instruction stringlengths 12 30k |
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So for the formula 1/x, If you were to add up all y values from x=1 to x=∞, wouldn't the sum approach a number because even though you are always adding, aren't you just adding smaller and smaller numbers? Wouldn't this mean that it approached a certain number? |
Why does 1/x diverge? |
If $\mathcal{F}_1 \subset \mathcal{F}_2 \subset \dotsb$ are sigma algebras, what is wrong with claiming that $\cup_i\mathcal{F}_i$ is a sigma algebra?
It seems closed under complement since for all $x$ in the union, $x$ has to belong to some F_i, and so must its complement.
It seems closed under countable union, ... |
I was considering the group of automorphisms on a vector space $\mathbb{Q}^3$ with the binary operation of addition. Is there a general description of the elements of this group? At first, it seemed to be that it would consist of bijective linear transformations, but I'm not sure how informative that is. Is there a spe... |
Is there a particular characterization of $Aut(\mathbb{Q}^3)$? |
I'm actually looking for more intuition behind what's going on than a simple answer. I understand that the Borel sets are the smallest sigma algebra that contain all of the open sets. I can see how open intervals and their unions are in this set, and how it makes sense to measure them. It's hard to see why you'd wan... |
Why are the sets of rational and irrational numbers Borel sets (over the reals)? |
Can anyone recommend good, free online articles or books to learn Discrete mathematics? When I google'd for them, I came across few resources..but don't know whether they are good to start learning with. It's better to know a tutorial or a book that get you started in Discrete Mathematics. Can readers of SE Mathematics... |
Given coordinates of hypotenuse, how can I calculate coordinates of other vertex? |
For any continuous scalar field $f$ on the sphere, you can define a continuous scalar field $g$ as $g(p) = f(p) - f(\bar{p})$, where $\bar{p}$ is the antipodal point of $p$. Unless $g$ is zero everywhere, there is a point $q$ for which $g(p) > 0$ and $g(\bar{q}) = -g(q) < 0$. As $g$ is continuous, a closed contour sepa... |
**Motivation**: Assume one has a continued fraction (c.f.) $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $, where $a_n,b_n$ are integers. **I am looking for a c.f. transformation that can transform it into a new c.f. in such a way that its elements $a_{n}^{*},b_{n}^{*}$ are integers as ... |
Here is a question that I can't find the answer to in my notes or my textbook.
Is there any relation between $\text{Sym}_n$ and $\text{Sym}_k$ where $k < n$? Is $\text{Sym}_k \subset \text{Sym}_n$? Since $k! | n!\text{ } \forall k < n $ they satisfy Lagrange's theorem, but of course that doesnt guarantee the exist... |
Here is a question that I can't find the answer to in my notes or my textbook.
Is there any relation between $\text{Sym}_n$ and $\text{Sym}_k$ where $k < n$? Is $\text{Sym}_k \subset \text{Sym}_n$? Since $k! | n!\text{ } \forall k < n $ they satisfy Lagrange's theorem, but of course that doesnt guarantee the exist... |
**Motivation**: Assume one has a continued fraction (c.f.) $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $, where $a_n,b_n$ are integers. **I am looking for a c.f. transformation that can transform it into a new c.f. in such a way that its elements $a_{n}^{*},b_{n}^{*}$ are integers as ... |
I have a series of inequalities:
$$y_{1min} \leq y_{1}(x) \leq y_{1max}$$
$$y_{2min} \leq y_{2}(x) \leq y_{2max}$$
$$..$$
$$y_{2min} \leq y_{n}(x) \leq y_{2max}$$
The question is, is there a **general method** that allows me to find the range of $x$ that satisfies the above inequalities?
Things are easy i... |
I have a series of inequalities:
$$y_{1min} \leq y_{1}(x) \leq y_{1max}$$
$$y_{2min} \leq y_{2}(x) \leq y_{2max}$$
$$..$$
$$y_{nmin} \leq y_{n}(x) \leq y_{nmax}$$
Note that $x\in\mathbb{R}$
The question is, is there a **general method** that allows me to find the range of $x$ that satisfies the above ineq... |
The CASIO fx-7400G calculator is very quick at doing $(1+1/1000000)^{1000000}$, giving an output of 2.718280469.
Mathematica (v7) takes over a minute, running on an Intel Core2 Duo processor, and its output is this:
![alt text][1]
[1]: https://i.stack.imgur.com/dP0ac.jpg
With n=10000000, my patient ran ... |
For a start, suppose we can find the interval $I_i\subseteq\mathbb{R}$ such that $y_{imin}\leq y_i(x)\leq y_{imax}$. Then the interval $I$ which satisfies all the constraints is
$I=\bigcap_{i=1}^n I_i$
Thus we can concentrate on finding $I_i$ separately for each $1\leq i\leq n$. Here I'm sure we will need some c... |
**Motivation**: Assume one has a continued fraction (c.f.) $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $, where $a_n,b_n$ are integers. **I am looking for a c.f. transformation that can transform it into a new c.f. in such a way that its elements $a_{n}^{*},b_{n}^{*}$ are integers as ... |
A rigid transformation consists of a rotation and a translation. It can preserve all the structures of inner product spaces, so it is a [morphism][1]/[isometry][2] between inner product spaces.
I was wondering what are the most special spaces between which rigid transformations preserve the structures of these space... |
I have a series of inequalities:
$$y_{1min} \leq y_{1}(x) \leq y_{1max}$$
$$y_{2min} \leq y_{2}(x) \leq y_{2max}$$
$$..$$
$$y_{nmin} \leq y_{n}(x) \leq y_{nmax}$$
Note that $x\in\mathbb{R}$
The question is, is there a **general method** that allows me to find the range of $x$ that satisfies the above ineq... |
For a start, suppose we can find the interval $I_i\subseteq\mathbb{R}$ such that $y_{imin}\leq y_i(x)\leq y_{imax}$. Then the interval $I$ which satisfies all the constraints is
$I=\bigcap_{i=1}^n I_i$
Thus we can concentrate on finding $I_i$ separately for each $1\leq i\leq n$. Here I'm sure we will need some c... |
For a start, suppose we can find the interval $I_i\subseteq\mathbb{R}$ such that $y_{imin}\leq y_i(x)\leq y_{imax}$. Then the interval $I$ which satisfies all the constraints is
$I=\bigcap_{i=1}^n I_i$
Thus we can concentrate on finding $I_i$ separately for each $1\leq i\leq n$. Here I'm sure we will need some c... |
I have a series of inequalities:
$$y_{1min} \leq y_{1}(x) \leq y_{1max}$$
$$y_{2min} \leq y_{2}(x) \leq y_{2max}$$
$$..$$
$$y_{nmin} \leq y_{n}(x) \leq y_{nmax}$$
Note that $x\in\mathbb{R}$
The question is, is there a **general method** that allows me to find the range of $x$ that satisfies the above ineq... |
[Goursat's Lemma][1] characterizes the subgroups of direct products. Is there a similar characterization for the subgroups of semidirect products? What about if I'm only interested in the normal subgroups?
[1]: http://en.wikipedia.org/wiki/Goursat%27s_lemma |
What are the subgroups of a semidirect product? |
I have a series of inequalities:
$$y_{1min} \leq y_{1}(x) \leq y_{1max}$$
$$y_{2min} \leq y_{2}(x) \leq y_{2max}$$
$$..$$
$$y_{nmin} \leq y_{n}(x) \leq y_{nmax}$$
Note that $x\in\mathbb{R}$
The question is, is there a **general method** that allows me to find the range of $x$ that satisfies the above ineq... |
**Motivation**: Assume one has a continued fraction (c.f.) $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $, where $a_n,b_n$ are integers. **I am looking for a c.f. transformation that can transform it into a new c.f. in such a way that its elements $a_{n}^{*},b_{n}^{*}$ are integers as ... |
**Motivation**: Assume one has a continued fraction (c.f.) $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $, where $a_n,b_n$ are integers. **I am looking for a c.f. transformation that can transform it into a new c.f. in such a way that its elements $a_{n}^{*},b_{n}^{*}$ are integers as ... |
An affine transformation is a linear transformation followed by a translation. They are [morphism][1] between affine spaces.
A rigid transformation consists of a rotation and a translation. I was wondering what are the spaces between which the rigid transformations can become morphisms?
what are the spaces be... |
I have a series of inequalities:
$$y_{1min} \leq y_{1}(x) \leq y_{1max}$$
$$y_{2min} \leq y_{2}(x) \leq y_{2max}$$
$$..$$
$$y_{nmin} \leq y_{n}(x) \leq y_{nmax}$$
Note that $x\in\mathbb{R}$
The question is, is there a **general method** that allows me to find the range of $x$ that satisfies the above ineq... |
The CASIO fx-7400G calculator is very quick at doing $(1+1/1000000)^{1000000}$, giving an output of 2.718280469.
Mathematica (v7) takes over a minute, running on an Intel Core2 Duo processor, and its output is this:
![alt text][1]
[1]: https://i.stack.imgur.com/dP0ac.jpg
With n=10000000, my patience ran... |
I am reading about active contours models. It is said that this can be broadly classified as either parametric active contour models and geometric active contour models. The first one are represented explicitly as parameterized curves in a Lagragian framework, while the geometric active contours are represented implici... |
What is the name of this kind of polynomial? |
So first off all, this WAS homework. Submitted it about 3 hours ago.
Let $A$ be an $m x m$ matrix. then show that if rank A = m, then Ax = 0 has a unique solution.
My roommate said this is wrong but here is how I solved it.
very simply, i stated that if the rank of A is m that means the number of leading o... |
So first off all, this WAS homework. Submitted it about 3 hours ago.
Let $A$ be an $m \times m$ matrix. then show that if rank A = m, then Ax = 0 has a unique solution.
My roommate said this is wrong but here is how I solved it.
very simply, i stated that if the rank of A is m that means the number of lead... |
**Motivation**: Assume one has a continued fraction (c.f.) $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $, where $a_n,b_n$ are integers. **I am looking for a c.f. transformation that can transform it into a new c.f. in such a way that its elements $a_{n}^{*},b_{n}^{*}$ are integers as ... |
I have all points where my curve pass through, but I need to get the coordinates of the control points to be able to draw the curve. How can I do to calculate this points? |
I was reading this book "Rational points on Elliptic curves" by J.Silverman, and J.Tate, 2 prominent figures in Number theory and was very intrigued after reading the first couple of pages.
The connection between Algebra and Geometry is displayed in a professional manner in that book. One can generally ask this ques... |
**Motivation**: Assume one has a continued fraction (c.f.) $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $, where $a_n,b_n$ are integers. **I am looking for a c.f. transformation that can transform it into a new c.f. in such a way that its elements $a_{n}^{*},b_{n}^{*}$ are integers as ... |
I was reading this book "Rational points on Elliptic curves" by J.Silverman, and J.Tate, 2 prominent figures in Number theory and was very intrigued after reading the first couple of pages.
The connection between Algebra and Geometry is displayed in a professional manner in that book. One can generally ask this ques... |
How do I find a function from a differential equation? |
While searching on the net i found this article. See whether it helps or not: http://www.springerlink.com/content/l272110h87u05667/fulltext.pdf
If one can't access this then try: dx.doi.org/10.1007/BF01058705 ams.org/mathscinet-getitem?mr=1148867
Thanks to Jack. |
While searching on the net i found this article. See whether it helps or not: http://www.springerlink.com/content/l272110h87u05667/fulltext.pdf
If one can't access this then try: [dx.doi.org/10.1007/BF01058705][1] or [ams.org/mathscinet-getitem?mr=1148867][2]
Thanks to Jack.
[1]: http://dx.doi.org/10.100... |
Do generalized hypergeometric functions
$$ pFq(a_1,\ldots,a_p; b_1, \ldots,b_q; z) $$
with $p = q+1$ always possess a singularity at $z=1$, independent of the their parameters $a_1,\ldots,a_p$ and $b_1,\ldots,b_q$ under the provision that all cases of finite polynomials are excluded? Can someone mention a reliable s... |
What are the benefits of using a conjugate linear inner product in a complex vector space vs a simple linear inner product? That is, why do we demand that (y,x) = (x,y)* as opposed to (y,x)=(x,y)? Of course, this ensures that (x,x) is real and thus makes an easy definition of norm, but is that necessary?
|
Is complex conjugation needed for valid inner product? |
**Motivation**: Assume one has a continued fraction (c.f.) $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $, where $a_n,b_n$ are integers. **I am looking for a c.f. transformation that can transform it into a new c.f. in such a way that its elements $a_{n}^{*},b_{n}^{*}$ are integers as ... |
Prompted by today's Minute Math question on the MAA site (<a href="http://amc.maa.org/mathclub/5-0,problems/T-problems/T-web,ia/2005web/tb05-12-ia.shtml">http://amc.maa.org/mathclub/5-0,problems/T-problems/T-web,ia/2005web/tb05-12-ia.shtml</a>), I started thinking about the probability that the sum of the numbers rolle... |
What's the probability that a sum of dice is prime? |
For (1.), Rosser and Schoenfeld (sp?) non-asymptotic estimates bound $\pi(n)$ between functions of the form $x/(\log x + C)$ and this should be enough. |
For (1.), Rosser and Schoenfeld (sp?) non-asymptotic estimates bound $\pi(n)$ between functions of the form $x/(\log x + C)$ and this should be enough.
For (2) your integrals are rapid decaying and incredibly close to the integrals on the whole real line. O($n$) standard deviations from the average is quite an unli... |
I want to point out that there are, topicwise, different paths from the most suggested standard one:
As noted by "jmoy", time is limited - the comments telling you to learn some physics or programming first or on the way seem to ignore this somewhat. Sure it is good and will pay off to have these things as backgrou... |
I have a series of inequalities:
$$y_{1min} \leq y_{1}(x) \leq y_{1max}$$
$$y_{2min} \leq y_{2}(x) \leq y_{2max}$$
$$..$$
$$y_{nmin} \leq y_{n}(x) \leq y_{nmax}$$
Note that $x\in\mathbb{R}$
The question is, is there a **general method** that allows me to find the range of $x$ that satisfies the above ineq... |
I have a series of inequalities:
$$y_{1min} \leq y_{1}(x) \leq y_{1max}$$
$$y_{2min} \leq y_{2}(x) \leq y_{2max}$$
$$..$$
$$y_{nmin} \leq y_{n}(x) \leq y_{nmax}$$
Note that $x\in\mathbb{R}$
The question is, is there a **general method** that allows me to find the range of $x$ that satisfies the above ineq... |
A common way to define a group is as the group of structure-preserving transformations on some structured set. For example, the symmetric group on a set $X$ preserves no structure: or, in other words, it preserves only the structure of being a set. When $X$ is finite, what structure can the *alternating* group be sai... |
What structure does the alternating group preserve? |
The torsion of a curve in $\mathbb{R}^3$ indicates how much it twists around. The curl of a vector field indicates how much the vector field twists around. Is there a relation between the curl of a vector field and the torsion of a curve through that vector field at a given point? |
Fix a finite group $G$, and look at all its irreducible representations/$\mathbb{C}$. It is said in Serre's book that "there cannot be any $\mathbb{C}$-linear relation between the matrix coefficients of these irreducible representations (of course we fix a set of bases first), because of the orthogonality formulas". I ... |
The paper "<a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.13.9738">Curvature Measures of 3D Vector Fields and Their Applications</a>" by Weinkauf et al. describes how to compute the curvature and torsion of the tangent curves in terms of the spatial derivatives of the underlying vector field.
Going... |
Since this looks like homework, I won't tell you how to do it with the binomial theorem and will tell you how to do it bijectively instead. How many ways are there to choose a committee of $k$ people out of $n$ people, split the committee into Democrats and Republicans, then elect a President from that committee, for ... |
Before I begin, note that this is significantly outside what I've studied, so if it's a load of crap just let me know.
Up till now, everything I've covered assumes the same domain for all variables.
I'm trying to define an arbitrary one-to-one relationship between elements in two different sets, and want each of ... |
Since this looks like homework, I won't tell you how to do it with the binomial theorem and will tell you how to do it bijectively instead. How many ways are there to choose a committee of $k$ people out of $n$ people, split the committee into Democrats and Republicans, then elect a President from that committee, for ... |
How does one find a simple expression of the one below applying the binomial theorem:
$$\sum_{k=1}^n k \cdot 2^k{ n \choose k}$$
**Edit:**
$\frac{d}{dx}(x^k)=kx^{k-1}$
$(1 + x)^n = \sum_{k=0}^n { n \choose k}x^k = \sum_{k=0}^n { n \choose k}\frac{d}{dx}(x^k) = \sum_{k=0}^n { n \choose k}kx^{k-1} $
$n(1+x... |
How does one find a simple expression of the one below applying the binomial theorem:
$$\sum_{k=1}^n k \cdot 2^k{ n \choose k}$$
**Edit:**
$\frac{d}{dx}(x^k)=kx^{k-1}$
$(1 + x)^n = \sum_{k=0}^n { n \choose k}x^k = \sum_{k=0}^n { n \choose k}\frac{d}{dx}(x^k) = \sum_{k=0}^n { n \choose k}kx^{k-1} $
$n(1+x... |
How does one find a simple expression of the one below applying the binomial theorem:
$$\sum_{k=1}^n k \cdot 2^k{ n \choose k}$$
**Edit:**
$\frac{d}{dx}(x^k)=kx^{k-1}$
$(1 + x)^n = \sum_{k=0}^n { n \choose k}x^k = \sum_{k=0}^n { n \choose k}\frac{d}{dx}(x^k) = \sum_{k=0}^n { n \choose k}kx^{k-1} $
$n(1+x... |
**Motivation**: Assume one has a continued fraction (c.f.) $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $, where $a_n,b_n$ are integers. **I am looking for a c.f. transformation that can transform it into a new c.f. in such a way that its elements $a_{n}^{*},b_{n}^{*}$ are integers as ... |
How to prove and find the partial numerators and denominators formulae of these two generalized continued fractions given their approximants? |
You may go directly to QUESTION 1.
**Motivation**: Assume one has a continued fraction (c.f.) $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $, where $a_n,b_n$ are integers. **I am looking for a c.f. transformation that can transform it into a new c.f. in such a way that its elements ... |
How to prove and find the partial numerators and denominators formulae of two generalized continued fractions given their approximants? |
How to prove and find the partial numerators and denominators formulae of two generalized continued fractions? |
How to prove the partial denominators formula of a generalized continued fraction and find the formula for the elements of another ? |
I apologise for the really basic question. This didn't really fit on any other StackExchange website so the Maths one was the closest one where I could ask.
Really Basic Question- What are 4 sided shapes called again?
Like how triangles are 3 sided shapes, octagons are 8 sided shapes, ... What are the 4 sided on... |
What Are 4 Sided Shapes Called Again? |
Given `$X_i, Y_i \in SO_3, i=1..n, n>2$`, find `$A,B$` that satisfies `$AX_iB=Y_i$`.
For `$n=1$`, there are an infinite number of solutions, a trivial one is:
`$B=I$`
`$A=Y_1X_1^T$`
For `$n=2$`, there should be exactly one solution. One could do the following:
Compute `$A$` from the equation where `$... |
You may go directly to QUESTION 1.
**Motivation**: Assume one has a continued fraction (c.f.) $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $, where $a_n,b_n$ are integers. **I am looking for a c.f. transformation that can transform it into a new c.f. in such a way that its elements ... |
Given $X_i, Y_i \in SO_3, i=1..n, n>2$, find $A,B$ that satisfies $AX_iB=Y_i$.
For $n=1$, there are an infinite number of solutions, a trivial one is:
$B=I$
$A=Y_1X_1^T$
For $n=2$, there should be exactly one solution. One could do the following:
Compute $A$ from the equation where $i=1$: $A=Y_1B^{-1... |
How to formulate such problem mathematicaly? (line continuation search) |
The problem asks us to calculate:
$ \sum_{i = 0}^{n}(-1)^n \binom{n}{i} \binom{n}{n-i}$
The way I tried solving is:
The given sum is the coefficient of $x^n$ in $ (1+x)^n(1-x)^n $, which is $ (1 - x^2)^n $.
The coefficient of $x^n$ in $(1 -x^2)^n$ is $ (-1)^{n/2} \binom{n}{n/2} $.
Am I doing it right? |
$\mathbf{C}\mathbf{B}-\mathbf{B}\mathbf{D}=\mathbf{0}$ is a [Sylvester equation][1], for which algorithms like [Bartels-Stewart][2] or [Hessenberg-Schur][3] apply.
I haven't seen your more general problem in my notes, but I'll update this answer when I find something.
[1]: http://en.wikipedia.org/wiki/Sylveste... |
Assume $k,n$ are positive integers. The sum
$\dfrac{1}{k+1}+\dfrac{1}{k+2}+\dfrac{1}{k+3}+\ldots +\dfrac{1}{k+n-1}+\dfrac{1}{k+n}$
is not an integer.
The case $k=0$ is proved in this question "[Is there an elementary proof that $\sum_{k=1}^{n}1/k$ is never an integer?][1]"
[1]: http://math.stackexcha... |
###Background:
I was studying Theorem 2.3.3 from Introduction to Graph Theory by W. B. West. The main idea of his proof is as follows:
<br>
<b>T</b>, resulting tree.
<br>
<b>T<sup> *</sup></b>, spanning tree of minimum weight.
<br>
Let, T $\neq$ T<sup> *</sup>, then there are edges in T<sup> *</sup> that are not... |
If T is a tree and one joins two vertices of this tree that are not already joined by an edge with a new edge, one gets a unique circuit C. Now, what happens if one removes an edge from C different from the edge just added to T? One gets a new tree. |
Call a finite group A **affine** if it has a normal, self-centralizing, complemented, elementary abelian subgroup V. Such a group A is a semi-direct product G⋉V where V is a vector space of dimension n over GF(p) and G is a group of matrices in GL(n,p). The elements of A can be written as matrices [ g, v ; 0, 1 ] whe... |
When is an affine group complete? |
Suppose $\mu$ is a measure. Is there any difference in meaning between the notation
$\int f(x)d\mu(x)$
and the notation
$\int f(x) \mu(dx)$?
Many thanks.
|
The problem asks us to calculate:
$ \sum_{i = 0}^{n}(-1)^i \binom{n}{i} \binom{n}{n-i}$
The way I tried solving is:
The given sum is the coefficient of $x^n$ in $ (1+x)^n(1-x)^n $, which is $ (1 - x^2)^n $.
The coefficient of $x^n$ in $(1 -x^2)^n$ is $ (-1)^{n/2} \binom{n}{n/2} $.
Am I doing it right? |
You may go directly to QUESTION 1.
**Motivation**: Assume one has a continued fraction (c.f.) $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $, where $a_n,b_n$ are integers. **I am looking for a c.f. transformation that can transform it into a new c.f. in such a way that its elements ... |
I have coordinates for 4 vertices/points that define a plane and the normal/perpendicular.
The plane has an arbitrary rotation applied to it.
How can I 'un-rotate'/translate the points so that the plane has rotation 0 on x,y,z ?
I've tried to get the plane rotation from the plane's normal:
rotationX = ata... |
How can I remove rotations from points defining a plane ? |
You can go directly to QUESTION 1.
**Motivation**: Assume one has a continued fraction (c.f.) $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $, where $a_n,b_n$ are integers. **I am looking for a c.f. transformation that can transform it into a new c.f. in such a way that its elements ... |
I spent a little time looking around the Wikipedia and Wolfram articles on Penrose Tiling, the Domino Problem, Wang Tiles, etc., but I'm having a little trouble getting into them. A lot of these resources are fairly non-linear in that they often have circular dependencies of concepts that you need to know.
For exam... |
Suppose $\mu$ is a measure. Is there any difference in meaning between the notation
$\int f(x)d\mu$
and the notation
$\int f(x) \mu(dx)$?
Many thanks.
|
When teaching abstract vector spaces for the first time, it is handy to have some really weird examples at hand, or even some really weird non-examples that may illustrate the concept. For example, a physicist friend of mine uses "color space" as a (non) example, with two different bases given essentially {red, green, ... |
Vivid examples of vector spaces? |
For my homework, I have been asked to rationalise and simplify this surd;
11
_______
3root3 + 7
Each time I do this I get the wrong answer. The method I am using is;
(11) x (3root3 - 7)
(3root3 - 7) x (3root3 - 7)
I ended up with 33 + 11root3 - 77
_________________
9 + 3... |
For my homework, I have been asked to rationalise and simplify this surd;
$$\frac{11}{3\sqrt{3}+7}$$
Each time I do this I get the wrong answer. The method I am using is;
(11) x (3root3 - 7)
(3root3 - 7) x (3root3 - 7)
I ended up with 33 + 11root3 - 77
_________________
9 + ... |
For my homework, I have been asked to rationalise and simplify this surd;
$$\frac{11}{3\sqrt{3}+7}$$
Each time I do this I get the wrong answer. The method I am using is;
$$ \frac{11}{3\sqrt3-7} \times \frac{3\sqrt3-7}{3\sqrt3-7} $$
I ended up with $\frac{33+11\sqrt3-77}{9+3+21+7\sqrt3-21-7\sqrt3}$
This ... |
Do you mean rationalise
$$
\frac{11}{3\sqrt{3}-7} \qquad \text{?}
$$
And are you sure you're trying
$$
\frac{11}{3\sqrt{3}-7} \cdot \frac{3\sqrt{3} + 7}{3\sqrt{3} + 7}\qquad \text{?}
$$
In general, [Wikipedia][1] is almost always of great help too.
(Sorry for so many questions, but I can hardly under... |
Solving the integral $\int\limits_{0}^{\infty} \frac{\sin{x}}{x} \ dx = \frac{\pi}{2}? |
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