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In [[mathematics]], a '''one-parameter group''' or '''one-parameter subgroup''' usually means a [[continuous (topology)|continuous]] [[group homomorphism]]
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:φ : '''R''' → ''G''
 
from the [[real line]] '''R''' (as an [[Abelian group|additive group]]) to some other [[topological group]] ''G''. That means that it is not in fact a [[group (mathematics)|group]], strictly speaking; if φ is [[injective]] then φ('''R'''), the image, will be a subgroup of ''G'' that is isomorphic to '''R''' as additive group.  
 
==Discussion==
 
That is, we start knowing only that
 
:φ (''s'' + ''t'') = φ(''s'')φ(''t'')
 
where ''s'', ''t'' are the 'parameters' of group elements in ''G''. We may have  
 
:φ(''s'') = ''e'', the [[identity element]] in ''G'',
 
for some ''s'' ≠ 0. This happens for example if ''G'' is the [[Circle group|unit circle]] and
 
:φ(''s'') = ''e''<sup>''is''</sup>.
 
In that case the [[kernel (algebra)|kernel]] of φ consists of the integer multiples of 2π.
 
The [[action (group theory)|action]] of a one-parameter group on a set is known as a [[flow (mathematics)|flow]].
 
A technical complication is that φ('''R''') as a [[subspace topology|subspace]] of ''G'' may carry a topology that is [[finer topology|coarser]] than that on '''R'''; this may happen in cases where φ is injective. Think for example of the case where ''G'' is a [[torus]] ''T'', and φ is constructed by winding a straight line round ''T'' at an irrational slope.
 
Therefore a one-parameter group or one-parameter subgroup has to be distinguished from a group or subgroup itself, for the three reasons
#it has a definite [[parametrization]],
#the group homomorphism may not be injective, and
#the induced topology may not be the standard one of the real line.
 
==Examples==
Such one-parameter groups are of basic importance in the theory of [[Lie group]]s, for which every element of the associated [[Lie algebra]] defines such a homomorphism, the [[exponential map]]. In the case of matrix groups it is given by the [[matrix exponential]].
 
Another important case is seen in [[functional analysis]], with ''G'' being the group of [[unitary operator]]s on a [[Hilbert space]]. See [[Stone's theorem on one-parameter unitary groups]].
 
In his 1957 monograph ''Lie Groups'', [[P. M. Cohn]] gives the following theorem on page 58:
:Any connected 1-dimensional Lie group is analytically isomorphic either to  the additive group of real numbers <math> \mathfrak{R}</math>, or to <math> \mathfrak{T}</math>, the additive group of real numbers mod 1. In particular, every 1-dimensional Lie group is locally isomorphic to '''R'''.
 
==Physics==
In [[physics]], one-parameter groups describe [[dynamical systems]].<ref>Zeidler, E. ''Applied Functional Analysis: Main Principles and Their Applications''. Springer-Verlag, 1995.</ref> Furthermore, whenever a system of physical laws admits a one-parameter group of [[derivative|differentiable]] [[symmetry group|symmetries]], then there is a [[conservation law|conserved quantity]], by [[Noether's theorem]].
 
In the study of [[spacetime]] the use of the [[unit hyperbola]] to calibrate spacio-temporal measurements has become common since [[Hermann Minkowski]] discussed it in 1908. The [[principle of relativity]] was reduced to arbitrariness of which diameter of the unit hyperbola was used to determine a [[world-line]]. Using the parametrization of the hyperbola with [[hyperbolic angle]], the theory of [[special relativity]] provided a calculus of relative motion with the one-parameter group indexed by [[rapidity]]. The ''rapidity'' replaces the ''velocity'' in kinematics and dynamics of relativity theory. Since rapidity is unbounded, the one-parameter group it stands upon is non-compact. The rapidity concept was introduced by [[E.T. Whittaker]] in 1910, and named by [[Alfred Robb]] the next year. The rapidity parameter amounts to the length of a [[versor#Hyperbolic versor|hyperbolic versor]], a concept of the nineteenth century. Mathematical physicists [[James Cockle (lawyer)|James Cockle]], [[William Kingdon Clifford]], and [[Alexander Macfarlane]] had all employed in their writings an equivalent mapping of the Cartesian plane by operator (cosh ''a''&nbsp;+&nbsp;''r''&nbsp;sinh&nbsp;''a''), where ''a'' is the hyperbolic angle and ''r'' <sup>2</sup>&nbsp;=&nbsp;+1.
 
== See also ==
* [[Integral curve]]
* [[One-parameter semigroup]]
* [[Noether's theorem]]
 
==References==
<references/>
 
[[Category:Lie groups]]
[[Category:One]]
[[Category:Topological groups]]

Latest revision as of 01:37, 25 May 2014

Many folks have this habit of doing all the stuff by themselves, irrespective of how important or simple they are! These folks won't allow others interfere inside their matters. While this stance will function inside different regions of life, it is very really not the greatest way to reply when you want to fix the Windows registry. There are some jobs like removing spywares, virus and also obsolete registry entries, which are best left to professional softwares. In this particular article I will tell we why it happens to be important to fix Windows registry NOW!

You can reformat the computer to create it run quicker. This usually reset your computer to whenever you initially used it. Always remember to back up all files plus programs before doing this because this may remove a files from a database. Remember before we do this we want all of the motorists plus installation files plus this ought to be a last resort when you are shopping for slow computer strategies.

Over time the disk will furthermore get fragmented. Fragmentation causes your computer to slow down because it takes windows much longer to locate a files place. Fortunately, a PC has a built inside disk defragmenter. We can run this program by clicking "Start" - "All Programs" - "Accessories" - "System Tools" - "Disk Defragmenter". We may have the choice to choose that drives or partition you want to defragment. This action might take you several time thus it is very advised to do this on a regular basis thus because to avoid further fragmentation and to accelerate the windows XP computer.

Windows errors may be caused by any amount of factors, however, there's almost usually one cause. There's a hidden part of your program which is responsible for making 90% of all Windows errors, plus it's called the 'registry'. This really is the central database for your system and is where a computer shops all its system files and settings. It's a important piece of Windows, that is demands to be capable to function. However, it's furthermore one of the largest causes of problems on your PC.

The 2nd step to fixing these errors is to employ a program called a "tuneup utilities 2014" to scan by a computer and fix some of the registry mistakes which may additionally be leading to this error. A registry cleaner is a software program that might scan through the computer and repair any of the difficulties which Windows has inside, permitting the computer to "remember" all the settings it has whenever it loads up. Although the registry is continually being employed to help load up a large amount of programs on a PC, it's continually being saved incorrectly - leading to a large amount of errors to be formed. To fix this issue, it's suggested you download a registry cleaner from the Internet plus install it on the Pc, permitting Windows to run smoother again.

Active X controls are used across the whole spectrum of computer plus internet technologies. These controls are referred to as the building blocks of the internet plus because the glue that puts it all together. It is a standard that is selected by all programmers to create the internet more beneficial and interactive. Without these control standards there would basically be no public web.

Your registry is the spot all your significant configurations for hardware, software and consumer profile configurations and needs are stored. Every time 1 of these aspects is changed, the database then starts to expand. Over time, the registry will become bloated with unwanted files. This causes a general slow down however, in extreme cases could result significant jobs and programs to stop functioning all together.

Most people make the mistake of striving to fix Windows registry by hand. I strongly suggest we don't do it. Unless you are a computer expert, I bet we will invest hours plus hours learning the registry itself, let alone fixing it. And why could you waste the precious time inside understanding plus fixing something you understand nothing about? Why not let a smart plus professional registry cleaner do it for we? These software programs would be able to do the job inside a far better technique! Registry cleaners are quite affordable as well; you pay a once fee plus employ it forever. Additionally, many pro registry cleaners are surprisingly reliable and simple to use. If you need more information on how to fix Windows registry, merely see my website by clicking the link under!