instruction stringlengths 12 30k |
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Can someone please point me in the direction of any theory on graphs where the edges are not scalars but represent some relation between the nodes that is a simple function of a single variable (simple, say piecewise linear).
In particular, I'm interested in various basic graph properties and also thinking of the g... |
If I were you I would start by looking for similar threads.. ;)
[Could you recommend some classic textbooks on ordinary/partial differential equation?][1]
[Good 1st PDE book for self study][2]
[1]: http://math.stackexchange.com/questions/3335/could-you-recommend-some-classic-textbooks-on-ordinary-partia... |
From some reading, I've noticed that $\sup(\emptyset)=\min(S)$, but $\inf(\emptyset)=\max(S)$, given that $\min(S)$ and $\max(S)$ exist, where $S$ is the universe in which one is working. Is there some inherent reasoning/proof as to why this is? It seems strange to me that an upper bound of a set would be smaller than ... |
Is anything known about these integrals? Textbook suggestions are welcome
$f(n,p)=\int_{x=-0.5}^p \frac{n!}{x!(n-x)!} dx$
$n>0, p\le n+0.5$
For instance, as n grows $2^n-f(n,n+1/2)$ seems to be positive and monotonically decreasing...how would I find the limit?
<pre>
ListPlot[Table[2^n - NIntegrate[n!/(x! ... |
If memory serves, ten years ago to the week (or so), I taught first semester freshman calculus for the first time. As many calculus instructors do, I decided I should ask some extra credit questions to get students to think more deeply about the material. The first one I asked was this:
1) Recall that a function $... |
Is there a function with a removable discontinuity at every point? |
No.
Please see theorem 5.63 in this link:
http://books.google.co.in/books?id=6l_E9OTFaK0C&pg=PA250&lpg=PA250&dq=Function+with+Removable+discontinuities+at+every+point+on+R&source=bl&ots=tpEaf09M-9&sig=gaTEbOI5J7nY8OzR5pRIl6t_CQE&hl=en&ei=GgJ-TPazEIjovQO_5LWDDg&sa=X&oi=book_result&ct=result&resnum=3&ved=0CCIQ6AE... |
No.
Please see theorem 5.63 in this link:
http:// books.google.com/books?id=6l_E9OTFaK0C&pg=PA249 |
If memory serves, ten years ago to the week (or so), I taught first semester freshman calculus for the first time. As many calculus instructors do, I decided I should ask some extra credit questions to get students to think more deeply about the material. The first one I asked was this:
1) Recall that a function $... |
If memory serves, ten years ago to the week (or so), I taught first semester freshman calculus for the first time. As many calculus instructors do, I decided I should ask some extra credit questions to get students to think more deeply about the material. The first one I asked was this:
1) Recall that a function $... |
I really like mathematics, but I am not good at learning it. I find it takes me a long time to absorb new material by reading on my own and I haven't found a formula that works for me. I am hoping a few people out there will tell me how they go about learning math so I can try out their systems.
I need to know bas... |
How Do You Go About Learning Mathematics? |
If memory serves, ten years ago to the week (or so), I taught first semester freshman calculus for the first time. As many calculus instructors do, I decided I should ask some extra credit questions to get students to think more deeply about the material. The first one I asked was this:
1) Recall that a function $... |
Von Neumann is credited with saying that you never understand mathematics, but merely get used to it. To an extent, I find this to be true. But what is it that you're getting used to when you learn mathematics?
1. You get used to the sorts of objects and properties-of-objects which are the subject of study. An excel... |
No.
Please see theorem 5.63 in this link:
http://books.google.com/books?id=6l_E9OTFaK0C&pg=PA249 |
I subdivide a cubic bezier curve at a given *t* value using de Casteljau’s algorithm, which yields two cubic bezier curves. Afterwards I “scale” the second curve (proportionally).
I’d like to reconnect or approximate the two curves to/with a single curve in a third step. Is that possible?
 Isn't this false for $\kappa=1$? The left side is $\lambda$ and ... |
Let $N \subseteq M$ be a subobject in an abelian category (say, modules). A complement of $N$ in $M$ is then defined to be a subobject $Q \subseteq M$ which is maximal with respect to the condition $Q \cap N = 0$.
I've got just a short question what maximality means here: Does it mean that every other $Q'$ with $Q' ... |
No.
Please see theorem 5.63 in this link:
http://books.google.com/books?id=6l_E9OTFaK0C&pg=PA249
You can also refer this article, which gives more information: S.S.Kim, Amer.Math.Monthly 106 (1999), 258-259 |
The presentation of the dihedral group $D_{2n}$ is
$D_{2n}=< r,s | r^{n}=s^{2}=1, rs=sr^{-1}>$.
Why is it incorrect to conclude that $r=s=1$ ? In other words why doesn't this group presentation describe the trival group. |
Let $R$ be the coordinate ring of an affine complex variety (i.e. finitely generated, commutative, reduced $mathbb{C}$ algebra) and $M$ be an $R$ module.
Let $s\in M$ be an element, such that $s\in \mathfrak{m}M$ for every maximal ideal $\mathfrak{m}$. Does this imply $s=0$? |
Does a section that vanihes at every point vanish? |
I'm learning sequences right about now, and I'm having a really hard time finding the formula for a given sequence of numbers. I'm worried that, at exam time I'll be spending as much time finding the formula as I am now.
For example, one problem gave me:
$$\langle a_n \rangle = \\{2, 8, 18, 32, 50\\}$$
Okay, ... |
How to Determine a Formula for a Sequence of Numbers? |
> **Possible Duplicate:**
> [How can I write an equation that matches any sequence?](http://math.stackexchange.com/questions/656/how-can-i-write-an-equation-that-matches-any-sequence)
<!-- End of automatically inserted text -->
I'm learning sequences right about now, and I'm having a really hard time finding the ... |
I think the following works:
Here is a sketch, I will fill in the details later if required.
Let g(x) = Lt f(x). Then g(x) is continuous.
Let h(x) = f(x) - g(x). Then Lt h(x) exists and is zero everywhere.
All we now need to show is that h(c) = 0 for some c.
For this consider any point x0.
By existenc... |
first question on math.stackexchange :)
I'm studying for a Cryptography - Communication Security exam, and it involves a certain quantity of number theory - finite field theory, so be warned: this is my first encounter with these topics, and you'll have to be extra-clear with me :)
I thought I was doing pretty we... |
Does a section that vanishes at every point vanish? |
Let $R$ be the coordinate ring of an affine complex variety (i.e. finitely generated, commutative, reduced $\mathbb{C}$ algebra) and $M$ be an $R$ module.
Let $s\in M$ be an element, such that $s\in \mathfrak{m}M$ for every maximal ideal $\mathfrak{m}$. Does this imply $s=0$? |
I think the following works:
Here is a sketch, I will fill in the details later if required.
Let $g(x) = \lim_{t\rightarrow x} f(t)$. Then we can show that $g(x)$ is continuous.
Let $h(x) = f(x) - g(x)$. Then $\lim_{t \rightarrow x} h(t)$ exists and is $0$ everywhere.
All we now need to show is that $h(c) =... |
first question on math.stackexchange :)
I'm studying for a Cryptography - Communication Security exam, and it involves a certain quantity of number theory - finite field theory, so be warned: this is my first encounter with these topics, and you'll have to be extra-clear with me :)
I thought I was doing pretty we... |
First: a Galois field is a finite field. A finite field will have $p^n$ elements for some prime $p$ and some positive integer $n$ (in fact, there is, up to isomorphism, one and only one finite field with $p^n$ elements). If a field has $p^n$ elements, then $x+x+\cdots+x=0$ ($p$ summands) for all $x$, and no smaller pos... |
Can someone please point me in the direction of any theory on graphs where the edge weights are not scalars but represent some relation between the nodes that is a simple function of a single variable (simple, say piecewise linear).
In particular, I'm interested in various basic graph properties and also thinking o... |
Lets say that you accept the definition of a convex set in higher dimensions, like a sphere in $\mathbb{R}^3$. The question I seek to provide insight into is why convex functions in one variable are defined as opening up instead of down, since this seems like an arbitrary definition. This is because, depending on how y... |
First, (2): if you define a cardinal to be an ordinal that is not bijectable with any strictly smaller ordinal, then suppose that $\alpha$ is the ordinal supremum of $\kappa^{\theta}$, $\theta\lt\lambda$. Let $\beta$ be a strictly smaller ordinal than $\alpha$. By the definition of supremum, there must exist a $\theta\... |
In the question
"$r^{n}=s^{2}=1, rs=sr^{-1}$ ... why is it incorrect to *conclude* that $r=s=1$ ? ... why doesn't this group presentation *describe* the trivial group"
the key words are "conclude" and "describe". You are in effect asking what logic applies to statements about $r$ and $s$.
The answer is tha... |
In the question
"$r^{n}=s^{2}=1, rs=sr^{-1}$ ... why is it incorrect to *conclude* that $r=s=1$ ? ... why doesn't this group presentation *describe* the trivial group"
the key words are "conclude" and "describe". You are in effect asking what logic applies to statements about $r$ and $s$.
The answer is tha... |
The presentation of the dihedral group $D_{2n}$ is
$$D_{2n}= \langle r,s \mid r^{n}=s^{2}=1, rs=sr^{-1} \rangle.$$
Why is it incorrect to conclude that $r=s=1$ ? In other words why doesn't this group presentation describe the trivial group. |
In the question
"$r^{n}=s^{2}=1, rs=sr^{-1}$ ... why is it incorrect to *conclude* that $r=s=1$ ? ... why doesn't this group presentation *describe* the trivial group"
the key words are "conclude" and "describe". You are in effect asking what logic applies to statements about $r$ and $s$.
The answer is tha... |
This was an aside in lecture today.
Proposition: For a set $X$ and its power set $P(X)$, any function $f\colon P(X)\to X$ has at least two sets $A\neq B\subseteq X$ such that $f(A)=f(B)$.
I can see how this would be true if $X$ is a finite set, since $|P(X)|\gt |X|$, so by the pidgeonhole principle, at least two... |
Why does the set of all singleton sets not exist? |
Must a polynomial function $f \in \mathbb{R}[x_1, \ldots, x_n]$ that's lower bounded by some $\lambda \in \mathbb{R}$ have a global minimum over $\mathbb{R}^n$? |
Does a polynomial that's bounded below have a global minimum? |
Lets say that you accept the definition of a convex set in higher dimensions, like a sphere in $\mathbb{R}^3$. The question I seek to provide insight into is why convex functions in one variable are defined as opening up instead of down, since this seems like an arbitrary definition. This is because, depending on how y... |
This problem first came to me in high school, and a couple times since, and I even assigned it for extra credit in one of my calculus classes after I became a teacher. So I know the solution. What I am looking for is other WAYS to obtain the solution. I've been told there exists a solution using only arithmetic, but... |
The [nine point circle][1].
Three sets of three points, each of which obviously determines a circle. That these three constructions _always_ give the **same** circle!?
[1]: http://en.wikipedia.org/wiki/Nine-point_circle |
Lets say that you accept the definition of a convex set in higher dimensions, like a sphere in $\mathbb{R}^3$. The question I seek to provide insight into is why convex functions in one variable are defined as opening up instead of down, since this seems like an arbitrary definition. This is because, depending on how y... |
Let $\kappa$ a cardinal of cofinality $\omega$; let $C \subseteq \kappa$ be a unbounded countable subset. Why is then $C$ closed (and thus a c.u.b.)? This means that if $\delta < \kappa$ is a limit ordinal, and $C \cap \delta$ is unbounded in $\delta$, then $\delta \in C$.
I doubt that this is true; however, it's a... |
The question of whether one set of equations ($r^{n}=s^{2}=1$ and $rs=sr^{-1}$) implies another ($r=s=1$) gets at a more basic conceptual point: *what logic applies to statements about $r$ and $s$* (or other variables whose values are group elements)?
The answer is that assertions about $r$ and $s$ are understood ... |
$\frac{a}{b} = \frac{p^{2}\alpha}{\beta} + \frac{1}{p}$ where $(\alpha,\beta) = (p,\beta) = 1$
It follows immediately from this I believe.
So the exercise helps.
Related: http://en.wikipedia.org/wiki/Wolstenholme%27s_theorem
Also, a hint for the exercise: consider $f(x) = (x-1)(x-2)\cdots(x-p+1)$. Does $p^... |
Let $pi(x)$ be the number of primes not greater than $x$.
<a href="http://en.wikipedia.org/w/index.php?title=Prime-counting_function">Wikipedia article</a> says that $pi(10^{23}) = 1,925,320,391,606,803,968,923$.
The question is how to calculate $pi(x)$ for large $x$ in a reasonable time? What algorithms do exist... |
The question in the title is a consequence of the [axiom of regularity][1].
In fact If there was such a set $X$, then $\{X\}\in X$. But obviously we always have $X\in\{X\}$. Hence, we would be able to construct an infinite downward chain of sets which contradicts the axiom of regularity.
[1]: http://books.go... |
The question in the title is a consequence of the [axiom of regularity][1].
In fact If there was such a set $X$, then $\\{X\\}\in X$. But obviously we always have $X\in\\{X\\}$. Hence, we would be able to construct an infinite downward chain of sets which contradicts the axiom of regularity.
[1]: http://book... |
Let $G$ be a finite group, and let $p^{\alpha} \mid |G|$, where $p$ is a prime. Now does this imply $p^{\alpha} \mid |Aut(G)|$?
Clearly if $|G| \leq 2$, then the Automorphism group is the trivial group, so one can see that this need not be true for $\alpha =1$. I am curious to know for higher powers of $\alpha$. |
Let $p$ be a prime, then does $p^{\alpha} \mid |G| \Longrightarrow p^{\alpha} \mid Aut(G)$? |
Let $p$ be a prime, then does $p^{\alpha} \mid |G| \Longrightarrow p \mid Aut(G)$? |
Let $G$ be a finite group, and let $p^{\alpha} \mid |G|$, where $p$ is a prime. Now does this imply $p \mid |Aut(G)|$?
Clearly if $|G| \leq 2$, then the Automorphism group is the trivial group, so one can see that this need not be true for $\alpha =1$. I am curious to know for higher powers of $\alpha$. |
If A and B are square matrix such that AB = I where I is identity matrix. Show that BA = I. I do not understand anything more than the following.
1. Elementary row operations.
2. Linear dependence.
3. Row reduced forms and their relations with original matix.
If Entries of matrix are not from a mathema... |
So you want to find a proof of this well-known fact, which avoids the usual "indirect" proofs? I've also pondered over this some time ago.
By adding some identity matrix in a block, I think we may reduce to quadratic matrices. Now we have the following general assertion:
Let $M$ be a finite-dimensional $K$-alge... |
So you want to find a proof of this well-known fact, which avoids the usual "indirect" proofs? I've also pondered over this some time ago.
By adding some identity matrix in a block, I think we may reduce to quadratic matrices. Now we have the following general assertion:
Let $M$ be a finite-dimensional $K$-alge... |
So you want to find a proof of this well-known fact, which avoids the usual "indirect" proofs? I've also pondered over this some time ago.
By adding some identity matrix in a block, I think we may reduce to quadratic matrices. Now we have the following general assertion:
Let $M$ be a finite-dimensional $K$-alge... |
So you want to find a proof of this well-known fact, which avoids the usual "indirect" proofs? I've also pondered over this some time ago.
The claim is also true for non-square matrices. Anyway, by adding some identity matrix in a block, I think we may reduce to square matrices. Now we have the following general as... |
as a function of a real variable, apparently.
Part of the freedom in choosing a proof is that you get to choose what definition of $e^{ix}$ to start from -- do you use a differential equation? a power series? a definition in terms of trig functions?
Another bit of freedom is that you get to choose what definition of ... |
What is your favorite proof that e^ix has a period of 2pi? |
What is your favorite proof that $e^{ix}$ has a period of $2\pi$? |
Let A and B be two matrices which can be multiplied.
Then **$\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B))$**
I proved **$\operatorname{rank}(AB) \leq \operatorname{rank}(B)$** interpreting AB as a composition of linear maps, observing that $\operatorname{ker}(B) ... |
So you want to find a proof of this well-known fact, which avoids the usual "indirect" proofs? I've also pondered over this some time ago.
We have the following general assertion:
Let $M$ be a finite-dimensional $K$-algebra, and $a,b \in M$ such that $ab=1$, then $ba=1$. [For example, $M$ could be the algebra o... |
Singular value decomposition ([SVD](http://en.wikipedia.org/wiki/Singular_value_decomposition)) and principal component analysis (PCA) are two eigenvalue methods used to reduce a high-dimensional dataset into fewer dimensions while retaining important information. Articles online say that these methods are 'related' bu... |
Let $\kappa$ ba weakly inaccessible cardinal. Why are there $\kappa$ regular cardinals $\lambda < \kappa$? I've tried a recursive construction, but I don't know what to do in the limit step. Supremum does not work, since then we loose regularity. |
It's a **simple consequence of the pigeonhole principle**. Here's an excerpt from my old [post][1]:
Recall (below) $\rm\;\; BA = I \Rightarrow AB = I\;$ reduces to: $\rm\;A\;$ injective $\rm\;\Rightarrow A \;$ surjective. $\;$ But
**THEOREM** $\;$ $\rm\;A\;$ injective $\rm\;\Rightarrow A\;$ surjective, $\:$ ... |
It's a **simple consequence of the pigeonhole principle**. Here's an excerpt from my old [post][1]:
Recall (below) $\rm\;\; BA = I \Rightarrow AB = I\;$ reduces to: $\rm\;A\;$ injective $\rm\;\Rightarrow A \;$ surjective. $\;$ But
**THEOREM** $\;$ $\rm\;A\;$ injective $\rm\;\Rightarrow A\;$ surjective, $\:$ ... |
It's a **simple consequence of the pigeonhole principle**. Here's an excerpt from my old [post][1]:
Recall (below) $\rm\;\; BA = I \Rightarrow AB = I\;$ reduces to: $\rm\;A\;$ injective $\rm\;\Rightarrow A \;$ surjective. $\;$ But
**THEOREM** $\;$ $\rm\;A\;$ injective $\rm\;\Rightarrow A\;$ surjective, $\:$ ... |
For a periodic ( and not so periodic) function, it is always possible to use fourier series to find out the frequencies contained in the function.
But what about function that cannot be expressed in mathematical terms? For example, this graph ( accelerogram):
![alt text][1]
Is there anyway to apply a sort of... |
This queston can be regarded as a variant of this one:
http://math.stackexchange.com/questions/3799/does-a-section-that-vanihes-at-every-point-vanish
Let $X,Y$ be varieties, $M$ be an $\O_X$-Module and
$$\pi:X\times Y\rightarrow X$$
be the provjection. Let $s$ be a local section of $M$, which vannishes along e... |
Some years ago I was interested in the following Markov chain
whose state space is the positive integers. The chain begins at state "1",
and from state "n" the chain next jumps to a state uniformly
selected from {n+1,n+2,...,2n}.
As time goes on, this chain goes to infinity, with occasional
large jumps. In any ... |
This queston can be regarded as a variant of this one:
http://math.stackexchange.com/questions/3799/does-a-section-that-vanihes-at-every-point-vanish
Let $X,Y$ be varieties, $M$ be an $\O_X$-Module and
$$\pi:X\times Y\rightarrow X$$
be the provjection. Let $s$ be a local section of $M$, which vanishes along ea... |
**HINT** $\rm\;\;\displaystyle p^4\:\big|\:ap-b = \:bp\bigg(\frac{a}{b} - \frac{1}{p}\bigg)\;\;$ via $\rm\;\; p^2|bp\;\;$ and $\rm\displaystyle\; p^2\:\big|\:\bigg(\frac{a}{b} - \frac{1}{p}\bigg)\;\;$ via [Wolstenholme][1]
[1]:http://books.google.com/books?id=rey9wfSaJ9EC&pg=PA113 |
**It follows simply by the pigeonhole principle**. Here is an excerpt from my [old post][1]:
Recall (below) $\rm\; BA \:=\:\: I \:\;\Rightarrow\; AB \:=\: I\;\;$ easily reduces to:
**THEOREM** $\;$ $\rm\;A\;$ injective $\rm\;\Rightarrow\; A\;$ surjective, $\:$ for linear $\rm\:A\:$ on finite dim vector spa... |
In the question
"$r^{n}=s^{2}=1, rs=sr^{-1}$ ... why is it incorrect to *conclude* that $r=s=1$ ? ... why doesn't this group presentation *describe* the trivial group"
the key words are "conclude" and "describe". You are in effect asking what logic applies to statements about $r$ and $s$.
The answer is tha... |
This queston can be regarded as a variant of this one:
http://math.stackexchange.com/questions/3799/does-a-section-that-vanihes-at-every-point-vanish
Let $X,Y$ be varieties, $M$ be an $\mathcal{O}_X$-Module and
$$\pi:X\times Y\rightarrow X$$
be the projection. Let $s$ be a local section of $M$, which vanishes ... |
In the question
"$r^{n}=s^{2}=1, rs=sr^{-1}$ ... why is it incorrect to *conclude* that $r=s=1$ ? ... why doesn't this group presentation *describe* the trivial group"
the key words are "conclude" and "describe". You are in effect asking what logic applies to statements about $r$ and $s$.
The answer is tha... |
There's the obvious approach of proceeding based on taking the logarithm of both sides of your regression formula:
$\ln\;y=\ln\left(ax^b\right)$
$\ln\;y=\ln\;a+b\ln\;x$
which can be seen to be an expression of the form
$v=k+bu$
where $v=\ln\;y$, $u=\ln\;x$, and $k=\ln\;a$.
Now a linear regression in t... |
In the question
"$r^{n}=s^{2}=1, rs=sr^{-1}$ ... why is it incorrect to *conclude* that $r=s=1$ ? ... why doesn't this group presentation *describe* the trivial group"
the key words are "conclude" and "describe". You are in effect asking what logic applies to statements about $r$ and $s$.
The answer is tha... |
**It follows simply by the pigeonhole principle**. Here is an excerpt from my [old post][1]:
Recall (below) $\rm\; BA \:=\:\: I \:\;\Rightarrow\; AB \:=\: I\;\;$ easily reduces to:
**THEOREM** $\;$ $\rm\;A\;$ injective $\rm\;\Rightarrow\; A\;$ surjective, $\:$ for linear $\rm\:A\:$ on finite dim vector spa... |
In the question
"$r^{n}=s^{2}=1, rs=sr^{-1}$ ... why is it incorrect to *conclude* that $r=s=1$ ? ... why doesn't this group presentation *describe* the trivial group"
the key words are "conclude" and "describe". You are in effect asking what logic applies to statements about $r$ and $s$.
The answer is tha... |
In the question
"$r^{n}=s^{2}=1, rs=sr^{-1}$ ... why is it incorrect to *conclude* that $r=s=1$ ? ... why doesn't this group presentation *describe* the trivial group"
the key words are "conclude" and "describe". You are in effect asking what logic applies to statements about $r$ and $s$.
The answer is tha... |
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 cancelling any factors of $\rm\:A\;$ from $\rm... |
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... |
I am currently prepping for uni having been a few years out of the studying loop (programming as it happens). Anyway, I've been re-reading my A-level notes/exercises and looking through OpenCourseWare stuff and I noticed this assertation in my notes:
Given $a \cos\theta \pm b \sin\theta = c$ then:
$a \sin\theta \... |
Relation between "harmonic form" and fourier series? |
No.
Please see theorem 5.63 in this link:
http://books.google.com/books?id=6l_E9OTFaK0C&pg=PA249
You can also refer this article, which gives more information: S.S.Kim, Amer.Math.Monthly 106 (1999), 258-259
[I don't see why this article is relevant to the question at hand, since it does not discuss "pre... |
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... |
How can I count the number of colored combinations in a set of regions? |
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... |
I have a data set of $C_i$ vectors which are to be described by the following:
$C_i = X F_i$
where $C$ and $F$ are $(n\times 1)$ vectors and $X$, an $(n\times n)$ matrix, has the pattern:
$\begin{align}
\mathsf X &= \begin{bmatrix}
1 & R_{1,2} & \dots & \dots & R_{1,n} \\\
R_{2,1} & 1 & R_{2,3} & \dots ... |
How can I solve this matrix? |
If I understand correctly, you're trying to correlate $n$ separate equations for $n$ different values of $C_i$ and $F_i$ into a solution for $X$; is that correct? If so, then the easiest way to go is to turn your set of vector equations into a matrix equation! Simply write $C = \[ C_1 C_2 C_3 \ldots C_n \]$ (assuming... |
No.
Please see theorem 5.63 in this link:
http://books.google.com/books?id=6l_E9OTFaK0C&pg=PA249
|
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... |
On my homework today, we had to find all the normal subgroups of $D_{n}$, the dihedral group of order 2n. I solved the problem by looking at how the conjugacy classes change based on whether n is even or odd and then constructed the normal subgroups as unions of the conjugacy classes.
I have 2 questions:
(i) Is t... |
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