Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
How to solve $\left|\frac{1 + a + bi}{1 + b - ai}\right| = 1$ I have a problem with solving following equation:
$$\left|\frac{1 + a + bi}{1 + b - ai}\right| = 1$$ (where $a$, $b$ are real numbers and $i$ is an imaginary unit)
I tried to simplify its left side to something like $c + di$ but I don't know any method to a... | Hint:
Use the fact that for any complex number $z_1$ and $z_2$, $$\left|\frac{z_1}{z_2}\right|=\frac{|z_1|}{|z_2|}$$ Then try and rearrange.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/961183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Find the maximum of $\frac{1}{1+|x|}+\frac{1}{1+|x-a|}$ Let $a>0$. Show that the maximum value of the function
$$f(x)= \frac{1}{1+|x|}+\frac{1}{1+|x-a|}$$
is $$\frac{2+a}{1+a}.$$
really need some help with this thing
| Study what happens in each of the three regions $\{x<0\}$, $\{0\le x\le a\}$ and $\{x>a\}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/961252",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
A continuous function defined on an interval can have a mean value. What about a median? A function can have an average value
$$\frac{1}{b-a}\int_{a}^{b} f(x)dx$$
Can a continuous function have a median?
How would that be computed?
| Sure a continuous function can have a median, as for if it always has a median I am unsure of.
Let's suppose we have a some function $f$ and it is continuous on $[a,b]$. We want to find its median $m$. We know that if $m$ is a median than if we take any value $c \in [a,b] $ at random then the $P(c \leq m) = $$P(c \geq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/961302",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 4,
"answer_id": 2
} |
Incredible Blackjack Hand Last Saturday night I played at Bally's in Atlantic City and got a hand I could not believe. Dealer had 9 and I was dealt 2 8s. I split the 8s and was given a third card. It was an 8 so I split them again. The next card I was dealt was a fourth 8. This has happened to me three other times in m... | Calculations from symmetricuser were all perfect when trying to calculate the probability of taking, for example, 6 8's in a row. Even though you ask for 6 cards in a row, of any denomination, so you are not forcing the number. Therefore, you have to multiply that for all the 13 possible denominations.
$$13\times \frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/961377",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 6,
"answer_id": 4
} |
How to prove: If $x \geq 0 $ and $x \leq \epsilon$, for all $\epsilon > 0$, then $x = 0$? I am trying to prove this problem for my homework. I am having some difficulty with this, because we are just supposed to use several ordered field axioms, the four order axioms, and several basic facts about the real numbers. If ... | Suppose that $x>0$. Now choose an $\epsilon>0$ such that $\epsilon\in(0,x)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/961467",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Is O(n) a proper class or a set? Is $O(n)$ as the collection of all functions that are bounded above by $n$ a proper class or just a set?
What about $O(\infty)$?
| Since $\Bbb R$ is a set, we know that $\Bbb{R\times R}$ is a set, so $\mathcal P(\mathbb{R\times R})$ is a set.
Therefore the collection of all functions from $\Bbb R$ to itself is a set. In particular any definable subcollection of a set is a set. For example, all the functions which are $O(n)$ or otherwise.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/961592",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
A sequence converges iff the tail converges A sequence converges iff its tail converge.
This statement is obviously true. But how will one prove this? I appreciate any help!
I already did try to write it out, but the indexing is a lot of trouble. I try to index it right so that I can prove this.
| You want to prove that a sequence, say $(a_n)$, converges iff its tail converges.
$(\rightarrow)$ Suppose $(a_n)$ converges. Let $(a_n)$ converge to $a$. Then, by the definition of the limit point of a function, $\forall \epsilon > 0, \exists N \in \mathbb{N} \textrm{ s.t. } \color{blue}{n \geq N} \implies \left|a_n -... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/961684",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Prove that the algebra does not vanish and that the algebra separates Show that the algebra generated by {sin(x),cos(x)} does not vanish on the
set [0, 2π]. Also, show that this algebra separates over this interval.
So I know the definition of an algebra and what it means for it to not vanish for any x on interval. ... | Since sin(x) $\in$ A, $sin^2(x)\in A$, similarly $cos^2(x)\in A$. Thus $sin^2(x) + cos^2(x) = 1 \in A$. $A$ vanishes at no point on $[0, 2\pi]$.
I am stuck on the second part..
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/961751",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Find correspondences of nodes between two graphs I have two graphs (actually two street maps, one very fine grained and exact, the other coarser and maybe a bit faulty w.r.t. the nodes coordinates and the topology) and I want to find corresponding nodes and or edges between those two graphs. The nodes in the graph have... | This is also known as the network alignment problem. Here is an example algorithm for dealing with this problem: GHOST
More details can be found in this bachelor's thesis:Survey on the Graph Alignment Problem and a Benchmark of
Suitable Algorithms
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/961942",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Fourier parseval prove for misunderstanding of second negative exponent sign See the picture below:
I know if the sign is not '-', the following derivation can not continue，but I really want to know why $$e^{itx}\cdot e^{i\tau x}=e^{i(t-\tau)x}$$
How it can be that? I really want to know.
plus $\hat{f}(t)$ is the stan... | Thaks to Paul，I know what I was confusing:$|f(x)|^2=f(x)\overline{f(x)}$So,the second bracket on the first line should have the negative sign before $i\tau x$ that be $-i\tau x$.
To describe it simply.That is:due to $e^{i\tau x}$ being the complex number,so taking the conjugate make the $e^{i\tau x}$ become $e^{-i\tau ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/962044",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Is there a linear transformation with the given kernel and image? I have this problem :
$U=\{(x,y,z,t)\in R^4 | y+z+t=0\}$
$W = \{(x,y,z,t) \in R^4 | x+y=0$ and $z=2t\}$
Is there a linear transformation?
$T : R^4 \rightarrow R^4$
That $\operatorname{Im} T = U$ and $\ker T = W$?
I don't really know where to start, I... | We have $\dim im (T)+\dim \ker(T)=4$ by the dimension theorem. Since $\dim U=3$ and $\dim W=2$ this would yield $2+3=4$, which is impossible.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/962161",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Graph of $\arcsin(\cos(x))$ How to draw graph of $\arcsin(\cos(x))$ or even $\arcsin(\sin(x))$ without use of graphing calculator , its sort of confusing me from long time. It gives pointed curves when drawing from calculator , why does it looks like that? Why it isn't just linear graph? Can someone shed some light her... | Keep in mind that the domain of the arcsin function is $[-1,1]$ and its range is $[-\pi/2,\pi/2]$. This is important because even though $\sin x$ and $\arcsin x$ are inverse functions, it's not correct to say that $\arcsin(\sin x)=x$ for all $x$.
This might explain why your graphing calculator is giving you "pointed" ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/962260",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Change of variable in differential equation legitimate? Just a general question ( I don't want to solve this ODE, I just want to understand why this is legitimate to do or not):
Assuming we have the ODE $$y'(x) - \cos(x) y(x)=0$$ on $[0,2\pi]$
Am I allowed to make the substitution $z = \cos(x)$? This substitution is d... | It is a diffeomorphism from $(0,\pi)$ to $(-1,1)$. Do your computations, and when you finish, check that the solution is valid everywhere.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/962443",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Derivative of matrix inversion function? Let's say I have a function $f$ which maps any invertible $n\times n$ matrix to its inverse. How do I calculate the derivative of this function?
| To calculate the directional derivatives of the map, at a fixed matrix $A$:
Suppose you change entry $i,j$ of your matrix $A$ by adding a small $t$ (small enough that the result is still invertible). Let $E_{i,j}$ be the matrix with a $1$ in the $i,j$th position and $0$ elsewhere.
Note that
$$
(A + tE_{i,j})(A + tE_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/962579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Field extension-Why does this hold? $K\leq E$ a field extension, $a\in E$ is algebraic over $K$.
Could you explain me why the following holds??
$$K\leq K(a^2)\leq K(a)$$
| Well by definition $K \leq K(a^2)$.
Then since $a^2 \in K(a)$ ($K(a)$ is closed under multiplication), $K(a^2) \leq K(a)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/962674",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Does the zeroth root exist? Definition of $Nth$ root:
$3rd$ order inverse group $1$ hyperoperation.
Division is how many times you can subtract a certain divisor from the dividend before it becomes negative.
Likewise Nth root is the result of repeated division by a certain divisor before it becomes $1$ or a decimal. Th... | No this is usually not defined. One definition of the $n$th root is
$$
\sqrt[n]{x} = x^{\frac{1}{n}}.
$$
So for a fixed $x>1$ you see that
$$
\lim_{n\to 0^+} x^{\frac{1}{n}} = \infty.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/962807",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 0
} |
Simplifying a characteristic equation when one eigenvalue is known This is either trivial, or difficult; if the former I should be embarrassed to ask it. Anyway... I have a 4x4 matrix of non-zero integer values, for which the determinant is zero. Given that therefore (at least) one eigenvalue is zero, is there a way ... | Let $\{e_1,e_2,e_3,e_4\}$ be the canonical basis of $\mathbb{R}^4$.
Find the eigenvector associated to $0$. Let $v_1=(a,b,c,d)$ be this eigenvector. If $a\neq 0$ then $\{v_1,e_2,e_3,e_4\}$ is a basis of $\mathbb{R}^4$.
Let $V\in M_{4\times 4}=(v_1,e_2,e_3,e_4)=\left(\begin{array}{cccc}
a & 0 & 0 & 0 \\
b & 1 & 0 & 0 \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/962911",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Indefinite integral with trig components The following integral has me stumped. Any help on how to go about solving it would be great.
$\int\frac{\cos\theta}{\sin2\theta - 1}d\theta$
| If you set $\theta=\frac{\pi}{4}-x$, the integral becomes
$$
\int \frac{\cos(\frac{\pi}{4}-x)}{\sin(\frac{\pi}{2}-2x)-1}\,(-dx)
= -\int \frac{\frac{1}{\sqrt2}\cos x + \frac{1}{\sqrt2}\sin x}{\cos 2x -1}\,dx
= \frac{-1}{\sqrt2} \int\frac{\cos x+\sin x}{(1-2\sin^2 x)-1} \, dx
= \frac{1}{2\sqrt2}\left( \int \frac{\cos x \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/963039",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Verify the following combinatorial identity: $\sum_{k=0}^{r} \binom{m}{k}\binom{n}{r-k} = \binom{m+n}{r}$ $$\sum_{k=0}^{r} \binom{m}{k}\binom{n}{r-k} = \binom{m+n}{r}$$
Nice, so I've proven some combinatorial identities before via induction, other more simple ones by committee selection models.... But this one is weird... | Note that there exist different $m+n$ balls and two bags.
One bag contains $m$ balls and the other has $n$.
What is the number of choosing $r$ balls ? Clearly $$ _{m+n}C_r$$
But we divided into two bag. : The number of choosing $k$ balls in the first bag is $ _mC_k $ And we must choose $r-k$ balls in second. The numb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/963164",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 0
} |
What do "canonical" and "natural" mean exactly? "Canonical" and "natural" are two words frequently seen in mathematical literature. For example, we often find "there is no canonical/natural way to", "it's canonical/natural to". So I'd like to know what is exactly "canonical/natural" way and also some examples for expla... | "Canonical" can mean simple in appearance or utility. For example, The Jordan Canonical Form is a transformation of a matrix so that it's block diagonal and all the blocks are upper triangular. It's simple in appearance (most of the elements are 0, that's good). And it's a nice form to be able to use. For example, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/963266",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Tricky sequences and series problem For a positive integer $n$, let $a_{n}=\sum\limits_{i=1}^{n}\frac{1}{2^{i}-1}$. Then are the following true:
$a_{100} > 200$ and
$a_{200} > 100$?
Any help would be thoroughly appreciated. This is a very difficult problem for me. :(
| Clearly, for $k>1$,
$$
\int_k^{k+1}\frac{dx}{x}<\frac{1}{k}<\int_{k-1}^{k}\frac{dx}{x}.
$$
Hence
$$
n\log 2=\log (2^n)=\int_1^{2^n}\frac{dx}{x}<1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{2^n-1}<1+\int_1^{2^n-1}\frac{dx}{x}=1+\log(2^n-1),
$$
and thus
$$
n\log 2<a_n<1+\log(2^n-1)<1+n\log 2.
$$
So
$$
a(100)<1+100\log 2<1+1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/963372",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
} |
Find a line that is perpendicular from x=10 and one point, (4,5) The title is quite self-explainary ;)
I know that the answer is $y=5$, but I am not sure how does one person come to that conclusions showing all the steps. Can you please help me here?
| I had to do my own version, using @Tharindu's knowledge for my assignment. Here it is if anyone wants a more detailed version. The coordinates are different, since I had a different question:
We need to find an equation that will create a line that is perpendicular to the equation $x=3.58$, given that one point on the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/963502",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Uniform limit of real valued functions on a compact space. Is the union of their images necessarily compact? Let $K$ be a compact space with $f_n$, $f$ continuous functions $K \to \mathbb{R}$ such that $f_n \to f$ uniformly.
Is $\mathrm{im}f \cup \bigcup \mathrm{im}(f_n) \subseteq \mathbb{R}$ necessarily compact?
For $... | If I am not mistaken, another kind of solution.
a) Let $E$ be the space of continuous fonctions on $K$ with values in $\mathbb{R}$, equipped with the "Sup" norm. Let $F=E\times K$. Then the application $\phi$: $F\to \mathbb{R}$ defined by $\phi((g,y))=g(y)$ is continuous. To see why, note that if $(g_0, y_0)$ is given,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/963580",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Solving the euclidian distance squared to kernelize a Lagrangian dual Homework question, looking for a hint on the following problem:
I'm trying to solve this dual lagranging form (which could potentially be wrong already, but let's assume it is right)
$\boldsymbol{x}$ is a set of datapoints in R^2, $\{\alpha_i\}$ are ... | Got a hint to solve $\sum^N_{i=1} \alpha_i(\boldsymbol{x}_i - \boldsymbol{S})^T(\boldsymbol{x}_i - \boldsymbol{S})$ instead.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/963673",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Does $f(x)=ax$ intersect $g(x)=\sqrt{x}$ It maybe a stupid question but I want to be sure how to explain it formally.
Does $f(x)=ax$ intersect $g(x)=\sqrt{x}$, when $x>0$ and $a>0$ (however small it is)
I think it does.
The derivative of $f(x)$ is constant, positive. And the derivative of $g(x)$ tends to $0$. So there... | If there is a point of intersection, it will satisfy the equation $$ax = \sqrt x\implies (ax)^2 = x \iff a^2x^2 - x = 0 \iff x(a^2x - 1) = 0\;$$
Indeed, the graphs intersect, when $x = 0$ and when $a^2x-1=0 \iff x = \frac 1{a^2}$. Since we are interested in only $x\gt 0$, the point of intersection you are looking for i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/963819",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 0
} |
Existence of right and left identity in minimalistic algebraic structure Let $(A,\cdot)$ be some algebraic structure in which there exists elements $e_r,e_l$ such that
$$e_l\cdot x = x, \forall x\in A$$
$$x\cdot e_r = x, \forall x\in A$$
By definition, if $(A,\cdot)$ is a monoid or a group then we must have $e_r = e_l... | By the defining property of $e_l$, we have
$$e_l\cdot e_r = e_r.$$
And by the defining property of $e_r$, we have
$$e_l\cdot e_r = e_l.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/963896",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
About Integration $ \int \frac{\tanh(\sqrt{1+z^2})}{\sqrt{1+z^2}}dz $ How to calculate the following integral
$$
\int \frac{\tanh(\sqrt{1+z^2})}{\sqrt{1+z^2}}dz
$$
Is there any ways to calculate those integral in analytic? (Is $[0,\infty]$, case the integral is possible?)
How about using numerical method? Is there are ... | For $\int_0^\infty(1-\tanh\cosh x)~dx$ ,
Similar to How to evaluate $\int_{0}^\infty \frac{{x^2}}{e^{\beta {\big(\sqrt{x^2 + m^2}}- \nu\big)} + 1} dx$,
$\int_0^\infty(1-\tanh\cosh x)~dx$
$=\int_0^\infty\left(1-\dfrac{1-e^{-2\cosh x}}{1+e^{-2\cosh x}}\right)~dx$
$=\int_0^\infty\dfrac{2e^{-2\cosh x}}{1+e^{-2\cosh x}}~dx$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/963995",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Number of Integer Solutions Problem An elevator in the Empire State Building starts at the basement with 57 people (not including the elevator operator) and discharges them all by the time it reaches the 86th floor. In how many ways could the operator have perceived the people leaving the elevator if they all looked di... | I have another result. Start with a small example. You have 2 distinguishable urns (floors) and 3 numbered balls (people).
The possible ways putting the balls into the urns:
$\text{urn 1} \ | \ \text{urn 2}$
$\text{1,2} \ \ \ | \ \ \ \ \text{3}$
$\text{1,3} \ \ \ | \ \ \ \ \text{2}$
$\text{3,2} \ \ \ | \ \ \ \ \text{1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/964110",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Algebraic Structures: Does Order Matter? (I want to link to similar question with a very good answer: Question about Algebraic structure?)
An algebraic structure is an ordered tuple of sets. One of these is called the underlying set, and the others are operations of various arity. Since operations are functions, which ... | The reason is really silly in some sense. Since all the objects ($G,+$ and so on) are sets, how can you know which one is the group, and which one is the addition?
You can say, well, $+\subseteq G^2\times G$. But there are sets $X$ such that $X^2\times X\subseteq X$. So it's really not so obvious as much as we might w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/964206",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Find all kinds of homomorphisms from $(\Bbb Z_n, +)$ to $(\Bbb C^*, \times)$. Find all kinds of homomorphisms from $(\Bbb Z_n, +)$ to $(\Bbb C^*, \times)$ and from $(\Bbb Z, +)$ to $(\Bbb C^*, \times)$.
Explain why they are the complete collection.
My intuition is:
1) we can construct $\phi:\Bbb Z_n \to \Bbb C^*$ the... | From the idea of Sami Ben Romdhane
For $\Bbb Z$, I have got $\phi (0)=1$ & $\phi (1)=a\in \Bbb C^*$ then $\phi (n)=a^n$. These are all homomorphisms.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/964361",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Integral $\int_0^1\frac{x^{42}}{\sqrt{x^4-x^2+1}}\operatorname d \!x$ Could you please help me with this integral?
$$\int_0^1\frac{x^{42}}{\sqrt{x^4-x^2+1}} \operatorname d \!x$$
Update: user153012 posted a result given by a computer that contains scary Appel function, and Cleo gave much simpler closed forms for power... | Odd case: The change of variables $x^2=t$ transforms the integral into
$$\mathcal{I}_{2n+1}=\int_0^1\frac{x^{2n+1}dx}{\sqrt{x^4-x^2+1}}=\frac12\int_0^1\frac{t^ndt}{\sqrt{t^2-t+1}}$$
Further change of variables $t=\frac12+\frac{\sqrt3}{4}\left(s-\frac1s\right)$ allows to write $t^2-t+1=\frac3{16}\left(s+\frac1s\right)^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/964438",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 3,
"answer_id": 2
} |
Prove the gcd $(4a + b, a + 2b) $ is equal to $1$ or $7$. So in the question it says to let $a$ and $b$ be nonzero integers such that $\gcd(a,b) = 1$.
So based on that I know that $a$ and $b$ are relatively prime and that question is basically asking if the GCD divides $7$.
So I tried to prove through the Euclidean alg... | $7$ has a geometric interpretation here.
Take integers $u$ and $v$ such that $av-bu=1$. Then the parallelogram in $\mathbb Z^2$ defined by the vectors $(a,b)$ and $(u,v)$ has area $1$.
The linear transformation $T(x,y)= (4x + y, x + 2y)$ sends that parallelogram to another parallelogram having area $7=\det T$.
Hence $7... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/964537",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Find 3rd and 4th co-ordinates for a square given co-ordinates of two points? To construct a square we need 4 points . In my problem 2 points are given we can find 3rd and 4th point .
e.g. A (1,2) B(3,5) what should be the co-ordinate of 3rd (C) and 4th (D) points . Please provide a formula to calculate third point and... | Let $A = (x_A, y_A), B = (x_B, y_B), \Delta x = x_B - x_A, \Delta y = y_B - y_A$.
Then
$$x_C = x_B \pm \Delta y$$
$$y_C = y_B \mp \Delta x$$
$$x_D = x_A \pm \Delta y$$
$$y_D = y_A \mp \Delta x$$
In your case:
$$x_A = 1, y_A = 2$$
$$x_B = 3, y_B = 5$$
$$\Delta x = 2, \Delta y = 3$$
Then
$$x_C = 0, 6; y_C = 7, 3$$
$$x_D ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/964625",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Can't get this implicit differentiation I've been working at this implicit differentiation problem for a little over an hour now, and I, nor my friends can figure it out. The question reads "Find the equation of the tangent line to the curve (a lemniscate) $2(x^2+y^2)^2=25(x^2−y^2)$ at the point (3,1). Write the equati... | We have
$2(x^2+y^2)^2=25(x^2−y^2)$.
Since
$(x^2)' = 2 x \ dx$,
I would differentiate this as
$2(2(x^2+y^2)(x^2+y^2)')
=25(2x\ dx - 2y\ dy)
$
or
$4(x^2+y^2)(2x\ dx-2 y\ dy)
=25(2x\ dx - 2y\ dy)
$
or
$8(x(x^2+y^2)dx-y(x^2+y^2)dy)
=50x\ dx-50y\ dy
$
or
$(8x(x^2+y^2)-50x)dx
=(8y(x^2+y^2)-50y)dy
$
or
$\dfrac{dy}{dx}
=\dfra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/964720",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Real Analysis - Lebesgue integrable functions Let $E$ be a measurable set. Suppose $f \geq 0$ and let $E_k=\{x \in E_k|f(x) \in (2^k, 2^{k+1}] \} $ for any integer $k$. If $f$ is finite almost everywhere, then $\bigcup E_k = \{x \in E |f(x)>0 \},$ and the sets $E_k$ are disjoint.
Prove that $f$ is integrable $\iff$ $\... | We have $$\int_E fdm =\int_{\{x\in E : f(x)>0\}} fdm=\int_{\bigcup_{k\in\mathbb{Z} }E_k } fdm =\sum_{k\in\mathbb{Z}} \int_{E_k} fdm \leq \sum_{k\in\mathbb{Z}} 2^{k+1} m(E_k )\\ =2\cdot\sum_{k\in\mathbb{Z}} 2^{k} m(E_k ).$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/964819",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Lebesgue space and weak Lebesgue space Let $1\le p<\infty$.
We define the weak Lebesgue space $wL^p(\mathbb{R}^d)$ as the set of all measurable functions $f$ on $\mathbb{R}^d$ such that
\begin{equation}
\|f\|_{wL^p}=\sup_{\gamma>0} \gamma (\{x\in \mathbb{R}^d : |f(x)|>\gamma \})^{1/p}<\infty.
\end{equation}
By Chebysh... | On $\mathbb R^n$, the answer is negative: if $p\ne q$, there is a function in $L^q$ that is not in $wL^p$, e.g.,
$$f(x)=\frac{|x|^{-n/q}}{1+\log^2|x|}\tag{*}$$
On a subset of finite measure, Lebesgue spaces are nested: $L^p\subset L^q$ if $p\ge q$. Therefore, we have the inequalities
$$\|f\|_{L^p(K)} \le C_1\|f\|_{wL^q... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/964955",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 1,
"answer_id": 0
} |
How to approach $\displaystyle\lim_{x\to0} (\cos x)^{\frac{1}{x^2}}$ I'm working through some apparently tricky limits (for a basic fellow like me), and I'm not sure how to treat the following situation:
$$\displaystyle\lim_{x\to0} (\cos x)^{\frac{1}{x^2}}$$
How does one deal with powers which include $x$ when evaluati... | Setting $x=2h$ $$\lim_{h\to0}(1-2\sin^2h)^{\frac1{4h^2}}=\left(\lim_{h\to0}\left[1+(-2\sin^2h)\right]^{-2\sin^2h}\right)^{-\frac12\left(\lim_{h\to0}\frac{\sin h}h\right)^2}$$
The inner limit converges to $e$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/965025",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Sum/Product of two natural numbers is a natural number I wanted to prove that the sum and the product of two natural numbers is a natural number. Intuitively it's clear to my why that is true, however I couldn't prove it.
So our lecturer first defined what an inductive set is. Then he defined the Natural numbers as the... | I'm going to do something I rarely do, namely defining $\Bbb N$ to include $0$ as an element. It makes the treatment herein a little easier or at least a little more Peano-conventional; feel free to adjust it, as an exercise, to start $\Bbb N$ at $1$.
From $$0\in\Bbb N,\,\forall n\in\Bbb N (Sn\in\Bbb N),\,a+0=a,\,a+Sb=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/965178",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
a question about field extensions and tower formula if $K|F$ is a field extension & $a_1,a_2,...,a_n$ are the elements of $K$ which are algebraic on $F$ , we know that $[F(a_1,a_2,...,a_n):F]=<\Pi_{i=1}^n[F(a_i):F]$,it can be proved by induction on $n$.
is it also true that $ (\Pi_{i=1}^n[F(a_i):F])$ is divisible by $[... | Hint: Consider the case $F=\Bbb{Q}$, $a_1=\root3\of2$, $a_2=\omega\root3\of2$, where $\omega=(-1+i\sqrt3)/2$ is a primitive third root of unity.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/965245",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proving $\frac{\sin x + \sin 2x + \sin3x}{\cos x + \cos 2x + \cos 3x} = \tan2x$ I need to prove:
$$
\frac{\sin x + \sin 2x + \sin3x}{\cos x + \cos 2x + \cos 3x} = \tan2x
$$
The sum and product formulae are relevant:
$$
\sin(A + B) + \sin (A-B) = 2 \sin A \cos B \\
\sin(A + B) - \sin (A-B) = 2 \cos A \sin B \\
\cos(A + ... | \begin{align}
\frac{\sin x + \sin 2x + \sin3x}{\cos x + \cos 2x + \cos 3x}&=\frac{\sin x + \sin 3x + \sin2x}{\cos x + \cos 3x + \cos 2x}\\
&=\frac{2\sin 2x\cos x+ \sin 2x}{2\cos 2x\cos x + \cos 2x}\\
&=\frac{\sin2x(2\cos x+1)}{\cos 2x(2\cos x + 1)}\\
& = \tan2x
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/965329",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Generate evenly spaced points on 2D graph I want to draw dots on an image that is W by W pixels. The image is stored as a 1-D array of pixels. The pixel (x,y) is at array index x + y * W. I am thinking that I can use a fixed step size, N, and draw a dot at every Nth pixel. What value of N will produce equilateral tria... | I doubt if there is a simple answer to this question. Below is a plot of the "goodness" of the 10 triangles that can be formed using the first five points. Goodness is min side length over max side length. W = 100, and the first point is at (50,0). The horizontal axis is N, from 600 to 1200.
You can see there are 10 p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/965386",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Contour integral with cauchy Calculate integral
$$\oint_{\gamma} \frac{e^{2i z}}{z^4}-\frac{z^4}{(z-i)^3}dz$$
when $\gamma$ is circles $S(0,6)$ parameterization once rotated over space $[2\pi]$.
Is there more to it than just calculate it with cauchy.
\begin{align}
\oint_{\gamma} \frac{e^{2i z}}{z^4}dz- \oint_{\gamma}\f... | Your use of $\;n_i\;$ is odd. The order of the derivative should be put directly, for example
$$\oint\limits_\gamma\frac{e^{2iz}}{z^4}dz=\frac{2\pi i}{3!}\frac{d^3}{dz^3}\left(e^{2iz}\right)_{z=0}=\frac{\pi i}3(-8i)=\frac{8\pi}3$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/965494",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Computing dot products of linear combinations of unit vectors.
For any unit vectors $v$ and $w$, find the dot products (actual numbers) of:
a) $v$ and $-v$
b) $v+w$ and $v-w$
c) $v-2w$ and $v+2w$
I have worked part a :
a) $(v) \cdot (-v) = \cos (180) = -1$
Not getting any ideas on how to work part b and c... Any help... | Use the fact that dot products distribute over addition and dot products are commutative and scalars can be pulled out from either vector. For any scalar $c$, we have that:
\begin{align*}
(\vec v + c \vec w) \cdot (\vec v - c \vec w)
&= \vec v \cdot \vec v - c(\vec v \cdot \vec w) + c(\vec v \cdot \vec w) - c^2(\vec w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/965568",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
Inequality proof of integers My question is from Apostol's Vol. 1 One-variable calculus with introduction to linear algebra textbook.
Page 36. Exercise 7. Let $n_1$ be the smallest positive integer $n$ for witch the inequality $(1+x)^n>1+nx+nx^2$ is true for all $x>0$. Compute $n_1$, and prove that the inequality is tr... | $$1+(n+1)x+(n+1)x^2=(1+nx+nx^2)+(x+x^2)<(1+x)^n+x(1+x)<$$
(inductive hypothesis for first inequality)
$$<(1+x)^n+x(1+x)^n=(1+x)^{n+1}$$
($x>0$ for second inequality)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/965641",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Signature of a finite covering space Suppose $\tilde{M}\rightarrow M$ is a finite (k-fold) covering of the smooth, oriented, compact 4-manifold $M$. Is there a relation between the signatures (http://en.wikipedia.org/wiki/Signature_(topology)) of $M$ and $\tilde{M}$?
I have a line of reasoning involving the Hirzebruch ... | Let $M$ (and hence $\widetilde{M}$) be a closed, connected, oriented, smooth manifold of dimension $4n$, and $\pi : \widetilde{M} \to M$ a smooth $k$-sheeted covering.
As $\pi$ is a smooth covering map, it is a local diffeomorphism, so $\pi^*TM \cong T\hat{M}$. In particular,
$$\pi^*p_i(M) = \pi^*p_i(TM) = p_i(\pi^*TM... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/965795",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Quaternions as a Lie algebra, its derivations Let $\mathbb{H}$ be the algebra of quaternions. It can be proven that each derivation $D:\mathbb{H}\to \mathbb{H}$ is inner that is of the form $\mathrm{ad}x$ for some $x\in \mathbb{H}$. I am to prove that the Lie algebra of derivations of $\mathbb{H}$ is a Lie algebra $\ma... | If we denote $Q\in\mathbb H$ by $q_0+q$, where $q_0$ is the scalar part and $q$ is the vector part, then:
$$
\begin{split}
[P,Q]&=PQ-QP\\
&=\underbrace{p_0q_0-p\cdot q}_{\text{scalar}}+\underbrace{p_0q+q_0p+p\times q}_{\text{vector}}-(\underbrace{q_0p_0-q\cdot p}_{\text{scalar}}+\underbrace{q_0p+p_0q+q\times p}_{\text{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/965882",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Why this is true using Riemann-Roch theorem Let $C$ be a curve of genus $g$ over an algebraically closed field $k$ and $K=k(C)$ the field of rational functions of $C$. Consider $P$ a point at $C$.
What I know:
For each $r\in \mathbb N$, we have $l(rP)\le l((r+1)P)$, because we have the following inequality betweeen the... | Notice that it follows from Riemann-Roch Theorem that $$l(D) = deg(D) + 1 - g$$ if $deg(D) \geq 2g-1$.
Skecthing the proof, if you have $h \in l(W-D)-\lbrace 0 \rbrace$ then $(h) \geq D-W$ and it follows that $$0 = deg((h)) \geq deg(D-W) > 0$$
which is a contradiction.
Take $D = (2g -1)P$ and you will have the result... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/965976",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Jacobian of parametrized ellipsoid with respect to parametrized sphere I'm not even sure how best to phrase this question, but here goes. Given $\theta$ (elevation) and $\phi$ (azimuth), the unit sphere can be parametrized as
$
x = \cos(\theta)\sin(\phi) \\
y = \cos(\theta)\cos(\phi) \\
z = \sin(\theta).
$.
A general ... | The map $(x,y,z)\mapsto (ax,by,cz) = (X,Y,Z)$ takes three variables to three variables, rather than the two variables of your parametrization. The Jacobian matrix of this three-dimensional transformation is
$$J = \begin{bmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{bmatrix}.$$
(So the answer to your first question... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/966079",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Unit vectors in $\Bbb R^n$ Suppose that $x$ and $y$ are unit vectors in $\Bbb R^n$. Show that if $\left\Vert{{x+y}\over2}\right\Vert=1$ , then $x =y$.
Please enlighten me for this problem!
Thanks in advance!
| $$1=1^2=||\frac{x+y}{2}||^2=\frac{1}{4}<(x+y),(x+y)>$$
$$4=<x,x>+2<x,y>+<y,y>=2+2<x,y>$$
$$<x,y>=1$$ So x and y are the same unit vector. Otherwise $|<x,y>|<1$ or $<x,y>=-1$ if $y=-x$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/966159",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Let $f$ be a morphism of chain complexes. Show that if $ker(f)$ and $coker(f)$ are acyclic, then $f$ is a quasi-isomorphism. Let $f$ be a morphism of chain complexes. Show that if $ker(f)$
and $coker(f)$ are acyclic, then $f$ is a quasi-isomorphism. Is the converse true?
I am self reader of homology algebra and I stuc... | We have the exact sequences
$$(1) \quad 0 \to \ker f \to C \to \mathrm{im }f \to 0$$
and
$$(2) \quad 0 \to \mathrm{im }f \to D \to \mathrm{coker }f \to 0.$$
The long exact sequence of homology coming from $(1)$, along with the assumption that $\ker f$ is acyclic, shows that $C \to \mathrm{im} f$ is a quasi-isomorphi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/966253",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Why John Tukey set 1.5 IQR to detect outliers instead of 1 or 2? To define outliers, why we cannot use:
Lower Limit: Q1-1xIQR
Upper Limit: Q3+1xIQR
OR
Lower Limit: Q1-2xIQR
Upper Limit: Q3+2xIQR
| As I recall, Prof. Michael Starbird, in one of his lectures in the recorded series, Joy of Thinking: The Beauty and Power of Classical Mathematical Ideas, answers this question. Dr. Starbird reports having attended the very conference presentation in which Tukey introduced this test, and during which Tukey himself was ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/966331",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 4,
"answer_id": 1
} |
System of two equations with two unknowns - can't get rid of $xy$ The system is:
$x^2 + 2y^2 + 3xy = 12$
$y^2 - 3y = 4$
I try to turn $x^2 + 2y^2 + 3xy$ into $(x + y)^2 + y^2 + xy$ , but it's a dead end from here. Can anyone please help?
| If you look at the second part $$12 y^2 - 3y = 4$$ it is just a quadratic equations the roots of which being $$y_{\pm}=\frac{1}{24} \left(3\pm\sqrt{201}\right)$$ Now, consider $x^2 + 2y^2 + 3xy = 4$ where $y$ is a known parameter and you get another quadratic in $x$ the roots of which being $$x=\frac{1}{2} \left(\pm\sq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/966424",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
implicit equation for elliptical torus I just wondering what the implicit equation would be if an ellipse with major axis a and minor axis b, rotating about the Z axis with a distance of $R_0$. The $R_0$>a and $R_0$>b which means the rotation will result in a non-degenerate torus.
My aim is to determine if some points... | You can obtain this as follows.
If you start with a slice where $y = 0$, you begin with the equation
$$
\frac{z^2}{a^2} + \frac{(x - R_0)^2}{b^2} - 1 = 0
$$
However, this doesn't give you the rotated version; to rotate it about the $z$-axis, simply replace the $x$ by $\sqrt{x^2 + y^2}$, yielding
$$
\frac{z^2}{a^2} + \f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/966503",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Which is the best way to find the complexity? I want to find the asymptotic complexity of the function:
$$g(n)=n^6-9n^5 \log^2 n-16-5n^3$$
That's what I have tried:
$$n^6-9n^5 \log^2 n-16-5n^3 \geq n^6-9n^5 \sqrt{n}-16n^5 \sqrt{n}-5 n^5 \sqrt{n}=n^6-30n^5 \sqrt{n}=n^6-30n^{\frac{11}{2}} \geq c_1n^6 \Rightarrow (1-c_1)n... | For a log-polynomial expression $\sum a_kn^{i_k}\log^{j_k}(n)$, take the highest power of $n$ and in case of equality take the highest power of the $\log$.
In your case, the dominant term is $n^6$, and
$$\frac{g(n)}{n^6}=1-\frac{9\log^2 n}n-\frac{16}{n^5}-\frac5{n^3}.$$
The limit is clearly $1$.
Also note that the seco... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/966835",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Norm of an element in a C*-algebra The following is a part of a proof in Takasaki's Operator theory:
Let $\epsilon>0$ and $A$ is a C*-algebra. For an $x\in A$, put $h=x^*x$ and $u_\epsilon=(h+\epsilon)^{-1}h$. We have then $$||x(1-u_\epsilon)^\frac{1}{2}||=||\epsilon x(h+\epsilon)^\frac{-1}{2}||$$$$~~~~~~~~~~~~~~~~~~~~... | I think it's a typo and my way is correct.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/966925",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
If $p$ is prime and $\sigma(p^k) = n$, then $p\mid (n-1)$ If $p$ is prime and $\sigma(p^k) = n$, then $p\mid (n-1)$.
proof: Suppose $\sigma(p^k) = [p^{k+1} -1]/(p-1) = n$.
Then $n-1 = [p^{k+1} -1]/(p-1) - 1= [p^{k+1} -1 - (p-1)] /(p-1) = [p^{k+1} - p]/(p-1) = p(p^k -1)/(p-1)$ then let $m = (p^k -1)/(p-1)$ be an inte... | It is correct. $m$ is an integer because
$$ \dfrac{p^k-1}{p-1}=p^{k-1}+p^{k-2}+\ldots+p+1.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/967054",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Show convergence in distribution by the continuity theorem So the problem I'm about to solve is to show that:
$X \in \Gamma(a,b)$. Show that
\begin{equation}
\frac{X-E[X]}{\sqrt{Var(X)}} \xrightarrow{d} N(0,1)
\end{equation}
as $a \rightarrow \infty$, by using the continuity theorem.
I've made it this far:
\begin{equat... | Hint: You could consider
$$\exp(-x) = 1 - x + \dfrac{x^2}{2!} - \dfrac{x^3}{3!} +\cdots$$
and
$$(1+x)^k = 1 + kx + \dfrac{k(k-1)}{2!}x^2+ \cdots$$
and
$$\lim_{y \to \infty} \left(1-\frac{x}{y}\right)^y =\exp(-x)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/967121",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
The ring of idempotents Let $R$ be a commutative ring. Then its ring of idempotents $I(R)$ consists of the idempotent elements of $R$, with the same multiplication as in $R$, but with the new addition $x \oplus y := x+y-2xy$. (This addition might look a bit mysterious, but when we identify idempotent elements with the ... | In a boolean ring every element is idempotent and, since the characteristic is $2$, we have $x\oplus y=x+y$. So, for a boolean ring $B$, $I(B)=B$. If $i$ denotes the embedding functor $\mathsf{Bool}\to\mathsf{CRing}_2$, the category of rings with characteristic $2$. This suggest there is an adjunction between $i$ and $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/967237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 0
} |
Inhomogeneous Wave Equation Derivation This is an assignment question which I've been working on to solve the inhomogeneous wave equation $u_{tt} - c^{2}u_{xx} = f(x,t)$.
I separated the equation out into a system of two equations:
$u_{t} + cu_{x} = v$ and $v_{t} - cv_{x} = f(x,t)$.
It says to solve the first different... | The one dimensional wave equation is easy because as you did you can factor it into (setting $c=1$) $(\partial_t-\partial_x)(u_t+u_x)=f$, or calling $v\equiv u_t+u_x$ we have $v_t-v_x=f$, or $\frac{dv(s,x+t-s)}{ds}=f(s,x+t-s)$, so that integrating we get $v(t,x)-v(0,x+t)=\int_0^t f(s',x+t-s')\,ds'$. Now as for the $u_t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/967406",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Can we have $\sum_{n\leq [x]}e^{-\sqrt{\frac{\log x}{r}}}\ll \frac{x}{e^{c \sqrt{\log x}}}$ for some constant $c>0$, where $x>1.$ Let positive interger $n$ is square-free, that is $n=p_1p_2\cdots p_r$ some $r$.
Can we have
$$\sum_{n\leq [x]}e^{-\sqrt{\frac{\log x}{r}}}\ll \frac{x}{e^{c \sqrt{\log x}}}$$
for some const... | No.$$\frac{x}{e^{\sqrt{\log x/\log\log x}}}\ll\sum_{n\le x}e^{-\sqrt{\log x/r}}$$holds. This can easily be seen by the Hardy-Ramanujan theorem.
$|w(n)-\log\log n|<(\log\log n)^{1/2+\epsilon}$ for all $n\le x$ except $o(x)$ numbers by the Hardy-Ramanujan theorem. And the number of square-free numbers less than $x$ are $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/967553",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Show that there do not exist 3 $\times$ 3 matrices $A$ over $\mathbb{Q}$ such that $A^8 = I $and $A^4 \neq I.$. Show that there do not exist 3 $\times$ 3 matrices $A$ over $\mathbb{Q}$ such that $A^8 = I $and $A^4 \neq I.$.
I am aware that the minimal polynomial of $A$ divides $(x^8−1)=(x^4−1)(x^4+1)$.If the minimal po... | Hint: Prove that $x^4+1$ is irreducible in $\Bbb{Q}[x]$. What factors of $x^4+1$ can thus occur as factors of the minimal polynomial of a $3\times3$ matrices with entries from $\Bbb{Q}$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/967686",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Union of Sequences My Analysis professor mentioned the following theorem: If a sequence $(a_n)$ is the union of finitely many disjoint subsequences, and if all the subsequences converge to $l$,then sequence $(a_n)$ converges to $l$.
I'm not entirely sure whether this language is appropriate. Could someone correct it/he... | Sequences are functions which in turn are sets and union of sets is nothing weird. The union of functions need not be a function. A sufficient, but not necessary condition for the union of functions to be a function is that the domains are disjoint.
As the OP has pointed out in this comment, the union of disjoint subse... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/967805",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Can this be solve using modular arithmetic? $k$ is prime $\Rightarrow$ $8k+1$ is prime Is the following statement true or false?
$\forall k \in \mathbb{N}, k$ is prime $\Rightarrow$ $8k+1$ is prime
The answer is that the statement is false because if $k=7$, then $k$ is prime but $8k+1=57$ is not prime.
Is there are ... | There are infinitely many primes $p\equiv1\pmod 3$ (Dirichlet's theorem). And for these primes, $8p+1\equiv 0\pmod 3$, so $8p+1$ is not prime.
I am not sure if this is what you are asking for.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/968034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
What properties does a rank-$1$ matrix have? I have seen a lot of papers mentioning that a certain matrix is rank-$1$. What properties does a rank-$1$ matrix have?
I know that if a matrix is rank-$1$ then there are no independent columns or rows in that matrix.
| Any rank one matrix $A$ can always be written $A=xy^\intercal$ for vectors $x$ and $y$. More precisely...
Proposition: A matrix in $\mathbb{C}^{n\times n}$ has rank one if and only if it can be written as the outer product of two nonzero
vectors in $\mathbb{C}^{n}$ (i.e., $A=xy^{\intercal}$).
Proof. This follows fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/968126",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
"answer_id": 0
} |
Evaluate $\int_{0}^1 x^{p}(\log x)^q dx$ Evaluate
$$\int_{0}^1 x^{p}(\log x)^q dx$$
for $p \in \mathbb{N}$ and $q \in \mathbb{N}$.
| For $\mathbb{R} \ni p>-1$ and $q \in \mathbb{R}$ by setting $x = e^{-t/(p+1)}$ we get
$$\int_0^1 x^{p}(\log x)^q\,dx=\frac{(-1)^q}{(p+1)^{q+1}} \cdot \int_0^\infty e^{-t}t^q \, dt.$$
By definition of the upper incomplete gamma function we get for
$$\int_0^\infty e^{-t}t^q \, dt = \Gamma(1+q).$$
Using this we get
$$\int... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/969282",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
Game Theory - First move vs second move advantage? This question came up in a lunchtime discussion with coworkers. None of us are professional mathematicians or teachers of math, and we weren't sure how to get the answer. I apologize in advance if my question is not rigorous or uses the wrong terminology.
Is there any... | I'm sure there's plenty of practical solutions out there. Here's a trivial solution that proves the existence of such games:
Let's say we play a game where starting from 0, each person gets to add a number from 1 to 9 to the running total. The winner is the person who makes the total be ten.
In this case, the secon... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/969360",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Overlapping Probability in Minesweeper I play a lot of minesweeper in my spare time. Sometimes I'll be in the situation where probabilities of hitting a mine overlap:
Situation on a board
*
*There's one mine in the squares A, B, C, and D (25% chance per square).
AND
*There's one mine in the squares D and E (50% ... | Sorry for this being a long answer. Actual probabilities are at the end if you want to jump straight there.
To work this out we need the underlying probability of any random square being a mine, knowing nothing about the square. Let's call this probability $q$.
$$q = \dfrac{\text{#mines in the grid}}{\text{#squares in ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/969589",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Efficiently calculating the 'prime-power sum' of a number.
Let $n$ be a positive integer with prime factorization $p_1^{e_1}p_2^{e_2}\cdots p_m^{e_m}$. Is there an 'efficient' way to calculate the sum $e_1+e_2+\cdots +e_m$?
I could always run a brute force algorithm to factor $n$ and then calculate the sum directly, ... | As reply to @Roger's post I'm writing this answer. It's possible in $\mathcal{O}(\sqrt{n})$. In fact it will be just a little bit optimized brute-force algorithm.
We should notice, if we found $d$, such that $\left(\nexists d' \in \mathbb{Z}\right) \left( 2 \leq d' <d \wedge d' \mid n \right) \Rightarrow d \in \mathbb{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/969682",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
No Lorentzian metric on $S^{2}$ In the book Riemannian Geometry by Gallot et al there is a remark at the beginning that there is no Lorentzian metric on $S^{2}$. Is it a difficult theorem? Or there is an easy solution? Any hint/idea how to prove this?
| A Lorentzian structure would produce in every tangent space a light cone. In a $2$-dim case like $S^2$ that would be a pair of $1$-dimensional subspaces. Since $S^2$ is simply connected you can smoothly choose them apart. So now you have two $1$-dimensional linear distributions (subspace in the tangent space at each po... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/969759",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Taylor Expansion of complex function $ \frac{1}{1 - z - z^2} $ Knowing that
$$c_{n+2}=c_{n+1}+c_n, \quad n \geq 0,$$
the problem is to prove that we have
$$ \frac{1}{1 - z - z^2}=\sum^{+\infty}_{n=0} c_nz^n.$$
I have tried to transform the function into the form:
$$ A(\frac{1}{z-b_1}+\frac{1}{b_2-z})$$
Then expa... | You may start from
$$c_{n+2}=c_{n+1}+c_n, \quad n\geq0. \tag1$$
Multiply $(1)$ by $z^{n+2}$ and sum over $n\geq 0$ then you obtain
$$
\begin{align}
\sum^{+\infty}_{n=0} c_{n+2}z^{n+2}&=\sum^{+\infty}_{n=0} c_{n+1}z^{n+2}+\sum^{+\infty}_{n=0} c_{n}z^{n+2}\\
\sum^{+\infty}_{n=0} c_{n+2}z^{n+2}&=z\sum^{+\infty}_{n=0} c_{n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/969846",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Prove $\operatorname{stab}_G(g \cdot x) = g\operatorname{stab}_G(x)g^{-1}$ Suppose a group $G$ acts on a set $X$. The stabilizer in $G$ of $x \in X$ is
$$
\operatorname{stab}_G(x) = \{a \in G : a \cdot x = x \}
$$ For each $g \in G$, let
$$
g\operatorname{stab}_G(x)g^{-1} = \{gag^{-1} : a \in \operatorname{stab}_G(x)\... | Let $a\in\operatorname{stab}_G(g\cdot x)$; then $a\cdot(g\cdot x)=g\cdot x$, that is
$$
(ag)\cdot x=g\cdot x
$$
or else
$$
(g^{-1}ag)\cdot x=x
$$
Therefore $b=g^{-1}ag\in\operatorname{stab}_G(x)$, which means that
$$
a=gbg^{-1}
$$
where $b\in\operatorname{stab}_G(x)$. This proves one of the inclusions; can you show the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/969905",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
complexification of $SO(2)$ While computing the complexification of Lie group $SO(2)$, I get the result is all the matrix of the following form
$$\left(\begin{array}{cc}
\frac{e^{t-\sqrt{-1}\theta}+e^{-t+\sqrt{-1}\theta}}{2} & \frac{e^{-t+\sqrt{-1}\theta}-e^{t-\sqrt{-1}\theta}}{2\sqrt{-1}} \\
-\frac{e^{-t+\sqrt{-1}\th... | Consider the map $\mathbb{C}^{\ast}\rightarrow SO_2(\mathbb{C})$ given by
$$
t\mapsto \begin{pmatrix}
\frac{t+t^{-1}}{2} & \frac{i(t-t^{-1})}{2} \cr
-\frac{i(t-t^{-1})}{2} & \frac{t+t^{-1}}{2}
\end{pmatrix}
$$
This is a group isomorphism. So we have a simple description.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/970019",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
A question about a conditional expectation in C*-algebra Let $\Gamma$ be a discrete group. Consider a conditional expectation $\Phi: B(l^{2}(\Gamma))\rightarrow l^{\infty}(\Gamma)$ defined by
$$\Phi(T)=\sum_{g\in \Gamma}e_{g,g}Te_{g,g},$$
where $e_{g,g}$ is the rank-one projection onto $\delta_{g}\in l^{2}(\Gamma)$ (wh... | Note that $\Phi$ has norm one and $\Phi\circ \Phi = \Phi$. Then use Tomiyama's Theorem (Theorem II.6.10.2 from Blackadar's book Operator algebras; thanks Martin!). Actually, the proof of this theorem will show you how to prove the module property.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/970104",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Demonstrating that a function is monotonically increasing/decreasing my question is more of a conceptual one, but i'll use the problem i'm stuck on to keep things clear. I am confused about how to demonstrate whether a function is strictly monotonically increasing or decreasing etc. (i'm using the wrong brackets becaus... | You can see that your function is monotonically for $x<0$ increasing here:
http://www.wolframalpha.com/input/?i=1%2F%28x%5E2%29
For the left (right) side of $0$, you can show strictly increasing (decreasing) by the sign of the derivative:
$$f(x)'=\frac{-2}{x^3}$$
So $f$ is strictly increasing for $x<0$ and strictly dec... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/970183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
How to find all solutions of $\tan(x) = 2 + \tan(3x)$ without a calculator? Find all solutions of the equation $\tan(x) = 2 + \tan(3x)$ where $0<x<2\pi$.
By replacing $\tan(3x)$ with $\dfrac{\tan(2x) + \tan(x)}{1-\tan(2x)\tan(x)}$
I've gotten to $\tan^3 (x) - 3 \tan^2 (x) + \tan(x) + 1 =0$.
I am not sure how to procee... | Let $\tan x=t$. Then, we have
$$t=2+\frac{3t-t^3}{1-3t^2}\Rightarrow (t-2)(1-3t^2)=3t-t^3$$$$\Rightarrow t^3-3t^2+t+1\Rightarrow (t-1)(t^2-2t-1)=0.$$
Hence, we have
$$\tan x=t=1,1\pm\sqrt 2\Rightarrow x=\frac{\pi}{4}+n\pi,\frac{3}{8}\pi+n\pi,-\frac{\pi}{8}+n\pi$$
where $n\in\mathbb Z$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/970284",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Possible eigenvalues of a matrix $AB$ Let matrices $A$, $B\in{M_2}(\mathbb{R})$, such that $A^2=B^2=I$, where $I$ is identity matrix.
Why can be numbers $3+2\sqrt2$ and $3-2\sqrt2$ eigenvalues for the Matrix $AB$?
Can be numbers $2,1/2$ the eigenvalues of matrix $AB$?
| Note that
$$
\pmatrix{1&0\\0&-1}
\pmatrix{a& b\\-(a^2-1)/b & -a} =
\pmatrix{
a & b\\
(a^2 - 1)/b & a
}
$$
Every such product is similar to a matrix of this form or a triangular matrix with 1s on the diagonal. The associated characteristic polynomial is
$$
x^2 - 2ax + 1
$$
Check the possible roots of this polynomial. Th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/970409",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Verify statement about conjugates in symmetric group At http://planetmath.org/simplicityofthealternatinggroups it states the following.
Let $\pi$ be a permutation
written as disjoint cycles
\[
\pi = (a_1, a_2, \ldots, a_k)(b_1, b_2, \ldots, b_l)\ldots
(c_1,\ldots, c_m)
\]
It is easy to check that for every permutatio... | Applying on $\sigma(a_i)$ suppose $(\sigma \pi (\sigma)^{-1})\sigma(a_i)= \sigma \pi(a_i)=\sigma(a_{i+1})$. Now you are clear I think.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/970573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Growth of $\sum_{x=1}^{n-1} \left\lceil n-\sqrt{n^{2}-x^{2} } \right\rceil$ I'm interested in the growth of $$f(n):=\sum_{x=1}^{n-1} \left\lceil n-\sqrt{n^{2}-x^{2} } \right\rceil \quad \text{for}\quad n\rightarrow\infty $$
Progress
(From comments) I've got
$$\frac{f(n)}{n^2} \ge 1-n^{-1} (1+\sum\limits_{x=1}^{n-1} \... | We can write
$$
\frac{f(n)}{n^2} = \frac{\delta_n}{n} + \frac{1}{n} \sum_{k=0}^{n-1} \left(1-\sqrt{1-k^2/n^2}\right),
$$
where $0 \leq \delta_n \leq 1$. When $n$ is not the hypotenuse of a pythagorean triple we have the formula
$$
\delta_n = 1 - \frac{1}{n} - \frac{1}{n} \sum_{k=0}^{n-1} \left\{n - \sqrt{n^2-k^2}\righ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/970654",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Deducing $(\lnot B) \to A$ from $\lnot A \to B$ using Hilbert deductive system As the title says, I've been trying to prove this:
$(\lnot A \to B) \vdash (\lnot B) \to A)$
but unfortunately keep winding up with crazy long steps and then I have no idea where to go. The only axioms I can use are:
\begin{gather}
A \to (B ... | First, Ax4 is provable from the first three axioms.
Second, I'll suggest that you prove this handy formula: {(A→B)→[(C→A)→(C→B)]}. You can prove it from axioms 1 and 2 only, and you only need to use modus ponens/detachment three times to prove it.
Now you can hopefully solve this yourself. If note, the following info... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/970760",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Differential equation of inclined plane I'm having some trouble with the equation $$\frac{d}{dt}\dot{x}=g\sin\Theta \implies \dot{x}(t)=\dot{x}(t=0)+\int_0^t dt'\:g\sin\Theta=\dot{x_0}+g\:t\sin\Theta $$
which appears in page 4 of http://www.astro.caltech.edu/~golwala/ph106ab/ph106ab_notes.pdf. Because of the appearence... | This is just the application of the fundamental theorem of calculus.
That is, for a continuous function $f(x) = \frac{d}{dx}F(x)$ on an interval $(a,b)$, $$\int_a^b f(x)\,dx = F(b)-F(a).$$
For this example, start by integrating both sides from $0$ to $t$
$$\int_0^t \frac{d}{dt} \dot{x}\,dt =\int_0^t g\sin \Theta\, dt.$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/970859",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove that $10 | (n^a - n^b)$. $n$ is a positive integer. Prove that there exists positive integers $a$ and $b$, $(a > b)$ such that $10 | (n^a - n^b)$.
I have tried to prove this by induction on $n$, but I get stuck at the induction step trying to prove it for $n = k + 1$, not knowing how to expand $((k + 1)^a - (k +... | As suggested in the comment by Akiva Weinberger, take $a=5$ and $b=1$.
Then $(k+1)^a-(k+1)^b=(k+1)^5-(k+1)=(k+1)[(k+1)^4-1]$
$=(k+1)(k^4+4k^3+6k^2+4k)=(k+1)k(k^3+4k^2+6k+4)$.
This is divisible by $2$ because $(k+1)k$ is,
and modulo $5$ this is the same as $(k+1)k(k-1)(k^2+1)$, so it is divisible by $5$ too,
and theref... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/970910",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
If $f(x)$ belong to set of probability distributions, then what can be deduced about $\frac{1}{a} f(\frac{1}{a}\cdot x)$ My question is in the context of probability distributions, whose Fourier transforms (characteristic function) almost always exit.
If $f(x)$ be some function such that $ \int_{-\infty}^\infty f(x) \,... | Hint: use change of variable $z=\frac{x}{a}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/971018",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove that if $a|b$ and $a|c$ then $a|(sb+tc)$ for all $s, t \in \mathbb{Z}$ Would this be the same thing as saying "Prove that if $a|b$ and $a|c$ then $a|(sb+tc)$ for any $s, t \in \mathbb{Z}$"? I can do the proof for any integers $s$ and $t$, but if any and all mean the same thing, then I can do this proof. If not, h... | In response to your question about any and all, they generally mean the same thing. The symbol $\forall$ can be read "for all" or "for every" or "for any" or "for each". These all mean the same thing.
This is how you could write the proof, with meticulous rigor.
Let $a,b,c \in \mathbb{Z}$ with $a \mid b$ and $a \mid ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/971191",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Is a path connected covering space of a path connected space always surjective? If $X$ is a path connected topological space, a covering space of $X$ is a space $\tilde{X}$ and a map $p:\tilde{X} \to X$ such that there exists an open cover $\left\{ U_\alpha \right\}$ of $X$ where $p^{-1}(U_\alpha)$ is a disjoint union ... | This is my attempt to prove it.
Let's assume there is some $x\in X$ such that $x\not\in p(\tilde{X})$. Then given the open cover of the definition, $\{U_a\}_{a\in A }$, there exist some $a$ such that $x\in U_a$.
Let's say $U\subseteq \tilde{X}$ is a preimage of $U_a$, $p(U)\cong U_a$ (pick any such $U$). Since $p|_U$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/971253",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Count the number of a kind of matrix I want to count the number of $M\times N$ matrices with $0s$ and $1s$ which have exactly $k$ $1s$ and of which each column and each row has a least one 1. It is a little difficult for me. Could anyone help me ? Thanks in advance!
| The number $\lambda(M, N, k)$, $0 \leq k \leq MN$, of $M \times N$ matrices with precisely $k$ entries $1$ and all other entries $0$ in each column is $${{MN}\choose{k}}.$$
We must subtract from this the number of matrices that fail the row/column condition. Note that any such matrix has a zero row or column, and delet... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/971346",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
A function equal to its integration? It is asked that I find a function such that $$10-f(x)=2\int_0^xf(t)dt.$$ I tried giving a new function F(x) such that ${dF(x)\over dx}=f(x)$, but all I got was a new equation $$F(x)=10x-2\int_0^xF(t)dt.$$ So how do we find such function. Thanks in advance! (I am new to differential... | Differentiate the equation with respect to $t$ and use the Fundamental theorem of calculus: We obtain :
$$ - \frac{ d f }{dt } = 2 f(t) $$
Next, write this equation as follows:
$$ \frac{df}{f} = - 2 dt \implies \int \frac{df}{f} = - 2 \int dt \implies \ln f = -2t + C \implies f(t) = e^{-2t + C} $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/971457",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Dirac delta of x Just so there are no misunderstandings let me first ask whether it is true that:
$$
\int_{-\infty}^{\infty}x\delta(x)\mathrm{d}x=0.
$$
If that is not true, then I don't know anything about the Dirac delta distribution and I will be off to correct this :)
Otherwise I have this question: Why can we take ... | The Dirac delta $\delta$ distribution can be paired with a Schwarz function, but we can make sense of pairing it with more general objects too. For example, for a continuous function $f$, $$\int f(x) \delta(x) \,dx = f(0),$$ which in particular confirms your computation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/971562",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Definition of direct sum of modules? I have just started studying modules, and am trying to figure out the definition of the direct sum of modules but I'm having trouble since different sources seem to give different definitions, for example:
MIT says:
The direct sum of the Mλ is the subset of restricted vectors:
$\big... | For finitely many summands everything agrees. Then $A\oplus B$ is what you wrote, and this holds in the same way for finitely many summands. For infinitely many summands there are differences, and we need to distinguish between $\oplus M_i$ and $\prod M_i$. This is expalined in books on rings and modules. A good exampl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/971683",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Positive Linear Functional on $C[0,1]$ I have an exercise which seems to be missing some information. Or it could be that I really don't need that information at all. Please let me know what you think and give a solution if possible. Thank you in advance.
"A linear functional $f$ on $X = C[0,1]$ is called positive if $... | Actually, you can even give an explicit bound.
Let $C := \langle f,1 \rangle$, where I abused notation and wrote $1$ for the constant function equal to $1$.
Then, for any $g$ with $\|g\| \leq 1$, you must have $|\langle f,g\rangle| \leq C$. Indeed, as both $1+g$ and $1-g$ are nonnegative, you must have $C \pm \langle f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/971777",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Substitution method for solving recurrences piece wise function I don't know how to use the substitution method for the following function:
piece wise function:
$T(n) = c$, if $n=0$
$T(n) = d$, if $n=1$
$T(n)=2T(n-1)-T(n-2)+1$, if $n > 1$
| Let $T(n)=f(n)+an+b$ where $a,b$ are arbitrary constants
$$\implies f(m)+am+b=2[f(m-1)+a(m-1)+b]-[f(m-2)+a(m-2)+b]+1$$
$$\iff f(m)-2f(m-1)+f(m-2)=a+1$$
Set $a+1=0$ and use this
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/971849",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proving a Recursion Using Induction I am trying to prove the following recursion.
$$a(n) = \left\{\begin{matrix}
n(a(n-1)+1) & \text{if } n \geq 1\\
0 & \text{if } n = 0
\end{matrix}\right.$$
is the series definition of $a(n)$. using this, I need to prove that
$$ a(n) = n!\bigg(\frac{1}{0!} + \frac{1}{1!} + \cdots + \... | For $m\ge1,$
If $a(m)=m!\left(\sum_{r=0^{m-}}\frac1{r!}\right)$
\begin{align}
a(m+1) & =(m+1)[a(m)+1] \\[6pt]
& =(m+1)[m!\left(\sum_{r=0^{m-}}\frac1{r!}\right)+1] \\[6pt]
& =(m+1)!\left(\sum_{r=0^{m-}}\frac1{r!}\right)+m+1 \\[6pt]
& =(m+1)!\left(\sum_{r=0^{m-}}\frac1{r!}\right)+\frac{(m+1)!}{m!}
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/971936",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
matrix representation of a permutation in GAP I have wrote the following lines in GAP:
$ gap> G:=PSL(4,4);\\
<permutation~group~of~size~987033600~with~2~generators>\\
gap> p:=SylowSubgroup(G,5);;\\
gap> e:=Elements(p);;\\
gap> e[2];\\
(2,23,27,31,35)(3,25,26,30,37)(4,24,29,33,36)(5,22,28,32,34)(6,51,43,10,79)(7,50,42,... | If you form PSL, you get the permutation action on the projective line.
To get the matrix representation work with SL instead:
gap> G:=SL(4,4);
SL(4,4)
gap> p:=SylowSubgroup(G,5);
<group of 4x4 matrices of size 25 over GF(2^2)>
gap> e:=Elements(p);;
gap> e[2];
[ [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ], [ Z(2)^0, Z(2)^0, 0*Z... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/972000",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Inverse of a function from $\mathbb{Z}^+ \times \mathbb{Z}^+ \to \mathbb{Z}^+$ Given the function $f: \mathbb{Z}^+ \times \mathbb{Z}^+ \to \mathbb{Z}^+$
$f(a,b) = b^2 +a$, if $b>a$
or $a^2 +a + b$, if $b<a$,
which associates $a,b \to z$, find its inverse, which associates $z \to a, b$.
| If you look back at the origin of the problem, this function is a bijection from $\mathbb N \times \mathbb N$ to $\mathbb N$.
To find the inverse, start by finding a $k$ so that $k^2 \leq z < (k+1)^2$.
Than see what happens depending if $z-k^2$ is larger or smaller than $k$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/972152",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Probability of failure, given condition (elementary probability) A machine needs at least 2 of its 3 parts to work correctly. The probability of part 1 failing is 1/2, the probability of part 2 failing is p and the probability of part 3 failing is also p. We are asked to find the probability that the machine fails. We ... | Maybe this helps. The probability that the machine fails is the probability that parts 1 and 2 fail and 3 doesn't fail, or parts 1 and 3 fail and 2 doesn't fail, or parts 2 and 3 fail and 1 doesn't fail, or all three parts fail. These are all disjoint events, so you can sum the probabilities for each one. For example, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/972221",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Show that the recursive sequence converges and find its limit Let $x_1>2$ and $x_{n+1} := 1 + \sqrt{x_n - 1}$ for all $n \in \mathbb{N}$. Show that $\{x_n\}$ is decreasing and bounded below, and find its limit.
I showed that it's bounded below ($x_n>2$ for all $n$) using induction. But then I don't know how to do the r... | If you already prove that $x_n> 2$, then $$\dfrac{x_{n+1}-1}{x_n -1} = \dfrac{1}{\sqrt{x_n -1}} < 1 $$
$$\implies x_{n+1} -1 < x_n -1$$
So $x_n$ is decreasing and bounded below. The limit point is already in the other answers.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/972294",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Propositional logic and distributive law I am having trouble trying to understand how this question passes from this point
$$
( ( p\vee q )\wedge (p \vee \neg r ) \wedge (\neg q \vee \neg r ) ) \vee ( \neg p \vee r )
$$
to this
$$
(p\vee q \vee \neg p \vee r)\wedge(p \vee \neg r \vee \neg p \vee r)\wedge(\neg q \... | The first formula is of the form
$$(A\land B\land C)\lor D$$
while the second one is
$$(A\lor D)\land(B\lor D)\land(C\lor D)\,.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/972382",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How do I find this partial derivative I have the following function u(x,y) defined as:
$$u(x,y) = \frac {xy(x^2-y^2)}{(x^2+y^2)}$$ when x and y are both non zero,
and $u(0,0)=0$
I want to compute its partial derivative $u_{xy}$ at (0,0).
How do I do this?
| You will need $\frac{\partial u}{\partial y}(x,y)$ and $\frac{\partial u}{\partial y}(0,0)$. The first one you can get using the quotient rule. And for the other, remember that by definition, we have: $$\frac{\partial u}{\partial y}(0,0) = \lim_{t \to 0} \frac{u(0, 0+t) - u(0,0)}{t}$$
With this in hands, we use the def... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/972501",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Number of Curvature Maxima of a 2D Cubic Bezier curve I am trying to prove that a standard cubic Bezier curve can only have at most 2 curvature maxima over $t \in [0,1]$. Assuming that no 3 adjacent control points are colinear, the curvature will either have 2 true local maxima, and the curvature at the endpoints will ... | Have you tried writing out the formula for the curvature directly in terms of the polynomials? Note that a standard Bezier curve is really just a way of writing a general cubic curve, so "Bezier" is a red herring here: you're really asking if an arbitrary cubic curve can have more than two curvature extreme on its who... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/972574",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Orthogonal Vectors in a 2D Lattice with minimum area I came across an interesting problem in my research (not a mathematician). Here it goes:
Suppose, there is a 2D lattice $\Lambda$ in the X-Y plane with basis vectors $\vec{a}$ and $\vec{b}$, which are not orthogonal to each other. I want to find a rectangle in this l... | THIRD answer: It turns out the condition I gave in my second answer is necessary and sufficient for existence; I can also show a two parameter family, I'm afraid to minimize could be algorithmic but not formulaic. i will have time for that aspect later. Given Gram matrix with $A,B,C$ as before,
we have the equation
$$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/972657",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Given the following derivatives, find the integrals
Find the derivatives of $\ln(x+\sqrt{x^2+1})$ and $\arcsin(x)$, and use the result to find the integrals of the following functions:
*
*$$ \dfrac{1}{ \sqrt{ \pm x^2 \pm a^2 }} $$
*$$ \sqrt{\pm x^2 \pm a^2} $$
Except for the cases where both are minuses. $a$ is a ... | Hint. You may use an integration by parts for the second family:
$$
\begin{align}
\int \sqrt{\pm x^2 \pm a^2} \:{\rm{d}}x&=x\sqrt{\pm x^2 \pm a^2}-\int \frac{x \times\pm x }{\sqrt{\pm x^2 \pm a^2}}\:{\rm{d}}x\\
&=x\sqrt{\pm x^2 \pm a^2}-\int \frac{\pm x^2}{\sqrt{\pm x^2 \pm a^2}}\:{\rm{d}}x\\
&=x\sqrt{\pm x^2 \pm a^2}-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/972735",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.