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## Reproducible Notebooks

Millions of Jupyter notebooks are spread over the internet - machine learning, astrophysics,biology, economy, you name it. "What a great age for reproducible science!" - Or that's what you think until you try to actually run these notebooks. Then you realize that having understandable high-level code alone is not enough to reproduce something on a computer.

Below the superficial charm of a well-written notebook lurks a messy web of low-level dependencies. A notebook doesn't know how to set those up. Dozens to hundreds of libraries can be called and all of them can potentially influence the final results. Setting them up correctly is painful, requiring a different strategy for each programming language. In a previous post, we used a Docker container to circumvent this problem but a Docker container is difficult to customize. Thus it is often non-trivial to run an arbitrary notebook on a machine different from the one it was written on.

This blog post is about defining and reproducing the compute environments below the surface of a Jupyter notebook. We will show how to setup configurable and reproducible Jupyter environments that include JupyterLab with extensions, the classical notebook, and configurable kernels for various programming languages. These environments are declarative, which means that their content - not the steps required to install them - is defined in a few lines of code. A single command installs and runs the compute environment from the configuration file.

Under the hood, we use the Nix package manager, which enables us to build up such composable environments with simplicity. The Jupyter environments can also be containerized with a single Nix command as a Docker image. Such Docker images can then be served with JupyterHub to give multiple users access to a well-defined reproducible environment. With this setup, installation of a Jupyterlab environment with multiple kernels such as in the following image is easy and robust:

A reproducible compute environment is certainly not the only thing that is required to reproduce a notebook. Just think of a random number generator that could be called in one of the code cells. If the seed is not properly set, the notebook will look differently after each run. Reproducible compute environments are therefore only a necessary first step on the road to full reproducibility.

## Declarative Environments

How does such a reproducible compute environment look like? Here is a simple example. First, make sure that Nix is installed on your computer. This works on any Linux and macOS, and won't modify any files except create one single directory in your filesystem. Then, write a shell.nix file with the following contents:

let
jupyter = import (builtins.fetchGit {
url = https://github.com/tweag/jupyterWith;
rev = "10d64ee254050de69d0dc51c9c39fdadf1398c38";
}) {};

ihaskell = jupyter.kernels.iHaskellWith {
name = "haskell";
packages = p: with p; [ hvega formatting ];
};

ipython = jupyter.kernels.iPythonWith {
name = "python";
packages = p: with p; [ numpy ];
};

jupyterEnvironment = jupyter.jupyterlabWith {
kernels = [ ihaskell ipython ];
};
in
jupyterEnvironment.env

and run nix-shell --command "jupyter lab".

After downloading and building of all dependencies, this should open a shell and launch JupyterLab. That's it, all the kernels are installed, and all libraries are accessible to them. Although the first run can take quite a while, subsequent runs are instant because all build steps are automatically cached. More examples of JupyterWith, including Jupyterlab extensions, Docker images, and many other kernels can be found in the README of the project.

## Behind the scenes: Nix

Packaging Jupyter with kernels from many different language ecosystems is complicated. Language specific package managers can only handle their own subsystems and often rely on libraries that are provided by the underlying operating system. If one manages to set all of these up together, the outcome will be conflict-prone and difficult to change. This is where the Nix package manager enters the scene.

Nix is a package manager whose packages are written in the Nix language, which can be thought of as a simple configuration language but with functions. It is used to describe derivations, build recipes that describe every step necessary to build a binary or library from source. A derivation knows its dependencies, source code locations and hashes, build scripts, environment variables and so on. Nix is powerful because we can describe derivations programmatically, as we did for example with the jupyterlabWith, iPythonWith and iHaskellWith functions. We can also programmatically compose different derivations, our own or the ones from the enormous Nixpkgs repository, to generate new ones. For example, different kernels that are themselves composite packages can be combined with Jupyterlab into a single application. The output is a big dependency web that links the final build output, i.e. Jupyterlab with kernels to all of its dependencies as shown in this dependency graph:

Packages and their dependencies are represented as interconnected nodes in this graph. Such a graph is called the closure of JupyterWith, that is the set of all derivations that are required to run the Jupyterlab. This means that the environment generated by JupyterWith includes not only Jupyter and its kernels, but also all system packages that are required. This guarantees a high degree of reproducibility.

The final JupyterLab environment is complete to a point that we can simply copy it into a base docker Alpine image to run it. No further installation or configuration is required.

## Behind the scences: packaging Jupyterlab

Let's look in more detail into the packaging process of JupyterLab. A JupyterLab environment is composed of three parts:

• the JupyterLab frontend is essentially a browser application that consists of several bundled Typescript/Javascript core and extension modules;
• the Jupyter server is a Python application that intermediates interactions between kernel compute environments and frontends;
• kernels such as IPython or IHaskell execute code that is send from a frontend such as JupyterLab via the Jupyter server.

The Jupyter server is easy to install with Nix because it is just a set of Python packages that have to be put together. Nix already has a fairly well developed Python ecosystem which can be used for this.

Setting up the kernel environments is a bit more tricky. Kernels are independent executables that require independent package sets and environment variables. As is common in Nix, we wrap the kernel executable into a shell script that locally sets all required environment variables. This wrapper script is then exposed to Jupyter with a kernel spec file. For example, the file for the IHaskell kernel looks as follows:

{
"display_name" = "Haskell";
"language" = "haskell";
"argv" = [
"/path/to/ihaskellShell"
"kernel"
"{connection_file}"
];
"logo64" = "logo-64x64.svg";
}

Jupyter is made aware of the location of these files through the JUPYTER_PATH environment variable. For kernels, all we need to do is building packages and setting environment variables. Nix handles this wonderfully.

When dealing with Jupyterlab extensions we face a problem since Jupyterlab expects to manage them itself. Usually, they are installed with the command jupyter labextension install which is a wrapper of the Javascript package manager yarn. Yarn determines compatible versions for the JupyterLab core modules and the required extension modules together with all their Javascript dependencies. This resolver step is difficult to reproduce with Nix, since Nix is not about finding compatible versions. Even worse: it doesn't even make sense to use the precise versions that yarn determined and feed them into Nix. Every combination of extensions can have a different set of compatible versions, which means that we can't pre-resolve all of them.

We therefore pre-build JupyterLab into a custom, self-contained folder with all required extensions. This folder is the starting point of the Nix derivation and can be referred to from the shell.nix file.

## Conclusions

Jupyter is a showcase example for the power of a configuration language like Nix combined with the Nixpkgs collaborative project to describe all open source packages out there. With a few lines of Nix code, we can pull together dependencies from various language ecosystems in a robust way and build configurable compute environments and applications on top of them. Such configurable compute environments can be used and distributed in various ways, including Docker containers. This is a boon for reproducible science: declaratively specify your dependencies and compose your dependency descriptions to incrementally build larger and more complex notebooks that you know will run now and forever in the future.

Suggestions and PRs on JupyterWith are welcome!


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# Is a complex number with transcendental imaginary part, transcendental?

A complex number that has transcendental real part is always transcendental? How about in the case of imaginary part?

-

The answer to your first question, "A complex number that has transcendental real part is always transcendental?" is yes, since the conjugate of an algebraic number $r$ is a root of the same polynomial that witnesses that $r$ is algebraic. But then the sum of $r$ and its complex conjugate (twice the real part) is algebraic as well -- the algebraic numbers form a field, so they are closed under addition, multiplication, inverses...
Similarly, if the imaginary part of a number is transcendental, then the number is transcendental, by essentially the same argument (now you would consider the difference between $r$ and its conjugate).
On the other hand, if the real part of a number is transcendental, we cannot conclude that the imaginary part is transcendental as well. For example, look at $\pi$, or $\pi+i$. Similarly, if the imaginary part is transcendental, we cannot conclude anything about the real part.

format: plain


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# TASM include'ing

This topic is 5477 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

## Recommended Posts

I'm using TASM for a class, and we're learning from Irvine's book, which uses MASM. Thus, for things specific to TASM, I have to wait for the professor to give us packets on the commands/parameters. I've emailed my professor, but she only "gets" her email when she needs something from me. All I have to work with at present is a batch file I've written from the first packet, which is primarily this: tasm/l/n/z gn tlink/3/m/v gn gn I'm now trying to do the equivalent of "#include" and I'm having some problems. What exactly is the syntax for this presuming I use
call Random
in my main file and have something like this procedure:
;--------------------------------------------------------------
Random  proc
;
; Returns an unsigned pseudo-random 8-bit integer
; in AL,in the range 0 - 9.
; Last update: 9/23/04
;--------------------------------------------------------------
.data
.code
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;          Get system seconds             ;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
mov ah,2Ch
int 21h
mov al,dh ;; is now 0-59 (dh is seconds)

cmp al,10

DECR_LOOP:
sub al,10
cmp al,9
jc DONE

jmp DECR_LOOP

DONE:
ret
Random  endp

I've tried
include proc.asm
after the .stack/.model directives which removes some errors but not all. What parameters am I missing in the bat file and in the proc.asm?

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What errors are you getting, specifically?
Just stating that you are getting errors isn't very helpful. Please try and be more detailed in your descriptions of your problem. It will make both your life and ours easier. It will also ensure that you will get a much more correct answer.

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Quote:
 Original post by WashuWhat errors are you getting, specifically?Just stating that you are getting errors isn't very helpful. Please try and be more detailed in your descriptions of your problem. It will make both your life and ours easier. It will also ensure that you will get a much more correct answer.

Yeah, I know. I was just hoping someone could give me a skeleton set up, as that's essentially what I need to get moving.

Anyway, per request, ....

Actually, I'm not sure what happened but it's working now...

I think I've found it, I may've had .code commented out in proc.asm

Anyway, I seem to recall having to pass .obj files as params to masm's linker a few years ago. Maybe I'm totally wrong, but this is working.

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Quote:
 Original post by kordovaI think I've found it, I may've had .code commented out in proc.asm

Yes, that would do it
Quote:
 Anyway, I seem to recall having to pass .obj files as params to masm's linker a few years ago. Maybe I'm totally wrong, but this is working.

Only if you used /C to not perform the linking step. Then in the final build you would do something like: link a+b+c,out;

For those cases you declare the exports (in a header) and define the functions in other files. Generally including the whole file is a big PITA. Including a header with the exports is much cleaner, less error prone too.

Thanks a lot!

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Find the value(s) of ‘k’ for which the roots of the following equations are real and equal. NCERT Class 11 revision notes maths ch 5 Complex Numbers and Quadratic Equations are prepared by the expert teachers at Vedantu. Complex Numbers Lesson 33 Today’s Vocabulary imaginary unit i pure imaginary number complex number complex conjugates rationalizing the denominator Learn Pure Imaginary Numbers In your math studies so far, you have worked with real numbers. In other words, there is no intersection of the graph of this function and the xaxis. (0) zero is examined to be both real. There's also a bunch of ways to solve these equations! Watch this tutorial and get introduced to quadratic equations!. First Complex Number ( N1 ): a= b= a+b*i. Rational numbers can be classified as natural numbers (N), whole numbers (W), integers (Z), and/or quotients of these. There is a thin line difference between both, complex number and an imaginary number. 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In this video I show you how we can use this fact to find the quadratic equation knowing one of the roots. This chapter covers graphing quadratic functions, solving quadratic equations, and functions, solving quadratic equations, and comparing linear, exponential, and quadratic models. c = 0, the value of which can be used to determine the number of solutions of a quadratic equation. Solving quadratic equation. When a real number, a, is added to an imaginary number, a + bi is said to be a complex number. Have your paper edited by your writer as many times as you need, until it’s perfect. Solving quadratic equations using the quadratic formula 3. How Would You Solve This Hard Letter Math. 156 completing the square, p. To find the quadratic mean of 1, 2 and 2, you square then all: 1, 4, 4. A quadratic is an algebraic expression having 2 as the highest power of its varia. However, in this case the roots. (0) zero is examined to be both real. The Complex Number Z Is Purely Imaginary If. 3: Solve quadratic equations using square roots. Chapter 9: Imaginary Numbers Conceptual. ax2 + bx + c = 0 is the standard form of a quadratic equation where a, b and c are variables and a ≠ 0. But if there are other symbolic parameters in the equation, then the solutions will typically be functions of these parameters. Diagonalization. Students will understand what it means to solve a quadratic equation. Imaginary Numbers The square root of a negative real number is called an imaginary number, e. Completing the square. 3 x 2 = 100 - x 2 Solution: Step 1. √-47 is usually written as i √47 indicating it's an imaginary number. Finally, students will solve quadratic functions over the set of complex numbers. Imaginary numbers have the form bi where b is a non-zero (real number) and i is the imaginary unit, defined as the square root of − 1. An equation is a mathematical expression presented as equality between two elements with How to solve an equation step by step? The calculation steps of the solver are not shown because they do not correspond to the steps a human would make. It "cycles" through 4 different values each time we multiply:. And these imaginary roots, combined with the real number, the -b, in the numerator of the quadratic formula make the solutions complex (unless b = 0 in which case the solutions are pure imaginary numbers). You can use various methods. Solving Quadratics Equations y Simplified Radical Form Using the Square Root to find Exact S01Utions Practice Problems 4. Imaginary numbers and quadratic equations. All fields have the same properties in relation with systems of linear equations, but quadratic equations behave differently from field to field. Source code: For real or complex numbers. The following calculator can be used to simplify ANY expression with complex numbers. Its solution may be presented as x = √a. Topics to study include (but are not limited to) how to graph transformations and parabolas, to write the quadratic equation given a graph or information, to be able to perform operations on imaginary numbers, find the discriminant, calculate values of i, solve quadratic equations, and write equations of word problems modeled with quadratics. Learning Target: natural numbers: whole numbers: 80% of the students will be able to solve !!+16=0. 1 Solving Quadratic Equations Graphically, p. In those cases, or if you're uncertain whether the line actually crosses the y-axis in this particular point you can calculate b by solving the equation for b and then substituting x and y with one of your two points. where x1 and x2 are the roots of the quadratic equation ax2 + bx + c = 0. The latter is impossible because a is a real number and the first equation would imply that a 2 = −1. a, b, and c can also be imaginary numbers. 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How are imaginary numbers used in solving quadratic equations? How can solutions of this type be represented graphically? Can Euler's equation be used to find any number raised to a complex power? How is it possible that all real numbers raised to an imaginary power map to the complex. Solve linear, quadratic and polynomial systems of equations with Wolfram|Alpha. The imaginary numbers are a subset of the complex numbers. pure imaginary number. For example: $x^{2}$ - 2ix - 1 = 0 has both roots equal to 'i'. We have negative 3x squared plus 10x minus 3 is equal to 0. Implicit Differentiation. Perform operations with complex numbers. x2 + 6x + 9. b = 0 the number is a real number. You can solve a quadratic equation using the quadratic formula. Videos are created by fellow teachers for their students using the guided notes from the unit. Interesting Property. Solving Quadratic Congruences. In chapter 2 we established rules for solving equations Upon completing this section you should be able to solve equations involving signed numbers. We can then multiply these numbers together to get an The traditional method would be to work out values for A, B, and C and plug them into the quadratic formula. The square root of a negative number is a complex number. You just need to fill in the boxes "around" the equals signs. 3  Notes  Solving Quadratics with Imaginary Numbers. If a solution is not possible explain why. Proceedings Number Theory Eger 1996, (Györy, Pethö, Sos eds. Today they have important uses in electronics, engineering, and even biology. The number of equations in the system To change the signs from "+" to "-" in equation, enter negative numbers. b = 0 the number is a real number. Imaginary numbers are used to help us work with numbers that involve taking the square root of a negative number. These free lessons are cross-referenced to help you find related material, and the "Search" box on every page is available to help you find whatever math content you're looking for. Given two complex numbers, be able 'teo write in standard form A, Their surn Their difference C. Solve a linear system of equations with multiple variables, quadratic, cubic and Write each equation on a new line or separate it by a semicolon. Using the Quadratic Formula: Scientific Notation: Imaginary Numbers: Values of Symbols for Which Fractions are Undefined: Graphing Equations in Three Variables: Writing Fractions as Decimals: Solving an Equation with Two Radical Terms: Solving Linear Systems of Equations by Elimination: Factoring Trinomials: Positive Rational Exponents. Its solution may be presented as x = √a. notebook 6 January 11, 2017 Jan 199:49 PM Solving a Quadratic by Completing the Square 1. PROPERTIES OF SQUARE ROOTS (a > 0, b > 0) Product Property:ab p Quotient Property: a b. Jul 24, 2015 - Quadratic Equations and Complex Numbers (Algebra 2 Curriculum - Unit 4) DISTANCE LEARNINGUPDATE: This unit now contains a Google document with: (1) Links to instructional videos. This page will show you how to solve two equations with two unknowns. Recognize that the number of atoms of each element is conserved in a chemical reaction. All in due course. Just in case you will need advice on beginning algebra as well as multiplying and dividing fractions, Mathpoint. All equations can be solved by factoring, however, you could easily have the students solve by completing the square or the quadratic formula as well. The constants ‘a’, ‘b’ and ‘c’ are called the coefficients. π,2,35, etc). I can use the discriminant to determine the number and type of roots of a. The name of the property that states:If ab= 0, then a = 0 or b = 0, The form that a quadratic equation must be written in in order to use the factoring method, How would this equation be factored:x2 - 9x + 18 = 0, How would this equation be factored?x2 + 6x - 40 = 0. Given x 2 - 4 x + 13 = 0 The discriminant Δis given by Δ= b 2 - 4 a c = (-4) 2 - 4(1)(13) = -36; Since the discriminant is negative, the square root of the discriminant is a pure imaginary number. Real numbers. 9x2 + 25 = 0 Solve the equation. The calculator uses the quadratic formula to find solutions to any quadratic equation. Just in case you will need advice on beginning algebra as well as multiplying and dividing fractions, Mathpoint. The first thing you have to do when given the quadratic equation is bring all the terms to one side so that you have a zero on the other side of the equals to. Now, let's start the completing-the-square process. For example, 5i is an imaginary number, and its square is −25. a unique quadratic function. ObjectivesDefine and use imaginary and complexnumbers. Imaginary numbers are used to help us work with numbers that involve taking the square root of a negative number. Understand solving equations as a process of reasoning and explain the reasoning. The formula is: $\frac{ -b \pm \sqrt{b^2 -4ac}}{2a }$ The quadratic formula calculator below will solve any quadratic equation that you type in. There is one main place that complex numbers will pop up on you: The Quadratic Formula! Sometimes, you'll get negative numbers under that radical. Mathematical Practices 6 Attend to precision. This page will show you how to solve two equations with two unknowns. Indefinite Integral Rules. 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Imaginary Numbers: Solving Quadratic Equations Using the Quadratic Formula: Solving Quadratic Equations: Algebra: Order of Operations: Dividing Complex Numbers: Polynomials: The Appearance of a Polynomial Equation: Standard Form of a Line: Positive Integral Divisors: Dividing Fractions: Solving Linear Systems of Equations by Elimination: Factoring. The set of complex numbers is introduced and students will operate with the imaginary number i. c = 0, the value of which can be used to determine the number of solutions of a quadratic equation. This is of the form ax 2 + bx + c = 0. Question 8. "So we seek two numbers of the form -B/2±z, where z is a single unknown quantity," he says. Algebra-help. These free lessons are cross-referenced to help you find related material, and the "Search" box on every page is available to help you find whatever math content you're looking for. A 7-3 Solving Any Quadratic Equation. SOLVING QUADRATIC EQUATIONS Block: Standard Form For Complex Numbers can be written in the form a + bi (where a and b are real numbers). You can also learn the "official" version about the formula for solving a quadratic equation also at Purplemath. These free lessons are cross-referenced to help you find related material, and the "Search" box on every page is available to help you find whatever math content you're looking for. Solving linear congruences is analogous to solving linear equations in calculus. Well, the quadratic equation is all about finding the roots and the roots are basically the values of the variable x and y as the case may be. Equation on a Calculator References Demo Slide 1. A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e. 146 root of an equation, p. Solving (1) for y, we have y = x 2 /a, and by substitution into (2), we get y 2 = (x 2 /a) 2 = x 4 /a 2 = xb, which yields: x 3 = a 2 b. b^2 - 4ac, describes the roots of a quadratic equation: if it is greater than or equal to 0, there are two real. This program works for all positive real numbers. I need help with this problem, i am new with the ''complex world''. The map d ↦ Q(√d) is a bijection from the set of all square-free integers d ≠ 0,1 to the set of all quadratic fields. Free downlod 10th maths books pdf files, square a number by a fraction, simple mathematics for dummies. The answer you come up with is a valid "zero" or "root" or "solution" for "ax 2 + bx + c = 0", because, if you plug it back into the quadratic, you'll get zero after you simplify. The complex number system includes the real numbers and the imaginary numbers. Just in case you need assistance on mathematics or even decimals, Algebra-help. can be written as the complex number. This was solved by Girolamo Cardano, who divided 10 into two. It took a supercomputer and millions of hours to finally solve it. A Quadratic Equation with Imaginary Solutions We have seen that a quadratic equation can be expressed in the general form: ax +bx+c=O, a All quadratic equations can be solved using the quadratic formula: _ b ± b2 If the discriminant is negative, there are no real solutions. 4 determine the solution set for linear and quadratic inequalities; Graphical and algebraic solutions. It is of importance largely because multiple questions in the other chapters are based on quadratic equations, which you cannot solve until and unless you have. in a+bi complex number i is called the imaginary unit,in given expression "a" is the real part. We can help you solve an equation of the form "ax2 + bx + c = 0" Just enter the values of a, b and c below How Does this Work? The solution(s) to a quadratic equation can be calculated using the Quadratic Formula : The "±" means we need to do a plus AND a minus. These are sometimes called pure imaginary numbers. to find the solutions to quadratic equations. b^2 - 4ac, describes the roots of a quadratic equation: if it is greater than or equal to 0, there are two real. Roots of quadratic equation ax2 + bx + c. In other words: A quaternion squares to −1 if and only if it is a vector quaternion with norm 1. However, some equations such as x2 + x + 1 = 0 do not have real solutions. Now we return to the quadratic with real coeﬃcients, x2 + 2ax+ b= 0. Pure imaginary numbers – numbers in the form bi – where i= −1. We can now solve both of these equations trivially. where x1 and x2 are the roots of the quadratic equation ax2 + bx + c = 0. In fact, the new numbers allow the solution of any quadratic equation, and first saw light in this application. Upon completing this goal the student will be able to: * solve quadratic equations by graphing, factoring, and completing the square. The imaginary unit “i” is used to represent: i 1 and i2 1 Ex. The complex number equation calculator returns the complex values for which the quadratic equation is zero. Where we begin It all started at a meeting of the National Union of Teachers. All solutions are prepared by subject matter experts of Mathematics at. Simplify each of the following. My assignment is to make the quadratic equation, and get imaginary numbers but I'm having an extremely difficult time getting there. Quadratic Functions part 2 of 2 TOPjCS Perfect Squares Cubes Chart Algebra 2 page I. The graph of this quadratic equation is a parabola. Quadratic Equation Calculator. Imaginary roots. Example: Solve the following quadratic equation by taking the square roots of both sides. 3: Simplify i13 Ex. Online equations solver. Quadratic Equations, Cubic and Higher Order Equations : Plots, Factorization, Formulas. Imaginary numbers have a similar story. 2: Add, subtract, and multiply complex numbers. Polynomials of degree three and four are known as cubic and biquadratic polynomials (i) To find the common root of two equations, make the coefficient of second degree term in the two equations equal and subtract. The solutions to the quadratic formula would be imaginary numbers. First way is to find out the roots then add 1 to them and then the form new equation. The square roots of negative numbers are imaginary. Write a C program to calculate the root of a Quadratic Equation. Our writers know exactly Unit 4 Solving Quadratic Equations Homework 4 Pure Imaginary Numbers what points to highlight to make your writing suitable and convincing for the admission board. Imaginary numbers are complex numbers where a = 0 and b ≠ 0. Negative number were needed to solve a + x = b, even when a > b. We have negative 3x squared plus 10x minus 3 is equal to 0. This is what i thought: Given a complex number: $z= a + bi$ where $a,b\in. Core Pure 2. In this section, we will learn how to find We have already seen that completing the square is a useful method to solve quadratic equations. Solution of a Quadratic Equation The equation ax2 + bx + c = 0, where a, b and c are numbers (real or complex. 156 completing the square, p. A quadratic equation has two solutions. All in due course. A quadratic is an algebraic expression having 2 as the highest power of its. Find the zeros of each function. Activity 2 The need for complex numbers Solve if possible, the following quadratic equations by factorising or by using the quadratic formula. I think it was one of my math teachers who said, when asked about whether imaginary numbers were real things, “they’re as real as any other number!” One little quibble, though – the voltage at the wall socket in North America is about 110V rms, so the peak voltage (assuming a perfect sinusoid) is sqrt(2)*110, or about 156V. So this "solution to the equation" is not an x-intercept. Keyword-suggest-tool. 2x 2 + x – 6x – 3 = x 2 + 5x. An equation is a mathematical expression presented as equality between two elements with How to solve an equation step by step? The calculation steps of the solver are not shown because they do not correspond to the steps a human would make. Roots of Quadratic Equations and the Quadratic Formula. Implicit Differentiation. In those cases, or if you're uncertain whether the line actually crosses the y-axis in this particular point you can calculate b by solving the equation for b and then substituting x and y with one of your two points. Imaginary numbers have a similar story. 7) 10n2 - n - 8 = 08) 8p2 - 12p + 7 = 0 9) 2r2 + 2r + 6 = 0 10) 11r2 - 5r - 12 = 7 11) -14 + a = -3a2 12) -5 = 11b2 - 2b 13) 3n2 + 10n = -12 - 8n2 + 10n14) r2 - 2r - 4 = 2r2 + 8 Find the discriminant of each quadratic equation then state the number and type of solutions. be a real quadratic number field of discriminant. For this purpose, we find the factors of this function. You can solve a quadratic equation using the quadratic formula. COmpleting the Square the QuadratiC Formula Practice Problems 4. The imaginary_part function calculates online the imaginary part of a complex number. Perform operations with complex numbers. Free ordinary differential equations (ODE) calculator - solve ordinary differential equations (ODE) step-by-step. The Venn diagram shows some examples of complex numbers. Imaginary Number Rules Suppose, a pure quadratic equation is x*x= a, where the value of a is known. Because of this we can think of the real numbers as being a subset of the complex numbers. In this video the tutor shows how to simplify a quadratic formula result. Learn - Program to find power of a number. Imaginary number definition is - a complex number (such as 2 + 3i) in which the coefficient of the imaginary unit is not zero —called also imaginary. Second Complex Number ( N2 ): c= d= c+d*i. Perform operations with complex numbers. Discriminant 10. Condition Number. 38: Solve. Imaginary numbers just simplify the mathematics in this case. int main() {. Imaginary part on the left = imaginary part on the right: 4y=8. "Complex" numbers have two parts, a "real" part (being any "real" number that you're used to dealing with) and an imaginary" part (being any number with an "i" in it). Quadratic Equation. Roman numbers. ( where i represents an imaginary. (3 - 5i) - (6 + 7i) = (3 - 6) + (-5 - 7)i = -3 - 12i. f(x) = x2 + 8x – 24 3. 15) -4m2 + 4m - 1 = 016) 10n2 - 2n = 0. Solve the linear equations with variables in numerator and denominator, check the solution and determine the conditions of solvability. why imaginary numbers were invented. Quadratic equation – 2nd degree. Exercise 1a Task 36 And 37 Further Pure Mathematics 1. Imaginary part of complex number: imaginary_part. These complex roots will be expressed in the form a + bi. Quadratic Equation is the most important chapter from a IIT -JEE main & advance point of view. Students will develop skills in solving quadratic equations. Square roots of negative numbers are called imaginary numbers and will be discussed later in this section. Free quadratic equation calculator - Solve quadratic equations using factoring, complete the square and the quadratic formula step-by-step. Solve equations of the form$ax^2 + bx + c = 0\$ show help ↓↓ examples ↓↓. doc Author: E0022430 Created Date: 2/9/2010 12:03:19 PM. Type the equations here. WORDS NUMBERS ALGEBRA An imaginary number is the square root of a negative number. Real numbers – a combination of rational and irrational numbers. Solving quadratic equations. Next, let’s take a look at a complex number that has a zero imaginary part, $z = a + 0i = a$ In this case we can see that the complex number is in fact a real number. Since the real and complex parts of these numbers are completely separable, they can also be considered to be ordered pairs of real numbers, representing points or vectors in R 2. x 2 - 3 = 0. All non-imaginary numbers are real. A complex. It outputs pure imaginary numbers that cannot be shown. When a real number, a, is added to an imaginary number, a + bi is said to be a complex number. The square of an imaginary number bi is −b2. Two roots will be displayed, equivalent to the solutions found "by hand" using the Quadratic Formula. Organizing topic Rational and Radical Equations. The image is built on the theory of self-similarity and the operation of iteration. Where a ≠ 0, b and c are a constant numbers. Imaginary Unit. 2 Complex Numbers with answers 5 If b ≠ 0, then a + bi is an imaginary number. Just in case you will need advice on beginning algebra as well as multiplying and dividing fractions, Mathpoint. Introduction To Quadratic Equations / Basic Of Quadratic Equations. Name: Unit 4: Solving Quadratic Equations Directions: Solve the equations. 156 imaginary number, p. a, b, and c can also be imaginary numbers. To solve these, the rule is that you need the same number of equations as you have unknowns. Either two distinct real solutions, one double real solution or two imaginary solutions. can be written as the complex number. b = 0 the number is a real number. Normally, it is impossible to solve one equation for two unknowns. Addition Subtraction Multiplication Division Square Root Modulus (absolute value). Online equations solver. Solved name unit 4 solving quadratic equations date b chegg com equation with pure imaginary solutions you nonmonic roots nagwa Algebra 2 4 Notes Example 3 Equation With Pure Imaginary Numbers You. These are all quadratic equations in disguise: In disguise. i is a number that is not a real number. A complex number is any sum of a real number and imaginary number. Solve quadratic equations by factoring. number is a pure imaginary number however when. There are more advanced formulas for expressing roots of cubic and quartic polynomials, and also a number of numeric methods for approximating roots of arbitrary polynomials. Solve Quadratic Equation. They are defined separately from the negative real numbers in that they are a square root of a negative real number (instead of a positive real number). There are always two roots for any quadratic equation, although sometimes they may coincide. (The degree of the equation is 2) (The highest exponent of a variable is 2). And you might say, hey, wait Sal. Hence verified. What is Quadratic Equation? # An equation of the form is known as a quadratic equation, where a, b and c are numbers and a is not equal to 0. All solutions are prepared by subject matter experts of Mathematics at. Are there other "imaginary" numbers like [;i;], where there defined by some property that no other integer has? For instance …. A quadratic equation has two solutions. A complex. High quality Imaginary Numbers gifts and merchandise. The following calculator can be used to simplify ANY expression with complex numbers. Can I solve quadratic equations by using the quadratic formula? Can I use the discriminant to determine the number. Before going to class, some students have found it helpful to print out Purplemath's math lesson for that day's topic. pure imaginary numbers: square roots of negative real numbers: complex number: any number that can be written in the form a+bi, where a and b are real imaginary numbers and i is the imaginary unit: completing the square: a method that can be used to solve all quadratic equations by using the Square Root Property and manipulating the equation. The square of an imaginary number bi is −b2. Note: A quadratic equation is a polynomial equation of degree 2. The set of complex numbers is introduced and students will operate with the imaginary number i. Free quadratic equation calculator - Solve quadratic equations using factoring, complete the square and the quadratic formula step-by-step. First we write the characteristic equation: \[{k^2} + 4i = 0. Exercise 1a Task 36 And 37 Further Pure Mathematics 1. Imaginary numbers. ), Gruyter 1998, 123-146 arxiv: On Gauss Bounds of quadratic extensions of imaginary quadratic Euclidean number fields Publ. I think it was one of my math teachers who said, when asked about whether imaginary numbers were real things, “they’re as real as any other number!” One little quibble, though – the voltage at the wall socket in North America is about 110V rms, so the peak voltage (assuming a perfect sinusoid) is sqrt(2)*110, or about 156V. A wonderful algebra tutor that has made equation solving easy for me. x2 - 3x + 1 = 0. Over the years, scientists have Mathematician Makes Quadratic Equations Easier. Math 2 Name: _____ Solving Quadratic Equations Worksheet #4 Solve the following quadratics with complex numbers: 1. Solve cubic or quartic equations with real coefficients. Concepts factored form FOIL method Lesson 4. Javascript application that finds solutions to quadratic modular equations. Note that, in some disciplines, the symbol j is used in place of i. You can change the value of a, b and c in the above program and test this program. We have negative 3x squared plus 10x minus 3 is equal to 0. Complex Imaginary Numbers Practice Problems 4. But just how do we simplify these pure imaginary numbers? Let's take a closer look at the first example. We call α the real part of z because, if β = 0, then z = α is just a real number. to solve for the length of a side of a right triangle (Pythagorean Theorem). it works fine until I get to an equation that involves the negative discriminant and with that, the solution involves Is there a way to solve for the imaginary number?. The roots belong to the set of complex numbers, and will be called "complex roots" (or "imaginary roots"). To solve a system of linear equations using Gauss-Jordan elimination you need to do the following steps. If the product of two numbers is zero then one or both of the numbers must also be equal to zero. Jan 25, 2015 - This Maze has 42 question (6 by 7). Remember that the calculator program is. The number i is just as normal as other. Real numbers are complex numbers where b = 0. The friction and heat transfer coefficient for a flat plate can be determined by solving the conservation of mass, momentum, and energy equations. equation_straight_line equation_tangent_line euclidean_division exp expand expand_and_simplify expand_log expand_trigo factorial factoring_calculator fraction_calculator gcd imaginary_part inequality_solver integral_calculator inverse_matrix is_even is_odd is_odd_or_even_function lcm. 156 complex number, p. Substitute the value of y to the first equation, we have. Pg 280, #1-16. Addition Subtraction Multiplication Division Square Root Modulus (absolute value). We can now solve both of these equations trivially. 15) -4m2 + 4m - 1 = 016) 10n2 - 2n = 0. NUMBER AND TYPE OF SOLUTIONS OF A QUADRATIC EQUATION Consider the quadratic equation ax2 + bx + c = 0. Notice that this product eliminates For the complex number we define its complex conjugateto be The multiplication of complex conjugates results in a real number. (a) x 2 −1=0 (b) x2 −x −6 =0 (c) x 2 −2x −2 =0 (d) x2 −2x +2 =0 You should have found (a), (b) and (c) straightforward to solve. The number -2i is a pure imaginary number. FACTORING: Find the roots of the following. Finally, students will solve quadratic functions over the set of complex numbers. com - Your collection of math tasks. Unit 3 - Quadratic Functions. Natural Whole umben Wr Numben Rftva(oomaxs l, 2131 C h — OAS -RYO. This technique can solve 'perfect square' quadratic equations like this I hope this short insights video has been useful to you to help explain to your learners the types of equations 'completing the square' solves and a very visual way to explain how to use the 'completing the square' method. What is Quadratic Equation? # An equation of the form is known as a quadratic equation, where a, b and c are numbers and a is not equal to 0. A guy named Cardano developed complex numbers off the base of the imaginary number i 1, the solution to our “easy. So the question is: The equation z^4 - 8z^3 + 20z^2 - 72z + 99 = 0 has a pure imaginary root. The calculator accepts numbers of up to 1000 digits, but notice that the modulus n should be factored (some large numbers cannot be factored in a reasonable amount of time). Solving quadratic equations by factoring 2. A pure imaginary number is any number which gives a negative result when it is squared. Be careful though: 0i is real, because 0i = 0. The "standard" format for complex numbers is "a + bi"; that is, real-part first and i-part last. Return only those solutions for which every subexpression of the original equation represents a real number. For nonnegative real numbers a and b, √ =√ ∙√ and √ Ô Õ =√ Ô √ Õ, ≠0. 5-3 Solving Quadratic Equations by Factoring. Following rules are followed in imaginary number – i = √-1; i 2 = -1; i 3 = -i; i 4 = +1; i 4n = 1; i 4n-1 = -i; Imaginary Number Chart One of the most interesting properties of i is when you multiply. Imaginary unit. Complex Numbers and Quadratic equations. You can use the imaginary unit to write the square root of any negative number. can be written as the complex number. Addition Subtraction Multiplication Division Square Root Modulus (absolute value). Fields are number systems in which every linear equation has a solution, such as the set of all rational numbers Q or the set of all real numbers R. Imaginary Part. It is possible to multiply imaginary numbers and obtain a real number. Every other imaginary number is just a multiple of i. There are two ways of solving it. Complex numbers are numbers that are part real number and part imaginary number. A quadratic equation MUST have at least one x^2 term; is allowed (but not required) to have x and constant terms; cannot have any other term types. Quadratic Equation Calculator The calculator will solve the quadratic equation step by step either by completing the square or using the quadratic formula. Complex numbers are not real, unless b = 0. Euler proposed letting by represented by i Fact #1: Fact #2: Why? Because Euler said so! Why? Because if: Complex numbers should be written in the form of a + bi 2i is an "pure" imaginary number 1 + 2i-- is a complex. Linear equation has one, two or three variables but not every linear system with 03 equations. pure imaginary numbers: square roots of negative real numbers: complex number: any number that can be written in the form a+bi, where a and b are real imaginary numbers and i is the imaginary unit: completing the square: a method that can be used to solve all quadratic equations by using the Square Root Property and manipulating the equation. Given two complex numbers, be able 'teo write in standard form A, Their surn Their difference C. Core Pure 2. Solution : 2x 2 + 5x. This occurs when we multiply and Use the FOIL method. We can now solve both of these equations trivially. Check 5x2 + 90 = 0 0 02 0 5(18)i +90 90(–1) +90 5x2 + 90 = 0. And there's many ways to solve this. In algebraic number theory, a quadratic field is an algebraic number field K of degree two over Q, the rational numbers. How Would You Solve This Hard Letter Math. Add to each side of the equation to makea. Students study the fundamental operations involving whole numbers, fractions, decimals and percents, ratio and proportion, interpretation of graphs, geometry, and introductory algebra, including axioms and properties of the real number system, fundamental operations involving algebraic expressions, and first and second degree equations and. Complex Numbers. Quadratic Equation: ax2 + bx + c =0.   B Check the sign. An equivalent form is b2 — 4ac. Free ordinary differential equations (ODE) calculator - solve ordinary differential equations (ODE) step-by-step. f(x) = x2 + 8x – 24 3. Diagonalization. Solving Quadratic Equation on a Calculator Lesson 15. That is, anything of the form a + bi, where a and b themselves are real. The Venn diagram shows some examples of complex numbers. Implicit Function or Relation. Venn diagrams may be used. It can perform both classification and transform (for LDA). Quadratic equation usually used to find the unknown number(s) of x in the equation. I need help with this problem, i am new with the ''complex world''. ©p W2T0J1m6r fKWuitLaC `SsoXfltSwJairseh [LBLrCt. Solving quadratic equations. For example, $$3i$$ is an imaginary number, and its square is $$−9$$. Online Equation Solver. But you cannot graph a complex number on the x,y-plane. Following rules are followed in imaginary number – i = √-1; i 2 = -1; i 3 = -i; i 4 = +1; i 4n = 1; i 4n-1 = -i; Imaginary Number Chart One of the most interesting properties of i is when you multiply. Each method also provides information about the corresponding quadratic graph. The square root of a negative number is a complex number. This post is an introductory discussion on the congruence equations of the form where the modulus is an Specific algorithms for solving quadratic congruence eqautions with odd prime moduli are discussed in this subsequent post. Any number in the form of a+-bi, where a and b are real numbers and b not equal 0 is considered a pure imaginary number. In the rst century, the Chinese mathematician Sun-Tsu asked: There are certain things whose number is unknown. x2 + 6x + 9. a, b and c. For example, it is not possible to find a real solution of. ObjectivesDefine and use imaginary and complexnumbers. (0) zero is examined to be both real. The real part is the a-value and the imaginary part is hi. be an imaginary numbers(i). Type the equations here. Rational numbers can be classified as natural numbers (N), whole numbers (W), integers (Z), and/or quotients of these. Finally, students will solve quadratic functions over the set of complex numbers. A complex number is often designated as z. Further Pure 1 Complex Numbers Page 7. It is of importance largely because multiple questions in the other chapters are based on quadratic equations, which you cannot solve until and unless you have. Quadratic Formula. Solve the linear equations with variables in numerator and denominator, check the solution and determine the conditions of solvability. c = 0, the value of which can be used to determine the number of solutions of a quadratic equation. They write the quadratic equation for 7 problems from the solution sets. Imaginary Numbers: Solving Quadratic Equations Using the Quadratic Formula: Solving Quadratic Equations: Algebra: Order of Operations: Dividing Complex Numbers: Polynomials: The Appearance of a Polynomial Equation: Standard Form of a Line: Positive Integral Divisors: Dividing Fractions: Solving Linear Systems of Equations by Elimination: Factoring. Now, let's start the completing-the-square process. Topics include simplifying expressions with addition, multiplication,and division, solving multi-step equations, solving inequalities of various types, and more View the lesson. Because the square root of a negative is an imaginary number, where i is the imaginary unit. Unit 2: Quadratic Equations Lesson 5: The Imaginary Number i Essential Question: What is an imaginary number? Standard: N-CN. The square of an imaginary number bi is −b2. The example below illustrates how this formula applies to the quadratic equation x2 - 2x - 8. I am writing a program for the dreaded Quadratic Equation. √-2, √-5 etc. Organizing topic Rational and Radical Equations. com Imaginary numbers and quadratic equations sigma-complex2-2009-1 Using the imaginary number iit is possible to solve all quadratic equations. The first thing you have to do when given the quadratic equation is bring all the terms to one side so that you have a zero on the other side of the equals to. The complex number system includes the real numbers and the imaginary numbers. Solving quadratic equations using the quadratic formula 3. Example 2B: Solving a Quadratic Equation with Imaginary Solutions 5x2 + 90 = 0 Check It Out! Example 2b x2 + 48 = 0 Solve the equation. When solving word problems, some common quadratic equation applications include projectile motion problems and Geometry area problems. * The powers of i can always be simplified to 1, i, -1, -i (by using clock 4 math, but I recommend you use the calculator feature ipart) Complex Numbers * Complex numbers are a combination of real and imaginary. Colebrook equation calculator solving for Darcy friction factor given absolute roughness, diameter and Reynolds number. com - Your collection of math tasks. 1 Know there is a complex number i such that i2 = –1, and every complex number has the form a + bi with a and b real. float a, b, c; float root1, root2, imaginary; float discriminant; printf("Enter values of a, b, c of quadratic equation (aX^2 + bX + c): "); scanf("%f%f%f", &a, &b, &c). Quadratic Equations, Cubic and Higher Order Equations : Plots, Factorization, Formulas. (The degree of the equation is 2) (The highest exponent of a variable is 2). Pure real numbers then correspond to the horizontal axis, pure imaginary numbers correspond to the vertical axis. Example: 3i If a ≠0 and b ≠ 0, the complex number is a Example 2 SOLVING QUADRATIC EQUATIONS FOR COMPLEX SOLUTIONS Solve each equation. 3 apply solution techniques of equations to solve real life problems; Worded problems including quadratic equations, supply and demand functions and equations of motion in a straight line. For example, the complex number 0 - 2i can be written as -2i. The 'svd' solver is the default solver used for LinearDiscriminantAnalysis, and it is the only available solver for QuadraticDiscriminantAnalysis. Videos are created by fellow teachers for their students using the guided notes from the unit. x2 - 3x + 1 = 0. Tool/solver to resolve one or more equations. Each method also provides information about the corresponding quadratic graph. This calculator is designed to give a value, even if complex, for the data entered. An attempt to solve the equation gives x2 =−1 and x =± −1 Since i = −1, the solutions of the equation x2 +1 =0 can be written as i and −i. Fluid Mechanics - Turbulent Pipe Flow. Quadratic Equation Solver. To solve the two equations for the two variables x and y, we'll use SymPy's solve() function. When solving word problems, some common quadratic equation applications include projectile motion problems and Geometry area problems. Take the arithmetic mean of that: 3. The imaginary_part function calculates online the imaginary part of a complex number. Well, "simplify" at the expense of Solving linear differential equations The famous LRC circuit (or damped harmonic oscillator) is also It's more complicated than the quadratic equation, and really people usually only give a formula that. The answer you come up with is a valid "zero" or "root" or "solution" for "ax 2 + bx + c = 0", because, if you plug it back into the quadratic, you'll get zero after you simplify. , an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. Mandelbro's equation is a very simple equation with only two variables, 'Z' and 'C'. Then zw = (ac−bd)+(bc+ad)i =(ac−bd)−(bc+ad)i =(a−bi)(c−di)=zw. First Complex Number ( N1 ): a= b= a+b*i. Let us solve some more examples using this method. I can use the discriminant to determine the number and type of roots of a. Solve quadratic equations step-by-step. Warm UpSimplify each expression. Where a ≠ 0, b and c are a constant numbers. quadratic. The quadratic formula is usually written in the form b2 — 4ac If a, b and c are rational coefficients, then 4ac. Rational Expressions. Beginning millions of years ago with ancient “ant odometers” and moving through time to our modern-day quest for new dimensions, prolific polymath Clifford Pickover covers 250 milestones in mathematical history. Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more. i is "a" solution to the quadratic equation. There are several methods you can use to solve a quadratic equation: Factoring Completing the Square Quadratic Formula Graphing All methods start with setting the equation equal to zero.


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# I Some confusion about Kepler’s formula

1. Jan 10, 2018

### JohnnyGui

Hello,

If I understand correctly, according to Kepler, a planet orbiting a star should cover a constant surface area of its ellipsoid or circular orbit per time unit, regardless of the distance of that planet to the star.

I have tried to prove this by using a scenario in which a planet is orbiting in a perfect circle around a star, so that the distance is the radius $r$. I would expect that changing the distance $r$ would still make the planet cover a constant area of the circle per time unit through changing its orbital velocity.

The time period $P$ in which this planet orbits a full circle around the star is defined by:
$$\sqrt{\frac{4\pi^2\cdot r^3}{G(M+m)}}=P$$
Dividing the circumference of the circle $2\pi r$ by that formula would give the orbital velocity per second. Divding that orbital velocity again by the circumference of the circle would give the fraction of the circumference that the planet covers per second. Multiplying that fraction by by the surface area of the circle $\pi r^2$ would give the surface area that the planet orbits per second.

Rewriting all this in the formula and simplifying would eventually give me:
$$\sqrt{\frac{r \cdot G(M+m)}{4}} = A$$
In which $A$ is the covered surface area per second. This formula still shows that $A$ is dependent on the distance $r$. From what I understand, this shouldn’t be the case since a planet revolving around a star should cover a constant area regardless of distance.

I have a feeling that this "constant area per unit time" rule only applies within the orbit of the planet itself, not when changing its distance. Within a circular orbit, $A$ would indeed be constant.
Furthermore, if the orbit of a planet $1$ is an ellips with a semi-major axis $a_1$, then the planet only sweeps equal areas per time unit within that particular ellips with that semi-major axis $a_1$. An equally massive planet $2$ orbiting in an ellips but has a different semi-major axis $a_2$ would not sweep the same area per time unit as planet $1$, but it will sweep equal areas within its own elliptical orbit with semi-major axis $a_2$.
Is this the cause of my misunderstanding?

2. Jan 10, 2018

### Orodruin

Staff Emeritus
It will cover a constant area, but this area does not need to be the same as that of the lower orbit. The area covered generally depends on the angular momentum and Kepler's third law ultimately boils down to conservation of angular momentum. If you have two orbits with different angular momentum, the area covered per unit time will be different.

For a circular orbit the third law is a trivial statement just saying that the orbital velocity remains constant.

This is a misunderstanding of Kepler's third law. It does not say that this area is independent of the orbit. It says that given a particular orbit, the area covered per time will be constant.

Essentially, yes.

3. Jan 10, 2018

### stefan r

picture from wikipedia:

4. Jan 11, 2018

### JohnnyGui

@Orodruin : Thanks for your explanation. I know now where my misunderstanding was. @stefan r : Thanks for the illustration.

One other question; I know that 2 massive objects, e.g. a star and a planet, orbit around a center of mass, the center of mass almost being in the center of the star if that star is way more massive than the planet. In case of an elliptical orbit, that center of mass is in one of the 2 foci of the ellips. So the true distance between the planet and the star is not actually the same as the distance between the planet and the focus of the ellips.
Does the planet sweep equal areas per unit time with respect to its distance to the star or with respect to its distance to the center of mass? I assume it's the latter?

5. Jan 11, 2018

### jbriggs444

Right. The two bodies orbit the combined center of mass in lock-step, each in its own elliptical path. The mutual attraction always has a line of action through the center of mass, so angular momentum of either about the center of mass is conserved and Kepler's law upheld.

6. Jan 12, 2018

### stefan r

7. Jan 13, 2018

### JohnnyGui

Got it. Thanks for the confirmation.
Thanks again @stefan r for the helpful illustrations and info page!

How about the formula for orbital velocity $\frac{G(M+m)}{r}=v^2$ and gravitational force $\frac{G \cdot M \cdot m}{r^2} = F$.
Is it correct that the $r$ in both formulas represent the true distance between the 2 masses and not the distance to the center of mass?

8. Jan 15, 2018

### stefan r

That statement reads ambiguous. You have, for example, the center of Pluto's mass, the center of Charon's mass, and the center of the Pluto-Charon system's mass.

$r$ is the distance between "2 point masses" acting on each other. The gravity of a sphere(or spherically symmetric solid) will act like the gravity of a point mass.

$r$ is not the distance between the closest point on the surface of 2 planets. $r$ is the distance between the center of mass of object #1 and the center of mass of object #2.

9. Jan 15, 2018

### JohnnyGui

Apologies, with "true distance" I meant the distance between the centers of 2 masses. I'm leaving the barycenter out of this.

If $r$ is the distance between 2 point masses like you say, then in case of a circular orbit like the following in which the masses don't differ too much from each other...

...then the constant of Kepler's formula $\frac{p^2}{r^3}$ would be different from when $M$ is way more massive than $m$, correct?
This is because the distance $r$ between the 2 masses would have to be transformed into the distance between mass $m$ to the barycenter, $r_{bc(m)}$ in order to calculate its period $p_m$ correctly, by dividing the circumference of its circular orbit with radius $r_{bc(m)}$ by its orbital speed $v_m$.

So I'd reckon that in case of a circular orbit between mass $M$ and mass $m$ that don't differ a lot from each other (like in the above illustration) you'd have to calculate the distanc between mass $m$ to the barycenter $r_{bc(m)}$ first from $r$, in order to deduce Kepler's constant correctly from Newton's centrifugal force formula.

I deduced that $r_{bc(m)}$ can be calculated through $r$ by:
$$r_{bc(m)} = r \cdot \frac{M}{M+m}$$
This means that the period $p$ of mass $m$ should be its circular orbit's circumference with radius $r_{bc(m)}$ divided by its orbital speed calculated through Newton's centrifugal force formula:
$$p_m = \frac{2\pi \cdot r \cdot \frac{M}{M+m}}{\sqrt{\frac{G(M+m)}{r}}}$$
This eventually gives me:
$$\frac{p_m^2}{r^3} = \frac{4\pi^2 \cdot (\frac{M}{M+m})^2}{G(M+m)} = \frac{4\pi^2 \cdot M^2}{G(M+m)^3}$$
Here, $r$ is the distance between the centers of the 2 masses. Is this formula then correct?

Last edited: Jan 15, 2018
10. Jan 16, 2018

### JohnnyGui

Please scratch what I said above, I didn't realise that Newton's formula for orbital velocity was already corrected to get the distance to the barycenter So I kind of double corrected it.

However, I have another problem. I verified that $r \cdot \frac{M}{M+m} = r_{bc(m)}$.
The equation between gravitational force and centripetal force in case of $M$ being way larger than $m$ is:
$$\frac{GMm}{r^2} = \frac{mv^2_m}{r}$$
Here, the $r$ is the distance between $M$ and $m$. If these 2 masses don't differ a lot from each other, the $r$ in the right equation needs to be converted to the distance of $m$ to the barycenter; $r_{bc(m)}$. Substituing $r$ with $r \cdot \frac{M}{M+m}$ gives:
$$\frac{GMm}{r^2} = \frac{m \cdot v^2_m (M+m)}{r \cdot M}$$
Rewriting it to get $v_m$ gives me:
$$\sqrt{\frac{GM^2}{r(M+m)}} = v_m$$
This conclusion is correct according to this link . However, this link shows a different relationship for calculating $v_m$, saying that $\sqrt{\frac{G(M+m)}{r}} = v_m$.

Which one is correct?

11. Jan 21, 2018

### JohnnyGui

UPDATE: Ok I learned that one formula represents the relative velocity between the 2 masses and the other one the velocity with respect to the center of mass. I was able to derive the one from the other since the relative velocity w.r.t. the masses is the sum of both velocities of $m$ and $M$ w.r.t. the center of mass.

I am now trying to find a way how one can derive the period of a mass in an elliptical orbit. I think I have found a way to deduce this.

I learned that one can calculate the velocity in an elliptical orbit, as a function of distance between the masses $r$ and the semi-major axis $a$ of the ellips, as follows:
$$v = \sqrt{GM(\frac{2}{r} - \frac{1}{a})}$$
I took a first step and thought of a very large $M$ w.r.t. $m$ so that $M$ is basically in one of the focus points of the ellips ($r$ is thus the distance from $m$ to the focus point). I think that it's possible to calculate the period of $m$ by calculating a small piece of the length of an ellips $dL$ and using the velocity formula to know how much time the mass $m$ would do over that piece of $dL$. Doing this for every piece $dL$ of the ellips and adding all those time intervals would give the total period for $m$.

Since the graph formula of half an ellips is $y=\sqrt{b^2(1-\frac{x^2}{a^2}}$, I figured that the distance from $m$ to the focus point $r$ is the hypotenuse of a triangle in the graph with its 2 short sides $y$ and $[ae - x]$ where $e$ is the eccentricity. In that case, $r$ for e.g..the domain $x ≥ 0$ is then calculated with Pythagoras theorem as:
$$r=\sqrt{y(x)^2 + [ae - x]^2}$$
$$x ≥ 0$$
I reckon that calculating two $r$'s (e.g. $r_1$ and $r_2$) merely differing from each other by $dr$, and then finding the x,y-coordinates of their intersections with the ellips, would get you a triangle with sides $dy$ and $dx$ and its hypotenuse $dL$ which is a small length piece of the ellips. By using the Pythagoras theorem this would mean that $\sqrt{dy^2 + dx^2} = dL$.
Then, using the velocity formula above for an ellips, one can calculate how long $m$ would to over that piece of $dL$ at a distance $r$ that is in between $r_1$ and $r_2$ (half of $d\theta$ subtended by $r_1$ and $r_2$ from the focus point). Doing this for each piece of $dL$ of half an ellips for the domain $-a ≤ x ≤ a$, adding their corresponding time intervals, and then multiplying that total time duration by 2 (to get the time duration for the other half of the ellips) would give the period of mass $m$.

This is probably a very inconvenient way to calculate the period of $m$. My question is though, would one theoretically deduce the correct period of mass $m$ with my procedure if $M$ is almost at an ellips focus point? (very large $M$)?

Last edited: Jan 21, 2018
12. Jan 23, 2018

### JohnnyGui

I have made an illustration to clarify my idea how one could calculate the circumference of an ellips.

The x and y coordinates of every distance $r$ and $(r - dr)$ from the focus to any point on the circumference of the ellips can be calculated using the formula for the ellips $y = \sqrt{b^2(1-\frac{x^2}{a^2})}$ and Pythagoras theorem. The difference in x- and y coordinates $dx$ and $dy$ can then be used to calculate a piece of the circumference $dL$ through Pythagoras theorem again. I am exaggerating the magnitude of $dL$ in the picture above.

Dividing each calculated $dL$ by the formula for the speed $v = \sqrt{GM(\frac{2}{r} - \frac{1}{a})}$ in which $r$ is the midpoint between $r$ and $r - dr$ would give the small time interval over $dL$. Doing this for each piece of the circumference of the ellips and adding all those time intervals together would give the time duration for mass $m$ to travel half the ellips as shown in the picture. So multiplying this by 2 would give the period of $m$. This is again if $M$ is so large that it is basically in the focus point of the ellips (green dot in the picture).

I was wondering if this is a correct technique to calculate the correct time period for a mass $m$. My technique sounds like a summation instead of an integration but I'm not sure regarding this.

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Paper

### Robust Binary Fused Compressive Sensing using Adaptive Outlier Pursuit

We propose a new method, {\it robust binary fused compressive sensing} (RoBFCS), to recover sparse piece-wise smooth signals from 1-bit compressive measurements. The proposed method is a modification of our previous {\it binary fused compressive sensing} (BFCS) algorithm, which is based on the {\it binary iterative hard thresholding} (BIHT) algorithm... (read more)

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#### Vol. 7, No. 1, 2014

 Recent Issues
 The Journal About the Journal Editorial Board Subscriptions Editors’ Interests Scientific Advantages Submission Guidelines Submission Form Editorial Login Ethics Statement ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print) Author Index To Appear Other MSP Journals
The nonlinear Schrödinger equation ground states on product spaces

### Susanna Terracini, Nikolay Tzvetkov and Nicola Visciglia

Vol. 7 (2014), No. 1, 73–96
##### Abstract

We study the nature of the nonlinear Schrödinger equation ground states on the product spaces ${ℝ}^{n}×{M}^{k}$, where ${M}^{k}$ is a compact Riemannian manifold. We prove that for small ${L}^{2}$ masses the ground states coincide with the corresponding ${ℝ}^{n}$ ground states. We also prove that above a critical mass the ground states have nontrivial ${M}^{k}$ dependence. Finally, we address the Cauchy problem issue, which transforms the variational analysis into dynamical stability results.

##### Keywords
NLS, stability, stability of solitons, rigidity, ground states
Primary: 35Q55
Secondary: 37K45


text:
## Journal of Applied Probability

### An application of the backbone decomposition to supercritical super-Brownian motion with a barrier

#### Abstract

We analyse the behaviour of supercritical super-Brownian motion with a barrier through the pathwise backbone embedding of Berestycki, Kyprianou and Murillo-Salas (2011). In particular, by considering existing results for branching Brownian motion due to Harris and Kyprianou (2006) and Maillard (2011), we obtain, with relative ease, conclusions regarding the growth in the right-most point in the support, analytical properties of the associated one-sided Fisher-Kolmogorov-Petrovskii-Piscounov wave equation, as well as the distribution of mass on the exit measure associated with the barrier.

#### Article information

Source
J. Appl. Probab. Volume 49, Number 3 (2012), 671-684.

Dates
First available in Project Euclid: 6 September 2012

https://projecteuclid.org/euclid.jap/1346955325

Digital Object Identifier
doi:10.1239/jap/1346955325

Mathematical Reviews number (MathSciNet)
MR3012091

Zentralblatt MATH identifier
1278.60127

Subjects
Primary: 60J68: Superprocesses 35C07: Traveling wave solutions

#### Citation

Kyprianou, A. E.; Murillo-Salas, A.; Pérez, J. L. An application of the backbone decomposition to supercritical super-Brownian motion with a barrier. J. Appl. Probab. 49 (2012), no. 3, 671--684. doi:10.1239/jap/1346955325. https://projecteuclid.org/euclid.jap/1346955325

#### References

• Addario-Berry, L. and Broutin, N. (2011). Total progeny in killed branching random walk. Prob. Theory Relat. Fields 151, 265–295.
• Aïdékon, E., Hu, Y. and Zindy, O. (2011). The precise tail behavior of the total progeny of a killed branching random walk. Preprint. Available at http://arxiv.org/abs/1102.5536v1.
• Berestycki, J., Kyprianou, A. E. and Murillo-Salas, A. (2011). The prolific backbone decomposition for supercritical superdiffusions. Stoch. Process. Appl. 121, 1315–1331.
• Dynkin, E. B. (1991). Branching particle systems and superprocesses. Ann. Prob. 19, 1157–1194.
• Dynkin, E. B. (1991). A probabilistic approach to one class of nonlinear differential equations. Prob. Theory Relat. Fields 89, 89–115.
• Dynkin, E. B. (1993). Superprocesses and partial differential equations. Ann. Prob. 21, 1185–1262.
• Dynkin, E. B. (2001). Branching exit Markov systems and superprocess. Ann. Prob. 29, 1833–1858.
• Dynkin, E. B. (2002). Diffusions, Superdiffusions and Partial Differential Equations. American Mathematical Society, Providence, RI.
• Dynkin, E. B. and Kuznetsov, S. E. (2004). $\mathbb{N}$-measures for branching exit Markov systems and their applications to differential equations. Prob. Theory Relat. Fields 130, 135–150.
• El Karoui, N. and Roelly, S. (1991). Propriétés de martingales, explosion et représentation de Lévy–Khintchine d'une classe de processus de branchement à valeurs mesures. Stoch. Process. Appl. 38, 239–266.
• Engländer, J. and Pinsky, R. G. (1999). On the construction and support properties of measure-valued diffusions on $D\subseteq \mathbb{R}^{d}$ with spatially dependent branching. Ann. Prob. 27, 684–730.
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• Fitzsimmons, P. J. (1988). Construction and regularity of measure-valued Markov branching processes. Israel J. Math. 64, 337–361.
• Harris, J. W., Harris, S. C. and Kyprianou, A. E. (2006). Further probabilistic analysis of the Fisher–Kolmogorov–Pretrovskii–Piscounov equation: one sided travelling-waves. Ann. Inst. H. Poincaré Prob. Statist. 42, 125–145.
• Kametaka, Y. (1976). On the nonlinear diffusion equation of Kolmogorov–Petrovskii–Piskunov type. Osaka J. Math. 13, 11–66.
• Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.
• Kyprianou, A., Liu, R.-L., Murillo-Salas, A. and Ren, Y.-X. (2012). Supercritical super-Brownian motion with a general branching mechanism and travelling waves. Ann. Inst. H. Poincaré Prob. Statist. 48, 661–687.
• Le Gall, J.-F. (1999). Spatial Branching Processes, Random Snakes and Partial Differential Equations. Birkhäuser, Basel.
• Maillard, P. (2012). The number of absorbed individuals in branching Brownian motion with a barrier. To appear in Ann. Inst. H. Poincaré Prob. Statist..
• Neveu, J. (1988). Multiplicative martingales for spatial branching processes. In Seminar on Stochastic Processes (Prog. Prob. Statist. 15), eds E. Çinlar, K. L. Chung and R. K. Getoor, Birkhaüser, Boston, MA, pp. 223–241.
• Pinsky, R. G. (1995). K-P-P-type asymptotics for nonlinear diffusion in a large ball with infinite boundary data and on ${\bf R}^d$ with infinite initial data outside a large ball. Commun. Partial Differential Equat. 20, 1369–1393.
• Salisbury, T. and Verzani, J. (1999). On the conditioned exit measures of super Brownian motion. Prob. Theory Relat. Fields 115, 237–285.
• Salisbury, T. S. and Verzani, J. (2000). Non-degenerate conditionings of the exit measure of super Brownian motion. Stoch. Process. Appl. 87, 25–52.
• Sheu, Y.-C. (1997). Lifetime and compactness of range for super-Brownian motion with a general branching mechanism. Stoch. Process. Appl. 70, 129–141.
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# Extremely confused on finding eigenvectors

Jorge Cantu
Extremely confused on finding eigenvectors? Below I have a picture that gives the matrice and the eigenvectors. How did the solution find these eigenvectors??
i.e. the eigenvalues are 7 and -2

http://tinypic.com/r/2liii68/9
http://tinypic.com/view.php?pic=2liii68&s=9#.VkY_YfmrSUk

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text:
Quarter Hollow Sphere - MapleSim Help

Home : Support : Online Help : MapleSim : MapleSim Component Library : Magnetic : Shapes : Leakage : componentLibrary/magnetic/shapes/leakage/quarterHollowSphere

Quarter Hollow Sphere

Description

The Quarter Hollow Sphere component models the leakage flux through a quarter hollow sphere air-gap.  The equations are

 ${V}_{m}=\mathrm{Φ}\cdot {R}_{m}$ ${V}_{m}\equiv {V}_{{m}_{p}}-{V}_{{m}_{n}}$ ${\mathrm{Φ}}_{p}+{\mathrm{Φ}}_{n}=0$ ${R}_{m}=\frac{1}{{G}_{m}}$ ${G}_{m}={\mathrm{μ}}_{0}\cdot 0.25\cdot t$

Connections

 Name Description Color ${\mathrm{port}}_{p}$ Positive magnetic port Solid orange ${\mathrm{port}}_{n}$ Negative magnetic port Open orange

Variables

 Symbol Units Description Modelica ID ${V}_{m}$ $A$ Magnetic potential across component V_m ${V}_{{m}_{x}}$ $A$ Magnetic potential at port_x.V_m ${\mathrm{Φ}}_{x}$ $\mathrm{Wb}$ Magnetic flux into ${\mathrm{port}}_{x}$, port_x.Phi $\mathrm{Φ}$ $\mathrm{Wb}$ Magnetic flux through component Phi ${R}_{m}$ ${H}^{-1}$ Magnetic reluctance Rm ${G}_{m}$ $H$ Magnetic permeance Gm

Parameters

 Symbol Default Units Description Modelica ID $t$ $0.01$ $m$ thickness of shell t

Initial Conditions

 Symbol Units Description Modelica ID ${V}_{\mathrm{m0}}$ $A$ Magnetic potential across component V_m ${\mathrm{Φ}}_{0}$ $\mathrm{Wb}$ Magnetic flux through component Phi ${R}_{\mathrm{m0}}$ ${H}^{-1}$ Magnetic reluctance Rm ${G}_{\mathrm{m0}}$ $H$ Magnetic permeance Gm

 See Also The components described in this topic are from the Modelica Standard Library. To view the original documentation, which includes author and copyright information, click here.