A set Failed to parse (Missing texvc executable; please see math/README to configure.): C

and its dual cone Failed to parse (Missing texvc executable; please see math/README to configure.): C^*

.

A set Failed to parse (Missing texvc executable; please see math/README to configure.): C

and its polar cone Failed to parse (Missing texvc executable; please see math/README to configure.): C^o

. The dual cone and the polar cone are symmetric to each other with respect to the origin.

Dual cone and polar cone are closely related concepts in convex analysis, a branch of mathematics.

Dual cone[link]

The dual cone Failed to parse (Missing texvc executable; please see math/README to configure.): C^*

of a subset Failed to parse (Missing texvc executable; please see math/README to configure.): C
in a linear space Failed to parse (Missing texvc executable; please see math/README to configure.): X

, e.g. Euclidean space Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb R^n , with topological dual space Failed to parse (Missing texvc executable; please see math/README to configure.): X^*

is the set

Failed to parse (Missing texvc executable; please see math/README to configure.): C^* = \left \{y\in X^*: \langle y , x \rangle \geq 0 \quad \forall x\in C \right \},

Failed to parse (Missing texvc executable; please see math/README to configure.): C^*

is always a convex cone, even if Failed to parse (Missing texvc executable; please see math/README to configure.): C 
is neither convex nor a cone.  

When Failed to parse (Missing texvc executable; please see math/README to configure.): C

is a cone, the following properties hold:[1]

• A non-zero vector Failed to parse (Missing texvc executable; please see math/README to configure.): y

is in Failed to parse (Missing texvc executable; please see math/README to configure.): C^*
if and only if both of the following conditions hold: (i) Failed to parse (Missing texvc executable; please see math/README to configure.):  y 
is a normal at the origin of a hyperplane that supports Failed to parse (Missing texvc executable; please see math/README to configure.): C 

. (ii) Failed to parse (Missing texvc executable; please see math/README to configure.): y

and Failed to parse (Missing texvc executable; please see math/README to configure.): C 
lie on the same side of that supporting hyperplane.  

• Failed to parse (Missing texvc executable; please see math/README to configure.): C^*

is closed and convex.

• Failed to parse (Missing texvc executable; please see math/README to configure.): C_1 \subseteq C_2

implies Failed to parse (Missing texvc executable; please see math/README to configure.): C_2^* \subseteq C_1^*

.

• If Failed to parse (Missing texvc executable; please see math/README to configure.): C

has nonempty interior, then Failed to parse (Missing texvc executable; please see math/README to configure.): C^* 
is pointed, i.e. Failed to parse (Missing texvc executable; please see math/README to configure.): C^* 
contains no line in its entirety.

• If Failed to parse (Missing texvc executable; please see math/README to configure.): C

is a cone and the closure of Failed to parse (Missing texvc executable; please see math/README to configure.): C 
is pointed, then Failed to parse (Missing texvc executable; please see math/README to configure.): C^* 
has nonempty interior.

• Failed to parse (Missing texvc executable; please see math/README to configure.): C^{**}

is the closure of the smallest convex cone containing Failed to parse (Missing texvc executable; please see math/README to configure.): C 

.

A cone is said to be self-dual if Failed to parse (Missing texvc executable; please see math/README to configure.): C = C^* . The nonnegative orthant of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{R}^n

and the space of all positive semidefinite matrices are self-dual.

Polar cone[link]

The polar of the closed convex cone Failed to parse (Missing texvc executable; please see math/README to configure.): C

is the closed convex cone Failed to parse (Missing texvc executable; please see math/README to configure.): C^o,

and vice-versa.

For a set Failed to parse (Missing texvc executable; please see math/README to configure.): C

in Failed to parse (Missing texvc executable; please see math/README to configure.): X

, the polar cone of Failed to parse (Missing texvc executable; please see math/README to configure.): C

is the set

Failed to parse (Missing texvc executable; please see math/README to configure.): C^o = \left \{y\in X^*: \langle y , x \rangle \leq 0 \quad \forall x\in C \right \}.

^[2]

It can be seen that the polar cone cone is equal to the negative of the dual cone, i.e. Failed to parse (Missing texvc executable; please see math/README to configure.): C^o=-C^* .

For a closed convex cone Failed to parse (Missing texvc executable; please see math/README to configure.): C

in Failed to parse (Missing texvc executable; please see math/README to configure.): X

, the polar cone is equivalent to the polar set for Failed to parse (Missing texvc executable; please see math/README to configure.): C .^[3]

See also[link]

• Bipolar theorem
• Polar set

References[link]

• Goh, C. J.; Yang, X.Q. (2002). Duality in optimization and variational inequalities. London; New York: Taylor & Francis. ISBN 0-415-27479-6. 

• Boltyanski, V. G.; Martini, H., Soltan, P. (1997). Excursions into combinatorial geometry. New York: Springer. ISBN 3-540-61341-2. 

• Ramm, A.G.; Shivakumar, P.N.; Strauss, A.V. editors (2000). Operator theory and its applications. Providence, R.I.: American Mathematical Society. ISBN 0-8218-1990-9. 

Dr. Eric S. Chivian is the founder and Director of the Center for Health and the Global Environment (CHGE) at Harvard Medical School,^[1] where he is also an assistant clinical professor of psychiatry.

Between 1980 and 2000, he was a staff psychiatrist in the MIT Medical Department.^[2] Dr. Chivian was the co-founder (with Professors Bernard Lown, Herbert Abrams and James Muller), treasurer, and member of the board of directors of the organization International Physicians for the Prevention of Nuclear War,^[3] which won the Nobel Peace Prize in 1985 for its efforts to highlight the implications of nuclear conflict for global health.^[4] In the mid 1980s, he directed the first scientific survey (under the auspices of the American Academy of Arts and Sciences and MIT’s Center for International Studies) of American and Soviet teenagers’ attitudes about the prospect of nuclear war and their concerns for the future. He was the lead author of a seminal paper on this study in The New England Journal of Medicine [Chivian E. et al.. “American and Soviet Teenagers’ Concerns about Nuclear War and the Future.” The New England Journal of Medicine; 1988. 319(7): 407-413].^[5] He was the senior editor and author for the book Last Aid: The Medical Dimensions of Nuclear War, published by W.H. Freeman and Company in 1983.^[6]

In the early 1990s, Dr. Chivian became involved in efforts to create a greater awareness of the impacts which environmental degradation has on human health and well-being. His second book (as senior editor and lead author) was Critical Condition: Human Health and the Environment.^[7] The book, published by MIT Press^[8] in 1993, was one of the first books on this topic for a general audience (later editions were published in German, Spanish, Japanese, Chinese and Persian).

Chivian founded the Center for Health and the Global Environment at Harvard Medical School in 1996, with the mission 'To help people understand that our health, and that of our children, depends on the health of the environment and that we must do everything we can to protect it.'^[9] The Center is designated an official "Collaborating Center" of the United Nations Environment Programme. Dr. Chivian developed and directed the Harvard Medical School course "Human Health and Global Environmental Change", which he taught for a decade and which has been disseminated to 65 other medical schools, colleges, and universities in the U.S. and abroad); he also designed an intensive annual course on the environment and health for the U.S. Congress and led and participated in numerous congressional briefings.^[10]

Dr. Chivian is recognized as a leading proponent of efforts to create a greater awareness amongst policy makers and the public of the importance of biodiversity to human health, an issue on which he has contributed several book chapters, and published many scientific papers, literature reviews, and editorials.^[11] Dr. Chivian is the senior editor and author, with CHGE associate Dr. Aaron Bernstein, of Sustaining Life: How Human Health Depends on Biodiversity,^[12] published in June 2008 by Oxford University Press and co-sponsored by the United Nations Development Programme, the United Nations Environment Programme, the Secretariat of the U.N. Convention on Biological Diversity, and the World Conservation Union (IUCN). The book was launched at U.N. Headquarters, the Smithsonian Institution, and at the U.N. Convention of Biological Diversity Conference of the Parties meeting in Bonn, Germany in 2008.^[13] The book includes contributions from over 100 leading biodiversity and health scientists, and has been widely praised for its detailed coverage of a broad range of issues in a manner accessible to the general public.^[14] Sustaining Life was named best biology book for 2008 by the Library Journal along with E. O. Wilson’s and Bert Hölldobler’s Superorganism. ^[15]

Dr. Chivian was also involved in setting up the First International Conference on Health and Biodiversity (COHAB 2005) which he opened in Galway, Ireland in 2005. This was the first global meeting of its kind to bring UN agencies, scientists, NGOs, policy makers, economists, indigenous and local community representatives and the private sector together to explore the wide-ranging implications of biodiversity loss for human health and well-being. The conference, and the global programme for Co-operation on Health and Biodiversity which was established as a result, were largely based on Chivian's work and the concepts behind the Sustaining Life project.^[16]

In recent years, Dr. Chivian has worked to explore common ground between scientific and religious perspectives on environmental issues. Together with the Rev. Richard Cizik, then Vice President for Governmental Affairs of the National Association of Evangelicals, Dr. Chivian was named by Time Magazine in 2008 as one of the 100 Most Influential People in the World, for their work in organizing scientists and Evangelicals to join together in efforts to protect the global environment.^[17]

Dr. Chivian runs Pairidaeza Farm, an almost fully organic orchard in central Massachusetts growing heirloom apples, peaches, pears, Asian pears, apricots, plums, cherries and grapes.^[18]

References[link]

External links[link]

• Notes On The Road interviews Dr. Eric Chivian: On Backyard Biodiversity, Oslo & The Next Ten Years
• Profile on the website of the Centre for Health and the Global Environment at Harvard Medical School
• Joint press release from UNDP, UNEP, SCBD and IUCN on the launch of Sustaining Life
• Video of a discussion of the Sustaining Life publication at the Wilson Centre, Washington DC, January 2009

The Holidays
The Holidays
Background information
Origin	Sydney, Australia
Genres	Indie rock, Pop rock, Soul
Years active	2006 - present
Labels	Liberation (Australia), Rallye Label (Japan)
Website	Official Website Myspace
Members
Simon Jones Will Magnus Alex Kortt Andrew Kerridge

The Holidays are an Australian indie pop/soul band. Formed in Sydney in 2006, the band consists of Simon Jones, Will Magnus, Alex Kortt and Andrew Kerridge. The band is most notably known for their Australian Music Prize nominated debut album Post Paradise.

Biography[link]

The Holidays formed in late 2006 and quickly established themselves in the Sydney music scene as an upbeat, energetic addition. Simon Jones & Will Magnus had attended high school together, previously playing in cover bands in their schools' band competition. They were often up against the band Roger Explosion in these competitions, who went on to become the Australian electro-pop band Art vs. Science.

Within their first year as a band together, The Holidays toured with Jamie T,^[1] with The View,^[2] with Ben Kweller,^[3] and recorded an EP of demos which quickly sold out at gigs and received regular airplay on radio stations both in Australia^[4] and in the USA. In November 2007 The Holidays were featured on the Next Crop segment of national youth radio station Triple J,^[5] soon followed by their signing with independent Australian record label Liberation Music, in early 2008.

The Holidays released their debut EP on 19 April 2008 to positive reviews^[6] which was followed by a substantial national tour to promote the release. They released their second EP "When The Ship Goes Down" in October 2008, which showed the band's natural progression with song-writing. Recorded with Wayne Connolly of The Vines and Josh Pyke fame, this EP was followed by a 20 date national tour, co-headlining with Brisbane band Yves Klein Blue.

The Holidays' debut album Post Paradise soon followed, after several months of writing and experimenting. Drawn away from the simplicity of their earlier indie guitar manifestation, Post Paradise showed the bands development with both the groove and layers of a song, combined with their recognizable pop hooks. This album differed from previous recordings by The Holidays' in that it was self-produced and recorded in various locations around Sydney and New South Wales.

"Moonlight Hours", the first single from their debut album was taken to radio in November 2009, and added to high rotation on national youth radio station Triple J. "Golden Sky" was the second single from The Holidays debut album, released in May 2010, which was followed by an east coast tour of Australia to promote the release. There is a music video for both these singles, made by Melbourne company Moop Jaw.

"Post Paradise" was released in Australia on 24 September 2010, in conjunction with third single, Broken Bones, which was added to radio nationally. The album has received very positive reviews since its release, which included amongst other reviews, being shortlisted for the Australian Music Prize and winning the Red Bull Award for Best Debut Album. Post Paradise also received "Album of the Week" in Drum Media, Beat Magazine, BMA Mag & The Brag,^[7] 4.5 stars in The Herald Sun, The Daily Telegraph & The Courier Mail, as well as positive reviews in The Age & The Sydney Morning Herald. Post Paradise won The Age Readers Choice Best Album of 2010,^[8] and was the Triple J Feature Album for the week commencing 26 December 2010.

The Holidays toured Post Paradise substantially in 2010 and 2011, including shows at SXSW in Austin, Texas, The Great Escape Festival in Brighton, UK, and Australian festivals including Groovin' the Moo, Laneway Festival and Splendour in the Grass. The Holidays' current live show incorporates a fifth touring member, David Zucker as percussionist, as well occasional backing vocalists.

Members[link]

• Simon Jones - Vocals, Guitar, Keys, Production
• Will Magnus - Lead Guitar, Keys
• Alex Kortt - Bass Guitar
• Andrew Kerridge - Drums, Percussion, Supreme Aficionado

Touring[link]

• Toured with Jamie T in May 2007
• Toured with The View in June 2007
• Toured with Ben Kweller in October 2007
• Played nationally at the 2008 St Jerome's Laneway Festival
• Toured with The Wombats in March 2008
• Toured with Lightspeed Champion in July 2008
• Played at Homebake 2008
• Played at Falls Festival 2008
• Toured with Bluejuice in April 2010
• Played at St Jerome's Laneway Festival 2011 across Australia
• Played at SXSW in Austin, Texas in March 2011
• Played at the Groovin' the Moo Festival across Australia in May 2011
• Toured with Cut Copy in Australia in May 2011
• Played at The Great Escape Festival in Brighton, and Dot to Dot Festival across the UK in May 2011
• Played at Splendour In The Grass in August 2011.
• Played at the inaugural Harvest Festival in Melbourne, Sydney & Brisbane in November 2011 along with The Flaming Lips, Portishead and Tv on the Radio.

Discography[link]

Albums[link]

• Post Paradise (Australia - 24 September 2010, Japan - January 2011)

EPs[link]

• The Holidays (19 April 2008)
• When The Ship Goes Down (11 October 2008)
• Golden Sky (Australia - May 2010, UK - August 2010 )

Singles[link]

• Moonlight Hours (December 2009)
• Golden Sky (May 2010)
• Broken Bones (September 2010)
• 2 Days (February 2011)

Awards & Nominations[link]

• 2010 The Age EG Award - Readers Choice Best Album of 2010 - Won
• 2010 Australian Music Prize - Nominated
• 2010 Australian Music Prize - Red Bull Award for Best Debut Album - Won
• 2011 Jägermeister Independent Music Awards - Breakthrough Independent Artist of the Year - Nominated

Further information[link]

• The Holidays song 'Telephone' was featured on an Australian television advertising campaign for Clearasil in late 2007.
• The Holidays song 'Golden Sky' was used in the Channel 7 summer 2010/2011 advertising campaign.
• Simon Jones collaborated & sung vocals with Australian dance duo Bag Raiders on the track 'Not Over' from their eponymous debut album.

References[link]