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Elder Barsanuphius (1845–1913), a disciple of Elder Anatole (Zertsalov) and Elder Nektary, was a highly cultured man and a successful colonel before entering the monastic life at a relatively late age. Due to his purity of heart and spiritual sobriety, he was transformed by God into a grace-filled Elder virtually overnight. The period before the first World War and the Russian Revolution was a complex one. The world was seething in unprecedented passions, and one could clearly sense the approach of darkness. With a sympathetic soul Elder Barsanuphius attended to the world, and the Lord parted a curtain before him, showing him the horrors of the future.
"We live in a difficult time; we are weak and infirm but we have to encourage others. And here I am in the same condition, weak and infirm. I myself need help and consolation, but according to the obligation of my rank I have to comfort those who turn to me. They come to Optina from a world that no longer moans, as before, but howls; they come seeking consolation and soothing…. According to their faith they receive what they are seeking and leave in a state of peace." — Elder Barsanuphius
People from the world, thirsting for consolation in their sorrows, streamed to Optina. Elder Barsanuphius, in frail health but strengthened by the power of the Lord, received them, conversed with them, confessed them, and directed them to the only true path. He was a man of extraordinary spiritual vision, and could see clearly into the hearts of those who came to him.
"Elders are called clairvoyant, indicating that they're able to see the future. Yes, great grace is given to eldership—this is the gift of discernment. This is the greatest gift given by God to man. They have, in addition to physical eyes, spiritual eyes, before which the human soul is revealed. Before a man thinks of something, before a thought arises in him, they see it with their spiritual eyes, and they even see the reason for such a thought arising. And nothing is hidden from them…." — Elder Barsanuphius
Presented here, for the first time in English, is a complete biography of this great Elder, including his talks on a variety of subjects, excerpts from the journal of his cell-attendant (the future Elder Nikon, himself a great Optina Elder and a subsequent victim of the communist terror), and selections from his spiritual poetry.840 pp., illustrated, full-color cover, paperback
The Ancient Chants: All-Night Vigil - Based on the Melodies of the Valamo Monastery Performed by Orthodox Singers Valery Petrov, conductor This musical manuscript is the result of seven years'...
During the dark night of Soviet totalitarianism, holy men and women shone like stars throughout the many lands that comprised the U.S.S.R. One of these was Elder Vitaly (Sidorenko). (1928–1992)....
An engaging, illustrated guide to confession for children. This 100-page, easy-to-use aid will help a child understand and prepare for confession. Designed for both younger and older children, this book...
Saint Theophan the Recluse, while from a different era and country, has an uncanny ability to communicate with modern Westerners. Excerpted from his classic work The Path to Salvation, Raising Them Right provides...

subset: commoncrawl


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The Australian newspaper's edition detailing Tuesday morning's terror raids was available in Melbourne from 1.30am yesterday, putting the lie to claims that the paper could not have compromised a police operation to arrest a suspected terrorist cell operating in the Melbourne suburbs.
The Victoria Police Commissioner, Simon Overland, claimed both his men and the operation to arrest the suspected terrorists were put in a situation of "unacceptable risk" by the story, entitled 'Army Base Terror Plot Foiled' on the front page of The Australian's second edition, yesterday. Overland claimed that the paper was available advance of the raids across several northern Melbourne suburbs at about 4.30am.


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Alleged drone misuse at Stack Rocks reported to police
Pembrokeshire Coast project in exciting UK partnership launch
Pembrokeshire Coast Trust celebrates at Strumble Head lighthouse
Welsh language tour to reveal Carew Castle's rich history
National Park Authority Members' allowances and expenses published
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Points were typical up until the time we began doing the evening change a whole lot. In the start we were yet when a month doing the evening change, after that it began obtaining like two times in a week.
It so occurred that day came faster than later on and also we had a couple of glasses of white wine as well as cooled out in a bar. Well, I was not reviewing a lot right into all these points, actually she was over 45 years old as well as I might claim her butt was a little also level for me! You would certainly have presumed, I am a butt guy, I like them soft as well as rounded as well as absolutely nothing even more will certainly make me difficult than to see the wobble to the side and also back, as they stroll in securely suitable pants.
This is likewise a real tale, as you would certainly have thought; I function in the clinical area, so a couple of years from the initial tale when I had my initial experience, I was functioning with a fully grown registered nurse. She was 45 years old or around there as well as had 4 children. Her children were all ladies beginning with 21,17, 12 and also after that 8 years.
I did the recipes early as well as waited with a container of white a glass of wine. Quickly we were right into the food as well as the red wine, she was amazed by my skill with food preparation and also quickly since of the flavors was taking a lot more gulps from the red wine glass.
Well, I was not reviewing a lot right into all these points, truly she was over 45 years old as well as I might claim her butt was a little also level for me!
One night, she stated that it was her birthday celebration. I claimed, "Wow, satisfied birthday celebration, however we must commemorate." After that she was functioning the following day early in the early morning as well as I likewise had a very early beginning, so we chose to go for a beverage one of the list below days.
Currently I would certainly define her thoroughly. Blond hair to her shoulders, excellent reasonable skin, no imperfections from the sunlight or blemishes, no additional pounds on her body other than the butt. I would certainly state 178cm and also weight 65kg.
This is likewise a real tale, as you would certainly have thought; I operate in the clinical area, so a couple of years from the very first tale when I had my initial experience, I was dealing with a fully grown registered nurse. She was 45 years of ages or around there as well as had 4 youngsters. Her children were all women beginning with 21,17, 12 and after that 8 years.
Well, the regimen of dance to some tracks and also even more white wine obtained her in the state of mind for partying, so we struck the community. I was not actually taking a look at her at all sex-related now, however quickly I was done, suggesting hungover and also needed to make it house. I was not exactly sure if I educated her I was leaving, there was a haziness to occasions as so occurs when you have intoxicated over your limitation in the beverage division.
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Head Cashier Part-Time
It can be a smile, remembering a name, or maybe even just a wave at the end of the day. Perhaps it's an unwavering desire to know about every product and service found within our walls and on our shelves. But whatever it is for you, we want to empower you to feel confident in unleashing it. Because at the Home Depot, the world's largest home improvement retailer, we know that the difference in our associates is what makes the difference for our customers. Discover how you can help others achieve their home improvement dreams. Apply to work at The Home Depot today.
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The front end is the last opportunity we have to thank our customers and show them we appreciate their business. The Head Cashier is accountable to ensure this occurs by carrying out the responsibilities listed in this description as well as holding the Front End associates accountable to effectively carry out their duties and responsibilities.
Position Responsibilities:
We're looking for the type of associates who can tap into their inner orange. Overseeing work activity within his/her department, motivating the team to consistently demonstrate Customer FIRST behaviours.
- Driving awareness of current promotions by providing Front End associates with promotion locations so they can effectively direct customers , communicating store and Home Depot objectives to associates
- Training and supervising Cashiers (including cashiers in all remote cashier locations) and Lot Associates
- Setting clear performance expectations as well as providing on the job feedback
- Providing outstanding service to both external and internal Customers by ensuring efficient Front End and Remote Cashier operations
- Handling customer complaints in a professional manner
- Providing a safe working and shopping environment by following all safety policies & standards
- Completing specified safety training, immediately correcting hazards ; unsafe conditions or reporting conditions to the Manager On Duty, and working safely as not to endanger themselves, co-workers, vendors, or customers
Experience/Knowledge Required: The kind of orange we are looking for will possess: Minimum time in position and performance management code requirements of V2 (or P3 in previous system)
- Ability to work a flexible schedule including evenings and weekends Excellent communication skills and customer service skills
- Knowledge of all policies, procedures and duties related to customer transactions, cash handling, the service desk and lot
- 1 year of supervisory experience
- Excellent decision making ability and problem solving skills


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Episode 51
Manage episode 233887562 series 1256409
Author Mike Flynn reminds us that confidence is intentional by talking about his new book Master the Key: A Story to Free Your Potential, Find Meaning and Live Life on Purpose. Sponsored by 4Over.
52 episodes available. A new episode about every 22 days averaging 0 mins duration .


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Vricon wins Army award for One World TerrainJune 14, 2019 - - The Army Contracting Command - Orlando awarded Vricon a contract for the One World Terrain capability of the Common Synthetic Environment. The Common Synthetic Environment is the platform enabler of the Synthetic Training Environment that... Read More »
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Vricon and Cesium partner to bring users streaming 3D geodataJune 4, 2019 - - Announced at this year's GEOINT 2019 Symposium in San Antonio, Texas, Vricon and Cesium are providing 3D digital surface models through Cesium ion, starting with highly detailed datasets for six cities. The 3D Tiles free... Read More »
Vricon's Assaad al Skaff presenting at 5G World 2019!June 18, 2019 - - On 12 June at 2:20 pm, Vricon's Assaad al Skaff will share his insights on 3D geodata maps at 5G World 2019 in London. He'll be supporting the Here & Now track at the conference.... Read More »
Vricon to move Swedish offices to Ebbepark in 2021June 18, 2019 - - Vricon signed an agreement to expand its offices to the new Ebbepark office park under construction in Linköping. The agreement with Vricon is another step in establishing Ebbepark as a hub for business ideas, innovation,... Read More »
Major telco company partners with Vricon to accelerate 5G rollouts across the USANovember 14, 2018 - - [caption id="attachment_819" align="alignleft" width="300"] Vricon 3D building and vegetation vectors in Atlanta, USA. The Vricon Telco Suite leverages Big-Data and automation to reduce project cost and speed up implementation for large areas. Nationwide deliveries are... Read More »
Vricon featured in Defence Web – Africa's leading defence news portalOctober 26, 2018 - - Defence Web interviews Vricon Senior Director for Business Development, Tommy Hultin, at the African Aerospace and Defence Conference. Tommy discusses the various geodata use cases for government agencies in the African market. Check out the full... Read More »
Vricon announces contract directly supporting first respondersOctober 11, 2018 - - 11 October 2018—McLean, VA. Vricon announces a new contract to support the North Central Texas Council of Governments (NCTCOG) Regional 9-1-1 Program. This represents Vricon's first purchase by a regional government entity, as well as the first contract... Read More »
Vricon's disruptive 3D GEOINT technology fuels company growthJune 18, 2018 - - Vricon, a global leader in 3D data and solutions, today announced that it has more than doubled its year-over-year sales growth rate and set a record pace for new customer acquisition. While greatly increasing and... Read More »
Experience 3D analytics at GEOINT 2018 SymposiumApril 24, 2018 - - Vricon and Esri, the global leader in spatial analytics, are teaming to bring the best of 3D analysis to warfighters, analysts, and users at this year's GEOINT 2018 Symposium in Tampa, Florida. Working with 3D... Read More »
Vricon unveils new solution at GEOINT 2018: Vricon Precision 3D Registration (P3DR)April 23, 2018 - - Vricon is pleased to announce that it is unveiling the next in a series of GEOINT solutions — Vricon Precision 3D Registration (P3DR) — at this year's GEOINT 2018 Symposium in Tampa, Fla. Vricon P3DR... Read More »


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NUFORC Home Page
Web Report Indexes : by Event Date | by Location | by Shape | by Posting Date
|Sighting Report
|Occurred : 6/11/2018 21:20 (Entered as : 06/11/18 21:20)
Reported: 6/12/2018 9:45:51 AM 09:45
Posted: 6/15/2018
Location: Bethpage, NY
Shape: Formation
Duration:30 minutes +
|Clusters of orange lights seen over Nassau County, LI, for period of over 30 minutes.
Smoking cigar on my terrace, noticed a large number initially (10+) of bright orange dots rising from the horizon to the south/southwest in small clusters, eventually fading from view. Additional clusters of these lights rose from the horizon every few minutes for the next half hour or so. They seemed to be quite a distance away from my position.
Took 2 pictures with my Samsung phone, also 2 videos, video quality very noisy.


text:
Hawkins Construction chief executive Gary Walker and INZIDE Commercial co-owner Steve Aschebrock have joined the board of the Green Building Council.
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My Little Pony: Lyra and Bon Bon and the Mares from S.M.I.L.E. by G. M. Berrow
My Little Pony: Lyra and Bon Bon and the Mares from S.M.I.L.E. G. M. Berrow ebook
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Publisher: Little, Brown Books for Young Readers
Page: 160
MLP:FIM Imageboard - Image #1039900 - bon bon, book, equestria daily, g.m. Is Now Available for News, Videos, Pictures which are all about Ponies! I don't own MLP, so please don't C&D! Images tagged lyra and bon bon and the mares of s.m.i.l.e. You share, find and discover new art and media surrounding the show My Little Pony: Friendship is Magic. Size of this preview: 330 My Little Pony chapter books (licensed by Hasbro) are a series of novellas based on the show See full list > Categories: Lyra Heartstrings images. My Little Pony: Lyra and Bon Bon and the Mares from S.M.I.L.E. Next on our list of ponies just happens to be Princess Luna as the Winter My Little Pony: Lyra and Bon Bon and the Mares from S.M.I.L.E. This was also the case with a book called: Lyra and Bon Bon and the mares from S.M.I.L.E. Berrow, Paperback, bol.com prijs 8,49, Nog niet verschenen - reserveer een exemplaar. MLP - Bon-Bon and Lyra by JoeHellser.deviantart.com on @deviantART. Archive with best MLP creations. MLP - Bon-Bon and Lyra MLP - Bon Bon Super Cute Smile Puppet Rigs More Mlp Fim, Lyra Hearts, Candy, Forever Brony, Candy Mare, Mlp Vortex, Bon Bon Mlp. Size: 843x540 | Tagged: bon bon, book, equestria daily, g.m. Lyra and Bonbon and the Mares from S.M.I.L.E. File:Lya and Bon Bon and the Mares from SMILE book cover.jpg. My Little Pony: Lyra and Bon Bon Paperback. Lyra looked over at Bon Bon and put on a pretend-hurt expression. My Little Pony: Daring Do and the Forbidden City of Clouds · Cover Image Not My Little Pony: Lyra and Bon Bon and the Mares from S.M.I.L.E. Discussions in /r/mylittlepony.
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School Day Tours
We invite all families interested in Gateway School to visit, ask questions, and get to know our community!
Schedule a private tour to experience our classrooms in action. You will have a guided tour of our Kindergarten through 8th grade and co-curricular specialist classes. Visiting our classrooms in session provides you with an opportunity to get a real feel for the teaching, learning, and community at Gateway. We recommend that tours be attended by adults only; prospective students will have the opportunity to visit Gateway School during the application process.
Thank you for your interest in Gateway School! We look forward to getting to know you when you come for a visit.


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Stunning silver moon Christmas tree decoration. This personalised reflective mirror bauble captures the twinkle of your fairy lights and the engraved moon glows with light.
We love this little moon disc bauble and how it hangs on the Christmas tree changing in the twinkle of fairy lights. You can have it personalised with a name to make an extra special moon bauble for your family Christmas tree.
It has cut out craters and engraved moon details. The light glows through the frosted engraving and the mirrored part shines with reflections of all the sparkles on your Christmas tree.
We have other quirky moon themed goodies in the Newton and the Apple shop. including personalised moon phase gifts.
MADE FROM: The moon is made from silver mirror acrylic. This is not a glass mirror so it does not have sharp edges. The ribbon is a luxurious silver grey grosgrain.
DIMENSIONS: This personalised silver moon bauble decoration has a diameter of 73mm. Perfect size for a Christmas tree.
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New Listing - 6/7/2019
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21 India Street, #28-K, Brooklyn, NY, New York 11222. Be the first resident in this 1124sf, 2 Bedroom, 2 Bathroom condo with 90sf private terrace, in the hottest New Development in Greenpoint. Unparalleled views, overlooking the East River, Empire State Building and Midtown Manhattan, One World Trade and Downtown Manhattan, and the Williamsburg Bridge. The open kitchen and dining/living area features wall-to-wall floor-to-ceiling windows, Miele appliances, and Statuario marble counter/backsplash. King-size master bedroom features 2 walls of windows, and en suite bathroom with double sinks, oversize stall shower, Dornbracht fixtures, and full-length mirror/medicine cabinet. 2nd bedroom also king-size, and 2nd bath features marble and soaking tub. High ceilings, wide-plank walnut floors, great closet space, individual-room climate control, and Bosch washer and dryer in-unit.The Greenpoint is like nothing else in Brooklyn. Enjoy 30,000 square feet of exclusive amenities, including 24hr concierge, condo-only elevators, valet parking on-site, fitness center with basketball court, co-working space, lounge, and sundeck. Commute to anywhere you need to be with the subway close by, and the ferry adjoining the building. Contact us now, and let's get you your new home.
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Classic Twinbrook Forest Contemporary home has been beautifully updated throughout and modernized for the way we live today. Updated kitchen with Corian counter top, stainless steel appliances and breakfast bar. Two bedrooms and full bath on the main level while the entire upper level is devoted to the master bedroom retreat. Lots of closet space, cathedral ceilings and a fabulous bath with jetted tub, separate shower and double vanity sink. Big fenced backyard with deck is a great place for the kiddos and pups to play. Close to Montgomery County Offices or Rockville town center with movies, restaurants, shopping, shopping, shopping! Great commuter location for Bethesda and Washington DC too! Priced-to-sell! UNITED STATES BANKRUPTCY SALE.
Single Family, Single Family-Detached
2 Full
0.15 Acres
Gas Water Heater, Public Water Service
Vinyl Siding
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1,309 sq. ft
Yes -1 space
Twinbrook Forest
School Information
Montgomery County Public Schools
Detailed Room Information
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Module - Language, Identity and culture Play- Shafana and Aunt Sarrinah, Alana Valentine which is the prescribed text 1. How does the play Shafana and Aunt Sarrinah invite you to question cultural assumptions? 2. How does the language used in play Shafana and Aunt Sarrinah express the complexities of personal, Social and Cultural identity? - have an introduction, body and conclusion. - Structure your paragraph effectively eg. STEEL
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The featured performer for this event is the Underside Pattern. The band consists of Colin Langstraat, Jonah Falk, as well as Craig and Corey Gomez. This band considers themselves as, "time travelers from the 70's" Their infectious melodies and harmonies are perfect for every generation. Make sure you come to the Patio and check this band out!
Start Date:
September 4, 2019
End Date: September 4, 2019
Start Time: 4:30 PM
End Time: 6:30 PM
Cost: $Free
Website: Event Website Link
Relocation of the Red Deer Downtown Business Association The Red Deer Downtown Business Association is pleased to announce that it will be moving locations to 5009 50 Avenue on Little Gaetz Avenue on June 6.Read More


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Saints Fun Facts: St. Barbara
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St. Barbara
Saint Barbara, also known as the Great Martyr Barbara, was an early Christian Greek martyr born mid-third century in Heliopolis, Phoenicia. Barbara was the daughter to a rich pagan named Dioscorus. After the death of Barbara's mother, Dioscorus dedicated his life to his only daughter. Barbara was known for being extremely beautiful. As a way of "protecting" her, Dioscorus shielded her from the world by locker her high up within a tower. He allowed only her pagan teachers to see her. Barbara ... continue reading
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St. Addal
A disciple of Christ sent by St. Thomas to the court of King Abgar the Black, the second century Osroene ruler. Legendary accounts claim Abgar wrote to Christ asking Jesus to cure him of an intolerable and incurable illness. Abgar's court was in Edessa in Asia Minor (modern Turkey). Addal cured Abgar and converted the king and his people to the faith. One of these was Addai, who became Addal's successor. Addal is also supposed to have sent another disciple, Man, to various sites along the ... continue reading
St. Ado of Vienne
An archbishop and scholar, Ado was born in Sens and educated at the Benedictine abbey of Ferrieres. Abbot Lupus Servatus, an outstanding humanist of the time, trained Ado, and was impressed with the obvious holiness of the young man. A noble by birth, Ado renounced his inheritance and became a Benedictine, in time assigned to the monastery of Prum, near Trier, Germany. Ado's holiness made him enemies, and he was forced to leave Prum. He went to Rome on a pilgrimage and remained there for two ... continue reading
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An elder Mexican man makes his way to Mass in the early morning twilight of December 9, 1531. He is a peasant, a simple farmer and laborer, and he has no education. Born under Aztec rule, he is a convert to Catholicism, and each step he takes this morning is a step ... continue readingMore Female Saints
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St. Gabriel is an angel who serves as a messenger for God to certain people. He is one of the three archangels. Gabriel is mentioned in both the Old and the New Testaments of the Bible. First, in the Old Testament, Gabriel appears to the prophet Daniel to explain his ... continue reading
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The holy men and women are also called the "Protomartyrs of Rome." They were accused of burning Rome by Nero , who burned Rome to cover his own crimes. Some martyrs were burned as living torches at evening banquets, some crucified, others were fed to wild animals. ... continue reading
St. Francis of Assisi
Founder of the Franciscan Order, born at Assisi in Umbria, in 1181. In 1182, Pietro Bernardone returned from a trip to France to find out his wife had given birth to a son. Far from being excited or apologetic because he'd been gone, Pietro was furious because she'd ... continue reading
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Pregnancy milestone cards include mommy's favourite food, baby's first kick as well as my first photo with mommy/daddy and grandparents. Such a gorgeous way to track your pregnancy and baby's birth


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Fantasy Coin, LLC; a Virginia based gaming components company was founded in 2012 with the intent to have the best interest of the gamer community in mind while designing products. High quality yet affordable coin sets for many board games, LARP, RP, small film production, and just for fun.
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What makes a word beautiful? Is it merely how it sounds? Or does a word's meaning affect its aesthetic effect? Max Beerbohm had some helpful thoughts about gondola, scrofula, and other words in his essay "The Naming of Streets." Several years ago, Grant wrote a column on this topic for The New York Times. This is part of a complete episode.
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- Had the Radish (episode #1527) 06/21/2019: Your first name is very personal, but what if you don't like it? For some people, changing their name works out great but for others... [more]
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text: Hey all, Just curious, for those of you who are matching your SX(es) to a deckhand rod, how are you mounting them? included/default/factory clamp? Cork Puppy? Trigger Grip (like the one sold at Charkbait)? I was using the clamp that came with mine but don't like how tight I have to screw them down. I had one come loose too years ago when fighting a yellow - that was no fun lol. I asked Mark @ Charkbait and he said the trigger grip they sell may not be the best fit. Will the cork puppy be the best bet for this then?


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Sexi undertøy swingers gangbang4 5, in the music video Meister appears nude in some parts, which created a stir in media. "Linni med ny singel". 1 2 "Fikk sparken etter sexvideo". Giftet seg i all hemmelighet - Her er Linnis hjerteknuser External links Edit Retrieved from " p?titleLinni_Meister oldid ". "Se puppene til Linni". 7 Her professional music career began in summer 2007 after her participation in Robinsonekspedisjonen 2007, when she recorded a cover of the Destiny's Child song " Survivor which was released by CC Records. 10 The video was uploaded to various pornographic websites within hours of being stolen. "BB-Kristian og Linni gir ut hjemmeporno".
1 3, meister is most known abroad for her music single "My Ass which was released as promotion for the horror-comedy film. 8 It was panned by critics, amongst them the main Norwegian newspaper. The film contains images from their trip abroad in 2006. Archived from the original on June 1, 2009. 1 2 3 4 "Slaktet og sensurert". Retrieved March 19, 2011.
TV2 youth program, waschera, where she tried out a number of jobs to more or less success. A b "Fikk sparken etter sexvideo". 1 2 "Robinson exit". The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files. 11 Meister tried to make a deal with pornography producer Thomas Rocco Hansen so she could earn money for the film, but the deal never happened and the situation lead to Meister being sacked from Team Aylar.


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Read more: Aging Network Planning Tools, Aging / General Issues, Health, HIV & Aging, Housing / Homelessness / Long Term Care, Aging Providers, LGBT Organizations, LGBT Older Adults
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May 2019 | Vicki Stanley, SAGECare Trainer
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Year Built: 1993.


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Rashad Jennings' 180 Round Table
Our first annual Rashad Jennings' 180 Table will take place on Thursday, July 11th from 6:30-8:30pm at E.C. Glass High School in Lynchburg, Virginia. This event will be a panel discussion, talking about multiple topics regarding the community. The 180 Table will be interactive and audience engaging. Come on out to join us at this free of charge and open to the public event!


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Part of a series of articles on
The mathematical constant e
Natural logarithm · Exponential function
Applications in:
compound interest · Euler's identity & Euler's formula ·
half-lives & exponential growth/decay
Defining e:
proof that e is irrational ·
representations of e · Lindemann–Weierstrass theorem
John Napier · Leonhard Euler
Schanuel's conjecture
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for any real number x,
where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively, with the argument x given in radians. This complex exponential function is sometimes denoted cis(x) ("cosine plus i sine"). The formula is still valid if x is a complex number, and so some authors refer to the more general complex version as Euler's formula.[1]
Euler's formula is ubiquitous in mathematics, physics, and engineering. The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics."[2]
It was Johann Bernoulli who noted that[3]
And since
the above equation tells us something about complex logarithms.
Bernoulli, however, did not evaluate the integral.
Bernoulli's correspondence with Euler (who also knew the above equation) shows that Bernoulli did not fully understand complex logarithms.
Euler also suggested that the complex logarithms can have infinitely many values.
Meanwhile, Roger Cotes, in 1714, discovered that
(where "ln" means natural logarithm, i.e. log with base e).[4]
We now know that the above equation is true modulo integer multiples of 2\pi i, but Cotes missed the fact that a complex logarithm can have infinitely many values due to the periodicity of the trigonometric functions.
It was Euler (presumably around 1740) who turned his attention to the exponential function instead of logarithms, and obtained the correct formula now named after him.
It was published in 1748, and his proof was based on the fact that the two Taylor series on each side of the formula are equal.[5]
Neither of these mathematicians saw the geometrical interpretation of the formula: the view of complex numbers as points in the complex plane arose only some 50 years later (see Caspar Wessel).
This formula can be interpreted as saying that the function eix traces out the unit circle in the complex number plane as x ranges through the real numbers. Here, x is the angle that a line connecting the origin with a point on the unit circle makes with the positive real axis, measured counter clockwise and in radians.
The original proof is based on the Taylor series expansions of the exponential function ez (where z is a complex number) and of sin x and cos x for real numbers x (see below). In fact, the same proof shows that Euler's formula is even valid for all complex numbers x.
A point in the complex plane can be represented by a complex number written in
cartesian coordinates. Euler's formula provides a means of conversion between cartesian coordinates and polar coordinates. The polar form simplifies the mathematics when used in multiplication or powers of complex numbers. Any complex number z = x + iy can be written as
\phi \, is the argument of z—i.e., the angle between the x axis and the vector z measured counterclockwise and in radians—which is defined up to addition of 2π. Many texts write tan−1(y/x) instead of atan2(y,x) but this needs adjustment when x ≤ 0.
Now, taking this derived formula, we can use Euler's formula to define the logarithm of a complex number. To do this, we also use the definition of the logarithm (as the inverse operator of exponentiation) that
and that
both valid for any complex numbers a and b.
Therefore, one can write:
for any z ≠ 0. Taking the logarithm of both sides shows that:
and in fact this can be used as the definition for the complex logarithm. The logarithm of a complex number is thus a multi-valued function, because \phi is multi-valued.
Finally, the other exponential law
which can be seen to hold for all integers k, together with Euler's formula, implies several trigonometric identities as well as de Moivre's formula.
Euler's formula provides a powerful connection between analysis and trigonometry, and provides an interpretation of the sine and cosine functions as weighted sums of the exponential function:
The two equations above can be derived by adding or subtracting Euler's formulas:
and solving for either cosine or sine.
These formulas can even serve as the definition of the trigonometric functions for complex arguments x. For example, letting y = ix, we have:
Complex exponentials can simplify trigonometry, because they are easier to manipulate than their sinusoidal components. One technique is simply to convert sinusoids into equivalent expressions in terms of exponentials. After the manipulations, the simplified result is still real-valued. For example:
\cos x\cdot \cos y & = \frac{(e^{ix}+e^{-ix})}{2} \cdot \frac{(e^{iy}+e^{-iy})}{2} \\
& = \frac{1}{2}\cdot \frac{e^{i(x+y)}+e^{i(x-y)}+e^{i(-x+y)}+e^{i(-x-y)}}{2} \\
& = \frac{1}{2} \bigg[ \underbrace{ \frac{e^{i(x+y)} + e^{-i(x+y)}}{2} }_{\cos(x+y)} + \underbrace{ \frac{e^{i(x-y)} + e^{-i(x-y)}}{2} }_{\cos(x-y)} \bigg] \
Another technique is to represent the sinusoids in terms of the real part of a more complex expression, and perform the manipulations on the complex expression. For example:
\cos(nx) & = \mathrm{Re} \{\ e^{inx}\ \}
= \mathrm{Re} \{\ e^{i(n-1)x}\cdot e^{ix}\ \} \\
& = \mathrm{Re} \{\ e^{i(n-1)x}\cdot (e^{ix} + e^{-ix} - e^{-ix})\ \} \\
& = \mathrm{Re} \{\ e^{i(n-1)x}\cdot \underbrace{(e^{ix} + e^{-ix})}_{2\cos(x)} - e^{i(n-2)x}\ \} \\
& = \cos[(n-1)x]\cdot 2 \cos(x) - \cos[(n-2)x] \
This formula is used for recursive generation of cos(nx) for integer values of n and arbitrary x (in radians).
See also Phasor arithmetic.
In the language of topology, Euler's formula states that the imaginary exponential function t\mapsto e^{it} is a (surjective) morphism of topological groups from the real line \mathbb R to the unit circle \mathbb S^1. In fact, this exhibits \mathbb R as a covering space of \mathbb S^1. Similarly, Euler's identity says that the kernel of this map is \tau\mathbb Z, where \tau = 2\pi. These observations may be combined and summarized in the commutative diagram below:
In differential equations, the function eix is often used to simplify derivations, even if the final answer is a real function involving sine and cosine. The reason for this is that the complex exponential is the eigenfunction of differentiation. Euler's identity is an easy consequence of Euler's formula.
In electronic engineering and other fields, signals that vary periodically over time are often described as a combination of sine and cosine functions (see Fourier analysis), and these are more conveniently expressed as the real part of exponential functions with imaginary exponents, using Euler's formula. Also, phasor analysis of circuits can include Euler's formula to represent the impedance of a capacitor or an inductor.
The exponential function ex for real values of x may be defined in a few different equivalent ways (see Characterizations of the exponential function). Several of these methods may be directly extended to give definitions of ez for complex values of z simply by substituting z in place of x and using the complex algebraic operations. In particular we may use either of the two following definitions which are equivalent. From a more advanced perspective, each of these definitions may be interpreted as giving the unique analytic continuation of ex to the complex plane.
For complex z
Using the ratio test it is possible to show that this power series has an infinite radius of convergence, and so defines ez for all complex z.
Various proofs of the formula are possible.
Here is a proof of Euler's formula using power series expansions
as well as basic facts about the powers of i:[6]
i^0 &{}= 1, \quad &
i^1 &{}= i, \quad &
i^2 &{}= -1, \quad &
i^3 &{}= -i, \\
i^4 &={} 1, \quad &
i^5 &={} i, \quad &
i^6 &{}= -1, \quad &
i^7 &{}= -i,
and so on. Using now the power series definition from above we see that for real values of x
e^{ix} &{}= 1 + ix + \frac{(ix)^2}{2!} + \frac{(ix)^3}{3!} + \frac{(ix)^4}{4!} + \frac{(ix)^5}{5!} + \frac{(ix)^6}{6!} + \frac{(ix)^7}{7!} + \frac{(ix)^8}{8!} + \cdots \\[8pt]
&{}= 1 + ix - \frac{x^2}{2!} - \frac{ix^3}{3!} + \frac{x^4}{4!} + \frac{ix^5}{5!} - \frac{x^6}{6!} - \frac{ix^7}{7!} + \frac{x^8}{8!} + \cdots \\[8pt]
&{}= \left( 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \frac{x^8}{8!} - \cdots \right) + i\left( x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \right) \\[8pt]
&{}= \cos x + i\sin x \ .
In the last step we have simply recognized the Maclaurin series for cos(x) and sin(x). The rearrangement of terms is justified because each series is absolutely convergent.
An alternative proof[7] starts from the limit definition of e^z:
Substitute z=ix, and let n be a very large integer. Then consider the sequence:
(The last element of the sequence approaches eix.) If the points of this sequence are plotted in the complex plane (see animation at right), they roughly trace out the unit circle, with each point being x/n radians counterclockwise of the previous point. (This statement is more and more accurate as n increases. The proof is based on the rules of trigonometry and complex-number algebra.[7]) Therefore, in the limit n→∞, the last point in the sequence, (1 + ix/n)n, is the point on the unit circle of the complex plane located x radians counterclockwise from +1, that is the point cos x + i sin x. Therefore, eix = cos x + i sin x.
Another proof[8] is based on the fact that all complex numbers can be expressed in polar coordinates. Therefore for some r and \theta depending on x,
Now from any of the definitions of the exponential function it can be shown that the derivative of e ^{ix} is i e ^{ix}. Therefore differentiating both sides gives
Substituting r (\cos(\theta) + i \sin(\theta)) for e^{ix} and equating real and imaginary parts in this formula gives \textstyle \frac{dr}{dx} = 0 and \textstyle \frac{d\theta}{dx} = 1. Together with the initial values r(0) = 1 and \theta(0) = 0 which come from e^{i0} = 1 this gives r=1 and \theta=x. This proves the formula e^{ix} = 1(\cos(x)+i \sin(x)).


text:
Autech's version of the Leaf features exclusive styling updates and will go on sale in Japan next month.
Interest is sedans remains high, even though Ford and GM are abandoning some segments.
The cabin of this Nissan GT-R features blood red leather with black inserts, ambient lighting and plenty more.
If you can't beat the SUVs, you might want to kinda join them instead of quit.
Both the NSX and the GT-R are two of the fastest-accelerating cars in the world.
Renault's Chairman and CEO also get from one seat in Nissan's new board.
The system allows hands-off driving on highways and will debut in Japan this fall.
Following its debut at Overland Expo West, the Ultimate Parks Titan will be donated to the Grand Canyon Conservancy.
While the Destination Frontier is a one-off, Nissan says fans could build their own for around $40,000.
Ghosn and his camp argue that the ultra-expensive party was held for business purposes.
Overall, sports cars were the most deadliest type of vehicle as they were responsible for the highest rate of fatalities.
With a single owner since new, original condition and a $5,500, it's for sale in Canada.


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KATIE Price has come under fire because of the limited sizes available in her new fashion range.
The 41-year-old former glamour model recently launched her new summer capsule for JYY London, which includes two piece co-ords, tracksuits and denim jackets.
5 Katie Price has come under fire for the sizing in her new fashion rangeCredit: Instagram
But customers are furious the collection is only available in sizes eight to 14, especially as the star has had liposuction and has recently been promoting weight loss drinks.
The description for the Crochet Metallic Short Set reads: "One size gits UK 8-14."
While other garments listed a large as the equivalent to a dress size 12 to 14.
Shoppers have vented their frustration on Twitter, with one writing: "A large is considered a 12/14 really? No wonder girls think they are fat."
5 The former glamour model's collection with JYY London is only available in sizes 8 to 14Credit: Instagram
5 Katie's customers are furious over the sizing and have asked if it comes in plus sizeCredit: Instagram
A second tweeted: "It's a joke, making girls feel even more insecure about ourselves."
A third added: "Couldn't agree more. I nearly chocked on my chocolate when I saw it."
The sizing has also swayed people from purchasing swimwear too.
A separate shopper wrote: "I was going to order a bikini but I'm not bothering because of the sizing."
5 Katie often promotes weight loss drinks and has undergone lipo in the pastCredit: Instagram
While another questioned if there items were available to buy in plus size.
They asked: "Do you do this in plus sizes?"
To which another social media user replied: "Don't take this wrong, but define plus size? As according to her a size 12 is a large. It's a joke."
Katie Price strips down and dons leopard print swimsuit on sizzling shoot in Spain
COBBLES ICON dead Coronation Street star Irene Sutcliffe dies aged 88 BUZZ KILL BGT in major gaffe as leaderboard puts the WRONG acts through to the final BARRY THE HATCHET Gavin and Stacey's James and Mat back on track ahead of Christmas special ExclusiveNO TO PORNO Sheridan Smith quits Channel 4's Adult Material series on British porn industry GOT YOUR NUMBER BGT blunder as magician predicts WRONG numbers – but Ant & Dec don't notice FERRY TALE ENDING Chloe Ferry reveals she's QUIT Geordie Shore after split from Sam Gowland ExclusiveGOOD NEIGHBOUR Woman who collapsed was stunned to find she was treated by Jason Donovan wonder web BGT judge Amanda Holden wears a SPIDERWEB on her boob ExclusivePAYING THE PRICE Katie Price 'furious' as former best pal claims she's 'having a breakdown' PicturedBASQUE TO HER BEST Cheryl stuns in sexy lace corset & reflects on 'crazy' Girls Aloud years
However, Katie's collection is not the only one with limited sizes, as other lines by other celebrities also stop at a dress size 14, when the average dress size is a 16.
Katie often shares images of her sporting merchandise from the label, but customers are even more frustrated because Katie has recently undergone liposuction to slim down.
The mum of five has recently returned from a trip to Turkey where she underwent a face lift, and bum lift, but she has also previously opened up about having other cosmetic treatments, including lipo and a string of boob jobs.
Got a story? email firstname.lastname@example.com or call us direct on 02077824220.
We pay for videos too. Click here to upload yours.


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Home of the Cougars!
(A) Wednesday Jan 8 @ 6:30PM
vs Kearsarge HS
@ Kearsarge HS
(A) Friday Feb 14 @ 7:00PM
vs Stevens HS
@ Stevens HS
ConVal Regional High School
184 Hancock Rd.
Peterborough, NH 03458
Privacy Policy Terms of Use
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