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Intimate Details Of Runtown's Love story With Girlfriend Adut Akech
By chika Udeh
There are only a handful of celebrities whose love lives remains private despite their illustrious career. Those whose relationship continues to hit the timeline are either celebrated or criticized for some obvious reasons. Runtown's love story is one of such stories we all admire on our television screens. Runtown and his girlfriend Adut Akech are nothing less than a relationship goal to some of his fans and celebrities alike.
What started as a joke on the singer's comments section has finally transformed into a serious relationship. The couple has both fallen head over heels for each other and can't stop expressing their love for each other as they continue to share PDA moments of themselves on their separate Instagram pages. As a result, this has sparked many questions as fans want to know more about the pair's love life and how far they plan to remain together. If you are also keen to know more about Runtown's love story with his foreign girlfriend, these interesting facts may be of help.
How Runtown's Love Story with Adut Akech Began
A post shared by Runtown HQ (@runtown.hq)
Runtown's love story cannot be complete without mentioning his relationship with his popular girlfriend, Adut Akech. What appears to be a major topic on various tabloids began in 2019 when the Nigerian singer-songwriter and producer Douglas Jack Agu (aka Runtown) and Adut were seen together in a viral video. The relationship between the two was not made official for several months until a question and answer session hosted by the artist had a curious fan asking him about his girlfriend.
In his reply, Runtown said she is from South Sudan. This automatically confirmed the rumor about Runtown and Adut being in a relationship. Talking about how the two met, it appears that the South Sudanese-born model made the first move that sparked off their relationship. In one of Runtown's comments sections on Instagram, Adut requested the singer's mobile number. Apparently, the Nigerian artist forwarded his number.
Soon after what seemed like just a flirty move by both parties, photos of the pair together began to flood the internet, and it generated questions of if the couples were an item? They both dropped pieces of clues and hints for their fans to pick up to show they were in a relationship as they regularly posted pictures and videos of themselves publicly displaying their affection for each other but still not coming out to confirm their relationship.
What we Know About Runtown's Girlfriend Adut Akech
A post shared by Adut Akech Bior (@adutakech)
Though not as popular as her Nigerian boyfriend, Adut had a strong following on social media, which indicates that she has relevance in showbiz. However, it is true that her relationship with Runtown further brought her to the limelight.
Born on December 25, 1999, Adut is the second of five children born to her South Sudanese parents, who relocated to Kenya shortly after her birth. Adut's elder sister is Kim Akech, while her younger siblings are Alakiir Akech, Yar Akech, Bior Akech, and Akoul Akech. She lived with her family in a refugee camp in Kakuma before relocating to Adelaide, Australia as a South Sudanese refugee. At the time of their relocation to Australia, Adut was already 7 years old.
Adut used a different name in Australia because the school teachers found it difficult to pronounce her birth name. She went by her mother's name Mary. However, she reverted to her original name when she kicked off her career as a model. At 16, she was signed to Chadwick Models, a modeling agency run by her mother. She made her first appearance on the runway at a local fashion show hosted by her aunt.
Adut has had a lucrative career as a model and has grown into an A-list model. She claimed the crown of Model of the Year at the British Fashion Awards in 2019 and appeared on the cover of the British Vogue September 2019 edition. Adut is also a two-time winner of Model of the Year by Model.com.
How Often Were the Pair Seen together?
Runtown with his girlfriend, Adut Akech
Couples like Runtown and Adut cannot avoid being stalked by fans who are eager to have the smallest clue about their affairs. On their part, the couple never shied away from feeding their fans with content that would interest them. The pair are often seen enjoying constant baecations in several luxurious places around the world from Paris to London to California and a host of other top go-to spots.
During one of their many trips to France, Runtown offered his model girlfriend a promise ring that officially sparked their relationship. During the artist's 31st birthday in 2020, they were also seen spending quality time together in Greece. Aside from their romantic trips, Adut has featured in a couple of Runtown's songs. When the video of Runtown's single Kini Issue dropped, she was featured in it as the singer's main vixen.
Has the Couple Been in Any Scandal?
It is not often unusual to have celebrity couples getting involved in scandals, and while many survive it, some are merely lucky to escape. For the likes of Runtown and Adut Akech, they've rarely been mentioned to be involved in any known controversy. The two might be quite public with their relationships, but even so, they tend to settle their scores privately to avoid any public scandals.
As far as criticism goes, a few thought their relationship wouldn't last this long as they judged based on the fact that most celebrity couples don't last long together. Many also feared the two wouldn't last long together following the huge age gap between them. There is a 10 years age gap between Runtown and Adut Akech. Runtown would be 32 later this year while Adut will clock 22 this year.
Meanwhile, the only known scandal involving the music artist was with the Nigerian UK-based model, Abdul-Kareem Jamila. The romance scandal has Jamila accusing Runtown of not picking her call since he abandoned her in the United Kingdom after his birthday celebration. Also attached to her message was the ultrasound scan of what she described as their three-month-old baby.
Runtown and Adut Akech Were Never Engaged
There have been many rumors about Runtown and Adut Akech being engaged since Adut made a video she shared on her Instagram story of her and her friends screaming joyfully with a huge diamond ring on her finger. The video fueled the speculations about a possible union between the two.
Speculations grew further when Runtown shared a picture of the pair with Adut wearing the same diamond ring. However, Runtown's management quickly debunked the rumor by claiming that the ring the singer and his girlfriend shared was nothing but a promise ring.
Their Relationship Ended Sometime in 2021
Like many other celebrity relationships that were quick to hit the rock while it just began, Runtown's love story with Adut Akech never stood the test of time as many would expect. In July 2021, the South Sudanese model revealed that her relationship with the Nigerian singer has ended. Adut said so while responding to questions from her fans on Instagram. When asked if she was in a relationship, the 22-year-old popular supermodel said she's now single and ready to mingle.
A Peek Into Runtown's Past Relationships
Before meeting Adut, Runtown was involved in a relationship with a couple of women, some of whom we are familiar with. A good example of these women is Selina Leath. Selina is a model and actress based in the United States. Selena is older than Runtown by two years. During the course of their relationship, they welcomed a son named Zamar.
Before Runtown's relationship with the US-based model Selina, he was rumored to be dating a co-music star, Cynthia Morgan. The rumor about them being an item went viral when the two were caught together at an undisclosed eatery. However, the two were quick to debunk the rumor.
chika Udeh


This Easy Grilled Pork Recipe is made with thin slices of pork belly and marinated using 3 simple ingredients. I used a cast iron skillet as I only cooked a pound of meat and don't really want to start the grill. It worked just fine and turned out really good.
You can grill, pan sear or fry and serve hot with spicy vinegar and kalamansi for dipping.
Rinse pork slices and pat dry. Set aside.
In a wide container, combine soy sauce, kalamansi juice and minced garlic. Mix well.
Add pork slices to the marinade. Mix well and refrigerate for a hour or two.
Once marinated, drain and you can either grill or fry pork belly slices.
Serve hot with spicy vinegar and lots of steam rice. Enjoy.


Maitreyi College was established by Government of Delhi in 1967. University of Delhi (DU) provides affiliation to this college for offering courses at Under Graduate (UG) and Post Graduate (PG) level. At present various courses in arts, science, commerce and applied sciences are offered.
The dedicated faculty members, well-stocked library, sophisticated laboratories and scope for discussions; academic and non academic topics makes Maitreyi College, one of the Hot Colleges under Delhi University. College gives first preference to academics, though, for overall development of learners, co curricular and extra curricular activities are also focused. Maitreyi College is already reputable for remarkable achievements in fields of sports, NCC, NSS and Cultural activities. In academics, it makes preferred choice among admission seekers.
Cut Offs B.A B.Com. B.Com.


Today was a fantastic day in Milwaukee, WI for Summer Spark presented by University School of Milwaukee. Summer Spark is a two day professional development conference with a host of keynotes and workshops for teachers and facilitated by teachers.
Google Innovator, Jeff Heil said, "We are all here because we want to do what's best for our students and to be better teachers for the students we serve." This is the underlining mission of Summer Spark, to support teachers with engaging, relevant and fun professional development. Today I had the opportunity to learn from amazing educators that helped to refocus my teaching purpose and passion.
The keynote was presented by Tom Murray(@thomascmurray), co-author of Learning Transformed. He introduced eight key ideas about teaching today to help students succeed in the future. The main point is that stand and deliver method of instruction doesn't work today and that we need to make learning personal. Questions are more important than answers and we need to support students and allow them to figure things out, flip and twist in order to engage and create – not regurgitate answers. He used the analogy of a Rubik's Cube . . .
The rest of the day I was on a gamification kick and attended three workshops on games and game building.
Melissa Pilakowski (@mpilakow) shared the Top Ten Games for any classroom. From Jenga writing and Scattegories to Gimkit and March Madness Brackets, Melissa had a game and gaming ideas to inspire students and make learning fun. Some digital games are helpful for basic vocabulary concepts and formative assessments like Peardeck's Flashcard Factory and Vocabulary Dominoes, where others were great for argumentative and or creative writing like the card game Fun Employed and Storiumedu.
And the games didn't stop there, Stephanie Crawford (@MrsCford_tweets) presented a session on Engaging and Empowering Mini Games that ignite the classroom by providing hands on assessment and critical thinking. Take out the play doh, legos, and dice and let's have some fun. We had different challenges and in a short amount of time were given four types of challenge that promoted collaboration, creativity, and fun.
Michael Matera's (@mrmatera) session on getting started with gamification was a reminder of the elements needed to sustain playfulness with challenge and purpose. In gamifying you classroom you need to choose a theme, create epic learning experiences, and set up the game mechanics. Matera's Master Chef challenge is one I would like to replicate in my own classroom. Twenty students were selected to compete in this challenge and working on teams of four or five, students had to answer questions from a mystery box correctly. Strips of paper with the assessment questions on them were put into a box and students had to select a question and write our their answers on a team answer sheet to the questions. The teamwork and random selection made this game exciting for the students and fun.
The last workshop of the day was lead by Shelley Burgess (@burgess_Shelley), co-author of Lead Like a Pirate. We began by sharing our top three movies and then discussed what these movies say about us as educators. She reminded us that our job is to "raise human potential, and that raising test scores is not the end all, be all." Education today is a people business and about relationships." She asked us what type of germs are we spreading throughout the school and spoke about inspiring and supporting adults and students in our schools.
Of course the day was not over after a keynote and three workshops, there was more fun and collaboration to be had. Throughout the conference we were put into teams and played Goosechase, a digital scavenger hunt taking pictures, videos and sharing ideas to compete against each others. Before the end of the day we met in our teams to create a teacher superhero modeled from the super hero teams we are on. Lastly, trivia night was the most challenging trivia I have ever played. Despite the crazy questions and questions that stumped my team it has been an inspiring day and I have many ideas for the wild and wonderful first day of school come September.
The following blog post was written by Julie Randles for ISTE's EdTekHub. The original post can be found here.
Awarding badges is more than a way to recognize student accomplishments. For educator Michele Haiken, badges also offer a way to give students a self-paced learning experience.
"I looked to my gaming experience and I borrowed the idea of badging as I re-examined my curriculum to find ways that students could work independently and in a self-paced environment to meet learning targets," says Haiken, a teacher at Rye Middle School in New York.
And with that new benefit in mind, Haiken was hooked.
Consider reversing curriculum design. Haiken found the best way to get started with badging was to "backward design" some of her curriculum. She started with her targets for students by semester's end – say meeting Common Core standards or her own standards – and then created self-paced learning projects.
She took this approach in both an English class and a speech and debate elective, making the first 10 weeks of class self-paced and requiring students to complete three badges by the end of the quarter. It all began with asking herself what she wanted students to be able to do in 10 weeks and what smaller pieces could she create that show evidence of learning?
Revise or re-rig. If the backward design approach is too much to bite off, Haiken suggests revising current curriculum to include opportunities for students to master learning levels to earn badges.
She took this approach for a dystopian reading unit where all students were reading different novels. The entire class met to discuss broad themes in all dystopian novels, but when students met in smaller reading groups or worked independently, Haiken provided badge-based activities that let her know individual students understood the texts they were reading.
Build in opportunities for reflection and revision. Adding badging into the learning mix is a great way to encourage students to slow down, understand concepts and use old knowledge to build new knowledge.
It's also a good way to address the 2016 ISTE Standards for Students, which expect students to use technology to take an active role in choosing, achieving and demonstrating competency in their learning goals.
Students used the knowledge they gained from that reflection to created their own speeches, and earn their next badge.
"I would send notes through Google Classroom so they could revise or improve; so it wasn't one and done and their work showed a synthesis of old knowledge and new knowledge." Forcing students to improve their work before they could earn the next badge helped drive home the importance of revision and reflection.
Try badges for motivation. Badges can also help create a positive classroom culture. Consider awarding badges to students who have gone above and beyond as "super helpers" or to encourage acts of collaboration, character and citizenship.
Educators interested in learning more about how to use badges to recognize mastery and achievement can join Haiken for the ISTE Professional Learning Series webinar "Improving Student Achievement with Classroom Badges" on April 26.
Hear about badging ideas, criteria and ways to organize them in their classrooms.
Get resources for designing and distributing digital and physical badges.
Learn how other educators are using badges across content areas and grade levels.
ISTE members can sign up now for the ISTE Professional Learning Series that includes the webinar "Improving Student Achievement with Classroom Badges." Not a member? Join ISTE today.
Old School games are a great way to bring gaming into any content area. Whether playing Jeopardy, Who Wants to be A Millionaire, or Jenga, these types of games build collaboration and can help students deepen their content knowledge. One of my "go to" games with my students is Bingo. Here are a few ways that I have adapted Bingo for learning and assessment.
1. Text Dependent Questions – I will fill an entire bingo board with text dependent questions or problems and students have a specific time to fill out the Bingo board. You might utilize this as a homework assignment for the week (each night complete one row or column), assessments (A = complete the entire board correctly, B = complete 4 rows of Bingo, C = 3 rows of Bingo), or an in class activity. Below is a class activity that I use to review Chapter 7 & 8 in To Kill A Mockingbird.
2. Pursuit – Give students a Bingo board with situations or actions and students are required to find specific textual details (or direct quotes) that highlights the situation. I recently made a Bingo board like this for MidSummer Night's Dream Act 3. The pursuit gave students a mission to uncover key events and show their understanding while reading the play in class.
3. Picture Bingo & Empty Bingo Boards – Use pictures instead of text or give students a word bank to fill in their own Bingo Board. Then, ask questions related to the words in the word bank or images.
4. Persuasive Bingo – When I taught speech and debate I created five different Bingo Boards with a variety of persuasive speaking tasks: Persuade your parents to increase you allowance, persuade your sibling to do your chores, persuade your teacher to give you an extra day to complete an assignment. The key was that the students couldn't bully, blackmail, or bribe to achieve Bingo. When a number and letter was called the students had to persuade the entire class effectively in order for it to count.
Bingo is fun and interactive. Bingo boards can be adapted for any content area or grade level. Plus, they are easy to make. Depending on the task created for students the questions can tap into Bloom's questioning, critical thinking, and allow teachers to assess student understanding.
Why a badge driven curriculum?
Students are able to monitor their own learning and take responsibility for their work and understanding. Students are aware of the learning expectations and the work they need to master in order to successfully pass the course. Students know the end goal and are awarded badges for completing specific learning targets and challenging and extraordinary tasks.
I have transformed my speech and debate elective class for middle school students this semester into a module based independent study where students complete different tasks to show mastery and earn badges.
The first ten weeks of the semester students are to complete three different badges for students to work towards and show their understanding and knowledge of public speaking. The second half of the semester students will focus on debating skills and participate in different types of debate.
Read about the Great Speakers Are Made, Not Born Badge expectations here.
This badge is designed to help students utilize public speaking and writing skills by crafting a non fiction speech. Throughout this bade students are working on structure, word choice, and literary devices in their own writing.
Once students earn the Words are Powerful Badge they can work on the Presentation Guru Badge. This badge requires students to put together all that they have learned about the qualities of great speakers and writing strong speeches. The final part of this badge is for students to write and present a TED style talk.
Here are some of the specific learning targets and extraordinary tasks students will set out to complete this semester.
Write the title of the story AND give the episode a "new" creative title.
Write 3 things you learned about the speaker telling the story.
Write 3 new questions you have you want to ask the storyteller, and WHY you want to know the answers to these questions.
How did the speakers make you feel? Engage You In the Topic? Encourage You to Continue Listening?
What were some of the rhetorical moves the speakers utilized to successfully present the information in the podcast?
What other things did you notice about the podcast that are worth mentioning and pointing out to other listeners?
Stories have a change. The main character (you!) has to change in some way from beginning to end.
Stories have stakes. Why did this moment matter to you?
Know where your story is heading. Steer clear of meandering endings!
Be YOURSELF. This is not a monologue, a standup routine, or a rant.
#3: Podcast Your Story – Record on a Podcast your story for others to hear*. Turn in your audio podcast on Google Classroom.
* Almost everyone needs to practice reading aloud their story to make it exciting and interesting in terms of the words as well as one's vocal presentation. Your voice needs to be loud, clear, and authentic.
RadioLab Style Podcast – RadioLab is a show on NPR that presents topics related to science through engaging conversations, media clips, and investigative journalism. Check out http://radiolab.org for more information and to listen to a few podcasts before you get started.
Choose a ONE HOUR episode of Radiolab on http://radiolab.org. Download or listen online to the ENTIRE episode.
Write the title of the episode AND give the episode a "new" creative title.
Write 3 things you learned about the topic in the podcast.
Write 3 new questions you now have about the topic, and WHY you want to know the answers to these questions.
What were some of the rhetorical moves the speakers/hosts utilized to successfully present the information in the podcast?
Select and Research a topic of your choice (most RadioLab Episodes are science related).
Write a script for a 3-5 Minute Mini-Radiolab Episode sharing insight, research, and findings.
You will also turn in a short, 1-2 page reflective paper AFTER delivering your podcast; the reflection paper should describe the process that went into developing your project and your thoughts about the performance.
Complete the Data Collection Worksheet* For each TED Talk you view. You are to view 3 or more Talks.
This semester I am teaching a section of Speech and Debate for 8th graders and I wanted to create a self guided course that requires students to research, write, speak, and reflect continuously throughout the course. There are three key modules for students to show their mastery of public speaking utilizing a badge system.
Think about what the Girl Scouts and Boy Scouts have been doing all along, recognizing mastery and achievement with badges. Teachers can do the same by recognizing student accomplishments and mastery with badges. Badges go beyond grades because they are given out at the teacher's discretion for more than just academic achievement. Students work toward completing different badges to show mastery of a concept, standard or skill.
In the public speaking elective I designed, students have a great deal of choice – to choose topics and projects using a badging system. This approach gives students both freedom and responsibility as they will be required to manage their time wisely and work independently (in community) toward their personal course goals.
Summarize the key ideas presented in each speech and the rhetorical devices used in each speech to make these iconic speeches. Use direct textual evidence (3-4) to support your claims.
Write a paragraph about the speakers themselves. What strategies do MLK and JFK emulate in their public speaking? Identify four or more public speaking strategies using direct textual evidence to support your claims.
What makes MLK and JFK "Great Speakers?" Reflect on the elements of these speakers and their speeches that you hope to emulate this semester in Speech and Debate. Use specific details and examples. These will in turn be your personal speaking goals for the semester.
#3: Who is "great speaker" of our time?
Identify a speaker today who you think emulates the qualities of great speakers you identified in JFK and MLK.
Find a video that showcases this speaker and include the video link that highlights the elements of public speaking utilized by this person. Post this video on Google Classroom for all to view, reflect, and review. (Select a video that is 3-10 minutes in length).
Write an analysis (1-2 pages) describing the strategies and rhetorical devices presented by this speaker. Use direct textual evidence to support your claim. Include specific information describing the aspects of this speaker you aspire to when speaking in public.
Using one of the following Content Curation/Social Bookmarking sites: Pinterest, ThingLink, or Symbaloo, find nine or more (9+) research and public speaking videos (3), research articles (3), and interviews with public speaking teachers or figures (3).
Annotate what can be found at each site and/or video and highlight one (or more), key quotes and a strategy for public speaking highlighted in the resource.
Describe in 2-3 sentences how this content can help you to be a better public speaker.
You can read about the expectations for the Words Are Powerful Badge and Presentation Guru Badge in this post.
The following post is a guest blog post I wrote for ISTE's EdTekHub and was published on 1/5/2017.
Many of our students are among the 155 million Americans playing video games regularly, and you might be, too. That's a good thing. When my 11-year-old is playing video games, he's using many skills – facts and information are tools to solve problems in context, and he gains actionable feedback he uses to win the game. When he fails to level up, he doesn't give up, but continues playing until he progresses to the next level. He also seeks information online to help him find Easter eggs hidden throughout the game. He teaches his friends how to power up with each level of the game. Failure is a source of feedback and learning, collaboration is necessary, and learning and assessment are tightly integrated.
How can we use this pervasive and engaging gaming phenomenon to redesign and supercharge the learning experience?
Adapt old-school games for classroom use. Scavenger hunts, bingo, dice games, Connect Four and Scrabble have been around for decades and can be adapted for classroom learning. Put vocabulary words on bingo cards and see if students can match the words after hearing the definitions. Working in groups, students can play Scrabble by spelling out answers to content-specific questions. Using the app Goose Chase, create digital scavenger hunts by sending students off to take pictures, create a video, or search for an answer online related to a specific topic.
Play digital games. Students love playing Kahoot!, Quizizz and Quizlet. These free platforms allow teachers to create multiple-choice questions that players answer on their own devices. Teachers can also choose from the thousands of quizzes already shared on these sites or create content-specific questions to use as pre-assessments, quizzes or exit tickets. Breakout EDU also has a collection of digital games, puzzles and ciphers that promote critical thinking.
Create a quest. A quest is a mission with an objective. Every year my students participate in an adventure quest based on the weekly current events reading. Students who correctly answer a specific text-dependent question earn points. The student with the most points after six weeks wins a prize. I post additional questions on Remind and Twitter to allow students to earn extra points. Quests can also be independent projects or activities for the students who have finished their work.
Battle it out with a boss battle. In gaming, a "boss" is a villain who the hero must defeat to save the day. Think of the monster at the end of each level in the original Super Marios Bros. who must be defeated before moving to the next level. On the gamification platform Classcraft, teachers can create their own boss battles using questions from any content area. Teachers can also create boss battles using Google Forms or Google Slides, creating their own unique fictional boss.
Earn a badge for mastery. The Girl Scouts and Boy Scouts recognize mastery and achievement with badges. Teachers can do the same by rewarding student accomplishments and mastery with badges, which go beyond grades because they represent more than just academic achievement. Students work toward completing different badges to show mastery of a concept, standard or skill. Badges can be presented digitally using Classbadges or can be displayed for all to see once students have earned a specific badge.
Gamification is about transforming the classroom environment and regular activities into a game. It requires creativity, collaboration and play. There are numerous ways to bring games and game playing into the content area classroom to promote learning and deepen student understanding. Whether teachers are looking to bring some aspect of gaming into their class or use a game platform across the curriculum, they can use gamification elements to enhance learning and student engagement, tap into Common Core State Standards and address the 2016 ISTE Standards for Students.


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Dorothy Smith Harlow: An Active Community Volunteer
... Stratford and wintered in Scottsdale, AZ. Dorothy is survived by her children: Patricia Harlow (Alan Sussman) of Greenwich, William S. (Rhonda) Harlow of Miami, FL, Mary (John) Gulland of the Isle of Man, England, and Barbara (Jerry) Logan of ...
http://stratford.patch.com/groups/obituaries/p/dorothy-smith-harlow-an-active-milford-community-volunteer
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... said Foley, the former U.S. ambassador to Ireland under President George W. Bush and a private equity manager from Greenwich. Danbury Mayor Mark Boughton, who is looking to wrest the endorsement away from Foley, also presented himself as a model of ...
http://m.ctpost.com/local/article/A-tale-of-two-nominating-conventions-for-governor-5482267.php
Hero cops honored with Clarke Award
Greenwich Police Chief James Heavey, left, presented the Greenwich Lions Club's annual John A. Clarke Memorial Award to Officers Francesco DiPietro, center, and Shawn Fox during a luncheon in Greenwich, Conn.
http://www.greenwichtime.com/local/article/Hero-cops-honored-with-Clarke-Award-5482242.php
Billionaires buying more $100M homes in 2014
... purchased the East Hampton home earlier this spring, timber magnate John Rudey sold his Copper Beech estate in Greenwich, Connecticut, which he had owned since the 1980s, to an unknown buyer for $120 million. At the time of the sale, Copper Beech ...
http://therealdeal.com/blog/2014/05/15/billionaires-buying-more-100m-homes-in-2014/
Ronto to build 120-unit luxury hi-rise on vacant Bonita Bay tract
... planned, but because of the economic downturn, only six were built. In November, Wheelock Street Capital LLC, a Greenwich, Conn.-based real estate investment firm, partnered with Ronto to buy the final two adjacent parcels for a total of almost ...
http://www.naplesnews.com/news/2014/may/15/ronto-build-120-unit-luxury-hi-rise-bonita-bay-lee/?partner=RSS
Foley says he's raised enough to qualify for funds
Greenwich businessman and Republican gubernatorial candidate Tom Foley says he's raised enough in contributions to qualify for a grant under Connecticut's public financing system.
http://www.greenwichtime.com/news/article/Foley-says-he-s-raised-enough-to-qualify-for-funds-5481567.php
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... covers 10 acres. "It's a real struggle to start a business. There's a reason why they come to Bridgeport and not Greenwich -- we have a lot of programs," said Sam Pierpont, principal in the innovation center. Commenting that in 2013, the city's ...
http://m.ctpost.com/news/article/Bridgeport-mayor-city-staff-stress-opportunities-5481459.php
Nick Packer takes over as Torus Insurance CEO
... operating officer of Enstar Group Ltd., the companies said Thursday. Hamilton, Bermuda-based Enstar together with Greenwich, Connecticut-based Stone Point Capital L.L.C acquired Torus at the beginning of April. Mr. O'Donohoe is leaving after "having ...
http://www.businessinsurance.com/article/20140515/NEWS04/140519891?CSAuthResp=1%3A173536852693886%3A109186%3A64%3A24%3Aapproved%3A00FB4EFD83832A5F2270C6CD41F79A29
Sole interest: Greenwich author shares her fantasy shoes in new book
Artist Douglas Wilsons "Automata VI" is among the shoes featured in Greenwich resident Jane Gershon Weitzman's book "Art & Sole," which was recently released.
http://www.greenwichtime.com/news/articleGallery/Sole-interest-Greenwich-author-shares-her-5480595.php
Celebrity Real Estate: Marv Albert, Ron Howard
... his wife and family to an amazing estate in Armonk, N.Y., about an hour north of New York City and close to tony Greenwich, Conn. But with the kids grown, and now 60 years old, Howard says it's time to sell the family spread. The man known as Opie ...
http://www.recordpub.com/real%20estate/2014/05/15/celebrity-real-estate-marv-albert-ron-howard
GE Antares supports PE-backed Merex Group's buy of Kellstrom Defense
... at www.merexinc.com. About Dubin Clark: Dubin Clark & Company, Inc. is a private investment firm headquartered in Greenwich, Connecticut with a 30-year history of investing in control-position, middle-market buyouts and building businesses in ...
http://www.pehub.com/2014/05/ge-antares-supports-pe-backed-merex-groups-buy-of-kellstrom-defense/?utm_source=feedburner&utm_medium=feed&utm_campaign=Feed%3A+pehub%2Fblog+%28PE+HUB+Blog%29
GE Antares Capital Finances Merex Group's Acquisition of Kellstrom Defense Aerospace, Inc.
... is available at www.merexinc.com . Dubin Clark & Company, Inc. is a private investment firm headquartered in Greenwich, Connecticut with a 30-year history of investing in control-position, middle-market buyouts and building businesses in partnership ...
http://www.businesswire.com/portal/site/topix/index.jsp?ndmViewId=news_view&newsId=20140515005052&newsLang=en&ndmConfigId=1000639&vnsId=41
William H. Willis, Jr., 86, Longtime Greenwich Resident
... the American Museum of Natural History in 1953. He founded William Willis Worldwide with offices in New York and Greenwich. He was the founding chairman of the World Search Group. Early in his career, he was in industry with AMF and Owens Corning ...
http://greenwich.dailyvoice.com/obituaries/william-h-willis-jr-86-longtime-greenwich-resident
Several Greenwich Residents Face Domestic Violence Related Charges
Several Greenwich residents are facing a variety of charges including disorderly conduct, assault and strangulation in connection with a series of arguments and assaults investigated by Greenwich Police.
http://greenwich.patch.com/groups/police-and-fire/p/several-residents-face-domestic-violence-related-charges
American Industrial Partners buys SERVA From Wexford Capital
... ("Wexford"), an SEC registered investment advisor with over $4 billion of assets under management headquartered in Greenwich, CT. Founded in 1991, SERVA is a leading designer and manufacturer of fracking equipment, coiled tubing equipment, and well ...
http://www.pehub.com/2014/05/american-industrial-partners-buys-serva-from-wexford-capital/?utm_source=feedburner&utm_medium=feed&utm_campaign=Feed%3A+pehub%2Fblog+%28PE+HUB+Blog%29
Greenwich's Temple Sholom Hosts Father, Son Rabbis
Greenwich's Temple Sholom will welcome the return of Rabbinic Intern Erez Sherman and his father, author Rabbi Charles Sherman, for a series of activities.
http://greenwich.dailyvoice.com/neighbors/greenwichs-temple-sholom-hosts-father-son-rabbis
Greenwich Shines Spotlight On Fraud Prevention To Educate Seniors
There was a fashion show Wednesday in Greenwich Town Hall -- but the models weren't leggy young beauties sporting the latest in haute couture.
http://greenwich.dailyvoice.com/police-fire/greenwich-shines-spotlight-fraud-prevention-educate-seniors
Greenwich Man Facing Charges In Domestic Dispute
A 38-year-old Greenwich man was arrested after taking a phone out of a woman's hand during a domestic dispute early Wednesday, police said.
http://greenwich.dailyvoice.com/police-fire/greenwich-man-facing-charges-domestic-dispute
Greenwich Police Arrest Two on DWI Charges
Greenwich Police arrested two motorists in the past week on driving while intoxicated charges.
http://greenwich.patch.com/groups/police-and-fire/p/greenwich-police-arrest-two-on-dwi-charges_af80a6bb
Guiding Eyes' White Plains Event Features Eli Manning
... hold its golf classic on Monday, June 9, at Mount Kisco Country Club in Mount Kisco and Fairview Country Club in Greenwich,Conn. Guiding Eyes representatives anticipate the classic will raise more than $700,000 for its programs and services, ...
http://whiteplains.dailyvoice.com/news/guiding-eyes-white-plains-event-features-eli-manning
Sacred Heart lacrosse team downs Greenwich High
Greenwich's Carolyn Paletta carries the ball as Convent of the Sacred Heart's Emma Novick defends during the girls lacrosse game at Greenwich High School on Wednesday, May 14, 2014.
http://www.greenwichtime.com/sports/article/Sacred-Heart-lacrosse-team-downs-Greenwich-High-5479068.php
Dorothy Smith Harlow: An Active Community Voluntee...
exit-arrow-visit|article-5482267|mobile_simple_lis...
Ronto to build 120-unit luxury hi-rise on vacant B...
Foley says he's raised enough to qualify for funds...
exit-arrow-visit|article-5481459|mobile_home_categ...
Sole interest: Greenwich author shares her fantasy...
GE Antares supports PE-backed Merex Group's buy of...
GE Antares Capital Finances Merex Group's Acquisit...
William H. Willis, Jr., 86, Longtime Greenwich Res...
Several Greenwich Residents Face Domestic Violence...
American Industrial Partners buys SERVA From Wexfo...
Greenwich's Temple Sholom Hosts Father, Son Rabbis...
Greenwich Shines Spotlight On Fraud Prevention To ...
Guiding Eyes' White Plains Event Features Eli Mann...

redpajama_set_name: RedPajamaCommonCrawl


Thank you so much again! Dnyaneshwar Chaudhari 21 December at The book is in the 17th edition currently is around pages.
Notify me of new posts via email. Rahul "-" 30 Shivcharltra at Akash Patil 1 November at I have downloaded page by page and it took hours to download pages.
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Suraj Khopade 28 January at Goodreads helps you keep track of books you want to read. Sanket Walavalkar 17 January at Shivcharitra in marathi PDF sent on your email-id. Unknown 2 September at Unknown 9 June at Vaibhav Mahajan 8 November at Just enter the search keyword and download the publication for free. Please mail the pdf at amardeepchaudhari08 gmail.
Plz share me pdf file of Shiv charitra Email-Email-tusharbachhav gmail. Hemant Jaitapkar 5 August at Please send on shantanu. Nilesh Gaikwad 9 September at Anonymous 20 Babasahebb at Manapasun Abhari ahe Rajendra: Ranapratap Patil 14 November at A Classic and yet it's a must read.
Return to Book Page. Ankush Chavan 6 November at Vivek Namaskar Aplya mehenatila amchaa mujraa Shivaji Maharajanche thor vichar apan aplya acharnaat aanu!!!!!
Anil Shinde 22 August at Mangesh 11 June at Dear vivek sir, mala pan plz send kara PDF on bhushanpilot12 gmail. But direct downloading facility is not provided. Babasaheb's passion for his subject- shines through this magnum opus.


α + 12C elastic and inelastic to the Hoyle state (0+2, 7.65 MeV) differential cross-sections were measured at the energies 60 and 65 MeV with the aim of testing the microscopic wave function  widely used in modern structure calculations of 12C. Deep rainbow (Airy) minima were observed in all four curves. The minima in the inelastic angular distributions are shifted to the larger angles relatively those in the elastic ones, which testify the radius enhancement of the Hoyle state. In general, the DWBA calculations failed to reproduce the details of the cross sections in the region of therainbow minima in the inelastic scattering data. However, by using the phenomenological density with rms radius equal 2.9 fm, we can reproduce the Airy minimum positions.


text:
Pole vaulter places 9th in World University Games
By Derek Bouma
After staying in a village with elite athletes from around the world and walking out to an opening ceremony in front of a crowd of about 50,000, Mike Woepse could only describe the experience as Olympic-like.
This July, the rising senior pole vaulter was selected to represent the United States in the 2013 World University Games in Kazan, Russia, with athletes varying in age from 18 to 27.
"To be in front of all these people doing my favorite thing in the world and to have the opportunity to do it wearing a USA uniform was an unbelievable experience," Woepse said.
Although he described his ninth place finish out of 11 pole vaulters as disappointing, Woepse did not let the results taint the experience and instead viewed it as motivation.
"Just to have a taste of it makes you hungry for it," Woepse said. "I've always dreamed of being in this situation so it was a dream come true, but it also made me realize that these goals I have are not too far out of reach."
After coming back from Russia, Woepse will only rest for one week before he begins training for the NCAA season. He has already created a mental list of goals for himself that includes making the national team.
Having worked with Woepse the past three years, volunteer jumps coach Anthony Curran said that Woepse is one of the most committed athletes he's ever coached and has no doubts that he can reach his loftiest goals.
"He has everything it takes; obviously since there's only three people in the entire United States that makes the Olympic team, it's always going to be a tough thing to do," Curran said. "But if there's anybody who has the ability, the strength, the power and the mind, it's Mike Woepse."
As a teammate and roommate for the past three years, rising senior hammer thrower Alec Faldermeyer said he couldn't have been happier for Woepse to make the World University Games, and that he also sees a bright future for the pole vaulter.
"Obviously he's going to be enjoying a professional career after this, I would imagine, with how talented he is, so that is just a big stepping stone in the right direction for him," Faldermeyer said.
Both Faldemeyer and Curran cited hard work as the main reason why they believe Woepse can reach his goals.
"Nobody, nobody outworks Mike, nobody," Faldermeyer said. "I've never met a person who's worked harder every single day. No matter how he's feeling, he's out doing his full workout, going above and beyond the call."
Woepse said he does not plan to stop after this summer's glimpse into what could be his future.
"It ended up being 10 times more of an experience than I could ever imagine it being," Woepse said. "I'll remember it for the rest of my life and something that I'll chase to be able to do again for the rest of my life."
Mike Woepse 2
FROM: http://dailybruin.com/2013/07/22/pole-vaulter-places-9th-in-world-university-games/
Green Wave Track & Field, Cross Country signing classes announced
Isinbayeva 'positive' after comeback


The answer we should all have is, "of course." But, do you really answer "yes" every morning when you look in the mirror and ask yourself this question?
The reality is that you are not sure if you are actually closing 100 per cent of your possible sales. And to compound the situation, you may not really know how to measure this to improve sales. Marketing encompasses four main components — product, place, price and promotion. The sales function is part of the place component. By reviewing each of these four marketing pillars, you can better understand how your customers would answer yes to this important question.
Begin with an assessment of your product. Fundamentally, if your product or service is not highly sought, there may be trouble. Just because you have built that proverbial better mousetrap, does not mean that people will beat a path to your door or internet store. If you do not offer something more compelling than great service or friendly staff (your competitors likely offer the same thing), you will become a commodity, and selected exclusively on low cost.
The answer we should all have is, "of course." But, do you really answer "yes" every morning when you look in the mirror and ask yourself this question?
When was the last time you spoke with your customers in detail about how your product was used, and what benefits people actually receive from using it? Have you seen this in action? Where is your proof? When a customer is selecting your product or service, they are also assessing the quality and knowledge of your staff, and the level of support provided before, during and after a sale.
Next is a review of the place component of your product or service. Place isn't always a physical "place." This is your distribution channel, which includes your sales force, other sales channels such as your physical store or plant, and any online sales platforms you may have. Again, there are some important questions to ask yourself. Have you actually tried to place an order at any of these touch points? Are you absolutely certain that each step is rock solid and no one will ever say no? Exceptional sales professionals ask insightful questions to uncover the true needs of the customer by getting the customer to talk about their situation. More importantly, they listen carefully for the answers to create the strongest possible connection between the problem your product or service is designed to address and the value-added solution it provides.
The third component to consider is your pricing policy. This is often overlooked, except when your customer says your price is too high. How do you know this for certain? Have you assessed the price of your product or service against your competitors? Have you considered the impact of market demand and industry supply? Do you actually generate cash from operations? How about your late payment and collection policies? Are these factors properly customer-focused?
The final component is a review of your promotion efforts. This is the marketing component that typically gets the most attention because it includes your traditional advertising, social and digital advertising, and any public relations activities that you may undertake. Have you ever reviewed all of your messaging for consistency and relevance? Corporate social responsibility activities are also included in this area. Do you have a structured corporate giving program that guides where you want to invest to make your company a great place to work and improve the quality of life in your community?
These initial questions should be carefully considered as the beginning of a marketing audit. Starting with great questions and applying them in a structured approach will help you uncover whether your organization is truly aligned for people to want to buy from you. Sales expert Jeffrey Gitomer famously stated that "people don't like to be sold to, but they like to buy."
Having confidence in your product or service, in the people that deliver it, and the messaging that you use, are all part of a customer's buying process. When you have all these elements properly aligned, there is a much greater chance that your customers will select you when their need arises. If any of the four Ps are out of sync, then your chances of sales success are reduced.
Tim's bits: Remember to keep the customer at the centre of all your decisions. Manage customer expectations within your operating environment to ensure you can profitably deliver what the customer expects. Business guru Tom Peters, author of In Search of Excellence, stated that "winners focus obsessively on their customers. Losers don't."
Do you consider the product, place and price together to help craft promotional messaging that works? After working through a structured review of how people buy from you, you are probably doing many things well. Create a plan to fill the gaps. You should now be able to look in the mirror and answer "yes!" to the question with pride.
Tim Kist is a certified management consultant who works with organizations to improve their overall performance by being truly customer focused. He can be reached at tim@tk3consulting.ca.


You are here: Home / Events / Atlanta Events / HENNESSY & M BAR Present The FALCONS vs. PATRIOTS Super Bowl Viewing This SunDAY 2/5/17… When Falcons Score We POUR!! No Cover….Ever!
HENNESSY & M BAR Present The FALCONS vs. PATRIOTS Super Bowl Viewing This SunDAY 2/5/17… When Falcons Score We POUR!! No Cover….Ever!
WATCH THE BIG GAME & STAY FOR THE AFTER PARTY!!
to…Experience Score & Pour NFL Sundays at M BAR.
at the 9th year of a stellar NFL career?
We'll be watching closely at M BAR.
of course the SUPER BOWL!!
not a bad seat in the house.
BTW, WHEN THE FALCONS SCORE WE POUR!!!!
SUPER BOWL FESTIVIES START AT 5:00PM!!
All Other Seating Is First Come, First Serve….
So PLEASE Get There Early!!!
WE WILL SEE YOU THERE! CHEERS!
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text:
\section{Introduction}
Compared to complex algebraic geometry, real algebraic geometry has a flexible nature, thus offering many different approaches to the subject. For example, in the real case there is no analogue of Hartogs' extension theorem. This means, in particular, that given an analytic function germ $f: (\RR^2, 0) \rightarrow \RR$, we can construct an analytically distinct $g: (\RR^2, 0) \rightarrow \RR$ such that $f$ and $g$ are isomorphic on a complement of the origin.

Thus, for a qualitative study of real singularities, we have a chance to introduce a suitable equivalence relation taking into account this phenomenon. Blow-analytic equivalence, originally introduced by Kuo \cite{original}, is one possibility.

For function germs, several classification results are known, cf. \cite{Koike} \cite{Fukui}. In this article, we shall study the classification of germs of \textit{embedded} plane curve singularities up to blow-analytic homeomorphism, instead.

The first result in this direction appeared in \cite{Kuo} in 1998 and it showed how a straight line is blow-analytically equivalent to a cusp (in fact, it is equivalent to any unibranched germ of plane curve, no matter how singular). The peculiar behaviour of this example ---known as the Kobayashi--Kuo example--- suggests that the blow-analytic equivalence of embedded curves has strong topological nature.

Here we aim to continue and expand the blow-analytic classification of embedded plane curve singularities. Firstly, we recall the basic definitions and constructions ---which can be found in \cite{Kuo} \cite{Kob}--- along with previous results in the subject, mostly concerning the classification of unibranched and bibranched germs of plane curves.

The study of the tribranched case occupies sections four and five, where we compute an upper estimate for the number of blow-analytic equivalence classes depending on a combinatorial invariant $\mu'$ defined in section two, and then we proceed to find representatives for each class for $\mu' \leq 2$.

Our main result concerns the general $n$-branched case and is stated in section three. We show the ``local'' finiteness of the blow-analytic classification, that is, the finiteness of the number of blow-analytic equivalence classes for fixed values of the number of branches and of the above mentioned discrete invariant.

We conclude this article with some topological considerations about the tools used, which will be further addressed in a future study \cite{new}.\\

We would like to thank the referee for the useful comments and suggestions made during the revision process.

\section{Blow-analytic invariants}
\begin{definition}
A function $f: \RR^2 \rightarrow \RR$ is \textit{blow-analytic} if there exists a composition of blow-ups $\beta = \beta_1 \circ \cdots \circ \beta_m : X \rightarrow \RR^2$ such that the composition $f \circ \beta$ is analytic.

A homeomorphism $h: \RR^2 \rightarrow \RR^2$ is \textit{blow-analytic} if $h$ and its inverse $h^{-1}$ both have blow-analytic components.
\end{definition}

In this article, we study the classification of embedded germs of real plane curves up to blow-analytic homeomorphism. We say that two germs of curves $(C, 0)$, $(D, 0)$ in $\RR^2$ with an isolated singularity at the origin are \textit{blow-analytically equivalent} if there exist a blow-analytic homeomorphism $h: \RR^2 \rightarrow \RR^2$ that carries $(C, 0)$ to $(D, 0)$.\\

It follows from the definition that two analytically equivalent curves are also blow-analytically equivalent. On the other hand, there are known examples of blow-analytically equivalent curves that are not even $\mathcal{C}^1$-equivalent or bi-Lipschitz equivalent (\cite{Koike}). In this sense, the classification of singular curves up to blow-analytic homeomorphism offers a more flexible approach than the analytic one.\\

Now, let $X$ be a surface which is a tubular neighbourhood of the union of compact smooth curves $\{E_j\}_{j=1}^m$ intersecting transversally. We construct the weighted dual graph $\varGamma$ associated to $X$ by drawing a vertex $v_i$ for each central curve $E_i$, and connecting two vertices by an edge if and only if the corresponding curves intersect. To each vertex we assign as weight the $\mathbb{Z}/2\mathbb{Z}$-valued self-intersection number of the corresponding curve. In figures of the graphs $\varGamma$, we represent odd curves as white vertices and even curves as black vertices.

\begin{definition}
We say that $X$ is \textit{smoothly contractible} if it is a surface obtained from $(\RR^2, 0)$ by a finite sequence of blow-ups and blow-downs.
\end{definition}

It follows easily from the definition that if $X$ is smoothly contractible, then its dual graph $\varGamma$ is a tree.

Let $A = (a_{ij})$ be the $\mathbb{Z}/2\mathbb{Z}$-valued intersection matrix associated to $\varGamma$, i.e., the matrix whose entries are the $\mathbb{Z}/2\mathbb{Z}$-valued intersection numbers $a_{ij} = E_i \cdot E_j$. It has been proven in \cite{Kuo} that $X$ is smoothly contractible if and only if the determinant of $A$ is $1$. Since the information about the intersection matrix is captured by $\varGamma$, we also say that $\varGamma$ is \textit{smoothly contractible} if and only if the determinant of $A$ is $1$.\\

In the unibranched case, this invariant has been used to completely classify plane curve singularities up to blow-analytic homeomorphism.

\begin{theorem}[\cite{Kuo}]
All unibranched germs of plane curves are blow-analytically equivalent to a line.
\end{theorem}

Next, assume that $(C,0)$ has more than one branches and set $C = \bigcup_{i=1}^n C_i$ its irreducible decomposition. Let $X$ be a good resolution of $\RR^2$ at the origin, i.e., an embedded resolution which is a composition of successive blow-ups and blow-downs such that the support of the total transform of $C$ is simple normal crossing. We define $\varGamma^*$ to be the extension of $\varGamma$ obtained by adding a vertex for each component of the strict transform and an edge where a non-compact component (i.e., a component of the strict transform) intersects an exceptional curve. It follows from the goodness of the resolution that $\varGamma^*$ is a tree.

Blow-analytic equivalence of curve germs determines an equivalence relation for triplets $(X,\, \cup^n_{i=1} \tilde{C_i}, \, \cup_j E_j)$ (where $\tilde{C_i}$ is the strict transform of $C_i$), which induces an equivalence relation for trees $\varGamma$ and $\varGamma^*$. In case of ambiguity, we specify which curve germ corresponds to the dual graphs by writing $\varGamma(C)$ and $\varGamma^*(C)$.\\

Let $C = \bigcup_{i=1}^n C_i$ and $C' = \bigcup_{i=1}^n C'_i$ be two blow-analytically equivalent plane curve germs, then the blow-analytic homeomorphism $h:(\RR^2, \, C, \,0) \rightarrow (\RR^2,\, C', \,0)$ induces a bijection $\bar{h}: \{1, 2, \, \ldots \,, n\} \rightarrow \{1, 2, \,\ldots \,, n\}$ such that $h(C_i) = C'_{\bar{h}(i)}$.

Let $(X, \, \cup^n_{i=1} \tilde{C_i}, \, \cup_j E_j)$ and $(X', \, \cup^n_{i=1} \tilde{C'_i}, \, \cup_{j'} E'_{j'})$ be good embedded resolutions of $(C, 0)$ and $(C', 0)$ respectively, and let $(\tilde{X}, \, \cup^n_{i=1} \tilde{\tilde{C_i}}, \, \cup_{\tilde{j}} \tilde{E}_{\tilde{j}})$ be a common good resolution which dominates $(X, \, \cup^n_{i=1} \tilde{C_i}, \, \cup_j E_j)$ and $(X', \, \cup^n_{i=1} \tilde{C'_i}, \, \cup_{j'} E'_{j'})$. Consider a path $\gamma_{ij}$ in the exceptional set of $X$ connecting the strict transforms of $C_i$ and $C_j$, with $i\neq j$. We restrict ourselves to minimal paths, i.e., those $\gamma_{ij}$ which, amongst all paths connecting the strict transforms of $C_i$ and $C_j$, go through the minimum number of exceptional curves. Each path $\gamma_{ij}$ has a lift in $\tilde{X}$, and thus an image $\gamma'_{\bar{h}(i)\bar{h}(j)}$ in $X'$.

In the dual graph, $\gamma_{ij}$ (resp. $\gamma'_{\bar{h}(i)\bar{h}(j)}$) determines a path $\gamma^*_{ij}$ (resp. $(\gamma'_{\bar{h}(i)\bar{h}(j)})^*$) in $\varGamma^*(C)$ (resp. $\varGamma^*(C')$) between the vertices corresponding to the strict transforms of $C_i$ and $C_j$ (resp. $C'_{\bar{h}(i)}$ and $C'_{\bar{h}(j)}$). Since $\varGamma^*$ is a tree, for fixed $i,j$ there is a unique path $\gamma^*_{ij}$ in the dual graph corresponding to all minimal paths $\gamma_{ij}$ in the resolution.

\begin{lemma}\label{paths}
Let $\varGamma_{ij}(C)$ (resp. $\varGamma^*_{ij}(C)$) be the graph obtained by removing all vertices in $\gamma^*_{ij}$ and the connecting edges from $\varGamma(C)$ (resp. $\varGamma^*(C)$), and let $\Delta_{ij}(C)$ be the set of connected components $G$ in $\varGamma_{ij}(C)$ such that $\mu(G)\neq 0$, where $\mu(G)$ is the corank of the $\mathbb{Z}/2\mathbb{Z}$-valued intersection matrix associated to $G$. Let $\Delta^*_{ij}(C)$ denote the natural extension of $\Delta_{ij}(C)$ in $\varGamma^*(C)$.

Then, a blow-analytic homeomorphism $h:(\RR^2,\, C, \,0) \rightarrow (\RR^2,\, C',\, 0)$ induces a bijection $\hat{h}: \Delta^*_{ij}(C) \rightarrow \Delta^*_{\bar{h}(i)\bar{h}(j)}(C')$. In particular, $\mu(\varGamma_{ij}(C)) = \mu(\varGamma_{\bar{h}(i)\bar{h}(j)}(C'))$.
\end{lemma}

\begin{proof} As before, take a good resolution $(\tilde{X}, \, \cup^n_{i=1} \tilde{\tilde{C_i}}, \, \cup_{\tilde{j}} \tilde{E}_{\tilde{j}})$ dominating $(X, \, \cup^n_{i=1} \tilde{C_i}, \, \cup_j E_j)$ and $(X', \, \cup^n_{i=1} \tilde{C'_i}, \, \cup_{j'} E'_{j'})$, and let $\beta_k$ be a step in the sequence of blow-ups from $X$ to $\tilde{X}$.

For any $k$, if the centre of $\beta_k$ is a point in $\gamma_{ij}$, then the exceptional curve $E_k$ intersects the lift of $\gamma_{ij}$, therefore it does not contribute to $\varGamma^*_{ij}(C)$.

If the centre of $\beta_k$ is not in $\gamma_{ij}$ but on a curve belonging to $\gamma^*_{ij}$, after the blow-up an isolated odd vertex is added to $\varGamma^*_{ij}(C)$, thus creating a new connected component which is smoothly contractible and does not contribute to $\Delta_{ij}(C)$.

Finally, if the centre of $\beta_k$ is not on any curve in $\gamma^*_{ij}$, the vertex corresponding to the exceptional curve $E_k$ extends one of the connected components $G$ of $\varGamma_{ij}(C)$. Let us call $G'$ the extended component. Slightly abusing the notation, let $A' =~ (E'_p \cdot E'_q )$ be the $\mathbb{Z}/2\mathbb{Z}$-valued intersection matrix associated to $G'$. By a change of basis, $A' \approx_{\mathbb{Z}}~ \left( \begin{array}{cc}
1 & 0 \\
0 & A \end{array} \right)$, where $A$ is the $\mathbb{Z}/2\mathbb{Z}$-valued intersection matrix associated to $G$. Clearly, $\mu(G') = \mu(G)$, so $\beta_k$ preserves the corank of the connected components of $\varGamma_{ij}(C)$.

Since this holds for each step $\beta_k$ in the blow-up sequence, there exists a bijection between the elements of $\Delta^*_{ij}(C)$ and the elements of its lift in $\tilde{X}$. Furthermore, since $\gamma'_{\bar{h}(i)\bar{h}(j)}$ is the image of $\gamma_{ij}$ in $X'$, we have a bijection $\hat{h}: \Delta^*_{ij}(C) \rightarrow \Delta^*_{\bar{h}(i)\bar{h}(j)}(C')$. \phantom{blah}
\end{proof}

Let $I = \{ I_k : k=1, \, \ldots \,,p\}$ denote a partition of $\{1, 2, \, \ldots \,, n\}$ (i.e., $I_1 \cup \cdots \cup I_p = \{1, \,\ldots \,, n\}$ and $I_k\cap I_l=\emptyset$ if $1\leq k < l \leq p$). By considering the union of minimal paths $\gamma_{ij}$ between two components $\tilde{C}_i$ and $\tilde{C}_j$ of the strict transform with $i, j\in I_k$ and $i \neq j$, the proof of the above lemma can be generalised to a partition on the set of strict transform components. Namely, we have that a blow-analytic homeomorphism $h:(\RR^2,\, C, \,0) \rightarrow (\RR^2,\, C', \,0)$ induces a bijection $\hat{h}: \Delta_I(C) \rightarrow \Delta_{\bar{h}(I)}(C')$ (where $\bar{h}(I) = \{\bar{h}(I_k) : k=1, \, \ldots \,, p\}$) and, in particular, the corank $\mu(\varGamma_I)$ is a blow-analytic invariant.\\

When $I = \{ \{1\}, \{2\}, \, \ldots \,, \{n\} \}$, $\mu_I$ generalises the invariant $\mu$ defined in \cite{Kob}, although the author uses a different method to prove its invariance.

When $I = \{1, 2,\, \ldots \,, n\}$, we write $\mu_I$ as $\mu'$ for convenience. As we shall see, the value of $\mu'$ bounds from below the least number of components in the exceptional divisor of a good resolution of any curve germ in that equivalence class.\\

In the case of bibranched singularities, $\mu'$ provides the following classification:

\begin{theorem}[\cite{Kob}]
Bibranched germs of plane curves have isomorphic resolution graphs if and only if they have the same $\mu'$.
\end{theorem}

Our aim is to study the classification of singularities with three branches. In the following sections, firstly we present some general results for $n$-branched germs and then focus on the tribranched case.

\section{Finiteness}

We approach the problem of the classification of embedded plane curve singularities by providing a classification of the dual graphs of their resolutions. Namely, given a smoothly contractible graph $\varGamma$, we perform blow-ups and blow-downs to simplify $\varGamma$ and reduce it to a standard form, without changing the blow-analytic equivalence class of the corresponding embedded curve germ $(C,0)$. This method allows us to make easy combinatorial computations and graphic representations.

Two blow-analytically equivalent germs have by definition a pair of isomorphic dual graphs. It should be noted that the converse is not true: in fact, we can explicitly construct examples of non-equivalent singularities with isomorphic dual graphs, as shown in the last section of this paper. However, to any dual graph correspond only a finite number of blow-analytically distinct embedded plane curve germs.\\

In what follows, we denote $Q$ an even vertex with valency $1$ in $\varGamma^*$, where the \textit{valency} of a vertex is the number of edges incident to it. We call a vertex \textit{extremal} if it has valency $1$ in $\varGamma$, and we call \textit{special vertex} a vertex with valency $3$ or more in $\varGamma^*$. We remark that a configuration is not smoothly contractible if $\varGamma$ contains two vertices of type $Q$ attached to the same vertex. In fact, if the graph contains such a part, then the determinant of the $\mathbb{Z}/{2\mathbb{Z}}$-valued intersection matrix associated to $\varGamma$ vanishes (\cite{Kuo}).\\

Let $X$ be a good resolution and $\varGamma^*$ its extended dual graph. The operations listed below are a composition of blow-ups and blow-downs of $X$, expressed for simplicity in the graph language.
\begin{itemize}
	\item[C1] (\textit{Contraction $1$}): contract an odd vertex with valency $1$ in $\varGamma^*$;
	\item[C2] (\textit{Contraction $2$}): contract an odd vertex with valency $2$ in $\varGamma^*$;
	\item[C3] (\textit{Contraction $3$}): remove two adjacent even vertices, each having valency at most $2$ in $\varGamma^*$, by first blowing up at the intersection of the two exceptional curves and then performing C2 three times (contracting the newly created exceptional curve last);
	\item[M1] (\textit{Modification $1$}): if a vertex of type $Q$ is attached to an odd vertex, change the parity of the latter as shown in \cite{Kob}; namely, perform a blow-up at the point where the even curve in $Q$ intersects the odd curve and then contract the extremal odd vertex.
\end{itemize}

Given a graph $\varGamma^*$ as above, perform contractions C1, C2 and C3 repeatedly, until no more contractions can be made. Since the size of the graph is finite and each contraction decreases the number of vertices in $\varGamma$, after a finite number of steps $\varGamma$ will be minimal under C1, C2 and C3. Next, apply M1 wherever it is possible. If $n=0$ (i.e., the embedded curve germ is an isolated point), the minimal graph under the above operations is an odd vertex with valency $0$, which we further contract, obtaining the empty graph.

The resulting surface $X$ is blow-analytically equivalent to the original one, and its dual graph is reduced to a \textit{standard form} of $\varGamma$.

\begin{proposition}
A standard form of $\varGamma$ satisfies the following properties:
\begin{itemize}
	\item[\normalfont{P1}] All non-special vertices are even;
	\item[\normalfont{P2}] All special vertices adjacent a vertex of type $Q$ are even;
	\item[\normalfont{P3}] The segment between two special vertices is at most one even vertex;
	\item[\normalfont{P4}] There are exactly $\mu'$ vertices of type $Q$.
\end{itemize}
\end{proposition}
\begin{proof}
P1 follows from the fact that any odd non-special vertex has been contracted by C1 or C2. P2 is a consequence of M1. P3 follows from P1 and by C3. To prove P4, set $I =\{1, 2, \, \ldots \, , n\}$ and consider the extremal vertices of each connected component in $\varGamma_I$. By P1, they are all even. Moreover, by C3 they can only be part of a path of length $1$ and two of them cannot be connected to the same vertex, since $\varGamma$ is smoothly contractible. Thus $\varGamma_I$ contains only even vertices, all disconnected, and exactly $\mu'$ of them, since each contributes to the corank by $1$.
\end{proof}

An arbitrary $\varGamma$ can always be reduced as shown above. Therefore, we shall restrict our attention to the easier task of classifying standard configurations.

\begin{proposition}\label{branches}
For an $n$-branched embedded plane curve germ, a standard form of $\varGamma$ has at most $\mu' + n$ extremal vertices.
\end{proposition}
\begin{proof}
Since we assume $\varGamma$ to be a standard form, its extremal vertices must be either even vertices (corresponding to vertices of type $Q$) or vertices adjacent to at least a non-compact component. There are exactly $\mu'$ vertices of the first kind and at most $n$ of the second kind, thus there are at most $\mu' + n$ extremal vertices.
\end{proof}

\textit{Remark}. The number of extremities could be strictly less than $\mu' + n$. In fact, more than one non-compact components could be adjacent to the same extremal vertex, or it could also happen that some non-compact components are attached to non-extremal vertices.\\

We can gain additional information about standard forms by looking at the weights of extremal vertices and at the vertices to which they are connected.\\

Consider the case where an extremal vertex is adjacent to a non-compact component. If the extremal vertex is odd, then it must be adjacent to at least two non-compact components (otherwise it has valency $2$ in $\varGamma^*$ and can be smoothly contracted).

Now, assume that the extremal vertex is even and adjacent to exactly one non-compact component. Then the preceding vertex $v$ must be a special vertex (if it were a even vertex with valency $2$, $\varGamma$ could be further reduced by C3 without losing normal crossingness) and, to avoid configurations which are not smoothly contractible, there cannot be a vertex of type $Q$ attached to $v$. Thus $v$ must be either adjacent to a non-compact component or have valency at least $3$ in $\varGamma$.

The above considerations prove the following lemma.
\begin{lemma}\label{extremal}
There are only four kinds of extremal vertices in a standard form:
\begin{itemize}
	\item vertices of type $Q$;
	\item vertices adjacent to at least two non-compact components;
	\item even vertices adjacent to exactly one non-compact component and preceded by another vertex adjacent to a non-compact component;
	\item even vertices adjacent to exactly one non-compact component and preceded by a vertex with valency at least $3$ in $\varGamma$.
\end{itemize}
\end{lemma}

We now prove our main result.

\begin{maintheorem}
The number of blow-analytic equivalence classes of $n$-branched germs of plane curves with $\mu' = k$ is finite for any $k$ in $\mathbb{N}$.
\end{maintheorem}
\begin{proof}
Given a germ of plane curve $(C,0)$, take a good resolution of the embedded singularity and consider its dual graph $\varGamma$. By the process described above, the tree $\varGamma$ can be reduced to its standard form, which, by Proposition \ref{branches}, has at most $k + n$ branches. Furthermore, the length of each branch is limited by the properties of standard forms. Since the number of trees of a finite size is finite, it follows that there are only a finite number of standard forms, given $n$ and $k$.

Observe that the number of smooth surfaces $X$ corresponding to a given standard form, as well as the number of choices for the positions of the $n$ non-compact components on $X$, is finite up to diffeomorphism. Thus, only finitely many blow-analytic equivalence classes of embedded plane curve singularities exist for fixed $n$ and $k$ in $\mathbb{N}$.
\end{proof}

\textit{Remark}. The theorem states the ``local'' finiteness of the blow-analytic classification, that is, for fixed values of $n$ and $\mu'$. Globally, the classification is infinite. In fact, it is easy to find standard configurations (and thus blow-analytic equivalence classes) for any number of branches and any $\mu'$ in $\mathbb{N}$.

\section{An upper bound}

While it is difficult to recover a generating formula for the exact number of blow-analytic equivalence classes given the number of branches $n$ and the value of $\mu'$, an upper bound to the number of standard forms can be estimated using combinatorial methods and some observations about the shape of $\varGamma$.

\begin{proposition}
In the tribranched case, the number of standard forms of $\varGamma$ with $\mu'=k$ is less than or equal to
\[
(k^3 -2k^2 -k +11) 2^{k-2}.
\]
\end{proposition}

\begin{proof}
Consider the topological structure of the minimal subtree connecting all non-compact components in $\varGamma$. We call this the \textit{trunk} of $\varGamma$.

In the tribranched case, there are only four possible shapes for the trunk of a standard form:

\begin{figure}[H]
\begin{subfigure}[h]{0.49\textwidth}
\centering
\includegraphics[scale=0.25]{TypeA.png}
\caption*{Type A}
\end{subfigure}
\begin{subfigure}[h]{0.49\textwidth}
\centering
\includegraphics[scale=0.25]{TypeB.png}
\caption*{Type B}
\end{subfigure}

\begin{subfigure}[h]{0.49\textwidth}
\centering
\includegraphics[scale=0.25]{TypeC.png}
\caption*{Type C}
\end{subfigure}
\begin{subfigure}[h]{0.49\textwidth}
\centering
\includegraphics[scale=0.25]{TypeD.png}
\caption*{Type D}
\end{subfigure}
\end{figure}

\noindent where $\times$ represents a non-compact component, the grey vertices can be either even or odd exceptional curves, and waved edges between two vertices represent finite chains of even curves connecting them.\\

For $\mu' = k$, standard forms of $\varGamma$ can be obtained by adding $k$ vertices of type $Q$ to the trunks above.

Observe that, in order to avoid not smoothly contractible configurations, two vertices of type $Q$ cannot be attached to the same vertex. This implies that graphs of type A exist only for $k = 0$ and $k=1$.\\

In what follows assume $k>2$ for simplicity. The formula still holds for $k=0,1,2$, as shown by the computations in the next section.\\

\textit{Type B}. By Lemma \ref{extremal}, the vertex $a$ cannot be extremal, so there must be a vertex of type $Q$ attached to it.

If another vertex of type $Q$ is attached to $b$, the remaining $k-2$ vertices of type $Q$ must be placed in the middle. The segment between each pair of special vertices (if it exists) is at most one even vertex, so $2^{k-1}$ different configurations are obtained this way.

Similarly, if $b$ is extremal, then $k-1$ vertices of type $Q$ are attached to the edge of the trunk, which gives $2^k$ configurations.

Furthermore, $b$ can be either odd or even, so there are
\[
2 (2^{k-1} + 2^k) = 3 \cdot 2^k
\]
configurations of type B.\\

\textit{Type C}. If vertices of type $Q$ are attached to both $a$ and $b$, there are
\[
\sum_{\alpha + \beta =k-2}(2^{\alpha+1}2^{\beta+1}) + \sum_{\alpha + \beta = k-3}(2^{\alpha+1}2^{\beta+1}) =
\]
\[
=(k-2)(k-1) 2^{k-1} + (k-2)(k-3) 2^{k-2} = (3k^2 -11k +10) 2^{k-2}
\]
configurations, where the two terms in the sum count separately whether there is a vertex of type $Q$ attached to $c$ or not.

On the other hand, if $b$ is an extremal vertex, then, by Lemma \ref{extremal}, the right edge is empty. Since $k>2$, there must be one vertex of type $Q$ attached to $a$ and the other $k-1$ to the left edge.
This gives
\[
2^{k-1} + 2^{k-2} = 3 \cdot 2^{k-2}
\]
new configurations.

Since $c$ can be either odd or even, the total number of configurations of type C is
\[
2 [(3k^2 -11k +10)2^{k-2} + 3 \cdot 2^{k-2}] = (3k^2 -11k +13) 2^{k-1}.
\]

\textit{Type D}. First consider the case in which $a$, $b$ and $c$ each have a vertex of type $Q$ attached to them. This gives
\[
\sum_{\alpha + \beta + \gamma = k-3}(2^{\alpha+1}2^{\beta+1}2^{\gamma+1}) + \sum_{\alpha + \beta + \gamma = k-4}(2^{\alpha+1}2^{\beta+1}2^{\gamma+1}) =
\]
\[
= \frac{1}{3}(k-3)(k-2)(k-1) 2^{k-1} + \frac{1}{3}(k-4)(k-3)(k-2) 2^{k-2}
\]
configurations.

Next, assume that $c$ is extremal. By Lemma \ref{extremal}, this means that the downward edge is empty and there cannot be vertices of type $Q$ attached to $d$, else not smoothly contractible configurations arise. So we have
\[
\sum_{\alpha + \beta = k-2}(2^{\alpha+1}2^{\beta+1}) = (k-2)(k-1)2^{k-1}
\]
configurations.

Observe that if two of the vertices in the trunk are extremal, Lemma \ref{extremal} implies that the corresponding edges are empty, which leads to configurations that are not smoothly contractible. So the previous two cases cover all possible configurations.

Since there are two colour choices for the vertex $d$, in total there are
\[
(k-2)(k^2 -3k +4)2^{k-1}
\]
configurations of type D.\\

Adding the numbers obtained for each type, we get the upper bound
\[
(k^3 -2k^2 -k +11) 2^{k-2}.
\]
\end{proof}

\textit{Remark.} The above formula is merely an upper estimate of the number of standard forms for $\mu'=k$. In fact, the number includes some not smoothly contractible configurations as well as pairs of configurations which are blow-analytically equivalent (in the pair, one configuration is a standard form, to which the other can be reduced).

\section{Explicit classification of tribranched germs}

In this section, we restrict our attention to tribranched germs of plane curves and determine explicitly a standard form for each blow-analytic equivalence class, for low values of the invariant $\mu'$.

\begin{proposition}[\cite{Kob}] A germ of a tribranched plane curve with $\mu'=~0$ is blow-analytically equivalent to one of the following:
\begin{figure}[H]
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[scale=0.25]{01.png}
\caption*{$(\{xy(x-y)=0\},0)$}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[scale=0.25]{02.png}
\caption*{$(\{xy(x-y^2)=0\},0)$}
\end{subfigure}
\end{figure}
\end{proposition}

The following results provide a classification of the dual graphs of good resolutions with $\mu'=1, 2$.\\

Up to this point, we are not able to prove in general the uniqueness of standard forms in a given blow-analytic equivalence class. In Propositions \ref{k1} and \ref{k2}, we use the invariants $\mu_I$ to show that the standard forms listed in the statements are in fact blow-analytically distinct.\\

It may happen that two blow-analytically non-equivalent germs share the same graph standard form. However, to each standard form correspond at most a finite number of blow-analytic equivalence classes of plane curve germs, so we feel that a classification of the dual graphs is still a strong one from the blow-analytic point of view.

\begin{proposition}\label{k1}
The dual graph of any good resolution of a tribranched plane curve germ with $\mu' = 1$ is blow-analytically equivalent to exactly one of the following standard forms:
\begin{figure}[H]
\centering
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[scale=0.25]{11.png}
\caption*{$A_2: (\{y(y-x^2)(y+x^2)=0\},0)$}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[scale=0.25]{12.png}
\caption*{$B_1: (\{x(y - x)(y^2 - x^3) = 0\}, 0)$}
\end{subfigure}
\end{figure}

\begin{figure}[H]
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[scale=0.25]{13.png}
\caption*{$B_4: (\{y(y - x^2)(y - x^4) = 0\}, 0)$}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[scale=0.25]{14.png}
\caption*{$C_2: (\{y(y - x^2)(y^2 - x^5) = 0\}, 0)$}
\end{subfigure}
\end{figure}
\end{proposition}

\begin{proof}
Consider a tribranched germ of plane curve $(C,0)$ and assume $\mu'=~1$. Take a good resolution of $(C,0)$, construct its dual graph $\varGamma$ and reduce it to a standard from as described in section two.

Since $(C,0)$ is tribranched, the trunk of the reduced $\varGamma$ must be of type $A$, $B$, $C$ or $D$. Furthermore, the assumption $\mu'=1$ implies that $\varGamma$ contains exactly one vertex of type $Q$.

Draw all configurations with $\mu'=1$ for each type, remembering that a segment between two special vertices is at most one even vertex and using Lemma \ref{extremal} for the extremal vertices. Then, $\varGamma$ must be blow-analytically equivalent to one of the following configurations:

\begin{figure}[H]
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[scale=0.18]{a101.png}
\caption*{$A_1$}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[scale=0.18]{a102.png}
\caption*{$A_2$}
\end{subfigure}

\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[scale=0.18]{a103.png}
\caption*{$B_1$}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[scale=0.18]{a104.png}
\caption*{$B_3$}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[scale=0.18]{a105.png}
\caption*{$B_5\; (\mu\neq 0)$}
\end{subfigure}

\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[scale=0.18]{a106.png}
\caption*{$B_2\; (\mu\neq 0)$}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[scale=0.18]{a107.png}
\caption*{$B_4$}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[scale=0.18]{a108.png}
\caption*{$B_6\; (\mu\neq 0)$}
\end{subfigure}

\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[scale=0.18]{a109.png}
\caption*{$C_1$}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[scale=0.18]{a110.png}
\caption*{$C_3\; (\mu\neq 0)$}
\end{subfigure}

\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[scale=0.18]{a111.png}
\caption*{$C_2$}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[scale=0.18]{a112.png}
\caption*{$C_4\; (\mu\neq 0)$}
\end{subfigure}
\end{figure}

Observe that all configurations of type $D$ with $\mu'=1$ are not smoothly contractible, thus cannot be the dual graph of a resolution. For the same reason, we also cross out of the list all configurations with $\mu\neq 0$.

For the remaining configurations, $A_1$, $B_3$ and $C_1$ are blow-analytically equivalent to $A_2$, $B_4$ and $C_2$ respectively. Only $4$ graphs are left and they are those of the statement.

Finally, the equation of a representative for each configuration can be found by contracting all exceptional curves (possibly performing blow-ups if no odd curves are present).

To show that the four configurations are blow-analytically distinct, label $\{1, 2, 3\}$ the vertices corresponding to the three non-compact components and consider the triplets $\{\mu(\varGamma_{12}), \mu(\varGamma_{13}), \mu(\varGamma_{23}) \}$, which are blow-analytic invariants by Lemma \ref{paths}. We have:
\begin{gather*}
A_2, B_4: \{1,1,1\} \qquad B_1:\{0,1,1\} \qquad C_2: \{0,1,2\}.
\end{gather*}
Since the coranks are not sufficient to distinguish between $A_2$ and $B_4$, we look explicitly at the sets $\Delta^*_{ij}$ for $1 \leq i < j \leq 3$:
\begin{align*}
\Delta^*_{12}(A_2) = \Delta^*_{13}(A_2) & = \Delta^*_{23}(A_2)= \lbrace \bullet, \times \rbrace ; \\
\Delta^*_{12}(B_4) = \Delta^*_{13}(B_4) & = \lbrace \bullet, \times \rbrace , \qquad \Delta^*_{23}(B_4) = \lbrace \includegraphics[scale=0.14]{a107b2.png} \rbrace.
\end{align*}

There is no bijection between $\Delta^*_{ij}(A_2)$ and $\Delta^*_{23}(B_4)$ for any choice of $ij$, so we conclude that no blow-analytic homeomorphism exists between plane curve germs having good resolutions equivalent to $A_2$ and $B_4$ respectively.
\end{proof}

\begin{proposition}\label{k2}
The dual graph of any good resolution of a tribranched plane curve germ with $\mu' = 2$ is blow-analytically equivalent to exactly one of the following standard forms:
\end{proposition}
\begin{figure}[H]
\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[scale=0.24]{21.png}
\caption*{$B_2$}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[scale=0.24]{22.png}
\caption*{$B_5$}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[scale=0.24]{23.png}
\caption*{$B_{12}$}
\end{subfigure}
\end{figure}

\begin{figure}[H]
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[scale=0.24]{24.png}
\caption*{$C_1$}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[scale=0.24]{25.png}
\caption*{$C_4$}
\end{subfigure}

\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[scale=0.24]{26.png}
\caption*{$C_8$}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[scale=0.24]{27.png}
\caption*{$D_2$}
\end{subfigure}
\end{figure}

\begin{proof}
The proof is similar to that of the previous proposition. For each type, draw all reduced configurations in which exactly two vertices of type $Q$ appear. The dual graph of any resolution of a tribranched singularity with $\mu'=2$ is blow-analytically equivalent to one of the graphs in the list below. Again, notice that there are no smoothly contractible configurations of type $A$.

\begin{figure}[H]
\centering
\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[scale=0.18]{a201.png}
\caption*{$B_1$}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[scale=0.18]{a202.png}
\caption*{$B_3 \; (\mu\neq 0)$}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[scale=0.18]{a203.png}
\caption*{$B_5$}
\end{subfigure}

\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[scale=0.18]{a204.png}
\caption*{$B_2$}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[scale=0.18]{a205.png}
\caption*{$B_4 \; (\mu\neq 0)$}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[scale=0.18]{a206.png}
\caption*{$B_6 \; (\mu\neq 0)$}
\end{subfigure}
\end{figure}

\begin{figure}[H]
\centering
\begin{subfigure}{0.325\textwidth}
\centering
\includegraphics[scale=0.18]{a207.png}
\caption*{$B_7 \; (\mu\neq 0)$}
\end{subfigure}
\begin{subfigure}{0.325\textwidth}
\centering
\includegraphics[scale=0.18]{a208.png}
\caption*{$B_9 \; (\mu\neq 0)$}
\end{subfigure}
\begin{subfigure}{0.325\textwidth}
\centering
\includegraphics[scale=0.18]{a209.png}
\caption*{$B_{11}$}
\end{subfigure}

\begin{subfigure}{0.325\textwidth}
\centering
\includegraphics[scale=0.18]{a210.png}
\caption*{$B_8 \; (\mu\neq 0)$}
\end{subfigure}
\begin{subfigure}{0.325\textwidth}
\centering
\includegraphics[scale=0.18]{a211.png}
\caption*{$B_{10} \; (\mu\neq 0)$}
\end{subfigure}
\begin{subfigure}{0.325\textwidth}
\centering
\includegraphics[scale=0.18]{a212.png}
\caption*{$B_{12}$}
\end{subfigure}
\end{figure}

\begin{figure}[H]
\begin{subfigure}{0.25\textwidth}
\centering
\includegraphics[scale=0.18]{a213.png}
\caption*{$C_1$}
\end{subfigure}
\begin{subfigure}{0.35\textwidth}
\centering
\includegraphics[scale=0.18]{a214.png}
\caption*{$C_3$}
\end{subfigure}
\begin{subfigure}{0.35\textwidth}
\centering
\includegraphics[scale=0.18]{a215.png}
\caption*{$C_5 \; (\mu\neq 0)$}
\end{subfigure}

\begin{subfigure}{0.25\textwidth}
\centering
\includegraphics[scale=0.18]{a216.png}
\caption*{$C_2 \; (\mu\neq 0)$}
\end{subfigure}
\begin{subfigure}{0.35\textwidth}
\centering
\includegraphics[scale=0.18]{a217.png}
\caption*{$C_4$}
\end{subfigure}
\begin{subfigure}{0.35\textwidth}
\centering
\includegraphics[scale=0.18]{a218.png}
\caption*{$C_6 \; (\mu\neq 0)$}
\end{subfigure}
\end{figure}

\begin{figure}[H]
\begin{subfigure}{0.25\textwidth}
\centering
\includegraphics[scale=0.18]{a219.png}
\caption*{$C_7$}
\end{subfigure}
\begin{subfigure}{0.35\textwidth}
\centering
\includegraphics[scale=0.18]{a220.png}
\caption*{$C_9 \; (\mu\neq 0)$}
\end{subfigure}
\begin{subfigure}{0.35\textwidth}
\centering
\includegraphics[scale=0.18]{a221.png}
\caption*{$C_{11} \; (\mu\neq 0)$}
\end{subfigure}

\begin{subfigure}{0.25\textwidth}
\centering
\includegraphics[scale=0.18]{a222.png}
\caption*{$C_8$}
\end{subfigure}
\begin{subfigure}{0.35\textwidth}
\centering
\includegraphics[scale=0.18]{a223.png}
\caption*{$C_{10} \; (\mu\neq 0)$}
\end{subfigure}
\begin{subfigure}{0.35\textwidth}
\centering
\includegraphics[scale=0.18]{a224.png}
\caption*{$C_{12} \; (\mu\neq 0)$}
\end{subfigure}
\end{figure}

\begin{figure}[H]
\begin{subfigure}{0.25\textwidth}
\centering
\includegraphics[scale=0.18]{a225.png}
\caption*{$D_1$}
\end{subfigure}
\begin{subfigure}{0.35\textwidth}
\centering
\includegraphics[scale=0.18]{a226.png}
\caption*{$D_3 \; (\mu\neq 0)$}
\end{subfigure}
\begin{subfigure}{0.35\textwidth}
\centering
\includegraphics[scale=0.18]{a227.png}
\caption*{$D_5 \; (\mu\neq 0)$}
\end{subfigure}

\begin{subfigure}{0.25\textwidth}
\centering
\includegraphics[scale=0.18]{a228.png}
\caption*{$D_2$}
\end{subfigure}
\begin{subfigure}{0.35\textwidth}
\centering
\includegraphics[scale=0.18]{a229.png}
\caption*{$D_4 \; (\mu\neq 0)$}
\end{subfigure}
\begin{subfigure}{0.35\textwidth}
\centering
\includegraphics[scale=0.18]{a230.png}
\caption*{$D_6 \; (\mu\neq 0)$}
\end{subfigure}
\end{figure}

Next, remove all configurations having $\mu \neq 0$, as they are not smoothly contractible.

Finally, observe that some of the remaining configurations are pairwise blow-analytically equivalent (namely, $B_1, B_{11}, C_3, C_7$ and $D_1$ are equivalent to $B_2, B_{12}, C_4, C_8$ and $D_2$ respectively).

Again, we label $\{1, 2, 3\}$ the vertices corresponding to the three non-compact components and consider the values of the invariants $\{\mu(\varGamma_{12}), \mu(\varGamma_{13}), \mu(\varGamma_{23}) \}$ to show that several of the configurations are non-equivalent. In fact, we have the following:
\begin{align*}
B_2, B_{12}, C_4 &: \{1,2,2\} \qquad B_5 :\{0,2,2\} \qquad C_1: \{1,1,2\}\\
 C_8 &: \{0,2,3\} \qquad D_2 : \{1,1,3\}. \qquad \phantom{C_1: \{1,1,2\}}
\end{align*}
To further distinguish between $B_2, B_{12}$ and $C_4$, we look explicitly at the sets $\Delta^*_{ij}$ for $1 \leq i < j \leq 3$:
\begin{align*}
& \Delta^*_{12}(B_2) = \Delta^*_{13}(B_2) = \lbrace \bullet , \bullet , \times \rbrace , \qquad \Delta^*_{23}(B_2) = \lbrace \bullet , \times \rbrace ;\\
& \Delta^*_{12}(B_{12}) = \Delta^*_{13}(B_{12}) = \lbrace \bullet , \bullet , \times \rbrace , \qquad \Delta^*_{23}(B_{12}) = \lbrace \includegraphics[scale=0.14]{a212b2.png} \rbrace; \\
& \Delta^*_{12}(C_4) = \lbrace \bullet , \times \rbrace , \; \Delta^*_{12}(C_4) = \lbrace \bullet , \includegraphics[scale=0.14]{a217b2.png} \rbrace , \; \Delta^*_{23}(C_4) = \lbrace \bullet , \bullet , \times \rbrace.
\end{align*}

Since we do not have the bijections implied by Lemma \ref{paths}, we can say that $B_2, B_{12}, C_4$ define different blow-analytic equivalence classes.
\end{proof}

\section{From graphs to germs}

Blow-ups and blow-downs are local transformations, so, in particular, they do not change the order in which the semi-branches intersect the boundary of a small of circle around the origin. We represent this piece of information in a \textit{chord diagram} by drawing vertices on $S^1$ where the semi-branches intersect such boundary, and joining two vertices if they belong to the same local analytic component.

For example, in the tribranched case, there are five possible chord diagrams:
\begin{figure}[H]
\centering
\includegraphics[scale=0.18]{abcabc.png}\hspace{5mm}\includegraphics[scale=0.18]{abcacb.png}\hspace{5mm}\includegraphics[scale=0.18]{aabbcc.png}\hspace{5mm}\includegraphics[scale=0.18]{aabccb.png}\hspace{5mm}\includegraphics[scale=0.18]{aabcbc.png}.
\end{figure}

Since chord diagrams are blow-analytic invariants, we can prove that two configurations are non-equivalent by showing that they have different chord diagrams.\\

This invariant does not add new information to the classification of standard forms in Proposition \ref{k1}. In fact, to each configuration corresponds exactly one chord diagram in the following way:
\[
A_2, B_4: \aabccb \qquad B_1, C_2: \aabcbc.
\]

As we consider configurations with a larger value of $\mu'$, however, a new phenomenon appears. For example, the standard form $B_2$ in Proposition \ref{k2}
\begin{figure}[H]
\centering
\includegraphics[scale=0.2]{img1.png}
\end{figure}
\noindent has two possible chords diagrams: \aabbcc and \aabccb. Resolutions corresponding to this dual graph are smooth surfaces diffeomorphic to a chain of four cylinders intersecting transversally, and different choices for the respective positions of the strict transform components give different chord diagrams.

This means that we can have two blow-analytically distinct embedded plane curve germs with the same dual graph, as shown below:

\begin{figure}[H]
\centering
\begin{subfigure}[h]{0.49\textwidth}
\centering
\includegraphics[scale=0.25]{diagram1.png}
\caption*{$(\{(y^3-x^4)(y^2+x^3)(y^2-x^3)=0\}, 0)$}
\end{subfigure}
\begin{subfigure}[h]{0.49\textwidth}
\centering
\includegraphics[scale=0.25]{diagram2.png}
\caption*{$(\{(y^3-x^4)(2y^2-x^3)(y^2-x^3)=0\}, 0)$}
\end{subfigure}
\end{figure}

The complete list of chord diagrams for the standard forms in Proposition \ref{k2} is as follows:
\begin{align*}
B_2, C_1, C_4, D_2: \aabbcc , \aabccb  \qquad  B_5, C_8: \aabcbc \qquad B_{12}: \aabccb.
\end{align*}

Thus, to each of the configurations $B_2, C_1, C_4, D_2$ correspond pairs of plane curve germs which are blow-analytically non-equivalent.\\

We remark that the induced equivalence of dual graphs is weaker than the blow-analytic equivalence of embedded plane curve germs. This follows from the fact that some topological information is lost in the passage from a resolution to its dual graph, namely, we lose track of the respective position of the strict transform components. One should pay attention to this kind of phenomena when passing from the equivalence of dual graphs $\varGamma$ to the blow-analytic classification of germs.

However, for each standard form, there is only a finite possibility of equivalence classes of germs. In some cases, as above, we can distinguish the classes by using chord diagrams, which are determined solely by the order of the branches near the origin.

redpajama_set_name: RedPajamaArXiv


text:
Doncaster is an unincorporated community in Charles County, Maryland, United States. Doncaster is located at the junction of Maryland routes 6 and 344,  west of La Plata.

References

Unincorporated communities in Charles County, Maryland
Unincorporated communities in Maryland

redpajama_set_name: RedPajamaWikipedia


text:
Q: What is the difference between /usr/include/linux and /usr/include/x86_64-linux-gnu/ I'm very new to native C programming and now I'm trying to configure my IDE to set up include paths correctly. Since I would like to navigate by sys/xxx.h files I added
/usr/include/x86_64-linux-gnu/

to my include path. But there is also /usr/include/linux which seems contains the same headers also. So what is the difference between them? And which one should I use actually?

A: I can tell you immediately that the directories /usr/include/linux and /usr/include/asm should never be included in a list of system header directories.  The headers in those directories are meant to be used as #include <linux/whatever.h> or #include <asm/whatever.h>, not as #include <whatever.h>.
The rest of the answer to this question depends on exactly which "distribution" of Linux you are using, so I can't just say it.  Fortunately, there is a way to get the compiler to tell you.  Run this command (exactly as shown) in a terminal window:
LC_ALL=C gcc -v -xc -E /dev/null 2>&1 |
    sed -ne '/search starts here/,/End of search list/p'

You will get output that looks something like this, but the details may be slightly different:
#include "..." search starts here:
#include <...> search starts here:
 /usr/lib/gcc/x86_64-linux-gnu/8/include
 /usr/local/include
 /usr/lib/gcc/x86_64-linux-gnu/8/include-fixed
 /usr/include/x86_64-linux-gnu
 /usr/include
End of search list.

The directories listed are the directories you should configure your IDE to look for system headers in, for purpose of looking up declarations and whatnot.  However, you should not configure your IDE to pass any of these directories to the compiler as -I directories.  It already knows to use them, it doesn't need to be told again, and telling it again can mess things up (for instance, the order of the above directories matters).
As an application programmer you don't need to worry about which headers "belong" in which directories.  That's entirely the compiler and C library developers and Linux distribution maintainers' task.

redpajama_set_name: RedPajamaStackExchange


FF5 Advance not working on Gameboid?
Discussion in 'Android Games Discussions' started by DMFA, Sep 18, 2011.
I can't get Final Fantasy 5 Advance to work on Gameboid. Once I get through the opening scene to where Bartz is about to get on his chocobo, it resets every time.
I've tried it in every language. The app runs every other game just fine. The ROM file plays just fine on my computer's GBA emulator.
it keeps restarting in random places throughout the game...what I had to is start a game on the computer using Visual Boy Advance, save it after that point, and pick it up and play from there.
I still can't save/load anything the usual way after that point, but since it's an emulator all anyone really does is save states, anyway.


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A little less than two months ago, the LebanTeks plant in Lebane started producing woolen yarn and now the first shipment has been exported.
According to the director of LebanTeks, Besim Minkara, over 23 tons of woolen yarn have been exported from Lebane to foreign markets so far, namely, to Bulgaria, Albania and Turkey.
He adds that around three tons of quality woolen yarn, varied in color, are produced each day and adds that, in addition to the said markets, they also plan to export to Russia.
– Three trucks carrying around 23 tons of woolen yarn have left for Turkey. We plan to ship two truck loads each week, and in addition to Turkey, we've also received an order from Russia. I am very pleased with the results so far and I hope for an even bigger success in the future – Minkara said.
However, although it was said on the occasion of the opening that 350 people would be employed there initially, Minkara says that around 130 workers are employed at the plant for now. He adds that another hundred will be employed in 2018, which still amounts to fewer jobs than announced.
The Lebane plant opened in late December. As said at the time, the Turkish investor invested EUR 500,000 in the facility.