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# Tag Info

## Hot answers tagged typography

10

I think the best way to proceed is to stop using \hline -- and, while you're at it, ditch all vertical bars -- and to use the line-drawing macros of the booktabs package. For the table at hand, \toprule and \bottomrule should be used. If, for some reason, you simply must use vertical bars (and hence can't use \toprule and \bottomrule), I suggest you load ...

10

I'd like to add our main project report, which was created the last semester at a 2-year technical college. Main Report as pdf (dropbox link) Stripped-down example/template (needs XeLaTex) (Note: This template bears some marks from being taken from a paper written under a constant time-pressure with multiple revisions, and was never meant to be shown off ...

10

The problem is not in diacritics, it is in LaTeX tables. You can give rows some vertical padding with the cellspace package, which ensure a minimal vertical distance between a cell and the above and below rules. \documentclass{article} \usepackage[utf8]{inputenc} \usepackage{array, cellspace} \setlength\cellspacetoplimit{3pt} ...

8

You can \usepackage{lmodern} to get a version somewhat in between. In particular, the ring on Å is a circle (though somewhat thicker than the original). The accents are less slanted as well, especially for Ő and Ű. \documentclass[12pt]{article} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{lmodern} \begin{document} A\'A\AA ...

7

Insert a large \strut. \documentclass{article} \begin{document} \begin{table}[!h] \begin{tabular}{|c|c|c|c|c|c|} \hline {\Large\strut}\"Ostersund & \AA m\aa l & Oxel\"osund & Alings\aa s & \'Orlaith &O\'O\"O\H{O}\\ \hline \end{tabular} \end{table} \end{document}

3

As mentioned in some of the comments, the text editor you use and its configuration are crucial in making text readable or alienating. The stuff you can act on: Font. In my opinion, fixed-width fonts give the best results because the letters are evenly spaced across all lines, which makes it easier on the eyes (there is some kind of vertical symmetry), and ...

3

One can use lmodern, but redefine the functions \', \H, and \", though that may have some danger associated with it. As wipet points out, this approach breaks hyphenation, which in itself is enough to recommend against its use. \documentclass[12pt]{article} \usepackage[T1]{fontenc} \usepackage{lmodern,stackengine,scalerel,graphicx} ...

3

The fact you're using babel-french has nothing to do with the problem. Straight quotes are very rarely used in typography, so when you type the ' character, you get a curly glyph that serves both as a closing quote and as apostrophe. If you want a straight quote, you can use the glyph provided by textcomp. \documentclass{report} \usepackage[T1]{fontenc} % ...

3

Well you can do it. But you need to remove the quote first from the list of reserved characters. But if you uncomment the l'enfer you will see why csquotes did put the character in the list (and so prevents you from using it as a \MakeInnerQuote character). I would advise you to avoid such tricks. They bite you when you don't expect them. Using \enquote ...

2

csquotes decides which type of quotation marks to use based on the language setting. For example, in British typography it is custom to have single quote marks as the outer quote, and double as inner quote marks. For American English it's the opposite. Hence, \documentclass{article} \usepackage[british]{babel} \usepackage{csquotes} \MakeOuterQuote{"} ...

2

OPTION 1: Row specific padding One can add a \stackgap to a given element of the line, which pads the element vertically by the amount of the optional argument (default 3pt). The MWE below shows 3 rows that have been padded with 1pt, 3pt, and 5pt, respectively. \documentclass{article} \usepackage{stackengine} \begin{document} \begin{table}[!h] ...

2

Here is a solution (I used erewhon, a clone and extension of Adobe Utopia). \documentclass{book} \usepackage[utf8]{inputenc} \usepackage{geometry} \usepackage[explicit]{titlesec} \usepackage{array, graphicx} \usepackage[tracking=true]{microtype} \usepackage{fourier, erewhon} \usepackage{lipsum}% just to generate text for the example ...

2

An important question to ask is, what do the subscripts represent? If "sub" is of the form i or j and serves to index elements of the vector named a, it's customary not to typeset the indices in bold. Thus, you'd write \mathbf{a}_i and \mathbf{b}_j. If, on the other hand, "sub" forms an integral part of the name of the vector itself, it's more common to ...

2

You can use the calligra font (see http://www.ctan.org/pkg/calligra and http://www.ctan.org/pkg/calligra-type1). A potential alternative is the use of Zapf Chancery (a widely available font, use the package chancery from the psnffss bundle: http://www.ctan.org/pkg/psnfss) that is probably cursive enough for you purposes.

Only top voted, non community-wiki answers of a minimum length are eligible


text:
What is the correct way to determine the amount of difference between two proportions?

Suppose I am testing the effectiveness of a new piece of machinery. The current machine has a success rate of 20%, that is, for every hundred widgets produced, 20 can be sold.

Now I've developed a new prototype and I want to know if it performs better. Additionally, it only makes sense to invest in the new machine if it is 10% better, i.e. a 22% success rate.

So I set up an experiment and collect this data:

machine     attempts   successes
---------------------------------
Original      n1         k1
Prototype     n2         k2

Assume that
* alpha= 5%
* beta= 20%
* I'd like to use the smallest sample size possible.

My understanding is that normally I'd want a 2-tailed difference of proportions test. But that test would only tell me if the two machines performed differently, not if one was 10% or more better.

How do I determine what sample size to use, what is the appropriate test(s) to use, and how do I report the magnitude of the difference?

-
 How large are $n_1$ and $n_2$? Tests for comparing proportions tend to lose power if the sample sizes differ too much. – MånsT Jul 20 '12 at 6:03 If the new machine has a $21.8\%$ success rate, you would like to reject it unless the original really only succeeds about $19.8\%$ of the time? Does testing the original cost the same as testing the prototype? – Douglas Zare Jul 20 '12 at 6:16 @MånsT n1 and n2 could be between 1,000 to 100,000 if necessary. I can keep them roughly equal if that helps power. – dan Jul 20 '12 at 13:46 @Douglas Zare testing both machines costs the same. The business case is that the cost of fully developing the prototype would only pay off if I can expect a 10% improvement. – dan Jul 20 '12 at 14:05 @dan: I have some recommendations, but I don't have the time to post them now. I'll write an answer on Sunday or Monday unless threre already is a thorough answer by then! – MånsT Jul 20 '12 at 20:17
show 1 more comment

The standard hypothesis test only tells you that you are significantly better but not by the magnitude. To show that the magnitude is greater than a specified $\Delta$ requires a larger sample size. Instead of testing $p_1-p_2>0$ you test $p_1-p_2>\Delta$.

format: plain


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Making Beautiful Maps Programmatically with Rayshader

(See the bottom of the page for a description on how I generated the above figure–and read the rest of the article to see how to use rayshader and R to make beautiful maps yourself. I'm also available to create custom maps--contact me for a quote.)

My most memorable experience with cartography occurred in Rome last October where I found myself briskly walking down an amazing map-covered hallway in the Vatican museum, towards the Sistine Chapel, looking around and thinking to myself “I sure hope there’s a way to get back here.” My wife and I bought tickets to have breakfast at the Vatican–which was really a roundabout way to see the Sistine Chapel before the crowds arrived. We had quickly finished our (very rushed, very un-Italian) breakfast, and started towards the Sistine Chapel as soon as we were allowed. On our way, we entered a 350 foot-long hallway that was absolutely covered in beautiful maps, illustrating the entirety of the Italian peninsula in forty large-scale frescos. I slowed down my pace, but there was no time to waste–we bought these tickets for a reason. But I wanted to get back.

Figure 1: The Gallery of Maps: Okay, the Sistine Chapel is far more breathtaking. But these frescos were impressive in their own way.

And I did get back to what I later found out was (aptly named) the Gallery of Maps. They don’t make it easy–you can’t go backwards through the museum, and I had to ask a security guard for permission to loop back around. But seeing the Italian landscape represented like it was the first few pages of a fantasy novel was incredible. The maps themselves were detailed and beautifully illustrated, and it was a nice example of combining cartography as art. The mountains, sea, hills of Italy, painted and beautifully illustrated--It got me thinking: what if there was an R package that could do just that?

... I kid, of course--no real-life human would take in the grandeur of the Vatican and think: “THIS HAS INSPIRED ME... TO DEVELOP SOFTWARE 🤖.” But there's something about a beautiful, detailed map that's a joy to behold. And I have developed an R package that easily produces beautiful stylized maps, and I’m going to show you how to do it as well.

Figure 2: Example of the hillshaded maps that rayshader can produce with the new features added in the latest version (spherical texture mapping, lambertian reflectance, ambient occlusion, water detection).

The initial version of rayshader was simple–it took a elevation matrix, traced lines towards a light source at each point, and subtracted light from that point if it ran into the ground in the process. It had the bare minimum number of features you could have to call it a raytracer: rays and, well, traces. But once it was working and making pretty pictures, I whipped up a blog post and released it to the public. And the response immediately drove the development of rayshader to a very cool place.

Ask the internet how to do something and you’re likely to get a bunch of digital shrugs and maybe—if you’re lucky—a passive aggressive link to a Wikipedia page. Tell the internet you’ve done something—and they’ll tell you exactly what you can do to improve it. By telling you everything (they thought 🤔) you did wrong. There was some great initial positive feedback, but also a bunch of notes from a few people describing what they thought was missing. One person was aghast that my raytracer didn’t have Lambertian reflectance, which is the technical term for the fact that surfaces pointing towards the light are brighter than those askew. Reasonable suggestion, I thought, so--I’ll implement it.

Figure 3: A hillshade with just raytracing (ray_shade), Lambertian reflectance (lamb_shade), and the combined shadow map. The two shadow maps were combined with a new function, add_shadow, that automatically rescales shadow maps so they don't completely overwhelm the original image.

Implementing Lambertian reflectance involves determining which direction the surface is pointing and taking the dot product between that and the light vector. The intensity is then equivalent to the cosine of the angle difference: zero (the surface directed right at the light) means it has maximum brightness, and 90 degrees or more means the intensity is zero. Looking for an existing function in the R ecosystem that would do this, I found none--but luckily for me, years and years of TA-ing Intro to Physics during my PhD immediately let me see the solution.

Ask the internet how to do something and you’re likely to get a bunch of digital shrugs and maybe—if you’re lucky—a passive aggressive link to a Wikipedia page. Tell the internet you’ve done something—and they’ll tell you exactly what you can do to improve it.

With an elevation matrix on a square grid, we can think of each point having four vectors pointing to it: one from below, one from the left, one from the right, and one from above 1. The x and y components of each of these vectors are fixed because the grid is fixed–the only component that changes is the z. This component is simply the elevation difference between the two points. By taking the cross product of these vectors, we use the right hand rule to get a vector that points perpendicularly to the surface defined by those vectors. We get two of these surfaces for every point, and we average those vectors to get a single normal vector for that point.

Figure 4: We want to determine the surface normal at each position–basically, which direction in the sky the surface is oriented towards. We take the cross product of the vectors pointing from the adjacent points, giving us two normal vectors. We average those two normal vectors to get the direction of the surface at that point.

This worked and the effect was cool–but more importantly, it set me up to easily implement everything that came next. When I posted rayshader to Twitter, @ozjimbob quickly combined it with his spherical texture mapping to create a truly gorgeous map. I decided that this would be the niche rayshader would fill–a package to create beautiful topographic maps, with a combination of texture mapping and raytracing.

I couldn’t use his code, as it was too slow on the extremely large elevation matrices I was using. So I had to implement it myself–but luckily, I had already done most of the work. Spherical texture mapping isn't too hard to explain: imagine you stretched a texture across a sphere surrounding you in the sky. Each point on the surface is facing towards a point on that sphere. All you need to do is figure out which pixel on the sphere your surface is facing, and then color that point appropriately. The color is entirely a function of the normal vector–which we already have a function to calculate! We don’t even need to call any trigonometric functions; once we have the normal vector, the mapping between surface and sphere is just a few multiplications and divisions away2.

Here, let’s assume we have a square, $$NxN$$ array representing our texture. The image can either be imported (using an R package like png or jpeg), but more conveniently rayshader now has a create_texture function that programmatically creates texture maps given five colors: a highlight, a shadow, a left fill light, a right fill light, and a center color for flat areas. The user can also optionally specify the colors at the corners, but create_texture will interpolate those if they aren't given. We need to map the x, y, and z-components of the normal vector to positions on the texture map. We map our z-normal onto the distance from the center R, and the x and y-normals ($$n_x$$ and $$n_y$$) tell us from what angle around the texture map we sample.

$1 = n_x^2 +n_y^2 +n_z^2$ $r = \sqrt{n_x^2 +n_y^2}$$R = (1-n_z)$$x_{\text{coord}} = R \sin(\theta) = R \frac{-n_x}{r} = R\frac{-n_x}{\sqrt{1-n_z^2}}$ $y_{\text{coord}} = R\cos(\theta) = R \frac{n_y}{r} = R\frac{n_y}{\sqrt{1-n_z^2}}$ In order to rotate the direction of the highlight from due north to an angle $$\theta_{\text{light}}$$, we apply a rotation transformation to the above coordinates. We also add the coordinates of the center of the texture, and apply the floor function to get the integer coordinates of the color on the texture.
$X_{\text{image}} = \mathrm{floor}\big( x_{\text{coord}}\cos{\theta_{\text{light}}} - y_{\text{coord}}\sin{\theta_{\text{light}}} + \frac{N}{2}\big)$$Y_{\text{image}} = \mathrm{floor}\big(x_{\text{coord}}\sin{\theta_{\text{light}}} + y_{\text{coord}}\cos{\theta_{\text{light}}} + \frac{N}{2}\big)$

This is implemented in a function in rayshader called sphere_shade. By inputting the elevation matrix and a texture, you can create beautifully colored maps on which you can layer more shadows from rayshader's other functions. rayshader also includes 7 built-in palettes: Four inspired by by the palettes of Eduard Imhof c("imhof1","imhof2","imhof3",imhof4"), a desert palette "desert", a black and white palette "bw", and one called… 🦄 “unicorn” 3.

Figure 5: A surface, the hillshade of that surface, and the texture used in spherical mapping. This figure shows the We break a vector down into it’s component parts, and use those parts to determine from where on the sphere you sample the color. The z-component tells us how far out from the origin we are looking–when it’s pointing straight up (z-component length 1, flat surface), you sample from the center, and when it’s pointing straight sideways (z-component length 0, vertical surface) you are sampling from the edge of the circle. The x-component and y-component tell us what angle around the circle we are sampling from.

We now have a beautiful textured, rayshaded map (with Lambertian reflectance--thanks, Hacker News commentator!)–but we can improve it even more. The amount of light hitting a surface is not just related to the direct light, but also scattered light from the atmosphere. We can model this with an ambient occlusion model, where the amount of light hitting the surface is also modulated depending on the amount of sky visible from that point. This makes valleys darker than flat areas, due to a much smaller amount of the sky visible, even when you’re in direct sunlight. And we can reuse existing code; we simply use the ray_shade function to raytrace out a short distance in a circle around that point, and check for surface intersections. If no rays intersect the surface, it means the full sky is visible from that point. Otherwise, that point is darkened for each intersecting ray.

Figure 6: With ambient occlusion, areas within valleys are darkened, while ridges and large flat areas stay at full brightness.

There's still something extremely important missing, however: water. Simply filling in the areas with water with the same color we use for relatively flat surfaces would deprive our map of its most basic job: topographic maps, at their most basic level, should tell us at least two things: where there is land, and where there is water. Bodies of water naturally give maps a sense of scale and breaks up maps into human relatable chunks. Remove the distinction between land and water--well, kid, that's a nice bump map you have there. So we need some automated way of detecting and adding water to our maps.

Thankfully, we (yet again) have already done most of the work to build a water-detection algorithm when we figured out how to calculate the surface normals. Water is flat, so its surface normal is always pointing straight upwards. If we extract all of the areas that are pointing straight up (with a little wiggle room defined by the user), and then only extract contiguous areas larger than a user-defined minimum area (in order to avoid classifying every small flat area as water), you can easily extract the bodies of water from a map. We can extract this area, and then use the function add_water with a color argument to add a water layer to your map. Matching water colors are provided for each of the built-in palettes included in sphere_shade: just provide the same palette name that you provided when building the original texture map (yes, all palettes have a matching water type... 🦄 ). Otherwise supply your own hexadecimal color. This should be done after calling sphere_shade, but before adding any shadows to your map, otherwise the water will overwrite the shadows you added.

Bodies of water naturally give maps a sense of scale and breaks up maps into human relatable chunks. Remove the distinction between land and water--well, kid, that's a nice bump map you have there.

Finally, we combine the spherical texturing, raytraced shadows, lambertian reflectivity, ambient occlusion, and water layers all together to create our final image. To do this, we element-wise multiply our R, G, and B channels in the spherical texture map with our shadow matrices. By default, our shadow matrix is scaled from 0 to 1 (0 is complete darkness, 1 is full illumination), but we don’t want to cover up all of the details in the completely shaded areas (0 in all color channels is black). We then rescale the shadow matrices from from 0-1 to a much smaller range (the default in rayshader is 0.7-1.0), so that completely shaded areas won’t be black, just darker. Rayshader has an add_shadow function that does just that: input a shadow map and the map to be shaded, and it rescales and multiplies them correctly. This works at any stage of the process: you can use add_shadow to combine shadow matrices as well as texture maps, in any order.

elmat %>%
plot_map()

Figure 7: Each function, with its associated image, in the latest version of rayshader. With the add_shadow and add_water functions, we can easily add more layers to our hillshade.

One of the lessons I’ve learned during all of this is the importance of releasing as soon as you have a “minimally viable product.” I had a few ideas of where next to take the project, but seeing other people use it immediately drove the development to a place I never even considered (although I'm still going to develop those ideas I originally had in mind). If you want to try it yourself but don't have any topographic data on hand (and let's face it--the volcano data set is pretty boring), here's the test data set I've been playing with--it includes water (zip file, 620Kb). Or you could try the elevatr package by Jeff Hollister. It can easily load topographic data from Amazon's tile service, and you can get rolling as easy as this:

library(rayshader)
library(elevatr)
library(magrittr)

#from the .tif file
eltif = raster::raster("~/path/to/file/dem_01.tif")
elmat1 = matrix(raster::extract(eltif,raster::extent(eltif),buffer=10000),nrow=ncol(eltif),ncol=nrow(eltif))

#or from the elevatr package
elevation <- get_elev_raster(lake, z = 11, src = "aws")

elmat2 = matrix(raster::extract(elevation,raster::extent(elevation),buffer=10000),nrow=ncol(elevation),ncol=nrow(elevation))

elmat2 %>%
plot_map()

As for the latest version of rayshader, see the link below--I look forward to seeing what cool maps you can make with it!

X

Creation of the featured Lake Mead figure:

The creation of this figure started by going to the USGS map viewer website and downloading topographic data for the area. This gave me the size of the lake at full capacity, which I was then able to extract the lake surface with the detect_water function in rayshader. I then found bathymetric (undersea elevation) data of Lake Mead collected from sonar data gathered in 2001 by the USGS (thanks again!). This gave me the topographic information of the area sans water, although of much lower resolution than the actual topographic map. You might notice jumps in the animation: those occur every 30 feet, which was the sampling resolution of the sonar.

I then aligned the bathymetric map with the lake surface extracted from the other dataset. To raise the water level, I replaced everything under the lake surface and under a set height with that height. This just involved subsetting the elevation matrix for those heights under the set limit, and replacing them if they also were in the lake area (defined by the mask)  elmat[mask == 1 & elmat < height] = height. I did this in half-meter increments from no water (232m above sea level) to full depth (374m above sea level), and output an image for each height.

I then scraped the historical monthly lake levels from this US Bureau of Reclamation website to determine the water level since the Hoover Dam was built. I then wrote a script to create images in order of the the correct height for each date. The first few years when the lake was filling was much more interesting than the following 80 years, so I interpolated between those data points to slow down the animation and emphasize that portion.

I extracted the volume of the lake by taking the sum of the total difference between the new adjusted lake height map and the original map (which gives the elevation difference in meters), and multiplying it by 100 m^2 (10mx10m area per data point).

For the aesthetic portion, I used the “desert” built-in palette in sphere_shade. I used the detect_water function to determine where the waterline was, and replaced the color on that area with the "imhof2" palette (there's a matching lighter palette for "desert", but it didn't look as nice with the ambient occlusion layer added under the water). I added a raytraced layer with add_shadow, ray_shade, and ambient_shade. This would take a long time to do for every single frame, so I also used the lake surface to tell ray_shade to only update the raytraced shadows under the lake surface. The ambient_shade layer I only calculated once and added on top of the water layer, to show the topographic features below the surface. Finally, I combined the map with plots using ImageMagick showing the water level as a function of date, and total water volume (computed by taking the volume of water under the water level). I then used ffmpeg to create a video out of all 1759 images.

Tyler is an avid R user, physicist, comedian, writer, programmer, and data enthusiast.

1. June 30, 2018

Great work Tyler,
thank you for the detailled description.
I really loved to look over it.
I did put my R installation a bit to sleep i favour of python. Your Text makes me put the dust away and try to,play with maps and shader with R.
Thomas

2. […] Previous Post Draining Lake Mead: Making Beautiful Maps with Rayshader […]

3. Dr.Duleep Kumar Samuel
October 5, 2018

> library(magrittr)
>
> #Here, I load a map for the River Derwent in Tasmania with the raster package:
> localtif = raster::raster(“tasmania.tif”)
Error in .local(.Object, …) :

Error in .rasterObjectFromFile(x, band = band, objecttype = “RasterLayer”, :
Cannot create a RasterLayer object from this file. (file does not exist)
>
> #And convert it to a matrix:
> elmat = matrix(raster::extract(localtif,raster::extent(localtif),buffer=1000),
+ nrow=ncol(localtif),ncol=nrow(localtif))
Error in raster::extract(localtif, raster::extent(localtif), buffer = 1000) :


text:
Some fixed point theorems for $\theta$-$\phi$ ${C}$-contractions

Volume 10, Issue 11, pp 5723--5733 Publication Date: November 15, 2017       Article History
• 833 Views

Authors

Dingwei Zheng - College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi, 530004, P. R. China. Xinhe Liu - College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi, 530004, P. R. China. Gengrong Zhang - College of Mathematics and Computationl Science, Hunan First Normal University, Changsha, Hunan, 410205, P. R. China.

Abstract

In this paper, we introduce the notion of $\theta$-$\phi$ ${C}$-contraction and establish some fixed point and coupled fixed point theorems for these mappings in the setting of complete metric spaces and ordered metric spaces. The results presented in the paper improve and extend some well-known results. Also, we give an example to illustrate them.

Keywords

• Fixed point
• coupled fixed point
• complete metric space
• $\theta$-$\phi$ ${C}$-contraction

•  47H10
•  54H25

References

• [1] F. Bojor , Fixed point theorems for Reich type contractions on metric spaces with a graph, Nonlinear Anal., 75 (2012), 3895–3901.

• [2] D. W. Boyd, J. S. W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc., 20 (1969), 458–464.

• [3] F. E. Browder , On the convergence of successive approximations for nonlinear functional equations, Indag. Math., 71 (1968), 27–35.

• [4] S. K. Chatterjea, Fixed-point theorems , C. R. Acad. Bulgare Sci., 25 (1972), 727–730.

• [5] B. S. Choudhury , Unique fixed point theorem for weakly C-contractive mappings, Kathmandu Univ. J. Sci. Eng. Technol., 5 (2009), 6–13.

• [6] B. S. Choudhury, A. Kundu, A coupled coincidence point result in partially ordered metric spaces for compatible mappings, Nonlinear Anal., 73 (2010), 2524–2531.

• [7] J. Dugundji, A. Granas, Weakly contractive maps and elementary domain invariance theorem, Bull. Soc. Math. Gréce (N.S.), 19 (1978), 141–151.

• [8] Z. M. Fadail, A. G. B. Ahmad, G. S. Rad, V. Ozturk, S. Radenović , Some remarks on coupled, tripled and n-tupled coincidence and fixed points theorems in ordered abstract metric spaces, Far East J. Math. Sci., 97 (2015), 809–839.

• [9] T. Gnana Bhaskar, V. Lakshmikantham , Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal., 65 (2006), 1379–1393.

• [10] J. Harjani, B. López, K. Sadarangani , Fixed point theorems for weakly C-contractive mappings in ordered metric spaces, Comput. Math. Appl., 61 (2011), 790–796.

• [11] J. R. Jachymski , Equivalence of some contractivity properties over metrical structures, Proc. Amer. Math. Soc., 125 (1997), 2327–2335.

• [12] M. Jleli, B. Samet, A new generalization of the Banach contraction principle, J. Inequal. Appl., 2014 (2014), 8 pages.

• [13] R. Kannan, Some results on fixed points, II, Amer. Math. Monthly, 76 (1969), 405–408.

• [14] J. Matkowski, Integrable solutions of functional equations, Dissertationes Math. (Rozprawy Mat.), 127 (1975), 68 pages.

• [15] H. Piri, P. Kumam , Some fixed point theorems concerning F-contraction in complete metric spaces, Fixed Point Theory Appl., 2014 (2014), 11 pages.

• [16] S. Radenović , Remarks on some coupled fixed point results in partial metric spaces, Nonlinear Funct. Anal. Appl., 18 (2013), 39–50.

• [17] S. Radenović, A note on tripled coincidence and tripled common fixed point theorems in partially ordered metric spaces, Appl. Math. Comput., 236 (2014), 367–372.

• [18] S. Radenović, Bhaskar-Lakshmikantham type results for monotone mappings in partially ordered metric spaces, Int. J. Nonlinear Anal. Appl., 5 (2014), 96–103.

• [19] S. Radenović, Coupled fixed point theorems for monotone mappings in partially ordered metric spaces, Kragujevac J. Math., 38 (2014), 249–257.

• [20] S. Radenović, Remarks on some coupled coincidence point results in partially ordered metric spaces, Arab J. Math. Sci., 20 (2014), 29–39.

• [21] B. Samet , Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces, Nonlinear Anal., 72 (2010), 4508–4517.

• [22] B. Samet, C. Vetro, P. Vetro, Fixed point theorems for $\alpha-\psi$-contractive type mappings, Nonlinear Anal., 75 (2012), 2154–2165.

• [23] W. Shatanawi, Fixed point theorems for nonlinear weakly C-contractive mappings in metric spaces, Math. Comput. Modelling, 54 (2011), 2816–2826.

• [24] D. Wardowski , Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl., 2012 (2012), 6 pages.

• [25] D.-W. Zheng, Z.-Y. Cai, P. Wang , New fixed point theorems for $\theta-\phi$ contraction in complete metric spaces, J. Nonlinear Sci. Appl., 10 (2017), 2662–2670.


text:
# Production of J/psi and Upsilon mesons in pp collisions at sqrt(s) = 8 TeV

Abstract : The production of J/psi and Upsilon mesons in pp collisions at sqrt(s) = 8 TeV is studied with the LHCb detector. The J/psi and Upsilon mesons are reconstructed in the mu+mu- decay mode and the signal yields are determined with a fit to the mu+mu- invariant mass distributions. The analysis is performed in the rapidity range 2.0
Document type :
Journal articles

http://hal.in2p3.fr/in2p3-00818152
Contributor : Sabine Starita <>
Submitted on : Friday, April 26, 2013 - 10:34:11 AM
Last modification on : Tuesday, April 20, 2021 - 12:00:09 PM

### Citation

R. Aaij, Y. Amhis, S. Barsuk, O. Callot, J. He, et al.. Production of J/psi and Upsilon mesons in pp collisions at sqrt(s) = 8 TeV. Journal of High Energy Physics, Springer, 2013, 06, pp.064. ⟨10.1007/JHEP06(2013)064⟩. ⟨in2p3-00818152⟩

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Magnetic Force Between Parallel Currents Video Lessons

Concept

# Problem: The force per meter between the two wires of a jumper cable being used to start a stalled car is 0.225 N/m a. What is the current in the wires, given they are separated by 2.00 cmb. Is the force attractive or repulsive

###### FREE Expert Solution

Magnetic force per unit length between two parallel conductors:

$\overline{)\frac{\mathbf{F}}{\mathbf{l}}{\mathbf{=}}\frac{{\mathbf{\mu }}_{\mathbf{0}}{\mathbf{i}}_{\mathbf{1}}{\mathbf{i}}_{\mathbf{2}}}{\mathbf{2}\mathbf{\pi }\mathbf{r}}}$, where i1 and i2 are the currents in the two parallel wires, r is the distance between the parallel wires, and μ0 is the permeability of free space.

a.

Both the wires have the same magnitude of the current.

i1 = i2 = i

###### Problem Details

The force per meter between the two wires of a jumper cable being used to start a stalled car is 0.225 N/m

a. What is the current in the wires, given they are separated by 2.00 cm

b. Is the force attractive or repulsive


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# Math Help - real hard problem

1. ## real hard problem

Suppose that $lim_{x\rightarrow a}$ $f(x)$ exist and is positive. Show that for any positive integer n we have
$
\lim_{x\rightarrow a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x\rightarrow a} {f(x)}}
$
.

2. Originally Posted by mancillaj3
Suppose that $lim_{x\rightarrow a}$ $f(x)$ exist and is positive. Show that for any positive integer n we have
$lim_{x\rightarrow a} \frac{n}{}\sqrt{f(x)}$ $= \frac{n}{}\sqrt{lim_{x\rightarrow a} {f(x)}}$ .
Those are meant to be n'th roots, right? $\lim_{x\rightarrow a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x\rightarrow a} {f(x)}}$ ?

The slick way to do this is to say that $\sqrt[n]{f(x)}$ is just the composition of the function f with the n'th root function $g(t) = t^{1/n}$. The function g(t) is continuous at all points t>0 (because in fact it is differentiable, with derivative $\tfrac1n t^{(1/n)-1}$). And the composition of two continuous functions is continuous. So the function $\sqrt[n]{f(x)}$ is continuous at $x=a$ (where $f(a)$ is defined to be $\lim_{x\to a}f(x)$).

3. thank you, is there another way to approach it?

4. Originally Posted by mancillaj3
is there another way to approach it?
You could use the identity $s^n-t^n = (s-t)(s^{n-1} + s^{n-2}t + \ldots + st^{n-2} + t^{n-1})$. This tells you that $|s-t| = \frac{|s^n-t^n|}{s^{n-1} + s^{n-2}t + \ldots + st^{n-2} + t^{n-1}} \leqslant \frac{|s^n-t^n|}{t^{n-1}}$ provided that s and t are positive.

Now suppose that $\lim_{x\to a}f(x) = \ell>0$. Put $s = \sqrt[n]{f(x)}$ and $t = \sqrt[n]\ell$ in the above inequality, to get $\bigl|\sqrt[n]{f(x)} - \sqrt[n]\ell\bigr| \leqslant \ell^{(1-n)/n}|f(x)-\ell|$. From that, you should be able to show that $\lim_{x\to a}\sqrt[n]{f(x)} = \sqrt[n]\ell$.


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Ai Cheat Sheet
Search…
⌃K

# Poisson Distribution

It is a discrete distribution function describing the probability that an event will occur a certain number of times in a fixed time (or space) interval.
It is used to model count-based data, like the number of emails arriving in your mailbox in one hour or the number of customers walking into a shop in one day, for instance. It can be used to predict how many times an event might occur in a given time period.
• The number (k) of hits on a website in one hour with an average hit rate of 6 hits per hour is poisson distributed.
• Insurance companies to conduct risk analysis (eg. predict the number of car crash accidents within a predefined time span) to decide car insurance pricing.
The Poisson distribution is parametrized by the expected number of events λ (pronounced “lambda”) in a time or space window. The distribution is a function that takes the number of occurrences of the event as input (the integer called k in the next formula) and outputs the corresponding probability (the probability that there are k events occurring).

### The Probability Mass Function

$P(k;\lambda) = \frac{\lambda^k e^{-\lambda}}{k!}$
for k = 0, 1, 2, ... The formula of P(k; λ) returns the probability of observing k events given the parameter λ which corresponds to the expected (sometimes called average) number of occurrences in that time slot.
Poisson Process: The timing of the next event is completely independent of when the previous event happened.

### Example

Say we do a Bernoulli trial every minute for an hour, each with a success probability of 0.1. We would do 60 trials, and the number of successes is Binomially distributed, and we would expect to get about 6 successes. This is just like the Poisson story we discussed, where we get on average 6 hits on a website per hour.
So, the Poisson distribution with arrival rate equal to np approximates a Binomial distribution for n Bernoulli trials with probability p of success (with n large and p small). Importantly, the Poisson distribution is often simpler to work with because it has only one parameter instead of two for the Binomial distribution.


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# Quantum algorithms and approximating polynomials for composed functions with shared inputs

Mark Bun1, Robin Kothari2, and Justin Thaler3

1Boston University
2Microsoft Quantum
3Georgetown University

### Abstract

We give new quantum algorithms for evaluating composed functions whose inputs may be shared between bottom-level gates. Let $f$ be an $m$-bit Boolean function and consider an $n$-bit function $F$ obtained by applying $f$ to conjunctions of possibly overlapping subsets of $n$ variables. If $f$ has quantum query complexity $Q(f)$, we give an algorithm for evaluating $F$ using $\tilde{O}(\sqrt{Q(f) \cdot n})$ quantum queries. This improves on the bound of $O(Q(f) \cdot \sqrt{n})$ that follows by treating each conjunction independently, and our bound is tight for worst-case choices of $f$. Using completely different techniques, we prove a similar tight composition theorem for the approximate degree of $f$.
By recursively applying our composition theorems, we obtain a nearly optimal $\tilde{O}(n^{1-2^{-d}})$ upper bound on the quantum query complexity and approximate degree of linear-size depth-$d$ AC$^0$ circuits. As a consequence, such circuits can be PAC learned in subexponential time, even in the challenging agnostic setting. Prior to our work, a subexponential-time algorithm was not known even for linear-size depth-3 AC$^0$ circuits.
As an additional consequence, we show that AC$^0 \circ \oplus$ circuits of depth $d+1$ require size $\tilde{\Omega}(n^{1/(1- 2^{-d})}) \geq \omega(n^{1+ 2^{-d}} )$ to compute the Inner Product function even on average. The previous best size lower bound was $\Omega(n^{1+4^{-(d+1)}})$ and only held in the worst case (Cheraghchi et al., JCSS 2018).

### ► References

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### Cited by

[1] Scott Aaronson, Daniel Grier, and Luke Schaeffer, "A Quantum Query Complexity Trichotomy for Regular Languages", arXiv:1812.04219.

[2] Kamil Khadiev and Liliya Safina, "Quantum Algorithm for Dynamic Programming Approach for DAGs. Applications for Zhegalkin Polynomial Evaluation and Some Problems on DAGs", arXiv:1804.09950.

The above citations are from SAO/NASA ADS (last updated successfully 2022-10-04 13:47:49). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2022-10-04 13:47:47).


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TheInfoList

A flywheel is a mechanical device which uses the conservation of
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational equivalent of linear momentum. It is an important quantity in physics because it is a conserved quantity—the total angular momentum of a close ...
to store
rotational energyRotational energy or angular kinetic energy is kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion (physics), motion. It is defined as the work (physics), work needed to accelerate a body o ...
; a form of kinetic energy proportional to the product of its
moment of inertia The moment of inertia, otherwise known as the mass moment of inertia, angular mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceleration about a rotational ax ...

and the square of its
rotational speed Rotational speed (also known as speed of revolution or rate of rotation), of an object rotating around an axis is the number of turns of the object divided by time, specified as revolutions per minute Revolutions per minute (abbreviated rpm ...
. In particular, if we assume the flywheel's moment of inertia to be constant (i.e., a flywheel with fixed mass and
second moment of area The second moment of area, or second area moment, or quadratic moment of area and also known as the area moment of inertia, is a geometrical property of an area Area is the quantity that expresses the extent of a two-dimensional region, ...
revolving about some fixed axis) then the stored (rotational) energy is directly associated with the square of its rotational speed.. Since a flywheel serves to store mechanical energy for later use, it is natural to consider it as a
kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion (physics), motion. It is defined as the work (physics), work needed to accelerate a body of a given mass from rest to its stated velocity. Having gaine ...
analogue of an . Once suitably abstracted, this shared principle of energy storage is described in the generalized concept of an accumulator. As with other types of accumulators, a flywheel inherently smoothes sufficiently small deviations in the power output of a system, thereby effectively playing the role of a
low-pass filter A low-pass filter (LPF) is a filter that passes signals with a frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also referred to as temporal frequency, which emphasizes the contrast to spatial freq ...

with respect to the mechanical velocity (angular, or otherwise) of the system. More precisely, a flywheel's stored energy will donate a surge in power output upon a drop in power input and will conversely absorb any excess power input (system-generated power) in the form of rotational energy. Common uses of a flywheel include: * Smoothing the power output of an energy source. For example, flywheels are used in
reciprocating engine , internal combustion, gasoline engine, gasoline piston engine. A reciprocating engine, also often known as a piston engine, is typically a heat engine (although there are also pneumatic motor, pneumatic and hydraulic motor, hydraulic reciprocatin ...
s because the active torque from the individual pistons is intermittent. * Energy storage systems * Delivering energy at rates beyond the ability of an energy source. This is achieved by collecting energy in a flywheel over time and then releasing it quickly, at rates that exceed the abilities of the energy source. * Controlling the orientation of a mechanical system,
gyroscope A gyroscope (from Ancient Greek γῦρος ''gûros'', "circle" and σκοπέω ''skopéō'', "to look") is a device used for measuring or maintaining Orientation (geometry), orientation and angular velocity. It is a spinning wheel or disc in ...

and
reaction wheel A reaction wheel (RW) is a type of flywheel used primarily by spacecraft for three-axis attitude control, which does not require Rocket engine, rockets or external applicators of torque. They provide a high pointing accuracy, and are particularly u ...
Flywheels are typically made of steel and rotate on conventional bearings; these are generally limited to a maximum revolution rate of a few thousand RPM.; "Flywheels move from steam age technology to Formula 1"; Jon Stewart , 1 July 2012, retrieved 2012-07-03 High energy density flywheels can be made of carbon fiber composites and employ
magnetic bearing A magnetic bearing is a type of bearing that supports a load using magnetic levitation. Magnetic bearings support moving parts without physical contact. For instance, they are able to levitate a rotating shaft and permit relative motion with ver ...

s, enabling them to revolve at speeds up to 60,000 RPM (1 kHz)., "Breakthrough in Ricardo Kinergy ‘second generation’ high-speed flywheel technology"; Press release date: 22 August 2011. retrieved 2012-07-03

# Applications

Flywheels are often used to provide continuous power output in systems where the energy source is not continuous. For example, a flywheel is used to smooth fast angular velocity fluctuations of the
crankshaft A crankshaft is a shaft Shaft may refer to: Rotating machine elements * Shaft (mechanical engineering), a rotating machine element used to transmit power * Line shaft, a power transmission system * Drive shaft, a shaft for transferring torque ...

in a reciprocating engine. In this case, a crankshaft flywheel stores energy when torque is exerted on it by a firing
piston A piston is a component of reciprocating engine , internal combustion piston engine. A reciprocating engine, also often known as a piston engine, is typically a heat engine (although there are also pneumatic Pneumatics (from Greek ‘wind, ...

, and returns it to the piston to compress a fresh charge of air and fuel. Another example is the
friction motor A friction motor is a simple mechanism to propel toy cars, trucks, trains, action figures and similar toys. The motor consists of a large flywheel which is connected to the drive wheels of the toy via a very low gear ratio, so that the flywheel revo ...

which powers devices such as
toy car A model vehicle or toy vehicle is a Physical model, miniature representation of an automobile. Other miniature motor vehicles, such as trucks, buses, or even ATVs, etc. are often included in this general category. Because many miniature vehicles ...

s. In unstressed and inexpensive cases, to save on cost, the bulk of the mass of the flywheel is toward the rim of the wheel. Pushing the mass away from the axis of rotation heightens
rotational inertia The moment of inertia, otherwise known as the mass moment of inertia, angular mass, or most accurately, rotational inertia, of a rigid body In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), kn ...
for a given total mass. A flywheel may also be used to supply intermittent pulses of energy at power levels that exceed the abilities of its energy source. This is achieved by accumulating energy in the flywheel over a period of time, at a rate that is compatible with the energy source, and then releasing energy at a much higher rate over a relatively short time when it is needed. For example, flywheels are used in
power hammer Power hammers are mechanical forging hammers that use a non-muscular power source to raise the hammer preparatory to striking, and accelerate it onto the work being hammered. Also called "Open Die Power Forging Hammers." They have been used by ...
s and
riveting machines A rivet is a permanent mechanical fastener. Before being installed, a rivet consists of a smooth cylinder (geometry), cylindrical shaft with a head on one end. The end opposite to the head is called the ''tail''. On installation, the rivet is ...
. Flywheels can be used to control direction and oppose unwanted motions. Flywheels in this context have a wide range of applications from
gyroscope A gyroscope (from Ancient Greek γῦρος ''gûros'', "circle" and σκοπέω ''skopéō'', "to look") is a device used for measuring or maintaining Orientation (geometry), orientation and angular velocity. It is a spinning wheel or disc in ...

s for instrumentation to
ship stability ''Ship stability'' is an area of naval architecture and ship design that deals with how a ship behaves at sea, both in still water and in waves, whether intact or damaged. Stability calculations focus on centers of gravity, centers of buoyancy, t ...

and satellite stabilization (
reaction wheel A reaction wheel (RW) is a type of flywheel used primarily by spacecraft for three-axis attitude control, which does not require Rocket engine, rockets or external applicators of torque. They provide a high pointing accuracy, and are particularly u ...
), to keep a toy spin spinning (
friction motor A friction motor is a simple mechanism to propel toy cars, trucks, trains, action figures and similar toys. The motor consists of a large flywheel which is connected to the drive wheels of the toy via a very low gear ratio, so that the flywheel revo ...

), to stabilize magnetically levitated objects (
Spin-stabilized magnetic levitationImage:Levitron-levitating-top-demonstrating-Roy-M-Harrigans-spin-stabilized-magnetic-levitation.ogg, 300px, A brief demonstration and small explanation about the ''Levitron'' branded levitating top device which demonstrates the spin-stabilized magnet ...
) Flywheels may also be used as an electric compensator, like a synchronous compensator, that can either produce or sink reactive power but would not affect the real power. The purposes for that application are to improve the power factor of the system or adjust the grid voltage. Typically, the flywheels used in this field are similar in structure and installation as the synchronous motor (but it is called synchronous compensator or synchronous condenser in this context). There are also some other kinds of compensator using flywheels, like the single phase induction machine. But the basic ideas here are the same, the flywheels are controlled to spin exactly at the frequency which you want to compensate. For a synchronous compensator, you also need to keep the voltage of rotor and stator in phase, which is the same as keeping the magnetic field of rotor and the total magnetic field in phase (in the rotating frame reference).

# History

The principle of the flywheel is found in the
Neolithic The Neolithic period is the final division of the Stone Age, with a wide-ranging set of developments that appear to have arisen independently in several parts of the world. It is first seen about 12,000 years ago when the first developments of ...
spindle and the
potter's wheel , Germany ) , image_map = , map_caption = , map_width = 250px , capital = Berlin , coordinates = , largest_city = capital , languages_type = Official language , languages = German language, German , demonym = Germans, German , ...

, as well as circular sharpening stones in antiquity.Lynn White, Jr., "Theophilus Redivivus", ''Technology and Culture'', Vol. 5, No. 2. (Spring, 1964), Review, pp. 224–233 (233) In early 11th century
Ibn Bassal Ibn Bassal ( ar, ابن بصال) was an 11th-century Andalusian Arab The Arabs (singular Arab ; singular ar, عَرَبِيٌّ, ISO 233 The international standard are technical standards developed by international organizations (intergover ...
pioneered the used on flywheel in noria and saqiya. The use of the flywheel as a general mechanical device to equalize the speed of rotation is, according to the American medievalist
Lynn White Lynn Townsend White Jr. (April 29, 1907 – March 30, 1987) was an American historian. He was a professor of medieval In the history of Europe The history of Europe concerns itself with the discovery and collection, the study, organiza ...
, recorded in the ''De diversibus artibus'' (On various arts) of the German artisan
Theophilus Presbyter Theophilus Presbyter (Floruit, fl. c. 1070–1125) is the pseudonymous author or compiler of a Latin text containing detailed descriptions of various Middle Ages, medieval arts, a text commonly known as the ''Schedula diversarum artium'' ("List of v ...
(ca. 1070–1125) who records applying the device in several of his machines. In the
Industrial Revolution The Industrial Revolution was the transition to new manufacturing processes in Great Britain, continental Europe Mainland or continental Europe is the contiguous continent of Europe, excluding its surrounding islands. It can also be ...
,
James Watt James Watt (; 30 January 1736 (19 January 1736 OS) – 25 August 1819) was a Scottish Scottish usually refers to something of, from, or related to Scotland, including: *Scottish Gaelic, a Celtic Goidelic language of the Indo-European langu ...

contributed to the development of the flywheel in the
steam engine from Stott Park Bobbin Mill, Cumbria, England A steam engine is a heat engine In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energ ...

, and his contemporary James Pickard used a flywheel combined with a crank to transform reciprocating motion into rotary motion.

# Physics

A flywheel is a spinning wheel, or disc, or rotor, rotating around its symmetry axis. Energy is stored as
kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion (physics), motion. It is defined as the work (physics), work needed to accelerate a body of a given mass from rest to its stated velocity. Having gaine ...
, more specifically
rotational energyRotational energy or angular kinetic energy is kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion (physics), motion. It is defined as the work (physics), work needed to accelerate a body o ...
, of the
rotor ROTOR was an elaborate air defence radar Radar is a detection system that uses radio waves to determine the range, angle, or velocity of objects. It can be used to detect aircraft An aircraft is a vehicle that is able to flight, fly ...
: * $E_k = \frac I \omega^2$ where: * $E_k$ is the stored
kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion (physics), motion. It is defined as the work (physics), work needed to accelerate a body of a given mass from rest to its stated velocity. Having gaine ...
, * ω is the
angular velocity In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. " ...

, and * $I$ is the
moment of inertia The moment of inertia, otherwise known as the mass moment of inertia, angular mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceleration about a rotational ax ...

of the flywheel about its axis of symmetry. The moment of inertia is a measure of resistance to
torque In physics and mechanics Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical objects, more specifically the relationships among force, matter, and motion. Forces applied to objects result in Dis ...

applied on a spinning object (i.e. the higher the moment of inertia, the slower it will accelerate when a given torque is applied). * The moment of inertia for a solid cylinder is $I = \frac mr^2$, * for a thin-walled empty cylinder is $I = m r^2$, * and for a thick-walled empty cylinder is $I = \frac m\left(^2 + ^2\right)$, where $m$ denotes mass, and $r$ denotes a radius. When calculating with units, the units would be for mass,
kilograms The kilogram (also kilogramme) is the SI base unit, base unit of mass in the International System of Units (SI), the metric system, having the unit symbol kg. It is a widely used measure in science, engineering and commerce worldwide, and is oft ...
; for radius, metres; and for angular velocity,
radian The radian, denoted by the symbol \text, is the SI unit for measuring angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''sides'' of the angle, sharing a common endpoint, called the ''verte ...

s per
second The second (symbol: s, also abbreviated: sec) is the base unit of time Time is the indefinite continued sequence, progress of existence and event (philosophy), events that occur in an apparently irreversible process, irreversible succession ...
and the resulting energy would be in
joule The joule ( ; symbol: J) is a derived unit of energy In physics, energy is the physical quantity, quantitative physical property, property that must be #Energy transfer, transferred to a physical body, body or physical system to perform W ...

s. Increasing amounts of rotation energy can be stored in the flywheel until the rotor shatters. This happens when the
hoop stress In mechanics Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical objects, more specifically the relationships among force, matter, and motion. Forces applied to objects result in Displacement (vecto ...
within the rotor exceeds the
ultimate tensile strength Ultimate tensile strength (UTS), often shortened to tensile strength (TS), ultimate strength, or F_\text within equations, is the maximum stress that a material can withstand while being stretched or pulled before breaking. In brittle tensil ...
of the rotor material. * $\sigma_t = \rho r^2 \omega^2 \$ where: * $\sigma_t$ is the tensile stress on the rim of the cylinder * $\rho$ is the density of the cylinder * $r$ is the radius of the cylinder, and * $\omega$ is the
angular velocity In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. " ...

of the cylinder. A flywheel powered by electric machine is common. The output power of the electric machine is approximately equal to the output power of the flywheel. The output power of a synchronous machine is: $P=\left(V_i\right)\left(V_t\right)\left \left( \frac\right \right)$ where: * $V_i$ is the voltage of rotor winding, which is produced by field interacting with the stator winding * $V_t$ is stator voltage * $\delta$ is the torque angle (angle between two voltages)

# Material selection

Flywheels are made from many different materials; the application determines the choice of material. Small flywheels made of lead are found in children's toys. Cast iron flywheels are used in old steam engines. Flywheels used in car engines are made of cast or nodular iron, steel or aluminum. Flywheels made from high-strength steel or composites have been proposed for use in vehicle energy storage and braking systems. The efficiency of a flywheel is determined by the maximum amount of energy it can store per unit weight. As the flywheel's rotational speed or angular velocity is increased, the stored energy increases; however, the stresses also increase. If the hoop stress surpass the tensile strength of the material, the flywheel will break apart. Thus, the tensile strength limits the amount of energy that a flywheel can store. In this context, using lead for a flywheel in a child's toy is not efficient; however, the flywheel velocity never approaches its burst velocity because the limit in this case is the pulling-power of the child. In other applications, such as an automobile, the flywheel operates at a specified angular velocity and is constrained by the space it must fit in, so the goal is to maximize the stored energy per unit volume. The material selection therefore depends on the application. The table below contains calculated values for materials and comments on their viability for flywheel applications. CFRP stands for
carbon-fiber-reinforced polymer Carbon fiber reinforced polymer (American English American English (AmE, AE, AmEng, USEng, en-US), sometimes called United States English or U.S. English, is the set of varieties of the English language native to the United States. Curr ...
, and GFRP stands for glass-fiber reinforced polymer. The table below shows calculated values for mass, radius, and angular velocity for storing 250 J. The carbon-fiber flywheel is by far the most efficient; however, it also has the largest radius. In applications (like in an automobile) where the volume is constrained, a carbon-fiber flywheel might not be the best option.

# Table of energy storage traits

For comparison, the energy density of petrol (gasoline) is 44.4 MJ/kg or 12.3 kWh/kg.

## High-energy materials

For a given flywheel design, the kinetic energy is proportional to the ratio of the
hoop stress In mechanics Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical objects, more specifically the relationships among force, matter, and motion. Forces applied to objects result in Displacement (vecto ...
to the material density and to the mass: * $E_k \varpropto \fracm$ $\frac$ could be called the specific tensile strength. The flywheel material with the highest specific tensile strength will yield the highest energy storage per unit mass. This is one reason why
carbon fiber Carbon fiber reinforced polymer (American English American English (AmE, AE, AmEng, USEng, en-US), sometimes called United States English or U.S. English, is the set of varieties of the English language native to the United States. Curr ...

is a material of interest. For a given design the stored energy is proportional to the hoop stress and the volume: it is true. * $E_k \varpropto \sigma_tV$

# Design

## Rimmed

A rimmed flywheel has a rim, a hub, and spokes. Calculation of the flywheel's moment of inertia can be more easily analysed by applying various simplifications. For example: * Assume the spokes, shaft and hub have zero moments of inertia, and the flywheel's moment of inertia is from the rim alone. * The lumped moments of inertia of spokes, hub and shaft may be estimated as a percentage of the flywheel's moment of inertia, with the majority from the rim, so that $I_\mathrm=KI_\mathrm$ For example, if the moments of inertia of hub, spokes and shaft are deemed negligible, and the rim's thickness is very small compared to its mean radius ($R$), the radius of rotation of the rim is equal to its mean radius and thus: * $I_\mathrm=M_\mathrmR^2$

## Shaftless

A shaftless flywheel eliminates the annulus holes, shaft or hub. It has higher energy density than conventional design but requires a specialized magnetic bearing and control system. The specific energy of a flywheel is determined by * $\frac = K \frac$ In which $K$ is the shape factor, $\sigma$ the material's tensile strength and $\rho$ the density. A typical flywheel has a shape factor of 0.3. Better designs, such as the shaftless flywheel, have a shape factor close to 0.6, the theoretical limit is about 1.

## Superflywheel

The first superflywheel was patented in 1964 by the Soviet-Russian scientist Nurbei Guilia. A superflywheel consist of a solid core (hub) and multiple thin layers of high-strength flexible materials, such as special steels, carbon fiber composites, glass fiber, or graphene, wound around it. Compared to conventional flywheels, superflywheels can store more energy and are safer to operate In case of failure, superflywheel does not explode or burst into large shards, like a regular flywheel, but instead splits into layers. The separated layers then slow a superflywheel down by sliding against the inner walls of the enclosure, thus preventing any further destruction. Although the exact value of energy density of a superflywheel would depend on the material used, it could theoretically be as high as 1200 Wh (4.4 MJ) per kg of mass for graphene superflywheels.

*
Dual-mass flywheel A dual-mass flywheel (DMF or DMFW) is a rotating mechanical device that is used to provide continuous energy (rotational energyRotational energy or angular kinetic energy is kinetic energy In physics, the kinetic energy of an object is the ene ...
*
Flywheel energy storage Flywheel energy storage (FES) works by accelerating a rotor (flywheel) to a very high speed and maintaining the energy in the system as rotational energy. When energy is extracted from the system, the flywheel's rotational speed is reduced as a co ...
*
Accumulator (energy)An accumulator is an energy storage Energy storage is the capture of energy produced at one time for use at a later time to reduce imbalances between energy demand and energy production. A device that stores energy is generally called an Accumulat ...
*
Diesel rotary uninterruptible power supply Most forms of uninterruptible power supply (UPS) can be either powered by battery (electricity), battery or flywheel energy. These are ready for immediate use at the instant that the mains electricity fails, but the relatively small and finite amou ...
*
List of moments of inertia Moment of inertia, denoted by , measures the extent to which an object resists rotational acceleration about a Rotation around a fixed axis, particular axis, and is the rotational analogue to mass (which determines an object's resistance to Linear a ...
*
Clutch Single dry-clutch friction disc. The splined hub is attached to the disc with springs to damp chatter. A clutch is a mechanical device that engages and disengages power transmission Transmission may refer to: Science and technology * Power ...

*
Fidget Spinner A fidget spinner is a skill toy, toy that consists of a bearing (mechanical), ball bearing in the center of a multi-lobed (typically two or three) flat structure made from metal or plastic designed to spin along its axis with very little effort. F ...

# References

* * * https://pserc.wisc.edu/documents/general_information/presentations/presentations_by_pserc_university_members/heydt_synchronous_mach_sep03.pdf


text:
# Math Help - Need help

1. ## Need help

How do I solve the followinf equation using the elmination method?

p = 28 +q
3p = -9q+120

2. Originally Posted by mathdummy22
How do I solve the followinf equation using the elmination method?

p = 28 +q
3p = -9q+120
First rewrite it as

$\begin{array}{rcrcr}p&-&q&=&28\\3p&+&9q&=&120\end{array}$

I noticed that if you divide the second equation through by 3, you end up with this system:

$\begin{array}{rcrcr}p&-&q&=&28\\p&+&3q&=&40\end{array}$

I would like to eliminate p. Thus, multiply the first equation by -1 and then add the equations together:

$\begin{array}{rcrcr}-p&+&q&=&-28\\p&+&3q&=&40\end{array}\implies 4q=12\implies \color{red}\boxed{q=3}$

Can you find p?

--Chris

3. will p = 31 ?


text:
Coupled Mode Theory
Get Coupled Mode Theory essential facts below. View Videos or join the Coupled Mode Theory discussion. Add Coupled Mode Theory to your PopFlock.com topic list for future reference or share this resource on social media.
Coupled Mode Theory

Coupled mode theory (CMT) is a perturbational approach for analyzing the coupling of vibrational systems (mechanical, optical, electrical, etc.) in space or in time. Coupled mode theory allows a wide range of devices and systems to be modeled as one or more coupled resonators. In optics, such systems include laser cavities, photonic crystal slabs, metamaterials, and ring resonators.

## History

Coupled mode theory first arose in the 1950s in the works of Miller on microwave transmission lines,[1] Pierce on electron beams,[2] and Gould on backward wave oscillators.[3] This put in place the mathematical foundations for the modern formulation expressed by H. A. Haus et al. for optical waveguides.[4][5]

In the late 1990s and early 2000s, the field of nanophotonics has revitalized interest in coupled mode theory. Coupled mode theory has been used to account for the Fano resonances in photonic crystal slabs[6] and has also been modified to account for optical resonators with non-orthogonal modes.[7]

## Overview

The oscillatory systems to which coupled mode theory applies are described by second order partial differential equations (e.g. a mass on a spring, an RLC circuit). CMT allows the second order differential equation to be expressed as one or more uncoupled first order differential equations. The following assumptions are generally made with CMT:

• Linearity
• Time-reversal symmetry
• Time-invariance
• Weak mode coupling (small perturbation of uncoupled modes)
• Energy conservation

## Formulation

The formulation of the coupled mode theory is based on the development of the solution to an electromagnetic problem into modes. Most of the time it is eigenmodes which are taken in order to form a complete base. The choice of the base and the adoption of certain hypothesis like parabolic approximation differs from formulation to formulation. The classification proposed by [8] of the different formulation is as follows:

1. The choice of starting differential equation. some of the coupled mode theories are derived directly from the Maxwell differential equations [9][10] (here) although others use simplifications in order to obtain a Helmholtz equation.
2. The choice of principle to derive the equations of the CMT. Either the reciprocity theorem [9][10] or the variational principle have been used.
3. The choice of orthogonality product used to establish the eigenmode base. Some references use the unconjugated form [9] and others the complex-conjugated form.[10]
4. Finally, the choice of the form of the equation, either vectorial [9][10] or scalar.

When n modes of an electromagnetic wave propagate through a media in the direction z without loss the power transported by each mode is described by a modal power Pm. At a given frequency ?.

${\displaystyle P^{\omega }(z)=\sum _{m}^{n}P_{m}^{\omega }(z)={\frac {1}{4}}\sum _{m}^{n}N_{m}^{\omega }\left|a_{m}^{\omega }(z)\right|^{2}\,\!}$

where Nm is the norm of the mth mode and am is the modal amplitude.

## References

1. ^ S.E.Miller,"Coupled wave theory and waveguide applications.", Bell System Technical Journal, 1954
2. ^ J. R. Pierce, "Coupling of modes of propagations", Journal of Applied Physics, 25, 1954
3. ^ R.W. Gould, "A coupled mode description of the backward-wave oscillator and the Kompfner dip condition" I.R.E. Trans. Electron Devices, vol. PGED-2, pp. 37-42, 1955.
4. ^ Haus, H., et al. "Coupled-mode theory of optical waveguides." Journal of Lightwave Technology 5.1 (1987): 16-23.
5. ^ H. A. Haus, W. P. Huang. "Coupled Mode Theory."Proceedings of the IEEE, Vol 19, No 10, October 1991.
6. ^ S. Fan, W. Suh, J. Joannopoulos, "Temporal coupled-mode theory for the Fano resonance in optical resonators," JOSA A, vol. 20, no. 3, pp. 569-572, 2003.
7. ^ W. Suh, Z. Wang, and S. Fan, "Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities," Quantum Electronics, IEEE Journal of, vol. 40, no. 10, pp. 1511-1518, 2004
8. ^ Barybin and Dmitriev,"Modern Electrodynamics and Coupled-mode theory",2002
9. ^ a b c d Hardy and Streifer, "Coupled mode theory of parallel waveguides", Journal of Lightwave Technology,1985
10. ^ a b c d A. W. Snyder and J. D. Love, "Optical waveguide Theory",Chapman and Hall, 1983

## See also

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