In mathematics, the distributive property of binary operations is a generalization of the distributive law, which asserts that the equality

`{\displaystyle x\cdot (y+z)=x\cdot y+x\cdot z}`

gives

is always true in elementary algebra. For example, in elementary arithmetic, one has

`{\displaystyle 2\cdot (1+3)=(2\cdot 1)+(2\cdot 3).}`

gives

Therefore, one would say that multiplication distributes over addition.

This basic property of numbers is part of the definition of most algebraic structures that have two operations called addition and multiplication, such as complex numbers, polynomials, matrices, rings, and fields. It is also encountered in Boolean algebra and mathematical logic, where each of the logical and (denoted

`{\displaystyle \,\land \,}`

gives

)and the logical or denoted

`{\displaystyle \,\lor \,}`

gives

) distributes over the other.**Definition**

Given a set

`{\displaystyle S}`

gives

and two binary operators`{\displaystyle \,*\,}`

gives

and`{\displaystyle \,+\,}`

gives

on`{\displaystyle S,}`

gives

i) the operation

`{\displaystyle \,*\,}`

gives

is left-distributive over (or with respect to)

`{\displaystyle \,+\,}`

gives

if, given any elements`{\displaystyle x,y,{\text{ and }}z}`

gives

of`{\displaystyle S,}`

gives

`{\displaystyle x*(y+z)=(x*y)+(x*z);}`

gives

iii) the operation

`{\displaystyle \,*\,}`

gives

is right-distributive over`{\displaystyle \,+\,}`

gives

if, given any elements`{\displaystyle x,y,{\text{ and }}z}`

gives

of`{\displaystyle S,}`

gives

`{\displaystyle (y+z)*x=(y*x)+(z*x);}`

gives

iii) and the operation

`{\displaystyle \,*\,}`

is

is distributive over`{\displaystyle \,+\,}`

gives

if it is left- and right-distributive.When

`{\displaystyle \,*\,}`

gives

is commutative, the three conditions above are logically equivalent.**Meaning**

The operators used for examples in this section are those of the usual addition

`{\displaystyle \,+\,}`

gives

and multiplication`{\displaystyle \,\cdot .\,}`

gives

If the operation denoted

`{\displaystyle \cdot }`

gives

is not commutative, there is a distinction between left-distributivity and right-distributivity:(left-distributive)

`{\displaystyle a\cdot \left(b\pm c\right)=a\cdot b\pm a\cdot c\qquad {\text{ (left-distributive) }}}`

gives

(right-distributive)

`{\displaystyle (a\pm b)\cdot c=a\cdot c\pm b\cdot c\qquad {\text{ (right-distributive) }}.}`

gives

In either case, the distributive property can be described in words as:

To multiply a sum (or difference) by a factor, each summand (or minuend and subtrahend) is multiplied by this factor and the resulting products are added (or subtracted).

If the operation outside the parentheses (in this case, the multiplication) is commutative, then left-distributivity implies right-distributivity and vice versa, and one talks simply of distributivity.

One example of an operation that is "only" right-distributive is division, which is not commutative:

`{\displaystyle (a\pm b)\div c=a\div c\pm b\div c.}`

gives

In this case, left-distributivity does not apply:

`{\displaystyle a\div (b\pm c)\neq a\div b\pm a\div c}`

gives

The distributive laws are among the axioms for rings (like the ring of integers) and fields (like the field of rational numbers). Here multiplication is distributive over addition, but addition is not distributive over multiplication.

Examples of structures with two operations that are each distributive over the other are Boolean algebras such as the algebra of sets or the switching algebra.

Multiplying sums can be put into words as follows: When a sum is multiplied by a sum, multiply each summand of a sum with each summand of the other sum (keeping track of signs) then add up all of the resulting products.

]]>**Gist**

Tarbela Dam, giant rock-fill dam on the Indus River, Pakistan. Built between 1968 and 1976, it has a volume of 138,600,000 cubic yards (106,000,000 cubic m). With a reservoir capacity of 11,098,000 acre-feet (13,690,000,000 cubic m), the dam is 469 feet (143 m) high and 8,997 feet (2,743 m) wide at its crest. Tarbela Dam is one of two main structures (the other is Mangla Dam on the Jhelum River) in the Indus Basin project, which resulted from the Indus Waters Agreement between India and Pakistan. Together with their subsidiary dams, Tarbela and Mangla were built to control seasonal fluctuations on the Indus River.

**Summary**

The Indus River basin extends from the Himalaya Mountains that form the northeastern boundary of Pakistan to the alluvial plains of Sindh near the Arabian Sea coastline. Tarbela Dam is part of the Indus Basin Project, which resulted from a water treaty signed in 1960 between India and Pakistan. This treaty guaranteed Pakistan water supplies independent of upstream control by India. Designed primarily for water storage rather than power generation, the dam was completed in 1977.

Turquoise waters of the Indus River (to the south of the dam) reflect the high proportion of silt and clay suspended in waters released by the spillways (chutes on either of side of the main dam). With a volume of 142,000,000 cubic meters, the Tarbela Dam is the largest earth and rock fill dam in the world and stands 147 meters above the Indus riverbed. Its reservoir occupies an area of 37 square kilometers. While the dam has fulfilled its purpose in storing water for agricultural use in Pakistan, there have been environmental consequences to the Indus river delta. Reduction of seasonal flooding and reduced water flows to the delta have decreased mangrove stands and the abundance of some fish species.

**Details**

Tarbela Dam is an earth-filled dam along the Indus River in Pakistan's Khyber Pakhtunkhwa province. It is located mainly in the Haripur Tehsil Ghazi.

It is about 20 km (10 mi) from the city of Swabi KPK, 105 km (65 mi) northwest of Islamabad, and 125 km (80 mi) east of Peshawar. It is the largest earth-filled dam in the world. The dam is 143 metres (470 ft) high above the riverbed and its reservoir, Tarbela Lake, has a surface area of approximately 250 square kilometres (97 sq mi).

The Tarbela Dam is located on the Indus River near the village of Tarbela in Bara, approximately 30 kilometers from the town of Attock. Positioned where the Indus River emerges from the foothills of the Himalayas and enters the Potwar Plateau, the dam features a reservoir for water storage. The average annual flow available is 101 billion cubic meters (3221 m^3/sec). It stands 143 meters tall and covers an area of 243 square kilometers. It has a storage capacity of 11.9 billion cubic meters of water and has nine gates to control the outflow of water. The dam was completed in 1976 and was designed to utilize water from the Indus River for irrigation, flood control, and the generation of hydroelectric power by storing flows during the monsoon period while subsequently releasing stored water during the low flow period in winter. The installed capacity of the 4,888 MW Tarbela hydroelectric power stations will increase to 6,418 MW after completion of the planned fifth extension financed by Asian Infrastructure Investment Bank and the World Bank. Then, it will be the 12th largest hydroelectric dam in the world, for electricity production capacity.

**Project description**

The dam is at a narrow spot in the Indus River valley, named after the town of Tarbela in the Haripur District of the Hazara Division within the Khyber Pakhtunkhwa province of Pakistan.

The main dam wall, built of earth and rock fill, stretches 2,743 metres (8,999 ft) from the island to river right, standing 148 metres (486 ft) high. A pair of concrete auxiliary dams spans the river from the island to river left. The dam's two spillways are on the auxiliary dams rather than the main dam. The main spillway has a discharge capacity of 18,406 cubic metres per second (650,000 cu ft/s) and the auxiliary spillway, 24,070 cubic metres per second (850,000 cu ft/s). Annually, over 70% of water discharged at Tarbela passes over the spillways and is not used for hydropower generation.

Five large tunnels were constructed as part of the outlet works. Hydroelectricity is generated from turbines in tunnel 1 through 3, while tunnels 4 and 5 were designed for irrigation use. Both tunnels are to be converted to hydropower tunnels to increase Tarbela's electricity-generating capacity. These tunnels were originally used to divert the Indus River while the dam was being constructed.

MA hydroelectric power plant on the right side of the main dam houses 14 generators fed with water from outlet tunnels 1, 2, and 3. There are four 175 MW generators on tunnel 1, six 175 MW generators on tunnel 2, and four 432 MW generators on tunnel 3, for a total generating capacity of 3,478 MW.

Tarbela Reservoir is 80.5 kilometres (50.0 mi) long, with a surface area of 250 square kilometres (97 sq mi). The reservoir initially stored 11,600,000 acre-feet (14.3 km3) of water, with a live storage of 9,700,000 acre-feet (12.0 km^3), though this figure has been reduced over the subsequent 35 years of operation to 6,800,000 acre-feet (8.4 km^3) due to silting. The maximum elevation of the reservoir is 1,550 ft (470 m) above MSL and the minimum operating elevation is 1,392 ft (424 m) above MSL. The catchment area upriver of the Tarbela Dam is spread over 168,000 square kilometres (65,000 sq mi) of land largely supplemented by snow and glacier melt from the southern slopes of the Himalayas. There are two main Indus River tributaries upstream of the Tarbela Dam. These are the Shyok River, joining near Skardu, and the Siran River near Tarbela.

**Background**

Tarbela Dam was constructed as part of the Indus Basin Project after signing of the 1960 Indus Waters Treaty between India and Pakistan. The purpose was to compensate for the loss of water supplies of the eastern rivers (Ravi, Sutlej and Beas) that were designated for exclusive use by India per terms of the treaty. By the mid-1970s, power generation capacity was added in three subsequent hydro-electrical project extensions which were completed in 1992, installing a total of 3,478 MW generating capacity.

**Construction**

Construction of Tarbela Dam was carried out in three stages to meet the diversion requirements of the river. Construction was undertaken by the Italian firm Salini Impregilo.

**Stage 1**

In the first stage, the Indus River was allowed to flow in its natural channel, while construction works commenced on the right bank where a 1,500 feet (460 meters) long and 694.8 feet (211.8 meters) wide diversion channel was being excavated along with a 105 feet (32 meters) high buttress dam that was also being constructed. Stage 1 construction lasted approximately 2½ years.

**Stage 2**

The main embankment dam and the upstream blanket were constructed across the main valley of the river Indus as part of the second stage of construction. During this time, water from the Indus river remained diverted through the diversion channel. By the end of construction works in stage 2, tunnels had been built for diversion purposes. Stage 2 construction took 3 years to complete.

**Stage 3**

Under the third stage of construction, works were carried out on the closure of the diversion channel and construction of the dam in that portion while the river was made to flow through diversion tunnels. The remaining portion of upstream blanket and the main dam at higher levels was also completed as part of stage 3 works, which were concluded in 1976.

An area of about 260 square kilometers and about 82,000 acres (33,000 ha) of land was acquired for construction. The large reservoir of the dam submerged 135 villages, which resulted in the displacement of a population of about 96,000 people, many of whom were relocated to townships surrounding the Tarbela Reservoir or in adjacent higher valleys.

For the land and built-up property acquired under the Land Acquisition Act of 1984, a cash compensation of Rs 469.65 million was paid to those affected. In the absence of a national policy, the resettlement concerns of the people displaced by the Tarbela Dam were addressed on an ad hoc basis. In 2011, many such people had still not been resettled or given land in compensation for their losses by the government of Pakistan, in accordance with its contractual obligations with the World Bank.

**Lifespan**

Because the source of the Indus River is glacial meltwater from the Himalayas, the river carries huge amounts of sediment, with an annual suspended sediment load of 200 million tons. Live storage capacity of Tarbela reservoir had declined more than 33.5 per cent to 6.434 million acre feet (MAF) against its original capacity of 9.679 MAF because of sedimentation over the past 38 years. The useful life of the dam and reservoir was estimated to be approximately 50 years. However, sedimentation has been much lower than predicted, and it is now estimated that the useful lifespan of the dam will be 85 years, to about 2060.

Pakistan plans to construct several large dams upstream of Tarbela, including the Diamer-Bhasha Dam. Upon completion of the Diamer-Bhasha dam, sediment loads into Tarbela will be decreased by 69%.

**Project benefits**

In addition to fulfilling the primary purpose of the dam, i.e., supplying water for irrigation, Tarbela Power Station has generated 341.139 billion kWh of hydro-electric energy since commissioning. A record annual generation of 16.463 billion kWh was recorded during 1998–99. Annual generation during 2007–08 was 14.959 billion kWh while the station shared peak load of 3702 MW during the year, which was 23.057% of total WAPDA system peak.

**Tarbela-IV Extension Project**

Tarbela dam extension-IV was planned in June, 2012, and PC-1 was developed for the project. US ambassador Richard Olson offered aid for construction of this project during his visit to Pakistan, in March, 2013. In September 2013, Pakistan's Water and Power Development Authority signed a Rs. 26.053 billion contract with Chinese firm Sinohydro and Germany's Voith Hydro for executing civil works on the 1,410 MW Tarbela-IV Extension Project. Construction commenced in February 2014, and was completed in February 2018.

This project was constructed on Tunnel No. 4 of Tarbela Dam. It consists of three turbine-generator units, each with a capacity of 470 MW. The project is expected to provide an average of 3.84 billion units of electricity annually to the National Grid. It is intended to help supplement electricity supply during the high-demand summer months.

Annual benefits of the project were estimated at Rs. 30.7 billion. On an annual basis, over 70% of water passing through Tarbela is discharged over spillways, while only a portion of the remaining 30% is used for hydropower generation.

The Water and Power Development Authority in Pakistan says the third and last unit at its 1,410-MW Tarbela 4th Extension Hydropower Project has been synchronized with the National Grid. With this extension, the installed capacity of the Tarbela Hydel Power Station has increased to 4,888 MW.

**Financing**

The project's cost was initially estimated to be $928 million, but the cost was revised downwards to $651 million. The World Bank had agreed to provide an $840 million loan for the project in June 2013.

The loan had two components: The first is a $400 million International Development Association loan, which will be lent as a concessional loan at low interest rates. The second portion consists of a $440 million from the World Bank's International Bank for Reconstruction and Development. Pakistan's Water and Power Development Authority was to provide the remaining $74 million required for construction, before the project's cost was downwardly revised by $277 million. Interest costs for the loans are estimated to cost $83.5 million.

Because of revised lower costs to $651 million from $928 million, the World Bank permitted Pakistani officials to expedite completion of the project by 8 months at a cost of an additional $51 million. Pakistani officials were also permitted to divert $126 million towards the Tarbela-V Extension Project.

**Tarbela-V Extension Project**

The Tarbela Dam was built with five original tunnels, with the first three dedicated to hydropower generation, and the remaining two slated for irrigation use. The fourth phase extension project uses the first of the two irrigation tunnels, while the fifth phase extension will use the second irrigation tunnel. Pakistan's Water and Power Development Authority sought expressions of interest for the Tarbela-V Extension Project in August 2014, and was given final consent for construction in September 2015.

The hydropower project of tunnel 5 has two major components: power generation facilities and power evacuation facilities. The major works included under the project are modifications to tunnel 5 and building a new power house and its ancillaries to generate about 1,800GWh of power annually, a new 50 km of 500kV double-circuit transmission line from Tarbela to the Islamabad West Grid Station for power evacuation, and a new 500kV Islamabad West Grid Station.

Construction commenced in August 2021 and will require an estimated 3.5 years for completion. The project will require the installation of three turbines with a capacity of 510 MW each in Tarbela's fifth tunnel which was previously dedicated to agricultural use. Upon completion, the total power generating capacity of Tarbela Dam will increase to 6,418 MW.

**Financing**

In November 2015, the World Bank affirmed that it would finance at least $326 million of the project's estimated $796 million cost which includes $126 million of funding that was diverted from the $840 million fourth phase extension project after costs for that project were revised downwards. In September 2016, the World Bank approved an additional financing of $390 million for the fifth extension hydropower project of Tarbela dam that will support the scaling up of the power generation capacity by adding 1,530 megawatts to the existing tunnel 5.

The project will be financed by the International Bank for Reconstruction and Development (IBRD), with a variable spread and 20-year maturity, including a six-year grace period. This will be the first World Bank-supported project in South Asia to be jointly financed with the Asian Infrastructure Investment Bank (AIIB) which will be providing $300m and the Government of Pakistan $133.5m. The total cost of the project is $823.5m.

]]>

**Gist**

Pulsars are very compact stars that radiate radio waves with very regular variations. In 1974 Russell Hulse and Joseph Taylor discovered a pulsar comprised of two stars in very close proximity that rotate around each other. Hulse and Taylor could demonstrate that the stars’ radiation and movements correspond with Einstein’s general theory of relativity. Among other things, this theory predicts that the pulsar would emit energy in the form of gravitational waves, which should result in slowly declining intervals. Taylor was able to confirm this in 1978.

**Summary**

Russell Alan Hulse (born November 28, 1950, New York, New York, U.S.) is an American physicist who in 1993 shared the Nobel Prize for Physics with his former teacher, the astrophysicist Joseph H. Taylor, Jr., for their joint discovery of the first binary pulsar.

Hulse studied at Cooper Union College in New York City (B.S., 1970) and earned a Ph.D. degree in physics (1975) from the University of Massachusetts at Amherst, where he was a graduate student under Taylor. Using the large radio telescope at Arecibo, Puerto Rico, they discovered dozens of pulsars, which are rapidly spinning neutron stars that emit rapid, regular bursts of radio waves. Irregularities in the radio emissions of the pulsar PSR 1913 + 16 led them to deduce that the pulsar had a companion neutron star with which it was locked in a tight orbit. This discovery was made by Taylor and Hulse in 1974.

PSR 1913 + 16 proved doubly important because it provided the first means of detecting gravity waves. The two stars’ enormous interacting gravitational fields were affecting the regularity of the radio pulses, and by timing these and analyzing their variations, Taylor and Hulse found that the stars were rotating ever faster around each other in an increasingly tight orbit. This orbital decay is presumed to occur because the system is losing energy in the form of gravity waves. This finding, as reported by Taylor and Hulse in 1978, afforded the first experimental evidence for the existence of the gravitational waves predicted by Albert Einstein in his general theory of relativity.

In 1977 Hulse changed fields from astrophysics to plasma physics and joined the Plasma Physics Laboratory at Princeton University. There he conducted research associated with the Tokamak Fusion Test Reactor, an experimental nuclear-fusion facility. In 2004 Hulse began teaching at the University of Texas at Dallas, where he founded the Science and Engineering Education Center.

**Details**

Russell Alan Hulse (born November 28, 1950) is an American physicist and winner of the Nobel Prize in Physics, shared with his thesis advisor Joseph Hooton Taylor Jr., "for the discovery of a new type of pulsar, a discovery that has opened up new possibilities for the study of gravitation".

**Biography**

Hulse was born in New York City and graduated from the Bronx High School of Science and the Cooper Union. He received his PhD in physics from the University of Massachusetts Amherst in 1975.

While working on his PhD dissertation, he was a scholar in 1974 at the Arecibo Observatory in Puerto Rico of Cornell University. There he worked with Taylor on a large-scale survey for pulsars. It was this work that led to the discovery of the first binary pulsar.

In 1974, Hulse and Taylor discovered binary pulsar PSR B1913, which is made up of a pulsar and black companion star. Neutron star rotation emits impulses that are extremely regular and stable in the radio wave region and is nearby condensed material body gravitation (non-detectable in the visible field). Hulse, Taylor, and other colleagues have used this first binary pulsar to make high-precision tests of general relativity, demonstrating the existence of gravitational radiation. An approximation of this radiant energy is described by the formula of the quadrupolar radiation of Albert Einstein (1918).

In 1979, researchers announced measurements of small acceleration effects of the orbital movements of a pulsar. This was initial proof that the system of these two moving masses emits gravitational waves.

**Later years**

After receiving his PhD, Hulse did postdoctoral work at the National Radio Astronomy Observatory in Green Bank, West Virginia. He moved to Princeton, where he has worked for many years at the Princeton Plasma Physics Laboratory. He has also worked on science education, and in 2003 joined the University of Texas at Dallas as a visiting professor of physics and of mathematics and science education.

In 1993, Hulse and Taylor shared the Nobel Prize in Physics for the discovery of the first binary pulsar.

Hulse was elected a Fellow of the American Association for the Advancement of Science in 2003, and is cited in the American Men and Women of Science.

In 2004, Hulse joined University of Texas at Dallas and became the Founding Director of UT Dallas Science and Engineering Education Center (SEEC).

In July 2007 Hulse joined the Aurora Imaging Technology advisory board.

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1. I went through a period when I felt my film characters were having more fun than I was. It might partly explain why I ended up tattooed or doing certain extreme things in my life. - Angelina Jolie

2. If there is any secret to my success, I think it's that my characters are very real to me. I feel everything they feel, and therefore I think my readers care about them. - Sidney Sheldon

3. I think people throughout the world identify with my characters. - Sidney Sheldon

4. I mean I tried to transform myself through characters throughout my career. - Charlize Theron

5. I was of the generation where most of the Disney princesses and female characters were not girls that I admired. They just weren't characters I looked up to and identified with. - Angelina Jolie

6. They really stay just characters to me. I look at them, and I don't see always the same person up there. And hopefully, people will see that too. Because it's very easy to bore people, and that's a killer. So hopefully that won't happen. - Charlize Theron

7. People say that to me and I think what unites all my characters is that they are hurt; it's most accurate to say I play characters that are hurt but are responding to their environment. - Ashley Judd

8. Acting allows me to explore new worlds, to discover characters by delving into their lives, and ultimately to become someone else entirely. - Pierce Brosnan.

]]>Because he's always spotted!

* * *

What kind of flower doesn't sleep at night?

The Day-zzz.

* * *

Why do birds fly south for the winter?

Its easier than walking!

* * *

What kind of key opens a banana?

A monkey!

* * *

Did you hear about the vampire bicycle that went round biting people's arms off?

It was a vicious cycle.

* * *]]>

#9905. What does the term in Chemistry Organosulfur chemistry mean?

#9906. What does the term in Chemistry Organic redox reaction mean?

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]]>#2628. What does the medical term Urinary tract infection mean?

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]]>Prove that the two equations have one solution if the following two conditions are satisfied:

B*D = E*A

and

A*E^2 - D*B^2 = 4*A*D*(F-C)

As you know, this is simply a special case.

Your answer is right always. I mean, your condition is satisfied in this case too.

.]]>