Copyright © Philip M. Parker, INSEAD. Terms of Use.

Power

Specialty Definition: Power

Top     

Specialty Definition: Electric power

(From Wikipedia, the free Encyclopedia)

Electric power, often known as power or electricity, involves the production and delivery of electrical energy in sufficient quantities to operate domestic appliances, office equipment, industrial machinery and provide sufficient energy for both domestic and commercial lighting, heating, cooking and industrial processes.

History

Although electricity had been known to be produced as a result of the chemical reactions that take place in an electrolytic cell since Alessandro Volta reported doing so in 1800, its production by this means was, and still is, expensive. In 1831, Michael Faraday devised a machine that generated electricity from rotary motion, however it took almost 50 years for the technology to reach a commercially viable stage. In 1878, Thomas Edison developed and sold a commercially viable replacement for gas lighting and heating using locally generated and distributed direct current electricity. In Edison's direct current system, generating stations needed to be close to or on the consumer's premises. To combat losses, and the voltage drops at end of the distribution, extra power generating stations needed to be installed. As Edison was not able to produce a system that permitted multiple generators to be connected together, expansion of his system required whole new generating stations to be constructed. The need for additional power plants is primarily explained by Ohm's law: as losses increase in proportion to the square of the current, or load, and in proportion to the resistance, having long cable runs in the Edison system meant using dangerous voltages in some places, or expensive and large cables or both.

Nikola Tesla, who had worked for Edison for a short time and appreciated the electrical theory in a way that Edison did not, devised an alternative system using alternating current. Tesla realised that while doubling the voltage would halve the current and reduce losses by three-quarters, only an alternating current system allowed the transformation between voltage levels in different parts of the system. He went on to develop the overall theory of his system, devising theoretical and practical alternatives for all of the direct current appliances then in use, and patented his novel ideas in 1887, in thirty separate patents.

In 1888, Tesla's work came to the attention of George Westinghouse, who owned a patent for a transformer and had been operating an alternating current lighting plant in Great Barrington, Massachusetts since 1886. While Westinghouse's system could use Edison's lights and had heaters, it did not have a motor. With Tesla and his patents, Westinghouse built a power system for a gold mine in Teluride in 1891, with a water driven 100 horsepower generator powering a 100 horsepower motor over a 2.5 mile (4 km) power line. Then, in a deal with General Electric, which Edison had been forced to sell, Westinghouse's company went on to construct a power station at the Niagara Falls, with three 5,000 horsepower Tesla generators supplying electricity to an aluminium smelter at Niagara and the town of Buffalo 22 mile (35 km) away. The Niagara power station commenced operation on April 20 1895. Its opening set the scene for the electric power industry for over a hundred years.

Electric Power Today

Today, Tesla's alternating-current electric power system is still the primary means of delivering electrical energy to consumers throughout the world. While high-voltage direct current (HVDC) is used to transmit large quantities of electricity over long distances, the bulk of electricity generation, transmission, distribution and supply takes place using alternating current.

In many countries, electric power companies own the whole infrastructure from generating stations to transmission and distribution infrastructure. For this reason, electric power is viewed as a natural monopoly. The industry is generally heavily regulated, often with price controls and is frequently government-owned and operated. In some countries, wholesale electricity markets operate, with generators and retailers trading electricity in a similar manner to shares and currency.

See also

• Electric power transmission
• Electricity distribution
• Distributed generation
• Electricity retailing
• Electricity generation
• Auxiliary power
• Power control
• Power factor
• Power transmission
• Uninterruptible power supply
• Electrical generator
• Electrical bus
• New Zealand Electricity Market
• Electricity market
• Transformer

Further reading:

• The Man Who Invented the Twentieth Century - Nicola Tesla, Forgotten Genius of Electricity, Robert Lomas, Headline Book Publishing, London, 1999.

  Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Electric power."

Top     

Electricity distribution

(From Wikipedia, the free Encyclopedia)

Electricity distribution is the penultimate process in the delivery of electric power, i.e. the part between generation and user consumption. Other processes in power delivery are transmission and retailing.

In the early days of electricity generation, direct current (DC) generators were connected to loads at the same voltage. This imposed limitations on how far the distribution system could extend because of the voltage drop. It also meant that cables and lines had to be made from relatively large diameter copper in order to carry the high currentss required to meet the demand of distributed load. (Power lost in generating heat in a conductor is proportional to the square of the current ie Losses = I²R. These losses can be reduced by reducing the resistance (R) of the conductor, hence increasing the diameter; or, more effectively, by reducing the current (I).)

The adoption of alternating current (AC) for electricity generation dramatically changed the situation. Power transformers could be used to raise the voltage from the generators and reduce it to supply loads. Increasing the voltage reduced the current and hence the size of conductors and distribution losses, making it more economic to distribute power over long distances. The ability to transform to extra-high voltages enabled generators to be located far from loads and transmission systems to interconnect generating stations and distribution networkss.

In North America, early distribution systems were single phase and used a voltage of 2200 volts corner-grounded delta. Over time, this was gradually increased to 2400 volts. As cities grew and the use of three-phase power became more widespread, most 2400 volt systems were upgraded to 2400/4160Y three-phase systems, which also benefitted from better surge suppression due to the grounded neutral. Some city and suburban distribution systems continue to use this range of voltages, but most have been converted to 7200/12470Y.

European systems used higher voltages, generally 3300 volts to ground, in support of the 220Y/380 volt power systems used in those countries. In the UK, urban sysytems progressed to 6.6 kV and then 11 kV, the most common distribution voltage.

Rural Electrification systems, in contrast to urban systems, tend to use higher voltages because of the longer distances covered by those distribution lines. 7200 volts is commonly used in the United States. Other voltages are occasionally used in unusual situations or where a local utility simply has engineering practices that differ from the norm.

In New Zealand, Australia and South Africa, single wire earth return systems (SWER) are used to electrify remote rural areas.

Electricity industry reform has led to the creation of electricity markets through the separation of contestable retailing from distribution, a natural monopoly and the separation of the monopoly transmission from generation. It also led to the development of new terminology to describe the distributor such as Line company, Wires Business and Network Company.

See also Distributed generation.

U.S. and U.K. terminology

U.S.	U.K.
Grounded	Earthed
Wye or Y	Star

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Electricity distribution."

Top     

Exponentiation

(From Wikipedia, the free Encyclopedia)

In mathematics, exponentiation is a process of repeated multiplication, in much the same way that multiplication is a process of repeated addition. For example, 3⁴ equals 3 × 3 × 3 × 3 equals 81. Here, 3 is the base, 4 is the exponent (written as a superscript), and 81 is 3 raised to the 4th power. Notice that the base 3 appears 4 times in the repeated multiplication, because the exponent is 4. In contexts where superscripts are not available, such as computer languages and e-mail, 3⁴ is commonly written "3^4".

Raising 10 to a power is easy: for example 10⁷ = 10,000,000 with seven zeros. Exponentiation with base 10 is often used in the physical sciences to describe large or small numbers in scientific notation; for example, 299792458 can be written as 2.99792458 × 10⁸ and then approximated as 2.998 × 10⁸ if this is useful. SI prefixes are also used to describe small or large quantities, but even these are based on powers of ten; for example, the prefix kilo means 10³ = 1000, so a kilometre is 1000 metres.

Exponents with base 2 are used in computer science; for example, there are 2ⁿ possible values for a variable that takes n bits to store in memory. A kilobyte usually stands for 2¹⁰ = 1024 bytes, but sometimes also for 10³ = 1000 bytes; the term kibibyte has been suggested for the former meaning.

Exponents with base e (a transcendental number approximately equal to 2.71828) are described by the exponential function exp x = e^x.

We define exponentiation of a positive real number x with a negative exponent by

x^-n = 1/xⁿ

and with a fractional exponent as

So for instance 10⁻³ = 0.001 and 8^2/3 = 4. x^y where y is an arbitrary real number can then be defined by continuity.

Exponentiation of real numbers, and even complex numbers, can also be understood with the aid of the exponential function and its inverse, the natural logarithm; in general, we can define

x^y = exp (y ln x).

For more on exponents in real and complex numbers, and other situations relevant to mathematical analysis, see Exponential function. That article also lists certain exponential laws (more general than the algebraic laws listed below) that apply in these situations.

Exponentiation in abstract algebra

Exponentiation can also be understood purely in terms of abstract algebra, if we limit the exponents to integers.

Specifically, suppose that X is a set with a power-associative binary operation, which we will write multiplicatively. In this very general situation, we can define xⁿ for any element x of X and any nonzero natural number n, by simply multiplying x by itself n times; by definition, power associativity means that it doesn't matter in which order we perform the multiplications.

Now additionally suppose that the operation has an identity element 1. Then we can define x⁰ to be equal to 1 for any x. Now xⁿ is defined for any natural number n, including 0.

Finally, suppose that the operation has inverses. Then we can define x^-n to be the inverse of xⁿ when n is a natural number. Now xⁿ is defined for any integer n.

In particular, xⁿ is defined for any integer n and any element x of a group. However, because we need only power associativity and not general associativity, the concept of exponentiation also makes sense in some other useful situations, such as the nonzero octonions.

Exponentiation in this purely algebraic sense satisfies the following laws (whenever both sides are defined):

• x^m+n = x^mxⁿ
• x^m-n = x^m/xⁿ
• x^-n = 1/xⁿ = (1/x)ⁿ
• x⁰ = 1
• x¹ = x
• x^-1 = 1/x
• (x^m)ⁿ = x^mn

Here, we use a division slash ("/") to indicate multiplying by an inverse, in order to reserve the symbol x^-1 for raising x to the power -1, rather than the inverse of x. However, as one of the laws above states, x^-1 is always equal to the inverse of x, so the notation doesn't matter in the end.

If in addition the operation is commutative and alternative, then we have some additional laws:

• (xy)ⁿ = xⁿyⁿ
• (x/y)ⁿ = xⁿ/yⁿ

Here, alternativity is a condition stronger than power associativity but weaker than general associativity. So in particular, this law is satisfied in an Abelian group, such as the multiplicative group of elements from a given field that are distinct from zero.

Notice that in this algebraic context, 0⁰ is always equal to 1. In some contexts involving calculus, it may be more useful to leave 0⁰ undefined.

However, when exponentiation is purely algebraic, that is when the exponents are taken only to be integers, then it's generally most useful to let 0⁰ be 1, just like every other case of x⁰. For example, if you expand (0 + x)ⁿ using the binomial theorem, you'll want to use 0⁰ = 1.

If we take this whole theory of exponentiation in an algebraic context but write the binary operation additively, then "exponentiation is repeated multiplication" can be reinterpreted as "multiplication is repeated addition". Thus, each of the laws of exponentiation above has an analogue among laws of multiplication.

When one has several operations around, any of which might be repeated using exponentiation, it's common to indicate which operation is being repeated by placing its symbol in the superscript. Thus, x^*n is x * ··· * x, while x^#n is x # ··· # x, whatever the operations * and # might be.

Exponential notation is also used, especially in group theory, to indicate conjugation. That is, g^h = h^-1gh, where g and h are elements of some group. Although conjugation obeys some of the same laws as exponentiation, it's not an example of repeated multiplication in any sense. A quandle is an algebraic structure in which these laws of conjugation play a central role.

Exponentiation over sets

The above algebraic treatment of exponentiation builds a finitary operation out of a binary operation. In more general contexts, one may be able to define an infinitary operation directly on an indexed set.

For example, in the arithmetic of cardinal numbers, it makes sense to say

for any index set I and cardinal numbers k_i. By taking k_i = k for every i, this can be interpreted as a repeated product, and the result is k^I. In fact, this result depends only on the cardinality of I, so we can define exponentiation of cardinal numbers so that k^l is k^I for any set I whose cardinality is l.

This can be done even for operations on sets or sets with extra structure. For example, in linear algebra, it makes sense to index direct sums of vector spaces over arbitrary index sets. That is, we can speak of

where each V_i is a vector space. Then if V_i = V for each i, the resulting direct sum can be written in exponential notation as V^(+)I, or simply V^I with the understanding that the direct sum is the default. We can again replace the set I with a cardinal number k to get V^k, although without choosing a specific standard set with cardinality k, this is defined only up to isomorphism. Taking V to be the field R of real numbers (thought of as a vector space over itself) and k to be some natural number n, we get the vector space that is most commonly studied in linear algebra, the Euclidean space Rⁿ.

If the base of the exponentiation operation is itself a set, then by default we assume the operation to be the Cartesian product. In that case, S^I becomes simply the set of all functions from I to S. This fits in with the exponentation of cardinal numbers once gain, in the sense that |S^I| = |S|^|I|, where |X| is the cardinality of X. We also have |PX| = 2^|X|, where PX is the power set of X. (This is where the term "power set" comes from.)

Note that exponentiation of cardinal numbers doesn't match up with exponentiation of ordinal numbers, which is defined by a limit process. In the ordinal numbers, a^b is the smallest ordinal number greater than a^c for c < b when b is a limit ordinal, and of course a^b+1 := a^ba.

In category theory, we learn to raise any object in a wide variety of categories to the power of a set, or even to raise an object to the power of an object, using the exponential.

External link

• sci.math FAQ: What is 0⁰?

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Exponentiation."

Top     

IBM POWER

(From Wikipedia, the free Encyclopedia)

POWER is a RISC CPU architecture designed at IBM. The name, arguably, stands for Performance Optimization With Enhanced RISC. POWER series CPUs are used as the main CPU in many of IBM's servers, minicomputers, workstations, and supercomputers. The POWER architecture was used to develop (and remains very similar to) the PowerPC architecture, used in all Apple Macintosh computers, some IBM workstations, as well as a number of embedded applications.

The POWER design is descended directly from the earlier IBM 801 CPU, widely considered to be the first true RISC chip design. It was used in a number of applications inside IBM hardware, but did not become public until they released the poorly-performing IBM PC/RT in the mid-1980s.

At about the same time the PC/RT was being released, IBM started the America Project, to design the most powerful CPU on the market. They were interesting primarily in fixing two problems in the 801 design in the resulting POWER design:

1. the 801 required all instructions to complete in one clock cycle, which eliminated floating point instructions
2. although the decoder was pipelined as a side effect of these single-cycle operations, they didn't use superscalar effects

Floating point became a focus for the America Project, and IBM were able to use new algorithms developed in the early 1980s that could support 64-bit double-precision multiplies and divides in a single cycle. The FPU portion of the design was separate from the instruction decoder and integer parts, allowing the decoder to send instructions to both the FPU and ALU (integer) units at the same time. IBM complemented this with a complex instruction decoder which could be fetching one instruction, decoding another, and sending one to the ALU and FPU at the same time, resulting in one of the first superscalar CPU designs in use.

The system used thirty-two 32-bit integer registers and another thirty-two 64-bit floating point registers, each in their own unit. The branch unit also included a number of "private" registers for its own use, including the program counter.

The 801 was a simple design, and an overcorrection to its simplicity resulted in the POWER design being more complex than most RISC CPUs. For instance, the POWER (and PowerPC) instruction set includes over 100 op-codes of variable length, many of which are variations on others. This compares (for instance) with the ARM which has only 34 instructions.

Another interesting feature of the architecture is a virtual address system which maps all addresses into a 52-bit space. In this way applications can share memory in a "flat" 32-bit space, and all of the programs can have different blocks of 32-bits each.

The first POWER1 CPUs consisted of three chips; branch, integer and floating point. These were wired together on a largish motherboard to produce a single system. POWER1 was used primarily in the RS/6000 series of workstations.

POWER2 was a product-improved POWER1 and was the longest-lived of the POWER series, released in 1993 and still in use five years later. It added a second floating-point unit, 256k of cache and 128-bit floating point math.

POWER3 followed in 1998, moving to a full 64-bit implementation, while remaining completely compatible with the POWER instruction set. This had been one of the goals of the PowerPC project and the POWER3 was the first of the IBM processors to take advantage of it. It also added a third ALU and a second instruction decoder, for a total of eight functional units.

The latest implementation is the POWER4 series which places two complete CPU cores (otherwise similar to the POWER3) on a single chip, speeds it up, and adds high-speed connections to up to three other pairs of POWER4 CPUs. They can be placed together on a motherboard to produce an 8-cpu SMP building block. When processing requires high throughput instead of high code complexity, one of a pair of cores can be turned off so that the remaining cores have the entire bus and L3 cache to themselves. The POWER4, even in single form, is considered by many to be the most powerful CPU available.

IBM plans to roll out the "POWER5" processor in 2004. Ravi Arimilli, IBM's chief microprocessor designer has said; "The Power5 chip is more of a midrange or low-end design that can drive up to the high end and then down to things like blades.". A 64-bit RISC design that will help IBM cover the entire board as far as high and low end server CPUs are being announced by Intel and AMD. This is no low-end chip however, and is slated to slowly phase out the POWER4.

The PowerPC was essentially a POWER1 CPU with some of the more basic instructions emulated in microcode, using a bus interface based on the Motorola 88000 design. This allowed IBM to use the CPU in a number of workstation machines, changing only the motherboard. Since then the PowerPC and POWER architechtures have diverged somewhat, but remain compatible at the instruction level.

The IBM RS64 family of processors is based on PowerPC (and thus POWER) and has been used in the RS/6000 and AS/400 product lines. It is optimized for commercial workloads, and does not have the floating point power expected in the POWER line. It is now mostly replaced by the POWER4.

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "IBM POWER."

Top     

Power

(From Wikipedia, the free Encyclopedia)

This article discusses power in physics. For alternative uses, see Power (mathematics), Power (international), Electric power, Statistical power and Power (sociology).

In physics, power is the amount of work done per unit of time. This is equivalent to the rate of change of the energy in a system, or the time rate of doing work, as defined by

,

where

• P is power
• E is energy or work
• t is time.

The units of power are therefore work divided by time (e.g. foot-pounds per minute, ergs per second or joules per second). The SI unit of power is the watt, which is equal to one joule per second.

Non-SI units of power include horsepower (HP), Pferdstarke (PS) and the cheval vapeur (CV). One unit of horsepower is equivalent to 33,000 foot-pounds per minute, or the power required to lift 550 pounds one foot in one second, and is equivalent to about 746 watts.

The power consumption of a human is on average roughly 100 watts, ranging from 85 W during sleep to 800 W while playing a strenuous sport.

For direct current (DC) and voltage

In electrical engineering, the instantaneous power consumed by a two-terminal electrical device is the product of the voltage across the terminals and the current passing through the device. That is,

where I is the instantaneous or average direct current (DC) and U is the instantaneous or average voltage. If I is in amperes and U is in volts then P is in watts.

For sinusoidal alternating current (AC) and voltage

The average power consumed by a two-terminal electrical device is a function of the root mean square values of the sinusoidal voltage across the terminals and the sinusoidal current passing through the device. That is,

where I is the root mean square value of the sinusoidal alternating current (AC) and U is the root mean square value of the sinusoidal alternating voltage. φ is the phase angle between the voltage and the current sine functions. If I is in amperes and U is in volts then P is in watts.

This can also be called the effective power, as compared to the larger apparent power which is expressed in volt-amperes reactive (VAR) and does not include the term due to the current and voltage being out of phase.

The efficient transfer of electrical power is governed by the maximum power theorem.

External Links

• Power, from Hypertext Physics
• Tony R. Kuphaldt: Lessons In Electric Circuits -- Volume II: Power factor. Power in resistive and reactive AC circuits

  Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Power."

Top     

Power (international)

(From Wikipedia, the free Encyclopedia)

In the context of International relations and diplomacy such states are called Powers, Regional Powers, Great Powers or Super powers which can dominate other sovereign states.

In the field of political theory, Niccolo Machiavelli theorised early and influentially on the mechanisms of gaining and retaining political power, publishing The Prince in 1513.

Power is usually defined as the ability to impose one's will on others, or to pursue one's interests on the expense of others'. Violence or other kinds of force, or the threat of such force, can be used to exercise power (coercion).

Political analysis often personifies nation states as powers, discussing superpowers, great powers, second-order powers and "European powers", for example, with convenient simplicity as manifestations of Realpolitik. In Western thought these terms are generally qualitative terms. In current Chinese political thinking national power can be measured quantitatively using an index known as comprehensive national power. Chinese political thought also distinguishes between various forms of national power. In particular, hard power (military power) stands in contradistinction to soft power (economic or cultural or persuasive power).

Quotation

'Power tends to corrupt, and absolute power corrupts absolutely'.
(Attributed to Lord Acton, 1887.)

See also:

• Power (sociology)

  Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Power (international)."

Top     

Power (sociology)

(From Wikipedia, the free Encyclopedia)

Sociologists usually define power as the ability to impose one's will on others, even if those others resist in some way. The imposition need not involve coercion (force or threat of force); "power" used in the sociological sense is a separate concept from physical power or political power and in some ways is closer to what is called "influence" in everyday English.

More generally, it can be defined as the real or perceived ability or potential to bring about significant change, usually in people’s lives through the actions of others.

The exercise of power seems to be endemic to people as social and gregarious beings.

Analysis and operation of power

Power manifests itself in a relational manner: one cannot meaningfully say (pace advocates of empowerment) that a particular social actor "has power" without also specifying the other parties to the social relationship.

Power almost always operates reciprocally, but usually not equally reciprocally. To control others, one must have control over things that they desire or need, but one can rarely exercise that control without a measure of reverse control - larger, smaller, or equal - also existing. For example, an employer usually wields considerable power over his workers because he has control over wages, working conditions, hiring and firing. The workers, however, hold some reciprocal power: they may leave, work more or less diligently, group together to form a union, and so on.

Because power operates both relationally and reciprocally, sociologists speak of the balance of power between parties to a relationship: all parties to all relationships have some power: the sociological examination of power concerns itself with discovering and describing the relative strengths: equal or unequal, stable or subject to periodic change. Sociologists usually analyse relationships in which the parties have relatively equal or nearly equal power in terms of constraint rather than of power.

Even in structuralist social theory, power appears as a process, an aspect to an ongoing social relationship, not as a fixed part of social structure.

Types and sources of power

Power may be held through:

• Delegated authority (e.g. in the democratic process)
• Personal or group charisma
• Ascribed power(acting on perceived,or assumed abilities whether these are tested or otherwise)
• Ability or skills (the power of medicine to bring about health)
• Knowledge (granted or withheld, shared or kept secret)
• Money (control of labour, ownership)
• Force (violence, military might, coercion).
• Moral suasion
• Application of non-violence
• Operation of group dynamics
• Influence of tradition

Theories of power

The thought of Friedrich Nietzsche underlies much 20th century analysis of power. Nietzsche disseminated ideas on the "will to power", which he saw as the domination of other humans as much as the exercise of control over one's environment.

Some schools of psychology, notably that associated with Alfred Adler, place power dynamics at the core of their theory (where orthodox Freudians might place sexuality).

In the Marxist tradition, Antonio Gramsci elaborated the role of cultural hegemony in ideology as a means of bolstering the power of capitalism and of the nation-state.

One of the broader modern views of the importance of power in human activity comes from the work of Michel Foucault. Feminist analysis of the patriarchy often concentrates on issues of power: note the "Rape Mantra": Rape is about power, not sex.

Deconstruction often works to reveal hidden power structures and relationships.

See also:

• Class
• Authority
• Status
• Charisma
• Power (international).

  Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Power (sociology)."

Top     

Power, Montana

(From Wikipedia, the free Encyclopedia)

Power is a town located in Teton County, Montana. As of the 2000 census, the town had a total population of 171.

Geography

Power is located at 47°42'55" North, 111°41'13" West (47.715367, -111.687054)¹. According to the United States Census Bureau, the town has a total area of 3.9 km² (1.5 mi²). 3.9 km² (1.5 mi²) of it is land and 0.66% is water.

Demographics

As of the census of 2000, there are 171 people, 68 households, and 51 families residing in the town. The population density is 44.0/km² (114.1/mi²). There are 71 housing units at an average density of 18.3/km² (47.4/mi²). The racial makeup of the town is 97.08% White, 0.00% African American, 1.17% Native American, 0.00% Asian, 0.00% Pacific Islander, 0.00% from other races, and 1.75% from two or more races. 0.58% of the population are Hispanic or Latino of any race. There are 68 households out of which 33.8% have children under the age of 18 living with them, 66.2% are married couples living together, 7.4% have a female householder with no husband present, and 25.0% are non-families. 22.1% of all households are made up of individuals and 7.4% have someone living alone who is 65 years of age or older. The average household size is 2.51 and the average family size is 2.94. In the town the population is spread out with 30.4% under the age of 18, 4.1% from 18 to 24, 30.4% from 25 to 44, 19.9% from 45 to 64, and 15.2% who are 65 years of age or older. The median age is 37 years. For every 100 females there are 90.0 males. For every 100 females age 18 and over, there are 91.9 males. The median income for a household in the town is $38,036, and the median income for a family is $39,286. Males have a median income of $27,083 versus $13,125 for females. The per capita income for the town is $16,527. 18.6% of the population and 8.9% of families are below the poverty line. Out of the total people living in poverty, 48.6% are under the age of 18 and 0.0% are 65 or older.

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Power, Montana."

Top     

Product topology

(From Wikipedia, the free Encyclopedia)

In topology, the cartesian product of topological spaces is turned into a topological space in the following way. Let I be a (possibly infinite) index set and suppose X_i is a topological space for every i in I. Set X = Π X_i, the cartesian product of the sets X_i. For every i in I, we have a canonical projection p_i : X -> X_i. The product topology on X is defined to be the coarsest topology (i.e. the topology with the fewest open sets) which turns all the maps p_i into continuous maps.

Explicitly, the topology on X can be described as follows. A subset of X is open if and only if it is a union of (possibly infinitely many) intersections of finitely many sets of the form p_i^-1(O), where i in I and O is an open subset of X_i. This implies that, in general, not all products of open sets need to be open in X.

We can describe a basis for the product topology in a simple way using the bases of the constituting spaces X_i. Suppose that for each i in I we choose a set Y_i which is either the whole space X_i or a basis element in that space, and let B be the product of the Y_i. Then, as long as X_i = Y_i, that is, we choose the entire space, for all but finitely many i in I, B will be a basis element of the product space, and a complete basis is generated in this way. In particular, this means that a finite product of spaces X has a simple basis given by products of bases in the X_i.

Examples

If one starts with the standard topology on the real line R and defines a topology on the product of n copies of R in this fashion, one obtains the ordinary Euclidean topology on Rⁿ.

The Cantor set is homeomorphic to the product of countably many copies of the discrete space {0,1} and the space of irrational numbers is homeomorphic to the product of countably many copies of the natural numbers, where again each copy carries the discrete topology.

Properties

The product topology is also called the topology of pointwise convergence because of the following fact: a sequence (or net) in X converges if and only if all its projections to the spaces X_i converge. In particular, if one considers the space X = R^I of all real valued functions on I, convergence in the product topology is the same as pointwise convergence of functions.

An important theorem about the product topology is Tychonoff's theorem: any product of compact spaces is compact. This is easy for finite products, but the statement is (surprisingly) also true for infinite products, when the proof requires the axiom of choice in some form.

The product space X, together with the canonical projections, can be characterized by the following universal property: If Y is a topological space, and for every i in I, f_i : Y -> X_i is a continuous map, then there exists precisely one continuous map f : Y -> X such that p_i\ o f = f_i for all i in I. This shows that the product space is a product in the sense of category theory.

To check whether a given map f : Y -> X is continuous, one can use the following handy criterion: f is continuous if and only if p_i o f is continuous for all i in I. In other words, if we write f as a tuple of its components, f=(f_i)_{i in I}, then f is continuous if and only if each of the f_i is. Checking whether a map g : X -> Z is continuous is usually more difficult; one tries to use the fact that the p_i are continuous in some way.

Relation to other topological notions

• Separation
  • Every product of T₀ spacess is T₀
  • Every product of T₁ spacess is T₁
  • Every product of Hausdorff spaces is Hausdorff
  • Every product of Regular spaces is Regular
  • Every product of Tychonoff spaces is Tychonoff
  • A product of normal spaces need not be normal
• Compactness
  • Every product of compact spaces is compact (Tychonoff's theorem)
  • A product of locally compact spaces need not be locally compact
• Connectedness
  • Every product of connected (resp. path-connected) spaces is connected (resp. path-connected)
  • Every product of hereditarily disconnected spaces is hereditarily disconnected.

Please add more results like these

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Product topology."

Top     

Statistical power

(From Wikipedia, the free Encyclopedia)

The power of a statistical test is the probability that the test will reject a false null hypothesis – that is, that it will not make a Type II error. The higher the power, the greater the chance of obtaining a statistically significant result when the null hypothesis is false.

Statistical tests attempt to use data from samples to determine if differences or similarities exist in a population. For example, to test the null hypothesis that the mean scores of men and women on a test do not differ, samples of men and women will be drawn, the test administered to them, and the mean score in each group compared with a statistical test. If the populations of men and women have different mean scores but the test of the sample data concludes that there is no such difference, a Type II error has been made.

Statistical power depends on the significance criterion, the size of the difference or the strength of the similarity (that is, the effect size) in the population, and the sensitivity of the data.

A significance criterion is a statement of how unlikely a difference must be, if the null hypothesis is true, to be considered significant. The most commonly used criteria are probabilities of 0.05, 0.01, and 0.001. If the criterion is 0.05, the probability of the difference must be less than 0.05, and so on. The greater the effect size, the greater the power. Calculation of power requires that researchers determine the effect size they want to detect.

Sensitivity can be increased by using statistical controls, by increasing the reliability of measures (as in psychometric reliability), and by increasing the size of the sample. Increasing sample size is the most commonly used method for increasing statistical power.

Although there are no formal standards for power, most researchers who assess the power of their tests use 0.80 as a standard for adequacy.

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Statistical power."

Top     

Abbreviations & Acronyms: Power

Top     

Top     

Synonyms within Context: Power

Top     

Top     

Modern Usage: Power

Top     

Commercial Usage: Power

Top     

Image Slideshow: Power

Photos: Power More pictures...

Illustrations: Power More pictures...

Computer Images: Power More pictures...

Top     

Photo Album: Power

Thumbnail	Description & Credit	Thumbnail	Description & Credit
	Shown is an older white woman, wearing sunglasses and a hat, power walking outside. Credit: Unknown photographer/artist.		Here a field worker runs a power sprayer on a truck as it emits an insecticide. Credit: CDC.
	AH-56 Cheyenne Full Power Test. Credit: NASA.		Skylab Shroud in Plum Brook Space Power Facility. Credit: NASA.
	The Space Power Facility. Credit: NASA.		Power source for hoisting steel Jack up truck and use "fifth wheel" This could be dangerous - at least one leg broken by getting wrapped up Safety modifications made to system after individual severely injured Triangulation party of Paul A. Smith. Credit: Coast & Geodetic Survey Historical Image Collection.
	Where horse power and horsepower converge Triangulation party of E. O. Heaton. Credit: Coast & Geodetic Survey Historical Image Collection.		Looking east to electric power plant on the south shore of Lake Michigan. Credit: America's Coastlines.
	FAIRWEATHER moored at Marine Corps Air Station in Kaneohe Bay. During Hurricane Iwa. Power knocked out to Marine Corps Base. FAIRWEATHER cooked Thanksgiving turkeys for Marine Corps families. Credit: America's Coastlines.		The bowels of South Pole Station - the heating and power plant. Credit: Paths Less Taken - NOAA at the Ends of the Earth.
Source: pictures compiled by the editor from various references; see picture credits.

Top     

Digital Photo Gallery: Power
 

"East coast power outage" by Flavio Masson Commentary: "Full moon in between Union Square residential towers - photo taken on Aug 14, 2003 during the east coast power outage."	"Power Lines" by Dan Willis Commentary: "Power lines to my house."
Source: photographs selected by the editor, with permission from the photographers.

Top     

Sounds Captioned with "Power".

Play	Caption	Play	Caption
	A low power chord played on a distorted electric guitar.		Heavy voltage power being turned on.
	A power chord bent in an upward manner on an electric guitar.
Source: compiled by the editor from various references; see credits.

Top     

Familiar Quotations: Power

Top     

Historic Usage: Power

Top     

Use in Literature: Power