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hmm...okay, maybe I'm starting to see it right about now...or at lease I think I'm starting to see something resembling what you are describing.....
Basically, suppose I shift the parabola to the right 10 units. The new x intercepts will be 10 and 60. The new midpoint will be 35, but the difference between the intercepts and midpoint will still be 25. So choosing the point (10, 0) to solve the vertex formula, I end up with the same equation $0 = a(25)^{2}+ 7.5$. So, we arrive at the same scaling factor because intercepts 0 and 50 are equivalent to 10 and 60 when it comes to the intercept minus the midpoint.
So $w = i - m$ will always be the same regardless of where you shift the parabola horizontally, so $w^{2} = (i - m)^{2}$ is equivalent to the value you get when you choose the x - intercept as your point to plug into the vertex formula. Kinda?
16. Sep 27, 2015
### HallsofIvy
Staff Emeritus
First, "apex" of a parabola is just another word for the "vertex" of a parabola, those less often used. It is the "point" of the parabola just as the apex of an angle is the "point" of the angle.
I am not sure what you are trying to do with your last post. Initially, you asked how to find a curve such that the "area under it" (I presume you mean between the curve and the x-axis) is 250. The simplest curve that cuts the x-axis twice, so has an "area under it", is a parabola. You can start with pretty much any parabola. For example, since 0 and 1 are easy numbers, x(1- x)= x- x^2 is a parabola that cuts the x-axis at 0 and 1 and rises above it so has an "area under it".
It is easy to calculate that area: $\int_0^1 x- x^2 dx= \left[\frac{x^2}{2}- \frac{x^3}{3}\right]_0^1= \frac{1}{2}- \frac{1}{3}= \frac{1}{6}$.
That area is, of course, not 250! 250 is $\frac{250}{\frac{1}{6}}= 1500$. So multiply the original function by 1500!
The parabola $1500(x- x^2)= 1500x- 1500x^2$ has area under the curve and above the x-axis 250.
17. Sep 27, 2015
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17. Sep 27, 2015
### Ssnow
Depends what kind of function you want consider, if it is exponential $y=e^{x}$ you can consider $\int_{0}^{b}e^{x}d\,x=250$ so $e^{b}-1=250$. The function $e^{x}$ for $x\in [0,\ln{251}]$ is an example ...
18. Sep 28, 2015
### Ocata
Hi HallsofIvy,
Per jbriggs444's post #2 and subsequent guidance, I understand the step by step logic to come up with a general parabola that cuts the x-axis at a certain points.
For instance, given the example you provided, x = 0 and x = 1.
I would start with (x-0)(x-1) = $(x^{2} - x)$ = $x(x - 1)$
and for an upside down parabola, $-(x^{2}-x)$ = $-x^{2} + x$ = $x - x^{2}$ = $x(1 - x)$
Now, it has taken me a few steps to arrive at the general parabola, but your example with jbriggs444 example for the interval between 0 and 50 has revealed a quick way to generate a general parabola.
For example, given any parabola that cuts the x-axis at 0 and some point $x_{1}$, a general parabola is:$x(x-x_{1})$ or $-(x(x_{1}-x))$
Thank you, by studying the two different sets of x intercepts, [0 and 1] and [0 and 50], both arriving at a function with of the same form, I can now formulate a general function of a parabola step by step or by knowing the final form that the function should have.
And I believe I'm comfortable with coming up with a scaling factor to generate a parabola with a specific area between the curve and the x axis.
What I am still have a bit of uncertainty about is how jbrigg444 was able to create a function of a parabola the way he described, given only the vertex (h,k) and of a given area between the curve and x axis.
Afterwards, I found a method of simply choosing any point on the curve and plugging it into the vertex form of the parabola and solving for the coefficient.
Last edited: Sep 29, 2015
19. Sep 29, 2015
### Ocata
Actually Ssnow, I'm also interested in the sin function. | {
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### Ocata
Actually Ssnow, I'm also interested in the sin function.
Suppose a sin function crosses the x axis at x = 0 and then at x = 50 and the area under that portion of the curve is 250. How would I find the function that represents these parameters?
20. Sep 29, 2015
### Staff: Mentor
One arch of the sine function has intercepts at x = 0 and x = $\pi$. Two of the several kinds of transformations you can apply are compressions and expansions, which make each arch narrower or wider, respectively. The graph of y = sin(2x) represents a compression toward the y-axis of the basic, untransformed function by a factor of 2, so that the intercepts mentioned before are now at x = 0 and x = $\pi/2$.
The graph of $y = \sin(\frac 1 3 x)$ represents an expansion away from the y-axis of the untransformed function by a factor of 3. The intercepts of the transformed graph are now at x = 0 and x = $3\pi$.
I leave it to you to figure out what the multiplier needs to be so that the intercepts will be at x = 0 and x = 50. | {
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# Toronto Math Forum
## MAT244--2018F => MAT244--Lectures & Home Assignments => Topic started by: Xinyu Jiao on September 25, 2018, 08:58:39 PM
Title: Can there exists infinite number of solutions given initial conditions.
Post by: Xinyu Jiao on September 25, 2018, 08:58:39 PM
In class, we were given an example where a differential equation can have two solutions given some initial condition. Specifically, the equation was $y' = y^\alpha$ with $0<\alpha<1$, and initial condition $y(0) = 0$. This shows that it's not unique, because it does not satisfy some condition which I do not understand.
My question is, can there be a differential equation (of order 1) such that given an initial condition, can acquire an infinite number of solutions? The answer to this question should be able to shed light as to the mechanism through which the equation acquires more than one solution.
Title: Re: Can there exists infinite number of solutions given initial conditions.
Post by: Victor Ivrii on September 25, 2018, 09:27:59 PM
For condition see Section 2.8 of the textbook or this Lecture Note (https://q.utoronto.ca/courses/56504/files/1311954?module_item_id=332283)
Yes, this equation $y'=3 y^{2/3}$ (I modified it for simplicity) has a general solution $y=(x-c)^{3}$ but also a special solution $y=0$. Thus problem $y'=3 y^{2/3}$, $y(0)=0$ has an infinite number of solutions. Restricting ourselves by $x>0$ we get solutions y=\left\{\begin{aligned} &0 &&0<x<c,\\ &(x-c)^3 && x\ge c\end{aligned}\right. with any $c\ge 0$ and similarly for $x< 0$.
This happens because this Lipschitz condition is violated at each point of the solution $y=0$. | {
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This happens because this Lipschitz condition is violated at each point of the solution $y=0$.
Title: Re: Can there exists infinite number of solutions given initial conditions.
Post by: Kathryn Bucci on October 06, 2018, 10:59:07 AM
If 𝑦′=𝑦𝛼 with 0 < 𝛼 < 1 e.g. 𝑦′=3𝑦2/3= f(t,y), then ∂f/∂y=2y-1/3 is not continuous at (0,0).
According to theorem 2.4.2 (existence and uniqueness for 1st order nonlinear equations), both f and ∂f/∂y have to be continuous on an interval containing the initial point (0,0) - ∂f/∂y is not continuous there so you can't infer that there is a unique solution.
Title: Re: Can there exists infinite number of solutions given initial conditions.
Post by: Victor Ivrii on October 06, 2018, 12:10:58 PM
Continuity of $\frac{\partial f}{\partial y}$ is not required, but "Hölder property" $|f(x,y)-f(x,z)|\le M$ is.
There is a notion of the singular solution | {
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# Prove: If $A \subseteq C$ and $B \subseteq D$, then $A \cap B \subseteq C \cap D$
Is the form and correctness of my elementwise proof of this correct? I don't have any other way of getting feedback for my proofs and I want to improve.
Proof. Suppose $A, B, C, D$ are sets such that $A \subseteq C$ and $B \subseteq D$ and let $x \in A \cap B$. It has to be shown that $x \in C \cap D$.
$x \in A \cap B$ means that $x \in A$ and $x\in B$. Because $A \subseteq C$, $x \in C$ and because $B \subseteq D$, $x \in D$. Thus, $x \in C \cap D$.
Thus, if $A \subseteq C$ and $B \subseteq D$, then $A \cap B \subseteq C \cap D$.
-
This is excellent. – Brian M. Scott Oct 21 '12 at 15:18
Thanks! I just fixed an error that I made in the title. Should I elaborate more on where $x \in A \cap B$ comes from? It comes from $A \cap B \subseteq C \cap D$, correct? – highphi Oct 21 '12 at 15:22
@BrianM.Scott Realized that too. Deleted my comment before seeing you reply. – hwhm Oct 21 '12 at 15:23
No need to say any more: the reason for choosing $x\in A\cap B$ initially is clear just from the inclusion that you’re trying to prove. – Brian M. Scott Oct 21 '12 at 15:24
You’re very welcome, and I agree with what Asaf wrote in the answer below. – Brian M. Scott Oct 21 '12 at 15:37
This is a very well written proof. You state your assumptions and what you wish to prove, then you use the definitions to prove that.
There is nothing more to add, and nothing to reduce. Incidentally today I had the first class of the semester and this is exactly what I tried to teach my students. If they all write such proofs by the end of the month, I should be proud of my work.
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MODULE 1: Introduction
Proof-Writing Guidelines
1
Introduction
A (mathematical) proof of a statement is simply an argument, presented as an essay, that convinces the reader of the validity of the statement. Therefore, whether a proof is acceptable or not (e.g. whether it has the right level of detail, whether it is written in good style, etc.) is a subjective matter without very strict rules. This makes the task of writing good proofs (i.e. proofs that are correct and easily understood and verified) a difficult one.
As you will notice when you read this document, there are a lot of similarities between writing a correct computer program with good style and writing a correct mathematical proof with good style. Unfortunately, however, writing a good proof is arguably harder since there is no compiler to point out all the compile-time errors you have (e.g. syntax errors or type-checking errors). Furthermore, you cannot run a mathematical proof on a computer to see if it produces the expected output. All the errors in a mathematical proof must be caught by you, the author of the proof, and this requires an extra level of attention to detail.
Our goal in this document is to give you suggestions on how to write correct and clearly presented proofs, and help you more easily catch logical flaws or gaps in your arguments. Most of our suggestions are stylistic in nature, because a proof that is written using good style is a proof that exposes its bugs (if there are any). So by following our guidelines, you will make it easier for yourself and everyone else to understand and validate your proof.
Mathematics, sciences and arts all have deep, complicated and beautiful ideas. Our rate of progress as a species depends on clear communication of those ideas so they can quickly spread and evolve. We hope that you will seek clarity of thought and effective communication of ideas, not just in the proofs you write, but in all of your endeavors in life.
2
Guideline Points | {
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2
Guideline Points
This section contains the guideline points (10 of them) that we ask you to follow when you write your proofs. Before we start, below is an example of a “proof” that violates almost all the guideline points. Feel free to refer back to this example as you go through this section.
Exercise ($$2^n > n$$)
Show that $$2^n > n$$ for all integers $$n \geq 1$$.
Proposed solution
(Line numbers are added for easy referencing.)
1. $$F_n =$$$$2^n > n$$
2. $$F_1 =$$$$2 > 1$$$$\checkmark$$
3. $$F_n \implies F_{n+1}:$$
4. $$2^{n+1} = 2 \cdot 2^n > 2\cdot n \text{ (induction) } \geq n+1$$ because $$n \geq 1$$
5. Therefore proved.
Keep in mind that the above example is really a toy example. Some of the points we make below have more meaning and value in the context of more sophisticated proofs, which are the kinds of proofs you will be writing in CS251.
2.1
Is the basic structure right?
A proof is not a calculation or a sequence of mathematical symbols. A proof is an essay! It is an argument written in some human language (which, in this course, is English). This means that your proof should consist of paragraphs, and every paragraph should consist of full English sentences. Every word and mathematical notation must be part of a complete sentence. The only purpose of mathematical notation/symbols is to make your arguments clear and concise. Do not equate mathematical notation with rigor or formalism. You can write a completely rigorous and formal proof using just English words.
Pictures and/or diagrams are highly encouraged when they help clarify your argument. However, a picture or a diagram does not replace an actual argument that needs to be clearly specified in English.
2.2
Are you starting the right way? | {
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2.2
Are you starting the right way?
Often, the most important step in coming up with a proof is making sure that you understand exactly what assumptions are given to you and what statement needs to be derived. To make these things absolutely clear to you (the author of the proof) and the reader, we expect the first paragraph of your proof to include a restatement of the assumptions given and what needs to be derived. Unpack the definitions of technical terms if appropriate.
Another important component of the first paragraph is stating your general proof strategy (e.g. proof by contradiction, proof by induction, etc.) For proofs by contradiction, explicitly negate the statement that you are trying to prove and assume it. For proofs by contrapositive, the contrapositive of the statement should be explicitly laid out. For proofs by induction, the parameter being inducted on should be clear.
2.3
When someone reads your proof, they will read it like they read any other essay. Therefore your proof should have a good flow and should be easy to read out loud even with the mathematical notation interspersed in the sentences. In particular, the mathematical notation you use or the diagrams you draw should not break the flow.
2.4
Is the purpose of every sentence clear? | {
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2.4
Is the purpose of every sentence clear?
The purpose of every sentence should be clear as you are reading it. Otherwise, the sentence can be very confusing for the reader and break the flow of the proof. A common example that violates this point is to make a statement without clarifying whether it follows from the previous statements or assumptions, or whether it is a statement that will be proved later on. With this in mind, whenever you write a sentence, make sure that it is clear if the sentence is (i) an assumption, (ii) a statement that follows from or combines previously established statements or assumptions, (iii) a claim that will be proved later, (iv) a sentence setting up a goal, (v) a sentence introducing a new variable, terminology, definition etc. (vi) part of an example or illustration, (vii) a digression, or (viii) something else.
A strategy that helps a lot is to set explicit and clear goals, and say what you will do before doing it.
2.5
Are you giving the right level of detail?
What is the right level of detail to provide when writing a proof? This is an important question whose answer depends on the audience of the proof. It is possible to write a proof that has all the right ingredients, but does not have the proper justifications for each step. So then the reader has to check the details themselves and verify that the proof is indeed correct. These kinds of proofs are actually not uncommon in, say, computer science or mathematics publications. The author may knowingly give just enough detail in a proof so that the gaps can be figured out and verified by an expert in the field. In CS251, however, we do not accept such proofs. No extra effort/verification from the reader should be necessary. For this reason, it is better to err on the side of caution, and spell things out as much as you feel is reasonable. | {
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To help with this, our recommendation is that you view your audience as a classmate who does not know how to prove the statement you are presenting the proof for. Keep your audience in mind when presenting your proofs and provide the appropriate level of detail in your arguments. Your classmate should be able to read your proof from start to finish once, and be convinced that your argument is correct.
One thing that can be hard to appreciate is that when you think hard on a problem, you develop a lot of intuition in the process, and certain things that were not too trivial in the beginning start becoming obvious to you. This is great, because it signals that you are acquiring a deeper understanding of the problem. However, when it is time to write down your proof, you have to keep in mind that the reader has not gone through the mental process that you have gone through trying to come up with the proof. So the intuitions you have built are not necessarily accessible to your reader. And the things that are obvious to you may not be obvious to the reader. It is not always easy, but try to put yourself in the shoes of your reader and don’t skip over details that can be crucial to understanding your argument.
Related to the point above, if in your proof you are using a word that is synonymous to “obvious”, double check (with your audience in mind) that it is justified. In particular, an obvious statement should be such that a proof of it springs to mind immediately with no effort. When in doubt about whether a statement qualifies as obvious or not, ask the course staff.
2.6
Should you break up your proof? | {
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2.6
Should you break up your proof?
When you first learn programming, one thing that will be emphasized over and over again is the value of breaking up your problem into smaller parts and using lots of helper functions. If you write a function that is too long or does more than one task, that is a good indication that you should consider breaking it up into smaller helper functions. Liberal use of helper functions makes your program easier to understand and debug.
In the above sense, mathematical proofs are similar to computer programs. Your proof should be divided up into lemmas and/or claims when appropriate. This will make the proof much easier to understand and it will be easier to spot and fix any potential errors. Even if you are not defining new lemmas/claims, each component of your argument should be separated into different paragraphs, and the high level organization should be easy to identify.
2.7
In programming, you are used to the idea that every object (piece of data) has a type/class. For instance, many programming languages have an integer type and a string type. The type of the data determines what kinds of operations you can apply on the data. You can, for example, multiply two integers, but you cannot multiply two strings. If you try to do any operation that conflicts with the data type, then you will get a compile-time error, and your code will not run.
Another way to get a compile-time error is by trying to access a variable that has not been defined and initialized. And in statically typed programming languages, you need to explicitly specify the type of a variable when you declare it. This way, the compiler can type-check your program before running it. | {
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Similar to programming, every mathematical object has a type (e.g. it could be an integer, a set, a function, etc.). And as in programming, the type of an object determines what kind of operations you can apply to the object. For example, you can add an element to a set, but you cannot add an element to a function. If you do not respect these constraints, we say that your argument does not type-check. A type-checking error often leads to a nonsensical sentence (even though the author may not realize this).
With these in mind, make sure that your proofs always type-check. You must act as a compiler to find anything that might violate type constraints. In order to make this easier, whenever you learn about a new definition, make a note of the type of the object being defined. This is one of the most important parts of a definition. And when referencing an object in your proof, be aware of its type and make sure that your proof type-checks.
In addition to the above, remember that all the variables in your proofs should be introduced properly. In particular, it should be clear if a variable represents an arbitrary object (e.g. in the context of a “for all” quantifier), a specific object known to exist (e.g. in the context of a “there exists” quantifier), or something else (e.g. a specific value). In all cases, the variable must have a specific type, and it should be clear what the type is.
Here is a suggestion to keep in mind. If in a sentence you are referring to a variable, consider preceding the variable with its type if you feel that this makes things clearer and does not add too much redundancy. For example, if x and y are variables referring to strings, you can consider changing a sentence like “Concatenating x and y...” to “Concatenating string x and string y...”.
2.8 | {
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2.8
As you develop an argument in a proof, you will want to refer back to (and use) a previously established statement or an assumption. Implicitly using previous statements or assumptions can make it hard to follow the logic of the proof and put unnecessary burden on the reader. Therefore, avoid implicit references and make sure it is clear which part(s) of the proof you are using to establish the current step of your proof. Whenever you make a reference, you want it to be very easy for the reader to spot the thing that is being referred to. For example, if there is an important equation that will be referenced later on in the proof, it is a good idea to put that equation on a separate line and label it. This way, you can use the equation’s label to refer to it, and the fact that it is on a separate line by itself will make it very easy for the reader to identify it.
In addition to the above, please be careful when you use a word like “it”, “this”, or “these” in your proof. While you are writing your proof, when you use such a word, you know perfectly well what that word refers to. However, ask yourself whether it would also be perfectly clear to the reader. In general, it is a good idea to try to avoid such words as much as possible, even if this means a certain level of repetitiveness is introduced in the used words.
2.9
Where are you using the assumptions? | {
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2.9
Where are you using the assumptions?
When you are asked to write a proof, you are usually given certain assumptions that you take as true (which is your starting point), and you have a target statement that you want to derive. As pointed out in the “Are you starting the right way?” section above, it helps you and the reader to explicitly lay out the given assumptions as well as the statement that needs to be derived. Once your proof is complete, ask yourself where in the proof the assumptions are being used. If it turns out that you are not using a given assumption, you should raise your alertness level and check: is the statement even true without that assumption? (There may be some exceptions, but almost always, the answer will be no.) If it is not true, and you do not use the assumption in your proof, then your proof is wrong.
When someone reads your proof to verify it, they will be trying to spot where the assumptions are used in the proof. This is a quick sanity check that the proof is not missing an essential component. In order to make this check easy for the reader (and also for yourself), you should make clear in your write-up where and how the assumptions are being used in your argument. In the rare case that a given assumption is not needed for the proof, mention this explicitly.
2.10
Is the proof idea clear?
As mentioned before, a proof is an argument that convinces the reader that a certain statement is true. After reading the proof, the reader may walk away with a clear and intuitive understanding of why the statement is true. Or, they may not, even if they are perfectly convinced that the statement is indeed true.
We do not want to get into a philosophical discussion about what it means to understand why something is true. But hopefully we can agree that certain proofs have more explanatory content than others. | {
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We strongly encourage you to always write proofs with strong explanatory content! For this reason, if the motivation/intuition behind the proof is not transparent, consider adding a paragraph or two explaining the main ideas that make the proof work. This can be either added within the proof itself, or can be a separate “Proof Idea” section preceding the proof.
Summary
1. Is the basic structure right?
2. Are you starting the right way?
4. Is the purpose of every sentence clear?
5. Are you giving the right level of detail?
6. Should you break up your proof?
9. Where are you using the assumptions?
10. Is the proof idea clear?
3
An Example
Let’s now come back to the example from the beginning of Section (Guideline Points) and see how it fares. For ease of reference, we reproduce the “proof” here:
Proposed solution
1. $$F_n =$$$$2^n > n$$
2. $$F_1 =$$$$2 > 1$$$$\checkmark$$
3. $$F_n \implies F_{n+1}:$$
4. $$2^{n+1} = 2 \cdot 2^n > 2\cdot n \text{ (induction) } \geq n+1$$ because $$n \geq 1$$
5. Therefore proved.
• Is the basic structure right? It is quite easy to see that there is a gross violation of this point. The argument does not consist of sentences. In fact, there is not a single fully formed English sentence.
• Are you starting the right way? We are missing a restatement of what the given assumptions are and what needs to be derived. (Yes, this is kind of pedantic with the toy example, but still valuable in the context of more complicated examples.) Furthermore, even though the proof is supposed to be a proof by induction, we only learn about this towards the end of the proof.
• Did you read your proof out loud? If we attempt to read the argument out loud, we hear an incomprehensible sequence of words.
• Is the purpose of every sentence clear? What is the purpose of the first line? Is it a definition? Is it a claim? Is it an assumption? We can also ask these questions for lines 2 to 4. | {
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• Are you giving the right level of detail? Implicit in the “proof” is the claim that “$$2\cdot n \geq n+1$$ because $$n\geq 1$$”. This claim is true, and would qualify as an obviously true statement. However, there is a step that is being skipped, which if included, would make the claim more immediately obvious. In particular, saying “$$2 \cdot n = n + n \geq n + 1$$, where the inequality follows because $$n \geq 1$$.” is preferable. Again, in the context of this toy example, our point may come across as too pedantic, but it is good to err on the side of caution and spell things out as much as possible. In another context/proof, a step that you skip might cause the reader a headache.
• Should you break up your proof? This proof does not require any “helper functions”. It is short and simple enough.
• Does your proof type-check? The variable $$n$$ is not introduced properly. From the problem statement, we can infer that $$n$$’s type is a natural number. However, the variable should still be declared properly. For instance, $$F_n$$ is supposed to be the statement “$$2^n > n$$”, but here $$n$$ is undefined/unquantified.
• Are your references clear? There seems to be a reference to an induction hypothesis, but it is poorly presented since there is no indication that the proof is by induction until there is an attempt to make a reference to the induction hypothesis.
• Where are you using the assumptions? The part “because $$n \geq 1$$” is a reference to an assumption that is given by the problem statement, so we expect the author to point this out explicitly. Once again, this point may seem pedantic, but in longer and more complicated proofs, these things really do make a difference.
• Is the proof idea clear? As the given problem and its proof are quite simple, a ‘proof idea’ section is not necessary.
Here is an example of how the proof can be written using good style.
Good solution | {
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Here is an example of how the proof can be written using good style.
Good solution
We will prove that for all integers $$n \geq 1$$, $$2^n > n$$. The proof is by induction on $$n$$.
Let’s start with the base case which corresponds to $$n=1$$. In this case, the inequality $$2^n > n$$ translates to $$2^1 > 1$$, which is indeed true.
To carry out the induction step, we want to argue that for all $$n \geq 1$$, $$2^n > n$$ implies $$2^{n+1} > n+1$$. We do so now. For an arbitrary $$n \geq 1$$, assume $$2^n > n$$. Multiplying both sides of the inequality by $$2$$, we get $$2^{n+1} > 2n$$. Note that since we are assuming $$n \geq 1$$, we have $$2n = n+n \geq n+1$$. Therefore, we can conclude that $$2^{n+1} > n + 1$$, as desired.
A remark is in order. It is widely accepted that mathematical proofs can be completely formalized in a way that can be mechanically verified, e.g. by a computer. However, writing proofs in a completely formal way is like writing computer programs using machine language. Mathematicians do not communicate proofs in that level of detail.
However, this example is in fact not made up. It was a turned-in solution in CS251 many years ago.
Avoid using long paragraphs as they hurt the clarity of the exposition.
According to Steven Pinker, a cognitive scientist, psychologist, linguist, and a popular science author, the biggest reason why many intelligent people write very poorly is “the curse of knowledge”. People have a hard time imagining what it is like for someone else to not know something that they know. | {
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# How do I solve $32x \equiv 12 \pmod {82}$?
I am able to solve simpler linear congruences, for example $3x \equiv 2 \pmod 5$. What I would do in this case is use that $0 \equiv 10 \pmod 5$ and then utilising a theorem: $3x \equiv 12 \pmod 5$. Then I can divide by $3$ leaving me $x \equiv 4 \: \left( \mathrm{mod} \: {\frac{5}{\mathrm{GCD}(5,3)}} \right) \quad \Longleftrightarrow x \equiv 4 \pmod{5}$ which means the solution is $x = 4k + 5$ where $k \in \mathbb{Z}$.
But I cannot apply the same method to this congruence: $$32x \equiv 12 \pmod {82}$$ This is how far I got: $$8x \equiv 3 \: \left( \mathrm{mod} \: \frac{82}{\mathrm{GCD}(82, 4)} \right)$$ $$\Updownarrow$$ $$8x \equiv 3 \pmod {41}$$
What could I do next? Please provide solutions without the Euclidean algorithm.
EDIT:
What I found later is that I can say that $$0 \equiv 205 \pmod {41}$$ And then I can add it to the congruence in question and divide by $8$. So I guess my question is essentially 'How can I find a number that is a multiple of $41$ (the modulus) and which, if added to $3$ gives a number that is divisible by $8$?'
I reckon the Euclidean algorithm is something which gives an answer to these kinds of questions?!
• It's just too bad that the Euclidean algorithm is the perfect way to tackle these problems. – Lord Shark the Unknown Sep 9 '17 at 11:28
• I'd suggest turning it into algebraic equations without congruence. – user451844 Sep 9 '17 at 11:30
• @LordSharktheUnknown I am aware of that, but as I am a complete beginner in the topic, I'd firstly like to find solutions which do not involve the algorithm which I am not familiar with yet. – bertalanp99 Sep 9 '17 at 11:37
• In that case you may just observe that as $\gcd(5,41)=1$ $$8x\equiv3\pmod{41}\Leftrightarrow40x\equiv15\pmod{41}.$$ Here $40\equiv-1$, so... This amounts to replacing Extented Euclidean Algorithm with a "lucky" observation. – Jyrki Lahtonen Sep 9 '17 at 11:39
• Please see my edit – bertalanp99 Sep 9 '17 at 11:40 | {
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Hint :you can do like this $$\quad{8x \equiv 3 \pmod {41}\\ 8x \equiv 3+41 \pmod {41}\\8x \equiv 44 \pmod {41} \div4 \\ 2x \equiv 11 \pmod {41}\\2x \equiv 11+41 \pmod {41}\\2x \equiv 52 \pmod {41}\div 2\\x \equiv 26 \pmod {41}\\x=41q+26}$$
You just have to find the inverse of $8$ modulo $41$.
The general method uses the extended Euclidean algorithm, but in the particular case, it's much simpler: from $5\cdot 8=40\equiv -1\mod 41$, you get at once that $8^{-1}\equiv -5\mod 41$, so $$x\equiv -5\cdot 3=-15\equiv 26\mod 41.$$
basically you are asking how to solve $\frac 1n \mod m$ where $\gcd(m,n)=1$
where $\frac 1n \mod m$ is notation for the $x$ so that $n*x \equiv 1\mod m$.
Let $m = k + qn$ then $nx = 1 + Zm = 1 + Z(k + qn)$ implies
$x = \frac 1n + \frac {Zk}n + Zq$ where $n|Zk + 1$
Ex. $x \equiv \frac 17 \mod 67$. As $67 = 9*7 + 4$ then
$x = \frac {1+Z*4}{7} + 9$
Which means we have to find $Z \equiv -\frac 14 \mod 7$.
Oh... I guess this is Euclid's algorithm.
$7 = 3 + 4$
So $-Z = \frac{1+ 3Y}4 + 1$
Which is clearly $Y =1$ and $-Z \equiv 2\mod 7$ and $Z \equiv -2 \mod 7$
and $x = \frac {1 + 4*(-2)}7 + (-2)*9 \equiv -19 \mod 67$.
And indeed $7*(-19) \equiv -133= (-1)*67 + 1 \equiv 1 \mod 67$
... Yeah, you need Euclid's alogrithm.
OK, without the Euclidean algorithm, you are looking for some $k$ such that $8$ divides $41k+3$, which gives you a new equation in smaller numbers (which is a Euclidean algorithm idea too, but still): $$41k+3\equiv 0 \bmod 8 \\ 41\equiv 1 \bmod 8 \\ k+3 \equiv 0 \bmod 8 \\ k\equiv 5 \bmod 8$$
So then you have your $41\times 5 = 205$ to make the divisibility.
$\, 8x = 3\!+\!41n\!\iff\!\bmod 8\!:\ 0\equiv 3\!+\!41n\equiv 3\!+\!n\!\iff\! n\equiv -3\,$ so $\,x\equiv \frac{3+41(-3))}8\equiv -15\equiv 26$
Alternatively $\bmod 41\!:\ x\equiv \dfrac{3}{8}\equiv \dfrac{15}{40}\equiv\dfrac{15}{-1}\equiv 26\$ by Gauss's algorithm | {
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Alternatively $\bmod 41\!:\ x\equiv \dfrac{3}{8}\equiv \dfrac{44}{8}\equiv \dfrac{11}{2}\equiv\dfrac{52}{2}\equiv 26\$ by adding $\pm 41$ to simplify divisions
Alternatively $\bmod 41\!:\ x\equiv \dfrac{1}8\,\dfrac{3}1\equiv \dfrac{-40}8\ \dfrac{3}1\equiv (-5)3\equiv -15$
Remark The first method essentially uses a single step of the (extended) Euclidean algorithm, and the second is a special case of that for prime moduli. The other methods are ad-hoc - they try to massage the fractions by adding small multiples of the modulus to make quotients exact / simpler.
Beware $\$ Modular fraction arithmetic is well-defined only for fractions with denominator coprime to the modulus. See here for further discussion.
$$32x\equiv12\pmod{82}$$ is equivalent to $$16x\equiv6\pmod{41}$$ and since $(41,2)=1$, we can divide by $2$ $$8x\equiv3\pmod{41}$$ Then noting that $5\cdot8\equiv-1\pmod{41}$, we get that $36\cdot8\equiv1\pmod{41}$. Multiplying both sides by $36$ yields $$\bbox[5px,border:2px solid #C0A000]{x\equiv26\pmod{41}}$$ Note: When looking for the inverse of $a$ mod $m$ it is often a good idea to see if $a\mid(m-1)$ or $a\mid(m+1)$; if either of these hold, they give a quick inverse mod $m$: $a^{-1}\equiv-\frac{m-1}a$ or $a^{-1}\equiv\frac{m+1}{a}$. Above, it was noted that $8|(41-1)$ leading to $8^{-1}\equiv-\frac{41-1}8\pmod{41}$.
Alternate Method of Finding the Inverse of $\boldsymbol{8\pmod{41}}$ | {
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Alternate Method of Finding the Inverse of $\boldsymbol{8\pmod{41}}$
Since $41$ is prime and $(41,8)=1$, we know, by Fermat's Little Theorem, that $8^{39}\equiv8^{-1}\pmod{41}$. We can compute $8^{39}\pmod{41}$ using the Square and Multiply Algorithm. $39=100111_\text{two}$, therefore, \begin{align} 8^1&\equiv8&\pmod{41}\\ 8^2&\equiv23&\pmod{41}&&\text{square}\\ 8^4&\equiv37&\pmod{41}&&\text{square}\\ 8^8&\equiv16&\pmod{41}&&\text{square}\\ 8^9&\equiv5&\pmod{41}&&\text{multiply}\\ 8^{18}&\equiv25&\pmod{41}&&\text{square}\\ 8^{19}&\equiv36&\pmod{41}&&\text{multiply}\\ 8^{38}&\equiv25&\pmod{41}&&\text{square}\\ 8^{39}&\equiv36&\pmod{41}&&\text{multiply}\\ \end{align} Therefore, $8^{-1}\equiv36\pmod{41}$. | {
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• You should elaborate on how you "note that $5\cdot 8\equiv -1$" else it amounts to pulling the inverse out of a hit - which is good for magic, but not for math. – Bill Dubuque Sep 10 '17 at 18:33
• @BillDubuque: Is it magic to realize that $5\cdot8=40$? Be careful when casting stones. This is why I added the second method. – robjohn Sep 10 '17 at 18:41
• If you don't say how you "note" that then it is magic, not math. There are of course many ways to compute such inverses, so it is strange that you want to exhibit the inverse but not say how you computed it. Helping someone improve an answer is certainly not "casting stones". In case you may not have noticed, I often give constructive feedback on answers where results are pulled out of a hat. – Bill Dubuque Sep 10 '17 at 19:17
• Certainly, in general, I would not suggest an answer by inspection, however, in this answer you seem to suggest that inspection is a valid approach. However, since I knew that was not a good general strategy, I supplied the alternate, non-Euclidean Algorithm, approach. Furthermore, I have added a description of how I came up with $5\cdot8\equiv-1\pmod{41}$, but as this doesn't work in general, I originally opted to describe the Fermat's Little Theorem approach. – robjohn Sep 10 '17 at 22:12
• Inspection (or brute-force) is much easier there being mod $3$ vs. mod $41$ (and there I say by "inspection or Euclid"). Glad to see that you added an explanation. I call this method the easy inverse case, It is equivalent to the numerator twiddling I do with fractions in my answer, i.e. try $\ 1/a\equiv (1\pm km)/a\pmod{\!m}\,$ for small $k$ to try to make the division exact (i.e. try the first few steps of a brute-force search). It often proves handy for greatly simplifying CRT calculations. – Bill Dubuque Sep 10 '17 at 22:43 | {
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Subsequences and convergence. Do all subsequences have to converge to the same limit for the sequence to be convergent?
Prove that $a_n$ converges if and only if:
$a_{2n},a_{2n+1},a_{3n}$ all converge
I thought this was an easy generic question until I read the hint which said: Note: It is not required that the three sub-sequences have the same limit. This needs to be shown
This is what is confusing me because I have found two sources stating something different:
Proposition 4.2. A sequence an converges to L ∈ R if and only if every subsequence converges to L.
and
Let $a_n$ be a real sequence. If the subsequence $a_{2n}$ converges to a real number L and the subsequence $a_{2n+1}$ converges to the same number L, then $a_n$ converges to L as well.
So my question is: for a sequence $a_n$ to converge does it's subsequences have to converge to the same limit? (I suspect not) and if the answer is no can you help me prove why?
• If a sequence converges its subsequences must converge to the same limit. This follows directly from the definition of the limit of a sequence. – John Douma Dec 1 '15 at 21:53
If a sequence has two subsequences that do not both converge to the same limit, then the sequence does not converge.
This can be proven using the $\epsilon-\delta$ definition of convergence:
• Let $a_n$ converge to $L$, and let $\{a_{n_k}\}_{k\to\infty}$ be a subsequence of $a_n$.
• Let $\epsilon > 0$.
• Then, because $a_n$ converges to $L$, there exists some $N$ such that if $n>N$, then $|a_n - L|<\epsilon$.
• Because $n_k$ is an increasing sequence of integers (by definition of a subsequence), there exists such $K$ that $n_K > N$.
• Then, if $k > K$, we have $n_k > n_K > N$, and from the previous point, we get that $|a_{n_k} - L|<\epsilon$, meaning that $a_{n_k}$ converges to $L$. | {
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However, your case is special in that there is an overlap of sequences, for example $a_6$ is in the first and third sequence, and $a_9$ is in the second and third sequence. In fact, the third sequence alternates between the first two sequences, and you can use this to prove all three limits must be equal.
• Ok so for $a_{2n}$ and $a_{2n+1}$ I can say we are either in the odd or the even case and then $a_{3n}$ goes between the two cases. – babylon Dec 1 '15 at 22:10
• @babylon Yup. And that, forces the sequences to converge to the same number. Then you need to prove that the whole sequence also converges to the same number. – 5xum Dec 1 '15 at 22:16
I think the word "required" in the hint is confusing. Focus instead on the last sentence:
Note: It is not required that the three sub-sequences have the same limit. This needs to be shown
In other words, you need to show that as a result of the given hypotheses, the three sub-sequences do have the same limit. You need to do that precisely because it is a necessary condition for $a_n$ to converge. | {
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# Scipy Optimization Example
This short example shows you how to use the scipy minimize function to identify model parameters. This example is set up similarly to the linear least squares example for consistency. Being able to define pretty much anything in a python function, however, gives you great power to customize this, as opposed to a specific approach like linear least squares.
First, we import all the necessary modules
%matplotlib inline
import numpy
import numpy.random
import matplotlib.pyplot as plt
import numpy.linalg
import scipy.optimize
Next create the x matrix
x = numpy.r_[-10:10:.5]
Next, define a y vector based on some model. Note: you can use any of these models or add them together. We scale the output in this example to eliminate the natural weighting of each of these functions over the given range.
#y = x
y = x**2
#y = x**3
#y = numpy.sin(x)
y /= y.max()
Add some noise to y:
rand = numpy.random.randn(*y.shape)/10
y_rand = y + rand
plt.plot(x,y)
plt.plot(x,y_rand,'o')
[<matplotlib.lines.Line2D at 0x7f860d882a30>]
Create an A matrix consisting of several different models
A = numpy.array([(x),(x)**2,(x)**3,numpy.sin(x)]).T
Now create a function that outputs the sum of squared error of each model applied to the given x. In this case we are solving for the weighting coefficients, $k$, as in the linear least squares example:
def myfunc(k):
# make sure our coefficients are in the form of a numpy array
k = numpy.array(k)
# generate y* = Ak^T
y_model = A.dot(k.T)
# sum the square of the error of our model against the input data, y_rand
error = ((y_model-y_rand)**2).sum()
#return the error
return error
Create an initial guess for each of the weights. In this case we just give each coefficient the value of 1 as an initial guess
ini = [1]*A.shape[1]
ini
[1, 1, 1, 1]
Now, call the minimize function. The first value should be the function you are trying to minimize, and the For more information see the optimization function page | {
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sol = scipy.optimize.minimize(myfunc,ini)
sol
fun: 0.29336548348717617
hess_inv: array([[ 2.35078351e-03, -1.23710160e-05, -3.31813423e-05,
4.86251949e-04],
[-1.23710160e-05, 6.46844761e-06, 3.00084328e-07,
-3.66529338e-05],
[-3.31813423e-05, 3.00084328e-07, 5.62458836e-07,
-1.91576150e-05],
[ 4.86251949e-04, -3.66529338e-05, -1.91576150e-05,
2.78470176e-02]])
jac: array([3.7252903e-09, 3.7252903e-09, 3.7252903e-09, 0.0000000e+00])
message: 'Optimization terminated successfully.'
nfev: 65
nit: 9
njev: 13
status: 0
success: True
x: array([-1.13654781e-03, 1.09398078e-02, -2.22292008e-05, 1.93916411e-02])
sol.x contains the solution for $k$
k_optimum = sol.x
k_optimum
array([-1.13654781e-03, 1.09398078e-02, -2.22292008e-05, 1.93916411e-02])
xx = numpy.r_[:4]
labels = '$x$','$x^2$','$x^3$','$\sin(x)$'
f = plt.figure()
ax.bar(xx,k_optimum)
ax.set_xticks(xx)
ax.set_xticklabels(labels)
[Text(0, 0, '$x$'),
Text(1, 0, '$x^2$'),
Text(2, 0, '$x^3$'),
Text(3, 0, '$\\sin(x)$')]
Now generate $y^*$
y_model = A.dot(k_optimum.T)
Plot the model against the input data
fig = plt.figure()
a = ax.plot(x,y_rand,'.')
b = ax.plot(x,y_model)
ax.legend(a+b,['data','model'])
<matplotlib.legend.Legend at 0x7f860ce494f0>
And plot the residual as well
plt.figure()
plt.plot(x,y_model-y_rand)
[<matplotlib.lines.Line2D at 0x7f860cdb2d00>]
Now try other models, higher resolution data, and different domains | {
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# Algorithm to find a basis of a quotient space $R^n/R^m$.
I have a set of $$m$$ vectors $$\{x_i\}$$, $$x_i \in R^n$$. How can I obtain a basis for $$R^n/span(\{x_i\})$$?
Find a basis of $$\text{span(\{x_i\})}=:W$$, say $$\{x_1,x_2,... ,x_k\}$$, and a basis of $$\mathbb{R}^n$$ of the form $$\{x_1,x_2,...,x_k\}\cup\{y_1,y_2,...,y_{n-k}\}$$. Then the classes $$y_j+W, 1\leq j\leq n-k$$, are a basis of $$\mathbb{R}^n/W$$.
• Yes, but how do we obtain the $\{y_i\}$?. In 3D (n=3), if m=2, we can take the cross product. If m=1, and $x_0=[a,b,c]$ then take $y_0=[−b,a,0]$ and $y_1=x_0 \times y_0$ (with some linear independence assumptions). Does this algorithm generalize? – Scott Oct 19 '18 at 8:56 | {
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## Polar Coordinates Pdf | {
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Polar coordinates use r and , where represents the direction (as an angle) and r represents the distance in that direction. 2 (No Test this week) 10. To use polar coordinates to specify a point, enter a distance and an angle separated by an angle bracket (<). 10 (Intro to Polar packet): 1-12 all. The location of a point is expressed according to its distance from the pole and its angle from the polar axis. To find the polar angle t, you have to take into account the sings of x and y which gives you the quadrant. 2 , 53 o) to rectangular coordinates to three decimal places. My data set is defined in (R, theta) coordinates. Examples Convert ( 6;2) to polar coordinates Solution: r = p ( 6)2 +22 = p 40 ˇ6:325 tan = 1 3, so we find tan 1 1 3 ˇ 18:4 , but is in the second quadrant, so ˇ161:6 Convert r = 10, = 276 to Cartesian coordinates. Two points are specified using polar coordinates. The spherical polar coordinate system is like the polar coordinate system, except an additional angle variable is used, frequently labeled as phi (φ). In polar coordinates, the position of the point of contact of the ball at times t and t = 0 respectively are (r,θ) and (r 0, θ 0). 6 Velocity and Acceleration in Polar Coordinates 2 Note. All four types are used in CNC applications, for different machines and different kinds of work. Math 232 Calculus III Brian Veitch Fall 2015 Northern Illinois University 10. Definition of Polar Coordinates. Polar coordinates with polar axes. The points shown has Cartesian coordinates (√2, √2) and polar coordinates (2,45), with the angle measured in degrees. edu is a platform for academics to share research papers. They plot and label points and identify alternative coordinate pairs for given points. There are other possibilities, considered degenerate. You will then need something like the Free Printable Polar Coordinate Graph Paper. To plot the coordinate, draw a circle centered on point O with that radius. Department of Mathematics - University of | {
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draw a circle centered on point O with that radius. Department of Mathematics - University of Houston. Defining Polar Coordinates. ? $\endgroup$ – Will Jun 10 '15 at 20:41. The coordinate system in such a case becomes a polar coordinate system. If a curve is given in polar coordinates , an integral for the length of the curve can be derived using the arc length formula for a parametric curve. By printing out this quiz and taking it with pen and paper creates for a good variation to only playing it online. 7 7, 6 ⎛⎞π ⎜⎟ ⎝⎠ 2. We convert from polar coordinates to rectangular coordinates and from rectangular coordinates to polar coordinates. Use rectangular coordinates when the number is given in rectangular form and polar coordinates when polar form is used. The polar coordinate system provides an alternative method of mapping points to ordered pairs. When you drag the red point, you change the polar coordinates $(r,\theta)$, and the blue point moves to the corresponding position $(x,y)$ in Cartesian coordinates. Two different polar coordinates, say (r 1,θ 1) and (r 2,θ 2), can map to the same point. In this section we will introduce polar coordinates an alternative coordinate system to the 'normal' Cartesian/Rectangular coordinate system. For clockwise rotation, it decreases. Math 232 Calculus III Brian Veitch Fall 2015 Northern Illinois University 10. Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. Polar Coordinates x is a function of r and ; y is a function of r and. Polar coordinates use an angle measurement from a polar axis, which is usually positioned as horizontal and pointing to the right. Introduction to polar coordinates. r = secθcscθ ⇒ 24. Polar coord unit vectors and normal. Examples on Converting Polar and Rectangular Coordinates Example 1 Convert the polar coordinates (5 , 2. Until now, we have worked in one coordinate system, the Cartesian coordinate | {
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coordinates (5 , 2. Until now, we have worked in one coordinate system, the Cartesian coordinate system. The area of a region in polar coordinates defined by the equation $$r=f(θ)$$ with $$α≤θ≤β$$ is given by the integral $$A=\dfrac{1}{2}\int ^β_α[f(θ)]^2dθ$$. Thus, to nd. Then a number of important problems involving polar coordinates are solved. Differentiatingur anduθ with respectto time t(and indicatingderivatives with respect to time with dots, as physicists do), the Chain Rule gives. txt) or read online for free. Our sailing "Polar" is a diagram showing boatspeed across a range of wind angles and wind speeds, displayed in polar coordinates. Graphing in Polar Coordinates Jiwen He 1 Polar Coordinates 1. Definition The polar coordinates of a point P ∈ R2 is the ordered pair (r,θ), with r > 0 and θ ∈ [0,2π). 42008 S3 Q3 The point P(acos ;bsin ), where a>b>0, lies on the ellipse x2 a2 + y2 b2 = 1: The point S( ea;0), where b2 = a2(1 e2), is a focus of the ellipse. 3) Instead of using (x;y), we describe a point by (r; ) in the polar coordinates where ris its dis-tance from the origin and is the angle it makes with the positive x axis. 2 Calculus In The Polar Coordinate System Contemporary Calculus 4 Area in Rectangular Coordinates (Fig. Polar Coordinates (r,θ) Polar Coordinates (r,θ) in the plane are described by r = distance from the origin and θ ∈ [0,2π) is the counter-clockwise angle. To convert polar coordinates to rectangular coordinates use the formulas: x = r cos y = r sin To convert rectangular coordinates to polar coordinates use the following formulas: r = √x2 + y2 θ = tan-1 (when x > 0) θ = tan-1 + π (when x < 0) y x y x (OR + 180o if it's in degrees). So depending upon the flow geometry it is better to choose an appropriate system. Multiplication and division in polar form Introduction When two complex numbers are given in polar form it is particularly simple to multiply and divide them. 5 Test Review Polar. 2_practice_solutions. r=8sin(θ) Example: The | {
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to multiply and divide them. 5 Test Review Polar. 2_practice_solutions. r=8sin(θ) Example: The graph of 2 /3 is shown below. To find the coordinates of a point in the polar coordinate system, consider Figure 7. Using di erent names for the radial coordinate, on the other hand, causes few problems. For example, the coordinates of [2, π] do not satisfy the equation. The card10id is a special kind of limaçon. The reference point (analogous to the origin of a Cartesian coordinate system) is called the pole, and the ray from the pole in the reference direction is the polar axis. as a function of. For example in Lecture 15 we met spherical polar and cylindrical polar coordinates. In polar coordinates rectangles are clumsy to work with, and it is better to divide the region into wedges by using rays. polar coordinates project - Free download as Word Doc (. This system divides the earth into latitude lines, which indicate how far north or south of the equator a location is, and longitude lines, which indicate how far east or west of the prime meridian a location is. 4 The Reference 21 4. Displacements in Curvilinear Coordinates. r is the radius, and θ is the angle formed between the polar axis (think of it as what used to be the positive x-axis) and the segment connecting the point to the pole (what used to be the origin). Unique cylindrical coordinates. Normally, angle x is. In the Menu Bar, choose Layer > Merge Layers. 6 Cylindrical and Spherical Coordinates A) Review on the Polar Coordinates The polar coordinate system consists of the origin O;the rotating ray or half line from O with unit tick. With the polar grid paper, you can locate someone's exact location. 4 1 x y a The magnitudeof a determines the spread of the parabola: for j a very small, the curve is narrow, and as j a gets large, the parabola broadens. theta: variable to map angle to (x or y) start: Offset of starting point from 12 o'clock in radians. So far, we have described plane curves by giving: y. What | {
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starting point from 12 o'clock in radians. So far, we have described plane curves by giving: y. What is the. Introduction of Polar Coordinates. Convert the following equation of a circle to polar coordinates: 2x2 +3x+2y2 + −5y = 7 7. Let r1 denote a unit vector in the direction of the position vector r , and let θ1 denote a unit vector perpendicular to r, and in the direction of increasing θ, see Fig. The coordinates are displayed in the form (r, ). There are some aspects of polar coordinates that are tricky. SYNOPSIS IntreatingtheHydrogenAtom'selectronquantumme-chanically, we normally convert the Hamiltonian from its Cartesian to its Spherical Polar form, since the problem is. Please update your bookmarks accordingly. Polar Coordinates The system was introduced by Newton to more easily describe curves in the rectangular coordinate system and consequently perform Calculus with those curves. Graphs of Polar Equations. Rectangular form to polar form Change x2 + y2 – 2y = 0 to polar form. Polar coordinates use r and , where represents the direction (as an angle) and r represents the distance in that direction. The use of r for the spherical radial coordinate can be confused with the radial coordinate in polar or cylindrical coordinates, but computations requiring both at the same time are rare. Example Sketch the curve described by the polar equation. (If r was negative, then we would head in the opposite direction. We find from the above equations that dur dθ = −(sinθ)i +(cosθ)j = uθ duθ dθ = −(cosθ)i−(sinθ)j = −ur. Cartesian coordinates need two lines within an orthogonal system. To view the value of θ. A consensus was reached that planetocentric coordinates should be used and that the selected Lunar Coordinate System should be compatible with the one used within the PDS for Clementine data. The polar axis is usually horizontal and directed toward the right. 5 Polar Coordinates. 2 The naddplot Command: Coordinate Input. See figure -1. The points shown has Cartesian | {
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Coordinates. 2 The naddplot Command: Coordinate Input. See figure -1. The points shown has Cartesian coordinates (√2, √2) and polar coordinates (2,45), with the angle measured in degrees. Some properties of polar coordinates. Notice that this solution can be transformed back into rectangular coordinates but it would be a mess. Polar Form of an Ellipse—C. This can happen in the following ways: (a) It can happen if r 2 = r 1 and θ 2 = θ 1 ± 2πn for any. Then each point in the plane can be assigned polar coordinates as follows. This is a subtle point but you need to keep that in mind. My data set is defined in (R, theta) coordinates. 5 Test Review Polar. Polar Coordinates x is a function of r and ; y is a function of r and. Navy are to declare the ability to operate and deploy the F-35 in 2016 and 2018 respectively, and full-rate production of the aircraft is to begincapability”) in 2016 and 2018 respectively, and full-rate production decision of the program is planned for 2019. A line through the pole, making angle 0 with the polar axis, has an equation. To use polar coordinates to specify a point, enter a distance and an angle separated by an angle bracket (<). r = tanθ ⇒ 10. The origin is the vertex of the parabola. PHYS 419: Classical Mechanics Lecture Notes POLAR COORDINATES A vector in two dimensions can be written in Cartesian coordinates as r = xx^ +yy^ (1) where x^ and y^ are unit vectors in the direction of Cartesian axes and x and y are the components of the vector, see also the flgure. Consider this exam question to be reminded how well this system works for circular motion:. Solution This time we find x and y from the polar coordinates. But many teachers might prefer that you measure angles by yourself using a protractor on blank paper. The location of P in the plane can also be described using polar coordinates. (5, 960°) SOLUTION: Let P(r, θ) = (5, 960°). In the polar coordinate system, points are represented by ordered pairs of the form (r; ), where | {
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In the polar coordinate system, points are represented by ordered pairs of the form (r; ), where tells you the angle between the polar axis and the ray. First try to convert to x and y coordinates, by multiplying by r if necessary and/or a suitable trig substitution. 10 (Intro to Polar packet): 1-12 all. Thus, in this coordinate system, the position of a point will be given by the ordered. Also remember that there are three types of symmetry - y-axis, x-axis, and origin. Examples on Converting Polar and Rectangular Coordinates Example 1 Convert the polar coordinates (5 , 2. If f: [a;b]! Rbe a continuous function and f(x) ‚ 0 then the area of the region between the graph of f and the x-axis is. We also know. • θis measured from an arbitrary reference axis • e r and eθ are unit vectors along +r & +θdirns. Cylindrical Coordinates. HPC - Polar Coordinates Unit Test Sample Open Response Answer Key - Page 2. 21 Locating a point in polar coordinates Let’s look at a specific example. , the z coordinate is constant), then only the first two equations are used (as shown below). 5) i Real Imaginary 6) (cos isin ) Convert numbers in rectangular form to polar form and polar form to rectangular form. There are a total of thirteen orthogonal coordinate systems in which Laplace’s equation is separable, and knowledge of their existence (see Morse and Feshbackl) can be useful for solving problems in potential theory. The fixed point is called the pole and the fixed line is called the polar axis. This article is about Spherical Polar coordinates and is aimed for First-year physics students and also for those appearing for exams like JAM/GATE etc. Note that this definition provides a logical extension of the usual polar coordinates notation, with remaining the angle in the – plane and becoming the angle out of that plane. This allows you to fully utilize the paper size that you have on hand. In polar coordinates, angles are labeled in either degrees or radians (or both). The reference | {
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hand. In polar coordinates, angles are labeled in either degrees or radians (or both). The reference point (analogous to the origin of a Cartesian coordinate system) is called the pole, and the ray from the pole in the reference direction is the polar axis. Use Page 2. 1 De ning Polar Coordinates oT nd the coordinates of a point in the polar coordinate system, consider Figure 1. Spherical coordinates system (or Spherical polar coordinates) are very convenient in those problems of physics where there no preferred direction and the force in the problem is spherically symmetrical for example Coulomb's Law due to point. Example contributed by Armin Moser. Thus, in this coordinate system, the position of a point will be given by the ordered. Professional Publications, Inc. De nition (polar coordinate system). State three other pairs of polar coordinates for each point where —2m < 9 < 2m. A point P in the plane, has polar coordinates (r; ), where r is the distance of the point from the origin and is the angle that the ray jOPjmakes with the positive x-axis. The area of a region in polar coordinates defined by the equation $$r=f(θ)$$ with $$α≤θ≤β$$ is given by the integral $$A=\dfrac{1}{2}\int ^β_α[f(θ)]^2dθ$$. b) In the rotated system of Cartesian coordinates (X r, Y r) the X r-axis is parallel to the direction of vector c, defined by initial position r 0 = Xi +Yj and velocity V 0 = Ui. Recall from trigonometry that if x, y, r are real numbers and r 2 = x 2 + y 2, then there is a unique number θ with 0 ≤ θ < 2π such that. Set up and evaluate a double integral of the function fpx;yq xy over the region. Polar coordinates with polar axes. y x x r y θ. 30 Coordinate Systems and Transformation azimuthal angle, is measured from the x-axis in the xy-plane; and z is the same as in the Cartesian system. Find a different pair of polar coordinates for each point such that 0 ≤ ≤ 180° or 0 ≤ ≤ π. Instead, we design P-RSDet which is an anchor-free detector modeled in polar | {
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0 ≤ ≤ 180° or 0 ≤ ≤ π. Instead, we design P-RSDet which is an anchor-free detector modeled in polar coor-dinates. Math 215 Examples Double Integrals in Polar Coordinates. In this section, we will focus on the polar system and the graphs that are generated directly from polar coordinates. Stewart Calculus 7e Solutions Chapter 10 Parametric Equations and Polar Coordinates Exercise 10. In polar coordinates, the position of the point of contact of the ball at times t and t = 0 respectively are (r,θ) and (r 0, θ 0). You can select different variables to customize these graphing worksheets for your needs. txt) or read online for free. 6) continued… If the particle is constrained to move only in the r – plane (i. 2 Slopes in r pola tes coordina When we describe a curve using polar coordinates, it is still a curve in the x-y plane. Polar coordinates use a different kind of graph instead, based on circles: The center point of the graph (or "origin" in a rectangular grid) is the pole. ) 𝜃 is an angle from the polar axis to the line segment from the pole to P. Polar coordinate lines. 1 Þ Locate each of the following points on the polar coordinate system. Thus, to nd. Then we count out a distance of three units along the. In the equation = 5ˇ 4, ris free, so we plot all of the points with polar representation r;5ˇ 4. 3D surface with polar coordinates¶ Demonstrates plotting a surface defined in polar coordinates. Multiplication and division in polar form Introduction When two complex numbers are given in polar form it is particularly simple to multiply and divide them. What is the distance between polar coordinates #(-2, 300^circ)# and #(2, 10^circ)#? What's the difference in finding the distance between two polar coordinates and two rectangular See all questions in Finding Distance Between Polar Coordinates. Double integrals in polar coordinates (Sect. Complete the back of Graphing Roses Revisited and also p. A general system of coordinates uses a set of parameters to define a | {
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Roses Revisited and also p. A general system of coordinates uses a set of parameters to define a vector. You should pay attention to the following: 1. Before we can start working with polar coordinates, we must define what we will be talking about. Two points are specified using polar coordinates. A consensus was reached that planetocentric coordinates should be used and that the selected Lunar Coordinate System should be compatible with the one used within the PDS for Clementine data. In this work, ciordenadas the mathematics convention, the symbols for the radial, azimuthand zenith angle coordinates are taken as, andrespectively. Plot each point in the complex plane. The divergence We want to discuss a vector fleld f deflned on an open subset of Rn. Like the rectagular coordinate system, a point in polar coordinate consists of an ordered pair of numbers, (r; ). 2 We can describe a point, P, in three different ways. In spherical polar coordinates we describe a point (x;y;z) by giving the distance r from the origin, the angle anticlockwise from the xz plane, and the angle ˚from the z-axis. Until now, we have worked in one coordinate system, the Cartesian coordinate system. Polar coordinates When you were first introduced to coordinate systems you will have used cartesian coordinates. Example contributed by Armin Moser. This Precalculus video tutorial provides a basic introduction into polar coordinates. Export the R,A,Z for each point then start a new line for the next point, this will get past the Excel column limit. Polar Coordinates Graphs of Polar Equations An equation expressed in terms of polar coordinates is called a polar equation. 2 (No Test this week) 10. Find a formula for. The old vvvv nodes Polar and Cartesian in 3d are similar to the geographic coordinates with the exception that the angular direction of the longitude is inverted. The value of r can be positive, negative, or zero. We will now look at graphing polar equations. 1] can lie on a curve given by | {
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negative, or zero. We will now look at graphing polar equations. 1] can lie on a curve given by a polar equation although the coordinates. 2) Convert the following to polar coordinates: :4,150° ; (‐6, 2) 3) Typical Polar Graphs: Make sure you watch the Application Walk Through Video to see how you should graph these. Angle t is in the range [0 , 2Pi) or [0 , 360 degrees). Polar Coordinates Identify the curve by finding a Cartesian equation for the curve. New Music Updates in your inbox! Enter your email address:. 4 Polar Equations Polar coordinate system is a plane with point O, the pole and a ray from O, the polar axis. I Calculating areas in polar coordinates. In its basic form, Newton's Second Law states that the sum of the forces on a body will be equal to mass of that body times the rate of. Before we can start working with polar coordinates, we must define what we will be talking about. Draw a horizontal line to the right to set up the polar axis. µ = tan¡1 ‡y x ·, if x > 0, 3. Unique cylindrical coordinates. The X and Y relative coordinates are signed numbers. This is the result of the conversion to polar coordinates in form. It is a two-dimensional coordinate system in which each point is at a definite distance from the reference point. r (x ;y)=( rcos( ) sin( )) =ˇ 6 =ˇ 3 Polar coordinates are related to ordinary (cartesian) coordinates by the formulae x = r cos( ) y = r sin( ) r = p x 2+ y = arctan(y=x):. Until now, we have worked in one coordinate system, the Cartesian coordinate system. In Lemma we have seen that the vector r(t) × r˙(t) = C is a constant. Exploring Space Through Math. Polar Curves Curves in Polar Coordinate systems are called Polar Curves, which can be written as r = f(µ) or, equivalently, as F(r;µ) = 0. Different microphones have different recording patterns depending on their purpose. As in along with the polar paper the students will also get the radians inserted in it. to describe using polar coordinates. Then each point in the plane | {
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get the radians inserted in it. to describe using polar coordinates. Then each point in the plane can be assigned polar coordinates as follows. The activity is designed as a puzzle sort and match. com, a free online dictionary with pronunciation, synonyms and translation. In this case, the path is only a function of F r = ma. Allows students to discover what polar coordinates are and how math and art can work together. This substitution would result in the Jacobian being multiplied by 1. Formula Sheet Parametric Equations: x= f(t); y= g(t); t Slope of a tangent line: dy dx = dy dt dx dt = g0(t) f0(t) Area: Z g(t)f0(t)dt Arclength: Z p (f0(t))2 + (g0(t))2dt Surface area: Z p 2ˇg(t) (f0(t))2 + (g0(t))2dt Polar Equations: r= f( ); Polar coordinates to cartesian: x= rcos( ); y= rsin( ) Cartesian coordinates to polar: r= p x2 + y2. Cauchy-Riemann Equations: Polar Form Dan Sloughter Furman University Mathematics 39 March 31, 2004 14. Show the angle θ between two lines with slopes m 1 and m 2 is given by the equation tanθ = m 2 −m 1 1−m 2m 1 I’ve added some more information to the diagram, based on the hint to include the angle the lines make with the x-axis. Arc length and surface area of parametric equations. A polar equation is an equation that tells about the details of the relation between the origin and the coordinates. The 2-D polar coordinates #P ( r, theta)#, r = #sqrt (x^2 + y^2 ) >= 0#. Until now, we have worked in one coordinate system, the Cartesian coordinate system. This creates a visual bias that does not portray actual data. TrigCheatSheet. (Angles may be in degrees or radians) 8. In this unit we explain how to convert from Cartesian co-ordinates to polar co-ordinates, and back again. You can use absolute or relative polar coordinates (distance and angle) to locate points when creating objects. For each point in the coordinate plane, there is one representation, but for each point in the polar plane, there. We would like to be able to compute slopes and | {
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but for each point in the polar plane, there. We would like to be able to compute slopes and areas for these curves using polar coordinates. An Introduction to Polar Coordinates Polar coordinates are used in many, many fields even at an introductory level. Let (r,θ) denote the polar coordinates describing the position of a particle. Infinitely many angles, and r can also be negative. Then a number of important problems involving polar coordinates are solved. Convert the following equation to polar coordinates: y = − 4 3 x 6. State three other pairs of polar coordinates for each point where —2m < 9 < 2m. By default, angles increase in the counterclockwise direction and decrease in the clockwise direction. 31) Polar coordinates can be calculated from Cartesian coordinates like. If you are looking for basic graph paper, then the Graph Paper Template is the resource you need. com, a free online dictionary with pronunciation, synonyms and translation. Math 2300 Practice with polar coordinates (c) r= 3sin2 0 1 2 3 0 ˇ=2 ˇ 3ˇ=2 Solution: The graph hits the origin at = ˇ 2 and = ˇ, = 3ˇ 2, and = 2ˇ. The method of setting, water coordinates in the AutoCAD by. as a function of. 11) ( , ), ( , ) 12) ( , ), ( , ) Critical thinking question: 13) An air traffic controller's radar display uses polar coordinates. Figure 3: Relationship between coordinate plane and polar plane determine this is shown in gure 3. However, we can use other coordinates to determine the location of a point. Is the point that coordinates are just labels to keep track of where all the points on the manifold are, so within a given patch we are free to choose any coordinate system we like (although in practice we would choose one that suited the problem at hand), not just Cartesian or spherical polar etc. Preview Activity 11. Polar Coordinates Polar coordinates of a point consist of an ordered pair, r θ( , ), where r is the distance from the point to the origin, and θ is the angle measured in standard | {
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, ), where r is the distance from the point to the origin, and θ is the angle measured in standard position. r is a directed distance from the pole to P. Also, you have a DeltaMath assignment that is due Thursday morning. ) 𝜃 is an angle from the polar axis to the line segment from the pole to P. doc), PDF File (. In this handout we will find the solution of this equation in spherical polar coordinates. Since the axis of the parabola is vertical, the form of the equation is Now, substituting the values of the given coordinates into this equation, we obtain Solving this system, we have Therefore, y 5 or 5x2 14x 3y 9 0. Angles are measured relative to the wind, and shown as "true wind angle" or TWA. theta# determines the direction. Polar coordinates are in the form r, , where is the independent variable. The origin is the vertex of the parabola. k = 5 Since k is odd, we need to replace r with -r to obtain the correct polar coordinates. 11, page 636. 4 The Reference 21 4. For each point in the coordinate plane, there is one representation, but for each point in the polar plane, there are infinite representations. 4 5, 4 S SS S SS · ¸ rr ¹ · ¸ r ¹ Yes, there are infinitely many polar coordinates for a given pair of rectangular coordinates. do not satisfy the equation. Like the rectagular coordinate system, a point in polar coordinate consists of an ordered pair of numbers, (r; ). Rectangular form to polar form Change x2 + y2 - 2y = 0 to polar form Solution : Use: r2 = x2 + y2 and y = r sin(θ). Once we've moved into polar coordinates $$dA \ne dr\,d\theta$$ and so we're going to need to determine just what $$dA$$ is under polar coordinates. Polar coordinates When you were first introduced to coordinate systems you will have used cartesian coordinates. Preview Activity 11. • Polar–Rectangular conversions where coordinates of points in polar coordinates, say bearings and distances, are converted to rectangular coordinates. 1 Polar form of the Cauchy-Riemann Equations Theorem | {
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are converted to rectangular coordinates. 1 Polar form of the Cauchy-Riemann Equations Theorem 14. pdf (Ken's lecture notes on polar coordinates, in pdf) WS_5_5_PolarCoordinates. the standard n-dimensional polar coordinates. In polar coordinates, lines occur in two species. r = sin(3θ) ⇒ 22. coordinates. For example, think of a circle of radius centred on the point. Polar Rectangular Regions of Integration. Representing Polar Coordinates Well, as you already know, a point in the Rectangular or Cartesian Plane is represented by an ordered pair of numbers called coordinates (x,y). A system of coordinates in which the location of a point is determined by its distance from a fixed point at the center of the coordinate space. Example (FEIM): A 2500 kg truck skids with a deceleration of 5 m/s2. Test multiples of 180. the given equation in polar coordinates. There are approximately 20 problems on this. I Calculating areas in polar coordinates. Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. Points are. Polar Coordinates-measures the distances (and direction) from the origin (radius)& the circle •• (r, f), (radius): •• ndusionf Rectangular Coordinates deal with horizontal & vertical distances, whereas polar coordinates deal with diagonal & circular distances. the part of the solution depending on spatial coordinates, F(~r), satisfies Helmholtz’s equation ∇2F +k2F = 0, (2) where k2 is a separation constant. Convert the following equation to polar coordinates: y = − 4 3 x 6. com, a free online dictionary with pronunciation, synonyms and translation. 2_practice_solutions. (If r was negative, then we would head in the opposite direction. r=−2sinθ Identify the polar graph (line, circle, cardioid, limacon, rose): If a circle, name the center (in polar coordinates) and the radius. polar coordinate system synonyms, polar coordinate system pronunciation, polar coordinate system | {
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polar coordinate system synonyms, polar coordinate system pronunciation, polar coordinate system translation, English dictionary definition of polar coordinate system. This point will be labeled with rectangular coordinates instead of polar coordinates. All four types are used in CNC applications, for different machines and different kinds of work. many polar representations in addition to the standard one in the picture above where r >0 and 02≤θ< π. To convert the point (x, y, z) from rectangular to cylindrical coordinates we use: 222 y. b) In the rotated system of Cartesian coordinates (X r, Y r) the X r-axis is parallel to the direction of vector c, defined by initial position r 0 = Xi +Yj and velocity V 0 = Ui. Look it up now!. The point N is the foot of the perpendicular from the origin, O, to the tangent to the ellipse at. When we defined the double integral for a continuous function in rectangular coordinates—say, $$g$$ over a region $$R$$ in the $$xy$$-plane—we divided $$R$$ into subrectangles with sides parallel to the coordinate axes. ) for polar coordinates are shown. 2 Slopes in r pola tes coordina When we describe a curve using polar coordinates, it is still a curve in the x-y plane. The point with rectangular coordinates (-1,0) has polar coordinates (1,pi) whereas the point with rectangular coordinates (3,-4) has polar coordinates (5,-0. Polar coordinates use an angle measurement from a polar axis, which is usually positioned as horizontal and pointing to the right. Therefore, in rectangular coordinates, r=sin( ) is written as p x2 + y2=y/ p x2 + y2. ) The graph of = , where is a constant, is the line of inclination. 1 Review: Polar Coordinates The polar coordinate system is a two-dimensional coordinate system in which the position of each point on the plane is determined by an angle and a distance. tan y x θ = y r = sinθ 2 2 2 r x y = + Example 1: Convert the polar coordinate 2 2, 3 π to rectangular form. Press b and choose Trace⎮ Trace Settings to | {
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the polar coordinate 2 2, 3 π to rectangular form. Press b and choose Trace⎮ Trace Settings to adjust the trace step. In the polar coordinate system, a circle centered at the origin with a radius a units has equation r = a 7KH dartboard has a radius of 225 mm, so its boundary equation is r = 225. The method of setting, water coordinates in the AutoCAD by. 1 r =4secθ r =4secθ ⇒ r secθ =4 ⇒ 4cos(θ) ⇒ x =4 Thus,theCartesianequationisx =4. 4, - 14 A point in polar coordinates is given. To convert polar coordinates to rectangular coordinates use the formulas: x = r cos y = r sin To convert rectangular coordinates to polar coordinates use the following formulas: r = √x2 + y2 θ = tan-1 (when x > 0) θ = tan-1 + π (when x < 0) y x y x (OR + 180o if it's in degrees). I Formula for the area or regions in polar coordinates. In a rectangular coordinate system, we were plotting points based on an ordered pair of (x, y). Math 126 Worksheet 5 Polar Coordinates Graphing Polar Curves The aim of this worksheet is to help you familiarize with the polar coordinate system. The spherical polar coordinate system is like the polar coordinate system, except an additional angle variable is used, frequently labeled as phi (φ). In fact, we will look at how to calculate the area given one polar function, as well as when we need to find the area between two polar curves. 2 , 53 o) to rectangular coordinates to three decimal places. Polar Coordinates (r-θ)Ans: -0. So far, we have described plane curves by giving: y. In the polar coordinate system, the ordered pair will now be (r, θ). In this fun Polar Coordinates, No Prep, Interactive Activities for PreCalculus and Trigonometry your students practice both graphing polar coordinates and also finding equivalent forms of polar coordinates. Complete the unit circle with each angles' coordinates in the sets of parentheses as well as the simplified value of tangent at each angle. In polar coordinates, if ais a constant, then r= arepresents a circle. | {
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of tangent at each angle. In polar coordinates, if ais a constant, then r= arepresents a circle. Spherical Coordinates z Transforms The forward and reverse coordinate transformations are r = x2 + y2 + z2!= arctan" x2 + y2,z # $% &= arctan(y,x) x = rsin!cos" y =rsin!sin" z= rcos! where we formally take advantage of the two argument arctan function to eliminate quadrant confusion. To convert from Polar coordinates to Cartesian coordinates, draw a triangle from the horizontal axis to the point. The coordinate systems allow the geometrical problems to be converted into a numerica. ) 𝜃 is an angle from the polar axis to the line segment from the pole to P. For example, we will review trigonometric concepts, such as trigonometric identities and real valued functions with points on the coordinate plane, when learning the polar coordinate system. 3 1 x y a Figure 11. The magnetic turbulence is confined near the auroral zone and is similar to that seen at higher altitudes by HEOS-2 in the polar cusp. 1 POLAR COORDINATES Polar coordinate system: a pole (fixed point) and a polar axis (directed ray with endpoint at pole). [email protected] Polar Coordinates T NOTES MATH NSPIRED ©2015 Texas Instruments Incorporated education. Here we provide you with free printable graph paper pdf. 3: Double Integrals in Polar Coordinates We usually use Cartesian (or rectangular) coordinates (x;y) to represent a point P in the plane. Student information Link. NCT program example to show how G81 drilling cycle can be used to drill in a circle using G15 G16 Polar Coordinate Commands and G81 Drilling Cycle. It has been accepted for inclusion in Chemistry Education Materials by an authorized administrator of [email protected] In the first two cases,. 3) Instead of using (x;y), we describe a point by (r; ) in the polar coordinates where ris its dis-tance from the origin and is the angle it makes with the positive x axis. Unique cylindrical coordinates. I Double integrals in disk sections. Double | {
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the positive x axis. Unique cylindrical coordinates. I Double integrals in disk sections. Double Integrals in Polar Coordinates 1. Applications [ edit ] Polar coordinates are two-dimensional and thus they can be used only where point positions lie on a single two-dimensional plane. Find a formula for. Its graph is the circle of radius k, centered at the pole. 4 2D Elastostatic Problems in Polar Coordinates Many problems are most conveniently cast in terms of polar coordinates. Convert the following rectangular coordinates to polar form. The equations are easily deduced from the standard polar triangle. 4) I Review: Polar coordinates. b) In the rotated system of Cartesian coordinates (X r, Y r) the X r-axis is parallel to the direction of vector c, defined by initial position r 0 = Xi +Yj and velocity V 0 = Ui. Conversion: Rectangular to Polar/ Polar to Rectangular 2011 Rev by James, Apr 2011 1. To this end, first the governing differential equations discussed in Chapter 1 are expressed in terms of polar coordinates. Mon Nov 11 - I retaught graphing roses and then we began converting from polar form to rectangular and rectangular to polar. The 50 -point section has a radius of 6. Coordinates in AutoCAD. I Calculating areas in polar coordinates. Corrective Assignment. The axial coordinate or height z is the signed distance from the chosen plane to the point P. 7) Partition the domain x of the rectangular coordinate function into small pieces ∆x. The fact that a single point has many pairs of polar coordinates can cause complications. (As a teacher, one of my favorite questions on homework or exams will be to ask what happens when $$r$$ is negative. In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. Introduction to the Polar Coordinate System A polar coordinate system consists of a fixed point (called the pole or | {
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the Polar Coordinate System A polar coordinate system consists of a fixed point (called the pole or origin) and a ray from the origin (called the polar axis). We will look at polar coordinates for points in the xy-plane, using the origin (0;0) and the positive x-axis for reference. In this polar coordinates worksheets, students change ordered pairs from rectangular form to polar form. [2] Polar Coordinate System, Summary article about the polar coordinate system. Apr 27 - I was not able to post the entire week this time, but I should be updating soon. 2 Slopes in r pola tes coordina When we describe a curve using polar coordinates, it is still a curve in the x-y plane. Click on the tags below to find other worksheets in the same. Find the volume of the region bounded by the paraboloid z= 2 4x2 4y2 and the plane z= 0. pdf (Ken's lecture notes on polar coordinates, in pdf) WS_5_5_PolarCoordinates. The polar coordinates (r,θ) are related to the usual rectangular coordinates (x,y) by by x = r cos θ, y = r sin θ The figure below shows the standard polar triangle relating x, y, r and θ. See Large Polar Graph Paper. the report the polar coordinates of each hit(you can get the polar values at that time), to export the data into a CSV file it is probably easier to create your own utility to do that. Integration in polar coordinates Polar Coordinates Polar coordinates are a different way of describing points in the plane. Coordinates in AutoCAD. In Polar Coordinate System, the references are a fixed point and a fixed line. Convert the following equation of a circle to polar coordinates: 2x2 +3x+2y2 + −5y = 7 7. Polar Form of an Ellipse—C. Cylindrical and spherical coordinates Recall that in the plane one can use polar coordinates rather than Cartesian coordinates. Consider this exam question to be reminded how well this system works for circular motion:. In fact, we will look at how to calculate the area given one polar function, as well as when we need to find the area between | {
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how to calculate the area given one polar function, as well as when we need to find the area between two polar curves. pdf (Worksheet practicing this material, in pdf) WS_Soln_5. There are a total of thirteen orthogonal coordinate systems in which Laplace’s equation is separable, and knowledge of their existence (see Morse and Feshbackl) can be useful for solving problems in potential theory. ;) 21) ( , ), ( , ) 22) ( , ). Concentric Circles: 17 vs 13 Polar Radians. Frame of Reference In the polar coordinate system, the frame of reference is a point O that we call the pole and a ray that. To find the polar angle t, you have to take into account the sings of x and y which gives you the quadrant. We find from the above equations that dur dθ = −(sinθ)i +(cosθ)j = uθ duθ dθ = −(cosθ)i−(sinθ)j = −ur. The area of a region in polar coordinates defined by the equation with is given by the integral ; To find the area between two curves in the polar coordinate system, first find the points of intersection, then subtract the corresponding areas. An angle is considered positive if measured in the counterclockwise direction from the polar axis, and negative if measured in the. 7) Partition the domain x of the rectangular coordinate function into small pieces ∆x. 2) Convert the following to polar coordinates: :4,150° ; (‐6, 2) 3) Typical Polar Graphs: Make sure you watch the Application Walk Through Video to see how you should graph these. Using standard trigonometry we can find conversions from Cartesian to polar coordinates and from polar to Cartesian coordinates Example. Let r1 denote a unit vector in the direction of the position vector r , and let θ1 denote a unit vector perpendicular to r, and in the direction of increasing θ, see Fig. Polar Coordinates. WaterproofPaper. 31) Polar coordinates can be calculated from Cartesian coordinates like. 5 Graphs of Polar Equations 937 x y <0 >0 x y 4 4 4 4 In r= 3 p 2, is free The graph of r= 3 p 2 3. Find the volume of the region | {
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x y <0 >0 x y 4 4 4 4 In r= 3 p 2, is free The graph of r= 3 p 2 3. Find the volume of the region bounded by the paraboloid z= 2 4x2 4y2 and the plane z= 0. L-01 (Cartesian and Polar coordinates ). The polar coordinate system will be useful for many problems you encounter at MIT, such as those involving circular motion or radial forces. 3 mm, so its boundary equation is r = 6. Watch today's lesson and complete pp. My data set is defined in (R, theta) coordinates. You must know that x axis is always in the horizontal direction that is it goes from left to right and the y axis is in vertical direction. coordinates. Such definitions are called polar coordinates. 1 Polar Coordinates - PRACTICE. 2 Slopes in r pola tes coordina When we describe a curve using polar coordinates, it is still a curve in the x-y plane. Preview Activity 11. Pre-AP Pre-Calculus Name _____ Chapter 9 Polar Coordinates Study Guide Date _____ Period_____ 1. 1 DEFINITION OF CYLINDRICAL COORDINATES A location in 3-space can be defined with (r, θ, z) where (r, θ) is a location in the xy plane defined in polar coordinates and z is the height in units over the location (r, θ)in the xy plane Example Exercise 11. Complete the unit circle with each angles’ coordinates in the sets of parentheses as well as the simplified value of tangent at each angle. In Lemma we have seen that the vector r(t) × r˙(t) = C is a constant. (As a teacher, one of my favorite questions on homework or exams will be to ask what happens when $$r$$ is negative. 4 (Circular motion). In polar coordinates, every point is located around a central point, called the pole, and is named (r,nθ). Use double integrals in polar coordinates to calculate areas and volumes. You should pay attention to the following: 1. png 488 × 468; 81 KB. There are approximately 20 problems on this. Precalculus Examples. Apr 11, 2014 - Explore brittanykaye911's board "polar coordinates", followed by 154 people on Pinterest. Consider Figure 13. 4 Polar Coordinate | {
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"polar coordinates", followed by 154 people on Pinterest. Consider Figure 13. 4 Polar Coordinate System Blank; 6. 5 Systems of Linear Inequalities; 7. Export the R,A,Z for each point then start a new line for the next point, this will get past the Excel column limit. a polar equation is the set of all points in the plane that can be described using polar coordinates that satisfy the equation. Cartesian coordinates need two lines within an orthogonal system. Find polar coordinates for the point with rectangular coordinates 00,. The arc length of a polar curve defined by the equation with is given by the integral. The polar coordinate system provides an alternative method of mapping points to ordered pairs. units away from the last point entered. Suppose f is defined on an neighborhood U of a point z 0 = r 0eiθ 0, f(reiθ) = u(r,θ)+iv(r,θ), and u r, u θ, v r, and v θ exist on U and are continuous at (r 0,θ 0). The polar coordinate system (r, θ) and the Cartesian system (x, y) are related by the following expressions: With reference to the two-dimensional equ ations or stress transformation. In this section we will look at converting integrals (including dA) in Cartesian coordinates into Polar coordinates. Note that this definition provides a logical extension of the usual polar coordinates notation, with remaining the angle in the – plane and becoming the angle out of that plane. In Polar Coordinate System, the references are a fixed point and a fixed line. I Double integrals in disk sections. rectangular coordinates ⇒ polar coordinates polar coordinates ⇒ rectangular coordinates N=√ T2+ 2 U, 𝜃= P T= N K O𝜃 U= N O𝑖𝜃 The angle, θ, is measured from the polar axis to a line that passes through the point and the pole. Equilibrium equations in polar coordinates Hooke’s Law in polar coordinates √ Miner’s rule Crack Propagation √ √ @ A @ A Strain displacement Equations in Polar Coordinates Airy stress function in polar Coordinates Fracture mechanics √ von Mises effective | {
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Polar Coordinates Airy stress function in polar Coordinates Fracture mechanics √ von Mises effective stress: for 2-D √ Maximum Distortion Energy Theory:. Angle t is in the range [0 , 2Pi) or [0 , 360 degrees). 3 WS Polar Coordinates (Answers). Pre-Calculus Worksheet Name: _____ Section 10. A region R in the xy-plane is bounded below by the x-axis and above by the polar curve defined by 4 1 sin r T for 0 ddTS. Polar Graph Paper – free line graph paper polar in petech pregenerated files usually i put the most useful outputs here i still did that but i also tried odd things what happens when you divide a circle by 365 25 and also 12 5 free printable polar graph paper in pdf polar graph paper radians this is an advanced form of paper that will be available by us as in along with the polar paper the. 1 The Axis-Environments. Area of regions in polar coordinates (Sect. Spherical-polar coordinates. Department of Mathematics - University of Houston. Input the Cartesian coordinates of P (1, 1), x first. We recall that the Dirichlet problem for for circular disk can be written in polar coordinates with 0 r R, ˇ ˇ as u= u rr+ 1 r u r+ 1 r2 u = 0 u(R; ) = f( ): 6. 2 (pdf) S&Z 11. So depending upon the flow geometry it is better to choose an appropriate system. We need to show that ∇2u = 0. Cartesian coordinate system: start with xand yaxes. Watch today's lesson and complete pp. Polar Graph Paper – free line graph paper polar in petech pregenerated files usually i put the most useful outputs here i still did that but i also tried odd things what happens when you divide a circle by 365 25 and also 12 5 free printable polar graph paper in pdf polar graph paper radians this is an advanced form of paper that will be available by us as in along with the polar paper the. The 2-D polar coordinates #P ( r, theta)#, r = #sqrt (x^2 + y^2 ) >= 0#. In this section we see that in some circumstances, polar coordinates can be more useful than rectangular coordinates. So let us first set us a | {
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polar coordinates can be more useful than rectangular coordinates. So let us first set us a diagram that will help us understand what we are talking about. Polar coordinates. Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. The use of r for the spherical radial coordinate can be confused with the radial coordinate in polar or cylindrical coordinates, but computations requiring both at the same time are rare. 1 Review: Polar Coordinates The polar coordinate system is a two-dimensional coordinate system in which the position of each point on the plane is determined by an angle and a distance. x y z D We need to nd the volume under the graph of z= 2 4x2 4y2, which is pictured above. pdf from MATH 111 at American Public University. This is one application of polar coordinates, represented as (r, θ). Professional Publications, Inc. [2] Polar Coordinate System, Summary article about the polar coordinate system. I also presume length judgements in polar coordinates are more difficult. Example of finding the polar coordinates of a point Give the four basic polar coordinates of points A, B, C, and D shown in the figure. 5) i Real Imaginary 6) (cos isin ) Convert numbers in rectangular form to polar form and polar form to rectangular form. The Laplacian in Spherical Polar Coordinates C. Cylindrical coordinates are defined with respect to a set of Cartesian coordinates, and can be converted to and from these coordinates using the atan2 function as follows. based and Cartesian coordinates modeling. Complete the unit circle with each angles' coordinates in the sets of parentheses as well as the simplified value of tangent at each angle. We can thus regard f as a function from Rn to Rn, and as such it has a derivative. Find the volume of the region bounded by the paraboloid z= 2 4x2 4y2 and the plane z= 0. Coordinates were specified by the distance from the pole and the angle | {
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2 4x2 4y2 and the plane z= 0. Coordinates were specified by the distance from the pole and the angle from the polar axis. For example the graph of the equation x2 + y2 = a we know to be a circle, if a > 0. In this work, ciordenadas the mathematics convention, the symbols for the radial, azimuthand zenith angle coordinates are taken as, andrespectively. A Cartesian coordinate system is the unique coordinate system in which the set of unit vectors at different points in space are equal. To use polar coordinates to specify a point, enter a distance and an angle separated by an angle bracket (<). Notice that this solution can be transformed back into rectangular coordinates but it would be a mess. Solution: The function that we need to use in this example is G, which converts the pair of rectangular coordinates (x,y) into the polar coordinates (r,!). The coordinates of a point determine its location. 1 Equilibrium equations in Polar Coordinates One way of expressing the equations of equilibrium in polar coordinates is to apply a change of coordinates directly to the 2D Cartesian version, Eqns. In this unit we explain how to convert from Cartesian co-ordinates to polar co-ordinates, and back again. L-01 (Cartesian and Polar coordinates ). 3 mm, so its boundary equation is r = 6. Polar Coordinates-measures the distances (and direction) from the origin (radius)& the circle •• (r, f), (radius): •• ndusionf Rectangular Coordinates deal with horizontal & vertical distances, whereas polar coordinates deal with diagonal & circular distances. By printing out this quiz and taking it with pen and paper creates for a good variation to only playing it online. If we restrict rto be nonnegative, then = describes the. ) for polar coordinates are shown. The distance of these lines passing throw the origin or pole is called radians. Plane Curvilinear Motion Polar Coordinates (r -θ) The particle is located by the radial distance r from a fixed point and by an angular measurement θto the | {
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is located by the radial distance r from a fixed point and by an angular measurement θto the radial line. 11, page 636. 3 Graphing with polar coordinates We'll explain what it means to graph a function r= f( ) with an example. 1] can lie on a curve given by a polar equation although the coordinates. Homework 2: Spherical Polar Coordinates Due Monday, January 27 Problem 1: Spherical Polar Coordinates Cartesian coordinates (x,y,z) and spherical polar coordinates (r,θ,ϕ) are related by x = r sinθ cosϕ y = r sinθ sinϕ z = r cosθ. 4 2D Elastostatic Problems in Polar Coordinates Many problems are most conveniently cast in terms of polar coordinates. The divergence We want to discuss a vector fleld f deflned on an open subset of Rn. 1 Cylindrical coordinates If P is a point in 3-space with Cartesian coordinates (x;y;z) and (r; ) are the polar coordinates of (x;y), then (r; ;z) are the cylindrical coordinates of P. In this note, I would like to derive Laplace’s equation in the polar coordinate system in details. But many teachers might prefer that you measure angles by yourself using a protractor on blank paper. GRAPHING IN POLAR COORDINATES SYMMETRY Recall from Algebra and Calculus I that the concept of symmetry was discussed using Cartesian equations. Example contributed by Armin Moser. Double integrals in polar coordinates (Sect. Unit Vectors The unit vectors in the cylindrical coordinate system are functions of position. The polar coordinate system provides an alternative method of mapping points to ordered pairs. Tangents of polar curves. In certain problems, like those involving circles, it is easier to define the location of a point in terms of a distance and an angle. Applications [ edit ] Polar coordinates are two-dimensional and thus they can be used only where point positions lie on a single two-dimensional plane. Convert the following equation of a circle to polar coordinates: 2x2 +3x+2y2 + −5y = 7 7. NCT program example to show how G81 drilling cycle can be used | {
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coordinates: 2x2 +3x+2y2 + −5y = 7 7. NCT program example to show how G81 drilling cycle can be used to drill in a circle using G15 G16 Polar Coordinate Commands and G81 Drilling Cycle. A Cartesian coordinate system is the unique coordinate system in which the set of unit vectors at different points in space are equal. We are used to using rectangular coordinates, or xy-coordinates. In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuthal angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to. Polar Coordinates. Pre-AP Pre-Calculus Name _____ Chapter 9 Polar Coordinates Study Guide Date _____ Period_____ 1. 180 Spoke Radians. pdf), Text File (. In Lemma we have seen that the vector r(t) × r˙(t) = C is a constant. x2 24y 96 0 x2 4 6 y 4 x h 2 4p y k 25. Coordinates in AutoCAD. Also remember that there are three types of symmetry - y-axis, x-axis, and origin. 12/9- Polar and rectangular coordinates VECTORS Re-TEST THIS WEEK 12/10- converting polar to rectangular equations 12/11- exploration of special polar equations 12/12- Group Project - finish special polar graphs 12/13 - review 12/16 - Test day PROJECT DUE 12/19 5/13 - review TEST 12/16 PROJECT DUE: 12/19. On questions 7-10, you should write your answers in degrees. This can happen in the following ways: (a) It can happen if r 2 = r 1 and θ 2 = θ 1 ± 2πn for any. TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES 3 Notice how easy it is to nd the area of an annulus using integration in polar coordinates: Area = Z 2ˇ 0 Z 2 1 rdrd = 2ˇ[1 2 r 2]r=2 r=1 = 3ˇ: [We are nding an area, so the function we are integrating is f= 1. 1 De ning Polar Coordinates oT nd the coordinates of a point in the polar coordinate system, consider Figure | {
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Polar Coordinates oT nd the coordinates of a point in the polar coordinate system, consider Figure 1. a) , Š ‹%$ 1 b) , Œ % # \$ 1 c) ,Œ % & % 1 d) , Œ " (' 1. 686 CHAPTER 9 POLAR COORDINATES AND PLANE CURVES The simplest equation in polar coordinates has the form r= k, where kis a positive constant. | {
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NumPy: Eigenvalues & Eigenvectors
In this tutorial, we will explore NumPy's numpy.linalg.eig() function to deduce the eigenvalues and normalized eigenvectors of a square matrix.
Let $A$ be a square matrix.
In Linear Algebra, a scalar $\lambda$ is called an eigenvalue of matrix $A$ if there exists a column vector $v$ such that
$$Av = \lambda v$$
and $v$ is non-zero. Any vector satisfying the above relation is known as eigenvector of the matrix $A$ corresponding to the eigen value $\lambda$.
We take an example matrix from a Schaum's Outline Series book Linear Algebra (4th Ed.) by Seymour Lipschutz and Marc Lipson1.
Given the matrix
$$A = \begin{bmatrix} 3 & 1 \\ 2 & 2 \end{bmatrix},$$
the column vector
$$v = \begin{bmatrix} 1 \\ -2 \end{bmatrix}$$
is its eigenvector corresponding to the eigenvalue $\lambda = 1$ as
$$Av = \begin{bmatrix} 3 & 1 \\ 2 & 2 \end{bmatrix} \begin{bmatrix} 1 \\ -2 \end{bmatrix} = \begin{bmatrix} 1 \\ -2 \end{bmatrix} = 1v$$
Also,
$$u = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$$
is its another eigenvector corresponding to the eigenvalue $\lambda = 4$ as
$$Au = \begin{bmatrix} 3 & 1 \\ 2 & 2 \end{bmatrix} \begin{bmatrix} 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 4 \\ 4 \end{bmatrix} = 4 \begin{bmatrix} 1 \\ 1 \end{bmatrix} = 4u$$
NumPy has the numpy.linalg.eig() function to deduce the eigenvalues and normalized eigenvectors of a given square matrix. And since the returned eigenvectors are normalized, if you take the norm of the returned column vector, its norm will be 1.
So, take the cue from here. Since the returned eigenvectors are NORMALIZED, they may not always be the same eigenvectors as in the texts you are referring.
Note the two variables w and v assigned to the output of numpy.linalg.eig(). The first variable w is assigned an array of computed eigenvalues and the second variable v is assigned the matrix whose columns are the normalized eigenvectors corresponding to the eigenvalues in that order.
import numpy as np | {
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import numpy as np
a = np.array([[3, 1], [2, 2]])
w, v = np.linalg.eig(a)
print(w)
print(v)
Executing the above Python script, the output is as follows:
Here we will explain the output. The first printed array is w, which constitutes the eigenvalues. The second printed matrix below it is v, whose columns are the eigenvectors corresponding to the eigenvalues in w. Meaning, to the w[i] eigenvalue, the corresponding eigenvector is the v[:,i] column in matrix v.
In NumPy, the ith column vector of a matrix v is extracted as
v[:,i]
So, the eigenvalue
• w[0] goes with v[:,0]
• w[1] goes with v[:,1]
We will now check if the condition
$$Av = \lambda v$$
holds here.
The LHS of the above equation $Av$ here is
np.dot(a,v[:,0])
and the RHS part $\lambda v$ is
np.dot(w[0],v[:,0])
So if the returned eigenvalues and eigenvectors are correct, the following line of script should return True
print(np.allclose(np.dot(a,v[:,0]),np.dot(w[0],v[:,0])))
Also, just to see if the returned eigenvectors are normalized, use the numpy.linalg.norm() function to cross-check them. The below script should return 1.0 in both the print() statements.
print(np.linalg.norm(v[:,0]))
print(np.linalg.norm(v[:,1]))
Notes
• 1) Seymour Lipschutz and Marc Lipson, Linear Algebra. McGraw-Hill Companies, Inc, 2009. Chapter 9: Diagonalization: Eigenvalues and Eigenvectors, p. 297, Ex. 9.5.
• The eigenvectors returned by the numpy.linalg.eig() function are normalized. So, you may not find the values in the returned matrix as per the text you are referring. | {
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# Thread: *sigh*... need to prove somthing...
1. ## *sigh*... need to prove somthing...
I'm told that gcd(a,n)=1 and gcd(b,n)=1, a,b,n are natural.
I need to prove that gcd(ab,n)=1.
I don't succeed :-\ Tried millions of things!
2. Originally Posted by aurora
I'm told that gcd(a,n)=1 and gcd(b,n)=1, a,b,n are natural.
I need to prove that gcd(ab,n)=1.
I don't succeed :-\ Tried millions of things!
Say $\gcd(ab,n)=d > 1$. Then $d|ab$ and $d|n$. Note $\gcd(d,a)=1$ since $d|n$ and that would imply $\gcd(a,n) \geq d > 1$ a contradiction, so $\gcd(d,a)=1$. But then $d|ab \implies d|b$. And so $\gcd(b,n)\geq d > 1$ a contradiction. Thus, $d=1$.
3. Originally Posted by aurora
I'm told that gcd(a,n)=1 and gcd(b,n)=1, a,b,n are natural.
I need to prove that gcd(ab,n)=1.
I don't succeed :-\ Tried millions of things!
so we have $ra+sn=kb+\ell n=1,$ for some integers $r,s,k,\ell.$ thus $(kr)ab + (ksb + \ell)n=1. \ \ \square$
4. Thank you very much!
However, NonCommAlg, correct me if I'm mistaken - but the fact that you find this sort of linear presentation of a and n that equals 1 doesn't mean that 1 is their gcd...
5. Originally Posted by aurora
Thank you very much!
However, NonCommAlg, correct me if I'm mistaken - but the fact that you find this sort of linear presentation of a and n that equals 1 doesn't mean that 1 is their gcd...
You are mistaken..
If gcd(n, ab) = d > 0 then d|n and d|ab => d| (kr)ab + (ksb + l)n => d|1 => d = 1
However note that NonCommAlg's trick only works for gcd of 1
6. Originally Posted by Isomorphism
You are mistaken..
If gcd(n, ab) = d > 0 then d|n and d|ab => d| (kr)ab + (ksb + l)n => d|1 => d = 1
However note that NonCommAlg's trick only works for gcd of 1
I agree. However, in my algebra class, my professor would require that you show the extra step (what you just did), so i see where aurora is coming from. i would have done it the way NonCommAlg did it. but i like the contradiction method, it seems elegant to me
7. Thank you guys | {
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7. Thank you guys
8. Originally Posted by Jhevon
I agree. However, in my algebra class, my professor would require that you show the extra step (what you just did), so i see where aurora is coming from. i would have done it the way NonCommAlg did it. but i like the contradiction method, it seems elegant to me
I like the contradiction method too. But I was telling aurora about the idea because its useful in an exam. I think its more mechanical.Contradiction may not hit at the right time
9. Originally Posted by Isomorphism
I like the contradiction method too. But I was telling aurora about the idea because its useful in an exam. I think its more mechanical.Contradiction may not hit at the right time
I agree. that's one of the reasons i like it. it is not so obvious at the moment you see the problem. NonCommAlg's way is something you would do on impulse to see how it works out, which is the charm of that method. good for tests when you can't think clearly or be fancy, but are racing against the clock and just need to get the job done | {
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# Prove the fractional field of an integral domain is the smallest field containing the integral domain
I have two questions about the fractional field of an integral domain.
Given an integral domain $D$:
1. Is there a difference between saying "the fractional field of $D$ is the smallest field containing $D$" or "the fractional field of $D$ is the smallest field containing an embedding of $D$"?
2. How do you prove that the fractional field is the smallest field containing $D$ (or an embedding of $D$, if there is a difference...)? Specifically, I want to show that if $F$ is any field containing $D$, then $F$ must contain the fractional field of $D$. | {
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• What is your definition of "fractional field of integer domain"? For me, it is precisely the minimal field containing the domain. – Timbuc Oct 11 '14 at 18:52
• @timbuc *integral domain. My definition is based on the construction. First, given an integral domain $D$, we define the set of ordered pairs where the second coordinate is non-zero. Then we define an equivalence relation where $(a,b) \sim (c,d)$ if $ad = bc$. Then we call the equivalence class containing $(a,b)$ as $\frac{a}{b}$. Then we define addition and multiplication on this set of equivalence classes in the same way as it is defined in $\mathbb{Q}$. This forms a field, what we call the fractional field. – layman Oct 11 '14 at 18:55
• Ok, but then it is trivial, isn't it? I mean, any field containing $\;D\;$ must contain all the multiplicative inverses of non-zero elements $\;d\in D\;$ and thus their product by any element in $\;D\;$ , and this means (by the definition!) that $\;d_1\cdot\frac1{d_2}:=\frac{d_1}{d_2}\;$ is in the field, for any $d_1,d_2\in D\;,\;\;d_2\neq 0\;$ , which means any such field contains the fractions field of $\;D\;$ . – Timbuc Oct 11 '14 at 18:57
• @Timbuc Well now I need to know the answer to my first question. Is there a difference between saying a field contains $D$ and saying it contains an embedding of $D$? – layman Oct 11 '14 at 19:00
• Well, formally there is, yet for most usual cases one doesn't usually pay attention to that slight difference. – Timbuc Oct 11 '14 at 19:01
Let $F'$ be a smallest field containing an embedding of $D$ ($f:D\to F'$), and $F$ a field of fraction of $D$.
We can extend $f$ to morphism of field $\tilde f:F\to F'$ by $\tilde f(a/b)=f(a)/f(b)$.
Now we have that $\tilde f(F)\subseteq F'$ and $\tilde f(F)$ containing an embedding of $D$ , by smallest property we have $\tilde f(F)=F'$.
So the tow fields $F$ and $F'$ are isomorphic.
Edit: If $F$ is any field containing $D$. And denote $K$ the field of fraction of $D$. | {
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Edit: If $F$ is any field containing $D$. And denote $K$ the field of fraction of $D$.
Let $a/b\in K$, $a\in D$ and $0\neq b\in D$, hence $a,b\in F$, it follow that $a$ and $1/b$ are in $F$ so $a. (1/b)=a/b\in F$. Thus $K\subseteq F$.
• Is your extension of $f$ well defined? – layman Oct 11 '14 at 19:26
• I guess it is since $f$ is a homomorphism, and if $a/b = c/d$, then $ad = bc$, so $f(a)f(d) = f(b)f(c)$ which implies $f(a)/f(b) = f(c)/f(d)$. – layman Oct 11 '14 at 19:28
• Yes $b\neq 0$ implies $f(b)\neq 0$ because $f(0)=0\neq f(b)$ ($f$ is injective). – Hamou Oct 11 '14 at 19:28
• I guess I would need to do a little bit more work because I would need to show the image of the field $F$ under the extension of $f$ is a field. – layman Oct 11 '14 at 19:29
• When $\frac ab\neq 0$, $\tilde f(a/b)\tilde f(b/a)=1$, hence $\tilde f(a/b)$ is invertible is in $\tilde f(F)$. – Hamou Oct 11 '14 at 19:32
The right (i.e. categorical) way to say this (without the ambiguities of words like "smallest", "containing", etc.) ought to be that the inclusion $\iota: D\to Q(D)$ has the following universal property:
If $K$ is a field, and $f: D\to K$ is any morphism of rings, then there is a unique morphism of fields $g : Q(D) \to K$ such that $f = g \circ \iota$.
(In particular, $Q(D)$ embeds into any field that $D$ embeds into.)
This property uniquely determines (up to isomorphism) not only $Q(D)$, but $\iota$ as well.
And it's easily proved, since $g(1/b)g(b)=g(1)$ forces $g(a/b) = f(a)/f(b)$, so this amounts to checking that $a/b \mapsto f(a)/f(b)$ is actually a homomorphism. | {
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# Finding the local maxima/minima of the following function
$$f: \mathbb{R}\rightarrow \mathbb{R}$$. Then determining if each of the solutions is a global maximum or a global minimum. $$f(x) = \frac{x^{4}}{4} - 2x^{3} + \frac{11}{2}x^{2} - 6x + 2$$ for all $$x \in \mathbb{R}$$.
I have used the synthetic division to find this equation: $$\left ( x^{2} - 5x + 6 \right )\left ( x - 1 \right ) = 0$$
So the critical points are $$x^{*} = 1,2,3$$.
I know that if $$x^{*} = 1$$, then $${f}''\left ( 1 \right ) = 2 > 0$$, so this is minimum.
If $$x^{*} = 2$$, then $${f}''\left ( 2 \right ) = 1 < 0$$, so this is maximum.
If $$x^{*} = 3$$, then $${f}''\left ( 3 \right ) = 2 > 0$$, so this is minimum.
How do I determine whether these solutions are local max/min and global max/min?
Are all of them global solutions as $$x \in \mathbb{R}$$?
EDIT: I have plugged the values of the $$x's$$ in the main function.
For $$x=1$$ and $$x=3$$, $$f(1)= -0.25 = f(3)$$, and for $$x=2$$, $$f(2)= 0$$.
So is $$x=1,3$$ a local minimum and $$x=2$$ a global maximum?
• Use the 2nd derivative. – Wuestenfux Oct 8 '18 at 7:05
• Local max/min: second derivative. Global Max/Min: compare values or find inequality. – Andreas Oct 8 '18 at 7:05
• Have used the second derivative. So are all of them local solutions? I think all of them are global solutions as $x \in \mathbb{R}$. – OGC Oct 8 '18 at 7:06
From the second derivative test the extremum points that you have found are all local. Note that $$\lim_{x\to \pm\infty}f(x)=+\infty$$, so $$x=1$$ is not a global maximum point. On the other hand, since $$f(1)=f(3)=-1/4$$, it follows that $$x=1$$ and $$x=3$$ are global minimum points.
• Without drawing the graph, how would I figure out that $x=1$ and $x=3$ are global minimum points? – OGC Oct 8 '18 at 7:19
• Since $f$ is differentiable in $\mathbb{R}$, and $\lim_{x\to \pm\infty}f(x)=+\infty$, any global minimum point is also a critical point. – Robert Z Oct 8 '18 at 7:23 | {
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# How do you solve this sort of definite integral?
Can someone walk me through how to do the following problem so I can attempt a few more practice problems?
If: $$\int_{1}^{5} f(x) dx = 12$$ and $$\int_{4}^{5} f( x) dx = 3.6$$ find: $$\int_{1}^{4} f( x) dx$$
Would it simply be $12 - 3.6$ ?
EDIT
If: $$\int_{0}^{9} f(x) dx = 37$$ and $$\int_{0}^{9} g( x) dx = 16$$ find: $$\int_{0}^{9} 2f(x)+3g(x) dx$$
Would this simply be: $2 \times 37 + 3 \times 16$?
-
Do you know any theorems or identities that relate the values of integrals of the same function with different limits? – MJD May 7 '12 at 4:18
If $a \leq c \leq b$, then $\int_a^b=\int_a^c+\int_c^b$. Use this profitably. – Ｊ. Ｍ. May 7 '12 at 4:19
As you said, it is simply 12-3.6. – MJD May 7 '12 at 4:19
@MarkDominus Wow, I didn't think it would be that simple. That's why I posted it up here. I'll post a second one up so it's not a waste of a question. – justcheckingin May 7 '12 at 4:21
Regarding the second problem: yes, that's right. – Brian M. Scott May 7 '12 at 4:26
This is done using additivity of integration on intervals i.e. if $c \in [a,b]$ and
$\displaystyle \int_a^b f(x) dx$, $\displaystyle \int_a^c f(x) dx$ and $\displaystyle \int_c^b f(x) dx$ are well- defined, then $$\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx$$ Hence, in your case, you have that $$\int_1^5 f(x) dx = \int_1^4 f(x) dx + \int_4^5 f(x) dx$$ Hence, we get that $$12 = \int_1^4 f(x) dx + 3.6$$ i.e. $$\int_1^4 f(x) dx = 8.4$$
For the second problem as Brian pointed out in the comments, it follows from linearity of integration i.e. $$\int \left( a f(x) + b g(x) \right)dx = a \int f(x)dx + b \int g(x) dx$$ provided all the integrals are well-defined.
-
$$\int_a^bf(x)dx+\int_b^cf(x)dx=\int_a^cf(x)dx$$
The integral in the interval $[1,4]$ is the difference: integral in $[1,5]$ - integral $[4,5]$. | {
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# General formula for iterated cumulative sum
Consider the sequence $$S_0$$ consisting of ones:
$$1,1,1,1,1,1,\ldots$$
Now compute the cumulative sum of this sequence, and call the resulting sequence $$S_1$$:
$$1,2,3,4,5,6,\ldots$$
Proceed iteratively to generate sequence $$S_2$$:
$$1,3,6,10,15,21,\ldots$$
then $$S_3$$:
$$1,4,10,20,35,56,\ldots$$
and so on.
It is well known that each sequence $$S_k$$ can be represented by a $$k$$-degree polynomial $$P_k(n)$$. For the above sequences the polynomials are $$P_0(n) = 1$$ $$P_1(n) = n$$ $$P_2(n) = \frac{n^2+n}{2}$$ $$P_3(n) = \frac{n^3+3n^2+2n}{6}$$
My question: Is there a general formula for coefficients of the polynomial $$P_k$$? Or more generally, is there a formula to compute $$S_k(n)$$ as a function of $$n$$ and $$k$$? I mean a closed formula $$S_k(n) = f(k,n)$$ (not an iterative procedure such as the construction method I just described).
If that helps, actually I'm not interested in $$S_k(n)$$, but rather in the quotient $$S_k(n)/S_k(n+1)$$.
• Just a test to see if I have understood: in $S_1$ should the last term be $21$? – User3773 Feb 26 '15 at 17:39
• @Cla Oops. Yes, sorry. Corrected! – Luis Mendo Feb 26 '15 at 17:40
The numbers in sequence $S_k$ are the binomial coefficients $\binom{m}{k}$; the $n$-th term of $S_k$ is $\binom{n-1+k}{k} = \frac{(n-1+k)!}{(n-1)!k!}$. One can prove this by using that $$\binom{i}{i} + \binom{i+1}{i} + \cdots + \binom{i+j}{i} = \binom{i+j+1}{i+1}$$ for any $i,j \geq 0$.
For $k=1$ we find $P_1(n) = \binom{n-1+1}{1} = \binom{n}{1} = n$, for $k=2$ we find $P_2(n) = \binom{n-1+2}{2} = \binom{n+1}{2} = \frac{n(n+1)}{2}$, for $k=3$ we find $P_3(n) = \binom{n-1+3}{3} = \binom{n+2}{3} = \frac{n(n+1)(n+2)}{6}$, etcetera.
• Wow. It was really simple! Could you provide any reference? – Luis Mendo Feb 26 '15 at 17:43
• Thanks a lot for your help. I've posted a related question here – Luis Mendo Feb 26 '15 at 18:47
We can generalize your problem to arbitrary initial sequences. | {
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We can generalize your problem to arbitrary initial sequences.
Let $a_n\,(n=0,1,2,3,...)$ be a sequence of numbers. Define its iterated partial sums using the recurrence $$S^{(0)}_n=a_n,\quad S^{(k+1)}_n=\sum_{m=0}^n S^{(k)}_m,\tag1$$ so that we have, for example, $$\small\begin{array} &S^{(0)}_0=\color{green}{a_0}, &S^{(0)}_1=\color{blue}{a_1}, &S^{(0)}_2=\color{maroon}{a_2},&...\\ S^{(1)}_0=\color{green}{a_0}, &S^{(1)}_1=\color{green}{a_0}+\color{blue}{a_1}, &S^{(1)}_2=\color{green}{a_0}+\color{blue}{a_1}+\color{maroon}{a_2},&...\\ S^{(2)}_0=\color{green}{a_0}, &S^{(2)}_1=\color{green}{a_0}+(\color{green}{a_0}+\color{blue}{a_1}), &S^{(2)}_2=\color{green}{a_0}+(\color{green}{a_0}+\color{blue}{a_1})+(\color{green}{a_0}+\color{blue}{a_1}+\color{maroon}{a_2}),&...\\ S^{(3)}_0=\color{green}{a_0}, &S^{(3)}_1=\color{green}{a_0}+(\color{green}{a_0}+(\color{green}{a_0}+\color{blue}{a_1})),&... \end{array}\tag2$$ Now we can prove by induction that the following formula holds: $$S^{(k+1)}_n=\sum_{m=0}^n\binom{m+k}k\,a_{n-m}.\tag3$$
• Vladimir, I've just given an approach which generalizes your solution even more. Please see my new answer. – Gottfried Helms Aug 29 '19 at 6:55
A more generalized solution, where even fractional iteration-heights $$h$$ for $$S_n^{(h)}$$ become possible, can be found using a matrix-ansatz. Consider the matrix-equation $$D \cdot A = S^{(1)}(A) \tag 1$$ where $$D$$ is the lower triangular unit-matrix $$D= \Tiny \begin{bmatrix} 1&.&.&.&\cdots\\1&1&.&.&\cdots\\1&1&1&.&\cdots\\ 1&1&1&1&\cdots\\ \vdots&\vdots&\vdots&\vdots&\ddots\\ \end{bmatrix} \tag 2$$ then of course \small \begin{align} D^2 \cdot A &= S^{(2)}(A) \\ D^3 \cdot A &= S^{(3)}(A) \\ \vdots \\ D^h \cdot A &= S^{(h)}(A) \\ \end{align} \tag 3 The $$h$$'th power of $$D$$ can be computed using $$L = \log(I + (D-I))$$ and $$\exp(L)$$ using the series-representation of this functions (which reduce to finite sums in the case of using $$D$$). We get formally | {
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$$\displaystyle \qquad \qquad \Large{D^h =}$$ $$\tag 4$$
and where we need only document the entries of the first column because of the schematic form of $$D^h$$:
$$\displaystyle \qquad \qquad S_n^{(h)} (A) = \sum_{c=0}^n D_{n,c} \cdot A[c] = \sum_{r=0}^n D_{n-r,0} \cdot A[r] \tag 5$$
The coefficients $$D_{r,0}$$ might look abscure, but can easily be described when factorials are extracted:
$$\displaystyle \qquad \qquad \large {S_5^{(h)}(A)=}$$ $$\tag 6$$
and even simpler
$$\displaystyle \qquad \qquad \large {S_5^{(h)}(A)=}$$ $$\tag 7$$
The coefficients in the previous representation are the unsigned Stirlingnumbers $$1$$'st kind and for integral $$h$$ this gives of course the appropriate binomial-expressions which are noted in the other answers and comments.
But we can easily insert fractional $$h$$ as well!
Concerning your last question, the quotient.
Example, let $$k=3$$ then the quotient can be found by multiple cancellations: $$P_3(n) = {n^3+3n^2+2n\over 6} = {(n+2)(n+1)n\over 6} \\\ P_3(n+1) = {(n+1)^3+3(n+1)^2+2(n+1)\over 6} = {(n+3)(n+2)(n+1)\over 6} \\\ {P_3(n+1)\over P_3(n)} ={{(n+3)(n+2)(n+1)\over 6}\over {(n+2)(n+1)n\over 6} } = {n+3\over n}$$ Thus in general: $${P_k(n+1)\over P_k(n)} ={{(n+k)\cdots(n+2)(n+1)\over k!}\over {(n+k-1)\cdots(n+1)n\over k!} } = {n+k\over n}$$ | {
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# Arcwise connected part of $\mathbb R^2$
Here's a question that I share: Show that if $D$ is a countable subset of $\mathbb R^2$ (provided with its usual topology) then $X=\mathbb R^2 \backslash D$ is arcwise connected.
• This is even true if you only require $D$ to have lebesgue-measure 0: math.stackexchange.com/questions/77791/… Jun 9 '12 at 9:20
• I have two further questions: 1) Is this result holds in higher dimension i.e., on $\mathbb{R}^n$, where $n \in \mathbb{N}$. 2) Can one characterize the topological space for which this kind of properties hold $?$ May 22 '13 at 18:47
• @TeresaLisbon I first asked it as new question but it was closed and repititive requests to open it were discarded in CURED chatroom and they suggested that new questions shouldn't be asked but bounty should be put on old question. Which I did now. Sep 23 '21 at 6:02
• Just for reference, the question that was closed as a duplicate is If A is countable subset of $\mathbb{R}^2$ , then $\mathbb{R}^2 - A$ is pathwise connected.. The relevant discussion in the chatroom CURED can be found here: Request for reopening //q/3973870. Sep 23 '21 at 6:57
• @No-One Got it, thanks. I think what I'll do, if I plan to answer this, is place your bounty text in my answer, to make sure that my answer targets your query, along with the question in general. Sep 23 '21 at 9:20
HINT: Not only is $\Bbb R^2\setminus D$ arcwise connected, but you can connect any two points with an arc consisting of at most two straight line segments.
Suppose that $p,q\in\Bbb R^2\setminus D$. There are uncountably many straight lines through $p$, and only countably many of those lines intersect $D$, so there are uncountably many straight lines through $p$ that don’t hit $D$. Similarly, there are uncountably many straight lines through $q$ that don’t hit $D$. Can you finish it from here? | {
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• Yes, I can : two of them at lest intersect. One auther way is to consider the bissector $\Delta$ of $[ab]$ and all straight lines $[p,\delta]\cup[\delta,q]$ where $\delta \in \Delta$, On of them at lest is a subset of $X$ Jun 9 '12 at 9:32
• @Mohamed: Exactly. (Or if you’re really lucky, one of them is the line through $p$ and $q$.) Jun 9 '12 at 9:37
• I have not thought about this one!! Thank's!! (Edit [p,q] not [a,b] ) Jun 9 '12 at 9:51
• From here. In order to show path $f:[0,1]\to \Bbb{R}^2-A$ is continuous.we can construct a function: $f(t)=f_1(t)$ ,if $t\in [0,1/2]$ and $f(t)=f_2(t)$ ,if $t\in [1/2,1]$. $f$ is continuous by pasting lemma. And we're done. Am I right?@Briam M Scott Jul 16 '20 at 11:57
• @AmanPandey: Yes, that’s right. Jul 16 '20 at 16:35
Any two points in $\mathbb R^2\setminus D$ are connected by uncountably many disjoint arcs of circles, of which only countably many may intersect $D$.
(This is adapted from one of my student's solutions to a related exam problem.)
Let $$p, q$$ be two distinct points in $$\mathbb{R}^2 \setminus D$$. If the straight line segment joining $$p$$ to $$q$$ lies in $$\mathbb{R}^2 \setminus D$$, then you are done: one possible parametrization of the path is $$\gamma(t) = (1 - t)p + tq$$, $$0 \leq t \leq 1$$.
So, suppose that the above does not work. Now, note that there are uncountably infinitely many straight lines passing through the point $$p$$. Since $$D$$ is only countable, in particular there are uncountably infinitely many straight lines passing through $$p$$ and not intersecting $$D$$. Fix any one such line $$L_1$$. Once more, there are uncountably infinitely many straight lines passing through $$q$$ that also intersect $$L$$. In particular, there are uncountably infinitely many such lines that also do not intersect $$D$$. Fix any one such line $$L_2$$. | {
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Suppose that $$L_1$$ and $$L_2$$ intersect at the point $$r$$. Then, you can take the path joining $$p$$ and $$q$$ to be the one that starts at $$p$$, traverses along the line $$L_1$$ to the point $$r$$, then traverses along the line $$L_2$$ to the point $$q$$. One possible parametrization of this path is $$\gamma(t) = \begin{cases} (1 - 2t)p + 2tr, & 0 \leq t \leq \tfrac{1}{2};\\ (2 - 2t)r + (2t - 1)q, & \tfrac{1}{2} \leq t \leq 1. \end{cases}$$ Note that the pasting lemma shows that $$\gamma$$ is indeed continuous.
As for how one arrives at these specific parametrizations, one first needs to know the parametric equation of a straight line. For instance, see Parametric form of a line for a quick refresher. Secondly, one needs to adjust the parameter $$t$$ by scaling and translating in order to get it to lie in the "correct" range for one's purposes. This is another linear change, so it should not be too difficult.
But, just to belabor the point further, here's one way you can mentally figure out the parametrization $$\gamma$$ in the second case. To join the points $$p$$ and $$r$$, we want to vary $$t$$ from $$0$$ to $$1/2$$ such that at $$t = 0$$ we are at $$p$$ and at $$t = 1/2$$ we are at $$r$$. So, if the parametrization for this part looks like $$\gamma(t) = f(t)p + g(t)r$$, then we want $$f(0) = 1$$ and $$f(1/2) = 0$$, and we also want $$g(0) = 0$$ and $$g(1/2) = 1$$. Can we find linear functions $$f(t) = \alpha_1 t + \beta_1$$ and $$g(t) = \alpha_2 t + \beta_2$$ that satisfy these conditions? Sure, just substitute those values of $$(t, f(t))$$ and $$(t, g(t))$$ that we wrote down for $$t = 0, 1/2$$ into the respective equations to solve for $$\alpha_i, \beta_i$$. Repeat the process for the line joining $$r$$ to $$q$$, and you are done.
• For completeness, you should at least state that your path is also injective, which implies that it defines an embedding $[0,1]\to X$, as required by the definition of an arcwise connected space. Sep 28 '21 at 17:57 | {
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Ok I will take a stab at clarifying Brian M. Scott's Answer instead of writing a fully different answer.
Let $$p,q\in \mathbb{R}-D$$. Then Define $$A:=\{\text{ Lines Through }p\}$$, $$B:=\{ \{\text{ Lines through }q\}$$.
One could write the definitions of these sets more formally if desired, but the meaning is clear. I claim the following: $$\exists l\in A, \text{ such that } l\cap D=\emptyset.$$ Similarly $$\exists n\in B, \text{ such that } l\cap D=\emptyset.$$ Lets us understand why this should be clear. Now define a mapping $$m:\mathbb{R} \to A$$ by $$m(x):=$$ the line in $$A$$ with slope $$x$$. This mapping is clearly $$1-1$$ and onto (almost see below*), as any line through a point $$p$$ is determined by its slope, and any slope uniquely determines a line through $$p$$. You could formalize this more by writing any line through $$p$$ in point slope form, but I leave that to the reader.
A similar argument follows for $$B$$, thus $$||A||=||B||=||\mathbb{R}||$$.
Now let us show that there are lines in $$A$$ and $$B$$ which do not contain points in $$D$$. define a map $$\phi:D\to A\times B$$, by $$\phi(d):=\{(l,m)\in A\times B| d=l\cap m\}$$. That is each point $$d\in D$$ gets mapped to the pair of lines in $$A$$ and $$B$$ which meet at $$d$$. This mapping is $$1-1$$, as any two points determine a unique line. By assumption $$D$$ is countable, therefor $$\phi$$ cannot be onto thus there exists a pair of lines, $$(l^*,m^*)$$, which do not contain any points in $$D$$. This follows as $$A\times B- \phi(D)\neq \emptyset$$.
we have a piecewise linear path $$p \to l^*\cap m^* \to q$$. QED.
The Crux of the argument, here is where we assert that the mapping $$\phi$$ cannot be onto, given the assumption that $$D$$ was countable. | {
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One small note, the map $$m$$ is not quite onto, as the vertical line through $$p$$ has undefined slope. However, this doesn't disturb the argument as the main idea is that $$A$$ and $$B$$ are both uncountable. We might also be able to circumvent this by taking a mapping from the extended reals.
• Great stuff, really nice answer, thanks for adding the clarification in the first line. Sep 25 '21 at 12:32 | {
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# Are at most $1/3$ vertices "kings"?
If $$G=(V,E)$$ is a finite, simple, undirected graph, and $$v\in V$$, we set $$N(v) = \{w\in V:\{v,w\}\in E\}$$, and $$\text{deg}(v)= |N(v)|$$. We say a vertex $$v\in V$$ is a king if $$\text{deg}(v) > \text{deg}(w)$$ for all $$w\in N(v)$$.
In the graph $$G=(\{0,1,2\}, \big\{\{0,1\}, \{1,2\}\big\})$$, one of the $$3$$ vertices is a king. Let $$\text{King}(G)$$ be the set of king vertices.
Question. Is it true that for any finite connected graph $$G=(V,E)$$ with $$|V|>1$$ we have $$|\text{King}(G)|/|V|\leq 1/3$$? If not, how large can this value get?
• A complete bipartite graph $G = U \cup V$ with $|U| = n$ and $|V| = n-1$ has $n - 1$ kings out of $2n - 1$ total vertices. Feb 28 at 14:16
• maybe related: mathoverflow.net/questions/343607/… Mar 1 at 15:14
For this discussion I am assuming we do not consider isolated vertices to be "Kings", even though technically your definition considers them to be so in a vacuous sense (I guess this convention goes back to Shakespeare). Otherwise of course one can make every vertex a king by having no edges whatseover. | {
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For the matching lower bound, observe that no two kings can be adjacent, and if there is at least one king, the set $$E'$$ of ordered pair edges $$(v,w)$$ in $$E$$ with $$v \in \mathrm{King}(G)$$ and $$w \not \in \mathrm{King}(G)$$ is non-empty. Now we do weighted double counting: \begin{align*} \# \mathrm{King}(G) &= \sum_{v \in \mathrm{King}(G)} 1 \\ &= \sum_{(v,w) \in E'} \frac{1}{d(v)}\\ &< \sum_{(v,w) \in E'} \frac{1}{d(w)} \\ &\leq \sum_{w \in V \backslash \mathrm{King}(G)} 1 \\ &= \#V - \# \mathrm{King}(G) \end{align*} hence $$\# \mathrm{King}(G) < \frac{1}{2} \# V.$$ Of course the same claim holds when there are no kings as long as the graph is not the empty graph. So this shows that the lower bound provided by the complete bipartite graph examples (adding an isolated vertex in the case when one wants an even number of vertices) are completely optimal: the maximal number of kings in a graph on $$n$$ vertices is $$\max( \lfloor \frac{n-1}{2} \rfloor, 0)$$.
This bound can also be viewed as quantifying a variant of the "friendship paradox". (Based on this connection, I propose "influencer" as a more modern and gender-neutral terminology alternative to "king".)
• There is already the gender-neutral "celebrity" which appears to be motivated by interpreting the vertex degree as the number of people by which a person is known. Mar 1 at 15:20
• Furthermore the terminology "king" is used with a different meaning in oriented tournaments (these are vertices with maximum out-degree). Mar 2 at 4:14
The fact that the number of kings is less than the number of non-kings is essentially equivalent to the following problem, proposed by Alexander Razborov to Tournament of Towns in 1990:
Given an $$m\times n$$ matrix, $$m. Some entries are starred, and each column contains a starred entry. Then there exists a star whose row contains more stars than its column. | {
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(Here columns correspond to kings, rows to non-kings, stars to edges, and the strictness of inequalities is different.)
I wanted to find a proof that uses Hall's marriage theorem [1] instead of double-counting.
Given a graph $$G=(V,E)$$, let $$K$$ be the set of kings in $$V$$, and $$R:=V \setminus K$$ the rest.
Claim: $$|K| < |R|$$.
Proof Let $$G=(V,E)$$ be a counterexample that minimizes the sum $$|V|+|E|$$, so $$|K| \ge |R|$$. Then $$G$$ is bipartite, since any edge between nodes in $$R$$ can be removed. If $$|K|>|R|$$ then removing one king would yield a smaller counterexample, so $$|K|=|R|$$. If there was a subset $$S$$ of $$K$$ where its neighborhood satisfies $$|N(S)|< |S|$$, then the induced graph on $$S \cup N(S)$$ would be a smaller counterexample. Thus the Hall condition is met in $$G$$. Removing from $$G$$ a perfect matching of $$K$$ to $$R$$ yields a smaller counterexample.
If we consider the complete bipartite graph $$K_{n,m},$$ where $$n,m$$ are natural numbers with $$n < m,$$ then the maximum degree of a vertex in this graph is $$=m$$ and there are at least $$n$$ many vertices having degree $$m$$, and so the number of "kings" (as per the definition above) in this graph is $$= n,$$ and the total number of vertices $$=n+m.$$
Now, since $$n,m$$ can be anything (with $$n \le m$$), so one can take $$n,m$$ to be such that $$n/(n+m) > \frac{1}{3},$$ and thus in this case we see that the number of kings in this graph is $$> \frac{1}{3} \cdot$$ the number of vertices in the graph.
For such graphs you can just set $$n=m-1$$ to see that you cannot get an "asymptotic" bound of $$< 1/2.$$
• If $n=m$ then there are no kings. The best this bipartite construction can do is $\lceil n/2\rceil$ many kings, as Nik Weaver has also pointed out. Feb 28 at 15:58 | {
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Why is the integral of $\frac1{x^2}$ from $1$ to $\infty$ not the same as the infinite sum from $1$ to $\infty$?
Studying series I am a bit confused on this point. The infinite sum of $1/x^2$ from $1$ to $\infty$ was proved by Euler to be $\pi^2$ divided by $6$:
$$\sum_{x=1}^\infty\frac 1 {x^2}=\frac {\pi^2} 6$$
But if I integrate from $1$ to $\infty$ of the same entity namely $1/x^2$ it is $1$. Correct..? Unless I did it wrong. $$\int_1^\infty\frac 1 {x^2}dx=1$$ How can this be since by integrating it seems we are adding a lot more numbers to cover the same area so we should by all means get the same thing or something at least as large as $\pi^2/6$?
• Why would anyone expect them to be the same? Dec 24, 2017 at 16:49
• @LordSharktheUnknown, please read his entire question. Dec 24, 2017 at 16:54
• The relationship between the sum and the integral is given by en.wikipedia.org/wiki/Euler–Maclaurin_formula. they differ because rectangles are not curved. The area in the rectangles is not the area under the curve. Dec 24, 2017 at 16:54
• @Sedumjoy, please read your edited question for how to write the integral and sum :). Dec 24, 2017 at 17:00
• I agree with @LordSharktheUnknown, it is not natural to expect that $$f(x)=\frac{1}{x^2},\qquad g(x)=\frac{1}{\lfloor x\rfloor ^2}$$ have the same integral over $(1,+\infty)$, also because $g(x)\geq f(x)$. Dec 24, 2017 at 17:55
Note that $$\int_1^\infty \frac{1}{x^2}\leq\sum_1^\infty \frac{1}{n^2}\tag{1}$$ by considering a Riemann sum with left endpoints. Here is a picture (for the case of $1/x$ but a similar picture can be drawn for this case as well). See this picture. Image credits go to Wikipedia. | {
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• +1 For the picture.. It illustrates the point better :) I will add it to my answer
– Ant
Dec 24, 2017 at 16:59
• no its the wrong graph, should be a graph of $\frac{1}{x^2}$, right? Dec 24, 2017 at 17:00
• @FoobazJohn, Just wondering, what program did you use to make this picture? Dec 24, 2017 at 17:00
• @mathreadler, yeah you're right, though the basic insight remains the same. Dec 24, 2017 at 17:01
• @mathreadler Yes I commented that it is for $1/x$ but a similar picture can be drawn for the case $1/x^2$. Dec 24, 2017 at 17:01
When you do the sum, you sort of approximate the area with rectangles of base length equal to $1$. Draw the function $1/x^2$ and draw the rectangles with base length 1; you'll see that the area under the rectangles is much bigger than the area under the function $1/x^2$
Here is an illustration for the function $1/x$, but it's essentially the same as in the $1/x^2$ case. (Thanks to @FoobazJohn) The integral adds a lot more numbers, but these numbers are multiplied by something very small. The end result is that the integral represents the area under the $1/x^2$ curve, which is less than the area under the rectangles.
Let us compare $1$ and $\int_1^2\frac1{x^2}\,\mathrm dx$. Since $\bigl(\forall x\in(1,2]\bigr):\frac1{x^2}<1$, $\int_1^2\frac1{x^2}\,\mathrm dx<1$. For the same reason, $\int_2^3\frac1{x^2}\,\mathrm dx<\frac14$, $\int_3^4\frac1{x^2}\,\mathrm dx<\frac19$, and so on. So$$1=\int_1^\infty\frac1{x^2}\,\mathrm dx<\sum_{n=1}^\infty\frac1{n^2}=\frac{\pi^2}6.$$
Note that it is not true that $\int_a^bf(x)\,\mathrm dx$ is the sum of all numbers $f(x)$ with $x\in[a,b]$. Instead, it is the average value of $f$ in $[a,b]$ times $b-a$.
• all three answers are very good in explaining this and I will pick one to close the problem but they are equally good....I feel stupid for not seeing this that I had to ask.... Dec 25, 2017 at 0:59 | {
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# Help Me Find Mistake in Exact Differential Equation
I am working on exact differential equations and I just cannot seem to understand them and was hoping to have my method checked and please provide feedback on that.
$$(18xy^2 - \sin(x))dx + (8 + 18x^2y)dy = 0;\,\, y(0) = 1$$ I first calculated the crossed partial derivatives for both terms to check they were equal which I found they both came out to be 36xy.
Then I have said that $$\frac{\partial u}{\partial x} = 18xy^2 - sin(x)$$ and therefore $$u = 9y^2 x^2 + cos(x) + h(y)$$
Now I have said that $$\frac{\partial u}{\partial y} = 18yx^2 + h'(y) = 8 + 18yx^2$$ which I believe is not a problem.
This says that h'(y) = 8 which I have then integrated with respect to y and found h(y) = 8y + C
I believe this means my final result for the answer which in our notes is always denoted as $$u(x,y) = 36xy + 8y + C$$
So far I cannot understand how these exact differential equations work and have been told my answer is wrong. I am yet to find the exact solution using the initial condition as I am unsure how to do that as well because I am confused by a few things with these problems however if possible could we please discuss the method how to solve these types of DEs and check to see if my answer is correct.
Thank you very much,
Michael
• Micheal, I have started the edits so take a look at the first question and hopefully you can fix the other equations? All details can be found here. – Chinny84 Mar 3 '16 at 10:18
• Thanks I wasn't sure until now how to properly do the formatting thank you. – Michael Mar 3 '16 at 10:46
• No problem. I think the site experience will become even better for you now :) – Chinny84 Mar 3 '16 at 10:52
$$\left(18xy(x)^2-\sin(x)\right)\space\text{d}x+\left(8+18x^2y(x)\right)\space\text{d}y=0$$
Let $\text{P}(x,y)=18xy^2-\sin(x)$ and $\text{Q}(x,y)=18x^2y+8$.
This is an exact equation, because $\frac{\partial\text{P}(x,y)}{\partial y}=36xy=\frac{\partial\text{Q}(x,y)}{\partial x}$. | {
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Define $f(x,y)$ such that $\frac{\partial f(x,y)}{\partial x}=\text{P}(x,y)$ and $\frac{\partial f(x,y)}{\partial y}=\text{Q}(x,y)$.
Then, the solution will be given by $f(x,y)=\text{C}$, where $\text{C}$ is an arbitrary constant.
Integrate $\frac{\partial f(x,y)}{\partial x}$ with respect to $x$ in order to find $f(x,y)$:
$$f(x,y)=\int(18y^2x-\sin(x))\space\text{d}x=9y^2x^2+\cos(x)+g(y)$$
Differentiate $f(x,y)$ with respect to $y$ in order to find $g(y)$:
$$\frac{\partial f(x,y)}{\partial y}=\frac{\partial}{\partial y}(9y^2x^2+\cos(x)+g(y))=18yx^2+\frac{\text{d}g(y)}{\text{d}y}$$
Substitute into $\frac{\partial f(x,y)}{\partial y}=\text{Q}(x,y)$:
$$18yx^2+\frac{\text{d}g(y)}{\text{d}y}=18yx^2+8$$
Solve for $\frac{\text{d}g(y)}{\text{d}y}$:
$$\frac{\text{d}g(y)}{\text{d}y}=8\Longleftrightarrow$$ $$\int\frac{\text{d}g(y)}{\text{d}y}\space\text{d}y=\int8\space\text{d}y\Longleftrightarrow$$ $$g(y)=8y$$
Substitute $g(y)$ into $f(x,y)$:
$$f(x,y)=9y^2x^2+8y+\cos(x)$$
The solution is $f(x,y)=\text{C}$:
$$9y^2x^2+8y+\cos(x)=\text{C}$$
$$9y(x)^2x^2+8y(x)+\cos(x)=\text{C}\Longleftrightarrow$$ $$y(x)=\frac{-4\pm\sqrt{16+9\text{C}x^2-9x^2\cos(x)}}{9x^2}$$
We know that $y(0)=1$ we can't use the solution with the '-' sign before the square root:
$$1=\lim_{x\to0}\frac{-4+\sqrt{16+9\text{C}x^2-9x^2\cos(x)}}{9x^2}\Longleftrightarrow$$ $$1=\frac{\text{C}-9}{72}\Longleftrightarrow$$ $$\text{C}=81$$
So the solution is:
$$y(x)=\frac{-4+\sqrt{81x^2-9x^2\cos(x)+16}}{9x^2}$$
• Thank you for the super clear and detailed answer! I think I should easily be able to follow this process through and apply it to any other problems :) – Michael Mar 3 '16 at 11:23
• @Michael You're welcome. Yes you can do that!! – Jan Mar 3 '16 at 11:46
HIT:
You found $\quad u = 9y^2 x^2 + \cos(x) + h(y)\quad$ which is correct.
You also found $\quad h(y) = 8y + C\quad$ which is correct.
Then obviously the result is not : $u(x,y) = 36xy + 8y + C$ but is : $$u(x,y) = 9y^2 x^2 + \cos(x) +8y+C$$ | {
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Then you can solve $u(x,y)$=constant for $y(x)$. | {
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# DOF of Natural Cubic Spline
I am curious as to what the answer to the below question is? The question specifies a modeler has a cubic spline with knots at {10, 20, 30, 50}. They realize their model is overfitting at the ends of the distribution and wants to impose an additional constraint that the curve before the first knot and after the last knot are linear, then to calculate the DOF of the new model.
My thought process is that the answer should be 4 DOF for the new natural cubic spline model. Below is how I arrived at my answer:
The cubic spline has 8 DOF: 4+(4)(4)-(4)(3)=8 where the first 4 is for the intercept, X, X^2, X^3; then add 4 terms for earch knot; subtract out the 3 constraints at each knot to account for continuity, and the first and second derivatives to be zero.
Then to make the cubic spline with 4 knots a natural cubic spline, subtract (2)(2)=4 to get 4 DOF since the interval below the lowest knot and above the largest knot need to be made linear.
I’ve been told the correct answer is 6 DOF and I don’t understand how. I’ve been told it has something to do with interior knots vs boundary knots. Can someone shed some light as to how the answer could possibly be 6? Is my answer of 4 wrong?
Let there be $$K$$ knots $$\xi_1 < \dots < \xi_K$$. I'll use $$x_\min$$ for the smallest observed $$x$$ value, and $$x_\max$$ is analogous.
I think the answer depends on whether or not the $$\xi_j$$ are assumed to be interior knots, and if we want the spline to change in $$[x_\min, \xi_1]$$ and $$[x_\max, \xi_K]$$ or not (and this also depends on whether or not $$x_\min < \xi_1$$ and/or $$\xi_K < x_\max$$). | {
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If we only care about behavior between $$\xi_1$$ and $$\xi_K$$, then we could use the following truncated power basis: $$h_j(x) = x^j, j=0,1,2,3$$ and $$h_j(x) = (x-\xi_{j-3})_+^3, \hspace{5mm} j = 4,\dots, K+3$$ leading to $$K+4$$ DoF. This represents four DoF coming from the global cubic that we start with and one DoF being added for every knot we pass.
Restricting this to a natural spline means we'll constrain $$\beta_2 = \beta_3 = 0$$ which frees up 2 DoF, and we'll need the coefficients of $$x^3$$ and $$x^2$$ to be zero when every basis is active, so this further frees up two DoF meaning there are now $$K$$ DoF.
In interpolation problems and smoothing splines I think this is the right way to account for DoFs, because we have every point as a knot so there aren't separate boundary knots from $$x_\min$$ and $$x_\max$$.
But if we are always thinking of the $$\xi_j$$ as interior knots then we would have two more regions where the spline in changing. This is where setting $$\xi_0 = x_\min$$ and $$\xi_{K+1} = x_\max$$ makes sense and really we now have $$K+2$$ knots and therefore the full cubic spline has $$K+6$$ DoF and the natural spline has $$K+2$$.
In Figure 7.5 of Introduction to Statistical Learning (in the question you linked) we can tell that they are using the $$K+2$$ DoF version because the spline is nonlinear on all of $$[x_\min, x_\max]$$, rather than just on $$[\xi_1, \xi_3]$$.
To answer the exact question: I can't tell which formulation they want but I think $$6$$ is more likely correct.
For a spline that is not interpolating or smoothing, which you have here, I think having $$\xi_1$$ and $$\xi_K$$ as interior knots makes sense so the $$K+2 = 6$$ answer makes sense. | {
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But in the exact wording of the question they say "the curve before the first knot and after the last knot will be linear". If they mean out of the four given knots, then that means an answer of $$K=4$$ is correct, but I'm guessing they worded this poorly and are thinking of $$x_\min$$ and $$x_\max$$ as the first and last knots respectively, so even though at face value this suggests an answer of $$4$$, with this boundary knot inclusion then again the answer is $$6$$. | {
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• thank you for your response! the source this question is based on is James' Intro to Statistical Learning. I've linked below another thread which is the exact question/confusion I'm having between my answer and what was deemed as the correct answer. Is there a way to know in the question asked that they want us to assume implicit bounds, making the total knots 6? stats.stackexchange.com/questions/396889/… – kres901567708 Oct 28 '19 at 16:15
• @kres901567708 I've just completely rewritten. I think in this question they are likely tacitly assuming boundary knots that are not any of the four listed and that makes $6$ correct (I was not doing this when I went through it before; I usually use smoothing splines where that doesn't make sense so I assumed no separate boundary knots but i think that was the wrong call here), but I also think the question is not clearly worded and without knowing that they likely want boundary knots I don't think $6$ vs $4$ DoF is clear. – jld Oct 28 '19 at 18:52
• thank you! Would you say in your opinion that I have a valid case to argue the question is unclear and 4 DOF should be an acceptable answer, or is there something in the question that gives away we should be including xmin and xmax as boundary knots? – kres901567708 Oct 29 '19 at 19:43
• @kres901567708 I think it’s unclear but as I’ve thought more about it, with these splines (ie not smoothing or interpolating) it probably is pretty much always the case that the specified knots are interior and wouldn’t be boundary knots. Like when the $\xi$ are thought of as tuning parameters, which I think is the case with a standard regression spline, they’re often put at quantiles of the data which would make them interior – jld Oct 30 '19 at 2:23 | {
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# $\lim_{x\to0^{+}} x \ln x$ without l'Hopital's rule
I have a midterm coming up and on the past exams the hard question(s) usually involve some form of $\lim_{x\to0^{+}} x \ln x$. However, we're not allowed to use l'Hopital's rule, on this year's exam anyways.
So how can I evaluate said limit without l'Hopital's rule? I got somewhere with another approach, don't know if it's useful:
1. $\lim_{x\to0^{+}} x \ln x = \lim_{x\to0^{+}} x^2 \ln (x^2) = L$
2. $= (\lim_{x\to0^{+}} 2x)(\lim_{x\to0^{+}} x \ln x)$
3. $= 0 * L$
Then I just need to prove that L is finite/exists (which means it must be 0)
• Observing that $x\ln x=\ln(x^x),$ this question is effectively a duplicate of this other one. – Cameron Buie Oct 11 '13 at 21:57
• I do not believe this should be closed, since it describes an interesting aproach to the problem that is absent elsewhere. – André Nicolas Oct 11 '13 at 22:10
• Lovely boldfaced typo. Have a sticky p key. Also shift. – André Nicolas Oct 11 '13 at 22:22
• @Raekye : That is a very clever approach. – Stefan Smith Oct 12 '13 at 1:54
• @AndréNicolas: Couldn't the approach be posted to the other post? I think it would make sense. – Najib Idrissi Oct 12 '13 at 2:55
The idea you described is a very nice one. We fill in the details.
We consider, as in the OP, $x^2\ln(x^2)$, that is, $(2x)(x\ln x)$. If we can show that $x\ln x$ is bounded near $0$, it will follow by Squeezing that $\lim_{x\to 0} x^2\ln(x^2)=0$, and therefore $\lim_{t\to 0^+}t\ln t=0$.
Let $f(x)=x\ln x$. Then $f'(x)=1+\ln x$. It follows that $f(x)$ is decreasing in the interval $(0,e^{-1})$. It reaches a minimum value of $-e^{-1}$ at $x=e^{-1}$.
Since $f(x)$ is negative in our interval, we have $|x\ln x|\le e^{-1}$ in the interval, and we have shown boundedness. | {
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• Ah, that makes sense. I thought I had to use the derivative to show "which direction the function is going" but couldn't spell it out. Thank you very much! – Raekye Oct 11 '13 at 22:12
• A deleted answer gives another way to show that $f(x)$ is bounded: $$0>x\ln(x)=x\int_1^x\frac1t\,dt\geq x(\frac1x(x-1))=x-1>-1$$ when $0<x<1$, so $|f(x)|\leq 2$ when $0<x<1$. – Jonas Meyer Oct 11 '13 at 22:15
• (Now that $f(x)=x\ln(x)$ instead of $2x\ln(x)$, the last line of my comment should say "$|f(x)|\leq 1$".) – Jonas Meyer Oct 11 '13 at 22:57
• Sorry about the little change, in checking for my usual typos I thought there was no point in dragging the $2$ around. – André Nicolas Oct 11 '13 at 22:59
• @Raekye: Please note that in my opinion the approach of user@17762 is "better." My answer was an exercise in pushing through your clever idea. – André Nicolas Oct 12 '13 at 0:27
Let $x=e^{-t}$ and note that as $x \to 0^+$, we have $t \to \infty$. Hence, $$L = \lim_{x \to 0} x \ln(x) = \lim_{t \to \infty} -te^{-t} = -\lim_{t \to \infty} \dfrac{t}{e^t}$$ Now recall that $e^t \geq \dfrac{t^2}2$, because $$e^t =\sum_{k=0}^{\infty}\frac{t^k}{k!} \geq \frac{t^2}{2}$$ Hence, we have $$\lim_{t \to \infty} \dfrac{t}{e^t} \leq \lim_{t \to \infty} \dfrac2t = 0$$ This gives us $L=0$.
• Ah this makes sense too. I selected Andre's answer though because he answered earlier. Thanks for your input though! – Raekye Oct 11 '13 at 22:17
• Can you explain why e^t >= t^2/2 ? I’m not good at math. – plhn Mar 15 '17 at 12:10 | {
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# How would I solve $x^2-4x=y^2-4y$ without knowing the answer beforehand?
The equation is $x^2-4x=y^2-4y$ in the case where $x\ne y$. The answer is $x+y=4$.
I can start from $x+y=4$ and create the equation very easily, and I can substitute $x+4=y$ into the equation and show both sides are equal easily. I just don't get how I would find the answer if I didn't know it before hand, and all I had was the equation? Any advice?
• I just realized, assuming that x and y aren't equal, then x must equal +/- (y-4) and y must equal +/- (x-4), otherwise there's no way x(x-4)=y (y-4). – Hockeyfan19 Nov 27 '17 at 2:11
• Be weary of the line of reasoning; it’s fallacious. Just because $ab = cd$ and $a \neq b$, doesn’t mean $a = c$. That trick only works when we have something like $ab = 0$, whence we can conclude $a$ or $b$ is zero. See the most recent answer to this question for the right application of this idea. It’s subtle. – Bob Krueger Nov 28 '17 at 1:49
\begin{align} x^2 - 4x &= y^2 - 4y \\ x^2 - y^2 &= 4x - 4y \\ (x-y)(x+y) &= 4(x-y) \\ x+y &= 4\end{align} where dividing by $x-y$ is allowed since $x \neq y$.
• It seems so obvious now, thanks for showing the way! I wasn't seeing it at all. – Hockeyfan19 Nov 26 '17 at 21:17
As an alternative, the solution that struck me first was completing the square in both $x$ and $y$. This is common when dealing with quadratics, especially once there are no $xy$ cross-terms. $$x^2 - 4x = y^2 - 4y$$ $$x^2 - 4x + 4 = y^2 - 4y + 4$$ $$(x-2)^2 = (y-2)^2$$ This means that either $x-2 = y-2$ or $x-2 = -(y-2)$, which means either $x = y$ or $x+y = 4$, as desired. | {
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• I actually saw that I could complete the square when I was working on it. I missed that it would progress towards the solution though. Thanks for the alternative approach! – Hockeyfan19 Nov 27 '17 at 1:38
• You're welcome. Look into quadratic forms and conic sections for more general problems of this type. – Bob Krueger Nov 27 '17 at 1:44
• This is the way I first went, so +1 ;) – Lamar Latrell Nov 27 '17 at 4:37
Let $c$ be the common value of $x^2-4x$ and $y^2-4y$. Then $x$ and $y$ are both roots of the polynomial $t^2-4t-c$. Since we are assuming $x$ and $y$ are distinct, they are all of the roots, so $t^2-4t-c$ factors as $(t-x)(t-y)$. Since $(t-x)(t-y)$ expands to $t^2-(x+y)t+xy$, comparing the coefficients of $t$ gives $x+y=4$.
(Conversely, if $x+y=4$, then since $x$ and $y$ are both roots of $(t-x)(t-y)=t^2-(x+y)t+xy=t^2-4t+xy$, $x^2-4x$ and $y^2-4y$ are both equal to $-xy$.)
• Very interesting approach, but where did the -4t come from in your polynomial? – Hockeyfan19 Nov 27 '17 at 2:04
• I'm not sure what you mean. Since $x^2-4x=c$, $x^2-4x-c=0$, and similarly for $y$. So $x$ and $y$ are roots of $t^2-4t-c$. – Eric Wofsey Nov 27 '17 at 2:32
• Okay so it comes from the original equation, I see now – Hockeyfan19 Nov 27 '17 at 2:43
• And you're using the t for clarities sake I think? – Hockeyfan19 Nov 27 '17 at 2:45
Write $x+y=d$. Then we have $y=d-x$ and so: $$(d-x)^2-4(d-x) = x^2-4x$$ so $$d^2-2dx-4d =-8x$$thus $$d(d-2x)-4(d-2x)=0$$ so $$(d-2x)(d-4)=0$$ If $d=2x$ we get $x=y$ which is impossible. So $d=4$ and we have $x+y=4$. | {
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941 views
An $n \times n$ array $v$ is defined as follows:
$v\left[i,j\right] = i - j$ for all $i, j, i \leq n, 1 \leq j \leq n$
The sum of the elements of the array $v$ is
1. $0$
2. $n-1$
3. $n^2 - 3n +2$
4. $n^2 \frac{\left(n+1\right)}{2}$
edited | 941 views
square matrix whose transpose is its negation; that is, it satisfies the condition −A = AT. If the entry in the i th row and j th column is aij, i.e. A = (aij) then the skew symmetric condition is aij = −aji
The sum of the $i^{th}$ row and $i^{th}$ column is $0$ as shown below. Since, the numbers of rows $=$ no. of columns, the total sum will be $0$.
$0$ $-1$ $-2$ $-3$ $-4$ $1$ $0$ $-1$ $-2$ $-3$ $2$ $1$ $0$ $-1$ $-2$ $3$ $2$ $1$ $0$ $-1$ $4$ $3$ $2$ $1$ $0$
edited by
this matrix is also a skew symmetric matrix, so definitely it's sum will be 0.
Let there are total N rows . You will find ∑ of elements of row i + ∑ of elements row (N-i+1) = 0.
So if N is even then
row 1 + row N =0
row 2 + row (N-1) =0
row 3 + row (N-2)=0
similarly row (N/2) + row (N/2+1) =0. (So total sum is 0)
But if N is odd then row ((N+1)/2) will have no corresponding rows BUT Ithe summation of elements of this row is 0 .
So for N = even or Odd , the sum of element is 0 .
If you look at code carefully it is very clear matrix getting defined is skew symmetric matrix.
Sum of all elements in skew symmetric matrix is 0.
i think we can easily get it without drawing matrix
as expression given v[i, j] = i - j
suppose i1-j1=k1 {for a particular index } so its opposit index shows j1-i1=-k1, for example if v[1,2]=x then v[2,1]=-x}
so we have cases here
1. for all i>j v[i, j] = i - j and jut for opposit indexof v[i, j] = i - j , v[j, i] = j - i =-(i-j) :: note this is for non diagonal elements
2. for all i=j v[i, j] = i - j=0 :: note this is for diagonal elaments
so total sum will be zero ucan easily get it by seeing above cases
ans is option (a) i.e 0
solution: suppose if n=3
then i<=3 and 1<=j<=3 | {
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ans is option (a) i.e 0
solution: suppose if n=3
then i<=3 and 1<=j<=3
i.e i=1,2,3, and j=1,2,3
=> v={i-j}=(0,-1,-2,1,0,-1,2,1,0}
sum of elements in v=0
edited by
@prakash What if i take, i=0,1,2. and j=1,2,3...? then it will be skew symmetric matrix..?
+1 vote
sum of all elements =n*( $\sum_{i=1}^{n}i-\sum_{j=1}^{n}j )= 0$ | {
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# Prove $(A \cap B) \cup (A \cap B')= A$ using Set Identities
I recently started a Discrete Mathematics course in college and I am having some difficulties with one of the homework questions. I need to learn this, so please guide me through at least two steps to get the ball rolling.
The question reads: Show that if $A$ and $B$ are sets, then: $(A \cap B) \cup (A \cap B')=A$
We are supposed to use set identities. I had a question prior, but it was simple: $(A \cap B \cap C)' = A'\cup B' \cup C'$ - Which would be one of De Morgan's laws.
I am at a loss. I have been reading the textbook and tried looking up some videos, but I am not sure exactly where to start. Any help you can provide, will be greatly appreciated!
Thanks, Kei | {
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Thanks, Kei
• I'm not sure what you're asking. Can you clarify what specifically you're confused with? Also I highly reccomend taking a look at "How to Prove It" as a supplementary text. It's a fantastic introduction to this subject. – lordoftheshadows Jan 23 '17 at 18:58
• Use distributivity! LHS is equal to $A \cap (B \cup B')$ – Crostul Jan 23 '17 at 18:58
• @lordoftheshadows: I wasn't sure on the first two steps to prove that the left side is indeed equal to the right side? If I am explaining it right. Our professor showed us an example in class such as this: (A ∪ (BnC)) = (C ∪ B) ∩ A = A ∩ (B ∩ C) - De Morgan Law = A ∩ (B u C) De Morgan Law = (B u C) ∩ A = Commutative Law = (C u B) ∩ A = Commutative Law So I am looking for the first two steps, I see that @Crostul had posted the first step. Thank you! I am going to see if I can take it from there. I do really appreciate the help! – Kei U. Jan 23 '17 at 19:00
• I have edited the question for formatting and some grammatical things for you, @KeiU. We will all see it when it is approved. Good luck! – The Count Jan 23 '17 at 19:03
• Thank you @TheCount! I appreciate that, I really should have looked over the question better, sorry! @Crostul: Do I distribute (A ∩ B) U (A ∩ B`) together? – Kei U. Jan 23 '17 at 19:05
The idea is to achieve get close $B$ and $B^c$. Then we use distributive property: $(A\cap B)\cup(A\cap B^c)=A\cap (B\cup B^c)=A\cap X=A$, with $X$ the universe | {
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• Hello, I really appreciate your answer you provided! I think where I was confused was, being able to reverse an identity. I was using the set identity, though not using it in reverse if that makes sense. This is what I did: I used Distributive law, Absorption law, and then Identity law. – Kei U. Jan 23 '17 at 19:24
• It would not let me edit the above comment. The last sentence should be a question. I wanted to ensure that I did use the right laws above, as you shown in your post Julio Maldonado Henriquez. Thank you! – Kei U. Jan 23 '17 at 19:33
Consider an element of $A$ - either it is in $B$, or it isn't, and thus is in the complement of $B$. Thus $A \subset (A \cap B)$ $\cup$ $(A \cap B')$. Now, try to argue on your own that the reverse "inclusion" holds: that we have $A \supset (A \cap B)$ $\cup$ $(A \cap B')$.
You can use identities such as $(A\cap B) \cup (A \cap C) = A \cap (B \cup C)$ to get $(A \cap B) \cup (A \cap B') = A \cap (B \cup B') = A \cap U = A$.
But I prefer to think of what it is saying. $A \cap B$ means "everything in A and in B" and $A \cap B'$ means "everything that is in A that is not in B" and $(A\cap B) \cup (A \cap B')$ means "every thing that is in A and B combined with everything that is not in B". Is there a logical reason that "everything in A and B combined with everything in A and not in B" would be "A"?
Well, I hope it should be obvious. Everything in A is either in B or not in B so combining the items of A that are not in B with those that are should give you all the items in A.
So the best way to express that idea directly would be:
$A = A \cap U = A \cap (B \cup B') = (A \cap B) \cup (A\cap B')$. Or if that's a little too abstract, I rather like to do an element by element proof: | {
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Let $x \in A$ either $x \in B$ or $x \in B'$. If $x \in B$ then $x \in A \cap B$. If $x \in B'$ then $x \in A \cap B'$. Either way $x \in A \cap B$ or $x \in A\cap B'$ so $x \in (A \cap B) \cup (A\cap B)$. So $A \subseteq (A\cap B) \cup (A \cap B)$. Likewise if $y \in (A \cap B) \cup (A\cap B)$ then either $y \in (A \cap B) \subset A$ or $y \in (A\cap B') \subset A$. Either way, $y \in A$ so $(A\cap B)\cup (A\cap B') \subseteq A$.
$A \subseteq (A\cap B) \cup (A \cap B)$ and $(A\cap B)\cup (A\cap B') \subseteq A$, so (A\cap B)\cup (A\cap B') = A$. A fourth way is the big guns. Let$x \in U$now one of four things might happen: 1)$x \in A$and$x \in B$. Then$x \in A$and$x \in A\cap B$and$x \in (A \cap B) \cup (A \cap B')$. 2)$x \in A$and$x \not \in B$. Then$x \in A$and$x \in B'$and$x \in A \cap B'$and$x \in (A \cap B) \cup (A \cap B')$3)$x \not \in A$and$x \in B$. Then$x \not \in A$and$x \not \in A \cap B$and$x \not \in A \cap B'$so$x \not \in (A \cap B) \cup (A \cap B')$. 4)$x \not \in A$and$x \not \in B$. Then$x \not \in A$and$x \not \in A \cap B$and$x \not \in A \cap B'$so$x \not \in (A \cap B) \cup (A \cap B')$. Looking at the four cases we see$x \in A \iff x \in (A \cap B) \cup (A \cap B')$. Thus$A$and$(A \cap B) \cup (A \cap B')$have precisely the same elements and neither has any element the other doesn't. In other words,$A = (A \cap B) \cup (A \cap B')\$. | {
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# Definition of random
Suppose that you has to guess given a set of numbers
• If they are random.
• The mathematical expectation
Is there a definition of randomness that allow this prove/test?
Is even possible? if so: How many value would be enough ?
Example:
If numbers come from a coin experiment, results could be coded as 0 or 1,
000100010110100011.....
Then how many "bits" are enough to "test" variable randomness?
According to the law of large numbers, the average of the results (adding the results and dividing by the number of trials) should become closer and closer to the expected value as more trials are performed.
But if we don't know where those numbers come from then we don't know what to expect!
What are the law of large numbers hypothesis?
Does the definition of random cointain a reference to average and to the expected value?
It is deeply confusing to me,
thanks in advance for any information | {
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It is deeply confusing to me,
thanks in advance for any information
• Random means that knowing the first $n$ terms of the sequence you cannot say anything about the $n+1$'th term at all besides that all possible combinations for that term to happen have the same probability. – Listing Jun 22 '11 at 19:38
• If you search on testing random number generators you can find a lot of information. It is a large and complicated topic. – Ross Millikan Jun 22 '11 at 20:14
• This is complicated topic indeed. For one thing: would you say that the digits of PI are random? – leonbloy Jun 22 '11 at 20:22
• @Hernan - You might want to look up Kolmogorov complexity, which looks at randomness from a computational viewpoint. – Unreasonable Sin Jun 22 '11 at 20:44
• @Ross Millikan thanks, as I see tests expects (or are aimed to) a certain distribution, it's like having a metadata about the data, as if randomness were not in a set of numbers, but in its interpretation @leonbloy it depends, if you already know numbers come from "PI" or from "a coin experiment" then that's not what I am asking about, because you have extra meta information about the data, the question is about the factibility of judge a set of data as "random", computable/non-computable random, is further discussion, but first I would like to know if e a definition of random is even possible – Hernán Eche Jun 22 '11 at 20:52
You may like to understand the randomness of you sequence heuristically as follows. This may help you to get more intuition.
Write favorite binary sequence you would like to test for randomness in a file and try to compress it (e.g. with zip, etc...).
Then take the ratio between the size of the compressed file and the size of the original one. If the ratio is close to 1, it means the sequence is quite random, and if the ratio is close to zero then it means that the sequence is not very random. | {
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