Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Calculate the derivate $dx/dy$ using $\int_0^x \sqrt{6+5\cos t} \, dt + \int_0^y \sin t^2 \, dt = 0$ I want to calculate $\frac{dx}{dy}$ using the equation below.
$$\int_0^x \sqrt{6+5\cos t}\;dt + \int_0^y \sin t^2\;dt = 0$$
I don't even know from where to start. Well I think that I could first find the integrals and t... | HINT: You have $$f(x)=\int_0^x\sqrt{6+5\cos t}\,dt=-\int_0^y\sin t^2 \,dt=g(y)\;.$$ What are $\dfrac{df}{dx}$ and $\dfrac{dg}{dy}$ according to the fundamental theorem? And when you have $\dfrac{df}{dx}$, what can you multiply it by to get $\dfrac{df}{dy}$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/101431",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Conditional probability. Targeting events
Electric motors coming off two assembly lines are pooled for storage in a common stockroom, and the room contains an equal number of motors from each line. Motors are periodically sampled from that room and tested. It is known that 10% of the motors from line I are defective a... | P(B) is not P(came from line 1) in this problem. You are being asked to calculate P(came from line 1 | is defective) so B is "is defective" and A is "came from line 1". You're right that P(AB) is "Came from line 1 and is defective", and if you know how to calculate P(B) correctly in this case then you're essentially ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Convergence of $\sum_{n=1}^\infty\frac{1}{2\cdot n}$ It is possible to deduce the value of the following (in my opinion) converging infinite series? If yes, then what is it?
$$\sum_{n=1}^\infty\frac{1}{2\cdot n}$$
where n is an integer. Sorry if the notation is a bit off, I hope youse get the idea.
| The series is not convergent, since it is half of the harmonic series which is known to be divergent$^1$.
$$\sum_{n=1}^{\infty }\frac{1}{2n}=\frac{1}{2}\sum_{n=1}^{\infty }\frac{1}{n}.$$
--
$^1$ The sum of the following $k$ terms is greater or equal to $\frac{1}{2}$
$$\frac{1}{k+1}+\frac{1}{k+2}+\ldots +\frac{1}{2k-1}+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/101618",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Equation of straight line I know,
$Ax + By = C$
is the equation of straight line but a different resource says that:
$y = mx + b$
is also an equation of straight line? Are they both same?
| Yes. That is, they both give the equation of a straight line and the equation of any non-vertical line can be written in either form.
If $B\ne 0$. Then you can write $Ax+By=C$ as
$$
By=-Ax+C
$$
and, since $B\ne0$, the above can be written
$$
y=-\textstyle{A\over B}x +{C\over B}.
$$
If $B=0$, the equation is $Ax=C$, w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/101681",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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When is $a^k \pmod{m}$ a periodic sequence? Let $a$ and $m$ be a positive integers with $a < m$. Suppose that $p$ and $q$ are prime divisors of $m$. Suppose that $a$ is divisible by $p$ but not $q$.
Is there necessarily an integer $k>1$ such that $a^k \equiv a \pmod{m}$?
Or is it that the best we can do is say th... | A nice presentation of such semigroup generalizations of the Euler-Fermat theorem and related number theory is the following freely available paper
S. Schwarz, The role of semigroups in the elementary theory of numbers, Math. Slovaca, Vol. 31 (1981) pp. 369–395.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/101755",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Finding a third point I have learnt that if we are given 3 points in the extended complex plane and their corresponding image points, we have a unique Möbius map that can perform the mapping. Suppose I have 2 orthogonally intersecting circles and I want to map them (individually) to the real and imaginary axes respecti... | Let one of the intersections be $p$. The inversion $z \to 1/(z-p)$ takes $p$ to $\infty$ and takes your two circles to straight lines intersecting orthogonally. Now just translate and rotate.
| {
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"url": "https://math.stackexchange.com/questions/101821",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Numerical analysis textbooks and floating point numbers What are some recommended numerical analysis books on floating point numbers? I'd like the book to have the following
*
*In depth coverage on the representation of floating point numbers on modern hardware (the IEEE standard).
*How to do arbitrary precision fl... | You could try the book written by J.M. Muller, N. Brisebarre:
*
*Handbook of Floating Point Arithmetic (amazon.com)
The literature of numerical mathematics concentrates on algorithms for mathematical problems, not on implementation issues of arithmetic operations.
How to compute the closest 32-bit floating point re... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Easiest way to perform Euclid's division algorithm for polynomials Let's say I have the two polynomials $f(x) = x^3 + x + 1$ and $g(x) = x^2 + x$ over $\operatorname{GF}(2)$ and want to perform a polynomial division in $\operatorname{GF}(2)$.
What's the easiest and most bullet proof way to find the quotient $q(x) = x ... | Polynomial long division is the way to go. Especially over a finite field where you don't have to worry about fractional coefficients (working over for instance the rational numbers these can get extremely unwieldy surprisingly soon). Over $\mathbb Z/2\mathbb Z$ you don't even have to worry about dividing coefficients ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Formulas for counting pairs in a set I have a few questions regarding Cartesian products that will help me optimize a complicated SQL query I'm working on.
Suppose I have 52 playing cards, and I want to know how many combinations of pairs (first two cards) a dealer can draw at the beginning. Obviously, this would be l... | The concepts you are looking for are known as "permutations" and "combinations."
*
*If you have $n$ items, and you want to count how many ordered $r$-tuples you can make without repetitions, the answer is someimtes written $P^n_r$, and:
$$P^{n}_{r} = n(n-1)(n-2)\cdots (n-r+1).$$
This follows from the "multiplication... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to prove that for $A\cap B\neq\varnothing$, $(\bigcap A)\cap(\bigcap B)\subseteq\bigcap(A\cap B)$? $A$ and $B$ are non empty sets with non empty intersection.
Prove that
$(\bigcap A)\cap(\bigcap B) \subseteq \bigcap (A\cap B).$
The definition of intersection of a set is something like this, if $M$ is a nonempty set... | For theorems like these, as Asaf wrote, expanding definitions and simplifying is the way to go. However, I do these kind of things more 'calculationally' using the rules of predicate logic.
In this case, we can easily calculate the elements $\;x\;$ of the left hand side:
\begin{align}
& x \in \bigcap A \;\cap\; \bigca... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/102071",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Tensoring with vector bundle is a dense endofunctor of $D^b(\text{coh }X) $? A functor $F:T\to R$ between triangulated categories is dense if every object of $R$ is isomorphic to a direct summand in the image of $F$.
Let $R=T=D^b(\text{coh }X)$ for a variety $X$ and consider the functor $-\otimes \mathcal{V}$, $\mathca... | What part of the claim you don't understand?
For any vector bundle $V$ the bundle $V\otimes V^\vee$ contains trivial 1-dimensional vector bundle (spanned by the section "Id"$\in V\otimes V^\vee$; the map in the opposite direction is the evaluation map). So any object $P\in D^b$ is a summand of the image of the object $... | {
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How can one prove that the cube root of 9 is irrational? Of course, if you plug the cube root of 9 into a calculator, you get an endless stream of digits. However, how does one prove this on paper?
| This is essentially the same proof I gave in my answer here.
Suppose $9^{\frac{1}{3}}$ is rational. Then $3^2n^3 = m^3$ for some natural numbers $n$ and $m$. On left side of the equation, the power of $3$ is of the form $3k + 2$ and on the right side it is of the form $3l$. This is a contradiction, because each integer... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find $DF$ in a triangle $DEF$ Consider we have a triangle $ABC$ where there are three points $D$, $E$ & $F$ such as point $D$ lies on the segment $AE$, point $E$ lies on $BF$, point $F$ lies on $CD$. We also know that center of a circle over ABC is also a center of a circle inside $DEF$. $DFE$ angle is $90^\circ$, $DE/... | Here is a diagram. I may or may not post a solution later.
Edit I will not post a solution since it appears to be quite messy. Please direct votes towards an actual solution.
| {
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"timestamp": "2023-03-29T00:00:00",
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Simple expressions for $\sum_{k=0}^n\cos(k\theta)$ and $\sum_{k=1}^n\sin(k\theta)$?
Possible Duplicate:
How can we sum up $\sin$ and $\cos$ series when the angles are in A.P?
I'm curious if there is a simple expression for
$$
1+\cos\theta+\cos 2\theta+\cdots+\cos n\theta
$$
and
$$
\sin\theta+\sin 2\theta+\cdots+\si... | Take the expression you have and multiply the numerator and denominator by $1-\bar{z}$, and using $z\bar z=1$:
$$\frac{1-z^{n+1}}{1-z} = \frac{1-z^{n+1}-\bar{z}+z^n}{2-(z+\bar z)}$$
But $z+\bar{z}=2\cos \theta$, so the real part of this expression is the real part of the numerator divided by $2-2\cos \theta$. But the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/102477",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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injection from double dual to finite-dimensional vector space (Note: I'm using the word "natural" to mean "without the need to choose a basis." I'm aware that there is a precise category-theoretic meaning of this word, but I don't have great intuition for it yet and am hoping, perhaps naively, it's not necessary to und... | For the sake of having an answer: no. Any good definition of "natural" would imply that this map also existed for infinite-dimensional vector spaces, which it doesn't. You shouldn't be able to do any better than "the inverse, when it exists, of the natural map $V \to (V^{\ast})^{\ast}$."
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/102540",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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100 Soldiers riddle One of my friends found this riddle.
There are 100 soldiers. 85 lose a left leg, 80 lose a right leg, 75
lose a left arm, 70 lose a right arm. What is the minimum number of
soldiers losing all 4 limbs?
We can't seem to agree on a way to approach this.
Right off the bat I said that:
85 lost a left ... | You can easily do it visually with a Venn diagram with the four sets of soliders with each limb. For mimimum number of soliders losing all four limbs, none of the inner sets overlap. So $100 - (15+20+25+30) = 10$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/102598",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "84",
"answer_count": 9,
"answer_id": 5
} |
Help to find the domain to this function $$
\sqrt{\log_\frac{1}{2}\left(\arctan\left(\frac{x-\pi}{x-4}\right)\right)}
$$
Please, could someone show me the steps to find the domain of this function?
It's the sixth time that I try to solve it, and I'm going to burn everything...
| I assume that you are talking about the so-called "natural domain" of a real valued function of real variable (a common concept in Calculus, at least in the U.S.): given a formula, such as the above, and no words about its domain, we assume the domain is to be taken to be a subset of the real numbers, and that this sub... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why do all circles passing through $a$ and $1/\bar{a}$ meet $|z|=1$ are right angles? In the complex plane, I write the equation for a circle centered at $z$ by $|z-x|=r$, so $(z-x)(\bar{z}-\bar{x})=r^2$. I suppose that both $a$ and $1/\bar{a}$ lie on this circle, so I get the equation
$$
(z-a)(\bar{z}-\bar{a})=(z-1/\... | I have a solution that relies on converting the complex numbers into ordered pairs although I believe there must be a solution with just the help of complex numbers.
Two circles intersect orthogonally, if their radii are perpendicular at the point of intersection. So, using this we can have a condition for orthogonalit... | {
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How can you find the number of sides on this polygon? I'm currently studying for the SAT. I've found a question that I can't seem to figure out.
I'm sure there is some logical postulate or assumption that is supposed to be made here. Here is the exact problem:
I don't really care for an answer, I would rather know s... | Hints:
The sum of the measures of the interior angles of an $n$-sided convex polygon is $(n-2)*180^\circ$. So, if the polygon is regular, each interior angle has measure $180^\circ-{360^\circ\over n}$. (You could also use the fact that the sum of the exterior angles of a convex polygon is $360^\circ$. An "exterior an... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 2
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Is this group a semidirect product? $G=\langle x,y,z:xy=yx,zxz^{-1}=x^{-1},zyz^{-1}=y^{-1}\rangle$, could you help me to understand if this group is a semidirect product of the type $\langle x,y\rangle\rtimes_\varphi\langle z\rangle$.
I was trying to prove that $\langle x,y\rangle\triangleleft G$ and $\langle x,y\rangl... | Showing that the intersection of two subgroups is trivial in a group described by generators and relations is a little tricky.
Clearly, it is enough to show that if $i,j,k$ are integers and $x^i y^j z^k = 1$, then $i=j=k=0$. This is of course equivalent to showing that if $i,j,k$ are not all zero, then $x^i y^j z^k \n... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is there a first-order-logic for calculus? I just finished a course in mathematical logic where the main theme was first-order-logic and little bit of second-order-logic. Now my question is, if we define calculus as the theory of the field of the real numbers (is it?) is there a (second- or) first-order-logic for calcu... | I take the view that the proper logical framework in which to do model theory for structures in analysis is continuous logic. For more information on the subject, look up the webpage of Ward Henson.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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"answer_id": 2
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Perfect squares always one root? I had an exam today and I was thinking about this task now, after the exam of course.
$f(x)=a(x-b)^2 +c$
Now, the point was to find C so that the function only has one root. Easy enough, I played with the calculator and found this. But I hate explanations like that, yes. You get a fe... | An alternative way to think about it is geometrically. The graph of $y=x^2$ is a parabola that opens up with vertex at the origin. The graph of
$$y = (x-b)^2$$
is then a horizontal shift by $b$ units (so $b$ units to the right if $b\geq 0$, and $|b|$ units to the left if $b\lt 0$) of the same graph. There is still only... | {
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How to test any 2 line segments (3D) are collinear or not? if we have two line segments in 3D, what would be the way to test whether these two lines are collinear or not? (I fogot to mentioned that my line segments are 3D. So, I edited the original post. Sorry for the inconveniences)
I wish to check the direction of th... | If the two line segments $AB$ and $CD$ are given by 4 distinct points A, B, C and D, it is also sufficient that both $AB \parallel CD$, $AC \parallel BD$ and $AD\parallel BC$.
To see if $A(a_1,a_2)B(b_1,b_2) \parallel C(c_1,c_2)D(d_1,d_2)$, you test whether or not $\vec{BA} = B-A $ and $\vec{DC} = C-D$ are linearly dep... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
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is there a connection between the following? Assume $A$ is $m \times n$ and $B$ is $m \times n$.
Is there a connection between the eigenvalues of $AB'$ and the eigenvalues of $B'A$?
One is an $m \times m$ and the other is $n \times n$.
($B'$ stands for the transpose of $B$)
| It seems easier for me to assume that $B$ is an $n \times m$ matrix. In that case, a classical argument shows that $AB$ and $BA$ have the same nonzero eigenvalues, not counting multiplicity. The case that these eigenvalues are distinct is dense in the general case, so $AB$ and $BA$ have the same nonzero eigenvalues cou... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Nomenclature of random variables $\{X=0, Y=0\}$ same as $\{X=0\}\cap \{Y=0\}$? just a small doubt.
My exercises keep oscillating their nomenclature on this small detail and I always have the other version.
Let $X,Y$ be random variables.
Is $\{X=0, Y=0\}$ the same as $\{X=0\}\cap \{Y=0\}$?
Another example.
Let $N$ be t... | $\{X=0,Y=0\}$ and $\{X=0\}\cap\{Y=0\}$ are the same thing. Both notations refer to
$$
\{\omega\in\Omega : X(\omega)=0\ \ \&\ \ Y(\omega)=0\} = \{\omega\in\Omega : X(\omega)=0\}\cap\{\omega\in\Omega : Y(\omega)=0\}.
$$
Your notation saying
$$
\begin{align}
X_n(w) := w_n = \{ & 0:=\text{user downloads no file}, \\... | {
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Does $\int_0^1 \sum_{n=0}^{\infty}x e^{-nx}\;dx = \sum_{n=0}^{\infty}\int_0^1 x e^{-nx}\;dx$? Does $$\int_0^1 \sum_{n=0}^\infty x e^{-nx}\;dx = \sum_{n=0}^\infty \int_0^1 x e^{-nx}dx$$ ?
This exercise leaves me stumped. On the one hand, it seems the series $\sum_{n=0}^\infty xe^{-nx}$ is not uniformly convergent in $[0... | You can use Fubini's theorem, but it seems overkill. Note that for all integer $N$ we have
$$\sum_{n=0}^N\int_0^1xe^{-nx}dx=\int_0^1\sum_{n=0}^Nxe^{-nx}dx\leq \int_0^1\sum_{n=0}^{+\infty}xe^{-nx}dx,$$
so
$$\sum_{n=0}^{+\infty}\int_0^1xe^{-nx}dx\leq \int_0^1\sum_{n=0}^{+\infty}xe^{-nx}dx.$$
For the reversed inequality... | {
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Conditional Probability Question Bowl A contains 6 red chips and 4 blue chips. Five chips are randomly chosen and transferred without replacement to Bowl B. One chip is drawn at random from Bowl B. Given that this chip is blue, find the conditional probability that 2 red chips and 3 blue chips are transferred from bowl... | There are $\frac{10!}{6!4!}$ (= 210) possible arrangements for the chips, and $\frac{5!}{2!3!}$ arrangements for the chips desired in bowl B. Any given arrangement of bowl B can occur for every corresponding arrangement in bowl A (also $\frac{5!}{2!3!}$ combinations)
The total number of possiblilities with the correct ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is the following derivative defined? I am new to this site so I am not sure if this is the right place to be asking this question but I will try anyway. I am reading an economics paper for my research and the author does the following:
$$\frac{\partial}{\partial C_t(j)} \int_0^1 P_t(j) C_t(j) dj = P_t(j)$$
I feel that ... | That derivative can be properly defined if and only if
there exists an appropriate pair of values $t$ and $j$ such that $\: C_t(j) = 0 \:$
and
all appropriate values of $t$ such that
[there exists an appropriate value of $j$ such that $\: C_t(j) = 0 \:$]
give the same value of the integral
and
$0$ is a limit point o... | {
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"timestamp": "2023-03-29T00:00:00",
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Circle and Line segment intersection I have a line segment (begin $(x_1,y_1)$, end $(x_2,y_2)$, with $D=5$, let’s say) and a circle (radius $R$, center $(x_3,y_3)$)
How can I check that if my line segment intersects my circle?
picture
http://kepfeltoltes.hu/120129/inter_www.kepfeltoltes.hu_.png
| The points $(x,y)$ on the line segment that joins $(x_1,y_1)$ and $(x_2,y_2)$ can be represented parametrically by
$$x=tx_1+(1-t)x_2, \qquad y=ty_1+(1-t)y_2,$$
where $0\le t\le 1$.
Substitute in the equation of the circle, solve the resulting quadratic for $t$. If $0\le t\le 1$ we have an intersection point, otherwise ... | {
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"url": "https://math.stackexchange.com/questions/103556",
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} |
Counterexample of G-Set
If every element of a $G$-set is left fixed by the same element $g$ of $G$,
then $g$ must be the identity $e$.
I believe this to be true, but the answers say that it's false. Can anyone provide a counter-example? Thanks!
| For concreteness, let $G$ be the group of isometries of the plane, and let $g$ be reflection in the $x$-axis. Let $S$ be the $x$-axis. Then $g(v)=v$ for every point $v$ in $S$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Every manifold is locally compact?
Theorem. Every Manifold is locally compact.
This is a problem in Spivak's Differential Geometry.
However, don't know how to prove it. It gives no hints and I don't know if there is so stupidly easy way or it's really complex.
I good example is the fact that Heine Borel Theorem, I w... | I do not think the above answers are completely right, since the "Hausdorff" condition in the definition of topological manifolds must be needed.
The key is to prove the following: If $V\subset U\subset X$, and X is Hausdorff, $\bar{V}_{U}$ is compact. Then $\bar{V}_{U}=\bar{V}$.
Proof: By definition, we only need to s... | {
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"source": "stackexchange",
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Proof that $\mathbb{Q}$ is dense in $\mathbb{R}$ I'm looking at a proof that $\mathbb{Q}$ is dense in $\mathbb{R}$, using only the Archimedean Property of $\mathbb{R}$ and basic properties of ordered fields.
One step asserts that for any $n \in \mathbb{N}$, $x \in \mathbb{R}$, there is an integer $m$ such that $m - 1 \... | Assume first that $x>0$, so that $nx>0$. By the Archimedean property there is a $k\in\mathbb{N}$ such that $k>nx$; let $m$ be the least such $k$. Clearly $m-1\le nx<m$. If $x=0$, just take $m=1$. Finally, if $x<0$, then $-nx>0$, so by the first part of the argument there is an integer $k$ such that $k-1\le -nx<k$, and ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If a coin is flipped 10 times, what is the probability that it will land heads-up at least 8 times? I absolutely remember learning this is middle school, yet I cannot remember how to solve it for the life of me. Something to do with nCr, maybe? ...
Thanks for any help.
| What we'd like to do is find a way to set the problem up in some way that we know how to solve it.
$P($At least $8$ heads) = $P(X \geq 8)$ where $X$ is the Random Variable associated with the number of heads attained.
Well, since $X$ can only have the values $0$ through $10$, perhaps we should split $P$ up:
$P(X \geq 8... | {
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Finding points in a grid with exactly k paths to them? Suppose that we begin at (0, 0) and are allowed to take two types of steps - steps one unit up and steps one unit to the right. For example, a legal path might be (0, 0) → (1, 0) → (2, 0) → (2, 1) → (3, 1).
Now, suppose that you are given a number k. Is there an ... | This sounds to me like a combinatorial problem.
Say you start in (x, y) and want to go to (x+3, y+3). If we represent all "up" movements by 'U' and all "right" movements by 'R', such a path could be UUURRR. The total number of possible paths would be all possible permutations of UUURRR, namely 6!/(3!3!) = 20.
An algori... | {
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Pythagorean Theorem Proof Without Words 6 Your punishment for awarding me a "Nice Question" badge for my last question is that I'm going to post another one from Proofs without Words.
How does the attached figure prove the Pythagorean theorem?
P.S. No, I will not go through the entire book page-by-page asking for help... | If you have a diameter of a circle and a point on the circle, the length of the altitude from that point to the diameter is the geometric mean of the lengths of the two parts of the diameter. Perhaps drawing in the two chords from the top point on the circle to the endpoints of the diameter and looking for similar tri... | {
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"source": "stackexchange",
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Groups of symmetries What are the symmetries of a solid rectangular box whose length, width and height are all different?
I get a group of order 4 by rotation 180, flipping along a vertical and horizontal axis and itself.
| Your contention that you'll have cyclic subgroup of order $4$ is quite natural but needs a little rethinking. Since, you consider the symmetries of a mattress like object (where symmetry is to mean any rigid motion in 3-space which will move a copy of the mattress in any fashion and place the copy back on the original)... | {
"language": "en",
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prove for all $n\geq 0$ that $3 \mid n^3+6n^2+11n+6$ I'm having some trouble with this question and can't really get how to prove this..
I have to prove $n^3+6n^2+11n+6$ is divisible by $3$ for all $n \geq 0$.
I have tried doing $\dfrac{m}{3}=n$ and then did $m=3n$
then I said $3n=n^3+6n^2+11n+6$ but now I am stuck.
| Here is a solution using induction:
Let $f(x)=x^3+6x^2+11x+6$
Since we want to see if it is divisible by 3 let us assume that $f(x)=3m$.
For the case where $x=0$, $f(0)=6$ which is divisible by 3.
Now that we have proved for one case let us prove for the case of $f(x+1)$
$$f(x+1)=(x+1)^3+6(x+1)^2+11(x+1)+6$$
$$= x^3+3... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Rotations and the parallel postulate. If we take a full rotation to be $360^\circ$, then it seems that we can prove the following
Starting from the red point, we walk clockwise along the triangle. At each vertex, we must turn through the green angles marked to proceed down the adjacent sides of the triangle. When we r... | Your picture, and perhaps your assumptions, are lying in the Euclidean plane. Take the same idea and put it on the sphere, where the parallel postulate is false, and we get something like the following:
Notice that, in this case, the sum of the exterior angles is $270^\circ$, not $360^\circ$.
However, in answer to you... | {
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Hofstadter's TNT: b is a power of 2 - is my formula doing what it is supposed to? If you've read Hofstadter's Gödel, Escher, Bach, you must have come across the problem of expressing 'b is a power of 2' in Typographical Number Theory. An alternative way to say this is that every divisor of b is a multiple of 2 or equal... | Your idea is sound, but the particular formula you propose
$$\neg\exists a:\exists a':\exists a'':( ((a\cdot a')=b) \land \neg (a=(a''\cdot SS0) \lor a=S0) )$$
does not quite express it. The problem is that the quantifier for $a''$ has too large scope -- what your formula says is that it will prevent $b$ from being a p... | {
"language": "en",
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"source": "stackexchange",
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Equivalence Class for Abstract Algebra Class Let
$$R_3= \{(a,b)\mid a,b \in \mathbb{Z}\text{ and there exists }k \in \mathbb{Z} \text{ such that }a-b=3k\}.$$
I know there is an equivalence relation but I'm not 100% on what it means to be an equivalence class for this problem. In class we got 3: $\{0,3,6,9,\ldots\}$ and... | I'll try to put it this way:
Define a relation $\sim$ on $\mathbb Z$, such that $a \sim b \iff \exists k \in \mathbb Z ~~ \text{such that}~~~~a-b=3k$
What does this say?
Integers $a$ and $b$ are related if and only if on their difference is a multiple of $3$. Since, the remainder when $a-b$ is divided by $3$ is the di... | {
"language": "en",
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Soccer and Probability MOTIVATION: I will quote Wikipedia's article on a soccer goalkeeper for the motivation:
Some goalkeepers have even scored goals. This most commonly occurs where a goalkeeper has rushed up to the opposite end of the pitch to give his team an attacking advantage in numbers. This rush is risky, as ... | In the case where you know the number of turns in advance, you can construct an optimal strategy in time $O(N^2)$ by reasoning backwards from the last round.
If, before the last turn, you find yourself with score $-0.5$, choose your strategy by comparing $p_1$ to $q_1$. On the other hand, if the score is $0.5$, compare... | {
"language": "en",
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Rudin's assertion that if $t = x/(1 + x)$ then $0 \leq t < 1$ I'm having trouble understanding one step in the proof of Theorem 1.21 in Rudin's Principles of Mathematical Analysis.
Theorem 1.21 For every real $x > 0$ and every integer $n > 0$ there is one and only one positive real $y$ such that $y^{n} = x$.
In the pro... | Notice that in this proof $x$ is a fixed positive real number, and that we are assuming $t=\frac{x}{x+1}$.
Since $x>0$, we have $x+1>0$ so $t=\frac{x}{x+1}>0$ hence $t>0$ thus $t\geq 0$. Furthermore, since $x<x+1$ and neither of these are $0$ we have $t=\frac{x}{x+1}<1$. Putting these together gives $0\leq t< 1$.
| {
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Number of critical points in a period of a periodic function I am interested in a relationship (if any) between the number of critical points of a periodic function $f$ of class $C^3([0,T])$ and the number of critical points of $f''$ in $[0,T]$.
| Consider a $C^2$ function $F\colon\mathbb{R}\to\mathbb{R}$ periodic of period $T>0$ and assume that $F$ has $N$ distinct zeroes $\{x_1,\dots,x_N\}\subset[0,T]$. By Rolle's theorem, $F'$ has at least $N-1$ zeroes in $(0,T)$, one in each interval $(x_i,x_{i+1})$, $1\le i\le N$.
*
*If $x_1=0$ (and hence $x_N=T$ ), then... | {
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Solve $\theta''+g\sin(\theta)=0$ I encountered the following differential equation when I tried to derive the equation of motion of a simple pendulum:
$\frac{\mathrm d^2 \theta}{\mathrm dt^2}+g\sin\theta=0$
How can I solve the above equation?
| replacing $\sin\theta$ by $\theta$ (physically assuming small angle deflection) gives you a homogeneous second order linear differential equation with constant coefficients, whose general solution can be found in most introductory diff eq texts (or a google search). this new equation represents a simple harmonic oscil... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/104626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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basic calculus proof - using theorems to prove stuff A function $f(x)$ is defined and continuous on the interval $[0,2]$ and $f(0)=f(2)$.
Prove that the numbers $x,y$ on $[0,2]$ exist such that $y-x=1$ and $f(x) = f(y)$.
I can already guess this is going to involve the intermediate value theorem.
So far I've defined... | if $g(0)$ is positive, $g(1)$ will be negative and vice versa, so the IVT provides a root. if both are zero, $g(0)=g(1)=0=f(1)-f(0)=f(2)-f(1)$ and you're done as well.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/104684",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Question regarding infinite Blaschke product According to Gamelin's $\textit{Complex Analysis}$, a finite Blaschke product is a rational function of the form $B(z)= e^{i \varphi} (\frac{z-a_1}{1-\bar{a_1} z} \cdots \frac{z-a_n}{1-\bar{a_n} z})$ where $a_1, ..., a_n \in \mathbb{D}$ and $0 \leq \varphi \leq 2\pi$. Simila... | Actually, the infinite Blaschke product, for $|a_n|\le1$ and $|z|<1$, is defined as
$$
e^{i\varphi}\prod_{n=1}^\infty\frac{|a_n|}{a_n}\frac{z-a_n}{\overline{a}_n z-1}\tag{1}
$$
The factor of $\;{-}\dfrac{|a_n|}{a_n}$ simply rotates $\dfrac{z-a_n}{1-\overline{a}_n z}$, which, for finite products, is incorporated into ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/104806",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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The differences between $\mathbb{R}/ \mathbb{Z}$ and $\mathbb{R}$
The cosets of $\mathbb{Z}$ in $\mathbb{R}$ are all sets of the form $a+\mathbb{Z}$, with $0 ≤ a < 1$ a real number. Adding such cosets is done by adding the corresponding real numbers, and subtracting 1 if the result is greater than or equal to 1. -- Ex... | $(a+\mathbb Z)+(b+\mathbb Z)$ is found by adding $a$ and $b$, the result of which is $a+b$. If $a+b<1$, then $(a+\mathbb Z)+(b+\mathbb Z)=(a+b)+\mathbb Z$. If $a+b\geq 1$, then $(a+\mathbb Z)+(b+\mathbb Z)=(a+b-1)+\mathbb Z$.
But this is only if you follow the stated convention of only listing representatives from $[... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/104846",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Existence of universal enveloping inverse semigroup (similar to "Grothendieck group") Context
In its simplest form, the Grothendieck group construction associates an abelian group to a commutative semigroup in a "universal way".
Now I'm interested in the following nilpotent commutative semigroup $N$ consisting of two e... | I now found out how to prove that no "universal enveloping regular semigroup" exists for the example given in the question. (The existence of the other two cases has already been proved in the answer by Martin Wanvik.)
Let
$a=\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$,
$b=\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/104893",
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Number of solutions of $x^2=1$ in $\mathbb{Z}/n\mathbb{Z}$ Next is what I have worked out to the moment.
$1$ and $-1$ are roots for all $n$.
$x \in \mathbb{Z}/n\mathbb{Z},\ $ $x^2\equiv1 \Leftrightarrow (x-1)(x+1)\equiv0 \Leftrightarrow \exists k \in \mathbb{Z}/n\mathbb{Z}: k(k+2)\equiv0 $.
But how can it be applied t... | (I'd have to check the details in the following but it provides some rough ideas.)
Write $n = \prod_i p_i^{\nu_i}$ and use the Chinese remainder theorem to obtain a system of equations
$$ x^2 \equiv 1 \pmod{p_i^{\nu_i}}$$
Of course, for every $p_i$, $x=\pm 1$ provides a solution. Based on some quick calculations, I thi... | {
"language": "en",
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show if $n=4k+3$ is a prime and ${a^2+b^2} \equiv 0 \pmod n$ , then $a \equiv b \equiv 0 \pmod n$ $n = 4k + 3 $
We start by letting $a \not\equiv 0\pmod n$ $\Rightarrow$ $a \equiv k\pmod n$ .
$\Rightarrow$ $a^{4k+2} \equiv 1\pmod n$
Now, I know that the contradiction will arrive from the fact that if we can show $a^2 \... | This is solution from Prasolov V.V. Zadachi po algebre, arifmetike i analizu (В. В. Прасолов. Задачи по алгебре, арифметике и анализу.) This book (in Russian) is freely available at http://www.mccme.ru/free-books/
The problem appears there as Problem 31.2.
We want to show that if $p=4k+3$ is a prime number and $p\mid ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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An ideal and all elements of the quotient ring
Defintion 1 An ideal I of R is an additive subgroup so that $$a\in R , s\in I \Rightarrow as,sa \in I$$
The ring R/I is called the quotient ring.
Example 1 : $R=\mathbb{Z}[x], I=nR (n\in \mathbb{Z})$. Then $R/I "=" (\mathbb{Z}/n\mathbb{Z})[x]$. Or $I=xR$. What are the... | "Example 1" seems to be three examples: (i) $R/I$ with $R=\mathbb{Z}[x]$, $I=nR$; (ii) $R/I$ with $I=xR$; and (iii) $R/I$ with $I=2R+xR$.
In the third case, $I$ consists of all polynomials with integer coefficients that have even constant term: all such polynomials can be written as a multiple of $2$ plus a multiple o... | {
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roots of complex polynomial - tricks What tricks are there for calculating the roots of complex polynomials like
$$p(t) = (t+1)^6 - (t-1)^6$$
$t = 1$ is not a root. Therefore we can divide by $(t-1)^6$. We then get
$$\left( \frac{t+1}{t-1} \right)^6 = 1$$
Let $\omega = \frac{t+1}{t-1}$ then we get $\omega^6=1$ which b... | Notice that $t=1$ is not a root. Divide by $(t-1)^6$.
If $\omega$ is a root of $z^6 - 1$, then a root of the original equation is given by $\frac{t+1}{t-1} = \omega$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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A better approximation of $H_n $ I'm convinced that
$$H_n \approx\log(n+\gamma) +\gamma$$ is a better approximation of the $n$-th harmonic number than the classical $$H_n \approx \log(n) +\gamma$$
Specially for small values of $n$. I leave some values and the error:
Just to make things clearer, I calculate the value... | The asymptotic expansion of the Harmonic numbers $H_n$ is given by
$$\log n+\gamma+\frac{1}{2n}+\mathcal{O}\left(\frac{1}{n^2}\right).$$
The Maclaurin series expansion of the natural logarithm tells us $\log(1+x)=x+\mathcal{O}(x^2)$, and we can use this in your formula by writing $\log(n+\epsilon)=\log n+\log(1+\epsilo... | {
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Union of uncountably many subspaces Supposing, $\{V_t\}, t > 0$ are an uncountable number of linear subspaces of $\mathbb{R}^n$. If $\bigcup_{t>0} V_t = \mathbb{R}^n$ is it true that $V_T = \mathbb{R}^n$ for some $T>0$?
Any help is appreciated. Thanks.
EDIT: I have forgot to add the condition that $V_t$ are increasing... | In general the answer is no. Consider family of subspaces of the form
$$
V_t=\{x\in\mathbb{R}^n:x_1\cos\frac{2\pi}{t+1}+x_2\sin\frac{2\pi}{t+1}=0\}
$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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chromatic number and subgraph
Prove that any graph $G$ with $n$ vertices and $ \chi(G)=k$ has a subgraph $H$ such that $ H \simeq \overline{K_p}$ where $p=n/k$ and $K_p$ is the complete graph with $n/k$ vertices.
My attempt: Because $ \chi(G)=k$ it must be $G \subseteq K_{p_1 p_2 \cdots p_k} $ where $\displaystyle{\... | If $\chi(G) = k$, it means we can color the graph with $k$ colors, $c_1, \ldots, c_k$. Each color class, $c_i$, consists of some vertices $V_i$. Necessarily, the vertices in $V_i$ are independent, or we could not color them all the same color, $c_i$.
Now, assume that every color class contains less than $n / k$ verti... | {
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"timestamp": "2023-03-29T00:00:00",
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Integrable function Please help me prove this.
Suppose $f$ is defined in$\left [ a,b \right ]$ ,$f\geq 0$, $f$ is integrable in$ [a,b]$ and $\displaystyle\int_{a}^{b}fdx=0$
prove:
$\displaystyle\int_{a}^{b}f(x)^2dx=0$
Thanks a lot!
| My first inclination would be to say that since $f\geq0$ and $\int_a^bfdx=0$, $f(x)=0$ for $x$ such that $a\leq x\leq b$. Because of this, $\int_b^af(x)^2dx=0$. You will probably need to fill in a fair amount of reasoning to make this hold up, though.
Also, suggestion - accept some answers to your questions. It ma... | {
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Proving that the "real part" function is continuous I want to use the definition of a limit,
$|f(z) - w_0| < \varepsilon$ whenever $0 < |z - z_0| < \delta$
to prove
$$\lim_{z \to z_0} \mathop{\rm Re}(z) = \mathop{\rm Re}(z_0)$$
By intuition this is obvious but I dont know how to show it using the defn. of a limit. This... | We have
$|z_0-z|^2=(\Re(z_0-z))^2+(\Im(z_0-z))^2\geq (\Re(z_0-z))^2$ so $|z_0-z|\geq |z_0-z|$. Now, check that $\delta=\varepsilon$ in the definition of the limit works.
| {
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"timestamp": "2023-03-29T00:00:00",
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} |
eigenvalues and eigenvectors of $vv^T$ Given a column vector $v$ in $\mathbb{R}^n$, what are the eigenvalues of matrix $vv^T$ and associated eigenvectors?
PS: not homework even though it may look like so.
| The columns of the matrix are $v_1v,\ldots,v_nv$ so if we take two column these one are linearly dependent, and so $vv^T$ has a rank of at most $1$. It's $0$ if $v=0$, and if $v\neq 0$, we have $\mathrm{Tr}A=|v|^2$ so the eignevalues are $0$ with multiplicity $n-1$ and $|v|^2$ with multiplicity $1$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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proof that if $AB=BA$ matrix $A$ must be $\lambda E$ Let $A \in Mat(2\times 2, \mathbb{Q})$ be a matrix with $AB = BA$ for all matrices $B \in Mat(2\times 2, \mathbb{Q})$.
Show that there exists a $\lambda \in \mathbb{Q}$ so that $A = \lambda E_2$.
Let $E_{ij}$ be the matrix with all entries $0$ except $e_{ij} = 1$.
$... | You wish to show that the center of the ring of matrices is the set of scalar matrices. Observe that the matrices commuting with diagonal matrices must be diagonal matrices. Hence the center must be contained in the set of diagonal matrices. Now a diagonal matrix that commutes with an arbitrary matrix must be a scalar ... | {
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"source": "stackexchange",
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Ring theory notation question I was wondering what the notation $R[a]$ really stands for, if $a\in K$, where $K$ is a ring and $R$ is a subring of $K$.
In my book they define $\mathbb{Z}[\sqrt2]=\{a+b\sqrt2|a,b \in \mathbb{Z}\}$.
So, my guess is that $R[a] = \{P(a)\mid P \in R[X]\}$. Since for $\mathbb{Z}[\sqrt2]$ thi... | By definition, $R[a]$ is the smallest subring of $K$ that contains both $R$ and $a$.
As you note, if $p(x)\in R[x]$, then $p(a)\in R[a]$ necessarily. Therefore,
$$\{p(a)\mid p(x)\in R[x]\} \subseteq R[a].$$
Conversely, note that
$$\{p(a)\mid p(x)\in R[x]\}$$
contains $a$ (as $p(a)$ where $p(x)=x$), contains $R$ (the co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/105807",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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The completion of a Boolean algebra is unique up to isomorphism Jech defines a completion of a Boolean algebra $B$ to be a complete Boolean algebra $C$ such that $B$ is a dense subalgebra of $C$.
I am trying to prove that given two completions $C$ and $D$ of $B$, then the mapping $\pi: C \rightarrow D$ given by $\pi(c)... | Edit: Ignore this answer. See Loronegro's.
Apparently I haven't done enough on Math.SE before, so I have to post this as an answer instead of a comment.
Hopefully this is helpful: first notice that $\pi c \le c$ for every $c\in C$ (the supremum of a bunch of things that are $\le c$ is also $\le c$). From this, you can ... | {
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"source": "stackexchange",
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What does integration do? I know that integrals are used to compute the area under a curve. Let's say I have $y = x^2$. It creates smaller rectangles and then add up the sum (assuming that rectangles are going infinitely in number and is like going to a limit).
But I recently encountered a problem in my mind. Suppose w... |
The key word here is instantaneous. Although the 2 graphs are "different". They are linked to each other through the "instantaneous area". What I mean is that if you take a point on your function and calculate the value of the area of the function from, say, 0 to that point. That value of area is the same as g(x). $(g... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How does $\exp(x+y) = \exp(x)\exp(y)$ imply $\exp(x) = [\exp(1)]^x$? In Calculus by Spivak (1994), the author states in Chapter 18 p. 341 that
$$\exp(x+y) = \exp(x)\exp(y)$$ implies
$$\exp(x) = [\exp(1)]^x$$
He refers to the discussion in the beginning of the chapter where we define a function $f(x + y) = f(x)f(y)$;... | I like (as I have done here before) to start with a functional equation
and derive properties of the function.
If $f(x+y) = f(x) f(y)$ and $f$ is differentiable (and non-zero somewhere),
$f(0) = 1$ and
$f(x+h)-f(x) = f(x)f(h)-f(x)
=f(x)(f(h)-1)
=f(x)(f(h)-f(0))
$
so
$$(f(x+h)-f(x))/h = f(x)(f(h)-f(0))/h.$$
Letting ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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If $A^2 = I$ (Identity Matrix) then $A = \pm I$ So I'm studying linear algebra and one of the self-study exercises has a set of true or false questions. One of the questions is this:
If $A^2 = I$ (Identity Matrix), then $A = \pm I$ ?
I'm pretty sure it is true but the answer says it's false. How can this be false (ma... | I know $2·\mathbb C^2$ many counterexamples, namely
$$A=c_1\begin{pmatrix}
0&1\\
1&0
\end{pmatrix}+c_2\begin{pmatrix}
1&0\\
0&-1
\end{pmatrix}\pm\sqrt{c_1^2+c_2^2\pm1}\begin{pmatrix}
0&-1\\
1&0
\end{pmatrix},$$
see Pauli Matrices $\sigma_i$.
These are all such matrices and can be written as $A=\vec e· \vec \si... | {
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"source": "stackexchange",
"question_score": "27",
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Prove that $n! > \sqrt{n^n}, n \geq 3$
Problem Prove that $n! > \sqrt{n^n}, n \geq 3$.
I'm currently have two ideas in mind, one is to use induction on $n$, two is to find $\displaystyle\lim_{n\to\infty}\dfrac{n!}{\sqrt{n^n}}$. However, both methods don't seem to get close to the answer. I wonder is there another me... | $(n!)^2 = (n \times 1) \times ((n-1)\times 2) \times \cdots \times (1 \times n) \gt n^n$
since $(n-1)\times 2 = 2n-2 \gt n$ iff $n \gt 2$.
Then take the square root.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
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"answer_id": 2
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Finding limit of fraction with square roots: $\lim_{r\to 9} \frac {\sqrt{r}} {(r-9)^4}$ I have been looking at this for five minutes, no clue what to do.
$$\lim_{r\to 9} \frac {\sqrt{r}} {(r-9)^4}$$
| The limit is $+\infty$ because the numerator approaches a positive number and the denominator approaches $0$ from above. Sometimes one says the limit "doesn't exist" when one means there is no real number that is the limit, so you could put it that way.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Trying to figure out a complex equality An answer to a comlex equation I was working on was
$$z = \frac{1}{2} + \frac{i}{2}$$
My teacher further developed it to be
$$e^{\frac{i\pi}{4}-\frac{1}{2}\ln{2}}$$
And here's what I tried:
$$z = \frac{1}{2} + \frac{i}{2} = z = \frac{1}{\sqrt{2}}e^{\frac{i\pi}{4}}
= e^{\frac{... | The mistake occurs here:
$$\frac{1}{\sqrt{2}}e^{\frac{i\pi}{4}}
= e^{\frac{1}{2}\ln{2}}e^{\frac{i\pi}{4}}.$$
In fact, we have
$$e^{\frac{1}{2}\ln{2}}=2^{\frac{1}{2}}=\sqrt{2}.$$
Therefore, we should have
$$\frac{1}{\sqrt{2}}=(\sqrt{2})^{-1}
= e^{-\frac{1}{2}\ln{2}}.$$
Mixing this, your answer matches with your te... | {
"language": "en",
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Finding the n-th root of a complex number I am trying to solve $z^6 = 1$ where $z\in\mathbb{C}$.
So What I have so far is :
$$z^6 = 1 \rightarrow r^6\operatorname{cis}(6\theta) = 1\operatorname{cis}(0 + \pi k)$$
$$r = 1,\ \theta = \frac{\pi k}{6}$$
$$k=0: z=\operatorname{cis}(0)=1$$
$$k=1: z=\operatorname{cis}\left(\fr... | Hint: Imagine that there is a unit circle on the Argand Plane. Now the roots will be the 6 equidistant points on that circle each having the argument as multiple of $\frac{2\pi}{6}=\frac{\pi}{3}$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is it possible to generate a unique real number for each fixed length sequence of real numbers? Let A be the set of all sequences of real numbers of size $n$. Does there exist an injection from A to R?
I know this is possible if we are only considering integers instead of real numbers; But I am not sure if it is possib... | You can easily create an injection even for infinite sequences of reals (an injective mapping from sequences to real numbers). So your request is too weak.
See Boas Primer of Real Functions Exercise 3.13 and its solution in the back.
Also see my thread about this exercise.
| {
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what is Prime Gaps relationship with number 6? Out of the 78499 prime number under 1 million. There are 32821 prime gaps (difference between two consecutive prime numbers) of a multiple 6. A bar chart of differences and frequency of occurrence shows a local maximum at each multiple of 6. Why is 6 so special?
| take any integer $n> 3$, and divide it by $6$. That is, write
$n = 6q + r$
where $q$ is a non-negative integer and the remainder $r$ is one of $0$, $1$, $2$, $3$, $4$, or $5$.
If the remainder is $0$, $2$ or $4$, then the number $n$ is divisible by $2$, and can not be prime.
If the remainder is $3$, then the number $n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/106417",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
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A question concerning maps of $G$-coverings I am having difficulties thinking about how an argument for the following exercise should proceed:
Let $p: Y \rightarrow X$ and $q: Z \rightarrow X$ be $G$-coverings (i.e., covering maps such that $X = Y /G = Z/G$ (quotient spaces)), with $X$ connected and locally path connec... | (Sorry I can't comment)
Show that each connected component of $Y$ contains a preimage of every $z\in Z$. Fix a connected component $Y_0$ of $Y$. Take a $y\in Y$. Using the transitivity of $G$, translate $y$ to $Y_0$. Then use the fact that you know the statement is true for $Y_0$.
| {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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intermediate step in proving old Ramsey lower bound Let $r(n,n)=r(n)$ be the usual Ramsey number of a graph. It is known that $$\frac{1}{e\sqrt{2}}n2^{n/2}<r(n)$$ as a lower bound for $r(n).$
Now, in the proof given in the book Erdős on Graphs by Graham and Chung, as an intermediate step this is given:
$$2^{\binom{m}{... | In fact, the inequality $(**)$ should be the other way around.
As Austin Mohr noted, Stirling's formula comes in handy here. The form that I will use is
$$n! \sim \biggl(\dfrac{n}{e}\biggr)^n \sqrt{2\pi n}. \tag{1}$$
Also, I assume that $m \to \infty$ and that $m - n \to \infty$.
We start by observing that the inequal... | {
"language": "en",
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"source": "stackexchange",
"question_score": "10",
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How to show a matrix is full rank? I have some discussion with my friend about matrix rank.
But we find that even we know how to compute rank, we don't know how to show the matrix is full rank.
How to show this?
| If you are talking about square matrices, just compute the determinant.
If that is non-zero, the matrix is of full rank.
If the matrix $A$ is $n$ by $m$, assume wlog that $m\leq n$ and compute all determinants of $m$ by $m$ submatrices. If one of them is non-zero, the matrix has full rank.
Also, you can solve the line... | {
"language": "en",
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"source": "stackexchange",
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Two ants on a triangle puzzle Last Saturday's Guardian newspaper contained the following puzzle:
Two soldier ants start on different vertices of an equilateral triangle. With each move, each ant moves independently and randomly to one of the other two vertices. If they meet, they eliminate each other. Prove that mu... | Note that the fact that it's a triangle is irrelevant. The ants could move to random vertices on any $n$-gon and the result would be the same.
Put another way, if two people repeatedly choose random integers from $1$ to $n$, they will eventually choose the same number.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/106654",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Does $\int_{1}^{\infty}\sin(x\log x)dx $ converge? I'm trying to find out whether $\int_{1}^{\infty}\sin(x\log x)dx $ converges, I know that $\int_{1}^{\infty}\sin(x)dx $ diverges but $\int_{1}^{\infty}\sin(x^2)dx $ converges, more than that, $\int_{1}^{\infty}\sin(x^p)dx $ converges for every $p>0$, so it should be co... | Since $x\log(x)$ is monotonic on $[1,\infty)$, let $f(x)$ be its inverse. That is, for $x\in[0,\infty)$
$$
f(x)\log(f(x))=x\tag{1}
$$
Differentiating implicitly, we get
$$
f'(x)=\frac{1}{\log(f(x))+1}\tag{2}
$$
Then
$$
\begin{align}
\int_1^\infty\sin(x\log(x))\;\mathrm{d}x
&=\int_0^\infty\sin(x)\;\mathrm{d}f(x)\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/106710",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
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An epimorphism from $S_{4}$ to $S_{3}$ having the kernel isomorphic to Klein four-group Exercise $7$, page 51 from Hungerford's book Algebra.
Show that $N=\{(1),(12)(34), (13)(24),(14)(23)\}$ is a normal subgroup
of $S_{4}$ contained in $A_{4}$ such that $S_{4}/N\cong S_{3}$ and
$A_{4}/N\cong \mathbb{Z}_{3}$.
I s... | Here is an approach:
Proof Idea: $S_4/N$ is a group with 6 elements. There are only two such groups, one is cyclic and the other is $S_3$, and $S_4/N$ cannot have elements of order $6$ thus must be $S_3$.
First it is easy to show that $N$ is normal in $S_4$. It follows that $S_4/N$ is a group with $6$ elements. Let us ... | {
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"url": "https://math.stackexchange.com/questions/106777",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Eigenvalues of a infinitesimal generator matrix Consider a Markov process on a finite state space $S$, whose dynamic is determined by a certain infinitesimal generator $Q$ (that is a matrix in this case) and an initial distribution $m$.
1) Is there anything general that can be said on the spectrum of the matrix $-Q$?
2... | The Gershgorin Circle Theorem can be used to construct a set of closed balls (i.e., circles) in the complex plane that are guaranteed to contain the eigenvalues of the matrix.
This theorem is not specific to generator matrices, so there may be some other related result that takes advantage of these matrices' specific s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/106832",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to directly prove that $M$ is maximal ideal of $A$ iff $A/M$ is a field? An ideal $M$ of a commutative ring $A$ (with unity) is maximal iff $A/M$ is a field.
This is easy with the correspondence of ideals of $A/I$ with ideals of $A$ containing $I$, but how can you prove it directly? Take $x + M \in A/M$. How can ... | From $(M,x)=A$ you can infer that there are $m\in M, y\in A$ so that $m+xy=1$. Thus, $xy+M=1+M$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/106909",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Why does $\sum_n\binom{n}{k}x^n=\frac{x^k}{(1-x)^{k+1}}$? I don't understand the identity $\sum_n\binom{n}{k}x^n=\frac{x^k}{(1-x)^{k+1}}$, where $k$ is fixed.
I first approached it by considering the identity
$$
\sum_{n,k\geq 0} \binom{n}{k} x^n y^k = \sum_{n=0}^\infty x^n \sum_{k=0}^n \binom{n}{k} y^k = \sum_{n=0}^\i... | You can work directly with properties of the binomial coefficient. For $k\ge 0$ let $$f_k(x)=\sum_{n\ge 0}\binom{n}kx^n\;.$$ Then
$$\begin{align*}
f_k(x)&=\sum_{n\ge 0}\binom{n}{k}x^n\\
&=\sum_{n\ge 0}\left[\binom{n-1}{k-1}+\binom{n-1}{k}\right]x^n\\
&=\sum_{n\ge 0}\left[\binom{n}{k-1}+\binom{n}k\right]x^{n+1}\\
&=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/106978",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Rittner equation I would like to know if the Rittner equation :
$$\partial_{t}{\varPhi(x,t)=k\partial_{xx}{\varPhi(x,t)}}-\alpha{\partial_{x}{\varPhi(x,t)}-\beta{\varPhi(x,t)}+g(x,t)}$$
can be solved using the Lax pair method or the Fokas method.
Thanks
| Setting $\psi(x,t) = e^{(\alpha^2/4k + \beta)t - \alpha x/2k} \Phi(x,t)$ and plugging in I get
$$
\begin{align*} \psi_t(x,t) - k \psi_{xx}(x,t) &= e^{(\alpha^2/4k + \beta)t - \alpha x/2k}g(x,t) \\
&\equiv f(x,t).
\end{align*}
$$
If the forcing $f(x,t)$ is in $L^1(\mathbf{R}\times \mathbf{R})$ then we can use the he... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Reflect a complex number about an arbitrary axis This should be really obvious, but I can't quite get my head round it:
I have a complex number $z$. I want to reflect it about an axis specified by an angle $\theta$.
I thought, this should simply be, rotate $z$ by $\theta$, flip it (conj), then rotate by $-\theta$.
But ... | Indeed, it's just $z^*$ rotated by $2\theta$... And it's almost the right answer! You know that a symmetry composed with a rotation is still a symmetry, and you know that an (orthogonal) symmetry is characterized by its fixed points. So you want to get a symmetry that fixes the axis spanned by $e^{i\theta}$ (as an $\ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/107125",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finite dimensional vector space with subspaces
Possible Duplicate:
Could intersection of a subspace with its complement be non empty.
Is it possible for a finite dimensional vector space to have 2 disjoint subspaces of the same dimension ?
Any help would be much appreciated.
| As David Giraudo points out, any subspace $U\subseteq V$ is going to contain the zero vector (it has to: the subspace is a vector space so is closed under multiplication by elements of the underlying scalar field, and zero is in the scalar field, so multiplying by it tells us the zero vector is in the subspace). In thi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/107194",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Essay about the art and applications of differential equations? I teach a high school calculus class. We've worked through the standard derivatives material, and I incorporated a discussion of antiderivatives throughout. I've introduced solving "area under a curve" problems as solving differential equations. Since i... | Differential equations is a rather immense subject. In spite of the risk of overwhelming you with the amount of information, I recommend looking in the Princeton Companion to Mathematics, from which the relevant sections are (page numbers are within parts)
*
*Section I.3.5.4 for an introductory overview
*Section I.... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Symmetries of the singular vectors of the line graph Consider the matrix $$ A = \left( \begin{matrix} 1/2 & 1/2 & 0 & 0 & 0 & 0 \\ 1/3 & 1/3 & 1/3 & 0 & 0 & 0 \\ 0 & 1/2 & 0 & 1/2 & 0 & 0 \\
\vdots & \ & \ddots & & \ddots & \\
0 & \cdots & 0 & 1/3 & 1/3 & 1/3 \\
0 & \cdots & 0 & 0 & 1/2 & 1/2 \end{matrix} \right) ... | The vectors (general vectors, not eigenvectors) with either kind of parity form $n/2$-dimensional subspaces. For instance, a basis for the even vectors is $(1,0,0,\dotsc,0,0,1)$, $(0,1,0,\dots,0,1,0)$, $\dotsc$
You've already explained that we can choose an eigenbasis for $A$ in which all vectors have definite parity. ... | {
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Does $\sum_{n=0}^{\infty}e^{-|x-n|}$ converge and uniformly converge? For $ 0 \leq x < \infty$, I'd like your help with deciding whether the following series converges and uniformly converges: $\sum_{n=0}^{\infty}e^{-|x-n|}$.
$$\sum_{n=0}^{\infty}\frac{1}{e^{|x-n|}}=\sum_{n=0}^{N}\frac{1}{e^{|x-n|}}+\sum_{i=0}^{\infty}... | Looking at the given series we see that uniform convergence is endangered by the fact that for arbitrary large $n$ we can find an $x$ where the $n$-th term is large, namely $\ =e^0=1$. Therefore we try to prove that the convergence is not uniform by exploiting this fact.
Put $s_n(x):=\sum_{k=0}^n e^{-|x-n|}$. By Cauchy... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/107416",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
badly approximated numbers on the real line form a meagre set Let $S$ be the set of real numbers $x$ such that there exist infinitely many (reduced) rational numbers $p/q$ such that $$\left\vert x-\frac{p}{q}\right\vert <\frac{1}{q^8}.$$
I would like to prove that $\mathbb{R}\setminus S$ is a meagre set (i.e. union of ... | The idea is to transform the quantifiers into unions/intersections.
For example, let $T$ be the same as $S$ but dropping the infinitely many assumption.
Consider $A_{\frac p q}=(-\frac 1 {q^8}+\frac p q, \frac p q +\frac 1{q^8})$ then $T=\bigcup_{\frac p q\in\mathbb Q}A_{\frac p q}$. Thus, $T$ is a countable union of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/107494",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
Generalization of Pythagorean triples Is it known whether for any natural number $n$, I can find (infinitely many?) nontrivial integer tuples $$(x_0,\ldots,x_n)$$ such that $$x_0^n + \cdots + x_{n-1}^n = x_n^n?$$
Obviously this is true for $n = 2$.
Thanks.
| These Pythagorean triples can appear in the most unexpected place.
If: $a^2+b^2=c^2$
Then alignment: $N_1^3+N_2^3+N_3^3+N_4^3+N_5^3=N_6^3$
$N_1=cp^2-3(a+b)ps+3cs^2$
$N_2=bp^2+3bps-3bs^2$
$N_3=ap^2+3aps-3as^2$
$N_4=-bp^2+3(2c-b)ps+3(3c-3a-2b)s^2$
$N_5=-ap^2+3(2c-a)ps+3(3c-2a-3b)s^2$
$N_6=cp^2+3(2c-a-b)ps+3(4c-3a-3b)s^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/107570",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
$k$-out-of-$n$ system probabilities
An engineering system consisting of $n$ components is said to be a $k$-out-of-$n$ system ($k \le n$) when the system functions if and only if at least $k$ out of the $n$ components function. Suppose that all components function independently of each other.
If the $i^{th}$ componen... | If you want something "prettier" you could take all the possible cases and write them using products and sums, such as $$\prod_{i=1}^4 p_i \left(1+\sum_{j=1}^4\frac{1-p_j}{p_j} + \sum_{k=1}^3 \sum_{l=k+1}^4 \frac{(1-p_k)(1-p_l)}{p_k \; p_l} \right)$$
or you could work out the probability that one or none work and subtr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/107626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Prove that a finite union of closed sets is also closed Let $X$ be a metric space. If $F_i \subset X$ is closed for $1 \leq i \leq n$, prove that $\bigcup_{i=1}^n F_i$ is also closed.
I'm looking for a direct proof of this theorem. (I already know a proof which first shows that a finite intersection of open sets is a... | It is sufficient to prove this for a pair of closed sets $F_1$ and $F_2$. Suppose $F_1 \cup F_2$ is not closed, even though $F_1$ and $F_2$ are closed. This means that some limit point $p$ of $F_1 \cup F_2$ is missing. So there is a sequence $\{ p_i\} \subset F_1 \cup F_2$ converging to $p$. By pigeonhole principle, at... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/107692",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 3,
"answer_id": 1
} |
Matrix multiplication, equivalent to numeric multiplication, or just shares the name? Is matrix multiplication equivalent to numeric multiplication, or do they just share the same name?
While there are similarities between how they work, and one can be thought of being derived from the other, I ask because they have di... | "Multiplication" is often used to describe binary operations that share only some of the properties of ordinary multiplication.
The case of matrix multiplication is special. There, multiplication is in some sense a generalization of ordinary multiplication. Let $M_n(a)$ be the $n\times n$ matrix whose diagonal entries... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/107827",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Prove or refute: If $f$ is Riemann integrable on $[0,1]$, then so is $\sin(f)$ I'd love your help with the following question: I need to prove or refute the claim that for a Riemann integrable function $f$ in $[0,1]$ also $\sin(f)$ is integrable on $[0,1]$.
My translation for this claim: If $\int_{0}^{1} f(x) dx < \inf... | I do not think your translation is correct (unless you meant Lebesgue, and not Riemann integrable). The concept of Riemann integrable and Lebesgue integrable are not the same.
Riemann integrable: $f\colon[a,b]\to\mathbb{R}$, $-\infty<a<b<+\infty$, $f$ bounded and the upper integral equal to the lower integral.
Lebesgu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/107872",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
$f \geq 0$ continuous, $\lim_{x \to \infty} f(x)$ exists, $\int_{0}^{\infty}f(x)dx< \infty$, Prove: $\int_{0}^{\infty}f^2(x)dx< \infty$ Something that bothers me with the following question: $f: [0, \infty] \to \mathbb{R}$,$f \geq 0$, $\lim_{x \to \infty} f(x)$ exists and finite, and $\int_{0}^{\infty}f(x)dx$ converg... | To answer your questions regarding the behaviour at $0$ first. Note that $f:[0;\infty[ \ \to \mathbb{R}$, hence $f$ is well-defined and continuous at $0$ - according to the assumptions you state. In particular this means that $\lim_{x\to 0}f(x) = f(0)$, so there is really no issue at $x=0$; your function is well-define... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/107923",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
If a series converges, then the sequence of terms converges to $0$. Following the guidelines suggested in this meta discussion, I am going to post a proposed proof as an answer to the theorem below. I believe the proof works, but would appreciate any needed corrections.
Theorem If a series $\sum_{n=1}^{\infty}a_n$ of r... | Another view of this may be useful.
First, recall a basic fact that if $a_n$ is a convergent sequence of numbers, then the sequence $b_n = a_{n+1} - a_n$ converges to $0$. This is easy to prove and does not require the notion of a Cauchy sequence.
Therefore, if the partial sums $s_n$ are convergent, then $b_n = s_{n+1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/107961",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "32",
"answer_count": 3,
"answer_id": 2
} |
Generalized "Duality" of Classical Propositional Logical Operations Duality in propositional logic between conjunction and disjunction, $K$ and $A$ means that for any "identity", such as $KpNp = 0$ (ignoring the detail of how to define this notion in propositional logic), if we replace all instances of $K$ by $A$, all ... | The resulting equation is also an identity. This is because any of the $16$ operations can be put in canonical disjunctive normal form using only $\land$, $\lor$, and $\lnot$. Then the replacement procedure described in the post becomes the standard one, and we are dealing with ordinary duality.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/108012",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Combinatorial interpretation of this identity of Gauss? Gauss came up with some bizarre identities, namely
$$
\sum_{n\in\mathbb{Z}}(-1)^nq^{n^2}=\prod_{k\geq 1}\frac{1-q^k}{1+q^k}.
$$
How can this be interpreted combinatorially? It strikes me as being similar to many partition identities. Thanks.
| The typical analytic proof is not difficult and is an easy consequence of Jacobi's triple product $$\sum_{n=-\infty} ^{\infty} z^{n} q^{n^{2}}=\prod_{n=1}^{\infty}(1-q^{2n})(1+zq^{2n-1})(1+z^{-1}q^{2n-1})$$ for all $z, q$ with $z\neq 0,|q|<1$. Let's put $z=-1$ to get the sum in question. The corresponding product is eq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/108077",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 3,
"answer_id": 2
} |
For $\sum_{0}^{\infty}a_nx^n$ with an infinite radius of convergence, $a_n \neq0$ , the series does not converge uniformly in $(-\infty , \infty)$. I'd like your help with the following question:
Let $\sum_{0}^{\infty}a_nx^n$ be a power series with an infinite
radius of convergence, such that $a_n \neq 0 $ , for i... | Hint: put $s_N(x):=\sum_{n=0}^Na_nx^n$. If the sequence $\{s_N\}$ is uniformly convergent on $\mathbb R$ then the sequence $\{s_{N+1}-s_N\}$ converges uniformly on $\mathbb R$ to $0$ so $\{a_{N+1}x^{N+1}\}$ converges uniformly to $0$. Do you see the contradiction?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/108160",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Where should the exponent be written in numbers with units of measurement? If you are to calculate the hypotenuse of a triangle, the formula is:
$h = \sqrt{x^2 + y^2}$
If you don't have any units for the numbers, replacing x and y is pretty straightforward:
$h = \sqrt{4^2 + 6^2}$
But what if the numbers are in meters?
... | Suppose you have been given $x$ and $y$ in metres, and you'd like to know the quantity, $z=\sqrt{x^2+y^2}$. Then, as you have predicted this quatity will be in metres.
Two things have been involved:
Homogeneity of Dimension
Two quantities of different dimensions cannot be added. This is one of the axioms of numerical ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/108200",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
What is the equation of an ellipse that is not aligned with the axis? I have the an ellipse with its semi-minor axis length $x$, and semi major axis $4x$. However, it is oriented $45$ degrees from the axis (but is still centred at the origin). I want to do some work with such a shape, but don't know how to express it... | Let the center of the ellipse be at $C = (x_c, y_c)$. Let the major axis be the line that passes through $C$ with a slope of $s$; points on that line are given by the zeros of $L(x,y) = y - y_c - s(x - x_c)$. Let the minor axis be the line perpendicular to $L$ (and also passing through $C$); points on that line are giv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/108270",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 1
} |
An application of the General Lebesgue Dominated convergence theorem I came across the following problem in my self-study:
If $f_n, f$ are integrable and $f_n \rightarrow f$ a.e. on $E$, then $\int_E |f_n - f| \rightarrow 0$ iff $\int_E |f_n| \rightarrow \int_E |f|$.
I am trying to prove (1) and the book I am using ... | Take $g_n = |f_n| + |f|$ and use the triangle inequality to get the bound.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/108313",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
continuous map of metric spaces and compactness
Let $f:X\rightarrow Y$ be a continuous map of metric spaces. Show that if $A\subseteq X$ is compact, then $f(A)\subseteq Y$ is compact.
I am using this theorem: If $A\subseteq X$ is sequentially compact, it is compact.
Also this definition: A set $A\subseteq X$ is seque... | Yet another formulation for topological spaces: If $f:X \to Y$ continuous and $f(x_\iota)$ is a net in $f(X)$, then $x_\iota$ has a converging subnet, say $x_\tau \to x$. Then $f(x_\tau) \to f(x)$, hence $f(x_\iota)$ has a converging subnet, so $f(X)$ is compact.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/108369",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
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