Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Is the support of a Borel measure measured the same as the whole space? Wikipedia says
Let (X, T) be a topological space. Let μ be a measure on the Borel σ-algebra on X. Then the support
(or spectrum) of μ is defined to be the set of all points x in X for
which every open neighbourhood Nx of x has positive measure... | For an example with a probability measure, consider the following standard counterexample: let $X = [0, \omega_1]$ be the uncountable ordinal space (with endpoint), with its order topology. This is a compact Hausdorff space which is not metrizable. Define a probability measure on the Borel sets of $X$ by taking $\mu(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/115078",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Arc Length Problem I am currently in the middle of the following problem.
Reparametrize the curve $\vec{\gamma } :\Bbb{R} \to \Bbb{R}^{2}$ defined by $\vec{\gamma}(t)=(t^{3}+1,t^{2}-1)$ with respect to arc length measured from $(1,-1)$ in the direction of increasing $t$.
By reparametrizing the curve, does this mean I... | Hint:
Substitute $(x-1)^{1/3}=t$. Your integral will boil down to $$\int_{-1}^1t\sqrt{4+9t^2}\rm dt$$
Now set $4+9t^2=u$ and note that $\rm du=18t~~\rm dt$ which will complete the computation. (Note that you need to change the limits of integration while integrationg over $u$.)
A Longer way:
Now integrate by parts wi... | {
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"url": "https://math.stackexchange.com/questions/115133",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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The derivative of a complex function.
Question:
Find all points at which the complex valued function $f$ define by $$f(z)=(2+i)z^3-iz^2+4z-(1+7i)$$ has a derivative.
I know that $z^3$,$z^2$, and $z$ are differentiable everywhere in the domain of $f$, but how can I write my answer formally? Please can somebody help?
N... | I'm not sure where the question is coming from (what you know/can know/etc.).
But some things that you might use: you might just give the derivative, if you know how to take it. Perhaps you might verify it with the Cauchy-Riemann equations. Alternatively, differentiation is linear (which you might prove, if you haven't... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/115198",
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"source": "stackexchange",
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Strengthened finite Ramsey theorem I'm reading wikipedia article about Paris-Harrington theorem, which uses strengthened finite Ramsey theorem, which is stated as "For any positive integers $n, k, m$ we can find $N$ with the following property: if we color each of the $n$-element subsets of $S = \{1, 2, 3,..., N\}$ wit... | Yes, your interpretation of the first formulation is correct.
In the second formulation the statement that $H$ is $f$-homogeneous simply means that every $m$-subset of $H$ is given the same color by $f$: in your notation, the induced coloring $$f\,':H^{(m)}\to\{0,\dots,c-1\}$$ is constant. However, the formulation is m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/115267",
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"source": "stackexchange",
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What on earth does "$r$ is not a root" even mean? Method of Undetermined Coeff Learning ODE now, and using method of Undetermined Coeff
$$y'' +3y' - 7y = t^4 e^t$$
The book said that $r = 1$ is not a root of the characteristic equation. The characteristic eqtn is $r^2 + 3r - 7 = 0$ and the roots are $r = -3 \pm \sqrt{3... | $1$ comes from the $e^t$ on the right side. If it was $e^{kt}$ they would take $r=k$.
| {
"language": "en",
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Complex Roots of Unity and the GCD I'm looking for a proof of this statement. I just don't know how to approach it. I recognize that $z$ has $a$ and $b$ roots of unity, but I can't seem to figure out what that tells me.
If $z \in \mathbb{C}$ satisfies $z^a = 1$ and $z^b = 1$ then
$z^{gcd(a,b)} = 1$.
| Hint $\:$ The set of $\rm\:n\in \mathbb Z$ such that $\rm\:z^n = 1\:$ is closed under subtraction so it is closed under $\rm\:gcd$.
Recall gcds may be computed by repeated subtraction (anthyphairesis, Euclidean algorithm)
| {
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Determining variance from sum of two random correlated variables I understand that the variance of the sum of two independent normally distributed random variables is the sum of the variances, but how does this change when the two random variables are correlated?
| Let's work this out from the definitions. Let's say we have 2 random variables $x$ and $y$ with means $\mu_x$ and $\mu_y$. Then variances of $x$ and $y$ would be:
$${\sigma_x}^2 = \frac{\sum_i(\mu_x-x_i)(\mu_x-x_i)}{N}$$
$${\sigma_y}^2 = \frac{\sum_i(\mu_y-y_i)(\mu_y-y_i)}{N}$$
Covariance of $x$ and $y$ is:
$${\sigma_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/115518",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is every invertible rational function of order 0 on a codim 1 subvariety in the local ring of the subvariety? I have been trying to read Fulton's Intersection Theory, and the following puzzles me.
All schemes below are algebraic over some field $k$ in the sense that they come together with a morphism of finte type to $... | The support of a Cartier divisor $D$ on $X$ is the union of all closed subvarieties $Z\subset X$ such that the local equation of $D$ at the generic point $z$ of $Z$ is not a unit of the local ring $O_{X,z}$. Note that $Z$ can be of codimension $>1$. However, let $f_Z$ be the local equation of $D$ and let $p\in\mathrm{S... | {
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confusion on legendre symbol i know that $\left(\frac{1}{2}\right)=1$ since $1^2\equiv 1 \pmod2$ now since
$3\equiv 1\pmod2$ we should have $\left(\frac{3}{2}\right)=\left(\frac{1}{2}\right)=1$ but on Maple i get that $\left(\frac{3}{2}\right)=-1$ why?
| The Legendre symbol, the Jacobi symbol and the Kronecker symbol are successive generalizations that all share the same notation. The first two are usually only defined for odd lower arguments (primes in the first case), whereas the Kronecker symbol is also defined for even lower arguments.
Since the distinction is mere... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/115624",
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What's the sum of $\sum_{k=1}^\infty \frac{2^{kx}}{e^{k^2}}$? I already asked a similar question on another post:
What's the sum of $\sum \limits_{k=1}^{\infty}\frac{t^{k}}{k^{k}}$?
There are no problems with establishing a convergence for this power series:
$$\sum_{k=1}^\infty \frac{2^{kx}}{e^{k^2}}$$
but I have probl... | $$\sum_{k=1}^{\infty}\frac{2^{kx}}{e^{k^{2}}} = -\frac{1}{2} + \frac{1}{2} \prod_{m=1}^{\infty} \left( 1 - \frac{1}{e^{2m}} \right) \left( 1+ \frac{ 2^x }{e^{2m-1} } \right) \left( 1 + \frac{1}{2^x e^{2m-1} }\right ). $$
| {
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Similarity Transformation Let $G$ be a subgroup of $\mathrm{GL}(n,\mathbb{F})$. Denote by $G^T$ the set of transposes of all elements in $G$. Can we always find an $M\in \mathrm{GL}(n,\mathbb{F})$ such that $A\mapsto M^{-1}AM$ is a well-defined map from $G$ to $G^T$?
For example if $G=G^T$ then any $M\in G$ will do th... | The answer is no, in general. For example, when ${\mathbb F}$ is the field of two elements, let $G$ be the stabilizer of the one-dimensional subspace of ${\mathbb F}^3,$ (viewed as column vectors, with ${\rm GL}(3,\mathbb{F})$ acting by left multiplication) consisting of vectors with $0$ in positions $2$ and $3$. Then ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Should I ignore $0$ when do inverse transform sampling? Generic method
*
*Generate $U \sim \mathrm{Uniform}(0,1)$.
*Return $F^{-1}(U)$.
So, in step 1, $U$ has domain/support as $[0,1]$, so it is possible that $U=0$ or $U=1$,
but $F^{-1}(0)=-\infty$. Should I reject the value $U=0$ and $U=1$ before applying step ... | In theory, it doesn't matter: the event $U=0$ occurs with probability $0$, and can thus be ignored. (In probabilistic jargon, it almost never happens.)
In practice, it's possible that your PRNG may return a value that is exactly $0$. For a reasonably good PRNG, this is unlikely, but it may not be quite unlikely enoug... | {
"language": "en",
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If we know the GCD and LCM of two integers, can we determine the possible values of these two integers? I know that $\gcd(a,b) \cdot \operatorname{lcm}(a,b) = ab$, so if we know $\gcd(a,b)$ and $\operatorname{lcm}(a,b)$ and we want to find out $a$ and $b$, besides factoring $ab$ and find possible values, can we find th... | If you scale the problem by dividing through by $\rm\:gcd(a,b)\:$ then you are asking how to determine coprime factors of a product. This is equivalent to factoring integers.
Your original question, in the special case $\rm\:gcd(a,b) = lcm(a,b),\:$ is much easier:
Hint $\:$ In any domain $\rm\:gcd(a,b)\ |\ a,b\ |\ lcm(... | {
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Multiple function values for a single x-value I'm curious if it's possible to define a function that would have more than two functionvalues for one single x-value.
I know that it's possible to get two y-values by using the root (one positive, one negative: $\sqrt{4} = -2 ; +2$).
Is it possible to get three or more fu... | Let's consider some multivalued functions (not 'functions' since these are one to one by definition) :
$y=x^n$ has $n$ different solutions $\sqrt[n]{y}\cdot e^{2\pi i \frac kn}$ (no more than two will be real)
The inverse of periodic functions will be multivalued (arcsine, arccosine and so on...) with an infinity of b... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find Double of Distance Between 2 Quaternions I want to find the geometric equivalent of vector addition and subtraction in 3d for quaternions. In 3d difference between 2 points(a and b) gives the vector from one point to another. (b-a) gives the vector from b to a and when I add this to b I find the point which is dou... | Slerp is exactly what you want, except with the interpolation parameter $t$ set to $2$ instead of lying between $0$ and $1$. Slerp is nothing but a constant-speed parametrization of the great circle between two points $a$ and $b$ on a hypersphere, such that $t = 0$ maps to $a$ and $t = 1$ maps to $b$. Setting $t = 2$ w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/116124",
"timestamp": "2023-03-29T00:00:00",
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Show that this limit is equal to $\liminf a_{n}^{1/n}$ for positive terms.
Show that if $a_{n}$ is a sequence of positive terms such that $\lim\limits_{n\to\infty} (a_{n+1}/a_n) $ exists, then this limit is equal to $\liminf\limits_{n\to\infty} a_n^{1/n}$.
I am not event sure where to start from, any help would be m... | I saw this proof today and thought it's a nice one:
Let $a_n\ge 0$, $\lim\limits_{n \to \infty}a_n=L$. So there are 2 options:
(1) $L>0$:
$$
\lim\limits_{n \to \infty}a_n=L
\iff \lim\limits_{n \to \infty}\frac{1}{a_n}=\frac{1}{L}$$
Using Cauchy's Inequality Of Arithmetic And Geometric Means we get:
$$\frac{n}{a_1^{-1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/116183",
"timestamp": "2023-03-29T00:00:00",
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proving by $\epsilon$-$\delta $ approach that $\lim_{(x,y)\rightarrow (0,0)}\frac {x^n-y^n}{|x|+|y|}$exists for $n\in \mathbb{N}$ and $n>1$ As the topic, how to prove by $\epsilon$-$\delta $ approach $\lim_{(x,y)\rightarrow (0,0)}\frac {x^n-y^n}{|x|+|y|}$ exists for $n\in \mathbb{N}$ and $n>1$
| You may use that
$$\left|\frac{x^n-y^n}{|x|+|y|}\right|\leq \frac{|x|^n-|y|^n}{|x|+|y|}\leq \frac{|x|}{|x|+|y|}|x|^{n-1}+\frac{|y|}{|x|+|y|}|y|^{n-1}\leq|x|^{n-1}+|y|^{n-1}.$$
Since you impose $x^2+y^2< \delta \leq 1$ you have $|x|, |y|<1\Rightarrow |x|^{n-1}<|x|,\ |y|^{n-1}<|y|.$
Then you have
$|x|^{n-1}+|y|^{n-1}<|x... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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RSA cryptography Algebra
This is a homework problem I am trying to do.
I have done part 2i) as well as 2ii) and know how to do the rest. I am stuck on 2iii) and 2vii).
I truly dont know 2vii because it could be some special case I am not aware of. As for 2iii) I tried to approach it the same way as I did 2ii in which ... | For $s$ sufficiently small, we can go from $b^2=n+s^2$ to $b^2\approx n$. Take the square root and you approximately have the average of $p$ and $q$. Since $s$ is small so is their difference (relatively), so we can search around $\sqrt{n}$ for $p$ or $q$. The part (iv) means absolute difference and should have written... | {
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What is the value of $\sin(x)$ if $x$ tends to infinity?
What is the value of $\sin(x)$ if $x$ tends to infinity?
As in wikipedia entry for "Sine", the domain of $\sin$ can be from $-\infty$ to $+\infty$. What is the value of $\sin(\infty)$?
| Suppose $\lim_{x \to \infty} \sin(x) = L$. $\frac{1}{2} > 0$, so we may take $\epsilon = \frac{1}{2}$.
let N be any positive natural number. then $2\pi (N + \frac{1}{4}) > N$ as is $2\pi (N+\frac{3}{4})$.
but $\sin(2\pi (N + \frac{1}{2})) = \sin(\frac{\pi}{2}) = 1$.
so if $L < 0$, we have a $y > N$ (namely $2\pi (N + \... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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The position of a particle moving along a line is given by $ 2t^3 -24t^2+90t + 7$ for $t >0$ For what values of $t$ is the speed of the particle increasing?
I tried to find the first derivative and I get
$$6t^2-48t+90 = 0$$
$$ t^2-8t+15 = 0$$
Which is giving me $ t>5$ and $0 < t < 3$, but the book gives a different ans... | Let's be careful. The velocity is $6(t^2-8t+15)$. This is $\ge 0$ when $t \ge 5$ and when $t\le 3$. So on $(5,\infty)$, and also on $(0,3)$, this is the speed. It is not the speed on $(3,5)$. There the speed is $-6(t^2-8t+15)$.
When $t > 5$ and also when $t< 3$, the derivative of speed is $6(2t-8)$, and is positive ... | {
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Divide inside a Radical It has been so long since I have done division inside of radicals that I totally forget the "special rule" for for doing it. -_-
For example, say I wanted to divide the 4 out of this expression:
$\sqrt{1 - 4x^2}$
Is this the right way to go about it?
$\frac{16}{16} \cdot \sqrt{1 - 4x^2}$
$16 ... | The correct way to do this, after fixing the mistake pointed out by Donkey_2009, is:
$\dfrac{2}{2} \cdot \sqrt{1-4x^2}$
$= 2 \cdot \dfrac{\sqrt{1-4x^2}}{2}$
$= 2 \cdot \dfrac{\sqrt{1-4x^2}}{\sqrt{4}} \qquad \Leftarrow$ applied $x = \sqrt{x^2}$
$= 2 \cdot \sqrt{\dfrac{1-4x^2}{4}} \qquad \Leftarrow$ applied $\frac{\sqrt... | {
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If $f(x)=f'(x)+f''(x)$ then show that $f(x)=0$
A real-valued function $f$ which is infinitely differentiable on $[a.b]$ has the following properties:
*
*$f(a)=f(b)=0$
*$f(x)=f'(x)+f''(x)$ $\forall x \in [a,b]$
Show that $f(x)=0$ $\forall x\in [a.b]$
I tried using the Rolle's Theorem, but it only t... | Hint $f$ can't have a positive maximum at $c$ since then $f(c)>0, f'(c)=0, f''(c) \le 0$ implies that $f''(c)+f'(c)-f(c) < 0$. Similarly $f$ can't have a negative minimum. Hence $f = 0$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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LTL is a star-free language but it can describe $a^*b^\omega$. Contradiction? Does the statement "LTL is a star-free language"(from wiki) mean that the expressiveness power of LTL is equivalent to that of star-free languages? Then why can you describe in LTL the following language with the star: $a^*b^\omega$?
$$\mathb... | Short answer: $a^*b^{\omega}$ describes a star-free language.
Longer answer:
In order to show that let's consider two definitions of a regular star-free language :
*
*Language has a maximum star height of 0.
*Language is in the class of star-free languages, which is defined as follows:
it's the smallest subset of $\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Can a basis for a vector space be made up of matrices instead of vectors? I'm sorry if this is a silly question. I'm new to the notion of bases and all the examples I've dealt with before have involved sets of vectors containing real numbers. This has led me to assume that bases, by definition, are made up of a number ... | Yes, you are right. A vector space of matrices of size $n$ is actually, a vector space of dimension $n^2$. In fact, just to spice things up: The vector space of all
*
*diagonal,
*symmetric and
*triangular matrices of dimension $n\times n$
is actually a subspace of the space of matrices of that size.
As with a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/116717",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Efficiently solving a special integer linear programming with simple structure and known feasible solution Consider an ILP of the following form:
Minimize $\sum_{k=1}^N s_i$ where
$\sum_{k=i}^j s_i \ge c_1 (j-i) + c_2 - \sum_{k=i}^j a_i$ for given constants $c_1, c_2 > 0$ and a given sequence of non-zero natural number... | I did not find a satisfying solution for this, so I will just re-iterate what I found: Using CPlex, the problem scales somewhat better. Sadly, it does not seem possible to tell CPlex that you have a feasible solution, only that you have a (claimed to be) optimal solution, which wastes effort.
| {
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"timestamp": "2023-03-29T00:00:00",
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Probability that, given a set of uniform random variables, the difference between the two smallest values is greater than a certain value Let $\{X_i\}$ be $n$ iid uniform(0, 1) random variables. How do I compute the probability that the difference between the second smallest value and the smallest value is at least $... | There's probably an elegant conceptual way to see this, but here is a brute-force approach.
Let our variables be $X_1$ through $X_n$, and consider the probability $P_1$ that $X_1$ is smallest and all the other variables are at least $c$ above it. The first part of this follows automatically from the last, so we must ha... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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The set of functions which map convergent series to convergent series Suppose $f$ is some real function with the above property, i.e.
if $\sum\limits_{n = 0}^\infty {x_n}$ converges, then $\sum\limits_{n = 0}^\infty {f(x_n)}$ also converges.
My question is: can anything interesting be said regarding the behavior of su... | Answer to the next question: no.
Let $f\colon\mathbb{R}\to\mathbb{R}$ be defined by
$$
f(x)=\begin{cases}
n\,x & \text{if } x=2^{-n}, n\in\mathbb{N},\\
x & \text{otherwise.}
\end{cases}
$$
Then $\lim_{x\to0}f(x)=f(0)=0$, $f$ transforms convergent series in convergent series, but $f(x)/x$ is not bounded in any open... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/116964",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Probability that sum of rolling a 6-sided die 10 times is divisible by 10? Here's a question I've been considering: Suppose you roll a usual 6-sided die 10 times and sum up the results of your rolls. What's the probability that it's divisible by 10?
I've managed to solve it in a somewhat ugly fashion using the followin... | The distribution of the sum converges to normal distribution with speed (if I remember correctly) of $n^{-1/2}$ and that error term could be dominating other terms (since you have got just constant numbers (=six) of samples out of resulting distribution). However, there is a small problem -- probability that your sum w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/117022",
"timestamp": "2023-03-29T00:00:00",
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"answer_count": 3,
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Coding theory (existence of codes with given parameters) Explain why each of the following codes can't exist:
*
*A self complementary code with parameters $(35, 130, 15)$. (I tried using Grey Rankin bound but 130 falls within the bound)
*A binary $(15, 2^8, 5)$ code. (I tried Singleton Bound but no help)
*A $10$-... | Let me elaborate on problem #2. As I said in my comment that claim is wrong, because there does exist a binary code of length 15, 256 words and minimum Hamming distance 5.
I shall first give you a binary $(16,256,6)$ code aka the Nordstrom-Robinson code.
Consider the $\mathbf{Z}_4$-submodule $N$ of $\mathbf{Z}_4^8$ gen... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/117086",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Probabilistic paradox: Making a scratch in a dice changes the probability? For dices that we cannot distinguish we have learned in class, that the correct sample space is $\Omega _1 = \{ \{a,b\}|a,b\in \{1,\ldots,6\} \}$, whereas for dices that we can distinguish we have $\Omega _2 = \{ (a,b)|a,b\in \{1,\ldots,6\} \}$... | The correct probability distribution for dice treats them as distinguishable. If you insist on using the sample space for indistinguishable dice, the outcomes are not equally likely.
However, if you are doing quantum mechanics and the "numbers" become individual quantum states, indistinguishable dice must be treated u... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/117154",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Applying Euler's Theorem to Prove a Simple Congruence I have been stuck on this exercise for far too long:
Show that if $a$ and $m$ are positive integers with $(a,b)=(a-1,m)=1$, then
$$1+a+a^2+\cdots+a^{\phi(m)-1}\equiv0\pmod m.$$
First of all, I know that
$$1+a+a^2+\cdots+a^{\phi(m)-1}=\frac{a^{\phi(m)-2}-1}{a-1},... | Hint: From $a^{\phi(m)}-1$ is congruent to 0 mod m and is congruent to the $(a^{\phi(m)}-1)/(a-1)$ mod m
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Given 5 children and 8 adults, how many ways can they be seated so that there are no two children sitting next to each other.
Possible Duplicate:
How many ways are there for 8 men and 5 women to stand in a line so that no two women stand next to each other?
Given 5 children and 8 adults, how many different ways can ... | The solution below assumes the seats are in a row:
This is a stars and bars problem. First, order the children (5! ways). Now, suppose the adults are identical. They can go in any of the places on either side or between of the children. Set aside 4 adults to space out the children, and place the other 4 in any arrangem... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Dummit Foote 10.5.1(d) commutative diagram of exact sequences.
I solved other problems, except (d): if $\beta$ is injective, $\alpha$ and $\gamma$ are surjective, then $\gamma$ is injective.
Unlike others, I don't know where to start.
| As the comments mention, this exercise is false as stated. Here's a counterexample: let $A$ and $B$ be groups, with usual inclusion and projection homomorphisms $$\iota_A(a) = (a,1),$$ $$\iota_B(b) = (1,b)$$ and $$\pi_B(a,b) = b.$$
Then the following diagram meets the stated requirements, except $\pi_B$ is not injecti... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Coin sequence paradox from Martin Gardner's book "An event less frequent in the long run is likely to happen before a more frequent event!"
How can I show that THTH is more likely to turn up before HTHH with a probability of
9/14, even though waiting time of THTH is 20 and HTHH, 18!
I would be very thankful if you coul... | Here is a both nontrivial and advanced solution.
Consider an ideal gambling where a dealer Alice tosses a fair coin repeatedly. After she made her $(n-1)$-th toss (or just at the beginning of the game if $n = 1$), the $n$-th player Bob joins the game. He bets $2^0\$$ that the $n$-th coin is $T$. If he loses, he leaves ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Ordered partitions of an integer Let $k>0$ and $(l_1,\ldots,l_n)$ be given with $l_i>0$ (and the $l_i's$ need not be distinct). How do I count the number of distinct tuples
$$(a_1,\ldots,a_r)$$
where $a_1+\ldots+a_r=k$ and each $a_i$ is some $l_j$. There will typically be a different length $r$ for each such tuple.
If ... | The generation function for $P_d(n)$, where no part appears more than $d$ times is given by
$$
\prod_{k=1}^\infty \frac{1-x^{(d+1)k}}{1-x^k} = \sum_{n=0}^\infty P_d(n)x^n.
$$
In your case $d=1$ and the product simplifies to
$$
\prod_{k=1}^\infty (1+x^k)= 1+x+x^2+2 x^3+2 x^4+3 x^5+4 x^6+5 x^7+6 x^8+8 x^9+10 x^{10}+12... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/117489",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Quadratic Diophantine equation in three variables How would one determine solutions to the following quadratic Diophantine equation in three variables:
$$x^2 + n^2y^2 \pm n^2y = z^2$$
where n is a known integer and $x$, $y$, and $z$ are unknown positive integers to be solved.
Ideally there would be a parametric solutio... | We will consider the more general equation:
$X^2+qY^2+qY=Z^2$
Then, if we use the solutions of Pell's equation: $p^2-(q+1)s^2=1$
Solutions can be written in this ideal:
$X=(-p^2+2ps+(q-1)s^2)L+qs^2$
$Y=2s(p-s)L+qs^2$
$Z=(p^2-2ps+(q+1)s^2)L+qps$
And more:
$X=(p^2+2ps-(q-1)s^2)L-p^2-2ps-s^2$
$Y=2s(p+s)L-p^2-2ps-s^2$
$Z=(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/117550",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Another residue theory integral I need to evaluate the following real convergent improper integral using residue theory (vital that i use residue theory so other methods are not needed here)
I also need to use the following contour (specifically a keyhole contour to exclude the branch cut):
$$\int_0^\infty \frac{\sqrt... | Close format for this type of integrals:
$$ \int_0^{\infty} x^{\alpha-1}Q(x)dx =\frac{\pi}{\sin(\alpha \pi)} \sum_{i=1}^{n} \,\text{Res}_i\big((-z)^{\alpha-1}Q(z)\big) $$
$$ I=\int_0^\infty \frac{\sqrt{x}}{x^3+1} dx \rightarrow \alpha-1=\frac{1}{2} \rightarrow \alpha=\frac{3}{2}$$
$$ g(z) =(-z)^{\alpha-1}Q(z) =\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/117619",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Find $y$ to minimize $\sum (x_i - y)^2$ I have a finite set of numbers $X$. I want to minimize the following expression by finding the appropriate value for y:
$$\sum\limits_{i=1}^n (x_i - y)^2$$
| This is one of those problems where you just turn the crank and out pops the answer. The basic optimization technique of "set the derivative equal to zero and solve" to find critical points works in its simplest form without issue here.
And as the others have mentioned, the special form of being quadratic allows you to... | {
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Proof of greatest integer theorem: floor function is well-defined I have to prove that
$$\forall x \in \mathbb{R},\exists !\,n \in \mathbb{Z} \text{ s.t. }n \leq x < n+1\;.$$
where $\exists !\,n $ means there exists a unique (exactly one) $n$.
I'm done with proving that there are at least one integers for the solutio... | The usual proof in the context of real analysis goes like this:
Let $A= \{ n \in \mathbb Z : n \le x \}$. Then $A$ is not empty. Indeed, there is $n\in \mathbb N$ such that $n>-x$, because $\mathbb N$ is unbounded. But then $-n\in A$.
Let $\alpha=\sup A$. Then there is $n\in A$ such that $\alpha-1<n\le\alpha$. But then... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/117734",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Name this paradox about most common first digits in numbers I remember hearing about a paradox (not a real paradox, more of a surprising oddity) about frequency of the first digit in a random number being most likely 1, second most likely 2, etc. This was for measurements of seemingly random things, and it didn't work... | It is Benford's Law
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Asymptotics of a solution Let $x(n)$ be the solution to the following equation
$$
x=-\frac{\log(x)}{n} \quad \quad \quad \quad (1)
$$
as a function of $n,$ where $n \in \mathbb N.$
How would you find the asymptotic behaviour of the solution, i.e. a function $f$ of $n$ such that there exist constants $A,B$ and $n_0\in... | Call $u_n:t\mapsto t\mathrm e^{nt}$, then $x(n)$ solves $u_n(x(n))=1$. For every $a$, introduce
$$
x_a(n)=\frac{\log n}n-a\frac{\log\log n}n.
$$
Simple computations show that, for every fixed $a$, $u_n(x_a(n))\cdot(\log n)^{a-1}\to1$ when $n\to\infty$. Thus, for every $a\gt1$, there exists some finite index $n(a)$ su... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Are $3$ and $11$ the only common prime factors in $\sum\limits_{k=1}^N k!$ for $N\geq 10$? The question was stimulated by this one. Here it comes:
When you look at the sum $\sum\limits_{k=1}^N k!$ for $N\geq 10$, you'll always find $3$ and $11$ among the prime factors, due to the fact that
$$
\sum\limits_{k=1}^{10}k!=3... | As pointed out in the comments, the case is trivial( at least if you know some theorems) if you fix $n>10$, instead of $n>p-1$ for prime $p$.
It follows from Wilson's Theorem, that if you have a multiple of $p$ at index $n= p-2$ you won't at index $p-1$ because it will decrease out of being one for that index. $p>12$ ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "18",
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Set of harmonic functions is locally equicontinuous (question reading in Trudinger / Gilbarg) I'm working through the book Elliptic Parial Differential Equations of Second Order by D. Gilbarg and N. S. Trudinger. Unfortunately I get stuck at some point. On page 23 they prove the following Theorem:
Let $u$ be harmonic ... | If $\{u_i\}_{i\in \mathcal{I}}$ is a bounded family of harmonic functions defined in $\Omega$ (i.e., there exists $M\geq 0$ s.t. $|u_i(x)|\leq M$ for $x\in \Omega$) then inequality:
$$\sup_{\Omega'}|D^\alpha u|\le \left(\frac{n|\alpha|}{d}\right)^{|\alpha|} \sup_{\Omega}|u|$$
with $|\alpha|=1$ implies:
$$\sup_{\Omega'}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proving an algebraic identity using the axioms of field I am trying to prove (based on the axioms of field) that
$$a^3-b^3=(a-b)(a^2+ab+b^2)$$
So, my first thought was to use the distributive law to show that
$$(a-b)(a^2+ab+b^2)=(a-b)\cdot a^2+(a-b)\cdot ab+(a-b)\cdot b^2$$
And then continuing from this point.
My probl... | Indeed you need distributive, associative and commutative laws to prove your statement.
In fact:
$$\begin{split}
(a-b)(a^2+ab+b^2) &= (a+(-b))a^2 +(a+(-b))ab+(a+(-b))b^2\\
&= a^3 +(- b)a^2+a^2b+(-b)(ab)+ab^2+(-b)b^2\\
&= a^3 - ba^2+a^2b - b(ab) +ab^2-b^3\\
&= a^3 - a^2b+a^2b - (ba)b +ab^2-b^3\\
&= a^3 -(ab)b +ab^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/118024",
"timestamp": "2023-03-29T00:00:00",
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closed form of a Cauchy (series) product I hope this hasn't been asked already, though I have looked around the site and found many similar answers.
Given:
Form Cauchy product of two series: $a_k\;x^k$ and $\tfrac{1}{1-x}=1+x+x^2+\cdots$.
So I come up with,
$\sum_{n=0}^{\infty}\;c_n = \sum_{n=0}^{\infty}\;\sum_{k+l=n... | Note that $\sum_{n\ge 0}nx^n$ is almost the Cauchy product of $\sum_{n\ge 0}x^n$ with itself: that Cauchy product is
$$\left(\sum_{n\ge 0}x^n\right)^2=\sum_{n\ge 0}x^n\sum_{k=0}^n 1^2=\sum_{n\ge 0}(n+1)x^n\;.\tag{1}$$
If you multiply the Cauchy product in $(1)$ by $x$, you get
$$x\sum_{n\ge 0}(n+1)x^n=\sum_{n\ge 0}(n+... | {
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Find the identity under a given binary operation I have two problems quite similar. The first:
In $\mathbb{Z}_8$ find the identity of the following commutative operation:
$$\overline{a}\cdot\overline{c}=\overline{a}+\overline{c}+2\overline{a}\overline{c}$$
I say:
$$\overline{a}\cdot\overline{i}=\overline{a}+\overli... | Hint $\ $ Identity elements $\rm\:e\:$ are idempotent $\rm\:e^2 = e\:$. Therefore
$\rm(1)\ \ mod\ 8\!:\ \ e = e\cdot e = 2e+2e^2\ \Rightarrow\ e\:(1+2e) = 0\ \Rightarrow\ e = \ldots\:$ by $\rm n^2 \equiv 1\:$ for $\rm\:n\:$ odd
$\rm(2)\ \ mod\ (9,9)\!:\ \ (a,b) = (a,b)^2 = (2a,-bb)\ \Rightarrow\ (-a, b\:(b+1)) = (0,0)... | {
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De Rham cohomology of $S^n$ Can you find mistake in my computation of $H^{k}(S^{n})$.
Sphere is disjoint union of two spaces:
$$S^{n} = \mathbb{R}^{n}\sqcup\mathbb{R^{0}},$$
so
$$H^{k}(S^n) = H^{k}(\mathbb{R}^{n})\oplus H^{k}(\mathbb{R^{0}}).$$
In particular
$$H^{0}(S^{n}) = \mathbb{R}\oplus\mathbb{R}=\mathbb{R}^{2}$$
... | You are wrong: $S^n$ is not the disjoint union $\mathbb R^n \sqcup \mathbb R^0$ - topologically.
Although $S^n$ is $\mathbb R^n$ with one point at infinity, the topology of this point at infinity is very different from that of $\mathbb R^0$.
| {
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"source": "stackexchange",
"question_score": "2",
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Existence theorems for problems with free endpoints? It is well known that the problem of minimizing
$$ J[y] = \int_{0}^{1} \sqrt{y(x)^2 + \dot{y}(x)^2} dx $$
with $y \in C^2[0,1]$ and $y(0) = 1$ and $y(1) = 0$ has no solutions. However, if we remove the condition $y(1) = 0$ and instead let the value of $y$ at $x = 1$ ... | Sure. You can take any smooth $f(x)$ with $f(0) = 0,$ then minimize
$$ \int_0^1 \sqrt{1 + \left( f(\dot{y}(x)) \right)^2} \; dx $$
with $y(0) = 73.$ The minimizer is constant $y.$
More interesting is the free boundary problem for surface area. Given a wire frame that describes a nice curve $\gamma$ in $\mathbb R^3,... | {
"language": "en",
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Prove the map has a fixed point Assume $K$ is a compact metric space with metric $\rho$ and $A$ is a map from $K$ to $K$ such that $\rho (Ax,Ay) < \rho(x,y)$ for $x\neq y$. Prove A have a unique fixed point in $K$.
The uniqueness is easy. My problem is to show that there a exist fixed point. $K$ is compact, so every se... | I don't have enough reputation to post a comment to reply to @андрэ 's question regarding where in the proof it is used that $f$ is a continuous function, so I'll post my answer here:
Since we are told that $K$ is a compact set. $f:K\rightarrow K$ being continuous implies that the $\mathrm{im}(f) = f(K)$ is also a comp... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "27",
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Modules with $m \otimes n = n \otimes m$ Let $R$ be a commutative ring. Which $R$-modules $M$ have the property that the symmetry map
$$M \otimes_R M \to M \otimes_R M, ~m \otimes n \mapsto n \otimes m$$
equals the identity? In other words, when do we have $m \otimes n = n \otimes m$ for all $m,n \in M$?
Some basic obs... | The question has an accepted answer at MathOverflow, and perhaps it is time to leave the Unanswered list.
| {
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"source": "stackexchange",
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De Rham cohomology of $S^2\setminus \{k~\text{points}\}$ Am I right that de Rham cohomology $H^k(S^2\setminus \{k~\text{points}\})$ of $2-$dimensional sphere without $k$ points are
$$H^0 = \mathbb{R}$$
$$H^2 = \mathbb{R}^{N}$$
$$H^1 = \mathbb{R}^{N+k-1}?$$
I received this using Mayer–Vietoris sequence. And I want only ... | It helps to use the fact that DeRahm cohomology is a homotopy invariant, meaning we can reduce the problem to a simpler space with the same homotopy type. I think the method you are trying will work if you can straighten out the details, but if you're still having trouble then try this:
$S^2$ with $1$ point removed ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Finding probability $P(X+Y < 1)$ with CDF Suppose I have a Cumulative Distribution Function like this:
$$F(x,y)=\frac { (x\cdot y)^{ 2 } }{ 4 } $$
where $0<x<2$ and $0<y<1$.
And I want to find the probability of $P(X+Y<1)$.
Since $x<1-y$ and $y<1-x$, I plug these back into the CDF to get this:
$$F(1-y,1-x)=\frac { ((1-... | We have the cumulative distribution function (CDF)
$$F_{X,Y}(x,y)=\int_0^y\int_0^x f_{X,Y}(u,v)dudv=\frac{(xy)^2}{4}.$$
Differentiate with respect to both $x$ and $y$ to obtain the probability density function (PDF)
$$f_{X,Y}(x,y)=\frac{d^2}{dxdy}\frac{(xy)^2}{4}=xy.$$
Finally, how do we parametrize the region given by... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Number of ways to pair 2n elements from two different sets Say I have a group of 20 people, and I want to split them to pairs, I know that the
number of different ways to do it is $\frac{(2n)!}{2^n \cdot n!}$
But let's say that I have to pair a boy with a girl?
I got confused because unlike the first option the number ... | If you have $n$ boys and $n$ girls, give them each a number and sort the pairs by wlog the boy's number. Then there are $n!$ possible orderings for the girls, so $n!$ ways of forming the pairs.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solve $ x^2+4=y^d$ in integers with $d\ge 3$
Find all triples of integers $(x,y,d)$ with $d\ge 3$ such that $x^2+4=y^d$.
I did some advance in the problem with Gaussian integers but still can't finish it. The problem is similar to Catalan's conjecture.
NOTE: You can suppose that $d$ is a prime.
Source: My head
| See a similar question that I asked recently: Nontrivial Rational solutions to $y^2=4 x^n + 1$
This question might also be related to Fermat's Last Theorem.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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How to come up with the gamma function? It always puzzles me, how the Gamma function's inventor came up with its definition
$$\Gamma(x+1)=\int_0^1(-\ln t)^x\;\mathrm dt=\int_0^\infty t^xe^{-t}\;\mathrm dt$$
Is there a nice derivation of this generalization of the factorial?
| Here is a nice paper of Detlef Gronau Why is the gamma function
so as it is?.
Concerning alternative possible definitions see Is the Gamma function mis-defined? providing another resume of the story Interpolating the natural factorial n! .
Concerning Euler's work Ed Sandifer's articles 'How Euler did it' are of value ... | {
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interchanging integrals Why does $$\int_0^{y/2} \int_0^\infty e^{x-y} \ dy \ dx \neq \int_0^\infty \int_0^{y/2} e^{x-y} \ dx \ dy$$
The RHS is 1 and the LHS side is not. Would this still be a legitimate joint pdf even if Fubini's Theorem does not hold?
| The right side,
$$\int_0^\infty \int_0^{y/2} e^{x-y} \ dx \ dy,$$
refers to something that exists. The left side, as you've written it, does not. Look at the outer integral:
$$
\int_0^\infty \cdots\cdots\; dy.
$$
The variable $y$ goes from $0$ to $\infty$. For any particular value of $y$ between $0$ and $\infty$, ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is my proof correct: if $n$ is odd then $n^2$ is odd?
Prove that for every integer $n,$ if $n$ is odd then $n^2$ is odd.
I wonder whether my answer to the question above is correct. Hope that someone can help me with this.
Using contrapositive, suppose $n^2$ is not odd, hence even. Then $n^2 = 2a$ for some integer $... | You need to show that $a/n$ is an integer. Try thinking about the prime factorizations of $a$ and $n$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Is $BC([0,1))$ ( space of bounded real valued continuous functions) separable? Is $BC([0,1))$ a subset of $BC([0,\infty))$? It is easy to prove the non-separability of BC([0,$\infty$)) and the separability of C([0,1]). It seems to me we can argue from the fact that any bounded continuous function of BC([0,$\infty$)) mu... | $BC([0,1))$ is not a subset of $BC([0,\infty))$; in fact, these two sets of functions are disjoint. No function whose domain is $[0,1)$ has $[0,\infty)$ as its domain, and no function whose domain is $[0,\infty)$ has $[0,1)$ as its domains. What is true is that $$\{f\upharpoonright[0,1):f\in BC([0,\infty))\}\subseteq B... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/119191",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Help Understanding Why Function is Continuous I have read that because a function $f$ satisfies
$$
|f(x) - f(y)| \leq |f(y)|\cdot|x - y|
$$
then it is continuous. I don't really see why this is so. I know that if a function is "lipschitz" there is some constant $k$ such that
$$
|f(x) - f(y)| \leq k|x - y|.
$$
But t... | You're right that the dependence on $y$ means this inequality isn't like the Lipschitz condition. But the same proof will show continuity in both cases. (In the Lipschitz case you get uniform continuity for free.) Here's how:
Let $y\in\operatorname{dom} f$; we want to show $f$ is continuous at $y$. So let $\epsilon... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Cauchy Sequence that Does Not Converge What are some good examples of sequences which are Cauchy, but do not converge?
I want an example of such a sequence in the metric space $X = \mathbb{Q}$, with $d(x, y) = |x - y|$. And preferably, no use of series.
| A fairly easy example that does not arise directly from the decimal expansion of an irrational number is given by $$a_n=\frac{F_{n+1}}{F_n}$$ for $n\ge 1$, where $F_n$ is the $n$-th Fibonacci number, defined as usual by $F_0=0$, $F_1=1$, and the recurrence $F_{n+1}=F_n+F_{n-1}$ for $n\ge 1$. It’s well known and not esp... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 8
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How to check convexity? How can I know the function $$f(x,y)=\frac{y^2}{xy+1}$$ with $x>0$,$y>0$ is convex or not?
| The book "Convex Optimization" by Boyd, available free online here, describes methods to check.
The standard definition is if f(θx + (1 − θ)y) ≤ θf(x) + (1 − θ)f(y) for 0≤θ≤1 and the domain of x,y is also convex.
So if you could prove that for your function, you would know it's convex.
The Hessian being positive semi-... | {
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How to prove $ \phi(n) = n/2$ iff $n = 2^k$? How can I prove this statement ? $ \phi(n) = n/2$ iff $n = 2^k $
I'm thinking n can be decomposed into its prime factors, then I can use multiplicative property of the euler phi function to get the $\phi(n) = \phi(p_1)\cdots\phi(p_n) $. Then use the property $ \phi(p) = p - ... | Edit: removed my full answer to be more pedagogical.
You know that $\varphi(p) = p-1$, but you need to remember that $\varphi(p^k) = p^{k-1}(p-1).$ Can you take it from here?
| {
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Represent every Natural number as a summation/subtraction of distinct power of 3 I have seen this in a riddle where you have to chose 4 weights to calculate any weight from 1 to 40kgs.
Some examples,
$$8 = {3}^{2} - {3}^{0}$$
$$12 = {3}^{2} + {3}^{1}$$
$$13 = {3}^{2} + {3}^{1}+ {3}^{0}$$
Later I found its also possib... | You can represent any number $n$ as $a_k 3^k + a_{k-1} 3^{k-1} + \dots + a_1 3 + a_0$, where $a_i \in \{-1,0,1\}$. This is called balanced ternary system, and as Wikipedia says, one way to get balanced ternary from normal ternary is to add ..1111 to the number (formally) with carry, and then subtract ..1111 without car... | {
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A smooth function f satisfies $\left|\operatorname{ grad}\ f \right|=1$ ,then the integral curves of $\operatorname{grad}\ f$ are geodesics $M$ is riemannian manifold, if a smooth function $f$ satisfies $\left| \operatorname{grad}\ f \right|=1,$ then prove the integral curves of $\operatorname{grad}\ f$ are geodesics.
| Well $\text{grad}(f)$ is a vector such that $g(\text{grad}(f),-)=df$, therefore integral curves satisfy
$$
\gamma'=\text{grad}(f)\Rightarrow
g(\gamma',X)=df(X)=X(f)
$$
Now let $X,Y$ be a vector fields
$$
XYf=Xg(\text{grad}(f),Y)=
g(\nabla_X\text{grad}(f),Y)+g(\text{grad}(f),\nabla_XY)=
g(\nabla_X\text{grad}(f),Y... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Martingales, finite expectation I have some uncertainties about one of the requirements for martingale, i.e. showing that $\mathbb{E}|X_n|<\infty,n=0,1,\dots$ when $(X_n,n\geq 0)$ is some stochastic process. In particularly, in some solutions I find that lets say $\mathbb{E}|X_n|<n$ is accepted, for example here (2nd s... | The condition $\mathbb E|X_n|\lt n$ is odd. What is required for $(X_n)$ to be a martingale is, in particular, that each $X_n$ is integrable (if only to be able to consider its conditional expectation), but nothing is required about the growth of $\mathbb E|X_n|$.
Consider for example a sequence $(Z_n)$ of i.i.d. cente... | {
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Numbers are too large to show $65^{64}+64^{65}$ is not a prime I tried to find cycles of powers, but they are too big. Also $65^{n} \equiv 1(\text{mod}64)$, so I dont know how to use that.
| Hint $\rm\ \ x^4 +\: 64\: y^4\ =\ (x^2+ 8\:y^2)^2 - (4xy)^2\ =\ (x^2-4xy + 8y^2)\:(x^2+4xy+8y^2)$
Thus $\rm\ x^{64} + 64\: y^{64} =\ (x^{32} - 4 x^{16} y^{16} + 8 y^{32})\:(x^{32} - 4 x^{16} y^{16} + 8 y^{32})$
Below are some other factorizations which frequently prove useful for integer factorization. Aurifeuille, Le ... | {
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"timestamp": "2023-03-29T00:00:00",
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Is $ f(x) = \left\{ \begin{array}{lr} 0 & : x = 0 \\ e^{-1/x^{2}} & : x \neq 0 \end{array} \right. $ infinitely differentiable on all of $\mathbb{R}$? Can anyone explicitly verify that the function $
f(x) = \left\{
\begin{array}{lr}
0 & : x = 0 \\
e^{-1/x^{2}} & : x \neq 0
\end{array}
... | For $x\neq 0$ you get:
$$\begin{split}
f^\prime (x) &= \frac{2}{x^3}\ f(x)\\
f^{\prime \prime} (x) &= 2\left( \frac{2}{x^6} - \frac{3}{x^4}\right)\ f(x)\\
f^{\prime \prime \prime} (x) &= 4\left( \frac{2}{x^9} - \frac{9}{x^7} +\frac{6}{x^5} \right)\ f(x)
\end{split}$$
In the above equalities you can see a path, i.e.:
$$... | {
"language": "en",
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Units and Nilpotents
If $ua = au$, where $u$ is a unit and $a$ is a nilpotent, show that $u+a$ is a unit.
I've been working on this problem for an hour that I tried to construct an element $x \in R$ such that $x(u+a) = 1 = (u+a)x$. After tried several elements and manipulated $ua = au$, I still couldn't find any clue... | If $u=1$, then you could do it via the identity
$$(1+a)(1-a+a^2-a^3+\cdots + (-1)^{n}a^n) = 1 + (-1)^{n}a^{n+1}$$
by selecting $n$ large enough.
If $uv=vu=1$, does $a$ commute with $v$? Is $va$ nilpotent?
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "31",
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Compute $\lim \limits_{x\to\infty} (\frac{x-2}{x+2})^x$ Compute
$$\lim \limits_{x\to\infty} (\frac{x-2}{x+2})^x$$
I did
$$\lim_{x\to\infty} (\frac{x-2}{x+2})^x = \lim_{x\to\infty} \exp(x\cdot \ln(\frac{x-2}{x+2})) = \exp( \lim_{x\to\infty} x\cdot \ln(\frac{x-2}{x+2}))$$
But how do I continue? The hint is to use L Hop... | you can use
$$\left( \frac{x-2}{x+2}\right)^x = \left(1 - \frac{4}{x+2}\right)^x$$
and $(1 + \frac ax)^x \to \exp(a)$,
HTH, AB
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that the following integral is divergent $$\int_0^\infty \frac{7x^7}{1+x^7}$$
Im really not sure how to even start this. Does anyone care to explain how this can be done?
| The only problem is in $+\infty$. We have for $x\geq 1$ that $1+x^7\leq 2x^7$ so $\frac{7x^7}{1+x^7}\geq \frac 72\geq 0$ and $\int_1^{+\infty}\frac 72dt$ is divergent, so $\int_1^{+\infty}\frac{7x^7}{1+x^7}dx$ is divergent. Finally, $\int_0^{+\infty}\frac{7x^7}{1+x^7}dx$ is divergent.
| {
"language": "en",
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Why do we look at morphisms? I am reading some lecture notes and in one paragraph there is the following motivation: "The best way to study spaces with a structure is usually to look at the maps between them preserving structure (linear maps, continuous maps differentiable maps). An important special case is usually th... | There is no short and simple answer, as has already been mentioned in the comments. It is a general change of perspective that has happened during the 20th century. I think if you had asked a mathematician around 1900 what math is all about, he/she would have said: "There are equations that we have to solve" (linear or... | {
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What exactly do we mean when say "linear" combination? I've noticed that the term gets abused alot. For instance, suppose I have
$c_1 x_1 + c_2 x_2 = f(x)$...(1)
Eqtn (1) is such what we say "a linear combination of $x_1$ and $x_2$"
In ODE, sometimes when we want to solve a homogeneous 2nd order ODE like $y'' + y' + y ... | It's a linear combination in the vector space of continuous (or differentiable or whatever) functions. $y_1$ and $y_2$ are vectors (that is, elements of the vector space in question) and $c_1$ and $c_2$ are scalars (elements of the field for the vector space, in this case $\mathbb{R}$). In linear algebra it does not ma... | {
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Sum of three consecutive cubes When I noticed that $3^3+4^3+5^3=6^3$, I wondered if there are any other times where $(a-1)^3+a^3+(a+1)^3$ equals another cube. That expression simplifies to $3a(a^2+2)$ and I'm still trying to find another value of $a$ that satisfies the condition (the only one found is $a=4$)
Is this im... | How about
$$\left(-\frac{1}{2}\right)^3 + \left(\frac{1}{2}\right)^3 + \left(\frac{3}{2}\right)^3 = \left(\frac{3}{2}\right)^3 ?$$
After all, the OP didn't specify where $a$ lives... (by the way, there are infinitely many distinct rational solutions of this form!).
Now for a more enlightened answer: no, there are no o... | {
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Proof that $\pi$ is rational I stumbled upon this proof of $\pi$ being rational (coincidentally, it's Pi Day). Of course I know that $\pi$ is irrational and there have been multiple proofs of this, but I can't seem to see a flaw in the following proof that I found here. I'm assuming it will be blatantly obvious to peop... | This "proof" shows that any real number is rational...
The mistake here is that you are doing induction on the sequence $\pi_n$ of approximations. And with induction you can get information on each element of the sequence, but not on their limit.
Or, put in another way, the proof's b.s. is on "therefore, by induction ... | {
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Dynamic Optimization - Infinite dimensional spaces - Reference request Respected community members,
I am currently reading the book "recursive macroeconomic theory" by Sargent and Ljungqvist. While reading this book I have realized that I do not always fully understand what is going on behind "the scenes".
In particul... | A fairly rigorous treatment with many economics applications is Stokey, Lucas and Prescott's (SLP) Recursive Methods in Economic Dynamics.
This MIT OCW course gives good additional readings. I find the ones on transversality conditions very important.
Standard mathematical treatments are Bertsekas's Dynamic Programmin... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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numerically solving differential equations $\frac{d^2 \theta}{dx^2} (1 + \beta \theta) + \beta \left(\frac{d \theta}{d x}\right)^2 - m^2 \theta = 0$
Boundary Conditions
$\theta=100$ at $x = 0$, $\frac{d\theta}{dx} = 0$ at $x = 2$
$\beta$ and $m$ are constants.
Please help me solve this numerically (using finite diffe... | Choose an integer $N$, let $h=2/N$ and let $\theta_k$ be the approximation given by the finite difference method to the exact value $\theta(k\,h)$, $0\le k\le N$. We get the system of $N-1$ equations
$$
\frac{\theta_{k+1}-2\,\theta_k+\theta_{k-1}}{h^2}(1+\beta\,\theta_k)+\beta\,\Bigl(\frac{\theta_k-\theta_{k-1}}{h}\Bi... | {
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Number of distinct limits of subsequences of a sequence is finite? "The number of distinct limits of subsequences of a sequence is finite?"
I've been mulling over this question for a while, and I think it is true, but I can't see how I might prove this formally. Any ideas?
Thanks
| No, the following is a counter-example: Let $E: \mathbb N \to \mathbb N^2$ be an enumeration of $\mathbb N^2$, and set $a_n = (E(n))_1$. Then $a_n$ contains a constant sub-sequence $a_{n_i} = k$ for every $k \in \mathbb N$.
| {
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Show that $\displaystyle{\frac{1}{9}(10^n+3 \cdot 4^n + 5)}$ is an integer for all $n \geq 1$ Show that $\displaystyle{\frac{1}{9}(10^n+3 \cdot 4^n + 5)}$ is an integer for all $n \geq 1$
Use proof by induction. I tried for $n=1$ and got $\frac{27}{9}=3$, but if I assume for $n$ and show it for $n+1$, I don't know what... | ${\displaystyle{\frac{1}{9}}(10^n+3 \cdot 4^n + 5)}$ is an integer for all $n \geq 1$
Proof by induction:
For $n=1, {\displaystyle{\frac{1}{9}}(10^1+3 \cdot 4^1 + 5) = \frac{27}{9} = 3}$, so the result holds for $n=1$
Assume the result to be true for $n=m$, i.e. $\displaystyle{\frac{1}{9}(10^m+3 \cdot 4^m + 5)}$ is a... | {
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"timestamp": "2023-03-29T00:00:00",
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How to prove a trigonometric identity $\tan(A)=\frac{\sin2A}{1+\cos 2A}$ Show that
$$
\tan(A)=\frac{\sin2A}{1+\cos 2A}
$$
I've tried a few methods, and it stumped my teacher.
| First, lets develop a couple of identities.
Given that $\sin 2A = 2\sin A\cos A$, and $\cos 2A = \cos^2A - \sin^2 A$ we have
$$\begin{array}{lll}
\tan 2A &=& \frac{\sin 2A}{\cos 2A}\\
&=&\frac{2\sin A\cos A}{\cos^2 A-\sin^2A}\\
&=&\frac{2\sin A\cos A}{\cos^2 A-\sin^2A}\cdot\frac{\frac{1}{\cos^2 A}}{\frac{1}{\cos^2 A}}\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Proof by contrapositive Prove that if the product $ab$ is irrational, then either $a$ or $b$ (or both) must be irrational.
How do I prove this by contrapositive? What is contrapositive?
| The statement you want to prove is:
If $ab$ is irrational, then $a$ is irrational or $b$ is irrational.
The contrapositive is:
If not($a$ is irrational or $b$ is irrational), then not($ab$ is irrational).
A more natural way to state this (using DeMorgan's Law) is:
If both $a$ and $b$ are rational, then $ab$ is rat... | {
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An example of an endomorphism Could someone suggest a simple $\phi\in $End$_R(A)$ where $A$ is a finitely generated module over ring $R$ where $\phi$ is injective but not surjective? I have a hunch that it exists but I can't construct an explicit example. Thanks.
| Let $R=K$ be a field, and let $A=K[x]$ be the polynomial ring in one variable over $K$ (with the module structure coming from multiplication). Then let $\phi(f)=xf$. It is injective, but has image $xK[x]\ne K[x]$.
| {
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Blowing up a singular point on a curve reduces its singular multiplicity by at least one Let $X$ be the affine plane curve given by $y^2=x^3$, and $O=(0,0)$. Then $X$ has a double singularity at $O$, since its tangent space at $O$ is the doubled $x$-axis. How do we see that, if $\widetilde{X}$ is the blow-up of $X$ at ... | Blowing up a cuspidal plane curve actually yields a nonsingular curve. So the multiplicity is actually reduced by more than one. This is e.g. Exercise 19.4.C in Vakil's notes "Foundations of Algebraic Geometry".
One can compute this quite easily in local charts following e.g. Lecture 20 in Harris' Book "Algebraic Geom... | {
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Graph Theory - How can I calculate the number of vertices and edges, if given this example An algorithm book Algorithm Design Manual has given an description:
Consider a graph that represents the street map of Manhattan in New York City. Every junction of two streets will be a vertex of the graph. Neighboring junction... | Every junction between an avenue and a street is a vertex. As there are $15$ avenues and (about) $200$ streets, there are (about) $15*200=3000$ vertices. Furthermore, every vertex has an edge along an avenue and an edge along a street that connect it to two other vertices. Hence, there are (about) $2*3000 = 6000$ edges... | {
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Is it mathematically correct to write $a \bmod n \equiv b$? This is not a technical question, but a question on whether we can use a particular notation while doing modular arithmetic.
We write $a \equiv b \bmod n$, but is it right to write $a \bmod n \equiv b$?
| It is often correct. $\TeX$ distinguishes the two usages: the \pmod control sequence is for "parenthesized" $\pmod n$ used to contextualize an equivalence, as in your first example, and the \bmod control sequence is for "binary operator" $\bmod$ when used like a binary operator (in your second example).
But in the la... | {
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"answer_id": 2
} |
Left or right edge in cubic planar graph Given a cubic planar graph, if I "walk" on one edge to get to a vertex, it it possible to know which of the other two edges is the left edge and which one is the right edge? Am I forced to draw the graph on paper, without edge crossing, and visually identify left and right edges... | My comment as an answer so it can be accepted:
The answer is no: This can't be possible, since you could draw the mirror image instead, and then left and right edges would be swapped, so they can't be determined by the abstract graph alone.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/121102",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Traveling between integers- powers of 2
Moderator Note: At the time that this question was posted, it was from an ongoing contest. The relevant deadline has now passed.
Consider the integers. We can only travel directly between two integers with a difference whose absolute value is a power of 2 and every time we do ... | It is easy to see that the function $s(n):=d(0,n)$ $\ (n\geq1)$ satisfies the following recursion:
$$s(1)=1,\qquad s(2n)\ =\ s(n), \qquad s(2n+1)=\min\{s(n),s(n+1)\}+1 \ .$$
In particular $s(2)=s(4)=1$, $s(3)=2$.
Consider now the numbers $$a_r:={1\over6}(4^r+2)\qquad (r\geq2)$$ satisfying the recursion
$$a_2=3,\qquad ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/121170",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 0
} |
If $A$ is a subset of $B$, then the closure of $A$ is contained in the closure of $B$. I'm trying to prove something here which isn't necessarily hard, but I believe it to be somewhat tricky. I've looked online for the proofs, but some of them don't seem 'strong' enough for me or that convincing. For example, they us... | I think it's much simpler than that. By definition #1, the closure of A is a subset of any closed set containing A; and the closure of B is certainly a closed set containing A (because it contains B, which contains A). QED.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/121236",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
"answer_count": 7,
"answer_id": 1
} |
Find the value of $(-1)^{1/3}$. Evaluate $(-1)^{\frac{1}{3}}$.
I've tried to answer it by letting it be $x$ so that $x^3+1=0$.
But by this way, I'll get $3$ roots, how do I get the actual answer of $(-1)^{\frac{1}{3}}$??
| Just put it like complex numbers: We know that $z=\sqrt[k]{m_\theta}$, so $z=\sqrt[3]{-1}$ $-1=1_{\pi}$ $\alpha_n=\dfrac{\theta+k\pi}{n}$ $\alpha_0=60$ $\alpha_1=180$ $\alpha_2=300$
So the answers are:
$z_1=1_{\pi/3}$
$z_2=1_{\pi}$
$z_3=1_{5\pi/3}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/121275",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Suppose two $n\times n$ matricies, $A$ and $B$, how many possible solutions are there. Suppose i construct a $n\times n$ matrix $C$, by multiplying two $n\times n$ matrices $A$ and $B$ i.e. $AB = C$. Given $B$ and $C$, how many other $A$'s can yield $C$ also i.e. is it just the exact $A$, infinitely many other $A$'s or... | In general, there could be infinitely many $A$.
Given two solutions $A_1B=C$ and $A_2B=C$, we see that $(A_1-A_2)B=0$
So, if there is at least one solution to $AB=C$, you can see that there are as many solutions to $AB=C$ as there are to $A_0B=0$
Now if $B$ is invertible, the only $A_0$ is $A_0=0$.
If $A_0B=0$ then $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/121349",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Eigenvalue or Eigenvector for a bidiagonal $n\times n$ matrix Let $$J =
\begin{bmatrix} a & b & 0 & 0 & \cdots & \cdots\\\\ 0 & a & b & 0 & \cdots & \cdots\\\\ \vdots & \vdots & \ddots & \cdots & \cdots & \cdots \\\\ \vdots & \vdots & \vdots & \ddots & \cdots & \cdots \\\\ \vdots & \vdots & \vdots &\ddots & a & b \\... | (homework) so some hints:
*
*The eigenvalues are the roots of ${\rm det}(A-xI) = 0.$
*The determinant of a triangular matrix is the product of all diagonal entries.
*How many diagonal entries does an $n\times n$ matrix have?
*How many roots does $(a - x)^n = 0$ have?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/121479",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Stability of trivial solution for DE system with non-constant coefficient matrix Given the arbitrary linear system of DE's
$$x'=A(t)x,$$
with the condition that the spectral bound of $A(t) $ is uniformly bounded by a negative constant, is the trivial solution always stable? All the $(2\times 2)$ matrices I've tried whi... | You can elongate a vector a bit over a short time using a constant matrix with negative eigenvalues, right? Now just do it and at the very moment it starts to shrink, change the matrix. It is not so easy to come up with an explicit formula (though some periodic systems will do it) but this idea of a counterexample is n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/121543",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Why does $PSL(2,\mathbb C)\cong PGL(2,\mathbb C)$ but $PSL(2,\mathbb R) \not\cong PGL(2,\mathbb R)$? Why does $PSL(2,\mathbb C)\cong PGL(2,\mathbb C)$ but $PSL(2,\mathbb R) \not\cong PGL(2,\mathbb R)$?
| You have surjective morphisms $xL(n,K)\to PxL(n,K)$ (whose kernel consists of the multiples of the identity) for $x\in\{G,S\}$, $n\in\mathbb N$ and and $K\in\{\mathbb C,\mathbb R\}$. You also have embeddings $SL(n,K)\to GL(n,K)$. Since the kernel of the composed morphism $SL(n,K)\to GL(n,K)\to PGL(n,K)$ contains (and i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/121619",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 5,
"answer_id": 1
} |
first order differential equation I needed help with this Differential Equation, below:
$$dy/dt = t + y, \text{ with } y(0) = -1$$
I tried $dy/(t+y) = dt$ and integrated both sides, but it looks like the $u$-substitution does not work out.
| This equation is not separable. In other words, you can't write it as $f(y)\;dy=g(t)\;dt$. A differential equation like this can be solved by integrating factors. First, rewrite the equation as:
$$\frac{dy}{dt}-y=t$$
Now we multiply the equation by an integrating factor so we can use the product rule, $d(uv)=udv+vdu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/121669",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Limit of $\arctan(x)/x$ as $x$ approaches $0$? Quick question:
I came across the following limit: $$\lim_{x\rightarrow 0^{+}}\frac{\arctan(x)}{x}=1.$$
It seems like the well-known limit:
$$\lim_{x\rightarrow 0}\frac{\sin x}{x}=1.$$
Can anyone show me how to prove it?
| We can make use of L'Hopital's rule. Since $\frac{d}{dx}\arctan x=\frac{1}{x^2+1}$ and $\frac{d}{dx}x=1$, we have
$$\lim\limits_{x\to0^+}\frac{\arctan x}{x}=\lim\limits_{x\to0^+}\frac{1}{x^2+1}=1.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/121721",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 7,
"answer_id": 1
} |
Deriving even odd function expressions What is the logic/thinking process behind deriving an expression for even and odd functions in terms of $f(x)$ and $f(-x)$?
I've been pondering about it for a few hours now, and I'm still not sure how one proceeds from the properties of even and odd functions to derive:
$$\begin{a... | This is more intuitive if one views it in the special case of polynomials or power series expansions, where the even and odd parts correspond to the terms with even and odd exponents, e.g. bisecting into even and odd parts the power series for $\:\rm e^{{\it i}\:x} \:,\;$
$$\begin{align}
\rm f(x) \ &= \ \rm\frac{f(x)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/121775",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Find the angle in a triangle if the distance between one vertex and orthocenter equals the length of the opposite side Let $O$ be the orthocenter (intersection of heights) of the triangle $ABC$. If $\overline{OC}$ equals $\overline{AB}$, find the angle $\angle$ACB.
| Position the circumcenter $P$ of the triangle at the origin, and let the vectors from the $P$ to $A$, $B$, and $C$ be $\vec{A}$, $\vec{B}$, and $\vec{C}$. Then the orthocenter is at $\vec{A}+\vec{B}+\vec{C}$. (Proof: the vector from $A$ to this point is $(\vec{A}+\vec{B}+\vec{C})-\vec{A} = \vec{B}+\vec{C}$. The vector ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/121843",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Hom of the direct product of $\mathbb{Z}_{n}$ to the rationals is nonzero. Why is $\mathrm{Hom}_{\mathbb{Z}}\left(\prod_{n \geq 2}\mathbb{Z}_{n},\mathbb{Q}\right)$ nonzero?
Context: This is problem $2.25 (iii)$ of page $69$ Rotman's Introduction to Homological Algebra:
Prove that
$$\mathrm{Hom}_{\mathbb{Z}}\left(\pr... | Let $G=\prod_{n\geq2}\mathbb Z_n$ and let $t(G)$ be the torsion subgroup, which is properly contained in $G$ (the element $(1,1,1,\dots)$ is not in $t(G)$, for example) Then $G/t(G)$ is a torsion-free abelian group, which therefore embeds into its localization $(G/t(G))\otimes_{\mathbb Z}\mathbb Q$, which is a non-zero... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/121924",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Residue of a pole of order 6 I am in the process of computing an integral using the Cauchy residue theorem, and I am having a hard time computing the residue of a pole of high order.
Concretely, how would one compute the residue of the function $$f(z)=\frac{(z^6+1)^2}{az^6(z-a)(z-\frac{1}{a})}$$ at $z=0$?
Although it i... | $$g(z)=\frac{1}{(z-a)(z-\frac{1}{a})}=\frac{\frac{1}{a-\frac{1}{a}}}{z-a}+\frac{\frac{-1}{a-\frac{1}{a}}}{z-\frac{1}{a}}$$
we know:
$$(a+b)^n =a^n+\frac{n}{1!}a^{n-1}b+\frac{n(n-1)}{2!}a^{n-2}b^2+...+b^n$$
$$ \text{As regards }: |a|<1 $$
Taylor series of f(z) is:
$$g(z)=\frac{\frac{1}{a-\frac{1}{a}}}{z-a}+\frac{\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/121977",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Dimension of subspace of all upper triangular matrices If $S$ is the subspace of $M_7(R)$ consisting of all upper triangular matrices, then $dim(S)$ = ?
So if I have an upper triangular matrix
$$
\begin{bmatrix}
a_{11} & a_{12} & . & . & a_{17}\\
. & a_{22} & . & . & a_{27}\\
. & . & . & . & .\\
0 & . & . & . & ... | I guess the answer is 1+2+3+...+7=28. Because every element in matrices in S can be a base in that space.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/122029",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Can One use mathematica or maple online? Is it possible to use some of these algebra packages online ?
I have some matrices that I would like to know the characteristic polynomial of.
Where could I send them to get a nicely factorised answer ?
My PC is very slow & it would be nice to use someone elses super powerful co... | For Mathematica, you can try Wolfram Alpha:
factor the characteristic polynomial of [[0, 3], [1, 4]]
For Sage, you can try Sagenb.org. There, you can do
import numpy
n=numpy.array([[0, 3],[1, 4]],'complex64')
m = matrix(n)
m.characteristic_polynomial().factor()
I'm not an expert on this in Sage, but the numpy appear... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/122133",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 7,
"answer_id": 0
} |
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