Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Limits calculus very short question? Can you help me to solve this limit? $\frac{\cos x}{(1-\sin x)^{2/3}}$... as $x \rightarrow \pi/2$, how can I transform this?
| Hint: let $y = \pi/2 - x$ and take the limit as $y \rightarrow 0$.
In this case, the limit becomes
$$\lim_{y \rightarrow 0} \frac{\sin{y}}{(1-\cos{y})^{2/3}}$$
That this limit diverges to $\infty$ may be shown several ways. One way is to recognize that, in this limit, $\sin{y} \sim y$ and $1-\cos{y} \sim y^2/2$, and t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/293560",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Find $\lim_{(x,y) \to (0,0)}\frac{x^3\sin(x+y)}{x^2+y^2}$ Find the limit $$\lim_{(x,y) \to (0,0)}\frac{x^3\sin(x+y)}{x^2+y^2}.$$
How exactly can I do this? Thanks.
| Using $|\sin z|\leq 1$ we find that the absolute value of your function is not greater than
$$
\frac{|x^3|}{x^2+y^2}\leq \frac{|x^3|}{x^2}=|x|.
$$
This is first when $x\neq 0$. Then observe that $|f(x,y)|\leq |x|$ also holds when $x=0$.
Can you conclude?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/293641",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Homogeneous Linear Equation General Solution I’m having some difficulty understanding the solution to the following differential equation problem.
Find a general solution to the given differential equation
$4y’’ – 4y’ + y = 0$
The steps I’ve taken in solving this problem was to first find the auxiliary equation and t... | The story behind what is going on here, is exactly the same we always see when search a Basis for a vector space over $V$ a field $K$. There; we look for a set of linear independent vectors which can generate the whole space. In the space of all solutions for an OE, we do the same as well. For any Homogeneous Linear OE... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/293725",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
When log is written without a base, is the equation normally referring to log base 10 or natural log? For example, this question presents the equation
$$\omega(n) < \frac{\log n}{\log \log n} + 1.4573 \frac{\log n}{(\log \log n)^{2}},$$
but I'm not entirely sure if this is referring to log base $10$ or the natural loga... | In many programming languages, log is the natural logarithm. There are often variants for log2 and log10.
Checked in C, C++, Java, JavaScript, R, Python
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/293783",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "49",
"answer_count": 6,
"answer_id": 5
} |
Need examples about injection (1-1) and surjection (onto) of composite functions The task is that I have to come up with examples for the following 2 statements:
1/ If the composite $g o f$ is injective (one-to-one), then $f$ is one-to-one, but $g$ doesn't have to be.
2/ If the composite $g o f$ is surjective (onto), ... | Consider maps $\{0\}\xrightarrow{f}\{0,1\}\xrightarrow{g}\{0\}$, or $\{0\}\xrightarrow{f}A\xrightarrow{g}\{0\}$ where $A$ is any set with more than one element.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/293851",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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kernel and cokernel of a morphism Let $\phi: A=\mathbb{C}[x,y]/(y^2-x^3) \to \mathbb{C}[t]=B $ be the ring morphism defined by $x\mapsto t^2, y \mapsto t^3.$ Let $f:Y=\text{Spec} B \to X=\text{Spec} A$ be the associated morphism of affine schemes.
Since $\phi$ is injective, then $f^{\sharp} : \mathcal{O}_X \to f_{\sta... | Hopefully you know that coherent sheaves over $\mathrm{Spec} A$ are equivalent to finitely generated $A$-modules. The structure sheaf $\mathcal O_X$ corresponds to $A$ as a module over itself and $f_\ast\mathcal O_Y$ corresponds to $\mathbb C[t]$ with the $A$-module structure given by restricting through $\phi$.
So $f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/293923",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
$\int_0^\infty\frac{\log x dx}{x^2-1}$ with a hint. I have to calculate $$\int_0^\infty\frac{\log x dx}{x^2-1},$$
and the hint is to integrate $\frac{\log z}{z^2-1}$ over the boundary of the domain $$\{z\,:\,r<|z|<R,\,\Re (z)>0,\,\Im (z)>0\}.$$
I don't understand. The boundary of this domain has a pole of the integrand... | Note
$$I(a)=\int_0^\infty\frac{\ln x}{(x+1)(x+a)}dx\overset{x\to\frac a x}
= \frac{1}{2}\int_0^\infty\frac{\ln a}{(x+1)(x+a)}dx= \frac{\ln^2a}{2(a-1)}
$$
Then
$$\int_0^\infty\frac{\ln x}{x^2-1}dx=I(-1)=-\frac14 [\ln(e^{i\pi})]^2=\frac{\pi^2}4
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/293990",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 5,
"answer_id": 4
} |
Normal Distribution Identity I have the following problem. I am reading the paper which uses this identity for a proof, but I can't see why or how to prove its true. Can you help me?
\begin{align}
\int_{x_{0}}^{\infty} e^{tx} n(x;\mu,\nu^2)dx &= e^{\mu t+\nu^2 t^2 /2} N(\frac{\mu - x_0 }{\nu} +\nu t )
\end{align}
... | *
*Substitute the expression for the Normal pdf.
*Gather together the powers of $e$.
*Complete squares in the exponent of $e$ to get the square of something plus a constant.
*Take the constant powers of $e$ out of the integral.
*Change variables to turn the integral into a integral of the standard normal pdf from ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/294068",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Permutation and Equivalence Let X be a nonempty set and define the two place relation ~ as
$\sigma\sim\tau$ if and only if $\rho^{-1}\circ\sigma\circ\rho=\tau$ for some permutation $\rho$
For reflexivity this is what I have:
Let x$\in$X such that $(\rho^{-1}\circ\sigma\circ\rho)(x)=\sigma(x)$
Than $\rho^{-1}(\sigma... | You assumed, I believe, what you were to prove. Look at your expression following "Let" and look at your expression following "Finally"...they say the same thing!
How about letting $\rho = \sigma$: all that matters to show is that for each $\sigma \in X$, $\sigma \sim \sigma$, there exists some permutation in $X$ satis... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/294129",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Can normal chain rule be used for total derivative? Chain rule states that $$\frac{df}{dx} \frac{dx}{dt} = \frac{df}{dt}$$.
Suppose that $f$ is function $f(x,y)$. In this case, would normal chain rule still work?
| The multivariable chain rule goes like this:
$$
\frac{df}{dt} = \frac{\partial f}{\partial x}\cdot \frac{dx}{dt}+ \frac{\partial f}{\partial y}\frac{dy}{dt}
$$
If you can isolate for $\dfrac{dy}{dx}$, then you can always just do implicit differentiation.
Let's do an example:
$$
f=f(x,y) = x^2 - y
$$
Where
$$
x(t) =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/294257",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Factoring Cubic Equations I’ve been trying to figure out how to factor cubic equations by studying a few worksheets online such as the one here and was wondering is there any generalized way of factoring these types of equations or do we just need to remember a bunch of different cases.
For example, how would you fact... | The general method involves three important rules for polynomials with integer coefficients:
*
*If $a+\sqrt b$ is a root, so is $a-\sqrt b$.
*If $a+ib$ is a root, so is $a-ib$.
*If a root is of the form $p/q$, then p is an integer factor of the constant term, and q is an integer factor of the leading coefficient.
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/294331",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Three random variables, two are independent Suppose we have 3 random variables $X,Y,Z$ such that $Y\perp Z$ and let $W = X+Y$. How can we infer from this that $$\int f_{WXZ}(x+y,x,z)\mathrm{d}x = \int f_{WX}(x+y,x)f_{Z}(z)\mathrm{d}x$$ Any good reference where I could learn about independence relevant to this question ... | After integrating out $x$, the left-hand side is the joint density for $Y$ and $Z$ at $(y,z)$, and the right-hand side is the density for $Y$ at $y$ multiplied by the density for $Z$ at $z$. These are the same because $Y$ and $Z$ were assumed to be independent.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/294394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How to compute conditional probability for a homogeneous Poisson process? Let $N$ be a homogeneous Poisson process with intensity $\lambda$. How do I compute the following probability:
$$P[N(5)=2 \, | \, N(2)=1,N(10)=3]?$$
| You know that exactly $2$ events occurred between $t=2$ and $t=10$. These are independently uniformly distributed over the interval $[2,10]$. The probability that one of them occurred before $t=5$ and the other thereafter is therefore $2\cdot\frac38\cdot\frac58=\frac{15}{32}$. The intensity $\lambda$ doesn't enter into... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Quantified Statements To English The problem I am working on is:
Translate these statements into English, where C(x) is “x is a comedian” and F(x) is “x is funny” and the domain consists of all people.
a) $∀x(C(x)→F(x))$
b)$∀x(C(x)∧F(x))$
c) $∃x(C(x)→F(x))$
d)$∃x(C(x)∧F(x))$
------------------------------------------... | This is because the mathematical language is more accurate than the usual language. But your answers are right and they are the same as the book.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/294519",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
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Show that monotone sequence is reducing and bounded by zero? Let $a_n=\displaystyle\sum\limits_{k=1}^n {1\over{k}} - \log n$ for $n\ge1$. Euler's Constant is defined as $y=\lim_{n\to\infty} a_n$. Show that $(a_n)^\infty_{n=1}$ is decreasing and bounded by zero, and so this limit exists
My thought:
When I was trying th... | Certainly it's decreasing. I get $1-0=1$, and $1 + \frac12 -\log2=0.8068528\ldots$, and $1+\frac12+\frac13-\log3=0.73472\ldots$.
$$
\frac{1}{n+1} = \int_n^{n+1}\frac{dx}{n+1} \le \int_n^{n+1}\frac{dx}{x} = \log(n+1)-\log n.
$$
$$
\frac1n = \int_n^{n+1} \frac{dx}{n} \ge \int_n^{n+1} \frac{dx}{x} = \log(n+1)-\log n.
$$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Distinction between "measure differential equations" and "differential equations in distributions"? Is there a universally recognized term for ODEs considered in the sense of distributions used to describe impulsive/discontinuous processes? I noticed that some authors call such ODEs "measure differential equations" whi... | As long as the distributions involved in the equation are (signed) measures, there is no difference and both terms can be used interchangeably. This is the case for impulsive source equations like $y''+y=\delta_{t_0}$.
Conceivably, ODE could also involve distributions that are not measures, such as the derivative of $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/294696",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
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Field extension of composite degree has a non-trivial sub-extension Let $E/F$ be an extension of fields with $[E:F]$ composite (not prime). Must there be a field $L$ contained between $E$ and $F$ which is not equal to either $E$ or $F$? To prove this is true, it suffices to produce an element $x\in E$ such that $F(x) \... | Let $L/\mathbb Q$ be a Galois extension with Galois group $A_4$, this is certainly possible although off the top of my head I can't remember a polynomial that gives you this extension. Now $A_4$ has a subgroup $H$ of index $4$ namely, the subgroup generated by a cycle, but this subgroup is not properly contained in ano... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/294780",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 2
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Counting primes of the form $S_1(a_n)$ vs primes of the form $S_2(b_n)$ Let $n$ be an integer $>1$. Let $S_1(a_n)$ be a symmetric irreducible integer polynomial in the variables $a_1,a_2,...a_n$. Let $S_2(b_n)$ be a symmetric irreducible integer polynomial in the variables $b_1,b_2,...b_n$ of the same degree as $S_1(a_... | I think the polynomial $2x^4+x^2y+xy^2+2y^4$ is symmetric and irreducible, but its values are all even, hence, it takes on at most $1$ prime value. If your other polynomial takes on infinitely many prime values --- $x^4+y^4$ probably does this, though proving it is out of reach --- then the ratio of primes represented ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/294849",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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what is the precise definition of matrix? Let $F$ be a field.
To me, saying 'Matrix is a rectangular array of elements of $F$' seems extremely terse. What kind of set is a rectangular array of elements of $F$?
Precisely, $(F^m)^n \neq (F^n)^m \neq F^{n\times m}$. I wonder which is the precise definition for $M_{n\tim... | In an engineering class where we don't do things very rigorously at all, we defined an $n\times m$ matrix as a linear function from $\mathbb{R}^m$ to $\mathbb{R}^m$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/294899",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 4
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half-angle trig identity clarification I am working on the following trig half angle problem. I seem to be on the the right track except that my book answer shows -1/2 and I didn't get that in my answer. Where did I go wrong?
$$\sin{15^{\circ}} = $$
$$\sin \frac { 30^{\circ} }{ 2 } = \pm \sqrt { \frac { 1 - \cos{30^{\c... | $$\sqrt { \dfrac { 1 - \dfrac {\sqrt 3 }{ 2 } }{ 2 } \times \dfrac22} = \sqrt{\dfrac{2-\sqrt3}{\color{red}4}} = \dfrac{\sqrt{2-\sqrt3}}{\color{red}2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/295038",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Convert $ x^2 - y^2 -2x = 0$ to polar? So far I got
$$r^2(\cos^2{\phi} - \sin^2{\phi}) -2 r\cos{\phi} = 0$$
$$r^2 \cos{(2\phi)} -2 r \cos{\phi} = 0$$
| You are on the right track. Now divide through by $r \ne 0$ and get
$$r \cos{2 \phi} - 2 \cos{\phi} = 0$$
or
$$r = 2 \frac{ \cos{\phi}}{\cos{2 \phi}}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/295162",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Is it true that, $A,B\subset X$ are completely seprated iff their closures are? If $A,B\subset X$ and $\overline{A}, \overline{B}$ are completely seprated, so also are $A,B$. since $A\subset \overline{A}$, $B\subset \overline{B}$ then, $f(A)\subset f(\overline{A})=0$ and $f(B)\subset f(\overline{B})=1$ for some contin... | HINT: Suppose that $A$ and $B$ are completely separated, and let $f:X\to[0,1]$ be a continuous function such that $f(x)=0$ for all $x\in A$ and $f(x)=1$ for all $x\in B$. Since $f$ is continuous, $f^{-1}[\{0\}]$ is closed, and certainly $A\subseteq f^{-1}[\{0\}]$, so ... ?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/295220",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Is there a way to solve $x^2 + 12y - 12x = 0$ for $x$? I'm doing some statistical analysis (random variate generation for simulation models) and I just ran the inverse transform of a CDF:
$$ F(x) = \begin{cases} (x-4)/4 & \text{for } x \in [2,3] \\ x - (x^2/12) & \text{for } x \in (3,6] \\ 0 & \text{otherwise}
\end{cas... | Try the quadratic equation formula:
$$x^2-12x+12y=0\Longrightarrow \Delta:=12^2-4\cdot 1\cdot 12y=144-48y=48(3-y)\Longrightarrow$$
$$x_{1,2}=\frac{12\pm \sqrt{48(3-y)}}{2}=6\pm 2\sqrt{3(3-y)}$$
If you're interested in real roots then it must be that $\,y\le 3\,$ ...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/295369",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Proving algebraic sets i) Let $Z$ be an algebraic set in $\mathbb{A}^n$. Fix $c\in \mathbb{C}$. Show that $$Y=\{b=(b_1,\dots,b_{n-1})\in \mathbb{A}^{n-1}|(b_1,\dots,b_{n-1},c)\in Z\}$$ is an algebraic set in $\mathbb{A}^{n-1}$.
ii) Deduce that if $Z$ is an algebraic set in $\mathbb{A}^2$ and $c\in \mathbb{C}$ then $Y=\... |
This answer was merged from another question so it only covers part ii).
$Z$ is algebraic, and hence the simultaneous solution to a set of polynimials in two variables. If we swap one variable in all the polynomials with the number $c$, you will get a set of polynomials in one variable, with zero set being your $Y$. ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
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Why aren't parametrizations equivalent to their "equation form"? Consider the parametrization $(\lambda,t)\mapsto (\lambda t,\lambda t^2,\lambda t^3)$. This is a union of lines (not sure how to visualize it precisely. I think it's a double cone). It doesn't appear that the $x$ or $z$ axis are in this parametrization (i... | When you say that you "solve" and obtain the relationship $y^2 = xz$ I image you just observed that $(\lambda t^2)^2 = (\lambda t)(\lambda t^3)$ correct? In this case what you have shown is that the set for which you have a parameterization is a subset of the set of solutions to $y^2 = xz$. What you have not shown is... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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with how many zeros it ends I have to calculate with how many zeros 1000! ends. This is wat I did, but I am not sure whether its good: I think I have to calculate how many times the product 10 is in 1000!. I found out the factor 10 is 249 times in 1000! using the fact that $s_p(n!)=\sum_{r=1}^{\infty}\lfloor \frac{n}{p... | First, let's note that $10$ can be factored into $2 \times 5$ so the key is to compute the minimum of the number of $5$'s and $2$'s that appear in $1000!$ as factors of the numbers involved. As an example, consider $5! = 120$, which has one $0$ because there is a single $5$ factor and a trio of $2$ factors in the prod... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/295643",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
For which Natural $n\ge2: \phi(n)=n/2$ For which Natural $n\ge2$ does this occur with?: $\phi(n)=n/2$
| Hint: $n$ is even, or $n/2$ wouldn't be an integer. Hence $n=2^km$ with $m$ odd and $k\ge1$. You have $\phi(2^km)=2^{k-1}\phi(m)$ which must equal $n/2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/295732",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
} |
Induction proof of $\sum_{k=1}^{n} \binom n k = 2^n -1 $
Prove by induction: $$\sum_{k=1}^{n} \binom n k = 2^n -1 $$ for all $n\in \mathbb{N}$.
Today I wrote calculus exam, I had this problem given.
I have the feeling that I will get $0$ points for my solution, because I did this:
Base Case: $n=1$
$$\sum_{k=1}^{1} \... | suppose that $$\sum_{k=1}^{n} \binom n k = 2^n -1 $$
then
$$\sum_{k=1}^{n+1} \binom {n+1}{k} =\sum_{k=1}^{n+1}\Bigg( \binom {n}{ k} +\binom{n}{k-1}\Bigg)=$$
$$=\sum_{k=1}^{n+1} \binom {n}{ k} +\sum_{k=1}^{n+1}\binom{n}{k-1}=$$
$$=\sum_{k=1}^{n} \binom {n}{ k}+\binom{n}{n+1}+\sum_{k=0}^{n} \binom {n}{k}=$$
$$=\sum_{k=1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/295802",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Graphs such that $|G| \ge 2$ has at least two vertices which are not its cut-vertices Show that every graph $G$, such that $|G| \ge 2$ has at least two vertices which are not its cut-vertices.
| Let $P$ be a maximal path in $G$. I claim that the end points of $P$ are not cut vertices.
Suppose that an end point $v$ of $P$ was a cut vertex. Let $G$ be separated into $G_1,\ G_2,\ \cdots,\ G_k$. It follows that any path from one component to another must pass through $v$ and namely such a path does not end on $v$... | {
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"url": "https://math.stackexchange.com/questions/295866",
"timestamp": "2023-03-29T00:00:00",
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"answer_id": 0
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Proving some basic facts about integrals Let $f$, $g$ be Riemann integrable functions on the interval $[a,b]$, that is $f,g \in \mathscr{R}([a,b])$.
(i) $\int_{a}^{b} (cf+g)^2\geq 0$ for all $c \in \mathbb{R}$.
(ii) $2|\int_{a}^{b}fg|\leq c \int_{a}^{b} f^2+\frac{1}{c}\int_{a}^{b} g^2$ for all $c \in \mathbb{R}^+$
I... | I will assume that the functions are real-valued.
For the first point, I will use that if an R-integrable function $h$ is nonegative on $[a,b]$, then $\int_a^bh(x)dx\geq 0$.
For the second point, I will use that if $h(x)\leq k(x)$ on $[a,b]$, then $\int_a^bh(x)dx\leq \int_a^bk(x)dx$. Note that the latter follows readil... | {
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"timestamp": "2023-03-29T00:00:00",
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When is $\| f \|_\infty$ a norm of the vector space of all continuous functions on subset S? Let S be any subset of $\mathbb{R^n}$. Let $C_b(S)$ denote the vector space of all bounded continuous
functions on S. For $f \in C(S)$, define $\| f \|_\infty = \sup_{x \in S} |f(x)|$
When is this a norm of the vector space of ... | For X a locally compact Hausdorff space such as an open or closed subset or non-empty intersection of open and closed subsets of R^n, the space of bounded continuous function on X with the sup norm, which is the same as the infinity norm or essential sup norm is complete. The sup or the essential sup is always a norm ... | {
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Evaluate $\lim_{x\to1^-}\left(\sum_{n=0}^{\infty}\left(x^{(2^n)}\right)-\log_2\frac{1}{1-x}\right)$
Evaluate$$\lim_{x\to1^-}\left(\sum_{n=0}^{\infty}\left(x^{(2^n)}\right)-\log_2\frac{1}{1-x}\right)$$
Difficult problem. Been thinking about it for a few hours now. Pretty sure it's beyond my ability. Very frustrating t... | This is NOT a solution, but I think that others can benefit from my failed attempt. Recall that $\log_2 a=\frac{\log a}{\log 2}$, and that $\log(1-x)=-\sum_{n=1}^\infty\frac{x^n}n$ for $-1\leq x<1$, so your limit becomes
$$\lim_{x\to1^-}x+\sum_{n=1}^\infty\biggl[x^{2^n}-\frac1{\log2}\frac{x^n}n\biggr]\,.$$
The series a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 4,
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Two Problem: find $\max, \min$; number theory: find $x, y$
*
*Find $x, y \in \mathbb{N}$ such that $$\left.\frac{x^2+y^2}{x-y}~\right|~ 2010$$
*Find max and min of $\sqrt{x+1}+\sqrt{5-4x}$ (I know $\max = \frac{3\sqrt{5}}2,\, \min = \frac 3 2$)
| Problem 1: I have read a similar problem with a good solution in this forum
| {
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Exponentiation when the exponent is irrational I am just curious about what inference we can draw when we calculate something like $$\text{base}^\text{exponent}$$ where base = rational or irrational number and exponent = irrational number
| $2^2$ is rational while $2^{1/2}$ is irrational. Similarly, $\sqrt 2^2$ is rational while $\sqrt 2^{\sqrt 2}$ is irrational (though it is not so easily proved), so that pretty much settles all cases. Much more can be said when the base is $e$. The Lindemann-Weierstrass Theorem asserts that $e^a$ where $a$ is a non-zero... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$p$ is an odd primitive, Show why there are no primitive roots of $\bmod 3p$ if $p$ is an odd primitive, Prove there are no primitive roots of $\bmod 3p$
Where I'm at: $a^{2(p-1)}=1 \pmod{3p}$ where a is a primitive root of $3p$ (by contradiction)
$(a/3p)=(a/3)(a/p)$ are the Legendre symbols, and stuck here..tried a c... | Note that when $p=3$ the theorem does not hold! 2 is a primitive root.
So supposing $p$ is not 3...
Since $p$ is odd let $p = 2^r k+1$ with $k$ odd.
The group of units is $\mod {3p}$ is $$(\mathbb Z/(3p))^\times \simeq \mathbb Z/(2) \times \mathbb Z/(2^r) \times (\mathbb Z/(k))^\times$$ by Sun Zi's theorem.
There can b... | {
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topology question about cartesian product I have a question about a proof I'm reading. It says:
Suppose $A$ is an open set in the topology of the Cartesian product $X\times Y$, then you can write $A$ as the $\bigcup (U_\alpha\times V_\alpha)$ for $U_\alpha$ open in $X$ and for $V_\alpha$ open in $Y$. Why is this?
(I g... | It isn't a random union, it is the union for all open $U,V$ such that $U\times V$ is contained in $A$. As a result we immediately have half of the containment, that the union is a subset of $A$.
To see why $A$ is contained in the union, consider a point $(x,y)$ in $A$. Since $A$ is open, there must be a basic open se... | {
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Given a point $x$ and a closed subspace $Y$ of a normed space, must the distance from $x$ to $Y$ be achieved by some $y\in Y$? I think no. And I am looking for examples. I would like a sequence $y_n$ in $Y$ such that $||y_n-x||\rightarrow d(x,Y)$ while $y_n$ do not converge.
Can anyone give a proof or an counterexampl... | This is a slight adaptation of a fairly standard example.
Let $\phi: C[0,1]\to \mathbb{R}$ be given by $\phi(f)=\int_0^{\frac{1}{2}} f(t)dt - \int_{\frac{1}{2}}^1 f(t)dt$. Let $Y_\alpha = \phi^{-1}\{\alpha\}$. Since $\phi$ is continuous, $Y_\alpha$ is closed for any $\alpha$.
Now let $\hat{f}(t) = 4t$ and notice that $... | {
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What are relative open sets? I came across the following:
Definition 15. Let $X$ be a subset of $\mathbb{R}$. A subset $O \subset X$ is said to be open in $X$ (or relatively open in $X$) if for each $x \in O$, there exists $\epsilon = \epsilon(x) > 0$ such that $N_\epsilon (x) \cap X \subset O$.
What is $\epsilon$ and $... | Forget your definition above. The general notion is:
Let $X$ be a topological space, $A\subset X$ any subset. A set $U_A$ is relatively open in $A$ if there is an open set $U$ in $X$ such that $U_A=U\cap A$.
I think that in your definition $N_\epsilon(x)$ is meant to denote an open neighborhood of radius $\epsilon$ of ... | {
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Example of a normal operator which has no eigenvalues Is there a normal operator which has no eigenvalues?
If your answer is yes, give an example.
Thanks.
| Example 1
'I think "shift operator or translation operator" is one of them.' – Ali Qurbani
Indeed, the bilateral shift operator on $\ell^2$, the Hilbert space of square-summable two-sided sequences, is normal but has no eigenvalues. Let $L:\ell^2 \to \ell^2$ be the left shift operator, $R:\ell^2 \to \ell^2$ the right s... | {
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Prove the relation of $\cosh(\pi/2)$ and $e$ Prove that:
$$\cosh\left(\frac{\pi}{2}\right)=\frac{1}{2}e^{-{\pi/2}}(1+e^\pi)$$
What I have tried.
$$\cosh\left(\frac{\pi}{2}\right)=\cos\left(i\frac{\pi}{2}\right)$$
$$=Re\{e^{i.i\frac{\pi}{2}}\}$$
$$=Re\{e^{-\frac{\pi}{2}}\}$$
Why is $e^{-\frac{\pi}{2}}$ not answer any wh... | $\cosh(x)$ is usually defined defined as $\frac{e^{x} + e^{-x}}{2}$. If you haven't some different definition, then it is quite straightforward:
$$\cosh\left(\frac{\pi}{2}\right)=\frac{e^{\frac{\pi}{2}} + e^{-\frac{\pi}{2}}}{2} = \frac{1}{2}e^{-\frac{\pi}{2}}(1 + e^x)$$
| {
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Field of characteristic 0 such that every finite extension is cyclic I am trying to construct a field $F$ of characteristic 0 such that every finite extension of $F$ is cyclic. I think that I have an idea as to what $F$ should be, but I am not sure how to complete the proof that it has this property. Namely, let $a\in ... | This is a fairly common question in algebra texts. Here's a series of hints taken from a prelim exam.
Let $F$ be a maximal subfield of $\bar{\mathbb Q}$ with respect to not containing $\sqrt{a}$. Let $F \subset E$ be a Galois extension. Show that $F(\sqrt{a})$ is the unique subfield of $E$ of degree $2$. Deduce that $... | {
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Are PA and ZFC examples of logical systems? Wikipedia says
A logical system or, for short, logic, is a formal system together with a form of semantics, usually in the form of model-theoretic interpretation, which assigns truth values to sentences of the formal language.
When we talk about PA or ZFC, are these logica... | When I talk about a logical system in the way that
A logical system or, for short, logic, is a formal system together with a form of semantics, usually in the form of model-theoretic interpretation
I understand that a logic $\mathcal{L}$ is a pair $(L,\models)$, where $L$ is a function, and the domain of $L$ is the ... | {
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Is the set of all bounded sequences complete? Let $X$ be the set of all bounded sequences $x=(x_n)$ of real numbers and let $$d(x,y)=\sup{|x_n-y_n|}.$$ I need to show that $X$ is a complete metric space.
I need to show that all Cauchy sequences are convergent.
I appreciate your help.
| HINT: Let $\langle x^n:n\in\Bbb N\rangle$ be a Cauchy sequence in $X$. The superscripts are just that, labels, not exponents: $x^n=\langle x^n_k:k\in\Bbb N\rangle\in X$. Fix $k\in\Bbb N$, and consider the sequence
$$\langle x^n_k:n\in\Bbb N\rangle=\langle x^0_k,x^1_k,x^2_k,\dots\rangle\tag{1}$$
of $k$-th coordinates o... | {
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Can I really factor a constant into the $\min$ function? Say I have $\min(5x_1,x_2)$ and I multiply the whole function by $10$, i.e. $10\min(5x_1,x_2)$. Does that simplify to $\min(50x_1,10x_1)$? In one of my classes I think my professor did this but I'm not sure (he makes very hard to read and seemingly bad notes), an... | Ys, that is legal as long as the constant is not negative. I.e., $10 \cdot \max(3, 5) = 10 \cdot 5 = 50$ is the same as $\max(10 \cdot 3, 10 \cdot 5) = 50$, but try multiplying by $-10$...
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Zero and Cozero-sets of $\mathbb{R}$ A subset $U$ of a space $X$ is said to be a zero-set if there exists a continuous real-valued function $f$ on $X$ such that $U=\{x\in X: f(x)=0\}$. and said to be a Cozero-set if here exists a continuous real-valued function $g$ on $X$ such that $U=\{x\in X: g(x)\not=0\}$.
Is it tru... | I just waant to know how $\phi$ and $\mathbb{R}$ are not zero set?
as if i take $f(x) = 0 \forall x$ and $g(x) = e^{x} + 1 \forall x$
both are cts.
Then $\phi$ and $\mathbb{R}$ are zero set.
| {
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How to show that $ n^{2} = 4^{{\log_{2}}(n)} $? I ran across this simple identity yesterday, but can’t seem to find a way to get from one side to the other:
$$
n^{2} = 4^{{\log_{2}}(n)}.
$$
Wolfram Alpha tells me that it is true, but other than that, I’m stuck.
| Take $\log_{2}$ of both sides and get
$$n^{2} = 4^{{\log_{2}} n}$$
$$2\log_{2}n =2\log_{2}n $$
| {
"language": "en",
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$P$ projector. prove that $\langle Px,x\rangle=\|Px\|^2.$ Let $X$ be a Hilbert space and $P \in B(X)$ a projector. Then for any $x\in X$:
$$\langle Px,x\rangle=\|Px\|^2.$$
My proof:
$$\|Px\|^{2}=\langle Px,Px\rangle=\langle P^{*}Px,x\rangle=\langle P^2x,x\rangle=\langle Px,x\rangle.$$
Is ok ? Thanks :)
| Yes, that is all.
$$ \quad \quad $$
| {
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Obvious Group and subgroup questions. My awesome math prof posted a practice midterm but didn't post any solutions to it :s
Here is the question.
Let $G$ be a group and let $H$ be a subgroup of $G$.
*
*(a) TRUE or FALSE: If $G$ is abelian, then so is $H$.
*(b) TRUE or FALSE: If $H$ is abelian, then so is $G$.
Par... | (b) Take the group of $(n \times n)$-matrices with $\mathbb{R}$-coefficients with usual matrix multiplication as G and let H be the subgroup of diagonal matrices. H ist abelian, but G is not abelian.
| {
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when $f(x)^n$ is a degree of $n$ why is useful to think $\sqrt{f(x)^n}$ as $n/2$? I have come across this question when doing problems from "Schaum's 3000 Solved Calculus Problems". I was trying to solve $$\lim_{x\rightarrow+\infty}\frac{4x-1}{\sqrt{x^2+2}}$$ and I couldn't so I looked the solution and solution said ... | Note that $$-|x|-\sqrt{2}\leq\sqrt{x^2+2}\leq|x|+\sqrt{2}$$
From the last inequality, you can conclude that the rate of growth of the function $\sqrt{x^2+2}$, is in some sense linear or in the language of the author, the degree of $\sqrt{f(x)}$ is something like $\frac{n}{2}$. Therefore the functions $4x-1$ and $\sqrt{... | {
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Is $A - B = \emptyset$? $A = \{1,2,3,4,5\}, B = \{1,2,3,4,5,6,7,8\}$
$A - B =$ ?
Does that just leave me with $\emptyset$? Or do I do something with the leftover $6,7,8$?
| $A - B = \emptyset$, because by definition, $A - B$ is everything that is in $A$ but not in $B$.
| {
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What is this math symbol called? My professors use a symbol $$x_0$$ and they pronounce it as x not or x nod, I am not sure what the exact name is because they have thick accents. I have tried looking this up on the Internet but I could not find an answer. Does anyone know what this is called?
| They actually call it x-naught. I believe it comes from British English. Kind of like how the Canadians call the letter z "zed". All it means is "x sub zero", just another way of saying the same thing. It does flow better though, I think. "sub zero" just takes so much more work to say. I do think "naught" and "not"... | {
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$T:\Bbb R^2 \to \Bbb R^2$, linear, diagonal with respect to any basis. Is there a linear transformation from $\Bbb R^2$ to $\Bbb R^2$ which is represented by a diagonal matrix when written with respect to any fixed basis?
If such linear transformation $T$ exists, then its eigenvector should be the identity matrix for a... | If the transformation $T$ is represented by the matrix $A$ in basis $\mathcal{A}$, then it is represented by the matrix $PAP^{-1}$ in basis $\mathcal{B}$, where $P$ is the invertible change-of-basis matrix.
Suppose that $T$ is represented by a diagonal matrix in any basis. Let $P$ be an arbitrary invertible matrix and... | {
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Construction of a triangle, given: side, sum of the other sides and angle between them. Given: $\overline{AB}$, $\overline{AC}+\overline{BC}$ and $\angle C$. Construct the triangle $\triangle ABC$ using rule and compass.
| Draw the side given AB
draw the angle given A
produce the adjacent side which equal to sum of given sides AP
connect remaining point from it PB
bisect that side PB
produce that side until cut AP
take the point of intersection C now you have triangle.
| {
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Probability with Chi-Square distribution What is the difference, when calculating probabilities of Chi-Square distributions, between $<$ and $\leq$ or $>$ and $\geq$.
For example, say you are asked to find P$(\chi_{5}^{2} \leq 1.145)$.
I know that this is $=0.05$ from the table of Chi-Square distributions, but what i... | The $\chi^2$ distributions are continuous distributions. If $X$ has continuous distribution, then
$$\Pr(X\lt a)=\Pr(X\le a).$$
If $a$ is any point, then $\Pr(X=a)=0$. So in your case, the probabilities would be exactly the same.
Many other useful distributions, such as the normal, and the exponential, are continuous.... | {
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Determining Ambiguity in Context Free Grammars What are some common ways to determine if a grammar is ambiguous or not? What are some common attributes that ambiguous grammars have?
For example, consider the following Grammar G:
$S \rightarrow S(E)|E$
$E \rightarrow (S)E|0|1|\epsilon$
My guess is that this grammar is n... | To determine if a context free grammar is ambiguous is undecidable (there is no algorithm which will correctly say "yes" or "no" in a finite time for all grammars). This doesn't mean there aren't classes of grammars where an answer is possible.
To prove a grammar ambiguous, you do as you outline: Find a string with two... | {
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What is the origin of the phrase "as desired" in mathematics? This is a sort of strange question that popped into my head when I was reading a paper. In writing mathematics, many authors use the phrase "as desired" to conclude a proof, usually written to indicate that one has reached the result originally stated. I kno... | From Wikipedia http://en.wikipedia.org/wiki/Q.E.D.:
Q.E.D. is an initialism of the Latin phrase quod erat demonstrandum, originating from the Greek analogous hóper édei deîxai (ὅπερ ἔδει δεῖξαι), meaning "which had to be demonstrated". The phrase is traditionally placed in its abbreviated form at the end of a mathemat... | {
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Trigonometric Identity $\frac{1}{1-\cos t} + \frac{1}{1+\cos t}$ I am just learning about trig identities, and after doing a few, I am stuck on this one:
$$
\frac{1}{1-\cos t} + \frac{1}{1+\cos t}.
$$
The only way to start, that I can think of is this:
$$
\frac{1}{1-(1/\sec t)} + \frac{1}{1+(1/\sec t)}.
$$
And from the... | Hint:
Use that
$$
\frac{1}{a}+\frac{1}{b}=\frac{a+b}{ab}
$$
along with the identity
$$
\sin^2t+\cos^2t=1.
$$
| {
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Existence of Matrix inverses depending on the existence of the inverse of the others.. Let $A_{m\times n}$ and $B_{n\times m}$ be two matrices with real entries. Prove that $I-AB$ is invertible iff $I-BA$ is invertible.
| Hint:$(I-BA)^{-1}=X$ (say), Now expand left side. we get $$X=I+BA+ (BA)(BA)+(BA)(BA)(BA)+\dots$$ $$AXB=AB+(AB)^2+(AB)^3+(AB)^4+\dots$$ $$I+AXB=I+(AB)+(AB)^2+\dots+(AB)^n+\dots=(I-AB)^{-1}$$
Check yourself: $(I+AXB)(I-AB)=I$, $(I-AB)(I+AXB)=I$
| {
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Intuitive meaning of immersion and submersion What is immersion and submersion at the intuitive level. What can be visually done in each case?
| First of all, note that if $f : M \to N$ is a submersion, then $\dim M \geq \dim N$, and if $f$ is an immersion, $\dim M \leq \dim N$.
The Rank Theorem may provide some insight into these concepts. The following statement of the theorem is taken from Lee's Introduction to Smooth Manifolds (second edition); see Theorem ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/297988",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "48",
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Groups with transitive automorphisms Let $G$ be a finite group such that for each $a,b \in G \setminus \{e\}$ there is an automorphism $\phi:G \rightarrow G$ with $\phi(a)=b$. Prove that $G$ is isomorphic to $\Bbb Z_p^n$ for some prime $p$ and natural number $n$.
| Hint 1: If $a, b \in G \setminus \{e\}$, then $a$ and $b$ have the same order.
Hint 2: Using the previous hint, show that $G$ has order $p^n$ for some prime $p$ and that every nonidentity element has order $p$.
Hint 3: In a $p$-group, the center is a nontrivial characteristic subgroup.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/298050",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Delta in continuity Let $f: [a,b]\to\mathbb{R}$ be continuous, prove that it is uniform continuous.
I know using compactness it is almost one liner, but I want to prove it without using compactness. However, I can use the theorem that every continuous function achieves max and min on a closed bounded interval.
I propo... | Let an $\epsilon>0$ be given and put
$$\rho(x):=\sup\bigl\{\delta\in\ ]0,1]\ \bigm|\ y, \>y'\in U_\delta(x)\ \Rightarrow\ |f(y')-f(y)|<\epsilon\bigr\}\ .$$
By continuity of $f$ the function $x\to\rho(x)$ is strictly positive and $\leq1$ on $[a,b]$.
Lemma. The function $\rho$ is $1$-Lipschitz continuous, i.e.,
$$|\rho(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/298136",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
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Product of numbers Pair of numbers whose product is $+7$ and whose sum is $-8$.
Factorise $x^{2} - 8x + 7$.
I can factorise but it's just I can't find any products of $+7$ and that is a sum of
$-8$. Any idea? Thanks guys!
Thanks.
| I do not understand why you are trying to factorise $x^2-8x+7$. I suggest you use viete's formulae. xy=2 and x+y=-3. Let's say you have a quadratic equation $x^2+ax+b$ Then the roots $x_1, x_2$ has the property $x_1+x_2=-a$ and $x_1.x_2=c$ So you have the quadratic equation $x^2-2x-3$ When you solve this, you get the a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How to find all polynomials with rational coefficients s.t $\forall r\notin\mathbb Q :f(r)\notin\mathbb Q$ How to find all polynomials with rational coefficients$f(x)=a_nx^n+\cdots+a_1x+a_0$, $a_i\in \mathbb Q$, such that $$\forall r\in\mathbb R\setminus\mathbb Q,\quad f(r)\in\mathbb R\setminus\mathbb Q.$$ thanks in ad... | The only candidates are those polynomials $f(x)\in\mathbb Q[x]$ that are factored over $\mathbb Q$ as product of first degree polynomials (this is because if $\deg f>1$ and $f$ is irreducible then all of its roots are irrationals.)
The first degree polynomials have this property. Can you see that these are all?
(Hint: ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Why does zeta have infinitely many zeros in the critical strip? I want a simple proof that $\zeta$ has infinitely many zeros in the critical strip.
The function $$\xi(s) = \frac{1}{2} s (s-1) \pi^{\tfrac{s}{2}} \Gamma(\tfrac{s}{2})\zeta(s)$$ has exactly the non-trivial zeros of $\zeta$ as its zeros ($\Gamma$ cancels al... | Hardy proved in 1914 that an infinity of zeros were on the critical line ("Sur les zéros de la fonction $\zeta(s)$ de Riemann" Comptes rendus hebdomadaires des séances de l'Académie des sciences. 1914).
Of course other zeros could exist elsewhere in the critical strip.
Let's exhibit the main idea starting with the Xi f... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
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Is this function differentiable at 0? I would like to know if this function is differentiable at the origin:
$$f(x) =
\left\{
\begin{array}{cl}
x+x^2 & \mbox{if } x \in \mathbb{Q}; \\
x & \mbox{if } x \not\in \mathbb{Q}. \end{array}
\right.$$
Intuitively, I know it is, but I don't know how to prove it.
Any ideas... | For continuity at any arbitrary point $c\in\mathbb{R}$ and considering sequential criteria(first consider a rational sequence converging to $c$ and then a irrational sequence converging to $c$ and equate the limit) of continuity at $c$ you need $c^2+c=c$ so $c^2=0$ so $c=0$, so only at $c=0$ the function is continuos,... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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A proof question about continuity and norm
Let $E⊂\mathbb{R}^{n}$ be a closed, non-empty set and $N : \mathbb{R}^{n}\to\mathbb{R}$ be a norm. Prove that the function $f(x)=\inf\left \{ N(x-a)\mid a\in E \right \}$, $f : \mathbb{R}^{n}→\mathbb{R}$ is continuous and $f^{-1}(0) = E$.
There are some hints:
$f^{-1}(0) = E... | The other answers so far are good, but here is an alternative hint for the first part.
Because $E$ is closed, its complement $E^c$ is open. A set in $\mathbb{R}^n$ is open if and only if the set contains an open ball around any point in the set. Thus, for any $x\in E^c$, there is some $r>0$ such that the open ball $B(x... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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What is inverse of $I+A$? Assume $A$ is a square invertible matrix and we have $A^{-1}$. If we know that $I+A$ is also invertible, do we have a close form for $(I+A)^{-1}$ in terms of $A^{-1}$ and $A$?
Does it make it any easier if we know that sum of all rows are equal?
| Check this question. The first answer presents a recursive formula to retrieve the inverse of a generic sum of matrices. So yours should be a special case.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Total divisor in a Principal Ideal Domain. Let $R$ be a right and left principal ideal domain. An element $a\in R$ is said to be a right divisor of $b\in R$ if there exists $x \in R$ such that $xa=b$ . And similarly define left divisor.
$a$ is said to be a total divisor of $b$ if $RbR = <a>_R \cap$ $ _R<a>$ .
How... | I'm going to assume $R$ contains $1$.
Also, my proof will only show that $RbR \subseteq aR \cap Ra$.
Most definitions of total divisors I've seen go like the following: An element $a$ in a ring $R$ is a total divisor of $b$ when $RbR \subseteq aR \cap Ra$. Whether this implies $RbR = aR \cap Ra$ in a ring that is bo... | {
"language": "en",
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How to check the continuity of this function defined as follows
The function $f:\Bbb R\to\Bbb R$ defined by $f(x)=\min(3x^3+x,|x|)$ is
(A) continuous on $\Bbb R$, but not differentiable at $x=0$.
(B) differentiable on $\Bbb R$, but $f\,'$ is discontinuous at $x=0$.
(C) differentiable on $\Bbb R$, and $f\,'$ is continu... | HINT: If $x<0$, $f\,'(x)=9x^2+1$, so $$\lim_{x\to 0^-}f\,'(x)=\lim_{x\to 0^-}\left(9x^2+1\right)=1\;.$$
Is this the same as $\lim_{x\to 0^+}f\,'(x)$? Are both one-sided limits equal to $f\,'(0)$?
| {
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"timestamp": "2023-03-29T00:00:00",
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Showing diffeomorphism between $S^1 \subset \mathbb{R}^2$ and $\mathbb{RP}^1$ I am trying to construct a diffeomorphism between $S^1 = \{x^2 + y^2 = 1; x,y \in \mathbb{R}\}$ with subspace topology and $\mathbb{R P}^1 = \{[x,y]: x,y \in \mathbb{R}; x \vee y \not = 0 \}$ with quotient topology and I am a little stuck.
I... | The easiest explicit map I know is:
$$(\cos(\theta), \sin(\theta))\mapsto [\cos(\theta/2):\sin(\theta/2)]$$
Note that although $\cos(\theta/2)$ and $\sin(\theta/2)$ depend on $\theta$ and not just $\sin(\theta)$ and $\cos(\theta)$, the map is well-defined so long as we use the same value of $\theta$ when computing coor... | {
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"source": "stackexchange",
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Proving that if $f: \mathbb{C} \to \mathbb{C} $ is a continuous function with $f^2, f^3$ analytic, then $f$ is also analytic
Let $f: \mathbb{C} \to \mathbb{C}$ be a continuous function such that $f^2$ and $f^3$ are both analytic. Prove that $f$ is also analytic.
Some ideas: At $z_0$ where $f^2$ is not $0$ , then $f^... | First rule out the case $f^2(z)\equiv 0$ or $f^3(z)\equiv 0$ as both imply $f(z)\equiv 0$ and we are done.
Write $f^2(z)=(z-z_0)^ng(z)$, $f^3(z)=(z-z_0)^mh(z)$ with $n.m\in\mathbb N_0$, $g,h$ analytic and nonzero at $z_0$. Then
$$(z-z_0)^{3n}g^3(z)=f^ 6(z)=(z-z_0) ^ {2m} h^2 (z)$$
implies $3n=2m$ (and $g^3=h^2$), hen... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/298951",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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For each $n \ge 1$ compute $Z(S_n)$ Can someone please help me on how to compute $Z(S_n)$ for each $n \ge 1$?
Does this basically mean compute $Z(1), Z(2), \ldots$? Please hint me on how to compute this.
Thanks in advance.
| Hint: $S_n$ denotes the symmetric group over a set of $n$ elements. It's the group of all posible permutations, so you have to find $Z(S_1),Z(S_2),...$ so you have to find the permutations that commute with every other permutations. That is the definition of $Z(G)$:
$$Z(G)=\lbrace g\in G,ga=ag\;\;\forall a\in G\rbrace$... | {
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Binomial-Like Distribution with Changing Probability The Question
Assume we have $n$ multiple choice questions, the $k$-th question having $k+1$ answer choices. What is the probability that, guessing randomly, we get at least $r$ questions right?
If no general case is available, I am OK with the special case $r = \left... | Let $U_k$ be an indicator random variable, equal to 1 if the $k$-th question has been guessed correctly. Clearly $(U_1, U_2,\ldots,U_n)$ are independent Bernoulli random variables with $\mathbb{E}\left(U_k\right) = \frac{1}{k+1}$.
The total number of correct guesses equals:
$$
X = \sum_{k=1}^n U_k
$$
The moment gen... | {
"language": "en",
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Recursion for Finding Expectation (Somewhat Lengthy) Preface:
Ever since I read the brilliant answer by Mike Spivey I have been on a mission for re-solving all my probability questions with it when possible. I tried solving the Coupon Collector problem using Recursion which the community assisted on another question of... | The main idea is to use probability generating functions. (If you don't know what that means, this will be explained later on in the solution)
We solve the problem in general, so replace $10$ by any non-negative integer $a$.
Let $p_{k, a}(i)$ be the probability of getting $k$ rolls with face $i$ when a fair dice is con... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Combinatorical meaning of an identity involving factorials While solving (successfully!) problem 24 in projectEuler I was doodling around and discoverd the foloowing identity:
$$1+2\times2!+3\times3!+\dots N\times N!=\sum_{k=1}^{k=N} k\times k!=(N+1)!-1$$
While this is very easy to prove, I couldn't find a nice and sim... | The number of ways you can sort a set of consecutive numbers starting from $1$ and none of which is larger than $N$ and then paint one of them blue is $(N+1)!-1$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is the diagonal set a measurable rectangle? Let $\Sigma$ denotes the Borel $\sigma$-algebra of $\mathbb{R}$ and $\Delta=\{(x,y)\in\mathbb{R}^2: x=y\}$. I am trying to clarity the definitions of $\Sigma\times\Sigma$ (the sets which contains all measurable rectangles) and $\Sigma\otimes\Sigma$ (the $\sigma$-algebra gene... | Let $x\neq y$. If there is a measurable rectangle containing $(x,x)$ and $(y,y)$, there must be a set $A$ containing both $x$ and $y$ and a set $B$ doing the same such that $(x,x)$ and $(y,y)$ are in $A\times B$. But then $(x,y)\in A\times B$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Show that the ideal of all polynomials of degree at least 5 in $\mathbb Q[x]$ is not prime
Let $I$ be the subset of $\mathbb{Q}[x]$ that consists of all the
polynomials whose first five terms are 0.
I've proven that $I$ is an ideal (any polynomial multiplied by a polynomial in $I$ must be at least degree 5), but I'... | Hint $\ $ For any prime ideal $\rm\,P\!:\,\ x^n\in P\:\Rightarrow\:x\in P.\:$ Thus $\rm\ x^5 \in I\,$ but $\rm\ x\not\in I\ \Rightarrow\ I\,$ is not prime.
Equivalently, $\rm\, R\ mod\ P\,$ has a nilpotent ($\Rightarrow$ zero-divisor): $\rm\, x^5\equiv 0,\ x\not\equiv 0,\,$ so it is not a domain.
Remark $\ $ Generally... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Calculating $\lim_{x\to\frac{\pi}{2}}(\sin x)^{\tan x}$ Please help me calculate: $\lim_{x\to\frac{\pi}{2}}(\sin x)^{\tan x}$
| Take $y=(\sin x)^{\tan x}$
Taking log on both sides we have,
$\log y=\tan x\log(\sin x)=\frac{\log(\sin x)}{\cot x}$
Now as $x\to \pi/2$, $\log(\sin x)\to 0$ and $\cot x\to 0$
Now you can use L'Hospital's Rule.
$$\lim_{x\to \pi/2}\frac{\log(\sin x)}{\cot x}=\lim_{x\to \pi/2}\frac{\cos x}{\sin x(-\csc^2 x)}=\lim_{x\to \... | {
"language": "en",
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Let the matrix $A=[a_{ij}]_{n×n}$ be defined by $a_{ij}=\gcd(i,j )$. How prove that $A$ is invertible, and compute $\det(A)$? Let $A=[a_{ij}]_{n×n}$ be the matrix defined by letting $a_{ij}$ be the rational number such that $$a_{ij}=\gcd(i,j ).$$ How prove that $A$ is invertible, and compute $\det(A)$? thanks in advanc... | There is a general trick that applies to this case.
Assume a matrix $A=(a_{i,j})$ is such that there exists a function $\psi$ such that
$$
a_{i,j}=\sum_{k|i,k|j}\psi(k)
$$
for all $i,j$.
Then
$$
\det A=\psi(1)\psi(2)\cdots\psi(n).
$$
To see this, consider the matrix $B=(b_{i,j})$ such that $b_{i,j}=1$ if $i|j$ and $b_... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "16",
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eigenvalues of a matrix with zero $k^{th}$ power For a matrix $A$, where $A^k=0$, $k\ge1$, need prove that $trace(A)=0$; i.e sum of eigenvalues is zero. How do you approach this problem?
| I assume your matrix is an $n\times n$ matrix with, say, complex coefficients.
Since $A^k=0$, the spectrum of $A$ is $\{0\}$ (or the characteristic polynomial of $A$ is $X^n$).
Next we can find an invertible matrix $P$ such that $PAP^{-1}$ is upper-triangular with $0$'s on the diagonal.
So
$$
\mbox{trace}A=\mbox{trac... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Variance for a product-normal distribution I have two normally distributed random variables (zero mean), and I am interested in the distribution of their product; a normal product distribution.
It's a strange distribution involving a delta function.
What is the variance of this distribution - and is it finite?
I know ... | Hint: We need to know something about the joint distribution. The simplest assumption is that $X$ and $Y$ are independent. Let $W=XY$. We want $E(W^2)-(E(W))^2$. To calculate $E((XY)^2)$, use independence.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/299829",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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convolution square root of uniform distribution I need to find a probability distribution function $f(x)$ such that the convolution $f * f$ is the uniform distribution (between $x=0$ and $x=1$). I would like to generate pairs of numbers with independent identical distributions, so that their sum is uniformly distribute... | Assume that $X$ is a random variable with density $f$ and that $f\ast f=\mathbf 1_{[0,1]}$. Note that the function $t\mapsto\mathbb E(\mathrm e^{\mathrm itX})$ is smooth since $X$ is bounded (and in fact, $X$ is in $[0,\frac12]$ almost surely). Then, for every real number $t$,
$$
\mathbb E(\mathrm e^{\mathrm itX})^2=\f... | {
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"timestamp": "2023-03-29T00:00:00",
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A question about Elementary Row Operation: Add a Multiple of a Row to Another Row The task is that I have to prove the following statement, using Linear Algebra arguments:
Given a matrix A, then:
To perform an ERO (Elementary Row Operation) type 3 :
(c * R_i) + R_k --> R_k (i.e. replace a row k by adding c-multiple... | Note: If $c$ is some non-zero scalar, then
*
*adding $cR_i$ to $R_k$ and replacing the original $R_{k\text{ old}}$ by $(R_k + cR_i)$
is the same as
*
*subtracting $−c⋅R_i$ from $R_k$ and replacing the old $R_k$ by the result $R_k - (-cR_i)$.
Since...$R_k + cR_i = R_k - (-cR_i)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/299985",
"timestamp": "2023-03-29T00:00:00",
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Using sufficiency to prove and disprove completeness of a distribution Let $X_1, \dots ,X_n$ be a random sample of size $n$ from the continuous distribution with pdf $f_X(x\mid\theta) = \dfrac{2\theta^2}{x^3} I(x)_{(\theta;\infty)}$ where $\theta \in \Theta = (0, \infty)$.
(1) Show that $X_{(1)}$ is sufficient for $\t... | For Part (1), great!
For Part (2), I'm unsure about that one.
For Part (3), Note: that the original distribution $f_{X}(x|\beta) = \frac{2*\theta^{2}}{x^{3}}*I_{(\theta,\infty)}(x)$ resembles a famous distribution, but this famous distribution has 2 parameters, $\alpha$ and $\beta$, where the value of $\beta = 2$, and ... | {
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"timestamp": "2023-03-29T00:00:00",
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Helix's arc length I'm reading this.
The relevant definitions are that of parametrized curve which is at the beginning of page 1 and the definition of arclength of a curve, which is in the first half of page 6.
Also the author mentions the helix at the bottom of page 3.
On exercise $1.1.2.$ (page 8) I'm asked to find t... | There are a number of ways of approaching this problem. And yes, you are correct, without the domain specified there is a dilemma here. You can give an answer for one complete cycle of $2\pi$. Depending on the context you may find it more convenient to measure arc length as a function of $z$-axis distance along the h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/300096",
"timestamp": "2023-03-29T00:00:00",
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example of morphism of affine schemes Let $X={\rm Spec}~k[x,y,t]/<yt-x^2>$ and let $Y={\rm Spec}~ k[t]$.
Let $f:X \rightarrow Y$ be the morphism determined by $k[t] \rightarrow k[x,y,t]/<yt-x^2>$.
Is f surjective> If f is surjective, why??
| I'm assuming your map of rings comes from the natural inclusion: $i:k[t]\rightarrow[x,y,t]\rightarrow k[x,y,t]/<yt-x^2>=A$.
A prime of $k[T]$ is of the form $(F(t))$ where $F(t)$ is an irreducible polynomial over $k$. Show that the $I=F(t)A$ is not the whole ring $A$ (This amounts to showing that $yt-x^2$ and $F(t)$ do... | {
"language": "en",
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Does the series $\sum\limits_{n=1}^\infty \frac{1}{n\sqrt[n]{n}}$ converge?
Does the following series converge? $$\sum_{n=1}^\infty \frac{1}{n\sqrt[n]{n}}$$
As $$\frac{1}{n\sqrt[n]{n}}=\frac{1}{n^{1+\frac{1}{n}}},$$
I was thinking that you may consider this as a p-series with $p>1$. But I'm not sure if this is correc... | Limit comparison test:
$$\frac{\frac{1}{n\sqrt[n]n}}{\frac{1}{n}}=\frac{1}{\sqrt[n]n}\xrightarrow[n\to\infty]{}1$$
So that both
$$\sum_{n=1}^\infty\frac{1}{n\sqrt[n] n}\,\,\,\text{and}\,\,\,\sum_{n=1}^\infty\frac{1}{n}$$
converge or both diverge...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/300243",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 4,
"answer_id": 0
} |
Let lim $a_n=0$ and $s_N=\sum_{n=1}^{N}a_n$. Show that $\sum a_n$ converges when $\lim_{N\to\infty}s_Ns_{N+1}=p$ for a given $p>0$. Let lim $a_n=0$ and $s_N=\sum_{n=1}^{N}a_n$.
Show that $\sum a_n$ converges when $\lim_{N\to\infty}s_Ns_{N+1}=p$ for a given $p>0$.
I've no idea how to even start. Should I try to prove th... | Put $s_n:=\epsilon_n|s_n|$ with $\epsilon_n\in\{-1,1\}$. Then from
$$\epsilon_n\epsilon_{n+1}|s_n|\>|s_{n+1}|=s_n\>s_{n+1}=:p_n\to p>0\qquad(n\to\infty)$$
it follows that $\epsilon_n=\epsilon_{n+1}$ for $n>n_0$. Assume $\epsilon_n=1$ for all $n> n_0$, the case $\epsilon_n=-1$ being similar.
The equation
$$s_n(s_n+a_{n+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/300309",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Prove the determinant of this matrix We have an $n\times n$ square matrix $\left(a_{i,j}\right)_{1\leq i\leq n, \ 1\leq j\leq n}$ such that all elements on main diagonal are zero, whereas the other elements are defined as follows:
$$a_{i,j}=\begin{cases}
1,&\text{if } i+j \text{ belongs to the Fibonacci numbers,}\\
0,&... | This is just a partial answer, too long to fit in a comment, written in order to start collecting ideas.
We have:
$$\det A=\sum_{\sigma\in S_n}\operatorname{sign}(\sigma)\prod_{i=1}^n a_{i,\sigma(i)},$$
hence the contribute of every permutation in $S_n$ belongs to $\{-1,0,1\}$. In particular, the contribute of a permut... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/300379",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "29",
"answer_count": 1,
"answer_id": 0
} |
Proving that for any odd integer:$\left\lfloor \frac{n^2}{4} \right\rfloor = \frac{(n-1)(n+1)}{4}$ I'm trying to figure out how to prove that for any odd integer, the floor of:
$$\left\lfloor \frac{n^2}{4} \right\rfloor = \frac{(n-1)(n+1)}{4}$$
Any help is appreciated to construct this proof!
Thanks guys.
| Take $n=2k+1$ then,
$\lfloor(n^2/4)\rfloor=\lfloor k^2+k+1/4\rfloor=k^2+k$
$\frac{(n-1)(n+1)}{4}=(n^2-1)/4=k^2+k=\lfloor(n^2/4)\rfloor$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/300493",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
how to find cosinus betwen two vector? i have task in linear-algebra.
Condition:
we have triangle angles
A(-4,2);
B(-1,6);
C(8,-3);
How to find cosinus between BA and BC vectors?
please help :( what is solution for this task?
| The dot product gets you just what you want. The dot product of two vectors $\vec u \cdot \vec v = |\vec u||\vec v|\cos \theta$ where $\theta$ is the angle between the vectors. So $\cos \theta =\frac{\vec u \cdot \vec v}{|\vec u||\vec v|}$. The dot product is calculated by summing the products of the components $\v... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/300551",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Construction of Hadamard Matrices of Order $n!$ I'm trying to get a hand on Hadamard matrices of order $n!$, with $n>3$. Payley's construction says that there is a Hadamard matrix for $q+1$, with $q$ being a prime power.
Since
$$
n!-1 \bmod 4 = 3
$$
construction 1 has to be chosen:
If $q$ is congruent to $3 (\bmod 4)... | I don't think a general construction for Hadamard matrices of order $n!$ is known. The knowledge about general construction methods for Hadamard matrices is quite sparse, the basic ones (see also the Wikipedia article) are:
1) If $n$ is a multiple of $4$ such that $n-1$ is a prime power or $n/2 - 1$ is a prime power $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/300635",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
If a function is uniformly continuous in $(a,b)$ can I say that its image is bounded? If a function is uniformly continuous in $(a,b)$ can I say that its image is bounded?
($a$ and $b$ being finite numbers).
I tried proving and disproving it. Couldn't find an example for a non-bounded image.
Is there any basic proof o... | Hint: Prove first that a uniformly continuous function on an open interval can be extended to a continuous function on the closure of the interval.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/300745",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
} |
Show that open segment $(a,b)$, close segment $[a,b]$ have the same cardinality as $\mathbb{R}$ a) Show that any open segment $(a,b)$ with $a<b$ has the same cardinality as $\mathbb{R}$.
b) Show that any closed segment $[a,b]$ with $a<b$ has the same cardinality as $\mathbb{R}$.
Thoughts:
Since $a<b$, $a,b$ are two di... | Consider the function $f:(0,1)\to \mathbb{R}$ defined as,
$$f(x)=\frac{1}{x}+\frac{1}{1-x}$$
Prove that $f$ is a bijective function.
Now, by previous posts, $(0,1)$ and $[0,1]$ have the same cardinality.
Consider the function $g:[0,1]\to[a,b]$, defined as,
$$g(x)=({b-a})x+a$$
Prove that $g$ is bijective function to co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/300815",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 2
} |
Intuition for scale of the largest eigenvalue of symmetric Gaussian matrix Let $X$ be $n \times n$ matrix whose matrix elements are independent identically distributed normal variables with zero mean and variance of $\frac{1}{2}$. Then
$$
A = \frac{1}{2} \left(X + X^\top\right)
$$
is a random matrix from GOE ensem... | The scaling follows from the Wigner semicircle law. Proof of the Wigner semicircle law is outlined in section 2.5 of the review "Orthogonal polynomials ensembles in probability theory" by W. König, Probability Surveys, vol. 2 (2005), pp. 385-447.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/300894",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
} |
Solving Recurrence $T(n) = T(n − 3) + 1/2$; I have to solve the following recurrence.
$$\begin{gather}
T(n) = T(n − 3) + 1/2\\
T(0) = T(1) = T(2) = 1.
\end{gather}$$
I tried solving it using the forward iteration.
$$\begin{align}
T(3) &= 1 + 1/2\\
T(4) &= 1 + 1/2\\
T(5) &= 1 + 1/2\\
T(6) &= 1 + 1/2 + 1/2 = 2\\
T(7) ... | The generating function is $$g(x)=\sum_{n\ge 0}T(n)x^n = \frac{2-x^3}{2(1+x+x^2)(1-x)^2}$$, which has the partial fraction representation
$$g = \frac{2}{3(1-x)} + \frac{1}{6(1-x)^2}+\frac{x+1}{6(1+x+x^2)}$$. The first
term contributes $$\frac{2}{3}(1+x+x^2+x^3+\ldots)$$, equivalent to $T(n)=2/3$ the second term contrib... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/300934",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Another trigonometric equation Show that :
$$31+8\sqrt{15}=16(1+\cos 6^{\circ})(1+\cos 42^{\circ})(1+\cos 66^{\circ})(1-\cos 78^{\circ})$$
| I don't think this is how the problem came into being. But, I think this to be a legitimate way.
$$(1+\cos 6^{\circ})(1+\cos 42^{\circ})(1+\cos 66^{\circ})(1-\cos 78^{\circ})$$
$$=(1+\cos 6^{\circ})(1+\cos 66^{\circ})(1-\cos 78^{\circ})(1+\cos 42^{\circ})$$
$$=(1+\cos 6^{\circ}+\cos 66^{\circ}+\cos 6^{\circ}\cos 66^{\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/301012",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How can a set be bounded and countably infinite at the same time? There is a theorem in my textbook that states,
Let $E$ be a bounded measurable set of real numbers. Suppose there is a bounded countably infinite set of real numbers $\Lambda$ for which the collection of translates of $E$, $\{\lambda + E\}_{\lambda \in ... | Hint: Consider $\Bbb Q$ intersected with any bounded set, finite or infinite. Since $\Bbb Q$ is countable, the new set is at most countable, and clearly can be made infinite; for example, $[0,1]\cap\Bbb Q$ is bounded and countable.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/301080",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Continuous function with zero integral Let $f$ be a continuous function on $[a,b]$ ($a<b$), such that $\int_{a}^{b}{f(t)dt}=0$
Show that $\exists c\in[a,b], f(c)=0$.
| Let $m=\min\{f(x)|x\in[a,b]\}, m=\max\{f(x)|x\in[a,b]\}$(We can get the minimum and maximum because $f$ is continuous on a closed interval) . If $m,M$ have the same sign it can be shown that the integral cant be zero (for example if both are positive, the the integral will be positive). If $m,M$ have different signs ap... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/301143",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Proof of the simple approximation lemma a) For the proof of the simple approximation lemma, our textbook says,
Let (c,d) be an open, bounded interval that contains the image of E, f(E), and $c=y_0 < y_1 < ... < y_n = d$ be a partition of the closed bounded interval [c,d] such that $y_k - y_{k-1} < \epsilon$ for $1 \le... | The definition of a function $f: A \to B$ being measurable is that for any measurable set $E \subseteq B$, $f^{-1}(E)$ is measurable, so this follows by definition.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/301278",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
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