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Algorithms is a monthly peer-reviewed open-access scientific journal of mathematics, covering design, analysis, and experiments on algorithms. The journal is published by MDPI and was established in 2008. The founding editor-in-chief was Kazuo Iwama (Kyoto University).. From May 2014 to September 2019, the editor-in-chief was Henning Fernau (Universität Trier). The current editor-in-chief is Frank Werner (Otto-von-Guericke-Universität Magdeburg).
==Abstracting and indexing==
The journal is abstracted and indexed in:
==See also==
Journals with similar scope include:
*ACM Transactions on Algorithms
*Algorithmica
*Journal of Algorithms (Elsevier)
==References==
==External links==
*
Category:Computer science journals
Category:Open access journals
Category:MDPI academic journals
Category:English-language journals
Category:Publications established in 2008
Category:Mathematics journals
Category:Monthly journals | Algorithms (journal) | [
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] |
Telecommunications in Antigua and Barbuda are via media in the telecommunications industry. This article is about communications systems in Antigua and Barbuda.
==Telephone==
Telephones – main lines in use: 37,500 (2006)
:country comparison to the world: 168
Telephones – mobile cellular: 110,200 (2006) (APUA PCS, Cable & Wireless, Digicel)
:country comparison to the world: 177
Telephone system:
domestic: good automatic telephone system
international: 3 fiber optic submarine cables (2 to Saint Kitts and 1 to Guadeloupe); satellite earth station – 1 Intelsat (Atlantic Ocean)
==Radio==
Radio broadcast stations: AM 4, FM 6, shortwave 0 (2002)
{| class="wikitable sortable"
|+ Radio Stations of Antigua and Barbuda
! Band / Freq.
! Call Sign
! Brand
! City of license
! Notes
|-
| AM 620
| V2C
| ABS Radio and TV
| Saint John's, Antigua
| ABS; 5 kW
|-
| AM 1100
| ZDK
| Radio ZDK
| Saint John's, Antigua
| Owner: Grenville Radio; 20 kW
|-
| AM 1160
| Unknown
| Radio Lighthouse
| Saint John's, Antigua
| 10 kW
|-
| AM 1580
| Unknown
| Unknown
| Judge Bay, Antigua
| 50 kW
|-
| FM 88.5
| Unknown
| Power FM
| Saint John's, Antigua
|
|-
| FM 89.7
| Unknown
| Catholic Radio
| Saint John's, Antigua
| 2 kW
|-
| FM 90.5
| V2C-FM
| ABS Radio and TV
| Saint John's, Antigua
| repeats AM 620
|-
| FM 91.1
| Unknown
| Observer Radio
| Saint John's, Antigua
|
|-
| FM 91.9
| Unknown
| Hitz 91.9
| Saint John's, Antigua
|
|-
| FM 92.3
| Unknown
| Radio Lighthouse
| Saint John's, Antigua
| repeats AM | Telecommunications in Antigua and Barbuda | [
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] |
1160
|-
| FM 92.9
| VYBZ-FM
| Vybz FM
| Saint John's, Antigua
|
|-
| FM 93.9
| Unknown
| Caribbean SuperStation
| Saint John's, Antigua
| repeats Caribbean SuperStation from Trinidad
|-
| FM 95.7
| Unknown
| Zoom Radio
| Saint John's, Antigua
|
|-
| FM 97.1
| ZDK
| Radio ZDK
| Saint John's, Antigua
| repeats AM 1100
|-
| FM 98.5
| Unknown
| Red Hot Radio
| Saint John's, Antigua
|
|-
| FM 99.1
| Unknown
| Hit Radio Music Power
| Saint John's, Antigua
|
|-
| FM 100.1
| Unknown (ZDKR-FM?)
| Sun FM
| Saint John's, Antigua
|
|-
| FM 101.5
| Unknown
| Second Advent Radio
| Saint John's, Antigua
| 20 watts
|-
| FM 102.3
| Unknown
| Variety Radio
| Saint John's, Antigua
|
|-
| FM 103.1
| Unknown
| Life FM
| Codrington, Barbuda
| 1 kW
|-
| FM 103.9
| Unknown
| Life FM
| Saint John's, Antigua
| repeats 103.1 Codrington
|-
| FM 104.3
| Unknown
| Nice FM
| Codrington, Barbuda
|
|-
| FM 107.3
| Unknown
| Crusader Radio
| Saint John's, Antigua
|
|-
| SW 3.255 mHz
| V2C
| ABS Radio and TV
| Saint John's, Antigua
| Repeats AM 620
|}
Radios: 36,000 (1997)
==Television==
Television broadcast stations: 2 (1997) (including ABS-TV)
Televisions: 31,000 (1997)
==Internet==
Internet Service Providers (ISPs): Cable & Wireless, Antigua Computer Technologies (ACT), Antigua Public Utilities Authority (APUA INET)
Internet hosts: 2,215 (2008)
:country comparison to the world: 140
Internet users: 60,000 (2007)
:country comparison to the world: 158
Country codes: AG
== Demographics ==
{| class="wikitable sortable"
|+Internet Users by Ethnicity
! rowspan="2" |Q48 Ethnic
! colspan="3" |Q55 Internet Use
|-
!Yes
!No
!Don't know/Not stated
|-
|African descendent
|47.42%
|50.74%
|1.84%
|-
|Caucasian/White
|83.27%
|16.03%
|0.70%
|-
|East Indian/India
|58.66%
|40.08%
|1.26%
|-
|Mixed (Black/White)
|64.34%
|33.35%
|2.31%
|-
|Mixed (Other)
|61.22%
|37.62%
|1.16%
|-
|Hispanic
|31.78%
|66.80%
|1.42%
|-
|Syrian/Lebanese
|60.77%
|36.76%
|2.48%
|-
|Other
|59.46%
|39.71%
|0.83%
|-
|Don't | Telecommunications in Antigua and Barbuda | [
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] |
know/Not stated
|20.58%
|70.61%
|8.81%
|-
!Total
!48.35%
!49.81%
!1.84%
|}
{| class="wikitable sortable"
|+Household internet access by ethnicity
! rowspan="2" |Q48 Ethnic
! colspan="3" |Q25 4 Internet access
|-
!No
!Yes
!Don't know/not declared
|-
|African descendent
|51.99%
|38.20%
|9.81%
|-
|Caucasian/White
|14.56%
|82.83%
|2.61%
|-
|East Indian/India
|36.63%
|54.93%
|8.44%
|-
|Mixed (Black/White)
|39.47%
|54.60%
|5.93%
|-
|Mixed (Other)
|41.76%
|51.53%
|6.71%
|-
|Hispanic
|67.68%
|20.56%
|11.76%
|-
|Syrian/Lebanese
|29.48%
|69.39%
|1.13%
|-
|Other
|34.02%
|58.96%
|7.02%
|-
|Don't know/Not stated
|53.86%
|27.58%
|18.56%
|-
!Total
!50.83%
!39.61%
!9.57%
|}
==See also==
*Antigua and Barbuda
*History of telecommunication
*List of telecommunications terminology
*Outline of telecommunication
==References==
==External links==
* Antigua and Barbuda, SubmarineCableMap.com
Antigua and Barbuda
Antigua | Telecommunications in Antigua and Barbuda | [
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] |
This article is about communications systems in Anguilla.
==Telephone==
Telephones – main lines in use: 6,200 (2002)
:country comparison to the world: 212
Telephones – mobile cellular: 1,800 (2002)
:country comparison to the world: 211
Telephone system:
Domestic: Modern internal telephone system
International: EAST CARIBBEAN FIBRE SYSTEM ECFS (cable system)
microwave radio relay to island of Saint Martin (Guadeloupe and Netherlands Antilles)
==Mobile phone (GSM)==
Mobile phone operators:
FLOW (Anguilla) Ltd. – GSM and UMTS 850 and 1900 MHz, LTE 700 MHz with Island-wide coverage
Digicel (Anguilla) Ltd. – GSM and UMTS 850 to 1900 MHz, LTE 700 MHz
Mobiles: ? (2007)
==Radio==
Radio broadcast stations: AM 3, FM 7, shortwave 0 (2007)
{| class="wikitable sortable"
|+ Radio Stations of Anguilla
! Band / Freq.
! Call Sign
! Brand
! City of license
! Notes
|-
| AM 690 kHz
| Unknown
| Caribbean Beacon
| The Valley
| Religious broadcaster
|-
| AM 1500 kHz
| Unknown
| Caribbean Beacon
| The Valley
| 2.5 kW repeater
|-
| AM 1610 kHz
| Unknown
| Caribbean Beacon
| The Valley
| 200 kW repeater
|-
| FM 92.9 MHz
| Unknown
| Klass 92.9
| The Valley
|
|-
| FM 93.3 MHz
| Unknown
| Rainbow FM
| The Valley
| Caribbean Music, News
|-
| FM 95.5 MHz
| Unknown
| Radio Anguilla
| The Valley
| Public broadcaster
|-
| FM 97.7 MHz
| Unknown
| Heart Beat Radio/Up Beat Radio
| The Valley
| 30 kW, Caribbean Music, News
|-
| FM 99.3 MHz
| ZNBR-FM
| NBR – New Beginning Radio / Grace FM
| The Valley
| 5 kW, Religious broadcaster
|-
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] |
FM 100.1 MHz
| Unknown
| Caribbean Beacon
| The Valley
| Religious broadcaster
|-
| FM 100.9 MHz
| Unknown
| CBN – Country Broadcast Network
| The Valley
| 3 kW
|-
| FM 103.3 MHz
| Unknown
| Kool FM
| The Valley
| Religious broadcaster, Urban Caribbean
|-
| FM 105.1 MHz
| ZRON-FM
| Tradewinds Radio
| The Valley
| 5 kW, Caribbean Music, News
|-
| FM 106.7 MHz
| unknown
| VOC – Voice Of Creation
| Sachasses
| Religious broadcaster
|-
| FM 107.9 MHz
| unknown
| GEM Radio Network
| The Valley
| Repeater (Trinidad)
|-
| SW 6090 kHz
| Unknown
| Caribbean Beacon
| The Valley
| Religious
|-
| SW 11775 kHz
| Unknown
| Caribbean Beacon
| The Valley
| Religious
|}
Radios: 3,000 (1997)
==Television==
Television broadcast stations: 1 (1997)
Televisions: 1,000 (1997)
==Internet==
Internet country code: .ai (Top level domain)
Internet Service Providers (ISPs): 2 (FLOW – , Digicel Anguilla – )
Internet hosts: 269 (2012)
:country comparison to the world: 192
Internet: users: 12,377 (2018)
:country comparison to the world: 206
==See also==
*Anguilla
* FLOW (Anguilla) Ltd.
== References ==
== External links ==
* Public Utilities Commission of Anguilla
* Anguilla, SubmarineCableMap.com
Category:Communications in Anguilla
Anguilla
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] |
Acoustic theory is a scientific field that relates to the description of sound waves. It derives from fluid dynamics. See acoustics for the engineering approach.
For sound waves of any magnitude of a disturbance in velocity, pressure, and density we have
:
\begin{align}
\frac{\partial \rho'}{\partial t} +\rho_0\nabla\cdot \mathbf{v} + \nabla\cdot(\rho'\mathbf{v}) & = 0 \qquad \text{(Conservation of Mass)} \\
(\rho_0+\rho')\frac{\partial \mathbf{v}}{\partial t} + (\rho_0+\rho')(\mathbf{v}\cdot\nabla)\mathbf{v} + \nabla p' & = 0 \qquad \text{(Equation of Motion)}
\end{align}
In the case that the fluctuations in velocity, density, and pressure are small, we can approximate these as
:
\begin{align}
\frac{\partial \rho'}{\partial t} +\rho_0\nabla\cdot \mathbf{v} & = 0 \\
\frac{\partial \mathbf{v}}{\partial t} + \frac{1}{\rho_0}\nabla p'& = 0
\end{align}
Where \mathbf{v}(\mathbf{x},t) is the perturbed velocity of the fluid, p_0 is the pressure of the fluid at rest, p'(\mathbf{x},t) is the perturbed pressure of the system as a function of space and time, \rho_0 is the density of the fluid at rest, and \rho'(\mathbf{x}, t) is the variance in the density of the fluid over space and time.
In the case that the velocity is irrotational (\nabla\times \mathbf{v} = 0), | Acoustic theory | [
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we then have the acoustic wave equation that describes the system:
:
\frac{1}{c^2}\frac{\partial^2 \phi}{\partial t^2} - \nabla^2\phi = 0
Where we have
:
\begin{align}
\mathbf{v} & = -\nabla \phi \\
c^2 & = (\frac{\partial p}{\partial \rho})_s\\
p' & = \rho_0\frac{\partial \phi}{\partial t}\\
\rho' & = \frac{\rho_0}{c^2}\frac{\partial \phi}{\partial t}
\end{align}
==Derivation for a medium at rest==
Starting with the Continuity Equation and the Euler Equation:
:
\begin{align}
\frac{\partial \rho}{\partial t} +\nabla\cdot \rho\mathbf{v} & = 0 \\
\rho\frac{\partial \mathbf{v}}{\partial t} + \rho(\mathbf{v}\cdot\nabla)\mathbf{v} + \nabla p & = 0
\end{align}
If we take small perturbations of a constant pressure and density:
:
\begin{align}
\rho & = \rho_0+\rho' \\
p & = p_0 + p'
\end{align}
Then the equations of the system are
:
\begin{align}
\frac{\partial}{\partial t}(\rho_0+\rho') +\nabla\cdot (\rho_0+\rho')\mathbf{v} & = 0 \\
(\rho_0+\rho')\frac{\partial \mathbf{v}}{\partial t} + (\rho_0+\rho')(\mathbf{v}\cdot\nabla)\mathbf{v} + \nabla (p_0+p') & = 0
\end{align}
Noting that the equilibrium pressures and densities are constant, this simplifies to
:
\begin{align}
\frac{\partial \rho'}{\partial t} +\rho_0\nabla\cdot\mathbf{v}+\nabla\cdot \rho'\mathbf{v} & = 0 \\
(\rho_0+\rho')\frac{\partial \mathbf{v}}{\partial t} | Acoustic theory | [
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] |
+ (\rho_0+\rho')(\mathbf{v}\cdot\nabla)\mathbf{v} + \nabla p' & = 0
\end{align}
===A Moving Medium===
Starting with
:
\begin{align}
\frac{\partial \rho'}{\partial t} +\rho_0\nabla\cdot\mathbf{w}+\nabla\cdot \rho'\mathbf{w} & = 0 \\
(\rho_0+\rho')\frac{\partial \mathbf{w}}{\partial t} + (\rho_0+\rho')(\mathbf{w}\cdot\nabla)\mathbf{w} + \nabla p' & = 0
\end{align}
We can have these equations work for a moving medium by setting \mathbf{w} = \mathbf{u} + \mathbf{v}, where \mathbf{u} is the constant velocity that the whole fluid is moving at before being disturbed (equivalent to a moving observer) and \mathbf{v} is the fluid velocity.
In this case the equations look very similar:
:
\begin{align}
\frac{\partial \rho'}{\partial t} +\rho_0\nabla\cdot\mathbf{v}+\mathbf{u}\cdot\nabla\rho' + \nabla\cdot \rho'\mathbf{v} & = 0 \\
(\rho_0+\rho')\frac{\partial \mathbf{v}}{\partial t} + (\rho_0+\rho')(\mathbf{u}\cdot\nabla)\mathbf{v} + (\rho_0+\rho')(\mathbf{v}\cdot\nabla)\mathbf{v} + \nabla p' & = 0
\end{align}
Note that setting \mathbf{u} = 0 returns the equations at rest.
==Linearized Waves==
Starting with the above given equations of motion for a medium at rest:
:
\begin{align}
\frac{\partial \rho'}{\partial t} +\rho_0\nabla\cdot\mathbf{v}+\nabla\cdot \rho'\mathbf{v} & = 0 \\
(\rho_0+\rho')\frac{\partial \mathbf{v}}{\partial t} + (\rho_0+\rho')(\mathbf{v}\cdot\nabla)\mathbf{v} + \nabla p' & = 0
\end{align}
Let us now take \mathbf{v},\rho',p' to all be small quantities.
In the case | Acoustic theory | [
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that we keep terms to first order, for the continuity equation, we have the \rho'\mathbf{v} term going to 0. This similarly applies for the density perturbation times the time derivative of the velocity. Moreover, the spatial components of the material derivative go to 0. We thus have, upon rearranging the equilibrium density:
:
\begin{align}
\frac{\partial \rho'}{\partial t} +\rho_0\nabla\cdot \mathbf{v} & = 0 \\
\frac{\partial \mathbf{v}}{\partial t} + \frac{1}{\rho_0}\nabla p' & = 0
\end{align}
Next, given that our sound wave occurs in an ideal fluid, the motion is adiabatic, and then we can relate the small change in the pressure to the small change in the density by
:
p' = (\frac{\partial p}{\partial \rho_0})_s\rho'
Under this condition, we see that we now have
:
\begin{align}
\frac{\partial p'}{\partial t} +\rho_0(\frac{\partial p}{\partial \rho_0})_s\nabla\cdot \mathbf{v} & = 0 \\
\frac{\partial \mathbf{v}}{\partial t} + \frac{1}{\rho_0}\nabla p' & = 0
\end{align}
Defining the speed of sound of the system:
:
c \equiv \sqrt{(\frac{\partial p}{\partial \rho_0})_s}
Everything becomes
:
\begin{align}
\frac{\partial p'}{\partial t} +\rho_0c^2\nabla\cdot \mathbf{v} & = 0 \\
\frac{\partial \mathbf{v}}{\partial t} | Acoustic theory | [
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] |
+ \frac{1}{\rho_0}\nabla p' & = 0
\end{align}
===For Irrotational Fluids===
In the case that the fluid is irrotational, that is \nabla\times\mathbf{v} = 0, we can then write \mathbf{v} = -\nabla\phi and thus write our equations of motion as
:
\begin{align}
\frac{\partial p'}{\partial t} -\rho_0c^2\nabla^2\phi & = 0 \\
-\nabla\frac{\partial\phi}{\partial t} + \frac{1}{\rho_0}\nabla p' & = 0
\end{align}
The second equation tells us that
:
p' = \rho_0 \frac{\partial \phi}{\partial t}
And the use of this equation in the continuity equation tells us that
:
\rho_0\frac{\partial^2 \phi}{\partial t} -\rho_0c^2\nabla^2\phi = 0
This simplifies to
:
\frac{1}{c^2}\frac{\partial^2 \phi}{\partial t^2} -\nabla^2\phi = 0
Thus the velocity potential \phi obeys the wave equation in the limit of small disturbances. The boundary conditions required to solve for the potential come from the fact that the velocity of the fluid must be 0 normal to the fixed surfaces of the system.
Taking the time derivative of this wave equation and multiplying all sides by the unperturbed density, and then using the fact that p' = \rho_0 \frac{\partial \phi}{\partial t} tells us that
:
\frac{1}{c^2}\frac{\partial^2 p'}{\partial t^2} -\nabla^2p' = 0
Similarly, | Acoustic theory | [
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] |
we saw that p' = (\frac{\partial p}{\partial \rho_0})_s\rho' = c^2\rho'. Thus we can multiply the above equation appropriately and see that
:
\frac{1}{c^2}\frac{\partial^2 \rho'}{\partial t^2} -\nabla^2\rho' = 0
Thus, the velocity potential, pressure, and density all obey the wave equation. Moreover, we only need to solve one such equation to determine all other three. In particular, we have
:
\begin{align}
\mathbf{v} & = -\nabla \phi \\
p' & = \rho_0 \frac{\partial \phi}{\partial t}\\
\rho' & = \frac{\rho_0}{c^2}\frac{\partial\phi}{\partial t}
\end{align}
===For a moving medium===
Again, we can derive the small-disturbance limit for sound waves in a moving medium. Again, starting with
:
\begin{align}
\frac{\partial \rho'}{\partial t} +\rho_0\nabla\cdot\mathbf{v}+\mathbf{u}\cdot\nabla\rho' + \nabla\cdot \rho'\mathbf{v} & = 0 \\
(\rho_0+\rho')\frac{\partial \mathbf{v}}{\partial t} + (\rho_0+\rho')(\mathbf{u}\cdot\nabla)\mathbf{v} + (\rho_0+\rho')(\mathbf{v}\cdot\nabla)\mathbf{v} + \nabla p' & = 0
\end{align}
We can linearize these into
:
\begin{align}
\frac{\partial \rho'}{\partial t} +\rho_0\nabla\cdot\mathbf{v}+\mathbf{u}\cdot\nabla\rho' & = 0 \\
\frac{\partial \mathbf{v}}{\partial t} + (\mathbf{u}\cdot\nabla)\mathbf{v} + \frac{1}{\rho_0}\nabla p' & = 0
\end{align}
====For Irrotational Fluids in a Moving Medium====
Given that we saw that
:
\begin{align} | Acoustic theory | [
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\frac{\partial \rho'}{\partial t} +\rho_0\nabla\cdot\mathbf{v}+\mathbf{u}\cdot\nabla\rho' & = 0 \\
\frac{\partial \mathbf{v}}{\partial t} + (\mathbf{u}\cdot\nabla)\mathbf{v} + \frac{1}{\rho_0}\nabla p' & = 0
\end{align}
If we make the previous assumptions of the fluid being ideal and the velocity being irrotational, then we have
:
\begin{align}
p' & = (\frac{\partial p}{\partial \rho_0})_s\rho' = c^2\rho' \\
\mathbf{v} & = -\nabla\phi
\end{align}
Under these assumptions, our linearized sound equations become
:
\begin{align}
\frac{1}{c^2}\frac{\partial p'}{\partial t} -\rho_0\nabla^2\phi+\frac{1}{c^2}\mathbf{u}\cdot\nabla p' & = 0 \\
-\frac{\partial}{\partial t}(\nabla\phi) - (\mathbf{u}\cdot\nabla)[\nabla\phi] + \frac{1}{\rho_0}\nabla p' & = 0
\end{align}
Importantly, since \mathbf{u} is a constant, we have (\mathbf{u}\cdot\nabla)[\nabla\phi] = \nabla[(\mathbf{u}\cdot\nabla)\phi], and then the second equation tells us that
:
\frac{1}{\rho_0} \nabla p' = \nabla[\frac{\partial\phi}{\partial t} + (\mathbf{u}\cdot\nabla)\phi]
Or just that
:
p' = \rho_0[\frac{\partial\phi}{\partial t} + (\mathbf{u}\cdot\nabla)\phi]
Now, when we use this relation with the fact that \frac{1}{c^2}\frac{\partial p'}{\partial t} -\rho_0\nabla^2\phi+\frac{1}{c^2}\mathbf{u}\cdot\nabla p' = 0, alongside cancelling and rearranging terms, we arrive at
:
\frac{1}{c^2}\frac{\partial^2 \phi}{\partial t^2} - \nabla^2\phi + \frac{1}{c^2}\frac{\partial}{\partial t}[(\mathbf{u}\cdot\nabla)\phi] + \frac{1}{c^2}\frac{\partial}{\partial t}(\mathbf{u}\cdot\nabla\phi) + \frac{1}{c^2}\mathbf{u}\cdot\nabla[(\mathbf{u}\cdot\nabla)\phi] = 0
We can write this in a familiar form as
:
[\frac{1}{c^2}(\frac{\partial}{\partial t} + \mathbf{u}\cdot\nabla)^2-\nabla^2]\phi = 0
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] |
differential equation must be solved with the appropriate boundary conditions. Note that setting \mathbf{u}=0 returns us the wave equation. Regardless, upon solving this equation for a moving medium, we then have
:
\begin{align}
\mathbf{v} & = -\nabla \phi \\
p' & = \rho_0(\frac{\partial}{\partial t} + \mathbf{u}\cdot\nabla)\phi\\
\rho' & = \frac{\rho_0}{c^2}(\frac{\partial}{\partial t} + \mathbf{u}\cdot\nabla)\phi
\end{align}
==See also==
* Acoustic attenuation
* Sound
* Fourier analysis
==References==
*
*
Category:Fluid dynamics
Category:Acoustics
Category:Sound | Acoustic theory | [
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] |
Ascorbic acid is an organic compound with formula , originally called hexuronic acid. It is a white solid, but impure samples can appear yellowish. It dissolves well in water to give mildly acidic solutions. It is a mild reducing agent.
Ascorbic acid exists as two enantiomers (mirror-image isomers), commonly denoted "" (for "levo") and "" (for "dextro"). The isomer is the one most often encountered: it occurs naturally in many foods, and is one form ("vitamer") of vitamin C, an essential nutrient for humans and many animals. Deficiency of vitamin C causes scurvy, formerly a major disease of sailors in long sea voyages. It is used as a food additive and a dietary supplement for its antioxidant properties. The "" form can be made via chemical synthesis but has no significant biological role.
==History==
The antiscorbutic properties of certain foods were demonstrated in the 18th century by James Lind. In 1907, Axel Holst and Theodor Frølich discovered that the antiscorbutic factor was a water-soluble chemical substance, distinct from the one that prevented beriberi. Between 1928 and 1932, Albert Szent-Györgyi isolated a candidate for this substance, which he called it "hexuronic acid", first from plants and later from animal adrenal glands. | Chemistry of ascorbic acid | [
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In 1932 Charles Glen King confirmed that it was indeed the antiscorbutic factor.
In 1933, sugar chemist Walter Norman Haworth, working with samples of "hexuronic acid" that Szent-Györgyi had isolated from paprika and sent him in the previous year, deduced the correct structure and optical-isomeric nature of the compound, and in 1934 reported its first synthesis.Story of Vitamin C's chemical discovery. Profiles.nlm.nih.gov. Retrieved on 2012-12-04.
In reference to the compound's antiscorbutic properties, Haworth and Szent-Györgyi proposed to rename it "a-scorbic acid" for the compound, and later specifically -ascorbic acid.. Part of the National Library of Medicine collection. Accessed January 2007 Because of their work, in 1937 the Nobel Prizes for chemistry and medicine were awarded to Haworth and Szent-Györgyi, respectively.
==Chemical properties==
===Acidity===
Ascorbic acid is a vinylogous carboxylic acid and forms the ascorbate anion when deprotonated on one of the hydroxyls. This property is characteristic of reductones: enediols with a carbonyl group adjacent to the enediol group, namely with the group –C(OH)=C(OH)–C(=O)–. The ascorbate anion is stabilized by electron delocalization that results from resonance between two forms:
:400px
For this reason, ascorbic acid is much more acidic than would be expected if the compound contained only isolated hydroxyl groups.
===Salts===
The ascorbate anion forms salts, | Chemistry of ascorbic acid | [
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such as sodium ascorbate, calcium ascorbate, and potassium ascorbate.
===Esters===
Ascorbic acid can also react with organic acids as an alcohol forming esters such as ascorbyl palmitate and ascorbyl stearate.
===Nucleophilic attack===
Nucleophilic attack of ascorbic acid on a proton results in a 1,3-diketone:
:Image:Ascorbic diketone.png
===Oxidation===
The ascorbate ion is the predominant species at typical biological pH values. It is a mild reducing agent and antioxidant. It is oxidized with loss of one electron to form a radical cation and then with loss of a second electron to form dehydroascorbic acid. It typically reacts with oxidants of the reactive oxygen species, such as the hydroxyl radical.
Ascorbic acid is special because it can transfer a single electron, owing to the resonance-stabilized nature of its own radical ion, called semidehydroascorbate. The net reaction is:
:RO• + → RO− + C6H7O → ROH + C6H6O6
On exposure to oxygen, ascorbic acid will undergo further oxidative decomposition to various products including diketogulonic acid, xylonic acid, threonic acid and oxalic acid.
Reactive oxygen species are damaging to animals and plants at the molecular level due to their possible interaction with nucleic acids, proteins, and lipids. Sometimes these radicals initiate chain reactions. Ascorbate can terminate these chain radical reactions by electron transfer. | Chemistry of ascorbic acid | [
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] |
The oxidized forms of ascorbate are relatively unreactive and do not cause cellular damage.
However, being a good electron donor, excess ascorbate in the presence of free metal ions can not only promote but also initiate free radical reactions, thus making it a potentially dangerous pro-oxidative compound in certain metabolic contexts.
Ascorbic acid and its sodium, potassium, and calcium salts are commonly used as antioxidant food additives. These compounds are water-soluble and, thus, cannot protect fats from oxidation: For this purpose, the fat-soluble esters of ascorbic acid with long-chain fatty acids (ascorbyl palmitate or ascorbyl stearate) can be used as food antioxidants.
===Other reactions===
It creates volatile compounds when mixed with glucose and amino acids in 90 °C.
It is a cofactor in tyrosine oxidation.
==Uses==
===Food additive===
The main use of -ascorbic acid and its salts is as food additives, mostly to combat oxidation. It is approved for this purpose in the EU with E number E300,UK Food Standards Agency: USA,US Food and Drug Administration: Australia, and New Zealand.Australia New Zealand Food Standards Code
===Dietary supplement===
Another major use of -ascorbic acid is as dietary supplement.
===Niche, non-food uses===
* Ascorbic acid is easily oxidized and so is used as a reductant in photographic developer solutions (among others) and | Chemistry of ascorbic acid | [
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] |
as a preservative.
* In fluorescence microscopy and related fluorescence-based techniques, ascorbic acid can be used as an antioxidant to increase fluorescent signal and chemically retard dye photobleaching.
* It is also commonly used to remove dissolved metal stains, such as iron, from fiberglass swimming pool surfaces.
* In plastic manufacturing, ascorbic acid can be used to assemble molecular chains more quickly and with less waste than traditional synthesis methods.
* Heroin users are known to use ascorbic acid as a means to convert heroin base to a water-soluble salt so that it can be injected.
* As justified by its reaction with iodine, it is used to negate the effects of iodine tablets in water purification. It reacts with the sterilized water, removing the taste, color, and smell of the iodine. This is why it is often sold as a second set of tablets in most sporting goods stores as Potable Aqua-Neutralizing Tablets, along with the potassium iodide tablets.
*Intravenous high-dose ascorbate is being used as a chemotherapeutic and biological response modifying agent. Currently it is still under clinical trials.
* It is sometimes used as a urinary acidifier to enhance the antiseptic effect of methenamine.
==Synthesis==
Natural biosynthesis of vitamin C occurs in many plants, and animals, | Chemistry of ascorbic acid | [
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by a variety of processes.
===Industrial preparation===
Eighty percent of the world's supply of ascorbic acid is produced in China.
Ascorbic acid is prepared in industry from glucose in a method based on the historical Reichstein process. In the first of a five-step process, glucose is catalytically hydrogenated to sorbitol, which is then oxidized by the microorganism Acetobacter suboxydans to sorbose. Only one of the six hydroxy groups is oxidized by this enzymatic reaction. From this point, two routes are available. Treatment of the product with acetone in the presence of an acid catalyst converts four of the remaining hydroxyl groups to acetals. The unprotected hydroxyl group is oxidized to the carboxylic acid by reaction with the catalytic oxidant TEMPO (regenerated by sodium hypochlorite — bleaching solution). Historically, industrial preparation via the Reichstein process used potassium permanganate as the bleaching solution. Acid-catalyzed hydrolysis of this product performs the dual function of removing the two acetal groups and ring-closing lactonization. This step yields ascorbic acid. Each of the five steps has a yield larger than 90%.
A more biotechnological process, first developed in China in the 1960s, but further developed in the 1990s, bypasses the use of acetone-protecting groups. A | Chemistry of ascorbic acid | [
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second genetically modified microbe species, such as mutant Erwinia, among others, oxidises sorbose into 2-ketogluconic acid (2-KGA), which can then undergo ring-closing lactonization via dehydration. This method is used in the predominant process used by the ascorbic acid industry in China, which supplies 80% of world's ascorbic acid.China's grip on key food additive / The Christian Science Monitor. CSMonitor.com (2007-07-20). Retrieved on 2012-12-04. American and Chinese researchers are competing to engineer a mutant that can carry out a one-pot fermentation directly from glucose to 2-KGA, bypassing both the need for a second fermentation and the need to reduce glucose to sorbitol.BASF’s description of vitamin C—developments in production methods . competition-commission.org.uk
There exists a -ascorbic acid, which does not occur in nature but can be synthesized artificially. To be specific, -ascorbate is known to participate in many specific enzyme reactions that require the correct enantiomer (-ascorbate and not -ascorbate). -Ascorbic acid has a specific rotation of [α] = +23°.
===Determination===
The traditional way to analyze the ascorbic acid content is the process of titration with an oxidizing agent, and several procedures have been developed.
The popular iodometry approach uses iodine in the presence of a starch indicator. Iodine is reduced by ascorbic acid, and, when all | Chemistry of ascorbic acid | [
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] |
the ascorbic acid has reacted, the iodine is then in excess, forming a blue-black complex with the starch indicator. This indicates the end-point of the titration.
As an alternative, ascorbic acid can be treated with iodine in excess, followed by back titration with sodium thiosulfate using starch as an indicator.
This iodometric method has been revised to exploit reaction of ascorbic acid with iodate and iodide in acid solution. Electrolyzing the solution of potassium iodide produces iodine, which reacts with ascorbic acid. The end of process is determined by potentiometric titration in a manner similar to Karl Fischer titration. The amount of ascorbic acid can be calculated by Faraday's law.
Another alternative uses N-bromosuccinimide (NBS) as the oxidizing agent, in the presence of potassium iodide and starch. The NBS first oxidizes the ascorbic acid; when the latter is exhausted, the NBS liberates the iodine from the potassium iodide, which then forms the blue-black complex with starch.
== See also ==
* Colour retention agent
* Erythorbic acid: a diastereomer of ascorbic acid.
* Mineral ascorbates: salts of ascorbic acid
* Acids in wine
==Notes and references==
== Further reading ==
*
*
*
*
*
== External links ==
*
*
*IPCS Poisons Information Monograph (PIM) 046
*Interactive 3D-structure of vitamin C with details | Chemistry of ascorbic acid | [
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] |
on the x-ray structure
Category:Organic acids
Category:Antioxidants
Category:Dietary antioxidants
Category:Coenzymes
Category:Corrosion inhibitors
Category:Furanones
Category:Vitamers
Category:Vitamin C
Category:Biomolecules
Category:3-Hydroxypropenals | Chemistry of ascorbic acid | [
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AMOS BASIC is a dialect of the BASIC programming language for the Amiga computer. Following on from the successful STOS BASIC for the Atari ST, AMOS BASIC was written for the Amiga by François Lionet with Constantin Sotiropoulos and published by Europress Software in 1990.
== History ==
AMOS competed on the Amiga platform with Acid Software's Blitz BASIC. Both BASICs differed from other dialects on different platforms, in that they allowed the easy creation of fairly demanding multimedia software, with full structured code and many high-level functions to load images, animations, sounds and display them in various ways.
The original AMOS was a BASIC interpreter which, whilst working fine, suffered the same disadvantages of any language being run interpretively. By all accounts, AMOS was extremely fast among interpreted languages, being speedy enough that an extension called AMOS 3D could produce playable 3D games even on plain 7 MHz 68000 Amigas. Later, an AMOS compiler was developed that further increased speed. AMOS could also run MC68000 machine code, loaded into a program's memory banks.The Creator, by François Lionet, 1990, "AMOS Basic includes special facilities which allow you to combine assembly language routines with your Basic programs."
To simplify animation of sprites, AMOS included the AMOS | AMOS (programming language) | [
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Animation Language (AMAL), a compiled sprite scripting language which runs independently of the main AMOS BASIC program. It was also possible to control screen and "rainbow" effects using AMAL scripts. AMAL scripts in effect created CopperLists, small routines executed by the Amiga's Agnus chip.
After the original version of AMOS, Europress released a compiler (AMOS Compiler), and two other versions of the language: Easy AMOS, a simpler version for beginners, and AMOS Professional, a more advanced version with added features, such as a better integrated development environment, ARexx support, a new user interface API and new flow control constructs. Neither of these new versions was significantly more popular than the original AMOS.
AMOS was used mostly to make multimedia software, video games (platformers and graphical adventures) and educational software.
The language was mildly successful within the Amiga community. Its ease of use made it especially attractive to beginners.
Perhaps AMOS BASIC's biggest disadvantage, stemming from its Atari ST lineage, was its incompatibility with the Amiga's operating system functions and interfaces. Instead, AMOS BASIC controlled the computer directly, which caused programs written in it to have a non-standard user interface, and also caused compatibility problems with newer versions of hardware.
Today, the language has declined in | AMOS (programming language) | [
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popularity along with the Amiga computer for which it was written. Despite this, a small community of enthusiasts are still using it. The source code to AMOS was released around 2001 under a BSD style license by Clickteam, a company that includes the original programmer.Amos & Stos » Main Download on clickteam.com (archived 2007)
On the 4 April 2019, François Lionet announced the release of AMOS2 on his website amos2.org. AMOS2 replaces STOS and AMOS together, using JavaScript as its code interpreter, making the new development system independent and generally deployed in internet browsers.
Amos 2 is now called AOZ Studio. Its website is at https://www.aoz.studio/.
== Software ==
Software written using AMOS BASIC includes:
* Miggybyte
* Scorched Tanks
* Games by Vulcan Software, amongst which was the Valhalla trilogy
* Amiga version of Ultimate Domain (called Genesia) by Microïds
* Flight of the Amazon Queen, by Interactive Binary Illusions
* Extreme Violence, included on an Amiga Power cover disk
* Jetstrike, a commercial game by Rasputin Software
== References ==
== External links ==
* Source code for AMOS Professional 68000 ASM from pianetaamiga.it (archived, ZIP)
* Source code for AMOS and STOS 68000 ASM from clickteam.com (archived, ZIP)
* The AMOS Factory (an AMOS support/community site)
* Amigacoding website (contains in-depth info and references | AMOS (programming language) | [
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] |
for AMOS - Archived version 22 Sep 2015)
* History of STOS and AMOS: how they came to be published in the UK
* Amos Professional group on Facebook (one of the members is AMOS' original developer François Lionet)
Category:BASIC programming language family
Category:Video game development software
Category:Amiga development software
Category:Software using the BSD license
Category:Programming languages created in 1990 | AMOS (programming language) | [
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In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have
*x(xy) = (xx)y
*(yx)x = y(xx)
for all x and y in the algebra.
Every associative algebra is obviously alternative, but so too are some strictly non-associative algebras such as the octonions.
==The associator==
Alternative algebras are so named because they are the algebras for which the associator is alternating. The associator is a trilinear map given by
:[x,y,z] = (xy)z - x(yz).
By definition, a multilinear map is alternating if it vanishes whenever two of its arguments are equal. The left and right alternative identities for an algebra are equivalent toSchafer (1995) p.27
:[x,x,y] = 0
:[y,x,x] = 0.
Both of these identities together imply that
:[x,y,x] = [x, x, x] + [x, y, x] - [x, x+y, x+y] = [x, x+y, -y] = [x, x, -y] - [x, y, y] = 0
for all x and y. This is equivalent to the flexible identitySchafer (1995) p.28
:(xy)x = x(yx).
The associator of an alternative algebra is therefore alternating. Conversely, any algebra whose associator is alternating is clearly alternative. By symmetry, any algebra which satisfies any two of:
*left alternative identity: x(xy) = (xx)y
*right alternative identity: (yx)x = y(xx)
*flexible identity: (xy)x = | Alternative algebra | [
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x(yx).
is alternative and therefore satisfies all three identities.
An alternating associator is always totally skew-symmetric. That is,
:[x_{\sigma(1)}, x_{\sigma(2)}, x_{\sigma(3)}] = \sgn(\sigma)[x_1,x_2,x_3]
for any permutation \sigma. The converse holds so long as the characteristic of the base field is not 2.
==Examples==
* Every associative algebra is alternative.
* The octonions form a non-associative alternative algebra, a normed division algebra of dimension 8 over the real numbers.
* More generally, any octonion algebra is alternative.
===Non-examples===
* The sedenions and all higher Cayley–Dickson algebras lose alternativity.
==Properties==
Artin's theorem states that in an alternative algebra the subalgebra generated by any two elements is associative.Schafer (1995) p.29 Conversely, any algebra for which this is true is clearly alternative. It follows that expressions involving only two variables can be written unambiguously without parentheses in an alternative algebra. A generalization of Artin's theorem states that whenever three elements x,y,z in an alternative algebra associate (i.e., [x,y,z] = 0), the subalgebra generated by those elements is associative.
A corollary of Artin's theorem is that alternative algebras are power-associative, that is, the subalgebra generated by a single element is associative.Schafer (1995) p.30 The converse need not hold: the sedenions are power-associative but not alternative.
The Moufang identities
*a(x(ay)) = (axa)y
*((xa)y)a = x(aya)
*(ax)(ya) = a(xy)a
hold in any alternative | Alternative algebra | [
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algebra.
In a unital alternative algebra, multiplicative inverses are unique whenever they exist. Moreover, for any invertible element x and all y one has
:y = x^{-1}(xy).
This is equivalent to saying the associator [x^{-1},x,y] vanishes for all such x and y. If x and y are invertible then xy is also invertible with inverse (xy)^{-1} = y^{-1}x^{-1}. The set of all invertible elements is therefore closed under multiplication and forms a Moufang loop. This loop of units in an alternative ring or algebra is analogous to the group of units in an associative ring or algebra.
Kleinfeld's theorem states that any simple non-associative alternative ring is a generalized octonion algebra over its center.Zhevlakov, Slin'ko, Shestakov, Shirshov. (1982) p.151
The structure theory of alternative rings is presented in.Zhevlakov, Slin'ko, Shestakov, Shirshov. (1982)
==Applications==
The projective plane over any alternative division ring is a Moufang plane.
The close relationship of alternative algebras and composition algebras was given by Guy Roos in 2008:Guy Roos (2008) "Exceptional symmetric domains", §1: Cayley algebras, in Symmetries in Complex Analysis by Bruce Gilligan & Guy Roos, volume 468 of Contemporary Mathematics, American Mathematical Society He shows (page 162) the relation for an algebra A with unit element e and an involutive anti-automorphism a \mapsto | Alternative algebra | [
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] |
a^* such that a + a* and aa* are on the line spanned by e for all a in A. Use the notation n(a) = aa*. Then if n is a non-singular mapping into the field of A, and A is alternative, then (A,n) is a composition algebra.
== See also ==
* Algebra over a field
* Maltsev algebra
* Zorn ring
==References==
*
*
==External links==
*
Category:Non-associative algebras | Alternative algebra | [
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Communications in Belgium are extensive and advanced. Belgium possesses the infrastructure for both mobile and land-based telecom, as well as having significant television, radio and internet infrastructure. The country code for Belgium is BE.
==Services==
===Mail===
Mail regulation is a national competency. Postal service in Belgium is in many cases performed by Belgian Post Group, a semi-private public company. Competitors include DHL and UPS.
Postal codes in Belgium consist of four digits which indicate regional areas, e.g. "9000" is the postal code for Ghent.
===Telephone===
The telephone system itself is highly developed and technologically advanced, with full automation in facilities that handle domestic and international telecom. Domestically speaking, the county has a nationwide cellular telephone system and an extensive network of telephone cables. Telephone regulation is a national competency.
The country code for Belgium is 32 and the international call prefix is 00.
A telephone number in Belgium is a sequence of nine or ten numbers dialled on a telephone to make a call on the telephone network in Belgium. Belgium is under a closed telephone numbering plan, but retains the trunk code, "0", for all national dialling.
====Fixed telephones====
There were 4.668 million land telephone lines in use in Belgium in 2007, a slight decrease on the 4.769 million | Telecommunications in Belgium | [
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in use in 1997.
The majority state-owned public telephone company of Belgium is Proximus. Some other or private operators exist, as Scarlet (Proximus) and Base (KPN).
====Mobile telephones====
Mobile telephone ownership has increased by nearly one thousand percent in the period 1997–2007, from 974,494 to 10.23 million.
There are three licensed mobile network operators (MNO) in Belgium, Proximus (Belgacom), Orange Belgium (Orange S.A.) and Telenet/Base and numerous mobile virtual network operators (MVNO).
A fourth license will be auctioned off by the government in January 2010.
===Internet===
There were 61 (2003) internet service providers in Belgium, serving 8.113 million internet users in 2009. The country code for Belgian websites is .be.
In September 2009 in Flanders there were 3,048,260 broadband internet customers (DSL and cable), of which 2,520,481 were residential users and 527,779 business users. Only 65,175 dial-up internet access accounts remained in the residential market and 9,580 in the business market.
====Internet providers====
=====xDSL Internet Providers=====
Belgium has numerous copper cable internet providers:
* Altercom *End service 2011
* Base
* Proximus
* Destiny
* Digiweb
* EDPnet
* Evonet
* Full Telecom
* Interxion
* iPFix
* LCL
* Mobistar (Orange S.A.) *End service : 2013
* Numericable (France Numericable)
* Perceval
* Portima
* Proximedia Group
* Scarlet (Belgacom)
* Verizon Business (Verizon Communications)
* Ergatel
Only Belgacom and Numericable currently offers fixed telephony and digital television in a | Telecommunications in Belgium | [
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] |
triple play formula. All other companies offer also fixed telephony in a duo play formula.
=====Cable Internet Providers=====
Belgium has three major fiberglass cable internet providers:
* Numéricable for the Brussels region (Ypso Holding)
* Telenet for the Flanders and Brussels regions (Liberty Global)
* VOO for the Walloon and Brussels regions (TECTEO)
*Orange Belgium use Telenet and VOO network combined
These companies all offer fixed telephony and digital television in a triple play formula.
* Interoute Managed Services
* Interxion
* LCL
* Nucleus
* Verizon Business (Verizon Communications)
These companies all offer specialised services.
=====Terrestrial Internet Providers=====
* Clearwire in Brussels, Ghent, Leuven, Aalst, Halle and Vilvoorde (Sprint Nextel)
* Perceval
=====Satellite Internet Providers=====
* Verizon Business (Verizon Communications)
=====ISP for public services=====
* The Brussels Regional Informatics Center (BRIC, Centre d'Informatique pour la Région Bruxelloise in French) offers Internet access to public administrations in the Brussels-Capital Region, relying directly on the national Belnet network and the IRISnet network.http://www.irisnet.eu/en?set_language=en regional
=====Not categorized=====
Other ISP are Chat.be, Connexeon, HostIT, Microsoft Belgium, Netlog, Ulysse, Ven Brussels, Rack66 (EUSIP bvba), WSD Hosting.
===Other===
The microwave relay network is, however, more limited. For international communications, Belgium has 5 submarine cables and a number of satellite earth stations, two of which are Intelsat, and one Eutelsat.
==References==
==External links==
* BIPT - Belgian Institute for Postal Services and Telecommunications
* ISPA | Telecommunications in Belgium | [
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] |
- Internet Service Providers Association of Belgium
* DNS - Domain Name System Belgium
* MAVISE - Belgian TV market
* Agoria - Federation of Belgian IT Employers
* Beltug - Federation of Belgian ICT Professionals
* UPP - Union of Belgian Periodical Press Publishers
* Febelma - Belgian Federation of Magazines
* VRM - Flemish Media Regulator (Dutch community)
* CSA - High Council for the Audiovisual Media (French community)
* MDGB - Germanic Media Council of Belgium (Germanic community) | Telecommunications in Belgium | [
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Telecommunications in Botswana include newspapers, radio, television, fixed and mobile telephones, and the Internet.
In addition to the government-owned newspaper and national radio network, there is an active, independent press (six weekly newspapers). Foreign publications are sold without restriction in Botswana. Two privately owned radio stations began operations in 1999. Botswana's first national television station, the government-owned Botswana Television (BTV), was launched in July 2000. It began broadcasting with three hours of programming on weekdays and five on weekends, offering news in Setswana and English, entertainment, and sports, with plans to produce 60% of its programming locally. The cellular phone providers Orange and MTN cover most of the country.
==Radio stations==
* 2 state-owned national radio stations; 3 privately owned radio stations broadcast locally (2007);
* AM 8, FM 13, shortwave 4 (2001).
==Television stations==
One state-owned and one privately owned; privately owned satellite TV subscription service is available (2007).
Television sets in use:
* 101,713 (2001);
* 98,568 (2003).
*173,327 (2006)
*297,233 (2008)
*297,971 (2011)
*365,650 (2014).
==Telephones ==
Main lines in use:
* 160,500 lines, 134th in the world (2012);"Botswana", World Fact Book, U.S. Central Intelligence Agency, 7 January 2014. Retrieved 21 January 2014.
* 136,900 (2006).
Mobile cellular in use:
* 3.1 million lines, 129th in the world (2012);
Telephone system
* | Telecommunications in Botswana | [
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] |
general assessment: Botswana is participating in regional development efforts; expanding fully digital system with fiber-optic cables linking the major population centers in the east as well as a system of open-wire lines, microwave radio relays links, and radiotelephone communication stations (2011);
* domestic: fixed-line teledensity has declined in recent years and now stands at roughly 7 telephones per 100 persons; mobile-cellular teledensity now pushing 140 telephones per 100 persons (2011);
* international: country code - 267; international calls are made via satellite, using international direct dialing; 2 international exchanges; digital microwave radio relay links to Namibia, Zambia, Zimbabwe, and South Africa; satellite earth station - 1 Intelsat (Indian Ocean) (2011).
==ISDB-T==
Features:
* Supports ISDB-T broadcast (13 segments).
* MPEG-2/ MPEG-4 AVC/ H.264 HD/ SD video.
* DiVX Compatible with 480i / 480p / 720p / 1080i/ 1080p video formats. Auto and manually scan all available TV and radio channels.
* Aspect ratio 16:9 and 4:3.
* 1000 channels memory.
* Parental control.
* Teletext / Bit map subtitle.
* Compliant with ETSI.
* Supported 7 days EPG function.
* VBI Teletext support 6 MHz software setting Auto / Manual program search.
* Multi language supported.
==Internet==
Internet top-level domain: .bw
Internet users:
* 241,272 users, 148th in the world; 11.5% of the population, 166th in the | Telecommunications in Botswana | [
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world (2012);Calculated using penetration rate and population data from "Countries and Areas Ranked by Population: 2012" , Population data, International Programs, U.S. Census Bureau, retrieved 26 June 2013"Percentage of Individuals using the Internet 2000-2012", International Telecommunications Union (Geneva), June 2013, retrieved 22 June 2013
* 120,000 users, 154th in the world (2009);
* 80,000 users (2007).
Internet broadband:
* 16,407 fixed broadband subscriptions, 134th in the world; 0.8% of the population, 143rd in the world;"Fixed (wired)-broadband subscriptions per 100 inhabitants 2012", Dynamic Report, ITU ITC EYE, International Telecommunication Union. Retrieved on 29 June 2013.
* 348,124 wireless broadband subscriptions, 102nd in the world; 16.6% of the population, 76th in the world."Active mobile-broadband subscriptions per 100 inhabitants 2012", Dynamic Report, ITU ITC EYE, International Telecommunication Union. Retrieved on 29 June 2013.
Internet hosts:
* 1,806 hosts (2012);
* 6,374 hosts (2008).
Internet IPv4 addresses: 100,096 addresses allocated, less than 0.05% of the world total, 47.7 addresses per 1000 people (2012).Select Formats , Country IP Blocks. Accessed on 2 April 2012. Note: Site is said to be updated daily.Population, The World Factbook, United States Central Intelligence Agency. Accessed on 2 April 2012. Note: Data are mostly for 1 July 2012.
Internet Service Providers:
* 11 ISPs (2001);
* 2 ISPs (1999).
ADSL has been | Telecommunications in Botswana | [
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] |
introduced in the following areas:
Gaborone, Tlkokweng, Mogoditsane, Phakalane, Francistown, Lobatse, Palapye, Maun, Kasane, Selibe-Phikwe, Letlhakane, Jwaneng, and Orapa.
==See also==
* Botswana
* Botswana TV
* Media of Botswana
* Internet in Botswana
* Botswana Internet Exchange
* Telephone numbers in Botswana
*Botswana Communications Regulatory Authority
==References==
*
==External links==
* Botswana Communications Regulatory Authority (BOCRA). | Telecommunications in Botswana | [
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] |
Country Code: +1284
International Call Prefix: 011 (outside NANP)
Calls from the British Virgin Islands to the US, Canada, and other NANP Caribbean nations, are dialled as 1 + NANP area code + 7-digit number. Calls from the British Virgin Islands to non-NANP countries are dialled as 011 + country code + phone number with local area code.
Number Format: nxx-xxxx
Telephones – main lines in use:
11,700 (2002)
Telephones – mobile cellular:
8,000 (2002)
Telephone system:
worldwide telephone service
general assessment:
worldwide telephone service
domestic:
NA
international:
Connected via submarine cable to Bermuda; the East Caribbean Fibre System (ECFS) submarine cable provides connectivity to 13 other islands in the eastern Caribbean (2007)
Radio stations:
AM 1, FM 5, shortwave 0 (2004)
* ZBVI 780 Tortola
* ZJKC-FM 90.9 Tortola (repeats WJKC 95.1 Christiansted, USVI)
* ZGLD-FM 91.7 Tortola
* ZCCR-FM 94.1 Todman's Peak
* ZWVE-FM 97.3 Tortola
* ZKNG-FM 100.9 Chalwell
* ZROD-FM 103.7 Tortola
* ZVCR-FM 106.9 Chalwell
Television stations:
1 (ZBTV), (plus one cable company) (1997)
Internet service providers (ISPs):
1 (1999)
Internet country code:
VG
Internet hosts:
465 (2008)
Internet users:
4,000 (2002)
:See also : British Virgin Islands
==Deregulation of the telephone market==
In 2006, the British Virgin Islands government undertook a deregulation of the telephone industry.Telecommunications Act (No 10 of 2006) Prior to 2006, in common with many other Caribbean countries, Cable & Wireless (Caribbean) had a statutory monopolyUnder the Cable and | Telecommunications in the British Virgin Islands | [
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Wireless (West Indies) Limited Telephones Act, 1975 (Cap 168) on telephone and other electronic communications services. However, in the 1990s, a local company called CCT Boatphone, which had previously provided radio boatphones to tourists on charter boats, expanded into cellular (mobile) telecommunications for land-based users. Although technically in breach of the statutory monopoly, CCT Boatphone was backed by a powerful collection of local interests known as the BVI Investment Club. Negotiations between Cable & Wireless and CCT Boatphone led to a split of the monopolies, with Cable & Wireless retaining a monopoly over fixed line and internet services, and CCT Boatphone keeping a de facto monopoly over cellular telephones.
In 2007 the government abolished the previously existing monopolies under an order made pursuant to the new legislation. The process proved politically fraught, and the government's Minister for Communications and Works, Alvin Christopher, ended up leaving the government and joining the opposition party as a result of the furore. The process was also criticised as cumbersome and slow, the initial deregulation having been announced in 2004, and taking no less than three years to come to fruition through delays in legislation and regulation.
Although there have been no | Telecommunications in the British Virgin Islands | [
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new entrants into the fixed line industry, the government issued three licences under the new regime to cellular telephone service providers. The existing provider, CCT Boatphone, obtained one licence. Bmobile, the cellular arm of Cable & Wireless, obtained a second. The third licence was obtained by BVI Cable TV, a local cable television service. The licence in favour of BVI Cable was controversial, as the Regulator had announced in advance that only three licences in total would be issued, and BVI Cable TV had crumbling cable television infrastructure, and was in no position to offer cellular telephone services (and to date, has not offered any cellular telephone services, or anything other than simple cable television). However, bmobile's main regional competitor, Digicel, was rejected for a licence. The decision was regarded as highly controversial in the local media.BVI Platinum news, 23 May 2007 and BVI Platinum news, 4 June 2007
Digicel then issued court proceedings against the Regulator, arguing that he had acted improperly by imposing an arbitrary limit of three licences (although no complaint was made about the decision to prefer BVI Cable TV's improbable licence over Digicel). Bmobile was joined to the suit | Telecommunications in the British Virgin Islands | [
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] |
as an interested party. High Court Judge Rita Joseph-Olivetti found in favour of Digicel and quashed the original decision.BVI Platinum News, 25 May 2007 Digicel commenced separate proceedings against Cable and Wireless (as bmobile's parent company) in the English courts, claiming that Cable & Wireless has unfairly stifled competition in several Caribbean jurisdictions.BBC World news, 19 July 2007 During the intervening period, bmobile has obtained a virtual stranglehold on the cellular telecommunications market in the British Virgin Islands by a combination of low prices and aggressive advertising, as well as significant investment in infrastructure and technology.BVI Platinum news, 8 June 2007 and BVI Platinum news, 25 June 2007
Digicel was finally granted a licence on 17 December 2007 and started operations in the BVI on 28 November 2008.Caribbean Net News: British Virgin Islands Jamaica Gleaner News – Digicel to take on two new markets – Investing US$15m in BVI roll-out, bids US$80m for Honduras licence – Friday | 21 December 2007 DIGICEL LAUNCHES IN BVI WITH US$17 MILLION INVESTMENT
==References==
British Virgin | Telecommunications in the British Virgin Islands | [
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] |
Telecommunications in Bulgaria include radio, television, fixed and mobile telephones, and the Internet.
==Radio and television==
* Radio broadcast stations: AM 31, FM 63, shortwave 2 (2001).
* Radio broadcast hours: 525,511 (2003).
* Television broadcast stations: 39 (2001).
* Television broadcast hours: 498,091 (2003).
==Telephones==
* Main lines in use: 1.6 million (2015 est).
* Mobile cellular: 8.98 million lines (2016).
* Telephone system:
** General assessment: an extensive but antiquated telecommunications network inherited from the Soviet era; quality has improved; the Bulgaria Telecommunications Company's fixed-line monopoly terminated in 2005 when alternative fixed-line operators were given access to its network; a drop in fixed-line connections in recent years has been more than offset by a sharp increase in mobile-cellular telephone use fostered by multiple service providers; the number of cellular telephone subscriptions now exceeds the population
** Domestic: a fairly modern digital cable trunk line now connects switching centers in most of the regions; the others are connected by digital microwave radio relay
** International: country code – 359; submarine cable provides connectivity to Ukraine and Russia; a combination submarine cable and land fiber-optic system provides connectivity to Italy, Albania, and North Macedonia; satellite earth stations – 3 (1 Intersputnik in the Atlantic Ocean region, 2 Intelsat in the | Telecommunications in Bulgaria | [
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Atlantic and Indian Ocean regions) (2007).
==Internet==
* Top-level domains: .bg"Communications: Bulgaria", World Factbook, U.S. Central Intelligence Agency, 28 January 2014. Retrieved 1 February 2014. and .бг (proposed, Cyrillic).
* Internet users:
** 4.1 million users (2016)
** 3.9 million users, 72nd in the world; 55.1% of the population, 74th in the world (2012);Calculated using penetration rate and population data from "Countries and Areas Ranked by Population: 2012" , Population data, International Programs, U.S. Census Bureau, retrieved 26 June 2013"Percentage of Individuals using the Internet 2000-2012", International Telecommunication Union (Geneva), June 2013, retrieved 22 June 2013
** 3.4 million users, 63rd in the world (2009);
** 1.9 million users (2007).
* Fixed broadband: 1.2 million subscriptions, 52nd in the world; 17.6% of population, 53rd in the world (2012)."Fixed (wired)-broadband subscriptions per 100 inhabitants 2012", Dynamic Report, ITU ITC EYE, International Telecommunication Union. Retrieved on 29 June 2013.
* Wireless broadband: 2.8 million, 55th in the world; 40.3% of the population, 41st in the world (2012)."Active mobile-broadband subscriptions per 100 inhabitants 2012", Dynamic Report, ITU ITC EYE, International Telecommunication Union. Retrieved on 29 June 2013.
* Internet hosts:
** 976,277 hosts, 47th in the world (2012);
** 513,470 (2008).
* IPv4: 4.2 million addresses allocated, 0.1% of the world | Telecommunications in Bulgaria | [
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] |
total, 589.7 addresses per 1000 people, 51st in the world (2012).Select Formats , Country IP Blocks. Accessed on 2 April 2012. Note: Site is said to be updated daily.Population, The World Factbook, United States Central Intelligence Agency. Accessed on 2 April 2012. Note: Data are mostly for 1 July 2012.
==See also==
* Internet Society – Bulgaria
* List of internet service providers in Bulgaria
* Bulgaria
==References== | Telecommunications in Bulgaria | [
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Telecommunications in Burkina Faso include radio, television, fixed and mobile telephones, and the Internet.
The telephony market in Burkina Faso is still relatively underdeveloped.
Although mobile penetration is just over 100%, it is still below the African average. Fixed-line telephony and internet connections are very low, due in large part to poor network infrastructure.
The government has a number of policies intended to improve the levels of investment and usage of networks but the impact of the SARS-Cov2 pandemic has hampered such efforts.
"Burkina Faso - Telecoms, Mobile and Broadband - Statistics and Analyses", Budde.com, Henry Lancaster, Peter Lange, 14 Jul 2020, Retrieved 06 April 2021.
Radio is the country's most popular communications medium. Use of telecommunications in Burkina Faso are extremely low, limited due to the low penetration of electricity, even in major cities. There were just 141,400 fixed line phones in use in 2012, in a country with a population of 17.4 million. Use of mobile phones has skyrocketed from 1.0 million lines in 2006 to 10 million in 2012. Internet use is also low, with only 3.7 users per 100 inhabitants in 2012, just over 643,000 users total. The Internet penetration rate in Africa as a whole was 16 users per 100 | Telecommunications in Burkina Faso | [
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inhabitants in 2013.
==Regulation and control==
The constitution and law provide for freedom of speech and of the press, and the government generally respects these rights in practice.
All media are under the administrative and technical supervision of the Ministry of Communications, which is responsible for developing and implementing government policy on information and communication. The Superior Council of Communication (SCC), a semiautonomous body under the Office of the President, monitors the content of radio and television programs, newspapers, and Internet Web sites to ensure compliance with professional ethics standards and government policy. The SCC may summon journalists and issue warnings for subsequent violations. Hearings may concern alleged libel, disturbing the peace, inciting violence, or violations of state security.
Journalists occasionally face criminal libel prosecutions and other forms of harassment and intimidation. In addition to the prohibition against insulting the head of state, the law also prohibits the publication of shocking images and lack of respect for the deceased. Although the government does not attempt to impede criticism, some journalists practice self-censorship.
The Burkinabé government, in its telecommunications development strategy, has stated its aim to make telecommunications a universal service accessible to all. A large portion of this strategy is the privatization of the National | Telecommunications in Burkina Faso | [
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Telecommunications Office (ONATEL), with an additional focus on a rural telephony promotion project. In 2006 the government sold a 51 percent stake in the national telephone company, ONATEL, and ultimately planned to retain only a 23 percent stake in the company.
==Radio and television==
* Radio stations: 2 AM, 26 FM, and 3 shortwave stations;"Communications: Burkina Faso", World Factbook, U.S. Central Intelligence Agency, 15 May 2008. Retrieved 11 February 2014 via the Internet Archive. state-owned radio runs a national and regional network; substantial number of privately owned radio stations; transmissions of several international broadcasters available in Ouagadougou (2007)."Communications: Burkina Faso", World Factbook, U.S. Central Intelligence Agency, 28 January 2014. Retrieved 11 February 2014.
* Television stations: 1 state-owned and 1 privately owned (2007).
Radio is the country's most popular communications medium. Dozens of private and community radio stations and a handful of private TV channels operate alongside their state-run counterparts. The BBC World Service, Voice of America, and Radio France Internationale are all on the air in the capital, Ouagadougou."Burkina Faso profile", BBC News, 14 August 2012. Retrieved 11 February 2014.
==Telephones==
* Calling code: +226
* International call prefix: 00Dialing Procedures (International Prefix, National (Trunk) Prefix and National (Significant) Number) (in Accordance with ITY-T Recommendation | Telecommunications in Burkina Faso | [
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E.164 (11/2010)), Annex to ITU Operational Bulletin No. 994-15.XII.2011, International Telecommunication Union (ITU, Geneva), 15 December 2011. Retrieved 2 January 2014.
* Main lines:
** 141,400 lines in use (2012);
** 94,800 lines in use, 144th in the world (2006).
* Mobile cellular:
** 10.0 million lines, 79th in the world (2012);
** 1.0 million lines, 123rd in the world (2006).
* Telephone system: system includes microwave radio relay, open-wire, and radiotelephone communication stations; fixed-line connections stand at less than 1 per 100 persons; mobile-cellular usage, fostered by multiple providers, is increasing rapidly from a low base (2011).
* Satellite earth stations: 1 Intelsat (Atlantic Ocean) (2011).
* Communications cables: Burkina Faso is linked to the global submarine cable network and the international Internet backbone through Senegal's Sonatel fibre-optic transmission network."Internet connectivity in Senegal", slide 15, Aminata Drame, ITU workshop on "Apportionment of revenues and international Internet connectivity", Geneva, Switzerland, 23–24 January 2012. Retrieved 11 February 2014.
==Early Development of the Mobile Market==
The state-run Office National Des Telecommunications (ONATEL) launched the first mobile network based on CDMA2000 technology in 1998.<BR>
Competition was introduced to the mobile telephone segment in 2000 with the introduction of new GSM network operators Celtel, Télécel Faso and ONATEL's Telmob. This pushed rates down | Telecommunications in Burkina Faso | [
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even as density and coverage area increased.<BR>
Use of mobile phones grew quickly in the 2000s, growing from 2,700 subscribers in 1998, to 1.0 million in 2006, to 10.0 million in 2012. and to 21.4 million in 2020."DONNEES DU MARCHE NATIONAL DE LA TELEPHONIE MOBILE - 3ème Trimestre 2020", ARCEP, 31 Dec 2020, Retrieved 06 April 2021.<BR>
ARPU remained low, however, as mobile subscribers adopted behaviours such as "flashing" to minimize their costs and Burkina Faso's ancient oral tradition and talking drum culture harmonized with the introduction of mobile phone technologies.Historical Dictionary of Burkina Faso, by Lawrence Rupley, Lamissa Bangali, Boureima Diamitani, (see Telecommunications) 2013, Third edition, Scarecrow Press, Inc. Additionally, mobile phone owners acquired status by being able to lend their phones to others in their communities.
==International Group Involvement==
In 2006, Maroc Telecom(itself part of Etisalat group) took a majority stake in ONATEL, which it increased to 61% in 2018"Maroc Telecom ups Onatel stake to 61%", Comms Update, 18 April 2018. Retrieved 06 April 2021. and from July 1, 2019 Maroc Telecom consolidated Onatel, Mauritel, Gabon Télécom, Sotelma, Casanet, AT Côte d'Ivoire, Etisalat Benin, AT Togo, AT Niger, AT Centrafrique, and Tigo Tchad in its accounts."Maroc Telecom_H1 2020 Consolidated | Telecommunications in Burkina Faso | [
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] |
Results", Globe Newswire, 20 July 2020, Retrieved 06 April 2021.<BR>
In January 2021, Maroc Telecom rebranded all of its African subsidiaries as Moov Africa."Maroc Telecom rebrands African subsidiaries as Moov Africa", Comms Update, 04 January 2021. Retrieved 06 April 2021.
In 2005 Celtel was acquired by the Kuwaiti Zain Group.
In 2010 Zain Group decided to sell most of the Celtel group to Indian group Bharti Airtel,"Zain Approves Airtel's Acquisition of Its African Operations", CIO, John Ribeiro, 25 March 2010, Retrieved 06 April 2021. which rebranded Celtel as Airtel Burkina Faso.
<BR>
In June 2016 Orange S.A. acquired the network and 4.6M subscribers of Airtel Burkina Faso."GSMA Mobile World Live: Orange completes Burkina Faso rebrand", Mobile World Live,Kavit Majithia, GSMA, 16 March 2017. Retrieved 06 April 2021. Following an ambitious network modernization plan, 9 months later the network rebranded as Orange Burkina Faso boasting a subscriber base of 6.3M.
According to the website of the Communication Regulator of Burkina Faso, at the end of 2020 the Mobile Telecommunications Market (21.4M subscriptions) was shared as follows:
* Orange BF S.A. 9,403,367 subscriptions (43.72%)
* Onatel S.A. | Telecommunications in Burkina Faso | [
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] |
9,086,709 subscriptions (42.24%)
* Télécel Faso S.A 2,946,469 subscriptions (13.70%)
==Internet==
* Top-level domain: .bf
* Internet users:
** 643,504 users, 127th in the world; 3.7% of the population, 194th in the world (2012);Calculated using penetration rate and population data from "Countries and Areas Ranked by Population: 2012" , Population data, International Programs, U.S. Census Bureau, retrieved 26 June 2013"Percentage of Individuals using the Internet 2000-2012", International Telecommunication Union (Geneva), June 2013, retrieved 22 June 2013
** 178,100 users, 144th in the world (2009);
** 80,000, 146th in the world (2006).
* Fixed broadband: 14,166 subscriptions, 139th in the world; 0.1% of population, 169th in the world (2012)."Fixed (wired)-broadband subscriptions per 100 inhabitants 2012", Dynamic Report, ITU ITC EYE, International Telecommunication Union. Retrieved on 29 June 2013.
* Wireless broadband: Unknown (2012)."Active mobile-broadband subscriptions per 100 inhabitants 2012", Dynamic Report, ITU ITC EYE, International Telecommunication Union. Retrieved on 29 June 2013.
* Internet hosts:
** 1,795, 164th in the world (2012);
** 193 hosts, 178th in the world (2007).
* IPv4: 32,512 addresses allocated, less than 0.05% of the world total, 1.9 addresses per 1000 people (2012).Select Formats , Country IP Blocks. Accessed on 2 April 2012. Note: Site is said | Telecommunications in Burkina Faso | [
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] |
to be updated daily.Population, The World Factbook, United States Central Intelligence Agency. Accessed on 2 April 2012. Note: Data are mostly for 1 July 2012.
* Internet Service Providers: 1 ISP (1999).
Internet use is low, but the sector began to improve following installation of a 22 Mbit/s fiber optic international link, a vast improvement over the previous 128 kbit/s link. Secondary access nodes began to appear in major cities, and cybercafés were providing Internet access to a broader spectrum of end users.
ONATEL's FasoNet is the country's leading wired Internet service provider, dominating the broadband market with its ADSL and EV-DO fixed-wireless offerings."Burkina Faso - Telecoms, Mobile and Broadband", BuddeComm, 29 January 2014. Retrieved 11 February 2014.
The mobile operators are offering data services using GPRS and EDGE technology, and third generation (3G) mobile broadband technology was not introduced until 2013 by Bharti Airtel."Airtel launches first 3.75 G service in Burkina Faso", IT News Africa, 28 May 2013. Retrieved 19 April 2015.
A March 2013 ITU Study on international Internet connectivity in sub-Saharan Africa reports that the Burkina Faso "Internet market is not sufficiently dynamic and competitive" and that the high costs for Internet capable mobile phones (more than six times the cost of | Telecommunications in Burkina Faso | [
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a basic mobile phone) and mobile Internet subscriptions (up to seven times the cost for basic mobile) limit the number of Internet users.Study on international Internet connectivity in sub-Saharan Africa, Abossé Akue-Kpakpo, International Telecommunication Union (ITU), March 2013. Retrieved 11 February 2014.
===Internet censorship and surveillance===
There are no government restrictions on access to the Internet; however, the Superior Council of Communication (SCC) monitors Internet Web sites and discussion forums to ensure compliance with existing regulations. For example, in May 2012 the SCC issued a warning to a Web site on which a user had allegedly insulted President Compaore in an Internet forum."Burkina Faso", Country Reports on Human Rights Practices for 2012, Bureau of Democracy, Human Rights and Labor, U.S. Department of State, 18 April 2013. Retrieved 11 February 2014.
The constitution and law provide for freedom of speech and of the press, and the government generally respects these rights in practice. The law prohibits persons from insulting the head of state or using derogatory language with respect to the office; however, individuals criticize the government publicly or privately without reprisal.
The constitution and law prohibit arbitrary interference with privacy, family, home, or correspondence, and the government generally respects these prohibitions in practice. In | Telecommunications in Burkina Faso | [
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cases of national security, however, the law permits surveillance, searches, and monitoring of telephones and private correspondence without a warrant.
==See also==
* Radio Télévision du Burkina, national broadcaster of Burkina Faso.
* Maroc Telecom, a 51% owner of ONATEL since December 2006.
* List of terrestrial fibre optic cable projects in Africa
* Media in Burkina Faso
* Economy of Burkina Faso
==References==
*
*
==External links==
* Autorité de régulation des communications électroniques (ARCE, Regulatory authority for electronic communications) , registrar for the .bf domain. | Telecommunications in Burkina Faso | [
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Baseball statistics play an important role in evaluating the progress of a player or team.
Since the flow of a baseball game has natural breaks to it, and normally players act individually rather than performing in clusters, the sport lends itself to easy record-keeping and statistics. Statistics have been kept for professional baseball since the creation of the National League and American League, now part of Major League Baseball.
Many statistics are also available from outside Major League Baseball, from leagues such as the National Association of Professional Base Ball Players and the Negro leagues, although the consistency of whether these records were kept, of the standards with respect to which they were calculated, and of their accuracy has varied.
==Development==
The practice of keeping records of player achievements was started in the 19th century by Henry Chadwick. Based on his experience with the sport of cricket, Chadwick devised the predecessors to modern-day statistics including batting average, runs scored, and runs allowed.
Traditionally, statistics such as batting average (the number of hits divided by the number of at bats) and earned run average (the average number of earned runs allowed by a pitcher per nine innings) have dominated attention in the statistical world of baseball. | Baseball statistics | [
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] |
However, the recent advent of sabermetrics has created statistics drawing from a greater breadth of player performance measures and playing field variables. Sabermetrics and comparative statistics attempt to provide an improved measure of a player's performance and contributions to his team from year to year, frequently against a statistical performance average.
Comprehensive, historical baseball statistics were difficult for the average fan to access until 1951, when researcher Hy Turkin published The Complete Encyclopedia of Baseball. In 1969, Macmillan Publishing printed its first Baseball Encyclopedia, using a computer to compile statistics for the first time. Known as "Big Mac", the encyclopedia became the standard baseball reference until 1988, when Total Baseball was released by Warner Books using more sophisticated technology. The publication of Total Baseball led to the discovery of several "phantom ballplayers", such as Lou Proctor, who did not belong in official record books and were removed.
==Use==
Throughout modern baseball, a few core statistics have been traditionally referenced – batting average, RBI, and home runs. To this day, a player who leads the league in all of these three statistics earns the "Triple Crown". For pitchers, wins, ERA, and strikeouts are the most often-cited statistics, and a pitcher leading his league in | Baseball statistics | [
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these statistics may also be referred to as a "triple crown" winner. General managers and baseball scouts have long used the major statistics, among other factors and opinions, to understand player value. Managers, catchers and pitchers use the statistics of batters of opposing teams to develop pitching strategies and set defensive positioning on the field. Managers and batters study opposing pitcher performance and motions in attempting to improve hitting. Scouts use stats when they are looking at a player who they may end up drafting or signing to a contract.
Some sabermetric statistics have entered the mainstream baseball world that measure a batter's overall performance including on-base plus slugging, commonly referred to as OPS. OPS adds the hitter's on-base percentage (number of times reached base by any means divided by total plate appearances) to their slugging percentage (total bases divided by at-bats). Some argue that the OPS formula is flawed and that more weight should be shifted towards OBP (on-base percentage). The statistic wOBA (weighted on-base average) attempts to correct for this.
OPS is also useful when determining a pitcher's level of success. "Opponent on-base plus slugging" (OOPS) is becoming a popular tool to evaluate a pitcher's actual performance. When | Baseball statistics | [
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analyzing a pitcher's statistics, some useful categories include K/9IP (strikeouts per nine innings), K/BB (strikeouts per walk), HR/9 (home runs per nine innings), WHIP (walks plus hits per inning pitched), and OOPS (opponent on-base plus slugging).
However, since 2001, more emphasis has been placed on defense-independent pitching statistics, including defense-independent ERA (dERA), in an attempt to evaluate a pitcher's performance regardless of the strength of the defensive players behind them.
All of the above statistics may be used in certain game situations. For example, a certain hitter's ability to hit left-handed pitchers might incline a manager to increase their opportunities to face left-handed pitchers. Other hitters may have a history of success against a given pitcher (or vice versa), and the manager may use this information to create a favorable
match-up. This is often referred to as "playing the percentages".
==Commonly used statistics==
Most of these terms also apply to softball. Commonly used statistics with their abbreviations are explained here. The explanations below are for quick reference and do not fully or completely define the statistic; for the strict definition, see the linked article for each statistic.
===Batting statistics===
* 1B – Single: hits on which the batter reaches first base safely without the contribution of | Baseball statistics | [
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a fielding error
* 2B – Double: hits on which the batter reaches second base safely without the contribution of a fielding error
* 3B – Triple: hits on which the batter reaches third base safely without the contribution of a fielding error
* AB – At bat: plate appearances, not including bases on balls, being hit by pitch, sacrifices, interference, or obstruction
* AB/HR – At bats per home run: at bats divided by home runs
* BA – Batting average (also abbreviated AVG): hits divided by at bats (H/AB)
* BB – Base on balls (also called a "walk"): hitter not swinging at four pitches called out of the strike zone and awarded first base.
* BABIP – Batting average on balls in play: frequency at which a batter reaches a base after putting the ball in the field of play. Also a pitching category.
* BB/K – Walk-to-strikeout ratio: number of bases on balls divided by number of strikeouts
* BsR – Base runs: Another run estimator, like runs created
* EQA – Equivalent average: a player's batting average absent park and league factors
* FC – Fielder's choice: times reaching base safely because a fielder chose to try for an out on another runner
* GO/AO – Ground | Baseball statistics | [
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ball fly ball ratio: number of ground ball outs divided by number of fly ball outs
* GDP or GIDP – Ground into double play: number of ground balls hit that became double plays
* GPA – Gross production average: 1.8 times on-base percentage plus slugging percentage, divided by four
* GS – Grand slam: a home run with the bases loaded, resulting in four runs scoring, and four RBIs credited to the batter
* H – Hit: reaching base because of a batted, fair ball without error by the defense
* HBP – Hit by pitch: times touched by a pitch and awarded first base as a result
* HR – Home runs: hits on which the batter successfully touched all four bases, without the contribution of a fielding error
* HR/H – Home runs per hit: home runs divided by total hits
* ITPHR – Inside-the-park home run: hits on which the batter successfully touched all four bases, without the contribution of a fielding error or the ball going outside the ball park.
* IBB – Intentional base on balls: times awarded first base on balls (see BB above) deliberately thrown by the pitcher. Also known as IW (intentional walk).
* ISO – Isolated power: a hitter's | Baseball statistics | [
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] |
ability to hit for extra bases, calculated by subtracting batting average from slugging percentage
* K – Strike out (also abbreviated SO): number of times that a third strike is taken or swung at and missed, or bunted foul. Catcher must catch the third strike or batter may attempt to run to first base.
* LOB – Left on base: number of runners neither out nor scored at the end of an inning
* OBP – On-base percentage: times reached base (H + BB + HBP) divided by at bats plus walks plus hit by pitch plus sacrifice flies (AB + BB + HBP + SF)
* OPS – On-base plus slugging: on-base percentage plus slugging average
* PA – Plate appearance: number of completed batting appearances
* PA/SO – Plate appearances per strikeout: number of times a batter strikes out to their plate appearance
* R – Runs scored: number of times a player crosses home plate
* RC – Runs created: an attempt to measure how many runs a player has contributed to their team
* RP – Runs produced: an attempt to measure how many runs a player has contributed
* RBI – Run batted in: number of runners who score due to a batter's action, except | Baseball statistics | [
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when the batter grounded into a double play or reached on an error
* RISP – Runner in scoring position: a breakdown of a batter's batting average with runners in scoring position, which includes runners at second or third base
* SF – Sacrifice fly: fly balls hit to the outfield which, although caught for an out, allow a baserunner to advance
* SH – Sacrifice hit: number of sacrifice bunts which allow runners to advance on the basepaths
* SLG – Slugging percentage: total bases achieved on hits divided by at-bats (TB/AB)
* TA – Total average: total bases, plus walks, plus hit by pitch, plus steals, minus caught stealing divided by at bats, minus hits, plus caught stealing, plus grounded into double plays [(TB + BB + HBP + SB – CS)/(AB – H + CS + GIDP)]
* TB – Total bases: one for each single, two for each double, three for each triple, and four for each home run [H + 2B + (2 × 3B) + (3 × HR)] or [1B + (2 × 2B) + (3 × 3B) + (4 × HR)]
* TOB – Times on base: times reaching base as a result of hits, walks, and hit-by-pitches (H + | Baseball statistics | [
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BB + HBP)
* XBH – Extra base hits: total hits greater than singles (2B + 3B + HR)
===Baserunning statistics===
* SB – Stolen base: number of bases advanced by the runner while the ball is in the possession of the defense
** CS – Caught stealing: times tagged out while attempting to steal a base
** SBA or ATT – Stolen base attempts: total number of times the player has attempted to steal a base (SB+CS)
** SB% – Stolen base percentage: the percentage of bases stolen successfully. (SB) divided by (SBA) (stolen bases attempted).
** DI – Defensive Indifference: if the catcher does not attempt to throw out a runner (usually because the base would be insignificant), the runner is not awarded a steal. Scored as a fielder's choice.
* R – Runs scored: times reached home plate legally and safely
* UBR – Ultimate base running: a metric that assigns linear weights to every individual baserunning event in order to measure the impact of a player's baserunning skill
===Pitching statistics===
* BB – Base on balls (also called a "walk"): times pitching four balls, allowing the batter to take first base
* BB/9 – Bases on balls per 9 innings pitched: base on balls multiplied by nine, divided | Baseball statistics | [
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] |
by innings pitched
* BF – Total batters faced: opponent team's total plate appearances
* BK – Balk: number of times pitcher commits an illegal pitching action while in contact with the pitching rubber as judged by umpire, resulting in baserunners advancing one base
* BS – Blown save: number of times entering the game in a save situation, and being charged the run (earned or not) which eliminates his team's lead
* CERA – Component ERA: an estimate of a pitcher's ERA based upon the individual components of his statistical line (K, H, 2B, 3B, HR, BB, HBP)
* CG – Complete game: number of games where player was the only pitcher for their team
* DICE – Defense-Independent Component ERA: an estimate of a pitcher's ERA based upon the defense-independent components of his statistical line (K, HR, BB, HBP) but which also uses number of outs (IP), which is not defense independent.
* ER – Earned run: number of runs that did not occur as a result of errors or passed balls
* ERA – Earned run average: total number of earned runs (see "ER" above), multiplied by 9, divided by innings pitched
* ERA+ – Adjusted ERA+: earned run average adjusted for the ballpark and the | Baseball statistics | [
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] |
league average
* FIP – Fielding independent pitching: a metric, scaled to resemble an ERA, that focuses on events within the pitcher's control – home runs, walks, and strikeouts – but also uses in its denominator the number of outs the team gets (see IP), which is not entirely within the pitcher's control.
** xFIP: This variant substitutes a pitcher's own home run percentage with the league average
* G – Games (AKA "appearances"): number of times a pitcher pitches in a season
* GF – Games finished: number of games pitched where player was the final pitcher for their team as a relief pitcher
* GIDP – Double plays induced: number of double play groundouts induced
* GIDPO - Double play opportunities: number of groundout induced double play opportunities
* GIR - Games in relief: games as a non starting pitcher
* GO/AO or G/F – Ground Out to Air Out ratio, aka Ground ball fly ball ratio: ground balls allowed divided by fly balls allowed
* GS – Starts: number of games pitched where player was the first pitcher for their team
* H (or HA) – Hits allowed: total hits allowed
* H/9 (or HA/9) – Hits allowed per 9 innings pitched: hits allowed times nine divided by | Baseball statistics | [
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innings pitched (also known as H/9IP)
* HB – Hit batsman: times hit a batter with pitch, allowing runner to advance to first base
* HLD (or H) – Hold: number of games entered in a save situation, recorded at least one out, did not surrender the lead, and did not complete the game
* HR (or HRA) – Home runs allowed: total home runs allowed
* HR/9 (or HRA/9) – Home runs per nine innings: home runs allowed times nine divided by innings pitched (also known as HR/9IP)
* IBB – Intentional base on balls allowed
* IP – Innings pitched: the number of outs a team gets while a pitcher is pitching divided by 3
* IP/GS – Average number of innings pitched per game started
* IR – Inherited runners: number of runners on base when the pitcher enters the game
* IRA – Inherited runs allowed: number of inherited runners allowed to score
* K (or SO) – Strikeout: number of batters who received strike three
* K/9 (or SO/9) – Strikeouts per 9 innings pitched: strikeouts times nine divided by innings pitched
* K/BB (or SO/BB) – Strikeout-to-walk ratio: number of strikeouts divided by number of base on balls
* L – Loss: number of games where pitcher | Baseball statistics | [
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was pitching while the opposing team took the lead, never lost the lead, and went on to win
* LOB% – Left-on-base percentage: LOB% represents the percentage of baserunners a pitcher does not allow to score. LOB% tends to regress toward 70–72% over time, so unusually high or low percentages could indicate that pitcher's ERA could be expected to rise or lower in the future. An occasional exception to this logic is a pitcher with a very high strikeout rate.
* OBA (or just AVG) – Opponents batting average: hits allowed divided by at-bats faced
* PC-ST – An individual pitcher's total game pitches [Pitch Count] and [ST] his no. of strikes thrown within that PC.
* PIT (or NP) – Pitches thrown (Pitch count)
* PFR – Power finesse ratio: The sum of strikeouts and walks divided by innings pitched.
* pNERD – Pitcher's NERD: expected aesthetic pleasure of watching an individual pitcher
* QOP – Quality of pitch: comprehensive pitch evaluation statistic which combines speed, location and movement (rise, total break, vertical break and horizontal break) into a single numeric value
* QS – Quality start: a game in which a starting pitcher completes at least six innings and permits no more than three earned runs
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RA – Run average: number of runs allowed times nine divided by innings pitched
* SHO – Shutout: number of complete games pitched with no runs allowed
*SIERA – Skill-Interactive Earned Run Average: another advanced stat that measures pitching. SIERA builds on FIP and xFIP by taking a deeper look at what makes pitchers better.
* SV – Save: number of games where the pitcher enters a game led by the pitcher's team, finishes the game without surrendering the lead, is not the winning pitcher, and either (a) the lead was three runs or fewer when the pitcher entered the game; (b) the potential tying run was on base, at bat, or on deck; or (c) the pitcher pitched three or more innings
* SVO – Save opportunity: When a pitcher 1) enters the game with a lead of three or fewer runs and pitches at least one inning, 2) enters the game with the potential tying run on base, at bat, or on deck, or 3) pitches three or more innings with a lead and is credited with a save by the official scorer
* W – Win: number of games where pitcher was pitching while their team took the lead and went on | Baseball statistics | [
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to win, also the starter needs to pitch at least 5 innings of work (also related: winning percentage)
* W + S – Wins in relief + saves.
* whiff rate: a term, usually used in reference to pitchers, that divides the number of pitches swung at and missed by the total number of swings in a given sample. If a pitcher throws 100 pitches at which batters swing, and the batters fail to make contact on 26 of them, the pitcher's whiff rate is 26%.
* WHIP – Walks and hits per inning pitched: average number of walks and hits allowed by the pitcher per inning
* WP – Wild pitches: charged when a pitch is too high, low, or wide of home plate for the catcher to field, thereby allowing one or more runners to advance or score
===Fielding statistics===
* A – Assists: number of outs recorded on a play where a fielder touched the ball, except if such touching is the putout
* CI – Catcher's Interference (e.g., catcher makes contact with bat)
* DP – Double plays: one for each double play during which the fielder recorded a putout or an assist.
* E – Errors: number of times a fielder fails to make | Baseball statistics | [
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] |
a play he should have made with common effort, and the offense benefits as a result
* FP – Fielding percentage: total plays (chances minus errors) divided by the number of total chances
* INN – Innings: number of innings that a player is at one certain position
* PB – Passed ball: charged to the catcher when the ball is dropped and one or more runners advance
* PO – Putout: number of times the fielder tags, forces, or appeals a runner and he is called out as a result
* RF – Range factor: 9*(putouts + assists)/innings played. Used to determine the amount of field that the player can cover
* TC – Total chances: assists plus putouts plus errors
* TP – Triple play: one for each triple play during which the fielder recorded a putout or an assist
* UZR – Ultimate zone rating: the ability of a player to defend an assigned "zone" of the field compared to an average defensive player at his position
===Overall player value===
* VORP – Value over replacement player: a statistic that calculates a player's overall value in comparison to a "replacement-level" player. There are separate formulas for players and pitchers
* Win shares: a complex metric that | Baseball statistics | [
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0.025278251618146896,
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0.4101432263851166,
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0.10900124162435532,
0.2887900769710541,
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0.09114963561296463,
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0.4344753324985504,
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0.3411276936531067,
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0.10208631306886673,
0.20148314535617828,
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0.501607358455658,
0.01913643442094326,
0.4029802680015564,
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0.18624599277973175,
-0.4052128195762634,
0.7385285496711731,
-0.10129150003194809,
0.747999906539917,
0.07352521270513535,
0.374184787273407,
-0.30986517667770386,
-0.07524151355028152,
-0.10542222857475281,
-0.3130072057247162,
-0.14859241247177124,
0.12205442786216736
] |
gauges a player's overall contribution to his team's wins
* WAR – Wins above replacement: a non-standard formula to calculate the number of wins a player contributes to his team over a "replacement-level player"
* PWA - Player Win Average: performance of players is shown by how much they increase or decrease their team's chances of winning a specific game
* PGP - Player Game Percentage: defined as, "the sum of changes in the probability of winning the game for each play in which the player has participated"
===General statistics===
* G – Games played: number of games where the player played, in whole or in part
* GS – Games started: number of games a player starts
* GB – Games behind: number of games a team is behind the division leader
* Pythagorean expectation: estimates a team's expected winning percentage based on runs scored and runs allowed
==MLB statistical standards==
It is difficult to determine quantitatively what is considered to be a "good" value in a certain statistical category, and qualitative assessments may lead to arguments. Using full-season statistics available at the Official Site of Major League BaseballMajor League Baseball Historical Statistics for the 2004 through 2015 seasons, the following tables show top ranges in various statistics, in | Baseball statistics | [
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] |
alphabetical order. For each statistic, two values are given:
*Top5: the top five players bettered this value in all of the reported seasons
*Best: this is the best of all of the players for all of the reported seasons
{| border="0" cellpadding="0" cellspacing="0"
|- valign="top"
|
{| class="wikitable" style="text-align:center"
|+Batting Statistics
|-
! Statistic !! Top5 !! Best
|-
! BA
| .322 || .372
|-
! HR
| 41 || 58
|-
! RBI
| 116 || 156
|-
! SLG
| .568 || .812
|-
! SB
| 42 || 78
|-
! OPS
| .974 || 1.422
|}
|
{| class="wikitable" style="text-align:center;"
|+Pitching Statistics
|-
! Statistic !! Top5 !! Best
|-
! CG
| 4 || 11
|-
! ERA
| 2.38 || 1.66
|-
! G
| 89 || 94
|-
! GS
| 34 || 35
|-
! IP
| 227.2 || 255
|-
! K
| 245 || 301
|-
! SHO
| 2 || 6
|-
! SV
| 44 || 62
|-
! W
| 19 || 24
|-
! WHIP
| 0.98 || 0.84
|}
|}
==See also==
*Baseball awards
* Cy Young Award winners
* Glossary of baseball
*Hank Aaron Award winners (best offensive performer)
*List of MLB awards
* MLB Most Valuable Player Award winners
* MLB Rookie of the Year Award winners
* Official Baseball Rules (OBR)
* List of pitches
* Rawlings Gold Glove Award winners
* Retrosheet
* Sabermetrics
*Silver Slugger Award winners
*Society for American Baseball Research (SABR)
* Strike zone
* Triple Crown in Major League Baseball
==References==
==Bibliography==
* Albert, Jim, and Jay M. Bennett. Curve Ball: Baseball, Statistics, and the Role of Chance in the Game. | Baseball statistics | [
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] |
New York: Copernicus Books, 2001. . A book on new statistics for baseball. MLB Record Book by: MLB.com
*Alan Schwarz, The Numbers Game: Baseball's Lifelong Fascination with Statistics (New York: St. Martin's, 2005). .
*The Official Site of Major League baseball – Baseball Basics: Abbreviations
== External links==
* Baseball 1 Stats
* Baseball Almanac
* Baseball-Reference.com
* Retrosheet | Baseball statistics | [
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In computer science, binary search, also known as half-interval search, logarithmic search, or binary chop, is a search algorithm that finds the position of a target value within a sorted array. Binary search compares the target value to the middle element of the array. If they are not equal, the half in which the target cannot lie is eliminated and the search continues on the remaining half, again taking the middle element to compare to the target value, and repeating this until the target value is found. If the search ends with the remaining half being empty, the target is not in the array.
Binary search runs in logarithmic time in the worst case, making O(\log n) comparisons, where n is the number of elements in the array. Binary search is faster than linear search except for small arrays. However, the array must be sorted first to be able to apply binary search. There are specialized data structures designed for fast searching, such as hash tables, that can be searched more efficiently than binary search. However, binary search can be used to solve a wider range of problems, such as finding the next-smallest or next-largest element in the array relative to | Binary search algorithm | [
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the target even if it is absent from the array.
There are numerous variations of binary search. In particular, fractional cascading speeds up binary searches for the same value in multiple arrays. Fractional cascading efficiently solves a number of search problems in computational geometry and in numerous other fields. Exponential search extends binary search to unbounded lists. The binary search tree and B-tree data structures are based on binary search.
== Algorithm ==
Binary search works on sorted arrays. Binary search begins by comparing an element in the middle of the array with the target value. If the target value matches the element, its position in the array is returned. If the target value is less than the element, the search continues in the lower half of the array. If the target value is greater than the element, the search continues in the upper half of the array. By doing this, the algorithm eliminates the half in which the target value cannot lie in each iteration.
=== Procedure ===
Given an array A of n elements with values or records A_0,A_1,A_2,\ldots,A_{n-1}sorted such that A_0 \leq A_1 \leq A_2 \leq \cdots \leq A_{n-1}, and target value T, the following subroutine uses binary search to find the | Binary search algorithm | [
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index of T in A.
# Set L to 0 and R to n-1.
# If L>R, the search terminates as unsuccessful.
# Set m (the position of the middle element) to the floor of \frac{L+R}{2}, which is the greatest integer less than or equal to \frac{L+R}{2}.
# If A_m < T, set L to m+1 and go to step 2.
# If A_m > T, set R to m-1 and go to step 2.
# Now A_m = T, the search is done; return m.
This iterative procedure keeps track of the search boundaries with the two variables L and R. The procedure may be expressed in pseudocode as follows, where the variable names and types remain the same as above, floor is the floor function, and unsuccessful refers to a specific value that conveys the failure of the search.
function binary_search(A, n, T) is
L := 0
R := n − 1
while L ≤ R do
m := floor((L + R) / 2)
if A[m] < T then | Binary search algorithm | [
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L := m + 1
else if A[m] > T then
R := m − 1
else:
return m
return unsuccessful
Alternatively, the algorithm may take the ceiling of \frac{L+R}{2}. This may change the result if the target value appears more than once in the array.
==== Alternative procedure ====
In the above procedure, the algorithm checks whether the middle element (m) is equal to the target (T) in every iteration. Some implementations leave out this check during each iteration. The algorithm would perform this check only when one element is left (when L=R). This results in a faster comparison loop, as one comparison is eliminated per iteration, while it requires only one more iteration on average. Procedure is described at p. 214 (§43), titled "Program for Binary Search".
Hermann Bottenbruch published the first implementation to leave out this check in 1962.
# Set L to 0 and R to n-1.
# While L \neq R,
## | Binary search algorithm | [
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] |
Set m (the position of the middle element) to the ceiling of \frac{L+R}{2}, which is the least integer greater than or equal to \frac{L+R}{2}.
## If A_m > T, set R to m-1.
## Else, A_m \leq T; set L to m.
# Now L=R, the search is done. If A_L=T, return L. Otherwise, the search terminates as unsuccessful.
Where ceil is the ceiling function, the pseudocode for this version is:
function binary_search_alternative(A, n, T) is
L := 0
R := n − 1
while L != R do
m := ceil((L + R) / 2)
if A[m] > T then
R := m − 1
else:
L := m
if A[L] = T then
return L
return unsuccessful
=== Duplicate elements ===
The procedure may return any index whose | Binary search algorithm | [
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] |
element is equal to the target value, even if there are duplicate elements in the array. For example, if the array to be searched was [1,2,3,4,4,5,6,7] and the target was 4, then it would be correct for the algorithm to either return the 4th (index 3) or 5th (index 4) element. The regular procedure would return the 4th element (index 3) in this case. It does not always return the first duplicate (consider [1,2,4,4,4,5,6,7] which still returns the 4th element). However, it is sometimes necessary to find the leftmost element or the rightmost element for a target value that is duplicated in the array. In the above example, the 4th element is the leftmost element of the value 4, while the 5th element is the rightmost element of the value 4. The alternative procedure above will always return the index of the rightmost element if such an element exists.
==== Procedure for finding the leftmost element ====
To find the leftmost element, the following procedure can be used:
# Set L to 0 and R to n.
# While L < R,
## Set m (the position of the middle element) to the floor of \frac{L+R}{2}, which is the greatest integer less than or | Binary search algorithm | [
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] |
equal to \frac{L+R}{2}.
## If A_m < T, set L to m+1.
## Else, A_m \geq T; set R to m.
# Return L.
If L < n and A_L = T, then A_L is the leftmost element that equals T. Even if T is not in the array, L is the rank of T in the array, or the number of elements in the array that are less than T.
Where floor is the floor function, the pseudocode for this version is:
function binary_search_leftmost(A, n, T):
L := 0
R := n
while L < R:
m := floor((L + R) / 2)
if A[m] < T:
L := m + 1
else:
R := m
return L
==== Procedure for finding the rightmost element ====
To find the rightmost element, the following procedure can be used:
# Set L to 0 | Binary search algorithm | [
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] |
and R to n.
# While L < R,
## Set m (the position of the middle element) to the floor of \frac{L+R}{2}, which is the greatest integer less than or equal to \frac{L+R}{2}.
## If A_m > T, set R to m.
## Else, A_m \leq T; set L to m+1.
# Return R - 1.
If R > 0 and A_{R-1}=T, then A_{R-1} is the rightmost element that equals T. Even if T is not in the array, n-R is the number of elements in the array that are greater than T.
Where floor is the floor function, the pseudocode for this version is:
function binary_search_rightmost(A, n, T):
L := 0
R := n
while L < R:
m := floor((L + R) / 2)
if A[m] > T:
R := m
else:
L := m + 1
return R - | Binary search algorithm | [
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1
=== Approximate matches ===
The above procedure only performs exact matches, finding the position of a target value. However, it is trivial to extend binary search to perform approximate matches because binary search operates on sorted arrays. For example, binary search can be used to compute, for a given value, its rank (the number of smaller elements), predecessor (next-smallest element), successor (next-largest element), and nearest neighbor. Range queries seeking the number of elements between two values can be performed with two rank queries.
* Rank queries can be performed with the procedure for finding the leftmost element. The number of elements less than the target value is returned by the procedure.
* Predecessor queries can be performed with rank queries. If the rank of the target value is r, its predecessor is r-1.
* For successor queries, the procedure for finding the rightmost element can be used. If the result of running the procedure for the target value is r, then the successor of the target value is r+1.
* The nearest neighbor of the target value is either its predecessor or successor, whichever is closer.
* Range queries are also straightforward. Once the ranks of the two values are known, the number of elements greater than or | Binary search algorithm | [
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equal to the first value and less than the second is the difference of the two ranks. This count can be adjusted up or down by one according to whether the endpoints of the range should be considered to be part of the range and whether the array contains entries matching those endpoints.
== Performance ==
In terms of the number of comparisons, the performance of binary search can be analyzed by viewing the run of the procedure on a binary tree. The root node of the tree is the middle element of the array. The middle element of the lower half is the left child node of the root, and the middle element of the upper half is the right child node of the root. The rest of the tree is built in a similar fashion. Starting from the root node, the left or right subtrees are traversed depending on whether the target value is less or more than the node under consideration.
In the worst case, binary search makes \lfloor \log_2 (n) + 1 \rfloor iterations of the comparison loop, where the \lfloor \rfloor notation denotes the floor function that yields the greatest integer less than or equal to the argument, | Binary search algorithm | [
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] |
and \log_2 is the binary logarithm. This is because the worst case is reached when the search reaches the deepest level of the tree, and there are always \lfloor \log_2 (n) + 1 \rfloor levels in the tree for any binary search.
The worst case may also be reached when the target element is not in the array. If n is one less than a power of two, then this is always the case. Otherwise, the search may perform \lfloor \log_2 (n) + 1 \rflooriterations if the search reaches the deepest level of the tree. However, it may make \lfloor \log_2 (n) \rfloor iterations, which is one less than the worst case, if the search ends at the second-deepest level of the tree.
On average, assuming that each element is equally likely to be searched, binary search makes \lfloor \log_2 (n) \rfloor + 1 - (2^{\lfloor \log_2 (n) \rfloor + 1} - \lfloor \log_2 (n) \rfloor - 2)/n iterations when the target element is in the array. This is approximately equal to \log_2(n) - 1 iterations. When the target element is not in the array, binary search makes \lfloor \log_2 (n) \rfloor + 2 - 2^{\lfloor \log_2 (n) \rfloor + 1}/(n + | Binary search algorithm | [
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1) iterations on average, assuming that the range between and outside elements is equally likely to be searched.
In the best case, where the target value is the middle element of the array, its position is returned after one iteration.
In terms of iterations, no search algorithm that works only by comparing elements can exhibit better average and worst-case performance than binary search. The comparison tree representing binary search has the fewest levels possible as every level above the lowest level of the tree is filled completely. Otherwise, the search algorithm can eliminate few elements in an iteration, increasing the number of iterations required in the average and worst case. This is the case for other search algorithms based on comparisons, as while they may work faster on some target values, the average performance over all elements is worse than binary search. By dividing the array in half, binary search ensures that the size of both subarrays are as similar as possible.
=== Space complexity ===
Binary search requires three pointers to elements, which may be array indices or pointers to memory locations, regardless of the size of the array. Therefore, the space complexity of binary search is O(1) in the word RAM model | Binary search algorithm | [
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of computation.
=== Derivation of average case ===
The average number of iterations performed by binary search depends on the probability of each element being searched. The average case is different for successful searches and unsuccessful searches. It will be assumed that each element is equally likely to be searched for successful searches. For unsuccessful searches, it will be assumed that the intervals between and outside elements are equally likely to be searched. The average case for successful searches is the number of iterations required to search every element exactly once, divided by n, the number of elements. The average case for unsuccessful searches is the number of iterations required to search an element within every interval exactly once, divided by the n + 1 intervals.
==== Successful searches ====
In the binary tree representation, a successful search can be represented by a path from the root to the target node, called an internal path. The length of a path is the number of edges (connections between nodes) that the path passes through. The number of iterations performed by a search, given that the corresponding path has length l, is l + 1 counting the initial iteration. The internal path length is the sum | Binary search algorithm | [
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of the lengths of all unique internal paths. Since there is only one path from the root to any single node, each internal path represents a search for a specific element. If there are n elements, which is a positive integer, and the internal path length is I(n), then the average number of iterations for a successful search T(n) = 1 + \frac{I(n)}{n}, with the one iteration added to count the initial iteration.
Since binary search is the optimal algorithm for searching with comparisons, this problem is reduced to calculating the minimum internal path length of all binary trees with n nodes, which is equal to:
I(n) = \sum_{k=1}^n \left \lfloor \log_2(k) \right \rfloor
For example, in a 7-element array, the root requires one iteration, the two elements below the root require two iterations, and the four elements below require three iterations. In this case, the internal path length is:
\sum_{k=1}^7 \left \lfloor \log_2(k) \right \rfloor = 0 + 2(1) + 4(2) = 2 + 8 = 10
The average number of iterations would be 1 + \frac{10}{7} = 2 \frac{3}{7} based on the equation for the average case. The sum for I(n) can be simplified to:
I(n) = \sum_{k=1}^n \left \lfloor \log_2(k) \right \rfloor = | Binary search algorithm | [
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(n + 1)\left \lfloor \log_2(n + 1) \right \rfloor - 2^{\left \lfloor \log_2(n+1) \right \rfloor + 1} + 2
Substituting the equation for I(n) into the equation for T(n):
T(n) = 1 + \frac{(n + 1)\left \lfloor \log_2(n + 1) \right \rfloor - 2^{\left \lfloor \log_2(n+1) \right \rfloor + 1} + 2}{n} = \lfloor \log_2 (n) \rfloor + 1 - (2^{\lfloor \log_2 (n) \rfloor + 1} - \lfloor \log_2 (n) \rfloor - 2)/n
For integer n, this is equivalent to the equation for the average case on a successful search specified above.
==== Unsuccessful searches ====
Unsuccessful searches can be represented by augmenting the tree with external nodes, which forms an extended binary tree. If an internal node, or a node present in the tree, has fewer than two child nodes, then additional child nodes, called external nodes, are added so that each internal node has two children. By doing so, an unsuccessful search can be represented as a path to an external node, whose parent is the single element that remains during the last iteration. An external path is a path from the root to an external node. The external path length is the sum of the lengths of all unique external paths. If | Binary search algorithm | [
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there are n elements, which is a positive integer, and the external path length is E(n), then the average number of iterations for an unsuccessful search T'(n)=\frac{E(n)}{n+1}, with the one iteration added to count the initial iteration. The external path length is divided by n+1 instead of n because there are n+1 external paths, representing the intervals between and outside the elements of the array.
This problem can similarly be reduced to determining the minimum external path length of all binary trees with n nodes. For all binary trees, the external path length is equal to the internal path length plus 2n. Substituting the equation for I(n):
E(n) = I(n) + 2n = \left[(n + 1)\left \lfloor \log_2(n + 1) \right \rfloor - 2^{\left \lfloor \log_2(n+1) \right \rfloor + 1} + 2\right] + 2n = (n + 1) (\lfloor \log_2 (n) \rfloor + 2) - 2^{\lfloor \log_2 (n) \rfloor + 1}
Substituting the equation for E(n) into the equation for T'(n), the average case for unsuccessful searches can be determined:
T'(n) = \frac{(n + 1) (\lfloor \log_2 (n) \rfloor + 2) - 2^{\lfloor \log_2 (n) \rfloor + 1}}{(n+1)} = \lfloor \log_2 (n) \rfloor + 2 - 2^{\lfloor \log_2 (n) \rfloor + 1}/(n + 1)
==== | Binary search algorithm | [
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Performance of alternative procedure ====
Each iteration of the binary search procedure defined above makes one or two comparisons, checking if the middle element is equal to the target in each iteration. Assuming that each element is equally likely to be searched, each iteration makes 1.5 comparisons on average. A variation of the algorithm checks whether the middle element is equal to the target at the end of the search. On average, this eliminates half a comparison from each iteration. This slightly cuts the time taken per iteration on most computers. However, it guarantees that the search takes the maximum number of iterations, on average adding one iteration to the search. Because the comparison loop is performed only \lfloor \log_2 (n) + 1 \rfloor times in the worst case, the slight increase in efficiency per iteration does not compensate for the extra iteration for all but very large n.
=== Running time and cache use ===
In analyzing the performance of binary search, another consideration is the time required to compare two elements. For integers and strings, the time required increases linearly as the encoding length (usually the number of bits) of the elements increase. For example, comparing a pair of 64-bit unsigned | Binary search algorithm | [
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integers would require comparing up to double the bits as comparing a pair of 32-bit unsigned integers. The worst case is achieved when the integers are equal. This can be significant when the encoding lengths of the elements are large, such as with large integer types or long strings, which makes comparing elements expensive. Furthermore, comparing floating-point values (the most common digital representation of real numbers) is often more expensive than comparing integers or short strings.
On most computer architectures, the processor has a hardware cache separate from RAM. Since they are located within the processor itself, caches are much faster to access but usually store much less data than RAM. Therefore, most processors store memory locations that have been accessed recently, along with memory locations close to it. For example, when an array element is accessed, the element itself may be stored along with the elements that are stored close to it in RAM, making it faster to sequentially access array elements that are close in index to each other (locality of reference). On a sorted array, binary search can jump to distant memory locations if the array is large, unlike algorithms (such as linear search and linear probing in | Binary search algorithm | [
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hash tables) which access elements in sequence. This adds slightly to the running time of binary search for large arrays on most systems.
== Binary search versus other schemes ==
Sorted arrays with binary search are a very inefficient solution when insertion and deletion operations are interleaved with retrieval, taking O(n) time for each such operation. In addition, sorted arrays can complicate memory use especially when elements are often inserted into the array. There are other data structures that support much more efficient insertion and deletion. Binary search can be used to perform exact matching and set membership (determining whether a target value is in a collection of values). There are data structures that support faster exact matching and set membership. However, unlike many other searching schemes, binary search can be used for efficient approximate matching, usually performing such matches in O(\log n) time regardless of the type or structure of the values themselves. In addition, there are some operations, like finding the smallest and largest element, that can be performed efficiently on a sorted array.
=== Linear search ===
Linear search is a simple search algorithm that checks every record until it finds the target value. Linear search can be done on a | Binary search algorithm | [
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linked list, which allows for faster insertion and deletion than an array. Binary search is faster than linear search for sorted arrays except if the array is short, although the array needs to be sorted beforehand. All sorting algorithms based on comparing elements, such as quicksort and merge sort, require at least O(n \log n) comparisons in the worst case. Unlike linear search, binary search can be used for efficient approximate matching. There are operations such as finding the smallest and largest element that can be done efficiently on a sorted array but not on an unsorted array.
=== Trees ===
A binary search tree is a binary tree data structure that works based on the principle of binary search. The records of the tree are arranged in sorted order, and each record in the tree can be searched using an algorithm similar to binary search, taking on average logarithmic time. Insertion and deletion also require on average logarithmic time in binary search trees. This can be faster than the linear time insertion and deletion of sorted arrays, and binary trees retain the ability to perform all the operations possible on a sorted array, including range and approximate queries.
However, binary search is | Binary search algorithm | [
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usually more efficient for searching as binary search trees will most likely be imperfectly balanced, resulting in slightly worse performance than binary search. This even applies to balanced binary search trees, binary search trees that balance their own nodes, because they rarely produce the tree with the fewest possible levels. Except for balanced binary search trees, the tree may be severely imbalanced with few internal nodes with two children, resulting in the average and worst-case search time approaching n comparisons. Binary search trees take more space than sorted arrays.
Binary search trees lend themselves to fast searching in external memory stored in hard disks, as binary search trees can be efficiently structured in filesystems. The B-tree generalizes this method of tree organization. B-trees are frequently used to organize long-term storage such as databases and filesystems.
=== Hashing ===
For implementing associative arrays, hash tables, a data structure that maps keys to records using a hash function, are generally faster than binary search on a sorted array of records. Most hash table implementations require only amortized constant time on average. However, hashing is not useful for approximate matches, such as computing the next-smallest, next-largest, and nearest key, as the only information given on a | Binary search algorithm | [
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failed search is that the target is not present in any record. Binary search is ideal for such matches, performing them in logarithmic time. Binary search also supports approximate matches. Some operations, like finding the smallest and largest element, can be done efficiently on sorted arrays but not on hash tables.
=== Set membership algorithms ===
A related problem to search is set membership. Any algorithm that does lookup, like binary search, can also be used for set membership. There are other algorithms that are more specifically suited for set membership. A bit array is the simplest, useful when the range of keys is limited. It compactly stores a collection of bits, with each bit representing a single key within the range of keys. Bit arrays are very fast, requiring only O(1) time. The Judy1 type of Judy array handles 64-bit keys efficiently.
For approximate results, Bloom filters, another probabilistic data structure based on hashing, store a set of keys by encoding the keys using a bit array and multiple hash functions. Bloom filters are much more space-efficient than bit arrays in most cases and not much slower: with k hash functions, membership queries require only O(k) time. However, Bloom filters suffer | Binary search algorithm | [
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from false positives.
=== Other data structures ===
There exist data structures that may improve on binary search in some cases for both searching and other operations available for sorted arrays. For example, searches, approximate matches, and the operations available to sorted arrays can be performed more efficiently than binary search on specialized data structures such as van Emde Boas trees, fusion trees, tries, and bit arrays. These specialized data structures are usually only faster because they take advantage of the properties of keys with a certain attribute (usually keys that are small integers), and thus will be time or space consuming for keys that lack that attribute. As long as the keys can be ordered, these operations can always be done at least efficiently on a sorted array regardless of the keys. Some structures, such as Judy arrays, use a combination of approaches to mitigate this while retaining efficiency and the ability to perform approximate matching.
== Variations ==
=== Uniform binary search ===
Uniform binary search stores, instead of the lower and upper bounds, the difference in the index of the middle element from the current iteration to the next iteration. A lookup table containing the differences is computed beforehand. For example, if | Binary search algorithm | [
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] |
the array to be searched is , the middle element (m) would be . In this case, the middle element of the left subarray () is and the middle element of the right subarray () is . Uniform binary search would store the value of as both indices differ from by this same amount. To reduce the search space, the algorithm either adds or subtracts this change from the index of the middle element. Uniform binary search may be faster on systems where it is inefficient to calculate the midpoint, such as on decimal computers.
=== Exponential search ===
Exponential search extends binary search to unbounded lists. It starts by finding the first element with an index that is both a power of two and greater than the target value. Afterwards, it sets that index as the upper bound, and switches to binary search. A search takes \lfloor \log_2 x + 1\rfloor iterations before binary search is started and at most \lfloor \log_2 x \rfloor iterations of the binary search, where x is the position of the target value. Exponential search works on bounded lists, but becomes an improvement over binary search only if the target value lies near | Binary search algorithm | [
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the beginning of the array.
=== Interpolation search ===
Instead of calculating the midpoint, interpolation search estimates the position of the target value, taking into account the lowest and highest elements in the array as well as length of the array. It works on the basis that the midpoint is not the best guess in many cases. For example, if the target value is close to the highest element in the array, it is likely to be located near the end of the array.
A common interpolation function is linear interpolation. If A is the array, L, R are the lower and upper bounds respectively, and T is the target, then the target is estimated to be about (T - A_L) / (A_R - A_L) of the way between L and R. When linear interpolation is used, and the distribution of the array elements is uniform or near uniform, interpolation search makes O(\log \log n) comparisons.
In practice, interpolation search is slower than binary search for small arrays, as interpolation search requires extra computation. Its time complexity grows more slowly than binary search, but this only compensates for the extra computation for large arrays.
=== Fractional cascading ===
Fractional cascading is a technique that speeds up | Binary search algorithm | [
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