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This property can be understood in terms of the continued fraction representation for the golden ratio:

The Fibonacci numbers occur as the ratio of successive convergents of the continued fraction for φ , and the matrix formed from successive convergents of any continued fraction has a determinant of +1 or −1. The matrix representation gives the following closed-form expression for the Fibonacci numbers:

Taking the determinant of both sides of this equation yields Cassini's identity ,

Moreover, since A n A m = A n + m for any square matrix A , the following identities can be derived (they are obtained from two different coefficients of the matrix product, and one may easily deduce the second one from the first one by changing n into n + 1 ),

In particular, with m = n ,

These last two identities provide a way to compute Fibonacci numbers recursively in O (log( n )) arithmetic operations and in time O ( M ( n )log( n )) , where M ( n ) is the time for the multiplication of two numbers of n digits. This matches the time for computing the n th Fibonacci number from the closed-form matrix formula, but with fewer redundant steps if one avoids recomputing an already computed Fibonacci number (recursion with Designer DESIGN Statement Swirl Coin Bracelet Gold Asos Get Authentic For Sale Online For Sale US2wmbTIL
). [28]

The question may arise whether a positive integer x is a Fibonacci number. This is true if and only if one or both of $5{x}^{2}+4$ or $5{x}^{2}-4$ is a perfect square . [29] This is because Binet's formula Quality From China Wholesale Statement Bag MAGNOLIA BEAUTY by VIDA VIDA Choice Classic Cheap Online Rv245wa
can be rearranged to give

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## Methodology Procedures

Projects covered include state-owned and private projects in the Infrastructure, Electric Power, Oil Gas (excluding upstream), Water and Waste, and Mining sectors.

Where applicable, a project is considered as such until the end of construction and ramp-up to full capacity (with the exception of mining projects, where the date of commissioning signifies the end of the project). Projects must have a:

a) Start date b) Estimate for completion c) Initial investment estimate

For state-owned projects, initial information about the start date and estimates for completion and investment are taken from original signed contracts, along with addendums and annexes.

If original contracts cannot be obtained, information is taken from public documents, presentations, news articles from BNamericas archives, and external sources.

In cases where contracts are subject to approval by legislators, the start date is the date of the law or legislative approval.

In the case of private projects, information is taken from public sources, such as stock exchange filings, annual reports, company presentations, third party research and press releases.

In judging the timing of investment decisions for private projects, great care was taken to differentiate between large projects receiving environmental approval and proceeding directly to construction, and smaller projects where the investment decision hinges on securing financing.

Best care is taken to ensure that recorded data is correct at the time of entry and that each entry is backed with a relevant source.

The greatest care possible was taken to ensure consistency of information in order make a like-for-like comparison in project costs. Insofar as the figures rely on disclosure by the organization or company responsible, the figures can be considered to be conservative in nature.

No attempt was made to adjust figures for inflation during the course of research or for consideration of the time value of money.

In the case of state-owned projects, validation is carried out where possible with either a member of the consortium, EPC contractor, a relevant state agency or advisors to either party.

Projects costs are measured in US dollars. Where project costs are measured in a local currency, amounts are converted to US dollars at the date of the relevant announcement, signature or report.

The severity of deviation from original estimates is defined according to set criteria.

Changes in scope beyond the start date are not accounted for.

Where guidance is non-specific as to dates for completion, the following is used. Project to be completed by 2018 = December 31, 2017 Project to be completed in 2018 = December 31, 2018

­

ABSTRACT: Pyogenic liver abscesses are relatively uncommon but potentially life-threatening. Most infections that lead to an abscess are associated with underlying biliary disease, or are due to hematogenous spread from a variety of nonbiliary sites. Although most infections are polymicrobial, monomicrobialabscesses do occur. These exceedingly rare monomicrobial infections typically occur in immunocompromised individuals or in the presence of periodontal disease. A recent case involved a solitary pyogenic liver abscess due toin an immunocompetent 44-year-old woman with no underlying risk factors. Despite the cryptogenic nature of the abscess, the patient was treated successfully with percutaneous drainage and antibiotics.

Clinicians should consider the possibility of pyogenic liver abscess when patients present with fever, right upper quadrant pain, and shortness of breath.

Formation of a pyogenic liver abscess (PLA) is potentially life-threatening, with an estimated incidence rate of 2.3 cases per 100 000 patients.[ 1,2 ] Despite recent advances in the investigation and management of such abscesses, the mortality rate still ranges from 2% to 12%.[ 3,4 ] Amebic liver abscess is the most common form of liver abscess worldwide, whereas PLA is the most common form in North America,[ The Medium leathertrimmed backpack Burberry Geniue Stockist Online Cheap Sale Fast Delivery Reliable Online Cheap Sale Low Shipping Fee KanGrKrc
] and is usually the result of polymicrobial infection.[ Leather wallet Alexander Wang Shopping Online Original CRm6X35
] It is believed that the introduction of pyogenic bacteria to the liver most often occurs during an intra-abdominal infection, particularly one involving biliary tract pathology.[ 1,3 ] Less often, bacteria spread by way of the portal vein or hematogenously during systemic infection,[ 1 ] with periodontal disease being recognized increasingly as a source of pathogens in this setting.[ 7-9 ]

Although monomicrobial infections are rare, they do occur. A recent case involved Fusobacterium nucleatum , a nonmotile, gram-negative, anaerobic bacterium that normally inhabits the oropharynx and is a significant contributor to the formation of periodontal plaques.[ 10 ] Anaerobic bacteria have long been known to play a part in PLA formation, with one case series identifying anaerobes in up to 45% of cases.[ 11 ] Typically, systemic F. nucleatum infections occur in immunocompromised individuals or are of odontogenic origin.[ 7-9,12 ]

Case data A 44-year-old woman presented to the emergency room with a 6-day history of fever, chills, rigor, vomiting, headache, and shortness of breath. She had diffuse abdominal pain that had become localized to the right upper quadrant, and had progressed to being sharp and steady in nature and worse with inspiration. Her surgical history included a total hysterectomy that had been complicated by an Escherichia coli bacteremia and sepsis secondary to a surgical site infection. Her medical history included chronic gastroesophageal reflux disease, which was controlled with esomeprazole. Her dental history was unremarkable.