→ Delivery date:about 25-31 days after the confirmation.
Note: we could do the different drained weight based on the client's requirements.
Unit 11C-2, Tower A of Haifu Center, No.597 Sishui Road, Huli District, Xiamen, Fujian Province, P.R.China.


Q. What types of furnishings do you offer?
A. We carry affordable home furnishings and accessories that offer a unique blend of modern, antique, and vintage with a twinge of European influence. We are actively involved in buying our pieces by the whole container, which allows us to share the savings with our clients. We also have chandeliers, wall sconces, and outdoor fixtures, along with designer table and floor lamps.
Q. Who is your customer?
Q. What are two trends we should look for this spring?
A. Mixed Metals: Platinum, gold, copper, steel—as long as they've got metallic sheen, they go together. It's bling like you've never seen before. While gold is not a brand-new trend, there are a lot of people who are still trying to get used to the idea of using it again in decorating. Texture: This was really big at market, including lots of faux fur—especially white fur. We saw it used on chairs and ottomans. It's so cozy and comfortable to relax in. We also are seeing chunky textures such as heavy knits and crocheted pillows and throws.
Q. If you could only redo one room in the house, what would it be?
A. It would have to be the place where family gathers. I have always advised clients on a limited budget to spend it in the one room where you will spend time together. Young couples will often spread out their purchases from room to room and never finish one space. This makes their home seem sparse, cold, and uninviting.
Q. The store really IS incredible. How else are you using this space?
A. We have started offering the showroom for special events on Thursday evenings. (We stay open later on Thursdays.) We've had book signings, women's groups, informal business presentations, decorating 101 classes, and tutorials on how to change the fabric on a dining room chair. Our customer's imagination is the only limit.
Urban Home Market showcases furnishings and accessories in 17, 000 square feet of retail space.
PAY ATTENTION TO THE DETAILS A few well-thought-out architectural details or accessories can turn a bland room into a grand room.
DON'T TIP THE SCALE Nothing makes a big space look out of scale more than a room full of tiny furnishings and accessories. For large walls, try grouping a collection of nicely framed art and a statement mirror. Remember to keep accessories focused and in proportion to the size of the room. Always group collections. A lot of small things scattered around can quickly become clutter rather than points of interest.
LIGHT UP YOUR LIFE The right (or wrong) lighting can instantly change the mood of a space. Ideally, a room should have at least three sources of lighting: natural, ambient, and task or accent lighting to add interest.
MIX IT UP A blend of eclectic furnishings and accessories makes a space look lived in. When using a palette of creams and grays, it is important to use a few pops of color.


When the US was in the midst of a recession (2007 – 2009), TBF saw a significant reduction in tag and release reports, and it comes as no surprise; the price of fuel was at an all-time high during 2008 as well as many people redirecting their expenditures towards other hobbies. In the years following the start of the recession, TBF saw large fluctuations in the amount of data coming in, but overall, the numbers remained fairly low. Participation from the captains, anglers, and mates who have long supported our mission via the Tag & Release Program began to make a comeback in 2012, and really began to boom in 2014. We are so excited to see cards pouring in from around the world, and also to see so many fishing crews are utilizing the self-entry Tag & Release Online database (www.tagbillfish.org). If you haven't already done so, please create a profile and test it out for yourself, but don't get too comfortable because we are in the process of making the data entry even more streamlined. While you're there generating your personal logbook, you can explore areas where fish have been caught around the world or look up marine weather wherever you plan to take your next trip.
Total number of fish reported tagged or released without a tag since 2009.
Take a look, the numbers speak for themselves, participation is on the rise. Thanks to all of your efforts, we're able to translate the valuable data we receive and utilize it to push for sound billfish conservation policy and advocate for angler rights and access. We know you're going out fishing anyway, and you're certainly not required to take the time out when things get crazy in the cockpit to tag a fish or write down the information necessary to report your releases. But the fact so many of you do take the time only goes to reinforce what we already know: our collective success can be attributed to this widely dispersed group of anglers, captains and crews, and the fervor and commitment with which specifically the billfish sportfishing community display to ensure a future for the species. We applaud you, and thank you for your continued support of TBF, because of which the 2014 Conservation Report will show numbers higher than the years before, and we hope this is a trend we will continue to enjoy for many years to come.


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What Not to Miss in New York During the Christmas Season
From left: The Lobster Club, courtesy S. Frances; Rockefeller Center; goop GIFT, courtesy Adrian Gaut
Few places are as festive over the holidays as New York City—a destination that turns into a virtual snow globe in winter, teeming with seasonal pop-ups, caroling and tree-lightings, a rich roster of shows and cultural events and more. Classic activities, like ice-skating at Rockefeller Center, taking in a performance of the Nutcracker, shopping at Bryant Park Winter Village and viewing the beautiful Christmas windows along Fifth Avenue, remain steadfast in their popularity, but there are also lots of new attractions getting attention. From a pop-up at one of our favorite boutiques on the Upper East Side that is ideal for purchasing stocking stuffers to Broadway comedy starring Amy Schumer and written by Steve Martin, here are the best things to do and see during this New York Christmas.
Contact Indagare for assistance planning a trip to New York City. Our specialists can book you at the hotel that is right for you and plan guided tours, great meals and special insider activities.
Seasonal Pop-ups: Holiday Shopping
goop GIFT, courtesy Adrian Gaut
No matter the holiday you celebrate, New York has an abundance of pop-ups this winter that are perfect for scooping up last-minute gifts. Leading the charge is goop GIFT (25 Bond St., through Dec. 24), a NoHo space teeming with Gwyneth Paltrow–approved finds organized into easy-to-shop stations for each type of person on your list (like the host, the traveler, the animal lover). On the Upper East Side, Fivestory—one of Indagare's favorite boutiques for high-end fashion—is hosting a Follain pop-up (through Jan. 15) where shoppers can scoop up the retailer's bestselling clean beauty products. Home décor enthusiasts should plan a visit to the Sugarplum Pop-Up (through Dec. 24) at the St. Regis, where interior designer Rebecca Gardener is selling a curated selection of delicate tableware, holiday gifts and entertaining staples. For a dose of girl-power and appropriate goodies for all the women in your life, check out the Bulletin Mini Mall (927 Broadway; through Jan. 15), a seasonal feminist boutique that exclusively sells products made by women and donates 10% of its proceeds to Planned Parenthood. Finally, Brooklynites needn't leave their nave to do any shopping thanks to FORM VINTAGE (through Dec. 20), a holiday pop-up at the Wythe Hotel that is selling jewelry, fashion and ceramics from local Brooklyn artists.
Fuel Up: Holiday Dining in New York
Courtesy The Lobster Club, S. Frances
Between shopping and museum-hopping, you'd be wise to plan out nearby meals at some of the city's best new eateries in the city's highest trafficked areas. A stylish newcomer, Blue Box Cafe at Tiffany's brings the brand's signature packaging to life with its beautiful robin's egg–blue space, where it serves tea and light café fare like avocado toast and truffled eggs. One of most anticipated openings of the year, Midtown's Lobster Club opened in November and serves Japanese-inspired brasserie fare from the team behind downtown juggernaut Carbone. Drawing critical acclaim is the elegant Cote, a new Korean steakhouse where patrons can warm up while cooking their own steaks at tabletop grills. For hearty cuisine and an early dinner after shopping in the West Village, Don Angie delivers with delicious lasagna for two and other upscale takes on comforting Italian classics.
Related: Top Tables New York
Culture Heatmap: The Coolest Winter Shows and Exhibits
Archers Shooting at a Herm by Michelangelo, Courtesy the Metropolitan Museum of Art
This winter, New York City is playing host to many cultural options that provide reason to head indoors and escape the chilly weather. One of this season's most impressive exhibitions is the spectacular Louis Vuitton show Volez, Voguez, Voyagez (through Jan. 7), which takes visitors on a visual journey through The House of Vuitton's travel legacy. For an early Christmas treat, surprise a loved one or friend with tickets to one of Indagare's exclusive VIP tours of the show on December 13 or 14, which will take participants behind the scenes with a Louis Vuitton Ambassador (contact us to learn more and sign up).
One could easily while away an entire day at the Met, which is currently showing Michelangelo: Divine Draftsmen and Designer (through Feb. 12), a must-see that displays a huge range of the artist's drawings, and a retrospective on David Hockney (through Feb. 25) showing some of his most iconic pieces. Unveiling much of the vast holdings of art dealer Eugene V. Thaw, Draw to Greatness: Master Drawings From the Thaw Collection (through Jan.7) at the Morgan Library and Museum features works by an array of artists including Vincent Van Gogh and Edgar Degas.
Courtesy the Metropolitan Museum of Art, David Hockney
Theater buffs have equally exciting reasons to plan a visit to the Big Apple. From House of Cards creator Beau Willimon, political drama The Parisian Woman (Hudson Theater) stars Uma Thurman and is set in Washington D.C after the 2016 presidential election. Time is running out to see Bette Midler as the star in the critically acclaimed Hello, Dolly! (Shubert Theater), as her last performance in the Broadway revival will be on January 14 (Bernadette Peters will assume the role in her absence). For serious laughs, purchase tickets to Meteor Shower, a hilarious new comedy from Steve Martin that stars Amy Schumer (through Jan. 21).
– Emma Pierce on November 30, 2017
Emma Pierce
PlusPicks
Elegant grandeur, luxurious, historic
The Wythe Hotel
Hip, industrial-chic, young

redpajama_set_name: RedPajamaCommonCrawl


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Mysteries of Jeju Island
Jo Injeung – 'Mysteries of Jeju Island'
B&W Jo Injeung, Mysteries of Jeju Island
Jo Injeung's series 'Mysteries of Jeju Island' – "a journey to the heart of the island, where the perpetual motion of nature is captured by the eye of the camera, discovering timelessness in a frozen moment."
Jo Injeung – Mysteries of Jeju Island
"Jeju Island is the biggest island and the smallest province of South Korea. It is a hidden gem in Asia with its pristine forests, volcanoes, and waterfalls; a World Heritage site; a true mecca for Korean travelers.
Jo Injeung's photographs not only capture the original beauty of Jeju Island but also make a reference to the concept of four elements, significant to the Korean culture. All four elements unite in Jeju-do, constituting the island's greatest mystery."
source Rosphoto.Museum and L'Œil de la Photographie


In early August 2014 FICO, there were several key points about the score that went uncovered, until now. Here's what you don't know about FICO 9, but really should.
A few years ago Experian began adding residential rental history to their credit reports. Recently TransUnion announced they'd be doing the same. That's great news for consumers who have little to no credit history BUT pay their apartment rent on time and want their credit reports to reflect them doing so. All FICO score versions prior to FICO 9 did not consider rental tradelines, even if they were on your credit report. FICO 9 will consider these accounts.
When FICO 9 Will Become Available?
The word on the street is that FICO 9 will be available sometime in the fall of 2014 or in early 2015 (as of today 8/4/15, it's currently not being utilized). Why the varying release dates? The answer is because FICO doesn't control when their scores will become commercially available because they don't sell their own scores. The credit bureaus sell FICO scores.
Because the FICO software is installed on the mainframes of the credit bureaus, the bureaus are the ones that will make the score available when they're ready to do so—not FICO.
There still seems to be some confusion about when a collection will be ignored by FICO 9. FICO 9 will ignore all collections as long as they have a "zero balance". Their press release suggests that all "paid" collections will be ignored. The assumption then became that only paid collections and not settled collections would be ignored, which is incorrect. According to FICO, any collection that has a zero balance will be ignored—regardless of whether it was "paid in full or settled". The collection agency community has to love this as it acts as an incentive to pay or settle your collections.
The challenge for FICO is explaining to lenders which of the two scores are actually the "right" score to use for underwriting. They have to say that the newer score is the valid score because it's their latest and greatest. But that would also suggest that the score the lender has been using for the past however many years was also horribly wrong about that particular consumer's risk level.
Yes, it's great news that collections that have a zero balance will be ignored. What nobody is talking about, however, is that the collections will still be physically present on the consumer's credit reports until they've hit their seven-year reporting limitations. And while the scoring model will ignore them, lenders certainly don't have to follow suit. It's entirely reasonable to expect that lenders will continue to have concerns about debts that have gone to collections, despite being paid or settled. And, they can still hold those against the consumer and deny them credit or saddle them with less beneficial terms.
Stay tuned for next topic on why there is a "credit score arms race happening." If you or someone you know needs help with their credit reports in Indianapolis" contact us for a free review today. We are Indianapolis's only Full Service Credit Consulting Company, InCreditable Advisors, please visit us at www.creditindy.com or give us a call at 317.202.1297 to schedule your appointment today.


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Q: Fn-lock with A1243 keyboard under windows? I am looking for a fn-lock function under Win 8.1/x64 for my apple keyboard. Most likely this has to be supported by the keyboard driver for the apple keyboard.

*

*Not all function keys work on my Apple keyboard as intended. F1/F2/F5/F6/F7 appear to try to fulfill their assigned functions. Keyboard is not defective. there are two to try and using the fn key, the codes are created as intended.

*IMHO this used to work, just creating F1-F12.

*This should not be a Bios configurable issue, as the standard F1-F12 are created on a normal windows keyaboard.

*This is not under Bootcamp, so the "configure bootcamp" option is not available.

*Key mappers do not permit me to remap function keys. Also the location of the fn-lock key for this keyboard is unclear. What is it on Macs?

I am therefore looking for either
a) a way to toggle fn-lock for the Apple keyboard under windows (without the bootcamp menu)
b) an option for a keyboard driver for the apple keyboard under Windows.
Thanks.

redpajama_set_name: RedPajamaStackExchange


House is on a cul-de-sac that backs up to the park!..
Watson Estates is a community that offers 2 and 3 bedroom manufactured homes and large grassy lots for rent. It is located on the outskirts of Chickasha, only a 35-minute..


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Stock market bulls have got it wrong, warns Nouriel Roubini
By Nick Clark
Global Research, April 22, 2009
The Independent 21 April 2009
Nouriel Roubini, the so-called "arch bear" economist who predicted the current financial crisis in 2006, added further gloom yesterday after he wrote off recent rises in global stock markets as no more than a dead cat bounce.
While an increasing number of analysts have in recent weeks urged investors to go back into equities, Mr Roubini, a professor at New York University's Stern School of Business who has emerged as one of the most respected economic voices in the wake of the credit crunch, warned yesterday that he didn't yet see a buying opportunity.
He holds little faith in the recent market rallies, which prompted some to suggest a recovery was underway. "I'm still cautious and bearish," he said. "I believe we are closer to a bottom in the stock market than a year ago, but this is a bear market rally."
Anthony Bolton, fund manager at Fidelity International, said last month that a bull phase had started, while analysts at Goldman Sachs have argued in recent weeks that "we are past the low point in the economic cycle".
However, Mr Roubini, dubbed "Dr Doom" for his warnings about financial meltdown, said there would be more bad news in the next few quarters.
In particular, the economist warned of further dangers ahead for the financial services industry in the US. "I see financial shocks in the months ahead. Some financial institutions are in so much trouble we may have to take them over," he said, before adding that losses in the industry could rise from $1 trillion to as high as $3.6 trillion.
Firms from across financial services will go out of business or be taken over, he said, particularly focusing on the bleak future for hedge funds.
Mr Roubini also disagrees with more optimistic forecasts for the US economy. In an interview published on Forbes.com yesterday, he said that the prediction of a 2 per cent growth rate next year was far too bullish. He called it at somewhere around 1 per cent. "So while we are going to be technically out of a recession, it is going to feel like a recession," he added.
He blamed weak recovery, deflation which would dog the US for the next two years, and financial shocks for the lower-than-expected growth.
Investors should stay on the sidelines, he cautioned, but added that there was light at the end of the tunnel. "I'm not a permabear. I believe that actually, if we do the right things, the US, Europe, Japan, but especially emerging markets can have a bright future of high economic growth."
Mr Roubini criticised market regulators, saying that while deregulation was positive "we took it to an extreme". He added: "Even financial markets need laws, institutions, rules; otherwise it is the law of the jungle."
Quoting from Oliver Stone's film Wall Street, he added: "Greed is good. There is nothing bad with greed. You know, that's what drives capitalism. But greed has to be contained by fear of losses and also realisation you are not going to be bailed out in bad times."
The original source of this article is The Independent
Copyright © Nick Clark, The Independent, 2009
Articles by: Nick Clark


Chase Allen has been featured in Coastal Living Magazine, Southern Living, Hilton Head Monthly, Charlston Home & Charlston Magazines.
"Iron Fish" Sculptures make for the perfect abstract wall art that really makes a statement. These vintage nautical decor pieces are sure to be a focal point where ever you place them.
Each of Chase Allen's signature handmade aquatic decorative sculptures are carefully oxidized, hand-forged, welded and shaped using traditional blacksmithing and self-taught methods, often using hand crafted tools. Each piece has a unique vintage finish that took Chase years to develop.


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PeopleGrove Administrators
Technical Setup
API and Webhook Documentation
Automated Data Integrations: From Your Database to PeopleGrove
Configuring G Suite (Google Apps) SSO with PeopleGrove
Custom Domain Setup
Data Imports
Email Deliverability
FERPA Compliance
Importing Data into PeopleGrove
Reilly Davis
What is FERPA?
What data does this cover?
How is FERPA data legally shared with or collected by PeopleGrove?
How does PeopleGrove ensure FERPA compliance with data once within the platform?
What can partners do to limit FERPA issues?
This document is intended for any PeopleGrove client or potential client who wants to review how PeopleGrove approaches FERPA data.
FERPA is a federal law that prohibits schools from disclosing personally identifiable information from a student's educational record to a third party without written consent from the parent. This right transfers to the student when the student turns 18 years old or matriculates at a college or university.
FERPA covers:
"Educational Records" - Records that directly relate to a student and maintained by an educational agency/institution or by a party acting for the educational agency
"Personal Information" - Direct identifiers (such as name, family name) and indirect identifiers (date of birth, mother's maiden name)
PeopleGrove does often collect or receive data that would be covered by FERPA when launching the platform and importing users.
For example, the following data could be covered by FERPA:
Name and email
School, major, degree, and graduation date
GPA or total credits
Gender, ethnicity, or other demographic data
Schools share data with PeopleGrove under the "school official" exception to FERPA. Under this exception, schools may share personally identifiable information from the educational record without parent/student consent as long as the vendor:
Performs a service or function for which the school would otherwise use its own employees (i.e. is acting as an outsourced service provider)
✔ PeopleGrove operates an an outsourced service provider
Is under the direct control of the school with regard to the collection and use of data
✔ Our contracts specify that our clients own the data, and have full control over the use of this data
Uses data only for authorized purposes and does not re-disclose personally identifiable information from the education record to other parties unless with consent of the school or permitted by FERPA
✔ PeopleGrove does not disclose this data to 3rd parties without consent and does not use personally identifiable data for any other purposes
With these conditions met, there is no legal concern with sharing student data with PeopleGrove after a contract has been signed.
When institutions disclose information to PeopleGrove that may be subject to FERPA, there is immediately no concern as we are acting as a "school official" and are not re-disclosing this data. However, once the PeopleGrove platform is configured and launched, PeopleGrove still needs to ensure that there is no unauthorized disclosure of personally identifiable information.
There are two ways in which this may be addressed.
Admin-Only Data
First, there is the concept of admin-only data within the PeopleGrove platform. This data may be stored in the PeopleGrove platform, but is not visible or accessible to anyone other than administrators of the platform - all of whom would also be school officials authorized to view this data. This data is only used for reporting and for the benefit of platform administrators.
For example, if a partner wanted to populate a student's class level and ethnicity so that reports could be pulled for meetings by ethnicity and class level, that would be imported into PeopleGrove as an admin-only field.
Public Profile Data
Second, there is public profile data. This data may be imported into the PeopleGrove platform and displayed on a user's profile, once the user has activated their account. For this data, PeopleGrove shows the user all data that has been populated during the signup process, meaning that the user has explicit control over the data that is shared within the community.
Since the user is explicitly consenting to this disclosure, it is permitted by FERPA.
The bottom line is that PeopleGrove will never disclose personally identifiable information without a user's consent. Since users opt-in to the platform, with a clear expectation of why they are opting in, FERPA should not be an issue.
Communicating with Students
Within the signup process, PeopleGrove allows our partners to insert a legally-binding document with an e-signature pad for all users to sign. This would be an appropriate place to add language letting students know that information they have provided or reviewed during the signup process could be disclosed with other members of the community.
This step would look like this:
This signup step and appropriate language should be enough to constitute informed disclosure by the student. You are able to view the signed agreements through the admin panel.
We believe that the steps outlined above are sufficient to ensure full FERPA compliance.
However, it is fully possible to launch the PeopleGrove platform without sharing any data with us in advance. If this is the case, all data is self-reported by users of the platform for the explicit purpose of sharing within the community, making the data not covered under FERPA. If your institution has an issue with sharing data in advance, that's not a problem.
This document does not constitute legal advice and has not been vetted by a lawyer. We request that all of our partners review this document with their legal counsel, as every situation is unique.
Privacy Policy and GDPR Compliance


The cast of THE TEMPEST. Photo by TOMMY THOMPSON. 2006.
Susan Heyward and David Loar in THE TEMPEST. Photo by TOMMY THOMPSON. 2006.
John Harrell and Susan Heyward in THE TEMPEST. Photo by TOMMY THOMPSON. 2006.
Sarah Fallon in THE TEMPEST. Photo by TOMMY THOMPSON. 2006.
Álvaro Mendoza and the cast of THE TEMPEST. Photo by TOMMY THOMPSON. 2006.
Paul Fidalgo and Matthew Sincell in THE TEMPEST. Photo by TOMMY THOMPSON. 2006.
John Harrell in THE TEMPEST. Photo by TOMMY THOMPSON. 2006.
Twelve years ago, Prospero, the Duke of Milan, was pushed from his throne by his brother, Antonio, with the help of Alonso, King of Naples, and the King's brother, Sebastian.
Cast adrift with his daughter, Miranda, Prospero landed on an island, where, by the use of his magic, he has ruled over the native Caliban and the spirit Ariel.
Hearing that his enemies are at sea, Prospero has told Ariel to raise a storm that will shipwreck them upon the isle and allow him to work his revenge.
On the island, the shipwrecked travelers are separated: Alonso, the King, searches for his son Ferdinand, feared drowned; Sebastian plots to kill his brother and usurp the crown of Naples.
The drunken butler, Stephano, and the jester, Trinculo, are persuaded by Caliban to kill Prospero.
Ferdinand meets Miranda and they fall in love.
Prospero sets heavy tasks to test Ferdinand and then presents the young couple with a betrothal masque celebrating chastity and the blessings of marriage.
Last season at the Globe I wrote in the program about that "scene in which Troilus and Cressida fall in love." By the time our rehearsals were over I had decided that they never were in love, simply infatuated. So program notes are hard – you never know what you are going to discover once rehearsals begin.
I'm fairly sure of this though: the longer I work on Shakespeare's plays the more I feel that clarity for the audience is achieved when the thoughts of each of the characters are fully realized. Shakespeare's words are incomparable, but there, before the words, are these thoughts that give rise to the words.
If we can bring theses "thoughts" to life, you will follow everything. I hope that when the play is over, you might even say "Well that was so clear; but you've modernized the text haven't you?" when we will have done no such thing!
"Clarity" doesn't mean explaining everything and erasing the play's ambiguities. Shakespeare is drawn to paradoxes and problems: jealousy, ambition, desire, unfaithfulness; weakness rather than strength; selfishness rather than putting others first. His characters are all multi-faceted and ambiguous in many ways.
So can you say that Prospero's a "good" man? He, like so many, is caught between opposing forces: a desire for revenge and the promptings of mercy. The magic powers he has cultivated and for which he is celebrated, have also been the cause of his downfall and expulsion from Milan. Is he really different from Sycora, who was marooned on what later becomes Prospero's island, pregnant with Caliban; marooned, because she also was a magician, and for which the people of Algiers expelled her?
The Tempest is the last play that Shakespeare wrote, apart from a couple of collaborations with John Fletcher. In common with the other late plays certain themes emerge: fathers and daughters; looking towards the next generation; and maybe, above all, a sense of forgiveness and reconciliation. For me, The Tempest is filled by a sense of the new, which touches almost every scene.
says Miranda, when she first sees Alonso and all his court; she, who until that moment, had only ever seen three other people and one of these, Ferdinand, she only met three hours before.
is the comic version of Miranda's, as he mistakenly takes the drunken Stephano for a god who has dropped down from heaven.
The Tempest is also filled with music: Ariel's music. Ariel is Prospero's spirit-helper but unlike Shakespeare's rather earth-bound fairies in A Midsummer Night's Dream, Ariel is all spirit. Yet, it is the spirit who, it seems to me, teaches the mortals how to behave with real humanity, each to other.
Now, will I still believe this in six weeks' time?


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Cricket League is a 3D multiplayer game that features quick two over matches, now up for pre-registration on Android
By Sumant Meena
| Oct 12, 2021
| Cricket League
Cricket League is a new mobile game for all cricket lovers, where they can enjoy playing quick matches in the comfort of their houses. It comes with 3D graphics and a multiplayer mode, which makes it even more fun. Cricket league is now available for pre-registration on Android to become the next MS Dhoni and beat their score.
This sports game has a 1v1 face-off with two overs; go field, bat, and bowl. If batting, bowling, and fielding isn't enough, you also have to be strategic, or you will lose. The controls are straightforward to learn, so you can quickly get the hang of it.
It also features many cricket stadiums; with venues from Mumbai to London. Each stadium also comes with different pitches, so make sure to study the environment properly before jumping into the action.
Winning a match will be crucial as it will earn you more coins that can be used to build your dream cricket team. It also has over 25 characters to add to your team.
Cricket League features quick two-over matches, which means you only get to play 12 balls in a game. You can choose to play against your friends or a random online player. Since the matches are quick to play and finish, it means you can pick it up anywhere anytime and kill time.
Cricket League is a mobile sports game where players get to experience a 3D world of cricket. The objective here is to build a dream team before taking on your friends and other online players to prove that your cricketing skills are the best. With just 12 balls to play in a match, each is crucial, and every run counts, you will need to be tactical if you want to win all your matches.
Cricket League is now up for pre-registration on Android. Interested players can pre-register for the title on Google Play. The game's release date has not been announced yet, but it is expected to release soon on both Android and iOS.
Interested in sports games? Check out our list of the Top 25 best sports games for Android phones and tablets.
Cookie Jam has announced a collaboration with The Price is Right that gets underway today
Minecraft: A few tips to help during your Snow Wars battles
Free Fire: A few tips to take note when exploring Alpine
Phobies lets you collect and battle creepy creatures in a hex-based battlefield, coming to mobile later this year
Warpath best units divided by classes


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Volume 4, Issue S2
Research Article| December 01 2006
The role of disease burden measures in future estimates of endemic waterborne disease
Glenn Rice
US Environmental Protection Agency, 26 W. Martin Luther King Dr. (MS-A110), Cincinnati OH 45268, USA
E-mail: rice.glenn@epa.gov
Matthew T. Heberling
Mary Rothermich
J. Michael Wright
Patricia A. Murphy
Michael F. Craun
Gunther F. Craun and Associates, 101 West Frederick Street, Suite 207, Staunton VA 24410, USA
Gunther F. Craun
J Water Health (2006) 4 (S2): 187–199.
Glenn Rice, Matthew T. Heberling, Mary Rothermich, J. Michael Wright, Patricia A. Murphy, Michael F. Craun, Gunther F. Craun; The role of disease burden measures in future estimates of endemic waterborne disease. J Water Health 1 December 2006; 4 (S2): 187–199. doi: https://doi.org/10.2166/wh.2006.023
The 1996 Safe Drinking Water Act amendments require the US Environmental Protection Agency and the Centers for Disease Control and Prevention to develop a national estimate of the occurrence of waterborne infectious disease that is attributable to public drinking water systems in the United States. Much of the information for developing the national estimate will be derived from epidemiologic data, and the primary outcome of this effort will be an estimate of the number of cases of gastrointestinal illness. While quantifying the number of these cases provides some measure of waterborne disease impact, the usefulness of this measure may be limited because the full spectrum of societal impact also involves consideration of the additional effects of these diseases such as hospitalization costs and lost productivity. If decision-makers wish to compare the impact of waterborne infectious diseases to the impact of some other public health concern (e.g. to aid in resource allocation decisions), then a comparison of case numbers may prove inadequate. Case numbers alone do not provide sufficient information about the severity of different illnesses. Society may value the avoidance of a few cases of severely debilitating illness more than it values the avoidance of many cases of mild illness. In order to compare disparate public health concerns, "burden of disease" measures that incorporate indicators of disease severity, costs, or societal values may prove essential for some types of decisions. We describe epidemiologic measures of severity, quality adjusted life years (QALYs), disability adjusted life years (DALYs), willingness-to-pay, and cost-of-illness methods commonly used for burden of disease estimates, and discuss how some of these summary measures of burden might be used for waterborne disease estimates.
DALY, disease burden, gastrointestinal illness, National estimate, QALY, willingness to pay
©2006 The Author(s)
This is an Open Access article distributed under the terms of the Creative Commons Attribution Licence (CC BY-NC-ND 3.0), which permits copying and redistribution for non-commercial purposes with no derivatives, provided the original work is properly cited (http://creativecommons.org/licenses/by-nc-nd/3.0/).


The final quarter of the year is here, and that means for many organizations the race is on to make the year-end number.
If you work with accounts on a regular basis, a few things you need to find out right now include what their hours of operations are going to be the last two weeks of the year.
With Christmas on a Wednesday, it's hard to know without asking what each of your customers might be doing.
Equally important is finding out in October where they stand with regard to their own numbers. Are they looking to make cuts in orders or looking to increase purchases? Waiting until they tell you can be too late for you to respond effectively, so you need to ask now.
New customers and prospects create their own set of opporutnities. Many companies shut down their Accounting Department to new vendors a week or so before the end of the year.
Last thing you want to do is close a sale, only to find out the new customer's Accounting Department won't process anything until after the new year starts.
Worse yet, with regard to year-end changes, is the common practice of many companies to simply stop taking appointments after Nov 1 to allow them to focus their own energies on making their own numbers.
Problem with this is companies don't advertise this. Typically, it's only found out when a salesperson is attempting to get an appointment.
Another important thing to know is how your own company is going to handle things at the end of the year. I'm always surprised at the number of salespeople who get surprised by things within their own company. Too many salespeople fail to pay attention to emails about changes in their own company.
Be a sales leader in Q4.
It starts by being pro-active in finding out information from your company, your customers and everyone else to ensure you can satisfy as many customers as possible and ulimtately make your year-end number.
PrevPreviousDo You Have a Victim Attitude?
With these great tips, any salesperson who reads this can be a sales leader in Q4. Any information that they can get about their companies, customers and prospects can be converted to useful information to promote sales. What they can do is to gather the data and map them so they can visualize their data more efficiently. This method allows them to see important information that they may have looked over before, information that will help them in their sales process along the way.


text:
Guzmán el Bueno é uma estação da Linha 6 e Linha 7 do Metro de Madrid.

História
A estação foi aberta ao público em 13 de janeiro de 1987, estendendo a Linha 6 da estação Cuatro Caminos até a Ciudad Universitaria, com dois corredores e três acessos externos.

Em 12 de fevereiro de 1999, a seção da Linha 7 entre as estações Canal e Valdezarza, que incluía a estação Guzmán el Bueno, foi inaugurada e aberta ao público.

Ligações externas

Guzmán el Bueno
Estações ferroviárias inauguradas em 1987
Estações ferroviárias inauguradas em 1999

redpajama_set_name: RedPajamaWikipedia


The US State Department added two operatives from the Islamic Movement of Uzbekistan and another from the Islamic Jihad Union to the list of Specially Designated Global Terrorists. The designation allows the US to freeze their assets and prohibits others from engaging in financial transactions with them.
The two Islamic Movement of Uzbekistan (IMU) operatives added to the terrorist list are Yassin Chouka and Mounir Chouka, two brothers from Germany. The Islamic Jihad Union (IJU) operative added to the list is Mevlut Kar. The Chouka brothers and Kar have "links to al Qaeda," according to the State Dept. press release announcing their designations.
The State Dept. described Mounir, who is also known as Abu Adam, and Yassin, also known as Abu Ibrahim, as "fighters, recruiters, facilitators and propagandists for the Islamic Movement of Uzbekistan," a terror group in Central Asia that is on the list of designated Foreign Terrorist Organizations.
Mounir also said that the IMU was involved in the May 19, 2010 suicide assault on Bagram Air Base. One US soldier was killed in the attack. According to the IMU propaganda tape, the Bagram assault was launched "in coordination and cooperation with other jihadi groups" and the assault team included "Turks, Tajiks, Arabs, Pashtuns, and Afghans." Bekkay Harrach, a German national who operated along the Afghan-Pakistani border with the IMU, is thought to have been killed while leading the assault on Bagram. Harrach led a team of 20 fighters made up from the ranks of al Qaeda, the Pakistani Taliban, and the Islamic Movement of Uzbekistan, according to a previous statement released by the IMU. The various terrorist groups carry out military operations in Afghanistan and Pakistan under the aegis of the Lashkar al Zil, or the Shadow Army [for more information, see LWJ report, Al Qaeda's paramilitary 'Shadow Army'].
One of the brothers was rumored to have been wounded in a US drone strike. In August 2011, the Victorious Sect, a terror group with ties to the IMU, issued a statement that Mounir may have been wounded in a US Predator airstrike along the Afghan-Pakistani border. The report was never confirmed.
The State Dept. described Kar as "a facilitator and recruiter for the Islamic Jihad Union," which is also listed as a Foreign Terrorist Organization. Kar has "provided more than 20 detonators to members of the IJU," according to State.
Kar, who is both a German and Turkish citizen, is also known as Mevlut Zikara, Mulfit Kar Iiyas, Abdullah the Turk, and Abu Obaidah al Turki. He was a member of the IJU cell that attempted to attack US military personnel and civilians in Germany in 2007. The targets of the attack included Ramstein Air Base and Frankfurt International Airport. Three members of the cell were arrested before the plot was executed.
In 2003 a Lebanese court convicted Kar in absentia to 15 years in prison "for attempting to establish an al Qaeda cell in Lebanon," the State Dept. designation notes.
"Following 9/11, Zikara allegedly helped members of al Qaeda living in Arab countries to reach Afghanistan via Iran, to fight with Taliban," according to The Australian.
i think you are wrong mike. I guess if you don't want to have them prosecuted or have any possible assets frozen that they may have or if you don't want their faces everywhere, you just do nothing. This way, at least these guys are more likely to get killed, have their money taken or have people sit in jails if they support these nuts.
They are already all over the jihadi media so this doesn't really built up their bona fides.


text:
\section{Introduction}

Given a graph $G$, a degree monotone path is a path $v_1v_2 \ldots v_k$ such that $deg(v_1) \leq deg(v_2) \leq \ldots \leq deg(v_k)$ or $deg(v_1) \geq deg(v_2 \geq \ldots \geq deg(v_k)$.  This notion, inspired by the well-known Erd{\H{o}}s-Szekeres Theorem \cite{eliavs2013higher,erdos1935combinatorial}, was introduced in \cite{updownhilldom} under the name of uphill and downhill path in relation to domination problems, also studied in \cite{deering2013uphill,downhill2,downhill}.  In \cite{updownhilldom}, the authors specifically suggested the study of the parameter $mp(G)$, which denotes the length of the longest degree monotone path in $G$.  This parameter was first studied by the authors in \cite{dmpclz}.  Links between this parameter and other classical paramaters such as the chromatic number and clique number, using in particular the Gallai-Roy Theorem \cite{dougwest} were explored, and lower bounds and upper bounds for $mp(G)$ were established in \cite{dmpclz}.  The close relation to Turan numbers \cite{bollobas2004extremal} was also studied and explained in  \cite{dmpclz}.

In this paper we consider another natural question related to the parameter $mp(G)$, that of the effect of graph operations on this parameter.  We consider both  operations on a single graph $G$  which produce a new graph $G'$, as well as  operations applied to two graphs to produce a new single graph.

In the first section, we consider operations involving edges, namely edge addition and deletion, subdivision and edge contraction while in the second section vertex addition and deletion is discussed.  For each operation we obtain sharp bounds on $mp(G')$, and give constructions which achieve these bounds.  In some cases, we consider the operation for a particular family of graphs which gives more interesting results.

We then consider the \emph{Cartesian product} and the \emph{graph join} for two graphs $G$ and $H$, where again we give sharp bounds and constructions which achieve these bounds.

Any graph theory terms not defined here can be found in \cite{dougwest}.

\section{Edge Operations}

\subsection{Edge Addition and Deletion}

We now look at the concept of adding/deleting an edge to or from a given graph $G$, and consider the effect of these operations on $mp(G)$.  We add an edge $e$ by connecting two vertices in $V(G)$ which are non-adjacent, and the resulting graph is denoted by $G+e$, while when we delete an edge $e$, the resulting graph is denoted by $G-e$.

\begin{theorem}
Given a graph $G$,
\begin{enumerate}
\item{$\frac{mp(G)+1}{3} \leq mp(G+e) \leq 3mp(G)$.}
\item{$\frac{mp(G)}{3} \leq mp(G-e) \leq 3mp(G)-1$.}
\end{enumerate}

In both cases, the bounds attained are sharp.
\end{theorem}

\begin{proof}

Let us first consider the right hand sides of both inequalities.

\noindent 1. \indent Let $G$ be a graph with $mp(G)=k \geq 1$.   If $k=1$ then $G$ is the empty graph with no edges, hence $mp(G+e)=2 \leq 3=3mp(G)$.  So let us assume that $k \geq 2$.

Suppose that $mp(G+e)=t \geq 3k+1$.  Consider $P=v_1,v_2,\ldots,v_t$, a longest degree monotone path in $G+e$ in non-decreasing order.  Observe that $e$ must have at least one of its vertices in $P$, otherwise $P$ was originally in $G$.

\medskip

\noindent \emph{Case 1.1}. \indent Let us assume that $e=(v_i,z)$, where $z$ is not on the path.  If $i=1$, then clearly $v_1,\ldots,v_t$ is also a degree monotone path in $G$ and hence $t \leq mp(G)=k <3k$, a contradiction.

If $i=t$ then $v_1,\ldots,v_{t-1}$ is a degree monotone path in $G$ and hence $t-1 \leq k$ which implies $t \leq k+1 < 3k$.

So let us assume $1 < i <t$.  Let us consider the paths  $v_1,\ldots,v_{i-1}$ and $v_i,\ldots,v_t$ --- both are non-decreasing degree monotone paths in $G$ and hence together they have length at most $2k$, implying that $t \leq 2k <3k$.

\medskip

\noindent \emph{Case 1.2}. \indent Now assume $e=(v_i,v_j)$, $i<j$.  Hence, in $G+e$, all vertices in $P$ have the same degree as they have in $G$ except for $v_i$ and $v_j$ whose degree has increased by one.  Consider the paths $P_1=v_1,\ldots,v_i$, $P_2=v_i,\ldots,v_{j-1}$ and $P_3=v_j,\ldots,v_t$.

$P_3$ is  clearly a  degree monotone path in $G$ since $deg_G(v_j)+1 = deg_{G+e}(v_j) \leq deg_{G+e}(v_{j+1})$ ( or $j=t$), hence $|V(P_3)| \leq k$.  Similarly $|V(P_2)| \leq k$.  So if $t \geq 3k+1$, $|V(P_1)| \geq k+2$ since $v_i$ is both in $P_1$ and $P_2$.  But $v_1,\ldots,v_{i-1}$ is a degree monotone path of length $k+1$ in $G$, contradicting the fact the $mp(G)=k$.  Hence $mp(G+e) \leq 3k$.

\medskip

For sharpness of the bound, consider $P_{3k}$, to which we add a leaf to vertices $v_2$ up to $v_{3k-2}$ except vertices $v_{k+1}$ and $v_{2k+1}$, and we connect $v_{3k-1}$ to a new vertex $z$ to which we add two leaves.  Thus the resulting graph $G_1^+$ has  $6k-2$ vertices.   Figure \ref{diag1} shows the construction for $k=4$. It is clear that $mp(G_1^+)=k$.  Now if we add the edge $e=(v_{k+1},v_{2k+1})$, these two vertices now have degree 3 also, and hence the path $v_1,v_2,\ldots,v_{3k-1},z$ is a degree monotone path so $mp(G_1^++e)=3k=3mp(G_1^+)$.

\begin{figure}[h!]
\centering
\includegraphics{diag1}
\caption{$G_1^+$  when $k=4$} \label{diag1}
\end{figure}

\noindent 2. \indent  Let $G$ be a graph with $mp(G)=k \geq 1$.   If $k=1$ then $G$ is the empty graph with no edges, hence we cannot delete edges.  Therefore assume $k \geq 2$ and suppose that $mp(G-e)=t \geq 3k$.  Let $P=v_1,\ldots,v_t$ be a degree monotone path of maximum length in non-increasing order.  Observe that $e$ must have at least one of its vertices in $P$, otherwise $P$ was originally in $G$.

\medskip

\noindent \emph{Case 2.1}. \indent  Let us first assume that $e=(v_i,z)$, where $z$ is not on the path.   If $i=1$, then clearly $v_1,\ldots,v_t$ is also a degree monotone path in $G$ and hence $t \leq mp(G)=k <3k$, a contradiction.

If $i=t$ then $v_1,\ldots,v_{t-1}$ is a degree monotone path in $G$ and hence $t-1 \leq k$ which implies $t \leq k+1 < 3k$.

So let us assume $1 < i <t$.  Let us consider the paths  $v_1,\ldots,v_{i-1}$ and $v_i,\ldots,v_t$ --- both are non-increasing degree monotone paths in $G$ and hence together they have length at most 2k, implying that $t \leq 2k <3k$.
\medskip

\noindent \emph{Case 2.2}. \indent Now assume $e=(v_i,v_j)$, $i<j$.  Hence, in $G-e$, all vertices in $P$ have the same degree as they have in $G$ except for $v_i$ and $v_j$ whose degree has decreased by one.  Consider the paths $P_1=v_1,\ldots,v_i,v_j$, $P_2=v_i,\ldots,v_{j-1}$ and $P_3=v_i,v_j,\ldots,v_t$.

$P_3$ is  clearly a  degree monotone path in $G$ since $v_i$ and $v_j$ have a larger (by 1) degree in $G$ and $deg(v_i) \geq deg(v_j)$.  Similarly $|V(P_2)| \leq k$.  So if $t \geq 3k$, $|V(P_1)| \geq k+3$ since $v_i$ and $v_j$ are also on $P_2$ and $P_3$.  But $v_1,\ldots,v_{i-1}$ is a degree monotone path of length $k+1$ in $G$, contradicting the fact the $mp(G)=k$.  Hence $mp(G-e) \leq 3k-1$.

\medskip

For sharpness, let us construct the graph $G_1^-$ as follows: we start with $P_{3k-1}$ and add three leaves to $v_1$and a leaf to $v_2$;  we then connect $v_{k+1}$ to $v_{2k+1}$ --- we call this edge $e$.  Figure \ref{diag2} shows the construction for $k=4$.  Clearly $mp(G_1^-)=k$ while $mp(G_1^- -e)=3k-1$ with the path $v_1,\ldots,v_{3k-1}$ and hence  $mp(G_1^--e)=3mp(G_1^-)-1$.

\begin{figure}[h!]
\centering
\includegraphics{diag2}
\caption{$G_1^-$  when $k=4$} \label{diag2}
\end{figure}

\bigskip

Let us now turn to the lower bounds in both cases.  These results are derived from those for the upperbounds.
\medskip

\noindent 1. \indent  Suppose for some graph $G$, $mp(G+e) < \frac{mp(G)+1}{3}$.  Then if we take $H=G+e$ so that $G=H-e$, we have $mp(H) < \frac{mp(H-e)+1}{3}$, which implies that $mp(H-e) > 3mp(H)-1$, contradicting the righthand side of part 2.

For sharpness, let $G_2^{+}$ be the graph $G_1^-$ without the edge $e$.  Then $mp(G_2^{+}+e)=k$ while $mp(G_2^{+})=3k-1$ giving $mp(G_2^{+}+e)=\frac{mp(G_2^{+})+1}{3}$.

\medskip

\noindent 2. \indent  Suppose that for some graph $G$, $mp(G-e) < \frac{mp(G)}{3}$.  Then let $H=G-e$ so that $G=H+e$ giving $mp(H) < \frac{mp(H+e)}{3}$ which implies $mp(H+e) > 3mp(H)$, contradicing the upper bound in part 1.

For sharpness of the bound, let $G_2^{-}$ be the graph $G_1^+$ with the added edge $e=(v_{k+1},v_{2k+1})$.  Then $mp(G_2^{-})=3k$ and $mp(G_2^{-}-e)=\frac{mp(G_2^{-})}{3}$.
\end{proof}

\subsection{Subdivision}

Given a graph $G$ and $e=(u,v) \in E(G)$, the subdivision of $e$ is the addition of a new vertex $w$ such that $(u,w)$ is an edge and $(w,v)$ is an edge but $(u,v)$ is no longer an edge.  Note that the degree of $u$ and $v$ remains the same, and the degree of $w$ is 2.  The graph obtained by subdividing $e$ is denoted by $G^*$.  Again, we look at the effect of this operation on the maximum length of a degree monotone path, $mp(G)$.

\begin{theorem}
Let $G$ be a graph and $e=(u,v)$ an edge in $G$ such that $G^*$ is the graph obtained by subdividing $e$ with the vertex $w$. Then \[\left \lceil \frac{mp(G)+1}{2} \right \rceil \leq mp(G^*) \leq mp(G)+1,\] and both bounds are sharp.
\end{theorem}

\begin{proof}
Let us first consider the upper bound.  Let $P$ be a degree monotone path of maximum length in $G^*$.  We consider the following cases:

\noindent \emph{Case 1.} \indent If $w \not \in P$, then $P$ is a degree monotone path in $G$ hence $mp(G^*) \leq mp(G)$.

\noindent \emph{Case 2.} \indent  If $w \in P$, then either $u$ or $v$ or both are in $P$ otherwise $P=\{w\}$ which is not maximal.  We consider these cases separately:
\begin{itemize}
\item{If both $u$ and $v$ are in $P$ then $P-\{w\} \cup (u,v)$ is a degree monotone path in $G$ hence $mp(G^*) \leq mp(G)+1$.}
\item{If only one of $u$ or $v$, say $u$, is in $P$ then $P-\{w\}$ is a degree monotone path in $G$ and again $mp(G^*) \leq mp(G)+1$.}
\end{itemize}

For sharpness of the bound, consider $G=P_n$ the path on $n$ vertices --- then $mp(G)=n-1$, and subdividing any edge gives $mp(G^*)=n=mp(G)+1$.

For the lower bound, let $P$ be a degree monotone path of maximum length in $G$ and let $P=v_1,v_2,\ldots,v_t,v_{t+1},\ldots,v_k$, such that $u=v_t$ and $v=v_{t+1}$.  Let us consider $e=(u,v)$ and the subdividing vertex $w$.

\noindent \emph{Case 1.} \indent If $e=(u,v)$ is not on the path $P$, then $P$ is a degree monotone path in $G^*$ hence $mp(G^*) \geq mp(G)$.

\noindent \emph{Case 2.} \indent If the edge $e=(u,v)$ is in $P$, then as $w$ subdivides $e$, we get $P^*=v_1,\ldots,v_t,w,v_{t+1},\ldots,v_k$ a path in $G^*$, where $v_t=u$ and $v_{t+1}=v$.

We assume without loss of generality that $P$ is a non-decreasing monotone path, and hence $deg(u) \leq deg(v)$.  The vertex $w$ has degree 2.  Now if $deg(u)>2$,  the paths $v_1,\ldots,v_t$ and the paths $w,v_{t+1},\ldots,v_k$ are degree monotone in $G^*$, while if $deg(u) \leq 2$, the path $P^*$ is degree montone in $G^*$.  Hence in $G^*$ there is a degree monotone path of length at least $\left \lceil \frac{k+1}{2} \right \rceil = \left \lceil \frac{mp(G)+1}{2} \right \rceil$.

For sharpness of the lower bound, let us take the path $P_n$ and add a leaf to the vertices $v_2$ up to $v_{n-1}$.  Hence $mp(G)=n-1$.  If we subdivide the edge $(v_{\lceil \frac{n}{2} \rceil},v_{\lceil \frac{n}{2} \rceil+1})$, we get $mp(G^*)=\left \lceil \frac{n}{2} \right \rceil= \left \lceil \frac{mp(G)+1}{2} \right \rceil$.
\end{proof}

\subsection{Edge Contraction}
In a graph $G$, \emph{contraction of an edg}e $e=(u,v)$ is the replacement of $u$ and $v$ with a single vertex $w$ adjacent (without multiple edges) to all vertices in $N(u) \backslash v \cup N(v) \backslash u$.  The resulting graph $G \cdot e$ has one less  vertex  than $G$.    In case a vertex $z \in V(G)$ is adjacent to both $u$ and $v$, the degree of $z$ in $G \cdot e$ decreases by one --- otherwise it remains the same as in $G$.  In view of this we first consider triangle-free graphs, in which case the degrees of the neighbours of $u$ and $v$ remain unchanged in $G \cdot e$, and $deg(w)=deg(u)+deg(v)-2$.
\begin{theorem}
Let $G$ be a triangle-free graph.  Let $e=uv$ be an edge of $G$ which is contracted to form $G \cdot e$ with new vertex $w$.  Then \[\frac{mp(G)}{3} \leq mp(G \cdot e) \leq 2mp(G).\]
\end{theorem}

\begin{proof}

Let us first consider the upper bound.  Clearly $mp(G) \geq 2$ as if $mp(G)=1$, $G$ has no edges and $G \cdot e$ is not defined.   Let $P=v_1v_2 \ldots v_k$ be a degree monotone path in non-decreasing order of maximum length in $G \cdot e$.  We know that $deg(w)=deg(u)+deg(v)-2$. Let us look at all the different possibilities.
\medskip

\noindent \emph{Case 1}: \indent Assume first that $deg(u)=deg(v)=1$ and hence $deg(w)=0$.  If this was the only edge in $G$, then $mp(G)=2$ while $mp(G \cdot e)=1$, and the upper bound holds.  If there are other edges in $G$, then $mp(G \cdot e)=mp(G)$, and again the upper bound holds.

\medskip

\noindent \emph{Case 2}: \indent Now assume $1 = deg(u) < deg(v)$ --- then $deg(w)=deg(v)-1$ in $G \cdot e$.  We consider the following cases:
\begin{itemize}
\item{Clearly, if $w$ is not a vertex in $P$, then $P$ is degree monotone in $G$ too, hence $mp(G \cdot e) \leq mp(G)$.}
\item{ If $w=v_1$ in $P$, then in $G$, $v_2 \ldots v_k$ is a degree monotone path, and hence $mp(G \cdot e) - 1 \leq mp(G)$ and hence $mp(G \cdot e) \leq mp(G)+1 \leq 2mp(G)$.}
\item{ If $w=v_k$, then $v_1 \ldots v_{k-1}v$ is degree monotone in $G$ and hence $mp(G \cdot e)  \leq mp(G)$.}

\item{If $w=v_j$ for $2 \leq j \leq k-1$, then $v_1 \ldots v_{j-1}v$  and $v_{j+1} \ldots v_k$ are degree monotone in $G$, and $\min\{\max\{j,k-j\}\}= \lceil \frac{k}{2} \rceil$.

If $k$ is odd, then $mp(G) \geq \frac{k+1}{2}$ and hence $mp(G \cdot e) \leq 2mp(G)-1$.

If $k$ is even, then $mp(G) \geq \frac{k}{2}$ and hence $mp(G \cdot e) \leq 2mp(G)$.}
\end{itemize}

\noindent \emph{Case 3}: \indent It remains to consider, without loss of generality, the case \[2 \leq deg(u) \leq deg(v) \leq deg(w)=deg(u)+deg(v)-2.\]  Again we consider each possibility.

\begin{itemize}
\item{If in $G$, $w$ is not a vertex in $P$, then $P$ is degree monotone in $G$ too, hence $mp(G \cdot e) \leq mp(G)$.}
\item{If $w=v_1$, then in $G$, either $u$ or $v$ is adjacent to $v_2$ --- since $deg(u) \leq deg(v) \leq deg(w)$ then either $uv_2 \ldots v_k$ or $vv_2 \ldots v_k$ is a degree monotone path in $G$, and hence $mp(G \cdot e) + 1 \leq mp(G)$ giving $mp(G \cdot e) \leq mp(G) -1 \leq 2mp(G)$.}
\item{If $w=v_k$, then $v_1 v_2 \ldots v_{k-1}$ is still a degree monotone path in $G$ hence $mp(G \cdot e) -1 \leq mp(G)$ that is $mp(G \cdot e) \leq mp(G) + 1 \leq 2mp(G)$.}
\item{If $w=v_j$, $2 \leq j \leq k-1$, then let us consider the following cases:
\begin{enumerate}
\item{If in $G$, $u$ is adjacent to both $v_{j-1}$ and $v_{j+1}$, and hence $v$ is not adjacent to either of these vertices since $G$ is triangle-free, then $v_1 \ldots v_{j-1}$ and $u,v_{j+1} \ldots v_k$ are degree monotone paths of length $j-1$ and $k-j+1$ respectively in $G$.  Again, $\min\{\max\{j-1,k-j+1\}\}= \lceil \frac{k}{2} \rceil$.

If $k$ is odd, then $mp(G) \geq \frac{k+1}{2}$ and hence $mp(G \cdot e) \leq 2mp(G)-1$.

If $k$ is even, then $mp(G) \geq \frac{k}{2}$ and hence $mp(G \cdot e) \leq 2mp(G)$.}

\item{ If in $G$, $u$ is adjacent to $v_{j-1}$ and $v$ is adjacent to $v_{j+1}$, then if $deg(v_{j-1}) \leq deg(u)$ then $v_1v_2,\ldots,v_{j-1},u,v,v_{j+1},\ldots v_k$ is degree monotone in $G$ and hence $mp(G \cdot e)+1 \leq mp(G)$ implying that $mp(G \cdot e) \leq mp(G)-1 \leq 2mp(G)$.

 If, on the other hand, $deg(v_{j-1}) > deg(u)$, then $deg(u) \leq deg(v) \leq deg(w) \leq deg(v_{j+1})$ and hence $v_1 v_2 \ldots v_{j-1}$  and $u,v,v_{j+1}\ldots v_k$ are  degree monotone paths in $G$ of length $j-1$ and $k-j+2$ respectively.   In this case we consider $\min\{\max\{j-1,k-j+2\}\}$.

 If $k$ is even this is equal to $\frac{k+2}{2}$, and hence $mp(G \cdot e) \leq 2mp(G)-2$.

 If  $k$ is odd, we get $\frac{k+1}{2}$, and hence $mp(G \cdot e) \leq 2mp(G)-1$.}
\item{If in $G$, $v$ is adjacent to $v_{j-1}$ and $u$ is adjacent to $v_{j+1}$, then $v_1v_2 \ldots v_{j-1}$ and $u v_{j+1} \ldots v_k$ are degree monotone paths of length $j-1$ and $k-j+1$ respectively in $G$, and again $\min\{\max\{j-1,k-j+1\}\}= \lceil \frac{k}{2} \rceil$.

If $k$ is odd, then $mp(G) \geq \frac{k+1}{2}$ and hence $mp(G \cdot e) \leq 2mp(G)-1$.

If $k$ is even, then $mp(G) \geq \frac{k}{2}$ and hence $mp(G \cdot e) \leq 2mp(G)$.}

\item{If in $G$, $v$ is adjacent to both $v_{j-1}$ and $v_{j+1}$, and $u$ is adjacent to neither since $G$ is triangle-free, then $v_1 \ldots v_{j-1}$ and $u,v,v_{j+1} \ldots v_k$ are  degree monotone paths in $G$ of length $j-1$ and $k-j+2$ respectively.  In this case we consider $\min\{\max\{j-1,k-j+2\}\}$.

 If $k$ is even this is equal to $\frac{k+2}{2}$, and hence $mp(G \cdot e) \leq 2mp(G)-2 $.

 If  $k$ is odd, we get $\frac{k+1}{2}$, and hence $mp(G \cdot e) \leq 2mp(G)-1 $.}
\end{enumerate}}
\end{itemize}
This bound is attained by the graph $G_1$ constructed as follows:  consider the path on $2k+1$ vertices --- we add a leaf to vertices $v_2$ up to $v_{2k}$, and to the vertices $v_k$ and $v_{2k}$ we add a second leaf.  Then $mp(G)=k$.  If we contract one of the edges joining $v_k$ and a leaf, then vertex $w$ has degree $3$ and hence $v_1v_2 \ldots v_{k-1}w v_{k+2}\ldots v_{2k}$ is degree monotone in $G \cdot e$ and has length $2k$, giving $mp(G \cdot e) = 2mp(G)$.  Figure \ref{contract1} shows the construction for $k=4$.

\begin{figure}
\centering
\includegraphics{contract1}
\caption{$G_1$  when $k=4$} \label{contract1}
\end{figure}

\bigskip

We now consider the lower bound.  Let $P=v_1 \ldots v_k$ be a degree monotone path in non-decreasing order of maximum length in $G$.  Let $e=uv$ be the edge contracted to vertex $w$ in $G \cdot e$, and without loss of generality, we assume $deg(u) \leq deg(v)$.  Clearly, if the vertices $u$ and $v$ are not on $P$, then $P$ is still degree monotone in $G \cdot e$ and hence $mp(G \cdot e) \geq mp(G)$.  So let us assume that $u$ and $v$ are on $P$, and hence $k=mp(G) \geq 2$.  If $k=2$, that is $P=uv$, then $mp(G \cdot e)\geq 1 \geq \frac{mp(G)}{2}$.  So let us assume that $k \geq 3$ and hence $deg(v) \geq 2$.  So let us assume that $u$ and $v$ are on $P$ and $mp(G) \geq 3$.
\medskip

\noindent \emph{Case 1.} \indent We first consider the case  in which either $u$ or $v$, but not both, are on $P$.  Without loss of generality, Let $u=v_j$ be on $P$.  Then in $G \cdot e$, $w$ is on $P$, and $v_1 \ldots v_{j-1}$ and $v_{j+1} \ldots v_k$ are degree montone in $G \cdot e$, of length $j$ and $k-j$ respectively.  Now $\min\{\max\{j,k-j\}\}=\lceil \frac{k}{2} \rceil$.

 If $k$ is even then $mp(G \cdot e) \geq \frac{mp(G)}{2} \geq \frac{mp(G)}{3}$.

If $k$ is odd then $mp(G \cdot e) \geq \frac{mp(G)+1}{2} \geq \frac{mp(G)}{3}$.
\medskip

\noindent \emph{Case 2.} \indent We now consider the case in which $u$ and $v$ are in $P$ but $e=uv$ is not in $P$.  Let $u=v_i$ and $v=v_j$ in $P$; in all cases $j-i \geq 3$, otherwise we have a copy of $K_3$ in $G$ --- we consider the following cases:
\begin{itemize}
\item{If $i=1$ and $j=k$, then $v_2 \ldots v_{k-1}w$ is degree monotone in $G \cdot e$, hence $mp(G \cdot e) \geq mp(G)-1 \geq \frac{mp(G)}{3}$ since $mp(G) \geq 2$.}
\item{If $i=1$ and $4 \leq j < k$, then $v_2 \ldots v_{j-1}w$ and $v_{j+1} \ldots v_k$ are degree monotone in $G \cdot e$.  Hence we need $\min\{\max\{j-1, k-j\}\}= \lfloor \frac{k}{2} \rfloor$. Then $mp(G \cdot e) \geq \lfloor \frac{mp(G)}{2} \rfloor \geq \frac{mp(G)}{3}$.}
\item{If $i>1$ and $j=k$, then $v_1 \ldots v_{i-1}$ and $v_{i+1} \ldots v_{k-1}w$ are degree monotone in $G \cdot e$.  Hence we need $\min\{\max\{i-1, k-i\}\}=\lfloor \frac{k}{2} \rfloor$. Then $mp(G \cdot e) \geq \lfloor \frac{mp(G)}{2} \rfloor \geq \frac{mp(G)}{3}$.}

\item{If $1 < i < j <k$, then $v_1 \ldots v_{i-1}w$, $v_{i+1} \ldots v_{j-1}w$ and $v_{j+1} \ldots v_k$ are degree monotone in $G \cdot e$, of lengths $i$, $j-1$ and $k-j$ respectively.  Let $k=3t+r$ where $0 \leq r \leq 2$.

 If $k-j > t$, then $k-j \geq t+1 \geq \frac{k}{3}$, and hence $mp(G \cdot e) \geq \frac{mp(G)}{3}$.

So let us assume that $k-j \leq t$ and hence $j \geq k-t=2t+r$.  Now $\max \{i,j-1\} \geq \frac{2t+r}{2}=t + \frac{r}{2}$.  If $r=0$ then $k \geq 3t$ hence $\max \{i,j-1\} = t = \frac{k}{3}$.  If $r=1$, then $k=3t+1$, then $\max \{i,j-1\} \geq t+1 = \frac{k+2}{3} > \frac{k}{3}$ since $\max \{i,j-1\}$ must be an integer.  Finally, if $r=2$, then $\max \{i,j-1\} \geq t+1 = \frac{k+1}{3} > \frac{k}{3}$.

Hence, in each case, $mp(G \cdot e) \geq \frac{mp(G)}{3}$.}
\end{itemize}

\noindent \emph{Case 3.} \indent
Finally, we consider the case in which $e=uv$ is in $P$.  We may assume that $mp(G) \geq 3$ since if$mp(G)=2$ then clearly $mp(G \cdot e) \geq 1 \geq \frac{mp(G)}{3}$.  This results in the following cases:
\begin{itemize}
\item{If $u=v_1$ and $v=v_2$, then in $G \cdot e$, $v_3 \ldots v_k$ is degree monotone, and hence $mp(G \cdot e) \geq mp(G) -2 \geq \frac{mp(G)}{3}$ for $mp(G) \geq 3$.}
\item{If $u=v_{k-1}$ and $v=v_k$, hence $2 \leq deg(u) \leq deg(v) \leq deg(w)$, then in $G \cdot e$, $v_1 \ldots v_{k-2}w$ is degree monotone since $deg(u) \leq deg(v) \leq deg(w)=deg(u)+deg(v)-2$.  Hence $mp(G \cdot e) \geq mp(G)-1 \geq \frac{mp(G)}{3}$ for $mp(G)\geq 3$.}
\item{If  $u=v_j$ and $v=v_{j+1}$ for $2 \leq j \leq k-2$ (so $k \geq 4$ otherwise we have one of the previous two cases),  then in $G \cdot e$, $v_1 \ldots v_{j-1}w$ and $v_{j+2} \ldots v_k$ are degree monone paths of lengths $j$ and $k-j-1$ respectively--- hence we must consider $\min\{\max\{j, k-j-1\}\}= \lceil \frac{k}{2} \rceil$.  Hence $mp(G \cdot e) \geq \lceil \frac{k}{2} \rceil \geq \frac{mp(G)}{3}$ as required.}

\end{itemize}

The bound is attained by the following construction to give graph $G_3$:  let us take the path on $3k+3$ vertices --- we add a leaf to the vertices $v_2$ up to $v_{3k+2}$ except for $v_{k+1}$ and $v_{2k+2}$, and we connect these two vertices by an edge $e$. We also attach three leaves to each of the leaves attached to $v_{k+2}$ and $v_{2k+1}$.  Finally we add 3 leaves to vertex $v_{3k+3}$.  Thus $deg(v_1)=1$, $deg(v_2)=deg(v_3) \ldots =deg(v_{3k+2})=3$ and $deg(v_{3k+3})=4$, hence $mp(G)=3k+3$.  Now let us contract the edge $e$.  Then  $deg(v_1)=1$, $deg(v_2)=deg(v_3) \ldots =deg(v_k)=deg(v_{k+2}) = \ldots = deg(v_{2k+1})=deg(v_{2k+3})=\ldots =deg(v_{3k+2})=3$ and $deg(v_{3k+3})=4$.  Vertex $w$ has degree 4.  Hence the possible degree monotone paths in $G \cdot e$ are $v_1 \ldots v_k w$, $v_{k+2} \ldots v_{2k+1}w$ and $v_{2k+3} \ldots v_{3k+3}$ --- all these are of length $k+1$ and therefore $mp(G_3 \cdot e)=\frac{mp(G_3)}{3}$.  Figure \ref{contract3} shows the construction for $k=3$.

\end{proof}

\begin{figure}[h!]
\centering
\includegraphics{contract3new}
\caption{$G_3$  when $k=4$} \label{contract3}
\end{figure}

The following proposition shows that if we consider contraction in graphs which are not triangle-free, the situation is very different.

\begin{theorem}

There exist arbitrarily large $K_4$-free graphs $G$ on $n$ vertices such that $mp(G)=4$ while $mp(G.e)=|V(G \cdot e)| =n-1$.
\end{theorem}

\begin{proof}
Consider the following construction:  $G$ is the graph constructed by taking a path $P=v_1 \ldots v_{4k}$ on $4k$ vertices where $k \geq 2$, and another edge $e=uv$.  We connect the vertices  $v_{2i} \in P$, $1 \leq i \leq 2k$ to both $u$ and $v$ --- we then connect $v_{2i-1} \in P$ for $1 \leq i \leq k$ to $u$, and  $v_{2i-1} \in P$ for $k+1 \leq i \leq 2k$ to $v$.  Finally we connect $v_1$ to $v_{4k}$.  Figure \ref{contract2} gives the construction for $k=2$.  $G$ is not triangle-free, but it is $K_4$-free.  One can see that in $P$, the vertices with even index have degree 4, while the vertices with odd index have degree 3, and $deg(u)=deg(v)=3k$.  Hence it is clear that $mp(G)=4$ for any value of $k \geq 2$.  Now if we contract $e=uv$ to a vertex $w$, all vertices in $P$  have degree 3, while $deg(w)=4k \geq 3$, and therefore $mp(G \cdot e) = 4k+1$.
\end{proof}
\begin{figure}[h!]
\centering
\includegraphics{contract2}
\caption{$G$  when $k=3$} \label{contract2}
\end{figure}

\section{Vertex Addition and Deletion}

Given a graph $G$, we can add a vertex $v$ and connect it to at least one vertex in $G$ to give the graph $G+v$.  On the other hand, given a graph $G$ and a vertex $v \in V(G)$, the graph $G-v$ is obtained by deleting the vertex $v$ and all its incident edges.  The effect on the maximum length of a degree monotone path can be seen through the following observations.

\begin{proposition}
Given a graph $G$ and $v \in V(G)$,
\begin{enumerate}
\item{$2\leq mp(G+v) \leq |V(G)|+1$}
\item{$1 \leq mp(G-v) \leq |V(G)|-1$}
\end{enumerate}
\end{proposition}
\begin{proof}
\mbox{ \\}

\noindent 1. \indent Consider the complete bipartite graph $G=K_{n,n+1}$ so that $mp(G)=2$.  We add a vertex $v_1$ and connect it to all the vertices in the larger part, to give $H=G+v_1=K_{n+1,n+1}$ --- now $mp(G+v_1)=mp(H)=|V(G)|+1$.  Now if we add another vertex $v_2$ and connect it to all the vertices in one part of the partition, we get $H+v_2=F=K_{n+1,n+2}$ and again $mp(H+v_2)=mp(F)=2$.
\medskip

\noindent 2. \indent For the upperbound, consider the graph $G=K_n$ --- then $mp(G)=n$, and $G-v=K_{n-1}$ hence $mp(G-v)=n-1=|V(G)|-1$.

For the lower bound consider $G=K_{1,m}$, $m \geq 1$.  Then deleting the vertex of degree $m$ gives $mp(K_{1,m}-v)=mp(G-v)=1$.
\end{proof}

In view of this general result, we consider vertex addition and deletion for the family of trees.  The following example shows that if one adds non-leaf vertices, the effect on  $mp(G)$ can be quite drastic.  Consider the tree $T$ constructed by taking a path on $2k+1$ vertices $(v_1 v_2 \ldots v_{2k+1})$, $k \geq 1$, and adding a leaf to the vertices $v_{2i}$, $1 \leq i \leq k$.  Clearly this tree has $mp(T)=2$.  Let us add a vertex $v$ and connect it to the vertices $v_{2i+1}$, $0 \leq i \leq k$.  So now all the vertices on the path have degree 3 except for $v_1$ and $v_{2k+1}$, and vertex $v$ has degree $k+1 \geq 2$.  Hence there is now a degree monotone path of length $2k+1$, that is $mp(T+v)=2k+1$.  Hence, in the sequel, we only consider the addition of leaves, so that the resulting graph is another tree.

\begin{theorem}
Let $T$ be a tree and $v \in V(T)$,
\begin{enumerate}
\item{If we add a vertex $v$ such that $T+v$ is also a tree, then \[\frac{mp(T)}{2} \leq mp(T+v) \leq 2mp(T).\]}
\item{If $v \in V(T)$ such that $T-v$ is a tree, then \[\frac{mp(T)}{2} \leq mp(T-v) \leq 2mp(T).\]}
\end{enumerate}

In both cases, the bounds attained are sharp.
\end{theorem}

\begin{proof}

Let us first consider the upper bound for each case.

\noindent 1. \indent If $T+v$ is a tree then clearly $v$ is a leaf otherwise it would form a cycle.  Let $P=v_1,v_2,\ldots,v_k$ be a degree monotone path of maximum length in $T+v$ in non-decreasing order.  Observe that $k \geq 2$ as $T+v$ contains at least one edge.  So we consider all possible cases.

\begin{itemize}
\item{If $v$ is not adjacent to any vertex in $P$ then $P$ is a degree monotone path in $T$ hence $mp(T+v) \leq mp(T)$.}
\item{If $v$ is adjacent to $v_1$, then $v,v_1,\ldots,v_k$ is a degree monotone path in $T+v$ contradicting the maximality of $P$.}
\item{If $v=v_1$ then $v_2,\ldots,v_k$ is a degree monotone path in $T$ of length $k-1$ so $k-1 \leq mp(T)$ and hence $2mp(t) \geq 2k-2 \geq k=mp(T+v)$ since $k \geq 2$.}
\item{If $v$ is adjacent to $v_k$, then $v_1,\ldots,v_{k-1}$ is a degree monotne path in $T$, hence again $2mp(t) \geq 2k-2 \geq k=mp(T+v)$ since $k \geq 2$.}
\item{If $v=v_k$, this implies that $deg(v)=1$ and that $k=2$, and since $T$ is not $K_1$, $mp(T) \geq 2$ hence $2mp(T) \geq 4 > 2=k=mp(T+v)$. (If $T=K_1$ then trivially $T+v=K_2$ so $mp(T+v)=2 =2 mp(T)$.)}
\item{ Let $v$ be adjacent to some $v_j$ for $2 \leq j \leq k-1$.  Note that $v$ can only be  $v_1$ or $v_k$ in $P$ since it is a leaf.  Consider $v_1,\ldots,v_{j-1}$ which is a degree monotone path in $T$ hence $j-1 \leq mp(T)$.  Similarly $v_j,\ldots,v_k$ is also a degree monotne path in $T$ with length at most $mp(T)$ hence by adding we get $mp(T+v) \leq 2 mp(T)$ as required.}
\end{itemize}

For sharpness, consider the tree $T_1^+$ constructed as follows:  take the path on $2k+1$ vertices $v_1,\ldots,v_{2k+1}$, where $k \geq 2$ --- we add a leaf to all the vertices except $v_1$,$v_{k+1}$ and $v_{2k+1}$.  Then $mp(T_1^+)=k$.  If we add a vertex $v$ and connect it to  vertex $v_{k+1}$, then $mp(T_1^++v)=2k=2mp(T_1^+)$.  Figure \ref{diag3} shows the construction for $k=4$.
\medskip

\begin{figure}[h!]
\centering
\includegraphics{diag3}
\caption{$T_1^+$  when $k=4$} \label{diag3}
\end{figure}
\noindent 2. \indent  Let $P=v_1,v_2,\ldots,v_k$ be a degree monotone path of maximum length in $T-v$ in non-decreasing order.  Not that if $k=1$ then $T-v$ has no edges which imples that $T=K_{1,m}$ for some $m \geq 1$, and so $mp(T)=2=2mp(T-v)$.

So let us assume that $k \geq 2$.  If in $T$ $v$ is not adjacent to any vertex of $P$, then $P$ is also a degree monotone path in $T$, hence $mp(T-v)=k \leq mp(T) < 2mp(T)$.

So we consider the case in which $v$ is adjacent to exactly one vertex in $P$.  We consider the different scenarios.

\begin{itemize}
\item{If $v$ is adjacent to $v_1$ in $T$, then $v_2,\ldots,v_k$ is a degree monotone path of length $k-1$ in $T$ hence $mp(T-v)=k \leq 2(k-1) \leq 2mp(T)$ since $k \geq 2$.}
\item{If $v$ is adjacent to $v_k$ in $T$ then $v_1,\ldots,v_k$ is also a degree monotone path in $T$,  hence again $mp(T-v)=k \leq  mp(T) < 2mp(T)$.}
\item{If $v$ is adjacent to some $v_j$ for $2 \leq j \leq k-1$, then consider in $T$ the path $v_1,\ldots,v_j$ --- this is degree monotone hence $j \leq mp(T)$.  Now $v_{j+1},\ldots,v_k$ is also a degree monotone path in $T$ and has length at most $mp(T)$.  Adding, we get $mp(T-v) \leq 2 mp(T)$ as required.}
\end{itemize}

For sharpness consider the graph $T_1^-$ constructed as follows --- we take a path on $2k+1$ vertices for $k \geq 2$ and we add a leaf to vertex $v_k$ and to vertex $v_{2k}$.  Then $mp(T_1^-)=k$, while if $v$ is the leaf attached to $v_k$, $mp(T_1^--v)=2k= 2mp(T_1^-)$.  Figure \ref{diag4} shows the construction for $k=4$.

\begin{figure}[h!]
\centering
\includegraphics{diag4}
\caption{$T_1^-$  when $k=4$} \label{diag4}
\end{figure}
\medskip

We now consider the lower bounds.
\medskip

\noindent 1. \indent  Assume $mp(T+v) < \frac{mp(T)}{2}$.  Let $T'=T+v$ hence $T=T'-v$ --- then $mp(T') < \frac{mp(T'-v)}{2}$ implying that $mp(T'-v) > 2mp(T')$, contradicting the upper bound in part 2.

For sharpness, let $T_2^+$ be the path on $2k+1$ vertices, $k \geq 2$, with a leaf added to vertex $v_{2k}$.  Then $mp(T_2^+)=2k$.  We add a vertex $v$ and connect it to the vertex $v_{k}$ --- then $mp(T_2^++v)=k=\frac{mp(T_2^+)}{2}$.

\medskip

\noindent 2. \indent  Assume $mp(T-v) < \frac{mp(T)}{2}$.  Let $T'=T-v$ hence $T=T'+v$ --- then $mp(T') < \frac{mp(T'+v)}{2}$ implying $mp(T'+v) > 2mp(T')$, contradicting the upperbound in part 1.

For sharpness, we construct the graph $T_2^-$ as follows:  take the path on $2k+1$ vertices for $k \geq 2$ and add a leaf to every vertex except the first and the last so that $mp(T_2^-)=2k$.  Consider the leaf $v$ connected to $v_{k+1}$ --- if we delete this vertex, $mp(T_2^--v)=k=\frac{mp(T_2^-)}{2}$.

\end{proof}

\section{Operations involving two graphs}
\subsection{Cartesian Product}

The Cartesian product $G \cartprod H$ of graphs $G$ and $H$ is a graph such that the vertex set of $G \cartprod H$ is the Cartesian product $V(G) \times V(H)$ and any two vertices $(u,u')$ and $(v,v')$ are adjacent in $G \cartprod H$ if and only if either
\begin{itemize}
\item{$u = v$ and $u'$ is adjacent with $v'$ in $H$, or}
\item{$u' = v'$ and $u$ is adjacent with $v$ in G.}
\end{itemize}

\begin{theorem}
Let $G$ and $H$ be two connected graphs.  Then \[ mp(G)+mp(H)-1 \leq mp(G \cartprod H)  \leq mp(G)mp(H)\] and both bounds are sharp.
\end{theorem}

\begin{proof}
 Let us first consider the lower bound.  Let $v_1 \ldots v_t$ be a longest degree monotone path in $G$, and let $u_1 \ldots u_s$ be a longest degree monotne path in $H$, both in non-decreasing order.

Consider the path in $G \cartprod H$ with vertex coordinates \[(v_1,u_1)(v_1,u_2)\ldots (v_1,u_s)(v_2,u_s)(v_3,u_s)\ldots (v_t,u_s).\]  This is clearly a degree monotone path in $G \cartprod H$ with $t+s-1$ vertices, and hence $mp(G \cartprod H) \geq mp(G) + mp(H) -1$.

Now let us consider the degree monotone path in in $G \cartprod H$ of maximum length $r= mp(G \cartprod H)$ with vertices $z_i$ for $1 \leq i \leq r$.  Let us label $z_i=(v_{a_i},u_{b_i})$.  So consider the vertices from $z_1$ to $z_i=(v_{a_i},u_{b_i})$: for vertex  $z_{i+1}$, either the $v$ coordinate or the $u$ coordinate will change but not both.

If the $v$ coordinate changes, then $z_{i+1}=(v_{a_{i+1}},u_{b_{i+1}})$ where $a_{i+1}>a_i$ and $b_{i+1}=b_i$, hence it follows that $deg(v_{a_{i+1}}) \geq deg(v_{a_i})$.

If the $u$ coordinate changes, then $z_{i+1}=(v_{a_{i+1}},u_{b_{i+1}})$ where $a_{i+1}=a_i$ and $b_{i+1}>b_i$, hence it follows that $deg(u_{b_{i+1}}) \geq deg(u_{b_i})$.

Now let us consider those vertices in which the index $a_{i+1}>a_i$, which implies that the corresponding vertices in $G$ are distinct --- it is clear that these vertices form a degree monotone paths in $G$.  Hence if the number of such vertices is $t$, $t \leq mp(G)$.

Similiarly, if we consider the vertices in which the index $b_{i+1}>b_i$, the corresponding vertices in $H$ are distinct and form a degree monotone path in $H$ --- if the number of such vertices is $s$, then $s \leq mp(H)$.

Now since for each move from $z_i$ to $z_{i+1}$, only one coordinate changes, we have at most $st$ coordinates, hence $mp(G \cartprod H) \leq mp(G)mp(H)$.

Now we look at construction which achieve these bounds.  Firstly, let $G=H=K_{1,m}$, where $m \geq 2$, and hence $mp(G)=mp(H)=2$. In $G \cartprod H$ there is one vertex of degree $2m$, $2m$ vertices of degree $m+1$ which are independent, and $m^2$ vertices of degree 2 which are independent.  Hence the longest degree monotone path has three vertices so  $mp(G \cartprod H) = 2= mp(G) + mp(H) -1$.

Now for the upper bound, let $G$ be a connected regular graph on $t$ vertices and let $H$ be a graph such that $mp(H)=s$.  Let the vertices of $G$ be $v_1,\ldots, v_t$, and let the vertices $u_1,\ldots,u_s$ in $H$ be vertices on a longest degree monotone path in $H$.

Now in $G \cartprod H$, consider the path \[(v_1,u_1),(v_2,u_1),\ldots,(v_t,u_1),(v_t,u_2),(v_{t-1},u_2),\ldots,(v_1,u_2),(v_1,u_3),\ldots(v_t,u_3),\ldots \] and we carry on in this fashion until we have used all the $st$ vertices.  This path is degree monotone and hence $mp(G \cartprod H) = mp(G)mp(H)$.
\end{proof}

\subsection{Graph join}

The \emph{graph join} $G+H$ of two graphs $G$ and $H$ with disjoint vertex sets, $V(G)$ and $V(H)$ and disjoint edge sets $E(G)$ and $E(H)$ is the graph such that
\begin{itemize}
\item{$V(G+H)=V(G) \cup V(H)$}
\item{$E(G+H)=E(G) \cup E(H) \cup \{xy: x \in V(G), y \in V(H)\}$}
\end{itemize}

We now consider degree monotone paths in $G+H$.

\begin{theorem}
Given two graphs $G$ and $H$, \[mp(G)+mp(H) \leq mp(G+H) \leq |V(G)| + |V(H)|,\] and both bounds are sharp.
\end{theorem}

\begin{proof}
The upper bound is trivial since for any graph $mp(G) \leq |V(G)|$.  So let us consider the lower bound.  Let $P=v_1,v_2,\ldots,v_t$ be a degree monotone path of maximum length in $G$ and let $P^*=u_1,u_2,\ldots,u_s$ be a degree monotone path of maximum length in $H$.  Let us rearrange $\{v_1,\ldots,v_t,u_1,\ldots,u_s\}$ in non-decreasing order according to their degrees in $G+H$, noting that $deg_{G+H}(v)=deg_G(v)+|V(H)|$ while $deg_{G+H}(u)=deg_H(u)+|V(G)|$.  Then it is clear that these vertices form a degree monotone path in $G+H$ of length $s+t$, hence $mp(G+H) \leq mp(G) + mp(H)$.

Let us consider the upper bound.  Let $G$ and $H$ be two graphs such that $|V(G)|=|V(H)|$ and $G$ and $H$ have the same degree sequence.  We write the vertices of $G+H$ in non-decreasing order of their degrees such that each vertex of the $G$ part is followed by the corresponding vertex in the $H$ part.  So for the degree monotone path in $G+H$, we start with the vertex of smallest degree in $G+H$ and alternately take vertices of this same degree from $H$ and $G$ until all vertices of this degree are included in the path:  we then move to the second smallest degree in $G$ and carry out the same procedure for every different degree in the sequence.  There is an even number of vertices in $G+H$ of each degree since $G$ and $H$ have the same number of vertices and the same degree sequence, so this alternating path can be continued until all vertices in $G+H$ have been included, which implies that $mp(G+H)=|V(G)|+|V(H)|$ in this case, achieveing the upper bound.

For the lower bound, consider $G=K_{1,m}$ and $H=K_k$.  So $mp(G)=2$ and $mp(H)=k$.  Then in $G+H$ there are $k+1$ vertices of degree $k+m$, namely the vertices of $H$ and the vertex of degree $m$ in $G$, and the  $m$ vertices of degree $k+1$ that are independent.  So if we can start the path with a vertex of degree $k+1$, and then we must move to a vertex of degree $k+m$, and all the vertices of this degree can be included in the path.  hence $mp(G+H)=1 + k+1=2 +k = mp(G)+mp(H)$.
\end{proof}

\section{Conclusion}

We have given sharp bounds for $mp(G')$ in terms of $mp(G)$ where $G'$ is obtained from $G$ by the most basic operations involving a single vertex or edge.  We have shown that the effect of edge contraction on $mp(G)$ in the case of $K_3$-free graphs  is bounded (above and below) by a multiplicative factor, while there exist $K_4$-free graphs for which $mp(G)=4$ while $mp(G \cdot e)=|V(G)|-1$.  This leads to the following questions:
\begin{enumerate}
\item{Is there a $K_4$-free graph $G$ such that $mp(G)=3$ and $mp(G \cdot e) \geq |V(G)|-1$?}
\item{Is there a characterization of $K_4$-free graphs for which $mp(G \cdot e)$ is bounded above and below by a multiplicative factor?}
\end{enumerate}

We have also obtained sharp bounds for $mp(G \times H)$ in terms of $mp(G)$ and $mp(H)$ where $\times$ is either the Cartesian product of the join of two graphs.  Repeating this for other products, such as the direct product, might be interesting.

Finally, the notion of edge addition gives rise to the question of what the minimum number of edges of a graph $G$ on $n$ vertices can be if adding any edge increses $mp(G)$.  This leads to a problem analogous to the saturation number of a graph \cite{faudree2011survey,kaszonyi1986saturated}, and we shall be considering this in a forthcoming paper.

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


text:
And /^I select the "(.*?)" category from (How They Help|Who They Help|What They Do)$/ do |category, cat_type|
  cat_id = case cat_type
    when 'What They Do'
      'what_id'
    when 'How They Help'
      'how_id'
    when 'Who They Help'
      'who_id'
    else
      raise "Unsupported category type"
  end
  select(category, :from => cat_id)
end

When /^I visit the proposed organisation show page for the proposed organisation that was proposed$/ do
  proposed_org = ProposedOrganisation.find_by(name: unsaved_proposed_organisation(User.first).name)
  visit proposed_organisation_path(proposed_org)
end
When(/^I visit "(.*?)"$/) do |path|
  visit path
end

def paths(location)
  {
      'home' => root_path,
      'sign up' => new_user_registration_path,
      'sign in' => new_user_session_path,
      'organisations index' => organisations_path,
      'proposed organisations index' => proposed_organisations_path,
      'new organisation' => new_organisation_path,
      'new proposed organisation' => new_proposed_organisation_path,
      'contributors' => contributors_path,
      'password reset' => edit_user_password_path,
      'invitation' => accept_user_invitation_path,
      'invite users to become admin of organisations' => organisations_report_path,
      'registered users' => users_report_path,
      'invited users' => invited_users_report_path,
      'volunteer opportunities' => volunteer_ops_path,
      'contributors' => contributors_path,
      'deleted users' => deleted_users_report_path
  }[location]
end

Then /^I (visit|should be on) the (.*) page$/ do |mode, location|
  location.downcase!
  raise "No matching path found for #{location}" if paths(location).nil?
  case mode
    when 'visit' then visit paths(location)
    when 'should be on' then current_path.should eq paths(location)
    else raise "unknown mode '#{mode}'"
  end
end

def find_record_for(object, schema, name)
  # all types begin as strings
  real_object = object.classify.constantize
  schema = schema.chomp('d').to_sym
  real_object.where(schema => name).first
end

Then /^I should be on the new organisation proposed edit page for the organisation named "(.*?)"$/ do |name|
  org = Organisation.find_by_name(name)
  url = new_organisation_proposed_organisation_edit_path org
  current_path.should eq url
end

Then /^I should be on the show organisation proposed edit page for the organisation named "(.*?)"$/ do |name|
  org = Organisation.find_by_name(name)
  prop_ed = org.edits.first
  url = organisation_proposed_organisation_edit_path org, prop_ed
  current_path.should eq url
end

Then /^I (visit|should be on) the new volunteer op page for "(.*?)"$/ do |mode, name|
  org = Organisation.find_by_name(name)
  url = new_organisation_volunteer_op_path(org)
  case mode
    when 'visit' then visit url
    when 'should be on' then current_path.should eq url
    else raise "unknown mode '#{mode}'"
  end
end

Then /^I (visit|should be on) the (edit|show) page for the (.*?) (named|titled) "(.*?)"$/ do |mode, action, object, schema, name|
  record = find_record_for(object, schema, name)
  url = url_for({
                    only_path: true,
                    controller: object.pluralize.underscore,
                    action: action,
                    id: record.id
                })
  case mode
    when 'visit' then visit url
    when 'should be on' then current_path.should eq url
    else raise "unknown mode '#{mode}'"
  end
end

And(/^the page should be titled "(.*?)"$/) do |title|
  page.should have_title(title)
end

And (/^I should see a full width layout$/) do
  within('#content') do
    page.should have_css('#one_column.span12')
  end
end

And (/^I should see a two column layout$/) do
  within('#content') do
    page.should have_css('#column1.span6')
    page.should have_css('#column2.span6')
  end
end

Then(/^the response status should be 404$/) do
  page.status_code.should == 404
end

Given /^I press "(.*?)"$/ do |button|
  click_button(button)
end

When /^I click "(.*)"$/ do |link|
  click_link(link)
end

When /^I click id "(.*)"$/ do |id|
  find("##{id}").click
  wait_for_ajax
end

When /^(?:|I )follow "([^"]*)"$/ do |link|
  click_link(link)
end

Then /^following Disclaimer link should display Disclaimer$/ do
  steps %Q{
When I follow "Disclaimer"
Then I should see "Disclaimer"
And I should see "Whilst Voluntary Action Harrow has made effort to ensure the information here is accurate and up to date we are reliant on the information provided by the different organisations. No guarantees for the accuracy of the information is made."
}
end

Then(/^the "([^"]*)" should be (not )?visible$/) do |id, negate|
  # http://stackoverflow.com/a/15782921
  # Capybara "visible?" method(s) are inaccurate!

  #regex = /height: 0/ # Assume style="height: 0px;" is the only means of invisibility
  #style = page.find("##{id}")['style']
  #sleep 0.25 if style # need to give js a moment to modify the DOM
  #expectation = negate ? :should : :should_not
  #style ? style.send(expectation, have_text(regex)) : negate.nil?

  elem = page.find("##{id}")
  negate ? !elem.visible? : elem.visible?
end

Then(/^the "([^"]*)" should be "([^"]*)"$/) do |id, css_class|
  page.should have_css("##{id}.#{css_class}")
end

When(/^I click link with id "([^"]*)"$/) do |id|
  page.find("##{id}").click
end

When(/^javascript is enabled$/) do
  Capybara.javascript_driver
end

And(/^I click tableheader "([^"]*)"$/) do |name|
  find('th', :text => "#{name}").click
  wait_for_ajax
end

Then(/^navbar button "(.*?)" should( not)? be active$/) do |button_text, negative|
  expectation_method = negative ? :should_not : :should
  within('.nav.nav-pills.pull-right') do
    page.send(expectation_method, have_css('li.active > a', :text => "#{button_text}"))
  end
end

Then(/^the page includes a hyperlink to "([^"]*)"$/) do |link|
  #page.should have_link link
  page.should have_xpath("//a[contains(.,'#{link}') and @href=\"#{link}\"]")
end

Then(/^the page includes email hyperlink "([^"]*)"$/) do  |link|
  page.should have_link link
end

Then(/^I should see "([^"]*)" in the flash error$/) do |message|
  page.should have_css('div#flash_error', :text => message)
end

Then(/^I should see "([^"]*)" in the flash$/) do |message|
  page.should have_css('div#flash_success', :text => message)
end

Then(/^I should( not)? see the call to update details for organisation "(.*)"/) do |negative, org_name|
    org = Organisation.find_by_name org_name
    expectation_method = negative ? :should_not : :should
    message = "You have not updated your details in over a year! Please click here to update now."
    page.send(expectation_method, have_css('div#flash_warning', :text => message))

    within(negative.nil? ? 'div#flash_warning' : 'body') do
      page.send(expectation_method, have_link("here", :href => edit_organisation_path(org)))
    end
end
Then(/^I should see an (active|inactive) home button in the header$/) do |active|
    active_class = (active == "active") ? ".active" : ""
    within('.nav.nav-pills.pull-right') do
      expect(page).to have_css("li#{active_class} > a[href='/']", :text => "Home")
    end
end

redpajama_set_name: RedPajamaGithub


text:
4 free apps to help college women feel safer
Jaleesa Jones
#YesAllWomen
In the manifesto, Rodger promised his revenge against all women for forcing him to live in what he called "an existence of loneliness, rejection and unfulfilled desires."
The hashtag also countered the #NotAllMen argument, which delegitimizes the experiences of sexual assault survivors by pinning these incidents to deviant men rather than the patriarchal culture that underpins male entitlement to the female body.
Perhaps one of the most chilling #YesAllWomen tweets was the following:
The amount interest in the hashtag points to a blatant truth: Sexual assault and street harassment are far too common. But with our phones constantly glued to our hands, there are tools available that can make us safer.
These four free apps could help prevent sexual assault, harassment or dangerous encounters in any situation -- even if you're just walking to your car in a dark parking lot.
1.bSafe A multi-feature app designed to quell safety concerns as you travel. While the app requires you to register one primary contact, users can add an unlimited number of friends. Highlights of the app include a "Follow Me" feature, which enables users' guardians or friends to track their journey in real-time using GPS technology, and an "I'm Here" feature, which alerts contacts when a user has arrived at their destination.
There is also an S.O.S. feature, and a timer feature which triggers an alarm when users do not check-in on time. A video recording begins once the alarm is activated to collect evidence of a possible crime.
2. Circle of 6Designed specifically to combat sexual assault and interpersonal violence, this app allows users to select six friends to form a trust circle. The app uses GPS location and comes with automated messages -- including "Come and get me. I need help getting home safely" and "Call and pretend you need me. I need an interruption" -- to provide interference in a potentially distressing situation.
There's also a built-in hotline, which allows users to specify the nature of their emergency.
3. Hollaback! The pervasiveness of street harassment can make sidewalks seem like gauntlets. With cat-callers leering and, at times, groping at women, it's easy to leave these claustrophobic spaces feeling afraid or ashamed.
Hollaback! seeks to project that shame back on street harassers. The app allows users to instantly report harassment and pinpoint exactly where it occurred. Users also have the option to upload a photo of the incident.
While Hollaback! may not have built-in alert systems like the others, it predicates on an important idea undergirding the anti-violence discussion: The importance of standing up and sharing our experiences.
4. GuardlyLooking for a more discreet app? Guardly may be your go-to. This one-tap app lets users add up to 15 contacts and create separate networks based on situation or location.
With GPS technology and network failure protection, Guardly works by allowing users to send one-way alerts to their contacts in the event of an emergency. College students should sign up with Guardly using their school email address. Upon registration, Guardly will connect students with campus police if they set off an emergency alert on campus.
While users can download Guardly for free, the app does offer a premium version, which can be accessed for $1.99 a month or $19.99 a year. Added benefits of the premium version include conference call capability, private instant messaging and continuous GPS tracking during calls.
While free apps are ideal, the following apps are also be worth checking out:1. LifeLine Response ($6.99) 2. Red Panic Button ($2.99) 3. React Mobile ($1.99/month, $19.99/year) 4. StaySafe ($6.99) 5. WatchOverMe ($3.99/month, $23.99/year) 6. MyForce ($11.99/month, $119.99/year) 7. OnWatch (Free 30-day trial, $2.99/month, $19.99/year)
Jaleesa Jones is a senior at the University of North Carolina at Chapel Hill. Your typical underachiever, Jaleesa is working towards a degree in communication studies with a concentration in media studies and production as well as minors in journalism and screenwriting. She is the current President and Campus Correspondent for Her Campus UNC, a branch of HerCampus.com, the number one online community for college women. She is also an arts reporter for the Daily Tar Heel and a member of the Carolina Association of Future Magazine Editors. In her spare time, she enjoys listening to spoken word poetry, binge-watching Law & Order: SVU and debating everything from respectability politics to celebrity news. You can follow her random outbursts on Twitter @newLEESonlife.
This article comes from The USA TODAY College Contributor network. The views expressed in this article do not necessarily reflect the views of USA TODAY. You understand that we have no obligation to monitor any discussion forums, blogs, photo- or video-sharing pages, or other areas of the Site through which users can supply information or material. However, we reserve the right at all times, in our sole discretion, to screen content submitted by users and to edit, move, delete, and/or refuse to accept any content that in our judgment violates these Terms of Service or is otherwise unacceptable or inappropriate, whether for legal or other reasons.


text:
Astoria Primary Care Opens
Columbia Memorial Hospital Grows Again
ASTORIA, Ore. – CMH Primary Care opens the doors of its new Astoria location on Oct. 1.
Demand for primary medical care has grown across the country. Since the CMH Primary Care Clinic opened in Warrenton five years ago, it's grown from one provider serving about 2,000 patients to six providers serving about 11,500 patients.
To meet this demand, Columbia Memorial Hospital built out a second Primary Care Clinic in Astoria. Three providers—Dr. Kevin Baxter, Dr. Anisa Richardson, and Mary Rizzo, FNP—will move their practice from the Warrenton clinic to the new CMH Astoria Primary Care, where they'll be joined by two new providers.
"Even after two expansions, we had out-grown the original clinic," said Chief Operating Officer Nicole Williams. "So rather than expand again, we made the decision to open a second location that would be more convenient for our patients in Astoria, Knappa and Pacific County."
New primary care providers
Rachel Flescher, PA-C
Elizabeth Erikson, DO
Wendy Hovden, NP
Rachel Flescher, PA, will join the new Astoria Primary Care Clinic later this month.
Flescher is a physician assistant. She is interested in family medicine, women's health and diabetes management.
She holds a bachelor's degree in exercise and sports science, pre-therapy and allied health from Oregon State University, and received her master's degree in physician assistant studies from Oregon Health & Science University. She grew up on the Oregon coast in Waldport, and loves small coastal towns, the outdoors, traveling, and reading.
Dr. Elizabeth Erikson recently joined the Warrenton Primary Care Clinic. She is a doctor of osteopathy. She enjoys practicing whole person and whole family care, and uses osteopathic manipulations and myofascial release techniques with patients.
Erikson earned a bachelor's degree in art history from Lewis & Clark College in Portland and her medical degree from A.T. Still University's School of Osteopathic Medicine in Mesa, Ariz. She is an avid skier, hiker and swimmer.
Clatsop County native Wendy Hovden, NP, will also be returning to the area to work with the CMH Primary Care team in November.
Hovden received her master's degree from Walden University in Minneapolis, Minn., and her bachelor's degree from Oregon Health & Science University. Before becoming a nurse practitioner, she was a registered nurse for 17 years. Her medical interests include preventative cardiovascular care and management of immediate healthcare needs.
Primary care is preventative care
Like eating well and staying active, seeing a primary care provider regularly is an important part of your health maintenance. People who visit their primary care provider regularly have better health outcomes, lower death rates and lower health care costs than people who don't, according to the American Academy of Family Physicians.
The Affordable Care Act requires most health plans to cover preventative health services at no cost to you. These services include many health screenings and immunizations. Your primary care provider serves as your guide to accessing many of these benefits.
Astoria Primary Care is in the Park Medical Building, across from Columbia Memorial Hospital. Call 503-338-4675 for more information.


It's no secret that I am a huge fan of NBC's hit drama THIS IS US. It's a family drama that really tugs on your heart strings!
If you are also a fan, you will want to tune in to a new drama that is premiering right after called THE VILLAGE.
Cast: Moran Atias, Dominic Chianese, Warren Christie, Frankie Faison, Jerod Haynes, Daren Kagasoff, Michaela McManus, Lorraine Toussaint and Grace Van Dien.
Tune in Tuesday, March 19th at 10PM EST for the premiere episode and let me know what you think!
1 comment on "The Village Premieres Tonight on NBC @nbcthevillage #TheVillage"
I just found This is Us on Hulu. I am still on Season 1. I will be watching this tonight too.