Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Existence of geodesic on a compact Riemannian manifold I have a question about the existence of geodesics on a compact Riemannian manifold $M$. Is there an elementary way to prove that in each nontrivial free homotopy class of loops, there is a closed geodesic $\gamma$ on $M$?
| Let be $[\gamma]$ nontrivial free homotopy class of loops and $l=\inf_{\beta; \beta\in[\gamma]}l(\beta)$ where $l(\beta)$ is a lenght of the curve $\beta.$ We will show that there is a geodesic $\beta$ in $[\gamma]$ such that $l(\beta)=l.$ Let be $\beta_n$ a sequence of loops in $[\gamma]$ such that $l(\beta_n)\t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/255247",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 1,
"answer_id": 0
} |
Dirichlet Series for $\#\mathrm{groups}(n)$ What is known about the Dirichlet series given by
$$\zeta(s)=\sum_{n=1}^{\infty} \frac{\#\mathrm{groups}(n)}{n^{s}}$$
where $\#\mathrm{groups}(n)$ is the number of isomorphism classes of finite groups of order $n$. Specifically: does it converge? If so, where? Do the residue... | According to a sci.math posting by Avinoam Mann which I found at http://www.math.niu.edu/~rusin/known-math/95/numgrps
the best upper bound is #groups(n) $\le n^{c(\log n)^2}$ for some constant $c$. That would indicate that your Dirichlet series diverges for all $s$, having arbitrarily large terms.
See also https://oei... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/255296",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
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Infinitely valued functions Is it possible to define a multiple integral or multiple sums to infinite order ? Something like $\int\int\int\int\cdots$ where there are infinite number of integrals or $\sum\sum\sum\sum\cdots$ . Does infinite valued functions exist (Something like $R^\infty \rightarrow R^n$ ) ?
| Yes, it is possible to define multiple integrals or sums to infinite order:
here is my definition:
for every function $f$ let
$$\int\int\int\cdots \int f:=1$$
and
$$\sum\sum\cdots\sum f:=1.$$
As you can see, I defined those objects.
But OK, I understand that you are looking for some definitions granting some usu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/255370",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
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Procedures to find solution to $a_1x_1+\cdots+a_nx_n = 0$ Suppose that $x_1, \dots,x_n$ are given as an input. Then we want to find $a_1,\ldots,a_n$ that satisfy $a_1x_1 + a_2x_2+a_3x_3 + a_4x_4+\cdots +a_nx_n =0$. (including the case where such $a$ set does not exist.)
How does one find this easily? (So I am asking fo... | Such $a_i$ do always exist (we can let $a_1 = \cdots = a_n = 0$) for example. The whole set of solutions is a $(n-1)$-dimensional subspace (the whole $k^n$ if $x_1 = \cdots = x_n= 0$) of $k^n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/255457",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
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Questions about $f: \mathbb{R} \rightarrow \mathbb{R}$ with bounded derivative I came across a problem that says:
Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function. If $|f'|$ is bounded, then which of the following option(s) is/are true?
(a) The function $f$ is bounded.
(b) The limit $\lim_{x\to\infty}f(x)$ ex... | (a) & (b) are false: Consider $f(x)=x$ $\forall$ $x\in\mathbb R$;
(c) is true: $|f'|$ is bounded on $\mathbb R\implies f'$ is bounded on $\mathbb R$ [See: Related result];
(d) is false: $f=0$ on $\mathbb R\implies${$x:f(x)=0$} $=\mathbb R$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/255652",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 4
} |
Proof by induction for Stirling Numbers I am asked this:
For any real number x and positive integer k, define the notation
[x,k] by the recursion [x,k+1] = (x-k) [x,k] and [x,1] = x.
If n is any positive integer, one can now express the monomial x^n as
a polynomial in [x,1], [x,2], . . . , [x,n]. Find a general fo... | I prefer the notation $x^{\underline k}$ for the falling power, so I’ll use that. You don’t want binomial coefficients in your expression: you want Stirling numbers of the second kind, denoted by $\left\{n\atop k\right\}$, and you want to show by induction on $n$ that
$$x^n=\sum_{k=1}^n\left\{n\atop k\right\}x^{\underl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/255730",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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The compactness of $\{x_n=cos nt\}_{n\in\mathbb{N}}\in L_2[-\pi,\pi]$ Is the set $\{x_n=\cos (nt): n\in\mathbb{N}\}$ in $L_2[-\pi,\pi]$closed or compact? I don't know how to prove it.
| As the elements are orthogonal, we have for $n\neq m$ that
$$\lVert x_n-x_m\rVert_{L^2}^2=2,$$
proving that the set cannot be compact (it's not precompact, as the definition doesn't work for $\varepsilon=1/2$).
But it's a closed set, as it's the orthogonal of the even square-integrable functions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/255792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Proof that an analytic function that takes on real values on the boundary of a circle is a real
Possible Duplicate:
Let f(z) be entire function. Show that if $f(z)$ is real when $|z| = 1$, then $f(z)$ must be a constant function using Maximum Modulus theorem
I'm having trouble proving that an analytic function that ... | You know that if $f=u+iv$, then $u$ and $v$ are harmonic. Now, by assumption you have that $v$ is zero on the boundary of the disk. But, by the two extrema principles, you know that the maximum and minimum of $v$ occur on the boundary of your disc, and so clearly this implies that $v$ is identically zero. Thus, $f=u$, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/255848",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
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Linear Algebra Proof
If A is a $m\times n$ matrix and $M = (A \mid b)$ the augmented matrix
for the linear system $Ax = b$.
Show that either $(i) \operatorname{rank}A = \operatorname{rank}M$, or
$(ii)$ $\operatorname{rank}A = \operatorname{rank}M - 1$.
My attempt:
The rank of a matrix is the dimension of its range sp... | Suppose the columns of $A$ have exactly $r$ linearly independent vectors. If $b$ lies in their span, then $\operatorname{rank} A=r=\operatorname{rank} M$. If not, then the columns of $A$ together with $b$ have exactly $(r+1)$ linearly independent vectors, so that $\operatorname{rank} A+1=r+1=\operatorname {rank} M$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/255898",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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Using the SLLN to show that the Sample Mean of Arrivals tends to the Arrival Rate for a simple Poisson Process Let $N_t = N([0,t])$ denote a Poisson process with rate $\lambda = 1$ on the interval $[0,1]$.
I am wondering how I can use the Law of Large Numbers to formally argue that:
$$\frac{N_t}{t} \rightarrow \lambda... | If $n\leqslant t\lt n+1$, then $N_n\leqslant N_t\leqslant N_{n+1}$ hence
$$
\frac{n}t\cdot\frac{N_{n}}{n}\leqslant\frac{N_t}t\leqslant\frac{n+1}t\cdot\frac{N_{n+1}}{n+1}.
$$
When $t\to\infty$, $\frac{n}t\to1$ and $\frac{n+1}t\to1$ because $n$ is the integer part of $n$ hence $t-1\lt n\leqslant t$. Furthermore, $\frac{N... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/255958",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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greatest common divisor is 7 and the least common multiple is 16940 How many such number-pairs are there for which the greatest common divisor is 7 and the least common multiple is 16940?
| Let the two numbers be $7a$ and $7b$.
Note that $16940=7\cdot 2^2\cdot 5\cdot 11^2$.
We make a pair $(a,b)$ with gcd $1$ and lcm $2^2\cdot 5\cdot 11^2$ as follows. We "give" $2^2$ to one of $a$ and $b$, and $2^0$ to the other. We give $5^1$ to one of $a$ and $b$, and $5^0$ to the other. Finally, we give $11^2$ to one ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/256035",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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How can this isomorphism be valid? How can $\mathbb{Z}_4 \times \mathbb{Z}_6 / <(2,3)> \cong \mathbb{Z}_{12} = \mathbb{Z}_{4} \times \mathbb{Z}_{3}$?
I am not convinced at the least that $\mathbb{Z}_{12}$ is isomorphic to $\mathbb{Z}_4 \times \mathbb{Z}_6 / <(2,3)>$
For instance, doesn't $<1>$ have an order of 12 in $... | I presume you mean ring isomorphism. If so, you should define what is the 0, the 1, the sum and the multiplication. Of course, those are rather implicit. But there lies your doubts. As @KReiser points out, the unit in $\mathbb{Z}_4 \times \mathbb{Z}_6 / \langle(2,3)\rangle$ is (the class of) (1,1).
In order to prove th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/256097",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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What is the difference between necessary and sufficient conditions?
*
*If $p\implies q$ ("$p$ implies $q$"), then $p$ is a sufficient condition for $q$.
*If $\lnot p\implies \lnot q$ ("not $p$ implies not $q$"), then $p$ is a necessary condition for $q$.
I don't understand what sufficient and necessary mean in this ... | Suppose first that $p$ implies $q$. Then knowing that $p$ is true is sufficient (i.e., enough evidence) for you to conclude that $q$ is true. It’s possible that $q$ could be true even if $p$ weren’t, but having $p$ true ensures that $q$ is also true.
Now suppose that $\text{not-}p$ implies $\text{not-}q$. If you know t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/256171",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 3,
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Prove divisibility law $\,b\mid a,c\,\Rightarrow\, b\mid ka + lc$ for all $k,l\in\Bbb Z$ We have to prove $b|a$ and $b|c \Rightarrow b|ka+lc$ for all $k,l \in \mathbb{Z}$.
I thought it would be enough to say that $b$ can be expressed both as $b=ka$ and $b=lc$. Now we can reason that since $ka+lc=2b$ and $b|2b$, it dire... | An alternative presentation of the solution (perhaps slightly less elementary than the already proposed answers) is to work in the quotient ring.
You can write that in $\mathbb{Z}/b\mathbb{Z}$
\begin{align*}
\overline{ka+lc}&=\bar{k}\bar{a}+\bar{l}\bar{c}\\
&=\bar{0}+\bar{0}\\
&=\bar{0}
\end{align*}
where $\bar{a}=\b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/256237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Square of the sum of n positive numbers I have a following problem
When we want to write $a^2 + b^2$ in terms of $(a \pm b)^2$ we can do it like that
$$a^2 +b^2 = \frac{(a+b)^2}{2} + \frac{(a-b)^2}{2}.$$
Can we do anything similar for $a_1^2 + a_2^2 + \ldots + a_n^2$ ?
I can add the assumption that all $a_i$ are positi... | Yes. You have to sum over all of the possibilities of $a\pm b\pm c$:
$$4(a^2+b^2+c^2)=(a+b+c)^2+(a+b-c)^2+(a-b+c)^2+(a-b-c)^2$$
This can be extended to n factors by:
$$\sum_{k=1}^n a_k^2=\sum_{\alpha=(1,-1,...,-1)\; |a_i|=1}^{(1,...,1)}\frac{\big(\sum_{i=1}^{n}\alpha_ia_i\big)^2}{2^{n-1}}$$
($\alpha$ is a multiindex wi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/256309",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
How to prove the given series is divergent? Given series
$$
\sum_{n=1}^{+\infty}\left[e-\left(1+\frac{1}{n}\right)^{n}\right],
$$
try to show that it is divergent!
The criterion will show that it is the case of limits $1$, so maybe need some other methods? any suggestions?
| Let $x > 1$. Then the inequality $$\frac{1}{t} \leq \frac{1}{x}(1-t) + 1$$ holds for all $t \in [1,x]$ (the right hand side is a straight line between $(1,1)$ and $(x, \tfrac{1}{x})$ in $t$) and in particular $$\log(x) = \int_1^x \frac{dt}{t} \leq \frac{1}{2} \left(x - \frac{1}{x} \right)$$ for all $x > 1$. Substitute... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/256359",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Use two solutions to a high order linear homogeneous differential equation with constant coefficients to say something about the order of the DE OK, this one utterly baffles me.
I am given two solutions to an nth-order homogeneous differential equation with constant coefficients. Using the solutions, I am supposed to p... | A related problem. We will use the annihilator method. Note that, since you are given two solutions of the ode with constant coefficients, then their linear combination is a solution to the ode too. This means the function
$$ y(x) = c_1 t^3 + c_2 te^{t}\sin(t) $$
satisfies the ode. Applying the operator $D^4((D-1)^2+1)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/256429",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Proving that cosine is uniformly continuous This is what I've already done. Can't think of how to proceed further
$$|\cos(x)-\cos(y)|=\left|-2\sin\left(\frac{x+y}{2}\right)\sin\left(\frac{x-y}{2}\right)\right|\leq\left|\frac{x+y}{2}\right||x-y|$$
What should I do next?
| Hint: Any continuous function is uniformly continuous on a closed, bounded interval, so $\cos$ is uniformly continuous on $[-2\pi,0]$ and $[0,2\pi]$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/256498",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Open affine neighborhood of points $X$ is a variety and there are $m$ points $x_1,x_2,\cdots,x_m$ on $X$. Can we find an open affine set which contains all $x_i$s?
| A such variety is sometimes called FA-scheme (finite-affine). Quasi-projective schemes over affine schemes (e.g. quasi-projective varieties over a field) are FA.
On the other hand, there are varieties which are not FA. Kleiman proved that a propre smooth variety over an algebraically closed field is FA if and only if ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/256648",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
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Infinite series question from analysis Let $a_n > 0$ and for all $n$ let $$\sum\limits_{j=n}^{2n} a_j \le \dfrac 1n $$ Prove or give a counterexample to the statement $$\sum\limits_{j=1}^{\infty} a_j < \infty$$
Not sure where to start, a push in the right direction would be great. Thanks!
| Consider sum of sums $\sum_{i=1}^\infty \sum_{k=i}^{2k} a_j$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/256699",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Is there any function in this way? $f$ is a function which is continous on $\Bbb R$, and $f^2$ is differentiable at $x=0$. Suppose $f(0)=1$. Must $f$ be differentiable at $0$?
I may feel it is not necessarily for $f$ to be differentiable at $x=0$ though $f^2$ is. But I cannot find a counterexample to disprove this. An... | Hint:
$$\frac{f(x)-1}{x}=\frac{f(x)^2-1}{x}\frac{1}{f(x)+1}\xrightarrow [x\to 0]{}...?$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/256785",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Give an example of a sequence of real numbers with subsequences converging to every real number Im unsure of an example
Give an example of a sequence of real numbers with subsequences converging to every real number
| A related question that you can try:
Let $(a_k)_{k\in\mathbb{N}}$ be a real sequence such that $\lim_k a_k=0$, and set $s_n=\sum_{k=1}^na_k$. Then the set of subsequential limits of the sequence $(s_n)_{n\in\mathbb{N}}$ is connected.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/256840",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
how does addition of identity matrix to a square matrix changes determinant? Suppose there is $n \times n$ matrix $A$. If we form matrix $B = A+I$ where $I$ is $n \times n$ identity matrix, how does $|B|$ - determinant of $B$ - change compared to $|A|$? And what about the case where $B = A - I$?
| As others already have pointed out, there is no simple relation. Here is one answer more for the intuition.
Consider the (restricting) codition, that $A_{n \times n}$ is diagonalizable, then $$\det(A) = \lambda_0 \cdot \lambda_1 \cdot \lambda_2 \cdot \cdots \lambda _{n-1} $$
Now consider you add the identity matrix.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/256969",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 6,
"answer_id": 1
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interesting matrix Let be $a(k,m),k,m\geq 0$ an infinite matrix then the set
$$T_k=\{(a(k,0),a(k,1),...,a(k,i),...),(a(k,0),a(k+1,1),...,a(k+i,i),...)\}$$is called angle of matrix
$a(k,0)$ is edge of $T_k$
$a(k,i),a(k+i,i),i>0$ are conjugate elements of $T_k$
$(a(k,0),a(k,1),...,a(k,i),...)$ is horizontal ray of $... | This is a very unnecessarily complicated, ambiguous and partly erroneous reformulation of a simple recurrence relation for the number $a(k,m)$ of partitions of $k$ with greatest part $m$. It works out if the following changes and interpretations are made:
*
*both instances of $k\gt1$ are replaced by $k\ge1$,
*"his ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/257039",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Limiting distribution Let $(q_n)_{n>0}$ be a real sequence such that $0<q_n<1$ for all $n>0$ and $\lim_{n\to \infty} q_n = 0$.
For each $n > 0$, let $X_n$ be a random variable, such that
$P[X_n =k]=q_n(1−q_n)^{k−1}, (k=1,2,...)$.
Prove that the limit distribution of
$\frac{X_n}{\mathbb{E}[X_n]}$
is exponential with p... | First we calculate the characteristic function of $X_n$:
$$\Phi_{X_n}(\xi) = \sum_{k=1}^\infty \underbrace{q_n}_{(q_n-1)+1} \cdot (1-q_n)^{k-1} \cdot e^{\imath \, k \cdot \xi} = - \sum_{k=1}^\infty (1-q_n)^k \cdot (e^{\imath \, \xi})^k+e^{\imath \, \xi} \sum_{k=1}^\infty (1-q_n)^{k-1} \cdot e^{\imath \, (k-1) \cdot \xi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/257092",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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countable group, uncountably many distinct subgroup? I need to know whether the following statement is true or false?
Every countable group $G$ has only countably many distinct subgroups.
I have not gotten any counter example to disprove the statement but an vague idea to disprove like: if it has uncountably many disti... | Let $(\mathbb{Q},+)$ be the group of the rational numbers under addition. For any set $A$ of primes, let $G_A$ be the set of all rationals $a/b$ (in lowest terms) such that every prime factor of the denominator $b$ is in $A$. It is clear that $G_A$ is a subgroup of $\mathbb{Q}$, and that $G_A = G_{A'}$ iff $A = A'$. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/257175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "34",
"answer_count": 3,
"answer_id": 1
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Evaluate $\lim_{n\to\infty}\sum_{k=1}^{n}\frac{k}{n^2+k^2}$ Considering the sum as a Riemann sum, evaluate $$\lim_{n\to\infty}\sum_{k=1}^{n}\frac{k}{n^2+k^2} .$$
| $$\sum_{k=1}^n\frac{k}{n^2+k^2}=\frac{1}{n^2}\sum_{k=1}^n\frac{k}{1+\left(\frac{k}{n}\right)^2}=\frac{1}{n}\sum_{k=1}^n\frac{\frac{k}{n}}{1+\left(\frac{k}{n}\right)^2}\xrightarrow [n\to\infty]{}\int_0^1\frac{x}{1+x^2}\,dx=\ldots$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/257248",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 2
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Conditions for matrix similarity Two things that are not clear to me from the Wikipedia page on "Matrix similarity":
*
*If the geometric multiplicity of an eigenvalue is different in two matrices $A$ and $B$ then $A$ and $B$ are not similar?
*If all eigenvalues of $A$ and $B$ coincide, together with their algebraic... | Intuitively, if $A,B$ are similar matrices, then they represent the same linear transformation, but in different bases. Using this concept, it must be that the eigenvalue structure of two similar matrices must be the same, since the existence of eigenvalues/eigenvectors does not depend on the choice of basis. So, to an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/257322",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
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Building a space with given homology groups Let $m \in \mathbb{N}$. Can we have a CW complex $X$ of dimension at most $n+1$ such that $\tilde{H_i}(X)$ is $\mathbb{Z}/m\mathbb{Z}$ for $i =n$ and zero otherwise?
| To expand on the comments:
The $i$-th homology of a cell complex is defined to be $\mathrm{ker}\partial_{i} / \mathrm{im}\partial_{i+1}$ where $\partial_{i+1}$ is the boundary map from the $i+1$-th chain group to the $i$-th chain group. Geometrically, this map is the attaching map that identifies the boundary of the $i... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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differentiability and compactness I have no idea how to show whether this statement is false or true:
If every differentiable function on a subset $X\subseteq\mathbb{R}^n$ is bounded then $X$ is compact.
Thank you
| Some hints:
*
*By the Heine-Borel property for Euclidean space, $X$ is compact if and only if $X$ is closed and bounded.
*My inclination is to prove the contrapositive: If $X$ is not compact, then there exists a differentiable function on $X$ which is unbounded.
*If $X$ is not compact, then either it isn't bounde... | {
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"source": "stackexchange",
"question_score": "1",
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Find $F'(x)$ given $ \int_x^{x+2} (4t+1) \ \mathrm{dt}$ Given the problem find $F'(x)$:
$$ \int_x^{x+2} (4t+1) \ \mathrm{dt}$$
I just feel stuck and don't know where to go with this, we learned the second fundamental theorem of calculus today but i don't know where to plug it in. What i did:
*
*chain rule doesn't r... | Let $g(t)=4t+1$, and let $G(t)$ be an antiderivative of $g(t)$.
Note that
$$F(x)=G(x+2)-G(x).\tag{$1$}$$
In this case, we could easily find $G(t)$. But let's not, let's differentiate $F(x)$ immediately. Since $G'(t)=g(t)=4t+1$. we get
$$F'(x)=g(x+2)-g(x)=[4(x+2)+1]-[4x+2].$$
This right-hand side ismplifies to $8$.
| {
"language": "en",
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"source": "stackexchange",
"question_score": "3",
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Can the graph of a bounded function ever have an unbounded derivative?
Can the graph of a bounded function ever have an unbounded derivative?
I want to know if $f$ has bounded variation then its derivative is bounded. The converse is obvious. I think the answer is "yes". If the graph were to have an unbounded deriva... | Oh, sure. I'm sure there are lots of examples, but because of the work I do, some modification of the entropy function comes to mind. Consider the following function:
$$
f:\mathbb{R}^n\rightarrow\mathbb{R}, \quad f(x) \triangleq \begin{cases} x \log |x| & x \neq 0 \\ 0 & x = 0 \end{cases}
$$
It is not difficult to veri... | {
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"source": "stackexchange",
"question_score": "25",
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Find the rank of Hom(G,Z)？ （1）
Prove that for any finitely generated abelian group G, the set Hom(G, Z) is a free Abelian group of finite rank.
（2）
Find the rank of Hom(G,Z) if the group G is generated by three generators x, y, z with relations
2x + 3y + z = 0,
2y - z = 0
| (i) Apply the structure theorem: write $$G \simeq \mathbb{Z}^r \oplus_i \mathbb{Z}/d_i$$
Now from here we compute $$Hom(\mathbb{Z}^r \oplus_i \mathbb{Z}/d_i, \mathbb{Z}) \simeq Hom(\mathbb{Z}^r, \mathbb{Z}) \oplus_i Hom(\mathbb{Z}/d_i, \mathbb{Z}) \simeq \mathbb{Z}^r$$
(ii) We just need to find the rank of the free par... | {
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Kullback-Liebler divergence The Kullback-Liebler divergence between two distributions with pdfs $f(x)$ and $g(x)$ is defined
by
$$\mathrm{KL}(F;G) = \int_{-\infty}^{\infty} \ln \left(\frac{f(x)}{g(x)}\right)f(x)\,dx$$
Compute the Kullback-Lieber divergence when $F$ is the standard normal distribution and $G$
is the nor... | I cannot comment (not enough reputation).
Vincent: You have the wrong pdf for $g(x)$, you have a normal distribution with mean 1 and variance 1, not mean $\mu$.
Hint: You don't need to solve any integrals. You should be able to write this as pdf's and their expected values, so you never need to integrate.
Outline: F... | {
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Properties of $\det$ and $\operatorname{trace}$ given a $4\times 4$ real valued matrix Let $A$, be a real $4 \times 4$ matrix such that $-1,1,2,-2$ are its eigenvalues. If $B=A^4-5A^2+5I$, then which of the following are true?
*
*$\det(A+B)=0$
*$\det (B)=1$
*$\operatorname{trace}(A-B)=0 $
*$\operatorn... | The characteristic equation of $A$ is given by $(t-1)(t+1)(t+2)(t-2)=0 $ which implies $t^{4}-5t^{2}+4=0$. Now $A$ must satisfy its characteristic equation which gives that $A^{4}-5A^{2}+4I=0$ and so we see that $B=A^{4}-5A^{2}+4I+I=0+I=I$. Hence, the eigenvalues of $(A+B)$ is given by $(-1+1),(1+1),(2+1),(-2+1)$ tha... | {
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Vector perpendicular to timelike vector must be spacelike? Given $\mathbb{R}^4$, we define the Minkowski inner product on it by $$ \langle v,w \rangle = -v_1w_1 + v_2w_2 + v_3w_3 + v_4w_4$$
We say a vector is spacelike if $ \langle v,v\rangle >0 $, and it is timelike if $ \langle v,v \rangle < 0 $.
How can I show that ... | The accepted answer by @user1551 is certainly good, but an intuitive physical explanation may be needed, I think.
A timelike vector in special relativity can be thought of as some kind of velocity of some object. And we can find a particular reference frame in which the object is at rest, i.e. with only time component ... | {
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Consider the quadratic form $q(x,y,z)=4x^2+y^2−z^2+4xy−2xz−yz $ over $\mathbb{R}$ then which of the following are true Consider the quadratic form $q(x,y,z)=4x^2+y^2-z^2+4xy-2xz-yz$ over $\mathbb{R}$. Then which of the followings are true?
1.range of $q$ contains $[1,\infty)$
2.range of $q$ is contained in $[0,\infty)$... | If you consider that for $x=0$ and $y=0$ we have that $q$ maps onto $(-∞,0]$ because $q(0,0,z)=-z^2$, and for $x=0$,$z=0$ we have that $q$ maps onto $[0,∞)$, then as a whole $q$ maps onto $(-∞,∞) = \mathbb{R}$.
| {
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"source": "stackexchange",
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Noetherian module implies Noetherian ring? I know that a finitely generated $R$-module $M$ over a Noetherian ring $R$ is Noetherian. I wonder about the converse. I believe it has to be false and I am looking for counterexamples.
Also I wonder if $M$ Noetherian imply that $R$ is Noetherian is true? And if $M$ Noetheria... | Let $R$ be a commutative non-Noetherian and let $\mathcal m$ be a maximal ideal. Then $R/\mathcal m$ is finitely generated and Noetherian - it only has two sub-$R$-modules.
Note that, even if $R$ isn't Noetherian, it contains a maximal ideal, by Krull's Theorem.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Finding a Pythagorean triple $a^2 + b^2 = c^2$ with $a+b+c=40$ Let's say you're asked to find a Pythagorean triple $a^2 + b^2 = c^2$ such that $a + b + c = 40$. The catch is that the question is asked at a job interview, and you weren't expecting questions about Pythagorean triples.
It is trivial to look up the answer.... | $$a^2=(c-b)(c+b) \Rightarrow b+c = \frac{a^2}{c-b}$$
$$a+\frac{a^2}{c-b}=40$$
For simplicity let $c-b=\alpha$.
then
$$a^2+\alpha a-40\alpha =0$$
Since this equation has integral solutions,
$$\Delta=\alpha^2+160 \alpha$$
is a perfect square.Thus
$$\alpha^2+160 \alpha =\beta^2$$
Or
$$(\alpha+80)^2=\beta^2+80^2 \,.$$
Th... | {
"language": "en",
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"source": "stackexchange",
"question_score": "10",
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Estimation with method of maximum likelihood Can anybody help me to generate the estimator of equation:
$$Y_i = \beta_0 + \beta_1X_{i1} + \beta_2X_{i2}+\cdots+\beta_4X_{i4}+\varepsilon_i$$
using method of maximum likelihood, where $\varepsilon_i$ are independent variables which have normal distribution $N(0,\sigma^2)$
| This is given by least squares estimation. To see this, write
$$
L(\beta, \sigma^2 | Y) = \prod_i (2\pi\sigma^2)^{1/2} \exp\left(\frac {-1} {2\sigma^2} (Y_i - \beta_0 - \sum_j \beta_j X_{ij})^2\right) = (2\pi\sigma^2)^{n/2} \exp\left(\frac {-1} {2\sigma^2} \sum_i(Y_i - \beta_0 - \sum_j \beta_j X_{ij})^2\right)
$$
Maxim... | {
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Category theory text that defines composition backwards? I've always struggled with the convention that if $f:X \rightarrow Y$ and $g:Y \rightarrow Z$, then $g \circ f : X \rightarrow Z$. Constantly reflecting back and forth is inefficient. Does anyone know of a category theory text that defines composition the other w... | I recall that the following textbooks on category theory have
compositions written from left to right.
*
*Freyd, Scedrov: "Categories, Allegories", North-Holland Publishing Co., 1990 .
*Manes: "Algebraic Theories", GTM 26, Springer-Verlag, 1976.
*Higgins: "Notes on Categories and Groupoids", Van Nostrand, 1971
(av... | {
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"source": "stackexchange",
"question_score": "10",
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How to do $\frac{ \partial { \mathrm{tr}(XX^TXX^T)}}{\partial X}$ How to do the derivative
\begin{equation}
\frac{ \partial {\mathrm{tr}(XX^TXX^T)}}{\partial X}\quad ?
\end{equation}
I have no idea where to start.
| By definition the derivative of $F(X)=tr(XX^TXX^T)$, in the point $X$, is the only linear functional $DF(X):{\rm M}_{n\times n}(\mathbb{R})\to \mathbb{R}$ such that
$$
F(x+H)=F(X)+DF(X)\cdot H+r(H)
$$
with $\lim_{H\to 0} \frac{r(H)}{\|H\|}=0$. Let's get $DF(X)(H)$ and $r(H)$ by the expansion of $F(X+H)$. But first we... | {
"language": "en",
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Smallest number of games per match to get $0.8$ chance to win the match. If the first person to win $n$ games wins the match, what is the smallest value of $n$ such that $A$ has a better than $0.8$ chance of winning the match?
For $A$ having a probability of $0.70$, I get smallest $n = 5$ (Meaning there must be $5$ game... | Using the same method as before, with A having a probability of winning a game, the probabilities of A winning the match are about $0.7$ for $n=1$, $0.784 $ for $n=2$, $0.837$ for $n=3$, $0.874$ for $n=4$ and $0.901$ for $n=5$.
So the answer is $n=3$ to exceed $0.8$.
$1−0.7^5$ is the answer to the question "What is the... | {
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Confused where and why inequality sign changes when proving probability inequality "Let A and B be two events in a sample space such that 0 < P(A) < 1. Let A' denote the complement of A. Show that is P(B|A) > P(B), then P(B|A') < P(B)."
This was my proof:
$$ P(B| A) > P(B) \hspace{1cm} \frac{P(B \cap A)}{P(A)} > P(B) $... | In general $P(B)=P(A)P(B|A) + P(A')P(B|A')$. What happens if $P(B|A)>P(B)$ and $P(B|A')\geq P(B)$?
Hint: Use $P(A)+P(A')=1$ and $P(A)>0$ and $P(A')\geq 0$ to get a contradiction.
Your proof was right up to (and including) this step:
$$P(A') > \frac{P(B \cap A')}{P(B)}$$
From here, multiply both sides by $\frac{P(B)}{P... | {
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Closed form for $\sum_{n=2}^\infty \frac{1}{n^2\log n}$ I had attempted to evaluate
$$\int_2^\infty (\zeta(x)-1)\, dx \approx 0.605521788882.$$
Upon writing out the zeta function as a sum, I got
$$\int_2^\infty \left(\frac{1}{2^x}+\frac{1}{3^x}+\cdots\right)\, dx = \sum_{n=2}^\infty \frac{1}{n^2\log n}.$$
This sum is ... | The closed form means an expression containing only elementary functions. For your case no such a form exists. For more informations read these links:
http://www.frm.utn.edu.ar/analisisdsys/MATERIAL/Funcion_Gamma.pdf
http://en.wikipedia.org/wiki/Hölder%27s_theorem
http://en.wikipedia.org/wiki/Gamma_function#19th-20th_... | {
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Solving for $y$ with $\arctan$ I know this is a very low level question, but I honestly can't remember how this is done. I want to solve for y with this:
$$
x = 2.0 \cdot \arctan\left(\frac{\sqrt{y}}{\sqrt{1 - y}}\right)
$$
And I thought I could do this:
$$
\frac{\sqrt{y}}{\sqrt{1 - y}} = \tan\left(\frac{x}{2.0}\right)... | So, since Henry told me that I wasn't wrong, I continued and got a really simple answer. Thanks!
x = 2 * arctan(sqrt(y)/sqrt(1 - y))
sqrt(y)/sqrt(1 - y) = tan(x/2)
1/(sqrt(1 - y) * sqrt(1/y)) = tan(x/2)
1/tan(x/2) = sqrt(1/y - 1)
1/(tan(x/2))^2 + 1 = 1/y
y = (tan(x/2))^2/((tan(x/2))^2 + 1)
Thanks again to Henry!
EDIT
... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why the Picard group of a K3 surface is torsion-free Let $X$ be a K3 surface. I want to prove that $Pic(X)\simeq H^1(X,\mathcal{O}^*_X)$ is torsion-free.
From D.Huybrechts' lectures on K3 surfaces I read that if $L$ is torsion then the Riemann-Roch formula would imply that $L$ is effective. But then if a section $s$ ... | If $L$ is torsion, then $L^k=O_X$ (tensor power). Since $X$ is K3 and because the first chern class of the trivial bundle vanishes, we have $c_1(X)=0$. Furthermore, since $X$ is regular, we get $h^1(O_X)=0$. Thus, $\chi(O_X)=2$.
Now the RRT says
$$\chi(L)=\chi(O_X) + \tfrac 12 c_1(L)^2$$
Thus, $\chi(O_X)=\chi(L^k)=\chi... | {
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Volume integration help A volume sits above the figure in the $xy$-plane bounded by the equations $y = \sqrt{x}$, $y = −x$ for $0 ≤ x ≤ 1$. Each $x$ cross section is a half-circle, with diameter touching the ends of the curves. What is the volume?
a) Sketch the region in the $xy$ plane.
b) What is the area of a cross-s... | The question has been essentially fully answered by JohnD: The picture does it all.
The cross section at $x$ has diameter $AB$, where $A$ is the point where the vertical line "at" $x$ meets the curve $y=\sqrt{x}$, and $B$ is the point where the vertical line at $x$ meets the line $y=-x$.
So the distance $AB$ is $\sqrt{... | {
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why a geodesic is a regular curve? In most definitions of the geodesic, it is required to be a regular curve,i.e. a smooth curve satisfying that the tangent vector along the curve is not 0 everywhere. I don't know why.
| Suppose $\gamma:[a,b]\to M$ be a smooth curve on a Riemannian manifold $M$ with Riemannian metric $\langle\cdot,\cdot\rangle$. Then we have
$$\tag{1}\frac{d}{dt}\langle\frac{d\gamma}{dt},\frac{d\gamma}{dt}\rangle=\langle\frac{D}{dt}\frac{d\gamma}{dt},\frac{d\gamma}{dt}\rangle+\langle\frac{d\gamma}{dt},\frac{D}{dt}\frac... | {
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Determine the PDF of $Z = XY$ when the joint pdf of $X$ and $Y$ is given The joint probability density function of random variables $ X$ and $ Y$ is given by
$$p_{XY}(x,y)=
\begin{cases}
& 2(1-x)\,\,\,\,\,\,\text{if}\,\,\,0<x \le 1, 0 \le y \le 1 \\
& \,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{otherwise.}
... | There are faster methods, but it can be a good idea, at least once or twice, to calculate the cumulative distribution function, and then differentiate to find the density.
The upside of doing it that way is that one can retain reasonably good control over what's happening. (There are also a number of downsides!)
So we... | {
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Evaluate $\lim\limits_{x \to \infty}\left (\sqrt{\frac{x^3}{x-1}}-x\right)$ Evaluate
$$
\lim_{x \to \infty}\left (\sqrt{\frac{x^3}{x-1}}-x\right)
$$
The answer is $\frac{1}{2}$, have no idea how to arrive at that.
| Multiply and divide by $\sqrt{x^3/(x-1)}+x$, simplify and take the limit.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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How to find $(-64\mathrm{i}) ^{1/3}$? How to find $$(-64\mathrm{i})^{\frac{1}{3}}$$
This is a complex variables question.
I need help by show step by step.
Thanks a lot.
| For any $n\in\mathbb{Z}$,
$$\left(-64i\right)^{\frac{1}{3}}=\left(64\exp\left[\left(\frac{3\pi}{2}+2\pi n\right)i\right]\right)^{\frac{1}{3}}=4\exp\left[\left(\frac{\pi}{2}+\frac{2\pi n}{3}\right)i\right]=4\exp\left[\frac{3\pi+4\pi n}{6}i\right]=4\exp \left[\frac{\left(3+4n\right)\pi}{6}i\right]$$
The cube roots in pol... | {
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"source": "stackexchange",
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Proof of convexity from definition ($x^Tx$) I have to prove that function $f(x) = x^Tx, x \in R^n$ is convex from definition.
Definition: Function $f: R^n \rightarrow R$ is convex over set $X \subseteq dom(f)$ if $X$ is convex and the following holds: $x,y \in X, 0 \leq \alpha \leq 1 \rightarrow f(\alpha x+(1-\alpha) y... | You can also just take the hessian and see that is positive definite(since this function is Gateaux differentiable) , in fact this means that the function is strictly convex as well.
| {
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"url": "https://math.stackexchange.com/questions/259415",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Intersection of a closed subscheme and an open subscheme of a scheme Let $X$ be a scheme.
Let $Z$ be a closed subscheme of $X$.
Let $U$ be an open subscheme of $X$.
Then $Y = U \cap Z$ is an open subscheme of $Z$.
Can we identify $Y$ with $U\times_X Z$?
| Yes. This doesn't have anything to do with closed subscheme. If $p: Z \to X$ is a morphism of schemes and $U \subset X$ is open subscheme, then the fibre product is $p^{-1}(U)$ with open subscheme structure.
| {
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Nature of D-finite sets. A is called D-finite if A is not containing countable subset.
With the above strange definition I need to show the following two properties:
*
*For a D-finite set A, and finite B, the union of A and B is D-finite.
*The union of two D-finite sets is D-finite.
By the way, can we construct su... | Hints only:
The first property may be shown directly.
The second however... Try showing what happens when the union of two sets is not D-finite.
Hope it helps.
| {
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Eigenvalues for LU decomposition In general I know that the eigenvalues of A are not the same as U for the decomposition but for one matrix I had earlier in the year it was. Is there a special reason this happened or was it just a coincidence? The matrix was
$A = \begin{bmatrix}-1& 3 &-3 \\0 &-6 &5 \\-5& -3 &1\end{bm... | It's hard to say if this is mere coincidence or part of a larger pattern. This is like asking someone to infer the next number to a finite sequence of given numbers. Whatever number you say, there is always some way to explain it.
Anyway, here's the "pattern" I see. Suppose
$$
A = \begin{pmatrix}B&u\\ v^T&\gamma\end{pm... | {
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"source": "stackexchange",
"question_score": "4",
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Number of Divisor which perfect cubes and multiples of a number n = $2^{14}$$3^9$$5^8$$7^{10}$$11^3$$13^5$$37^{10}$
How many positive divisors that are perfect cubes and multiples of $2^{10}$$3^9$$5^2$$7^{5}$$11^2$$13^2$$37^{2}$.
I'm able to solve number of perfect square and number of of perfect cubes.
But the extra ... | The numbers you are looking for must be perfect cubes. If you split them into powers of primes, they can have a factor $2^0$, $2^3$, $2^6$, $2^9$ and so on but not $2^1, 2^2, 2^4$ etc. because these are not cubes. The same goes for powers of $3, 5, 7$ and any other primes.
The numbers must also be multiples of $2^{10}... | {
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"source": "stackexchange",
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Evaluation of Derivative Using $\epsilon−\delta$ Definition Consider the function $f \colon\mathbb R \to\mathbb R$ defined by
$f(x)=
\begin{cases}
x^2\sin(1/x); & \text{if }x\ne 0, \\
0 & \text{if }x=0.
\end{cases}$
Use $\varepsilon$-$\delta$ definition to prove that the limit $f'(0)=0$.
Now I see that h should equ... | $$\left|{\dfrac{f(h)-f(0)}{h}}\right|=\left|{\dfrac{2h^2 \sin{\dfrac{1}{h}}}{h}}\right|=2 \left|{h \sin{\dfrac{1}{h}}}\right|<2\left|h\right|<\varepsilon.$$
Choose $\delta<\dfrac{\varepsilon}{2}.$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Use of $\mathbb N$ & $\omega$ as index sets Why all the properties of a sequence or a series or a sequence of functions or a series of functions remain unchanged irrespective of which of $\mathbb N$ & $\omega$ we are using as an index set? Is it because $\mathbb N$ is equivalent to $\omega$?
| It is because $\omega$ and $\mathbb N$ are just different names for the same set. Their members are the same, and so by the Axiom of Extensionality they are the same set.
| {
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"source": "stackexchange",
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Probability of a label appearing on a frazzle This is an exercise from a probability textbook:
A frazzle is equally likely to contain $0,1,2,3$ defects. No frazzle has more than three defects. The cash price of each frazzle is set at \$ $10-K^2$, where $K$ is the number of defects in it. Gummed labels, each representin... | It is not equally likely to go on any of the frazzles, because more labels will go to the frazzles with 0 defects than those with 3 defects, for example.
0,1,2,3 defects draws 10, 9, 6 and 1 labels respectively. So say you had 4 million frazzles. Since 0,1,2 or 3 defects are equally likely, suppose you have 1 million ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to prove the derivative of position is velocity and of velocity is acceleration? How has it been proven that the derivative of position is velocity and the derivative of velocity is acceleration? From Google searching, it seems that everyone just states it as fact without any proof behind it.
| The derivative is the slope of the function. So if the function is $f(x)=5x-3$, then $f'(x)=5$, because the derivative is the slope of the function. Velocity is the change in position, so it's the slope of the position. Acceleration is the change in velocity, so it is the change in velocity. Since derivatives are about... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
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Help with combinations The sum of all the different ways to multiply together $a,b,c,d,\ldots$ is equal to $$(a+1)(b+1)(c+1)(d+1)\cdots$$ right?
If this is true? why is it true?
| Yes it is true, give it a try...
Ok, sorry. Here's a little more detail:
We have the identity
$$
\prod_{j=1}^n ( \lambda-X_j)=\lambda^n-e_1(X_1,\ldots,X_n)\lambda^{n-1}+e_2(X_1,\ldots,X_n)\lambda^{n-2}-\cdots+(-1)^n e_n(X_1,\ldots,X_n).
$$
$\biggr[$use $\lambda=-1$ to get $ \prod_{j=1}^n ( -1-X_j)=(-1)^n\pro... | {
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Have I calculated this integral correctly? I have this integral to calculate:
$$I=\int_{|z|=2}(e^{\sin z}+\bar z)dz.$$
I do it this way:
$$I=\int_{|z|=2}e^{\sin z}dz+\int_{|z|=2}\bar zdz.$$
The first integral is $0$ because the function is holomorphic everywhere and it is a contour integral. As for the second one, I ha... | If $z = 2e^{i \theta}$, then $$\bar{z} dz = 2e^{-i \theta}2i e^{i \theta} d \theta = 4i d \theta$$
Hence, $$\int_{\vert z \vert = 2} \bar{z} dz = \int_0^{2 \pi} 4i d \theta = 8 \pi i$$
| {
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"source": "stackexchange",
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Why is the absolute value needed with the scaling property of fourier tranforms? I understand how to prove the scaling property of Fourier Transforms, except the use of the absolute value:
If I transform $f(at)$ then I get $F\{f(at)\}(w) = \int f(at) e^{-jwt} dt$ where I can substitute $u = at$ and thus $du = a dt$ (an... | Think about the range of the variable $t$ in the integral that gives the transform. How do the 'endpoints' of this improper integral transform under $t\to at$? Can you see how this depends on the sign of $a$?
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Equation to determine radius for a circle that should intersect a given point? Simple question. I tried Google but I don't know what search keywords to use.
I have two points on a $2D$ plane. Point 1 $=(x_1, y_1)$ and Point 2 $=(x_2, y_2)$.
I'd like to draw a circle around Point 1, and the radius of the circle should ... | The radius is simply the distance between the two points. So use the standard Euclidean distance which you should have learned.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the determinant of $A$ satisfying $A^{-1}=I-2A.$ I am stuck with the following problem:
Let $A$ be a $3\times 3$ matrix over real numbers satisfying $A^{-1}=I-2A.$
Then find the value of det$(A).$
I do not know how to proceed. Can someone point me in the right direction? Thanks in advance for your time.
| No such $A$ exists. Hence we cannot speak of its determinant.
Suppose $A$ is real and $A^{-1}=I-2A$. Then $A^2-\frac12A+\frac12I=0$. Hence the minimal polynomial $m_A$ of $A$ must divide $x^2-\frac12x+\frac12$, which has no real root. Therefore $m_A(x)=x^2-\frac12x+\frac12$. But the minimal polynomial and characteristi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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the Green function $G(x,t)$ of the boundary value problem $\frac{d^2y}{dx^2}-\frac{1}{x}\frac{dy}{dx} = 1$ the Green function $G(x,t)$ of the boundary value problem
$\frac{d^2y}{dx^2}-\frac{1}{x}\frac{dy}{dx} = 1$ , $y(0)=y(1)=0$ is
$G(x,t)= f_1(x,t)$ if $x≤t$ and $G(x,t)= f_2(x,t)$ if $t≤x$ where
(a)$f_1(x,t)... | Green's function is symmetric, so answer can be (b) and (d).
| {
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Can't argue with success? Looking for "bad math" that "gets away with it" I'm looking for cases of invalid math operations producing (in spite of it all) correct results (aka "every math teacher's nightmare").
One example would be "cancelling" the 6's in
$$\frac{64}{16}.$$
Another one would be something like
$$\frac{9}... | Slightly contrived:
Given $n = \frac{2}{15}$ and $x=\arccos(\frac{3}{5})$, find $\frac{\sin(x)}{n}$.
$$
\frac{\sin(x)}{n} = \mathrm{si}(x) = \mathrm{si}x = \boxed{6}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/260656",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "289",
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"answer_id": 21
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Showing that complicated mixed polynomial is always positive I want to show that
$\left(132 q^3-175 q^4+73 q^5-\frac{39 q^6}{4}\right)+\left(-144 q^2+12 q^3+70 q^4-19 q^5\right) r+\left(80 q+200 q^2-243 q^3+100 q^4-\frac{31 q^5}{2}\right) r^2+\left(-208 q+116 q^2+24 q^3-13 q^4\right) r^3+\left(80-44 q-44 q^2+34 q^3-\fr... | Because you want to show that this is always positive, consider what happens when $q$ and $r$ get really big. The polynomials with the largest powers will dominate the result.
You can solve this quite easily by approximating the final value using a large number of inequalities.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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$H_1\triangleleft G_1$, $H_2\triangleleft G_2$, $H_1\cong H_2$ and $G_1/H_1\cong G_2/H_2 \nRightarrow G_1\cong G_2$ Find a counterexample to show that if $ G_1 $ and $G_2$ groups,
$H_1\triangleleft G_1$, $H_2\triangleleft G_2$, $H_1\cong H_2$ and $G_1/H_1\cong G_2/H_2 \nRightarrow G_1\cong G_2$
I tried but I did not ha... | The standard counter example to that implication is the quaternion group $Q_8$ and dihedral group $D_4$.
Both groups have order $2^3=8$, so that every maximal group (i.e. one of order 4) is normal. The cyclic group of order 4 is contained in both groups and the quotient has order 2 in both cases. So all assertions are ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to integrate this $\int\frac{\mathrm{d}x}{{(4+x^2)}^{3/2}} $ without trigonometric substitution? I have been looking for a possible solution but they are with trigonometric integration..
I need a solution for this function without trigonometric integration
$$\int\frac{\mathrm{d}x}{{(4+x^2)}^{3/2}}$$
| $$\frac{1}{\left(4+x^2\right)^{3/2}}=\frac{1}{8}\cdot\frac{1}{\left(1+\left(\frac{x}{2}\right)^2\right)^{3/2}}$$
Now try
$$x=2\sinh u\implies dx=2 \cosh u\,du\implies$$
$$\int\frac{dx}{\left(4+x^2\right)^{3/2}}=\frac{1}{8}\int\frac{2\,du\cosh u}{(1+\sinh^2u)^{3/2}}=\frac{1}{4}\int\frac{du}{\cosh^2u}=\ldots $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/260831",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
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What exactly is infinity? On Wolfram|Alpha, I was bored and asked for $\frac{\infty}{\infty}$ and the result was (indeterminate). Another two that give the same result are $\infty ^ 0$ and $\infty - \infty$.
From what I know, given $x$ being any number, excluding $0$, $\frac{x}{x} = 1$ is true.
So just what, exactly, i... | I am not much of a mathematician, but I kind of think of infinity as a behavior of increasing without bound at a certain rate rather than a number. That's why I think $\infty \div \infty$ is an undetermined value, you got two entities that keep increasing without bound at different rates so you don't know which one is ... | {
"language": "en",
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"source": "stackexchange",
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"answer_id": 10
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Identity for central binomial coefficients On Wikipedia I came across the following equation for the central binomial coefficients:
$$
\binom{2n}{n}=\frac{4^n}{\sqrt{\pi n}}\left(1-\frac{c_n}{n}\right)
$$
for some $1/9<c_n<1/8$.
Does anyone know of a better reference for this fact than wikipedia or planet math? Also, d... | It appears to be true for $x > .8305123339$ approximately: $c_x \to 0$ as $x \to 0+$.
| {
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Proof for: $(a+b)^{p} \equiv a^p + b^p \pmod p$ a, b are integers. p is prime.
I want to prove:
$(a+b)^{p} \equiv a^p + b^p \pmod p$
I know about Fermat's little theorem, but I still can't get it
I know this is valid:
$(a+b)^{p} \equiv a+b \pmod p$
but from there I don't know what to do.
Also I thought about
$(a+b)^{p}... | First of all, $a^p \equiv a \pmod p$ and $b^p \equiv b \pmod p$ implies $a^p + b^p \equiv a + b \pmod p$.
Also, $(a+b)^p \equiv a + b \pmod p$.
By transitivity of modulo, combine the above two results and get $(a+b)^p \equiv a^p + b^p \pmod p$.
Done.
| {
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Proving an Entire Function is a Polynomial I had this question on last semesters qualifying exam in complex analysis, and I've attempted it several times since to little result.
Let $f$ be an entire function with $|f(z)|\geq 1$ for all $|z|\geq 1$. Prove that $f$ is a polynomial.
I was trying to use something about ... | Picard's Theorem proves this instantly; which states:
Let $f$ be a transcendental (non-polynomial) entire function. Then $f-a$ must have infinitely many zeros for every $a$ (except for possibly one exception, called the lacunary value).
For example, $e^z-a$ will have infinitely many zeros except for $a=0$ and so the l... | {
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"timestamp": "2023-03-29T00:00:00",
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Show $S = f^{-1}(f(S))$ for all subsets $S$ iff $f$ is injective Let $f: A \rightarrow B$ be a function. How can we show that for all subsets $S$ of $A$, $S \subseteq f^{-1}(f(S))$? I think this is a pretty simple problem but I'm new to this so I'm confused.
Also, how can we show that $S = f^{-1}(f(S))$ for all subsets... | $S \subseteq f^{-1}(f(S)):$ Choose $a\in S.$ To show $a\in f^{-1}(f(S))$ it suffices to show that $\exists$ $a'\in S$ such that $a\in f^{-1}(f(a'))$ i.e. to show $\exists$ $a'\in S$ such that $f(a)=f(a').$ Now take $a=a'.$
$S = f^{-1}(f(S))$ $\forall$ $A \subset S$ $\iff f$ is injective:
*
*$\Leftarrow:$ Let $f$ be ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/261157",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Understanding $\frac {b^{n+1}-a^{n+1}}{b-a} = \sum_{i=0}^{n}a^ib^{n-i}$ I'm going through a book about algorithms and I encounter this.
$$\frac {b^{n+1}-a^{n+1}}{b-a} = \sum_{i=0}^{n}a^ib^{n-i}$$
How is this equation formed? If a theorem has been applied, what theorem is it?
[Pardon me for asking such a simple question... | Multiply both sides by b-a, watch for the cancling of terms, and you will have your answer.
| {
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In the induction proof for $(1+p)^n \geq 1 + np$, a term is dropped and I don't understand why. In What is Mathematics, pg. 15, a proof of
$(1+p)^n \geq 1 + np$, for $p>-1$ and positive integer $n$
goes as follows:
*
*Substitute $r$ for $n$, then multiply both sides by $1+p$, obtaining:
$(1+p)^{r+1}\geq 1+rp+p+rp^2... | In $1.$ we have shown that $$(1+p)^{r+1}\geq 1+rp+p+rp^2$$
But we also know that $r > 1$ (because we're doing an induction proof from $1$ upwards); and obviously $p^2 \ge 0$ (because $p$ is real); so we know that $rp^2 \ge 0$. Therefore
$$1+rp+p+rp^2 \ge 1+rp+p$$
So putting these two together gives
$$(1+p)^{r+1}\geq 1+... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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n×n matrices A with complex enteries Let U be set of all n×n matrices A with complex enteries s.t. A is unitary. then U as a topological subspace of $\mathbb{C^{n^{2}}} $ is
*
*compact but not connected.
*connected but not compact.
*connected and compact.
*Neither connected nor compact
I am stuck on this probl... | For connectedness, examine the set of possible determinants, and whether or not you can find a path of unitary matrices between two unitary matrices with different determinants.
For compactness, look at sequences of unitary matrices and examine whether or not one can be constructed to not have a convergent subsequence.... | {
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"timestamp": "2023-03-29T00:00:00",
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How to calculate $\overline{\cos \phi}$ How do you calculate $\overline{\cos \phi}$? Where $\phi\in\mathbb{C}$.
I try to proof that $\cos \phi \cdot \overline{\cos \phi} +\sin \phi \cdot \overline{\sin \phi}=1$?
| $$
\cos(x+iy) = \cos x \cos (iy) - i \sin x \sin(iy)
$$
$$
\overline {\cos(x+iy)} = \cos x \cos (iy) + i \sin x \sin(iy) = \cos x \cos (-iy) - i \sin x \sin(-iy) = \cos(x-iy)
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/261508",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Testing Convergence of $\sum \sqrt{\ln{n}\cdot e^{-\sqrt{n}}}$ What test should i apply for testing the convergence/divergence of $$\sum_{n=1}^{\infty} \sqrt{\ln{n}\cdot e^{-\sqrt{n}}}$$
Help with hints will be appreciated.
Thanks
| The $n$-th term is equal to
$$\frac{\sqrt{\log n}}{e^{\sqrt{n}/2}}.$$
The intuition is that the bottom grows quite fast, while the top does not grow fast at all.
In particular, after a while the top is $\lt n$.
If we can show, for example, that after a while $e^{\sqrt{n}/2}\gt n^3$, then by comparison with $\sum \frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/261578",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding a dominating function for this sequence Let $$f_n (x) = \frac{nx^{1/n}}{ne^x + \sin(nx)}.$$
The question is: with the dominated convergence theorem find the limit
$$ \lim_{n\to\infty} \int_0^\infty f_n (x) dx. $$
So I need to find an integrable function $g$ such that $|f_n| \leq g$ for all $n\in \mathbf N$. I t... | We have
\begin{align}
\left|
\frac{nx^{1/n}}{ne^x + \sin(nx)}
\right|=
&
\frac{|x^{1/n}|}{|e^x + \sin(nx)/n|}
&
\\
\leq
&
\frac{\max\{1,x\}}{|e^x + \sin(nx)/n|}
&
\mbox{by } |x^{1/n}|\leq \max\{1,x\}
\\
\leq
&
\frac{\max\{1,x\}}{|e^x -\epsilon |}
&
\mbox{if } |e^x + \sin(nx)/n|\geq |e^x -\epsilon|
\\
\end{align}
No... | {
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"url": "https://math.stackexchange.com/questions/261628",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Working out digits of Pi. I have always wondered how the digits of π are calculated. How do they do it?
Thanks.
| The Chudnovsky algorithm, which just uses the very rapidly converging series $$\frac{1}{\pi} = 12 \sum^\infty_{k=0} \frac{(-1)^k (6k)! (13591409 + 545140134k)}{(3k)!(k!)^3 640320^{3k + 3/2}},$$ was used by the Chudnovsky brothers, who are some of the points on your graph.
It is also the algorithm used by at least one... | {
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"url": "https://math.stackexchange.com/questions/261694",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Constrain Random Numbers to Inside a Circle I am generating two random numbers to choose a point in a circle randomly. The circles radius is 3000 with origin at the center. I'm using -3000 to 3000 as my bounds for the random numbers. I'm trying to get the coordinates to fall inside the circle (ie 3000, 3000 is not in t... | Compare $x^2+y^2$ with $r^2$ and reject / retry if $x^2+y^2\ge r^2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/261754",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Class Group of $\mathbb{Q}(\sqrt{-47})$ Calculate the group of $\mathbb{Q}(\sqrt{-47})$.
I have this: The Minkowski bound is $4,36$ approximately.
Thanks!
| Here is another attempt. In case I made any mistakes, let me know and I will either try and fix it, or delete my answer.
We have Minkowski bound $\frac{2 \sqrt{47}}{\pi}<\frac{2}{3}\cdot 7=\frac{14}{3}\approx 4.66$. So let us look at the primes $2$ and $3$:
$-47\equiv 1$ mod $8\quad\Rightarrow\quad 2$ is split, i.e. $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/261828",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
The particular solution of the recurrence relation I cannot find out why the particular solution of $a_n=2a_{n-1} +3n$ is $a_{n}=-3n-6$
here is the how I solve the relation
$a_n-2a_{n-1}=3n$
as $\beta (n)= 3n$
using direct guessing
$a_n=B_1 n+ B_2$
$B_1 n+ B_2 - 2 (B_1 n+ B_2) = 3n$
So $B_1 = -3$, $B_2 = 0$
the part... | using direct guessing
$a_n=B_1 n+ B_2$
$B_1 n+ B_2 - 2 (B_1 (n-1)+ B_2) = 3n$
then
$B_1 - 2B_1 = 3$
$2 B_1 - B_2 =0$
The solution will be
$B_1 = -3, B_2=-6$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/261885",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
3-D geometry : three vertices of a ||gm ABCD is (3,-1,2), (1,2,-4) & (-1,1,2). Find the coordinate of the fourth vertex. The question is
Three vertices of a parallelogram ABCD are A(3,-1,2), B(1,2,-4) and C(-1,1,2). Find the coordinate of the fourth vertex.
To get the answer I tried the distance formula, equated AB=C... | If you have a parallelogram ABCD, then you know the vectors $\vec{AB}$ and $\vec{DC}$ need to be equal as they are parallel and have the same length. Since we know that $\vec{AB}=(-2,\,3,-6)$ you can easily calculate $D$ since you (now) know $C$ and $\vec{CD}(=-\vec{AB})$. We get for $\vec{0D}=\vec{0C}+\vec{CD}=(-1,\,1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/261946",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
Eigen-values of $AB$ and $BA$? Let $A,B \in M(n,\mathbb{C})$ be two $n\times n$ matrices. I would like know how to prove that eigen-value of $AB$ is the same as the eigen-values of $BA$.
| you can prove $|\lambda I-AB|=|\lambda I-BA|$ by computing the determinant of following
$$
\left(
\begin{array}{cc}
I & A \\
B & I \\
\end{array}
\right)
$$
in two diffeerent ways.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/262034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Solving the integral of a Modified Bessel function of the second kind I would like to find the answer for the following integral
$$\int x\ln(x)K_0(x) dx $$
where $K_0(x)$ is the modified Bessel function of the second kind and $\ln(x)$ is the natural-log. Do you have any ideas how to find?
Thanks in advance!
| Here's what Mathematica found:
Looks like an integration by parts to me (combined with an identity for modified Bessel functions).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/262180",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Card probabilities Five cards are dealt from a standard deck of 52. What is the probability that the 3rd card is a Queen?
What I dont understand here is how to factor in when one or both of the first two cards drawn are also Queens.
| All orderings of the $52$ cards in the deck are equally likely. So the probability the third card in the deck is a Queen is exactly the same as the probability that the $17$-th card in the deck is a Queen, or that the first card in the deck is a Queen: They are all equal to $\dfrac{4}{52}$.
The fact that $5$ cards wer... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/262238",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Show That This Complex Sum Converges For complex $z$, show that the sum
$$\sum_{n = 1}^{\infty} \frac{z^{n - 1}}{(1 - z^n)(1 - z^{n + 1})}$$
converges to $\frac{1}{(1 - z)^2}$ for $|z| < 1$ and $\frac{1}{z(1 - z)^2}$ for $|z| > 1$.
Hint: Multiply and divide each term by $1 - z$, and do a partial fraction decomposition,... | HINT: Use
$$
\frac{z^{n}-z^{n+1}}{(1-z^n)(1-z^{n+1})} = \frac{1}{1-z^n} - \frac{1}{1-z^{n+1}}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/262308",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Pigeon-hole Principle: Does this proof have a typo? This was an example of generalized pigeon-hole principle.
Ten dots are placed within a square of unit size. The textbook then shoes a box divided into 9 equal squares. Then there three dots that can be covered by a disk of radius 0.5.
The proof:
Divide our square int... | The proof is basically correct, but yes, there is a typo: the circumcircle of each of the four triangles has radius exactly $0.5$, not less than $0.5$. If $O$ is the centre of the square, and $A$ and $B$ are adjacent corners, the centre of the circumcircle of $\triangle AOB$ is the midpoint of $\overline{AB}$, from whi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/262363",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
What is Cumulative Distribution Function of this random variable? Suppose that we have $n$ independent random variables, $x_1,\ldots,x_n$ such that each $x_i$ takes value $a_i$ with success probability $p_i$ and value $0$ with failure probability $1-p_i$ ,i.e.,
\begin{align}
P(x_1=a_1) & = p_1,\ P(x_1=0)= 1-p_1 \\
P(x... | This answer is an attempt at providing an answer to a previous version of the
question in which the $x_i$ were independent Bernoulli random variables with
parameters $p_i$.
$P\{\sum_{i=1}^n x_i = k\}$ equals the coefficient of
$z^k$ in $(1-p_1+p_1z)(1-p_2+p_2z)\cdots(1-p_n+p_nz)$. This can be found by
developing the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/262426",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Let $f$ be a continuous function on [$0, 1$] with $f(0) =1$. Let $ G(a) = 1/a ∫_0^a f(x)\,dx$ then which of the followings are true? Let $f$ be a continuous function on [$0, 1$] with $f(0) =1$. Let $ G(a) = 1/a ∫_0^af(x)\,dx$ then which of the followings are true?
*
*$\lim_{(a\to 0)} G(a)=1/2$
*$\lim_{(a\to0)}... | Note that $G(a)$ is the mean (or average) value of the function on the interval $[0,a]$. Here’s an intuitive argument that should help you see what’s going on. The function $f$ is continuous, and $f(0)=1$, so when $x$ is very close to $0$, $f(x)$ must be close to $1$. Thus, for $a$ close to $0$, $f(x)$ should be close ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/262499",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
How can I solve this differential equation? How can I find a solution of the following differential equation:
$$\frac{d^2y}{dx^2} =\exp(x^2+ x)$$
Thanks!
| $$\frac{d^2y}{dx^2}=f(x)$$
Integrating both sides with respect to x, we have
$$\frac{dy}{dx}=\int f(x)~dx+A=\phi(x)+A$$
Integrating again
$$y=\int \phi(x)~dx+Ax+B=\chi(x)+Ax+B$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/262559",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
$\sqrt{(a+b-c)(b+c-a)(c+a-b)} \le \frac{3\sqrt{3}abc}{(a+b+c)\sqrt{a+b+c}}$ Suppose $a, b, c$ are the lengths of three triangular edges. Prove that:
$$\sqrt{(a+b-c)(b+c-a)(c+a-b)} \le \frac{3\sqrt{3}abc}{(a+b+c)\sqrt{a+b+c}}$$
| As the hint give in the comment says (I denote by $S$ the area of $ABC$ and by $R$ the radius of its circumcircle), if you multiply your inequality by $\sqrt{a+b+c}$ you'll get
$$4S \leq \frac{3\sqrt{3}abc}{a+b+c}$$
which is eqivalent to
$$a+b+c \leq 3\sqrt{3}\frac{abc}{4S}=3\sqrt{3}R.$$
This inequality is quite known... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/262619",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Escalator puzzle equation I'm trying to understand the escalator puzzle.
A man visits a shopping mall almost every day and he walks up an
up-going escalator that connects the ground and the first floor. If he
walks up the escalator step by step it takes him 16 steps to reach the
first floor. One day he doubles h... | Let $d$ be the distance traveled, which remains same in both the cases. if $v$ is the speed of the man and $x$ is the speed of elevator, in case 1 the number of steps taken is $$\frac d{v+x}=16$$ In case 2 it is $$\frac d{2v+x}=12$$ because now he is traveling at double the speed; eliminating $d$, we get $x=2v$; theref... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/262731",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Is this an equivalence relation (reflexivity, symmetry, transitivity) Let $\theta(s):\mathbb{C}\to \mathbb{R}$ be a well defined function. I define the following relation in $\mathbb{C}$.
$\forall s,q \in \mathbb{C}: s\mathbin{R}q\iff\theta(s)\ne 0 \pmod {2\pi}$ (and)
$\theta(q)\ne 0 \pmod {2\pi}$
The function $\pmo... | Your relation is
$$sRq\iff \theta(s)\not \equiv 0\text{ and }\theta(q)\not \equiv 0 \mod 2\pi$$
for $s,q\in \mathbb{C}$.
For symmetry:
$$sRq\iff \theta(s)\not \equiv 0\text{ and }\theta(q)\not \equiv 0 \mod 2\pi \iff qRs$$
For transitivity:
$$sRq\text{ and }qRp\iff \theta(s)\not \equiv 0\text{ and }\theta(q)\not \equiv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/262794",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Products of infinitely many measure spaces. Applications?
*
*What are some typical applications of the theory for measures on infinite product spaces?
*Are there any applications that you think are particularly interesting - that make the study of this worthwhile beyond finite products, Fubini-Tonelli.
*Are there ... | Infinite products of measure spaces are used very frequently in probability. Probabilists are frequently interested in what happens asymptotically as a random process continues indefinitely. The Strong Law of Large Numbers, for example, tells us that if $\{X_i\}_i$ is a sequence of independent, identically distribute... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/262843",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
What's the probability of a gambler losing \$10 in this dice game? What about making \$5? Is there a third possibility? Can you please help me with this question:
In a gambling game, each turn a player throws 2 fair dice. If the sum of numbers on the dice is 2 or 7, the player wins a dollar. If the sum is 3 or 8, the ... | [edit: Apparently I misread the question. The player starts out with 10 dollars and not five.]
Given that "rolling a 2 or 7" and "rolling a 3 or 8" have the same probability (both occur with probability 7/36), the problem of the probability of a player earning a certain amount of money before losing a different amount ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/262925",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
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