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Under either OSX 10.8.3 or 10.7.5, Java 7-17 runs under Firefox 19.02.
Under either OSX 10.8.3 or 10.7.5, Java 7-17 does not run under Safari 6.0.3.
Under either OSX 10.8.3 or 10.7.5, Java 7-17 does not run under Chrome 25.x (Chrome is 32 bit, Java is 64 bit only).
All replies
Paul_31 Mar 19, 2013 2:34 PM
in response to bodanmic
Have you updated to the latest version of Java? There seem to have been quite a few updates recently because of the security issues - there was one released last week, I believe.
Klaus1 Mar 19, 2013 3:50 PM
in response to bodanmic
Apple barred Java from running on Macs in order to safeguard users by blocking Java 7 Update 11 and adding it to the banned list in XProtect.
This was the second time in two weeks that Apple had blocked Oracle's code from running on Macs. The threat was so serious that the U.S. Department of Homeland Security had recommended that all Java 7 users disable or uninstall the software until a patch was issued. This time Java is blocked through Apple's XProtect anti-malware feature.
Java has come under fire as the means by which hackers have been able to gain control of computers. In April 2012 more than 600,000 Macs were reported to have been infected with a Flashback Trojan horse that was being installed on people's computers with the help of Java exploits. Then in August Macs were again at risk due to a flaw in Java, this time around, there was good news for Mac users: Thanks to changes Apple has made, most of us were safe from the threat.
Unwilling to leave its customers open to potential threats Apple decided it's safer to block Java entirely.
In order to block older versions of Flash, Apple has updated its "Xprotect.plist" file so that any versions that come before the current one (version 11.6.602.171) cannot be used on a Mac. Users who have older versions of Flash installed will be greeted with an alert that says "Blocked plug-in," and Safari will prompt the user to update to a newer version.
Macs running OS X Snow Leopard and beyond are affected.
UPDATE for those running Lion or Mountain Lion:
Oracle on Friday February 1 released a new version reportedly addressing vulnerabilities seen with the last build.
Apple disabled Java 7 through the OS X XProtect anti-malware system, requiring users to have at least version "1.7.0_10-b19" installed on their Macs. The release dated February 1 carries the designation "1.7.0_13-b20," meeting Apple's requirements.
Oracle "strongly recommends" applying the CPU fixes as soon as possible, saying that the latest Critical Patch Update contains 50 new security fixes across all Jave SE products.
Update for Snow Leopard users:
Apple issued update 12 for Java for OS 10.6:
Note: On systems that have not already installed Java for Mac OS X 10.6 update 9 or later, this update will configure web browsers to not automatically run Java applets. Java applets may be re-enabled by clicking the region labeled "Inactive plug-in" on a web page. If no applets have been run for an extended period of time, the Java web plug-in will deactivate.
If, after installing Java for OS X 2013-002 and the latest version of Java 7 from Oracle, you want to disable Java 7 and re-enable the Apple-provided Java SE 6 web plug-in and Web Start functionality, follow these steps:
Further update:
Apple issued this Java related security update No. 13 on February 19:
and Update No. 14 on March 4: http://support.apple.com/kb/DL1573
You should also read this:
The standard recommendation is for users to turn off Java except when they have to use it on known and trusted websites (like their bank). Javascript, which is unrelated despite the name, can be left on.
Further useful comment in this article:
Dave Devereaux-Weber Mar 21, 2013 10:12 AM
in response to Klaus1 Helpful
Under either OSX 10.8.3 or 10.7.5, Java 7-17 runs under Firefox 19.02.
Under either OSX 10.8.3 or 10.7.5, Java 7-17 does not run under Safari 6.0.3.
Under either OSX 10.8.3 or 10.7.5, Java 7-17 does not run under Chrome 25.x (Chrome is 32 bit, Java is 64 bit only).
Rogerio de Camargo Rocha Mar 21, 2013 10:49 AM
in response to bodanmic
I was experiencing the same thing in Safari 6.0.3. It worked fine under Firefox.
By some odd reason, the update disabled the Java plug-in under Safari and the inline link to activate it was never being displayed.
1. I had to uninstall the plug-in. How do I uninstall Java 7 for my Mac?
2. Reset all website notification warnings. Safari 6 (OS X Mountain Lion): Reset Safari
3. Reinstall it. java.com: Java + You
4. Visit Oracle's Java test site. How do I test whether Java is working on my computer?
Only then the "Activate Plug-in" inline link finally appeared.
I'm not sure which one of these steps exacltly did the trick, but this was the way I made it work again.
Hope it helps.
bodanmic Mar 23, 2013 10:54 AM
in response to bodanmic
Thanks for all your information. Due to my limited knowledge of the Mac-system, I didn't find a solution.
I realized:
- OS 10.8.2, Safari 6.0.2 and Java version 7 update 17.: Java works without any problem
- OS 10.8.3, Safari 6.0.3, Java version 7 update 17 (Build 1.7.0_17-b02): Java is blocked.
but Java works correctly with Firefox 19.0.2
Best regards
Paul_31 Mar 23, 2013 11:01 AM
in response to bodanmic
I don't have Java installed on my systems, so I'm a bit sketchy on updating it.
I think Apple have a version they keep updated (which is on their Software Updates page ( http://support.apple.com/downloads/ - page 2).
Also Oracle issue updates their version. All very confusing, to me at least.
In System Preferences do you have a Java Preference Pane? If so, click on that to see if it offers any updates.
Also, in your Applications folder do you have an app called Java Preferences. If so, launch that and see if any updates are offered.
danixdefcon5 Mar 26, 2013 12:21 PM
in response to bodanmic
It seems that XProtect got recently updated (maybe by the 10.8.3 OSX Update?) and is now blocking current-version Java:
# ls -l /System/Library/CoreServices/CoreTypes.bundle/Contents/Resources/XProtect.meta. plist
-rw-r--r-- 1 root wheel 755 Mar 23 10:20 /System/Library/CoreServices/CoreTypes.bundle/Contents/Resources/XProtect.meta. plist
Checking the file, I found
I'm not sure, but I do remember that one of the late-January Apple blockage for Java consisted in grabbing then-current Update 11's version, then adding one to the final number so that none of the current Java plugins would work. (Basically, disabled 'till Oracle releases a new version). I wouldn't be surprised if Update 17's version is 1.7.17.01 and that's why it won't work.
Thing is, I did edit the file manually to 1.7.16.02, saved, quit/restarted safari ... and it *still* doesn't work. Maybe iTunes is still using the Safari process?
danixdefcon5 Mar 26, 2013 8:45 PM
in response to danixdefcon5
Ummm... scratch that. Even with 1.7.17.02 on XProtect, it will work but after visiting the Oracle's Java site and clicking on the "inactive plugin" inline link.
Rogerio's steps 1, 3 and 4 are what I did to reenable the Java plugin. Resetting website warnings isn't needed.
hiro.igarashi Mar 27, 2013 11:50 PM
in response to danixdefcon5
I found modifying a value of MinimumPlugInBundleVersion to 1.7.0_17 solves this;
my modification is..
# vi /System/Library/CoreServices/CoreTypes.bundle/Contents/Resources/XProtect.meta. plist
<key>com.macromedia.Flash Player.plugin</key>
Please note that this modification might cause further defects on your mac.
Matt Wolanski Mar 28, 2013 9:12 AM
in response to Rogerio de Camargo Rocha
4. Visit Oracle's Java test site. How do I test whether Java is working on my computer?
That fixed it for me - was very strange - it was workign fine on another system and in FireFox - and after uninstall, reinstall, deleteing prefs, resetting Safari, reboot, etc - finally went to the link above and it said inactive plugin - clicked on that - got it reactivated and now works fine.
The site that I use which has a java applet was designed for use with windows so on the Mac it is a tad flaky.
Kjell Magnus Mar 30, 2013 1:16 PM
in response to Matt Wolanski
This worked for me too, tried on 3 machines now. Problem ws that not much really happened, when trying to verify or run applets. Following the link I got the inactive plugin. Click and there it went. Thanks!!!
Alessandro Cintoli Jun 23, 2013 11:17 PM
in response to Matt Wolanski
Matt, your suggestion helped me greatly when I experienced exactly the same problem after installing Java 7 Update 21, last June 18.
Apparently, the plug-in is disabled by default, you are not prompted to choose whether to run any applet in Safari or any other browser and you are led to think there is something wrong with your installation or Java version.
The plug-in test works around the problem by triggering the inactive plug-in and making it work normally.
To all Java users: remember that when you access a Java site, you are warned that some applets might be insecure. If you allow them to run, Safari might eventually block them anyway, because the applets are not signed or are known to have caused trouble. If you still wish to run them, do this:
1. Go to the Safari menu and select Preferences.
2. Click on the Security icon.
3. Make sure that the "Enable Java Script" and "Allow Java" options are checked.
4. Click on the "Manage Website Settings" button.
5. Make sure to choose "Allow Always" from the dropdown menu in the window that appears near the name of the site where the applets reside. If you only choose "Allow", some applets will not run.
Again, remember that using Java these days may pose a security threat. Whenever that happens, Apple will update the OS and disable Java.


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Miami Finally Getting to Host Latin Grammys; Show Avoids Controversy over Artists Living in Cuba
Peralta, Eyder, The Florida Times Union
Byline: Eyder Peralta, Times-Union staff writer
MIAMI -- The signs all over the city note that this time around, the Latin Grammys will finally take place in Miami. The lightpoles along the city's highways are lined with plastic banners announcing the Sept. 3 date. In front of the American Airlines Arena -- where the Latin Grammys are to be held tonight -- there is a monstrous billboard advertising the awards. Just around the corner, there's another with Juanes, the Colombian rocker, strutting with his electric guitar and touting his Grammy nominations.
And throughout the city, there is an awkward lull. There is little talk of protests, and the only reminders of the feud between the Latin Association of Recording Arts and Sciences and the exiled Cuban community in Miami are numbers that seem to almost magically appear on the long facades of one of Miami's most historic buildings, The Freedom Tower. The arena is one side of the street, its parking lots lined with freight trucks and staff members loitering on the long stairs. On the other side is the tower, a symbol of immigrant freedom, Miami's version of Ellis Island.
The gist of the battle is that, for years, the recording academy has nominated Cuban musicians -- musicians who still live in Cuba and have at times spoken out in favor of Fidel Castro's Communist regime -- and invited them to perform.
For the last two years, the Latin Grammys have been scheduled for Miami, but because of the threat of protests from the city's politically influential Cuban community, who consider support of Cuban-based artists to be support for Castro's regime, the show has been moved to Los Angeles.
The numbers on the Freedom Tower are supposed to be a peace offering of sorts. The Cuban American National Foundation owns the building, and the numbers are designed to be symbols of free speech, both in support of the crowd of protesters who are expected to make a showing tonight, and in support of the free speech of the musicians inside, CANF officials said in recent interviews. The numbers, which are well lighted, change regularly and stand for different things. The number one is in honor of the First Amendment to the Constitution, 143 stands for the estimated number of journalists killed in Cuba. CANF has been cagey about the meaning of all the numbers, saying they will release the full list at a later time.
This year, however, the controversy has been calmed somewhat by several factors. Cuban nominees like Ibrahim Ferrer of Buena Vista Social Club fame were not invited to perform at the ceremonies, which will be hosted by sitcom star George Lopez. And, they probably won't be able to attend the awards because so far, post-Sept. 11 delays have kept them from getting travel visas in time to enter the U.S.
Inside the arena Monday, the controversy seemed far away and rehearsals went as scheduled. The empty space was lighted brilliantly by thousands of lights on the stair-like banks. The 20,000-seat arena seemed dwarfed by the sheer size of the stage. Molotov, a Mexican rap-metal band that's nominated for four awards, including record of the year, was running through a sound check.
Their song Frijolero (Beaner), which earned them one Grammy nomination, is a biting tune laden with political commentary about racism, immigration and war. The band is scheduled to perform last tonight.
"The musical spectrum in radio and on television is governed by pop and that tends to have a very banal tendency and very meaningless content," said Paco Ayala, one of the band's two bass players. "And truth is, you are living a much more real situation in the streets of the Third World, in Mexico and Latin America, and it almost comes natural to talk about situations that make you uncomfortable and write social criticism with wit and irony. …


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Inhale by YulchaNyanWatch
©2013-2016 YulchaNyan
The Lady's beauty is intoxicating!
Hard to pick any one picture - her face is amazing in all of them! But this simple clean shot stands out!
The very best so far: your model looks great in every shot, yet this one has a motion and theme to it.
Wow, I love her lips!
Magnifique !
I'm drooling all over my desk. Amazing.
Red is definitely the color for her
Excellent photo as always.
Excellent photo as always.
I...I cant remember where did i see that awesome position! u have very beautiful face!
Beautiful She looks heavenly
Scent of roses.
Good name for this picture is "inhale"


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By SimplyBridal
This simple satin sheath dress is wondefully chic. The strapless neckline flows into a fitted bodice with an adorable peplum. The silhouette is simple enough to look great on all shapes, expecially petite and thin brides. This is a great choice for spring weddings in the garden, but also sophisticated enough for a fall banquet hall setting.


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Thanks guys! :D
Posted 12/28/2012
They're both really pretty! I slightly prefer Maeve to Neve, but Tessa is still my favorite. :-)
Proud Mommy to my twin boys (J & B)
Proud Mommy to my twin boys (J & B)
Blessed beyond measure to be a wife and mother...
Posted 12/28/2012
I prefer Maeve to Neve. The correct pronunciation of Neve is NEEV, so you're going to get that a lot, and for good reason. I think the pronunciation of Maeve is pretty straight-forward.
Good luck!
Good luck!
Mama to the Kindergamer (5/2002) and the Montessaur (7/2010)
... and introducing Baby Gretel! Margaret Meriwether, born 4/4/2013 at 37+1!
Posted 12/28/2012
Do NOT like either...they would never be pronounced correctly.
My- My 3.06, J 6.13, Baby girl #3 coming 10.30.15!
M and K <3 3.2011 <3
Posted 12/28/2012
I like both, but I agree that Neve will be prone to proununciation problems (I think you will have far less trouble with Maeve). I also really like Tessa Maeve.
Last edited 12/28/2012
I would definitely pronounce Neve as Nehv, like Neve Campbell.
I like Maeve, but I love Felicity and Tessa both so much more! I have them both on my list.
Morgan 6/10/13
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text: Stanford tops Oregon St. 81-73 Back to Article Comments Need to Register? X Remember Me Forgot your password? Curry (knee) won't play Tuesday, says Kerr Latest News Map shows SF tech buses committed over 800 traffic fouls Oakland cops sorry for 'insensitive' Cinco de Mayo message Tupac Shakur's mother, Afeni, dies in Marin Co. Ex-49er, Raider Stubblefield says rape charges are false Fremont company fined for paying workers under $2/hour Big win for pot industry: Feds drop bid to shut down Harborside Plastic and rubber found in Foster Farms chicken nuggets 29 million trees dead from bugs pose fire risk in Calif.


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Key Lime Cupcakes by DeathbypuddleWatch
©2014-2016 Deathbypuddle
TGIF everyone! Lets start off the weekend with Key Lime Cupcakes! yum!
Made by me!
Made by me!
ohh cool design!
Ooo yum! Makes me want to have a Margarita!
omg you have to give me this recipe! I've been looking for one forever! these look so good!
Me likey me wants
these.. my face.. now!
These look so yummy!
This food is also pretty. ^_^
Boom! Key Lime is one of my favorite flavors; really good job with the garnish; is it a cream cheese based frosting? Buttercream?
looks so delicious


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Jamaica Bay and the Rockaways
What is the Parks Inspection Program?
Learn how we rate parks
Bayswater Park
Dwight Ave., Seagirt Blvd. Bet. Beach 38 St. And Bay 32 St.
Beach 17 Playground
Seagirt Blvd Bet. B. 17 St. And B. 12 St.
Beach 59th St Playground
Rockaway Boardwalk Bet. B. 60 St. And B. 59 St.
Beach 9 Playground
Seagirt Blvd Bet. B. 12 St. And B. 9 St.
Beach Channel Park
Beach Channel Dr. Bet. B. 117 St. And B. 124 St.
Broad Channel American Park
Cross Bay Blvd. Bet. W. 20 Rd. And Beach Channel
Canarsie Park
Seaview Ave. Bet. Paerdegat Basin And E. 93 St., E. 102 St. And Fresh Creek Basin
Canarsie Park Zone 1 (Zone 0)
Seaview Ave, E 88 St, E 93 St, Skidmore Ave. Incl Soccer, Cricket Area West Of Little League Fields
Canarsie Park Zone 2 (Zone 0)
Seaview Ave, E 87 St, Paerdegat Basin
Canarsie Park Zone 3 (Zone 0)
Paerdegat Basin, Belt Pkwy: Natural Area And Water-Course, Sw Area Of Park
Canarsie Park Zone 4 (Zone 0)
Belt Pkwy, Canarsie Rd, Schenck St: Up To Soccer, Cricket Area
Canarsie Park-Canarsie Playground (Section 0)
Seaview Ave, E 80 St, Paerdegat Ave N
Canarsie Park-Dinapoli Playground (Section 0)
Seaview Ave, E 92 St
Dubos Point Wildlife Sanctuary
De Costa Ave. Bet. Sommerwille Basin And B. 65 St.,Bayfield Ave. Bet. B. 65 St. And B. 69 St.
Flight 587 Memorial Park
Beach 116 St., Ocean Promenade
Fresh Creek Nature Preserve
Flatlands Ave, Louisiana Ave., E 108 St. And Belt Parkway
Rockaway Pkwy, Schenck St, And Shore Pkwy Nw
Rockaway Pkwy, Schenck St, And Shore Pkwy Se
Huxley St & 149 Av
Hook Creek Park
Brookville Blvd., Huxleyy St. Bet. 149 Ave. And Hook Creek Basin
Idlewild Park
149 Ave., Rockaway Blvd., Jamaica Bay Bet. James Brown Pl. And Brookville Blvd.
Idlewild Park Zone 1 (Zone 0)
232 St, Springfield Ln, 148 Ave
Joseph T Mcguire Park - Mcguire Fields Tot Lot (Section 0)
Bergen Ave And Avenue W
Marine Park Zone 1 (Zone 0)
Fillmore Ave, Burnett St, E 32 St, Ave S, N Of Center Oval Containing Ballfields
Marine Park Zone 2 (Zone 0)
Ave S, E 33 St, Ave U, Path N Of Burnett St, E Of Center Oval Containing Ballfields
Directions to Jamaica Bay and the Rockaways
Know Before You Go
Marine Park
Marine Park's tennis courts will be closed for the season due to construction.
Jamaica Bay and the Rockaways Weather
- Major Habitat Restoration And Trail Development Project Gets Underway This Fall In Marine Park
- De Blasio Administration Launches Community Parks Initiative To Build More Inclusive And Equitable Park System
- More Than 30,000 New Yorkers Celebrate The Outdoors At Tenth Anniversary Of Adventures NYC
- It's My Park at Rockaway Beach and Boardwalk
- Sports and Fitness Day
- NYRR Open Run: Marine Park
- Seining
- Arts, Culture & Fun; Wildlife of New York
- Barbecuing Areas
- Baseball Fields
- Basketball Courts
- Bathrooms
- Beaches
- Bicycling and Greenways
- Bocce Courts
- Cricket Fields
- Dog-friendly Areas
- Eateries
- Fitness Equipment
- Football Fields
- Golf Courses
- Handball Courts
- Hiking Trails
- Kayak/Canoe Launch Sites
- Marinas
- Model Aircraft Fields
- Nature Centers
- Paddleboat Rentals
- Playgrounds
- Roller Hockey
- Skate Parks
- Soccer Fields
- Spray Showers
- Tennis Courts
- Volleyball Courts
- Wi-Fi Hot Spots
Know when to go:
View upcoming athletic area usage in
Bayswater Park
Beach 9 Playground
Canarsie Park
Idlewild Park
Marine Park
McGuire Fields
Rockaway Beach and Boardwalk
Rockaway Beach Boardwalk
Rockaway Community Park
Spring Creek Park


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One of the most powerful life lessons I learned was from a (literary) penguin. I can thank the lovable,...Read more
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Cecil's Pride: The Life of a Lion KingApril 22nd, 2016
In the spirit of Earth Day, we're excited to share Cecil's Pride: The True Story of a Lion King, a stunning new picture...Read more
New blog alert: Judy Newman at ScholasticApril 22nd, 2016
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How we're celebrating Poem in Your Pocket DayApril 21st, 2016
Every year, April is an exciting time to celebrate one of our favorite things here at Scholastic… Poetry!
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This year's 2016 Scholastic Art & Writing...Read more


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Portals_7th heaven by IvanLaliashviliWatch
©2013-2016 IvanLaliashvili
Illustration for game loading animation
Thank you all for your kind words, criticism and comments)))
Please click Download Image for original quality.
Adobe Photoshop CS5
Thank you all for your kind words, criticism and comments)))
Please click Download Image for original quality.
Adobe Photoshop CS5
I love this!!
A lovely evocation of dimensional portals - I've always been fascinated by the idea.
my the sun… IT LOOKS AMAZING! its simply gorgeous!
Хорошая работа.
wow beautiful
THIS IS A CRIME IF IT'S NOT AN ACTUAL GAME. lol That is frigging amazing! I LOVE it. Please tell me this will be a game please tell me this will be a game pllllleeeeeeasssseeee. lol
amazing work , this is so wonderful
great work!
great work!
Amazing work. I've been looking through a few of your piece and this is one of my favs
I love the lighting you've used and the shading. Just brilliant ^^
I love the lighting you've used and the shading. Just brilliant ^^
I really enjoy the texture and color of the grass.
Amazing! Perfect for a video game.
The best fantasy art, in my opinion. triggers stories in your imagination from what you see. This does that. Great job.
Brilliant compostion! Where did you the concept for this? Love it!
Love this
I think I've been through this Portal Before ? I Should leave Marks of where I've Been.
Fantastic. From the floating islands, the stairways through the clouds and the glowing portals, this is an atmospheric masterpiece. Makes you wonder what miracles await you if you step through one of these portals...
Wow, really stunning and vivid. I love the color on the grass and that fog/cloud effect.
Fantastic! Beautiful! I love your art!
* ¡Fantástico trabajo! Lo hemos destacado dentro de la carpeta "Featured nº5 o Destacados nº 5" en Special-Groups. (Es la carpeta donde se exhiben los mejores trabajos del grupo).
¡Por favor, queremos ver más trabajos tuyos en el grupo!
¡Por favor, queremos ver más trabajos tuyos en el grupo!
Invites to dream
Absolutely stunning
Love this one a lot. colourful.
Amazing scene! Really love the colors!
wonderful <3
ааа)) великолепные цвета)
There is much I would give to have even half the skill it musthave taken to make such a work of art.
Really awesome!
Very NEAT!!!!!!
Very nice work!
This brings me back to WC3, going from portal to portal with that madman illidan.
Потрясающе! Просто фантастически. Аж завораживает)
That's just fantastic!!!
Just like the portal arches from the PS1 Spyro games. I love this!
Gorgeous concept, colors and performance!
Nice work!
Gorgeous concept, colors and performance!
Nice work!
Nice detail it's amazing I love it
this is amazing love it
Dude! This is crazy awesome! Loving every bit of it
What game it for?
I can see this drawing like it was concept art for World of Warcraft


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"Living in the midwest" Ypsilanti by cmayda
Ypsilanti Travel Guide: 6 reviews and 2 photos
I teach at Eastern Michigan University, but am originally from Los Angeles. Therefore, it was important to me to live near where i worked for 2 reasons. My lifetime quota for traffic was filled when I left LA, and conserving energy should be a priority. So I live one mile from the school and often ride my bike there. But, though this is nice, it is not well received. No one rides a bike in Ypsilanti, or most of SE Michigan for that matter. Detroit is motor city, and though it is dying a horrible death, it could be saved if people thought beyond the short but glorious past of this part of America.
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Soaking up the sun in Dixon
A pelican suns itself on a log in the Rock River on Monday afternoon near Page Park in Dixon. Today's forecast calls for even nicer weather – for people and for birds. The National Weather Service is calling for sunny skies and temperatures reaching the upper 70s.
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May 3, 2016
Monday, February 6, 2012
ATLANTA (AP) -- Gov. Nathan Deal has announced a need-based scholarship to be privately funded and aimed at grooming Georgia middle school students for college.
AT&T joined the governor at Georgia Tech to announce the company will contribute $250,000 to the REACH Scholarship program.
Students who complete program requirements will receive a renewable yearly tuition scholarship of $2,500 to be used at any HOPE eligible institution. Georgia Tech is announcing it will match the $2,500 scholarship for REACH scholars who are accepted and enroll at the school.
Initially, 25 students from Bulloch, Douglas and Rabun counties will participate in the program, with additional counties to join later.
Students will have to meet certain guidelines, including getting mentors and signing a contract that they will stay out of trouble.
(Copyright 2012. Associated Press)
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I'm on my first round, I'm just wondering if any of you have been successful on your first round of 50mg clomid?
Success on first round of clomid 50mg cd5-9?
Comments (8)
I'm new to the group and to babycenter! Any buddies are sure welcomed :)
From what I have read BFP on your first round is common. I am on round two now. When did you start your clomid?
Round #1 3/24 (Fail), Round #2 4/17- (Fail)
Injectables (FSH) - Round #1 (BFN) Round#2 CD3 6/17 = BFP = MC Round#3 = BFP! Got my Rainbow!
.with my DD i got a BFP on my 1st round of 50mg clomid.
1 year later i tried again and BFP 1st try also, but that ended in a M/C
DD ( my clomid baby) 4/26/09
2 MMC and 1 CP
Emily (35) Christopher (45) - DS Samuel born November 23, 2010. TTC #2 - fingers crossed for 5th round of clomid
Good luck to all of you!!
Yes mam!!! 17 weeks today!!! Miracle drug!!! Although I do O on my own but with 3 mc's, I needed the extra help in the luteal phase!! First round of 50 worked for me. Good Luck!
DD ~ Logan Miles Norris 1-31-10
DD ~ Caroline Marie Norris 10-1-13
3 Angel Babies 2009, 2012, 2012
I started clomid on Friday the 26th and took my last pill today! I'm so hopeful that this will work for us! Good luck to the rest of you!!


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You are currently browsing the category archive for the 'paper' category.
Hoi Nguyen, Van Vu, and myself have just uploaded to the arXiv our paper "Random matrices: tail bounds for gaps between eigenvalues". This is a followup paper to my recent paper with Van in which we showed that random matrices of Wigner type (such as the adjacency matrix of an Erdös-Renyi graph) asymptotically almost surely had simple spectrum. In the current paper, we push the method further to show that the eigenvalues are not only distinct, but are (with high probability) separated from each other by some negative power of . This follows the now standard technique of replacing any appearance of discrete Littlewood-Offord theory (a key ingredient in our previous paper) with its continuous analogue (inverse theorems for small ball probability). For general Wigner-type matrices (in which the matrix entries are not normalised to have mean zero), we can use the inverse Littlewood-Offord theorem of Nguyen and Vu to obtain (under mild conditions on ) a result of the form
for any and , if is sufficiently large depending on (in a linear fashion), and is sufficiently large depending on . The point here is that can be made arbitrarily large, and also that no continuity or smoothness hypothesis is made on the distribution of the entries. (In the continuous case, one can use the machinery of Wegner estimates to obtain results of this type, as was done in a paper of Erdös, Schlein, and Yau.)
In the mean zero case, it becomes more efficient to use an inverse Littlewood-Offord theorem of Rudelson and Vershynin to obtain (with the normalisation that the entries of have unit variance, so that the eigenvalues of are with high probability), giving the bound
for (one also has good results of this type for smaller values of ). This is only optimal in the regime ; we expect to establish some eigenvalue repulsion, improving the RHS to for real matrices and for complex matrices, but this appears to be a more difficult task (possibly requiring some quadratic inverse Littlewood-Offord theory, rather than just linear inverse Littlewood-Offord theory). However, we can get some repulsion if one works with larger gaps, getting a result roughly of the form
for (one also has good results of this type for smaller values of ). This is only optimal in the regime ; we expect to establish some eigenvalue repulsion, improving the RHS to for real matrices and for complex matrices, but this appears to be a more difficult task (possibly requiring some quadratic inverse Littlewood-Offord theory, rather than just linear inverse Littlewood-Offord theory). However, we can get some repulsion if one works with larger gaps, getting a result roughly of the form
for any fixed and some absolute constant (which we can asymptotically make to be for large , though it ought to be as large as ), by using a higher-dimensional version of the Rudelson-Vershynin inverse Littlewood-Offord theorem.
In the case of Erdös-Renyi graphs, we don't have mean zero and the Rudelson-Vershynin Littlewood-Offord theorem isn't quite applicable, but by working carefully through the approach based on the Nguyen-Vu theorem we can almost recover (1), except for a loss of on the RHS.
As a sample applications of the eigenvalue separation results, we can now obtain some information about eigenvectors; for instance, we can show that the components of the eigenvectors all have magnitude at least for some with high probability. (Eigenvectors become much more stable, and able to be studied in isolation, once their associated eigenvalue is well separated from the other eigenvalues; see this previous blog post for more discussion.)
Kaisa Matomaki, Maksym Radziwill, and I have just uploaded to the arXiv our paper "An averaged form of Chowla's conjecture". This paper concerns a weaker variant of the famous conjecture of Chowla (discussed for instance in this previous post) that
as for any distinct natural numbers , where denotes the Liouville function. (One could also replace the Liouville function here by the Möbius function and obtain a morally equivalent conjecture.) This conjecture remains open for any ; for instance the assertion
is a variant of the twin prime conjecture (though possibly a tiny bit easier to prove), and is subject to the notorious parity barrier (as discussed in this previous post).
Our main result asserts, roughly speaking, that Chowla's conjecture can be established unconditionally provided one has non-trivial averaging in the parameters. More precisely, one has
Theorem 1 (Chowla on the average) Suppose is a quantity that goes to infinity as (but it can go to infinity arbitrarily slowly). Then for any fixed , we have
In fact, we can remove one of the averaging parameters and obtain
Actually we can make the decay rate a bit more quantitative, gaining about over the trivial bound. The key case is ; while the unaveraged Chowla conjecture becomes more difficult as increases, the averaged Chowla conjecture does not increase in difficulty due to the increasing amount of averaging for larger , and we end up deducing the higher case of the conjecture from the case by an elementary argument.
The proof of the theorem proceeds as follows. By exploiting the Fourier-analytic identity
(related to a standard Fourier-analytic identity for the Gowers norm) it turns out that the case of the above theorem can basically be derived from an estimate of the form
uniformly for all . For "major arc" , close to a rational for small , we can establish this bound from a generalisation of a recent result of Matomaki and Radziwill (discussed in this previous post) on averages of multiplicative functions in short intervals. For "minor arc" , we can proceed instead from an argument of Katai and Bourgain-Sarnak-Ziegler (discussed in this previous post).
The argument also extends to other bounded multiplicative functions than the Liouville function. Chowla's conjecture was generalised by Elliott, who roughly speaking conjectured that the copies of in Chowla's conjecture could be replaced by arbitrary bounded multiplicative functions as long as these functions were far from a twisted Dirichlet character in the sense that
(This type of distance is incidentally now a fundamental notion in the Granville-Soundararajan "pretentious" approach to multiplicative number theory.) During our work on this project, we found that Elliott's conjecture is not quite true as stated due to a technicality: one can cook up a bounded multiplicative function which behaves like on scales for some going to infinity and some slowly varying , and such a function will be far from any fixed Dirichlet character whilst still having many large correlations (e.g. the pair correlations will be large). In our paper we propose a technical "fix" to Elliott's conjecture (replacing (1) by a truncated variant), and show that this repaired version of Elliott's conjecture is true on the average in much the same way that Chowla's conjecture is. (If one restricts attention to real-valued multiplicative functions, then this technical issue does not show up, basically because one can assume without loss of generality that in this case; we discuss this fact in an appendix to the paper.)
(This type of distance is incidentally now a fundamental notion in the Granville-Soundararajan "pretentious" approach to multiplicative number theory.) During our work on this project, we found that Elliott's conjecture is not quite true as stated due to a technicality: one can cook up a bounded multiplicative function which behaves like on scales for some going to infinity and some slowly varying , and such a function will be far from any fixed Dirichlet character whilst still having many large correlations (e.g. the pair correlations will be large). In our paper we propose a technical "fix" to Elliott's conjecture (replacing (1) by a truncated variant), and show that this repaired version of Elliott's conjecture is true on the average in much the same way that Chowla's conjecture is. (If one restricts attention to real-valued multiplicative functions, then this technical issue does not show up, basically because one can assume without loss of generality that in this case; we discuss this fact in an appendix to the paper.)
Kevin Ford, Ben Green, Sergei Konyagin, James Maynard, and I have just uploaded to the arXiv our paper "Long gaps between primes". This is a followup work to our two previous papers (discussed in this previous post), in which we had simultaneously shown that the maximal gap
between primes up to exhibited a lower bound of the shape
for some function that went to infinity as ; this improved upon previous work of Rankin and other authors, who established the same bound but with replaced by a constant. (Again, see the previous post for a more detailed discussion.)
for some function that went to infinity as ; this improved upon previous work of Rankin and other authors, who established the same bound but with replaced by a constant. (Again, see the previous post for a more detailed discussion.)
In our previous papers, we did not specify a particular growth rate for . In my paper with Kevin, Ben, and Sergei, there was a good reason for this: our argument relied (amongst other things) on the inverse conjecture on the Gowers norms, as well as the Siegel-Walfisz theorem, and the known proofs of both results both have ineffective constants, rendering our growth function similarly ineffective. Maynard's approach ostensibly also relies on the Siegel-Walfisz theorem, but (as shown in another recent paper of his) can be made quite effective, even when tracking -tuples of fairly large size (about for some small ). If one carefully makes all the bounds in Maynard's argument quantitative, one eventually ends up with a growth rate of shape
thus leading to a bound
thus leading to a bound
on the gaps between primes for large ; this is an unpublished calculation of James'.
In this paper we make a further refinement of this calculation to obtain a growth rate
leading to a bound of the form
for large and some small constant . Furthermore, this appears to be the limit of current technology (in particular, falling short of Cramer's conjecture that is comparable to ); in the spirit of Erdös' original prize on this problem, I would like to offer 10,000 USD for anyone who can show (in a refereed publication, of course) that the constant here can be replaced by an arbitrarily large constant .
leading to a bound of the form
for large and some small constant . Furthermore, this appears to be the limit of current technology (in particular, falling short of Cramer's conjecture that is comparable to ); in the spirit of Erdös' original prize on this problem, I would like to offer 10,000 USD for anyone who can show (in a refereed publication, of course) that the constant here can be replaced by an arbitrarily large constant .
The reason for the growth rate (3) is as follows. After following the sieving process discussed in the previous post, the problem comes down to something like the following: can one sieve out all (or almost all) of the primes in by removing one residue class modulo for all primes in (say) ? Very roughly speaking, if one can solve this problem with , then one can obtain a growth rate on of the shape . (This is an oversimplification, as one actually has to sieve out a random subset of the primes, rather than all the primes in , but never mind this detail for now.)
Using the quantitative "dense clusters of primes" machinery of Maynard, one can find lots of -tuples in which contain at least primes, for as large as or so (so that is about ). By considering -tuples in arithmetic progression, this means that one can find lots of residue classes modulo a given prime in that capture about primes. In principle, this means that union of all these residue classes can cover about primes, allowing one to take as large as , which corresponds to (3). However, there is a catch: the residue classes for different primes may collide with each other, reducing the efficiency of the covering. In our previous papers on the subject, we selected the residue classes randomly, which meant that we had to insert an additional logarithmic safety margin in expected number of times each prime would be shifted out by one of the residue classes, in order to guarantee that we would (with high probability) sift out most of the primes. This additional safety margin is ultimately responsible for the loss in (2).
The main innovation of this paper, beyond detailing James' unpublished calculations, is to use ideas from the literature on efficient hypergraph covering, to avoid the need for a logarithmic safety margin. The hypergraph covering problem, roughly speaking, is to try to cover a set of vertices using as few "edges" from a given hypergraph as possible. If each edge has vertices, then one certainly needs at least edges to cover all the vertices, and the question is to see if one can come close to attaining this bound given some reasonable uniform distribution hypotheses on the hypergraph . As before, random methods tend to require something like edges before one expects to cover, say of the vertices.
However, it turns out (under reasonable hypotheses on ) to eliminate this logarithmic loss, by using what is now known as the "semi-random method" or the "Rödl nibble". The idea is to randomly select a small number of edges (a first "nibble") – small enough that the edges are unlikely to overlap much with each other, thus obtaining maximal efficiency. Then, one pauses to remove all the edges from that intersect edges from this first nibble, so that all remaining edges will not overlap with the existing edges. One then randomly selects another small number of edges (a second "nibble"), and repeats this process until enough nibbles are taken to cover most of the vertices. Remarkably, it turns out that under some reasonable assumptions on the hypergraph , one can maintain control on the uniform distribution of the edges throughout the nibbling process, and obtain an efficient hypergraph covering. This strategy was carried out in detail in an influential paper of Pippenger and Spencer.
In our setup, the vertices are the primes in , and the edges are the intersection of the primes with various residue classes. (Technically, we have to work with a family of hypergraphs indexed by a prime , rather than a single hypergraph, but let me ignore this minor technical detail.) The semi-random method would in principle eliminate the logarithmic loss and recover the bound (3). However, there is a catch: the analysis of Pippenger and Spencer relies heavily on the assumption that the hypergraph is uniform, that is to say all edges have the same size. In our context, this requirement would mean that each residue class captures exactly the same number of primes, which is not the case; we only control the number of primes in an average sense, but we were unable to obtain any concentration of measure to come close to verifying this hypothesis. And indeed, the semi-random method, when applied naively, does not work well with edges of variable size – the problem is that edges of large size are much more likely to be eliminated after each nibble than edges of small size, since they have many more vertices that could overlap with the previous nibbles. Since the large edges are clearly the more useful ones for the covering problem than small ones, this bias towards eliminating large edges significantly reduces the efficiency of the semi-random method (and also greatly complicates the analysis of that method).
Our solution to this is to iteratively reweight the probability distribution on edges after each nibble to compensate for this bias effect, giving larger edges a greater weight than smaller edges. It turns out that there is a natural way to do this reweighting that allows one to repeat the Pippenger-Spencer analysis in the presence of edges of variable size, and this ultimately allows us to recover the full growth rate (3).
To go beyond (3), one either has to find a lot of residue classes that can capture significantly more than primes of size (which is the limit of the multidimensional Selberg sieve of Maynard and myself), or else one has to find a very different method to produce large gaps between primes than the Erdös-Rankin method, which is the method used in all previous work on the subject.
It turns out that the arguments in this paper can be combined with the Maier matrix method to also produce chains of consecutive large prime gaps whose size is of the order of (4); three of us (Kevin, James, and myself) will detail this in a future paper. (A similar combination was also recently observed in connection with our earlier result (1) by Pintz, but there are some additional technical wrinkles required to recover the full gain of (3) for the chains of large gaps problem.)
Van Vu and I have just uploaded to the arXiv our paper "Random matrices have simple spectrum". Recall that an Hermitian matrix is said to have simple eigenvalues if all of its eigenvalues are distinct. This is a very typical property of matrices to have: for instance, as discussed in this previous post, in the space of all Hermitian matrices, the space of matrices without all eigenvalues simple has codimension three, and for real symmetric cases this space has codimension two. In particular, given any random matrix ensemble of Hermitian or real symmetric matrices with an absolutely continuous distribution, we conclude that random matrices drawn from this ensemble will almost surely have simple eigenvalues.
For discrete random matrix ensembles, though, the above argument breaks down, even though general universality heuristics predict that the statistics of discrete ensembles should behave similarly to those of continuous ensembles. A model case here is the adjacency matrix of an Erdös-Rényi graph – a graph on vertices in which any pair of vertices has an independent probability of being in the graph. For the purposes of this paper one should view as fixed, e.g. , while is an asymptotic parameter going to infinity. In this context, our main result is the following (answering a question of Babai):
Our argument works for more general Wigner-type matrix ensembles, but for sake of illustration we will stick with the Erdös-Renyi case. Previous work on local universality for such matrix models (e.g. the work of Erdos, Knowles, Yau, and Yin) was able to show that any individual eigenvalue gap did not vanish with probability (in fact for some absolute constant ), but because there are different gaps that one has to simultaneously ensure to be non-zero, this did not give Theorem 1 as one is forced to apply the union bound.
Our argument in fact gives simplicity of the spectrum with probability for any fixed ; in a subsequent paper we also show that it gives a quantitative lower bound on the eigenvalue gaps (analogous to how many results on the singularity probability of random matrices can be upgraded to a bound on the least singular value).
The basic idea of argument can be sketched as follows. Suppose that has a repeated eigenvalue . We split
for a random minor and a random sign vector ; crucially, and are independent. If has a repeated eigenvalue , then by the Cauchy interlacing law, also has an eigenvalue . We now write down the eigenvector equation for at :
Extracting the top coefficients, we obtain
If we let be the -eigenvector of , then by taking inner products with we conclude that
we typically expect to be non-zero, in which case we arrive at
In other words, in order for to have a repeated eigenvalue, the top right column of has to be orthogonal to an eigenvector of the minor . Note that and are going to be independent (once we specify which eigenvector of to take as ). On the other hand, thanks to inverse Littlewood-Offord theory (specifically, we use an inverse Littlewood-Offord theorem of Nguyen and Vu), we know that the vector is unlikely to be orthogonal to any given vector independent of , unless the coefficients of are extremely special (specifically, that most of them lie in a generalised arithmetic progression). The main remaining difficulty is then to show that eigenvectors of a random matrix are typically not of this special form, and this relies on a conditioning argument originally used by Komlós to bound the singularity probability of a random sign matrix. (Basically, if an eigenvector has this special form, then one can use a fraction of the rows and columns of the random matrix to determine the eigenvector completely, while still preserving enough randomness in the remaining portion of the matrix so that this vector will in fact not be an eigenvector with high probability.)
I've just uploaded to the arXiv my paper "The Elliott-Halberstam conjecture implies the Vinogradov least quadratic nonresidue conjecture". As the title suggests, this paper links together the Elliott-Halberstam conjecture from sieve theory with the conjecture of Vinogradov concerning the least quadratic nonresidue of a prime . Vinogradov established the bound
and conjectured that
for any fixed . Unconditionally, the best result so far (up to logarithmic factors) that holds for all primes is by Burgess, who showed that
for any fixed . See this previous post for a proof of these bounds.
and conjectured that
for any fixed . Unconditionally, the best result so far (up to logarithmic factors) that holds for all primes is by Burgess, who showed that
for any fixed . See this previous post for a proof of these bounds.
In this paper, we show that the Vinogradov conjecture is a consequence of the Elliott-Halberstam conjecture. Using a variant of the argument, we also show that the "Type II" estimates established by Zhang and numerically improved by the Polymath8a project can be used to improve a little on the Vinogradov bound (1), to
although this falls well short of the Burgess bound. However, the method is somewhat different (although in both cases it is the Weil exponential sum bounds that are the source of the gain over (1)) and it is conceivable that a numerically stronger version of the Type II estimates could obtain results that are more competitive with the Burgess bound. At any rate, this demonstrates that the equidistribution estimates introduced by Zhang may have further applications beyond the family of results related to bounded gaps between primes.
The connection between the least quadratic nonresidue problem and Elliott-Halberstam is follows. Suppose for contradiction we can find a prime with unusually large. Letting be the quadratic character modulo , this implies that the sums are also unusually large for a significant range of (e.g. ), although the sum is also quite small for large (e.g. ), due to the cancellation present in . It turns out (by a sort of "uncertainty principle" for multiplicative functions, as per this paper of Granville and Soundararajan) that these facts force to be unusually large in magnitude for some large (with for two large absolute constants ). By the periodicity of , this means that
must be unusually large also. However, because is large, one can factorise as for a fairly sparsely supported function . The Elliott-Halberstam conjecture, which controls the distribution of in arithmetic progressions on the average can then be used to show that is small, giving the required contradiction.
The implication involving Type II estimates is proven by a variant of the argument. If is large, then a character sum is unusually large for a certain . By multiplicativity of , this shows that correlates with , and then by periodicity of , this shows that correlates with for various small . By the Cauchy-Schwarz inequality (cf. this previous blog post), this implies that correlates with for some distinct . But this can be ruled out by using Type II estimates.
I'll record here a well-known observation concerning potential counterexamples to any improvement to the Burgess bound, that is to say an infinite sequence of primes with . Suppose we let be the asymptotic mean value of the quadratic character at and the mean value of ; these quantities are defined precisely in my paper, but roughly speaking one can think of
Thanks to the basic Dirichlet convolution identity , one can establish the Wirsing integral equation
for all ; see my paper for details (actually far sharper claims than this appear in previous work of Wirsing and Granville-Soundararajan). If we have an infinite sequence of counterexamples to any improvement to the Burgess bound, then we have
while from the Burgess exponential sum estimates we have
These two constraints, together with the Wirsing integral equation, end up determining and completely. It turns out that we must have
and then for , evolves by the integral equation
For instance
and then oscillates in a somewhat strange fashion towards zero as . This very odd behaviour of is surely impossible, but it seems remarkably hard to exclude it without invoking a strong hypothesis, such as GRH or the Elliott-Halberstam conjecture (or weaker versions thereof).
Tamar Ziegler and I have just uploaded to the arXiv our paper "Narrow progressions in the primes", submitted to the special issue "Analytic Number Theory" in honor of the 60th birthday of Helmut Maier. The results here are vaguely reminiscent of the recent progress on bounded gaps in the primes, but use different methods.
About a decade ago, Ben Green and I showed that the primes contained arbitrarily long arithmetic progressions: given any , one could find a progression with consisting entirely of primes. In fact we showed the same statement was true if the primes were replaced by any subset of the primes of positive relative density.
A little while later, Tamar Ziegler and I obtained the following generalisation: given any and any polynomials with , one could find a "polynomial progression" with consisting entirely of primes. Furthermore, we could make this progression somewhat "narrow" by taking (where denotes a quantity that goes to zero as goes to infinity). Again, the same statement also applies if the primes were replaced by a subset of positive relative density. My previous result with Ben corresponds to the linear case .
In this paper we were able to make the progressions a bit narrower still: given any and any polynomials with , one could find a "polynomial progression" with consisting entirely of primes, and such that , where depends only on and (in fact it depends only on and the degrees of ). The result is still true if the primes are replaced by a subset of positive density , but unfortunately in our arguments we must then let depend on . However, in the linear case , we were able to make independent of (although it is still somewhat large, of the order of ).
The polylogarithmic factor is somewhat necessary: using an upper bound sieve, one can easily construct a subset of the primes of density, say, , whose arithmetic progressions of length all obey the lower bound . On the other hand, the prime tuples conjecture predicts that if one works with the actual primes rather than dense subsets of the primes, then one should have infinitely many length arithmetic progressions of bounded width for any fixed . The case of this is precisely the celebrated theorem of Yitang Zhang that was the focus of the recently concluded Polymath8 project here. The higher case is conjecturally true, but appears to be out of reach of known methods. (Using the multidimensional Selberg sieve of Maynard, one can get primes inside an interval of length , but this is such a sparse set of primes that one would not expect to find even a progression of length three within such an interval.)
The argument in the previous paper was unable to obtain a polylogarithmic bound on the width of the progressions, due to the reliance on a certain technical "correlation condition" on a certain Selberg sieve weight . This correlation condition required one to control arbitrarily long correlations of , which was not compatible with a bounded value of (particularly if one wanted to keep independent of ).
However, thanks to recent advances in this area by Conlon, Fox, and Zhao (who introduced a very nice "densification" technique), it is now possible (in principle, at least) to delete this correlation condition from the arguments. Conlon-Fox-Zhao did this for my original theorem with Ben; and in the current paper we apply the densification method to our previous argument to similarly remove the correlation condition. This method does not fully eliminate the need to control arbitrarily long correlations, but allows most of the factors in such a long correlation to be bounded, rather than merely controlled by an unbounded weight such as . This turns out to be significantly easier to control, although in the non-linear case we still unfortunately had to make large compared to due to a certain "clearing denominators" step arising from the complicated nature of the Gowers-type uniformity norms that we were using to control polynomial averages. We believe though that this an artefact of our method, and one should be able to prove our theorem with an that is uniform in .
Here is a simple instance of the densification trick in action. Suppose that one wishes to establish an estimate of the form
for some real-valued functions which are bounded in magnitude by a weight function , but which are not expected to be bounded; this average will naturally arise when trying to locate the pattern in a set such as the primes. Here I will be vague as to exactly what range the parameters are being averaged over. Suppose that the factor (say) has enough uniformity that one can already show a smallness bound
whenever are bounded functions. (One should think of as being like the indicator functions of "dense" sets, in contrast to which are like the normalised indicator functions of "sparse" sets). The bound (2) cannot be directly applied to control (1) because of the unbounded (or "sparse") nature of and . However one can "densify" and as follows. Since is bounded in magnitude by , we can bound the left-hand side of (1) as
for some real-valued functions which are bounded in magnitude by a weight function , but which are not expected to be bounded; this average will naturally arise when trying to locate the pattern in a set such as the primes. Here I will be vague as to exactly what range the parameters are being averaged over. Suppose that the factor (say) has enough uniformity that one can already show a smallness bound
whenever are bounded functions. (One should think of as being like the indicator functions of "dense" sets, in contrast to which are like the normalised indicator functions of "sparse" sets). The bound (2) cannot be directly applied to control (1) because of the unbounded (or "sparse") nature of and . However one can "densify" and as follows. Since is bounded in magnitude by , we can bound the left-hand side of (1) as
The weight function will be normalised so that , so by the Cauchy-Schwarz inequality it suffices to show that
The left-hand side expands as
Now, it turns out that after an enormous (but finite) number of applications of the Cauchy-Schwarz inequality to steadily eliminate the factors, as well as a certain "polynomial forms condition" hypothesis on , one can show that
(Because of the polynomial shifts, this requires a method known as "PET induction", but let me skip over this point here.) In view of this estimate, we now just need to show that
Now we can reverse the previous steps. First, we collapse back to
One can bound by , which can be shown to be "bounded on average" in a suitable sense (e.g. bounded norm) via the aforementioned polynomial forms condition. Because of this and the Hölder inequality, the above estimate is equivalent to
By setting to be the signum of , this is equivalent to
This is halfway between (1) and (2); the sparsely supported function has been replaced by its "densification" , but we have not yet densified to . However, one can shift by and repeat the above arguments to achieve a similar densificiation of , at which point one has reduced (1) to (2).
Kevin Ford, Ben Green, Sergei Konyagin, and myself have just posted to the arXiv our preprint "Large gaps between consecutive prime numbers". This paper concerns the "opposite" problem to that considered by the recently concluded Polymath8 project, which was concerned with very small values of the prime gap . Here, we wish to consider the largest prime gap that one can find in the interval as goes to infinity.
Finding lower bounds on is more or less equivalent to locating long strings of consecutive composite numbers that are not too large compared to the length of the string. A classic (and quite well known) construction here starts with the observation that for any natural number , the consecutive numbers are all composite, because each , is divisible by some prime , while being strictly larger than that prime . From this and Stirling's formula, it is not difficult to obtain the bound
A more efficient bound comes from the prime number theorem: there are only primes up to , so just from the pigeonhole principle one can locate a string of consecutive composite numbers up to of length at least , thus
where we use or as shorthand for or .
What about upper bounds? The Cramér random model predicts that the primes up to are distributed like a random subset of density . Using this model, Cramér arrived at the conjecture
In fact, if one makes the extremely optimistic assumption that the random model perfectly describes the behaviour of the primes, one would arrive at the even more precise prediction
However, it is no longer widely believed that this optimistic version of the conjecture is true, due to some additional irregularities in the primes coming from the basic fact that large primes cannot be divisible by very small primes. Using the Maier matrix method to capture some of this irregularity, Granville was led to the conjecture that
(note that is slightly larger than ). For comparison, the known upper bounds on are quite weak; unconditionally one has by the work of Baker, Harman, and Pintz, and even on the Riemann hypothesis one only gets down to , as shown by Cramér (a slight improvement is also possible if one additionally assumes the pair correlation conjecture; see this article of Heath-Brown and the references therein).
This conjecture remains out of reach of current methods. In 1931, Westzynthius managed to improve the bound (2) slightly to
which Erdös in 1935 improved to
and Rankin in 1938 improved slightly further to
with . Remarkably, this rather strange bound then proved extremely difficult to advance further on; until recently, the only improvements were to the constant , which was raised to in 1963 by Schönhage, to in 1963 by Rankin, to by Maier and Pomerance, and finally to in 1997 by Pintz.
Erdös listed the problem of making arbitrarily large one of his favourite open problems, even offering ("somewhat rashly", in his words) a cash prize for the solution. Our main result answers this question in the affirmative:
Theorem 1 The bound (3) holds for arbitrarily large .
In principle, we thus have a bound of the form
for some that grows to infinity. Unfortunately, due to various sources of ineffectivity in our methods, we cannot provide any explicit rate of growth on at all.
We decided to announce this result the old-fashioned way, as part of a research lecture; more precisely, Ben Green announced the result in his ICM lecture this Tuesday. (The ICM staff have very efficiently put up video of his talks (and most of the other plenary and prize talks) online; Ben's talk is here, with the announcement beginning at about 0:48. Note a slight typo in his slides, in that the exponent of in the denominator is instead of .) Ben's lecture slides may be found here.
By coincidence, an independent proof of this theorem has also been obtained very recently by James Maynard.
I discuss our proof method below the fold.
I've just uploaded to the arXiv the D.H.J. Polymath paper "Variants of the Selberg sieve, and bounded intervals containing many primes", which is the second paper to be produced from the Polymath8 project (the first one being discussed here). We'll refer to this latter paper here as the Polymath8b paper, and the former as the Polymath8a paper. As with Polymath8a, the Polymath8b paper is concerned with the smallest asymptotic prime gap
where denotes the prime, as well as the more general quantities
In the breakthrough paper of Goldston, Pintz, and Yildirim, the bound was obtained under the strong hypothesis of the Elliott-Halberstam conjecture. An unconditional bound on , however, remained elusive until the celebrated work of Zhang last year, who showed that
The Polymath8a paper then improved this to . After that, Maynard introduced a new multidimensional Selberg sieve argument that gave the substantial improvement
unconditionally, and on the Elliott-Halberstam conjecture; furthermore, bounds on for higher were obtained for the first time, and specifically that for all , with the improvements and on the Elliott-Halberstam conjecture. (I had independently discovered the multidimensional sieve idea, although I did not obtain Maynard's specific numerical results, and my asymptotic bounds were a bit weaker.)
In Polymath8b, we obtain some further improvements. Unconditionally, we have and , together with some explicit bounds on ; on the Elliott-Halberstam conjecture we have and some numerical improvements to the bounds; and assuming the generalised Elliott-Halberstam conjecture we have the bound , which is best possible from sieve-theoretic methods thanks to the parity problem obstruction.
There were a variety of methods used to establish these results. Maynard's paper obtained a criterion for bounding which reduced to finding a good solution to a certain multidimensional variational problem. When the dimension parameter was relatively small (e.g. ), we were able to obtain good numerical solutions both by continuing the method of Maynard (using a basis of symmetric polynomials), or by using a Krylov iteration scheme. For large , we refined the asymptotics and obtained near-optimal solutions of the variational problem. For the bounds, we extended the reach of the multidimensional Selberg sieve (particularly under the assumption of the generalised Elliott-Halberstam conjecture) by allowing the function in the multidimensional variational problem to extend to a larger region of space than was previously admissible, albeit with some tricky new constraints on (and penalties in the variational problem). This required some unusual sieve-theoretic manipulations, notably an "epsilon trick", ultimately relying on the elementary inequality , that allowed one to get non-trivial lower bounds for sums such as even if the sum had no non-trivial estimates available; and a way to estimate divisor sums such as even if was permitted to be comparable to or even exceed , by using the fundamental theorem of arithmetic to factorise (after restricting to the case when is almost prime). I hope that these sieve-theoretic tricks will be useful in future work in the subject.
With this paper, the Polymath8 project is almost complete; there is still a little bit of scope to push our methods further and get some modest improvement for instance to the bound, but this would require a substantial amount of effort, and it is probably best to instead wait for some new breakthrough in the subject to come along. One final task we are performing is to write up a retrospective article on both the 8a and 8b experiences, an incomplete writeup of which can be found here. If anyone wishes to contribute some commentary on these projects (whether you were an active contributor, an occasional contributor, or a silent "lurker" in the online discussion), please feel free to do so in the comments to this post.
There are multiple purposes to this blog post.
The first purpose is to announce the uploading of the paper "New equidistribution estimates of Zhang type, and bounded gaps between primes" by D.H.J. Polymath, which is the main output of the Polymath8a project on bounded gaps between primes, to the arXiv, and to describe the main results of this paper below the fold.
The second purpose is to roll over the previous thread on all remaining Polymath8a-related matters (e.g. updates on the submission status of the paper) to a fresh thread. (Discussion of the ongoing Polymath8b project is however being kept on a separate thread, to try to reduce confusion.)
The final purpose of this post is to coordinate the writing of a retrospective article on the Polymath8 experience, which has been solicited for the Newsletter of the European Mathematical Society. I suppose that this could encompass both the Polymath8a and Polymath8b projects, even though the second one is still ongoing (but I think we will soon be entering the endgame there). I think there would be two main purposes of such a retrospective article. The first one would be to tell a story about the process of conducting mathematical research, rather than just describe the outcome of such research; this is an important aspect of the subject which is given almost no attention in most mathematical writing, and it would be good to be able to capture some sense of this process while memories are still relatively fresh. The other would be to draw some tentative conclusions with regards to what the strengths and weaknesses of a Polymath project are, and how appropriate such a format would be for other mathematical problems than bounded gaps between primes. In my opinion, the bounded gaps problem had some fairly unique features that made it particularly amenable to a Polymath project, such as (a) a high level of interest amongst the mathematical community in the problem; (b) a very focused objective ("improve !"), which naturally provided an obvious metric to measure progress; (c) the modular nature of the project, which allowed for people to focus on one aspect of the problem only, and still make contributions to the final goal; and (d) a very reasonable level of ambition (for instance, we did not attempt to prove the twin prime conjecture, which in my opinion would make a terrible Polymath project at our current level of mathematical technology). This is not an exhaustive list of helpful features of the problem; I would welcome other diagnoses of the project by other participants.
With these two objectives in mind, I propose a format for the retrospective article consisting of a brief introduction to the polymath concept in general and the polymath8 project in particular, followed by a collection of essentially independent contributions by different participants on their own experiences and thoughts. Finally we could have a conclusion section in which we make some general remarks on the polymath project (such as the remarks above). I've started a dropbox subfolder for this article (currently in a very skeletal outline form only), and will begin writing a section on my own experiences; other participants are of course encouraged to add their own sections (it is probably best to create separate files for these, and then input them into the main file retrospective.tex, to reduce edit conflicts. If there are participants who wish to contribute but do not currently have access to the Dropbox folder, please email me and I will try to have you added (or else you can supply your thoughts by email, or in the comments to this post; we may have a section for shorter miscellaneous comments from more casual participants, for people who don't wish to write a lengthy essay on the subject).
As for deadlines, the EMS Newsletter would like a submitted article by mid-April in order to make the June issue, but in the worst case, it will just be held over until the issue after that.
I've just uploaded to the arXiv the paper "Finite time blowup for an averaged three-dimensional Navier-Stokes equation", submitted to J. Amer. Math. Soc.. The main purpose of this paper is to formalise the "supercriticality barrier" for the global regularity problem for the Navier-Stokes equation, which roughly speaking asserts that it is not possible to establish global regularity by any "abstract" approach which only uses upper bound function space estimates on the nonlinear part of the equation, combined with the energy identity. This is done by constructing a modification of the Navier-Stokes equations with a nonlinearity that obeys essentially all of the function space estimates that the true Navier-Stokes nonlinearity does, and which also obeys the energy identity, but for which one can construct solutions that blow up in finite time. Results of this type had been previously established by Montgomery-Smith, Gallagher-Paicu, and Li-Sinai for variants of the Navier-Stokes equation without the energy identity, and by Katz-Pavlovic and by Cheskidov for dyadic analogues of the Navier-Stokes equations in five and higher dimensions that obeyed the energy identity (see also the work of Plechac and Sverak and of Hou and Lei that also suggest blowup for other Navier-Stokes type models obeying the energy identity in five and higher dimensions), but to my knowledge this is the first blowup result for a Navier-Stokes type equation in three dimensions that also obeys the energy identity. Intriguingly, the method of proof in fact hints at a possible route to establishing blowup for the true Navier-Stokes equations, which I am now increasingly inclined to believe is the case (albeit for a very small set of initial data).
To state the results more precisely, recall that the Navier-Stokes equations can be written in the form
for a divergence-free velocity field and a pressure field , where is the viscosity, which we will normalise to be one. We will work in the non-periodic setting, so the spatial domain is , and for sake of exposition I will not discuss matters of regularity or decay of the solution (but we will always be working with strong notions of solution here rather than weak ones). Applying the Leray projection to divergence-free vector fields to this equation, we can eliminate the pressure, and obtain an evolution equation
purely for the velocity field, where is a certain bilinear operator on divergence-free vector fields (specifically, . The global regularity problem for Navier-Stokes is then equivalent to the global regularity problem for the evolution equation (1).
purely for the velocity field, where is a certain bilinear operator on divergence-free vector fields (specifically, . The global regularity problem for Navier-Stokes is then equivalent to the global regularity problem for the evolution equation (1).
An important feature of the bilinear operator appearing in (1) is the cancellation law
(using the inner product on divergence-free vector fields), which leads in particular to the fundamental energy identity
This identity (and its consequences) provide essentially the only known a priori bound on solutions to the Navier-Stokes equations from large data and arbitrary times. Unfortunately, as discussed in this previous post, the quantities controlled by the energy identity are supercritical with respect to scaling, which is the fundamental obstacle that has defeated all attempts to solve the global regularity problem for Navier-Stokes without any additional assumptions on the data or solution (e.g. perturbative hypotheses, or a priori control on a critical norm such as the norm).
Our main result is then (slightly informally stated) as follows
Theorem 1 There exists an averaged version of the bilinear operator , of the form
for some probability space , some spatial rotation operators for , and some Fourier multipliers of order , for which one still has the cancellation law
and for which the averaged Navier-Stokes equation
admits solutions that blow up in finite time.
(There are some integrability conditions on the Fourier multipliers required in the above theorem in order for the conclusion to be non-trivial, but I am omitting them here for sake of exposition.)
Because spatial rotations and Fourier multipliers of order are bounded on most function spaces, automatically obeys almost all of the upper bound estimates that does. Thus, this theorem blocks any attempt to prove global regularity for the true Navier-Stokes equations which relies purely on the energy identity and on upper bound estimates for the nonlinearity; one must use some additional structure of the nonlinear operator which is not shared by an averaged version . Such additional structure certainly exists – for instance, the Navier-Stokes equation has a vorticity formulation involving only differential operators rather than pseudodifferential ones, whereas a general equation of the form (2) does not. However, "abstract" approaches to global regularity generally do not exploit such structure, and thus cannot be used to affirmatively answer the Navier-Stokes problem.
It turns out that the particular averaged bilinear operator that we will use will be a finite linear combination of local cascade operators, which take the form
where is a small parameter, are Schwartz vector fields whose Fourier transform is supported on an annulus, and is an -rescaled version of (basically a "wavelet" of wavelength about centred at the origin). Such operators were essentially introduced by Katz and Pavlovic as dyadic models for ; they have the essentially the same scaling property as (except that one can only scale along powers of , rather than over all positive reals), and in fact they can be expressed as an average of in the sense of the above theorem, as can be shown after a somewhat tedious amount of Fourier-analytic symbol manipulations.
If we consider nonlinearities which are a finite linear combination of local cascade operators, then the equation (2) more or less collapses to a system of ODE in certain "wavelet coefficients" of . The precise ODE that shows up depends on what precise combination of local cascade operators one is using. Katz and Pavlovic essentially considered a single cascade operator together with its "adjoint" (needed to preserve the energy identity), and arrived (more or less) at the system of ODE
where are scalar fields for each integer . (Actually, Katz-Pavlovic worked with a technical variant of this particular equation, but the differences are not so important for this current discussion.) Note that the quadratic terms on the RHS carry a higher exponent of than the dissipation term; this reflects the supercritical nature of this evolution (the energy is monotone decreasing in this flow, so the natural size of given the control on the energy is ). There is a slight technical issue with the dissipation if one wishes to embed (3) into an equation of the form (2), but it is minor and I will not discuss it further here.
where are scalar fields for each integer . (Actually, Katz-Pavlovic worked with a technical variant of this particular equation, but the differences are not so important for this current discussion.) Note that the quadratic terms on the RHS carry a higher exponent of than the dissipation term; this reflects the supercritical nature of this evolution (the energy is monotone decreasing in this flow, so the natural size of given the control on the energy is ). There is a slight technical issue with the dissipation if one wishes to embed (3) into an equation of the form (2), but it is minor and I will not discuss it further here.
In principle, if the mode has size comparable to at some time , then energy should flow from to at a rate comparable to , so that by time or so, most of the energy of should have drained into the mode (with hardly any energy dissipated). Since the series is summable, this suggests finite time blowup for this ODE as the energy races ever more quickly to higher and higher modes. Such a scenario was indeed established by Katz and Pavlovic (and refined by Cheskidov) if the dissipation strength was weakened somewhat (the exponent has to be lowered to be less than ). As mentioned above, this is enough to give a version of Theorem 1 in five and higher dimensions.
On the other hand, it was shown a few years ago by Barbato, Morandin, and Romito that (3) in fact admits global smooth solutions (at least in the dyadic case , and assuming non-negative initial data). Roughly speaking, the problem is that as energy is being transferred from to , energy is also simultaneously being transferred from to , and as such the solution races off to higher modes a bit too prematurely, without absorbing all of the energy from lower modes. This weakens the strength of the blowup to the point where the moderately strong dissipation in (3) is enough to kill the high frequency cascade before a true singularity occurs. Because of this, the original Katz-Pavlovic model cannot quite be used to establish Theorem 1 in three dimensions. (Actually, the original Katz-Pavlovic model had some additional dispersive features which allowed for another proof of global smooth solutions, which is an unpublished result of Nazarov.)
To get around this, I had to "engineer" an ODE system with similar features to (3) (namely, a quadratic nonlinearity, a monotone total energy, and the indicated exponents of for both the dissipation term and the quadratic terms), but for which the cascade of energy from scale to scale was not interrupted by the cascade of energy from scale to scale . To do this, I needed to insert a delay in the cascade process (so that after energy was dumped into scale , it would take some time before the energy would start to transfer to scale ), but the process also needed to be abrupt (once the process of energy transfer started, it needed to conclude very quickly, before the delayed transfer for the next scale kicked in). It turned out that one could build a "quadratic circuit" out of some basic "quadratic gates" (analogous to how an electrical circuit could be built out of basic gates such as amplifiers or resistors) that achieved this task, leading to an ODE system essentially of the form
where is a suitable large parameter and is a suitable small parameter (much smaller than ). To visualise the dynamics of such a system, I found it useful to describe this system graphically by a "circuit diagram" that is analogous (but not identical) to the circuit diagrams arising in electrical engineering:
The coupling constants here range widely from being very large to very small; in practice, this makes the and modes absorb very little energy, but exert a sizeable influence on the remaining modes. If a lot of energy is suddenly dumped into , what happens next is roughly as follows: for a moderate period of time, nothing much happens other than a trickle of energy into , which in turn causes a rapid exponential growth of (from a very low base). After this delay, suddenly crosses a certain threshold, at which point it causes and to exchange energy back and forth with extreme speed. The energy from then rapidly drains into , and the process begins again (with a slight loss in energy due to the dissipation). If one plots the total energy as a function of time, it looks schematically like this:
As in the previous heuristic discussion, the time between cascades from one frequency scale to the next decay exponentially, leading to blowup at some finite time . (One could describe the dynamics here as being similar to the famous "lighting the beacons" scene in the Lord of the Rings movies, except that (a) as each beacon gets ignited, the previous one is extinguished, as per the energy identity; (b) the time between beacon lightings decrease exponentially; and (c) there is no soundtrack.)
There is a real (but remote) possibility that this sort of construction can be adapted to the true Navier-Stokes equations. The basic blowup mechanism in the averaged equation is that of a von Neumann machine, or more precisely a construct (built within the laws of the inviscid evolution ) that, after some time delay, manages to suddenly create a replica of itself at a finer scale (and to largely erase its original instantiation in the process). In principle, such a von Neumann machine could also be built out of the laws of the inviscid form of the Navier-Stokes equations (i.e. the Euler equations). In physical terms, one would have to build the machine purely out of an ideal fluid (i.e. an inviscid incompressible fluid). If one could somehow create enough "logic gates" out of ideal fluid, one could presumably build a sort of "fluid computer", at which point the task of building a von Neumann machine appears to reduce to a software engineering exercise rather than a PDE problem (providing that the gates are suitably stable with respect to perturbations, but (as with actual computers) this can presumably be done by converting the analog signals of fluid mechanics into a more error-resistant digital form). The key thing missing in this program (in both senses of the word) to establish blowup for Navier-Stokes is to construct the logic gates within the laws of ideal fluids. (Compare with the situation for cellular automata such as Conway's "Game of Life", in which Turing complete computers, universal constructors, and replicators have all been built within the laws of that game.)


text:
Espero que les guste mucho mi nuevo trabajo las ropas que usan son del estilo victoriano mas esta moda data o puede datar del siglo XVII o XVIII.
I hope you like my new job much the clothes they wear are of Victorian fashion but this data or may date from the seventeenth or eighteenth century.
Here you can see my other CROSSOVER
I hope you like my new job much the clothes they wear are of Victorian fashion but this data or may date from the seventeenth or eighteenth century.
Here you can see my other CROSSOVER
No me imagino a Naruto y a Luffy comportándose como el Profesor Layton
Ichigo si es más "refinado"
Ichigo si es más "refinado"
Wow, they looks so suave!
they all look so handsome, but i'd choose ichigo.
My favourite is Ichigo. ^^ Because my favourite colours are black and red. xD
love this pic so much...
WOW! SO AWESOME! I love the idea!
muahaha that's awesome . ;D Had to say this ^^
Ahaha somehow it suits Ichigo best.
naruto plz
My three heroes
The costumes are superb. I love it. They are definitely closer to the 18th century than Victorian, but they are beautiful. Nicely done~!
One... Awesome!!!
nice job and the hands
Great Shonen Trinity
wowo this s awesomeness XD im not nuts about one-piece but naruto s pretty kul and of course ichigo s HOOOOT gr8 job on this pic
Muy impressivo *O*
Holy Crap on a Cracker!
This is amazing!
Holy Crap on a Cracker!
This is amazing!
te faltaron Natsu y Edward
*fangirls* ...*a lot*
Ichigo looks so handome as all ways~!
I like this cloths
I like this cloths
Naruto,One piece and Bleach....best anime's ive ever seen in my life *.*
Nice work!
That is so wrong, but I have to love it. Darn it.
Me encantan!!! Te quedaron muy bien!!!
Luffy is the best one done
wow they looks really similar to the characters in the anime. Truly amazing job!
Waw Ichigo looks hawt......
I 'll take Both of Naruto and Luffy ( most Luffy ) nice work
And, of course, Ichigo is the hottest ^^
lol it dont suit luffy as much as ichigo
wow cuute
Impossible! But still good.
cool! very creative
I like it
I like it
AWSOME!! my three fave cartoon characters!!!
luffy looks wired to the moon LOL
jajaja geneal!!!...prefiero que solo me atienda naruto jaja
te quedo geneal
te quedo geneal
Luxurious idea Ichigo awesome)))
Wow!~ Very awesome, I'm fall in love!~ <33
But I want hug only Luffy xD
But I want hug only Luffy xD


text:
Re: JetBlue: The Real Deal?
A reader is alarmed that my bag made it onto the flight without me:
if it were me I would contact DOHS (doh !) and FAA at once with the details. Heads should roll.That was a very, very bad thing that happened. That's so elementary it predates 9/11 by more than a decade. I can remember flights I took in Europeback in the 1980's where all baggage was removed from the plane, lined up on the tarmac and all passengers made to deplane and identify their bags. Unclaimed bag = no flight.
I had suspected this was a problem. I've now called TSA's press office for an answer. I'll let y'all know what I find out.
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