text:
HIV Peer Counselor in the Dominican Republic Shares Impact of His Work
By Alex Collins
Technical program manager, IntraHealth International
By Sarah Dwyer
Former communications manager, IntraHealth International
Alfredo Felix is a peer counselor with the Department of HIV at Jaime Mota Regional Hospital in Barahona, Dominican Republic. "I've always felt motivated to work in the community to inform people," he says. The area shares a border with Haiti and has a large immigrant population at risk for HIV.
Peer counselors like Alfredo play an important role in countering the effects of stigma, which can make it hard for people to seek information about HIV and follow through with treatment. Alfredo tells a story about someone he helped:"There was a patient who was a bit fragile and came in for his consultation, and we tried to educate him about his health condition. Later the patient returned home, and we lost contact with him. I thought to go out to the countryside, and there I was able to contact the patient and speak with him. I insisted that he come back to the health center. When he did return, he was even more fragile, and we worked with him again, educating him and getting him on medication. Now, the patient is better, he's stronger, and is thankful for the work [that we do].
"Alfredo recently took part in a workshop to improve the quality of services to eliminate mother-to-child transmission of HIV and congenital syphilis. The workshop was designed and delivered by the Directorate of STI/HIV/AIDS (DIGECITSS) and the health services network (REDES) along with CapacityPlus. Alfredo called it "very important, because it's intended to improve the quality of services that are offered, to train good human resources [for health], so that these human resources—by being trained and empowered—provide better services for the community.
"The workshop has since been replicated in seven out of the nine health regions in the country and reached over 200 health workers.
CapacityPlus provides assistance to the Ministry of Health to improve the human resources management systems that will strengthen the health workforce needed to contribute to an AIDS-free generation and the country's goals to reduce mother-to-child transmission of HIV to 2% or less and the incidence of congenital syphilis to 0.5 cases per 1,000 live births or less by 2015. The ministry's efforts have already led to improved access to health services through increased availability of skilled health workers, the elimination of traditional user fees, and an increase in national health insurance members receiving subsidized care. Alfredo's region received an influx of 234 new health workers, resulting in a 500% increase in the number of patient consultations for HIV and syphilis prevention, prenatal care, and family planning between 2012 and September 2013.
The coverage of services—including early detection of HIV and syphilis, family planning (particularly for HIV-positive women), and access to prenatal care for HIV-positive women—has risen in many regions. Increasing coverage of HIV services at the primary level results in improved coverage up the chain, with more HIV-positive individuals being referred for treatment and contraceptive options and more HIV-positive pregnant women and their babies cared for, to further the goal of an AIDS-free generation.
For Alfredo, the best part of his job "is when I see a patient who's in poor health and later when he comes [back] to us, we can see how he's been strengthened and recuperated." He shares his thoughts in this one-minute video.
Help CapacityPlus spread the word about strengthening the health workforce. Follow us on Twitter and like us on Facebook.
Photo by Alex Collins.This blog post was originally published on the CapacityPlus blog.
education & performance Management and Performance family planning HIV/AIDS Health Workers Dominican Republic
How Local Fistula Training Helped Prevent an Ebola Outbreak in Mali


text:
Kathlyn Williams (31 de mayo de 1879 – 23 de septiembre de 1960) fue una actriz teatral y cinematográfica estadounidense, activa en la época del cine mudo.

Inicios. Carrera
Su nombre completo era Kathleen Mabel Williams, y nació en Butte, Montana. Hija única, sus padres eran Joseph Edwin "Frank" Williams, propietario de una pensión, y Mary C. Boe (1846–1908) de origen galés y noruego. Muchas biografías sostienen que nació en 1888; sin embargo, en el censo de 1880 aparece con un año de edad.

Williams demostró desde joven interés por ser actriz. Miembro del Woman's Relief Corps, pudo demostrar su destreza vocal en recitales de carácter local. Aunque era conocida por tener una voz adecuada al canto, la principal vocación de Williams era la interpretación. Ella estudió en la Montana Wesleyan University (actual Rocky Mountain College) de Helena en los últimos años 1890, graduándose en 1901. En mayo de 1899 interpretó "The Gypsy Flower Girl" en la competición anual de su universidad. El 29 de mayo de 1900 recibió una medalla de oro por su actuación en "Old Mother Goose" en la Montana Wesleyan. Williams perdió a su padre hacia 1894, y su madre se casó con Fred Lavoie en 1895, divorciándose al siguiente año.

Con escasos recursos económicos en su familia, Williams dependía de terceras personas para poder completar sus estudios. Sus aspiraciones como actriz llamaron la atención de William A. Clark, un rico senador de Montana, que ayudó a financiar su educación y su preparación como actriz. Clark pagó su matrícula en la American Academy of Dramatic Arts de Nueva York. También recibió el apoyo de Richard "Uncle Dick" Sutton, propietario de varios teatros en Butte, donde Williams inició su carrera teatral. En 1902 Williams fue miembro de un grupo teatral llamado Norris & Hall and Company, con el cual interpretó a Phyllis Ericson en la popular obra "When We Were Twenty One", y con la que consiguió buenas críticas. La obra hizo una gira por los Estados Unidos hacia finales de 1903.

Williams empezó su carrera cinematográfica con la Selig Polyscope Company en Chicago, Illinois, debutando en 1908 bajo la dirección de Francis Boggs, siendo transferida en 1910 a los estudios que la compañía poseía en Los Ángeles. Williams fue "Cherry Malotte" en The Spoilers (1914), la primera película basada en la novela de Rex Beach, en un papel que más adelante sería interpretadao por Betty Compson (1930), Marlene Dietrich (1942), y Anne Baxter (1955). En 1916 fue la protagonista de un serial de aventuras en 13 episodios, The Adventures of Kathlyn. Aunque estuvo muy ocupada en los años del cine mudo,  cuando llegó el sonoro, y en parte también por su edad,  solamente rodó cinco cintas, la última de ellas en 1935. En sus últimas películas, Williams trabajó como actriz de carácter, dejando atrás sus trabajos como comediante y actriz de seriales.

Vida personal

Williams se casó tres veces. Aunque muchas biografías citan erróneamente a Victor Kainer como su primer marido, lo cierto es que era Otto H. "Harry" Kainer (1876–1952), que dirigía una empresa de importación y exportación en Wall Street, Nueva York. Se casaron el 2 de octubre de 1903, y tuvieron un hijo, Victor Hugo, nacido en 1905. La pareja se divorció en 1909, supuestamente por la desaprobación de Kainer hacia la carrera artística de su esposa. Tras la muerte de su madre en diciembre de 1908 y su divorcio, Williams decidió reactivar su carrera de actriz. En 1910 fue a vivir con su hijo a Los Ángeles, California, donde obtuvo con facilidad trabajo. El 4 de marzo de 1913 se casó con Frank R. Allen, también actor, pero el matrimonio duró únicamente poco más de un año.

Williams se casó el 2 de junio de 1916 con el ejecutivo de Paramount Pictures Charles Eyton, en Riverside, California. La pareja se había conocido unos diez años antes en Salt Lake City, Utah. Eyton era uno de los propietarios de la Oliver Morosco Photoplay Company. El 25 de febrero de 1922 falleció el hijo de Williams, ahora llamado Victor Eyton, a los 16 años de edad, a causa de las complicaciones de la gripe. Con el fin de intentar sobreponerse a su dolor, el matrimonio inició un viaje a Asia que duró cuatro meses. Finalmente se divorciaron en 1931.

El 29 de diciembre de 1949 Williams sufrió un accidente de tráfico en el que falleció su amiga Mary E. Rose. La actriz perdió su pierna derecha. Kathlyn Williams falleció a causa de un infarto agudo de miocardio en Hollywood, California, en 1960. Sus restos fueron incinerados, y las cenizas depositadas en el Crematorio Chapel of the Pines de Los Ángeles.

Por su contribución al cine, fue recompensada con una estrella en el Paseo de la Fama de Hollywood, en el 7038 de Hollywood Boulevard.

Selección de su filmografía

Actriz

1908
On Thanksgiving Day, de Francis Boggs

1909

The Politician's Love Story, de D.W. Griffith

Lines of White on a Sullen Sea, de D.W. Griffith

1910

Taming Wild Animals, de Francis Boggs
Gold Is Not All, de D.W. Griffith
A Romance of the Western Hills, de D.W. Griffith
Thou Shalt Not, de D.W. Griffith
The Fire Chief's Daughter, de Francis Boggs

Mazeppa, or the Wild Horse of Tartary, de Francis Boggs
Dora Thorne
Blasted Hopes
The Merry Wives of Windsor, de Francis Boggs
The Queen of Hearts

1911

Busy Day at the Selig General Office
The Curse of the Redman, de Francis Boggs
The Survival of the Fittest
The Man from the East
1861
The Witch of the Everglades, de Otis Turner
In Old California When the Gringos Came, de Francis Boggs
Back to the Primitive, de Francis Boggs y Otis Turner
Jim and Joe, de Otis Turner
The Rose of Old St. Augustine, de Otis Turner
Ten Nights in a Bar Room, de Francis Boggs
Captain Kate, de Francis Boggs y Otis Turner

Jealous George, de Otis Turner
Life on the Border, de Otis Turner
Dad's Girls, de Otis Turner
Wheels of Justice
The Wheels of Justice, de Otis Turner
The Two Orphans, de Otis Turner y Francis Boggs
Maud Muller, de Otis Turner
How They Stopped the Run on the Bank, de Otis Turner
Lost in the Jungle, de Otis Turner
The Inner Mind, de Otis Turner
Getting Married, de Colin Campbell
Paid Back, de Colin Campbell

1912

The Prosecuting Attorney, de Colin Campbell
The Horseshoe, de Otis Thayer
When Memory Calls, de Frank Beal
The Brotherhood of Man, de Frank Beal
Sons of the North Woods, de Frank Beal
Driftwood, de Otis Thayer
When the Heart Rules, de Richard Garrick
The Devil, the Servant and the Man, de Frank Beal
The Coming of Columbus, de Colin Campbell
The Stronger Mind, de Frank Beal

The Turning Point, de Frank Beal
The Girl with the Lantern, de Frank Beal
The Adopted Son, de Otis Thayer
On the Trail of the Germs, de William V. Mong
An Unexpected Fortune, de Otis Thayer
The Girl at the Cupola, de The Girl at the Cupola
As the Fates Decree, de Oscar Eagle
The House of His Master, de Lem B. Parker
Harbor Island, de Lem B. Parker

1913

The Lipton Cup: Introducing Sir Thomas Lipton, de Lem B. Parker
A Little Child Shall Lead Them, de Lem B. Parker
The Governor's Daughter, de Lem B. Parker
The Artist and the Brute, de Henry MacRae
Her Only Son, de Lem B. Parker
The Tide of Destiny, de Lem B. Parker
The Unwelcome Throne
The Love of Penelope, de Francis J. Grandon
I Hear Her Calling Me (from the Heart of Africa), de Francis J. Grandon
Two Men and a Woman, de Lem B. Parker
With Love's Eyes, de Lem B. Parker
A Wise Old Elephant, de Colin Campbell
The Burglar Who Robbed Death, de Lem B. Parker
Their Stepmother, de E.A. Martin
A Welded Friendship, de Lem B. Parker
Lieutenant Jones, de Lem B. Parker

The Stolen Melody, de Lem B. Parker
The Girl and the Judge, de Lem B. Parker
Woman: Past and Present, de Lem B. Parker
Mrs. Hilton's Jewels, de E.A. Martin
The Tree and the Chaff, de Lem B. Parker
Man and His Other Self, de Lem B. Parker
The Mansion of Misery, de Lem B. Parker
The Flight of the Crow, de E.A. Martin
The Child of the Sea, de Lem B. Parker
Two Too Many, de Henry MacRae
The Young Mrs. Eames, de Francis J. Grandon
The Conscience Fund, de Francis J. Grandon
Thor, Lord of the Jungles, de Colin Campbell
In the Midst of the Jungle, de Henry MacRae
The Adventures of Kathlyn, serial de Francis J. Grandon (1913-1914)

1914

The Spoilers, de Colin Campbell

Hearts and Masks, de Colin Campbell

1916

Sweet Lady Peggy
The Adventures of Kathlyn, de Francis J. Grandon

The Ne'er Do Well, de Colin Campbell
The Devil, the Servant, and the Man, de Frank Beal

1917

The Voice That Led Him, de Francis J. Grandon
A Man, a Girl, and a Lion, de Francis J. Grandon
Out of the Wreck, de William Desmond Taylor
The Cost of Hatred, de George Melford

The Highway of Hope, de Howard Estabrook
Big Timber, de William Desmond Taylor
The African Jungle, de E.A. Martin
Pioneer Days, de Oscar Eagle

1918

 We Can't Have Everything, de Cecil B. DeMille

The Whispering Chorus, de Cecil B. DeMille

1919
A Girl Named Mary, de Walter Edwards)

1920

Just a Wife, de Howard C. Hickman
The Tree of Knowledge, de William C. de Mille
The Prince Chap, de William C. de Mille

Conrad in Quest of His Youth, de William C. de Mille
The U.P. Trail, de Jack Conway

1921

Forbidden Fruit, de Cecil B. DeMille

Hush, de Harry Garson

1922
Clarence, de William C. deMille

1923
The Spanish Dancer, de Herbert Brenon

1924

The Enemy Sex, de James Cruze

When a Girl Loves, de Victor Halperin

1932
 Unholy Love, de Albert Ray

Guionista

The Last Dance, de Oscar Eagle (1912)
The Young Mrs. Eames, de Francis J. Grandon (1913)
The Leopard's Foundling, de Francis J. Grandon y Kathlyn Williams (1914)

The Strange Case of Talmai Lind, de William Robert Daly (1915)
A Sultana of the Desert, de Tom Santschi (1915)
Lost in Transit, de Donald Crisp (1917)

Directora
The Leopard's Foundling, codirigida con Francis J. Grandon (1914)

Referencias

Galveston Daily News, Screen Life in Hollywood, viernes, 20 de marzo de 1931, página 4.
New York Times, Kathlyn Williams, 25 de septiembre de 1960, página 86.
Reno Evening Gazette, Short Yarns About Many Favorites, sábado, 17 de junio de 1916, página 10.
The Anaconda Standard, "A Benefit Concert. Chance to Help a Young Girl Through College." 16 de agosto de 1900, página 4.
The Anaconda Standard, "Musical Entertainment: A Delightful Programme Prepared By the Woman's Relief Corps", 1 de septiembre de 1901, página 16.
The Anaconda Standard, "Butte Actress Loses Young Son", 7 de marzo de 1922, página 6.
The Bee, "Former Actress Sues For $136,615", sábado, 8 de abril de 1950, página 11.
New York Times, "Husband Must Pay Up", martes, 9 de mayo de 1905, página 2.
Pasaporte de 1924 en Ancestry.com afirma que nació en 1885.
Censo de Los Ángeles de 1910
Morning Olympian, "Kathlyn Wiliams Kainer", 7 de julio de 1905, página 2.
The Helena Independent, "For Honor and A Prize, University Students Will Take Part in Elocutionary Contest", 29 de mayo de 1899, página 8.
The Anaconda Standard, "Sad News Shocks Daughter: Rushes To Bedside Of Aged Pioneer Woman On Learning Plight For First Time", 12 de diciembre de 1908
The Trenton Times, "Stage Gossip", 15 de septiembre de 1902
Waterloo Reporter, "Kathlyn Williams Now Mrs. Allen", sábado, 29 de marzo de 1913, página 19.
Long Beach Independent, "Kathlyn Williams Estate Goes to Charity", 27 de octubre de 1960, página 8.
Nebraska State Journal, "On the Silver Screen. Among the Movies.", 26 de marzo de 1922, página 30.
Moberly Weekly Monitar, "Kathlyn Williams Sues for Divorce", 3 de julio de 1914, página 2.
San Jose Mercury News, "Promised Bride $10,000 A Year", 22 de mayo de 1905,
Los Angeles Times, "School Boys Bear Remains To Grave", 1 de marzo de 1922.
Logansport, Indiana, "Husband Failed to Pay Her For Leaving Stage", 23 de junio de 1905, página 5.
Dallas Morning News, "Amusements", 16 de enero de 1903
The Anaconda Standard, "Little Miss Butte Meets Success", 13 de julio de 1902.

Enlaces externos

 Artículos contemporáneos y entrevistas con Kathlyn Williams
 Kathlyn Williams en Virtual History

Nacidos en Butte (Montana)
Fallecidos en Hollywood
Actrices de Montana
Actores de cine de Estados Unidos
Actrices de cine mudo
Guionistas de cine de Estados Unidos
Paseo de la Fama de Hollywood
Fallecidos por infarto agudo de miocardio

redpajama_set_name: RedPajamaWikipedia


This afternoon offers the opportunity of doing laundry in preparation for "the packing" in preparation for my upcoming trip to Deutschland on Tuesday.
As usual, I'm conflicted – don't want to take the big suitcase, & also don't want to end up wishing that I'd brought more stuff with me. But, I usually end up taking about 2-3x as many shirts, pants, & warm stuff than I need. And every time I get back, I promise myself that I won't make the same mistake again. Which, of course, I inevitably do.
But this time will be different. Really. It could happen.
The 1st 4 days of my trip I will be in a place called "Gau Algesheim" – about a 2 hour train trip from Frankfurt. I'm headed there right after my arrival (early Wednesday a.m.) to spend some time with my friends Alex & Linda – something that I'm greatly looking forward to, as they are always 'highlights' for me to spend time with – for their wonderful personalities, great senses of humor, & determination to follow Christ, all of which are an inspiration to me.
After that, I'm off to Oberwesel – don't know anything about that, other than its by the Rhein River, which makes theBean more than a little jealous, as seeing lights on water, at night, is her favorite thing. I'll be there through the following Wednesday, 12 November, at which time its back to Frankfurt for meetings with friends at TPLF. Our sister church has a new pastor, Eddy Dueck, & I'll hopefully be able to spend some time with him during the last days.
I don't QUITE have my accommodations when I'm back in Frankfurt arranged. Which means that after 12 November, I don't know where I'll be staying, or with whom. And its not that I'm worried about not knowing, but I think I'd probably sleep better tonight, where I'll be sleeping then.
Ok. Time to load the washer.
About 3 years ago, I read one of the most influential books I've ever come across – called, Love Is A Choice. It made an impact because it brought to light & illustrated the human condition & how/why we connect with others… esp. in unhealthy, selfish, &/or destructive ways.
One of the things that stood out to me, (at least from the thoughts that ended up being provoked OUT of having read it,) was that in order for a friendship to be healthy & life-giving, both parties had to be 'givers' (my word;) nurturing, feeding, building up the relationship & the other person – a relationship where 1/both are 'takers' (again, my word) ends up with 1/both being taken advantage of, used & abused, all in the name of friendship.
1 persons constant tardiness or not showing up at all, & when they're questioned, saying, "Oh, I was busy," or offering up some other excuse. Always.
The blowing up in rage, pouting in petulance, avoiding in silent treatment upon saying "No" to a request (demand?), & seeing the behaviors (& acceptance) change only upon giving the "right" answer, "YES!" – do what I want.
1 person being the one that kept communication going – in emails, letters, phone calls, with the other always being 'too busy' or 'not good at" initiating contact.
Having been asked, "if you see something in what I'm doing, who I'm seeing, places I'm going that seem destructive, would you please tell me?" & actually believing it & doing it, to find that what you end up with is a person that 'dumps' you from the friendship because of what you've said, finding out too late that the fine sounding words spoken in a point of seeming relational intimacy were not quite what was meant.
hearing from someone just when they need something, & going weeks, months, & even years when they don't.
Not being able to talk about anything significant or important to being really real friends, because it just might upset the other person – meaning we wouldn't be able to maintain the minimal & surface-ey contact we have.
The crispness in the air is refreshing… makes me glad to live in Reno/Sparks, where we get to have all 4 seasons… The more I travel, the more I'm thankful for the change in temperature, the occasional frosty dew on the grass, & even the need to bundle up before going out. Even the brown that is the normal Nevada landscape is something I appreciate.
The last 3 months have been a time of reconnections – I've gotten more contacts from people from my past through Facebook in that time period that I had heard from in the previous 20 years. Very interesting to be able to hear from people who have been significant in my life at some point, & now to hear/see what is happening in whatever phase of life that they're in. And seeing the fruit of the choices that have been made too. Cool beans.
Diet Sierra Mist rocks the world. The DSM with Cranberry is on another level as well. Absolutely incredible. I shall pour me another.
The cause of a lot of offense, frustration, disappointment, & unmet expectations is an underlying belief that I might not have even identified, let alone articulated… someone owes me an explanation. Some thoughtfulness. Time. Respect.
It's easier to say, "I don't owe anyone anything," but I believe that might have to do with the selfishness of the statement, which speaks to the ability to live life with a reckless abandon in regards towards the feelings & perceptions of others.
…to CS Lewis, the guy in the picture, smoking his pipe. I love the way he communicates in his writings, without hesitating to move into the difficult areas of life. Pain. Failure. Temptation. Struggle. Hope. Joy.
He has been an inspiration to me … And every journey through Narnia. The Sci-Fi Trilogy. Mere Christianity. The Problem of Pain. Surprised by Joy… Brings a bit of joy to me.
Seems like there is a conspiracy of some sort… a conspiracy to antagonize me until I finally snap, & really lose it. Get angry. Throw stuff. Yell. Have a fit. This isn't a new thing. It happens all the time. A little something something pulls at me, twists me to give in to the rage, to "vent." To allow myself the indulgence of a rant.
Yet there's the whisper that says no. Don't. I don't need to. It won't bring the relief that it promises. The resulting frustrations & regrets far outweigh the false luxury of temporary giving away of control.
It doesn't really matter if it's been "dumb people," as I know that I am someone's dumb person. (BTW: please don't write me to confirm. I really, really would encourage you to KITY, which is my own acronym, just created for Keep it to Yourself…) Life is full of frustrations. Long lines. Computer updates that create snafus with the other programs on the computer. People that one has to interact with in order to transact business, eat a meal, fill a prescription, find necessary information… if I were to lose it every time I was justified in "losing it," I think that I'd be going off pretty much all the time. And the issue would be selfishness. Self-focus. Pride. Me issues.
And the indulgence of flesh gives place to the dark-side… provides a pivot point, a manipulation station the enemy can work from, in complete anonymity, hiding behind my well-developed sense of offense, wronged-ness, & me-ness.
Note: please go easy on the scoey-isms that have entered the blog. If you're reading, it's late. I'm tired. Frustrated. Bothered at several wasted hours. At dust & dirt. Silly dogs. Eyes that itch & burn & are really red, that EVERYONE is finding it necessary to point out to me.
Feeling a bit pensive today- & wanted to share the love.
How does one know if what they're doing is a success or a failure?
What constitutes a successful life?
Is it silly to think in terms of success & failure when it comes to evaluating a life/life decisions?
Is it accumulation of stuff? Wealth? Friendships? Philanthropy/benevolence/humanitarianism? Education? Accomplishments?
What standard will I be measured with? Against?
you say you can't take it with you?
we'll see about that, won't we?
What is it that makes a person hold onto their 'stuff' when its obvious that they're never going to even take the stuff out of the box its in, let alone use it? I have boxes in the garage that have been in every garage I've had since I got married. If they're not empty by November 30, they will be thrown away, no questions asked, no looking inside. Write it down.
We're approximately 53% unpacked & organized. Points of great need: closets; window coverings; hangage of posters & pictures.
Our 1st College Group of the semester is happening Sunday, 9/16 – we're celebrating our "comeback" with a great dinner & communion. I can't wait.
It's amazing how wonderful & smart I am when people like what I say/do; it's equally amazing how the same people can think I'm a flipping moron when I say/do something that they don't agree with.
Fantasy football is better played with friends. And a smak board.
My new toilet flushes very, very well. Bring it.
I don't think we really understand the significance of the cross.
A perfect day involves a comfy chair, an endless supply of great coffee, & a book. Gives me goosebumps thinking about it.
On that note, good friends are priceless. I love the people I work with, & am fortunate to have them as friends.


If you bought a home in Miami in 2005, we're sorry: over the following six years it depreciated in value by more than 54.3%.
And the rebound -- if there is a rebound -- won't come soon.
Between Q2 2011 and Q2 2016, Miami home prices will decline at an annualized rate of 0.7%, according to data provided by Fiserv Case Shiller.
Data provided by Fiserv Case Shiller Indexes.


Starting a trip lets you track your progress along a route. Start a trip by tapping the Start button on the Navigate, Track, Attitude or Trip Computer page.
Optionally tap the List button to choose a route to follow from the routes list page (see figure a, below). If no routes exist, use Manage Routes to create a new route.
When you're ready, tap the Play button in the application bar to start recording your trip. You will now see a red Recording indicator at the top of the Navigate, Track, Attitude and Trip Computer pages indicating that your trip is being recorded. To stop recording, tap the Stop button in the application bar.


It all begins with a pregnant Mama Elena chopping onions. From inside the womb, Tita begins to cry, and doesn't stop until she is pushed into this world by a torrid stream of tears. They overflow from the kitchen table (where she was birthed) to the kitchen floor. The ocean of tears is eventually evaporated by the sun, at which point Nacha sweeps them up and fills a 40-pound sack with them. And, of course, then they're used in cooking for a long time.
Ummm, ewww. While vivid and descriptive, it's actually pretty gross when you think about it. I mean, those tears were mixed with amniotic fluid and who-knows-what else. So, let's not think about it. Let's just take it as the beginning of a very luscious, very sensual, very dark and tragic film.
Tita's kitchen birth foreshadows what will be her life. Bound by an old family tradition, Tita is forbidden to marry. Since she is the youngest daughter, she is not allowed to marry, and it is her duty to care for her mother until her mother's death. So, Tita thrives under the beloved family cook's love and tutelage.
But as is bound to happen, she falls in love with the handsome Pedro. At first sight, they have an attraction, a feeling, a love. Pedro declares his feelings and wants to marry Tita. But of course, the stern (okay, downright nasty) Mama Elena forbids it, and insteads arranges for Tita's older sister Rosaura to marry Pedro instead. Pedro only agrees so that he can be close to Tita.
You can imagine how this plays out.
Tita seduces Pedro with her food. There's a bit of magic at play, similar to the food-magic displayed in Chocolat (another awesome "foodie" flick originating from a book). A wedding cake batter laced with tears leaves the entire guest list sobbing over their forks.
I don't want to give anything away, so I'll skip ahead 22 years, to the time when Pedro and Rosaura's son Alex marries John's daughter Esperanza. Rosaura has passed (I won't tell you how...you'll thank me), and Tita and Chencha are busy roasting chiles, shelling walnuts, and tearing apart pomegranates in preparation of a very traditional Mexican dish, Chiles en Nogadas.
The Chiles en Nogadas are eaten in delight until all that remains is are pools of sauce and stray pomegranate seeds, save for one chile. When asked for her heavenly Chiles en Nogadas recipe, Tita says that "the secret is to make them with lots of love", as witnessed by the mad dash of people to the bedroom in various states of undress.
This is just a glimpse into what is probably one of the most well-known movies-laced-with-food out there, Like Water for Chocolate. With a colorful cast of characters including Tita's other sister Gertrudis, Chencha the ranch maid, John who is married to Tita briefly, the nosy local gossiping ladies, although the film is at times dark and tragic, it is also filled with humor and hope. And food. Lots of food.
Poblanos chiles stuffed with a pork picadillo studded with peaches, apples, raisins, and nuts, and then cloaked in a creamy walnut sauce.
Roast the chiles over an open flame or under the broiler until black and blistered all over. Place them in a bowl and cover with a towel or plastic wrap (or put them in a large baggie and seal); allow to steam for 10 minutes, or until cool enough to handle.
Rub off the skin, taking care not to remove the stem. Cut a slit lengthwise in each poblano, reach inside and remove the seeds. Set aside.
Heat the oil in a large skillet over medium- high heat. Add onion and cook until tender, ~4 minutes. Add pork, garlic, sugar, cumin, salt, and white pepper to the pan; cook until meat is browned through, 8-10 minutes.
Add the tomatoes and bring to a boil. Reduce heat and allow to simmer gently until the mixture thickens up, 5-7 minutes, stirring occasionally. Stir in the apple, peach, raisins, walnuts, and almonds; and continue to simmer for another 5-8 minutes, until moist, but not soupy. Stir in the parsley, taste and adjust seasoning as needed. If not using immediately, refrigerate.
Stuff each prepared chile with about 2 heaping tablespoons (or a scant 1/4 cup) of filling, then wrap the chile over the filling and close. Line up on a sheet tray, and refrigerate until ready to use.
Place walnuts in a medium saucepan, and cover with water. Bring to a boil, and cook for 5 minutes; drain, Using a stiff-bristled brush, peel away as much skin as you can from walnuts.
Wipe out the same saucepan, then add the cream and walnuts and set over medium-high heat. When it just starts to simmer, cover then and let sit for about 30 minutes to soften the nuts. Transfer the cream and walnuts to a blender, add the remaining ingredients for the sauce, and purée until very smooth and thick, at least 2 minutes. Cover and refrigerate until ready to use. Reheat before using; if it seems too thick, thin with another drizzle or two of cream (or milk).
Heat 1 to 2 inches of oil in a deep-sided skillet over medium-high heat. I like to user a smaller amount of oil, and a large spoon to spoon hot oil over the top of the chile while it's cooking, but whichever you prefer is fine.
Fill a pie plate with flour. Beat the egg whites until soft peaks form, then whisk in the yolks, salt, and lemon juice. Dredge each stuffed chile in flour, shaking off the excess. Dip each floured chile in the egg batter.
Carefully set the battered chiles into the hot oil and fry, flipping once, and/or spoon hot oil over the top, until the outside is golden brown, and the inside is heated through. Using a slotted spoon or spatula, carefully transfer chiles to a rack to drain. Repeat as necessary.
Transfer to a serving platter or individual serving plates and ladle some (warm) sauce over the top. Strew some pomegranate seeds over the top, and add a sprig of parsley.
Tita did not dip her chiles in batter and fry them, but rather bathed the warm, stuffed chiles in walnut sauce while "naked". I went back and forth trying to decide which way to make them. I read differing on opinions on which way was the "right" way. It seems that they were traditionally battered and fried, but that these days they are just as often eaten without the dip in the hot oil, since it is healthier. I say make them whichever way you and your guests will enjoy them the most. And make them with lots of love.
This month's edition of Food 'n Flix is being hosted by Elizabeth at The Law Student's Cookbook, with her movie pick Like Water for Chocolate. Submissions are due by the end of the day on July 29th. Everybody is invited to join us!
If you missed the fun this month, you can join us next month when we'll be watching Cloudy with a Chance of Meatballs, chosen and hosted by Camilla at Culinary Adventures with Camilla.


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text:
.. meta::
    :author: Cask Data, Inc.
    :description: HTTP RESTful Interface to the Cask Data Application Platform
    :copyright: Copyright © 2014-2015 Cask Data, Inc.

.. _http-restful-api-dataset:

========================
Dataset HTTP RESTful API
========================

.. highlight:: console

The dataset API allows you to interact with datasets through HTTP. You can list, create,
delete, and truncate datasets. For details, see the
:ref:`CDAP Components, datasets section <datasets-index>`

Listing all Datasets
--------------------

You can list all datasets in CDAP by issuing an HTTP GET request to the URL::

  GET <base-url>/namespaces/<namespace>/data/datasets

.. list-table::
   :widths: 20 80
   :header-rows: 1

   * - Parameter
     - Description
   * - ``<namespace>``
     - Namespace ID

The response body will contain a JSON-formatted list of the existing datasets::

  {
     "name":"cdap.user.purchases",
     "type":"co.cask.cdap.api.dataset.lib.ObjectStore",
     "properties":{
        "schema":"...",
        "type":"..."
     },
     "datasetSpecs":{
        ...
     }
   }

.. _http-restful-api-dataset-creating:

Creating a Dataset
------------------

You can create a dataset by issuing an HTTP PUT request to the URL::

  PUT <base-url>/namespaces/<namespace>/data/datasets/<dataset-name>

with JSON-formatted name of the dataset type and properties in a body::

  {
     "typeName":"<type-name>",
     "properties":{<properties>}
  }

.. list-table::
   :widths: 20 80
   :header-rows: 1

   * - Parameter
     - Description
   * - ``<namespace>``
     - Namespace ID
   * - ``<dataset-name>``
     - Name of the new dataset
   * - ``<type-name>``
     - Type of the new dataset
   * - ``<properties>``
     - Dataset properties, map of String to String.

.. rubric:: HTTP Responses
.. list-table::
   :widths: 20 80
   :header-rows: 1

   * - Status Codes
     - Description
   * - ``200 OK``
     - Requested dataset was successfully created
   * - ``404 Not Found``
     - Requested dataset type was not found
   * - ``409 Conflict``
     - Dataset with the same name already exists

.. rubric:: Example
.. list-table::
   :widths: 20 80
   :stub-columns: 1

   * - HTTP Request
     - ``PUT <base-url>/namespaces/default/data/datasets/mydataset``
   * - Body
     - ``{"typeName":"co.cask.cdap.api.dataset.table.Table",`` ``"properties":{"dataset.table.ttl":"3600"}}``
   * - Description
     - Creates a dataset named *mydataset* of the type ``Table`` in the namespace *default*
       with the time-to-live property set to 1 hour

.. _http-restful-api-dataset-updating:

Updating an Existing Dataset
----------------------------

You can update an existing dataset's table and properties by issuing an HTTP PUT request to the URL::

	PUT <base-url>/namespaces/<namespace>/data/datasets/<dataset-name>/properties

with JSON-formatted properties in the body::

  {
     "key1":"value1",
     "key2":"value2"
  }

**Note:** The dataset must exist.

.. list-table::
   :widths: 20 80
   :header-rows: 1

   * - Parameter
     - Description
   * - ``<namespace>``
     - Namespace ID
   * - ``<dataset-name>``
     - Name of the existing dataset

.. rubric:: HTTP Responses
.. list-table::
   :widths: 20 80
   :header-rows: 1

   * - Status Codes
     - Description
   * - ``200 OK``
     - Requested dataset was successfully updated
   * - ``404 Not Found``
     - Requested dataset instance was not found

.. rubric:: Example
.. list-table::
   :widths: 20 80
   :stub-columns: 1

   * - HTTP Request
     - ``PUT <base-url>/namespaces/default/data/datasets/mydataset/properties``
   * - Body
     - ``{"dataset.table.ttl":"7200"}``
   * - Description
     - For the *mydataset* of type ``Table`` of the namespace *default*, update the dataset
       and its time-to-live property to 2 hours

Deleting a Dataset
------------------

You can delete a dataset by issuing an HTTP DELETE request to the URL::

  DELETE <base-url>/namespaces/<namespace>/data/datasets/<dataset-name>

.. list-table::
   :widths: 20 80
   :header-rows: 1

   * - Parameter
     - Description
   * - ``<namespace>``
     - Namespace ID
   * - ``<dataset-name>``
     - Dataset name

.. rubric:: HTTP Responses
.. list-table::
   :widths: 20 80
   :header-rows: 1

   * - Status Codes
     - Description
   * - ``200 OK``
     - Dataset was successfully deleted
   * - ``404 Not Found``
     - Dataset named *dataset-name* could not be found

.. rubric:: Example
.. list-table::
   :widths: 20 80
   :stub-columns: 1

   * - HTTP Request
     - ``DELETE <base-url>/namespaces/default/data/datasets/mydataset``
   * - Description
     - Deletes the dataset *mydataset* in the namespace *default*

.. _http-restful-api-dataset-deleting-all:

Deleting all Datasets
---------------------

If the property ``enable.unrecoverable.reset`` in ``cdap-site.xml`` is set to ``true``,
you can delete all Datasets (in a namespace) by issuing an HTTP DELETE request to the URL::

  DELETE <base-url>/unrecoverable/namespaces/<namespace>/datasets

.. list-table::
   :widths: 20 80
   :header-rows: 1

   * - Parameter
     - Description
   * - ``<namespace>``
     - Namespace ID

.. rubric:: HTTP Responses
.. list-table::
   :widths: 20 80
   :header-rows: 1

   * - Status Codes
     - Description
   * - ``200 OK``
     - All Datasets were successfully deleted
   * - ``403 Forbidden``
     - Property to enable unrecoverable methods is not enabled
   * - ``409 Conflict``
     - Programs are currently running in the namespace

This command will only work if all programs in the namespace are not running.

If the property ``enable.unrecoverable.reset`` in ``cdap-site.xml`` is not set to
``true``, this operation will return a Status Code ``403 Forbidden``. Note that this
operation can only be performed if all programs are stopped. If there's at least one
program that is running, this operation will return a Status Code ``409 Conflict``.

This method must be exercised with extreme caution, as there is no recovery from it.

Truncating a Dataset
--------------------

You can truncate a dataset by issuing an HTTP POST request to the URL::

  POST <base-url>/namespaces/<namespace>/data/datasets/<dataset-name>/admin/truncate

This will clear the existing data from the dataset. This cannot be undone.

.. list-table::
   :widths: 20 80
   :header-rows: 1

   * - Parameter
     - Description
   * - ``<namespace>``
     - Namespace ID
   * - ``<dataset-name>``
     - Dataset name

.. rubric:: HTTP Responses
.. list-table::
   :widths: 20 80
   :header-rows: 1

   * - Status Codes
     - Description
   * - ``200 OK``
     - Dataset was successfully truncated

Datasets used by an Application
-------------------------------

You can retrieve a list of datasets used by an application by issuing a HTTP GET request to the URL::

  GET <base-url>/namespaces/<namespace>/apps/<app-id>/datasets

.. list-table::
   :widths: 20 80
   :header-rows: 1

   * - Parameter
     - Description
   * - ``<namespace>``
     - Namespace ID
   * - ``<app-id>``
     - Application ID

.. rubric:: HTTP Responses
.. list-table::
   :widths: 20 80
   :header-rows: 1

   * - Status Codes
     - Description
   * - ``200 OK``
     - Request was successful

Datasets used by a Program
--------------------------

You can retrieve a list of datasets used by a program by issuing a HTTP GET request to the URL::

  GET <base-url>/namespaces/<namespace>/apps/<app-id>/<program-type>/<program-id>/datasets

.. list-table::
   :widths: 20 80
   :header-rows: 1

   * - Parameter
     - Description
   * - ``<namespace>``
     - Namespace ID
   * - ``<app-id>``
     - Application ID
   * - ``<program-type>``
     - Program type, one of ``flows``, ``mapreduce``, ``services``, ``spark``, or ``workflows``
   * - ``<program-id>``
     - Program ID

.. rubric:: HTTP Responses
.. list-table::
   :widths: 20 80
   :header-rows: 1

   * - Status Codes
     - Description
   * - ``200 OK``
     - Request was successful

Datasets used by an Adapter
---------------------------

You can retrieve a list of datasets used by an adapter by issuing a HTTP GET request to the URL::

  GET <base-url>/namespaces/<namespace>/adapters/<adapter-id>/datasets

.. list-table::
   :widths: 20 80
   :header-rows: 1

   * - Parameter
     - Description
   * - ``<namespace>``
     - Namespace ID
   * - ``<adapter-id>``
     - Adapter ID

.. rubric:: HTTP Responses
.. list-table::
   :widths: 20 80
   :header-rows: 1

   * - Status Codes
     - Description
   * - ``200 OK``
     - Request was successful

Programs using a Dataset
------------------------

You can retrieve a list of programs that are using a dataset by issuing a HTTP GET request to the URL::

  GET <base-url>/namespaces/<namespace>/data/datasets/<dataset-id>/programs

.. list-table::
   :widths: 20 80
   :header-rows: 1

   * - Parameter
     - Description
   * - ``<namespace>``
     - Namespace ID
   * - ``<dataset-id>``
     - Dataset ID

.. rubric:: HTTP Responses
.. list-table::
   :widths: 20 80
   :header-rows: 1

   * - Status Codes
     - Description
   * - ``200 OK``
     - Request was successful

redpajama_set_name: RedPajamaGithub


Description The U.S. Senate historian's office describes this episode in the Army-McCarthy hearings as follows: "In the spring of 1954, McCarthy picked a fight with the U.S. Army, charging lax security at a top-secret army facility. ... The army hired Boston lawyer Joseph Welch to make its case. At a session on June 9, 1954, McCarthy charged that one of Welch's attorneys had ties to a Communist organization. As an amazed television audience looked on, Welch responded with the immortal lines that ultimately ended McCarthy's career: "Until this moment, Senator, I think I never really gauged your cruelty or your recklessness." When McCarthy tried to continue his attack, Welch angrily interrupted, "Let us not assassinate this lad further, senator. You have done enough. Have you no sense of decency?" Overnight, McCarthy's immense national popularity evaporated. Censured by his Senate colleagues, ostracized by his party, and ignored by the press, McCarthy died three years later, forty-eight years old and a broken man."
Size p. 2426-2430 ; 24 cm.


text: planet Mars over Mt. Denali, or Mt. McKinley, tallest mountain in North America, interior of Alaska mars, planet, astronomy, universe, denali, mckinley, mountain, denali national park, national park, interior alaska, alaska, ak, north america, landscape, scenic, night, horizontal, sky, mountains, peaks, ridgeline, galleryfour, © Copyright Steven J Kazlowski & Left Eye Productions. Images included in this database are protected under national and international copyright laws. Images may be used for comping purposes only and remain the property of Steven J Kazlowski and Left Eye Productions until authorised for use by purchase. Terms and conditions may apply.

redpajama_set_name: RedPajamaCommonCrawl


01753537115 number has 0 comment and was searched 1 times.
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text:
Q: XML: Xpath to read only the first level of Items without knowing the other levels my xml looks like:
<item>
  <item>
  </item>
</item>
<item>
  <item>
  </item>
  <item>
  </item>
</item>
<item>
  <item>
  </item>
  <item>
  </item>
</item>

So in fact I have x level down of the same node as I have on every other node above...
How is the xpath to only get the first level of nodes here?

A: you can try the following:
//item/*[1]

brief explanation: this expression returns the first element under every node named item

redpajama_set_name: RedPajamaStackExchange


text:
\section{Introduction}

Inequalities involving trigonometric functions are used in many applications in various fields of mathematics such as
difference equations and inequalities \cite{R_P_Agarwal_1992}, theory of stability, theory of approximations, etc.
A method called {\it the natural approach}, proposed by {\sc Mortici} in \cite{Mortici_2011}, uses the idea of comparing functions to their corresponding {\sc Taylor} polynomials. This method has been successfully applied to prove and approximate
a wide category of trigonometric inequalities \cite{G_V_Milovanovic_2014}.

In this paper we extend the ideas of the natural approach
by comparing and replacing functions with  their corresponding power series.
In particular, we focus on the results of {\sc Mortici} in
\cite{Mortici_2011} related to {\sc Wilker}-{\sc Cusa}-{\sc Huygens}'s inequalities and
give generalizations and refinements of the inequalities stated in
Theorems 1, 2, 3, 4, 5 and 6 in that paper. They are cited below for readers  convenience.

\begin{statement}
\label{statement_1}
$(${\rm \cite{Mortici_2011}}, Theorem $1$$)$
For every $0<x<\pi/2$, we have
$$
\displaystyle - \frac{x^{4}}{15}\, < \,
\cos{x} - \left( \frac{\sin{x}}{x} \right)^{\!3}\!
\, < \,
- \frac{x^{4}}{15} + \frac{23x^{6}}{1890}.
$$
\end{statement}

\begin{statement}
\label{statement_2}
$(${\rm \cite{Mortici_2011}},  Theorem $2$$)$
For every $0<x<\pi/2$, we have
$$
\displaystyle - \frac{1}{180}x^{4}\, < \,
\frac{\sin{x}}{x} -\frac{ \cos{x} + 2}{3}\, < \,
- \frac{1}{180}x^{4} + \frac{1}{3780}x^{6}.
$$
\end{statement}

\begin{statement}
\label{statement_3}
$(${\rm \cite{Mortici_2011}}, Theorem $3$$)$
For every $0<x<\pi/2$, we have
$$
\displaystyle
3 + \left(\frac{3}{20}x^{4} - \frac{3}{140}x^{6} \right)\frac{1}{\cos{x}}\, < \,
2 \frac{\sin{x}}{x} \, + \, \frac{\mbox{\rm tan}\,x}{x}\, < \,
3 \, + \, \frac{3}{20}\, \frac{x^{4}}{\cos{x}}.
$$
\end{statement}

\begin{statement}
\label{statement_4}
$(${\rm \cite{Mortici_2011}}, Theorem $4$$)$ For every $0<x<\pi/2$, we have
$$
\displaystyle 2 + \left(\frac{8 \,x^{4}}{45}- \frac{8 \,x^{6}}{105}\right)\frac{1}{\cos{x}}\, < \,
\left(\frac{\sin{\,x}}{x}\right)^{\!2}\, + \, \frac{\tan{x}}{x}\, < \,
2 + \frac{8 \,x^{4}}{45}\frac{1}{\cos{x}}
.
$$
\end{statement}

\begin{statement}
\label{statement_5} $(${\rm \cite{Mortici_2011}}, Theorem $5$$)$ For every $0<x<\pi/2$, we have
$$
\displaystyle
\left(\frac{x}{\sin{x}}\right)^{\!2} + \frac{x}{\tan{x}} > 2 + \frac{2}{45}\,x^4.
$$
\end{statement}

\begin{statement}
\label{statement_6} $(${\rm \cite{Mortici_2011}}, Theorem $6$$)$ For every $0<x<\pi/2$, we have
$$\displaystyle
3 \frac{x}{\sin{x}} + \cos{x} > 4 + \frac{1}{10}\,x^4 + \frac{1}{210}x^6.
$$
\end{statement}

\medskip
\section{Preliminaries}

First, let us recall some of the well-known power series expansions that will be used in our proofs.

\smallskip
For $x \in R$ the following power series expansions hold:
\begin{equation}
\label{sin}
\mbox{\rm sin}\,x \,= \sum\limits_{k=0}^{\infty}{(-1)^{k}\displaystyle\frac{1}{(2k+1)!}\, x^{2k+1}},
\quad \quad
\mbox{\rm cos}\, x \,= \sum_{k=0}^{\infty}(-1)^{k}{\displaystyle\frac{1}{ \, (2\,k)!}\, x^{2k}}.
\end{equation}

Also, according to \cite{Gradshteyn-Ryzhik}, for $x \in R$ we have the following power series expansions:
\begin{equation}
\label{cos3} \mbox{\rm cos}^{3}\, x \,= \, \frac{1}{4}\, \sum_{k=1}^{\infty}{(-1)^{k}
 \displaystyle\frac{3^{2k} + 3 }{ \, (2\,k)!}\, x^{2k}},
\end{equation}
and
\begin{equation}
\label{sin3}
\sin^3 x
=
\frac{1}{4}
\sum_{k=1}^{\infty}{(-1)^{k+1}\displaystyle\frac{3^{2k+1}-3}{(2k+1)!}x^{2k+1}}
\end{equation}

For $\displaystyle x \!\in\! \left( 0, \displaystyle\frac{\pi}{2} \right)$, according to \cite{Gradshteyn-Ryzhik},
the following series expansions hold:
\begin{equation}
\label{cosec}
{\mathop{\rm cosec}\nolimits} (x) = \frac{1}{x} + \mathop \sum \limits_{k = 1}^{  \infty } \frac{{\left|
{{\mbox{\boldmath $B$}_{2k}}} \right|({2^{2k}} - 2)}}{{(2k)!}}{x^{2k - 1}},
\end{equation}
\begin{equation}
\label{cosec2}
{{\mathop{\rm cosec}\nolimits} ^2}(x) = \frac{1}{{{x^2}}} + \mathop \sum \limits_{k = 1}^{  \infty }
\frac{{\left| {{\mbox{\boldmath $B$}_{2k}}} \right|(2k - 1){4^k}}}{{(2k)!}}{x^{2k - 2}},
\end{equation}
and
\begin{equation}
\label{cotan} {\mathop{\rm cotan}\nolimits} (x) = \frac{1}{x} -
\mathop \sum \limits_{k = 1}^{ \infty } \frac{{\left| {{\mbox{\boldmath $B$}_{2k}}}
\right|{4^k}}}{{(2k)!}}{x^{2k - 1}}
\end{equation}
where $\mbox{\boldmath $B$}_{i}$ are {\sc Bernoulli}'s numbers.

\bigskip
\noindent
{\bf Theorem WD.}  $(\,${\rm \cite{Wu_Debnath_2009}}, Theorem $2$$)$

\smallskip
\noindent
{\em Suppose that $f(x)$ is a real function on $(a,b)$, and that $n$ is a positive integer such that $f ^{(k)}(a+),
f^{(k)}(b-)$,
$\left(k \!\in\! \{0,1,2, \ldots ,n\}\right)$ exist.

\medskip
\noindent
{\boldmath $(i)$} Supposing that  $(-1)^{(n)} f^{(n)}(x)$~is~in\-cre\-asing on $(a,b)$, then
for all $x \in (a,b)$ the following inequality holds$:$
\begin{equation}
\label{Debnath_Wu_first}
\begin{array}{c}
\displaystyle\sum_{k=0}^{n-1}{\mbox{\small $\displaystyle\frac{f^{(k)}(b\mbox{\footnotesize
$-$})}{k!}$}(x-b)^k}
+
\frac{1}{(a-b)^n}
{\bigg (}\!
f(a\mbox{\footnotesize $+$})
-
\displaystyle\sum_{k=0}^{n-1}{\mbox{\small $\displaystyle\frac{(a-b)^{k}f^{(k)}(b\mbox{\footnotesize
$-$})}{k!}$}
\!{\bigg )} (x-b)^{n}}
\\[2.0 ex]
<
f(x)
<
\displaystyle\sum_{k=0}^{n}{\frac{f^{(k)}(b\mbox{\footnotesize $-$})}{k!}(x-b)^{k}}.
\end{array}
\end{equation}
Furthermore, if $(-1)^{n} f^{(n)}(x)$ is decreasing on $(a,b)$, then the reversed inequality of {\rm
(\ref{Debnath_Wu_first})} holds.

\medskip
\noindent
{\boldmath $(ii)$} Supposing that  $f^{(n)}(x)$ is increasing on $(a,b)$, then for all $x \!\in\! (a,b)$ the following
inequality  holds:
\begin{equation}
\label{Debnath_Wu_second}
\begin{array}{c}
\displaystyle\sum_{k=0}^{n-1}{\mbox{\small $\displaystyle\frac{f^{(k)}(a\mbox{\footnotesize
$+$})}{k!}$}(x-a)^k}
+
\frac{1}{(b-a)^n}
{\bigg (}\!
f(b-)
-
\displaystyle\sum_{k=0}^{n-1}{\mbox{\small $\displaystyle\frac{(b-a)^{k}f^{(k)}(a\mbox{\footnotesize
$+$})}{k!}$}
\!{\bigg )} (x-a)^{n}}
\\[2.0 ex]
>
f(x)
>
\displaystyle\sum_{k=0}^{n}{\frac{f^{(k)}(a\mbox{\footnotesize $+$})}{k!}(x-a)^{k}}.
\end{array}
\end{equation}
Furthermore, if $f^{(n)}(x)$ is decreasing on $(a,b)$, then the reversed inequality~of~\mbox{\rm
(\ref{Debnath_Wu_second})} holds.}

\smallskip
\noindent
Let us mention that an interesting application of Theorem WD is~given~in~\cite{MRL_2017c},
see also~\cite{Milica_Makragic_2017}.

\break

\section{Main results}

\medskip
 We need the following Theorem  for the proofs of Theorems 1, 2, 3 and 4.

\smallskip
\begin{proposition}\label{Leibniz} ~\\
Let the series $\displaystyle f(x)= \sum_{k=1}^{\infty}(-1)^{k} A(k) x^{2k}$ converges for
$x \!\in\! (0,c), \,\, c \!\in\! R^{+}\!.$ Suppose the following statements are true$\,:$
\begin{itemize}
\item[$\,(i)\,$] If $c\!<\!1$ then the sequence $\left\{ A(k) \right\}_{k \in N}$ is a positive decreasing sequence
that converges to $0$.
\item[$(ii)$] If $c \!\geq\! 1$ then the sequence $\displaystyle \left\{ A(k) \right\}_{k \in N}$ is a positive sequence, \mbox{$\!\!\lim\limits_{k\rightarrow +\infty}\!\!c^kA(k)\!=\!0$} and $\displaystyle A(k) \!>\! c^2 A(k+1)$ for $k \!\geq\! 1$.
\end{itemize}

\noindent
Then for all $x \!\in\! (0, c)$ and for all $n \!\in\! N$ and $m \!\in\! N$ we have:
\begin{equation}
\label{Leibniz_03}
\displaystyle\sum_{k=1}^{2n}{(-1)^{k+1} A(k) \, x^{2k}}
<
f(x)
<
\displaystyle\sum_{k=1}^{2n+1}{(-1)^{k+1} A(k) \, x^{2k}}
\end{equation}
and
\begin{equation}
\label{Leibniz_04}
{\Big |}\,f(x) - \displaystyle\sum_{k=1}^{m}{(-1)^{k} A(k) \, x^{2k}}\,{\Big |}
<
A(m+1) \, x^{2m+2}
<
c^{2m+2}\, A(m+1).
\end{equation}
\end{proposition}
{\bf Proof.}
Suppose that $c \!<\! 1$. Then for every $x \!\in\! (0,c)$ the positive
sequence $\left\{A(k) \, x^{2k} \right\}_{k \in N}$ decreases monotonically and
\mbox{$\displaystyle \lim _{k\rightarrow \infty } A(k)\, x^{2k} =0$.} Thus,  assertions
(\ref{Leibniz_03}) and (\ref{Leibniz_04}) immediately follow from {\sc Leibniz}'s theorem for
the alternating series.

\smallskip
Suppose now that $c \!\geq\! 1$. We have
$$
\displaystyle
\begin{array}{lcl}\displaystyle
f(x) \!&\!\!=\!\!&\! \displaystyle \sum_{k=1}^{\infty}(-1)^{k+1} A(k) x^{2k}     \\[1.5 ex]
 \!&\!\!=\!\!&\! \displaystyle \sum_{k=1}^{\infty}(-1)^{k+1} A(k) \, c^{2k}\, \left(\frac{x}{c}\right)^{\!2k}.
\end{array}
$$
Let us introduce the substitution $t\!=\!\displaystyle\frac{x}{c}$ in the previous power series and consider the series
\begin{equation}
 \label{smena-t}
 \displaystyle
 \sum_{k=1}^{\infty}(-1)^{k+1} A(k) c^{2k} t^{2k} , \quad \mbox{for}\,\,\, t \!\in\! (0,1).
 \end{equation}
For the assumption $\displaystyle A(k) > c^2A(k+1)$ we have
$$
\displaystyle
A(k) > c^2 A(k+1)
\,\,\, \Longleftrightarrow \,\,\,
A(k)c^{2k} > c^{2k+2} A(k+1),
~~ \mbox{for} ~ k \!\geq\! 1.
$$
Hence, we conclude that for every $t \!\in\! (0,1)$  power series (\ref{smena-t}) satisfies {\sc Leibniz}'s
theorem for the alternating series, and for all $n, m \in N$ we have:
\begin{equation}
\label{1}
{\small
\displaystyle\sum_{k=1}^{2n}{(-1)^{k+1}\, A(k) \, c^{2k} \, t^{2k}}
<
\sum_{k=1}^{\infty}(-1)^{k+1}\, A(k) \,c^{2k}\, t^{2k}
<
\displaystyle\sum_{k=1}^{2n+1}{(-1)^{k+1} \, A(k) \, c^{2k} \, t^{2k}}
}
\end{equation}
and
\begin{equation}
\label{2}
\begin{array}{lcl}
{\Big |}\,
\displaystyle \sum_{k=1}^{\infty}(-1)^{k+1} A(k) c^{2k} t^{2k}
-
\displaystyle\sum_{k=1}^{m}{(-1)^{k+1}\, A(k) \, c^{2k}\, t^{2k}}\,{\Big |}
\!&\!\!<\!\!&\! A(m+1) \, c^{2m+2}\, t^{2m+2}   \\ [1.25 ex]
\!&\!\!<\!\!&\! A(m+1) \, c^{2m+2}.
\end{array}
\end{equation}
Returning the variable $x= t\, c$ to (\ref{1}) and (\ref{2}) gives the assertions of proposition.
\hfill $\Box$

\bigskip

\medskip
\subsection{ Refinements of the inequalities in Statement \mbox{1}}

\smallskip
We propose the following improvement and generalization of Statement 1:

\smallskip
\begin{theorem} ~\\
\label{improvement-s-1}
$(i)$ For every $\displaystyle \, x \!\in\! \left(0, \mbox{\small $\displaystyle\frac{\pi}{2}$}\right)$ and every $n \in N$, we have
\begin{equation}
\label{t-1}
\begin{array}{c}
\displaystyle
\sum_{k=2}^{2n} (-1)^{k} A(k) \, x^{2k}
<
\cos x - \left(\displaystyle\frac{\sin x}{x}\right)^{\!3}
<
\displaystyle
\sum_{k=2}^{2n+1}(-1)^{k} A(k) \, x^{2k},
\end{array}
\end{equation}
where
$\displaystyle
A(k) = \mbox{\small $\displaystyle \frac{3^{2k+3}\!-\!32k^3\!-\!96k^2\!-\!88k\!-\!27}{4\,(2k+3)!}$}.
$

\smallskip
\noindent $(ii)$ For every $x \!\in\! \left(0, \mbox{\small $\displaystyle\frac{\pi}{2}$}\right)$ and every $m \in N$,
we have the following error estimation:
\begin{equation}
\begin{array}{l}
{\Big |}\,
\cos x
-
\left(\displaystyle\frac{\sin x}{x}\right)^{\!3}
\!-
\displaystyle\sum_{k=1}^{m}{(-1)^{k}
A(k) \, x^{2k}}\,{\Big |}\,
<\, A(m+1) \, x^{2m+2}.
\end{array}
\end{equation}
\end{theorem}

\noindent
{\bf Examples. }   \\
Let $x \!\in\! \left(0, \mbox{\small $\displaystyle\frac{\pi}{2}$}\right)$. \\
For $n=1$ we get Statement~\ref{statement_1}. \\[0.5 ex]
For $n>1$ we have the following new results:

\begin{itemize}
\item Taking $n=2$ in (\ref{t-1}) gives:

$
- \mbox{\small $\displaystyle\frac{1}{15}$}x^{4}
+ \mbox{\small $\displaystyle\frac{23}{1890}$}x^{6}
- \mbox{\small $\displaystyle\frac{41}{37800}$}x^{8}\, < \,
\cos x \, - \left(\displaystyle\frac{\sin x}{x}\right)^{\!3} <
$                                                                                \\

\hfill
$
\, < \,
- \mbox{\small $\displaystyle\frac{1}{15}$}x^{4}
+ \mbox{\small $\displaystyle\frac{23}{1890}$}x^{6}
- \mbox{\small $\displaystyle\frac{41}{37800}$}x^{8}
+ \mbox{\small $\displaystyle\frac{53}{831600}$}x^{10}
$

\item Taking $n=3$ in (\ref{t-1}) gives:

$
\begin{array}{l}
\displaystyle
- \mbox{\small $\displaystyle\frac{1}{15}$}x^{4}
+ \mbox{\small $\displaystyle\frac{23}{1890}$}x^{6}
- \mbox{\small $\displaystyle\frac{41}{37800}$}x^{8}
+ \mbox{\small $\displaystyle\frac{53}{831600}$}x^{10}
- \mbox{\small $\displaystyle\frac{74677}{27243216000}$}x^{12} <                                              \\[0.5 em]
< \, \cos x - \left(\mbox{\small $\displaystyle\frac{\sin x}{x}$}\right)^{\!3} <                              \\[0.5 em]
<
-
\mbox{\small $\displaystyle\frac{1}{15}$}x^{4}
+
\mbox{\small $\displaystyle\frac{23}{1890}$}x^{6}
-
\mbox{\small $\displaystyle\frac{41}{37800}$}x^{8}
+
\mbox{\small $\displaystyle\frac{53}{831600}$}x^{10}
-
\mbox{\small $\displaystyle\frac{74677}{27243216000}$}x^{12}
+
\mbox{\small $\displaystyle\frac{989}{10897286400}$}x^{14}
\end{array}$

\item Taking $n=4$ in (\ref{t-1}) gives:

$
-
\mbox{\small $\displaystyle\frac{x^{4}}{15}$}
+
\mbox{\small $\displaystyle\frac{23 \,x^{6}}{1890}$}
-
\mbox{\small $\displaystyle\frac{41\,x^{8}}{37800}$}
+
\mbox{\small $\displaystyle\frac{53\,x^{10}}{831600}$}
-
\mbox{\small $\displaystyle\frac{74677\,x^{12}}{27243216000}$}
+
\mbox{\small $\displaystyle\frac{989\,x^{14}}{10897286400}$}
-
\mbox{\small $\displaystyle\frac{79649\,x^{16}}{33345696384000}$}
\, < \,
$ \\[0.5 em]
$
< \,
\cos x - \left(\mbox{\small $\displaystyle\frac{\sin x}{x}$}\right)^{\!3}
\, < \,
-
\mbox{\small $\displaystyle\frac{x^{4}}{15}$}
+
\mbox{\small $\displaystyle\frac{23\,x^{6}}{1890}$}
-
\mbox{\small $\displaystyle\frac{41\,x^{8}}{37800}$}
+
\mbox{\small $\displaystyle\frac{53\,x^{10}}{831600}$}
-
\mbox{\small $\displaystyle\frac{74677\,x^{12}}{27243216000}$}
\, +
$ \\ [0.5 em]
$
+ \,
\mbox{\small $\displaystyle\frac{989\,x^{14}}{10897286400}$}
-
\mbox{\small $\displaystyle\frac{79649\,x^{16}}{33345696384000}$}
+
\mbox{\small $\displaystyle\frac{454007\,x^{18}}{8869955238144000}$}
$

\item[\mbox{etc.}]
\end{itemize}

\bigskip
\noindent
{\bf Proof of Theorem~\ref{improvement-s-1}.}

\smallskip
Consider the function
$$
f(x) = \cos x - \left(\displaystyle\frac{\sin x}{x}\right)^{\!3},
\quad \mbox{for}\,\, x \!\in\! \left(0, \mbox{\small $\displaystyle\frac{\pi}{2}$}\right).
$$
Based on  power series expansion (\ref{sin}) and (\ref{sin3}) we have
$$
f(x)
=
\displaystyle\sum_{k=1}^{\infty}{(-1)^{k} A(k) \, x^{2k}}
=
\displaystyle\sum_{k=2}^{\infty}{(-1)^{k} A(k) \, x^{2k}},
$$
for all $x \!\in\! R$, where$:$
\begin{equation}
\label{Def_a_k}
\begin{array}{rcl}
A(k)
\!&\!\!=\!\!&\!
\displaystyle\frac{1}{4}\frac{3^{2k+3}-3}{(2k+3)!} - \displaystyle\frac{1}{(2k)!}  \\[2.1 ex]
\!&\!\!=\!\!&\!
\displaystyle\frac{3^{2k+3}-32k^3-96k^2-88k-27}{4(2k+3)!}.
\end{array}
\end{equation}
For $ c=\pi/2$ we have:
 $$ A(k)>0 \;\, \mbox{for} \; k\geq 2 \quad
 \mbox{and} \quad \displaystyle \lim _{k\rightarrow \infty }c^{2k}A(k) =0.$$
Also,
$$
\begin{array}{l}\displaystyle
c^{2}\,\mathrm{A}(k + 1) < \mathrm{A}(k) ~ \Longleftrightarrow ~  \\[0.25 em]
32\,k^{5} \!+\! 240\,k^{4} \!+\! (680 \!-\! 8\,c^{2})\,k^{3} \!+\!
(900 \!-\! 45\,c^{2} )\,k^{2} \!+\! (548 \!-\! {\displaystyle \frac{161\,c^{2}}{2}} )
\,k \!-\! 45\,c^{2} \!+\! {\displaystyle \frac{477}{4}}   \\[1.0 em]
\hspace{35mm} < \, {\displaystyle \frac{3\, (4\,c^{2}\,k^{2} + 18\,c^{2}\,k + 20\,c
^{2} - 9)\,}{4}\,9^{k+1}}.
\end{array}
$$
As the last inequality holds for $k \geq 1$, the assertions of Theorem~\ref{improvement-s-1} immediately follows
from Proposition~\ref{Leibniz}. \hfill $\Box$

\medskip
\subsection{ Refinements of the inequalities in Statement \mbox{2}}

\smallskip
We propose the following improvement and generalization of Statement 2:

\smallskip
\begin{theorem} ~\\
\label{improvement-s-2}
 $(i)$ For every $\displaystyle \, x \!\in\! \left(0, \mbox{\small $\displaystyle\frac{\pi}{2}$}\right)$ and every $n \in N$, we have
\begin{equation}
\label{t-2}
\begin{array}{c}
\displaystyle
\sum_{k=2}^{2n} (-1)^{k+1} B(k) \, x^{2k}
<
 \frac{\sin{x}}{x} -\frac{ \cos{x} + 2}{3}
<
\displaystyle
\sum_{k=2}^{2n+1}(-1)^{k+1} B(k) \, x^{2k},
\end{array}
\end{equation}
where
$\displaystyle
B(k) = {\displaystyle \frac{2}{3}}\,{\displaystyle \frac{k - 1}{(2
\,k + 1)\mathrm{!}}}.
$

\smallskip
\noindent
$\,(ii)$ For every $x \!\in\! \left(0, \mbox{\small $\displaystyle\frac{\pi}{2}$}\right)$ and every $m \in N$,
we have the following error estimation:
\begin{equation}
\begin{array}{l}
{\Big |}\,\displaystyle
\frac{\sin{x}}{x} -\frac{ \cos{x} + 2}{3}
\!-\!
\displaystyle\sum_{k=0}^{m}{(-1)^{k+1}
B(k) \, x^{2k}}\,{\Big |} \, < \,B(m+1) \, x^{2m+2}.
\end{array}
\end{equation}
\end{theorem}
{\bf Examples.} \\
Let $ x \!\in\! \left(0, \mbox{\small $\displaystyle\frac{\pi}{2}$}\right)$. \\
For $n = 1$ we get Statement~\ref{statement_2}. \\[1.0 ex]
For $n >1 $ we have the following new results:
\begin{itemize}
\item Taking $n=2$ in (\ref{t-2}) gives:

$
-
\mbox{\small $\displaystyle\frac{1}{180}$}\,x^{4}
+
\mbox{\small $\displaystyle\frac{1}{3780}$}\,x^{6}
-
\mbox{\small $\displaystyle\frac{1}{181440}$}\,x^{8}
<
\mbox{\small $\displaystyle\frac{\sin x}{x}$}
-
\mbox{\small $\displaystyle\frac{1}{3}$}\cos x
-
\mbox{\small $\displaystyle\frac{2}{3}$} <
$
$
-
\mbox{\small $\displaystyle\frac{1}{180}$}\,x^{4}
+
\mbox{\small $\displaystyle\frac{1}{3780}$}\,x^{6}
-
-\mbox{\small $\displaystyle\frac{1}{181440}$}\,x^{8}
+
\mbox{\small $\displaystyle\frac{1}{14968800}$}\,x^{10}
$

\item Taking $n=3$ in (\ref{t-2}) gives:

$
-
\mbox{\small $\displaystyle\frac{\,x^{4}}{180}$}
+
\mbox{\small $\displaystyle\frac{\,x^{6}}{3780}$}
-
\mbox{\small $\displaystyle\frac{\,x^{6}}{181440}$}
+
\mbox{\small $\displaystyle\frac{\,x^{10}}{14968800}$}
-
\mbox{\small $\displaystyle\frac{\,x^{12}}{1868106240}$}
< \!
\mbox{\small $\displaystyle\frac{\sin \,x}{x}$}\!
-\!
\mbox{\small $\displaystyle\frac{1}{3}$}\cos x
-\!
\mbox{\small $\displaystyle\frac{2}{3}$}\! <
$
$
-
\mbox{\small $\displaystyle\frac{\,x^{4}}{180}$}
+
\mbox{\small $\displaystyle\frac{\,x^{6}}{3780}$}
-
\mbox{\small $\displaystyle\frac{\,x^{8}}{181440}$}
+
\mbox{\small $\displaystyle\frac{\,x^{10}}{14968800}$}
-
\mbox{\small $\displaystyle\frac{\,x^{12}}{1868106240}$}
+
\mbox{\small $\displaystyle\frac{\,x^{14}}{326918592000}$}
$

\item Taking $n=4$ in (\ref{t-2}) gives:

$
-
\mbox{\small $\displaystyle\frac{1}{180}$}\,x^{4}
+
\mbox{\small $\displaystyle\frac{1}{3780}$}\,x^{6}
-
\mbox{\small $\displaystyle\frac{1}{181440}$}\,x^{8}
+
\mbox{\small $\displaystyle\frac{1}{14968800}$}\,x^{10}
-
\mbox{\small $\displaystyle\frac{1}{1868106240}$}\,x^{12}
+
\mbox{\small $\displaystyle\frac{1}{326918592000}$}\,x^{14} -
-\mbox{\small $\displaystyle\frac{1}{76218734592000}$}\,x^{16} <
\mbox{\small $\displaystyle\frac{\sin x}{x}$}
-
\mbox{\small $\displaystyle\frac{1}{3}$}\cos x
-
\mbox{\small $\displaystyle \frac{2}{3}$}
<
-
\mbox{\small $\displaystyle\frac{1}{180}$}\,x^{4}
+
\mbox{\small $\displaystyle\frac{1}{3780}$}\,x^{6}
-
\mbox{\small $\displaystyle\frac{1}{181440}$}\,x^{8}
+
+\mbox{\small $\displaystyle\frac{1}{14968800}$}\,x^{10}
-
\mbox{\small $\displaystyle\frac{1}{1868106240}$}\,x^{12}
+
\mbox{\small $\displaystyle\frac{1}{326918592000}$}\,x^{14}
-
\mbox{\small $\displaystyle\frac{1}{76218734592000}$}\,x^{16}
+
+\mbox{\small $\displaystyle\frac{1}{22808456326656000}$}\,x^{18}
$
\item[etc.]
\end{itemize}

\noindent
{\bf Proof of Theorem~\ref{improvement-s-2}.}

\smallskip
Consider the function $$f(x)= {\displaystyle \frac{\mathrm{sin}\,x
}{x}} - {\displaystyle \frac{1}{3}}\,\mathrm{cos}\,x -
{\displaystyle \frac{2}{3}}, \quad \mbox{for} \; x \in \left(0, \mbox{\small $\displaystyle\frac{\pi}{2}$}\right).$$

\noindent
Based on power series expansion (\ref{sin}) we have
$$
f(x)
= \displaystyle
- {\displaystyle \frac{2}{3}
} + {\displaystyle \sum _{k=0}^{\infty}}\,(-1)^{k + 1}\,  {\displaystyle \frac{2}{3}}\,{\displaystyle \frac{k - 1}{(2 \,k + 1)\mathrm{!}}} \,x^{2\,k}\, = \, {\displaystyle \sum _{k=2}^{\infty}}\,(-1)^{k + 1}\,  {\displaystyle \frac{2}{3}}\,{\displaystyle \frac{k - 1}{(2 \,k + 1)\mathrm{!}}} \,x^{2\,k}.
$$

The sequence $\left\{ B(k) \right\}_{k\in N, \,k \geq 2}$
satisfies the recurrence relation:
$$
B(k + 1)= {\displaystyle
\frac{k}{2\, (k^{2} - 1)\,(2\,k + 3)}}\, B(k).
$$
For $c=\pi/2$ we have:
$$
B(k)>0 \;\; \mbox{for} \;\; k\geq 2, \; \;
\quad\mbox{and} \;\; \quad \displaystyle \lim _{k\rightarrow \infty }c^{2k}B(k) =0.
$$
Also,
$$
\begin{array}{lcl}\displaystyle
c^{2}\,B(k + 1) < B(k)
\!&\!\!\Longleftrightarrow\!\!&\!
\left( {\displaystyle
\frac{c^{2}\, k}{2\, (k^{2} - 1)\,(2\,k + 3)}}\, - 1 \, \right)\cdot B(k) \, < \, 0 \\[1.1em]
\!&\!\!\Longleftrightarrow\!\!&\!
- {\displaystyle \frac{4\,k^{3} + 6\,k^{2} - (c^{2} + 4)\,k - 6}{2\,(k^{2} - 1)\,(2\,k + 3)}} \cdot B(k) \, < \, 0 \\[1.1em]
\!&\!\!\Longleftrightarrow\!\!&\!
- {\displaystyle \frac{2\,(2\,k^{3} - 3)+4 \, k \,(k-1) + k (2 \, k-c^{2})}{2\,(k^{2} - 1)(2\,k + 3)}} \cdot B(k) \, < \, 0.
\end{array}
$$
As the last inequality holds for every  $k \!\geq\! 2$, the assertions of Theorem~\ref{improvement-s-2}
 follow from Proposition~\ref{Leibniz}. \hfill $\Box$

\medskip
\subsection{ Refinements of the inequalities in Statement \mbox{3}}

\smallskip
We propose the following improvement and generalization of Statement 3:

\smallskip
\begin{theorem}
\label{improvement-s-3} ~\\
$(i)$ For every $ x \!\in\! \left(0, \mbox{\small $\displaystyle\frac{\pi}{2}$}\right)$ and every $n \in N$, we have
\begin{equation}
\label{t-3} \mbox{\small $3 + \displaystyle \frac{1}{\cos x} \!\! \sum\limits_{k=2}^{2n+1}{
(-1)^{k}\, C(k)\, x^{2k}}
 \, < \,
\displaystyle 2 \, \frac{\sin{x}}{x} + \frac{\mbox{\rm tan}\,x}{x}\, < \,
3 \, + \,\displaystyle \frac{1}{\cos{x}} \! \sum\limits_{k=2}^{2n}{
(-1)^{k}\, C(k) \, \, x^{2k}}$},
\end{equation}
where $C(k)= {\displaystyle 2\,\frac{4^{k} - 3\,k - 1}{(2\,k + 1)\mathrm{!}}
} $.

\smallskip
\noindent $(ii)$ For every $\displaystyle \, x \!\in\! \left(0, \mbox{\small $\displaystyle\frac{\pi}{2}$}\right)$
and every $m \in N, \, m\geq 2$, we have the following error estimation:
$$
\begin{array}{l}
{\Big |}\,\displaystyle
2 \, \frac{\sin{x}}{x} + \frac{\mbox{\rm tan}\,x}{x}
\!-\! \left(3 + \frac{1}{\cos \, x}
\displaystyle\sum_{k=2}^{m}{(-1)^{k+1}
C(k) \, x^{2k}} \right) \,{\Big |} \, < \, C(m+1)\, \frac{x^{2m+2}}{\cos \, x}.
\end{array}
$$
\end{theorem}

\noindent
{\bf Examples.} \\
Let $x \!\in\! \left(0, \mbox{\small $\displaystyle\frac{\pi}{2}$}\right)$. \\
For $n = 1$ we get Statement~\ref{statement_3}. \\[1.0 ex]
For $n > 1 $ we have the following new results:
\begin{itemize}
\item Taking $n=2$ in (\ref{t-3}) gives:

$
2
+
\mbox{\small $\displaystyle\frac{1}{\cos x}$}
\left(
\mbox{\small $\displaystyle\frac{3}{20}$}\,x^{4}
-
\mbox{\small $\displaystyle\frac{3}{140}$}\,x^{6}
+
\mbox{\small $\displaystyle\frac{3}{2240}$}\,x^{8}
-
\mbox{\small $\displaystyle\frac{1}{19800}$}\,x^{10}\right)
 \, < \,
2 \,
\mbox{\small $\displaystyle\frac{\sin x}{x}$}
+
\mbox{\small $\displaystyle\frac{\tan x}{x}$}
\, < \, \quad \quad
< 2
+
\mbox{\small $\displaystyle\frac{1}{\cos x}$}
\left(
\mbox{\small $\displaystyle\frac{3}{20}$}\,x^{4}
-
\mbox{\small $\displaystyle\frac{3}{140}$}\,x^{6}
+
\mbox{\small $\displaystyle\frac{3}{2240}$}\,x^{8}
\right)
$

\item Taking $n=3$ in (\ref{t-3}) gives:

$
2
+
\mbox{\small $\displaystyle\frac{1}{\cos x}$}
\left(
\mbox{\small $\displaystyle\frac{3}{20}$}\,x^{4}
-
\mbox{\small $\displaystyle\frac{3}{140}$}\,x^{6}
+
\mbox{\small $\displaystyle\frac{3}{2240}$}\,x^{8}
-
\mbox{\small $\displaystyle\frac{1}{19800}$}\,x^{10}
+
\mbox{\small $\displaystyle\frac{151}{115315200}$}\,x^{12}
-
\mbox{\small $\displaystyle\frac{101}{4036032000}$}\,x^{14}
\right)
<
$

$
<
2
\mbox{\small $\displaystyle\frac{\sin x}{x}$}
+
\mbox{\small $\displaystyle\frac{\tan x}{x}$}
<
2
+
\mbox{\small $\displaystyle\frac{1}{\cos x}$}
\left(
\mbox{\small $\displaystyle\frac{3}{20}$}\,x^{4}
-
\mbox{\small $\displaystyle\frac{3}{140}$}\,x^{6}
+
\mbox{\small $\displaystyle\frac{3}{2240}$}\,x^{8}
-
\mbox{\small $\displaystyle\frac{1}{19800}$}\,x^{10}
+
\mbox{\small $\displaystyle\frac{151}{115315200}$}\,x^{12}
\right)
$

\item Taking $n=4$ in (\ref{t-3}) gives:

$
\displaystyle 2+ \frac{1}{\cos{x}}\left({\displaystyle \frac{3}{20}}\,x^{4}
-
{\displaystyle \frac{3}{140}}\,x^{6}
+
{\displaystyle \frac{3}{2240}}\,x^{8}
-
{\displaystyle \frac{1}{19800}}\,x^{10}
+
{\displaystyle \frac{151}{115315200}}\,x^{12}
-
{\displaystyle \frac{101}{4036032000}}\,x^{14}
+\right.
$

$
+\left.
{\displaystyle \frac{7279}{19760412672000}}\,x^{16}
-
{\displaystyle \frac{809}{187723920384000}}\,x^{18}
\right)
\, < \,
\displaystyle 2 \, \frac{\sin{x}}{x} + \frac{\mbox{\rm tan}\,x}{x}\, < \,
$

$
2+ {\displaystyle \frac{1}{\cos{x}}}
\left({\displaystyle \frac{3}{20}}\,x^{4}
-
{\displaystyle \frac{3}{140}}\,x^{6}
+
{\displaystyle \frac{3}{2240}}\,x^{8}
-
{\displaystyle \frac{1}{19800}}\,x^{10}
+
{\displaystyle\frac{151}{115315200}}\,x^{12}
- \right.
$

$
\left.
-{\displaystyle \frac{101}{4036032000}}\,x^{14}
+
{\displaystyle \frac{7279}{19760412672000}}\,x^{16}\right)
$
\item[etc.]
\end{itemize}

\noindent
{\bf Proof of Theorem~\ref{improvement-s-3}.}

\smallskip
Consider the function
$$
f(x) = \frac{\sin{2 x}}{x}\,+\, \frac{\sin{x}}{x} - 3 \cos{x}
$$
for $x \!\in\! \left(0, \mbox{\small $\displaystyle\frac{\pi}{2}$}\right)$.
Based on  power series expansion (\ref{sin}) we have
$$
\begin{array}{lcl}f(x)
 \!&\!\!=\!\!&\!
\displaystyle \sum _{k=0}^{\infty}
\,(-1)^{k}\,\left({\displaystyle \frac{2^{(2\,k + 1)}}{(2\,k + 1)
\mathrm{!}}} + {\displaystyle \frac{1}{(2\,k + 1)\mathrm{!}}} - {\displaystyle \frac{3}{(2\,k)\mathrm{!}}} \right)\,x^{2\,k}, \\ [1.3em]
 \!&\!\!=\!\!&\! \displaystyle \sum _{k=2}^{\infty} \,(-1)^{k}\,{\displaystyle 2 \, \frac{\,4^{k} - 3\,k - 1}{(2\,k + 1)\mathrm{!}}}\,x^{2\,k},
\end{array}
$$
for $x \!\in\! \left(0, \mbox{\small $\displaystyle\frac{\pi}{2}$}\right)$.

\smallskip
For $c=\pi/2$ we have:
$$
C(k)>0 \;\; \mbox{for} \; k\geq 2, \;\; \quad \mbox{and} \; \; \quad \displaystyle\lim_{k\rightarrow \infty }c^{2k}C(k)=0.
$$
Also,
$$
\begin{array}{l}\displaystyle
c^{2}\, C(k + 1) < C(k) \,\, \Longleftrightarrow \\[1.0 ex]
-2 \left( 12\,k^{3} \!+\! 38\,k^{2} \!+\!
{\displaystyle \frac{(6\,c^{2} \!+\! 76)\,k}{2}}
\!+\! 4\,c^{2} \!+\! 12 \right)
\,<\,
4^{k + 1}\left(2\,(k+c)(k-c) \!+\! 5\,k \!+\! 3\right).
\end{array}
$$
As the last inequality holds for $k \geq 2$, the assertions of Theorem~\ref{improvement-s-2}
immediately follows from Proposition~\ref{Leibniz}.
\hfill $\Box$

\medskip
\subsection{ Refinements of the inequalities in Statement \mbox{4}}

\smallskip
We propose the following improvement and generalization of Statement 4:

\smallskip
\begin{theorem} ~\\
\label{improvement-s-4}
$(i)$ For every $x \!\in\! \left(0, \mbox{\small $\displaystyle\frac{\pi}{2}$}\right)$
and every $n \in N$, we have
\begin{equation}
\label{t-4}
\begin{array}{c}
\displaystyle
2 + \frac{1}{\cos x}\sum_{k=2}^{2n+1} (-1)^{k} D(k) \, x^{2k}
 <
 \left(\frac{\sin{\,x}}{x}\right)^{\!2} + \frac{\tan{x}}{x}\, < \,
\displaystyle 2 + \frac{1}{\cos x}
\sum_{k=2}^{2n}(-1)^{k} D(k) \, x^{2k},
\end{array}
\end{equation}
where
$\displaystyle
D(k) = {\displaystyle \frac{1}{4}}\,{\displaystyle \frac{
 - 9 + 3^{2k + 2} - 40\,k - 32\,k^{2}}{(2\,k + 2)\mathrm{!}}}.
$

\smallskip

\noindent $(ii)$ For every $x \!\in\! \left(0, \mbox{\small $\displaystyle\frac{\pi}{2}$}\right)$
and every $m \in N, m\geq 2$, we have the following
error estimation:
$$
\begin{array}{l}
{\Big |}\displaystyle
\left(\frac{\sin{\,x}}{x}\right)^{\!\!2} \! + \frac{\tan{x}}{x} - \left(2+\frac{1}{\cos \, x}
\displaystyle\sum_{k=2}^{m}{(-1)^{k+1}
D(k) \, x^{2k}}\right){\Big |} \, < \,D(m+1)\,\frac{x^{2m+2}}{\cos \, x}.
\end{array}
$$
\end{theorem}

\noindent
{\bf Examples.} \\
Let $x \!\in\! \left(0, \mbox{\small $\displaystyle\frac{\pi}{2}$}\right)$. \\
For $n = 1$ we get Statement~\ref{statement_4}. \\[1.0 ex]
For $n > 1$ we have the following new results:
\begin{itemize}
\item Taking $n=2$ in (\ref{t-4}) gives:

$
2
+
\mbox{\small $\displaystyle\frac{1}{\cos x}$}
\left(
\mbox{\small $\displaystyle\frac{8}{45}$}\,x^{4}
-
\mbox{\small $\displaystyle\frac{4}{105}$}\,x^{6}
+
\mbox{\small $\displaystyle\frac{19}{4725}$}\,x^{8}
-
\mbox{\small $\displaystyle\frac{37}{133650}$}\,x^{10}
\right)
<
\left(
\mbox{\small $\displaystyle\frac{\sin x}{x}$}
\right)^{\!2}
+
\mbox{\small $\displaystyle\frac{\tan x}{x}$}
<
$
$
<
2
+
\mbox{\small $\displaystyle\frac{1}{\cos x}$}
\left(
\mbox{\small $\displaystyle\frac{8}{45}$}\,x^{4}
-
\mbox{\small $\displaystyle\frac{4}{105}$}\,x^{6}
+
\mbox{\small $\displaystyle\frac{19}{4725}$}\,x^{8}
\right)
$

\item Taking $n=3$ in (\ref{t-4}) gives:

$
2
+
\mbox{\small $\displaystyle\frac{1}{\cos x}$}
\left(
\mbox{\small $\displaystyle\frac{8}{45}$}\,x^{4}
-
\mbox{\small $\displaystyle\frac{4}{105}$}\,x^{6}
+
\mbox{\small $\displaystyle\frac{19}{4725}$}\,x^{8}
-
\mbox{\small $\displaystyle\frac{37}{133650}$}\,x^{10}
+
\mbox{\small $\displaystyle\frac{283}{20638800}$}\,x^{12}
-
\mbox{\small $\displaystyle\frac{3503}{6810804000}$}\,x^{14} \right)
<
$
$
<
\left(
\mbox{\small $\displaystyle\frac{\sin x}{x}$}
\right)^{\!2}
+
\mbox{\small $\displaystyle\frac{\tan x}{x}$}
\, < \,
2
+
\mbox{\small $\displaystyle\frac{1}{\cos x}$}
\left(
\mbox{\small $\displaystyle\frac{8}{45}$}\,x^{4}
-
\mbox{\small $\displaystyle\frac{4}{105}$}\,x^{6}
+
\mbox{\small $\displaystyle\frac{19}{4725}$}\,x^{8}
-
\mbox{\small $\displaystyle\frac{37}{133650}$}\,x^{10}
+
\mbox{\small $\displaystyle\frac{283}{20638800}$}\,x^{12}\right)\!
$

\item Taking $n=4$ in (\ref{t-4}) gives:

$
2
+
\mbox{\small $\displaystyle\frac{1}{\cos x}$}
\left(
\mbox{\small $\displaystyle\frac{8}{45}$}\,x^{4}
-
\mbox{\small $\displaystyle\frac{4}{105}$}\,x^{6}
+
\mbox{\small $\displaystyle\frac{19}{4725}$}\,x^{8}
-
\mbox{\small $\displaystyle\frac{37}{133650}$}\,x^{10}
+
\mbox{\small $\displaystyle\frac{283}{20638800}$}\,x^{12}
-
\mbox{\small $\displaystyle\frac{3503}{6810804000}$}\,x^{14}
+
\right.
$

$
\left.
+\mbox{\small $\displaystyle\frac{189169}{12504636144000}$}\,x^{16}
-
\mbox{\small $\displaystyle\frac{18917}{52797352608000}$}\,x^{18}
\right)
\, < \,
\left(
\mbox{\small $\displaystyle\frac{\sin x}{x}$}\right)^{\!2}
+
\mbox{\small $\displaystyle\frac{\tan x}{x}$}
\, <$

$
<
2
+
\mbox{\small $\displaystyle\frac{1}{\cos x}$}
\!\left(
\mbox{\small $\displaystyle\frac{8}{45}$}\,x^{4}
-
\mbox{\small $\displaystyle\frac{4}{105}$}\,x^{6}
+
\mbox{\small $\displaystyle\frac{19}{4725}$}\,x^{8}
-
\mbox{\small $\displaystyle\frac{37}{133650}$}\,x^{10}
+
\mbox{\small $\displaystyle\frac{283}{20638800}$}\,x^{12} -
\right.
$

$
\left.
-\mbox{\small $\displaystyle\frac{3503}{6810804000}$}\,x^{14}
+
\mbox{\small $\displaystyle\frac{189169}{12504636144000}$}\,x^{16}\right)\!
$

\item[etc.]

\end{itemize}

\medskip
\noindent
{\bf Proof of Theorem~\ref{improvement-s-4}.}

\smallskip
Consider the function
$$
f(x)
=
{\displaystyle \frac{\mathrm{cos}\,x
-
\mathrm{cos}^{3}\,x}{x^{2}}}
+
{\displaystyle \frac{\mathrm{sin}\,x}{x}}
-
2\,\mathrm{cos}\,x , \quad \mbox{for} \; x \!\in\! \left(0, \mbox{\small $\displaystyle\frac{\pi}{2}$}\right).
$$
Based on  power series expansion (\ref{cos3}) we have
$$
\begin{array}{lcl}
f(x) \!&\!\!=\!\!&\! {\displaystyle \sum _{k=0}^{\infty}}
\,(-1)^{k}\left( - {\displaystyle \frac{1}{(2\,k + 2)\mathrm{!}}
} + {\displaystyle \frac{1}{4}}\,{\displaystyle \frac{3^{(2\,
k + 2)} + 3}{(2\,k + 2)\mathrm{!}}} + {\displaystyle \frac{1}{(
2\,k + 1)\mathrm{!}}} - {\displaystyle \frac{2}{(2\,k)\mathrm{!
}}} \right)x^{2\,k} \\ [1.1em]
 \!&\!\!=\!\!&\! {\displaystyle \sum _{k=0}^{\infty}}\,(-1)^{k} \,{\displaystyle \frac{
9^{k + 1} - (32\,k^{2} + 40\,k +9)}{4 \, (2\,k + 2)\mathrm{!}}}\,x^{2\,k} \\ [1.1em]
 \!&\!\!=\!\!&\! {\displaystyle \sum _{k=2}^{\infty}}\,(-1)^{k}\,{\displaystyle \frac{
9^{k + 1} - (32\,k^{2} + 40\,k +9) }{4 \, (2\,k + 2)\mathrm{!}}}\,x^{2\,k}.
\end{array}
$$
For $ c=\pi/2$ we have:
$$
D(k)>0 \;\; \mbox{for} \; k\geq 2, \;\; \quad
\mbox{and} \; \; \quad \displaystyle \lim _{k\rightarrow \infty }c^{2k}D(k) =0.
$$
Also,

\smallskip
$
\begin{array}{l}\displaystyle
c^{2}\, D(k + 1) < D(k) ~ \Longleftrightarrow \\[0.5 ex]
{\displaystyle 32\,k^{4} + 152\,k^{3} + ( 245- 8 c^{2}
)\,k^{2} + \left({\displaystyle \frac{303}{2}} - 26\,c^{2}\right)\,k + 27
 - {\displaystyle \frac{81\,c^{2}}{4}} } \\[0.5 ex]
\hfill <
{\displaystyle \frac{1}{4}}\,{\displaystyle 9^{k+1}\,( 4 \,k^{2} + 14\,k + 12-
81\,c^{2} )}.
\end{array}
$

\medskip
\noindent
As the last inequality holds for $k \geq 2$ the assertions of Theorem~\ref{improvement-s-2}
immediately follows from Proposition~\ref{Leibniz}.
\hfill $\Box$

\medskip
\subsection{ Refinements of the inequalities in Statement \mbox{5}}

\smallskip
We prove the following generalization of Statement 5:

\begin{theorem} ~\\
\label{improvement-s-5}
$\!\!$For every $x \!\in\! \left(0, \mbox{\small $\displaystyle\frac{\pi}{2}$}\right)$ and $m \in N$, $m \ge 2$, the
following inequalities hold:
\begin{equation}
\label{t-5}
\begin{array}{c}\displaystyle
2 + \mathop \sum \limits_{k = 2}^{m - 1} \frac{{\left| {{\mbox{\boldmath $B$}_{2k}}} \right|(2k - 2){4^k}}}{{(2k)!}}{x^{2k}} +
{\left( {\frac{{2x}}{\pi}} \right)^{\!2n}}
\left( {
\frac{{{\pi^2}}}{4}
- 2 - \mathop \sum \limits_{k = 2}^{m - 1} \frac{{\left| {{\mbox{\boldmath $B$}_{2k}}} \right|(2k - 2){4^k}}}{{(2k)!}}{{\left(
{\frac{\pi}{2}} \right)}^{2k}}} \right)      \\[3.0 ex]
> \, \displaystyle {\left( {\frac{x}{{\sin x}}} \right)^2} + \frac{x}{{\tan x}}\, > \, 2 + \sum \limits_{k = 2}^m \frac{{\left| {{\mbox{\boldmath $B$}_{2k}}} \right|(2k - 2){4^k}}}{{(2k)!}}{x^{2k}}
\end{array}
\end{equation}
where $\mbox{\boldmath $B$}_{i}$ are {\sc Bernoulli}'s numbers.
\end{theorem}

\noindent
{\bf Proof of Theorem~5.}

Consider the function
$$
f(x)
=
{\displaystyle \left( {\frac{x}{{\sin\,x}}} \right)^{\!2}} + \frac{x}{{\tan\,x}},
\quad \mbox{for} \;\; x \!\in\! \left(0, \mbox{\small$\displaystyle\frac{\pi}{2}$}\right).
$$
Based on  series expansion (\ref{cosec2}) and (\ref{cotan}), we have
$$
\begin{array}{lcl}
f(x)
\!&\!\!=\!\!&\!
\displaystyle {x^2}\left( {\frac{1}{{{x^2}}} + \displaystyle \sum \limits_{k = 1}^{  \infty }
\frac{{\left| {{\mbox{\boldmath $B$}_{2k}}} \right|(2k - 1){4^k}}}{{(2k)!}}{x^{2k - 2}}} \right) + x\left( {\frac{1}{x} - \mathop
\sum \limits_{k = 1}^{  \infty } \frac{{\left| {{\mbox{\boldmath $B$}_{2k}}} \right|{4^k}}}{{(2k)!}}{x^{2k - 1}}} \right)\\[1.1em]
\!&\!\!=\!\!&\!
\displaystyle 2 + \mathop \sum \limits_{k = 1}^{  \infty } \frac{{\left| {{\mbox{\boldmath $B$}_{2k}}} \right|(2k -
1){4^k} - {4^k}}}{{(2k)!}}{x^{2k}}      \\[1.1em]
\!&\!\!=\!\!&\!
\displaystyle 2 + \mathop \sum \limits_{k = 2}^{  \infty } \frac{{\left| {{\mbox{\boldmath $B$}_{2k}}} \right|(2k -
2){4^k}}}{{(2k)!}}{x^{2k}}\\ [1.1em]
\!&\!\!>\!\!&\! \displaystyle 2 + \mathop \sum \limits_{k = 2}^m \frac{{\left| {{\mbox{\boldmath $B$}_{2k}}} \right|
(2k - 2){4^k}}}{{(2k)!}}{x^{2k}}.
\end{array}
$$
Since all coefficients are positive, by applying  Theorem WD we get the inequalities in the statement of
the theorem.
\hfill $\Box$

\medskip
\noindent
{\bf Examples.} \\
For $x \!\in\! \left( {0, \displaystyle\frac{\pi}{2} } \right)$ and
$
f(x)\!=\!{\left(\displaystyle\frac{x}{{\sin x}} \right)^{\!2}} + \displaystyle\frac{x}{{\tan x}},
$
we show the inequalities for $m \!=\! 2, 3,$ $4, 5$.
\begin{itemize}
\item For $m=2$:
\[\displaystyle
2
+
{\left( {\mbox{\small $\displaystyle\frac{2}{\pi}$}} \right)^{\!4}}\!\left(
-2
+
\mbox{\small $\displaystyle\frac{{{\pi^2}}}{4}$} \right){x^4}
> f(x) >
2
+
\mbox{\small $\displaystyle\frac{2}{{45}}$}{x^4}
\]
On the right-hand side we see the inequality from Statement 5.
\item For $m = 3$:
\[\displaystyle
2
+
\mbox{\small $\displaystyle\frac{2}{{45}}$}{x^4}
+
{\left({\mbox{\small $\displaystyle\frac{2}{\pi}$}}\right)^{\!6}}\!\left(
-2
+
\mbox{\small $\displaystyle\frac{{{\pi^2}}}{4}$} -
\mbox{\small $\displaystyle\frac{{{\pi^4}}}{{360}}$}\right){x^6}
> f(x) >
2
+ \mbox{\small $\displaystyle\frac{2}{{45}}$}{x^4}
+
\mbox{\small $\displaystyle\frac{8}{{945}}$}{x^6}.
\]
\item For $m = 4$:\\
$\begin{array}{c}\displaystyle
2
+
\mbox{\small $\displaystyle\frac{2}{{45}}$}{x^4}
+
\mbox{\small $\displaystyle\frac{8}{{945}}$}{x^6}
+
{\left(\mbox{\small $\displaystyle\frac{2}{\pi}$}\right)^{\!8}}\!\left(
-2
+
\mbox{\small $\displaystyle\frac{{{\pi^2}}}{4}$}
-
\mbox{\small $\displaystyle\frac{{{\pi^4}}}{{360}}$}
-
\mbox{\small $\displaystyle\frac{{{\pi^6}}}{{7560}}$} \right){x^8} >
\displaystyle f(x) \\ [1.3em]
\hfill >
\displaystyle
2
+
\mbox{\small $\displaystyle\frac{2}{{45}}$}{x^4}
+
\mbox{\small $\displaystyle\frac{8}{{945}}$}{x^6}
+
\mbox{\small $\displaystyle\frac{2}{{1575}}$}{x^8}.
\end{array}
$
\item For $m = 5$: \\
\[
\begin{array}{c}\displaystyle
2
\!+\!
\mbox{\small $\displaystyle\frac{2}{{45}}$}{x^4}
\!+\!
\mbox{\small $\displaystyle\frac{8}{{945}}$}{x^6}
\!+\!
\mbox{\small $\displaystyle\frac{2}{{1575}}$}{x^8}
\!+\!
{\left( \mbox{\small $\displaystyle\frac{2}{\pi}$}
\right)^{\!10}}\!\left(
\!-\!
2
\!+\!
\mbox{\small $\displaystyle\frac{{{\pi^2}}}{4}$}
\!-\!
\mbox{\small $\displaystyle\frac{{{\pi^4}}}{{360}}$}
\!-\!
\mbox{\small $\displaystyle\frac{{{\pi^6}}}{{7560}}$}
\!-\!
\mbox{\small $\displaystyle\frac{{{\pi^8}}}{{201600}}$}
\right){x^{10}} >  \\[3.0 ex]
> f(x) >
2
\!+\!
\mbox{\small $\displaystyle\frac{2}{{45}}$}{x^4}
\!+\!
\mbox{\small $\displaystyle\frac{8}{{945}}$}{x^6}
\!+\!
\mbox{\small $\displaystyle\frac{2}{{1575}}$}{x^8}
\!+\!
\mbox{\small $\displaystyle\frac{{16}}{{93555}}$}{x^{10}}.
\end{array}
\]
\end{itemize}
\noindent {\bf Remark.}\\
Let us notice that Theorem WD allows for the  approximation error to be estimated.
The difference between the right-hand side and the left-hand side of the double inequality in
Theorem~\ref{improvement-s-5} can be represented by the following function:
\[
\displaystyle
{R_n}\left( x \right)
=
\left(\!{f\left( {\frac{\pi}{2}} \right)
- 2
- \mathop \sum \limits_{k = 1}^n \frac{{\left| {{\mbox{\boldmath $B$}_{2k}}} \right|(2k - 2){4^k}}}{{(2k)!}}{{\left( {\frac{\pi}{2}} \right)}^{2k}}} \right)
\!{\left( {\frac{{2x}}{\pi}} \right)^{\!2n }}
\]
The maximum values of these functions are reached at $\mbox{\small $\displaystyle\frac{\pi}{2}$}$.
They are shown in the following table for $n = 3,4,5,6$:
$$
\begin{array}{|c|c|c|c|c|c|c|} \hline
n \!&\! 3 \!&\! 4 \!&\! 5 \!&\! 6 \\ \hline
R_{n}(x) \!&\! 0.0696524 \!&\! 0.0225863 \!&\! 0.00694574 \!&\! 0.00206173 \\ \hline
\end{array}
$$

\smallskip
\subsection{ Refinements of the inequalities in Statement \mbox{6}}

\smallskip
We propose the following generalization of Statement 6.

\begin{theorem} ~\\
\label{improvement-s-6 }
$\!\!$
For every $x \!\in\! \left( {0,\displaystyle\frac{\pi}{2}} \right)$ and $m \in N$, $m \ge 3$, the following inequality holds:
\begin{equation}
\label{t-6}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\begin{array}{c}
\displaystyle
4 \!+\! \mathop \sum \limits_{k = 1}^{m - 1} \frac{3{\left| {{\mbox{\boldmath $B$}_{2k}}} \right|({2^{2k}} \!-\! 2)
\!+\! {{(-1)}^k}}}{{(2k)!}}{x^{2k}} \!+\! {{{\left(\displaystyle\frac{2x}{\pi}\right)^{\!\!2m}}}}\!\!\left(\!
{f\!\left(\frac{\pi}{2}\right) \!- 4 -\! \displaystyle \sum \limits_{k = 1}^{m - 1} \frac{3{\left|
{{\mbox{\boldmath $B$}_{2k}}}\right|({2^{2k}} \!-\! 2) \!+\! {{( - 1)}^k}}}{{(2k)!}}{{\left( {\frac{\pi}{2}} \right)}^{\!2k}}} \right) \\[1.3em]
> \displaystyle 3\frac{x}{{\sin x}} \!+\! \cos x > 4 + \mathop \sum \limits_{k = 1}^m \frac{{3\left| {{\mbox{\boldmath $B$}_{2k}}}
\right|({2^{2k}} \!-\! 2) \!+\! {{( - 1)}^k}}}{{(2k)!}}{x^{2k}}.
\end{array}
\end{equation}
where $\mbox{\boldmath $B$}_{i}$ are {\sc Bernoulli}'s numbers.
\end{theorem}
{\bf Proof of Theorem~6.}
Consider the function
$$
f(x)= \displaystyle 3\frac{x}{{\sin x}} + \cos x, \quad \mbox{for} \; x \!\in\! \left(0, \mbox{\small $\displaystyle\frac{\pi}{2}$}\right).
$$
Based on the series expansion (\ref{sin}) and (\ref{cosec}), we have
\[\displaystyle f(x)
= 4 + \displaystyle \sum \limits_{k = 1}^{ \infty } \frac{{3\left| {{\mbox{\boldmath $B$}_{2k}}} \right|({2^{2k}} - 2) + {{( -
1)}^k}}}{{(2k)!}}{x^{2k}}.\]
It is easy to verify that $3\left| {{\mbox{\boldmath $B$}_{2k}}} \right|({2^{2k}} \!-\! 2) > 1$ for $k\geq2$, and that it is equal to $1$
for $k\!=\!1$, therefore all corresponding coefficients are positive. Now, using  Theorem WD, we get the inequalities
in the statement of the theorem. \hfill $\Box$\\

\medskip
\noindent
{\bf Examples.} \\
For $x \!\in\! \left(0, \mbox{\small $\displaystyle\frac{\pi}{2}$}\right)$
and
$
f(x)=3\displaystyle\frac{x}{{\sin x}} + \cos x,
$
we show the inequalities for $m = 3, 4,$ $5, 6$.

\begin{itemize}
\item For $m = 3$:
\[
4
+
\mbox{\small $\displaystyle\frac{1}{{10}}$}{x^4}
+
{\left({\mbox{\small $\displaystyle\frac{2}{\pi}$}} \right)^{\!6}}\!\left(
-
4
+
\mbox{\small $\displaystyle\frac{{3\pi}}{2}$}
-
\mbox{\small $\displaystyle\frac{{{\pi^4}}}{{160}}$} \right){x^6}
> f(x) >
4
+
\mbox{\small $\displaystyle\frac{1}{{10}}$}{x^4}
+
\mbox{\small $\displaystyle\frac{1}{{210}}$}{x^6}.
\]
On the right-hand side we see the inequality from Statement 6.

\item For $m = 4$:
\[
4
+
\mbox{\small $\displaystyle\frac{1}{{10}}$}{x^4}
+
\mbox{\small $\displaystyle\frac{1}{{210}}$}{x^6}
+
{\left({\mbox{\small $\displaystyle\frac{2}{\pi}$}}\right)^{\!8}}\!\left(
-4
+
\mbox{\small $\displaystyle\frac{{3\pi}}{2}$}
-
\mbox{\small $\displaystyle\frac{{{\pi^4}}}{{160}}$}
-
\mbox{\small $\displaystyle\frac{{{\pi^6}}}{{13440}}$}\right){x^8}
> f(x) >
4
+
\mbox{\small $\displaystyle\frac{1}{{10}}$}{x^4}
+
\mbox{\small $\displaystyle\frac{1}{{210}}$}{x^6}
+
\mbox{\small $\displaystyle\frac{{11}}{{16800}}$}{x^8}.
\]

\item For $m = 5$:
\[
\begin{array}{c}\displaystyle
4 +
\mbox{\small $\displaystyle\frac{1}{{10}}$}{x^4}
+
\mbox{\small $\displaystyle\frac{{{1}}}{{210}}$}{x^6}
+
\mbox{\small $\displaystyle\frac{{11}}{{16800}}$}{x^8}
+
{\left({\mbox{\small $\displaystyle\frac{2}{\pi}$}}\right)^{10}}\!\left(
- 4
+
\mbox{\small $\displaystyle\frac{{3\pi}}{2}$}
-
\mbox{\small $\displaystyle\frac{{{\pi^4}}}{{160}}$}
-
\mbox{\small $\displaystyle\frac{{{\pi^6}}}{{13440}}$}
-
\mbox{\small $\displaystyle\frac{{11{\pi^8}}}{{4300800}}$} \right){x^{10}} >                                 \\[3.0 ex]
> f(x) >
\displaystyle
4
+
\mbox{\small $\displaystyle\frac{1}{{10}}$}{x^4}
+
\mbox{\small $\displaystyle\frac{1}{{210}}$}{x^6}
+
\mbox{\small $\displaystyle\frac{{11}}{{16800}}$}{x^8}
+
\mbox{\small $\displaystyle\frac{{53}}{{831600}}$}{x^{10}}.
\end{array}
\]

\item For $m = 6:$
\[\begin{array}{c}\displaystyle
4
\!+\!
\mbox{\small $\displaystyle\frac{{{x^4}}}{{10}}$}
\!+\!
\mbox{\small $\displaystyle\frac{{{x^6}}}{{210}}$}
\!+\!
\mbox{\small $\displaystyle\frac{{11{x^8}}}{{16800}}$}
\!+\!
\mbox{\small $\displaystyle\frac{{53{x^{10}}}}{{831600}}$}
\!+\!
{\left( {\mbox{\small $\displaystyle\frac{2}{\pi}$}} \right)^{\!12}}
\!\left(
-4
\!+\!
\mbox{\small $\displaystyle\frac{{3\pi}}{2}$}
\!-\!
\mbox{\small $\displaystyle\frac{{{\pi^4}}}{{160}}$}
\!-\!
\mbox{\small $\displaystyle\frac{{{\pi^6}}}{{13440}}$}
\!-\!
\mbox{\small $\displaystyle\frac{{11{\pi^8}}}{{4300800}}$}
\!-\!
\mbox{\small $\displaystyle\frac{{53{\pi^{10}}}}{{851558400}}$}
\right){x^{12}} >    \\[3.0 ex]
> f(x) >
\displaystyle
4
+
\mbox{\small $\displaystyle\frac{{{x^4}}}{{10}}$}
+
\mbox{\small $\displaystyle\frac{{{x^6}}}{{210}}$}
+
\mbox{\small $\displaystyle\frac{{11{x^8}}}{{16800}}$}
+
\mbox{\small $\displaystyle\frac{{53{x^{10}}}}{{831600}}$}
+
\mbox{\small $\displaystyle\frac{{117911{x^{12}}}}{{18162144000}}$}.
\end{array}
\]
\end{itemize}
{\bf Remark.}
The difference between the right-hand side and the left-hand side of the double inequality in Theorem 6 can be
represented by the following function:

\[
{R_n}\left( x \right)
=
\left( {f\left( {\frac{\pi}{2}} \right) - 4
+ \mathop \sum \limits_{k = 1}^n \frac{{3\left| {{\mbox{\boldmath $B$}_{2k}}} \right|({2^{2k}} - 2) + {{( -
1)}^k}}}{{(2k)!}}{{\left( {\frac{\pi}{2}} \right)}^{2k}}} \right)\!{\left( {\frac{{2x}}{\pi}} \right)^{\!2n}}.
\]
The maximum values of these functions are reached at $\mbox{\small$\displaystyle\frac{\pi}{2}$}$;
they are shown in the following table for $n = 3, 4, 5, 6$
$$
\begin{array}{|c|c|c|c|c|c|c|} \hline
n \!&\! 3 \!&\! 4 \!&\! 5 \!&\! 6 \\ \hline
R_{n}(x) \!&\! 0.0320502 \!&\! 0.00778176 \!&\! 0.00195321 \!&\! 0.000488251 \\ \hline
\end{array}
$$

\medskip
\section{Conclusion}

\smallskip
\qquad

The  idea to compare and replace functions with their
corresponding power series  to get more accurate approximations  was used in
 \cite{MRL_2017a}, \cite{MRL_2017b} and  \cite{MRL_2017c}.
 Following  the same idea, in this paper we extended the natural approach.
 We proposed and proved new inequalities which represent
refinements and generalizations of the inequalities stated in
\cite{Mortici_2011}, related to {\sc Wilker}-{\sc Cusa}-{\sc Huygens}'s inequalities.

\smallskip
Note that proofs of the new inequalities (\ref{t-1}), (\ref{t-2}), (\ref{t-3}), (\ref{t-4}), (\ref{t-5})
and (\ref{t-6}) for any fixed $n,\, m \in N$ can be obtained by substituting $x \!=\! \sin t$
for $\displaystyle t \!\in\! \left[0, \mbox{\small $\displaystyle\frac{\pi}{2}$}\right]$
and using the methods and algorithms developed in  \cite{Malesevic2016} and \cite{Lutovac2017}.
However, our approach provides proofs for the approximation of the corresponding function
by the inequality of an arbitrary degree.

The  results of the present research
 can be used to  verify  as well as to refine   a  broad
category of inequalities.
For example, {\sc Bercu} (\cite{GB_2017}, Theorem 2.4) proved  the following inequalities
$$
\!\!\!
\begin{array}{rl}
  \!&\! \left(\displaystyle\frac{x}{\sin x}\right)^2 + \displaystyle\frac{x}{\tan x}                              \\[2.0 ex]
> \!&\!
\mbox{\small $\displaystyle\frac{
\mbox{\footnotesize $11220$}\,x^{10}
\!-\!
\mbox{\footnotesize $205560$}\,x^8
\!-\!
\mbox{\footnotesize $14256000$}\,x^6
\!+\!
\mbox{\footnotesize $512179200$}\,x^4
\!-\!
\mbox{\footnotesize $3157056000$}\,x^2
\!+\!
\mbox{\footnotesize $13716864000$}}{
\mbox{\footnotesize $242$}\,x^{12}
\!-\!
\mbox{\footnotesize $8580$}\,x^{10}
\!+\!
\mbox{\footnotesize $25560$}\,x^8
\!-\!
\mbox{\footnotesize $1080000$}\,x^6
\!+\!
\mbox{\footnotesize $103680000$}\,x^4
\!-\!
\mbox{\footnotesize $1578528000$}\,x^2
\!+\!
\mbox{\footnotesize $6858432000$}}$}                                                                          \\[2.0 ex]
> \!&\! 2 + \displaystyle\frac{2}{45}x^4 > 2,
\end{array}
$$
 for every  $x \in (0, b)$ where $b=\mbox{\small $\sqrt{\displaystyle\frac{240-6\sqrt{1090}}{17}}$} = 1.5701 \ldots
< \displaystyle\frac{\pi}{2}$.

According to  Theorem~5,  taking $m=7$ in (\ref{t-5}) gives
$$\left(\displaystyle\frac{x}{\sin x}\right)^{\!2} + \displaystyle\frac{x}{\tan x} \,>\, P_{7}(x), ~~~ \mbox{for} ~ x \in \left(0, \dfrac{\pi}{2}\right)$$
where $$
P_7(x)
=
\displaystyle 2 + \mathop \sum \limits_{k = 2}^7 \frac{{\left| {{\mbox{\boldmath $B$}_{2k}}} \right|
(2k - 2){4^k}}}{{(2k)!}}{x^{2k}}.
$$
Thus {\sc Bercu}'s inequality is reduced to the following decidable problem (see \cite{Tarski_1951}, \cite{Poonen_2014}):
for every $x \in \left(0, \mbox{\small $\displaystyle\frac{\pi}{2}$}\right)$  it is true that

\vspace*{-3.5 mm}

$$
\!\!\!\!\!\!\!\!
\begin{array}{r}
  P_7(x)\! >\!
\mbox{\small $\displaystyle\frac{
\mbox{\footnotesize $11220$}\,x^{10}
\!-\!
\mbox{\footnotesize $205560$}\,x^8
\!-\!
\mbox{\footnotesize $14256000$}\,x^6
\!+\!
\mbox{\footnotesize $512179200$}\,x^4
\!-\!
\mbox{\footnotesize $3157056000$}\,x^2
\!+\!
\mbox{\footnotesize $13716864000$}}{
\mbox{\footnotesize $242$}\,x^{12}
\!-\!
\mbox{\footnotesize $8580$}\,x^{10}
\!+\!
\mbox{\footnotesize $25560$}\,x^8
\!-\!
\mbox{\footnotesize $1080000$}\,x^6
\!+\!
\mbox{\footnotesize $103680000$}\,x^4
\!-\!
\mbox{\footnotesize $1578528000$}\,x^2
\!+\!
\mbox{\footnotesize $6858432000$}}$}.
\end{array}
$$
The above inequality is a refinement of the inequality obtained by  {\sc Bercu}.
Moreover,  the above inequality   shows  that {\sc Bercu}'s inequality   holds true over an  extended interval
$\left(0, \mbox{\small $\displaystyle\frac{\pi}{2}$}\right)$.

Similarly,  Theorems 1, 2, 3, 4, 5 and 6 can be applied to other results and inequalities from \cite{GB_2016}, \cite{GB_2017}, as well as  to a broad category  of analytical inequalities.

\bigskip
\noindent \textbf{Acknowledgement.} The research of the first, second
and third authors was supported in part by the Serbian Ministry of
Education, Science and Technological Development, under Projects ON
174032, III 44006  and TR 32023 and ON 174033 respectively.

\bigskip
\noindent \textbf{Competing Interests.} The authors would like to
state that they do not have any competing interests in the subject
of this research.

\bigskip
\noindent \textbf{Author's Contributions.} All the authors
participated in every phase of the research conducted for this
paper.

\bigskip

redpajama_set_name: RedPajamaArXiv


Denver, CO — The United States District Court of Colorado has ordered the U.S. Forest Service to fully disclose documents pertaining to its decision to approve a controversial land exchange on Wolf Creek Pass in southwestern Colorado. The Court ruled that the Forest Service did not search for public records, including emails and other communications between the agency's Regional and Washington Offices and the developers. The 20-page ruling also confirmed the Forest Service failed to justify its decision to withhold thousands of pages from public scrutiny.
The court order came in response to a lawsuit filed by Denver-based conservation group Rocky Mountain Wild. That suit claimed that the Forest Service had unlawfully withheld public records and communications requested through a February 27, 2014 Freedom of Information Act request. The Court's order gives the U.S. Forest Service until October 30, 2015 to conduct a search and to fully disclose documents pertaining to its decision.
The land exchange at the center of this controversy was approved by Rio Grande Forest Supervisor, Dan Dallas, in May of this year, and would trade 205 federal acres for 177 acres of private land within the boundaries of the Rio Grande National Forest. The exchange gives highway access to the private land, making it possible for the owner/developer to construct a proposed "village" for 8,000 tourists. Conservation organizations and local communities oppose this development, siting the environmental and economic ramifications for downstream communities, and the direct impacts to wildlife in the area.
Opposition to the development has steadily grown, attracting local businesses, skiers, ranchers, local landowners, downstream water users, hunters, anglers, and conservationists. An online petition started this summer has garnered over 79,000 signatures urging the Chief of the U.S. Forest Service, Tom Tidwell, to protect the pass.
Friends of Wolf Creek is a coalition of conservation organizations fighting to keep Wolf Creek Pass wild. The coalition includes: Rocky Mountain Wild, San Juan Citizens Alliance, and San Luis Valley Ecosystem Council. More information can be found at www.friendsofwolfcreek.org.
A federal appeals court Tuesday dismissed the latest attempt from the would-be developers of the Village at Wolf Creek to reinstate a land exchange rejected by a federal judge last year.


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