Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
$\operatorname{lcm}(n,m,p)\times \gcd(m,n) \times \gcd(n,p) \times \gcd(n,p)= nmp \times \gcd(n,m,p)$, solve for $n,m,p$? $\newcommand{\lcm}{\operatorname{lcm}}$
I saw this in the first Moscow Olympiad of Mathematics (1935), the equation was :
$$\lcm(n,m,p)\times \gcd(m,n) \times \gcd(n,p)^2 = nmp \times \gcd(n,m,p)$$
... | Using the standard trick: $$d=\gcd(n, m, p), u=\frac{\gcd(n, m)}{\gcd(n, m, p)}, v=\frac{\gcd(n, p)}{\gcd(n, m, p)}, w=\frac{\gcd(m, p)}{\gcd(n, m, p)}$$
we may write$\newcommand{\lcm}{\operatorname{lcm}}$
$$n=duvn_1, m=duwm_1, p=dvwp_1$$
where
$$\gcd(vn_1, wm_1)=\gcd(un_1, wp_1)=\gcd(um_1, vp_1)=1$$
This gives
$$\gcd... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/790526",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Equivalent definitions of vector field There are two definitions of a vector field on a smooth manifold $M$.
*
*A smooth map $V:M \rightarrow TM, \forall p \in M:V(p) \in T_p M$.
*A linear map $V:C^{\infty}(M) \rightarrow C^{\infty}(M), \forall f,g:V(fg)=fV(g)+gV(f)$
I can't undestand why they are equivalent. We mu... | This depends heavily on your definition of the tangent space $T_{p}M$, and thus the tangent bundle $TM$. There are several equivalent ways of defining it. Which book are you following?
If your definition of the tangent space $T_{p}M$ is a vector space of linear maps $V : C^{\infty}(p)\to\mathbb{R}$ that satisfy the Lei... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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"answer_id": 0
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simplify $\sqrt[3]{11+\sqrt{57}}$ I read in a book (A Synopsis of Elementary Results in Pure and Applied Mathematics) that the condition to simplify the expression $\sqrt[3]{a+\sqrt{b}}$ is that $a^2-b$ must be a perfect cube.
For example $\sqrt[3]{10+6\sqrt{3}}$ where $a^2-b
=(10)^2-(6 \sqrt{3})^2=100-108=-8$ and $\sq... | It is not a sufficient condition (I don't know if it's necessary). Not all expressions of the form $\sqrt[3]{a+\sqrt{b}}$, satisfying the condition that $a^2-b$ is a perfect cube, can be simplified.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/790738",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
How to evaluate $\cos(\frac{5\pi}{8})$? I'm sorry I don't know the way to input pie (3.14) don't have symbol on my pc
| Recall the identity
$$\cos 2\theta=2\cos^2 \theta-1.\tag{1}$$
Let $\theta=\frac{5\pi}{8}$. Then $\cos 2\theta=-\frac{1}{\sqrt{2}}$. To finish, note that $\cos(5\pi/8)$ is negative.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/790803",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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} |
How do I prove $\csc^4 x-\cot^4x=(1+\cos^2x)/\sin^2x$ How do I prove $\csc^4 x-\cot^4x=\dfrac{(1+\cos^2x)}{\sin^2x}$
Do you start from RHS or LHS? I get stuck after first few steps-
| The LHS:
$$\frac{1}{\sin^4 x}-\frac{\cos^4 x}{\sin^4 x}=\frac{1-\cos^4 x}{\sin^4 x}=\frac{(1-\cos^2 x)(1+\cos^2 x)}{\sin^4 x}=\frac{1+\cos^2 x}{\sin^2 x}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/790921",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Why are cosine and sine used to solve this differential equation? $$
\frac{d^2 u}{dt^2}+\lambda u =0
$$
Why are cosine and sine used to solve this differential equation of second order?
| Apart from the mathematical theory, and assuming $\lambda >0$ since otherwise no sine/cosine is involved, this is a rough argument: you are looking for a function whose second derivative is a negative multiple of the function itself. Since the second derivative of the sine is minus the sine and the second derivative of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/791016",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What is the radius of the circle?
Please help with this grade nine math problem. How does one calculate the radius if the two sides of the right angle triangle are 85cm. The sides of the triangle are tangent to the circle.
|
It's useful to realize that the "left" and "right" radia, as drawn in the above picture, will be parallel to the respective cathetae.
Then you get:
$$C=\sqrt{A^2+A^2}=\sqrt{2}A$$
The height of the triangle is then:
$$h=\sqrt{A^2-\left(\frac{C}{2}\right)^2}=\sqrt{A^2-\frac{A^2}{2}}=\frac{1}{\sqrt{2}}A$$
Define x-axis a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/791227",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
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Corollary of identity theorem without connectedness assumption The identity theorem has a corollary, which is often stated as "If $U$ is a connected domain, $f,g$ are analytic in $U$, and the set of points where $f$ and $g$ coincide has a limit point in $U$, then $f=g.$"
The proof runs by showing that the set $L$ of li... | Where have you seen it stated without connectedness?. For what i understand, A domain = an open connected set. "Specifically, if two holomorphic functions f and g on a domain D agree on a set S which has an accumulation point c in D then f = g on all of D.". You're still using connectedness but instead of f and g being... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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combinatorical acquaintanceship problem Group of people went hiking.
It's given that if we pick any four of them, than at least one knows everybody in that quad.
We have to prove, than in group everybody knows everybody, except at most 3 persons.
I tried to sketch a problem for the case when group size is 5, to get s... | Hint:
You cannot have four distinct people with "$A$ and $B$ do not know each other" and "$C$ and $D$ do not know each other", as it contradicts "if we pick any four of them, than at least one knows everybody in that quad".
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/791341",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$f\circ g=g\circ f$ + increasing $\Rightarrow$ common fixed point.
Let $f,g:\mathbb [a,b] \to \mathbb [a,b]$ be monotonically increasing functions
such that $f\circ g=g\circ f$
Prove that $f$ and $g$ have a common fixed point.
I found this problem in a problem set, it's quite similar to this Every increasing funct... | Let $A=\{x \in [a,b]/ x \leq f(x) \; \text{and} \; x \leq g(x) \}$
*
*$a\in A$
*let $u=\sup A$
*Let us prove that $f(u)$ and $g(u)$ are upper bounds for $A$
Indeed let $x\in A$.
Then $x\leq u$. Hence $f(x) \leq f(u)$, thus $x\leq f(x) \leq f(u)$ and finally $x\leq f(u)$
In the same way, $x\leq g(u)$
*
*Theref... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/791535",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Projection onto the column space of an orthogonal matrix The projection of a vector $v$ onto the column space of A is
$$A(A^T A)^{-1}A^T v$$
If the columns of $A$ are orthogonal, does the projection just become $A^Tv$? I think it should because geometrically you just want to take the dot product with each of the column... | No. If the columns of $A$ are orthonormal, then $A^T A=I$, the identity matrix, so you get the solution as $A A^T v$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/791657",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
The inequality about recurrence sequence Sequence $(x_n)$ is difined
$x_1=\frac {1}{100}, x_n=-{x_{n-1}}^2+2x_{n-1}, n\ge2$
Prove that
$$\sum_{n=1}^\infty [(x_{n+1}-x_n)^2+(x_{n+1}-x_n)(x_{n+2}-x_{n+1})]\lt \frac {1}{3} $$
I found relation $(1-x_n)=(1-x_{n-1})^2$
I don't know what to do next.
There is a real number whi... | A direct proof (note that I've shifted indices from starting at 1 to 0):
First, notice that $x_n\to1$ is the only possible limit. ($x=-x^2+2x \implies (x-1)^2=0$)
[edit]
The obvious mistake in my algebra was pointed out -- $x^2=x$ so $x=0$ or $1$. Recentering about $x=0$ doesn't change the recurrence, but recentering a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/791751",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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How and in what context are polynomials considered equal? There's two notions of equivalent polynomials floating around, one saying that $f = g$ iff they're equivalent as maps, and the other saying $f = g$ iff they're equal on each coefficient when written in standard form.
I'm interested in polynomials over a finite... | In abstract algebra polynomials over a ring $R$ are defined as formal sums
$$
\sum_{k=0}^N a_k X^k
$$
where $X$ is a formal variable and all $a_k\in R$. To make this precise, we can also model polynomials as sequences $(a_0, a_1, \dots)$ where all but finitely many $a_i$ are zero. Addition and multiplications is then g... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Richardson Extrapolation Matlab Code: Example and try out code included. currently I am studying Numerical Methods in Matlab and I need a Matlab code which would calculate Richardson Extrapolation using data given in a table, respectively for x and f(x).
For example: Use the table below to calculate f'(0) as accurately... | Your find method is returning empty matrices because it's looking for values of y equal to some condition.
For instance, find(y == xp) looks for values of y equal to xp and it returns the index. You haven't told us what xp is, but chances are, there aren't any values of y that equal xp.
Furthermore, find returns the in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/791949",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Finding the eigenvalue and eigenvector of a matrix Confirm by multiplication that x is an eigenvector of A, and find the corresponding eigenvalue.
Given:
\begin{align}
A = \begin{pmatrix} 1&2\\3&2\\\end{pmatrix}, &&
x = \begin{pmatrix} 1\\-1\\\end{pmatrix}
\end{align}
I know: $Ax = \lambda x$
My work:
I know $\lambda I... | The directions are confirm by multiplication. All you need do is compute $Ax$ for the given $A$ and $x$ then compare that result to the given $x$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/792012",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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How to finish this integration? I'm working with the integral below, but not sure how to finish it...
$$\int \frac{3x^3}{\sqrt[3]{x^4+1}}\,dx = \int \frac{3x^3}{\sqrt[3]{A}}\cdot \frac{dA}{4x^3} = \frac{3}{4} \int \frac{dA}{\sqrt[3]{A}} = \frac{3}{4}\cdot\quad???$$
where $A=x^4+1$ and so $dA=4x^3\,dx$
| $$\dfrac{1}{\sqrt[\large 3]{A}} = \dfrac 1{A^{1/3}} = A^{-1/3}$$
Now use the power rule.
$$ \frac{3}{4} \int A^{-1/3}\,dA = \frac 34 \dfrac {A^{2/3}}{\frac 23} + C = \dfrac 98 A^{2/3} + C$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/792086",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Integral $\int \operatorname{sech}^4 x \, dx$ How we can solve this?$\newcommand{\sech}{\operatorname{sech}}$
$$
\int \sech^4 x \, dx.
$$
I know we can solve the simple case
$$
\int \sech \, dx=\int\frac{dx}{\cosh x}=\int\frac{dx\cosh x}{\cosh ^2x}=\int\frac{d(\sinh x)}{1+\sinh^2 x}=\int \frac{du}{1+u^2}=\tan^{-1}\sin... | Note that
$$
\int \DeclareMathOperator{sech}{sech}{\sech}^4x\,dx=\int{\sech}^2{x}\cdot(1-\tanh^2x)\,dx
$$
Letting $u=\tanh x$ gives $du={\sech}^2x$ so
$$
\int{\sech}^4x\,dx=\int(1-u^2)\,du=u-\frac{u^3}{3}+C=\tanh x-\frac{1}{3}\tanh^3x+C
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/792160",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
If the sum of two independent random variables is in $L^2$, is it true that both of them are in $L^1$? Let $X$ and $Y$ be two independent random variables. If $\mathbb E(X+Y)^2 < \infty$, do we have $\mathbb E |X| < \infty$ and $\mathbb E |Y| < \infty$?
What I actually want is that $X$ and $Y$ are both in $L^2$, i.e., ... | If they have the finite mean x=$E X < \infty$ and $y= E Y <\infty$ then yes
$E(X+Y)^2 = E(X^2)+E(Y^2)+2 E(X)*E(Y) = E(X^2)+E(Y^2)+2xy < \infty$. If no, I am affraid one can found a pathological case when this is not true.
Let me update myself. I think the previous answer given by David Giraudo is correct. Let me ju... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Why is $\cos (90)=-0.4$ in WebGL? I'm a graphical artist who is completely out of my depth on this site.
However, I'm dabbling in WebGL (3D software for internet browsers) and trying to animate a bouncing ball.
Apparently we can use trigonometry to create nice smooth curves.
Unfortunately, I just cannot see why.
I ca... | Wrong unit.
You talk about deg, while the function obviously expects rad, which gives -0.4.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/792365",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "123",
"answer_count": 9,
"answer_id": 8
} |
Word for "openness"/"closedness" of an interval What word properly completes the phrase
the radius of convergence does not depend on the $\text{______}$ of the interval
to mean that it doesn't matter whether $(a, b)$, $[a, b)$, $(a, b]$, or $[a, b]$ is the correct answer?
*
*Openness and closedness don't really see... | "The radius of convergence does not depend on whether the interval is open, closed, or neither."
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/792485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Curse of Dimensionality ... as illustrated by Christopher Bishop I'm reading Christopher Bishop's book "Neural Networks for Pattern Recognition". I'm on pg 7 about curse of dimensionality.
Here is the relevant part:
For simplicity assume the dimensionality we are working with is 3. Now "divide the input variables $x_... | Let's be explicit and consider $M=3$. If $d=1$, then you divide a segment into three subsegments. For example you divide the interval $[0,1]$ into $[0,1/3)$, $(1/3,2/3]$, and $(2/3,1]$.
If $d=2$, then you divide a square into thirds horizontally and also vertically, so there are $9$ areas. If $d=3$, then you divide a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/792559",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Taylor series of a division-by-zero equation I need to calculate taylor series of $(\frac1{t^3}+\frac3{t^2})^{1/3} - \sqrt{(\frac1{t^2}-\frac2{t})}$ at $t = 0$
to calculate limit $(\frac1{t^3}+\frac3{t^2})^{1/3} - \sqrt{(\frac1{t^2}-\frac2{t})}$ as $t \rightarrow 0$
I got division-by-zero error where $t = 0$. however, ... | Remember that when $x$ is small compared to $1$, $(1+x)^n \simeq (1+n~x)$. So $$(1+3t)^{1/3} \simeq 1+t$$ $$(1-2t)^{1/2} \simeq 1-t$$ and then $$\frac{1}{t}(1+3t)^{1/3}-\frac{1}{t}(1-2t)^{1/2} \simeq \frac{1}{t} (1+t)-\frac{1}{t} (1-t)=2$$.
If you have needed to go further, you could have used the binomial expansion... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/792664",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Covolution (space) over compact Lie groups Let $G$ be a compact Lie group. Is there any way one can characterize the functions $\phi$ of the form $\phi=\psi\ast \psi^\ast$ in $C^\infty(G)$ where $\psi\in C^\infty(G)$? Here as usual $\psi^*(x)=\overline{\psi(x^{-1})}$.
Another (perhaps easier) question is whether the a... | The only thing that comes to my mind about this type of functions is as follows:
Given $f\in L^2(G)$, the function $f\ast \tilde{f}$, where $\tilde{f}(g)=\overline{f(g^{-1})}$, is a function of positive type associated with the left regular representation of $G$. For the involved terminology and definitions see Appendi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/792737",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Unbounded sequence with convergent subsequence I'm just wondering if anyone knows any nice sequences that are unbounded themselves, but have one or more convergent sub-sequences?
| There are plenty.
Take any convergent sequence, say $a_n \to a \in \mathbb R$. Then take any unbounded sequence, say $b_n \to \infty$. Then define $$ c_n = \begin{cases} a_n & \text{n even} \\ b_n & \text{n odd.} \end{cases}$$
Then $c_n$ is unbounded, but has a convergent sequence. Notice that you can generalize this: ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/792840",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Why are two statements about a polynomial equivalent? I am reading a claim that the following two statements are equivalent.
*
*One of the roots of a polynomial $v(t)$ is a $2^j$-th root of unity, for some $j$.
*The polynomial $v(t)$ is divisible either by $1-t$ or by $1+t^{2^{j-1}}$, for some $j$.
We know that t... | A $2^j$-th root of unity is a root of a polynomial $P$ if and only if the minimal polynomial of that root is a factor of $P$. The minimal polynomials of roots of unity are called cyclotomic polynomials, and it's easy to see that for $j=0$, this is $1-t$, and for $j > 0$, it is $1 + t^{2^{j-1}}$ :
$$\Phi_{2^j} = \prod_... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why does $ x^2+y^2=r^2 $ have uncountably many real solutions? What is exactly the reason the equation of a cirle of radius $ r $ and centered at the origin has uncountably many solutions in $\mathbb { R} $?
| The mapping $f\colon\mathbb{R}\to\mathbb{R}^2$ defined by
$$
f(t)=\left(r\frac{1-t^2}{1+t^2},r\frac{2t}{1+t^2}\right)
$$
is injective and its image is the circle with center at the origin and radius $r$, except for the point $(-r,0)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/793018",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
average number of rolls of a die between appearance of a side I saw this might have been duplicated in places here -- I think this might be a variation on the coupon collector problem -- but I wanted to be sure and understand how to do the calculation.
I have an n-sided die. I want to know what the average number of r... | Let n denote any face number other than k. At the outset or after a $k$ has turned up, we roll the die until a $k$ reappears. The possibilities are:
$$k, nk, nnk, nnnk, ...$$
If $p$ is the probability of $k$ and $q = (1-p)$ is the probability of $n$, then the expected waiting time is
$$E(N) = \sum_{j=1}^{\infty} jq^{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Finding determinant for matrix using upper triangle method Here is an example of a matrix, and I'm trying to evaluate its determinant:
$$
\begin{pmatrix}
1 & 3 & 2 & 1 \\
0 & 1 & 4 & -4 \\
2 & 5 & -2 & 9 \\
3 & 7 & 0 & 1 \\
\end{pmatrix}
$$
When applying first row operation i get:
$$
\begin{pmatrix}
1 & 3 & 2 & 1 \\
0 ... | You just multiplied a row with $\frac {1}{-2}$! This will change the value of determinant. What you can do is take $-2$ common from a row and write it outside.
Consider a $1\times 1$ matrix $A=[1]$.
$det(A)=1$
Apply $R_!\to2R_1$
$A=[2]$
$det(A)=2$
Can you see why you cannot do it?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/793217",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find all the singularities of $f(z)= \frac{1}{z^4+1}$ and the associated residue for each singularity I know that there are poles at
$$\Large{z=e^{\frac{i\pi}{4}}},$$
$$\Large{z=-e^{\frac{i\pi}{4}}},$$
$$\Large{z=e^{\frac{i3\pi}{4}}},\text{ and}$$
$$\Large{z=-e^{\frac{i3\pi}{4}}}$$
I am having trouble with the residues... | Like N3buchadnezzar just said the residues are given by
$$\mathrm{Res}\left(\frac{f(z)}{g(z)},z_k\right) = \frac{f(z_k)}{g'(z_k)}$$
In your case the algebra involved in the calculation may lead to many errors if you consider the residues as you listed them. I suggest you to write the singularities of $\frac{1}{z^4+1}$... | {
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Why does $1 \cdot 0=0$ not stand? A set $G$ together with an operation $*$ is called group when it satisfies the following properties:
*
*$a*(b*c)=(a*b)*c, \forall a,b,c \in G$
*$ \exists e \in G: e*a=a*e=a, \forall a \in G$
*$\forall a \in G \exists a' \in G: a'*a=e=a*a'$
$$$$
$$(\mathbb{Z}, \cdot ) \text{ is n... | There isn't any failure in terms in property $(2)$. $1$ is certainly the identity, and it does stand that $0\times 1 = 1\times 0 = 0$.
But, zero creates another problem:
Consider property $(3)$ asserting that for every element in a group, its inverses exists and is in the group, too. This is where things "go bad" for... | {
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Dividing into 2 teams In how many ways can $22$ people be divided into $ 2 $ cricket teams to play each other?
Actual answer : $\large \dfrac{1}{2} \times \dbinom{22}{11}$
My approach :
Each team consists of $11$ members. Number of ways to select a team of $11$ members = $ \dbinom{22}{11}$
Number of ways by which $22$... | This is because when you choose $ \large11 $ people out of $ \large 22 $ people, there is a complementary team formed on the other side, that is, the other $ \large 11 $ people also form a team. So we overcount by a factor of $ \large 2 $, that is, we count every time twice.
For example, let $ \large 1, 2, 3, 4 $ be t... | {
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Convergence $I=\int_0^\infty \frac{\sin x}{x^s}dx$ Hi I am trying to find out for what values of the real parameter does the integral
$$
I=\int_0^\infty \frac{\sin x}{x^s}dx
$$
(a) convergent and (b) absolutely convergent.
I know that the integral is convergent if $s=1$ since
$$
\int_0^\infty \frac{\sin x}{x}dx=\fra... | $$\varphi_1(\alpha) =\int_0^\infty \frac{\sin t}{t^\alpha}\,dt\tag{I}$$
case $\alpha\gt 0$
Near $t=0$, $\sin t\approx t.$ Which yields, $\frac{\sin t}{t^{\alpha}}\approx \frac{1}{t^{\alpha -1}}$ and the convergence of the integral in (I) holds nearby $t=0$ if and only if $\alpha<2 $.
Now let take into play the cas... | {
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"source": "stackexchange",
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Is infinity a real or complex quantity? Since I was interested in maths, I have a question. Is infinity a real or complex quantity? Or it isn't real or complex?
| The question is a bit meaningless. "The infinite" is a philosophical concept. There are a wide variety of very different mathematical objects that are used to represent "the infinite", and now that we're in the realm of mathematics and not philosophy, I can make the concrete mathematical claim that no, those objects ar... | {
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Product of CW complexes question I am having trouble understanding the product of CW complexes. I know how to actually do the computations and all, I just don't understand how exactly it works.
So here's my questions specifically: If $X,Y$ are CW-complexes then say $e,f$ are $p,q$ cells on $X,Y$ respectively, then we k... | You can think of $D^n$ as the homeomorphic cube $I^n$. This way, the product
$$\left(D^k\times D^l,\ \partial D^k× D^l\cup D^k×∂D^l\right)\\
\cong\left(I^k×I^l,\ ∂I^k×I^l\cup I^k×∂I^l\right)\\
=\left(I^{k+l},∂\left(I^k×I^l\right)\right)\\
\cong \left(D^{k+l},∂\left(D^k×D^l\right)\right)$$
The homeomorphism between $D^k... | {
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Sufficient conditions for a meromorphic function to be rational I know that rational functions are meromorphic, but under what conditions are meromorphic functions rational? I know that the automorphisms of the Riemann sphere are rational, but are there any more general conditions that ensure rationality?
| The given function should have a finite number of poles on the Riemann sphere with the counting done with multiplicity given by the order of the pole.
First consider the complex plane, i.e., the Riemann sphere without the point at infinity. Then by multiplying the meromorphic function, $f(z)$ with an entire function $... | {
"language": "en",
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Gift advice: present for high school graduate interested in math I am a PhD student in mathematics who recently found out that I will be attending my girlfriend's cousin's high school graduation party. I have never met the cousin, but hear that he is very interested in mathematics and is hoping to major in mathematics ... | I second Kaj Hansen's suggestion of "what is mathematics" and I'd suggest also "Gödel, Escher, Bach, en eternal golden braid", it deals with very interesting topics (Gödel's incompleteness theorem, formal systems and similar things) in a very accessible and entertaining way.
I read both of them 2 years ago, when I was ... | {
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Summation Notation Confusion I am unclear about what the following summation means given that $\lambda_i: \forall i \in \{1,2,\ldots n\}$:
$\mu_{4:4} = \sum\limits_{i=1}^{4} \lambda_i + \mathop{\sum\sum}_{1\leq i_1 < i_2 \leq 4}(\lambda_{i_1} + \lambda_{i_2}) + \mathop{\sum\sum\sum}_{1\leq i_1 < i_2 <i_3 \leq 4}(\lambd... |
I understand how this term expands
$\sum\limits_{i=1}^{4} \lambda_i = \lambda_1 + \lambda_2 + \lambda_3 + \lambda_4$.
But, I don't understand what how this term expands
$\mathop{\sum\sum}_{1\leq i_1 < i_2 \leq 4}(\lambda_{i_1} + \lambda_{i_2})$
The subscript is just another way of indicating the domain of the indices... | {
"language": "en",
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Find all ordered triples $(x,y,z)$ of prime numbers satisfying equation $x(x+y)=z+120$ This question was from my Math Challenge II Number Theory packet, and I don't get how to do it. I know you can distribute to get $x^2+xy=z+120$, and $x^2+xy-z=120$, but that's as far as I got. Can someone explain step by step?
| If $x = 2$, the left side is even - hence, $z $ must also be $2$.
If $x$ is an odd prime and $y$ is also odd, the left side is again even, implying that $z = 2$.
So the interesting case is when $x$ is an odd prime and $y = 2$; in this case, we have that
$$x(x + 2) = z + 120$$
Upon adding $1$ to both sides and factoring... | {
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Let $\sum^{\infty}_{n=1} \frac {z^{n+1}} {n(n+1)}$ be a power series. Show sum-function $g(z)$ is continuous on $|z|\le 1$. Let $\sum^{\infty}_{n=1} \frac {z^{n+1}} {n(n+1)}$ be a power series.
I've shown that radius of convergence is $R=1$.
I've a theorem saying that the sum-function $g(z)=\sum^{\infty}_{n=1} \frac {z... | This is only a partial answer taking into account comments and answers to comments.
I suppose that you noticed that $$f(z)=\sum^{\infty}_{n=1} \frac {z^{n+1}} {n(n+1)}$$ is the antiderivative of $$\sum^{\infty}_{n=1} \frac {z^{n}} {n}=-\log (1-z)$$ So, integration by parts leads to $$f(z)=z+(1-z) \log (1-z)$$
| {
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Product measure with a Dirac delta marginal Let $(S,\mathcal F)$ be a measurable space, and let $\nu \in\mathcal P(S,\mathcal F)$ be a probability measure on $(S,\mathcal F)$. Fix some $x\in S$ and consider Dirac measure $\delta_x$. Would like to prove
If $\mu \in \mathcal P(S×S,\mathcal F\otimes \mathcal F)$ and has ... | For any measurable set $B\subset S$, $\mu(S\times B)=\delta_x(B)=\mathbb{1}_B(x)$. In particular, $\mu(S\times\{x\})=1$, and if $x\notin B$,
$$\mu(A\times B)=0=\nu(A)\delta_x(B),\qquad A\in\mathcal{F}$$
for $\mu(A\times B)\leq\mu(A\times(S\setminus\{x\})=0$.
Suppose now that $x\in B$. Then, $\delta_x(B)=1$, and $\mu(A... | {
"language": "en",
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Use power series $\sum^{\infty}_{n=1} \frac {z^{n+1}} {n(n+1)}$ to show $\sum^{\infty}_{n=1} \frac {1} {n(n+1)} =1$. Consider $\sum^{\infty}_{n=1} \frac {z^{n+1}} {n(n+1)}$ (power series). I've found that the sum-function $g(z) := \sum^{\infty}_{n=1} \frac {z^{n+1}} {n(n+1)}$ is defined and continuous on $|z| \le 1$.
L... | Hint
When $x$ goes to $0$, $x\log(x)$ has a limit of $0$ and, so, when $x$ goes to $1$, $(1-x)\log(1-x)$ has also a limit of $0$.
| {
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What remainder does $34!$ leave when divided by $71$? What is the remainder of $34!$ when divided by $71$?
Is there an objective way of solving this?
I came across a solution which straight away starts by stating that
$69!$ mod $71$ equals $1$ and I lost it right there.
| From $$69!=1\mod 71\Rightarrow 34!36!=-1\mod 71$$ Multiplying both sides by $4$ and noting that $35\cdot 2=-1\mod 71,\ 36\cdot 2=1\mod 71$, we get $$(34!)^2=4\mod 71\Rightarrow x^2=4\mod 71$$ where $34!=x\mod 71$ So, $$71|(x-2)(x+2)\Rightarrow x+2=71, or \ x=2\Rightarrow x=69\ or\ 2$$ since $1\le x\le 70$.
| {
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$6^{(n+2)} + 7^{(2n+1)}$ is divisible by $43$ for $n \ge 1$ Use mathematical induction to prove that 6(n+2) + 7(2n+1) is divisible by 43 for n >= 1.
So start with n = 1:
6(1+2) + 7(2(1)+1) = 63 + 73 = 559 -> 559/43 = 13. So n=1 is divisible
Let P(k): 6(k+2)+7(2k+1) , where k>=1
Show that P(k+1): 6((k+1)+2) + 7(2(k+1)+1... | The sequence $A_n = 6^{n+2} + 7^{2n+1}$ satisfies a two term linear recurrence relation with integer coefficients. Specifically it satisfies $A_n = 55A_{n-1} - 294A_{n-2}$ but we don't actually care what the relation is. If $43$ divides $A_n$ for two consecutive $n$ then by induction it must divide every term after t... | {
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A question about optimal codes Recall that a code attaining any bound is called an optimal code. Is the dual code of an optimal code also an optimal code?
| It depends on the bound and on the code - A code is said to be optimal with respect to a particular bound.
For example, the dual of a linear MDS code is another linear MDS code, so the dual and the original linear code both meet the singleton bound (recall a MDS code is one which meets the singleton bound and thus is ... | {
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Prove that if $ u \cdot v = u \cdot w $ then $v = w$ I've tried putting it up as:
$$ [u_1 v_1 + \ldots + u_n v_n] = [u_1 w_1 + \ldots + u_n w_n] $$
But this doesn't make it immediately clear...I can't simply divide by $u_1 + \ldots + u_n$ as these ($u$, $v$ and $w$) are vectors...
Any hints?
| $$
u\cdot v=u\cdot w
$$
Others have shown how to show that $v=w$ if one assumes the above for all values of $u$.
To show that it's now true if one just assumes $u$, $v$, $w$ are some vectors, let's look at the circumstances in which it would fail. Recall that $u\cdot v = \|u\| \|v\|\cos\theta$ where $\theta$ is the an... | {
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Is there a way in matrix math notation to show the 'flip up-down', and 'flip left-right' of a matrix? Title says it all - is there an accepted mathematical way in matrix notation to show those operations on a matrix?
Thanks.
| $$\begin{pmatrix} 0 & 1 \\ 1 & 0\end{pmatrix} \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} 0 & 1 \\ 1 & 0\end{pmatrix} = \begin{pmatrix} d & c \\ b & a\end{pmatrix}.$$
In general, left-multiplying by the anti-diagonal identity matrix swaps all rows. Right-multiplying swaps columns.
| {
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Convergence of the series $\sum \frac{(-1)^{\sqrt{n}}}{n}.$ I'm looking for some help to show that:
$$\sum {(-1)^{\lfloor \sqrt{n}\rfloor}\over n} < \infty$$
| After clarification, it seems that the goal is to prove that the sequence $(S_n)$ converges, where, for every $n\geqslant1$,
$$
S_n=\sum_{k=1}^n\frac{(-1)^{\lfloor k\rfloor}}k.
$$
To do so, consider, for every $n\geqslant1$,
$$
T_n=\sum_{k=n^2}^{(n+1)^2-1}\frac{(-1)^{\lfloor k\rfloor}}k=(-1)^n\sum_{k=n^2}^{(n+1)^2-1}\f... | {
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Evaluating the following integral: $\int\frac1{x^3+1}\,\mathrm{d}x$ How to integrate
$$\int\frac1{x^3+1}~\mathrm{d}x$$
Is it possible to use Taylor expansion?
| If $x^3 + 1=0$ then $x^3=-1$ so $x=-1$, at least if $x$ is real.
If you plug $-1$ in for $x$ in a polynomial and get $0$, then $x-(-1)$ is a factor of that polynomial.
So you have $x^3+1=(x+1)(\cdots\cdots\cdots\cdots)$.
The second factor can be found by long division or other means. It is $x^2-x+1$.
Can that be facto... | {
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Suppose $A$, $B$, and $C$ are sets, and $A - B \subseteq C$. Then $A - C \subseteq B$. I know how to prove it by contradiction, but I am wondering if it's possible to prove it directly. I tried doing that, but so far no results. Is it not possible to prove it directly?
Thanks.
| We have
$$A-B=A\cap B^c\subset C\Rightarrow C^c\subset A^c\cup B$$
hence
$$A-C=A\cap C^c\subset A\cap (A^c\cup B)=A\cap B\subset B$$
| {
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Divergence test for $\sum_{n=1}^{\infty}\ln (1+\frac{1}{n})^n$. I am trying to prove that this is divergent
$$\sum_{n=1}^{\infty} \left(1+\dfrac{1}{n}\right)^n$$
by finding the limit of
$$\ln \left(1+\dfrac{1}{n}\right)^n$$
I know its $e$ and I am trying to arrive at that value by this
$$\ln y = n \ln(1 + \dfrac{1}{n... | But if you want to prove that diverges is not most easy:
$$\sum (1+1/n)\leq \sum (1+1/n)^n?$$
| {
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How to go about proving that $\cos^2 x$ is everywhere differentiable? My first line of reasoning was to try directly evaluating $$\lim\limits_{h \to 0}\frac{\cos^2 (x+h) - \cos^2 (x)}{h}$$ and showing such a limit existed for any x, but when $\cos^2(x)$ evaluates to zero (e.g. when $x = \frac{\pi}{2}$), then directly e... | HINT:
$$\cos^2B-\cos^2A=1-\sin^2B-(1-\sin^2A)$$
Using Prove $ \sin(A+B)\sin(A-B)=\sin^2A-\sin^2B $, this becomes $$\sin^2A-\sin^2B=\sin(A+B)\sin(A-B)$$
So, $$\cos^2(x+h)-\cos^2x=\sin(2x+h)\sin(-h)=-\sin(2x+h)\sin(h)$$
| {
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A matrix $M$ that commutes with any matrix is of the form $M=\alpha I$ I feel like this is probably a simple proof but I can't quite come up with it in an elegant way nor could I find it here.
Prove that if a matrix $M$ commutes with any matrix then $M$ is of the form $M=\alpha I$.
Proving the contrapositive seems like... | Here's somewhat of an overkill answer for what it is worth.
A normal matrix is a matrix that is unitarily similar to a diagonal matrix. Another characterization is that a matrix $M$ is normal iff $M^* M = M M^*$.
If $M$ commutes with all matrices then it is clear it is normal. From this we have $M = UDU^*$ for some un... | {
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Crossings in an Eulerian Trail Exercise 11.2 in Graph Theory by Harary says
Every plane eulerian graph contains an eulerian trial that never crosses itself.
What does it mean for a trail to not cross itself? The book does not give a formal definition of this notion.
| I don't know the "formal" definition, but informally it means just what you would think. If you regard the Eulerian trail as a curve in the plane, the curve does not cross itself, in the sense that the graphs of $y=0$ and $y=x$ cross at the origin, but the graphs of $y=0$ and $y=x^2$ touch without crossing.
For instanc... | {
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Conjugate to the Permutation How many elements in $S_{12}$ are conjugate to the permutation $$\sigma=(6,2,4,8)(3,5,1)(10,11,7)(9,12)?$$
How many elements commute with $\sigma$ in $S_{12}$?
I believe I use the equation $n!/|K_{\sigma}|$ for the second question, but I'm not sure. Is anyone aware of how to do these?
| Two permutations are conjugate if they have the same cycle structure (same number of cycles with same lengths).
Given a permutation with this cycle structure, you won't change it if you rotate each cycle as much as you want.
So, there are $4$ "rotated cycles" for the first one, $3$ for the next two cycles, and $2$ for ... | {
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How do I evaluate this definite integral which blows up at lower limit? I have an integral of the form
$$\int^{\infty}_{0}{\frac{2a^2-x^{2} }{a^{2}+x^{2}}e^{\frac{-x^{2}}{b^2}}xdx}.$$
On substitution of $x^2=t$ and simplifying, I get integral of the form $$\int^{\infty}_{0}{\frac{e^{-t}}{t}dt}$$ which blows up as $ t ... | That integral is a well known special function, the Exponential Integral . You didn't do the substitution right, though.
If $b$ is $1$, the integral is $\int_0^\infty \frac{a^2-x^2}{a^2+x^2} e^{-x^2} x dx = \int_0^\infty \frac{a^2 - t}{a^2+t} e^{-t} dt$. Then, you $u$-substitute $u=a^2+t$ and shift the bounds from $u=a... | {
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Possibility of publishing First little background. I have master degree in mathematics. Then I decided to continue to study PhD level. After some years I cancel study (reason was in some things in my life). Now I am returning back to mathematics. I have job, but I do mathematics in my free time.
Is it possible to publi... | Of course you can. But since you want to publish something maybe it would be better to consider re-entering a PhD program in order to spend more time doing math. There you will have more opportunities to do research and publish your work.
| {
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} |
Differentiability and Lipschitz Continuity I've seen some questions on this site that are similar to the following, but not precisely the same.
Let $f:[a,b]\to\mathbb{R}$ be a differentiable function and assume $f'$ is continuous in $[a,b]$. Prove that $f$ is Lipschitz continuous. What is the best possible Lipschitz... | More or less... More directly, $|f(x_2)-f(x_1)| = |f'(c)| |x_2-x_1| \leq \max_{c \in [a,b]} |f'(c)| |x_2-x_1|$. Hence $A=\|f'\|_\infty = \max_{c \in [a,b]} |f'(c)|$ is one possible Lipschitz constant.
On the other hand, if $f(x)=mx+q$, then $\|f'\|_\infty=m$, and clearly $m$ is also the best Lipschitz constant.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why the geodesic curvature is invariant under isometric transformations? As I know the geodesic curvature
$$
\kappa_g = \sqrt{\det~g} \begin{vmatrix} \frac{du^1}{ds} & \frac{d^2u^1}{ds^2} + \Gamma^1_{\alpha\beta} \frac{du^\alpha}{ds} \frac{du^\beta}{ds} \\ \frac{du^2}{ds} & \frac{d^2u^2}{ds^2} + \Gamma^2_{\alpha\bet... | $ \kappa_g$ depends purely on the coefficients of the first fundamental form ( of surface theory FFF) and their derivatives, second fundamental form SFF coefficients are not involved.
It is invariant in isometric mappings ( bending transformations) like lengths,angles, $K$ Gauss curvature , integral curvature etc. Lio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/796042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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order of groups Question: Suppose $\operatorname{ord}(g)=20$. Find elements $h,k\in G$ such that $\operatorname{ord}(h)=4$, $\operatorname{ord}(k)=5$, and $hk^{-1}=k^{-1}h=g$.
I can't seem to find anything in my notes on how to complete this question. Can someone help hint how to find the solution to this question plea... | Hint: What are the orders of $g^5$ and $g^4$?
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Interesting "real life" applications of serious theorems As student in mathematics, one sometimes encounters exercises which ask you to solve a rather funny "real life problem", e.g. I recall an exercise on the Krein-Milman theorem which was something like:
"You have a great circular pizza with $n$ toppings. Show that... | The $n=2$ case of the Borsuk-Ulam theorem can be visualized by saying there exists some pair of antipodal points on the Earth with equal temperatures and barometric pressures. Of course, this is assuming that temperature and pressure vary continuously.
Ramsey's theorem says that, if given a sufficiently large complete... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/796236",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "64",
"answer_count": 17,
"answer_id": 4
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Expectation of Continuous variable. Given the probability density function
$$
f(x) =
\begin{cases}
\frac{cx}{3}, & 0 \leq x < 3, \\
c, & 3 \leq x \leq 4, \\
0 & \text{ otherwise}
\end{cases}
$$
I have found $c$ to be $0.4$ and $E(X)$ to be $2.6$. But I'm being asked to find $E(3X - 5)$ and I'm unsure of what to do.
| $$\mathbf E(3\mathbf X-5)=3\mathbf E(\mathbf X)-5=3(2.4)-5=2.2$$
| {
"language": "en",
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"source": "stackexchange",
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Identically distributed and same characteristic function If $X,Y$ are identically distributed random variables, then I know that their characteristic functions $\phi_X$ and $\phi_Y$ are the same. Does the converse also hold?
| Yes, it is true as a consequence of the inversion formula
$$\mu(a,b) +\frac 12\mu(\{a,b\}) = \frac 1{2\pi}\lim_{ t\to +\infty}\int_{-T}^T\frac{e^{ita} -e^{itb} }{it}\varphi_\mu (t)\mathrm dt,$$
valid for $a\lt b$.
If $\mu$ and $\nu$ have the same characteristic function, then $\mu([a,b])=\nu([a,b])$ for each $a\lt b$. ... | {
"language": "en",
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"source": "stackexchange",
"question_score": "2",
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How to prove ${\bf u}\cdot{\bf v}=|{\bf u}|\cdot|{\bf v}|\cos\theta$, if $\theta$ is the angle between $|{\bf u}|$ and $|{\bf v}|$ This is a snippet from my book.
How did they get from $|{\bf u}|^2={\bf u}\cdot{\bf v}=|{\bf u}||{\bf v}|\frac{|{\bf u}|}{|{\bf v}|}$?
| I dont understand the comments from your book, but I assume the question is how to prove the equivalence of the two definitions of dot product, $\mathbf{u}\cdot \mathbf{v}=u_1v_1+u_2v_2+u_3 v_3$, and the equation $\mathbf{u}\cdot \mathbf{v}=|\mathbf{u}||\mathbf{v}| \cos \theta$.
This is a consequence of the cosine law,... | {
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"timestamp": "2023-03-29T00:00:00",
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Direct proof that $n!$ divides $(n+1)(n+2)\cdots(2n)$ I've recently run across a false direct proof that $n!$ divides $(n+1)(n+2)\cdots (2n)$ here on math.stackexchange. The proof is here prove that $\frac{(2n)!}{(n!)^2}$ is even if $n$ is a positive integer (it is the one by user pedja, which got 11 upvotes). The proo... | Here is a more direct number theoretical type proof that if $a \ge 0$ and $a_1+a_2+\cdots+a_r = n$ that $\frac{n!}{a_1!a_2! \cdots a_r!}$ is an integer.
This reduces to proving $\sum \left \lfloor \frac{n}{p_i} \right \rfloor \ge \sum \left \lfloor \frac{a_1}{p_i} \right \rfloor + \sum \left \lfloor \frac{a_1}{p_i}... | {
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"source": "stackexchange",
"question_score": "10",
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$\mathbb{Q}(\sqrt[3]{2}, \zeta_{9})$ Galois group How do I calculate the degree of $\mathbb{Q}(\sqrt[3]{2}, \zeta_{9})$ over $\mathbb{Q}$. Should it be 18, as $[\mathbb{Q}(\sqrt[3]{2}):\mathbb{Q}] = 3$, and $[\mathbb{Q}(\zeta_{9}):\mathbb{Q}] = 6$?
However $(\sqrt[3]{2})^{3} \in \mathbb{Q}(\zeta_{9})$, how this affect... | $\newcommand{\Q}{\mathbb{Q}}\newcommand{\Size}[1]{\lvert #1 \rvert}$$\sqrt[2]{2}$ has minimal polynomial
$f = x^{3}-2$ over $\Q$. You have to show that $f$ is also the minimal
polynomial over $F = \Q(\zeta_{9})$, that is, that $f$ is irreducible in
$F[x]$, and since $f$ has degree $3$, it is enough to show
that $f$ ha... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How find this minimum of the value $f(1)+f(2)+\cdots+f(100)$ Give the positive integer set $A=\{1,2,3,\cdots,100\}$, and define function
$f:A\to A$ and
(1):such for any $1\le i\le 99$,have
$$|f(i)-f(i+1)|\le 1$$
(2): for any $1\le i\le 100$,have $$f(f(i))=100$$
find the minium of the value
$$f(1)+f(2)+f(3)+f(4)+\cdots... | Claim: to achieve the minimum, f(n) is a non decreasing function. Suppose not, take the natural construction $f^*(n)$ where we smooth out the decreasing part, show that it satisfies the conditions and has a smaller sum.
Claim: Suppose that The image of $f(n)$ consists of $k$ elements. Then, because we have a non decre... | {
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"source": "stackexchange",
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"answer_id": 2
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Inverse Laplace Through Complex Roots I have been asked to apply inverse laplace to this:
$$ \frac{(4s+5)}{s^2 + 5s +18.5} $$
What I have done is;
I found the roots of denominator which are : $$ (-5-7i)/2 $$ and $$ (-5+7i)/2 $$
Then I factorized the denominator as :
$$ \frac{(4s+5)}{(s+\frac{(5+7i)}{2})(s + \frac{(5-7i... | I checked carefully your calculations and they are perfectly correct ! Congratulations. May be your professor would prefer $\frac{35}{49}$ to be replaced by $\frac{5}{7}$ !!
You are totally correct with the fact that sines and cosines would be obtained using Euler's identity. So, factor $e^{-\frac {5}{2}t}$ and give $$... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Do prime numbers satisfy this? Is this true that $n\log\left(\frac{p_n}{p_{n+1}}\right)$ is bounded, where $p_n$ is the $n$-th prime number?
| Seems unbounded:
Let $g_n = p_{n+1} - p_n$ be the prime gap, then Westzynthius's result (see link below) states that $\lim\sup \left[ g_n/(\log p_n) \right] = \infty$, hence
$$\lim \sup n \log(p_{n+1}/p_n) = \lim \sup n \log (1 + g_n/p_n) = \lim \sup n g_n/ p
_n = \lim \sup g_n/\log n = \infty$$
http://en.wikipedia... | {
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"source": "stackexchange",
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From any list of $131$ positive integers with prime factor at most $41$, $4$ can always be chosen such that their product is a perfect square Author's note:I don't want the whole answer,but a guide as to how I should think about this problem.
BdMO 2010
In a set of $131$ natural numbers, no number has a prime factor gr... | HINT - though I haven't followed through a solution ...
Finding four numbers all at once could be hard. Sometimes divide and conquer goes along with pigeonhole - using pigeonhole to find (disjoint) pairs which give the right parity on some fixed subset of the prime factors, and then using it again to find two pairs whi... | {
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Question about calculating at Uniform distribution A train come to the station $X$ minuets after 9:00, $X\sim U(0,30)$.
The train stay at the station for 5 minutes and then leave.
A person reaches to the station at 9:20.
Addition:
There was no train when the person came to the station
What is the probability that he di... | Hint: figure out for which values of $X$ will the condition (not missing the train) work.
For example, if the train waits 30 minutes, then the probability (not missing)
is 1.
If it's hard to think in continuous terms, imagine that the train comes at an integer time, $0 \le i \le 30$. What happens then?
| {
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"timestamp": "2023-03-29T00:00:00",
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Nuking the Mosquito — ridiculously complicated ways to achieve very simple results Here is a toned down example of what I'm looking for:
Integration by solving for the unknown integral of $f(x)=x$:
$$\int x \, dx=x^2-\int x \, dx$$
$$2\int x \, dx=x^2$$
$$\int x \, dx=\frac{x^2}{2}$$
Can anyone think of any more exampl... | Here is a major number-theoretical nuking. By a result of Gronwall (1913) the Generalized Riemann Hypothesis (GRH) implies that the only quadratic number fields $\,K$ whose integers have unique factorization are $\,\Bbb Q[\sqrt {-d}],\,$ for $\,d\in \{1,2,3,7,11,19,43,67,163\}.\,$ Therefore, if $\,K$ is not in this lis... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/798215",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "66",
"answer_count": 21,
"answer_id": 3
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Second derivative of $\arctan(x^2)$ Given that $y=\arctan(x^2)$ find $\ \dfrac{d^2y}{dx^2}$.
I got
$$\frac{dy}{dx}=\frac{2x}{1+x^4}.$$
Using low d high minus high d low over low squared, I got
$$\frac{d^2y}{dx^2}=\frac{(1+x)^4 \cdot 2 - 2x \cdot 4(1+x)^3}{(1+x^4)^2}.$$
I tried to simplify this but didn't get the answe... | Alternatively,
$ \large \tan (y) = x^2 \Rightarrow \sec^2 (y) \cdot \frac { \mathrm{d}y}{\mathrm{d}x} = 2x $
$ \large \Rightarrow \frac { \mathrm{d}y}{\mathrm{d}x} = \frac {2x}{\sec^2 (y)} = \frac {2x}{\tan^2 (y) + 1} = \frac {2x}{x^4 + 1} $
$ \large \Rightarrow \frac { \mathrm{d^2}y}{\mathrm{d}x^2} = \frac {2(x^4+ 1... | {
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"url": "https://math.stackexchange.com/questions/798277",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to prove such an elementary inequality The inequality is the following:
$a^ \theta b^ {1-\theta}$ $\leq$ $[\theta ^ \theta (1-\theta)^ {1-\theta}]^{1/p}(a^p+b^p)^{1/p}$, where $\theta \in [0,1]$, $a,b$ are nonnegative.
This inequality is used to give a sharper constant in the proof of an embedding theorem in Sobole... | By the AM/GM inequality,
$$ \left(\frac a\theta\right)^\theta \left(\frac b{1-\theta}\right)^{1-\theta}
\le \theta\left(\frac a\theta\right) + (1-\theta)\left(\frac b{1-\theta}\right)
= a+b
$$
Now replace $a$ and $b$ with $a^p$ and $b^p$.
| {
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"url": "https://math.stackexchange.com/questions/798464",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Please, help with this integration problem Consider the region bounded by the curves $y=e^x$, $y=e^{-x}$, and $x=1$. Use the method of cylindrical shells to find the volume of the solid obtained by rotating this region about the y-axis.
I drew the corresponding graph. I'm confused by the fact that the area is rotating ... | Rotating about the vertical line $x=1$ is in principal no different than rotating about the $y$-axis, which after all is just the vertical line $x=0$. The radius of a shell when rotating about the $y$-axis is the distance of $x$ from $0$, which is $r=|x-0|=|x|$; this further simplifies to $|x|=x$ if $0\leq x$. The only... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 0
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Definition of Cyclic subgroup
The above is a theorem from my book. What I don't understand is the second sentence when it says $b$ generates $H$ with $n/d$ elements. I thought that since $b = a^s$ generates $H$, it would have $s$ elements, meaning $H = \{e, a, a^2, \dots, a^s \}$? I've found some counterexamples to ... | Another proof
Let $k=o\langle b\rangle=o\langle a^s\rangle$ be the order of H
and according to the definition,
$$a^n=e$$
and
$$b^k=(a^s)^k=a^{ks}=e$$
so that
$$n\mid ks$$
where k is the minimum possible value
$$ks\equiv 0 \pmod n$$
and finally
$$k=\frac{n}{\text{gcd}(n,s)}=n/d$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Tell whether $\dfrac{10^{91}-1}{9}$ is prime or not? I really have no idea how to start. The only theorem considering prime numbers I know of is Fermat's little theorem and maybe its related with binomial theorem.
Any help will be appreciated.
| Just think through the actual number.
$10^{91}$ is a $1$ with $91$ $0$'s after it.
$10^{91}-1$ is therefore $91$ $9$'s in a row.
$\frac{10^{91}-1}{9}$ is therefore $91$ $1$'s in a row.
Due to the form of this number, $x$ $1$'s in a row will divide it, where $x$ is a divisor of $91$.
For example $1111111$ is a divisor, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/798778",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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$\text{Im}(z)$ in equation I'm having trouble with this equation:
$$\text{Im}(-z+i) = (z+i)^2$$
After a bit of algebra i've gotten:
$$1-\text{Im}(z) = z^2 + 2iz - 1$$
But i have no clue where to go from here, how do i get rid of the "$\text{Im}$"?
| Hint
Write $z=a+i~b$ in which $a$ and $b$ are real numbers. So $$Im(-z+i)=Im(-a+i(1-b))=1-b$$
Since John's answer came while I was typing, just continue the way he suggests (this is what I was about to write).
Continuation of my answer
The right hand side is $$(z+i)^2=(a+i(b+1))^2=a^2-(b+1)^2+2a(b+1)i$$ So the equatio... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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b such that Ax = b has no solution having found column space $A:=\begin{bmatrix}
2 & 6 & 0 \\
3 & 1 & 3 \\
1 & 0 & 0 \\
4 & 8 & 1
\end{bmatrix}$
I've found the basis for the column space by doing row reduction (i.e. basis is just the columns vectors of A in this case), and the null space only has the trivial s... | Denote $A\in\mathbb{F}^{m\times n}$ by
$$
A=\begin{bmatrix}a_{11} & a_{12} & . & . & . & & & a_{1n}\\
a_{21} & & & & & & & a_{2n}\\
. & & . & & & & & .\\
. & & & . & & & & .\\
. & & & & . & & & .\\
\\
\\
a_{m1} & a_{m2} & . & . & . & & & a_{mn}
\end{bmatrix}
$$
and
$$
x=\begin{bmatrix}x_{1}\\... | {
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"timestamp": "2023-03-29T00:00:00",
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Limits in cofinite topology/why is the limit of x_n = n equal to 1 in the cofinite topology. Just reading about topological spaces for my exam, and I was wondering if anybody could explain exactly how limits work in the cofinite topology. So I am aware of the topological definition of a limit:
$ Let~(X, \tau)$ be a top... | Take any open neighbourhood $U$ of $1$. Since $\Bbb R\setminus U$ must be finite, $U$ contains all but a finite number of terms of the sequence $x_n$. Therefore $x_n\to 1$. You can see $1$ is not special at all.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Colloquialisms in Math Terminology What are some of your favorite colloquial sounding names for mathematical objects, proofs, and so on? For example, manifolds are often described using an atlas and a neighborhood describes a small set surrounding a point.
| I can't believe no one has yet mentioned the following. These were quite common, at least at one time (1970s).
abstract nonsense -- refers to (Eilenberg/MacLane) category theory
invariant under change of notation -- refers to differential geometry, especially when tensors are involved
proof by diagram chasing -- refers... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Probability of drawing 6 specific letters from a set of 144 with 21 draws I know similar questions have been asked before, but I am struggling to find a method to calculate the probability for the situation below:
I have a set of 144 scrabble tiles with the following letter distribution.
2 x , K, Q, X, Z
3 x B, C, F,... | There are a total of ${144\choose 21}=8943919960348862933558400$ ways to choose 21 tiles from the 144 available.
Let $I=\{12,6,8,18,13,9\}$ be the counts of tiles corresponding
to the letters I,S,N,E,A,T. Then the number of bad samples, i.e, missing one
of the required letters, of size 21 can be worked out using the... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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How to calculate the square root of a number? By searching I found few methods but all of them involve guessing which is not what I want.
I need to know how to calculate the square root using a formula or something. In other words how does the calculator calculate the square root?
| The easiest way to find $\sqrt[n]{a}$ for integer $n$ and $a>0$ efficiently is to use the Newton-Raphson approximation to invert the function $f : x \mapsto x^n - a$. But one must be careful with choosing the right starting point, so that the iteration will converge quadratically. Quadratic convergence means that at ea... | {
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Isn't this wrong? This worksheet
This question:
$$w^2 - w \leq 0$$
This answer:
$$(-\infty, -1] \cup [0, 1]$$
Isn't this wrong ? At $w = -2$, it becomes: $(-2)^2 - (-2)$, which is $4 + 2$, which is $\geq 0$. But might be that I must be wrong somewhere. Please correct me. Thanks.
| $w^2-w\le 0$
$w(w-1)\le 0$
$0\le w\le 1$
The answer given is wrong.
| {
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Algebraic solution for the intersection point(s) of two parabolas I recently ran through an algebraic solution for the intersection point(s) of two parabolas $ax^2 + bx + c$ and $dx^2 + ex + f$ so that I could write a program that solved for them. The math goes like this:
$$
ax^2 - dx^2 + bx - ex + c - f = 0 \\
x^2(a -... | You lost a factor $4$ somewhere. You can simply rewrite your problem as
$$(a-d)x^2+(b-e)x+(c-f)=0$$
and use the standard formula for a quadratic equation, i.e.
$$x=-\frac{b-e}{2(a-d)}\pm\sqrt{\frac{(b-e)^2}{4(a-d)^2}-\frac{c-f}{a-d}}$$
Before evaluating this equation, you need to check if $a-d=0$, in which case
$$x=\fr... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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} |
Fourier transform of 1/cosh How do you take the Fourier transform of
$$
f(x) = \frac{1}{\cosh x}
$$
This is for a complex class so I tried expanding the denominator and calculating a residue by using the rectangular contour that goes from $-\infty$ to $\infty$ along the real axis and $i \pi +\infty$ to $i \pi - \infty... | To calculate the residue, we use the formula
\begin{equation*}
\text{Res}_{z_0}f=\lim_{z\rightarrow z_0}(z-z_0)f(z)
\end{equation*}
Then we replace $z_0$ by $i\pi/2$
\begin{equation*}
\begin{split}
(z-z_0)f(z)&=e^{-2\pi iz\xi}\frac{2(z-z_0)}{e^{\pi z}+e^{-\pi z}} \\ &=2e^{-2\pi iz\xi}e^{\pi z}\frac{2(z-z_0)}{e^{２\pi z}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/799616",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 5,
"answer_id": 3
} |
Definite Trig Integrals: Changing Limits of Integration $$\int_0^{\pi/4} \sec^4 \theta \tan^4 \theta\; d\theta$$
I used the substitution: let $u = \tan \theta$ ... then $du = \sec^2 \theta \; d\theta$.
I know that now I have to change the limits of integration, but am stuck as to how I should proceed.
Should I sub the ... | $\sec^4\theta = (1 + \tan^2\theta)\cdot \sec^2\theta$, then substitute $u = \tan\theta$ to get:
$$I = \displaystyle \int_{0}^1 (u^6 + u^4) du = \left.\dfrac{u^7}{7} + \dfrac{u^5}{5}\right|_{0}^1 = \dfrac{12}{35}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/799711",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Describe the Galois Group of a field extension I'm struggling to understand the basics of Galois theory. One of the things I don't understand is how to actually derive automorphisms of a field extension. Let's say you had a simple problem:
$x^2-3$ over $\mathbb{Q}$ has splitting field $\mathbb{Q}(\sqrt{3}$) correct?
O... | It is important to remember that these automorphisms fix the base field. In particular, they fix the polynomial $x^2 - 3$, so they must send roots of that polynomial to other roots.
They are also field maps and, since they fix the base field, they are in particular linear over the base field - Q, in your example . Sin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/799797",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Conditions under which a group homomorphisms between products of groups arises as a product of homomorphisms Let $\phi: G\times H\to G\times H$ be a group homomorphism. Under what conditions can we write $\phi=(f,g)$, where $f:G\to G$ and $g:H\to H$, where $f$, $g$ are group homomorphisms?
| $\phi : G \times H \to G \times H$ is determined uniquely by its values on the natural embeddings of $G$ and $H$, since $\phi(g,h) = \phi(g,1) \phi(1,h)$.
We restrict $f$ to these natural embeddings to get functions $f_G : G \to G \times H$ and $f_H : H \to G \times H$. Then the function $(f_G, f_H) : G\times H \to G \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/799897",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Real number comparisons: must a number be less than or equal to or greater than another number? I've been reading Knuth's Surreal Numbers recently and came up with this question about real numbers.
Is is true that among all three relationships (=, >, <), a real number must be of one, and only one relationship with anot... | Yes - this is called the trichotomy property (
http://en.wikipedia.org/wiki/Trichotomy_%28mathematics%29 ) . It can be easier to see depending on how you define the real numbers. For instance: http://en.wikipedia.org/wiki/Dedekind_cut (see ordering of cuts part way down the page)
http://en.wikipedia.org/wiki/Trichotom... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/799970",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Proof of power series uniform convergence on compact set I proved:
If a power series converges pointwise on a compact set then it converges uniformly.
Please could somebody check my proof?
My idea is to use Abel's theorem:
Let $g(x) = \sum_{n=0}^\infty a_n x^n$ be a power series that converges at the point $x=R > 0$.... | Your proof is correct. Presentation may be improved by preceding it with a lemma: if a series converges uniformly on each of the sets $E_1,\dots,E_m$, then it converges uniformly on $\bigcup_{i=1}^m E_i$. (That is, uniformity of convergence is preserved under finite unions.)
Then you have $K\subseteq E_1\cup E_2$ where... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/800195",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Show that a map is not an automorphism in an infinite field How should I show that a map $f(x) = x^{-1}$ for $x \neq 0$ and $f(0) = 0$ is not an automorphism for an infinite field?
Thanks for any hints.
Kuba
| An elemantary way;
Assume that $\phi$ is an automorphism of $F$ as you defined.Notice that if $\phi(x)=x$ then $x=1$ or $x=-1$ or $x=0$.
Now let $r$ be any nonzero elements of $F$ then set $x=r+\dfrac 1r$.
So we have, $\phi(x)=\phi(r+\dfrac 1r)=\phi(r)+\phi(\dfrac 1r)=\dfrac 1r+r=x$ which means that
$r$ must be a root ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/800245",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
matrix representation of linear transformation For a set $N$ let $id_N:N \rightarrow N$ be the identical transformation. Be $V:=\mathbb{R}[t]_{\le d}$. Determine the matrix representation $A:=M_B^A(id_V)$ of $id_V$ regarding to the basis $A=\{1,t,...,t^d\}$ and $B=\{1,(t-a),...,(t-a)^d\}$.
I know, that i have to write... | Do the binomial expansion
$$
(z + a)^{k}
=
a^{k} + \binom{k}{1} a^{k-1} z + \dots + \binom{k}{i} a^{k-i} z^{i} + \dots + z^{k},
$$
and then substitute $t = z + a$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/800326",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Example of a bounded lattice that is NOT complete I know that every complete lattice is bounded. Is there a simple example for a bounded lattice that is not complete?
Thank you
| Update: My answer below is wrong! (Thanks to bof for pointing that out.) I will leave the answer here because I think my mistake and bof's comment could maybe be instructive.
Let $\mathbb{N}$ be the set of natural numbers. Let $\mathcal{P}_{fin}(\mathbb{N})$ denote the collection of finite subsets of $\mathbb{N}$. Then... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/800429",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 2
} |
If $\sum{a_n}$ converges does that imply that $\sum{\frac{a_n}{n}}$ converges? I know if $\sum{a_n}$ converges absolutely then $\sum{\frac{a_n}{n}}$ converges since $0\le \frac{|a_n|}{n} \le |a_n| $ for all $n$ so it converges absolutely by the basic comparison test and therefore converges. However, I cannot prove the ... | Yes;
A theorem found in "Baby'' Rudin's book: If $\sum a_{n}$} converges and $\lbrace{ b_{n} \rbrace}$ monotonic and bounded then $\sum a_{n}b_{n}$ converges. See:
Prob. 8, Chap. 3 in Baby Rudin: If $\sum a_n$ converges and $\left\{b_n\right\}$ is monotonic and bounded, then $\sum a_n b_n$ converges.
Here, we take $b_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/800490",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 4,
"answer_id": 2
} |
Extrema of $x+y+z$ subject to $x^2 - y^2 = 1$ and $2x + z = 1$ using Lagrange Multipliers
Find the extrema of $x+y+z$ subject to $x^2 - y^2 = 1$ and $2x + z = 1$ using Lagrange multipliers.
So I set it up:
$$
1 = 2x\lambda_1 + 2\lambda_2 \\
1 = -2y\lambda_1 \\
1 = \lambda_2
$$
Plug in for $\lambda_2$:
$$
1 = 2x\lambd... | The constraints define two curves in ${\mathbb R}^3$ as follows: The constraint $x^2-y^2=1$ defines a hyperbolic cylinder $Z$ consisting of two sheets, which can be parametrized as follows:
$$(t,z)\mapsto(\pm\cosh t,\sin t, z)\qquad(-\infty<t<\infty, \ -\infty<z<\infty)\ .$$
Intersecting these two sheets with the plane... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/800575",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Linear independence of two functions, how to solve I have a problem. I need to show that $\{f,g\}$ is a linearly independent set in the vector space of all functions from $\mathbb{R}^{+}$ to $\mathbb{R}$, where
$$f(x)=x$$
$$g(x)=\frac1{x}$$
First (and least important), is there a standard notation for this vector spac... | The standard notation for the vector space of all functions $\Bbb R^+\to\Bbb R$ is $\Bbb R^{\Bbb R^+}$. This is sort of awkward and I've also seen the notation $\mathcal F(0,\infty)$ used. As long as you define your notation properly I don't think notation matters much here.
As for showing that $f(x)$ and $g(x)$ are li... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/800828",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 2
} |
Probability question - (Probably) Bayes' Rule and Total Probability Theorem I just took a probability final exam and was fairly confident in my solution of 28/31, but I wanted to be sure... because according to http://www.stat.tamu.edu/~derya/stat211/SummerII02/Final.Summer02.doc which has it as the second question, th... | I get .68, probably rounding but what happened was you fell off right at start and the checkers accepted your term definitions which are incorrect. A=1/4 B=7/10 and P(A/~B)=1/10
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/800925",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Does the series $\sum_{n\ge0}\frac{x^n\sin({nx})}{n!}$ converge uniformly on $\Bbb R$? The series $$\sum_{n\ge0}\frac{x^n\sin({nx})}{n!}$$ converges uniformly on each closed interval $[a,b]$ by Weierstrass' M-test because $$\left|\frac{x^n\sin({nx})}{n!}\right|\le\frac{\max{(|a|^n,|b|^n)}}{n!}.$$
But does this series c... | To rephrase Paul's answer: if the series converges uniformly, then the sequence of functions $(f_n)_{n\geq1}$ with $f_n(x)=\dfrac{x^n\sin(nx)}{n!}$ converges uniformly to zero.
If $n\geq1$, let $\xi_n=\pi\left(2n+\frac1{4n}\right)$. Then $\xi_n>n$, so that $(\xi_n)^n>n!$ and $\sin(n\xi_n)= \sin \left(\pi \left(2n^2+\t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/801110",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Matrices and Complex Numbers Given this set:
$$
S=\left\{\begin{bmatrix}a&-b\\b&a\end{bmatrix}\middle|\,a,b\in\Bbb R\right\}
$$
Part I:
Why is this set equivalent to the set of all complex numbers a+bi (when both are under multiplication?)
There is one matrix that corresponds to a specific complex number. Can this exam... | Looking in the comments since I've posted, the following answer appears to be expanding on Jyrki's idea.
There exists a homomorphism $\phi:S \rightarrow \mathbb{C}$ defined as follows:
$$\begin{bmatrix}
a & -b \\[0.3em]
b & a \\[0.3em]
\end{bmatrix} \mapsto (a + bi)$$
Of course, you will want to prove t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/801204",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Estimate the number of roots of an analytic function Let $f : \mathbb{C}\longrightarrow \mathbb{C}$ be analytic with $0 \not = f(0)$. Suppose we have normalized $f$ such that $|f(0)| = 1$. Suppose that $f$ has $n$ roots (including repeated roots) and they are all in $B_{\frac 12}(0)$. Is it possible to estimate $n$ in ... | By Jensen's formula, $n\leq \frac{\log M}{\log2}$, where $M = \sup\limits_{\vert z\vert =1}\vert f(z)\vert$.
https://en.wikipedia.org/wiki/Jensen%27s_formula
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/801312",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
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