Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Quasiconformal mappings: Metric deffinition In the lectures notes http://users.jyu.fi/~pkoskela/quasifinal.pdf (Prof. Koskela has made them freely available from his webpage, so I am guessing is OK that I paste the link here) Quasiconformality is defined by saying that $\displaystyle \limsup_\limits{r \rightarrow 0} \f... | To answer 1), he's only using the definition of the $\mathbb{R}^2$ derivative. $Df(x) : \mathbb{R}^2 \to \mathbb{R}^2$ is a linear transformation and is characterized by the formula
$$\lim_{h \to 0} \frac{f(x+h) - f(x) - Df(x)(h)}{|h|} = 0
$$
Fix $|h|=r$ for very small $r$ and you will see from this that the ratio $L_f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1158970",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How many ways to reach a given tennis-score? Let's say a tennis player wins a set with a game score of 6-3. In how many different ways can we reach this score?
Assuming H means the home-player won the game and A means the away-player won the game, one permutation would be HHHHHAAAH.
Note that the winner of the set will... | As you saw yourself, you’re looking for the number of permutions of five H’s and three A’s. That’s a string of $8$ symbols, and the A’s can occupy any $3$ of the $8$ positions, so the desired number is simply the number of ways to choose $3$ things from a set of $8$ things. This is the binomial coefficient
$$\binom83=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1159063",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Seemingly simple high school combinatorics proof doesn't add up Prove that $\binom{n}{n-2}\binom{n+2}{n-1}$ is an integer for all $n\in\mathbb Z^+$.
My take on this:
Recall: $${n\choose k} = \frac{n!}{k!(n-k)!}$$
So our problem reads (simplified): $$\frac{n!}{(n-2)!2!} \cdot \frac{(n+2)!}{(n-1)3!}$$
But simply substitu... | By definition $\dbinom{n}k=0$ when $k<0$, so
$$\binom{1}{-1}\binom{3}0=0\cdot1=0\;.$$
Added: In the edited question you’ve reduced the case $n\ge 2$ to showing that
$$\frac{n(n+2)!}{12(n-2)!}$$
is an integer. Do a bit more cancellation to get
$$\frac{n^2(n+2)(n+1)(n-1)}{12}\;.$$
Now show that any product of four conse... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1159144",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find Inverse of ideal $I= \langle 3, 1+2\sqrt{-5} \rangle$ Question- Find Inverse of ideal $I = \langle 3, 1 + 2\sqrt{-5} \rangle$ of $O_K$ (ring of integers of algebraic number field $K$), where $K = \mathbb{Q}(\sqrt{-5})$.
First of all can $I$ be simplified more than $I = \langle 3, 1 + 2\sqrt{-5} \rangle = \langle 3... | Hints:
*
*$I=\langle 3,1-\sqrt{-5}\rangle$.
*If $J=\langle 3,1+\sqrt{-5}\rangle$, then prove that $IJ=\langle 3\rangle$.
*Deduce that $I^{-1}$ is $J$ times a principal fractional ideal.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1159279",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Confusion regarding limiting variances (Casella, Statistical Inference, 2nd edition, example 10.1.8) In the book Statistical Inference (George Casella 2nd ed.), page 470, there is an example:
$\bar{X}_n$ is the mean of $n$ iid observations, and E$X=\mu$, $\operatorname{Var}X=\sigma^2$. "If we take $T_n=1/\bar{X}_n$, we... | In the example, the mean $\overline{X}_n$ is taken of $n$ iid normal observations. Therefore, $\overline{X}_n$ also has a normal distribution with mean $\mu$ and variance $\sigma^2/n$; its probability density function is therefore $f(x)=\frac1{\sqrt{2\pi\sigma^2/n}}e^{\frac{-(x-\mu)^2}{2\sigma^2/n}}$. When we try to co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1159500",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove the sum of the even Fibonacci numbers Let $f_n$ denote the $nth$ Fibonacci number. Prove that
$f_2\:+\:f_4\:+...+f_{2n}=f_{2n+1}-1$
I am having trouble proving this. I thought to use induction as well as Binet's formula where,
$f_n=\frac{\tau^2-\sigma^2}{\sqrt5}$ where $\tau=\frac{1+\sqrt5}{2}$ and $\sigma=\frac... | Using the recurrence relation for $f_n$ we find
\begin{align}f_2 + f_4 + \cdots + f_{2n} = (f_3 - f_1) + (f_5 - f_3) + \cdots + (f_{2n+1} - f_{2n-1}),
\end{align}
which telescopes to $f_{2n+1} - f_1 = f_{2n+1} - 1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1159572",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Does having positive second derivative at a point imply convexity in some neighborhood? Suppose that I have a real valued function of a single variable $f(x)$ which is twice differentiable in some open interval $I$.
Then, I know from calculus that if $f''(x) >0 $ on $I$, then $f$ is convex on $I$.
But, what I am wonder... | No, having $f''(x_0)>0$ does not imply that $f$ is convex in some neighborhood of $x_0$. Take the function
$$
g(x) = x+2x^2\sin(1/x),\quad g(0)=0
$$
which, as shown here, is not increasing in any neighborhood of $0$ despite $g'(0)=1$. (It oscillates rapidly on small scales.)
Then let $f(x)=\int_0^x g(t)\,dt$. This fun... | {
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"timestamp": "2023-03-29T00:00:00",
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Updated: Prove completely $\int^\infty_0 \cos(x^2)dx=\frac{\sqrt{2\pi}}{4}$ using Fresnel Integrals Prove completely $\int^\infty_0 \cos(x^2)dx=\frac{\sqrt{2\pi}}{4}$
I've tried substituting $ x^2 = t $ but could not proceed at all thereafter in integration. Any help would be appreciated.
I should mentioned at the star... | As is common, use $f(z)=e^{-iz^2}=\cos(z^2)-i\sin(z^2)$
Now $$\int_{-\infty}^{\infty}e^{-iz^2}{\rm d}z=\int_{-\infty}^{\infty}e^{-\left(e^{i\pi/4}z\right)^2}{\rm d}z=\frac1{e^{i\pi/4}}\int_{-\infty}^{\infty}e^{-x^2}{\rm d}x=e^{-i\pi/4}\sqrt{\pi}=\sqrt{\frac{\pi}2}-i\sqrt{\frac{\pi}2}$$
Now, since $f(z)$ is even:
$$\int... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1159818",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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IMO problem number theory Determine the greatest positive integer $k$ that satisfies the following property.
The set of positive integers can be partitioned into $k$ subsets $A_1,A_2,A_3,\ldots,A_k$ such that for all integers $n\ge 15$ and all $i\in\{1,2,\ldots,k\}$ there exist two distinct elements of $A_i$ such that ... | The greatest such number is $3$
First we are going to show that there exists a construction for $k = 3$ and then we are going to prove that $k \not\ge 4$
PART 1
We build the 3 sets $A_1,A_2\ and\ A_3$ as:
$$
\begin{align}
\\&A_1 = \{1,2,3\} \cup \{3m\ |\ m \ge\ 4\}
\\&A_2 = \{4,5,6\} \cup \{3m - 1\ |\ m \ge\ 4\}
\\&A_3... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Non-trivial examples of power series which are uniformly convergent on $[0,1)$ and left-continuous at $x = 1$ The question is motivated by a more extensive problem that needs a formal proof, but I am not interested in help on the proof itself, but I'd like to see some examples of such power series.
I put non-trivial in... | Using Abel's Theorem you can come up with lots of examples, for example
$$\sum_{n=1}^\infty\frac{x^n}{n^p}\;,\;\;\forall\,p>1\;,\;\;\text{with convergence radius}\;\;R=1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1160017",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Mutual difference of vectors squared, does it have a name? Given a set of $n$ vectors $\def\vv{\vec{v}} \vv_i$ with the additional property that they all have the same absolute value $||\vv_i||=c$, define the average of the vectors as $\vv = \frac{1}{n}\sum_{i=1}^n \vv_i$.
If all $\vv_i$ are identical we have $||\vv||... | Meanwhile I learned that the right hand side is identical to the variance of the $\\v_i$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1160118",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Sum of Exponents and Primes Is it necessary that for every prime $p$, there exists at least one of $1+p$, $1+p+p^2$, $1+p+p^2+p^3$, ... that is a prime? Is it also true that an infinite number of $1+p$, $1+p+p^2$, $1+p+p^2+p^3$, etc. are primes?
I am thinking of using something similar to Euclid's proof of the infinitu... | For the case as simple as $p=2$, we don't even know the answer to the last question. Primes of this form are called Mersenne primes: http://en.wikipedia.org/wiki/Mersenne_prime.
For the first question note that $1+p+\cdots+p^n=11\cdots 1$ ($n+1$ ones) in base $p$. The primes of this form are called repunit primes. It i... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Show that the function $f(x, y)$ = $xy$ is continuous. How do I show that $xy$ is continuous?
I know that the product of two continuous functions is continuous but how do I show that $x$ is continuous and $y$ is continuous?
| The function $f(x,y) = x$ is continuous since given $\epsilon > 0$ and $(a,b)\in \Bbb R^2$, setting $\delta = \epsilon$ makes
$$|f(x,y) - f(a,b)| = |x - a| = \sqrt{(x - a)^2} \le \sqrt{(x - a)^2 + (y - b)^2} < \epsilon$$
whenever $\sqrt{(x - a)^2 + (y - b)^2} < \delta$. Similarly, the function $g(x,y) = y$ is continu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1160354",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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A group whose automorphism group is cyclic Is there an Abelian group $A$ which is not locally cyclic whose automorphism group is cyclic ?
| This MathOverflow question cites this paper, which says there are torsion-free groups of any finite rank of without automorphism group $C_2$, and which in turn cites this paper in which Theorem III says there are Abelian groups of all finite ranks $\ge 3$
with automorphism group $C_2$. Rank $>1$ means they can't be loc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1160453",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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When is the supremum\infimum an accumulation point? Trying to show that if a sequence converges, it either has a maximum, a minimum or both, I reached a dead-end. Assuming it is not constant, it is still bounded and its supremum and infimum aren't equal. Then I assumed that the supremum and infimum are not in the seque... | Consider the sequence $\{\mathbf{x}_n\}_{n\in\mathbb{N}} \subseteq A$, where $A \subseteq \mathbb{R}^n$. If this sequence is convergent, then it is bounded. Also, the sequence must converge to $\mathbf{x} \in \overline{A}$. Show that for all $\epsilon > 0$, there are finitely many $\mathbf{x}_n \notin N(\mathbf{x}, \ep... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1160568",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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The congruence $(34\times 10^{(6n-1)}-43)/99 \equiv -1~\text{ or }~5 \pmod 6$ Trying to prove this congruence:
$$ \frac{34\times 10^{(6n-1)}-43}{99} \equiv-1~\text{ or }~5 \pmod 6,\quad
n\in\mathbb{N}$$
Progress
Brought it to the form
$$34\times 10^{6n-1}-43\equiv -99\pmod{6\cdot 99}$$
but how to proceed further? $6\cd... | ${\ 6\mid f_{n+1}-f_n =\, \dfrac{34\cdot(\color{#c00}{10^6\!-1}) 10^{6n\!-1}}{99}\ }$ by ${\ 3\cdot 99\mid \color{#c00}{10^6\!-1} = 999999 = 99\cdot\!\!\!\!\!\!\!\!\!\!\!\underbrace{10101}_{\equiv\, 1+1+1\pmod 3}}$
So $\,{\rm mod}\ 6\!:\ f_{n+1}\!\equiv f_n\,\overset{\rm induct}\Rightarrow\,f_n\equiv f_1\equiv -1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1160646",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $x=\cos t,y=\cos(2t+\pi/3)$ find an analytical relation between $x$ and $y$. I'm having a bit of trouble figuring this out. At the moment this is the near solution I have:
$$y=\frac12(2\cos^2 t-1)-\sqrt{3}\sin t\cos t.$$
I should be just about to solve it but find myself stuck. I appreciate any hint on how to "elimi... | You're almost there. You have correctly used that
$$
\cos\left(2t+\frac{\pi}{3}\right)=\frac{1}{2}\cos2t-\frac{\sqrt{3}}{2}\sin2t
$$
so
$$
2y=2\cos^2t-1-2\sqrt{3}\sin t\cos t=2x^2-1-2x\sqrt{3}\sin t.
$$
Thus
$$
2x\sqrt{3}\sin t=2x^2-1-2y.
$$
Hence
$$
12x^2(1-x^2)=(2x^2-1-2y)^2.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1160740",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Predictable Processes in Brownian Setting Maybe it's a silly question. I've been reading Protter's book on stochastic integration. And all the integrands are required to be predictable.
But from what I can recall, in the traditional stochastic calculus in Brownian setting (i.e., integration with respect to a Brownian m... | It is a bit subtile but in the particular case of the Brownian motion, you doen't need to assume the integrand is predicable. However it is only because you consider a particular $L^2$ space where your adapted process is a.s. equal to a predicable one. Thus, the loss of generality is an illusion.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1160846",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proof: Number theory: Prove that if $n$ is composite, then the least $b$ such that $n$ is not $b$-pseudoprime is prime. I'm looking to prove this, but not too sure how:
If $n$ is composite, then the least $b$ such that $n$ is not $b$-psp is prime.
Thanks!
| Let $b$ be the minimum such element and assume it is not prime.Then $b=cd$ and $b^{n-1}\not\equiv 1 \bmod n$ . Then $b^{n-1}=c^{n-1}d^{n-1}\not \equiv 1$. So one of $b^{n-1}$ and $c^{n-1}$ is not $1\bmod n$. Contradicting the minimality of $b$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1160949",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Differentiating this problem $\frac{2t^{3/2}}{\ln(2t^{3/2}+1)}$ How does one differentiate the function
$$y(t)=\frac{2t^{3/2}}{\ln(2t^{3/2}+1)}.$$
I am still tying to understand MathJaX and not sure what is wrong with the expression.
Anyways,
How do I start/process solving this? Do i take the ln of both side? If so I g... | In this case the quotient rule is probably the best option. The symmetry of the $2 t^{2/3}$ in the top and bottom makes me suspect that some things might end up canceling out.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1161066",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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What is an example of a nonmetrizable topological space? Find topological spaces $X$ and $Y$ and a function $f:X\to Y$ which is not
continuous, but which has the property that $f(x_n)\to f(x)$ in $Y$ whenever $x_n\to x$ in $X$
I know this is true if $X$ is metrizable, so I want a counterexample when $X$ is not metrizab... | You want an example of a function $f:X\to Y$ that is sequentially continuous but not continuous. Let $X=\omega_1+1$, the space of all ordinals up to and including the first uncountable ordinal, with the order topology, and let $Y=\{0,1\}$ with the discrete topology. Let
$$f:X\to Y:\alpha\mapsto\begin{cases}
0,&\text{if... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1161175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why line integral of f(x.y)=(x.y) is not zero along the circle? I am asked to determine whether f(x.y)=(x.y) is gradient or not.
It is clear that there exists a function g whose derivative with respect to x and y is equal to first and second component of f.
So, f is gradient.
If f is gradient, then on closed path , it... | $$\int_C f(x,y)d\vec x=\int_0^{2\pi}f(\cos \theta,\sin\theta)\cdot d(\cos\theta,\sin \theta)=\int_0^{2\pi}(\cos\theta,\sin\theta)\cdot (-\sin\theta,\cos\theta)d\theta=\int_0^{2\pi}\underbrace{(-\cos\theta\sin\theta+\cos\theta\sin\theta)}_{=0}d\theta=0.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1161258",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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On the definition of clone of relations I am reading A short introduction to clones and I am stuck at this definition ($A$ is a set and $R_A$ the set of finitary relations on $A$)
Definition A subset $R\subseteq R_A$ is called a clone of relations on $A$ if:
(i) $\varnothing\in R$
(ii) $R$ is closed under general super... | Here are some questions:
1) What is $k$?
$k$ is the natural number that is the arity of the relation mentioned in part (ii) of the definition.
2) What does it mean that $\phi_i$ is a map? What is a map from a number to a number?
map=function.
3) I know what the composition of two or many relations is, but what is meant... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1161505",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Irreducible representations of $C(T,B(X))$ Let $T$ be a compact topological space, $X$ a finite-dimensional Hilbert space, $B(X)$ the algebra of operators in $X$, and $C(T,B(X))$ the $C^*$-algebra of continuous maps from $T$ into $B(X)$ (with the poinwise algebraic operations and the uniform norm). I think, all irreduc... | Strictly speaking, the answer is no. The following is true: every irreducible representation is unitarily equivalent to a "point-evaluation". That is, there exists $t\in T$ and a unitary operator $U\in B(X)$ such that
$$
\pi:C(T,B(X))\to B(X),\qquad \pi(f)x=(U^*f(t)U)x,\qquad x\in X,\ f\in C(T,B(X)).
$$
It follows fro... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1161661",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How come $\left(\frac{n+1}{n-1}\right)^n = \left(1+\frac{2}{n-1}\right)^n$? I'm looking at one of my professor's calculus slides and in one of his proofs he uses the identity:
$\left(\frac{n+1}{n-1}\right)^n = \left(1+\frac{2}{n-1}\right)^n$
Except I don't see why that's the case.
I tried different algebraic tricks and... | HINT:
$1+\frac{2}{n-1}=\frac{n-1}{n-1}+\frac{2}{n-1}=\frac{n+1}{n-1}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1161746",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
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Exotic bijection from $\mathbb R$ to $\mathbb R$ Clearly there is no continuous bijections $f,g~:~\mathbb R \to \mathbb R$ such that $fg$ is a bijection from $\mathbb R$ to $\mathbb R$.
If we omit the continuity assumption, is there such an example ?
Notes: to follow from Dustan's comments:
Notes: By definition $fg~:~... | Let $f(x)=x$, and define $g$ piecewise by
$$
g(x) =
\begin{cases}
-x ,& x \in \cdots \cup (-16,-8] \cup (-4,-2] \cup \cdots
&= \bigcup_k -[2 \cdot 4^k, 4 \cdot 4^k) \\
4x ,& x \in \cdots \cup (-8,-4] \cup (-2,-1] \cup \cdots
&= \bigcup_k -[4^k, 2 \cdot 4^k) \\
0 ,& x = 0 \\
x ,& x \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1161829",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
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Inverse of continuous function If we have continuous function $f$ between two topological spaces, such that it is one to one and onto, is it true that we conclude $f^\rm{-1}$ is continuous?
| No. As a counterexample, take
$
f:[0,1) \to S^1
$
given by
$$
f(x) = (\cos(2\pi x),\sin(2\pi x))
$$
or, depending on your preferred definition of $S^1$,
$$
f(x) = e^{2 \pi i x}
$$
However, it is useful to note that if $f:X \to Y$ is one to one and onto with $X$ compact and $Y$ Hausdorff, then $f^{-1}$ must be continuou... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1161912",
"timestamp": "2023-03-29T00:00:00",
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} |
Find the equation of a line parallel to the y-axis, that goes through the point $(\pi,0)$ I have been trying to do this problem and I am very confused.
I know the gradient is infinity when any line is parallel to the y-axis, therefore, $y = \infty \cdot x + c$, right ($y = mx + c$ being the general equation a straight ... | Is this in $\mathbb{R}^2$? If so, the equation $y=mx+b$ will not help you, as the slope of any line parallel to the $y$-axis is undefined. Instead, a vertical line (parallel to $y$-axis) has the equation $x=a$, where $a$ is the $x$-intercept of the line. This should clarify the (correct) answer provided by your teacher... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1162131",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
About multivariable quadratic polynomials Say one has a polynomial function $f : \mathbb{C}^n \rightarrow \mathbb{C}$ such that it is quadratic in any of its variables $z_i$ (for $i \in \{ 1,2,..,n\}$). Then it follows that for any $i$ one can rewrite the polynomial as $f = A_i (z_i - a_i)(z_i - b_i)$, where the consta... | You can't expect the $A_i,a_i,b_i$ to be constant. One could have $$f=z_1^2+z_2^2=(z_1+iz_2)(z_1-iz_2).$$
What about if $$f=z_1^2+z_2^2+z_3^2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1162216",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
You construct a rectangular Box with volume K cm^3 Prove that a cube uses the least amount of material to construct the box
| Let $x$, $y$ and $z$ the lenghts of the sides of the box. If we suppose this box is closed, we must to minimize the function $f(x,y,z)=2xy+2xz+2yz$ restricted to $xyz=K$, we can write the volume of the box, $V$, as follows
$$V=2\left(xy+\frac{K}{y}+\frac{K}{x}\right)$$
If we fix $x$ then we have $V_y=2\left(x-\frac{K}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1162347",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
Can we predict the past? Can we use probability rules to predict the occurrence of an event which has already happened in the past or already formed?
For example, hemoglobin is a protein formed of $141$ amino acids connected like a chain with specific order, the first amino acid is Leucine (we have only $20$ types of ... | Probabilities can represent a state of limited knowledge of events that have
already happened. This can occur, for example, in games of cards after the
hands have already been dealt but before players have revealed their cards through play.
Each player has perfect knowledge of his or her own hand but only probabilistic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1162419",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Find the locus of $w$ $$
\text{Find the locus of $w$, where $z$ is restricted as indicated:} \\
w = z - \frac{1}{z} \\ \text{if } |z| = 2
$$
I have tried solving this by multiplying both sides by $z$, and then using the quadtratic equation. I get $z = \frac{w \pm \sqrt{w^2+4}}{2}$.
I then set $0 \leq w^2+4 $ But I sti... | let $z = 2(\cos t + i \sin t)$. Then $$w= u + iv= z - \frac 1 z=2(\cos t + i \sin t ) - \frac 1 2(\cos t - i\sin t). $$ that gives you $$u = \frac 3 2\cos t, v = \frac 52 \sin t, \, \text{ which is an ellipse } \frac 49 u^2 + \frac 4 {25} v^2 = 1. $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1162573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Integral of an inverse Let $f(x)=x^3−2x^2+5$. Then find the integral
$$\int_{37}^{149} \! f^{-1}(x) \, \mathrm{d}x$$ I know the inverse theorem for differentiation.( I don't think we can apply it here). Is there other theorem for integration.(I am not finding the inverse and then integrating).
| Here $f(x)$ is a cubic polynomial. $f'(x)=x(3x-4)>0$ for $x>37$. So for $x>37, f$ is increasing and hence it is a one to one and onto function from $37$ to $149$. So $f^{-1}$ exists on $[37,149]$. Now use change of variables.
Let $x=f(t)$ for $x\in [37,149]$. Then $f^{-1}(x)=t$. Then, $dx=f'(t)dt$. So, $f^{-1}(x) dx=tf... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1162786",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Limit of an unknown cubic expression Let $f(x)=ax^3+bx^2+cx+d$ and $g(x)=x^2+x-2$. If $$\lim_{x \to 1}\frac{f(x)}{g(x)}=1$$ and $$\lim_{x \to -2}\frac{f(x)}{g(x)}=4$$ then find the value of $$\frac{c^2+d^2}{a^2+b^2}$$ Since the denominator is tending to $0$ in both cases, the numerator should also tend to $0$, in order... | What about? $$f(x)=-x^3+x^2+4x-4$$
So:
$$\frac{c^2+d^2}{a^2+b^2}=32/2=16$$
Let us make ansatz that $$f(x)=a(x+\alpha)(x-1)(x+2)$$
So:
$$a(1+\alpha)=1,a(\alpha-2)=4\implies a=-1,\alpha=-2$$
so:
$$f(x)=-(x-1)(x+2)(x-2)=-x^3+x^2+4x-4$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1162864",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
$Y=X+Z$ and $X \perp Z$ and $Y$, $X$, $Z$ continuous. Is $X \perp Z|Y$? Let $Y=X+Z$. Assume that $Y$, $X$, $Z$ are continuously distributed. Let $X \perp Z$. Prove / disprove that $X \perp Z | Y$.
I've seen a bunch of these types of problems on math.SE (and almost all disprove by counterexample) but in all problems I'v... | Other than in degenerate examples, intuitively, it seems obvious that $X,Z$ are dependent given $Y$ because given both $Z$ and $Y$, the value of $X$ is determined exactly. This applies for both discrete and continuous random variables.
In the continuous case, for any distributions of $X,Z:\;\;$ $f_{X,Z\mid Y}(x,z\mid y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1162948",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Inclusion relation between two summability methods Let $0\leq x<1$ and $s_n$ be a sequence of partial sums of the series $\sum_{n=0}^{\infty}a_n$. It is called that the series $\sum_{n=0}^{\infty}a_n$ is $(A)$ or Abel summable to $s$ if $$\lim_{x\to1^-}(1-x)\sum_{n=0}^{\infty}s_nx^n=s,$$ and the series $\sum_{n=0}^{\in... | $$
\begin{matrix}
\text{Let} &
F(x) = \sum\limits_{n=0}^\infty \frac{s_n}{n+1} x^{n+1}
&\text{so}&
F'(x) = \sum\limits_{n=0}^\infty s_n x^n \\
\text{and} &
G(x) = -\log(1-x)
&\text{so}&
G'(x)=\frac1{1-x} \\
\text{then} &
-\dfrac1{\log (1-x)}\sum\limits_{n=0}^\infty \dfrac{s_n}{n+1}x^n=\dfrac{F(x)}{G(x)}
&\text{and}&
(1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1163052",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Contour integration of cosine of a complex number I am trying to find the value of
$$ -\frac{1}{\pi}\int_{-\pi/2}^{\pi/2} \cos\left(be^{i\theta}\right) \mathrm{d}\theta,$$
where $b$ is a real number.
Any helps will be appreciated!
| Because cosine is an even function you may write the integral as
$$-\frac1{2 \pi} \int_{-\pi/2}^{3 \pi/2} d\theta \, \cos{\left ( b e^{i \theta} \right )} = -\frac{1}{i 2 \pi} \oint_{|z|=1} dz \frac{\cos{b z}}{z} $$
which, by the residue theorem or Cauchy's theorem, is
$$-\frac{1}{i 2 \pi} i 2 \pi \cos{0} = -1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1163150",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
} |
Solving recurrence equation with generating indices of positive indices I don't know how to solve recurrence equation with positive indices like $$a_{n+2} + 4a_{n+1}+
4a_n = 7$$ by generating functions.
How to solve such kind of problems.
| Hint. You may just multiply both sides of the following relation by $x^{n+2}$ and summing it:
$$
a_{n+2} + 4a_{n+1}+4a_n = 7 \tag1
$$ to get
$$
\sum_{n=0}^{\infty}a_{n+2}x^{n+2} + 4x\sum_{n=0}^{\infty}a_{n+1}x^{n+1}+4x^2\sum_{n=0}^{\infty}a_n x^n= 7x^2\sum_{n=0}^{\infty}x^n
$$ or
$$
\sum_{n=2}^{\infty}a_{n}x^{n} + 4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1163250",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
How to prove this limit is $0$? Let $f:[0,\infty )\rightarrow \mathbb{R}$ be a continuous function such that:
*
*$\forall x\ge 0\:,\:f\left(x\right)\ne 0$.
*$\lim _{x\to \infty }f\left(x\right)=L\:\in \mathbb{R}$.
*$\forall \epsilon >0\:\exists x_0\in [0,\infty)$ that $0<f\left(x_0\right)<\epsilon $.
P... | First note that $f(x) >0$ for all $x$. This follows from 3. and the fact that $f$ is
continuous (and so $f([0,\infty))$ is connected). It follows from this that
$L \ge 0$.
Let $\alpha(x) = \min_{t \in [0,x]} f(t)$. Since $[0,x]$ is compact, we see that
$\alpha(x) >0$ for all $x$. Property 3. shows that $\lim_{x \to \in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1163331",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
} |
On a property of split short exact sequences Let $A_{\bullet}, B_\bullet$ and $C_\bullet$ be three short exact sequences of groups (not necessarily abelian) out of which $A_\bullet$ and $B_\bullet$ are split. Assume that there is again a short exact sequence, $$0 \to A_{\bullet} \to B_{\bullet} \to C_{\bullet} \to 0$$o... | $\newcommand{\ZZ}{\mathbb{Z}}$
Here is what I believe to be a counterexample, where all groups are $\ZZ$-modules (i.e., abelian groups):
$A_1 = 2 \ZZ$. $A_2 = \left\{\left(i,j\right) \in \ZZ \times \ZZ \mid 4 \mid 2i+j\right\}$ (the first "$\mid$" here is a "such that" symbol, while the second one is a "divides" sign).... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1163460",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
convergence of a numerical series I would like to study the convergence of the numerical serie
$$
S_n=\sum_{k= 1}^n u_k=\sum_{k= 1}^n \frac{1}{\left(\sqrt[k]{2}+\log k\right)^{k^2}}.
$$
I tried the Cauchy rule (i.e. evaluate $\lim_{k\rightarrow +\infty}(u_k)^{\frac 1 k}$ but there is no issue.
| $$\frac1{\left(\sqrt[k]2+\log k)\right)^{k^2}}<\frac1{\sqrt[k]{2}^{k^2}}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1163560",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
H-function for the following integral I stumbled upon the integral $\int\limits_0^{+\infty} u^\nu\exp(-au-bu^\rho)du$, $\Re(a)>0,\,\,\Re(b)>0,\,\,\rho>0$. I cannot find any way to represent it using the Fox-H function. Any hints? PS: This is the Krätzel function, right? The one known as reaction rate integral.
| Expand the factor $exp(-bu^\rho)$ into a Taylor series. You get:
$\int_0^\infty u^\nu exp(-au-bu^\rho)du=\sum_{i=0}^\infty \frac{1}{i!} (-b)^i \int_0^\infty u^{\nu+i \rho} exp(-au)du$. With the Substitution $v=au$ you get the following series of Gamma functions:
$\int_0^\infty u^\nu exp(-au-bu^\rho)du=\sum_{k=0}^\infty... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1163645",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
linear least square problems solved using LU decomposition I have been given this datafile
For which I have to solve Ax=b. In which A is a matrix, x a vector and b a vector.
The datafile consist of 2 vector one with the X-coordinates, and the other one with y-coordinates. I don't how i based from that shall create a ... | Here is how you get the matrix. You substitute the pairs $(x_i,y_i)$ in the model
$$ y = \beta_0+\beta_1 x +\beta_2 x^2 + \epsilon $$
to get the system of equations
$$ y_i = \beta_0+\beta_1 x_i +\beta_2 x_i^2 + \epsilon, \quad i=1,2,\dots, n $$
then you will get the matrix.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1163715",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Trees have a vertex of degree $1$ Prove that any tree has a vertex of degree $1$.
Let graph $G=(V,E)$ have $n$ vertices and $m$ edges where $m<n$. We need to prove that the minimum degree of, $\delta (G)=1$
Since G is connected then there exists a path from $u$ to $v$ such that $u,v \in V$.
Is what I have said so far... | This is trivial: every tree has n-1 edges, where n is the number of vertices. If every vertices has degree at least 2 the sum of the degree is at least 2n.So there are at least n edges. Impossible.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1163824",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
In how many ways you can write $2015$ as a $20x+15y$? In how many ways you can write $2015$ as a $20x+15y$ where $x$ and $y$ are natural numbers?
So probably I can do it using Euclid's algorithm but right now I am not sure how to do it. Could anyone explain step-by-step how to do exercises like this? I will be so grate... | This might be overkill, but it couldn't do any harm.
Let
$$\begin{align*}
f : \mathbb Z \times \mathbb Z &\rightarrow \mathbb Z \\
(x,y) &\mapsto 20x + 15y
\end{align*}$$
We are looking for solutions to $f(x,y) = 2015$.
Suppose you've found two solutions $(x,y)$ and $(X,Y)$. Then $f(X,Y) - f(x,y) = 0 = f(X-x,Y-y)$ and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1163922",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Cheeger inequalities for nonregular graphs I'm looking for a reference for something I thought was easy and well known.
There are (at least) two definitions of expander graphs. There is a combinatorial definition via edge expansion, and an algebraic definition using the spectral gap.
Neither of these definitions requi... | For graphs that are not regular, the right matrix to look at is $A_G' := D^{−1/2}A_GD^{−1/2}$. (Here $D^{−1/2}$ is simply the diagonal matrix whose $(i; i)$th entry is $(\deg(i))^{−1/2}$. We assume there are no isolated vertices, so none of the degrees is zero).
This matrix is sometimes called the normalized adjacency ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1164023",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Is there a difference between the calculated value of Pi and the measured value? The mathematical value of Pi has been calculated to a ridiculous degree of precision using mathematical methods, but to what degree of precision has anyone actually measured the value of Pi (or at least the ratio of diameter to circumferen... | There's an underlying error in the question, namely the assumption that being in a curved space would result in a "different measured value of $\pi$".
What happens in a curved space is that the ratio between a circle's circumference and diameter is no longer the same for all circles. More precisely, the ratio will depe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1164156",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Complex Analysis: Confirmation for a question in my textbook? I'm being told that $$\frac{\exp{(1+i3\pi})}{\exp{(-1+i\pi /2})}= \exp(2)i$$
I keep getting $-\exp(2)i$. I have no idea how they didn't get that to be negative.
| Relatively straightforward :
$$\frac{\exp{(1+i3\pi})}{\exp{(-1+i\pi /2})}= \frac{e \exp({i3\pi})}{e^{-1}\exp{(i\frac{\pi}{2}})} = \frac{e\times(-1) }{e^{-1}\times i} = -e^2 \frac{1}{i}$$
But $\frac{1}{i} = -i$ (because $-1 = i\times i$ then you divide by $i$) so you get
$$\frac{\exp{(1+i3\pi})}{\exp{(-1+i\pi /2})}=ie... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1164260",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
How to compute the summation of a binomial coefficient/ show the following is true $\sum\limits_{k=0}^n \left(2k+1\right) \dbinom{n}{k} = 2^n\left(n+1\right)$.
I know that you have to use the binomial coefficient, but I'm not sure how to manipulate the original summation to make the binomial coefficient useful.
| Consider
$$f(x)=(1+x)^n=\sum_{k=0}^n\binom{n}{k}x^k$$
so
$$f'(x)=\sum_{k=1}^{n}k\binom{n}{k}x^{k-1}$$
Hence
$$f'(1)=\sum_{k=1}^{n}k\binom{n}{k}=\sum_{k=0}^{n}k\binom{n}{k}$$
On the other hand
$$f'(x)=n(1+x)^{n-1}\Rightarrow f'(1)=n2^{n-1}$$
It follows that
$$\sum_{k=0}^{n}k\binom{n}{k}=n2^{n-1}$$
Similarly we have
$$\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1164366",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Mathematical induction for inequalities: $\frac1{n+1} + \frac1{n+2} + \cdots +\frac1{3n+1} > 1$ Prove by induction: $$\frac1{n+1} + \frac1{n+2} + \cdots +\frac1{3n+1} > 1$$
adding $1/(3m+4)$ as the next $m+1$ value proves pretty fruitless. Can I make some simplifications in the inequality that because the $m$ step is t... | More generally
(one of my favorite phrases),
let
$s_k(n)
=\sum\limits_{i=n+1}^{kn+1} \frac1{i}
$.
I will show that
$s_k(n+1)>s_k(n)$
for $k \ge 3$.
In particular,
for $n \ge 1$
$s_3(n)
\ge s_3(1)
=\frac1{2}+\frac1{3}+\frac1{4}
=\frac{6+4+3}{12}
=\frac{13}{12}
> 1
$.
$\begin{array}\\
s_k(n+1)-s_k(n)
&=\sum\limits_{i=n+2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1164493",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Integral of $\frac{e^{-x^2}}{\sqrt{1-x^2}}$ I am stuck at an integral $$\int_0^{\frac{1}{3}}\frac{e^{-x^2}}{\sqrt{1-x^2}}dx$$
My attempt is substitute the $x=\sin t$, however there may be no primitive function of $e^{-\sin^2 t}$.
So does this integral has a definitive value? If does, how can we solve it? Thank you!
| We have:
$$ I = \int_{0}^{\arcsin\frac{1}{3}}\exp\left(-\sin^2\theta\right)\,d\theta \tag{1}$$
but since:
$$ \exp(-\sin^2\theta) = \frac{1}{\sqrt{e}}\left(I_0\left(\frac{1}{2}\right)+2\sum_{n\geq 1}I_n\left(\frac{1}{2}\right)\cos(2n\theta)\right)\tag{2} $$
we have:
$$ I = e^{-1/2}\left(\arcsin\frac{1}{3}\right) I_0\lef... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1164662",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Evaluate $ \int \frac{e^x}{\left({1+\cos(x)}\right)} dx$ Background: I was in the process of solving some interesting integrals from this site, only to find out I needed a lot more practice before becoming familiar with special functions.
So while doing some problems, I encountered some difficulty with one particular ... | The integrand does not possess an elementary antiderivative. This can be shown using either Liouville's theorem or the Risch algorithm. However, doing so requires advanced knowledge of abstract algebra. Alternately, expand $~\dfrac1{1+\cos x}~$ into its binomial series, then switch the order of summation and integrat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1164780",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Prove that: $\lim\limits_{x \to +\infty}\frac{(\log x)^b}{x^a} = 0$ and $\lim\limits_{x \to +\infty}\frac{x^b}{e^{ax}} = 0$ If $a > 0$ and $b > 0$, show that
$$\lim_{x \to +\infty}\frac{(\log x)^b}{x^a} = 0 \tag{1}$$
and
$$\lim_{x \to +\infty}\frac{x^b}{e^{ax}} = 0 \tag{2}$$
Attempts:
$(1)$
Given that
$$\log x = \fr... | You must know something more, if you cannot use De l'Hospital. For instance
$$
\lim_{x \to +\infty} \frac{x^b}{e^x}=0 \quad\hbox{for every $b>0$}
$$
or
$$
\lim_{x \to +\infty} \frac{x}{e^{ax}}=0 \quad\hbox{for every $a>0$}.
$$
Indeed,
$$
\frac{x^b}{e^{ax}}=\left( \frac{x}{e^{\frac{a}{b}x}} \right)^b
$$
or
$$
\frac{x^b}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1164895",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Find a linear operator s.t. $A^{2}=A^{3}$ but $A^{2}\neq A$? From Halmos's Finite-Dimensional Vector Spaces, question 6a section 43, the section after projections. Find a linear transformation A such that $A^{2}(1-A)=0$ but A is not idempotent (I remember A is idempotent iff it is a projection).
I had no luck.
| (Too long for a comment.) You have no luck because you depend on luck.
It really doesn't take much to solve the problem. If $A\ne A^2=A^3$, there must exist a vector $x$ so that when $y=Ax$ and $z=A^2x$, we have $y\ne z$ but $Az=z$. In other words, by applying $A$ repeatedly on $x$, we get the following chain of iterat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1165126",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Find : $ dy/dx, y=\sqrt{4x^2 - 7x - 2}$ The problem says Find $dy/dx, y=\sqrt{4x^2 - 7x - 2}$
So far I changed it to $(4x^2 - 7x - 2)^{1/2}$
I don't know where to go from there.
| hint: Use the chain rule: $\sqrt{u(x)}' = \dfrac{u'(x)}{2\sqrt{u(x)}}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1165243",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If we have a square matrix thats invertible, do its row and column space coincide? If we have a square matrix thats invertible, do its row and column space coincide?
Regarding an nxn invertible matrix:
-The row space of the matrix is R^n
-The column space of the matrix is R^n
-The rank of the matrix is n
Is this a suff... | $\newcommand{\Reals}{\mathbf{R}}$The row space and column space of an $n \times n$ matrix are not generally equal, e.g.,
$$
A = \begin{pmatrix}
0 & 1 \\
0 & 0
\end{pmatrix},\quad
\text{row space} = \{0\} \times \Reals,\quad
\text{column space} = \Reals \times \{0\}.
$$
The row space and column space of an $n \times n$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1165349",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How can I demonstrate that $x-x^9$ is divisible by 30? How can I demonstrate that $x-x^9$ is divisible by $30$ whenever $x$ is an integer?
I know that $$x-x^9=x(1-x^8)=x(1-x^4)(1+x^4)=x(1-x^2)(1+x^2)(1+x^4)$$
but I don't know how to demonstrate that this number is divisible by $30$.
| Let's factor $x^9-x$ like you have done:
$$
x^9-x=(x-1)x(x+1)(x^2+1)(x^4+1).\tag{$*$}
$$
Let's look at the RHS. The product of the first 2 terms is divisible by $2$ because it consists of 2 consecutive integers. Similarly, the product of the first 3 terms is divisible by $3$. Now, if you had
$$
(x-2)(x-1)x(x+1)(x+2)
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1165438",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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Factoring Polynomials in Fields I always have problems to factorize polynomials that have no linear factors any more. For example ($x^5-1$) in $\mathbb{F}_{19}$. It's easy to find the root 1 and to split it. ($x^5-1$) = ($x-1$) * ($x^4$+$x^3$+$x^2$+x+1).
I think the last part must split into two irreducible polynomials... | There is a trick to see that the polynomial is reducible. The multiplicative group $\mathbb{F}_{19^2}^*$ of the field of $19^2$ elements has $19^2 - 1 \equiv 0 \mod 5$ elements, so it contains a fifth root of unity. So the minimal polynomial of a fifth root of unity over $\mathbb{F}_{19}$, which divides $x^5 - 1$, has ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1165576",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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Show that for any two Cauchy sequences of rational numbers, their difference is a equivalent to a sequence of nonnegative numbers Let $a_n$ and $b_n$ be Cauchy sequences of rational numbers, either $b_n-a_n$ or $a_n-b_n$ is a sequence of nonnegative numbers.
I don't really understand how this is true, I think first we ... | This sentence with
either $b_n-a_n$ or $a_n-b_n$ is a sequence of nonnegative numbers
is not true in this form: you can modify the first some elements of a sequence anyhow without affecting its being Cauchy. What is true, is either
*
*one of the sequences $b_n-a_n$ and $a_n-b_n$ is equivalent to (=has the same lim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1165669",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Prove that lim$_{n → \infty} \int_{0}^{3} \sqrt(sin \frac{x}{n} + x + 1) dx$ exists and evaluate it Prove that the following limits exist and evaluate them.
c) $\lim_{n \to \infty} \int_{0}^{3} \sqrt{\sin \frac{x}{n} + x + 1}\ \text dx$
I need to use the following theorem from analysis;
Suppose $f_n \to f$ uniformly on... | I am not sure that this could be the answer you expect; so, please forgive me if I am off-topic.
When $n$ is large, we can approximate $\sin(x)$ by its Taylor expansion built at $x=0$. The problem is that only the first term of the expansion can be used (otherwise we should bump on nasty elliptic integrals).
$$I_n=\int... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1165941",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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If $c_{n} > 0$ then $\sum_{0}^{n}c_{k}x^{k} > 0$ for some $x \in \mathbb{R}$? Let $n \geq 1$ be an integer and let $c_{0}, \dots, c_{n} \in \mathbb{R}$. If $c_{n} > 0,$ is there necessarily an $x \in \mathbb{R}$ such that
$$\sum_{0}^{n}c_{k}x^{k} > 0?$$
I just realized that for a while I had implicitly taken this for ... | Yes.
Suppose $x>0$ and consider $f(x) = {\sum_{k=0}^n c_k x^k \over x^n} = \sum_{k=0}^n c_k {1 \over x^{n-k}}$. Then $\lim_{x \to \infty} f(x) = c_n>0$, hence there is some $M$ such that if $x \ge M$, then $f(x) >0$. Hence
${\sum_{k=0}^n c_k x^k } >0$ for $x \ge M$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1166053",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What kind of mathematics is required by game theory? I want to learn about game theory, but I do not know if I have the necessary background to do so.
What kind of mathematics does game theory involve the most? What are some of the things that an undergrad in mathematics might not have seen which arises in game theory?... | Perhaps the main thing that you will come up with and with which you will not be familiar as an undergraduate student are fixed point theorems (functional analysis) and linear or/and dynamic programming.
Generally, for an undergraduate course in game theory you will mostly need to be familiar with the following:
*
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Series Convergence $\sum_{n=1}^{\infty} \frac{1}{n} \left(\frac{2n+2}{2n+4}\right)^n$ I have to show if this series converges or diverges. I tried using asymptotics, but it's not formally correct as they should work only when the arguments are extremely small. Any ideas?
$$
\sum_{n=1}^{\infty} \frac{1}{n} \left(\frac{2... | Hint: Put $n=k-2$ and simplify the fraction inside the parentheses.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1166192",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 1
} |
Find base vectors and dim Find base vectors and dim of a space described by the following system of equation:
$$2x_1-x_2+x_3-x_4=0 \\ x_1+2x_2+x_3+2x_4=0 \\ 3x_1+x_2+2x_3+x_4=0$$
I did rref of the matrix and as a result i get:
$$\begin{pmatrix}
1 & 2 & 1 & 2 \\
0 & -5 & -1 & -5 \\
0 & 0 & 0 & 0
\end{pmatrix} $$ Th... | My definition of RREF is different from yours it seems, and I calculated RREF form of augmented matrix to be:
$$ \left(
\begin{array}{cccc|c}
1 & 0 & 3/5 & 0 & 0\\
0 & 1 & 1/5 & 1 & 0\\
0 & 0 & 0 & 0 & 0
\end{array}
\right)
$$
From this you set $x_3=t, x_4=s$ where $t,s$ are (scalar) parameters, and you can describe yo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1166277",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Complex solutions of equations in Maple Is it possible to compute all complex solutions of the equation
$$
e^z = 1
$$
in Maple?
That is, I need Maple print all solutions $z=2\pi k I$.
What procedure do I have to use?
Thank you very much in advance!
| It is done by
solve(exp(z)=1,z,AllSolutions=true);
The output will be
2*I*Pi*_Z1~
The _Z1 represents some constant, and the tilde implies that there is some assumption on the constant, which in this case means that it is an integer.
getassumptions(_Z1);
tells you that it must be an integer.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1166406",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Barycentric coordinates in a triangle - proof I want to prove that the barycentric coordinates of a point $P$ inside the triangle with vertices in $(1,0,0), (0,1,0), (0,0,1)$ are distances from $P$ to the sides of the triangle.
Let's denote the triangle by $ABC, \ A = (1,0,0), B=(0,1,0), C= (0,0,1)$.
We consider triang... | Let $A_i$ $\>(1\leq i\leq3)$ be the vertices of your triangle $\triangle$, and let $P=(p_1,p_2,p_3)$ be an arbitrary point of $\triangle$. Then the cartesian coordinates $p_i$ of $P$ satisfy $p_1+p_2+p_3=1$, and at the same time we can write
$$P=p_1A_1+p_2A_2+p_3A_3\ ,$$
which says that the $p_i$ can be viewed as well... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 2,
"answer_id": 1
} |
Solve $\sin(z) = 2$ There are a number of solutions to this problem online that use identities I have not been taught. Here is where I am in relation to my own coursework:
$ \sin(z) = 2 $
$ \exp(iz) - \exp(-iz) = 4i $
$ \exp(2iz) - 1 = 4i \cdot \exp (iz) $
Then, setting $w = \exp(iz),$ I get:
$ w^2 - 4iw -1 = 0$
I can ... | Setting $w=e^{iz},$ we need to solve the equation $w^2-4iw-1=0.$ The solutions to this quadratic equation are $w=i(2+\sqrt 3)$ and $w=i(2-\sqrt 3).$
Let's deal with the first solution. We need to find $z=x+iy$ such that $e^{iz}= e^{ix}e^{-y}= i(2+\sqrt 3).$ This implies $\cos x =0.$ As you point out, that has solution ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1166562",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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Prove that a number that consists of $3^n$ ones is divisible by $3^n$ I can't even make sense of this question. Isn't this just like asking, "Prove that 3 is divisible by 3." Isn't any number divisible by itself? Is this all there is to this question—it seems like there must be more to it.
| We can use induction. I prefer to show that the number that "consists of" $3^n$ $9$'s is divisible by $9\cdot 3^n$.
The number whose decimal representation consists of $3^n$ consecutive $9$'s is $10^{3^n}-1$.
For the induction step, note that $10^{3^{k+1}}-1=x^3-1$ where $x=10^{3^k}$. This factors as $(x-1)(x^2+x+1)$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1166675",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show $17$ does not divide $5n^2 + 15$ for any integer $n$ Claim: $17$ does not divide $5n^2 + 15$ for any integer $n$.
Is there a way to do this aside from exhaustively considering $n \equiv 0$, $n \equiv 1 , \ldots, n \equiv 16 \pmod{17}$ and showing $5n^2 + 15$ leaves a remainder of anything but $0$. It's easy but te... | ${\rm mod}\ 17\!:\ 5(n^2\!+3)\equiv0\,\Rightarrow\,n^2\equiv-3\,\Rightarrow\,n^4\equiv 9\,\Rightarrow\, n^8\equiv -4\,\Rightarrow\,n^{16}\equiv -1$ contra little Fermat
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1166877",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 2
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Prove that the equation $x^{2}-x\sin(x)-\cos(x)=0$ has only one root in the closed interval $(0,\infty)$. Here's the graph
(http://www.wolframalpha.com/input/?i=%28x%5E2%29-xsenx-cosx%3D0).
The part I'm having trouble with is proving that the root is unique.
I can use the intermediate value theorem to find the interva... | $f(x)=x^2-x \sin x -\cos(x)$ and $f'(x)=2x-x \cos x=x(2-\cos x)$. Clearly $2-\cos x>0$ and for $x>0$ we have that $f'(x)> 0$ $x>0$, therefore $f(x)$ is strctly increasing. now since $f(0)<0$ and $f(\infty)>0$, $f(x)$ just has one root on $x>0$.
This is a graphs of $f(x)$
and here is $f'(x)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1166972",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
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How is N defined? I understand that $\mathbb{Z}$ and $\mathbb{Q}$ are defined as ...
$$\mathbb{Z} = \mathbb{N} \cup \{ -n \mid n \in \mathbb{N} \}$$
$$\mathbb{Q} = \left\{ \frac{a}{b} \mid a,b \in \mathbb{Z} \right\}$$
... but how is $\mathbb{N}$ defined?
(and how is the order on these sets defined?)
| You can make a construction of natural numbers based on set theory by defining $0:=\{\,\}=\varnothing$ and the successor of $n$ $($denoted by $S(n))$ as $S(n)=n\cup\{n\}$. And so for instance :
*
*$1:=S(0)=\varnothing\cup\{\varnothing\}=\{\varnothing\}$.
*$2:=S(1)=\{\varnothing\}\cup\{\{\varnothing\}\}=\{\{\varnoth... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why is the ring of integers initial in Ring? In Algebra Chapter 0 Aluffi states that the ring $\Bbb{Z}$ of integers with usual addition and multiplication is initial in the category Ring. That is for a ring $R$ with identity $1_{R}$ there is a unique ring homomorphism $\phi:\Bbb{Z}\rightarrow{R}$ defined by $\phi(n)\ma... | Any such morphism $f$ satisfies $f(n)=f(\sum_{i=1}^n 1) = \sum_{i=1}^n f(1) = \sum_{i=1}^n 1_R = n 1_R$. You have the unicity. The existence is trivial, just define it this way, by the previous formula. So you have the definition for any ring $R$ with unit. ;-) Note that the image of $\mathbf{Z}$ is always in $R$'s cen... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1167176",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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"answer_id": 2
} |
Is $\sqrt{x^2} = x$? Does the $\sqrt{x^2}$ always equal $x$? I am trying to prove that $i^2 = -1$, but to do that I need to know that $\sqrt{(-1)^2} = -1$. If that is true then all real numbers are imaginary, because an imaginary number is any number that can be written in terms of $i$. For example, 2 can be written as... | *
*It is not true that $\sqrt{x^2} = x$. As a very simple example, with $x=-2$, we obtain
$$ \sqrt{(-2)^2} = \sqrt{4} = 2 \ne -2. $$
In general, if $x \in \mathbb{R}$, then $\sqrt{x^2} = |x|$. Things get more complicated when you start working with complex numbers, but I think that a discussion of "branches of the s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1167315",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Inequality involving exponential of square roots How can I show that:
$$ 2e^\sqrt{3} \leq 3e^\sqrt{2} $$
? (that's all I have)
Thank you so much!
| Let $f(x)=\frac{e^{x}}{x^2}$. Then $$f'(x)=e^x\left(\frac{x-2}{x^3}\right)$$
So $f'(x)<0$ for $0<x<2$. Then $f(\sqrt{3})<f(\sqrt{2})$ and your result follows.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1167459",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Have I obtained the proper solution to this PDE? I'm a little stuck on this.
Consider $ u_t -(1+t^2)u_x = \phi(x,t) \quad u(x,0)=u_0(x)$
Via the method of characteristics, the total derivative of $u(x,t)$ is
$$\frac{du}{dt} = \dfrac{\partial u}{\partial t} + \frac{dx}{dt}\frac{\partial u}{\partial x}\>. $$
Therefore,... | Follow the method in http://en.wikipedia.org/wiki/Method_of_characteristics#Example:
$\dfrac{dt}{ds}=1$ , letting $t(0)=0$ , we have $t=s$
$\dfrac{dx}{ds}=-(1+t^2)=-1-s^2$ , letting $x(0)=x_0$ , we have $x=x_0-s-\dfrac{s^3}{3}=x_0-t-\dfrac{t^3}{3}$
$\dfrac{du}{ds}=\phi(x,t)=\phi\left(x_0-s-\dfrac{s^3}{3},s\right)$ , le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1167600",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
$\sum_{n=1}^{\infty}\frac{n^2}{a^2_{1}+a^2_{2}+\cdots+a^2_{n}}$is also convergent? Let sequence $a_{n}>0$, $n\in N^{+}$, and such
$\displaystyle\sum_{n=1}^{\infty}\dfrac{1}{a_{n}}$ convergent. Show that
$$\sum_{n=1}^{\infty}\dfrac{n^2}{a^2_{1}+a^2_{2}+\cdots+a^2_{n}}$$ is also convergent?
Jack A related result: maybe I... | The Polya-Knopp's inequality (that is an instance of Hardy's inequality for negative exponents) states that for any $p\geq 1$ and for every positive sequence $\{a_n\}_{n\in\mathbb{N}}$ we have:
$$ \frac{N^{\frac{p+1}{p}}}{(p+1)\left(a_1^p+\ldots+a_N^p\right)^{1/p}}+\sum_{n=1}^N \left(\frac{n}{a_1^p+\ldots+a_n^p}\right)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1167832",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 1,
"answer_id": 0
} |
Prove that $n^3+2$ is not divisible by $9$ for any integer $n$ How to prove that $n^3+2$ is not divisible by $9$?
| Suppose, $\exists n\in \mathbb{N}$ such that $n^3+2\equiv 0 \pmod{9}\implies n^{6}\equiv 4\pmod{9}$, which is not true since\begin{array}{rl}
n^6\equiv 1\pmod 9 & \mbox{if $n$ and $9$ are relativley prime, by Euler's theorem, since $\phi(9)=6$.}\\
n^6\equiv 0\pmod 9 & \mbox{otherwise}
\end{array}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1167968",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 2
} |
How to evaluate $\sum_{n=1}^\infty \frac{1}{n 2^{2n}}$?
Evaluate $$\sum_{n=1}^\infty \frac{1}{n2^{2n}}$$
I'd be glad for guidance, because, frankly, I don't have clue here.
| We could write this just as well as
$$
\sum_{n=1}^\infty \frac{1}{n}(1/4)^n
$$
Consider the function
$$
f(x) = \sum_{n=1}^\infty \frac{1}{n}x^n \quad x \in (-1,1)
$$
noting that $f(0) = 0$.
We evaluate
$$
f'(x) = \sum_{n=0}^\infty x^n = \frac{1}{1-x} \quad x \in (-1,1)
$$
It follows that
$$
f(4) = f(0) + \int_0^{1/4} f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1168084",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Algorithm that decides whether collection of primes exists which satisfies 3 equations Suppose someone gives you a list of $n$ positive numbers $(a_1, \ldots , a_n)$, together with an upper limit $N$ and asks you to find prime numbers $p_1, \ldots ,p_n$ in the range $2, \ldots , N$ satisfying
$p_2 = p_1 + a_1$ and $p_3... | Here is some pseudo-code for my brutish approach to it.
Let A = {a1, a2, ..., an}
Let sieve = {p | p is prime and less than N}
If ((size of sieve < size of A) OR (some a[i] == 1 for i > 1))
return False;
Let primeDiffs = {sieve[i] - sieve[i-1] | i in [2..(size of sieve)]}
For i in [1..(size of primeDiffs - size... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1168206",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Explicit formula for IFS fractal dimesnion Is there an explicit formula for finding the box counting dimension of an arbitrary IFS fractal, such as the IFS fern or any other random IFS fractal? If not, is there at least a summation, or recurrence relation that could find the fractal dimension? An example of how this wo... | For graph-directed IFS of similarities satisfying an open set condition with every cycle contracting, it's possible to compute the Hausdorff dimension:
Hausdorff Dimension in Graph Directed Constructions
R Daniel Mauldin, S C Williams
Transactions of AMS, vol309 no2, October 1988, 811-829
A regular IFS of can be consid... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1168300",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Notation for Average of a Set? In particular, I have some set $S = \{s_1, s_2, s_3, ..., s_n\}$ and a subset $S^\prime$, and I want to denote the average of the elements in $S^\prime$. I would generally just use $\frac{\sum\limits_{i=1}^n s_i}{n}$, but $S^\prime$ only contains some of the elements of $S$ and so this wo... | Physicists may use $\langle S'\rangle$; statisticians might write $\bar{S'}$. But since you mention $\displaystyle\sum_{i=1}^n s_i /n$, I suggest $\displaystyle\sum_{s\in S}s/|S|$ or $\displaystyle\sum_{s\in S'}s/|S'|$, as the case may be. In some contexts one could write $\displaystyle\sum S/|S|$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1168533",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
"answer_id": 0
} |
Geometry problem (triangle) How to solve the following problem
Let $P$ be a point inside or outside (but not on) of a triangle $\Delta ABC$. Prove that $PA +PB +PC$ is greater than half of the perimeter of the triangle.
That is, show
$$
PA+PB+PC > 1/2(A+B+C)
$$
| Just use the fact that the sum of two sides of a triangle is always bigger then the third side. For example, if $P$ was inside then you look at the (small) triangle $\Delta PBC$, there
$$PB+PC \geq BC$$
Similarly we get
$$PA+PC \geq AC$$
and
$$PB+PA \geq AB.$$
Now add all the inequalities.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1168611",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How Do I solves these types of limits such as $\lim_{x\to\infty}\frac{7x-3}{2x+2}$ I guess the best question I have is how does my strategy change when I get a limit such as
$$\lim_{x\to\infty}\dfrac{7x-3}{2x+2}$$
What is essential to have as an understanding to solve these problems? Help welcomed.
| The method you should be taking is to take advantage of algebra, and in some cases L'Hospitals rule.
So for this question, we have $\lim_{x \to \infty} $ $ \frac{7x-3}{2x+2} $
We I'm sure you can see as x approached infinity we have the case of infinity/infinity so it is valid to use the rule. So (7x-3)'=(7) and (2x+2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1168671",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Does $\|u\|=\|u^\ast\|$ imply $\|uh\| = \|u^\ast\|$? Let $H$ be a Hilbert space and $u \in B(H)$ and let $u^\ast$ denote its adjoint. I know that $\|u\|=\|u^\ast\|$. But now I am wondering:
Does $\|u\|=\|u^\ast\|$ imply $\|uh\| = \|u^\ast h\|$ for all $h\in
H$?
At first I thought that yes but on second thought I can... | Consider $H=\mathbb{R}^2$, and $u\in B(H)$ is given by the matrix $\begin{bmatrix} 1& 1\\
0 &1\end{bmatrix}$, then $u^*$ is the transpose $\begin{bmatrix} 1& 0\\
1 &1\end{bmatrix}$. Let $h=\begin{bmatrix} 1\\
2 \end{bmatrix}$, then
$$uh=\begin{bmatrix} 3\\
2 \end{bmatrix},\quad u^*h=\begin{bmatrix} 1\\
3 \end{bmatrix}$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1168739",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find $\lim_{x \to \infty} \left(\frac{x^2+1}{x^2-1}\right)^{x^2}$ How to calculate the following limit?
$$\lim\limits_{x \to \infty} \left(\frac{x^2+1}{x^2-1}\right)^{x^2}$$
| The qty in bracket tends to 1 as x→infinte and power tends to infinity u can easily prove that Lt(x→a){f(x)}^(g(x)) if f(a)→1 and g(a)→∞ then its equal to limit of e^(f(x)-1)(g(x)) as x→a so here it is.. e^(2/(x^2-1))(x^2) limit as x→ ∞ giving e^2 .. !
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1168828",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 7,
"answer_id": 6
} |
Completion of borel sigma algebra with respect to Lebesgue measure There are two ways of extending the Borel $\sigma$-algebra on $\mathbb{R}^n$, $\mathcal{B}(\mathbb{R}^n)$, with respect to Lebesgue measure $\lambda$.
*
*The completion $\mathcal{L}(\mathbb{R}^n)$ of $\mathcal{B}(\mathbb{R}^n)$ with respect to $\lamb... | From the definition of the outer measure $\lambda^{*}$, you can show that if $A\in \mathcal{L}'$ then there's a $G_{\delta}$ set $B$ so that $A\subseteq B$ and $\lambda^{*}(B\setminus A)=0$. After that, the answer to this question is an easy yes.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1168953",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 2,
"answer_id": 1
} |
Example of equivalent metrics on the same set such that uniform continuity of some function is not preserved Give example of a set $X$ and two metrics $d_1,d_2$ on $X$ such that $(X,d_1)$ and $(X,d_2)$ are topologically equivalent but there exist a function $f:X \to X$ which is uniformly $d_1$ continuous but not unifor... | Let $X=(0,1)$, and let $d_1$ be the usual Euclidean metric on $X$. For $x,y\in X$ let $$d_2(x,y)=\left|\frac1x-\frac1y\right|\;.$$
*
*Verify that $d_2$ is a metric on $X$ and is topologically equivalent to $d_2$.
*Consider the function $f(x)=1-x$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1169040",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$\psi$ is upper semicontinous $\Longleftrightarrow\{z:\psi(z)Let $\Omega\subseteq\Bbb C$ be open. A function $\psi:\Omega\to[-\infty,+\infty[$ is called upper semicontinous if $\psi(z_0)\ge\limsup_{z\to z_0}\psi(z)\;\;\forall z_0\in\Omega$.
How can I show that $\psi$ is upper semicontinous IFF $\{z:\psi(z)<c\}$ is open... | Suppose $\psi$ is upper semicontinuous. Fix $c\in \Bbb R$ and let $X_c := \{z : \psi(z) < c\}$. Let $z$ be a limit point of $\Bbb C \setminus X_c$. Then $z = \lim_{n\to \infty} z_n$ for some sequence $z_n$ in $\Bbb C \setminus X_c$. Since $\psi(z_n) \ge c$ for all $n \in \Bbb N$, $\limsup_{n\to \infty} \psi(z_n) \ge c$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1169151",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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$L_1 \cap L_2$ is dense in $L_2$? We were talking about Fourier series the other day and my professor said that the requirement that a function be in $L_1 \cap L_2$ wasn't a huge obstacle, because this is dense in $L_2$. Why is this true?
| An element $f$ of $\mathbb L^2$ can be approximated for the $\mathbb L^2$ norm by a linear combination of characteristic function of measurable sets of finite measure. Such a function is integrable, hence the function $f$ can be approximated for the $\mathbb L^2$ norm by an element of $\mathbb L^1\cap \mathbb L^2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1169249",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proof of an Limit Using the formal definition of convergence, Prove that $\lim\limits_{n \to \infty} \frac{3n^2+5n}{4n^2 +2} = \frac{3}{4}$.
Workings:
If $n$ is large enough, $3n^2 + 5n$ behaves like $3n^2$
If $n$ is large enough $4n^2 + 2$ behaves like $4n^2$
More formally we can find $a,b$ such that $\frac{3n^2+5n}{... | Perhaps simpler:
With the Squeeze Theorem:
$$\frac34\xleftarrow[x\to\infty]{}\frac{3n^2}{4n^2}\le\frac{3n^2+5n}{4n^2+2}\le\frac{3n^2+5n}{4n^2}=\frac34+\frac54\frac1{n}\xrightarrow[n\to\infty]{}\frac34+0=\frac34$$
With arithmetic of limits:
$$\frac{3n^2+5n}{4n^2+2}=\frac{3n^2+5n}{4n^2+2}\cdot\frac{\frac1{n^2}}{\frac1{n^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1169336",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Strange Double sum 3 Could you explain how to get the following double sum:
$$\sum _{j=0}^{\infty } \sum _{k=0}^{\infty } \frac{2 (-1)^{k+j}}{(j+1)^2 k!\, j! \left((k+1)^2+(j+1)^2\right)}=(\gamma -\text{Ei}(-1))^2$$ where $\text{Ei}$ is the ExpIntegral?
| Hint:
$$\begin{align}
S
&=\sum_{k=0}^{\infty}\sum_{j=0}^{\infty}\frac{2(-1)^{k+j}}{k!j!(j+1)^2\left[(k+1)^2+(j+1)^2\right]}\\
&=\sum_{k=0}^{\infty}\sum_{j=0}^{\infty}\frac{(-1)^{k+j}}{k!j!(j+1)^2\left[(k+1)^2+(j+1)^2\right]}+\sum_{k=0}^{\infty}\sum_{j=0}^{\infty}\frac{(-1)^{k+j}}{k!j!(j+1)^2\left[(k+1)^2+(j+1)^2\right]... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1169404",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Proof of boundedness of a function Let $|x|<1$ and $f(x)=\displaystyle\frac{e^{\frac{1}{1+x}}}{(|x|-1)^{-2}}.$
Is $f(x)$ bounded?
| $|f(x)| \leq e^{\frac{1}{1+x}}$ because the denominator is smaller than 1.
Only for $x=-1$ you will get $\infty$ for the exponential function, but this value is excluded. Hence $f(x)$ is bounded for $|x| < 1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1169518",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Can all linear transformations be calculated with a matrix? For all finite vector spaces and all linear transformations to/from those spaces, how can you prove/show from the definition of a linear transformation that all linear transformations can be calculated using a matrix.
| The most general way to show this is to show the dual space of linear transformations on a finite dimensional vector space V over a field F is isomorphic to the vector space of m x n matrices with coeffecients in the same field F. However, although the proof isn't difficult,it needs considerably more machinery then jus... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1169591",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 2
} |
Showing a set is Dense in Metric Space $(\Omega, d)$ Let $(\Omega, d)$ be a metric space, and let $A,B \subseteq \Omega$ such that
*
*$A \subseteq \overline{B}$ and
*$A$ is dense in $\Omega$.
Show that $B$ is also dense in $\Omega$.
Here is my attempt at a solution:
Let $x \in \overline{B}$. Then there is a sequ... | Your alternative solution is correct and is definitely the way to go. There are some problems with your first attempt; I’ll quote part of it with comments.
Let $x \in \overline{B}$.
You’re trying to show that $B$ is dense in $\Omega$, so this is the wrong place to start: if you’re going to use this sequences approac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1169712",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How to find $y$ from $y' = e^{2x}-e^x y$? The problem asks me to find $y(x)$ from the equation
$$y' = e^{2x}-e^x y$$
The $y'$ is $dy/dx$ right, so wouldn't the correct step be to integrate right away? If not, should I change some terms before integrating? I'm fairly new to this, and am unaware of rules so please be cle... | You can use the method of integrating factors. First write
$$y' + e^x y = e^{2x}.$$
An integrating factor for the equation is $\exp(\int e^x\, dx) = e^{e^x}$. So we multiply both sides by $e^{e^x}$.
$$e^{e^x}y' + e^x e^{e^x}y = e^{2x}e^{e^x}.$$
The left hand side is $(e^{e^x} y)'$. So
$$(e^{e^x}y)' = e^{2x}e^{e^x}.$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1169802",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Why is the tangent of 22.5 degrees not 1/2? Sorry for the stupid question, but why is the tangent of 22.5 degrees not 1/2?
(Okay... I get
that that the tangent of 45 degrees is 1 ("opposite" =1, "adjacent" =1, 1/1 = 1. Cool. I am good with that.) Along those same lines, if the "opposite" drops to 1/2 relative to th... | The basic error being made is the assumption that you are converting one unit into another unit such as when you convert meters into yards. Degrees are units but tangent represents a ratio.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1169871",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 8,
"answer_id": 7
} |
Properties of resolvent operators I am asked to prove the identities of $(12)$ and $(13)$, which are given on page 438 of the textbook PDE Evans, 2nd edition as follows:
THEOREM 3 (Properties of resolvent operators).
(i) If $\lambda,\mu \in \rho(A)$, we have $$R_\lambda - R_\mu=(\mu-\lambda)R_\lambda R_\mu \quad \text... | Hint: For the first inequality, look at
$$
(\lambda I - A)(R_{\lambda}-R_{\mu})(\mu I-A)
$$
on $D(A)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1169977",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
$A ⊂ B$ if and only if $A − B = ∅$ I need to prove that $A ⊂ B$ if and only if $A − B = ∅$.
I have the following "proof":
$$ A \subset B \iff A - B = \emptyset$$
proof for $\implies:$
$$\forall x \in A, x \in B$$
Therefore,
$$A - B = \emptyset$$
proof for $\impliedby$:
If $$A - B = \emptyset$$
then
$$\forall x \in B, x... | One might proceed more directly by noting that $A\subseteq B$ is equivalent to $$\forall x,(x\in A)\implies(x\in B),$$ which is equivalent to $$\forall x,\neg(x\in A)\vee(x\in B).$$ The negation of this is $$\exists x:(x\in A)\wedge\neg(x\in B),$$ which is equivalent to $$\exists x:(x\in A)\wedge(x\notin B),$$ which is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1170190",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
} |
Equivalence of another formula for the number of $r$-combinations with repetition allowed
Basically it means choosing r things out of n, where order doesn't matter, and you are allowed to pick a thing more than once. For example, $\{1, 1, 2\}$ out of $\{1, 2, 3, 4\}$.
I managed to find another solution:
$$
{n \choose ... | As has already been pointed out, unfortunately the two solutions are not equivalent.
However, if we use Pascal's Rule:-
$${n \choose k} = {n-1 \choose k-1} + {n-1 \choose k}$$
and apply this $r$ times to
${r+n-1 \choose r}$ the following solution can be shown to be equivalent:-
$${r-1 \choose r-1}{n \choose r} + {r-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1170237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
Show that Möbius transformation $S$ commute with $T$ if $S$ and $T$ have the same fixed point. Let $T$ be a Möbius Transformation such that $T$ is not the identity. Show that Möbius transformation $S$ commute with $T$ if $S$ and $T$ have the same fixed point.
Here is what I know so far
1) if $T$ has fixed points says ... | Assume that $S$ and $T$ are two Moebius transformations of the extended $z$-plane $\overline{\Bbb C}$ having the same fixed points $z_1$, $z_2\in{\Bbb C}$, $\>z_1\ne z_2$. The we can introduce "temporarily" in $\overline{\Bbb C}$ a new complex coordinate
$$w:={z-z_1\over z-z_2}\ .$$
The point $z=z_1$ gets the $w$-coord... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1170332",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
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