Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Is Fibonacci sequence the minimum of unique pairwise sum sequence? Let $(a_n)_{n=1}^\infty$ be a strictly increasing (condition added per earlier answer of Amitesh Datta) sequence of natural numbers where all pairwise element sums are unique. Can anyone prove or disprove whether the Fibonacci sequence $(f_n)_{n=1}^\inf... | No. The Mian-Chowla sequence is such a sequence and it begins 1, 2, 4, 8, 13... ; it also grows only polynomially fast, no faster than n3.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/455389",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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inversely proportional term in task let us consider following problem:
The amount of time taken to paint a wall is inversely proportional to the number of painters working on the job. If it takes 3 painters 5 days to complete such a job, how many days longer will it take if there are only 2 painters working?
so let us ... | Let $d(n)$ be the number of days required when there are $n$ painters. We’re told that $d(n)$ is inversely proportional to $n$, so there is a constant $c$ such that $$d(n)=\frac{c}n\;.$$ We’re also told that $d(3)=5$, so $$5=d(3)=\frac{c}3\;,$$ and therefore $c=3\cdot5=15$. Therefore $$d(2)=\frac{15}2=7.5\;.$$ Thus, tw... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that $\sin3\alpha \sin^3\alpha + \cos3\alpha \cos^3\alpha = \cos^32\alpha$
Show that $\sin3\alpha \sin^3\alpha + \cos3\alpha \cos^3\alpha = \cos^32\alpha$
I have tried $\sin^3\alpha(3\sin\alpha - 4 \sin^3\alpha) = 3\sin^4\alpha - 4\sin^6\alpha$ and $\cos^3\alpha(4\cos^3\alpha - 3\cos\alpha) = 4\cos^6\alpha - 3\c... | \begin{align}
L.H.S=& \sin 3\alpha \sin \alpha\sin^2\alpha+\cos 3\alpha\cos \alpha \cos^2 \alpha\\
\ =& \frac{1}{2}\left(\sin 3\alpha \sin \alpha (1-\cos 2\alpha)+\cos 3\alpha\cos \alpha(1+\cos 2\alpha)\right)\\
\ =& \frac{1}{2}\left(\sin 3\alpha \sin \alpha+ \cos 3\alpha\cos \alpha\right)+\frac{1}{2}\left(\cos 3\alpha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/455518",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Material derivative of a material vector field
On page 12 of An Introduction to Theoretical Fluid Dynamics, following the introduction of a material vector field $v_i(\mathbf a,t)=J_{ij}(\mathbf a,t)V_j(\mathbf a)$ the author wrote:
$$
\frac{\mathrm D \mathbf v}{\mathrm D t}
=
\left. \frac{\partial \mathbf... | For some clarity the author has made the following calculation (I will explicitly give the variables that $\mathbf{v}$ depends on in each equation to avoid confusion)
$$\dfrac{\mathbf{Dv}}{\mathbf{D}t} = \dfrac{\text{d}\mathbf{v}(\mathbf{x}(t),t)}{\text{d}t}= \dfrac{\text{d}\mathbf{v}(\mathbf{a},t)}{\text{d}t} = \dfrac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/455610",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
} |
Advanced undergraduate(?) Real Analysis book which is concise and lots of interesting problems I have gone through the other book recommendations on Real Analysis, but I think my requirements and background is slightly different. I am a Physics undergrad teaching myself Pure math. My journey is pure math has been highl... | I like Kolmogorov and Fomin "Introductory Real Analysis" - which gives lots of examples and has plenty of good problems. But I'm not sure what kind of problem you are looking for.
If you are after challenging integrals and limits etc Hardy's "Pure Mathematics" has lots of those. I also think Apostol's Mathematical Anal... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/455735",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to check if two inequalities express the same thing? Suppose:
$$
\delta_1>0 \\
\delta_2>0 \\
\delta=\text{min}(\delta_1,\delta_2)
$$
If we know that the following is true:
$$
a-\delta_2<a-\delta<x<a \hspace{2cm} \text{ or } \hspace{2cm} a<x<a+\delta<a+\delta_1
$$
Can we conclude that the followng is true too?
$$
a... | Yes. if you chose $|x-a|<\delta $.
maybe i dont get this question but your inequality is true for this specific $a$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/455824",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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$f: X \to Y$ is a smooth map and $\omega$ is a $p$-form on $Y$, what is $\omega[f(x)]$? I am reading Guillemin and Pollack's Differential Topology Page 163:
If $f: X \to Y$ is a smooth map and $\omega$ is a $p$-form on $Y$, define a $p$-form $f^*\omega$ on $X$ as follows: $$f^*\omega(x) = (df_x)^*\omega[f(x)].$$
So m... | $\omega[f(x)]$ is the value of the form $\omega$ at the point $f(x)$. It is a vector in the $p$-th exterior power of the cotangent space at $f(x)$. (Remember that a $p$-form is a smooth function assigning to each point of a manifold such a vector.) I guess you were just confused by the use of square brackets, which ... | {
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What exactly does linear dependence and linear independence imply? I have a very hard time remembering which is which between linear independence and linear dependence... that is, if I am asked to specify whether a set of vectors are linearly dependent or independent, I'd be able to find out if $\vec{x}=\vec{0}$ is the... | I've found the best way to understand this is as follows: a set of vectors is linearly dependent if you can write one of them in terms of the others.
When you multiply a matrix by a vector, $A\vec{x}$, that's shorthand for "multiply each column of $A$ by the corresponding entry of $\vec{x}$, and then add them together.... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Pullback expanded form.
Definition. If $f: X \to Y$ is a smooth map and $\omega$ is a $p$-form on $Y$, define a $p$-form $f^*\omega$ on $X$ as follows: $$f^*\omega(x) = (df_x)^*\omega[f(x)].$$
According to Daniel Robert-Nicoud's nice answer to $f: X \to Y$ is a smooth map and $\omega$ is a $p$-form on $Y$, what is $\... | I suppose generally a $p$-form is a sum of such terms, but if we can understand how one such element pulls-back then linearity extends to $\sum_{i_1, \dots , i_p}\alpha_{i_1,\dots , i_p} dy^{i_1} \wedge \cdots \wedge dy^{i_p}$. That said, to calculate $\gamma$ you just have to sort out the sign needed to arrange the in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/456086",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Can one zero-pad data prior to Fourier transformation, then reverse the change afterwards? Suppose I have a set of $n$ points $\underline{x}\in\mathbb{C}^n$ with $n \in \mathbb{P}$ ($n$ is prime), and I want to find the Fourier transform of $\underline{x}$.
There are some prime-length Fourier algorithms out there, but ... | They are the same Fourier transform but just sampled at different locations. If your signal vector is of length $N$ and sampled at $f_s$ Hertz, then an $N$-point DFT will have a sample spacing of $f_s / N$ Hertz/sample with the samples ranging from $-f_s/2$ to $+f_s/2 - f_s/N$. If you zeropad your data out to $M>N$ sam... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What do $R$, $R^2$ and $R^{-1}$ represent? Question
What do $R$, $R^2$ and $R^{-1}$ represent when $R$ is a relation on the set of natural numbers?
I'm doing some homework, but the $R^2$ and $R^{-1}$ notation confuses me.
Does $R^2 = R*R$?
Does $R^{-1} = \frac{1}{R}$?
| $R^{-1}$ is the inverse relation: $R^{-1}=\{\langle \ell,k\rangle:\langle k,\ell\rangle\in R\}$. In other words, you get $R^{-1}$ from $R$ by turning each ordered pair in $R$ around. If $R=\{\langle n,n+1\rangle:n\in\Bbb N\}$, so that $R$ contains such pairs as $\langle 1,2\rangle$ and $\langle 7,8\rangle$, then $R^{-1... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Are real coefficients also complex coefficients? Can $x^2 + x +1$ be called a polynomial with complex coefficients?
I know that all real numbers are complex numbers, so does this hold here as well?
| The polynomial $x^2 + x +1$ can certainly be called a polynomial with complex coefficients, but moreover the idea of doing so has important mathematical applications. For example, when proving that a real symmetric matrix has a real eigenvalue, it is very convenient to extend the scalars to $\mathbb{C}$, find an eigen... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/456300",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Convergence in metric and in measure Let $\mu$ be a finite measure on $(X, A)$, with the semimetric
$$ d(f,g) = \int \frac{|f-g|}{1+ |f-g|}d\mu$$
on all real-valued, A-measurable functions.
Show that $$\lim_n d(f_n, f) = 0$$ holds iff
$(f_n)$ converges to $f$ in measure.
I know that convergence in mean implies conv... | First, a remark: the map $x\mapsto \frac x{1+x}$ is increasing over the set of non-negative real numbers, and is bounded by $1$.
If $f_n\to f$ in measure, fix $\varepsilon$ and integrate over $\{|f_n-f|>\varepsilon\}$ and $\{|f_n-f|\leqslant \varepsilon\}$.
Conversely, if $d(f_n,f)\to 0$, then $\frac{\varepsilon}{1+\v... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/456355",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Does the Pigeonhole principle apply in this problem? I came accross this problem a while ago at school during a math contest. I dont remember the exact instruction (word for word) but it went something like :
Randomed A and B, 2 natural integer $\in [0,100]$ , Start a Fibonacci-like sequence.
A + B => C
B + C => D
etc.... | Look at the sequence of couples of consecutive terms. Observe that given two numbers $(a,b)$, the rest of the sequence is fully determined; therefore, it is sufficient to prove that no matter what are the initial two numbers, as some point you will fall back to two successive terms of the sequence $(a^\prime,b^\prime)$... | {
"language": "en",
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How to find the vertex of a 'non-standard' parabola? $ 9x^2-24xy+16y^2-20x-15y-60=0 $ I have to find out the vertex of a parabola given by:
$$ 9x^2-24xy+16y^2-20x-15y-60=0 $$
I don't know what to do. I tried to bring it in the form:
$$ (x-a)^2 + (y-b)^2 = \dfrac {(lx+my+n)^2} {l^2+m^2} $$
but failed in doing so. Is the... | Think of the standard equations of a parabola you know $y=x^2$ or $y^2=4ax$ - something squared = something linear, and the squared quantity and the linear quantity represent axes at right-angles to one another.
The vertex occurs where the squared quantity is equal to zero, ie on the axis of symmetry.
Now notice that t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/456476",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Logic problem, proposed by Euler A professor of mine asked me this challenge.
"(Proposed by Euler) A person bought horses and oxen. Paid 31 shields per horse and 20 per ox, and found that all oxen cost 7 shields more than all the horses. How many horses and oxen were bought? "
And I is not getting ...
Called C the amou... | Yes, that's right.
Note that this linear Diophantine equation has infinitely many solutions... I suppose the "most likely" solution is the one with positive number of horses and oxen that costs the least total.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/456545",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that $pTq \longleftrightarrow |p| = |q|$ is Equivalence Relation on $A$ set of all point in the plane I want to prove that this relation is equivalence relation on A
*
*$A$ set of all points in the plane
*$pTq \longleftrightarrow |p| = |q|$ , |p| is the distance from origin.
about transitivity, there are c... | Notice that this relation $T$ is defined by the equality relation which's the most natural equivalence relation so $T$ will inherit the same properties and then it's also an equivalence relation.
Remark You can use this method for all relation defined by
$$xRy\iff f(x)=f(y)$$
For the class equivalence of $x$:
$$[x]=\{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/456602",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Probability density function for radius within part of a sphere I would like to find the probability density function for radius within a given section of a sphere. For example, suppose I specify $\pi / 4 < \theta < \pi / 3$ and $\pi /7 < \phi < \pi /5 $ and $1 < r < 4$. If I select a point at random from within thi... | Define $V:[1,4]\rightarrow\mathbb{R}$ by
$$
V(r):=\int_{\pi/4}^{\pi/3}\int_{\pi/7}^{\pi/5}\int_1^r\rho^2\sin\phi\,d\rho\,d\phi\,d\theta.
$$
(Here, we have used the spherical coordinate transformation where $\rho$ is distance to the origin, $\theta$ is the angle formed in the $(x,y)$-plane, and $\phi$ is measured down f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/456687",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Can $p^{q-1}\equiv 1 \pmod {q^3}$ for primes $pFor prime $q$ can it be that
$$
p^{q-1}\equiv 1 \pmod{q^k}
$$
for some prime $p<q$ and for $k\ge 3$?
There doesn't seem to be a case with $k=3$ and $q<90000$, and I also checked for small solutions with $3<k\le 20$ and found none.
If we remove the condition $p<q$ then the... | Let $w>1$ be any integer and let $q$ be an odd prime and $w^{q-1}$ $\equiv 1 \pmod {q^3}$. Let v be a primitive root mod $q^3$ where $v^h$ $\equiv w \pmod {q^3}$. So $v^{h(q-1)}$ $\equiv 1\pmod {q^3}$. Therefore h=$q^2 k$ ; k >= 1. Assume k> 1 , then $w^{(q-1)/k}$ $\equiv 1\pmod {q^3}$ ; $v^{q^2 k-k}$ $\equiv(w/v^k)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/456744",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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Derivative of an integral $\sqrt{t}\sin t dt$ I need to find the derivative of this function. I know I need to separate the integrals into two and use the chain rule but I am stuck.
$$y=\int_\sqrt{x}^{x^3}\sqrt{t}\sin t~dt~.$$
Thanks in advance
| Hint
By the chain rule we prove easly:
If
$$F(x)=\int_{u(x)}^{v(x)}f(t)dt$$
then
$$F'(x)=f(v(x))v'(x)-f(u(x))u'(x)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/456826",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Clifford Algebras What would be the best source to learn Clifford Algebras from? Anything online would suffice or any textual sources for that matter..
I'm interested in doing a project in the subject, but I'm not sure where to begin to learn. What would be the prerequisites to ensure a stable foundation?
Thanks ... | Clifford algebras arise in several mathematical contexts (e.g., spin geometry, abstract algebra, algebraic topology etc.). If you're just interested in the algebraic theory, then the prerequisites would probably be a solid background in abstract algebra. For example, I think linear algebra and ring theory are prerequis... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/456968",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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prove that a connected graph with $n$ vertices has at least $n-1$ edges Show that every connected graph with $n$ vertices has at least $n − 1$ edges.
How can I prove this? Conceptually, I understand that the following graph has 3 vertices, and two edges:
a-----b-----c
with $a$, $b$ and $c$ being vertices, and $\{a,b\}... | Hint: Let $\Gamma$ be a connected graph. If $T \subset \Gamma$ is a maximal subtree, then $|E(\Gamma)| \geq |E(T)|$ and $|V(\Gamma)|=|V(T)|$. (Where $E(\cdot)$ and $V(\cdot)$ is the set of edges and vertices respectively.)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/457042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "30",
"answer_count": 8,
"answer_id": 5
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Compact variety which is not projective While reading Andreas Gathmann's notes on Algebraic Geometry, I stumbled upon this statement: "Projective varieties form a large class of “compact” varieties that do admit such a unified global description. In fact, the class of projective varieties is so large that it is not eas... | Such example does not exist in dimension 1. For dimension 3, see the appendix B in Hartshorne, Example 3.4.1
As Liu pointed out below, there is a list discussing related questions in Hartshorne Chapter II Beneath Remark 4.10.2.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/457154",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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How to prove $\int_{-\infty}^{+\infty} f(x)dx = \int_{-\infty}^{+\infty} f\left(x - \frac{1}{x}\right)dx?$ If $f(x)$ is a continuous function on $(-\infty, +\infty)$ and $\int_{-\infty}^{+\infty} f(x) \, dx$ exists. How can I prove that
$$\int_{-\infty}^{+\infty} f(x) \, dx = \int_{-\infty}^{+\infty} f\left( x - \frac{... | We can write
\begin{align}
\int_{-\infty}^{\infty}f\left(x-x^{-1}\right)dx&=\int_{0}^{\infty}f\left(x-x^{-1}\right)dx+\int_{-\infty}^{0}f\left(x-x^{-1}\right)dx\\
&=\int_{-\infty}^{\infty}f(2\sinh\theta)\,e^{\theta}d\theta+\int_{-\infty}^{\infty}f(2\sinh\theta)\,e^{-\theta}d\theta\\
&=\int_{-\infty}^{\infty}f(2\sinh\th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/457231",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "75",
"answer_count": 7,
"answer_id": 0
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The contradiction method used to prove that the square root of a prime is irrational The contradiction method given in certain books to prove that sqare root of a prime is irrational also shows that sqare root of $4$ is irrational, so how is it acceptable?
e.g. Suppose $\sqrt{4}$ is rational,
$$\begin{align}
\sq... | The step from $4\mid p^2$ to $4\mid p$ is wrong. For example, take $p=6$. Then $4\mid 6^2$ but it is not true that $4\mid 6$.
In general, $q\mid p^2$ implies $q\mid p$ only for squarefree $q$. A number $q$ is "squarefree" if it is not divisible by any square larger than 1.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/457291",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Can $\sqrt{n} + \sqrt{m}$ be rational if neither $n,m$ are perfect squares? Can the expression $\sqrt{n} + \sqrt{m}$ be rational if neither $n,m \in \mathbb{N}$ are perfect squares? It doesn't seem likely, the only way that could happen is if for example $\sqrt{m} = a-\sqrt{n}, \ \ a \in \mathbb{Q}$, which I don't thin... | Nice way to see thinks
Assume that, $$(\sqrt{n}+\sqrt{m})=\frac{p}{q}$$
Then we have
$$(\sqrt{n}+\sqrt{m})=\frac{p}{q}\in\Bbb Q \implies
n+m+2\sqrt{nm} =(\sqrt{n}+\sqrt{m})^2 =\frac{p^2}{q^2}\in\Bbb Q\\\implies \sqrt{nm} =\frac{n+m}{2}+\frac{p^2}{2q^2}\in\Bbb Q $$
But if $ nm $ is not a perfect square then $\sqrt{n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/457382",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "41",
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Resources to help an 8yo struggling with math Friends of mine asked me for suggestion for one of their children (age 8) who had bad scores at the local Star test (the family is based in California).
Both parents work, so they have also limited time/energies to go through math exercise with the kid (or may have time onl... | In my personal opinion getting him interested in mathematics is the best way to get him to get better at it hands down. When I was a kid I played math games for kids on my computer and I would also compete against my mom to see who could answer basic arithmetic question (mabye TMI).
In other words if you can get the ki... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/457446",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
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Prove the triangle inequality I want to porve the triangle inequality:
$x, y \in \mathbb{R} \text { Then } |x+y| \leq |x| + |y|$
I figured out that probably the cases:
*
*$x\geq0$ and $y \geq 0$
*$x<0$ and $y < 0$
*$x\geq0$ and $y < 0$
*$x<0$ and $y \geq 0$ <- Here I am not sure...
have to be proven. However, I... | If both $x$ and $y$ are $0$ or $x=-y$ then the inequality is clear. Otherwise we note that for $x,y\in\mathbb{R}$ $x\le|x|$ and similarly $y\le|y|$, which follows from the definition of the absolute value.
This tells us that $x+y\le|x|+|y|$ which implies $\frac{x+y}{|x|+|y|}\le1$ since $|x|+|y|>0$.
Thus, $|x+y|=|\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/457513",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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find the inverse of $x^2 + x + 1$ In $\mathbb{F}_2[x]$ modulo $x^4 + x + 1$
find the inverse of $x^2 + x + 1$
not 100% sure but here what i have:
user euclid algorithm:
$x^4 + x + 1 = (x^3 + 1)(x + 1) + x$
$(x^3 + 1) = x * x * x + 1$
$1 = (x^3 + 1) - x * x * x $
| Using that $\;x^4=x+1\; $ in $\,\Bbb F_2[x]/(x^4+x+1)\;$ , prove that
$$x^2+x=(x^2+x+1)^{-1}\;\;\; (\text{ further hint:}\;(x^2+x+1)^3=1)$$
| {
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Do the densities of a Uniform[0,1/n] random variable converge pointwise to zero? I'm trying to think of densities that converge pointwise to a function that is not a density. It seems to me that the only way this is possible is if the densities converge to some constant.
Here is something that I thought of but doesn't ... | For example, you can use the normals mean $0$, variance $n^2$. These flatten out nicely as $n\to\infty$. It is easy to see that
$$\lim_{n\to\infty} \frac{1}{n\sqrt{2\pi}}e^{-x^2/2n^2}=0$$
for all $x$.
Your proposed example of density function $f_n(x)$ equal to $n$ on $\left(0,\frac{1}{n}\right)$ and $0$ elsewhere als... | {
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Definite integral of unknown function given some additional info Given
*
*$f$ integrable on [0,3]
*$\displaystyle\int_0^1 f(x)\,\mathrm{d}x = 1$,
*$f(x+1) = \frac{1}{2}f(x)$ for all x $\in [0, 2]$
How can I find $\displaystyle\int_0^3 f(x)\,\mathrm{d}x$ ?
I tried breaking it into as follows:
$\displaystyle\i... | Hint:
$$\int_1^2f(t)\,dt=\int_0^1f(x+1)\,dx={1\over 2}\int_0^1f(x)\,dx={1\over 2}$$
| {
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Counting Real Numbers Forgive me if this is a novice question. I'm not a mathematics student, but I'm interested in mathematical philosophy.
Georg Cantor made an argument that the set of rational numbers is countable by showing a correspondence to the set of natural numbers. He did this by scanning rational numbers in ... | You only counted a subset of the reals, namely, the set including the integers as well as reals with one decimal place. You cannot count the reals, as you would have to count to an infinite number of decimal places, as some reals have no fractional representation.
To 'count' as you propose, you would need the top headi... | {
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Integrating $\int_0^\infty \frac{\log x}{(1+x)^3}\,\operatorname d\!x$ using residues I am trying to use residues to compute $$\int_0^\infty\frac{\log x}{(1+x)^3}\,\operatorname d\!x.$$My first attempt involved trying to take a circular contour with the branch cut being the positive real axis, but this ended up cancell... | Consider the integral
$$\oint_C dz \frac{\log^2{z}}{(1+z)^3}$$
where $C$ is a keyhole contour in the complex plane, about the positive real axis. This contour integral may be seen to vanish along the outer and inner circular contours about the origin, so the contour integral is simply equal to
$$\int_0^{\infty} dx \fr... | {
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If $f$ is an entire function with $|f(z)|\le 100\log|z|$ and $f(i)=2i$, what is $f(1)$? Let $f$ be an entire function with $|f(z)|\le 100\log|z|,\forall |z|\ge 2,f(i)=2i, \text{ Then} f(1)=?$
I have no idea how to solve this one!
$g(z)={f(z)\over \log|z|}$ Then Can I say $g$ is constant by Liouville Theorem?
| You can't directly use Liouville's theorem, since dividing $f$ by $\log \lvert z\rvert$ or $\log z$ doesn't produce an entire function.
But you can use the Cauchy estimates to show that that bound on $\lvert f(z)\rvert$ actually implies that $f$ is constant, for $R > 2$ and $\lvert z\rvert \leqslant \frac{R}{2}$, you c... | {
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Express spherical coordinates with different centers in terms of each other. Imagine that you have two spheres with a distance $R$ from one center to the other one. Now, it is well known how one would get the cartesian position vector of each point in sphere 1 by using spherical coordinates with the reference frame bei... | Let the origin of your first spherical coordinates be the origin of the first Cartesian system being used, and the origin of your second spherical coordinates be at $X_o, Y_o, Z_o$ in the same Cartesian system.
Use spherical polar coordinates, with $\phi$ being the azimuth angle in the $X-Y$ plane, and $\theta$ being t... | {
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Prove the inequality $\sum_{1\le iPlease demonstrate this is true
This is the exercise:
$$\sqrt{a_1a_2}+\sqrt{a_1a_3}+\ldots+\sqrt{a_1a_n}+\sqrt{a_2a_3}+\ldots+\sqrt{a_{n-1}a_n}<\frac{n-1}2(a_1+a_2+a_3+\ldots+a_n).$$
I tried to solve it, but I couldn't do anything right.
This is my idea:
$\sqrt{a_1a_2}<\frac{a_1+a_2}2... | Use the AM-GM inequality, which states that:
$$\sqrt{a_ia_j}\le\frac{a_i+a_j}{2}$$
Then we have the following:
$$\sum_{i< j}\sqrt{a_ia_j}\le\sum_{i< j}\frac{a_i+a_j}{2}=\frac{1}{2}\sum_{i< j}a_i+a_j\\=\frac{1}{2}[(a_1+a_2)+(a_1+a_3)+\ldots+(a_1+a_n)+(a_2+a_3)+\ldots+(a_{n-1}+a_n)]$$
Now notice that in the sum on the ri... | {
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Dimension of $\mathbb{Q}\otimes_{\mathbb{Z}} \mathbb{Q}$ as a vector space over $\mathbb{Q}$ The following problem was subject of examination that was taken place in June. The document is here. Problem 1 states:
The tensor product $\mathbb{Q}\otimes_{\mathbb Z}\mathbb{Q}$ is a vector space
over $\mathbb{Q}$ by multi... | There is another solution using category theory : tensor product is the fiber coproduct in the category of commutative rings. $\mathbb{Z} \rightarrow \mathbb{Q}$ is an epimorphism since it's a localization.
If $A \rightarrow B$ is an epimorphism, and $B \rightarrow C, B \rightarrow D$ are any morphisms, then $C \coprod... | {
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Is The *Mona Lisa* in the complement of the Mandelbrot set. Here is a description of how to color pictures of the Mandelbrot set, more accurately the complement of the Mandelbrot set. Suppose we have a rectangular array of points. Say the array is $m$ by $n$. Suppose also we have a number of color names. Now suppose we... | I think yes
Consider the sequence of "westernmost" islands increasing in period:
Here are some examples, you can see them increasing in hairyness / spinyness.
Period 20:
Period 30:
Period 40:
Period 50:
Zooming in near to a sufficiently hairy high-period island, you can get very nearly parallel spines:
There is a... | {
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Solution Manual for Chapters 13 and 14, Dummit & Foote I bought the third edition of "Abstract Algebra" by Dummit and Foote. In my opinion this is the best "algebra book" that has been written.
I found several solution manual but none has solutions for Chapters 13 and 14 (Field extensions and Galois theory respectively... | If anyone is interested, I made a full solution manual for Chapter 13 - Field Theory.
You can find it here
https://positron0802.wordpress.com/algebra-dummit-chapter-13-solution/.
| {
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Sports competition team gaming Let $A,B,C,D,E,F$ be six teams in a sports competition, and each team will play exactly once with another team.
Now we know that Team $A,B,C,D,E$ had already played $5,4,3,2,1$ games, correspondingly.
So, how do I figure out which team haven't played a game with team $B$ yet?
| A has played all 5 other teams and E has only played one team. That means E must have only played A, so the only team B hasn't played must be E.
| {
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Conditions under which $BA = I_{n}$, where $A\in\mathbb{C}^{m\times n}$ and $B\in\mathbb{C}^{n\times m}$ Let $A\in\mathbb{C}^{m\times n}$ . I want to to know what conditions can I apply on the matrix $B\in\mathbb{C}^{n\times m}$ such that product $BA = I_{n}$ or matrix $B$ is the left inverse of the matrix $A$.
Pleas... | I think that, as @OwenSizemore correctly suggested, if $n\leq m$ and A is a full rank matrix, then we can think of some Moore-Penrose Pseudoinverse type of solution.
If not, then I'm not sure if this is even possible. Given a vector $\mathbf{v}$ such that $\mathbf{Ax=v}$ we can't really get $\mathbf{x}$ since given a ... | {
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Does there exists a theorem like this? Statement: Suppose $T\subseteq \mathbb{N}$, then all $x^i,i\in T$ generate a dense linear subspace of $C^0[a,b]$ iff $\sum_{i\in T} 1/i$ is divergent.
I heard it somewhere a long time ago, so there may be minor errors, but the meaning goes like this. I heard it was called a "Berns... | An excellent reference, not mentioned in the Wikipedia article, is section 4.2 of Polynomials and Polynomial Inequalities by Borwein and Erdélyi. On 35 pages of that section the authors collect a huge number of variations of the theorem (and then return to it in 4.4 in the setting of rational functions).
Here is a samp... | {
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Example of semiprime ring The ring is semiprime if $x\in R, xyx=0$ for all $y\in R$ implies $x=0$ or equivalently for $x\neq0$ exists $y_{0}\in R$ such that $xy_{0}x\neq0$.
I found an example of semiprime ring. However, I am not sure that if I understand properly.
$R=\left\{ \begin{pmatrix}a & 0\\
0 & b
\end{pmatrix},\... | The example you gave is indeed semiprime, but it is a complicated way to look at the ring $F_2^2$.
Given $(a,b)$ nonzero, one of $(a,b)(1,0)(a,b)$ or $(a,b)(0,1)(a,b)$ is nonzero.
Any semisimple or Von Neumann regular ring or prime ring is going to be semiprime.
I recommend that you try to show that the ring of linear ... | {
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solving the differential equation$y'+\sin y+x \cos y+x=0.$ How to solve the following differential equation?
$$y'+\sin y+x \cos y+x=0.$$
| HINT:
$$y'+\sin y+x \cos y+x=0 \implies \frac{dy}{dx}+\sin y=-x(1+\cos y)$$
$$\implies \frac{dy}{dx}\cdot\frac1{1+\cos y}+\frac{\sin y}{1+\cos y}=-x$$
Using $\cos 2z=2\cos^2z-1,\sin2z=2\sin z\cos z,$
$$\frac12\sec^2\frac y2 \frac{dy}{dx}+\tan\frac y2=-x$$
$$\implies \frac{d\left(\tan\frac y2\right)}{dx}+\tan\frac y2 \... | {
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Prove that if $AB$ is invertible then $B$ is invertible. I know this proof is short but a bit tricky. So I suppose that $AB$ is invertible then $(AB)^{-1}$ exists. We also know $(AB)^{-1}=B^{-1}A^{-1}$. If we let $C=(B^{-1}A^{-1}A)$ then by the invertible matrix theorem we see that since $CA=I$(left inverse) then $B$ i... | $\;AB\;$ invertible $\;\implies \exists\;C\;$ s.t.$\;C(AB)=I\;$ , but using associativity of matrix multiplication:
$$I=C(AB)=(CA)B\implies B\;\;\text{is invertible and}\;\;CA=B^{-1}$$
| {
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$\sum_{k=0}^{n/2} {n\choose{2k}}=\sum_{k=1}^{n/2} {n\choose{2k-1}}$, Combinatorial Proof: How am I supposed to prove combinatorially:
$$\sum_{k=0}^{n/2} {n\choose{2k}}=\sum_{k=1}^{n/2} {n\choose{2k-1}}$$
$${n\choose{0}}+{n\choose{2}}+{n\choose{4}}+\dots={n\choose{1}}+{n\choose{3}}+{n\choose{5}}+\cdots$$
Absolutely clue... | The question as currently posed can be answered by looking at the symmetry of the rows of Pascal's triangle corresponding to odd $n$ (which have an even number of elements). By definition
$\large{n\choose{k}\large}=\frac{n!}{k!(n-k)!}$.
Therefore ${n\choose{0}}={n\choose{n}}$, ${n\choose{1}}={n\choose{n-1}}$, and in ... | {
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Topologist's Sine Curve not a regular submanifold of $\mathbb{R^2}$? I am trying to work out the details of following example from page 101 of Tu's An Introduction to Manifolds:
Example 9.3. Let $\Gamma$ be the graph of the function $f(x) =
\sin(1/x)$ on the interval $]0, 1[$, and let $S$ be the union of
$\Gamma$ a... | One simple way to see that $S$ is not a regular submanifold around $p$ is that it is not locally Euclidean: while there exist open subsets of $\mathbb{R}$ with as many connected components as you like, there are no points like $p$ in $\mathbb{R}$, that is, with every neighborhood not connected.
So, in the sentence "sin... | {
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Computing Brauer characters of a finite group I am studying character theory from the book "Character Theory of Finite Groups" by Martin Isaac. (I am not too familiar with valuations and algebraic number theory.)
In the last chapter on modular representation theory, Brauer characters, blocks, and defect groups are intr... | This is a difficult question and you would probably need to learn more theory in order to understand the different methods available.
But one method that is often used in practice is to calculate the representations and then just find the Brauer character directly from the matrices of the representations. Of course, yo... | {
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contour integral with integration by parts Is there a complex version of integration-by-part? I saw someone used it but didn't find it in textbook. I tested integrals $\int_{\mathcal{C}}\frac{\log(x+1)}{x-2}\mathrm{d}x$ and $\int_{\mathcal{C}}\frac{\log(x-2)}{x+1}\mathrm{d}x$, where $\mathcal{C}$ encloses both -1 and 2... | Integration by parts is just the product rule and the Fundamental Theorem of Calculus. But you need well-defined analytic functions on your contour, which you don't have here.
| {
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An application of fixed point theorem I want to use the fixed point method to solve the equation to find $y$: $$ y = c_1 y^3 - c_2 y$$where $c_1, c_2$ are real valued constants. So I designed $$ y_{k+1} = c_1 y_k^3 - c_2 y_k$$ to approximate $y$. But I don't know what to do next. Also I want to know the convergence for... | Do you want to do this on a computer or by hand? Approximating things by hand usually makes little sense, so suppose by computer.
If so, then the thing to do is first to put some value of $y_0$ (probably close to $0$, or else $y_k \to \infty$ as $k \to \infty$, but also not precisely $0$, or else $y_k = 0$ for all $k$)... | {
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How to find cubic non-snarks where the $\min(f_k)>6$ on surfaces with $\chi<0$? Henning once told me that,
[i]t follows from the Euler characteristic of the plane that the average face degree of a 3-regular planar graph with $F$ faces is $6-12/F$, which means that every 3-regular planar graph has at least one face wit... | For orientable surfaces, here's a representative element of a family of non-snarky cubic graphs on an $n$-torus with $4n-2$ vertices, $6n-3$ edges and a single $(12n-6)$-sided face.
If it is a problem that some pairs of vertices have more than one edge going between them, that can easily be fixed with some local rearr... | {
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How much money should we take? I'm a new user so if my question is inappropriate, please comment (or edit maybe).
We want to define a dice game. We will be the casino and a customer will roll dice. I will assume customer is man. He can stop whenever he wants and when he stopped he can take the money as much as sum of h... | Let $f(n)$ be the expected final win for a player already having a balance of $n$ and employing the optimal strategy. Trivially, $f(n)\ge n$ as he might decide to stop right now.
However, if the optimal strategy tells him to play at least once more, we find that $f(n)=\frac16\cdot 0+\sum_{k=2}^6 \frac16f(n+k)$. Thus
$$... | {
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Are quantifiers a primitive notion? Are quantifiers a primitive notion? I know that one can be defined in terms the other one, so question can be posed, for example, like this: is universal quantifier a primitive notion? I know, that $\forall x P (x) $ can be viewed as a logical conjunction of a predicate $ P $ being ... | Note for example in PA that even if $P(0)$ and $P(1)$ and $P(2)$ and ... are all theorems, it may happen that $\forall n\colon P(n)$ is not a theorem. Thus $\forall n\colon P(n)$ is in fact something different from $P(0)\land P(1)\land P(2)\land \ldots$ even if one were to accept such an infinte string as a wff (which... | {
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$X$ homeomorphic to $f(X)$
Let $X$, and $Y$ be topological spaces, and let $f:X\rightarrow Y$ be a continuous and one-to-one map.
When is $X$ homeomorphic to $f(X)$?
| Well... when $f$ is open (or closed). A nice criterion is: $X$ compact, $Y$ Hausdorff, then $f$ is a closed map. Indeed let $C\subset X$ be closed, then $C$ is compact. The continuous image of a compact set is compact, so $f(C)\subset Y$ is compact, and thus closed.
Note: I interpreted your question as: "When is $f$ an... | {
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Examples of uncountable sets with zero Lebesgue measure I would like examples of uncountable subsets of $\mathbb{R}$ that have zero Lebesgue measure and are not obtained by the Cantor set.
Thanks.
| Let $(r_n)_{n\in\mathbb{N}}$ be a dense sequence in $\mathbb{R}$ (it could, for example, be an enumeration of the rationals). For $k \in \mathbb{N}$, let
$$U_k = \bigcup_{n\in\mathbb{N}} (r_n - 2^{-(n+k)},\, r_n + 2^{-(n+k)}).$$
$U_k$ is a dense open set with Lebesgue measure $\leqslant 2^{2-k}$, thus
$$N = \bigcap_{k\... | {
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Average of all 6 digit numbers that contain only digits $1,2,3,4,5$ How do I find the average of all $6$ digit numbers which consist of only digits $1,2,3,4$ and $5$?
Do I have to list all the possible numbers and then divide the sum by the count? There has to be a more efficient way, right?
Thank you!
| Don't want to completely give it away, but there are $5^6$ of these numbers as the first through sixth digits can all take on five different values. I'm sure there's something slicker you could do, but it should be easy to then sum them all up by evaluating the sum
$$
\sum_{a=1}^5 \sum_{b=1}^5 \sum_{c=1}^5 \sum_{d=1}^... | {
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How to check a set of ring is a subring? To check a subset of a given ring is a subring,
is it enough to check that the subset is closed under induced operations(multiplication and addition) or
do I also need to show that it contains 0 and additive inverses of each element?
| You do need to show that it contains an additive inverse for each of its elements. (For example, $\mathbb{N}$ is not a subring of $\mathbb{Z}$ though it is closed under addition and multiplication.) Provided that you know the subset is nonempty, this together with it being closed under addition will then imply that $0$... | {
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$(a^{n},b^{n})=(a,b)^{n}$ and $[a^{n},b^{n}]=[a,b]^{n}$? How to show that $$(a^{n},b^{n})=(a,b)^{n}$$ and $$[a^{n},b^{n}]=[a,b]^{n}$$ without using modular arithmetic? Seems to have very interesting applications.$$$$Try: $(a^{n},b^{n})=d\Longrightarrow d\mid a^{n}$ and $d\mid b^n$
| Show that the common divisors of $a^n$ and $b^n$ all divide $(a,b)^n$ and that any divisor of $(a,b)^n$ divides $a^n$ and $b^n$ (the proofs are pretty straight forward). It might be useful to consider prime factorization for the second direction.
Similarly, show that $[a,b]^n$ is a multiple of $a^n$ and $b^n$ but that... | {
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Evaluate the integral $\int_{0}^{+\infty}\frac{\arctan \pi x-\arctan x}{x}dx$ Compute improper integral : $\displaystyle I=\int\limits_{0}^{+\infty}\dfrac{\arctan \pi x-\arctan x}{x}dx$.
| We have
$$\int_a^b \dfrac{\arctan \pi x-\arctan x}{x}dx=\int_{\pi a}^{\pi b}\dfrac{\arctan x}{x}dx-\int_{ a}^{ b}\dfrac{\arctan x}{x}dx\\=\int_{ b}^{\pi b}\dfrac{\arctan x}{x}dx-\int_{ a}^{ \pi a}\dfrac{\arctan x}{x}dx$$
and since the function $\arctan$ is increasing so
$$\arctan( b)\log\pi=\arctan( b)\int_b^{\pi b}\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/460307",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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Distributing persons into cars We want to distribute $10$ persons into $6$ different cars knowing that each car can take three persons. How many ways have to do it. The order of the person inside the same car is not important and the car can be empty.
| If we put $i$ people in the 1-st car, there's $\binom{10}{i}$ ways to do this. Once this is done, we put $j$ people in the 2-nd car, and there's $\binom{10-i}{j}$ ways to do this. And so on, until we get to the final car, where we attempt to put in all of the unassigned passengers. If there's more than 3, we discard... | {
"language": "en",
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Question involving exponential tower of 19 Consider:
$$
y = \underbrace{19^{19^{\cdot^{\cdot^{\cdot^{19}}}}}}_{101 \text{ times}}
$$
with the tower containing a hundred $ 19$s. Take the sum of the digits of the resulting number. Again, add the digits of this new number and get the sum. Keep doing this process till you ... | $$10^0a_0+10^1a_1+10^2a_2+\ldots+10^na_n=(a_0+a_1+\ldots+a_n)+\text{a multiple of }9.$$
Therefore taking the sum of the digits of a number gives you a number that leaves the same remainder, in the division by $9$, as the one you had before (and it is also smaller as long as the original number is not $<10$). Therefore ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Folliation and non-vanishing vector field. The canonical foliation on $\mathbb{R}^k$ is its decomposition into parallel sheets $\{t\} \times \mathbb{R}^{k-1}$ (as oriented submanifolds). In general, a foliation $\mathcal{F}$ on a compact, oriented manifold $X$ is a decomposition into $1-1$ immersed oriented manifolds $... | I assume that your manifold and foliation are smooth and foliation is of codimension 1, otherwise see Jack Lee'a comment. Then pick a Riemannian metric on $X$ and at each point $x\in M$ take unit vector $u_x$ orthogonal to the leaf $F_x$ through $x$: There are two choices, but since your foliation is transversally orie... | {
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What would have been our number system if humans had more than 10 fingers? Try to solve this puzzle. Try to solve this puzzle:
The first expedition to Mars found only the ruins of a civilization.
From the artifacts and pictures, the explorers deduced that the
creatures who produced this civilization were four-legg... | The correct answer is (a) 10.
There is no comment which number system the given answers refer to. As all other numbers refer to the Martian number system, we can safely assume the answers refer to the Martian number system as well.
| {
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"url": "https://math.stackexchange.com/questions/460729",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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difference between expected values of distributions let us assume two distributions $p$ and $p'$ over the set of naturals $N$.
Is it true the following property?
$\sum_{n \in N} p(n) \cdot n \le \sum_{n \in N} p'(n) \cdot n$
IFF
for all $0 \le u \le 1$
$\sum_{n \in N} p(n) \cdot u^{n} \ge \sum_{n \in N} p'(n) \cdot u... | Call $X$ a random variable with distribution $p$ and $Y$ a random variable with distribution $p'$, then one considers the assertions:
*
*$E[X]\leqslant E[Y]$
*$E[u^X]\geqslant E[u^Y]$ for every $u$ in $[0,1]$
Assertion 2. implies assertion 1. because $X=\lim\limits_{u\to1}\frac1{1-u}\cdot(1-u^X)$ and the limit i... | {
"language": "en",
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Proof of strong Holder inequality Let $a>1$ and $f,g :\left(0,1\right) \rightarrow \left(0,\infty\right)$ measurable functions, $B$ a measurable subset of $\left(0,1\right)$ such that $$\left(\int_{C} f^2 dt\right)^{1/2} \left(\int_{C} g^2 dt\right)^{1/2} \geq a \int_{C} fg dt$$ for all $C$ measurable subset of $B$. Pr... | Yes, it's true in more general situations.
Let $\mu$ a positive measure on $X$, and $f,\, g \colon X \to (0,\,\infty)$ measurable. Let $a > 1$. Then every measurable $B$ with $\mu(B) > 0$ contains a measurable $C \subset B$ with
$$\left(\int_C f^2\,d\mu\right)^{1/2} \left(\int_C g^2\,d\mu\right)^{1/2} < a\cdot \int_C f... | {
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How to prove Disjunction Elimination rule of inference I've looked at the tableau proofs of many rules of inference (double-negation, disjunction is commutative, modus tollendo ponens, and others), and they all seem to use the so-called "or-elimination" (Disjunction Elimination) rule:
$$(P\vdash R), (Q\vdash R), (P \lo... | The rules of Disjunction Elimination and Constructive dilemma are interchangable.
You can proof Disjunction Elimination from Constructive Dilemma and
You can proof Constructive Dilemma from Disjunction Elimination.
So whichever you have you can prove the other.
Your second question is Disjunction Elimination an axiom?
... | {
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Proving a Set is NOT a vector space Before I begin, I will emphasis I DO NOT want the full solution. I just want some hints.
Show that the set $S=\{\textbf{x}\in \mathbb{R}^3: x_{1} \leq 0$ and $x_{2}\geq 0 \}$ with the usual rules for addition and multiplication by a scalar in $\mathbb{R}^3$ is NOT a vector space by ... | Absolutely! A single counterexample is all you need. Nice work.
In general elements of $S$ will not have additive inverses in $S$. (Can you determine the exceptions?) Otherwise, the axioms are satisfied.
Geometrically speaking, I recommend that you focus on the lack of additive inverses. Note that if $A$ is a set of ve... | {
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Conjugates of $12^{1/5}+54^{1/5}-144^{1/5}+648^{1/5}$ over $\mathbb{Q}$ After much manual computation I found the minimal polynomial to be $x^5+330x-4170$, although I would very much like to know if there's a clever way to see this. I suspect there is from seeing that the prime factorisations of the four integers are a... | Let $x$ be your number.
You can immediately see that $x \in \Bbb Q(12^{1/5}, 54^{1/5})$ (or even $x \in \Bbb Q(2^{1/5},3^{1/5})$, from your remark about the prime factors).
It is not too hard to show that its normal closure is $\Bbb Q(\zeta_5,12^{1/5},54^{1/5}) ( = \Bbb Q(\zeta_5,2^{1/5},3^{1/5}))$, and from there you ... | {
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Proof a $2^n$ by $2^n$ board can be filled using L shaped trominoes and 1 monomino Suppose we have an $2^n\times 2^n$ board. Prove you can use any rotation of L shaped trominoes and a monomino to fill the board completely.
You can mix different rotations in the same tililng.
| Forgive me if what is below is ‘old hat’ or easily found on some other site. I only found this site whilst solving the similar problem: for which $n$ can an $n\times n$ square be covered without overlapping by T shapes each comprising four small squares. The fact that an nxn square can be almost covered by L shapes com... | {
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Where did this statistics formula come from: $E[X^2] = \mu^2 + \sigma^2$ I am studying statistics and I need some guidance as to where this formula came from. All I know is that $\displaystyle E[X^2] = x^2 \sum_{i=0}^n p_{i}(x)$
| Edit: Thanks to the Did's comment and as a alternative answer.
You can use the following definition:
If $X$ is any random variable with distribution $F_{X}(x)$, then
$$\mu_{X}=\int \limits_{0}^{+\infty}\left({1-F_{X}(x)}\right)dx-\int\limits_{-\infty}^{0}F_{X}(x)dx,$$
$$\sigma_{X}^{2}=\int \limits_{0}^{+\infty} 2x \lef... | {
"language": "en",
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A basic question on the definition of order In the first chapter of Rudin's analysis book "order" on a set is defined as follows :
Let $S$ be a set. An order on $S$ is a relation, denoted by $<$, with the following two properties :
(i) If $x \in S$ and $y \in S$ then one and only one of the statements
$$ x <... | The root of your difficulty seems to be that people use different conventions for defining "order". The first issue arises in the defintion of "partial order", by which some people mean a reflexive, transitive, and antisymmetric relation $\leq$, while other people mean an irreflexive, transitive relation $<$. Given a ... | {
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Proving that pullback objects are unique up to isomorphism In Hungerford's Algebra he defines a pullback of morphisms $f_1 \in \hom(X_1,A)$ and $f_2 \in \hom(X_2,A)$ as a commutative diagram
$$\require{AMScd}
\begin{CD}
P @>{g_1}>>X_2\\
@V{g_2}VV @V{f_1}VV \\
X_2 @>{f_2}>> A
\end{CD}$$
satisfying the universal property... | As said by Martin Brandenburg in the comments, you stated the universal property wrong (not relevant anymore since the edit of the original post).
A pullback of $f_1\colon X_1 \to A,f_2\colon X_2 \to A$ is a diagram
$$ \require{AMScd}
\begin{CD}
P @>{g_1}>>X_1\\
@V{g_2}VV @V{f_1}VV \\
X_2 @>{f_2}>> A
\end{CD} $$
satis... | {
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Starting with $\frac{-1}{1}=\frac{1}{-1}$ and taking square root: proves $1=-1$ In this blog post, RJ. Lipton mentions an example of common mathematical traps. In particular, that ``square root is not a function''. He shows the following trap:
Start with:
$\frac{-1}{1}=\frac{1}{-1}$, then take the square root of both s... | Without going into complex analysis, I think this is the simplest way I can explain this. Let $f(x) = \sqrt{x}$. Note that the (maximal) domain of $f$ is the set of all non-negative numbers. And how is this defined? $f(x) = \sqrt{x}$ is equal to a non-negative number $y$ such that $y^2=x$. In this sense, square root is... | {
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Rudin 4.22 Theorem Could you help me understand
why 1. f(H) = B and
why 2. $\bar A$ $\cap$ B is empty and
why 3. $\bar G$ $\cap$ H is empty?
| In 2. (@Antoine's solution) the closure is understood in subspace topology.
In 3. An indirect proof is (technically) simpler.
Assume that $x\in \overline G\cap H$. Then $f(x)\in f(\overline G)\subseteq \overline A$, and $f(x)\in f(H)=B$, which is a contradiction because $\overline A\cap B=\emptyset$. In fact, $f(H)\s... | {
"language": "en",
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Injective function $f(x) = x + \sin x$ How can I prove that $f(x) = x + \sin x$ is injective function on set $x \in [0,8]$?
I think that I should show that for any $x_1, x_2 \in [0,8]$ such that $x_1 \neq x_2$ we have $f(x_1) \neq f(x_2)$ id est $x_1 + \sin x_1 \neq x_2 + \sin x_2$. But I don't know what can I do next.... | Hint: What is the sign of $f'(x)$. What does it tell you about $f(x)$?
Since $f'(x)\ge0$, the function is non-decreasing. So if $f(a)=f(b)$ for some $a<b$, then $f(x)$ would have to be constant on the interval $[a,b]$. This would imply $f'(x)=0$ for each $x\in[a,b]$. But there is no non-trivial interval such that $f'... | {
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How many values of $n<50$ exist that satisfy$ (n-1)! \ne kn$ where n,k are natural numbers? How many natural numbers less than 50 exist that satisfy $ (n-1)! \ne kn$ where n,k are natural numbers and $n \lt 50$ ?
when n=1
$0!=1*1$
when n=2
$1!\ne2*1$
...
...
...
when n=49
$48!=\frac{48!}{49}*49$
Here $k = \frac{48!}{4... | If you're only interested in the case of up to n = 49 then the answer isn't very enlightening... the answer is 16.
As the hints above suggest, you need to look at what happens when n is a prime, although you also need to take care in this case with n = 4 (can you see why?).
| {
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How to solve the Riccati's differential equation I found this question in a differential equation textbook as a question
The equation
$$
\frac{dy}{dx} =A(x)y^2 + B(x)y +C(x)
$$
is called Riccati's equation
show that if $f$ is any solution of the equation, then the transformation
$$
... | The question is wrongly posed, which is probably why you don't comprehend it. Here's the correct one:
Let $y$ and $f$ be solutions to the above diff. equation such that $y=f+1/v$ for some function $v(x)$. Show that $v$ satisfies a linear diff. equation.
The solution is provided by Amzoti.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is the MLE strongly consistent and asymptotically efficient for exponential families? It is known that the Maximum Likelihood Estimator (MLE) is strongly consistent and asymptotically efficient under certain regularity conditions. By strongly consistent I mean that $\hat{\theta}_{MLE} \rightarrow \theta$ almost surely.... | In a paper I read a while ago, which can be found here, the authors propose a set of conditions of eigenvalues which are weaker than the usual regularity conditions. They show that under this set of conditions, the usual MLE are strongly consistent for (nonlinear) exponential families.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why does factoring eliminate a hole in the limit? $$\lim _{x\rightarrow 5}\frac{x^2-25}{x-5} = \lim_{x\rightarrow 5} (x+5)$$
I understand that to evaluate a limit that has a zero ("hole") in the denominator we have to factor and cancel terms, and that the original limit is equal to the new and simplified limit. I unde... | One of definitions of $\lim_{x \to A} f(x) = B$ is:
$$\forall_{\varepsilon > 0}\exists_{\delta > 0}\forall_{0 < \left|x - A\right| < \delta}\left|f(x) - B\right| < \varepsilon$$
The intuition is that we can achieve arbitrary 'precision' (put in bounds on y axis) provided we get close enough (so we get the bounds on x a... | {
"language": "en",
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Show that $f:I\to\mathbb{R}^{n^2}$ defined by $f(t)=X(t)^k$ is differentiable Let $I$ be a interval, $\mathbb{R}^{n^2}$ be the set of all $n\times n$ matrices and $X:I \to\mathbb{R}^{n^2}$ be a differentiable function. Given $k\in\mathbb{N}$, define $f:I\to\mathbb{R}^{n^2}$ by $f(t)=X(t)^k$. How to prove that $f$ is di... | Hint: The entries of $X(t)$ are obviously differentiable, and the entries of $f(t)$ are polynomials in the entries of $X(t)$.
| {
"language": "en",
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"source": "stackexchange",
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Possible values of difference of 2 primes Is it true that for any even number $2k$, there exists primes $p, q$ such that $p-q = 2k$?
Polignac's conjecture talks about having infinitely many consecutive primes whose difference is $2k$. This has not been proven or disproven.
This is a more general version of my question ... | In short, this is an open problem (makes Polignac seem really tough then, doesn't it?).
The sequence A02483 of the OEIS tracks this in terms of $a(n) =$ the least $p$ such that $p + 2n = q$. On the page it mentions that this is merely conjectured.
In terms of a positive result, I am almost certain that Chen's theorem s... | {
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What is wrong in saying that the map $F:\mathbb{S}^1 \times I \to \mathbb{S}^1$ defined by $F(z,t)=z^{t+1}$ is a homotopy from $f$ to $g$? Let $\mathbb{S}^1$ be the unit circle of complex plane and $f,g:\mathbb{S}^1 \to \mathbb{S}^1$ be two maps defined by $f(z)=z$ and $g(z)=z^2$. What is wrong in saying that the map $... | Raising a complex number to a non-integer power is more complicated than you're realizing. The "function" $z^{t}$ is really multivalued when $t\notin\mathbb{Z}$, and even after choosing a branch, it won't be continuous.
| {
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Infinite Limit Problems Can some one help me to solve the problems?
*
*$$\lim_{n\to\infty}\sum_{k=1}^n\left|e^{2\pi ik/n}-e^{2\pi i(k-1)/n}\right|$$
*$$ \lim_{x\to\infty}\left(\frac{3x-1}{3x+1}\right)^{4x}$$
*$$\lim_{n\to\infty}\left(1-\frac1{n^2}\right)^n $$
(Original scan of problems at http://i.stack.imgur.com/... | For the first one, consider a geometric interpretation. Recall that when $a$ and $b$ are complex numbers, $|a-b|$ is the distance in the plane between the points $a$ and $b$. Peter Tamaroff says in a comment that the limit is $2\pi$, which I believe is correct.
Addendum: The points $e^{2\pi ik/n}$ are spaced evenly ar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/462529",
"timestamp": "2023-03-29T00:00:00",
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How to approximate unknown two-dimensional function? I have a surface defined by values on a two-dimensional grid. I would like to find an approximate function which would give me a value for any arbitrary point within certain range of xs and ys.
My general idea is to construct some sort of polynomial, and then tweak i... | You might be interested in Lagrange interpolation (link only talks about one-variable version). Just input 100 points from your values and you'll probably get something good. You may need more points for it to be accurate at locations of high variance. If you insist on using an evolutionary algorithm, you might use it ... | {
"language": "en",
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Finding a Hopf Bifucation with eigenvalues I am trying to show that the following 2D system has a Hopf bifurcation at $\lambda=0$:
\begin{align}
x' =& y + \lamb... | Step 1: As jkn stated, the first step is to find the equilibrium such that $\dot x=\dot y=0$. It is easy to obtain the equilibrium is $(0, 0)$.
Step 2: Compute the eigenvalue around the equilibrium $(0, 0)$.
$$J= \left(\begin{array}{cc}
\lambda &1\\
... | {
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"timestamp": "2023-03-29T00:00:00",
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Mathematical Games suitable for undergraduates I am looking for mathematical games for an undergraduate `maths club' for interested students. I am thinking of things like topological tic-tac-toe and singularity chess. I have some funding for this so it would be nice if some of these games were available to purchase, al... | Ticket to Ride is a very easy game to learn and can lead to some interesting discussions of graph theory. On a more bitter note, a game of Settlers of Catan never fails to provide a wonderful example of the difference between theoretical and empirical probability.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/462723",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 7,
"answer_id": 3
} |
Finding all bifurcations in a 2D system I want to find all bifurcations for the system:
\begin{align}
x' =& -x+y\\ ... | Notice that x is real only when the discriminant 1-4λ^2 > 0. I.e. The curves of x' and y' do not intersect at points when 1-4λ^2 > 0 does not hold.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/462805",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Using the FTOC to find the derivative of the integral. I'm apologizing ahead of time because I don't know how to format an integral.
If I have the following integral: $$\int_{x^2}^5 (4x+2)\;dx.$$
I need to find the derivative, so could I do the following?
Multiply the integral by -1 and swap the limits of integration s... | You have
$$
f(x) = \int_{x^2}^5 4t + 2 \; dt
$$
and you want to find $f'(x)$.
First note that
$$
f(x) = -\int_5^{g(x)} 4t + 2\; dt
$$
where $g(x) = x^2$. So $f(x)$ is the composition of two functions:
$$
f(x) = h(g(x)).
$$
where
$$\begin{align}
h(x) &= -\int_5^x 4x + 2\; dt \quad \text{and}\\
g(x) &= x^2.
\end{align}
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/462871",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Does the graph of $y + |y| = x + |x|$ represent a function of $x$? The question is whether or not the graph $y + |y| = x + |x|$ represents a function of $x$. Explain why.
It looks like a weird graph but it would probably be a function because if you say $f(x) = y$ (you get a $y$ value)?
| It is a function only if for every $x$ you only have one $y$ value satisfying the equation.
Now look for example the value $x=-1$ and $y=-1$ or $x=-1$ and $y=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/462933",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Show that if $a$ has order $3\bmod p$ then $a+1$ has order $6\bmod p$. Show that if $a$ has order $3\bmod p$ then $a+1$ has order $6\bmod p$. I know I am supposed to use primitive roots but I think this is where I am getting caught up. The definition of primitive root is "if $a$ is a least residue and the order of $a\... | Note that if $a=1$, $a$ has order $1$. Thus, we can assume $a\ne1$. Furthermore, $p\ne2$ since no element mod $2$ has order $3$. Therefore, $-1\ne1\pmod{p}$.
$$
\begin{align}
(a+1)^3
&=a^3+3a^2+3a+1\\
&=1+3a^2+3a+1\\
&=-1+3(1+a+a^2)\\
&=-1+3\frac{a^3-1}{a-1}\\
&=-1
\end{align}
$$
Therefore, $(a+1)^3=-1$ and $(a+1)^6=1$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/463010",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Using Van-Kampen in $S^1\times S^1$ as cylinder I was trying to use Van-Kampen theorem to compute $S^1 \times S^1$ using its representation as a cylinder.
$U$= from a little below the middle circle to the upper part of the cylinder.
$V$= from a little above the middle circle to the bottom part of the cylinder.
Then $U... | The product $S^1\times S^1$ is not a cylinder. It's a torus.
So what you've done is a correct computation of the fundamental group, but of a wrong space.
In general if you want to compute the fundamental group of a product of spaces, you don't need van Kampen. The group $\pi_1(A\times B)$ is just going to be the produc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/463054",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Inverse of a function $e^x + 2e^{2x}$ The function is $f(x) = e^x+2e^{2x}$
So to find the inverse I went
$$WTS: x = ____$$
$$y = e^x+2e^{2x}$$
$$ log3(y)=log3(3e^{2x})$$
$$ log3(y) = 2x$$
$$ log3(y)=5x$$
$$ x=\frac{log3(y)}{2}$$
Am i correct?
| No, you are wrong, $log$ is not a linear function, set $g(x)=e^x$, $2g(x)^2+g(x)-y=0$，
solving this equation, since $y>0$, $g(x)=e^x=-{1\over4}+{1\over 2}\sqrt{{1\over4}+2y}.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/463110",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Prove that $\sum\limits_{i=1}^{n-k} (-1)^{i+1} \cdot \frac{(k-1+i)!}{k! \cdot(i-1)!} \cdot \frac{n!}{(k+i)! \cdot (n-k-i)!}=1$
Prove that, for $n>k$, $$\sum\limits_{i=1}^{n-k} (-1)^{i+1} \cdot \frac{(k-1+i)!}{k! \cdot(i-1)!} \cdot \frac{n!}{(k+i)! \cdot (n-k-i)!}=1$$
I found this problem in a book at the library of m... | Note first that two factorials nearly cancel out, leaving us with $\frac1{k+i}=\int_0^1t^{k+i-1}\mathrm dt$, hence the sum to be computed is
$$
S=\sum_{i=1}^{n-k}(-1)^{i+1}n{n-1\choose k}{n-k-1\choose i-1}\int_0^1t^{k+i-1}\mathrm dt
$$
Second, note that
$$
\sum_{i=1}^{n-k}(-1)^{i+1}{n-k-1\choose i-1}t^{k+i-1}=t^k\sum_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/463197",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
Estimate $\displaystyle\int_0^\infty\frac{t^n}{n!}e^{-e^t}dt$ accurately. How can I obtain good asymptotics for $$\gamma_n=\displaystyle\int_0^\infty\frac{t^n}{n!}e^{-e^t}dt\text{ ? }$$
[This has been already done] In particular, I would like to obtain asymptotics that show $$\sum_{n\geqslant 0}\gamma_nz^n$$
converges ... | $$
\begin{align}
\sum_{n=1}^\infty\gamma_nz^n
&=\sum_{n=1}^\infty\int_0^\infty\frac{t^nz^n}{n!}e^{-e^t}\,\mathrm{d}t\\
&=\int_0^\infty e^{tz}e^{-e^t}\,\mathrm{d}t\\
&=\int_0^\infty e^{t(z-1)}e^{-e^t}\,\mathrm{d}e^t\\
&=\int_{1}^\infty u^{z-1}e^{-u}\,\mathrm{d}u\\
&=\Gamma(z,1)
\end{align}
$$
The Upper Incomplete Gamma ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/463272",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 2
} |
Numerical inversion of characteristic functions I have a need to use the FFT in my work and am trying to learn how to use it. I am beginning by attempting to use the FFT to numerically invert the characteristic function of a normal distribution. So I have discretised the integral using the trapezoidal rule (I know, thi... | Please have a look the section on the numerical inversion of characteristic functions in the dissertation found here:
http://wiredspace.wits.ac.za//handle/10539/9273
It maybe helpful.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/463317",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
What's the difference between theorem, lemma and corollary? Can anybody explain me what is the basic difference between theorem, lemma and corollary?
We have been using it for a long time but I never paid any attention. I am just curious to know.
| Terence Tao (Analysis I, p. 25, n. 4):
From a logical point of view, there is no difference between a lemma, proposition,
theorem, or corollary - they are all claims waiting to be proved. However, we use
these terms to suggest different levels of importance and difficulty.
A lemma is an easily proved claim which i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/463362",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "118",
"answer_count": 10,
"answer_id": 6
} |
how did he conclude that?integral So the question is : Find all continuous functions such that $\displaystyle \int_{0}^{x} f(t) \, dt= ((f(x)^2)+C$.
Now in the solution, it starts with this, clearly $f^2$ is differentiable at every point ( its derivative is $f$). So $f(x)\ne0$? I have no idea how he concluded that, th... | I interpret the problem as follows:
Find all continuous functions $f:\ \Omega\to{\mathbb R}$ defined in some open interval $\Omega$ containing the origin and satisfying the integral equation
$$\int_0^x f(t)\ dt=f^2(x)+ C\qquad(x\in\Omega)$$
for a suitable constant $C$.
Assume $f$ is such a function and that $f(x)\ne 0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/463446",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Coin chosen is two headed coin in this probability question I have a probability question that reads:
Question:
A box has three coins. One has two heads, another two tails and the last is a fair coin. A coin is chosen at random, and comes up head. What is the probability that the coin chosen is a two headed coin.
My... | For such a small number of options its easy to count them
The possible outcomes are:
heads or heads using the double head coin
tails or tails using the double tail coin
heads or tails using the fair coin
All these outcomes are equally likely. How many of these are heads and of those how many use the double headed coi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/463517",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 9,
"answer_id": 0
} |
Zeroes about entire function Given an entire function $f(z)$ which satisfies $|f(z)|=1, \forall z \in \mathbb{R}$, the problem asks to show that there exists an entire function $g(z)$ such that $f(z)=\exp(g(z))$.
The only thing need to show is that $f(z)$ admits no zeros on $\mathbb{C}$ so that we can define $g(z)$ by ... | Consider the function $h(z) = \overline{f(\overline{z})}$. That is an entire function too, and hence so is $k(z) = f(z)\cdot h(z)$. On the real line, you have
$$k(x) = f(x)\cdot h(x) = f(x) \overline{f(x)} = \lvert f(x)\rvert^2 = 1,$$
hence $k \equiv 1$. That guarantees that $f$ has no zeros.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/463664",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
What is the product of this by telescopic method? $$\prod_{k=0}^{\infty} \biggl(1+ {\frac{1}{2^{2^k}}}\biggr)$$
My teacher gave me this question and said that this is easy only if it strikes the minute you read it. But I'm still thinking. Help!
P.S. This question is to be attempted by telescopic method.
| The terms of the product are $(1+1/2)(1+1/4)(1+1/16)(1+1/256)\cdots$ with each denominator being the square of the previous denominator.
Now if you multiply the product with $(1-1/2)$ you see telescoping action:
$(1-1/2)(1+1/2)=1-1/4$
$(1-1/4)(1+1/4)=1-1/16$
$(1-1/16)(1+1/16)=1-1/256$
Do you see the pattern developing?... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/463758",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 4,
"answer_id": 0
} |
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