Angular momentum coupling

In mathematics, an Euler brick, named after Leonhard Euler, is a cuboid whose edges and face diagonals all have integer lengths. A primitive Euler brick is an Euler brick whose edge lengths are relatively prime.

Properties

Alternatively stated, an Euler brick is a solution to the following system of Diophantine equations:

$\{\begin{array}{l}{a}^2+b^2=d^2\\ a^2+c^2=e^2\\ b^2+c^2=f^2\end{array}$

Euler found at least two parametric solutions to the problem, but neither give all solutions.

Given an Euler brick with edges (a, b, c), the triple (bc, ac, ab) constitutes an Euler brick as well.

Examples

The smallest Euler brick, discovered by Paul Halcke in 1719, has edges $(a,b,c)=(44,117,240)$ and face diagonals 125, 244, and 267.

Some other small primitive solutions, given as edges (a, b, c) — face diagonals (d, e, f), are below:

• (85, 132, 720) — (157, 725, 732);
• (140, 480, 693) — (500, 707, 843);
• (160, 231, 792) — (281, 808, 825);
• (240, 252, 275) — (348, 365, 373).

Perfect cuboid

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Here is my homepage ... new launch ec A perfect cuboid (also called a perfect box) is an Euler brick whose space diagonal also has integer length.

In other words, the following equation is added to the system of Diophantine equations defining an Euler brick:

$a^2+b^2+c^2=g^2.$

Template:Asof, no example of a perfect cuboid had been found and no one has proven that none exist. Exhaustive computer searches show that, if a perfect cuboid exists, one of its edges must be greater than 3·10¹². Furthermore, its smallest edge must be longer than 10¹⁰.

Some facts are known about properties that must be satisfied by a primitive perfect cuboid, if one exists, based on modular arithmetic: Template:Cn

• One edge, two face diagonals and the body diagonal must be odd, one edge and the remaining face diagonal must be divisible by 4, and the remaining edge must be divisible by 16
• 2 edges must have length divisible by 3 and at least 1 of those edges must have length divisible by 9
• 1 edge must have length divisible by 5.
• 1 edge must have length divisible by 7.
• 1 edge must have length divisible by 11.
• 1 edge must have length divisible by 19.
• 1 edge or space diagonal must be divisible by 13.
• 1 edge, face diagonal or space diagonal must be divisible by 17.
• 1 edge, face diagonal or space diagonal must be divisible by 29.
• 1 edge, face diagonal or space diagonal must be divisible by 37.

Solutions have been found where the space diagonal and two of the three face diagonals are integers, such as:

$(a,b,c)=(672,153,104).$

Solutions are also known where all four diagonals but only two of the three edges are integers, such as:

$(a,b,c)=(18720,\sqrt{211773121},7800)$

and

$(a,b,c)=(520,576,\sqrt{618849}).$

Perfect parallelepiped

A perfect parallelepiped is a parallelepiped with integer-length edges, face diagonals, and body diagonals, but not necessarily with all right angles; a perfect cuboid is a special case of a perfect parallelepiped. In 2009, a perfect parallelepiped was shown to exist, answering an open question of Richard Guy. Solutions with only a single oblique angle have been found.

Notes

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References

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  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
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  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang