Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
LCM of randomly selected integers What is the expected LCM of 21 randomly selected positive integers under 10000000?
How would someone even approach this problem?
EDIT: The positive integers are chosen with replacement.
| Let $1 \leq a_1, \dots, a_n \leq N$ be number chosen uniformly at random. What is their least common multiple?
What is the highest power of $2$ dividing any of the $a_1, \dots, a_n$?
*
*All the numbers are odd with probability $(1 - \frac{1}{2})^n$
*All the numbers are odd or even (but not divisible by 4) with pro... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1410576",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Cauchy's root test for series divergence Just a question regarding determining the divergent in this example :$$\sum{ 1 \over \sqrt {n(n+1)}} $$ is divergent. It explains the reason by saying that $a_n$ > $1 \over n+1$. If I am not wrong it uses the root test but should not we have the $a_n \ge 1$ but how does their re... | Or still simpler, with equivalences, since it is a series with positive terms:
$$\frac1{\sqrt{n(n+1)}}\sim_\infty\frac 1n,$$
which diverges, hence the original series diverges.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1410637",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Area of a sphere bounded by hyperplanes Say we have a sphere in d-dimensional space, and k hyperplanes (d-1 dimensional) all passing through the origin. Is there a way to calculate (or approximate) the area of the surface of the sphere enclosed by the half-spaces \begin{align*}w_1 \cdot x &\leq 0 \\
w_2 \cdot x &\leq 0... | Seven years late here. But this citation might be useful if someone (probably someone else) is looking into this problem.
Cho, Y., & Kim, S. (2020). Volume of Hypercubes Clipped by Hyperplanes and Combinatorial Identities. In The Electronic Journal of Linear Algebra (Vol. 36, Issue 36, pp. 228–255). University of Wyomi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1410799",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
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Prove that for any integers $x,y,z$ there exist $a,b,c$ such that $ax+by+cz=0$ It is rather obvious that for any 3 coprime integers $x,y,z$ there exist 3 non-zero integers $a,b,c$ such that:
$$ax+by+cz=0$$
Any simple argument to prove it?
| Assume $z\neq 0$. One of them needs to be for $x,y,z$ to be coprime.
Find $u,v\neq 0$ so that $ux+vy\neq 0$. Then let $a=zu,b=zv,c=-(ux+vy)$.
There is always such $u,v$ unless one of $x,y$ is zero. Assume $x=0$ and $y\neq 0$ then choose $a=1,b=z,c=-y$.
If $z=1$ and $x=y=0$, then you can't find a solution with non-zero ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1410920",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Fastest way to perform this multiplication expansion? Consider a product chain:
$$(a_1 + x)(a_2 + x)(a_3 + x)\cdots(a_n + x)$$
Where $x$ is an unknown variable and all $a_i$ terms are known positive integers.
Is there an efficient way to expand this?
| The constant term is easy to compute, it's just the product of the $a_i$. The coefficient of $x^{n-1}$ is $\sum_i a_i$, so that is also easy.
Assume for a moment that $n = 2k$ is even, then the coefficient of $x^k$ is the sum of all possible products consisting of $k$ different $a_i$. There are $\binom{n}{k} \approx 2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1411025",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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A question about matrix kernels and Kronecker products Let us define
$$
v:=v_A\otimes v_B\quad (*)
$$
where $v_A$ is a fixed vector in $\mathbb{R}^{d_A}$, $v_B$ is any vector in $\mathbb{R}^{d_B}$ and $\otimes$ denotes the Kronecker product. To rule out trivial cases assume $d_A,d_B>1$.
My question: Suppose that $v$, d... | No. Consider $v_A=(1,0,0)^T,\ v_B\in\mathbb R^2$ and
$$
C=\pmatrix{
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&1&0&0&1\\
0&0&0&1&1&0\\
0&0&0&1&1&0\\
0&0&1&0&0&1}.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1411104",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proof of Brouwer fixed point theorem using Stokes's theorem $\omega$ is the volume form on the boundary
B -ball
$f\colon B \to \partial B$
$$
0 < \int_{\partial B}\omega = \int_{\partial B} f^*(\omega) = \int_{B} df^*(\omega) = \int_B f^*(d\omega) = \int_B(0) = 0
$$
What I do not understand, is why on the left, we may ... | In the equation
$$
\int_{\partial \Omega} \alpha = \int_{\Omega} d\alpha
$$
it is assumed that $\alpha$ is defined on $\Omega$. If $i\colon\partial\Omega\to\Omega$ is the inclusion, it induces a restriction map $i^*$ on differential forms in the opposite direction. Really the $\alpha$ on the left-hand side is $i^*... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1411178",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show that from a group of seven people whose (integer) ages add up to 332 one can select three people with the total age at least 142. I need help with this problem, and I was thinking in this way:
$$ x_{1} + x_{2} + x_{3} + x_{4} + x_{5} + x_{6} + x_{7} = 332 $$
and I need to find three of these which sum is at least ... | You can form $\binom{7}{3}$ triplets, i.e. 35 triplets. Every men participates in $\binom{6}{2}$ triplets, i.e 15. So the sum of all 35 triplets is $15 \times 332 = 4980$, which means the average is $\frac{4980}{35} = 142.427...$ so there has to be at least one triplet with total age of at least 142. Actually since all... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1411280",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
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"answer_id": 2
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Why is the estimate of the order of error in Trapezoid converging to $2.5$? The integral in question is:
$\int_{0}^{\infty} \frac{x^{3/2}}{\cosh{(x)}}dx$
I coded a program to compute $p$, an estimate of the order of the error for the Trapezoid method of numerical integration. From class, I know that the order of the er... | Some random babblings:
Let $f(x)
=\frac{x^{3/2}}{\cosh(x)}
$.
For large $x$,
$f(x)$ is essentially zero,
so you might be losing
many significant digits
in subtraction,
especially if
$I_{2n}$ is quite close to
$I_n$.
Also,
at the origin,
$f(x)
\approx x^{3/2}
$,
so there is no problem there.
$f'(x)
=\frac{(3/2)x^{1/2}\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1411425",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Is there a Markov-type inequality for the Median? Markov's theorem states that $P(|X| \geq a) \leq \frac{E[|X|]}{a}$. Is there an similar type of inequality that involves the median (somehow I doub't it, but I make no claim to comprehensive knowledge of probability inequalities).
What if we restrict $X$ to a non-negati... | If $X$ is a random variable with finite variance $\sigma^2$, we have that the distance between the median and $\mathbb{E}[X]$ is at most $\sigma$ by Cantelli's inequality, hence the answer is affirmative under slightly stronger assumptions ($X\in L^2$ instead of just $X\in L^1$).
That assumptions gives, for instance:
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1411522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Number of integer functions satisfying three constraints I am trying to understand how many functions $\mathbb{Z^+}\to \mathbb{Z^+}$ which satistfy the three following constraints exist:
*
*For every $n \in \mathbb{Z^+}$
$$f(f(n))\leq\frac{n+f(n)}{2}$$
*For every $n \in \mathbb{Z^+}$
$$f(n)\leq f(n+1)$$
*For every... | The constant function $f(n)=1$ is indeed the only solution, even if we weaken the first condition to: $f(f(n)) \le C(n+f(n))$ for some fixed $C>0$ (no matter how large).
As a proof-of-concept, we prove: there is no such solution with $f(1)>1$. Note that if $f(1)>1$ then $f(n+1)>f(n)$ (strictly) for all $n$, because of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1411599",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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calculating the characteristic polynomial I have the following matrix: $$A=\begin{pmatrix}
-9 & 7 & 4 \\
-9 & 7 & 5\\
-8 & 6 & 2
\end{pmatrix}$$
And I need to find the characteristic polynomial so I use det(xI-A) which is $$\begin{vmatrix}
x+9 & -7 & -4 \\
9 & x-7 & -5\\
8 & -6 & x-2
\end{vmatri... | You could use $\displaystyle\begin{vmatrix}x+9&-7&-4\\9&x-7&-5\\8&-6&x-2\end{vmatrix}=\begin{vmatrix}x+2&-7&-4\\x+2&x-7&-5\\2&-6&x-2\end{vmatrix}$ $\;\;\;$(adding C2 to C1)
$\displaystyle\hspace{2.6 in}=\begin{vmatrix}0&-x&1\\x+2&x-7&-5\\2&-6&x-2\end{vmatrix}$$\;\;\;$(subtracting R2 from R1)
$\displaystyle\hspace{2.6 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1411724",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Let the plane V be defined by $ax + by + cz + d = 0$; with $a, b, c, d \in \mathbb{R}$ and the vector $(a; b; c)$ a unit vector. I am battling to get my mind around some of the concepts involving vectors in $3$-space. This question asks me whether the following statements are True or False:
(A) The line $(a; b; c)$ is ... | For part (A) your reasoning is correct. A line with direction vector $\mathbf{v}$ is parallel to a plane with normal vector $\mathbf{n}$ if and only if $\mathbf{v}$ is orthogonal to $\mathbf{n}$, that is, if and only if $\mathbf{v}\cdot\mathbf{n} = 0$. In this case the direction vector of the line is the same as the no... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1411928",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
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How can the trigonometric equation be proven? This question :
https://math.stackexchange.com/questions/1411700/whats-the-size-of-the-x-angle
has the answer $10°$. This follows from the equation
$$2\sin(80°)=\frac{\sin(60°)}{\sin(100°)}\times \frac{\sin(50°)}{\sin(20°)}$$
which is indeed true , which I checked with Wolf... | $$\begin{align}\frac{\sin 80^\circ \sin 20^\circ\sin 100^\circ }{\sin 50^\circ}&=\frac{\sin 80^\circ \sin 20^\circ\cdot 2\sin 50^\circ \cos 50^\circ }{\sin 50^\circ}\\&=2\sin 80^\circ\sin 20^\circ \cos 50^\circ\\&=2\left(-\frac 12(\cos 100^\circ-\cos 60^\circ)\right)\cos 50^\circ\\&=(\cos 60^\circ-\cos 100^\circ)\cos 5... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1412021",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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Inequality problem: Application of Cauchy-Schwarz inequality Let $a,b,c \in (1, \infty)$ such that $ \frac{1}{a} + \frac{1}{b} + \frac{1}{c}=2$. Prove that:
$$
\sqrt {a-1} + \sqrt {b-1} + \sqrt {c-1} \leq \sqrt {a+b+c}.
$$
This is supposed to be solved using the Cauchy inequality; that is, the scalar product inequality... | We apply the Cauchy Schwarz Inequality to the vectors $x = \left({\sqrt {\dfrac{a-1}{a}} , \sqrt {\dfrac{b-1}{b}} , \sqrt {\dfrac{c-1}{c}}}\right) $ and $y = \left({\dfrac{1}{\sqrt{bc}},\dfrac{1}{\sqrt{ac}}, \dfrac{1}{\sqrt{ab}} }\right)$ in $\Bbb R^3$.
Then, $x \cdot y \le \lVert x\rVert \lVert y\rVert$ yields,
$$ \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1412095",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 0
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$W$ a subset of $\mathbb{R}^5$ consisting of all vectors an odd number of the entries in which are equal to $0$. Is $W$ a subspace of $\mathbb{R}^5$? Let $W$ be the subset of $\mathbb{R}^5$ consisting of all vectors an odd number of the entries in which are equal to $0$. Is $W$ a subspace of $\mathbb{R}^5$?
I'm not sur... | No. For instance, let $v=(0,1,1,1,1)$ and $w=(1,0,1,1,1)$. Then $v,w\in W$ but $v+w=(1,1,2,2,2)\notin W$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1412177",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that the locus of the poles of tangents to the parabola $y^2=4ax$ with respect to the circle $x^2+y^2-2ax=0$ is the circle $x^2+y^2-ax=0$. Prove that the locus of the poles of tangents to the parabola $y^2=4ax$ with respect to the circle $x^2+y^2-2ax=0$ is the circle $x^2+y^2-ax=0$.
I have encountered this questi... | In the following part of the question,
the locus of the poles of $\left(\text{tangents to the parabola $y^2=4ax$}\right)$ with respect to the circle $x^2+y^2−2ax=0$
note that the $\left(\quad\right)$ part is the polar (with respect to the circle $x^2+y^2-2ax=0$). (in other words, the question means that the tangen... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1412476",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
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Methods of finding the difference to the nearest integer? The question asked to find the "smallest value of n such that $(1+2^{0.5})^n$ is within 10^-9 of a whole number." I'm unsure of the approach to the question.
The question was in the chapter of 'binomial expansion' in the textbook.
Thanks for your time!
| For such problems, the concept of conjugated terms is important.
If you look at the binomial expansion of $(1+\sqrt{2})^n$ and $(1-\sqrt{2})^n$ you will see that all the terms including $\sqrt{2}$ in an odd power will have opposite signs. Hence
$$(1+\sqrt{2})^n+(1-\sqrt{2})^n$$ is always an integer.
But fortunately $|1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1412555",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Minesweeper probability I ran into the situation pictured in the minesweeper game below. Note that the picture is only a small section of the entire board.
Note: The bottom right $1$ is the bottom right corner tile of the board and all other tiles have been marked/deemed safe.
What we know:
*
*There are exactly $2$... | You know
*
*exactly one of A and C is a bomb
*exactly one of B and D is a bomb
*exactly one of A and B is a bomb
So, as you say, there are two possibilities
*
*A and D are bombs while B and C are not
*B and C are bombs while A and D are not
As far as I can tell, these have the same likelihood
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1412652",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 4,
"answer_id": 3
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Spaces $X$ in which every subset is either open or closed, and only $\varnothing$ and $X$ are clopen Let $(X, \tau)$ be a topological space. Then $X, \varnothing \in \tau$ and are both clopen.
But I wonder if it is possible to construct a topological space $X$ in which all subsets are either open or closed, but $X$ and... | Pick any $X$ and $a \in X$.
Define $Y \subset X$ to be open if and only i f$X= \emptyset$ or $a \in Y$. It follows that a set $Y$ is closed if and only if $Y=X$ or $a \notin Y$.
It is easy to show that this is a topology which has the required properties.
It is likely that there is no separable topology with these prop... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1412725",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 4
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Is the intersection of the following closed and open set closed? Generally? Ok, I have been informed that the below lemma is incorrect. I needed it to prove the following statement. Could someone else provide a proof?
Statement: If m(E) is finite, there exists a compact set K with $K\subset E$ and $m(E-K)\le \epsilon$.... | (The OP drastically changed the question after this answer was posted.)
I'm not sure if you intended the $n$ in the superscript of $\mathbb{R}^n$ to be the same as the $n$ in $B_{n}(0)$.
Nevertheless: Your question asks about the intersection of a closed and open set, so I assume that by "ball" you mean an open ball.
T... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1412788",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Subspaces of an infinite dimensional vector space It is well known that all the subspaces of a finite dimensional vector space are finite dimensional.
But it is not true in the case of infinite dimensional vector spaces. For example in the vector space $\mathbb{C}$ over $\mathbb{Q}$, the subspace $\mathbb{R}$ is infini... | You won't find anything like that. If $V$ is an infinite dimensional vector spaces, then it has an infinite basis. Any proper subset of that basis spans a proper subspace whose dimension is the cardinality of the subset. So, since an infinite set has both finite and infinite subsets, every infinite dimensional vector s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1412881",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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What's the answer to this limit question? Can anyone find the limit to this one?
$\lim_{n \rightarrow \infty} (\sum_{i=(n+1)/2}^n {n \choose i} \times 0.51^i \times 0.49^{n-i})$
When I plot it, it seems to me to approach 1, which makes me feel that a limit should exist and it should be 1. But I can't solve this mathema... | Consider tossing an unfair coin $2n-1$ times, where we assign the value $+1$ to head, which has probability $p=0.51$ and $-1$ to tails, with probability $0.49$. So we have a sequence of IID variables $X_1, \dots,X_{2n-1}$, and we denote the $S_n = \sum_{i=1}^{2n-1} X_i$. Furthermore, we have $$\mu = \mathbb{E} X_1 = 0.... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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In the $x + 1$ problem, does every positive integer $x$ eventually reach $1$? I know that the more famous $3x + 1$ problem is still unresolved. But it seems to me like the similar $x + 1$ problem, with the function
$$f(x) = \begin{cases} x/2 & \text{if } x \equiv 0 \pmod{2} \\ x + 1 & \text{if } x \equiv 1 \pmod{2}.\en... | The variant you have described is not hard. For example one can give a formal proof by strong induction.
Here is the induction step. Suppose that we ultimately end up at $1$ if we start at any number $u\lt k$. We show that we end up at $1$ if we start at $k$.
If $k$ is even, in one step we are below $k$, and we are fin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1413097",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Why is $p_n \sim n\ln(n)$? I know that the prime number theorem states that the number of primes less than $x$ is approximately $\frac{x}{\ln(x)}$. However, why does this mean that $p_n \sim n\ln(n)$? (where $p_n$ is the $n$-th prime). If we replace $x$ with $p_n$ in the original equation, we have that $\pi(p_n) \ln({p... | Made a jpeg, see if it comes out readable: Theorem 3 says that putting in the extra $\log \log n$ term is quite good.
Theorem 1. We have $$\frac{x}{\log x}\left(1 + \frac{1}{2 \log x}\right) < \pi(x)\qquad \text{for}\,59\leqq x, \tag{3.1}\label{3.1}$$ $$\pi(x) < \frac{x}{\log x}\left(1 + \frac{3}{2 \log x}\right) \qqu... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 0
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Sketching phase portrait of an ellipse I have a system of linear ODE's as follows:
$$\frac{dx}{dt} = y, \frac{dy}{dt} = -4x$$
which has solution
$$\begin{bmatrix}x\\y\end{bmatrix} = \alpha\begin{bmatrix}\cos2t\\-2\sin2t\end{bmatrix} + \beta\begin{bmatrix}\sin2t\\2\cos2t\end{bmatrix}$$
And I'm having some issues trying... | $$
\frac{d}{dt}\{ 4x^{2}+y^{2}\} = 8x\frac{d x}{dt}+2y\frac{dy}{dt}=8xy-8yx=0.
$$
Therefore, $4x^{2}+y^{2}=C$ is constant in time.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1413265",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Find groups that contain elements $a$ and $b$ such that $|a|=|b|= 2$ and $|ab|=5$ Find groups that contain elements $a$ and $b$ such that $|a|=|b|= 2$ and $|ab|=5$
My thoughts:
$|a|=|b|=2\implies a^2=e$ and $b^2=e$
I see that the group cannot be abelian as the order wont be greater than $2$ : $(ab)^2 = a^2b^2=e$
Not ... | Hint: A regular pentagon can be rotated and reflected
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1413388",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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invariant properties between p-group and it's automorphism Let $G$ be a p-group and $Aut(G)$ be group of automorphisms of $G$
which properties of $G$ can help us with studding $Aut(G)$?
for example If $G$ is infinite/finite does this guaranty $Aut(G)$ be infinite/finite?
or knowing order of $G$ can help us with making ... | Any infinite torsion group has infinite automorphism group. This is (allegedly) proved in R. Baer, Finite extensions of abelian groups with minimum condition, Trans. Amer. Math.
Soc. 79 (1955), 521-540.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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We all use mathematical induction to prove results, but is there a proof of mathematical induction itself? I just realized something interesting. At schools and universities you get taught mathematical induction. Usually you jump right into using it to prove something like
$$1+2+3+\cdots+n = \frac{n(n+1)}{2}$$
However... | Suppose that $ 1 \in S \text{ and that } n \in S \Rightarrow n+1 \in S$, and suppose that there is a smallest number $N$ such that $N+1\notin S$...
Supposing that any subset of natural numbers has a minimal element, the above proves the principle of induction in set theory, whether some people like it or not.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "49",
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Finding the minimal polynomial of a linear operator Let $P=\begin{pmatrix}
i & 2\\
-1 & -i
\end{pmatrix}$ and $T_P\colon M_{2\times 2}^{\mathbb{C}} \to M_{2\times 2}^{\mathbb{C}}$ a linear map defined by $T_P(X)=P^{-1}XP$. I need to find the minimal polynomial of $T_P$.
I was able to find the minimal polynomial using... | It's given as a hint that $P^2+3I = 0$, so:
$$
\begin{align}
&P^2 = -3I \\
&P\cdot-\frac{1}{3}P = I \\
\therefore \quad &P^{-1} = -\frac{1}{3}P
\end{align}
$$
So, $T_P$ is actually
$$
T_P(X)=-\frac{1}{3}PXP
$$
Note that $P^2=-3I$ and we have two $P$s in the above expression for $T_P$, so it might be useful to calculate... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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limit of sum $\dfrac{(-1)^{n-1}}{2^{2n-1}}$ What is: $$\sum^{\infty}_{n=1}\dfrac{(-1)^{n-1}}{2^{2n-1}}$$
I have done a Leibniz convergence test and proved that this series converges, but I do not know how to find the limit. Any suggestions?
| We can appeal to the sum of a geometric series as follows:
$$\begin{align}
\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{2^{2n-1}}&=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2^{2n+1}}\\\\
&=\frac12\sum_{n=0}^{\infty}\left(\frac{-1}{4}\right)^n\\\\
&=\frac12\frac{1}{1+\frac14}\\\\
&=\frac25
\end{align}$$
| {
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Finding Maximum in a Set of Numbers If I have a set of $n$ numbers: $(a_1,..., a_n)$, then how can I find the two maximum numbers in the set? Suppose that all the numbers are positive integers.
| Completely edited (Misread your question, and you changed your question). The minimum of two numbers is given by:
$\min(a,b) = \large \frac{|a+b|- |a-b|}2$
Of three numbers thus:
$$\min(a,b,c) = \min(\min(a,b),c) = \frac{\left| \frac{|a+b|- |a-b|}2 + c \right| - \left| \frac{|a+b|- |a-b|}2 - c\right| }2 $$
Since you wa... | {
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"source": "stackexchange",
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Branch cut of $\sqrt{z}$ In my complex analysis book, the author defines the $\sqrt{z}$ on the slit plane $\mathbb{C}\setminus (-\infty,0]$. I understand this is done because $z^2$ is not injective on the entire complex plane, so we restrict the domain of $z^2$ to $(\frac{-\pi}{2},\frac{\pi}{2})$, and then $z^2$ maps b... | The domain of $w=f(z)=z^2$ is not restricted to $(\frac{-\pi}{2},\frac{\pi}{2})$ to obtain an inverse. Rather, the interval $[0,\pi ]$ (or $[\pi ,2\pi ]$ is obtained as the image of a bijection from the slit $w$ plane. This map is then a branch of the inverse.
The procedure is as follows:
it's no harder to look at th... | {
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"source": "stackexchange",
"question_score": "2",
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Can 720! be written as the difference of two positive integer powers of 3? Does the equation:
$$3^x-3^y=720!$$
have any positive integer solution?
| If $$720!=3^x-3^y=3^y\left(3^{x-y}-1\right)$$ and since the power of $3$ dividing $720!$ is $$\left\lfloor\frac{720}{3}\right\rfloor+\left\lfloor\frac{720}{9}\right\rfloor+\left\lfloor\frac{720}{27}\right\rfloor+\left\lfloor\frac{720}{81}\right\rfloor+\left\lfloor\frac{720}{243}\right\rfloor=240+80+26+8+2=356\text{,}$$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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Intuition for a ring homomorphism? A map $f: A \to B$ between two rings $A$ and $B$, is called a ring homomorphism if $f(1_A) = 1_B$, and one has $f(a_1 + a_2) = f(a_1) + f(a_2)$ and $f(a_1 \cdot a_2) = f(a_1) \cdot f(a_2)$, for any $a_1, a_2 \in A$.
My question is, what is the intuition for ring homomorphisms? My gras... | Is $482350923581014689 + 51248129432153$ an even or odd number? Of course it is even because adding two odd numbers gives an even number right? You didn't have to calculate.
Is $1254346425649847 \times 64341341232606942$ an even or odd number? Well, an odd number times an even number must be even. Again, you didn't hav... | {
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Project point on plane - Unique identfier? I have a number of planes (in $\mathbb{R}^3$), each represented by a point $\vec{P_i}$ which lies within each plane and the normal vector $\vec{n_i}$.
If I project a point $\vec{Q}$ (which does not lie within any of the planes) onto each plane, the resulting projected points a... | By projecting $\vec Q$ onto eaqch plane, I assume you mean orthogonal projection. So the question is, can one point projected onto two different planes result in the same image point. For that to be possible, the planes in question have to intersect, otherwise they don't have any common points at all. But if they inter... | {
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Is it possible disconnected graph has euler circuit? I have doubt !
Wikipedia says :
An Eulerian graph is one in which all vertices have even degree;
Eulerian graphs may be disconnected.
What I know :
Defitition of an euler graph
"An Euler circuit is a circuit that uses every edge of a graph exactly once. ▶ An... | The other answers answer your (misleading) title and miss the real point of your question.
Yes, a disconnected graph can have an Euler circuit. That's because an Euler circuit is only required to traverse every edge of the graph, it's not required to visit every vertex; so isolated vertices are not a problem. A graph i... | {
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Evaluate the summation $\sum_{k=1}^{n}{\frac{1}{2k-1}}$ I need to find this sum $$\sum_{k=1}^{n}{\frac{1}{2k-1}}$$ by manipulating the harmonic series.
I have been given that $$\sum_{k=1}^{n}{\frac{1}{k}} = \ln(n) + C$$ where $C$ is a constant. I have given quite a bit of thought to it but have not been able to arrive... | $$\sum_{k=1}^{n}\frac{1}{2k-1}=\sum_{k=1}^{n}\frac{1}{2k}+\sum_{k=1}^{n}\frac{1}{2k(2k-1)}=\frac{H_n}{2}+\log(2)-\sum_{k>n}\frac{1}{2k(2k-1)}$$
hence it follows that your sum behaves like $\frac{1}{2}\log(n)+O(1)$.
Through the above line and the asymptotics for harmonic numbers, we can say even more:
$$\sum_{k=1}^{n}\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Polynomial tending to infinity Take any polynomial $(x-a_1)(x-a_2)\ldots(x-a_n)$ with roots $a_1, a_2,\ldots,a_n$ where we order them so that $a_{i+1}>a_i$ is increasing so $a_n$ is the biggest root. It doesn't matter whether the polynomial is of odd or even degree or whether there is a minus sign in front of highest ... | It is the product of positive and strictly increasing functions.
Alternatively: The derivative of $p$ has degree $n-1$ and at least one root in each interval $(a_i,a_{i+1})$ (Rolle). Hence it has no root besides these, especially it does not change signs beyond $x=a_n$.
| {
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"source": "stackexchange",
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Closure set of closure set intersection - formal languages Can we say...
Given L1 and L2.
Is the following true?
$$
L_{1}^{*} \cap L_{2}^{*} = (L_{1}^{*} \cap L_{2}^{*})^{*}
$$
I think it is true but I can't be sure. What permutation of either wouldn't be contained in each other. Sorry I don't know how to format on Mat... | Let $L$ be a language of the free monoid $A^*$. The answer easily follows form the fact that $L = L^*$ if and only if $L$ is a submonoid of $A^*$.
Now $L_1^*$ and $L_2^*$ are submonoids of $A^*$ and hence their intersection is a submonoid of $A^*$. Thus $L_1^* \cap L_2^*$ is a submonoid of $A^*$ and it is equal to its ... | {
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Isolate Costs in NPV equation Hey can anyone help with this? This is the classic NPV equation:
NPV = -CapEx + ∑ (Revenue − Costs) / (1+Discount)^i
The partial sum is from i = 0 to n years.
For my purposes all the elements are know except costs.
I need to isolate costs in this equation. Is this possible?
Thanks,
Mike... | Let $\frac{1}{\texttt{1+discount}}=a$. The equation becomes
$NVP=-CapEx+\sum_{i=0}^n\texttt{(Revenue-Costs)}\cdot a^i $
$NVP=-CapEx+\texttt{(Revenue-Costs)}\cdot\sum_{i=0}^n a^i $
$\sum_{i=0}^n a^i $ is the partial sum of a geometric series.
$\sum_{i=0}^n a^i =\frac{1-a^{n+1}}{1-a}$ Therefore
$NVP=-CapEx+\texttt{(Reve... | {
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Limit $n \rightarrow \infty \frac{n}{e^x-1} \sin\frac{x}{n}$ I am just working through some practice questions and cannot seem to get this one. Plugging this into wolfram alpha I know the limit should be $\frac{x}{e^x-1}$, but I am having a bit of trouble working through this.
I have tried to rewrite the limit as $\fra... | With equivalents, this is trivial: $\;\sin \dfrac xn\sim \dfrac xn\enspace(n\to\infty)$, hence
$$\frac n{\mathrm e^x -1}\,\sin \frac xn\sim\frac n{\mathrm e^x -1}\frac xn=\frac x{\mathrm e^x -1}.$$
| {
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"timestamp": "2023-03-29T00:00:00",
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Show that $k[x,y]/(xy-1)$ is not isomorphic to a polynomial ring in one variable.
Let $R=k[x,y]$ be a polynomial ring ($k$, of course, is a field). Show that
$R/(xy-1)$ is not isomorphic to a polynomial ring in one variable.
| Suppose $R\simeq K[T]$, where $K$ is a commutative ring. Then $K$ is an integral domain and since $\dim K[T]=1$ we get $\dim K=0$ (why?). It follows that $K$ is a field. So $k[X,X^{-1}]\simeq K[T]$. Now use this answer.
| {
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"source": "stackexchange",
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Is $V$ a simple $\text{End}_kV$-module? Let $V$ be a finite-dimensional vector space over $k$ and $A = \text{End}_k V$. Is $V$ a simple $A$-module?
| Simply look at the definition: Recall that a module $M$ over a ring $R$ is simple if the only non-zero submodule of $M$ is $M$ itself. Equivalently, for any $x\neq 0$ in $M$, the submodule $Rx=\left\{rx:r\in R\right\}$ generated by $x$ is $M$.
Now in your case: Take a nonzero $x\in V$ and any $y\in V$. Is it true that ... | {
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How to calculate the closed form of the Euler Sums We know that the closed form of the series
$$\sum\limits_{n = 1}^\infty {\frac{1}{{{n^2}\left( {\begin{array}{*{20}{c}}
{2n} \\
n \\
\end{array}} \right)}}} = \frac{1}{3}\zeta \left( 2 \right).$$
but how to evaluate the following series
$$\sum\limits_{n = 1}^... | Using the main result proved in this question,
$$ \sum_{n\geq 1}\frac{x^n}{n^2 \binom{2n}{n}}=2 \arcsin^2\left(\frac{\sqrt{x}}{2}\right)\tag{1}$$
it follows that:
$$ \sum_{n\geq 1}\frac{H_n}{n^2 \binom{2n}{n}}=\int_{0}^{1}\frac{2 \arcsin^2\left(\frac{\sqrt{x}}{2}\right)-\frac{\pi^2}{18}}{x-1}\,dx \tag{2}$$
and:
$$ \sum... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$(a,b) \mathbin\# (c,d)=(a+c,b+d)$ and $(a,b) \mathbin\&(c,d)=(ac-bd(r^2+s^2), ad+bc+2rbd)$. Multiplicative inverse? Let $r\in \mathbb{R}$ and let $0\neq s \in \mathbb{R}$. Define operations $\#$ and $\&$ on $\mathbb{R}$ x $\mathbb{R}$ by $(a,b) \mathbin\#(c,d)=(a+c,b+d)$ and $(a,b) \mathbin\&(c,d)=(ac-bd(r^2+s^2), ad+... | The addition and multiplication formulas somewhat remind of the formulas for complex number. We suspect that a map of the form
$$ \phi\colon (x,y)\mapsto \alpha x+\beta y$$
for suitable $\alpha,\beta\in\mathbb C$ turns out to be a field isomorphism. As we want $(1,0)\mapsto 1$, we conclude $\alpha=1$.
Then
$$ \phi((a,... | {
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"timestamp": "2023-03-29T00:00:00",
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Similarities Between Derivations of Fourier Series Coefficients (odd, even, exponential) I've recently started learning about Fourier series from this set of tutorials: (http://lpsa.swarthmore.edu/Fourier/Series/WhyFS.html) and a few other sources.
While chugging through the material, I noticed and was intrigued by the... | Yes, there is something profound. Spaces of real and complex valued functions can be viewed as infinite dimensional vector spaces since they can be added and multiplied by numbers. In some of these spaces you can introduce scalar product (which allows you to introduce distances and angles). What you are doing when you ... | {
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Give an example of a structure of cardinality $\omega_2$ that has a substructure of $\omega$ but no substructure of $\omega_1$
Give an example of a structure of cardinality $\omega_2$ that has a substructure of $\omega$ but no substructure of $\omega_1$
This is from Hodges' A Shorter Model Theory.
My idea is to take... | HINT: Using only relations won't work, since given a language with only relation symbols, and a structure to the language, every subset of the domain defines a substructure. So using functions is necessary here.
Let $\{f_\alpha\mid\alpha<\omega_2\}$ be a list of $\omega_2$ unary function symbols. For sake of concretene... | {
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Vector space over an infinite field which is a finite union of subspaces
Let $V=\bigcup_{i=1}^n W_i$ where $W_i$'s are subspaces of a vector space $V$ over an infinite field $F$. Show that $V=W_r$ for some $1 \leq r \leq n$.
I know the result "Let $W_1 \cup W_2$ is a subspace of a vector space $V$ iff $W_1 \subseteq ... | The general result is that if $\lvert K\rvert\ge n$, $V$ cannot be the union of $n$ proper subspaces (Avoidance lemma for vector spaces).
We'll prove that if $V=\displaystyle\bigcup_{i=1}^n W_i$ and the $W_i$s are proper subspaces of $V$, then $\;\lvert K\rvert\le n-1$.
We can suppose no subspace is contained in the u... | {
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"timestamp": "2023-03-29T00:00:00",
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How does uniqueness of the additive inverse imply that $-(ax) = (-a)x$?
How does uniqueness of the additive inverse imply that $-(ax) = (-a)x$?
In my title, I should be clear that the additive inverse should be unique. But how does this help?
I dont even get why uniqueness of a negative number matters... Is there a t... | Hint: Use distributivity to show that $ax+(-(ax))=0$, then apply uniqueness.
| {
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Prove that $1^2 + 2^2 + ..... + (n-1)^2 < \frac {n^3} { 3} < 1^2 + 2^2 + ...... + n^2$ I'm having trouble on starting this induction problem.
The question simply reads : prove the following using induction:
$$1^{2} + 2^{2} + ...... + (n-1)^{2} < \frac{n^3}{3} < 1^{2} + 2^{2} + ...... + n^{2}$$
| If you wish a more direct application of induction, we have that if
$$
1^2+2^2+\cdots+(n-1)^2 < \frac{n^3}{3}
$$
then
$$
1^2+2^2+\cdots+n^2 < \frac{n^3+3n^2}{3}
< \frac{n^3+3n^2+3n+1}{3} = \frac{(n+1)^3}{3}
$$
Similarly, if
$$
1^2+2^2+\cdots+n^2 > \frac{n^3}{3}
$$
then
$$
1^2+2^2+\cdots+(n+1)^2 > \fr... | {
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"timestamp": "2023-03-29T00:00:00",
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Largest Number that cannot be expressed as 6nm +- n +- m I'm looking to find out if there is a largest integer that cannot be written as $6nm \pm n \pm m$ for $n,m$ elements of the natural numbers. For example, there are no values of $n,$m for which $6nm \pm n \pm m = 17.$ Other numbers which have no solution are $25... | This question is equivalent to the Twin Prime Conjecture.
Let's look at the three cases:
$$6nm+n+m =\frac{(6n+1)(6m+1)-1}{6}\\
6nm+n-m = \frac{(6n-1)(6m+1)+1}{6}\\
6nm-n-m = \frac{(6n-1)(6m-1)-1}{6}
$$
So if $N$ is of one of these forms, then:
$$6N+1=(6n+1)(6m+1)\\
6N-1=(6n-1)(6m+1)\\
6N+1=(6n-1)(6m-1)$$
If $6N-1$ and ... | {
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Geometric proof of $QM \ge AM$ Prove by geometric reasoning that:
$$\sqrt{\frac{a^2 + b^2}{2}} \ge \frac{a + b}{2}$$
The proof should be different than one well known from Wikipedia:
DISCLAIMER: I think I devised such proof (that is not published anywhere to my knowledge), but I believe it is too complex (and not that... | What about something like this:
$\hspace{80pt}$
The two right triangles have common base, which is also the diameter
of the red circle. The endpoints of the black diameter are also
the foci of the ellipses, and their size is determined by the tip of the respective triangle.
Observe, that the minor radius of the blue el... | {
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"timestamp": "2023-03-29T00:00:00",
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Calculate simple expression: $\sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}}$ Tell me please, how calculate this expression:
$$
\sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}}
$$
The result should be a number.
I try this:
$$
\frac{\left(\sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}}\right)\left(\sqrt[3]{\left(2 + \sq... | $(\sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}} )^3 \\
=(\sqrt[3]{2 + \sqrt{5}})^3+(\sqrt[3]{2 - \sqrt{5}} )^3+3(\sqrt[3]{2 + \sqrt{5}} ) (\sqrt[3]{2 - \sqrt{5}} )(\sqrt[3]{2 + \sqrt{5}} +\sqrt[3]{2 - \sqrt{5}} ) $
S0 $s^3=4-3s$ From this we get S = 1
| {
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"timestamp": "2023-03-29T00:00:00",
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When do circles on the sides of a triangle intersect?
If you have a non degenerate triangle and two of the sides are chords of two respective circles, then under what conditions do the circles intersect at two distinct points?
I'm having trouble understanding why this condition holds in a step of a proof of a theorem... | Let we say that $\Gamma_C$ is a circle through $A,B$ and $\Gamma_B$ is a circle through $A,C$. Since $\Gamma_B$ and $\Gamma_C$ meet at $A$, they have for sure a second intersection point, unless they are tangent at $A$. That implies that their centres $O_B,O_C$ and $A$ are collinear, so if $P,Q$ are the remote points o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1416839",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Combinatorics question on group of people making separate groups If there are $9$ people, and $2$ groups get formed, one with $3$ people and one with $6$ people (at random), what is the probability that $2$ people, John and James, will end up in the same group?
I'm not sure how to do this. So far, I've got:
The total n... | P(both in same group) = P(both in group of 3) + P(both in group of 6)
$$ =\frac{{3\choose 2}+{6\choose 2}}{9\choose 2} = \frac12$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1416939",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How is $\mathbb N$ actually defined? I know perfectly well the Peano axioms, but if they were sufficient for defining $\mathbb N$, there would be no controversy whether $0$ is a member of $\mathbb N$ or not because $\mathbb N$ is isomorphic to a host of sub- and supersets of $\mathbb Z$.
So it seems that mathematicians... | In the set theory course I took, natural numbers were not defined by Peano's axioms. One can define an inductive set in this manner: $I$ is inductive set if
$$\emptyset\in I\ \wedge\ \forall x\in I\ ((x\cup \{x\})\in I) $$
We say that $n$ is natural number if $n\in I$ for any inductive set $I$. Some examples of natura... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Find $\sum\limits_{n=1}^{\infty}\frac{n^4}{4^n}$ So, yes, I could not do anything except observing that in the denominators, there is a geometric progression and in the numerator, $1^4+2^4+3^4+\cdots$.
Edit: I don't want the proof of it for divergence or convergence only the sum.
| \begin{align}
n^4 = {} & \hphantom{{}+{}} An(n-1)(n-2)(n-3) \\
& {} + B n(n-1)(n-2) \\
& {} + C n(n-1) \\
& {} + D n
\end{align}
Find $A,B,C,D$. Then
\begin{align}
n^4 x^n & = An(n-1)(n-2)(n-3) x^n + \cdots \\[10pt]
& = A\frac{d^4}{dx^4} x^n + B \frac{d^3}{dx^3} x^n + \cdots
\end{align}
Then add over all values of $n$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Proving $[(P\lor Q)\land(P\to R)\land(Q\to R)]\to R$ is a tautology without using a truth table?
$$[(P\lor Q)\land(P\to R)\land(Q\to R)]\to R\tag{1}$$
How can I prove that $(1)$ is a tautology without using a truth table?
I used the identity $$(P\to R)\land(Q\to R)\equiv(P\lor Q)\to R$$ but from there I get stuck and... | $$
\begin{array}{ll}
&(P\vee Q) \wedge (P \Rightarrow R) \wedge (Q \Rightarrow R)\\
\equiv&\hspace{1cm}\{ \text{ by the tautology mentioned in the question }\}\\
&(P\vee Q) \wedge ( (P\vee Q) \Rightarrow R) \\
\equiv&\hspace{1cm}\{ \text{ } A \wedge (A \Rightarrow B ) \equiv A \wedge B, \text{ see below }\}\\
&(P\vee Q... | {
"language": "en",
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"source": "stackexchange",
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a bijection is an injective (one-to-one) , surjective (onto) map between sets. if S = (0, 1) and T =R, find a map from S to T which is a bijection is an injective (one-to-one) , surjective (onto) map between sets. if S = (0, 1) and T =R, find a map from S to T which is
my effort
1)
(a) f(x) = x is a one to one funct... | For part c), possible answers are $\displaystyle f(x)=\frac{2x-1}{x(x-1)},\;f(x)=\frac{\ln 2x}{x-1},\;$ or $f(x)=\tan((x-\frac{1}{2})\pi)$,
which is equivalent to your answer.
For part b), possible answers are $\displaystyle f(x)=\frac{(4x-1)(2x-1)(4x-3)}{x(1-x)}\;$ or
$\hspace{2.1 in}f(x)=\tan((\pi(2x-\frac{1}{2}))$... | {
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"timestamp": "2023-03-29T00:00:00",
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Deriving the formula of the Surface area of a sphere My young son was asked to derive the surface area of a sphere using pure algebra. He could not get to the right formula but it seems that his reasoning is right. Please tell me what's wrong with his logic.
He reasons as follows:
*
*1.Slice a sphere into thin circl... | An interesting proof may go through the following lines:
*
*If $S(R)$ and $V(R)$ are the surface area and volume of a sphere with radius $R$, we have $S(R)=\frac{d}{dR} V(R)$ by triangulating the surface and exploiting $S(\alpha R)=\alpha^2 S(R), V(\alpha R)=\alpha^3 V(R)$ for any $\alpha > 0$, so the problem boils ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 1
} |
Modular Arithmetic and prime numbers With respect to the maths behind the Diffie Hellman Key exchange algorithm.
Why does:
(ga mod p)b mod p = gab mod p
It might be fairly obvious, but what basic maths guarantees this? Why does the first modulo p expression disappear from the LHS in the expression on the RHS.
Apologi... | We can actually use the binomial theorem:
Observe that if we write $g^a=m+pk$, where $m\equiv g^a\mod p$, then $g^{ab}=(m+pk)^b=\sum_{n=0}^b \binom{b}{n} m^n(pk)^{n-b}\equiv m^b\mod p$, as every term except for the $n=b$ term will we divisible by $p$. Therefore, $(g^a\mod p)^b\mod p\equiv m^b\mod p$ and on the other ha... | {
"language": "en",
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Number of divisors $d$ of $n^2$ so that $d\nmid n$ and $d>n$ I just wanted to share this nutshell with you guys, it is a little harder in this particular case of the problem:
Find the number of divisors $d$ of $a^2=(2^{31}3^{17})^2$ so that $d>a$. What is the solution for the general case?
| I think the number of divisors is (17+1)(31+1)-1=575.
d/a^2 and d>a.
{1,2,2^2,...,2^31}{1,3,3^2,...,3^17}={(1,1),(1,3),(1,3^2),...,(1,2),(1,2^2),...}
From (e,f) then ef is one divisor of a.
d=ef*a is devisor of a^2.
| {
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"source": "stackexchange",
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Residue of $g(z)g'(z)$
I know how to use the residue theorem and the winding number to find the residue of $f$, but I have no idea how to relate the residue of $g$ to that of $g\cdot g'$, especially without knowing anything about the order of the poles.
| Hint: We can apply a substitution write
$$
\int_{\gamma} g(z)g'(z)\,dz=
\int_{g(\gamma)} u\,du
$$
Of course, the function $\phi(u) = u$ is entire.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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there does not exist a perfect square of the form $7\ell+3$ I have been trying to prove that there does not exist a perfect square of the form $7\ell+3$. I've tried using $n$ as even or odd, and I'm getting stuck. Can someone put me on the path? Is this an equivalence class problem?
| For any integer $a,$
$$a\equiv0,\pm1,\pm2,\pm3\pmod7\implies a^2\equiv0,1,4,9\equiv2$$
$$\implies a^2\not\equiv3(=7-4),5(=7-2),6(=7-1)\pmod7$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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checking definition of bounded linear function involves operator maps between different spaces Let $H$ and $K$ be two Hilbert spaces. Let $T:K\to H$ be a bounded linear operator. Denote the inner products on $H$ and $K$ by $\langle\cdot,\cdot\rangle_H$, $\langle\cdot,\cdot\rangle_K$. Fix any $y\in H$; then the linear m... | Hint: How do you prove that $y^*\colon H\to {\bf C}$ defined by $y^*(y'):=\langle y',y\rangle$ is bounded? Or, if you know what an adjoint operator is, just notice that $\Phi$ is just $T^*(y^*)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1418120",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Currency conversion using available rates I have the following rates available:
USD -> USD = 1
USD -> EUR = 0.887662
USD -> GBP = 0.654514
I want to calculate the following rates
GBP -> EUR
EUR -> GBP
Using only the first 3 rates.
My process so far has been:
1 USD / 0.654514 GBP = 1.527851
So
1 GBP = 1.53 USD
1 USD ... | We know the exchange rates $\frac{$}{€}=0.887662$ and $\frac{$}{£}=0.654514$. So we have
$$
\frac{€}{£}=\frac{$}{£}\times\frac{€}{$}=\frac{$}{£}\times \frac{1}{$/€}=0.654514\times \frac{1}{0.887662}=\frac{0.654514}{0.887662}=0.737346
$$
and
$$
\frac{£}{€}= \frac{1}{€/£}=\frac{1}{0.737346}=1.356215
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1418245",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is the Lie algebra morphism induced by surjective Lie group morphism also surjective? Let $G$ be a matrix lie group and $\Pi : G\to \Pi(G)$ a surjective Lie group morphism. Let $\mathfrak g$ and $\mathfrak h$ be the respective Lie algebras of $G$ and $\Pi(G)$. Then there is a unique morphism of Lie algebras $\pi : \m... | Using a dimension argument and Sard theorem (or anything to assure that the identity is a regular value of $\Pi$ ), the dimension of G is the sum of the dimensions of the kernel and image of $\Pi$ and the same holds for $\pi$ so it remains to show that the dimensions of the kenerls coincide, using that $\Pi ( e^X) = e^... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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verification that simplification in textbook is incorrect I am using a textbook that asks for the following expression to be simplified:
${9vw^3 - 7v^3w - vw^2}$
The answer given is:
${8vw^2 - 7v^3w}$
which seems wrong because I did not think you could subtract 2 similar terms with different polynomials.
| The answer given is wrong: $9vw^3$ and $-vw^2$ are not like terms, so they cannot be added to make a single term. Maybe the problem was supposed to be $9vw^2 - 7v^3w - vw^2$?
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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An online calculator that can calculate a sum of binomial coefficients Is there any online calculator that can calculate
$$\dfrac{\sum_{k=570}^{770} \binom{6,700}{k}\binom{3,300}{1,000-k}}{\binom{10,000}{1,000}}
$$
for me? There are a few binomial coefficient calculators but for the sum in the numerator there are not u... | Typing in the following input into WolframAlpha:
Sum[Binomial[6700,k]Binomial[3300,1000-k]/Binomial[10000,1000],{k,570,770}]
yields an exact result followed by the approximate decimal value, which to 50 digits is
$$0.99999999999855767906391784086205133574169750988724.$$
The same input in Mathematica will yield the sa... | {
"language": "en",
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Show that $\csc^n\frac{A}{2}+\csc^n\frac{B}{2}+\csc^n\frac{C}{2}$ has the minimum value $3.2^n$ Show that in a $\Delta ABC$, $\sin\frac{A}{2}\leq\frac{a}{b+c}$
Hence or otherwise show that $\csc^n\frac{A}{2}+\csc^n\frac{B}{2}+\csc^n\frac{C}{2}$ has the minimum value $3.2^n$ for all $n\geq1$.
In this problem, I have p... | As a different approach, Lagrange multipliers work here...the constraint function $$A+B+C=\pi$$ has gradient $(1,1,1)$. Hence at an "internal" critical point the three partials of your function must be equal. Note that the boundary doesn't matter...the boundary will have one or more angles equal to $0$ and the functi... | {
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Proof - Uniqueness part of unique factorization theorem The uniqueness part of the unique factorization theorem for integers says that given any integer $n$, if $n=p_1p_2 \ldots p_r=q_1q_2 \ldots q_s$ for some positive integers $r$ and $s$ and prime numbers $p_1 \leq p_2 \leq \cdots \leq p_r$ and $q_1 \leq q_2 \leq \cd... | The OP can simply state in their proof that they 'arrived' at the following 'situation':
$\tag 1 n = p_1p_2 \ldots p_r \text{ and } n=q_1q_2 \ldots q_s \text{ and } p_i \neq q_j \text { for any integers } i \text{ and } j$
In words, there exist an integer $n$ with two prime factorizations containing no common prime fac... | {
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"timestamp": "2023-03-29T00:00:00",
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Proving convergence of a series and then finding limit I need to show that $\sqrt{2}$, $\sqrt{2+\sqrt{2}}$, $\sqrt{2+\sqrt{2+\sqrt{2}}},\ldots$
converges and find the limit.
I started by defining the sequence by $x_1=\sqrt{2}$ and then $x_{n+1}=\sqrt{2+x_n}$. Then I proved by induction that the sequence is increasing ... | Prove that this sequence is increases and bounded. Hence it convergence to some point $p\in \mathbb{R}$ and make limit transition in $x_{n+1}=\sqrt{2+x_n}$
| {
"language": "en",
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The Galois connection between topological closure and topological interior [Update: I changed the question so that $-$ is only applied to closed sets and $\circ$ is only applied to open sets.]
Let $X$ be a topological space with open sets $\mathcal{O}\subseteq 2^X$ and closed sets $\mathcal{C}\subseteq 2^X$. Consider t... | Put $A=B=S$, where $S$ is any subset of $X$. Then
$$
\overline S\subseteq S \iff S\subseteq\mathring S
$$
hence
$$
S \text{ is closed} \iff S \text{ is open}
$$
Conversely, assume that every open set is closed (which implies that any closed set is open). If $A$ and $B$ are subsets of $X$ such that $\overline A ⊆ B$, th... | {
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Log function solve for x The function is defined by $y=f(x)=3e^{{1\over3}x+1}$
Solve for $x$ in terms of $y$
My answer:
$$x={\ln({y\over3})-1\over3}$$
Is this the correct way to go about this question?
Update. Finding the inverse and range.
Would the inverse of this function be:
$$f^{-1}(x)=3(\ln({x\over3})-1)$$
and it... | The steps are right. Just don't put the 3 in the denominator at last, multiply 3.
| {
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What is a mock theta function? We define a mock theta function as follows:
A mock theta function is a function defined by a $q$-series convergent
when $|q|<1$ for which we can calculate asymptotic formulae when $q$
tends to a rational point $e^{2\pi ir/s}$ of the unit circle of the
same degree of precision as th... | Since 2008, the number theory community has decided on a bit more "scientific" definition of mock theta function. However, given the context in which you asked the question, I don't assume you want the technical definition of the holomorphic part of a weak Maas form (whatever that means I just need to say this to kee... | {
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"timestamp": "2023-03-29T00:00:00",
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If $\lambda=$ measure of a set and all $G_k$'s are open sets, then : $\lambda ( \cup_{k=1}^{\infty} G_k ) \le \sum _{k=1}^{\infty}\lambda ( G_k)$ I just started reading the book Lebesgue Integration on Euclidean Spaces by Frank jones, in which the author gives a result and it's proof as : the If $\lambda$ denotes the m... | I don't see a problem. In the example you give, assuming $P = I_1 \cup I_2 \cup I_3$, we would take $P_1 = P_2 = \emptyset$, $P_3 = I_1 \cup I_2$, and $P_4 = I_3$, and then $P_5 = P_6 = \dots = \emptyset$.
Maybe you are misinterpreting the sentence "Since each $I_j$ is contained in one of $G_1, G_2, \dots$." What Jon... | {
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Checking whether the result is positive definite or positive semi-definite with two methods Given, $$A = \begin{bmatrix}
1 &1 & 1\\
1&1 & 1\\
1& 1& 1
\end{bmatrix}.$$
I want to see if the matrix $A$ positive (negative) (semi-) definite.
Using Method 1:
Define the quadratic form as $Q(x)=x'Ax$.
Let $x \in \math... | First is wrong. $Q(x)$ doesn't mean $A$ is positive definite.
Moreover $rank(A)=1$. Since $A$ is not of full rank, it can not be positive definite.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1419348",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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can real line be written as a disjoint unions of set with cardinality 5 Can the real line be written as a disjoint union of sets with cardinality 5?
I tried using to write it as a set of sets.. but I didn't end up to a good position.
| If we allow the axiom of Choice, then one can show that the cardinality of $\mathbb R \times 5$ is equal to the cardinality of $\mathbb R$.
Now, $card(\mathbb R \times 5)$ is the same as $card(\sum_{i \in \mathbb R}5) $. This indicates that there exists a correspondence between $\mathbb R$ and $\mathbb R$ disjoint set... | {
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"timestamp": "2023-03-29T00:00:00",
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Subgroups of generalized dihedral groups A generalized dihedral group, $D(H) := H \rtimes C_2$, is the semi-direct product of an abelian group $H$ with a cyclic group of order $2$, where $C_2$ acts on $H$ by inverting elements.
I know that the total number of subgroups of $D(H)$ is the number of subgroups of $H$ plus ... | Let $K$ be a subgroup of $D(H)$.
If $K$ is not contained in $H$, then $K$ contains some $g \not\in H$. Then $g$ is an involution and $D(H) = H \rtimes \langle g \rangle$. Hence $K = (H \cap K) \rtimes \langle g \rangle$ and $K \cong D(H \cap K)$.
| {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "3",
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What does it mean for a polynomial to be positive semi-definite? I'm reading proofs about Cauchy-Schwarz inequality.
Some of them use the argument of a polynomial being positive semi-definite as the starting argument.
That is, they argue that
$$\forall x,y \in\mathbb{R^n}, a\in\mathbb{R}\\\langle {ax+y,ax+y} \rangle$$
... | It means that the expression $$\langle ax+y, ax+y\rangle$$
is always greater than or equal to zero.
This is true because the inequality $\langle x,x\rangle \geq 0$ is true for any element $x$ in your vector space.
| {
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Proof by induction for "sum-of" Prove that for all $n \ge 1$:
$$\sum_{k=1}^n \frac{1}{k(k+1)} = \frac{n}{n+1}$$
What I have done currently:
Proved that theorem holds for the base case where n=1.
Then:
Assume that $P(n)$ is true. Now to prove that $P(n+1)$ is true:
$$\sum_{k=1}^{n+1} \frac{1}{k(k+1)} = \frac{n}{n+1} + n... | Since the other answer doesn't use induction, here's an induction proof.
Base case: We use the smallest value for $n$ and check that the form works. The smallest value is $n=1$, in this case, the sum on the LHS is
$$
\sum_{k=1}^1\frac{1}{k(k+1)}=\frac{1}{1(1+1)}=\frac{1}{2}.
$$
Also, the RHS is
$$
\frac{1}{1+1}=\frac{... | {
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Lines and planes - general concepts I've come across a book that has this general questions about lines and planes. I can't agree with some of the answers it presents, for the reasons that I'll state below:
True or False:
*
*Three distinct points form a plane - BOOK ANSWER: True - MY ANSWER: False, they cannot b... | (1) For three points to be on a plane, they need not be on the same line. Draw a triangle on a sheet of paper, and you will see what I mean.
(2) If they are coincident lines, they are not called 'intersecting' lines. Intersecting lines meet at one point ONLY.
(4) You are right. (However, even if they were all coincid... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1419875",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Riemann integral on trigonometric functions I have to calculate Riemann integral of function $g:[0;\pi/4]\rightarrow\mathbb{R}$ (on interval $[0;\pi/4]$) given as $g(x)=\frac{\tan(x)}{(\cos(x)^2-4)}$. Function $g$ is continous on interval $[0;\pi/4]$ so it is enough to calculate $\int_0^{\frac{\pi}{4}}{\frac{\tan(x)}{(... | substitue $u = cosx$, then you will have $du = -sinx dx$. Then you will have 1 in numerator and some polynomial in denominator, which can be fractionized as $u * (u - 2) * (u + 2)$. Then your going to separate it to partial fractions. Rest is simple integration... Don't forget to change integration limits.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1419956",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Independence between conditional expectations Suppose $(\Omega, F, P)$ is a sample space, $X$ and $Y$ random variables, and $N$ and $M$ sub sigma algebras of $F$.
*
*I know that $E(X\mid N)$ and $E(X\mid\{\emptyset, \Omega\})$ are independent. Generally, if $N$ and $M$ are independent, will $E(X\mid N)$ and $E(X\mi... | To answer your second question: Let $X, Y$ be independent random variables that assume the values $0$ and $1$ each with probability $\frac{1}{2}$. Setting $N = X + Y$ you get
$$E[X \mid X + Y] = E[Y \mid X + Y] = \frac{X + Y}{2}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1420093",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Nilpotent ideal and ring homomorphism In "Problems and Solutions in Mathematics", 2nd Edition, exercice 1308
Problem statement
Let $I$ be a nilpotent ideal in a ring $R$, let $M$ and $N$ be $R$-modules, and let
\begin{equation}
f : M \rightarrow N
\end{equation}
be an $R$-homomorphism. Show that if the induced map
\beg... | *
*It is false. It is even meaningless: a priori, $IM$ is not contained in $N$. What is true is that $f(M)+IN=N$.
*It is what is written just afterwards: $\;(IN+f(M))/f(M)$.
*It is derived from the correct relation in question $1$.
Once you have $N/f(M)=I\cdot N/f(M)$, you deduce repeatedly:
$$N/f(M)=I\cdot N/f(M)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1420224",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
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What is a surjective function? I am a 9th grader self-studying about set theory and functions. I understood most basic concepts, but I didn't understand what is a surjective function. I have understood what is an injective function, and if I know what is a surjective function, I think I could understand what is a bijec... | Basically, if a function $f: A \to B$ is surjective, it means that every element of the set $B$ can be "obtained" through the function $f$. Formally, like you said, it means that for every $b \in B$, there exists some $a \in A$ such that $f(a) = b$. Phrased differently, $f$ being surjective means that the range of $f$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1420284",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 8,
"answer_id": 4
} |
Multiplying Single Digit Numbers to Get Product >1000 This is yet another Alice and Bob problem.
Alice and Bob are playing a game on a blackboard. Alice starts by writing the number $1$. Then, in alternating turns (starting with Bob), each player multiplies the current number by any number from $2$ to $9$ (inclusive) ... | bob can pick between 2 and 9.
7,8 and 9 dont work as Ross Milikan pointed out as any number more than 56/9 loses.
For 4 and 6 which from bobs perspective are the winning numbers. if he picks between 4 and 6, Alice must pick between 8 and 54, so Bob can then pick between 56 and 111. Whatever Alice then picks, Bob can mu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1420383",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Show that $\mathbb{Q}(\sqrt{2})$ is the smallest subfield of $\mathbb{C}$ that contains $\sqrt{2}$ This was an assertion made in our textbook but I have no idea how to show that either statement is true. Also would like to show that that $\mathbb{Q}(\sqrt{2})$ is strictly larger than $\mathbb{Q}$, which was the second... | $\mathbb{Q}(\sqrt{2})$ is by definition the smallest subfield of $\mathbb{C}$ that contains both $\mathbb{Q}$ and $\sqrt{2}$. The surprising thing (perhaps) is that any element of $\mathbb{Q}(\sqrt{2})$ can be written in the form $a + b\sqrt{2}$, with $a,b \in \mathbb{Q}$. To prove this, you have to show the followin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1420471",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Create a rectangle with coordinates (latitude and longitude) I have two points on a map, I want to create a rectangle where the two points are the line that intersect the rectangle. I understand that there is no true rectangle on a sphere, but the areas I am dealing with are small, the length of the rectangles are no m... | The simplest case: Let the two points be $(x_1,y_1)$ and $(x_2,y_1)$ and the thickness of rectangle is $t$. The coordinates of the rectangle are:
$(x_1,y_1+t/2),(x_1,y_1-t/2),(x_2,y_1-t/2),(x_2,y_1+t/2)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1421538",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Solving Functional Equations by Limits ... Let $f(x)$ be a continuous function and satisfying the equation :
$f(2x) - f(x) = x$. Given $f(0)=1$ ; Find $f(3)=?$
My teacher solves this as :
$$f(x) - f(x/2) = x/2$$
$$f(x/2) - f(x/4) = x/4$$
.
.
.
$$f(x/2^{n-1}) - f(x/2^{n}) = x/ 2^{n-1}$$
...........…......
Add them up... | I would say,
search function $f(x) = x+\mu(x), \quad \mu(x)$ continuous for $x\in R,\,\mu(0)=1$:
$f(2x)-f(x)=x\Rightarrow 2x + \mu(2x) - x - \mu(x)=x\Rightarrow \mu(2x)=\mu(x)$
$\Rightarrow \mu(x) \equiv$ const $= \mu(0) = 1$
$\Rightarrow f(x)=x+1\Rightarrow f(3)=4$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1421644",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
A program to visualize Linear Algebra? I am asking here because I believe you have some idea of a good visualizer 3d program to see what are really: eigenvectors, subspaces, rowspaces, columnspaces and just answers on normal matrix multiplication visualized.
It's easy to calculate, but it's hard to visualize I think. ... | It depends how you want to go about it. With this I mean that you need to structure the way you want to visualize things for example eigen vectors a good way to visualize them is by doing 3D or 2D orthogonal transformations a very cool application with great visual implications is Principal components analysis for exam... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1421743",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Linear Independence for functions defined by integration I came across this problem while doing some work. I'm been unable to make any progression on it. Any suggestions would be greatly appreciated.
Given that the set of strictly positive and continuous functions $$f_i(x,y) >0, \quad i=1,\dots,n$$
are linearly indepe... | No, not in general. For $n$ a natural number greater than or equal to $3$, let
$$f_i(x,y) = \left\{ \begin{array}{l l} (i+1)(xy)^i + (i+1)y^i & \text{for $i \in \{1,...,n-1\}$}\\
\sum_{j=1}^{n-1} (j+1)(xy)^j + (n-1)& \text{for $i=n$}\end{array}\right.$$
Then the the functions $f_1,...,f_n$ are linearly independent. Let... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1421844",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Derivative of integral in interval Let $$F(x)=\int_{2}^{x^3}\frac{dt}{\ln t}$$ and $x$ is in $(2,3)$. Find $F'(x)$. Can somebody give me idea how to do this?
Thank you
| Recall that:
$$F(u) = \int_a^u f(t)\ dt\implies \frac{d}{du}F(u) = f(u)$$
We are now interested in
$$\frac{d}{dx}F(u(x)) = \frac{d}{dx}\int_a^{u(x)}f(t)\ dt = \left(\frac{d}{du}F(u(x))\right)\frac{du}{dx} = f(u)\frac{du}{dx} = \dots$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1421947",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
How to prove what element in $\mathbb{Z}_n$ you get when the elements of $\mathbb{Z}_n$ are summed? Based on trial and error I found that when $n$ is odd, the sum of the elements of $\mathbb{Z}_n$ is zero in $\mathbb{Z}_n$. When $n$ is even, the sum of the elements of $\mathbb{Z}_n$ is $n/2$ in $\mathbb{Z}_n$. Here is ... | Pair $x\in\mathbb{Z}_n$ with $n-x$. Their sum is clearly $n\equiv 0\mod{n}$. If $n$ is odd, each $x$ pairs with $n-x\ne x$, while if $n$ is even, $\frac{n}{2}$ pairs with itself. The result follows.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1422054",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Convergent Operator, weakly convergent sequence => weakly convergent? Suppose we have a Hilbert space $X$, a weakly convergent sequence $u_k\rightharpoonup u$ and a convergent operator $T_k \rightarrow T$ in the norm of $\mathcal{L}(X)$ (bounded, linear operators).
Is the assertion $T_k u_k \rightharpoonup Tu$ correct?... | As noted by @Razieh Noori, if $f\in X^*$, then $f\circ T\in X^*$. Since $u_k\rightharpoonup u \Rightarrow \exists M:\|u_k\|\leq M\quad \forall k\quad$. Also $T_k\rightarrow T \Rightarrow$ for $\epsilon>0\quad\|T_k-T\|_{op}<\frac{\epsilon}{2 M \|f\|_*}$ for $k>K_1$.
Then
$|f\circ T_ku_k-f\circ T u|\leq |f\circ T_k u_k-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1422140",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Sum of two harmonic alternating series
Evaluate the series $$\sum_{n=1}^\infty (-1)^{n+1}\frac{2n+1}{n(n+1)}.$$
I've simplified it to the form $$\sum_{n=1}^\infty (-1)^{n+1}\frac{1}{n+1} + \sum_{n=1}^\infty (-1)^{n+1}\frac{1}{n}$$
and I've proved that both parts converge. However, I'm having trouble finding the limi... | $$
\sum_{n=1}^{\infty}{\frac{{(-1)}^{n+1}}{n+1}}=\sum_{n=0}^{\infty}{\frac{{(-1)}^{n+1}}{n+1}}+1\\
=\sum_{n=1}^{\infty}{\frac{{(-1)}^{n}}{n}}+1
$$
So summing the two summation yields:
$$
\sum_{n=1}^{\infty} {{(-1)}^{n+1}\frac{2n+1}{n(n+1)}}=\sum_{n=1}^{\infty}{\frac{{(-1)}^{n+1}}{n+1}}+\sum_{n=1}^{\infty}{\frac{{(-1)}^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1422244",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
Succinct Proof: All Pentagons Are Star Shaped
Question: What is a succinct proof that all pentagons are star shaped?
In case the term star shaped (or star convex) is unfamiliar or forgotten:
Definition Reminder: A subset $X$ of $\mathbb{R}^n$ is star shaped if there exists an $x \in X$ such that the line segment from... | This is several years late, but I think I have a nicer proof. I thought I would post it.
Let's believe we can triangulate the 5-gon into three triangles. Note that these three triangles all must have a vertex $v$ in common: To see this, remove one of the triangles $T$, s.t. we have a 4-gon left. Then whichever way we t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1422305",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 4,
"answer_id": 2
} |
Projection from triangle to spherical triangle Consider a triangle, $T$, in $\mathbb{R}^3$ with vertices $(0,0,1), (0,1,0)$, and $(1,0,0)$. Let $S$ denote the sphere centered at the origin with radius 1 and let $S_1$ denote the portion of the sphere in the same quadrant at the triangle. We can define a map $f: T \right... | Notice under the direct mapping, any point $p$ on $T$ get mapped to a point $f(p)$ on $S$ which is a scalar multiple of $p$. So for any point $q = (x,y,z) \in S$, the inverse mapping $f^{-1}$ will send $q$ to a point which is a scalar multiple of $q$. This means there exists a function $\lambda : S \to \mathbb{R}$ such... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1422388",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Conditional expectation of iid nonnegative random variables I am studying Ross's book, stochastic processes. There is the following lemma:
Let $Y_1, Y_2, ... , Y_n$ be iid nonnegative random variable. Then,
$E[Y_1+ \cdots +Y_k | Y_1+\cdots+Y_n=y] = \frac{k}{n} \cdot y, \quad k=1,\cdots,n$
But, I really can't underst... | Informally, as all the $Y_i$ are identically distributed, the fact that they sum to $y$ implies that their (conditional) expectation must be $\frac yn$. To see this formally, note that we certainly have equality of all the $E[Y_i\;|\;Y_1+...+Y_n=y]$, call the common value $E$. Then
$$nE=E\left[Y_1\;|\;Y_1+Y_2+...+Y_n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1422465",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
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