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As a dedicated walker, I get an opportunity to observe joggers every day as they come slogging past me. And based upon years of study I can confirm that there is one obvious difference between female joggers and male joggers.
It's not speed, or running style — it's attire.
Women runners clearly have paid attention to what they are wearing. Their outfits are colorful and stylish, and they've clearly put time and thought into the clothing effort. Often their running shoes match their headwear, or gloves. Earlier this week I saw a young woman runner sporting a terrific color-coordinated ensemble with bright white shoes, striped red and white leggings, a red fleece jacket, and bright white earmuffs and gloves. She looked like she was jogging to the mall to get her picture taken with Santa Claus.
The guy joggers? Not so much. They've apparently just rolled out of bed and rummaged through the dirty laundry pile to try to find their "lucky" sweatpants and a ratty sweatshirt. They pretty much look like crap as they go huffing and puffing around Schiller Park, and they obviously don't care about it one bit.
Since I have never been a jogger, I can't speak to motivation when it comes to running, but I can certainly see how selecting the right outfit might actually contribute to the exercise effort. Careful attention to clothing is bound to help with the mental preparation for the run, and if you feel like you look ready for a good run it's more likely that you are ready for a good run — just like kids who wore Red Ball Jets sneakers back in the '60s felt like they could "run faster and jump higher" in those shoes than in others.
The guy joggers need to up their game.
Yesterday I got one of the endless number of emails trying to sell me something that bombard my inbox. This one was trying to sell me "Barbie PJs." The picture showed what looked to me like standard PJs that were pink with a silhouette of a Barbie head on the top. "Hmmm," I thought idly, "I wonder if Kish would like those, or hate them." And then I hit the delete button.
I wish I could effectively communicate to that company, and others that try to sell me women's clothing, how absolutely unlikely I am to buy anything they're offering. I haven't bought Kish any kind of garment — or footwear, hats, you name it — for more than three decades, because I long ago learned that I have no sense of fashion and really don't know what she likes and what she doesn't like on the apparel front. In short, if an item can be donned or doffed, I'm far out of my depth.
This profound condition of clothing cluelessness became clear when I tried to buy Kish some clothes one long-ago Christmas, and each purchase — boots, a blouse, a winter cap — was a miserable failure that she looked at quizzically. "How did you happen to buy purple boots?" she asked after opening one of the presents. "I thought purple was your favorite color," I stammered in response. "No, it's green," she said.
Fortunately, I had retained all of the receipts for the ill-advised gifts, so she was able to return them and get some things she really liked and wanted — and we moved forward with the implicit understanding that I would never again try to buy clothes for her. In fact, I've always suspected that the "returns" department at stores was created by a department store proprietor who, after totally flubbing some gift for his wife or girlfriend, realized that there was a desperate need for a special area where puzzled women could discreetly return the reckless clothing purchases of misguided males.
So don't try to sell me "Barbie PJs," or poofy fashion scarves, or knee-high boots. Those kinds of purchases fall entirely into the "Kish self-purchase category." I'll happily buy her objects, or even perfume if I receive sufficiently explicit instruction that can be communicated to the helpful saleswoman at the perfume department at Nordstrom's. Attire, however, is in the danger zone.
The act of tying a tie is a simple one — and also a pain for those of us who toil in jobs where we still are expected to wear a piece of fabric cinched around our necks — but that doesn't make its successful accomplishment any less satisfying.
For most of us unfortunates, the act of tying your tie to get ready for work is as rote as tying your shoe or starting the car in the morning. The process is so automatic and ingrained you don't even think about the individual steps.
II don't know the name of my tie-knotting technique and whether it produces a Windsor knot, a Four-in-hand, or something else. I just know that the chosen cravat is placed over my shoulders with the wide end on one side and the narrow on the other, and the relative length of each is adjusted by instinct. The wide end then is looped around the narrow, popped through a hole directly under my chin, and flopped on top of the narrow end and drawn down to make a reasonably acceptable knot. The last step is to tug down the narrow end until the gap between the tie and the shirt collar is closed and the button is no longer visible. Voila!
If I can accomplish this and avoid the dreaded "Oliver Hardy" look — where the narrow end is longer than the wide end, which ends up flapping forlornly on the belly — while also having the wide end reach belt level, the operation was a success. Extra points if I meet those goals and also produce the perfectly centered dimple.
It's the little things, especially on a Thursday morning.
My work winter coat is a navy blue cloth greatcoat that extends down to about knee level. It's sturdy and warm and reasonably professional looking — and also seems to magically attract every strand of white dog hair in our household.
When you live in a house with two dogs, dealing with dog hair is just part of life. When we're gone our dogs jump up on chairs, flop down on rugs, and leave their fur behind. You can brush their coats regularly, sweep and vacuum repeatedly, and flap out rugs until you can flap no more, but dog hairs are always going to be there, ready to leap onto any item of dark clothing and make you look like a vagabond who's been sleeping with a pack of strays in a downtown alley.
In our household, we deal with the dog-fur-on-clothing issue by owning approximately two dozen adhesive rollers designed specifically to remove hair from garments. (Of course, the plastic handles of the rollers have all been chewed to smithereens by our dogs, which is just another fun thing about life with dogs.) Although the rollers are designed to remove hairs and pick up most random items, they don't do an especially good job on dog hairs. The only real way to remove dog hairs from your coat is to exercise your fine motor skills and individually remove them, hair by stinking hair.
That is because dog hairs are clingier than your two-year-old at his first terrifyingly large family reunion. Dog hairs have a special bonding property that makes them stick — well, doggedly — to any dark item of clothing. In fact, I wouldn't be surprised if the NSA is attempting to build listening devices into strands of dog fur to assist in its surveillance programs. Because the NSA apparently monitors just about everything, and dog hairs can be found just about anywhere, it seems like a match made in heaven.
When the dogs and I stepped outside this morning, my breath was faintly visible and a distinct chill was in the air. When we got home I checked the weather app on my iPhone and saw that the temperature is supposed to reach the 50s and perhaps touch the low 60s today. I therefore declare this First Sweater Day.
It is not officially recognized, of course, but it is important nevertheless. It marks the first day that I can wear a sweater to work. It has to be on a Friday, when the casual dress rule prevails, and it can't be declared too early, when wearing a sweater would leave me uncomfortably warm. It arrives in Columbus at a different date than it would in, say, Minneapolis or Portland, Maine, where First Sweater Day probably came weeks ago. But when the air outside has begun to acquire that gelid feel, and leaves are scattered on the sidewalks and roadways, and pumpkins appear on the porches of your neighbors, you know that First Sweater Day is here.
First Sweater Day is one of those real-life demarcations of the seasons, just like the start of school or the beginning of the fall TV season used to be. Donning a sweater today will help me to stave off wearing any kind of overcoat for a while, and from now on, as we move more deeply into the layering season, sweaters will be a staple of the wardrobe.


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Are We in the Throes of an Evolutionary Process!
A thousand goals have there been hitherto, for a thousand peoples have there been. Only the fetter for the thousand necks is still lacking; there is lacking the one goal. As yet humanity hath not a goal. But pray tell me, my brethren, if the goal of humanity be still lacking, is there not also still lacking- humanity itself? - Nietszche
By Hwaa Irfan
The impetus to continue life as we have understood it is prevalent for many of us, especially those who are not ready for change. Our perceptions of the world has we have been entrained to believe by modern science and medicine infused with politics and self interested entities, has given us no reason to believe otherwise, unless we step off the conveyor belt of our lives and take an overdue deep intake of breathe.
After all what does it matter if the universe as we have understood it is in the midst of change? This is the life we have been given to make the most of, but are we!
The trouble with that belief is that it is rooted in a system's interest that we perpetuate it, and not rise above it, because to do so would make the system falter. The system is not interested in the individual discovering their true purpose to our potential unless it serves the system. It is not interested in the emotional, psychological and ethical sustainability of our respective lives; it is only interested in being sustained. As long as we remain alluded to the idea that our addictive habits are actually expressions of freedom, the system or the matrix that constructs our lives for us does not care. This is no more evident as various countries around the world feed into that system and react accordingly to thwart any attempt to change the status quo throughout 2011 and onwards.
Yet there is one important truth that is occurring as we pretend to live and breathe, the change we desire in our hearts is supported by the universal changes taking place. For a growing number, this is enough inspiration to encourage us towards seeking that change both on a personal and societal level.
Change Afoot in the Universe
Many things are happening, some not evidenced and some are, but it does not change the fact that it is happening.
The most evident for those who do not wish to see, is the collective action of a group of youth who go beyond the self posessed stereotype and called for a dream that represents everyone. They were not youth calling for employment and fare access to education as the corporate media likes to purport, but youth wiser than than system that seeks to thwart them. They inspired many around the world when they rose up and called for serious change on January 25th 2011 in Egypt. They reach across faiths, and have been honest about who they are, and see the real change requires a change of hearts and minds in order to faciltate change on a domestic level, and that takes time. Their message has yet to be comprehended as the governance of the country seeks to pull back the country from the inspiring act of the youth that magically brought an individuated people into unison again, being responsible and accountable to one another. That "pulling back" is feared based, and onyl serves to benefit the elite minded, and elite aspiring, and is plunging the country into a level of discord at all levels. However there is hope, because for the many who listened only to the governance of the country, they are beginning to realize that the call of those youth is beginning to prove itself.
The Washington Consensus
The global economic system is shaped by the Washington Consensus, but at no other time have we witnessed a mass resignation from one of the key entities, the World Bank. For whatever reason, the result is the same as between 2011 – 2012 to date, there have been 358 resignations from this overly esteemed initiative, and related entities, set up to control the global economic order in service of the global elite.
New Blood Types and More!
Besides blood types A, B, AB, and O, two more have been discovered. Langereis and Junior blood types have been found amongst some Japanese. The discovery is a result of blood transfusion problems that occur as a result of mother-fetus incompatibility. These two blood group proteins were discovered a while ago. But dont stop there, becuase the International Blood Transfusion Society recognizes 28 additional blood types!
A growing number of cases reveal that there is greater diversity in human DNA tan previously thought. It has always been assumed that DNA, and its "driver" RNA are identical in protein sequence. But these cases tell another tale. Just as The Establishment is set to register all our DNA on a massive database, nature says so close but so far. Known sincw the 1980s, they have yet to fathom what process is at play that makes the RNA produce a sequence that is not identical to DNA. Twenty-seven people with this 'anomaly' were analyzed by the Howard Hughes Medical Institute. Our genetic blue print does not read like a computer chip afterall!
New Life Forms
If one believes in the Universal Creator, the question would not arise that if we are on the precipice of extinction or Armageddon that the world is coming to an end. If one is connected with one's true self, and the unity of life, then one would reply if that is the case, why all these new life forms then? Yes, many species have passed out of existence, but many new life forms are coming into existence – at least one could say that experts are becoming aware of new life forms.
Embracing the unity of all life makes one aware of the connectedness of all things so that we may learn.
Apart from the Indigo children who are children (some are adults now) who have special attributes that serve the sacredness of life in various fields, in 2010, scientists discovered arsenic-using microbe was discovered at Mono Lake in California. In homeopathy the characteristic of an arsenic personality in a state of balance is one who places trust and honest relationships high on their agenda, which is the underlying call from the mass protests around the world that genuinely arise from the people.
Geologically, arsenic is found where there are copper-gold deposits.
The finding of arsenic using life forms completel challenges the understanding of the chemistry of life: chemistry of life: oxygen, carbon, hydrogen, nitrogen, phosphorus and sulphur .Nasa-funded astrobiologists have now added arsenic to this chemistry of life. In the same year, researchers found arsenic (As2O3) to be effective in the treatment of leukemia (acute promyelocytic leukemia). Leukemia s a blood cancer, and in homeopathy, the psyche of a sufferer has a pattern of suppression, feels unworthy, and has unfulfilled potentials, a common characteristic amongst those of the developed world.
In New Guinea, 1,060 new species were discovered from 1998 – 2008, a signifiant year, and 615 new species were discovered in Madagascar from 1999 – 2010. One of the findings in New Guinea included an new specie of dolphin, the snub-fin dolphin, Orcaella heinsohni.
Evolution of the Universe
It has now been established by science that we are living in an expanding universe, and that there is life beyond the 9 planets that we have come to know and love. As our Earth becomes increasingly bombarded with solar radiation, meteors and the like, challenging the Big Bang theory, which requires a static universe. It has now been established that "We are not alone" and that hundreds if not thousands of other galaxies exist has revealed by the Hubble Space telescope.
Meteors and the like contain elements that are new to our Earthly environment, and as such is adding elements that will influence the environment along with increased exposure to solar radiation-ultra viole radiation that will play a role in the synthesis of these elements into our environment.
To add to the intrigue, there is the argument by New Agers, and metaphysicians understanding that we are moving away from functioning at the limited 3D level, i.e. seeing is believing, and being challenged by the universe to move towards a 5D world. Modern physicists are now saying that there is a subtle 5th force at play in the universe with the increasingly discovery of finer particles – some faster than the speed of light. We are familiar with 4 forces: gravity, electromagneticism, and the short range strong nuclear force and the weak nuclear force.
Life is stranger than fiction, and much more interesting. To believe that Nature can be defined, labeled, and exploited is small minded, including that of human nature itself is limited by the nature of our minds, not by nature itself! Even down to how we perceive ourselves over-rides our true and higher potential, and the world we have become addicted to is merely a reflection of that. When a bridge os formed between our higher ideals and aspirations, and our ability to mainfest them, then and only then can we begin to live without harming one's self, others, and the very environment in which we live. All negative things born out of fear, only tells us that which we must relinquish, and that is fear it self!
Carroll. S. "Are There Mysterious Forces Lurking in Our Atoms and Galaxies?" http://discovermagazine.com/2011/nov/12-out-there-mysterious-fifth-force-atoms-galaxies/article_view?b_start:int=1&-C=
Connor, S. "Science Grapples With The Concept That Alien Life May Be Among." ushttp://www.independent.co.uk/news/science/science-grapples-with-the-concept-that-alien-life-may-be-among-us-2149758.html
"Final Update: 358 Resignations from World Banks, Investment Houses, Money Funds." http://consciouslifenews.com/final-update-358-resignations-world-banks-investment-houses-money-funds/1126008/
Helias, V et al. "ABCB6 is dispensable for erythropoiesis and specifies the new blood group system Langereis." Nature Genetics, 2012; 44 (2): 170 DOI: 10.1038/ng.1069.
Li, M. et al. Widespread RNA and DNA Sequence Differences in the Human Transcriptome. Science, May 19, 2011 DOI: 10.1126/science.1207018
Wake Forest University Baptist Medical Center (2010, November 12). Arsenic early in treatment improves survival for leukemia patients, study finds. from http://www.sciencedaily.com /releases/2010/11/101111121630.htm
Who or What Gave You Life?
The Patterns of Our Lives
The Laws of Three: Concealment and Attraction
From the Symbolic Ascension to the Ascension of Our Lives
Ordinary Women Doing the Impossible
The Echo of Life
Can You Change Your Mind Just Like That!
The Lesson That Cannot Be Taught
Nature Helps Our Brain Connect
Deep Thinking
And He Shows Us His Signs!
What Frequency Am I Traveling on Right Now!
The Intelligent Heart
The High Price of Suppressing Compassion
Heavenly Signs: Stronger Ultraviolet Rays from our Sun
101 Resignations from the World Bank*
Shift of the Ages: Siberia's Earthquake
Heavenly Signs: A Cosmic Triangle
Heavenly Signs: Unexplained Happenings
Sanity to Be Redefined
The Children of God's Revolution 1
Aligning with the Universal Changes: Your Mind
This entry was posted on March 25, 2012, in Activism, Environment, Forum, Life, Self Development, Spirituality, Universe and tagged 2012, activism, Allah, environment, global governance, health, jihad an nafs, lifestyles, self development, society, spirituality, universe, well-being.	1 Comment

redpajama_set_name: RedPajamaCommonCrawl


Every spring the Berkeley Branch holds a story contest for Bay Area Fifth Graders.
Teachers in Contra Costa and Alameda Counties are invited to submit one story per student by any number of students. The entry fee is $1 per submission. Entry fees and donations help the club to offer this contest.
up to 12 Fourth Place prizes of $20.
We will honor the teacher of the grand-prize winner with a cash award of $50. Teachers are entitled to submit an unlimited number of entries.
Only teachers may submit entries; entries from students or parents are not accepted.
Stories must be original and the student's own work.
Only one entry per student may be submitted; teachers may submit any number of entries.
Entries may not exceed 1,000 words in length.
Names of entrants (first name and last initial) may be posted on the comments of this page on receipt.
All participants are invited to the awards ceremony in June, date TBA.
Names of the winners will be announced by May 15th in local newspapers and on our website.
CWC may publish stories for educational purposes only.
Awards Ceremony will be held on Sunday, June 2, 2012.
We suggest teachers consult www.coreygreen.com, setonhill.edu, or other guides to elements of the short story, to help students produce their best work.
Stories must be typed and printed on one side only of standard white 8-1/2 x 11-inch paper.
Please, no fancy paper, covers, or artwork.
First page must include title of the story and the word count.
Do not write the author's name or other identification anywhere on the story.
The contest will be judged by California Writers Club members through a process known as "blind reading." This means the judges will not know the name or school of the author of the story.
Creative Writing Scholarships for High School Juniors and Seniors!
Notre Dame de Namur University is offering a $20,000 creative writing scholarship to a California high school junior or senior who wins the San Mateo County Fair Literary Contest.
To qualify, the student must have a 3.0 GPA or above, submit a poem, short story, or essay (limited to one entry per applicant) that has not been previously entered in a San Mateo County Fair competition, and enroll and be accepted to Notre Dame de Namur within one year of graduating from high school.
The opportunity to become a published author by Sand Hill Review Press is also being offered by the San Mateo County Fair Literary Anthology 2012.
Enter a short story, poetry, essay, and sponsored contest submissions. All entries must be submitted in .doc or .docx file.
Every entrant will have at least one piece published, which will be available for purchase at the Fair between June 9 to June 17 and also on www.Amazon.com.
For entry forms or more details about both literary opportunities, visit www.sanmateocountyfair.com, call 650-574-3247 or email literary@smeventcenter.com. The submission deadline for both is April 16.
The Occupy Movement is raising concerns about social inequality in our country. From your perspective, are there improvements you would make and inequalities you would end within your school, community, and/or nation that would help you, your family, or others? Describe.
Interested students whose teachers are not participating may call the office at 925-935-7850 to receive an application. Cash awards will be offered!
Students may address this issue through art, essay, or creative writing. Selections will be announced in April, and awardees, their teachers, and families will be feted in May with a celebratory dinner, during which students will present their work.
Many thanks to the George Miller Youth Fund and Ludell Deutscher for their generous donation to this project. Anyone else who wishes to donate should please contact the office.


Picking Crissyfield Armchair ByOphelia & Co.
Wonderful Allen Armchair ByKlaussner Furniture is necessary have in virtually any home. You desire for the greatest items, and also you want to make sure you in no way pay too much on their behalf. Seems a little bit difficult, proper? Nicely, this information is here to aid. Continue reading and look for some terrific expert techniques for locating the discounts on home furniture items you can expect to really like. Before purchasing any kitchen cabinets, wide open all of the compartments and search inside. You're not simply being sure that each of the compartments are designed to very last and open without any hitches, you also want to make certain that the inner of the drawers have some form of concluding too. When buying Allen Armchair ByKlaussner Furniture, calculate your bedroom before you go buying. There is absolutely nothing worse than selecting the excellent group of Allen Armchair ByKlaussner Furniture only to discover that it does not fit into your master bedroom when you obtain it property. Alternatively, effectively measure your areas dimensions before heading noting in which doorway and windows are located.
An elegant and stylish dining table can acquire the most out of any dinner you like with family or friends. Within the specific the event of buddies, they will most likely judge you by how neat and elegant your dining area- dining room table- appears. Are you thinking of buying new furniture for your kitchen or dining area? This article will help you sample the top best ten contemporary dining tables in 2018 evaluations. A few of the factors that established the search positions for these products consist of, but not limited to, flexibility, style, comfort, size, materials used, and set up table includes chairs. The types of materials mainly regarded as are steel and wooden while the size is a stride of how lots of people a particular desk can hold easily.
Carolina Mild Gray Material Sectional Sofa is well known because of its spectacular comfort and ease and remarkable style. It has a lift-up style that enables for several arrangement choices to suit your needs.The set functions gentle fabric and pushes cushions to bring about the desired comfort and ease. More importantly, its of the great size to accommodate you along with the group of buddies. Depending on the space and your preferred shape, you are able to mix and match the I seats to create an impressive form. Its highly recommended for all those with small rooms and requires a couch.
The term this is actually the comfort and sophisticated look. This established has a tufted backrest and nail mind accent that provides the sophisticated appear. Its gray color contributes to its remarkable appear. It can match as much as 8 inches mattress. That on your own lets you know how comfy the established could be. Particularly, it does not include a mattress. So you may need to buy one. The trundle features a castor to allow quick access. You may also draw the trundle that people take a seat on. Just to mention, it should take assembling, however its super easy to put together.
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If youre searching for some severe relaxation, this loveseats solid pillow back and rolled hands will deliverbut at a higher price tag. Each chair cushioning contains 30 individual pocketed circles for springtime and assistance, and high resilience seat foam for an extra coating of plushness and sturdiness. In addition, the froth cushions are topped with super soft, blended lower feathers. For your additional comfort, you will find 3 standard positions: seated, a small lie down by having an raised footrest, along with a complete recline with an elevated foot rest. It does not hurt that this reclining loveseat looks lux, either. Its all-leather-based upholstery and ornamental nail head cut give it a vintage look that can work with lots of home dcor styles, from conventional to nation to arts and crafts. Plus, leather stands up nicely against everyday use, children and pet hair and nails. Just be sure you condition the leather every six to 12 months.
Typically now used in homes having a living room or open up floor plan, they are usually large configurations that seat lots of visitors. In smaller sized rooms, they works well for seats in an area that has a strange part or any other room restriction. The ability to mix corner units, end units and reclining areas based on room and person choice tends to make this couch design extremely versatile. Sectional sofas are also available in a multitude of designs, from extremely modern or extremely luxe, to much more family-pleasant modern designs.
This set is upholstered in a push and comfortable distinctive cushioned purple velvet. This provides it the amazing comfort that it is loved for. Again it supports chaise style seating for adequate comfort and ease. Furthermore, the sinus springtime foundation increases the durability and comfort. Furthermore, the couch has hardwood frames that increase its sturdiness and sturdiness. The couch has soft cushions on both the chair and also the back again. But still, the sofa can move from a sitting position to a lying down placement easily. That said, just to replicate, the sofa is extremely soft and gives exceptional comfort and ease.
This set of RTA sofas is available in different upholstery smoothness. A number of them include glued leather-based, linen, corduroy, linen consistency, polyester. You just need to find the right tapestry in importance for your balanced requirements. Accessible materials provide an easy-to-thoroughly clean component. A characteristic that implies its versatility to face up to food and drinks. The extravagant chair and back again soft cushions election in support of a cutting-advantage look. The sq . shaped arm that starts just above them enhances the elegance. You have the comforts and soft styles that hit your entrance.
This sofa is worked with the bell-shaped contour along with the upper armrest from the cushioning. Additionally, the top is supported by obstructed corner sides. The thighs, nevertheless, have became a member of the phony wood covering. The couch provides a comfy and extravagant sensation. This is actually the direct consequence of extremely flexible padding inside the rayon furniture. You can sit down easily for the moment to relax. Each of the tones provided with this couch shows its noteworthy element, Versatile style. The red-colored tone could be ideal for each subjects with mild and darkish shades. That continues to be continuous for various shades that it offers as well. The couch is situated in room. Regardless of whether its Thanksgiving holiday or perhaps a large family reunion, you can trust it. Race-watching while at the same time collapsing in your couch with your buddies is really a legitimate choice.
This is actually the subsequent couch on our list. It is extra comfortable and incredibly well suited for small areas or attic living. It's upholstered in polyester fabric, potential customers inset control keys that provide a stylish gemstone-tufted style. It is made from durable supplies and also the legs are constructed with long lasting wooden to add to its sturdiness. The loveseat has an java stained wood legs and non-marking foot hats. It offers an appropriate foam padding and polyester material furniture that makes it extremely magnificent. It has a longue place that gives an exceptional room for relaxing.
If the Chesterfield is the elegant man among sofas, the Cabriole is the grande dame. Known for its uncovered wooden and stylish thighs, reminiscent of the Louis XV period, the cabriole also offers a distinctive outline. It had been additionally a well-liked form within the work of Wallington Parsons Chair (Set of 2) maker Jones Chippendale. Typically, the rear is all one constant piece without soft cushions and has a stylish bending line. This particular cabriole version consists of jewels in the tufts for additional glamour. It is a sofa design that can be as basic or luxurious as you wish. Upholstered with a luxurious materials like velvet produces a much various style feeling than if it were padded inside a much more muted, textural natural. The main aura of elegance derives from its general shape and lithe thighs meaning it'll always give a refined atmosphere to a room.
Because of so many options, its easy to understand why you would wish to accomplish some research prior to committing to a settee. The right choice will take you years of comfort and ease and recollections. There are many issues to take into consideration before making such a purchase. The Ashley Kennard Leather-based Complement Power Reclining Sofa in Lotion is a great choice for most of us. While the light colour may be a problem, the overall value and high-quality materials used to get this to piece are worthwhile. The design is functional even though its modern, its made to fit into more traditional styles as well. Our prime customer reviews also make this a high option. A great back-up if you are worried about an easy color may be the Homelegance Dual Reclining Sofa in Dark brown Glued Leather. This can be a sensible and traditional design, particularly appealing to those who have children. Whichever sofa youve chosen is a superb investment. Take a moment and get your order started now!


If you require new forklift truck attachments or if your own is damaged and in need of factory repair, B&B has an extensive range available for long and short-term hire. This includes paper roll clamps, double pallet handlers, keg clamps, carton clamps, bale clamps, push pull devices, rotators, sideshifts, crane jibs, carpet booms, brick and block clamps all maintained to the highest standards and available at competitive rates.
B&B Service and support all hire attachments in the field using its own network of service engineers.
B&B'S comprehensive workshop facilities ensure all hire attachments are maintained to the highest level ensuring minimum down time.
Please contact the main office for all enquiries regarding pricing and availability.
Please note that this section is still currently under development and will continue to be updated. For a complete list of available attachments please contact the main office.


A two part series of workshops to deepen and expand knowledge in Hawaiian cultural practice from masters in storytelling, chant, leimaking, kapa making, lauahala, and more. September 2006—April 2007. For more information and for registration for the workshops.
A group of Hawaiian cultural practitioners located on Maui and O'ahu who expand their various cultural practices through study and hands-on learning. The practices learned then become the foundation for participation in ceremonies, cultural gatherings, and protocols.
A two and four year comprehensive Hawaiian cultural practices certificate that is currently being developed by Kauahea. It is anticipated to begin in Fall 2007. For more information contact: hokulani@kauahea.org.


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*When I type backspace after test, it didn't work, it appended spaces on the shell

Here's the environment variables ,
any ideas for the bug ?
RUBY_MAJOR=2.2
RUBY_VERSION=2.2.2
RUBY_DOWNLOAD_SHA256=5ffc0f317e429e6b29d4a98ac521c3ce65481bfd22a8cf845fa02a7b113d9b44
GEM_HOME=/usr/local/bundle
BUNDLE_APP_CONFIG=/usr/local/bundle
LANG=en_US.UTF-8
LANGUAGE=en_US:en
LC_ALL=en_US.UTF-8
HOME=/root
LOGNAME=root
SHLVL=1
PWD=/associate-app
OLDPWD=/associate-app
ZSH=/root/.oh-my-zsh
PAGER=less
LESS=-R
LSCOLORS=Gxfxcxdxbxegedabagacad
_=/usr/bin/env

DOCEKRFILE   (here's my docker file)
FROM rails:onbuild
RUN apt-get update && apt-get -y upgrade
RUN apt-get -y -qq --force-yes install \
        build-essential \
        tree \
        locales \
        ruby-dev \
        vim \
        vim-scripts \
        git \
        git-flow\
        curl \
        zsh \
        sudo

# Install Zsh
RUN git clone git://github.com/robbyrussell/oh-my-zsh.git ~/.oh-my-zsh \
      && cp ~/.oh-my-zsh/templates/zshrc.zsh-template ~/.zshrc \
      && chsh -s /bin/zsh

RUN sed -i -E "s/^plugins=\((.*)\)$/plugins=(\1 git git-flow ruby tmux)/" ~/.zshrc

# Make ssh dir
RUN mkdir /root/.ssh/
# Copy over private key, and set permissions
ADD id_rsa /root/.ssh/id_rsa
# Create known_hosts
RUN touch /root/.ssh/known_hosts
# Add bitbuckets key
RUN ssh-keyscan bitbucket.org >> /root/.ssh/known_hosts

RUN echo "alias ls='ls --color=auto'" >> /etc/zsh/zshrc && \
    echo "alias ll='ls -halF'" >> /etc/zsh/zshrc && \
    echo "alias ls='ls --color=auto'" >> /etc/profile && \
    echo "alias ll='ls -halF'" >> /etc/zsh/zshrc

# copy GEMFILE
WORKDIR /tmp
COPY Gemfile Gemfile
COPY Gemfile.lock Gemfile.lock
RUN bundle install

RUN mkdir /associate-app
WORKDIR /associate-app
ADD . /associate-app

RUN locale-gen en_US.UTF-8
ENV LANG en_US.UTF-8
ENV LANGUAGE en_US:en
ENV LC_ALL en_US.UTF-8
CMD echo 'LC_ALL=en_US.UTF-8' >> ~/.zshrc && \
    echo 'LANG=C.UTF-8' >> ~/.zshrc && \
    echo 'LC_CTYPE=en_US.UTF-8' >> ~/.zshrc && \
CMD ["zsh"]

stty -a
speed 38400 baud; rows 40; columns 164; line = 0;
intr = ^C; quit = ^\; erase = ^?; kill = ^U; eof = ^D; eol = <undef>; eol2 = <undef>; swtch = <undef>; start = ^Q; stop = ^S; susp = ^Z; rprnt = ^R; werase = ^W;
lnext = ^V; flush = ^O; min = 1; time = 0;
-parenb -parodd -cmspar cs8 -hupcl -cstopb cread -clocal -crtscts
-ignbrk -brkint -ignpar -parmrk -inpck -istrip -inlcr -igncr icrnl ixon -ixoff -iuclc -ixany -imaxbel -iutf8
opost -olcuc -ocrnl onlcr -onocr -onlret -ofill -ofdel nl0 cr0 tab0 bs0 vt0 ff0
isig icanon iexten echo echoe echok -echonl -noflsh -xcase -tostop -echoprt echoctl echoke

A: Your local settings are inconsistent
LANG=en_US.UTF-8
LANGUAGE=en_US:en
LC_ALL=en_US.UTF-8

That is, LANGUAGE is not using UTF-8, while the other settings are doing this.
That may be part of the problem.
Also, some terminals implement the iutf8 setting for stty.  That tells the terminal driver how to adjust the cursor position for multibyte (UTF-8) characters.  You might check to see if stty -a shows -iutf8 (feature disabled), and enable it if it is not, e.g.,
stty -a
stty iutf8

When the feature is available, the terminal driver knows that it should take into account whether the text uses UTF-8 (and may have multiple bytes per column).  If it does not, it will count each byte as a column, and when erasing the "last" character, may do something like this:
backspace space space

producing the effect which is shown above.
Finally, it would be nice to know the actual terminal in use.  For instance, rxvt does not support UTF-8; xterm can be configured with/without UTF-8. The value for TERM is largely irrelevant, since your shell is likely ignoring it in favor of built-in behavior.

redpajama_set_name: RedPajamaStackExchange


text:
\section{Introduction}

Let $\mathbb{F}_q$ be the finite field of $q$ elements with characteristic $p$.  Let $\gamma$ be a fixed primitive element of $\mathbb{F}_q$ throughout the paper.
 The {\it value set} of a polynomial $g$ over $\mathbb{F}_q$ is the set $V_g$ of images  when we view $g$ as a mapping from $\mathbb{F}_q$ to itself.
 Clearly $g$ is a {\it permutation polynomial (PP)} of $\mathbb{F}_q$ if and only if the
cardinality $|V_g|$ of the value set $V_g$ is $q$.
Asymptotic formulas such as $|V_g| = \lambda(g) q + O(q^{1/2})$, where $\lambda(g)$ is a constant depending only on certain Galois groups associated to $g$,
can be found in Birch and Swinnerton-Dyer \cite{BW}  and Cohen \cite{Cohen:70}.
Later, Williams \cite{Williams} proved that almost all polynomials $g$ of degree $d$ satisfy  $\lambda(g) = 1 -\frac{1}{2!} + \frac{1}{3!} + \cdots + (-1)^{d-1} \frac{1}{d!}$.

There are also several results on explicit upper bound for $|V_g|$ if $g$ is not a PP over $\mathbb{F}_q$; see for example \cite{GurWan,Tu,W1}.
 Perhaps the most well-known result is due to  Wan \cite{W1} who proved that if a polynomial $g$ of degree $d$ is not a PP then

\begin{equation}\label{Wan's bound}
 \vert V_g \vert \leq q -{q-1\over d}.
\end{equation}

On the other hand, it is easy to see that $|V_g|\ge \lceil q/d \rceil$ for any polynomial $g$ over $\mathbb{F}_q$  with degree $d$. The polynomials achieving this lower bound are called   {\it minimal value set polynomials}. The classification of minimal value set polynomials over $\mathbb{F}_{p^k}$ with $k\leq 2$ can be found in  \cite{Carlitzetal:61,Mills:64}, and in \cite{BC} for all the minimal value set polynomials in $\mathbb{F}_q[x]$ whose value set is a subfield of $\mathbb{F}_q$. See  \cite{DasMul,WSC} for further results on lower bounds of $|V_g|$ and \cite{GomezMadden:88} for some classes of polynomials with small value sets.  More recently, algorithms and complexity in computing $|V_g|$ have been studied in \cite{CHW}.

We note that all of  these results mentioned above relate $|V_g|$ to the degree $d$ of $g$.  It is also well known that every polynomial $g$ over $\mathbb{F}_q$ such that $g(0) =b$
has the form $ax^rf(x^s)+b$ with some positive integers
 $r, s$ such that $s\mid q-1$. There are different ways to choose $r, s$ in the form
$ax^rf(x^s)+b$.   However,  in \cite{AGW:09},  the concept of the index of a polynomial was first introduced
and any  non-constant  polynomial $g\in\mathbb{F}_q[x]$ of
degree $\leq q-1$  can be
written {\it uniquely} as
$g(x) = a(x^rf(x^{(q-1)/\ell}))+b$
with  index $\ell$ defined below.
Namely, write
$$g(x)=a(x^n+a_{n-i_1} x^{n-i_1}+\cdots+a_{n-i_k} x^{n-i_k})+b,$$
where $a,~a_{n-i_j}\neq 0$, $j=1, \dots, k$.  The case that $k=0$ is trivial. Thus, we shall assume that
$k\geq 1$. Write $n-i_k=r$, the vanishing order of $x$ at $0$ (i.e., the lowest degree of $x$ in $g(x)-b$ is $r$).
Then $g(x)=a\left(x^r f(x^{(q-1)/\ell}) \right)+b,$ where $f(x)=
x^{e_0}+a_{n-i_1} x^{e_1}+\cdots+ a_{n-i_{k-1}}x^{e_{k-1}} + a_{r}
$,
 $$\ell=\frac{q-1}{\gcd(n-r,n-r-i_1,\dots, n-r-i_{k-1}, q-1)} := \frac{q-1}{s},$$
and $\gcd(e_0, e_1, \dots, e_{k-1}, \ell)=1 .$ The integer $\ell=\frac{q-1}{s}$ is called the
  {\it index} of $g(x)$.  From the above definition of index $\ell$, one can see that the greatest common divisor condition makes $\ell$ minimal among those possible choices.

Clearly, the study of the value set of $g$ over $\mathbb{F}_q$ is equivalent to studying the value set $x^rf(x^{(q-1)/\ell})$ over $\mathbb{F}_q$ with index $\ell$.
Recently Mullen, Wan and Wang \cite{MWW2} used an index approach to study the upper bound of the value set for any polynomial which is not a PP. They proved that if $g$ is not a PP then
\begin{equation}
|V_g| \leq q- \frac{q-1}{\ell}.
\end{equation}
This result improves Wan's result when the index $\ell$ of a polynomial is strictly smaller
than the degree $d$.  We note that the index $\ell$ of a polynomial is always smaller than the degree $d$ as long as $\ell \leq \sqrt{q} -1$.

The above result is obtained through a study of cyclotomic mapping polynomials which were studied earlier in  \cite{Evans:92,NW:05,Wang}. The index of a polynomial  is closely related to the concept of the least index of a cyclotomic mapping polynomial.  Recall that $\gamma$ is a fixed primitive element of $\mathbb{F}_q$. Let $\ell \mid q-1$ and the set of all nonzero $\ell$-th powers be $C_0$. Then $C_0$ is a subgroup of $\mathbb{F}_q^*$ of
index $\ell$. The elements of the factor group $\mathbb{F}_q^*/C_0$
are the {\it cyclotomic cosets}
$$ C_i := \gamma^i C_0, \  \  \  i = 0, 1, \cdots, \ell-1.$$

For any $a_0, a_1, \cdots, a_{\ell-1} \in \mathbb{F}_q$ and a positive integer $r$,   the  {\it $r$-th order cyclotomic mapping $f^{r}_{a_0, a_1, \cdots,
a_{\ell-1}}$ of index $\ell$ }  from $\mathbb{F}_q$ to itself  (see Niederreiter and Winterhof in \cite{NW:05} for $r=1$ or  Wang \cite{Wang}) is defined by
\begin{equation}\label{CycloMappingDef}
f^{r}_{a_0, a_1, \cdots,
a_{\ell-1}} (x) =
\left\{
\begin{array}{ll}
0, &   \mbox{if} ~ x=0; \\
a_i x^{r},   &   \mbox{if} ~x \in C_i, ~ 0\leq i \leq \ell-1. \\
\end{array}
\right.
\end{equation}

It is shown that  $r$-th order cyclotomic mappings of index $\ell$ produce the polynomials of the form  $x^r f(x^s)$ where $s =\frac{q-1}{\ell}$. Indeed,  the polynomial presentation is given by
\begin{equation*}
g(x) =\frac{1}{\ell} \sum_{i=0}^{\ell-1} a_i x^{r} \sum_{j=0}^{\ell-1} \zeta^{-ji}x^{js},
\end{equation*}
where $\zeta =\gamma^s$ is a fixed  primitive $\ell$-th root. On the other hand, as we mentioned earlier, each polynomial $f(x)$ such that $f(0)=0$ with index $\ell$ can be written as $x^r f(x^{(q-1)/\ell})$, which is an $r$-th order cyclotomic mapping with the least index $\ell$ such that $a_i = f(\zeta^i)$ for $i=0, \ldots, \ell-1$.

In this paper, we are interested in the probability distribution of the value set size of a random $r$-th order cyclotomic mapping polynomial for any given index $\ell$ and any positive integer $r$, as defined in Equation~(\ref{CycloMappingDef}). Thus this enables us to derive the probability distribution of the size of value set of a random polynomial of degree $d \leq q-1$ over a finite field.  In Section~\ref{Methodology} we outline our method
and a crucial result on normal distribution which is used in this paper. Essentially, we are interested in the distribution of the  size of the union of subsets, namely,  the distribution of the random variable $X_{t \ell}=|\cup_{j=1}^\ell A_j|$ where $A_j = g(C_{j-1})$ for $j=1, \ldots, \ell$ and $t=(r, s)$.  In Section~\ref{random mappings}, we first consider a simplified model such that none of $a_i$'s in Equation~(\ref{CycloMappingDef}) is zero.  Hence $a_i$'s are chosen independently at random from $\mathbb{F}_q^*$. This means that the zero is not contained in any one of the subsets $A_j$'s. In particular, in Theorem~\ref{Thm1} we obtain the distribution of the number of missing values for a random $r$-th order cyclotomic mapping $f^{r}_{a_0, a_1, \cdots,
a_{\ell-1}} $ such that none of $a_i$'s is zero. Moreover, in Theorem~\ref{normaldistribution},  we show that this distribution is asymptotically normal.

In Section~\ref{random polynomials}, we study any random $r$-th order cyclotomic mapping polynomial by  choosing $a_i$'s in Equation~(\ref{CycloMappingDef}) independently at random from $\mathbb{F}_q$. The probability distribution of value set size is given in Theorem~\ref{valuesetexclude0}.
As a consequence, for $\ell = q-1$, we obtain the exact probability distribution of the value set size of a random polynomial over $\mathbb{F}_q$ with degree at most $q-1$.
 In particular, we have the following corollaries to Theorem~\ref{valuesetexclude0}.

\begin{corollary}\label{Smallwithzero} Let $g(x)$ be a random polynomial of degree at most $q-1$ over $\mathbb{F}_q$ with $g(0) =0$. Then
$${\mathbb P}(|V_g|= k+1)= \ {q-1\choose k}  \sum_{j=0}^{k} (-1)^{k-j} {k\choose j}\left(\frac{1+j}{q}\right)^{q-1}. $$
Consequently, for $k=o(q)$, we have
$${\mathbb P}(|V_g|= k+1)\sim \frac{1}{k!}(q-1)^k\left(\frac{k+1}{q}\right)^{q-1}.
$$
\end{corollary}

This proves that if $k > 1$ is small compared to $q$ then the number of polynomials over $\mathbb{F}_q$ with degree less than or equal to $q-1$ such that the value set size is $k$ is always exponential in $q$. Moreover, we have

\begin{corollary}\label{randomnormaldistribution}
Let $g(x)$ be any random polynomial of degree at most $q-1$ over finite field $\mathbb{F}_q$ with $g(0) =0$. Let $Y_q=q-|g(\mathbb{F}_q)|$ denote the number of missing nonzero values in the value set of $g$. Let $\mu_{q} = q/e$ and $\sigma_q^2 = (e^{-1} - 2e^{-2})q$. Then the distribution of $(Y_q-\mu_q)/\sigma_q$ tends to the standard normal, as $q\to \infty$.
\end{corollary}

Finally in Section~\ref{union of random sets} we study a variation of our model used in Section~\ref{random mappings}. We consider the case when each subset $A_i$ is  chosen uniformly at random from all $m_i$-subsets of
a given $n$-set for $i=1, \ldots, \ell$. This extends a result by
Barot and Pe\~{n}a \cite{BP} and David \cite{Da}. We  also show that the size of the complement of $\cup_{i=1}^{\ell}A_i$ is asymptotically normal.

\section{Methodology}\label{Methodology}

In \cite{MWW}, we obtained the following formula for the cardinality of the value set for an arbitrary polynomial.

\begin{proposition}[Proposition 2.3 in \cite{MWW}]\label{valuesetCyclo}
Let $g(x) = a x^rf(x^s) + b $ ($a \neq 0$) be any polynomial over
$\mathbb{F}_q$ with index $\ell = \frac{q-1}{s}$ and let $\gcd (r, s)=t$.  Let $\gamma$ be a fixed primitive element of $\mathbb{F}_q$.
Then
\[|V_g| = c \frac{s}{t}  + 1,~or~ |V_g| = (c-1) \frac{s}{t}+1, \]
where  $c = |\{ ( \gamma^{ir} f(\gamma^{s i}) )^{s/t} \mid i = 0, \ldots, \ell-1 \}|$.
\end{proposition}

As discussed earlier, it is sufficient to assume that  $a=1$ and $b=0$ in Proposition~\ref{valuesetCyclo}. That is, we can view $g(x)$ as a $r$-th order cyclotomic mapping polynomial with the least index $\ell$.   In this case,
we have $g(x) = a_i x^r$ when $x\in C_i$, where $a_i = f(\gamma^{s i})$ for $i=0, \ldots, \ell-1$.
Recall that $C_0$ is the subgroup of $\mathbb{F}_q^*$ consisting of all the $\ell$-th powers of $\mathbb{F}_q^*$ and we let $T_0$ be the subgroup of $\mathbb{F}_q^*$ consisting of all the $t\ell$-th powers. Hence
$T_i$ with $0\leq i \leq t\ell-1$ give all the cyclotomic cosets of index $t\ell$.
We also note that $x^r$ maps $C_0$ onto $T_0$ which contains  $\frac{s}{t}$ distinct elements.  So $x^r$ maps each coset $C_i = \gamma^i C_0$ onto $\gamma^{ir} T_0$.  Therefore $g$ maps $C_i$ onto  $ \gamma^{ir} f(\gamma^{s i})  T_0$, which could be either the set  $\{0\}$ (if $a_i = f(\gamma^{s i}) =0$) or one of the nonzero cyclotomic cosets of index $t\ell$. We observe that $c$ is the number of  distinct cyclotomic cosets of the form $\gamma^{ir}  f(\gamma^{s i}) T_0$, possibly along with the subset $\{0\}$ if one of $a_i$'s is zero. Hence we have  $|V_g| = c \frac{s}{t}  + 1$ or $(c-1) \frac{s}{t}+1$, the latter happens when  some of $a_i$'s in $g(x) = a_i x^r$ equal $0$.

Therefore the value set problem  for a random $r$-th order cyclotomic mapping polynomial (or random polynomial) $g$, essentially requires us to study the number $c$ in Proposition~\ref{valuesetCyclo}, the size of union of some cyclotomic cosets and possibly the subset $\{0\}$ if $a_i$'s take zero. More specifically, for $0\leq i \leq \ell-1$, each $C_i$ is mapped to $A_{i+1} =g(C_i)$ which is one of $T_0, \ldots, T_{t\ell -1}$ or $\{0\}$. Then $c$ is the number of distinct $A_j$'s  ($1\leq j\leq \ell$) and the value set size is either $c\frac{s}{t} + 1$ or $(c-1)\frac{s}{t} +1$.
More generally, we are interested in the distribution of the random variable $X_{t \ell}=|\cup_{j=1}^\ell A_j|$, while $A_j$ are  chosen independently according to a given distribution depending on that $a_i$ is chosen independently at random from $\mathbb{F}_q$. Similar problems have been studied
in \cite{BP,Da} when each $A_j$ is chosen uniformly at random from all $k$-subsets of a given $n$-set.

Let $n=t\ell$ and let $D_0=\{0\}$ and $D_{j} = T_{j-1}$ for $1\leq j\leq t\ell -1$.  Let $Y_{n}$ be the the number of $D_1, \ldots, D_{n-1}$ which are not in $\cup_{j=1}^\ell A_j$ for a random $r$-th order cyclotomic mapping polynomial with index $\ell$ such that $(r, s)=t$.   We will derive exact probability distributions of $Y_n$ and show that they are asymptotically normal. Throughout the paper, we shall use $(Y_n)_k$ to denote the falling factorial $Y_n(Y_n-1)(Y_n-2)\cdots(Y_n-k+1)$. We use ${\mathbb P}$, ${\mathbb E}$, ${\mathbb V}$ to denote the probability, expectation, and variance of a random variable, respectively.

The main tool for deriving the probability distribution of $Y_n$ in the paper is through the sieve method and  the falling factorial moments.  Let
$B_1, \ldots, B_{n}$ be $n$ events in a probability space and $\mathcal{N} =\{1, \ldots, n\}$.
We note that ${\mathbb P}(Y_n=k)$ is the probability that exactly $k$ of the $B_j$ occur.  Define
$$S_h=\sum_{J\subset \mathcal{N},|J|=h}{\mathbb P}(\cap_{j\in J}B_j).$$
Then the well-known sieve formula (See e.g. \cite[Theorm~10]{Bo}) gives
\begin{eqnarray}
{\mathbb P}(Y_n=k)&=&\sum_{h=k}^n(-1)^{h-k}{h\choose k}S_h,\label{pk}\\
S_k&=&\sum_{h=k}^n{h\choose k}{\mathbb P}(Y_n=h).\label{Sk}
\end{eqnarray}
Hence
\begin{equation}\label{factorial}
{\mathbb E}((Y_n)_k)=k!S_k.
\end{equation}

We also  need the following
result \cite[Theorem~1]{GW} in order to show that $Y_n$ is asymptotically normal.

\begin{lemma}\label{normal}
Let $s_n>-\mu_n^{-1}$ and
$$\sigma_n=\sqrt{\mu_n+\mu_n^2s_n},$$
where $\mu_n\to \infty$ as $n\to \infty$.
Suppose that $$\mu_n=o(\sigma_n^3),$$ and a sequence $Y_n$ of nonnegative random variables satisfies
$${\mathbb E}((Y_n)_k)\sim \mu_n^k\exp\left(\frac{k^2s_n}{2}\right),$$
uniformly for all integers $k$ in the range $c\mu_n/\sigma_n\le k\le c'\mu_n/\sigma_n$ for some constants
$c'>c>0$. Then $(Y_n-\mu_n)/\sigma_n$ tends in distribution to the standard normal as $n\to \infty$.
\end{lemma}

\section{Sizes of value sets of cyclotomic mapping polynomials with nonzero branches} \label{random mappings}

In this section, we study of the value set size of a random $r$-th order cyclotomic mapping polynomial with nonzero branches (i.e., none of $a_i$'s is zero).  This means that we   choose $a_i$ in (\ref{CycloMappingDef}) independently at random from $\mathbb{F}_q^*$ and it leads to the following model.
Let $n=t\ell$ and $D_1,D_2,\ldots D_{t\ell}$ be pairwise disjoint subsets of $\mathbb{F}_q^*$
such that $|D_i|=s/t$ for all $1\le i\le t\ell$.  Because $a_i$ is chosen independently at random from $\mathbb{F}_q^*$,  this means that   $A_1,A_2,\ldots, A_{\ell}$ are chosen independently and uniformly at random from $\{D_1,D_2,\ldots, D_{t\ell}\}$. We are interested in the distribution of $X_n=|\cup_{j=1}^{\ell}A_j|$. This is closely related to the distributions of $\ell$ labeled balls into $t\ell$ labeled boxes. Let $Y_n$ be the number of empty boxes in a random distribution of $\ell$ labeled balls
into $t\ell$ labeled boxes. We note $X_n=n-(s/t)Y_n$. We first prove the following
\begin{theorem}\label{Thm1}
Let $Y_n$ be the number of empty boxes in a random distribution of $\ell$ labeled balls into $n = t\ell$ labeled boxes.
We have
\begin{eqnarray*}
{\mathbb E}((Y_n)_k)&=&(t\ell)_k\left(\frac{t\ell-k}{t\ell}\right)^{\ell},\\
{\mathbb P}(Y_n=k)&=&{t\ell\choose k}\frac{1}{(t\ell)^{\ell}}\sum_{j=1}^{t\ell-k}(-1)^{t\ell-k-j}{t\ell-k\choose j}j^{\ell}.
\end{eqnarray*}
\end{theorem}
\noindent {\bf Proof }
Let $\mathcal{N} = \{1, 2, \ldots, t\ell\}$.  Let $B_j$ be the event that box $j$ is empty.
We have
$$S_k= \sum_{J\subset \mathcal{N},|J|=k}{\mathbb P}(\cap_{j\in J}B_j) = {t\ell\choose k}{\mathbb P}(\cap_{j=1}^k B_j)={t\ell\choose k}\left(\frac{t\ell -k}{t\ell}\right)^{\ell}.$$
It follows from (\ref{factorial}) that
$${\mathbb E}((Y_n)_k)=(t\ell)_k\left(\frac{t\ell-k}{t\ell}\right)^{\ell}.$$

Using Equation~(\ref{pk}), we obtain
\begin{eqnarray*}
{\mathbb P}(Y_n=k)&=&\sum_{h=k}^{n} (-1)^{h-k} { h \choose k} S_h \\
&=& \sum_{h=k}^{t\ell} (-1)^{h-k} { h \choose k}  {t\ell\choose h}\left(\frac{t\ell -h}{t\ell}\right)^{\ell} \\
&=&  {t\ell\choose k} \sum_{h=k}^{t\ell} (-1)^{h-k} {t\ell-k\choose h-k} \left(\frac{t\ell -h}{t\ell}\right)^{\ell} \\
&=& {t\ell\choose k} \frac{1}{(t\ell)^\ell} \sum_{j=0}^{t\ell-k} (-1)^{t\ell -j-k} {t\ell-k\choose j} j^{\ell}\\
\end{eqnarray*}
\qed

Next we obtain

\begin{theorem}\label{normaldistribution}
Suppose $t=o(\ell^{1/5})$ as $n = t \ell \to \infty$. Define $$\mu_n=te^{-1/t}\ell,~\sigma_n^2=te^{-2/t}(e^{1/t}-1-1/t)\ell.$$
Then the distribution of $(Y_n-\mu_n)/\sigma_n$ tends to the standard normal, as $n\to \infty$.
\end{theorem}
\noindent {\bf Proof } From Theorem~\ref{Thm1}, we have, as $\ell\to \infty$,
\begin{eqnarray}
{\mathbb E}(Y_n)&=&t\ell\left(1-\frac{1}{t\ell}\right)^{\ell}\nonumber\\
&=&t\ell\exp\left(\ell\ln\left(1-\frac{1}{t\ell}\right)\right)\nonumber\\
&=&t\ell\exp\left(-\ell\left(\frac{1}{t\ell}+\frac{1}{2t^2\ell^2}+O\left(\frac{1}{(t\ell)^{3}}\right)\right)\right)\nonumber\\
&=&t\ell\exp\left(-\frac{1}{t}-\frac{1}{2t^2\ell}+O\left(\frac{1}{t^3\ell^2}\right)\right)\nonumber\\
&\sim&\mu_n, \label{mean} \nonumber \\
\end{eqnarray}
\begin{eqnarray}
{\mathbb V}(Y_n)&=&{\mathbb E}(Y_n(Y_n-1))+{\mathbb E}(Y_n)-({\mathbb E}(Y_n))^2\nonumber\\
&=&(t\ell)(t\ell-1)\left(1-\frac{2}{t\ell}\right)^{\ell} + t\ell \left(1-\frac{1}{t\ell}\right)^{\ell} - (t\ell)^2 \left(1-\frac{1}{t\ell}\right)^{2\ell} \nonumber\\
&=&(t\ell)(t\ell-1)\exp\left(\ell\ln\left(1-\frac{2}{t\ell}\right)\right)+ t\ell\exp\left(\ell\ln\left(1-\frac{1}{t\ell}\right)\right) \nonumber\\
&&-(t\ell)^2\exp\left(2\ell\ln\left(1-\frac{1}{t\ell}\right)\right) \nonumber\\
&=& (t\ell)(t\ell-1)\exp\left(-\ell\left(\frac{2}{t\ell} + \frac{2}{(t\ell)^2}+ O\left( \left(\frac{2}{t\ell}\right)^3 \right)\right)\right) \nonumber \\
&& + t\ell\exp\left(-\ell\left(\frac{1}{t\ell} + \frac{1}{2(t\ell)^2}+ O\left( \left(\frac{1}{t\ell}\right)^3 \right)\right)\right) \nonumber \\
&&  -(t\ell)^2\exp\left(-2\ell\left(\frac{1}{t\ell} + \frac{1}{2(t\ell)^2}+ O\left( \left(\frac{1}{t\ell}\right)^3 \right)\right)\right) \nonumber\\
&=& (t\ell)(t\ell-1)\exp\left(-\ell\left(\frac{2}{t\ell}\right)\right)\exp\left(-\frac{2\ell}{(t\ell)^2}+ O\left( \left(\frac{2}{t\ell}\right)^3 \right)\right) \nonumber \\
&& + t\ell \exp\left(-\ell\left(\frac{1}{t\ell}\right)\right) \exp\left(- \frac{\ell}{2(t\ell)^2}+ O\left( \left(\frac{1}{t\ell}\right)^3 \right)\right) \nonumber\\
&& - (t\ell)^2\exp\left(-2\ell\left(\frac{1}{t\ell}\right)\right) \exp\left(- \frac{\ell}{(t\ell)^2}+ O\left( \left(\frac{1}{t\ell}\right)^3 \right)\right) \nonumber\\
\end{eqnarray}
\begin{eqnarray}
&\sim& (t\ell)(t\ell-1)\exp\left(-\ell\left(\frac{2}{t\ell}\right)\right) \left(1-\frac{2\ell}{(t\ell)^2}\right) \nonumber \\
&& + t\ell \exp\left(-\ell\left(\frac{1}{t\ell}\right)\right) \left(1 - \frac{\ell}{2(t\ell)^2} \right) \nonumber\\
&& - (t\ell)^2\exp\left(-2\ell\left(\frac{1}{t\ell}\right)\right) \left(1- \frac{\ell}{(t\ell)^2}\right) \nonumber\\
&\sim& t \ell e^{-\frac{1}{t}} - t \ell e^{-\frac{2}{t}} - \ell e^{-\frac{2}{t}} \nonumber \\
&\sim&\sigma_n^2.\label{variance}
\end{eqnarray}
Under the assumption that $t=o(\ell^{1/5})$, we can verify that $\frac{\mu_n^2}{\sigma_n^6} \to 0$ as $n \to \infty$.
We note $\mu_n/\sigma_n$ is of the order $\sqrt{t^3\ell}$. Hence for $k$ in the range specified in Lemma~\ref{normal}, we have
$k^2=O(t^3\ell)$. Therefore
\begin{eqnarray*}
(t\ell)_k&=&(t\ell)^k\prod_{j=1}^{k-1}\left(1-\frac{j}{t\ell}\right)\\
&=&(t\ell)^k\exp\left(\sum_{j=1}^{k-1}\ln\left(1-\frac{j}{t\ell}\right)\right)\\
&=&(t\ell)^k\exp\left(-\sum_{j=1}^{k-1}\left(\frac{j}{t\ell}+\frac{j^2}{2t^2\ell^2}+O\left(\frac{j^3}{t^3\ell^3}\right)\right)\right)\\
&\sim&(t\ell)^k\exp\left(-\frac{k^2}{2t\ell}\right),\\
\left(\frac{t\ell-k}{t\ell}\right)^{\ell}&=&\exp\left(\ell\ln\left(1-\frac{k}{t\ell}\right)\right)\\
&=&\exp\left(-\ell\left(\frac{k}{t\ell}+\frac{k^2}{2t^2\ell^2}+O\left(\frac{k^3}{t^3\ell^3}\right)\right)\right)\\
&\sim&\exp\left(-\frac{k}{t}-\frac{k^2}{2t^2\ell}\right).
\end{eqnarray*}
It follows from (\ref{factorial}) and  (\ref{mean}) that
$${\mathbb E}((Y_n)_k)\sim (t\ell)^k\exp\left(-\frac{k}{t}-\frac{k^2}{2t\ell}-\frac{k^2}{2t^2\ell}\right)
\sim \mu_n^k\exp\left(-\frac{k^2(t+1)}{2t^2\ell}\right).
$$
Now the result follows from Lemma~\ref{normal} and the estimation
$$s_n=\frac{\sigma_n^2-\mu_n}{\mu_n^2}=\frac{{\mathbb E}(Y_n(Y_n-1))}{\mu_n^2}-1
=-\frac{1}{t\ell}-\frac{1}{t^2\ell}+O\left(\frac{1}{t^2\ell^2}\right) > - \mu_n^{-1},$$
as $\frac{t}{\ell} \to 0$ when $\ell \to \infty$.
\qed

The following corollary follows immediately from Theorem~\ref{Thm1}, by noting $s=t=1$ and $X_n=n-Y_n$.
\begin{corollary}\label{Small}
Under the assumption of Theorem~\ref{Thm1} and $X_n= n-Y_n$, we have
$${\mathbb P}(X_n=h)=\frac{1}{n^n}{n\choose h}\sum_{j=1}^h(-1)^{h-j}{h\choose j}j^n.
$$
In particular,
$${\mathbb P}(X_n=n)=\frac{n!}{n^n},
$$
and for  $h=o(n)$,
$${\mathbb P}(X_n=h)\sim \frac{1}{h!}n^h\left(\frac{h}{n}\right)^{n}.
$$
\end{corollary}

Consider $\ell=n=q-1$ and $t=1$.  Because an $r$-th order cyclotomic mapping polynomial $f^{r}_{a_0, \ldots, a_{q-2}}(x)$ always maps $0$ to $0$,  Corollary~\ref{Small}
implies

\begin{corollary}\label{Smallwithoutzero} Let $g(x)$ be any random $r$-th order cyclotomic mapping polynomial $f^{r}_{a_0, \ldots, a_{q-2}}(x)$ over finite field $\mathbb{F}_q$ such that none of $a_i$'s is zero. Then the probability of $\mid V_f \mid = h+1$ is given by
$${\mathbb P}(X_{q-1}=h)= \frac{1}{(q-1)^{q-1}}{q-1\choose h}\sum_{j=1}^h(-1)^{h-j}{h\choose j}j^{q-1}.$$
In particular, the probability of such a random cyclotomic mapping polynomial $f^{r}_{a_0, \ldots, a_{q-2}}(x)$ is a permutation polynomial is
$${\mathbb P}(X_{q-1}=q-1)=\frac{(q-1)!}{(q-1)^{q-1}},
$$
and for $h=o(q)$, the probability of such a random cyclotomic mapping polynomial $f^{r}_{a_0, \ldots, a_{q-2}}(x)$ with a value set size $h+1$ is
$${\mathbb P}(X_{q-1}=h)\sim \frac{1}{h!}(q-1)^{h}\left(\frac{h}{q-1}\right)^{q-1}.
$$
\end{corollary}

\section{Sizes of value sets of random polynomials }\label{random polynomials}

Recall $q-1 = \ell s$ and $r$ is a positive integer  such that $(r, s) = t$.
In this section we consider the value set size for any random $r$-th order cyclotomic mapping polynomial $f^{r}_{a_0, \ldots, a_{\ell-1}}(x)$ with index $\ell$ over finite field $\mathbb{F}_q$. Namely, $a_0, \ldots, a_{\ell-1}$ are independently chosen at random from $\mathbb{F}_q$.
 We note that the value set problem for any random polynomial with degree at most $q-1$ is in fact the value set problem for a random $r$-th order cyclotomic mapping polynomial with index $\ell=q-1$.

As discussed in Section~\ref{Methodology}, we are interested in the distribution of $X_n=|\cup_{j=1}^{\ell}A_j|$.
However, we need to include the element 0 in our analysis similar to those in Section~\ref{random mappings}. Because each $a_i$ in (\ref{CycloMappingDef}) is chosen independently at random from $\mathbb{F}_q$, this leads to the following model.

Let us define $D_0=\{0\}$ and $D_j = T_{j-1}$ for $1\leq j \leq t\ell$.
The distribution of each random set $A_i$ is ${\mathbb P}(A_i=D_0)=\frac{1}{q}$,  and
$${\mathbb P}(A_i=D_j)=\frac{1}{t\ell}\left(1-\frac{1}{q}\right)=\frac{s}{tq},~j=1,\ldots,\ell.$$
As in the case discussed earlier, we define the events $B_j= \cap_{i=1}^\ell \{A_i\ne D_j\}$, $0\le j\le t\ell$.
 Let $Y_{t\ell}$ be the number of $B_1,B_2,\ldots, B_{t\ell}$ (note $B_0$ is excluded) which occur.
Then ${\mathbb P}(Y_{t\ell}=k)$ is the probability that exactly $k$ of $t\ell$ cyclotomic sets $T_j$'s of index $t \ell$ are not in the value set of $g$. Then we have
\begin{lemma}\label{randomExpected}
 Let $q-1 = \ell s$ and $r$ be a positive integer  such that $(r, s) = t$.  Let $g(x)$ be any random $r$-th order cyclotomic mapping polynomial $f^{r}_{a_0, \ldots, a_{\ell-1}}(x)$ with index $\ell$ over finite field $\mathbb{F}_q$. Let $Y_{t\ell}$ be the number of cyclotomic sets of index $t \ell$ not contained in the value set of $g$. Then
\begin{eqnarray*}
{\mathbb E}((Y_{t\ell})_k)&=&(t\ell)_k\left(1-\frac{sk}{tq} \right)^{\ell},\\
{\mathbb P}(Y_{t\ell}=k)&=&{t\ell\choose k}  \sum_{j=0}^{t\ell - k} (-1)^{t\ell -j-k} {t\ell-k\choose j}\left(\frac{1}{q} + \frac{sj}{tq}\right)^{\ell}.
\end{eqnarray*}
\end{lemma}
\noindent {\bf Proof } Let $\mathcal{N} = \{1, 2, \ldots, t\ell\}$. Define
$$
S_k=\sum_{J\subset \mathcal{N},|J|=k}{\mathbb P}(\cap_{j\in J}B_j).
$$
Because $A_1, \ldots, A_{\ell}$ are mutually independent, we have
$$
S_k={t\ell\choose k}{\mathbb P}(\cap_{j=1}^k B_j)={t\ell\choose k}\left(\frac{1}{q}+\frac{s(t\ell-k)}{tq}\right)^{\ell}={t\ell\choose k}\left(1-\frac{sk}{tq} \right)^{\ell}.
$$

Using Equation~(\ref{pk}), we obtain

\begin{eqnarray*}
{\mathbb P}(Y_{t\ell}=k)&=&\sum_{h=k}^{t\ell} (-1)^{h-k} { h \choose k} S_h \\
&=& \sum_{h=k}^{t\ell} (-1)^{h-k} { h \choose k}  {t\ell\choose h}\left(\frac{1}{q}+\frac{s(t\ell-h)}{tq}\right)^{\ell} \\
&=&  {t\ell\choose k} \sum_{h=k}^{t\ell} (-1)^{h-k} {t\ell-k\choose h-k} \left(\frac{1}{q}+\frac{s(t\ell-h)}{tq}\right)^{\ell} \\
&=& {t\ell\choose k}  \sum_{j=0}^{t\ell - k} (-1)^{t\ell -j-k} {t\ell-k\choose j}\left(\frac{1}{q} + \frac{sj}{tq}\right)^{\ell}.
\end{eqnarray*}

 \qed

Using the above lemma, we can obtain the distribution of $X_{t\ell} = |V_g|$.
\begin{theorem}\label{valuesetexclude0}
 Let $q-1 = \ell s$ and $r$ be a positive integer  such that $(r, s) = t$.  Let $f(x)$ be any random $r$-th order cyclotomic mapping polynomial $f^{r}_{a_0, \ldots, a_{\ell-1}}(x)$ with index $\ell$ over $\mathbb{F}_q$.
 Then
 $${\mathbb P}(X_{t\ell}=1+ks/t) = {t\ell\choose k}  \sum_{j=0}^{k} (-1)^{k -j} {k\choose j}\left(\frac{1}{q} + \frac{sj}{tq}\right)^{\ell}.
  $$
\end{theorem}
\noindent {\bf Proof }  Indeed,
\begin{eqnarray*}
{\mathbb P}(X_{t\ell}=1+ks/t)&=&{\mathbb P}(\{Y_{t\ell}=t\ell-k\})\\
&=& {t\ell\choose k}  \sum_{j=0}^{k} (-1)^{k -j} {k\choose j}\left(\frac{1}{q} + \frac{sj}{tq}\right)^{\ell}.
\end{eqnarray*}

We now obtain an application of the above results on random polynomials with degree at most $q-1$. It is sufficient to study the distribution of the subclass of random polynomials $g(x)$ such that $g(0) =0$ with degree $\leq q-1$, because any polynomial $f(x)$ such that $f(0) =b$ can be written as $g(x) +b$ with $g(0) =0$ and vice versa.  So we can view any random polynomial $g$ of degree at most $q-1$ with $g(0)=0$ and index $\ell$ as an $r$-th order cyclotomic mapping polynomial with the least index $\ell$, which is therefore a cyclotomic mapping polynomial with index $q-1$. Because there are $q^{q-1}$ such polynomials with $g(0) =0$, they correspond to all the cyclotomic mapping polynomials with index $q-1$. So a random polynomial with degree less than or equal to $q-1$ and $g(0)=0$ is a random $r$-th order cyclotomic mapping polynomial with index $\ell=q-1$ for any $r \geq 1$.  Therefore,  Lemma~\ref{randomExpected} implies that any random polynomial with degree $q-1$ has expected value set size $(1-\frac{1}{q})^{q-1}\sim \frac{q}{e}$. This verifies William's result \cite{Williams} saying that almost all the polynomials of degree $q-1$ is a general polynomial.  Moreover, applying Theorem~\ref{valuesetexclude0} to the case $\ell =q-1$ (hence $s=t=1$),  we obtain exact probability distribution of the value set size for a random polynomial over finite field $\mathbb{F}_q$ in Corollary~\ref{Smallwithzero}.
Moreover, we can drive Corollary~\ref{randomnormaldistribution} that the distribution of $(Y_{t\ell}-u_{t\ell})/\sigma_{t\ell}$ tends to the standard normal distribution as well, as $\ell \to \infty$ and $t = o(\ell^{1/5})$,  following the same arguments as in the proof of Theorem~\ref{normaldistribution}.

\rmv
Again, when $\ell =q-1$ and $s=t=1$, we have

\begin{corollary}\label{Smallwithzero} Let $g(x)$ be a random polynomial of degree at most $q-1$ over $\mathbb{F}_q$ with $g(0) =0$. Then the probability of $|V_g| = k+1$  is given by
$$\ {q-1\choose k}  \sum_{j=0}^{k} (-1)^{k-j} {k\choose j}\left(\frac{1+j}{q}\right)^{q-1},  $$
and  the probability of any random polynomial with $g(0)=0$ is a permutation polynomial is
$$ \left(\frac{1}{q}\right)^{q-1} \sum_{j=0}^{q-1} (-1)^{q-1 -j} {q-1\choose j} j^{q-1}  =  \frac{(q-1)!}{q^{q-1}}.$$
In particular, for $k=o(q)$, the probability that $|V_g|=k+1$ is asymptotic to
$$ \frac{1}{k!}q^k\left(\frac{k+1}{q}\right)^{q-1}.
$$
\end{corollary}
\noindent {\bf Proof }  The proof is essentially the same as that of Theorem~\ref{valuesetexclude0}.  \qed

This proves that if $k > 1$ then the number of polynomials over $\mathbb{F}_q$ with degree less than or equal to $q-1$ such that the value set size is $k$ is always exponential in $q$.

Moreover, we can show that the distribution of $(Y_{t\ell}-u_{t\ell})/\sigma_{t\ell}$ tends to the standard normal distribution as well, as $\ell \to \infty$ and $t = o(\ell^{1/5})$,  following the same arguments as in the
proof of Theorem~\ref{normaldistribution}. In particular, we have

\begin{corollary}
Let $g(x)$ be a random polynomial over finite field $\mathbb{F}_q$ with $g(0) =0$. Let $Y_{q-1} $ denote the size of missing nonzero values in the value set of $g$. Let $\mu_{q-1} = q/e$ and $\sigma_{q-1} = (e^{-1} - 2e^{-2})q$. Then the distribution of $(Y_{q-1}-\mu_{q-1})/\sigma_{q-1}$ tends to the standard normal distribution, as $q\to \infty$.
\end{corollary}

\section{Size of the Union of Random Sets}\label{union of random sets}

Recall $\mathcal{N}=\{1, \ldots, n\}$.
In Section~\ref{random mappings}, we considered
$|\cup_{i=1}^{\ell} A_i|$  where each $A_i$ a random cyclotomic cosets of index $\ell$. In Section~\ref{random polynomials},
we allowed the possibility that $A_i$ could be $\{0\}$, a subset with a different size from cyclotomic cosets. Earlier,   Barot and Pe\~{n}a \cite{BP} considered the probability distribution of
$|\cup_{i=1}^{\ell} A_i|$ where each $A_i$ is chosen independently and uniformly at random from $\mathcal{P}_{m}$ where $\mathcal{P}_m$ is the set of all $m$-subsets of $\mathcal{N}$.
In this section, we consider $|\cup_{i=1}^{\ell} A_i|$ such that $m_i$'s can be distinct. Let $X_n=|\cup_{i=1}^{\ell} A_i|$ and $Y_n=n-X_n$. In the following we establish the distributions of  $X_n$ and $Y_n$ and we show that the $Y_n$  is asymptotically normal. This generalizes the following result by Barot and Pe\~{n}a \cite{BP} because we allow $m_i$'s to be distinct.

\begin{theorem}[Barot and Pe\~{n}a, 2001]
Let $\mathcal{N}=\{1, \ldots, n\}$.  Let $X_n=|\cup_{i=1}^{\ell} A_i|$ where each $A_i$ is chosen independently and uniformly at random from $\mathcal{P}_{m}$ where $\mathcal{P}_m$ is the set of all $m$-subsets of $\mathcal{N}$ and $Y_n=n-X_n$.
We have
\begin{eqnarray*}
{\mathbb P}(X_n=i) &=&\frac{{n\choose i}}{{n\choose m}^\ell}\sum_{j=0}^{i-m}(-1)^{j}{i\choose j}{i-j\choose m}^\ell,\\
{\mathbb E}(X_n) &=&n\left(1-\left(1-\frac{m}{n}\right)^\ell\right),\\
{\mathbb V}(X_n) &=&n(n-1)\left(1-\frac{m}{n}\right)^\ell \left(1-\frac{m}{n-1}\right)^\ell -\left({\mathbb E}(X_n)\right)^2+{\mathbb E}(X_n).
\end{eqnarray*}
\end{theorem}

First of all, we extend the above results to general $m_i$'s.

\begin{lemma} Let $\mathcal{N}=\{1, \ldots, n\}$ and  $\mathcal{P}_{m_i}$ be the set of all $m_i$-subsets of $\mathcal{N}$
for $1\leq i\leq \ell$. Let $X_n$ be the random variable for the size of $\cup_{i=1}^{\ell} A_i$ where $A_i$ is a random set chosen independently and uniformly from $\mathcal{P}_{m_i}$ and $Y_n=n-X_n$. We have
\begin{eqnarray}
{\mathbb E}((Y_n)_k)&=&(n)_k\prod_{j=1}^\ell \frac{(n-m_j)_k}{(n)_k},\\
{\mathbb P}(Y_n=k)&=&\sum_{h=k}^n(-1)^{h-k}{h \choose k}{n\choose h}\prod_{j=1}^\ell \frac{(n- m_j)_h}{(n)_h},\\
{\mathbb P}(X_n=i)&=&{n\choose i}\sum_{h=0}^{i}(-1)^h{i\choose h}\prod_{j=1}^\ell \frac{ {i-h \choose m_j}}{{n \choose m_j}}.
\end{eqnarray}
\end{lemma}
\noindent {\bf Proof }  For each $j\in \mathcal{N}$, let $B_j$ denote the event that $j\notin \cap_{i=1}^\ell A_i$. We note that ${\mathbb P}(Y_n=k)$ is the probability that exactly $k$ of the $B_j$'s occur.
Since $A_1,A_2,\ldots,A_\ell$ are mutually independent, we have
\begin{eqnarray*}
S_k&=&{n\choose k}{\mathbb P}(B_1\cap B_2\cap \cdots \cap B_k)\\
&=&{n\choose k}\prod_{j=1}^\ell {\mathbb P}(1\notin A_j,2\notin A_j,\ldots,k\notin A_j)\\
&=&{n\choose k}\prod_{j=1}^\ell \frac{(n-m_j)_k}{(n)_k},
\end{eqnarray*}
and hence
$${\mathbb E}((Y_n)_k)=k!S_k=(n)_k\prod_{j=1}^\ell \frac{(n-m_j)_k}{(n)_k}.
$$
Now from (\ref{pk}) we obtain
${\mathbb P}(Y_n=k) =\sum_{h=k}^n(-1)^{h-k}{h \choose k}{n\choose h}\prod_{j=1}^\ell \frac{(n- m_j)_h}{(n)_h}$.
Finally,
\begin{eqnarray*}
{\mathbb P}(X_n=i)&=&{\mathbb P}(Y_n = n-i) \\
&=& \sum_{h=n-i}^n(-1)^{h-n+i}{h \choose n-i}{n\choose h}\prod_{j=1}^\ell \frac{(n- m_j)_h}{(n)_h}\\
&=&{n\choose i}\sum_{h=0}^{i}(-1)^h{i\choose h}\prod_{j=1}^\ell \frac{ {i-h \choose m_j}}{{n \choose m_j}},
\end{eqnarray*}
where the last equality holds after the substitution $h: = h-n+i$. \qed.

Next we show that $Y_n$ is asymptotically normal. The condition on $u_i$ in the following theorem can be relaxed considerably, but we
use the current form for the sake of simplicity.
\begin{theorem}
Let $u_j=m_j/n$. Suppose $\ell \to \infty$, $u_j=O(1/\ell)$ uniformly for all $1\le j\le \ell$, and $\sum_{i=1}^\ell u_i>c$ for some positive constant $c$.
Then $Y_n$ is asymptotically normal with mean and variance, respectively, equal to
\begin{eqnarray*}
\mu_n&=&n\prod_{i=1}^\ell (1-u_i),\\
\sigma_n^2&=&n\left(1-\left(1+\sum_{i=1}^\ell u_i\right)\prod_{i=1}^\ell (1-u_i)\right)\prod_{i=1}^\ell (1-u_i).
\end{eqnarray*}
\end{theorem}
\noindent {\bf Proof } For $k=O(\sqrt{n})$,
We have
\begin{eqnarray*}
{\mathbb E}((Y_n)_k)&=&(n)_k\prod_{i=1}^\ell \frac{(n-m_i)_k}{(n)_k}\\
&=&n^k\exp\left(-\frac{k(k-1)}{2n}+O\left(\frac{k^3}{n^2}\right)\right)\prod_{i=1}^\ell \prod_{j=0}^{k-1}\left(1-u_i-\frac{ju_i}{n-j}\right)\\
&=& \mu_n^k\exp\left(-\frac{k(k-1)}{2n}+O\left(\frac{k^3}{n^2}\right)\right)\prod_{i=1}^\ell \exp\left(\sum_{j=0}^{k-1}
\ln\left(1-\frac{ju_i}{(1-u_i)(n-j)}\right)\right)\\
&=&\mu_n^k\exp\left(-\frac{k(k-1)}{2n}+O\left(\frac{k^3}{n^2}\right)-\sum_{i=1}^\ell \sum_{j=0}^{k-1}\frac{ju_i}{(1-u_i)(n-j)}\right) \\
&=& \mu_n^k\exp\left(-\frac{k(k-1)}{2n}-\sum_{i=1}^\ell \sum_{j=0}^{k-1}\frac{ju_i}{(1-u_i)n}+O\left(\frac{k^3}{n^2}\right)\right) ~~~~ (because ~j \leq k ~\sim \sqrt{n})\\
&=& \mu_n^k\exp\left(-\frac{k(k-1)}{2n}\left(1+\sum_{i=1}^\ell u_i\right)
+O\left(\frac{k^3}{n^2}+\frac{k^2}{n\ell}\right)\right). ~~~(note ~ u_j = O(1/\ell))
\end{eqnarray*}
In particular we have
$$
{\mathbb E}((Y_n)_2)=\mu_n^2\exp\left(-\frac{1}{n}\left(1+\sum_{i=1}^\ell u_i\right)+O\left(\frac{1}{n^2}+\frac{1}{n\ell}\right)\right),
$$
and for $k$ of the order $\sqrt{n}$,
$$
{\mathbb E}((Y_n)_k)\sim \mu_n^k\exp\left(-\frac{k^2}{2n}\left(1+\sum_{i=1}^\ell u_i\right)\right).
$$
It follows that
\begin{eqnarray*}
s_n&=&\frac{\sigma_n^2-\mu_n}{\mu_n^2}\\
&=&\frac{{\mathbb E}(Y_n(Y_n-1))}{\mu_n^2}-1\\
&=&\exp\left(-\frac{1}{n}\left(1+\sum_{i=1}^\ell u_i\right)+O\left(\frac{1}{n^2}+\frac{1}{n\ell}\right)\right)-1\\
&=&-\frac{1}{n}\left(1+\sum_{i=1}^\ell u_i\right)+O\left(\frac{1}{n^2}+\frac{1}{n\ell}\right),
\end{eqnarray*}
and
$$
{\mathbb E}((Y_n)_k)\sim \mu_n^k\exp\left(\frac{k^2s_n}{2}\right).
$$
Now the theorem follows from Lemma~\ref{normal}. \qed

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redpajama_set_name: RedPajamaArXiv


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text:
Hicks, Britton M.
Britton primarily practices in the areas of Social Security Disability, Wills, Powers of Attorney, and Estate Administration, however she also has a background in Family Law, Juvenile Law and Unemployment Law. Britton is a member of the National Organization of Social Security Claimant's Representatives. Her love of Social Security law was sparked when she was successful in helping her uncle receive Social Security benefits. Since then, she's been practicing in the area for ten years. In addition to private practice, Britton has also worked as a Hearings Officer for the Unemployment Compensation Review Commission from 2006-2009, and the Ohio Legislative Service Commission from 2003-2006.
She is a 2003 Graduate of Capital University Law School, where she earned distinction as a Member of the Order of the Curia and the CALI Award for Excellence in Adoption Law, and a proud 2000 graduate of Marshall University, where she received a B.A. in History, with a minor in Political Science.
Britton was born and raised in Huntington, West Virginia. Her family now lives in Grove City, Ohio, where she is an active member of Our Lady of Perpetual Help parish. She is also an active member of Grove City Rotary, focusing her attention on developing a love of service and volunteering in middle and high school students.
Capital University Law School, Columbus, Ohio
Honors: Member of the Order of the Curia
Honors: Received for CALI Award for Excellence in Adoption Law
Honors: Past President of the Womens Law Association of Capital University
Honors: With Distinction
Marshall University, Huntington, West Virginia
B.A. - 2000
Major: History
Minor: Political Science
Pro-Bono Activities
Member, Our Lady of Perpetual Help
National Organization of Social Security Claimant's Representatives (NOSSCR), Member, 2009 to Present
Current Employment Position
Past Positions
Ohio Legislative Service Commission, Attorney, 2003 to 2006
National Center for Adoption Law and Policy, 2001 to 2003
Unemployment Compensation Review Commission, Hearing officer, 2006 to 2009


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Official publication of Indian Association of 0ccupational Health
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Article in PDF (214 KB)
Year : 2019 | Volume : 23 | Issue : 3 | Page : 101
Dealing with occupational lung diseases – A few challenges
Bobby Joseph
Occupational Health Services, Department of Community Health, St. John's Medical College, Bangalore, Karnataka, India
Date of Submission 09-Dec-2019
Date of Acceptance 09-Dec-2019
Date of Web Publication 16-Dec-2019
Dr. Bobby Joseph
Occupational Health Services, Department of Community Health, St. John's Medical College, Bangalore - 560 034, Karnataka
DOI: 10.4103/ijoem.IJOEM_282_19
Joseph B. Dealing with occupational lung diseases – A few challenges. Indian J Occup Environ Med 2019;23:101
Joseph B. Dealing with occupational lung diseases – A few challenges. Indian J Occup Environ Med [serial online] 2019 [cited 2020 Jan 23];23:101. Available from: http://www.ijoem.com/text.asp?2019/23/3/101/273037
Experiences of practitioners of occupational medicine will surely point to an underrecognition and subsequent underreporting of occupational lung diseases (OLDs). Although history will tell us that the father of occupational medicine, Bernardino Ramazzini, first reported hypersensitivity pneumonitis among wheat reapers exposed to overheated grains, medical practice has scarcely been able to keep pace with the understanding of exposure to newer molecules and their outcomes in specific occupational groups. On the other hand, traditionally, reference books on OLDs – while dealing with pneumoconiosis, hypersensitivity pneumonitis, occupational asthma, and pulmonary function tests – rarely consider infectious diseases as one among those that are probably occupational in origin.
This issue of the IJOEM hopes to bring these two issues – OLDs and infectious diseases of the lung that are of occupational origin – to the attention of the readers. The report on Gujjar lung is a case in point – though identified first in 1991 as a pneumoconiosis attributable to biomass combustion at home, larger studies quantifying its prevalence and listing its risk factors are rare. The consequence of which is alarming from a public health point of view, wherein the disease is misdiagnosed as pulmonary tuberculosis and unnecessarily treated for the same. This phenomenon, which is probably true of many unrecognized "pneumoconioses," has serious implications in the fight against tuberculosis. This has been the case with silicosis, where the lack of the facilities and skills to rule out tuberculosis, fear of consequences of the labeling of a case as silicosis, and even the well-known constraints in the management of the disease often lead to a "trial" of anti-tuberculosis treatment.
The issue of tuberculosis as an occupational-acquired disease begs consideration. While specific pneumoconiosis can be directly linked to being occupationally involved in an industry where the specific causative agent is present, it is difficult to prove whether tuberculosis in health workers is community-acquired or occupationally acquired. The mere incidence or prevalence of the disease in larger numbers than in other occupations is not sufficient to warrant it being called an occupational lung disease. Advanced statistical methods and mathematical models make it possible to estimate the burden of occupationally acquired pulmonary tuberculosis. However, as occupational physicians, in dealing with individual employees, it is important that we identify tuberculosis and other infectious diseases in as effective and efficacious a manner as is possible. The article in this issue on QuantiFERON-TB Gold In-Tube Test versus Tuberculin Skin Test focuses on this problem – where there is the likelihood of individual establishments in different geographic locations having to take decisions depending on factors related to cost and feasibility.
This issue has two articles that deal with unique medical consequences of OLDs – skin thickening and spontaneous pneumothorax, while another article deals with the broader quality of life of an occupational group with poor lung function.
In the coming year, selected issues of this journal will focus on specific occupational health themes – occupational health of health workers, occupational health of workers in unorganized sectors, and health of working women. We encourage researchers to submit articles to these thematic issues of the journal.
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© 2005 - Indian Journal of Occupational and Environmental Medicine
Online since 1st April '05


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australia gold mines leading companies that deal with gold Jun
Best Gold Stocks for 2021 | The Motley Fool
Jun 23, 2021· The VanEck Vectors Gold Miners ETF holds the stocks of large gold mining companies. It's one of the largest gold ETFs, with nearly $16 billion in assets as of mid-2021. At the time of this ...
Northern Star, Saracen create $11.5bn Australian gold ...
Oct 06, 2020· The combined company, with mines in Australia and Alaska, will produce 2 million ounces of gold a year from fiscal 2027. It is also expected …
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Sep 23, 2021· The company announced in early August financial results for Q2 FY 2021, the three-month period ended June 30, 2021. Equinox Gold reported net …
Santos signs new gas deal with Gold Fields | Santos
Santos, Western Australia's biggest domestic gas supplier, is pleased to announce that it has entered into a new gas supply agreement with Gold Fields Limited for its three gold mines in Western Australia. Santos will supply nearly 5.5 petajoules of natural gas from its Varanus Island gas plant over three years, starting on 1 July 2020.
Australia tops China's gold production
Aug 30, 2021· US-owned Newmont Corporation's Boddington mine in Western Australia delivered 700,000 ounces in the year to June 30, up 36,000oz, making it Australia's second-largest goldmine in terms of output.. Newcrest Mining's Cadia East operation in NSW is the largest, having delivered almost 765,000oz.. Kirkland Lake Gold's Fosterville mine in Victoria produced about 592,000oz and …
What are the Best Gold ETFs on the ASX in 2021?
GDX provides exposure to the ~50 companies involved in mining gold and silver. It is unhedged with a large focus on North America charging 0.53% per year and has ~$418m in assets. MNRS is a hedged version that invests in ~50 companies engaged in gold, silver or other metal mining.
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Jul 13, 2021· Best Gold Mining Stocks to Invest In. 15. New Gold Inc. ( NYSE: NGD) Number of Hedge Fund Holders: 14. New Gold Inc. (NYSE: NGD) is a Canada-based mining company …
Which are the best gold stocks on the ASX?
Jun 10, 2021· In 2020, gold mine production in Australia reached 328 tonnes. Australia is expected to overtake China as the world's largest gold producer in 2021, producing 384 tonnes, as per the report. On that note, let us look at a few of the leading gold producers listed on the Australian Securities Exchange (ASX).
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May 31, 2021· Northern Star was a newcomer to the top 10 list, while Polymetal and Kirkland Lake have dropped out of the quarterly ranking. Kitco ranked the top 10 largest gold mining companies by production in Q1 2021. Newmont was the top gold producer in Q1 2021. The company's attributable gold production was in line compared to the prior year quarter ...
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Mar 26, 2020· Few sectors of the global economy are flourishing in these uncertain times like Australian gold mining. A combination of factors including gold's status as a safe haven investment, global monetary policy in response to the coronavirus threat and exchange rate movements have this week driven the Australian dollar gold price to a record of almost $2750 an ounce.
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May 04, 2021· It retained its top spot among the world's largest gold mining companies despite an almost 9% year-on-year production decline. It has operations across Africa, Australia, North America and South America, and boasts mine reserves totalling 94.2 million attributable ounces of gold. It also mines copper, silver, zinc and lead.
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Oct 11, 2021· Comino is Australia's largest gold producer, with approximately five million ounces being produced annually. Gold Mining Australia, or Goldeneze, is another company that is extremely popular with both mining engineers and prospectors. They have operations in South Africa, Canada, Norway, the United States, and Brazil.
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Nov 06, 2019· The company also has exposure to silver at its owned Prominent Hill mine in South Australia, where the metal is produced as a by-product of its copper-gold mining operation. During the September quarter, OZ recorded the operation milling 2.36Mt of ore, grading 1.18% copper, 0.5g/t gold and 3.15g/t silver.
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Australia: Santos signs new gas deal with Gold Fields
Jun 16, 2020· Santos, Western Australia's biggest domestic gas supplier, has entered into a new gas supply agreement with Gold Fields Limited for its three gold mines in Western Australia.. Santos will supply nearly 5.5 petajoules of natural gas from its Varanus Island gas plant over three years, starting on 1 July 2020. Santos Managing Director and Chief Executive Officer, Kevin Gallagher said Santos ...
Gold Royalty and Ely Gold merge to create new royalty ...
Jun 21, 2021· Ely Gold Royalties is a Nevada focused gold royalty company. Its current portfolio includes royalties at Jerritt Canyon, Goldstrike and Marigold, three of Nevada's largest gold mines, as well as the Fenelon mine in Quebec, operated by Wallbridge Mining. Gold Royalty is a gold-focused royalty company. Gold Royalty's diversified portfolio ...
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Let's say, for example, that the price of gold is $1,200 per ounce. A gold company might be able to mine gold at a cost of $1,000 per ounce. Gold companies generally measure this by their all-in sustaining cost (AISC) per ounce. So in over-simplified terms for the purpose of example, they make $200 per ounce in profit at current prices.
Deadly attack at Burkina Faso gold mine has Australian ...
Nov 08, 2019· A Perth-based mining services company with workers at the Canadian-owned the Boungou mine has confirmed none of its Australian staff were killed or injured in a deadly attack on a convoy yesterday.
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Sep 16, 2020· The two companies began pouring gold at the Western Australian site on 30 June 2019 and as at 30 June 2020, had produced 230,590 ounces with an all-in sustaining cost (AISC) of production. This incorporates not only the "cash cost" of production, but all the costs that allow production to be sustained of $1,186 per attributable ounce.
Super mines: Australia's biggest mining projects
Mar 07, 2019· Australia's largest gold mine has been owned by Newmont since 2009, when it purchased a controlling share in the project from AngloGold Ashanti and Newcrest. The company has since invested heavily in the Western Australian open pit mine, which reached two million ounces of gold produced by 2012, and ramped up production to 787,000oz of gold ...
Perseus Mining (ASX:PRU) increases gold production in June ...
Jul 21, 2021· Company shares are up 0.65 per cent to trade at $1.54 at 1:20 pm AEST; Perseus Mining (PRU) has announced a 16 per cent quarter-on-quarter increase in gold production for the June quarter. Perseus' three operating gold mines, Edikan in Ghana, and Sissingué and Yaouré in Côte d'Ivoire, produced a combined total of 102,788 ounces of gold in ...
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Jul 21, 2020· Coming in third on this top gold-mining companies list is AngloGold Ashanti, which produced 102.1 tonnes of gold in 2019. That represents a small decrease from the 106.1 tonnes it produced in 2018.
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Aussie Mine provides industry and financial analysis on Australia's mid-tier mining sector as represented by the mid-tier 50 (MT50) – the largest ASX mining companies with a market capitalisation of less than $5 billion as at 30 June 2020. largest commodity group in the MT50, comprising 44% of the MT50 by number. However, MT50
Jun 24, 2016· Government overlooks mining for financial assistance (0) July 15, 2021 Australia's mining industry incurred a tariff input penalty of $53 million in …
Oct 16, 2020· The top 10 biggest gold mines in Australia produced 1,268 koz of gold in June Quarter 2020, a 3% decline year-over-year (1,311 koz). With 237 koz of gold mined in June Quarter 2020, Newcrest's Cadia East mine was the largest gold producing center in Australia, followed by Newmont's Boddington (168 koz) and Kirkland Lake's Fosterville mine ...
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Jun 22, 2020· Gold Fields Limited is a globally diversified company with nine active mines. Projects are located in Australia, Peru, South Africa, West Africa, the Philippines and Chile. Gold Fields ranks as the seventh largest gold producer in the world. [11] Gold Fields Limited is a publicly traded company, with shares offered primarily on the JSE.
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Robins and Parson Named D2CCA Softball All-Region
May 15, 2017 by Staff Reports
MURFREESBORO, N.C. – The Division II Conference Commissioner's Association released on Monday, May 15 All-Atlantic Regional selections with Kayla Robins and Brittany Parson getting an All-Regional nods.
Kayla Robins was named to the D2CCA All-Atlantic Region First Team after a career senior season. The senior pitcher picked up CIAA Pitcher of the Year, CIAA First-Team, and NFCA All-Region honors.
Robins finished her three-year career with Chowan posting a 53-20 record with a 1.74 ERA in 471.1 innings. The Mathews County, Va. native struck out 680 batters including 367 punch outs this season.
Brittany Parson was named to the D2CCA All-Atlantic Region Second Team pacing the team with a .349 batting average as a freshman. Parson picked up CIAA Rookie of the Year and CIAA Rookie Team honors.
During the season, Parson led the team with doubles (12), home runs (11), RBI (54), along with her .349 batting average.


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