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Why is the following true:? If $(X,d)$ is a compact metric space and $f: X \rightarrow X$ is non-expansive (i.e $d(f(x),f(y)) \leq d(x,y)$) then $f$ is an isometry ?
|
Why is the following true?
>If $(X,d)$ is a compact metric space and $f: X \rightarrow X$ is non-expansive (i.e $d(f(x),f(y)) \leq d(x,y)$) then $f$ is an isometry.
|
Are there any interesting examples of second order stationary processes on ${\mathcal R}^2$ or ${\mathcal R}^3$ that are not isotropic? The book I am looking at has no examples. |
Why is the following true?
>If $(X,d)$ is a compact metric space and $f: X \rightarrow X$ is non-expansive (i.e $d(f(x),f(y)) \leq d(x,y)$) and surjective then $f$ is an isometry.
|
Yes there are linear algebra and graph theoretic results for this game, for any arrangement of switches and starting from all off position.
Notice that you can construct a simple graph out of the arrangement of lights, by taking the lights to be vertices, and making them adjacent in the graph if they are neighbours.... |
It is my question when I was in Senior High School. Up to now, I have no idea about the correct explanation. I just accept it by faith :D
Here is the question:
![tangent line diagram][1]
If the symbol $\Delta x \to 0$ does not mean $\Delta x =0$, how can the points $A$ and $B$ get overlapped which in turn, ho... |
Approaching to zero, but not equal to zero, then why do the points get overlapped? |
In fact more is true. For a compact, metric space, $f$ locally non-expansive $\implies$ $f$ local isometry.
This is theorem 4.2 in the paper linked below.
http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.pjm/1102970464 |
Basically, what you have put your finger on is the original difficulty with <a href="http://en.wikipedia.org/wiki/Infinitesimals">infinitesimals</a>, and the basis on which Bishop Berkeley made some of his famous objections (mind you, Berkeley didn't really care about the logical foundations of Calculus; he was engaged... |
It is my question when I was in Senior High School. Up to now, I have no idea about the correct explanation. I just accept it by faith :D
Here is the question:
![tangent line diagram][1]
If the symbol $\Delta x \to 0$ does not mean $\Delta x =0$, how can the points $A$ and $B$ get overlapped which in turn, ho... |
Given a Noetherian, local ring $(R,m)$, can we always find a system of parameters whose images in the cotangent space $m/m^2$ are linearly independent. We can do this in the regular case, by just choosing a basis for the cotangent space and looking at their preimages in $m$ under the canonical map. By Nakayama's lemma ... |
Basically, what you have put your finger on is the original difficulty with <a href="http://en.wikipedia.org/wiki/Infinitesimals">infinitesimals</a>, and the basis on which Bishop Berkeley made some of his famous objections (mind you, Berkeley didn't really care about the logical foundations of Calculus; he was engaged... |
Basically, what you have put your finger on is the original difficulty with <a href="http://en.wikipedia.org/wiki/Infinitesimals">infinitesimals</a>, and the basis on which Bishop Berkeley made some of his famous objections (mind you, Berkeley didn't really care about the logical foundations of Calculus; he was engaged... |
Basically, what you have put your finger on is the original difficulty with <a href="http://en.wikipedia.org/wiki/Infinitesimals">infinitesimals</a>, and the basis on which Bishop Berkeley made some of his famous objections (mind you, Berkeley didn't really care about the logical foundations of Calculus; he was engaged... |
Basically, what you have put your finger on is the original difficulty with <a href="http://en.wikipedia.org/wiki/Infinitesimals">infinitesimals</a>, and the basis on which Bishop Berkeley made some of his famous objections (mind you, Berkeley didn't really care about the logical foundations of Calculus; he was engaged... |
my Calculus II class is nearing the end of the quarter and we've just started differential equations to get ready for Calculus III. In my homework, I came upon these problems.
One of the problems was:
Find the general solution to the differential equation dy/dt = t^3 + 2t^2 - 8t.
The teacher just said to integ... |
my Calculus II class is nearing the end of the quarter and we've just started differential equations to get ready for Calculus III. In my homework, I came upon these problems.
One of the problems was:
Find the general solution to the differential equation
$$\frac{dy}{dt} = t^3 + 2t^2 - 8t.$$
The teacher just... |
Hint 1:
The first equation is very easy, so :
$$\frac{dy}{dt}=t^3 + 2t^2 - 8t \Rightarrow dy = (t^3 + 2t^2 - 8t) dt \Rightarrow $$
$$ \int dy = \int (t^3 + 2t^2 - 8t) dt $$ ( which is the left side ? )
Hint2:
For the other.
$$\frac{dy}{dt}=y^3 + 2y^2 - 8y \Rightarrow \frac{1}{y^3 + 2y^2 - 8y}dy = ... |
my Calculus II class is nearing the end of the quarter and we've just started differential equations to get ready for Calculus III. In my homework, I came upon these problems.
One of the problems was:
Find the general solution to the differential equation
$$\frac{dy}{dt} = t^3 + 2t^2 - 8t.$$
The teacher just... |
In the first question, you are given the derivative in terms of the variable. But in the second question, you are given an expression for the derivative that involves *the function*. For instance, it would be one thing if you were told $\frac{dy}{dx} = x$ (which would mean that $y = \frac{1}{2}x^2 + C$), and a complete... |
The difference is that the main variable is $y$ and you separate the variables to integrate and obtain the solution
Hint 1:
The first equation is very easy, so :
$$\frac{dy}{dt}=t^3 + 2t^2 - 8t \Rightarrow dy = (t^3 + 2t^2 - 8t) dt \Rightarrow $$
$$ \int dy = \int (t^3 + 2t^2 - 8t) dt $$ ( which is th... |
The difference is that the main variable is $y$ and you separate the variables to integrate and obtain the solution
**Hint 1:**
The first equation is very easy, so :
$$\frac{dy}{dt}=t^3 + 2t^2 - 8t \Rightarrow dy = (t^3 + 2t^2 - 8t) dt \Rightarrow $$
$$ \int dy = \int (t^3 + 2t^2 - 8t) dt $$ ( which i... |
I know from an online calculator <http://www.numberempire.com/derivatives.php> that
$\lim_{x \to \infty} \left ( \frac{x+2}{x} \right )^{x}=e^{2}$. How do you calculate this step by step? |
This is more or less by definition of $e$, depending on which definition you use. Do you know the definition $e=\lim_{n\rightarrow\infty} (1+\frac{1}{n})^n$? Then set $x=2n$ and you are done. |
Three planes *A*, *B* and *C* intersect at point P. The dihedral angle between *A* and *B* is $\theta$ and the dihedral angle between *B* and *C* is $\psi$.
1. Solve for the dihedral angle between by *A* and *C*.
2. Planes *A* and *B* intersect plane *C* in two lines that form angle QPR. That is, point Q lies i... |
Find a step-by-step derivation here (and not only for this special problem):
http://www.wolframalpha.com/input/?i=limit+((x%2B2)/x)^x+x-%3Eoo
(if the link doesn't work copy and paste the whole line)
...or go to http://www.wolframalpha.com directly and type:<br>
"limit ((x+2)/x)^x x->oo"
Click on "Show st... |
Could someone give me an example of a (smooth) $n$-manifold $(n=2, 3)$ which cannot be embedded (or immersed) in $\mathbb R^4$?
Thanks in advance!
S. L. |
I [tried][1] (and [tried again][2]) to improve the Wikipedia article but the edits were promptly reverted as if well-known solutions of Lights Out are "original research", despite providing several citations.
If any Stack Exchange members are Wikipedians who can assist in restoring the article, it would be greatly a... |
In Linear Algebra, when computing an inner product $<x,y> = y^*Wx$, what is the name of the matrix W?
If it doesn't have a name, where can I find a practical explanation of how to construct it for a particular problem or space?
Is there a text on the subject that explains this simply? A note on my background,... |
What is the name of the matrix used to weight an inner product? |
In Linear Algebra, when computing an inner product $<x,y> = y^*Wx$, what is the name of the matrix W?
If it doesn't have a name, where can I find a practical explanation of how to construct it for a particular problem or space?
Is there a text on the subject that explains this simply? A note on my background,... |
I finding for correct proof of this statement: If group $G/Z(G)$ is cyclic, then $G$ is commutative.
Proof: $G/Z(G)$ is isomorphic to $In(G)$ that is cyclic, and than for every inner isomorphism $\gamma_a$ is $\gamma_a \gamma_b = \gamma_{ab} = \gamma_{ba} = \gamma_b \gamma_a$ and therefore for every $a,b \in G$ is $... |
How to prove DeMorgan's Law?
$$A - (B \cup C) = (A - B) \cap (A - C)$$
$$A - (B \cap C) = (A - B) \cup (A - C)$$ |
How to prove DeMorgan's law? |
A lot of textbooks said it was hard for human to accept zero when it was first introduced.
How could it be? It seems to me as natural as postive integer which represent there is no elements at all. |
What's the hard part of zero? |
In Linear Algebra, when computing an inner product $<x,y> = y^*Wx$, what is the name of the matrix W?
If it doesn't have a name, where can I find a practical explanation of how to construct it for a particular problem or space?
Is there a text on the subject that explains this simply? A note on my background,... |
Maybe you're familiar with the fact that if $A\subseteq B$ and $B\subseteq A$, then $A=B$?
Since you're viewing $A,B,C$ as sets, you can prove these by showing the set on the left of $=$ is a subset of the set on the right of $=$, and vice versa.
For example, suppose $x\in A-(B\cup C)$. So $x\in A$, but $x\not\in... |
How to prove DeMorgan's Law?
$$A - (B \cup C) = (A - B) \cap (A - C)$$
$$A - (B \cap C) = (A - B) \cup (A - C)$$
**EDIT**:
Here is what I have tried so far:
Considering the first equation, assuming $x \in A - (B \cup C)$ then $x \in A$ and $x \not\in B$ and $x \not\in C$, while the right hand means ($x \in A... |
How to prove DeMorgan's Law?
$$A - (B \cup C) = (A - B) \cap (A - C)$$
$$A - (B \cap C) = (A - B) \cup (A - C)$$
**EDIT**:
Here is what I have tried so far:
Considering the first equation, assuming $x \in A - (B \cup C)$ then $x \in A$ and $x \not\in B$ and $x \not\in C$, while the right hand means ($x \... |
Of course it seems natural to you; you grew up in the modern world, where everyone accepts zero. More importantly, people now accept the abstract concept of numbers and are capable of divorcing them from the things that they represent. This is a sophisticated point of view. From a more naive point of view, a number ... |
In Linear Algebra, when computing an inner product $<x,y> = y^*Wx$, what is the name of the matrix W?
If it doesn't have a name, where can I find a practical explanation of how to construct it for a particular problem or space?
Is there a text on the subject that explains this simply? A note on my background,... |
I stumbled across this paragraph:
> Hence, user b cannot decrypt C
> directly. But using e and d , user b
> can factor N.
How is it possible to speedup to prime factorization when knowing e and d?
|
How possible methods for finding groups of given order we have?
Sylow theorems - and next? We know something about center of group, about possible orders of these elements and subgroups. But what is most effective way, to find all non-isomorphic groups of given order? Is there any other strong "weapon" for this (lik... |
I have only learnt a bit of calculus and does not know what does manifold or metric space mean. But now I want to know what does tangent vector or tangent plane mean.
Here is what my textbook told me.
Assuming the formula for surface $\Sigma$ is $$F(x,y,z) = 0$$
Suppose $X_0 = (x_0, y_0, z_0)$ is a point on the... |
Help understanding the definition of tangent vector or tangent plane? |
For $n > 2$, is it possible to embed $S^1 x ... \mathrm{n\ times} ... x S^1$ into $\mathbb R^{n+1}$?
|
Here is what my textbook told me:
Assuming the formula for surface $\Sigma$ is $$F(x,y,z) = 0$$
Suppose $X_0 = (x_0, y_0, z_0)$ is a point on the surface $\Sigma$ and we assuming F(x,y,z) is differentiable and $$\mathbf{J}F(X_0) = (\frac{\partial F(X_0)}{\partial x}, \frac{\partial F(X_0)}{\partial y}, \frac{\part... |
For $n > 2$, is it possible to embed $\underbrace{S^1 \times \cdots \times S^1}_{n\text{ times}}$ into $\mathbb R^{n+1}$?
|
I have a uniform grid of square cells and a point on that grid. I want to calculate the intersection of cells on the grid, with a circular area centered on the point, with a radius R.
Does anyone know of an efficient way of doing this?
Thanks! |
Follow-up to <a href="http://math.stackexchange.com/questions/12193/field-reductions-part-two">Field reductions. part two</a>
For a field $K$, an element $a \in K$, and $K(\setminus a)$ the set of field reductions (maximal subfields of $K$ not containing $a$) are there any interesting equivalence classes on $K(\setm... |
What is the limit of the series 1/(2n)! for n in [0, infinity) ?
I've ground out the sum of the 1st 1000 terms to 1000 digits using Python,
(see http://tutoree7.pastebin.com/pv0i1XQb ), but how would a mathematician calculate the limit? And what is it?
No, this isn't homework. I'm 73. Just curious.
Thanks |
What is limit of 1/(2n)! for n in [0, infinity) ? |
I stumbled across this paragraph in a paper:
> Hence, user b cannot decrypt C
> directly. But using e and d , user b
> can quickly factor N.
How is it possible to speedup the prime factorization when knowing e and d?
|
I stumbled across this paragraph in a paper:
> Hence, user b cannot decrypt C
> directly. But using e and d , user b
> can quickly factor N.
How is it possible to speedup the prime factorization when knowing e (public key) and d (private key)?
**For clarification:**
RSA provides us with these equations... |
If you don't like semi-direct products, then you will not get very far in your endeavour. I recommend that you play around with them until you start liking them. The reason is that the single most powerful result in trying to classify all finite groups of given order is the so-called Schur-Zassenhaus theorem:
**Schu... |
Follow-up to <a href="http://math.stackexchange.com/questions/12193/field-reductions-part-two">Field reductions. part two</a>
For a field $K$, an element $a \in K$, and $K(\setminus a)$ the set of field reductions (maximal subfields of $K$ not containing $a$) are there any interesting equivalence classes on $K(\setm... |
Here's how to finish without assuming the Sylow 5-subgroup is normal.
(Assume by way of contradiction:) If $n=2$, so $g^{-1} f g = ff$, then: $$g^{-2} f g^2 = g^{-1}( g^{-1} f g ) g = g^{-1}( ff )g = g^{-1}( fg g^{-1}f)g = (g^{-1}fg) (g^{-1}fg) = (ff) (ff) = f^4.$$ Every time you apply $g^{-1}( * )g$, you double the... |
Follow-up to <a href="http://math.stackexchange.com/questions/12193/field-reductions-part-two">Field reductions. part two</a>
For a field $K$, an element $a \in K$, and $K(\setminus a)$ the set of field reductions (maximal subfields of $K$ not containing $a$) are there any interesting equivalence classes on $K(\setm... |
Follow-up to <a href="http://math.stackexchange.com/questions/12193/field-reductions-part-two">Field reductions. part two</a>
For a field $K$, an element $a \in K$, and $K(\setminus a)$ the set of field reductions (maximal subfields of $K$ not containing $a$) are there any interesting equivalence classes on $K(\setm... |
Follow-up to <a href="http://math.stackexchange.com/questions/12193/field-reductions-part-two">Field reductions. part two</a>
For a field $K$, an element $a \in K$, and $K(\setminus a)$ the set of field reductions (maximal subfields of $K$ not containing $a$) are there any interesting equivalence classes on $K(\setm... |
Some hints: what is the third angle? Draw a picture. Can you spot some symmetry?
If not, then decode the following hint with [http://www.rot13.com/index.php][1].
But please, spend some time trying first.
Ersyrpg gur gevnatyr va gur evtug natyr. Jung vf fb fcrpvny va gur erfhygvat gevnatyr? Gel znxvat hfr bs gur rk... |
What is limit of $\sum_{n=0}^{\infty}\frac{1}{(2n)!} $ ? |
The problem was that math started out of necessity. Shepherds needed to count how many sheep they own, parents needed to count how many children they had, etc. For most simple mathematics, zero is simply a place-holder. Who needs to count zero sheep? Even the most basic forms of counting--using your fingers--who ha... |
Basically, what you have put your finger on is the original difficulty with <a href="http://en.wikipedia.org/wiki/Infinitesimals">infinitesimals</a>, and the basis on which Bishop Berkeley made some of his famous objections (mind you, Berkeley didn't really care about the logical foundations of Calculus; he was engaged... |
What are the necessary and sufficient conditions on a vector field $F$ for the divergence $\nabla\cdot F$ to exist at a given point.
**EDIT**
In [Divergence][1] in the second line under the heading "**Application in Cartesian coordinates**", why is it assumed that $\vec{F}$ to be a continuously differentiable v... |
Although I am not (by any stretch) an expert on finite simple groups, let me flesh out my above comment.
Consider the following **QCFSG** (i.e., "qualitative" CFSG): with only finitely many exceptions, every finite simple group has prime order, is alternating, or is one of the finitely many known infinite families o... |
What are the necessary and sufficient conditions on a vector field $F$ for the divergence $\nabla\cdot F$ to exist at a given point.
**EDIT**
In [Divergence][1] in the second line under the heading "**Application in Cartesian coordinates**", why is it assumed that $\vec{F}$ to be a continuously differentiable v... |
The Abramowitz–Stegun Handbook of Mathematical Functions claims on page 787 that all the roots are simple: http://convertit.com/Go/ConvertIt/Reference/AMS55.ASP?Res=150&Page=787
|
This is a follow-up to the question
[on field reductions][1].
There isn't a unique largest subfield of $\mathbb{R}$ not containing $a$,
so let $\{\mathbb{R}(\setminus a)\}$ denote the set of maximal subfields of $\mathbb{R}$ not containing $a$,
What is the cardinality of $\{\mathbb{R}(\setminus a)\}$ ?
From... |
Original post on Mathoverflow [here](http://mathoverflow.net/questions/47487/probability-of-the-maximum-levy-stable-random-variable-in-a-list-being-greater).
Given a list of identical and independently distributed [Levy Stable](http://en.wikipedia.org/wiki/Stable_distribution) random variables, $(X_0, X_1, \dots, X_... |
This is a follow-up to the question
[on field reductions][1].
There isn't a unique largest subfield of $\mathbb{R}$ not containing $a$,
so let $\{\mathbb{R}(\setminus a)\}$ denote the set of maximal subfields of $\mathbb{R}$ not containing $a$,
What is the cardinality of $\{\mathbb{R}(\setminus a)\}$ ?
From... |
What you are asking for is a special case of the joint distribution of the maximum and sum of a sequence of random variables. The fact that the variables are stable simplifies this and allows us to compute the joint distribution in the limit as n goes to infinity. This limit can be described in terms of the joint distr... |
This is a follow-up to the question
[on field reductions][1].
(EDIT: Originally this question used the notation $\{\mathbb{R}(\setminus a)\}$, but now uses $\mathbb{R}(\setminus a)$ instead.)
There isn't a unique largest subfield of $\mathbb{R}$ not containing $a$,
so let $\mathbb{R}(\setminus a)$ denote the ... |
What you are asking for is a special case of the joint distribution of the maximum and sum of a sequence of random variables. The fact that the variables are stable simplifies this and allows us to compute the joint distribution in the limit as n goes to infinity. This limit can be described in terms of the joint distr... |
What are the necessary and sufficient conditions on a vector field $F$ for the divergence $\nabla\cdot F$ to exist at a given point.
**EDIT**
In [Divergence][1] in the second line under the heading "**Application in Cartesian coordinates**", why is it assumed that $\vec{F}$ to be a continuously differentiable v... |
What is the Fourier transform of the product of two functions? |
I have a couple questions that I'd like answered:
(1) How do I evaluate this integral?
(2) Is it possible to do a substitution? What about $u=e^{i \cdot t}$?
(3) What if 5 and 3 are replaced by other constant integers?
(4) What should be the most effective method to handle 3, using the fewest calcula... |
I'll place something as an answer in the interest of not having questions with no answers.
I don't know what the cardinality of $\{\mathbb{R}(\setminus a)\}$ is, but I would be surprised if it is not uncountable. I'll see if I can think about it some in the next couple of days.
As for your final paragraph, the di... |
I have a couple questions that I'd like answered:
(1) How do I evaluate this integral?
(2) Is it possible to do a substitution? What about $u=e^{i \cdot t}$?
(3) What if 5 and 3 are replaced by other constant integers?
(4) What should be the most effective method to handle question (3), using the few... |
I'll place something as an answer in the interest of not having questions with no answers.
I don't know what the cardinality of $\{\mathbb{R}(\setminus a)\}$ is, but I would be surprised if it is not uncountable. I'll see if I can think about it some in the next couple of days.
As for your final paragraph, the di... |
Are there any interesting examples of second order stationary processes on ${\mathcal R}^2$ or ${\mathcal R}^3$ that are not isotropic? The book I am looking at has no examples.
Update : I asked this [question][1] in mathoverflow, apparently such examples are not easy to come by.
[1]: http://mathoverflow.net/... |
I'll place something as an answer in the interest of not having questions with no answers.
I don't know what the cardinality of $\{\mathbb{R}(\setminus a)\}$ is, but I would be surprised if it is not uncountable. I'll see if I can think about it some in the next couple of days.
As for your final paragraph, the di... |
In Linear Algebra, when computing an inner product $<x,y> = y^*Wx$, what is the name of the matrix W?
If it doesn't have a name, where can I find a practical explanation of how to construct it for a particular problem or space?
Is there a text on the subject that explains this simply? A note on my background,... |
Are there any interesting examples of second order stationary processes on ${\mathcal R}^2$ or ${\mathcal R}^3$ that are not isotropic? The book I am looking at has no examples.
Update : I asked this [question][1] in mathoverflow, apparently such examples are not easy to come by.
Update: Processes with [anisotrop... |
I have a system that is modeled by the following differential equation:
db/dt = j(t)*ha + k*(Ta-b(t))
where db/dt is the rate of temperature change, j(t) is an input, ha, k, Ta are all constants, and b(t) is an output. Note that this is newtonian cooling with a heating input, j(t)*ha.
I want to find the trans... |
I have data for points of a 3D grid. The points of the grid are generated from three nonorthogonal vectors: each grid point has coordinates $\mathbf{q}_{ijk} = i \mathbf{a} + j \mathbf{b} + k \mathbf{c}$, where $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ are nonorthogonal, noncoplanar vectors. (In cristallography, it's... |
Suppose I is an index set (not necessarily countable). Say we have a family of subsets $\{C_{i}: i \in I\}$ and suppose that this family is a cover of a space $X$, that is, for each $x$ in $X$ we have $x \in C_{j}$ for some $j \in I$.
Can we say the following? for each $x \in X$ there exists some natural number $n... |
Let's say that $f(x)=f^{1}(x)$ and that $f(f(x))=f^{2}(x)$. Moreover, $f^{n}(x)$ is the n-th iterate of $f(x)$, for $n \in \mathbb{N}$. I'm curious about extending iteration to larger number sets. For $n \in \mathbb{R}$, there's the concept <a href="http://en.wikipedia.org/wiki/Flow_(mathematics)">flow</a>. I don't und... |
Explicit examples of functions with flow? |
Suppose I is an index set (not necessarily countable). Say we have a family of subsets $\{C_{i}: i \in I\}$ and suppose that this family is a cover of a space $X$, that is, for each $x$ in $X$ we have $x \in C_{j}$ for some $j \in I$.
Can we say the following? for each $x \in X$ there exists some natural number $n$ ... |
Given a topological space (with nice enough conditions, maybe Hausdorff, compactly generated, or CW complex, I'm not sure) $X$ and a subspace $A\subset X$, is it true that $H^n(X,A)\cong H^n(X\backslash A)$ whenever there is an open set containing $A$ which can be retracted to $A$? Can this be shown easily using the M... |
Question: (a) Let $f: V \rightarrow W$ with $ V,W \simeq \mathbb{R}^{3}$ given by: $$f(x_1, x_2, x_3) = (x_1 - x_3, 2x_1 -5x_2 -x_3, x_2 + x_3).$$
Determine the matrix of $f$ relative to the basis $\{(0,2,1),(-1,1,1),(2,-1,1)\}$ of $V$ and $\{(-1,-1,0),(1,-1,2),(0,2,0)\}$ of $W$.
(b) Let $n \in \mathbb{N}$ and $U... |
Let's say that $f(x)=f^{1}(x)$ and that $f(f(x))=f^{2}(x)$. Moreover, $f^{n}(x)$ is the n-th iterate of $f(x)$, for $n \in \mathbb{N}$. I'm curious about extending iteration to larger number sets. For $n \in \mathbb{R}$, there's the concept <a href="http://en.wikipedia.org/wiki/Flow_(mathematics)">flow</a> (I think?). ... |
Question: (a) Let $f: V \rightarrow W$ with $ V,W \simeq \mathbb{R}^{3}$ given by: $$f(x_1, x_2, x_3) = (x_1 - x_3, 2x_1 -5x_2 -x_3, x_2 + x_3).$$
Determine the matrix of $f$ relative to the basis $\{(0,2,1),(-1,1,1),(2,-1,1)\}$ of $V$ and $\{(-1,-1,0),(1,-1,2),(0,2,0)\}$ of $W$.
(b) Let $n \in \mathbb{N}$ and $U... |
Let's say that $f(x)=f^{1}(x)$ and that $f(f(x))=f^{2}(x)$. Moreover, $f^{n}(x)$ is the n-th iterate of $f(x)$, for $n \in \mathbb{N}$. I'm curious about extending iteration to larger number sets. For $n \in \mathbb{R}$, there's the concept <a href="http://en.wikipedia.org/wiki/Flow_(mathematics)">flow</a> (I think?). ... |
I have calculate radius, `R`, for a point traveling at velocity, `V`. If we start at point, `(X,Y)`, what is the position of the point at time, `t`.
For example
R = 52
V = 250
theta = 1 degrees
Starting Point = (0,0)
Where is the point at t = 2?
|
The following (famous) question is perhaps in the spirit of the OP's question:
If $M$ and $N$ are closed aspherical $n$-manifolds with isomorphic fundamental groups, are $M$ and $N$ homeomorphic? (This is known as the Borel Conjecture.)
Scott Carter's example illustrates the need for the assumption that the mani... |
I have a radius, `R`, for a point traveling at velocity, `V`. If we start at point, `(X,Y)`, what is the position of the point at time, `t`.
For example
R = 52
V = 250
theta = 1 degrees
Starting Point = (0,0)
Where is the point at t = 2?
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Let's say that $f(x)=f^{1}(x)$ and that $f(f(x))=f^{2}(x)$. Moreover, $f^{n}(x)$ is the n-th iterate of $f(x)$, for $n \in \mathbb{N}$. I'm curious about extending iteration to larger number sets. For $n \in \mathbb{R}$, there's the concept <a href="http://en.wikipedia.org/wiki/Flow_(mathematics)">flow</a> (I think?). ... |
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