Kirchhoff–Love plate theory: Difference between revisions

Revision as of 13:52, 16 December 2013

In music theory, a nondominant seventh chord is a chord which is a diatonic chord that is a seventh chord but that does not possess dominant function^[2] and thus is not a dominant seventh chord.

Since the V and [[Leading-tone|viiTemplate:Music]] chords are the dominant function chords,^[2] the "major minor seventh" V⁷ and "half-diminished seventh" [[leading-tone seventh chord|viiTemplate:Music⁷]] are the dominant seventh chords. Since the nondominant function chords are I, i, ii, iiTemplate:Music, iii, III, IV, iv, vi, and VI,^[2] the nondominant seventh chord qualities include the augmented major seventh chord, major seventh chord, minor major seventh chord, minor seventh chord, and major minor seventh chords that do not possess dominant function, such as, in melodic minor, IV $_{m}^{7}$ .

To analyze seventh chords indicate the quality of the triad; major: I, minor: ii, half-diminished: viiTemplate:Music, or augmented: III+; and the quality of the seventh; same: ⁷, or different: $_{M}^{7}$ or $_{m}^{7}$ .^[2] In macro analysis indicate the root and chord quality, and add ⁷, thus a seventh chord on ii in C major (minor minor seventh) would be d⁷.^[1]

As with dominant seventh chords, nondominant seventh chords usually progress according to the circle progression, thus III+ $_{M}^{7}$ resolves to vi or VI,^[4] for example.

File:"Try To Remember" nondominant seventh chords.png

Nondominant seventh chords are, "found in large number," in popular music and jazz ("a legacy from the romantic period"), such as in this example from "Try To Remember" (The Fantasticks) by Harvey Schmidt (lyrics: Tom Jones)^[4] My name: Lindsey Gavin
My age: 28
Country: Sweden
Home town: Vemdalen
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Look into my weblog :: http://www.hostgator1centcoupon.info/. Note the circle progression derived root motion by fourths/fifths.

When possible, as in circle progressions, resolve the seventh of nondominant seventh chords down by step to the third of the following chord.^[3]

Sources

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

In computational complexity theory, the potential method is a method used to analyze the amortized time and space complexity of a data structure, a measure of its performance over sequences of operations that smooths out the cost of infrequent but expensive operations.^[5]^[6]

Definition of amortized time

In the potential method, a function Φ is chosen that maps states of the data structure to non-negative numbers. If S is a state of the data structure, Φ(S) may be thought of intuitively as an amount of potential energy stored in that state;^[5]^[6] alternatively, Φ(S) may be thought of as representing the amount of disorder in state S or its distance from an ideal state. The potential value prior to the operation of initializing a data structure is defined to be zero.

Let o be any individual operation within a sequence of operations on some data structure, with S_before denoting the state of the data structure prior to operation o and S_after denoting its state after operation o has completed. Then, once Φ has been chosen, the amortized time for operation o is defined to be

T_{\mathrm {amortized} }(o)=T_{\mathrm {actual} }(o)+C\cdot (\Phi (S_{\mathrm {after} })-\Phi (S_{\mathrm {before} })),

where C is a non-negative constant of proportionality (in units of time) that must remain fixed throughout the analysis. That is, the amortized time is defined to be the actual time taken by the operation plus C times the difference in potential caused by the operation.^[5]^[6]

Relation between amortized and actual time

Despite its artificial appearance, the total amortized time of a sequence of operations provides a valid upper bound on the actual time for the same sequence of operations. That is, for any sequence of operations $o_{0},o_{1},\dots$ , the total amortized time $\sum _{i}T_{\mathrm {amortized} }(o_{i})$ is always at least as large as the total actual time $\sum _{i}T_{\mathrm {actual} }(o_{i})$ . In more detail,

\sum _{i}T_{\mathrm {amortized} }(o_{i})=\sum _{i}\left(T_{\mathrm {actual} }(o_{i})+C\cdot (\Phi (S_{i+1})-\Phi (S_{i}))\right)=\left(\sum _{i}T_{\mathrm {actual} }(o_{i})\right)+C\cdot (\Phi (S_{\mathrm {final} })-\Phi (S_{\mathrm {initial} }))\geq \sum _{i}T_{\mathrm {actual} }(o_{i}),

where the sequence of potential function values forms a telescoping series in which all terms other than the initial and final potential function values cancel in pairs, and where the final inequality arises from the assumptions that $\Phi (S_{\mathrm {final} })\geq 0$ and $\Phi (S_{\mathrm {initial} })=0$ . Therefore, amortized time can be used to provide accurate predictions about the actual time of sequences of operations, even though the amortized time for an individual operation may vary widely from its actual time.

Amortized analysis of worst-case inputs

Typically, amortized analysis is used in combination with a worst case assumption about the input sequence. With this assumption, if X is a type of operation that may be performed by the data structure, and n is an integer defining the size of the given data structure (for instance, the number of items that it contains), then the amortized time for operations of type X is defined to be the maximum, among all possible sequences of operations on data structures of size n and all operations o_i of type X within the sequence, of the amortized time for operation o_i.

With this definition, the time to perform a sequence of operations may be estimated by multiplying the amortized time for each type of operation in the sequence by the number of operations of that type.

Example

A dynamic array is a data structure for maintaining an array of items, allowing both random access to positions within the array and the ability to increase the array size by one. It is available in Java as the "ArrayList" type and in Python as the "list" type. A dynamic array may be implemented by a data structure consisting of an array A of items, of some length N, together with a number n ≤ N representing the positions within the array that have been used so far. With this structure, random accesses to the dynamic array may be implemented by accessing the same cell of the internal array A, and when n < N an operation that increases the dynamic array size may be implemented simply by incrementing n. However, when n = N, it is necessary to resize A, and a common strategy for doing so is to double its size, replacing A by a new array of length 2n.^[7]

This structure may be analyzed using a potential function Φ = 2n − N. Since the resizing strategy always causes A to be at least half-full, this potential function is always non-negative, as desired. When an increase-size operation does not lead to a resize operation, Φ increases by 2, a constant. Therefore, the constant actual time of the operation and the constant increase in potential combine to give a constant amortized time for an operation of this type.

However, when an increase-size operation causes a resize, the potential value of n prior to the resize decreases to zero after the resize. Allocating a new internal array A and copying all of the values from the old internal array to the new one takes O(n) actual time, but (with an appropriate choice of the constant of proportionality C) this is entirely cancelled by the decrease of n in the potential function, leaving again a constant total amortized time for the operation.

The other operations of the data structure (reading and writing array cells without changing the array size) do not cause the potential function to change and have the same constant amortized time as their actual time.^[6]

Therefore, with this choice of resizing strategy and potential function, the potential method shows that all dynamic array operations take constant amortized time. Combining this with the inequality relating amortized time and actual time over sequences of operations, this shows that any sequence of n dynamic array operations takes O(n) actual time in the worst case, despite the fact that some of the individual operations may themselves take a linear amount of time.^[6]

Applications

The potential function method is commonly used to analyze Fibonacci heaps, a form of priority queue in which removing an item takes logarithmic amortized time, and all other operations take constant amortized time.^[8] It may also be used to analyze splay trees, a self-adjusting form of binary search tree with logarithmic amortized time per operation.^[9]

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro. Template:Chords

↑ ^1.0 ^1.1 ^1.2 ^1.3 Benward & Saker (2003). Music: In Theory and Practice, Vol. I, p.230. Seventh Edition. ISBN 978-0-07-294262-0.
↑ ^2.0 ^2.1 ^2.2 ^2.3 ^2.4 Benward & Saker (2003), p.229.
↑ ^3.0 ^3.1 Benward & Saker (2003), p.233-34.
↑ ^4.0 ^4.1 Benward & Saker (2003), p.232.
↑ ^5.0 ^5.1 ^5.2 Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
↑ ^6.0 ^6.1 ^6.2 ^6.3 ^6.4 Template:Introduction to Algorithms
↑ Goodrich and Tamassia, 1.5.2 Analyzing an Extendable Array Implementation, pp. 139–141; Cormen et al., 17.4 Dynamic tables, pp. 416–424.
↑ Cormen et al., Chapter 20, "Fibonacci Heaps", pp. 476–497.
↑ Goodrich and Tamassia, Section 3.4, "Splay Trees", pp. 185–194.

[B&S_230-1] 1.0 ^1.1 ^1.2 ^1.3 Benward & Saker (2003). Music: In Theory and Practice, Vol. I, p.230. Seventh Edition. ISBN 978-0-07-294262-0.

[B&S_229-2] 2.0 ^2.1 ^2.2 ^2.3 ^2.4 Benward & Saker (2003), p.229.

[B&S_233-3] 3.0 ^3.1 Benward & Saker (2003), p.233-34.

[B&S_232-4] 4.0 ^4.1 Benward & Saker (2003), p.232.

[gt-ad-5] 5.0 ^5.1 ^5.2 Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.

[clrs-6] 6.0 ^6.1 ^6.2 ^6.3 ^6.4 Template:Introduction to Algorithms

[7] Goodrich and Tamassia, 1.5.2 Analyzing an Extendable Array Implementation, pp. 139–141; Cormen et al., 17.4 Dynamic tables, pp. 416–424.

[8] Cormen et al., Chapter 20, "Fibonacci Heaps", pp. 476–497.

[9] Goodrich and Tamassia, Section 3.4, "Splay Trees", pp. 185–194.

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+[[Image:Incomplete dominant seventh chord in C major.png|thumb|right|Dominant seventh (V<sup>7</sup>) and incomplete dominant seventh (vii{{music|dim}}) in C major: G7 and b{{music|dim}} chords {{audio|Incomplete dominant seventh chord in C major.mid|Play}}.]]
+[[Image:III+7M chord in C minor.png|thumb|right|[[augmented major seventh chord|III+]]<math>_M^7</math> chord in C [[Minor scale#Harmonic and melodic minor|harmonic]] or ascending [[Minor scale|melodic minor]]<ref name="B&S 230">Benward & Saker (2003). ''Music: In Theory and Practice, Vol. I'', p.230. Seventh Edition. ISBN 978-0-07-294262-0.</ref> {{audio|III+7M chord in C minor.mid|Play}}.]]
+[[Image:Major seventh chord on F.png|thumb|right|Major seventh chord on F {{audio|Major seventh chord on F.mid|Play}}. IV<sup>7</sup> in C major.<ref name="B&S 229">Benward & Saker (2003), p.229.</ref>]]
+[[Image:Minor major seventh chord on C.png|thumb|right|Minor major seventh chord on C.<br/> i<math>_M^7</math> in C [[Minor scale#Harmonic and melodic minor|melodic]] or [[minor scale|ascending melodic minor]].<ref name="B&S 230"/>]]
+[[Image:Minor seventh chord on C.png|thumb|right|Minor-minor (i<sup>7</sup>) seventh chord on C<ref name="B&S 230"/> {{audio|Minor seventh chord on C.mid|Play}}.]]
+[[Image:Nondominant seventh chord resolution.png|thumb|right|Nondominant seventh chord resolution along a circle progression, the seventh resolves down by step to the third of the next chord: I<sup>7</sup>-IV<ref name="B&S 233"/> {{audio|Nondominant seventh chord resolution.mid|Play}}. B resolves to A.]]
+In [[music theory]], a '''nondominant seventh chord''' is a [[chord (music)|chord]] which is a [[diatonic and chromatic|diatonic chord]] that is a [[seventh chord]] but that does not possess [[dominant (music)|dominant]] [[diatonic function|function]]<ref name="B&S 229"/> and thus is ''not'' a [[dominant seventh chord]].
+Since the [[dominant (music)|V]] and [[Leading-tone|vii{{music|dim}}]] chords are the dominant function chords,<ref name="B&S 229"/> the "major minor seventh" V<sup>7</sup> and "half-diminished seventh" [[leading-tone seventh chord|vii{{music|halfdim}}<sup>7</sup>]] are the dominant seventh chords. Since the nondominant function chords are [[tonic (music)|I]], i, [[supertonic|ii]], ii{{music|dim}}, [[mediant|iii]], III, [[subdominant|IV]], iv, [[submediant|vi]], and VI,<ref name="B&S 229"/> the nondominant seventh chord qualities include the [[augmented major seventh chord]], [[major seventh chord]], [[minor major seventh chord]], [[minor seventh chord]], and major minor seventh chords that do not possess dominant function, such as, in [[Minor scale#Melodic minor scale|melodic minor]], [[subdominant|IV]]<math>_m^7</math>.
+To [[musical analysis|analyze]] seventh chords indicate the quality of the triad; [[major chord|major]]: I, [[minor chord|minor]]: ii, half-diminished: vii{{music|halfdim}}, or [[augmented chord|augmented]]: III+; and the quality of the [[seventh (chord)|seventh]]; same: <sup>7</sup>, or different: <math>_M^7</math> or <math>_m^7</math>.<ref name="B&S 229"/> In [[macro analysis]] indicate the [[root (chord)|root]] and [[Major and minor|chord quality]], and add <sup>7</sup>, thus a seventh chord on ii in C major (minor minor seventh) would be d<sup>7</sup>.<ref name="B&S 230"/>
+As with dominant seventh chords, nondominant seventh chords usually progress according to the [[circle progression]], thus [[augmented major seventh chord|III+]]<math>_M^7</math> resolves to vi or [[subdominant|VI]],<ref name="B&S 232">Benward & Saker (2003), p.232.</ref> for example.
+[[Image:"Try To Remember" nondominant seventh chords.png|thumb|center|330px|Nondominant seventh chords are, "found in large number," in popular music and jazz ("a legacy from the romantic period"), such as in this example from "[[Try To Remember]]" (''[[The Fantasticks]]'') by [[Harvey Schmidt]] (lyrics: [[Tom Jones (writer)|Tom Jones]])<ref name="B&S 232"/> {{audio|"Try To Remember" nondominant seventh chords.mid|Play}}. Note the circle progression derived root motion by fourths/fifths.]]
+When possible, as in circle progressions, [[resolution (music)|resolve]] the seventh of nondominant seventh chords down by [[steps and skips|step]] to the [[third (chord)|third]] of the following chord.<ref name="B&S 233">Benward & Saker (2003), p.233-34.</ref>
+[[Image:ii7-V7-I turnaround in C.png|thumb|center|330px|[[ii-V-I turnaround|ii<sup>7</sup>-V<sup>7</sup>-I turnaround]] in C {{audio|ii7-V7-I turnaround in C.mid|Play}}.]]
+==See also==
+*[[Irregular resolution]]
+*[[Dominant ninth]], etc.
+*[[Dominant seventh flat five chord]]
+==Sources==
+{{reflist}}
+{{Degrees|state=expanded}}
+{{Chords|state=expanded}}
+[[Category:Seventh chords]]

Kirchhoff–Love plate theory: Difference between revisions

Revision as of 13:52, 16 December 2013

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Definition of amortized time

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Kirchhoff–Love plate theory: Difference between revisions

Revision as of 13:52, 16 December 2013

See also

Sources

Definition of amortized time

Relation between amortized and actual time

Amortized analysis of worst-case inputs

Example

Applications

References

Navigation menu

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