instruction stringlengths 12 30k |
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Say I have a triangle with vertices `(0,0), (2,4), (4,0)` that I want to rotate along the origin. Rotation by multiples of 90 is simple. However, I want to rotate by something a bit more complicated, such as 54 degrees. How do I figure out where the vertices would be then? |
I propose the following lemma and its proof. It is related to row-null and column-null matrices - i.e. matrices whose rows and columns both sum to zero. Could you please give your opinion on the plausibility of the lemma, and the validity of the proof?
**Lemma:** Let $Z\in\text{GL}(n,\mathbb{R})$ be a general $n\tim... |
If $\rm\;A\;$ is a linear map on a finite dimensional vector space $\rm\: V\;$ over a field $\rm\;K\:,\:$ then trivially (see below) by finite dimensionality there is a nonzero polynomial $\rm\;p(x)\in K[x]\;$ such that $\rm\;p(A) = 0\;$. We may assume $\rm\:p(0) \ne 0\;$ by canceling any factors of $\rm\:A\;$ from $\r... |
I propose the following lemma and its proof. It is related to row-null and column-null matrices - i.e. matrices whose rows and columns both sum to zero. Could you please give your opinion on the plausibility of the lemma, and the validity of the proof?
**Lemma:** Let $Z\in\text{GL}(n,\mathbb{R})$ be a general $n\tim... |
Back in the day, before approximation methods like splines became vogue in my line of work, one way of computing the area under an empirically drawn curve was to painstakingly sketch it on a piece of graphing paper (usually with the assistance of a French curve) along with the axes, **painstakingly** cut along the curv... |
A 1-1 function is called injective. What is an n-1 function called ?
I'm thinking about homomorphisms. So perhaps homojective ?
Onto is surjective. 1-1 and onto is bijective.
What about n-1 and onto ? Projective ? Polyjective ?
I think n-m and onto should be hyperjective as in hypergroups.
|
A 1-1 function is called injective. What is an n-1 function called ? |
What are some interesting applications of the concept of homomorphism?
Example: If there is a homorphism from a ring R to a ring r then a solution to a polynomial equation in R gives rise to a solution in r. e.g. if f:R->r and X^2+Y^2=0 then f(X^2+Y^2)=f(0), f(X^2)+f(Y^2)=0, f(X)^2+f(Y)^2=0, x^2+y^2=0 |
I was droodling a bit and a given moment I drew the following construction:
![alt text][1]
[1]: https://i.stack.imgur.com/XwnQg.png
It appears that the three blue intersections are collinear (red line), no matter how I draw the construction lines. If this is always true, I assume that this a know fact ... |
What are some interesting applications of the concept of homomorphism?
Example: If there is a homorphism from a ring $R$ to a ring $r$ then a solution to a polynomial equation in $R$ gives rise to a solution in $r$. e.g. if $f:R \rightarrow r$ and $X^2+Y^2=0$ then $f(X^2+Y^2)=f(0), f(X^2)+f(Y^2)=0, f(X)^2+f(Y)^2=0, ... |
In my opinion, this is not really a math question. Which procedure is correct depends what you're going to do with your calculation.
As a matter of definition, the integral indeed measures the signed area (positive area minus negative area), as you suggest. So your approach is computing an approximation to the def... |
First there was arithmetic with numerical calculations (i.e. one unknown on one side of an equation). Then algebra with manipulations of variables (many unknowns anywhere in an equation). Then systems are studied that differ from ordinary arithmetic but share some of the same properties (equations where the unknowns re... |
What's more general than category theory? |
I've got a path that is made up of multiple points - i.e. 0,0 0,50 50,50 70,20
If I just draw this line on the screen it looks quite harsh as it puts a sharp angle at the joining of each point.
Hence I was wondering what the bezier curves algorithm/method would look like which I could call that automatically c... |
I have a set of RGB colour values detected by a camera $C_{i_{RGB}}$ which are to be described by the following:
$C_{i_{RGB}} = X F_{i_{rgb}}$
where $F_{i_{rgb}}$ is the component incident light at the detector, where the components $rgb$ are fixed at narrow bands and $X$, an $(n\times 3)$ matrix, is a descriptio... |
I learned that the volume of a sphere is $\frac{4}{3}\pi r^3$, but why? The $\pi$ kind of makes sense because its round like a circle, and the $r^3$ because it's 3-D, but $\frac{4}{3}$ is so random! How could somebody guess something like this for the formula? |
Can anyone give me an example of a vector space $V$ such that there is no norm which is complete on $V$? |
Define $R(0)=0=\emptyset, R(n+1)=P(R(n))$ and $R(\omega) = \cup_{n < \omega} R(n)$. Thus $R(\omega)$ is the set of all sets, which are build out of finitely many braces and $0$.
Consider the following relation $E$ on $\omega$: If and only if the $n$th number in the binary representation of $m$ is $1$, then $n E m$.... |
In Exercise 5 (f) of Angus Taylor's Advanced calculus (p. 659) one is asked to find the value of the following integral if convergent:
$I:=\displaystyle\underset{R}{\int \int \int }\dfrac{x^{2}y^{2}z^{2}}{r^{17/2}}dV$
where $R$ is the unit sphere $x^{2}+y^{2}+z^{2}\leq 1$ and $r^{2}=x^{2}+y^{2}+z^{2}$.
Observi... |
If $\rm\:B\:$ is a linear map on a finite dimensional vector space $\rm\: V\:$ over field $\rm\:K\:,\:$ then trivially (cf. note below) by finite dimensionality there is a nonzero polynomial $\rm\:p(x)\in K[x]\;$ such that $\rm\:p(B) = 0\:$. We may assume $\rm\:p(0) \ne 0\;$ by canceling any factors of $\rm\:B\;$ from ... |
This is such a basic question I'm sure, but I've been trying to find a robust solution to it for a while.
Lets say I have the numbers 0-100 in series.
As the number increases, I want another number say 1000, to decrease. so that when you got to 100, it was at 0
0 = 1000
50 = 500
100 = 0
What is this k... |
**It follows simply by the pigeonhole principle**. Here is an excerpt from my [old post][1]:
Recall (below) $\rm\; AB \:\:=\:\:\: I \:\;\Rightarrow\; BA \:\:=\:\: I\;\;\:$ easily reduces to:
**THEOREM** $\;$ $\rm\;\;B\;$ injective $\rm\;\Rightarrow\:\: B\;$ surjective, $\:$ for linear $\rm\:B\:$ on finite ... |
My nephew was folding laundry, and turning the occasional shirt right-side-out. I showed him a "trick" where I turned it right-side-out by pulling the whole thing through a sleeve instead of the bottom or collar of the shirt. He thought it was really cool (kids are easily amused, and so am I).
So he learned that yo... |
Ok so I'm a programmer, I'm not a mathematician, I've got a minor in math but I didn't even do particularly well at it so please bear with some possibly really stupid thoughts. Just please try to explain to me why what I'm asking is stupid so that I don't keep making a fool of myself.
I recently found myself thinki... |
Is there a Mathematical system which is stochastic at its core? |
My nephew was folding laundry, and turning the occasional shirt right-side-out. I showed him a "trick" where I turned it right-side-out by pulling the whole thing through a sleeve instead of the bottom or collar of the shirt. He thought it was really cool (kids are easily amused, and so am I).
So he learned that yo... |
I hope this question is suitable for the site.
I recently had to work with Mathematica, and the experience was, to put it kindly, unpleasing. I do not have much experience with similar programs, but I remember not liking much Matlab or Maple either. The result is that I am a mathematician who likes programming, but ... |
As you mention “series” in the question I was wondering if something as simple as the term “Arithmetic Progression” is what you're after.
Your first arithmetic progression being $$a_n=n$$ for n=0,1,...,100 and your second being
$$b_n=1000-10n=1000-10a_n.$$
Given your chosen name, I was wondering if your intended... |
I propose the following lemma and its proof. It is related to row-null and column-null matrices - i.e. matrices whose rows and columns both sum to zero. Could you please give your opinion on the plausibility of the lemma, and the validity of the proof?
**Lemma:** Let $Z\in\text{GL}(n,\mathbb{R})$ be a general $n\tim... |
I was trying to obtain this result:
integrate e^|x| dx
The result is `integral e^(|x|) dx = 1/2 e^(-x) ((e^x-1)^2 sgn(x)-2 e^x+e^(2 x)-1)+constant` which is fine but I would like to see the 2part solution for `x>0` and `x<0`, I know you can see that because just after before the final output I can see it but ... |
wolframalpha, how to format the output? |
I have been thinking a little bit about the binary decimal expansion of inverse prime numbers; and I have a few questions.
I found [this neat table](http://brad.chalfan.net/binq/qtob.html) which lists the binary expansion of many fractions, and I was trying to find some patterns.
Here are my questions, for brevit... |
I have been thinking a little bit about the binary decimal expansion of reciprocal prime numbers; and I have a few questions.
I found [this neat table](http://brad.chalfan.net/binq/qtob.html) which lists the binary expansion of many fractions, and I was trying to find some patterns.
Here are my questions, for bre... |
Does it exist a free interactive geometry software, like GeoGebra, which works for 3D geometry? I would be able to draw spheres, great circles, and so on. |
I was going to ask this on Stack Overflow, but finally decided this was more math than programming. I may still turn out to be wrong about that, but...
Given a number represented in binary, it's fairly easy to derive a decimal representation using integer division and remainders to extract digits in reverse order. Y... |
How to calculate the number of decimal digits for a binary number? |
Assume you have a triangle in the plane defined by its three vertices at $(x_0,y_0)$, $(x_1,y_1)$, and $(x_2,y_2)$. Is there a general expression for the moments of the triangle, where the moments are defined as averages by the following expression:
$$\langle x^n y^m \rangle = \frac{1}{A}\int_\triangle x^n y^m \\, d... |
Assume you have a triangle in the plane defined by its three vertices at $(x_0,y_0)$, $(x_1,y_1)$, and $(x_2,y_2)$. Is there a general expression for the moments of the triangle, where the moments are defined as averages by the following expression:
$$\langle x^n y^m \rangle = \frac{1}{A}\int_\triangle x^n y^m \\, d... |
Your question is undoubtedly philosophical in nature: it is predicated on the premise "it's solipsism to talk about anything that cannot be put to use in predicting an outcome". But [solipsism][1] is a philosophical term (and more fundamentally, an inherently philosophical -- as opposed to mathematical or scientific -... |
Is a prime number in the decimal system still a prime when converted to a different base?
|
Is a prime number still a prime when in a different base? |
We know that a group $G$ cannot be written as the set theorectic union of two of its proper subgroups. Also $G$ can be written as the union of 3 of its proper subgroups if and only if $G$ has a homomorphic image, a non-cyclic group of order 4?
In this paper http://www.jstor.org/stable/2695649 by M.Bhargava, it is sh... |
A mall has sweep stakes where they give you a coupon for every X dollars you spend (amount is not important). You fill the coupon with your contact details and put it in the box. Every two weeks, they draw 10 coupons, give their owners prizes, and discard the rest of the coupons.
Now, thanks to some very big purchas... |
Probability of winning sweep stakes, put all coupons in one box or split them between boxes? |
Taken from : [answers.yahoo.com/question/index?qid=20100904034737AAfGkYF][1]
--------
A mall has sweep stakes where they give you a coupon for every X dollars you spend (amount is not important). You fill the coupon with your contact details and put it in the box. Every two weeks, they draw 10 coupons, give their... |
We know that a group $G$ cannot be written as the set theoretic union of two of its proper subgroups. Also $G$ can be written as the union of 3 of its proper subgroups if and only if $G$ has a homomorphic image, a non-cyclic group of order 4?
In this paper http://www.jstor.org/stable/2695649 by M.Bhargava, it is sho... |
One of my math professors gave me the following challenge. It isn't graded, it's just for fun.
Consider the function:
$f_n(x)=x+3^3x^3+5^3x^5+...+(2n-1)^3x^{2n-1}, x \epsilon (0, 1)$
I want to find which of the following functions $f_n$ is getting close to as $n$ gets larger:
$a)\frac{x(x+1)(x^4+22x^2+1)}{(... |
One of my math professors gave me the following challenge. It isn't graded, it's just for fun.
Consider the function:
$f_n(x)=x+3^3x^3+5^3x^5+...+(2n-1)^3x^{2n-1}, x \epsilon (0, 1)$
I want to find which of the following functions $f_n$ is getting close to as $n$ gets larger:
$ \displaystyle a)\frac{x(x+1)(... |
We know that a group $G$ cannot be written as the set theoretic union of two of its proper subgroups. Also $G$ can be written as the union of 3 of its proper subgroups if and only if $G$ has a homomorphic image, a non-cyclic group of order 4?
In this paper http://www.jstor.org/stable/2695649 by M.Bhargava, it is sho... |
I'm doing some old exams and found following question:
> Set $S={{1,2,3}}$ is given. Provide an
> example of binary operation in set S,
> binary relation in set S and a
> function $f:S\rightarrow R$.
So, I'm thinking about $+_4$ as operation, but wouldn't it be a relation too? I can take ordered pair from $S$ ... |
Your question is undoubtedly philosophical in nature: it is predicated on the premise "it's solipsism to talk about anything that cannot be put to use in predicting an outcome". But [solipsism][1] is a philosophical term (and more fundamentally, an inherently philosophical -- as opposed to mathematical or scientific -... |
I was trying to obtain this result:
integrate e^|x| dx
[The result](http://www.wolframalpha.com/input/?i=integrate+e%5e|x|+dx) is
$$ \int e^{|x|} dx = \frac12 e^{-x} \left( (e^x-1)^2 \operatorname{sgn}(x) -2e^x +e^{2x} - 1 \right) \color{gray}{\\,+ \text{constant}} $$
which is fine but I would like to ... |
Wolfram Alpha: How to show functions with abs() and sgn() as a piecewise function? |
Consider the Banach space $\ell^1(\mathbb N)$.
The sequence $\(e_n\)_\{n\in\mathbb N\}$ feels like a kind of basis because every element $a\in\ell^1(\mathbb N)$ can be written as an absolutely convergent infinite linear combination $\sum_{n\in\mathbb N}a(n)e_n$ in a unique way.
(Here $e_n$ denotes the vector whos... |
Is there a notion of basis for Banach spaces? |
Consider the Banach space $\ell^1(\mathbb N)$.
The sequence $\(e_n\)_{n \in \mathbb N}$ feels like a kind of basis because every element $a\in\ell^1(\mathbb N)$ can be written as an absolutely convergent infinite linear combination $\sum_{n\in\mathbb N}a(n)e_n$ in a unique way.
(Here $e_n$ denotes the vector whos... |
Consider the Banach space $\ell^1(\mathbb N)$.
The sequence $(e_n)\_{n\in\mathbb N}$ feels like a kind of basis because every element $a\in\ell^1(\mathbb N)$ can be written as an absolutely convergent infinite linear combination $\sum_{n\in\mathbb N}a(n)e_n$ in a unique way.
(Here $e_n$ denotes the vector whose $... |
Let $X = \mathcal{C}([0,1],\mathbb{R})$, be the ring of all continuous valued functions $f:[0,1] \to \mathbb{R}$. For $x \in [0,1]$, let $M_{x} = \\{ f \in M \ | \ f(x)=0\\}$. One can show by using compactness of $[0,1]$ that every maximal ideal is of this form.
**Extending the Question to Entire functions**: Let ... |
I try to solve for the specific function
$f(x) = \frac{2-2a}{x-1} \int_0^{x-1} f(y) dy + af(x-1)$
It looks similar to the function used to find the [Renyi's parking constant][1] because it came out from a simple generalization of that problem.
The skill I have gained in my differential class can't even solve
$f... |
How to solve DE that relate values of derivatives at different points? |
Here is a list of some applications of the exponential function.
1) The exponential mapping in Lie theory.
I put this first because in my mind, this is the most important application. My intuition tells me that this must be the most fundamental, or deep, way of thinking about the exponential function. I have ... |
Is there a (deep) relationship between these various applications of the exponential function? |
Here is a list of some applications of the exponential function.
1) The exponential mapping in Lie theory.
I put this first because my intuition tells me that this must be the most fundamental, or deep, way of thinking about the exponential function. I have often been misled by my intuition however, and the ma... |
Alas, the accepted answer is misleading (and arguably incorrect). The reason that primality (or any other purely arithmetic property) is preserved in radix representation is simply that such representation faithfully preserves all of the arithmetic operations on integers. More precisely, $\:$ if $\rm\;n\to r(n)\;$ is t... |
Alas, the accepted answer is misleading (and arguably incorrect). The reason that primality (or any other purely arithmetic property) is preserved in radix representation is simply that such representation faithfully preserves all of the arithmetic operations on integers. More precisely, $\:$ if $\rm\;n\to r(n)\;$ is t... |
Alas, the accepted answer is misleading (and arguably incorrect). The reason that primality (or any other purely arithmetic property) is preserved in radix representation is simply that such representation faithfully preserves all of the arithmetic operations on integers. More precisely, $\:$ if $\rm\;n\to r(n)\;$ is t... |
Let p_i denote the i-th prime number. (Thus (p1,p2,p3,...) = (2,3,5,...).) Find the smallest positive integer k such that the product n = p1 p2 · · · pk satis- fies σ(n) > 3n. Is there any positive integer m < n satisfying σ(m) > 3m? |
Let $p_i$ denote the $i^{th}$ prime number.
(Thus $(p_1,p_2,p_3,...) = (2,3,5,...)$.)
Find the smallest positive integer $k$ such that the product $n = p_1 \cdot p_2 \cdots p_k$ satisfies $\sigma(n) > 3n$.
Is there any positive integer $m < n$ satisfying $\sigma(m) > 3m$? |
Let $p_i$ denote the $i^{th}$ prime number.
Find the smallest positive integer $k$ such that the product $n = p_1 \cdot p_2 \cdots p_k$ satisfies $\sigma(n) > 3n$.
Is there any positive integer $m < n$ satisfying $\sigma(m) > 3m$? |
Let $p_i$ denote the $i^{th}$ prime number.
Find the smallest positive integer $k$ such that the product $n = p_1 \cdot p_2 \cdots p_k$ satisfies $\sigma(n) > 3n$.
Is there any positive integer $m < n$ satisfying $\sigma(m) > 3m$?
---
**Taken From** (Rosen), at the end of the chapter under challenge p... |
I've recently started making a small fractal app in Javascript using the famous Mandelbrot bulb (z = z^2 + c). I've been trying to find the best method of coloring the points on the complex plane, and I've come across some very interesting ideas:
http://linas.org/art-gallery/escape/escape.html
http://en.wikibooks... |
I am working through Theorem 6 in Chapter 8 of Hoffman and Kunze. (The section is <em>Linear Functionals and Adjoints</em>.)
The statement is <blockquote> Let $V$ be a finite-dimensional inner product space, and $f$ a linear functional on $V$. Then there exists a unique vector $\beta$ in $V$ such that $f(\alpha)... |
I'm trying to learn predicate logic.
And I'm looking for some good resources on it:
I've seen that I learn better when I can program
So I was wondering if there was a 'predicate logic' programning language?
Maybe an interactive tutorial?
Maybe lots of examples, something like: Predicate Logic by Example?... |
What are good resources for learning predicate logic / predicate predicate calculus? |
I am working through Theorem 6 in Chapter 8 of Hoffman and Kunze. (The section is <em>Linear Functionals and Adjoints</em>.)
The statement is <blockquote> Let $V$ be a finite-dimensional inner product space, and $f$ a linear functional on $V$. Then there exists a unique vector $\beta$ in $V$ such that $f(\alpha)... |
This question may smell like my older question http://math.stackexchange.com/questions/1778/a-probability-game, but this time my intentions are different.
The above problem came from my friend. Or so I thought, it really came from me incorrectly remembering his problem. The problem he actually gave me was:
Suppos... |
I'm currently interested in the following result:
Let $f: X \to Y$ be a fpqc morphism of schemes. Then there is an equivalence of categories between quasi-coherent sheaves on $Y$ and "descent data" on $X$. Namely, the second category consists of quasi-coherent sheaves $\mathcal{F}$ on $X$ with an isomorphism $p_{1}... |
Do cokernels in RingSpc automatically lead to descent? |
I'm currently interested in the following result:
Let $f: X \to Y$ be a fpqc morphism of schemes. Then there is an equivalence of categories between quasi-coherent sheaves on $Y$ and "descent data" on $X$. Namely, the second category consists of quasi-coherent sheaves $\mathcal{F}$ on $X$ with an isomorphism $p_{1}... |
I'm currently interested in the following result:
Let $f: X \to Y$ be a fpqc morphism of schemes. Then there is an equivalence of categories between quasi-coherent sheaves on $Y$ and "descent data" on $X$. Namely, the second category consists of quasi-coherent sheaves $\mathcal{F}$ on $X$ with an isomorphism $p_{1}... |
How do you show the ring of formal laurent series is well-defined? |
**Question:** To prove that the Ring of Complex Entire functions is neither Artinian nor noetherian.
Proof: Clearly $R$ is not Artinian because it is a commutative integral domain which is not a field, and $R$ is not noetherian because its not a factorisation domain.
Is there a proof of this theorem using th... |
There is a theorem that states that if $f$ is analytic in a domain $D$, and the closed disc {$ z:|z-\alpha|\leq r$} contained in $D$, and $C$ denotes the disc's boundary followed in the positive direction, then for every $z$ in the disc we can write:
$$f(z)=\frac{1}{2\pi i}\int\frac{f(\zeta)}{\zeta-z}d\zeta$$
My q... |
See this SO thread: [**Calculating which item is next in a percentage distribution**][1].
To summarize,
you calculate
max ((rand1 * 10), (rand2 * 20), (rand3 * 30), (rand4 * 40))
for four random numbers in a range (say `1-100`).
If the `max` is
rand1 * 10, you choose 'A'
rand2 * 20, you ... |
How to calculate the expected distribution of results using random numbers? |
When for some number a: a(n-1)=n(n+1)/2 this is only possible when there is at least one stick of length n+1 in the collection. Otherwise the division is always possible.
This can be prved with complete induction. The case n=1 is trivial - it must be one stick of length 1. Suppose the assumption holds for n. Now con... |
As Chandru1 pointed out, a beautiful particular case of Cauchy's integral formula gives us a very particular *winding number*. For the sake of simplicity, let us look only at closed paths on the boundary of the unit disc, $\gamma : [0,1] \longrightarrow S^1$. Then, the *winding number* of $\gamma$ is the number of time... |
As Chandru1 pointed out, a beautiful particular case of Cauchy's integral formula gives us a very particular *winding number*. For the sake of simplicity, let us look only at closed paths on the boundary of the unit disc, $\gamma : [0,1] \longrightarrow S^1$. Then, the *winding number* of $\gamma$ is the number of time... |
As Chandru1 pointed out, a beautiful particular case of Cauchy's integral formula gives us a very particular *winding number*. For the sake of simplicity, let us look only at closed paths on the boundary of the unit disc, $\gamma : [0,1] \longrightarrow S^1$. Then, the *winding number* of $\gamma$ is the number of time... |
As Chandru1 pointed out, a beautiful particular case of Cauchy's integral formula gives us a very particular *winding number*. For the sake of simplicity, let us look only at closed paths on the boundary of the unit disc, $\gamma : [0,1] \longrightarrow S^1$. Then, the *winding number* of $\gamma$ is the number of time... |
As Chandru1 pointed out, a beautiful particular case of Cauchy's integral formula gives us a very particular *winding number*. For the sake of simplicity, let us look only at closed paths on the boundary of the unit disc, $\gamma : [0,1] \longrightarrow S^1$. Then, the *winding number* of $\gamma$ is the number of time... |
What you've done is correct, but I think it's better to work without coordinates; just with the definition of norm in terms of the dot product:
$$
\| A \| = +\sqrt{A\cdot A} \ .
$$
Then you may observe that, since $\|A \| \geq 0$,
$$
\|A+B\| = \|A -B\| \ \Longleftrightarrow \ \|A +B\|^2 = \|A-B\|^2 .
... |
I've been struggling with figuring out how to add powers of _i_.
For example, the result of _i^3 + i^4 + i^5_ is _1_. But how do I get the result of _i^3 + i^4 + ... + i^50_? Writing it all down would be pretty mundane...
It has to do something with division by 4, since the "power cycle" of _i_ repeats every four... |
I've been struggling with figuring out how to add powers of $i$.
For example, the result of $i^3 + i^4 + i^5$ is $1$. But how do I get the result of $i^3 + i^4 + ... + i^50$? Writing it all down would be pretty mundane...
It has to do something with division by 4, since the "power cycle" of $i$ repeats every four... |
I've been struggling with figuring out how to add powers of $i$.
For example, the result of $i^3 + i^4 + i^5$ is $1$. But how do I get the result of $i^3 + i^4 + ... + i^{50}$? Writing it all down would be pretty mundane...
It has to do something with division by 4, since the "power cycle" of $i$ repeats every fo... |
As Chandru1 pointed out, a beautiful particular case of Cauchy's integral formula gives us a very particular *winding number*. For the sake of simplicity, let us look only at closed paths on the boundary of the unit disc, $\gamma : [0,1] \longrightarrow S^1$. Then, the *winding number* of $\gamma$ is the number of time... |
Consider the closed interval $[0,1]$, there is $\frac{2}{3} \in [0,1]$ where $p=2$ and $q=3$. Similarly consider $[2,3]$, one can have $\frac{5}{2} \in [2,3]$ where $p=5$ and $q=2$. Does every interval of the form $[a,b]$, where $a,b \in \mathbb{R}$ contain a rational of this kind. If yes, how can we prove it? |
In the book *Elliptic Curves - McKean & Moll* we are given the outline for a proof of Eulers addition theorem:
> The (projective) quartic $\mathbf y^2 = (1-\mathbf x^2)(1-k^2 \mathbf x^2)$ has four intersections with the quadratic $\mathbf y = 1 + a \mathbf x + b \mathbf x^2$. One is located at $(\mathbf x, \mathbf ... |
As Chandru1 pointed out, a beautiful particular case of Cauchy's integral formula gives us a very particular *winding number*. For the sake of simplicity, let us look only at closed paths on the boundary of the unit disc, $\gamma : [0,1] \longrightarrow S^1$. Then, the *winding number* of $\gamma$ is the number of time... |
Explain why calculation divergent series could cause paradox? |
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