Big O notation: Difference between revisions

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Undid revision 642215650 by Quietbritishjim (talk)Hopelessly confused explanation. An author "forced to make clear what variable the function depends on", simply writes the variable.
 
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'''Radius of gyration''' or '''gyradius''' is the name of several related measures of the size of an object, a surface, or an ensemble of points. It is calculated as the [[root mean square]] distance of the objects' parts from either its center of gravity or a given axis.
And how much of those feelings are caused by pressure from the society in general. Overall, most patients report long-term satisfaction with their implants even when they have to be re-operated for complications or aesthetic reasons. If you are taking breast enlargement pills that cause strange behaviour to your body, you must discontinue using it imediately. As a medicinal plant, Watercress has been traditionally considered a diuretic, expectorant, purgative, stimulant, stomachic and tonic. For some women a breast enlargement surgery alone is not enough. <br><br>In addition, problems include leak or rupture of the implant requiring additional surgery to change the previous one, pain, seroma (fluid retention), infections and a condition called capsular contracture. Yes, they all have well-endowed breasts, may it be open or artificially enhanced. Breast Massage can be used as natural breast cancer treatment. We come in all different shapes and sizes and we can only play the cards that. For bigger and developed figures, perfect brands should be applied, so that required target can be approached. <br><br>The progress of it also depends on how your body is taking the phytoestrogens that promotes hormones that is elemental to the growth and volume of your breasts. Every time you go back to the same bar, they are there waiting to expose their necks for your pleasure. Hold one weight in each hand and extended your arms out at shoulder level. These exercises are just part of the stay fit regime more and more people are embracing all over the world. Think about it, your breast is already stretched out of its magnitude so why would you want to stretch it even more using a breast enlargement pump. <br><br>whether it is done through breast enlargement creams and pills or breast surgery. Alternatively, with both hands holding the breast meat should be bent and very gently wrung. What the surgery does is that it makes them feel much younger. Estrogen-based breast augmentation products may cause a temporary enhancement as many women have discovered when using birth control pills. Though the desired size varies, the reason beneath remains the same—bigger than usual. <br><br>Using a breast enlargement cream is another way to stimulate breast growth, most breast creams contain an herb called Pueraria Mirifica which mimics natural estrogen production, when applied directly to the breast it not only stimulates breast growth making the breasts larger, it can also help women reduce sagging in their breasts while at the same time making them fuller and perkier. The patients own tissue is used to reconstruct the breast. If you use implants the surgery will be a combination of enlargement and lift. Once this is done, use a dry eye shadow brush and use the same color to apply more to the same spot so as to create that pop. You must have patience and wait for them to fade away naturally. <br><br>Spending your bucks on fruitless endeavors is never a great concept. Keep your ankles crossed while doing this exercise, breast enlargement, not to lift too much weight. The venous drainage of the breast is mainly to the axillary vein, but there is some drainage to the internal thoracic vein and the intercostal veins. Breast augmentation has become increasingly popular in recent times, and with celebrities who have had breast implants before and after pictures being splashed across gossip magazines on an almost weekly basis,  it seems more and more women are seeing it as a viable option when it comes to enhancing their appearance. Just be sure you keep these sessions because the probability of infection raises in case the drain stays in place longer.<br><br>If you beloved this report and you would like to obtain far more facts concerning how do i make my breasts bigger naturally ([http://www.lucky-house.info/ www.lucky-house.info]) kindly go to our own web-page.
 
==Applications in structural engineering==
In [[structural engineering]], the two-dimensional radius of gyration is used to describe the distribution of [[cross section (geometry)|cross sectional]] area in a column around its centroidal axis. The radius of gyration is given by the following formula 
 
:<math>R_{\mathrm{g}}^{2} = \frac{I}{A},</math>
or
:<math>R_{\mathrm{g}} = \sqrt{ \frac {I} {A} },</math>
 
where I is the [[second moment of area]] and ''A'' is the total cross-sectional area. The gyration radius is useful in estimating the stiffness of a column. However, if the principal moments of the two-dimensional [[gyration tensor]] are not equal, the column will tend to [[buckling|buckle]] around the axis with the smaller principal moment. For example, a column with an [[ellipse|elliptical]] cross-section will tend to buckle in the direction of the smaller semiaxis.
 
It also can be referred to as the radial distance from a given axis at which the mass of a body could be concentrated without altering the rotational inertia of the body about that axis.
 
In [[engineering]], where people deal with continuous bodies of matter, the radius of gyration is usually calculated as an integral.
 
==Applications in mechanics==
The radius of gyration about a given axis (<math>r_{\mathrm{g}\text{ axis}}</math>) can be computed in terms of the [[mass moment of inertia]] <math>I_\text{axis}</math> around that axis, and the total mass ''m'';
 
:<math>r_{\mathrm{g}\text{ axis}}^{2} = \frac{I_\text{axis}}{m},</math>
or
:<math>r_{\mathrm{g}\text{ axis}} = \sqrt{ \frac{I_\text{axis}}{m}},</math>
<math>I_\text{axis}</math> is a [[scalar (mathematics)|scalar]], and is not the moment of inertia [[tensor]].
<ref>
See for example
{{Citation | last = Goldstein | first = Herbert | author-link = Herbert Goldstein
| title = Classical Mechanics | place= Reading,  Massachusetts
| publisher = Addison-Wesley Publishing Company
| year = 1950 | edition = 1st }} equation 5-30 </ref>
 
==Molecular applications==
In [[polymer physics]], the radius of gyration is used to describe the dimensions of a [[polymer]] [[ideal chain|chain]]. The radius of gyration of a particular molecule at a given time is defined as:
:<math>
R_{\mathrm{g}}^{2} \ \stackrel{\mathrm{def}}{=}\  \frac{1}{N} \sum_{k=1}^{N} \left( \mathbf{r}_{k} - \mathbf{r}_{\mathrm{mean}} \right)^{2},
</math>
 
where <math>\mathbf{r}_{\mathrm{mean}}</math> is the [[mean]] position of the monomers.
As detailed below, the radius of gyration is also proportional to the root mean square distance between the monomers:
 
:<math>
R_{\mathrm{g}}^{2} \ \stackrel{\mathrm{def}}{=}\  \frac{1}{2N^{2}} \sum_{i,j}
\left( \mathbf{r}_{i} - \mathbf{r}_{j} \right)^{2}.
</math>
 
As a third method, the radius of gyration can also be computed by summing the principal moments of the [[gyration tensor]].
 
Since the chain [[Chemical structure|conformations]] of a polymer sample are quasi infinite in number and constantly change over time, the "radius of gyration" discussed in polymer physics must usually be understood as a mean over all polymer molecules of the sample and over time. That is, the radius of gyration which is measured is an ''average'' over time or [[Ensemble average|ensemble]]:
 
:<math>
R_{\mathrm{g}}^{2} \ \stackrel{\mathrm{def}}{=}\  \frac{1}{N} \langle \sum_{k=1}^{N} \left( \mathbf{r}_{k} - \mathbf{r}_{\mathrm{mean}} \right)^{2} \rangle,
</math>
 
where the angular brackets <math>\langle \ldots \rangle</math> denote the [[ensemble average]].
 
An entropically governed polymer chain (i.e. in so called theta conditions) follows a random walk in three dimensions. The radius of gyration for this case is given by
 
:<math>R_{\mathrm{g}} = \frac{1}{ \sqrt 6\ } \ \sqrt N\ a.</math>
 
Note that although <math>aN</math> represents the [[contour length]] of the polymer, <math>a</math> is strongly dependent of polymer stiffness and can vary over orders of magnitude. <math>N</math> is reduced accordingly.
 
One reason that the radius of gyration is an interesting property is that it can be determined experimentally with [[static light scattering]] as well as with [[Small-angle neutron scattering|small angle neutron-]] and [[Small-angle X-ray scattering|x-ray scattering]]. This allows theoretical polymer physicists to check their models against reality.
The [[hydrodynamic radius]] is numerically similar, and can be measured with [[Dynamic Light Scattering]] (DLS).
 
===Derivation of identity===
To show that the two definitions of <math>R_{\mathrm{g}}^{2}</math> are identical,
we first multiply out the summand in the first definition:
 
:<math>
R_{\mathrm{g}}^{2} \ \stackrel{\mathrm{def}}{=}\ 
\frac{1}{N} \sum_{k=1}^{N} \left( \mathbf{r}_{k} - \mathbf{r}_{\mathrm{mean}} \right)^{2} =
\frac{1}{N} \sum_{k=1}^{N} \left[ \mathbf{r}_{k} \cdot \mathbf{r}_{k} +
\mathbf{r}_{\mathrm{mean}} \cdot \mathbf{r}_{\mathrm{mean}}
- 2 \mathbf{r}_{k} \cdot \mathbf{r}_{\mathrm{mean}} \right].
</math>
 
Carrying out the summation over the last two terms and using the definition of <math>\mathbf{r}_{\mathrm{mean}}</math> gives the formula
 
:<math>
R_{\mathrm{g}}^{2} \ \stackrel{\mathrm{def}}{=}\ 
-\mathbf{r}_{\mathrm{mean}} \cdot \mathbf{r}_{\mathrm{mean}} +
\frac{1}{N} \sum_{k=1}^{N} \left( \mathbf{r}_{k} \cdot \mathbf{r}_{k} \right).
</math>
 
==Notes==
{{Reflist}}
 
==References==
* Grosberg AY and Khokhlov AR. (1994) ''Statistical Physics of Macromolecules'' (translated by Atanov YA), AIP Press. ISBN 1-56396-071-0
* Flory PJ. (1953) ''Principles of Polymer Chemistry'', Cornell University, pp. 428-429 (Appendix C o Chapter X).
 
==External links==
* [http://www.amesweb.info/SectionalPropertiesTabs/SectionalProperties.aspx Online Sectional Properties Calculator]
 
[[Category:Solid mechanics]]
[[Category:Polymer physics]]

Latest revision as of 01:45, 13 January 2015

And how much of those feelings are caused by pressure from the society in general. Overall, most patients report long-term satisfaction with their implants even when they have to be re-operated for complications or aesthetic reasons. If you are taking breast enlargement pills that cause strange behaviour to your body, you must discontinue using it imediately. As a medicinal plant, Watercress has been traditionally considered a diuretic, expectorant, purgative, stimulant, stomachic and tonic. For some women a breast enlargement surgery alone is not enough.

In addition, problems include leak or rupture of the implant requiring additional surgery to change the previous one, pain, seroma (fluid retention), infections and a condition called capsular contracture. Yes, they all have well-endowed breasts, may it be open or artificially enhanced. Breast Massage can be used as natural breast cancer treatment. We come in all different shapes and sizes and we can only play the cards that. For bigger and developed figures, perfect brands should be applied, so that required target can be approached.

The progress of it also depends on how your body is taking the phytoestrogens that promotes hormones that is elemental to the growth and volume of your breasts. Every time you go back to the same bar, they are there waiting to expose their necks for your pleasure. Hold one weight in each hand and extended your arms out at shoulder level. These exercises are just part of the stay fit regime more and more people are embracing all over the world. Think about it, your breast is already stretched out of its magnitude so why would you want to stretch it even more using a breast enlargement pump.

whether it is done through breast enlargement creams and pills or breast surgery. Alternatively, with both hands holding the breast meat should be bent and very gently wrung. What the surgery does is that it makes them feel much younger. Estrogen-based breast augmentation products may cause a temporary enhancement as many women have discovered when using birth control pills. Though the desired size varies, the reason beneath remains the same—bigger than usual.

Using a breast enlargement cream is another way to stimulate breast growth, most breast creams contain an herb called Pueraria Mirifica which mimics natural estrogen production, when applied directly to the breast it not only stimulates breast growth making the breasts larger, it can also help women reduce sagging in their breasts while at the same time making them fuller and perkier. The patients own tissue is used to reconstruct the breast. If you use implants the surgery will be a combination of enlargement and lift. Once this is done, use a dry eye shadow brush and use the same color to apply more to the same spot so as to create that pop. You must have patience and wait for them to fade away naturally.

Spending your bucks on fruitless endeavors is never a great concept. Keep your ankles crossed while doing this exercise, breast enlargement, not to lift too much weight. The venous drainage of the breast is mainly to the axillary vein, but there is some drainage to the internal thoracic vein and the intercostal veins. Breast augmentation has become increasingly popular in recent times, and with celebrities who have had breast implants before and after pictures being splashed across gossip magazines on an almost weekly basis, it seems more and more women are seeing it as a viable option when it comes to enhancing their appearance. Just be sure you keep these sessions because the probability of infection raises in case the drain stays in place longer.

If you beloved this report and you would like to obtain far more facts concerning how do i make my breasts bigger naturally (www.lucky-house.info) kindly go to our own web-page.

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