Lexicographic order topology on the unit square: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
Newer edit →
en>Helpful Pixie Bot
m ISBNs (Build KE)
 
en>Tobias Bergemann
Grammar, links
Line 1: Line 1:
Almost certainly the most prevalent query asked on blogs, forums and social web-sites connected to throwing knives is, What is the ideal throwing knife for beginners?” It really is a wonderful question, but there is no 1 correct way to answer it. It depends on your budget. In the mid-sixties a locking attribute was added to the buck knife as a safety feature which keeps the blade from accidentally closing. The addition of most knife units could be the chef's blade.  Also referred to as the French blade, originally this knife was utilized mainly for cutting foods.  I go to gun shows a lot and generally see a table full of knives advertised as Any Knife $5 or $10″. There is A purpose the knife is $5 or $10. Then verify out the boning knife guide!<br><br><br><br>I'm not advocating spending thousands of dollars on a set unless you can afford it, but you actually will need to get into the hundred to two hundred dollar range prior to you get real high quality blades. Come across a set you can afford, then obtain other knives to add to it as you can afford them. It has lately come to my interest that folks have a hard time locating a very good knife set in the $one hundred to $200 price variety.<br><br>MAC knives appear to be one particular of the finest kept secrets of the consumer kitchen knife marketplace. But it really is a big kitchen knife world out there and everybody's welcome to their personal opinion! I have a Cutco hand-me-down butcher's knife from my Mom which I enjoy to use for slicing up unwieldy slabs of meat like sides of salmon or flank steak. But for standard, day-to-day use, I do not consider a Cutco chef's knife would hold up as properly as the brands and models I've advised in this write-up.<br><br>A forged knife is made from heated steel and then its blade is shaped. A forged knife is typically heavier than a stamped knife and numerous cooks think it is also much more sturdy. With its heavy heft, a German knife is very easily used to slice via meats or vegetables. This moves the center of gravity back toward the deal with, giving the knife a heavier really feel. There are, having said that, a number of American knife producers that make use of either forging or stamped goods.  Look at the hilt of the knife.<br><br>Without the need of additional ado, the Ulitity Knife Guide is made to assist you have an understanding of how you take this out of the kitchen. The carving knife guide will enable you understand extra about what it requires to turn into top rated dog in the planet of carving the thick juicy meat slabs. This is the boning knife and its ready to debone anything you throw at it. Lengthy versatile and capable to cut into something and almost everything. If you are searching for great quality knife sets, Chicago Cutlery knives are the best decision.<br><br>It is is an off the wall brand that you have never ever heard of ahead of, then it definitely doesn't matter if they give you a lifetime guarantee if they will be out of small business in a year or if will expense you much more to send back your knife then it was to purchase it in the first spot. When you are getting your knife or knife set these 5 guidelines will enable you get the ideal knife for your money and meals preparation tasks. My favorite is this red set due to the fact no matter where you place it in your kitchen it stands out.<br><br>I almost certainly use about four diverse types of knife in a day, and I am cooking and preparing allof meals and dishes every single day. I do have more than four kitchen knives which I use occasionally, but for you at dwelling, getting 4 kinds of kitchen knife will be additional than adequate. Your first Kitchen Knife, most chefs would agree with us, need to be a excellent eight -ten inch chef's knife. Most chefs look at the classic chef's knife additional Most versatile. The right knife is a protected knife.<br><br>I seriously do not imply to sound ungrateful since I know he just didn't understand how essential a excellent sharp knife was to me. Soon after explaining, he was fine with my returning the set and purchasing the Wusthof Knife. I applied a a further gift certificate and money to acquire my paring knife a Short time later. Should you beloved this information and you want to be given more info regarding [http://www.thebestkitchenknivesreviews.com/best-knife-set-reviews-top-kitchen-sets/ best Knife sets brand] kindly go to our own site. It so happened that a lady helped me that had the knife that I purchased. I waited  Best Knife Set For Butchering Deer literally years to ultimately get a excellent knife.  A watermelon knife sounds intriguing.
{{Multiple issues
| notability=October 2010
| original research=October 2010
}}
One of the causes of [[attenuation]] of [[radio propagation]] is the [[Absorption (electromagnetic radiation)|absorption]] by the [[atmosphere]]. There are many well known facts on the phenomenon and qualitative treatments in [[textbook]]s.<ref name="ReferenceA">''Antennas and radiowave propagation''. R. E. Collin. Mc Graw Hill, 1985</ref> A document published by the [[International Telecommunication Union]]  (ITU)
<ref name="ReferenceC">''ITU recommendation ITU-R'' pp.&nbsp;676–8, 2009</ref>
provides some basis for a quantitative assessment of the attenuation.  That document describes a simplified model along with semi-empirical formulas based on [[data fitting]].  It also recommended an [[algorithm]] to compute the attenuation of radiowave propagation in the atmosphere. [[NASA]] also published a study on a related subject.<ref name="ReferenceD">''http://trs-new.jpl.nasa.gov/dspace/handle/2014/41145''. NASA progress report</ref> Free software from [[CNES]] based on ITU-R recommendations is available [http://logiciels.cnes.fr/PROPA/fr/logiciel.htm for download] and is available to the public.
 
==The model and the ITU recommendation==
 
The document ITU-R P.676-8 of the [[ITU-R]] section considers the atmosphere as being divided into spherical homogeneous layers; each layer has a constant [[refraction index]].  By the use of [[trigonometry]], a couple of formulas and an algorithm were derived.
 
Through the use of an [[invariant]], the same results can be directly derived:
 
[[File:Optinv.png]]
 
An [[incident ray]] at A under the angle Φ hits the layer B at the angle&nbsp;''&theta;''. From basic [[Euclidean geometry]]:
 
: <math> |CK| = R\sin u = r\sin\theta. \, </math>
 
By [[Snell's law]] (or [[René Descartes]]' law from the French point of view!) :
 
: <math> n_1 \sin\Phi = n_2\sin u, \, </math>
 
so that
 
: <math> n_1 R\sin\Phi = n_2 r \sin\theta = INV \, </math>
 
'''Notes:'''
 
* One proof <ref name="ReferenceA"/> starts from the [[Fermat's principle]]. As a result one gets the proof of the Snell’s law along with this invariance. This invariant is valid in a more general situation; the spherical radius is then replaced by the [[Radius of curvature (mathematics)|Radius of curvature]] at points along the ray. It is also used in equation (4) of the 2005 NASA's report<ref name="ReferenceD"/> in an application of satellite tracking.
 
* The assumption of the refraction index varying with the latitude is not strictly compatible with the notion of layers. However the variation of the index is very small, this point is usually ignored in practice.
 
The ITU recommended algorithm consists of launching a ray from a [[radio source]], then at each step, a layer is chosen and a new [[incidence angle]] is then computed. The process is iterated until the altitude of the target is reached. At each step, the covered distance ''dL'' is multiplied by a specific attenuation [[coefficient]] ''g'' expressed in&nbsp;dB/km. All the increments ''g''&nbsp;''dL'' are added to provide the total attenuation.
 
Note that the algorithm does not guaranty that the target is actually reached. For this, a much harder [[boundary value problem]] would have to be solved.
 
==The eikonal equation==
 
This equation is discussed in the references.<ref>''Microwave and optical ray geometry''. S. Cornbleet, Wiley, 1984</ref><ref>''Light transmission optics''. Detrich Marcuse, Van Nostrand, 1982</ref><ref name="ReferenceB">''Methods in Electromagnetic Wave Propagation''. D. S. Jones, Oxford, 1987</ref> The equation is highly non-linear. Given that a smooth data fitting curve n(altitude) is  provided by the ITU<ref>''ITU recommandation ITU-R'' pp.&nbsp;835–4, 2009</ref> for the refraction index n, and that the values of n  differs from 1 only by something of the order&nbsp;10<sup>&minus;4</sup>, a [[numerical solution]] of the [[eikonal equation]] can be considered. Usually the equation is presented under the self-adjoint form, a more tractable equation for the ray head position vector '''r'''<ref name="ReferenceB"/> is given in generic parametric form:
 
: <math> \ddot{\mathbf{r}} = n\, \mathrm{grad}\, n </math>
 
==Implementations==
 
Three implementations to compute the attenuations exist:
 
* Take the ray to be a straight line.
* Use the optical invariant and apply the ITU recommendationITU.<ref name="ReferenceC"/>
* Solve the eikonal equation.
 
The first two are only of 1st order approximation (see [[Orders of approximation]]). For the [[eikonal equation]], many numerical schemes are available.<ref name="ReferenceB"/> Here only a simple second order scheme was chosen. For most standard configurations of source-target, the three methods differ little from each other. It is only in the case of rays grazing the ground that the differences are meaningful. The following was used for testing:
 
At the latitude of 10°, when a ray starts at 5&nbsp;km altitude with an elevation angle of&nbsp;&minus;1° to hit a target at the same longitude but at latitude 8.84° and altitude 30&nbsp;km. At 22.5&nbsp;GHz, the results are:
 
[[File:The3results.png]]
 
Note that 22.5&nbsp;GHz is not a practical frequency<ref name="ReferenceA"/> but it is the most suitable for algorithms comparison. In the table, the first column gives the results in dB, the third gives the distance covered and the last gives the final altitude. Distances are in km. From the altitude 30&nbsp;km up, the attenuation is negligible. The paths of the three are plotted:
 
[[File:The3methods.png]]
 
The linear path is the highest on the figure, the eikonal is the lowest.
 
'''Note''': A [[MATLAB]] version for the uplink ([[Telecommunications link]]) is available from the ITU<ref name="ReferenceC"/>
 
==The boundary value problem==
 
When a point S communicates with a point T, the orientation of the ray is specified by an elevation angle. In a naïve way, the angle can be given by tracing a straight line from S to T. This specification does not guaranty that the ray will reach T: the variation of refraction index bends the ray trajectory. The elevation angle has to be modified<ref name="ReferenceD"/> to take into account the bending effect.
 
For the Eikonal equation, this correction can be done by solving a [[boundary value problem]]. As the equation is of second order, the problem is well defined. In spite of the lack of a firm theoretical basis for the ITU method, a trial-error by dichotomy (or [[Binary search algorithm|binary search]]) can also be used. The next figure shows the results of numerical simulations.
 
[[File:Compare.JPG|Compare]]
 
The curve labeled as bvp is the trajectory found by correcting the elevation angle. The other two are from a fix step and a variable steps (chosen in accordance to the ITU recommendations<ref name="ReferenceB"/>) solutions without the elevation angle correction. The nominal elevation angle for this case is -0.5 degree. The numerical results obtained at 22.5&nbsp;GHz were:
 
[[File:Numerical compare.gif]]<br />
<br />
Note the way the solution bvp bents over the straight line. A consequence of this property is that the ray can reach locations situated below the horizon of S. This is consistent with observations.<ref>''ITU recommandation ITU-R'' pp.&nbsp; 834-6, 2007</ref> The trajectory is a [[Concave function]] is a consequence of the fact that the gradient of the refraction index is negative, so the Eikonal equation implies that the second derivative of the trajectory is negative. From the point where the ray is parallel to ground, relative to the chosen coordinates, the ray goes down but relative to ground level, the ray goes up.
 
Often engineers are interested in finding the limits of a system. In this case, a simple idea is to try some low elevation angle and let the ray reach the desired altitude. This point of view has a problem: if suffice to take the angle for which the ray has a tangent point of lowest altitude. For instance with the case of a source at 5&nbsp;km altitude, of nominal elevation angle -0.5 degree and the target is at 30&nbsp;km altitude; the attenuation found by the boundary value method is 11.33 dB. The previous point of view of worst case leads to an elevation angle of -1.87 degree and an attenuation of  170.77 dB. With this kind of attenuation, every system would be unusable! It was found also for this case that with the nominal elevation angle, the distance of the tangent point to ground is 5.84&nbsp;km; that of the worst case is 2.69&nbsp;km. The nominal distance from source to target is 6383.84&nbsp;km; for the worst case, it is 990.36&nbsp;km.
 
There are many numerical methods to solve boundary value problems.<ref name="ReferenceI">''Initial Value Methods for Boundary Value Problems''. Mayer. Academic Press, 1973</ref> For the Eikonal equation, due the good behavior of the refraction index just a simple [[Shooting method]] can be used.
 
==Conclusions==
 
Of the three methods, the linear and the ITU methods require some coding since they are not presented as differential equations. These methods do not benefit from the help of standard numerical packages; however, only high school mathematics are required to understand the methods. The more technical eikonal equation can be solved using standard differential equations solvers offered by a few numerical software packages mentioned in the Wikipedia [[List of numerical analysis software]] and it offers a higher precision order.
 
The attenuation mechanism as described here is only one amongst many others others.<ref name="ReferenceA"/><ref name="ReferenceJ">''Field Guide to Atmospheric Optics (SPIE Vol. FG02)''. Larry C. Andrews. SPIE Publications, 2004</ref> The full problem is much more complex.
 
==See also==
*[[Ray tracing (physics)]]
 
== References ==
 
<!--- See [[Wikipedia:Footnotes]] on how to create references using <ref></ref> tags which will then appear here automatically -->
{{Reflist}}
{{Use dmy dates|date=October 2010}}
 
== External links ==
* [http://www.itu.int/publications/sector.aspx?lang=en&sector=1 ITU Publications]
* [http://trs-new.jpl.nasa.gov/dspace/handle/2014/41145 JPL Publication 09-14]
 
{{DEFAULTSORT:Radiowave Attenuation In The Atmosphere, Computation Of}}
[[Category:Applied mathematics]]
[[Category:Communication]]
[[Category:Atmospheric sciences]]

Revision as of 10:12, 11 February 2013

Template:Multiple issues One of the causes of attenuation of radio propagation is the absorption by the atmosphere. There are many well known facts on the phenomenon and qualitative treatments in textbooks.[1] A document published by the International Telecommunication Union (ITU) [2] provides some basis for a quantitative assessment of the attenuation. That document describes a simplified model along with semi-empirical formulas based on data fitting. It also recommended an algorithm to compute the attenuation of radiowave propagation in the atmosphere. NASA also published a study on a related subject.[3] Free software from CNES based on ITU-R recommendations is available for download and is available to the public.

The model and the ITU recommendation

The document ITU-R P.676-8 of the ITU-R section considers the atmosphere as being divided into spherical homogeneous layers; each layer has a constant refraction index. By the use of trigonometry, a couple of formulas and an algorithm were derived.

Through the use of an invariant, the same results can be directly derived:

An incident ray at A under the angle Φ hits the layer B at the angle θ. From basic Euclidean geometry:

By Snell's law (or René Descartes' law from the French point of view!) :

so that

Notes:

  • One proof [1] starts from the Fermat's principle. As a result one gets the proof of the Snell’s law along with this invariance. This invariant is valid in a more general situation; the spherical radius is then replaced by the Radius of curvature at points along the ray. It is also used in equation (4) of the 2005 NASA's report[3] in an application of satellite tracking.
  • The assumption of the refraction index varying with the latitude is not strictly compatible with the notion of layers. However the variation of the index is very small, this point is usually ignored in practice.

The ITU recommended algorithm consists of launching a ray from a radio source, then at each step, a layer is chosen and a new incidence angle is then computed. The process is iterated until the altitude of the target is reached. At each step, the covered distance dL is multiplied by a specific attenuation coefficient g expressed in dB/km. All the increments g dL are added to provide the total attenuation.

Note that the algorithm does not guaranty that the target is actually reached. For this, a much harder boundary value problem would have to be solved.

The eikonal equation

This equation is discussed in the references.[4][5][6] The equation is highly non-linear. Given that a smooth data fitting curve n(altitude) is provided by the ITU[7] for the refraction index n, and that the values of n differs from 1 only by something of the order 10−4, a numerical solution of the eikonal equation can be considered. Usually the equation is presented under the self-adjoint form, a more tractable equation for the ray head position vector r[6] is given in generic parametric form:

Implementations

Three implementations to compute the attenuations exist:

  • Take the ray to be a straight line.
  • Use the optical invariant and apply the ITU recommendationITU.[2]
  • Solve the eikonal equation.

The first two are only of 1st order approximation (see Orders of approximation). For the eikonal equation, many numerical schemes are available.[6] Here only a simple second order scheme was chosen. For most standard configurations of source-target, the three methods differ little from each other. It is only in the case of rays grazing the ground that the differences are meaningful. The following was used for testing:

At the latitude of 10°, when a ray starts at 5 km altitude with an elevation angle of −1° to hit a target at the same longitude but at latitude 8.84° and altitude 30 km. At 22.5 GHz, the results are:

File:The3results.png

Note that 22.5 GHz is not a practical frequency[1] but it is the most suitable for algorithms comparison. In the table, the first column gives the results in dB, the third gives the distance covered and the last gives the final altitude. Distances are in km. From the altitude 30 km up, the attenuation is negligible. The paths of the three are plotted:

The linear path is the highest on the figure, the eikonal is the lowest.

Note: A MATLAB version for the uplink (Telecommunications link) is available from the ITU[2]

The boundary value problem

When a point S communicates with a point T, the orientation of the ray is specified by an elevation angle. In a naïve way, the angle can be given by tracing a straight line from S to T. This specification does not guaranty that the ray will reach T: the variation of refraction index bends the ray trajectory. The elevation angle has to be modified[3] to take into account the bending effect.

For the Eikonal equation, this correction can be done by solving a boundary value problem. As the equation is of second order, the problem is well defined. In spite of the lack of a firm theoretical basis for the ITU method, a trial-error by dichotomy (or binary search) can also be used. The next figure shows the results of numerical simulations.

Compare

The curve labeled as bvp is the trajectory found by correcting the elevation angle. The other two are from a fix step and a variable steps (chosen in accordance to the ITU recommendations[6]) solutions without the elevation angle correction. The nominal elevation angle for this case is -0.5 degree. The numerical results obtained at 22.5 GHz were:

File:Numerical compare.gif

Note the way the solution bvp bents over the straight line. A consequence of this property is that the ray can reach locations situated below the horizon of S. This is consistent with observations.[8] The trajectory is a Concave function is a consequence of the fact that the gradient of the refraction index is negative, so the Eikonal equation implies that the second derivative of the trajectory is negative. From the point where the ray is parallel to ground, relative to the chosen coordinates, the ray goes down but relative to ground level, the ray goes up.

Often engineers are interested in finding the limits of a system. In this case, a simple idea is to try some low elevation angle and let the ray reach the desired altitude. This point of view has a problem: if suffice to take the angle for which the ray has a tangent point of lowest altitude. For instance with the case of a source at 5 km altitude, of nominal elevation angle -0.5 degree and the target is at 30 km altitude; the attenuation found by the boundary value method is 11.33 dB. The previous point of view of worst case leads to an elevation angle of -1.87 degree and an attenuation of 170.77 dB. With this kind of attenuation, every system would be unusable! It was found also for this case that with the nominal elevation angle, the distance of the tangent point to ground is 5.84 km; that of the worst case is 2.69 km. The nominal distance from source to target is 6383.84 km; for the worst case, it is 990.36 km.

There are many numerical methods to solve boundary value problems.[9] For the Eikonal equation, due the good behavior of the refraction index just a simple Shooting method can be used.

Conclusions

Of the three methods, the linear and the ITU methods require some coding since they are not presented as differential equations. These methods do not benefit from the help of standard numerical packages; however, only high school mathematics are required to understand the methods. The more technical eikonal equation can be solved using standard differential equations solvers offered by a few numerical software packages mentioned in the Wikipedia List of numerical analysis software and it offers a higher precision order.

The attenuation mechanism as described here is only one amongst many others others.[1][10] The full problem is much more complex.

See also

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. 30 year-old Entertainer or Range Artist Wesley from Drumheller, really loves vehicle, property developers properties for sale in singapore singapore and horse racing. Finds inspiration by traveling to Works of Antoni Gaudí.

External links

  1. ↑ 1.0 1.1 1.2 1.3 Antennas and radiowave propagation. R. E. Collin. Mc Graw Hill, 1985
  2. ↑ 2.0 2.1 2.2 ITU recommendation ITU-R pp. 676–8, 2009
  3. ↑ 3.0 3.1 3.2 http://trs-new.jpl.nasa.gov/dspace/handle/2014/41145. NASA progress report
  4. ↑ Microwave and optical ray geometry. S. Cornbleet, Wiley, 1984
  5. ↑ Light transmission optics. Detrich Marcuse, Van Nostrand, 1982
  6. ↑ 6.0 6.1 6.2 6.3 Methods in Electromagnetic Wave Propagation. D. S. Jones, Oxford, 1987
  7. ↑ ITU recommandation ITU-R pp. 835–4, 2009
  8. ↑ ITU recommandation ITU-R pp.  834-6, 2007
  9. ↑ Initial Value Methods for Boundary Value Problems. Mayer. Academic Press, 1973
  10. ↑ Field Guide to Atmospheric Optics (SPIE Vol. FG02). Larry C. Andrews. SPIE Publications, 2004
Retrieved from "https://en.formulasearchengine.com/index.php?title=Lexicographic_order_topology_on_the_unit_square&oldid=26009"