Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
How to prove an Inequality I'm beginner with proofs and I got the follow exercise:
Prove the inequality $$(a + b)\Bigl(\frac{1}{a} + \frac{4}{b}\Bigr) \ge 9$$ when $a > 0$ and $b > 0$. Determine when the equality occurs.
I'm lost, could you guys give me a tip from where to start, or maybe show a good resource for beg... | We can play with the inequality: let's suppose it was true for now, and perform a series of reversible steps and see what the inequality would imply. So we begin by multiplying everything out, which gives:
$1 + 4 + \dfrac{b}{a} + \dfrac{4a}{b} \geq 9 \Leftrightarrow \dfrac{b}{a} + \dfrac{4a}{b} \geq 4$. (1)
Now, when w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/207521",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 8,
"answer_id": 3
} |
counting the number of loop runs I have the following loop structure:
for $i_1=1$ to $m$
for $i_2=i_1$ to $m$
$\vdots$
for $i_n=i_{n-1}$ to $m$
Of course, all indices $i_k$ are integers, and $m$ and $n$ are also positive integers.
How can I count how many times the inner loop will run?
| This is a problem in which it really helps to look at some small examples or to start with the simplest version and work up (or both!). The simplest version is $n=0$, which isn’t very interesting. The next simplest is $n=1$, which is also pretty trivial. The first interesting case is $n=2$, but it turns out to be helpf... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/207590",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Prove $ \int_{cX} \frac{dt}{t} = \int_{X} \frac{dt}{t}$ for every Lebesgue measurable set $X$ Let $c>0$. Let $X \subseteq (0,\infty)$ be a Lebesgue measurable set. Define $$ cX := \{ cx \mid x \in X \}. $$ Then $$ \int_{cX} \frac{dt}{t} = \int_{X} \frac{dt}{t}$$
Now I can prove this for $X$ an interval and, thus, any s... | Suppose $m$ and $n$ are non-negative measures and $c$ is a positivie number and $n=m/c$. Can you show that
$$
\int_A f\,dm = \int_A (c f)\,dn\text{ ?}
$$
If you can, let $A=cX$, $m=$ Lebesgue measure, $f(t)=1/t$. Then find a one-to-one correspondence between $A=cX$ and $X$ such that the value of $1/t$ for $t\in X$ is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/207734",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Size of factors in number ring Let $R$ be the ring ${\mathbb Z}[\sqrt{2}]$. For $z\in R$, $z=x+y\sqrt{2}\in R$ with $x$ and $y$ in $\mathbb Z$, put $\|z\|={\sf max}(|x|,|y|)$ and
$$D_z=\left\{ (a,b) \in R^2 : ab=z,\ a \text{ and } b \text{ are not units in } R\right\}$$
and
$$
\rho(z)=
\begin{cases}
0,& \text{ if} \... | If $x$ is even in $z=x+y\sqrt 2$ then $z$ is a multiple of $\sqrt 2$, hence $\rho(z)\le 1$.
On the other hand, if $|N(z)|=|x^2-2y^2|$ is the square of a prime $p$ (and $x+y\sqrt 2$ is not a prime in $R$), then we must have $N(a+b\sqrt2)=a^2-2b^2=\pm p$, which implies $a^2\ge p$ or $b^2\ge \frac2p$, hence $||a+b\sqrt 2|... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Expected value of applying the sigmoid function to a normal distribution Short version:
I would like to calculate the expected value if you apply the sigmoid function $\frac{1}{1+e^{-x}}$ to a normal distribution with expected value $\mu$ and standard deviation $\sigma$.
If I'm correct this corresponds to the following... | Apart from the the MacLaurin approximation, the usual way to compute that integral in Statistics is to approximate the sigmoid with a probit function.
More specifically $\mathrm{sigm}(a) \approx \Phi(\lambda a)$ with $\lambda^2=\pi/8$.
Then the result would be:
$$\int \mathrm{sigm}(x) \, N(x \mid \mu,\sigma^2) \, dx
\a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/207861",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 3,
"answer_id": 1
} |
Eigenvalues of a rectangular matrix I've read that the singular values of a matrix are equal to the $$\sigma=\sqrt{\lambda_{K}}$$ where $\lambda$ are the eigenvalue but I'm assuming this only applies to square matrices. How could I determine the eigenvalues of a non-square matrix. Pardon my ignorance.
| Eigenvalues aren't defined for rectangular matrices, but the singular values are closely related: The right and left singular values for rectangular matrix M are the eigenvalues of M'M and MM'.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/207991",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Modular Arithmetic order of operations In an assignment, I am given $E_K(M) = M + K \pmod {26}$. This formula is to be applied twice in a formal proof, so that we have $E_a(E_b(M)) =\ ...$. What I'm wondering is; is the original given formula equal to $(M + K)\pmod{26}$, or $M + (K \mod{26})$? This will obviously make ... | I suspect that what is meant is $(M+K)\bmod 26$, where $\bmod$ is the operator, especially if this is in a cryptographic context. More careful writers reserve the parenthesized notation $\pmod{26}$ for the relation of congruence modulo $26$, using it only in connection with $\equiv$, as in $27\equiv 53\pmod{26}$.
Thus,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/208028",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
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What's the probability of losing a coin tossing gambling with a wealthy man? Imagine you have $k$ dollars in your pocket and you are gambling with a wealthy man (with infinitely much money). The rule is repeatedly tossing a coin and you win $\$1$ if it's a head, otherwise you lose $\$1$. Now, what's the probability tha... | The probability is $1$. This is the gambler’s ruin problem, and see also one-dimensional random walks.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Integral of sinc function multiplied by Gaussian I am wondering whether the following integral
$$\int_{-\infty}^{\infty} \frac{\exp( - a x^2 ) \sin( bx )}{x} \,\mathrm{d}x$$
exists in closed form. I would like to use it for numerical
calculation and find an efficient way to evaluate it.
If analytical form does not exi... | We assume $a,b>0$.
Then
$$\begin{eqnarray*}
\int_{-\infty}^\infty dx\, e^{-a x^2}\frac{\sin b x}{x}
&=& \int_0^b d\beta \, \int_{-\infty}^\infty dx\, e^{-a x^2} \cos \beta x \\
&=& \int_0^b d\beta \, \mathrm{Re} \int_{-\infty}^\infty dx\, e^{-a x^2+i \beta x} \\
&=& \int_0^b d\beta \, \mathrm{Re}\, \sqrt{\frac{\pi}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/208250",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
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Intersection and Span Assume $S_{1}$ and $S_{2}$ are subsets of a vector space V.
It has already been proved that span $(S1 \cap S2)$ $\subseteq$ span $(S_{1}) \cap$ span $(S_{2})$
There seem to be many cases where span $(S1 \cap S2)$ $=$ span $(S_{1}) \cap$ span $(S_{2})$ but not many where span $(S1 \cap S2)$ $\not=... | HINT: Let $S_1=\{v\}$, where $v$ is a non-zero vector, and let $S_2=\{2v\}$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Combination with repetitions. The formula for computing a k-combination with repetitions from n elements is:
$$\binom{n + k - 1}{k} = \binom{n + k - 1}{n - 1}$$
I would like if someone can give me a simple basic proof that a beginner can understand easily.
| To make the things more clear let there are $n$ different kinds of drinks available in a restaurant, and there are $m$ people including yourself, at your birthday party. Each one is to be served only one 'drink' as one desires/orders without restriction that two or more can ask for different or the same drink as per in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/208377",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "37",
"answer_count": 9,
"answer_id": 8
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Can vectors be inverted? I wish to enquire if it is possible to solve the below for $C$.
$$B^{-1}(x-\mu) = xc $$
Here obviously $B$ is an invertible matrix and both $c$ and $\mu$ are column vectors.
Would the solution be $$x^{-1}B^{-1}(x-\mu) = c $$ is it possible to invert vectors ?
How about if it was the other way
$... | Vectors, in general, can't be inverted under matrix multiplication, as only square matricies can inverses. However, in the situation you've described, it's possible to compute $c$ anyway, assuming the equation is satisfied for some $c$. If we multiply both sides by $X^T$, the result is $x^T B^{-1} (x-\mu) = x^T x c = |... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/208447",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
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"answer_id": 1
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Extrapolating to derive an $O(h^3)$ formula The Forward difference formula can be expressed as:
$$f'(x_0)={1 \over h}[f(x_0+h)-f(x_0)]-{h\over 2}f''(x_0)-{h^2 \over 6}f'''(x_0)+O(h^3)$$
Use Extrapolation to derive an an $O(h^3)$ formula for $f'(x_0)$
I am unsure how to begin but From what I have seen in the textbook, e... | Given: $$f'(x_0)={1 \over h}[f(x_0+h)-f(x_0)]-{h\over 2}f''(x_0)-{h^2 \over 6}f'''(x_0) + O(h^3) \tag{1}$$
Replace $h$ with $2h$ and simplify: $$f'(x_0)={1 \over {2h}}[f(x_0+2h)-f(x_0)]-hf''(x_0)-{{2}\over{3}h^2}f'''(x_0) + O(h^3) \tag{2}$$
Subtract $\frac{1}{2} (2)$ from $(1)$: $$f'(x_0) - \frac{1}{2}f'(x_0)={1 \over ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/208505",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Distribution function given expectation and maximal variance I'm working on the following problem I got in a hw but I'm stuck. It just asks to find the distribution function of a random variable $X$ on a discrete probability spaces that takes values in $[A,B]$ and for which $Var(X) = \left(\frac{B-A}{2}\right)^{2}.$
I ... | Let $m=\frac12(A+B)$ and $h=\frac12(B-A)$. The OP indicates in a comment how to prove that any random variable $X$ with values in $[A,B]$ and such that $\mathrm{Var}(X)=h^2$ is such that $\mathbb E(X)=m$ and $\mathbb E((X-m)^2)=h^2$.
Starting from this point, note that $|X(\omega)-m|\leqslant h$ for every $\omega$ sin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/208565",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is a hexagon? Having a slight parenting anxiety attack and I hate teaching my son something incorrect.
Wiktionary tells me that a Hexagon is a polygon with $6$ sides and $6$ angles.
Why the $6$ angle requirement? This has me confused.
Would the shape below be also considered a hexagon?
| Yes, It is Considered as a Hexagon. There is a difference between an Irregular Hexagon and a Regular Hexagon. A regular hexagon has sides that are segments of straight lines that are all equal in length. The interior angles are all equal with 120 degrees. An irregular hexagon has sides that may be of different lengths.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/208603",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "40",
"answer_count": 11,
"answer_id": 3
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What is the limit of $\left(2\sqrt{n}\left(\sqrt{n+1}-\sqrt{n}\right)\right)^n$ as $n \to \infty$? I'd would like to know how to get the answer of the following problem:
$$\lim_{n \to \infty} \left(2\sqrt{n}\left(\sqrt{n+1}-\sqrt{n}\right)\right)^n$$
I know that the answer is $\frac{1}{e^{1/4}}$, but I can't figure out... | Expanding the Taylor series of $\sqrt{1+x}$ near $x=0$ gives $ \sqrt{1+x} = 1 + \frac{x}{2} - \frac{x^2}{8} + \mathcal{O}(x^3).$ Thus $$ \left( 2\sqrt{n} ( \sqrt{n+1} - \sqrt{n} )\right)^n =2^n n^{\frac{n+1}{2}} \left(\sqrt{1+\frac{1}{n}}-1 \right)^n$$
$$=2^n n^{\frac{n+1}{2}} \left( \frac{1}{2n} - \frac{1}{8n^2}+ \mat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/208715",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 1
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Can we prove that a statement cannot be proved? Is it possible to prove that you can't prove some theorem? For example: Prove that you can't prove the Riemann hypothesis. I have a feeling it's not possible but I want to know for sure. Thanks in advance
| As far as I know the continuum hypothesis has been proved "independent" from the ZFC axioms. So you can assume it true or false.
This is very different from just true undecidible statements (as the one used by Gödel to prove his incompleteness theorem). Gödel statements is true even though it is not derived by the axio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/208761",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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"answer_id": 5
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Asymptotic growth of $\sum_{i=1}^n \frac{1}{i^\alpha}$? Let $0 < \alpha < 1$. Can somebody please explain why
$$\sum_{i=1}^n \frac{1}{i^\alpha} \sim n^{1-\alpha}$$
holds?
| We have for $x\in [k,k+1)$ that $$(k+1)^{—\alpha}\leq x^{-\alpha}\leq k^{—\alpha},$$
and integrating this we get
$$(k+1)^{—\alpha}\leq \frac 1{1-\alpha}((k+1)^{1-\alpha}-k^{\alpha})\leq k^{—\alpha}.$$
We get after summing and having used $(n+1)^{1-\alpha}-1\sim n^{1-\alpha}$, the equivalent $\frac{n^{1-\alpha}}{1-\alp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/208827",
"timestamp": "2023-03-29T00:00:00",
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How do I find the variance of a linear combination of terms in a time series? I'm working through an econometrics textbook and came upon this supposedly simple problem early on.
Suppose you win $\$1$ if a fair coin shows heads and lose $\$1$ if it shows tails. Denote the outcome on toss $t$ by $ε_t$. Assume you want t... | The variance of $w_t$ is 0.25, for the reasons you explained. But conditioning on $ε_{t−3}=ε_{t−2}=1$ changes the problem since now, $w_t=$ 0.25$ε_t+$ 0.25$ε_{t−1}+$ some constant, hence the conditional variance is only 0.125.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Is $\{a^m!a^n : n > 0, m > 0, n > m\}$ a context free language? I'm trying to construct a context-free grammar for the language $L = \{a^m!a^n : n > 0, m > 0, n > m\}$, but I'm not sure how to ensure that the right trail of $a$s is longer than the left one.
Is there a way I could include the left side in the right side... | We could look at the language as strings of the form $a^n!a^ka^n.$ Guided by this insight we could use the CFG with productions $S\rightarrow aSa\ |\ a!Ta,\quad T\rightarrow aT\ |\ a$. We're generating strings of the form $a^n\dots a^n$ and then finishing off by placing $!a^k$ in the middle. This idiom of "constructing... | {
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"timestamp": "2023-03-29T00:00:00",
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For every real number $a$ there exists a sequence $r_n$ of rational numbers such that $r_n$ approaches $a$. How to prove that for every real number $a$ there exists a sequence $r_n$ of rational numbers such that $r_n \rightarrow a$.
| By Riemann's series theorem it follows, that for every $x \in \mathbb{R}$ there is an arrangement $\sigma:\mathbb{N} \to \mathbb{N}$ such that $\sum_{n=1}^\infty \frac{(-1)^{\sigma(n+1)}}{\sigma(n)} = x$. Note that $b_n := \sum_{k=1}^n \frac{(-1)^{\sigma(n+1)}}{\sigma(n)}$ is a rational sequence.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/209001",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 5,
"answer_id": 4
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Positive Definite Matrix Determinant
Prove that a positive definite matrix has positive determinant and
positive trace.
In order to be a positive determinant the matrix must be regular and have pivots that are positive which is the definition. Its obvious that the determinant must be positive since that is what a p... | We will solve it by assuming a function which is +ve definite and then using continuity definition of ϵ-δ.
So,if you know ϵ-δ continuity definition of a function,then only consider this solution,otherwise skip it.
A is +ve definite.lets define a function :
f(x)= |xA + (1-x)I| for 0<=x<=1 -----(1), here I is the identi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/209082",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 2
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Does a power series vanish on the circle of convergence imply that the power series equals to zero? Let $f(z)=\sum_{n=0}^{\infty} a_n z^n$ be a power series, $a_n, z\in \mathbb{C}$. Suppose the radius of convergence of $f$ is $1$, and $f$ is convergent at every point of the unit circle.
Question:If $f(z)=0$ for every $... | It seems to me that this is a particular case of an old Theorem from Cantor (1870), called Cantor's uniqueness theorem. The theorem says that if, for every real $x$,
$$\lim_{N \rightarrow \infty} \sum_{n=-N}^N c_n e^{inx}=0,$$
then all the complex numbers $c_n$'s are zero.
You can google "Uniqueness of Representation b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/209171",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
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Modular arithmetic congruence class simple proof I have the following question but I'm unsure of how it can be approached by a method of proof. I'm new to modular arithmetic and any information on how to solve this would be great for me.
(b) Let $t,s\in\{0,1,2,3,4,5\}$. In $\mathbb Z_{25}$, prove that $[t]\,[s]\neq[24... | First note that $24\equiv -1\pmod{25}$ and hence we are trying to show that $ts\not\equiv-1\pmod{25}$. Suppose for contradiction that $ts\equiv -1\pmod{25}$, then multiplying through by $-1$ we get $-ts\equiv 1\pmod{25}$ so $t$ or $s$ is invertible, say $t$ with inverse $-s$. Therefore $\gcd(t,25) = 1$ (why?), and h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/209274",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Lebesgue measuarable sets under a differentiable bijection Let $U,V \subseteq \mathbb{R}^{n}$ be open and suppose $A\subseteq U$ are (Lebesgue) measurable. Suppose $\sigma \in C^{1} (U,V)$ be a bijective differentiable function. Then does it follow that $\sigma(A)$ is (Lebesgue) measurable?
I've tried work on it, but s... | The answer is yes. Let me call your differentable bijection $f$...
Hint : Every Lebesgue measurable set is the union of a $F_{\sigma}$ and a set of measure zero. Now, use the fact that the image by $f$ of any $F_{\sigma}$ is Lebesgue measurable (why?) and that $f$ maps sets of measure zero to sets of measure zero...
ED... | {
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Identity proof $(x^{n}-y^{n})/(x-y) = \sum_{k=1}^{n} x^{n-k}y^{k-1}$ In a proof from a textbook they use the following identity (without proof):
$(x^{n}-y^{n})/(x-y) = \sum_{k=1}^{n} x^{n-k}y^{k-1}$
Is there an easy way to prove the above? I suppose maybe an induction proof will be appropriate, but I would really lik... | I see no one likes induction.
For $n=0$,
$$
\frac{x^0-y^0}{x-y}=\sum_{1 \le i \le 0}x^{0-i}y^{i-1}=0.
$$
Assume for $n=j$ that the identity is true.
Then, for $n=j+1$,
$$
\begin{align}
\sum_{1 \le i \le j+1}x^{j+1-i}y^{i-1}&=\left(\sum_{1 \le i \le j}x^{j+1-i}y^{i-1}\right)+y^j\\
&=\left(x\sum_{1 \le i \le j}x^{j-i}y^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/209467",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 4
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Prove that $n!e-2< \sum_{k=1}^{n}(^{n}\textrm{P}_{k}) \leq n!e-1$ Prove that $$n!e-2 < \sum_{k=1}^{n}(^{n}\textrm{P}_{k}) \leq n!e-1$$ where $^{n}\textrm{P}_k = n(n-1)\cdots(n-k+1)$ is the number of permutations of $k$ distinct objects from $n$ distinct objects and $e$ is the exponential constant (Euler's number).
| Denote the expression in question by $S$
$$
\sum_{k=0}^{n-1}\frac{1}{k!}=e-\sum_{k=n}^{\infty}\frac{1}{k!}=e-\frac{1}{n!}\bigg(1+\frac{1}{n+1}+\frac{1}{(n+1)(n+2)} + \cdots \bigg)\\
\leq e-\frac{1}{n!} \bigg(1+\frac{1}{n+1} + \frac{1}{(n+1)^2} +\cdots \bigg)\\
=e-\frac{1}{n!} \sum_{k=0}^{\infty} \bigg(\frac{1}{n+1} \b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/209619",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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} |
Almost Sure Convergence Using Borel-Cantelli I am working on the following problem:
Let $(f_n)$ be a sequence of measurable real-valued functions on $\mathbb{R}$. Prove that there exist constants $c_n > 0$ such that the series $\sum c_n f_n$ converges for almost every $x$ in $\mathbb{R}$. (Hint: Use Borel-Cantelli Lem... | Assume that $f_n$ is Lebesgue almost everywhere finite, for every $n$ (otherwise the result fails).
For every $n$, the interval $[-n,n]$ has finite Lebesgue measure hence there exists $c_n\gt0$ such that the Lebesgue measure of the Borel set
$$
A_n=\{x\in[-n,n]\,\mid\,c_n\cdot|f_n(x)|\gt1/n^2\}
$$
is at most $1/n^2$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Which function grows faster: $(n!)!$ or $((n-1)!)!(n-1)!^{n!}$? Of course, I can use Stirling's approximation, but for me it is quite interesting, that, if we define $k = (n-1)!$, then the left function will be $(nk)!$, and the right one will be $k! k^{n!}$. I don't think that it is a coincidence. It seems, that there ... | For $(nk)!$ your factors are $1,2,3,\dots, k$ then $k+1, \dots, 2k,2k+1 \dots, k!$.
For $k! k^{n!}$ your factors are $1,2,3,\dots, k$ but then constant $k,\dots,k$.
So every factor of (nk)! is > or = to each factor of k!k^(n!)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/209856",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Homomorphism between $A_5$ and $A_6$ The problem is to find an injective homomorphism between the alternating groups $A_5$ and $A_6$ such that the image of the homomorphism contains only elements that leave no element of $\{1,2,3,4,5,6\}$ fixed. I.e., the image must be a subset of $A_6$ that consists of permutations wi... | What do you think of this homomorphism:
f:A5->A6
f(x)=(123)(456)x(654)(321)
This is a homomorphism because
f(xy)=(123)(456)xy(654)(321)
=(123)(456)x(654)(321)(123)(456)y(654)(321)
=f(x)f(y)
Because (321)(123)=e=(654)(456).
And it is injective because if f(x)=f(y) then,
(123)(456)x(654)(321)=(123)(456)y(654)(321)
And ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/209897",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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Question about the global dimension of End$_A(M)$, whereupon $M$ is a generator-cogenerator for $A$ Let $A$ be a finite-dimensional Algebra over a fixed field $k$. Let $M$ be a generator-cogenerator for $A$, that means that all proj. indecomposable $A$-modules and all inj. indecomposable $A$-modules occur as direct sum... | The global dimension of a noetherian ring with finite global dimension is equal to the supremum of the projective dimensions of its simple modules. This is proved in most textbooks dealing with the subject. For example, this is proved in McConnell and Robson's Noncommutative Noetherian rings (Corollary 7.1.14)
If the r... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Example of a *-homomorphism that is faithful on a dense *-subalgebra, but not everwhere
Let $A,B$ be C*-algebras and let $\varphi: A \to B$ be a $*$-homomorphism. Suppose that $\ker( \varphi) \cap D = \{0\}$ where $D$ is a dense $*$-subalgebra of $A$. Does it follow that $\varphi$ is injective?
I'm pretty sure the an... | If $A=C[0,2]$ and $B=C[0,1]$, define $\phi:A\to B$ to be the restriction map $\phi(f)=f|_{[0,1]}$. Let $D\subset A$ be the algebra of polynomial functions on $[0,2]$.
Then $\ker(\phi)\cap D=\{0\}$ because no nonzero polynomial function vanishes on $[0,1]$. However, $\phi$ is not injective because for example it sends ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/210100",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
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Second derivative "formula derivation" I've been trying to understand how the second order derivative "formula" works:
$$\lim_{h\to0} \frac{f(x+h) - 2f(x) + f(x-h)}{h^2}$$
So, the rate of change of the rate of change for an arbitrary continuous function. It basically feels right, since it samples "the after $x+h$ and t... | The only problem is that you’re looking at the wrong three points: you’re looking at $x+2h,x+h$, and $x$, and the version that you want to prove is using $x+h,x$, and $x-h$. Start with $$f\,''(x)=\lim_{h\to 0}\frac{f\,'(x)-f\,'(x-h)}h\;,$$ and you’ll be fine.
To see that this really is equivalent to looking at $$f\,''(... | {
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"url": "https://math.stackexchange.com/questions/210264",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "36",
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Modules with projective dimension $n$ have not vanishing $\mathrm{Ext}^n$
Let $R$ be a noetherian ring and $M$ a finitely generated $R$-module with projective dimension $n$. Then for every finitely generated $R$-module $N$ we have $\mathrm{Ext}^n(M,N)\neq 0$. Why?
By definition, if the projective dimension is $n$ thi... | Take $R=\mathbb Z\times\mathbb Z$, consider the elements $e_1=(1,0)$, $e_2=(0,1)\in R$, and the modules $M=R/(2e_1)$ and $N=Re_2$. Show that the projective dimension of $M$ is $1$ and compute $\operatorname{Ext}_R^1(M,N)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/210304",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
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A holomorphic function $f$, injective on $\partial D$, must be injective in $\bar{D}$?
Prove: If $f$ is holomorphic on a neighborhood of the closed unit disc $\bar{D}$, and if $f$ is one-to-one on $\partial D$, then $f$ is
one-to-one on $\bar{D}$.
(Greene and Krantz's Function Theory of One Complex Variable (3rd), ... | Some hints:
The function $f$ restricted to $\partial D$ is an injective continuous map of $\partial D\sim S^1$ into ${\mathbb C}$. By the Jordan curve theorem the curve $\gamma:=f(\partial D)$ separates ${\mathbb C}\setminus\gamma$ into two connected domains $\Omega_{\rm int}$ and $\Omega_{\rm ext}$, called the interi... | {
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"url": "https://math.stackexchange.com/questions/210396",
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"source": "stackexchange",
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How to show that linear span in $C[0,1]$ need not be closed
Possible Duplicate:
Non-closed subspace of a Banach space
Let $X$ be an infinite dimensional normed space over $\mathbb{R}$. I want to find a set of vectors $(x_k)$ such that the linear span of $(x_k)$ of vectors is not closed.
I feel like the set $P$, whic... | You can take your favourite convergent sequence of polynomials (e.g. partial sums of $\exp x = \displaystyle \sum_{n = 0}^\infty \frac{x^n}{n!}$) and then prove that the limit is not in the span.
This proves the span is not sequentially closed. Since $X$ is normed, it follows that the span isn't closed.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/210449",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove by induction $\sum_{i=1}^ni^3=\frac{n^2(n+1)^2}{4}$ for $n\ge1$ Prove the following statement $S(n)$ for $n\ge1$:
$$\sum_{i=1}^ni^3=\frac{n^2(n+1)^2}{4}$$
To prove the basis, I substitute $1$ for $n$ in $S(n)$:
$$\sum_{i=1}^11^3=1=\frac{1^2(2)^2}{4}$$
Great. For the inductive step, I assume $S(n)$ to be true and ... | Here's yet another way:
You have $\frac{n^2(n+1)^2}{4}+(n+1)^3$ and you want $\frac{(n+1)^2(n+2)^2}{4}$
So manipulate it to get there;
$\frac{n^2(n+1)^2}{4}+(n+1)^3 =$
$\frac{(n^2 + 4n + 4)(n+1)^2 - (4n + 4)(n+1)^2}{4}+(n+1)^3 =$
$\frac{(n+2)^2(n+1)^2}{4}- \frac{ (4n + 4)(n+1)^2}{4}+(n+1)^3 =$
$\frac{(n+2)^2(n+1)^2}{4}... | {
"language": "en",
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"source": "stackexchange",
"question_score": "3",
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upper bound of exponential function I am looking for a tight upper bound of exponential function (or sum of exponential functions):
$e^x<f(x)\;$ when $ \;x<0$
or
$\displaystyle\sum_{i=1}^n e^{x_i} < g(x_1,...,x_n)\;$ when $\;x_i<0$
Thanks a lot!
| Since you suggest in the comments you would like a polynomial bound, you can use any even Taylor polynomial for $e$.
Proposition. $\boldsymbol{1 + x + \frac{x^2}{2!} + \cdots + \frac{x^n}{n!}}$ is an upper bound for $\boldsymbol{e^x}$ when $\boldsymbol{n}$ is even and $\boldsymbol{x \le 0}$.
Proof.
We wish to show $f(x... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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What does the value of a probability density function (PDF) at some x indicate? I understand that the probability mass function of a discrete random-variable X is $y=g(x)$. This means $P(X=x_0) = g(x_0)$.
Now, a probability density function of of a continuous random variable X is $y=f(x)$. Wikipedia defines this functi... | 'Relative likelihood' is indeed misleading. Look at it as a limit instead:
$$
f(x)=\lim_{h \to 0}\frac{F(x+h)-F(x)}{h}
$$
where $F(x) = P(X \leq x)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/210630",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
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"answer_id": 4
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Proof that Quantile Function characterizes Probability Distribution The quantile function is defined as $Q(u)= \inf \{x: F(x) \geq u\}$.
It is well known the distribution function characterizes the probability distribution in the following sense
Theorem Let $X_{1}$ and $X_{2}$ be two real valued random variables with ... | Changed since version discussed in first five comments:
The key line in the proof of Corollary 1.2 in Severini's Elements of Distribution Theory book is
Hence by part (iii) of Theorem 1.8, $F_2(x_0) \ge F_1(x_0)$ so that $F_1(x_0) \lt F_2(x_0)$ is impossible.
As you say, $F_1(x_0) \lt F_2(x_0)$ in fact implies $F_... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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If Same Rank, Same Null Spaces? "If matrices B and AB have the same rank, prove that they must have the same null spaces."
I have absolutely NO idea how to prove this one, been stuck for hours now. Even if you don't know the answer, any help is greatly appreciated.
| I would begin by showing that the null space of $B$ is a subspace of the null space of $AB$. Next show that having the same rank implies they have the same nullity. Finally, what can you conclude when a subspace is the same dimension as its containing vector space?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/210731",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solve $3\log_{10}(x-15) = \left(\frac{1}{4}\right)^x$ $$3\log_{10}(x-15) = \left(\frac{1}{4}\right)^x$$
I am completely lost on how to proceed. Could someone explain how to find any real solution to the above equation?
| Put
\begin{equation*}
f(x) = 3\log_{10}(x - 15) - \left(\dfrac{1}{4}\right)^x.
\end{equation*}
We have $f$ is a increasing function on $(15, +\infty)$.
Another way,
$f(16)>0 $ and $f(17)>0$. Therefore the given equation has only solution belongs to $(16,17)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/210810",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Two convergent sequences in a metric space. Question: Let {$x_n$} and {$y_n$} be two convergent sequences in a metric space (E,d). For all $n \in \mathbb{N}$, we defind $z_{2n}=x_n$ and $z_{2n+1}=y_n$. Show that {$z_n$} converges to some $l \in E$ $\longleftrightarrow$ $ \lim_{n \to \infty}x_n$= $\lim_{n \to \infty}y_n... | Hint Remember the fact that "every convergent sequence is a Cauchy sequence".
$(\Rightarrow)$ Assume $\lim_{n \to \infty} z_n=l$, then notice that
$$ |x_n-l|=|z_{2n}-l|=|(z_{2n}-z_n)+(z_n-l)|\leq |z_{2n}-z_n|+|z_n-l|<\dots $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/210869",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 3
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$E$ is measurable, $m(E)< \infty$, and $f(x)=m[(E+x)\bigcap E]$ for all $x \in \mathbb{R}$
Question: $E$ is measurable, $m(E)< \infty$, and $f(x)=m[(E+x)\bigcap E]$ for all > $x \in \mathbb{R}$. Prove $\lim_{x \rightarrow \infty} f(x)=0$.
First, since measure is translation invariant, I'm assuming that $(E+x)\bigcap... | Well, it seems you are a bit confused about which object lives where..
By translation invariance, we indeed have $m(E)=m(E+x)$, but not $E=E+x$ as you wrote. Also, $\{1,2,3\}\cap\{2,3,4\}$ has two common elements, not just one:)
The hint in one of the comments to consider $E_n:=E\cap[-n,n]$ is a great idea, because in ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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seminorm & Minkowski Functional It is known that if $p$ is a seminorm on a real vector space $X$, then the set
$A= \{x\in X: p(x)<1\}$ is convex, balanced, and absorbing. I tried to prove that the Minkowski functional $u_A$ of $A$ coincides with the seminorm $p$.
Im interested on proving that $u_A$ less or equal to $p... | What you proved is that $u_A(x)\leq s$ for every $s\in(p(x),\infty)$. In other words,
$$
u_A(x)\leq p(x)+\varepsilon
$$
for every $\varepsilon>0$. This implies that $u_A(x)\leq p(x)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/211081",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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question regarding metric spaces let X be the surface of the earth for any two points on the earth surface. let d(a,b) be the least time needed to travel from a to b.is this the metric on X? kindly explain each step and logic, specially for these two axioms d(a,b)=0 iff a=b and triangle inequality.
| This will generally not be a metric since the condition of symmetry is not fulfilled: It usually takes a different time to travel from $a$ to $b$ than to travel from $b$ to $a$. (I know that because I live on a hill. :-)
The remaining conditions are fulfilled:
*
*The time required to travel from $a$ to $b$ is non-ne... | {
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Finding Tangent line from Parametric I need to find an equation of the tangent line to the curve $x=5+t^2-t$, $y=t^2+5$ at the point $(5,6)$.
Setting $x=5$ and $y = 6$ and solving for $t$ gives me $t=0,1,-1$. I know I have to do y/x, and then take the derivative. But how do I know what $t$ value to use?
| You have
i) $x=5+t^2-t$ and ii) $y=t^2+5$ and $P=(5,6)$.
From $P=(5,6)$ you get $x=5$ and $y=6$.
From ii) you get now $t^2=1\Rightarrow t=1$ or $t=-1$ and from i) you get (knowing that $t\in${1,-1} ):$\ $ $t=1$
Your curve has the parametric representation
$\gamma: I\subseteq\mathbb{R}\rightarrow {\mathbb{R}}^2: t\maps... | {
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Cross Product of Partial Orders im going to have a similar questions on my test tomorrow. I am really stuck on this problem. I don't know how to start. Any sort of help will be appreciated. Thank you
Suppose that (L1;≤_1) and (L2;≤_2) are partially ordered sets. We define a partial order ≤on the set L1 x L2 in the most... | I will talk about part a, then you should give the other parts a try. They will follow in a similar manner (i.e. breaking $\leq$ into its components $\leq_1$ and $\leq_2$).
To show $(L_1 \times L_2, \leq)$ is a partial order, we need to show it is reflexive, anti-symmetric, and transitive.
Reflexivity: Given any $(x,y)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/211266",
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"source": "stackexchange",
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Number of prime divisors of element orders from character table. From wikipedia:
It follows, using some results of Richard Brauer from modular representation theory, that the prime divisors of the orders of the elements of each conjugacy class of a finite group can be deduced from its character table (an observation o... | I think I probably wrote the quoted passage in Wikipedia. If we let $\pi$ be a prime ideal of $\mathbb{Z}[\omega]$ containing $p,$ where $\omega$ is a primitive complex $|G|$-th root of unity, then it is the case that two elements $x$ and $y$ of $G$ have conjugate $p^{\prime}$-part if and only if we have $\chi(x) \equ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prime, followed by the cube of a prime, followed by the square of a prime. Other examples? The numbers 7, 8, 9, apart from being part of a really lame math joke, also have a unique property. Consecutively, they are a prime number, followed by the cube of a prime, followed by the square of a prime. Firstly, does this oc... | If you'll settle for a prime, cube of a prime, square of a prime in arithmetic progression (instead of consecutive), you've got $$5,27=3^3,49=7^2\qquad \rm{(common\ difference\ 22)}$$ and $$157,\ 343=7^3,\ 529=23^2 \qquad \rm{(common\ difference\ 186)}$$ and, no doubt, many more where those came from. A bit more ex... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/211379",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Inverse of a Positive Definite
Let K be nonsingular symmetric matrix, prove that if K is a positive
definite so is $K^{-1}$ .
My attempt:
I have that $K = K^T$ so $x^TKx = x^TK^Tx = (xK)^Tx = (xIK)^Tx$ and then I don't know what to do next.
| inspired by the answer of kjetil b halvorsen
To recap, matrix $A \in \mathbb{R}^{n \times n}$ is HPD (hermitian positive definite), iff $\forall x \in \mathbb{C}^n, x \neq 0 : x^*Ax > 0$.
HPD matrices have full rank, therefore are invertible and $A^{-1}$ exists. Also full rank matrices represent a bijection, therefore ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/211453",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Set of points of continuity are $G_{\delta}$ Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function. Show that the points at which $f$ is continuous is a $G_{\delta}$ set.
$$A_n = \{ x \in \mathbb{R} | x \in B(x,r) \text{ open }, f(x'')-f(x')<\frac{1}{n}, \forall x',x'' \in B(x)\}$$
I saw that this proof was already... | Here's a slightly different approach.
Let $G$ be the set of points where $f$ is continuous, $A_{n,x} = (x-\frac{1}{n}, x+ \frac{1}{n})$ is an open set where $f$ is continuous, and $A_n = \bigcup_{x \in G} A_{n,x}$.
Since $A_n$ is union of open sets which is open, $\bigcap_{n \in \mathbb{N}} A_n $ is a $G_\delta$ set. W... | {
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"url": "https://math.stackexchange.com/questions/211511",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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Riemann-Stieltjes integral, integration by parts (Rudin) Problem 17 of Chapter 6 of Rudin's Principles of Mathematical Analysis asks us to prove the following:
Suppose $\alpha$ increases monotonically on $[a,b]$, $g$ is
continuous, and $g(x)=G'(x)$ for $a \leq x \leq b$. Prove that,
$$\int_a^b\alpha(x)g(x)\,dx=G(b)\... | Compare with the following theorem,
Theorem: Suppose $f$ and $g$ are bounded functions with no common discontinuities on the interval $[a,b]$, and the Riemann-Stieltjes integral of $f$ with respect to $g$ exists. Then the Riemann-Stieltjes integral of $g$ with respect to $f$ exists, and
$$\int_{a}^{b} g(x)df(x) = f(b)g... | {
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"url": "https://math.stackexchange.com/questions/211552",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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What is the number of all possible relations/intersections of n sets? If n defines number of sets, what is the number of all possible relations between them? For example, when n = 2:
1) A can intersect with B
2) A and B can be disjoint
3) A can be subset of B
4) B can be subset of A
that leaves us with 4 possible relat... | Disclaimer: Not an answer®
I'd like to think about this problem not as sets, but as elements in a partial order. Suppose all sets are different. Define $\mathscr{P} = \langle\mathscr{P}(\bigcup_n A_n), \subseteq\rangle$ as the partial order generated bay subset relation on all "interesting" sets.
Define the operation $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/211645",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 2
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Why not write $\sqrt{3}2$? Is it just for aesthetic purposes, or is there a deeper reason why we write $2\sqrt{3}$ and not $\sqrt{3}2$?
| Certainly one can find old books in which $\sqrt{x}$ was set as $\sqrt{\vphantom{x}}x$, and just as $32$ does not mean $3\cdot2$, so also $\sqrt{\vphantom{32}}32$ would not mean $\sqrt{3}\cdot 2$, but rather $\sqrt{32}$. An overline was once used where round brackets are used today, so that, where we now write $(a+b)^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/211695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 4,
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How to find the eigenvalues and eigenvector without computation? The given matrix is
$$
\begin{pmatrix}
2 & 2 & 2 \\
2 & 2 & 2 \\
2 & 2 & 2 \\
\end{pmatrix}
$$
so, how could i find the eigenvalues and eigenvector without computation?
Thank you
| For the eigenvalues, you can look at the matrix and extract some quick informations.
Notice that the matrix has rank one (all columns are the same), hence zero is an eigenvalue with algebraic multiplicity two. For the third eigenvalue, use the fact that the trace of the matrix equals the sum of all its eigenvalues; sin... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Are there any memorization techniques that exist for math students? I just watched this video on Ted.com entitled:
Joshua Foer: Feats of memory anyone can do
and it got me thinking about memory from a programmers perspective, and since programming and mathematics are so similar I figured I post here as well. There are ... | For propositional logic operations, you can remember their truth tables as follows:
Let 0 stand for falsity and 1 for truth. For the conjunction operation use the mnemonic of the minimum of two numbers. For the disjunction operation, use the mnemonic of the maximum of two numbers. For the truth table for the materia... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 6,
"answer_id": 3
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linear algebra problem please help let $V=\mathbb{R}^4$ and let $W=\langle\begin{bmatrix}1&1&0&0\end{bmatrix}^t,\begin{bmatrix}1&0&1&0\end{bmatrix}^t\rangle$. we need to find the subspaces $U$ & $T$ such that $ V=W\bigoplus U$ & $V=W \bigoplus T$ but $U\ne T$.
| HINT: Look at a simpler problem first. Let $X=\{\langle x,0\rangle:x\in\Bbb R\}$, a subspace of $\Bbb R^2$. Can you find subspaces $V$ and $W$ of $\Bbb R^2$ such that $\Bbb R^2=X\oplus V=X\oplus W$, but $V\ne W$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/212017",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Arcsine law for Brownian motion Here is the question:
$(B_t,t\ge 0)$ is a standard brwonian motion, starting at $0$.
$S_t=\sup_{0\le s\le t} B_s$. $T=\inf\{t\ge 0: B_t=S_1\}$.
Show that $T$ follows the arcsinus law with density
$g(t)=\frac{1}{\pi\sqrt{t(1-t)}}1_{]0,1[}(t)$.
I used Markov property to get the fol... | Let us start from the formula $\mathbb P(T\lt t)=\mathbb P(\hat S_{1-t}\lt S_t)$, where $0\leqslant t\leqslant 1$, and $\hat S_{1-t}$ and $S_t$ are the maxima at times $1-t$ and $t$ of two independent Brownian motions.
Let $X$ and $Y$ denote two i.i.d. standard normal random variables, then $(\hat S_{1-t},S_t)$ coinci... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/212072",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Derivative wrt. to Lie bracket. Let $\mathbf{G}$ be a matrix Lie group, $\frak{g}$ the corresponding Lie algebra,
$\widehat{\mathbf{x}} = \sum_i^m x_i G_i$ the corresponding hat-operator ($G_i$ the $i$th basis vector of the tangent space/Lie algebra $\frak{g}$) and $(\cdot)^\vee$ the inverse of $\widehat{\cdot}$:
$$(X... | I may be misunderstanding your question, but it seems like you are asking for the derivative of the map
$$
\mathfrak g \to \mathfrak g, ~~ a \mapsto [a, b]
$$
where $b \in \mathfrak g$ is fixed. Since $\mathfrak g$ is a vector space, the derivative at a point can be viewed as a map $\mathfrak g \to \mathfrak g$. But ... | {
"language": "en",
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"source": "stackexchange",
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A subset of a compact set is compact? Claim:Let $S\subset T\subset X$ where $X$ is a metric space. If $T$ is compact in $X$ then $S$ is also compact in $X$.
Proof:Given that $T$ is compact in $X$ then any open cover of T, there is a finite open subcover, denote it as $\left \{V_i \right \}_{i=1}^{N}$. Since $S\subset ... | According to the definition of the compact set, we need every open cover of set K contains a finite subcover. Hence, not every subsets of compact sets are compact.
Why closed subsets of compact sets are compact?
Proof
Suppose $F\subset K\subset X$, F is closed in X, and K is compact. Let $\{G_{\alpha}\}$ be an open co... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "31",
"answer_count": 6,
"answer_id": 4
} |
What does it mean for something to be true but not provable in peano arithmetic? Specifically, the Paris-Harrington theorem.
In what sense is it true? True in Peano arithmetic but not provable in Peano arithmetic, or true in some other sense?
| Peano Arithmetic is a particular proof system for reasoning about the natural numbers. As such it does not make sense to speak about something being "true in PA" -- there is only "provable in PA", "disprovable in PA", and "independent of PA".
When we speak of "truth" it must be with respect to some particular model. In... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/213253",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 5,
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Proofs with limit superior and limit inferior: $\liminf a_n \leq \limsup a_n$ I am stuck on proofs with subsequences. I do not really have a strategy or starting point with subsequences.
NOTE: subsequential limits are limits of subsequences
Prove: $a_n$ is bounded $\implies \liminf a_n \leq \limsup a_n$
Proof:
Let $a... | Hint: Think about what the definitions mean. We have
$$\limsup a_n = \lim_n \sup \{ a_k \textrm{ : } k \geq n\}$$
and $$\liminf a_n = \lim_n \inf \{ a_k \textrm{ : } k \geq n\}$$
What can you say about the individual terms $\sup \{a_k \textrm{ : } k \geq n\}$ and
$\inf \{a_k \textrm{ : } k \geq n\}$ ?
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove an inequality with a $\sin$ function: $\sin(x) > \frac2\pi x$ for $0$$\forall{x\in(0,\frac{\pi}{2})}\
\sin(x) > \frac{2}{\pi}x $$
I suppose that solving $ \sin x = \frac{2}{\pi}x $ is the top difficulty of this exercise, but I don't know how to think out such cases in which there is an argument on the right si... | Here is a simple solution.
Let $0 < x < \pi/2$ be fixed. By Mean value theorem
there exists $y \in (0, x)$ such that
$$\sin(x)-\sin(0)= \cos(y)(x-0).$$
Thus $$\frac{\sin(x)}{x}= \cos(y).$$
Similarly there exists $z \in (x, \pi/2)$ such that
$$\sin(\pi/2)-\sin(x)= \cos(z)(\pi/2-x).$$
Thus $$\frac{1-\sin(x)}{\pi/2- x}= ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/213382",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Tenenbaum and Pollard, Ordinary Differential Equations, problem 1.4.29, what am I missing? Tenenbaum and Pollard's "Ordinary Differential Equations," chapter 1, section 4, problem 29 asks for a differential equation whose solution is "a family of straight lines that are tangent to the circle $x^2 + y^2 = c^2$, where $c... | I'll assume the point $P=(x,y)$ lies on the circle $x^2+y^2=c^2$ in the first quadrant. The slope of the tangent at $P$ is $y'$ as you say. You need to express the $y$ intercept.
Extend the tangent line until it meets the $x$ axis $A$ and the $y$ axis at $B$, and call the origin $O$. Then the two triangles $APO$ and $... | {
"language": "en",
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"question_score": "3",
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Solving trigonometric equations of the form $a\sin x + b\cos x = c$ Suppose that there is a trigonometric equation of the form $a\sin x + b\cos x = c$, where $a,b,c$ are real and $0 < x < 2\pi$. An example equation would go the following: $\sqrt{3}\sin x + \cos x = 2$ where $0<x<2\pi$.
How do you solve this equation wi... | Riffing on @Yves' "little known" solutions ...
The above trigonograph shows a scenario with $a^2 + b^2 = c^2 + d^2$, for $d \geq 0$, and we see that
$$\theta = \operatorname{atan}\frac{a}{b} + \operatorname{atan}\frac{d}{c} \tag{1}$$
(If the "$a$" triangle were taller than the "$b$" triangle, the "$+$" would become "... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/213545",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 6,
"answer_id": 2
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definite and indefinite sums and integrals It just occurred to me that I tend to think of integrals primarily as indefinite integrals and sums primarily as definite sums. That is, when I see a definite integral, my first approach at solving it is to find an antiderivative, and only if that doesn't seem promising I'll c... | Also, read about how Feynman learned some non-standard
methods of indefinite integration
(such as differentiating under
the integral sign)
and used these to get various integrals
that usually needed complex integration.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/213606",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 3,
"answer_id": 1
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Get the equation of a circle when given 3 points Get the equation of a circle through the points $(1,1), (2,4), (5,3) $.
I can solve this by simply drawing it, but is there a way of solving it (easily) without having to draw?
| Big hint:
Let $A\equiv (1,1)$,$B\equiv (2,4)$ and $C\equiv (5,3)$.
We know that the perpendicular bisectors of the three sides of a triangle are concurrent.Join $A$ and $B$ and also $B$ and $C$.
The perpendicular bisector of $AB$ must pass through the point $(\frac{1+2}{2},\frac{1+4}{2})$
Now find the equations of the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/213658",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "33",
"answer_count": 16,
"answer_id": 0
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The intersection of a line with a circle Get the intersections of the line $y=x+2$ with the circle $x^2+y^2=10$
What I did:
$y^2=10-x^2$
$y=\sqrt{10-x^2}$ or $y=-\sqrt{10-x^2}$
$ x+ 2 = y=\sqrt{10-x^2}$
If you continue, $x=-3$ or $x=1$ , so you get 2 points $(1,3)$, $(-3,-1)$
But then, and here is where the problems... | Let the intersection be $(a,b)$, so it must satisfy both the given eqaution.
So, $a=b+2$ also $a^2+b^2=10$
Putting $b=a+2$ in the given circle $a^2+(a+2)^2=10$
$2a^2+4a+4=10\implies a=1$ or $-3$
If $a=1,b=a+2=3$
If $a=-3,b=-3+2=-1$
So, the intersections are $(-3,-1)$ and $(1,3)$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to find subfields of a non-Galois field extension? Let $K/F$ be a finite field extension.
If $K/F$ is Galois then it is well known that there is a bijection
between subgroups of $Gal(K/F)$ and subfields of $K/F$.
Since finding subgroups of a finite group is always easy (at least
in the meaning that we can find ever... | In the inseparable case there is an idea for a substitute Galois correspondence due, I think, to Jacobson: instead of considering subgroups of the Galois group, we consider (restricted) Lie subalgebras of the Lie algebra of derivations. I don't know much about this approach, but "inseparable Galois theory" seems to be ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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Volume of Region R Bounded By $y=x^3$ and $y=x$ Let R be the region bounded by $y=x^3$ and $y=x$ in the first quadrant. Find the volume of the solid generated by revolving R about the line $x=-1$
| The region goes from $y=0$ to $y=1$. For an arbitrary $y$-value, say, $y=c$, $0\le c\le1$, what is the cross section of the region at height $c$? That is, what is the intersection of the region with the horizontal line $y=c$? What do you get when you rotate that cross-section around the line $x=-1$? Can you find the ar... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Matching in bipartite graphs I'm studying graph theory and the follow question is driving me crazy. Any hint in any direction would be appreciated.
Here is the question:
Let $G = G[X, Y]$ be a bipartite graph in which each vertex in $X$ is of odd degree. Suppose that any two distinct vertices of $X$ have an even numbe... | Hint for one possible solution:
Consider the adjacency matrix $M\in\Bbb F_2^{|X|\times|Y|}$ of the bipartite graph, i.e.
$$M_{x,y}:=\left\{ \begin{align} 1 & \text{ if }x,y \text{ are adjacent}
\\ 0 & \text{ else} \end{align} \right. $$
then try to prove, it has rank $|X|$, and then, I think, using Gaussian elimina... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/213923",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Prove $\lim _{x \to 0} \sin(\frac{1}{x}) \ne 0$ Prove $$\lim _{x \to 0} \sin\left(\frac{1}{x}\right) \ne 0.$$
I am unsure of how to prove this problem. I will ask questions if I have doubt on the proof. Thank you!
| HINT
Consider the sequences $$x_n = \dfrac1{2n \pi + \pi/2}$$ and $$y_n = \dfrac1{2n \pi + \pi/4}$$ and look at what happens to your function along these two sequences. Note that both sequences tend to $0$ as $n \to \infty$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/214010",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 0
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Find the limit without l'Hôpital's rule Find the limit
$$\lim_{x\to 1}\frac{(x^2-1)\sin(3x-3)}{\cos(x^3-1)\tan^2(x^2-x)}.$$
I'm a little rusty with limits, can somebody please give me some pointers on how to solve this one? Also, l'Hôpital's rule isn't allowed in case you were thinking of using it. Thanks in advance.
| $$\dfrac{\sin(3x-3)}{\tan^2(x^2-x)} = \dfrac{\sin(3x-3)}{3x-3} \times \left(\dfrac{x^2-x}{\tan(x^2-x)} \right)^2 \times \dfrac{3(x-1)}{x^2(x-1)^2}$$
Hence, $$\dfrac{(x^2-1)\sin(3x-3)}{\cos(x^3-1)\tan^2(x^2-x)} = \dfrac{\sin(3x-3)}{3x-3} \times \left(\dfrac{x^2-x}{\tan(x^2-x)} \right)^2 \times \dfrac{3(x-1)(x^2-1)}{x^2(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/214076",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Probability distribution for sampling an element $k$ times after $x$ independent random samplings with replacement In an earlier question ( probability distribution of coverage of a set after `X` independently, randomly selected members of the set ), Ross Rogers asked for the probability distribution for the coverage o... | What is not as apparent as it could be in the solution on the page you refer to is that the probability distributions in this kind of problems are often a mess while the expectations and variances can be much nicer. (By the way, you might be confusing probability distributions on the one hand, and expectations and vari... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/214132",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the probability mass function. Abe and Zach live in Springfield. Suppose Abe's friends and Zach's friends are each a random sample of 50 out of the 1000 people who live in Springfield.
Find the probability mass function of them having $X$ mutual friends.
I figured the expected value is $(1000)(\frac{50}{1000})^2 =... | The expected value is right, since:
$$E[X]=\frac{1}{\binom{1000}{50}}\sum_{k=1}^{50}k\binom{50}{k}\binom{950}{50-k}=\frac{50}{\binom{1000}{50}}\sum_{k=0}^{49}\binom{49}{k}\binom{950}{49-k}$$
$$\frac{\binom{1000}{50}}{50}\,E[X]=[x^{49}]\left((1+x)^{49} x^{49} (1+x^{-1})^{950}\right)=[x^{49}]\left((1+x)^{999}x^{-901}\rig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/214220",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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function from A5 to A6
Possible Duplicate:
Homomorphism between $A_5$ and $A_6$
Why is it true that every element of the image of the function $f: A_5\longrightarrow A_6$ (alternating groups) defined by $f(x)=(123)(456) x (654)(321)$ does not leave any element of $\{1,2,3,4,5,6\}$ fixed (except the identity)?
| I think you are asking why the following is true:
Every element of $A_6$ of the form $(123)(456)x(654)(321)$, with $x$ a nontrivial element of $A_5$, leaves no element of $\{1,2,3,4,5,6\}$ fixed.
This is actually false: let $x = (12)(34)$. Then $f(x)$ fixes either $5$ or $6$, depending on how you define composition ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/214285",
"timestamp": "2023-03-29T00:00:00",
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Bounding the Gamma Function I'm trying to verify a bound for the gamma function
$$ \Gamma(z) = \int_0^\infty e^{-t}t^{z - 1}\;dt. $$
In particular, for real $m \geq 1$, I'd like to show that
$$ \Gamma(m + 1) \leq 2\left(\frac{3m}{5}\right)^m. $$
Knowing that the bound should be attainable, my first instinct is to split... | I'll prove something that's close enough for my applications; in particular, that $$\Gamma(m + 1) \leq 3\left(\frac{3m}{5}\right)^m.$$
Let $0 < \alpha < 1$ be chosen later. We'll split $e^{-t}t^m$ as $(e^{-\alpha t}t^m)e^{-(1 - \alpha)t}$ and use this to bound the integral.
First, take a derivative to find a maximum f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/214422",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 1
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Show $m^p+n^p\equiv 0 \mod p$ implies $m^p+n^p\equiv 0 \mod p^2$ Let $p$ an odd prime. Show that $m^p+n^p\equiv 0 \pmod p$ implies $m^p+n^p\equiv 0 \pmod{p^2}$.
| From little Fermat, $m^p \equiv m \pmod p$ and $n^p \equiv n \pmod p$. Hence, $p$ divides $m+n$ i.e. $m+n = pk$.
$$m^p + n^p = (pk-n)^p + n^p = p^2 M + \dbinom{p}{1} (pk) (-n)^{p-1} + (-n)^p + n^p\\ = p^2 M + p^2k (-n)^{p-1} \equiv 0 \pmod {p^2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/214497",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Generators and cyclic group concept These statements are false according to my book. I am not sure why though
*
*In every cyclic group, every element is a generator
*A cyclic group has a unique generator.
Both statements seem to be opposites.
I tried to give a counterexample
*
*I think it's because $\mathbb{Z}... | Take $Z_n$. This group is cyclic and the generators are $\phi(n)$ = all the numbers that are relatively prime to $n$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Trigonometric bounds Is there a nice way to show:
$\sin(x) + \sin(y) + \sin(z) \geq 2$ for all $x,y,z$ such that $0 \leq x,y,z \leq \frac{\pi}{2}$ and $x + y + z = \pi$?
| Use the following inequality: $$\sin(x) \geq x\frac{2}{\pi} , x \in [0,\pi/2]$$
And to prove this inequality, Consider the function:
$ f(x) = \frac{\sin(x)}{x} $ if $x \in (0, \pi/2]$ and $f(x) = 1$ if $x=0$. Now show $f$ decreases on $[0,\pi/2]$. Hint: Use Mean Value Theorem.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/214624",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Find pair of polynomials a(x) and b(x) If $a(x) + b(x) = x^6-1$ and $\gcd(a(x),b(x))=x+1$ then find a pair of polynomials of $a(x)$,$b(x)$.
Prove or disprove, if there exists more than 1 more distinct values of the polynomials.
| There is too much freedom. Let $a(x)=x+1$ and $b(x)=(x^6-1)-(x+1)$. Or else use $a(x)=k(x+1)$, $b(x)=x^6-1-k(x+1)$, where $k$ is any non-zero integer.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/214705",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Sobolev differentiability of composite function I was wondering about the following fact: if $\Omega$ is a bounded subset of $\mathbb{R}^n$ and $u\in W^{1,p}(\Omega)$ and $g\in C^1(\mathbb{R},\mathbb{R})$ such that $|g'(t)t|+|g(t)|\leq M$, is it true that $g\circ u \in W^{1,p}(\Omega)$?
If $g'\in L^{\infty}$ this would... | You get $g' \in L^\infty$ from the assumptions, since $|g'(t)| \le M/|t|$, and $g'$ is continuous at $0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/214893",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Uncountability of basis of $\mathbb R^{\mathbb N}$ Given vector space $V$ over $\mathbb R$ such that the elements of $V$ are infinite-tuples. How to show that any basis of it is uncountable?
| Take any almost disjoint family $\mathcal A$ of infinite subsets of $\mathbb N$ with cardinality $2^{\aleph_0}$.
Construction of such set is given here.
I.e. for any two set $A,B\in\mathcal A$ the intersection $A\cap B$ is finite.
Notice that
$$\{\chi_A; A\in\mathcal A\}$$
is a subset of $\mathbb R^{\mathbb N}$ which ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/214984",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Irreducibility of $X^{p-1} + \cdots + X+1$
Can someone give me a hint how to the irreducibility of $X^{p-1} + \cdots + X+1$, where $p$ is a prime, in $\mathbb{Z}[X]$ ?
Our professor gave us already one, namely to substitute $X$ with $X+1$, but I couldn't make much of that.
| Hint: Let $y=x-1$. Note that our polynomial is $\dfrac{x^p-1}{x-1}$, which is
$\dfrac{(y+1)^p-1}{y}$.
It is not difficult to show that $\binom{p}{k}$ is divisible by $p$ if $0\lt k\lt p$. Now use the Eisenstein Criterion.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/215042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 4,
"answer_id": 2
} |
Continuity of $f \cdot g$ and $f/g$ on standard topology. Let $f, g: X \rightarrow \mathbb{R}$ be continuous functions, where ($X, \tau$) is a topological space and $\mathbb{R}$ is given the standard topology.
a)Show that the function $f \cdot g : X \rightarrow \mathbb{R}$,defined by
$(f \cdot g)(x) = f(x)g(x)$
is cont... | The central fact is that the operations $$p:\ {\mathbb R}^2\to{\mathbb R},\quad (x,y)\mapsto x\cdot y$$ and similarly $q:\ (x,y)\mapsto {\displaystyle{x\over y}}$ are continuous where defined and that $$h:\ X\to{\mathbb R}^2,\quad x\mapsto\bigl(f(x),g(x)\bigr)$$ is continuous if $f$ and $g$ are continuous.
It follows t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/215109",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Cartesian product of a set Question:
What is A $\times$ A , where A = {0, $\pm$1, $\pm$2, ...} ?
Thinking:
Is this a set say B = {0, 1, 2, ... } ?
This was in my homework can you help me ?
| No, $A\times A$ is not a sequence. Neither is $A$: it’s just a set.
By definition $A\times A=\{\langle a_1,a_2\rangle:a_1,a_2\in A\}$. Thus, $A\times A$ contains elements like $\langle 1,0\rangle$, $\langle -2,17\rangle$, and so on. Indeed, since $A$ is just the set of all integers, usually denoted by $\Bbb Z$, $A\time... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/215177",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Show that $\liminf \limits _{k\rightarrow \infty} f_k = \lim \limits _{k\rightarrow \infty} f_k$ Is there a way to show that $\liminf \limits _{k\rightarrow \infty} f_k = \lim \limits _{k\rightarrow \infty} f_k$. The only way I can think of is by showing $\liminf \limits _{k\rightarrow \infty} f_k = \limsup \limits _{k... | The following always holds: $\inf_{k\geq n} f_k \leq f_n \leq \sup_{k\geq n} f_k$. Note that the lower bound in non-decreasing and the upper bound is non-increasing.
Suppose $\alpha = \liminf_k f_k = \limsup_k f_k$, and let $\epsilon>0$. Then there exists a $N$ such that for $n>N$, we have $\alpha -\inf_{k\geq n} f_k <... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/215258",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Why is $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$? It seems as if no one has asked this here before, unless I don't know how to search.
The Gamma function is
$$
\Gamma(\alpha)=\int_0^\infty x^{\alpha-1} e^{-x}\,dx.
$$
Why is
$$
\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}\text{ ?}
$$
(I'll post my own answer, but I know ... | This is a "proof". We know that the surface area of the $n-1$ dimensional unit sphere is
$$
|S^{n-1}| = \frac{2\pi^{\frac{n}2}}{\Gamma(\frac{n}2)}.
$$
On the other hand, we know that $|S^2|=4\pi$, which gives
$$
4\pi = \frac{2\pi^{\frac32}}{\Gamma(\frac32)} = \frac{2\pi^{\frac32}}{\frac12\Gamma(\frac12)}.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/215352",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "76",
"answer_count": 11,
"answer_id": 10
} |
What gives rise to the normal distribution? I'd like to know if anyone has a generally friendly explanation of why the normal distribution is an attractor of so many observed behaviors in their eventuality. I have a degree in math if you want to get technical, but I'd like to be able to explain to my grandma as well
| To my mind the reason for the pre-eminence can at best be seen in what must be the most electrifying half page of prose in the scientific literature, where
Clark Maxwell deduces the distribution law for the velocities of molecules
of an ideal gas (now known as the Maxwell-Boltzmann law), thus founding the discipline o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/215432",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 1
} |
Function that sends $1,2,3,4$ to $0,1,1,0$ respectively I already got tired trying to think of a function $f:\{1,2,3,4\}\rightarrow \{0,1,1,0\}$ in other words:
$$f(1)=0\\f(2)=1\\f(3)=1\\f(4)=0$$
Don't suggest division in integers; it will not pass for me. Are there ways to implement it with modulo, absolute value, and... | Look the example for Lagrange Interpolation, then it is easy to construct any function from any sequence to any sequence.
In this case :
$$L(x)=\frac{1}{2}(x-1)(x-3)(x-4) + \frac{-1}{2}(x-1)(x-2)(x-4)$$
wich simplifies to:
$$L(x)=\frac{-1}{2}(x-1)(x-4)$$
which could possibly explain Jasper's answer, but since the meth... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/215487",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 10,
"answer_id": 9
} |
Prove that the language $\{ww \mid w \in \{a,b\}^*\}$ is not FA (Finite Automata) recognisable. Hint: Assume that $|xy| \le k$ in the pumping lemma.
I have no idea where to begin for this. Any help would be much appreciated.
| It's also possible — and perhaps simpler — to prove this directly using the pigeonhole principle without invoking the pumping lemma.
Namely, assume that the language $L = \{ww \,|\, w \in \{a,b\}^*\}$ is recognized by a finite state automaton with $n$ states, and consider the set $W = \{a,b\}^k \subset \{a,b\}^*$ of wo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/215553",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Differentiability at 0 I am having a problem with this exercise. Please help.
Let $\alpha >1$. Show that if $|f(x)| \leq |x|^\alpha$, then $f$ is differentiable at $0$.
| Use the definition of the derivative. It is clear that $f(0)=0$. Note that if $h\ne 0$ then
$$\left|\frac{f(h)-0}{h}\right| \le |h|^{\alpha-1}.$$
Since $\alpha\gt 1$, $|h|^{\alpha-1}\to 0$ as $h\to 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/215633",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Solve logarithmic equation I'm getting stuck trying to solve this logarithmic equation:
$$
\log( \sqrt{4-x} ) - \log( \sqrt{x+3} ) = \log(x)
$$
I understand that the first and second terms can be combined & the logarithms share the same base so one-to-one properties apply and I get to:
$$
x = \frac{\sqrt{4-x}}{ \sqrt... | Fine so far. I would just use Wolfram Alpha, which shows there is a root about $0.89329$. The exact value is a real mess. I tried the rational root theorem, which failed. If I didn't have Alpha, I would go for a numeric solution. You can see there is a solution in $(0,1)$ because the left side is $-4$ at $0$ and $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/215732",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
How to convert a formula to CNF? I am trying to convert the formula: $((p \wedge \neg n) \vee (n \wedge \neg p)) \vee z$. I understand i need to apply the z to each clause, which gives: $((p \wedge \neg n) \vee z) \vee ((n \wedge \neg p) \vee z)$. I know how to simplify this, but am unsure how it will lead to the answe... | To convert to conjugtive normal form we use the following rules:
Double Negation:
*
*$P\leftrightarrow \lnot(\lnot P)$
De Morgans Laws
*
*$\lnot(P\bigvee Q)\leftrightarrow (\lnot P) \bigwedge (\lnot Q)$
*$\lnot(P\bigwedge Q)\leftrightarrow (\lnot P) \bigvee (\lnot Q)$
Distributive Laws
*
*$(P \bigvee (Q\bi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/215790",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How do you prove that proof by induction is a proof? Are proofs by induction limited to cases where there is an explicit dependance on a integer, like sums? I cannot grasp the idea of induction being a proof in less explicit cases. What if you have a function that suddenly changes behavior? If a function is positive up... | This math.SE question has a lot of great answers as far as induction over the reals goes. And as Austin mentioned, there are many cases in graph theory where you can use induction on the vertices or edges of a graph to prove a result. an example:
If every two nodes of $G$ are joined by a unique path, then $G$ is connec... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/215846",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 6,
"answer_id": 3
} |
Proving that floor(n/2)=n/2 if n is an even integer and floor(n/2)=(n-1)/2 if n is an odd integer. How would one go about proving the following. Any ideas as to where to start?
For any integer n, the floor of n/2 equals n/2 if n is even and (n-1)/2 if n is odd.
Summarize:
[n/2] = n/2 if n = even
[n/2] = (n-1)/2 if n ... | You should set $n=2m$ for even numbers, where $m$ is an integer. Then $\frac n2=m$ and the floor of an integer is itself. The odd case is similar.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/215909",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Solve the Relation $T(n)=T(n/4)+T(3n/4)+n$
Solve the recurrence relation: $T(n)=T(n/4)+T(3n/4)+n$. Also, specify an asymptotic bound.
Clearly $T(n)\in \Omega(n)$ because of the constant factor. The recursive nature hints at a possibly logarithmic runtime (because $T(n) = T(n/2) + 1$ is logarithmic, something similar... | $T(n)=T\left(\dfrac{n}{4}\right)+T\left(\dfrac{3n}{4}\right)+n$
$T(n)-T\left(\dfrac{n}{4}\right)-T\left(\dfrac{3n}{4}\right)=n$
For the particular solution part, getting the close-form solution is not a great problem.
Let $T_p(n)=An\ln n$ ,
Then $An\ln n-\dfrac{An}{4}\ln\dfrac{n}{4}-\dfrac{3An}{4}\ln\dfrac{3n}{4}\equiv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/215984",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
$ \lim_{x \to \infty} \frac{xe^{x-1}}{(x-1)e^x} $ $$ \lim_{x \to \infty} \frac{xe^{x-1}}{(x-1)e^x} $$
I don't know what to do. At all.
I've read the explanations in my book at least a thousand times, but they're over my head.
Oh, and I'm not allowed to use L'Hospital's rule. (I'm guessing it isn't needed for limits of ... | $$\lim_{x\to\infty}\frac{xe^{x-1}}{(x-1)e^x}=\lim_{x\to\infty}\frac{x}{x-1}\cdot\lim_{x\to\infty}\frac{e^{x-1}}{e^x}=1\cdot\frac{1}{e}=\frac{1}{e}$$
the first equality being justified by the fact that each of the right hand side limits exists finitely.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/216049",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Prove DeMorgan's Theorem for indexed family of sets. Let $\{A_n\}_{n\in\mathbb N}$ be an indexed family of sets. Then:
$(i) (\bigcup\limits_{n=1}^\infty A_n)' = \bigcap\limits_{n=1}^\infty (A'_n)$
$(ii) (\bigcap\limits_{n=1}^\infty A_n)' = \bigcup\limits_{n=1}^\infty (A'_n)$
I went from doing simple, straightforward in... | If $a\in (\bigcup_{n=1}^{\infty}A_{n})'$ then $a\notin A_{n}$ for any $n\in \mathbb{N}$, therefore $a\in A_{n}'$ for all $n\in \mathbb{N}$. Thus $a\in \bigcap_{n=1}^{\infty}A_{n}'$. Since $a$ was arbitrary, this shows $(\bigcup_{n=1}^{\infty}A_{n})' \subset \bigcap_{n=1}^{\infty}A_{n}'$. The other containment and th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/216149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
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