formulasearchengine - User contributions [en]

Potential of mean force

2014-12-18T09:46:21Z

160.45.66.55: simulation is different from experiment (is different from theory)

Anxiety is a part of everyone's day-to-day life, and is associated with all possible events and instances. In extreme cases, catatonic posturing is seen. Everyone tells you take a deep breath and calm down for everything else. Panic disorder Your children unexpectedly panics, or have and anxiety attack for no reason at all. This is maybe against to what you have already learned about your condition. Panic disorders are a problem of the mind. The work performed in therapy through behavior modification and the effects of medicine, has allowed suffers to manage and not live in constant fear of when the next attack will happen. Many people have just one attack and never have another. There are 6 kind of [http://treasureflorida.com/bookmarks/view/5412/the-true-way-of-panic-attack-treatment-during-pregnancy cure anxiety attacks] disorder with each one of them has its own typical characteristics. I would strongly state, face it and get rid of depression with the help of your family members and loved ones. Often the most intense symptom during an active attack is the fear that you're going to go crazy or are about to die. It is [http://mateistanbul.com/blogs/post/26535 Web Page] important that these symptoms are treated early on so as to ensure that the attacks don't lead to panic disorder symptoms. If your child is prescribed Xanax, be sure to offer no more than the doctor has prescribed. 6: Remember that you can not hurt anxiety attacks . Where they become particularly troubling is when the same thoughts continually return, causing us to dwell on them constantly, time and time again. 

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Microwave cavity

2014-07-30T17:02:25Z

160.45.20.112: /* Rectangular cavity */

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Heegaard splitting

2014-05-21T14:21:09Z

160.45.112.75: /* References */

My name is Xavier and I am studying Dramatic Literature and History and Athletics and Physical Education at Wyssachen / Switzerland. Also visit my web page: [http://free-minecraft-project.hackservices.com free minecraft project]

Geometric median

2014-01-21T18:46:42Z

160.45.40.209: /* Implicit formula */

The '''MEMO Model''' (version 6.2) is a [[Lagrangian and Eulerian coordinates|Eulerian]] non-hydrostatic prognostic [[mesoscale meteorology|mesoscale]] model for wind flow simulation. It was developed by the [[Aristotle University of Thessaloniki]] in collaboration with the [[Universität Karlsruhe]]. The MEMO Model together with the photochemical [[Atmospheric dispersion modeling|dispersion model]] MARS are the two core models of the [[Europe]]an Zooming Model (EZM). This model belongs to the family of models designed for describing atmospheric transport phenomena in the local-to-regional scale, frequently referred to as mesoscale [[air pollution]] models.

==History==
Initially, EZM was developed for modelling the transport and chemical transformation of pollutants in selected European regions in the frame of the EUROTRAC sub-project EUMAC and therefore it was formerly called the EUMAC Zooming Model (EUROTRAC, 1992). EZM has evolved to be one of the most frequently applied mesoscale air pollution model systems in Europe. It has been already successfully applied for various European airsheds including the [[Upper Rhine]] valley and the areas of [[Basel]], [[Graz]], [[Barcelona]], [[Lisbon]], [[Madrid]], [[Milano]], [[London]], [[Cologne]], [[Lyon]], [[The Hague]], [[Athens]] (Moussiopoulos, 1994; Moussiopoulos, 1995) and [[Thessaloniki]]. More details are to be found elsewhere (Moussiopoulos 1989), (Flassak 1990), (Moussiopoulos et al. 1993).

== Model equations ==
The prognostic mesoscale model MEMO describes the dynamics of the [[Planetary boundary layer|atmospheric boundary layer]]. In the present model version, air is assumed to be unsaturated. The model solves the [[continuity equation]], the [[momentum]] equations and several [[transport equation]]s for scalars (including the thermal energy equation and, as options, transport equations for water vapour, the turbulent kinetic energy and [[pollutant]] concentrations).

== Transformation to terrain-following coordinates ==
The lower boundary of the model domain coincides with the ground. Because of the inhomogeneity of the terrain, it is not possible to impose boundary conditions at that boundary with respect to [[Cartesian coordinate system|Cartesian coordinates]]. Therefore, a transformation of the vertical coordinate to a terrain-following one is performed. Hence, the originally irregularly bounded physical domain is mapped onto one consisting of unit cubes.

== Numerical solution of the equation system ==
The discretized equations are solved numerically on a staggered grid, i.e. the scalar quantities <math>\rho</math>, <math>p</math> and <math>\theta</math> are defined at the cell centre while the velocity components <math>u</math>, <math>v</math> and <math>w</math> are defined at the centre of the appropriate interfaces.

Temporal discretization of the prognostic equations is based on the explicit second order [[Linear multistep method#Adams.E2.80.93Bashforth methods|Adams-Bashforth scheme]]. There are two deviations from the Adams-Bashforth scheme: The first refers to the implicit treatment of the nonhydrostatic part of the mesoscale pressure perturbation <math>p_{{nh}}</math>. To ensure non-divergence of the flow field, an elliptic equation is solved. The elliptic equation is derived from the continuity equation wherein velocity components are expressed in terms of <math>p_{{nh}}</math>. Since the elliptic equation is derived from the discrete form of the continuity equation and the discrete form of the pressure gradient, conservativity is guaranteed (Flassak and Moussiopoulos, 1988). The discrete pressure equation is solved numerically with a fast elliptic solver in conjunction with a generalized conjugate gradient method. The fast elliptic solver is based on fast [[Fourier analysis]] in both horizontal directions and [[Gaussian elimination]] in the vertical direction (Moussiopoulos and Flassak, 1989).

The second deviation from the explicit treatment is related to the turbulent diffusion in vertical direction. In case of an explicit treatment of this term, the stability requirement may necessitate an unacceptable abridgement of the time increment. To avoid this, vertical turbulent diffusion is treated using the second order [[Crank–Nicolson method]].

On principle, advective terms can be computed using any suitable advection scheme. In the present version of MEMO, a 3-D second-order total-variation-diminishing (TVD) scheme is implemented which is based on the 1-D scheme proposed by Harten (1986). It achieves a fair (but not any) reduction of numerical diffusion, the solution being independent of the magnitude of the scalar (i.e., preserving transportivity).

== Parameterizations ==
Turbulence and radiative transfer are the most important physical processes that have to be parameterized in a prognostic mesoscale model. In the MEMO model, radiative transfer is calculated with an efficient scheme based on the emissivity method for longwave radiation and an implicit multilayer method for shortwave radiation (Moussiopoulos 1987).

The diffusion terms may be represented as the divergence of the corresponding fluxes. For turbulence parameterizations, K-theory is applied. In case of MEMO turbulence can be treated either with a zero-, one- or two-equation turbulence model. For most applications a one-equation model is used, where a conservation equation for the turbulent kinetic energy is solved.

== Initial and boundary conditions ==
In MEMO, initialization is performed with suitable diagnostic methods: a mass-consistent initial wind field is formulated using an objective analysis model and scalar fields are initialized using appropriate interpolating techniques (Kunz, R., 1991). Data needed to apply the diagnostic methods may be derived either from observations or from larger scale simulations.

Suitable boundary conditions have to be imposed for the wind velocity components <math>u</math>, <math>v</math> and <math>w</math>, the potential temperature <math>\theta</math> and pressure <math>p</math> at all boundaries. At open boundaries, wave reflection and deformation may be minimized by the use of so-called ''radiation conditions'' (Orlanski 1976).

According to the experience gained so far with the model MEMO, neglecting large scale environmental information might result in instabilities in case of simulations over longer time periods.

For the nonhydrostatic part of the mesoscale pressure perturbation, homogeneous [[Neumann boundary condition]]s are used at lateral boundaries. With these conditions, the wind velocity component perpendicular to the boundary remains unaffected by the pressure change.

At the upper boundary, Neumann boundary conditions are imposed for the horizontal velocity components and the potential temperature. To ensure non-reflectivity, a radiative condition is used for the hydrostatic part of the mesoscale pressure perturbation
<math>p_h</math> at that boundary. Hence, vertically propagating internal gravity waves are allowed to leave the computational domain (Klemp and Durran 1983). For the nonhydrostatic part of the mesoscale pressure perturbation, homogeneous staggered [[Dirichlet conditions]] are imposed. Being justified by the fact that nonhydrostatic effects are negligible at large heights, this condition is necessary, if singularity of the elliptic pressure equation is to be avoided in view of the Neumann boundary conditions at all other boundaries.

The lower boundary coincides with the ground (or, more precisely, a height above ground corresponding to its aerodynamic roughness). For the nonhydrostatic part of the mesoscale pressure perturbation, inhomogeneous Neumann conditions are imposed at that boundary. All other conditions at the lower boundary follow from the assumption that the [[Monin-Obukhov Length|Monin-Obukhov]] similarity theory is valid.

The one way interactive nesting facility is possible within MEMO. Thus, successive simulations on grids of increasing resolution are possible. During these simulations, the results of the application to a coarse grid are used as boundary conditions for the application to the finer grid (Kunz and Moussiopoulos, 1995).

== Grid definition ==
The governing equations are solved numerically on a staggered grid. Scalar quantities as the temperature, pressure, density and also the cell volume are defined at the centre of a grid cell and the velocity components <math>u</math>, <math>v</math> and <math>w</math> at the centre of the appropriate interface. Turbulent fluxes are defined at different locations: Shear fluxes are defined at the centre of the appropriate edges of a grid cell and normal stress fluxes at scalar points. With this definition, the outgoing fluxes of momentum, mass, heat and also turbulent fluxes of a grid cell are identical to incoming flux of the adjacent grid cell. So the numerical method is conservative.

== Topography and surface type ==
For calculations with MEMO, a file must be provided which contains orography height and surface type for each grid location The following surface types are distinguished and must be stored as percentage:

* water (type: 1)
* arid land (type: 2)
* few vegetation (type: 3)
* farmland (type: 4)
* forest (type: 5)
* suburban area (type: 6)
* urban area (type: 7)

Only surface types 1 to 6 have to be stored. Type 7 is the difference between 100% and the sum of types 1 to 6. If the percentage of a surface type is 100%, then write the number 10 and for all other surface types the number 99.

The [[orography]] height is the mean height for each grid location above sea level in meter.

== Meteorological data ==
The prognostic model MEMO is a set of partial differential equations in three spatial directions and in time. To solve these equations, information about the initial state in the whole domain and about the development of all relevant quantities at the lateral boundaries is required.

===Initial state===
To generate an initial state for the prognostic model, a diagnostic model (Kunz, R., 1991) is applied using measured temperature and wind data. Both data can be provided as:
* surface measurements i.e. single measurements directly above the surface (not necessary)
* upper air soundings (i.e., soundings that consist of two or more measurements at different heights) at a constant geographical location is required (with at least one sounding for temperature and wind velocity).

===Time-dependent boundary conditions===
Information about quantities at the lateral boundaries can be taken into account as surface measurements and upper air soundings. Therefore, a key word and the time when boundary data is given must occur in front of a set of boundary information.

== Nesting facility ==
In MEMO, a one-way interactive nesting scheme is implemented. With this nesting scheme a coarse grid and a fine grid simulation can be nested. During the coarse grid simulation, data is interpolated and written to a file. A consecutive fine grid simulation uses this data as lateral boundary values.

== See also ==

*[[Bibliography of atmospheric dispersion modeling]]
*[[Atmospheric dispersion modeling]]
*[[List of atmospheric dispersion models]]
*[[Air pollution dispersion terminology]]
*[[Useful conversions and formulas for air dispersion modeling]]

== References ==

* EUROTRAC (1992), Annual Report 1991, Part 5.
* Flassak, Th. and Moussiopoulos, N. (1988), ''Direct solution of the Helmholtz equation using Fourier analysis on the CYBER 205'', Environmental Software 3, 12–16.
* Harten, A. (1986), ''On a large time-step high resolution scheme'', Math. Comp. 46, 379–399.
* Klemp, J.B. and Durran, D.R. (1983), ''An upper boundary condition permitting internal gravity wave radiation in numerical mesoscale models'', Mon. Weather Rev.111, 430–444.
* Kunz, R. (1991), ''Entwicklung eines diagnostischen Windmodells zur Berechnung des Anfangszustandes fόr das dynamische Grenzschichtmodell MEMO'', Diplomarbeit Universitδt Karlsruhe.
* Kunz R. and Moussiopoulos N. (1995), ''Simulation of the wind field in Athens using refined boundary conditions'', Atmos. Environ. 29, 3575–3591.
* Moussiopoulos, N. (1987), ''An efficient scheme to calculate radiative transfer in mesoscale models'', Environmental Software 2, 172–191.
* Moussiopoulos, N. (1989), ''Mathematische Modellierung mesoskaliger Ausbreitung in der Atmosphδre'', Fortschr.-Ber. VDI, Reihe 15, Nr. 64, pp. 307.
* Moussiopoulos N., ed. (1994), ''The EUMAC Zooming Model (EZM): Model Structure and Applications'', EUROTRAC Report, 266 pp.
* Moussiopoulos N. (1995), ''The EUMAC Zooming Model, a tool for local-to-regional air quality studies'', Meteorol. Atmos. Phys. 57, 115–133.
* Moussiopoulos, N. and Flassak, Th. (1989), ''A fully vectorized fast direct solver of the Helmholtz equation'' in ''Applications of supercomputers in engineering: Algorithms, computer systems and user experience'', Brebbia, C.A. and Peters, A. (editors), Elsevier, Amsterdam 67–77.
* Moussiopoulos, N., Flassak, Th., Berlowitz, D., Sahm, P. (1993), ''Simulations of the Wind Field in Athens With the Nonhydrostatic Mesoscale Model MEMO'', Environmental Software 8, 29–42.
* Orlanski, J. (1976), ''A simple boundary condition for unbounded hyperbolic flows'', J. Comput. Phys. 21, 251–269.

==External links==
* [http://air-climate.eionet.eu.int/databases/MDS/index_html Model Documentation System]
* [http://air-climate.eionet.europa.eu/ European Topic Centre on Air and Climate Change (ETC/ACC)]
* [http://pandora.meng.auth.gr/mds/showlong.php?id=20 Long description of model 'MEMO']

{{Atmospheric, Oceanographic and Climate Models}}

[[Category:Atmospheric dispersion modeling]]

[[Category:Environmental engineering]]
[[Category:Environmental science software]]
[[Category:Numerical climate and weather models]]
[[Category:Wind]]

JPEG

2014-01-21T14:29:44Z

160.45.40.209: /* Patent issues */

In [[physics]], '''jerk''', also known as '''jolt''', '''surge''', or '''lurch''', is the rate of change of [[acceleration]]; that is, the [[derivative]] of acceleration with respect to time, the [[second derivative]] of [[velocity]], or the third derivative of [[position (vector)|position]]. Jerk is defined by any of the following equivalent expressions:

:<math>\vec j=\frac {\mathrm{d} \vec a} {\mathrm{d}t}=\frac {\mathrm{d}^2 \vec v} {\mathrm{d}t^2}=\frac {\mathrm{d}^3 \vec r} {\mathrm{d}t^3}</math>
where
:<math>\vec a</math> is acceleration,
:<math>\vec v</math> is velocity,
:<math>\vec r</math> is position,
:<math>\mathit{t}</math> is time.

Jerk is a vector, and there is no generally used term to describe its scalar magnitude (e.g., "speed" as the scalar magnitude for velocity).

The [[SI base unit|SI units]] of jerk are [[metre]]s per [[second]] cubed (metres per second per second per second, m/s3, or m·s−3)=100 [[Gal (unit)|Gal]] per second in [[CGS]] units. There is no universal agreement on the symbol for jerk, but ''j'' is commonly used. [[Newton's notation]] for the derivative of acceleration <math>(\dot{a})</math> can also be used, especially when "surge" or "lurch" is used instead of "jerk" or "jolt".

If acceleration can be felt by a body as the force (hence pressure) exerted by the object bringing about the acceleration on the body, jerk can be felt as the change in this pressure. For example a passenger in an accelerating vehicle with zero jerk will feel a constant force from the seat on his or her body; whereas positive jerk will be felt as increasing force on the body, and negative jerk as decreasing force on the body.

== Applications ==
Normally concerning forces, speed and acceleration are used for analysis. For example, the "jerk" produced by falling from [[outer space]] to the Earth is not particularly useful given the [[gravitational acceleration]] changes very slowly. Sometimes the analysis has to extend to jerk for a particular reason.

Jerk is often used in [[engineering]], especially when building [[roller coaster]]s.<ref name="thetartan2007">{{cite web|url=http://thetartan.org/2007/4/16/scitech/work |title=How Things Work: Roller Coasters - The Tartan Online |publisher=Thetartan.org |date=2007-04-16 |accessdate=2013-09-15}}</ref> Some precision or fragile objects — such as passengers, who need time to sense stress changes and adjust their muscle tension or suffer conditions such as [[Whiplash (medicine)|whiplash]] — can be safely subjected not only to a maximum acceleration, but also to a maximum jerk.<ref name="thetartan2007"/> Even where occupant safety isn't an issue, excessive jerk may result in an uncomfortable ride on [[elevator]]s, [[tram]]s and the like, and engineers expend considerable design effort to minimize it. Jerk may be considered when the excitation of vibrations is a concern. A device that measures jerk is called a "jerkmeter".

Jerk is also important to consider in [[manufacturing]] processes. Rapid changes in acceleration of a cutting tool can lead to premature tool wear and result in uneven cuts. This is why modern motion controllers include jerk limitation features.

In mechanical engineering, jerk is considered, in addition to velocity and acceleration, in the development of cam profiles because of [[tribology|tribological]] implications and the ability of the actuated body to follow the cam profile without [[Machining vibrations|chatter]].<ref>Blair, G., "Making the Cam", ''Race Engine Technology '' 10, September/October 2005</ref>

===Third-order motion profile===
In motion control, a common need is to move a system from one steady position to another (point-to-point motion). Following the fastest possible motion within an allowed maximum value for speed, acceleration, and jerk, will result in a third-order motion profile as illustrated below:

[[File:Third-order motion profile.svg]]

The motion profile consists of up to seven segments defined by the following:<ref>There is an idealization here that the jerk can be changed from zero to a constant non-zero value instantaneously. However, since in [[classical mechanics]] all forces are caused by [[smooth function|smooth]] [[field (physics)|fields]], all derivatives of the position are continuous. On the other hand, this is also an idealization; in [[quantum field theory]] particles do change momentum discontinuously.</ref>
# acceleration build-up, with maximum positive jerk
# constant maximum acceleration (zero jerk)
# acceleration ramp-down, approaching the desired maximum velocity, with maximum negative jerk
# constant maximum speed (zero jerk, zero acceleration)
# deceleration build-up, approaching the desired deceleration, with maximum negative jerk
# constant maximum deceleration (zero jerk)
# deceleration ramp-down, approaching the desired position at zero velocity, with maximum positive jerk
If the initial and final positions are sufficiently close together, the maximum acceleration or maximum velocity may never be reached.

=== Jerk systems ===
A ''jerk system'' is a system whose behavior is described by a ''jerk equation'', which is an equation of the form {{harv|Sprott|2003}}:

:<math>\frac{\mathrm{d}^3 x}{\mathrm{d} t^3}=
f\left(\frac{\mathrm{d}^2 x}{\mathrm{d} t^2},\frac{\mathrm{d} x}{\mathrm{d} t},x\right)
</math>

For example, certain simple electronic circuits may be designed which are described by a jerk equation. These are known as ''jerk circuits''.

One of the most interesting properties of jerk systems is the possibility of [[chaos theory|chaotic behavior]]. In fact, certain well-known chaotic systems, such as the [[Lorenz attractor]] and the [[Rössler map]], are conventionally described as a system of three first-order differential equations, but which may be combined into a single (although rather complicated) jerk equation.

An example of a jerk equation is:
:<math>
\frac{\mathrm{d}^3 x}{\mathrm{d} t^3}+A\frac{\mathrm{d}^2 x}{\mathrm{d} t^2}+\frac{\mathrm{d} x}{\mathrm{d} t}-|x|+1=0.
</math>

Where ''A'' is an adjustable parameter. This equation has a chaotic solution for A=3/5 and can be implemented with the following jerk circuit:

[[Image:JerkCircuit01.png|frameless|upright=1.4|center]]

In the above circuit, all resistors are of equal value, except <math>R_A=R/A=5R/3</math>, and all capacitors are of equal size. The dominant frequency will be <math>1/2\pi R C</math>. The output of [[operational amplifier|op amp]] 0 will correspond to the x variable, the output of 1 will correspond to the first derivative of x and the output of 2 will correspond to the second derivative.

==Non-calculus explanation==
{{unreferenced section|date=July 2013}}
Jerk can be difficult to conceptualize when it is defined in terms of calculus. When a [[force]] (push or pull) is applied to an object, that object starts to move. As long as the force is applied, the object will continue to speed up. When described in these terms, we are oversimplifying slightly. We think along the lines that there is no force on the object, then suddenly there is a force on the object. We do not think about how long it takes to apply the force.

However, in truth, the application of force does not instantly happen. A change always happens over time. Jerk is the change in acceleration over time. Typically, the time of contact where a force is applied is a split second.

If you push on a wall, it takes a fraction of a second before you apply the full push. Your fingertips will [[wikt:squoosh|squoosh]] slightly as you begin to push. How long the squooshing takes determines the jerk. If you push on a wall very slowly, you can actually feel your push increasing. In such a case, the jerk is very low, because the change in force is happening over a relatively long time of several seconds. Jerk happens when a force is applied and removed. But the whole time a force is acting consistently on an object, there is no jerk. (This is because the acceleration is constant when there is a constant force.)

How quickly the force starts its push or pull determines the yank and subsequently the jerk. In most applications, it is not important how quickly the force is applied, and thus we typically think of forces being applied instantaneously. A familiar example of jerk is the rate of application of brakes in an automobile.

[[Image:Jerk Diagram.png|frameless|upright=1.4|center]]

An experienced driver gradually applies the brakes, causing a slowly increasing deceleration (small jerk). An inexperienced driver, or a driver responding to an emergency, applies the brakes suddenly, causing a rapid increase in deceleration (large jerk). The sensation of jerk is noticeable, causing the passenger’s head to jerk forward.

=== Equations ===

* <math>Y = \frac{\Delta F}{\Delta t}</math> (yank: force per unit time)

* <math>j = \frac{\Delta a}{\Delta t}</math> (jerk: acceleration per unit time)

* <math>j = \frac{y}{m}</math>, following from [[Newton's second law]] divided by <math>\Delta t</math> and the above two relations: <math>\frac{\Delta F}{\Delta t} = m \frac{\Delta a}{\Delta t} \implies y = m j</math>

* <math>j = \frac{\Delta F}{m \Delta t}</math>

The higher the force or acceleration, the higher the jerk. The shorter the time of change in acceleration, such as a rollercoaster 'whipping' around a corner, the higher the jerk.
For uniform jerk, the following equation can be applied:

* <math>a = a. + jt </math>
* <math>v = u + a.t + \frac{1}{2}jt^2 </math>
* <math>s = ut + \frac{1}{2}a.t^2 + \frac{1}{6}jt^3 </math>

where
a : final acceleration
a. : initial acceleration
j : jerk (change in acceleration)
v : final velocity
u : initial velocity
s : distance/displacement
t : time taken

==See also==
* [[Abraham–Lorentz force]], a force in electrodynamics whose magnitude is proportional to jerk
* [[Shock (mechanics)]]
* [[Wheeler-Feynman absorber theory]]

==Notes==
{{Reflist}}

==References==
*{{cite book | author = Sprott JC | title = Chaos and Time-Series Analysis | publisher = Oxford University Press | year = 2003 | isbn = 0-19-850839-5 |ref=CITEREFSprott2003 }}
*{{cite journal |title=Some simple chaotic jerk functions |url=http://sprott.physics.wisc.edu/pubs/paper229.pdf |format=PDF |accessdate=2009-09-28 |doi=10.1119/1.18585 |journal=Am J Phys |volume=65 |issue=6 |pages=537–43 |year=1997 |author=Sprott JC |bibcode = 1997AmJPh..65..537S }}
*{{cite journal |title=Making the Cam |url=http://www.profblairandassociates.com/pdfs/Camshaft%20RET%20010.pdf |format=PDF |accessdate=2009-09-29 |journal=Race Engine Technology |issue=010 |year=2005 |author=Blair G }}

==External links==
*[http://math.ucr.edu/home/baez/physics/General/jerk.html What is the term used for the third derivative of position?], description of jerk in the [http://math.ucr.edu/home/baez/physics/index.html Usenet Physics FAQ].
*[http://www.pmdcorp.com/news/articles/html/Mathematics_of_Motion_Control_Profiles.cfm Mathematics of Motion Control Profiles]

{{Kinematics}}

{{DEFAULTSORT:Jerk (Physics)}}
[[Category:Classical mechanics]]
[[Category:Physical quantities]]
[[Category:Acceleration]]
[[Category:Time]]

Object theory

2013-05-21T12:07:26Z

160.45.152.6: /* See also */ : Adding Metatheory, Object language.

In [[mathematics]], the '''(formal) complex conjugate''' of a [[complex numbers|complex]] [[vector space]] <math>V\,</math> is the complex vector space <math>\overline V</math> consisting of all formal [[complex conjugate]]s of elements of <math>V\,</math>. That is, <math>\overline V</math> is a vector space whose elements are in [[bijection|one-to-one correspondence]] with the elements of <math>V\,</math>:
:<math>\overline V = \{\overline v \mid v \in V\},</math>
with the following rules for [[addition]] and [[scalar multiplication]]:
:<math>\overline v + \overline w = \overline{\,v+w\,}\quad\text{and}\quad\alpha\,\overline v = \overline{\,\overline \alpha \,v\,}.</math>
Here <math>v\,</math> and <math>w\,</math> are vectors in <math>V\,</math>, <math>\alpha\,</math> is a complex number, and <math>\overline\alpha</math> denotes the complex conjugate of <math>\alpha\,</math>.

More concretely, the complex conjugate vector space is the same underlying ''real'' vector space (same set of points, same vector addition and real scalar multiplication) with the conjugate [[linear complex structure]] ''J'' (different multiplication by ''i'').

==Antilinear maps==
If <math>V\,</math> and <math>W\,</math> are complex vector spaces, a function <math>f\colon V \to W\,</math> is [[antilinear]] if
:<math>f(v+v') = f(v) + f(v')\quad\text{and}\quad f(\alpha v) = \overline\alpha \, f(v)</math>
for all <math>v,v'\in V\,</math> and <math>\alpha\in\mathbb{C}</math>.

One reason to consider the vector space <math>\overline V</math> is that it makes antilinear maps into [[linear map]]s. Specifically, if <math>f\colon V \to W\,</math> is an antilinear map, then the corresponding map <math>\overline V \to W</math> defined by
:<math>\overline v \mapsto f(v)</math>
is linear. Conversely, any linear map defined on <math>\overline V</math> gives rise to an antilinear map on <math>V\,</math>.

One way of thinking about this correspondence is that the map <math>C\colon V \to \overline V</math> defined by
:<math>C(v) = \overline v</math>
is an antilinear bijection. Thus if <math>f\colon \overline V \to W</math> is linear, then [[Function composition|composition]] <math>f \circ C\colon V \to W\,</math> is antilinear, and ''vice versa''.

==Conjugate linear maps==
Any linear map <math>f \colon V \to W\,</math> induces a '''conjugate linear map''' <math>\overline f \colon \overline V \to \overline W</math>, defined by the formula
:<math>\overline f (\overline v) = \overline{\,f(v)\,}.</math>
The conjugate linear map <math>\overline f</math> is linear. Moreover, the [[identity function|identity map]] on <math>V\,</math> induces the identity map <math>\overline V</math>, and
:<math>\overline f \circ \overline g = \overline{\,f \circ g\,}</math>
for any two linear maps <math>f\,</math> and <math>g\,</math>. Therefore, the rules <math>V\mapsto \overline V</math> and <math>f\mapsto\overline f</math> define a [[functor]] from the [[category theory|category]] of complex vector spaces to itself.

If <math>V\,</math> and <math>W\,</math> are finite-dimensional and the map <math>f\,</math> is described by the complex [[matrix (mathematics)|matrix]] <math>A\,</math> with respect to the [[basis of a vector space|bases]] <math>\mathcal B</math> of <math>V\,</math> and <math>\mathcal C</math> of <math>W\,</math>, then the map <math>\overline f</math> is described by the complex conjugate of <math>A\,</math> with respect to the bases <math>\overline{\mathcal B}</math> of <math>\overline V</math> and <math>\overline{\mathcal C}</math> of <math>\overline W</math>.

==Structure of the conjugate==
The vector spaces <math>V\,</math> and <math>\overline V</math> have the same [[dimension of a vector space|dimension]] over the complex numbers and are therefore [[isomorphism|isomorphic]] as complex vector spaces. However, there is no [[natural isomorphism]] from <math>V\,</math> to <math>\overline V</math>. (The map <math>C\,</math> is not an isomorphism, since it is antilinear.)

The double conjugate <math>\overline{\overline V}</math> is naturally isomorphic to <math>V\,</math>, with the isomorphism <math>\overline{\overline V} \to V</math> defined by
:<math>\overline{\overline v} \mapsto v.</math>
Usually the double conjugate of <math>V\,</math> is simply identified with <math>V\,</math>.

== Complex conjugate of a Hilbert space ==
Given a [[Hilbert space]] <math>\mathcal{H}</math> (either finite or infinite dimensional), its complex conjugate <math>\overline{\mathcal{H}}</math> is the same vector space as its [[continuous dual space]] <math>\mathcal{H}'</math>.
There is one-to-one antilinear correspondence between continuous linear functionals and vectors.
In other words, any continuous [[linear functional]] on <math>\mathcal{H}</math> is an inner multiplication to some fixed vector, and vice versa.

Thus, the complex conjugate to a vector <math>v</math>, particularly in finite dimension case, may be denoted as <math>v^*</math> (v-star, a [[row vector]] which is the [[conjugate transpose]] to a column vector <math>v</math>).
In quantum mechanics, the conjugate to a ''ket vector'' <math>|\psi\rangle</math> is denoted as <math>\langle\psi|</math> – a ''bra vector'' (see [[bra-ket notation]]).

==See also==
* [[Linear complex structure]]

==References==
* Budinich, P. and Trautman, A. ''The Spinorial Chessboard''. Spinger-Verlag, 1988. ISBN 0-387-19078-3. (complex conjugate vector spaces are discussed in section 3.3, pag. 26).

[[Category:Linear algebra]]
[[Category:Vectors|Vector space]]

Double layer forces

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160.45.33.182:

The '''Meissner equation''' is a linear [[ordinary differential equation]] that is a special case of [[Hill differential equation|Hill's equation]] with the periodic function given as a square wave.<ref>{{cite book |
title=Analysis of periodically time-varying systems |
author=Richards, J. A. |
isbn=9783540116899 |
lccn=82005978 |
year=1983 |
publisher=Springer-Verlag
}} </ref> <ref>
{{cite article | author=E. Meissner |
title=Ueber Schüttelerscheinungen in Systemen mit periodisch veränderlicher Elastizität |
journal=Schweiz. Bauzeit. |
volume=72 |
number=11 |
year=1918 |
pages=95–98
}}
</ref> There are many ways to write the Meissner equation. One
is as

: <math> \frac{d^2y}{dt^2} + (\alpha^2 + \omega^2 \sgn \cos(t)) = 0 </math>

or

: <math> \frac{d^2y}{dt^2} + ( 1 + r f(t;a,b) ) y = 0 </math>

where
: <math> f(t;a,b) = -1 + 2 H_a( t \mod (a+b) ) </math>
and <math> H_c(t) </math> is the Heaviside function shifted to <math>c</math>. Another version is

: <math> \frac{d^2y}{dt^2} + \left( 1 + r \frac{\sin( \omega t)}{|\sin(\omega t)|} \right) y = 0. </math>

The Meissner equation was first studied as a toy problem for certain resonance problems. It is also useful for understand resonance problems in evolutionary biology.

Because the time-dependence is piecewise linear, many calculations can be performed exactly, unlike for the [[Mathieu equation]]. When <math> a = b = 1</math>, the [[Floquet theory|Floquet exponents]] are roots of the quadratic equation

: <math> \lambda^2 - 2 \lambda \cosh(\sqrt{r}) \cos(\sqrt{r}) + 1 = 0 .</math>

The determinant of the Floquet matrix is 1, implying that origin is a center if <math> |\cosh(\sqrt{r}) \cos(\sqrt{r})| < 1 </math>
and a saddle node otherwise.

== References ==
{{reflist}}

[[Category:Ordinary differential equations]]

Axiom schema of replacement

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