, time-varying target sets and obstacles have been considered in several studies on the reachability of
deterministic systems, allowing reachability analysis to be used for more complex engineering problems.
For instance, a time-varying obstacle can be regarded as a time-varying unsafe region outside of which the evolution trajectory
of the system should be kept \cite{i4,a15,a16}.

However, in some engineering problems, it is quite difficult to model the system accurately.
In these problems, due to the uncertainty of the systems themselves and external disturbances,
the state transitions of the systems are no longer deterministic, but take the form of probability distributions.
In such cases, the reachability problem can no longer be simply answered by "yes" or "no".
Although some studies have discussed the reachability of nondeterministic systems and
proposed the definition of probabilistic reachable set \cite{a17,a18,a19},
these studies assume that the target set is time-invariant and do not take into account obstacles.

Motivated by the previous works mentioned above, the following contributions are made in this paper:
\begin{itemize}
    \item [(1)] In this paper, we adopt a discrete time point of view to study reachability problems of nondeterministic systems,
    in which we consider time-varying target sets and obstacles, and refine the definition of probabilistic reachable sets.
    \item [(2)] A method based on recursion and grid interpolation is proposed for computing probabilistic reachable sets. In this method,
    the probabilistic reachable set is represented as a level set of a scalar function approximated by grid interpolation.
\end{itemize}

This paper unfolds as follows: Section 2 mathematically describes the problem.
The principle of the method for computing probabilistic reachable sets is presented in Section 3.
Section 4 introduces the implementation of the method.
A numerical examples are given in Section 5 to illustrate the validity of the proposed method.
The results are summarized in Section 6.

\section{Problem Statement}
Consider the following discrete-time control system with uncertainty:
\begin{align}\label{eq1}
    s_{k+1}\sim \mathcal{F}\left(.|s_k,u_k\right)
\end{align}
where $s_k\in \mathbb{R}^n$ and $u_k$ are the system state ant control input at time $k$, respectively,
and $u_k$ is selected from a prescribed set $\mathcal{U}$.
$\mathcal{F}\left(.|s_k,u_k\right)$ denotes a probability distribution which is related to $s_k$ and $u_k$,
and for any $s\in\mathbb{R}^n$, $\mathcal{F}\left(s|s_k,u_k\right)$ represents the probability density of
this probability distribution at $s$.

\begin{definition}[Control policy]\label{def1}
    The sequence of control inputs $(u_0,u_1,...)$ can be determined by a control policy which can be represented by a mapping
    $\mathcal{E}(.,.):\mathbb{R}^n\times \mathbb{N}\to \mathcal{U}$, and $u_k=\mathcal{E}(s_k,k)$.
\end{definition}

Let $\mathbb{Z}_{k_1}^{k_2}$ ($k_1$, $k_2$ are integers and $k_2\geq k_1$) denote the set of all integers between $k_1$ and $k_2$,
i.e., $\mathbb{Z}_{k_1}^{k_2}=\{k_1,k_1+1,...,k_2\}$ and let $\mathcal{D}$ denote the set of all probability distributions on $\mathbb{R}^n$.
Then, given the state $s_{k_1}$ at time $k_1$ and control policy $\mathcal{E}(.,.)$, the evolution of system
(\ref{eq1}) in $\mathbb{Z}_{k_1}^{k_2}$ can be expressed as a probability distribution over time
$\phi_{k_1}^{k_2}\left(.|.,s_{k_1},\mathcal{E}\right):\mathbb{Z}_{k_1}^{k_2}\to \mathcal{D}$, and
$\phi_{k_1}^{k_2}\left(.|k,s_{k_1},\mathcal{E}\right)$ denotes the distribution of the system state at time $k$,
while $\phi_{k_1}^{k_2}\left(s|k,s_{k_1},\mathcal{E}\right)$ denotes the probability density of the system state reaching state $s$ at time $k$.
Denote by $\mathcal{A}_k$ and $\mathcal{B}_k$ the target set and the obstacle at time $k$, respectively. Consequently,
given a time horizon $T\in\mathbb{N}$ and a control policy $\mathcal{E}(.,.)$,
The evolution of system (\ref{eq1}) initializing from any state $s_0\in \mathbb{R}^n$
has a probability of reaching the target set at some time $k\in \mathbb{Z}_{0}^T$ and avoiding obstacles before
reaching the target set, and this probability is expressed mathematically as:
\begin{align}\label{eq2}
    \mathrm{P}\left(s_0|{T},{\mathcal{E}}\right)=
    \mathrm{P}\left[ \left(\exists k\in\mathbb{Z}_0^T,
            \phi_0^T(.|k,s_0,\mathcal{E} )\in \mathcal{A}_k \right)
            \land
            \left(\forall l\in \mathbb{Z}_0^k, \phi_0^T(.|l,s_0,\mathcal{E} )\notin \mathcal{B}_l \right) \right]
\end{align}
where "$\land$" is the logical operator "AND".
Then, given a positive real number $\gamma\in [0,1]$, the probabilistic reachable set can be defined:
\begin{definition}[$\gamma-$probabilistic reachable set]\label{def2}
    \begin{align}\label{eq3}
        \mathcal{R}_\gamma^{\mathcal{E}}(\mathcal{A},\mathcal{B},T)=\left\{ s_0\in\mathbb{R}^n|\mathrm{P}\left(s_0|{T},{\mathcal{E}}\right)\geq \gamma   \right\}
    \end{align}
\end{definition}
It is clear that the value of the expression of the probability in Eq. (\ref{eq2}) depends on the control policy adopted.
By finding the optimal control policy one can maximize this probability and the maximum probabilistic reachable set can be defined:
\begin{definition}[Maximum $\gamma-$probabilistic reachable set]\label{def3}
    \begin{align}\label{eq4}
        \mathcal{R}_\gamma^{\mathrm{max}}(\mathcal{A},\mathcal{B},T)=\left\{ s_0\in\mathbb{R}^n|\max_{\mathcal{E}} \mathrm{P}\left(s_0|{T},{\mathcal{E}}\right)\geq \gamma   \right\}
    \end{align}
\end{definition}
In the next section, we show that $\mathcal{R}_\gamma^{\mathcal{E}}(\mathcal{A},\mathcal{B},T)$ and
$\mathcal{R}_\gamma^{\mathrm{max}}(\mathcal{A},\mathcal{B},T)$ can be computed by a backward recursive procedure.

\section{Method to compute probabilistic reachable sets}
Let $\mathbb{I}_{\mathcal{A}_k}(.):\mathbb{R}^n\to \{0,1\}$ and $\mathbb{O}_{\mathcal{B}_k}(.):\mathbb{R}^n\to \{0,1\}$
be the indicator functions of target set $\mathcal{A}_k$ and obstacle $\mathcal{B}_k$, respectively, i.e.:
\begin{align}\label{eq5}
  \mathbb{I}_{\mathcal{A}_k}(s)=\begin{cases}
    1,s\in \mathcal{A}_k\\
    0,s\notin \mathcal{A}_k
  \end{cases}\quad \quad
  \mathbb{O}_{\mathcal{B}_k}(s)=\begin{cases}
    0,s\in \mathcal{B}_k\\
    1,s\notin \mathcal{B}_k
  \end{cases}
\end{align}
Then the following theorem can be proved.

\begin{thm}\label{thm1}
    Fix a control policy $\mathcal{E}(.,.)$ and let the scalar function $V_k^{\mathcal{E}}(s_k)$
    denote the probability that the evolution of the system from state $s_k$ at time $k$
    reaches the target set at some time in $\mathbb{Z}_k^T$ and avoids the obstacle, i.e.:
    \begin{align}
        \begin{split}
        V_k^{\mathcal{E}}(s_{k})=  \mathrm{P} \left[ \left(\exists k_1\in \mathbb{Z}_k^T,\phi_k^T(.|k_1,s_k,\mathcal{E})\in \mathcal{A}_{k_1}\right) \land
        \left(\forall k_2\in \mathbb{Z}_k^{k_1},\phi_k^{k_1}(.|k_2,s_k,\mathcal{E})\notin \mathcal{B}_{k_2}\right)  \right]
        \end{split}
    \end{align}
    If
    \begin{align}
        \begin{split}
            V_{k-1}^{\mathcal{E}}(s_{k-1})=\min \left\{\mathbb{O}_{\mathcal{B}_{k-1}}(s_{k-1}),
            \max\left\{\mathbb{I}_{\mathcal{A}_{k-1}}(s_{k-1}),
            \mathbb{E}\left[ V_k^{\mathcal{E}}\left( \mathcal{F}\left(.|s_{k-1},\mathcal{E}(s_{k-1},k-1) \right) \right)\right]  \right\} \right\}
        \end{split}
    \end{align}
    Then
    \begin{align}\label{eq8}
        \begin{split}
        V_{k-1}^{\mathcal{E}}(s_{k-1})=  \mathrm{P} \left[ \left(\exists k_1\in \mathbb{Z}_{k-1}^T,
        \phi_{k-1}^T(.|k_1,s_{k-1},\mathcal{E})\in \mathcal{A}_{k_1}\right) \land
        \left(\forall k_2\in \mathbb{Z}_{k-1}^{k_1},\phi_{k-1}^{k_1}(.|k_2,s_{k-1},\mathcal{E})\notin \mathcal{B}_{k_2}\right)  \right]
        \end{split}
    \end{align}
\end{thm}

\noindent
\textbf{Proof.} The event
\begin{align}
    \begin{split}
        \left(\exists k_1\in \mathbb{Z}_{k-1}^T,
        \phi_{k-1}^T(.|k_1,s_{k-1},\mathcal{E})\in \mathcal{A}_{k_1}\right) \land
        \left(\forall k_2\in \mathbb{Z}_{k-1}^{k_1},\phi_{k-1}^{k_1}(.|k_2,s_{k-1},\mathcal{E})\notin \mathcal{B}_{k_2}\right)
    \end{split}
\end{align}
and the event
\begin{align}
    \begin{split}
        &\left\{ \left( s_{k-1}\in \mathcal{A}_{k-1} \right)\lor
        \left[ \left(\exists k_1\in \mathbb{Z}_{k}^T,\phi_k^T(.|k_1,s_{k-1},\mathcal{E})\in \mathcal{A}_{k_1}\right) \land
        \left(\forall k_2\in \mathbb{Z}_k^{k_1},\phi_{k-1}^{k_1}(.|k_2,s_{k-1},\mathcal{E})\notin \mathcal{B}_{k_2}\right)  \right] \right\}  \\
        &\land \left( s_{k-1}\notin \mathcal{B}_{k-1} \right)
    \end{split}
\end{align}
are equivalent, where "$\lor$" is the logical operator "OR".
For the sake of brevity, we let
\begin{align}\label{eq11}
    \begin{split}
        &e_0^{\mathcal{E}}=\left(\exists k_1\in \mathbb{Z}_{k-1}^T,
        \phi_{k-1}^T(.|k_1,s_{k-1},\mathcal{E})\in \mathcal{A}_{k_1}\right) \land
        \left(\forall k_2\in \mathbb{Z}_{k-1}^{k_1},\phi_{k-1}^{k_1}(.|k_2,s_{k-1},\mathcal{E})\notin \mathcal{B}_{k_2}\right)\\
        &e_1= s_{k-1}\in \mathcal{A}_{k-1}\\
        &e_2^{\mathcal{E}}= \left(\exists k_1\in \mathbb{Z}_{k}^T,\phi_{k-1}^T(.|k_1,s_{k-1},\mathcal{E})\in \mathcal{A}_{k_1}\right) \land
        \left(\forall k_2\in \mathbb{Z}_k^{k_1},\phi_{k-1}^{k_1}(.|k_2,s_{k-1},\mathcal{E})\notin \mathcal{B}_{k_2}\right)\\
        &e_3=s_{k-1}\notin \mathcal{B}_{k-1}
    \end{split}
\end{align}
Therefore, $V_{k-1}^{\mathcal{E}}(s_{k-1})$ can be expanded as follows:
\begin{align}
    \begin{split}
        V_{k-1}^{\mathcal{E}}(s_{k-1})=\mathrm{P} \left(e_0^{\mathcal{E}}\right)
        = \left\{ 1- \left[ 1-\mathrm{P}\left( e_1 \right) \right] \times
        \left[1-\mathrm{P}\left( e_2^{\mathcal{E}}  \right) \right] \right\}\times
        \mathrm{P}\left( e_3 \right)
    \end{split}
\end{align}
According to the following equation, the state space can be divided into three parts:
\begin{align}
    \mathbb{R}^n = \mathcal{B}_{k-1} \cup \left( \mathcal{A}_{k-1}-\mathcal{B}_{k-1} \right) \cup \complement_{\mathbb{R}^n} \left( \mathcal{A}_{k-1}\cup\mathcal{B}_{k-1} \right)
\end{align}
where $\mathcal{A}_{k-1}-\mathcal{B}_{k-1}=\left\{s\in\mathbb{R}^n|s\in \mathcal{A}_{k-1} \land s\notin \mathcal{B}_{k-1} \right\}$.
These three parts are discussed separately below.

\noindent
\textbf{Case 1.} $s_{k-1}\in \mathcal{B}_{k-1}$:
\begin{align}
    \begin{split}
    &s_{k-1}\in \mathcal{B}_{k-1}\Longrightarrow \mathrm{P}\left( e_3 \right)=0 \Longrightarrow
    \mathrm{P} \left(e_0^{\mathcal{E}}\right)=0
    \end{split}
\end{align}
and
\begin{align}
    \begin{split}
        &s_{k-1}\in \mathcal{B}_{k-1}\Longrightarrow \mathbb{O}_{\mathcal{B}_{k-1}}(s_{k-1})=0 \Longrightarrow\\
        &\mathbb{O}_{\mathcal{B}_{k-1}}(s_{k-1}) \leq \mathbb{I}_{\mathcal{A}_{k-1}}(s_{k-1}) \land
        \mathbb{O}_{\mathcal{B}_{k-1}}(s_{k-1}) \leq\mathbb{E}\left[ V_k^{\mathcal{E}}\left( \mathcal{F}\left(.|s_{k-1},
        \mathcal{E}(s_{k-1},k-1) \right) \right)\right] \Longrightarrow \\
        &\min \left\{\mathbb{O}_{\mathcal{B}_{k-1}}(s_{k-1}),
        \max\left\{\mathbb{I}_{\mathcal{A}_{k-1}}(s_{k-1}),
        \mathbb{E}\left[ V_k^{\mathcal{E}}\left( \mathcal{F}\left(.|s_{k-1},\mathcal{E}(s_{k-1},k-1) \right) \right)\right]  \right\} \right\}=0
    \end{split}
\end{align}
Therefore, Eq. (\ref{eq8}) holds when $s_{k-1}\in \mathcal{B}_{k-1}$.

\noindent
\textbf{Case 2.} $s_{k-1}\in \mathcal{A}_{k-1}- \mathcal{B}_{k-1}$:
\begin{align}
    \begin{split}
    &s_{k-1}\in \mathcal{A}_{k-1}-\mathcal{B}_{k-1}\Longrightarrow \mathrm{P}\left( e_1 \right)=1 \land \mathrm{P}\left( e_3 \right)=1 \Longrightarrow
    \mathrm{P} \left(e_0^{\mathcal{E}}\right)=1
    \end{split}
\end{align}
and
\begin{align}
    \begin{split}
        &s_{k-1}\in \mathcal{A}_{k-1}- \mathcal{B}_{k-1}\Longrightarrow \mathbb{I}_{\mathcal{A}_{k-1}}(s_{k-1})=1 \land
        \mathbb{O}_{\mathcal{B}_{k-1}}(s_{k-1})=1 \Longrightarrow\\
        &\mathbb{O}_{\mathcal{B}_{k-1}}(s_{k-1}) = \mathbb{I}_{\mathcal{A}_{k-1}}(s_{k-1})
         \geq\mathbb{E}\left[ V_k^{\mathcal{E}}\left( \mathcal{F}\left(.|s_{k-1},
        \mathcal{E}(s_{k-1},k-1) \right) \right)\right] \Longrightarrow \\
        &\min \left\{\mathbb{O}_{\mathcal{B}_{k-1}}(s_{k-1}),
        \max\left\{\mathbb{I}_{\mathcal{A}_{k-1}}(s_{k-1}),
        \mathbb{E}\left[ V_k^{\mathcal{E}}\left( \mathcal{F}\left(.|s_{k-1},\mathcal{E}(s_{k-1},k-1) \right) \right)\right]  \right\} \right\}=1
    \end{split}
\end{align}
Therefore, Eq. (\ref{eq8}) holds when $s_{k-1}\in \mathcal{A}_{k-1}- \mathcal{B}_{k-1}$.

\noindent
\textbf{Case 3.} $s_{k-1}\notin \mathcal{A}_{k-1} \cup \mathcal{B}_{k-1}$:
\begin{align}
    \begin{split}
    &s_{k-1}\notin \mathcal{A}_{k-1} \cup \mathcal{B}_{k-1}\Longrightarrow \mathrm{P}\left( e_1 \right)=0 \land \mathrm{P}\left( e_3 \right)=1 \Longrightarrow
    \mathrm{P} \left(e_0^{\mathcal{E}}  \right)=\mathrm{P} \left(e_2^{\mathcal{E}}  \right)\\
        &=\mathrm{P}\left[ \left. \left(\exists k_1\in \mathbb{Z}_k^T,\phi_k^T(.|k_1,s_k,\mathcal{E})\in \mathcal{A}_{k_1}\right) \land
        \left(\forall k_2\in \mathbb{Z}_k^{k_1},\phi_k^{k_1}(.|k_2,s_k,\mathcal{E})\notin \mathcal{B}_{k_2}\right) \right|
        s_k\sim \mathcal{F}\left(.|s_{k-1},u_{k-1}\right)  \right]\\
        &=\int_{\mathbb{R}^n} V_k^{\mathcal{E}}(s)\mathcal{F}\left(s|s_{k-1},u_{k-1}\right)ds=\mathbb{E}\left[ V_k^{\mathcal{E}}\left( \mathcal{F}\left(.|s_{k-1},\mathcal{E}(s_{k-1},k-1) \right) \right)\right]
    \end{split}
\end{align}
and
\begin{align}
    \begin{split}
        &s_{k-1}\notin \mathcal{A}_{k-1}\cup \mathcal{B}_{k-1}\Longrightarrow \mathbb{I}_{\mathcal{A}_{k-1}}(s_{k-1})=0 \land
        \mathbb{O}_{\mathcal{B}_{k-1}}(s_{k-1})=1 \Longrightarrow\\
        &\mathbb{O}_{\mathcal{B}_{k-1}}(s_{k-1})
         \geq\mathbb{E}\left[ V_k^{\mathcal{E}}\left( \mathcal{F}\left(.|s_{k-1},
        \mathcal{E}(s_{k-1},k-1) \right) \right)\right] \geq \mathbb{I}_{\mathcal{A}_{k-1}}(s_{k-1}) \Longrightarrow \\
        &\min \left\{\mathbb{O}_{\mathcal{B}_{k-1}}(s_{k-1}),
        \max\left\{\mathbb{I}_{\mathcal{A}_{k-1}}(s_{k-1}),
        \mathbb{E}\left[ V_k^{\mathcal{E}}\left( \mathcal{F}\left(.|s_{k-1},\mathcal{E}(s_{k-1},k-1) \right) \right)\right]  \right\} \right\}\\
        &=\mathbb{E}\left[ V_k^{\mathcal{E}}\left( \mathcal{F}\left(.|s_{k-1},\mathcal{E}(s_{k-1},k-1) \right) \right)\right]
    \end{split}
\end{align}
Therefore, Eq. (\ref{eq8}) holds when $s_{k-1}\notin \mathcal{A}_{k-1}\cup\mathcal{B}_{k-1}$.

\rightline{$\square$}

Theorem \ref{thm1} reveals a recursive formula, for which the terminal condition can be derived from the following equation:
\begin{align}
    \begin{split}
        V_T^{\mathcal{E}}(s_{T})=  &\mathrm{P} \left[ \left(\exists k_1\in \mathbb{Z}_T^T,\phi_T^T(.|k_1,s_T,\mathcal{E})\in \mathcal{A}_{k_1}\right) \land
        \left(\forall k_2\in \mathbb{Z}_T^{k_1},\phi_T^{k_1}(.|k_2,s_k,\mathcal{E})\notin \mathcal{B}_{k_2}\right)  \right]\\
        =&\mathrm{P} \left( s_T\in \mathcal{A}_T \land s_T\notin \mathcal{B}_T  \right)=
        \min \left[ \mathbb{I}_{\mathcal{A}_T}\left( s_T \right),\mathbb{O}_{\mathcal{B}_T}\left( s_T \right)  \right]
    \end{split}
\end{align}
Consequently, the complete backward recursive procedure is as follows:
\begin{align}
    \begin{split}
        &V_T^{\mathcal{E}}(s_{T})=\min \left[ \mathbb{I}_{\mathcal{A}_T}\left( s_T \right),\mathbb{O}_{\mathcal{B}_T}\left( s_T \right)  \right]\\
        &V_{k-1}^{\mathcal{E}}(s_{k-1})=\min \left\{\mathbb{O}_{\mathcal{B}_{k-1}}(s_{k-1}),
        \max\left\{\mathbb{I}_{\mathcal{A}_{k-1}}(s_{k-1}),
        \mathbb{E}\left[ V_k^{\mathcal{E}}\left( \mathcal{F}\left(.|s_{k-1},\mathcal{E}(s_{k-1},k-1) \right) \right)\right]  \right\} \right\}\\
        &...\\
        &V_{0}^{\mathcal{E}}(s_0)=\left\{\mathbb{O}_{\mathcal{B}_{0}}(s_{0}),
        \max\left\{\mathbb{I}_{\mathcal{A}_{0}}(s_{0}),
        \mathbb{E}\left[ V_1^{\mathcal{E}}\left( \mathcal{F}\left(.|s_{0},\mathcal{E}(s_{0},0) \right) \right)\right]  \right\} \right\}
    \end{split}
\end{align}
According to Theorem \ref{thm1}, $V_{0}^{\mathcal{E}}(s_0)=\mathrm{P}\left(s_0|T,\mathcal{E}\right)$, and the
$\gamma-$probabilistic reachable set can be characterized by the $\gamma-$level set of $V_{0}^{\mathcal{E}}(.)$, i.e.:
\begin{align}
    \mathcal{R}_\gamma^\mathcal{E}\left(\mathcal{A},\mathcal{B},T\right)=
    \left\{s_0\in\mathbb{R}^n|V_{0}^{\mathcal{E}}(s_0)\geq\gamma \right\}
\end{align}

\begin{thm}\label{thm2}
    Let the scalar function $V_k^{\mathrm{max}}(s_k)$
    denote the probability that, under the optimal control policy, the evolution of the system from state $s_k$ at time $k$
    reaches the target set at some time in $\mathbb{Z}_k^T$ and avoids the obstacle, i.e.:
    \begin{align}
        \begin{split}
        V_k^{\mathrm{max}}(s_{k})=\max_{\mathcal{E}}  \mathrm{P} \left[ \left(\exists k_1\in \mathbb{Z}_k^T,\phi_k^T(.|k_1,s_k,\mathcal{E})\in \mathcal{A}_{k_1}\right) \land
        \left(\forall k_2\in \mathbb{Z}_k^{k_1},\phi_k^{k_1}(.|k_2,s_k,\mathcal{E})\notin \mathcal{B}_{k_2}\right)  \right]
        \end{split}
    \end{align}
    If
    \begin{align}
        \begin{split}
            V_{k-1}^{\mathrm{max}}(s_{k-1})=\min \left\{\mathbb{O}_{\mathcal{B}_{k-1}}(s_{k-1}),
            \max\left\{\mathbb{I}_{\mathcal{A}_{k-1}}(s_{k-1}),
            \max_{u_{k-1}\in\mathcal{U}} \mathbb{E}\left[ V_k^{\mathrm{max}}\left( \mathcal{F}\left(.|s_{k-1},u_{k-1} \right) \right)\right]  \right\} \right\}
        \end{split}
    \end{align}
    Then
    \begin{align}\label{eq25}
        \begin{split}
        V_{k-1}^{\mathrm{max}}(s_{k-1})= \max_{\mathcal{E}}  \mathrm{P} \left[ \left(\exists k_1\in \mathbb{Z}_{k-1}^T,
        \phi_{k-1}^T(.|k_1,s_{k-1},\mathcal{E})\in \mathcal{A}_{k_1}\right) \right. \land \\
        \left.\left(\forall k_2\in \mathbb{Z}_{k-1}^{k_1},\phi_{k-1}^{k_1}(.|k_2,s_{k-1},\mathcal{E})\notin \mathcal{B}_{k_2}\right)  \right]
        \end{split}
    \end{align}
\end{thm}

\noindent
\textbf{Proof.} The proof of Theorem \ref{thm2} is similar to that of Theorem \ref{thm1}.
Here, we continue with the notations in Eq. (\ref{eq1}).
Eq. (\ref{eq25}) can be expanded as follows:
\begin{align}
    V_{k-1}^{\mathrm{max}}(s_{k-1})=\max_{\mathcal{E}} \mathrm{P}\left(e_0^{\mathcal{E}}\right)
    =\left\{ 1- \left[ 1-\mathrm{P}\left( e_1 \right) \right] \times
    \left[1- \max_{\mathcal{E}} \mathrm{P}\left( e_2^{\mathcal{E}}  \right) \right] \right\}\times
    \mathrm{P}\left( e_3 \right)
\end{align}
Case 1 ($s_{k-1}\in \mathcal{B}_{k-1}$) and Case 2
($s_{k-1}\in\mathcal{A}_{k-1}- \mathcal{B}_{k-1}$) are exactly the same as those in Theorem \ref{thm1},
and will not be repeated here. Case 3 is discussed below.

\noindent
\textbf{Case 3.} $s_{k-1}\notin \mathcal{A}_{k-1} \cup \mathcal{B}_{k-1}$:
\begin{align}
    \begin{split}
    &s_{k-1}\notin \mathcal{A}_{k-1} \cup \mathcal{B}_{k-1}\Longrightarrow \mathrm{P}\left( e_1 \right)=0 \land \mathrm{P}\left( e_3 \right)=1 \Longrightarrow
    \max_{\mathcal{E}}\mathrm{P} \left(e_0^{\mathcal{E}}  \right)=\max_{\mathcal{E}}\mathrm{P} \left(e_2^{\mathcal{E}}  \right)\\
    &=\max_{\mathcal{E},u_{k-1}}\mathrm{P}\left[ \left. \left(\exists k_1\in \mathbb{Z}_k^T,\phi_k^T(.|k_1,s_k,\mathcal{E})\in \mathcal{A}_{k_1}\right) \land
    \left(\forall k_2\in \mathbb{Z}_k^{k_1},\phi_k^{k_1}(.|k_2,s_k,\mathcal{E})\notin \mathcal{B}_{k_2}\right) \right|
    s_k\sim \mathcal{F}\left(.|s_{k-1},u_{k-1}\right)  \right]\\
    &=\max_{u_{k-1}}\int_{\mathbb{R}^n} V_k^{\mathrm{max}}(s)\mathcal{F}\left(s|s_{k-1},u_{k-1}\right)ds=
    \max_{u_{k-1}}\mathbb{E}\left[ V_k^{\mathrm{max}}\left( \mathcal{F}\left(.|s_{k-1},u_{k-1} \right) \right)\right]
    \end{split}
\end{align}
and
\begin{align}
    \begin{split}
        &s_{k-1}\notin \mathcal{A}_{k-1}\cup \mathcal{B}_{k-1}\Longrightarrow \mathbb{I}_{\mathcal{A}_{k-1}}(s_{k-1})=0 \land
        \mathbb{O}_{\mathcal{B}_{k-1}}(s_{k-1})=1 \Longrightarrow\\
        &\mathbb{O}_{\mathcal{B}_{k-1}}(s_{k-1})
         \geq\mathbb{E}\left[ V_k^{\mathcal{E}}\left( \mathcal{F}\left(.|s_{k-1},
        \mathcal{E}(s_{k-1},k-1) \right) \right)\right] \geq \mathbb{I}_{\mathcal{A}_{k-1}}(s_{k-1}) \Longrightarrow \\
        &\min \left\{\mathbb{O}_{\mathcal{B}_{k-1}}(s_{k-1}),
        \max\left\{\mathbb{I}_{\mathcal{A}_{k-1}}(s_{k-1}),
        \max_{u_{k-1}}\mathbb{E}\left[ V_k^{\mathrm{max}}\left( \mathcal{F}\left(.|s_{k-1},u_{k-1} \right) \right)\right]  \right\} \right\}\\
        &=\max_{u_{k-1}}\mathbb{E}\left[ V_k^{\mathrm{max}}\left( \mathcal{F}\left(.|s_{k-1},u_{k-1} \right) \right)\right]
    \end{split}
\end{align}
Therefore, Eq. (\ref{eq25}) holds when $s_{k-1}\notin \mathcal{A}_{k-1}\cup\mathcal{B}_{k-1}$.

\rightline{$\square$}

Similarly, Theorem \ref{thm2} indicates that, the maximum $\gamma-$probabilistic reachable set
can be represented by the $\gamma-$level set of $V_0^{\mathrm{max}}(.)$, i.e.:
\begin{align}
    \mathcal{R}_\gamma^{\mathrm{max}}(\mathcal{A},\mathcal{B},T)=\left\{s_0\in\mathbb{R}^n|V_{0}^{\mathrm{max}}(s_0)\geq\gamma \right\}
\end{align}
and $V_0^{\mathrm{max}}(.)$ can be obtained by the following recursive formula:
\begin{align}\label{eq30}
    \begin{split}
        &V_T^{\mathrm{max}}(s_{T})=\min \left[ \mathbb{I}_{\mathcal{A}_T}\left( s_T \right),\mathbb{O}_{\mathcal{B}_T}\left( s_T \right)  \right]\\
        &V_{k-1}^{\mathrm{max}}(s_{k-1})=\min \left\{\mathbb{O}_{\mathcal{B}_{k-1}}(s_{k-1}),
        \max\left\{\mathbb{I}_{\mathcal{A}_{k-1}}(s_{k-1}),
        \max_{u_{k-1}\in\mathcal{U}} \mathbb{E}\left[ V_k^{\mathrm{max}}\left( \mathcal{F}\left(.|s_{k-1},u_{k-1} \right) \right)\right]  \right\} \right\}\\
        &...\\
        &V_{0}^{\mathrm{max}}(s_0)=\left\{\mathbb{O}_{\mathcal{B}_{0}}(s_{0}),
        \max\left\{\mathbb{I}_{\mathcal{A}_{0}}(s_{0}),
        \max_{u_{0}\in\mathcal{U}}\mathbb{E}\left[ V_1^{\mathrm{max}}\left( \mathcal{F}\left(.|s_{0},u_0 \right) \right)\right]  \right\} \right\}
    \end{split}
\end{align}
In addition, Eq. (\ref{eq30}) reveals the expression for the optimal control policy:
\begin{align}\label{eq31}
    \mathcal{E}^*\left(s_k,k\right)=\arg\max_{u_k \in \mathcal{U}} \mathbb{E} \left[ V_{k+1}^{\mathrm{max}}\left( \mathcal{F}\left(.|s_{k},u_k \right) \right)\right]
\end{align}

\section{Method implementation}
For any $k\in \mathbb{Z}_0^{T-1}$, the analytic forms of the expressions of
$V_k^{\mathcal{E}}(.)$ and $V_k^{\mathrm{max}}(.)$ are difficult to obtain.
This section introduces a method based on grid interpolation
to approximate these functions. First, a rectangular
computational domain, denoted as $\mathcal{S}$, needs to be specified in
the state space. Then, divide $\mathcal{S}$ into a Cartesian grid structure.
The values of function $V_k^{\mathcal{E}}(.)$ or $V_k^{\mathrm{max}}(.)$ at the grid points are stored in
an array with the same dimensions as the state space, and $V_k^{\mathcal{E}}(.)$ or $V_k^{\mathrm{max}}(.)$
can be approximated by the grid interpolation.

In addition, for any $k\in \mathbb{Z}_0^{T-1}$,
both of the recursive procedures described above involve computing the expectation value
$\mathbb{E}\left[ V_k^{\mathcal{E}}\left( \mathcal{F}\left(.|s_{k-1},\mathcal{E}(s_{k-1},k-1) \right) \right)\right]$ or
$\mathbb{E}\left[ V_k^{\mathrm{max}}\left( \mathcal{F}\left(.|s_{k-1},u_{k-1} \right) \right)\right]$.
In the current study, these expectation values will be calculated by using Monte Carlo method.
Denote the sampling result from probability distribution $\mathcal{F}\left(.|s_{k-1},\mathcal{E}(s_{k-1},k-1) \right)$ or
$\mathcal{F}\left(.|s_{k-1},u_{k-1} \right)$ as $\{s^1,...,s^m\}$, then the expectation values
can be estimated by the following equations:
\begin{align}
    \begin{split}
        &\mathbb{E}\left[ V_k^{\mathcal{E}}\left( \mathcal{F}\left(.|s_{k-1},
        \mathcal{E}(s_{k-1},k-1) \right) \right)\right]=\frac{1}{m}\sum_{i=1}^m V_k^{\mathcal{E}}\left(s^i\right)  \\
        &\mathbb{E}\left[ V_k^{\mathrm{max}}\left( \mathcal{F}\left(.|s_{k-1},u_{k-1} \right) \right)\right]
        =\frac{1}{m}\sum_{i=1}^m V_k^{\mathrm{max}}\left(s^i\right)
    \end{split}
\end{align}

Given all the techniques described above, the complete algorithm can be obtained.
As an example, take the maximum probabilistic reachable set of a two-dimensional system,
where the system state is denoted as $s=[x,y]^\mathrm{T}$,
the pseudocode of the proposed method
is shown in Algorithm \ref{alg1}.

\begin{algorithm}[htbp]
	\label{alg1}
    \caption{Method to compute maximum probabilistic reachable set}
	\begin{algorithmic}[1]
	  \State \textbf{Inputs:}Time horizon $T$, dynamic system $s_{k+1}\sim \mathcal{F}(.|s_k,u_k)$,
        a real number $\gamma\in [0,1]$, computational domain $\mathcal{S}=[x_d,x_u]\times [y_d,y_u]$, grids number $N_x\times N_y$;
	  \State Construct two $N_x\times N_y$ arrays, denoted as $\mathcal{V}$ and $\mathcal{V}'$;
	  \State $\displaystyle{\delta x\leftarrow \frac{ x_u-x_d }{ N_x-1 },\delta y\leftarrow \frac{ y_u-y_d }{ N_y-1 }}$;
      \For{$i_x\leftarrow 0,...,N_x-1 $} \textcolor{gry}{\quad\quad\quad\quad\quad $\backslash\backslash$ The terminal condition of the recursive formula.}
        \For{$i_y\leftarrow 0,...,N_y-1 $}
            \State $s_T\leftarrow \left[x_d+i_x\delta x,y_d+i_y\delta y    \right]^\mathrm{T}$;
            \State $\mathcal{V}[i_x][i_y]\leftarrow \min\left[ \mathbb{O}_{\mathcal{B}_T}\left(s_T\right),\mathbb{I}_{\mathcal{A}_T}\left(s_T\right)   \right]$
        \EndFor
      \EndFor

	  \For{$k \leftarrow N-1,...,0$}
      \For{$i_x\leftarrow 0,...,M_x-1 $}
        \For{$i_y\leftarrow 0,...,M_y-1 $}
          \State $s_k\leftarrow \left[x_d+i_x\delta x,y_d+i_y\delta y    \right]^\mathrm{T}$;
          \If{$s_k \in \mathcal{B}_k$}\textcolor{gry}{\quad\quad\quad\quad\quad $\backslash\backslash$ $s_k \in \mathcal{B}_k \Longrightarrow V_k(s_k)=\mathbb{O}_{\mathcal{B}_k}(s_k)=0$}
            \State $\mathcal{V}'[i_x][i_y]\leftarrow 0$;
            \State Continue;
          \EndIf
          \If{$s_k \in \mathcal{A}_k$}\textcolor{gry}{\quad\quad\quad\quad\quad $\backslash\backslash$ $s_k \in \mathcal{A}_k-\mathcal{B}_k \Longrightarrow V_k(s_k)=\mathbb{I}_{\mathcal{A}_k}(s_k)=1$}
            \State $\mathcal{V}'[i_x][i_y]\leftarrow 1$;
            \State Continue;
          \EndIf
          \State Construct a bilinear interpolation function $\widehat{V}(.)$ using $\mathcal{V}$;
          \State $v^*\leftarrow 0$;
          \For{$u_k\in \mathcal{U}$}
            \State Sample from probability distribution $\mathcal{F}(.|s_k,u_k)$, denote the samples as $\{s^1,...,s^m\}$;
            \State $v\leftarrow \frac{1}{m} \sum_{i=1}^m \widehat{V}(s^i)$;
            \If{$v^*<v$}
              \State $v^*\leftarrow v$;
            \EndIf
          \EndFor
          \State $\mathcal{V}'[i_x][i_y]\leftarrow v^*$;
          \textcolor{gry}{\quad\quad\quad\quad\quad $\backslash\backslash$
          $s_k \notin \mathcal{A}_k\cup \mathcal{B}_k \Longrightarrow V_k^{\mathrm{max}}(s_k)=\displaystyle{\max_{u_k}} \mathbb{E} \left[ V_{k+1}^{\mathrm{max}}( \mathcal{F}(.|s_k,u_k) ) \right]$}
        \EndFor
      \EndFor
      \State Copy $\mathcal{V}'$ to $\mathcal{V}$;
    \EndFor
    \State Return $\left\{s\in\Omega|\widehat{V}(s)\geq \gamma \right\}$;
    \end{algorithmic}
\end{algorithm}

\section{Numerical examples}
A flight vehicle moves in a plane, the vehicle
is modeled as a simple mass point with fixed linear velocity $v=1$ and controllable heading angular velocity.
Its motion in still air is determined by the following equation:
\begin{align}
	\left\{\begin{array}{l}
		\dot{x}=v \cos \theta \\
		\dot{y}=v \sin \theta \\
		\dot{\theta}=u
	\end{array}\right.
\end{align}
where $[x,y]^\mathrm{T} \in \mathbb{R}^2$ and $\theta\in (-\pi,\pi]$ are the position and heading angle of the vehicle respectively,
and $u\in \mathcal{U}=[-1,1]$ is the control input.
There exists a wind field in the plane, which is determined by the following vector field:
\begin{align}
	\left[
		\begin{array}{c}
			w_x(x,y)\\w_y(x,y)
		\end{array}
	\right]=\left[
		\begin{array}{c}
			\displaystyle{-y-0.1y^3} \\ \displaystyle{x+0.1x^3}
		\end{array}
	\right]
\end{align}
Then the dynamics of the vehicle in the wind field can be described by:
\begin{align}\label{eq35}
	\dot{s}=f(s,u)=\left[
		\begin{array}{c}
			v \cos \theta +w_x(x,y)\\ v\sin \theta+w_y(x,y) \\ u
		\end{array}
	\right]
\end{align}
The vehicle selects a control input from $\mathcal{U}$ every $\Delta t=0.1$ time,
and the control input is fixed as a constant during each time step.
The uncertainty of the vehicle's position accumulated in one time step is represented by a uniform distribution
over a circular region with radius $r=0.1$. Therefore, the time discrete form of the system is as follows:
\begin{align}
    s_{k+1}\sim \mathcal{F}\left(.|s_k,u_k\right)=F(s_k,u_k)+\mathscr{U}_r(.)
\end{align}
where $F(s_k,u_k)$ is derived by solving Eq. (\ref{eq35}) using Runge-Kutta method,
$\mathscr{U}_r(.)$ is a probability distribution whose probability density at $s=[x,y,\theta]^\mathrm{T}$ is:
\begin{align}
    \mathscr{U}_r(s)=\begin{cases}
        \displaystyle{ \frac{1}{\pi r^2}},& x^2+y^2\leq r^2\\
        0,& x^2+y^2> r^2
    \end{cases}
\end{align}

The target set and obstacle at the $k$th time step are:
\begin{align}
    \begin{split}
      &\mathcal{A}_k=\left\{[x,y,\theta]^\mathrm{T} \bigg|
      x\in \left[ 2 \cos\left( \frac{k\pi}{40}  \right)-\frac{1}{2},2 \cos\left( \frac{k\pi}{40}  \right)+\frac{1}{2}  \right],
        y\in \left[ 2 \sin\left( \frac{k\pi}{40}  \right)-\frac{1}{2},2 \sin\left( \frac{k\pi}{40}  \right)+\frac{1}{2}  \right] \right\}\\
      &\mathcal{B}_k=\left\{[x,y,\theta]^\mathrm{T} \bigg|
      x\in \left[ 2 \cos\left( \frac{k\pi}{40}+\pi  \right)-\frac{1}{2},2 \cos\left( \frac{k\pi}{40} +\pi  \right)+\frac{1}{2}  \right],\right.\\
      &\quad \quad \quad \quad \quad \quad \quad \left. y\in \left[ 2 \sin\left( \frac{k\pi}{40}+\pi \right)-\frac{1}{2},2 \sin\left( \frac{k\pi}{40}+\pi   \right)+\frac{1}{2}  \right] \right\}
    \end{split}
\end{align}
and the time horizon is $T=23$, the given probability is $\gamma=0.6$.
The problem is depicted visually in Fig. \ref{fig1}.
\begin{figure}[htbp]
    \centering
    \includegraphics[width=0.5\textwidth]{fig/fig1.pdf}
    \caption{Sketch of the numerical example.}
    \label{fig1}
\end{figure}

\subsection{Computation of probabilistic reachable set}
Assume that the control policy aims to point the vehicle's heading to the center of the target set.
That is, the control policy is fixed at:
\begin{align}\label{eq39}
    \mathcal{E}\left(s_k,k\right)=\min\left\{1,\max\left[-1, \displaystyle{ \text{atan2}\left[\sin\left( \frac{k\pi}{40}  \right)-y_k, 2 \cos\left( \frac{k\pi}{40}  \right)-x_k \right]-\theta_k }  \right]\right\}
\end{align}

where the definition of function $\text{atan2}$ is \cite{a20}:
\begin{align}
    \begin{split}
      \text{atan2}(\Delta y,\Delta x)=
      \begin{cases}
          \arctan \frac{\Delta y}{\Delta x}, &\Delta x>0\\
          \arctan \frac{\Delta y}{\Delta x}+\pi, &\Delta x<0 \text{ and } \Delta y\geq 0\\
          \arctan \frac{\Delta y}{\Delta x}-\pi, &\Delta x<0 \text{ and } \Delta y< 0\\
          \frac{\pi}{2}, &\Delta x=0 \text{ and } \Delta y>0\\
          -\frac{\pi}{2}, &\Delta x=0 \text{ and } \Delta y<0
      \end{cases}
    \end{split}
\end{align}

The solver setups of this example are listed in Table \ref{tb1}.
\begin{table}[htbp]
	\centering
	\caption{Solver settings for the computation of probabilistic reachable set}
	\label{tb1}
	\begin{tabular}{lc}
	\toprule[1pt]
	\textbf{Parameter} 			  			& \textbf{Setting}  \\ \toprule
	Computational domain $\mathcal{S}$         	&      $[-4,4]\times [-4,4] \times [-\pi,\pi]$      \\
	Number of grid points $N_x\times N_y \times N_\theta$   &      $201\times 201 \times 201$     \\
	Time horizon $T$              				&      $23$      \\
    Probability $\gamma$              				&      $0.6$      \\
	Sample size for the Monte Carlo method $m$            	&      10000     \\ \bottomrule[1pt]
	\end{tabular}
\end{table}
The computation result of the $\gamma-$probabilistic reachable set is shown in Fig. \ref{fig2}(a).
In order to verify the correctness of the result, some points in slice $\theta=0$
of the state space are chosen as the initial states, for each initial state, 1000 times of evolution under control policy
\ref{eq39} are simulated and the counts of avoiding obstacles and reaching the target set are recorded.
Fig. \ref{fig2}(b) shows the simulation results,
where the evolutions starting from the points marked by the green circles reach the target set and avoid the obstacles more than 600 times.
\begin{figure}[htbp]
    \centering
    \subfigure[$\gamma-$probabilistic reachable set.]{\includegraphics[width=0.4\textwidth]{fig/fig2a.png}}\ \ \ \ \ \ \ \ \ \  \ \ \ \ \
    \subfigure[Simulation results in slice $\theta=0$.]{\includegraphics[width=0.4\textwidth]{fig/fig2b.pdf}}
    \caption{Computation results of $\gamma-$probabilistic reachable set.}
    \label{fig2}
\end{figure}
It can be seen that the outline of $\gamma-$probabilistic
reachable sets and the border of the area marked
by the green circles almost coincide, which indicates the accuracy of the computation results.

\subsection{Computation of Maximum probabilistic reachable set}
From line 24 of Algorithm \ref{alg1},
it is clear that the computation of the maximum probability reachable set requires traversing the set $\mathcal{U}$.
In this example, since the set $\mathcal{U}$ is a continuous set, it is impractical to traverse this set.
Therefore, the $\mathcal{U}$ is replaced by the following set consisting of some discrete points in the computation procedure.
\begin{align}
    \bar{\mathcal{U}}=\left\{-1+\frac{1}{10}i \bigg| i=0,1,...,20 \right\}
\end{align}
The solver setups are also shown in Table \ref{tb1}.
The computation result of the maximum $\gamma-$probabilistic reachable set is shown in Fig. \ref{fig3}(a).
During the computation, the functions $V^{\mathrm{max}}_k(.)$ are saved for any $k\in \mathbb{Z}_0^T$.
These functions can be used to implement the control policy (\ref{eq31}).
In this subsection, the initial states inside slice $\theta=0$ are also simulated to validate the correctness of the result.
In these simulations, the control policy is chosen as Eq. (\ref{eq31}).
Fig. \ref{fig3}(b) displays the simulation results, which also illustrate the accuracy of the proposed method.
\begin{figure}[htbp]
    \centering
    \subfigure[Maximum $\gamma-$probabilistic reachable set.]{\includegraphics[width=0.4\textwidth]{fig/fig3a.png}}\ \ \ \ \ \ \ \ \ \  \ \ \ \ \
    \subfigure[Simulation results in slice $\theta=0$.]{\includegraphics[width=0.4\textwidth]{fig/fig3b.pdf}}
    \caption{Computation results of maximum $\gamma-$probabilistic reachable set.}
    \label{fig3}
\end{figure}

\section{Conclusions}
In this paper, we study the reachability problems of non-deterministic discrete-time systems.
Based on the existing works, the time-dependent target set and obstacle are taken into account and
the definition of probabilistic reachable set is refined.
Two types of probabilistic reachable sets are discussed, namely, probabilistic
reachable set with a given control policy and maximum probabilistic reachable set.
To compute these probabilistic reachable sets, a numerical method is proposed.
In the proposed method, the probability reachable set is represented as
a non-zero level set of a scalar function that is approximated by backward recursion and grid interpolation.
The scalar function generated by each recursive step can also be used to design optimal control policy.
The paper concludes with some examples to verify the effectiveness of the proposed method.

The proposed method has some potential for
improvement. For example, the computation of expectation values using Monte Carlo method
is computationally expensive
The sample size required grows as the accuracy requirement increases.
New mechanisms for computing expectations will be developed in our future work.

\section*{Acknowledgements}
\par The authors gratefully acknowledge support from National Defense Outstanding Youth Science Foundation (Grant No. 2018-JCJQ-ZQ-053),
and Central University Basic Scientific Research Operating Expenses Special Fund Project Support (Grant No. NF2018001).
Also, the authors would like to thank
the anonymous reviewers, associate editor, and editor for
their valuable and constructive comments and suggestions.

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv