text:
4-група Кляйна є найменшою нециклічною групою. Названа на честь німецького математика Фелікса Кляйна оскільки вона зустрічається в його роботі «Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade» 1884 року.

Визначення
4-групою Кляйна називається прямий добуток  двох циклічних груп порядку 2, чи будь-яка ізоморфна група. Дана група має чотири елементи. Порядок кожного елемента за винятком одиничного рівний 2. Якщо позначити елементи групи (1,i, j,k) то таблиця Келі даної групи матиме вигляд:

 {| border="1" cellpadding="11" style="border-collapse: collapse; border: 1px #aaa solid;"
!style="background:#efefef;"| *
!style="background:#efefef;"| 1
!style="background:#efefef;"| i
!style="background:#efefef;"| j
!style="background:#efefef;"| k
|-
!style="background:#efefef;"| 1
| 1 || i || j || k
|-
!style="background:#efefef;"| i
| i || 1 || k || j
|-
!style="background:#efefef;"| j
| j || k || 1 || i
|-
!style="background:#efefef;"| k
| k || j || i || 1
|}
Як і будь-яка інша група 4-група Кляйна є підгрупою групи перестановок. Її циклічний запис:
 V4 = { (1), (12)(34), (13)(24), (14)(23) }

Властивості
 4-група Кляйна є єдиною групою порядку 4.

Скінченні групи

redpajama_set_name: RedPajamaWikipedia


text:
Detroit Lions notes: Nick Fairley works with…
Detroit Lions notes: Nick Fairley works with second string
By Paula Pasche | The Oakland Press
PUBLISHED: August 11, 2014 at 7:24 p.m. | UPDATED: June 17, 2021 at 7:45 a.m.
ALLEN PARK >> Defensive tackle Nick Fairley lined up with the twos (the second string) Monday at Detroit Lions training camp, but coach Jim Caldwell said not to read too much into it.
"Was there a depth chart that said anything about the twos?" Caldwell said. "We haven't put out a depth chart. When we do we will, but we haven't yet. So what you see moreso than anything else is we decide how we rep guys. It doesn't necessarily mean that they're one, two, three or whatever it might be. It depends on the situation. So don't read a whole lot into it."
Fairley started Saturday's game against the Browns.
"Just like everybody else, he could be better," Caldwell said of Fairley's performance in that game.
C.J. Mosley got reps with the starters in Monday's practice. He was asked if he's the starter now.
"I don't know, I don't worry about that, I just go in when they call me," Mosley said.
A few hours after Caldwell's press conference a new depth chart was released with Fairley still listed as the starter.
The Lions play their second preseason game at 10 p.m. on Friday at the Oakland Raiders.
Stafford jinxed
Matthew Stafford threw his first interception during training camp late in Monday's practice.
His pass was a little high, bounced off the hands of Brandon Pettigrew and into the hands of cornerback Bill Bentley.
When Stafford was told on Friday that offensive coordinator Joe Lombardi had mentioned his interception-free camp, Stafford's initial reaction was that he had been jinxed.
Calvin Johnson practiced on Monday after he sat out Saturday night's Lions' preseason opener. Also returning to practice were Theo Riddick who suffered an abdominal injury in the game and Tahir Whitehead who left the game and appeared to be having a concussion test on the sideline.
Missing from Monday's practice was right guard Larry Warford, safety Don Carey, along with Ziggy Ansah and TJ Jones who are on the physically unable to perform list.
Practice was moved indoors due to the torrential rain. Since it was moved, it was closed to the public. The final practice set to be open to fans is at 2 p.m. on Wednesday.
Follow Paula Pasche @paulapasche

redpajama_set_name: RedPajamaCommonCrawl


text:
Fraud and Corruption, Private Sector
Greensill / Gupta – Might this Be a Whole New Type of "Invoice Fraud"?
We've written a couple of times about the Greensill affair, and now more is emerging about another key player in the financial scandal. Greensill in effect lent billions to Sanjeev Gupta, creator of the GFG Alliance of steel businesses. That appears to have been based on both financing the invoices where GFG owed money to their suppliers, and also making early payment to gupta's firms where GFG invoiced its own customers.
But the Financial Times, which has been instrumental in exploring matters, reports that Grant Thornton, the administrator for Greensill, has contacted some GFG "customers". Clearly, they in theory owe Greensill money. However, "some of them say they did no business with Gupta". In other cases, there are allegations that the customers were friends or associates of Gupta.
If this is true, it seems that Greensill was advancing money to GFG based on their invoices which had in theory been issued. Greensill would collect the money owed from the customers in line with payment terms. So note this is financing Gupta based on its sales, rather than improving its cash flow by helping on the purchase side. But if these invoices – or some of them – were fake – then we have a real fraud, and Greensill obviously won't be able to collect its debts. Maybe Greensill was an innocent victim, being told by GFG these were real customers and real debts. Or maybe not.
Anyway, this link with supply chain finance is for me potentially a new type of invoice-related fraud. I must admit I did not cover this in Bad Buying, but it might be in the 2nd edition / follow-up!
The more usual invoice frauds that I describe in my book fall into three categories.
Fake invoices are created, submitted and authorised by someone inside the organisation. The money is paid to firms (probably set up for this purpose) which the insider(s) controls.
Fake or inaccurate invoices are submitted by an external party, either "on spec" in the hope that the internal systems are poor and they get paid, or to be authorised by an accomplice internally. The supplier may even be genuine, but the amount invoiced may not reflect the actual goods supplied or work done.
Invoice mis-direction, where the fraudster persuades the firm to pay a genuine invoice to the fraudsters bank account rather than to the real supplier's account.
"Fake invoice" fraud by insiders happens in the private sector, in government, and even in the charity sector. And it can be the most unlikely people – as in this case (taken from my book), where the former head of counter-fraud at Oxfam, the charity that fights poverty globally, was jailed after stealing more than £64,000 from the organisation.
Edward McKenzie-Green, 34, defrauded the organisation while investigating fellow charity workers in earthquake-hit Haiti. He filed fake invoices from bogus companies, making £64,612 in nine months before resigning because of unrelated disciplinary proceedings. The scheme was discovered after an internal inquiry was launched to investigate allegations that he'd behaved unprofessionally while leading a team in Haiti in 2011.
He agreed to resign, was given a £29,000 "golden handshake", but then investigators unearthed 17 fraudulent invoices from two companies under his control. An audit of his own counter-fraud department revealed payments to "Loss Prevention Associates" and "Solutions de Recherche Intelligence" in 2011. Investigators contacted the supposed head of one company, Keith Prowse, for an explanation of invoices for 'intelligence investigation', 'surveillance equipment' and 'Haiti Confidential'. But there was no Mr Prowse – that was, in fact, Mackenzie-Green. (The "real" Keith Prowse founded a very successful corporate hospitality firm in the UK).
McKenzie Green got two years in jail and Judge Wendy Joseph QC told him: "You have taken from those who desperately need it substantial sums of money. Worse, you have undermined the public confidence in a charitable institution. You were head of a department set up to counter fraud. This was a profound abuse of the trust invested in you."
We suspect that the magnitude of the Gupta / Greensill affair might dwarf the Oxfam case and most of the others in the book, except perhaps for the Petrobras / Odebrecht scandal in Latin America, where fake invoicing was only a small part of the wider fraud and corruption picture. In any case, it will be interesting to see what emerges in the Gupta case over the coming months.
April 6, 2021 /2 Comments/by Peter Smith
Tags: Invoicing, Supply chain finance, Greensill Capital, Gupta
https://badbuying.com/wp-content/uploads/2021/04/Money-2-sepia.jpg 848 1280 Peter Smith http://badbuying.wpengine.com/wp-content/uploads/2020/03/bad-buying-300x66.png Peter Smith2021-04-06 12:07:272021-04-06 12:07:29Greensill / Gupta – Might this Be a Whole New Type of "Invoice Fraud"?
Bad Buying Reviewed in The Times – and Why "Sophisticated Frauds" Usually Aren't
Greensill Capital – When Good Supply Chain Finance Turns Bad
Retroactive Inducements – Greensill Capital, David Cameron and a Threat to Democracy & Capitalism?
Supply Chain Finance Fraud Isn't New, Neither is Dodgy Accounting
Procurement Fraud in the NHS – Could It Happen In Your Organisation?
BBC Panorama Looks Into the Greensill Affair - Bad Buying says:
[…] to the Greensill supply chain finance (SCF) scandal, the excellent BBC Panorama programme earlier this month dug further into the affair, including the […]
Supply Chain Finance Fraud Isn't New, Neither is Dodgy Accounting - Bad Buying says:
[…] my last article about fraud related to supply chain finance (which came to mind because of the emerging Greensill / […]
Retroactive Inducements – Greensill Capital, David Cameron and a Threat... Supply Chain Finance Fraud Isn't New, Neither is Dodgy Accounting


text:
Q: Grouping Adjacent Items using XSLT 1.0 I'm hoping to get some advice on how to group adjacent elements with the same name in an unordered list using XSLT 1.0.
Here's some example XML:

  <TextContent>

lorem

    <VisualContent id="1" />
    <VisualContent id="2" />
    <VisualContent id="3" />

ipsum

    <VisualContent id="4" />

dolor

    <VisualContent id="5" />
  </TextContent>

Here's the output I'd like:

  <HtmlContent>

lorem

ipsum

dolor

  </HtmlContent>

Unfortunately, XSLT 1.0 is a strict requirement for this one. Any suggestions would be greatly appreciated.

A: I have skipped the part where <TextContent> turns into <HtmlContent> and <VisualContent id="n" /> becomes  because the question is difficult enough without these distractions.
The method I have chosen looks at the first preceding sibling whose name is not the same as the current element's name. The unique ID of that sibling is the key by which the adjacent elements of same name can be grouped:
<?xml version="1.0" encoding="utf-8"?>
<xsl:stylesheet version="1.0"
xmlns:xsl="http://www.w3.org/1999/XSL/Transform">
<xsl:output method="xml" version="1.0" encoding="utf-8" indent="yes"/>

<xsl:key name="prevByName" match="TextContent/*" use="generate-id(preceding-sibling::*[not(name()=name(current()))][1])" />

<xsl:template match="/">

<TextContent>
    <xsl:for-each select="Article/TextContent/*[generate-id()=generate-id(key('prevByName', generate-id(preceding-sibling::*[not(name()=name(current()))][1]))[1])]">
        <xsl:variable name="myGroup" select="key('prevByName', generate-id(preceding-sibling::*[not(name()=name(current()))][1]))" />
        <xsl:choose>
            <xsl:when test="count($myGroup) > 1">

                    <xsl:for-each select="$myGroup">

<xsl:copy-of select="."/>

                    </xsl:for-each>

            </xsl:when>
            <xsl:otherwise>
                <xsl:copy-of select="."/>
            </xsl:otherwise>
        </xsl:choose>
    </xsl:for-each>
</TextContent>

</xsl:template>
</xsl:stylesheet>

When applied to your input example, the following result is produced:
<?xml version="1.0" encoding="utf-8"?>

   <TextContent>

lorem

            <VisualContent id="1"/>

            <VisualContent id="2"/>

            <VisualContent id="3"/>

ipsum

      <VisualContent id="4"/>

dolor

      <VisualContent id="5"/>
   </TextContent>

EDIT:
Here's a modified version that only groups adjacent "VisualContent" elements:
<?xml version="1.0" encoding="utf-8"?>
<xsl:stylesheet version="1.0"
xmlns:xsl="http://www.w3.org/1999/XSL/Transform">
<xsl:output method="xml" version="1.0" encoding="utf-8" indent="yes"/>

<xsl:key name="prevByName" match="TextContent/*" use="generate-id(preceding-sibling::*[not(name()=name(current()))][1])" />

<xsl:template match="/Article/TextContent">

<TextContent>
        <xsl:apply-templates select="*"/>
    </TextContent>

</xsl:template>

<xsl:template match="TextContent/*[not(self::VisualContent)]">
    <xsl:copy-of select="."/>
</xsl:template>

<xsl:template match="VisualContent[generate-id()=generate-id(key('prevByName', generate-id(preceding-sibling::*[not(name()=name(current()))][1]))[1])]">
<xsl:variable name="myGroup" select="key('prevByName', generate-id(preceding-sibling::*[not(name()=name(current()))][1]))" />
    <xsl:choose>
        <xsl:when test="count($myGroup) > 1">

                <xsl:for-each select="$myGroup">

<xsl:copy-of select="."/>

                </xsl:for-each>

        </xsl:when>
        <xsl:otherwise>
            <xsl:copy-of select="."/>
        </xsl:otherwise>
    </xsl:choose>
</xsl:template>

</xsl:stylesheet>

This could probably use some streamlining, but I think the principle is clear and I need to get some sleep...

A: I am also new to XSLT.
This is not your expected output, I could not get the
, but see if you can build on top of this. I will also try.
<xsl:stylesheet version="2.0"
    xmlns:xsl="http://www.w3.org/1999/XSL/Transform">
    <xsl:output method="xml" indent="yes" encoding="UTF-8" />
    <xsl:template match="/">

            <HtmlContent>
                <xsl:apply-templates />
            </HtmlContent>

    </xsl:template>

    <xsl:template match="VisualContent">

            <xsl:element name="img">
                <xsl:attribute name="date-id"><xsl:number/></xsl:attribute>
            </xsl:element>

    </xsl:template>

    <xsl:template match="p">
        <xsl:element name="p">
            <xsl:apply-templates />
        </xsl:element>
    </xsl:template>
</xsl:stylesheet>

This is the output:

    <HtmlContent>

lorem

ipsum

dolor

    </HtmlContent>

A: After finding out, that I need a current version of xsltproc (1.1.29), I further changed the style sheet. Grouping worked only for children of TextContent, but I needed grouping to work on any nesting level.
Here is my input:

    <TextContent>

lorem

        <VisualContent id="1" />
        <VisualContent id="2" />
        <VisualContent id="3" />

ipsum

        <VisualContent id="4" />

dolor

sit

        <VisualContent id="5" />
    </TextContent>

amet

    <VisualContent id="6" />
    <VisualContent id="7" />

consectetur

    <VisualContent id="8" />

I use the following style sheet:
<?xml version="1.0" encoding="utf-8"?>
<xsl:stylesheet version="1.0"
xmlns:xsl="http://www.w3.org/1999/XSL/Transform">
    <xsl:output method="xml" version="1.0" encoding="utf-8" indent="yes"/>

    <xsl:template match="/ | @* | node()">
        <xsl:copy>
            <xsl:apply-templates select="@* | node()" />
        </xsl:copy>
    </xsl:template>

    <xsl:key name="prevByName" match="*/*" use="generate-id(preceding-sibling::*[not(name()=name(current()))][1])" />

    <xsl:template match="VisualContent">
        <xsl:if test="generate-id()=generate-id(key('prevByName', generate-id(preceding-sibling::*[not(name()=name(current()))][1]))[1])">
            <xsl:variable name="myGroup" select="key('prevByName', generate-id(preceding-sibling::*[not(name()=name(current()))][1]))" />
            <xsl:choose>
                <xsl:when test="count($myGroup) > 1">

                        <xsl:for-each select="$myGroup">

                                <xsl:copy-of select="."/>

                        </xsl:for-each>

                </xsl:when>
                <xsl:otherwise>
                    <xsl:copy-of select="."/>
                </xsl:otherwise>
            </xsl:choose>
        </xsl:if>
    </xsl:template>

</xsl:stylesheet>

Now, grouping works on any element level. :-)
<?xml version="1.0" encoding="utf-8"?>

    <TextContent>

lorem

<VisualContent id="1"/>

<VisualContent id="2"/>

<VisualContent id="3"/>

ipsum

        <VisualContent id="4"/>

dolor

sit

        <VisualContent id="5"/>
    </TextContent>

amet

<VisualContent id="6"/>

<VisualContent id="7"/>

consectetur

    <VisualContent id="8"/>

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I still cannot wrap my mind around that the fact 2014 is almost over. Ringing in this year down in sunny St.Barths practically feels like yesterday.
I've been going through all of this year's posts to select the best looks of 2014. Of course, there is so many aspects of blogging that I love, but having my own sartorial diary at keyboard distance is ranking high on the list. It's wild to look back on how my style has evolved over this year (and 2013 as well..). I am forever grateful to have this outlet in my life and for each of you for reading it. 2 1/2 years ago, I had maybe twenty readers- if that- and now to thing that you are from around the world, means so much to me.
Return on the 31st for a more personal reflection of 2014... until then, let's get ready for 2015.
I love your outfits and you have such good photography too! Have a happy new year Sloane!
Loving the outfits from January and August! Happy New Years!
August is one of my favorites as well!!
I'm pretty sure me taking 40 minutes to edit a picture and procrastinate takes the cake.


Including those for sweets and beverages.
Their analysis also exposed that long-term smoking is certainly too viagra mechanism of action.
According to a new report from the British federal government.
We are going to need to up our education, so that the nurses and doctors and vets who prescribe antibiotics do it knowing the dangers and advantages, and think about that balance, and in addition spend time with individuals explaining why they're not prescribing them, Davies stated. AusBiotech and GlaxoSmithKline contain the annual competition as an integral initiative to encourage even more students to pursue analysis and professions in bio-sciences also to reward early-profession successes, recognising high accomplishment, unique innovations and work. We have to reward those that do decide to devote their lives to analyze. With respect to GSK I'd like to congratulate all finalists on the achievements so far.
It is important to reduce this unwanted fat before it gets tough to control.


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Dr. Satya Sharma steps down as Executive Director for CEWIT
On the afternoon of November 3rd, over fifty invited guests and colleagues gathered at the Center of Excellence in Wireless and Information Technology (CEWIT) building to say farewell to CEWIT's longtime leader, Dr. Satya Sharma. Dr. Sharma announced in July he was stepping down but stayed on through the CEWIT 2022 conference held in October.
Dr. Sharma has been the sole Director for CEWIT since the center's inception in 2003, and oversaw the physical construction of the CEWIT building as the anchor building in the Stony Brook Research and Development Park and the approval of CEWIT as one of the original New York State designated Centers of Excellence, which was passed in the NYS legislature in 2008.
During the time of Dr. Sharma's leadership, he helped in building CEWIT to be a very powerful institution that ranks as one of the two centers of its kind in the world. Its mission of world class research, working with companies, and creating the leaders of tomorrow in information technology is as relevant today as it was when CEWIT was created.
Before joining Stony Brook, Dr. Sharma was Senior Vice President at Symbol Technologies from 1995 to 2003, overseeing various divisions of the company including World-Wide Operations, Mobile Computing & Wireless Engineering, and Quality & Process Improvements. Prior to his tenure at Symbol, he was Director of AT&T Power Systems and an adjunct professor at Southern Methodist University. Dr. Sharma has managed technology, led product development, managed marketing and financial functions, led operations, and taken a leadership role in organizational transformation. He led Lucent Technologies to win the Deming Prize in 1994, making it the first and still the only American manufacturing company to have this honor. While leading Symbol's Mobile Computing & Wireless Engineering, the company won the National Medal of Technology in 2000. Dr. Sharma also led Lucent and Symbol to win the Shingo Prize in 1992 and 2003 respectively. Dr. Sharma has more than 70 technical publications and conducts research in a wide variety of disciplines including wireless and mobile computing, quality management, and materials science.
Dr. Sharma felt that "every institution needs new blood, new ideas and a new source of oxygen for its continuing growth." With that in mind, he decided to retire from his position. While Dr. Sharma will be missed at CEWIT, he plans to stay in the entrepreneurial arena, and continue to work with students whom he devoted most of his life to. You can still contact him at his Stony Brook email address since he will remain as a Research Professor with Mechanical Engineering Department.
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This module, developed by Lisa Bell, has been designed to enable students to learn and practice a variety of fieldwork techniques in their school grounds. It is aimed at Key Stage 3 (KS3), and would fit into a unit of study on ecosystems; a standard component of the KS3 scheme of work.
Students will be presented with a map of the school grounds, showing several different sites. The fieldwork will involve them investigating both the soils and the microclimate at each site. They will then analyse and compare their findings, and will use thinking skills and independent research to make informed decisions about which is the most suitable site for a vegetable garden.
As an extension to the task, students make links between the variations they have found in their small scale study and variations on a biome-scale. They research the different abiotic features of different biomes and then use this information to decide on the best location for different Educational Eden-Zones within the school grounds.


Made from lightweight canvas and open cotton mesh for coolness, this hat is the perfect fishing companion.
The extra broad brim which is 7 cm wide gives greater protection from the sun. The top of the hat is also made from canvas and the sides are made from cotton mesh to ensure your head remains protected but cool.
The leather flyband can be used to display your favourite fishing flies.
A decorative fishing fly is attached on the right hand side of the flyband to compliment the style of this hat.
Made in Ashburton, New Zealand - Selke Enterprises specialise in quality hats that are handcrafted by a dedicated team of skilled artisans.
The colour of the hat is khaki as pictured.
Available in sizes - XS, S, M, L, XL and XXL.


text:
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The 106th edition of the world's most famous bicycle race, one of cyclinings three Grand Tours, starts this year in Brussels to honor the 50th anniversary of the first Tour de France win of Eddy Merckx. The modern edition of the race consist of 21 day-long stages over a 23-day period (with 2 rest days) […]
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Marriage She married Elias Allen Wright, son of Leander Allen Wright and Clara Conger, on September 15, 1908.
[S1710] Ancestry.com, 1920 United States Federal Census (Provo, Utah: Ancestry.com Operations Inc., 2012), Year: 1920; Census Place: Seattle, King, Washington; Roll: T625_1929;Page: 2A; Enumeration District: 248; Image: 327.
[S1803] Ancestry.com, 1910 United States Federal Census (Provo, Utah: Ancestry.com Operations Inc., 2012), Year: 1910; Census Place: Seattle Ward 3, King, Washington; Roll:T624_1658; Page: 8A; Enumeration District: 0083; FHL microfilm:1375671.
[S1801] Ancestry.com, 1930 United States Federal Census (Provo, Utah: Ancestry.com Operations Inc., 2012), Year: 1930; Census Place: Seattle, King, Washington; Roll: 2499; Page:3A; Enumeration District: 148; Image: 652.0; FHL microfilm: 2342233.
[S1780] Ancestry.com, U.S., Sons of the American Revolution Membership Applications,1889-1970 (Provo, Utah: Ancestry.com Operations Inc., 2011), Volume: 310; SAR Membership Number: 61850.
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[S1800] Ancestry.com, 1940 United States Federal Census (Provo, Utah: Ancestry.com Operations Inc., 2012), Year: 1940; Census Place: Seattle, King, Washington; Roll: T627_4379;Page: 8B; Enumeration District: 40-178.


Newly remodeled modern bungalow fully-furnished available for short or longer term lease. Includes all utilities, internet/WIFI and cable. Washing machine & dryer included. Private fenced back yard (pets are welcome). Popular/vibrant Virginia Highland restaurants scene. Only 1 block from the Beltline in-between Ponce City Market and Piedmont Park. Walking distance to parks, shopping, and coffee/tea cafes. Drive 5 minutes to Atlanta Botanical Garden. This adorable bungalow boasts a private second story master suite. Master bedroom closet include lazy Susan for shoes.


The fourth Sustainable Development Goal (SDG4) focuses on education and aims to 'ensure inclusive and equitable quality education and promote lifelong learning opportunities for all'. One of the targets – Target 4.7 – in the new global education goal truly captures the transformative aspirations of the new Sustainable Development Agenda. It focuses on the moral purposes of education, asking us all to think about why we are learning and for what aim. The target also promotes the importance of lifelong learning, and does not specify the education levels or age groups to which its themes apply.
The best photo will be awarded with $500; second best with $250 and third with $100!
Anyone between the age of 18-35 is eligible to enter the competition. (Photo ID will be required from the winning candidates. Members of UNESCO staff, Permanent Delegations or National Commissions (either permanent or temporary, current or former), interns, contractors, sponsors, or an immediate family member of any of the foregoing, may participate, but are not eligible to win.
enter message to recipient: "2017 Photo Contest". Photos will not be accepted without the description text.
Please make sure you also upload a completed entry form with your photo(s).


It is 2018 and still don't have a website ?
There are around 14 million internet subscribers in Australia. 87% of Australians access the internet daily 10 hours a day. Thus your new customer is on internet but you are still waiting for them to visit your old-school designed website.
Let's work together to develop a new website for your business that you will show your customer. You go to the meetings either with your tablet or desktop to show your online business on your website. Forget about presentations, your website shows your portfolio.
Don't have a website? Don't worry, you will have one in 2 weeks. Just get in touch!


text:
Sophie Massalovitch is a freelance journalist who regularly writes for the woman's magazine 'Avantages' and the economics magazine 'Challenges'. She is an experienced journalist and very often publishes articles about Croatia in those two magazines. The article in 'Avantages' is an attractive tourism article describing Dubrovnik as a city that "reinvents the pleasure of strolling". The article published in 'Challenges' was aimed at promoting Split in wintertime suggesting the readers to take December's special - deal flights to Split. Sophie Massalovitch is a great connoisseur of Croatia, of its history, its culture and its tourist offer who has published around ten articles about Croatia in the last four years and contributed to many books and guides issued about Croatia.
Ignas Krupavicius is a journalist, presenter and editor of the TV talk show Good Morning, Lithuania (Labas rytas, Lietuva), a morning program of Lithuanian National Radio and Television that covers short travel reports. In October 2013 Ignas Krupavicius filmed a series of TV reports about Croatia where he presented viewers with inspiring destinations by introducing the life of the everyday man. The stories were broadcasted during October and November 2013 at Good Morning, Lithuania and will continue during 2014 at the same TV show.


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What can you tell job seekers about Pacific Mobile Structures's Gym Membership?"
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Ethiopia: No Liberalisation of the State Telecoms Monopoly
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Hailemariam Desalegn, Ethiopia's Prime Minister has rejected the breaking calls of for the state-monopoly in telecoms services. He said that their governmetn needs the funding of revenue for unrelated railway project.
In order to access the World Trade Organisation (WTO) the liberalisation of the telecoms market in the country is required. WTO has been stalled for a decade by the telecoms monopoly held by Ethio Telecom.
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Hailemariam Desalegn, also said that one may think that the Ethiopian government can get money from taxation but there was no way to get as much money from taxation and the sector therefore remains with them for years to come.
This has so far been argued, mainly by GSMA that boosting telecoms participation through lower prices thanks to competition can grow the overall economy and taxes from that more than offset any losses a government suffers from lack of a state-monopoly.
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text:
\section{Introduction}

It is well known that the series play important role in mathematics, physics and applications. When numerical
ingredients of the infinite series are rational numbers then they can be treated in any  $p$-adic as well as in real
number field, because rational numbers are endowed by real and $p$-adic norms simultaneously. In particular,
a real divergent series  deserves $p$-adic investigation when its $p$-adic sum is a rational for a rational argument.

Terms of many series in string theory, quantum field theory and quantum mechanics contain factorials. Such series are usually divergent in
the real case but convergent in $p$-adic ones. Motivated by this reason, different $p$-adic aspects of the series with factorials
are considered in \cite{bd1,bd2,bd3,bd4,bd5,bd6,bd7,bd8,bd9,bd10,bd11} and many summations are performed  in rational points. Also, notions of $p$-adic number
field invariant summation, rational summation, and adelic summation are introduced.

Note that $p$-adic numbers and $p$-adic analysis have been successfully applied in modern mathematical physics (from strings to complex systems
and the universe as a whole) and in some related fields (in particular in bioinformation systems, see, e.g. \cite{bd13}), see \cite{freund,vvz} for an early review and \cite{bd12} for a recent one.

In this paper we are interested in $p$-adic invariant summation of a class of infinite functional series which terms contain $n!$, i.e.
$\sum n! P_k (n; x) x^n ,$ where $P_k (n; x)$ are polynomials in $x$ of degree $k ,$ and which coefficients depend on $n .$ We  show that there exist
polynomials $P_k (n; x)$ for any degree $k ,$ such that for any $x \in \mathbb{Z}$ the corresponding sums are also  rational numbers. Moreover, we have found
recurrent relations how to calculate all ingredients of such  $P_k (n; x)$. The obtained summation formula is a generalization of previously
derived one when $x = 1$ (see \cite{bd10}) in $\sum n! P_k (n; x) x^n .$  Many results are illustrated by some simple examples.

 All necessary general information on $p$-adic series can be found in standard books on $p$-adic analysis, see, e.g. \cite{schikhof}.

\section{$p$-Adic Functional Series with Factorials}

We consider $p$-adic functional series of the form \begin{align}
S_k (x) = \sum_{n = 0}^{+\infty} n! \, P_k (n; x)\, x^n = P_k (0;x) + 1!\, P_k (1;x)\, x  + 2!\, P_k (2;x)\, x^2 + \dots \,, \label{2.1}
\end{align}
where
\begin{align} \label{2.2}
& P_k (n;x) =C_k (n)\, x^k + C_{k-1} (n)\, x^{k-1} + \dots + C_1(n)\, x + C_0(n)\,,  \\
& n! = 1\cdot 2 \cdots n, \, (0!=1), \quad k \in \mathbb{N}_0 = \mathbb{N}\cup\{ 0 \}, \quad x \in \mathbb{Q}_p \nonumber
\end{align}
and $C_j (n), \, 0 \leq j \leq k ,$ are some polynomials in $n$ with integer
coefficients. Mainly we are interested for which polynomials $P_k(n; x)$ we have that if $x \in \mathbb{Z}$ then the sum of the series
(1) is $S_k (x) \in \mathbb{Z}$, i.e. $S_k (x)$ is also an integer the same in all $p$-adic cases. Since polynomials $P_k(n; x)$ are determined
by polynomials  $C_j (n), \, 0 \leq j \leq k ,$ it means that one has to find these $C_j (n), \, 0 \leq j \leq k .$  Our task is to find connection
between polynomial $P_k(n; x)$ and sum $S_k (x),$ which are also a polynomial.

\subsection{Convergence of the $p$-Adic Series}

Note that necessary and sufficient condition for $p$-adic power series
to be convergent \cite{schikhof,vvz} is
\[
f(x) = \sum_{n=1}^{+\infty} a_n x^n ,   \quad a_n \in \mathbb{Q} \subset \mathbb{Q}_p ,
\quad x \in \mathbb{Q}_p ,  \quad |a_n x^n|_p \to 0 \,\, as \,\, n
\to \infty .
\]
Necessary condition follows like in the real case, and sufficient condition is a consequence
of ultrametricity. Recall that  ultrametric distance satisfies the strong triangle inequality
$
d(x,y) \leq max \{d(x,y), d(y,z)\}.
$   $p$-Adic distance $d_p (x,y) = |x-y|_p$ is ultrametric one, where $|\cdot|_p$ is $p$-adic norm.
According to the Cauchy criterion, the sufficiency follows from $|a_n + a_{n+1} + \cdots + a_{n+m}|_p \leq max_{n\leq i\leq n+m} |a_i|_p = |a_n|_p \to 0$ when
$n \to \infty .$

 The functional series (1) contains $n!$, hence to investigate its convergence one has to know $p$-adic norm of $n! .$
 First, let us find a power $M(n)$ by which  prime $p$ is contained in $n!$  (see, e.g. \cite{vvz}). Let $n = n_0 + n_1 p +
... + n_r p^r $ and $ s_n = n_0 + n_1 + ... + n_r $ denotes the sum of digits in expansion of a natural number $n$ in base $p$. We also denote by $[x]$
the integer part of a number $x$.   Then
\begin{align} M(n) &= \Big[\frac{n}{p}\Big] + \Big[\frac{n}{p^2}\Big] +
\cdots + \Big[\frac{n}{p^r}\Big] \nonumber \\
&= \frac{n-n_0}{p} + \frac{n-n_0 -n_1 p}{p^2} + \cdots +
\frac{n-n_0 -n_1 p  - \cdots - n_{r -1} p^{r-1}}{p^r} \nonumber \\
& = \frac{n}{p} \Big(1 + \frac{1}{p} + \cdots + \frac{1}{p^{r-1}}\Big) -
\frac{n_0}{p} \Big(1 + \frac{1}{p} + \cdots + \frac{1}{p^{r-1}}\Big) \nonumber \\
& - \frac{n_1}{p} \Big(1 + \frac{1}{p} + \cdots + \frac{1}{p^{r-2}}\Big) -
\frac{n_2}{p} \Big(1 + \frac{1}{p} + \cdots + \frac{1}{p^{r-3}}\Big)\nonumber \\
& - \cdots - \frac{n_{r-1}}{p} .  \label{2.3}
 \end{align}

 Performing summation in \eqref{2.3},  it follows

\begin{align} M(n) &= n \frac{1-p^{-r}}{p-1} - n_0 \frac{1-p^{-r}}{p-1}
- n_1 \frac{1-p^{-r+1}}{p-1} - n_2 \frac{1-p^{-r+2}}{p-1}
\nonumber \\ &- \cdots - n_{r-1} \frac{1-p^{-1}}{p-1} -
n_r\frac{1}{p-1} + n_r\frac{1}{p-1} \nonumber \\ &= \frac{n -
s_n}{p-1} \label{2.4}
\end{align}

 Hence, one obtains
 \begin{align} &n! = m\, p^{M(n)} = m\,  p^{\frac{n -s_n}{p-1}} ,\quad p \nmid  m , \nonumber  \\
 &|n!|_p = p^{-\frac{n -s_n}{p-1}} . \label{2.5} \end{align}

Consider $p$-adic norm of the general term in \eqref{2.1} with $P_k (n; 1)$, i.e.

\begin{align}  |n! P_k(n; 1)  x^n|_p \leq  |n! x^n|_p = p^{- \frac{n - s_n}{p-1}} |x|_p^n =
\Big( p^{- \frac{n - s_n}{n(p-1)}} |x|_p  \Big)^n \nonumber \end{align}
Then, because convergence requires $|n! P_k(n; 1)  x^n|_p \to 0$ as $n \to 0,$ one has the following relations:
\begin{align}
p^{- \frac{n - s_n}{n(p-1)}} |x|_p < 1, \quad  |x|_p < p^{
\frac{n - s_n}{n(p-1)}} \to p^{\frac{1}{p-1}} \,\, as \,\, n \to
\infty . \nonumber
\end{align}

 Hence, the region of convergence $D_p$ of the power series $\sum P_k(n; 1)n!x^n$
is $D_p =\mathbb{Z}_p$, i.e. $|x|_p \leq 1$, because norms in
$\mathbb{Q}_p$ are of the form $p^\nu , \,\, \nu \in \mathbb{Z}$
and $1 < p^{\frac{1}{p-1}}. $ The region of convergence $|x|_p \leq 1$ is valid also
for the series \eqref{2.1}, because $|P_k(n; x)|_p \leq |P_k(n; 1)|_p$ when $|x|_p \leq 1.$

 Since $\bigcap_p \mathbb{Z}_p = \mathbb{Z}$, it means that the
infinite series $\sum P_k(n; x) n! x^n$ is simultaneously convergent for all
integers and all $p$-adic norms.

\section{Summation at Integer Points}

We are interested now in determination of the polynomials $P_k(n; x)$ and the corresponding sums $S_k (x) = Q_k (x)$ of the infinite series
\eqref{2.1}, where
\begin{align}
Q_k (x)= q_k \, x^k \, + \, q_{k-1} \, x^{k-1} \, + \cdots + \, q_1 \, x^1 \, + \, q_0   \label{3.1}
\end{align}
are also some polynomials related to $P_k(n; x) ,$  so that $P_k(n; x) $ and $Q_k (x)$ do not depend on concrete
$p$-adic consideration and that they are valid for all $x \in \mathbb{Z} .$

The simplest illustrative case \cite{schikhof} of $p$-adic invariant summation of the infinite series is

\begin{align} \sum_{n \geq 0} n! \, n = 1! 1 + 2! 2 + 3! 3 + ... = - 1 \label{3.2} \end{align}
and obtains from \eqref{2.1} taking  $x = 1, \, \, P_1(n; 1) = n,$ which gives $Q_1 (1) = - 1.$
 To prove  \eqref{3.2}, one can use any one of the following two properties:
\[  n! n = (n+1)! - n!  \,, \quad \, \qquad \sum_{n = 1}^{N-1} n! \, n  = -1 + N! , \]
where $n! \to 0$ as $n \to \infty .$

In the sequel we shall develop and apply the corresponding method of the definite partial sums.

\subsection{The Definite Partial Sums}

\begin{theorem}
Let
\begin{equation}
A_k (n; x) = a_k(n) x^k + a_{k-1}(n) x^{k-1} + \cdots + a_1(n) x + a_0(n)  \label{3.3}
\end{equation}
be a polynomial with coefficients $a_j (n) ,\,\, 0\leq j \leq k,$ which are polynomials in $n$ with integer coefficients.
Then there exists such polynomial $A_{k-1} (N; x), \,\, N \in \mathbb{N},$ that equality
\begin{align} \sum_{n=0}^{N-1} n! \, [ n^k x^k + U_k (x) ] x^n = V_k (x) + N! x^N A_{k-1}(N; x)   \label{3.4}  \end{align}
holds, where
\begin{align} U_k(x) = x A_{k-1} (1; x) - A_{k-1} (0; x) \,, \quad V_k(x) = - A_{k-1} (0; x).  \label{3.5}\end{align}
\end{theorem}

\begin{proof}
Let us consider summation of the following finite power series:
\[ S_k(N; x) = \sum_{n=0}^{N-1} n! n^k  x^n  = \delta_{0k} + \sum_{n=1}^{N-1} n! n^k  x^n, \quad   k \in \mathbb{N}_0  .  \]

We derive the following recurrent formula:
\begin{align}
S_k(N; x) &= \delta_{0k} + \sum_{n=0}^{N-2} (n+1)! (n+1)^k x^{n+1} = \delta_{0k} + x +
\sum_{n=1}^{N-1} n! (n+1)^{k+1} x^{n+1} - N! N^k x^N \nonumber \\
&= \delta_{0k} + x + x \sum_{n=1}^{N-1} n! x^n \sum_{\ell =0}^{k+1} \Big(\begin{aligned} k&+1 \\
&\ell \end{aligned}\Big) n^\ell - N! N^k x^N \nonumber \\ &= \delta_{0k} +
x S_0(N;x) + x  \sum_{\ell =1}^{k+1} \Big(\begin{aligned} k&+1\\
& \ell \end{aligned}\Big) S_\ell (N; x) - N! N^k x^N  .  \label{3.6}
\end{align}

The above recurrent formula gives possibility to calculate sum
$S_{k+1} (N; x)$ knowing all preceding sums $S_{\ell} (N; x), \,
\, \ell = 1, 2, ..., k$ as function of $S_{0} (N; x)$. These sums
have the form
\[ \sum_{n=0}^{N-1} n! \, [ n^k x^k + U_k (x) ] x^n = V_k (x) + N! x^N A_{k-1}(N; x) , \, \, k \in \mathbb{N} ,  \]
which can be rewritten  as
\begin{align}  S_k (N; x) = - U_k(x) S_0 (N; x) x^{-k} + V_k (x) x^{-k} + N! A_{k-1} (N; x) x^{N-k} , \, \, k \in \mathbb{N} , \label{3.7} \end{align}
where $U_k (x)$ and  $V_k (x)$ are some polynomials in x of the
degree  $k$, and $A_k (N; x)$ is a polynomial in $x$ which
coefficients of $x^n$ are polynomials in $N$ of degree $n$.

Substituting the above expressions of the sums \eqref{3.7} into the recurrence formula  \eqref{3.6}, we
obtain recurrence formulas for $U_{k+1} (x), V_{k+1} (x)$ and
$A_{k}(N; x):$
\begin{equation}
\sum_{\ell =1}^{k+1} \Big(\begin{aligned}k&+1\\ &\ell\end{aligned}\Big) x^{k-\ell+1} \, U_\ell(x) - U_k(x) - x^{k+1} = 0 , \label{3.8}
\end{equation}
\begin{equation}
\sum_{\ell =1}^{k+1} \Big(\begin{aligned}k&+1\\ &\ell\end{aligned}\Big) x^{k-\ell+1} \, V_\ell(x) - V_k(x)  = 0 , \label{3.9}
\end{equation}
\begin{equation}
\sum_{\ell =1}^{k+1} \Big(\begin{aligned} k&+1\\ &\ell\end{aligned}\Big) x^{k-\ell+1} \, A_{\ell-1}(N;x) - A_{k-1}(N;x) - N^kx^{k} = 0 . \label{3.10}
\end{equation}

Putting $N = 0$ in \eqref{3.10}, it follows that $V_k (x)$ is proportional to $A_{k-1} (0; x) .$  After explicit  calculation of  $A_0 (0; x)$ and $V_1 (x) ,$ we conclude that
$V_k (x) = -  A_{k-1} (0; x)$ for all $k \in \mathbb{N} .$  On the other hand, equality $U_k(x) = x A_{k-1} (1; x) - A_{k-1} (0; x)$ in \eqref{3.5} obtains if we replace $N =1$ in \eqref{3.10} and multiply it by $x$, and then subtracting  from such expression
recurrence relation \eqref{3.10} with $N= 0 .$

\end{proof}

It is worth emphasizing that equality \eqref{3.4} is valid in real and all $p$-adic cases. The central role in this equality plays
polynomial $A_{k} (N; x) $ which is solution of the recurrence relation \eqref{3.10}.
 When $N \to \infty$ in \eqref{3.4}, the term with polynomial
$A_{k-1} (N; x)$ $p$-adically disappears giving the sum of the following $p$-adic infinite functional series:
\begin{equation}
 \sum_{n=0}^{\infty} n! \, [ n^k x^k + U_k (x) ] x^n = V_k (x) .    \label{3.11}
\end{equation}
This equality has the same form for any $k \in \mathbb{N} ,$ and polynomials $U_k (x)$ and $V_k (x)$ separately have the same values in all $p$-adic cases for any
$x \in \mathbb{Z} .$ In other words, nothing depends on $p$-adic properties in \eqref{3.11} if $x \in \mathbb{Z}$, i.e. this is $p$-adic invariant
result. This result gives us the possibility to present a general solution of the problem posed on $p$-adic invariant summation of the series \eqref{2.1}.

\begin{theorem}
The functional series \eqref{2.1}  has $p$-adic invariant sum
\begin{equation}
\sum_{n=0}^{\infty} n! \, P_k (n; x) x^n = Q_k (x)   \label{3.12}
\end{equation}
if
\begin{equation}
P_k (n; x) = \sum_{j =1}^k C_j [n^j x^j + U_j (x)]    \quad \text{and} \quad   Q_k (x) = \sum_{j =1}^k C_j \, U_j (x),  \label{3.13}
\end{equation}
where $C_j, \, x \in \mathbb{Z} .$
\end{theorem}

Recurrent formulas \eqref{3.8}--\eqref{3.10} enable to calculate these polynomials for all $k \in \mathbb{N}$,
knowing initial expressions of  $U_1 (x),  V_1 (x)$ and $A_0 (N; x)$, which can be
obtained from  \eqref{3.6} and they are: $U_1(x) = x-1, \, V_1(x) =
-1$ and $A_0 (N; x) = 1$.

For the first five values of degree $k$ we have obtained the following explicit expressions.

\begin{itemize}
\item  $k = 1$
\begin{align}
&U_1(x) = x-1, \nonumber\\
&V_1(x) = -1, \nonumber \\
&A_0 (n; x) = 1.  \label{3.14}
\end{align}
\item  $k = 2$
\begin{align}
&U_2(x) = - x^2+ 3x -1, \nonumber\\
&V_2(x) =  2x - 1, \nonumber\\
&A_1(n; x) = (n -2) x  + 1 .   \label{3.15}
\end{align}
\item $k = 3$
\begin{align}
&U_3(x) =  x^3 - 7x^2  + 6x -1, \nonumber\\
&V_3(x) = -3 x^2 +5 x -1, \nonumber\\
&A_2(n; x) = (n^2 -3n +3)x^2 + (n -5) x  + 1 .  \label{3.16}
\end{align}
\item   $k = 4$
\begin{align}
&U_4(x) = -x^4 + 15x^3 - 25 x^2+ 10x -1, \nonumber\\
&V_4(x) =  4x^3 -17x^2 + 9x -1, \nonumber\\
&A_3(n; x) = (n^3 - 4n^2 +6n -4)x^3 +(n^2 -7n +17)x^2 +(n -9) x  + 1 .  \label{3.17}
\end{align}
\item    $k = 5$
\begin{align}
&U_5(x) = x^5 -31x^4 + 90 x^3 - 65 x^2+ 15x -1,  \nonumber \\
&V_5(x) = -5 x^4 + 49x^3 - 52 x^2 + 14x -1,   \nonumber\\
&A_4(n; x) =  (n^4 -5n^3 +10n^2 -10n +5)x^4 +  (n^3 - 9n^2 +31
n-49)x^3 \nonumber \\ & \qquad  \qquad + (n^2 -12n +52)x^2 + (n -14) x + 1 .  \label{3.18}
\end{align}
\item  $k = 6$
\begin{align}
&U_6(x) = -x^6 + 63x^5 -301 x^4 +350 x^3 - 140 x^2 +21 x -1, \nonumber \\
&V_6(x) = 6 x^5 -129x^4+246x^3 - 121 x^2 + 20 x -1,   \nonumber  \\
&A_5(n; x) = (n^5 -6 n^4 +15 n^3 - 20 n^2 +15 n -6) x^5 + (n^4 - 11n^3  \nonumber \\ & \qquad \qquad
+ 49n^2 - 111n  +129)x^4 + (n^3 - 15n^2 + 88 n - 246) x^3 \nonumber \\ & \qquad \qquad + (n^2 - 18 n +121) x^2 + (n - 20) x + 1 .   \label{3.18a}
\end{align}
\end{itemize}

It is already noted that polynomial $A_k (n; x)$ plays central role in $p$-adic invariant summation of the series \eqref{2.1}.
$A_k (n; x)$ as well as $U_k (x)$ and $V_k (x)$ can be written it the compact form
\begin{equation}
A_k (n; x) = \sum_{\ell = 0}^k A_{k\ell} (n) \, x^\ell , \quad U_k (x)= \sum_{\ell = 0}^k  U_{k\ell} \, x^\ell ,   \quad V_k (x) = \sum_{\ell = 0}^k  V_{k\ell} \, x^\ell ,      \label{3.18b}
\end{equation}
where $A_{k\ell} (n)$ is polynomial in $n$ of degree $\ell$ with $n^\ell$ as the term of highest degree.

Then the following properties hold:
\begin{itemize}
\item  $A_k(n; 0) = A_{k0} = 1 ,  \, \, k = 0, 1, 2, ...$
\item  $A_{kk}(1) = (-1)^k , \, \,  k = 0, 1, 2, ...$
\item  $ U_{k0} = V_{k0} =  -1 , \, \, k =  1, 2, ...$
\item  $U_{kk}= (-1)^{k+1} , \, \,  k =  1, 2, ... $
\item  $V_{kk} = (-1)^{k} k , \, \,  k =  1, 2, ...$
\end{itemize}

\subsection{Connection with Bernoulli Numbers and Some \\ Simple Examples}

Using the Volkenborn integral one can make connection of our summation formulas with the Bernulli numbers. By definition the Volkenborn integral \cite{schikhof} is
$$
\int_{\mathbb{Z}_p} f(x)\, dx = \lim_{x \to \infty} p^{-n} \sum_{j=0}^{p^n -1} f(j) ,
$$
where $f$ is a $p$-adic continuous function. One of its properties is
\begin{equation}
\int_{\mathbb{Z}_p} x^n \, dx  = B_n ,  \quad n = 0, 1, 2, ... , \label{3.19a}
\end{equation}
where $B_n$ are the Bernoulli numbers which can be defined by the recurrent relation
$$
\sum_{j=0}^{n-1}\Big(\begin{aligned} &n \\ &j \end{aligned} \Big) B_j =0 , \, \, B_0 = 1 . \nonumber
$$
Note that $B_1 = -\frac{1}{2}, \quad B_3 = B_5 = ... = B_{2n+1} = 0 .$

As an illustration of the above summation formula \eqref{3.11}, we present three simple (k = 1, 2, 3) examples
including connection with the Bernoulli numbers.

\begin{itemize}

\item  $ k = 1$
\begin{align}
 &\sum_{n=0}^{\infty} n! \, [(n + 1)x  - 1]\, x^n = -1 ,  \quad x \in \mathbb{Z} , \\   \label{3.19}
&\sum_{n=0}^{\infty} n! \, n \,  x^n = -1 ,  \quad x = 1, \nonumber  \\
&\sum_{n=0}^{\infty} n! \, (-1)^n \,(n + 2)  =  1 ,  \quad x = - 1, \nonumber \\
&\sum_{n=0}^{\infty} n! \, [(n + 1) B_{n+1}  - B_n] = -1 . \nonumber
\end{align}

\item $k = 2$
\begin{align}
 &\sum_{n=0}^{\infty} n! \, [(n^2 - 1) x^2 + 3x - 1]\, x^n = 2x -1 ,  \quad x \in \mathbb{Z} , \\   \label{3.20}
&\sum_{n=0}^{\infty} n! \, (n^2 + 1) = 1 ,  \quad x = 1 ,     \nonumber \\
&\sum_{n=0}^{\infty} (-1)^n \, n! \, (n^2 - 5) = -3 ,  \quad x = -1 , \nonumber \\
&\sum_{n=0}^{\infty} n! \, [(n^2 - 1) B_{n+2} + 3 B_{n+1} - B_n] = -2 . \nonumber
\end{align}

\item $k = 3$
\begin{align}
 &\sum_{n=0}^{\infty} n! \, [(n^3 + 1) x^3  - 7 x^2 + 6x - 1]\, x^n = -3x^2 + 5x -1 ,  \quad x \in \mathbb{Z} , \\   \label{3.21}
&\sum_{n=0}^{\infty} n! \, (n^3 - 1) x^3  = 1 ,  \quad x = 1 ,   \\
&\sum_{n=0}^{\infty} (-1)^n \, n! \, (n^3 + 15) =  9 ,  \quad x = -1 ,    \\
&\sum_{n=0}^{\infty} n! \, [(n^3 + 1) B_{n+3}  - 7 B_{n+2} + 6 B_{n+1} - B_n] = -4 .
\end{align}
\end{itemize}

The above series with the Bernoulli numbers are $p$-adic convergent, because $|B_n|_p \leq p$ (see \cite{schikhof}, p. 172).

\section{Concluding Remarks}
\bigskip

The main results presented in this paper are Theorem 1 and Theorem 2. These results are  generalizations of an earlier
result for $x = 1$ \cite{bd10} (see also \cite{bd9,bd11} for some partial generalizations).

Finite series with their sums  \eqref{3.4} are valid for real and $p$-adic numbers. When $n \to \infty$ the corresponding infinite
series are divergent in real case, but are convergent and have  the same sums in all $p$-adic cases. This fact can be used to extend
these sums to the real case. Namely, the sum of a divergent series depends on the way of its summation and here it can be used rational sum
of the series valuable in all $p$-adic number fields. This way of summation of real divergent series was introduced for the first time in
\cite{bd2} and called adelic summability. How this adelic summability is important depends on its practical future use in some concrete examples.

The simplest infinite series with $n!$ is $\sum n! .$ It is convergent in all $\mathbb{Z}_p ,$ but has not $p$-adic invariant sum. Even it is not
known so far does it has a rational sum in any $\mathbb{Z}_p .$  Rationality of this series and $\sum n! n^k x^n$ was discussed in \cite{bd9}. The series
$\sum n!$ is also related to Kurepa hypothesis which  states $(!n, n!) = 2, \quad 2 \leq n \in \mathbb{N} ,$  where $!n = \sum_{j=0}^{n-1} j!$. Validity
of this hypothesis is still an open problem in number theory. There are many equivalent statements to the Kurepa hypothesis, see  \cite{bd10} and references
therein. From $p$-adic point of view, the Kurepa hypothesis reads: $\sum_{j=0}^{\infty} j!  = n_0 + n_1 p + n_2 p^2 + \cdots ,$ where  digit $n_0 \neq 0$ for all primes $p \neq 2 .$

When $x = 1$, then \eqref{3.11} becomes
\begin{equation}
 \sum_{n=0}^{\infty} n! \, [ n^k x^k + u_k ] x^n = v_k ,    \label{3.22}
\end{equation}
where $u_k = U_k (1)$ and $v_k = V_k (1)$ are some integers. Equality \eqref{3.22} was introduced in \cite{bd5}, and properties of $u_k$ and $v_k$ are investigated in series of papers by Dragovich (see references \cite{bd8,bd9,bd10,bd11}). In \cite{murty} relationships  of $u_k$ with the Stirling and the Bell numbers are established, and $p$-adic irrationality of $\sum_{n \geq 0} n! n^k$ was discussed (\cite{subedi1,subedi2,alexander}). Note that the following sequences are related to some real (combinatorial) problems \cite{sloane}:
\begin{align}
&A_{k-1} (0;1) = -V_k (1) = - v_k : \, \, 1, -1, -1, 5, -5, -21, ...        \quad  \text{see  \, A014619} \\
&A_{k-1} (0;-1) = -V_k (-1) = - \bar{v}_k : \, \, 1, 3, 9, 31, 121, 523, ...   \quad  \text{see  \, A040027} \\
&A_{k-1} (1;1) - A_{k-1}(0;1) = U_k (1) = u_k : \, \, 0, 1, -1, -2, 9, -9, ...  \quad  \text{see  \, A000587} \\
&A_{k-1} (1;-1) + A_{k-1}(0;-1) = -U_k (-1) = -\bar{u}_k : \, \, 2, 5, 15, 52, 203, ... \quad  \text{see  \, A000110}.
\end{align}

There are many possibilities how results obtained in this paper can be generalized in future research of $p$-adic invariant summation of
some real divergent series. Various  aspects of the sequence of polynomials $A_k (n; x)$ deserve to be analyzed.
It would be also interesting to investigate relations between the Bernoulli numbers using
the above finite sums of the form \eqref{3.4}.

\section*{Acknowledgments}
Work on this paper was supported by Ministry of Education, Science and Technological Development of the Republic of Serbia, projects: OI 174012, TR 32040 and
TR 35023.

\bibliographystyle{amsplain}

redpajama_set_name: RedPajamaArXiv


Bhopal (Madhya Pradesh) [India], Apr 4 (ANI): Ahead of the Lok Sabha polls, senior Congress leader Digvijaya Singh took a potshot at the NDA-government, which he said was resorting to talking on "nationalism, terrorism, and Hindu-Muslim" as it did not have any answers to the issue of unemployment in the country.
"BJP will talk about nationalism, terrorism, and Hindu-Muslim because they don't have an answer to the issue of unemployment. They don't have an answer to what they did for the poor, how they have destroyed the Indian economy and about their promise to deposit Rs 15 lakhs into accounts," Singh said addressing party workers here on Wednesday.
Singh urged party workers to inform first-time voters about the work done by the Congress party.


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text:
P&G launches 1 Like = 1 Day charity Facebook campaign
P&G is launching 1 Like = 1 Day, a Facebook campaign to drive awareness of its global safe drinking water initiative.
By Rosie Baker	 25 Jan 2012 4:50 pm
Under the campaign, P&G, which owns brands like Pampers, Ariel and Pantene, has pledged to provide one day of clean drinking water for children in developing nations for every Facebook Like.
P&G will also share stories, photos, and videos from the field so that consumers can see how it works with humanitarian, NGO, and emergency relief partners worldwide.
As part of the digital initiative P&G has launched a blog and webpage so that consumers can keep up to date with developments around the world.
The Children's Safe Drinking Water (CSDW) Programme, which sees P&G provides packets of powdered water purifier it developed in partnership with the U.S. Centers for Disease Control and Prevention to 65 nations where clean water is not readily available, has been running for seven years. This is the first social media activity to promote the initiative and engage consumers.
The FMCG firm claims CSDW has helped save more than 22,000 lives by purifying 4bn litres of water. It recently announced that it would rebrand the packets with the P&G brand for the first time.
Bob McDonald, P&G's chairman, president and CEO, says: "As we work toward our goal to save one life every hour by 2020, we want others to play a part and share in our journey. This is one simple, but meaningful way for people all over the world to help us touch and improve lives by sharing clean drinking water."
The initiative launches today (25 January).
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Christa Carone is taking personal control of redefining Xerox – click here to read the main profile See what other marketers had to ask Christa Carone Marketing Week (MW): What challenges does Xerox face in the UK? Darrell Minards (DM): While Xerox is on this journey of transformation, we need to shift perceptions of the […]
Former Diageo exec Slay to head up Carlsberg UK
Carlsberg, the world's fourth biggest brewer, has hired Benet Slay, former managing director for Diageo Northern Europe, to head up its UK business.
'Marketers are the UK's happiest office workers'
Marketing professionals are the happiest office workers in the UK and are less likely to fall out with their colleagues, according to a new study commissioned by Mars Drinks UK.


text:
On 'POV,' a Swiss Approach to Educating Immigrant Students
By Mark Walsh — August 16, 2015 2 min read
In the United States, school districts have relied on a variety of approaches to educating and counseling immigrant students. Still, there are potential lessons to be learned from how a country much less vast and diverse than our own deals with the issue.
In "Neuland," an 87-minute documentary debuting Monday night on the "POV" series on PBS (check local listings), filmmaker Anna Thommen examines a program that helps young migrants in Switzerland learn one of the nation's four official languages (German), as well as mathematics and job and life skills.
Switzerland seeks to preserve its national identity by instilling shared values and a common language among these high school age students, who are asylum seekers from world trouble spots such as Afghanistan, Cameroon, Serbia, and Venezuela.
These students in Basel encounter teacher Christian Zingg, a middle-age man who comes off at first like he's seen it all before and that life will go on regardless of whether his students succeed or not. But Mr. Zingg soon enough reveals himself to care deeply about whether the students make it.
He tells them he could be making more money and have an easier job teaching at a traditional Swiss high school, but the two-year integration program is much more motivating for him.
The film pays extra attention to Ehsanullah, whose family borrowed $20,000 to get him to Switzerland, where he lives in an immigrant camp of some sort and must appeal an initial denial of asylum. What's more, he must send money back to Afghanistan so his family can repay his benefactors, or else they could lose their land.
Ehsanullah displays evidence of cutting himself out of anxiety, and keeping him in the school program isn't easy when he finds restaurant work that is allowing him to save money to send to his family.
Meanwhile, Ismail and Nazlije, a brother and sister who left Serbia for Switzerland, seem to pick up German faster than other students but have different levels of motivation for finding jobs. Nazlije is a good student and would like to be a primary school teacher, but Mr. Zingg has to gently recalibrate her expectations of getting such a job in her new country. Ismail seems disappointed to learn that even a house painting job requires math skills.
One fascinating tradition that appears to be in widespread use in the country is something called "Taster Week," in which students seek short-term apprenticeships with potential employers. Mr. Zingg helps his charges role-play the steps just to contact employers, not to mention job expectations.
When the students seem discouraged about their prospects, Mr. Zingg invokes a soccer analogy. Basel Football Club of the European Champions League had evidently just defeated the mighty Manchester United team, a David-beats-Goliath story.
"Of course, your chances are poor," he tells them. "Of course, you are Basel FC. But take that chance. The you'll have achieved something that no one else thought possible."
"Neuland," or New Territory, is focused fully on Mr. Zingg's class, and we don't get much in the way of context for Switzerland's immigration policies or challenges. But that's OK. Some ideas are world class.
[UPDATE: At Education Week's Global Learning blog, Heather Singmaster looks at the "Neuland" documentary in a broader context of Switzerland's system of guidance counseling and what lessons that system may provide.)
Contributing Writer, Education Week
Mark Walsh covers education law and the U.S. Supreme Court for Education Week.
A version of this news article first appeared in the Education and the Media blog.


text:
Viola's Heart Strings – Sent to the Corner
The moment the latch of the door clicked, he spun on his heels. "You do not fraternise with the servants. You do not talk to them beyond giving them your orders. You do not laugh with them. Do you understand me?" he thundered, throwing his hat and gloves on a nearby chair.
Viola shrank, surprised by his outburst, then she rallied, indignant that his abrupt appearance ruined her fun. She straightened her shoulders. "I didn't fraternise. I engaged in conversation. Am I to walk about this house like a ghost?"
He paced across the floor, halting by his desk. "It is the subject of your chatter that ails me. You know I'm French in lineage. How does it appear to the servants to hear you speak unkindly of my people?"
Voila covered her mouth. She had forgotten Anton's ancestry. "I merely joked," she attempted to dilute her mischief, hoping to wipe out the stern expression from his face
Anton tapped the table with his finger. "Joke. Childish banter. Very well, I will see you disciplined in the same vein. Go stand in the corner and lift up your skirts."
Viola's feet glued themselves to the carpet beneath her. The hesitation wasn't intentional—his request stunned her. She didn't think her behaviour warranted disciplining. She recalled what he had told her in the bedchamber and those words echoed about her head, the determination for her to be obedient. The same attitude her governess had instilled in her as a child. She remembered the dismay when sent to the corner of the nursery of her father's house and the shameful tears, which had streamed down her cheeks. At least back then, her tender years had warranted such punishments. How would a spell in the corner of his study correct her behaviour when it belittled her, transporting her into a childlike state?
She hadn't considered how her silliness might garner her husband's displeasure. An impetuous response to his directive would be to march out of the room. Would he drag her back, admonish her within earshot of the servants. Did Anton have it in him to scandalise their marriage weeks after the wedding?
Awash with contrary emotions, she fisted her hands into tight balls and glared at her husband. He held her gaze, matching her glower with one of his own. Their wills battled against each other, and each second of his unblinking stare left her knees weakened and her eyes blurred with tears, all sparked by her tempestuous revolt. She didn't want this discord in their relationship. What she craved was his approval. The romantic heart within her had to see past his stiff words to the other man she married, the one who exhibited kindness and made love to her.
The thought of all he was to her triggered her reluctant surrender. Stumbling forward, she shuffled towards a corner, the one farthest from his desk. Once there, she fretted, uncertain, and her arms locked rigidly to her sides, unable to enact his command regarding her skirts.
"Do it," he said softly.
The humiliation gnawed. What could he possibly gain from seeing her degraded? She stared at her feet, the hem of her dress resting on her bootlaces. It seemed a long way down to where the petticoats lurked beneath. However, with an awkward stiffness, she carried out his instructions, lifting up her skirts and petticoats to reveal her white drawers with their fancy lacework. She clutched her petticoats to her chest. Bared in the bedroom hadn't caused her grief, but downstairs in his study, the embarrassment grew and she turned away from her husband, not wishing to see his face.
"Lower your drawers," he hissed.
Good grief! Her pulse quickened and to her mortification, her sex responded as if wired to her husband's voice. Why oh why, was she filled with lust? The soft hiss of his demanding, but oh so seductive tone added to her dilemma. She wanted to comply, but the dignified woman, whom her virtuous governess had taught shouted inside her head not do to it, not to debase herself for such a trivial act of humour.
"Please, Anton," she whispered and squashed her legs together. She glanced over her shoulder.
Anton remained by his desk with a pensive expression. He fiddled with a fountain pen, his gaze darting about the table. "If you are ashamed by my request, it is serving a purpose, is it not? When you disgrace yourself with the servants—that is how I feel, inside." He looked up and held her attention as she swayed between obedience and defiance. "Don't make me come over there and do it myself."
While holding the skirts in one hand, she looped the loose cord around her forefinger of the other. She tugged and released the bowtie, however, her hips held the waistband in place. The drawers merely sagged about her thighs. She froze again. To release them, she would have to let go of her skirts or wriggle them down with her hips. Only her stockings covered her skin below the knee. How could she bend without showing him her naked cleft? She hesitated, unable to comply—she was quite convinced something between her legs had begun to flow.
"Very well," he snapped.
She heard his footsteps approach and the click of his knees as he crouched down. He yanked on the waistband with both hands, pulling down the underwear and exposing her bare bottom. Viola gasped as she felt the rush of cool air over her flesh. Mortification rose up to her mouth, a bitter taste of discomfort at her revealing pose. Her sex, between her legs, seemed to have a mind of its own. While horrified by her predicament, her silly pussy behaved otherwise. Burying her face in her hands, she let out a sob.
He crouched down by Viola's semi-clad legs, pausing to reflect on his secret passion. Anton's admiration for the female bottom held no limitations. Once again, fantasies haunted his waking thoughts, concocting a daydream centred on his wife's charming posterior.
The colouration changes he could instigate if he so desired, the marks he could leave. He pictured the quivering flesh rebounding from his blows, creating those telling features, which were his preference. The pain she would sense, but he would manage and deliver it as he saw appropriate. He'd never be harsh or demonstratively cruel towards to her, he had the skills to determine the right level of humiliation, leaving her with a sore behind, which would help teach her to be less wayward with her witty tongue.
None of it was possible. Her innocence and lack of understanding of his ways would tarnish the trust between them, unlike another, who had eagerly begged him to discipline her with a rod until he had turned her bottom into rows of red stripes. No, he would not rush down that path again and have the consequences of his actions thrown back in his face. Viola would suffer other methods in his efforts to shape a better wife.
The hardest part would be holding back and not fucking her, as yet again she enticed him with her flesh and how he loved her luscious figure. Anton gritted his teeth and concentrated on keeping his hands off her. Rising, he walked backwards and slumped into a chair, not once taking his eyes off her curves, the narrow waist and the delectable firmness of her buttocks.
He breathed heavily, sucking air into his lungs as he traced the outline of the apex of her thighs, using only his vision and imagination. He clenched a fist on his lap, tucking the forefinger into the palm, digging the shallow nail into his skin. That finger would be the one to explore her nether region, poke and probe her and judge her readiness. He would have to keep his distance for the rest of the day, to ensure she recognised her time in the corner was not supposed to be sexual in origin. However, the reality gnawed at him. She looked divine—a picture of temptation.
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text:
Honorary Member: Zeta Nu at University of Maryland, College Park
Home » Members » Honorary Member: Zeta Nu at University of Maryland, College Park
By Tessa Marconi on April 3, 2019.
Dr. Robert Matera received his Bachelors degree magna cum laude from Carleton College and his Ph.D. in Classics from the University of Southern California with a dissertation on Propertius. He is an outstanding teacher in Latin and Classics who is much loved by his students here at Maryland and who has inspired a number of them to become Classics majors or to go on to graduate study.


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3.This headband is the perfect accessory to complete your cat costume for Halloween or themed parties.
4.The battery is 1pc CR2032,Irreplaceable.
5.Ten LEDs,accept custom led color.


A compressed working schedule allows an employee to work longer hours on some days and meet their required weekly hours sooner. The employee can then take a day (or part-day) off with the time they have accrued.
Common compressed working schedules include a 4-day week and a 9-day fortnight.
In a 4-day week, an employee works their required weekly hours over 4 days.
Note: The example below shows a 36.25 hours per week roster.
In a 9-day fortnight, an employee works their required fortnightly hours over 9 days.
Another option for employees is to work a compressed weekly schedule and accrue enough time for a half-day off.
When allowing compressed hours of work, be careful not to breach the relevant industrial award or agreement that covers your business.
Learn more about hours of work on the Fair Work Ombudsman website.
For health and safety reasons, some positions may not be suitable for a compressed working schedule; and some may be more suited to a 9-day fortnight rather than a 4-day week. For example, heavy keyboarding positions are not suited to extended hours because of the risk of repetitive strain injuries.
Sick and carer's leave is paid at an employee's base pay rate for each hour (or part of an hour) they take.
Learn more about payment for sick and carer's leave.
more efficient use of workplace facilities or equipment (through increased spread of hours).
Employees on a compressed working schedule will continue to have public holidays and sick leave debited at the applicable daily hours rate (e.g. 7.6 hours). This means that if leave is taken on a day when the employee would normally have worked for longer hours (in order to be able to take a scheduled day off), the employee will need to make other arrangements.
taking a part-day or half-day off instead of a full day.
Regardless of the option chosen, the employee should not be working outside of the normal spread of allowable working hours for health and safety reasons.
If an employee changes from a standard 5-day work week to a compressed schedule, any important responsibilities specific to their day/s off should be reassigned to a colleague.


text:
Co-author of Online Free Speech Petition in China Wins Nobel
On Oct. 8, the 2010 Nobel Peace Prize was awarded to imprisoned Chinese dissident Liu Xiaobo. Liu is a longtime democracy advocate and one of the authors of Charter 08, a manifesto calling for democratization and political reform in China. The Chinese government's reaction to the news was to promptly delete it from search engines and microblogging sites.
CCIA has long opposed Internet censorship efforts by foreign governments as sabotaging what should be the greatest tool for freedom in the world. The Internet enables the widespread dissemination within a society of information and ideas that are the building blocks of democracy.
China's Great Firewall seeks to keep out inconvenient information and prevent such building blocks from taking shape. If democracy is to be brought about through the efforts of men like Mr. Liu who "dream of things that never were and ask why not?" the free flow of information lets them know what already is in other countries and helps them envision what could someday be in China.
Internet censorship is not only about keeping foreign content out. The Internet is also a great enabler of information sharing and discussion within China. Such open and robust debates are not something alien to Chinese culture. There is a long history of it going all the way back to the Hundred Schools of Thought in ancient times.
More recent efforts to embrace that history include Charter 08, an online petition that was issued in December 2008, calling for greater human rights and democratic freedom in China. Modeled after Charter 77, the anti-Soviet document published in Czechoslovakia in 1977, Charter 08 calls for such positions as guaranteeing human rights, democratic elections, freedom of association, freedom of assembly, freedom of expression, and freedom of religion. Mr. Liu was a principal author of this blueprint for a free and democratic China.
Charter 08 was widely disseminated online and signed by 10,000 Chinese citizens online. When a group of reformist Communist Party elders (including a former secretary to Mao Zedong) published a letter last week calling for an easing of restrictions on freedom of speech, they did so on the microblogging service of China's largest news portal. The Internet is already functioning as a modern-day printing press –even in China.
As the free world salutes China's most prominent freedom activist, it is appropriate for those of us in the Internet industry to reflect once again on the potential power of a free Internet to aid him in his struggle, and how we can counter efforts to undermine that power.
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If your New Year's resolution was to see more of the world in 2017, we've already done some of your research: Here are the 15 best countries to visit for your next trip. What's that? Can we be more specific? Surely. Those precious vacation days are not to be squandered!
2017 brings with it a host of brand-new attractions, buzzed-about hotels, museum openings, and one-of-a-kind adventures to be had. (A night on a super-luxe passenger train through the Andes? Why not.) This is the world's newest and coolest stuff worth squeezing into your travel itinerary this year -- or at the very least, filing away in the "definitely someday" folder.
In case 2017 turns out to be just as god-awful as 2016, plan to spend the first half of 2018 as a citizen of the sea. The maiden voyage of the Viking Sun will be a 151-day odyssey from Miami to London, stopping in 66 ports in 35 countries. For the NBD price of 31,000 pounds (which Google pegs at 38,192.31 US dollars), you'll go from Cuba through the Panama Canal, down to New Zealand and Australia, and through the South Pacific to Tahiti and much of southeast Asia. Stops in Egypt and Tunisia are on the European itinerary before the trip ends in Greenwich. And for those long, long days at sea, the ship has a retractable roof pool and in-room flat-screens with plenty of movies on demand.
Dubai likes to brag about having the largest of everything, but its sand-stuffed brother to the west will soon boast the world's largest hotel. The 12-tower, 10,000-room Abraj Kudai cost $3.5 billion and is targeted toward the estimated 15 million Muslims who make the pilgrimage to the holy city every year. It'll boast 70 restaurants, designated sections for visiting royalty, and one of the world's largest domes on top, which will house the ballrooms and conference center.
Writers have the enviable (or unenviable?) task of telling the world's best stories; now the American Writers Museum aims to tell their stories. Here you'll not only see cool artifacts like furniture from the homes of famous authors and original manuscripts of classic literature (On the Road FTW!), but interactive exhibits will give visitors an inside look at the writing processes of some of our best minds. Not sure if that exhibit will include an open bar or LSD, but there will be a children's gallery and some vocab-bending word games.
The Andes Mountains are a magical sight to see. Actually traveling through them? Unless you're big into treacherous drives on semi-paved roads, it can be a little dicey. If you've got the dough, South America's first luxury sleeper train makes the journey as memorable as the destination, starting at $462 per person for one night. Heading from Cusco through the Andes, the train will traverse one of the highest train routes in the world, reaching almost 15,000ft before heading into the floating islands of Lake Titicaca and Arequipa. It'll offer one- and two-night trips, with an open-air car up front to take in the views, rotating seasonal menus, luxe suites for 68 passengers, and an open bar.
London's fabled Victoria and Albert Museum was a pioneer in showcasing Chinese art to Europe. Now it's helping to launch a sprawling new cultural center in Shenzhen, where it and the Guanfu art museum will each have its own permanent exhibition space. The center will also boast a multipurpose theater, a rooftop garden, restaurants, shops, and plenty of public programming.
This is like the Disneyland of Italian food. Maybe it doesn't have Spaghetti Mountain, or any rides really, but Eataly World will be a 30-acre park devoted to the "farm-to-fork" experience, featuring active farms and livestock, markets, cooking workshops, and educational programs -- all devoted to enthusiasts of Italian food and wine. There will also be 25 restaurants to enjoy all the food you've just seen, creating an international gathering place for all those interested in the cuisine of the boot.
This spring, everyone's favorite volcanic zone is finally opening an entire interactive science center devoted to tectonic activity. The new Lava Centre sits on Iceland's Ring Road, at the base of some of the island's most impressive volcanoes, and will feature an earthquake simulator, artifacts from historic eruptions, and educational exhibits that demonstrate the power of the island's volcanoes. A 360-degree viewing platform will also offer views of the Katla, Hekla, and Eyjafjallajökull volcanoes, and will serve as a jumping-off point for tours and treks into the mountains.
If you're into stuff like wine tours and bourbon trails, you'll love this caffeine-fueled equivalent. Major coffee producers like the El Carmen Estate will be opening up their roasting houses and coffee forests for agritourism, so visitors to El Salvador can see how red-and-green berries end up as their morning pick-me-up. And, yes, tasting is part of the experience.
You may not have heard of the Harland and Wolff drawing office on Queen's Island in Belfast, but you've definitely heard of the ships that were built there. The birthplace of the world's most famous ocean liner (we're talking about the Titanic, dude) has sat vacant since 1989, and reopens this year as a boutique hotel devoted to Belfast's industrial and shipbuilding history. Billed as the "world's most authentic Titanic-themed hotel," there will be no mention of Jack Dawson or Celine Dion playing in the lobby, but you can expect to learn all about the famous ship, the people who built it, and how such a monstrous vessel gets constructed.
This is a five-day trek through the most spectacular wilderness of Southern Australia, where you'll hike and camp along the new 38-mile trail. Yes, you'll see plenty of hopping marsupials, but they're almost a sideshow to the stunning South Ocean vistas you'll find as you wind your way over seaside cliffs and past isolated beaches and stunning rock formations like the Remarkable Rocks and Admirals Arch. And there are loads of freshwater lagoons along the way to take a dip when the sun gets too hot.
Sure, you learned a little about the Boston Tea Party in seventh-grade history class, but it's been pretty much a brain dump since then. Get patriotic and re-educate yourself at this new interactive museum, opening two blocks from Independence Hall. On display you'll find George Washington's headquarters tent, firearms used in the war, soldier diaries, camp equipment, and battle-front-line simulators.


text:
Kiwis halves decision proves costly
Author Play The Ball
Publish date Nov 2, 2015
benji marshall england international kiwis new zealand peta hiku stephen kearney tuimoala lolohea
International Game
Kiwis coach Stephen Kearney's decision to leave Benji Marshall out of his touring squad has been exposed as England eased to a 26-12 win in the first test in Hull.
With test regulars Shaun Johnson, Kieran Foran and Thomas Leuluai all unavailable for the tour, former test captain and Dally M Player of the Year runner-up Marshall loomed as the obvious replacement to provide experience and class in the halves. But for some unexplained reason Kearney overlooked the Dragons half, opting instead to play rookie Warriors utility Tuimoala Lolohea in the all-important number seven jumper, alongside makeshift five-eighth Pita Hiku, while Marshall remained Down Under and underwent surgery that could have been delayed until after the tour.
The Kiwis attack was stunted throughout the match while the kicking game was abysmal. The memories of a superb Four Nations tournament and break-through ANZAC Test win in the past year are fading fast as the number one ranked side in the world stare down the barrel of a series defeat unless they can find a significant turn-around at the Olympic Stadium in London next weekend.
Kearney has never properly addressed the reasons for leaving Marshall out. The best he has managed is to suggest there were no personal issues between the pair and that the decision to overlook him was based around form and a plan to pick players who he can develop before the next World Cup in 2017.
Form can't surely be the issue here – Marshall enjoyed a better 2015 NRL season than ANY Kiwi half, leading the Dragons to the post-season and finishing second only to Johnathan Thurston in the Dally M Player of the Year standings. The idea of developing halves ahead of the next World Cup certainly has merit but playing Hiku out of position makes little sense. Lolohea is clearly a talent and Kearney could still have partnered the Warriors rookie up with Marshall in the halves on this tour and instigated his development in a position he's played only a handful of first grade games in.
Kearney's move to play Lolohea and Hiku together in the halves is akin to a basketball coach selecting a power forward at point guard and wondering why the team's offense wasn't able to click.
Despite possessing a forward pack and outside backs that are more than capable of matching anything England can throw up, the disadvantage of playing without territory on the back of a weak kicking game and without the play-making ability to construct tries leaves the Kiwis too much to do.
A series loss in England just erodes the ground the Kiwis have worked so hard to gain in the past 12 months.
Reactions: ShaunJohnson7 and OMG
Play The Ball
This article has been written by Dale Budge. Dale is a well known rugby league commentator and journalist, having covered the sport for over 10 years.
Website : playtheball.co.nz
Twitter : @playtheball
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text:
SCHEDULE 13D
(Amendment No. 2)
PERNIX THERAPEUTICS HOLDINGS, INC.
(Name of Issuer)
Common Stock, par value $0.01 per share
(Title of Class of Securities)
71426V108
(CUSIP Number)
Cetus Capital III, L.P.
Cetus Capital II, LLC
Littlejohn Opportunities Master Fund LP
SG Distressed Fund, LP
8 Sound Shore Drive
(Name, Address and Telephone Number of Person
Authorized to Receive Notices and Communications)
Copy to:
Shon Glusky, Esq.
Jason Schendel, Esq.
Sheppard Mullin Richter & Hampton LLP
(Date of Event Which Requires Filing of this Statement)
If the filing person has previously filed a statement on Schedule 13G to report the acquisition that is the subject of this Schedule 13D, and is filing this schedule because of Rule 13d-1(e), 13d-1(f) or 13d-1(g), check the following box. o
Note: Schedules filed in paper format shall include a signed original and five copies of the schedule, including all exhibits. See §240.13d-7 for other parties to whom copies are to be sent.
* The remainder of this cover page shall be filled out for a reporting person's initial filing on this form with respect to the subject class of securities, and for any subsequent amendment containing information which would alter disclosures provided in a prior cover page.
The information required on the remainder of this cover page shall not be deemed to be "filed" for the purpose of Section 18 of the Securities Exchange Act of 1934 ("Act") or otherwise subject to the liabilities of that section of the Act but shall be subject to all other provisions of the Act (however, see the Notes).
CUSIP No. 71426V108
Name of Reporting Persons
Check the Appropriate Box if a Member of a Group
SEC Use Only
Check if Disclosure of Legal Proceedings Is Required Pursuant to Items 2(d) or 2(e) o
Citizenship or Place of Organization
Beneficially
Person With
Sole Voting Power
Shared Voting Power
Sole Dispositive Power
Shared Dispositive Power
Aggregate Amount Beneficially Owned by Each Reporting Person
Check Box if the Aggregate Amount in Row (11) Excludes Certain Shares o
Percent of Class Represented by Amount in Row (11)
Type of Reporting Person
(1) Based on 61,127,615 shares of common stock outstanding as of March 3, 2016, as reported in the Annual Report on Form 10-K filed by the Issuer on March 10, 2016.
This Amendment No. 2 to Schedule 13D ("Amendment No. 2") amends the statement on Schedule 13D filed with the Securities and Exchange Commission (the "SEC") on March 3, 2014, as amended by Amendment No. 1 thereto filed with the SEC on August 12, 2015 (together with this Amendment No. 2, this "statement") relating to shares of common stock, par value $0.01 per share (the "Shares") of Pernix Therapeutics Holdings, Inc., a Maryland corporation (the "Issuer"). Items 2, 3, and 5 are hereby amended and supplemented as set forth below.
Item 2. Identity and Background
(a) This statement is filed by the individuals and entities listed below, all of whom together are referred to herein as the "Reporting Persons".
(i) Cetus Capital III, L.P., a Delaware limited partnership ("Cetus III"), whose general partner is Littlejohn Associates V, L.L.C., a Delaware limited liability company ("Littlejohn V").
(ii) Cetus Capital II, LLC, a Delaware limited liability company ("Cetus II"), is a single member LLC whose only member is Littlejohn Fund IV, L.P., a Delaware limited partnership, and whose manager is Littlejohn Managers LLC, a Delaware limited liability company.
(iii) Littlejohn Opportunities Master Fund LP, a Cayman Islands limited partnership ("Littlejohn Opportunities Master Fund"), whose general partner is Littlejohn Opportunities GP LLC, a Delaware limited liability company.
(iv) SG Distressed Fund, LP, a Delaware limited partnership ("SG Distressed Fund"), whose general partner is Littlejohn Opportunities GP.
(b) The address of the principal business office of each of the Reporting Persons is 8 Sound Shore Drive, Suite 303, Greenwich, CT 06830.
(c) Cetus III, Cetus II, Littlejohn Opportunities Master Fund and SG Distressed Fund are private investment vehicles engaged in investing in debt and equity instruments.
(d) — (e) During the last five years, none of the Reporting Persons or, to their knowledge, any of the directors or executive officers of the Reporting Persons, has been convicted in a criminal proceeding (excluding traffic violations or similar misdemeanors) or was a party to a civil proceeding of a judicial or administrative body of competent jurisdiction as a result of which such person was or is subject to a judgment, decree, or final order enjoining future violations of, or prohibiting or mandating activities subject to, federal or state securities laws, or finding any violation with respect to such laws.
(f) Cetus III is a Delaware limited partnership. Cetus II is a Delaware limited liability company. Littlejohn Opportunities Master Fund is a Cayman Islands limited partnership. SG Distressed Fund is a Delaware limited partnership.
Item 3. Source and Amount of Funds or Other Consideration
Item 3 is hereby supplemented by adding the following additional disclosure:
A total of approximately $968,811.28 was paid for the 331,648 Shares purchased on the open market since August 6, 2015, the date of the Reporting Persons' most recent Schedule 13D amendment.
Item 5. Interest in Securities of the Issuer
(a) — (b) The information contained on the cover page of this statement is hereby incorporated by reference.
All calculations of percentage ownership in this statement are based on 61,127,615 shares outstanding as of March 3, 2016, as reported in the Issuer's annual report on Form 10-K filed with the SEC on March 10, 2016. As of the date hereof, the Reporting Persons are the beneficial owners of 3,165,444 Shares, which constitutes 5.18% of the Issuer's outstanding shares of common stock. Cetus III has the sole power to vote and sole power to dispose of
233,545 Shares. Cetus II has the sole power to vote and sole power to dispose of 2,241,177 Shares. Littlejohn Opportunities Master Fund has the sole power to vote and sole power to dispose of 622,898 Shares. SG Distressed Fund has the sole power to vote and sole power to dispose of 67,824 Shares. The information set forth in Item 6 of this statement as amended or supplemented is hereby incorporated herein by reference.
(c) On February 21, 2014, (i) Cetus II acquired $8,812,500 principal amount of the Notes, which were convertible into 2,447,916.9 Shares at Cetus II's option; (ii) Littlejohn Opportunities Master Fund acquired $2,437,500 principal amount of the Notes, which were convertible into 677,083.4 Shares at Littlejohn Opportunities Master Fund's option; and (iii) SG Distressed Fund acquired $1,250,000 principal amount of the Notes, which were convertible into 347,222.3 Shares at Cetus II's option; in each case, from the Issuer in a private placement transaction.
The Reporting Persons conducted 32 open market sales in the ordinary course of business within the past 60 days, as set forth in the table attached below:
Common Stock Sold
Price Per Share ($),
before sales commissions
LJ Opportunities Fund, LP
Cetus Capital III, LP
Except as set forth above, there have been no transactions with respect to the Shares during the sixty days prior to the date of this Schedule 13D by the Reporting Persons or, to their knowledge, by any executive officer or director of the Reporting Persons.
(d) No person other than the Reporting Persons is known to have the right to receive or the power to direct the receipt of dividends from, or proceeds from the sale of, such Shares.
(e) Not applicable.
Item 7. Material to be Filed as Exhibits
Description of Exhibit
Joint Filing Agreement.
After reasonable inquiry and to the best of my knowledge and belief, the undersigned certifies that the information set forth in this statement is true, complete and correct.
Dated: April 15, 2016
By: Littlejohn Associates V, L.L.C., its general partner
/s/ Robert E. Davis
Robert E. Davis, authorized signatory
By: Littlejohn Fund IV, L.P., manager
By: Littlejohn Associates IV, L.L.C., its general partner
By: Littlejohn Opportunities GP LLC, its general partner
EX-99.1 2 a16-8537_1ex99d1.htm EX-99.1
JOINT FILING AGREEMENT
In accordance with Rule 13d-1(k) under the Securities Exchange Act of 1934, as amended, the undersigned agree to the joint filing on behalf of each of them of a Statement on Schedule 13D (including any and all amendments thereto) with respect to the to the Common Stock, par value $0.01 per share, of Pernix Therapeutics Holdings, Inc., and further agree that this Joint Filing Agreement shall be included as an Exhibit to such joint filings.
The undersigned further agree that each party hereto is responsible for the timely filing of such Statement on Schedule 13D and any amendments thereto, and for the accuracy and completeness of the information concerning such party contained therein; provided, however, that no party is responsible for the accuracy or completeness of the information concerning any other party, unless such party knows or has reason to believe that such information is inaccurate.
This Joint Filing Agreement may be signed in counterparts with the same effect as if the signature on each counterpart were upon the same instrument.
IN WITNESS WHEREOF, the undersigned have executed this Agreement as of April 15, 2016.