instruction stringlengths 12 30k |
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It suffices to find a path to the 0 function from any other continuous function.
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be any continuous function and define $F(x,t) = tf(x)$. Assuming we can prove this is, in fact, continuous, note that $F(x,1) = f(x)$ and $F(x,0)$ is the $0$ function.
So, why is it continuous... |
It is often said that differential forms (sections of an exterior power of the cotangent bundle) are the things that you can integrate. But unless I'm being thoroughly dense differential forms are not the only things that you can integrate, c.f. the arclength form (on a 2d manifold) $ds=\sqrt{dx^2+dy^2}$, the unsigned ... |
What is the theory of non-linear forms (as contrasted to the theory of differential forms)? |
How can we prove that it is posible to arrange numbers 1,2,3,4.........n in a row so that the average of any two of these numbers never appears between them? |
There are two components to the original question:
* Why do exponential functions generally (the class $x \mapsto ab^x$ for $b>0$) have the property of being eigenfunctions of the derivative operator? That is: why do these functions f have the property $\frac{\mathrm d}{\mathrm dx} f(x) = \lambda f(x)$?
* Why are... |
I haven't studied properly the theory of infinities yet.
Let A_0 denote the set of natural numbers. Let A_i+1 denote the set whose elements are all the subsets of A_i for i=0,...,n,...
I understand well that the cardinality of A_i+1 is always greater than the cardinality of A_i for all i in N={0,1,...,n,...}.
... |
I would like to know whether the number of infinity sets of different cardinalities is uncountable or countable. Is the proof simple?
Thanks.
|
I'm making a game which you can see here, if you are on Windows or Linux: http://insertnamehere.org/birdsofprey/
If you click and hold your mouse on a bird, you can see I'm just swinging the bird back and forth in pendulum motion. I would like to, instead, implement a more realistic motion, where the movement of you... |
The mouse velocity is [(current mouse position) - (previous mouse position)]/timestep (presumably you're using a constant timestep of 10ms or something?) and then the acceleration is similarly [(current mouse velocity) - (previous mouse velocity)]/timestep. (At least, approximately.)
Once you have the acceleration, ... |
If you haven't been using "The Art and Craft of Problem Solving" by Zeitz, you should. Like, right at this moment. :)
For neat problems... you might want to look into "Which Way Did The Bicycle Go?" by Konhauser, Wagon and Velleman. |
I am interested in the Gelfand transformation
$$
\Phi\colon\ell^1(\mathbb Z)\to\mathcal C(\mathbb T),\quad a\mapsto\sum_{n\in\mathbb Z}a_n z^n.
$$
This is an injective homomorphism of Banach algebras. It is neither isometric nor surjective. However, its image---the Wiener algebra $W$ consisting of all continuous fu... |
Personal experiences as an elementary school math resource teacher and a 1st grade teacher have led me to believe that mathematics learning should be engaging and fun at any level (including the graduate level), but clearly it is of the utmost importance to effectively engage students at an early age, so as to hopefull... |
How can I find the number of *k*-permutations of *n* objects, where there are *x* types of objects, and r<sub>1</sub>, r<sub>2</sub>, r<sub>3</sub> ... r<sub>x</sub> give the number of each type of object?
Example:
> I have 20 letters from the alphabet. There are some duplicates - 4 of them are *a*, 5 of them are... |
Is there a comprehensive resource listing binomial identities? I am more interested in combinatorial proofs of such identities, but even a list without proofs will do. |
A comprehensive list of binomial identities? |
"*Life is open book.*"
With the advent of widely accessible, inexpensive (or even free) computational tools and Computer Algebra Systems (TI-89, Wolfram|Alpha, etc.), much of what traditionally comprises a high school math curriculum can now easily be done by almost everyone. Factoring polynomials, solving inequali... |
What should the high school math curriculum consist of? |
Contrary to the responses thus far (and with no disrespect intended to their authors), I think we can say quite unequivocally that there are ***strictly more*** than countably-many sets with distinct infinite cardinalities, and that (ipso facto) there are uncountably many such sets.
The answers before mine fixate on... |
The Banach--Tarski theorem applies only in the case of three or more dimensions. In 3D, there are five regular solids, two of them being not at all obvious, and the 4D case is also interesting; but the higher-dimensional cases each yield just three "solids", only one of which isn't obvious. In dynamics, particles tend ... |
Why is the 3D case so rich? |
Contrary to the responses thus far (and with no disrespect intended to their authors), I think we can say quite unequivocally that there are ***strictly more*** than countably-many sets with distinct infinite cardinalities, and that (ipso facto) there are uncountably many such sets.
The answers before mine fixate on... |
Prove that $m = S_m (\mod 3)$? |
i know the method of solving the equation like this |2x+1|=|3x+9| but the problem is if the same equation would be like this | 2x+1 | = x | 3x+9 |, how can i solve this ???????????? |
how can i solve this? |
When I was first learning about the axiom of choice, a very helpful and fun exercise was to try to find a rule that maps every nonempty subset of $\mathbb R$ to one of its members. No matter how hard you try, it was impossible to find one. (If you try the same for the nonempty subsets of $\mathbb N$, it is pretty eas... |
I know the method of solving the equation like this $|2x+1|=|3x+9|$ but the problem is if the same equation would be like this $| 2x+1 | = x | 3x+9 |$, how can I solve this? |
Can anyone help me out here? Can't seem to find the right rules of divisibility to show this. |
In a recent discussion, someone told me tuples in the modern meaning (in particular, tuples are heterogeneous: that is, different elements of a tuple can belong to different sets/have different types) first appeared in Codd's [tuple calculus][1]. I was surprised it would be so late, but [searching Google Books][2] bef... |
When did the term "tuple" get its current meaning? |
Given:
A = (0,0)
B = (0,-10)
Using the angle to C, **how are the coordinates at C calculated?** |
> **Possible Duplicate:**
> [Question about solving absolute values](http://math.stackexchange.com/questions/2887/question-about-solving-absolute-values)
<!-- End of automatically inserted text -->
I know the method of solving the equation like this $|2x+1|=|3x+9|$ but the problem is if the same equation would be... |
Given:
A = (0,0)
B = (0,-10)
AB = AC
Using the angle between AB and AC, **how are the coordinates at C calculated?** |
> A pedestrian was at 5 am in the
> morning from A to B, a distance of 30
> km. At 7 am a bike rider, that his
> speed was 2 times greater than the
> pedestrian's speed, rode from A to B
> too. After some time, the bike rider reached the pedestrian, and continued his way. Two hours after the meeting **the pedestri... |
I'm going to be a bit contrary: is this really true? Or is it just that we started studying the 3D case before we could study anything higher-dimensional, and that our brains are obviously optimized for thinking in three dimensions? |
I am interested in continuous functions which are not smooth, i.e. which have only finitely many derivatives (for now just taking the case f:R->R). As far as I can tell, there are only 2 features or properties of such a function which can make it non-smooth.
1. The derivative of some order diverges to infinity at ... |
I am interested in continuous functions which are not smooth, i.e. which have only finitely many derivatives (for now just taking the case f:R->R). As far as I can tell, there are only 2 features or properties of such a function which can make it non-smooth.
1. The derivative of some order diverges to infinity at ... |
I am interested in continuous functions which are not smooth, i.e. which have only finitely many derivatives (for now just taking the case f:R->R). As far as I can tell, there are only 2 features or properties of such a function which can make it non-smooth.
1. The derivative of some order diverges to infinity at ... |
> At 5am, a pedestrian starts walking from A to B, a distance of 30 km. At 7 am a bicyclist whose speed is 2 times the pedestrian's speed, started riding from A to B, too. After some time, the bicyclist met and passed the pedestrian. Two hours after the meeting **the pedestrian** reached his destination (B). (The bicyc... |
If I want to show a topological subspace is closed in an ambient space, does it suffice to know what happens on an open cover of the ambient space? More specifically,
>Let $X$ is a topological space with a given open cover $\{ U_i \}$. Suppose that $Z \subset X$ is a set such that $Z \cap U_i$ is closed in $U_i$ for... |
Is whether a set is closed or not a local property? |
>At 5am, a pedestrian starts walking from A to B, a distance of 30 km.
Let speed of pedestrian be $v$. At time $t$, pedestrian is at distance $vt$. (Where $t$ denotes time since pedestrian started.)
>At 7 am a bicyclist whose speed is 2 times the pedestrian's speed, started riding from A to B, too.
Speed of cy... |
An absolute value expression such as $|ax-b|$ can be rewritten in two cases as $|ax-b|=\begin{cases}
ax-b & \text{ if } x\ge \frac{b}{a} \\\\
b-ax & \text{ if } x< \frac{b}{a}
\end{cases}$, so an equation with $n$ separate absolute value expressions can be split up into $2^n$ cases, but is there a better way?
F... |
What is the best way to solve an equation involving multiple absolute values? |
This was part of a [homework problem][1] from J.B. Conway's complex analysis text which I was assigned long ago but didn't get. A few years later I was a TA for a course where the problem was assigned. I still didn't know how to solve it, nor did any of my students. If you can give me a simple answer, my gratefulnes... |
What is the image near the essential singularity of z sin(1/z)? |
It can (apparently) be shown that the equation
$\sin z = Az$ has infinitely many roots for any complex number $A$.
(For instance see this: http://books.google.com/books?id=DYCOCBCBwoIC&pg=PA270&lpg=PA270&source=bl&ots=-idQQ8Me9b&sig=xMkSpCl5nTnxFLnLW2ZhcHthKeA&hl=en&ei=eAVzTOzfF4P58AbctaybDQ&sa=X&oi=book_result&c... |
It can (apparently) be shown that the equation
$\sin z = Az$ has infinitely many roots for any complex number $A$.
(For instance see this: http://books.google.com/books?id=DYCOCBCBwoIC&pg=PA270&lpg=PA270&source=bl&ots=-idQQ8Me9b&sig=xMkSpCl5nTnxFLnLW2ZhcHthKeA&hl=en&ei=eAVzTOzfF4P58AbctaybDQ&sa=X&oi=book_result&c... |
It can (apparently) be shown that the equation
$\sin z = Az$ has infinitely many roots for any complex number $A$.
(For instance see this: http://books.google.com/books?id=DYCOCBCBwoIC&pg=PA270)
Given any $\delta > 0$ there must be such a root with $|z| > \frac{1}{\delta}$, otherwise we can show that $\sin z -... |
Let $p$ be a prime number. Prove that every algebraic integer of the $p$th cyclotomic field can be represented as a sum of (finitely many) distinct units of the ring of algebraic integers of the field.
Reference: http://www.artofproblemsolving.com/Forum/resources.php?c=2&cid=152&year=1977&sid=151602f87027a7ce87d3aa9... |
Let $p >5$ be a prime number. Prove that every algebraic integer of the $p$th cyclotomic field can be represented as a sum of (finitely many) distinct units of the ring of algebraic integers of the field.
Reference: http://www.artofproblemsolving.com/Forum/resources.php?c=2&cid=152&year=1977&sid=151602f87027a7ce87d3... |
Miklos Schweitzer is a _very hard_ contest.
Anyway, solution for this (and other problems) can be found in the book:
Contests in Higher Mathematics, published by Springer.
Google books has it:
http://books.google.com/books?id=2wwXImJ2HocC
And this particular problem's solution appears here:
http://boo... |
Serre's criterion for affineness (Hartshorne III.3.7) states that:
> Let $X$ be a noetherian scheme. Suppose $H^1(X, \mathcal{F})= 0$ for every quasi-coherent sheaf on $X$. Then $X$ is affine.
There is a more general statement (EGA II.5.2.1) that reaches the same conclusion with the hypothesis that $X$ is a separ... |
Why do we need noetherianness (or something like it) for Serre's criterion for affineness? |
Here is a time sample: $Q = \\{(t_i, x_i) | 0 \leq x_i \leq x_{i+1}, 1 \leq i \leq n\\}$
and rules:
(1) $T_1 \leq t_{i+1} - t_i < T_2$ where $T_1, T_2 > 0$
(2) $x_i$ comes with error:
$x_i = \lfloor x_{t_i} \rfloor$ where $\lfloor x \rfloor$ - whole part of $x$, $x_{t_i}$ - voluntary unknown law of object ... |
Here is a time sample: $Q = \\{(t_i, x_i) | 0 \leq x_i \leq x_{i+1}, 1 \leq i \leq n\\}$
and rules:
(1) $T_1 \leq t_{i+1} - t_i < T_2$ where $T_1, T_2 > 0$
(2) $x_i$ comes with error:
$x_i = \lfloor x(t_i) \rfloor$ where $\lfloor x \rfloor$ - whole part of $x$, $x(t_i)$ - voluntary unknown law of object mo... |
By "isomorphism" I mean any bijective, structure-preserving map.
(Please accept my advance apology if this question is out of bounds. I sense that it's borderline, but I'm hoping it'll be considered in better taste than the typical "What is your favorite X?" question. I think a collection of great isomorphisms would... |
What is your favorite isomorphism? |
I was having a look at [this question on SO][1].
From what I know, the centroid is the center of mass of an object. so, by definition its position is given by a simple mean of the positions of all the points in the object.
For a polygon, it only has mass at the vertices. So, the centroid should be given by the ar... |
Why doesn't a simple mean give the position of a centroid in a polygon? |
The **derivative product rule** is merely a special case of the **congruence product rule** for rings, i.e.
**PRODUCT RULE** $\rm\quad\;\: a\equiv a\_0,\; b\equiv b\_0 \;\Rightarrow\;\; ab \equiv a\_0b\_0$
**Proof**: $\rm\quad\; ab-a\_0b\_0 \equiv a(b-b\_0)+(a-a\_0)b\_0 \equiv 0 \quad$ **QED**
Thus $\rm\quad\q... |
I am interested in continuous functions which are not smooth, i.e. which have only finitely many derivatives (for now just taking the case f:R->R). As far as I can tell, there are only 2 features or properties of such a function which can make it non-smooth.
1. The derivative of some order diverges to infinity at ... |
My nephew was folding laundry, and turning the occasional shirt right-side-out. I showed him a "trick" where I turned it right-side-out by pulling the whole thing through a sleeve instead of the bottom or collar of the shirt. He thought it was really cool (kids are easily amused, and so am I).
So he learned that yo... |
By "isomorphism" I mean any structure-preserving map with a structure-preserving inverse.
(Please accept my advance apology if this question is out of bounds. I sense that it's borderline, but I'm hoping it'll be considered in better taste than the typical "What is your favorite X?" question. I think a collection of... |
Personal experiences as an elementary school math resource teacher and a 1st grade teacher have led me to believe that mathematics learning should be engaging and fun at any level (including the graduate level), but clearly it is of the utmost importance to effectively engage students at an early age, so as to hopefull... |
Here algebraic group means affine algebraic group in both instances. Also I'm mainly interested in groups over $\mathbb{C}$. In fact I'm taking $\pi_1(G)$ to mean the fundamental group of $G_{an}$, the analytification. So I guess my question only applies to the base field being either $\mathbb{C}$ or $\mathbb{R}$. ... |
Is the universal cover of an algebraic group an algebraic group? |
I am interested in the Gelfand transformation
$$
\Phi\colon\ell^1(\mathbb Z)\to\mathcal C(\mathbb T),\quad a\mapsto\sum_{n\in\mathbb Z}a_n z^n.
$$
This is an injective homomorphism of Banach algebras. It is neither isometric nor surjective. However, its image---the Wiener algebra $W$ consisting of all continuous fu... |
[This question][1] made me realize I had a misconception about the cyclotomic integers: I thought the units were exactly the roots of unity. There are only finitely many units but infinitely many integers so the question is impossible to solve unless there are more units. So what are the units of cyclotomic integers?
... |
What are the units of cyclotomic integers? |
From [Wikipedia][1]:
> First, we can use Einstein notation in
> linear algebra to distinguish easily
> between vectors and covectors: upper
> indices are used to label components
> (coordinates) of vectors, while lower
> indices are used to label components
> of covectors. However, vectors
> themselves (not ... |
I am aware of gauge transformations and covariant derivatives as understood in Quantum Field Theory and I am also familiar with deRham derivative for vector valued differential forms.
I thinking of the gauge field A of the gauge group G as a Lie(G) valued 1-form on the manifold.
But I can't see why under a gau... |
Can anyone kindly give some reference on taking trace of vector valued differential forms?
Like if $A$ and$B$ are two vector valued forms then I want to understand how/why this equation is true?
$dTr(A\wedge B) = Tr(dA\wedge B) - Tr(A\wedge dB)$
|
If G is a simply connected Lie Group then why is every G-bundle over an orientable 3-manifold trivial? (Why is orientability important?)
|
Is there a proof that the ratio of a circle's diameter and the circumference is the same for all circles, that doesn't involve some kind of limiting process, e.g. a direct geometrical proof? |
Are you sure that orientability is necessary? The result is proved in lemma 4.1.1. [here][1] and I do not see where orientability is used.
[1]: http://www.math.uni-bonn.de/people/himpel/himpel_cstheory.pdf "here" |
The **derivative product rule** is merely a special case of the **congruence product rule** for rings, i.e.
**PRODUCT RULE** $\rm\quad\:\:\:\: a\equiv a\_1,\; b\equiv b\_1 \;\Rightarrow\;\; ab \equiv a\_1 b\_1$
**Proof**: $\rm\quad\; ab-a\_1 b\_1 \equiv a(b-b\_1)+(a-a\_1)b\_1 \equiv 0 \quad$ **QED**
Thus $\rm\... |
Does a limit that has the value of infinite exist or not?
I've recently come across certain sources that say that if the value of a limit is infinite, then that limit does not exist. This contradicts what my calculus teachers and lecturers taught me however, that a limit doesn't exist if the right hand limit and lef... |
We have a set of items I = {i_1, i_2, ..., i_n}. Each of these items has what we call a _p_ value, which is some real number. We want to choose a subset of I, call it I', of size m (for some m with 1 <= m <= n) such that the average of the _p_ values of the items in I' falls within some specified range, [p_l, p_u]. ... |
We have a set of items $I = \\{i_1, i_2, ..., i_n\\}$. Each of these items has what we call a _p_ value, which is some real number. We want to choose a subset of $I$, call it $I'$, of size $m$ (for some $m$ with $1 \le m \le n$) such that the average of the _p_ values of the items in $I'$ falls within some specified ... |
One solution I was considering was to choose items for I' successively such that the average _p_ value of the items selected so far always falls within some upper and lower bound.
Obviously, we could just choose p_l and p_u as our lower and upper bounds, respectively, and this would satisfy the average _p_ value c... |
> **Possible Duplicate:**
> [-1 is not 1, so where is the mistake?](http://math.stackexchange.com/questions/438/1-is-not-1-so-where-is-the-mistake)
<!-- End of automatically inserted text -->
I'm trying to understand the exact point of failure in the following reasoning:
$$1 = \sqrt{1} = \sqrt{(-1)(-1)} = \sqr... |
Obvious it depends on the definition of "exists". Some authors explicitly work over the extended real line with $\pm\infty$ adjoined, so that such infinite limits do explicitly "exist" as first-class values. But there is no consensus. One needs to pay attention to the author's definitions and conventions.
Perhaps it... |
Obviously it depends on the definition of "exists". Some authors explicitly work over the extended real line with $\pm\infty$ adjoined, so that such infinite limits do explicitly "exist" as first-class values. But there is no consensus. One needs to pay attention to the author's definitions and conventions.
Perhaps ... |
Consider the sequence defined as
$x_1 = 1$
$x_{n+1} = \sin x_n$
I think I was able to show that the sequence $\sqrt{n} x_{n}$ converges to $\sqrt{3}$ by a tedious elementary method which I wasn't too happy about.
(I think I did this by showing that $\sqrt{\frac{3}{n+1}} < x_{n} < \sqrt{\frac{3}{n}}$, don't ... |
((4^n)-1)/(n-1) > (4*10^6)
What is an easy way to solve it by hand? I need to find the minimum integral value of `n` to satisfy this equation. |
How to solve this equation? |
$$ \frac{4^n-1}{n-1} > 4 × 10^6 $$
What is an easy way to solve it by hand? I need to find the minimum integral value of `n` to satisfy this equation. |
Suppose $f:[0,1]->R$ is defined as f(x) = 1 if x is irrational and f(x)=0 if x is rational.
I know from Real Analysis that $\int_{0}^{1} f =1$, but intuitively it seems that the "area under curve" is 0.
Is there some intuitive explaination as to why $\int_{0}^{1} f =1$ ? |
Suppose $f:[0,1]->R$ is defined as f(x) = 1 if x is irrational and f(x)=0 if x is rational.
I know from Real Analysis that $\int_{0}^{1} f =1$, but intuitively it seems that the "area under curve" is 0.
Is there some intuitive explaination as to why $\int_{0}^{1} f =1$ (without using measure theory)? |
Some of this answer is implicit in many people's comments, but let's make it explicit.
Here's a question. It seems to you that the area under the curve --- of Dirichelt's function --- is zero. But Dirichelt's function is not a "curve", precisely because it is discontinuous. What then do you mean?
Let's consider t... |
Here's an alternative method that is due to Gauss. Scale the congruence so to reduce the leading coefficient. We seek a multiple of $\:25\:$ that is smaller $\pmod{109}.\;$ Clearly $4\:$ works, since $\; 4*25\equiv 100 \equiv -9 \;$ has smaller absolute value than $25$. Scaling by $4$ yields $-9 x \equiv 12.\;$ Similar... |
Here's an alternative method that is due to Gauss. Scale the congruence so to reduce the leading coefficient. We seek a multiple of $\:25\:$ that is smaller $\pmod{109}.\;$ Clearly $4\:$ works, since $\; 4*25\equiv 100 \equiv -9 \;$ has smaller absolute value than $25$. Scaling by $4$ yields $-9 x \equiv 12.\;$ Similar... |
How can I convert a hexadecimal number, for example `0x1A03` to its octal value?
I know that one way is to convert it to decimal and then convert it to octal
0x1A03 = 6659 = 0o15003
- Is there a simple way to do it without the middle step (conversion to decimal)?
- Why do we tend to convert it to Ba... |
How to convert a hexadecimal number to an octal number? |
Deciphering this explanation for 'shrinking a set of nodes'? |
I would like help with the following problem (chapter 9, #4) from Lee's Smooth Manifolds [its not homework, I'm reading it and I got stuck on this one]
If a Lie group $G$ acts smoothly and freely on a smooth manifold $M$ and the orbit space $M/G$ has a smooth manifold structure such that the quotient map $\pi: M\to ... |
> **Possible Duplicate:**
> [-1 is not 1, so where is the mistake?](http://math.stackexchange.com/questions/438/1-is-not-1-so-where-is-the-mistake)
<!-- End of automatically inserted text -->
I'm trying to understand the exact point of failure in the following reasoning:
$1 = \sqrt{1} = \sqrt{(-1)(-1)} = ... |
Can anyone kindly give some reference on taking trace of vector valued differential forms?
Like if $A$ and$B$ are two vector valued forms then I want to understand how/why this equation is true?
$dTr(A\wedge B) = Tr(dA\wedge B) - Tr(A\wedge dB)$
One particular case in which I am interested in will be when $A$ ... |
How can I convert a hexadecimal number, for example `0x1A03` to its octal value?
I know that one way is to convert it to decimal and then convert it to octal
0x1A03 = 6659 = 0o15003
- Is there a simple way to do it without the middle step (conversion to decimal or conversion to binary)?
- Why do we ... |
Draw some graph with vertex set $V$. Mark off a subset of those vertices and call it $S$. You now have two types of vertices: those in $S$ and those outside of $S$, and you have three types of edges: those connecting vertices of $S$, those connecting vertices outside of $S$, and those on the "boundary", connecting vert... |
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