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Curiosity Will Make You Open This. What Else Will It Do For You?
Curiosity. It's probably the reason you're reading this.
If I tell you there's a mind-blowing, life-revelation at the end of this blog, you'll probably want to scroll down and read it (I bet some people didn't even make it to the end of that sentence without scrolling to check – you curious cats)*.
The second part of my blog is named after my passion for asking questions and discovering new things. Like the Compassion intro series, I wanted to start by penning a couple of blogs exploring what curiosity is and how it can benefit us.
What does curiosity mean to you?
Think about this for 30 seconds. What comes to mind? Maybe it's the hours or even nights you've lost on Youtube? Maybe it's a place you've visited or a food you've tried? I'd love to hear from you.
Well, The Cambridge Dictionary defines curiosity as:
an eager wish to know or learn about something
Why does it matter?
A million reasons. Essentially, when I'm curious about something, it keeps the fire in my belly burning. This helps every aspect of my life. It helps my mental health, gives me a sense of achievement, makes me a more interesting person and a better friend, daughter, brother, parent etc… I've written more about that here.
Two Types of Curiosity: Inward and Outward
In my mind, there are two kinds of curiosity and we need them both!
Outward: Curiosity about the world around you; people, environment, skills.
This is probably the type of curiosity that came to mind when you initially thought of the word just now. If you've ever wanted to know the life story of that fitty (as my mum would say) that just passed you in the street, learnt to make a foreign dish or lost an entire weekend to documentaries, this is outward curiosity. It's the desire to understand things outside of yourself.
I'll call this 'Curiosity Living'. It can include anything that takes your fancy but here are some broad categories:
- Food
- Hobbies
- Reading
- Travel
- Shopping
- Blogging
- People
- Business
I'll probably be writing a little about each of these as I really want compassionandcuriosity.com to be a look at this messy thing called life and how we can inject a little bit of those two crucial values into every aspect of it.
Inward: Curiosity about yourself and the way you do things
I'll call this 'Curious Thinking'. Ever found yourself taking a Buzzfeed quiz to find out what kind of sandwich you are? There's no judgement here, we've all spent a shameful amount of time on pointless exercises like this. What you may not fully appreciate is just how powerful this curiosity thinking can be when directed towards slightly more helpful sources.
Being curious about our own thoughts and behaviours is actually a crucial part of mindfulness. It can help to:
- recognise what fills us up and brings us down
- better connect with others
- break habits
- manage anxiety and mental health
- better align our actions with our values
Live a furiously curious life.
Overall, the more curious we are, the more we deepen our understanding of ourselves and the world around us. A lot of what scares us does so because we don't understand it. If we open up and embrace curiosity, fear won't disappear but it won't paralyse us or prevent us doing things that really matter. I'll be brutally honest about my own curiosities and I hope you'll share yours with me too!
*There isn't by the way. The life-changing magic is interwoven throughout this series so you'll have to grit your teeth and read it all! Mwuahahaha.


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Independent Asean activists will boycott a traditional meeting with regional leaders in a sideline session of the Asean Summit in Phnom Penh next month after the Cambodian government set conditions for nominating their representatives.
Asean non-profit organisation representatives were to meet Asean leaders on April 3 in a related session of the summit in Phnom Penh on April 3-4 under the so-called interface dialogue.
Phnom Penh has sidestepped the nomination of representatives by telling its Asean colleagues to nominate their own people instead of the Asean Civil Society Conference and the Asean People's Forum (ACSC/APF) as normal.
NGOs said Cambodia's action has undermined the process of independent groups galvanising local support and coordinating functions with other Asean counterparts.
Suntaree Saeng-ging, secretary-general of the Thai NGO Coordination Committee, told the Bangkok Post the Asean NGOs have decided to proceed with their own plans with an ACSC/APF meeting at the Lucky Star Hotel in Phnom Penh on March 29. The theme of transforming Asean into a people-centred community will have as its keynote speaker a renowned political prisoner from Myanmar, Min Ko Naing. But they will boycott the government interface.
"Why do we have to be their decoration and simply let ourselves be used as a justification that Asean is participatory and people-centred, despite their negligence towards the significance of transparency and accountability for the people's voices," Ms Suntaree said.
An activist source said the Cambodian Council of Ministers was responsible for the nomination issue, not the Foreign Ministry.
The Council of Ministers has assigned two Cambodian representatives _ a woman, Prak Sokhany, executive director of Cambodian Civil Society Partnership, and a youth _ to meet at the interface dialogue on April 3.
Phnom Penh also told its Asean partners to select two people from their respective NGOs. In Thailand's case, the Social Development and Human Security Ministry has nominated Chalida Tacharoensak from the People's Empowerment Foundation, sources said.
This has undermined the fragile unity and poor coordination among NGOs in all Asean countries, another source said. –Bangkok Post
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What time is it when you have a toothache? 2:30! (Tooth hurty). All joking aside, an emergency dentist in Leesburg may not be able to take you immediately even if it is 2:30, because the office may be closed this time of year for the holidays. In that case, you'll be glad to know that there are a few home remedies. Call my office first and speak to the answering service, and then try one of these at-home treatments.
Just in case the pain is due to a piece of food trapped between two teeth, floss around the sore tooth. Never use a sharp object in your mouth in an attempt to dislodge debris. I've seen dire consequences!
Rinse with Saltwater
Mix ½ teaspoon of table salt in eight ounces of lukewarm water. Swish this around in your mouth and spit it out. Don't swallow the saltwater.
OTC Pain Relievers and Anesthetics
If a child has a toothache, administer a regular dose of acetaminophen. For an adult, you can choose ibuprofen, acetaminophen or aspirin. If you choose aspirin, don't believe that old wives' tale that says to put the tablet directly on the tooth or your gums. It doesn't work and you could irritate tender gum tissue.
Anesthetics, or pain-relieving gels and liquids, may be applied directly to your sore tooth and gums. The benzocaine in these products will numb your mouth for a time. A word of caution—these are meant to be used for a short time only.
A Cold Compress
An ice pack against your cheek may help if your face is swollen. It can also lessen the pain if you've chipped, broken or dislodged a tooth. Swelling may also be a sign that you have an abscess, which is a serious infection deep in the tooth's root. In this case, you may also have a fever. This is a true dental emergency in Leesburg. Call my office immediately or head to the nearest hospital emergency room.
Clove Oil
This natural remedy has been around for ages and is known for numbing pain. Rub some of the oil directly on the sore tooth, or soak a cotton ball or gauze pad and dab it on the tooth and gums. Some claim this natural oil is as effective as benzocaine.
If you or a member of the family experiences a toothache this holiday season, please call my office right away. If I'm not available, the answering service will instruct you about what to do. And be sure to maintain your regular oral healthcare routine in order to help avoid dental problems this holiday season!
Meet the Doctor
Dr. David Groy has years of experience and extensive training. He and his team of dental hygienists take care of smiles of all ages with preventive dentistry, cosmetic treatments, restorative care and emergency dentistry. Call today for an appointment.


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On Monday we went to London Zoo and contrary to my pessimistic expectation it didn't rain. In fact the weather was rather nice. We went with our friend C and her son, Matty's friend from school, taking advantage of the way we could all fit into our car because C's husband, an airline pilot, is currently stuck in Washington because of the Icelandic volcano.(Every volcanic aeroplane death cloud has a silver lining, apparently.)
London Zoo has changed since my youth. Gone are the elephants and Polar bears circling confined enclosures, going slowly mad. Instead the emphasis is on conservation with Gorilla Kingdoms and Rain Forest experiences. I particularly enjoyed the Night Life experience, which was not a disco but a chance to see all sorts of nocturnal creatures knocking about naturally.
I am slightly suspicious that many of the terrariums in the reptile house were empty despite signs saying they contained a Taiwanese Spitting snake, a Borneo Bouncing lizard or a Completely Invisible Sleeping newt. Occasionally I did spot a snake but it might quite easily have been a rubber one from the zoo souvenir shop. Telling me how cunningly they used camouflage didn't fool me.
Sam, having been to see the film How To Train Your Dragon was keen to see the Komodo dragons. He was disappointed to find they neither flew nor breathed fire but was slightly mollified when I told him how they could kill a buffalo. Matty found it difficult to choose a favourite animal, being torn between the giraffes and "those catty things with spots near the tigers."
It was a good final day of the school holidays.
Until next time. . .


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Virginia's Governor Knows Something About Attractive Businesses That Ohio's Governor Doesn't
Yes, Virginia, not all state governors are equally bright. In fact, Virginia, some governors can't see the future clearly because their heads are stuck so far up their fundamental ideology that what should be a no-brainer turns into a brainer.
Case in point, what makes a state attractive to business? Ohio's term-limited chief executive, John Kasich, seems oblivious to the fact that when companies shop states, factors like taxes, utilities, labor force and location become decision points in their site selection process. But so do other factors, like whether the social climate is right for the company's managers and employees and their families.
Mr. Kasich campaigned on a promise to create lots of jobs, but he has fallen short 47 out of 48 months from even meeting the national job creation average. He boasted on the presidential campaign trail this year that he learned how business works during his half-dozen years working as a Wall Street banker for Lehman Brothers, the storied Wall Street banking house whose collapse triggered the national economic calamity now known as the Great Recession. While at Lehman Brothers, John Kasich befriended fellow Lehmanite John Minor. When citizen Kasich took over as state chief executive 2010, his first bill was to create a private agency responsible for bringing jobs to the state. He installed Mr. Minor, who's drawing hundreds of thousands in salary, in charge of it.
For all his talk about know how business works, what Gov. Kasich doesn't see clearly is the toll his record of signing into law 18 bills that hurt or harm access women have to their constitutionally guaranteed right to health care, including abortion if that's their choice, has on business attraction.
Yes, pro-growth policies can boost the image of Ohio as a place for business, but that image can be equally blackened by bad bills that create a counter social landscape reflective of lawmakers who don't understand that the message a state sends to companies on how their female employees and family members will be treated is as potent as their tax rates and regulatory stances.
Virginia Gov. Terry McAuliffe knows that as he just schooled Gov. Kasich in the fine art of image building. In stark contrast to Gov. Kasich signing into law a 20-week ban on abortions in the lame-duck session just ended, Gov. McAuliffe warned his Republican lawmakers that he would veto similar legislation banning abortion after 20 weeks of pregnancy if it were to arrive on his desk.
Whereas Ohio's governor signed the bill behind closed doors, saying nothing about it when he did, Virginia's chief executive called such a bill a "socially divisive" proposal that hurts the state's image, the AP reported.
In Virginia as in Ohio, Republican legislators see the election of Donald Trump as a sign to push forward with more harmful measures aimed at limiting a woman's access to certain health decisions. Their calculus is that Donald Trump and Congressional Republicans will add an extreme justice to the Supreme Court who will side with court conservatives and one day overturn Roe v Wade, the 1973 case that made access to abortion a constitutional right.
McAuliffe, a Democrat, told The Associated Press on Tuesday that he wanted to send a clear message to the Republican-controlled General Assembly not to "waste time" trying to be part of that effort. McAuliffe said he doesn't normally comment on proposed legislation until after it passes both chambers, but he needed to make clear to companies looking to invest in Virginia that the legislation had no hope of passage.
"I can't sit back and have that sitting out the same time I am traveling the globe recruiting businesses to Virginia," McAuliffe said as he prepared to go on an important recruiting trip this weekend, the AP reported. "If there's something that would be damaging toward business, and to our image around the country and the globe, I'll veto it, you bet I will."
The AP reports that another state that doesn't get it is ready to follow Gov. Kasich's lead. Kentucky Republicans will introduce a 20-week abortion ban, along with a requirement that women undergo ultrasounds before having abortions. Like Ohio, Kentucky Republicans enjoy overwhelming House and Senate majorities and control of the governor's office, so similar bills are considered likely to pass.
Gov. Kasich crows that he talks to CEOs every day, but what would he say to a female CEO who asked him about the social climate in Ohio? He would be hard pressed to find a good explanation about why she should subject herself and her female employees and family members to the harsh anti-choice measures the governor has allowed to become law.
It's no secret that CEOs like to play one state against another to get the best deal. If she said she could get as good or a better deal from Virginia or another state on tax incentives or labor training programs or rail or road improvements, but said her decision would be based on which state offers a friendly family climate conducive to her company's workers, what would Gov. Kasich say?
If John Kasich wants to make Ohio great again he should stop signing bad bills into law and follow Gov. McAuliffe's lead of warning his legislature not to send him more bills that give the state's image a black eye.
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The Law Shop is now closed. Please click here to find out more.
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Q: I have a drain which is shared with one neighbour. They wish to come onto my land and connect to my drains as they would like to use the drainage system for a new toilet. What should I do?
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Badminton Wales is the sport's governing body in Wales. Our goal is to develop badminton as a sport for all, and to encourage the best players to progress through the structure at club, county and International level.
Badminton Wales Team
The Welsh Badminton Union was incorporated as a Limited Company on the 8th December 2003. The Union elected a Board of Directors who are responsible for the running of the company. Each Director is appointed for a maximum period of 3 years.
Badminton Wales History
History - A brief look back in time to the beginning of badminton in Wales
Roll of Honour - A list of the significant achievements by Welsh players in recent times.
Life Members - A list of those members who have been honoured with the Badminton Wales Life Member and Meritorious award for an outstanding contribution to Badminton in Wales. The IBF (now BWF) Meritorious award is the highest honour to be presented to a volunteer from a member nation for an outstanding contribution to the sport of Badminton.
National Winners - The Senior Welsh National Championships were first played in 1937 in Llandudno, North Wales at the delightful Craigside Hydro hotel, a rather grand hotel with its own badminton courts. The singles and level doubles events were not introduced until 1960 and 1959 respectively.
Welsh International Winners - In 1928 the North Wales Badminton Union transformed itself into the Welsh Badminton Union and the annual Welsh Championships were instituted, the first being played in season 1927-28 at Wrexham.


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Manklalo Disowns Togbe Atatim
From Samuel Mensa Torbizo, Kumasi
The Manaklalo of Mafi Traditional area in the Central Tongu District of the Volta Region, Togbe Asafo Brentoa, is fuming with anger, over the sub chief's decision to arrogate to himself the title "Chief of the Mafi Traditional Area" and pledge the support of the Mafi people to a political party in the forthcoming presidential election.
The Manklalo, said the attitude of Togbe Atatim, a sub chief of Mafi Avedo within the same traditional area, is not only deceptive, but could spark chieftaincy dispute to derail the peace and development agenda in the area.
The Herald's information is that, per the setting of the Mafi traditional system, the Manklalo, who is also the chief of Mafi Kumase is the over lord of the Mafi people.
Togbe Atatim , who was described as the paramount chief of the Mafia Traditional Area, in a meeting with Nana Akufo Addo; the flag bearer of the New Patriotic Party (NPP) at Adidome during his visit to the region , pledged his support on behalf of the people and other chiefs in the three Tongu Constituencies, namely South ,Central and North.
Togbe Atatim, was captured by both social and mainstream media, including Joy FM as saying "Akufo-Addo Will Not Disappoint Us – Volta Chief".
The story had read " Togbe Atatim IV, chief of the Mafi Traditional Area has indicated that the chiefs and people South Central and North Tognu are confidence that the 2016 presidential candidate of the NPP Nana Addo Dankwa Addo is capable of rising to the occasion when given the opportunity in this year's elections".
Togbe Atatim IV, in the story was said to have commended the NPP flag bearer for the healthy campaign he is running and admonished other candidates in this year's elections to follow suit.
Continuing, the story said, perceiving strongholds of political parties, should not have characteristics based on tribalism, opportunism and elitism and urged the electorate not to pay heed to politicians, who thrive on tribalism.
He mentioned that "these are negative features which can plunge the nation into chaos. In this context, therefore, the chiefs would want to unequivocally and unreservedly state that the 2016 elections should be devoid of tribalism and action by dishonest politicians aimed luring votes".
Togbe Atatim, obviously out to get recognition before Nana Addo, lashed at the current administration for failing to complete projects, started by the NPP under President Kuffour.
Additionally the chief, urged Nana Addo to fund the development of market structures at Sogakofe, Adidome and other places in the district to offer jobs for the unemployed.
But in a rebuttal, the Manklalo Asafo Brentoa, cautioned him to desist from posing himself as the chief of Mafi Traditional area.
Togbe Brentoa, in telephone interview with The Herald , said nobody would begrudge Togbe Atatim over why he had a meeting with the NPP flagbearer, but the title he has arrogated to himself, was misleading.
Giving the chronology to the events prior to Nana Addo 's visit to the Constituency, the Manklalo disclosed that, it was the NPP chairman for the area , who brought him an invitation on Thursday evening, indicating Nana Addo would be in the Central Tongu on Friday, but he told him that because of his busy schedule, he would not be able to attend the meeting.
"I was shock to hear that major radio station such as Joy Fm aired the without crosschecking that Togbe Atatim who is only a sub chief of Avedo without a linage to the paramountcy claimed to be the chief of the Mafi traditional area ,I did not delegate any of my subjects to represent me so the whole thing looked to as a " silent coup, " he remarked .
He said, prior to the 2012 elections when he received early invitation for similar event, he led all the chiefs to meet Nana Addo, and even met him later again in Accra.
Togbe Asafo Brentoa insisted that "I want to say without reservation that if Togbe Atatim wants to do politics nobody would prevent him but he must stop parading himself as the overlord of the Mafi Traditional area by dragging our names in the mud".
He stressed that "I would like to use your platform to calm the nerves all the Mafi descendants both home and abroad on any disaffection the irresponsible behavior of Togbe Atatim might have caused them. We won't allow any group or individual for their selfish interest to sway attention from our development agenda''.


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I'd been hoping that, for some bizarre reason, nobody would want to travel between London and Vancouver on Easter Monday, and I'd be offered a cheap, if not free, upgrade from Air Canada. Business class had been available – but at more than £800 I didn't think I could justify the cost versus the amount of champagne and cheese I could eat during an eight hour flight.
Even Air Canada's business lounge at Heathrow was maintaining standards : "I'm sorry, we don't allow economy passengers in, even for a fee." They were less choosey at the United lounge, where anyone could come in for $50US. Given I'd checked in three hours before the flight, I did a quick currency conversion in my head, looked at the piles of pastries and free drinks around the corner, and concluded : "Sod it. I'm on holiday."
Stuff for free is always good, even if you know you're probably paying for it. Take, for instance, the daily free wine tasting hour at the Listel Hotel in Vancouver. It was a nice surprise, having just checked in, to be told that the big event was just about to begin. Though to start with, it was just me, the roaring fire and the rhino.
I'm soon joined by a couple who are having a quiet night away from home; home being a suburb just ten miles from downtown Vancouver. They like to just "get away from it" – whatever it is – and enjoy a lazy evening on the town. And stepping out onto Robson Street, it's easy to see why. Unlike many places in the United States, this major city's downtown area consists of roads that are no more than four lanes wide, making them easy to negotiate.
Although it's officially only night one of the trip, the eight hour time difference means I've effectively been awake for about 24 hours. And after a couple of glasses of British Columbia's best, there's only one place to regroup my mind, body and soul. The nearest Irish pub.
Doolin's just off Granville Street isn't originally, quirky or clever. But it does have a fine selection of beers and a menu that's as familiar as any Irish bar in any city. And it also has an extremely welcoming and polite staff.
Roger is also happy to be there. He's a businessman from Nova Scotia, doubling up as an insurance agent for the shipping trade (and with the port of Halifax that's a lot of shipping) and what he himself described as "succession planning" – or to you and me, what to do with your old money if it's tied up in a small business.
"It's a generational thing," explains Roger. "You have these guys with really successful family companies, but kids who want nothing to do with it." His job, it seems, is to provide timely financial advice for those needing it. And, naturally, taking his own cut. Along with Roger are two local girls who work as bartenders. It's Monday, the quietest night of the week. So they're spending their night off in someone else's bar.
It's still relatively early but I'm beat. And tomorrow promises a busy day involving totem poles, a brewery and ice hockey. In that order.


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Wolfgang Lutz
Founding Director
Research Group: Migration, Education & Environment
Phone: +43 1 313 36 - 7702
E-Mail: lutz(at)iiasa.ac.at
Wolfgang Lutz is Founding Director of the Wittgenstein Centre for Demography and Global Human Capital, a collaboration between the International Institute for Applied Systems Analysis (IIASA), the Vienna Institute of Demography of the Austrian Academy of Sciences (VID/ÖAW) and the Vienna University of Economics and Business (WU). He joined IIASA in October 1985, where he is Program Director of the World Population (POP) Program. Since 2002 he has also been Director of the Vienna Institute of Demography and since 2008, Full Professor of Applied Statistics (part time) at the WU. He is also a Professorial Research Fellow at the Oxford Martin School for 21st Century Studies.
Professor Lutz studied philosophy, theology, mathematics and statistics at the Universities of Munich, Vienna and Helsinki and holds a PhD in Demography from the University of Pennsylvania (1983) and a second doctorate (Habilitation) in Statistics from the University of Vienna.
He has worked on family demography, fertility analysis, population projection and the interaction between population and environment. He has been conducting a series of in-depth studies on population–development–environment interactions in Mexico, several African countries and Asia. He is the author of a series of world population projections produced at IIASA and has developed approaches for projecting education and human capital. He is also principal investigator at the Asian MetaCentre for Population and Sustainable Development Analysis. Lutz is author and editor of 28 books and more than 200 refereed articles, including seven in "Science" and "Nature". In 2008 he received an ERC Advanced Grant, in 2009 the Mattei Dogan Award of the IUSSP and in 2010 the Wittgenstein Prize, the highest Austrian science award. He is elected full member of the Austrian Academy of Sciences and of the German National Academy Leopoldina as well as a member of the Committee on Population of the US National Academy of Sciences. Furthermore he is a member of the Finnish Society of Sciences and Letters (Societas Scientiarum Fennica) and the World Academy of Sciences (TWAS).
Areas of Expertise
- Population Projections
- Human Capital
Research Areas
Curriculum Vitae


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Drug Information Residency Stipends and Benefits
Stipend and Benefits
Residents receive a stipend of $47,500 per year. Additionally, each year residents receive the following benefits:
- Health insurance
- Eight paid legal holidays (at a minimum each resident is required to staff one major and one minor)
- 10 days of paid time off
- Funding for professional meetings
Residents are not required to use vacation time to attend meetings. As employees of UW Hospitals and Clinics, residents receive full health, vision and dental insurance benefits and full access to the University of Wisconsin's recreational, educational and cultural facilities.


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MPT Group Philosophy
Our groups encourage caregivers and loved ones to share in the development of their child's social skills, fine and gross motor skills, cognitive abilities, understanding of language and gain independence - all while having FUN together! Babies and toddlers learn, grow, and thrive by following a routine. Our classes provide just the right amount of structured activities to keep children stimulated while helping to improve attention span and listening skills.
Communication Station-Runs Year Round
When: Tuesdays 9:30am-10:30am & 11:00am-12:00pm (Northminster Presbytarian Church) see address below
Wednesdays 4:30pm-5:30pm (Morton Public Library) see address below
For: Children 18-35 months-Caregiver must be present
Where: Northminster Presbyterian Church 10720 N Knoxville Ave, Peoria, IL 61615
Morton Public Library 315 W. Pershing Street Morton, IL 61550
Please call 309.282.6704 to register
Led By: Peoria Group-Speech-Language Pathologists, Emma Lane M.S. CCC-SLP/L and Kelly Ryon M.S., CCC-SLP/L
Morton Group-Speech-Language Pathologists, Brittany Hoffmire M.S. CCC-SLP/L and Makenzie Bogner M.S. CF-SLP/L
Communication Station is a Speech and Language Group for children 18-35 months and includes 1-2 field trips every 12 weeks. Each week the group will target a new theme which is incorporated into the activities. This group is open to all children regardless of their communication ability. We will teach a total communication approach which includes a combination of pictures, sign language, sounds, and words to help your child initiate communication with peers and adults. While children will learn how to shape vocalizations into meaningful communication, parents will learn how to guide their child in the use of these techniques into their everyday routines. A snack time and sensory exploration (trampoline, peanut balls, rice tables etc.) is also provided. Children and families are welcome to join this class at any point in the year!!
Lunch Bunch-Summer Feeding Group **Summer Only**
Is your child a Picky Eater?! Do you dread mealtime whether at home or at a restaurant?! We can help! We want to work with your child to increase their exposure to a variety of foods and experience food in a safe and fun enviornment! During this group we will work on exploration of different food groups to help children feel comfortable and have an overall positive experience with food! We will work on oral motor, sensory, and behavior skills to help them during mealtimes. The group is open to all children regardless of their eating ability.
Group will include:
*Gross Motor and Sensory Activities and Sensory Play with food/non food items
*Oral Motor Activities
*Pretend play, food story time, and crafts with foods
*Social models from peers and therapists
*Caregiver handouts
Led By: Vicky Jordan, MS CCC-SLP/L Pediatric Feeding Therapist/Speech-Language Pathologist and Kelly Ryon, MS CCC-SLP/L Pediatric Feeding Therapist/Speech-Language Pathologist.
For: Children 3-5 years. *We will make exceptions if your child is within 3 months of the age requirements* Minimum of 2 children/Maximum 5 children. Caregivers do not stay.
When: There are 2, 4-week sessions being offered. Each session is 60 minutes. You can sign up for both or just 1 session.
Session 1: Meets every WEDNESDAY June 5th-June 26th, 2019 11:00am-12:00pm
Session 2: Meets every WEDNESDAY July 10th-July 31st, 2019 11:00am-12:00pm
Where: Mueller Pediatric Therapy 411 E. Washington Street East Peoria, IL 61611
Cost: $205.00 per 4 week session. Total is due at time of signup. We would be happy to accept any private insurance and bill them as a courtesy for you. You are responsible for contacting our office with your information prior to the start of the camp. A refund will be issued if applicable after processing private insurance. Otherwise, no refunds or pro-rating for this camp is available.
Sign up here:
Advanced Communication Station-SUMMER Speech-Language Group **Summer Only**
Come join the fun with your child while we engage in music, singing & dancing, sensory exploration, interactive story-time, themed activities/crafts, socialization opportunities with peers, and snack-time!! Children of all developmental levels are welcome!! A wonderful opportunity for any child to enrich their Speech-Language skills through hands on experiences.
Led by: Emma Lane, MS CCC-SLP/L Pediatric Speech-Language Pathologist.
For: 3-5 years Minimum of 4 children/Maximum 6 children. We will make exceptions if your child is within 3 months of the age requirements. *Caregiver must stay with child.*
When: 9:30am-10:30am Meets every MONDAY June 17th-July 29th, 2019 (NO CLASS July 1st)
Where: Mueller Pediatric Therapy 411 E. Washington Street East Peoria, IL
Cost: $90.00 for 6 week session
Sign up here:
Say & Play-Speech Sound Summer Group **Summer Only**
Say & Play is designed to help children work on specific speech sounds they are having difficulty producing. Your child may already be working on these sounds during the school year and this group will help bridge the summer gap to ensure your child continues to make progress on their sounds and speech intelligibility!! Children will have fun and be involved in games and activities while working on their speech sounds!! Child does NOT have to have an IEP to participate. Weekly homework will be provided.
Children may already have a diagnosis of:
Apraxia of Speech
Articulation Disorder
Phonological Disorder
Led by: Brittany Hoffmire, M.S. CCC-SLP/L, Licensed Speech-Language Pathologist with almost a decade of experience working with children.
Where: Mueller Pediatric Therapy-411 E. Washington Street East Peoria, IL 61611
When: June 11th-August 6th, 2019 8 week Session meeting every TUESDAY (no session July 2nd) . Min-2 kids Max-4 kids per group.
9:30-10:00am 3-5 year olds
10:15-10:45am 6-9 year olds
Cost: $105.00 paid in full at time of sign up to reserve your spot. No prorated or refunds available.
Sign up here:
Social Sprouts- Young Children's Social Skills Group **Summer Only**
This Summer we can help your child improve their social skills!! Does your preschooler or kindergartner have difficulty making and keeping friends? Would they benefit from recognizing social cues? Does your child need help expressing emotions appropriately? Maybe they have been diagnosed with Social Anxiety Disorder, Asperger's Syndrome, Autism, or AD/HD? Or are they simply shy or socially awkward? Are there issues with your child's behavior being disruptive at home, school, or daycare? Some children need a little extra help learning social skills! Through a series of lessons, activities, games, role-play, community field trips, our group leaders will teach your child to enhance his/her social skills and self-esteem. This summer group will place an emphasis on practicing conversational and play skills with peers.
Led by Carrie Reinboldt LCPC, Licensed Clinical Professional Counselor with Mueller Pediatric Therapy. Not only does she have extensive professional experience working with children needing help with various social skills, she also has a mother's perspective as she has a 8 year old son with Autism.
Led by Emma Lane M.S. CCC-SLP/L, Licensed Speech-Language Pathologist with Mueller Pediatric Therapy. Emma has been working with children who struggle with appropriate social skills for almost a decade. She also has extensive experience professionally working with children who are diagnosed with Autism and AD/HD.
The following topics will be covered:
Play and Conversational Skills:
• Greeting people • Conversational skills • Listening • Working together and playing with others in a group • Taking turns and sharing
Understanding Social Skills:
• Keeping hands to oneself and respecting personal space • Proper eye contact • Personal space • Speaking with an appropriate tone and volume • How to respond in a socially appropriate way to personal triggers
For: Children ages 4 years- 7 years. Caregivers do not stay.
When: 6 week session June 17th -July 29th, 2019 Every MONDAY 11:00am-12:00 pm (No Class July 1st) Class minimum is 4 children and maximum is 6 children.
Where: Mueller Pediatric Therapy. At least 1 of the sessions will be community outings at To Be Determined locations within the area. This will allow the therapists to work on social skills in different settings.
Cost: $405.00 for 6 week session. Total is due at time of signup. We would be happy to accept any private insurance and bill them as a courtesy for you. You are responsible for contacting our office with your information prior to the start of the camp. A refund will be issued if applicable after processing private insurance. Otherwise, no refunds or pro-rating for this camp is available.
Sign up here:
Master Printing and Grip Group-A Handwriting Camp **Summer Only**
This camp is designed to be a creative and fun way to learn and develop handwriting skills!! Master Printing and Grip Group is a handwriting camp that utilizes the "Handwriting Without Tears" curriculum in conjunction with other therapeutic strategies. The curriculum utilizes a hands-on, multi-sensory approach to writing.
- Fine motor labs to develop strong fine motor skills through activities such as coloring, cutting
- Pencil grasp development with suggestions of activities or grips when necessary
- Movement and music activities
- Workbook practice
- Small Groups and 1:1 instruction time
- Distribution / discussion of home activities
We will focus on pencil grasp, letter placement, legibility, spacing and letter formation for upper and lowercase letters using direct teaching and fun activities!
Led By: Kelcy Suydam, OTRL-Occupational Therapist and Brooke Pfanz COTA/L-Occupational Therapy Assistant. Our Therapists have over 20 years of experience working with children needing assistance with developing their handwriting skills. Both Therapists are trained in the Handwriting Without Tears curriculum.
For: Children going into Kindergarten - 4th grade. Each child will be taught at their cognitive and developmental readiness for handwriting. Caregivers do not stay.
When: 6 week session June 6th - July 18th, 2019 Every THURSDAY 10:00am-11:00 am. No class July 4th, 2019. Class minimum is 4 children and maximum is 8 children.
Where: Mueller Pediatric Therapy 411 E. Washington Street, East Peoria, IL 61611
Cost: $205.00 for 6 week session. Total is due at time of signup. We would be happy to accept any private insurance and bill them as a courtesy for you. You are responsible for contacting our office with your information prior to the start of the camp. A refund will be issued if applicable after processing private insurance. Otherwise, no refunds or pro-rating for this camp is available.
Sign up here:


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Compare at $675
Since 1898. When Salvatore Ferragamo died in 1960 he had realised the great dream of his life: to create and produce the most beautiful shoes in the world. His family was left with the task of carrying on and fulfilling the plan that Salvatore had nurtured in his final years: transforming Ferragamo into a great fashion house. Today the Ferragamo family still manages the company: a factor that sets it apart from other brands and highlights its Italian roots.


text:
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United States District Court, W.D. Oklahoma
April 16, 2018
ALAN MCKOWN, Movant,v.UNITED STATES DEPARTMENT OF JUSTICE, Respondent.
MILES-LaGRANGE UNITED STATES DISTRICT JUDGE
the Court is movant Alan McKown's ("McKown")
Motion to Quash Administrator Subpoena and Application to
Enjoin Governmental Authority from Obtaining Financial
Records Pursuant to Customer Challenge Provisions of the
Right to Financial Privacy Act of 1978 ("Motion to
Quash"), filed December 14, 2017. On January 17, 2018,
the United States filed its response. Based upon the
parties' submissions, the Court makes its determination.
Customer Notice dated December 8, 2017 was mailed to McKown
by the United States Department of Justice
("DOJ") pursuant to the Right to Financial
Privacy Act of 1978, 12 U.S.C. §§ 3401-3422
("RFPA"). The Customer Notice informed McKown
that the DOJ has issued a subpoena duces tecum dated November
27, 2017 ("Subpoena") to American National
Bank.[1]McKown is a customer of American National
Bank in his individual capacity as a joint account holder
with his spouse and in his capacity as an
officer/shareholder/member of other entities. The Subpoena
seeks financial records to investigate possible violations of
state and federal law related to health care fraud.
now moves this Court to quash the Subpoena and enjoin the DOJ
from obtaining his financial records. McKown asserts that the
Subpoena is improper on its face based upon the timing of the
requested production and the date the Customer Notice was
mailed to him. McKown further asserts the financial records
at issue are not sought for a lawful purpose relevant to the
investigation. Finally, McKown asserts that the financial
records sought are not reasonably described nor is the scope
of the investigation reasonably tailored to put McKown on
notice of what in fact may be relevant to a legitimate
3410 of the RFPA provides district courts with the following
standard to assess the merits of a motion to quash:
If the court finds that the applicant is not the customer to
whom the financial records sought by the Government authority
pertain, or that there is a demonstrable reason to believe
that the law enforcement inquiry is legitimate and a
reasonable belief that the records sought are relevant to
that inquiry, it shall deny the motion or application, and,
in the case of an administrative summons or court order other
than a search warrant, order such process enforced. If the
court finds that the applicant is the customer to whom the
records sought by the Government authority pertain, and that
there is not a demonstrable reason to believe that the law
enforcement inquiry is legitimate and a reasonable belief
that the records sought are relevant to that inquiry, or that
there has not been substantial compliance with the provisions
of this chapter, it shall order the process quashed or shall
enjoin the Government authority's formal written request.
12 U.S.C. § 3410(c).
to the RFPA, a copy of the subpoena must be served upon the
customer or mailed to his last known address on or before the
date on which the subpoena was served on the financial
institution. See 12 U.S.C. § 3405(2). Because
the Subpoena has not been served on American National Bank,
the Court finds that the Customer Notice mailed to McKown was
timely and the Subpoena should not be quashed on this basis.
Lawful purpose
RFPA provides that in order for financial records to be
obtained, there must be "reason to believe that the
records sought are relevant to a legitimate law enforcement
inquiry". 12 U.S.C. § 3405(1). The term "law
enforcement inquiry" is defined as "a lawful
investigation or official proceeding inquiring into a
violation of, or failure to comply with, any criminal or
civil statute or any regulation, rule, or order issued
pursuant thereto." 12 U.S.C. § 3401(8).
Additionally, pursuant to 18 U.S.C. § 3486(a), the
United States Attorney General or his designee is authorized
to issue an administrative subpoena requiring "the
production of any records or other things relevant to the
investigation" of a ...
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text:
Capricorn compatibility with the various zodiac signs
Compatibility with Aries: Both have a strong will, perseverance, determination; both do not make the vital mistakes. If you thought it was rather the situation for the battle than love, perhaps you're right. They are compatible in bed, but will argue about money, about who gets to make decisions about friends, career. It will be a little jealousy fuel to the fire. Relationship is good, but marriage is less likely.
Compatibility with Taurus: Both are a homebody, value money; believe that security is a concern. In the Taurus has patience, and Capricorn is ready to operate for a common goal - the mutual pleasure. In sexual terms the couple could conceive anything - all realized. Capricorn's ambitions perfectly coincide with the purposefulness of Taurus. Excellent prospects in all respects.
Compatibility with Gemini: Opposing sexual nature. The Gemini are impulsive and fate. Capricorn - slow, loyal, cautious. The initial attraction of opposites can not last long.
Compatibility with Cancer: Cancer is a shy sexually, but Capricorn wants to take the lead. Most frankness on both sides at first will be useful. Any understanding that emerged in the bedroom, you will need to overcome other obstacles. They are the antithesis of Zodiac with all the ensuing consequences. Capricorn will be too demanding and dominating for sensitive Cancer. It will be short duration relationship.
Compatibility with Leo: Leo believes that Capricorn is full of love and the squandering it sparingly. Capricorn is also not much of a partner for the Leo because of the unimaginative. Capricorn prefers more fire under the sheets. They are not very suited to each other physically; both are independent labels that tend to dominate outside the bedroom. During the relationship of these differences, however, may be negligible, but in a marriage - is unlikely.
Compatibility with Virgo: Practicality Capricorn and accuracy of the Virgo are in perfect harmony. Possible misunderstanding of sexual violence, but otherwise they are so well understand each other, that probably is not their concern. Both are dependent, conservative, who understand - a good foundation for long-term alliance.
Compatibility with Libra: The charm and sexual attraction at first attracted by Capricorn. However, in more familiar Libra seems to be too selfish and do not meet the physical needs of Capricorn. Not being able to express own feelings. Capricorn is looking for other partners. Relationship is not a simple one. Marriage is not complicated, only if there are no financial opportunities that meet Capricorn.
Compatibility with Scorpio: Successful sexual relationships. Scorpio is more inventive, and Capricorn - methodical, but they are in this area are compatible, and success in the bedroom opens up great opportunities. Both have a strong will, but Scorpio is inclined to dominate. Capricorn realizes that Scorpio is a sense of ownership by virtue of his own love. Good relationship, successful marriages.
Compatibility with Sagittarius: Capricorn likes to have sex in a comfortable atmosphere. Sagittarius is willing to devote to this even in the freshly dug grave. Capricorn is cautious and conservative and worried about security. Sagittarius is the player. He is extravagant and irresponsible. Sagittarius will be irritated by watching stingy and binding Capricorn. Not the best relationship. It is unsuccessful marriage.
Compatibility with Capricorn: Their romantic life quickly becomes a routine. Possible success if both tend to receive less than originally hoped. None of the experiments did expand their horizons. On the other hand, both are hard working, serious, thrifty. Relationship and marriage may not be too happy, but can satisfy them.
Compatibility with Aquarius: In the process, find out how quickly the novel becomes a friendship. In intimate matters difficult to cope with Aquarius. Aquarius prefers unorthodox sex, and Capricorn - the traditional. He is freedom-loving, resourceful. Aquarius can not long be in love in practical Capricorn. A satisfactory relationship and marriage will require efforts.
Compatibility with Pisces: Pisces will try to inflate Capricorn, because it's in their nature, but the Capricorn is able to cope with this and take the lead in bed. Pisces will soon want to follow him. Pisces, in addition, loving and sincere and will make the happiness of Capricorn. These very different people complement each other emotionally. A good forecast for the relationship or more long-term relationships.
Copyright © 2010


text:
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text:
One million more young adults live with parents than 20 years ago as they can't get their own home
One in four 20-34 year-olds now live at home - a total of 3.4 million and a third higher than the late 1990s
ONE million more young aduflts are living with their parents than two decades ago, an alarming study into the housing crisis has found.
One in four 20-34 year-olds now live at home - a total of 3.4 million and a third higher than the late 1990s, according to analysis by the Civitas think tank.
And when youngsters do finally move out they're much less likely to live alone as many of those moving out share the costs of running a home with friends or a partner.
There are 1.3 million 20-34 year-olds living solo now compared to a high of 1.8 million in 2002.
The report blames the growing trend in youngsters living with parents on soaring rents, household bills and the decline in social housing.
This "collapse in single living" partly explains why average household sizes - which had been falling for most of the 20th century - plateaued in the 2000s and have even started rising in some places.
A previous report from Lloyds Bank found the average house across UK cities in 2018 equated to 7.2 times average annual earnings, making the cost of buying a home the least affordable since 2007.
The proportion of "single-person" households has plateaued at about 30 per cent, in contrast to some other parts of Europe where single living has been increasing - to more than 35 per cent of all households in France and the Netherlands and more than 40 per cent in Germany and Denmark.
Civitas boss Daniel Bentley said: "An important consequence of the housing crisis that has gone largely unnoticed has been depressed household formation.
most read in politics
"As owner-occupation and social housing have each become more difficult to enter, hundreds of thousands of young adults have taken one look at the high rents in the private rented sector and decided to stay with their parents a bit longer instead.
"And those who have moved out have been much more inclined than in the 1990s to share, either with a partner or others."
He added: "Building new homes in line with household growth during this period would entrench the under-supply of housing for decades to come."
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How the world's leading farmers are responding to global changes, pp. 143-151.
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Petso Mokhatla, Ntsikane Maine and Wim T Nell
Sustainable farmer settlement in South Africa, pp. 57-63.
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text:
By James Case
Tyrone Duncan, a professor of mathematics at the University of Kansas, received the 2013 W.T. and Idalia Reid Prize for "fundamental contributions to nonlinear filtering, stochastic control, and the relation between probability and geometry." The prize was awarded at the SIAM Annual Meeting and control conference in San Diego, where Duncan gave the prize lecture, "Some Solvable Stochastic Control Problems." For readers who missed the talk, James Case considers highlights.
Optimal control theory began to diverge from the calculus of variations in the aftermath of World War II, when it was found that problems of the form
\[\tag{P} \mathrm{minimize}~_{u(\cdot)\in\mathfrak{U}}J=K(T,x(T))\\ + \int^{T}_{0} L(t,x(t),u(t))dt\\ \dot{x}(t)=f(t,x(t),u(t)); u(t) \in U \]
offered modeling flexibility unmatched by the older theory. If \(f(t,x(t),u(t)) \equiv u(t)\), P reduces to a problem in the calculus of variations. Here \(T\) is the first instant at which \(x(t)\) belongs to a specific "terminal manifold" \(\mathfrak M\) in space–time, \(U\) is a (frequently compact) subset of Euclidean space, and \(\mathfrak U\) is a class of admissible control histories \(u(t);0 \leq t \leq T\) taking values in \(U\). Early milestones include Donald Bushaw's Princeton thesis (1953), in which he devised a method for solving a narrow class of two-dimensional "bang-bang" control problems, L.S. Pontryagin's multidimensional maximum principle (published in the West in 1962), and R.E. Kalman's solution of the (also multidimensional) linear-quadratic "tracking" problem (1960). Initially, the problems considered were all deterministic. Only later was it realized that quite similar methods could be used to treat stochastic problems, in which a random process—most commonly Gaussian white noise \(W(t)\)—is included among the arguments of \(f\).
Kalman's linear-quadratic problems form the most exhaustively (and fruitfully) studied class of optimal control problems. In them one seeks—for a fixed termination time \(T\)—to keep the output "history" \(y(t); 0 \leq t \leq T\) of a specific system from wandering uncomfortably far from a predetermined path \(z(t); 0 \leq t \leq T\) without recourse to unduly costly control histories \(u(t); 0 \leq t \leq T\). The instantaneous output \(y(t)\) is related to the instantaneous state \(x(t)\) of a linear system \(\dot{x}(t) = A(t)x(t) + B(t)u(t)\) by the relation \(y(t) = C(t)x(t)\), where \(x\), \(y\), and \(u\) are column vectors of (constant) dimension \(N\), \(n\), and \(m\), respectively; for each \(t \in [0,T]\), \(A(t)\), \(B(t)\), and \(C(t)\) are conformable real matrices. In this case
½ < [y(t)-z(t)], Q(t)[y(t)-z(t)]>+\\
½ <u(t),r(t)u(t)>
This reduces to (2) when \(y(t)\) is replaced by \(C(t)x(t)\) in \(K\) and \(L\). The latter evidently grows with the vectors to be multiplied by the symmetric matrices \(Q(t)\), \(R(t)\), and \(M\), of which \(R(t)\) is assumed to be positive-definite for each \(t \in [0,T]\), while \(M\) and \(Q(t)\) need only be positive-semi-definite. Thus, \(J\) is invariably non-negative, vanishing only in contrived circumstances. Following Kalman, Duncan confined his remarks to the special case \(z(t) \equiv 0; C(t) \equiv I\), so that the controller's conflicting aims are to make both \(x(t)\) and \(u(t)\) small.
To obtain the complete solution of such an optimal control problem, it suffices to solve the first-order (typically nonlinear) Hamilton–Jacobi partial differential equation
V_{t}+\mathrm{min}_{(u \in R^{m})}H(t,x,u,\bigtriangledown V(t,x))=0
satisfied by the value function \(V(t,x) = \mathrm{inf}_{u(\cdot)\in\mathfrak{U}}J\). Ordinarily, the class \(\mathfrak{U}\) of admissible control histories \(u(t);0 \leq t \leq T\) must include an abundance of discontinuous functions to transform the foregoing inf into an attained minimum. Here
H(t,x,u,p) = L(t,x,u)+ < p, f(t,x,u),>
and the HJE is to be solved subject to the boundary condition
for all \((t,x) \in \mathfrak{M}\). These linear-quadratic tracking problems are by far the largest class of optimal control problems for which the foregoing program can be carried to completion. For less tractable problems, it is usually necessary either to "synthesize" the complete solution from partial solutions obtained by satisfying, for example, Euler equations or the maximum principle, or to approximate.
As Kalman demonstrated, the solution of the HJE for the linear-quadratic tracking problem is \(V(t,x) = ½ < x,P(t)x >\), where \(P(t)\) is the unique positive-definite symmetric solution of the matrix Riccati initial value problem (IVP)
Accordingly, the optimal feedback control is \(u^{*}(t) = –R^{–1}(t)B(t)P(t)x(t)\), and, once calculated, the solution \(P(t); 0 \leq t \leq T\) can be used repeatedly to compute any number of optimal pairs \((x^{*}(t),u^{*}(t)); 0 \leq t \leq T\) associated with initial states \(x(0) = x_{0}\) of interest. Should the matrices \(A\), \(B\), \(C\), \(Q\), \(R\), and \(M\) be composed entirely of constants, one can set \(T = \infty\) and employ \(u^{*}(t) = –R^{–1}BPx(t)\), where \(P\) is the unique positive-definite symmetric solution of the algebraic Riccati equation obtained from the above by setting \(\dot{P} = 0\). In that case it turns out that—under suitable conditions—the eigenvalues of the matrix \(\hat{A} = A – BR^{–1}BP\) all have negative real parts, so that the "controlled system" \(\dot{x} = \hat{A}x\) is asymptotically stable.
Linear-quadratic models are better known for their tractability than for their ability to mimic real-world situations. Nevertheless, they are frequently employed in practice and have been generalized in countless directions. Among the most interesting generalizations are those in which the (now random) state variable \(X(t)\) evolves according to the stochastic relation
while the "objective functional" \(J\) is replaced by its expected value \(\hat{J} = E(J)\). It is a remarkable fact—for which Duncan provided a partial explanation—that as long as \(W\) remains Markovian, the previous feedback control \(u^{*}(t) = –R^{–1}(t)B(t)P(t)x(t)\) is again optimal! If, however, the noise process \(W(t)\) is less well-behaved, the optimal control becomes
where \(\Sigma(t)\) is a certain linear functional (stochastic integral) of \(W(t)\) representing the minimum least-squares expected error prediction of the response of the system in question to future noise. Duncan has developed explicit constructions for \(\Sigma(t)\) under various assumptions on the noise process \(W(t)\). If the noise is standard (Gaussian) Brownian motion, for instance, the value function \(V(t,x)\) is again well defined and satisfies the (now second-order) Hamilton–Jacobi–Bellman PDE
\mathrm{min}_{u \in R^{m}}H(t,x,u,\bigtriangledown V(t,x))=0,
along with the boundary condition \(V(t,x) = K(t,x)\) for all \((t,x) \in \mathfrak{M}\). Solving for the unknown value function \(V(\cdot,\cdot)\) is then feasible. But if \(W\) is an arbitrary square-integrable process with continuous sample paths and a "filtration" \(\mathcal{F}(t);0 \leq t \leq T\), other techniques are called for.
A few words on filtration: The study of stochastic differential equations—and thus of stochastic optimal control—ordinarily takes place in a measurable space \((\Omega, \mathcal{F})\) in which \(\mathcal{F}\) denotes a \(\sigma\)-algebra of measurable subsets of \(\Omega\), to be thought of as "events" that might or might not occur. A filtration \(\mathcal{F}(t);0 \leq t \leq T\) on \(\mathcal{F}\) is then a collection of \(\sigma\)-subalgebras of \(\mathcal{F}\) indexed such that \(\mathcal{F}(s) \subset \mathcal{F}(t)\) whenever \(s < t\). \(\mathcal{F}(t)\) is thus a collection of events—growing with the passage of time—that occur at or before time \(t\). The requirement that \(\mathfrak{U}\) consist of control histories \(u(t); 0 \leq t \leq T\) that are \(\mathcal{F}(t)\)-measurable for each \(0 \leq t leq T\) then ensures that the elements of \(\mathfrak{U}\) are "non-anticipative" in the sense that they do not employ as yet unavailable information.
Rather than solve more HJB equations, Duncan resorts to a variant of the age-old "complete the square" argument to establish the optimality of the feedback control \(u^{*}(t)\) mentioned earlier, provided that
\Sigma(t)=\int^T_t \Phi_P(s,t)P(s)dW(s),
\(\Phi_{P}\) being the solution to the (linear) matrix IVP
Such solutions are often called "fundamental" solutions of (11).
Perhaps the most interesting special case of the foregoing result is one that Duncan has explored since 2006 in a series of papers with B. Pasik-Duncan, in which \(W(t)\) is a "standard fractional Brownian motion." A real-valued process\((\mathbb{B}(t);t \geq 0)\) is a standard fractional Brownian motion with Hurst parameter \(H \in (0,1)\) if it is a Gaussian process with continuous sample paths satisfying \(E[\mathbb{B}(t)] = 0\) and \(E[\mathbb{B}(s)\mathbb{B}(t)] = ½(t^{2H} + s^{2H} – |t – s|^{2H})\) for all \(s,t > 0\). The formal derivative \(d\mathbb{B}/dt\) is then called "fractional" Gaussian noise. Fractional Brownian motion differs from standard Brownian motion in that, for \(r < s < t\), successive increments \(\mathbb{B}(s) – \mathbb{B}(r)\) and \(\mathbb{B}(t) – \mathbb{B}(s)\) are no longer independent. Increments are positively correlated with Hurst coefficients \(H > ½\) and negatively correlated with \(H < ½\). Duncan pointed out that time series with correlated increments occur in the empirical study of turbulence, hydrology, earthquakes, telecommunications, and economic data, to name but a few, and displayed a formula analogous to the above for \(\Sigma(t)\) involving a fractional integral if \(H > ½\) and a fractional derivative if \(H < ½\). Stock market crashes seem to occur when the successive increments in key financial time series cease to be independent and become positively correlated.
Duncan also exhibited a few extensions of linear-quadratic regulator theory to infinite-dimensional state spaces, to cases in which the objective functional is \(\hat{J} = \mathrm{exp}(\int^{T}_{0}L(t,x,u)dt)\), and to two-player, zero-sum linear-quadratic stochastic differential games. (He did not mention many-player differential game theory, either stochastic or deterministic, as developed during the 1960s and 1970s.) Finally, he considered the control of Brownian motions in rank-one symmetric spaces, such as the (compact) two-sphere and the (noncompact) real hyperbolic space of dimension 2. In the latter, the optimal feedback control involves the hyperbolic tangent of the state variable. In each case, he was able to give specific formulas for the main objects of interest, and argued by example that, though much is known, much remains to be learned about the control of even quite simple stochastic processes.
James Case writes from Baltimore, Maryland.
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