Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
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Explanation of the first step in the proof of Vitali theorem. The theorem and the first part of its proof is given below:
But I do not understand the first statement in the proof, why countable subadditivity of outer measure lead us to suppose that $E$ is bounded? could anyone explain this for me, please?
| Theorem 17: Any set of real numbers with positive outer measure contains a subset that fails to be measurable.
Theorem 17': Any bounded set of real numbers with positive outer measure contains a subset that fails to be measurable.
The part of the proof that follows the first sentence proves theorem 17'. We therefore ju... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3359562",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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TRUE OR FALSE : Events in a partition cannot be independent (assumption: every event in partition has nonzero probability) I need help solving this T or F question.
I got the following but unsure if correct.
Let A and B be two partitions in set X.
$$P(A \cap B) = 0$$
Let's assume A and B are independent, then:
$$P(A)P(... | Yes. Let me formalize a bit. Consider a partition, i.e. some events $A_1, \dots, A_n$ such that $A_i \cap A_j = \emptyset$ if $i \ne j$ and $\cup_{i=1}^n A_i = \Omega$. Then, for all $i \in \{1,\dots,n\}$, for all $j \ne i$, $A_j \subset A_i^c$ (where $B^c = \Omega \setminus B$ for any event $B$). So, as you mentioned,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3359726",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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"Generalised eigenvectors" $Ax=\lambda Bx$ Proof of B-orthogonality? I came across this in a textbook, and it is not the usual definition of generalized eigenvectors I've seen.
The generalised eigenvectors of matrices A and B are vectors that satisfy
$Ax=\lambda Bx$
and $\lambda$ is the corresponding generalised eigen... | If $B$ is not positive definite, then the generalized eigenvalues may not be real, for example: if $A = \left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right)$ and $B = \left(\begin{smallmatrix}1&0\\0&-1\end{smallmatrix}\right)$, then since $B$ is invertible, the generalized eigenvalues are the eigenvalues of $B^{-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3359876",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solve $(x+1)(y+1)(z+1)=144$ in primes "Solve $(x+1)(y+1)(z+1)=144$ in primes".
So far, I have concluded that the solutions are $(x,y,z)=(2,3,11)$ or $(2,5,7)$ and their permutations. I worked like this:
*
*$x \equiv 0\mod 2\Rightarrow x+1=3, 144=2^4*3^2 \Leftrightarrow (y+1)(z+1)=48=2^4*3$
*
*$y \equiv 0\mod 2\Ri... | $144 = 2^4*3^2$
If $x,y,z$ are prime then $a=x+1,b=y+1,c=z+1 \ge 3$ so we will only consider factors at least $3$. If, wolog, $x+1, y+1 \ge 3$ then $z+1 \le \frac {144}9 = 16$.
So we need to only consider triplets of factors between $3$ and $16$.
The factors of $144$ are of the form $2^4*3^2$ and are $1,2,4,8,16, 3,6,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3360021",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why does this alternate Fibonacci relation appear to be true? So the normal fibonacci relation is famously as follows:
$$\begin{aligned}F_0 &= 0 \\
F_1 &= 1 \\
F_n &= F_{n-1} + F_{n-2}\end{aligned}$$
My friend appears to have discovered this alternate relation that allows you to skip by 3s, but I can't find it anywhere... | That alternative definition "forgets" the terms $F_n$ when $n$ is not a multiple of three. So $F_5$, say, could be a crazy elephant.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3360168",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Books that provide justifications of perturbations and asymptotic methods. I am looking for books which provide justifications (proofs of appropriate theorems ) of various perturbation methods.
In particular I would like to study about justification of matched asymptotic expansions, multiply scales,
WKB, Poincare metho... | I would say that a nice source to base your search on is
C. Kuehn, Multiple Time Scale Dynamics, Springer (2015), ISBN 978-3-319-12315-8, [link].
The book is very extensive concerning perturbation methods. In addition, it not only contains proofs, but also an extensive literature section, which guides you to literature... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3360317",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Calculating the curvature of product manifold $\mathbb{S}^2 \times \mathbb{R}$ I've read that $\mathbb{S}^2 \times \mathbb{R}$ is one of the model geometries of Thurston which has non constant curvature. I took it to mean that the manifold $(\mathbb{S}^2 \times \mathbb{R}, g)$ has non constant sectional curvature (wher... | The fact that $X$ and $Y$ are orthonormal says nothing about whether or not $X_1$ and $X_2$ are orthonormal. Thus, the condition $\langle R(X_1,Y_1)Y_1, X_1\rangle =1$ does not need to hold. In fact, $X_1$ and $Y_1$ could be linearly dependent, in which case the curvature is $0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3360460",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
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Bounded operator with an inverse on a dense invariant subspace Let $X$ and $Y$ be two Banach spaces such that $X$ is continuously embedded in $Y$. We suppose also that $X$ is dense in $Y$. Let $A\, \colon \, Y \to Y$ be a bounded operator that maps $X$ into itself such that the restriction $A_{|X} \colon X \to X$ is a ... | This is a nice question because everybody should rather immediately think that the answer must be negative (such a theorem would look much too good) -- and nevertheless it is far from obvious (at least for me) how to find a counterexample.
I believe that an example is contained in the paper https://doi.org/10.1007/BF0... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$\int_{-\infty}^{\infty}x e^{-x^2 + x(i+1)}dx$ how to? I am having some problems in solving the integral reported in the title, which is:
$$\int_{-\infty}^{\infty}x e^{-x^2 + x(i+1)}dx$$
As from the general theory of Gaussian integrals I have been trying to write the known integral
$$\int_{-\infty}^{\infty} e^{-x^2 - ... | Hint:
$$\int_{x=-\infty}^\infty xe^{-x^2+(1+i)x}dx\\
=\int_{x=-\infty}^\infty\left(x-\frac{1+i}2\right)e^{((1+i)/2)^2-(x-(1+i)/2)^2}dx
+\frac{1+i}2\int_{x=-\infty}^\infty e^{((1+i)/2)^2-(x-(1+i)/2)^2}dx\\
=e^{i/2}\int_{x=-\infty}^\infty ze^{-z^2}dz
+\frac{1+i}2e^{i/2}\int_{x=-\infty}^\infty e^{-z^2}dz.$$
The first inte... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3360666",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 3
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Prove $\frac{a}{b^{2}+1} + \frac{b}{c^{2}+1} + \frac{c}{a^{2} + 1} \ge \frac{3}{2}$ $a,b,c > 0$ and $a+b+c=3$, prove
$$ \frac{a}{b^{2} + 1} + \frac{b}{c^{2}+1} + \frac{c}{a^{2}+1} \ge 3/2 $$
Attempt:
Notice that by AM-Gm
$$\frac{a}{b^{2} + 1} + \frac{b}{c^{2}+1} + \frac{c}{a^{2}+1} \ge 3\frac{\sqrt[3]{abc}}{\sqrt[3]{... | Another way.
Your inequality is a sixth degree. We can reduce this degree by the Bacteria's method.
Indeed, by C-S, Murhead, Rearrangement and SOS(here it's also a Tangent Line method) we obtain: $$\sum_{cyc}\frac{a}{b^2+1}=\sum_{cyc}\frac{a^2(a+c)^2}{a(a+c)^2(b^2+1)}\geq\frac{\left(\sum\limits_{cyc}(a^2+ab)\right)^2}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3360763",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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How to find the following limit involving the given integration
I have tried the problem and got $\lim_{x\to0}4\frac{f(x)}{x}=4$. From this how to proceed?
Since $\displaystyle\lim_{x\to0}4\frac{f(x)}{x}=4$, so $\displaystyle\lim_{x\to0}\frac{1}{4}(4\frac{f(x)}{x})=\frac{1}{4}\lim_{x\to0}4\frac{f(x)}{x}=1\implies\lim_... | Let $g(x) = f(x)-x$. Then $g(x) \to 0$ as $x \to 0$, and $12\frac{g(4x)}{4x}-10\frac{g(2x)}{2x}+2\frac{g(x)}{x} \to 0$ as $x \to 0$. I'll show $\frac{g(x)}{x} \to 0$ as $x \to 0$ (which gives $f(x)/x \to 1$).
I'll show $\frac{g(x)}{x} \to 0$ as $x \to 0^+$. The following argument (easily modified) works for $x \to 0^-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3360892",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Find rational $\frac{p}{q}$ such that $\frac{1}{3000}<|\sqrt{2}-\frac{p}{q}|<\frac{1}{2000}$
Find rational $\frac{p}{q}$ such that $\frac{1}{3000}<|\sqrt{2}-\frac{p}{q}|<\frac{1}{2000}$
My Attempt
take a sequence which converges to $\sqrt{2}$ : $p_1=1+\frac{1}{2}, p_{n+1}=1+\frac{1}{1+p_n}$
I find how to calculate th... | Try $$ \frac{58}{41}.$$
It is well-known that the continued fraction of $\sqrt 2$ is
$$ 1+\frac1{2+\frac1{2+\frac1{2+\frac1{2+\ldots}}}}$$
Numerically(!), we find the continued fractions for $\sqrt 2+\frac1{2000}$ an $\sqrt 2+\frac1{2000}$:
$$ 1+\frac1{2+\frac1{2+\frac1{2+\frac1{2+\frac1{\color{red}1+\frac1{\ldots}}}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3360994",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Color a 1000 by 1000 grid with 0s and 1s Show that one could remove 990 rows with a 1 remaining in every column, OR delete 990 columns with 0 in remaining in every row.
My approach: let $r_i$ be the number of 1s in row i and $c_i$ be the number of 0s in column i but then I am stuck. Any hint would be appreciated!
| Not a solution, too long for comments. I was thinking on the problem and decided to write my thoughts; my first idea was to use the pideonhole principle and it would be straightforward; it is more complicated than I thought.
A second idea is to try to use some linear indepedence argument, considering rows and columns a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3361240",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that $X_{\tau}$ is a random variable. Let ${Y_{n}, n \geq 0}$ be real-valued random variables on $(\Omega, \mathcal{F}, P)$ that satisfy
$\lim _{x \rightarrow \infty} X_{n}(\omega) = \infty$
for every $\omega \in \Omega$, and let $B < \infty$ be a real number. Prove that the integer-valued
quantity
$$τ (ω) := \in... | Hint: Write $$X_{\tau} = \sum_{n=0}^{\infty} X_n 1_{\{\tau=n\}}$$
*
*What do you know about the measurability of $X_n$ and $1_{\{\tau=n\}}$?
*What do you know about measurability of products, sums and limits of measurable functions?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3361368",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Computing $\underset{x\rightarrow0}{\lim}\big(a^{x}+b^{x}-c^{x}\big)^\frac{1}{x}$ A friend asked me to help him with calculating a certain hideous limit:
$$\underset{x\rightarrow0}{\lim}\big(a^{x}+b^{x}-c^{x}\big)^\frac{1}{x},\space\space0<a,b,c\in\mathbb{R}$$
I came up with a solution (and wolfram alpha confirmed), bu... | Apply $\ln$ to the expression to get
$$\tag 1 \frac{\ln(a^x+b^x-c^x)}{x}.$$
Let $f(x) = \ln(a^x+b^x-c^x).$ Then $(1)$ equals
$$\frac{f(x)-f(0)}{x}.$$
The limit of this as $x\to 0$ is, by definition, $f'(0).$ Let's compute:
$$f'(x) = \frac{1}{a^x+b^x-c^x}\cdot (\ln a\cdot a^x + \ln b\cdot b^x -\ln c\cdot c^x).$$
Thus $f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3361483",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
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Integrate $\int{\frac{x^2-1}{x^4+3x^3+5x^2+3x+1}}dx$ The answer of this integral is
$$\int{\frac{x^2-1}{x^4+3x^3+5x^2+3x+1}}dx$$
$$=\frac{2}{\sqrt{3}}arctan(\frac{2}{\sqrt{3}}(x+\frac{1}{x})+\sqrt{3})+C$$
But I can't figure out how could I solve it. I tried to use partial fraction, but the denominator
$$x^4+3x^3+5x^2... | \begin{align}
&\int{\frac{x^2-1}{x^4+3x^3+5x^2+3x+1}}dx\\
=&\int \frac{ 1 - \frac{1}{x^2}}{x^2+3x+5+\frac{3}{x}+\frac{1}{x^2}}dx
=\int \frac{ d\left( x+\frac{1}{x}\right) }{\left( x+\frac{1}{x}+\frac{3}{2}\right)^2 + \frac 34 } \\
=&\ \frac{2}{\sqrt{3}}\arctan\left[\frac{2}{\sqrt{3}}\left(x+\frac{1}{x}\right)+\sqrt{3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3361579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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The number of the group homomorphsim , $\phi : S_3 \to S_3$ Surely, $S_3 = \langle(1,2), (1,2,3)\rangle$
And Put $\alpha = (1,2)$ and $\beta = (1,2,3)$ for homomorphism, $\phi : S_3 \to S_3$
Say $\phi(\alpha) =f(\in S_3)$ and $\phi(\beta) =r(\in S_3)$
Then, $\vert f \vert $ and $\vert r \vert$ should be divisor of the... | When classifying homomorphisms of a group, generally one cannot prescribe the orders of images of generators independently.
Indeed: If $\operatorname{ord}(\phi((12))) = 1$, then $\phi((12)) = 1$ (the identity permutation). But in a symmetric group any conjugacy class consists precisely of all permutations of a given cy... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3361706",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Behaviour of $x^n$, $ln(x)$, and $e^x$ as $x\to \infty$ In the chapter "Limits of a Function", I came across the following property:
As $x\to \infty$, $\ln(x)$ increases much slower than any positive power of $x$ where as $e^x$ increases much faster than any positive power of $x$.
So the following properties hold good... | 4) follows by applying L'Hopital's Rule $n$ times. (You will end up with $\lim_{x\to \infty} \frac {n!} {e^{x}}$ which is $0$). 2) is same as 4) with $x$ changed to $\ln x$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3361802",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Different solution for same contour integral $\int_{0}^{\infty}\frac{\cos(x)}{1+x^2}dx$ This question might be silly but I am quite puzzled by this problem.
In this exercise I am required to solve the following integral
$$\int_{0}^{\infty}\frac{\cos(x)}{1+x^2}dx$$
which I will call $A$ from now on. Since $f(x)= f(-x)$ ... | Unfortunately your choice of function and curve don't work and is why proof of contours vanishing or not vanishing are important (I wish physicists would take note here).
Using the residue theorem and equating it to your integral relies on the integral along the circular arc vanishing. But at the top of the arc
$$\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3361906",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove: Let $x,y \in \mathbb{R}$. If $x^{2}=y^{2}$, then $x=\pm{y}$. Prove: Let $x,y \in \mathbb{R}$. If $x^{2}=y^{2}$, then $x=\pm{y}$.
My attempt: If $x^{2}=y^{2}$ then, $\sqrt{x^{2}}=\sqrt{y^{2}}$ $\rightarrow$ $\pm{x}=\pm{y}$ $\rightarrow$ $x=\pm{y}$ and $-x=\pm{y}$. And I think that would leave me $x=y$, $x=-y$, $-... | It's generally a bad idea to "take square roots of both sides." It overcomplicates things, as the correct way to handle it is $\sqrt{x^2}=|x|$. Avoid the absolute value so you can avoid cases. Here's how.
Realize that from $x^2=y^2$ you get $x^2-y^2=0$. Now factor to get $(x+y)(x-y)=0$. But this means at least one ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3362020",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Use coordinate method to solve a pretty hard geometry problem Here is a hard geometry problem for my homework.
Let $D$ be a point inside $\Delta ABC$ such that $\angle BAD=\angle BCD$ and $\angle BDC=90^\circ$. If $AB=5,BC=6$ and $M$ is the midpoint of $AC$, find the length of $DM$.
(Taken from HK IMO Prelim 2012)
T... | Let $\angle BDA = \alpha$ and apply the sine rule to the triangle $\triangle ABD$,
$$\frac{\sin \alpha}{\sin\angle BAD}=\frac{5}{6\sin\angle BCD}$$
Given that $\beta = \angle BAD = \angle BCD$, we immediately get
$$\sin \alpha = \frac 56$$
Now, express the coordinates of $A(-x,-y)$ as follows,
$$x=AB\sin \angle ABD = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3362125",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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Is $p \lor (q \land r)\equiv p \land (q \lor r)$? Before writing, I'm not good at English. As I developed the law of distribution, I felt strange.
$$\begin{align}
p \land (q \lor r)& \equiv (p \land q) \lor (p \land r)\\
&\equiv X \lor (p\land r)\tag{$X=p \land q$}\\
&\equiv (X \lor p) \land (X \lor r)\\
&\equiv [(p \l... | At first you make a mistake in third rows from down to up.
Then it's true that in general $A \equiv A \land B$ isn't true for independent $A,B$.
But in your case they are dependent in value!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3362250",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
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How was row reduction used to obtain the result in this example? In the example below, I obtained a different result for $(2)$, and don't know how to reproduce the given one. Here is my result:
$$ \begin{bmatrix}a \\ b \\ c \\d\end{bmatrix} = \begin{bmatrix} -2 + 2r \\ -2 + r \\ -1 \\ r \end{bmatrix}$$
I verified my re... | Note that in your calculation, you use the last variable $d$ as the "free parameter" (you have the equation $d = r$ and then you express $a,b,c$ in terms of $r$). However, in the calculation attached from the book, they use the first variable $a$ as the free parameter. To convert your answer to the book's answer, start... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3362382",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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What happens when matrix is moved to the other side of the equation? I have been doing several questions related to Matrix.
I have a question ask me to simplify the expression (knowing that A,X,B are squared matrix with the same order):
$AXB=(AB)^2$
$AXB=A^2B^2$
Then I am thinking where should I put A and B when I mov... | There are a couple errors here. First note that $(AB)^2 = (AB)\cdot (AB) = ABAB$, not $A^2B^2$.
The bigger problem is with how you think about what you're doing. There is no mathematical notion of "moving one thing from one side of the equation to the other." There are only things which happen to look like they do this... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3362492",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How many integral solutions does $2x + 3y + 5z = 900$ have when $ x, y, z \ge 0$? Solution: Let $2x + 3y = u.$ Then we must solve $\begin{align} u + 5z = 900 \tag 1 \\ 2x + 3y = u \tag 2 \end{align}$
For $(1),$ a particular solution is $(u_0, z_0) = (0, 180).$ Hence, all the integral solutions of $(1)$ are $\begin{case... | You can solve this problem by creating equals Ax = b. Where A - transition matrix by x and b.
Let's consider this example.
Introduce F(m, i) where F(m, 1) - count of solutions 2x = m, F(m, 2) - count of solutions 2x + 3y = m, F(m, 3) - count of solution 2x + 3y + 5z = m ...
The following equations are valid:
F(m, 1) = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3362584",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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A not that obvious proof of vector property I want to show that if $\vec a \cdot \vec b=0$, then there must be some vector $\vec x$ to let $\vec x\times\vec b=\vec a$ which $\vec a ,\vec b,$ and $\vec x$ are not $\vec 0$ in $\Bbb R^3$.
The reverse is obvious. If $\vec x\times\vec b=\vec a$ then $\vec b\cdot(\vec x\ti... | Sketch:
*
*The case where $\vec{a}$ is $\vec{0}$ is trivial, so this case can be dealt with easily.
*The case where $\vec{b}$ is $\vec{0}$ is a counterexample (unless $\vec{a}$ also equals $\vec{0}$), so we must assume that $\vec{b}\not=\vec{0}$.
*Observe that $\vec{a}\cdot\vec{b}=0$ means that $\vec{a}$ and $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3362862",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that $f$ is a homeomorphism Let $f: (M,d) \rightarrow (N,\rho)$ be a one-one and onto mapping. Prove that the following are equivalent:
*
*$f$ is a homeomorphism.
*$g: N\rightarrow \mathbb{R}$ is contiuous if and only if $gof: M
\rightarrow \mathbb{R}$ is continuous.
To show $1 \implies 2$ I have proceed... | A hint for $\ 2\implies 1\ $ is to take $\ g(y) = \rho\left(y,f(x_0)\right)\ $ for any fixed $\ x_0\in M\ $, and show that the continuity of $\ g\circ f\ $ implies the continuity of $\ f\ $ at $\ x_0\ $.
Edit: I originally missed the OP's statement that the proof of the continuity of $\ g\ $ following from $1$ and the ... | {
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"timestamp": "2023-03-29T00:00:00",
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Is there a line homomorphism to a disk? This is a question from an interview. I am confused about this problem. I said yes in that interview because I remember something about Hilbert curve(I mean, is there a line that can fill a square completely?), but I am not sure. Is it right?
| No.
Let a line $L$ be homeomorphic to a disk $B(x,R)$
Then excluding a point $y \in L$ we will have that $B(x,R) \setminus \{point\} $ will be homeomorphic to $L\setminus \{y\}$.
This cannot be true because then line minus a point is disconnected and the disk minus a point is still connected.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Convergence of $x_{n+1} = \frac{(x_n)^3 + 3ax_n}{3(x_n)^2+a} $
Let $a \geq 0$ and let $(x_n)_n$ be a sequence such that $$x_{n+1} = \frac{(x_n)^3 + 3ax_n}{3(x_n)^2+a} $$ with $x_0 \geq 0$. Does this sequence converge to $\sqrt{a}$ for every $x_0 \geq 0$?
If $x_0 > \sqrt{a}$, then we can easily prove that the derivat... | The map $\displaystyle\;\varphi: x \mapsto \frac{x^3 + 3ax}{3x^2 + a}$ has $3$ fixed points: $0, \pm \sqrt{a}$.
Instead of the sequence $x_{n+1} = \varphi(x_n)$, one can study auxiliary sequence of the form $z_n = f(x_n)$ where $f(x)$ is a rational function over $x$
depending on the fixed points of $\varphi$.
If one i... | {
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Which is the importance of Young’s tableaux in mathematics? I don’t know much about combinatorics, I’m just getting started on this. I want to know, why Young’s tableaux are important? and why it is important to relate them to matrices?
Thank you very much.
| Any polynomial representation of $GL(n,\mathbb{C})$ has a weight-space decomposition which is basically simultaneous eigenspaces for the diagonal matrices (i.e, maximal torus). Now a lot of information about the representation is encoded in the dimensions of these weight spaces. So we are interested in finding out the ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How find the function $f(f(x)+xy)=f(x)+xf(y),\forall x,y\in R$ Find all function $f:R\to R$ and such
$$f(f(x)+xy)=f(x)+xf(y),\forall x,y\in R$$
Let $x=y=0$,then $$f(f(0))=f(0)$$
| Let denote that $f^0(x) = x$ and that for positive integer $n$, $f^n(x)=f(f^{n-1}(x))$.
It is easily proven by direct calculation that for every positive integer $n$ and every real $x$, $f^n(x)=f(x)+xf(0)$.
This fact implies that $f(f(x)-x)=f(x)-x +f(x)f(0)-xf(0)$.
But substituting $y = -1$, for every real $x$ provides... | {
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$ab+ac+bc \equiv 1 \bmod abc$ or "easy chinese remainder theorem problems" When teaching students about the Chinese remainder theorem, it is traditional to ask them questions like: "An integer $n$ is equivalent to $r_1 \bmod m_1$, to $r_2 \bmod m_2$ and to $r_3 \bmod m_3$. Compute $n \bmod m_1 m_2 m_3$." For example,
... | The trick is nice, but I am afraid it is hard to find any tuples, except for (2,3,5).
I tried brute force with 3 mods. For mods below 1000, only (2,3,5) satisfied.
However, even if you have only 1 pair that has $m_1\,m_2 \bmod m_3 = ±1$, it helps.
Example, $x ≡ a \bmod 5, x ≡ b \bmod 7, x ≡ c \bmod 9$
Since $5\times... | {
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How can I find an approximation of a smooth function $R^2 \to R^4$ such that when restricted to $S^1$ it is an immersion? I am having trouble with the following exercise:
Given a smooth function $f: R^2 \to R^4$ and $S^{1}$ embedded in $R^2$, then $\forall \epsilon >0$ there exists a smooth function $f_{1}$ such that $... | Here's a sketch of an elementary solution if you know the transversality theorem (e.g., see Guillemin & Pollack or "Parametric transversality theorem" here).
(1) Observe that you only need to perturb $f$ near $S^1$ to obtain the result. So let $\rho\colon \Bbb R^2\to\Bbb R$ be a smooth function that is $1$ in a small n... | {
"language": "en",
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Problem regarding bacteria increase modeled by formula Studying "Basic Mathematics" p.358, by Serge Lang, came across this exercise.
I don't know how to start with it. Any insight would be appreciated.
| Hint:
First note that $C=B(0)=10^6$.
Next, you can rewrite this equation as $\;\dfrac{B(t)}{B(0)}=\mathrm e^{kt}$, and you have to find $k$. You're given that
$$\frac{B(12)}{B(0)}=2. $$
WWhen ytou have $k$, you'll just have to solve
$$\frac{B(t)}{B(0)}=10.$$
Can you end the calculations?
| {
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Proof that a function is nonzero given constraints on $x$ Let $f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}, \;\left(a_{n} \neq 0\right) .$
Let $A=\max \left\{\left|a_{0}\right|,\left|a_{1}\right|, \ldots,\left|a_{n}\right|\right\}$.
Let $f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}, \left(a_{n} \neq... | It is obvious that $B=nA/{a_n}\ge n\ge1$
For $|x|>B$, we have $\frac{|f(x)|}{a_n|x|^n}\ge|1|-|\frac{a_{n-1}}{a_n}x^{-1}|-...-|\frac{a_0}{a_n}x^{-n}|\gt 1-\frac B n (B^{-1}+B^{-2}+...+B^{-n})\ge1-\frac Bn(nB^{-1})=0$
| {
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Find all $n$ for which $3n^2+3n+1$ is a perfect square. Find all natural numbers $n$ for which $3n^2+3n+1$ is a perfect square.
I used discriminant method but failed. Then I found upper and lower bounds of this expression:
Lower:$(n+1)^2$
Upper:$(2n)^2$
But, this too does not seem to be useful. Please help me.
| Cool. I'll just fully answer the question.
Starting with $ \ m^2=3n^2+3n+1\iff (2m)^2-3(2n+1)^2=1 \quad $. Lets call $ \ p=2m, \ \ $ $q=2n+1 \ \ $. So it's now $$p^2-3q^2=1$$
By a quick inspection the smallest solution is: $(p_0,q_0)=(2,1) \quad $. It can be used to find all others. Use it to write the number $1... | {
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Evaluate $\hat f(x):=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(t)e^{-itx}dt$ for $f(t)=\sum_{n=0}^{\infty}\frac{1}{2^n}1_{[-2,2]}(x-2n)$
$\hat f(x):=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(t)e^{-itx}dt$, for $f \in L^1(\mathbb R)$
Let $f(t)=\sum_{n=0}^{\infty}\frac{1}{2^n}1_{[-2,2]}(t-2n)$, where $1_{[.,.]}$... | If you are also required to be rigorous you have to justify the interchange of the integral and the infinite sum. You can do this by observing that your sereis converges in $L^{1}$ norm and $s_n \to f$ in $L^{1}$ norm implies $\hat {s_n} \to \hat {f}$ pointwise. [Take $s_n$ to be the n-th partial sum of the series]. ... | {
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Perpendicular bisector that pass through a fixed point Points A and B fixed, and point C moves on circle such that ABC acute triangle. $AT = BT$ and $TM \perp AC, \, TN \perp BC$. How can I proove that all the middle perpendiculars (perpendicular bissector) to $MN$ passes through a fixed point?
| Hint:Let's mark perpendicular bisector as (PB).
If C is coincident on A or B then M and N will be coincident on A and B respectively and the (PB) of MN is exactly the (PB) of AB.Also when the triangle is isosceles MN is parallel with AB, so again (PB) of MN and AB are coincident. That is the point is on the (PB) of AB... | {
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Limit of this expression when n tends to infinity The limit is equal to: $$\lim_{n\to\infty} n^2 \int_0^1 \frac{1}{(1+x^2)^n} dx$$
P.S. What should be the better approach for such kinds of problems?
| This might be a bit overkill but at least it works to show the divergence of the sequence and gives a nice additional result:
*
*Consider first $I_n = \int_0^1\frac{dx}{(1+x^2)^n}$.
*To bring in somehow the exponential function susbtitute $nx^2 = y$:
$$I_n =\frac{1}{2\sqrt{n}}\underbrace{\int_0^n\frac{dy}{\sqrt{y}\... | {
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$\int \sin x \cos x \ dx \neq \frac {\sin^2x}{2}$ if we use the identity $\sin A \cos B = \frac12[\sin(A-B) + \sin(A+B)]?$ How is it that we can get two different answers for an integral depending on whether we apply an identity or not?
Typically, $$\int \sin x \cos x \ dx = \frac {\sin^2x}{2}+C~.$$
However if we appl... | You forgot the arbitrary constant in your second method.
The non-constant part of the first result may be transformed as follows: $$\frac{\sin^2x}{2}=\frac{1-\cos 2x}{4}=-\frac14\cos 2x+C,$$ since $C$ is arbitrary.
| {
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How to substitute a complex number in a complex function? I know for the regular cases, but what I am after is something like this:
$\lim\limits_{z \to 1+5i} ix+y$
is it: i(1) + (5i) = 6i, or: i(1) + 5 = 5+i
?!
| We usually define $z=x+yi$ so $x,\,y\in\Bbb R$, making the answer $5+i$.
| {
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Let $A,B\subseteq \Bbb R $such that A and B are bounded, show that AB is bounded $AB=\lbrace ab \mid a\in A , b\in B \rbrace$.
Let $M$ is an upper bound of $A$ and $N$ is an upper bound of $B$.
Then can we say that $MN$ is upper bound of $AB$?
So how can we prove that $AB$ is bounded?
| No $MN$ won’t be the upper bound of $AB$. Take the example of $A=[-1,0]$ and $B=\{1\}$.
However if $m$ is such that $\vert a \vert \le m$ for all $a \in A$ and $n$ is such that $\vert b \vert \le n$ for all $b \in B$, $mn$ is such that $\vert x \vert\le mn$ for all $x \in AB$. That proves that $AB$ is indeed bounded.
| {
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Prove S is a subspace of $R^3$ Let S={(a,b,c)$\in$$R^3$|2a+b-3c=0}. Prove S is a subspace of $R^3$
I just asked another question like this earlier and this is just to confirm im on the right track when approaching problems like these. After starting this problem, i am finding myself confused when it comes to making sur... | $$2a_1+b_1-3c_1=0$$ $$2a_2+b_2-3c_2=0$$
Thus $2(a_1+a_2)+(b_1+b_2)-3(c_1+c_2)=0+0=0$
So $u+v \in S$
Similarly prove that $au \in S ,\forall a \in \Bbb{R}$
| {
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Prove $y = z$ if $y^p = z^p$ where $p \in \mathbb{N}$ and $y, z \in \mathbb{R}_{++}$ As above. I am beginner in analysis. I don't know how rigorously you have to prove things. To me it's obvious.
| Let $f(x)=x^p$ and $p\in\mathbb N^*$ then $f'(x)=px^{p-1}>0$ for $x>0$.
So $f \nearrow$ strictly on $(0,+\infty)$, which implies injectivity.
Thus $f(x)=f(y)\implies x=y$.
| {
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Vectorization of a diagonal matrix Is there any representation for the vectorization of a diagonal matrix $\text{vec}(\text{Diag}(x))$?
For example, for two elements $$\text{vec}(\begin{bmatrix} x_1 & 0 \\ 0 &x_2 \end{bmatrix}) = \begin{bmatrix} 1 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 1 \end{bmatrix} \begin{pmatrix} x_1 \\ x_2 ... | For $x\in{\mathbb R}^{n}\,$ you can express the matrix
$A\in{\mathbb R}^{n^2\times n}\,$ in index notation
$$A_{ij} = \begin{cases}
{\tt1}\quad{\rm if}\;(i+n=j+nj) \\
{\tt0}\quad{\rm otherwise} \\
\end{cases}
$$
This translates verbatim into any language which permits array comprehensions.
For example, to reproduce the... | {
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Prove divergence of a product of sequences using contradiction If a sequence converges (not to zero) and another sequence diverges, prove by contradiction that the product of these sequences is divergent.
This is a past test paper question that I am looking at and I really have no idea how to write the proof. My own w... | If $(a_n)$ converges to $L\neq 0$, then you have $\vert a_n \vert \ge \vert L \vert /2 >0$ for $n$ large enough. Let say $n\ge N$.
Now if the sequence $(b_n)$ diverges, either it is not bounded and therefore $(a_n b_n)$ is not bounded either and therefore diverges. And if $(b_n)$ is bounded (and doesn’t converge), it h... | {
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Expected number of "trips" for spatial Poisson process
The appearance of an object of interest in a region of area $x$ units$^2$ follows a (two-dimensional) spatial Poisson process $N$ with constant rate $\lambda$ units$^{-2}$. An observer, who wishes to witness the object, can only view a fixed amount of total area, ... | The probability of observing the object on the $i$th trip is equal to
$$P(N(B_i) \ge 1) = 1-P(N(B_i) = 0) = 1 - e^{-\lambda \text{area}(B_i)}$$
since the areas are disjoint, and for a Poisson process the number of objects in disjoint intervals are independent, the number of trips till you observe the object once follow... | {
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Why isn't $\lim\limits_{x\to 3}=6$ true in these conditions? Let $\left(u_n\right)$ be a succession with general term:
$$u_n=3+\frac{(-1)^n}{n}$$
and $h$ a real function such that $\lim h\left(u_n\right)=6$.
This is a multiple choice question and I'm pretty sure the right answer is the following:
If $h$ is continuous... | The point is this:
$\lim_{n\to\infty}h(x_n)=6$ means that the sequence $(h(x_n))_{n\in\mathbb{N}}$ (with the given sequence $(x_n)_{n\in\mathbb{N}}$) converges to 6.
But $\lim_{x\to 3}h(x)=6$ means, that for every sequence $(x_n)_{n\in\mathbb{N}}\subset \mathbb{R}\setminus \{3\}$ that converges to $3$ it holds that the... | {
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(elementary?) proof of simplicial homology groups of $\Delta_n$ Using the (trivial) CW-complex structure of $\Delta_n$, I would like to compute the homology groups of $\Delta_n$. It's obvious that (for $k \leq n$) $C_k \simeq \mathbb{Z}^{{n+1 \choose k+1}}$, and that $im \delta_{k+1} \subset ker \delta_k$. It's the rev... | You should remember that it is difficult to construct cellular homology without singular homology (you must figure out how to define degrees of maps of spheres) and then it is even harder to show it is even a homeomorphism invariant.
However, if you wish to sweep these things under the rug: the n-simplex is homeomorphi... | {
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Determine values for which the general solution converges Textbook problem.
Given the following general solution to a recurrence relation
$$z_n = \alpha(1+\sqrt{3})^n + \beta(1-\sqrt{3})^n$$
For which values $\alpha, \beta$ does the solution converge? And determine the order of the rate of convergence for these ... | You are correct.
For $a \neq 0$ we have that $z_n \to +\infty$ or $-\infty$ since $(1+\sqrt{3})>1$
Also $(1-\sqrt{3})<1 $
Thus the sequence converges $\forall b \in \Bbb{R}$ and for $a=0$
And converges to zero ,for every such value.
| {
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Is $\infty + (\infty/\infty)$ indeterminate? I know $ (\infty/\infty)$ is indeterminate, but it can't be less than $0$.
So can you assume $\infty + (\infty/\infty)$ is determinate because $\infty + n$ where $n\ge 0$ is still $\infty$ ?
The equation this question is based off of is $$\lim_{n \to \infty} \frac{n \log ... | This is a very interesting question... I believe you are correct. Below is a proof.
*
*Let $lim_{x \rightarrow a}[f(x)/g(x)]$ be some arbitrary indeterminate function such that $lim_{x \rightarrow a}[f(x)/g(x)]=\infty/\infty$ where $x\in \mathbb{R}$ and $a$ is a finite real number.
*Also let $lim_{x \rightarrow b}[... | {
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Let $f:[0,n]\to \Bbb R$ be continuous with $f(0)=f(n)$. Then there are $n$ pairs of numbers $x,y$ such that $f(x)=f(y)$ and $y-x\in\Bbb N$.
Theorem. Let $f:[0,n]\to \Bbb R$ be continuous with $f(0)=f(n)$ ($n\in\Bbb N$). Then there exist (at least) $n$ distinct pairs of numbers $x,y$ which satisfy $f(x)=f(y)$ and $y-x\... | I thought it would be nice to present a direct proof, so pardon me for answering an old question.
First extend $f$ periodically, then for $1\le j<n$, define $g_j(x)=f(x)-f(x+j)$. Notice $g_j$ is $n$-periodic because $f$ is. I will show each $g_j$ has at least 2 roots in $[0,n)$.
First suppose $g_j$ has no roots. Since ... | {
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Show that each $k\in \mathbb{N}$ can be represent as $ k = S_m(a)+S_n(b)$ Let $a,b$ are non negative integers and $m,n$ are positive integers.
Let define
$$S_m(a)=0^m+1^m+2^m+...+a^m$$
and $$S_n(b)=0^n+1^n+2^n+...+b^n$$
Question
Show that each $k\in \mathbb{N}$ can be represent as
$$ k = S_m(a)+S_n(b)$$
Example
$5... | It appears that this claim is untrue. There are quite a few numbers which cannot be expressed in this manner, the smallest being $52$.
Observe the terms $S_n(k) \leqslant 52$
First powers = $\{0,1,3,6,10,15,21,28,36,45\}$
Second powers = $\{0,1,5,14,30\}$
Third powers = $\{0,1,9,36\}$
Fourth powers = $\{0,1,17\}$
Fifth... | {
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$\int (f)^n \, \textrm dx$ where $f$ is a polynomial and $n$ is a positive integer Besides expanding the integrand, is there some general method for solving indefinite integrals of the form $\int (f)^n \, \textrm dx$ where $f$ is a polynomial and $n$ is a positive integer? For example, $$\int (x^2 +x)^{100} \, \textrm ... | $\int(x^2+x)^{100}~dx$
$=\int x^{100}(x+1)^{100}~dx$
$=\int x^{100}\sum\limits_{n=0}^{100}C_n^{100}x^n~dx$
$=\int\sum\limits_{n=0}^{100}C_n^{100}x^{n+100}~dx$
$=\sum\limits_{n=0}^{100}\dfrac{C_n^{100}x^{n+101}}{n+101}+C$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3366510",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Find the smallest positive integer x where 149|(x^2-69^3) $$
\text{Find the smallest positive integer x where 149|}\left( \text{x}^2-69^3 \right)
$$
I almost do not spot any clue about this question.
It is from PUMaC-CHINA, 2019.8.17
| This follows the comment by J. W. Tanner and the answer by sirous.
All congruences are mod $149$.
We have $x^2-69^3 \equiv x^2+36 =x^2+6^2 \equiv (x-6j)(x+6j)$, if $j^2\equiv -1$.
Write $149 = 7^2 + 10^2$. Then $10j \equiv 7$ and so $j\equiv 105$.
Thus $x^2-69^3 \equiv (x-34)(x+34)$ and so $34$ is the smallest positiv... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Simple property of arrow notation
Assume that $\kappa \to (\lambda)^r_s$ holds.
Prove that if $r'\le r$, $\kappa \to (\lambda)^{r'}_s$
I proved similar cases like when $s'\le s$ etc. but I have no idea about this case.
Actually I can't understand why that statment has to be true.
Any help or suggestion is welcome!
| Given a coloring of the $r'$-sets, define a coloring of the $r$-sets as follows. If $x_1\lt x_2\lt\cdots\lt x_r$, give $\{x_1,\dots,x_r\}$ the same color as $\{x_{r-r'+1},\dots,x_r\}$; i.e., a set of size $r$ gets the same color as the set of its $r'$ biggest elements.
Suppose $H$ is a monochromatic set for the new col... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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} |
Evaluate $\frac{1}{m_1} +\frac{1}{n_1} +\frac{1}{m_2} +\frac{1}{n_2}+...+\frac{1}{m_{2011}} +\frac{1}{n_{2011}}$ When $a=1,2,3,...,2010,2011$ the roots of the equation $x^2 -2x-a^2-a=0$ are $(m_1,n_1 ), (m_2,n_2 ), (m_3,n_ 3),..., (m_{2010},n_{2010} ), (m_{2011},n_{2011 }) $ respectively. Evaluate $\frac{1}{m_1} +\frac... | Hint:
$$
\frac{-2}{a(a+1)} = \frac2{a+1} - \frac 2a
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3366852",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Difference between a vector function and parametric equations What is the difference between a vector function and parametric equations? Both are capable of describing plane/space curves and both can indicate direction. The only notable difference I see is that a vector function includes a measured distance between a g... | What do you mean that a vector function includes a measured distance (magnitude)?
Vector functions and parametric equations are essentially the same thing.
Sometimes, if you're working in a different coordinate system, like polar coordinates, you might see that they include the magnitude $r$ in their functions.
For ex... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3366963",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Dense $G_\delta$ of open interval This is not a hard question, I just want to make sure whether it is possible or not. Let $I$ be an open interval of $\mathbb R$. The question is it always possible to find $G_\delta$ dense subsets of $I$? I know it true to find countable dense of $I$ as it consider as a second countabl... | Take any countable subset $C$ of $I$ and then $I\setminus C$ is a dense $G_\delta$. Or use the complement of a homeomorphic copy of a Cantor set (the standard Cantor set lies in $[0,1]\subseteq(-1,1)$ and all open intervals are homeomorphic, so we always have them).
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What are the ways to bound function for using the theorem of dominated convergence? I need to bound $f(x,n)=\frac{\sqrt{n}}{1+n\ln(1+x^2)}$, on $(0,1)$. I tried to use the fact that $a^2+b^2\geqslant 2ab$, but integral $\int\limits_0^1 \frac{1}{\sqrt{\ln(1+ x^2)}}\ dx$ is not convergent so I can't use theorem of domina... | You do not need to bound it.
$f(x,n)$ is decreasing sequence of functions and $\int_0^1f(x,1)dx <+\infty$
Let $g(x,n)=f(x,1)-f(x,n)$
$f(x,n) \to 0 ,\forall x \in (0,1)\Longrightarrow g(x,n) \to f_1(x,n), \forall x \in (0,1)$
Thus $\int_0^1g(x,n)dx \to \int_0^1 f(x,1)dx \Longrightarrow \int_0^1 f(x,n)dx \to 0$
By monoto... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Is every group the unit group of some ring? Let the functor $F\colon\bf Ring\rightarrow\bf Grp$ send the ring $A$ to its group of units $A^\times,$ and the ring homomorphism $f\colon A\rightarrow B$ to the group homomorphism $f^\times\colon A^\times\rightarrow B^\times:a\mapsto f(a)$.
I was curious about this functor, ... | Your statement about $\mathbb{F}_2[G]$ is incorrect. Consider when $G = \mathbb{Z}_5$, generated by some element $a$ with $a^5 = e$. Then,
$$(e + a^2 + a^3)(e + a + a^4) = (e + a^2 + a^3) + (a + a^3 + a^4) + (a^4 + a + a^2) = e + (a+a) + (a^2+a^2) + (a^3+a^3) + (a^4+a^4) = e$$
So, the unit group of $\mathbb{F}_2[G]$ in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3367423",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 3,
"answer_id": 0
} |
What is the minimum number of groups of three coins you can flip in order to flip every coin so that it is heads-down?
Question: Suppose that there are seven coins arranged in a circle. Every coin is heads-up. Your goal is to flip every coin so that it is heads-up. You may, however, only flip groups of three adjacent ... | HINT, as requested.
I don't think doing things mod $7$ helps. Instead I would do it mod $2$, to represent the fact that if you flip the same coin twice, that is equivalent to doing nothing. Since you want to flip every coin from Heads to Tails, any solution with $x$ moves (i.e. flipping $x$ groups) must satisfy:
$$3x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3367499",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Texts on Mathematical Billiards I want to study the theory of mathematical billiards, and was looking for a text for self-study. If you have a text in mind that is more general i.e. on dynamical systems as a whole but still contains some material on billiards, that would be great too. I'm an undergraduate with backgrou... | I think these are some books you might find interesting, regarding mathematical billiards at a relatively introductory level
*
*An Introduction to Mathematical Billiards, by Utkir A. Rozikov;
*Chaotic Billiards, by Nikolai Chernov;
*Geometry and billiards, by Sergei Tabachnikov.
Different approaches and fascinat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3367595",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Convolutions Support. It is known that:
$$
\operatorname{supp}(u *v) \subset \operatorname{supp}(u) + \operatorname{supp}(v)
$$
Where:
$$
\operatorname{supp}(u) = \overline{\{x \in \mathbb{R}^n: u(x) \neq 0\}}
$$
And:
$$
(u*v)(x)=\int_\limits{\mathbb{R}^n} u(x-y)v(y) dy
$$
I need an example of two functions $u,v$ such ... | Let $u$ be any non-negative function with compact support which has the value $1$ on some open set. Let $v(x)=e^{-x^{2}}$. Then $(u*v)(x) >0$ for all $x$. So $u*v$ does not have compact support.
For an explicit example let $u(x)=1$ for $0 \leq x \leq 1$, $0$ for $x \geq 1+\frac 1 n$as well as for $x \leq -\frac 1 n$, $... | {
"language": "en",
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"source": "stackexchange",
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Proving that the sublevel set of a quadratic function is convex
Let
$$C := \{ x \in \mathbb{R} : 2x^2 \le 1 \}$$
Prove that $C$ is convex.
I started with the definition: for $ x_1, x_2 \in C$
$$ \lambda x_1 + (1-\lambda)x_2 = \dots $$
but didn't make any progress. How should I approach this problem?
| Hint: I'd rewrite the set as
$$C = \{x\in{\Bbb R}\mid -1/\sqrt 2\leq x\leq 1/\sqrt 2\} = \left[-\frac{1}{\sqrt 2};\frac{1}{\sqrt 2}\right].$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $f:(a,b)\to \mathbb{R}$ is differentiable in a point $x\in (a,b)$, show that $\lim\limits_{h\to 0}\frac{f(x+h)-f(x-h)}{2h}$ exists
If $f:(a,b)\to \mathbb{R}$ is differentiable in a point $x\in (a,b)$,
show that $\lim\limits_{h\to 0}\frac{f(x+h)-f(x-h)}{2h}$ exists and
that it is equal to $f'(x)$.
I figured tha... | Hint:
Just consider
$$\frac{f(x+h)-f(x-h)}{2h} =\frac{1}{2}\left(\frac{f(x+h)-f(x)}{h} + \frac{f(x)-f(x-h)}{h}\right) $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3368036",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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How to prove this inequality with Big O term? Let $s= s_0-\zeta_0^{-1/2}b^{-1/6}(1+\mathcal{O}(\sqrt{b}))$
where $\zeta_0 = \left(\frac{3\pi}{4}\right)^{2/3}$ and $s_0 = b^{-2/3}\zeta_0.$ Note that here $b$ is a parameter.
We define $\omega(s) = \exp\left(\frac{2}{3}s_{+}^2\right)$ where $s_{+} = \max(0,s).$ I want to ... | The second inequality never holds.
The first inequality is equivalent to $\frac 23s_+^2\le \frac{\pi}{2b}-\frac{1}{\sqrt{b}}+D$ for some constant $D$. Since $\frac{\pi}{2b}-\frac{1}{\sqrt{b}}\ge -\frac 1{4\pi}$ for each $b\ge 0$, if $D\ge -\frac 1{4\pi}$ then the inequality holds for $s_+=0$. So it remains to consider... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If $A \in \mathbb{R}^{n\times n}$ and $\textbf{x}\in \mathbb{R}^{n}$why $A = \frac{1}{2} A + \frac{1}{2} A^\top $ is not always true? If $A \in \mathbb{R}^{n\times n}$ and $\textbf{x}\in \mathbb{R}^{n}$,
It is possible to prove that
$$
\textbf{x}^\top A \textbf{x} = \textbf{x}^\top(\frac{1}{2} A + \frac{1}{2} A^\top)... | It is true that if
$$x^T A y = x^T B y$$
for all vectors $x,y$, then $A=B$. The proof is by plugging in basis vectors $e_i$ (with a $1$ in the $i$th entry, and zeroes elsewhere):
$$A_{ij} = e_i^T A e_j = e_i^T B e_j = B_{ij}.$$
Now if you are forced to multiply by the same vector on both sides, instead of two arbitrary... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3368309",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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the conjugates of r-cycles are r-cycles. Let $0<r \leq n$ be positive integers, and consider an $r$-cycle $\sigma(j_1\ j_2...j_r) \in S_n.$ I want to show that $\tau \sigma \tau^{-1}=(\tau(j_1)\tau(j_2)...\tau(j_3))$.
Let $\tau \in S_n.$ We can write $\sigma$ as $\sigma=(j_1j_r)(j_1j_{r-1})...(j_1j_2)$, so \begin{align... | You don't need to split it into a product of transpositions. It is much easier. If $1\leq k<r$ then where does $\tau\sigma\tau^{-1}$ send $\tau(j_k)$?
$\tau\sigma\tau^{-1}(\tau(j_k))=\tau\sigma(j_k)=\tau(j_{k+1})$
And similarly $\tau\sigma\tau^{-1}(\tau(j_r))=\tau(j_1)$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$n^\pi$ and $n^e$ for positive integer $n\ge2$ Is it known if $n^\pi$ or $n^e$ are integer for some integer $n\ge 2$?
Gelfond - Schneider's theorem does not answer the question because both base and exponent should be algebraic and haven't found nothing about this in internet.
I don't expect a proof, it seems extremely... | I think Schanuel's conjecture should imply that $n^e$ is never an integer. Here is the idea of how that could go:
Suppose $n^e = m$ then $e \log(n) = \log(m)$.
This would imply $\mathbb{Q}(1, \log(n), \log(m), e, n, m)$ has transcendence degree at most 2, violating the conjecture (assuming we rule out the case that $1$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is it possible to have $\int_0^\infty f(x)\,dx$ convergent and $|f|=1$ everywhere? I came across a problem that asked if it is posible for a function to be Riemann integrable function in $[0,+\infty)$ but also $|f(x)|\geq 1$ for all $x\geq 0$.
At first I thought it was imposible, but I realized that only holds for con... | Here is a more elementary example:
Let $\phi$ be a $1$-periodic function such that $\phi(x)=-1$ for $x \in [0,{1 \over 2})$ and $\phi(x) = 1$ for $x \in [{1 \over 2},1)$.
Note that for an integer $n$ and $x \in [0,1]$, we have $|\int_n^{n+x} \phi(2^kt)dt| \le {1 \over 2^{k+1}}$.
Define $f(x) = \sum_{n=0}^\infty 1_{[n,n... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Can we obtain isomorphic semi-direct groups by different homomorphism given? A semi-direct product of two finite groups is determined by the homomorphism from one group to the other's automorphism group. I wonder if it is possible to have two different homomorphism that turns out to have the same resulting semi-direct ... | The tl;dr is yes! As Arturo mentioned in the comments, every (nontrivial)
$\varphi : \mathbb{Z}/p \to \text{Aut}(\mathbb{Z}/q)$ (when $p \mid q-1$) gives rise to the same semidirect product (the unique nonabelian group of order $pq$).
As for your second question, involving a "general pattern", the answer is a (somewha... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Potentially overthinking transformations of functions... need confirmation So this may be really simple. The question is
Write the equation of the transformed function $y=x^2$ after the following transformation in the order given.
*
*a vertical compression by a factor of $\frac13$ followed by a transformation of $3$... | For any function $y=f(x)$, transformations can be described by the equation
$$y=af[k(x-c)]+d$$
where $|a|$ is the vertical dilation, $|k|$ is the horizontal dilation, $c$ is the horizontal translation, and $d$ is the vertical translation. If $a<0$, there is a vertical reflection, and if $k<0$, there is a horizontal re... | {
"language": "en",
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Extreme confusion with the product of distributions Let $l_1$ and $l_2$ be two distributions in disjoint variables $x_1, ..., x_n$ and $y_1, ..., y_m$. Then it is said to be possible to define a product distribution.
However, I am fundamentally confused. Distributions are in fact linear functionals on the space of smoo... | If you want to multiply $\delta(x_1)$ and $\delta(x_2)$ you first need to make them into functions acting on the space space, so you multiply $\delta(x_1)Id(x_2)$ and $Id(x_1)\delta(x_2)$, where $Id$ is just the identity map. With that interpretation you get $\delta(x_1)\delta(x_1)=\delta(x_1, x_2)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3369069",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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Max eigenvalue of symmetric matrix and its relation to diagonal values I saw few questions about it, but still can't understand.
Let $A$ be a symmetric matrix and $\lambda_{\max}$ its largest eigenvalue. Is the following true for all $A$?
$$
\lambda_{\max} \ge a_{ii} \forall i
$$
That is, is the largest eigenvalue of ... | Let $\lambda_1,\cdots, \lambda_n$ be eigenvalues of $A $ in increasing order,i.e. $\lambda_n$ is the maximum eigenvalue.
By Min-max theorem $$\lambda_n=\max_{x\in \mathbb{R^n},||{x}||=1 }x^TAx=\sum x_ix_j a_{ij} $$
while on the other hand$$a_{ii}=e_i^TAe_i $$
It follows that $$\lambda_n\ge a_{ii} \qquad \forall i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3369267",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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For a stopping time does $E[(\tau \wedge t)1_A]=E[(\tau \wedge s)1_A] $ for $s \le t $ and any $A \in \mathcal F_{\tau \wedge s }$? I have the following question as stated in the title:
For a stopping time $\tau $ does $E[(\tau \wedge t)1_A]=E[(\tau \wedge s)1_A] $ for $s \le t $ and any $A \in \mathcal F_{\tau \wedge... | We have
\begin{alignat*}{2}
\mathbb{E}[(\tau \wedge t)\mathbf{1}_A] & = \mathbb{E}\Big[\mathbb{E}\left[(\tau \wedge t)\mathbf{1}_A\;|\;\mathcal F_{\tau \wedge s }\right]\Big]\quad\text{by the tower property}\\
& = \mathbb{E}\Big[\mathbb{E}\left[(\tau \wedge t)\;|\;\mathcal F_{\tau \wedge s }\right]\mathbf{1}_A\Big]\qua... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3369535",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Evaluating $\lim_{n\to \infty}\left(\frac{1}{\sqrt{n^2}}+\frac{1}{\sqrt{n^2+1}}+\frac{1}{\sqrt{n^2+2}}+\dots +\frac{1}{\sqrt{n^2+2n}}\right)$
Question:
Evaluate the following limit: $$\lim_{n\to \infty}\left(\frac{1}{\sqrt{n^2}}+\frac{1}{\sqrt{n^2+1}}+\frac{1}{\sqrt{n^2+2}}+\dots +\frac{1}{\sqrt{n^2+2n}}\right)$$
My ... | $$\lim_{n\rightarrow \infty} \sum_{k=0}^{2n} \frac{1}{n} \frac{1}{\sqrt{1+k/n^2}}= \int_{0}^{2} dx =2.$$ Here $\frac{k}{n^2}-\rightarrow 0.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3369675",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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$X$ an integral element over $K[X^d]$ I'm currently redoing some homework for my algebra test and I came across a task that wants me to find a noether normalization of $k[X]$, where $k$ is a field. The proof I know uses the argument, that $X$ is an integral element over $k[X^d]$ for $d$ element of IN. However I don't r... | It is a solution of the monic polynomial equation
$$y^d-X^d=0$$
where the indeterminate is $y$ and $X^d$ is the element of the ring.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3369808",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Small question regarding left invariant vectorfields Usually, one writes out the left invariance of vector fields $X$ on a Lie group $G$ as $$(L_x)_*X=X$$for every $x$.
However, I have trouble understanding this equality, since both sides does not map to the same domain. Evaluating the right hand side in $y$ gives an ... | The derivative of the left multiplication map, $L_g(x)=gx$ gives an isomorphism of the tangent spaces. See this.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3369911",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Show that the limit is in the set I'm having some trouble proving the following :
Let $B = \{(x,y)\in \mathbb{R}^2 : e^x - sin(y)\leq 0\}$ and let $(x_{k},y_{k})_{k \in \mathbb{N}}$ a sequence of elements in B. We suppose that it exists $(x,y) \in \mathbb{R^2}$ such as $\left \| (x_{k},y_{k}) -(x,y)\right \|_{1}\right... | You probably know that exponential and sine are continuous functions. Hence we have $e^{x_k}\to e^x$ and $\sin(y_k)\to \sin(y)$ when $k\to\infty$. Also, since the sequence $(x_k,y_k)$ contains elements of $B$ we have $e^{x_k}-\sin(y_k)\leq 0$ for all $k\in\mathbb{N}$. By taking $k\to\infty$ we get $e^x-\sin(y)\leq 0$. ... | {
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"answer_id": 1
} |
Integration of function nonnegative at $\mathbb{N}$ I'm having a little trouble to write down my solution of exercise 4G of Bartle's book The Elements of Integration and Lebesgue Measure. Here is the problem:
Let X = $\mathbb{N}$ and $\mathcal{A} = 2^X$ be $\sigma$-Algebra and define the measure to be the counting me... | Denote $v$ the counting measure.
Then $$\int_{\{n\}}f(k)dv=\int_{\{k \in \Bbb{N}:k=n\}}f(k)dv=f(n)\int_{\{n\}}dv=f(n)v(\{n\})=f(n)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3370157",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Halting problem within a finite time interval? Let the finite time halting problem to be one where the program is counted as "not halting" if the Turing machine takes more than a given time interval to run the program.
Is this finite time halting problem undecidable?
| This subject has annoying terminological redundancy - e.g. "decidable," "computable," and "recursive" all mean the same thing in this context. Below I'm using "computable" exclusively, since "recursive" is slightly old-fashioned and "decidable" does significant double-duty elsewhere in logic.
It is computable.
Precisel... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3370296",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
How to show that this graph is planar?-Formal Proof How to show that this graph is planar?
I am unable to write a formal proof
In the book the answer is given that this graph is planar.
Can you kindly say how to prove that this graph has no subgraph homeomorphic to $K_5$ or $K_{3,3}$?
I find that this graph has two ve... | A graph is planar if there exist at least one drawing of this graph (called an embedding) on the plane with no crossing edges.
Therefore as explained by Bercy, you just need to show one drawing of this graph with no crossing edges. To do so take you vertex labelled 0, and pull it out of the other vertices, outside the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3370416",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
} |
Definite Integral $\int{x^2+1 \over x^4+1}$ Evaluate
$$\int_0^{\infty}{x^2+1 \over x^4+1}$$
I tried using Integration by parts ,
$$\frac{{x^3 \over 3 }+x}{x^4+1}+\int\frac{{x^3 \over 3 }+x}{(x^4+1)^2}.4x^3.dx$$
First term is zero
But it got me no where.
Any hints.
| *
*Divide the numerator by $x^2$ and you have $1+\frac{1}{x^2}$
*Divide the denominator by $x^2$ and you have $x^2+\frac{1}{x^2} =\bigl(x-\frac{1}{x}\bigr)^{2}+2$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3370526",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Diophantine equation from the Latvian Baltic Way team selection competition 2019 So here is the problem statement:
Find all integer triples $(a, b, c)$ such that
$(a-b)^3(a+b)^2 = c^2 + 2(a-b) + 1$
The only things I have so far figured out is that (-1, 0, 0) and (0, 1, 0) are solution, gcd((a-b), c) = 1 and that c must... | Equation, $(a-b)^3(a+b)^2 = (c^2 + 2(a-b) + 1)$
Thanks @Piquito, for reviewing my previous answer.
By mistake I solved "OP' equation as a quartic rather than a quantic.
If we put the condition, $(a=b+c)$ then we get:
$c^3(2b+c)^2=(c+1)^2$
Which has solution at, $(b,c)=[(1/2),(1)]$
And so we get, $(a,b,c)=[(3/2),(1/2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3370630",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 2
} |
Rigorous, concise, self-contained, systematic presentation of calculus (real and complex analysis) I'm looking for a book recommendation satisfying the above requirements, with the presentation accessible to graduate students. Ideally, it would develop real and complex analysis axiomatically, rigorously prove all the m... | Rudin's Principles of Mathematical Analysis is exactly what I was looking for. Thanks to J. E. Greilhuber for the recommendation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3370746",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Is $PGL(n,\Bbb{R}) \cong SL(n+1, \Bbb{R})$ for even n? Is the following claim correct?
Claim: If n is even, $PGL(n,\Bbb{R}) \cong SL(n+1, \Bbb{R})$.
Proof:
Recall $PGL(n,\Bbb{R}) \cong GL(n+1, \Bbb{R})/Z$, where $Z = \{M | M=\alpha I, \alpha \in \Bbb{R}\}$.
Define a homomorphism $ \phi : GL(n+1,\Bbb{R}) \to SL(n+1, \... | The proof looks good to me.
Another way of framing this picture is to observe that for a finite-dimensional vector space $V$ over the field $\Bbb F$, the sequences
$$0 \to Z(SL(V)) \to SL(V) \to PSL(V) \to 0$$
and
$$0 \to PSL(V) \to PGL(V) \to \Bbb F^* / (\Bbb F^*)^{\dim V} \to 0$$
are exact. (A sequence $0 \to A \stac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3370836",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Informative proof that any real-valued symmetric matrix only has real eigenvalues I am looking for an informative proof that any real-valued symmetric matrix only has real eigenvalues. By informative, I mean that there is an explanation accompanying the proof, rather than just a copy-and-paste job, which is not informa... | I found the proof in http://pi.math.cornell.edu/~jerison/math2940/real-eigenvalues.pdf to be informative and educational.
The Spectral Theorem states that if $A$ is an $n \times n$ symmetric matrix with real entries, then it has $n$ orthogonal eigenvectors. The first step of the proof is to show that all the roots of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3370991",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
If $A+B+C=\pi$, prove that $\cos (A-B) \cos (B-C) \cos (C-A)\ge 8\cos A \cos B \cos C$ If $A+B+C=\pi$, prove that $\cos (A-B) \cos (B-C) \cos (C-A)\ge 8\cos A \cos B \cos C$
I know this is true for acute angle triangle.
I want to know whether it is true for every real $A,B,C$ such that $A+B+C=\pi.$
| Result to be established :
$$\begin{matrix}A+B+C=\pi \ \implies\\ \ \cos (A-B) \cos (B-C) \cos (C-A)\ge 8\cos A \cos B \cos C\end{matrix}\tag{*}$$
I would like to give here a variation on the excellent idea of Michael to use the following parameterization of a "triangle shape", i.e., a triangle known by its angles) :
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3371107",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
I had to show that a' + b' = a + b mod n, when a mod n = a' and b mod n = b' I had to show the following:
For $\,\, n,a,b,a',b' \in N$;
if: $ a'\equiv a\mod{n}$ and $ b'\equiv b\mod{n} $
than $ a' + b' = \,(a+b)\mod{n}$
My try:
$\frac{a}{n} = k +a'$ , where $k \in N $ and $k$ is divisable by $n$.
$\frac{b}{n} = p + b'$... | $a\equiv a'$ and $b\equiv b'$ mod $n$ means that $a$ and $a'$ give the same remainder when divided by $n$, or equivalently, their difference $a-a'$ is a multiple of $n$. The same applies to $b$ and $b'$. So you have
$$
a-a'=kn;\qquad b-b'=rn
$$
for some integers $r$ and $k$. If you add the two equations above and arran... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3371219",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Proof review: Symmetric matrices have real eigenvalues This document provides the following proof:
The Spectral Theorem states that if $A$ is an $n \times n$ symmetric matrix with real entries, then it has $n$ orthogonal eigenvectors. The first step of the proof is to show that all the roots of the characteristic poly... | The dot product can be indicated by
$$\vec w \cdot \vec v$$
or equivalently
$$\vec w^T\vec v$$
or also
$$\langle \vec w,\vec v\rangle$$
and we can move the scalar factor $\lambda$ in any position, that is
$$\lambda\vec w \cdot \vec v=\vec w \cdot \lambda\vec v=\langle \lambda\vec w,\vec v\rangle=\cdots$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3371317",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
$\mathbb{Q}$ - linear function from $\mathbb{R}$ to $\mathbb{R}$ with kernel $\mathbb{Q}$ I read somewhere that it is possible to construct a $\mathbb{Q} $ - linear function $f:\mathbb{R} \rightarrow \mathbb{R}$ with $\ker f = \mathbb {Q}$ . Can someone enlighten me on that matter? I thought it might be an easy matter ... | More of the same:
Assume you have $f$ a non-zero $\mathbb{Q}$-linear map from $\mathbb{R}$ to $\mathbb{Q}$. Consider a $\beta$ so that $\alpha\colon=f(\beta)\ne 0$. The map $$p\colon \mathbb{R} \to \mathbb{Q}\\
x\mapsto \frac{1}{\alpha} f(\beta x)$$ is a $\mathbb{Q}$-linear projection. Then $p'\colon= 1_{\mathbb{R... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3371407",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Prove Variance of Gaussian by Differentiating Integral of N(x) =1 by $\sigma^2$ and rearranging The normalized Gaussian distribution is defined as:
$$N(x|\mu,\sigma^2)=\frac{1}{(2\pi\sigma^2)^{1/2}}exp\bigg(\frac{-1}{2\sigma^2}(x-\mu)^2\bigg)$$
Prove that:
$$Var[x]=E[(X-\mu)^2]=\sigma^2$$
by differentiating both sides ... | $$\int_{-\infty}^{\infty} N(x|\mu,\sigma^2)~dx=1$$
Now Differentiate both sides by $\sigma$:
$$\int_{-\infty}^{\infty} \frac{d}{d\sigma}N(x|\mu,\sigma^2)~dx=\frac{d}{d\sigma}1$$
$$\int_{-\infty}^{\infty} \frac{d}{d\sigma}N(x|\mu,\sigma^2)~dx=0~~~~~(1)$$
Now Working just on the differentiation:
$$\frac{d}{d\sigma}N(x|\m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3371526",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Writing improper integral and infinite sum with limit or not? I just wondering what is actually the correct notation of these two operators?
1) Writing the improper integral
Which one is correct?
a) $\displaystyle\int_0^{\infty}f(x) \mathbb dx$
b) $\displaystyle\lim_{b \to \infty}\displaystyle\int_0^{b}f(x) \mathbb dx$... | $\sum_{k=0}^\infty b_k$ is exactly the definition of $\lim_{n\to\infty}\sum_{k=0}^n b_k$ when the limit exists. And $\int_0^\infty f(x)dx$ is the definition of $\lim_{b\to\infty}\int_0^b f(x)dx$ when $f$ is Riemann integrable in $[0,b]$ for all $b>0$ and the limit exists. So you can use any of these notations, but of c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3371864",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How small can "spanning sumsets" of $[n]$ be? Let $[n]$ denote the natural numbers $1$ through $n$. Let's say a subset $X \subset [n]$ is a spanning sumset if $\{x+y: x,y \in X\} = [n] \setminus \{1\}$. I'm interested in studying spanning sumsets of minimal possible size. In particular, is there either an asymptotic or... | Let $S_k=\{1,2,3,...,k,2k,3k,4k,...,(k-1)k\}$.
Then $S_k$ is a spanning set for $[k^2]$.
For a given $n$ if we let $k=\lceil \sqrt{n} \rceil$, then $S_k$ will be a spanning set of $[n]$ with size $2k-2\approx 2\sqrt{n}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3371925",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Angle of visible part of circle I have a circle and a point in 2D.
The point lies outside of the circle.
Given the distance between the point and circle center and the radius of the circle, what is the angle of the circle that the point can 'see'?
When the point is infinitely far away, it will see 180 degrees or 1 pi ... | We have that indicating with:
*
*radius $R$
*distance form the centre $ d$
$$\alpha = 2\arccos \left(\frac R d\right)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3372017",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Solving a factorial equation Prove that the only solution to $k! +m! =n!$ is
$k = 1, m = 1, n = 2$
How would you go about this? I can't seem to figure out where to start.
| If you assume, wolog, $k\le m$ and you divide by $k!$ you get:
$1 + [(k+1).....m]= [(k+1).....m][(m+1)....n]$
Which is only possilbe if $[(k+1).....m] =1$.
Which is $k < m$ means $k=0$ and $m=1$.
......
Just finesse a bit allow for $k = m$ and divide allow "$[(k+1).....m]$" to be $1$ you get, if you divide by $k!=m!$:... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3372103",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
How to recalculate a lerp percentage value, so that it returns the same value, even when the max lerp value is adjusted? I currently have a lerp function, which is
$$
y = p \cdot x_2 + (1 - p) \cdot x_1,
$$
where
*
*$x_1$ is the min lerp value,
*$x_2$ is the max lerp value,
*$p$ is the percentage to lerp betw... | Let $f(a,b,t)=tb+a(1-t)$. Then, we are looking for a value $t_2$ such that $f(a,b_1,t_1)=f(a,b_2,t_2)$ where $a$ represents the source value, $b_1$ represents the destination value, $t_1$ represents the interpolation factor, and $b_2$ and $t_2$ represent the new source and interplation factor respectively. If we substi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3372204",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find all real matrices such that $X^{3}-4X^{2}+5X=\begin{pmatrix} 10 & 20 \\ 5 & 10 \end{pmatrix}$ The following question come from the 1998 Romanian Mathematical Competition:
Find all matrices in $M_2(\mathbb R)$ such that $$X^{3}-4X^{2}+5X=\begin{pmatrix} 10 & 20 \\ 5 & 10 \end{pmatrix}$$
Can you guy please help me... | Let $p$ be the polynomial in question and $R$ be the right hand side. Note that $R$ is equivalent to $D=\operatorname{diag} (20,0)$.
Let $V^{-1}RV = D$, then since $V^{-1}p(X)V = p(V^{-1}XV) = D$, we can look for solutions
to $p(X)=D$ and then conjugate back to get the original solutions.
Note that $De_1 = 20 e_1, D e_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3372342",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 2
} |
Compute in a closed form the following sum : $\sum_{n=1}^{+\infty}\frac{\Gamma^{4}(n+\frac{3}{4})}{(4n+3)^{2}\Gamma^{4}(n+1)}$ Today Im going to find the closed form of :
$\sum_{n=1}^{+\infty}\frac{\Gamma^{4}(n+\frac{3}{4})}{(4n+3)^{2}\Gamma^{4}(n+1)}$
My attempt :
We know that : $\Gamma(z)=\int_0^{+\infty}t^{n-1}e^... | As one could expect, the result must involve hypergeometric function.
A CAS gave for the infinite summation
$$\frac{\Gamma \left(\frac{3}{4}\right)^4 \left(3136 \left(\,
_5F_4\left(\frac{3}{4},\frac{3}{4},\frac{3}{4},\frac{3}{4},\frac{3}{4};1,1,1,\frac{7}{4};1\right)-1\right)-243 \,
_6F_5\left(\frac{7}{4},\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3372415",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
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