Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Visual representation of the domain and range of a function? An excerpt in the book "College Algebra by Michael Sullivan is that:
When the graph of a function is given, its domain may be viewed as the shadow created by the graph on the x- axis by vertical beams of light. Its range can be viewed as the shadow created b... | Consider a particular point on the function. If you cast a vertical light beam through that point it would cast a shadow on the x-axis according to the x value of that point. Since the domain is the set of all possible x values of a function, imagine repeating the vertical light beams on every point on the function. Th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2800183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
Epsilon delta proof for $\lim_\limits{x\to a} \sqrt{x} = \sqrt{a}$ where $a>0$ $f(x) = \sqrt{x}$. For right limit, $x-a > 0$. Let $x-a < \delta$ then $(\sqrt{x}-\sqrt{a}) \lt \frac{\delta}{\sqrt{x}+\sqrt{a}}$. Then chose we must choose $\epsilon $ so that $\epsilon > \frac{\delta}{\sqrt{x}+\sqrt{a}}$. Is this correct? ... | For $x >0$, $a>0$:
$|√x-√a| =\dfrac {|x-a|}{√x+√a}\lt$
$\dfrac{|x-a|}{√a}.$
Let $\epsilon >0$ be given.
Choose $\delta \le \epsilon √a.$
Then
$|x-a| \lt \delta$ implies
$|√x-√a| \lt \dfrac{|x-a|}{√a} \lt $
$\dfrac{\delta}{√a} \le\epsilon.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2800305",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Understanding Galois theory, swapping roots I am trying to understand the rudiments of galois theory but I have a hard time answering the following question: Suppose we have $f(X)=x^5-2$ over $\mathbb{Q}$. What is $Gal(E_f: \mathbb{Q})$?
After seeing similar questions here I am more confused: Isn't it true that since $... | First note that the splitting field of $f$ is given by $L = \mathbb{Q}(\sqrt[5]{2}, \zeta_5)$, where $\zeta_5$ is the primitive root of unity.
Now we know that any automorphism on $L$ is uniquely determined by the action on the adjoined elements $\sqrt[5]{2}$ and $\zeta_{5}$. Additionally we know that if $\alpha_1$ an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2800435",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Add together numbers to make final number? I have this problem where I want to add some numbers together to make a final number. I ask how many ways are there to make this final number using exactly k numbers? Here is the condition:
(1) The numbers are in a consecutive sequence but with a number missing like 1,3,4 or 2... | Your question is similar to this one if not the sameFinding all possible combinations of numbers to reach a given sum
i wanted to comment but couldnt so i had to put it as an answer
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2800525",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Hypothesis testing - Experiment with replacement Exercise :
A box contains 4 balls out of which $\theta$ are white and $4-\theta$ are black. Suppose that you want to check the null hypothesis $H_0 : \theta =2$ against the alternative $H_1 : \theta \neq 2$. You draw 2 balls with replacement from the box and if they are... | Assuming this is a test of drawing two balls with replacement......
A type I error is rejecting the null when you shouldn't. The only scenario where this can occur is when there are two white and two black and two of the same color are chosen in the test. The probability of that is $0.5$. The first choice can be either... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2800627",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Permutation and characteristic polynomial of a matrix I have a matrix $A\in \mathbb{R}^{n\times n}$, after some manipulations I find:
$$\tilde{A}=\begin{pmatrix} 0 & I_n&0&\cdots&0\\0&0&I_n&\cdots&0\\ \vdots&\vdots& \vdots&\ddots &\vdots\\0&0&0&\cdots&I_n\\ A & 0 &0 &\cdots &0 \end{pmatrix};\; \mathcal{A}=\begin{pma... | I see no easy way to relate the eigenvalues of $\tilde A$ and $\mathcal A$ directly. However, both of these have eigenvalues that are neatly related to those of $A$.
Characteristic polynomial of $\mathcal A$: Note that the vectors of the form
$$
v = \pmatrix{x_1\\\vdots \\ x_{k-1} \\ 0}, \quad v = \pmatrix{0\\ \vdots\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2800736",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Presentation of the holomorph of $\mathbb Z/5 \mathbb Z$ When I look up the presentation of the holomorph of $\mathbb Z/5 \mathbb Z$ it reads like the following:
$\left\langle a,b \mid a^5 = 1, b^4 = 1, bab^{-1} = a^2\right\rangle$
See https://groupprops.subwiki.org/wiki/General_affine_group:GA(1,5)
The automorphisms o... | Let let $x := (1,\iota)$ and $y := (0, \psi^3)$ as motivated by the comment of Tobias Kildetoft above.
*
*$y^2 = (0, \psi^3) \star (0, \psi^3) = (0 + \psi^3(0), \psi^3\circ\psi^3) = (0+0,\psi^2) = (0,\psi^2)$
*$y^3 = (0, \psi^5) = (0, \psi)$
*$y^4 = (0, \psi^4) = (0,\iota)$
*$y\star x \star y^{-1} = (0, \psi^3)\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2800873",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Is this complex exponential inequality true? Is it true that $|f(z)|\leq M$ implies that $|e^{f(z)}|\leq e^{M}$. If not, what is a simple counterexample?
| For $f=u+iv$ we have $u(z)\le |f(z)|\le M$ and
$$
|e^{f(z)}|=|e^{u(z)}e^{iv(z)}|=e^{u(z)}\le e^M
$$
since $e$ is increasing.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2800954",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Finding $\frac1{1+1^2+1^4}+\frac2{1+2^2+2^4}+\cdots+\frac n{1+n^2+n^4}$.
Find an expression for
$$\frac1{1+1^2+1^4}+\frac2{1+2^2+2^4}+\cdots+\frac n{1+n^2+n^4}.$$
This was given in the chapter for APs. However, I do not see how this relates to them. I tried using telescopic sums, but I am not proficient in them and w... | Let $a_k = k/(1+k^2+k^4)$.
Then:
$$a_k =
\underbrace{\frac{k(1-k)}{2(1-k+k^2)}}_{b_k}
+
\underbrace{\frac{k(1+k)}{2(1+k+k^2)}}_{c_k}.$$
Now observe that
$$b_{k+1} = \frac{(k+1)(-k)}{2(-k+(k+1)^2)}
=
\frac{-k(1+k)}{2(1+k+k^2)} = -c_k.$$
Can you conclude?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2801070",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Find the following limit: $\lim \limits_{x,y \to 0,0} \frac{x+y-\frac{1}{2}y^2}{\sin\left(y\right)+\log\left(1+x\right)}$ Recently I came upon a limit which confused me. The reason is that when I try to solve the following limit using polar coordinates I get a constant which I do not know if it gives me information.
Le... | Note that
*
*$x=0,\, y=t\to 0 \implies \frac{x+y-\frac{1}{2}y^2}{\sin\left(y\right)+\log\left(1+x\right)}=\frac{t-\frac{1}{2}t^2}{\sin\left(t\right)}\to 1$
*$x=-t+\frac12t^2,\, y=t,\, t\to 0 \implies \frac{x+y-\frac{1}{2}y^2}{\sin\left(y\right)+\log\left(1+x\right)}
=\frac{-t+\frac12t^2+t-\frac12t^2}{\sin\left(t\ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2801203",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Why is $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+2\alpha {{x}_{1}}{{x}_{2}}>0$ for all $({{x}_{1}},{{x}_{2}},{{x}_{3}})\ne (0,0,0)$ if and only if $|\alpha |<1$? Why is $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+2\alpha {{x}_{1}}{{x}_{2}}>0$ for all $({{x}_{1}},{{x}_{2}},{{x}_{3}})\ne (0,0,0)$ if and only if $|\alpha |<1$? I tried solving t... | There are probably easier ways to do it. But whatever, consider the matrix representing this quadratic form: $$A = \begin{pmatrix} 1 & \alpha & 0 \\ \alpha & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}.$$Your condition is equivalent to the matrix $A$ being positive-definite, which by Sylvester's Criterion happens if and only if al... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2801314",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Annihilation and direct sums On
http://www.math.harvard.edu/~shlomo/docs/Advanced_Calculus.pdf
page 87, ex. 3.8, we read:
Given $V=M\oplus N$, prove that $V^*= M^0 \oplus N^0$, being $V^*$ the dual space of $V$ and $M^0$ the annihilator of M (same for $N$).
If $V$ is finite, I can see that $N$ is isomorphic to $M^0$ an... | Just use the definitions. If one desires to check that $V^* = M^0 \oplus N^0$, there are two things to be done.
1) $M^0 \cap N^0 = 0$. Let $f \in M^0 \cap N^0$. Then $f|_M = 0$ and $f|_N = 0$ gives $f|_{M \cup N} = 0$. But $f$ is linear, so it follows that $f|_{V}= f|_{M\oplus N} = f|_{{\rm span}(M\cup N)} = 0$.
2) $V^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2801389",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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How do I use the given vector equation to resolve vector $p$ into a parallel and perpendicular component? I am working on the following problem:
Here's what I've done so far:
I know that dotting the first component with q should equal one to show that it is parallel and dotting the second component with q should equ... |
I know that dotting the first component with q should equal one to
show that it is parallel
Not in general:
$$
p_\parallel \cdot q
= \lVert p_\parallel \rVert \lVert q \rVert \cos \angle(p_\parallel, q)
= \lVert p_\parallel \rVert
$$
and dotting the second component with q
should equal to 0 to show that it is e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2801497",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Evaluating $\int_0^1\frac{\ln^2(1+x^2)}{x^4}dx$ I want to evaluate $$\int_0^1\frac{\ln^2(1+x^2)}{x^4}dx$$
My attempt: Letting
$$I(\alpha,\beta)=\int_0^1\frac{\ln(1+\alpha^2x^2)\ln(1+\beta^2x^2)}{x^4}dx$$
$$
\begin{aligned}
I_{12}''(\alpha,\beta)&=\int_0^1\frac{4\alpha\beta}{(1+\alpha^2x^2)(1+\beta^2x^2)}dx\\
&=\frac{4\... | Taking integration by parts,
$$ \int_{0}^{1} \frac{\log^2(1+x^2)}{x^4} \, dx
= -\frac{1}{3}\log^2 2 + \frac{4}{3}\int_{0}^{1} \frac{\log(1+x^2)}{x^2(1+x^2)} \, dx. $$
Now
$$ \int_{0}^{1} \frac{\log(1+x^2)}{x^2(1+x^2)} \, dx
= \int_{0}^{1} \frac{\log(1+x^2)}{x^2} \, dx - \int_{0}^{1} \frac{\log(1+x^2)}{1+x^2} \, dx, $$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2801621",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
The unit group of a finite dimensional associative algebra is a Lie group? I am reading Serre's "Lie algebras and Lie groups" p.103.
Let $k$ be a complete valued field(for example $\mathbb{R}$, $\mathbb{C}$, or $\mathbb{Q}_p$) and $R$ be a finite dimensional associative $k$-algebra.
Surely $R$ is an additive Lie group.... | *
*For $x$ of norm $<1$, $(\displaystyle\sum_{k=0}^n x^k)_n$ is a Cauchy sequence (by the triangle inequality) , thus by completeness of $k$ and finite dimensionality of $R$, it converges in $R$. The open neighbourhood is $||x||<1$.
*Multiplication $R\times R\to R$ is bilinear so it must be smooth, thus so is its res... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2801717",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Does this matrix equality hold? How to prove it? $B \bullet A=B\bullet U\Lambda U^T=U^TBU\bullet A$? Suppose $A,B$ are all symmetric matrices, How tp prove that $B \bullet A=B\bullet U\Lambda U^T=U^TBU\bullet A$, in which $\bullet$ is the inner product of a matrix defined as $A\bullet B=\sum_{i,j=1}A_{ij}B_{ij}$. $\Lam... | Presumably, we're given that $A = U\Lambda U^T$.
Note that $A \bullet B = \operatorname{Tr}(A^TB)$. With that in mind, we note that
$$
B \bullet A = \operatorname{Tr}(B^TA) = \operatorname{Tr}([B^TU\Lambda] U^T) =
\operatorname{Tr}(U^T[B^TU \Lambda]) =
\operatorname{Tr}([U^TBU]^T\Lambda) = [U^TBU] \bullet \Lambda
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2801934",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Determine the Taylor series for $f(x) = x^3 \cdot \ln{\sqrt{x}}$ around the point $a = 1$ and determine its radius of convergence. As the question states:
"Determine the Taylor series for $f(x) = x^3 \cdot \ln{\sqrt{x}}$ around the point $a = 1$ and determine its radius of convergence."
I have consulted this related q... | $$f(x+1)=(x+1)^3\ln(\sqrt{x+1})=\frac{(x+1)^3}{2}\ln(x+1)$$
Now write
$$\ln(x+1) = \sum\limits_{n=1}^\infty \frac{(-1)^{n+1}x^n}{n} $$
And multiply the infinite sum by the polynomial.
Then you can switch back to $f(x)$. Your approach is good in some cases, but often it's just guessing.
Easiest way to derive the Taylor ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2802024",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 1
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Intersection of surfaces is a curve if intersection of tangent planes is line I think I need a hint in solving the following exercise:
Let $S_1$, $S_2 \subseteq \mathbb R^3$ be surfaces (i.e. 2-dimensional submanifolds), with nonempty intersection $\Gamma := S_1 \cap S_2.$ Show that, if $V :=T_pS_1 \cap T_pS_2$ is a l... | Let $p \in S_1\cap S_2$.
Show that $T_pS_1 \cap T_pS_2$ $($and here I'm guessing the tangent planes are thought of as embedded in $\mathbb R^3)$ is a line if and only if $S_1$ and $S_2$ cross (rather than touch) at $p$. Can you conclude?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2802161",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Limit of a function with two variables existance I have the function $f(x,y)=\frac{x^3}{y^2}e^{\frac{-x^2}{y}}$, and I have to show whether the following limit exists: $\lim\limits_{(x,y)\to(0,0)}f(x,y)$. If I set $y=x$, then $\lim\limits_{(x,x)\to(0,0)}f(x,x)=\lim\limits_{x \to 0}f(x,x)=0$; and if I set $y=x^2$ the l... | Yes it is correct since
*
*$x=0 \implies \frac{x^3}{y^2}e^{\frac{-x^2}{y}}=0$
*$y=x^2 \quad x\to 0^+ \implies \frac{x^3}{y^2}e^{\frac{-x^2}{y}}=\frac{1}{xe}\to +\infty$
the limit doesn't exist.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2802265",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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What is the series representation of $\int_{0}^{t} \exp(-x^n)\, dx$ where $n$ is a positive integer? I have searched in the web to get series representation of $$\int_{0}^{t} e^{-x^n}\,dx$$ where $n$ is a positive integer, but I haven’t succeeded; really I want its representation as hypergeometric series.
| Integrating the Power Series
We have the power series
$$
e^{-x^n}=\sum_{k=0}^\infty\frac{(-1)^kx^{kn}}{k!}\tag1
$$
which leads to the alternating power series
$$
\begin{align}
\int_0^t e^{-x^n}\,\mathrm{d}x
&=\sum_{k=0}^\infty\frac{(-1)^kt^{kn+1}}{(kn+1)k!}\\
&=\bbox[5px,border:2px solid #C0A000]{t\,{}_1F_1\!\left(\tfr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2802575",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
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The $p^{th}, q^{th},r^{th}$ term of an A.P are in G.P . Find the ratio of the $p^{th}$ and the $r^{th}$ term. I took the $p^{th}, q^{th},r^{th}$ term as $t_p,t_q,t_r$ respectively. Then $t_r = t_p * c^2$ (If c is the common ratio). and the common difference of the A.P is $\frac{t_r -t_p}{r-p}$. But I think I need to re... | Hint:
If $a,d(\ne0)$ be the first term & the common difference of the AP
$$\{a+(p-1)d\}\{a+(r-1)d\}=\{a+(q-1)d\}^2$$
$a^2+ad(p+r-2)+(r-1)(p-1)d^2=a^2+2ad(q-1)+(q-1)^2d^2$
As $d\ne0$
$$a(p+r-2)+(r-1)(p-1)d=2a(q-1)+(q-1)^2d$$
Express $a$ in terms of $d$
Now we need $$\dfrac{a+(p-1)d}{a+(r-1)d}$$
Replace the value of $a$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2802713",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Show that a compact operator is bounded
Definition: A linear operator $T: V \to W$ is compact if and only if the image of the unit ball in $V$ is precompact (= every sequence
has a cauchy subsequence $\iff $ totally bounded).
Prove: Let $T: V \to W$ be a compact linear operator. Show that $T$ is
bounded.
My attempt:
... | It follows directly from the definitions. Since $T$ is compact, the image $T(V_1)\subset W$ of the closed unit ball of $V$ is precompact. Consider the cover $W\subset \bigcup_n W_n$, where $W_n$ is the ball of radius $n$. As $\overline{T(V_1)}$ is compact, the cover has a finite subcover, which means that there exists ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2802862",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Difference between $x\in[0,1]$ and $x\in \{0,1\}$? If I write
$$
x\in [0,1] \tag 1
$$
does it mean $x$ could be ANY number between $0$ and $1$?
Is it correct to call $[0,1]$ a set? Or should I instead write $\{[0,1]\}$?
Q2:
If I instead have
$$
x\in \{0,1\} \tag 2
$$
does it mean $x$ could be only $0$ OR $1$?
| $[0,1]$ is (defined as) the set $\{ x \in \mathbb{R} : 0 \leq x \leq 1 \}$, i.e. it is a set that contains every real number between $0$ and $1$ (inclusive). It contains an uncountable number of elements.
$\{0,1\}$ is a set containing 2 elements: $0$, and $1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2802959",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
Can two different models of arithmetic have non-comparable views of peano arithmetic? For a given model of arithmetic $M$, we say that models view of peano arithmetic, $V(M)$, is $\{\phi : M \models (PA \vdash \phi) \}$.
For example the view of the standard model is $\{\phi : PA \vdash \phi \}$. On the other hand, for ... | First of all, I believe you would be interested in a paper by Kikuchi and Kurahashi, "Illusory Models of Peano Arithmetic". They explore related questions in detail in that paper. In their notation, what you call "V(M)" they call $\mathrm{Thm}_{\mathsf{PA}}(M)$.
Let's first note that if $M \models \lnot \mathrm{Con}(\m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2803063",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
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The nth prime calculator How to prove the correctness of the following algorithm :
Input: $n$ : the ordinal number
Output: $x$ : the nth prime number
$b_1=6$ , $b_2=6$ , $b_3=6$
If $n==1$ , then $x=2$ , else $x=3$
$k=4$ , $m=3$
While $m \leq n$ do:
$\phantom{5}$ $b_4=b_1+ \operatorname{lcm}(k-2,b_1)$
$\phantom{5}$ $a=... | Updated 11.06.18
The proposed calculator enumerates the prime numbers on the pass.
Let us construct the similar algorithm based on the ideas of Eratosthenes sieve.
m = 1;
x = 2;
b = 2;
while (m < n)
{
for(k = 3; ; k = k+2)
{
if(gcd(b,k)==1)
{
x = k;
m = m + 1;
b = b * x;
}
}
}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2803156",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Find a fraction $\frac{m}{n}$ which satisfies the given condition Find a fraction such that all of $\frac{m}{n}$, $\frac{m+1}{n+1}$, $\frac{m+2}{n+2}$, $\frac{m+3}{n+3}$, $\frac{m+4}{n+4}$, $\frac{m+5}{n+5}$ are reducible by cancellation. Condition: $m≠n$.
What I tried was... I wrote $$\frac{m}{n}=k$$ Then, I replaced ... | I have obtained a very interesting form of fraction which will follow all the conditions. Unfortunately I am not seasoned in number theory and Hence my solution is lengthy, confusing and has a large no. Of variable, and has a few assumptions. It would be helpful if someone could suggest a better and shorter solution. T... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2803295",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Problem with central limit theorem Let ($\xi_{k}$)$_{k \in \mathbb{N}}$ be a sequence of i.i.d random variables with expectation $\mu$ and variance $\sigma^2$ $\in (0, \infty)$.
We define $X_k$ = $\xi_k$ - $3\xi_{k+1}$ + $\xi_{k+2}$, $S_n$ = $\sum_{k=1}^n X_k$, $k,n \in \mathbb{N}$
Compute for $x \in \mathbb{R}$ the li... | Your intuition, in this case, is off the target. But that's a very interesting mistake you made (in a good sense!).
The problem is that the version of the CLT you want to use is valid for sequences of independent random variables. And the $X_k$ are not independent. For instance, if $\xi_3$ is large and positive, then $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2803481",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Convergence of $\sum_{k=1}^\infty\frac{\operatorname{rad}(k!)}{\operatorname{lcm}(1,2,\ldots,k)}$ I did experiments with a Pari/GP program that suggest that the numerical series
$$\sum_{k=1}^\infty\frac{\operatorname{rad}(k!)}{\operatorname{lcm}(1,2,\ldots,k)}\tag{1}$$
is convergent, where for an integer $n>1$ $$\oper... | $\sum_{k=1}^\infty\frac{\operatorname{rad}(k!)}{\operatorname{lcm}(1,2,\ldots,k)}
$
From its definition,
$rad(k!)
=\prod_{p \le k} p
$
so
$\ln rad(k!)
=\sum_{p \le k} \ln p
=\theta(k)
$
and
$\ln(\operatorname{lcm}(1,2,\ldots,k))
=\psi(k)
$,
where
$\theta$ and $\psi$
are the two
Chebychev
prime counting functions
(http... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2803544",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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An interesting series converging to a constant Let $K>0$ be a constant. Suppose $\{z_n\}_{n=1}^\infty$ is a non-decreasing positive sequence. Then the series
$$\sum_{n=1}^\infty\frac{z_n}{(K+z_1)(K+z_2)\cdots(K+z_n)}K^n=K$$
This is a quite interesting result as the series is convergent and the limit doesn't depend on ... | The series can be transformed into a telescoping series in the following way:
$$
\sum_{n=1}^{\infty}\frac{z_n}{\prod_{j=1}^n(K+z_j)}K^n=\sum_{n=1}^{\infty}\left(\frac{K^n}{\prod_{j=1}^{n-1}(K+z_j)}-\frac{K^{n+1}}{\prod_{j=1}^n(K+z_j)}\right)=K
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2803637",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Circumference inscribed in a square What is the area of the hatched region, knowing that the arc AC is 1/4 of the circumference with a center in D?
I've tried using algebra to solve this but it seemed insufficient, I thought of using integrals to find the area, by finding the analytic geometry equations for the circle... |
Sector 1 Angle $= 2\cdot \sin^{-1}(\sqrt(\frac{31}{128})\cdot 2)= 2.78617$ rad
Sector 2 Angle $= 2\cdot \sin^{-1}(\sqrt(\frac{31}{128})) = 1.02906$ rad
Shaded Area = Area Chord Segment 1 – Area Chord Segment 2
Area Seg 1 = Area Sector 1 - Area Triangle 1
$\frac{2.78617}{2π}\cdot π(\frac{a}{2})^2 - (\sqrt(\frac{31}{12... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2803741",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Why is $\sqrt{x^2}=|x|$ instead of $\pm x$? $\sqrt{a}=\pm a$ for any $a$. $x^2$ always removes the negative, meaning that it will result in a positive number for $a$, but that doesn’t change the ambiguity of the square root operation. Thus I would think that $\sqrt{x^2}=\pm x$, where you select which output to use base... | If $x\in\mathbb{R};x\ge 0$, then $\sqrt{x}$ is defined as "the non-negative real number which, when squared, equals to $x$.
Saying this is correct:
$\sqrt{x^2}=|x|=\begin{cases}x, \text{if }x\ge 0\\-x, \text{if }x\le 0\end{cases}$
I will give you another example first before considering your question:
$x^2-1=0\Leftrig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2803851",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Which integers can be written as $x^2+2y^2-3z^2\ $?
For which integers $n$ has the diophantine equation $$x^2+2y^2-3z^2=n$$ solutions ?
These theorems
https://en.wikipedia.org/wiki/15_and_290_theorems
do not apply because the given quadratic form is not positive (or negative) definite. It seems that the quadratic for... | We claim that any $n\in\mathbb{Z}$ can be written as $x^2+2y^2-3z^2$.
Note that any such integer $n$ is $0$ or it is equal to a $4^a\cdot 2^b\cdot d$ where $a$ is a non negative integer, $b\in\{0,1\}$ and $d$ is a signed odd number. Then we consider the following cases.
0) If $n=0$ then let $x=0$, $y=0$ and $z=0$.
1)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2803972",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
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Linear Transformation $T\colon V\rightarrow V$ defined by $T(y)=y+e^{-2t}\frac{dy}{dt}$
Consider the set of functions $B=\left\{ {1,e^{2t},e^{4t}}\right\}$. It is given that B is a linearly independent set. Write $V=\text{span}(B)$ and let $T\colon V\rightarrow V$ be the linear transfomration defined by $$T(y)=y+e^{-2... | Since teh detreminant of $M$ is 1 there can be only one solution. Equate $T(a+be^{2t}+ce^{4t})$ to $1+2e^{2}+3e^{4t}$ and you will get $a=13, b=-10,c=3$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2804105",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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If $f(x) \le M$ and $\lim_{x \to a} f(x) = A$, prove that $A \le M$. I am an adult software developer who is trying to do a math reboot. I am working through the exercises in the following book.
Ayres, Frank , Jr. and Elliott Mendelson. 2013. Schaum's Outlines Calculus Sixth Edition (1,105 fully solved problems, 30 pro... | When presenting this proof we usually use the factor $\frac{1}{2}$ to make it very clear that $f(x) \geq M + \varepsilon / 2 > M$, and not have to bother with strict inequalities. For instance sometimes in defining the continuity of $f$ at point $a$ you could end your sentence by using a large inequality, those two def... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2804301",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Clear Definition of Two Graphs Being Isomorphic I am wanting to deal with canonical graph homorphisms and other graph homorphisms much more deeply, but I need a definition for isomorphism between graphs to start. I know the definition of two groups being isomorphic, but I do not know how to tell if two graphs are isomo... | Consider the following definition of group isomorphism: it is a bijection $f:G_1\to G_2$ such that
$$uvw^{-1}=e_{G_1} \iff f(u)f(v)f(w)^{-1}=e_{G_2}\quad \forall u,v,w\in G_1.$$
And now compare graphs:
$$uv\in E(G_1) \iff f(u)f(v)\in E(G_2)\quad \forall u,v\in V(G_1).$$
Not much difference really.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2804411",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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What is the best way to guess a number in a limited number of guesses? A random integer is picked from 0 to 100. You can make 5 guesses at what the number is, and after each guess, you are told if your guess was too high or too low. What strategy maximizes your probability of guessing the number?
| You can work your way back from the end of the game to optimize the strategy. At each stage, you know that the number is in some interval. On the last guess, you'll just guess any number in the interval, and your chance to win is one over the length of the interval. On the penultimate guess, you have some interval $[i,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2804503",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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The area between the graphs $f_t(x)=e^{x-t}-t$ and $g_t(x)=\log(x+t)+t$ is irrational only if $t$ is irrational.
Find the smallest $t \in \mathbb{R}$ such that the graphs of $f_t:x\mapsto \exp(x-t)-t$ and $g_t:x\mapsto\log(x+t)+t$ intersect. Note that $\log$ is meant as the natural logarithm.
Hint: I guess it's $\fr... | Assume it to be true, then
If $t \in \mathbb{Q} \Rightarrow $ Area between $f$ and $g$ $\in \mathbb{Q}$
Then specially for $t = 0$
$$
\int_{0}^{\infty} e^{x}- \log(x) dx = \left[x+ e^x + x (-\log (x)) \right]_{0}^{\infty} = \\
\lim_{x \to \infty} x + e^x + x (-\log (x)) - \lim_{x \ \to 0 } x+ e^x + x (-\log (x)) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2804643",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Are there intractable problems? Obviously, there is no problem in NP which can be shown to be intractable (by which I mean: not in P).
Is there a problem (outside of NP) which can be shown to be intractable (not lying in P)?
| What you want is the Time Hierarchy Theorem, which broadly speaking states that for any "reasonable" function $f$ there are problems that take about $f(n)$ time to decide.
Concretely, each of the known EXPTIME-complete problems would be an example of something that has been proved not to lie in P.
Here is a related que... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2804787",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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The inverse of a continuous one-to-one function that is defined on a connected set is not always continuous I am trying to find a function $f:B \subset \Bbb R^n \rightarrow \Bbb R^m$ for $B$ a connected set that is continuous, one-to-one where $f^{-1} = f(B) \rightarrow B$ is discontinuous. The hint I have been given i... | I was working on this, managed to prove the opposite (If wrong, please point out the mistake)
I claim that $f^{-1}$ is continuous. Let's call it $g$ for convenience of notation, i.e. $g\circ f (x) = x$ $\forall x \in B$
Enough to show that $g^{-1}(U)$ is open in $f(B)$ if $U$ is open in $B$
Now, $g^{-1}(U) = \{ x : g(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2805020",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Other methods for solving homogeneous differential equation I want to find a general solution of the following homogeneous equation
$$\frac{dy}{dt}=\frac{3t+12y}{t+14y}$$
I have tried using the substitution $z = y/t$ to make the equation separable, but then it gets a little bit tedious to solve.
$$\frac{dy}{dt}=\fra... | My teacher in ODEs show me this "rarely unknown"substitution method, it Works with homogeneous and almost homogeneous ODEs.
1° Let $y=p^S$ and $t=q^R$, respectively $dy=Sp^{S-1}dS$, $dt=Rq^{R-1}dR$; R and S are constants. Then substitute in the ODE:
$$\frac{dy}{dx}=\frac{3t+12y}{t+14y}\rightarrow\frac{Sp^{S-1}dS}{Rq^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2805155",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Geometric construction to divide a segment Given a segment AB, I would like to construct using only straightedge and compass, a point C on the segment AB such that $\frac{AC}{CB}$ is equal to $\frac{\phi}{2}$, where $\phi$ is the golden ratio, $\phi = 1.61803..$ .
The Wikipedia article on the golden ratio offers a cons... | Render
$\frac{1+\phi /2}{\phi /2}=\sqrt{5}$
So the entire length divided by the shorter piece is $\sqrt{5}$.
Start with the given segment $AB$. Bisect this segment to identify the midpoint $M$ and construct a circle $Z$ centered at $M$, passing through $A$ and thus also through $B$. Construct the perpendicular to $AB... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Convergence of improper integral $\to\pi$ $$\int_0^\pi\frac{\sin^2x}{\pi^2-x^2}\mathrm dx$$
$\sin$ is converging, but the rest has infinite limit in this segment, while $ \to\pi$, so Abel doesn't work. Dirichlet is not working either. I tried to bound this integral, but unsuccessfully. Not sure what else to use. Any t... | Split the integral in half and apply $x\mapsto\pi-x$ to the $x>\pi/2$ integral. You'll see each integral ends up with an integrand whose numerator and denominator are respectively $O(x^2)$ and $O(x^{0\,\text{or}\,1})$ for small $x$, and the integrals converge.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2805489",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Let $ A\in M_n(\mathbb{C}) $ be an invertible and non-diagonalizable matrix. Prove that for all $k\ge 1 \Rightarrow A^k$ is not diagonalizable.
Let $ A\in M_n(\mathbb{C}) $ be an invertible and non-diagonalizable matrix. Prove that for all $k\ge 1 \Rightarrow A^k$ is not diagonalizable.
Hi all. Since $A$ is over $\ma... | We can assume that $A$ is in the Jordan form.
$A$ is not diagonalizable so $\exists\lambda \in \sigma(A)$ such that the $A$ has a block $J_j(\lambda)$ with $j \ge 2$, which is equivalent to $\dim\ker (A - \lambda I) < \dim\ker (A - \lambda I)^2$.
We have
$$J_j(\lambda) = \begin{pmatrix}
\lambda & 1 & 0 & \dots &... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2805691",
"timestamp": "2023-03-29T00:00:00",
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Visualisation of dot/cross product If this question is a duplicate / the answer already exists I am very sorry.
I recently started working with vectors in 3 dimensions. My visualisation of the dot product is the following:
Lets say we have the vector v(0,0,1) with the origin O. The dot product between v and the vector ... | Cross product between two vectors
$$\vec A=(x_1,x_2,x_3)\;,\;\;\;\vec B=(y_1,y_2,y_3)$$
is defined as
$$\vec C=\vec A\times \vec B:=\begin{vmatrix}e_1&e_2&e_3\\x_1&x_2&x_3\\y_1&y_2&y_3\end{vmatrix}=(x_2y_3-x_3y_2\,,\,x_3y_1-x_1y_3\,,\,x_1y_2-x_2y_1)$$
and it turns out to be a vector $\vec C$ orthogonal to the plane spa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2805790",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Connected, compact, orientable smooth manifold with finite fundamental group The problem is prove that a connected, compact, orientable smooth manifold $M$ with finite fundamental group has $H^{1}_{dR}(M)=0$. The hint is to apply
Suppose $\tilde M$ and $M$ are smooth $n$-manifolds and $\pi:\tilde M\to M$
is a smoot... | I honestly don't know what I was thinking when I wrote that hint -- it doesn't seem to be useful at all.
A more appropriate hint would have been something like this: "If $\omega$ is a closed $1$-form on $M$, let $\widetilde\omega$ be the pullback of $\omega$ to the universal cover of $M$, and let $\tilde f$ be a poten... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2806050",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Doubt related to the $\epsilon-\delta$ definition of limit $$\text{If}\quad\lim\limits_{x\to a}{f(x)}=L\quad\text{then:}$$
$$\text{If}\quad\forall (\epsilon > 0)\;\exists (\delta>0\;\text{and}\;\forall x\quad((x\neq a\;\text{and}\;|x-a|<\delta)\quad\Rightarrow\quad|f(x)-L|< \epsilon)).$$
I have understood the intiution... | There is nothing wrong if you put $\leq$ instead of $<$, its just that conventionally a topology is described by its open sets, even if it can also be defined in terms of the closed sets. Here, behind the idea of taking less than, the topological aspect plays a big role so that the definition can easily be extended in ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2806162",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Can the integers $n$ expressible as $2a^2-3b^2$ be classified? Let $S$ be the set of integers $n$ that are expressible with $$n=2a^2-3b^2$$ with integers $a,b$
Can $S$ be classified by an iff-condition that allows to decide whether $n\in S$ , when the factorization of $n$ is known ?
I found out several partial result... | The integer solutions of $2 x^2 - 3 y^2 = n$ are invariant under the mapping
$(x,y) \to (5x+6y, 4x+5y)$. Thus if there is an integer solution, there must be one with $\sqrt{n/2} \le x \le 5 \sqrt{n/2}$ (if $n > 0$), or $\sqrt{-n/3} \le y \le 6 \sqrt{-n/3}$ (if $n < 0$).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2806271",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
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Determine the arithmetic and geometric sequence given the relationships betweeen the first four terms No matter what I try, I can't solve this problem. I'm almost done but I need to get just one more thing to be able to finish it.
Problem
The first, second and fourth term of the arithmetic and of the geometric sequenc... | Here we know that $r\neq1$ otherwise all terms would be equal and it would not be possible to have the third term of the AP being greater than the third term of the GP.
Let $T_i$ be the $i-$th term of the AP.
$$\begin{align}
\frac {T_4-T_2}{T_2-T_1}&=\frac {4-2}{2-1}\\
\frac {r^3-r}{r-1}&=2
&&\scriptsize (T_i=ar^{i-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2806363",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to show the double sum identity $\sum ^{n}_{i=1}\sum ^{i}_{j=1}i-j$ = $\dfrac {1}{6}n\left( n-1\right) \left( n+1\right) $ I am not sure how to go about showing this: $\sum ^{n}_{i=1}\sum ^{i}_{j=1}i-j$ = $\dfrac {1}{6}n\left( n-1\right) \left( n+1\right) $
It is a bit like the formula for $\sum ^{n}_{i=1}i^{2}$
an... | I would suggest an inductive proof to that identity. Assume the induction hipotesis
$$
\sum^{n}_{i=1}\sum^{i}_{j=1}i-j= \dfrac{1}{6}n\left( n-1\right) \left( n+1\right).
$$
It's easy to verify identity for $n=1$, $n=2$ and $n=3$. Consider the scheme.
$$
\begin{array}{rl}
\sum^{n}_{i=1}\sum^{i}_{j=1}i-j=&0
\\
+&0+1
\\
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2806535",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Expanding $\prod_{m=1}^{n-1}(1+a_m+b_m)$. Quite a complicated product in the form below occured in my research. I'm having trouble getting started in evaluating it.
Let $n\in\Bbb N\setminus \{1\}$, $(a_m)_{m\in\overline{1, n-1}}\in \Bbb R^{n-1}$, and $(b_m)_{m\in\overline{1, n-1}}\in \Bbb R^{n-1}$. Consider the produc... | $$\begin{align}\prod_{m=1}^{n-1}(1+c_m)&=(1+c_1)(1+c_2)\cdots(1+c_{n-1})\\&=1+\sum_{\text{cyc}}c_1+\sum_{\text{cyc}}c_1c_2+\cdots+\prod_{i=1}^{n-1}c_i\end{align}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2806668",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Basel Problem - Area of $\frac 16$ of Circle with Radius $\sqrt{\pi}$. There are several proofs to the solution of the well-known Basel Problem, i.e.
$$\sum_{n=1}^\infty \frac 1{n^2}=\frac {\pi^2}6$$
Is is possible to create a geometrical interpretation of this identity in the form of the area of $\frac 16$ of a circl... |
TOPIC: Area of $\frac{1}{6}$ of Circle with Radius $\sqrt{\pi}$
Last time I show how to get an easy approximation for radius via Pythagorean theorem:
$$ (\sqrt{\pi})^2\geq1.7^2+0.5^2
$$
and have give a suggestion to do different- let me show for $\pi\leq\frac{22}{7}$ (by using intercept theorem for $\frac{\sqrt{7}}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2806799",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 3,
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Cone of tangents of $S=\{(x,y):y\ge-x^3\}$ Find the cone of tangents for the set $S$ at the point $\overline x=(0\ 0)^t$.
$$S=\{(x,y):y\ge-x^3\}$$
The cone of tangents of $S$ at $\overline x$ is the set of directions $d$ such that $d=\lim_{k\to\infty}\lambda_k(x_k-\overline x),\ \lambda_k>0,x_k\in S$ for all $k$ and $x... | There are several first-order cones that appear in context of KKT conditions and constraint qualifications (like feasible directions, attainable directions etc), but all those cones, including the tangent cone, are between two standard cones for the linearized problem that are easy to calculate (here $I$ is the set of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2806898",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Find rectangular box which has biggest volume Exercise: Find rectangular box with surface area 10 that has minimum volume
My solution: We know that surface area is $$S= 2(wl +hl+hw)$$and volume $$V=whl$$ ,where w-width,h-heigh, l=length.
We will use lagrange function$$L = whl - \lambda(2(wl +hl+hw)-10)$$.
Now we will f... | As an alternative and to check let use AM-GM inequality
$$\frac {S} 6= \frac{wl +hl+hw}{3}\ge\sqrt[3]{w^2l^2h^2}\implies V^{\frac23}\le\frac {5}3\implies V\le\sqrt{\left(\frac{5}3\right)^3}$$
with equality for $$wl=hl=hw \implies w=l=h=\sqrt{\frac{5}3}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2806970",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Can you find the result of a linear transformation by projecting a given vector onto known eigenvectors? I have a practice exercise I'm struggling with. Essentially, it gives you the eigenvalues and eigenvector for a given linear transformation and then asks you to find the result of the transformation on some other ve... | Yes, this is the right approach; the idea is to appeal to linearity. First decompose the given vector into a linear combination of the eigenvectors, then:
$$\begin{align}
A\left(c_1\begin{bmatrix}-1\\-1\\-2\end{bmatrix} + c_2\begin{bmatrix}1\\1\\1\end{bmatrix} + c_3\begin{bmatrix}-1\\-4\\-3\end{bmatrix}\right) &= c_1\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2807104",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Show that the Euler characteristic of a chain complex is equal to the Euler characteristic of its homology Let $C_*$ be a chain complex such that each $C_i$ is a torsion-free, finite-range abelian group with $C_i=0$ for all $i<0$. Suppose that $C_i=0$ for all $i$ is large enough. The Euler characteristic of $C_*$ chain... | Your chain complex looks like
$$0\to C_n\to C_{n-1}\to\cdots\to C_0\to0$$
where the $C_i$ are nonzero outside this range. Proceed by induction
on $n$. Call this complex $\mathbf C$. Let $\mathbf{C}'$ be the subcomplex
$$0\to0\to C_{n-1}\to\cdots\to C_0\to0.$$
Then there is a short exact sequence of complexes
$$0\to\mat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2807224",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
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Maxima & Minima Word Problem Problem: Given the following profit-versus-production function for a certain commodity: $P=200000-x-(\frac{1.1}{1+x})^8$. Where P is the profit and x is the unit of production. Determine the maximum profit.
Solution: Taking its first derivative, $\frac{dP}{dx} = -1-8(\frac{1.1}{1+x}^7) * (\... | By Second Derivative test we can decide function is maximum or minimum at point x. http://mathworld.wolfram.com/SecondDerivativeTest.html
http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-maxmin-2009-1.pdf
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2807337",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Two variants of proof for Baire's category theorem One of the variants of the so called Baire's Category Theorem (BCT) says that
Given a (possibly nonempty) complete metric space $(X,d)$ and a system of open dense subsets $(O_n)$ of $X,$ then $\bigcap_{n \in \mathbb{N}} O_n$ is dense in $X.$
Most proofs of this theor... | I think his conclusion at that point is wrong or incomplete.
Let's consider the example
$$
X=[0,1], d(x,y)=|x-y|,
O_n = (0,1).
$$
Then $O_n$ is open and dense for every $n\in\mathbb N$.
However, we can assume that the construction of $x_n$ and $r_n$ yields
$$
x_n= 2^{-n},
r_n = 2^{-n}.
$$
Then $B_{r_{n+1}}(x_{n+1}) \su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2807469",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Formula for the floor function I found the following formula for the floor function:
$$\lfloor x \rfloor = -\frac12+x+\frac{\arctan(\cot\pi x)}{\pi}$$
for all $x$ not an integer.
My question is where I can find the proof of this formula.
| Consider
$$f(x) = \frac{1}{\pi}\int_{0}^{\cot(\pi x)}\frac{dy}{1+y^2}=\frac{\arctan(\cot(\pi x))}{\pi},$$
with the latter equation obtained by substituting $y=\tan u.$ Since $x\to \cot(\pi x)$ is manifestly periodic of period $1,$ and $f$ integrates to $0$ over one period (since $\cot$ is an odd function), $f$ also i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2807610",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 2
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Question about Spivaks Inverse function theorem proof In the opening lines of Spivaks "Calculus on Manifolds" proof of the inverse function theorem he writes the equation below. let $\lambda = Df(a)$
$$D(\lambda^{-1}\circ f)(a) = D(\lambda^{-1})(f(a))\circ Df(a) = \lambda^{-1}\circ Df(a)$$
I don't see how $ D(\lambda^{... | It's simply because $\lambda$ is a linear map. So, its inverse is also a linear map and the derivative of a linear map at every point of its domain is again that linear map.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2807744",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Making a substitution in a PDE - how did the author get to this result?
How did the author get from $k_t = k^2 k_{\theta \theta} + k^3$ to that form I outlined? I've tried computing $k_{t}$ and $k_{\theta \theta}$ using $k(\theta, t)^2 = A(\theta) + B(t)$ and substituting that in the PDE, but it was all a mess and not... | Differentiating $k(\theta,t)^2=A(\theta)+B(t)$ twice with respect to $\theta$ and once with respect to $t$ yields
\begin{eqnarray*}
2kk_\theta&=&A'\;,\\
2k_\theta^2+2kk_{\theta\theta}&=&A''\;,\\
2kk_t&=&B'\;.
\end{eqnarray*}
Solve the first equation for $k_\theta$, substitute the result into the second equation, solve ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2807998",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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The implication: $x+\frac{1}{x}=1 \implies x^7+\frac{1}{x^7}=1$ $$x+\frac{1}{x}=1 \implies x^7+\frac{1}{x^7}=1$$
The graphs/range: $\quad y_1(x)=x+\frac{1}{x}, \quad y_7(x)=x^7+\frac{1}{x^7} \quad$
and do not touch the line $\quad y=1\quad$. The relation $\quad x+\frac 1 x=1 \quad$ appears to only be true for certain ... | You shouldn't be trying to find where $y_1=y_2$, but whether a root of $y_1$ is necessarily a root of $y_2$.
That is,
$$\forall x\Big(y_1(x)=0\implies y_2(x)=0\Big)$$
or equivalently,
$$\forall x\Big(y_1(x)\neq0\lor y_2(x)=0\Big)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2808056",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
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Closed form / computationally efficient method for binomial + factorial sum I'm trying to evaluate the sum given below
$$
S(x,y) = \frac{1}{k!}\sum_{j=0}^k \binom{k}{j} (-x)^j y^{k-j} j!
$$
where $x,y > 0$. Is there any clever closed form way of writing this sum? If not, maybe there is a clever way of writing it as a p... | Since ${k \choose j} = {k \choose k-j} = \frac{k!}{j!(k-j)!}$, upon making the substitution $j \leftrightarrow k-j$ you can write your sum as
$$ S(x,y) = (-x)^k \sum_{j=0}^k \frac{1}{j!} \left( - \frac{y}{x} \right)^j = (-x)^k S(1, -y/x) \; .$$
Now we recognize $S(u) := S(1, u) = \sum_{j=0}^k \frac{u^j}{j!}$ as the $k$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2808154",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Convex sum order If I have a strictly convex function $f(x)$ with $f''(x)>0$
and if I know that for some $a\le b \le c$ and $x \le y \le z$ I have
$$a+b+c = x+y+z$$
$$f(a)+f(b)+f(c)=f(x)+f(y)+f(z)$$
can I conclude that at least one of the following must be true about the order?
$$ a \le x \le y \le b \le c \le z$$
$$ ... | In geometry a triangle $(ABC)$ on the plane $xOy$ with coordinates $A=(a_1,a_2)$, $B=(b_1,b_2)$, $C=(c_1,c_2)$ has its gravity center (baricenter) the point
$$
G=\left(\frac{a_1+b_1+c_1}{3},\frac{a_2+b_2+c_2}{3}\right).
$$
Hence what you ask is:
Given a curve (function) $c_f:y=f(x)$ or $M=(x,f(x))$, with positive cur... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2808323",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Find the limit of $(\tan(x) + \sec (x))^{1/\sin(x)}$ $$ \lim_{x \rightarrow 0} [\tan(x) + \sec(x)]^{\csc(x)} = e $$
how to arrive at e, according to wolfram alpha, that this is the answer?
| Note that:
$$\begin{align}\lim_{x \rightarrow 0} [\tan(x) + \sec(x)]^{\csc(x)} = &\lim_{x \rightarrow 0} \left[\frac{\sin x}{\cos x} + \frac{1}{\cos x}\right]^{1/\sin x} =\\ &\lim_{x \rightarrow 0} \frac{[1+\sin x]^{1/\sin x}}{[\cos x]^{1/\sin x}} =\\
&\frac{\lim_{x \rightarrow 0}[1+\sin x]^{1/\sin x}}{\lim_{x \rightar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2808420",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 2
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The number of functions $ f : \left \{ 1, 2, . . . , 10 \right \} \rightarrow \left \{ 1, 2, . . . , 10 \right \}$ such that $f(x) \neq x$ for all $x$ Question
The number of functions $ f : \left \{ 1, 2, . . . , 10 \right \} \rightarrow \left \{ 1, 2, . . . , 10 \right \}$ such that $f(x) \neq x$ for all $x$ is
Ap... | Case 1: If functions are not bijective.
It's a simple case for every $i^{th}$ element ,$1\le i\le 10$, you have 9 possibilities to choose from for the output.
Hence total number of functions=$ 9^{10}$
Case 2: If functions are bijective
On close observation you might notice that what you need is exactly the number o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2808547",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Uniform convergence of $\sum_{n=1}^\infty \frac{1}{n^3+x}$ How do is show that for $-1<x<1$ the series $\sum_{n=1}^\infty \frac{1}{n^3+x}$ converges uniformly.
For $x\geq0$ I can make a convergent Majorant series, choosing $M_n = 1/n^3$, but I can't seem to determine a convergent Majorant series for $-1<x<0$.
| Applying Weierstrass. For every $r \in (-1,1)$ you can consider $A_r = (r,1)$. Now apply M-test using that $\forall x \in A_r$
$$
\frac{1}{n^3+x} \leq \frac{1}{n^3+r}
$$
And obviously
$$
\sum_{n=1}^\infty \frac{1}{n^3+r} < \infty
$$
You can check it by comparation with $\sum_n 1/n^3$. So you prove uniform convergence i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2808735",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find a short expression for the long sum First of all, I'm quite new here, so sorry if this is not asked in the correct place.
The sum is $$X={{100}\choose{1}}+3\cdot{{100}\choose{3}}+5\cdot{{100}\choose{5}}+...+97\cdot{{100}\choose{97}}+99\cdot{{100}\choose{99}}$$ I have noticed that I can get a much simpler sum:
$... |
We obtain
\begin{align*}
\color{blue}{\sum_{n=0}^{24}\binom{100}{2n+1}}&=\frac{1}{2}\sum_{n=0}^{49}\binom{100}{2n+1}\tag{1}\\
&=\frac{1}{2}\left(1\cdot\sum_{n=0}^{49}\binom{100}{2n+1}+0\cdot \sum_{n=1}^{50}\binom{100}{2n}\right)\tag{2}\\
&=\frac{1}{2}\sum_{n=0}^{100}\frac{1-(-1)^n}{2}\binom{100}{n}\tag{3}\\
&=\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2808850",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Someone's Inequality ? $(a + b)^2 \le 2(a^2 + b^2) $ For real $a, b$ then $(a + b)^2 \le 2(a^2 + b^2) $
This fairly trivial inequality crops up a lot in my reading on (Lebesgue) integration, is it named after someone ? It extends rather obviously for positive reals to $a^2 + b^2 \le (a + b)^2 \le 2(a^2 + b^2) $.
Proof... | This inequality can as weel be seen as a particular case of the equivalence between $p$-norms in $\Bbb R^n$. Indeed for $1<p\leq q<\infty$, it holds
$$\|x\|_q \leq \|x\|_p \leq n^{1/p-1/q}\|x\|_q$$
In the particular case $n=2$, $p=1$ and $q=2$ we get
$$(|a|+|b|)=\|(a,b)\|_1\leq 2^{1-1/2}\|(a,b)\|_2 =\sqrt{2(a^2+b^2)},$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2808947",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Fourier transform of $\lvert x^2-1\rvert $ One of the exercises of my assignment was to determine the Fourier transform of function
$$f(x)=\lvert x^2-1\rvert$$
The domain wasn't specified. First I was puzzled since $f$ isn't a $L^1$ function.
If I were to calculate
$$\mathcal{F}(f)(\xi)=\int_\mathbb{R} \lvert x^2-1\... | The Fourier transform of a distribution is defined as
$$(\mathcal F[f], \phi) = (f, \mathcal F[\phi]).$$
That is, the action of $\mathcal F[f]$ on $\phi$ is given by the rhs of this identity, where we know that $\mathcal F[\phi]$ is well-defined and is again a valid test function.
We can find the transforms of $1$ and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2809075",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Courant–Fischer Theorem Proof [Meyer]. Intersection of two subspaces with same dimension. Suppose we have 2 subspaces $\mathcal{A} \subseteq \mathbf{R}^n$ and $\mathcal{B}\subseteq \mathbf{R}^n$ that have the same dimension, say dim $\mathcal{A}=$ dim $\mathcal{B}=l>0$. Is it true that they have a non null intersection... | I'm not sure of your question, hence I'll give two answers.
1. Two vector subspaces of $\Bbb R^n$ always have non-empty ($\ne \varnothing$) intersection. Indeed, by definition, every vector subspace must contain the zero vector.
2. But their intersection can be exactly $\{0\}$ (thus having dimension $0$). Simply take t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2809189",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Predicate formulas on the natural numbers using only $\le$. We want to find predicate formulas about the natural numbers using only the $\le$ predicate and no constants. For instance, the following predicate formula defines equality:
$[x=y] ::= [x \le y \; \land y \le x].$
And then, using that we can define $[x>0]$:
... | $x=y+1$ iff both $x>y$ and there is no $z$ with $x>z$ and $z>y$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2809328",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Show that infinite Cartesian product is uncountable let $A_i=\{0,1\} \forall i \in \mathbb Z^+$ and $A^{\omega}= \prod _\limits{i \in \mathbb Z^+}A_i$ thus $A^\omega=\{(a_i)|a_i=1,2 \forall i\in \mathbb Z^+\}$
now here is my problem
let $(\underline x_n)_{n \in \mathbb Z^+}$ where $\underline x_n=(x_{nk})_{k \in \math... | Given any function $f \colon \mathbb{Z}^+ \to A^\omega$, let $\underline{x}_n = f(n)$. Then use (a) to show that $f$ can't be surjective.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2809420",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
prove that $S_{\tau \land n} \to S_{\tau}$ in $L^1$ for a random walk with $E\tau^{1/2} < \infty$
Let $X_i$ be a sequence of iid $L^2$ RVs with $EX_i = 0$ and define the martingale $S_n = \sum_1^n X_i$. I want to show that if $\tau$ is a stopping time and $E\tau^{1/2} < \infty$, then $ES_{\tau} =0$.
I have been give... | First of all: Your calculation of $A_n$ is not correct; note that
$$A_n = \sum_{k=1}^n \mathbb{E}(S_k^2 \color{red}{-S_{k-1}^2} \mid \mathcal{F}_{k-1})$$
and then you will end up with $$A_n = \sum_{k=1}^n \mathbb{E}(X_k^2) = n \mathbb{E}(X_1^2). \tag{1}$$
If we consider the stopped process $Y_n := S_{n \wedge T}$, the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2809505",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
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Denote Riemann integral as sum of infinite series One of the standard definitions of Riemann Integral is as follows:
Let $f$ be bounded on $[a, b]$. For any partition $P = \{x_{0}, x_{1}, x_{2}, \ldots, x_{n}\}$ of $[a, b]$ and any choice of points $t_{k} \in [x_{k - 1}, x_{k}]$ the sum $$S(P, f) = \sum_{k = 1}^{n}f(t... | YES.
Set $$
P_n=\left\{t_0=0,t_1=\frac{1}{n},t_2=\frac{2}{n},\ldots,t_n=1\right\}.
$$
Then
$$
\frac{1}{n}\sum_{k=1}^nf\left(\frac{k}{n}\right)=S(P_n,f), \quad \text{with $t_k=\frac{k}{n}$}.
$$
If we choose an arbitrary $\varepsilon>0$, then, according to the definition in the OP, there exists a $\delta>0$, such that i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2809810",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solve $\log_3(x^2+2x+1)=\log_2(x^2+2x)$ Solve $\log_3(x^2+2x+1)=\log_2(x^2+2x)$
I have tried to do to as followed:
$\log_3(x^2+2x+1)=\frac{\log_3(x^2+2x)}{\log_3(2)}$
$\iff\log_3(x^2+2x+1).\log_3(2)=\log_3(x^2+2x)$
Is it possible to proceed this way? Or should one approach this differently?
| If $\log_3(x^2+2x+1)=\log_2(x^2+2x)=y$
$f(y)=3^y-2^y=1$
Now $f(y)$ is an increasing function in $[0,\infty)$ and decreasing in $(-\infty,0]$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2809917",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Inequality $x^2\leq y$ I have a question about how to handle this inequality:
$$x^2\leqslant y \to x\leqslant \pm\sqrt{y}$$ or it should be $$\sqrt{x^2}\leq \sqrt{y}\Rightarrow$$
either $$x\leq\sqrt{y}$$
or
$$-x\leq\sqrt{y}\Rightarrow x\geq-\sqrt{y}$$ so$$-\sqrt{y}\leq x\leq\sqrt{y}$$
Is my way of thinking is corr... | I assume you work in $\mathbb{R}$. If $y$ is negative, then no $x$ satisfies the inequality. If $y$ is non-negative, then $-\sqrt{y} \leq x \leq \sqrt{y}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2810026",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Determining the point from which the most area in a polygon is visible I am wondering about the following problem: Given a polygon and the set of points $S$ inside it, what are the point(s) in $S$ from which the most area in $S$ is visible? Furthermore, what is the maximum visible area?
Here, I define $q$ to be visible... | This is not an answer to your question, but an answer to a related question
that you might find interesting. The paper below computes
"the maximum clique in the visibility graph $G$ of a simple polygon $P$ in time
$O(n^2 e)$,
where $n$ and $e$ are number of vertices and edges of $G$ respectively."
Ghosh, Subir Kumar,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2810150",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 1
} |
Independence of shifted squares mod p Given an odd prime $p$, let $S\subseteq \mathbb F_p$ be the set of quadratic residues modulo $p$. Given $a,b\in \mathbb F_p$ we write $aS+b$ for the set $$aS+b:=\{t\in\mathbb F_p:\ t=ax^2+b \text{ for some $x\in\mathbb F_p$}\}.$$
What is the cardinality of $S\cap ((S+1)\cup (2-S))... | $S \cap (S+1)$ is approximately $\frac 14 \operatorname{Card}\{(x,y) \in \Bbb F_p^2 \mid x^2 = y^2+1 \}$.
This set is an algebraic curve of dimension $1$ (and of genus $0$) so it has $p+O(1)$ points. And so $\operatorname{Card}(S \cap (S+1)) = p/4+O(1)$
Similarly, $S \cap (2-S)$ will be related to the curve $x^2 = 2-y^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2810286",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that this stochastic process is a.s. strictly positive. Consider the one-dimensional SDE
$$
dX_t = f(X_t)dt + \sigma(X_t)dW_t,\quad X_0 = 1,
$$
where $W_t$ is a Brownian motion under the measure $\mathbb{P}$, in its natural filtration. Suppose that $f(x)-\sigma^2(x)/2x=0$ and $\sigma^2(x)\le x^2$. Show that $X_t>0... | Define stopping times by
$$T_k := \inf\{t \geq 0; X_t \leq k^{-1}\}, \qquad k \in \mathbb{N}$$
and
$$T := \inf\{t \geq 0; X_t \leq 0\}.$$
Following the reasoning in your question we find that
$$\mathbb{E} \left( \left| \log(X_{t \wedge T_k}) \right|^2 \right) \leq t.$$
By the continuity of the sample paths of $(X_t)_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2810419",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
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} |
A student must pick 5 classes from 12 courses, if he must have at least one WH class or USH class, how many different choices does he have? The World History course and United States history course is already part of the 12 courses he must choose from. It is required to have at least one of these classes in his new cla... | We will take "At least one USH or WH" to mean that you could take all five courses from this set. This is odd, but it is the best semantic match for the question wording. If it is supposed to mean one or both, this approach wont work.
In this case, the "stars and bars" technique applies. Ken Ribet is a better teacher... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2810493",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 4
} |
General question about Self-adjoint operators on Hilbert Spaces so if I have a self-adjoint operator $K \in B(H,H)$. Then I found a theorem that said that
$A$ is invertible if and only if $(A^*Ax,x) \geq c||x||$ and $(AA^*x,x) \geq c||x||$ for all $x$.
Then does that mean that if I have a self adjoint operator like $K... | In an infinite-dimensional vector space, there's a big difference between $(Ax,Ax)\geq c\|x\|^{2}$ for all $x\in H$ and $(Ax,Ax)>0$ for all nonzero $x\in H$. For an example, consider $\ell^{2}(\mathbb{N})$. Then if $A(e_{i})=2^{-i}e_{i}$ for all $i\geq 0,$ we can see that the inverse must be $A^{-1}(e_{i})=2^{i}e_{i}$,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2810598",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Spivak's Calculus: Proofs concerning Pascal's Triangle Problem 3 of Chapter 2 in Spivak's Calculus poses 5 problems associated with Pascal's Triangle. The first of these asks you to prove that $\binom{n+1}{k}$ = $\binom{n}{k-1} + \binom{n}{k}$ which was fairly straightforward.
The second task was to prove by induction ... | For convenience, let $S_n = \{x_1, x_2, x_3, \dots, x_n\}$ represent a set of $n$ distinct objects.
Lets use $_nC_k$ to represent the number of ways of selecting $k$ objects from $S_n$.
So what does $_{n+1}C_k$ look like?
An element of such a selection will either include the object $x_{n+1}$ or it will not. We alread... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2810715",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Is $Q + A$ postive semidefinte? Let $Q$ be a Matrix and $V \subseteq \Bbb R^n$ be a vector subspace on which $Q$ is positive semidefinite i.e, $$\langle x , Q x \rangle \ge 0 ~~~~~ \quad \forall x \in V $$
Prove or provide a counter example
There exist a matrix $A$ such that $Q + A$ is positive semidefinite (on wh... | If $P$ is a projection onto $V$, then for any $x \in \mathbb{R}^n$ we have
$$\langle x, P^T QPx \rangle = \langle Px, QPx \rangle \ge 0$$
since $Px \in V$, so $P^T QP$ is positive semidefinite. Moreover if $x \in V$ then $Px = x$, so
$$\langle x, P^T QPx \rangle = \langle Px, QPx \rangle = \langle x, Qx \rangle.$$
So p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2810818",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Maximal ideals in a group ring when the ring is a field Is there any simple characterization of the maximal ideals in a group ring $R[G]$ when $R$ is a field, perhaps in terms of maximal subgroups of $G$?
| The maximal ideals of $R[G]$ are in canonical bijection with the isomorphism classes of simple $R[G]$-modules. In one direction, if $I$ is a maximal ideal, then $R[G]/I$ is a simple ring, hence has a unique simple module up to isomorphism, which you can lift to a simple module of $R[G]$. In the other direction, given a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2811094",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Well definedness of factor group multiplication I have been reading on this site and the internet, and I do not quite understand the comments which are being made with respect to the problem of proving that factor group multiplication is well defined. According to John Fraleigh, A First Course in Abstract Algebra, 7th ... |
(x′y′)H=(x′H)(y′H)=(xH)(yH)=(xy)H
This is known as "proof by notation". It's a trap you have to be careful about falling into. Simply because you declare some expression to represent a value, doesn't mean that the value exists or is well-defined.
For instance, suppose we define f(p) to be the area of triangles with ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2811426",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Proof that if $f:S^{n} \to \mathbb{R} $ is continuous, then is not injective I have to prove that if $f:S^{n} \to \mathbb{R} $ (where $S^{n}= \{(x_1, ..., x_{n+1})\in\mathbb{R}^{n+1} | x_1^2+...+x_{n+1}^2=1\} $ is continuous, then is not injective. If possible, I would like it to be proven by using connectivity argume... | Since $S^n$ is compact, if $f$ was injective, it would be a homeomorphism onto $f(S^n)$. But $S^n$ is compact and connected and the only subsets of $\mathbb R$ which are compact and connected are the the intervals $[a,b]$. However, if you remove one point from the middle of this interval, it becomes disconnected. No po... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2811562",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
How many trees over ${1,2,3,...n}$ with conditions I’m stuck on this question in graph theory.
The question is:
How many labeled trees are there over $V={0,1,2,...n}$
with which vertices 1,2,3 are leaves, and distance between any two of these leaves is 3 or more.
I tried using Cayley theorem but I don’t know how to app... | Like every other question about counting trees, this can be answered using Prüfer codes.
Each tree with vertex set $\{1,2,\dots,n\}$ corresponds to a unique Prüfer code, which is a sequence of $n-2$ elements of $\{1,2,\dots,n\}$. Moreover, a vertex of degree $k$ in the tree appears $k-1$ times in the Prüfer code.
So to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2811683",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
gradient of least squares loss function derivation I am trying to derive the derivative of the loss function from least squares. If I have this (I am using ' to denote the transpose as in matlab)
(y-Xw)'(y-Xw)
and I expand it
=(y'- w'X')(y-Xw)
=y'y -y'Xw -w'X'y + w'X'Xw
=y'y -y'Xw -y'Xw + w'X'Xw
=y'y -2y'Xw + w'X'Xw
N... | Here is a piece of background information that we must be clear on at the beginning. If $F:\mathbb R^p \to \mathbb R^q$ is differentiable at a point $z$, then $F'(z)$ is a $q \times p$ matrix.
I assume $X$ is a real $m \times n$ matrix and $y$ is an $m \times 1$ column vector. Let $g:\mathbb R^m \to \mathbb R$ be defi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2811811",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove or show a counter example for: $\forall a, b, c \in \mathbb Z$, if $ab\mid c$ then $a\mid c$ and $b\mid c$ I'm working on my Discrete Mathematics homework and they are asking me this:
Prove or show a counterexample for: $\forall a, b, c \in \mathbb Z$, if $ab\mid c$ then $a\mid c$ and $b\mid c$
I'm not completely... | Hint: If $ab\mid c$, then, since $a\mid ab$, …
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2811914",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
prove there exists a unique polynomial $p\in P_{m}(\mathbb{F})$ such that $p(z_j) = w_j$ Suppose $m$ is a nonnegative integer, $z_1,\cdots, z_{m+1}$ are distinct elements of $\mathbb{F}$, and $w_1, \cdots, w_{m+1} \in \mathbb{F}$. Prove there exists a unique polynomial $p\in P_{m}(\mathbb{F})$ such that $p(z_j) = w_j$ ... | The question is about to find a certain polynomial of degree $\le m$, and these form a vector space.
The standard basis of $P_m(\Bbb F) $ is $1,x, x^2,\dots, x^m$, which makes coordinates from the coefficients.
The determinant of (the matrix written in the standard basis of) the given transformation is called Vanderm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2811988",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Where does the following formula come from? Where does the following formula come from?
For a Laurent polynomial $f(z)=\sum a_j z^j$ and a positive integer $n$ we have $$\sum_{k\equiv \alpha\pmod n} a_k=\frac1n\sum_{\omega:\omega^n=1} \omega^{-\alpha}f(\omega).$$
I hope someone can answer this question or give some ref... | This formula comes from Thomas Simpson's Series Multisection Theory. Speak in the concrete，
a multisection of the series of an analytic function
$$f(z) = \sum_{n=0}^\infty a_n\cdot z^n.$$
has a closed-form expression in terms of the function $f(x)$:
$$\sum_{m=0}^\infty a_{qm+p}\cdot z^{qm+p} = \frac{1}{q}\cdot \su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2812120",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Simplify $\arccos{\sqrt{ 2\over 3}}−\arccos{\frac{\sqrt6+1}{2\sqrt3}}$
Prove $$\arccos{\sqrt{ 2\over 3}}−\arccos{\dfrac{\sqrt6+1}{2\sqrt3}} = \dfrac\pi6$$
How to proceed with this question? I have tried changing them to $\arctan$ and applying $\arctan a- \arctan b$ but ended up getting some numbers which cant be sim... | Let $$\alpha = \arccos{\sqrt2\over \sqrt3}\;\;\;\;{\rm and}\;\;\;\;\beta =\arccos{\sqrt6+1\over 2\sqrt3}$$
so $$ \cos \alpha = \sqrt{2\over 3}\;\;\;\;{\rm and}\;\;\;\;\cos \beta = {\sqrt{6}+1\over 2\sqrt{3}}$$
and
$$ \sin \alpha = \sqrt{1-{2\over 3}} ={1\over \sqrt{3}}\;\;\;\;{\rm and}\;\;\;\;\sin \beta = \sqrt{1-{7+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2812246",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Find $f(3)$ if $f(f(x))=3+2x$ A function $f\colon \mathbb{R} \to \mathbb{R}$ is defined as $f(f(x))=3+2x$
Find $f(3)$ if $f(0)=3$
My try:
Method $1.$ Put $x=0$ we get $f(f(0))=3$ $\implies$ $f(3)=3$
Method $2.$ Replace $x$ with $f(x)$ we get
$$f(f(f(x)))=3+2f(x)$$ $\implies$
$$f(3+2x)=3+2f(x)$$
Put $x=0$
$$f(3)=9$$
I f... | Not only is $f(f(x))=3+2x$ injective, it is bijective, and $f(f(x))\not=x$ except when $x=-3$
so you must have $f(x)\not =x$ and thus must have have $f(f(x)) \not =f(x)$, except for the case $x=-3$ in which case you must have $f(-3)=-3$
in particular you cannot have $f(0)=f(f(0))=3$
In fact you can have $f(0)$ with an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2812415",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Function with positive Fourier coefficients decaying as $\frac{1}{n}$ Is there an example (I'm not looking for a sufficient or necessary condition but just for an example) of a bounded Rieman-integrable function $f\colon [-\pi,\pi]\rightarrow\mathcal{R}$ with Fourier coefficients
$c_n(f) = \int_{-\pi}^\pi{f(t)e^{-int}... | No, there is no such function.
Suppose $f(t)\sim\sum_nc_ne^{int}$ where $c_n\ge0$ and $\sum c_n=\infty$. Then $f$ is not bounded (hence not Riemann integrable).
Not-quite proof: Let $t=0$: it follows that $f(0)=+\infty$.
Of course that's not quite a proof, since there's no reason a priori that the series should con... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2812497",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Find real part of $\frac{1}{1-e^{i\pi/7}}$ How can you find
$$\operatorname{Re}\left(\frac{1}{1-e^{i\pi/7}}\right).$$
I put it into wolframalpha and got $\frac{1}{2}$, but I have no idea where to begin. I though maybe we could use the fact that $$\frac{1}{z}=\frac{\bar{z}}{|z|^2},$$ where $\bar{z}$ is the conjugate of ... | Let
$$z=\frac1{1-e^{it}}$$
where $t$ is real and $e^{it}\ne1$.
Then
$$z+\overline z=\frac1{1-e^{it}}+\frac1{1-e^{-it}}
=\frac1{1-e^{it}}+\frac{e^{it}}{e^{it}-1}=
\frac{1-e^{it}}{1-e^{it}}=1.$$
Therefore the real part of $z$ equals $1/2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2812616",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
Minimize $ \mbox{tr} ( X^T A X ) + \lambda \mbox{tr} ( X^T B ) $ subject to $ X^T X = I $ - Linear Matrix Function with Norm Equality Constraint We have the following optimization problem in tall matrix $X \in\mathbb R^{n \times k}$
$$\begin{array}{ll} \text{minimize} & \mbox{tr}(X^T A X) + \lambda \,\mbox{tr}(X^T B)\\... | EDIT 1. We assume that $\lambda=1$ (change $B$ with $\lambda B$) and $A$ is only symmetric (the non-negativity has nothing to do here). $M_{n,k}$ denotes the real $n\times k$ matrices.
Let $f:X\in M_{n,k}\rightarrow tr(X^TAX)+ tr(X^TB)$. Since $Z=\{X;X^TX=I_k\}$ is compact, then the minimum of $f$ is reached in a poin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2812691",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Convergence or not of infinite series: $\sum^{\infty}_{n=1}\frac{n}{1+n^2}$ How can we prove that the series $\displaystyle \sum^{\infty}_{n=1}\frac{n}{1+n^2}$ is convergent or divergent?
Solution I try:
$$\lim_{m\rightarrow \infty}\sum^{m}_{n=1}\frac{n}{1+n^2}<\lim_{m\rightarrow \infty}\sum^{m}_{n=1}\frac{n}{n^2}$$
Di... | Another method is the integral test: $$\int_1^\infty\frac{x}{1+x^2}dx=\frac{1}{2}[\ln (1+x^2)]_1^\infty =\infty.$$(Note that $\frac{x}{1+x^2}=\frac{1}{x+1/x}$ is maximised at $x=1$, by the AM-GM inequality.)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2812845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 2
} |
How to solve this multiple-absolute-value equation using regions-in a number line method? How to solve this multiple-absolute-value equation using three-region number line?
I can solve it with combination of giving each absolute value a negative sign and leaving it as it is. There are four combinations. The method usi... | Case 1: Let $x<-2$ therefore $$|2x+4|-|3-x|=-2x-4-(3-x)=-1$$which yields to valid answer $x=-6$
Case 2: Let $-2\le x\le3$ therefore $$|2x+4|-|3-x|=2x+4-(3-x)=-1$$which yields to valid answer $x=-\dfrac{2}{3}$
Case 3: Let $x>3$ therefore $$|2x+4|-|3-x|=2x+4+(3-x)=-1$$which yields to invalid answer $x=-8$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2812939",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Minimizing a sum of decaying exponentials subject to constraints Consider the following problem
\begin{align*}
\min_{x}\quad\sum_i\sum_j&\exp(-a_{ij}x_{ij})\\
\text{subject to}\quad\sum_i\sum_j&x_{ij}=n\\
&x_{ij}\ge0\,\,\forall i,j
\end{align*}
where $x\in\mathbb{R}^{I\times J}$ is the optimization variable and each $a... | Note, that since you are not asked to find the optimal solution, but only prove that the given one is optimal, you don't need to solve the KKT system. You only need to substitute the given solution and show that there exist $\lambda, \mu$ such that the the KKT system holds. From convexity and Slater any point is optima... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2813092",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
What is the relationship between different theorems all called Hilbert's Nullstellensatz? The following statements are all named the Hilbert's Nullstellensatz, but they appear at first to be completely unrelated to each other. What is the relationship between them exactly?
*
*(Theorem 1.3A on page 4 of Hartshorne's ... | Number 4. is sometimes called Zariski's lemma. The relation to Nullstellensatz is that if $k$ is algebraically closed, then the residue field is a finite algebraic extension of $k$, which must be $k$ itself. From this follows 3.; it is not hard to see that $k[x_1, \dots, x_n]/ \mathfrak{m} = k$ implies that $\mathfrak... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2813188",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 1,
"answer_id": 0
} |
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