A great piece from @The Chalk Face by John Kuhn who covers both the year in review AND what to do in 2014.
Will the Common Core be perceived by the masses as a sensible set of standards to guide instruction in 2014, or will progressives across America view it as another armament in the artillery trained on public education and teachers, while conservatives across the nation view it as a federal takeover of a traditional state responsibility and an attempt to brainwash the children?
1. Be active online, in the papers, and in your state capital. In the blogosphere, in the halls of your legislative bodies, in the letters-to-the-editor section, and during every single election, public education supporters can't afford to sit back.
2. Be active locally. The corporate reformers aren't merely interested in statewide and national elections. They have found more bang for their buck at the local level.
3. Embrace your expertise. I am a huge proponent of the Network for Public Education and The Educators Room because organizations like these put educators in places of engagement and efficacy. The motto of The Educators Room is "empowering teachers as the experts," and that is exactly the thing that is needed.
4. Join others. Relatedly, if you are serious about protecting the promise of public education, you have little choice but to join others in holding back the tide of corporate reform. There is diversity in the pro-public education camp. If you are progressive, there is a place for you.
5. Be great. The best defense of the public education system is a strong public education system.
More news rolls in for updating charter schools in Washington State.
First, three Charter Commissioners have expiring terms and are up for re-appointment (apparently all three want to stay).
Commissioners Chris Martin, Margit McGuire and Kevin Jacka have expiring terms. It's early to say who should stay and who should go but so far these three seem capable. Margit McGuire is the most thoughtful of the these three and definitely should stay.
The Commission welcomes public comments and asks that all comments be delivered to the Executive Director, Joshua Halsey, by January 11, 2014. Comments can be emailed or mailed to Director Halsey.
As well, public forums for each charter application are starting to be scheduled. The first one I know of is next Wednesday, January 8th in Yakima for the Sunnyside Charter Academy. According to the Daily Sun News, Charter Schools of Sunnyside is seeking as many charter supporters to come to the forum to show the Charter Commission (who will have a couple of members in attendance at each charter forum) that there is community support (or conversely, you can show up to protest).
Each forum will have a presentation - probably about 10 minutes - by the applicant with time for public comment (about 30 minutes). Public comment is limited to two minutes and the speakers will be randomly selected by lottery prior to the meeting. (I would assume that if there are just 10 speakers, all would be allowed to speak.) The charter applicants are not required to answer questions but you could certainly work those questions into any comment that you may want to make.
The Yakima forum will be Perry Technical Institute at 6:30 p.m.
The next meeting for the Charter Commission will be in Seattle on Thursday, January 30, 2014. The time is TBD but it will be held at Bethaday Community Learning Space, 605 SW 108th Street. Meeting minutes from December.
Goodbye to 2013 - Hello to 2014!
Nice idea for the new year - a Good Times Jar project. A good way to keep track of what matters to your family and showing gratitude for the good things (big and small) in your lives. Even little victories count.
Start this January with an empty jar. Throughout the year write the good things that happened to you on little pieces of paper. On December 31st, open the jar and read all the amazing things that happened to you that year.
I've been pacing myself to go through all the charter school applications. At about 566 pages each, it's a long slog.
But I can confirm that I see none of them - submitted to the Charter Commission - that are applying to be conversion schools. What is somewhat troubling is that one of them - Pioneer School which currently exists as a private school - didn't even seem to understand the question. Have they read the law? It's unclear.
I also note the presence of several ex/current Microsoft or Gates Foundation employees in a couple of applications.
I was also quite surprised to see former Washington State legislator, Dawn Mason, on the application coming from the Gulen group. I note that Gulen is in the news as their supporters in Turkey are presumed to be behind the effort to overthrow that country's secular government.
In the US, it runs one of the largest networks of charter schools – purportedly secular – with links to more than 100 schools. In Turkey, it controls a media and business empire that includes the newspaper Zaman, the country's highest-selling daily.
Gulen followers, who were often clean-shaven, Western-educated, and English-speaking, defied the stereotypes of Islamists in Turkey during the 1990s and early 2000s, which some analysts believe allowed them to enter the judiciary and police without attracting the attention of the secularist establishment.
It would seem prudent for our state to steer clear of any charter applicant with this kind of confusion around who leads their charter group and what their goals are and what their commitment is to U.S. public education.
Upon learning there was no kindergarten due to a double levy loss, Patt volunteered for the school levy campaign in 1960.
By the late 1960s Patt was a leading expert in state education funding and finance. She served on the League of Women Voters Education Committee, Seattle Schools Condition Study, Seattle Citizens for School Support, and the Greater Seattle PTA.
Elected in 1973 as Region 6 School Board Representative, Patt championed stable school funding by initiating Seattle School District v. The State of Washington, also known as the Doran decision.
In 1982, she testified on educational finance to Congress as a member of the National School Boards Association.
She co-authored the first "Student Rights and Responsibilities" handbook for Seattle schools with fellow activist Margaret Ceis after observing mistreatment of African-American students while monitoring halls.
Patt believed that Seattle schools were best served by implementing their own mandatory desegregation plan rather than a court-imposed one.
She survived a recall effort, and in 1978, the Seattle School Board was selected "Citizens of the Year" by the Municipal League.
Her commitment extended to special education (the Washington Association for Retarded Children, Northwest School), gifted education (Northwest Gifted Child Association past president), the development of alternative schools, and establishing alignment of community colleges within the K-16 system.
She said that while desegregating schools and weathering two teacher strikes as board vice-president and president were hard, closing schools was the hardest thing she did.
Remembrances may be made to the Patt Sutton Scholarship through the Alliance 4 Education 509 Olive Way, Suite 500, Seattle, WA 98101-2556.
I am humbled by her efforts.
What a remarkable, tireless and good person we have lost in our community.
A new Florida bill would ensure that the state's high school students have a little more time to catch up on their sleep.
Filed by state Rep. Matt Gaetz, a Republican, the bill would make it impossible for any Florida high schools to begin the school day before 8 a.m. starting with the 2014-2015 school year, reports the Tampa Tribune.
A recent Education Week story highlighted this issue, with a look at some research on the subject and efforts to highlight the value of later start times. Sleep deprivation is considered a widespread health problem for adolescents, it notes. While the exact benefits of adequate sleep are hard to pinpoint, a May 2012 study found that, for middle school students in Wake County, N.C., a later start time correlated with a 2 to 3 percentile point jump in standardized math and reading test scores.
According to the Education Week story, experts recommend that high-school-age young people get around nine hours of sleep—not an easy feat for those students whose classes begin at 7:30 a.m.
This debate isn't unique to Florida. In August, in fact, U.S. Secretary of Education Arne Duncan said that arguments in favor of later school start times made sense.
And earlier this year, Education Week reported that the Fairxfax County, Va. school district created an "opt-out program" for its seniors that allowed them to start their school day as late as 10:30 a.m.
Do you have a girl? Two fascinating looks at how we show girls - via advertising - what's important.
From Dove soap (seemingly a business leader in changing how women and girls perceive beauty), a one-minute zip through what a young girl may see in a lifetime of advertising.
From Upworthy, a look at how media portrayed women this past year - not good.
School violence - a high schooler's account of the experience of the Aprapahoe High School shooting ( our most recent high school shooting that left one wonderful girl dead).
Quizno's for lunch, easy classes right after; it was looking like it was going to be a great friday.
My entire life had changed in the course of 30 seconds.
Hug your child tonight and pledge to NOT have this happen anymore. It starts in Washington State.
Support Moms Demand Action and support I-594 for background checks on firearms sales and transfers. (Before anyone says something like, "It wouldn't have matter at Aprapahoe High School", that's NOT the point. The point is we need to do something to make it less possible to happen.
New York State, and in particular, NYC, is a great place to consider what has happened and what changes may be coming for ed reform. Call it the ghost of ed reform today, the future and, of course, the past.
Update: some of you may have missed the link below to a video of a kindergarten teacher teaching math via Common Core standards. I'm putting it in this thread as well.
New York State's Education department has a website to promote its work called EngageNY. @The Chalk Face education blog has a story on Common Core videos that are being used.
I personally found the one in the story - for kindergarteners - to be somewhat disturbing. There is something so mechanical and detached about the way this teacher is providing the teaching. I also learned that there's two ways to think/learn about math. The teacher does not state the name of the first way (as she starts) but she calls the second way, "the math way." Can a teacher or someone who knows about teaching math explain why the students would need to learn to think of math in two different ways?
What is interesting as well are the comments to this story. Is warmth and engagement just as important as teaching and learning? Does it matter more at one grade level than another?
First, a link to a great interview in the NY Times with Liping Ma, a former teacher and principal in China, who writes about the differences in how China and the US teach math. Important reading if you care about math instruction (thanks to reader Dan Dempsey).
The writing below is from a member of the SPS math committee who does not want to be named but wanted to give input on what is happening with the work. This person is careful to state that he/she does NOT speak for the committee but is reflecting on their work.
I appreciate the input on Common Core standards.
Publisher: "Seattle, like many other districts currently conducting curricula reviews, has rigid guidelines based on the Common Core. The boilerplate publisher applications are exacting to the letter of the CCSS, but not necessarily the spirit of the movement. Thus, our currently available series would not pass the muster despite the obvious parallels between [our framework] and the Standards for Mathematical Practice. Looking forward, we will be releasing a new CCSS-aligned edition [of our product] this coming Spring and therefore will be much better positioned to market to schools and districts looking for 'CCSS Stamped' programs."
I can give you my impression of the email based on my experience as a member of the Math Adoption committee, but recognize I am not an expert. I first confronted the common core (CC) standards in detail when I first joined the committee.
Interesting news about the necessity of dental X-rays for kids from the Los Angeles Times.
The Chicago Sun-Times is reporting that Chicago Public Schools is making computer science a core subject.
In the next three years, every high school will offer a foundational computer science course, and within five years, CPS plans to be the first urban district offering kindergarten through eighth-grade computer courses, officials said.
"The new bilingual is knowing computer code writing, and what we're setting up today, while it's a good foundation, the fact is that in the UK and in China, computer science and computer coding is now fundamental to elementary school education, and we're playing catch-up to that effort," the mayor added.
The Chicago Teachers Union said the initiative partnering with the nonprofit code.org for free computer science curriculum and teacher training is positive on its face, but pointed out most schools still struggle for basic resources.
It's nice to see that more and more people - especially educators - are rising up to fight back against ed reform that does not work.
The poorest Americans need equity, but our nation offers them accountability instead. They need bread, but we give them a stone. We address the soft bigotry of low expectations so that we may ignore the hard racism of inequity. Standardized tests are a poor substitute for justice.
So I say to Arne Duncan and President Obama, go ahead and label me. I will march headlong into the teeth of your horrific blame machine and I will teach these kids. You give me my scarlet letter and I will wear it proudly, because I will never cull the children who need education the most so that my precious scores will rise.
I will not race to the top. I will stop like the Good Samaritan and lift hurting children out of the dirt. Let me lose your race, because I'm not in this for the accolades. I'm not in it for the money. I'm in it because it's right. I am in it because the children of Perrin, Texas need somebody like me in their lives.
Congressmen, politicians, if you want children that are lush, stop firing the gardeners and start paying the water bill. Politicians, your fingerprints are on these children. What have you done to help them pass their tests?
And an interesting article from the Mother Crusader blog about the NEXT generation of wealthy philanthropists and their view on helping public education.
A group of young heirs in the Philadelphia chapter of Resource Generation has released a statement that decries any reliance on philanthropy for the funding of public schools; instead, they say, rich people should pay more in taxes.
We are a group of people in our 20s and 30s with inherited wealth and class privilege who believe that philanthropy has played a role in contributing to the crisis. Current forms of philanthropy are not leading to the transformation of public schools our city needs.
NORAD, the Northern American Aerospace Defense Command, has this Santa tracker. Santa IS on his way (apparently over Mount Everest at this time - thank goodness he dresses for the weather).
NORAD Tracks Santa is an annual Christmas-themed entertainment program, which has existed since 1955, produced under the auspices of the North American Aerospace Defense Command (NORAD). Every year on Christmas Eve, "NORAD Tracks Santa" purports to track Santa Claus as he leaves the North Pole and delivers presents to children around the world.
Also, the USDA, United States Department of Agriculture, has issued a special livestock permit to Mr. S. Clause to allow all his reindeer into U.S. territory. In the spirit of the season, they have waived the normal application fees and disease testing for his reindeer.
To all our readers, thank you for your support of public education and this blog. Merry Christmas and happy holidays to all (and to all a good night).
The state Charter School Commission received 19 applications from 18 organizations hoping to launch the type of alternative public schools that already exists in most other states.
Also Monday, teams of experts hired by the state commission will start poring over the extensive applications, which include details such as curriculum and instructional models, background information about school leaders and members of each school's governing board, projected enrollments, plans for serving students with special needs and a school discipline policy.
The commission plans to hold public forums in January, when charter applicants, parents and others can offer comments on the proposed charters. Dates and locations for those forums are to be discussed at Thursday's commission meeting.
SPS is looking to hire a new Director of Project Management (basically a tech director) with a fairly wide-ranging salary (from $94k to $142k annually).
I'm From the West; How about You?
Fun quiz that you and your family might enjoy (especially if you and your partner are from different places in the country) - from the NY Times, a dialect finder.
What I found interesting is that I hesitated in a couple of places because words/phrases I used in my childhood, I don't always use now because I live somewhere else. But soda is always soda to me.
No "This Week in Seattle Schools" today. Instead a round-up of stories - happy and sad - about parenting.
Let's go with happy first (and you can skip sad if you want to).
visit the Gingerbread Villlage at the Sheraton, 1400 Sixth Avenue. Donations accepted for Juvenile Diabetes. Free.
Seattle Center Winterfest. Lots of activities, not open on Christmas. Free.
Holiday Carousel at Westlake. Fee.
Model Train Festival in Tacoma at the Washington State History Museum. Fee.
Seattle Public Library events today and tomorrow. Libraries are closed on Christmas Eve and Christmas Day. Note: on Friday, December 27th/December 28th there is a Children's film festival of animated short films at the Central Library.
I recommend a walk around Green Lake to look for bald eagles and blue herons. The bald eagles can sometimes be found way up top the highest trees. The blue herons like to hang out at the water center on the southwest side of the lake by the docks.
I also recommend Candy Cane Lane in Ravenna for lights but please park a couple of blocks away and walk in. Much better. Also, Olympic Manor out in Ballard is a great place to see lights. There is also Steve Pool's online resource for great holiday lighting.
I called and left a message for two different people at OSPI today to get a comment about OSPI signing an agreement with the Seattle Times for student and teacher data (it was November 13, 2013 to be exact). No one called back.
I called and talked to Jim Simon, who is the managing editor at the Times. He's very busy, says that KUOW got it wrong on the personally identifiable student or staff-related data and they were preparing a statement. That was about 10 am this morning and yet, the Times remains silent, both in sending anything out via e-mail or at their newspaper.
(Update: Mr. Simon phoned me late Friday afternoon. He told me that they had talked to KUOW about the inaccuracy of their story in saying that personally identifiable information would be given to the Times via the agreement with OSPI. He said it was a "significant" update. I told him that was his view because, while the update is true, it does not mean that students could not be identified other ways.
He said that they had requested this agreement to get data to have a "richer" analysis for their stories. He said that this was not an "unusual" agreement. I told him in my research I had not found many like it and he said he didn't know what was going on in the rest of the country.
I asked about who would get the data and he said he didn't remember everyone's name on the agreement. I asked if any were researchers and he said no, they were reporters but one, Justin Mayo, is a data analyst. As per my reporting below, Mr. Mayo has requested "demographics" on students and that a rather large amount of data.
I mentioned that the Times got the grant from the Gates Foundation for the Times' Education Lab and now are leveraging it to get the agreement with OSPI. He said the Times had been upfront about the grant and that they would not be told by the Foundation what to write about.
I asked why this story - between the only daily newspaper in the state and the state education department - had not appeared in their newspaper. He said there would be a blog post. There is and boy, is it buried. There is no link to it anywhere on the front page, despite it being new. I asked if, as the editor of a newspaper, that this was not a newsworthy item. He said they chose to put it on their blog area and that's part of the newspaper.
An irony here about their blog piece is two-fold. First, the Times itself NEVER reported on this agreement and yet decides the only time they WILL is to bad-mouth KUOW. Two, is that the blog piece says that the Times asked for data from OSPI to "spot trends that might be newsworthy.
It's almost as if it didn't happen.
But hey, it's the holiday weekend and folks are busy and distracted.
I'm sure that's exactly what they are hoping will happen.
So sure, let's embrace the season (and hopefully, our loved ones) and eat and watch movies and take walks, etc.
But folks, let's vow that your students' information is not available to anyone and everyone who asked.
Let's vow to tell our Superintendent and our School Board that we want to see real teeth to these agreements. Not "may" or "can" audit safety of our service providers who get this information but "shall" and "will." And that there will be no data going out that is not thoroughly vetted as to who is getting it, why and for what use.
On this subject of parental notification, I want to tell you about the Operations Committee meeting that I attended on October 28, 2013 where data privacy was discussed. I talked about this before but I didn't tell you how disturbing the discussion got.
Apparently, I did not hear/understand Bob Westgard of SPS Logistics correctly yesterday in my reporting on coming changes in transportation.
First, the total number of students that may be impacted is 1,000 with under 300 being students at Option schools.
Second, the standardization of times are for ARRIVAL, not bell times.
4. Option Elementary / K-8 Schools – Option School Elementary and K-8 students whose transportation service address is within the boundaries of their service area or linked service area and outside of the designated walk boundaries are eligible for transportation. District arranged transportation is provided for those students attending an elementary or K-8 Option School in their service area or linked service area. ORCA cards may be provided for 6th through 8th grade students who live within the boundaries of Seattle Public School District choosing a school outside of their service area, if they live farther than 2.0 miles of the school.
a. Students who require specialized transportation services as determined by their Individualized Education Program (I.E.P.).
b. Students requiring medical transportation as approved by District Health Services.
One thing to point out is that yesterday Director Blanford asked why there are 11 different transportation situations and the answer was "well, there used to be 26." If you go to the last page of this particular document, you see many exceptions and there's your answer. The district chooses to have a more complicated system than most districts.
Here's the Board Action Report for this proposal.
Transportation Standards Options for FY15. Trying to get to just three arrival times which would affect 21 elementary schools, 1 middle school, 4 high schools and 10 K-8s.
Hey, former Governor Mike Huckabee - remember him - says to FOX News recently "Common Core is dead." Well, not so fast, Mike.
The American Girl dolls, long a fun way for little girls to immerse themselves in play associated with American history and culture, have now met Common Core. A new American Girl doll school backpack set comes complete with everything a doll will need in school, including a Common Core-aligned math textbook, published by Common Core industry giant Pearson—the same Pearson whose charitable foundation is paying a settlement of $7 million for allegations of illegally helping its corporate parent develop Common Core-aligned courses.
Great debate on Common Core between Ed Week's Anthony Cody and Maria Ferguson of the Center on Education Policy at George Washington University, and Michael Petrilli of the Thomas B. Fordham Institute.
Seattle Schools running 2 hours late (but being parents, you already know that).
I have seen no statement from the Seattle Times or OSPI on their student data sharing agreement to share Seattle students' data. Nothing like a newspaper not reporting the news. Want to let Randy Dorn at OSPI know?
1. bell times could change for about 21 elementaries, 5 regular K-8s, 5 Option K-8s, 1 middle school and 4 high schools. It's not a significant change for the overwhelming majority. but there will be a change.
2. There could be transportation changes affecting up to 1300 students in the district. The most basic change would be for Option students who would be told that transportation will no longer be given and you have a spot at your neighborhood school. If there is no spot at your neighborhood school, you can stay at your Option school with transportation. There was no discussion on parents paying to use the school bus if they want to keep that transportation.
3. Some older grandfathering is to expire but new grandfathering will kick in for some of the new changes due to growth boundaries. Neither was cited in specific.
4. There was some discussion about getting this info out to parents but yes, the staff needs this to all get approved by the end of January. This blog is your early warning system. The Board members seemed worried about parents getting this word in a timely manner.
Update: to be clear, for Option school students, your transportation would ONLY be eliminated if you live OUTSIDE THE SERVICE AREA (NOT just if you attend an Option school). My apologies for not being clear on that. If you live in your middle school service area of your linked option school, you WILL still receive transportation. My (huge) error.
Update: You'll note on the upper right of our webpage, we've included the e-mails for senior staff. Given that so many of them are new and it can be difficult to find these addresses, we included them for your convenience.
A reader asked a simple question - who are the ed reformers?
That's a more complicated question than it seems. But let's break it down by who's who nationally, at the state level and in our city.
Let me just say that there are education bloggers out there who have done the lion's share of work to compile this information so credit goes to them.
The real top gun is, of course, the Gates Foundation. Nearly every single ed reform flows from their money in one way or another. They even help fund the also-wealthy Broad Foundation (another major player). The third member of this trifecta would be Alice Walton of the Walton Foundation (but I think she gets her guidance from Gates).
Diane Ravitch, in the Huffington Post from Oct of this year, salutes writer Mercedes Schneider who went through and researched who gets Gates dollars. As Diane says, it almost becomes "who didn't get Gates money?" This link has ALL of Schneider's posts in one handy place.
In short, the organizations included in Parts One thru Six are a comprehensive listing of the $173.5 million in Gates funding designated for CCSS as of October 4, 2013.
I learned of this yesterday but was waiting. KUOW has done a good job showing the issues around student data privacy.
- To be clear, SPS did NOT know this was happening and was not asked.
- No data has been released as of today.
- OSPI said the data it planned to give the Times is not available through a public records request.
KUOW obtained a copy of the two-year agreement between the Office of Superintendent of Public Instruction and The Seattle Times, signed last month, which authorizes eight Times journalists to work with, but not publish, confidential student and staff information, including names and Social Security numbers.
"Wow," said Seattle Public Schools Superintendent Jose Banda. "I wasn't aware of [this agreement], and I don't think any of my staff was aware that this was being considered and approved."
"This is really disconcerting for us, because we've been assuring families that we are really mindful about following [data privacy] rules," Banda said.
The contract outlines measures the Times must take to secure confidential data it receives, including allowing OSPI to inspect Times facilities and requiring any confidential information to be returned or destroyed when the contract expires.
One sour note - KUOW did not explain the "grant-funded" Education Lab project at Seattle Times is thru the Gates Foundation.
The agenda for the Board's Operations Committee meeting was finally posted yesterday and now, updated today. I'm hoping to see the new Board members there as this is the last committee meeting for the year and a good way to get up-to-speed.
*Their Linked-in page says: Brainbox Consulting is a Seattle-based business intelligence consulting firm specializing in delivering data visualizations, big data analytics, dashboards, and data integration solutions.
The Times continues its march towards becoming less of a newspaper and more of an mouthpiece for ed reform.
Folks, I have personally heard from a number of Times' editors/writers who continue to claim that there is a solid wall between editorial and reporting. That may be true in some areas but that is not true for education reporting especially on ed reform.
- they lean the reporting towards ed reform, doing things like only getting a comment from an ed reform person and not anyone directed associated with the opposition. This has happened several times.
- they mislead readers (see Lynne Varner saying that the Washington State Charter Schools Association will "review" the charter applications which would lead the average reader to believe the WSCSA will be the ones picking the winners. They aren't - it's the Washington State Charter Commission (and the lone district authorizer, Spokane) who will decide but Varner doesn't mention them. The Times is content to allow readers to believe that WSCSA is somehow the main player. They aren't - they are just another Gates-funded group.
And, in today's piece by Linda Shaw, who is normally a good reporter, she does more of the same. Her piece is about deciphering what the charter law ruling may mean.
I had posted a link to a 5-part series at the New York Times that followed a young girl and her homeless family living in NYC. It is devastating reading.
Mayor Bloomberg weighed in. Now, he has the biggest city in the country to run and I absolutely don't expect everything to work well particularly for the poor if only because that's not the way of the world. But a certain level of sanitation and safety, especially in shelters with babies and children, is a must. He said nothing about the issues raised in the story about those problems.
"It's fair to say that New York City has done more than any city to help the homeless and we should be very proud of that," declared the mayor, who went on to express optimism that the city's public schools system would help Dasani break the cycle of poverty.
"This kid was dealt a bad hand. I don't know quite why. That's just the way God works. Sometimes some of us are lucky and some of us are not," he said.
Well, there's honesty for you. Except that most people reading that might think he's either shockingly blunt or related to Marie Antoinette. Of course, he probably knows Dasani and her family probably don't read the NY Times and won't hear his words.
Fitzgerald: The rich are different from you and me.
Standardized Curriculum - Reformers push the idea of standardized curricula, not only across classrooms within a school, but across schools within a district, across districts within a state, and, with the Common Core, across all states in the country. Standardized Curricula leads directly to standardized instruction - teachers essentially working off a script and pacing guide and being judged on their "fidelity of implementation". This boils down to every classroom in the country being on page 56 on the same day. This includes the ideas of vertical and horizontal articulation. Not only does this rob teachers of their autonomy and thereby de-professionalize teaching, but it is antithetical to differentiated instruction and therefore it worsens education for students. This is the first step towards delivering teaching through some process other than face-to-face contact between a teacher and a student who have a relationship.
A lot of Education Reform is focused on the idea of increasing productivity. The productivity cap is created by the requirement of a student-teacher relationship. The only way to increase productivity is to by-pass that relationship.
But that relationship is essential to learning. Teaching is a creative, improvisational act that is driven by the dynamic and unpredictable interaction between two real, live human beings. The students respond to the teacher and the teacher then responds to each student in an ongoing back-and-forth. This dialog can - and does - follow a myriad of paths that cannot be predicted or pre-programmed. It is like the universe of all possible chess games. Teachers get better over time because they have more experience and learn what moves to make in response to the students moves.
Anyway, standardized curricula de-professionalizes teaching and is a step on the path to replace teachers with machines. It worsens education but promises savings.
On the heels of that statement was this story about Rocketship, the highly touted charter school based in California from AlterNet. It's called, "Major Charter School Chain's Classrooms Look Like Cubicles for Telemarketers."
From Silicon Valley, the Rocketship chain of charter schools is hoping to expand across the country. It's backed by some of the biggest names in the tech world and claims high test scores.
I will be prime sponsor in House of AG Bob Ferguson's open government agenda bill (HB 2121).
We've spent the past year collaborating and refining the bill to require that public officials take an on-line training to learn their obligations to the public to meet in open public meetings, and ensure that the public has access to public records. Watch the AG's news conference here from TVW.
I will also be introducing a constitutional amendment to undo the recent State Supreme Court decision giving Nixon-like Executive Privilege to the Governor to hide public records and corporate lobbying (the decision was not about Gov Inslee, who pledges not to claim Executive Privilege).
The 3rd piece of my open government agenda is HB 1197 to guarantee that you have a right to comment at a city / county or other council or state agency board meeting BEFORE the council or agency takes action. Many councils and board don't take comment until AFTER they have voted, or don't allow public comment at all!
I believe transparency IS the most important issue in a democratic society and plan to support these efforts. If you agree, please let your own legislators know. Here's a link to contacting them.
A fairly definitive list of every Christmas movie and tv show made. It would be a hard choice to decide which is the "best" but I'd go with It's a Wonderful Life (for drama), A Christmas Story (comedy) and Die Hard (for action).
According to the LA Times, Governor Jerry Brown of California, pushing back against standardized testing, says he is still "haunted" by a final exam he took in high school (and one particular question).
If you have not read this 5-part series on a homeless girl in the NY Times, it's very good (and sobering) reading.
Here is a comment, pulled out as the start of a new thread.
My review of the proposals made by Education Reform Organizations shows that all of their efforts, in the end, are either about reducing the costs of education - and therefore keeping a few tax dollars in their pockets - or about directing some of those tax dollars into their friends' pockets. It's all about their money, not the students. I wish someone could prove me wrong about that.
...then get down to the John Stanford Center and review the proposed ideas for math.
Seven new Mathematics programs for Kindergarten through Grade 5 use are on display in the Professional Library on the second floor of the John Stanford Center. These are the textual materials that are being reviewed by our Adoption Committee for potential use by the district in the coming years.
You may visit the display of materials any time that the Stanford Center is open between now and January 8, 2014. There are Public Review Forms if you wish to add comments or suggestions for the committee's consideration.
Please use your voice to help us choose new Math materials for our elementary students.
Thursday, December 19, 2013, 8:30 AM to 3:30 PM in the Professional Library of the John Stanford Center for Educational Excellence.
- one, at the webpage there is a link to Common Core math. That is a huge consideration if you wonder how materials got picked for this list.
- two, yes, not great timing to say to come down during the busiest holiday season of the year.
First, the Seattle Times continues to try to use smoke and mirrors on their readers on this issue.
Their latest editorial's headline says "A charter schools victory at the state Supreme Court." The reality?
It was a decision by a King County court (although, yes, it is very likely to head to the Supreme Court).
This, like Lynne Varner's statement in her opinion piece that it is the Gates' puppet charter group that will review charter applications, is a gross error that should be correct and yet, the Times is allowing both to stand.
Very bad journalistic form and it points to their agenda in both reporting and editorializing.
Just to be completely clear, the judge specifically ruled that charter schools - when and if they are created - will not be common schools.
This ruling - which is not disputed by the state attorney general or the Seattle Times - means that charter schools cannot be funded with any of the money designated for common schools. That money includes capital funds for school construction and all of the operating funds that the state legislature has ever approved for public schools in the past.
The judge made it clear that the legislature - if they wanted to provide funding for charter schools - would have to make special provision for them. Again, that ruling is not disputed by the state attorney general, by charter school supporters in the state legislature or by the Seattle Times.
All of the supporters of I-1240 claimed that charter schools would be common schools, and the judge has ruled that they are not. That proved false one of the primary assertions of the initiative's supporters. That's worthy of mention.
Finally, the judge said that there may be other elements of the new law that are unconstitutional but that it is too soon to try those provisions because they haven't happened yet. She made it very clear that they could (and should) be tried once they have occurred.
The judge didn't invalidate the law all at once, but she made it clear that this law is very, very much at risk of being invalidated piece by piece.
Seattle Special Education PTSA Meeting, 7 pm at Rm 2700, School District main offices at 2445 3rd Ave S (3rd and Lander).
Audit&Finance Committee Meeting (Quarterly Audit Meeting) from 4-6 p.m. Agenda.
I see some troubling and yet vital info in this presentation but I think that will be a separate thread.
JAMS meeting from at the Jane Addams library at 6:30 p.m.
Operations Committee Meeting from 4-6 pm. Agenda not yet available.
Community meeting with Director Patu (yes, this is still on) from 10 am to noon at Cafe Vita.
I know there was a JAMS meeting on December 11th and someone requested a thread to talk about that one.
Also, there is another Jane Addams' Middle School "launch/intro" at the JAMS library this Wednesday Dec. 18th at 6:30pm.
Meet the wonderful principal, Paula Montgomery, and hear about the exciting work being done. Lots of Q&A, and a tour of JAMS too!
United Way of King County is recruiting people of color for the 2014 session of Project LEAD (Leadership, Effectiveness and Diversity) and we need your help to get the word out. The application deadline is Wednesday, December 18th, 2013. More information on Project LEAD, including the class schedule and application is available here.
I am an opponent of corporatist, millionaire- and billionaire-backed Education Reform, but I am no supporter of the status quo. I want to see radical change in our public education system, just not the changes that the Gates Foundation, the Broad Foundation, or the WalMart heirs support. I support changes that directly address the failures in the system and are proven effective.
So, just to make it perfectly clear, I will list a number of education reforms that I absolutely support. These reforms address the fundamental flaws in our education system. And what are those flaws?
Our schools, here in Seattle and across the country, do a great job of educating students who come to them prepared, supported, and motivated. They always have. However, our schools, here in Seattle and across the country, do a dreadful job of educating students who arrive at school without preparation, support, or motivation. They always have. The solution, to me, is obvious: We - as a society - must provide the required preparation, support, and motivation when they are missing.
Let's review the Times coverage of the Washington charter school law this week.
On the charter law court ruling, they manage to confuse readers with two different headlines but settled on one that makes it sound like it was upheld but " questions remain." Then, they only quote the pro-charter side (going to their go-to source, Lisa Macfarlane). They do quote the plaintiffs' lawyer but that's not the same as going to sources who oppose charters (like some of the actual plaintiffs).
Next comes Lynne Varner and other mistake-ridden education opinion along with her usual shot at the teachers' union.
First, she hilariously says that Judge Rietschel's opinion is "antiquated" because it comes from our "constitutional framers." (I'm guessing that she would never agree with Supreme Court justice Scalia who is a strict constitutionalist.) The problem is that we have the Constitution that exists and until that is changed, she's stuck with it.
For all of the bellyaching by charter opponents about the need for more money for education, my guess is they will find plenty of cash to continue fighting charters.
What? The WEA's money is to be used in support of their teachers. That they use the money to fight measures they believe are not good for teachers AND public education is their right. The WEA does NOT fund education so her juxtaposition of their money with state funding for education is puzzling.
Meanwhile, the Washington State Charter Schools Association continues to review 21 applications vying to become one of eight charter schools that can open in fall 2014.
Again, what? I note she doesn't say this group will decide but writing it like this would leave any reader to believe that group is the one who will make the decisions.
The Times just doesn't care if they are factual or fair.
Anyone who has given thought to Common Core standards and has a opinion you might want to pass along/talk about, please shoot me an e-mail at sss.westbrook@gmail.com. I'll forward your e-mail on. Please send it to me by noon if possible.
Haven't checked other news sources but I suspect the charter law decision is generating a lot of discussion.
Both Board Director community meetings (Carr and Martin-Morris) are cancelled for tomorrow.
Someone, quick, get me a lawyer!
Naturally, I go out of town and this ruling comes down and I am scrambling to get info and understand it.
She said that under the court case Bryan, "the legislature 'by any designation or defintion' establish a common school that does not meet the minimum constitutional criteria. Bryan has NOT been overruled.
"A charter school cannot be defined as a common school because it is not under the control of the voters of the school district.
Considering the requirements the charter schools must comply with, namely educational goals. et al, the court holds that the charter school act meets the definition of a general and uniform school system. The Plaintiffs have not made a sufficient showing for facial invalidity on this ground."
So charters are not - financially - common schools but they are legally?
What I believe I am seeing as I read the ruling is that the Court can't go on "this may happen" or "may be true" - the Court needs to see it play out in real time and THEN rule on if what happens is unconstitutional.
It is unclear to me what it all means for funding. Are charters NOT eligible for state dollars of any kind or only levy dollars? And, if levy dollars, both operations and capital or only capital?
I suspect an appeal may come.
Update: Both sides are claiming something of a victory per remarks in the Times. I honestly think it is quite up in the air how charters can be funded and the plaintiffs have said they will go to a higher court. I don't think it is a comfortable spot for any charter applicant or would-be applicant to be in.
P.S. I don't say this a lot but I did say - repeatedly - that part or all of this thing would be overturned. And it was.
In one of those likely "it's only a matter of time" stories, The Oregonian reports that Tim King, who is a charter school chain operator in Oregon was recently banned - for life -in a court settlement from ever "running, advising, or otherwise getting financially involved with any charter school in Oregon." He and a partner also must pay the state of Oregon $475k and dismantle their schools.
You can imagine the repercussions to students and parents at those schools.
The pair had been accused of engaging in "racketeering, money laundering and other fraud in operating charter schools in Oregon from 2007-2010." The Oregon DOJ had initially wanted $20M but backed off because "the state now believes most of the $17M that went to them was legitimately used to educate" Oregon students.
The Oregon Department of Education provided start-up grants of up to $450,000 per charter school. It also paid about $6,000 a year for each student enrolled, relying on the charter school operators to document the number. The state now says those records were "erroneous, false and misleading."
The Board retreat was last Saturday. I was one of two observers along with reader Cecilia.
As I previously mentioned, I felt a lot of good energy and reason for hope from this new team. There were a couple of people around the table that felt the need to point out how many new people there were but there is nothing to be done except regroup and move on. Rapid turnover is not good for any organization but I personally feel this is a good team (as does the Superintendent).
I also mentioned a couple of things that the Superintendent said that stuck out. One, that staff needs to not be overloaded because they have a lot on their plates already. He said, "our system lends itself to that..." He said "If we want to hold onto people, we need to set aside peripheral things and work on specifics."
I agree BUT the Board has to be able to ask for data and discussion so they can do their work. More on this later on in the discussion.
There was a survey done of the Cabinet and I am hoping to see this as there was some discussion about it.
They went around the table and talked about successes and challenges. There was mention again of the turnover but the Strategic Plan and its successful implementation seems high in people's minds. New director Stephan Blanford said it was a challenge for him to run as he is an introvert and you need to be an extrovert to run. Legal counsel Ron English struck a somber note saying that staff had prepared for the Board meeting on the vote for the new Growth Boundaries and "yet it was a 7 /2 meeting even with preparation."
A great story from Huffington Post on the "Buddy Bench."
Second-grader Christian Bucks, of York, Pa., knew that some of his classmates felt lonely during recess, and he decided to do something about it. His simple, utterly heartwarming solution was to install a playground "buddy bench."
As reported by the York Daily Record, a buddy bench is a designated seating area where students feeling lonely or upset can seek camaraderie. The area is customarily painted in bright, inviting colors.
Christian told the Record that he hopes the bench will help "grow our dream circle of friends."
The state Senate majority caucus congratulated itself Tuesday for surviving a year and vowed to pursue legislation that did not pass last session, including changes to K-12 education and workers compensation.
When asked about meeting a state Supreme Court mandate to increase funding for education, Tom, the Senate majority leader, said during a news conference it's all about prioritizing spending within existing resources.
"We should never have a conversation that we need new revenue for education," he said.
Clarity around the title "interim"
Seattle Public Schools bureaucrats love to play word games. Their favorite word game is to re-define words. They do it for a variety of reasons. They have re-defined "task force" and "advisory committee" to evade control by Board policy. They have re-defined "program" and "service" to expand superintendent authority. They have changed the definition of "curriculum" at least six times to confuse the Board and discredit activists. Today we learn the new, revised meaning of the word "interim".
A person appointed to a position on an interim basis has the job for a year. The interim appointee gets a mid-year review and, if they pass that review, they are appointed to the job on a continuing basis. For example, interim principals are appointed to schools without any consultation with the school community. That interim principal gets a mid-year evaluation and, with a positive evaluation, they are then appointed as the full principal of the school. Here's another example. Stephen Martin was appointed as the interim supervisor for Advanced Learning. He will have a mid-year review and, if his performance is found satisfactory, the job will be his.
By re-defining "interim" in this way the District can now hire people for jobs without anyone knowing that's what they are doing. Did your school get appointed that principal who has been bounced from building to building? Don't worry - it's just an interim appointment. Except that the interim becomes permanent if there are no disasters in the first four months on the job.
This reminds me of the way that the District uses the word "draft" - as in "This isn't the time to protest; this is just a draft version. No decisions have been made yet." Followed immediately by the adoption of the draft. This typically comes with claims that it is too late to make any changes and that the time to speak up was when the document was in its draft form. There are actually published documents that still say draft on them. It is a technique to defer - and eventually evade - debate. Likewise these rules around interim appointments are a technique to defer discussion - until never.
The public is going to have its say. The District gets to decide the timing. If the District allows the pulbic to have its say before the decision, then it is input. If the public isn't allowed to have any say until after the decision, then it is complaint. So the District gets to decide whether they get input or complaints. They choose to get complaints.
"1:58 PM: Seattle Police are searching for a man reported to have jumped out of the bushes near the east side of West Seattle High School, in the Walnut/Stevens vicinity. Police say the man was reported to have grabbed the arm of a passing female, who hit him and ran into the school. Police were not able to tell us if she is a student. She was not hurt. The man was described as white, 30s-40s, medium build, about 5-11, in a blue zipup jacket and black pants.
Today, one of our students was off campus during lunch when she was grabbed by a male stranger. She was able to break away and ran back to school, where she immediately reported it to our school administration. We called 9-1-1 and the police responded quickly.
Please be assured that we are committed to keeping our students safe while they are at school. We will continue to remind students to report these types of incidents to us immediately, and we also encourage you to discuss personal safety with your student."
This new webpage at SPS is a welcome addition. It has an FAQ page (chock-full of info), Building Rentals page and a Contacts page. The main page, called PTSA/PTO Resources, also has links for forms/documents, policies and procedures and info for various groups/individuals.
I am also glad to see PTOs given the same rank and recognition as PTA/PTSAs. It's the right thing to do.
As many of you know, the Seattle School Board on Nov. 20 approved boundary changes for the 2014-15 school year to help address our growing enrollment and to plan for three new middle schools.
We have heard from several families -- especially from parents of 7th grade Hamilton International Middle School APP students and parents of Eckstein Middle School 7th graders – asking that their students stay at their current school next year, and not move to the new Jane Addams Middle School (JAMS).
However, we will not be "grandfathering" – or guaranteeing a seat – for students who are currently in grade 6 or 7 at Hamilton and Eckstein. The new boundary changes mean these students will move to Jane Addams Middle School next fall.
I understand it is a lot to ask of our students to change schools in their last year of middle school. Please know that we do not take decisions like this lightly.
When opening a new middle school, districts have two options. One is to create new boundaries and assign students living in the neighborhood to the new school. The other option is to create a 6th grade roll up, where the middle school starts with only incoming 6th graders assigned to the new school, adding 7th grade the following year and then in the third year, grade 8. There was widespread reaction from the community against a 6th grade roll up, so the School Board approved starting JAMS as a comprehensive 6-8th grade middle school.
"I want to assure you that Jane Addams Middle School will offer an appropriate range of courses, such as advanced math and science classes, music, and world languages, which both Hamilton and Eckstein offer. Paula Montgomery, the planning principal, is working with families to ensure JAMS is off to a fantastic start and is currently hiring a team of teacher leaders committed to developing a first-class school. This is an exciting time to be a part of opening a new school. I want to thank all the families who have already begun to work together to develop a middle school, including APP, and create a foundation for the future. For more information on the great work the JAMS community is already doing, please visit http://jamsplans.blogspot.com .
Do We Care about Gun Violence in this Country?
City to Ease Pot Buffer Near Schools?
Is There Such a Thing as "Casual Sexism"?


text:
\section{Introduction}
Local Lipschtz geometry of complex algebraic sets has been intensively studied in the last years. One of the recent works on this subject, the paper \cite{BirbrairFLS:2016} (see also \cite{Sampaio:2016}),  somehow, showed a kind of rigidity of such a local geometric structure of algebraic sets. Indeed, it was proved  that Lipschitz regular complex algebraic germs of sets in $\C^n$, that is, germs of complex algebraic sets in $\C^n$ which are bi-Lipschitz homeomorphic to the germ of $\R^d$ at some point, are analytically smooth (see Theorem 3.2 in \cite{BirbrairFLS:2016} and Theorem 4.2 in \cite{Sampaio:2016}). From another way, looking to scrutinize global Lipschitz geometry of such sets in some sense, we arrived on the notion of subsets of $\C^n$ to be Lipschitz regular at infinity,  that means, subsets that outside a compact subset are bi-Lipschitz homeomorphic to the complement of a Euclidean ball in some $\R^d$.

The aim of this paper is to provide a proof that any complex algebraic set in $\mathbb{C}^n$ which is Lipschitz normally embedded at infinity and its tangent cone at infinity is a linear subspace of $\mathbb{C}^n$ must be an affine linear subspace of $\mathbb{C}^n$. In particular, complex algebraic subsets of $\C^n$ which are Lipschitz regular at infinity are affine linear subspace of $\C^n$. The main ingredients of these proofs stand on the notion of tangent cone at  infinity and the inner distance on path connected Euclidean subsets which we are going to address in the sections \ref{cones} and \ref{inner distance} respectively.

Let us observe that these results are somehow a Liouville or Bernstein type theorem as, for instance, a celebrated theorem due to Bombieri, De Giorgi and Miranda which says that entire positive minimal graph of functions in Euclidean spaces must be horizontal affine hyperplane. Finally, we get an application of the main results which is somehow  like  Theorem J in ([5], p. 180) (see also
[11] and [7]), namely, we prove that pure dimension analytic subsets of $\C^n$ which is  Lipschitz Normally Embedded and with tangent cone at infity being a linear subspace must be linear subspaces of $\C^n$ themself. Notice that, we address complex analytic sets in $\mathbb{C}^n$ which are not necessarily graph of smooth functions; a priori, they are not supposed  even smooth.

\bigskip

\section{Preliminaries}\label{preliminaries}
All the subsets of $\R^n$ (or $\C^n$) are considered equipped with the induced Euclidean metric.

\begin{definition}
Let $X\subset\R^n$ and $Y\subset\R^m$. A mapping $f\colon X\rightarrow Y$ is called {\bf Lipschitz} if there exists $\lambda >0$ such that is
$$\|f(x_1)-f(x_2)\|\le \lambda \|x_1-x_2\|$$ for all $x_1,x_2\in X$. A Lipschitz mapping $f\colon X\rightarrow Y$ is called
{\bf bi-Lipschitz} if its inverse mapping exists and is Lipschitz.
\end{definition}

\begin{definition}
Let $X\subset \R^n$ and $Y\subset\R^m$ be two subsets. We say that $X$ and $Y$ are {\bf bi-Lipschitz homeomorphic at infinity}, if there exist compact subsets $K\subset\R^n$ and $\widetilde K\subset \R^m$ and a bi-Lipschitz homeomorphism $\phi \colon X\setminus K\rightarrow Y\setminus \widetilde K$.
\end{definition}

\begin{definition}
A subset $X\subset\R^n$ is called {\bf Lipschitz regular at infinity} if $X$ and $\R^k$ are bi-Lipschitz homeomorphic at infinity, for some $k\in\N$.
\end{definition}

\begin{example}\label{real_example} Let $X\subset\R^3$ be defined by $ X = \{ (x,y,z)\in\R^3 \ : \ x^2+y^2=z^3 \}.$ We see that $X$ is an algebraic subset of $\R^3$ with an isolated singularity at $0\in\R^3.$ By using the mapping $\pi\colon X\rightarrow\R^2$; $\pi (x,y,z)=(x,y)$, it is easy to see that $X$ is Lipschitz regular at infinity.
\end{example}

\begin{example} Let $Y\subset\R^3$ be defined by $ Y = \{ (x,y,z)\in\R^3 \ : \ x^2+y^2=z \}.$ We see that $Y$ is a smooth algebraic subset of $\R^3$. From another way,  $Y$ is not Lipschitz regular at infinity. For instance, one can use Theorem \ref{tg_cones} to see that $Y$ is not Lipschitz regular at infinity.
\end{example}

At this moment we are ready to state one of the main results of the paper.

\begin{theorem}\label{main theorem}
Let $X\subset\C^n$ be a complex algebraic subset. If $X$ is Lipschitz regular at infinity, then $X$ is an affine linear subspace of $\C^n$.
\end{theorem}

We are going to prove this theorem in the Section \ref{main}. Notice that Theorem \ref{main theorem} does not hold true (with same assumptions) for real algebraic sets (cf. Example \ref{real_example} above).

\section{Inner distance}\label{inner distance}
Given a path connected subset $X\subset\R^m$ the
\emph{inner distance}  on $X$  is defined as follows: given two points $x_1,x_2\in X$, $d_X(x_1,x_2)$  is the infimum of the lengths of paths on $X$ connecting $x_1$ to $x_2$. As we said in the beginning of Section \ref{preliminaries} all the sets considered in this paper are supposed to be equipped with the Euclidean induced metric. Whenever we consider the inner distance, we emphasize it clearly.

\begin{definition}[See \cite{BirbrairM:2000}] A subset $X\subset\R^n$ is called {\bf Lipschitz normally embedded} if there exists $\lambda >0$ such that
$$d_X(x_1,x_2)\le \lambda \|x_1-x_2\|$$
for all $x_1,x_2\in X$.
\end{definition}

\begin{proposition}
If a closed unbounded subset $X\subset\R^n$ is Lipschitz regular at infinity, then there exists a compact $K\subset\R^n$ such that each connected component of $X\setminus K$ is Lipschitz normally embedded.
\end{proposition}

\begin{proof}Let $X\subset\R^n$ be  a closed and unbounded subset. Let us suppose that $X$ is Lipschitz regular
at infinity, that is, there exist compact subsets $K_1\subset\R^k$ and $K_2\subset\R^n$ and a bi-Lipschitz homeomorphism $\psi \colon \R^k\setminus K_1\rightarrow X\setminus K_2$. Without loss generality, one can suppose that $K_1$ is a  Euclidean closed ball. Let us denote $Y=\R^k\setminus K_1$, $Z=X\setminus K_2$.

First, let us  suppose that $k>1$. Since there are positive constant $\lambda_1 <\lambda_2$ such that:
$$ \lambda_1\|p-q\|\leq \|\psi(p)-\psi(q)\|\leq \lambda_2 \|p-q\|, \, \forall p,q\in Y,$$
it follows that
$$\lambda_1d_Y(p,q)\leq d_Z(\psi(p),\psi(q))\leq \lambda_2 d_Y(p,q), \, \forall p,q\in Y.$$
On the other hand,  $d_Y(p,q)\leq \pi|p-q|$, for all $p,q\in Y$ and, by those inequalities above, it follows that
$$d_Z(\psi(p),\psi(q))\leq\frac{\lambda_2\pi}{\lambda_1}\|\psi(p)-\psi(q)\|, \, \forall \ p,q\in Y.$$
Therefore,
$$d_Z(x,y)\leq\frac{\lambda_2\pi}{\lambda_1}\|x-y\|, \, \forall \ x,y\in Z.$$
In other words, $Z$ is Lipschitz normally embedded.

In the case where $k=1$, $\R\setminus K_1$ has two connected components which we denote  by $Y_1$ and $Y_2$. Actually, $Y_1$ and $Y_2$ are half-lines and, therefore, for each $i=1,2$, $d_{Y_i}(p,q)\leq |p-q|$, for all $p,q\in Y_i$. Likewise as it was done above, we have
$$d_{Z_i}(x,y)\leq\frac{\lambda_2}{\lambda_1}\|x-y\|, \, \forall \ x,y\in Z_i,$$
where $Z_1=\psi(Y_1)$ and $Z_2=\psi(Y_2)$ are the connected components of $Z$. Hence, the proposition is proved.
\end{proof}

Let us finish this section pointing out the following result which we are going to use in the proof of Theorem \ref{main theorem}

\begin{corollary}\label{neighborhood}
Let $X\subset\C^n$ be a complex algebraic subset. If $X$ is Lipschitz regular at infinity, then there exists a compact subset $K\subset\C^n$ such that $X\setminus K$ is Lipschitz normally embedded.
\end{corollary}

\section{Tangent cone at infinity}\label{cones}
Let us start this section recalling two well known results about semialgebraic sets, namely, the Monotonicity Theorem and Curve Selection Lemma.

\begin{lemma}[Theorem 1.8 in \cite{HaP:2017}]\label{monotonicity}
Let $f\colon (a,b)\to \R$ be a semialgebraic function. Then, there are $a=a_0<a_1<...<a_k=b$ such that, for each $i=0,...,k-1$, the restriction $f|_{(a_i,a_{i+1})}$ is analytic and either constant, strictly increasing or strictly decreasing.
\end{lemma}

\begin{lemma}[Theorem 2.5.5 in \cite{BochnakCR:1998}]\label{selection_curve}
Let $X$ be a semialgebraic subset of $\R^n$ and $x\in\R^n$ being a non-isolated point of $\overline{X}$. Then, there
exists a continuous semialgebraic mapping $\gamma\colon [0,1]\to \R^n$ such that $\gamma(0)=x$ and $\gamma((0,1])\subset X$.
\end{lemma}

\begin{definition}
Let $X\subset \R^m$ be an unbounded subset. We say that $E\subset \R^{m}$ is {\bf a tangent cone of $X$ at infinity} if there are a sequence $\{t_j\}_{j\in \N}$ of positive real numbers and a subset $D\subset \mathbb{S}^{m-1}$ such that $\lim\limits_{j\to \infty} t_j=+\infty $, $E=\{tv;\, v\in D\mbox{ and }t\geq 0\}$ and $D=\lim \frac{1}{t_j} X_{t_j}$, where $X_{t_j}=X\cap \{x\in \R^m; \|x\|=t_j\}$ for all $j\in \N$ and the limit is the Hausdorff's limit. When $X$ has a unique tangent cone at infinity, we denote it by $C_{\infty }(X)$ and we call $C_{\infty }(X)$ {\bf the tangent cone of $X$ at infinity}.
\end{definition}

\begin{proposition}\label{selection_lemma}
Let $Z\subset \R^n$ be an unbounded semialgebraic set and let $E$ be a tangent cone of $Z$ at infinity. A vector $v\in\R^n$ belongs to $E$ if and only if there exists a continuous semialgebraic  curve $\gamma\colon (\varepsilon ,+\infty )\to Z$  such that $\lim\limits _{t\to +\infty }|\gamma(t)|=+\infty $ and $\gamma(t)=tv+o_{\infty }(t),$ where $g(t)=o_{\infty }(t)$ means $\lim\limits _{t\to +\infty }\frac{g(t)}{t}=0$.
\end{proposition}
\begin{proof}
Let us consider the semialgebraic mapping $\phi:\R^n\setminus\{0\}\to \R^n\setminus\{0\}$ given by $\phi(x)=\frac{x}{\|x\|^2}$ and denote $X=\phi(Z\setminus \{0\})$.  Since $Z$ is an unbounded set,  $0\in \overline{X}$. Let $\rho:\mathbb{S}^{n-1}\times [0,+\infty )\to \R^n\setminus \{0\}$ be the mapping given by $\rho(x,t)=tx$. We see that $\rho|_{\mathbb{S}^{n-1}\times (0,+\infty )}:\mathbb{S}^{n-1}\times (0,+\infty )\to \R^n\setminus \{0\}$ is a semialgebraic homeomorphhism with inverse mapping $\rho^{-1}:\R^n\setminus \{0\}\to \mathbb{S}^{n-1}\times (0,+\infty )$ given by $\rho^{-1}(x)=(\frac{x}{\|x\|},\|x\|)$. Therefore, the set $Y=\rho^{-1}(X)\subset \mathbb{S}^{n-1}\times [0,+\infty )$ and $\overline{Y}$ are semialgebraic sets.

Since $E$ is a tangent cone of $Z$ at infinity, there are a sequence $\{t_j\}_{j\in \N}$ of positive real numbers and a subset $D\subset \mathbb{S}^{m-1}$ such that $\lim\limits_{j\to \infty} t_j=+\infty $, $E=\{tv;\, v\in D\mbox{ and }t\geq 0\}$ and $D=\lim \frac{1}{t_j} X_{t_j}$, where $X_{t_j}=X\cap \{x\in \R^m; \|x\|=t_j\}$ for all $j\in \N$. Let $w\in E$ and we are going to consider two cases:

\bigskip

\noindent 1) Case $w\not=0$. By definition of $E$, there exists a sequence  $\{z_k\}_{k\in\N}\subset X\setminus\{0\}$ such that $\|z_k\|=t_k$ for all $k\in \N$ and $\lim \limits_{k\to \infty }\frac{z_k}{s_k}=w$, where $s_k=\frac{t_k}{\|w\|}$ for all $k\in \N$. Thus, for each $k\in \N$, let us define $x_k=\phi(z_k)$. In this case,  $v:=\lim\limits _{k\to \infty } s_k\cdot x_k= \frac{w}{\|w\|^2}$. In particular, $\lim\limits _{k\to \infty } \frac{x_k}{\|x_k\|}=\frac{w}{\|w\|}=\frac{v}{\|v\|}$ and $u=(\frac{v}{\|v\|},0)\in \overline{Y}$. Then by Curve Selection Lemma (Lemma \ref{selection_curve}), there exists a continuous semialgebraic curve $\beta:[0,\delta)\to \overline{Y}$ such that $\beta(0)=u$ and $\beta((0,\delta ))\subset Y$. By writing $\beta(t)=(x(t),s(t))$, we get $s:[0,\delta )\to \R$ is a semialgebraic and non-constant function such that $s(0)=0$ and $s(t)>0$ if $t\in (0,\delta )$. By Lemma \ref{monotonicity}, one can suppose that $s$ is analytic in the domain $(0,\delta)$ and  strictly increasing. Hence, $s:[0,\delta/2]\to [0,\delta' ]$ is a semialgebraic homeomorphism, where $\delta' =s(\frac{\delta}{2})$. Let us define $\alpha:[0,r)\to \overline{X}$ by
$$
\alpha(t)=\rho\circ \beta\circ s^{-1}(t\|v\|)=\rho(x(s^{-1}(t\|v\|)),s(s^{-1}(t\|v\|)))=t\|v\|x(s^{-1}(t\|v\|)),
$$
where $r=\min\{\frac{\delta'}{\|v\|},\delta'\}$. Therefore,
$$
\lim\limits _{t\to 0^+}\frac{\alpha(t)}{t}=\lim\limits _{t\to 0^+}\frac{t\|v\|x(s^{-1}(t))}{t}=\lim\limits _{t\to 0^+}\|v\|x(s^{-1}(t))=\|v\|x(0)=v,
$$
and $\alpha(t)=tv+o(t)$. Finally, by defining $\gamma:(\frac{1}{r},+\infty )\to Z$ in this way $\gamma(t)=\phi^{-1}(\alpha(\frac{1}{t}))$, we get
\begin{eqnarray*}
\gamma(t) &=& \frac{\frac{1}{t}v+o(\frac{1}{t})}{\|\frac{1}{t}v+o(\frac{1}{t})\|^2}=t\textstyle{\frac{v}{\|v\|^2}}+o_{\infty }(t)\\
		  &=& tw+o_{\infty }(t).
\end{eqnarray*}
Since $\gamma$ is a composition of semialgebraic mappings,  $\gamma$ is a semialgebraic mapping as well.

\bigskip

\noindent 2) Case $w=0$. In this case,  let $\{x_k\}_{k\in\N}\subset Z$ be a sequence satisfying $\|x_k\|=t_k$ for all $k\in \N$ (this sequence exists, because $Z$ is unbounded). Thus, $\{\frac{x_k}{\|x_k\|}\}_{k\in\N}$ is a convergent sequence. Let $v\in  E$ be the limit of this sequence, i.e., $\lim\limits _{k\to \infty }\frac{x_k}{\|x_k\|}=v$. Likewise as it was done in the Case 1, one can show that there exists a continuous semialgebraic curve $\gamma\colon (\varepsilon,+\infty )\to Z$ such that $\gamma(t)=tv+o_{\infty }(t)$. Let us define $\widetilde\gamma\colon (\varepsilon^{2}, +\infty )\to Z$ by $\widetilde{\gamma}(t)=\gamma(t^{\frac{1}{2}})$. Thus, we have $\widetilde{\gamma}(t)=o_{\infty }(t)=tw+o_{\infty }(t)$.

\bigskip

We have proved that  $E\subset\{v\in\R^n;\, \exists \gamma:(\varepsilon,+\infty )\to Z$ semialgebraic s.t.  $\lim\limits _{t\to +\infty }|\gamma(t)|=+\infty$ and $\gamma(t)=tv+o_{\infty }(t)\}$. Since the other inclusion is obvious, we have finished the proof.
\end{proof}
In particular, Proposition \ref{selection_lemma} tell us that if $Z\subset \R^n$ is an unbounded semialgebraic set, then $Z$ has a unique tangent cone at infinity.

The next result is a kind of version at infinity of  Theorem 3.2 in \cite{Sampaio:2016}, where the second named author of this paper proved that  bi-Lipschitz homeomorphic subanalytic subsets have  bi-Lipschitz homeomorphic tangent cones.

\begin{theorem}\label{tg_cones}
Let $A\subset \R^m$ and $B\subset\R^n$ be unbounded semialgebraic subsets. If $A$ and $B$ are bi-Lipschitz homeomorphic at infinity, then their tangent cones at infinity $C_{\infty }(A)$ and $C_{\infty }(B)$ are bi-Lipschitz homeomorphic.
\end{theorem}

\begin{proof}
Let $\widetilde K_1\subset\R^m$ and $\widetilde K_2\subset \R^n$ be compact subsets such that there exists a bi-Lipschitz homeomorphism $\phi\colon A\setminus \widetilde K_1\to B\setminus \widetilde K_2$. Let us denote $X=A\setminus \widetilde K_1$ and $Y=B\setminus  \widetilde K_2$. By taking $\R^N=\R^m\times\R^n$ and doing the following identifications:
$$X \leftrightarrow X\times\{ 0\} \ \mbox{ and } \ Y \leftrightarrow \{ 0\}\times Y $$ one can suppose that $X,Y\subset\R^N$ and there exists a bi-Lipschitz map $\varphi\colon \R^N\to\R^N$ such that $\varphi(X)=Y$ (see Lemma 3.1 in \cite{Sampaio:2016}). Let $K>0$ be a constant such that
\begin{equation}
\frac{1}{K}\|x-y\|\leq \|\varphi(x)-\varphi(y)\|\leq K\|x-y\|, \quad \forall x,y\in \R^N.
\end{equation}

For each $k\in\N$, let us  define the mappings $\varphi_k,\psi_k:\R^N\to \R^N$ given by $\varphi_k(v)=\frac{1}{k}\varphi(kv)$ and $\psi_k(v)=\frac{1}{k}\varphi^{-1}(kv)$. For each integer $m\geq 1$, let us define $\varphi_{k,m}:=\varphi_k|_{\overline{B}_m}:\overline{B}_m\to \R^N$ and $\psi_{k,m}:=\psi_k|_{\overline{B}_{mK}} :\overline{B}_{mK}\to \R^N$, where $\overline{B}_r$ denotes the Euclidean closed ball of radius $r$ and with center at the origin in $\R^N$.
Since
$$
\frac{1}{K}\|x-y\|\leq \|\varphi_{k,1}(x)-\varphi_{k,1}(y)\|\leq K\|x-y\|, \quad \forall x,y\in \overline{B}_1,\,\, \forall k\in \N
$$
and
$$
\frac{1}{K}\|u-v\|\leq \|\psi_{k,1}(u)-\psi_{k,1}(v)\|\leq K\|u-v\|, \quad u,v\in \overline{B}_K,\,\, \forall k\in \N,
$$
there exist a subsequence $\{k_{j,1}\}_{j\in \N}\subset \N$ and Lipschitz mappings $d\varphi_1:\overline{B}_1\to \R^N$ and $d\psi_1: \overline{B}_K\to \R^N$ such that $\varphi_{k_{j,1},1}\rightrightarrows d\varphi_1$ uniformly on $\overline{B}_1$ and $ \psi_{k_{j,1},1}\rightrightarrows d\psi_1$ uniformly on $\overline{B}_K$ (notice that $\{\varphi_{k,1}\}_{k\in \N}$ and $\{\psi_{k,1}\}_{k\in \N}$ have uniform Lipschitz constants). Furthermore, it is clear that
$$
\frac{1}{K}\|u-v\|\leq \|d\varphi_1(u)-d\varphi_1(v)\|\leq K\|u-v\|, \quad \forall u,v\in \overline{B}_1
$$
and
$$
\frac{1}{K}\|z-w\|\leq \|d\psi_1(z)-d\psi_1(w)\|\leq K\|z-w\|, \quad \forall z,w\in \overline{B}_K.
$$
Likewise as above, for each $m>1$, we have
$$
\frac{1}{K}\|x-y\|\leq \|\varphi_{k,m}(x)-\varphi_{k,m}(y)\|\leq K\|x-y\|, \quad x,y\in \overline{B}_m,\,\, \forall k\in \N
$$
and
$$
\frac{1}{K}\|u-v\|\leq \|\psi_{k,m}(u)-\psi_{k,m}(v)\|\leq K\|u-v\|, \quad u,v\in \overline{B}_{mK},\,\, \forall k\in \N.
$$
Therefore, for each $m>1$, there exist a subsequence $\{k_{j,m}\}_{j\in \N}\subset\{k_{j,m-1}\}_{j\in \N}$ and Lipschitz mappings $d\varphi_m\colon \overline{B}_m\to \R^N$ and $d\psi_m\colon \overline{B}_{mK}\to \R^N$ such that $\varphi_{k_{j,m},m}\rightrightarrows d\varphi_m$ uniformly on $\overline{B}_m$ and $ \psi_{k_{j,m},m}\rightrightarrows d\psi_m$ uniformly on $\overline{B}_{mK}$ with $d\varphi_m|_{\overline{B}_{m-1}}=d\varphi_{m-1}$ and $d\psi_m|_{\overline{B}_{(m-1)K}}=d\psi_{m-1}$. Furthermore,
\begin{equation}\label{diflipum}
\frac{1}{K}\|u-v\|\leq \|d\varphi_m(u)-d\varphi_m(v)\|\leq K\|u-v\|, \quad \forall u,v\in \overline{B}_m
\end{equation}
and
\begin{equation}\label{diflipdois}
\frac{1}{K}\|z-w\|\leq \|d\psi_m(z)-d\psi_m(w)\|\leq K\|z-w\|, \quad \forall z,w\in \overline{B}_{mK}.
\end{equation}

Let us define $d\varphi,d\psi:\R^N\to \R^N$ by $d\varphi(x)=d\varphi_m(x)$, if $x\in \overline{B}_m$ and $d\psi(x)=d\psi_m(x)$, if $x\in \overline{B}_{mK}$ and, for each  $j\in \N$, let $t_j=n_j=k_{j,j}$.

\bigskip

\noindent {\it Claim 1.} $\varphi_{n_j}\rightarrow d\varphi$ and $\psi_{n_j}\rightarrow d\psi$ uniformly on compact subsets of $\R^N$.

\bigskip

Let $F\subset \R^N$ be a compact subset. Let us take  $m\in \N$ such that $F\subset \overline{B}_m\subset \overline{B}_{mK}$. Thus, $\{n_j\}_{j>m}$ is a subsequence of $\{k_{j,m}\}_{j\in \N}$ and, since  $\varphi_{k_{j,m},m}\rightrightarrows d\varphi_m$ uniformly on $\overline{B}_m$ and $ \psi_{k_{j,m},m}\rightrightarrows d\psi_m$ uniformly on $\overline{B}_{mK}$, it follows that $\varphi_{n_j}\rightrightarrows d\varphi$ and $\psi_{n_j}\rightrightarrows d\psi$ uniformly on $F$.

\bigskip

\noindent {\it Claim 2.} $d\varphi:\R^N\to \R^N$ bi-Lipschitz homeomorphism and $d\psi= (d\varphi)^{-1}$.

\bigskip

It follows from inequalities (\ref{diflipum}) and (\ref{diflipdois}) that $d\varphi,d\psi:\R^N\to \R^N$ are Lipschitz mappings. Therefore, it is enough to show that $d\psi= (d\varphi)^{-1}$. In order to do that, let $v\in \R^N$ and $w=d\varphi(v)=\lim \limits_{j\to \infty }\frac{\varphi(t_jv)}{t_j}$. Thus,
$$
\begin{array}{lllll}
\|d\psi(w)-v\|&=&\bigg\|\lim \limits_{j\to \infty }\frac{\psi(t_jw)}{t_j} - v\bigg\| &=&\lim \limits_{j\to \infty }\bigg\|\frac{\psi(t_jw)}{t_j} - \frac{t_jv}{t_j}\bigg\|\\
				    &=&\lim \limits_{j\to \infty }\frac{1}{t_j}\bigg\|\psi(t_jw) - t_jv\bigg\| &=&\lim \limits_{j\to \infty }\frac{1}{t_j}\bigg\|\psi(t_jw) - \psi(\varphi(t_jv))\bigg\|\\
				    &\leq &\lim \limits_{j\to \infty }\frac{K}{t_j}\bigg\|t_jw - \varphi(t_jv)\bigg\| &\leq &\lim \limits_{j\to \infty }K\bigg\|w - \frac{\varphi(t_jv)}{t_j}\bigg\|\\
					&=&0.& &
\end{array}
$$

Then, $d\psi(w)=d\psi(d\varphi(v))=v$, for all $v\in \R^N$, i.e., $d\psi\circ d\varphi={\rm id}_{\R^N}$. Analogously, one can show that $d\varphi\circ d\psi={\rm id}_{\R^N}$.

\bigskip

\noindent {\it Claim 3.} $d\varphi(C_{\infty }(X))= C_{\infty }(Y)$.

\bigskip

By  Claim 2, it is enough to verify that $d\varphi(C_{\infty }(X))\subset C_{\infty }(Y)$. In order to do that, let  $v\in C_{\infty }(X)$. Then, there is $\alpha\colon (\varepsilon, \infty)\to X$ such that $\alpha(t)=tv+o_{\infty }(t)$. Thus, $\varphi(\alpha(t))=\varphi(tv)+o_{\infty }(t)$, since $\varphi$ is a Lipschitz mapping. However, $\varphi(t_jv)=t_jd\varphi(v)+o_{\infty }(t_j)$ and then
$$
d\varphi (v)=\lim \limits_{j\to \infty }\varphi_{n_j}(v)=\lim \limits_{j\to \infty }\frac{\varphi(t_jv)}{t_j}=\lim \limits_{j\to \infty }\frac{\varphi(\alpha(t_j))}{t_j}\in C_{\infty }(Y).
$$

Therefore, $d\varphi:C_{\infty }(X)\to C_{\infty }(Y)$ is a bi-Lipschitz homeomorphism. We finish the proof by remarking that $C_{\infty }(A)=C_{\infty }(X)$ and $C_{\infty }(B)=C_{\infty }(Y)$.

\end{proof}

Let us finish this section pointing out the following result which we are going to use in the proof of Theorem \ref{main theorem}

\begin{corollary}
Let $X\subset\C^n$ be a complex algebraic subset. If $X$ is Lipschitz regular
at infinity, then the tangent cone of $X$ at infinity is Lipschitz regular at infinity.
\end{corollary}

\section{Proof of Theorem \ref{main theorem}}\label{main}

Let us begin this section by recalling some basic facts about degree of complex algebraic sets. More precisely, we are going to see that affine linear subspaces of $\C^n$ are characterized among algebraic subsets of $\C^n$ by being those  of degree 1. In such a direction, let $\iota\colon\C^n\hookrightarrow \mathbb{P}^n$ be the  embedding given by $\iota(x_1,...,x_n)=[1:x_1:...:x_n]$ and let $p\colon \C^{n+1}\setminus\{0\}\to \mathbb{P}^n$ be the projection mapping given by $p(x_0,x_1,\dots,x_n)=[x_0:x_1:...:x_n]$.

\begin{remark}
 {\rm Let $A$ be an algebraic set in $\mathbb{P}^n$ and $X$ be an algebraic set in $\C^n$. Then $\widetilde{A}=p^{-1}(A)\cup \{0\}$ is a homogeneous complex algebraic set in $\C^{n+1}$ and the closure $\overline{\iota(X)}$ of $\iota(X)$ in $\mathbb{P}^n$ is an algebraic set in $\mathbb{P}^n$.}
\end{remark}
\begin{definition}
 Let $A$ be an algebraic set in $\mathbb{P}^n$. We define {\bf the degree of} $A$ by ${\rm deg}(A)=m(\widetilde{A},0)$, where $m(\widetilde{A},0)$ is the multiplicity of $\widetilde{A}$ at $0\in\C^{n+1}$.
\end{definition}
\begin{definition}
 Let $X$ be a complex algebraic set in $\C^n$. We define {\bf the degree of} $X$ by $ {\rm deg}(X) ={\rm deg}(\overline{\iota(X)})$.
\end{definition}

\begin{proposition}\label{linear_prop}
Let $X\subset \mathbb{\C}^n$ be an algebraic subset. Then,  ${\rm deg}(X)=1$ if and only if $X$ is an affine linear subspace of $\mathbb{C}^n$.
\end{proposition}

\begin{proof}
 Let $A=\overline{\iota(X)}$ be the closure of $\iota(X)$ in $\mathbb{P}^n$. By definition, ${\rm deg}(X)={\rm deg}(A)$ and ${\rm deg}(A)=m(\widetilde{A},0)$, where $\widetilde{A}=p^{-1}(A)\cup \{0\}$. Thus, ${\rm deg}(X)=1$ if and only if $m(\widetilde{A},0)=1$. However, as $\widetilde{A}$ is a homogeneous complex algebraic set in $\C^{n+1}$, $m(\widetilde{A},0)=1$ if and only if $\widetilde{A}$ is a complex linear subspace. In order to finish the proof, we remark that $\widetilde{A}$ is a complex linear subspace in $\C^{n+1}$ if and only if $A$ is a complex projective plane in $\mathbb{P}^n$.
\end{proof}

From now on, we are going to prove of the main results of the paper.

\begin{theorem}\label{proposition A}
Let $X\subset\C^n$ be a pure $d$-dimensional algebraic subset such that $C_{\infty}(X)$ is a complex linear subspace of $\C^n$. If there exists a compact subset $K\subset\C^n$ such  that $X\setminus K$ is Lipschitz normally embedded, then $X$ is an affine linear subspace of $\C^n$.
\end{theorem}
\begin{proof} Since $C_{\infty }(X)$ is a linear subspace of $\C^n$, one can consider the orthogonal projection $\pi\colon \C^n\rightarrow C_{\infty }(X).$ Let us choose linear coordinates $(x,y)$ in $\C^n$ such that $$C_{\infty }(X)=\{(x,y)\in\C^n;\,y=0\}.$$ Notice that the restriction of the orthogonal projection $\pi$ to $X$ has the following properties:
\begin{enumerate}
\item [1)] There exist a compact subset $K\subset \C^n$ and a constant $C>0$ such that $\|y\|\leq C\|x\|$ for all $(x,y)\in X\setminus K$.
\item [2)] If $\gamma :(\varepsilon,\infty )\rightarrow X$ is an arc such that $\lim\limits_{t\to +\infty}\|\gamma(t)\|=+\infty $ and $\pi\circ\gamma(t)=tv+o_{\infty}(t)$, then $\gamma(t)=tv+o_{\infty}(t)$.
\end{enumerate}

Indeed, if 1) is not true, there exists a sequence $\{(x_k,y_k)\}\subset X$ such that $\lim\limits_{k\to+\infty}\|(x_k,y_k)\|=+\infty$ and $\|y_k\|> k\|x_k\|$. Thus, up to a subsequence,  one can suppose that $\lim\limits_{k\to+\infty}\frac{y_k}{\|y_k\|}=y_0$. Since $\frac{\|x_k\|}{\|y_k\|}< \frac{1}{k}$,  $(0,y_0)\in C_{\infty }(X)$, which is a contradiction, because $y_0\not=0$. Therefore, 1) is true.

Now, in order to prove 2), let us write $\gamma(t)=(x(t),y(t))$. By 1), there exists $t_0>0$ such that  $\|y(t)\|\leq C\|x(t)\|$ for all $t\geq t_0$, since $\lim\limits_{t\to +\infty}\|\gamma(t)\|=+\infty$.  Thus, since $\frac{x(t)}{t}$ is bounded, $\frac{y(t)}{t}$ is bounded. Let us suppose that $y(t)\not=o_{\infty}(t)$. Then, there exist a sequence $\{t_k\}_{k\in \N}$ and $r>0$ such that $t_k\to +\infty$ and $\frac{\|y(t_k)\|}{t_k}\geq r$ for all $k$. Since $\left\{\frac{y(t_k)}{t_k}\right\}_{k\in \N}$ is bounded, up to a subsequence, one can suppose  that $\lim\limits_{k\to +\infty}\frac{y(t_k)}{t_k}=y_0$. Therefore, $\lim\limits_{k\to +\infty}\frac{\gamma(t_k)}{t_k}=(v',y_0)\in C_{\infty }(X)$, where $v=(v',0)$. However, this is a contradiction, since $\|y_0\|\geq r>0$ and this implies that $y_0\not=0$. Then, $y(t)=o_{\infty}(t)$ and, therefore, $\gamma(t)=tv+o_{\infty}(t)$.

Let $L=\pi^{-1}(0)$. One can see that $\overline{\iota(X)}\cap\overline{\iota(L)}\cap (\mathbb{P}^n\setminus \iota( \C^n))=\emptyset $. Therefore, $\pi|_{ X}\colon X\rightarrow C_{\infty }(X)$ is a ramified cover with degree equal to ${\rm deg} (X)$ (see \cite{Chirka:1989}, Corollary 1 in the page 126). Moreover, the ramification locus of $\pi|_X$ is a codimension $\geq 1$ complex algebraic subset $\Sigma$ of the linear space $ C_{\infty }(X)$.

Let us suppose that the degree ${\rm deg} (X)$ is greater than 1. Since $\Sigma$ is a codimension $\geq 1$ complex algebraic subset of the space $C_{\infty }(X)$, there exists a unit tangent vector $v_0\in C_{\infty }(X)\setminus C_{\infty }(\Sigma)$,
where $C_{\infty }(\Sigma) $ is the tangent cone at infinity of $\Sigma$.

Since $v_0$ is not tangent to $\Sigma$ at infinity, there exist positive real numbers $\lambda$ and $R$ such that
$$C_{\lambda,R}=\{v\in C_{\infty }(X); \ \|v-tv_0\|< \lambda t, \ \forall \ t>R \}$$
does not intersect the set $\Sigma$. Since we have assumed that the degree ${\rm deg} (X)\geq 2$, we have at least two different liftings $\gamma_1(t)$ and $\gamma_2(t)$ of the half-line $r(t)=tv_0$,  i.e. $\pi(\gamma_1(t))=\pi(\gamma_2(t))=tv_0$.  Since $\pi$ is the orthogonal projection on $C_{\infty }(X)$ and the vector $v_0$ is the unit tangent vector at infinity to the images
$\pi\circ \gamma_1$ and $\pi\circ \gamma_2$, then $v_0$ is the tangent vector at infinity to the arcs $\gamma_1$ and $\gamma_2$. By construction, we have ${\rm dist}(\gamma_i(t),{\pi|_ X}^{-1}(\Sigma))\geq \lambda t$ for $i=1,2$, where by dist we mean the Euclidean distance.

On the other hand, any path in $X$ connecting $\gamma_1(t)$ to $\gamma_2(t)$ is the lifting of a loop, based at the point $tv_0$, which is not contractible in $C_{\infty }(X)\setminus {\Sigma}$. Thus, the length of such a path must be at least  $2\lambda t$. It implies that the inner distance, $\rm{d_{X}}(\gamma_1(t),\gamma_2(t))$, in $X$, between $\gamma_1(t)$ and $\gamma_2(t)$, is at least $2\lambda t$. But, since $\gamma_1(t)$ and $\gamma_2(t)$ are tangent at infinity, that is,
$$\frac{\|\gamma_1(t)-\gamma_2(t)\|}{t}\to 0 \ \mbox{ as } \ t\to +\infty ,$$
and $\lambda>0$, we obtain that $X$ is not Lipschitz normally embedded at infinity. Otherwise there will be $\widetilde C >0$ and a compact subset $K\subset\C^n$ such that:
$$d_X(x_1,x_2)\leq \widetilde  C \|x_1-x_2\| \quad \mbox{for all} \quad  x_1,x_2\in X\setminus K,$$
hence:
\begin{eqnarray*}
2\lambda &\leq & \frac{\rm{d_{X}}(\gamma_1(t),\gamma_2(t))}{t} \\
&\leq &\widetilde C \frac{\|\gamma_1(t)-\gamma_2(t)\|}{t}\to 0 \ \ \mbox{as} \ t\to +\infty,
\end{eqnarray*}
which is a contradiction. We have concluded that ${\rm deg}(X)=1$ and, by Proposition \ref{linear_prop}, it follows that $X$ is an affine linear subspace.

\end{proof}

Notice that the assumptions in  Theorem \ref{proposition A} are sharp in the sense that, in order to get the same conclusion, none of those assumptions can be removed, as we can see bellow.
\begin{example}
The plane complex curve $X=\{(x,y)\in \C^2;\, y=x^2\}$ has linear tangent cone at infinity; $X$ is not an affine linear subset of $\mathbb{C}^2$. As another example, $Z=\{(x,y,z)\in \C^3;\, x^2+y^2+z^2=0\}$. We see that $Z$ is Lipschitz normally embedded; $Z$ is not an affine linear subset of $\C^3$.
\end{example}

As an application of Theorem \ref{proposition A}, we obtain a result like Theorem J in (\cite{GreeneW:1979}, p. 180) (see also \cite{SiuY:1977} and \cite{Itoh:1979}).

\begin{corollary}\label{analytic_version}
Let $X\subset\C^n$ be a pure $d$-dimensional analytic subset. Suppose $X$ has a unique tangent cone at infinity and it is a $d$-dimensional complex linear subspace of $\C^n$. If $X$ is closed and Lipschitz Normally Embedded at infinity, then $X$ is an affine linear subspace of $\C^n$.
\end{corollary}
\begin{proof} Let us choose linear coordinates $(x,y)$ in $\C^n$ such that $C_{\infty }(X)=\{(x,y)\in\C^n;\,y=0\}$ and consider the orthogonal projection $\pi\colon \C^n\rightarrow C_{\infty }(X).$

\noindent {\bf Claim.} There exist positive constants $C$ and $\rho$ such that $X\subset \{(x,y);\|y\|< C\|x\|\}\cup B_{\rho}$.

Indeed, since $C_{\infty }(X)=\{(x,y)\in\C^n;\,y=0\}$, by the proof of Theorem \ref{proposition A}, there exist a positive constant $C$ and a compact $K\subset \C^n$ such that $\|y\|< C\|x\|$ for all $(x,y)\in X\setminus K$. Let $\rho$ be a positive number such that $K\subset B_{\rho}$. Thus, we have that $X\subset \{(x,y);\|y\|< C\|x\|\}\cup B_{\rho}$.

Now, by Theorem 2 in (\cite{Chirka:1989}, page 77), $X$ is an algebraic set and the proof follows from Theorem \ref{proposition A}.
\end{proof}

Let us remark that the real version of Corollary \ref{analytic_version} does not hold true in general, as it is shown bellow.

\begin{example}
The set $X=\{(x,y)\in\R^2;\,y=\sin{x}\}$ is Lipschitz Normally embedded, since it is bi-Lipschitz homeomorphic to $\R$ and, moreover, $X$ has a unique tangent cone at infinity with $C_{\infty }X=\{(x,y)\in\R^2;\,y=0\}$. However, $X$ is not an algebraic set and, in particular, it is not a linear subspace of $\R^2$.
\end{example}

Let $X\subset\C^n$ be a complex algebraic subset. Let $\mathcal{I}(X)$ be the ideal of $\C[x_1,...,x_n]$ given by the polynomials which vanishes on $X$. For each $f\in\C[x_1,...,x_n]$, let us denote by $f^*$ the homogeneous polynomial composed of the monomials in $f$ of maximum degree.

\begin{proposition}[Theorem 1.1 in \cite{LeP:2016}]
Let $X\subset\C^n$ be a complex algebraic subset. Then, $C_{\infty }(X)$ is the affine variety $V(\langle f^*;\, f\in \mathcal{I}(X)\rangle) $.
\end{proposition}

It follows from above proposition that the tangent cone at infinity of a complex algebraic subset of $\C^n$ is a homogeneous complex algebraic subset and, therefore, it is a complex cone in the following sense: an algebraic subset of $\C^n$ is called a \emph{complex cone} if it is a union of one-dimensional complex linear subspaces of $\C^n$. The next result was proved by David Prill in \cite{Prill:1967}:

\begin{lemma}[Theorem in \cite{Prill:1967}] \label{prill} Let $V\subset \C^n$ be a complex cone. If $0\in V$ has a neighborhood homeomorphic to a Euclidean ball, then $V$ is a linear subspace of $\C^n$
\end{lemma}

At this moment, we are ready to give a proof of Theorem \ref{main theorem}
\begin{proof} [Proof of Theorem \ref{main theorem}]

Let us suppose that the complex algebraic subset $X\subset\C^n$ is Lipschitz regular at infinity. Thus, let $h\colon U\rightarrow \R^N\setminus B$ be a bi-Lipschitz homeomorphism, where is an open neighborhood $U$ of the infinity in $X$ (i.e. a complement of a compact subset in X) and $B\subset\R^N$ is a closed Euclidean ball centered at the origin $0\in\R^N$. Let $C_{\infty }(X)$ be the tangent cone at infinity of $X$.
It comes from Theorem \ref{tg_cones} that there exists a bi-Lipschitz homeomorphism $dh\colon C_{\infty }(X) \rightarrow C_{\infty }(\R^N\setminus B)=\R^N$. In particular, $C_{\infty }(X)$ is a topological manifold. By Prill's Theorem (Lemma \ref{prill}), it follows that $C_{\infty }(X)$ is a complex linear subspace of $\C^n$. By Corollary \ref{neighborhood}, there exists a compact subset $K$ such that $X\setminus K$ is Lipschitz normally embedded and, by Theorem \ref{proposition A}, it follows that $X$ is an affine linear subspace of $\C^n$.
\end{proof}

redpajama_set_name: RedPajamaArXiv


Take a look at the most powerful and responsive Speed Triple ever - the all new Speed Triple RS. With a major engine development, lighter weight and cutting edge technological advancements the iconic 1050cc Triple engine revs harder for longer, improving on its already legendary agility, razor-sharp handling, and pinpoint precision.
Raising the game to a new high with a totally addictive ride, delivering all of the Speed Triple's characteristic nose-down focused attitude together with superior handling, a higher standard of finishing, and a phenomenal enhanced triple soundtrack. This is the greatest Speed Triple ever.
Full-colour 5-inch TFT instrument display that presents both essential and additional rider information from the on-board computer, with six screen styles to choose from, all with high and low contrast settings, adjustable for optimum visibility.
Ergonomically-designed switch cubes with stylish backlighting for easy fingertip use under any lighting condition to access functions such as TFT instruments, riding modes, and the new cruise control, significantly improving long distance rideability.
Easily accessed through the left hand switch cube, choose from five riding modes, Rain, Road, Sport, Rider-configurable and Track mode. Each altering the throttle map, ABS and traction control to suit the conditions or desired riding style.
Optimised cornering ABS and traction control delivers optimum braking and torque control, improving both straight line and cornering performance. This is supported by a new inertial measurement unit (IMU) which continually calculates lean angle.
The Speed Triple's new advanced keyless ignition and steering lock system recognises the proximity of your fob before activation is initiated. For additional security, you can turn off the transmitting signal to disable the keyless function when not in use.
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The RS comes with high specification 43mm Öhlins NIX30 upside down forks and TTX36 twin tube rear monoshock RSU, fully adjustable for pre-load, compression and rebound damping. There's extra comfort too, with a new rider seat fitted with 3D net inserts.
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EURO 4 Standard: 119 g/km **CO2 and fuel consumption are measured according to regulation 168/2013/EC. Figures on fuel consumption are derived from specific test conditions and are for comparative purposes only. They may not reflect real driving results.