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I'm trying to understand a proof in "Combinatorial Group Theory" by Lyndon & Schupp, page 26, Lemma 4.7.
Here's my problem:
$F$ is a free group with $X$ as basis. $x \in X$. $\alpha \in IA(F)$ - meaning it's an automorphism of $F$ which induces the identity automorphism on $F/[F,F]$. $k \in F_n$ - a term in the d... |
I'm trying to understand a proof in "Combinatorial Group Theory" by Lyndon & Schupp, page 26, Lemma 4.7.
Here's my problem:
$F$ is a free group with $X$ as basis. $x \in X$. $\alpha \in IA(F)$ - meaning it's an automorphism of $F$ which induces the identity automorphism on $F/[F,F]$. $k \in F_n$ - a term in the d... |
Since b, x, y, z, ... integers greater than 1, how can we prove that
gcd (b ^ x - 1, b ^ y - 1, b ^ z - 1 ,...)= [b ^ gcd (x, y, z, .. .)] - 1?
Thank you!
Paulo Argolo |
This is initially a funny question, because I've found this on old notes but I do not find/recover my own derivation... But then the question is more general.
Q1:
I considered the function
$ f(x) = \frac {2x^2+3x}{(x+1)^2} $
I expressed this by a powerseries $ f_1(x) = -3x + 4x^2 - 5x^3 + 6x^4 - \ldot... |
From what I can see, perhaps you have the R and Xw matrix switched around. If you had Xw*R, then I think it would be okay. Did you derive this formula or did you find it somewhere? |
This is just a simple doubt I wanted to clear. I'm calculating the Hamming and Euclidean distance between two columns in a matrix, where each column is storing time information in seconds of when a particular event occurred. When I calculate the Hamming and Euclidean distance between the two columns, I get back a certa... |
Let's say I have an "S" shaped curve made up of 50 points. I am only given the data points, and there is no simple or obvious equation for each point on the curve.
So given only the data points, how would I be able to calculate the slope of the curve at a particular data point? |
Suppose you have a function $h(x)$ which is differentiable, and you consider the function $\mathcal{F}(x) = h(x)f(x)$.
Since $h$ and $f$ are both continuous on $[a,b]$, so is $\mathcal{F}$.
Since $h$ and $f$ are both differentiable on $(a,b)$, so is $\mathcal{F}$.
Since $f(a)=f(b)=0$, then $\mathcal{F}(a)=h(a... |
What follows is too long to be a comment. It is also very useful.
Suppose first that $X_i$, $i=1,2$, are independent ${\rm N}(0,\sigma_i^2)$ variables. Their characteristic functions are ${\rm E}[{\rm e}^{{\rm i}zX_i} ] = {\rm e}^{\sigma _i^2 ( - z^2 /2)}$. Suppose now that $X_i$, $i=1,2$, are independent Poisson va... |
I am trying to upper bound $\sum_{l=1}^\infty (1 - n! 2^{-ln} {2^l \choose n})$.
I tried to use the simple ${2^l \choose n} \ge (\frac{2^l}{n})^n$, but then $l$ disappears and it diverges. Any ideas? |
This is initially a funny question, because I've found this on old notes but I do not find/recover my own derivation... But then the question is more general.
Q1:
I considered the function
$ f(x) = - \frac {2x^2+3x}{(x+1)^2} $
I expressed this by a powerseries $ f_1(x) = -3x + 4x^2 - 5x^3 + 6x^4 - \l... |
Given $c \in \mathbb{R}$ and $m,n \in \mathbb{N}$, is it possible to combine the inequations
- $n < m$
- $\frac{8}{c} < m$
into one inequation?
I argued that due $n < m$ and $\frac{8}{c} < m$, there has to be one $n$ between $m$ and $\frac{8}{c}$, being $n = \left\lceil \frac{8}{c} \right\rceil$, so that ... |
Let's say that we have three points: $p = (x_p,y_p)$, $q = (x_q,y_q)$ and $a = (x_a,y_a)$. How can i find point $b$ which is reflection of $a$ across a line drawn through $p$ and $q$? I know it's simple to calculate, when we have $p$, $q$ etc. But I want to do this in my program, and I'm not sure, how to compute this.
... |
Nope. What if $m=2$, $c=7$?
There's really no way of finding out the what the relationship b/w $n$ and $8/c$ is without additional information.
Also, the task of finding an $n$ that satisfies certain properties is very different from the task of determining whether $n$ satisfies certain properties. (It's the diff... |
Given $c \in \mathbb{R}$ and $m,n \in \mathbb{N}$, is it possible to combine the inequations
- $n < m$
- $\frac{8}{c} < m$
into one inequation?
I argued that due $n < m$ and $\frac{8}{c} < m$, there has to be one $n$ between $m$ and $\frac{8}{c}$, being $n = \left\lceil \frac{8}{c} \right\rceil$, so that ... |
Nope. What if $m=2$, $c=7$?
There's really no way of finding out the what the relationship b/w $n$ and $8/c$ is without additional information.
Also, the task of finding an $n$ that satisfies certain properties is very different from the task of determining whether $n$ satisfies certain properties. (It's the diff... |
So, you need to show that for every $c>0 \in \mathbb{R}$ there exists $n \in \mathbb{R}^+$ such that $|f(x)|<c|g(x)|$ for all $x>n$. Given a $c>0$, let $n=\sqrt{\frac{8}{c}}$.
Then...
$x>n \implies |f(x)|<c|g(x)|$
That help at all? |
I've been playing around with this equation:
$\displaystyle\int_{-\pi}^\pi{\displaystyle\frac{1-e^{3it}}{1-e^{it}}dt}$
Now it seems to me that we can (possibly) split the integral into four seperate integrals; one for each quadrant of the complex plane. Doing this, we can determine the complex part if we know th... |
This is a follow up of http://math.stackexchange.com/questions/11587/upper-bound-binomial-sum
I was working on the problem in the above thread and noticed an interesting thing. I wanted to try and improve the bound Derek had (which was a quadratic $n$).
If we reformulate the problem as Derek has (since for this w... |
I am currently a mathematics graduate student at Western Kentucky University in Bowling Green, KY. I am looking for some kind of summer opportunity to participate in during summer 2011.
Does anyone have any suggestions of good opportunities or a good list of opportunities?
I would really appreciate it!
I am mo... |
Question: Let $v = (1,1,0,1) \in \mathbb{R}^{4}$ Let $$V:= \{f: \mathbb{R}^{4} \rightarrow \mathbb{R}^{2} \mbox{ linear } | f(v) = 0\}.$$
(a) Show that $V$ forms a vectorspace.
(b) Determine dim$(V)$. (Tip: Describe the elements in $V$ by extending $v$ to a basis of $\mathbb{R}^{4}$).
My attempt: for (a): ok i... |
The length of any curve that goes from A to B (other than the line segment AB) is greater than the length of that segment. This statement may be a theorem? Or, necessarily, is an axiom?
It seems to me that any attempt to build a theorem eventually turning into begging.
Thank you very much!
Paulo Argolo |
Lower curve from A to B: axiom ou theorem? |
How to decompose function like x^2+x-6 ? |
This is really several questions. In increasing order of depth:
1. Why multiply polynomials? Because they are functions, and multiplication is a natural or desirable operation on functions.
2. Why multiply functions? Because the values of the functions are integers, or rational numbers, or real numbers, or ob... |
I was trying to wrap my brain around this but could not think of anything. I am very poor at maths but trying to learn of it.
Edit:
Exact question is
If x^2+x-6 is a composite function f(g(x)); then figure out f(x) and g(x) ?
To me information looks incomplete to figure out such details. |
This is a follow up of http://math.stackexchange.com/questions/11587/upper-bound-binomial-sum
I was working on the problem in the above thread and noticed an interesting thing. I wanted to try and improve the bound Derek had (which was a quadratic $n$).
If we reformulate the problem as Derek has (since for this w... |
You are right that you are kind of assuming what you want, because you cannot even write down $f'(c)$ until you prove it exists; this can be fixed if you replace $f'(c)$ by a specific limit, and the specific limit you want is probably $B$. But even so ran into a bit of an alley, so let me help you out.
Here is the beg... |
This is a follow up of http://math.stackexchange.com/questions/11587/upper-bound-binomial-sum
I was working on the problem in the above thread and noticed an interesting thing. I wanted to try and improve the bound Derek had (which was a quadratic $n$).
If we reformulate the problem as Derek has (since for this w... |
Prove the following: $\displaystyle a^{n}-b^{n} = (a-b) \sum\limits_{k=0}^{n-1} a^{k}b^{n-1-k}$.
So one could use induction on $n$? Could one also use trichotomy or some type of combinatorial argument? |
Can we build a combinatorial argument along these lines?
Say if we have say $n$ students to be allotted in $a$ rooms with $b$ of the $a$ room being non air-conditioned.
You could use geometric series to conclude the result as well.
The right hand side is $(a-b) b^{n-1} \displaystyle \sum_{k=0}^{n-1} (\frac{a}{... |
Can we build a combinatorial argument along these lines?
Say if we have say $n$ students to be allotted in $a$ rooms with $b$ of the $a$ room being non air-conditioned. (Assume the students are distinguishable)
The total number of ways is $a^n$.
Suppose all students are allotted in the $b$ rooms, the number of... |
Can we build a combinatorial argument along these lines?
Say if we have say $n$ students to be allotted in $a$ rooms with $b$ of the $a$ room being non air-conditioned. (Assume the students are distinguishable)
The total number of ways is $a^n$.
Suppose all students are allotted to the $b$ rooms, the number of... |
This is a follow up to a question I asked earlier. I had previously asked that if a space was regular if and only if any closed set $Z$ was the intersection of all open sets containing $Z$, but the backwards implication turned out to be untrue. I'm now curious to see if this new bi-implication is fully true.
My def... |
A topological space is regular if and only if any closed set $Z$ is the intersection of its closed neighborhoods? |
Can we build a combinatorial argument along these lines?
Say if we have say $n$ students to be allotted in $a$ rooms with $b$ of the $a$ room being non air-conditioned. (Assume the students are distinguishable so we can order the student as $1,2,3,...,n$)
The total number of ways is $a^n$.
Suppose all students... |
This is a follow up to a question I asked earlier. I had previously asked that if a space was regular if and only if any closed set $Z$ was the intersection of all open sets containing $Z$, but the backwards implication turned out to be untrue. I'm now curious to see if this new bi-implication is fully true.
My def... |
I'm trying to classify (at least as fully as possible) the values of "t" in (0,pi/2) for which the following equation has a solution for some natural number, n.
(1/n)sin(nt)=(1/(n+2))sin[(n+2)t]
Does anyone know of any results that will help me? |
when is (1/n)sin(nt)=(1/(n+2))sin[(n+2)t] ? |
I agree that you don't know that there is a closed neighborhood of $p$ contained in $Z^c$.
However, this perhaps will work: Since $p$ is not in the intersection $\cap \mathcal{N}$, there is a closed set $V$ with $Z\subseteq V^{\circ}$ and with $p\notin V$. Since $V$ is closed, then $V^c$ is an open set containing $p... |
I'm trying to classify (at least as fully as possible) the values of $t$ in $(0,\pi/2)$ for which the following equation has a solution for some natural number, $n$.
$$\frac1{n}\sin(nt)=\frac1{n+2}\sin((n+2)t)$$
Does anyone know of any results that will help me? |
When is $\displaystyle \frac1{n}\sin(nt)=\frac1{n+2}\sin((n+2)t)$? |
This is a follow up of http://math.stackexchange.com/questions/11587/upper-bound-binomial-sum
I was working on the problem in the above thread and noticed an interesting thing. I wanted to try and improve the bound Derek had (which was a quadratic in $n$).
If we reformulate the problem as Derek has (since for thi... |
When is $\frac1{n}\sin(nt)=\frac1{n+2}\sin((n+2)t)$? |
I have a few weeks off from school soon, and I was hoping to self-study a bit of algebra. I don't think this question has been asked on here before, but does anyone have any suggestions for algebra books which are particularly suited to self-study?
---
Here is some background and motivation, if it's helpful.
... |
Good books for self-studying algebra? |
Lower curve from A to B: axiom or theorem? |
Assume f has a finite derivative and |f'(x)| ≤ y < 1 for all x in (a,b)
f is continuous and a ≤ f(x) ≤ b for all x in [a,b]
Prove f has a unique fixed point in [a,b]
So far I have for every c in (a,b) |f'(c)| ≤ y
=> lim x->c |f(x) - f(c)|/|x-c| ≤ y
=> lim x->c |f(x) - f(c)| ≤ y lim x->c |x-c|
Would t... |
[A Survey of Modern Algebra](http://www.amazon.com/gp/product/1568814542/ref=pd_lpo_k2_dp_sr_2?pf_rd_p=486539851&pf_rd_s=lpo-top-stripe-1&pf_rd_t=201&pf_rd_i=0828403309&pf_rd_m=ATVPDKIKX0DER&pf_rd_r=1SK7RY784NFNKY3FWD1Y) by Birkhoff and Maclane. |
Assume $f$ has a finite derivative and $|f'(x)| \leq y < 1$ for all $x \in (a,b)$
$f$ is continuous and $a \leq f(x) \leq b$ for all $x \in [a,b]$. Prove $f$ has a unique fixed point in $[a,b]$.
So far I have for every c in (a,b) |f'(c)| ≤ y
=> lim x->c |f(x) - f(c)|/|x-c| ≤ y
=> lim x->c |f(x) - f(c)| ≤ y li... |
Is there a fundamental relationship between circuits and cycles in an arbitrary graph? Are they mutually independent objects?
Thanks |
As is well known, if $x,y$ are natural numbers then their GCD $(x,y)$ has a representation $(x,y) = ax + by$ where $a,b$ are natural. Now let's prove that if $(x,y)=1$ then $(xy,x+y) = 1$. Starting from $ax+by=1$, $$1 = (ax+by)^2 = a^2x^2 + b^2y^2 + 2abxy = (a^2x+b^2y)(x+y) - (a-b)^2xy.$$ The reverse implication is tri... |
A proof system for GCD? |
I was wondering if we have a finite group G with subgroups H and K. With $X:=G/H$.
Is there a simple way of finding $X^K$ and when will $X^K=\emptyset$?
Also does this have any baring on when $G/H≅ G/K$?
(This is the first question I've ever asked here so apologies for errors etc.) |
You have two questions, the explicit one about why you would want to multiply polynomials, and an implicit one in your final paragraph about what multiplication by a non-integer might mean or why we would care to multiply by a non-integer in the first place.
To address the last one first: once you have multiplicati... |
I was wondering if we have a finite group $G$ with subgroups $H$ and $K$. With $X:=G/H$.
Is there a simple way of finding $X^K$ and when will $X^K=\emptyset$?
Also does this have any baring on when $G/H≅ G/K$?
(This is the first question I've ever asked here so apologies for errors etc.) |
> $\lim_{x\to c} |f(x) - f(c)|\leq y\ \lim_{x\to c} |x-c|$
>
> Would that be the definition of a contractive map in R?
No, this is just the statement that $f$ is continuous at $c$, because the right-hand side is $0$. The fixed point theorem will apply, but to show that $f$ is contractive you will want to use the... |
Quote from Don Zagier (Mathematicians: An Outer View of the Inner World):
" I like explicit, hands-on formulas. To me they have a beauty of their own. They can be deep or not. As an example, imagine you have a series of numbers such that if you add 1 to any number you will get the product of its left and right ne... |
What is the connection of the sequence 3, 4, 5/3, 2/3, 1 with deep topics? |
What would be an example of a matrix that would not have a $LDL^T$ decomposition? This is a trivial case but I was thinking of a zero matrix which would result in L being an identity matrix and D would be a zero matrix. When you compute $LDL^T$ it still gives you the original zero matrix so I don't believe this is corr... |
I'm assuming that $G/H$ represents the left cosets of $H$ in $G$, that is the sets of the form $gH$ with $g\in G$, with the action of $G$ on $X$ being given by $x\cdot(gH) = xgH$ (left multiplication), and that $X^K$ is the set of fixed points of $K$; that is, all $gH\in X$ such that for all $k\in K$, $k(gH)=gH$.
If... |
I'm assuming that $G/H$ represents the left cosets of $H$ in $G$, that is the sets of the form $gH$ with $g\in G$, with the action of $G$ on $X$ being given by $x\cdot(gH) = xgH$ (left multiplication), and that $X^K$ is the set of fixed points of $K$; that is, all $gH\in X$ such that for all $k\in K$, $k(gH)=gH$.
If... |
I have no idea what you did with the limit:
\begin{align*}
\lim{x\to c}\frac{f(x,y)-f(c,y)}{x-c} &= \lim_{x\to c}\frac{\frac{xy}{x^2+y^2} - \frac{cy}{c^2+y^2}}{x-c}\\\
&= \lim_{x\to c}\frac{xy(c^2+y^2)-cy(x^2+y^2)}{(c-y)(x^2+y^2)(c^2+y^2)}\\\
&= \lim_{x\to c}\frac{(x-c)(y^3-xyc)}{(x-c)(x^2+y^2)(c^2+y^2)}\\\
&= \li... |
I have no idea what you did with the limit. For $y\neq 0$ or $c\neq 0$, we have:
\begin{align*}
\lim{x\to c}\frac{f(x,y)-f(c,y)}{x-c} &= \lim_{x\to c}\frac{\frac{xy}{x^2+y^2} - \frac{cy}{c^2+y^2}}{x-c}\\\
&= \lim_{x\to c}\frac{xy(c^2+y^2)-cy(x^2+y^2)}{(c-y)(x^2+y^2)(c^2+y^2)}\\\
&= \lim_{x\to c}\frac{(x-c)(y^3-xyc)... |
How can I prove that $a^x$ < $a^{x + \Delta x}$, where $a$ is a constant (in my case, $a=2$)?
I don't want to use a graph, of course. |
How can I prove that $a^x$ < $a^{x + \Delta x}$, where $\Delta x > 0$, $a$ is a constant, and $a > 1$ (in my case, $a=2$)?
I don't want to use a graph, of course. |
It all depends on what you know. Do you know that if $a\gt 1$ and $b\gt 0$ then $a^b\gt 1$? If so, $2^{x+\Delta x}-2^x = 2^x(2^{\Delta x}-1)$ is a product of positive numbers. |
Let $x$ be a positive rational number of the form $\displaystyle x = \sum\limits_{k=1}^{n} \frac{a_k}{k!}$ where each $a_k$ is a nonnegative integer with $a_k \leq k-1$ for $k \geq 2$ and $a_n >0$. Prove that $a_1 = [x]$, $a_k = [k!x]-k[(k-1)!x]$ for $k = 2, \dots, n$, and that $n$ is the smallest integer such that $n... |
This is a follow up of http://math.stackexchange.com/questions/11587/upper-bound-binomial-sum
I was working on the problem in the above thread and noticed an interesting thing. I wanted to try and improve the bound Derek had (which was a quadratic in $n$).
If we reformulate the problem as Derek has (since for thi... |
As far as I am concerned, probability **distribution** function is for discrete random variables while probability **density** function is for continuous random variables. To find the probability value of continuous random variable, we have to take the total area under the function which differ from discrete random var... |
$\textbf{EDIT:}$
**So Mike has proved that $S(n)$ in fact diverges.**
**More interestingly, he has been able to quantify the divergence of $S(n)$. i.e. $S(n) > \lfloor\log_{2}(n-1)\rfloor$.**
**The conjecture is now that $\displaystyle \lim_{n \rightarrow \infty} (2 \log_{2}(n) - S(n)) = \alpha$ where $\alph... |
The division of the [chromatic scale][1] in 7 natural notes (white keys in a piano) and 5 [accidental][2] ones (black) seems a bit arbitrary to me.
Apparently, adjacent notes in a piano (including white or black) are always separated by a semitone. Why the distinction, then? Why not just have scales with 12 notes? (... |
Mathematical difference between white and black notes in a piano? |
The first thing you have to understand is that notes are not uniquely defined. Everything depends on what tuning you use. I'll assume we're talking about <a href="http://en.wikipedia.org/wiki/Equal_temperament">equal temperament</a> here. In equal temperament, a half-step is the same as a frequency ratio of $\sqrt[1... |
The first thing you have to understand is that notes are not uniquely defined. Everything depends on what tuning you use. I'll assume we're talking about <a href="http://en.wikipedia.org/wiki/Equal_temperament">equal temperament</a> here. In equal temperament, a half-step is the same as a frequency ratio of $\sqrt[1... |
The division of the [chromatic scale][1] in 7 natural notes (white keys in a piano) and 5 [accidental][2] ones (black) seems a bit arbitrary to me.
Apparently, adjacent notes in a piano (including white or black) are always separated by a semitone. Why the distinction, then? Why not just have scales with 12 notes? (... |
The first thing you have to understand is that notes are not uniquely defined. Everything depends on what tuning you use. I'll assume we're talking about <a href="http://en.wikipedia.org/wiki/Equal_temperament">equal temperament</a> here. In equal temperament, a half-step is the same as a frequency ratio of $\sqrt[1... |
For Riemann zeta function and $L$-functions of number fields, the width of critical strip is $1$. For $L$-functions of modular forms of weight $k$, the width of the critical strip is $k$.
Why is there a variation in the width of the critical strip for various $L$-functions? Is there a conceptual explanation or an ... |
Why is width of critical strip what it is? |
For proving the quadratic reciprocity, Gauss sums are very useful. However this seems an ad-hoc construction. Is this useful in a wider context? What are some other uses for Gauss sums?
|
The context is the definition of Hecke Größencharakter:
http://en.wikipedia.org/wiki/Hecke_character
This is supposed to generalize the Dirichlet $L$-series for number fields. Dirichlet characters are characters of the multiplicative groups of $\mathbb Z/p\mathbb Z$. An appropriate generalization would be instead... |
Let X be a topological space and A be an abelian group. Give A the discrete topology. For any open set U of X, Let $\cal A(U)$ be the group of all continuous aps of U into A. Thus with the usual restriction maps we obtain a sheaf $\cal A$.
SO, why for every connected open set U, $\cal A(U)\cong A$ for all U?
And, ... |
The first thing you have to understand is that notes are not uniquely defined. Everything depends on what tuning you use. I'll assume we're talking about <a href="http://en.wikipedia.org/wiki/Equal_temperament">equal temperament</a> here. In equal temperament, a half-step is the same as a frequency ratio of $\sqrt[1... |
The first thing you have to understand is that notes are not uniquely defined. Everything depends on what tuning you use. I'll assume we're talking about <a href="http://en.wikipedia.org/wiki/Equal_temperament">equal temperament</a> here. In equal temperament, a half-step is the same as a frequency ratio of $\sqrt[1... |
Is there a rational number (with denominator not greater than 200) between 15/106 and 16/113? |
Is there a rational number (with denominator not greater than 200) between 15/106 and 16/113? |
Some hints:
For your first question: $A$ has the discrete topology, so elements of $A$ are open and closed. Then, what properties does the preimage of a single element under a continuous map $U \to A$ have? Recall the definition of connectedness. What does this imply for continuous maps $U \to A$?
For your second... |
Gauss sums are not an ad-hoc construction! I know two ways to motivate the definition, one of which requires that you know a little Galois theory and the other which is totally mysterious to me.
Here is the Galois-theoretic explanation. Let $\zeta_p$ be a primitive $p^{th}$ root of unity, for $p$ prime. The <a hr... |
The Manin-Drinfeld theorem asserts that for a modular curve $X_0(N)$ and Jacobian $J_0(N)$ with the former being embdedded in the latter under the map that takes $i\infty$ to $0$, the cusps are torsion.
The proof Manin-Drinfeld theorem seems to use Hecke operators and thus seems to be valid only for congruence sub... |
**HINT** $\ \ $Google "farey sequence" and "mediant" and / or see my many sci.math posts on these topics, e.g. [this post][1], then [this post.][2].
[1]:http://groups.google.com/groups?selm=y8zpsy7ixpf.fsf%40nestle.csail.mit.edu
[2]:http://groups.google.com/groups?selm=y8zwv4xtxj3.fsf%40nestle.ai.mit.edu |
**HINT** $\ \ $Google "farey sequence" and "mediant" and / or see my many sci.math posts on these topics, e.g. see [this post][1] which explains the lucid lattice geometric viewpoint, and see [this post][2] on how to apply this to perform Farey mediant binary searching to find the simplest fraction(s) between two fract... |
**HINT** $\ \ $Google "farey sequence" and "mediant" and / or see my many sci.math posts on these topics, e.g. see [this post][1] which explains the lucid lattice geometric viewpoint, and see [this post][2] on how to apply this to perform Farey mediant binary searching to find the simplest fraction(s) between two fract... |
If I have an arc (which comes as part of an ellipse), can I represent it (or at least closely approximate) by cubic Bezier curve? And if yes, how can I calculate control points for that Bezier curve? |
Can elliptic arc be represented by cubic Bezier curve? |
Hint: What functions do you know whose second derivative is proportional to the function? There should be a two-dimensional solution space to the homogeneous equation (where the RHS is set to zero) |
**HINT** $\ \ $Google "farey sequence" and "mediant" and / or see my many sci.math posts on these topics, e.g. see [this post][1] which explains the lucid lattice geometric viewpoint, and see [this post][2] on how to apply this to perform Farey mediant binary searching to find the simplest fraction(s) between two fract... |
How elliptic arc can be represented by cubic Bezier curve? |
In English (I think this is universal anyway) we use the 1s, 10s, and 100s in a cycle. One, one thousand, one million; twenty two, twenty two thousand, twenty two million; one hundred and forty six, one hundred and forty six thousand, one hundred and forty six million, etc.
Who decided to cycle every (3x+1)th digit? |
Even for planar 3-connected graphs, which are the vertex-edge graphs that arise from convex 3-dimensional polyhedra, one can have all four possibilities: non-HC and non-Eulerian graphs, non-HC and Eulerian graphs; HC and Non-Eulerian graphs; HC and and Euler graphs, where HC means has a Hamiltonian circuit, and Euleria... |
Factoring a rational function as:
$$\frac{(x-a)(x-b)\cdots(x-z)}{(x-A)(x-B)\cdots(x-Z)}$$
Clearly displays the x-intercepts (roots) and vertical asymptotes (poles) of a rational function.
Similarly, dividing the denominator D into the numerator N to get N = DQ+R and writing N/D as Q + R/D clearly displays the ... |
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