Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
$\lim_{a \to \infty}\prod_{i=0}^{a} x_i = \infty$ for $x_i > 1$? For $x_i > 1$, where $i$ is an index and $x_i$ real number, $\lim_{a \to \infty}\prod_{i=0}^{a} x_i = \infty$ always? How does one prove/disprove this?
| It is relatively easy to find a counterexample showing that this is not true. Simply choose any increasing bounded sequence $(p_n)$ such that $p_0=1$. I.e., you have $p_n<p_{n+1}$ for each $n$ and, since the sequence is bounded, there exists a finite limit $\lim\limits_{n\to\infty} p_n=P$.
Now you can put $x_0=1$ and
$... | {
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"url": "https://math.stackexchange.com/questions/1936927",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is this a sufficient statistic for uniform distribution? I'm having trouble proving that given a uniform distribution $X_i \sim U(0,\theta)$ with $\theta$ unknown, the statistics $2\bar X$ and $\bar X$ are not sufficient; any ideas?
Thanks for your help.
| Here's a hint: $ \operatorname{E}(2\bar X) = \theta,$ so $2\bar X$ is an unbiased estimator of $\theta$, but if, for example, $(X_1,X_2,X_3) = (1,2,12)$ then the estimate of $\theta$ is actually smaller than the largest of the three observations. That shows there is more information about $\theta$ in the sample than t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1937018",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Radius-4 circle skewered on one line and touching another Find the equation of the circle with radius $4$ units, whose centre lies on the line $4x+13y=32$ and which touches the line $4x+3y+28=0$.
My Approach:
Radius $r=4$ units
Let $P(h,k)$ be the centre of the circle. Then $4h+13k=32$.
Please help me to move further.
| The equation of a circle with center in $(a,b)$ and radius $4$ is
$$
\left( {x - a} \right)^{\,2} + \left( {y - b} \right)^{\,2} = r^{\,2} = 16
$$
Now you must have
$$
\left\{ \begin{gathered}
4a + 13b = 32\quad \text{(center}\,\text{on}\,\text{the}\,\text{line}\,(\text{a))} \hfill \\
4x + 3y + 28 = 0\quad \text... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1937148",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
What is the value of $x$ in $222^x−111^x∗7=111^x$? Can anyone help me on this? It is for a 8th grader.
What is the value of $x$ in $222^x-111^x*7=111^x$?
I know the equation can be rearranged as $222^x=111^x*7-111^x=6*111^x$. Then what is next?
| Actually you got the first step backwards.
$222^x - 7 \cdot 111^x = 111^x$
$222^x = 8 \cdot 111^x$
$(2 \cdot 111)^x = 8 \cdot 111^x$
$2^x \cdot 111^x = 8 \cdot 111^x$
$2^x = 8$
$x = 3$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1937258",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Manifold with Ricci curvature not bounded below Could somebody please show an example (or give a reference to one) of a connected complete Riemannian manifold whose Ricci curvature is not bounded by below?
I guess there are standard examples to this, somewhere...
| Consider $g=dr^2+ f(r)^2 d\theta^2$ on $\mathbb{R}^2$ So if we have suitable sequences $x_n,\ y_n$ and $f$ s.t. $$ x_n<y_n<x_{n+1},\ f(x_n)>f(y_n)<f(x_{n+1}) $$
then Gaussian curvatures around $r=y_{n}$ go to $-\infty$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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In Linear Algebra, what is a vector? I understand that a vector space is a collection of vectors that can be added and scalar multiplied and satisfies the 8 axioms, however, I do not know what a vector is.
I know in physics a vector is a geometric object that has a magnitude and a direction and it computer science a v... | Just to help and understand the change of concept from physics to linear algebra about vectors, without pretending to be rigorous.
Consider that in physics (Newtonian) you consider an euclidean space, so you can speak in terms of magnitude. In linear algebra we want to be able and define a vector in broader terms, in a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1937464",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "84",
"answer_count": 9,
"answer_id": 1
} |
Showing that $\Bbb R[x] / \langle x^2 + 1 \rangle$ is isomorphic to $\Bbb C$ question
Show that $\Bbb R[x] / \langle x^2 + 1 \rangle$ is isomorphic to $\Bbb C$.
Let $\phi$ be the homomorphism from $\Bbb R[x]$ onto $\Bbb C$ given by
$f(x) \rightarrow f(i)$ (that is, evaluate a polynomial in $\Bbb R[x]$
at $i$). Th... | $⟨x^2+1⟩$ is an irreducible polynomial over $\mathbb R$, then it is a maximal ideal.Because of that $\mathbb R[x]/⟨x2+1⟩$ is a field. All elements in $⟨x^2+1⟩$ given by $a+bx$. Now we could easily define an isomorphism between $\mathbb C$ and $\mathbb R[x]/⟨x^2+1⟩$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1937513",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 3
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How many triangles with whole number leg lengths are there such that area and the perimeter is equal? I've tried to use Heron's formula to approach the problem , but it doesn't make any sense .I also tried to guess the lengths and I got two triangles , one of them is (5,12,13) and the second is (6,8,10).
So,I hope you ... | Trying with Pythagorean triplets with sides $ (2mn, m^2-n^2,m^2+n^2),$
your condition leads to
$$ 2 m^2 + 2 mn = mn ( m^2-n^2) ; \quad n (m-n) = 2; $$
or
$$ n = (m + \sqrt{ m^2-8})/2 $$
which gives an infinite set including
$$ (m,n) = (3,2), \sqrt2 (2,1),.. $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1937588",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Proving a graph is planar - mutually tangent circles in a plane
Let $C_1,\dots ,C_n$ be circles in the plane with pairwise disjoint
interiors. Define the tangency graph to have $n$ vertices such
that vertices are adjacent if the corresponding circles are tangent to
each other. Prove this graph is planar.
It loo... | It's obvious: Take the centers $v_i$ $(1\leq i\leq n)$ of the circles as vertices and connect two vertices by a straight segment if the corresponding circles are tangent to each other.
Or have I missed something?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1937741",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Using L'Hopital's Rule to show limit is 0? I am trying to show for any non-negative integer $n$, $\lim \limits_{x \rightarrow 0+} \frac{e^{-{1 /x}}}{x^n}=0$. For $n=0$ this follows directly since $1/x \rightarrow +\infty$ as $x\rightarrow 0+$. For $n>0$, I notice the limit has indeterminate form $0/0$ but applying L'... | Actually, this limit can be done through a simple substitution. As indicated we assume $x\to0^+$ for your limit. Setting $x=1/t$, the limit becomes $\lim \limits_{t \rightarrow \infty} \frac{t^n}{e^t}$. After applying L'Hospital's Rule sufficient amount of times (it's an infinity over infinity situation), your polynomi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Choosing toppings for pizza Problem: Ordering a "deluxe" pizza means you have four choices from 15 available toppings. How many combinations are possible if toppings cannot be repeated? If they can be repeated? Assume the order in which the toppings are selected does not matter.
If the toppings cannot be repeated, then... | This may not be the most efficient way of doing this, but you can consider it as separate cases.
*
*If all toppings are distinct, then you have $C_4^{15}$ combinations.
*If there are three distinct toppings, you have $3 \cdot C_3^{15}$ combinations (because we have $C_3^{15}$ choices for toppings and then $3$ choic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1937959",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Unable to get a terminal at the start for the GNF The grammar is:
S-> AA|a
A-> SA|b
If I substitute one rule in another, there would always remain a non-terminal at the start. How do I get the terminal at the start?
GNF has productions of form:
A->xB
Where x is a single terminal and B can be a combination of non-term... | To transform a grammar in GNF you have to remove direct and indirect left recursion. here you have indirect left recursion so the strategy is to make it direct by replacing the first $A$ in the rule of $S$. You obtain the following equivalent grammar:
$$S\to SAA|bA|a $$
$$A\to SA | b $$
You now have a direct recursions... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Do there exist $m,n$ such that $6 = 2(2m+1)^2/(2n+1)^2$? Can the two numbers $n$ and $m$ exists such that
$$6=\frac{2(2m+1)^2}{(2n+1)^2}$$
where $\gcd(m,n) = 1$?
| By the comments above, it suffices to show that $\sqrt{3}$ is not a rational number. We proceed by contradiction. So suppose that $\sqrt{3}=\frac{a}{b}$ with $a,b$ integers such that $\text{gcd}(a,b)=1$. (We may assume this without loss of generality). Then $3=\frac{a^2}{b^2}$, hence $3a^2=b^2$. It follows that $3\mid ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1938203",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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In a finite field product of non-square elements is a square I came across one problem in a finite field as follows:
Let $F$ be a finite field. Show that if $a, b\in F$ both are non-squares, then $ab$ is a square.
I wanted to prove it by using the idea of Biquadratic field extension. But there is no biquadratic exte... | Hint: I assume, $a,b,\neq 0$ otherwise it is obvious. $F^*$ is cyclic. Suppose it is generated by $x$, you can write $a=x^n, b=x^m$, $n,m$ odd. Then $n+m$ is even.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1938366",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 5,
"answer_id": 0
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Why are these sets equal? how can I formally see that these following sets are equal? For $X_1,\ldots,X_n,\ldots$ random variables with values in $[-\infty,\infty]$, then:
$\{\inf_n X_n < a\} = \bigcup_n\{X_n < a\}$ and $\{\sup_n X_n>a\} = \bigcup_n
\{X_i > a\}$
I have also difficulties to see the following equality $\... | Suppose $\inf_n X_n < a.$ Remember that "inf" means the largest lower bound. That means nothing larger than that can be a lower bound. Thus $a$ is not a lower bound of $\{X_n: n\}.$ To say that $a$ is not a lower bound of that set means $\exists n\ X_n<a,$ and that's the same as saying the event $\bigcup_n \{X_n<a\}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1938591",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is there a formula for $\int {x^n e^{-x} dx}$ I saw in this question that
$$
\int {x^n e^x dx} = \bigg[\sum\limits_{k = 0}^n {( - 1)^{n - k} \frac{{n!}}{{k!}}x^k } \bigg]e^x + C.
$$
and I was wondering if we can get some formula like that for
$$
\int {x^n e^{-x} dx}
$$
when $n \in \mathbb N$. I already know that
$... | The general formula is simply $n\to z$ in which $z\in\mathbb{R}$:
$$\int_0^{+\infty}\ t^{z-1}e^{-t}\ \text{d}t = \Gamma(z)$$
Possibly avoid the poles: $z = 0, -1, -2, \ldots$.
If you are talking about a general formula for the indefinite integral, then the series expansion is what you are searching for.
Just expand the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1938682",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show that a certain set of elements is a basis of the free module $\mathbf{Z}[\xi]$ Let $\xi$ be a $p$-th root of unity for $p$ a prime. It is well-known that $\mathbf{Z}[\xi]$ is a free $\mathbb Z$-module.
Now I'd like to show that $1, (1-\xi)^2, ..., (1-\xi)^{p-1}$ is a basis of $\mathbf{Z}[\xi]$.
Since I know th... | It is not sufficient to show that a set is linearly independent to show that it is an integral basis. For instance, consider $ \mathbf Z[\sqrt{2}] $ which is a free $ \mathbf Z $-module of rank $ 2 $ - the set $ \{ 1, 2 \sqrt{2} \} $ is linearly independent over $ \mathbf Z $, but it is not an integral basis of $ \math... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1938774",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that $\{m,n\} =\{k,l\}$
Let $n,m,k,l$ be positive integers and $p$ and odd prime with $0 \leq n,m,k,l \leq p-1$ and $$n+m \equiv k+l \pmod{p}, \quad n^2+m^2 \equiv k^2+l^2 \pmod{p}.$$ Prove that $\{m,n\} =\{k,l\}$.
We can rearrange the third condition to get $(n-l)(n+l) \equiv (k-m)(k+m) \pmod{p}$. How do we co... | First, observe that $n^2 + 2mn + m^2 \equiv (n+m)^2 \equiv (k+l)^2 \equiv k^2 + 2kl + l^2$, so $kl \equiv mn$ (since $p$ is odd, 2 is a unit).
If $k \equiv 0$, then $0 \equiv kl \equiv mn$. Without loss of generality we have $m \equiv 0$, and thus $n \equiv l$. The claim follows then from $0 \le m,n,k,l \le p-1$.
Other... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"answer_id": 2
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Get covariance of pixels in an image? What is the best way to calculate the covariance matrix for a small square of pixels? Assume it is a $3\times 3$ square of grayscale values.
I have read that in some cases, you can get the covariance matrix by inverting the Hessian. So, I was thinking of using a gradient operator t... | From my tests, it appears that inverting the Hessian does not give the correct covariance matrix. I'm not sure under what circumstances the inverted Hessian will give the correct answer.
What I ended up doing is assigning each pixel becomes a vector $x$ based on its pixel coordinates:
$$
\begin{matrix}
(-1,-1)&(0,-1)&(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1939040",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Minimizing Perimeter of a quadrilateral Let us say that we are given a quadrilateral where the diagonals are congruent and fixed at a certain length, and the angle between the two diagonals are fixed. How would you prove that the minimum perimeter is achieved when the quadrilateral is a rectangle? I know it when diagon... | Call $a,b,c,d$ the vertices and view them as vectors in $\mathbb R^2$. Call $m,n$ the midpoints of the two diagonals $\overline{ac}$ and $\overline{bd}$. Finally, suppose that the baricenter $(a+b+c+d)/4$ is the origin. Then it is easy to check that $m=-n$ (because the baricenter coincides with the midpoint between $m$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1939164",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Explanations about some Mittag-leffler partial fraction expansions Is it possible to show where the following series come from?
$$\sum _{k=1}^{\infty } \left(\frac{1}{\pi ^2 k^2}-\frac{2}{(x-2 \pi k)^2}-\frac{2}{(2 \pi k+x)^2}\right)+\left(-\frac{2}{x^2}-\frac{1}{6}\right)=\frac{1}{\cos (z)-1}$$
$$\sum _{k=1}^{\infty... | As I commented earlier, I have a problem with the first expression. So, since Maple said that it is correct, I suppose I am wrong but I would like to know where.
Let me consider $$S_1=\sum _{k=1}^{\infty } \frac{1}{\pi ^2 k^2}\qquad S_2=\sum _{k=1}^{\infty }\frac{1}{(x-2 \pi k)^2}\qquad S_3=\sum _{k=1}^{\infty }\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1939284",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is the value of the nested radical $\sqrt[3]{1+2\sqrt[3]{1+3\sqrt[3]{1+4\sqrt[3]{1+\dots}}}}$? The closed-forms of the first three are well-known,
$$x_1=\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\dots}}}}\tag1$$
$$x_2=\sqrt[3]{1+\sqrt[3]{1+\sqrt[3]{1+\sqrt[3]{1+\dots}}}}\tag2$$
$$x_3=\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\dots... | To answer this question we need to find the system that is error resilient.
We can write the equation as
$$y(x)=x\sqrt[3]{1+(x+1)\sqrt[3]{1+(x+2)\sqrt[3]{1+...}}}$$
from where we have
$$y(x)=x\sqrt[3]{1+y(x+1)}$$
or
$$y(x-1)=(x-1)\sqrt[3]{1+y(x)}$$
Now we need to estimate how this function behaves and we can easily see... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1939394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Probability of a painted cube being reassembled into itself Suppose 27 cubes are stacked together, suspended in the air, to form a larger cube.
The cube is then painted on all the exposed surfaces and dried.
The smaller cubes are then randomly permuted in both spatial position and spatial orientation to form another la... | There are $27$ cubies of four types:
*
*one body cubie (B),
*six face cubies (F),
*twelve edge cubies (E), and
*eight vertex cubies (V).
We can represent the cubie types occupying the $27$ cubie positions by a $27$-letter word using the letters B, F, E, and V with the multiplicities above. The number of distin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1939512",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Separable closure and normality Let $K/F$ be a normal algebraic extension and let $L = (K/F)^{sep}$ be the subfield of elements of $K$ which are separable over $F$ (this is also called the separable closure of $F$ in $K$). Is $L/F$ necessarily normal?
I really do not have a clue about this question so any hints will be... | Take $a\in L$ and let $p$ be its minimal polynomial. Since $a\in K$ and $K$ is normal, $p$ is a product of linear factors :
$$p=\prod_{j=1}^n(X-a_j),$$
(where $a=a_1$) and by definition of $L$ all the $a_j$'s are distinct. They have the same minimal polynomial $p$, so all of them are in $L$; hence $L$ is normal.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1939612",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How many homomorphisms are there from $D_6$ to $D_5$? I know that:
$D_6$={$e,a,a^{2},a^{3},a^{4},a^{5},b,ab,a^{2}b,a^{3}b,a^{4}b,a^{5}b$}, with $a^{6}=e$ and $ba^{k}b=a^{-k}$.
$D_5$={$e,r,r^{2},r^{3},r^{4},s,rs,r^{2}s,r^{3}s,r^{4}s$}, with $r^{5}=e$ and $sr^{k}s=r^{-k}$.
Let $\varphi:D_6\rightarrow D_5$, a homomorphism... | Hint: In light of the first isomorphism theorem, you want to find normal subgroups $N\triangleleft D_6$ and then find subgroups of $D_5$ isomorphic to $D_6/N$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1939704",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Understanding proof that connected graph with $V=E+1$ is acyclic This is an excerpt proving that if a graph $G$ is connected and has one more vertex than edge, then it is acylic.
Suppose $|V|=n$ and that $G$ has a $k$-cycle. This cycle has $k$ vertices and edges, hence $G$ has $n-k$ additional vertices. Each of these ... | Let $v$ be a vertex not belonging to the cycle $c_1c_2\cdots c_k$. Then by connectedness there exists a path the form $v_0v_1\cdots v_r$ with $v_0=v$ and $v_r=c_i$. Among all such paths for $v$, pick one that minimizes $r$. As $v$ is not in the cycle, $r\ge 1$. Associate $v$ with the first edge $vv_1$ of one such shor... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1939814",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Fixed-point iteration and continuity of parameters Let $X$ a compact set and $A\subseteq \mathbb{R}$. Consider a continuous function $f\colon X\times A\to X$ and construct a fixed-point iteration as follows
$$
x_{k+1}=f(x_k,a),\quad x_0\in X, a \in A.\quad (\star)
$$
My question: If $(\star)$ admits a unique fixed poi... | Suppose $X$ is metric (with metric $d_X$), and consider the following parameterized family of optimization problems:
$$
\max_{x \in X} \, -d_X\big(x, f(x,a)\big) \tag{$\ast$}
$$
By construction, $d_X\big(x, f(x,a)\big)$ is continuous in $(x,a)$, and given $X$ is compact, it is straightforward to verify the other condit... | {
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"url": "https://math.stackexchange.com/questions/1939968",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Commutativity among the elements in the quotient group $G$ and the ful group $H$ under homomorphism $H \overset{r}{\to} G$ Say the full group is $H$, and we pick up a normal subgroup $N$, and we define the quotient group $$G=H/N.$$
There is a group homomorphism $r$ from $H$ to $G$
$$
H \overset{r}{\to} G.
$$
My questio... | Since $r$ is a group homomorphism, your relations are in general not true: Try a pair $(h_1,h_2)$ of (non-commuting elements), of a non-abelian group $H$, such that: $h_1h_2h_1^{-1}h_2^{-1}\notin N$, for providing counterexamples:
$$
r(h_1h_2)=r(h_1)r(h_2)=r(h_2)r(h_1)=r(h_2h_1)\Leftrightarrow \\r(h_1)r(h_2)r(h_1)^{-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1940149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Let $f:[0,1]\to\Bbb{R}$ a continuous function which is differentiable in $(0,1)$, but is not in $0$ and $1$ I need to draw a graphic which the function is differentiable in whole $(0,1)$ but is not in $0$ and $1$.
I just can imagine the function $f(x)=cotg(\frac{x}{\pi})$ but this $f$ is not well-defined in $[0,1]$, de... | Oh, I missed the restriction that the domain of $f$ is $[0,1]$. If $f$ is differentiable on $(0,1)$ and continuous on $[0,1]$ and nonexistent on $(-\infty,0)$ and $(1, \infty)$ this is impossible unless $\lim f'(x) = \pm \infty$.
$f'(0) = \lim_{h\rightarrow 0^+}\frac{f(h) - f(0)}{h}$.
As $f$ is continuous $f(0) = lim_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1940267",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that $\tan(x+\frac{\pi}8)>\mathrm{e}^x+\ln x$ for $x\in(0,\frac{3\pi}8)$
Prove that $$\tan\left(x+\frac{\pi}8\right)> \mathrm{e}^x+\ln x,\quad x\in\left(0,\frac{3\pi}8\right).$$
By plotting the graph I found that this is indeed true (in fact I found this inequality through plotting), but how can I prove this? T... | To prove that $$\tan(x+\frac{\pi}8)>e^x+\ln x \qquad\ x\in(0,\frac{3\pi}8)$$ I suppose that it is sufficient to show that function $$F(x)=\tan(x+\frac{\pi}8)-e^x-\ln x $$ is always positive in the given range.
The function is positive infinite at both ends. So, let us show that $F(x)$ goes through aminimum value which ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1940384",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
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Uniqueness unitization of a non unital $C^*$-algebra I am trying to show that the unitization of a non-unital AF $C^*$-algebra, $A$ , is again an AF $C^*$-algebra.
In order to do so, I tried to claim the following:
Let $A, B$ be $C^*$ algebras and suppose that $A$ is an ideal in $B$ and $B/A$ is isomorphic to $\Bbb{... | When $A$ is not unital, we have that $A^+=A\oplus\mathbb C$ with the product $$(a_1,\lambda_1)(a_2,\lambda_2)=(a_1a_2+\lambda_2a_1+\lambda_1a_2,\lambda_1\lambda_2).$$
Consider the map $\phi:A^+\to B$ given by
$$
\phi(a,\lambda)=a+\lambda I.
$$
This is clearly a $*$-homomorphism. Since $A$ is non-unital, this map is o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1940472",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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As $\frac1{(1-x)^2} = {1+2x+3x^2 +...nx^{n-1}} + \frac{x^n}{(1-x)^2} + \frac{nx^n}{(1-x)}$, is this is a function of only $x$ or both $x$ and $n$? $$\frac1{(1-x)^2} = {1+2x+3x^2 +...nx^{n-1}} + \frac{x^n}{(1-x)^2} + \frac{nx^n}{(1-x)}$$
Now, the LHS seems to be a function of only $x$, whereas the RHS seems to be a func... | If you rewrite as:
$$\lim\limits_{n\to\infty}\sum\limits_{i=1}^n ix^{i-1}$$
the '$n$' is just an index.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1940583",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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Union of metric spaces Is the union of two metric spaces a metric space?
I tried it but could't define a suitable metric on intersection. Can somebody help me to understand it?
| (1). If $(X_1,d_1), (X_2,d_2)$ are metric spaces and $X_1\cap X_2=\emptyset$ we can define a metric $d_3$ on $X_1\cup X_2$ by $d_3(x_1,x_2)=1$ when $x_1\in X_1 ,x_2\in X_2,$ and $d_3(x,y)=d_1(x,y)$ when $x,y\in X_1,$ and $d_3(x,y)=d_2(x,y)$ when $x,y\in X_2.$ Then the subspace topologies on $X_1$ and $X_2,$ as subspace... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1940700",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Non trivial solutions to $ y'' + 2y' + ay = 0$ For which values of $a$ does the following equation has non trivial solutions
$y'' + 2y' + ay = 0 , \space y(0) = y(\pi ) = 0$
The characteristic equation is:
$$x^2+2x+a = 0$$
and I have found the roots to be
$$x_1 = \sqrt{(1-a)}-1$$
$$x_2 = -\sqrt{(1-a)}-1$$
I have tried ... | Hint. Consider the case when $1-a>0$, $1-a<0$, $1-a=0$.
Note that when $1-a<0$, then the general solution is
$$y(x)=Ae^{-x}\cos(\sqrt{a-1}x)+Be^{-x}\sin(\sqrt{a-1}x).$$
The condition $y(0)=0$ implies that $A=0$ and $y(\pi)=0$ implies
$$Be^{-\pi}\sin(\sqrt{a-1}\pi)=0$$
The solution is non trivial if $B\not=0$, therefor... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1940808",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Burgers' equation with boundary conditions Consider the signaling problem
$u_t + c(u)u_x= 0, t> 0, x> 0$
$u(x, 0) = u_0, x> 0,$
$u(0, t) = g(t), t> 0,$
where $c$ and $g$ are given functions and $u_0$ is a positive constant. If $c
(u) > 0$,
under what conditions on the signal $g$ will no shocks form? Determine the
solut... | $$u_t+C(u)u_x=0\quad\text{where }C(u)\text{ is a given function}$$
GENERAL SOLUTION :
The system of characteristic differential equations is :
$$\frac{dt}{1}=\frac{dx}{C(u)}=\frac{du}{0}$$
A first equation of characteristic cuves comes from $du=0\quad\to\quad u=c_1$ .
A second equation of characteristic cuves comes fro... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1940946",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Why is cross product defined in the way that it is?
$\mathbf{a}\times \mathbf{b}$ follows the right hand rule? Why not left hand rule? Why is it $a b \sin (x)$ times the perpendicular vector? Why is $\sin (x)$ used with the vectors but $\cos(x)$ is a scalar product?
So why is cross product defined in the way that it... | This may be a bit too deep but let $V$ be a finite dimensional vector space with basis $v_1,...,v_n$. We say $(v_1,...,v_n)$ is an oriented basis for $V$. We can define an equivalence class on orientations of $V$ by $[v_1,...,v_n] \sim [b_1,...,b_n] \iff [v_1,...,v_n] = A[b_1,...,b_n]$ (where $A$ is the transition matr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1941044",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "60",
"answer_count": 13,
"answer_id": 2
} |
Problem on rearrangement inequality I read that the rearrangement inequality deals with sorted sequences of real numbers. We have $-5>-6$ and $3>2$ , hence by rearrangement inequality we have $-15>-12$ which is obviously false.
What am I missing out?
| In this context the rearrangement inequality says that
$$
x_1 = -6 < -5 = x_2 \text{ and } y_1 = 2 < 3 = y_2
$$
implies that
$$
-28 = -18 -10 = x_1 y_2 + x_2 y_1 \le x_1 y_1 + x_2 y_2 = -12 -15 =-27,
$$
which is true. It does not say what you're asserting.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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a divides bc if and only if
For all integers $a,b,c$,
$$\;a\mid bc \iff \frac{a}{\gcd(a,b)}∣c.$$
Can anyone help me in proving above statement?
I thought I'd start with trying to prove $\Leftarrow.$
$$\frac{a}{\gcd(a,b)}\mid c \Rightarrow$$
I know that $\gcd(a, b) = ax+by = d$ but I am lost as to how to make use... | $ a\mid bc \iff a\mid ac,bc\iff a\mid (ac,bc)=(a,b)c\iff a/(a,b)\mid c\ $ by gcd Distributive Law
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1941259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Graph theory problem: n-dimensional cube Let $Q_n$ be the $n-dimensional$ cube graph: Its vertices are all the $n-tuples$ of $0$ and $1$ with two vertices being adjacent if they dier in precisely one position.For example, in $Q_3$, the vertices $(1,0,0)$ and $(1,0,1)$ are adjacent because they differ only in the third... | Hint: Think of the defining properties of a bipartite graph. In particular, what do you know about the length of cycles in a bipartite graph?
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Probability Unfair Coins I have two coins, yielding heads at $P(H) = a$ and the other with $P(H) = b$.
Is it possible for the following to be equal? If so, what should be values for $a$ and $b$?
(a) $p_1$=neither is head
(b) $p_2$=exactly one of the coins is a head
(c) $p_3$=both are heads
I am not sure where I am goi... | To satisfy (a),(b) and (c) you need first (a)=(c), which as you say, means
$$a+b=1$$
This also means that $1-2ab=ab$ (from (b)=(c)), so $ab=\frac 13$.
Substitute $a=1-b$, we get $b(1-b)=\frac 13$.
However $b(1-b)$ has a local maxima at $b=\frac 12$, and this equates to $\frac 14$, meaning that no real solution is possi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1941491",
"timestamp": "2023-03-29T00:00:00",
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Evaluation of Half Dirac Delta Function $\delta^{(1/2)}_\mu(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}|k|^{1/2}e^{ikx}e^{-\mu|k|}dk$ which provides the half-Dirac delta distribution in the limiting case of $\mu\to0$.
I know the solution of $\delta^{(1/2)}_\mu=\frac{1}{2\pi}\int_{-\infty}^{\infty}|k|^{1/2}e^{ikx}e^{-\mu|k... | As I said you need some complex analysis for showing that
$$\int_0^\infty t^{a-1} e^{-zt}dt = \Gamma(a) z^{-a}, \qquad \text{Re}(z) > 0,\ \text{Re}(a) > 0$$
The proof is that for $\text{Re}(z) > 0$ the LHS and the RHS are analytic in $z$, and for $z \in (0,\infty)$ they are equal (change of variable $u = zt$), hence... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1941585",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What is the quotient group o mod the identity I am trying to prove something using the first isomorpism theorem, which basically says if $G,H$ are groups, and $f:G\to H$ is a group homomorphism, then $G/ker(f)\cong f(H)$. In this case, suppose $f$ is an epimorphism meaning $f(G)=H$.
I want to show that two groups are i... | You are correct in that if $f:G\to H$ is a surjective homomorphism (epimorphism) then $G/\text{Ker} f \cong H$. It is perfectly valid to find a map $f$ with $\text{Ker}f = \{e_G\}$. The then valid conclusion is that $G /\{e_G\} \cong G \cong H$.
It is worth noting however that in your search for such a map, showing tha... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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mimimum value of expression $a^2+b^2$
If $a,b$ are two non zero real numbers and $ab(a^2-b^2) = a^2+b^2,$ Then $\min(a^2+b^2)$
$\bf{My\; Try::}$ We can write it as $$ab=\frac{a^2+b^2}{a^2-b^2}\Rightarrow a^2b^2=\frac{(a^2+b^2)^2}{(a^2+b^2)^2-4a^2b^2}$$
Now Put $a^2+b^2=u$ and $a^2b^2=v,$ Then expression convert into ... | If we let $z = a + b i$ then we get
$$ab = \frac{ z^2 - \bar{z}^2}{4i}$$
$$a^2 - b^2 = \frac{ z^2 + \bar{z}^2}{2}$$
$$a^2 + b^2 = z \bar{z}$$
Thus your original equation can be rewritten as
$$\frac{Im(z^4)}{4} = \frac{z^4 - \bar{z}^4}{8 i} = \frac{(z^2 - \bar{z}^2)(z^2 + \bar{z}^2)}{8i} = z \bar{z}$$
We have $\frac{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1941844",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 0
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Showing there exists no subintervals without digit. Let $A$ be the subset of $(0,1)$ containing the real numbers which have a 3 in their decimal expansion.
As part of a larger result in an exercise in my text, I'm trying to show that $B = [0,1] \setminus A$ doesn't have any open interval $(x,y)$ with $0 \leq x < y \le... | Consider the standard (i.e., we prefer things like $0.7000\ldots$ over 0.6999\ldots$ $) decimal digit representations $0.x_1x_2x_3\ldots$ and $0.y_1y_2y_3\ldots$ of $x$ and $y$. (The special case $y=1.0000\ldots$ is not covered by this, but not difficult).
As $x\ne y$, there is a first index $n$ such that the $x_n\ne y... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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$ \int\frac{2+\sqrt{x}}{\left(x+\sqrt{x}+1\right)^2}dx$ $$I=\int\frac{2+\sqrt{x}}{\left(x+\sqrt{x}+1\right)^2}dx$$
I can't think of a substitution to solve this problem, by parts won't work here. Can anyone tell how should I solve this problem?
| Let $$I = \int\frac{2+\sqrt{x}}{(x+\sqrt{x}+1)^2}dx = \int \frac{2+\sqrt{x}}{x^2\left(1+x^{-\frac{1}{2}}+x^{-1}\right)^2}dx$$
So $$I = \int\frac{2x^{-2}+x^{-\frac{3}{2}}}{\left(1+x^{-\frac{1}{2}}+x^{-1}\right)^2}dx$$
Put $\left(1+x^{-\frac{1}{2}}+x^{-1}\right) = t\;,$ Then $\displaystyle \left(-\frac{1}{2}x^{-\frac{3}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1942038",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Does negative transpose sign mean inverse of a transposed matrix or transpose of an inverse matrix? I want to know meaning of $$H^{-T}$$Is it same with $$(H^{-1})^T$$or $$(H^T)^{-1}$$
| $H^{-1}$ is defined such that $I=H^{-1}H=HH^{-1}$, taking the transpose of this equation yields $$I=I^T=(H^{-1}H)^T=H^T(H^{-1})^T$$ Therefore $(H^{-1})^T$ is the inverse of $H^T$, so $$(H^{-1})^T=(H^T)^{-1}$$
So yes, $H^{-T}$ it is the same as both.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1942211",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Use congruence's to find the reminder when $2^{50}$ and $41^{65}$ are divided by $7$ Use congruence's to find the reminder when $2^{50}$ and $41^{65}$ are divided by 7
$2^{50}$
$50=(7)^2+1$
$2^{50}=2^{7\cdot7+1}$
and I'm not sure where to go from here?
| Note that, $$\begin{align} & 2^3\equiv 1 \pmod7 \\ \implies & (2^3)^{16}\equiv 1^{16} \pmod7 \\ \implies & 2^{48}\equiv 1 \pmod7 \\ \implies & 2^{48}\cdot 2^2\equiv 1\cdot 2^2 \pmod7 \\ \implies & \color{blue}{2^{50}\equiv 4 \pmod7}\end{align}$$
Also note that $$\begin{align} & 41\equiv -1\pmod7 \\ \implies & 41^{65}\e... | {
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"timestamp": "2023-03-29T00:00:00",
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A sum involving binomial coefficients, powers and alternating signs How to prove that
$$
\sum_{k=0}^n(-1)^{n-k}{n \choose k} k^n = n!
$$
using mathematical induction? Please, do not use definition of Stirling number etc. algebra tricks.
| Suggested intro: maybe these solutions don't meet the OP's specific requirements, but I think that they're worth making available anyway.
$\color{red}{n!}$ is the number of bijective functions from $A=\{1,2,\ldots,n\}$ to $A$.
Let we say that a function $f:A\to A$ has type $k$ if $|f(A)|=k$. The number of functions wi... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Are they the same function? $y = x^2/x$ and $y = x$ Are they the same function?
$$y=\frac{x^2}{x}$$
and
$$y=x$$
For the first function, if we don't divide both the numerator and the denominator by x, then the domain of it is the real line except the point x = 0, which is different from the domain of the second function... | The are not the same function, since they have different domains.
For two function to be the same they should have the same domain and the same mapping rules.
| {
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"url": "https://math.stackexchange.com/questions/1942489",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Existence of measurable set $A\subseteq{\mathbb R}$ which is locally uncountable and so is its complement Is there a measurable set $A\subseteq{\mathbb R}$ such that $|A\cap I|$ and $|A^\complement\cap I|$ are both uncountable for any open interval $I$?
| The set of real numbers whose decimal representation has a finite number of ones (let's agree a number can't end with infinite 9's, even if it doesn't really change anything.)
It is measurable because you can write it as a countable union of countable intersections of intervals (it's kind of tedious to write down); it ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1942586",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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properties of relations I'm trying to do some chapter problems on equivalence relations. I'm stuck in the second section "properties of relations."
Question: Let $A=\{a,b,c,d\}$. Give an example of a relation $R$ on $A$ that is neither reflexive, symmetric ,or transitive.
What I tried doing was writing out all the pair... | Well, the relation you give is indeed neither reflexive nor transitive, but there are many ways to get such a relation. "Cancelling out pairs that don't match the laws" is not a well defined procedure. (Well, it is defined for reflexivity, but it does not do what you want: You would be left with $\{(a,a),(b,b),(c,c)... | {
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Prime subfield is either isomorphic to $\mathbb{Q}$ or $F_p$ I'm trying to prove the following statement:
Let $F$ be a field. The intersection of all subfields of $F$ is a subfield which is isomorphic to $\mathbb{Q}$ if $\operatorname{char}(F)=0$, and isomorphic to $F_p$ if $\operatorname{char}(F)=p$.
I assume I need t... | Define a map $f:\Bbb{Z}\to{F}$ by $f(n)=n.e$ , where $e$ is the unit element of $F$.
$f$ is homomorphism:
$$f(m+n)=(m+n).e=m.e+n.e=f(m)+f(n)$$
and
$$f(mn)=(mn).e=(m.e)(n.e)=f(m)f(n)$$.
By Fundamental theorem of homomorphism,
$$f(\Bbb{Z})\cong \frac{\Bbb{Z}}{\operatorname{Ker}{f}}\subseteq F$$.
Where $\operatorname{Ker... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1942821",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Algebra transitive action (Isaacs 4.1) I'm working on the following questions and I'm unable to make any progress on it. Can anyone offer any insight?
Let $G$ act transitively on a set $\Omega$ and suppose $G$ is finite. Define an action of $G$ on $\Omega \times \Omega$ by putting $(\alpha, \beta) \cdot g =(\alpha\cdo... | With character theory this can be proved rather easily. Put $\chi(g)=\#\{\alpha \in \Omega: \alpha^ g=\alpha\}$. Then the function $\chi$ is called the permutation character of the action of $G$ on $\Omega$. It is a standard fact (basically the Burnside-Cauchy-Frobenius Lemma) that the number of orbits of $\Omega$ unde... | {
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"timestamp": "2023-03-29T00:00:00",
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Is this rewrite correct? If x mod 9 = 1, can I rewrite the equation as x = 1 mod 9. If not, what is the correct way?
I know this sounds silly. But please help me.
| There are two possible meanings of "mod" here:
One is a binary operation: $a \operatorname{mod} n$ is a number; it is the unique integer $r$ with $0 \le r < n$ such that $a - r$ is a multiple of $n$. If interpreted this way, the two equations aren't equivalent (for example, $x = 10$ is a solution of the former but not ... | {
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Prove that $\{ \sum_{n=0}^k z^n \}_{k=0}^\infty$ does not converge uniformly? So as the title says, can anyone prove that $\{ \sum_{n=0}^k z^n \}_{k=0}^\infty$ does not converge uniformly 0n the disk $D(0,1)$? I think it would converge uniformly to $1/(1-z)$ since it is a geometric series, but professor posed the probl... | Realize that $\sum_{k=0}^{\infty} x = 1/(1-x)$ is unbounded at $x=1$ on $(0,1)$. Also realize that $\sum_{k=0}^n x$ is bounded by $n$ on $(0,1)$. From this, you can conclude that convergence cannot be uniform (do you see how?).
| {
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"timestamp": "2023-03-29T00:00:00",
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Question about mathematical logic with set and inclusion relation Let $L = \{ \subseteq \}$ and let $M$ be the L-structure whose universe is $P^\mathbb{N}$, the set of all subsets of $\mathbb{N}$, and where $\subseteq$ is interpreted in the usual way.
1) Show that for all integer $n$, there is a formula $\psi_n[x]$ suc... | Regarding the first question:
$$\psi_n[x] \equiv\exists y_1 \dots \exists y_n \ (\bigwedge_{i=1}^n y_i \subseteq x) \wedge (\bigwedge_{1 \le i < j \le n} y_i \neq y_j)$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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The definitions of finitely presented modules? Let $M$ be a module over a ring or an algebra $A$.I have seen three definitions of finitely presented modules:
(1) A module $M$ is called finitely presented if there is an exact sequence of $A$-modules: $0 \rightarrow L \rightarrow F \rightarrow M \rightarrow 0$, where $F$... | Suppose $$P_2\to P_1\to A\to 0$$ is as in (3). Since $P_1$ is finitely generated and projective, there is a finitely generated module $Q$ such that $P_1\oplus Q=F$ is a finite rank free module. Now take the direct sum of our exact sequence and the exact sequence $$Q\stackrel{1_Q}\to Q\to 0\to 0$$ to get another exact... | {
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$G$ is a group that is transitive on $\{1,2,\dots,n\}$, and let $H_i$ be the subgroup of $G$ that leaves $i$ fixed. Prove that $|G|=n|H_i|$. A subgroup $H$ of the group $S_n$ is called transitive on $B=\{1,2,\dots,n\}$ if for each pair $i,j$ of elements of $B$ there exists an element $h\in H$ such that $h(i)=j$.
Suppos... | Hint: Use the orbit-stabilizer theorem.
| {
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Why is the maximum likelihood estimator $y_{\operatorname{max}}$ instead of $y_{\operatorname{min}}$ Use the method of maximum likelihood to estimate the parameter $\theta$ in the uniform pdf,
$f_y(y ;\theta)=\frac{1}\theta$ where $0\leq y\leq\theta$
According to the solution manual $\theta_e=y_{\operatorname{max}}$, h... | Well if you think about it your answer does not make much sense!
Maybe better to see it as follows:
Let $\hat \theta$ be the MLE, then the likelihood of all points $y_i$ that would be greater than $\hat\theta$ would be zero according to that model, and the entire likelihood would be zero. Therefore, in order to maximiz... | {
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Can you get the asymptotics of the following integral? I am interested in the big $N$ asymptotics of the following integral
$$
\int_0^{\infty}dx\,e^{-2xN}\frac{1-(2x)^N}{1-2x}
$$
I have considered applying Laplace's method in some way, but I cannot make it work. Can anybody do better than me?
| $\dfrac{1-(2x)^N}{1-2x} = \sum_{j=0}^{N-1} (2x)^j$ so your integral is
$$ \sum_{j=0}^{N-1} \int_0^\infty dx\; e^{-2xN} (2x)^j = \sum_{j=0}^{N-1} \dfrac{j!}{2N^{1+j}}$$
Write this as $\dfrac{1}{2N}\sum_{j=0}^\infty a_j(N)$, so $a_0(N) = 1$. Note that for each $j$,
$a_j(N) \le a_j(j+1) = j!/(j+1)^j$, and $\sum_{j=0}^\... | {
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Show that $\mathrm{span}(v_1,\dots,v_k) = \mathrm{span}(v_1,\dots,v_k,v)$ Show that $\mathrm{span}(v_1,\dots,v_k) = \mathrm{span}(v_1,\dots,v_k,v)$ if and only if $v$ is in $\mathrm{span}(v_1,\dots,v_k)$
I am thinking that if $v$ is in $\mathrm{span}(v_1,\dots,v_k$) it must be one of the elements and can be written $\m... | Prove the implication two ways.
First prove the forward direction $\implies $:
Let $V=\text{span}(v_1, \ldots, v_k)$ and $W = \text{span}(v_1, \ldots, v_k, v)$ and assume $V=W$. We will prove that $v \in V$. This is trivial since $v \in W = V$ and thus $v \in V$.
Now we prove the backwards implication $\impliedby$:
As... | {
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How to prove that $(a\cos\alpha)^n + (b\sin\alpha)^n = p^n$ under the following conditions? How to prove that $(a\cos\alpha)^n + (b\sin\alpha)^n = p^n$ when then line $x\cos\alpha + y\sin\alpha = p$ touches the curve $$\left (\frac{x}{a} \right )^\frac{n}{n-1} + \left (\frac{y}{b} \right )^\frac{n}{n-1}=1$$
What I've ... | At the tangent point (x, y), the normal of the two curves are parallel, so we can get
$$\{\cos \alpha, \sin \alpha \}//\{\frac{x^\frac{1}{n-1}}{a^\frac{n}{n-1}}, \frac{x^\frac{1}{n-1}}{a^\frac{n}{n-1}}\} $$
so we get the equation:
$$\frac{x^\frac{1}{n-1}}{a^\frac{n}{n-1}\cos \alpha}=\frac{x^\frac{1}{n-1}}{a^\frac{n}{n... | {
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Practice PSAT Question About Rational Functions This is taken directly from a PSAT Practice Test:
$f(x)=\dfrac{2x-4}{2x^2+2x-4}$
A rational function is defined above. Which of the following is an equivalent form that displays values not included in the domain as constants or coefficients?
A) $f(x)=\dfrac{x-2}{x^... | The answer should be (b). The domain does not include -2 and 1. Answer (a) has "-2" as a constant (for x-2 and x^2+x-2) and "1" as a coefficient (for x^2(="1"*x^2) and x terms). No values other than "-2" and "1" as constants or coefficients are in (a).
Though (b) has "-2" as a constant for x-2 and "1" as a coefficien... | {
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Is it possible to draw a regular pentagon on a regular 3D grid by only connecting the intersection points? Think of it as infinite rows of cubes side by side and infinite rows on top of each other.
Like this, but as many cubes as needed
Let's say the distance between the points (the edges of the cubes) are of 10 units.... | No.
Suppose you have found such a regular pentagon $ABCDE$
then, since the lattice points are well.. on a lattice, the points $A' = A + \vec{BC}, B' = B + \vec{CD}, \ldots , E' = E + \vec{AB}$ are also on the lattice.
And they also form a regular pentagon, but one whose side is smaller than the side of the original pe... | {
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function spaces in linear algebra I am in upper division linear algebra, and i need help in proving a function space as a vector space. I just need help proving 2 particular vector space axioms.
Axiom 1: For each pair of elements a,b in F(field), and each element x in V (Vector space), ab(x)= a(bx).
Axiom 2:For each el... | It's generally a bad idea to try and work with a vector space when you haven't got a good idea of how it's defined. What do we mean by scalar multiplication? Addition of two vectors? The way it's usually set up is as follows:
If $V$ is a set of functions from $X$ to $Y$, where $Y$ is a vector space over a field $F$, th... | {
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Determining which functions are one to one
So I know for sure that $x^3$ is one to one, being that the entire span of real number $y$'s can be found by some x and it passes the vertical line test. Graphically, its pretty clear. I know that $x^2$ is also one to one though not onto, since no negative $y$'s can be obtain... | The easiest way is to look at the kernel for the not-so obvious ones. I should note that this works for only linear maps.
(b) The kernel is $\{ (x,x,x) \}$, so no not injective
(c) No, clearly.
(d) The kernel of this is $\{(x,-x) \}$, so no not injecive
(e) Yes the kernel of this is trivial (can you see why?)
| {
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Prove Bonferroni’s inequality I have read other solutions regarding proof of Bonferroni’s inequality.
However, is my derivation correct?
Suppose that E and F are two events of a sample space S.
Conclude: P(EF) ≥ P(E) + P(F) − 1
Proof:
P(EF)= 1 - P((EF)^c)
= 1 - P(E^c ∪ F^c)
= 1 - (P(E^c) + P(F^c))
= 1 ... | You seem to assume that $E^c$ and $F^c$ are disjoint in writing
$$ 1 - P(E^c \cup F^c) = 1 -[P(E^c) + P(F^c)].$$
(Also, you don't write any inequalities in your proof. Though maybe you meant to use an inequality at precisely this step...)
A simple proof notes that in general we have, $$P(E \cap F) = P(E) + P(F) - P(... | {
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Prove Ux$\cdot$Uy = x$\cdot$y $\forall$ x,y $\in$ $\mathbb{R}^n$ where U is Isometric transformation Suppose that U is a linear transformation from $\mathbb{R}^n$ into $\mathbb{R}^m$ that is isometric, i.e., $\lVert Ux\rVert$ =$\lVert x\rVert$ for all x $\in$ $\mathbb{R}^n$
a) Prove that Ux$\cdot$Uy = x$\cdot$y $\qquad... | You're right to look at $||U(x+y)||^2$, but I suggest trying to use the polarization identity
$$ x\cdot y=\frac{1}{4}\Big(||x+y||^2-||x-y||^2\Big)$$
| {
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How would you calculate the surface of the part of the paraboloid $z=x^2+y^2$ with $1 \le z \le 4$? Do you calculate it like done below? can you calculate it in another way?
$z=x^2+y^2$
Let
$x=\sqrt{z}\cos\theta$
$y=\sqrt{z}\sin\theta$
$z=z$
where $\theta\in[0,2\pi]$ and $z\in[1,4]$
$ \dfrac{\partial{x}}{\partial{\thet... | Your result is correct. By using polar coordinates, we obtain
$$\iint_S dS=\int_{\rho=1}^2 \int_0^{2\pi} \sqrt{1+f_x^2+f_y^2} \,(d\theta\,\rho d\rho)=2\pi\int_{\rho=1}^2 \sqrt{1+4\rho^2} \,\rho d\rho\\
=\dfrac{\pi}{6}\left[(1+4\rho^2)^{3/2}\right]_{\rho=1}^2
=\dfrac{\pi}{6}(17\sqrt{17}-5\sqrt{5})$$
where $f(x,y)=x^2+y... | {
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Series $\sum_{n=2}^\infty \frac{14^n}{3^{3n+4}(3n+7)}$ Convergence or Divergence Using The Ratio Test I am trying to determine if the following series converges or diverges by using the ratio test, which I believe can be summarized as the following:
$$L=\lim_{n\to \infty} \left| \frac{a_{n+1}}{a_n} \right|$$
If $L < 1... | Ratio test:
$$\frac{14^{n+1}}{3^{3n+7}(3n+10)}\cdot\frac{3^{3n+4}(3n+7)}{14^n}=\frac{14}{3^3}\cdot\frac{3n+7}{3n+10}\xrightarrow[n\to\infty]{}\frac{14}{27}\cdot1=\frac{14}{27}<1$$
$\;n\,-$ th root test:
$$\sqrt[n]{\frac{14^n}{3^{3n+4}(3n+7)}}=\frac{14}{3^3}\cdot\frac1{\sqrt[n]{3^4(3n+7)}}\xrightarrow[n\to\infty]{}\frac... | {
"language": "en",
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why two solutions to DE are contradictory?
Solution to the given differential equation $$\frac{dx}{dt}=4.9-0.196x$$
is given by
a) $x=25+ke^{-0.196t}\qquad$ b) $x=50+ke^{-0.196t}\qquad$ c) $x=50-ke^{-0.196t}\qquad$
d) $x=25-ke^{-0.196t}\qquad$ where k is some constant
my try:
method(1)
$$\frac{dx}{4.9-0.196x}=dt$... | They are both correct.
$k$ is any real number.
Observe that the set $$\{5k | k \in \mathbb R\}$$ is the same as $$\{-5k | k \in \mathbb R\}$$
| {
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How is $\frac{dt}{dx}$ just the reciprocal of $\frac{dx}{dt}$ ?? Came across this problem. Derivative of a parametric.
$$x=2\cos(t)$$
$$y=2\sin(t)$$
.....Hence,
$$\frac{dx}{dt}=-2\sin(t)$$
$$\frac{dy}{dt}=2\cos(t)$$
As we know, chain rule states: $$\frac{dy}{dx} = \frac{dy}{dt} \cdot \frac{dt}{dx}$$
Apparently, this... | They are the same thing, but one is expressed in terms of $t$ and the other is in terms of $x$.
$$\frac{dt}{dx} = -\frac{1}{\sqrt{4-x^2}} = -\frac{1}{\sqrt{4-(2\cos t)^2}} = -\frac{1}{2\sqrt{1-\cos^2 t}} = -\frac{1}{2 \sin t}$$
where the last step uses the Pythagorean identity: $\sin^2 t + \cos^2 t = 1$.
For parametri... | {
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Solving two curves Let two curves be:-
S1: $x^2/9+y^2/4=1$
S2: $y^2=2x$
Now, on solving these two ,by substituting $y^2$ as $2x$ from S2 in S1, I get two values viz. $x=3/2$ and $x=-6$.
But we can clearly see from the graph that the curves intersect at two distinct points in the first and fourth quadrant. So, should no... | In general, when you intersect two conics, you can have $4$ intersection points, that's why you see a ghost.
Imagine if you pinched the parabola and pulled it to the left through the other side of the ellipse. Then you would get $4$ intersections, and the other solution for $x$ would be relevant.
| {
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Finding limit of $\left(1 +\frac1x\right)^x$ without using L'Hôpital's rule. Is there a way to find this limit without using L'Hôpital's rule. Just by using some basic limit properties.
$$\lim_{x\to\infty}\left(1+\frac1x\right)^x=e$$
| In the comments it has been pointed out that this usually serves as a definition of e. But I think I know what you're asking. Usually in introductory calculus classes to evaluate this limit you let $$y= \lim_{x\to\infty}\left(1+\frac{1}{x}\right)^x,$$ and then take the natural log of both sides to get $$\ln(y) = \lim... | {
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How to derive the closed form solution of geometric series I have the following equation:
$$g(n) = 1 + c^2 + c^3 + ... + c^n$$
The closed form solution of this series is $$g(n) = \frac{c^{n+1} -1}{c-1}$$
However, I am having a difficult time seeing the pattern that leads to this.
$$ n =0 : 1 $$ $$ n =1 : 1 + c $$ $$ ... | Let $S=1+c+c^2+...+c^n$ then $c.S=c+c^2+...c^{n+1}$, subtract them.
| {
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If two matrices have the same column space and null space, are they the same matrix? If two matrices have the same column space and null space, are they the same matrix? I am thinking no because if A=[1 2;2 1] and B=[2 1;1 2] then they have the same column space (I think) but they are not identical
| This fails even in one dimension: $1$ and $2$ have the same column and null spaces. You can easily find other examples in higher dimensions. For example $I$ and $2I$.
In fact, all invertible matrices have the same column and null spaces, yet there are many different invertible matrices.
| {
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Efficiently computing Schur complement I would like to compute the Schur complement $A-B^TC^{-1}B$, where $C = I+VV^T$ (diagonal plus low rank). The matrix $A$ has $10^3$ rows/columns, while $C$ has $10^6$ rows/columns. The Woodbury formula yields the following expression for the Schur complement:
$$A-B^TB+B^TV(I+V^TV)... | Let us decompose the matrix $VV^T$ into its eigenvectors. Thus we have,
$VV^T = \Sigma u_iu_i^T$ where $u_i$ is defined where $ u_i = \sqrt[2]\lambda_i \alpha_i$ where $\lambda_i$ is an eigenvalue of $VV^T$ and $\alpha_i$ is the corresponding eigenvector. Since the matrix $VV^T$ is positive definite its eigenvectors w... | {
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Bounding sum of reciprocals of the square roots of the first N positive integers I am trying to derive the following inequality:
$$2\sqrt{N}-1<1+\sum_{k=1}^{N}\frac{1}{\sqrt{k}}<2\sqrt{N}+1,\; N>1.$$
I understand for $N\rightarrow \infty$ the summation term diverges (being a p-series with p=1/2), which is consistent w... | To get the lower bound, you can convert the sum to an integral. $\frac 1{\sqrt k} \gt \int_k^{k+1}\frac 1{\sqrt x}dx=2\sqrt x|_k^{k+1}=2(\sqrt {k+1}-\sqrt k)$, so $\sum_{k=1}^N \frac 1{\sqrt k} \gt \int_1^N \frac 1{\sqrt x}dx=2\sqrt N -2$ Then to get the upper limit you have the same approach. $\frac 1{\sqrt k} \lt \... | {
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Prove that the Dihedral group $D_n$ is isomorphic to $Z_n \rtimes_{\psi} Z_2$ I consider the following map $\psi : Z_2 \rightarrow Aut(Z_n)$ where we map
the identity element 0 to the identity map and $1 \mapsto \theta : Z_n \rightarrow Z_n$ where $\theta(x) = -x$. I am not sure how to proceed further. Any help would ... | Perfectly good answer was already given by Galena Rupp, but I will try to give intuition why the given isomorphism really is a natural choice.
First of all, let us start with general semidirect product $N\rtimes_\psi H$. Instead of ordered pair $(n,h)$, I will simply write $nh$ (where concatenation is multiplication in... | {
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Who discovered the following formula for finding the next consecutive square? I know I'm not the first to discover this while being bored in seventh-grade math "learning" about perfect squares. The formula I discovered is this:
Sorry guys, I tried expressing it algebraically but I'm not very good with numbers, so I'll ... | I'm not sure who first discovered this, but notice this: $(n+1)^2=n^2+2n+1$. The $2n+1$ is an odd number which is being added to the square of $n$ to obtain the square of $n+1$. Just like you discovered yourself.
Using the method of induction, one may prove without any doubt that $n^2=1+3+5+\cdots+(2n-3)+(2n-1)$.
You... | {
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Probability of an event happening if at least one of 2 events happen So I'm doing a probability question, the probability of event A happening is $0.4$, event B is $0.7$. What is the probability of only event A happening given that at least one of the two events happen?
What I have tried so far is getting the probabili... | You want to find $P(A \cap \neg B|A \cup B)$.
$$P(A \cap \neg B|A \cup B) = \frac{P((A \cap \neg B) \cap (A \cup B))}{P(A \cup B)} = \frac{P(A \cap \neg B)}{P(A \cup B)} = \frac{P(A) P(\neg B)}{P(A \cup B)} = \frac{P(A) P(\neg B)}{1-P(\neg A)P(\neg B)} = \frac{0.4 \times 0.3}{1-0.6 \times 0.3} = \frac{0.12}{0.82} \appr... | {
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ODE for mixing problem
A jar of hot water is in a sink, and we are pouring water into the jar with a rate of $a$ gallons per minute. The water spills over the jar at a rate of $a$ gallons per minute after being thoroughly mixed. Write the ODE for $x(t),$ where the $x(t)$ is the temperature of water in the jar at time ... | My attempt.
Over a period of time $\Delta t$, a quantity $a \Delta t$ will flow to the jar, from the source at temperature $T_{source}$. The jar has volume $L$, and for simplicity I define $n = L / a \Delta t$.
Assuming that the updated temperature is just a (volume)-weighted average, the updated temperature $T_1$ wi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1946071",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Direct sums of invariant subspaces Let $A$ be a complex $n\times n$ matrix, with its Jordan carnonical form as $J=diag(J_1,\cdots,J_s)$. Then there exists an invertible matrix $P$ such that $P^{-1}AP=J$. It is easy to verify that $\Bbb C^n=V_1\oplus \cdots \oplus V_s$ with $V_i$ is the columns of $P$ correspoing to $J_... | This is false. Let $A$ be the $2\times 2$ identity matrix. Then $V_1 = span{\begin{pmatrix} 1 \\ 0 \end{pmatrix}}$, and $V_2 = span{\begin{pmatrix} 0 \\ 1 \end{pmatrix}}$ are the invariant subspaces corresponding to the Jordan blocks of $A$, and $\mathbb{C}^2 = V_1 \oplus V_2$.
Let $V = span{\begin{pmatrix} 1 \\ 1 \end... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1946186",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Why are (S)pinor representations *restrictions* of Clifford algebra representations? Firstly, just to clarify my notation:
Let $Cl(V,q)$ denote the Clifford Algebra of a quadratic vector space $(V,q)$ and denote by $Cl(V,q)_{0\vert 1}$ the even/odd part in the $\mathbb{Z}_2$-grading of $Cl(V,q) = Cl(V,q)_0 \oplus Cl(V,... | I suspect that there is a matter of terminology involved here.
When speaking about "spinors" we do not exactly mean the elements of the group $Spin(V,q)$ but rather the elements of special vector spaces upon which Clifford algebras act. In other words, the terminology spinors imply the elements of some specific (Clif... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1946332",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Prove the transitivity of $<$ relation in natural numbers using the classical definition and the associativity of addition In the book Notes on Mathematical Analysis the set $\mathbb{N}$ of natural numbers was defined using the classical Peano's axioms. Then $<$ was defined as if exist $k\in \mathbb{N}$ such that $b=S^... | Long comment
From : $a < b$ and $b < c$, we have :
$b=S^k(a)$ and $c=S^l(b)$.
By substitution : $c=S^l[S^k(a)]$.
Now we need addition; $S(a)=a+1$, form first axiom for addition and from the hypotheses : $S^n(a)=a+n$ we have : $S^{n+1}(a)=S[S^n(a)]=S[(a+n)]=(a+n)+1$.
By asociativity : $(a+n)+1=a+(n+1)$ and thus we con... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1946422",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Product of Levi-Civita symbol is determinant? I am confused with how can one write product of Levi-Civita symbol as determinant? I want to prove 'epsilon-delta' identity and found this questions answers it. But I am stuck at product of Levi-Civita symbol
Proof relation between Levi-Civita symbol and Kronecker deltas in... | This identity relates the product of the volumes spanned by two sets of three vectors (with a minus sign if the sets are oppositely oriented) in terms of the inner products of the individual vectors.
In particular, the determinant can be understood as computing the product of the volumes by projecting one set of vector... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1946493",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 2
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Power sets in set theory What is the set $\mathcal{P}(\mathcal{P}(\mathcal{P}( \emptyset )))$?
Would it be $\{\{\{ \emptyset\}\}\}$?
I understand that $\mathcal{P}(\{a,b,c\}) = \{ \{ \}, \{a\}, \{b\}, \{c\}, \{a,b\}, \{a,c\}, \{b,c\}, \{a,b,c\} \}$.
| For every set $A$ we have $A\subseteq A$ so that $A\in\wp(A)$. So also for $\varnothing$. If $B\neq\varnothing$ then some $b$ exists with $b\in B$. So in that case $B\subseteq\varnothing$ would lead to $b\in\varnothing$, wich is evidently not true. We conclude now that $\varnothing$ is the only subset of $\varnothing$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1946604",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Help with calculating $(3+5i)^6$ I want to calculate $(3+5i)^6$ with $i = \sqrt{-1}$, but somehow I'm doing something wrong.
How I calculate it:
Say $z = x + yi = 3+5i$, so $z^6$ has to be calculated.
$$|z| = \sqrt{3^2+5^2} = \sqrt{34}$$
$$\text{Arg}(z) = \arctan\left(\frac{y}{x}\right) = \arctan\left(\frac{5}{3}\right... | Generalize the problem, when $\text{z}\in\mathbb{C}$ and $\text{n}\in\mathbb{R}$:
$$\text{z}^\text{n}=\left(\Re\left[\text{z}\right]+\Im\left[\text{z}\right]i\right)^\text{n}=\left(\left|\text{z}\right|e^{\left(\arg\left(\text{z}\right)+2\pi k\right)i}\right)^\text{n}=\left|\text{z}\right|^\text{n}e^{\text{n}\left(\arg... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1946697",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Proof of 1 = 0, use of ill-formed statements In his book "Analysis 1", Terence Tao writes:
A logical argument should not contain any ill-formed
statements, thus for instance if an argument uses a statement such
as x/y = z, it needs to first ensure that y is not equal to zero.
Many purported proofs of “0=1” or ot... | Start with the assumption
$$x = 0$$
Divide both sides by $x$ to get
$$x/x=0/x$$
and thus
$$1=0$$
That's the general scheme. Of course it generally gets more obfuscated, for example by starting with the assumption $a+b=c$ and then later dividing both sides with $c-a-b$.
| {
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"url": "https://math.stackexchange.com/questions/1946824",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Transfinite Numbers in Set Theory Since my school doesn't teach higher-level things like set theory or transfinite numbers, I have been learning on my own. There are large gaps in my knowledge of higher-level math because of this.
Where would transfinite numbers fit in this sequence?
$$\mathbb S \supseteq \mathbb O \su... | They would fit nowhere. In fact, the "transfinite numbers", more commonly called ordinals, extend the natural numbers, but do not contain the negative integers. So, the closes thing we can say is that
$$
\mathbb S \supseteq \mathbb O \supseteq \mathbb H \supseteq \mathbb C \supseteq \mathbb R \supseteq \mathbb Q \sups... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1947079",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why does this derivative equation hold? $$\frac{d(Q/x)}{dx} = \frac{x(\frac{dQ}{dx})-Q(\frac{dx}{dx})}{x^2}$$
Assume $Q$ is a function of $x$. This equation is in my microeconomics textbook, but I don't know how we can get from the left-hand side to the right-hand side. Can someone please explain?
| It is known as the quotient rule. The more general derivative is given as:
$$\frac{d}{dx}\frac fg=\frac{f'g-fg'}{g^2}$$
Inputting $f=Q$ and $g=x$ gives
$$\frac{d}{dx}\frac Qx=\frac{Q'x-Qx'}{x^2}=\frac{x\left(\frac{dQ}{dx}\right)-Q\require{cancel}\cancel{\frac{dx}{dx}}}{x^2}$$
$$=\frac{x\left(\frac{dQ}{dx}\right)-Q}{x^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1947145",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What is $\bigcap_{n=1}^{\infty}[\frac{1}{3n}, 1+\frac{1}{n})$ What is $\bigcap_{n=1}^{\infty}[\frac{1}{3n}, 1+\frac{1}{n})$
So, just from thinking about it logically, I got $\bigcap_{n=1}^{\infty}[\frac{1}{3n}, 1+\frac{1}{n})=[\frac{1}{3}, 1)$.
However, I'm not sure about whether the bound on the right should be closed... | $1\in [1/3n, 1+1/n)$ for every $n\in \mathbb N.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1947314",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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For a compact set K, does $K\supset f(K)\supset f^{2}(K)\supset...$ hold for function f? Let $K$ be a compact normed space and $f:K\rightarrow K$ such that $$\|f(x)-f(y)\|<\|x-y\|\quad\quad\forall\,\, x, y\in K, x\neq y.$$
Is it true that $K\supset f(K)\supset f^{2}(K)\supset...$?
How to show it?
| If this were true, then all points of the space would be fixed: every set of the form $\{x\}$ is compact.
(and then the map would be the identity, which is impossible...)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1947427",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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How was this geometry problem created?
This is a standard High School Olympiad problem and for an experienced problem solver a quite easy solve. But how was this problem created. To pose a problem, I believe is much harder, than to solve a posed problem.
Here the problem poser had to first make the figure up and then... | Let me show you another solution ;)... Maybe this is how they came up with this problem. Manipulating midpoints, orthogonality and parallel lines.
Draw the perpendicular of edge $AC$ at point $A$ and let it intersect the line $BN$ at point $P$. Since $\angle \, CBP = 90^{\circ} = \angle \, CAP$, it follows that quadri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1947507",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 1
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Find positive integer $x,y$ such that $7^{x}-3^{y}=4$ Find all positive integers $x,y$ such that $7^{x}-3^{y}=4$.
It is the problem I think it can be solve using theory of congruency. But I can't process somebody please help me .
Thank you
| Let us go down the rabbit hole. Assume that there is a solution with $ x, y > 1 $, and rearrange to find
$$ 7(7^{x-1} - 1) = 3(3^{y-1} - 1) $$
Note that $ 7^{x-1} - 1 $ is divisible by $ 3 $ exactly once (since $ x > 1 $): the contradiction will arise from this.
Reducing modulo $ 7 $ we find that $ 3^{y-1} \equiv 1 $, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1947650",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Very general object-free categories? Is there a name for categories $\mathcal C$ such that $\text{Obj}(\mathcal C)$ coinside with $\text{Mor}(\mathcal C)$?
In the diagram below the morphisms are
$a\overset{a}{\to}a$, $a\overset{b}{\to}c$, $d\overset{c}{\to}c$,
$a\overset{d}{\to}e$, $e\overset{e}{\to}f$, $c\overset{f}... | Interpreted in a certain way what you are asking is standard. A category $C$ is a collection $C_1$ together with two maps $s,t:C_1\to C_1$ and a map $m$ from $C_1\times_{\langle s,t\rangle} C_1 =\{(f,g)\,|\,s(f)=t(g)\}$ to $C_1$ satisfying:
$st=t$, $ts=s$, $m(m(f,g),h))=m(f,m(g,h))$ and $m(f,s(f))=f=m(t(f),f)$.
Note t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1947747",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
All alternative solution for an equation I'm looking for all alternative solutions of this
$$x'=x(x-1)(x+1)$$
But I absolutely don't know what I have to do!
Thanks!
| Let we start by noticing that
$$\frac{1}{(z-1)z(z+1)} = \frac{1/2}{z-1}-\frac{1}{z}+\frac{1/2}{z+1}\tag{1}$$
by partial fraction decomposition/the residue theorem. It follows that the separable DE
$$ \frac{x'}{(x-1)x(x+1)} = 1 \tag{2}$$
leads to
$$ \frac{1}{2}\log(x-1)-\log(x)+\frac{1}{2}\log(x+1) = t+C \tag{3} $$
or t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1947843",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
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How can I know whether the point is a maximum or minimum without much calculation?
Find the maximum and minimum of this function and state whether they
are local or global:
$$f: \mathbb{R} \ni x \mapsto \frac{x}{x^{2}+x+1} \in \mathbb{R}$$
\begin{align*}
f'(x)&= \frac{-x^{2}+x}{\left(x^{2}+x+1\right)^{2}}\\
f'(x)&... | Your expression for the derivative is wrong, but I'll let you sort that out. What is important is that:
*
*$f(0)=0$;
*there are just two values of $x$ for which $f'(x)=0$, and one of them is positive, the other negative;
*$f(x)$ tends to zero as $x$ tends to $\pm\infty$;
*$f(x)$ has the same sign as $x$.
These ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1947917",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Is there a series approximation in terms of $n$ for the sum of the harmonic progression : $\sum_{k=0}^{n}\frac{1}{1+ak}$? When $a=1$ the sum is given by $ H_{n} $ and we have : $$H_{n}=log(n)+γ+\frac{1}{2n}-\frac{1}{12n^2}+\frac{1}{120n^4} \hspace{0.5cm}.\hspace{.1cm}.\hspace{.1cm}. $$ Does any representation of a simi... | As Felix Marin answered,$$\sum_{k = 1}^{n}{1 \over 1 + ak}=\frac{H_{n+\frac{1}{a}}-H_{\frac{1}{a}}}{a}$$ Now, using the asymptotics $$H_m=\gamma +\log \left({m}\right)+\frac{1}{2 m}-\frac{1}{12
m^2}+\frac{1}{120 m^4}+O\left(\frac{1}{m^5}\right)$$ $$H_{n+\frac{1}{a}}=\gamma +\log \left({n+\frac{1}{a}}\right)+\frac{1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1948139",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Find the values of $\alpha $ satisfying the equation(determinant)
Find the values of $\alpha $ satisfying the equation
$$\begin{vmatrix}
(1+\alpha)^2 & (1+2\alpha)^2 & (1+3\alpha)^2\\
(2+\alpha)^2& (2+2\alpha)^2 & (2+3\alpha)^2\\
(3+\alpha)^2& (3+2\alpha)^2 & (3+3\alpha)^2
\end{vmatrix}=-648\alpha $$
I used ... | Hint write it as a product of two determinants after taking $\alpha,\alpha^2$ common from one of the determinants to get $\alpha=\pm 9$ or to continue your method use $R_1\to R_1-R_2$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1948216",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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