Happy Ending problem: Difference between revisions

Revision as of 02:18, 27 July 2013

In general, homogeneity is defined as the quality or state of being homogeneous (of the same or similar nature, from Greek ὀμός meaning 'same'). It also means having a uniform structure throughout. For instance, a uniform electric field (which has the same strength and the same direction at each point) would be compatible with homogeneity (all points experience the same physics). A material constructed with different constituents can be described as effectively homogeneous in the electromagnetic materials domain, when interacting with a directed radiation field (light, microwave frequencies, etc.)^[1]^[2] In physics, homogeneous usually means describing a material or system that has the same properties at every point of the space; in other words, uniform without irregularities. In physics, it also describes a substance or an object whose properties do not vary with position. For example, an object of uniform density is sometimes described as homogeneous.^[3]^[4] Another related definition is simply a substance that is uniform in composition.^[5]

Mathematically, homogeneity has the connotation of invariance, as all components of the equation have the same degree of value whether or not each of these components are scaled to different values, for example, by multiplication or addition. Cumulative distribution fits this description. "The state of having identical cumulative distribution function or values".^[1]^[2]

Context

The definition of homogeneous strongly depends on the context used. For example, a composite material is made up of different individual materials, known as "constituents" of the material, but may be defined as a homogeneous material when assigned a function. For example, asphalt paves our roads, but is a composite material consisting of asphalt binder and mineral aggregate, and then laid down in layers and compacted.

In another context, a material is not homogeneous in so far as it composed of atoms and molecules. However, at the normal level of our everyday world, a pane of glass, or a sheet of metal is described as glass, or stainless steel. In other words, these are each described as a homogeneous material.

A few other instances of context are: Dimensional homogeneity (see below) is the quality of an equation having quantities of same units on both sides; Homogeneity (in space) implies conservation of momentum; and homogeneity in time implies conservation of energy.

Homogeneous alloy

In the context of composite metals is an alloy. A blend of a metal with one or more metallic or nonmetallic materials is an alloy. The components of an alloy do not combine chemically but, rather, are very finely mixed. An alloy might be homogeneous or might contain small particles of components that can be viewed with a microscope. Brass is an example of an alloy, being a homogeneous mixture of copper and zinc. Another example is steel, which is an alloy of iron with carbon and possibly other metals. The purpose of alloying is to produce desired properties in a metal that naturally lacks them. Brass, for example, is harder than copper and has a more gold-like color. Steel is harder than iron and can even be made rust proof (stainless steel).^[6]

Homogeneous cosmology

Homogeneity, in another context plays a role in cosmology. From the perspective of 19th-century cosmology (and before), the universe was infinite, unchanging, homogeneous, and therefore filled with stars. However, German astronomer Heinrich Olbers asserted that if this were true, then the entire night sky would be filled with light and bright as day; this is known as Olbers' paradox. Olbers presented a technical paper in 1826 that attempted to answer this conundrum. The faulty premise, unknown in Olbers' time, was that the universe is not infinite, static, and homogeneous. The Big Bang cosmology replaced this model (expanding, finite, and inhomogeneous universe). However, modern astronomers supply reasonable explanations to answer this question. One of at least several explanations is that distant stars and galaxies are red shifted, which weakens their apparent light and makes the night sky dark.^[7] However, the weakening is not sufficient to actually explain Olbers' paradox. Many cosmologists think that the fact that the Universe is finite in time, that is that the Universe has not been around forever, is the solution to the paradox.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. The fact that the night sky is dark is thus an indication for the Big Bang.

Translation invariance

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By translation invariance, one means independence of (absolute) position, especially when referring to a law of physics, or to the evolution of a physical system.

Fundamental laws of physics should not (explicitly) depend on position in space. That would make them quite useless. In some sense, this is also linked to the requirement that experiments should be reproducible. This principle is true for all laws of mechanics (Newton's laws, etc.), electrodynamics, quantum mechanics, etc.

In practice, this principle is usually violated, since one studies only a small subsystem of the universe, which of course "feels" the influence of rest of the universe. This situation gives rise to "external fields" (electric, magnetic, gravitational, etc.) which make the description of the evolution of the system depending on the position (potential wells, etc.). This only stems from the fact that the objects creating these external fields are not considered as (a "dynamical") part of the system.

Translational invariance as described above is equivalent to shift invariance in system analysis, although here it is most commonly used in linear systems, whereas in physics the distinction is not usually made.

The notion of isotropy, for properties independent of direction, is not a consequence of homogeneity. For example, a uniform electric field (i.e., which has the same strength and the same direction at each point) would be compatible with homogeneity (at each point physics will be the same), but not with isotropy, since the field singles out one "preferred" direction.

Consequences

In Lagrangian formalism, homogeneity in space implies conservation of momentum, and homogeneity in time implies conservation of energy. This is shown, using variational calculus, in standard textbooks like the classical reference [Landau & Lifshitz] cited below. This is a particular application of Noether's theorem.

Dimensional homogeneity

As said in the introduction, dimensional homogeneity is the quality of an equation having quantities of same units on both sides. A valid equation in physics must be homogeneous, since equality cannot apply between quantities of different nature. This can be used to spot errors in formula or calculations. For example, if one is calculating a speed, units must always combine to [length]/[time]; if one is calculating an energy, units must always combine to [mass]•[length]²/[time]², etc. For example, the following formulae could be valid expressions for some energy:

E_{k} = \frac{1}{2} m v^{2}; E = m c^{2}; E = p v; E = h c / λ

if m is a mass, v and c are velocities, p is a momentum, h is Planck's constant, λ a length. On the other hand, if the units of the right hand side do not combine to [mass]•[length]²/[time]², it cannot be a valid expression for some energy.

Being homogeneous does not necessarily mean the equation will be true, since it does not take into account numerical factors. For example, E = m•v² could be or could not be the correct formula for the energy of a particle of mass m traveling at speed v, and one cannot know if h•c/λ should be divided or multiplied by 2π.

Nevertheless, this is a very powerful tool in finding characteristic units of a given problem, see dimensional analysis.

Theoretical physicists tend to express everything in natural units given by constants of nature, for example by taking c = ħ = k = 1; once this is done, one partly loses the possibility of the above checking.

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.

@@ Line 1: / Line 1: @@
-The writer is called Irwin. To collect badges is what her family members and her enjoy. North Dakota is her beginning location but she will have to move one working day or an additional. Hiring is his profession.<br><br>my web site [http://premium.asslikethat.com/user/CSalerno at home std test]
+In general, '''homogeneity''' is defined as the quality or state of being ''[[Homogeneity and heterogeneity|homogeneous]]'' (of the same or similar [[Nature (innate)|nature]], from Greek ὀμός meaning 'same'). It also means having a uniform structure throughout. For instance, a uniform [[electric field]] (which has the same strength and the same direction at each point) would be compatible with homogeneity (all points experience the same physics). A material constructed with different constituents can be described as effectively homogeneous in the [[electromagnetic field|electromagnetic]] materials domain, when interacting with a directed [[EM radiation|radiation field]] (light, microwave frequencies, etc.)<ref name=homogeneity>[http://www.merriam-webster.com/dictionary/homogeneity Homogeneity]. Merriam-webster.com</ref><ref name=Homogeneous>[http://www.merriam-webster.com/dictionary/Homogeneous Homogeneous]. Merriam-webster.com</ref> In physics, homogeneous usually means describing a material or system that has the same properties at every point of the space; in other words, uniform without irregularities. In physics, it also describes a substance or an object whose properties do not vary with position. For example, an object of uniform density is sometimes described as homogeneous.<ref name=homogeneous-physics>
+{{cite encyclopedia
+|last=  Rennie, Richard
+|first=Science Online
+|title= Homogeneous (physics)
+|publisher= The Facts On File Dictionary of Atomic and Nuclear Physics.
+|year=2003
+|url =http://www.fofweb.com/activelink2.asp?ItemID=WE40&SID=5&iPin=ffdatom0854&SingleRecord=True
+|quote =Describing a material or system that has the same properties in any direction; i.e. uniform without irregularities.}} (accessed November 16, 2009).</ref><ref name=homogeneous-physics-2>
+Tanton, James.
+"homogeneous."
+Encyclopedia of Mathematics.
+New York: Facts On File, Inc., 2005.
+Science Online. Facts On File, Inc.
+"A polynomial in several variables p(x,y,z,…) is called homogeneous [...] more generally, a function of several variables f(x,y,z,…) is homogeneous [...] Identifying homogeneous functions can be helpful in solving differential equations [and] any formula that represents the mean of a set of numbers is required to be homogeneous. In physics, the term homogeneous describes a substance or an object whose properties do not vary with position. For example, an object of uniform density is sometimes described as homogeneous."
+[http://www.fofweb.com/activelink2.asp?ItemID=WE40&SID=5&iPin=EMATH0406&SingleRecord=True James. homogeneous (math).]
+(accessed: 2009-11-16)</ref>
+Another related definition is simply a substance that is uniform in composition.<ref name=substance>
+{{Cite encyclopedia
+  | title =Kristi Lew, "homogenous"| encyclopedia =''Science Online''| volume =Database
+  | pages =1| publisher =Facts On File, Inc.| year =2008| accessdate =2009-12-31}}</ref>
+Mathematically, homogeneity has the connotation of [[invariant (physics)|invariance]], as all components of the [[equation]] have the same degree of value whether or not each of these components are scaled to different values, for example, by multiplication or addition. Cumulative distribution fits this description. "The state of having identical cumulative distribution function or values".<ref name=homogeneity/><ref name=Homogeneous/>
+==Context==
+The definition of homogeneous strongly depends on the context used. For example, a [[composite material]] is made up of different individual materials, known as "''constituents''" of the material, but may be defined as a homogeneous material when assigned a function. For example, [[asphalt]] paves our roads, but is a composite material consisting of asphalt binder and mineral aggregate, and then laid down in layers and compacted.
+In another context, a material is not homogeneous in so far as it composed of [[atom]]s and [[molecule]]s.  However, at the normal level of our everyday world, a pane of glass, or a sheet of metal is described as glass, or stainless steel. In other words, these are each described as a homogeneous material.
+A few other instances of context are: ''Dimensional homogeneity'' (see below) is the quality of an equation having quantities of same units on both sides; ''Homogeneity (in space)'' implies [[Conservation law|conservation of momentum]]; and ''homogeneity in time'' implies [[Conservation law|conservation of energy]].
+===Homogeneous alloy===
+In the context of composite metals is an alloy. A blend of a metal with one or more metallic or nonmetallic materials is an alloy. The components of an alloy do not combine chemically but, rather, are very finely mixed. An alloy might be homogeneous or might contain small particles of components that can be viewed with a microscope. Brass is an example of an alloy, being a homogeneous mixture of copper and zinc. Another example is steel, which is an alloy of iron with carbon and possibly other metals. The purpose of alloying is to produce desired properties in a metal that naturally lacks them. Brass, for example, is harder than copper and has a more gold-like color. Steel is harder than iron and can even be made rust proof (stainless steel).<ref>Rosen, Joe. "Alloy." Encyclopedia of Physics. New York: Facts On File, Inc., 2004. Science Online. Facts On File, Inc. accessed 2009-11-16</ref>
+===Homogeneous cosmology===
+Homogeneity, in another context plays a role in [[cosmology]]. From the perspective of 19th-century cosmology (and before), the [[universe]] was [[Infinity|infinite]], unchanging, homogeneous, and therefore filled with [[star]]s. However, German astronomer [[Heinrich Olbers]] asserted that if this were true, then the entire night sky would be filled with light and bright as day; this is known as [[Olbers' paradox|Olbers' paradox.]] Olbers presented a technical paper in 1826 that attempted to answer this conundrum. The faulty premise, unknown in Olbers' time, was that the universe is not infinite, static, and homogeneous. The [[Big Bang]] cosmology replaced this model (expanding, finite, and [[Inhomogeneous cosmology|inhomogeneous universe]]). However, modern astronomers supply reasonable explanations to answer this question. One of at least several explanations is that distant stars and [[galaxies]] are [[red shift]]ed, which weakens their apparent light and makes the night sky dark.<ref>Todd, Deborah, and Joseph A. Angelo Jr. "''Olbers, Heinrich Wilhelm Matthäus''." A to Z of Scientists in Space and Astronomy. New York: Facts on File, Inc., 2005. Science Online. Facts On File, Inc. [http://www.fofweb.com/activelink2.asp?ItemID=WE40&SID=5&iPin=AZSSA0100&SingleRecord=True Olbers, Heinrich Wilhelm Matthäus] (accessed 2009-11-16)</ref> However, the weakening is not sufficient to actually explain Olbers' paradox. Many cosmologists think that the fact that the Universe is finite in time, that is that the Universe has not been around forever, is the solution to the paradox.{{citation needed|date=April 2013}} The fact that the night sky is dark is thus an indication for the Big Bang.
+==Translation invariance==
+{{main|translational invariance}}
+By [[translation (geometry)|translation]] invariance, one means independence of (absolute) position, especially when referring to a law of physics, or to the evolution of a physical system.
+Fundamental laws of physics should not (explicitly) depend on position in space. That would make them quite useless. In some sense, this is also linked to the requirement that experiments should be [[reproducible]].
+This principle is true for all laws of mechanics ([[Newton's laws]], etc.), electrodynamics, quantum mechanics, etc.
+In practice, this principle is usually violated, since one studies only a small subsystem of the universe, which of course "feels" the influence of rest of the universe. This situation gives rise to "external fields" (electric, magnetic, gravitational, etc.) which make the description of the evolution of the system depending on the position ([[potential well]]s, etc.). This only stems from the fact that the objects creating these external fields are not considered as (a "dynamical") part of the system.
+Translational invariance as described above is equivalent to [[translational invariance|shift invariance]] in [[system analysis]], although here it is most commonly used in [[linear]] systems, whereas in physics the distinction is not usually made.
+The notion of [[isotropy]], for properties independent of direction, is not a consequence of homogeneity. For example, a uniform electric field (i.e., which has the same strength and the same direction at each point) would be compatible with homogeneity (at each point physics will be the same), but not with [[isotropy]], since the field singles out one "preferred" direction.
+===Consequences===
+In [[Lagrangian]] formalism, homogeneity in space implies conservation of [[momentum]], and homogeneity in time implies conservation of [[energy]]. This is shown, using [[variational calculus]], in standard textbooks like the classical reference [Landau & Lifshitz] cited below. This is a particular application of [[Noether's theorem]].
+==Dimensional homogeneity==
+As said in the introduction, ''dimensional homogeneity'' is the quality of an equation having quantities of same units on both sides. A valid equation in [[physics]] must be homogeneous, since equality cannot apply between quantities of different nature. This can be used to spot errors in formula or calculations. For example, if one is calculating a [[speed]], units must always combine to [length]/[time]; if one is calculating an [[energy]], units must always combine to [mass]•[length]²/[time]², etc. For example, the following formulae could be valid expressions for some energy:
+:<math>E_k = \frac 12 m v^2 ;~~ E = m c^2 ;~~ E = p v ; ~~ E = hc/\lambda</math>
+if ''m'' is a mass, ''v'' and ''c'' are velocities, ''p'' is a [[momentum]], ''h'' is [[Planck's constant]], ''λ'' a length. On the other hand, if the units of the [[right hand side]] do not combine to [mass]•[length]<sup>2</sup>/[time]<sup>2</sup>, it cannot be a valid expression for some energy.
+Being homogeneous does not necessarily mean the equation will be true, since it does not take into account numerical factors. For example, ''E = m•v<sup>2</sup>'' could be or could not be the correct formula for the energy of a particle of mass ''m'' traveling at speed ''v'', and one cannot know if ''h•c''/λ should be divided or multiplied by 2π.
+Nevertheless, this is a very powerful tool in finding characteristic units of a given problem, see [[dimensional analysis]].
+[[Theoretical physicist]]s tend to express everything in [[natural units]] given by [[constants of nature]], for example by taking ''c'' = ''ħ'' = ''k'' = 1; once this is done, one partly loses the possibility of the above checking.
+==See also==
+*[[Translational invariance]]
+==References==
+{{reflist|2}}
+==Further reading==
+*Landau - Lifschitz: "Theoretical Physics - I. Mechanics", Chapter One.
+{{DEFAULTSORT:Homogeneity (Physics)}}
+[[Category:Dimensional analysis]]
+[[Category:Concepts in physics]]

Happy Ending problem: Difference between revisions

Revision as of 02:18, 27 July 2013

Contents

Context

Homogeneous alloy

Homogeneous cosmology

Translation invariance

Consequences

Dimensional homogeneity

See also

References

Further reading

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Happy Ending problem: Difference between revisions

Revision as of 02:18, 27 July 2013

Context

Homogeneous alloy

Homogeneous cosmology

Translation invariance

Consequences

Dimensional homogeneity

See also

References

Further reading

Navigation menu

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