Generators info

In kinematics, Euler’s rotation theorem states that, in three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a rotation about a fixed axis through that point. The theorem is named after Leonhard Euler.

In mathematical terms, this is a statement that, in 3D space, any two coordinate systems with a common origin are related by a rotation about some fixed axis. This also means that the product of two rotation matrices is again a rotation matrix. A (non-identity) rotation matrix has a real eigenvalue which is equal to unity. The eigenvector corresponding to this eigenvalue is the axis of rotation connecting the two systems.

Applications

Generators of rotations

Suppose we specify an axis of rotation by a unit vector [x, y, z] , and suppose we have an infinitely small rotation of angle Δθ  about that axis. To first order the rotation matrix ΔR  is represented as:

<math>\Delta R =

 \begin{bmatrix}
 1&0&0\\
 0&1&0\\
 0&0&1
 \end{bmatrix}

+

 \begin{bmatrix}
 0 & z&-y\\
 -z& 0& x\\
 y &-x& 0
 \end{bmatrix}\,\Delta \theta

= \mathbf{I}+\mathbf{A}\,\Delta \theta.
</math>

A finite rotation through angle θ about this axis may be seen as a succession of small rotations about the same axis. Approximating Δθ  as θ/N where N  is a large number, a rotation of θ about the axis may be represented as:

<math>R =\left(\mathbf{1}+\frac{\mathbf{A}\theta}{N}\right)^N

\approx e^{\mathbf{A}\theta}.</math>

It can be seen that Euler’s theorem essentially states that all rotations may be represented in this form. The product <math>\mathbf{A}\theta</math> is the “generator” of the particular rotation. Analysis is often easier in terms of these generators, rather than the full rotation matrix. Analysis in terms of the generators is known as the Lie algebra of the rotation group.

Quaternions

It follows from Euler’s theorem that the relative orientation of any pair of coordinate systems may be specified by a set of four numbers. Three of these numbers are the direction cosines that orient the eigenvector. The fourth is the angle about the eigenvector that separates the two sets of coordinates. Such a set of four numbers is called a quaternion.

While the quaternion as described above, does not involve complex numbers, if quaternions are used to describe two successive rotations, they must be combined using the non-commutative quaternion algebra derived by William Rowan Hamilton through the use of imaginary numbers.

Rotation calculation via quaternions has come to replace the use of direction cosines in Aerospace applications through their reduction of the required calculations, and their ability to minimize round-off errors. Also, in computer graphics the ability to perform spherical interpolation between quaternions with relative ease is of value.

See also

• Euler pole
• Euler angles
• Euler-Rodrigues parameters
• Rotation representation

A turbo generator is a turbine directly connected to an electric generator for the generation of electric power. Large Steam powered turbo generators (steam turbine generators) provide the majority of the world’s electricity and are also used by steam powered, turbo-electric ships.

Smaller turbo-generators with gas turbines are often used as auxiliary power units. For base loads diesel generators are usually preferred, since they offer much better fuel-efficiency and are also more reliable, but on the other hand they are much heavier and need more space.

The efficiency of larger gas turbine plants can be enhanced, if the hot exhaust gases are used to generate steam which drives another turbo generator.

Turbo generators were also used on steam locomotives as a power source for coach lighting and heating systems.

External Links

• http://www.srdrives.com/turbo-generator.shtml

In kinematics, Euler’s rotation theorem states that, in three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a rotation about a fixed axis through that point. The theorem is named after Leonhard Euler.

In mathematical terms, this is a statement that, in 3D space, any two coordinate systems with a common origin are related by a rotation about some fixed axis. This also means that the product of two rotation matrices is again a rotation matrix. A (non-identity) rotation matrix has a real eigenvalue which is equal to unity. The eigenvector corresponding to this eigenvalue is the axis of rotation connecting the two systems.

Applications

Generators of rotations

Suppose we specify an axis of rotation by a unit vector [x, y, z] , and suppose we have an infinitely small rotation of angle Δθ  about that axis. To first order the rotation matrix ΔR  is represented as:

<math>\Delta R =

 \begin{bmatrix}
 1&0&0\\
 0&1&0\\
 0&0&1
 \end{bmatrix}

+

 \begin{bmatrix}
 0 & z&-y\\
 -z& 0& x\\
 y &-x& 0
 \end{bmatrix}\,\Delta \theta

= \mathbf{I}+\mathbf{A}\,\Delta \theta.
</math>

A finite rotation through angle θ about this axis may be seen as a succession of small rotations about the same axis. Approximating Δθ  as θ/N where N  is a large number, a rotation of θ about the axis may be represented as:

<math>R =\left(\mathbf{1}+\frac{\mathbf{A}\theta}{N}\right)^N

\approx e^{\mathbf{A}\theta}.</math>

It can be seen that Euler’s theorem essentially states that all rotations may be represented in this form. The product <math>\mathbf{A}\theta</math> is the “generator” of the particular rotation. Analysis is often easier in terms of these generators, rather than the full rotation matrix. Analysis in terms of the generators is known as the Lie algebra of the rotation group.

Quaternions

It follows from Euler’s theorem that the relative orientation of any pair of coordinate systems may be specified by a set of four numbers. Three of these numbers are the direction cosines that orient the eigenvector. The fourth is the angle about the eigenvector that separates the two sets of coordinates. Such a set of four numbers is called a quaternion.

While the quaternion as described above, does not involve complex numbers, if quaternions are used to describe two successive rotations, they must be combined using the non-commutative quaternion algebra derived by William Rowan Hamilton through the use of imaginary numbers.

Rotation calculation via quaternions has come to replace the use of direction cosines in Aerospace applications through their reduction of the required calculations, and their ability to minimize round-off errors. Also, in computer graphics the ability to perform spherical interpolation between quaternions with relative ease is of value.

See also

• Euler pole
• Euler angles
• Euler-Rodrigues parameters
• Rotation representation

Polyphase coils are electrical coils (phases) connected together in a polyphase system such as a generator or motor. In modern systems the number of phases is usually three, or a multiple of three. Each phase carries a sinusoidal alternating current whose phase is delayed relative to one of its neighbours and advanced relative to its other neighbour. The phase currents are separated in time evenly within each period of the alternating current. For example, in a three-phase system, the phases are separated from each other by one-third of the period.

Coil construction

Like all coils used in electrical machinery, polyphase coils (made from insulated conducting wire) are wound around ferromagnetic armatures with radial projections and maximum core-surface exposure to the magnetic field.

The windings are physically separated around the circumference of an electrical machine. The result of such an arrangement is a rotating magnetic field that is used to convert electrical power to rotary mechanical work, or vice versa.

Polyphase motors and generators

Compared to single-phase motors and generators, polyphase motors are simpler, because they do not require external circuitry (using capacitors and inductors) to produce a starting torque. Polyphase machines can deliver constant power over each period of the alternating current, eliminating the pulsations found in a single-phase machine as the current passes through zero amplitude.

History

The use of polyphase coils in electrical power systems was pioneered by the engineers Nikola Tesla, Galileo Ferraris, and Michail Dolivo-Dobrovolsky.

See also

• Induction motor

Let G be a deterministic polynomial time function from N^<ω to N^<ω with stretch function

l: N → N,

so that if x has length n then G(x) has length l(n). Then let G_n be the distribution on strings of length l(n) defined by the output of G on a randomly selected string of length n selected by the uniform distribution.

Then we say G is a pseudorandom generator if

{G_n}_{n ∈ N}

is pseudorandom.

In effect, G translates a random input of length n to a pseudorandom output of length l(n). Assuming

l(n) > n,

this expands a random sequence (and can be applied multiple times, since G_n can be replaced by the distribution of G(G(x))).

Often, we are concerned not with the behavior of G on all strings, but only on strings of some prescribed length. This case allows a slightly easier definition:

A function <math>G_l: \left \{0,1\right\}^n \rightarrow \left \{0,1\right\}^m\,</math> with <math>n < m\,</math> is a pseudorandom generator if

• <math>G_l\,</math> can be computed in <math>poly(n)\,</math> time, and
• <math>G_l(x)\,</math> is pseudorandom.

It is an open problem whether or not pseudorandom generators exist. It is known that if one-way functions or hard-core predicates exist, then pseudorandom generators exist. It is also known that if

l(n) > n,

there is some other pseudorandom generator with

l(n) > p(n)

for any polynomial, p(n). This follows from the following theorem:

Theorem: If there is a pseudorandom generator

<math>G_l: \left \{0,1\right\}^{n} \rightarrow \left \{0,1\right\}^{n+1}\,</math>

then for any <math>m = poly(n) \,</math>, there is a pseudorandom generator

<math>G_l: \left \{0,1\right\}^n \rightarrow \left \{0,1\right\}^m\,</math>

Pseudorandom generators have numerous applications. In cryptography, a simple application
is providing an efficient analog of `one time pads’. It is well known that in order to
encrypt a message m in a way that the cipher text provides no information on the plaintext,
the key k used should be random over strings of length |m|. Then m can be encrypted
via <math>c=k\oplus m</math>. This operation is very costly in terms of key length.
Key length can be reduced if we compromise on semantic security.
Then, given G, which expands by a polynomial <math>n^{c+1}</math>, then a sequence of
<math>n^c</math> messages of length n can be encrypted by xor-ing each with
the corresponding area of G(k) (inspired the idea of stream chipers).
Pseudorandom generators may also be used to construct symmetric key cryptosystems,
where any polynomial number of messages can be `safely’ encrypted under the same key,
that is, the polynomial <math>n^c</math> is not apriority known at time of key generation.
Such a construction can be based on a generalization of pseudo random generators, called pseudorandom functions. A family of pseudorandom functions (PRF’s) is a collection of
efficiently computable keyed functions, which `act randomly’
in the scene that no efficient algorithm can distinguish between an oracle to a function corresponding to a random key, and an oracle to a random function.
It’s known that if PRG’s exist, than so do PRF’s (for more details see pseudorandom function). One application of PRF’s is to understanding learning theory. Loosely speaking, given
a sequence of examples <math>(x_1,f(x_1)),(x_2,f(x_2)),\ldots,(x_m,f(x_m)))</math> e.t.c,
the goal is to efficiently find a succinct representation of a function f out of a given class of functions consistent with the examples. PRF families (if exist) are a natural example of a class of functions with small representation size, but are not learnable.

Another application is to derandomizing algorithms.
A nice pseudorandom generator is a pseudorandom number generator,

<math>G:\{0,1\}^n\rightarrow\{0,1\}^m</math>

with

<math>n=O(\log m)\,</math>.

If a nice pseudorandom generator exists, then P=BPP.
In fact, this strong derandomization result follows assuming the existence of a weaker type of
pseudorandom generators, Nisan-Wigderson type generator with exponential stretch. Their definition weakens the definition
of PRG above in two essential ways. First, it allows <math>G_l</math> to run in exponential in n time. Another important difference is that the output distribution is only required to be indistinguishable from uniform for circuits of size S’(n) for some fixed exponential S’ which is smaller than S, as opposed to generators as in the definition above.
It’s easy to see that the existence of nice pseudorandom generators of this kind
for some polynomial S(n) is sufficient to imply P=BPP, and follows from plausible hardness assumptions (that some problems in EXP don’t have sub exponential circuits). In a nutshell, the idea is to replace the randomness used by a BPP algorithm A,
by G(s), where s is a short (O(log(n))) random string. By pseudorandomness of G, the behavior
of A on any given x will not change much, so we can count the number of 1’s output by A obtained iterating over the s, and answer according to the majority. That is, <math>A(x,\cdot)</math> can be viewed as a non uniform distinguisher of proper size.
For more details on this result and other derandomization results see BPP.

For more on these and other applications of PRG’s, see chapters 10,17 in a draft of a book by Aurora and Barak available at http://www.cs.princeton.edu/theory/complexity/

‘Tohatsu Corporation’ of Tokyo, Japan, was founded in 1932. It manufactures and sells outboard motors, pleasure boats, portable fire pumps, small fire trucks, pumps for construction and drainage, refrigeration units for transportation and also does real estate property management in Japan.

The company was originally named Takata Motor Research Institute, and manufactured railcars. Research and development of high-speed, portable engine generators and radio-controlled generators began immediately and were brought to production in 1930. During the 1930s and 40s, Tohatsu consolidated its product line and moved its corporate office to Tokyo. In 1950, production and sales of motorcycles began. 1955 brought aggressive growth to Tohatsu. Capital increased to 150 million yen and production on a new line of engines started. Sales offices were established in Fukuoka, Nagoya, Tokyo, Sendai and Sapporoand. Dealers were set up throughout Japan. 1956 ushered in the production of the first Tohatsu outboards (1.5 hp).

Tohatsu Outboards

Tohatsu Outboards are produced by Tohatsu Corporation. 1956 ushered in the production of the first Tohatsu Outboards (1.5 hp). Since then, Tohatsu outboards have capably served a variety of marketplaces: commercial fishing, marine transport, recreation and competition. They are the second largest producer of outboards in the world and a leader in innovative design with the environmentally conscious TLDI series of Two-stroke Low pressure Direct Injection outboards that meet current EPA regulations for the U.S. Mercury outboards from 30 hp and below are rebadged Tohatsu’s and all Nissan outboard engines in North America are Tohatsus with a Nissan decal.

External links

• Tohatsu America Corp. - N. America Corporate site
• Tohatsu Corp. - Japanese Corporate site
• Tohatsu Corp. - English Corporate site

Fleming’s right hand rule (for generators) shows the direction of induced current flow when a conductor moves in a magnetic field.

The right hand is held with the thumb, first finger and second finger mutually at right angles, as shown in the diagram .

• The Thumb represents the direction of Motion of the conductor.

• The First finger represents the direction of the Field.

• The Second finger represents the direction of the induced or generated Current (in the classical direction, from positive to negative).

There is also a Fleming’s left hand rule (for electric motors). The appropriately-handed rule can be recalled from the letter “g“, which is in “right” and “generator”.

These mnemonics are named after British engineer John Ambrose Fleming, who invented them.

See also

• Fleming’s left hand rule for motors
• Right hand grip rule
• FBI mnemonics

External links

• Diagrams at magnet.fsu.edu
• Diagram and interactive exercises

In mathematics, the HNN extension is a basic construction of combinatorial group theory.

Introduced in a 1949 paper Embeddings of Groups by Graham Higman, B. H. Neumann and Hanna Neumann, it embeds a given group G into another group G’ , in such a way that two given isomorphic subgroups of G are conjugate (through a given isomorphism) in G’ .

Construction

Let <math>G</math> be a group with
presentation <math>G=\langle S\mid R\rangle</math>, and let
<math>\alpha</math> be an isomorphism between two subgroups <math>H</math> and <math>K</math> of <math>G</math>.
Let <math>t</math> be a new symbol not in <math>S</math>, and define

<math>

G*_{\alpha} = \langle S,t \mid R, tht^{-1}=\alpha(h), \forall h\in H\rangle
</math>
The group <math>G*_{\alpha}</math> is called the HNN extension of
<math>G</math> relative to <math>\alpha</math>. The original group G is called the base group for the construction, while the subgroups <math>H</math> and <math>K</math> are the associated subgroups. The new generator <math>t</math> is called the stable letter.

Key properties

Since the presentation for <math>G*_{\alpha}</math> contains all the generators and relations from the presentation for
<math>G</math>, there is a natural homomorphism, induced by the identification of generators, which takes <math>G</math> to <math>G*_{\alpha}</math>. Higman, Neumann and Neumann proved that this morphism is injective, that is, an embedding of <math>G</math>
into <math>G*_{\alpha}</math>. A consequence is that two isomorphic subgroups of a given group are always conjugate in some overgroup; the desire to show this was the original motivation for the construction.

Applications

In terms of the fundamental group in algebraic topology, the HNN extension is the construction required to understand the fundamental group of a topological space X that has been ‘glued back’ on itself by a mapping f. That is, HNN extensions stand in relation of that aspect of the fundamental group, as free products with amalgamation do with respect to the Seifert-van Kampen theorem for gluing spaces X and Y along a connected common subspace. Between the two constructions essentially any geometric gluing can be described, from the point of view of the fundamental group.

The idea of HNN extension has been extended to other parts of abstract algebra, including Lie algebra theory.

Generalizations

HNN extensions are elementary examples of fundamental groups of graphs of groups, and as such are of central importance in Bass-Serre theory.

A wood gas generator is a wood-fueled gasification reactor mounted on an internal combustion engine, to provide a wood gas, a form of syngas.

Shortages of petroleum-based fuels during World War II resulted in very large numbers of such generators, often improvised. Commercial generators were in production before and after the war, for use in special circumstances or in distressed economies: production skyrocketed during the war.

Gasification was an important & familiar 19th century technology, and its potential and practical applicability to internal combustion engines was well-understood, from the earliest days of their development. Town gas was produced as a local business, and experience in the trade was very widespread: most practicing technical people would know a good deal about it. When stationary internal combustion engines became available, they were commonly fueled by town gas during the early 20th century.

By the time World War II arrived, town gas production had been displaced by petroleum gas, but older people remembered both town gas and gas-fueled engines, and wood gas generators were in active production.

World War II era wood gas generators were of the “Imbert” type. FEMA has published a book, “Construction of a Simplified Wood Gas Generator for Fueling Internal Combustion Engines in a Petroleum Emergency”, in March 1989, describing a different design called the “stratified downdraft gasifier”. It solves several drawbacks of earlier types, and should be used to guide contemporary projects instead of the earlier designs.

Usually, wood gas generators burn wood, but improvements to efficiency and energy-density are possible, by using charcoal instead.

The disadvantages of wood gas generators are: large size, slow to start (build a fire), and the batch-burn operation (of some designs). When not carefully designed and used deaths have occurred due to the fact that wood gas contains a large percentage of deadly Carbon Monoxide(CO) gas.

Modern wood gas generators

With raising oil price, wood gas generators come back. A project about the energy future of Europe was held in 2005 in Güssing, Austria with contribution of European Union research furtherance. The project consisted of a power plant with a wood gas generator and a gas engine to convert the wood gas into 2 MW electric power and 4.5 MW heat.

At the wood gas power plant are also 2 containers for experiments with wood gas. One container is an experiment to convert wood gas, using the Fischer-Tropsch process, to a diesel fuel like liquid. By October 2005, it was possible to convert 5 kg wood into 1 litre fuel.

See Also

• “Holzbrenner Strength through Joy Wagon” (Volkswagen Beetle, 1940-1945)

External links

• Wood gas generator power plant in Güssing Austria
• Experiment to make a Diesel like liquid fuel out of wood gas

The Korea General Machinery Trading Corporation is a large North Korean machine company. The company is headquartered in the Tongdaewon District near the capital, Pyongyang.

The company imports steel, chemical raw stock and some machine tools and turns out machine tools, metal parts, gears, electric motors, generators, hydroelectric generators, pumps, valves, mining equipment, rolling stock and other machinery.

Taean Heavy Machine Complex

An important part of Korea General Machinery is the Taean Heavy Machine Complex in Tongdaewon. This is where its hydroelectric generators and other thermal power generating equipment is manufactured, including turbines, motors, transformers, etc. Taean manufactures different sizes classes of hydroelectric equipment in its Tongsuse Class including a 50,000 kVA generator. The factory complex is up-to-date in its computerization.

Huichon Machine Tool Factory

Huichon Tool-Machine Factory|Huichon Machine Tool Factory in Huichon is North Korea’s leading manufacturer of heavy-duty machine tools for domestic use and for export. The 50-year old factory group is involved in machine tool production processes including steel-making, casting, processing, assembly, painting and packing. Product is produced on serial basis and small lot basis.

Its output of precision machine tools includes an assortment of spline-grinding machines and industrial lathes.

See Also

• Economy of North Korea

External link

• Naenara, North Korea’s Web Portal

In group theory, a central extension of a group G is a short exact sequence of groups

<math>1\rightarrow A\rightarrow E\rightarrow G\rightarrow 1</math>

such that A is in Z(E), the center of the group E.

Examples of central extensions can be constructed by taking any group G and any abelian group A, and setting E to be A×G. This kind of split example (a split extension in the sense of the extension problem, since G is present as a subgroup of E) isn’t of particular interest. More serious examples are found in the theory of projective representations, in cases where the projective representation cannot be lifted to an ordinary linear representation.

Similarly, the central extension of a Lie algebra is an exact sequence

<math>0\rightarrow \mathfrak{a}\rightarrow\mathfrak{e}\rightarrow\mathfrak{g}\rightarrow 0</math>

such that <math>\mathfrak{a}</math> is in the center of <math>\mathfrak{e}</math>.

If the group G is a Lie group, then a central extension of G is a Lie group as well, and the Lie algebra of a central extension of G is a central extension of the Lie algebra of G. In the terminology of theoretical physics, the generators of E not included in G are called central charges. These generators are in the center of the Lie algebra of E; by Noether’s theorem, generators of symmetry groups correspond to conserved quantities, referred to as charges.

In Lie group theory, central extensions arise in connection with algebraic topology. Suppose G is a connected Lie group that is not simply connected. Its universal cover G* is again a Lie group, in such a way that the projection

π: G* → G

is a group homomorphism, and surjective. Its kernel is (up to isomorphism) the fundamental group of G; this is known to be abelian (see H-space). This construction gives rise to central extensions.

A special case is that of the metaplectic groups. These stand in relation with the symplectic groups, in the same way that the spinor groups do with the special orthogonal groups. The case of SL₂(R) involves a fundamental group that is infinite cyclic. Here the central extension involved is well known in modular form theory, in the case of forms of weight ½. A projective representation that corresponds is the Weil representation, constructed from the Fourier transform, in this case on the real line. Metaplectic groups also occur in quantum mechanics.

See also

• Virasoro algebra

In theoretical physics, the Haag-Lopuszanski-Sohnius theorem shows that the possible symmetries of a consistent 4-dimensional quantum field theory do not only consist of internal symmetries and Poincaré symmetry, but can also include supersymmetry as a nontrivial extension of the Poincaré algebra. This significantly generalized the Coleman-Mandula theorem.

One of the important results is that the fermionic part of the Lie superalgebra has to have spin-1/2 (spin 3/2 or higher are ruled out)

History

Prior to the Haag-Lopuszanski-Sohnius theorem, the Coleman-Mandula theorem was the strongest of a series of no-go theorems, stating that the symmetry group of a consistent 4-dimensional quantum field theory is the direct product of the internal symmetry group and the Poincaré group.

In 1975, Rudolf Haag, Jan Łopuszański, and Martin Sohnius published their proof that weakening the assumptions of the Coleman-Mandula theorem by allowing both commuting and anticommuting symmetry generators, there is a nontrivial extension of the Poincaré algebra, namely the supersymmetry algebra.

Importance

What is most fundamental in this result (and thus in supersymmetry), is that there can be an interplay of spacetime symmetry with internal symmetry (in the sense of “mixing particles”): the supersymmetry generators transform a bosonic particle into a fermionic one, but the anticommutator of two such transformations yields a translation in spacetime. Precisely such an interplay seemed excluded by the Coleman-Mandula theorem, which stated that (bosonic) internal symmetries cannot interact non-trivially with spacetime symmetry.

This theorem was also an important justification of the previously found Wess-Zumino model, an interacting four-dimensional quantum field theory with supersymmetry, leading to a renormalizable theory.

See also

• Supergravity
• S-matrix

References

• R. Haag, J. T. Lopuszanski and M. Sohnius, “All Possible Generators Of Supersymmetries Of The S Matrix”, Nucl. Phys. B 88 (1975) 257.

A bio-nano generator is a nanoscale electrochemical device, like a fuel cell or galvanic cell, but drawing power from blood glucose in a living body, much the same as how the body generates energy from food. To achieve the effect, an enzyme is used that is capable of stripping glucose of its electrons, freeing them for use in electrical devices.

The average person’s body could, theoretically, generate 100 watts of electricity using a bio-nano generator. However, this estimate is only true if all food was converted to electricity, and the human body needs some energy consistently, so possible power generated is likely much lower.

The electricity generated by such a device could power devices embedded in the body (such as pacemakers), or sugar-fed nanorobots.

A similar technology was presented in the Matrix series of science fiction major motion pictures, with robots shown enslaving mankind for its bio-energy. At the time, bio-nano generators were unheard of in popular culture, and the plot device of “a form of fusion” was used to make the harnessing of bio-electricity more plausible to the audience.

Much of the research done on bio-nano generators is still experimental, with Panasonic’s Nanotechnology Research Laboratory among those at the forefront.

External links

• Article detailing the technology and its development

Fleming’s right hand rule (for generators) shows the direction of induced current flow when a conductor moves in a magnetic field.

The right hand is held with the thumb, first finger and second finger mutually at right angles, as shown in the diagram .

• The Thumb represents the direction of Motion of the conductor.

• The First finger represents the direction of the Field.

• The Second finger represents the direction of the induced or generated Current (in the classical direction, from positive to negative).

There is also a Fleming’s left hand rule (for electric motors). The appropriately-handed rule can be recalled from the letter “g“, which is in “right” and “generator”.

These mnemonics are named after British engineer John Ambrose Fleming, who invented them.

See also

• Fleming’s left hand rule for motors
• Right hand grip rule
• FBI mnemonics

External links

• Diagrams at magnet.fsu.edu
• Diagram and interactive exercises

Tyranny, released in 2001, is the third album by alternative/punk rock band, The Generators.

Track listing

1. “Down In The City”
   
2. “Murder”
3. “Keep On Running”
4. “Hijacked”
5. “Summer Of Unrest”
6. “Tyranny”
7. “Dead At 16″
8. “Rats”
9. “Us Against Them”
10. “All Night Long”
11. “Suburban Bitch”
12. “Coming Down”

Credits

• Doug Dagger - Vocals
• Doosky - Lead, Rhythm Guitars, Backup Vocals
• Mike Snow - Lead, Rhythm Guitars, & Backup Vocals
• Dirty Ernie - Drums, Backup Vocals
• Rich Mouser - Bass, Back up Vocals

Future Sound Corporation is an Italian-based hard trance/hardstyle record label fronted by DJ Ben and Radium (who work together as Trance Generators). Artists that have released work through Future Sound include Radium, Trance Generators, Jon The Baptist, Force 9, Atomic Alliance, DJ Chuck-E and The Elite.

See also

• List of record labels

External links

• Official site
• Future Sound Corporation on discogs.com

Fleming’s left hand rule (for electric motors) shows the direction of the thrust on a conductor carrying a current in a magnetic field.

The left hand is held with the thumb, index finger and middle finger mutually at right angles.

• The First finger represents the direction of the Field.

• The Second finger represents the direction of the Current (in the classical direction, from positive to negative).

• The Thumb represents the direction of the Thrust or resultant Motion.

There also exists Fleming’s right hand rule (for generators). The appropriately-handed rule can be recalled by remembering that the letter “g” is in “right” and “generator”.

Both mnemonics are named after British engineer John Ambrose Fleming who invented them.

Other mnemonics also exist that use a left hand rule or a right hand rule for predicting resulting motion from a pre-existing current and field.

Right-handed variant

A right-handed variant of this rule is as follows:
If the current through a wire is in the direction of the thumb of the right hand, and the direction of an external magnetic field on the wire is represented by the fingers, then the force experienced by the wire will be in the direction the palm of the right hand is facing.

Right-handed variants are useful for people who write with their left hand.

See also

• Fleming’s right hand rule
• Right hand rule
• FBI mnemonics

External links

• Overview at ac.wwu.edu (Right Hand - ‘FIB’)
• Diagrams at magnet.fsu.edu. See figure #9. (Left Hand - ‘FBI’)
• Description at starprof.com (Right Hand - ‘IBF’ - palm alternative)

HappyDoc is a documentation generator for the Python programming language. It can produce code documentation in HTML, XML, SGML, or PDF, and is written by Doug Hellmann.

See also

• Comparison of documentation generators

External links

• HappyDoc

In mathematics, a symmetric function of multiple variables is one that is invariant under permutation of its variables; that is, the value of the function does not depend on the order of the n-tuple of arguments.

In most contexts, the term refers to a polynomial with this property: a symmetric polynomial.
The theory of symmetric polynomials is part of the theory of polynomial equations, and also a substantial chapter of combinatorics. Examples of symmetric polynomials are

X₁ + X₂ + … + X_n ,

X₁³ + X₂³ + … + X_n³ ,

and

X₁X₂…X_n .

If P(x) is a polynomial with roots

α₁, α₂, …, α_n ,

a symmetric function of the roots of P means

S(α₁, α₂, …, α_n)

where S is a symmetric function of n variables.

There is a direct relation between certain symmetric functions of the roots and the coefficients in the polynomial P. Assume for simplicity that P is monic, so that

P(x) = xⁿ + a_n−1xⁿ⁻¹ + … + a₀ = (x − α₁)(x − α₂)…(x − α_n),

where the a_i are in a field K and the α_i are in the algebraic closure of K. The sum of all the α_i equals −a_n−1 and their product equals (−1)ⁿa₀. In fact, it is classical algebra (Viète’s formulas) that the intermediate coefficients of P are plus or minus the sums of the products of the roots taken j at a time, for 1 < j < n. (The sign is alternately + and −.) These formulae are the basis of the traditional theory of equations. In symbols the formulas say:

a_n−j = (−1)^n−j Σ α_k(1)…α_k(j)

with the summation taken over all index sequences

1 ≤ k(1) < … < k(j) ≤ n.

These sums for j = 1, 2, …, n are called the elementary symmetric functions of the roots, because the jth elementary symmetric polynomial, written σ_j, is given by the same formula, but in indeterminates X_i. A basic theorem states that any symmetric polynomial function S of n variables can be expressed as a polynomial in the elementary symmetric functions. In the solution of polynomial equations, the symmetric polynomials of the roots lie in K.

The polynomial relations underlying that assertion are universal (independent of choice of P); and, if we work with the symmetric polynomials created from a monomial, we can eliminate dependence on K, too, to get formulae with integer coefficients. Putting this more algebraically, we can define a subring Symm(n) of Z[X₁, X₂, …, X_n] consisting of the integral symmetric polynomials (those invariant under the action of the symmetric group on indices); and then assert that the formulae for σ_j, for which we retain the notation, are ring generators of Symm(n). What is more, they are independent generators (no algebraic relations hold), so that Symm(n) is abstractly also a polynomial ring on n generators. A great deal of attention was paid, in older algebra textbooks, to algorithmic procedures expressing the procedural content of this (which has been stated as an existence theorem but has computational content).

The most important single application is to the power sums α₁^k + α₂^k + … + α_n^k, in terms of the a_j. The formulae for doing this are attributed to Isaac Newton. They were encountered in K-theory too, where they underlie the Adams operations.

They also support the theory of the Newton polygon, part of the theory of ramification. In Newton’s case the point was to work with a_j in a formal power series ring; here passage to the algebraic closure is the theory of Puiseux expansions in fractional powers, and the Newton polygon is a device for computing the required exponents.

See also

• Newton’s identities
• Symmetric matrix

A signal generator, also known variously as a test signal generator, function generator, tone generator, arbitrary waveform generator, or frequency generator is an electronic device that generates repeating electronic signals (in either the analog or digital domains). They are generally used in designing, testing, troubleshooting, and repairing electronic or electroacoustic devices; though they often have artistic uses as well.

There are many different types of signal generators, with different purposes and applications (and at varying levels of expense); in general, no device is suitable for all possible applications.

Traditionally, signal generators have been embedded hardware units, but since the age of multimedia-PCs, flexible, programmable software tone generators have also been available.

General purpose signal generators

Function generators

A function generator is a device which produces simple repetitive waveforms. Such devices contain an electronic oscillator, a circuit that is capable of creating a repetitive waveform. (Modern devices may use digital signal processing to synthesize waveforms, followed by a digital to analog converter, or DAC, to produce an analog output). The most common waveform is a sine wave, but sawtooth, step (pulse), square, and triangular waveform oscillators are commonly available as are arbitrary waveform generators (AWGs). If the oscillator operates above the audio frequency range (>20 kHz), the generator will often include some sort of modulation function such as amplitude modulation (AM), frequency modulation (FM), or phase modulation (PM) as well as a second oscillator that provides an audio frequency modulation waveform.

Function generators are typically used in simple electronics repair and design; where they are used to stimulate a circuit under test. A device such as an oscilloscope is then used to measure the circuit’s output. Function generators vary in the number of outputs they feature, frequency range, frequency accuracy and stability, and several other parameters.

Arbitrary waveform generators

Arbitrary waveform generators, or AWGs, are sophisticated signal generators which allow the user to generate arbitrary waveforms, within published limits of frequency range, accuracy, and output level. Unlike function generators, which are limited to a simple set of waveforms; an AWG allows the user to specify a source waveform in a variety of different ways. AWGs are generally more expensive than function generators, and are often more highly limited in available bandwidth; as a result, they are generally limited to higher-end design and test applications.

Special purpose signal generators

In addition to the above general-purpose devices, there are several classes of signal generators designed for specific applications.

Tone generators and audio generators

A tone generator is a type of signal generator optimized for use in audio and acoustics applications. Tone generators typically include sine waves over the audio frequency range (20 Hz–20 kHz). Sophisticated tone generators will also include sweep generators (a function which varies the output frequency over a range, in order to make frequency-domain measurements), multitone generators (which output several tones simultaneously, and are used to check for intermodulation distortion and other non-linear effects), and tone bursts (used to measure response to transients). Tone generators are typically used in conjunction with sound level meters, when measuring the acoustics of a room or a sound reproduction system, and/or with oscilloscopes or specialized audio analyzers.

Many tone generators operate in the digital domain, producing output in various digital audio formats such as AES-3, or SPDIF. Such generators may include special signals to stimulate various digital effects and problems, such as clipping, jitter, bit errors; they also often provide ways to manipulate the metadata associated with digital audio formats.

The term synthesizer is used for a device that generates audio signals for music, or that uses slightly more intricate methods.

Video signal generators

A video signal generator is a device which outputs predetermined video and/or television waveforms, and other signals used to stimulate faults in, or aid in parametric measurements of, television and video systems. There are several different types of video signal generators in widespread use. Regardless of the specific type, the output of a video generator will generally contain synchronization signals appropriate for television, including horizontal and vertical sync pulses (in analog) or sync words (in digital). Generators of composite video signals (such as NTSC and PAL) will also include a colorburst signal as part of the output. Video signal generators are available for a wide variety of applications, and for a wide variety of digital formats; many of these also include audio generation capability (as the audio track is an important part of any video or television program or motion picture).

External Links

Signal Generators

Applications for signal generators

Unit Generators (or ugens) are the basic formal unit in many MUSIC-N-style computer music programming languages. They are sometimes called opcodes (particularly in Csound), though this expression is not accurate in that these are not machine-level instructions.

Unit Generators form the building blocks for designing synthesis and signal processing algorithms in software. For example, a simple unit generator called OSC could generate a sinusoidal waveform of a specific frequency (given as an input or argument to the function or class that represents the unit generator). ENV could be a unit generator that delineates a breakpoint function. Thus ENV could be used to drive the amplitude envelope of the oscillator OSC through the equation OSC*ENV. Unit generators often use predefined arrays of values for their functions (which are filled with waveforms or other shapes by calling a specific generator function).

The Unit Generator theory of sound synthesis was first developed and implemented by Max Mathews and his colleagues at Bell Labs in the 1950s.

The A2W reactor is a naval reactor used by the United States Navy to provide electricity generation and propulsion on warships. The A2W designation stands for:

• A = Aircraft carrier platform
• 2 = Second generation core designed by the contractor
• W = Westinghouse was the contracted designer

This nuclear reactor was used in the world’s first nuclear-powered aircraft carrier, the USS Enterprise (CVN-65). The four propulsion plants on Enterprise each contain two reactors, numbered 1A-1B, 4A - 4B, 2A - 2B, and 3A - 3B (numbered as they are located from fore to aft). Each propulsion plant is capable of operating on one reactor plant through most of the power range required to propel the ship at speeds in excess of 33 knots (61 km/h) (with a possible maximum speed up to approximately 35 knots; higher speeds such as the 40-50 knots sometimes rumored would rapidly become impossible due to hydrodynamic drag even if the reactors were capable of delivering enough power). Both reactors would be on-line to simultaneously provide maximum ship speed and plane launching capability.

Design and operation

The reactors are pressurized water reactors fueled by highly-enriched (upwards of 93%) U-235. Light water is used as both neutron-moderator and reactor coolant. Hafnium Control rods are used to control the operation of the reactor. Extracting the rods to a calculated height allows the reactor to reach “criticality” - the point at which the nuclear fission reactions reach a self-sustaining level. Thereafter, steam flow (from the steam generators) regulates reactor power as explained below. The control rods are “shimmed” in or out to regulate average coolant temperature or lowered to the bottom of the reactor vessel to shut the reactor down (either done in a slow controlled manner or dropped rapidly during what is referred to as a SCRAM to immediately shut the reactor down.

Much of the reactor power control during steady state operation comes as a result of the coolant water’s negative temperature coefficient. The “power” of the reactor is determined by the number of fission events that takes place in the fuel at any given moment. As the water heats up, it expands and becomes less dense which provides fewer molecules per volume to moderate the neutrons, hence fewer neutrons are slowed to the required thermal energies to sustain thermal fission. Conversely, when the coolant water temperature decreases, its density increases and a greater number of neutrons reach the required thermal energy, increasing the number of fissions per unit of time, creating more heat. This has the effect of allowing “steam demand” to control reactor power, requiring little intervention by the Reactor Operator for changes in the power demanded by the ship’s operations.

The hot water from the reactors is sent, via large pipes, into heat exchangers called steam generators. There the heat from the reactor coolant water is transferred, through tube walls, to water being fed into the steam generators from a separate feed system. In the A1W and A2W systems, the pressurized water reactor coolant is kept between 525 and 545 °F (274 and 285 °C). In the steam generators, the water from the feed system is converted to steam at 535 °F (279 °C) and a pressure of about 600 lb/in² (4 MPa). Once the reactor coolant water has given off its heat in the steam generators, it is returned, via large electric pumps (four per reactor), to the reactors to repeat the cycle.

Saturated steam at 600 lb/in² (4 MPa) is channeled from each steam generator to a common header, where the steam is then sent to the main engine, electrical generators, aircraft catapult system, and various auxiliaries. The main propulsion turbines are double-ended, in which the steam enters at the center and divides into two streams as it enters the actual turbine wheels, expanding and giving up its energy as it does so, causing the turbine to spin at high speed. The main shaft enters a reduction gear in which the high rotational velocity of the turbine shaft is stepped down to a usable turn rate for propelling the ship. The expended steam from the main engine and other auxiliaries enters condensers to be cooled into water and recycled to the feed system.

TwinText is a commercially available Source Code Documentation Tool. It can generate HTML documents from a variety of programming languages, including [[C++]], C, Java, IDL, PHP, C#, and Visual Basic. It runs on Windows.

See also

• Comparison of documentation generators

External links

• TwinText Homepage

In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable),

<math>

 f_n(X) \partial_X^n + \cdots + f_1(X) \partial_X + f_0(X).

</math>

More precisely, let F be a field, and let F[X] be the ring of polynomials in one variable, X, with coefficients in F. Then each f_i lies in F[X]. ∂_X is the derivative with respect to X. The algebra is generated by X and ∂_X.

The Weyl algebra is an example of a simple ring that is not a matrix ring over a division ring. It is also a noncommutative example of a domain, and an example of an Ore extension.

You can also construct the Weyl algebra as a quotient of the free algebra on two generators, X and Y, by the ideal generated by the single relation

YX − XY − 1.

The Weyl algebra is the first in an infinite family of algebras, also known as Weyl algebras. The n-th Weyl algebra, A_n, is the ring of differential operators with polynomial coefficients in n variables. It is generated by X_i and <math>\part_{X_i}</math>.

Weyl algebras are named after Hermann Weyl, who introduced them to study the Heisenberg uncertainty principle in quantum mechanics. It is a quotient of the universal enveloping algebra of the Lie algebra of the Heisenberg group, by setting the element 1 of
the Lie algebra equal to the unit 1 of the universal enveloping algebra.

One may give an abstract construction of the algebras A_n in terms of generators and relations.
We do so in a more sophisticated way: Start with an
abstract vector space V (of dimension 2n) equipped with a symplectic form <math>\omega</math>.
Define the Weyl algebra W(V) to be

<math>W(V) := T(V) / (\!( v \otimes w - w \otimes v - \omega(v,w), \text{ for } v,w \in V )\!),</math>

where the notation <math>(\!( )\!)</math> means “the ideal generated by”. In other words, <math>W(V)</math> is the algebra generated by V subject
only to the relation <math>vw - wv = \omega(v,w)</math>. Then, W(V) is isomorphic to <math>A_{n}\,</math> (it does not depend on the choice
of <math>\omega</math>). In this form, one sees that W(V)
is a quantization of the symmetric algebra Sym(V). If V is over a field of characteristic zero,
then W(V) is naturally isomorphic to the symmetric algebra Sym(V) equipped with the deformed Moyal product (considering the symmetric
algebra to be polynomial functions on <math>V^*</math>, where the variables span the vector space V, and replacing <math>i \hbar</math> in
the Moyal product formula with 1). The isomorphism is given by
the symmetrization map from Sym(V) to W(V):
<math>a_1 \cdots a_n \mapsto \frac{1}{n!} \sum_{\sigma \in S_n} a_{\sigma(1)} \otimes \cdots \otimes a_{\sigma(n)}</math>. If one prefers to have the
<math>i \hbar</math> and work over the complex numbers, one could have instead defined the Weyl algebra above as generated by X_i and
<math>i \hbar \part_{X_i}</math> (as is frequently done in quantum mechanics).

Thus, the Weyl algebra is a quantization of the symmetric algebra, which is essentially the same as the Moyal quantization (if for the latter one restricts to polynomial functions), but
the former is in terms of generators and relations (considered to be differential operators) and the latter is in terms of a deformed multiplication.
In the case of exterior algebras, the analogous quantization to the Weyl one is the Clifford algebra.

For more details about this quantization in the case <math>n=1</math> (and an extension using the Fourier transform to integrable (”most”)
functions, not just polynomial functions), see Weyl quantization.

References

• M. Rausch de Traubenberg, M. J. Slupinski, A. Tanasa, Finite-dimensional Lie subalgebras of the Weyl algebra, (2005) (Classifies subalgebras of the one dimensional Weyl algebra over the complex numbers; shows relationship to SL(2,C))

Orion Platinum is a music production suite developed by Synapse Audio Software.

The software supports all stages of music production, from composing/arranging music to mixing and mastering it. It includes multitrack audio recording, generators, effects and a mixing desk with sub busses. It supports both traditional linear sequencing and pattern-based sequencing. Supported external interfaces include VST, DX and ReWire. Audio output resolutions range from 16 Bit/44.1kHz up to 32 Bit/192 kHz, Stereo.

Features

Generators

Orion Platinum comes with the following internal generators (as of version 7):

Toxic III

Licensed by Maxx Claster

Toxic II

Licensed by Maxx Claster. Despite Toxic III being introduced in v7, Toxic II is still included.

Tomcat

Sampler

Ultran

Wasp XT

History

Wasps roots come from a free standalone software-synth, the TS-404. It was coded by Richard Hoffman (who is now the main driving force behind Synapse Audio), and was later licensed to Image-Line Software for FL Studio. The TS-404 was later developed into the first version of Wasp. Wasp was included in early versions Orion and also licensed to Image Line who sold it as an add on plugin for Fruity Loops and as a separate VSTi plugin.

Wasp was continually developed by Synapse Audio / Sonic Syndicate as part of Orion. A major update to Wasp, Wasp XT, was released with the first version of Orion Platinum in 2002. Image-Line didn’t update their version of Wasp until FL Studio v6 was released in early 2006 (also changing the name to Wasp XT). It is not clear whether the later Image-Line version uses the same code as the Synapse-Audio version.

Drums

Monobass

Emulation of TB-303

XR-909

Emulation of Roland TR-909

Wavefusion

Screamer

Plucked String

MIDI Out

External links

• Orion Platinum
• Synapse Audio Software

This is an article about nuclear power plant equipment. For other uses, see steam generator.

Steam generators are heat exchangers used to convert water into steam from heat produced in a nuclear reactor core. They are used in pressurized water reactors between the primary and secondary coolant loops.

In commercial power plants steam generators can measure up to 70 feet in height and weigh as much as 800 tons. Each steam generator can contain anywhere from 3,000 to 16,000 tubes, each about three-quarters of an inch in diameter. The coolant is pumped, at high pressure to prevent boiling, from the reactor coolant pump, through the nuclear reactor core, and through the tube side of the steam generators before returning to the pump. This is referred to as the primary loop. That water flowing through the steam generator boils water on the shell side to produce steam in the secondary loop that is delivered to the turbines to make electricity. The steam is subsequently condensed via cooled water from the tertiary loop and returned to the steam generator to be heated once again. The tertiary cooling water may be recirculated to cooling towers where it sheds waste heat before returning to condense more steam. Once through tertiary cooling may otherwise be provided by a river, lake, ocean. This primary, secondary, tertiary cooling scheme is the most common way to extract usable energy from a controlled nuclear reaction.

These loops also have an important safety role because they constitute one of the primary barriers between the radioactive and non-radioactive sides of the plant as the primary coolant becomes radioactive from its exposure to the core. For this reason, the integrity of the tubing is essential in minimizing the leakage of water between the two sides of the plant. There is the potential that if a tube bursts while a plant is operating; contaminated steam could escape directly to the secondary cooling loop. Thus during scheduled maintenance outages or shutdowns, some or all of the steam generator tubes are inspected by eddy-current testing.

In other types of reactors, such as the pressurised heavy water reactors of the CANDU design, the primary fluid is heavy water. Liquid metal cooled reactors such as the in Russian BN-600 reactor also use heat exchangers between primary metal coolant and at the secondary water coolant.

Boiling water reactors do not use steam generators, as steam is produced in the pressure vessel.

Types

Westinghouse and Combustion Engineering designs have vertical U-tubes with inverted tubes for the primary water. Canadian, Japanese, French, and German PWR suppliers use the vertical configuration as well. Russian VVER reactor designs use horizontal steam generators, which have the tubes mounted horizontally. Babcock and Wilcox plants (e.g., Three Mile Island) have smaller steam generators that force water through the top of the OTSGs (once-through steam generators; counter-flow to the feedwater) and out the bottom to be recirculated by the reactor coolant pumps. The horizontal design has proven to be less susceptible to degradation than the vertical U-tube design.

See also

• Nuclear power plant
• Power station
• Steam turbine

TwinText is a commercially available Source Code Documentation Tool. It can generate HTML documents from a variety of programming languages, including [[C++]], C, Java, IDL, PHP, C#, and Visual Basic. It runs on Windows.

See also

• Comparison of documentation generators

External links

• TwinText Homepage

In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable),

<math>

 f_n(X) \partial_X^n + \cdots + f_1(X) \partial_X + f_0(X).

</math>

More precisely, let F be a field, and let F[X] be the ring of polynomials in one variable, X, with coefficients in F. Then each f_i lies in F[X]. ∂_X is the derivative with respect to X. The algebra is generated by X and ∂_X.

The Weyl algebra is an example of a simple ring that is not a matrix ring over a division ring. It is also a noncommutative example of a domain, and an example of an Ore extension.

You can also construct the Weyl algebra as a quotient of the free algebra on two generators, X and Y, by the ideal generated by the single relation

YX − XY − 1.

The Weyl algebra is the first in an infinite family of algebras, also known as Weyl algebras. The n-th Weyl algebra, A_n, is the ring of differential operators with polynomial coefficients in n variables. It is generated by X_i and <math>\part_{X_i}</math>.

Weyl algebras are named after Hermann Weyl, who introduced them to study the Heisenberg uncertainty principle in quantum mechanics. It is a quotient of the universal enveloping algebra of the Lie algebra of the Heisenberg group, by setting the element 1 of
the Lie algebra equal to the unit 1 of the universal enveloping algebra.

One may give an abstract construction of the algebras A_n in terms of generators and relations.
We do so in a more sophisticated way: Start with an
abstract vector space V (of dimension 2n) equipped with a symplectic form <math>\omega</math>.
Define the Weyl algebra W(V) to be

<math>W(V) := T(V) / (\!( v \otimes w - w \otimes v - \omega(v,w), \text{ for } v,w \in V )\!),</math>

where the notation <math>(\!( )\!)</math> means “the ideal generated by”. In other words, <math>W(V)</math> is the algebra generated by V subject
only to the relation <math>vw - wv = \omega(v,w)</math>. Then, W(V) is isomorphic to <math>A_{n}\,</math> (it does not depend on the choice
of <math>\omega</math>). In this form, one sees that W(V)
is a quantization of the symmetric algebra Sym(V). If V is over a field of characteristic zero,
then W(V) is naturally isomorphic to the symmetric algebra Sym(V) equipped with the deformed Moyal product (considering the symmetric
algebra to be polynomial functions on <math>V^*</math>, where the variables span the vector space V, and replacing <math>i \hbar</math> in
the Moyal product formula with 1). The isomorphism is given by
the symmetrization map from Sym(V) to W(V):
<math>a_1 \cdots a_n \mapsto \frac{1}{n!} \sum_{\sigma \in S_n} a_{\sigma(1)} \otimes \cdots \otimes a_{\sigma(n)}</math>. If one prefers to have the
<math>i \hbar</math> and work over the complex numbers, one could have instead defined the Weyl algebra above as generated by X_i and
<math>i \hbar \part_{X_i}</math> (as is frequently done in quantum mechanics).

Thus, the Weyl algebra is a quantization of the symmetric algebra, which is essentially the same as the Moyal quantization (if for the latter one restricts to polynomial functions), but
the former is in terms of generators and relations (considered to be differential operators) and the latter is in terms of a deformed multiplication.
In the case of exterior algebras, the analogous quantization to the Weyl one is the Clifford algebra.

For more details about this quantization in the case <math>n=1</math> (and an extension using the Fourier transform to integrable (”most”)
functions, not just polynomial functions), see Weyl quantization.

References

• M. Rausch de Traubenberg, M. J. Slupinski, A. Tanasa, Finite-dimensional Lie subalgebras of the Weyl algebra, (2005) (Classifies subalgebras of the one dimensional Weyl algebra over the complex numbers; shows relationship to SL(2,C))

Event generators are software libraries that generate simulated high-energy particle physics eventsM. L. Mangano & T. J. Stelzer, Ann. Rev. Nucl. Part. Sci. 55, 555 (2005).M. A. Dobbs et al., hep-ph/0403045..

Despite the simple structure of the tree-level perturbative quantum field theory description of the collision and decay processes in an event, the observed high-energy process usually contains significant amount of modifications, like photon and gluon bremsstrahlung or loop diagram corrections, that usually are too complex to be easily evaluated in real calculations directly on the diagrammatic level. Furthermore, the non-perturbative nature of QCD bound states makes it necessary to include information that are outside the perturbative quantum field theory, and well beyond present ability of computation in lattice QCD. And in collisional systems more complex than a few leptons and hadrons (e.g. heavy-ion collisions), the collective behavior of the system would involve a phenomenological description that also cannot be easily obtained from the fundamental field theory by a simple calculus.

Any realistic test of the underlying physical process in a particle accelerator experiment, therefore, requires an adequate inclusion of these complex behaviors surrounding the actual process. Based on the fact that in most processes, a factorization of the full process into individual problems is possible (which means a negligible effect from interference), these individual processes are calculated separately, and the probabilistic branching between them are performed using Monte Carlo methods.

The final-state particles generated by event generators can be fed into the detector simulation, allowing a precise prediction and verification for the entire system of experimental setup. However, as the detector simulation is usually a complex and computationally expensive task, simple event analysis techniques are also performed directly on event generator results.

A typical hadronic event generator simulates the following subprocesses:

• Initial-state composition and substructure
• Initial-state showers
• The hard process
• Resonance decay
• Final-state showers
• Accompanying semi-hard processes
• Hadronization and further decay

A typical heavy-ion event generator usually can be less strict in simulating the rare and rather negligible processes found in a hadronic generator, but would need to simulate the following subprocesses, in addition to those in a hadronic generator:

• Nuclear initial-state
• High multiplicity, soft processes
• In-medium energy loss
• Collective behavior of the medium (not handled properly by any generators sofar)

Partly due to historic reasons, most event generators are written in FORTRAN 77, with a few [[C++]] generators slowly emerging in recent years. The Particle Data Group maintains a standard for designating Standard Model particles and resonances with integer codes in event generators (also known as the “PDG code”).

List of event generators

The major event generators that are used by current experiments are:

Hadronic event generators

• PYTHIA (formerly Pythia/Jetset)
• HERWIG
• ISAJET

Heavy ion event generators

• HIJING

Neutrino event generators

• GENIE

Specialized event generators

• AcerMC – LHC background processes
• ALPGEN – multiple parton processes
• Ariadne – QCD cascade with Color Dipole Model
• MC@NLO – parton shower with next-to-leading-order QCD matrix elements
• JIMMY – multiple parton processes

“Meta-generator”

• CompHEP – automatic evaluation of tree level matrix elements for event generation or export into other event generators

References

External links

• 2006 Monte Carlo Number Scheme, from the 2006 Review of Particle Physics.
• List of Monte Carlo Programs – from DESY

Thunder Force is a scrolling shooter computer game released by Technosoft in 1983. It is the first game in the Thunder Force series. Known versions of it were released on the following Japanese based computers: Sharp X1, Sharp MZ-1500, NEC PC-6001 mkII, and NEC PC-8801 mkII. In 1984, it was released for the Fujitsu FM-7, and NEC PC-9801 computers as Thunder Force Construction, featuring an add-on that allowed players to create custom made areas.

Story

The ORN Empire (antagonists of the game) has built a large asteroid fortress named the Dyradeizer to oppose the Galaxy Federation. In addition to its high firepower capabilities, Dyradeizer is supported by shield generators hidden in various locations by ORN, which render the fortress invisible. In an attempt to destroy Dyradeizer, the Galaxy Federation sends their specially designed fighter, the FIRE LEO (controlled by the player), to locate and destroy the shield generators and defeat Dyradeizer.

Gameplay

The structure of the game consists of overhead, free-directional scrolling areas and the player’s ship is armed with main shot to shoot airborne targets and a bomb shot to shoot ground enemies. Gameplay consists of flying the FIRE LEO over ORN occupied areas while destroying enemy base installations and turrets. Each area has a certain number of shield generators hidden under the ground based enemy targets; in order for an area to be completed, the shield generators must be found and destroyed. After doing so, the Dyradeizer will temporally appear and the player must destroy a certain section of it. Once this section is destroyed, the Dyradeizer will disappear and the player will be taken to the next area to repeat the process.

Graphic and sound wise, Thunder Force is very crude and modest compared to its successors, and is the most obscure game of the series (at least from a non-Japanese perspective).

Download and Play The Game Here

http://www.funtimes.us/Genesis_Games/T/20

The Lorentz group of theoretical physics has a variety of representations, corresponding to particles with integer and half-integer spins in relativistic quantum mechanics. These representations are normally constructed out of spinors.

The group may also be represented in terms of a set of functions defined on the Riemann sphere. these are the Riemann P-functions, which are expressible as hypergeometric series. An important special case is the subgroup SO(3), where these reduce to the spherical harmonics, and find practical application in the theory of atomic spectra.

Finding representations

According to general representation theory of Lie groups, one first looks for the representations of the complexification of the Lie algebra of the Lorentz group. A convenient basis for the Lie algebra of the Lorentz group is given by the three generators of rotations J^k=ε^ijkL_ij and the three generators of boosts Kⁱ=L_it where i, j, and k run over the three spatial coordinates and ε is the Levi-Civita symbol for a three dimensional spatial slice of Minkowski space. Note that the three generators of rotations transform like components of a pseudovector J and the three generators of boosts transform like components of a vector K under the adjoint action of the spatial rotation subgroup.

This motivates the following construction: first complexify, and then change basis to the components of A = (J + i K)/2 and B = (J – i K)/2. In this basis, one checks that the components of A and B satisfy separately the commutation relations of the Lie algebra sl₂ and moreover that they commute with each other. In other words, one has the isomorphism

<math>\mathfrak{so}(3,1)\otimes\mathbb{C} \cong \mathfrak{sl}_2(\mathbb{C})\oplus \mathfrak{sl}_2(\mathbb{C}).</math>

The utility of this isomorphism comes from the fact that sl₂ is the complexification of the rotation algebra, and so its irreducible representations correspond to the well-known representations of the spatial rotation group; for each j in Z/2, one has the (2j+1)-dimensional spin-j representation spanned by the spherical harmonics with j as the highest weight. Thus the irreps of the Lorentz group are simply given by an ordered pair of half-integers (m,n) which fix representations of the subalgebra spanned by the components of A and B respectively.

Properties of the (m,n) irrep

Since the angular momentum operator is given by J = A + B, the highest weight of the rotation subrepresentation will be m+n. So for example, the (1/2,1/2) representation has spin 1. The (m,n) representation is (2m+1)(2n+1)-dimensional.

Common reps

• (0,0) the Lorentz scalar representation. This representation is carried by relativistic scalar field theories.
• (1/2,0) is the left-handed Weyl spinor and (0,1/2) is the right-handed Weyl spinor representation.
• (1/2,0) ⊕ (0,1/2) is the Dirac spinor representation.
• (1/2,1/2) is the vector representation. The electromagnetic vector potential lives in this rep. It is a 1-form field.
• (1,0) is the self-dual 2-form field representation and (0,1) is the anti-self-dual 2-form field representation.
• (1,0) ⊕ (0,1) is the representation of a parity invariant 2-form field. The electromagnetic field tensor transforms under this representation.
• (1,1/2) ⊕ (1/2,1) is the Rarita-Schwinger field representation.
• (1,1) is the spin-2 representation of the traceless metric tensor.

Full Lorentz group

The (m,n) representation is irreducible under the restricted Lorentz group (the identity component of the Lorentz group). When one considers the full Lorentz group, which is generated by the restricted Lorentz group along with time and parity reversal, not only is this not an irreducible representation, it is not a representation at all, unless m=n. The reason is that this representation is formed in terms of the sum of a vector and a pseudovector, and a parity reversal changes the sign of one, but not the other. The upshot is that a vector in the (m,n) representation is carried into the (n,m) representation by a parity reversal. Thus (m,n)⊕(n,m) gives an irrep of the full Lorentz group. When constructing theories such as QED which is invariant under parity reversal, Dirac spinors may be used, while theories that do not, such as the electroweak force, must be formulated in terms of Weyl spinors.

See also

• Poincaré group
• Wigner’s classification

Reference

• V. Bargmann, Irreducible unitary representations of the Lorenz group. Ann. of Math. 48 (1947), 568-640.

Stanford Research Systems is a maker of general test and measurement instruments. The company was founded in 1980, is privately held, and is not affiliated with Stanford University.

Stanford Research Systems (SRS) manufactures all of their products at their Sunnyvale, California facility.
SRS produces scientific and engineering instruments for a number of different fields. Many of the products fall under the general category of “signal recovery.” These products are primarily sold to industry, university, and government labs, or sold as OEM components to other manufacturers.

Electronic Products

• Analog PID controllers
• Lock-in amplifiers
• Low-noise preamplifiers
• High voltage power supplies
• Gated integrators and boxcar averagers
• Synthesized function and clock generators
• Digital delay and pulse generators
• Frequency counters
• FFT spectrum analyzers
• LCR meters
• Thermocouple monitors
• Programmable filters
• Compact rubidium (atomic) frequency standards

Other Products

• Quartz Crystal Microbalances
• a melting point apparatus
• a nitrogen laser
• an optical chopper
• Vacuum gauges and controllers
• Residual Gas Analyzers (quadrupole mass spectrometers) and controllers
• Cryogenic temperature measurement instrumentation

External links

• Stanford Research Systems

Berkeley Yacc is a reimplementation of the Unix parser generator Yacc, originally written by Robert Corbett in 1990. It has the advantages of being written in ANSI C and being public domain software.

Installation

A very simple tutorial that explains how to compile and install Berkeley Yacc on Linux can be found here.

Let G be a deterministic polynomial time function from N^<ω to N^<ω with stretch function

l: N → N,

so that if x has length n then G(x) has length l(n). Then let G_n be the distribution on strings of length l(n) defined by the output of G on a randomly selected string of length n selected by the uniform distribution.

Then we say G is a pseudorandom generator if

{G_n}_{n ∈ N}

is pseudorandom.

In effect, G translates a random input of length n to a pseudorandom output of length l(n). Assuming

l(n) > n,

this expands a random sequence (and can be applied multiple times, since G_n can be replaced by the distribution of G(G(x))).

Often, we are concerned not with the behavior of G on all strings, but only on strings of some prescribed length. This case allows a slightly easier definition:

A function <math>G_l: \left \{0,1\right\}^n \rightarrow \left \{0,1\right\}^m\,</math> with <math>n < m\,</math> is a pseudorandom generator if

• <math>G_l\,</math> can be computed in <math>poly(n)\,</math> time, and
• <math>G_l(x)\,</math> is pseudorandom.

It is an open problem whether or not pseudorandom generators exist. It is known that if one-way functions or hard-core predicates exist, then pseudorandom generators exist. It is also known that if

l(n) > n,

there is some other pseudorandom generator with

l(n) > p(n)

for any polynomial, p(n). This follows from the following theorem:

Theorem: If there is a pseudorandom generator

<math>G_l: \left \{0,1\right\}^{n} \rightarrow \left \{0,1\right\}^{n+1}\,</math>

then for any <math>m = poly(n) \,</math>, there is a pseudorandom generator

<math>G_l: \left \{0,1\right\}^n \rightarrow \left \{0,1\right\}^m\,</math>

Pseudorandom generators have numerous applications. In cryptography, a simple application
is providing an efficient analog of `one time pads’. It is well known that in order to
encrypt a message m in a way that the cipher text provides no information on the plaintext,
the key k used should be random over strings of length |m|. Then m can be encrypted
via <math>c=k\oplus m</math>. This operation is very costly in terms of key length.
Key length can be reduced if we compromise on semantic security.
Then, given G, which expands by a polynomial <math>n^{c+1}</math>, then a sequence of
<math>n^c</math> messages of length n can be encrypted by xor-ing each with
the corresponding area of G(k) (inspired the idea of stream chipers).
Pseudorandom generators may also be used to construct symmetric key cryptosystems,
where any polynomial number of messages can be `safely’ encrypted under the same key,
that is, the polynomial <math>n^c</math> is not apriority known at time of key generation.
Such a construction can be based on a generalization of pseudo random generators, called pseudorandom functions. A family of pseudorandom functions (PRF’s) is a collection of
efficiently computable keyed functions, which `act randomly’
in the scene that no efficient algorithm can distinguish between an oracle to a function corresponding to a random key, and an oracle to a random function.
It’s known that if PRG’s exist, than so do PRF’s (for more details see pseudorandom function). One application of PRF’s is to understanding learning theory. Loosely speaking, given
a sequence of examples <math>(x_1,f(x_1)),(x_2,f(x_2)),\ldots,(x_m,f(x_m)))</math> e.t.c,
the goal is to efficiently find a succinct representation of a function f out of a given class of functions consistent with the examples. PRF families (if exist) are a natural example of a class of functions with small representation size, but are not learnable.

Another application is to derandomizing algorithms.
A nice pseudorandom generator is a pseudorandom number generator,

<math>G:\{0,1\}^n\rightarrow\{0,1\}^m</math>

with

<math>n=O(\log m)\,</math>.

If a nice pseudorandom generator exists, then P=BPP.
In fact, this strong derandomization result follows assuming the existence of a weaker type of
pseudorandom generators, Nisan-Wigderson type generator with exponential stretch. Their definition weakens the definition
of PRG above in two essential ways. First, it allows <math>G_l</math> to run in exponential in n time. Another important difference is that the output distribution is only required to be indistinguishable from uniform for circuits of size S’(n) for some fixed exponential S’ which is smaller than S, as opposed to generators as in the definition above.
It’s easy to see that the existence of nice pseudorandom generators of this kind
for some polynomial S(n) is sufficient to imply P=BPP, and follows from plausible hardness assumptions (that some problems in EXP don’t have sub exponential circuits). In a nutshell, the idea is to replace the randomness used by a BPP algorithm A,
by G(s), where s is a short (O(log(n))) random string. By pseudorandomness of G, the behavior
of A on any given x will not change much, so we can count the number of 1’s output by A obtained iterating over the s, and answer according to the majority. That is, <math>A(x,\cdot)</math> can be viewed as a non uniform distinguisher of proper size.
For more details on this result and other derandomization results see BPP.

For more on these and other applications of PRG’s, see chapters 10,17 in a draft of a book by Aurora and Barak available at http://www.cs.princeton.edu/theory/complexity/

Regular temperament is any tempered system of musical tuning such that each frequency ratio is obtainable as a product of powers of a finite number of generators, or generating frequency ratios. The classic example of a regular temperament is meantone temperament, where the generating intervals are usually given in terms of a slightly flattened fifth and the octave.

If the generators are all of the prime numbers up to a given prime p, we have what is called p-limit just intonation. Sometimes some irrational number close to one of these primes is substituted (an example of tempering) to favour other primes, as in twelve tone equal temperament where 3 is tempered to 2^19/12 to favour 2, or in quarter-comma meantone where 3 is tempered to 2·5^1/4 to favor 2 and 5.

In mathematical terminology, the products of these generators defines a free abelian group. The number of independent gen