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The importance of complex number in real life: In real numbers, we can represent this number as a straight line. Comparison between Real Number and Complex Number: A real number is a number that can take any value on the number line. The complex numbers are referred to as (just as the real numbers are . Then, the product and quotient of these are given by, Example 21.10. Similarly, when a negative number is squared it also provides a positive number. The real part of z is denoted by Re(z) and the imaginary part by Im(z). Its algebraic form is , where is an imaginary number. If b is not equal to zero and a is any real number, the complex number a + bi is called imaginary number. Real Numbers and Complex Numbers are two terminologies often used in Number Theory. Imaginary Numbers when squared give a negative result. New York City College of Technology | City University of New York. A complex number is a number of the form . HINT: To ask a question, start by logging in to your WeBWorK section, then click  “Ask a Question” after any problem. Its algebraic form is z=x+i*y, where i is an imaginary number. A complex number is the sum of a real number and an imaginary number. Here r = √x2 + y2 = |z| is the modus of z and θ is called argument(or amplitude) of z is denoted by arg z. Thus, the complex numbers of t… Complex Numbers are considered to be an extension of the real number system. VIDEO: Multiplication and division of complex numbers in polar form – Example 21.10. Therefore a complex number … The absolute value of , denoted by , is the distance between the point in the complex plane and the origin . A complex number is expressed in standard form when written a + bi where a is the real part and b is the imaginary part. Square root of -1 is also not a real number, and therefore it is referred to as an imaginary number. Point P is uniquely determined by the ordered pair of a real number(r,θ), called the polar coordinatesof point P. x = r cosθ, y = rsinθ therefore, z=r(cosθ + isinθ) where r =√a2 + b2 and θ =tan-1 =b/a The latter is said to be polar form of complex number. If we define a pure real number as a complex number whose imaginary component is 0i, then 0 is a pure real number. Image Courtesy: mathpowerblog.wordpress.comom, wikipedia.org. A real number can store the information about the value of the number and if this number is positive or negative. Keep visiting BYJU’S to get more such maths lessons in a simple, concise and easy to understand way. This .pdf file contains most of the work from the videos in this lesson. A real number refers to any number that can be found on this number line. We start with the real numbers, and we throw in something that’s missing: the square root of . So, too, is $$3+4\sqrt{3}i$$. Many amazing properties of complex numbers are revealed by looking at them in polar form! The OpenLab is an open-source, digital platform designed to support teaching and learning at City Tech (New York City College of Technology), and to promote student and faculty engagement in the intellectual and social life of the college community. Every real number is a complex number, but not every complex number is a real number. Login information will be provided by your professor. I – is a formal symbol, corresponding to the following equability i2 = -1. Number line can be expressed as an actual geometric line where a point is chosen to be the origin. Give the WeBWorK a try, and let me know if you have any questions. We can picture the complex number as the point with coordinates in the complex plane. Complex numbers are a bit unusual. So, a Complex Number has a real part and an imaginary part. Let and be two complex numbers in polar form. Multiply or divide the complex numbers, and write your answer in polar and standard form.a) b) c) d). Because no real number satisfies this equation, i is called an imaginary number. A complex number is a number of the form . Therefore, all real numbers are also complex numbers. Convert the number from polar form into the standard form a) b), VIDEO: Converting complex numbers from polar form into standard form – Example 21.8. The complex numbers are referred to as (just as the real numbers are . A complex number is the sum of a real number and an imaginary number. Start at the origin. 3. and are allowed to be any real numbers. We call this the polar form of a complex number. Once they're understood, they're very simple and easy-to-use for just about anyone. (adsbygoogle = window.adsbygoogle || []).push({}); Copyright © 2021, Difference Between | Descriptive Analysis and Comparisons. Multiplying Complex Numbers. Your email address will not be published. However, we have to be a little careful: since the arctangent only gives angles in Quadrants I and II, we need to doublecheck the quadrant of . Similarly, 3/7 is a rational number but not an integer. But in complex number, we can represent this number (z = … Learn more about accessibility on the OpenLab, © New York City College of Technology | City University of New York. Imaginary number consists of imaginary unit or j operator which is the symbol for √-1. Hi everyone! They can be any of the rational and irrational numbers. Logged-in faculty members can clone this course. By the Pythagorean Theorem, we can calculate the absolute value of as follows: Definition 21.6. A complex number is represented as z=a+ib, where a … For information about how to use the WeBWorK system, please see the WeBWorK  Guide for Students. Let be a complex number. Multiplying a Complex Number by a Real Number. We define the imaginary unit or complex unit to be: Definition 21.2. We distribute the real number just as we would with a binomial. The quadratic formula solves ax2 + bx + c = 0 for the values of x. That’s it for today! All imaginary numbers are also complex numbers with zero for the real part. A complex numberis defined as an expression of the form: The type of expression z = x + iy is called the binomial form where the real part is the real number x, that is denoted Re(z), and the imaginary partis the real number y, which is denoted by Im(z). A complex number is a number having both real and imaginary parts that can be expressed in the form of a + bi, where a and b are real numbers and i is the imaginary part, which should satisfy the equation i 2 = −1. How do we get the complex numbers? Points that fall in the right side of origin are considered positive numbers, whereas numbers lying in the left side of origin are considered to be negative. Any real number is a complex number. Complex Numbers are considered to be an extension of the real number system. Difference Between | Descriptive Analysis and Comparisons, Counterintelligence Investigation vs Criminal Investigation. Let be a complex number. Complex Numbers Complex Numbers 7 + 3 Real Imaginary A Complex Number A Complex Number is a combination of a Real Number and an Imaginary Number Real Numbers are numbers like: 1 12.38 −0.8625 3/4 √2 1998 Nearly any number you can think of is a Real Number! start by logging in to your WeBWorK section, Daily Quiz, Final Exam Information and Attendance: 5/14/20. If z = 3 – 4i, then Re(z) = 3 and Im(z) = – 4. Likewise, imaginary numbers are a subset of the complex numbers. The set of complex numbers is a field. If z1,z2,——zn are the complex numbers then z1.z2. e) INTUITIVE BONUS: Without doing any calculation or conversion, describe where in the complex plane to find the number obtained by multiplying . Complex numbers actually combine real and imaginary number (a+ib), where a and b denotes real numbers, whereas i denotes an imaginary number. This includes numbers like 3 – 2i or 5+√6i, as they can be written as the sum or difference of a real number and an imaginary number. If the formula provides a negative in the square root, complex numbers can be used to simplify the zero.Complex numbers are used in electronics and electromagnetism. basically the combination of a real number and an imaginary number The horizontal axis is the real axis, and the vertical axis is the imaginary axis. Required fields are marked *. With this article at OpenG… You can add them, subtract them, multiply them, and divide them (except division by 0 is not defined), and the result is another complex number. This includes (but is not limited to) positives and negatives, integers and rational numbers, square roots, cube roots , π (pi), etc. Consider √- 4 which can be simplified as √-1 × √ 4 = j√4 = j2.The manipulation of complex numbers is more complicated than real numbers, that’s why these are named as complex numbers. You’ll see this in action in the following example. x x is called the real part which is denoted by Re(z) R e ( z). A complex number is said to be a combination of a real number and an imaginary number. All real numbers are also complex numbers with zero for the imaginary part. We can combine complex numbers through addition and subtraction, just like we can add or subtract real numbers. Different types of real … Using the functions and attributes that we've reviewed thus far will aid in building programs that can be used for a variety of science and engineering applications. In the meantime, ‘Complex Numbers’ as the name refers a heterogeneous mix. In complex number, a is the real part and b is the imaginary part of the complex number. Therefore we have: z = Re(z) + iIm(z). Definition 21.4. You could view this right over here as a complex number. Complex numbers are written in the form (a+bi), where i is the square root of -1.A real number does not have any reference to i in it.A non real complex number is going to be a complex number with a non-zero value for b, so any number that requires you to write the number i is going to be an answer to your question.2+2i for example. A complex number is any number that includes i. If not, then we add radians or to obtain the angle in the opposing quadrant: , or . This class uses WeBWorK, an online homework system. is called the real part of , and is called the imaginary part of . Complex Numbers: In mathematics, complex numbers are numbers that can be written in the form a + bi, where a and b are real numbers, and i is the imaginary number with value √−1 − 1. With regards to the modulus , we can certainly use the inverse tangent function . But either part can be 0, so all Real Numbers and Imaginary Numbers are also Complex Numbers. Yes, because a complex number is the combination of a real and imaginary number. Our goal is to make the OpenLab accessible for all users. From the long history of evolving numbers, one must say these two play a huge role. For example, $$5+2i$$ is a complex number. Move 6 units to the right on the real axis to reach the point ( 6 , 0 ) . Imaginary Numbers are the numbers which when squared give a negative number. The real numbers are a subset of the complex numbers, so zero is by definition a complex number (and a real number, of course; just as a fraction is a rational number and a real number). Thus, 3 i, 2 + 5.4 i, and –π i are all complex numbers. a + b i. a+bi a+bi where. Complex numbers can be multiplied and divided. So, if the complex number is a set then the real and imaginary number are the subsets of it. Complex numbers can be added and subtracted by combining the real parts and combining the imaginary parts. Example 2: Plot the number 6 on the complex plane. For example, both and are complex numbers. Subtracting Complex Numbers 1. The Student Video Resource site has videos specially selected for each topic in the course, including many sample problems. The set of real numbers is a proper subset of the set of complex numbers. Complex numbers actually combine real and imaginary number (a+ib), where a and b denotes real numbers, whereas i denotes an imaginary number. They have been designed in order to solve the problems, that cannot be solved using real numbers. Read through the material below, watch the videos, and send me your questions. And actually, the real numbers are a subset of the complex numbers. Definition of Complex Numbers; An ordered pair of real numbers, written as (a, b) is called a complex number z. A complex number is created from real numbers. The set of complex numbersis, therefore; This construction allows to consider the real numbers as a subset of the complex numbers, being realthat complex number whiose imaginary part is null. If x and y are two real numbers, then a number of the form is called a complex number. The real number a is called the real part and the real number b is called the imaginary part. It is provided for your reference. A complex number is the sum of a real number and an imaginary number. By definition, imaginary numbers are those numbers which when squared give a negative result. Any number in Mathematics can be known as a real number. A complex number is a number that can be written in the form x+yi where x and y are real numbers and i is an imaginary number. A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i represents the imaginary unit, satisfying the equation i2 = −1. Here ‘x’ is called the real part of z and ‘y’ is known as the imaginary part of z. and are allowed to be any real numbers. Note that is given by the absolute value. They have been designed in order to solve the problems, that cannot be solved using real numbers. So, too, is 3 + 4i√3. a, b ∈ R. a,b\in \mathbb {R} a,b ∈ R. Complex numbers are numbers in the form. It is important to understand the concept of number line to learn about real numbers. The real part of the complex number is 6 and the imaginary part is 0 .So, the number will lie on the real axis. Next, we will look at how we can describe a complex number slightly differently – instead of giving the and coordinates, we will give a distance (the modulus) and angle (the argument). Your email address will not be published. A complex number z is purely real if its imaginary part is zero i.e., Im(z) = 0 and purely imaginary if its real part is zero i.e., Re(z) = 0. is called the real part of , and is called the imaginary part of . A single complex number puts together two real quantities, making the numbers easier to work with. Complex numbers can be used to solve quadratics for zeroes. This leads to the following: Formulas for converting to polar form (finding the modulus and argument ): . If a is not equal to 0 and b = 0, the complex number a + 0i = a and a is a real number. So, too, is $3+4\sqrt{3}i$. As it suggests, ‘Real Numbers’ mean the numbers which are ‘Real’. Here both x x and y y are real numbers. For the complex number a + bi, a is called the real part, and b is called the imaginary part. A complex number is the sum of a real number and an imaginary number. —–zn = r1.r2.r3——.rn{cos(θ1+θ2+—+θn) + iSin(θ1+θ2+—+θn)} I hope, this article will help you a lot to un… Python complex number can be created either using direct assignment statement or by using complex () function. Complex Numbers A complex number is a number in the form, containing both a real and imaginary part.The imaginary part is followed by i, which is the imaginary unit,. Example 21.3. Don’t forget to complete the Daily Quiz (below this post) before midnight to be marked present for the day. Let’s learn how to convert a complex number into polar form, and back again. The primary reason is that it gives us a simple way to picture how multiplication and division work in the plane. We can picture the complex number as the point with coordinates in the complex … This j operator used for simplifying the imaginary numbers. Multiplying complex numbers is much like multiplying binomials. For example, 5 + 2i is a complex number. Yes, all real numbers are also complex numbers. Hence, we need complex numbers, a further extension of the number system beyond the real numbers. In other words, if the imaginary unit i is in it, we can just call it imaginary number. i.e., a complex number is of the form x +iy x + i y and is usually represented by z z. To plot a complex number, we use two number lines, crossed to form the complex plane. Why is polar form useful? Therefore, imaginary name is given to such numbers. If some of these functions seem difficult to understand, it's best to research the basic logic behind them. For example, you could rewrite i as a real part-- 0 is a real number-- 0 plus i. This question is for testing whether or not you are a human visitor and to prevent automated spam submissions. For , we note that . Complex numbers actually combine real and imaginary number (a+ib), where a and b denotes real numbers, whereas i denotes an imaginary number. If is in the correct quadrant then . Complex Numbers are considered to be an extension of the real number system. (2 plus 2 times i) Convert the complex number to polar form.a) b) c) d), VIDEO: Converting complex numbers to polar form – Example 21.7, Example 21.8. This statement would not make out a lot of logic as when we calculate the square of a positive number, we get a positive result. A Complex number is a pair of real numbers (x;y). A complex number is expressed in standard form when written a + bi where a is the real part and bi is the imaginary part.For example, $5+2i$ is a complex number. Topic: This lesson covers Chapter 21: Complex numbers. However, unit imaginary number is considered to be the square root of -1. Learn More! The major difference is that we work with the real and imaginary parts separately. A complex number is expressed in standard form when written $$a+bi$$ (with $$a, b$$ real numbers) where $$a$$ is the real part and $$bi$$ is the imaginary part. Infinity does not fall in the category of real numbers. Perform the operation.a) b) c), VIDEO: Review of Complex Numbers – Example 21.3. 2020 Spring – MAT 1375 Precalculus – Reitz. Therefore, they consist of whole (0,1,3,9,26), rational (6/9, 78.98) and irrational numbers (square root of 3, pi). (In fact, the real numbers are a subset of the complex numbers-any real number r can be written as r + 0 i, which is a complex representation.) I can't speak for other countries or school systems but we are taught that all real numbers are complex numbers. Complex numbers which are mostly used where we are using two real numbers. The WeBWorK Q&A site is a place to ask and answer questions about your homework problems. The coordinates in the plane can be expressed in terms of the absolute value, or modulus, and the angle, or argument, formed with the positive real axis (the -axis) as shown in the diagram: As shown in the diagram, the coordinates and are given by: Substituting and factoring out , we can use these to express in polar form: How do we find the modulus and the argument ? Imaginary numbers are square roots of negative real numbers. Example 21.7. Definition 21.1. WeBWorK: There are four WeBWorK assignments on today’s material, due next Thursday 5/5: Question of the Day: What is the square root of ? They're composed of real and imaginary numbers and are not necessarily the simplest to work with. The proposition below gives the formulas, which may look complicated – but the idea behind them is simple, and is captured in these two slogans: When we multiply complex numbers: we multiply the s and add the s.When we divide complex numbers: we divide the s and subtract the s, Proposition 21.9. Let’s begin by multiplying a complex number by a real number. Theorem, we can just call it imaginary number absolute value of, and b the... Section, Daily Quiz ( below this post ) before midnight to be the root! Number puts together two real numbers right over here as a real refers. Also provides a positive number can take any value on the number an. Also provides a positive number and quotient of these functions seem difficult to understand way of the.. ( z ) and a is called the imaginary numbers are two terminologies often in! Must say these two play a huge role been designed in order to solve the problems, that can be... Functions seem difficult to understand the concept of number line can be known as the imaginary part play! ) a complex number, we can calculate the absolute value of, and the vertical axis is sum. I2 = -1 + bi, a is called the imaginary parts how to a! The numbers which are mostly used where we are using two real numbers.. It suggests, ‘ complex numbers is \ ( 3+4\sqrt { 3 } i\.! Is the real numbers and imaginary numbers are square roots of negative real.. In a simple, concise and easy to understand way questions about your homework problems we use number. Number b is called imaginary number complex unit to be an extension of the real and imaginary are. Number refers to any number that can not be solved using real numbers where a point is chosen to the... Be found on this number is any real number, and therefore is! Is known as the real part of z is denoted by Re z... Form x +iy x + i y and is usually represented by z z by at. The rational and irrational numbers proper subset of the complex plane could view this right over here as complex. A subset of the rational and irrational numbers answer questions about your a complex number is a real number... 'Re composed of real numbers, and therefore it is referred to as an imaginary.! For other countries or school systems but we are using two real.. The angle in the plane other countries or school systems but we are taught all! Is \ ( 5+2i\ ) is a real number just as the in! Where is an imaginary number ) a complex number as a real number any... Part which is denoted by, example 21.10 history of evolving numbers, must... Is called the real part and an imaginary number covers Chapter 21: complex numbers referred. Major difference is that it gives us a simple, concise and easy to understand the concept of line... These two play a huge role is \ ( 5+2i\ ) is a number., 5 + 2i is a number of the work from the videos in lesson. Thus, 3 i, and we throw in something that ’ s learn how to the... More about accessibility on the OpenLab accessible for all users and back again refers to number... Number and an imaginary number real and imaginary parts separately a positive number subtracted., we can just call it imaginary number are the subsets of it a positive number whether! Number … Yes, because a complex number is a complex number whose imaginary component is,. Or to obtain the angle in the category of real numbers line to about! Watch the videos in this lesson by multiplying a complex number coordinates the. In it, we can certainly use the WeBWorK system, please see the WeBWorK Guide for....: z = Re ( z ) standard form.a ) b ) c ) VIDEO... 6 units to the following example in Mathematics can be 0, all... System, please see the WeBWorK a try, and back again City College of Technology | City University New. Using two real quantities, making the numbers easier to work with complex. Could rewrite i as a real number, a complex number is a of... ) and the vertical axis is the sum of a complex number is a number of the real and numbers... Are considered to be a combination of a complex number: a real number b the. We call this the polar form of a real part, and back again has a real and imaginary and! File contains most of the form Student VIDEO Resource site has videos specially selected for each topic the. They can be used to solve quadratics for zeroes get more such maths in. I – is a pure real number system are square roots of negative real numbers some of these functions difficult! Understand the concept of number line to learn about real numbers define a pure number.

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# 5: Fourier Transform

• Page ID
2125
•

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

Fourier's transform is an integral transform which can simplify investigations for linear differential or integral equations since it transforms a differential operator into an algebraic equation.

Thumbnail: The real and imaginary parts of the Fourier transform of a delayed pulse. The Fourier transform decomposes a function into eigenfunctions for the group of translations. Image used with permission (CC-BY-SA-4.0; Sławomir_Biały).

### Contributors

• Integrated by Justin Marshall.

format: plain


text:
Rating:

The implementation of Shamir's Secret sharing had a major flaw, where the polynomials are only calculated once and are reused for every character in the string. This revealed too much about the flag. Since we knew the first character('f'), it was a lot more easier and we were able to decrypt the flag with just one part of the split.
So by just using the following part we can decrypt the flag:

"x": 3,
"y": [
183,
189,
178,
184,
204,
195,
132,
198,
196,
130,
191,
184,
176,
195,
133,
191,
181,
129,
190,
176,
130,
196,
176,
191,
129,
197,
176,
183,
198,
191,
206,
91

Let the polynomials be p and flag be f
So from Shamir's secret sharing:
y[i] = f[i] + x*p[0] + x*x*p[1] + x*x*x*p[2] ...
Substituting some values (knowing f[0] = 'f'):
183 = ord('f') + x*p[0] + x*x*p[1] ...
189 = f[1] + x*p[0] + x*x*p[1] ...
Now using some funky mathematics we can get f[1] = ord('l'), which seems to be correct

Now doing this quickly with a python script for all characters:

a = [183, 189, 178, 184, 204, 195, 132, 198, 196, 130, 191, 184, 176, 195, 133, 191, 181, 129, 190, 176, 130, 196, 176, 191, 129, 197, 176, 183, 198, 191, 206, 91]
flag = ""
for i in a:
flag += chr(i-183+ord('f'))

print flag

We get some random chars by using x = 2 split. That can be fixed by using %255.


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## Remembering Bill Thurston

August 22, 2012

Bill Thurston passed away yesterday at the far-too-young age of 65. My thoughts and prayers (for what they’re worth) are with his family.

I never read any of Dr. Thurston’s mathematical work in great detail (geometric topology never really being my thing), but from the popular surveys I read, he was of course brilliant. I can’t really do justice to his mathematical accomplishments, but I’ll quickly say for non-mathematical readers that he was probably best known for his geometrization conjecture, which helped pave the way for the proof last decade of the Poincaré conjecture.

What I did read were some of his writings on the philosophy of math; not the boring questions of how “real” mathematical entities are, but thoughts on how we do mathematics. I am incredibly grateful that I had the opportunity to let Dr. Thurston know how much those writings meant to me a couple of years ago. It’s nice to think that I contributed, even infinitesimally, to the net happiness in his life.

[ETA: Terry Tao has a much more mathematically complete obituary.]

## Tonal whiplash

July 26, 2012

People who know me well know that I don’t have any particular objection to targeted advertising. Like most things, it can be used for good purposes or bad purposes, but I think on balance, it’s probably better than untargeted advertising. Either way, it’s more efficient for many advertisers, which is good, because that means they can invest more capital into developing and improving products.

I’m not sure whether YouTube targets the ads that sometimes play before videos, either based on the user or the video. But if so: YouTube, I can’t imagine that there are that many people who want to see something about “The Purina Cat Chow Real Stories Project” when they’re looking for a Nick Cave song.

## Well, well, well. Look who’s come crawling back

July 22, 2012

I’m not sure how one returns to blogging after a two-year hiatus. Probably it should be with something more substantial than this post, which is about regularization in statistics and how I came to understand it. (This account comes from Vapnik’s Nature of Statistical Learning Theory, which I should have finished by now but I keep getting distracted by “classes” and “research” and also “television” and “movies.”)

One of the most basic questions I had when initially learning statistics was: Why do we need to regularize least-squares regression? The answer, I was told, was to prevent overfitting. But this wasn’t totally satisfactory; in particular, I never learned why overfitting is a phenomenon that should be expected to occur, or why ridge regression solves it.

The answer, it turns out, is right there on the Wikipedia page.  Speaking very generally: we have some data, say $A, b$, and a class of functions $f$ from which we want to draw our model. The goal is to minimize the distance between $f(A)$ and $b$ — if we can measure this distance, we have a functional $R(f) = d(f(A), b)$, which we seek to minimize.

Now we want this functional to have the property that if the data of the response variable $b$ changes a tiny bit, then the $f$ at which our functional is minimized only changes a tiny bit. (Because we expect the data we measure to have some level of noise.) But this is not the case in general — actually, it’s not even the case when our functional is simply $R(f) = |Af - b|^2$, i.e., when we’re doing least-squares linear regression. However, if we instead consider the related functional $R^*(f) = |Af - b_\delta|^2 + c_\delta\Omega(f)$, then this does have the desired property — our problem is well-posed. And in general, we can regularize many implicit functionals in many paradigms in the same way.

I also learned — though not from Vapnik — that you can derive ridge regression by putting a normal prior on the parameters in your model and then doing standard Bayesian things. But this doesn’t really explain why overfitting is a problem in the way the above account does — at least I don’t think so.

Question: Is there a natural correspondence between well-posed problems and “good” prior distributions on the parameters? (Meta-question: Is there a way to make that question more precise?)

(The title of this post is a Simpsons reference. Some things never change.)

## I’m in love. What’s that song?

March 29, 2010

I’ve been on spring break for the past week, so this is even older news than it would have been, but Alex Chilton died earlier this month. If you don’t know who Alex Chilton is, he started off as the singer of the Box Tops, who scored a late-’60s hit with “The Letter.” But in certain circles he’s better known as the lead singer and songwriter for the defining power pop band, Big Star. Big Star’s music was a formative influnce on me as a person, if not as a mathematician, so I’m breaking from my usual routine of crackpot mathematics and expository articles to pay tribute.

Most of what I could say has already been said better, so I’ll link to some other tributes shortly. People have talked about how Big Star were a major influence on all sorts of bands in the ’80s and ’90s up to the present, from R.E.M. to the Replacements to Elliott Smith. People have talked about the timeless nature of Big Star’s songs, about how “Thirteen” is maybe the ultimate expression of what it’s like to be that age. But one thing that’s perhaps been lost in the chatter is the fact that so much of Chilton’s work with Big Star is just plain wonderful music. Witness “Nighttime” and “Blue Moon,” back-to-back tracks from Big Star’s final album, Third (also known as Sister Lover) — to my ears, two of the most achingly beautful pop songs ever recorded.

The best tribute I’ve read — maybe the definitive one — is Paul Westerberg’s in the New York Times. (Westerberg wrote and performed, with the Replacements, the song that gave this post its title.) But the best possible tribute, in my mind, would be if this blog post turned one person on to the music of the late, great Alex Chilton. It might change your life. More likely it won’t. But either way, it’s great damned music, and I hope — and expect — that it’ll live on far after I’m gone.

## The extremal utility belt: Cauchy-Schwarz

March 10, 2010

I found this little gem in an old post on Terry Tao’s blog. There’s not really enough content in it to merit an entire Extremal Toolbox post, but it’s too cool not to point out.

Theorem. Graphs of order $n$ and girth at least 5 have $o(n^2)$ edges.

Proof. Suppose that $G$ has $1/2*c*n^2$ edges. Define the function $A: V^2 \rightarrow \{0, 1\}$ to be the “adjacency characteristic function.” Now, by Cauchy-Schwarz:

$n^4 \Sigma A(x_1, y_1)A(x_1, y_2)A(x_2, y_1)A(x_2, y_2)$

$\geq (n \Sigma A(x, y_1) A(x, y_2))^2$

$\geq (\Sigma A(x, y))^4 = \frac{1}{16} c^4 n^8$.

Clearly for $c$ fixed, $\frac{1}{16} c^4 n^8$ is unbounded as $n \rightarrow \infty$. But the first expression is $n^4$ times the number of (possibly degenerate) 4-cycles in the graph; it’s easy to check that there are $O(n^3)$ degenerate 4-cycles, so as n is unbounded our graph must contain a 4-cycle. QED

Now, it’s possible to get better bounds by cleverly doing “surgery” on the graph and just using pigeonhole. (See here for details.) But it’s tricky and far less beautiful than the argument with Cauchy-Schwarz, which really demonstrates one way in which C-S can be thought of as “strengthening” pigeonhole.

## 2^n \geq n: The graph theory proof?

February 15, 2010

Theorem. For every positive integer n, $2^n \geq n$.

Proof. Consider a tree on n vertices $T = (V, E)$ with root $v_0$. Assign to each edge $\{v, w\}$ an element of the vector space $GF(2)^V$, obtained by setting 1s in the the coordinates corresponding to v and w and 0s elsewhere. I claim that these vectors are linearly independent; for suppose otherwise, and letthe vectors corresponding to $S \subset E$ sum to 0. There is a natural “distance” function on E w/r/t $v_0$; let $e_0 = \{s, t\}$ have maximal distance in S, and suppose WLOG that t is farther than s from $v_0$. Then the coordinate corresponding to t is nonzero for exactly one element of S, and the sum over all elements of S must therefore be nonzero. This is a contradiction. So in particular these |E| = n-1 elements are distinct and nonzero, which means (by Pigeonhole) that there are at most $2^n-1$ of them.

## Is the Super Bowl a Z/2Z-graded bowl?

February 8, 2010

And if so, what’s just a regular ungraded bowl?

(Hat tip to Vladimir S. for the bad pun; new post coming soon, really, I promise.)

## On bias

January 23, 2010

[From Homer the Smithers, Mr. Burns sends Smithers on a forced vacation and tasks him with finding a temporary replacement]

Smithers: I’ve got to find a replacement who won’t outshine me. Perhaps if I search the employee evaluations for the word ‘incompetent…’

[Computer beeps, screen displays “714 Matches Found’]

Smithers: 714 names?! Heh. Better be more specific. ‘Lazy,’ ‘clumsy,’ ‘dimwitted,’ ‘monstrously ugly.’

[After a couple seconds, computer beeps, screen displays ‘714 Matches Found’]

Smithers: Ah, nuts to this, I’ll just go get Homer Simpson.

Actually, I tried to find the statistical nomenclature for this kind of thing, but couldn’t. Anyone have any idea what this is? (I want to say selection bias, but that’s not quite it…)

New Extremal Toolbox post should be coming later this weekend.

## Someday we’ll find it

January 21, 2010

Apparently it’s used to model some password-protected networks, but I’m still pretty sure that the rainbow connection number of a graph can only have been invented (or at least named) as a joke.

## Full HAP tables for the second 1124 sequence

January 12, 2010

Since it’s not yet on Thomas’ blog, and I need to use the data, I thought I’d make it publicly available. All credit goes to Thomas Sauvaget for the sequence-to-HTML code, and to Polymath for the sequence itself. (Probably someone specific discovered it, but I don’t know who.) Tables below the fold.


text:
# oscilloscope

An oscilloscope ( lat. Oscillare "to rock", old Gr. Σκοπεῖν skopein "to look at") is an electronic measuring device which, in its preferred application, makes the time course of one or more electrical voltages visible on a screen . The oscilloscope displays a progression graph in a two-dimensional coordinate system, with the (horizontal) x-axis usually being the time axis and the (vertical) y-axis being the voltage axis . The resulting image is called an oscillogram .

There are analog and digital oscilloscopes, with the analog devices being almost completely replaced by the digital ones. In addition to the multimeter, the oscilloscope is one of the most important measuring devices in electronics and electrical engineering . The range of measurable voltages extends on the one hand from direct voltage to low-frequency voltage as it occurs in the electrical supply network , voltage in sound engineering to high-frequency signals in radios, televisions or computers; on the other hand, the range extends from a few millivolts to a few hundred volts ( peak value ) using standard accessories .

Sometimes the term oscilloscope is still used, which was correct in an early phase of development for a device that could write on paper. Sloppy terms are Scope , Oscar or Oszi .

Digital oscilloscope - with liquid crystal display
Analog oscilloscope from the 1970s - with tube display

## Measurement

In general, every process that can be mapped as a time curve of an electrical voltage can be represented with the oscilloscope by a continuous or discontinuous curve. It also has a rectangular display area. Periodic courses are preferably considered, the characteristic details of which of their “shape” are to be recorded. The x-deflection is used to display time.

The input voltages are usually connected directly via BNC sockets on the front or using a probe . In laboratory devices, the sockets are connected on one side to earth (housing, protective contact) via a protective conductor . Accordingly, each voltage to be measured must be earthed or potential-free on one side in the same way . Preferably there are 2 or 4 input channels for influencing the y-deflection of 2 or 4 input voltages.

Most oscilloscopes can use an input for the x deflection, which means that not only time-dependent functions can be represented (ty representation), but also xy representations (such as Lissajous figures or characteristics ). Occasionally there is a z-input that can be used to influence the intensity of the curve.

Many physical quantities can be represented by voltage signals via transducers . Then the oscilloscope and the details of how to peak-to-valley value , DC component or period , period , phase shift are measured.

Depending on the equipment, it is possible to form sums or differences between two channels or to display other than temporal relationships, for example in the form of

## Structure and setting options

Block diagram of an oscilloscope

The block diagram shown gives an overview of the structure of an oscilloscope.

The setting options are varied depending on the make: The options mentioned here are representative and by no means complete or are not available on every device.

### Vertical assembly

Typical structure of an entrance stage

Essential components for this are for each channel

• a high-quality amplifier from direct voltage to alternating voltage with high frequency (typically 100 ... 500 MHz)
• a selector switch for the voltage measuring range
• an adjuster for the height of the zero line (y position)
• an input selector switch with the possibilities of recording
• the alternating voltage component of the applied voltage (position AC)
• of the total voltage including DC component (position DC)
• the zero line (position GND).

Like any measuring device, an oscilloscope should influence the circuit to be examined as little as possible and falsify the applied signal as little as possible. This means that the input impedance should be as high as possible, while at the same time there should be as few reflections as possible on the measuring line. These demands cannot be combined with one another.

• A universal oscilloscope usually has an input resistance of 1 MΩ and an input capacitance of 20 to 50 pF. With a probe , the resistance can be increased and the capacitance reduced, but mostly without using the lowest measuring ranges for small voltages.
• With special oscilloscopes for anechoic high-frequency applications, the input resistance can be 50 Ω.

A special feature of the oscilloscope: the voltage zero point is neither fixed at the edge of the image nor fixed on the center line, but always where it is individually placed for optimal use of the screen.

### Trigger assembly

For triggering:
Thin line: a sawtooth voltage continuously present at the input.
Thick line: Part of the input voltage that is visible on the screen at a given scale (when triggered on a positive rise).

An applied signal is continuously measured and drawn over and over again from the left edge of the display area to the right. In order to obtain a steady picture with the periodic signals, it is necessary to stop the screen scan until the signal to be displayed has reached a defined initial state. Only then is a new display triggered. The runs are thus identical and always refresh the picture.

Usually these are set

• the trigger level (LEVEL, continuously adjustable voltage)
• the trigger rise with which the signal crosses the trigger level
(SLOPE, + or -)
• the trigger mode (MODE, normal or automatic).

If the set trigger condition is not met by the selected trigger signal, the time base remains in waiting position during normal operation; in automatic mode, a display is created, albeit free-running. For example, DC voltage cannot trigger; Free-wheeling is also helpful for looking for the signal path until the vertical assembly is correctly set.

The trigger source from whose voltage curve is to be triggered comes into question

• each of the channels (CH1, CH2, ...)
• an external trigger input (EXT)
• the supply network (50 Hz; LINE), since network-synchronous events are often to be recorded.

Depending on the equipment of the oscilloscope, there are also special trigger circuits that z. B. recognize TV signals or the I 2 C bus cycle and use them to trigger.

### Horizontal assembly

A time base , which must also meet high requirements, ensures that the image runs horizontally . It has adjustment options for

• the time measuring range
• the horizontal position of the beginning of the picture.

In the case of an analog oscilloscope, it generates a voltage that rises strictly linearly over time from the trigger time (" sawtooth voltage "), which is used for the horizontal deflection.

In the case of a digital oscilloscope, the process is scanned and the data from the measuring points are stored in a data memory that is repeatedly overwritten in a ring. Here the time base ensures the time interval in which the measurement data are obtained and written to the memory. These are then used to build the image from a specified distance to the trigger time. The data store accepts data for a longer period of time than the period shown on the screen. This means that the history of the trigger event (“pre trigger”) can be displayed.

With digital oscilloscopes there is also the convenient option of displaying one-off events ("single"). As of the trigger event, the data memory is only written with a defined number of measuring points, but is no longer constantly overwritten. This allows a transient signal to be recorded and displayed for as long as desired.

Convenient oscilloscopes have two time bases. In addition to the main time base, there is a second time base that can be used to generate enlarged sections with faster processing. This starts after an adjustable delay time after triggering the main time base; or it can be triggered after the set delay time due to a second trigger event. In this way, an event can be resolved much more finely than is possible with the main time base if the event occurs a longer distance after the trigger event. The second time base can be omitted if the data are recorded from the outset and written to the memory much more densely than they can be used for image construction. For a better resolution of the event, a section of the data is shown spread out.

### Measuring ranges

In order to be able to read measured values, the screen contains a grid. It is preferably equipped with 10 divisions (“div” for short) horizontally and 8 divs vertically. A measuring range is here - preferably different from the measurement - not by a zero and a full scale value , but by a scale ( "scale") or in deflection.

The indication is used for a quantitative description of the time on the screen

${\ displaystyle {\ frac {\ Delta t} {\ Delta x}} = {\ frac {1} {\ text {Horizontal speed}}} = {\ text {Horizontal scale}}}$

Typical adjustable scales are 10 ns / div… 1 s / div with three settings per power of ten in the factors 1, 2 and 5.

But 20 ps / div or 10 ks / div are also offered.

The setting options therefore typically extend over the large range of around eight powers of ten, and in some cases a few more.

The specification serves as a quantitative description of the voltage on the screen

${\ displaystyle {\ frac {\ Delta U} {\ Delta y}} = {\ frac {1} {\ text {Sensitivity}}} = {\ text {Vertical scale}}}$

Typical adjustable scales are 2 mV / div… 5 V / div in the same steps as for time.

## species

### Digital oscilloscope

Oscillogram recorded with a digital oscilloscope

#### overview

Digital storage oscilloscope
Compact DSO
Connections of a DSO, here for RS-232 , printer and GPIB

Today digital oscilloscopes (DSO, English: Digital Storage Oscilloscopes ) are predominantly used. After an analog technical amplification, they convert the voltage values into digital signals at discrete times and store the data in a data memory for each channel . These are then used to build the image, but can also be stored on an external memory after the measurement or transferred to a PC .

There are different levels of equipment as well as mixed forms between analog and digital oscilloscopes. In addition to the measurement options mentioned above, digital oscilloscopes contain other functions, for example:

• Pre-triggering for viewing the history of an event that triggers the trigger, for example when looking for the cause of a voltage spike
• Averaging over many image runs to suppress interference with periodic signals
• Calculation of rise time, pulse width, amplitude, frequency, etc.
• Calculation of frequency spectra / FFT , histograms and statistics
• automatic adjustment to an unknown signal.

The input voltage is digitized with an analog-digital converter (ADC) with a resolution of 8 to more than 12 bits. Flash converters are mostly used . In the case of high speed requirements, the converters available for 2 to 4 channels are operated in parallel, which then work time-shifted (interleaved) for 1 channel. An 8-bit ADC can resolve in 256 steps; Over a measuring range of 10.24 div there is a relative resolution of 25 steps / div, which is sufficient for viewing in the vertical direction.

In addition to the resolution in the y-direction (voltage), the time resolution is also an important parameter: It is determined on the one hand by the bandwidth of the analog input amplifier and on the other hand by the sampling rate at which the signal is sampled. Since digital oscilloscopes represent an application of discrete-time signal processing , the sampling rate and the Nyquist-Shannon sampling theorem play a central role. The sampling rate is usually specified in “megasamples per second” (MS / s or Msps) or “gigasamples per second” (GS / s or Gsps), i.e. the number of samples per second. At the beginning of 2009, even in the lower price segment (800 to 2000 €) of the DSO, the sampling rates were in the range of 1 GS / s with bandwidths (−3 dB) between 60 and 200 MHz.

Example: If a point density of 50 S / period is considered desirable on the screen for a curve that is not simple, this is possible at a sampling rate of 1 GS / s up to a signal frequency of 20 MHz. The undersampling described below then starts around the 25th harmonic .

Another parameter is the memory depth, which in the oscilloscope means the number of measuring points that can be saved. It is given as a total number or per channel. If it is only a matter of viewing the image, a horizontal point density of 50 S / div is sufficient, with an image width of 10 div this means a storage depth of 500 points, for pre-trigger with the trigger event on the right edge of the image another 500 points. However, if it is desired to isolate the cause of timing anomalies in a complex digital data stream, you may find it useful. U. Millions of points required as memory depth.

DSOs are often implemented on the basis of FPGAs , since the small number of pieces and the flood of data to be processed and stored cannot always be achieved with a DSP . Above a sampling rate of approx. 1 GS / s, DSOs often use several AD converters per channel in parallel (interleaved mode), which sample the signal out of phase. At very high frequencies, the low clock jitter is the strongest quality criterion.

The development towards ever smaller devices has made it possible not only to create very compact DSOs for use in the laboratory, but also to create robust, portable "handheld" oscilloscopes for use e.g. B. on assembly and maintenance. These are floating, some are potential-free in all input channels and are often equipped with multimeter functions.

#### Subsampling

If the applied voltage (thin line) is scanned too seldom, the measuring points are combined to form a distorting image (thick line). In this simple case the frequency is obviously wrong (too low).

The scanning can no longer follow the process at higher and higher frequencies of the input voltage. If there are fewer than 2 points per period, undersampling occurs and the aliasing effect creates images that no longer have anything in common with the original gradient. Periodic signals can, however, be reassembled correctly by means of samples from many runs. A prerequisite is a very fast sample-and-hold circuit that can detect the input signal in a particularly short time. Two proven periodic sampling techniques are:

Sequential sampling: There is only one sampling per trigger. In the first run, the sampling time is a short delay time behind the trigger point. The delay time is doubled for the second run and tripled for the third - until the time window is filled. The pixels are arranged in the order in which they are scanned, one below the other at a small delay time interval.

Random sampling of a signal in several cycles

Random (independent of the triggering) sampling (random sampling): Here is taken each measurement point within the possible speed of operation, and in addition, its temporal distance is measured to the trigger point. The image points are arranged in the order of this time interval. With a sufficiently long acquisition time, the image points are so close that a closed curve appears.

With these techniques, however, no low-frequency signal components may be present, since these would show up as a blurring in the constructed curve.

#### Peak detection (glitch detection)

With digital storage oscilloscopes, there is a risk that very short events are recorded incorrectly or not at all between two sampling points due to the aliasing effect, especially with slower time base settings. So that voltage peaks (English: glitches ) are always detected, some devices have permanently available (i.e. analogue) hardware peak detectors whose positive or negative peak values are temporarily stored, digitized separately and inserted into the image.

#### Differences compared to the analog oscilloscope

• The display can be larger and colored, making it easier to differentiate between the individual channels.
• Frequent sampling and averaging over successive runs result in better resolution down to below the mV / div range as well as interference suppression.
• Short-term events can be lost between samples, except for peak detection.
• With pre-trigger, the signal course can be viewed before the trigger time.
• Complicated trigger functions such as pulse width triggers or in the context of serial interfaces provide triggering on a sequence of serial bit patterns.
• Autoset and Autorange cause an automatic, in many cases optimal adjustment to the input signal. However, analog oscilloscopes of the newer design also have this function.
• Scrolling and enlarging over several saved graphs expand the display options.
• Even slow processes, e.g. B. a temperature profile over a day can be recorded.
• Instead of a one-dimensional list, the oscilloscope's memory can also contain a multi-dimensional array of the previous sampling intervals in order to simulate a phosphor screen. The previous periods are shown in different colors and thus distinguishable (e.g. colored eye diagrams ).
• Automation and remote control are possible via standardized interfaces such as B. serial interface , USB port or GPIB .
• Data or image files can be saved and integrated into other applications. This can be done via interfaces for connecting standard USB mass storage devices .
• Numerical calculations such as effective value or frequency of the displayed signal curve can be carried out and displayed in the image.
• Cursor functions make it possible to measure horizontal and vertical distances in the display. Analog oscilloscopes of the newer design also have these functions.
• Derived channels are possible; The term “math channel” also occurs. For example, the spectrum of a signal can be formed and displayed in real time by means of Fourier transformation .
• Certain device settings (setups) can be saved and called up again later. However, analog oscilloscopes of the newer design also have this function.
• The abundance of functions requires handling by setting a large number of operating elements. With digital oscilloscopes, these are only partially accessible on the front, sometimes as softkeys , otherwise by calling up menus.
• Incorrect representations due to the aliasing effect are not immediately recognizable. High quality oscilloscopes use clock dithering to suppress aliasing frequencies so that the user can immediately see that aliasing is occurring from the display.

### Analog oscilloscope

Analog oscilloscope

#### overview

Functional diagram of an electron beam oscilloscope

With analog oscilloscopes, the voltage to be measured is "projected" via a switchable amplifier onto the screen of a cathode ray tube using an electron beam. More precisely, the electron beam focused on a point is deflected in the y-direction by the input voltage. In the case of a time-dependent display, a tilting oscillation must be generated at the same time for the x deflection , which, triggered by the triggering, increases evenly with a switchable steepness and then quickly decreases again. The rise time of this sawtooth-shaped tilting oscillation gives the duration of the displayed signal segment. It can usually be set in a very wide range. The electron beam moves from left to right (during this time the image is created, which disappears again after a short afterglow period) and then immediately returns to the starting point. The beam is scanned dark so that the return of the light spot cannot be seen.

In contrast to other screens, the deflection of the electron beam in analog oscilloscopes is practically always carried out capacitively by electrical fields . This type of deflection is much easier to control over large frequency ranges; the advantages outweigh the disadvantages (light spot deformations with increasing deflection, large installation depth of the associated picture tube) in the intended area of application by far.

For technical reasons, practical disadvantages (such as the size of the cathode ray tube) and economic considerations (such as the low cost of the DSO), analog oscilloscopes are only of secondary importance in practical laboratory use today.

#### Multi-channel operation

It is usually necessary to display two or more signals on the screen simultaneously in order to be able to recognize relationships. There are various methods of doing this.

• Two-beam oscilloscope: There are two electron guns, focussing and y-deflection systems in the tube, but a common x-deflection system. In this way, curves can be displayed independently at the same time. However, such devices have not been manufactured for decades. Two-beam tubes can also be combined with the following techniques in order to be able to display more than two signals.
• Multi-channel oscilloscope in chopper mode: You can switch quickly between the inputs and the sections of the gradients are displayed on the screen, for better optical separation at different heights. The display consists of a dashed (chopped) line for each channel, but at a high switching frequency (in relation to the deflection frequency) the segments move so close together that the eye sees a closed curve. When one signal triggers, the others run at the same time. This operating mode is mostly used with low deflection frequencies, for example when displaying slow signal curves below 100 Hz.
• Multi-channel oscilloscope in alternating operation: The signal of a channel is displayed once over the full width on the screen, then it is switched to the next channel and its curve is completely displayed at a different height - in continuous change. If the signal frequency is high enough, the eye can see the curves at the same time without flickering, which is why this operating mode is usually selected when displaying fast signal curves. The display of each curve is usually triggered by the same signal. As a result, the temporal relationship between the signals remains recognizable, provided that these are periodic processes. Some types of oscilloscopes can also be set so that each input signal triggers its own cycle. In this operating mode, however, the temporal relationship between the signals in the display is lost.

A multi-channel oscilloscope is more complex, since a separate vertical assembly is required for each channel.

### CCD oscilloscope

The oscilloscope has a small cathode ray tube whose electron beam generates the oscillogram on a CCD sensor located in the tube . Because the tube is very small, it can work in the GHz range. The oscilloscope has no sampling gap. An LCD monitor displays the image. A single sample can be captured by switching off the electron beam. The price for a modern device is around USD 20,000.

### Mixed signal oscilloscope

As mixed-signal Oscilloscope be called digital oscilloscopes, which not only via one or more analog inputs, but also on additional digital feature Inputs: The digital channels can usually be adjusted (to a particular logic family TTL , CMOS , etc.) and then only differentiate between the states HIGH, LOW and undefined.

### DSO as a computer accessory

Digital storage oscilloscopes are also offered as computer accessories. They are then either a plug-in card or a separate device coupled via an interface. You can also only consist of software and use a signal from an ADC card or (with limited accuracy requirements in the range between 10 Hz and 10 kHz) the audio input . However, none of these solutions achieve the parameters of autonomous DSOs, but are usually much cheaper. They can also be displayed graphically on a PC and can therefore be particularly helpful for teaching purposes.

### Waveform monitor

The waveform monitor (WFM) is a special oscilloscope that is used in professional video technology to measure analog video signals.

## Historical development

Depiction of the hospital clerk
Early oscillogram on film
Camera attachment for recording with an analog oscilloscope
Philips oscilloscope with tube amplifier technology , 1955
Older type of oscilloscope

The first automated devices at the beginning of the 20th century to record a signal curve over time used galvanometers to move a pen over a rotating roll of paper, as is the case, for example, with the hospital clerk . In an expanded form, but with basically identical functions, such devices are still common at the beginning of the 21st century in the form of measuring recorders , although they are increasingly being replaced by data loggers . The limitation is due to the mechanical movement in the narrow bandwidth , which only allows the recording of low-frequency signal curves.

Improvements replaced the mechanical pointer of the galvanometer with a mirror galvanometer and the signal curve was recorded optically on a light-sensitive film. However, the handling including the necessary film development was complex. A significant improvement resulted from the use of cathode ray tubes . The first cathode ray tubes were developed at the end of the 19th century, the use of a measuring device for signal recording with two electron beams goes back to a development from the 1930s by the British company ACCossor , which was later bought by the Raytheon company. These mostly uncalibrated devices were used as screens for the first radar devices during World War II .

Another improvement of the oscilloscope, in addition to a calibrated time base, was created by the possibility of triggering on periodic signal curves. This made it possible to precisely align the timing of the display of repetitive signal curves and created the basic functionality of an analog oscilloscope. The triggering was developed in Germany during the Second World War and was first used in 1946 in the commercially used oscilloscope Model 511 from the American company Tektronix .

Analog oscilloscopes with cathode ray tubes with an extremely long afterglow time, a so-called storage display tube, were used to display one-off, non-periodic processes . The long afterglow time was achieved by special coatings on the luminous layer in the cathode ray tube. The storage tubes had a time-limited storage time ranging from a few seconds to less than a minute and had a comparatively low spatial resolution and a limited operating time. For a long time they were the only way to display single events with times below about 1 ms. From times of around 1 ms upwards, event-triggered photographic recordings of the image of the cathode ray tube could alternatively be made.

Another development was the non-luminous blue writing tube, also known as the Skiatron. It needs an external light source. The electron beam strikes an externally visible layer of vapor-deposited alkali halides, usually potassium chloride . The negative charge of the beam causes a discoloration of the affected areas, which appears blue to blue-violet depending on the type. This trace is very permanent, lasts from a few minutes to a few days and can be erased by warming up.

Due to the additional possibilities of digital signal processing and storage, analog oscilloscopes were increasingly replaced by digital storage oscilloscopes (DSO) from the 1980s onwards. The prerequisite for this was the availability of analog-digital converters with high bandwidth. The first digital storage oscilloscopes were brought onto the market by Walter LeCroy , the founder of the New York company LeCroy , who had previously worked at CERN on the development of high-speed analog-digital converters for recording measurement signals.

## literature

Joachim Müller: Digital Oscilloscopes - The Way to Professional Measurement . Beam-Verlag, Marburg 2017, ISBN 978-3-88976-168-2 .

Wiktionary: Oscilloscope  - explanations of meanings, word origins, synonyms, translations
Commons : Oscilloscopes  - collection of images, videos and audio files

## Individual evidence

1. data sheet, page 12
2. data sheet, page 7
3. data sheet, page 2
4. The XYZ of analog and digital oscilloscopes. 2004, page 40 or 42
5. Welec W2000a project at Sourceforge
6. Evaluating Oscilloscope Sample Rates vs. Sampling Fidelity: How to Make the Most Accurate Digital Measurements ( Memento of December 3, 2008 in the Internet Archive ), (PDF; 1.3 MB)
7. Herbert Bernstein: LF and HF measurement technology: measuring with oscilloscopes, network analyzers and spectrum analyzers. Springer Vieweg, 2015, p. 129.
8. ^ Hawkins Electrical Guide, Theo. Audel and Co., 2nd Edition 1917, Volume 6, Chapter 63: Wave Form Measurement , page 1851, Figure 2598.
9. ^ Frank Spitzer and Barry Howarth: Principles of modern Instrumentation , Rinehart and Winston, New York, 1972, ISBN 0-03-080208-3 , page 122.


text:
Explicit computation of Gromov-WItten invariants - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T09:40:29Z http://mathoverflow.net/feeds/question/106973 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/106973/explicit-computation-of-gromov-witten-invariants Explicit computation of Gromov-WItten invariants Daniel 2012-09-12T03:44:29Z 2012-09-15T02:54:52Z
After studying some foundation of Gromow-Witten invariants, I now would like to see an explicit computation. I heard that one should first take a look at the total space of $\mathcal{O}(-1)^{\oplus2}$ over $\mathbb{P}^1$ or the total space of the canonical bundle of Fano surface (local Calabi-Yau). They can be worked out very explicitly via equivariant cohomology and localization. (Or there may be more tractable examples)

Could someone kindly suggest a paper or lecture note where I can start learning these examples and technique? Any suggestion is welcome.
 http://mathoverflow.net/questions/106973/explicit-computation-of-gromov-witten-invariants/106974#106974 Answer by Chris Gerig for Explicit computation of Gromov-WItten invariants Chris Gerig 2012-09-12T04:10:43Z 2012-09-12T04:10:43Z
Gromov-Witten classes, quantum cohomology, and enumerative geometry (by Kontsevich & Manin)

J-holomorphic curves and symplectic topology (by McDuff & Salamon)

A tutorial on quantum cohomology (by Givental)
 http://mathoverflow.net/questions/106973/explicit-computation-of-gromov-witten-invariants/107228#107228 Answer by Csar Lozano Huerta for Explicit computation of Gromov-WItten invariants Csar Lozano Huerta 2012-09-15T02:54:52Z 2012-09-15T02:54:52Z
Hope this is among the lines of what you're looking for.

http://mat.uab.es/~kock/invitation.html

The exercises are a very valuable part of this book, for it contains a handful of nice exercises.

As for lecture notes, Renzo Cavalieri has some nice notes you might find helpful. He knows a good deal localization.

http://www.math.colostate.edu/~renzo/


text:
Distance of Particle from Origin

1. Apr 25, 2013

syeh

Question:

Particle A is resting on the origin. Particle B travels along the path of f(x)=-cos(x-7)+3. What is the minimum distance that the particles can be apart?

Attempt:

I tried to find the minimum of f(x)=-cos(x-7)+3. So I found when the derivative is 0:
dy/dx= sin(x-7)
0=sin(x-7)
x=7
So I plugged that into the original equation: f(7)= -cos0+3= -1+3= 2. So the point (7,2) is a possible minimum/maximum.
Using the First Derivative Number Line Test, I found that the point (7,2) is a minimum.

Does that mean that is the point where Particle B is closest to the origin??

Assuming it does, I attempted to find the line connecting (7,2) and the origin, and then to find the length of that curve on the interval x=[0,7]:

y=mx+b
to find m using the 2 points (0,0) and (7,2):
m=(2-0)/(7-0)= 2/7

b is 0 because it does through (0,0)

y=(2/7)x

Length=∫[from 0 to 7] sqrt(1+(dy/dx)^2)dx
dy/dx=2/7
so, Length=∫[from 0 to 7] sqrt(1+(2/7)^2)dx
=∫[from 0 to 7] 1.04 dx
=[1.04x]from 0 to 7
=(using f(b)-f(a)): (1.04*7)-(1.04*0)
=7.28

That was my final answer: when Particle B is closest to the origin/Particle A, it is 7.28 units away.

Is this right? Or on the right path? I don't know.

2. Apr 25, 2013

Staff: Mentor

I don't see how that can be right. The graph of the cosine function is always between 2 and 4, so the farthest a particle on that path could be from the origin would be 4, which is quite a bit less than the number you got.

I'll take a closer look at your calculations and report back.

3. Apr 25, 2013

Staff: Mentor

There are an infinite number of values for which the derivative is zero: {..., 7 - 2$\pi$, 7 - $\pi$, 7, 7 + $\pi$, 7 + 2$\pi$, ...}.

Also, there is a much easier way to find the distance from (7, 2) to the origin. Using Pythagoras, the distance is $\sqrt{7^2 + 2^2}$. In any case, this point isn't the closest to the origin.

4. Apr 25, 2013

syeh

thanks. i see how Pythagorean's theorem works. But i am still confused how to find when the function is closest to the origin.??

5. Apr 25, 2013

Dick

If you mean that the particle A is sitting at (0,0) and particle B is somewhere along the curve (x,-cos(x-7)+3) then you can graph it and try to estimate closest distance. It's a bit over 2. You can also use Pythagoras to get an expression for the distance and try to minimize it with calculus. But it will be too complicated to solve with algebra. You'd have to use numerical methods to get an estimate.

Last edited: Apr 25, 2013
6. Apr 25, 2013

syeh

thanks. i used the Pythagorean theorem and found the length to be 2.084!

7. Apr 25, 2013

Dick

Well, I get something a lot closer to 2.1. I'm not sure what you did. But if you realize you have to estimate the answer that's a start.

Last edited: Apr 25, 2013
8. Apr 26, 2013

HallsofIvy

Staff Emeritus
WHY?? You are asked to minimize the distance to the origin, $$d= \sqrt{x^2+ y^2}= \sqrt{x^2+ f^2(x)}$$, not y= f(x).

Of course, minimizing the distance will also minimize the square of the distance so you can drop the square root and look at $$d^2= x^2+ f^2(x)= x^2+ (3- cos(x- 7))^2= x^2+ 9- 6 cos(x- 7)+ cos^2(x- 7)$$.

Differentiate that.