Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Show $\int_0^\infty xe^{-ax^2}dx=\frac{1}{2a}$ I tried integrating by parts but I always arrive at an expression other than $\frac{1}{2a}$ which contains $\sqrt{\frac{\pi}{a}}$ from the Gaussian integral $\int_0^\infty e^{-ax^2}dx=\frac12 \sqrt{\frac{\pi}{a}}$. Is there some kind of trick to evaluating this integral?
| Substitute $x^2=y$. Then
$$\int_0^\infty xe^{-ax^2}\,dx=\frac12\int_0^\infty e^{-ay}\,dy.$$
| {
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"timestamp": "2023-03-29T00:00:00",
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What is the right definition of a cycle? I've seen this in all introductory courses on graphs, but every time it bugs me : the definition of a cycle is usually wrong.
In the last course I have seen they define paths in the obvious way, adding edges inbetween vertices.
Then they say " a cycle is a non-trivial path whos... | A cycle is either:
*
*a simple graph (= no double edges, no loops) with 1 component and all vertices having vertex degree 2
*a graph with 2 vertices and two edges between them
*a graph with 1 vertex and a loop
| {
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A matrix is a product of nilpotent matrices iff its not invertible I just completed a homework problem which proves the following result: a matrix (with coefficients in some field) is a product of nilpotent matrices iff its not invertible.
The proof was broken into several parts and was quite involved. I'm wondering if... | Following is a proof for fields that allow a Jordan-type canonical form. Write $A=P^{-1}TP$ with $T$ in Jordan form (but this can be relaxed somewhat). $A$ is not invertible if, and only if, at least one of the diagonal elements is zero. Taking it to be the last eigenvalue for simplicity, note the following example dec... | {
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Shortest distance from point to curve I could use some help solving the following problem. I have many more like this but I figured if I learn how to do one then I can figure out the rest on my own. Thanks in advance!
A curve described by the equation $y=\sqrt{16x^2+5x+16}$ on a Cartesian plane. What is the shortest d... | Since distance is positive and the square root function is increasing, it suffices to find the smallest value the squared distance between $(x,y)$ on the curve and the point $(2,0)$ can take. This is
$$ L(x) = (x-2)^2 + (y-0)^2 = (x-2)^2+y^2 = x^2-4x+4 + 16x^2+5x+16 = 17x^2+x+20. $$
A minimum can only occur if $L'(x)=0... | {
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What is the distribution of 1/($X$+1)? I have a problem that I'm having trouble figuring out the distribution with given condition.
It is given that 1/($X$+1), where $X$ is an exponentially distributed random variable with parameter 1.
Original Problem:
What is the distribution of 1/($X$+1), where $X$ is an expoential... | $X$ is not "writen" as $e^{-x}$. The probability density function of $X$, called $f_X(x)$, is equal to $e^{-x}~\big[x\geqslant 0\big]$.
The cummulative distribution function of $X$ is: $$\begin{align}F_X(x) ~&=~ \mathsf P(X\leqslant x) \\[1ex] &=~ (1-e^{-x}) ~\big[x\geqslant 0\big]\end{align}$$
Let $Y:=1/(1+X)$. Th... | {
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Finding the limit of a sequence If $x_1$ and $y_1$ be positive numbers, $x_{n+1}$ $=$ $1\over 2$$(x_n+y_n)$ and $2\over y_{n+1}$ $=$ $1\over x_n$ $+$ $1\over y_n$ $\forall$ $n$ $\ge$ $1$. Show that the sequences $x_n$ and $y_n$ are monotonic and converge to a common limit $l$ where $l^2$ $=$ $x_1y_1$.
I proved that the... | Hint:
$$\require{cancel}
x_{n+1} y_{n+1} = \frac{x_n+y_n}{2}\cdot\frac{2}{\cfrac{1}{x_n}+\cfrac{1}{y_n}}=\frac{\cancel{x_n+y_n}}{\bcancel{2}}\cdot\frac{\bcancel{2} \,x_n y_n}{\cancel{x_n+y_n}} = x_n y_n
$$
| {
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Find $\lim_{n \to \infty} \frac{1}{n}\sum_{k=1}^{n} \ln\left(\frac{k}{n} + \epsilon_n\right)$ if $\epsilon_n>0$ and $\epsilon_n\to0$
Let $\epsilon_{n}$ be a sequence of positive reals with $\lim\limits_{n \rightarrow \infty} \epsilon_{n}=0$. Then find $$\lim_{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n} \ln\left(\... | Via mean value theorem we can see that $$\log\left(\frac{k} {n} +\epsilon_{n} \right) = \log\frac{k} {n} +\frac{n} {k} \epsilon_{n} +o(\epsilon_{n}) $$ and hence the sum in question is equal to $$\frac{1}{n}\sum_{k=1}^{n}\log\frac{k}{n}+\epsilon_{n}\sum_{k=1}^{n}\frac{1}{k}+o(\epsilon_{n})$$ First sum tends to $\int_{0... | {
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I do not know how to approach this coin problem, where to start? When you throw a coin, you can either win $10$ dollars if heads or lose $5$ dollars if tails. After $100$ throws, you win $895$ dollars. Is this a fair coin? What probabilities can you associate with each side of the coin?
Can you please explain how to ap... | For a fair coin, the number of heads after 100 throws should be $N\cdot p=50$, and the standard deviation of this number should be $\sqrt{Npq}=5$. By the stated rules, the win with $n$ out of $100$ heads is $10n-5(100-n)=15n-500$. We conclude that $n=\frac{895+500}{15}=93$. This differs by $\frac{93-50}{5}=8.6$ standar... | {
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Calculate $\lim\limits_{n\to\infty}\dfrac{\sum\limits_{k=0}^n\log\binom{n}{k}}{n^2}$ How to prove if the following limit exists?
If it exists, what's the value?
$$\lim\limits_{n\to\infty}\dfrac{\sum\limits_{k=0}^n\log\binom{n}{k}}{n^2}$$
Thanks!
| $\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,}
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Have these disk/washer problems been set up correctly? Q1: R is the region between $f(x) = x^{2}$ and $g(x) = x + 2$. Find the volume $V_1$ of the region generated by revolving about the line $y = - 3$
Q2: R is the region between $f(x) = 3x$, and $g(x) = 3x$, Find the volume $V_2$ of the region generated by revolving a... | For $V_1$, you have the inner and outer radii reversed: on the interval $[-1,2]$, $g(x) \ge f(x)$, consequently your evaluation of $V_1$ will result in a negative number.
For $V_2$, I cannot verify your integral, because you have stated $f(x) = g(x) = 3x$, meaning there is no region enclosed by these two functions. H... | {
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For any set $A$, there is some $x$ not in $A$. I am working through a set theory text and am having trouble giving a formal proof of the above statement. So far I have:
Suppose the contrary. Then there exists a set $A$ such that $x\in A$ for all $x$. Thus $A$ is the set of all sets, which gives rise to Russell's parado... | A more direct (but very similar) proof can be concieved: even though Russell's paradox is often thought as a limitative result, the argument it uses is precisely a proof of your result:
Let $x$ be the subset $\{a \in A \ | \ a \notin a\}$ of $A$. $x \in A$ is absurd, hence $x \notin A$.
Note that assuming fondation, $... | {
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Prob. 8, Chap. 5, in Rudin's PMS, 3rd ed: If $f^\prime$ is continuous on $[a, b]$, then $f$ is uniformly differentiable on $[a,b]$ Here is Prob. 8, Chap. 5, in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition:
Suppose $f^\prime$ is continuous on $[a, b]$ and $\varepsilon > 0$. Prove that there... | Yes, I verify that your attempt is correct.
To summarize, the key ideas for proving the scalar case are indeed uniform continuity and the mean value theorem:
*
*the mean value theorem allows one to replace the Newton quotient in the conclusion with a derivative;
*the uniform continuity of $f'$ gives a desired $\delt... | {
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Every graph has a core I'm currently going through Godsil/Royle's chapter about graph homomorphisms and using the following definitions I struggle to follow through the proof of the existence of a core of a graph.
A graph $X$ is a core if every homomorphism $f$ from $X$ to itself is a bijection.
A subgraph $Y$ is a co... | You've shown that there exists a $Y$ which is a minimal subgraph of $X$ such that there exists a homomorphism $f:X\rightarrow Y$. Now, suppose that there exists a homomorphism $g:Y\rightarrow Y$ which is not a bijection. This means that $g(Y)$ has less vertices than $Y$. However, homomorphisms are closed under composit... | {
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Creating a function F(x) such that all outputs exist between -5 and 5 Take this sample graph illustration:
As $X$ approaches negative infinity the output approaches $-5$
As $X$ approaches positive infinity the output approaches $5$
From what I recall this would be leveraging $\log, \ln$ or $e$ but I'm failing to remem... | The simplest is $$f(x)=\frac{10}{1+e^{-x}}-5$$
It is a simple logistic function subtracted by 0.5 multiplied by $10$.
| {
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Graph Theory: Prove that G must be connected Let $G$ be a simple graph with $n$ vertices and $ \frac 1 2 (n-1)(n-2)+1$ edges. Prove that $G$ must be connected.
Thank you!
| [Edited - thanks to M. Vinay who spotted the initial mistake!]
Suppose for contradiction that the graph is disconnected. Then it is possible to partition the graph into two subgraphs (disconnected from one another), with $k$ vertices and $n- k$ vertices respectively.
By a simple counting argument, you should be able to... | {
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Homomorphism Between Non-Abelian Group and Abelian Group I know that when finding homomorphisms between groups, for a cyclic group to any other group, then the homomorphism is completely determined by where you send the generator. However, I have two questions regarding homomorphisms between non-abelian groups and abel... | $C_2$ is isomorphic to a subgroup of $S_3$ so that should help with the second question. As to the first, the commutator is generated by products of the form $xyx^{-1}y^{-1}$ so any homomorphism into an abelian group would allow these elements to commute, and this cancel. That is $$\phi(xyx^{-1}y^{-1})=\phi(x)\phi(y)\... | {
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A subring of the upper triangular matrices I want to compute the Jacobson radical, the right socle, and the left socle of the ring of $3\times 3$ upper triangular matrices with entries in $\mathbb Z_4$ and such that the entries on the main diagonal are equal.
I know that the Jacobson radical of the ring of $3\times 3$... | Well, you have that $\begin{bmatrix}0&b&c\\ 0& 0& d\\ 0&0&0\end{bmatrix}$ is a nilpotent ideal, and that $\begin{bmatrix}2&0&0\\ 0& 2& 0\\ 0&0&2\end{bmatrix}$ is a central nilpotent, so the Jacobson radical must contain at least $\begin{bmatrix}2a&b&c\\ 0& 2a& d\\ 0&0&2a\end{bmatrix}$ for any choice of $a,b,c\in \mathb... | {
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The square of any odd number is $1$ more than a multiple of $8$ I'm taking a single variable calculus course and asked following :
Say wether the following is a valid proof or not :
the square of any odd number is 1 more than a multiple of $8$.
Proof : By the division theorem any number can be expressed in one of t... | The proof is valid.
(1) Since $2n-1=2(n-1)+1$, you don't have to consider $2n-1$ and $2n+1$ separately.
(2) Now consider only $2n+1$. We further divide it into two cases: (i) if $n$ is even, then $n=2q$ for some $q$ and so $2n+1=4q+1$. (ii) if $n$ is odd, then $n=2q+1$ for some $q$ and so $2n+1=4q+3$.
All possible case... | {
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Semidefinite programming, SDP, eigenvalues If I have an $n\times n$ hermitian matrix $A$ and I want to find all the eigenvalues of $A$, i.e $\{\lambda_{i}\}$, $i=1,...,n$ where $\lambda_{i+1}>\lambda_{i}$, if I only know the biggest eigenvalue (found using SDP), i.e $\lambda_{n}$, my question is:
How can I transform $\... | Set $A' = A - \lambda_n v_n v_n^*$ where $v_n$ is a normalized eigenvector for $\lambda_n$. The spectrum of $A'$ is that of $A$, except $\lambda_n$ is now replaced with 0.
| {
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Interesting integral: $\int_0^1{\frac{nx^{n-1}}{x+1}}dx$ Find the value of $$\int_0^1{\frac{nx^{n-1}}{x+1}}dx.$$
I had no luck while integrating it. I also tried differentiating w.r.t n but still couldn't reach anywhere. Need help.
| Put $y=1+x$ and the integral becomes
$$ \int_{1}^2 \frac{n(y-1)^{n-1}}{y} \, dy = \int_1^2 \sum_{k=0}^n n\binom{n-1}{k} (-1)^{n-k-1} y^{k-1} \, dy = \left[ n(-1)^{n-1}\log{y} + \sum_{k=1}^n \frac{n}{k} \binom{n-1}{k} (-1)^{n-k-1} y^k \right]_1^2 \\
= n(-1)^{n-1}\log{2} + \sum_{k=1}^n \frac{n}{k} \binom{n-1}{k} (-1)^{n-... | {
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Accessing irreducible representations in GAP character table I'm writing an algorithm in GAP to automate some easy calculations related to elements of prime order in groups that appear in the ATLAS.
To automate the process, I need to access the character values for some irreducible complex representations of small deg... | If the character table is in the ATLAS (or related) you can access it immediately by its name:
c:=CharacterTable("L4(3)");;
The long time you observe is presumably a call such as
gap> d:=CharacterTable(PSL(4,3));;
gap> Irr(d);
which stems from calculating the character table afresh from first principles. In this seco... | {
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Finding the asymptotes of an odd function The question gave the graph shown below, giving one asymptote as $y=f(x)=\frac{1}{2}x - 3$. Graph:
I was asked to complete the rest of the graph and draw on any missing asymptotes, and this is what I drew:
For the other asymptote which I drew I wasn't sure what its equation ... | the given asymptote means
$$\lim_{x\to -\infty}(f (x)-(\frac {1}{2}x-3))=0$$
but $f $ is odd $ (f (-x)=-f (x)) $.
replacing $x $ by $-x $, we get
$$\lim_{x\to+\infty}(f (-x)-(\frac {-1}{2}x-3))=0$$
$$\implies$$
$$\lim_{+\infty}(f (x)-(\frac {1}{2}x+3))=0$$
thus the other asymptote is
$$y=\frac {1}{2}x+3$$
| {
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Is this proof reasonable, given this information? Suppose I have the figure in the image marked 'Original':
Visually, that figure appears to be a parallelogram.
Would the following proof that $\bigtriangleup BCA \cong \bigtriangleup DAC$ be valid?
*
*$BC \parallel AD$, because Diagram
*$\angle BAC \cong \angle DCA... | Your step (5) says it's using side-angle-side, but the pictures show the configuration is side-side-angle, which does not imply congruence precisely because of this kind of counterxample.
| {
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If $\alpha$ is a root, then $\sigma(\alpha)$ is a root There is the well-known proposition (for a specific reference: see section 14.1 of Dummit/Foote):
Proposition. If $\sigma \in \text{Aut}(K/F)$ and $f(x) \in F[x]$ has $\alpha \in K$ as a root, then $f(x)$ also has $\sigma(\alpha)$ as a root.
This got me wondering ... | The reasoning is correct, but we can make a stronger observation. The proposition you mentioned can be strengthened as follows: If $\sigma \in \operatorname{Aut}(K/F)$, $f \in F[x]$, and $\alpha \in K$, then $\sigma(\alpha)$ has the same multiplicity as a root of $f$ as does $\alpha$.
Proof:
For any $p \in K[x]$, let $... | {
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Is this the correct homotopy rel $\{0,1\}$? Concerning the problem circled in red:
Would this work as the homotopy rel $\dot I$?
$F(t_1, t_2) =
\begin{cases}
\sigma(e_1), & \text{if $t_1=0,1$ or $t_2=1$} \\
(\sigma_0 * \sigma_1^{-1}) * \sigma_2[(1-t_2)t_1 + t_2], & \text{otherwise}
\end{cases}$
And also, how would you... | Your homotopy is not a homotopy. Its image always lies on $\partial \Delta^2 \cong S^1$, and $S^1$ is not contractible. More specifically, we have for any $0<t_2<1$,
$$\lim_{t_1 \to 0} F(t_1,t_2) = \sigma_0\sigma_1^{-1}\sigma_2(t_2) \not= \sigma(e_1) = F(0,t_2).$$
I think you have the answer, but you just don't recogni... | {
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Prove that $\sum_{k=0}^n \frac{(-1)^{(n+k)}\sum_{r=0}^n a_rk^r}{k!(n-k)!}=a_n\quad \forall n \in \mathbb{N_0} $ Can Someone prove that the following equation is true? I tried to prove it using induction but I didn't get very far.
$$\sum_{k=0}^n \frac{(-1)^{(n+k)}\sum_{r=0}^n a_rk^r}{k!(n-k)!}=a_n\quad \forall n \in \ma... | $\begin{array}\\
s_n
&=\sum_{k=0}^n \frac{(-1)^{(n+k)}}{k!(n-k)!}\sum_{r=0}^n a_rk^r\\
&=\frac{(-1)^n}{n!}\sum_{k=0}^n \frac{(-1)^{k}n!}{k!(n-k)!}\sum_{r=0}^n a_rk^r\\
&=\frac{(-1)^n}{n!}\sum_{k=0}^n (-1)^{k}\binom{n}{k}\sum_{r=0}^n a_rk^r\\
&=\frac{(-1)^n}{n!}\sum_{r=0}^n\sum_{k=0}^n (-1)^{k}\binom{n}{k} a_rk^r\\
&=\f... | {
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Is $SO(n)$ diffeomorphic to $SO(n-1) \times S^{n-1}$ There is a fibration $SO(n-1) \mapsto SO(n) \mapsto S^{n-1}$, from basically taking the first column of the matrix in $\mathbb{R}^n$. Is this fibration trivializable?
| If $n=3$, the Hopf fibration is the composition $S^3=Spin(3)\rightarrow SO(3)\rightarrow S^2$, so if $SO(3)\rightarrow S^2$ is trivial, so the hopf fibration will be flat and this is not true.
https://en.wikipedia.org/wiki/Hopf_fibration#Geometric_interpretation_using_rotations
| {
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Prove that this function is zero for all real number Let be $f$ a continuous function on $[0,2\pi]$. If $\int_0^{2\pi}f(x)e^{-inx}dx=0$ for all $n\in\mathbb{N}$, then $f(x)=0$ for all $x\in [0,2\pi]$.
I was thinking this problem a lot but I don't know what I'm not seeing. I think, by the Stone-Weierstrass theorem I hav... | The example $f(x)=e^{-ix}$ shows that the question is not correct as stated. Instead, we must require that $\int_0^{2\pi}f(x)e^{-inx}\;dx=0$ for all $n\in\mathbb{Z}$.
If we make this assumption, then it follows that
$$ \int_0^{2\pi}f(x)\overline{p(x)}\;dx=0 $$
for any trigonometric polynomial
$$ p(x)=\sum_{n=-N}^Nc_ne^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2268042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Computing maximum dimension of a vector subspace given that it's every element is a symmetric matrix and is closed under matrix multiplication
Q. Let $S$ be a subspace of the vector space of all $11 \times 11$ real matrices such that (i) every matrix in $S$ is symmetric and (ii) $S$ is closed under matrix multiplicati... | If $A$ and $B$ are symmetric, then $AB$ symmetric means $AB=(AB)^T=B^TA^T=BA$. All the matrices therefore commute. A symmetric
real matrix is diagonalisable, and pairwise commuting real matrices
are simultaneously diagonalisable. So the dimension of such a space is
at most $11$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2268156",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Integral of $f_n$ from $0$ to $1$ is zero Let $f_n(t)$ be defined as $$f_n(t)=\begin{cases}1, \text{if}\,\,t\in[\frac{p}{2^k},\frac{p+1}{2^k})\\ 0, \text{otherwise}\end{cases}$$ where $n=2^k+p,\,\,0\le p<2^k$. Then,
a)what can be the value of $\lim\sup f_n(t)$ and $\lim\inf f_n(t)$?
b)Is $\int_0^1|f_n(t)|\to0$ when $... | Hints:
a) For a fixed $t \in [0,1]$ and some time $N$, is there an $n > N$ so that $f_n(t) > 0$?
b) Write out an explicit bound to formalize your idea. What is the value of integral at the stage when the interval has length $2^{-n}$?
Bonus: Think about $b$ for the function $2^k f_n$ instead ($n = 2^k + p$). What's the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2268241",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Limit $\lim_{x\rightarrow \infty}x \ln x+2x\ln \sin \left(\frac{1}{\sqrt{x}} \right)$ I think that this limit should be not defined
$$\lim_{x\rightarrow \infty}x \ln x+2x\ln \sin \left(\frac{1}{\sqrt{x}} \right)$$
| You can do it without substitution,
$$\lim_{x\to \infty}\left(x\ln x+2x\ln\sin\left(\frac{1}{\sqrt{x}}\right)\right)$$
$$=\lim_{x\to \infty}2x\left(\ln\sqrt{x}+\ln\sin\left(\frac{1}{\sqrt{x}}\right)\right)$$
$$=\lim_{x\to \infty}2x\ln\left(\sqrt{x}\sin\left(\frac{1}{\sqrt{x}}\right)\right)$$
$$=\lim_{x\to \infty}2x\ln\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2268330",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Maximum of $x^3+y^3+z^3$ with $x+y+z=3$ It is given that, $x+y+z=3\quad 0\le x, y, z \le 2$ and we are to maximise $x^3+y^3+z^3$.
My attempt : if we define $f(x, y, z) =x^3+y^3 +z^3$ with $x+y+z=3$ it can be shown that,
$f(x+z, y, 0)-f(x,y,z)=3xz(x+z)\ge 0$ and thus $f(x, y, z) \le f(x+z, y, 0)$. This implies that $f... | $$
\begin{eqnarray}
&(x+y+z)^3 &=& x^3 + y^3 + z^3 + 3(x+y)(y+z)(z+x)\\
\implies & x^3 + y^3 + z^3 &=& (x+y+z)^3 - 3(x+y)(y+z)(z+x)\\
&& = & 27 - 3(x+y)(y+z)(z+x)
\end{eqnarray}
$$
Now, $x^3+y^3+z^3$ is maximum when $t = (x+y)(y+z)(z+x)$ is minimum. Now since $x$, $y$ and $z$ are each non-negative, therefore $t$ is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2268524",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
} |
How to show a bound on expected sum of squared martingale increments? I am reading a paper with the following set up. Let $(\Omega, \mathcal{F},P)$ be a probability space, $B \in \mathcal{F}$, and $\{\mathcal{F}_n : n \in \mathbb{N} \}$ a filtration with $\mathcal{F}_n \uparrow \mathcal{F}$.
Let $q_n = P(B \mid \mathca... | Increments of a martingale are orthogonal hence
$$\mathbb E\left[\sum_{n=1}^{+\infty}\left(q_{n+1}-q_n\right)^2 \right]
=\mathbb E\left[\left(\sum_{n=1}^{+\infty}q_{n+1}-q_n\right)^2 \right].$$ By the martingale convergence theorem, $$\sum_{n=1}^{+\infty}q_{n+1}-q_n=\mathbb P\left(B\mid\mathcal F\right)-\math... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2268614",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Area between $y^2=2ax$ and $x^2=2ay$ inside $x^2+y^2\le3a^2$ I need to find the area between $y^2=2ax$ and $x^2=2ay$ inside the circle $x^2+y^2\le3a^2$. I know it's an integral but I can't seem to find the right one.
| The parabolas will intersect the circle at the points $(a,\sqrt{2}a)$ and $(\sqrt{2}a,a)$ giving the following region:
So you want to find
$$ \int_0^a\sqrt{2ax}-\frac{x^2}{2a}\,dx+\int_a^{\sqrt{2}a}\sqrt{3a^2-x^2}-\frac{x^2}{2a}\,dx $$
In polar coordinates you can use the symmetry of the region and find the area by ev... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2268705",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Uniformly continuous function Let $f:[0,1] \cup \{-1\} \to \Bbb R$ be defined by,
$f(x)=1 , \forall x \in [0,1]$;
$=0$ ,for
$x=-1$.
Is the function uniformly continuous?
*
*here the domain is closed and bounded. Thus compact. And also f(x) is continuous. Thus f should be uniformly continuous. But here how the de... | The "same" reason you saw it is continuous. The function is uniformly continuous on $[0,1]$ clearly. If $x,y \in \{ -1 \}$, then $|x-y| = 0$ and $|f(x)-f(y)| = 0$. So $f|_{\{-1\}}$ is such that for every $\varepsilon > 0$ and every $\delta > 0$ we have $|x-y| < \delta$ implying $|f(x)-f(y)| < \varepsilon$; this is even... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2268881",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Consider a measurable function $f:[0,1]\rightarrow \mathbb{R}$ such that $f(x)\geq ||f||_2$. So I am stuck on this question and I don't really know how to go about it. The question is
Let $X=[0,1]$ and $\mu = \lambda$ be the Lebesgue measure.
Consider a measurable function $f:[0,1]\rightarrow \mathbb{R}$ such that $f... | Using Holder's inequality
$$
\|f\|_2 = \int_0^1 \|f\|_2 \leq \int_0^1 f \leq \left(\int_0^1 |f|^2\right)^{1/2} = \|f\|_2.
$$
Hence the non-negative function $g(x) := f(x) - \|f\|_2$ satisfies $\int_0^1 g = 0$, so that $g(x) = 0$ a.e. in $[0,1]$, i.e. $f(x) = \|f\|_2$ a.e. in $[0,1]$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2269008",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
The conjecture: $\binom{2n-1}{n-1} \equiv 1 \pmod{n^3} \quad \Longrightarrow \quad n \in \mathbb{P}$ I recall seeing the following conjecture somewhere, but I cannot find the reference any more. Where can I find more information about this conjecture? Does it have a name?
Conjecture: For any natural number $n$ it holds... | This is known as Wolstenholme's theorem.
For a prime $p > 3$ we have:
$$\binom{2p-1}{p-1} \equiv 1 \pmod{p^3}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2269128",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Will $2$ linear equations with $2$ unknowns always have a solution? As I am working on a problem with 3 linear equations with 2 unknowns I discover when I use any two of the equations it seems I always find a solution ok. But when I plug it into the third equation with the same two variables , the third may or may not... | Each linear equation represents a line in the plane. Most of the time two lines will intersect in one point, which is the simultaneous solution you seek. If the two lines have exactly the same slope, they may not meet so there is no solution or they may be the same line and all the points on the line are solutions. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2269220",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 0
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If $\sin x + \sin^2 x =1$ then find the value of $\cos^8 x + 2\cos^6 x + \cos^4 x$ If $\sin x + \sin^2 x =1$ then find the value of $\cos^8 x + 2\cos^6 x + \cos^4 x$
My Attempt:
$$\sin x + \sin^2 x=1$$
$$\sin x = 1-\sin^2 x$$
$$\sin x = \cos^2 x$$
Now,
$$\cos^8 x + 2\cos^6 x + \cos^4 x$$
$$=\sin^4 x + 2\sin^3 x +\sin^... | Hint:
$$\cos^8x+2\cos^6x+\cos^4x=(\cos^4x+\cos^2x)^2$$
Now as $\cos^2x=\sin x,\cos^4x=(\cos^2x)^2=?$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2269298",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 4
} |
Prove if $x > 3$ and $y < 2$, then $x^{2} - 2y > 5$ My solution is:
Multiply $x > 3$ with $x$, yielding $x^{2} > 9$
Multiply $y < 2$ with $2$, yielding $2y < 4$
Thus, based on the above $2$ yielded inequalities, we can prove that if $x > 3$ and $y < 2$, then $x^{2} - 2y > 5$.
Is this a correct proofing steps?
| As a slightly extended version of Michael Rozenberg's answer, this can very simply be written down as:$%
\newcommand{\calc}{\begin{align} \quad &}
\newcommand{\op}[1]{\\ #1 \quad & \quad \unicode{x201c}}
\newcommand{\hints}[1]{\mbox{#1} \\ \quad & \quad \phantom{\unicode{x201c}} }
\newcommand{\hint}[1]{\mbox{#1} \unico... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2269402",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
} |
Defining a matrix in Magma with finite field entries Consider the following matrix
$$
G:=\left[
\begin {array}{cccccccc}
1&0&0&0&\alpha&\alpha+1&1&1\\
0&1&0&0&1&\alpha&\alpha+1&1\\
0&0&1&0&1&1&\alpha&\alpha+1\\
0&0&0&1&\alpha+1&1&1&
\alpha\end {array}
\right]
$$
where entries of matrix $G$ come from finite fiel... | At First, we should define the finite field $GF(2^8)$ by our polynomial as follows
$$
K<x>:=ExtensionField< GF(2), z | z^8+z^4+z^3+z+1 >;
$$
After that we have to define matrix space over the finite field $K$, in the following form
$$
M := KMatrixSpace(K, 4, 8);
$$
Now, we define our matrix as shown
$$
G := M ! [1,0,0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2269557",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Show $\forall_{z \in \mathbb{C}, n \in \mathbb{N}} : \exists_{u,v \in \mathbb{Q}} |z - (u+ vi)| < \frac{1}{n}$
Show $\forall_{z \in \mathbb{C}, n \in \mathbb{N}} : \exists_{u,v \in \mathbb{Q}} |z - (u+ vi)| < \frac{1}{n}$
In other words, show that you can make an infinitely small complex number (> 0) by using rational... | Hint: choose $u\in\mathbb Q$ such that $\left|a-u\right|\leqslant 1/(2n)$ and $v\in\mathbb Q$ such that $\left|b-v\right|\leqslant 1/(2n)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2269861",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is this inequality on Schatten p-norm and diagonal elements true? Let $A=[a_{ij}]\in\mathbb R^{m\times n}$ be a matrix with $m\ge n$, and $\Vert A\Vert_p$ be the Schatten p-norm of $A$.
It is known that $\Vert A\Vert_1\ge \sum_i \vert a_{ii}\vert$ and $\Vert A\Vert_2^2=\Vert A\Vert_F^2\ge \sum_i \vert a_{ii}\vert^2$.
C... | This is true, and is a consequence of the following majorization result. Let $d_1,\dots,d_n$ be the main diagonal entries of $A$ ordered so that $|d_1|\ge |d_2| \ge \dots\ge |d_n|$. Also let $\sigma_1\ge \sigma_2 \ge \dots \ge \sigma_n$ be the singular values of $A$. The aforementioned (weak) majorization inequality is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2269975",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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How do I show that coordinate basis spans the tangent space? A tangent space $T_{m,M}$ is defined as the set of all linear derivations at a point $m$ on a manifold $M$. Linear derivations are operators that satisfy the Leibniz rule, i.e. $f,g \in F_{m,M}, O(f,g) = f*O(g)+g*O(f)$ where $F_{m,M}$ is the space of smooth f... | This is half of the content of Proposition 3.2 in Lee's Introduction to Smooth Manifolds. In summary, and using the notation of the question here:
Hint The given definition of derivation at local, so by choosing smooth coordinates of the given manifold $M$ centered at the given point $m \in M$, we need only prove the c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2270093",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Limit of 1/x as x approaches infinity Why is $\lim_{x\to\infty} \frac{1}{x}$ equal to $0$, when really the limit appears to be an infinitesimal quantity?
I am trying to understand why there is no distinction between 0 and an infinitesimal quantity in the context of limits.
| In standard real analysis/calculus, there are no infinitesimal quantities. Everything is formulated in terms of real numbers. What $\lim_{x\to \infty} f(x) = c$ means is that for all $\varepsilon > 0$ there exists $x_o\in \mathbb{R}$ such that whenever $x>x_0$, we have that $\vert f(x)-c\vert < \varepsilon$. In words, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2270193",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$\epsilon - \delta$ proof for $\frac{x^2 - 16}{x + sin x}$ limit I'm having difficulty writing an $\epsilon - \delta$ proof for the following limit:
$\lim_{x\to 4} \frac{x^2-16}{x+\sin x} = 0$
I've factored it to $\frac{(x+4)(x-4)}{x+\sin x} = 0$
and guessed that I need $\delta = \frac{2}{5}\epsilon$ for $|x-4| < \delt... | $$|x-4| < \delta$$
$$4-\delta < x < 4+ \delta$$
$$3 - \delta< x+ \sin x < 5 + \delta$$
$$\frac{1}{5+\delta} < \frac{1}{x+\sin x} < \frac{1}{3-\delta}$$
If $\delta < 1$, $-\delta > -1$, $3-\delta > $2, $\frac{1}{3-\delta} < \frac12$
$$\left| \frac{x^2-16}{x+\sin x}\right| \leq \frac12 |x^2-16|$$
Also, if $\delta < 1$, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2270291",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Prove if 2 divides $a^2$, then 2 divides $a$. If 2 divides $a^2$, then 2 divides a.
I know that 2 divides $a^2$ means there is some integer $n$ such that $a^2 = 2n$,
and similarly, 2 divides $a$ means there is some integer $m$ such that $a = 2m$
I thought I could rewrite $a^2 = 2n$ into this $= a = 2(n/a)$ but I don'... | By division algorithm, $a=2q+r$ where $r=0\ \text{or}\ 1$ and $q\in\mathbb{Z}$. Now $a^2=4q^2+4qr+r^2$. Since $2|a^2$ it follows that $2|r^2$, whence $r=0$.
OR use the fact that if $p$ is a prime such that $p$ divides $ab$ then $p$ divides $a \ \text{or}\ b$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2270529",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 9,
"answer_id": 2
} |
By transfer principle, is the set of hypernatural the set of naturals? In Jech, we learn that $x=\mathbb{N}$ is a $\Delta_0$-formula. Can you tell me what is wrong with the following reasoning ?
Let $\phi(x)$ be the formula $x=\mathbb{N}$
The tranfer principle tells us that $\forall y\in\mathbf{S},\phi^{\mathbf{S}}(y)\... | Jech proves that "$x=\mathbb{N}$" can be expressed by a $\Delta_0$ formula in ZFC. That is, there is some $\Delta_0$ formula $\phi(x)$ such that ZFC proves there is exactly one set satisfying $\phi(x)$ (and that set is what we think of intuitively as $\mathbb{N}$).
However, you are not working in ZFC. You are working... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2270633",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Probability of area of a rectangle (uniform distribution) Given $X\sim U(0,2)$ and $Y\sim U(0,3)$ which are length and width of a rectangle (respectively). I want to find the probability of the area of the rectangle less than 1.
The hint is the joint density $f(x,y) = 1/6$ for $0\leq x \leq 2$ and $0 \leq y \leq 3$.
S... | Hint:
Try to understand whether the following is true and evaluate it.
\begin{align}
Pr(XY < 1) = \int_0^2 \int_0^{\min(3,\frac1x)}f(x,y)dydx
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2270766",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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How to calculate $p$ value bound for $\chi^2$ test Consider a hypothesis test concerning the variance from a normal population with $H_0: \sigma_2=339.7$ and $H_a: \sigma_2<339.7$. Select bounds on the $p$ value for $n=11$ and test statistic $\chi^2=1.36$.
A) $0.025\leq p\leq0.05$
B) $0.0001\leq p\leq 0.001$
C) $p\leq ... | Just to make sure the rational and computations are clear:
To test $H_0: \sigma = 339.7$ against $H_a: \sigma < 339.7,$
one uses the test statistic
$$Q_{obs} = \frac{(n-1)S^2}{\sigma_0^2},$$
where $S$ is the sample standard deviation of your normal
sample of size $n = 11$ and $\sigma_0 = 339.7.$
You do not give the nu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2270909",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Linea Algebra: I do not understand this definition of a Vector Space I'm studying linear algebra, I'm reviewing Vector Spaces, and i came across the definition of this Real Vector Space $( \mathcal P_n (\mathbb R), +, \cdot\mathbb R)$:
where the 'addition' and 'multiplication' operations are defined as follows:
And I... | The source of your confusion may be this: the symbols $+$ and $\cdot$ [although suppressed] are being used in two different ways.
First of all we have them in the set of polynomials, just the usual multiplications $a_j x^j$ and so on, and usual additions $a_0+a_1 x +a_2 x^2$ and so on.
But then we want to make this in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2271035",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
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Unitary operator that leaves a dense subspace invariant Let $\mathcal{H}$ be a Hilbert space, $\mathcal{D}$ be a dense subspace of $\mathcal{H}$ and $U$ be an unitary operator on $\mathcal{H}$.
Suppose that $U\mathcal{D}\subseteq \mathcal{D}$. Can we say that $U\mathcal{D}= \mathcal{D}$?
If this is not true, do you kno... | Suppose that base is $\{e_n, n \in\mathbb{Z}\}$ consider the shift operator $S(e_n)=e_{n+1}$ and $D=\{x=(x_i)_{i\in\mathbb{Z}}$ with $x_i\neq 0, i<0$. $S(D)\subset D$, but $x=(x_n)$ with $x_n=1/n, i\neq 0, x_0=0$ is not in $S(D)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2271105",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Poisson process: How to find the probability of an event occurring during a sub-interval given it occurred during a bigger interval Title doesn't make a lot of sense due to the complicated explanation and the fact I tried to be brief.
Basically, I have a Poisson process, and I have been given the information that an ev... | let's say that number of events in 10 seconds is poisson with parameter $\lambda$
then you should know that for 30 seconds it is poisson with $3\lambda$
then P(1 event in 30 seconds) = $3\lambda exp(-3\lambda)$
p(1 event 20-30 seconds only) = P(0 events 0-20)P(1 event 20-30) = $exp(-2\lambda) \times \lambda exp(-\lambd... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2271283",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Constructive proof for existence of uncountable sets? I stumbled upon this question by trying to prove that the rationals $\Bbb Q$ are not uncountable, but not by using the knowledge that they are already provably countable. I think I forbid myself using a proof by contradiction. But then, can I even show that the real... | Your question "Can I show the existence of an infinite set with no bijection to N N without using proof by contradiction?" is ill-posed because it does not clarify the nature of "existence" you have in mind. This is the gist of constructivist objections to classical mathematics as formulated for instance by Errett Bis... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2271415",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Which event has a higher probability? Which event has a higher probability?
$24$ rolls of 2 dice at once we get at least 2 $1$s
or
one roll of 4 dice at once we get at least one $1$?
| Hint. $1$ minus the probability that in 24 rolls of 2 dice we never get 2 ones (at once): $1-\left(\frac{6^2-1}{6^2}\right)^{24}$.
1 minus the probability that in 1 roll of 4 dice we never get one: $1-\left(\frac{6-1}{6}\right)^{4}$.
Which number is greater?
P.S. Both numbers are quite close to $0.5$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Show that $a+b+c+\sqrt {\frac {a^2+b^2+c^2}{3}}\le4$, Let $a,b,c$ be positive real numbers such that $a^2+b^2+c^2+abc=4.$ Show that $$a+b+c+\sqrt {\frac {a^2+b^2+c^2}{3}}\le4.$$
| Let $a=\frac{2x}{\sqrt{(x+y)(x+z)}}$ and $b=\frac{2y}{\sqrt{(x+y)(y+z)}},$ where $x$, $y$ and $z$ are positives.
Hence, $c=\frac{2z}{\sqrt{(x+z)(y+z)}}$ and we need to prove that
$$\sum_{cyc}\frac{2x}{\sqrt{(x+y)(x+z)}}+\sqrt{\frac{4}{3}\sum_{cyc}\frac{x^2}{(x+y)(x+z)}}\leq4$$ or
$$\sum_{cyc}x\sqrt{y+z}+\sqrt{\frac{1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2271863",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Recurrence relation with limit How do I find the lim for a series that has the following recurrence relation
$$ a_{0}=2 $$ $$a_1=16 $$ and $$a_{n+1}^2=a_na_{n-1} $$ .I applied $\ln$ and I substituted $$ b_n=\ln (a_n) $$ and so I found the following recurrence relation $$2b_{n+1}=b_n+b_{n-1}$$ .What do I do next? Can s... | The Ansatz $b_n = x^n$ leads to $2x^2 - x - 1 = 0$ which implies $x = 1,-1/2$. Hence the general solution is $$b_n = c_1 + c_2 \left(-\frac{1}{2} \right)^n.$$
Using the initial conditions we see that $c_1 + c_2 = \ln 2$ and $c_1 - \frac{c_2}{2} = \ln 16$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2272007",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Probability of equal no. of red/black cards from selection - simulation vs. answers discrepancy Following reading this thread: "Probability of drawing exactly 13 black & 13 red cards from deck of 52", I created a simple simulation using Excel/VBA to help my son grasp the concept - he's only 7 but wanted to know more...... | The chance of getting five red and five black is $\frac {{26 \choose 5}^2}{52 \choose 10} \approx 0.2735$, very close to your simulation. The $26 \choose 5$s are the number of ways to choose five of the $26$ red (black) cards.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2272117",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Regarding the sum $\sum_{p \ \text{prime}} \sin p$ I'm very confident that
$$\sum_{p \ \text{prime}} \sin p $$
diverges. Of course, it suffices to show that there are arbitrarily large primes which are not in the set $\bigcup_{n \geq 1} (\pi n - \epsilon, \pi n + \epsilon)$ for sufficiently small $\epsilon$. More stro... | My answer here only includes partial results. First, we use Vinogradov's inequality:
Let $\alpha$ be a real number. If integers $a$ and $q$ satisfies $(a,q)=1$ and
$$
\left| \alpha - \frac aq \right| \leq \frac 1{q^2},
$$
then
$$
\sum_{n\leq N} \Lambda(n) e^{2\pi i \alpha n} = O\left( (Nq^{-1/2} +N^{4/5} + N^{1... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the sum $\sum_{k=1}^\infty {\frac{6^k}{\left(3^k-2^k\right) \left(3^{k+1}-2^{k+1}\right)}}$ I need to find the sum,
$$\sum_{k=1}^\infty {\frac{6^k}{\left(3^k-2^k\right) \left(3^{k+1}-2^{k+1}\right)}}$$
I have tried to break the terms into partial fractions (method of differences) but am not able to do so. How to p... | First we can try to split things into two pieces:
$$\frac{6^k}{\left(3^k-2^k\right) \left(3^{k+1}-2^{k+1}\right)} = \dfrac{A}{3^k-2^k} + \dfrac{B}{3^{k+1}-2^{k+1}}$$
So we have $A \cdot (3^{k+1}-2^{k+1}) + B \cdot (3^k-2^k) = 6^k$ which can be arranged to $3^k (3 A + B) - 2^k (2 A + B) = 6^k$.
If we make $2A+B=0$ and ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find a continuous function having following properties.
Which of the following statements is/are true?
*
*There exists a continuous map $f:ℝ\to ℝ$
such that $f(ℝ)=\mathbb{Q} $.
*There exists a continuous map $f:ℝ\to ℝ$
such that $f(ℝ)=\mathbb{Z} $.
*There exists a continuous map $f:ℝ\to ℝ^2$
such ... | Hints for 3: Don't try to find a bijective function; it doesn't exist. What is the name of the set where $x^2+y^2=1$? Have you studied that set before?
Hints for 4: Don't try to find a nice formula, like a polynomial. You can define a function in English if you want. What do you need the function to do?
| {
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"question_score": "5",
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The integer values of $\sum_{d|n}\frac{\sigma(n/d)^d}{d}$. I was looking for integer values of the $$\sum_{d|n}\frac{\sigma(n/d)^d}{d}$$
where the $\sigma(n)$ is a divisors sum of $n$.
And amazingly has found that the only integer values for $n<1500000$ are:
$$1, 39, 793$$
So I assume that these are the only integer nu... | Let $$f(n) =\sum_{d | n} \frac{\sigma(n/d)^d}{d}$$
If $p,q$ are two different primes then
$$f(pq) =\sigma(pq) + \frac{p(p+1)^q + q (q+1)^p + 1}{pq}$$
For $f(pq) \in \mathbb{Z}$
we need $p(p+1)^q + q (q+1)^p + 1 \equiv 0 \bmod p$ and $q$
$$\implies \qquad q (q+1)\equiv -1 \bmod p, \qquad p (p+1)\equiv -1 \bmod q$$
(by ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Efficient algorithm for finding period of Markov chain What is the least time consuming way to find a period of state of irreducible Markov chain? I wondering if there is an algorithm which does not use matrix multiplication?
| There is an efficient algorithm based on breadth-first search here:
http://cecas.clemson.edu/~shierd/Shier/markov.pdf
Given a dense N x N matrix the algorithm would be worst-case O(n^2) whereas computing all matrix powers would be O(n^4) which is most likely much slower for matrices of any significant size.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2272873",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Tile Edge Challenge I've been having a problem with these problems for a week now so I thought I'd post it on MSE. Read the text below. . I've also typed it up:
Annabel made a shape by placing identical square tiles in a frame as shown in the diagram above. The tiles are arranged in columns. Each column touches the bas... | If the shape is convex, since the height ant go up equal to the height ant go down and side length ant go is side,
therefore
Formula of convex type is 「Total=HighestVertical×$2$+LongestSide」
b: Lost pieces are $(3,3,1)(1,3,3)$.
c: $(5,5,5,5,5,5,5,5,5,4)$ is $20$.
d: Here are 4 ways
$(7,7,,,7,4)
(8,8,,,8,1)
(9,9... | {
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A Question From Shiryaev I was studying about $\lambda$-systems from Shiryaev where I countered the following statement about equivalence of conditions for defining a $\lambda$-system.
I can check the equivalence of the second set of conditions given the first. However, going the other way, I am not able to show condi... | Hint: If $A,B \in \mathscr{L}$ and $A \subseteq B$, then can you show (In my notation, $B^c$ is the complement of $B$)
a) $B^c \in \mathscr{L}$
b)$B^c\bigcup A \in \mathscr{L}$ (Since $A\subseteq B \iff A\cap B^c = \phi$).
c)$B^c\bigcup A = (B\setminus A)^c$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Number of real solutions $2\cos(\frac{x^2+x}{2})=2^x+2^{-x} $ The number of real solutions of $2\cos(\frac{x^2+x}{2})=2^x+2^{-x} $ is
(1) 0
(2) 1
(3) 2
(4) infinitely many .
My work :
$$
1\geq \cos\left(\frac{x^2+x}{2}\right)=\frac{2^x+2^{-x} }{2}\geq 1
\qquad \text{by (AM-GM).}
$$
So $\frac{x^2+x}{2}=2n\pi$ for a... | hint
$$-2\leq 2\cos(\frac {x^2+x}{2})\leq 2$$
$$2^x+2^{-x}\geq 2$$
the root must satisfy
$$2^x+2^{-x}=2$$ which gives $x=0$.
the unique root.
| {
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"url": "https://math.stackexchange.com/questions/2273239",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Calculate $\sqrt{2i}$ I did:
$\sqrt{2i} = x+yi \Leftrightarrow i = \frac{(x+yi)^2}{2} \Leftrightarrow i = \frac{x^2+2xyi+(yi)^2}{2} \Leftrightarrow i = \frac{x^2-y^2+2xyi}{2} \Leftrightarrow \frac{x^2-y^2}{2} = 0 \land \frac{2xy}{2} = 1$
$$\begin{cases}
\frac{x^2-y^2}{2} = 0 \\
xy = 1\\ \end{cases} \\
=\... | Write $2i=2e^{i \pi/2+2k\pi}$. Then square root to get: $\sqrt{2} e^{i\pi/4+k\pi}$. So your roots are $\sqrt{2}e^{i\pi/4}$ and $\sqrt{2}e^{3i\pi/4}$. Which are $\pm(1+i)$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Using Cramer's Rule to Derive Dot Product I'm hoping to derive an equation for the dot product using Cramer's rule. Here I'm going to try in $\mathbb{R}^2$ and will generalize once I get this first issue cleared.
I'm hoping to arrive at an expression for the dot product by first asking how to describe some vector $\m... | This only works if the two basis vectors are orthonormal. Putting that in your assumption presupposes the conclusion. However, there is another way to see it: this should define a new inner product, not necessarily the dot product, in which they are orthonormal.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Number of solutions for $x_1+ x_2+ x_3 + \cdots + x_k=n$, where $0\leq x_i\leq p$ for all $i$ I know that for $1\leq x_i\leq p$ the answer will be the coefficient of $x^n$ in $(x + x^2 + x^3 + ... + x^p)^k$. But what will be the answer for the constraint $0 \leq x_i \leq p?$
Also, how can I generate a definite formul... | For any fixed $p,k$ you look at the generating function $((1-x^{p+1})/(1-x))^k$ and the coefficient of $x^n$ is the answer $a(n)$. The generating function factors as $(1-x^{p+1})^k (1-x)^{-k}$ and each of these involves binomial coefficients. Thus the product coefficients is given by a sum involving products of binomia... | {
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Union of connected space
If $\{A_\alpha, \alpha \in I\}$ is a collection of conected spaces and $\cap A_\alpha \neq \emptyset$ then $\cup A_\alpha$ is connected.
My proof: If $\cup A_\alpha$ is not connected then we can find $V,W\neq\emptyset$ such that $V\cup W = \cup A_\alpha$ and $V\cap W = \emptyset$. Suppose $\c... | Suppose $\cup_{\alpha} A_{\alpha}$ is not connected and let $V$ and $W$ be two nonempty disjoint sets so that $V\cup W = \cup_{\alpha} A_{\alpha}$. Let $x\in \cap_{\alpha} A_{\alpha}$. Then $x \in V$ or $x \in W$ but not both. Wolog let $x \in V$.
Let $y \in W$, then $y \in A_i$ for some set $A_i$. And $x = A_i$ be... | {
"language": "en",
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If $f(x)=x^{n-1}\log x$, then the $n$-th derivative of $f$ is equals? The options are:
A) $\dfrac{(n-1)!}{x}$;
B) $\dfrac{n}{x}$;
C) $(-1)^{n-1}\dfrac{(n-1)!}{x}$;
D) $\dfrac{1}{x}$
My attempt:
$$f'(x)= (n-1)x^{n-2}\log x+ x^{n-2}$$
$$f''(x)=(n-2)(n-1)x^{n-3}\log x+ (n-1)x^{n-3}+(n-2)x^{n-3}$$
But I fail to see any pat... | Here is the pattern:
The second summand in $f'(x)$, is $x^{n-2}$. This summand will not survive $n-1$ more derivatives, and so, you may ignore it. This leaves you with $$f'(x)=(n-1)x^{n-2}\log x+(\mathrm{irrelevant\; stuff}).$$
Likewise, for the second derivative you have$$f''(x)=(n-2)(n-1)x^{n-3}\log x +(\mathrm{irrel... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How am I computing $\int e^{x}\ln(1+e^{x})\,dx$ incorrect? I am stuck on this problem.
Problem Evaluate $$\int e^{x} \ln (1+e^{x})$$
Attempt
Integration by Parts:
Let $u=\ln(1+e^x), \ dv = \int e^{x} \ dx$ and we have $\ du = \frac{e^x}{1+e^x}$ and $v=e^x$
$$\int u \ dv= uv - \int v \ du$$
Thus $$I=\ln(1+e^x) e^x - \in... | You should just let $u=1+e^x$, $du=e^x\,dx$
Then
\begin{eqnarray} \int e^{x}\ln(1+e^{x})dx&=&\int\ln(u)\,du \\
&=&u\ln(u)-u+c
\end{eqnarray}
and take it from there.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Money Interest Problem
Jake has loaned 20000 pesos from the bank with an interest of 10% every year. If Jake completes the payment within 3 years, what amount should be paid to the bank?
Could someone explain to me the entire solution?
| Amount= $Principal+Interest$
(assumption, Interest is calculated at initial principal, i.e. simple interest)
$Interest$=$Principal\times Rate\times years$
=$20,000\times0.1\times 3$
Total money= $20,000+6,000$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2274217",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Domain of $f(x)=\sqrt{\lfloor x\rfloor-1+x^2}$ I drew the number line and tested with different values, getting the correct domain $(-\infty,-\sqrt3)\cup[1,\infty)$. However, how do I solve this faster by manipulating the function?
| You must find the values of $x$ such that $\lfloor x\rfloor-1+x^2\geq0$.
The easy part:
*
*It's true for $x\geq1$, since $x^2\geq1$ and $\lfloor x\rfloor>0$.
*For $0\leq x<1$, it's false, because $\lfloor x\rfloor=0$ and $x^2<1$.
Now, the case $x<0$. First notice that for $x\in[n,n+1[$, for integer $n$, you have $\... | {
"language": "en",
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Integral with $+i\epsilon$ prescription involving residue theorem? Consider the integral $$I = \int_{-1}^{1} \frac{\text{d}x}{(x + \xi - i\epsilon) (x- \xi + i \epsilon)}$$ where $\xi$ is valued in $[-1,1]$. If I want to note the contribution of this integral at the point $x=\xi$ does the $+i\epsilon$ prescription allo... | It is not clear to me what your question means.
But why not calculate the integral
$$f(\xi ,\epsilon )=\int_{-1}^1 \frac{1}{(\xi +x-i \epsilon ) (-\xi +x+i \epsilon )} \, dx$$
explicitly?
Writing the integrand as
$$\frac{1}{(\xi +x-i \epsilon ) (-\xi +x+i \epsilon )}=\frac{1}{a^2+x^2}$$
with
$$a\to \epsilon +i \xi$$
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Convergence question from Stromberg Classical Analysis I have been trying to find whether the following series converge. If so, to what limit?
Define $x_0=0$ and $x_1=1$, and
$$
x_{n+1~}=\frac{1}{n+1}x_{n-1}+\frac{n}{n+1}x_n \qquad n\geq 1
$$
There are other ways to show convergence, but I showed that $x_{2n-2}... | It seems to me that your guessed limit is
$$
1 + \sinh(1) - \cosh(1).
$$
Just compare the expansions
$$
\cosh(x)=\sum_{n=1}^\infty \frac{x^{2n}}{(2n)!}, \quad \sinh(x) = \sum_{n=1}^\infty \frac{x^{2n+1}}{(2n+1)!}.
$$
You should try to prove it, maybe by induction.
| {
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What is wrong with this "proof"? $1$ is always an eigenvalue for $I + A$ ($A$ is nilpotent)?
Consider the nilpotent matrix $A$ ($A^k = 0$ for some positive $k$).
It is well known that the only eigenvalue of $A$ is $0$.
Then suppose $\lambda$ is any eigenvalue of $I + A$ such that
$(I + A) \mathbf{v} = \lambda \ma... | You have proven that if $A$ is nilpotent, then the eigenvalue of $A+I$ (NOT the eigenvalues of $A$) is equal to $1$. There's nothing wrong with the proof.
| {
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"url": "https://math.stackexchange.com/questions/2274656",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Proving an absence of rational and integer solutions. Prove $$(x+y\sqrt2)^2+(z+t\sqrt2)^2=5+4\sqrt2$$
has no solution in rational $(x,y,z,t)$
Prove $$(5+3\sqrt2)^m=(3+5\sqrt2)^n$$
has no solution for positive integers $(m,n)$
How do I approach these kinds of problems? I'm not sure where to start. Also, what are some ... | Hint for the second part: Take norms: $N(a+b\sqrt2)=a^2-2b^2$. The key property is $N(\alpha\beta)=N(\alpha)N(\beta)$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Completing the square of $x^2 - mx = 1$ is not giving me the right answer. This is my attempt
$$
\begin{align}
x^2 - mx &= 1 \\
x^2 - mx - 1 &= 0 \\
\left(x^2 - mx + \frac{m^2}{4} - \frac{m^2}{4}\right) - 1 &= 0 \\
\left(x^2 - mx + \frac{m^2}{4}\right) - \frac{m^2}{4} - 1 &= 0 \\
\left(x^2 - mx + \f... | (Not an answer, just a long comment.)
Your actual question has already been answered, but I want to point out another mistake, namely when you go from
$$\left(x - \frac{m}{2}\right)^2 = \frac{m^2 - 4}{4}$$
to
$$\sqrt{\left(x - \frac{m}{2}\right)^2} = \sqrt{\frac{m^2 - 4}{4}}.$$
At this point, there should be $\pm$ sign... | {
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Truncated taylor series inequality I came across the following fact in a paper and am having trouble understanding why it is true:
Consider the error made when truncating the expansion for $e^a$ at the $K$th term. By choosing $K = O(\frac{\log N}{\log \log N})$, we can upper bound the error by $1/N$, in other words
$$\... | What we are asked to prove is this: there is a constant $C$ such that if
$$ K \ge C \frac{\log N}{\log \log N} $$
then
$$ \sum_{j=K+1}^\infty \frac{a^j}{j!} \le \frac1N. \tag1$$
Now clearly the left hand side of (1) grows to infinity as $a \to \infty$, so it must be that $C$ depends upon $a$. Suppose $K \ge 2a$ (which... | {
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} |
Proving Variance of Normal Distribution My question is as follows: using the standard integral $$\int_{-\infty}^{\infty}e^{-ax^2}dx=\sqrt{\frac{\pi}{a}}$$
prove directly from the definition that the variance of the normal distribution,
$$f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{\frac{(x-\mu)^2}{2\sigma^2}},$$ is $\sigma^2$. ... | Your substitution step is lacking an extra factor of $\sigma$ because you should have written $$\begin{align*}
\frac{1}{\sqrt{2\pi} \sigma} \int_{x=-\infty}^\infty (x-\mu)^2 e^{-(x-\mu)^2/(2\sigma^2)} \, dx &= \frac{1}{\sqrt{2\pi} \sigma} \int_{y=-\infty}^\infty (\sigma y)^2 e^{-y^2/2} \sigma \, dy \\
&= \frac{\sigma^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2275273",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How can I show that $P(A \cap B) \gt P(A) + P(B) - 1$ How can I show that $P(A \cap B) \gt P(A) + P(B) - 1$
I know that
$P(A \cap B)= P(A)P(B)$
But I don't see how that can help me get to that inequality.
Can someone give me a hint on how to start this?
| The inequality is not correct, for example, let $A$ be the event that you get a head and $B$ be the even that you get a tail.
$$P(A \cap B)=0$$
$$P(A)+P(B)-1=0$$
$$0> 0$$ which is a contradiction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2275407",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 2
} |
Let $A, B$ be unitary rings and let $f$ be a surjective homomorphism of rings. Then $f(Jac(A)) \subseteq Jac(B)$ Let $A, B$ be unitary rings, and let $J(A)$, $J(B)$ denote the Jacobson's radical of $A$ and $B$. Let $f$ be a surjective homomorphism of rings. Then:
$$f(J(A)) \subseteq J(B)$$
Attempt: It suffices to prove... | *
*Yes, it is true, for the reason below:
*Yes, this is true as well. By the correspondence theorem, the ideals of $ B $ correspond bijectively to the ideals of $ A $ containing $ \ker f $ in an inclusion respecting manner, so a maximal ideal of $ B $ pulls back to a maximal ideal of $ A $. Note that this is not true... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2275538",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
How do you prove the statement: $A^c=(A\cup B)^c\cup (B \setminus A)$ I've been stuck on this question for a while and the problem is basically I just don't know how to prove it, I tried converting it to $\vee$ and $\wedge$ symbols then I did some research and found mathematically it doesn't make sense to compare them,... | The standard way to show that two sets are equal is to show that one is contained in the other. On the left, you have everything that is not in $A$. On the right, everything that is not in ($A$ or $B$) together with everything in $B$ but not $A$. So if $x$ is not in $A$ then show that either $x \in (A \cup B)^c$ or $x ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2275659",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Proving that sequences with the pattern aaaaaaa don't include a perfect square I need help with the following problem:
Which of the following sequences doesn't have perfect square:
A) $11, 111, 1111, \dots$
B) $33, 333, 3333, \dots$
C) $44, 444, 4444, \dots$
D) $77, 777, 7777, \dots$
I proved that the first sequence ... | For 1 you are done.
For 2 not divisible by 4.
For 3 no perfect square ends with digit 3.
For 4 just divide by 4 and get back to the case with 1.
For 5 not divisible by 25.
For 6 not divisible by 4.
For 7 no perfect square ends in digit 7.
For 8 not divisible by 16.
For 9 divide by 9 and get back to 1.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2275879",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
What is the largest $a$ for which all the solutions to the equation $3x^2+ax-(a^2-1)=0$ are positive and real? Problem: What is the largest $a$ for which all the solutions to the equation $3x^2+ax-(a^2-1)=0$ are positive and real?
Attempt: Solving the equation for $x$ I get $$x_{1,2}=-\frac{a}{6}\pm\sqrt{\frac{13a^2-12... | The equation is : $3x^2+ax-(a^2-1)=0$
*
*First condition is which you identified, that the discriminant must be positive.
*Note that the abscissa of vertex of this parabola is $\dfrac{-b}{2a}$ . For both roots to be positive, this value must be positive
*Furthermore, since the parabola is upward, $f(0) > 0$ impli... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2276023",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
How to find period of a real function $f$ given the functional equation $\sqrt{3}f(x) = f(x-1) + f (x+1) $? If a periodic function satisfies the equation $\sqrt{3}f(x) = f(x-1) + f (x+1) $ for all real $x$ then prove that fundamental period of the function is $12$.
Here fundamental period means the smallest positive r... | Define the linear operator $T$ by
$$Tf(x):=- √3f(x) + f(x-1) + f(x+1)$$
and $E$ by
$$Ef(x)=f(x+1).$$
One can see that $T=E-\sqrt{3}+E^{-1}$.
Consider solving $$x-\sqrt{3}+x^{-1}=0$$
$$\implies x^2-2x\frac{\sqrt{3}}{2}+1=0$$
$$\implies\left(x-\frac{\sqrt{3}}{2} \right)^2=-\frac{1}{4}$$
$$\implies x = \frac{\sqrt{3}\pm i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2276088",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
} |
Find all natrual numbers that are $13$ times bigger than the sum of their digits.
Find all natural numbers that are $13$ times bigger than the sum of their digits.
I had a solution and just wanted to verify it. By solving the equation $$13(a_1+a_2+a_3+\dots a_n)=\overline{a_1a_2\dots a_n},$$ we can get $n=3$ and by ... | An $n$-digit number is $\ge 10^{n-1}$, but the sum of its digits $\le 9n$. This give us the inequality $10^{n-1}\le 13\cdot 9n=117n$, which easily leads to $n\le 3$.
Then $13\cdot(a+b+c)=100a+10b+c$ leads to $87a=3b+12c$, or $29a=b+4c$.
*
*With $a=0$, we arrive at $b=c=0$.
*With $a=1$, we arrive at $b+4c=29$, he... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2276159",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Given hypotenuse, find the other two sides. Note that we are only interested in integral pythagorean triplets, we are given the hypotenuse $c$, how can I efficiently find the other two sides of the right angled triangle. I need something better than the bruteforce approach of iterating over all lengths $a$ below $c$, a... | How about using the standard formula for generating Pythagorean triples? Solve $c = m^2 + n^2$ for $m$ and $n$. Then you have $a = m^2 - n^2$ and $b = 2mn$. (If $m$ and $n$ are co-prime and of opposite parity, the triple is primitive, otherwise not.) This requires less brute force than the approach you wanted to avoid... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2276258",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
$\alpha : H^{n}(\mathrm{Hom}(K,G)) \to \mathrm{Hom}(H_{n}(K),G)$ is an isomorphism Let $G$ is divisible group and abelian , $K$ is a chain complex . The map $\alpha$ in the title above takes idea from inner product . To be more specific , $x \in H_{n}(K)$ and $u \in H^{n}(\mathrm{Hom}(K,G))$ the inner product $<u,x>$ i... | By the universal coefficient theorem, the obstruction to isomorphism
will be an Ext group: $\text{Ext}^1(H_{n-1}(X),G)$. But because
$G$ is divisible, this Ext group vanishes, as divisible groups are
injective in the category of Abelian groups.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2276352",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Why is the following inequality true: $\frac{k!}{k^k} > e^{-k}$ I stumbled upon the following inequality in a scientific paper which estimates a lower bound for $\frac{k!}{k^k}$ for $k \in \mathbb{N}$:
$$\frac{k!}{k^k} > e^{-k}$$
They did not explain why this holds true, and I could not find any answer by myself yet.
| Use the Taylor series:
$$e^k = 1+k+\frac{k^2}{2!} +\cdots+\frac{k^k}{k!} +\cdots.$$
Because all terms on the right are positive, we have
$$e^k > \frac{k^k}{k!},$$
then just take reciprocals of both sides.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2276404",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
is this sequence Cauchy in the Banach space of continuous functions with infinity norm? In the Banach space $(C[0,1], ||\cdot||_{\infty})$ where $||\cdot||_{\infty} = max_{[0,1]}|f(x)|$ let the sequence of functions $\{f_n(x)\}$ be given by $f_n(x) = \frac{n\sqrt{x}}{1+nx}$.
State whether the sequence is Cauchy in this... | Suppose $(f_n)$ is Cauchy in $C([0,1]).$ Then there exists $N\in \mathbb N$ such that $\|f_n-f_N\| < 1$ for $n>N.$ This $N$ is now fixed. It follows that
$$\tag 1 \|f_n\| \le \|f_n-f_N\| + \|f_N\| < 1 + \|f_N\|$$
for $n>N.$ But notice $f_n(1/n)=\sqrt n/2.$ Thus $\|f_n\|\to \infty,$ violating $(1),$ contradiction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2276551",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Expectation from a colour-matching withdrawal game
A person draws 3 balls from a bag containing 3 white, 4 red and 5 black balls. He is offered 10 bucks, 5 bucks, 2 bucks if he draws 3 ball of same color, 2 balls of same color and 1 ball of each color respectively. Find how much he expects to earn.
I tried to solve t... | You had a good start. The expected value is calculated from the sum of the probability and the value associated with that probability.
So, the expected value here would be the following:
$10$ $ 3 \choose 3 $ $12 \choose 3$^-1 + $10$$4 \choose 3$ $2 \choose 3$^-1 + $10$ $5 \choose 3$ $12 \choose 3$^-1 + $5$$ 3 \choos... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2276658",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Representing lower order B-Splines as higher order B-splines I have tried to figure out how B-splines of degree $p - 1$ can be represented as linear combinations of B-splines of degree $p$.
Definitions:
*
*Given a set of increasing real values $t = (t_i)_{i = 1}^{p+n+1}$, the $i$th B-spline of degree $p$ is defined ... | In general the Degree Elevation Algorithm can express a spline $S=\sum \limits _{j} c_{j} B_{j,p}$ in terms of B-Splines of order $p+1$, i.e. computes the coefficients $c^*_{i} $ such that $S=\sum \limits _{j} c_{j} B_{j,p}=\sum \limits _{i} c^*_{i} B_{i,p+1}$. You can find the details of the algorithm for example in "... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2276752",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Sum of entries of a matrix For a matrix $A \in \mathbb{R}^{n \times n}$, it is clear that the sum of all the entries of $A$ can be expressed as
$$\vec{1}^{T} A \vec{1} = \sum \limits_{i,j} A_{i,j}$$
Now suppose $A,B \in \mathbb{R}^{n \times n}$ are symmetric matrices. Then by the above expression, it is clear that the ... | Let $X$ be the square matrix whose each element is $1$.
(Is there canonical notation for this?)
This is a symmetric matrix.
$\DeclareMathOperator{\tr}{tr}$
The sum of elements of $A$ is $s(A)=\tr(AX)$.
For symmetric $A$ and $B$ the sum of elements in the product is
$$
s(AB)=\tr(ABX)=\tr((ABX)^T)=\tr(X^TB^TA^T)=\tr(XBA)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2276877",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
} |
Area of triangle and determinant The area of a $\vartriangle ABC$ with given vertices $(a,a^2),(b,b^2),(c,c^2)$ is $\frac{1}{4}$ $sq. units$ and area of another $\vartriangle PQR$ with given vertices $(p,p^2),(q,q^2),(r,r^2)$ is $3$ $sq. units$.
Then what is the value of
$$
\begin{vmatrix}
(1+ap)^2 & (1... | Let $A, B, C$, $P, Q, R$ be the $6$ column vectors
$$
\begin{cases}
A^T = (1, \sqrt{2}a, a^2),\\
B^T = (1, \sqrt{2}b, b^2),\\
C^T = (1, \sqrt{2}c, c^2)
\end{cases}
\quad\text{ and }\quad
\begin{cases}
P^T = (1, \sqrt{2}p, p^2),\\
Q^T = (1, \sqrt{2}q, q^2),\\
R^T = (1, \sqrt{2}r, r^2)
\end{cases}
$$
Using identites of t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2276979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
How would I find the number of distinct homomorphisms and isomorphisms mapping the Klein four group to the Klein four group? How would I find the number of distinct homomorphisms and isomorphisms from the Klein four group to the Klein four group?
Thank you
| Here’s an approach that perhaps is more advanced, maybe even too advanced.
Your group is a two-dimensional vector space $V$ over the field $k=\Bbb F_2$ with two elements. Every homomorphism $V\to V$ is automatically a $k$-linear map, so to count these, we need only count the $2\times2$ matrices over $K$, so sixteen in ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2277128",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
If $ x^4+x^3+x^2+x+1=0$ then what is the value of $x^5$ If $$x^4+x^3+x^2+x+1=0$$ then what's the value of $x^5$ ??
I thought it would be $-1$ but it does not satisfy the equation
| Well, we have $x^5-1=(x-1)(x^4+x^3+x^2+x+1)=(x-1)0=0$ so that $x^5=1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2277392",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Proving an interesting property of absolute values: $\frac{|x+y|}{1+|x+y|}\le\frac{|x|}{1+|y|}+\frac{|y|}{1+|y|}$
Question
If $x$ and $y$ are real numbers, show that
$$\frac{|x+y|}{1+|x+y|}≤\frac{|x|}{1+|y|}+\frac{|y|}{1+|y|}$$
I am having difficulty in proving this equation. I don't know where to start. Your help wi... | We know that $|x+y|\leq|x|+|y|$. You need to prove $$\frac{|x+y|}{1+|x+y|}\leq\frac{|x|}{1+|x|}+\frac{|y|}{1+|y|}$$ $$or,~|x+y|.(1+|x|)(1+|y|)\leq(1+|x+y|)(|x|.(1+|y|)+|y|.(1+|x|))$$
by cross multiplication. By expanding we get
$$|x+y|\leq|x|+|y|+2|x||y|+|x||y||x+y|$$
which is obviously true since $2|x||y|+|x||y||x+y|\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2277665",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
$f$ will always reach 1 if $\lim \limits_{x \to \infty}f(x)=1^-$? Suppose an algorithm A, has success probability of 10%.
One experiment is composed by a (possibly infinite) number of executions of A.
In one experiment, the success probability of at least one execution after x trials can be found by:
$P[x\ge1]=1-(\fra... | Let B be the algorithm running A until it succeeds.
There are different ways of looking at this
*
*There is a case where B runs forever (if A keeps failing)
*The probability of this case (that B runs forever), is $0$
*For any $k$, the probability, that B will require more than $k$ calls to $A$, is greater than $0$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2277791",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
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