Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Simple permutations: approach and proof explanation for an exercise not very clear Set $A = (a_1, a_2, a_3), B = (b_1, b_2, b_3)$ and $C = (c_1, c_2)$ are assigned.
How many are the ordered sequences formed by $5$ distinct elements containing $2$ elements of $A$, two elements of $B$, and one of $C$?
Related question: I... | Your computation assumes that the five element sequence has the $a$'s first, the $b$'s second, and the $c$ last. If you are allowed to mix them up, it is easier to choose the two $a$'s without order in ${3 \choose 2}=3$ ways, the two $b$'s without order in ${3 \choose 2}=3$ ways and the $c$ in ${2 \choose 1}=2$ ways f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3792239",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Show that $ G $ is isomorphic to the direct product of $ H $ and $ K $. Let $ G $ be a group of order $ 20 $. Suppose that $ G $ has a subgroup $ H $ of order $ 4 $ and a subgroup $ K $ of order $ 5 $ such that $ hk = kh $ for all $ h \in H $ and $ k \in K $. Show that $ G $ is isomorphic to the direct product of $ H $... | Hint: It suffices to check that the following three properties are satisfied: $1)H\cap K=\emptyset \quad 2)G=HK$ and $3)$the elements of $H$ and $K$ commute.
These properties are easy to verify: $1)$ follows from Lagrange. $2)$ does also, after noting that $HK\le G$ since both $H$ and $K$ are normal. For $H$ and $K... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3792397",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Word Problems regarding Probability This post is going to be lengthy. I am studying Probability to recall my knowledge about it before I take a class in Stats this college. The thing is, the textbook I have didn't provide any solution which could help me determine if my answers were correct or not. Anyway, here are the... | Ok, here we go!
I'll give you some answers and working, and leave some for you:
*
*This depends on the wording. If the books are all distinct, then there are $7! = 7*6*5*4*3*2*1 = 5040$ arrangements. But, if bio books are identical and math books are identical, there are$ \frac{7!}{5!*2!} = \frac{5040}{240} =$ 21.
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3792510",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why the reals with the operation $x \bullet y = \sqrt[3]{x^3 + y^3}$ is a group? The operation above is a group for the real numbers, since 0 is the identity element, and the negative of any real number is its inverse, as it can be observed trivially. Associativity is less trivial, but it holds.
In fact, if we substitu... | For an arbitrary bijection $f\colon \mathbf R \to \mathbf R$, the operation $x*y = f^{-1}(f(x) + f(y))$ is a group law on $\mathbf R$. All this says is that if you rename each real number $x$ as $f(x)$ then you can convert the original group law $+$ into a group law $*$ so that $f$ is an isomorphism from $(\mathbf R, *... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3792604",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 1
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How to construct an algebra / field that is infinitely countable? It is well-known that a $\sigma$-algebra / $\sigma$-field can only be finite or uncountable infinite, but how to construct an example of algebra / field that is infinitely countable? This is actually a question from Billingsley's Probability and Measure ... | An example is the collection of finite unions of intervals in $\Bbb{R}$ with end points in $\Bbb{Q}\cup \{\infty, -\infty\}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3792707",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Conditional expectation with multiple conditioning For any r.v.s $X$ and $Y$:
$$E(Y|E(Y|X)) = E(Y|X)$$
But I cannot seem to be able to prove this. I tried using Adam's Law with extra conditioning ($E(Y|X) = E(E(Y|X,Z)|Z)$) but I don't seem to get anywhere with it.
What I tried is the following:
$$g(X) = E(Y|X)$$
$$E(Y|... | Given the level of formality of the book, I think what the exercise is going for is primarily conceptual. I.e., what does the below conditioning mean:
E(Y|E(Y|X), X) | X = x
This represents the expectation of Y, if I know what the expectation of Y given X would be... and now I also know X! So I'm exactly in the situati... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3792932",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Congruent sets of an arithmetic sequence and a geometric sequence Suppose we have a $a,d,$ and $q$ such that $a \neq 0, d \neq 0.$ Then, let $M = \{a, a + d, a + 2d\}$ and $N = \{a, aq, aq^2\}.$ Given that $M = N,$ find the value of $q.$
(A) $\frac12$
(B) $\frac13$
(C) $-\frac14$
(D) $-\frac12$
(E) $-2$
I immediately ... | One could also work with the difference between terms, rather than eliminating it. The two possible orderings of the elements in the set are
$ \ \{ a \ , \ aq \ = \ a + d \ , \ aq^2 \ = \ a + 2d \} \ $ or $ \ \{ a \ , \ aq \ = \ a + 2d \ , \ aq^2 \ = \ a + d \} \ \ . $ [Initially, it "feels like" the second arrange... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3793023",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Density of Borel set at 0 The Lebesgue density theorem says that if $E$ is a Lebesgue measurable set, then the density of $E$ at almost every element of $E$ is 1 and the density of $E$ at almost every element not in $E$ is 0.
However, is it true that for each $t$ strictly between 0 and 1, there is a Borel set $E$ that ... | Consider a sequence of numbers $r_n \searrow 0$ such that $\frac{r_{n-1}}{r_n} \to 1$. Let $\theta$ be a measure preserving map from $(0,r_1]$ to $\mathbb R^2$ that takes $(\pi r_{n}^2,\pi r_{n-1}^2] \subset \mathbb R$ to $\{x \in \mathbb R^2: r_n < |x| \le r_{n-1}\}$. Then let $A$ be a 'piece of pie' centered at the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3793108",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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Inequality for function of $\arctan(x)$ I want to show that $$f(x) = \frac{1}{\arctan(x)} - \frac{1}{x} $$ is increasing on $(0, \infty)$. I can see this clearly by plotting it, but I'm struggling to write it out rigorously. It obviously suffices to show its derivative is always positive in this range (which is also ... | Consider instead $ \displaystyle g(x) = \arctan{x} - \frac{x^2}{1 + x^2}$. Note that $g(0) = 0$, so it suffices to show that $g'(x) = 0$ for $x \ge 0$.
Now, $\displaystyle g'(x) = \frac{2[(1 + x^2)\arctan{x} - x]}{(1 + x^2)^2}$. It thus suffices to consider $$h(x) = \arctan{x} - \frac{x}{(1 + x^2)},$$ and show that $h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3793223",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
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Continuous function with upper dini derivative greater than 0 implies function is increasing
Let $f$ be continuous on $[a,b]$ with $\bar D f \geq 0$ (upper Dini derivative of $f$) on $(a,b)$. Show that $f$ is increasing on $[a,b]$.
Hint: Show this is true for $g$ with $\bar D g \geq \epsilon > 0$ on $[a,b]$. Apply thi... | How do you know that $2$ holds? In fact, this is the gist of the proof, unless I am misreading your question, you need to do a bit of work. (Drawing a picture will help!) First suppose that $\bar D f >0$ on $(a,b)$. If there are $a<c<d<b$ such that $f(c)>f(d)$ then we may choose $f(c)>\mu>f(d)$. Let $S=\{t\in (c,d):f(t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3793363",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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$f$ is Lipschitz if and only if there exists $L\geq0$ such that $|f'(x)|\leq L$ Let $f:[a,b]\to\mathbb{R}$ be absolutely continuous.
Prove that $f$ is Lipschitz if and only if there exists $L\geq0$ and a set $E\subset[a,b]$ such that, $m(E)=0$ and $f$ is differentiable at each $x\in[a,b]\setminus E\quad$ with $|f'(x)|... | Since $f$ is absolutely continuous, we have that
$$ f(x) - f(y) = \int_x^y f'(t) \, dt .$$
Then
\begin{align}
\text{$f$ is Lipschitz with constant $L$} &\Leftrightarrow
-L (y-x) \le \int_x^y f'(t) \, dt \le L (y-x) \text{ for all $x < y$} \\
&\Leftrightarrow \int_x^y (L - f'(t)) \, dt \ge 0 \text{ and } \int_x^y (L + ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3793507",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Find the sum of series: $\sum_{n=0}^{\infty}\frac{x^{2n}}{(2n)!}$ I have some trouble with series theory. The specific questions are as follows:
\begin{equation}
\sum_{n=0}^{\infty}\frac{x^{2n}}{(2n)!}
\end{equation}
My idea is just like this:
Since $e^x=\sum_{n=0}^{\infty}\frac{x^{n}}{n!}$,
\begin{align}
\sum_{n=0}^{\... | What you did wrong was changing $(2n)!$ to $2^nn!$.
You were correct that $e^x=\sum\limits_{n=0}^{\infty}\dfrac{x^{n}}{n!}$,
so $\cosh x = \dfrac{e^x+e^{-x}}2=\dfrac{\sum\limits_{n=0}^{\infty}\frac{x^{n}}{n!}+\sum\limits_{n=0}^{\infty}\frac{(-x)^{n}}{n!}}2=\dfrac{\sum\limits_{n=0}^{\infty}\frac{x^{n}}{n!}\left(1+(-1)^n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3793598",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to calculate $\int _{-\infty }^{\infty }\frac{x\sin \left(x\right)}{1+x^4}\,dx$ I want to calculate $\int _{-\infty }^{\infty }\frac{x\sin \left(x\right)}{1+x^4}\,dx$, but I don’t want to use complex analysis. How can I calculate it?
I tried
$$I\left(t\right)=\int _{-\infty }^{\infty }\frac{x\sin \left(tx\right)}{... | With $I\left(t\right)=\int _{-\infty}^{\infty }\frac{x\sin \left(tx\right)}{1+x^4}\:dx$, you have
$I’’’’(t)+I(t)= 0$, along with all the initial conditions
$$I(0)=0, \>\>\>I’(0)=\int_{-\infty}^\infty \frac{x^2}{1+x^4}dx =\frac\pi{\sqrt2} ,\\ I’’(0)=-\pi, \>\>\>
I’’’(0)=\int_{-\infty}^\infty \frac{1}{1+x^4}dx =\frac\p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3793751",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 0
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Probability of meeting in the grid. Aubrey leaves home at $(0,0)$ to walk to school at $(1,5)$ and travels one block (one unit) north or east every minute; she does not leave the rectangle $0 \leq x \leq1$, $0\leq y \leq5$. Xander leaves school at the same time and is headed back toward home else at $(0,0)$ and travels... | First, note that the distance between home and school is $6$ units. Since they start at the same time and are moving with the same speed ($1$ unit per minute) , they can only meet after three minutes. So, the possible meeting points are $(0,3)$ and $(1,2)$. Let's find the probability that the meet in the first one. For... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3793866",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Is it possible to write a metric a space as a countable disjoint union of compact sets? Let $ (X,d)$ be a metric space and let $\mu $ be a Radon $\sigma$-finite measure on the Borel $\sigma$-algebra. I read that it's possible to find countable disjoint compact sets $\lbrace K_n\rbrace_{\mathbb{N}}$ and a $\mu$-null set... | The key assumption here is that $\mu$ is a Radon measure, meaning it is inner regular with respect to compact sets. Without this assumption, this is not true, not even if $\mu$ is finite (for instance, there are metric spaces supporting continuous measures in which all compact sets are finite).
Write $X=\bigcup_n X_n$,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3793962",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Show that the solution of $\frac{\rm d}{{\rm d}t}X^x(t)=v(t,X^x(t))$, $X^x(0)=x$, is differentiable in $x$ Let $X^x$ be the solution of$^1$ \begin{align}\frac{\rm d}{{\rm d}t}X^x(t)&=v(t,X^x(t))\\ X^x(0)&=x\end{align} and $$T_t(x):=X^x(t).$$
Assuming that $v$ is differentiable in the second argument, can we show that ... | Define an augmentation of the ODE system via
$$
\frac{d}{dt}U^x(t)=\frac{\partial v}{\partial x}(t,X^x(t))\,U^x(t), ~~~ U^x(0)=I.
$$
Then use Grönwall or similar to find a bound for $E^{x,Δx}(t)=X^{x+\Delta x}(t)-X^x(t)-U^x(t)\Delta x$.
Assume that the following considerations are restricted to a compact domain so tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3794233",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Obtaining a Quotient Space of $\mathbb{R}^2$ Using stereographic projection of a sphere, $S^2$, we can obtain the one point compactification of $\mathbb{R}^2$ is sphere, i.e. $S^2$ can be thought of as $\mathbb{R}^2 \cup \{ \infty \}$. Now I am wondering how can $S^2$ be obtained by quotienting $\mathbb{R^2}$.
I have a... | $\mathbb R^2$ has no boundary. But that doesn't matter, we can still get $S^2$ by quotienting $\mathbb R^2$: take the open unit disc and collapse every point which is not in the disc into a single point. This point is essentially the pole needed for the one-point compactification of the unit disc.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3794356",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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cardinality of functions from N to N using Schröder–Bernstein theorem Im trying to prove that $\left|A\right|=\aleph$ for the following
group: $A=\left\{f\in\mathbb{N}\rightarrow\left.\mathbb{N}\right|\forall n\le m\ .f\left(n\right)\le f\left(m\right)\right\}$ using Schröder–Bernstein theorem
To prove that $\left|A\... | Hint: to get an injection of $\Bbb{P}(\Bbb{N})$ into $A$, define $f(X)$ to the function such that $f(0) = 0$ and:
$$f(x + 1) = \left\{ \begin{array}{l@{\quad}l}
f(x) & \mbox{$x \not\in X$}\\
f(x) + 1 & \mbox{$x \in X$}
\end{array}\right.$$
Now show how you can recover $X$ from $f(X)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3794462",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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The minimal poly of $\sqrt[3]{2}$ over $\Bbb{Q}$ is equal to $\det(T_a - xI)$ where $T_a$ is a matrix over $\Bbb{Q}$ that represents mult. by $a$. Let $K/F$ be a field extension of degree $n \in \Bbb{N}$ and for each $a \in K$ define $L_a(x) = a x$. Then $L_a(x)$ is an $F$-linear transformation of $K$ as a vector spac... | I think you've computed $T_a$ incorrectly. I assume you're using the ordered basis $(1,\sqrt[3]{2},\sqrt[3]{4})$ for $K$ as $\mathbb Q$-vector space (edit: I see that you are). So applying $L_a$ to the first basis vector gives $L_a(1)=\sqrt[3]{2}$. In terms of the coordinate vectors relative to this ordered basis, this... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3794597",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Pseudoinverse of a diagonal matrix Let matrix $A \in \Bbb R^{n \times n}$ have $k$ diagonal elements, where $k < n$, and rest of the elements are zero. I am trying to find the pseudoinverse of $A + \lambda I$ when $\lambda$ approaches zero.
Then $\frac{1}{a_i + \lambda}$ would be the diagonal elements for $i$ going fro... | The problem is that the pseudo inverse is not a continuous function on the space of matrices as exactly you've shown. Consider the 1d matrix $(x)$ for $x\in\mathbb R$. Then the pseudo-inverse map is
$$
(x)\mapsto\begin{cases}1/x&\text{ if }x\neq 0,\\0&\text{ otherwise.} \end{cases}
$$
This is not a continuous at zero,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3794718",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Line in polar coordinates I just wanted to clarify something. A line in polar coordinates has the parameterization of $\theta = k\pi$ for $k \in \mathbb{R}$ right? Or am I missing something?
| Hint: Let $x = r \cos \theta$ and $y= r \sin \theta$ and see if you can find the polar equations for
$$y = mx + b \\ y = x + b \\ y = x$$ For the third equation, what do you notice?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3794791",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
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A Lie algebra with two of its Lie bracket are zero while the third is not. Is it possible to construct a system of three vector field u,v and w an $R^{3}$ such that [u,v]=0=[u,w], but [v,w]$\neq$0?
I tried to solve it by applying the Jacobi identity property which makes the 2nd and 3rd term vanish so I am left with [u[... | Yes, consider the 2 dimensional non commutative Lie algebra, $[a,b]=a$, add $c$, in the center $[c,a]=[c,b]=0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3794908",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Integrating a 'twisted' rational function For $x\in [0,1]$, let
$$
P_n (x) = \prod_{k=1}^{n} (x^k+1)^{(-1)^k}.
$$For example, $\displaystyle{P_4(x) = \frac{(x^2+1)(x^4+1)}{(x+1)(x^3+1)}}$. Of note: $P_n(1)=1/2$ if $n$ is odd and $1$ if $n$ is even, so we cannot expect uniform convergence on $[0,1)$. I am interested in ... | The infinite product $$P(x) = \prod_{k=1}^\infty (x^k+1)^{(-1)^k}$$
converges to a nonzero value if $|x| < 1$ because
$$\sum_{k=1}^\infty \log \left((x^k+1)^{(-1)^k}\right) = \sum_{k=1}^\infty (-1)^k \log(x^k+1)$$
converges. Its Maclaurin series coefficients are OEIS sequence A083365. According to that, $P(x) = \psi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3795183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 1,
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Understanding the proof of: Every convex function is continuous I am trying to understand the following proof:
Theorem 2.10. If $f$ is a convex function defined on an open interval $(a, b)$ then $f$ is continuous on $(a, b)$
Proof. Suppose $f$ is convex on $(a, b),$ and let $[c, d] \subseteq(a, b) .$ Choose $c_{1}$ an... | Since the author found to numbers $\alpha$ and $\beta$ such that you always have, when $c\leqslant x<y\leqslant d$,$$\frac{f(y)-f(x)}{y-x}\leqslant\alpha$$and$$\frac{f(y)-f(x)}{y-x}\geqslant\beta,$$then the set$$\left\{\frac{f(y)-f(x)}{y-x}\,\middle|\,c\leqslant x<y\leqslant d\right\}$$is bounded and therefore the set$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3795317",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A sum of series with the inverse squared central binomial coefficient A nice challenge by Cornel Valean:
Show that
$$2\sum _{n=1}^{\infty }\frac{2^{4 n}}{\displaystyle n^3 \binom{2 n}{n}^2}-\sum _{n=1}^{\infty }\frac{2^{4 n}}{\displaystyle n^4 \binom{2 n}{n}^2}+\sum _{n=1}^{\infty }\frac{2^{4 n} H_n^{(2)}}{\displaystyl... | An excellent answer was already given (the chosen one), but good to have more ways in place.
A solution by Cornel Ioan Valean
Instead of calculating all three series separately, we might try to calculate them all at once. So, we have that
$$2\sum _{n=1}^{\infty }\frac{2^{4 n}}{\displaystyle n^3 \binom{2 n}{n}^2}-\sum... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3795555",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
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Prove that this number is divisible by 7 Without using induction, how can it be proved that 7 divides $3^{2n+1}+2^{n+2}$ for each $n\in\mathbb{N}$?
I tried to expand it using $\frac{x^{n+1}-1}{x-1}=1+x+..+x^n$ but I had no success.
It would be great if more than one proof is provided.
| \begin{eqnarray*}
\sum_{n=0}^{\infty} (3^{2n+1}+2^{n+2})x^n = \frac{3}{1-9x}+\frac{4}{1-2x} = \frac{ \color{red}{7} (1-6x)}{(1-9x)(1-2x)}.
\end{eqnarray*}
This function clearly has integer coefficients
\begin{eqnarray*}
\frac{ (1-6x)}{(1-9x)(1-2x)}=(1-6x) \left( 1 +9x+81x^2+ \cdots \right) \left( 1 +2x+4x^2+ \cdots ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Calculate Grade Points and Credits needed to reach a specific GPA I'm attempting to come up with a formula that solves the amount of Grade Points and Credits Needed to achieve a certain GPA (On a $4.0$ scale). The formula for GPA is Grade Points / Credits.
As an example, let's say I have obtained $18$ Grade Points from... | Let $P$ represent the points you currently have, and $C$ represent the credits taken. As you pointed out, your current GPA would be $\frac{P}{C}$.
Then let $P_s$ represent the points you will earn in the next semester, and $C_s$ be the number of credits you will take next semester. After the semester is over, your GPA ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If $f: \mathbb N \rightarrow \mathbb N \times \mathbb N $ such as $f(n)=(n,n+1)$ Is it surjective and/or injective? If $f: \mathbb N \rightarrow \mathbb N \times \mathbb N $ such as $f(n)=(n,n+1)$ Is it surjective and/or injective?
I know that it is surjective $\Leftrightarrow \forall (a,b) \in \mathbb N \times \mathbb... | Consider $(1,1)\in\mathbb{N}\times\mathbb{N}$. Suppose for contradiction that there exists $n\in\mathbb{N}$ with $f(n)=(n,n+1)=(1,1)$. Then reading the first entry, we get $n=1$. Reading the second entry, we get $n+1=1\implies n=0$. Clearly we can't have $n=1$ and $n=0$ at the same time. Contradiction. Hence $f$ is not... | {
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"timestamp": "2023-03-29T00:00:00",
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product $\prod \limits_{i=2}^{\infty} (1+\frac{1}{(p_i -2)p_i})$ for primes $p_i$ I want to calculate the product $\alpha= \prod \limits_{i=2}^{\infty} (1+\frac{1}{(p_i -2)p_i})$ for all primes $p_i >2$.
I calculated this product first with computer and get for the first primes under ten millions $\alpha=1.514780119260... | Recall that if $\{ a_n \}$ is positive and $a_n \to 0$ then $\prod (1 + a_n)$ and $\sum a_n$ converge/diverge together. Given that $p_n \sim n \log n$ convergence is clear. So at least that.
As for a closed form: this is actually the reciprocal of the twin primes constant. This appears in many, many conjectures about t... | {
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Proof that the set $S = \{(x, y)\in \mathbb{R}^2 \mid y=x^2\}$ is closed. Our professor defined that a closed set is a set whose complement is open.
Based on this definition, how do I prove that the set $S = \{(x, y)\in \mathbb{R}^2 \mid y=x^2\}$ is closed?
It makes intuitive sense to me, but I'm unable to pen down a p... | I give a solution using sequence characterization:
$A\subseteq \mathbb{R}^n$ is closed if and only if for each sequence $(x_n)\subseteq A$ such that $x_n \to x$ then $x \in A$.
Let $z_n=(x_n,y_n)$ sequence in $S$ such that $z_n \to (x,y)$. So $y_n=x_n^2, \forall n\in \mathbb{N}$.
How $z_n \to (x,y)$ we have $x_n \to x$... | {
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find the distance of point $P(0,0,1)$ from the level surface $f(x,y,z)=0$ of $f(x, y, z)=\left(z-x^{2}\right)(z+3 y)$ given $$f(x, y, z)=\left(z-x^{2}\right)(z+3 y)$$
I am asking to find the distance of point $p=(0,0,1)$ from the level surface $f(x, y, z)= 0$.
The idea of what I am asked is pretty simple but How should... | Consider:
Objective function $x^2+y^2+(z-1)^2$, which is the square of
the distance from the position $p$, and constrained to
$(z-x^2)(z+3y)=0$.
So, your auxiliary function is $F=x^2+y^2+(z-1)^2+\lambda(z-x^2)(z+3y)$.
Then the equations
$$\frac{\partial F}{\partial x}=0,$$
$$\frac{\partial F}{\partial y}=0,$$
$$\frac{\... | {
"language": "en",
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Why is $|-a+\sqrt{a^2-1}|<1<|-a-\sqrt{a^2-1}|$ (where $a>1$) true? Why does $|-a+\sqrt{a^2-1}|<1<|-a-\sqrt{a^2-1}|$ (where $a>1$) hold? I understand that $a>1 \implies 1<|-a-\sqrt{a^2-1}|$ and that $|-a+\sqrt{a^2-1}|<|-a-\sqrt{a^2-1}|$
But I can't see why $a>1 \implies |-a+\sqrt{a^2-1}|<1$.
Does anyone see why? Thank y... | We define $f(a)=-a+\sqrt{a^2-1}$. You can easily see that its derivative is defined for $a \in [1,+\infty[$ and $f'(a)=\frac{a-\sqrt{a^2-1}}{\sqrt{a^2-1}} >0$ in this interval (just consider the numerator).
Then $f$ is increasing on the interval $[1,+\infty[$, $f(1)=-1$ and $\lim_{x\to \infty} f(x)=0$ (as $\sqrt{a^2-1}... | {
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How to prove that $\sum_{n=0}^{\infty} (-1)^n \ln \frac{3n+2}{3n+1}=\frac{1}{2} \ln 3$ The sum $$\sum_{n=0}^{\infty} (-1)^n \ln \frac{3n+2}{3n+1}=\frac{1}{2} \ln 3$$ has been encountered in the post below:
How can I prove $\int_{0}^{1} \frac {x-1}{\log(x) (1+x^3)}dx=\frac {\log3}{2}$
I would like to know as to how this... | Not a solution but a starting point. Use that $\ln(x)$ has the property that $\ln(ab) = \ln(a) + \ln(b)$. So we can rewrite the partial sum as $$\sum_{n=0}^N (-1)^n\ln\Big(\frac{3n+2}{3n+1}\Big) = \ln\Big(\prod_{n=0}^N \Big(\frac{3n+2}{3n+1}\Big)^{(-1)^n}\Big)$$
Since $x\to \ln(x)$ is a continuous function, then if we ... | {
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If $M$ is a standard class model of ZFC isomorphic to $V$, then is $M = V$? Consider the following statement: (T) "If $M$ is a standard class model of ZFC isomorphic to $V$, then $M = V$." The statement (T) is equivalent to: "If the transitive collapse of a standard class model $M$ of ZFC is equal to $V$, then $M = V$.... | No. Define $F:V\to V$ by $\in$-recursion as $F(x)=\{F(y):y\in x\}\cup\{\emptyset\}$. Clearly $F(x)$ is nonempty for all $x$. Also, $F$ is injective: if $F(x)=F(x')$, then by induction on $\max(\operatorname{rank}(x),\operatorname{rank}(x'))$ we may assume $F$ is injective on $x\cup x'$. Since $F(x)=F(x')$ we must h... | {
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Find height of irregular trapezoid with known angles and surface area
KNOWN:
*
*Length DC
*Alpha
*Beta
*Surface S
NEEDED:
*
*Height h
For an algorithm, I require a way to solve this for any trapezoid. Sort of like this question (Given a known isosceles Trapezoid find height of another with same angles & one b... | This seems like a problem best done using trig. Consider:
Draw a vertical line upward from $D$ to a point $E$ on $AB$. Do the same downward from $B$ to $F$ on $CD$.
We know $\overline{DE}$ and $\overline{BF}$ are equal to h. $\overline{BE}$ and $\overline{DF}$ are some unknown distance $d$.
As you noted, the area is th... | {
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For $A$ any subset of vector space $X$, is it true that $A+A=2A$? It seems a trivial question:
Prove/disprove: If we have a vector space $X$, then for any subset $A$ of $X$, we have $A+A =2A$.
It seems that $2A$ is always subset of $A+A$, but I don't think $A+A$ is subset of $2A$.
I am thinking in set of integers mod... | Consider $A = \{v, -v\}$ for some vector $v\ne 0$.
| {
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In what direction should the airplane fly not to miss the airport? Here is the question:
A plane sights an airport at $[N30°E]$ and is travelling at a speed of 500km/hr. There is a wind from $[N30°W]$ at a speed of 25km/hr. Determine the reading the airplane must travel in order not to miss the airport.
Options are:
a)... | I would leave a comment but I do not have the reputation to do so.
I believe your problem is with the wind vector.
The wind is coming from $N30°W$, which means it is blowing in a direction $S60°E$. This would make both components of the wind vector negative.
| {
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Solver for finding fixpoints of a boolean system Intro: The Problem
My problem relates to solving a system of equations that find the fixpoint of a studied boolean system ($F(X)=X$).
A Simple Example
Let $\bar{x}=\{x_1,x_2,x_3\} \in \{0,1\}$ be some boolean variables of interest.
Let $F=\{f_1,f_2,f_3\}$ be the update f... | So it turns out that ASP can be used to solve this problem! Here I provide a possible encoding of the problem in a file named fp.lp:
% variables
var(1..3).
... | {
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$(a+1)(b+1)(c+1)\leq4$ for triangle sides $a,b,c$ with $ab+bc+ac=1$
Given that $a,b,c$ are the lengths of the three sides of a triangle, and $ab+bc+ac=1$, the question is to prove $$(a+1)(b+1)(c+1)\leq4\,.$$
Any idea or hint would be appreciated.
This is Problem 6 of Round 1 of the BMO (British Mathematical Olympiad)... | OK first let's expand the bracket
$(a+1)(b+1)(c+1)=abc+ab+ac+bc+a+b+c+1$.
Now we know that $ab+ac+bc=1$ so we actually need $abc+a+b+c+1 \leq 3$ or $abc+a+b+c \leq{2}$.
Since $a,b$ and $c$ form the sides of a triangle, we know that $a \leq b+c$ and $b \leq a+c$ and $c \leq a+b$.
I found it hard to progress from here an... | {
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Check the convergence of the series $\displaystyle{\sum_{n=1}^{+\infty}\frac{\left (n!\right )^2}{\left (2n+1\right )!}4^n}$ I want to check if the following series converge or not.
*
*$\displaystyle{\sum_{n=1}^{+\infty}\frac{\left (n!\right )^2}{\left (2n+1\right )!}4^n}$
I suppose we have to find here an upper boun... | Some hints:
For first we can use Raabe's test
$$n\left(\frac{a_n}{a_{n+1}}-1 \right) = \frac{n}{2(n+1)}$$
For second
$$\frac{1}{2\sqrt{n}} \leqslant \frac{1}{2} \frac{3}{4} \cdots \frac{2n-1}{2n} \leqslant \frac{1}{\sqrt{2n}}\quad (1)$$
Proof:
For $n=1$ we have $\frac{1}{2} \leqslant \frac{1}{2} \leqslant \frac{1}{\... | {
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How to describe the unit ball as an Intersection of hyperplanes? How can one describe the unit ball in $\mathbb{R}^{3}$ as an intersection of supporting halfspaces?
| Let $B_3\subset \mathbb{R}^3$ denote the unit ball. By supporting halfspace, I assume you mean an affine half space containing $B_3$ such that its boundary contains at least one boundary point of $B_3$. Taking
$$B_3=\bigcap_{|x|=1}\{x+z:\langle z,x\rangle\leq 0\}$$
should work. Here is a good picture of the lower dimen... | {
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Are Riemann integrable functions the pointwise limit of continuous functions?
Given a function $f$ that is Riemann integrable on $[a,b]$, does there exist a sequence of continuous functions $\{f_n\}_{n=1}^\infty$ that converges to $f$ pointwise everywhere on $[a,b]$?
If I just require pointwise almost everywhere, thi... | Everywhere? no. Almost everywhere, yes.
A pointwise limit of a sequence of continuous functions is said to be a function of Baire class $1$. Baire proved many properties of such functions. In particular, if $E$ is a nonempty perfect set, then the restriction of $f$ to $E$ has a point of continuity.
Consider the foll... | {
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Two elements in the set differ by 3 or more
Call a set of integers sparse if any two elements in the set differ by at least 3. Find the number of sparse subsets of $\{1, 2, 3, \dots, 12\}.$ (Both $\emptyset$ and one-element sets are sparse, to my understanding.) For example, {$1, 5, 11, 12$} is a sparse set, since $1$... | Let $S_n$ be the set of sparse subsets on $\{1..n\}$. Then $S_0 = \{\emptyset\}$, $S_1 = \{\emptyset, \{1\}\}$, $S_2 = \{\emptyset, \{1\}, \{2\}\}$, and in general
$S_{n + 3} = S_{n + 2} \cup \{A \cup \{n + 3\} : A \in S_n\}$
Define $F_n = |S_n|$. Then we see that $F_0 = 1$, $F_1 = 2$, $F_2 = 3$, and $F_{n + 3} = F_{n ... | {
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Reference request: Sheaf for the Zariski topology In my course notes page 62 I've read the following: let $(E,0)$ be an elliptic curve over an arbitrary scheme $S$, then $U\rightarrow\ker(\,0^*_U:\operatorname{Pic}(E_U)\rightarrow \operatorname{Pic}(U))$ is a sheaf for the Zariski topology.
What is "sheaf for the Zaris... | You can define a sheaf $\cal F$ with values in a category $\mathbf{C}$ on any topological space $X$ by letting ${\cal F}(U)$ be an object in $\mathbf{C}$ for any open set $U\subset X$ so that the usual axioms are met.
E.g. see this Wikipedia entry.
To fix ideas you may think that $\mathbf{C}$ is the category of groups,... | {
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Triangle greater than (probability) This one is a follow up of my previous question. But a different problem. And this one should have a more interesting answer. I don't really know how to approach this problem nonetheless reach a solution, so again help is appreciated.
Question:
You have a circle with radius $R$. If t... | This isn't an answer, but just a simulation. I get the approximate value
$$P(A\geq \frac{1}{5}) \approx 0.45$$
Here is my Sage-code if someone wants to check it. It agrees with the mean value of mathworld
def randPt():
r = random()**0.5 #sqrt to make it uniform
a = random()*2*float(pi)
return (r*cos(a), r*s... | {
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Finite field with nonprime cardinality In my textbook for discrete mathematics the following is stated:
Theorem: $\mathbb Z_p$ is a field if and only if $p$ is prime.
In the following we denote the field with $p$ elements by $GF(p)$ rather than $Z_p$. As explained later, "$GF$" stands for Galois field. Galois discover... | Yes, there are finite fields other than $\Bbb Z_p$. The cardinal of such a field is always the power of a prime number. And, yes, there is a field with $4$ elements. It can be defined as $\Bbb Z_2[x]/\langle x^2+x+1\rangle$.
| {
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Form an 8 letter word using A,B,C,D,E, if the letters in the word must appear in alphabetical order Form an 8 letter word using A,B,C,D,E, where each letter can be used multiple times. How many words can I form if the letters in the word must appear in alphabetical order?
For example: AABBDDDE is acceptable, BBBACCCE i... | Observe that the type of word that you want is univocally decided by the numbers of letters A,B,C,D and E. Then the problem is the same of ask in how many ways can you write 8 as sum of 5 numbers, and the answer is ${12\choose 8}$, do you know why?
| {
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Morley rank of group This is an example from S.Buechler's book Essential Stability Theory.
Let $M$ be the group $\bigoplus_{i<\omega}(\mathbb{Z}_4)_{i}$ with $\mathbb{Z}_4=\mathbb{Z}/4\mathbb{Z}$. Suppose $M^{*}$ is the monster model of $Th(M)$. My questions are the following:
*
*Why $2M^{*}$ is a vector space over $... | For the first question: $M$ is an abelian group of exponent $4$, and hence so is $M^*$. It follows that $2M^*$ is an abelian group of exponent $2$. An abelian group of prime exponent $p$ is a vector space over the $p$-element field. (More generally, an abelian group of exponent $n>0$ is a $\mathbf Z/n\mathbf Z$-module.... | {
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Continuous, injective map between annuli, but with a "hole" in the image In $\mathbb R^n$ let $B_r$ be the open ball with center zero and radius $r$. For $r\in (0,1)$ let $A_r = \overline{B_1}\setminus B_r$. Let $r,s\in (0,1)$ and assume that $F : A_r\to A_s$ is continuous and injective such that $F(\partial B_1) = \pa... | No, that's not possible. Let's pick $r = s$, and in fact work with an annulus $A$ of inner radius $1$ and outer radius $2$. And let's pick a point $P$ in the set $U$, so that $P$ is a point of $A$ such that $P \notin F(A)$. So we have our map,
$$
F : A \to A
$$
whose image misses $P \in A \subset \Bbb R^2$. Define
$$
\... | {
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"timestamp": "2023-03-29T00:00:00",
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How do I prove that $\log(1/b)^x= -\log b^x$? How do I prove that $\log(1/b)^x= -\log b^x$?
I am working on logarithm properties. And I have come across one of the power rules, where the base is a fraction. I'm struggling to make assumptions and prove that the two equations are equal. Someone to help.
|
You want to show:
$$\log(1/b)^x= -\log b^x$$
We need to use two Theorems for this.
$\bullet~$Theorem 1: Let $a, b > 0$. Then $\log(a/b) = \log(a) - \log(b)$
Proof: Let, $u = \log a$ and $v = \log b$. Let the base be $10$ wlog. Therefore from the definition of $\log$, we have that
$$ \log a = u \implies 10^u = a \quad... | {
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Lemma of open mapping theorem Kreyszig Revising the Proof of the Kreyszig Open Mapping Theorem, being more specific the proof of the lemma 4.12-3, the central idea is find a open ball $B(0,r)$ such that exists $B(0,\delta)\subseteq T(B(0,r))$.
In the book they prove it for the case $r = 1$,
however it is valid for any ... | Suppose that $$\tag1\sum_k\|x_k\|=\sum_k2^{-k}.$$ Then
$$
0=\sum_k\tfrac1{2^k}-\sum_k\|x_k\|=\sum_k\big(\tfrac1{2^k}-\|x_k\|\big).
$$
This last series has non-negative terms; actually, all of its terms are positive. That is, the sequence of partial sums is positive and increasing, so its limit cannot be zero. The contr... | {
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Sum of an empty set and a finite set. Can we define the sum of an empty set and a finite set$?$
For example -
If $A= \{1,2\} , B= \emptyset$
Then what is $A+B$.
My intuition says it should be $A$.
But I couldn't find any proper reason behind it.
| If by $A+B$ you mean $\{\,a+b\mid a\in A,b\in B\,\}$, then if $B$ is empty there is no $b\in B$ so there are no elements in $A+B$. "$B$ is empty" is very different from "zero is in $B$".
| {
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Find the angles of triangle $NPQ$.
$ABC$ is a triangle. $ACM$ and $BCN$ are equilateral triangles where $M$ and $N$ are at the outside of the triangle. $P$ is center of $ACM$. $Q$ is midpoint of AB. Then find the angles of the triangle $NPQ$.
I need the solution using homothety. I have already solved the problem, but... | Let $D$ be a midpoint of $BC$. Since $$\angle PCN = \angle QDN = 90+\gamma$$ and $${PC \over QD} = {CN\over DN} = {2\over \sqrt{3}}$$ we see that $\triangle PCN\sim \triangle QDN$, so the spiral similarity at $N$ takes $\triangle PCN$ to $\triangle QDN$. But this spiral similarity induces new spiral similarity which h... | {
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"answer_id": 0
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The wrong way of finding the average distance between two points on a circle I was trying to find the average distance between two points on a circle and got the following result.
Why is my method wrong?
| Let $P_1$ be fixed while $P_2$ moves around a circle. The distance between these points is:
$$s=2R\sin{\alpha \over 2}$$
...where $\alpha$ represents the central angle corresponding to points $P_1,P_2$
Because of symmetry we can check only one half of the circle to calculate the average distance:
$$d=\frac{\int sdl}{\i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3799608",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
The solution of the indefinite integral contains an error (I know correct answer), but I cannot find it There is an exercise on indefinite integral in some infinitesimal calculus book:
$$
\int \sqrt{x^{2} +1} \cdot dx
$$
The solution uses the first substitution x = sinh u and after some transformations the book gets th... | You made a mistake going from $u$ to $z$: if $z = e^{2u}$, then $dz = 2e^{2u}\,du$, where you take it to be $2e^u\,du$. In fact, if you look at just what happens when you go from $t$ to $z$, you have replaced $t^2$ by $z$, but also replaced $dt$ by $dz$, where it should be $2\sqrt z\,dz$.
You could have stopped at $t$:... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3799770",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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One point compactification of $\mathbb{R}^{n}$ is homeomorphic to $S^{n}$ I'd like to understand the first proof given to me of the fact that the one point compactification of $\mathbb{R}^{n}$ is homeomorphic to $\mathbb{S}^{n}$.
The proof goes as follows : there is an initial remark about $i: \mathbb{R}^{n} \longrigh... | Let $K$ be a compact Hausdorff space, $a\in K$, $K'=K\setminus\{a\}$
and $K'^+=K'\cup\{\infty\}$ be the one-point compactification of $K'$. Then $\phi:K'^+\to K$ given by inclusion on $K'$ and $\phi(\infty)=a$ is a homeomorphism.
One just has to prove that $\phi$ is continuous, since then $\phi$ is a continuous
bijecti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3799861",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Show that $\{x\}\in \mathbb{B}(X)$ for every $x\in X$ Let $(X, \tau)$ be a Hausdorff space, and let $\mathbb{B}(X)$ be the Borel $\sigma$ algebra on $X$. The question is,
Is it true that, if $x\in X$, then $\{x\}\in \mathbb{B}(X)$?
The reason why I ask is because of the previous post I made; the answer shows that one... | Since $X$ is Hausdorff $\{x\}$ (a singleton) is closed and $U = X \setminus \{x\}$ is open. Since $B(X)$ is a $\sigma$-algebra, it's closed under taking the complement:
$$
B(X) \ni X \setminus U = X \setminus ( X \setminus \left\{ x\right\} ) = \left\{ x \right\}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3800015",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Complex Analysis proof problem I have an feeling this bad proof because I am assuming my conclusion and I go into repeating. This isn't allowed in mathematics, so I need help to refix my steps.
So the proof I am proving is |ab| = |a||b|
So I started with:
Let a,b $\in \mathbb{C}$
I claimed $|ab|^{2} = |a|^{2} |b|^{2}$ ... | $|ab|^{2}=(ab)(ab)^{*}=aba^{*}b^{*}=(aa^{*}) (bb^{*})=|a|^{2}|b|^{2}$. Now take square root.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3800195",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Solving $\left(\frac{x}{10}\right)^{\log(x)-2}<100$ How to solve the following inequality?
$$\left(\frac{x}{10}\right)^{\log(x)-2}<100$$
The solution given is $x\in(1, 1000)$
I considered some things in my solving, but I couldn't get the solution to the problem. I would like to know if those assumptions were wrong.
Fir... | More directly, one can write the sequence of equivalent inequalities $$10^{(\log(x)-1)(\log(x)-2)} = \left(\frac{x}{10}\right)^{\log(x)-2}<100=10^2 \\ (\log(x)-1)(\log(x)-2) < 2 \\ \log(x)(\log(x)-3) < 0 \\ 0 < \log(x) < 3 \\ 1 < x < 1000$$
As for your solution, it is fine until
$$x^{\log(\frac{x}{100})}<x \Longleftri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3800287",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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A question on linear maps itself Here is just a sample problem.
Suppose that $V$ is finite dimensional and $S, T \in \mathcal{L}(V)$. Prove that $ST = I$ if and only if $TS = I$
Proof : Suppose that $ST = I$. The identity map $I$ is invertible, so by problem 3.22 both $S$ and $T$ are invertible. Multiply $ST = I$ on th... | Maybe not a good explanation but since $V$ is a vector space over $\mathbb K$ of finite dimension, say , $n$ , we can always view $L(V)$ as matrix algebra $M_n(\mathbb K)$.
The isomorphism is constructed by calculating $f(v_j)$ where $\{v_1,...,v_n\}$ is the basis of $V$, and decomposing them into $f(v_j)=a_{1j}v_1+...... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3800382",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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If $f(x)=\sin^{-1} (\frac{2x}{1+x^2})+\tan^{-1} (\frac{2x}{1-x^2})$, then find $f(-10)$ Let $x=\tan y$, then
$$
\begin{align*}\sin^{-1} (\sin 2y )+\tan^{-1} \tan 2y
&=4y\\
&=4\tan^{-1} (-10)\\\end{align*}$$
Given answer is $0$
What’s wrong here?
| Let $\tan^{-1}\dfrac{2x}{1-x^2}=u\implies-\dfrac\pi2<u<\dfrac\pi2$
$\tan u=\dfrac{2x}{1-x^2}$
$\implies\sec u+\sqrt{1+\left(\dfrac{2x}{1-x^2}\right)^2}=\dfrac{1+x^2}{|1-x^2|}$
$\sin u=\dfrac{\tan u}{\sec u}=\text{sign of}(1-x^2)\cdot\dfrac{2x}{1+x^2}$
$\implies u=\sin^{-1}\left(\text{sign of}(1-x^2)\cdot\dfrac{2x}{1+x^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3800521",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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There are given $n$ points on plane. Prove that there are not more than $n$ pairs of vertices, distance between which is exactly $d$ $\textbf{Source:}$I found this question in aopslink
As you can see in this link it doesn't mention any source either.
$\textbf{Question:}$There are given $n$ points on plane. Let $d$ be ... | Let $G$ denote the graph on the $n$ vertices, where two vertices share an edge if and only if the distance between them is $d$. Let $k$ denote the number of edges in $G$. We wish to show that $k\leq n$.
Let $G'$ denote the graph obtained by repeatedly removing all vertices $v\in G$ with $\deg v\leq1$, so that the numbe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3800648",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Complete first order theories While studying Hodges' A shorter model theory I came across this observation:
Given a first order language $L$, we say that an $L$-theory $T$ is complete if $T$ has models and any two of its models are elementary equivalent. [...] the compactness theorem implies that any complete theory i... | This is indeed a mistake. As Nagase says, it's not present in the original ("big") model theory book. My suspicion is that Hodges added it after mixing up two notions of completeness: "satisfiable and all models are elementarily equivalent" versus "contains each sentence or its negation." Using the latter sense of comp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3800805",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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You sample from a uniform distribution $[0, d]$ $n$ times. What is your best estimate of $d$, using on the variance of the samples drawn? Suppose I have $X_1, X_2, ..., X_n$ where $X \sim \text{Uniform}[0,d]$.
Since, $E[X]$ = $\frac{d}{2}$, an obvious estimator for $d$ would be $2\cdot\bar{X}$, where $\bar{X}$ is the... | With a random sample from $\mathsf{Unif}(0,\delta),$ if you insist on using the sample variance $S^2$ to estimate $\delta$ you can do it, but it isn't the best way to estimate $\delta.$
Notice that the variance of $\mathsf{Unif}(0,\delta)$ is
$\sigma^2 = \delta^2/12,$ so the method of moments estimator
is $\tilde \delt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3800899",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Evaluating Multivariable Limit $\lim\limits_{(x,y) \to (0,2)} \frac{\sin(xy)}{x}$ Question: Evaluate the limit $$\lim\limits_{(x,y) \to (0,2)} \frac{\sin(xy)}{x}$$
My first thought is that the limit looks a lot like the single variable $\lim\limits_{x \to 0} \frac{\sin(x)}{x} = 1$. Regardless of what $y$ is (as long as... | Your idea is correct but we need some correction, indeed since $xy\to 0$ we have that
$$\lim\limits_{(x,y) \to (0,2)} \frac{\sin xy}{x}=\lim\limits_{(x,y) \to (0,2)} \frac{\sin xy}{xy}\cdot \frac{xy}{x}=\lim\limits_{(x,y) \to (0,2)} \frac{\sin xy}{xy}\cdot y=1\cdot2=2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3801046",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Updates of Serge Lang — Differential manifolds I have Serge Lang — differential manifolds. An interesting read. But the book is 50 years old. Are there newer books that give a better and more comprehensive treatment of the material, or this the best of its kind?
| I don't have much experience with Lang's book, but some other books that are in vogue amongst graduate students right now are:
*
*John Lee, Introduction to Smooth Manifolds
*Loring Tu, An Introduction to Manifolds
*Guillemin and Pollack, Differential Topology
*Milnor, Topology from the Differential Viewpoint
*Do ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3801180",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Counterexample to: If $|f''(x)|\leq m$, then $|f''(0)|+|f''(a)|\leq am$
Attempt towards a counter-example:
Consider $f:[0,1]\to \mathbb{R}$, $f(x)=-(x-\frac{1}{2})^4$.
$f$ takes its largest value at $x=0.5 \in [0,1]^o$, i.e. the interior of $[0,1]$.
$f''(x)=-12(x-\frac{1}{2})^2$, and $|f''(x)|\leq 3$, $\forall x \in ... | Your counterexample is correct. The statement is (trivially) correct if $a \ge 2$ and wrong if $a < 2$. As a counterexample one can choose any twice-differentiable function $f: [0, a] \to \Bbb R$ which has a maximum in the interior of the interval and where $f''$ attains its maximum both at $x=0$ and $x=a$. So another ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3801325",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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The derivative $\frac{\mathrm d}{\mathrm dx} x^x=x^x\left(\ln x+1\right)$ is problematic for $x<0$ To take the derivative of $x ^ x$, we write
$$\dfrac {\mathrm d}{\mathrm dx} x^x=\dfrac {\mathrm d}{\mathrm dx} e^{\ln x^x}=\dfrac {\mathrm d}{\mathrm dx} e^{x\ln x}= e^{x\ln x}× \dfrac {\mathrm d}{\mathrm dx}(x\ln x)=x^x... | The differentiability of a function can only be found if it is continuous in an interval $(a,b)$.
$x^x$ is continuous only for $x > 0$. For $x<0$, the graph can only be drawn for some discrete points. Differentiability is not defined for this part of the graph.
$$\frac{\mathrm d (x^x)}{\mathrm{d}x}=x^x(\ln x+1)\quad \f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3801432",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Differential multi-variable function limit using polar coordinates does not work Given:
$$ f(x,y) = \frac{xy^3}{x^2 + y^6}$$
Is it differentiable at $(0,0)$ ?
I said no, as it is not even continuous by the path $x = y^3$
However, when we go to polar view that is: $x = r \cos(\theta) , y = r \sin(\theta)$
we get that $$... | The problem is that : $\lim_{r \rightarrow 0^+} \frac{r^2 \cos(\theta) \sin^3 (\theta)}{(\cos^2(\theta) + r^4 \sin^4(\theta)}$ is not always determinate contrary to what you probably think. What happens if $\theta$ assumes a value which makes denominator $0$ as $r\to 0$. You have indeterminate form $(0/0)$.
As it tur... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3801584",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Interchanging calculus operations with pi product In the domain of convergence, we can interchange derivatives, sums and integrals but what about $ \prod$ and previous operations?
For example,
$$ \frac{d}{dx} \sum_{j=1}^{n} f_j(x) = \sum_{j=1}^{n} \frac{d}{dx}f_j(x)$$
and,
$$ \frac{d}{d u^i} \int_{a}^{b} F(u_1,u_2,u_3... | The formula for the derivative of a product gives you
$$\frac{d}{dx} \prod_{j=1}^{n} f_i(x)= \sum_{j=1}^n f'_j(x)\prod_{i=1 \\ i\neq j}^{n} f_i(x).$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3801717",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Computing the limit of a sinc-like function Compute $\displaystyle{\lim_{k\to∞} \int_0^\infty \frac{k\sin(x/k)}{x^{3/2}} dx}$ .
Let $f_k = \frac{k\sin(x/k)}{x^{3/2}}$. by L'Hopital, $\displaystyle{\lim_{k\to \infty} f_k(x)} = \frac{1}{x^{3/2}}$, which is not Lebesgue integrable on $(0, 1]$ since the improper Riemann in... | Note that $0\le \sin (x)\le x$ for $x\ge 0$. So, $\lim_{k\to \infty}\frac{k\sin(x/k)}{x^{3/2}}=\frac1{x^{1/2}}$.
Aside, enforcing the substitution $x/k\mapsto x$, we find that
$$\int_0^\infty \frac{k\sin(x/k)}{x^{3/2}}\,dx=\sqrt{k}\int_0^\infty \frac{\sin(x)}{x^{3/2}}\,dx=\sqrt{2\pi k}\to \infty$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3801817",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Relationship between the symmetric difference of two sets and their intersection How do we prove that the union of symmetric difference of two sets and their intersection is the same as the union of the two sets?
i.e. $(A\backslash B) \cup (A\cap B) \cup (B\backslash A) = A\cup B$
where $A, B$ are two sets?
I know how ... | Maybe you were looking for a "computational" proof:
$$\underbrace{(A\backslash B)\cup(A\cap B)}_{A}\cup(B\backslash A)=A\cup (B\backslash A)=A\cup B.$$
In the first equality, I am using the fact that for any sets $A$ and $B$, we have
$$A=(A\backslash B)\cup (A\cap B).$$
Intuitively, this means that everything in $A$ ei... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3801970",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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one-relator groups which are free-by-cyclic I am reading an article of Baumslag:
Baumslag, Gilbert, "Finitely generated cyclic extensions of free groups are residually finite." Bull. Austral. Math. Soc. 5 (1971), 87–94.
and he mentions that many one-relator groups, in particular, fundamental groups of surfaces, are fre... | The orientable surface group is free-by-cyclic: if the standard generators are $x_1,...,x_g, y_1,...,y_g$ then the homomorphism onto the cyclic group $\langle x_1\rangle$ which kills all other generators is onto and its kernel is of infinite index, whence free.
The group $\langle a,b,c| c^n=[a,b]\rangle =\langle a,b,c|... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is $P(a)$ logically equivalent to $\forall y [(y=a) \rightarrow P(y)]$? I am not entirely sure if my proof is correct. I would appreciate if somebody helped.
$(\rightarrow)$
Assume $P(a)$. Let an arbitrary $y$. Let $y=a$. Since $P(a)$ and $y=a$, then $P(y)$. Since $y$ is arbitrary, then $\forall y [(y=a) \rightarrow P(... | Your first proof is valid.
Semantically: $P(a)$ holds exactly when "anything that is $a$ satisfies $P$."
Your second is not quite correct. $\Gamma$ cannot be a set of formula, rather it must be a well formed formula to be used the way you are using it. Otherwise the proof is okay.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3802222",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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How to show this inequality holds, when proving self-similar processes are non-differentiable? In the book Elementary Stochastic Calculus by T. Mikosch (1998), there is a result which shows that Brownian motion is a self-similar process and therefore it is nowhere differentiable. In the proof of this result, there is a... | Look at the section Borel-Cantelli Lemmas in Durret's Book (Probability Theory and Examples) an application of Fatou's Lemma ensure that $P(\limsup_n An)\geq \limsup_n P(A_n)$. So if you write $A_n$ for the set if LHS of the inequality and $B_n$ for the another then $$\lim_n P(A_n)=P(\lim_n A_n)=P(\limsup_n A_n)\geq P(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3802328",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If $d\mid nm$ and $\gcd(n, m)= 1$ then exist $d_1, \,d_2$ such that $d=d_1d_2$ and $d_1\mid n,\,d_2\mid m$ (without Fund. Theorem of Arit) We want to prove that if $d\mid nm$ and $\gcd(n,m)=1$ then $d=d_1d_2$ where $d_1\mid n$ and $d_2\mid m$ and $\gcd(d_1,d_2)=1$
We already proved it using Fundamental Theorem of Arith... | We can use the following two facts:
Lemma 1:
Given $m,n \in \mathbb{N}$, if $gcd(m,n) = 1$, then there exists, $x,y \in \mathbb{N}$, such that $xm + yn = 1$
Lemma 2:
For, $m, n \in \mathbb{N}$, if there exists $x, y \in \mathbb{N}$, such that $xm + yn= 1$, then $gcd(m,n) = 1$.
Proof:
Now we can show that if $d_1 = gcd(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3802477",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to identify the coefficients in a series expansion on a non-orthogonal basis? The solution of a PDE lead to a series expansion of the form
$$
\sum_{n=0}^\infty \left( A_n \cos \left( \lambda_n z\right) + B_n \sin \left( \lambda_n z \right) \right) = f(z) \, ,
$$
where $z \in [0,L]$ and $f(z)$ is a known function.
... | If this came from a self-adjoint PDE, and if you have endpoint conditions of the form
$$
Af(a)+Bf'(a)=0,\;\;\; Cf(b)+Df'(b)=0,
$$
then you can end up with trigonometric expansions where the periods are non-harmonic. But that does not mean they are not orthogonal, in which case the ODE solutions will sti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3802589",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to solve $ (y+u)\dfrac{\partial u}{\partial x} + (x+u)\dfrac{\partial u}{\partial y} = x+y$ via method of characteristics? How to solve $ (y+u)\dfrac{\partial u}{\partial x} + (x+u)\dfrac{\partial u}{\partial y} = x+y$ via method of characteristics?
My attempt.
These are equations with which I begin:
$\dfrac{dx}{ds... | $$ (y+u)\dfrac{\partial u}{\partial x} + (x+u)\dfrac{\partial u}{\partial y} = x+y$$
Charpit-Lagrange system of characteristic ODEs :
$$ds=\frac{dx}{y+u}=\frac{dy}{x+u}=\frac{du}{x+y}=\frac{dx-dy}{(y+u)-(x+u)}=\frac{dx+dy+du}{(y+u)+(x+u)+(x+y)}$$
$$\frac{dx-dy}{y-x}=\frac{dx+dy+du}{2(x+y+u)}$$
$$-\ln|x-y|=\frac12\ln|x+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3802816",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove $\frac{\mathbb Z[X,Y]}{(5,X^{2}-Y,XY+X+1)}$ is a field
Prove $\frac{\mathbb Z[X,Y]}{(5,X^{2}-Y,XY+X+1)}$ is a field.
I thought to prove that this is isomorphic with $\mathbb{\mathbb Z_{5}(X)}$, and because $5$ is prime it will follow that it's a field.
I wanted to use the first isomorphic theorem.
I wanted to u... | Hint:
$$
\frac{\mathbb Z[X,Y]}{\langle 5,X^{2}-Y,XY+X+1 \rangle}
\cong
\frac{\mathbb Z[X,X^2]}{\langle 5,0,X^3+X+1 \rangle}
\cong
\frac{\mathbb Z[X]}{\langle 5,X^3+X+1 \rangle}
\cong
\frac{\mathbb F_5[X]}{\langle X^3+X+1 \rangle}
$$
so it reduces to proving that $X^3+X+1$ is irreducible mod $5$, which is easy since the... | {
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"url": "https://math.stackexchange.com/questions/3802960",
"timestamp": "2023-03-29T00:00:00",
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prove that $xy+yz+zx\ge x\sqrt{yz}+y\sqrt{xz}+z\sqrt{xy}$ prove that $xy+yz+zx\ge x\sqrt{yz}+y\sqrt{xz}+z\sqrt{xy}$ if $x,y,z>0$
My try : dividing inequality by $\sqrt{xyz}$ and putting $\sqrt{x}=a,\sqrt{y}=b,\sqrt{z}=c$
we have to prove $$\sum_{cyc}\frac{ab}{c}\ge a+b+c$$
or
$$2\sum_{cyc}\frac{ab}{c}\ge 2(a+b+c)$$ usi... | $x,y,z>0$ and that the inequality is a symmetric expression implies that we can take without loss of generality, an ordering $x\ge y\ge z \implies xy\ge zx\ge yz \implies \sqrt{xy}\ge \sqrt{zx}\ge \sqrt{yz}$
so that the sequences $\{\sqrt{xy},\sqrt{zx},\sqrt{yz}\}, \{\sqrt{xy},\sqrt{zx},\sqrt{yz}\}$, (i.e. the same seq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3803069",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
What form of choice is "every Dedekind-finite set is finite" equivalent to? Halmos in his Naive Set Theory proves that every infinite set has a subset equivalent to $\omega$ using the axiom of choice with its full power. And this leads to the corollary that a set is infinite if and only if it is equivalent to some prop... | While answering this question: Strength of “Cofinite Choice”, I discovered that "every Dedekind-finite set is finite" is equivalent to the following "axiom of cofinite choice":
Let $A$ be a set of non-empty sets such that $(\bigcup A)\setminus X$ is finite for all $X\in A$. Then $A$ has a choice function.
See the lin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3803247",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Prove or disprove the following statement using the definition of big-Θ: NOTE: I am not provign big O here I am proving big-Θ
Prove or disprove the following statement using the definition of big-Θ:
$$n^2−4n = Θ(2^n)$$
so, by definition, $$T(N)=O(h(N))$$ and $$T(N)=Ω(h(N))$$ must both hold.
checking condition 1, $$2^n*... | This is correct, for $Ω$ as well.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3803408",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
If $\lim f(x)$ can theoretically be anywhere on the interval $0$ up to and including $\infty$, is the interval written $(0,\infty]$? Let's take $f(x)=a^x+b$ where $a\in\mathbb R$ and $b\in\mathbb R^+$. clearly $L=\lim_{x\to\infty}f(x)>0$, but is the interval written $L\in(0,\infty)$ because limits only approach infinit... | If you want to be able to write $L = \lim_{x\to\infty} f(x) = \infty$, then it would be false to state $L \in (0,\infty)$.
You would need to be able to write $L \in (0,\infty]$, but we also need to keep in mind that $L = \infty$ isn't a real number, so this only makes sense in the context of the extended real number li... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3803557",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Stone-Weierstrass theorem and Polynomials in multidimensional real space Stone-Weierstrass theorem on closed interval $[a, b]$ (in $\mathbb{R}$) states that any continuous function $f$ on $[a, b]$ can be approximated by polynomial function $p$, arbitrarily close to $f$.
From the above observation, I wonder if this can ... | Wikipedia quotes the Stone-Weierstrass theorem as
Stone–Weierstrass Theorem (real numbers). Suppose $X$ is a compact Hausdorff space and $A$ is a subalgebra of $C(X, \Bbb R)$ which contains a non-zero constant function. Then $A$ is dense in $C(X, \Bbb R)$ if and only if it separates points.
Which is to say, the only ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3803727",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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How to rewrite integral of L^2 function and conclude about convergence of series I've read the following assertions:
" Suppose $f \in L^2(\mathbb{R}). $ Then $$ \int_{-\frac{1}{2}}^{\frac{1}{2}} \sum_{k \in \mathbb{Z}} \vert f(x+k) \vert^2 dx = \int_{-\infty}^{\infty} \vert f(x) \vert^2 dx < \infty. $$ Thus, $ \sum_{k ... | We can write any real number $x \in \mathbb{R}$ as a sum $k + \epsilon$, with $k \in \mathbb{Z}, \epsilon \in \left[-\frac12, \frac12\right)$ in a unique way.
That's what the sum on the left does explicitly. If you replace the variable $x$ with $\epsilon$ in the left-hand-side, it should be even more clear.
The integra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3803863",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How do I convert this equation to the standard form of a circle? I'm looking for the radius of the sphere of this: $4x^2 + 4y^2 +4z^2 -16x - 24y + 8z= 44$.
I have to get it into standard form in order to find the radius. So I factored out a 4 and simplified it to:
$$x(x-4) + y(y-6) +z(z+2) =11$$
I am not sure what els... | For each of the variables $x, y, z$, you want to get something of the form $(x - x_0)^2$, by adding some constant if necessary. Of course, you also should add the same constant to the right-hand side.
So we want $(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = 11 + C = R^2$, for some number $C$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3804000",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Diophantine equation $x^2 + xy − 3y^2 = 17$ Determine all integer solutions to the equation $x^2 + xy − 3y^2 = 17$.
The previous part of the question was finding the fundamental unit in $\mathbb{Q}(\sqrt{13})$, which is $\varepsilon = \frac{3+\sqrt{13}}{2}$, so my guess is that I should factorise the equation in $\math... | Hint:
Complete the square: $x^2+xy+\frac14y^2-(3+\frac{1}4)y^2=17\implies(x+\frac12y)^2-\frac{13}4y^2=17$
$\implies (2x+y)^2-13y^2=68$. That's a Pell-type equation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3804117",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Vector parametrization of the line that passes through two vectors. How would I find the vector parametrization $()$ of the line $L$ that passes through the points $(2,1,4)$ and $(5,6,7)$?
So I found a directional vector: $(3,5,3)$.
What do I do next?
| You found a direction vector $(3,5,3)=(5,6,7)-(2,1,4)$,
so the line can be parametrized as $(2,1,4)+t(3,5,3)=(2+3t,1+5t,4+3t)$.
Note that $\mathbf r(0)=(2,1,4)$ and $\mathbf r(1)=(5,6,7)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3804223",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Lagrange Multipliers Maxima or Minima? If I evaluate a surface $f$$($$x$$)$, that is subjected to a constraint $g$$($$x$$)$, for it's maximum and minimum values using Lagrange Multipliers then how do I know that the solution that is found is maximum or minimum.
For example $f$$($$x$$)$= $x^2$+$y^2$+$z^2$
and $g$$($$x$$... | Since the point $(1,1,6 \sqrt{3})$ satisfies g and $f(1,1,6 \sqrt{3}) > f( \sqrt{3}, \sqrt{2},1) $ the point $( \sqrt{3}, \sqrt{2},1)$ must be a minima.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3804333",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Max limit on min magnitude of integer solution to underdetermined system of equations Given a system of equations on the form $\bar{a}_i\cdot\bar{x}=0\ \forall\ i\in\{1,\dots,d-1\}$, where $\bar{x}\in\mathbb{Z}^d\setminus\bar{0}$ must hold and $\bar{a}_i\in\{-L,-L+1,\dots,L-1,L\}^d\ \forall\ i\in\{1,\dots,d-1\}$ are kn... | Non-trivial solutions may not exist. Let $A$ be the $(d-1)\times d$ matrix whose $i$-th row is $\tilde{a}_i$ for each $i$. By assumption, the rank of $A$ is $d-1$. Therefore $\ker(A)$ is one-dimensional. More specifically, by relabelling the columns of $A$ if necessary, we may write
$$
A\tilde{x}=\pmatrix{B&v}\pmatrix{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3804421",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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} |
Prove: $ -\int_{-\infty}^{+\infty} \frac{e^{t}t^3}{(1-e^t)^3} dt = \pi^{2}$ $$
\mbox{Prove that:}\quad
-\int_{-\infty}^{+\infty}\frac{\mathrm{e}^{t}\, t^3}{\left(1 -\mathrm{e}^{t}\right)^{3}}\,\mathrm{d}t = \pi^{2}
$$
I tried to solve this, but it seems hard to me.
| Using the geometric series we deduce for $|x|<1$ that
$$\sum_{n\ge0}x^n=\frac1{1-x}\,\implies\,\sum_{n\ge1}nx^n=\frac x{(1-x)^2}\,\implies\,\sum_{n\ge2}n(n-1)x^n=\frac{2x^2}{(1-x)^3}$$
Now, split the integrand at $t=0$ and enforce $t\mapsto-t$ to obtain
\begin{align*}
-\int_{-\infty}^\infty\frac{t^3e^t}{(1-e^t)^3}\,{\r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3804538",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Quasi-polynomial time complexity and proof In my algorithm class, we were talking about a function $f(x)$ that grows faster than any polynomial ($x^c$), but is outgrown by any exponential ($c^x$), where $c > 1$. That is, a function $f(x)$, such that both the sequences $P_x = \frac{x^c}{f(x)}$ and $E_x = \frac{f(x)}{c^x... | Note that $$ n^{\log_c(n)} = \Big( c ^{\log_c (n)} \Big)^{\log_c(n)} = c^{(\log_c(n))^2}.$$
Since $(\log_c(n))^2 < n$ for large enough $n$, $\lim_{n\to \infty} \frac{n^{\log_c(n)}}{c^n} = 0$ follows.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3804718",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Bin Packing Problem with fixed size of bins I'm studying Bin Packing Problem for my thesis and I meet this definition of the decision verson of the problem in the book "Computers and Intractability" by Michael R. Garey and David S. Johnson:
INSTANCE: Finite set $U$ of items, a size $s(u) \in Z$ for each $u \in U$, a po... | With a fixed bin size you also have a fixed number of possible ways to (partially) fill a bin. Suppose there are $p$ ways to do that.
If you solve each of the $k$ bins separately, you'd get $p$ possibilities for each bin, and then $p^k$ possibilities alltogether. This is exponential, and not what we would like. Note th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3804912",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Expectation of Sinc of Standardnormal Distribution Does anyone happen to know how to compute the expectation of the Sinc of a standard normal random variable, i.e.
$\mathbb{E} \Big[ \frac{\sin X}{X} \Big]$ where $X \sim \mathcal{N}(0,1)$?
Thanks!
EDIT: I suppose I could have elaborated what I have tried, so here goes:
... | Using the series for $\sin(x)$,
\begin{align}
\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\frac{\sin(x)}{x}e^{-\frac{x^2}{2}}\,dx&=\sum_{k\ge 0}\frac{(-1)^k}{(2k+1)!}\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}x^{2k}e^{-\frac{x^2}{2}}\,dx \\
&=\sum_{k\ge 0}\frac{(-1)^k (2k-1)!!}{(2k+1)!}=\sqrt{\frac{\pi}{2}}\left(\oper... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3805101",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Problem books on harmonic analysis? I just got a copy of the book A Course In Abstract Harmonic Analysis by Gerald B Folland so I want a problem book that could be used with it?
Please give a list of problem books on Harmonic Analysis?
| An Intro to Harmonic Analysis by Yitzhak Katznelson has exercises following each chapter. It is easy to draw parallels between books for content like this.
Harmonic Analysis by Henry Helson is in the same boat. There are about 5 problems after each chapter’s section.
Lastly, Barry Simon’s book “Harmonic Analysis: A com... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3805223",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Prove that $\frac{1 - x^{n+1} }{n+1} \lt \frac{1-x^n}{n}$ given $n$ is a positive integer and $0 < x \lt 1$. Problem Statement: If $n$ is a positive integer and $0 < x \lt 1$, show that
$$ \frac{1 - x^{n+1} }{n+1} \lt \frac{1-x^n}{n}.$$
My Solution:
$$
\frac{ 1- x^{n+1} }{n+1} \lt \frac{1-x^n}{n} ~~~~\text{is true} \\
... | By your reasoning we need to prove that:
$$n(1+x^n+x^{2n}+...)<(n+1)(1+x^{n+1}+x^{2(n+1)}+...)$$ and it's not so clear, why it's true.
Another way:
We need to prove that:
$$nx^{n+1}-(n+1)x^n+1>0,$$ which is true by AM-GM:
$$nx^{n+1}+1\geq(n+1)\sqrt[n+1]{\left(x^{n+1}\right)^n\cdot1}=(n+1)x^n.$$
The equality occurs mayb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3805342",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
} |
Is essential spectrum not relevant to the topology on it Consider $F: \mathcal{D}(F)\subset X\rightarrow X$, we can define the essential spectrum as the set $\{\lambda\}$ s.t. the Fredholm index of $\lambda-F$ is not zero. Fredholm index can be written as
$ind\ \lambda-F= dim Ker(\lambda-F)-codim\ ran(\lambda-F)$ , w... | At least as a place-holder answer, in light of some comments: the usual definition/basic properties (whether something is part of the definition, or a basic property, depends on one's choice of logical order, and there is not a unique such...) of Fredholm operators on Banach spaces certainly does use the fact that the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3805460",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Terminology: Upper limit and lower limit. Let $(x_n)$ be a sequence of real numbers. It defines:
$\limsup\limits_{n \rightarrow \infty} x_n=\lim\limits_{n \rightarrow \infty} (\sup_{m≥n} x_m)=\inf_{n≥1}(\sup_{m≥n}x_m)$
How should I understand the notation of the definition? Specifically the $m≥n$ and $n≥1$ terms.
| The notation $\sup_{m\geq n} x_m$ is just a short way to write $\sup\{x_m: m\geq n\}=\sup\{x_n,x_{n+1},x_{n+2},...\}$, the supremum of the sequence $(x_m)$ starting from the $n$th element.
Similarly, if we let $y_n=\sup_{m\geq n} x_m$ for every $n\in\mathbb{N}$, then $\inf_{n\geq 1} y_n$ is just $\inf\{y_n: n\in\mathbb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3805581",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Let $f, g: X \to [-\infty, \infty]$ be measurable functions. Is it true that $f - g$ (where it's defined) is measurable? Let $f, g: X \to [-\infty, \infty]$ be measurable functions. Let $X'$ denote the set of all $x$ such that $f(x), g(x) \notin \{-\infty, \infty\}$. Then $X'$ is a measurable set.
Is it true that $h: X... | Yes, it is true.
$f-g$ is not defined in the set $B=\Big(\{f=\infty\}\cap\{g=\infty\}\Big)\cup\Big(\{f=-\infty\}\cap\{g=-\infty\}\Big)$. This is measurable set, for example
$$
\{f=\infty\}=\bigcup_{n\in\mathbb{N}}\{f>n\}
$$
Thus, for $a\in\mathbb{R}$,
$$
\begin{align}
\{f-g<a\}=(X\setminus B) \cap\bigcup_{q\in\mathbb{Q... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3805772",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to solve diffrential equations of Chemical Kinetics 3rd order reaction [Problem setting]
(ii) A chemical reaction is described by
$$
A+B+C \to^{k_1} D\\
D \to_{k_2} A+B+C
$$
If $x_1, x_2, x_3, x_4$ are the concentrations of A, B, C and D respectively, then write down the governing equations.
Hence deduce the equa... | From general principles of mass conservation, or the construction of the system by mass exchange terms that balance, you can easily see that
$$
3x+y=C=3x_0+y_0
$$
is a constant. Thus the system lives on the line $y=C-3x$. From the right side of the first equation we see that equilibrium points satisfy $k_1x^3-k_2y=0$. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3805920",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
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