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Actions Per Minute, commonly abbreviated as APM, is a term used mostly in real time strategy games. APM is commonly mistaken as AMP.

Contents 1 Actions per Minute 2 Speed in APM 3 Efficiency in APM 4 Accuracy in APM 5 References

Actions per Minute

Actions per minute is the number of actions completed within a minute of gameplay in a real time strategy game. High APM is usually associated with skill, because it indicates both that the player has the mental alertness and knowledge of the game to form a complex strategy, and that the player has the manual dexterity to execute that strategy at high speed. A high APM is also an indicator that the player has good “micro” skills. Software has been developed to analyze players’ APM in these games.BWChart FAQ

Speed in APM

Speed is one factor determining a player’s APM. It measures the rapidity at executing actions during gameplay. This includes the number of keystrokes and mouse clicks the player performs. The higher the speed rate, the higher the APM will be. Many players mistaken speed as to determining a player’s APM. Players will sometimes stroke two keys close together continually, or ‘trill’, at times of relative inactivity to increase the APM. This is usually employed so that the player displays a higher level of skill than they actually do. Sometimes though, players claim that this type of action is just a nervous reaction or a habit.

Efficiency in APM

Efficiency is another factor determining a player’s APM. Efficiency simply means how well a player can productively execute desired actions without wasting necessary time to execute other actions. This shows the valid APM during gameplay and does not include spammed actions. Any spammed actions during a gameplay will attribute to an increase of speed. Valid APM is generally measured and calculated by only the efficiency and accuracy of a player’s gameplay.

Accuracy in APM

Accuracy is the third and final factor determining a player’s APM. Accuracy is the coordination of precise mouse clicks and keystrokes. Greater accuracy will result in fewer mis-clicks and mis-strokes; thus the player’s efficiency increases resulting in a more accurate valid APM. Players who have greater accuracy can perform other vital tasks during gameplay and can multitask between two or more other actions without losing efficiency.

References

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Abiodun (reigned c. 1770–1789) was an eighteenth-century alaafin, or ruler of the Oyo people in what is now Nigeria. Coming to the throne shortly after the Oyo subjugation of neighboring Dahomey, Abiodun soon found himself embroiled in a civil war over the goals of the newly wealthy state.
Bashorun Gaha had used his power to pervert the constitutional terms of abdication in a bid to limit the powers of the Alaafin and gain more political power for himself. During Gaha’s power play, he had succeeded in removing three kings before Abiodun curtailed his excesses and had him burned.
In terms of trade, while Abiodun favored economic expansion for its own sake, his opponents favored using the wealth from Dahomey’s tribute to finance further military expansion. Abiodun soon proved victorious and pursued a policy of peaceful trade with the European merchants of the coast. This course significantly weakened the army, leaving his successor, Awole, facing a number of local revolts.

Abiodun’s reign is generally remembered as a time of peace and prosperity for the Oyo, though Nigerian playwright Femi Òsófisan portrays him as a despot in his play The Chattering and the Song (1973).

External links

• Oyo empire
• The dictating currents and the questioning of tyranny in Africa: an intertextual study of Fémi Osófisan’s Yungba-Yungba and the dance contest
  

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Norway was represented at the 1976 Summer Olympics in Montreal by the Norwegian Olympic Committee and Confederation of Sports.

Contents 1 Medals 1.1 Gold 1.2 Silver 2 Results by event 2.1 Archery 2.2 Athletics 2.3 Cycling

Medals

Norway finished in 21st position in the final medal rankings, with one gold medal and one silver medal.

Gold

• Alf Hansen and Frank Hansen — Rowing, Men’s Double Sculls

Silver

• Finn Tveter, Rolf Andreassen, Arne Bergodd, and Ole Nafstad — Rowing, Men’s Coxless Fours

Results by event

Archery

In the second appearance by the nation in the archery competition at the Olympics, Norway was represented by only one man. A veteran of the 1972 Summer Olympics, Jan Erik Humlekjær shot two points less than his performance of four years before. Nevertheless, he moved up eight places in the ranking.

Men’s Individual Competition:

• Jan Erik Humlekjær — 2337 points (→ 24th place)

Athletics

Men’s 1500 metres

• Lars Martin Kaupang

• Heat — 3:44.59 min (→ did not advance)

Women’s 1500 metres

• Grete Waitz

• Heat — 4:07.20 min (→ advanced to the semi final)
• Semi final — 4:04.80 min (→ did not advance)

Men’s 5000 metres

• Knut Kvalheim

• Heat — 13:20.60 min (→ advanced to the final)
• Final — 13:30.33 min (→ 9th place)

Men’s 10.000 metres

• Knut Børø

• Heat — 28:23.07 min (→ advanced to the final)
• Final — did not finish (→ no ranking)

Men’s High Jump

• Terje Totland

• Qualification — 2.16 m (→ advanced to the final)
• Final — 2.18m (→ 9th place)

• Leif Roar Falkum

• Qualification — 2.16 m (→ advanced to the final)
• Final — 2.10m (→ 14th place)

Women’s High Jump

• Astrid Tveit

• Qualification — 1.70 m (→ did not advance)

Men’s Discus Throw

• Knut Hjeltnes

• Qualification — 61.30 m (→ advanced to the final)
• Final — 63.06 m (→ 7th place)

Men’s Javelin Throw

• Terje Thorslund

• Qualification — 82.52 m (→ advanced to the final)
• Final — 78.24 m (→ 11th place)

• Bjørn Grimnes

• Qualification — 80.32 m (→ advanced to the final)
• Final — 74.88 m (→ 14th place)

Cycling

Men’s Individual Road Race

• Thorleif Andresen — 4:49:01 (→ 38th place)
• Geir Digerud — 5:04:42 (→ 55th place)
• Pål Henning Hansen — did not finish (→ no ranking)
• Stein Bråthen — did not finish (→ no ranking)

Men’s 1.000m Time Trial

• Harald Bundli — 1:08.093 (→ 7th place)

Men’s 4.000m Individual Pursuit

• Jan Georg Iversen — 7th place

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In mathematics and classical mechanics, canonical coordinates are a particular set of coordinates on the phase space, or equivalently, on the cotangent manifold of a manifold. Canonical coordinates arise naturally in physics in the study of Hamiltonian mechanics. As Hamiltonian mechanics is generalized by symplectic geometry and canonical transformations are generalized by contact transformations, so the 19th century definition of canonical coordinates in classical mechanics may be generalized to a more abstract 20th century definition in terms of cotangent bundles.

This article defines the canonical coordinates as they appear in classical mechanics. A closely related concept also appears in quantum mechanics; see the Stone-von Neumann theorem and canonical commutation relations for details.

Contents 1 Definition, in classical mechanics 2 Definition, on cotangent bundles 3 Formal development 4 Generalized coordinates 5 See also

Definition, in classical mechanics

In classical mechanics, canonical coordinates are the coordinates <math>q_i\,</math> and <math>p_i\,</math> in phase space that are used in the Hamiltonian formalism. The canonical coordinates satisfy the fundamental Poisson bracket relations:

<math>\{q_i, q_j\} = 0 \qquad \{p_i, p_j\} = 0 \qquad \{q_i, p_j\} = \delta_{ij}</math>

Canonical coordinates can be obtained from the generalized coordinates of the Lagrangian formalism by a Legendre transformation, or from another set of canonical coordinates by a canonical transformation.

Definition, on cotangent bundles

Canonical coordinates are defined as a special set of coordinates on the cotangent bundle of a manifold. They are usually written as a set of <math>(q^i,p_j)</math> or <math>(x^i,p_j)</math> with the x ’s or q ’s denoting the coordinates on the underlying manifold and the p ’s denoting the conjugate momentum, which are 1-forms in the cotangent bundle at point q in the manifold.

A common definition of canonical coordinates is any set of coordinates on the cotangent bundle that allow the canonical one form to be written in the form

<math>\sum_i p_i\,\mathrm{d}q^i</math>

up to a total differential. A change of coordinates that preserves this form is a canonical transformation; these are a special case of a symplectomorphism, which are essentially a change of coordinates on a symplectic manifold.

In the following exposition, we assume that the manifolds are real manifolds, so that cotangent vectors acting on tangent vectors produce real numbers.

Formal development

Given a manifold Q, a vector field X on Q (or equivalently, a section of the tangent bundle TQ) can be thought of as a function acting on the cotangent bundle, by the duality between the tangent and cotangent spaces. That is, define a function

<math>P_X:T^*Q\to \mathbb{R}</math>

such that

<math>P_X(q,p)=p(X_q)</math>

holds for all cotangent vectors p in <math>T_q^*Q</math>. Here, <math>X_q</math> is a vector in <math>T_qQ</math>, the tangent space to the manifold Q at point q. The function <math>P_X</math> is called the momentum function corresponding to X.

In local coordinates, the vector field X at point q may be written as

<math>X_q=\sum_i X^i(q) \frac{\partial}{\partial q^i}</math>

where the <math>\partial /\partial q^i</math> are the coordinate frame on TQ. The conjugate momentum then has the expression

<math>P_X(q,p)=\sum_i X^i(q) \;p_i</math>

where the <math>p_i</math> are defined as the momentum functions corresponding to the vectors <math>\partial /\partial q^i</math>:

<math>p_i = P_{\partial /\partial q^i}</math>

The <math>q^i</math> together with the <math>p_j</math> together form a coordinate system on the cotangent bundle <math>T^*Q</math>; these coordinates are called the canonical coordinates.

Generalized coordinates

In Lagrangian mechanics, a different set of coordinates are used, called the generalized coordinates. These are commonly denoted as <math>(q^i,\dot{q}^i)</math> with <math>q^i</math> called the generalized position and <math>\dot{q}^i</math> the generalized velocity. When a Hamiltonian is defined on the cotangent bundle, then the generalized coordinates are related to the canonical coordinates by means of the Hamilton–Jacobi equations.

See also

• symplectic manifold
• symplectic vector field
• symplectomorphism

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