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# Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/180

it is evident that this ratio is constant everywhere on the circle. This ratio is, by definition, the curvature of the circle, and we have

 (38) $K = \frac{1}{R}$.

The curvature of a circle equals the reciprocal of its radius.

101. Curvature at a point. Consider any curve. As in section,

 $\Delta \tau$ = total curvature of the arc PP', and $\frac{\Delta \tau}{\Delta s}$ = average curvature of the arc PP'.

More important, however, than the notion of the average curvature of an arc is that of curvature at a point. This is obtained as follows. Imagine P to approach P along the curve; then the limiting value of the average curvature $\left( = \tfrac{\Delta \tau}{\Delta s} \right)$ as P approaches P along the curve is defined as the curvature at P, that is,

Curvature at a point = $\lim_{\Delta s \to 0} \left( \tfrac{\Delta \tau}{\Delta s} \right) = \tfrac{d\tau}{ds}$.

 (39) ∴ $K = \frac{d\tau}{ds}$ = curvature.

Since the angle $\Delta \tau$ is measured in radians and the length of arc $\Delta s$ in units of length, it follows that the unit of curvature at a point is one radian per unit of length.

102. Formulas for curvature. It is evident that if, in the last section, instead of measuring the angles which the tangents made with OX, we had denoted by $\tau$ and $\tau + \Delta \tau$ the angles made by the tangents with any arbitrarily fixed line, the different steps would in no wise have been changed, and consequently the results are entirely in:dependent of the system of coordinates used. However, since the equations of the curves we shall consider are all given in either rectangular or polar coördinates, it is necessary to deduce formulas for K in terms of both. We have

 $\ \tan \tau$ $= \frac{dy}{dx}$, § 32 or $\ \tau$ $\arctan \frac{dy}{dx}$.


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There are several fonts with math support and perhaps you will find also an alternative font for the bread-and-butter type of your document. the example above, the Euro symbol was most recently used symbol, hence it's showing the (normal text) font (currently Calibri), with the Currency Symbols subset. And so, I don't think it would be quite useful, nor handy. Both shapes and colors can be used for symbols on maps. Greek Fonts. | free-fonts-download. NET Symbolics is a basic open source computer algebra library for. math #yourmathlabel a + b = c  Can be referenced with [#yourmathlabel] — rendered as 1, or (#yourmathlabel) — rendered as (1). Find the best 13 free fonts in the Math style. 1001 Free Fonts offers a huge selection of free fonts. Symbol Fonts Choose from over 560 symbols to add interest to web and print design projects of all kinds. It is the process of joining two metal parts. Technical Report on Math Font Encoding (Justin Ziegler) (332212 bytes, 93 pages, uses AMS and DC fonts) Math Font Group (Joint LaTeX3 Project / TUG Technical Working Group on Extended Math Font Encodings) Example of mathematical typesetting (contributed by Thierry Bouche). Math Symbol. Mathematical Symbols. Font Squirrel relies on advertising in order to keep bringing you great new free fonts and to keep making improvements to the web font generator. So now AMS-LaTeX is just a set of LaTeX packages. The world's largest free font site. But in normal copy text, I've found it necessary to use consistently Equation Editor inline formulas for all math expressions (provided that I use it for display formulas). Is there a way to show the mathematical symbol for power (X^Y) appropriately? Edit: The following question What HTML tags are allowed on Stack Exchange sites? provides a complete reference. 0 had already offered. A new question is presented daily on a variety of subjects including math, science, language arts, and history. If you want to help but don't know where then join the developer list and introduce yourself. Download Math stock photos. Mathematics. In addition, the collection includes a set of miscellaneous TeX fonts that augment the standard Computer Modern set. to add packages which modify the fonts used for math, such as '\\usepackage{newtxsf}' for sans-serif fonts, or '\\usepackage{fouriernc}' for serif fonts. Prepressure > Fonts > Font basics > List of character, symbol & sign names List of character, symbol & sign names Below is an overview of 75 frequently used characters, punctuation symbols or signs that are included in most fonts. (all the pages in this section need a unicode font installed - e. Lots symbols look similar but mean different things. Or Click here for a List of the 200 Greatest of All Time. Is there a mathematical symbol that means "For every element"? I want the meaning to have similar function as an iteration, say for loop. All humans exhibit this mathematical propensity, even little children. So What? If all you're doing is using with Word, the Insert Symbol tool may still be working for you. Arial Unicode MS, Doulos SIL Unicode, Lucida Sans Unicode - see: The International Phonetic Alphabet in Unicode. Financial math has as its foundation many basic finance formulas related to the time value of money. How to type all symbols characters for Windows, Mac, and in HTML. The driver informs the windows what fonts should be downloaded to the printer and the ones that not. Authors Top. It does seem that Word will render the symbol by "copy and paste". 1 Segoe UI Symbol icon font. Θ Theta, the 8th letter of Greek alphabet. On the expanded Text Formatting toolbar, click Symbol. Great Minds is a non-profit organization founded in 2007 by teachers and scholars who want to ensure that all students receive a content-rich education. For a complete set of universal proofreading symbols, the Writers' and Artists' Yearbook. As you can see in the example below, there are three formulas, all of them both starting and finishing with$\$ symbol. Math Symbol: All Imaginary Numbers. Download Math Symbols Normal font free! - FontZone. HV Math comprises the fonts necessary for mathematical typesetting (maths italic, math symbols and extensions) in normal and bold weights. How to write an equation with the align environment with the amsmath package is described in Advanced Mathematics. Note Windows supports TrueType fonts or fonts that are designed especially for Windows which can be purchased separately. MathJax renders mathematics dynamically so that formulas and symbols are nicely integrated into the surrounding text — with matching font size, margins, and baseline. The Easiest way to download fonts for free with awesome graphic samples, Lets quickly browse all fonts by today's categories - FontMagic Free Lord Ganpati Symbols. This video contain list of all mathematical basic symbols you need to know. It is a TrueType font, in files named symbol. I don't have much experience with field codes and can't tell how much the font issue matters there. This chart is intended to be used as an 'aide memoire' for some of the most common proofreading marks. Inserting Images; Tables; Positioning. 2 days ago · The Lexus symbol has a stylized L together with an oval that is around it. Thefileunicode-math-table. SMART Notebook Math Tools also recognizes many mathematical functions. It is licensed to purchasers of these products, and may not be redistributed without the express permission of Wolfram Research. Try, buy and download professional fonts for desktop, web and apps. This article shows several fonts for use in math mode. GRE Math Conventions 6 Example B: 1 d x p Example C: T is the tens digits of a two-digit positive integer, so T is an integer from 1 to 9. Code 2000 used to support all the currency symbols. With other editors I would normally just look the font up in the Font Book app and copy/paste the characters I need from there. Description. Find the best 13 free fonts in the Math style. The new font is saved in the Windows\Fonts folder. New fonts are added daily. Use InDesign’s Glyphs panel and choose Show: Math Symbols, to see all your options in the font you’re using. For example, Chinese, Japanese, Cyrillic, and Arabic fonts use Unicode to display more than the 255 characters that can fit in simple one-byte ASCII. The main difference is that enough symbols have been added to represent all necessary vowel sounds. The Select Equation utility of Kutool for Word can quickly select all equations, and then you can change the font style or font size for all equations in the document quickly. Purchase downloadable Adobe Type fonts for commercial use from best online collection. For instance, it's customary to represent real numbers with a blackboard bold font, or topological spaces with calligraphic font. Note: these may not. Similarly the symbols used to represent elementary particles are always roman. Formula Sheet and List of Symbols, Basic Statistical Inference. In addition, particulars related to certain financial instruments (bonds for example) are calculated using derivatives of these basic formulas. It would be inconvenient to have to exit math mode to type the text in normal roman type, so the most common function names are defined as special commands. ShareLaTeX includes a history of all of your changes so you can see exactly who changed what, and when. Texmaker, free cross-platform latex editor. It sounds like Adobe is attempting to use its default fonts instead of the document's fonts. Create the Path to Student Success. Many (but not all) graphemes that are part of a writing system that encodes a full spoken language are included in the Unicode standard, which also includes graphical symbols. Download 45038 fonts for Windows and Macintosh. In Unicode, the angle symbol is U+2220. Standards for Mathematical Practice » Reason abstractly and quantitatively. The following table lists MathType’s fonts. , conveniently arranged in 222 galleries. Noto Sans Symbols2. This site will not work if cookies are completely disabled. They also look good enough to be used for decorations. In a similar manner to the summation symbol, the product symbol can easily be added to a LaTeX document using the \prod notation. Try fonts from selection of high quality & professional desktop and web fonts. So What? If all you're doing is using with Word, the Insert Symbol tool may still be working for you. Download Free Fonts from Font Bundles. sample proportion.

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## Thursday, 14 August 2014

### Getting Gyroscope values from MPU-6050

Part 4 in my series on the MPU-6050, In the last entry we got the roll and pitch values out of the Accelerometer and displayed them on a little LabVIEW vi with some dials. In this entry I want to get to the gyroscope values and see if I can get them displaying on some dials too. After we have the gyroscope values we can look at combining the 2 sets of data and having a useful Roll,Pitch and Yaw sensor.

### How does a gyroscope work?

A gyroscope is a MEMS which contain small strips of metal that bend when the gyro twists and moves. By measuring the amount of bending the gryo can accurately report what angular velocity it is experiencing (1)

I found Pieter Jan's blog post on this very useful to understand this subject, all the maths is from there.

#### Well what is "Angular velocity"?

Angular velocity is "The rate of change of angular position" and it's units are "degrees per second". The gyroscope has 3 axis so the value we get when we poll each axis is going to be the angular velocity at the current time in deg/sec.

$$Angular\: Velocity = \: \dot{\theta}= \: \frac{\mathrm{d}\theta }{\mathrm{d} t}$$
where theta is just the angle.

We really want the angle not the rate of change of the angle so if you remember from calculus, to get rid of a derivative you integrate it giving us.

$$\theta(t)\:=\:\int_{0}^{t}\dot{\theta}\:dt$$
So all we need to do is integrate the 3 axis values and it will give us the current angle, now you ask yourself..

#### How do we integrate in software?

As we are using a microcontroller (digital system) we are going to need to make a software approximation of an integral. You may remember that an integral is just the area under a curve so what if we just took the current sample and multiplied it by the time since the last sample to give us the area of a rectangle and we just sum this and all other rectangles together to give us the area under the curve as shown in the photo below.

Where u(t) is our current value and T is the sample time. This is known as the rectangular approximation  it is good enough for what we need but as you can see from the diagram if the sample time is too long then bits of the graph may be overestimated (the red bits going over the curve). A more accurate approximation is the trapezoidal but we will just use rectangular for ease.

The pseudocode of what we are trying to achieve is going to be:
1. Start timer
2. Wait for timer to elapse
4. Multiply each by the sample time to get area of rectangle
5. Add area to sum of all previous areas to give us current position

Building on the project we made in the last part I am going to get the PSoC to constantly sample the gyroscope values and when it is polled with the character "g" at 9600 baud it will respond with the current integrated angles.

### What sample time should I choose?

I will finish this soon

$$E=mc^2$$

## Sunday, 20 July 2014

### Getting Roll, Pitch and Yaw from MPU-6050

Part 3 in our quest for Accelerometer and gyroscope values. In the previous entry we got the acceleration values in the form of G's and also the linear velocity in terms of degrees/s however these values aren't much use to us. Wouldn't it be better if we had roll,pitch and yaw and also angular position instead? Well that's what I am going to try to achieve in this post.

### Roll, Pitch and Yaw

For those that don't know what these are look at the diagram above, it shows that the roll is the rotation about the Y axis, the pitch is the rotation about the X axis and the yaw is the rotation about the Z axis.

#### How do we convert gravity into these values?

Well I am not going to go into the theory of how these values convert to roll and pitch because some nice mathematicians over on stackoverflow have done that for us.
The formulas to convert the acceleration values to pitch and roll are as follows.
Roll = atan2(Y, Z) * 180/PI;
Pitch = atan2(X, sqrt(Y*Y + Z*Z)) * 180/PI;
We use atan2 as it works over 4 quadrants, if we used atan then we would have problems with angles over 90 degrees. Don't worry about Yaw for now as we can't determine that using just the accelerometer, this is intuitive if you think about how Yaw is rotation about the Z axis. We will just read a constant 1G in the Z axis no matter how the accelerometer is rotated, We will need to augment our acceleration values with the Gyroscope ones to get the full picture which we will do in a later entry.

I have rewritten the PSoC code to respond to an interrupt on the UART. Whenever the PSoC receives the character 'a' it will respond with the acceleration values in comma seperated form of X,Y,Z.

So if you send 'a' it will give you back something like "0.1, 0.2, 1"  < these are all abitrary but something around 0 for X, 0 for Y and 1 for Z is expected if the MPU6050 is on a flat surface.

The new PSoC code can be found on my Github under "GettingG'sWithInterrupt" I have left the conversion to Roll and Pitch in the code but kept it commented out as I will be doing the conversion from LabVIEW but just uncomment it if you want the conversion on the PSoC.

### Visualising our values

Now this part is up to you are the PSoC code will work on any platform/language aslong as you poll it with the character 'a' at 9600 Baud. I personally used LabVIEW to do the polling as it has some nice inbuilt dials and bar graphs as can be seen in the screenshot below.

I have uploaded a Windows exe on my Github under /Examples/LabVIEW/Dials/EXE//MPU6050dials.exe of the Labview progam, please note you will have to install the LabVIEW runtime engine to run the EXE

I have also added the vi if you want to have a go at changing it.

Instructions for use
Change COM port to whatever your device is connected to, click write then read, you should see the text box fill up with values and you should see the LED blinking on the PSoC pioneer. If not you probably have a problem with your serial port connections.

In the next entry I will look at getting the Gyroscope values off the PSoC onto LabVIEW dials and then move onto combining the values and filtering them to get accurate non noisy values. You will notice at the minute the values jump around a lot, we need to get rid of this.

## Friday, 18 July 2014

### Processing data from MPU-6050

In the previous entry we got the MPU-6050 talking to the PSoC 4 over I2C and had it giving us raw values on a serial screen. Now let us take it further and see if we can turn those values into angles and such.

### Finding out the offsets

First of all we need to work out the offset errors and remove them from our data. Offsets are caused by a number of factors such as mechanical assembly, mounting, package damage and temperature fluctuations. So we need to work out what the offsets of our chip are and then remove them before processing the data.

I am getting all of my methods and theory from this application note from Freescale.

So what we need to do is to place the sensor level on a table (use a spirit level or something to make sure it is actually level), then we need to:

1. Measure the average acceleration/angle in the X, Y and Z axis over 10-20 samples
2. These values should be 0G in the X and Y axis and will be 1G in the Z axis
3. Anything over 0G in the X and Y and 1G in Z will be our offsets
So after keeping the device level and leaving a large (0.5 second) delay in between each sample (to let temperature stabilise) my average values are as shown below.

Test finished, offset values are shown below

AXoff:435, AYoff:-352, AZoff:15236 || GXoff:74, GYoff:131, GZoff:-126,

Don't worry if yours were different we are going to incorporate the calibration into our future code so that the device can auto zero regularly.

So from now on when we measure we will need to remove those offsets from our values before we process them.

The offset testing program is available on my Github under "CalibrateOffsets"

### Getting some G's

Now we know what our offsets are but we don't know what they mean in the form of angles and G's so lets do some scaling and conversion to get the real values.

#### Understanding the scaling

As Jeff Rowberg said in his own i2cdevlib forums
"The accelerometer and gyroscope measurements are explained in the MPU-6050 datasheet in the GYRO_CONFIG and ACCEL_CONFIG register descriptions (sections 4.4 and 4.5 on pages 14 and 15). The scale of each depends on the sensitivity settings chosen, which can be one of +/- 2, 4, 8, or 16g for the accelerometer and one of +/- 250, 500, 1000, or 2000 deg/sec for the gyroscope. The accelerometer produces data in units of acceleration (distance over time2), and the gyroscope produces data in units of rotational velocity (rotation distance over time). The output scale for any setting is [-32768, +32767] for each of the six axes. The default setting in the I2Cdevlib class is +/- 2g for the accel and +/- 250 deg/sec for the gyro. If the device is perfectly level and not moving, then:
• X/Y accel axes should read 0
• Z accel axis should read 1g, which is +16384 at a sensitivity of 2g
• X/Y/Z gyro axes should read 0
In reality, the accel axes won't read exactly 0 since it is difficult to be perfectly level and there is some noise/error, and the gyros will also not read exactly 0 for the same reason (noise/error)."
So first of all we need to know the sensitivity of our sensor. The default is +/-2G and +/-250deg/sec but where do we confirm we are running the default settings and how do we change the values.

If you look in mpu6050.c at line 66 you will find the function

Here you can see that we are running at the default selection of 2G for the accelerometer and 250 for the gyro, If you look at the defines in mpu6050.h you can change these to whatever you want.

So we know we are running at a sensitivity of +/-2G and +/- 250deg/s but how do our values correspond to those accelerations/angles.

These are both straight line graphs and we can work out from them that for 1G we will read 16384 and for 1degree/sec we will read 131.07 (Although the .07 will get ignore due to binary) these values were just worked out by drawing the straight line graph with 2G at 32767 and -2G at -32768 and 250/-250 at the same values.

So now we know our sensitivity values (16384 and 131.07) we just need to minus the offsets from our values and then devide by the sensitivity as it explains in the freescale application note.

These will work fine for the X and Y values but as the Z was recorded at 1G and not 0 we will need to minus off 1G (16384) before we divide by our sensitivity. This is explained in the freescale note.

I have written a project which calculates the values in G's and angles and it can be found on my GitHub under "GetG's"

## Thursday, 17 July 2014

### Getting started with MPU-6050 IMU

It has been a while but I have been struggling to get this thing working and hopefully my instructions might help someone else out there.

So the MPU-6050 is an IMU unit which has an accelerometer and a gyroscope built into it and communicates over I2C, it also has some complex processor in there too called a DMP which filters and processes all the results but I haven't got that working yet - Maybe as this blog develops I will post about how I got it up and running but let us just concentrate on the basics for now.

I am using the GY-521 breakout board which I bought off eBay for a couple of quid.

### Wiring

The board communicates over I2C and it has internal pull up resistors built in so all you have to do is connect the following. Note that the SDA and SCL lines will be specific to your microcontroller I am using a PSoC 4 Pioneer board but I also tested it on an Arduino UNO)

• VCC - 5V (I have not tested 3.3V)
• GND - 0V
• SDA - SDA on your microcontroller (P0[5] on PSoC 4)
• SCL - SCL on your microcontroller (P0[4] on PSoC 4)

Here it is connected to the PSoC 4

### Software

Here is where things get interesting because it is very easy to get communicating and get the raw values out of the sensor but actually understanding those values, filtering them and getting useful data from them is a different story.

Let us just start though by seeing if we can get some values out of the chip.

Jeff Rowberg has created a nice library for the MPU-6050 on his Github it is an extensive I2C library for many other development boards and is definitely worth checking out. I am going to be using it in our project except as I am using a PSoC and not an Arduino or a MSP430 I will be using an edited version of the i2cdevlib designed to work with the PSoC 4 Pioneer which a kind individual named Hernán Sánchez released on the PSoC forums.

I have altered his project to display the values in a more useful way and also to move a lot of the extraneous code out to another source file, I think this will help you understand more what is happening.

You can find my getting started code on my Github under Examples "Getting Started"

Just connect a serial monitor (PuTTY) to the relevant port and you should see the connection being made and the raw values being printed to the screen.

In the next post I am going to look at turning the values into angles and accelerations but for now, keep watching the skies.

## Wednesday, 2 January 2013

### Installing MongoDB on the Raspberry Pi

This will become a full post soon

$PATH=$PATH:/opt/mongo/bin/
\$ export PATH

### Node.js Raspberry Pi Tutorial 2 - Our first site

Well this is part 2 in a series of blogs I am making surrounding the Raspberry Pi and node.ns - Here is part 1

So in the last tutorial we installed node.js 0.8.16,

The aim of this tutorial is to create a site running on the Express framework.. Express is sort of like the defacto standard now for building web apps in node.js, there are more complex ones such as Meteor and Derby which feature loads more stuff.. and we might go into them in the future.. but they are built using express and the principles we are going to learn here anyway so it is worth starting at the bottom.. I think diving right into Meteor would be confusing..

### So what is express?

Express is the glue which holds our app together.. it is the framework of our app.. express is a series of tools bundled together which when initialized will spit out a ready to use app..

For example, say I wanted to create a website using node.js.. where would I start? Well first of all I know that I would need a HTTP server as that's just how the web works.. So I would look into nodes documentation and pull out its HTTP module and create a server..

So far so good.. Okay well, I am happy with my server just serving up "Hello World" or whatever the most basic example is and I have decided I want to add another page to my site which will serve up some other content.. let's go with a picture of a cat..
Well I only have the http server so I would have to set up some sort of request handler/router which would take the new page url (/cat) and figure out what functions to carry out and then actually execute those functions and pipe the image of the cat to the user.

I now have a server and a router.. (no split files or organisation yet just one file with some router code) say I want to parse the cookies of the user visiting my site so that they can stay logged in even when they leave the tab in their browser.. well I would have to look around in node.js libraries and would use the connect-cookie module and play around with that..
HTML or CSS templating? maybe pull in stylus or Jade and try and get the thing going..
This all sounds like a ball ache right? well that's where our good friend express comes into play..

Express just does all this for you.. all you have to do is type "express myApp" and it will create all the configuration, file organisation and module installation for you.. all you do is sat back and smoke a cigar. So are we agreed.. express is awesome? Okay let's install the thing.

### What is NPM?

NPM if you have never used it is this cool way to install packages in node.. it is much like apt in Raspbian, instead of typing "sudo apt-get install porno" you would type "npm install porno" and it would find the relevant porno for your needs.. the files get dumped in every directory you use the npm install command in.. for example if I was in a directory /bacon and I type npm install random-package, then npm would create a folder inside bacon called "node_modules" and inside that would be my random-package.. node always looks inside the node_modules folder..
/bacon/node_modules/random-package.

There is one other way to install packages in Node and that is globally so instead of just installing them for the current project like I showed above they are installed on the whole computer. This is especially useful for installing packages like express, as you are going to want to use them for every project you set up so it makes sense to be able to use them everywhere..

So that is what we are going to do, install express onto the whole computer but first of all we want to make sure our package manager NPM is up to date..

Type the above into any directory, this will tell NPM to update itself, once this is done type npm -v to make sure you are running an up to date version (1.1.69 is my version).

This will install express on the pi so we can use it in our projects.. the -g is the global install option which is mentioned above..

### Express installed.. let's make an App

Right for this tutorial we are just going to create the most basic Express app, there are some options you can tweak depending on your preferences, but as we are assuming you are all beginners lets not go into those just yet..

First of all find a directory you want to try out an express app in.. for this kind of thing I have a folder called "node-playground" where I test out all of this kind of stuff.. maybe try that.

The above commands make the directory we are going to play in, then the express command makes a separate directory called myApp (or anything you want to call it) with all the relevant files in..

If you have done it right you should see the following screen, I have listed the files inside the folder too

if you have something that looks like above then all is going well.. express it telling us to install the dependencies this app needs.. so we will do that in a second.. this also uses NPM,  (npm is used for everything pretty much so get used to it).. when you download or clone someones app you always have to let npm install the dependencies on your computer and compile them properly. So let's do it..

Change directory into the app if you haven't already like I have done above.. and then type

This should set loads of text going.. what npm is doing here is reading from the file called package.json for which modules express wants to use. It then goes through its libraries to find whatever versions are specified and downloads and installs them to a folder called node_modules.

I will go into package.json and how JSON works soon but for now just know that npm has installed everything express has asked for.. if you get any error which I am sure you won't.. maybe try installing the package build-essential (sudo apt-get install build-essential).. this just contains some basic tools for building packages which npm might want to use.

### That's it.. you have a finished website which will run on the Raspberry Pi

Seriously that's how easy it is.. install express.. npm install and then finally to actually run your app just type
node app.js
This tells node to run it.. if everything is working node should say.. "running on localhost:3000" if you go to the pi's ip and then type :3000 you should see a page saying "welcome to Express"..

eg my pi is on 192.168.0.11 so in chrome I type 192.168.0.11:3000 to see my amazing express app.

Well that's all well and good but how does all this work so I can make my own cool apps I hear you ask.. well full explanation is coming soon.. I am tired right now bed time!

## Tuesday, 1 January 2013

### Installing bcrypt with Windows 8 and Visual studio 2012

Okay so I was using this tutorial here.. which wanted me to use bcrypt.. I am using Windows 8 and Visual Studio 2012 (I have the 2010 C++ stuff installed too as TooLateNate says to on the node-gyp github. Anyway I couldn't get it running with: as bcrypt just wouldn't play ball.. but thanks to finding this thread here I managed to solve the problem with typing So if you are having trouble with anything and you have my problem just type npm install PACKAGE --msvs_version=2012 So for bcrypt happy trails


text:
# The Role of Centering in Dual Regression

Standard

Dual regression is maybe the simplest way to obtain subject-specific estimates of ICA-based resting-state networks (RSNs).  RSNs are regions of the brain that tend to act in a coordinated manner in the absence of a specific task, such as the three shown below (from a 50-component ICA of the Human Connectome Project).

Popular ICA toolboxes for fMRI analysis like GIFT and MELODIC can perform dual regression, but such a simple approach is also straightforward to implement one one’s own.  I end up doing this a lot (and would love to hear in the comments whether others do too or prefer to use existing implementations).  This post is about the important role of centering in dual regression, which is easy to overlook (at least for me) when implementing this simple technique.

In short, given a set of group-level independent components (ICs) each representing a resting-state network, dual regression estimates the subject-level ICs and their time courses by fitting two regression models.  In Regression 1, the group ICs are regressed against the subject’s fMRI data to estimate the subject-level time courses associated with each IC.  In Regression 2, those estimated time courses are regressed against the subject’s fMRI data to obtain estimates of subject-level ICs (spatial maps).  To formalize it a bit:

$T$ – number of volumes (time points) in fMRI timeseries

$V$ – number of vertices or voxels within the brain

$Q$ – number of ICs

$\mathbf{Y}_i\ ({T\times V})$ – fMRI timeseries for subject $i$

$\mathbf{S}_{grp}\ ({Q\times V})$ – group-level ICs

$\mathbf{S}_i\ ({Q\times V})$ – subject-level ICs (to be estimated)

$\mathbf{A}_i\ ({T\times Q})$ – subject-level IC time courses (to be estimated)

Based on the probabilistic ICA model $\mathbf{Y}_i =\mathbf{A}_i\mathbf{S}_i +\mathbf{E}_i$, dual regression fits the following two regression models:

Regression 1: $\mathbf{Y}_i' =\mathbf{S}_{grp}'\mathbf{A}_i' +\mathbf{E}_{i1}'$.

Note that $\mathbf{Y}_i$ has been transposed so that the $V$ locations are treated as data observations.  $\mathbf{S}_{grp}'$ is the design matrix, and we obtain the OLS estimates of the coefficients, $\hat{\mathbf{A}}_i$. This is a multivariate (and multiple) regression model, since the response $\mathbf{Y}_i'$ is a matrix with $T$ columns. In the absence of additional assumptions, this is just equivalent to fitting $T$ separate linear regression models, each with $V$ data observations.  (This makes it very computationally convenient, since we can use matrix operations to estimate the entire coefficient matrix in one step as $\hat{\mathbf{A}}_i' = (\mathbf{S}_{grp}\mathbf{S}_{grp}')^{-1}\mathbf{S}_{grp}\mathbf{Y}_i'$.)

Regression 2: $\mathbf{Y}_i = \hat{\mathbf{A}}_i\mathbf{S}_i +\mathbf{E}_{i2}$.

Note that in this model, $\mathbf{Y}_i$ is not transposed, so that the $T$ time points, rather than the $V$ locations, are treated as data observations.  $\hat{\mathbf{A}}_i$ serves as the design matrix, and we obtain the coefficient estimates $\hat{\mathbf{S}}_i$. Again, this can be estimated efficiently in one step using matrix operations as $\hat{\mathbf{S}}_i = (\hat{\mathbf{A}}_i'\hat{\mathbf{A}}_i)^{-1}\hat{\mathbf{A}}_i'\mathbf{Y}_i$.

Since neither regression model includes an intercept, both the data and design matrix must be centered in order to avoid model misspecification.  And since each model involves a different data matrix and design matrix, we need to worry about centering 4 different matrices. First, though, a brief review of the role of the intercept and centering in linear regression.

### Why center in linear regression?

The goal of simple linear regression is to describe the linear relationship between an explanatory variable x and a response variable y.  The left-hand scatterplot below shows a classic motivating example. In this model, the true intercept is not zero, and fitting the model without the intercept results in a flawed estimate of the relationship between x and y (red line). The model that does include the intercept (blue line) correctly captures the relationship.

Generally, an intercept is required in linear regression, unless (1) there is reason to believe that the intercept is actually zero in truth, or (2) both the response and (each) predictor have been centered.  An example of the former might be the relationship between gestational age of a fetus and its size (although that relationship would likely not be linear).  If the response and predictor(s) have been centered, however, the intercept is exactly zero and can therefore be excluded from the model.  The right-hand scatterplot below shows the same data after centering x and y, overlaid with the estimated linear regression lines with and without an intercept.  The two fits are now exactly the same, showing that the intercept can be dropped from the model.

### Dual regression requires centering across time and space

Coming back to dual regression, then, we should either include an intercept in each model OR properly center both the “data” and the “design matrix” in both regressions.  I find it simpler to do the latter and will therefore focus on centering.  A key point is that we need to center across observations in both regressions, where in Regression 1 the observations are the $V$ locations, and in Regression 2 the observations are the $T$ time points. Therefore, the data must be centered across both space (remove the global or gray matter signal) and time (remove the mean image, centering each time course at zero).

This double-centering can be performed in two steps, and it turns out that it doesn’t matter which is done first.  To see this, allow me to get a bit matrix-y for a moment.  Note that the mean across time for each location is $\tfrac{1}{T}\mathbf{1}_T'\mathbf{Y}_i$, and the data centered across time is $\tilde{\mathbf{Y}}_i = \mathbf{Y}_i - \tfrac{1}{T}\mathbf{1}_T\mathbf{1}_T'\mathbf{Y}_i =(\mathbf{I}_T - \tfrac{1}{T}\mathbf{1}_T\mathbf{1}_T')\mathbf{Y}_i$.  This thing $(\mathbf{I}_T - \tfrac{1}{T}\mathbf{1}_T\mathbf{1}_T')$ is a centering matrix, which is symmetric and idempotent.  Now to center the other way, we can do the same thing to $\tilde{\mathbf{Y}}_i$ after transposing.  The data centered across time then space is $\tilde{\tilde{\mathbf{Y}}}_i' = (\mathbf{I}_V - \tfrac{1}{V}\mathbf{1}_V\mathbf{1}_V')\tilde{\mathbf{Y}}_i' =(\mathbf{I}_V - \tfrac{1}{V}\mathbf{1}_V\mathbf{1}_V')\mathbf{Y}_i'(\mathbf{I}_T - \tfrac{1}{T}\mathbf{1}_T\mathbf{1}_T')$.  By doing it the other way (centering across space first, then time), we would have ended up with $(\mathbf{I}_T - \tfrac{1}{T}\mathbf{1}_T\mathbf{1}_T')\mathbf{Y}_i(\mathbf{I}_V - \tfrac{1}{V}\mathbf{1}_V\mathbf{1}_V')$, which is just $\tilde{\tilde{\mathbf{Y}}}_i$.  This also shows that the second centering does not somehow “mess up” the first centering, since centering in either order leads to the same result.

This double-centering prior to Regression 1 (along with centering the group ICs $\mathbf{S}_{grp}$ across locations, e.g. $\tilde{\mathbf{S}}_{grp}' =(\mathbf{I}_V - \tfrac{1}{V}\mathbf{1}_V\mathbf{1}_V')\mathbf{S}_{grp}'$) also happens to take care of the second regression, because $\hat{\mathbf{A}}_i$ ends up centered.  To see this, note that $\hat{\mathbf{A}}_i' = (\tilde{\mathbf{S}}_{grp}\tilde{\mathbf{S}}_{grp}')^{-1}\tilde{\mathbf{S}}_{grp}\tilde{\tilde{\mathbf{Y}}}_i'$.  This can be written $\hat{\mathbf{A}}_i' = (\tilde{\mathbf{S}}_{grp}\tilde{\mathbf{S}}_{grp}')^{-1}\tilde{\mathbf{S}}_{grp}\tilde{\mathbf{Y}}_i'(\mathbf{I}_T - \tfrac{1}{T}\mathbf{1}_T\mathbf{1}_T')$, where$\tilde{\mathbf{Y}}_i'$ is only centered across locations (required for Regression 1).  The part of $\hat{\mathbf{A}}_i'$ preceding the centering matrix is what we would have gotten if we had not also already centered $\mathbf{Y}_i$ across time also.  Therefore, by centering $\mathbf{Y}_i$ across time as well as space, our estimate of $\mathbf{A}_i$ is also centered across time.  So both the data and design matrix for Regression 2 are centered, and we’re good to go.

### What happens if we don’t center both ways?

Let’s see what happens if we fail to center the fMRI timeseries data both ways.  Using subject 100206 of the Human Connectome Project, we’ll use dual regression to estimate the subject-level IC’s associated with three group IC’s we showed at the beginning.  Each slideshow below shows one of the IC’s (motor, auditory and executive control), and toggles through the case when the data was centered across space only, time only, or both space and time.

The most blatantly obvious problem occurs when we center across space only, essentially removing the global or gray matter signal.  We haven’t removed the “mean image”, which is then essentially absorbed into the IC estimates.  When we center across time only, we remove the mean image but not the global signal.  In this case, we see more widespread “activation” in all three of the IC’s.  We’d also expect (perhaps even greater) contamination of the time courses associated with each IC in this case, since by failing to center across space, Regression 1 will be misspecified.

This slideshow requires JavaScript.

This slideshow requires JavaScript.

This slideshow requires JavaScript.

So in conclusion, before performing dual regression using your own implementation, center, center, and center some more!


text:
# Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled-resonator loops

## Abstract

Non-reciprocal components, which are essential to many modern communication systems, are almost exclusively based on magneto-optical materials, severely limiting their applicability. A practical and inexpensive route to magnetic-free non-reciprocity could revolutionize radio-frequency and nanophotonic communication networks. Angular-momentum biasing was recently proposed as a means of realizing isolation for sound waves travelling in a rotating medium1, and envisaged as a path towards compact, linear integrated non-reciprocal electromagnetic components2,3. Inspired by this concept, here we demonstrate a subwavelength, linear radio-frequency non-reciprocal circulator free from magnetic materials and bias. The scheme is based on the parametric modulation of three identical, strongly and symmetrically coupled resonators. Their resonant frequencies are modulated by external signals with the same amplitude and a relative phase difference of 120°, imparting an effective electronic angular momentum to the system. We observe giant non-reciprocity, with up to six orders of magnitude difference in transmission for opposite directions. Furthermore, the device topology is tunable in real time, and can be directly embedded in a conventional integrated circuit.

## Main

Early attempts to realize magnetic-free non-reciprocity were based on the non-reciprocal properties of transistors at microwave frequencies4, and on networks of electro-optical modulators at optical frequencies5,6,7,8. However, such approaches traded the absence of magnetic bias with other significant drawbacks, such as the strong nonlinearities and poor noise-performance of transistors, or the large size and complexity of the required electro-optical networks. More recently, non-reciprocity has been achieved in transistor-loaded metamaterials9,10 and nonlinear devices11,12,13,14. Also these solutions impose severe restrictions on the input power levels, generally degrading the signal quality because of noise or signal distortion. Another interesting approach to magnetic-free non-reciprocity has been introduced15, using asymmetric mode conversion in spatiotemporally modulated waveguides. This concept is especially attractive for integrated optical networks, as it may be fully realized in silicon photonics16. However, this technique and its variants17,18,19,20,21,22 lead to structures much larger than the wavelength, owing to the weak electro-optic or acousto-optic effects on which they rely, and require complex modulation schemes. In a similar context, the concept of a non-reciprocal device based on parametrically coupled resonators has been theoretically explored23.

An approach that can lead to compact, magnetic-free non-reciprocal devices with relaxed implementation complexity was recently introduced 1,2,3, based on angular-momentum biasing of a resonant ring. Angular momentum can be applied either by mechanically spinning a fluid, as proved for acoustic waves1, or, more conveniently for electromagnetic waves, by spatiotemporal modulation with a travelling wave, realizing an effective electronic spin2,3, as illustrated in Fig. 1a. The resonant nature of the modulated ring can substantially boost the otherwise weak electro-optic effects through which spatiotemporal modulation is typically achieved24,25, allowing the design of largely non-reciprocal devices with dimensions of the order of, or even smaller than the wavelength. Furthermore, in contrast to other approaches based on spatiotemporal modulation15,19, angular-momentum biasing is based on uniform modulation across the ring cross-section, thus significantly simplifying the fabrication process.

Inspired by these ideas, here we propose a device that provides strong magnetic-free non-reciprocity at the subwavelength scale by dynamically modulating three identical resonant circuit tanks arranged in a loop and strongly coupled to each other, as in Fig. 1b. The resonant frequencies of the individual tanks are temporally modulated in a circularly rotating fashion as ω1(t) = ω0 + δωm cos(ωmt), ω2(t) = ω0 + δωm cos(ωmt + 2π/3) and ω3(t) = ω0 + δωm cos(ωmt + 4π/3), where ω0 is the static value of the resonant frequency, δωm is the modulation amplitude and ωm is the modulation frequency, so that an effective electronic spin is imparted to the system. Without modulation, the loop supports two degenerate counter-rotating modes, similar to the uniform ring of Fig. 1a. However, when the modulation is switched on, this degeneracy is lifted and non-reciprocity is induced. As we show in the following, this solution markedly boosts the modulation efficiency of the device compared to the case of a single modulated resonant ring, as considered in refs 2, 3, largely relaxing the requirements in terms of the modulation intensity and subsequently improving the overall efficiency of the structure.

In the absence of modulation (δωm = 0), the loop of Fig. 1b supports three resonant states: a common state with state vector |c〉 = [1 1 1]T and frequency ωc = ω0 + 2κ, and two degenerate right- and left-handed states with state vectors |±〉 = [1 e±i2π/3 e±i4π/3]T and frequencies ω± = ω0κ, where κ is the coupling coefficient. The applied modulation mixes right- and left-handed states, producing two new hybrid states

where , ωR = ω± − Δω/2 and ωL = ω± + Δω/2, as analytically derived in the Methods for ωm ωcω±.

It is interesting to observe that both |R〉 and |L〉 consist of a dominant state at ωR, L and a secondary state, red- or blueshifted by ωm, as in the case of the uniformly modulated ring of refs 2, 3. However, despite this apparent resemblance in the form of |R〉 and |L〉, the mechanism that creates the frequency separation Δω between resonant states in the geometry analysed here is significantly different from the uniformly modulated ring of refs 2, 3. For a uniform resonant ring (Fig. 1a), Δω is the result of an azimuthally travelling wave modulation Δɛm cos(ωmtlmϕ), where Δɛm and lm are the modulation amplitude and azimuthal order, respectively. If lm = 2l, where l is the resonant order of the ring, δωm = ωlΔɛm/(2ɛ) (refs 2, 3), where ωl is the resonant frequency and ɛ the static permittivity. The ideal continuous modulation assumed in Fig. 1a (top) is difficult, if not impossible, to realize, and an angular discretization is typically required. A discrete modulation profile with N different modulation regions, as sketched in Fig. 1a (bottom), is equivalent to a continuous effective modulation with reduced effective amplitude Δɛm, eff = Δɛmsinc(2l/N) (ref. 3), revealing a fundamental trade-off between fabrication complexity, proportional to N, and non-reciprocal response. As an example, for N = 3—that is, the minimum value for which Δɛm, eff ≠ 0—and l = 50—a typical value in realistic high-Q micro-ring resonators at 1.55 μm—Δɛm, eff = 0.008Δɛm, implying that only 0.8% of the modulation signal is effectively used to generate non-reciprocity.

In contrast, in the composite resonant loop of Fig. 1b the frequency splitting is achieved by modulating the frequency of each resonator, which can be obtained by applying a uniform permittivity modulation with amplitude Δɛm all across each resonator24, leading to δωm = ω0Δɛm/(2ɛ), as shown in the Methods. Remarkably, in this topology the entire modulation signal is used to produce the frequency separation Δω, indicating that maximum (unitary) modulation efficiency is achieved in a simple fabrication scheme requiring only three independent modulation regions.

We realized the structure of Fig. 1b at RF using three basic LC circuit tanks, as in Fig. 2a, where the capacitance C is equally distributed on both sides of the inductance L to maintain a symmetric structure (see Supplementary Methods for a full schematic of the realized circuit). The resonance frequency modulation is achieved by means of capacitance modulation, commonly obtained in RF with varactor diodes. These diodes are biased by two signals, a static signal Vd.c., which provides the required reverse bias and controls the static capacitance, and a RF signal νm with frequency ωm and amplitude Vm, providing the time modulation. Assuming that the resonators are coupled to each other through capacitances Cc, as in Fig. 2b, the frequencies of the common and rotating states are and , respectively, where is the static resonance frequency of each tank. Then, if the amplitude of the capacitance modulation is ΔCm, the frequency modulation amplitude is found as δωm = ω±ΔCm/(2C). The frequency ωc should be designed to be as far as possible from ω± in order for the common mode not to affect the operation of the structure at ω±, at which non-reciprocity occurs. In the lumped-element circuit of Fig. 2b this condition is satisfied by taking Cc, or equivalently by coupling the tanks through short circuits, yielding ωc = 0 and .

The non-reciprocal response of the circuit of Fig. 2b is demonstrated by capacitively coupling it to three microstrip transmission lines, realizing a three-port device. Exciting the structure from, for example, port 1 at frequency ω± results in the excitation of |R〉 and |L〉 with the same amplitude and opposite phase φR = − φL, owing to the symmetrical distribution of these states around ω±. Then, the signals at ports 2 and 3 are proportional to $e i 2 π / 3 e i ϕ R +e i 4 π / 3 e i ϕ L$ and $e − i 2 π / 3 e i ϕ R +e − i 4 π / 3 e i ϕ L$, respectively, as the superposition of |R〉 and |L〉 at these ports. If δωm and ωm are selected so that φR = − φL = π/6, the signal at port 3 is identically zero, while the signal at port 2 is non-zero, routing the incident power from port 1 to port 2. Owing to the symmetry of the structure with respect to its ports, incident power from ports 2 and 3 is similarly routed to ports 3 and 1, thus realizing the functionality of a non-reciprocal circulator with infinite isolation. Notice that the above description assumes a weak excitation of the common state, which makes clear the importance of choosing its resonance frequency as far as possible from the resonance frequency of the rotating states.

The realized device was designed to resonate at 170 MHz with a Q-factor of about 10 for Vd.c. = 1.99 V and Vm = 0. The modulation frequency was set to 15 MHz, in order for the intermodulation by-products at frequencies ω ± ωm, created by the secondary substates of |R〉 and |L〉, to fall outside the resonance band, whose bandwidth is here around 10 MHz. The exact values of the circuit components and its full topology are provided in the Methods and Supplementary Methods. Figure 2c shows a photograph of the fabricated prototype. We underline here the deeply subwavelength size of the realized device (λ/75), simply based on three lumped resonant circuit tanks.

Without modulation, the signal is equally split at the two output ports, as expected from symmetry, and the system is fully reciprocal (Fig. 3a). When the modulation signal is switched on, the symmetry is broken and power is unequally split. By varying the modulation amplitude it is possible to find a value for which the signal entering port 1 is routed exclusively to port 2, corresponding to φR = − φL = π/6. This condition is satisfied for Vm = 0.6 V, as can be seen in Fig. 3b: at the resonance frequency of 170 MHz, power incident to ports 1, 2 and 3 is routed to ports 2, 3 and 1, respectively, demonstrating the operation of an ideal, magnetic-free, deeply subwavelength linear circulator. For comparison, Fig. 3c shows the S-parameters obtained using full-wave and circuit simulations: the agreement with the measurement is excellent.

To get a deeper insight into the effect of Vm on the device operation, Fig. 4a shows the transmission between ports 1 and 2 at resonance versus Vm. For Vm = 0, S21 = S12, as expected. Increasing Vm results in an increase of S21 and a decrease of S12 until Vm = 0.6 V, where S12 = 0. Past this point, S21 and S12 get closer, as expected when we depart from the destructive interference condition. For very large values of Vm, S21 and S12 both tend to zero, because the counter-rotating states move far from ω± and, therefore, are weakly excited at ω±. The magnitude of the asymmetry between S21 and S12 is measured by the isolation |S21/S12|, plotted in Fig. 4b on a logarithmic scale versus Vm. At the optimum modulation voltage Vm = 0.6 V, S21 is over four orders of magnitude larger than S12, indicating giant non-reciprocity, well above the levels of any commercial magnetic-based device.

Another unique property of the proposed device consists in its real-time tunability features. The biasing voltage Vd.c., which provides the reverse biasing condition for the varactor diodes, determines their static capacitance. Therefore, Vd.c. can be used to actively control the static resonance frequency of the LC tanks, and consequently the frequency band over which non-reciprocity occurs. Figure 4c shows the measured isolation versus frequency for Vd.c. varied between 1.73 V and 4.5 V. The non-reciprocal response of our device can be efficiently tuned between 150 MHz and 210 MHz, corresponding to a relative bandwidth of over 30%. Across all this range, our measured isolation is above 40 dB. This strong tuning capability is an additional advantage of the proposed device compared to conventional magnetic-based microwave circulators, and it may be exploited in scenarios requiring dynamic tuning to balance changes in temperature or in the environment. The electronic spin applied to the proposed coupled-resonator loop realizes the equivalent of a dynamically tunable, strongly biased ferromagnetic metamaterial substrate.

In addition to being an ideal replacement for microwave non-reciprocal components, with significant advantages in terms of size, integration, cost, linearity, tunability and noise reduction, our findings may become even more important when applied to different frequencies or other types of waves, such as light or sound. For instance, this concept may be disruptive for integrated nanophotonic technology, for which optical non-reciprocal components are critical for laser protection and signal routing. At visible frequencies, electro-optic modulation in silicon-based components is typically achieved by means of carrier injection/depletion24,25, which can provide relatively strong permittivity modulation, but is typically accompanied by significant loss and low modulation frequencies for large modulation amplitudes. These side effects impose limitations on the applicability of the principle of angular-momentum biasing in uniform micro-ring resonators2,3, as discussed before. In contrast, the concept presented here ensures maximum modulation efficiency, significantly relaxing the requirements in terms of modulation amplitude. This in turn allows large quality factors and large modulation frequencies, which translate into strong non-reciprocal response in deeply subwavelength devices. We also envisage the realization of the proposed rotating modulation of coupled-resonator loops in photonic crystal technology, for which high-Q-factor coupled cavities may be implemented and efficiently modulated26. Our study also represents a new demonstration of the exciting possibilities offered by dynamic modulation of coupled-resonator networks, with unique control over the flow of light, in the context of recently presented concepts of photonic topological edge states and effective magnetic fields for photons27,28,29.

## Methods

### Modes of the coupled-resonators loop.

The coupled-mode equations of the system of Fig. 1 read

where a1, a2 and a3 are the complex amplitudes of the three resonators. Equation (2) can be written in the more compact form

where is the state vector of the system and

In the absence of modulation, ω1, ω2 and ω3 are equal to ω0. Then, the eigenfrequencies and the corresponding state vectors of the system are found by the eigenvalues and eigenvectors of Ω, respectively. In particular, it can be shown that there are three states, a common state with resonant frequencyωc = ω0 + 2κ and state vector , a right-handed rotating state with resonant frequency ω+ = ω0κ and state vector , and a left-handed rotating state with resonant frequency ω = ω+ and state vector . Note that because the eigenvalues ω+ and ω are degenerate, the eigenvectors |+〉 and |−〉 are not the only ones corresponding to these eigenvalues. As a matter of fact, any linear combination of |+〉 and |−〉, such as the vectors and , are also valid eigenvectors of ω+ and ω. However, hereafter, we will use |+〉 and |−〉, because they bear an immediate physical meaning as counter-rotating states of the coupled-resonators loop, and they significantly simplify the mathematical analysis.

In the presence of modulation ω1(t) = ω0 + δωm cos(ωmt), ω2(t) = ω0 + δωm cos(ωmt + 2π/3) and ω3(t) = ω0 + δωm cos(ωmt + 4π/3) result in a time-dependent Ω. In this case, the eigenstates of the system cannot be found by the eigenvalues and eigenvectors of Ω, and a full solution of equation (3) is necessary. Because modulation constitutes a perturbation of the coupled system, it is convenient to express the eigenstates of the modulated system in terms of the eigenstates of the coupled non-modulated system. To this end, equation (3) is expressed in the basis of |c〉, |+〉 and |−〉, by multiplying |ψ〉 with the conjugate transpose of the matrix

with columns the state vectors |c〉, |+〉 and |−〉. In the new basis, , where ac, a+ and a are the complex amplitudes of the common, right- and left-handed states, respectively, and

where

If ωm | ωcω±|, coupling between the common and rotating states, corresponding to the first row and column of δΩ, can be neglected and equation (3) simplifies to

It can be seen that equation (4) is satisfied if

where A+, A and ω are constants. Then, equation (4) becomes

which is an eigenvalue problem with respect to ω. The eigensolutions of this problem yield the eigenstates of the modulated system, as given by equation (1).

### Modulation amplitude of the coupled-resonators system.

We assume a resonator whose modal distribution fully resides in materials with permittivity ɛ, permeability μ, and resonant frequency ω0. It can be shown from perturbation theory that a small change Δɛ of the permittivity results in the following change of the resonance frequency30

where E0 and H0 are the resonant electric and magnetic field, respectively, and integration is performed all over the volume of the resonator. Furthermore, it is known that at resonance the electric and magnetic energies are equal, hence ∫ ɛ |E0| 2dV = ∫ μ |H0| 2dV and

If, in addition, ɛ and Δɛ are uniform over the volume where most of the resonator’s energy is concentrated, ∫ Δɛ |E0| 2dV Δɛ∫ |E0| 2dV and ∫ ɛ |E0| 2dV ɛ∫ |E0| 2dV, showing that the frequency modulation, δωm, for a permittivity modulation amplitude Δɛm is equal to δωm = ω0Δɛ/(2ɛ).

### Description of the experimental set-up.

The complete experimental set-up is shown in Supplementary Fig. 2 and a list of the associated equipment is provided in Supplementary Table 2. A waveform generator provides the modulation signal, which is split into three equal parts by means of a power divider. The output signals are then led to three phase shifters, which provide the required phase difference of 120° for the modulation signals of the three coupled resonators. The phase shifters are powered by a d.c. source and their phase shift is controlled by potentiometers. The outputs of the phase shifters are connected to the low-pass ports of three diplexers, whose outputs are connected to the RF/modulation ports of the ring. The high-pass ports of two of the diplexers are connected to the VNA ports while the high-pass port of the third diplexer is terminated to a matched load. The diplexers combine the modulation and RF signals and at the same time provide infinite isolation between the RF and modulation paths. By rotating the diplexers, which are connected to the VNA ports, it is possible to measure all the S-parameters of the circuit. The d.c. signal for biasing of the varactors is provided by a d.c. source connected to ports 4, 5 and 6 of the ring.

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## Acknowledgements

The work was supported in part by AFOSR with YIP award No. FA9550-11-1-0009 and DTRA with YIP award No. HDTRA1-12-1-0022.

## Author information

Authors

### Contributions

N.A.E. performed the experiment. D.L.S. and N.A.E. designed the structure and conducted the numerical calculations and theoretical modelling. J.S. helped in the selection and modelling of the modulation varactors. A.A. directed and supervised the project. All authors have read and commented on the paper.

### Corresponding author

Correspondence to Andrea Alù.

## Ethics declarations

### Competing interests

The authors declare no competing financial interests.

## Supplementary information

### Supplementary Information

Supplementary Information (PDF 536 kb)

## Rights and permissions

Reprints and Permissions

Estep, N., Sounas, D., Soric, J. et al. Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled-resonator loops. Nature Phys 10, 923–927 (2014). https://doi.org/10.1038/nphys3134

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# 2.32: Cell Cycle

Difficulty Level: At Grade Created by: CK-12
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What is a cell's life like?

The eukaryotic cell spends most of its "life" in interphase of the cell cycle, which can be subdivided into the three phases, G1, S and G2. During interphase, the cell does what it is supposed to do. Though cells have many common functions, such as DNA replication, they also have certain specific functions. That is, during the life of a heart cell, the cell would obviously perform certain different activities than a kidney cell or a liver cell.

### The Cell Cycle

Cell division is just one of several stages that a cell goes through during its lifetime. The cell cycle is a repeating series of events that include growth, DNA synthesis, and cell division. The cell cycle in prokaryotes is quite simple: the cell grows, its DNA replicates, and the cell divides. In eukaryotes, the cell cycle is more complicated.

#### The Eukaryotic Cell Cycle

The diagram in Figure below represents the cell cycle of a eukaryotic cell. As you can see, the eukaryotic cell cycle has several phases. The mitotic phase (M) actually includes both mitosis and cytokinesis. This is when the nucleus and then the cytoplasm divide. The other three phases (G1, S, and G2) are generally grouped together as interphase . During interphase, the cell grows, performs routine life processes, and prepares to divide. These phases are discussed below. You can watch a eukaryotic cell going through these phases of the cell cycle at the following link: http://www.cellsalive.com/cell_cycle.htm .

The Eukaryotic Cell Cycle. This diagram represents the cell cycle in eukaryotes. The G1, S, and G2 phases make up interphase (I). The M phase includes mitosis and cytokinesis. After the M phase, two cells result.

#### Interphase

Interphase of the eukaryotic cell cycle can be subdivided into the following three phases, which are represented in Figure above :

• Growth Phase 1 (G1): during this phase, the cell grows rapidly, while performing routine metabolic processes. It also makes proteins needed for DNA replication and copies some of its organelles in preparation for cell division. A cell typically spends most of its life in this phase. This phase is sometimes referred to as Gap 1.
• Synthesis Phase (S): during this phase, the cell’s DNA is copied in the process of DNA replication .
• Growth Phase 2 (G2): during this phase, the cell makes final preparations to divide. For example, it makes additional proteins and organelles. This phase is sometimes referred to as Gap 2.

#### Control of the Cell Cycle

If the cell cycle occurred without regulation, cells might go from one phase to the next before they were ready. What controls the cell cycle? How does the cell know when to grow, synthesize DNA, and divide? The cell cycle is controlled mainly by regulatory proteins. These proteins control the cycle by signaling the cell to either start or delay the next phase of the cycle. They ensure that the cell completes the previous phase before moving on. Regulatory proteins control the cell cycle at key checkpoints, which are shown in Figure below . There are a number of main checkpoints.

• The G1 checkpoint, just before entry into S phase, makes the key decision of whether the cell should divide.
• The S checkpoint determines if the DNA has been replicated properly.
• The mitotic spindle checkpoint occurs at the point in metaphase where all the chromosomes should have aligned at the mitotic plate.

Checkpoints (arrows) in the eukaryotic cell cycle ensure that the cell is ready to proceed before it moves on to the next phase of the cycle.

#### Cancer and the Cell Cycle

Cancer is a disease that occurs when the cell cycle is no longer regulated. This may happen because a cell’s DNA becomes damaged. Damage can occur due to exposure to hazards such as radiation or toxic chemicals. Cancerous cells generally divide much faster than normal cells. They may form a mass of abnormal cells called a tumor (see Figure below ). The rapidly dividing cells take up nutrients and space that normal cells need. This can damage tissues and organs and eventually lead to death.

These cells are cancer cells, growing out of control and forming a tumor.

### Summary

• The cell cycle is a repeating series of events that cells go through. It includes growth, DNA synthesis, and cell division. In eukaryotic cells, there are two growth phases, and cell division includes mitosis.
• The cell cycle is controlled by regulatory proteins at three key checkpoints in the cycle. The proteins signal the cell to either start or delay the next phase of the cycle.
• Cancer is a disease that occurs when the cell cycle is no longer regulated. Cancer cells grow rapidly and may form a mass of abnormal cells called a tumor.

### Practice

Use this resource to answer the questions that follow.

• http://www.hippocampus.org/Biology $\rightarrow$ Biology for AP* $\rightarrow$ Search: The Cell Cycle
1. What is interphase?
2. What occurs during the S-phase?
3. What is G 0 ?
4. How long can nerve cells and liver cells remain in G 0 ?
5. Describe the outcomes of density-dependent inhibition.
6. What is a growth factor? Give an example.

### Review

1. Identify the phases of the eukaryotic cell cycle.

2. What happens during interphase?

3. Define cancer.

4. How might the relationship between cancer and the cell cycle be used in the search for causes of cancer?

5. Cells go through a series of events that include growth, DNA synthesis, and cell division. Why are these events best represented by a cycle diagram?

6. Explain how the cell cycle is regulated.

7. Why is DNA replication essential to the cell cycle?

### Vocabulary Language: English Spanish

cancer

cancer

Disease that occurs when the cell cycle is no longer regulated and cells divide out of control.
cell cycle

cell cycle

Repeating series of events that a cell goes through during its life, including growth, DNA, synthesis, and cell division.
interphase

interphase

Stage of the eukaryotic cell cycle when the cell grows, synthesizes DNA, and prepares to divide.
mitotic phase

mitotic phase

Stage of the eukaryotic cell cycle that includes mitosis and cytokinesis.
tumor

tumor

Abnormal mass of cells that may be cancerous.
DNA replication

DNA replication

Process of copying DNA prior to cell division.

Feb 24, 2012

Nov 11, 2014