instruction stringlengths 12 30k |
|---|
As a result of <a href="http://math.stackexchange.com/questions/9820/doubt-inequality-almost-everywhere/9821#9821">this question</a>, I've been thinking about the following condition on a topological space $Y$:
> For every topological space $X$, $E\subseteq X$, and continuous maps $f,g\colon X\to Y$, if $f$ and $g$ ... |
Does this property characterize a space as Hausdorff? |
As a result of <a href="http://math.stackexchange.com/questions/9820/doubt-inequality-almost-everywhere/9821#9821">this question</a>, I've been thinking about the following condition on a topological space $Y$:
> For every topological space $X$, $E\subseteq X$, and continuous maps $f,g\colon X\to Y$, if $E$ is dense... |
Certainly you can add them as elements of $V$. They should all have the same number of components as the dimension of $V$. But if $u_0 \in U$ why isn't $u_0+U$ just $U$? $U$ is a vector space, so is closed under addition. |
$K$ numbers are chosen with replacement form the numbers $1,2,3, \cdots N$.In how many ways we can choose the numbers so that the maximum number chosen is exactly $R \le N $ ? |
If $X$ is a Banach space, and $T:X \to X$ is a bounded linear operator with norm $< 1$, then $I-T$ has a bounded inverse defined by $(I-T)^{-1} = \sum_{n=0}^\infty T^n$.
Thinking in terms of a converse, if $T$ is any bounded linear operator defined on $X$, then does the existence of a bounded inverse $S=(I-T)^{-1}$... |
If $(I-T)^{-1}$ exists, can it always be written in a series representation? |
$K$ numbers are chosen with replacement form the numbers $1,2,3, \cdots N$.In how many ways we can choose the numbers so that the maximum number chosen is exactly $R \le N $ ?
The explanation given in my module is like this :
***Since the maximum number is $R$, each of $K$ numbers can be chosen in $R$ ways.so, th... |
If $X$ is a Banach space, and $T:X \to X$ is a bounded linear operator with norm < $1$, then $I-T$ has a bounded inverse defined by $(I-T)^{-1} = \sum_{n=0}^\infty T^n$.
Thinking in terms of a converse, if $T$ is any bounded linear operator defined on $X$, then does the existence of a bounded inverse $S=(I-T)^{-1}$... |
**EDIT:** My original answer was incorrect even with the interpretation I had in mind. Here is the correction.
I originally gave a formula for the number of ways to pick $K$ numbers so that the maximum is *at most* $R$, rather than exactly $R$. Moreover, I interpreted the question as asking for the number of differe... |
**EDIT:** My original answer was incorrect even with the interpretation I had in mind. Here is the correction.
I originally gave a formula for the number of ways to pick $K$ numbers so that the maximum is _at most_ $R$, rather than exactly $R$. Moreover, I interpreted the question as asking for the number of differe... |
The formula for finding the roots of a polynomial is as follows
$$x = \frac {-b +- \sqrt{ b^2 - 4ac }}{2a} $$
what happens if you want to find the roots of a polynomial like this simplified one
$$ 3x^2 + x + 24 = 0 $$
then the square root value becomes
$$ \sqrt{ 1^2 - 4 * 3 * 24 } $$
$$ = \sqrt{ -28... |
Is the average of i.i.d. random variables with zero mean and finite variance convergent to 0 in probability or $L^2$? How do I show this? I am just beginning to learn convergence of random variables, so an example would be helpful. |
The formula for finding the roots of a polynomial is as follows
$$x = \frac {-b \pm \sqrt{ b^2 - 4ac }}{2a} $$
what happens if you want to find the roots of a polynomial like this simplified one
$$ 3x^2 + x + 24 = 0 $$
then the square root value becomes
$$ \sqrt{ 1^2 - 4 * 3 * 24 } $$
$$ = \sqrt{ -2... |
The formula for finding the roots of a polynomial is as follows
$$x = \frac {-b \pm \sqrt{ b^2 - 4ac }}{2a} $$
what happens if you want to find the roots of a polynomial like this simplified one
$$ 3x^2 + x + 24 = 0 $$
then the square root value becomes
$$ \sqrt{ 1^2 - 4\cdot3\cdot24 } $$
$$ = \sqrt... |
One way to tessellate a 3D sphere is by iterated subdivision of an icosahedron. I am wondering whether this method gives a homogeneous surface density of vertices. To the eye, it seems to do so, and logic indicates that too (each face has the same are in the icosahedron, and faces in each subdivision are created of equ... |
Which tessellation of the sphere yields a constant density of vertices? |
How to prove that $N^{-1/2} W_N$ is an unitary matrix? |
**HINT:**
Expand
$$\sum_{n=m}^{m+N-1}h(n)$$
What do you know about $h$? |
If for a function $f(x)$ only its absolute value $|f(x)|$ and the absolute value $|\tilde f(k)|$ of its Fourier transform $f(k)=N\int f(x)e^{ikx}$ is known (with e.g. $N=1/2\pi$), can $f(x) = |f(x)|e^{i\phi(x)}$ and thus the phase function $\phi(x)$ be extracted? |
Can the phase of a function be extracted from only its absolute value and its Fourier transform's absolute value? |
If for a function $f(x)$ only its absolute value $|f(x)|$ and the absolute value $|\tilde f(k)|$ of its Fourier transform $\tilde f(k)=N\int f(x)e^{ikx}$ is known (with e.g. $N=1/2\pi$), can $f(x) = |f(x)|e^{i\phi(x)}$ and thus the phase function $\phi(x)$ be extracted? |
If for a function $f(x)$ only its absolute value $|f(x)|$ and the absolute value $|\tilde f(k)|$ of its Fourier transform $\tilde f(k)=N\int f(x)e^{ikx} dx$ is known, can $f(x) = |f(x)|e^{i\phi(x)}$ and thus the phase function $\phi(x)$ be extracted? (with e.g. $N=1/(2\pi)$) |
If for a function $f(x)$ only its absolute value $|f(x)|$ and the absolute value $|\tilde f(k)|$ of its Fourier transform $\tilde f(k)=N\int f(x)e^{-ikx} dx$ is known, can $f(x) = |f(x)|e^{i\phi(x)}$ and thus the phase function $\phi(x)$ be extracted? (with e.g. $N=1/(2\pi)$) |
Check out Chapter 1 of Paul Pollack's book "[Not Always Buried Deep][1]". He lists several elementary proofs for the infinitude of primes, some of which can be adapted to form simple iterative algorithms.
(Sorry, I can't get Latex to work here. Isn't it like MathOverflow? Anyway..)
Let S = {p1...pk} be the se... |
Check out Chapter 1 of Paul Pollack's book "[Not Always Buried Deep][1]". He lists several elementary proofs for the infinitude of primes, some of which can be adapted to form simple iterative algorithms.
(Sorry, I can't get Latex to work here. Isn't it like MathOverflow? Anyway..)
Let S = {p1...pk} be the se... |
If for a function $f(x)$ only its absolute value $|f(x)|$ and the absolute value $|\tilde f(k)|$ of its Fourier transform $\tilde f(k)=N\int f(x)e^{-ikx} dx$ is known, can $f(x) = |f(x)|e^{i\phi(x)}$ and thus the phase function $\phi(x)$ be extracted? (with e.g. $N=1/(2\pi)$)
As Marek already [stated](http://math.st... |
Let $\Omega = $ {$\omega_{i}$} be an ordered set of $n$ positive reals in the unit interval, $\omega_{1} \leq \cdots \leq \omega_{n} \leq 1$. Define the $n$-simplex $\Delta(\Omega; \(\mathbb{R}^{+})^{n})$ by the non-negative points $(x_{1}, \dots, x_{n}) \subset (\mathbb{R}^{+})^{n}$ which satisfy the inequality
\beg... |
Check out Chapter 1 of Paul Pollack's book "[Not Always Buried Deep][1]". He lists several elementary proofs for the infinitude of primes, some of which can be adapted to form simple iterative algorithms.
(Sorry, I can't get Latex to work here. Isn't it like MathOverflow? Anyway..)
Let S = {p1...pk} be the se... |
Let $\Omega = $ {$\omega_{i}$} be an ordered set of $n$ positive reals in the unit interval, $\omega_{1} \leq \cdots \leq \omega_{n} \leq 1$. Define the $n$-simplex $\Delta(\Omega; (\mathbb{R}^{+})^{n})$ by the non-negative points $(x_{1}, \dots, x_{n}) \subset (\mathbb{R}^{+})^{n}$ which satisfy the inequality
\begi... |
Well, let's go back to the definitions. I am sorry if I go too far into the details, but it is difficult to judge what you exactly understand based only on your question. So maybe I explain too much, but don't take this as an offense.
What exactly is $V/U$ ? I will explain this a little, since it might be the source... |
Well, let's go back to the definitions. I am sorry if I go too far into the details, but it is difficult to judge what you exactly understand based only on your question. So maybe I explain too much, but don't take this as an offense.
What exactly is $V/U$ ? I will explain this a little, since it might be the source... |
I have two sets (a,b) of data and I am restricted to using the
values in the sets once and only once when they are combined.
For example, the 2nd highest value possible from
a data set of a{3,2} and b{2,0} is a1+b2=3.
There are two possible ways of combining these two data sets:
(1) a1+b1 and a2+b2 which r... |
How many number of three digit numbers lying between 100 and 999 (inclusive) and having only two consecutive digits identical ? |
Let V be a vector space with a non-trivial subspaces U, W. $ U \neq W$. Consider $u_0 \in V$, with $u_0 \neq 0$ and $w_0 \in V$ with $w_0 \neq 0$ then we have $u_0 + U \in V/U$ and $w_0 + W \in V/W$. Is there any sensible way to define:
$(u_0 + U) + (w_0 + W)$?
I'm thinking "no" $(2,4) + (2,6,7)$ makes no sense.... |
As a result of <a href="http://math.stackexchange.com/questions/9820/doubt-inequality-almost-everywhere/9821#9821">this question</a>, I've been thinking about the following condition on a topological space $Y$:
> For every topological space $X$, $E\subseteq X$, and continuous maps $f,g\colon X\to Y$, if $E$ is dense... |
As a result of <a href="http://math.stackexchange.com/questions/9532/link-between-a-dense-subset-and-a-continuous-mapping">this question</a>, I've been thinking about the following condition on a topological space $Y$:
> For every topological space $X$, $E\subseteq X$, and continuous maps $f,g\colon X\to Y$, if $E$ ... |
Suppose given two manifolds $X$ and $Y$, both orientable of dimension $n$, and a map $f:X\to Y$. Is there a relationship between the degree of $f$ calculated with respect to homology (by which I mean the induced map on the top homology groups) and the degree of $f$ calculated with respect to cohomology (by which I mean... |
These are the probability densities of the assumed distributions, evaluated at the sample data. In this case, the distributions of height, weight and foot size are assumed to follow independent Gaussian distributions whose parameters are given in the second table. If you don't know the probability density function of a... |
I think it is desirable to have that $M_{m\times n}\left(\mathbb{K}\right)\not=M_{m'\times n'}\left(\mathbb{K}\right)$ if $m\not=m'$ or $n\not=n'$. In other words, the set of all $m\times n$ matrices on $\mathbb{K}$ should be different from the set of all $m'\times n'$ matrices on the same field when $m\not=m'$ or $n\n... |
I think it is desirable to have that $M_{m\times n}\left(\mathbb{K}\right)\not=M_{m'\times n'}\left(\mathbb{K}\right)$ if $m\not=m'$ or $n\not=n'$. In other words, the set of all $m\times n$ matrices on $\mathbb{K}$ should be different from the set of all $m'\times n'$ matrices on the same field when $m\not=m'$ or $n\n... |
How to convert the following partial differential equation (pde)
$$\frac{\partial V}{\partial t}=aV-as\frac{\partial V}{\partial s}-b^2s^2\frac{\partial^2 V}{\partial s^2}$$
into a pde of the form
$$\frac{\partial u}{\partial t}=x^2\frac{\partial^2 u}{\partial x^2}+cx\frac{\partial u}{\partial x}$$
by some change ... |
Let $$\cdots \rightarrow A_{n+1}\rightarrow^{f_{n+1}} A_n \rightarrow^{f_{n}} A_{n-1}\rightarrow \cdots $$ be an inverse system of finite dimensional vector spaces with the property that the $A_i$ are 'eventually constant', i.e., there is an $m$ such that the maps $f_i$ are isomorphisms for every $i\ge m$. Does it foll... |
Let $A,B \in \mathbb{R}^{n \times n}$ be two positive semi-definite matrices and let $a > 0$ be a constant.
Consider the following maximization problem
$$
\max_{x \in \mathbb{R}^n, \gamma}\ x^T \left(\frac{\gamma}{a + \gamma} A - \gamma B \right)x
\qquad \text{subject to } \gamma > 0, \|x\| = 1
$$
and suppose t... |
How to bound the change between optimal points when perturbing an objective function? |
I understand that for a number $a$ to have a multiplicative inverse in mod $n$, it must be coprime to $n$; therefore, all numbers that do not have a multiplicative inverse mod $n$ must share a factor with $n$.
Now, my prediction was that if $n$ is even, then the sum of all numbers that do not have a multiplicative i... |
Let $\Omega = $ {$\omega_{i}$} be an ordered set of $n$ positive reals in the unit interval, $\omega_{1} \leq \cdots \leq \omega_{n} \leq 1$. Define the $n$-simplex $\Delta(\Omega; (\mathbb{R}^{+})^{n})$ by the non-negative points $(x_{1}, \dots, x_{n}) \subset (\mathbb{R}^{+})^{n}$ which satisfy the inequality
\begi... |
As Laurent S points out, the sensible way to define $\mathbb{K}^0$ is as the one element set (either $\{0\}$ or $\{\emptyset\}$).
You could could likewise argue that since you want $M_{m\times n}(\mathbb{K})$ to correspond to linear transformations from $\mathbb{K}^n$ to $\mathbb{K}^m$, and since $\mathbb{K}^0$ is ... |
I am learning the basics of category theory, so this question is probably obvious to anyone who knows the subject.
The resources I've seen all take the following approach:
0) A category is a collection of objects and morphisms between those objects that satisfy some rules.
1) A functor is a morphism in the cat... |
Morphisms in the category of natural transformations ? |
Could someone provide a good reference for learning matrix calculus? I've recently moved to a more engineering-oriented field where it's commonly used and don't have much experience with it. |
Gene Howard Golub, Charles F. Van Loan book on "Matrix Computations" is regarded as the "Bhagavad Gita" for Matrix Algorithms.
http://books.google.com/books?id=mlOa7wPX6OYC&printsec=frontcover&dq=golub+van+loan&hl=en&src=bmrr&ei=UkbcTNWqOYi-sAPr6azlAw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDQQ6AEwAA#v=onepage&... |
Gene Howard Golub, Charles F. Van Loan book on "Matrix Computations" is regarded as the "Bhagavad Gita" for Matrix Algorithms.
http://books.google.com/books?id=mlOa7wPX6OYC&printsec=frontcover
There is also another book by "Gene Howard Golub, Gerard Meurant" on "Matrices, moments, and quadrature with applications... |
In response to user2969 :
My algorithm for is this something like this :
At first I have computed the gcd using Euclidean algorithm or you could use `__gcd()` a standard STL function,the C++ version of Euclidean algorithm looks like this :
int E_GCD(int a,int b){
return b ? gcd(b,(a%b)):a;
}
... |
In naieve set theory, you quickly run into existence trouble if you try to do meta-things things like take "the set of all sets". For example, does the set of all sets that don't contain themselves contain itself? (see, Russell)
On the other hand, people have no trouble casually talking about the "category of all ca... |
As Laurent S points out, the sensible way to define $\mathbb{K}^0$ is as the one element set (either $\{0\}$ or $\{\emptyset\}$).
You could likewise argue that since you want $M_{m\times n}(\mathbb{K})$ to correspond to linear transformations from $\mathbb{K}^n$ to $\mathbb{K}^m$, and since $\mathbb{K}^0$ is a perf... |
In response to user2969 :
My algorithm for is this something like this :
At first I have computed the gcd using Euclidean algorithm,the C++ version of of my implementation for the same looks like this :
int E_GCD(int a,int b){
return b ? gcd(b,(a%b)):a;
}
Now we need to compute the number o... |
Given a diffusion $dX_t = \mu(X_t)dt + \sigma(X_t)dW_t$, if one applies the transform
$f(x) = \int_a^x \frac{1}{\sigma(u)}du$, one can use Ito's lemma to show that
$df(X_t) = \left( \frac{\mu(X_t)}{\sigma(X_t)} - \frac{\sigma'(X_t)}{2} \right)dt + dW_t$.
Is there a way to explicitly invert this transform in th... |
How do I invert this transform? |
In response to user2969 :
My algorithm for is this something like this :
At first I have computed the gcd using Euclidean algorithm,the C++ version of of my implementation for the same looks like this :
int E_GCD(int a,int b){
return b ? gcd(b,(a%b)):a;
}
Now we need to compute the number o... |
In response to user2969 :
At first you need to computed the [gcd][1], it is always best to use [Euclidean algorithm][2] for this,my C++ implementation for the same :
int E_GCD(int a,int b){
return b ? gcd(b,a%b):a;
}
Now we need to compute the number of divisors of `N = E_GCD(A,B)`, this can b... |
Here's a sketch of my idea for proving that $g$ is continuous:
1. Show that $s$ and $t$ are witnesses that $Y$ is not $T_2$ iff any eventually nonconstant net converging to $s$ or $t$ converges to *both* $s$ and $t$.
2. Observe that $g$ is continuous on $E$ (why?).
3. Show that $g$ is continuous at $s$ by using th... |
If you base your mathematics on Set Theory, then you run into the same foundational issues in Category Theory as you do in Set Theory (since Categories will have to be modeled "with" Set Theory in some sense).
You can go the other way, though: you can base your mathematics with Category Theory, and develop the Categ... |
We all know that $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n}$ diverges.
Also, we know that $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^s}$ converges for $s>1$.
I was curious to see till what extent I can push the denominator so that it will still diverges.
So I took $\displaystyle \sum_{n=1}^{\infty} \fra... |
We all know that $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n}$ diverges.
Also, we know that $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^s}$ converges for $s>1$.
I was curious to see till what extent I can push the denominator so that it will still diverges.
So I took $\displaystyle \sum_{n=1}^{\infty} \fra... |
We all know that $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n}$ diverges.
Also, we know that $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^s}$ converges for $s>1$.
I was curious to see till what extent I can push the denominator so that it will still diverges.
So I took $\displaystyle \sum_{n=1}^{\infty} \fra... |
We all know that $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^s}$ converges for $s>1$ and diverges for $s<=1$. (Assume $s \in \mathbb{R}$).
I was curious to see till what extent I can push the denominator so that it will still diverges.
So I took $\displaystyle \sum_{n=1}^{\infty} \frac{1}{nlogn}$ and found that... |
We all know that $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^s}$ converges for $s>1$ and diverges for $s<=1$. (Assume $s \in \mathbb{R}$).
I was curious to see till what extent I can push the denominator so that it will still diverges.
So I took $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n\log n}$ and found th... |
We all know that $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^s}$ converges for $s>1$ and diverges for $s \leq 1$ (Assume $s \in \mathbb{R}$).
I was curious to see till what extent I can push the denominator so that it will still diverges.
So I took $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n\log n}$ and found... |
We all know that $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^s}$ converges for $s>1$ and diverges for $s \leq 1$ (Assume $s \in \mathbb{R}$).
I was curious to see till what extent I can push the denominator so that it will still diverges.
So I took $\displaystyle \sum_{n=2}^{\infty} \frac{1}{n\log n}$ and found... |
You say "Also since there is only one possible coset, this means that for all elements, a,b in G, then aH = bH and Ha = Hb, since each element in G acting on the subgroup must produce the same set." but this is not true. Let a be the identity and b not be in H.
But earlier you say "I know this means that there are ... |
In the derived category $D(C)$ of an abelian category $C$, one formally inverts quasi-isomorphisms. In the context of model categories, one inverts weak equivalences.
> What does one gain by doing so?
Is there a big-picture way to think about what is being done? |
Yoga of localization in categories? |
Let $G = \hbox{proj.lim.}_{\alpha} \{ G_{\alpha} , \varphi^{\alpha}_{\beta} \}$ be a projective limit of simple groups (i.e., each $\varphi^{\alpha}_{\beta}\colon G_{\alpha}\to G_{\beta}$ is a surjective group homomorphism between simple groups).
It is clear that if $\varphi^{\alpha}_{\beta}$ is not the trivial map ... |
Someone recently asked me why a negative * a negative is positive, and why a negative * a positive is negative, etc.
I went ahead and gave them a proof by contradiction like so:
Assume (-x) * (-y) = -xy
Then divide both sides by (-x) and you get (-y) = y
Since we have a contradiction, then our first assumpt... |
Why negative times negative = positive? |
Cyclic groups have at most one subgroup of any given finite index. Can we describe the class of all groups having such property?
Thank you! |
I know a proof that $\frac{(30n)!n!}{(15n)!(10n)!(6n)!}$ is an integer.
The proof goes as:
If a prime $p$ divides $(15n)!(10n)!(6n)!$, then the power of the prime dividing $(15n)!(10n)!(6n)!$ is lesser than the $(30n)!n!$. It is relatively easy to prove this.
$\textbf{My question is there a counting argument t... |
Does $\frac{(30n)!n!}{(15n)!(10n)!(6n)!}$ count something? |
I think the *x* and *y* get in the way a bit; you can see the crucial steps using just 1 and -1. What you've really shown is that (-1)(-1)=-1 leads to a contradiction. If we divide by -1, we get -1=1, which is not true!
Getting the right answer, (-1)(-1)=1, uses a couple more steps: First, you must agree that (1)... |
H is a subgroup of G and a,b are in G.
I was rather lost for a while but I think I may have
actually proved it, though I am unsure. . Does this make sense?
aH = bH
-> b is an element of aH
-> there exists an h in H s.t. ah = b
-> a-1ah = a-1b
-> h = a-1b
-> hb-1 = a-1bb-1
-> hb-1 = a-1
This means a-1... |
I'm trying to find a general solution for $\displaystyle\iint_{S}[[x+y]] dA$ where $S=[a,b] \times [c,d]$ and $[[x]]$ is the greatest integer function. I know that if $a,b,c,d \in \mathbb{Z}$, then the answer is just $\frac{(d-c)(b-a)(a+b+c+d-1)}{2}$ (proven below), but how do I account for the extra volume when $a,b,c... |
Well if I were to explain this in an intuitive way to someone (or at least try), I would like to think of an analogy with walking over the real line, by agreeing that walking left will be walking in the negative direction and walking right in the positive direction.
Then I will try to convey the idea that if you are... |
Why is the following inequality true: if $x \geq 0$ then $(1-\frac{x}{n})^{n} \leq e^{-x}$ ? here $n$ is a positive integer. Is there a quick way to see this? |
You can look at it this way:
$(1 - \frac{x}{n})^n = \sum_{k=0}^{n} C(n,k) (-1)^k \frac{x^k}{n^k}$
and
$e^{-x} = \sum_{k=0}^{+\infty} (-1)^k \frac{x^k}{k!}$
Is it clearer now ?
(hint: one of the sums is finite, the other not, and one of the terms grows quicker than the other) |
$H$ is a subgroup of $G$ and $a,b$ are in $G$.
I was rather lost for a while but I think I may have
actually proved it, though I am unsure. . Does this make sense?
$aH = bH$
$\rightarrow$ $b$ is an element of $aH$
$\rightarrow$ there exists an $h$ in $H$ s.t. $ah = b$
$\rightarrow$ $a^{-1}ah = a^{-1}b$
... |
$H$ is a subgroup of $G$ and $a,b$ are in $G$.
I was rather lost for a while but I think I may have
actually proved it, though I am unsure. . Does this make sense?
$aH = bH$
$\Rightarrow$ $b$ is an element of $aH$
$\Rightarrow$ there exists an $h$ in $H$ s.t. $ah = b$
$\Rightarrow$ $a^{-1}ah = a^{-1}b$
... |
I'm trying to find a general solution for $\displaystyle\iint_{S}[[x+y]] dA$ where $S=[a,b] \times [c,d]$ and $[[x]]$ is the greatest integer function. I know that if $a,b,c,d \in \mathbb{Z}$, then the answer is just $\frac{(d-c)(b-a)(a+b+c+d-1)}{2}$ (proven below), but how do I account for the extra volume when $a,b,c... |
Interesting calculus problems of medium difficulty? |
Why is the following inequality true: if $x \geq 0$ then $(1-\frac{x}{n})^{n} \leq e^{-x}$ ? here $n$ is a positive integer. Is there a quick way to see this? |
Given a sequence $(x_n)$ in a metric space $M$, call it *uniformly separated* if all pairwise distances $d(x_n,x_m)$ between distinct terms are uniformly bounded away from zero.
Suppose that a given sequence $(x_n)$ has the property that it has no Cauchy subsequence. Must it have a uniformly separated subsequence... |
If no Cauchy subsequence exists, must a uniformly separated subsequence exist? |
Without loss, you can assume $|u|\le|v|$ and replace $u$ by $uv$ to reduce the problem to the situation $v=1$ and $|u|\le1$. Unless you need $c_p$ explicitly, then it's clear, yes? |
I have been sick for my real analysis class and I am a having a tough time getting the book.
Could someone explain the what open, closed and compact sets are and how they are related with uniform continuity? I think the hinging point was a poor understanding of accumulation points.
Thanks |
If $C_0 + C_1/2 + \ldots + C_n/(n+1) = 0$, where each $C_i \in \mathbb{R}$, then prove that the equation $C_0 + C_1x + \ldots + C_nx^n = 0$ has at least one real root between 0 and 1. (Rudin, Ch 5 Exercise 4)
If $|f(x)| \leq |x|^2$ for all $x \in \mathbb{R}$, then prove that $f$ is differentiable at $x = 0$. (Spiv... |
I apologize for posting such an untechnical question, but with responses it could surely be posed in a better form.
I'm a math noob, but I've seen (as we all have) a few examples of "connections" between different areas of math by way of an equality or some other relating device. An example is:
e^iπ + 1 = 0
T... |
If for a function $f(x)$ only its absolute value $|f(x)|$ and the absolute value $|\tilde f(k)|$ of its Fourier transform $\tilde f(k)=N\int f(x)e^{-ikx} dx$ is known, can $f(x) = |f(x)|e^{i\phi(x)}$ and thus the phase function $\phi(x)$ be extracted? (with e.g. $N=1/(2\pi)$)
As Marek already [stated](http://math.st... |
The best site I have come across when it comes to these kind of problems, is
http://www.artofproblemsolving.com/Forum/index.php?
Regards |
A graph G is planar if and only if xxx.
What can xxx be substituted for? Note that this is from a topological POV so a graph is a 1-dim cw complex and I guess the fundamental group should be used somehow. |
When is a graph planar? |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.