text:
- All Categories
- Architects (881)
- Builders & Contractors (925)
- Cleaning (85)
- Consultants (444)
- Electricians (354)
- Energy Auditors (2432)
- Engineers (98)
- Finance (19)
- Flooring (62)
- Green Program Services (288)
- Interior Designers (205)
- Kitchen (50)
- Landscaping (118)
- Painting (32)
- Pest Control (18)
- Plastering (11)
- Plumbing (159)
- Real Estate (10535)
- Roofing (171)
- Solar & Alt Energy (184)
- Tile & Stone (22)
- Windows & Doors (28)
Green Green Program Services
HBA of Middle Tennessee
Brentwood TN – Service area: 100 miles
Founded in 1941, the HBAMT is an affiliate of the National Association of Home Builders (NAHB). A voluntary non-profit trade association, HBAMT is dedicated to promoting the American dream of homeownership in Middle Tennessee through affordable, quality housing.
HBA of Raleigh-Wake County Green Home Builders of the Triangle
Raleigh NC – Service area: 100 miles
The Home Builders Association of Raleigh-Wake County began in 1959 when 25 builders saw a need to organize in order to better represent the interest of the home builder on issues that affect our ability to provide affordable housing for all citizens of Wake County. Since then, the HBA has grown to over 3,200 members and is the second largest and one of the most active local home builders associations in the country. Our membership represents approximately 90 percent of all housing starts in Wake
HBA of Southwest Colorado
Durango CO – Service area: 100 miles
The Home Builders Association of Southwest Colorado is a non-profit professional association of over 150 member companies, all local businesses, whose diverse services include building homes, remodeling, architecture and design, real estate sales, banking and mortgage refinancing, insurance, subcontracting and material suppliers, etc. We are affiliated with the Colorado Association of Home Builders that has more than 3200 member companies and the National Association of Home Builders that has 216,000
Hd Supply Facilities Maintenance
San Diego CA – Service area: 200 miles
We provide a wide variety of green resources to help save you money, save resource, and ultimately save the earth. We offer over 2,800 environmentally preferable products and practices that will help you improve efficiencies to save money and resources. We focus on five main conservation categories:
1. Energy Efficiency
2. Water Conservation
3. Indoor Environmental Quality
4. Waste Reduction
5. Hazardous Material Disposal
Hickory-Catawba Valley HBA Catawba Valley Green Building Council
Hickory NC – Service area: 100 miles
The Hickory & Catawba Valley Home Builders Association is a non-profit, volunteer organization comprised of builders, developers, trade contractors, suppliers, service providers, and other businesses related to the building industry.
High Country Home Builders Association High Country Home Builders
Boone NC – Service area: 100 miles
Our purpose is to promote Green Building through education, training and special events. This website is an interactive tool to learn more about Green Building and the NAHB Green Building Program. You can also find out about upcoming events, local Certified Green Professionals and educational resources. In addition, there is a space to ask questions, share information and discuss green issues.
Hill Country Home Builders Association
Kerrville TX – Service area: 100 miles
We are an organization that teaches and promotes professional building and business standards for Builders, Developers, and Associates who are members of our Association, the Texas Association of Builders (TAB) and the National Association of Home Builders (NAHB).
Hilton Head Area Home Builders Association
Hilton Head Island SC – Service area: 100 miles
The Home Builders Association of Hilton Head is a local professional association whose membership includes residential & light commercial builders and remodelers from Hilton Head Island and southern Beaufort County.
Home Builder Association of Great Falls Great Falls Green Builid Initiative
Great Falls MT – Service area: 100 miles
The Home Builders Association of Great Falls is a non-profit Montana corporation – a trade association representing builders, developers and related industry professionals in Cascade County. HBA serves you within the building industry by working closely with MBIA and NAHB to provide comprehensive programs, including powerful legislative advocacy, effective business networking, money saving benefits, proven educational programs, up-to-date industry awareness, and valuable publications and resources.
Home Builders & Remodelers Association of Maine Build ME Green
Augusta ME – Service area: 100 miles
Our association is a leading voice for the housing industry in Maine. Members of the Home Builders and Remodelers Association of Maine are professionals who are committed to the promotion of the association. Our members are involved in all aspects of the housing industry including residential, commercial, and industrial construction. Many of our members are builders, remodelers, land developers, and builders of manufactured homes. We also have many associate members whose businesses supply and


text:
Politics, Supreme Court, and the Constitution
Legal scholars spoke about the current make up and operations of the U.S. Supreme Court, as well as its future direction. They also… read more
Legal scholars spoke about the current make up and operations of the U.S. Supreme Court, as well as its future direction. They also talked about the growing influence of conservative legal theory on the federal judiciary and Erwin Chemerinsky's book The Conservative Assault on the Constitution. Professor Eastman is the chairman of the Federalist Society's Federalism and Separation of Powers practice group. The discussion was moderated by Jim Newton, author of Justice for All: Earl Warren and the Nation He Made. They responded to questions from members of the audience.
"Is There a Conservative Assault on the Supreme Court?" was an "ALOUD at Central Library" program presented by the Los Angeles Library Foundation in the Mark Taper Auditorium of the Central Library. close
People in this video
- Kenneth Brecher President Library Foundation of Los Angeles
- Erwin Chemerinsky Founding Dean University of California, Irvine->School of Law
- John C. Eastman Dean (Former) Chapman University->School of Law
- Jim Newton Editor at Large Los Angeles Times->Editorial Page
- Louise Steinman Director Library Foundation of Los Angeles->Cultural Programs
Hosting Organization
Related Video
The Living Constitution Panel
A panel discussion was held about the U.S. Constitution and changes in its wording, interpretation, and implementation…
Future of Public Radio
Participants spoke about the 40-year history of National Public Radio (NPR). Among the topics they addressed were the…
Book Discussion on 1493: Uncovering the New World Columbus Created
Charles Mann, author of 1491, about the Americas before the arrival of Christopher Columbus, revisits the Americas…
Book Discussion on Hollywood Left and Right
Film historian Steven Ross presents a history of politics and the film industry. Mr. Ross examines the impact…

subset: commoncrawl


text:
National Computer Camp (NCC) at Fairfield University for boys and girls to get underway in late June
You might just find the future programmers of the next "Guitar Hero" at National Computer Camp (NCC) at Fairfield University this summer.
Now in its 32nd year, the camp, directed by Michael Zabinski, Ph.D., professor of engineering and physics at Fairfield, teaches kids to design web pages, take apart computers, navigate the Internet, and yes, even create video games. The primary focus of NCC is to instruct students about software applications and programming.
The co-ed camp is geared to children ages 8-18. Campers may attend one or multi-week sessions beginning June 21 and running through August.
Dr. Zabinski founded NCC to prepare young people for the digital age, and to foster interest in computers, engineering and science in an engaging, fun and interactive way. The age-appropriate curriculum is suitable for beginners as well as advanced learners.
Courses taught fall into the following seven categories: 2D and 3D video game design, digital video production, web page design, computer languages, software applications, preparation for Network + Certification, and preparation for A+ Hardware Certification.
Campers typically take one course for one week or longer, except for software applications. Residential and day programs are available. An optional sports program is also available. Tuition fees have been held to 2008 levels. For further information, call (203) 710-5771, e-mail firstname.lastname@example.com or visit www.NCCamp.com.
Media Contact: Meg McCaffrey, (203) 254-4000, ext. 2726, email@example.com
Posted on June 9, 2009
Vol. 41, No. 352


text:
Situated 18 minutes from downtown Sydney on the city's famous Northern Beaches, Novotel Sydney Manly Pacific is an ideal setting for picturesque meetings and events.
With a choice of 213 beachside rooms and suites, nine dynamic function rooms and space for up to 550 guests, our property is the premier Sydney events venue on Manly Beach. By taking advantage of breathtaking ocean views and contemporary design elements, Novotel Manly provides a memorable atmosphere for intimate board meetings, Sydney conferences, exhibitions, corporate events, social events and more.
We also offer an intimate space, Beachside Dojo – Japanese for "meeting place" – where guests can host parties and private banquets overlooking the sea.
Whether you want to exhibit, engage, entertain or educate, we have an innovative solution to suit your individual event and conference needs here at Manly Beach.
Conference venues and services at Novotel Sydney Manly Pacific include:
Learn more about Sydney meeting venue, Novotel Manly in our 2015 MICE Kit.
View renderings of the versatile function rooms at our Sydney events venue to see what space is right for your next gathering. Our Novotel Manly hotel features nine venues and an elegant ballroom with space for up to 550 guests, and many rooms offer ocean views and outdoor terraces.
Stay in touch and connected to all the news and happenings


text: This is the most horrible place to stay. The man that runs it is supper rude. Check in was supposed to be 11am we didn't get to check in until 2pm. We were told to come back in a hour several times. So after walks an the beach and pier we were ready to get a shower and rest. Well... More


text:
Tidy up and keep even the busiest entryway or hallway organized with this charming coat rack. This triple hook coat rack features a scalloped backplate and unique, spiral cone tips.
Ships in 1 - 2 Business Days
Product Details
- Material: Iron
- Design: Transitional
- Installation Type: Wall
- Length: 12"
- Width: 12"
- Height: 3-1/2"
- Depth: 2-1/2"
- Mounting Hardware Included: Yes
- Mounting Hardware Concealed: No
- Mounts From: Front
- Made of hand-forged iron.
- Rust finish is oxidized iron. May transfer onto some fabrics.
Write Your Own Review


text:
AdBlock blows! Please disable the add-on to ensure you get the most out of Tube8's features and functionality. Read about why you shouldn't use AdBlock. Close
Add to Favorite List:
Create a new folder:
Next »
Threesome Sex
Recognize a pornstar in this video?
Help make pornstars easier to find on Tube8 by telling us who is in this video.
Thanks! The pornstar names will be added to this video when enough users submit the same names.
You must enter a pornstar name that already exists in our database.
- Category: Hardcore
- Views: 41,838
- Added: 4 years ago
- Description: Three sexy jocks fucking hard.
- Tags: gay, jocks, studs, young, orgy, group, trio, sex
- Pornstars: Add Pornstar
- Favorite by: 39
- Featured: 4 years ago
- Report This Video
Video Comments
All Video Comments (0)


text:
Chevron's long record of developing crude oil and natural gas in the Middle East dates back to our earliest discovery, in Bahrain in 1932. Kuwait's Burgan Field, the world's second-largest onshore oil field, was discovered in 1938 by Gulf Oil, which later merged with Chevron. Following nationalization of this field by the government of Kuwait, Chevron subsidiaries continued to provide assistance to the Kuwait Oil Company via technical-service agreements through 2008.
The Chevron-led consortium competing to develop Kuwait's northern fields ended in May 2010. However, Chevron has an extensive history in the region and continues to be active in Kuwait, and we are one of two partners in the onshore Partitioned Zone (PZ) between Kuwait and Saudi Arabia.
Companies that later became part of Chevron have been producing oil on behalf of Saudi Arabia in the onshore PZ since 1949. Chevron has an agreement with Saudi Arabia to operate the kingdom's 50 percent interest in the hydrocarbon resources of the onshore PZ. The agreement was extended and amended in 2009; it now expires in 2039.
Development drilling, well workovers and facility enhancement programs are scheduled to continue through 2015 and are expected to partially offset field declines.
In 2009, steam injection began at the Large-Scale Steamflood Pilot (LSP) Project for the carbonate First Eocene reservoir at the Wafra Field. Steamflooding involves injecting steam into heavy oil reservoirs to heat the crude oil underground, which reduces its viscosity and allows its extraction through wells. This project was preceded by steam stimulation of some wells, followed several years later by a small-scale test. The entire development project is designed to determine the technical and economic viability of thermal-recovery projects in the Eocene reservoirs of the Wafra Field. In 2014, Chevron continued the steam injection pilot project in the target reservoir to better assess total recoverable volumes.
Development planning also continued on the follow-up to the LSP Project. The Wafra Steamflood Stage 1 Project, a full-field steamflood application in the Wafra Field First Eocene carbonate reservoir, has a planned design capacity of 80,000 barrels of crude oil per day. Front-end engineering and design work is expected to begin in the third quarter of 2015.
Work also continued on expanding the LSP Project to the Second Eocene carbonate reservoir. A final investment decision is planned for 2016.
A carbonate reservoir is an oil or gas trap formed in reefs, dolomite and certain types of limestone. Typically, carbonate reservoirs are highly fractured and not conducive to steamflooding on a large scale. However, the carbonate Eocene reservoirs at Wafra have properties that are promising for successful steamflooding.
The Central Gas Utilization Project is a facility construction project intended to increase the use of natural gas while eliminating natural gas flaring at the Wafra Field. As of early 2015, the development plan was being re-evaluated.
In July 2014, we began a 3-D seismic survey covering the entire onshore Partitioned Zone. It is one of the largest land seismic programs ever undertaken, covering 1.1 million acres (4,600 sq km).
Chevron licenses refining technology and supplies catalysts to Kuwait National Petroleum Co. refineries through our affiliates Chevron Lummus Global LLC and Advanced Refining Technologies LLC, respectively. Advance Refining Technologies has a minority interest in the Kuwait Catalyst Company.
Saudi Arabian Chevron (SAC) operates a storage and marine export facility at Mina Saud in the PZ.
In 2014, we marked our 12th consecutive year with no reductions or delays in scheduled tanker loading, no lost-time incidents and no spills attributable to SAC operations. A total of 60 tankers were loaded at the company's crude oil shipping terminals, handling a volume of nearly 35 million barrels of Ratawi and Eocene crude.
Chevron markets Caltex®-branded lubricants and coolants in Kuwait through authorized distributors.
Updated: May 2015
Cautionary Statement
Search open positions in Kuwait.
Go to Chevron Careers
The Chevron Way explains who we are, what we do, what we believe and what we plan to accomplish.
Kuwait (156 KB)
Chevron conducts operations in Saudi Arabia
Learn More
Our glossary of energy and financial terms
English (132 KB)
© 2001 - 2016 Chevron Corporation. All Rights Reserved. Terms of Use | Privacy Statement | Site Map
| Switch to Mobile
Connect with us:


text:
SpamAssassin 3.3.0 Splits Off Rules
With version 3.3.0, the free spam filter SpamAssassin has its first big release since May, 2007.
The new SpamAssassin version separates rules as a core component. To update rules, a new sa-update command is required. The behavior was already an option in version 3.2.x, but has now become the standard thanks to some positive feedback.
The new release also improves the plugin for DomainKeys Identified Mail (DKIM) that now supports Author Domain Signing Practices (ADSP) and many other signatures. The DKIM plugin is activated with all new installations, while the old DomainKeys plugin was retired.
Developers also improved IPv6 support. The new time_limit configuration option, defaulted at 300 seconds, attempts to gracefully terminate the checking after the given time limit and returns the results so far. Further details are in the comprehensive release notes. Special notes are available for users wishing to upgrade.
The SpamAssassin project, under the Apache Software Foundation umbrella for over six years, provides version 3.3.0 of its software in source code downloads in various binary formats, under Apache License 2.0.
Issue 184/2016
Buy this issue as a PDF
Linux Foundation Eliminates Individual Membership
But you can still be a non-voting "individual supporter" if you pay the money
Secret Backdoor Affects More Fortinet Firewalls
Several current systems could fall victim to the attack
Linus Announces Linux Kernel 4.4
Latest Linux engine comes with better graphics and support for Intel's new power-saving chips.
Mass Poem Greets Web Admins
Hackers send a message of beauty and liberation to server logs
Raspberry Pi Becomes a Thin Client
Citrix gets excited about new Pi-Powered XenDesktop client system
Microsoft Offers Linux Certification
Linux on Azure cert heralds a new era for Redmond.
CeBIT 2016: Call for Open Source Contributions
Proposals for presentations at the CeBIT Open Source Forum will be accepted through 24 January 2016.
Gone in a Flash
Adobe looks for a new start; renames its embattled Flash tool.
Rasp Pi Generates Weak SSH Keys
The Pi's popular Raspbian OS pursues secrecy without entropy.
Photon Controller Goes Open Source
VMware bids for a stake in the container industry with a bold effort to integrate containers with its classic virtualization system.


text:
John Galt Feb 5, 2013 7:43 PM
in response to ryanc522
First, make sure you have adequate backups. If your hard disk is failing this could be a sign. If it is Apple can replace it, but you must back up your data before approaching the Genius Bar.
Read: OS X Lion: If your Mac runs slowly
It is very basic, but be sure to read all the way to the end including the reference to run Disk Utility. The idea is sound, unfortunately Apple forgot to include any information in the linked document. Here is a useful one: Repair a disk using Disk Utility. To do this you must start your Mac from the Recovery System or Internet Recovery.
It is likely you have installed a few programs, some of which may not be fully compatible with Lion. If this is possible and you are interested in determining what you may have installed that could be contributing to this behaviour, read on:
Apple Support Communities contributor etresoft wrote a very useful app to quickly gather certain system information that may help point to a cause of this problem. Go to his website, download and run EtreCheck:
Etrecheck will take a moment to run as it collects its data.
Copy and paste its output in a reply. Do not be concerned about anything that says "Problem" or "failed".
EtreCheck was designed to remove any personal information (such as your computer's name and serial numbers) but if you see anything that looks like an email address or any other personal information that should not be divulged to others, please delete or obscure that information when you post the reply.
When you are finished with EtreCheck, quit the program. It occupies very little space, and you can keep it or drag it to the Trash as you wish.


text:
My Family
by Kristy Gordon
My family - Read/Gordon.MacLeod
love writing, love photography, love designing pages...this is more fun than scrapbooking! lol
Comment posted.
Sign in
or create an account to post a comment.
Published April 19, 2012
Kelsi & Andrew
Published November 28, 2011
LCVI in NY
Published June 22, 2011
Blurb Sites


text:
The best of what's newly streaming on Netflix includes a 'Beautiful' find
Great movies have the power to record emotions and send them back out into the audience, intact. Sometimes these can be the pondering of life or death, as in several of this week's movies. Sometimes they can be hope or guilt, the emotions that drive a certain "killer" toward a certain goal. Other times they can feel like nothing, like a lack of connection. Still other times they can make a monster seem human.
Other times, they can just make you laugh or give you a little thrill. Regardless of their intentions, all of this week's movies should make you feel at something.
Fruitvale Station (Netflix)Read more »


text:
William J. Larkin, Jr.
(R, C) 39th Senate District
George D. Maziarz
(R) 0 Senate District
Thomas F. O'Mara
(R, C, IP) 58th Senate District
Stephen M. Saland
(R) 0 Senate District
Catharine Young
(R, C, IP) 57th Senate District
Get updates on important news and legislation related to S3191A.
(R, C, IP) 61st Senate District
(R, IP) 0 Senate District
(R, C, IP) 50th Senate District
(R) 0 Senate District
(R, C, IP) 0 Senate District
Creates a small business electric energy tax credit; the term "business related electricity usage" shall refer to electrical power usage to further the economic activity of the taxpayer at the primary business location that is clearly delimited from any shared electrical power usage cost.
Open Legislation comments facilitate discussion of New York State legislation. All comments are subject to moderation. Comments deemed off-topic, commercial, campaign-related, self-promotional; or that contain profanity or hate speech; or that link to sites outside of the nysenate.gov domain are not permitted, and will not be published. Comment moderation is generally performed Monday through Friday.
By contributing or voting you agree to the Terms of Participation and verify you are over 13.


text:
4′ x 7′ Backdrop
3′ x 4′ Red Carpet
5′ x 8′ Backdrop
3′ x 5′ Red Carpet
7′ x 8′ Backdrop
3′ x 7′ Red Carpet
10′ x 8′ Backdrop
4′ x 10′ Red Carpet
Thank you for delivering so fast!!!!! The red carpet was a HIT!!!! Meagan McGough – The McGough Academy of Irish Dance
"I wanted to let you know that our press wall was a big hit and our event came off great! Thank you so much you helped make my life so much easier." -Tracy C.
"Thanks! I'm so glad you have the custom orange carpet! This will blow my client's mind when they receive it!" -Donna Cutting,www.donnacutting.com
...just wanted you to know we are very pleased with our purchase from you and feel it's a great investment. Thanks for your excellent service. Lucinda, NY -
Thank you so much for the amazing red carpet & backdrop! Robin C - francescasfancyflops.com -
Thanks for the nice back drop for the American Institute for Stuttering. You do nice work and it is always a pleasure working with you. - C. Madden Productions -
thank you for all your hard work this past week to get the Hand In Paw order to us before our event date! I also wanted to let you know the step and repeat was a huge success, everyone loved it! Will definitely be calling on you guys again soon. :-) Brian - Brian Batista Photography -


text:
I wish I could remember if this started when I upgraded to 2.8 or when I upgraded a specific plug in! I'm hoping someone else has had this happen and can help me out.
Since this weekend, I haven't been able to use the media upload, add tags, or set a date to schedule posts on my site in full Edit mode. Preview also won't show changes unless I save, which didn't used to be the case. I am able to add tags and change the date in Quick Edit mode, but the only way to upload pictures is to ftp them and then manually img src. Yuck!
Has anyone else dealt with this? Or do you have any idea what's going on? Posting is such a bummer now!


text:
Results 1 to 7 of 7
Thread: How do you justify...
-09-17-2006 #1raminsyr Guest
How do you justify...
punting with 3:45 left? You just knew we were never going to see the ball again:
4-8-STL41 (3:45) M.Turk punts 44 yards to SF 15, Center-C.Massey, fair catch by A.Battle.
San Francisco ***** at 03:38
1-10-SF15 (3:38) F.Gore right end pushed ob at SF 22 for 7 yards (F.Brown).
2-3-SF22 (3:32) F.Gore up the middle to SF 24 for 2 yards (T.Hargrove).
3-1-SF24 (2:47) F.Gore up the middle to SF 25 for 1 yard (D.Coakley).
1-10-SF25 (2:43) F.Gore up the middle to SF 29 for 4 yards (W.Witherspoon).
2-6-SF29 (2:00) F.Gore up the middle to SF 39 for 10 yards (O.Atogwe).
1-10-SF39 (1:54) F.Gore up the middle to SF 43 for 4 yards (D.Coakley).
2-6-SF43 (1:10) A.Smith kneels to SF 42 for -1 yards.
3-7-SF42 (:29) A.Smith kneels to SF 41 for -1 yards.
-09-17-2006 #2
Re: How do you justify...
Believe me I feel the pain !!!
Where was the Defensive aggressive play calling ??
-09-17-2006 #3
Re: How do you justify...
You justify it because 4th & 8 isn't easy to convert, because if it was easy, then we likely would have done it when it was 2nd & 8 and then 3rd & 8. They were counting on the D to make a stop and they didn't. Can't blame them because it's a rough position to be in.
-09-17-2006 #4
Re: How do you justify...
But you have to do something aggressive, when the whole team was playing passive. They were playing to no loose, and no playing to win.
-09-17-2006 #5
Re: How do you justify...
-09-17-2006 #6
Re: How do you justify...
I didn't think we would get the ball back either. We hadn't stopped their running attack all day. Perhaps it would've been a low percentage play,but at least we would of had one last chance to stay in the game.
-09-17-2006 #7
Re: How do you justify...
There was nothing wrong with showing a little confidence in the Defense, after all, they won last weeks' game.
3:45, two timeouts, and the two-minute warning is plenty of time to punt and get the ball back. We just didn't stop them.
Conversely, the offense has done nothing to make me think that they would've converted that 4th down play.


text:
- Year
- 2013 / 2012 / 2011 / 2010 / 2009 / 2008 / 2007 / 2006 / 2005 / 2004 / 2003 / 2002 / 2001 / >2000
- Month
- Jan / Feb / Mar / Apr / May / Jun / Jul / Aug / Sep / Oct / Nov / Dec
- Day
- 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 / 11 / 12 / 13 / 14 / 15 / 16 / 17 / 18 / 19 / 20 / 21 / 22 / 23 / 24 / 25 / 26 / 27 / 28 / 29 / 30 / 31
Articles published on April 23, 2009
2 American Idol finalists rolled off competition
By DERRIK J. LANG AP Entertainment Writer
Archbishop will not attend Xavier commencement, citing pro-abortion speaker
Michael Luke WWL-TV Eyewitness News
After four decades, Jazz Fest grows but honors roots
STACEY PLAISANCE Associated Press Writer
An expensive bailout in Louisiana
Board opposes opposes guns on La. college campuses
The Associated Press
Billups leads Nuggets to 2-0 lead over Hornets
Arnie Stapleton The Associated Press
Barrios residents oppose new subdivision
Naomi King Staff Writer
Baton Rouge man bicycling the length of La. 1
Lloyd J. Nelson III Staff Writer
Charlie Wi leads N.O. Zurich Classic pro golf tournament
MARY FOSTER AP Sports Writer
Car burglaries spike in neighborhoods
Matthew Pleasant Staff Writer
Damage to bridge in Galliano worse than expected
GM to temporarily shut 13 plants, cut production
TOM KRISHER AP Auto Writer
Gulf dead zone needs action now
Gulf Island's first-quarter earnings decline
The Associated Press
HLB's Jones signs with junior-college
Kelly McElroy Staff Writer
Heartbreak follows husband's bigamy charge
Raymond Legendre Staff Writer
Hornets drop Game 2, trail series 0-2
Is a greyhound right for you?
By LIZ BALMASEDA Cox Newspapers
In gym class, the waistline's the front line
By ALAN WECHSLER Albany Times Union
John Curtis boys, Isidore Newman girls take District 10-2A crown
Perry Pitre Sports Correspondent
Leno hospitalized with illness, cancels "tonight" show taping
LYNN ELBER AP Television Writer
LPB to air Jindal's address Monday to Legislature
The Associated Press
Local service providers fighting budget cuts
Daniel McBride
Lafourche pilot dies in plane crash
Nikki Buskey Staff Writer
LSU beats UL-Lafayette at Classic
From Staff Reports
Local players help Delgado to win
From Staff Reports
Labadieville crash victims identified
Raymond Legendre Staff Writer
Landrieu's hard choices
John Maginnis Louisiana Politics
Lafourche pilot dies in plane crash in Des Allemands
Lamar Bourgeois WWL-TV Eyewitness News
Mock car crash gives students a lesson on perils of drunken driving
Daniel McBride
Nicholls no-hits LSU-Alexandria
From Staff Reports
New Dan Brown novel coming in September
By HILLEL ITALIE AP National Writer
No big winners in Wednesday's Lotto, Powerball drawings
The Associated Press
Obama pledges protections for credit-card users
BEN FELLER Associated Press Writer
Our war in Afghanistan
John Whitehead Guest Columnist
Possible steel plant in St. James slowed by recession
ALAN SAYRE AP Business Writer
Police Report: April 16 - April 22
Pearl Rita Loretta Vinet Dauzat
Protests were the work of residents
Residents complain ditches cleaned poorly after storms
Naomi King
Southdown to hold annual jazz brunch
Secret tally has 87,215 Iraqis dead since 2005
KIM GAMEL Associated Press Writer
Second chance helps Webb fulfill NFL dream
Matt LeBlanc Staff Writer
Teen charged with trailer park murder
Robert Zullo Senior Staff Writer
Terrebonne Lady Tigers, South Lafourche boys win at 8-5A meet
Brent St. Germain Sports Editor
Two killed in Assumption wreck
Raymond Legendre Staff Writer
View full investigation of Houma Police
Wigged man robs convenience store
Matthew Pleasant Staff Writer
Wind still making area fishing tough
Chris Singleton Staff Writer
Zephyrs win second-straight game
From Staff Reports
Images of the DayTrumpet player Lionel Frebos plays at the French Quarter Festival in New Orleans, Friday, April 17, 2009. Ferbos, 97, one of the city's oldest actively working musicians, has never missed a Jazz Fest. The New Orleans Jazz and Heritage Festival starts Friday, April 24.
After two All-American seasons at Nicholls State, Lardarius Webb is expected to be picked anywhere from the third to the fifth round in this weekends NFL draft.
Direct support worker Moketta Daggs helps Terrebonne Association for Retarded Citizens client Charles Chucky LeBlanc Jr. this morning inside the kitchen at the TARC restaurant on Grand Caillou Road in Houma.


text:
Well, I'll probably get fired for this (Just kidding! Have a look @ Life At Microsoft - The Truth Revealed to see how working for MS REALLY looks like...) but I wanted to make you aware of some cool Photoshop tutorials. We might face the fact that paint isn't the first choice of serious photographers ;-)
PingBack from http://msdnrss.thecoderblogs.com/2008/03/06/cool-photoshop-tutorials-available-online-for-free-2/
I think there was a site that was doing paint.net tutorials that try to mimic the functionality of photoshop, obviously a free app can't do everything but.. I'll have to try to find the url again!
Hey Tony,
That would be awesome! Paint.NET comes in really handy when one wants to do some quick work without starting a full blown image processing software like PS.
Cool video, I laughed a lot! ;)


text:
Brad Lidge is thisclose to closing again.
There was no official switch by the Houston Astros on Tuesday, but the statistics are telling. The one-time Cherry Creek prep star lost his ninth-inning role two weeks into the season, only to rediscover his dominance April 23 at Philadelphia.
Since that outing - with runners on second and third, he struck out the side - Lidge has posted an 0.90 ERA with 27 strikeouts in 20 innings. He pitched the ninth last Saturday in a non-save situation, so it would surprise no one if he was out there again even without an announcement.
"I definitely want to close," said Lidge, who posted 104 saves for the Astros over the previous three seasons.
Blog: All Things Rockies
Troy Renck posts analysis, notes and minutiae on this blog devoted to the Colorado Rockies.
Lidge, who pitched the 8th inning on Tuesday, chooses his words carefully on the subject because he's a close friend of the man who replaced him, Dan Wheeler. Wheeler also would prefer to close, though he has worked as a solid bridge to Lidge in the past.
That Lidge is in position to reclaim his job is a function of improved mechanics and pitch selection. He is no longer flying open with his left shoulder during his delivery.
More important, he junked a cutter and is relying heavily on his four-seam fastball and a slider that ranks among baseball's most wicked.
"I have gone back to my bread and butter," he said. "May was probably the best month of my career. Success breeds confidence and I feel like everything is falling back into place right now."
Finley out, Barker in
In their attempt to create a veteran bench, the Rockies got old players instead of players of old. Two weeks after cutting ties with John Mabry, Colorado designated outfielder Steve Finley for assignment, clearing a roster spot for Triple-A slugger Sean Barker.
Finley hit just .181 in 94 at-bats, and became expendable with the emergence of Ryan Spilborghs. Barker went from watching the movie "Knocked Up" to getting called up. The call to The Show interrupted the show. And no, he didn't stick around for the ending.
"I have always taken pride in what I do. I am glad to get this opportunity," said Barker, who hit .333 with five home runs with the Sky Sox and has good speed.
Josh Fogg, after showing no problems running on his strained left groin, will start Thursday, pushing Taylor Buchholz back into the bullpen. Said Fogg, "It's been fun to watch this streak, but I feel bad because I haven't been able to help out." ... The Rockies will make a roster move Thursday to create a space for Fogg, with a reliever likely to be farmed out. ... Willy Taveras on playing his former team, "I think I did well over there. It wasn't my decision to get traded. The fans were great to me."


text:
Multi-channel Customer Engagement Falls Short
Customers engage with companies in multi-channel and cross-channel journeys, pausing them and resuming them over time. For example, according to the Customer Experience Board (CEB),
- 58% of callers have visited the web before calling, and
- 34% of callers are on the web while talking to a rep
Most companies today support multi-channel customer engagement. They support many channels: voice, email, chat; but they typically cannot support multiple channels simultaneously within a single interaction (multimodality), they cannot link interactions in a step-by-step workflow (orchestration), and they cannot manage the lifecycle of the customer journey (design, orchestrate, monitor, tune) to provide a personalized and context appropriate experience for each and every customer (journey management). For a customer to complete a single task—buy a product, resolve a question, understand a bill—it often requires multiple and disconnected interactions with a company. When a customer needs assisted service to supplement self-service, they typically must start over. In the case of voice, it's calling a contact center, using an IVR, and explaining their issue. In the case of chat, it's starting a dialog with an agent without context. These time-consuming and disconnected "channel shift" experiences are one of the leading causes of missed sales opportunities and high operating expense for companies and frustration for consumers.
Omnichannel Journey Orchestration is the Answer
Omnichannel customer engagement combines multimodality, orchestration, and journey management to provide multi-channel contact center best practices for great customer experience with compelling business outcomes.Leading companies with multichannel call centers are embracing omnichannel customer experience for competitive advantage. With omnichannel journey orchestration from Genesys, an organization has multimodal orchestration and multi-channel journey management capabilities that translate into both a positive customer experience and better business results. This approach transforms self-service channels by enabling an organization to present a customer with information in one channel, while offering assistance from another. With self and assisted service from Genesys, a customer on a mobile device could be presented with a visual IVR menu, chat interaction or mobile video session all within a single interaction.


text:
Tag "htc one specs"
Comparing the HTC One to the HTC One X+
After much hype and considerable fanfare, the HTC One is now an officially recognized smartphone. HTC today announced the Android handset for release in the coming weeks. With a name
What we expect: HTC M7
As one of the most rumored and anticipated devices of the year, the HTC M7 is expected to make its formal debut on February 19 at a joint press event


text:
Kudos to Miguel Ucovich for his well-stated letter regarding "Wineries are not community centers," (Reader Input, Feb. 27).
Currently there are several applications with the county for minor use permits to open event centers.
Because commercial businesses that are not primarily agricultural are not allowed in the rural residential zoning (FBX-10), these applicants have sited a provision allowing for "community centers." Unfortunately, this provision is not clearly defined, and extremely open-ended, allowing applicants to propose year-round public, commercial activities in areas otherwise restricted to residents and agriculture.
The community center provision is being abused to apply for event centers which is clearly inappropriate land use. This abuse allows relatively inexpensive agricultural land to support commercial business endeavors. These proposed facilities promise all the related issues that the zoning was intended to restrict: traffic on rural roads, noise, safety, lighting and little to no enforcement for permit violators.
Currently, wineries commonly violate the terms of the winery ordinance, and the county does not have the time or money for the proper enforcement owed the residents.
The rural residents pay the price for these profit-makers.
The event center itself is not evil. It can provide an influx of business to Placer County, but it needs to be located appropriately, on commercially zoned property, so that other surrounding businesses are more likely to benefit and rural residents do not have to forfeit the terms they originally invested their lives in.
The Board of Supervisors and Placer County residents need to take a long, hard look at this weakly defined provision and ask themselves "what was the original intent of the community center provision in the zoning code?"
It becomes apparent the original intent was to allow meeting places for community members to gather, residents that live acres away from their nearest neighbors, facilities such as grange halls and other not-for-profit facilities, to serve the community.
(Intent was) not to allow for misplaced event centers that compromise the quality of life and the very heart of what makes Placer County special. It's a big county. There is an appropriate place for a healthy diversity of endeavors, (just) find the right place.
Let's all take a high-level look at the county master plan and the language of the zoning, because the "community center" sheep's clothing vanishes to reveal the wolf.
Teresa Chaney, Newcastle


text:
Assembly's planned vote on Pacific High plans catches school district off guard. Reality show featuring Sitka fishery to premiere. Juneau, Ketchikan airports to get full-body scanners. FAA approves PBY landing 2 months after it was planned. Russian tanker headed to Nome with fuel. Police arrest suspected Juneau robber.
Thu Dec 29, 2011
Please read our comment guidelines.
Recent News
Send a Love Letter on the radio!
Is there someone in your life you'd like to appreciate over the air? Maybe it's a secret crush, your spouse of many years, or a friend who has always been there for you. Submit your love note to Raven Radio, and … more
Sitka Health Summit hosts networking event
Lloyd Platson with Sitka Counseling and Martha Pearson from SEARHC Health Promotion talk about the upcoming Sitka Health Summit networking event. Downloadable audio. more


text:
Narrow your search
The next version of the search giant's virtual reality goggles may be a lot more advanced than a folded piece of cardboard -- and not need a phone at all.
Apple was late to the music streaming game, but it's gaining ground faster than rivals.
The annual event tries to quash bullying and cruelty among kids on the Internet by enlisting their peers.
Criminals are increasingly attacking government agencies in hopes of stealing our money and information.
Declassified CIA memos from the 1960s show the United States was concerned about Chinese computing power and that other arms race with the Soviet Union -- the space race.
Technically Incorrect: President Enrique Peña Nieto seems to believe that the best microphone for good news is Twitter.
As Sanders and Clinton battle it out in the presidential debates, they're also competing for comic book fans, as stars of their own titles. Spoiler alert: Neither of them has mutant superpowers.
Technically Incorrect: In a new campaign ad, the leading Republican candidate insists that serious presidents don't watch sci-fi movies at times like this.
Technically Incorrect: Appearing in the popular Web series "Comedians in Cars Getting Coffee," the president and Jerry Seinfeld are almost funnier than Bill Gates and Seinfeld.
The Commander in Chief guest stars on season premiere of Jerry Seinfeld's Web series, "Comedians in Cars Getting Coffee."


text:
It doesn't matter if your house needs repainting or your website needs upgrading, a Work for Hire Agreement makes sure that everyone's happy once the project is finished.
Use the Work for Hire Agreement document if:
- You are hired as an independent contractor to complete a project.
- You are hiring an independent contractor to complete a project.
- You want to ensure the contractor has no rights of ownership over the work or product created under the agreement.
If you're looking to hire an independent contractor to complete work on a temporary basis, this document helps outline your expectations as to how the job will be completed. A Work for Hire Agreement allows you to determine timelines, work schedules, milestones and payment terms so that you worry less and enjoy the finished product more.
If you're hired as an independent contractor, a Work for Hire Agreement puts your obligations to clients in writing so that you're protected in the event of a disagreement. The document ensures you get paid and helps to maintain your professional relationships.
If you have legal needs outside of the Work for Hire Agreement, please check out our complete list of customizable service contracts. If you'd like more information about the hiring process, or employment forms, be sure to visit our Human Resources Guide.
Sample Work for Hire Agreement
More than just a template, our step-by-step interview process makes it easy to create a Work for Hire Agreement.
Save, sign, print, and download your document when you are done.
Sample Work for Hire Agreement
Our step-by-step interview process makes creating a printable Work for Hire Agreement easy.Make Document
Researched by lawyers
and legal staff.
We're here to answer your
Your document is free with your subscription.
We guarantee our service is
safe and secure.


text:
Daniel 6:2New King James Version (NKJV)
2 and over these, three governors, of whom Daniel was one, that the satraps might give account to them, so that the king would suffer no loss.
New King James Version (NKJV)
Scripture taken from the New King James Version®. Copyright © 1982 by Thomas Nelson. Used by permission. All rights reserved.
Already have a redemption code? Redeem it here.
Already redeemed? Refresh the page.
This is a content preview. Buy this title to see more.


text:
Attention Pink Floyd Fans
I'm not sure where they came from, but I stumbled upon a blog where you can download some incredible outtakes from Pink Floyd's Dark Side of the Moon recording sessions. It's also got this video with some hilariously dated interviews.
For that matter, this site has an alternate version of the entire Dark Side of the Moon album available for download.
And there are more demos and outtakes available here for download.
I won't lie: I love the Internet.
You must be signed in to leave a comment. Sign In / Register
Please keep your community civil. All comments must follow the NPR.org Community rules and Terms of Use. NPR reserves the right to use the comments we receive, in whole or in part, and to use the commenter's name and location, in any medium. See also the Terms of Use, Privacy Policy and Community FAQ.


text:
Orlando Jose
Funding via MiCredito
Lorenzo is currently planting beans and corn, most of which will be sold in Boaco. His wife works by selling "cosa de horno" (a bread made from sweet corn, cinnamon, and sugar). He is requesting a loan in order to buy compost, fertilizers, and pesticides to prevent the propagation of pests that could ruin his crops. One of his plans in the future is to buy land for planting vegetables. This will give him more production capacity and improve his earnings so that his family will have a better quality of life. His next goal going forward is to produce quality beans.
Success!! The loan was 100% repaid


text:
- This Hellcat-powered Jeep 6x6 is the coolest thing ever
- Got $40-Grand to Join the A-Team? Buy this A-Team Tribute Van
- Buy Fonzie's motorcycle on eBay, jump a shark with it
- Wild and Wacky Engines from the Grand National Roadster Show
- CAT Diesel Power, International Truck and Studebaker Sedan are Joined in Snapper Schomaker's Bad Buggy
Featured Cars
Auto Shows
Latest News
Microsoft Store Offers
- Microsoft Store | Sponsored: Find Valentine's Day gifts under $50 that they'll use all year long
- Microsoft Store | Sponsored: Warm their heart with the purple HP Stream Signature Edition Laptop
- Microsoft Store | Sponsored: Take your love to new heights with a drone (save $149)
- Microsoft Store | Sponsored: It's National Heart Month. Give the gift of health with a Band


text:
|Quick links:
Latest Team Rankings
Free Rivals Alerts
Member Services
|ShopMobileRadio RSSRivals.com Yahoo! Sports
|College Teams
|High Schools
September 13, 2012
Denard Robinson finds big targets in Gardner, Funchess
Midway through the second quarter of Michigan's 31-25 win over Air Force, senior quarterback Denard Robinson set up in the pocket and watched excitedly as freshman tight end Devin Funchess streaked across the field, slipping past linebackers and safeties until nothing stood between him and the goal line.
And the Robinson threw - it was an easy six points, and the quarterback is hoping many more plays will shape up similarly this year.
"He's a great athlete" Robinson said. "Sometimes, I wish he'd play wide receiver when I see him out there, because he's so fast. But I love him at tight end, because he'll get on a linebacker or safety and those guys usually can't cover as well as cornerbacks. I enjoy watching him going out there and making those guys look bad and run right past him."
The Wolverines' receiving corps was a major concern heading into this year, but the emergences of Funchess, junior wide receiver Devin Gardner and, to a lesser extent, junior wide receiver Jeremy Jackson, have helped Robinson find targets.
"Devin and Funchess going out there and getting the ball - it's always fun throwing the ball up to those guys," Robinson said. "And you have Roy Roundtree, he's one of the guys who plays big, and Jeremy Gallon, who plays big, and Jeremy Jackson is one of the guys who is taller.
"Devin [Gardner] is a phenomenal athlete, and he learns fast. So once he gets out there into space and he knows how to get out of his breaks, he's going to progress and get better every Saturday.
"I enjoy throwing to those guys, because they can go get the ball. That was a big question coming into this year. Everybody was saying, 'Oh, we don't know about the receivers.' But you guys have seen it. Those guys have potential. We just have to keep working and moving forward."
Although Robinson mentioned Roundtree as a gamebreaker, the fifth-year senior has caught just three passes for 17 yards through Michigan's first two games.
After finishing the 2010 season with 72 catches for 935 yards and seven touchdowns, Roundtree has totaled just 22 catches for 372 yards and two scores.
But Robinson thinks it's only a matter of time before Roundtree gets going.
"I missed him on a deep pass [against Air Force]," Robinson said. "That was my fault. I overthrew him, and I made a mistake. That was one of the plays we left on the field. That's a play we have to make happen next time."
"The one thing about Denard that is amazing is if something goes wrong in front of him, he can identify it quickly," Borges said. "He can tell you, he'll get on the phone and tell you right away, he goes, 'Blah blah blah missed his block right in front of me, I just couldn't see the throw. The safety jumped in front, that's why I backed out and did this.' It's amazing how accurate it is."
Robinson said that's one thing he worked on very hard coming into this season.
"Once I get off the field, I go to those headsets and say, 'OK, this is what I saw, and this is what we should probably do,' or something like that," Robinson said. "Sometimes, it's different, because it's year two, and I made a lot of strides in the summer, looking at film and having another year under my belt."
"Norfleet is one of the guys that brings energy every day I see him," Robinsons said. "Every time you see him on the football field, you love his vibe and the way he plays. He reminds me of Vincent Smith, because he's one of those guys that has that energy and he's a tough guy. When he goes out there and runs the ball, you can see he's so quick and he runs through the hole, and he hits them with all he has. I enjoy watching him play.
"He might be a little quicker than me. He's real quick. Every time I see him hit the hole, he's hitting it really quick. He makes a quick juke and it's fun watching him run the ball.
"He's small but he packs a punch. He's quick on his feet, and he's not scared to take on a block."
"Jeremy Clark, I always talk to him," Robinson said. "He's one of the guys that I think will have a very bright future. He's on the scout team right now, but I think he has lot of potential.
"I say, 'Make sure you don't get a big head. Just stay calm and your time will come.' One of the things I always tell them is to never give up on themselves. If you mess up one time, don't beat yourself down. That's something that I used to do as a freshman. You kind of dig yourself in a hole."


text:
April 24, 2012
Kevin Scarpati
Don't look now, but after a slow start, La Costa Canyon is moving up in the San Diego Section Baseball Rankings. Head coach Justin Machado always has his Mavericks ready to compete, and that's exactly what they did last week, taking two of three from ranked Rancho Buena Vista to move into the Top 10. ...More... To continue reading this article you must be a member. Sign Up Now!
Already a member? Click here to sign in


text:
This job posting is no longer available on Indeed.
Find similar jobs:Sales Marketing Associate jobs
- Hair Loss Control Clinic jobs
POSITION: SALES and MarketingEstablished Company w/25yrs in Bus.Unlimited earning potential
Established International Company in business for over 25 years, expanding the sales staff in the Albany market and is looking for talented Sales Consultants to establish new business in both the B2B and B2C markets.Opportunity for unlimited earning potential in a great work environment selling consumer services, as well as an unmatched opportunity to other businesses. The right candidate will earn a competitive salary plus residual commission and full benefits.QUALIFICATIONS:
. Must have reliable transportation
EDUCATION:Degree preferred but not required with comparable sales experience
Email this job to a friend


text:
America needs to invest more financial resources to help address a looming shortage of college graduates needed for the high-tech economy of tomorrow. Or do we?
Arizona Governor Janet Napolitano, current chair of the National Governor's Association, used her 2008 State of the State address to call for doubling the number of college graduates in Arizona by 2020. Napolitano proposed paying the tuition for students who graduate high school with a B average to accomplish this goal.
"We must recognize that higher education is something that all Arizona children will need to succeed," stated Governor Napolitano.
In the Carnegie Foundation's publication Change, Paul Barton wrote that the notion that the U.S. has a dire need for an ever increasing number of college graduates is a myth. "Confusion about the demand for college graduates runs throughout discussions of national workforce needs," Barton wrote.
According to Bureau of Labor Statistics data, only 29 percent of all jobs actually required a degree in 2004. The Bureau projects that of the top ten occupations with the largest growth from 2004 to 2014, seventy percent won't require a college education.
Interestingly, the U.S. Department of Education's National Education Longitudinal Study reports that 40 percent of its sample attained a two- or four-year degree or higher. Therefore, many people with college degrees have jobs that don't require them. So it really might be true when your cabbie says he has a Ph.D.
Barton's clear-eyed presentation of the data reveals a job market far more complex than simply an unmet demand for college-educated job applicants. For example, proponents of greater higher education funding often point to an increasing wage gap between the college educated and those who aren't.
Barton, however, notes that the wage gap is due largely to the falling earnings of high-school graduates and dropouts rather than to higher earnings for college graduates.
Rush Limbaugh defends Rubio's debate answer: He's absolutely right that Obama knows what he's doing | Allahpundit
Time for Team Hillary to break out the gender card for young liberal women who prefer Sanders | Allahpundit


text:
how to add a unique constraint of a sql table as foreign key reference to an another sql table in sql server 2005
In order to add FK constraint (in child table to parent table) you have to add unique constraint to parent table columns of relationship.
The parent table (in such FK relation) is frequently called (including by SSMS) as Primary Key table but PK is not must, unique key/constraint in parent table is enough (as PK is unique, it is particular case of unique constraint in parent table).
Drop TableA and TableB from answer by Matt, which is confusing for beginners,
Now, in order, to add FK
you should first create unique constraint on corresponding referenced column in parent table column:
Similarly for 2 columns foreign key constraint
You need to keep in mind that adding a FK on a column does not automatically put an index on that column. You'll need to do this in two steps.
Apologies but I'm not really sure what you're asking here. Giving more of an example with table definitions would help! I think you're saying you have two columns in TableA in a unique constraint named "Cons2cols", and you also want these two columns to be a FK to a two column PK / unqiue pair in TableB.
That works as follows, if you're creating the tables from scratch:
If the tables already exist, you can add in these same constraints after the fact:
Either way, TableA now has a unique, 2 column FK to another table.
Forget about the unique constraint for now. Just create your new foreign key on the two columns.


text:
Poll: Which of the brother's from Pruina do you like the best? Vote Now!
Author has written 15 stories for Life, Essay, Haiku, Humor, School, Humor, Western, Kids, Fantasy, Action, Romance, and Young Adult.
Jesus is my savior! (Contact me if you have any questions.)
I pretty much transfered from Fanfiction to here because I haven't written for myself in a long time. on Fanfiction.net I'm known to the world as "Fairyfinder". I used to be that on here as well, but I decided it was time for a switch. :)
So yeah, about me:
I'm just a regular bookworm teenage girl whose trying to be an author.
I like to think I'm nice and I'll never flame your story because really, why would I need to? Ranting may be fun, but not for an aspiring author. So, the worst I'll do is constructive (hopefully) criticism. If I offend you in any way PLEASE tell me! I will try my hardest to make it up to you.
I try to review every story I read, but sometimes I have to get off the computer in a hurry right after reading something, so I may not review.
I adore reviews!!!!!! Any review makes my day, whether it be flames or love! I'm a big girl, so don't worry too much about flaming me, I'll endure, but NEVER swear at me.
I LIKE TO DESIGN COVERS!! I'm not the best at it as I don't have too much experience, but if you would like to have me try and design for you I will do my very best! :)
Here is one of my rules:
No using swear words or cussing in any reviews. There are other ways of expressing your opinions.
Family tree of Jackal Pruina:
Nardia the Slayer's wings:
Part of the world of Pruina:


text:
Recipe by: Carol
Original recipe yields 14 servings
Based on a 2,000 calorie diet
Sort stores by
See how to make a moist and creamy coconut cake.
A moist, tender coffee cake with a sweet, crunchy pecan topping.
Satisfy your sweet tooth with this nutritious idea.
I did not add the apple into the cake, instead I peeled one apple and shaved half of it onto the top before baking. I also felt it was missing something so I made an orange glaze which improved ...


text: Comments (29) + Add comment Comments (29) Featured r21 + 1180d ago And so it begins.... Anyways, this was a pretty good review, made me laugh :D Glad to hear the SP isnt bad. Oh and this was for the vita version :) #1 Agree(18) | Disagree(3) | Report | Reply showtimefolks + 1180d ago Fanboys will still hate on this game no matter how great reviews it gets Anyway can't wait to play it #1.1 Agree(26) | Disagree(10) | Report | Reply r21 + 1180d ago Its to be expected, hardcore Nintendo fanboys will downplay this game and continue to call it a rip off. No matter how much hate it'll receive, Im still gonna buy, play and enjoy this game :D #1.1.1 (Edited 1180d ago ) | Agree(18) | Disagree(7) | Report Fishy Fingers + 1180d ago Yeah, just look at all those negative fanboy posts. Thank god you were here to defend it from, them? #1.1.2 (Edited 1180d ago ) | Agree(5) | Disagree(10) | Report MaxXAttaxX + 1180d ago This was the review for the Vita version? Still pretty good. ___ @above That's because some people have never seen a 4-player brawler before. So they call it a rip-off, despite the game playing differently. Haters like to ignore the fact that character mash-ups in fighting games have been around longer than the first SSB. It's like calling every 1 on 1 2D fighting game a Street Fighter clone. It's silly, but they'll have to learn the meaning of "genre" sooner or later. #1.1.3 (Edited 1180d ago ) | Agree(6) | Disagree(0) | Report Knight_Crawler + 1180d ago You mean like how PS3,360,PC and Wii fanboys hate on: Halo 4 COD Blk Ops 2 Zombie U LBP Racing Diablo 3 Yeah fanboys will always hate but end up loosing at the end as brand loyalty stops them from enjoying other great games. #1.1.4 Agree(5) | Disagree(0) | Report Godchild1020 + 1180d ago This review was informative and a good read. I have 6 more hours before I'm able to pick this game up and complain about how long Its taking for my day to end before I'm able to play. I might pick up the Vita version as well and bring my Vita, so I can play when no one is looking. Edit: I hope Sony releases a Soundtrack for this game. Just a note; The review is spoiler free. Like theres still anything to spoil, if you know who the boss is. #1.2 (Edited 1180d ago ) | Agree(2) | Disagree(3) | Report | Reply Sanquine90 + 1180d ago Cross buy... #1.2.1 Agree(4) | Disagree(1) | Report rezzah + 1180d ago Btw is there any difference between the vita and PS3 version? I might pick it up for vita if there is. #1.3 Agree(0) | Disagree(0) | Report | Reply DarkHeroZX + 1180d ago just buy the ps3 version and youll get the vita version free #1.3.1 Agree(3) | Disagree(1) | Report MaxXAttaxX + 1180d ago Only graphical. #1.3.2 Agree(2) | Disagree(0) | Report Nitrowolf2 + 1180d ago buy the PS3 version, you'll get both copies taht way #1.3.3 Agree(4) | Disagree(0) | Report rezzah + 1180d ago No point in owning the vita version if they are the same. So the digital version for free would be pointless. Thanks for you replies. #1.3.4 Agree(0) | Disagree(4) | Report Studio-YaMi + 1179d ago How is there "no point" in owning a Vita version when you can play the same game but anywhere you go ? 0_o It's not the same thing Vita = Portable device you can take anywhere with ease. But if you spend most of your time at home,I would understand then. #1.3.5 Agree(3) | Disagree(0) | Report rezzah + 1179d ago I mostly play my vita in bed, just seems the most conformable to me. Like reading a book. I've taken it outside, but would rather just listen to music. Also because it is exactly the same game. I don't have the time to play the same game twice. The vita and Ps3 have their own games to provide me with, so I play different games at once. #1.3.6 Agree(0) | Disagree(3) | Report Outside_ofthe_Box + 1180d ago I'm glad the reviewer didn't compare it to smash. Kudos. #1.4 Agree(2) | Disagree(0) | Report | Reply fresh2death93 + 1180d ago Read a review?!?!? hahahahah!! (dont mean to be an ass) #2 Agree(1) | Disagree(10) | Report | Reply Tonester925 + 1179d ago Did you pass English class? #2.1 Agree(1) | Disagree(0) | Report | Reply NeXXXuS + 1179d ago I don't think he did. #2.1.1 (Edited 1179d ago ) | Agree(0) | Disagree(0) | Report fresh2death93 + 1175d ago Me not wanting to read a video game review has nothing to do with English class... #2.1.2 Agree(0) | Disagree(0) | Report zchay + 1180d ago I agree with the score, but I would like to change 2 things Presentation should not be an 8, the menus are bland, and should have scored a 7 NOTHING was wrong with the graphics. Graphics should only grade a game down if it looks bad. Should have gotten a 9 for graphics. It looked great, that is all that matters. #3 Agree(1) | Disagree(12) | Report | Reply poopsack + 1180d ago The menus are nothing special, but they aren't exactly bad either, plus you have many options to customize your menu to your liking along with a little gamer card that represents you online. The graphics aren't bad either and for the most part are great, but these arent pushing incredible graphics either and he stated that at times he got lost in the action when it got to hectic, which has happened to me as well. I think his score is fair. #3.1 Agree(3) | Disagree(1) | Report | Reply WeAreLegion + 1180d ago If you think a game has gorgeous visuals and details environments, why would you give it an 8/10 in the graphics department? I mean, really? 8.5 is good, but...why? #4 Agree(4) | Disagree(3) | Report | Reply andibandit + 1180d ago I guess it has good graphics for the type of game it is. I dont see to many Chess games get 10's in the visual department either, altho they do look beautiful. #4.1 Agree(2) | Disagree(3) | Report | Reply xursz + 1179d ago Fair point. i remember playing the beta and thinking "Man this looks great on the vita". And an 8/10 is 'great' after all. #4.1.1 (Edited 1179d ago ) | Agree(0) | Disagree(0) | Report Zodiac + 1180d ago Three 8's and a 9 actually average out to 8.25, but that is still a great score. Please change it to 8.25 #5 (Edited 1180d ago ) | Agree(0) | Disagree(10) | Report | Reply rezzah + 1180d ago True but at least the rounding is minimal compared to some who go up or down full digit. #5.1 Agree(0) | Disagree(0) | Report | Reply WeAreLegion + 1180d ago Been playing the PS3 version for two hours. IT'S SO FUN!!! #6 Agree(9) | Disagree(0) | Report | Reply Veneno + 1180d ago I have never ever seen any reviewer that knows how to properly review a fighting game. #7 Agree(1) | Disagree(1) | Report | Reply Add comment You need to be registered to add comments. Register here or login Username Password Remember Rules & guidelines Login


text:
Gerald and Mary married on the 27th June, 1968 in Cork. In 2008 they have three sons-in-law, two daughters- in-law and nine grandchildren. This album is a celebration of the family on their 40th wedding anniversary. Shot by Barnaby Aldrick on 27th June 2008 at Hebden in the Yorkshire Dales.
I am a photographer based in Leeds, England specialising in capturing laid back portraits, stunning details and timeless moments.
Comment posted.
Sign in or create an account to post a comment.
Untitled Published May 29, 2010
John & Helen Published October 26, 2008
The Wedding of Lorna & Mick Published July 07, 2008
The Wedding of Lorna & Mick Published July 07, 2008
Amanda & James Published May 13, 2008
Vicky & Mark Published April 07, 2008
Bob & Ruth's Wedding Published January 17, 2008
Liz & Ed's Wedding Published December 26, 2007


text: Hi, I am new to this but was fortunate enough to get a Miyata one twelve frame for 50 bucks. I have been taking the time to get it rolling again granted I don't have the time or money to restore it to its original state, but I plan on styling it out and using it as a commuter. So far everything has been easy to come by but I can't seem to find a 7 speed downtube shifter anywhere online that isn't on ebay coming from Poland for way more than I want to spend. I know it was originally a six speed but I'm trying to make it a 7 speed. If anyone could point me in the right direction to finding a seven speed downtube friction/index shifter that would be awesome. Thanks for the help. Once the bike is finished I'll post some pictures


text:
Home » Courses » Mathematics » Single Variable Calculus » Video Lectures » Lecture 37: Infinite Series
Download the video from iTunes U or the Internet Archive.
Topics covered: Infinite series and convergence tests
Note: This video lecture was recorded in the Fall of 2007 and corresponds to the lecture notes for lecture 36 taught in the Fall of 2006.
Instructor: Prof. David Jerison
Lecture 37: Infinite Series
Lecture Notes (PDF - 1.4MB)
The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses visit MIT OpenCourseWare at ocw.mit.edu.
PROFESSOR: Today we are continuing with improper integrals. I still have a little bit more to tell you about them. What we were discussing at the very end of last time was improper integrals.
Now and these are going to be improper integrals of the second kind. By second kind I mean if they have a singularity at a finite place. That would be something like this. So here's the definition if you like. Same sort of thing as we did when the singularity was an infinity. So if you have the integral from to 1 of f of x. This is going to be the same thing as the limit. As a goes to from above. The integral from 8 to 1 of f of x d x.
The idea here is the same one that we had at infinity. Let me draw a picture of it. You have imagine a function which is coming down like this and here's the point 1. And we don't know whether the area enclosed is going to be infinite or finite until we cut it off at someplace a. And we let a go to from above. So really it's plus. So we're coming in from the right here. And we're counting up the area in this chunk. And we're seeing as it expands whether it goes to infinity or whether it tends to some finite limit.
Right so this is the example and this is the definition. And just as we did for the other kind of improper integral, we say that this converges-- so that's the key word here --if the limit is finite, exists maybe I should just say and diverges if not.
Let's just take care of the basic examples. First of all I wrote this one down last time. We're going to evaluate this one. The integral from to 1 of 1 over the square root of x. You almost don't notice the fact that it goes to infinity. This goes to infinity as x goes to 0. But if you evaluate it -- first of all we always write this as a power. Right? To get evaluation. And then I'm not even going to replace the by an a. I'm just going to leave it as 0. The antiderivative here is x to the 1/2 times 2. And then I evaluate that at and 1. And I get 2. 2 minus 0, which is 2.
All right so this one is convergent. And not only is it convergent but we can evaluate it.
The second example being not systematic but really giving you the principal examples that we'll be thinking about. Is this one here d x over x And this one gives you the antiderivative as the logarithm. Valuated at and 1. And now again you have to have this thought process in your mind that your really taking the limit. But this is going to be the log of 1 minus the log of 0. Really the log of from above. There is no such thing as the log from below.
And this is minus infinity. So it's minus minus infinity, which is plus infinity. And so this one diverges.
All right so what's the general? So more or less in general for powers anyway, if you work out this thing for d x over x to the p from to 1. What you're going to find is that it's 1 over 1 minus p. When p is less than one. And it diverges for p bigger than or equal to 1. Now that's the final result. If you carry out this integration it's not difficult.
All right so now I just want to try to help you to remember this. And to think about how you should think about it. So I'm going to say it in a few more ways. All right just repeat what I've said already but try to get it to percolate and absorb itself. And in order to do that I have to make the contrast between the kind of improper integral that I was dealing with before. Which was not as x goes to here but as x goes to infinity, the other side. Let's make this contrast.
First of all, if I look at the angle that we have been paying attention to right now. We've just considered things like this. 1 over x to the 1 half. Which is a lot smaller than one over x. Which is a lot smaller than say 1 over x squared. Which would be another example. This is as x goes to 0. So this once the smallest one. This one next smallest one. And this one is very large.
On the other hand it goes the other way at infinity. As x tends to infinity. All right so try to keep that in mind.
And now I'm going to put a little box around the bad guys here. This one is divergent. And this one is divergent. And this one is divergent. And this one is divergent. When the cross over point is 1 over x. When we get smaller than that, we get to things which are convergent. When we get smaller than it on this other scale, it's convergent.
All right so these guys are divergent. So they are associated with divergent integrals. The functions themselves are just tending towards-- well these tend to infinity, and these tend toward 0. So I'm not talking about the functions themselves but the integrals.
Now I want to draw this again here, not small enough. I want to try this again. So I'm just going to draw a picture of what it is that I have here. But I'm going to combine these two pictures. So here's the picture for example of y equals 1 over x. All right. That's y equals 1 over x. And that picture is very balanced. It's symmetric on the two ends.
If I cut it in half then what I get here is 2 halves. And this one has an infinite area. That corresponds to the integral from 1 to infinity, d x over x being infinite. And the other piece-- which this one we calculated last time. This is the one that we just calculated over here at example 2 has the same property. It's infinite. And that's the fact that the integral from to 1 is infinite. Right we lose on both ends.
On the other hand if I take something like-- I'm drawing it the same way but it's really not the same --y equals 1 over the square root of x. y equals one over x to the 1 half. And if I cut that in half here then the x to the 1 half is actually bigger than this guy. So this piece is infinite. And this part over here actually is going to give us an honest number. In fact this one is finite. And we just checked what the number is. It actually happens to have area 2. And what's happening here is if you would superimpose this graph on the other graph what you would see is that they cross. And this one sits on top. So if I drew this one in let's have another color here --orange let's say. If this were orange if I set it on top here it would go this way. OK and underneath the orange is still infinite. So both of these are infinite. On here on the other hand underneath the orange is infinite but underneath where the green is finite. That's a smaller quantity. Infinity is a lot bigger than 2. 2 is a lot less than infinity. All right so that's reflected in these comparisons here.
Now if you like if I want to do these in green. This guy good and this guy is good. Well let me just repeat that idea over here in this sort of reversed picture with y equals 1 over x squared. If I chop that in half then the good part is this end here. It's finite. And the bad part is this part of here which is way more singular. And it's infinite.
All right so again what I've just tried to do is to give you some geometric sense and also some visceral sense. This guy it's tail as it goes out to infinity is much lower. It's much smaller than 1 over x. And these guys trapped an infinite amount of area. This one traps only a finite amount of area.
All right so now I'm just going to give one last example which combines these two types of pictures. It's really practically the same as what I said before but I-- oh have to erase this one too.
So here's another example if you're in so let's take the following example. This is somewhat related to the first one that I gave last time. If you take a function y equals 1 over x minus 3 squared. And you think about it's integral.
So let's think about the integral from to infinity, d x over x minus 3 squared. And suppose you were faced with this integral. In order to understand what it's doing you have to pay attention just two places where it can go wrong. We're going to split into two pieces. I'm going say break it up into this one here up to 5, for the sake of argument. And say from 5 to infinity. All right. So these are the two chunks.
Now why did I break it up into those two pieces? Because what's happening with this function is that it's going up like this at 3. And so if I look at the two halves here. I'm going to draw them again and I'm going to illustrate with what colors we've chosen-- which are I guess red and green --what you'll discover is that this one here which corresponds to this piece here is infinite. And it's infinite because there's a square in the denominator. And as x goes to 3 this is very much like we shifted the 3 to 0. Very much like this 1 over x squared here. But not in this context. In the other context where it's going to infinity. This is the same as at the picture directly above with the infinite part in red. All right. And this part here This part is finite. All right. So since we have an infinite part plus a finite part the conclusion is that this guy converges. And this one diverges. But the total unfortunately diverges. Right because it's got 1 infinity in it. So this thing diverges. And that's what happened last time when we got a crazy number. If you integrated this you would get some negative number. If you wrote down the formulas carelessly. And the reason is that the calculation actually is nonsense. So you've gotta be aware if you encounter a singularity in the middle not to ignore it. Yeah. Question.
PROFESSOR: Why do we say that the whole thing diverges? The reason why we say that is the area under the whole curve is infinite. It's the sum of this piece plus this piece. And so the total is infinite.
PROFESSOR: We're stuck. This is an ill defined integral. It's one where you're red flashing warning sign should be on. Because you're not going to get the right answer by computing it. You'll never get an answer. Similarly you'll never get an answer with this. And you will get an answer with that. OK? Yeah another question.
PROFESSOR: So the question is if you have a little glance at an integral, how are you going to decide where you should be heading? So I'm going to answer that orally. Although you know but I'll say one little hint here. So you always have to check x going to infinity and x going to minus infinity, if they're in there. And you also have to check any singularity, like x going to 3 for sure in this case. You have to pay attention all the places where the thing is infinite. And then you want to focus in on each one separately. And decide what's going on it at that particular place. When it's a negative power. So remember d x over x as x goes to is bad. And d x over x squared is bad. D x over x cubed is bad. All of them are even worse. So anything of this form is bad and equals 1, 2, 3. These are the red box kind. All right.
That means that any of the integrals that we did in partial fractions which had a root, which had a factor of something, the denominator Those are all divergent integrals if you cross the singularly. Not a single one of them makes sense across the singularity.
If you have square roots and things like that then you can repair things like that. And there's some interesting examples of that. Such as with the arc sign function and so forth. Where you have an improper integral which is really OK. All right. So that's the best I can do. It's obviously something you get experience with. All right.
Now I'm going to move on and this is more or less our last topic. Yay but not quite. Well it's our penultimate topic. Right because we have one more lecture. All right.
So that our next topic is series. Now we'll do it in a sort of a concrete way today. And then we'll do what are known as power series tomorrow.
So let me tell you about series. Remember we're talking about infinity and dealing with infinity. So we're not just talking about any old series. We're talking about infinite series.
There is one infinite series which is probably, which is without question the most important and useful series. And that's the geometric series but I'm going to introduce it concretely first in a particular case. If I draw a picture of this sum. Which in principle goes on forever. You can see that it go someplace fairly easily by marking out what's happening on the number line. The first step takes us to 1 from 0. And then if I add this half, I get to 3 halves. Right, so the first step was 1 and the second step was a half.
Now see if I add this quarter in which is the next piece then I get some place here. But what I want to observe is that I can look at it from the other point of view. When I move this quarter I got half way to 2 here.
I'm putting 2 in green because I want you to think of it as being the good kind. Right. The kind that has a number. And not one of the red kind. We're getting there and we're almost there. So the next stage we get half way again. That's the eighth and so forth. And eventually we get to 2. So this sum we write equals two.
All right that's kind of a paradox because we never get to 2. This is the paradox that Zeno fussed with. And his conclusion with the rabbit and the hare. Oh no the rabbit and the tortoise. Sorry hare chasing-- anyway the rabbit chasing the tortoise. His conclusion-- I don't know if you're aware of this --but he understood this paradox. And he said you know it doesn't look like it ever gets there because they're infinitely many time between the time-- you know that the tortoise is always behind, always behind, always behind, always behind. So therefore it's impossible that the tortoise catches up right. So do you know what his conclusion was? Time does not exist. That was actually literally his conclusion. Because he didn't understand the possibility of a continuum of time. Because there were infinitely many things that happened before the tortoise caught up. So that was the reasoning. I mean it's a long time ago but you know he didn't believe in continuum.
All right. So anyway that's a small point. Now the general case here of geometric series is where I put in a number a instead of a half here. So what we had before. So that's 1 plus a plus a squared. Isn't quite the most general but anyway I'll write this down. And your certainly going to want to remember that the formula for this in the limit is 1 over 1 minus a.
And I remind you that there's only works when the absolute value is strictly less than 1. In other words when minus 1 is strictly less than a is less than 1. And that's really the issue that we're going to want to worry about now. What we're worrying about is this notion of convergence. And what goes wrong when there isn't convergence, when there's a divergence.
So let me illustrate divergence before going on. And this is what we have to avoid if we're going to understand series. So here's an example when a is equal to 1. You get 1 plus 1 plus 1 plus et cetera. And that's equal to 1 over 1 minus 1. Which is 1 over 0. So this is not bad. It's almost right. It's sort of infinity equals infinity.
At the edge here we managed to get something which is sort of almost right. But you know it's we don't consider this to be logically to make complete sense. So it's a little dangerous And so we just say that it diverges. And we get rid of this. So we're still putting it in red. All right. The bad guy here so this one diverges. Similarly if I take a equals minus 1, I get 1 minus 1 plus 1 minus one plus 1 because the odd and even powers in that formula alternate sign.
And this bounces back and forth. It never settles down. It starts at 1. And then it gets down to and then comes back up to 1, down to 0, back up to 1. It doesn't settle down. It bounces back and forth. It oscillates. On the other hand if you compare the right hand side. What's the right hand side? It's 1 over 1 minus minus 1. which is a half. All right. So if you just paid attention to the formula. Which is what we were doing when we integrated without thinking too hard about this. You get a number here but in fact that's wrong. Actually it's kind of an interesting number. It's halfway between the 2 between 1. So again there's some sort of vague sense in which this is trying to be this answer. All right. It's not so bad but we're still going to put this in a red box. All right. because this is what we called divergence. So both of these cases are divergent. It only really works when a is less than 1. I'm going to add one more case just to see that mathematicians are slightly curious about what goes on in other cases.
So this is 1 plus 2 plus 2 squared plus 2 to cube plus etc.. And that should be equal to-- according to this formula --1 over 1 minus 2. Which is negative 1. All right. Now this one it clearly wrong, right? This one it's totally Wrong It certainly diverges. The left hand side is obviously infinite. The right hand side is way off. It's negative 1.
On the other hand it turns out actually that mathematicians have ways of making sense out of these. In number theory there's a strange system where this is actually true. And what happens in that system is that what you have to throw out is the idea that is less than you 1. There is no such thing as negative numbers. So this number exists. And it's the additive inverse of 1. It has this arithmetic property but the statement that 1 is bigger than does not make sense.
So you have your choice either this diverges or you have to throw out something like this. So that's a very curious thing in higher mathematics. Which if you get to number theory there's fun stuff there. All right. OK but for our purposes these things are all out. All right. They're gone. We're not considering them. Only a between negative 1 and 1. All right.
Now I want to do something systematic. And it's more or less on the lines of the powers that I'm erasing right now. I want to tell you about series. Which are kind of borderline convergent. And then next time when we talk about powers series we'll come back to this very important series which is the most important one.
So now let's talk about some series general notations. And this will help you with the last bit. This is going to be pretty much the same as what we did for improper integrals. First of all I'm going to have capital S N which is the sum of a n, n equals to capital N. And this is what we're calling a partial sum.
And then the full limit which is capital S, if you like, and N equals to infinity is just the limit as N goes to infinity of the Sn. And then we have the same kind of notation that we had before. Which is there these two choices which is that if the limit exists. That's the green choice. And we say it converges. So we say the series converges.
And then the other case which is limit does not exist. And we can say the series diverges. Question.
PROFESSOR: The question was how did I get to this? And I will do that next time but in fact of course you've seen in high school. Right this a-- Yeah. Yeah. We'll do that next time. The question was how do we arrive-- sorry I didn't tell you the question --the question was how do we arrive at this formula on the right hand side here. But we'll talk about that next time. All right.
So here's the basic definition and what we're going to recognize about series. And I'm going to give you a few examples and then we'll do something systematic.
So the first example-- well the first example is the geometric series But the first example that I'm going to discuss now and in a little bit of detail is this sum 1 over n squared N equals 1 to infinity. It turns out that this series is very analogous-- and we'll develop this analogy carefully --the integral from 1 to x d x over x squared. And we're going to develop this analogy in detail later in this lecture. And this one is one of the ones-- so now you have to go back and actually remember this is one of the ones you really want to memorize. And you should especially pay attention to the ones with an infinity in them. This one is convergent. And this series convergent. Now it turns out that evaluating this is very easy. This is 1. It's easy to calculation. Evaluating this is very tricky. And Euler did it. And the answer with pie squared over 6. That's amazing calculation. And it was done very early in the history of mathematics. If you look at another example-- so maybe example two here -if you look at 1 over n cubed. n equals-- well you can't start here at by the way. I get to start where ever I want in these series. Here I start with 0. Here I started with 1.
And notice the reason why it was a bad idea to start with was that 1 over is undefined. Right so I'm just starting where it's convenient for me. And since I'm interested mostly in the tale behavior it doesn't matter to me so much where I start. Although if I want an exact answer I need to start exactly at n equals 1. All right. This one is similar to this integral here. All right. Which is convergent again. So there's a number that you get. And let's see what is it something like 2 thirds or something like that for this for this number. Or a third. What is it? No a half. I guess it's a half. This one is a half. You check that I'm not positive but anyway just doing it in my head quickly it seems to be a half. Anyway it's an easy number to calculate.
This one over here stumped mathematicians basically for all time. It doesn't have any kind of elementary form like this. And it was only very recently proved to be rational. People couldn't even couldn't even decide whether this was a rational number or not. But anyway that's been resolved it is an irrational number which is what people suspected. Yeah question.
PROFESSOR: Yeah sorry. OK I violated a rule of mathematics-- you said why is this similar? I thought that similar was something else. And you're absolutely right. And I violated a rule of mathematics. Which is that I uses this symbol for two different things. I should have written this symbol here. All right. I'll create a new symbol here. The question of whether this converges or this converges. These are this the same type of question. And we'll see why they're the same question it in a few minutes. But in fact the wiggle I use similar I used for the connection between functions. The things that are really similar are that 1 over n resembles 1 over x squared. So I apologize I didn't--
PROFESSOR: Oh you thought that this was the definition of that. That's actually the reason why these things correspond so closely. That is that the Riemann's sum is close to this. But that doesn't mean they're equal. The Riemann's sum only works when the delta x goes to 0. The way that we're going to get a connection between these two as, we will just a second, is with a Riemann sum with. What we're going to use is a Riemann's sum with delta x equals 1. All right and then that will be the connection between. Right that's absolutely right. All right.
So in order to illustrate exactly this idea that you've just come up with. And in fact that we're going to use. We'll do the same thing but we're going to do it on the example sum 1 over n. So here's example 3 and it's going to be sum 1 over n, n equals 1 to infinity. And what we're now going to see is that it corresponds to this integral here. And we're going to show therefore that thing diverges. But we're going to do more carefully. We're going to do this in some detail so that you see what the correspondence is between these quantities. And the same sort of reasoning applies to these other examples.
So here we go. I'm going to take the integral and draw the picture of the Riemann's sum. So here's the level one and here's the function y equals 1 over x. And I'm going to take the Riemann's sum. With delta x equals 1. And that's going to be closely connected to the series that I have. But now I have to decide whether I want to lower Riemann's sum or an upper Riemann's sum. And actually I'm going to check both of them because both of them are illuminating.
First will do the upper Riemann's sum. Now that's this staircase here. So we'll call this the upper Riemann's sum. And let's check what it's levels are. This is not to scale. This level should be a half. So if this is 1 and this is 2 and that level was supposed to be a half then this next level should be a third. That's how the Riemann's sums are working out.
And now I have the following phenomenon. Let's cut it off at the n. So that means that the integral if from 1 to n d x over x. And the Riemann's sum is something that's bigger than it. Because the areas are enclosing the area of the curved region. And that's going to be the area of the first box which is 1, plus the area of the second box which is a half plus the area of the third box which is a third. All the way up the last one but the last one starts today n minus 1. So it has 1 over n minus 1. There are not n boxes here. They're only n minus 1 boxes. Because the distance between 1 n is n minus one. Right so this is n minus one terms.
However if I use a notation for partial sum. Which is 1 plus 1 over 2 plus all the way up to n over n minus one plus 1 over n. In other words I go out to the n one which is what I would ordinarily do. Then this sum that I have here certainly is less than sn. Because there's one more term there. And so here I have an integral which is underneath this sum s n.
Now this is going to allow us to prove conclusively that the sum diverges. Why is that? Because this term here we can calculate. This is the log x evaluated at 1 and n. Which is the same thing as log n minus 0. All right the quantity log n minus log 1 which is 0. And so what we have here is that log n is less than s n. All right and clearly this goes to infinity right. As n goes to infinity this thing goes to infinity. So we're done. All right we've shown divergence. The way I'm going to use the lower Riemann's sum is to recognize that we've captured the rate appropriately. That is not only do have a lower bound like this but I have an upper bound which is very similar.
So if I use the upper-- remind oh sorry --the lower Riemann's sum again with delta x equals 1. Then I have that the integral from 1 to n of d x over x is bigger than-- Well what are the terms going to be if fit sit them underneath? If I fit them underneath I'm missing the first term. That is the box is going to be half height. It's going to be this lower piece. So I'm missing this first term. So it'll be a half plus a third plus it will keep on going. But now the last one instead of being 1 over n minus 1. It's going to be 1 over n. This is again a total of the n minus 1 terms. This is the lower Riemann's sum.
And now we can recognize that this is exactly equal to s n minus 1 minus the first term. So we missed the first term but we got all the rest of them. So if I put this to the other side remember this is the log n. All right. If I put this to the other side I have the other side of this bound. I have that f n is less than if I reverse it log n plus 1. And so I've trapped it on the other side. And here I have the lower bound. So I'm going to combine those together.
So all hold I have this correspondence here. It is the size of a log n is trapped between the size of f n which is relatively hard to calculate and understand exactly is trapped between log n and log n plus 1. Yeah question.
PROFESSOR: This step here is the step that you're concerned about. So this step is a geometric argument which is analogous to this step. All right it's the same type of argument. And in this case it's that the rectangle is on top and so the area represented on the right hand side is less than the area represented on this side. And this is the same type of thing except that the rectangle is underneath. So the sum of the areas of the rectangle is less than the area under the curve.
All right. So I've now trapped this quantity. And I'm now going to state the sort of general results. So here's what's known as integral comparison. It's this double arrow correspondence in the general case, for a very general case. There is actually even more cases where it works. But this is a good case and convenient.
And this is called integral comparison. And it comes with hypothesis but it follows the same argument that I just gave. If f of x is decreasing and it's positive then the sum f n, n equals 1 to infinity minus the integral from 1 to infinity of f of x d x is less than half of 1. That's basically what we showed. We showed that the difference between s n and log n was at most 1. All right. And the sum and the integral converge or diverge together. That is they either both converge or both diverge. This is the type of test that we like because then we can just convert the question of convergence over here to this question of convergence over on the other side.
Now I remind you that it's incredibly hard to calculate these numbers. Whereas these numbers are easier to calculate. Our goal is to reduce things to simpler things. In this case sums, infinite sums are much harder than infinite integrals.
All right so that the integral comparison. And now I have one last bit on comparisons that I need to tell you about. And this is very much like what we did with integrals. Which is so called limit comparison. The limits comparison says the following if f of n is similar to g of n. You will recall that means f of n over g of n tends to 1 as n goes to infinity. And we're in the positive case. So let's just say g n is positive. Then sum f n sum g n same this as above, either both converge or both diverge. All right. This is just saying that if they behave the same way in the tail, which is all we really care about, then they have similar behavior, similar convergence properties.
And let me give you a couple examples. So here's one example if you take the sum 1 over n squared plus 1 square root. This is going to be replaced by something simpler. Which is the main term here. Which is 1 over square root of n squared which we recognize as sum 1 over n which diverges. So this guy is one of the red guys.
On the red team now we have another example. Which is let's say the square root of n to the fifth minus n squared. Now if you have something where it's negative in the denominator you kind of do have to watch out that denominator makes sense. It isn't 0. So we're going to be careful and start this at n equals 2. I don't like 1 over as a term in my series. So I'm just going to be a little careful about how-- as I said I was kind of lazy here. I could have started this one at for instance. All right. So here's the picture.
Now this I just replace by it's main term which is 1 over n to the fifth square root. Which is sum 1 over n to the five halves which converges. All right. The power is bigger than 1. 1 is the divider for these things and it just misses. This one converges. All right so these are the typical ways in which these convergence processes are used. All right.
So I have one more thing for you. Which is an advertisement for next time. And I have this demo here which I will grab. But you will see this next time.
This is one of over 2,200 courses on OCW. Find materials for this course in the pages linked along the left.
MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum.
No enrollment or registration. Freely browse and use OCW materials at your own pace. There's no signup, and no start or end dates.
Knowledge is your reward. Use OCW to guide your own life-long learning, or to teach others. We don't offer credit or certification for using OCW.
Made for sharing. Download files for later. Send to friends and colleagues. Modify, remix, and reuse (just remember to cite OCW as the source.)
Learn more at Get Started with MIT OpenCourseWare
MIT OpenCourseWare makes the materials used in the teaching of almost all of MIT's subjects available on the Web, free of charge. With more than 2,200 courses available, OCW is delivering on the promise of open sharing of knowledge. Learn more »
© 2001–2015
Massachusetts Institute of Technology
Your use of the MIT OpenCourseWare site and materials is subject to our Creative Commons License and other terms of use.