Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Distance of the Focus of an Hyperbola to the X-Axis For a National Board Exam Review:
How far from the $x$-axis is the focus of the hyperbola $x^2 -2y^2 + 4x + 4y + 4$?
Answer is $2.73$
Simplify into Standard Form:
$$ \frac{ (y-1)^2 }{} - \frac{ (x+2)^2 }{-2} = 1$$
$$ a^2 = 1 $$
$$ b^2 = 2 $$
$$ c^2 = 5 $$
Hyperbola ... | $\frac{(x + 2)^2}{2} - \frac{(y - 1)^2}{1}; C = (-2, 1)$
$C = \sqrt{2 + 1} = \sqrt{3}$
Answer: $1 + \sqrt(3) = 2.73$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1400860",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Is there any example of a Category without generators? I was looking of an example of a category (a non trivial example) without generators (seperators). Is there any nice example? Or every category that we are using has generators?
| It depends on what you mean by "generators." There is a notion of a collection $S$ of objects being a "family of generators" of a category $C$, which means that the functors $\text{Hom}(s, -) : C \to \text{Set}, s \in S$ are jointly faithful. Any category has a (possibly large) family of generators given by taking ever... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1400936",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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What is a residue? I've heard of residues in complex analysis, contour integration, etc. but all I really know it to be is the $c_{-1}$ term in the Laurent series for a function. Is there some sort of intuition on what a "residue" actually is? The terminology makes it seem like something left over, or something like th... | The holomorphic functions have an extraordinary property: if you compute an integral along a path, the value of the integral does not depend on the path !
More precisely, if the function is holomorphic everywhere inside a closed path, the integral is just zero. But if the function has poles (zeroes at the denominator, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1400997",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
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Find a mistake type of math problems I am interested in the problems where the formulation of the problem has some kind of mistake in it and as a consequence gives unexpected answer.
Can't explain it better than this example:
For example: $4^2 = 4 \cdot 4 = 4+ 4 + 4+4$ (sum $4$ times)
Similarly: $$\frac{d}{dx}x^2=\frac... | Your equation does not make sense for many possible values of $x$. The equation
\begin{align}
x \cdot x &= \underbrace {x + x + \dots + x}_{x \text{ times}},
\end{align}
only holds when $x$ is a nonnegative integer. But $x$ is an indeterminant which can take on real values as well. So you can't write $x^2$ as the sum ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1401122",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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Finding the distance from the origin to the surface $xy^2 z^4 = 32$ using the method of Lagrange Multipliers Problem: Find the distance from the origin to the surface $xy^2z^4 = 32$.
Attempt: The Lagrange equation for this problem is $L(x,y,z, \lambda) = x^2 + y^2 + z^2 + \lambda (xy^2 z^4 - 32)$. Setting the first par... | A hint:
Multiply $(1)$ by $x$, $(2)$ by $y$, and $(3)$ by $z$ and look at the three equations you got.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1401261",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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Cutting a paper with the smallest number of cuts You want to cut a piece of paper of length $N$ to $N$ pieces of length 1.
It is not allowed to fold the paper, but if two or more previously-cut pieces of paper have the same length, it is allowed to put them one over the other.
If $N=2$ you need 1 cut.
If $N=3$ you nee... | Your problem is equivalent to that of finding the shortest addition chain for $N$. You can order the operations by descending lengths of the cut pieces, so you never need to cut the same length twice, so what you're doing is to prescribe for each length that occurs how to compose it from two smaller lengths, which is e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1401355",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Pretty easy equations of elements in a group Problem
$G$ is a group generated by $a,b\in G$ such that $a^5=e$, $aba^{-1}=b^2$ and $b\ne e$. I want to find the order of $b$.
Attempt
I tried to multiply the second equation from right by $a^{4}$: $$ba^{-1}=a^{4}b^{2}$$ Then $$ba^{-1}=(ba^{2})^{2}=ba^{2}ba^{2}$$
Then by mu... | Note that for every $m$ we have
$$ab^ma^{-1} = (aba^{-1})^m = b^{2m}.$$
Thus by induction we have
$$a^kba^{-k} = b^{2^k}\tag{$\ast$}$$
for all $k\in\mathbb{N}$. On the other hand, we have $a^5ba^{-5} = b$ since $a^5 = e$. Putting that and $(\ast)$ together reveals the order of $b$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1401457",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that $(a+b)(b+c)(c+a) \ge8$
Given that $a,b,c \in \mathbb{R}^{+}$ and $abc(a+b+c)=3$,
Prove that $(a+b)(b+c)(c+a)\ge8$.
My attempt: By AM-GM inequality, we have
$$\frac{a+b}{2}\ge\sqrt {ab} \tag{1}$$
and similarly
$$\frac{b+c}{2}\ge\sqrt {bc} \tag{2},$$
$$\frac{c+d}{2}\ge\sqrt {cd} \tag{3}.$$
Multiplying $(1)... | From $abc(a+b+c)=3$ we have $$a^2+a(b+c)=\frac {3}{bc}.$$ Therefore we have $$(a+b)(b+c)(c+a)\geq 8\iff$$ $$(a+b)(a+c)\geq \frac {8}{b+c}\iff$$ $$a^2+a(b+c)\geq -bc+ \frac {8}{b+c)}\iff$$ $$(\bullet ) \quad \frac {3}{bc}\geq -bc+\frac {8}{b+c}.$$ Now $b+c\geq 2\sqrt {bc},$ so $$\frac {8}{b+c}\leq \frac {4}{\sqrt {bc}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1401650",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 3
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Tangent planes to $2+x^2+y^2$ and that contains the $x$ axis I need to find the tangent planes to $f(x,y) = 2+x^2+y^2$ and that contains the $x$ axis, so that's what I did:
$$z = z_0 + \frac{\partial f(x_0,y_0)}{\partial x}(x-x_0)+\frac{\partial f(x_0,y_0)}{\partial y}(y-y_0) \implies \\ z = 2 + x_0^2 + y_0^2 + 2x_0(x-... | Taking advantage of the fact that the surface is a quadric, here’s a way to solve this without calculus. Working in homogeneous coordinates, all of the planes that contain the $x$-axis are of the form $[0:\lambda:\mu:0]$, for $\lambda$ and $\mu$ not both zero. These planes are tangent to the given surface iff they sati... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1401716",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Integrating unit impulse function Given that,
$$ \delta(t) = \begin{cases} \infty & \text{if } t = 0 \\
0 & \text{if } t \ne 0\\
\end{cases}$$
How is it that,
(A)
$$
\int_{-\infty}^\infty \delta(t) dt = 1
$$
(B)
$$
\int_{-\infty}^\infty f(t) \delta(t) dt = f(0)
$$
considering $f$ continuos at $t=0$
Thanks in advan... | Can I assume you're a physicist and avoid the mathematical complexities a bit?
The integral finds the area under functions. $\delta(x)$ is a really skinny and really tall rectangle. In fact, it's width is $\epsilon$ and its height is $\omega$ so the area is given by $ \epsilon \cdot \omega$. The variable $\epsilon={1 \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1401796",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Fibonacci proof question $\sum_{i=1}^nF_i = F_{n+2} - 1$ The sequence of numbers $F_n$ for $n \in N$ defined below are called the Fibonnaci numbers.
$F_1 = F_2 = 1$, and for $n \geq 2$, $F_{n+1} = F_n + F_{n-1}$.
Prove the following facts about the Fibonnaci numbers
For each of the following, $n \in N$
a) $\displayst... | Combinatorial Proof:
For each $n\in\mathbb{N}_0$, the number of ways to tile an $1$-by-$n$ array with $1$-by-$1$ squares and $1$-by-$2$ dominos is $F_{n+1}$. Hence, the number of ways to tile an $1$-by-$(n+1)$ array in such a manner with at least one domino is given by $F_{n+2}-1$.
Now, let $k$ be the earliest positi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1401868",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 3
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Is this linearly independent? What is the dimension span?
Consider the vectors in $\mathbb{R}^4$ defined by
$v_1 = (1,2,10,5), v_2 = (0,1,1,1), v_3 = (1,4,12,7)$.
Find the reduced row echelon form of the matrix which has these as its rows. What is its rank? Is $\{v_1 , v_2 , v_3 \}$ linearly independent? What is t... | The vectors are linearly dependent since $v_3=v_1+2v_2$. As you noted, you have the two leading ones, and it is clear that the vectors span a space with dimension two, the say way that $Span\{\vec i,\vec j\}$ spans two dimensions in $\Bbb R^3$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1402006",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Euler phi function and powers of two For small values of n$\ge$1 it appears that one has the inequality
$\phi(2^n)$ $\le$ $\phi(2^n + 1)$.
However, it seems unlikely that this would hold for all n .
Question: Are there any explicit values of n known for
which the inequality is violated ?
Thanks
| For each prime factor $p|2^n+1$ you get a reduction of $\frac {p-1}P$ in the $\varphi$ function. The $p$ you are looking for are ones for which $2^n\equiv -1 \bmod p$. Then for odd $k$, $2^{kn}\equiv -1 \bmod p$. So if we work with odd exponents we can collect primes - the question is can we collect enough small primes... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1402086",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Convergence and Limit of a Sequence I am looking at the sequence and I'm trying to see what happens as $n\to\infty$
$$
a_n = \frac{\prod_{1}^{n}(2n-1)}{(2n)^n} = \frac{1\cdot3\cdot5\cdot...\cdot(2n-1)}{(2n)^n}
$$
By inspection, I can see that the denominator increases faster than numerator which indicates that the sequ... | We have
$$
\frac{\prod_{1}^{n}(2k-1)}{(2n)^{n}} < \frac{2n-1}{(2n)^{2}} \to 0
$$
as $n$ grows.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1402185",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show that if $|G| = 30$, then $G$ has normal 3-Sylow and 5-Sylow subgroups.
Show that if $|G| = 30$, then $G$ has normal $3$-Sylow and $5$-Sylow subgroups.
Let $n_3$ denote the number of 3-Sylow subgroups and $n_5$ the number of $5$-Sylow subgroups. Then, by the third Sylow theorem, $n_3$ divides $10$ and $n_3 \equiv... | This is more delicate and difficult than most problems about Sylow subgroups, and it must be tackled by relating the 5-sylow subgroups and the 3-sylow subgroups. In general such a problem in group theory at the undergraduate level is either going to be trivially easy, or require this technique.
As you established $n_3 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1402245",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that $(2n+1)k_{n+1}=(2n+1)k_{n}+\cos^{2n+1} (x)$ Given that $$k_n=\int \frac{\cos^{2n} (x)}{\sin (x)} dx$$ Prove that $$(2n+1)k_{n+1}=(2n+1)k_{n}+\cos^{2n+1} (x)$$
I have tried to prove this is true by differentiating both sides with product rule:
$$2k_{n+1}+\frac{\cos^{2n+1} (x)}{\sin (x)}(2n+1)=2k_n+\frac{\cos^... | Verifying by differentiation is a good idea. We wish to prove that
$$k_{n+1}=k_n+\frac{1}{2n+1}\cos^{2n+1}(x).$$
Recall that indefinite integrals are only determined up to a constant. So we really need to show that the derivative of the left-hand side is equal to the derivative of the right-hand side. By the Fundamenta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1402317",
"timestamp": "2023-03-29T00:00:00",
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Of the three lines $x+\sqrt3y=0,x+y=1$ and $x-\sqrt3y=0$,two are equations of two altitudes of an equilateral triangle Of the three lines $x+\sqrt3y=0,x+y=1$ and $x-\sqrt3y=0$,two are equations of two altitudes of an equilateral triangle.The centroid of the equilateral triangle is
$(A)(0,0)\hspace{1cm}(B)\left(\frac{\s... | Notice, we have the following equations of the lines $$x+\sqrt 3y=0\iff y=\frac{-1}{\sqrt 3}x\tag 1$$
$$x+y=1\iff y=-x+1\tag 2$$
$$x-\sqrt 3y=0\iff y=\frac{1}{\sqrt 3}x\tag 3$$
Let the slopes of the above lines be denoted by $m_1=-\frac{1}{\sqrt 3}$, $m_2=-1$ & $m_3=\frac{1}{\sqrt 3}$ then the angles between them are c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1402436",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Finding range of $m$ in $x^2+mx+6$. Find the range of values of $m$ in the quadratic equation $x^2+mx+6=0$ such that both the roots of the equation $\alpha,\beta<1$.
My attempt - it is given that
$\alpha<1$ and $\beta<1$
$\rightarrow \alpha+\beta<2$
But $\alpha+\beta=-m$
Thus $m>-2$.
But this solution doesn't involve t... | You made a logical fallacy as old as time.
What you have proven is:
If $\alpha, \beta<1$, then $m>-2$
What you HAVE NOT PROVEN is:
If $m>-2$, then $\alpha, \beta < 1$.
You can easily see that the second statement is false, since if $m=0>-2$, the equation has no real solutions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1402501",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 0
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Decreasing sequence of sets: Power of natural numbers Let $P(N)$ be the set of all the possible subsets of natural numbers (power set of $N$). Suppose that we have a decreasing sequence of sets $S_n$, ie $S_{n+1} \subseteq S_n\;,\in P(N)$ such that they are all finite and a set $M$ such that $\#M \leq \#S_n$, for all $... | Let $S$ denote the intersection of the $S_n$. Start with a fixed $m$.
The set $S_m-S$ is finite and for every element $s\in S_m-S$ some $n_s$ exists with $s\notin S_{n_s}$.
Then $S_n=S$ if $$n\geq\max\{n_s|s\in S_m-S\}\in\mathbb N$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1402573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove $(1+ \tan{A}\tan{2A})\sin{2A} = \tan{2A}$. Prove the following statement
$$ (1+ \tan{A}\tan{2A})\sin{2A} = \tan{2A}. $$
On the left hand side I have put the value of $\tan{2A}$ and have then taken the LCM.
I got $\sin{2A}\cos{2A}$. How do I proceed?
Thanks
| Noting that
$$\cot \theta\tan\theta=1$$
We have
$$(1+\tan A\tan 2A)\sin 2A = (1+\tan A\tan 2A)\sin 2A\cot 2A\tan 2A$$
Now just show that
$$(1+\tan A\tan 2A)\sin 2A\cot 2A = 1$$
:)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1402684",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 4
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Multiple choice: $S = {x | 0 ≤ x < 280 ∧ x ≡ 3 (mod 7) ∧ x ≡ 4 (mod 8)}$ The question is:
Consider the following set of integers:
$$ S = \left\{x \left| 0 \le x < 280
∧ x \equiv 3 \mod 7
∧ x \equiv 4 \mod 8 \right. \right\}. $$
How many integers are there in S?
$0$? $1$? $2... | $x \equiv 3 \pmod{7}$ is equivalent to $x+4 \equiv 0 \pmod{7}$.
$x \equiv 4 \pmod{8}$ is equivalent to $x+4 \equiv 0 \pmod{8}$.
Thus we want to know which numbers $x$ are such that $x+4$ is divisible by both $7$ and $8$. Since $7$ and $8$ are coprime, it would be exactly those numbers such that $x+4$ is divisible by $7... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Completeness of Normed spaces. I want to prove the following proposition
If $(X,||\cdot||)$ and $(X,||\cdot||')$ are homeomorphic, then $(X,||\cdot||)$ is complete if and only if $(X,||\cdot||')$ is complete.
So, I only know the definition of homeomorphic, and can't figure out how only with that I can prove that propos... | Take a Cauchy sequence in $(X,||\cdot||)$ and show that is also a Cauchy sequence in $(X,||\cdot||')$.
This proves the assertion since convergence is a topological property which is preserved by homeomorphisms.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1402833",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
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probabilities of arrival time John and Mary arrive under the clock tower independently. Let X be John's arrival time and let Y be Mary's arrival time. If John arrives first and Mary is not there then he will leave. If Mary arrives first then she will wait up to one hour before leaving. John's arrival time, X, is expone... | The probability that Mary arrives first, and within an hour of John can be calculated as follows.
$$P[Y<X<Y+1] = \int_{0}^{3}\int_{0}^{y} \frac{2ye^{-x}}{9} = .822$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1402896",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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limit with greatest integer function How can I evaluate this limit?
$$\lim_{x\to-7} \frac{[x]^2+15[x]+56}{\sin(x+7)\sin(x+8)}$$ where $[x]$ denotes the greatest integer less than or equal to $x$
I could easily factor the numerator. But I cannot apply any of the standard limits due to $[x]$.
| The way the question is phrased makes me wonder if you've understood the question, either the nature of limits or the nature of the greatest-integer function. Your final sentence makes me suspect the latter.
If $x<-7$ then $\lfloor x\rfloor=-8$.
If $x>-7$ then $\lfloor x\rfloor= -7$.
It you need that explained to you ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1402962",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Is the following solvable for x? I have the following equation and I was wondering if I can solve for x given that it appears both as an exponent and a base:
$[\frac{1}{\sqrt {2\pi}.S}.e^{-\frac{(x-M)^2}{2S^2}}-0.5\frac{1}{\sqrt {2\pi}.S}.e^{-\frac{(x-M)^2}{2S^2}}.\frac{4.(\frac{x-N}{\sqrt {2}.D})}{\sqrt {\pi}.e^{-\fra... | Let's simplify the equation a bit so we can better see its form:
$$C_{1}e^{X_{1}} + C_{2}e^{X_{1}}\cdot \frac{X_{2}}{C_{3}e^{X_{3}} \sqrt{C_{4}e^{X_{3}} + C_{5}X_{3}}} + C_{6}e^{X_{4}} + C_{7}e^{X_{4}}\cdot \frac{X_{5}}{C_{8}e^{X_{6}} \sqrt{C_{9}e^{X_{6}} + C_{10}X_{6}}} = C_{11}$$
With constants as $C_{n}$ above and e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1403039",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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finding all solutions of an equation using counting lets say I have
$$ x_1 + x_2 + x_3 + x_4 = 17 $$
What are all the solutions for $$ x_i \ge 0? \quad \text{ where } i=1,2,3,4$$
How about if $$ x_i \ge 0 \text{ ?} $$ And if $$ x_i \gt 1 \quad\text{ where } i =1,2,3,4$$
I tried following the example of the book. For t... | Well this is a classic stars and bars problem. If $x_i \ge 0$ then you have:
$$\binom{17 + 4 - 1}{4-1} = \binom{20}{3} = 1140 \text{ solutions}$$
For the second problem write $x_i = 2 + k_i$, then the equation is reduced to $k_1 + k_2 + k_3 + k_4 = 9$, which can be easily solved simularly to the first problem.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1403106",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Angle of elevation and distance to an object A person walking a straight slope sees (from ground level) an Object across the valley at an angle of $45^{\circ}$, after another 50 meters walking the angle is $60^{\circ}$, How far away is the Object at that moment?
With this much given info I do not know how to approach s... | Hint...you need to draw a picture and use trigonometry
If $h$ is the height of the object and $x$ is the distance you want, you have$$\frac hx=\tan60$$ and $$\frac{h}{x+50}=\tan45$$
Eliminate $h$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1403167",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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} |
Area of quadrilateral: $S \leq \frac{(a+b)(c+d)}4 $ I got stuck on this problem:
Given a convex quadrilateral of area $S$ and sides $a$, $b$, $c$ and $d$, prove that: $$S \leq \frac{(a+b)(c+d)}4$$
What I've done so far was to proof that $$S \leq \frac{(a+c)(b+d)}4$$ using the relation that for a given triangle $ABC$,... | Let $s:=\frac{a+b+c+d}{2}$. By Bretschneider's Formula,
$$S=\sqrt{(s-a)(s-b)(s-c)(s-d)-abcd\cos^2\left(\frac{\alpha+\gamma}{2}\right)}\,,$$
where $\alpha$ and $\gamma$ are two opposite angles of the given convex quadrilateral. Hence,
$$S\leq \sqrt{(s-a)(s-b)(s-c)(s-d)}\;\leq \left(\frac{(s-a)+(s-b)}{2}\right)\left(\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1403262",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Decide if each is a basis for $P_2$. (a) $(x^2 + x - 1, 2x + 1, 2x - 1)$ I'm using Linear Algebra by Jim Hefferon (freely available, links below with solution).
I'm having trouble understanding Exercise 1.18 on page 117.
1.18 Decide if each is a basis for $P_2$.
(a) $(x^2 + x - 1, 2x + 1, 2x - 1)$
First, I try to prove... | First, set up the matrix so that every row is a vector. eg,
$$
\begin{bmatrix}x^2 & x & -1 \\0 & 2x & 1 \\ 0 & 2x & -1 \end{bmatrix}
$$
if you want to, you can simply represent the variables in a "normalized" basis by simply deciding that $\bar b_1 = x^2$. You can now use a "change of basis" from $\bar b_n$ to $\bar e_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1403345",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
$\int_{0}^{\pi}|\sqrt2\sin x+2\cos x|dx$ $\int_{0}^{\pi}|\sqrt2\sin x+2\cos x|dx$
MyAttempt
$\int_{0}^{\pi}|\sqrt2\sin x+2\cos x|=\int_{0}^{\pi/2}\sqrt2\sin x+2\cos x dx+\int_{\pi/2}^{\pi}|\sqrt2\sin x+2\cos x| dx$
I could solve first integral but in second one,i could not judge the mod will take plus sign or minus sig... | $\displaystyle A \sin \ x + B \cos \ x = \sqrt{A^2 + B^2}\sin( x + \phi )$, where $\displaystyle \cos \phi = \frac{A}{\sqrt{A^2 + B^2}}$.
In our case $\displaystyle \cos\phi=\frac{1}{\sqrt{3}} \Rightarrow \phi\in\left(0;\frac{\pi}{2}\right)$.
Thus: $\displaystyle \begin{aligned}\int\limits_0^\pi\left|\sqrt{2}\sin x+2\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1403526",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 4
} |
$\int_{0}^{1}\frac{\sin^{-1}\sqrt x}{x^2-x+1}dx$ $\int_{0}^{1}\frac{\sin^{-1}\sqrt x}{x^2-x+1}dx$
Put $x=\sin^2\theta,dx=2\sin \theta \cos \theta d\theta$
$\int_{0}^{\pi/2}\frac{\theta.2\sin \theta \cos \theta d\theta}{\sin^4\theta-\sin^2\theta+1}$ but this seems not integrable.Is this a wrong way?Is there a different ... | Let, $$I=\int_{0}^{1}\frac{\sin^{-1}(\sqrt x)}{x^2-x+1}dx\tag 1$$
Now, using property of definite integral $\int_{a}^{b}f(x)dx=\int_{a}^{b}f(a+b-x)dx$ we get
$$I=\int_{0}^{1}\frac{\sin^{-1}(\sqrt {1-x})}{(1-x)^2-(1-x)+1}dx$$
$$I=\int_{0}^{1}\frac{\sin^{-1}(\sqrt {1-x})}{x^2-x+1}dx\tag 2$$ Now, adding (1) & (2), we get... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1403619",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 2
} |
$\int_{0}^{1}\frac{1-x}{1+x}.\frac{dx}{\sqrt{x+x^2+x^3}}$ $$\int_{0}^{1}\frac{1-x}{1+x}.\frac{dx}{\sqrt{x+x^2+x^3}}$$
My Attempt:
$$\int_{0}^{1}\frac{1-x}{1+x}.\frac{dx}{\sqrt x\sqrt{1+x(1+x)}}$$
Replacing $x$ by $1-x$,we get
$$\int_{0}^{1}\frac{x}{2-x}.\frac{dx}{\sqrt{1-x}\sqrt{1+(1-x)(2-x)}}$$
Then I got stuck. Pleas... | The integral can also be found using a self-similar substitution of $u = \dfrac{1 - x}{1 + x}$.
Here we see that $x = \dfrac{1 - u}{1 + u}$ such that $dx = -\dfrac{2}{(1 + u)^2} \, du$.
Writing the integral as
$$I = -\int \frac{1 - x}{(1 + x) x \sqrt{x + 1 + \frac{1}{x}}} \, dx,$$
if we observe that
$$x + 1 + \frac{1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1403717",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
is this inner product positive-definite? $$\left \langle u, v \right \rangle = pu_{1}v_{1}+qu_{1}v_{2}+qu_{2}v_{1}+pu_{2}v_{2}\\\text{ for }\\ \text{p >0} \text{ and } p^{2}\geq q^{2}$$
The solution breaks down $$\left \langle u, u \right \rangle = pu_{1}u_{1}+qu_{1}u_{2}+qu_{2}u_{1}+pu_{2}u_{2} $$ into
$$p(u_{1}+\frac... | $p(u_1+\frac{q}{p} u_2)^2$ is nonnegative because it is the product of $p$ (which is strictly positive) with a square.
$(p-\frac{q^2}{p}) u_2^2$ is also nonnegative, being $\frac{1}{p} (p^2 - q^2) u_2^2$, and we're given that $p>0$ and $p^2 \geq q^2$.
Hence the sum is nonnegative.
When is it zero? Consider the second t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1403784",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove there exists a unique local inverse. I've been set this problem recently and I'm having a lot of trouble with it. Any help would be much appreciated!
Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function with continuous derivatives of all orders and suppose that, for some $x\in \mathbb{R}$ the derivative $f'(x... | $f'$ is nonzero on $D$ so $f$ is strictly monotonic. Thus it is one-to-one. This means that $f:D\to f[D]$ has an inverse $g:f[D]\to D$. $g$ is diffentiable with $g'(x)=\frac{1}{f'(g(x))}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1403896",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
$\int\frac{1+x^2}{x^4+3x^3+3x^2-3x+1}dx$ $$\int\frac{1+x^2}{x^4+3x^3+3x^2-3x+1}dx$$
I tried to solve it.
$$\int\frac{1+x^2}{x^4+3x^3+3x^2-3x+1}dx=\int\frac{1+x^2}{(x^2+1)^2+3x^3+x^2-3x+1}dx=\int\frac{1+x^2}{(x^2+1)^2+3x(x^2-1)+x^2+1}dx$$
But this does not seem to be solving.Please help.
| Please note that you have: $\int\frac{1+x^2}{x^4+3x^3+3x^2-3x+1}dx$, which on dividing numerator and denominator by $x^2$ becomes:
$\int\frac{1+1/x^2}{(x^2+1/x^2)+3+3(x-1/x)}dx$=$\int\frac{1+1/x^2}{(x-1/x)^2+5+3(x-1/x)}dx$
Now put x-1/x =t so that (1+1/$x^2$)dx=dt and thus you get:
$\int\frac{1}{t^2+5+3t}dt$;which can ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1403985",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 2
} |
Norm of a Block Matrix Let $X\in M_{m,n}(R)$ and $l=m+n$. Now consider the block matrix
$$
Y=\left[
\begin{array}{cc}
0 &X \\ \
X^T &0
\end{array}\right]
$$
where $Y\in M_l(R)$.
I want to show that $||X||=||Y||$ where for any $A\in M_{m,n}(R)$,
we define $||A||=max\{||Ax||: x\in R^n,||x||=1\}$ and $R^n$ has the stand... | Let $u\in\mathbb R^m$, $v\in\mathbb R^n$, and $w = (u,v)$. Then, we have
\begin{align}
\|Yw\| &= \left\| \begin{bmatrix} 0 & X \\ X^T & 0 \end{bmatrix} \begin{bmatrix} u \\ v \end{bmatrix} \right\| \\
&= \left\| \begin{bmatrix} Xv \\ X^T u \end{bmatrix} \right\| \\
&= \sqrt{ \|Xv\|^2 + \|X^T u\|^2 } \\
&\le \|X\| \sqr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1404043",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Why does Abel's identity imply either $W = 0$ or $W \neq 0$ everywhere? Let $y_1$ and $y_2$ be solutions to the linear differential equation $A(x)y'' + B(x)y' + C(x)y = 0 $ and let $W = W(y_1, y_2)$ be the Wronskian of the solutions. Why does Abel's identity
$\displaystyle W(y_1, y_2)(x) = W(y_1, y_2)(x_0)\cdot exp\le... | Suppose that $W$ were ever zero. Since the $\exp$ term is never zero, that means the $W(y_1, y_2)(x_0)$ term must be zero. But that means that $W$ is zero everywhere.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1404136",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Anti-symmetric if $AB= 1$ and $BA=0$ but every vertex has loops? I'm creating a directed graph from an adjacency list. The $0$ present that there is no relation while the $1$ represent that there is.
So i have a quick question regarding this.
Lets assume that $AB = 1$ that is that it has a connection.
$BA = 0$ which me... | Yes. A relation $\mathrel{R}$ is antisymmetric if
$$x\mathrel{R}y\quad\text{and}\quad y\mathrel{R}x\quad\text{implies}\quad x=y\;.$$
This means that if $x\ne y$, you can’t have both $x\mathrel{R}y$ and $y\mathrel{R}x$: you can have at most one of them. It says nothing at all about what you can (or must) have when $x=y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1404258",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Are there reasons not to use product of vectors as dot product? The dot product of two vectors is defined as following:
$$
\langle \vec v, \vec u \rangle =
\left<
\begin{pmatrix}
v_1 \\
v_2 \\
\dots \\
v_n
\end{pmatrix},
\begin{pmatrix}
u_1 \\
u_2 \\
\dots \\
u_n
\end{pmatrix}
\right>
= v_1 \cdot u_1 + v_2 \cdot u_2 +... | Strictly speaking, a row vector represents a linear form, i.e. a lineap map from $\mathbf R^n\to\mathbf R$. So they're different in essence.
However, to the vector $\vec v$, you can associate the linear map
\begin{align*}\varphi_{\vec v}\colon\mathbf R^n&\longrightarrow\mathbf R\\\vec u&\longmapsto \langle\vec v,\vec... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1404348",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Show that $\displaystyle{\int_{0}^{\infty}\!\frac{x^{a}}{x(x+1)}~\mathrm{d}x=\frac{\pi}{\sin(\pi a)}}$ Show that for $0<a<1$
$$\int_{0}^{\infty}\frac{x^{a}}{x(x+1)}~\mathrm{d}x=\frac{\pi}{\sin(\pi a)}$$
I want to solve this question by using complex analysis tools but I even don't know how to start. Any help would be g... | I know it's not technically a complex analysis route, but where would we be without the obligatory beta function route?
Rewrite
$$\begin{align}\int_0^{\infty} \frac{x^a}{x (x+1)} &= \int_0^{\infty} \frac{x^{a-1}}{x+1}\\ &= \int_1^{\infty} dx \frac{(x-1)^{a-1}}{x} \\ &= \int_0^1 \frac{dy}{y} \left (\frac1{y}-1 \right )... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1404434",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Show that if $\{X_n\}$ is a Markov Chain
Show that, if $\{X_n\}$ is a Markov Chain then
$$P(X_n=j\mid X_k=l,X_m=i)=P(X_n=j\mid X_m=i),0\leq k<m<n$$
What I did is
$$P(X_n=j\mid X_k=l,X_m=i)=\frac{P(X_n=j,X_k=l,X_m=i)}{P(X_k=l,X_m=i)}=\frac{\sum_{i_{n-1},\dots,i_0}P(X_n=j,X_{n-1}=i_{n-1},\dots,X_{m+1}=i_{m+1},X_m=i,\... | Yes. That's it.
A Markov chain is a sequence of random variables $\{X_k\}_{k\in\{0..n\}}$ representing $n+1$ subsequent states of a system, such that for all supported values $\{i_k\}_{k\in\{0..b\}}$, and $i_c$, where $0\leq a< b< c\leq n$ we have:
$$\mathsf P(X_c=i_c\mid \bigcap_{k\in\{0..b\}} X_k=i_k)= \mathsf P(X_c=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1404529",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How many subgroups are there in an elementary-$p$ group $C_p\times C_p$ has $1+1+(p-1)=p+1$ subgroups of order $p$ (all of which $\cong C_p$), so it has $1+(1+p)+1=3+p$ subgroups totally.
But how many subgroups in $C_p\times C_p \times C_p\times C_p$?
(its seems hard to count subgroups of order $p^2$ and order $p^3$)
F... | Subgroups of elementary abelian groups are also elementary abelian, so the problem in that context reduces to one of vector spaces: given a vector space $V$ of dimension $n$ over a finite field $\Bbb F_q$ (for us our $q$ is prime, but the question's difficulty is unchanged by generalizing), how many subspaces of dimens... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1404598",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Prove that the length of the common chord is $\frac{2ab\sin \theta}{\sqrt{a^2+b^2+2ab\cos \theta}}$ Two circles ,of radii $a$ and $b$,cut each other at an angle $\theta.$Prove that the length of the common chord is $\frac{2ab\sin \theta}{\sqrt{a^2+b^2+2ab\cos \theta}}$
Let the center of two circles be $O$ and $O'$ and ... | Let $A$ & $B$ be the centers of the circles with radii $a$ & $b$ respectively such that they have a common chord $MN=2x$ & intersecting each other at an angle $\theta$.
Let $O$ be the mid-point of common chord $MN$. Point $O$ lies on the line AB joining the centers of circle then we have $$MO=ON=x$$
In right $\triangl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1404695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
A sequence of numbers prime to $4n$ with $n$ being odd Let $n$ be odd. If I consider the sequence of $n$ numbers in the form $4k-1$ with $k$ running from $1$ to $n$ and take those with greatest common divisor with $4n$ being $1$ ( means those being prime to $4n$ ) I get exactly $\phi(n)$ of them.
I miss now a proof/arg... | The function $f_4: \mathbb{Z}_{n}\rightarrow \mathbb{Z}_{n},\;f_4(k) \equiv 4k-1 \pmod n$ is a bijection because $\gcd(4,n)=1$. The same applies if you substitute $4$ with any number $m$ with $\gcd(n,m)=1,\,$ and the additive constant $1$ can be replaced be any number.
Check this e.g. for $n=15, m=7, f(k) = 7k+4$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1404795",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to find the MacLaurin series of $\frac{1}{1+e^x}$ Mathcad software gives me the answer as: $$ \frac{1}{1+e^x} = \frac{1}{2} -\frac{x}{4} +\frac{x^3}{48} -\frac{x^5}{480} +\cdots$$
I have no idea how it found that and i don't understand. What i did is expand $(1+e^x)^{-1} $ as in the binomial MacLaurin expansion. I ... | $\begin{array}\\
f(x)
&=\frac{1}{1+e^x}\\
&=\frac{1}{2+(e^x-1)}\\
&=\frac12\frac{1}{1+(e^x-1)/2}\\
&=\frac12\sum_{n=0}^{\infty}(-1)^n\frac{(e^x-1)^n}{2^n}\\
&=\frac12\sum_{n=0}^{\infty}(-1)^n\frac{(x+x^2/2+x^3/6+...)^n}{2^n}\\
&=\frac12\sum_{n=0}^{\infty}(-1)^n(x/2)^n(1+x/2+x^2/6+...)^n\\
&=\frac12\sum_{n=0}^{\infty}(-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1404886",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 4
} |
A questions on the groups by a copy of $\Bbb Z$ Let $G$ be an abelian group and $H$ a subgroup of $G$ such that $G/H$ contains a copy of $\Bbb Z$. Is this true that $G$ contains a copy of $\Bbb Z$? ($\Bbb Z$ is the group of integer numbers)
| Another viewpoint using a bit simpler terms. Containing a copy of $\mathbb Z$ means that it has an element of infinite order, so we have $o(g+H)=\infty$ for some $g\in G$. Now, under the natural projection map, $o(\phi (g))|o(g)$so $\infty |o(g)$, hence $o(g)=\infty$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1404994",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Distribution of sum of 2 circular uniform random variables I hope you can help me resolve the following problem:
Let $\Phi_1$ and $\Phi_2$ circular uniform random variables such that $0\leq\Phi_i\leq 2\pi$ (with $i=1,2$). Then the probability density function (pdf) and the cumulative distribution function (cdf) are giv... | The immediate problem I see is the expression of the conditional probability in the integrand in the first step. To help you understand why it is not correct, consider the case where $\Phi_1 = 3\pi/2$, $\Phi_2 = \pi$. Then their sum is $$\Phi = \Phi_1 + \Phi_2 = 5\pi/2 \equiv \pi/2$$ where in the last step we have to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1405114",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
$\Gamma$ function is continuous on $]0,\infty [$ I would like to show that $\Gamma$ is continuous on $]0,\infty [$.
Let $x>0$,
$$|\Gamma(x+h)-\Gamma(x)|\leq\int_0^\infty t^{x-1}e^{-t}|t^h-1|dt=\underbrace{\int_0^1 t^{x-1}e^{-t}\underbrace{|t^h-1|}_{\leq |h|}dt}_{\underset{h\to 0}{\longrightarrow0 }}+\int_1^\infty t^{x-... | 1: Yes, your proof is correct:
2: For dominated convergence, you need a dominating function. Consider the function
$$
g(x) = \max\{t^{x + \delta}e^t,t^{x - \delta}e^t\}
$$
Where we can choose any $\delta \in (0, x)$. Once we know that this function is integrable, dominated convergence applies.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1405199",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Integrating $\int \sqrt{x+\sqrt{x^2+1}}\,\mathrm{d}x$ Integrating $$\int \sqrt{x+\sqrt{x^2+1}}\,\mathrm{d}x$$
Using substitution of $x=\tan \theta$, I got the required answer. But is there a more elegant solution to the problem?
| HINT
$$
\sqrt{ x + \sqrt{ x^2 + 1 } }
= \sqrt{ \frac{ x + i }{ 2 } } + \sqrt{ \frac{ x - i }{ 2 } }
$$
That would be enough simple to solve the integral...
We get
$$
\begin{eqnarray}
\int \sqrt{ x + \sqrt{ x^2 + 1 } } dx
&=& \int \left\{ \sqrt{ \frac{ x + i }{ 2 } } + \sqrt{ \frac{ x - i }{ 2 } }
\right\} d x\\\\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1405302",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 4,
"answer_id": 3
} |
Question of maps in Mayer-Vietoris sequence We obtain MV-seq. from short exact sequence
$$
0\to C_n(A\cap B) \to C_n(A)\oplus C_n(B)\to C_n(A+B)\to 0
$$
So map i wonder that map $H_n(A\cap B)\to H_n(A)\oplus H_n(B)$ maps $[a]$ to $([a],[-a])$. But for example in this case it's not quite clear why $1 \mapsto (2,-2)$. Wh... | Sam Nead has the correct suggestion here. In most instances of Mayer Vietoris you actually need to compute what the inclusion map induces or even, god forbid, what the snake homomorphism gives you. Here the boundary of a mobius strip includes into each mobius strip in such a way that it retracts onto the circle going a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1405410",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Solving a pair of ODEs I'm trying to solve a pair of ODEs for which I've obtained a solution. However, my problem is that my answer is slightly different from mathematica's answer.
$$ \frac{dA}{dt} = \theta - (\mu + \gamma)A, \ \ A(0) = G$$
$$ \frac{dT}{dt} = 2 \mu A - (\mu + \gamma)T, \ \ T(0) = B$$
Using an integra... | You forgot the integration constant. Following on from your integral for $T(t)$, we find
\begin{align}
T(t)e^{\alpha t} &= \int 2 \mu e^{\alpha t}\left(\frac{\theta}{\alpha} + \left(G -\frac{\theta}{\alpha}\right)e^{-\alpha t}\right)dt \\
&= \int 2 \mu e^{\alpha t}\left(\frac{\theta}{\alpha}\right) + 2 \mu \left(G -\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1405497",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Check Differentiability chech whether the function is differentiable at $x=0$
$$f(x)=\left\lbrace \begin{array}{cl}
\arctan\frac{1}{\left | x \right |}, & x\neq 0 \\
\frac{\pi}{2}, & x=0\\
\end{array}\right.$$
I feel that this is differentable at given point but I am unable to proceed with it.
| Simply use the identity
$$\arctan (1/x)=\frac{\pi}{2}\text{sgn}(x)-\arctan(x)$$
Then, the limit of the difference quotient is
$$\lim_{h\to0}\frac{\arctan(1/|h|)-\frac{\pi}{2}}{h}=-\lim_{h\to0}\frac{\arctan(|h|)}{h} \tag 1$$
We see that the limit in $(1)$ does not exist since the limit from the right side does not equa... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Differentiability of a two variable function $f(x,y)=\dfrac{1}{1+x-y}$ We're given the following function :
$$f(x,y)=\dfrac{1}{1+x-y}$$
Now , how to prove that the given function is differentiable at $(0,0)$ ?
I found out the partial derivatives as $f_x(0,0)=(-1)$ and $f_y(0,0)=1$ ,
Clearly the partial derivatives are ... | If all partial derivatives of a function (over all possible variables) are continuous at some point, then the function is differentiable at that point.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1405787",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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If $r:X\to A$ is a Retraction, Then $H_n(X)\cong H_n(A)\oplus H_n(X,A)$ $\DeclareMathOperator{\im}{Im}$
Let $A$ be a subspace of a topological space $X$ such that there is a retraction $r:X\to A$ of $X$ onto $A$.
Then $H_n(X)=H_n(A)\oplus H_n(X, A)$ for all $n$.
What I tried:
Let $i:A\to X$ be the inclusion map. Then... | For $A\subset X$ you have long exact sequence
$$
\dots\to H_n(A)\to H_n(X)\to H_n(X,A)\to\dots
$$
The composition $r_*\circ i_*:H_n(A)\to H_n(X)\to H_n(A)$ is identity, so we see that $i_*:H_n(A)\to H_n(X)$ is inclusion for all $n$. Thus, we can write
$$
0\to H_n(A)\to H_n(X)\to H_n(X,A)\to0,
$$
and $r_*$ gives us spl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1405880",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Solving $x^{2n} = \frac{1}{2^n}$ for $x$ What is the principle behind solving for a variable that is raised to another variable? I came across this problem doing infinite sums: I had to solve the equation
$$x^{2n} = \frac{1}{2^n}$$
for $x$. I posed the question in the online forum and the TA said the answer is
$$x =... | $x^{2n}=(x^2)^n=\frac{1}{2^n}$. So if $n\neq 0$, for $x\in \mathbb{R}$ you have $x^2=\frac{1}{2}$ then $x=\frac{1}{\sqrt{2}}$ or $x=-\frac{1}{\sqrt{2}}$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is an algorithm to find all primes up to $n$ that runs in $O(n)$ time fast? I kindly ask you if it is useful or fast for a prime number generator to run in $O(n/3)$ time?
I believe I have a way to generate all $P$ primes up to $n$, quickly and neatly, in $P$ comparisons and $n/3$ calculations.
| Calculating the primes up to $n$ in $O(n)$ time isn't particularly fast, and nor is it particularly slow. A naive Sieve of Eratosthenes works in time $O(n \log n \log \log n)$ and is very easy to implement. It can be sped up to run faster than $O(n)$ using some wheel techniques. I believe one can reduce the time to $O(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1406048",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Does the integral $\int_0^{\pi} \frac{dx}{\sin(2x)+\cos(3x)}$ exist? This link
http://www.wolframalpha.com/input/?i=integral%28x%3D0%2Cpi%2C1%2F%28sin2x%2Bcos3x%29%29
shows the visual representation of the integral
$$\int_0^{\pi} \frac{dx}{\sin(2x)+\cos(3x)}$$
Looking at the picture, I tend to believe that the integral... | Note at $x = {\pi \over 2}$, the denominator $\sin 2x + \cos 3x$ is zero. If $f(x) = \sin 2x + \cos 3x$, then $f'(x) = 2 \cos 2x - 3 \sin 3x$, so that $f'({\pi \over 2}) = 1$. Hence the function $f(x)$ behaves as ${1 \over x - {\pi \over 2}}$ near $x = {\pi \over 2}$; in particular, since $|{1 \over x - {\pi \over 2}}|... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1406157",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Showing $\bigcup_{k}\{(a_{1},...,a_{k}):a_{j}\in [0,1]\cap \mathbb{Q} ,\sum^{k} a_{j}=1\}$ is countable Showing $M=\bigcup_{k}\{(a_{1},...,a_{k}):a_{j}\in [0,1]\cap \mathbb{Q} ,\sum^{k} a_{j}=1\}$ is countable.
I find this hard to believe because say for fixed k and any $b,c\in [0,1]\cap \mathbb{Q}$, we have $(\frac... | Fix $k$. The set $\{(a_1,...,a_k); a_j\in [0,1]\cap \mathbb{Q}, \sum_ja_j=1\}\subset \mathbb{Q}^k$. Therefore it is countable. Countable union of countable sets in countable.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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multiple sets of complex roots of a number? I am not sure if this question was asked before but I couldn't find the right keywords to choose for searching.
So today I discovered a weird problem:
If we take this equation: $$x^2=1=e^{(0i)}$$
this equation has 2 solutions : $\{1, -1\}$
Until now everything is normal.
bu... | Certain exponent laws and properties go out the door when you start dealing with complex numbers. That's why you're getting to different solutions when going about this in two different way.
In general, if $n$ is an integer, then the solution set to the problem $z^n = 1$ is the set of roots
$\{1=e^{2\pi i * (0/n)},e^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1406339",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $y= |\sin x| + |\cos x|$, then $dy/dx$ at $x=2\pi/3$ is? If $$y= |\sin x| + |\cos x|,$$
then $dy/dx$ at $x = 2\pi/3$ is?
Ans: $(\sqrt{3} - 1)/2$
How can we differentiate modulus functions? Can anyone explain?
| you only need to worry about the behaviour in the vicinity of $x=2 \pi /3$ where $\sin x $ is positive and $ \cos x $ is negative so
$$y= \sin x - \cos x $$
$$y'= \cos x + \sin x $$
$$y'(2 \pi /3)= \cos (2 \pi /3) + \sin(2 \pi /3) = -\frac 12 + \frac{\sqrt 3}{2} = \frac{\sqrt 3 -1}{2}$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Write the co-ordinates of E such that the parallelogram ABCE is a rhombus. I'm unsure how to do this and it's always in my exams. (The original shape was a triangle and E was originally not a point)
A:(1,0) B:(0,8) C(7,4)
Gradient of AC:2/3
AC equation:2x - 3y - 2 = 0
Coordinates of the midpoint D of AC: 4,2
*
*AC... | The diagonals of a rhombus bisect each other at right angles (Proof)
If $O(p,q)$ is the midpoint of $AC, B E$
If $E(h,k)$ $$\dfrac{1+7}2=p=\dfrac{0+h}2$$ and $$\dfrac{0+4}2=q=\dfrac{8+k}2$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proof that transpose of Hadamard Matrix is also a Hadamard matrix The question is self explaining from the title, but let me elaborate it. In most of the articles/books I've read, fact that the transpose of Hadamard matrix is also a Hadamard matrix is used, but I was not able to find or deduce a proof for it. I can bas... | What you need is the fact, from elementary linear algebra, that, if a square matrix $A$ has a right inverse $B$ (i.e., $AB=I$) then $B$ is also a left inverse for $A$ (i.e., $BA=I$). In the case of an $n\times n$ Hadamard matrix, the definition in terms of orthogonal rows gives $\frac1n AA^\top=I$. So the linear algeb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1406656",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What's theoretical maximum information compression rate? Let's say I've got a random bit sequence s and a reversible function f(s), for which the following statement f'(f(s)) = s is true. What is the theoretical maximum average compression rate of such function?
IIRC, most if not all compression algorithms of today ten... | The question is answered by Shannon's source coding theorem.
For i.i.d. input, the theorem establishes that the minimum compressed bit rate is given by the entropy of the source.
For statistically dependent input bits the same result applies, but entropy has to be defined in a more general manner to take dependence int... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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existence of a sequence of continuous functions with two conditions
*
*$\displaystyle \int_0^1 \lim_{n\to\infty} f_n(x)\,dx = \lim_{n\to\infty}\int_0^1f_n(x)\,dx $
*There is no function $\,g:\left[0,1\right]\to \mathbb R\,$ lebesgue integrable such that $\,\left\lvert f_n (x)\right\rvert\le g(x)\,$ for $\,0\le x\le ... | Take the sequence:
$$f_n(x) = \begin{cases} n\qquad \mbox{if }\; \frac{1}{n+1}\leq x< \frac{1}{n} \\ 0 \qquad \mbox{otherwise}\end{cases}$$
We have $\int_{0}^1 f_n(x) dx = \frac{1}{n+1}$. So $\int_0^1 f_n(x) dx \to 0$.
What happens to $g(x)=\max\{f_n(x): n\in \mathbb{N}\}$?
Now you need to smoothen $f_n$ a little bi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1406830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to solve an exponential and logarithmic system of equations? $$
\left\{\begin{array}{c}
e^{2x} + e^y = 800
\\ 3\ln(x) + \ln(y) = 5
\end{array}\right.$$
I understand how to solve system of equations, logarithmic rules, and the fact that $\ln(e^x) = e^{\ln(x)} = x$. However, any direction I seem to go with this pro... | So here is another solution for your problem, again using Newton's method we get by using as initial value $x_0=3.2$ (it is very sensitive!) the following line of iterates
$$x_0=3.2$$
$$x_1=3.330187$$
$$x_2=3.303383$$
$$x_3=3.302208$$
$$x_4=3.302206$$
which gives us as an approximate solution
$$
f(3.302206)\approx-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1406917",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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Which is greater, $98^{99} $ or $ 99^{98}$? Which is greater, $98^{99} $ or $ 99^{98}$?
What is the easiest method to do this which can be explained to someone in junior school i.e. without using log tables.
I don't think there is an elementary way to do this. The best I could find was on Quora, in an answer by Michal... | The only "elementary" way I can think of is to write $99^{98} = (98 + 1)^{98}$ and then expand using the binomial expansion formula, and then show you get a sum of $99$ terms where each term is less than or equal to $98^{98}$, and the sum of the last two terms $98 + 1$ is strictly less than $98^{98}$. Then your sum is ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "28",
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"answer_id": 7
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Proof for parallelogram law of vector addition The Statement of Parallelogram law of vector addition is,
If two vectors are considered to be the adjacent sides of a parallelogram, then the resultant of two vectors is given by the vector that is a diagonal passing through the point of contact of two vectors.
But how d... | Addition of Vectors basically found their origin from the Triangular Law of Vector Addition
The triangle law of vectors basically is a process that allows one to take two vectors, draw them proportional to each other, connect them head to tail, then draw the resultant vector as a result of the third side that is miss... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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Prove an improper double integral is convergent I need to prove the following integral is convergent and find an upper bound
$$\int_{0}^{\infty} \int_{0}^{\infty} \frac{1}{1+x^2+y^4} dx dy$$
I've tried integrating $\frac{1}{1+x^2+y^2} \lt \frac{1}{1+x^2+y^4}$ but it doesn't converge
| Finding the exact value of $\int_0^\infty\frac{dx}{a^2+x^2}$ is just a calc I exercise. Let $a=\sqrt{1+y^4}$ and see what happens...
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Show That There Is One and (Essentially) Only One Field With 3 Elements My thought process is the following: By default, 0 and 1 have to be in this set of 3 elements since they are the neutral additive and neutral multiplicative elements, respectfully. So the set of the 3 elements that make up this field must be {0,1,x... | Your answer is correct, your reasoning is not. I realize you're new to abstract algebra, so I'll just say this: when we write down things like $1$ and $0$, we mean only that these things are the multiplicative and additive identities, respectively. We do NOT think of them as subsets of the real line. For example, we ca... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How can one compare these two 4-manifolds We would like to compare the following two real 4 dimensional manifolds:
1)$M$=The tangent bundle of $S^{2}$
2)$N$= The total space of the canonical line bundle over $\mathbb{C}P^{1}\simeq S^{2}$
Are these two manifolds homeomorphic? Are they diffeomorphic?
Some remarks: They... | Here's another (probably unnecessarily elaborate) proof that they are not diffeomorphic using Eliashberg's amazing theorem about which open $2$-handlebodies admit Stein structures and the adjunction inequality for Stein surfaces. I am sorry if this is not really accessible for the asker.
$M$ is the 4-manifold given by ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Is there any element of order $51$ in the group $U(103)$ Does there exist an element of order $51$ in the multiplicative group $U(103)$ ?
Now if the element exist say $x$ then it satisfies the equation $$x^{51}\equiv 1\pmod {103}$$ . Now $103$ being a prime it is clear that there is an e... | I suppose that $U(103)$ denotes the multiplicative group of the ring $\mathbb{Z}_{103}$.
Note that $\mathbb{Z}_{103}$ is a field; the multiplicative group of a finite field is cyclic; and a cyclic group of order $n$ contains elements of order $d$ whenever $d$ divides $n$ (take the $\frac{n}{d}$-th power of a generato... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$\lim_{y \rightarrow b} \lim_{x \rightarrow a} f \neq \lim_{(x,y)\rightarrow (a,b)} f \neq \lim_{x \rightarrow a} \lim_{y \rightarrow b} f$ Can someone give me an example to show that in general
$\lim_{y \rightarrow b} \lim_{x \rightarrow a} f(x,y) \neq \lim_{(x,y)\rightarrow (a,b)} f(x,y) \neq \lim_{x \rightarrow a} \... | Now since $\lim_{x\rightarrow a} \lim_{y\rightarrow b} f(x,y)$ and $\lim_{y\rightarrow b} \lim_{x\rightarrow a} f(x,y)$ both exist it is obvious that $\lim_{x\rightarrow a} f(x,y)$ and $\lim_{y\rightarrow b} f(x,y)$ have to exist. For if not then the former two limits cannot be be calculated.
Lemma - If $\lim_{(x,y)\ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1407614",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that $\left|(|x|-|y|)\right|\leq|x-y|$
Prove that $\left|(|x|-|y|)\right|\leq|x-y|$
Proof:
$$\begin{align} \left|(|x|-|y|)\right| &\leq|x-y| \\
{\left|\sqrt{x^2}-\sqrt{y^2}\right|}&\leq \sqrt{(x-y)^2} &\text{($\sqrt{a^2}=|a|)$}\\
\sqrt{\left(\sqrt{x^2}-\sqrt{y^2}\right)^2}&\leq \sqrt{(x-y)^2} &\text{($\sqrt{... | Because of the triangle inequality, there is: $\left| x+y \right| \le \left| x \right| +\left| y \right| $. Using this fact:
\begin{align}
\left| x \right| &= \left| (x-y)+y \right| \\
&\le \left| x-y \right| +\left| y \right| \\
\left| x \right| -\left| y \right| &\le \left| x-y \right| \tag{1}
\end{align}
Proceedin... | {
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"timestamp": "2023-03-29T00:00:00",
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Prove the trigonometric identity $\cos(x) + \sin(x)\tan(\frac{x}{2}) = 1$ While solving an equation i came up with the identity $\cos(x) + \sin(x)\tan(\frac{x}{2}) = 1$.
Prove whether this is really true or not. I can add that $$\tan\left(\frac{x}{2}\right) = \sqrt{\frac{1-\cos x}{1+\cos x}}$$
| An easy way to do that is:
sinx = sqrt((1+cosx)(1-cosx))
so LHS becomes:
cosx + sqrt((1+cosx)(1-sinx))sqrt((1-cosx)/1+cosx))
cosx + 1 - cosx = 1 = RHS
| {
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"url": "https://math.stackexchange.com/questions/1407794",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Does this game make you arbitrarily rich with probability one? We toss a coin. If it's heads we win $\$ 1$, otherwise we lose $ \$ 1$. Fix some large sum. Will we be winning this amount with probability one at some point? We assume that we have infinitely many throws and an infinite amount of money.
More formally (plea... | Let we say that a sequence of $2n$ throws is balanced if at any point we are not winning any dollar, but with the last throw we are losing zero. It is well-known that the number of balanced sequences of length $2n$ is given by the Catalan number:
$$ C_n = \frac{1}{n+1}\binom{2n}{n}. $$
If we say that a sequence of $2n$... | {
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"timestamp": "2023-03-29T00:00:00",
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A complex matrix with real eigenvalues Let $A$ be a $10\times 10$ matrix with complex entries and all eigenvalues non-negative real numbers and at least one eigenvalue strictly positive . Then there exist a matrix $B$ such that
$A$. $AB-BA=B$
$B$. $AB-BA=A$
$C$. $AB+BA=A$
$D$. $AB+BA=B$
Complex ma... | Of course A and D must be true: we can take $B = 0$.
Note that $\operatorname{trace}(AB - BA) = 0$ for any $A,B$ (why)? So, B can only be true if $A$ has a trace of $0$, which is necessarily not the case from the premise of the question.
The answer to C is yes. Try $B = \frac 12 I$.
| {
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"timestamp": "2023-03-29T00:00:00",
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How to prove by Mathematical Induction. I want to know how to prove this inequality by mathematical induction:
$a_k's$ are nonnegative numbers. Prove that$$a_1a_2\cdots a_n\leq \left(\frac{a_1+a_2+\cdots+a_n}{n}\right)^n.$$
In the inductive step, I tried using the inequality $ab\leq \frac{k}{k+1}a^\frac{k+1}{k}+\frac{1... | Maybe this is one of those cases in which an induction proof is easier if one makes the statement to be proved stronger, because then one has a stronger induction hypothesis to use. The inequality in the question is
$$
a_1^{1/n}\cdots a_n^{1/n} \le \frac 1 n \left(a_1 + \cdots+ a_n\right).
$$
These are a geometric mean... | {
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If points in a convex set $C$ escape to infinity roughly in direction $v$ then an infinite ray in that direction exists Let $C\subseteq \mathbb R^n$ a convex set.
Assume there is a sequence $\{c_k\}_{k\in\mathbb N}$ with $c_k\in C$, $|c_k|\to\infty$ such that $v:=\lim \frac 1{|c_k|}c_k$ exists. Does this imply that t... | You probably need that $C$ is closed.
Assume $0\in C$ without loss of generality. Otherwise, if $a\in C$, then $|c_k-a|\to \infty$ and
$$ \frac{c_k-a}{|c_k-a|} = \frac{c_k}{|c_k|}\frac{|c_k|}{|c_k-a|} - \frac{a}{|c_k-a|} \to v.$$
Let $r\in[0,\infty)$. As $|c_k|\to\infty$, for $k$ sufficiently large, we have $r < |c_k|$... | {
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"answer_id": 2
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What is the pre-requisite knowledge for generating my own integer sequence? I've recently come across the On-Line Encyclopedia of Integer Sequences and I'm completely fascinated by it; something about how easy integers are to grasp and yet how complex the sequences are. I find it intriguing and I'd love to be able to c... | You can come up with any sequence you like. If you look around OEIS, you'll find from the ubiquitous Fibonacci numbers to the Look and say sequence.
But consider that they prominently state that they have a huge backlog of proposed sequences, so you should make sure (a) it isn't a variant of something already in there,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1408442",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Complementary textbook algebraic geometry I don't know where to ask this or if it is allowed to do it, so please let me know any details for further questions of this kind.
I am taking an algebraic geometry class and am using the textbook "Ideals, Varieties, and algorithms" by David Cox et. al. I was wondering if any o... | You might like to take a look at Joe Harris: Algebraic Geometry.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1408525",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Where did I go wrong in my evaluation of the integral of cosine squared? $$\int{\cos^2(x)}dx$$
Where did I go wrong in my evaluation of this integral?
$$=x\cos^2x - \int-2x\sin(x)\cos(x)\,dx$$
$$=x\cos^2x + \int x\sin(2x)\,dx$$
$$=x\cos^2x + \left(\frac {-x\cos(2x)}2 -\int \frac{-\cos(2x)}2\,dx\right)$$
$$=x\cos^2x + \... | Using the formula $\displaystyle \bullet \; 2\cos^2 x = 1+\cos 2x$
So $$\displaystyle I = \frac{1}{2}\int 2\cos^2 xdx =\frac{1}{2}\int \left[1+\cos 2x\right]$$
So $$\displaystyle I = \frac{1}{2}\int 1 dx + \frac{1}{2}\int \cos 2x dx = \frac{1}{2}x+\frac{1}{4}\sin 2x+\mathcal{C}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1408634",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 6,
"answer_id": 3
} |
Calculate correct dose for pet if I have a solution that contains 750 mg of curcumin in 10 ml of water and I want to give a dose that is equal to 25 mg curcumin How many ml do I give?
Need to medicate my cat and I need to be sure to give correct dose. Thanks
| We apply the related ratios
$$\frac{750\,\,\text{mg}}{10\,\,\text{m}\ell}=\frac{25\,\,\text{mg}}{x}$$
whereupon solving for x gives
$$\bbox[5px,border:2px solid #C0A000]{x=\frac{10\,\,\text{m}\ell\times 25\,\,\text{mg}}{750\,\,\text{mg}}=\frac13\,\,\text{m}\ell}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1408695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Change of basis and inner product in non-orthogonal basis I have a vector, originally expressed in the standard coordinates system, and want to perform a change of basis and find coordinates in another basis, this basis being non-orthogonal.
*
*Let $B = \{e_1, e_2\}$ be the standard basis for $\Bbb R^2$.
*Let $B' = ... | What you want to do is change the basis of the vectors you are working with, then take the inner product on that. since $e_1', e_2'$ are a basis for $\mathbb{R}^2$, every vector $v$ = $v_1 e_1' + v_2 e_2'$ for unique values $v_1, v_2$. Then you set $\langle v, w \rangle = v_1 w_1 + v_2 w_2$
To find an explicit method o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1408827",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
Prove that $\frac{2^{x+1}+(x+1)^2}{2^x+x^2}\rightarrow 2$ as $x \rightarrow \infty$ Could someone please show me the proof that $\frac{2^{x+1}+(x+1)^2}{2^x+x^2}\rightarrow 2$ as $x \rightarrow \infty$
I have no idea where to begin with this one.
Thanks.
| Divide numerator and denominator by $2^n$. It's not hard to show that $\frac{x^2}{2^n}\to 0$ (use L'Hôpital's rule, for example) to see $$\lim_{x\to\infty}\frac{2^{x+1}+(x+1)^2}{2^x+x^2}=\lim_{x\to\infty}\frac{2+(x+1)^2/2^x}{1+x^2/2^x}=2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1408900",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
How is the Radius of Convergence of a Series determined? Consider $$\sum_{n=0}^{\infty}\frac{(-1)^nx^n}{(n+1)^2}$$ which by the ratio test the ratio of two consecutive terms converges to $|x|$ as $n\rightarrow \infty$ and has a radius of convergence equal to $1$.
Now consider $$\sum_{n=0}^{\infty}(-1)^n(2^n+n^2)x^n$$ w... | An interesting aspect is to evaluate the series of the question.
Consider
\begin{align}
S_{1}(x) &= \sum_{n=0}^{\infty} \frac{(-1)^{n} \, x^{n}}{(n+1)^{2}} \\
S_{2}(x) &= \sum_{n=0}^{\infty} (-1)^{n} \, (2^{n} + n^{2}) \, x^{n}.
\end{align}
The first series:
\begin{align}
\partial_{x} \left(x \, S_{1}(x) \right) &= \s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1408978",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
How to prove that the following sequence will never contains number greater than 3 You may now the following sequence:
1
11
21
1211
111221
312211
13112221
Explanation of the sequence (I've put an hint just for the ones who want to search a bit ;) )
Where, in each iteration you count the number of occurrence of a digi... | This is a typical proof by induction. The base case is just that the first term ($1$) does not contain any digit greater than $3$ nor more than $3$ consecutive, equal digits.
If some term had a digit greater than $3$, it would have had to come from more than $3$ (that is, $4$ or more) consecutive, equal digits in the p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1409062",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Calculate $\lim_{n\to\infty} (n - \sqrt {{n^2} - n} )$ Calculate limit:
$$\lim_{n\to\infty} (n - \sqrt {{n^2} - n})$$
My try:
$$\lim_{n\to\infty} (n - \sqrt {{n^2} - n} ) = \lim_{n\to\infty} \left(n - \sqrt {{n^2}(1 - \frac{1}{n}} )\right) = \lim_{n\to\infty} \left(n - n\sqrt {(1 - \frac{1}{n}})\right)$$
$$\sqrt {(1 -... | The reason is that inside a limit, you can't substitute an expression with the limit it is approaching. The expresion $\sqrt{1-1/n}$ does go to $1$, but it does so at a slow pace, while the expression $n$ goes to infinity at a certain pace as well, so $n\sqrt{1-1/n}$ does not necessarily grow arbitrarily close to $n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1409200",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 1
} |
Finding the rank of an non-invertible matrix I have a $3\times3$ matrix with three different eigenvalues $0,1, 2$.
The question is: what is the rank of this matrix? If the matrix was invertible, I could say that the rank was equal to $n=3$. But as zero is an eigenvalue of this matrix, this matrix does not satisfy the ... | *
*All eigenvalues are different, then the matrix is diagonalizable.
*The corresponding diagonal matrix has the eigenvalues on the diagonal, i.e.
$$
S^{-1}AS=D=\left[\matrix{2 & 0 & 0\\0 & 1 & 0\\0 & 0 & 0}\right].
$$
*The matrices $A$ and $D$ have the same rank.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1409311",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
The number of ordered pairs of positive integers $(a,b)$ such that LCM of a and b is $2^{3}5^{7}11^{13}$ I started by taking two numbers such as $2^{2}5^{7}11^{13}$ and $2^{3}5^{7}11^{13}$. The LCM of those two numbers is $2^{3}5^{7}11^{13}$. Similarly, If I take two numbers like $2^{3-x}5^{7-y}11^{13-z}$ and $2^{3}5^{... | Let $a = 2^{x_{1}}\cdot 5^{y_{1}}\cdot 11^{z_{1}}$ and $b = 2^{x_{2}}\cdot 5^{y_{2}}\cdot 11^{z_{2}}\;,$ Then Given $\bf{LCM(a,b)} = 2^{3}\cdot 5^{7}\cdot 11^{13}$
So Here $0\leq x_{1},x_{2}\leq 3$ and $0\leq y_{1},y_{2}\leq 7$ and $0\leq z_{1},z_{2}\leq 13.$
Here ordered pairs of $(x_{1},x_{2}) = \left\{(0,3),(1,3),(2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1409388",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Confused about rules for solving systems of linear equations Why are we allowed to add equations together/eliminate variables when solving systems of linear equations? I get that it works to find the solution but I don't understand why it works.
Also, why can't we do the same things with nonlinear systems of equations?... | If (x,y) is a solution to the first equation then the left side is 1, and if (x,y) is a solution to the second equation, then the left side is 5. When you add the two equations, the result on the left side must equal 6 only if (x,y) satisfies both equations i.e. the point of intersection. The same idea applies to a p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1409486",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
parallel resistors Consider the set $E_b = \left\{1, 1.2, 1.5, 1.8, 2.2, 2.7, 3.3, 3.9, 4.7, 5.6, 6.8, 8.2\right\}$. This is our base set. Let's define the set $E$ as follows:
$$ E = \left\{ 10^k e \mid k=0,1,2,\ldots, \text{for every} e \in E_b \right\}$$
and $\Omega$ as the class of all subsets of $E$. We are interet... | This is a partial answer, identifying some $n$ that cannot be synthesized with a finite subset of values from $E$.
The LCM of $E_{\text{b}}$ is $N=3541509972=2^2\cdot 3^3\cdot 7\cdot 11\cdot 13\cdot 17\cdot 41\cdot 47$. Thus $\frac{N}{e}$ is an integer for $e\in E_{\text{b}}$. For general $e\in E$, $\frac{N}{e}$ theref... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1409596",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 1,
"answer_id": 0
} |
Locally flat manifold from Frobenius, differential forms approach It is a known fact that a Manifold $M$ is locally flat if and only if the Riemann curvature tensor (Levi-Civita connection) vanishes. To show the if part is easy, but I am trying to follow the details of the proof given here but I am at a loss in reachin... | Following Élie Cartan, you want to think of your differential system $dR - R\omega = 0$ on $M\times SO(n)$. This $\mathfrak{so}(n)$-valued $1$-form is integrable, as you said, because of vanishing curvature. The integral manifolds of this differential system will locally be the graphs of functions $R\colon U\to SO(n)$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1409665",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
What do you call a space whose only compact sets are finite? What do you call a topological space where a subset is compact iff it's finite? Is there a technical name?
For example, take the discrete topology, or the countable complement topology.
| As others have noted, the term that you want is anticompact. This is an example of the kind of property studied by Paul Bankston in The total negation of a topological property, Illinois J. of Math., Vol. 23, Nr. 2 (1979), 241-252. I quote:
Let $K$ be a topological class. The spectrum $\operatorname{Spec}(K)$ of $K$ i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1409825",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Rearrangement and Cauchy Let $a_1, \ldots, a_n$ be distinct positive integers. I want to prove that
$$\frac{a_1}{1^2} + \frac{a_2}{2^2} + \cdots + \frac{a_n}{n^2} \geq \frac{1}{1} + \frac{1}{2} + \cdots + \frac{1}{n}.$$
I've been considering using Rearrangement and Cauchy Schwarz, but cannot make any progress
| First we quote (part of) the Rearrangement Inequality, in the notation used by Wikipedia:
$$x_ny_1 + \cdots + x_1y_n \le x_{\sigma (1)}y_1 + \cdots + x_{\sigma (n)}y_n\tag{1} $$
for every choice of real numbers
$$ x_1\le\cdots\le x_n\quad\text{and}\quad y_1\le\cdots\le y_n,$$
and every permutation
$$ x_{\sigma(1)},\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1409922",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
What can we say about open unit balls of sup-norm and integral-norm
Consider the normed linear spaces $X_1=(C[0,1], ||.||_1)$ and $X_{\infty}=(C[0,1],||.||_{\infty})$ , where $C[0,1]$ denotes the vector space of all continuous real valued functions on $[0,1]$ and $$||f||_1=\int_0^1|f(t)|\,dt.$$and $$||f||_{\infty}=\su... | Since it seems that you want to solve it yourself, I'll just give you a hint. If you want a more accurate answer, just leave a comment.
HINT: In a normed space the unit open balls is the set of all elements which have a norm strictly less than 1. Now, what does it mean that $\left\Vert f \right\Vert_1 < 1$? What does i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1410019",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Does $\lim\limits_{n \to +∞} \sum_{k=1}^n \frac{n\cdot \ln (k)}{n^2+k^2}$ diverge? Does the limit of this summation diverge?
$$\lim\limits_{n \to +∞} \sum_{k=1}^n \frac{n\cdot \ln (k)}{n^2+k^2}$$
Thanks!
| $$\sum\limits_{k = 1}^n {\frac{{n\ln k}}{{{n^2} + {k^2}}}} = \frac{1}{n}\sum\limits_{k = 1}^n {\frac{{\ln k}}{{1 + {{(\frac{k}{n})}^2}}}} = \frac{1}{n}\sum\limits_{k = 1}^n {\frac{{\ln (\frac{k}{n})}}{{1 + {{(\frac{k}{n})}^2}}}} + \frac{1}{n}\sum\limits_{k = 1}^n {\frac{{\ln n}}{{1 + {{(\frac{k}{n})}^2}}}} $$
When $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1410108",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Is $\sqrt[3]{-1}=-1$? I observe that if we claim that $\sqrt[3]{-1}=-1$, we reach a contradiction.
Let's, indeed, suppose that $\sqrt[3]{-1}=-1$. Then, since the properties of powers are preserved, we have: $$\sqrt[3]{-1}=(-1)^{\frac{1}{3}}=(-1)^{\frac{2}{6}}=\sqrt[6]{(-1)^2}=\sqrt[6]{1}=1$$ which is a clear contradict... | The notation $\sqrt[3]{-1}$ is a little bit ambiguous, since there are exactly three third roots of $-1$ over the complex numbers (in general, there are exactly $n$ $n-$roots of any complex number $z$ so the notation $\sqrt[n]{z}$ is ambiguous too).
Since
$$(-1)^3 = -1$$
$-1$ is one of those roots, but there are other ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1410185",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Calculating $\int_0^{\infty } \left(\text{Li}_2\left(-\frac{1}{x^2}\right)\right)^2 \, dx$ Do you see any fast way of calculating this one? $$\int_0^{\infty } \left(\text{Li}_2\left(-\frac{1}{x^2}\right)\right)^2 \, dx$$
Numerically, it's about
$$\approx 111.024457130115028409990464833072173251135063166330638343951498... | $$I=\int_0^\infty\operatorname{Li}_2^2\left(-\frac1{x^2}\right)\ dx\overset{IBP}{=}-4\int_0^\infty\operatorname{Li}_2\left(-\frac1{x^2}\right)\ln\left(1+\frac1{x^2}\right)\ dx$$
By using $$\int_0^1\frac{x\ln^n(u)}{1-xu}\ du=(-1)^n n!\operatorname{Li}_{n+1}(x)$$ setting $n=1$ and replacing $x$ with $-\frac1{x^2}$ we ca... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1410293",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 4,
"answer_id": 2
} |
Geodesics of Sasaki metric I would like to ask the community for a reference on the following question:
Let $(M,g)$ be a Riemannian manifold and $(T^1M,g_S)$ be the unit tangent bundle with the Sasaki metric. Is it true that the orbits of the geodesic flow $\varphi:T^1M\longrightarrow T^1M$ are geodesics of $(T^1M,g_S)... | (1) Consider a projection $\pi : (T_1M,G) \rightarrow (M,g) $ We
want to define a metric $G$. If $\widetilde{c_1} (t),\
\widetilde{c_2}(t) $ are curves starting at $(p,v)$ then we write
$\widetilde{c_i}(t)=(c_i(t),v_i(t)),\ c_i(0)=p,\ v_i(0)=v$ so that
let
$$ G(\widetilde{c_1}'(0),\widetilde{c_2}'(0)) =g(c_1'(0),c_2'(0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1410371",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 1,
"answer_id": 0
} |
bend measurement and calculating $\int_4^8 \sqrt{1+{\left(\frac{{x^2-4}}{4x}\right)^2}} $ How can i get the measure of this bend : $y=\left(\frac{x^2}{8}\right)-\ln(x)$ between $4\le x \le 8$. i solved that a bit according to the formula $\int_a^b \sqrt{1+{{f'}^2}} $:$$\int_4^8 \sqrt{1+{\left(\frac{x^2-4}{4x}\right)^2}... | You just need to expand the square:
$$
\int_{4}^{8}{\sqrt{1+\left(\frac{x^2-4}{4x}\right)^2}dx}=\int_{4}^{8}{\sqrt{1+\frac{x^4-8x^2+16}{16x^2}}dx}=\int_{4}^{8}{\sqrt{\frac{x^4+8x^2+16}{16x^2}}dx}=\int_{4}^{8}{\sqrt{\left(\frac{x^2+4}{4x}\right)^2}dx}=\int_{4}^{8}{\frac{x^2+4}{4x}dx}=\int_{4}^{8}{\frac{x}{4}+\frac{1}{x}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1410466",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
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