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Prince Chunk (also known as Princess Chunk and Captain Chunk, real name: Powder) (1998 – November 21, 2010) was a domestic shorthair cat, who at one time was alleged to weigh 44 pounds (20 kg), a world record. He was found in Voorhees, New Jersey, on July 26, 2008, by Animal Control, who nicknamed him "Captain Chunk". After a search for his owner, it was found that his name was Powder and that he was abandoned by Donna Oklatner, an elderly Voorhees resident. Oklatner claimed she could not afford to take care of the cat when she lost her home in foreclosure proceedings.

He was later adopted from the Camden County Animal Shelter in New Jersey by Donna Damiani of Blackwood, New Jersey, and in August 2008 his weight was already down to 22 pounds.

According to the Guinness World Records, the largest tabby cat was forty-six pounds and fifteen ounces. Since then, the publishers have dropped the category to prevent deliberate overfeeding and possible harm to cats.

He made television appearances on Live with Regis and Kelly, Good Morning America, Fox News, and MSNBC, based on his reported weight, but reliable sources of the cat's definitive weight in July 2008 are not available. One paper reported that "shelter officials said they were perplexed by the discrepancy", after the Damiani family found that the cat weighed just over 22 pounds, "days after" receiving him. The Damiani family later founded the Prince Chunk Foundation, a nonprofit that allows people to keep their pets in difficult financial situations by giving them free pet food and vet care.

Prince Chunk died in his sleep on November 21, 2010, due to heart disease.

See also
Meow (cat)
Obesity in pets
List of individual cats

References

External links
Camden County Animal Shelter

1998 animal births
2010 animal deaths
Individual cats in the United States

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Omnivision Entertainment includes production, talent development and branded entertainment divisions, and a distribution network that empowers cross-promotion across its owned-and-operated online channels, YouTube and other online, mobile and IPTV platforms.
Founded in 2007 by Rob Barnett and Warren Chao as My Damn Channel, the company has produced hundreds of hours of original programming, including comedy, music, unscripted reality and live variety programming. We are indebted to Rob Barnett for his vision and leadership over the years and he can be reached at his new venture RobBarnettMedia.com
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Warren oversees production and business partnerships for Omnivision Entertainment. Before starting the company he was VP of Business Development and Strategy at Gemstar/TV Guide, and prior to that he was a venture capitalist and corporate attorney in Silicon Valley. In addition to working with major advertising brands, Warren's favorite tasks at the company are writing comedy vignettes and developing new talent. His two favorite videos that he wrote with Matt Riotto are here: Fairy Tale Friday - Wizard of Oz and Thinking Man's Lamborghini Parody He can be reached at Warren@OmnivisionEnt.com
We are a full-service studio and production company and have our own distribution on YouTube through a network of owned and operated YouTube channels.
We also have one of the largest short-form comedy video libraries on the Internet available for license, spearheaded by award-winning shows including You Suck at Photoshop, Wainy Days, Horrible People, Cooking with Coolio, Mark Malkoff, Grace Helbig and more.
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redpajama_set_name: RedPajamaCommonCrawl


Yesterday's Poland-Greece 1-1 draw was one of the most exciting opening tournament matches that I can remember, possibly since Cameroon shocked Argentina at Italia 90. France losing to Senegal in 2002 had a similar shock factor, but yesterday's game had everything: red cards, missed penalties, goalkeeping errors and refereeing controversy, as well as a brilliantly-taken goal by Robert Lewandowski.
The Borussia Dortmund striker was named as the official man of the match, presumably by a committee who took the decision at half-time. Greece's second-half substitute Dimitris Salpingidis had a strong case for the award, after he scored Greece's eqiualiser, won the penalty that Poland's replacement keeper Przemyslaw Tyton saved from Giorgos Karagounis, and had a goal disallowed for a marginal offside with the assist.
The Greeks thoroughly deserved their point, even if the result threatened to take the edge off the hosts' opening party. But Warsaw's residents carried on partying anyway, with impromptu late-night gatherings at bars on the banks of the Vistula river that runs alongside the National Stadium.
Lewandowski is the undoubted star of the Poland squad. His chiselled looks feature in the shaving foam ad that is prominent in the ad breaks between TVP2's live match coverage.
Woijech Szczesny is the Polish representative in the crowd-surfing Pepsi ads that also feature Frank Lampard those well-know Euro stars Didier Drogba and Sergio Aguero. Forget the curse of Nike, welcome to the plague of Pepsi. Official tournament sponsors Coca-cola will probably be celebrating with something a little stronger.
I've been getting a few comments from people – I won't call them complaints, maybe constructive criticism! – about our Euro 2012 special edition. A few have highlighted the fact that we did not include Jetro Willems, the young Dutch left-back, in the Holland squad, or even in the "Also in Contention" list.
Willems, aged 18 years and 71 days, became the youngest player to feature at the European Championship when he played in Holland's defeat earlier today. He beat the previous record held by Enzo Scifo, who was 18 years and 115 days in France in 1984.
Willems has come from nowhere to international contention in a matter of weeks, so it's no surprise that we did not include him on a list that was compiled in early April – such are the deadlines for a monthly publication that was despatched to the printers before the end of April.
We try to provide a snapshot of the selection issues facing each national coach, providing people with the background issues to help analyse the final squads. If we miss the odd player, or late injury, so be it. That is the nature of the beast.
We don't claim to be predicting the final squads, but I'm always happy if we get 18 or more of the final 23. We tend to average around 19 or 20.
It's been a quiet day in Warsaw, it feels like a rest day with today's matches taking place in Ukraine. There's not the usual, mad travel between venues common in previous tournaments, as people opt for either Poland or Ukraine.
I had been planning to drive from Warsaw to Lviv for the Germany-Portugal game but the rental car office in Warsaw advised against. In Ukraine it's illegal to drive a rental car hire in Poland. "They won't let you in," was the no-nonsense advice even though Holiday Autos, the UK company who I made the booking through, had insisted I could take the car into Ukraine. Thanks, Holiday Autos. Suffice to say, you've had your last booking from me.


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BREAKING NEWS Thu 28 Jul 2022
Everton Complete McNeil Signing
By Everton
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Dwight McNeil has signed for Everton from Burnley for an undisclosed fee, the forward penning a five-year contract until the end of June 2027.
The 22-year-old becomes the Blues' third signing of the summer, following the arrival of his former Clarets teammate James Tarkowski and the loan capture of Portuguese defender Rúben Vinagre.
England Under-21 international McNeil moves to Merseyside after four years as a professional at Turf Moor, during which time he made more than 140 appearances and missed just two Premier League matches since December 2018.
Capable of playing on either wing, McNeil finished second for most successful dribbles in the English top-flight last season with 90 and was third for the highest number of crosses (199), having been fourth and fifth respectively in the previous two campaigns.
"It is an amazing feeling to sign. When I heard about Everton's interest I wanted to get it done straight away because of the Club and how good the team is," McNeil told evertontv.
"Speaking to the manager about what they want going forward, it really excited me and appealed to me. I wanted to be part of it. He knows me as a player, as does [Director of Football] Kevin Thelwell. I want to work hard and do well for the team and for the fans.
"I feel playing under a manager like Frank Lampard, he will help me to get my stats up – and that is what I want to do and improve my game.
"I am a direct runner with the ball. I like to get on the ball, cut inside and create chances for the team.
"I want to help the team as best as I can, work hard and do the defensive side of the game, too."
Born in Rochdale, McNeil joined Burnley's academy at the age of 14 and made his senior debut before his 19th birthday in a Premier League clash against AFC Bournemouth.
He earned his England Under-21s bow in October 2019 against Slovenia and has gone on to feature for the young Lions with his new Blues teammates Ben Godfrey and Tom Davies.
"Last season was difficult for the Club, but going into the last bit of the season, you saw from the players and fans the desire, mentality and togetherness they have," added McNeil.
"They came together and were united as one. That is massive for players and gives you that extra bit of confidence.
"This season we want to do better and I want to be part of the team to help them do that.
"I've played at Goodison a couple of times and you always knew it was always a difficult place to go because the fans are behind the lads from the start to the end. It is good and something I want to be part of.
"Now I am fortunate to be on the same side as the fans and am looking forward to getting going."
Everton Director of Football, Kevin Thelwell, said: "We are delighted to secure the signing of Dwight on a five-year contract.
"Despite being only 22, he has plenty of Premier League experience and the statistics prove we have brought in one of the best young talents in English football.
"We are confident that, working with Frank and his coaching staff, Dwight can become even better and enjoy a long, successful career with Everton."
Manager Lampard added: "I have been a big admirer of Dwight for some time. He has huge talent and is exactly the type of player we want to bring into the squad to help us improve."
BREAKING NEWS Welcome, Dwight News Men
Dyche Named New Everton Manager
Mon 30 Jan, 2023 1
Gordon Leaves Everton
Sun 29 Jan, 2023 2
New Manager Sets Out Plans For Everton
In Profile: Sean Dyche
Roof Expands At Everton Stadium
Wed 25 Jan, 2023 5
McNeil's 'Next Level' Everton Ambition
Thu 28 Jul, 2022 Winger aiming for significant contribution following Goodison transfer.
In Profile: Dwight McNeil
Thu 28 Jul, 2022 Everything you need to know about Everton's new young wing star.
Gallery: McNeil Is A Blue!
Thu 28 Jul, 2022 Images of our latest recruit!
WATCH: McNeil's Exclusive First Interview
Thu 28 Jul, 2022 New signing speaks to evertontv.
New Blue Dwight Chooses Number
Thu 28 Jul, 2022 McNeil confirms Everton shirt for 2022/23.
Fri 01 Jul, 2022
The Goodison Gang
SORENSEN ASSESSES CUP EXIT
Sun 29 Jan, 2023
HIGHLIGHTS: DERBY U18S 3-3 EVERTON U18S


Happy Wednesday! Did you catch my sneak peek on Monday? I'm very pleased to say that the painting is coming along nicely… In fact, I'm forcing myself not to finish the last wall (can I tell you what absolute agony it is for me to leave something partially finished?!) to write this tutorial for you.
And like my post on Editing in Lightroom, this is NOT a comprehensive Photoshop* tutorial. It is a simple explanation of what I do to 90% of my photographs that go on the blog. Today, we will pick up with an image just after I finished my Lightroom edits.
And so you know where we're going, here is the comparison to the final, post-Photoshop image. It's not as dramatic as the Lightroom transformation, but I still think it adds important depth to the photo.
For the record, I export my images from Lightroom already sized for my blog (long edge: 900 px). That way, when I open them in Photoshop, my computer is not struggling through giant originals. My best trick in Photoshop (unlike in Lightroom where I do everything manually) is presets.
One set of actions, to be exact.
Go here to the incredible Pioneer Woman's blog to download a set of free Photoshop actions. Her site includes instructions for downloading them properly. There are all sorts of fun ones, but the three main ones I use are Boost, Slight Lighten, and Sharpen This.
I usually start with a Boost (almost every image I use gets this). Sometimes, I lesson the opacity if the image is feeling too processed, but usually it is just the right amount of crispness.
Sometimes the intensity of the Boost has lost some of the lightness I desired. If so, I will then insert a Slight Lighten before (under the Boost layer on your Layers palette in the bottom right hand corner) my Boost. This is also a VERY easy way to lighten just a portion of your image (which is also possible to do in Lightroom, but I find it more cumbersome in that program). Run a Slight Lighten, then type ctrl+del (or command+del on Mac) to fill the layer mask black. This will remove your lightening, and you can now go in and lighten up just a portion of it if you use your brush tool. Set it to whatever size you want but a very low hardness (1-5%) and then just draw with white over the portions you want lightened.
Finally, if there are any details you need to be clearer or crisper, Sharpen This is a great tool (be sure to add it above the Boost layer). It works with another layer mask, so after you run the preset, you simply draw over the portion you want sharpened. I almost always lower the opacity of this level, though, because it is usually too processed looking.
See below how I sharpened the letters on the paint chips so they became more legible. It's subtle, but it makes a difference.
Once finished, I apply my watermark. Again, there is a way to do this in Lightroom, but I don't like the quality of the watermarks. In Photoshop, I keep a separate .PSD file with my watermark in various forms (all as texts), so I can just drag and drop the one I want for the image.
And that's it! Like I said with the Lightroom tutorial, I do not do all these things (except watermark) to every photo, but they are my usual toolbox for my images. The better my lighting, staging, and composition become, the less edits each image needs.
Obviously, these tutorials require Lightroom and Photoshop*, and while they are not free (like some other editing software), you can get them for just $10/month with the creative cloud*. I think it's totally worth it.
What do you think? Did you learn anything new here? I hope so! And that concludes our photography tutorials, unless you all have any more topics/ideas/questions. Don't hesitate to comment and ask! Monday should bring us back to our regular diy and decorating routine. Thank you all again for your kind words about my photography and your encouragement throughout this series!
Love it! I haven't used her actions in a long time. I'll have to check them out! I typically do a clean edit in Adobe Camera Raw & then just tweak a little more with the brightness & shadows once fully opened. Then I use my favorite sharpening action "Looking Sharp Baby" by Pure Photoshop actions. I really love these tips – as you know, photos of the home is my challenge.
These are really great tips. I am on the fence about paying for Photoshop again but I am in the midst of updating the house and it would be nice to have super cool photos.
I completely understand, Pamela! I've been really pleased with the monthly subscription to Photoshop instead of paying outright… Good luck with the updating!
Thank you so much for sharing your tutorial with your photo process! It has been so hard to find one that easily explains how to lighten and sharpen home photos.
You are so welcome, Elizabeth! I'm so glad you found it helpful.
Ha! Love it. If I didn't have LR to lighten things, I'm sure I'd be using curves much more, too!
Love love all of your tutorial!
I'm a new reader of your blog, and will come more often! Thanks for giving lots of helpful tutorials!


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[VIDEO HIGHLIGHT] UEFA CHAMPIONS LEAGUE FINAL: Liverpool Vs Tottenham 2 - 0
Admin, Empire Media NG Saturday, June 01, 2019 Sports , Video 0 Comments
. Kane missed the second leg of their quarter-final clash against Manchester City and both legs against Ajax along with the end of the Premier League campaign. However, his return to fitness has seen him recalled to lead the line at the expense of Lucas. The Brazilian's hat-trick against Ajax was the reason that Spurs were able to advance.
Pochettino will have had a lot thoughts about the move, but he has selected Kane, albeit short of match fitness.
Liverpool return to the Champions League final for the second year in a row. Last season their dreams of a sixth European title were ended by Real Madrid, although it came after Mohamed Salah was forced to withdraw in the first half and two goalkeeping errors from Loris Karius between the sticks. Jurgen Klopp's men staged an outstanding comeback of their own to beat Barcelona 4-0 in the second leg of their semi-final. The Reds were denied the Premier League crown, despite an excellent term. They will not want to miss out on another trophy opportunity this evening.
LIVERPOOL SUBS: Dejan Lovren, James Milner, Joe Gomez, Daniel Sturridge, Alberto Moreno, Adam Lallana, Alex Oxlade-Chamberlain, Simon Mignolet, Xherdan Shaqiri, Rhian Brewster, Divock Origi, Caoimhin Kelleher.
LIVERPOOL (4-3-3): Alisson, Trent Alexander-Arnold, Joel Matip, Virgil van Dijk, Andrew Robertson; Jordan Henderson, Fabinho, Georginio Wijnaldum; Mohamed Salah, Roberto Firmino, Sadio Mane.
TOTTENHAM SUBS: Davinson Sanchez, Erik Lamela, Victor Wanyama, Michel Vorm, Eric Dier, Kyle Walker-Peters, Fernando Llorente, Juan Foyth, Paulo Gazzaniga, Serge Aurier, Lucas Moura, Ben Davies.
TOTTENHAM (4-2-3-1): Hugo Lloris; Kieran Trippier, Toby Alderweireld, Jan Vertonghen, Danny Rose; Moussa Sissoko, Harry Winks; Dele Alli, Christian Eriksen, Son Heung-min; Harry Kane.
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Blog - Advent Devotional: The answer is JESUS!
Advent Devotional: The answer is JESUS!
Everything around us has become festive with Christmas lights and decoration, and although Hong Kong doesn't get snow it seems that snowmen abound. I have been enjoying the extra sparkle that comes this season but as we enter Advent, it is important to remember the reason we have to be festive and celebrate.
I was recently drawn to reflect on the reason to celebrate… The author of Proverbs 30 sets the words of the skeptic and the words of the believer back to back.
The skeptic then asks several questions.
climb into Heaven and take charge?
grab the winds and control them?
gather the rains in his bucket?
stake out the ends of the earth?
"Just tell me his name!" the skeptic challenges in Proverbs 30:4 (MSG).
The New Testament, however, answers the skeptic's questions directly, giving the name of the one the skeptic demands to know… And the answer to every one of his questions is Jesus, Jesus, Jesus, Jesus.
Jesus climbed the high mountain and captured the enemy. Ephesians 4:8.
When the skeptic around us (and even the skeptic within us) questions the existence of God and asks for proof, the answer is Jesus.
The life you have through Jesus!


text:
Babulal Khandelwal & Ors Vs. Balkishan D. Singhvi & Ors. [2008] INSC 1744 (16 October 2008)
IN THE SUPREME COURT OF INDIA CIVIL APPELLATE JURISDICTION CIVIL APPEAL NO. 6124 OF 2008 @ S.L.P. (C) NO.6846 of 2006 Babulal Khandelwal & Ors. ..Appellant Balkrishan D. Sanghvi & Ors. ...Respondents
ALTAMAS KABIR,J.
2. On 14th February, 2005, the Respondent No.1, Balkishan D. Sanghvi, filed suit No. 457 of 2005 in the Bombay High Court for administration of the Estate of his deceased parents Dwarkadas Sanghvi and Vimlaben Sanghvi, who were also the parents of the Respondents No.1, 2, 3 and 6. While the Respondents Nos. 2 1, 2 and 6 are the sons of the deceased, the Respondent No.3 is their daughter and the Respondents Nos. 4 and 5 are her husband and son respectively.
3. During the pendency of the suit, the Respondent No.1/Plaintiff filed Chamber Summons No.1270 of 2005 for impleading the Appellants herein as parties to the suit and to challenge the alienation of two properties, one to Prolific Consultancy Services (Mumbai) Pvt. Ltd. and the other to the Appellants herein. By judgment and order dated 13th February, 2006, the learned Single Judge of the Bombay High Court allowed the Chamber Summons and permitted the plaint to be amended as a result whereof the appellants herein stood impleaded as defendants Nos.7 to 12 in the suit on the ground that they were necessary parties to the suit, despite their objection that the Respondent No.8 is a company in which the Appellants were only 3 share-holders and that they were in no way related to any of the parties to the suit.
4. The amendment of the plaint, also had the effect of allowing the Respondent No.1 to challenge the transactions concluded by Dwarkadas Sanghvi and Vimlaben Sanghvi and the Appellants herein, before their death.
5. Questioning the legality of the order, Ms. Meenakshi Arora, learned advocate, submitted that in a suit for administration of the Estate of a deceased, transactions concluded by the deceased during his lifetime cannot be called into question. Consequently, those with whom such transactions had been entered into were not necessary parties in such suit. Ms. Arora submitted that the High Court had committed a fundamental error in holding that transactions entered into and concluded by the deceased during their lifetime could be questioned after their death in administration proceedings of their Estate, and thereby erroneously impleaded 4 the appellants as parties to the administration suit.
6. In support of her submissions, Ms. Arora firstly referred to the decision of this Court in Sunil Gupta vs. Kiran Girhotra, [2007 (8) SCC 506], where an almost identical question, as has been raised in this matter, had arisen in respect of a probate proceeding. The question posed in the said matter was whether a purchaser of a property belonging to the deceased testator should be impleaded as a party in a probate proceeding. While answering the said question, this Court held that since a probate can be granted only to an Executor appointed by a Will, a transferee of a property during the pendency of such a proceeding is not a necessary party.
7. As an offshoot of her main argument, Ms. Arora submitted that, in any event, in a probate proceeding the Court does not decide any question of title in respect of the Estate of 5 the deceased and any order passed in the proceedings acts as an order in rem. Ms. Arora submitted that even on such score, the appellants were not necessary parties to the administration suit and their impleadment in the suit by the High Court was wholly erroneous. In support of her said argument, Ms. Arora referred to and relied upon the judgment of this Court in the case of Chiranjilal Shrilal Goenka vs. Jasjit Singh & Ors. [1993 (2) SCC 507], wherein Ms. Arora's submissions are fully reflected with approval.
Reference was also made to the decision of this Court in (1) Ghulam Qadir vs. Special Tribunal and Ors.,[2002 (1) SCC 33] and (2) Krishna Kumar Birla vs. Rajendra Singh Lodha and Ors. [2008 (4) SCC 300], where the same views have been expressed.
8. Ms. Arora's submissions were opposed by Mr. Hariharan and Mr. Subramonium Prasad, learned advocates, who appeared for the respondent 6 No.1. It was submitted that when the Estate of a deceased is involved, the Court in seisin of the proceedings relating to the administration of the said Estate was entitled to scrutinize transactions which had taken place in respect of any property forming part of such Estate.
It was submitted that, in fact, such a course of action was also pragmatic as it would eliminate the prospect of multiplicity of proceedings.
9. Mr. Subramonium Prasad submitted that the High Court had quite correctly allowed the impleadment of the appellants in the suit filed by the respondent No.1 where all questions relating to and involving the Estate of Dwarkadas Sanghvi and Vimlaben Sanghvi would be resolved with finality.
10. Having considered the rival submissions of the parties, we are unable to accept Ms. Arora's objections to the impleadment of the appellants 7 as necessary parties in the administration suit filed by the respondent No. 1, particularly when both Dwarkadas Sanghvi and Vimlaben Sanghvi had died intestate.
11. It is well settled that in an administration suit, the Court, while considering the grant of authority to an individual having an interest in the Estate of the deceased to administer the Estate, has also to determine the extent of the estate of the deceased at the time of his death to facilitate the distribution of the estate to all the heirs of the deceased. It is equally well settled that during such enquiry the Court is not called upon to determine the right and title of the parties in the properties of the Estate, but to ascertain the extent of the properties of the Estate. The decision in Chiranjilal Shrilal Goenka's case (supra), applies to probate proceedings where it has been held that the probate Court does not decide any question of title or even the 8 existence of the property itself. In administration suits in respect of a person who dies intestate, the position is different. The Court while appointing an Administrator in an administration suit to administer the Estate of the decease, who dies intestate, may be required to examine transactions involving the properties of the Estate in order to determine the assets of the Estate as on the date of death of the owner thereof. Consequently, the impleadment of persons who may be involved in some transaction or the other concerning the Estate of the deceased, may become necessary for a decision in an administration suit. The High Court has noted this fact while allowing the prayer of the respondent No.1 for impleading the appellants as parties to the administration suit.
12. The decisions cited by Ms. Arora are in relation to Probate proceedings where either probate is to be granted or Letters of 9 Administration are to be given to an applicant having an interest in the Estate of the deceased. The circumstances in an administration suit where a person dies intestate are, however, different. The learned Single Judge of the High Court has correctly indicated that in an administration suit, the dispute between the parties relating to the title of the deceased in respect of his properties, can be gone into in an administration suit and that there is no bar to a Court determining the validity of transactions allegedly entered into by or on behalf of the deceased, whose Estate is to be administered. In Appendix "D" to the Code of Civil Procedure which deals with the forms of decrees in different suits, Form 17 indicates the form in which preliminary decrees in Administration Suits are to be passed.
Paragraph 3 of the Form deals with suits filed by the next of kin of the deceased who dies intestate, as in the instant case, and provides 10 for an inquiry to be made and account taken of what or of what share, if any, the plaintiff is entitled to as next-of-kin in the moveable properties of the Estate. However, if the moveable properties of the deceased are found to be insufficient for carrying out the objects of the suit, then by virtue of paragraph 10 of the Form the Court may order an inquiry as to what immovable property the deceased was seized of or entitled to at the time of his death and what encumbrances, if any, affect the immovable property of the deceased or any part thereof.
The said inquiry, in our view might also include the transactions with the appellants herein which had purportedly been concluded by the owners of the properties themselves during their life time, in order to ascertain whether the said properties continued to form part of the Estate of the deceased at the time of their death.
13. We, therefore, have little hesitation in holding that the High Court had not committed any error in allowing the amendments to the plaint which had the effect of impleading the appellants as parties to the administration suit filed by the respondent No.1 and permitting the respondent No.1 to question the transactions entered into by the owners of the Estate with third parties. The order of the High Court does not warrant any interference and the appeal must, therefore, be dismissed.
14. We, accordingly, dismiss the appeal and affirm the judgment and order of the High Court allowing the amendments to the plaint for impleading the appellants as parties to the administration suit filed by the respondent No.1 and for scrutinizing the transactions which were alleged to have been concluded by the parents of the respondent Nos. 1 to 4 during their lifetime.
15. There will, however, be no order as to costs.
_________________J. (Altamas Kabir)
_________________J


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Silversword (de Lohr Dynasty Book 7) by USA Today and Amazon All-Star best-selling author Kathryn Le Veque
Genre: Medieval Romance
ABOUT SILVERSWORD:
266 A.D. – Chadwick "Chad" de Lohr, son of Daniel de Lohr (SHADOWMOOR) is a hero at the Battle of Evesham. Saving the hostage Henry III from Simon de Montfort, he is riding high on a wave of royal praise at the conclusion of the battle. But that favor with the king is short-lived.
Henry, once again in power after the death of de Montfort, is determined to rid his country of the de Montfort-supporting rebels once and for all. This includes a campaign of harassment against the nobles who supported de Montfort through the years, namely, the House of de Shera (THE LORDS OF THUNDER TRILOGY). The House of the de Lohr and the House of de Shera are linked by blood, and when Henry orders that a de Shera heiress be removed from Newington Priory and placed in Henry's custody as a hostage to ensure the House of de Shera's behavior, Chad races to Newington to remove his distant cousin from the abbey before Henry can take her hostage.
Chad's noble act to save Alessandria de Shera is the first step in a chain of events that sees the House of de Lohr increasingly turned against the king. He is determined to take Alessandria back to Isenhall Castle where she can be protected by her warlord cousins, the LORDS OF THUNDER. It's a race through England as Chad and Alessandria stay one step ahead of Henry's men in their pursuit to capture her. Aided by Davyss de Winter (LESPADA) and Bose de Moray (THE GORGON), the journey of Chad and Alessandria takes them through life, death, great hardship, fear, and finally passion beyond what they could have ever imagined.
It's an eventual showdown between Henry, the LORDS OF THUNDER, and the House of de Lohr once Chad and Alessandria are cornered. Will Henry finally claim Alessandria? Or will Chad ultimately defeat the man he once saved?
SILVERSWORD EXCERPT:
Someone had her by the hair, pulling her up into the smoky air above. As Alessandria sputtered and tried to beat away the hand that held her, she could hear Chad's voice.
"Are you well?" he demanded. "My lady, are you injured?"
He sounded panicked. Alessandria's eyes were closed, water rushing in her face. "I am fine!" she shouted. "Let go of my hair!"
Chad instantly let her go and rushed to help Rhun and Jorden, who were pulling the burning blanket away from the door so Liselotte and Veronica could escape. He ushered his mother and sister out of the room, quickly, pausing in the chamber beyond to grab a bucket that was usually used to piss in. It still had some urine in the bottom of it but he didn't give it a second thought. Rushing back into the chamber where thick, black smoke was gathering near the ceiling, he went straight to the copper tub where Alessandria was trying desperately to cover herself.
"Sorry, my lady."
He said it swiftly, apologetically, as he dunked the bucket into her bathwater. Alessandria shrieked as some of the urine backwashed into the tub, watching as he threw the bucket onto the burning blanket, creating clouds of white steam and smoke from the doused flame. Meanwhile, Jorden had grasped a second bucket and also apologized to Alessandria before dropping the bucket into the bath and tossing the water onto the burning blanket.
Horrified at the fact that her bathwater was being depleted and her nakedness would be all that more apparent, Alessandria looked around with desperation for something to cover herself with and spied the stack of garments that Veronica had brought for her to wear. The closest thing she could get her hands on was red, and silk, and she yanked it into the tub with her, trying to cover herself up from the eyes of the men in the room.
But the attempt to cover herself made the situation go from bad to worse. With the water splashing and the silk in the tub with her, the dye ran and began to turn the water pink. Holding it up against her body as she was, the dye also ran onto her skin, turning it a lovely, blotchy shade of red.
Unfortunately, Alessandria didn't notice any of this right away; she was too involved watching the knights put out the fire on the blanket. More than half of it had burned. Jorden and Rhun were still stamping on it as Chad stood over them, watching the situation with a critical eye.
"For a blanket, that put out a hell of a lot of smoke," he muttered. "I thought the entire building was burning down."
Jorden stamped down the last of the embers, coughing as he did so. He tried to keep his head down, out of the fog of smoke overhead. "We are fortunate the entire room isn't ablaze," he said. "I have seen that before."
"So have I," Rhun said, pulling at the blanket to make sure all of the embers were out. "I have seen something smaller than this burn out entire keeps."
Chad, too, coughed as the smoke swirled around his head. "Now to get the smoke out of here," he said. Then, he turned to Alessandria. "My lady, if you…."
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Kathryn's Medieval Romance novels have been called 'detailed', 'highly romantic', and 'character-rich'. She crafts great adventures of love, battles, passion, and romance in the High Middle Ages. More than that, she writes for both women AND men – an unusual crossover for a romance author – and Kathryn has many male readers who enjoy her stories because of the male perspective, the action, and the adventure.
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text:
\section{Introduction}

The $n$-th hyperharmonic number of order $r$, $H_{n}^{(r)}$, is defined recursively as
\begin{equation*}
H_{n}^{(r)} = \sum_{k=1}^{n} H_{k}^{(r-1)} \,\,\text{and}\,\, H_{n}^{(1)} = H_n,
\end{equation*}
where $H_n$ is the ordinary harmonic number $1 + \frac{1}{2} + \cdots + \frac{1}{n}$. Here it is understood that $H_{n}^{(0)}= \frac{1}{n}$ for $n \geq 1$, and $H_0^{(r)} =0$ for $r \geq 0$. In 1996, Conway and Guy \cite[p. 258]{guy} provided the following identity
\begin{equation}\label{guy}
H_{n}^{(r)} = \binom{n+r-1}{r-1} \big( H_{n+r-1} - H_{r-1} \big),
\end{equation}
connecting the hyperharmonic numbers with the harmonic numbers. There exist various proofs of formula \eqref{guy} in the literature; see, e.g., \cite{benjamin,cere2} and \cite[pp. 227--229]{mezo}.

In this paper, we obtain a new formula for the sums of powers of the first $n$ positive integers, $S_k(n) = 1^k + 2^k + \cdots + n^k$, in terms of hyperharmonic numbers and Stirling numbers of the second kind $\genfrac{\{}{\}}{0pt}{}{k}{j}$. Specifically, in Section 2, we establish the following theorem.

\begin{theorem}\label{th:1}
For any integers $k \geq 0$ and $n \geq 1$, we have
\begin{equation}\label{th1}
S_{k}(n) = \frac{(-1)^{k+1}}{k+1} \sum_{j=1}^{k+1} (-1)^{j} j! \genfrac{\{}{\}}{0pt}{}{k+1}{j} \left( H_{j+1}^{(n)} -
\frac{1}{j+1} \right).
\end{equation}
\end{theorem}

As will become clear below, $H_{j+1}^{(n)}$ is a polynomial in $n$ of degree $j$. Then, in Section 3, using the explicit representation for the hyperharmonic numbers given in \eqref{frac}, we generalize formula \eqref{th1} to the sums of powers of the terms of an arithmetic progression with first term $r$ and common difference $m$
\begin{equation}\label{prog}
S_{k}^{r,m}(n) = \sum_{j=1}^{n} ( r + (j-1) m )^k,
\end{equation}
where $m$ and $r$ are assumed to be integer variables with $m \geq 1$ and $r \geq 0$. Furthermore, in Section 4, we express the Bernoulli polynomials $B_k(x)$ in terms of the hyperharmonic polynomials $\mathcal{H}_j(x):= H_{j+1}^{(x-1)}$, $j =0,1,\ldots,k$, and the Stirling numbers of the second kind (see equation \eqref{con2}). Finally, in Section 5, we extend the above formula in equation \eqref{th1} to negative values of $n$ by making use of the definition of hyperharmonic numbers of negative order set forth by Dil and Muniro\u{g}lu in \cite{dil3}.

\section{Proof of Theorem \ref{th:1}}

Next, we detail the proof of Theorem \ref{th:1}.

\begin{proof}
We start with the following polynomial formula for $S_k(n)$; see, e.g., \cite[Equation (7.5)]{gould2}:
\begin{equation}\label{cere}
S_k(n) = \sum_{j=1}^{k} a_{k,j} \binom{n+j}{j+1},  \quad k \geq 1,
\end{equation}
where the coefficients $a_{k,j}$ are given by
\begin{equation}\label{coeff}
a_{k,j} = (-1)^{k-j} j! \genfrac{\{}{\}}{0pt}{}{k}{j}.
\end{equation}

For convenience, we can think of $n$ as being a continuous variable. This is justified by the fact that, as is well-known, for fixed $k$ there is exactly one polynomial $S_k(x)$ in $x$ such that $S_k(x) = 1^k + 2^k + \cdots + x^k$ whenever $x$ is a positive integer (see, e.g., \cite[Theorem 1]{levy} and \cite{owens}). Keeping this in mind, we invoke the following elementary result according to which (see, e.g., \cite{owens,sher,wu}):
\begin{equation}\label{deriv}
S_{k}^{\prime}(n) = \frac{\text{d}S_k(n)}{\text{d}n} = k S_{k-1}(n) + (-1)^k B_k, \quad k \geq 1,
\end{equation}
where the $B_k$'s are the Bernoulli numbers \cite{apostol}. Thus, recalling the rule for the derivative of a product of functions $h_1(x), h_2(x),
\ldots, h_j(x)$,
\begin{equation*}
\frac{\text{d}}{\text{d}x} \left( \prod_{i=1}^{j} h_i(x) \right) = \left( \prod_{i=1}^{j} h_i(x) \right) \left( \sum_{i=1}^{j}
\frac{h_{i}^{\prime}(x)}{h_i(x)} \right),
\end{equation*}
and differentiating each side of equation \eqref{cere} with respect to $n$, we obtain
\begin{align*}
S_{k}^{\prime}(n) & = \frac{\text{d}}{\text{d}n} \left( \sum_{j=1}^{k} a_{k,j}\binom{n+j}{j+1} \right)  \\
& = \sum_{j=1}^{k} \frac{a_{k,j}}{(j+1)!} \frac{\text{d}}{\text{d}n} \left( \prod_{i=0}^{j} (n+i) \right)  \\
& = \sum_{j=1}^{k} \frac{a_{k,j}}{(j+1)!} \left( \prod_{i=0}^{j} (n+i) \right) \sum_{i=0}^{j} \frac{1}{n+i} \\
& = \sum_{j=1}^{k} a_{k,j} \binom{n+j}{j+1} \big( H_{n+j} - H_{n-1} \big).
\end{align*}
By virtue of identity \eqref{guy}, the last equation reduces to
\begin{equation}\label{pr2}
S_{k}^{\prime}(n) = \sum_{j=1}^{k} a_{k,j} H_{j+1}^{(n)}, \quad k \geq 1.
\end{equation}
Now, combining equations \eqref{pr2}, \eqref{deriv}, and \eqref{coeff}, and renaming the index $k$ as $k+1$, yields
\begin{equation}\label{pr3}
S_{k}(n) = \frac{(-1)^{k+1}}{k+1} \left( \sum_{j=1}^{k+1} (-1)^{j} j! \genfrac{\{}{\}}{0pt}{}{k+1}{j} H_{j+1}^{(n)} -
B_{k+1} \right), \quad k \geq 0.
\end{equation}
On the other hand, since $S_{k}(0) =0$, and noting that $H_{j+1}^{(0)} = \frac{1}{j+1}$, from \eqref{pr3} we deduce that
\begin{equation}\label{pr4}
B_{k+1} =  \sum_{j=1}^{k+1} (-1)^j \frac{j!}{j+1} \genfrac{\{}{\}}{0pt}{}{k+1}{j}.
\end{equation}
Therefore, from \eqref{pr4} and \eqref{pr3}, we finally get \eqref{th1}.
\end{proof}

\begin{remark}
Let $D_x$ be the derivative operator with respect to $x$, i.e., $D_x f(x) = \frac{\text{d}}{\text{d}x} f(x)$. The above proof of Theorem \ref{th:1} involves essentially an application of the formula giving the hyperharmonic number $H_n^{(r)}$ as the derivative of a binomial coefficient, namely (see \cite[Section 3]{cere2} and \cite[Proposition 11]{dil3})
\begin{equation*}
D_x \left. \binom{x+n+r-1}{n} \right|_{x=0} = H_n^{(r)},
\end{equation*}
which is in turn a generalization of the equation \cite[Equation (8)]{paule} (see also \cite[Equation (Z.60)]{gould})
\begin{equation*}
D_x \left. \binom{x+n}{n} \right|_{x=0} = H_n.
\end{equation*}
\end{remark}

\begin{remark}
Equation \eqref{pr4} is a well-known property of the Bernoulli numbers (see, e.g., \cite{qi}).
\end{remark}

\begin{remark}
Letting $n =1$ in equation \eqref{pr3} yields the identity
\begin{equation*}
B_k = (-1)^{k+1} k + \sum_{j=0}^{k} (-1)^j j! \genfrac{\{}{\}}{0pt}{}{k}{j} H_{j+1}, \quad k \geq 0.
\end{equation*}
\end{remark}

As a simple example illustrating Theorem \ref{th:1}, we may use equation \eqref{th1} to calculate $S_3(n)$.
\begin{example}
For $k =3$, equation \eqref{th1} reads as
\begin{align}\label{exam}
S_3(n) & = \frac{1}{4} \sum_{j=1}^{4} (-1)^j j! \genfrac{\{}{\}}{0pt}{}{4}{j} \left( H_{j+1}^{(n)} - \frac{1}{j+1} \right)  \notag \\
& = 6 H_{5}^{(n)} - 9 H_{4}^{(n)} + \frac{7}{2} H_{3}^{(n)} - \frac{1}{4} H_{2}^{(n)} + \frac{1}{120}.
\end{align}
In order to evaluate the involved hyperharmonic numbers $H_{j}^{(n)}$, it is useful to employ the following explicit formula derived in \cite[Theorem 1]{benjamin} and, additionally, in \cite[Theorem 5]{dil}
\begin{equation}\label{frac}
H_{j}^{(n)} = \sum_{t=1}^{j} \binom{n+j-t-1}{j-t} \frac{1}{t}, \quad n,j \geq 1,
\end{equation}
which gives $H_{j}^{(n)}$ as a weighted sum of the fractions $\frac{1}{1}, \frac{1}{2}, \ldots, \frac{1}{j}$. From \eqref{frac}, it is easily seen that $H_{j}^{(n)}$ is a polynomial in $n$ of degree $j-1$ with leading coefficient $\frac{1}{(j-1)!}$ and constant term $\frac{1}{j}$. Applying \eqref{frac}, we obtain
\begin{align*}
H_{2}^{(n)} & = n + \tfrac{1}{2},  \\
H_{3}^{(n)} & =\tfrac{1}{2}n^2 + n + \tfrac{1}{3},  \\
H_{4}^{(n)} & = \tfrac{1}{6}n^3 + \tfrac{3}{4}n^2 + \tfrac{11}{12}n + \tfrac{1}{4}, \\
H_{5}^{(n)} & = \tfrac{1}{24}n^4 +  \tfrac{1}{3}n^3 + \tfrac{7}{8}n^2 + \tfrac{5}{6}n + \tfrac{1}{5}.
\end{align*}
Substituting these expressions into \eqref{exam} and simplifying, we find that, as expected, $S_3(n) = \tfrac{1}{4} n^2 (n+1)^2$.
\end{example}

\begin{remark}
By using equation \eqref{frac} into \eqref{th1}, we can equivalently express $S_k(n)$ as a weighted sum of $\frac{1}{1}, \frac{1}{2}, \ldots,
\frac{1}{k+1}$ as follows
\begin{equation}\label{rem1}
S_k(n) = \frac{(-1)^{k+1}}{k+1} \sum_{t=1}^{k+1} V_{k,t}(n) \frac{1}{t},  \quad k \geq 0,
\end{equation}
where
\begin{equation}\label{rem2}
V_{k,t}(n) = \sum_{i=t}^{k+1} (-1)^i i! \genfrac{\{}{\}}{0pt}{}{k+1}{i} \binom{n+i-t}{i+1-t}.
\end{equation}
\end{remark}

\begin{remark}
As shown in \cite[Theorem 2]{benjamin}, the hyperharmonic numbers, $H_{n}^{(r)}$, and the $r$-Stirling numbers of the first kind,
$\genfrac{[}{]}{0pt}{}{n}{k}_r$, are related by
\begin{equation*}\label{benj}
H_{n}^{(r)} = \frac{1}{n!} \genfrac{[}{]}{0pt}{}{n+r}{r+1}_r.
\end{equation*}
Therefore, from Theorem \ref{th:1}, we can alternatively write $S_k(n)$ in the form
\begin{equation}\label{r1}
S_{k}(n) = \frac{(-1)^{k+1}}{k+1} \sum_{j=1}^{k+1} \frac{(-1)^{j}}{j+1} \genfrac{\{}{\}}{0pt}{}{k+1}{j}
\left( \genfrac{[}{]}{0pt}{}{n+j+1}{n+1}_n -  j! \right).
\end{equation}
This formula is to be complemented by the following one
\begin{equation}\label{r2}
\genfrac{[}{]}{0pt}{}{n+j+1}{n+1}_n = \sum_{i=1}^{j+1} (i-1)! \binom{j+1}{i} n^{\overline{j+1-i}},
\end{equation}
expressing $\genfrac{[}{]}{0pt}{}{n+j+1}{n+1}_n$ in terms of the rising factorials $n^{\overline{j+1-i}}$, $i =1,2,\ldots,j+1$. Hence,
substituting \eqref{r2} into \eqref{r1}, we get
\begin{equation*}
S_{k}(n) = \frac{(-1)^{k+1}}{k+1} \sum_{j=1}^{k+1} \frac{(-1)^{j}}{j+1} \genfrac{\{}{\}}{0pt}{}{k+1}{j}
 \sum_{i=1}^{j} (i-1)! \binom{j+1}{i} n^{\overline{j+1-i}}.
\end{equation*}
\end{remark}

\section{Generalization of Theorem \ref{th:1}}

Next, using the representation for the hyperharmonic numbers given in \eqref{frac}, we generalize Theorem \ref{th:1} to the arithmetic progression defined in \eqref{prog}.

\begin{theorem}
For any integers $k \geq 0$ and $n \geq 1$, we have
\begin{equation}\label{th2}
S_{k}^{r,m}(n) = (-1)^{k+1} \frac{m^k}{k+1} \sum_{t=1}^{k+1} \left( V_{k,t} \left(n-1+\frac{r}{m}\right) -
V_{k,t} \left(\frac{r}{m}-1 \right) \right) \frac{1}{t},
\end{equation}
where $V_{k,t}(n)$ is the polynomial given in \eqref{rem2}.
\end{theorem}
\begin{proof}
This follows in a rather straightforward way from the following simple but powerful result derived in \cite{griff}. Let $S_k(x)$ denote the unique polynomial in $x$ such that, for all $n \geq 1$, $S_k(n)$ gives us the sum of powers of the first $n$ positive integers (with $S_k(0) =0$). Then,
for any real number $x$, it turns out that \cite{griff}
\begin{equation}\label{th3}
\sum_{j=1}^{n} (j +x)^k = S_k(n+x) - S_k(x).
\end{equation}
Taking $x = \frac{r}{m} -1$ in \eqref{th3} yields
\begin{equation*}
S_{k}^{r,m}(n) = m^k \left( S_{k} \left(n-1+\frac{r}{m}\right) - S_{k}\left(\frac{r}{m}-1 \right) \right).
\end{equation*}
Hence, using the polynomial formula for $S_k(n)$ given in \eqref{rem1}, we get \eqref{th2}.
\end{proof}

\begin{remark}
Equation \eqref{th2} reduces to \eqref{rem1} when $r =m=1$.
\end{remark}

\section{Bernoulli polynomials}

In this section we derive an expression for the Bernoulli polynomials $B_k(x)$ involving the hyperharmonic polynomials $\mathcal{H}_j(x):= H_{j+1}^{(x-1)}$, $j =0,1,\ldots,k$, and the Stirling numbers of the second kind. Starting from the well-known relationship between $S_k(n)$ and $B_k(n)$, namely
\begin{equation*}
S_k (n) = \frac{1}{k+1} \big( B_{k+1}(n+1) - B_{k+1} \big), \quad k \geq 1,
\end{equation*}
it follows that $B_{k+1}^{\prime}(n+1) = (k+1) S_{k}^{\prime}(n)$. On the other hand, we have that \cite{apostol} $B_{k+1}^
{\prime}(n+1) = (k+1)B_{k}(n+1)$. Therefore, from \eqref{pr2}, we obtain that
\begin{equation}\label{con1}
B_{k}(n+1) = \sum_{j=0}^{k} a_{k,j} H_{j+1}^{(n)}, \quad k \geq 0.
\end{equation}
Clearly, the right-hand side of \eqref{con1} is a polynomial in $n$ of degree $k$. Hence, using equations \eqref{coeff}, \eqref{frac}, and \eqref{con1}, one can naturally extend $B_{k}(n+1)$ to a polynomial $B_k(x)$ in which $x$ takes any real value as follows
\begin{equation}\label{con2}
B_k(x) = \sum_{j=0}^{k} (-1)^{k-j} j! \genfrac{\{}{\}}{0pt}{}{k}{j} \mathcal{H}_{j}(x),  \quad k \geq 0,
\end{equation}
where
\begin{equation*}
\mathcal{H}_{j}(x):= H_{j+1}^{(x-1)} = \frac{1}{j} \left( x -  \frac{1}{j+1} \right) + \sum_{t=1}^{j-1}
\binom{x+j-t-1}{j+1-t} \frac{1}{t}, \quad j \geq 2,
\end{equation*}
with $\mathcal{H}_{0}(x) =1$ and $\mathcal{H}_{1}(x):= H_{2}^{(x-1)} = x - \frac{1}{2}$.

Note that, since $\mathcal{H}_{j}(0) = -\frac{1}{j(j+1)}$ for all $j \geq 1$, it follows from \eqref{con2} that
\begin{equation*}
B_k = (-1)^{k+1} \sum_{j=1}^{k} (-1)^j \frac{(j-1)!}{j+1}\genfrac{\{}{\}}{0pt}{}{k}{j}, \quad k \geq 1,
\end{equation*}
which is a variant of the identity in \eqref{pr4}. This formula for the Bernoulli numbers has recently been derived in \cite[Equation (3)]{jha}. On the other hand, using \eqref{con1} and the difference equation \cite{apostol}, $B_{k}(x+1) - B_{k}(x) = k x^{k-1}$, we obtain the following alternative
formula for $B_k(x)$:
\begin{equation*}\label{kx}
B_k(x) = \sum_{j=0}^{k} (-1)^{k-j} j! \genfrac{\{}{\}}{0pt}{}{k}{j} H_{j+1}^{(x)} - k x^{k-1},  \quad k \geq 0,
\end{equation*}
where
\begin{equation*}
H_{j+1}^{(x)} = \frac{1}{j+1} + \sum_{t=1}^{j} \binom{x+j-t}{j+1-t}\frac{1}{t}, \quad j \geq 1,
\end{equation*}
and $H_{1}^{(x)} =1$.

Let us further note that we can reverse \eqref{con2} to obtain
\begin{equation}\label{conv}
\mathcal{H}_k(x) = \frac{1}{k!} \sum_{j=0}^{k} \genfrac{[}{]}{0pt}{}{k}{j} B_j(x), \quad k \geq 0,
\end{equation}
where $\genfrac{[}{]}{0pt}{}{k}{j}$ are the (unsigned) Stirling numbers of the first kind. Incidentally, setting $x=0$ in \eqref{conv} allows
us to deduce the following recursive formula for the Bernoulli numbers:
\begin{equation*}
\sum_{j=1}^{k} \genfrac{[}{]}{0pt}{}{k}{j} B_j = - \frac{(k-1)!}{k+1}, \quad k \geq 1.
\end{equation*}
A proof of this last identity using the Riordan array method can be found in \cite[p. 288]{sprugnoli}.

We end this section with the following important observation.

\begin{remark}
The $j$-th degree polynomials $\mathcal{H}_j(x):= H_{j+1}^{(x-1)}$ introduced in this paper are closely related to the so-called {\it harmonic polynomials\/} $H_j(x)$ of degree $j$ in $x$ defined in \cite[Equation (28)]{cheon} by the ordinary generating function
\begin{equation*}
\frac{-\ln (1-t)}{t (1-t)^{1-x}} = \sum_{j=0}^{\infty} H_j(x) t^j,
\end{equation*}
where $H_j(0) = H_{j+1}$. Indeed, it turns out that
\begin{align*}
H_j(x) & =  H_{j+1}^{(1-x)} \quad \text{and}\quad H_j(x) = \mathcal{H}_j(2-x),
\intertext{or, conversely,}
H_{j+1}^{(x)} & =  H_j(1-x) \quad \text{and}\quad \mathcal{H}_j(x) = H_j(2-x).
\end{align*}
The harmonic polynomials $H_j(x)$, $j \geq 0$, have, in particular, the explicit representation (see \cite[Theorem 5.4]{cheon})
\begin{equation*}
H_j(x) = \sum_{t=1}^{j+1} \binom{j+1-t-x}{j+1-t} \frac{1}{t},
\end{equation*}
which can be recovered by letting $j \to j+1$ and $n \to 1-x$ in \eqref{frac}.
\end{remark}

\section{Extension of formula \eqref{th1} to negative values of $n$}

In \cite{mezo2}, Mez\H{o} defined the hyperharmonic function $H_z^{(w)}$ involving the Pochhammer symbol $(z)_w$, gamma $\Gamma(w)$ and digamma $\Psi(w)$ functions, as
\begin{equation*}
H_z^{(w)} = \frac{(z)_w}{z \Gamma(w)} \big( \Psi(z +w) - \Psi(w) \big),
\end{equation*}
where $w, z+w \in \mathbb{C} \setminus \mathbb{Z}^{-}$, and $\mathbb{Z}^{-} = \{0, -1, -2, \ldots\, \}$. Based on the hyperharmonic function, Dil \cite{dil2} presented formulas to calculate special values of $H_z^{(w)}$ subjected to the above restriction of $w, z+w \in \mathbb{C} \setminus \mathbb{Z}^{-}$. Subsequently, Dil and Muniro\u{g}lu \cite{dil3} showed a way to define ``negative-ordered hyperharmonic numbers''. According to \cite[Definition 25]{dil3}, for positive integers $n$ and $r$, the hyperharmonic number of negative order $H_n^{(-r)}$ can be defined by
\begin{equation}\label{def}
H_n^{(-r)} =\left\{
              \begin{array}{ll}
                \dfrac{(-1)^r r!}{n^{\underline{r+1}}}, & n >r \geq 1; \\[4mm]
                \displaystyle\sum_{i=0}^{n-1} (-1)^i \binom{r}{i} \dfrac{1}{n-i}, & r \geq n >1; \\[5mm]
                1, & n=1.
              \end{array}
            \right.
\end{equation}
Furthermore, as noted in \cite{dil3}, the identity (see \cite[Equation (1.43)]{gould2})
\begin{equation*}
\sum_{i=0}^{r} (-1)^i \binom{r}{i} \frac{1}{n-i} = \frac{(-1)^r}{(n-r)\binom{n}{r}},
\end{equation*}
ensures the consistency of the definition in \eqref{def}.

Therefore, for $n \geq 1$, we can use \eqref{def} to define $H_{j+1}^{(-n)}$ as follows
\begin{equation}\label{def2}
H_{j+1}^{(-n)} =
\left\{
  \begin{array}{ll}
   \displaystyle\sum_{i=0}^{j} \frac{(-1)^i}{j+1-i} \binom{n}{i}, &  j \geq 1 ; \\
    1, & j=0,
  \end{array}
\right.
\end{equation}
so that the extension of formula \eqref{th1} to negative values of $n$ can effectively be stated as
\begin{equation}\label{ext}
S_{k}(-n) = \frac{(-1)^{k+1}}{k+1} \sum_{j=1}^{k+1} (-1)^{j} j! \genfrac{\{}{\}}{0pt}{}{k+1}{j}
\sum_{i=1}^{j} \frac{(-1)^i}{j+1-i} \binom{n}{i},
\end{equation}
for $k \geq 0$ and $n \geq 1$.

\begin{remark}
It is a well-known fact that $S_1(n)$ is a factor of $S_k(n)$ for all $k \geq 1$, which means that $S_k(-1) =0$ for all $k\geq 1$. We can check from \eqref{ext} that the latter holds true. Indeed, setting $n=1$ in \eqref{ext} gives
\begin{equation*}
S_k(-1) = \frac{(-1)^{k}}{k+1} \sum_{j=1}^{k+1} (-1)^{j} (j-1)! \genfrac{\{}{\}}{0pt}{}{k+1}{j},
\end{equation*}
which is identically equal to zero for $k \geq 1$, according to the identity (A.17) in \cite{boya}.
\end{remark}

\begin{remark}
As we saw in the preceding section the harmonic and hyperharmonic polynomials are related by $H_j(x) = H_{j+1}^{(1-x)}$ or, $H_j(x) = H_{j+1}^{(-(x-1))}$. Since the definition given in \eqref{def2} is valid for any $n \geq 1$, we can licitly use \eqref{def2} to obtain the following representation of the harmonic polynomials introduced in \cite[Section 5]{cheon}
\begin{equation*}
H_j(x) = \sum_{i=0}^{j} \frac{(-1)^i}{j+1-i} \binom{x-1}{i}, \quad j \geq 1,
\end{equation*}
and $H_0(x) =1$. In particular, since $\binom{-1}{i} = (-1)^i$, the last formula yields, as it should be, $H_j(0) = H_{j+1}$.
\end{remark}

To close this paper, it is worth noting the symmetry property of the power sum polynomials $S_k(n)$, namely \cite{newsome}
\begin{equation}\label{sym}
S_k( -(n+1)) = (-1)^{k+1} S_k(n), \quad k \geq 1.
\end{equation}
Thus, using \eqref{th1}, \eqref{ext}, and \eqref{sym}, we can express $S_k(n)$ in the alternative form
\begin{equation*}
S_k(n) = \frac{1}{k+1} \sum_{j=1}^{k+1} (-1)^{j} j! \genfrac{\{}{\}}{0pt}{}{k+1}{j}
\sum_{i=1}^{j} \frac{(-1)^i}{j+1-i} \binom{n+1}{i},
\end{equation*}
for $k \geq 1$ and $n \geq 0$.

\section*{Acknowledgements}

The author is grateful to the anonymous referees for their valuable comments and suggestions which led to improvements to this paper. He would also like to thank one of the referees for providing some pertinent references.

\vspace{2mm}

redpajama_set_name: RedPajamaArXiv


Bohemian Nights at NewWestFest 2019 dates are August 9, 10 and 11. The 2019 festival lineup will be announced in early June, 2019.
Bohemian Nights is always looking for new, emerging and established Colorado artists and accepts submissions year-round.
Bohemian Nights at NewWestFest is a free, three-day music festival held each August in historic downtown Fort Collins. The 2019 festival will take place Aug. 9, 10 and 11. The 2019 festival lineup will be announced in early June, 2019. In addition to a large Colorado lineup, three renowned headliners anchor main stage performances each night.


text:
Spooky Vacation Destinations
With the chill of October comes Halloween, a day when indulgence in darker themes and paranormal activity is expected! Though Classic Journeys doesn't host any ghost-specific tours, we do pass through locations rich in spooky history. For those who like a little scare while on vacation, we've rounded up five destinations visited on our various tours, which are guaranteed to make your hair stand on-end just in time for Halloween.
Our Cotswolds tour rattles through foggy London, where, if you're brave enough, you'll have the opportunity to visit the famed Tower of London. The site of many grisly events throughout history, the Tower of London has served many purposes since its construction in 1078. Many were imprisoned, tortured or killed there, such as Anne Boleyn, Guy Fawkes and others seen as a threat to the British crown. Nowadays, visitors can tour its halls, including a torture chamber outfitted with a score of horrifying devices. And if you feel a sudden cold rush of air, that might just be Anne Boleyn's ghost– said to have roamed the halls for the last 481 years, seeking retribution following false accusations of treason and adultery, which resulted in her beheading on these hallowed grounds.
On our Charleston & Savannah Culture + Walking Tour creeps-and-crawls into Savannah in the week's second half. The city lays claim to being one of the most haunted in the United States. As the setting for the hit novel and movie Midnight in the Garden of Good and Evil, Savannah is positively rife with ghoulish locations, including the Mercer Williams House – site of the novel's murder. There is also the 17-Hundred-90 Inn, where Anna Powers leaped to her death from the third-floor balcony. Her ghost is said to still occupy room 208. Also popular are the Bonaventure Cemetery and The Olde Pink House, which is said to be haunted by the ghost of its builder James Habersham.
A stop on our Edinburgh & Scottish Highlands Culture + Walking Tour is the Edinburgh Castle, thought to be one of the most ghost-filled spots in all of Scotland. Over the years, various phantasmal sightings have been reported, including: French prisoners from the Seven Years War, a dog patrolling the castle's cemetery, a headless drummer, a phantom piper and prisoners who fought during the American Revolutionary War. Over the years, several professional paranormal studies have been done at the castle, with surprising results that support overwhelming paranormal activity within its walls.
In Nova Scotia, its far-flung, water-bound and windswept location makes it fertile ground for some of Canada's scariest stories. Some of the many legends swirling around the island include a gravestone that gives onlookers the symptoms of typhoid; an ever-burning ghost ship in Northumberland Strait; a grey nun at one of the hospitals, who comforts dying patients in their final hours; a haunted brewery; three ghosts who inhabit the 5 Fishermen restaurant; a ghoulish girl who follows tours on Citadel Hill; and a building in Halifax that is besieged by endless sounds of a baby's cries.
With its connection to Mexican culture, Santa Fe has a longstanding Day of the Dead tradition. Called Dia de los Muertos in Spanish, the holiday, which takes place from November 1-2, focuses on remembering loved ones who have passed. This is done through altars, songs, special foods, gravesite visits and other rituals to honor the dearly departed. As a world-renowned capital of art, Santa Fe boasts plenty of Day-of-the-Dead-inspired installations and pieces, including catrina statuettes and colorful sugar skulls. In particular, the Museum of International Folk Art hosts activities celebrating the day, and sets up a public altar laced in the holiday's signature flower: marigolds.


text:
Janelle Lynch
River 3
Impressió digital sobre paper fotogràfic sobre aluminio
Ed.3
Impresión digital sobre papel fotográfico montada en aluminio
126x157,5cm
Digital C-Print on aluminium
"FOSA COMÚN 5"
Digital C-Print
Fosa Común 8
"Serie Riu Llobregat"
Foto impresion lambda
CV - Biography:
Master Class in Photography, Bard College, Annandale-on-Hudson, NY, 2003
M.F.A., Photography and Related Media, School of Visual Arts, New York, NY, 1999
B.A. Liberal Arts, The New School, New York, NY 1995
Robert Morat Galerie, Berlin, Germany
Newark Museum, Newark, NJ
Photographic Resource Center, Boston
Radius, Santa Fe, New Mexico
Museo Archvio de la Fotografía, Mexico City, Mexico
Galería 3 Punts, Barcelona, Spain (catalog)
Museo de Arte Contemporaneo Salta, Argentina
Photofusion, London, England
Centro Cultural Banco de Brasil, Brasília, Brazil (catalog)
RO Galería de Arte, Buenos Aires, Argentina (catalog)
Gallery 44, Toronto, Canada
Furman University, Greenville, SC
University of California Berkeley, San Francisco Extension, San Francisco, CA
"Una inclinación Natural" 3 Punts Galeria, Barcelona, Spain
Wave Hill, Bronx, NY (catalog)
Book Award Exhibition, Recontre d'Arles, Arles, France
Book and Magazine Show, Photo Ireland, Dublin
Make Art Public, Montreal, Canada
Espaço Cultural Contemporâneo, Brasília, Brazil
Burchfield-Penney Museum, Buffalo, NY
Hudson River Museum, Yonkers, NY
Jersey City Museum, Jersey City, NJ
George Eastman House Museum, Rochester, NY
Festival International de Mode et de Photographie à Hyères, Hyères, France (catalog)
Centro Cultural Correios, Río de Janeiro, Brazil
Municipal Museum of Contemporary Art (Descubrimientos), Photo España, Madrid, Spain
PS 122 Gallery, New York, NY
Gallery TPW, Toronto, ON
The Center for Photography at Woodstock, Woodstock, NY
C. Grimaldis Gallery, Baltimore, MD
Rockefeller Arts Center, State University of NY at Fredonia, Fredonia, NY
Momenta Art, Brooklyn, NY
Pelham Art Center, Pelham, NY
Molloy College Art Gallery, Long Island, NY
Delaware Center for Contemporary Art, Wilmington, DE
Rochester Art Center, Rochester, MN
Herndon Gallery, Antioch College, Yellow Springs, OH
Weeks Gallery, Jamestown Community College (SUNY), Jamestown, NY
Visual Arts Gallery, New York, NY
Chautauqua Center for the Visual Arts, Chautauqua Institution, NY
Brooklyn Museum, Brooklyn, NY
Centro Cultural Banco de Brasil, Brasília, Brazil
Fundación Vilacasas, Barcelona, Spain
Museo de Arte Contemporáneo Salta, Argentina
Museum of Photographic Arts, SanDiego, CA
New-York Historical Society, New York, NY
New York Public Library, New York, NY
Villa Noailles, Hyères, France
GRANTS/HONORS
Kodak, Rochester, NY, 2011.
Commission, Wave Hill, Bronx, NY, 2011-2012.
Nominee, Prix Pictet, 2010.
Guest Artist, Facultad de Bellas Artes, Universidad de Barcelona, Barcelona, May 2009.
Finalist, Santa Fe Prize for Photography, March 2009.
Guest Artist, Centre de la Imatge i Technologia Multimedia, UPC, Barcelona, February 2009.
Visiting Artist, George Eastman House Museum, Rochester, NY, October 2007.
Air France, Paris, 2007.
Visiting Artist, Antioch College, Yellow Springs, OH, October 2000.
River is a series of waterscapes that explores themes of impermanence and loss through historical urban architecture. The images were made along the Hudson River in Manhattan between Canal and 65th Streets in areas that were part of New York's once-vital shipping industry and railroad transportation system. They contain remnants of the deteriorated maritime piers, piles that supported the piers, original railroad structures, as well as recent constructions that are part of the new Hudson River Park. Across the river, at the horizon line in some of the photographs, other historical structures can be seen in New Jersey such as the Erie Lackawanna Railroad terminal in Hoboken and the Colgate Clock in Jersey City. The architectural elements in the photographs conjure history of more than a century ago. A vast cultural shift is implied and the suggestion of evolution and change is imparted. The river and the architecture on its shores have been subjects in art and literature for nearly two centuries. In the mid-1850s, photography took the place of artist-produced prints of river scenery as well as natural and cultural landmarks. Some photographers followed the practice of their contemporaries and made images of romantic scenes while a greater movement happened toward documentary and action-based imagery. River continues this tradition while integrating a personal layer of meaning. Though the work examines impermanence through historical urban architecture, it is inspired by experiences related to mortality and individual loss. Related bodies of work include Harris Hill Road (1999-2001), which explores these themes through seasonal changes in the landscape of Western New York, where I was raised; and House/Home (2001), a series of photographs also made in Western New York which examines similar subjects through domestic interiors and architectural details. River expands these investigations.
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Extracts, transforms, and loads your Facebook data from the Download Your Information tool into an SQLite DB.
Use Facebook's Download Your Information tool using the format JSON and unzip the file.
Developed for Python 3. May work but not tested in Python 2.


text:
Home»Business»APGA blames fuel scarcity on corruption, urges Buhari to appoint substantive minister
APGA blames fuel scarcity on corruption, urges Buhari to appoint substantive minister
By NAN December 24, 2017 No Comments
Dr. Victor Oye, the National Chairman of the All Progressives Grand Alliance, has blamed the current fuel scarcity in the country on corruption and confusion in country's petroleum industry.
Oye said this in an interview with the News Agency of Nigeria in Awka while reacting to the situation on Saturday.
He said it was unfortunate that the cartel responsible for the hardship inflicted on Nigerians had not been defeated but were rather dictating the pace in the system.
Oye said there was the need for clear delineation of powers and functions of the offices of the Minister of State for Petroleum and that of the Group Managing Director of the Nigerian National Petroleum Cooperation.
Oye described the scarcity as a seasonal one when the poor suffered so that the rich could be richer, adding that it was beyond the control of filling station operators.
He said: "There are a lot of factors affecting fuel distribution in Nigeria and this bothers on sharp practices in the various agencies involved in the system like the Petroleum Products Pricing Regulatory Agency, Department of Petroleum Resources and the NNPC.
"We hear about corruption is responsible for the problem of fuel distribution in this country.
"What we have now is the consequence of the rot in the petroleum industry.
"When you keep pilfering a system, there will come a time when the entire system will feel the pressure.
"When we hear about billions of dollars about the size of our external reserve not being properly accounted for by NNPC, few persons are having money and and houses all over the world without questions about how they made this money.
"There is cartel in the oil industry.
"As we are groaning now, they are smiling to the bank.
"My definition of scarcity is the time of the year when the poor suffers for the rich to enjoy, that amount you pay on top of N145 per litre goes to the pockets of this cartel or cabals
"I am aware that Buhari said Dr. Ibe Kachikwu should end the scarcity as soon as possible, but it is not going to be possible.
"We need to know the powers available to the Minister of State for Petroleum and what powers are at the disposal of the NNPC GMD.
"The conflict between the two offices is accountable for what we are witnessing today."
The APGA chairman called on President Muhammadu Buhari to reduce the tasks in his hands by appointing a substantive Minister of Petroleum, who will supervise the ministry more closely with the capacity to act on policy issues.
Oye said: "The President cannot handle the position of Minister of Petroleum very close as he would have wished in the light of the numerous things at his hands.
"That position is sensitive.
"He needs to hand it over to a professional.
"Buhari is not getting younger.
"He just recovered from ill health and I am one of his prayer warriors so he should listen to people like us, he should unbundle himself and the system.
"Kachikwu is competent to do the job, but they are breathing down his neck, they are not allowing him to work because of reasons I cannot tell.
"The President needs political and will power to fight corruption.
"If you don't fight corruption, there is no way you can end fuel scarcity because some people make millions out of it.
"It is a seasonal business for them."
All Progressives Grand Alliance Dr Victor Oye Dr. Ibe Kachikwu Nigerian National Petroleum Corporation President Muhammadu Buhari
APC alleges orchestrated plot by PDP to unleash fake news to defame Tinubu, party


text:
2010 World Cup final: 12 years down the line
11 June 2022 – It was 12 years ago on this day when Spanish legend Andrés Iniesta scored the only goal of the match against the Netherlands in extra time to spearhead Spain to their maiden World Cup title at a packed FNB Stadium on 11 July 2010.
The South African Football Association (SAFA) remembers the momentous occasion and celebrates an event that is remembered the world over to this day.
If you ever want to see the power of sport, it was demonstrated on that day when South Africans and visitors from other parts of the globe descended on FNB Stadium to add their voices to the fantastic atmosphere that reveberated around the calabash. Many more partied in other parts of South Africa, on the continent, and indeed, around the globe on a day that remains etched in memory.
Iniesta's volleyed shot into the back of the net was the fitting cherry on the icing to a tournament that many describe as the best ever 12 years later.
SAFA President Dr Danny Jordaan said there were several milestones that were achieved during the 2010 World Cup tournament.
"People came together in unity, so credit goes to the fans for lifting the profile of the country," Dr Jordaan said. "The world still talks about the opening and closing ceremonies, Paul the Octopus [the octopus gained worldwide after predicting the results of international football matches, especially during the 2010 World Cup], the World Cup Kick-Off Celebration concert at the Orlando stadium, and of course, the vuvuzela. So it is a special day today and one of the standout days for SAFA."
Ghana striker Asamoah Gyan still remembers the tournament fondly after reaching the quarter-finals, eventually losing to Uruguay on penalties.


Oh snap, it's Dr.Pepper Snapple!
Come learn about the Brand Planning Process from Ryan Webb- Brand Planner of one of the largest soft drink companies in the United States. At this interactive speaker session, students are able to learn about the food industry and network with company representatives. Come and quench your thirst for marketing knowledge with Dr.Pepper Snapple!


At Storm Club Lacrosse, they strive to provide the best experience for each and every one of our players. As part of that, they put together this collection of recommended equipment for their players. You are welcome to choose from this selection as well as other selections that Hit the Net Sports has. We encourage you to check out the store which has a much larger selection than we can put online, as well as demo sticks to try in our Shooting Cage.


Vonage Holdings Corp. and Cisco Systems Inc.'s Linksys division are teaming up on a trade-in program.
Subscribers who sign up for Vonage's voice-over-Internet-protocol service can trade in their used routers for Linksys "Wireless-G Routers" with two phone jacks for broadband telephony (model "WRTP54G") free-of-charge after rebates.
"This is a great opportunity for anyone who has been thinking of cutting the cord to get a great deal on Vonage and a free wireless router, as well," Vonage senior vice president of sales Matthew Deatrick said in a prepared statement.


Cultural Centre Hall, Sale St, Orange NSW. Phone Howard on 02 6362 2368 or email him.
Kitchener St, Garran ACT. Phone Tony on 02 6255 4044.
309 King St, Newcastle. Enquiries to 02 9622 1399.
Postal and absentee auction. Bids must be in by 12:00PM CST 10 August 2015. View the Hallmark Website.
65 Northbourne Ave, Canberra City. Enquiries to edlins@tpg.com.au.
Queens Rd, Melbourne Victoria. Enquiries to APTA Website or direct to ANDA via email.
Free Entry. 2A Fitzroy St., Croydon NSW.
Cnr South Terrace and Sandgate St. Enquiries the PNS or Walter Bloom. There will be buying and selling, free valuations and advice on any aspect of coins, medals, tokens, banknotes and stamps.
Broadbeach, QLD. Enquiries to 07 55380 300.
Bathurst Stamp, Coin, Collectables & Lapidary Club Inc. Bathurst NSW. Enquiries to Alan McRae, 02 63315404 or via email.
Druitt St, Sydney NSW. Enquiries to APTA Website or direct to ANDA via email.
Cnr Albany Highway and George St, Cannington WA. Phone 08 9455 7197.
Cnr Leichhardt St and Wickham Tce. Brisbane QLD. Phone 07 3831 2599.
St Joseph's Church, Hill St. Orange NSW. Phone Howard on 02 6362 2368 or email him.
Cnr Victoria Drive and King Wiliam St, Adelaide. Free Admission.
Registrations can be made online at the NAA website.
66 Lygon St, Carlton VIC. Enquiries via email.
1022 Whitehorse Rd, Box Hill VIC 3120. Viewing at Downies Head Office 3 Redland Drive, Mitcham VIC 3132. Call 1300 788 358 or visit the Downies Website.
Community Hall, Higgins Road, East Bentleigh VIC. Phone Steele on 0439950084 or email him.
Corner West Row and Alinga Street, Canberra City. Enquiries to edlins@tpg.com.au.
Loeman Street, Strathmore, VIC. Phone Steele on 0439950084 or email him.


This is the story of Prince Jaime, who in 14th century broke with taboos and expressed his love for his Muslim male servant, Karim.
This long book is based on a real fact, that is the story of Prince Jaime and his love for his Muslim male servant Karim. It teaches children the vale of respect, equality, and freedom. A children's story on homosexuality, intended for both children and adults.


The phrase "lessons learned" is such a common one, yet people struggle with developing effective lessons learned approaches. The Lessons Learned Handbook is written for the project manager, quality manager or senior manager trying to put in place a system for learning from experience, or looking to improve the system they have. Based on experience of successful and unsuccessful systems, the author recognises the need to convert learning into action. For this to happen, there needs to be a series of key steps, which the book guides the reader through. The book provides practical guidance to learning from experience, illustrated with case histories from the author, and from contributors from industry and the public sector.
What is a 'lesson learned'?
Trial and error, or trial and success?
Will there always be an action?
What sort of actions are needed?
How do you decide the action?
Dr Nick Milton is a director and co-founder of Knoco Ltd - a Knowledge Management consultancy comprised of seasoned knowledge management practitioners, mentors, and coaches. Knoco Ltd has been delivering successful and sustained Knowledge Management implementation to clients since 1999.


text:
Jim Christian
Stage Directors and Choreographers Society (SDC)
Guest Director
Jim Christian is a native of Utah with over 45 years of professional experience as a director, teacher, actor, playwright, and producer in theatre, television and film. He has worked as a director/choreographer for the Tony Award-winning Utah Shakespearean Festival, Pioneer Theatre Company, Utah Opera Company, Utah Symphony, and theaters in California, Kentucky, Arizona, Montana, Florida, Wyoming, Tennessee, and Illinois. Jim is a member of the Society of Directors and Choreographers and The Dramatists Guild.
Events featuring Jim Christian
Elf The Musical
Based on the beloved holiday film, this hilarious fish-out-of-water comedy follows Buddy the Elf in his quest to find his true identity.
Elf The MusicalNovember 3, 2022 6:30pm