Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Help with Maclaurin series of $\cos(\ln(x+1))$? Hi I've almost completed a maths question I am stuck on I just can't seem to get to the final result. The question is:
Find the Maclaurin series of $g(x) = \cos(\ln(x+1))$ up to order 3.
I have used the formulas which I won't type out as I'm not great with Mathjax yet sor... | Since you also don't need very high orders, the straightforward calculation of derivatives is tractable, though not preferable computationally. If $f(x) = \cos \log(1+x)$, then $$f'(x) = -\frac{\sin \log (1+x)}{1+x}, \quad f'(0) = 0.$$ Then $$f''(x) = -\frac{\cos \log (1+x)}{(1+x)^2} + \frac{\sin \log(1+x)}{(1+x)^2},... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3080664",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Sylow 2 subgroups of S4 I am trying to find all the Sylow 2 subgroups of S4 using Sylow’s theorems. Now, I know that a Sylow 2 subgroup of S4 has size 8, and that there are either 1 or 3 of them (as the number of of Sylow 2-subgroups has form 1+2k and divides 3, the index).
Now my lecturer states “stabilisers of the 3 ... | 1) Yes, you've got it. There are three unique ways to write $\{1,2,3,4\}$ as a union of two sets of size two. These are $\{1,2\}\cup\{3,4\}$, $\{1,3\}\cup\{2,4\}$, and $\{1,4\}\cup\{2,3\}$. These should not be called "bijections of $\{1,2,3,4\}$", but this is what was probably meant. Let's look at the first one, the on... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3080776",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 1,
"answer_id": 0
} |
Are the letters $O$ and $\infty $ homeomorphic? I feel they are homeomorphic
But if we remove the intersection point from $\infty $ we get two disconnected components, right ?
| As you have realized, a useful topological invariant is the set of path components $\pi_0$. A continuous map $f\colon X\rightarrow Y$ must induce a map on path components $f\colon \pi_0(X)\rightarrow \pi_0(Y)$. This is because it respects the equivalence relation which defines path components; if $\beta\colon I\rightar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3080920",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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What does it mean to find the distance from the origin to a plane in $\mathbb R^3$? In do Carmo, one exercise gives a plane in $\mathbb R^3$, $ax +by +cz+d = 0$, and tells us to show that $|d|/\sqrt{a^2 + b^2 + c^2}$ measures the distance from the plane to the origin.
However, this seems a bit ambiguous since we don't ... | For the plane $ax+by+cz + d = 0$,
the normal vector is: $$\hat n = <a,b,c>$$
The vector from the plane to any arbitrary point is: $$\hat v = <(x-x_0),(y-y_0),(z-z_0)>$$
If we consider the origin in particular, $(x_0,y_0,z_0) = (0,0,0)$ so, $$\hat v = <(x),(y),(z)>$$
The MINIMUM distance from the origin to the plane is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3081076",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Proof verification: Show that $f$ is continuous if $f(\overline{A})\subset\overline{f(A)}$.
Let $X,Y$ be metric spaces and define $f: X\to Y$. Show that $f$ is continuous iff $f(\overline{A})\subset\overline{f(A)}$ for each $A\subseteq X$.
My proof: $\Rightarrow$ Let $f:X\to Y$ be continuous and $A\subseteq X$.
Let ... | A more direct approach from the definition of closure:
Let $f$ be continuous and suppose $y \in f[\overline{A}]$. So $y=f(x)$ with $x \in \overline{A}$. Now let $O$ be an open neighbourhood of $y$, then $f^{-1}[O]$ is an open neighbourhood of $x$, so $f^{-1}[O] \cap A$ is non-empty (as $x \in \overline{A}$), say that $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3081160",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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If union of n subspaces of V is a subspace of V, then one of the n subspaces must contain the other n-1 subspaces Help prove this? I can prove for n=2, but I'm stuck on proving it for general n. Thanks!
My proof for n=2
Forward direction: Consider A and B and A $\cup$ B is a subspace of V. We prove by contradiction by ... | Let $U_1, U_2, \dots, U_n$ be subspaces of $V$ such that their union is also a subset of $V$.
We can suppose that $U_1 \nsubseteq U_2 \nsubseteq \dots \nsubseteq U_{n-1} \nsubseteq U_n \nsubseteq U_1$, because if $U_i \subseteq U_j$ for $i \ne j$ then we use induction on $n$ to conclude that there is $k \in \{1, \dots,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3081296",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Differentiation under the integral sign - what transformations to use? Need some help with this integral
$$I (\alpha) = \int_1^\infty {\arctan(\alpha x) \over x^2\sqrt{x^2-1}} dx$$
Taking the first derivative with respect to $\alpha$
$$I' (\alpha) = \int_1^\infty { dx\over (1+\alpha^2 x^2) x\sqrt{x^2-1} }$$
What tran... | Substitute
$$u=\sqrt{x^2-1}\implies du=\frac{x}{\sqrt{x^2-1}}dx\implies dx=\frac{\sqrt{x^2-1}}{x}du$$
Then
$$\int { dx\over (1+\alpha^2 x^2) x\sqrt{x^2-1} }=\int { du\over x^2(1+\alpha^2 x^2)}=\int { du\over (u^2+1)(a^2u^2+a^2+1) }$$
Perform partial fraction decomposition
$$\int { du\over (u^2+1)(a^2u^2+a^2+1) }=\int\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3081395",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
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If $\ln(x)$ is gamma distributed, what is the distribution of $x$? Additionally, if someone could help calculate the mean and variance of $X$, that would be greatly appreciated.
| Just do derive what gt6989b said:
$$
\frac{f_Y(\ln x)}{x} = \frac{\beta^\alpha}{\Gamma(\alpha)x}\ln(x)^{\alpha-1}e^{-\beta\ln(x)} = \frac{\beta^\alpha}{\Gamma(\alpha)}\ln(x)^{\alpha-1}e^{-\beta}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3081506",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Why can't I make the substitution $ u = \sin (ax + b) $ to evaluate $ \int \sin (ax + b) \cos (ax + b) dx$? Evaluate $ \int \sin (ax + b) \cos (ax + b) dx$?
To do this, I started of by substituting $ u = \sin (ax + b) $. That made $ du = cos (ax + b) \cdot a $ and wrote the integral as $ \frac 1a \int u \ du $ to get t... | You can solve with the substitution
u=sin(ax+b) then,
du=acos(ax+b)dx so,
∫sin(ax+b)cos(ax+b)dx=∫(u/a)du=(u^2)/(2a)=sin^2(ax+b)/(2a)
which is same with your answer-1/4a
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3081656",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
For polynomial $g(x)$ satisfying $(g(a))^2+(g'(a))^2=0$, evaluate $\lim_{x\to a}\frac{g(x)}{g'(x)}\left\lfloor\frac{g'(x)}{g(x)}\right\rfloor$
If $g(x)$ is a polynomial function and
$$(g(\alpha))^2+(g'(\alpha))^2=0$$ then evaluate
$$\displaystyle \lim_{x\rightarrow \alpha}\frac{g(x)}{g'(x)}\bigg\lfloor \frac{g'(x)}{g... | $lim_{x\rightarrow a} \frac{g(x)}{g'(x)}=0$
You can apply L'Hospital's Rule, and get: $ lim_{x\rightarrow a} \frac{g(x)}{g'(x)}=lim_{x\rightarrow a} \frac{g'(x)}{g''(x)}$
if $g''(a)\neq 0$ you are done, otherwise you can keep going until you get $ lim_{x\rightarrow a} \frac{g^{(n)}(x)}{g^{(n+1)}(x)}=lim_{x\rightarrow ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3081913",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Any way to solve $|x-8| = |2-x|-6$ algebraically? Everything I've tried has given me $x = 2$ (which is obviously incorrect, since $-6 \neq 6$).
The actual answer is $x \geq 8$ which I obtained by observing a graph.
Would love assistance!
| For this equation $$|x-8| = |2-x| - 6$$ there are two modulus terms $|x-8|$ and $|2-x|$ so these two terms will behave differently at the points $x=2$ and $x=8$. Now break the terms in the following cases:
Case1:
$x>=8$ , this means
$-x\leq-8$ or $2-x\leq-6$ which reduces the equation to
$$ x-8 = x-2-6$$
which reduce... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3082011",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
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Show that lower semi-continuous function attains it's minimum. (Proof verification) (By contradiction) Let $f: [0,1]\to \mathbb{R}$ be a lower semi-continuous function, then
$$ \liminf_{x\to a} f(x) \geq f(a), \forall a \in [0,1]$$
I have to prove that $f$ attains its minimum on $[0,1]$, that is:
$\exists x_0 \in [0,1]... | Since $f$ is lower-semicontinuous, for each $x\in [0,1],$ there is an open interval $I_x\subseteq [0,1]$ such that $\inf\{f(y):y\in I_x\}\ge f(x)-1.$ The $I_x$ form an open cover of $[0,1]$ so passing to a finite subcover, we conclude that $f$ is bounded below.
So, letting $y=\inf\{f(x):x\in [0,1]\}$, we can find a seq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3082099",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Prove the inequality $\frac{e^x+e^{-x}}{2} \leq e^{x^2/2}$ for all real numbers $x$. How do I prove what's written in the title? I was able to get an incomplete proof for the case $x>2$. Here's my try: Use $e^x = \sum_{j=1}^{\infty} \frac{x^j}{j!}$. Now we can see that if $x$ is a real number, then: $$e^{x^2/2}-e^x/2-... | Your computation is flawed. The expansion of $\cosh{x}$ is $\sum_{j \geq 0}{\frac{x^{2j}}{(2j)!}}$, but the expansion of $e^{x^2/2}$ is $\sum_{j \geq 0}{\frac{x^{2j}}{2^j \cdot j!}}$.
So you just need to prove that $j!2^j < (2j)!$ for each $j$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3082185",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 0
} |
Unipotent matrix similar to an upper-triangular matrix "Any unipotent matrix is similar to an upper-triangular matrix with 1's on the diagonal"...
This is usually alleged, but I have no idea how to demonstrate that, starting with the definition : $A$ is unipotent if and only if there is $k\in \mathbb{N}$ so that $(A-I_... | Here I work over the complex field $\Bbb C$.
The main steps are (1.): show that $1$ is the only possible eigevalue of $A$; (2.) cast $A$ into Jordan form. To wit:
First, look at what the condition
$(A - I_n)^k = 0 \tag 1$
reveals about the eigenvalues of $A$: that they must all be $1$, for if
$A \vec x = \lambda \ve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3082463",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Why is there a unique ordinal $\alpha$ for every infinite cardinal $\kappa$ such that $\kappa = \aleph_\alpha$? For finding the $\alpha$ I literally can't get any further than writing out the definitions.
For the second part:
Suppose $\aleph_\alpha = \aleph_\beta$ and $\beta \neq \alpha$. Then either $\alpha \in \beta$... | First, use transfinite induction to prove that $\alpha\le \aleph_{\alpha}$ for all ordinal alpha.
From that we get $\kappa\le\aleph_{\kappa}$, so the following is well defined: $$\alpha=\min\{\beta\in On\mid \kappa\le \aleph_{\beta}\}$$
Now try proving that this $\alpha$ is indeed the alpha you are searching for.
For t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3082612",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$f(x) = (x-2)(x-4)(x-6) +2$ then $f$ has all real roots between $0$ and $6$. True or false? $f(x) = (x-2)(x-4)(x-6) +2$ then $f$ has all real roots between $0$ and $6$
$($ true or false$)?$
Here
$f(0) = -46$ and $f(6) = 2$ since function is continuous so it must have at least one root between $0$ and $6$, but how to ch... | Well,
$$\alpha >0 \to f(-\alpha)=2-(\alpha+2)(\alpha+4)(\alpha+6) < -46$$
and
$$f(6+\alpha)=2+(4+\alpha)(2+\alpha)(\alpha)> 2$$
So at the very least all its real roots are $\in (0,6)$
You just need to show all of its roots are real.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3082722",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
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Boundedness of a linear operator
Let $X$ be a real normed linear space of all real sequences which are eventually zero with the 'sup' norm and $T:X \to X$ be a bijective linear operator defined by $$T(x_1,x_2,x_3,....)=\left(x_1,\frac{x_2}{2^2},\frac{x_3}{3^2},....\right)$$
How to check whether $T$ and $T^{-1}$ is ... | *
*$\sup_n \{\frac{|x_n|}{n}\} \le \sup_n\{|x_n|\} =||x||.$
*Your considerations concerning $T^{-1}$ are correct.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3082810",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Do siblings share 50% of their genes? (This is my first question on Maths Stack Exchange, I'm not sure if this is a maths question or a physics question ... or something else.)
I was watching this video about psychology, and it it the presenter asserts (at about 5:45) that:
Identical twins share ... 100% of their gene... | Assuming the parents share no common alleles , then each twin will inherit $50\%$ of each parent's set. So for every gene, there is $50\%$ probability of both twins inheriting the same allele from a parent. There for the expectation is that the twins share half of their alleles.
However, since it is likely that the pa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3082903",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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It is true that $\overline{(0,\epsilon)\cup(\mathbb Q\cap(\epsilon,1))}=[0,1]$ for $\epsilon\in(0,1)$? Let $\epsilon\in(0,1)$ and $E=(0,\epsilon)\cup(\mathbb Q\cap(\epsilon,1))$. It is true that $\overline{E}=[0,1]$?
I know that $\overline{E}\subset[0,1]$, but how to show that $\overline{E}\supset[0,1]$?
I've tried to ... | The result is indeed true. Here is a way to show it.
Since $E$ is just a finite union of sets, we have
$$
\overline{E} = \overline{(0,\varepsilon) \cup (\mathbb{Q}\cap(\varepsilon,1))}=\overline{(0,\varepsilon)}\cup\overline{\mathbb{Q}\cap(\varepsilon,1)} = [0,\varepsilon] \cup \overline{\mathbb{Q}\cap(\varepsilon,1)}.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3083042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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When does the first repetition in $\;\lfloor x\rfloor, \lfloor x/2 \rfloor, \lfloor x/3\rfloor, \lfloor x/4\rfloor, \dots\;$ appear? Let $\lfloor x\rfloor$ denote the floor of $x$.
When does the first repetition in $\lfloor x\rfloor$, $\lfloor x/2\rfloor$, $\lfloor x/3\rfloor$, $\lfloor x/4\rfloor$, ... approximately ... | It's essentially the same as Jyrki Lahtonen's answer, but they invited me, so here's mine. Well, it's the same until the part where I go into detail about estimating where in that interval of potential values we actually get the first pair of equal values.
Let the sequence $a_n$, for $n=1,2,\dots$, be defined as $\left... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3083192",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 4,
"answer_id": 2
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Finding limit of $\sin(x^2/2)/\sqrt{2}\sin^2(x/2)$ as $x\rightarrow0$ Can anybody help me find the limit as $x$ tends to $0$, for $$\frac{\sin(x^2/2)}{\sqrt{2}\sin^2(x/2)}.$$ How can I simplify the expression or use equivalent transformations to find a limit (without using L'Hospital)?
| As $\dfrac{\sin x}x=1$, in multiplicative expressions you can replace $\sin x$ by $x$.
Hence
$$\lim_{x\to0}\frac{\sin\left(\dfrac{x^2}2\right)}{\sqrt2\,\sin^2\left(\dfrac x2\right)}
=\lim_{x\to0}\frac{\dfrac{x^2}2}{\sqrt2\,\left(\dfrac x2\right)^2}=\sqrt2.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3083288",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Smoothness is local Let us consider a map $f:M\to R$ where $M$ is a smooth manifold. If every point $p\in M$ has a neighborhood $U$ such that $f|_U$ is smooth, prove that $f$ is a smooth function.
My idea is to prove that any two coordinate charts from any two atlases are smoothly compatible (if $f|_U$ is smoothly than... | I think that is better to prove the statement by definition. Given $p\in M$ and $U$ neighborhood of $p$ such that $f|_U$ is smooth, there exists coordinate charts $(U\cap U_\alpha, \varphi_\alpha|_{U\cap U_\alpha})$ (where $(U_\alpha, \varphi_\alpha)$ is a chart of $M$) and $(V,\psi)$ of $p$ and $f|_U(p)=f(p)$ respecti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3083427",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Even holomorphic function on the punctured disk has a primitive Let $f$ such that $f$ is holomorphic on $\{z|0<|z|<1\}$, and $f$ is even.
I need to show that f has a primitive. Any ideas?
| Let $0 < r < 1$. Then we have
$$\int_{\partial B_r(0)} f(z)\, dz =
\int_0^{2\pi} f(re^(it)ire^{it}\, dt
= \int_0^{\pi} f(re^{it})ire^{it}\, dt + \int_\pi^{2\pi} f(re^{it})ire^{it}\, dt.$$
For the second integal we have
$$ \int_\pi^{2\pi} f(re^{it})ire^{it}\, dt
= \int_0^{\pi} f(re^{i(t+ \pi)})ire^{i(t+ \pi) }\, dt
=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3083547",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Discrete Spherical Symmetry Group Take two spheres each having a certain number (say 5) of identical dots on them. What is the approach to proving/disproving that they are equivalent under the set of spherical rotations?
One could label the points: say (A,B,C,D,E),(1,2,3,4,5)
Align (A,1),(X,Y) with (X,Y) being success... | Well, you could calculate the various point-to-point distances in either setting. When the arrangements would be different then the set of distances also will come out different. You not even are forced to calculate them all: the first mismatch here is enough to deside.
--- rk
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Does $\partial B(x_0, r) \subseteq \{x \in X : d(x_0,x) = r \}$ hold in an arbitrary metric space? Let $X$ be a metric space, and let $B(x_0, r)$ denote the open ball of radius $r$ centred at $x_0 \in X$.
Does the statement $\partial B(x_0, r) \subseteq \{x \in X : d(x_0,x) = r \}$ hold true always?
A similar question ... | Let $(X,d)$ be a metric space, and let $B(x,r)\subseteq X$ be an open ball. Since $\text{Int}\left(B(x,r)\right)=B(x,r)$ (an open ball is an open set) and $\text{Cl}\left(B(x,r)\right)\subseteq\{y \in X :d(y,x)\leq r\}$ (a closed ball is a closed set), it follows that we have
\begin{aligned}\partial B(x,r)=\text{Cl}\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3083769",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Exponential series with $k$ as base I've tried to understand why $\displaystyle\sum_{k=0}^{\infty} \frac{k^x}{k!}$ for lets say $x = 4$ equals $15e$.
It's clear why $\displaystyle\sum_{k=0}^{\infty} \frac{x^k}{k!} = e^x$ and that $\displaystyle\sum_{k=0}^{\infty} \frac{1^k}{k!}=e$
It's also unclear for me why $\displa... | It is very more simple than you think, it is only a recursive propertie. When $x=1$
$$
\sum_{k=0}^{\infty}\frac{k}{k!}=\sum_{k=1}^{\infty}\frac{k}{k!}
$$
$$
\sum_{k=0}^{\infty}\frac{k}{k!}=\sum_{k=1}^{\infty}\frac{k}{(k-1)!\,k}
$$
$$
\sum_{k=0}^{\infty}\frac{k}{k!}=\sum_{k=1}^{\infty}\frac{1}{(k-1)!}
$$
$$
\sum_{k=0}^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3083891",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Is the textbook solution wrong by a sign? Laurent Series
Find the Laurent series of $\frac{e^z}{z^2 -1}$ about $z = 1$.
Here is my solution:
Factor denominator
$\frac{e^z}{(z-1)(z+1)}$
let $w = z - 1$, and so $z = w + 1$, substitute in
$\frac{e^{w+1}}{w(w+2)}$
Do partial fraction decomposition to get rid of the expon... | You considered $e^{w+1}$ as a rational function (which is not!). Instead you should expand it at $w=0$ as
$$e^{w+1}=e\cdot e^w=e \left(1+w+\frac{w^2}{2}+\frac{w^3}{6}+\dots\right).$$
Hence, after decomposing the rational function $\frac{1}{w(w+2)}$,
$$\frac{e^{w+1}}{w(w+2)}=e^{w+1}\left(\frac{1}{2w} - \frac{1/4}{1+w/2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3084133",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Is there any property shared by all possible matrices $M$ such that $A=M\cdot B$, with $M$ being lower triangular and A and B 1 dimension arrays I'm a little rusty on my linear algebra, but I would like to explore the solutions for the matrix $M$ that transforms $A$ into $B$.
I generated code to find some solutions for... | Note that you can look at each row of $M$ independently. For row $i$ you have one equation and $i$ unknowns, so you expect typically for there to be an $i-1$ dimensional space of solutions for that row.
More specifically, let $b_i$ be the $i$th entry of $b$, $[a]_i$ the vector consisting of the first $i$ entries of $a$... | {
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CW-Structure of Spin(n) I cannot find any information about the CW-Structure of $Spin(n)$ groups. Clearly $\pi_1=\pi_2=0$ and I think $H_3(Spin(n))=\mathbb Z$ $(n\geq 5)$, so the $3$-skeleton is $S^3$. What is the $4$-skeleton?
| For each $n$ there is a fibration sequence
$$Spin_n\rightarrow Spin_{n+1}\rightarrow S^n$$ covering the corresponding fibration sequence $SO_n\rightarrow SO_{n+1}\rightarrow S^n$. The point is that the inclusion $Spin_n\hookrightarrow Spin_{n+1}$ is $(n-1)$-connected, so the $4$-skeleton of $Spin_n$ is the same as the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3084420",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Math competition problem involving ratios of areas. What's a good way to attack similar problems? 9 lines each separately partition a square into two quadrilaterals with areas having the ratio 2: 3. Show that 3 of these lines intersect at the same point.
Any answer or hints is greatly appreciated, as I'm very puzzled b... | You ask : "What is a good way to attack such problems" ?
Instead of giving a solution that would not differ much from the solutions given by @Aretino or @Jaap Scherphuis, I will stress a feature that is often useful in these issues. I will call it pompously "the principle of area balance". Take a look at the following ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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For which values of $x$ the matrix is invertible? The following matrix has coefficients in $\Bbb Z_{11}$:
$\left(\begin{matrix}
1 & 0 & 3 & 0 & 5 \\
0 & 3 & 0 & 5 & 0 \\
3 & 0 & x & 0 & 7 \\
0 & 5 & 0 & 7 & 0 \\
5 & 0 & 7 & 0 & 9
\end{matrix}\right)$
To determine for which values of $x$ it is invertible, I tried to fin... | The original matrix $A$ will not be invertible if and only if there is a nonzero vector $v=(v_1,\ldots,v_5)^T$ such that $Av=0$.
By the pattern of zeros of $A$ we see that the equations from $Av=0$ for $v_2,v_4$ are independent of those for $v_1,v_3,v_5$. Moreover we have $3v_2+5v_4=0=5v_2+7v_4$, which are independent ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3084734",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Inner product on a sequence and its limit I am stuck on a question, and it seems like I'm missing a really obvious Cauchy-Schwarz application or something, but I am left scratching my head.
Let $(x_n):n \in \mathbb{N}$ be a sequence in a Hilbert space $H$. Let $x$ satisfy $\|x_n\|\to \|x\|$ and $\langle x,x_n\rangle \t... | Also the left term tends to zero as it is
$$
\langle x_n, x_n - x \rangle = ||x_n||^2 -\langle x_n,x \rangle \to ||x||^2 - ||x||^2 = 0
$$
where I used both the hypotheses.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Probability of picking a number from a set of unique integers Suppose I have a set of $k$ integers such that every number is unique.
Let $A = \{1,2,3,4,...,k\}$
Now suppose that we rearrange these numbers to a random permutation in the set. I want to find the probability of finding a fixed number $x$ at any position o... | Hint:
You are wrong at the second place in the set.
$P(x=2)=P(x\neq1 )\cdot P(x=2|x\neq1)=(1-\frac{1}{k})\cdot \frac{1}{k-1}=\frac{k-1}{k}\cdot \frac{1}{k-1}=\frac{1}{k} $
This is because you already know it is not in the first place.
Now try to look for the third one, and then the fourth..
And you can ask yourself, wh... | {
"language": "en",
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Use the Central Limit Theorem to deduce that if $λ$ is large, then $X$ approximately has a normal distribution.
The time instants of incoming requests at a data server can be modelled with a Poisson process.
Let $X$ be the number of requests in one hour and let $λ$ be the intensity (requests per hour) of the Poisso... | Consider two Poisson iid variables of parameter $\lambda=1$.
Their pdf is $$p_1(k)=\frac1{ek!}.$$
The pdf of the sum of these variables is given by
$$p_2(k)=\sum_{i+j=k}\frac1{e^2i!j!}=\frac1{e^2k!}\sum\binom ki=\frac{2^k}{e^2k!}$$ which is simply a Poisson law of parameter $\lambda=2$.
More generally, you could show t... | {
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Why does this math trick work? 35 by 11 is 385 because 3+5 is 8, so it's the digit in the middle.
Same for:
72 by 11 is 792 because 7+2 is 9, so it's the digit in the middle.
I see it works because 35 by 10 is 350, or 72 by 10 is 720. The 0 is replaced with the extra digit. The last digit is 5 by 1 or 2 by 1, so it sta... | When performed in written calculation,
$$ab\times11$$ is
$$\ \ \ \ ab\\ab\\\ \ \overline{acb}$$
so that the digits are $a,a+b,b$. This breaks when there is a carry, e.g. $76\times11=836$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Finding Dual Basis from a basis in $\mathbb R^2$ I am given the fact that these two vectors form the basis B of $\mathbb R^2$:
$$
B=\{\begin{bmatrix} 2 \\ 1 \\ \end{bmatrix} \begin{bmatrix} 3 \\ 1 \end{bmatrix}\}
$$
and then asked to find the dual basis or the basis of $(\mathbb R^2)^*$.
I would really appreciate if a... | $(\mathbb{R}^2)^*$ consists of linear maps $\ell:\mathbb{R}^2\to \mathbb{R}$ which have standard matrix representation
$$ \begin{bmatrix}
a&b
\end{bmatrix}$$
where $a=\ell(e_1)$ and $b=\ell(e_2)$ for $e_1,e_2$ the standard basis. Let's write your basis vectors as $b_1,b_2$ respectively. The dual basis is the basis ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find the sum of $x_1+x_2+x_3$ of intercept points Suppose that the straight line $L$ meets the curve $y=3x^3-15x^2+7x-8$ in three points $(x_1,y_1)$, $(x_2,y_2)$ and $(x_3,y_3)$. Then $x_1+x_2+x_3=?$
A) 3 $\quad$ B) 4 $\quad$ C) 5 $\quad$ D) 6 $\quad$ E) 7
At the beginning, my main idea is to use ... | Let $y=ax+b$ the equation of the straight line $L$. Then $x_1,x_2,x_3$ are solutions of the equation
$3x^3-15x^2+(7-a)x-8-b=0$.
Then Vieta says: $x_1+x_2+x_3= - \frac{-15}{3}=5.$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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What are the completions of first-order group theory? A completion of some theory $T$ (i.e. set of first order statements $T$) is a consistient theory $T' \supseteq T$ such that for every first order statement $\phi$, either $\phi \in T'$ or $\lnot \phi \in T'$.
For example, the completions of the theory of algebraical... | This is totally intractible. For instance, the complete theory of every finite group is a completion (and these completions are distinct for non-isomorphic finite groups since elementarily equivalent finite structures are isomorphic), so describing all the completions is at least as hard as classifying finite groups. ... | {
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How to prove combinatorial $\sum_{k=0}^{n/2} {n\choose2k} = \sum_{k=0}^{n/2 - 1} {n\choose2k+1} = 2^{n-1}$ I have problems solving the following formula for even positive integers $n$:
$$\sum_{k=0}^{n/2} {n\choose 2k} = \sum_{k=0}^{n/2 - 1} {n\choose 2k+1} = 2^{n-1}$$
I tried to prove it by induction but it didn't work... | Hint:
You can get the result by expanding these two expressions using the binomial theorem
$$(1-1)^n=0\ \ \ \text{ and } \ \ \ (1+1)^n=2^n.$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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How to transition from context-free grammar $G$ to to context-free grammar which starts and ends with specific letters?
Given context-free grammar $G$ whose terminal letters are $\{a,b,c,d\}$ how can we transition to context-free grammar which contains the words from the language that $G$ creates but which start with ... | Note that with the constuction from the question "How to define a grammar which creates a language from words of another grammar without one of the letters?" you see how messengers can be sent inside the derivation (or derivation tree) to perform certain actions or to check certain properties of the tree.
In your case ... | {
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"source": "stackexchange",
"question_score": "1",
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Easy example of a herbrand structure Can someone give me an easy example of a Herbrand structure?
I can't really visualise the difference between a Herbrand and a normal structure.
| Example
Consider the very simple FOL formula : $R(c)$.
The domain of the Herbrand structure is :
the set of all ground terms [i.e. closed terms] of the language.
In the above case, we have only the individual constant $c$ as gorund term. Thus, the domain is $H = \{ c \}$.
With it, we define the Herbrand interpretat... | {
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How to find the value $\sum_{n\geq2}^{\infty}(-1)^{n+1}\frac{n}{n^2-1}$ converges to? How to find the value this sum converges to?$$\sum_{n\geq2}^{\infty}(-1)^{n+1}\frac{n}{n^2-1} $$
I've tried writing it like this
$$\sum_{n\geq2}^{\infty}(-1)^{n+1}·n·\Bigg(\frac{1/2}{n-1}-\frac{1/2}{n+1}\Bigg) $$
and writing a few ter... | Note that$$(-1)^{n+1}\frac n{n^2-1}=\frac12\times\frac{(-1)^{n+1}}{n-1}+\frac12\times\frac{(-1)^{n+1}}{n+1}.$$But$$\sum_{n=2}^\infty\frac{(-1)^{n+1}}{n-1}=\sum_{n=1}^\infty\frac{(-1)^n}n=-\log(2)$$and$$\sum_{n=2}^\infty\frac{(-1)^{n+1}}{n+1}=\sum_{n=3}^\infty\frac{(-1)^n}n=\left(\sum_{n=1}^\infty\frac{(-1)^n}n\right)+1... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Homogeneous or non - homogeneous $?$ The second order differential equation is given by -
$ \frac{d^{2}y}{dx^{2}} + \sin (x+y) = \sin x$
Is this a homogeneous differential equation $?$
Well, I guess this is not a homogeneous differential equation since the form of this equation is not $a(x)y'' + b(x)y' +c(x)y = 0$.
Bu... | You are correct, as it is not a linear ODE, it is neither homogeneous nor inhomogeneous.
The cited characterization is most likely based on the fact that $y=0$ is a solution, but that is only a necessary condition for linearity, not a sufficient one.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Defining a tricky function: $(A \rightarrow \mathcal{P}(B)) \rightarrow\mathcal{P}(A \rightarrow B)$ How would I define a function of the form:
\begin{align*}
\phi: (A \rightarrow \mathcal{P}(B)) \rightarrow\mathcal{P}(A \rightarrow B)
\end{align*}
I know what behaviour I want, I'm just struggling to define it. For exa... | The elements of $\phi(f)$ are exactly those $g \colon A \to B$ for which
$$\forall a \in A: g(a) \in f(a).$$
They are choice functions: for every $a \in A$, they choose an element $g(a) \in f(a)$. The fact such a function, in general, exists at all is the axiom of choice.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Complex analysis proof triangle inequality: Given: $|z+w|^2=|z|^2+|w|^2+2Re(z\bar w)$
Prove:$|z+w|\leq |z|+|w|$
Work done so far:
Let $z=x+iy$ and $w=a+bi$, then:
$$|x+iy+a+ib|=|z+w|=\sqrt{(x+a)^2+(y+b)^2}$$
$$\sqrt{x^2+y^2}+\sqrt{a^2+b^2}=|z|+|w|$$
Squaring it I get,
$$x^2+y^2+2|z||w|+a^2+y^2$$
After this I am lost, ... | HINT: Please do not write out real and imaginary parts. Just use the inequality you were given. Compare $|z+w|^2$ and $(|z|+|w|)^2$, and recall what you know (or can prove) if $a,b\ge 0$ and $a^2\le b^2$.
| {
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Inequality. ${{\sqrt{a}+\sqrt{b}+\sqrt{c}} \over {2}} \ge {{1} \over {\sqrt{a}}} + {{1} \over {\sqrt{b}}} + {{1} \over {\sqrt{c}}}$ Question. If ${{a} \over {1+a}}+{{b} \over {1+b}}+{{c} \over {1+c}}=2$ and $a$, $b$, $c$ are all positive real numbers, prove that
$${{\sqrt{a}+\sqrt{b}+\sqrt{c}} \over {2}} \ge {{1} \over... | Now, use $$\sqrt{x(y+z)}-\sqrt{y(x+z)}=\frac{x(y+z)-y(x+z)}{\sqrt{x(y+z)}+\sqrt{y(x+z)}}=\frac{z(x-y)}{\sqrt{x(y+z)}+\sqrt{y(x+z)}}.$$
| {
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"url": "https://math.stackexchange.com/questions/3086636",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Show that the function $H(x, y) = x^2 + y^2 + |x − y|^{-1}$ achieves its global minimum somewhere on the set $\{(x, y) \in \mathbb{R}^2 : x \ne y\}$. Show that the function $H(x, y) = x^2 + y^2 + |x − y|^{-1}$ achieves its global minimum somewhere on the set $\{(x, y) \in \mathbb{R}^2 : x \ne y\}$.
I kind of understand... | What this question is really asking is if the function has a global minimum at all, as it isn't even defined on the set $x=y$ as you mentioned.
The set $\{(x,y)\in R^2:x\ne y\}$ is open, and therefore ay local (and thus the global) minimum must have the partial derivatives equal $0$. The function is symmetric, so we as... | {
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Compute $\int_C ze^{\sqrt{x^2+y^2}} \mathrm ds$
Compute $\int_C ze^{\sqrt{x^2+y^2}} \mathrm ds$ where
$$C:x^2+y^2+z^2=a^2, x+ y=0, a \gt 0$$
At first I thought to parametrize this as: $x=a \cos t , y=a \sin t, z =0$, but then the integral will result in $0$ and this might not be true.
| The curve $C$ is a circle in the plane $x+y=0$ centered at the origin with radius $a$, so it is symmetric with respect to the plane $z=0$. Moreover the integrand is odd with respect to $z$and therefore, by symmetry, the given integral $\int_C ze^{\sqrt{x^2+y^2}} \mathrm ds$ is zero.
BTW a convenient parametrization for... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Solve the equation |x-1|=x-1 Solve the equation:$|x-1|=x-1$
My solution:
Case 1 :$ x\ge1$, Hence $x-1=x-1$, therefore infinite solution
Case 2 :$ x<1$, Hence $1-x=x-1$,$x=1$, hence no solution
But the solution i saw concept used is $ x\le1$ in lieu of $ x<1$
Hence final answer is $[1,\infty]$, is this concept correct
| Your answer is right, apart from the square bracket pointed out in @ElevenEleven's comment ($\infty$ can't be the upper end of a closed interval).
Another way to get it is to note that $|a|=a$ only when $a\geq 0$, so for $a=x-1$,
$$|x-1|=x-1$$ implies $$x-1\geq 0$$
which gives you the answer without needing to consider... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 2
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Compact operator by proving Ascoli-Arzelà I need to prove that this operator satisfies Ascoli-Arzelà's hypothesis.
$T: C^0[0,1] \rightarrow C^0[0,1] $, defined $Tu(x)=\int_0^x a(x,t) u(t)dt$, where $a(x,t)=C^0 ([0,1] \times [0,1])$.
Equiboundedness is okay, I need to prove equicontinuity: taken $\{ u_n \} \subset C^0... | You need to prove that for every $\varepsilon > 0$ and $x \in [0,1]$ there is a $\delta > 0$ such that $|x-y| < \delta$ implies that $|Tu_n(x) - Tu_n(y)| < \varepsilon$. The kind of upper bound you exhibit is insufficient to do this (assuming from context that $C$ is a positive constant) since for $\varepsilon = C/2$ y... | {
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Riemann-integration problem Here is the exercise
Let $f:[a,b]\rightarrow \mathbb{R}$ be Riemann-integrable. Prove that $f^+$, $f^-$ and $|f|$ are also Riemann-integrable, when
$$f^+=\begin{cases}
f(x) & f(x)\geq 0 \\
0 & otherwise
\end{cases}$$
$$f^-=\begin{cases}
-f(x) & f(x)\leq 0 \\
0 ... | We prove the Riemann integrability of $f^+$. A similar proof can be done for $f^-$.
As $f$ is Riemann-integrable, for all $\epsilon>0$ there exists step functions $h \leq f\leq g $ such that $\int h -\int g <\epsilon$.
Now define $h^+ = \max(h,0)$ and $g^+ = \max(g,0)$. You can verify that:
*
*$h^+, g^+$ are step fu... | {
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Investigate the convergence of $ \sum_{n=1}^{\infty } (-1)^{n}\frac{n+2}{n(n+1)} $ I am supposed to investigate the convergence of $ \sum_{n=1}^{\infty } (-1)^{n}\frac{n+2}{n(n+1)} $. I'm unsure whether to use Leibniz' criterion or a comparison test and I really can't start. Thanks
| You can directly use Leibniz' criterion, but you have to show the absolute value of the general term is non-increasing. Here, I suggest an (arguably) simpler way to see what's happening and prove convergence:
You have
$$
\frac{n+2}{n(n+1)} = \frac{1}{n+1} + \frac{2}{n(n+1)}
$$
and therefore
$$
\sum_{n=1}^\infty (-1)^n ... | {
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Calculate the value of $\int_0^\infty \frac{\sqrt{x}\cos(\ln(x))}{x^2+1}\,dx$ I'm asked to evaluate the integral $\displaystyle\int_0^\infty \frac{\sqrt{x}\cos(\ln(x))}{x^2+1}\,dx$.
I tried defining a funcion $f(z)=\frac{e^{(1/2+i)\operatorname{Log}(z)}}{z^2+1}$, taking $\operatorname{Log}$ with a branch cut along the ... | As @Adrian suggested, define $\log z =\log |z|+i\arg(z)$ where $\arg(z)\in (0,2\pi)$ and let the contour be a keyhole contour.
Then
$$
\left|\int_{\gamma_R}\frac{e^{(1/2+i)\log z}}{z^2+1}\,dz\right|\le \int_{\gamma_R}\frac{e^{1/2 \log|z|-\arg(z)}}{R^2-1}\,|dz|\le C\frac{R^{3/2}}{R^2-1}\stackrel{R\to\infty}\longrighta... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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point cloud in complex plane I want to draw the point cloud represented by the following term.
$$M_{4}=\left\{z \in \mathbb{C} : | z-1|=\frac{1}{2}| z-j|\right\}$$
$j$ equals $i$, the imaginary square root of $-1$.
I have made several attempts to get a solution for the equation. This is the one that looks the most pro... | I am sorry if I am giving you a bum steer. I believe by point cloud we are looking for the locus of points described by the above equation. I don't know TEX. Six steps to solution. First, multiply both sides by of (z+i). Simplify. Second, add (iz)^2 to both sides. Simplify. Third, add the imaginary unit to both ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How this inequality is derived? Let $T$ ∶ $ℓ_2$ → $ℓ_2$ be defined by
$T((x_1,x_2,...,x_n...))$=$(X_2-X_1, X_3-X_2,...,X_{n+1}-x_n,...)$
Then I have find the norm of $T$.
Here is the answer to this question: https://math.stackexchange.com/a/1647794/581242
I am not able to see how this first inequality is derived.
$\|... | It appears that there was a small typo. Use the inequality $|a-b|^{2} \leq 2(a^{2}+b^{2})$ . The final result is correct but the first inequality is wrongly stated.
| {
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"timestamp": "2023-03-29T00:00:00",
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What is the definition of $e^{ix}$? This might seem silly, but in proving Euler Formula taught in Calculus classes, we make the assumption that
$\frac{d}{dx}e^{ix} = ie^{ix}$
However $e^{g}$ Pre-Euler’s Formula, only takes in real numbers for g. If we tried to use the Chain Rule where $g = ix$ we have no definition fo... | When I was teaching, I always used the series definition, as explained in @ErikParkinson’s answer. But you may also define
$$
e^z=\lim_{n\to\infty}\left(1+\frac zn\right)^n\,.
$$
When you look at this closely, the formula $e^{it}=\cos t+i\sin t$ becomes very reasonable.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3087919",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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if $f$ has Newton Polygon consisting of one segment $(0,0)$ to $(n,m)$ with $m,n$ coprime, then $f$ cannot be factored Let $f(X)\in 1+ X\mathbb{Z}_p[X]$ have Newton Polygon consisting of one segment joining $(0,0)$ to $(n,m)$ with $m,n$ coprime. I have to show that $f(X)$ cannot be factored as a product of two polynomi... | If you understand that every root $\rho$ of $f$ satisfies $v(\rho)=-m/n$, then you see that this will happen for both of $g$ and $h$ if $f=gh$.
Now, what can the Newton polygon of $g$ be? It will be of width $r$ for some integer with $0<r<n$, since neither $g$ nor $h$ is constant. And the right-hand vertex? Since the s... | {
"language": "en",
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Finding sum of none arithmetic series I have a question to find the sum of the following sum:
$$
S = \small{1*1+2*3+3*5+4*7+...+100*199}
$$
I figured out that for each element in this series the following holds:
$$
a_n = a_{n-1} + 4n - 3
$$
But I don't know where to go from here, I tried subtracting some other series b... | $a_n=\sum_{r=1}^n(4r-3)+a_0=\dfrac n2(1+4n-3)+a_0=2n^2-n+a_0$
$$\sum_{n=1}^ma_n=2\sum_{n=1}^mn^2-\sum_{n=1}^mn+a_0\sum_{n=1}^m1$$
Here $a_0=0$
Alternatively,
$$a_m=b_m+a+bm+cm^2$$
$$4n-3=a_n-a_{n-1}=b_n-b_{n-1}+b+c(2n-1)=b_n-b_{n-1}+2c(n)+b-c$$
WLOG set $2c=4,b-c=-3\iff c=b+3$ to find $b_n=b_{n-1}$
set $a=0$ so that $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3088304",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 2
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Apply the Implicit Function Theorem to find a root of polynomial
Caculate the value of the real solution of the equation $x^7+0.99x-2.03$, and give a estimate for the error.
The hint is: use the Implicit Function Theorem. I dont know how to use the IFT in this case, I'm not familiarized with this.
I think in construc... | Let $F(x,y,z)=x^7+y\,x-z$; then $F(1,1,2)=0$. We have
$$
\frac{\partial F}{\partial x}=7\,x^6+y\implies\frac{\partial F}{\partial x}(1,1,2)\ne0.
$$
By the IFT, you can solve for $x$ in the equation $F(x,y,z)=0$ on a neighborhood of $(1,1,2)$. That is, there is a $C^1$ function $\phi(y,z)$ such that $\phi(1,2)=1$ and $F... | {
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Exercise in abstract algebra Assume that $M$ is a set, and that $R$ is a ring. Let $\cal{F}$ be the collection all functions $M\rightarrow{R}$. Prove that the following two statements are equivalent.
*
*$\cal{F}$ is a field.
*$M$ is a singleton (i.e. consisting one element), and $R$ is a field.
The implication 2. $... | Suppose M has at least two distinct elements : $m_1$ and $m_2$.
Then, consider $f :M\to R$ such that $f(m_1)=1$ and $f(m)= 0$ otherwise; and $g:M\to R$ such that $g(m_2)=1$ and $g(m) =0$ otherwise.
Then, as functions (that is, as elements of $F$), $f$ and $g$ are non zero but $fg=0$.
So $F$ can't be a field.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3088526",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Small question on Pythagoras theorem The projections-of-the-legs over the hypotenuse should add up to the hypotenuse $c$.
Is there any alternative way to prove below?
$$a\cos \alpha + b\sin \alpha = \sqrt{a^2+b^2}$$
| Well, $$\cos \alpha = \frac ac\quad \&\quad \sin \alpha = \frac bc$$
so $$a\cos \alpha +b\sin \alpha = \frac 1c \times (a^2+b^2)=\frac {c^2}c=c$$
Of course, that last step requires the Pythagorean Theorem ("PT"). It is worth remarking that, without using PT the argument shows $$a\cos \alpha +b\sin \alpha = \frac {a^2+... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to prove the following formula using an indirect proof I need to prove that the premise $A \to (B \vee C)$ leads to the conclusion $(A \to B) \vee (A \to C)$. Here's what I have so far.
From here I'm stuck (and I'm not even sure if this is correct). My idea is to use negation intro by assuming the opposite and com... | Hint: if you assume $A \to (B \lor C)$, $\lnot(A \to B)$ and $A$, then you can conclude $B \lor C$ and $\lnot B$. Can you take it from there?
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Translate this sentence to predicate logic The question given asks to translate to predicate logic:
Every positive real number has a unique positive real root.
My solution to this problem is to separate it into the appropriate quantifiers.
C(x) = "Every positive real number x"
S(x) = "x has a positive real root"
the ... | Couple of things:
First, you don't want to define:
$C(x)$ = "Every positive real number $x$"
The 'every' needs to be taken care of by a quantifier so that should not be part of the meaning of this formula. Indeed, the formula $C(x)$ should simply express a property of $x$, just like your:
$S(x)$ = "$x$ has a positiv... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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$\lim_n{f(\frac{x}{n})}=0$, for all $x\in (0,1)$ implies $lim_{x\to0}f(x)=0$ (edit, $f$ is not continuous) Let be $f:(0,1)\rightarrow \mathbb{R}$. For all $x\in (0,1)$ I have that $\lim_n{f(\frac{x}{n})}=0$.
It's true that $lim_{x\to0}f(x)=0$?
And if $f$ is continuous?
For the second point I just use the definition of... | I will provide a proof for the continuous case. This result requires Baire Category Theorem. If $\epsilon >0$ then $(0,1)=\cup_n A_n$ where $A_k=\{x:|f(\frac x n )| \leq \epsilon \forall k \geq n\}$. Since $(0,1)$ is of second category it follows that there is some interval $(a,b)$ and some $n_0$ such that $|f(\frac x ... | {
"language": "en",
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Can a simple closed curve in a compact surface be dense? I do not see an argument immediately that it cannot be, but it feels dubious. Does genus have anything to do with it?
| A simple closed curve in a surface $X$ is a continuous injection $f:S^1\to X$. Since $S^1$ is compact, the image of $f$ is compact and hence closed. So, the image cannot be dense (the image cannot be all of $X$ since $f$ is a homeomorphism to its image).
More generally, the same argument applies to any Hausdorff spac... | {
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Making $x$ the subject of $x^3-2x+y=1$ How to I make x the subject of this formula?
$$x^3-2x+y=1\tag1$$
My attempt:
I apply complete the square
$$(x-1)^3+y-1=1\tag2$$
$$(x-1)^3=2-y\tag3$$
$$x-1=\sqrt[3]{2-y}\tag4$$
$$x=1+\sqrt[3]{2-y}\tag5$$
but my teacher said it is wrong!
Can you please help? Thank in advance!
| You make a mistake in the first step when you attempt to complete the square. Remember: you are working with an $x^3$, not an $x^2$, and because of this, any "completing of the cube" would require an $ax^2$ term as well. Unfortunately, due to the fact that $x^3 - 2x + y = 1$ is not one-to-one, there is no easy way to r... | {
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$\mathscr{B} = \{ [a, b) | a< b \in \mathbb{R} \}$ is a basis for a Topology in $\mathbb{R}$ I just want to ask if my proof for this problem is correct.
$$\mathscr{B} = \{ [a, b) | a< b \in \mathbb{R} \}$$ is a basis for a Topology in $\mathbb{R}$ .
Here is my proof:
*
*Let $x \in \mathbb{R}$. Choose $B \in \maths... | Simpler and direct is
$[a,b) \cap [r,s) = [\max(a,r), \min(b,s))$.
In addition, it is necessary to note that every $r \in \mathbb{R}$ is in some base set. $[r, r+1)$ for example.
| {
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Purpose of Variance I know how to calculate variance, in probability and experimental data ect but, what I cant seem to understand is the real world application of it. And just wondering if someone could give an explantation of it importance in statistics and experimental data, by giving an example.
| For Practical Example I can say,
Let's say that you've to a coke plan a party in your house, there are 100(s) of guest coming to your house. You want to give them a Coca Cola but the problem is that each of them has their different capacity. A kid might need just a glass, but an adult would need more than that.
So, in ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Probability that sum of integer reciprocals is larger than a fixed number. Suppose $n$ numbers are drawn independently from the list of $m$ integers $\{1,2,3,\ldots ,m\}$ uniformly at random. Denote these $n$ picks as $x_1,x_2,\ldots x_n$. Note that $n\geq m$ is possible. Fix a positive integer $C$. I am trying to dete... | This is actually more of a consideration than an answer,
but wishfully it may be of some help.
We have $n$ discrete uniform i.i.d. random variables $X_k$, ranging from $1$ to $m$,
and we want to find the distribution of the sum of their inverse.
To the scope of finding an approximation for high values of $m$ and $n$,... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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proof that e is the sum of the reciprocals of factorials Ok so we know that:
e= lim n→∞(1+1/n)^n.
and we know by binomial theorem, that
lim n→∞ of $\sum_{k=0}^n {n \choose k} (1/n)^k = (1+ \frac{1}{n})$
To simplify further to $\sum_{k=0}^n \frac{1}{k!} = e$
we must evaluate the following limit:
n→∞${n \choose k} \frac... | We have $$e=\lim_{n\to\infty}\sum_{k=0}^n\frac{\binom{n}{k}}{n^k}=\lim_{n\to\infty}\sum_{k=0}^\infty\frac{\binom{n}{k}}{n^k}=\lim_{n\to\infty}\sum_{k=0}^\infty\frac{1}{k!},$$where with $k$ fixed$$\lim_{n\to\infty}\frac{\binom{n}{k}}{n^k}=\frac{1}{k!}\lim_{n\to\infty}\prod_{j=1}^{k-1}\left(1-\frac{j}{n}\right)=\frac{1^{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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If $(a,b)$ is a multiple of $(c,d)$, show that $(a,c)$ is a multiple of $(b,d)$ I need help with this problem:
If $(a,b)$ is a multiple of $(c,d)$ with $abcd\neq0$, show that $(a,c)$ is a multiple of $(b,d)$. This is suprisingly important: call it a challenge question. You could use numbers first to see how $a,b,c$ and... | First note the following: $(a,b) = (a,(\frac{b}{a}) a)$. [As $abcd \not =0$ we can assume that $\frac{b}{a}$ exists]
So for some scalar $x$ we note: $(c,d) = x(a,b) = (xa,xb)$ $=(xa,x(\frac{b}{a}) a)$. Thus $c$ can be written $c=xa$ and $d$ can be written $d=x(\frac{b}{a}) a$.
Thus $(b,d) = (\frac{b}{a} a, x(\frac{b}{a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Periodic solutions of a planar ODE Linearizing the equation of the two body problem at a circular solution I came accross the following planar second order system of differential equations
$$
\begin{cases}
2\frac{1}{\omega^2} \ddot{u}=3u\cos2\omega t+3v\sin2\omega t+u\\
2\frac{1}{\omega^2}\ddot{v}=3u\sin2\omega t-3v\co... | If we define $\eta = u + i v$, then the set of equations can be rewritten as
$$
0 = - \frac{2}{\omega^2} \ddot{\eta} + 3 e^{2 i \omega t} \bar{\eta} + \eta.
$$
If $\eta$ is periodic with period $2 \pi/\omega$, then it will be expressible as a power series of the form
$$
\eta = \sum_{m = - \infty}^\infty a_m e^{i m \om... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Solve the recurrence relation $a_n=6a_{n-1}-9a_{n-2}-8$ for $n\geq2$, $a_0=0$, $a_1=1$ My task:
$a_n=6a_{n-1}-9a_{n-2}-8$ for $n\geq2$, $a_0=0$, $a_1=1$
My solution
$x^{2}-6x+9$
$\Delta=0$
$x_0=3 $
So I am gonna use following formula:
$a_n=ar^{n}+bnr^{n}$
$a_n=a*(3)^{n}+bn*3^{n}$
$-8$ is the problem, so I am looking f... | I think that your choice of $c$ is wrong
From $$b_n=6(b_{n-1}-c)-9(b_{n-2}-c)-8+c=6b_{n-1}-9b_{n-2}-8+4c$$
this gives you $c=2$ and $b_n=a_n+2$
$$b_0=2, b_1=3$$ after solving $$b_n=a*3^{n}+bn*3^{n}$$ you get $a=2 ,b=-1$
| {
"language": "en",
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Algorithm for optimal assignment of tasks to a team of people Is there an algorithm to get a team of people to complete a certain number of tasks the fastest, where the time taken to complete a certain task is different for different people?
Each task must be done fully by one person (eg can't have person A do the firs... | This problem is $NP$-complete, even in the case of two people and where both people take the same amount of time for each task (i.e.: $n = 2$ and $a_i = b_i$ for all $i=1,\ldots,m$) since this is essentially the PARTITION problem. As such, it is unlikely that you will find a polynomial-time algorithm that solves the pr... | {
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"timestamp": "2023-03-29T00:00:00",
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Will the following sequence ever repeat? I'm unsure if the notation used by the author is common, so I will define some terms before stating the problem.
{$0, 1$}$^\infty$ is the set of all functions $f:\mathbb{N} \rightarrow ${$0, 1$}.
{$0, 1$}$^n$ is the set of all functions $f:$ {$1, ..., n$}$ \rightarrow ${$0, 1$}.... | Let $R_n$ be the event that the second block of $n$ bits matches the first block of $n$ bits, that is, the event that $ww\prec S$ for some $w\in\{0,1\}^n$, and let $N=N(S)=\sum_{n\ge1}\mathbb 1_{R_n}(S)$ be the number of $n$ such that $S\in R_n$, that is, the number of $n$ such that $ww\prec S$ for $w$ of length $n$. ... | {
"language": "en",
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Show local convexity Given a metric space $(X,d)$ where $X=C([0,1])$ and $d$ is defined as: $$d(f,g)=\inf \{\epsilon : \mu \{x \in [0,1] : |f(x)-g(x)|>\epsilon\}<\epsilon\}$$ ($\mu$ is a Lebesgue measure).
I want to show this space is not locally convex.
If it is not locally convex then $\exists u=\{g|d(f,g)<\epsilon ... | It is easier to consider the space $Y$ of step functions, i.e., linear combinations of indicator functions of intervals with the same metric (which btw describes the convergence in measure). I claim that the only convex neighbourhood $U$ of $0$ is the full space $Y$: Indeed $U$ contains some ball $B_\varepsilon=\{f\in ... | {
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Show that $n^{23}+6n^{13}+4n^{3}$ is a multiple of $11$ I was checking the following Fermat's little theorem exercise:
Show that $n^{23}+6n^{13}+4n^{3}$ is a multiple of $11$
I've started by stating each congruence individually suposing that each $n,6n$ and $4n$ are primes with $11$, for the first one I have:
$$n^{1... | Fermat's little theorem only applies when prime $p$ does not divide $a$ in $a^{p-1} \equiv 1 \pmod p$.
In your expression, if $n$ is a multiple of $11$ the theorem doesn't apply but the expression is trivially a multiple of $11$.
If $n$ is not a multiple of $11$ the theorem applies. Now note:
$n^{10} \equiv 1 \pmod{11}... | {
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Expected value of $\sin(x)$ I have a uniformly distributed random variable $ \omega $ in the range $[\frac\pi2, \frac\pi{-2}]$. Then I have the function $ s = \sin(\omega) $ I want to calculate the expected value of this function $ s $.
So far I know that the uniformly distributed random variable can be written as $$... | The expected value of any random variable $s(\omega)$ where $\omega$ is having the probability distribution function $f(\omega)$ is given by:
$$ E(s(\omega)) = \int_{-\infty}^{\infty} s(\omega)f(\omega)d\omega$$
since $\omega$ is distributed uniformly in the interval $[-\pi/2,\pi/2]$ we have $$f(\omega) = \frac{1}{(\pi... | {
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Strictly positive inner product for a pair of non-zero, positive operators. Let $ A,B $ be non-zero positive operators on a infinite-dimensional separable Hilbert space $(H , \langle \cdot, \cdot \rangle)$. I am required to prove that there exists $u' \in H$ such that
\begin{alignat*}{2}
\langle Au' , u'\rangle >0 \ \... | Let $v,w\in H$ be as you have defined them. For $t\in[0,1]$ put $x_t=tu+(1-t)w$, and define $f,g:[0,1]\to [0,\infty)$ by
$$f(t)=\langle Ax_t,x_t\rangle,\quad g(t)=\langle Bx_t,x_t\rangle.$$
Note that $f$ and $g$ are non-zero polynomials (of degree at most $2$). Argue that there is some point $t_0\in[0,1]$ such that ... | {
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how to prove a spanning set of polynomial I am struggling so much understanding this concept of subspace and span.
The question is, Given that
$P2:W=\{(x+1)(ax+b)| a,b \in R\}$
show that
$\{x^2+x, x^2+2x+1\}$
is a spanning set of $W$.
I don't know if I got this concept right, but I've tried to do things by letting
$... | Hint: First, $W$ is a $2$-dimensional vector space (easy to see).
Now, $\{x^2+x,x^2+2x+1\}$ is linearly independent (easy to see)
Let $a=1,b=0$. We get $x^2+x$. Now let $a=1,b=1$. We get $x^2+2x+1$.
Thus $W=\operatorname{span} \{x^2+x,x^2+2x+1\}$.
.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3090901",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Unitary Central Character by Schur's Lemma Consider an irreducible smooth representation $\pi$ of the group $G=GL_n(\mathbb{Q}_p)$ with center $Z$. Does there exist a unitary central character for $\pi$?
More precisely, is there a (quasi-)character $\omega: G \to \mathbb{C}^{\times}$ such that $\pi \otimes \omega$ whe... | There is a central character $\omega:Z \rightarrow \mathbb{C}^{\times}$ by Schur, but it need not be unitary. E.g., consider $∣\text{det}∣$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3091005",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Is there arbitrage?
An economist writes a 1-period expectation model for valuing options.
The model assumes that the stock starts at S and moves to $2S$ or
$\frac{1}{2}S$ in 1 year's time with equal probability. Strike is equal to $K$
Assume rates are zero.
I found that the value of the call option using the eco... | First of all, economists think in terms of risk premia. There's a concept in the economic branch of asset pricing, called "stochastic discount factor", which differentiates economists from mathematicians.
That being said, the question asks you to find the value of the option today. As an economist, you will attach 1/2 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3091155",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Bound and limsup for cumulative sum of a random walk It is well-known from the law of iterated logarithm, that, if $X_k$ are symmetric Bernoulli random variables $\pm 1$, then $S_n= X_1 + X_2 + ... + X_n$ has this property:
$$\limsup_{n \to \infty} \frac{S_n}{\sqrt{2 n \log \log n}} = 1 \qquad \text{a.s.}$$
giving an ... | For any $\epsilon>0$, since $|T_i-T_{i-1}| \le i$ using Azuma's inequality
\begin{align}
\sum_n P\left( \left|T_n \right|\ge \epsilon n^{\alpha}\right)
\le \sum_n 2 \exp\left( \frac{-\epsilon^2 n^{2\alpha}}{2 \sum_{i=1}^n i^2} \right)
= \sum_n 2 \exp\left( \frac{-3 \epsilon^2 n^{2\alpha-1}}{ (n+1)(2n+1)} \right)
\le \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3091444",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Concerning the identity in sums of Binomial coefficients Let be the following identity
$$\sum_{k=1}^{n}\binom{k}{2}=\sum_{k=0}^{n-1}\binom{k+1}{2}=\sum_{k=1}^{n}k(n-k)=\sum_{k=0}^{n-1}k(n-k)=\frac16(n+1)(n-1)n$$
As we can see the partial sums of binomial coefficients are expressed in terms of $3$-rd order polynomial $P... | I would say that you have a good answer already. But their are other possible answers which seem reasonable. Further restriction might force the favored solution above.
In the case $k=3$ (which is the only one I will discuss in any detail)
$$\sum_{s=1}^n\binom{s}3= \\ \sum_{s=1}^ns\binom{n-s}2=\sum_{s=1}^n(n-s)\bin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3091598",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Formula for the sequence 0,3,8,15,24 ... Out of my own interest I've been practicing finding formula for for sequences and I've been having trouble finding one for the nth term for this sequence.
0,3,8,15,24 ...
Clearly you add 5,7,9,11 ... to the previous number but if anyone had some insight about how to express thi... | The solution for $ a(n) $ is this
$ a(n) = n(n+2) = (n+1)^2 - 1. $
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3091713",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 10,
"answer_id": 3
} |
Matrix function that gives a scalar I have the following function:
$$f(z) = z\vec{b}^T[I-zA]^{-1}\vec{1},$$
where $z$ is a complex scalar with $Re(z)<0$ (for simplicity, WLOG, we can take $z$ to be real), $b$ is a vector, $1$ is a vector of ones. $I$ is the identity matrix, and $A$ is some arbitrary matrix so that $I-z... | So, assuming that $z \in \mathbf{R}$ we have
*
*$z = 0 \implies f(z) = 0$
*$z \neq 0 \implies b^T(I - zA)^{-1}\cdot \mathbf{1} = 0$
Now, using the formula of inversion of the sum of matrices (see, e.g. here) we get
$$
b^T\left(I + \frac{z}{1 - \text{tr}(A)z}A\right)1 = 0 \implies b^T\cdot \mathbf{1} + \frac{z}{1 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3091826",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Literature concerning Hawkes Processes I am looking into an introdoction in Hawkes processes (self-exciting processes). Are there books or lecture notes that explain the math behind it, or a chapter in a more general book? I haven't found anything good so far, so every recommendation is appreciated.
| Hawkes Processes by Patrick J. Laub, Thomas Taimre, Philip K. Pollett is a short article that introduces Hawkes Processes.
For the more mathematical theory, this can be found in Daley and Vere-Jones' An Introduction to the Theory of Point Processes: Volume I. Note however that they are focused on the general theory of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3091933",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Do we have an explicit expression of this function? Let $\Delta^{n-1} \equiv \{(p_1, \ldots, p_n) \, |\, p_i \geq 0, \sum_i p_i = 1\}$ be the $n-1$ dimensional simplex. Define $f : \Delta^{n-1} \times \Delta^{n-1} \rightarrow \Delta^{n-1}$ such that $$ f(p, q) = \Big(\frac{p_1 q_1}{\sum_i p_i q_i}, \ldots, \frac{p_n q_... | No, we don't because there is no such function for $n\ge 2$. Suppose
$$
\exists h:\Delta^{n-1}\to \Bbb R\ \ : \ \ h(f(p,q))=\sum_{i=1}^n p_iq_i.
$$ For $p=q=(\frac{1}{2},\frac{1}{2},0,0,\ldots,0)$, it holds $f(p,q)=(\frac{1}{2},\frac{1}{2},0,0,\ldots,0)$ and $$h\left(\frac{1}{2},\frac{1}{2},0,0,\ldots,0\right)=h(f(p,q)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3092065",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Asymptotics of Hypergeometric $_2F_1(a;b;c;z)$ for large $|z| \to \infty$? I found this list of asymptotics of the Gauss Hypergeometric function $_2F_1(a;b;c;z)$ here on Wolfram's site for large $|z| \to \infty$
In particular there is a general formula for $|z| \to \infty$
$$
_2F_1(a;b;c;z) \approx \frac{\Gamma(b-a)\Ga... | Converting ${_2\hspace{-1px}F_1}$ to the Meijer G-function, we obtain
$${_2\hspace{-1px}F_1}(a, b; c; z) =
\frac {\Gamma(c)} {\Gamma(a) \Gamma(b)}
G_{2, 2}^{1, 2} \left( -z \middle| {1 - a, 1 - b \atop 0, 1 - c} \right) = \\
\frac {\Gamma(c)} {2 \pi i \Gamma(a) \Gamma(b)} \int_{\mathcal L} \frac
{\Gamma(y + a) \Gamma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3092190",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
prove $\int_0^\infty \frac{\log^2(x)}{x^2+1}\mathrm dx=\frac{\pi^3}{8}$ with real methods Context: I looked up "complex residue" on google images, and saw this integral. I, being unfamiliar with the use of contour integration, decided to try proving the result without complex analysis. Seeing as I was stuck, I decided ... | Pretty straightforward this integral can be related to the Dirichlet Beta Function $\beta(s)$ and its integral representation which is given by
$$\beta(s)~=~\frac1{\Gamma(s)}\int_0^\infty \frac{t^{s-1}}{e^{t}+e^{-t}}\mathrm dt$$
Therefore, enforce the substitution $x=e^{-t}$ within your integral $J$ to obtain
\begin{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3092412",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 5,
"answer_id": 3
} |
How to find $x_n$ from $x_{n+1} = \frac{x_n}{1-a+a x_n}$? For $n\geq 0$ let $x_{n+1} = \frac{x_n}{1-a+a x_n}$, where $a\in (0,1)$.
I would like to know if it is possible to express $x_n$ as a function of $a$ and $x_0$ for all $n\geq0$.
| Note that $$\frac1{x_{n+1}}=\frac{1-a}{x_n}+a$$ which implies that $$\frac1{x_{n+1}}-1=(1-a)\left(\frac1{x_n}-1\right)$$ hence $$\frac1{x_n}-1=(1-a)^n\left(\frac1{x_0}-1\right)$$ from which an explicit formula for $x_n$ in terms of $n$, $a$ and $x_0$ follows.
(Of course, there is some well known theory behind all this.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3092537",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
But what is a continuous function? I have a very basic problem. I am confused about "continuous function" term.
What really is a continuous function? A function that is continuous for all of its domain or for all real numbers?
Let's say:
$\ln|x|$ - the graph clearly says it's continuous for all real numbers except for... | The exact answer depends on your chosen definition of "function" (there is more than one). For most uses, a function is regarded as being continuous on an interval $(a,b)$ if for every number $c$ in $(a,b)$, $f(x)=\lim_{x\to c} f(x)$.
In your example $f(x)=\ln{x}$ is continuous on the interval $(0,\infty)$ and either u... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3092659",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 2
} |
For complex vectors $z_1$ and $z_2$, How do I show that if $|z_1+z_2|=|z_1|+|z_2|$ then the vectors $z_1$ and $z_2$ are parallel or antiparallel. In my complex analysis class, we went over a geometric proof of this with the triangle inequality, but I'm trying to find a more algebraic proof. I'm also trying not to use A... | I will answer the question in the heading (not including $0$).
Let $z_k=x_k+iy_k$ Square both sides and remove common terms to get: $(x_1+iy_1)(x_2-iy_2)+(x_1-iy_1)(x_2+iy_2)=(2(x_1x_2+y_1y_2))=2\sqrt{(x_1^2+y_1^2)(x_2^2+y_2^2)}$. Now square both sides and eliminate common terms to get $2x_1x_2y_1y_2=x_1^2y_2^2+x_2^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3092752",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
} |
Integral $\int\frac{1}{1+x^3}dx$
Calculate$$\int\frac{1}{1+x^3}dx$$
After calculating the partial fractions I got:
$$\frac{1}{3}\int\frac{1}{x+1}dx+\frac{1}{3}\int\frac{2-x}{x^2-x+1}dx=\frac{1}{3}\ln(x+1)+\frac{1}{3}\int\frac{2-x}{x^2-x+1}dx$$
I have no idea on how to proceed. Am I missing a substitution or something... | Alternative approach: Partial fractions.
Recall that for any complex $z$, and any $n=1,2,...$
$$z^{1/n}=|z|^{1/n}\exp\left[\frac{i}{n}(2\pi k+\arg z)\right],\qquad k=0,1,...,n-1$$
Then plug in $z=-1$ and $n=3$ to see that $\arg z=\arg(-1)=\pi$ so that in fact,
$$1+x^3=\prod_{k=0}^2\left(x-\exp\frac{i\pi(2k+1)}3\right)$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3092884",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
How to check convergence of sequence in complete metric space.
Let $\{x_n\}$ and $\{y_n\}$ be two sequences in a complete metric space $(X,d)$ such that
*
*$d(x_n,x_{n+1})\le\frac{1}{n^2}$
*$d(y_n,y_{n+1})\le \frac{1}{n}$ , for all $n\in \mathbb{N}.$
Then which sequence would converge? Justify.
Now sin... | Your reasoning is good.
For 2, use the triangle inequality to show that, for $m < n$,
$$d(x_n, x_m) \le \sum_{k = m}^{n - 1} d(x_k, x_{k+1}).$$
Cauchiness follows from the convergence of $\sum \frac{1}{n^2}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3093009",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
The set of all possible values of $AC$ is an open interval $(m,n)$
Side $\overline{AB}$ of $\triangle ABC$ has length $10$. The bisector of angle $A$ meets $\overline{BC}$ at $D$, and $CD = 3$. The set of all possible values of $AC$ is an open interval $(m,n)$. What is $m+n$?
A) 16 B) 17 C) 18 D) 19 E) 20
Could someo... | I believe you have the correct range for $x$ of $2 \lt x \lt 10$, but that is for $\overline{BD}$. However, the question asks for $\overline{AC}$. Since $\overline{AC} = \frac{30}{x}$, this means it's range is $3 \lt \overline{AC} \lt 15$, so $m = 3$ and $n = 15$, giving the correct answer of $m + n = 3 + 15 = 18$, i.e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3093173",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Polar plots of $\sin(kx)$ The plots of $\sin(kx)$ over the real line are somehow boring and look essentially all the same:
For larger $k$ you cannot easily tell which $k$ it is (not only due to Moiré effects):
But when plotting $\sin(kx)$ over the unit circle by
$$x(t) = \cos(t) (1 + \sin(kt))$$
$$y(t) = \sin(t) (1 +... | I did not grasp exactly what you are asking, however it might be of interest to know that in "old times" electrical engineers were used to visualize phase and frequency of a sinusoidal wave by feeding it to the $x$ axis of an oscilloscope in combination to a known and tunable signal (sinusoidal, triangular , ..) fed to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3093359",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 2,
"answer_id": 1
} |
If $f:M \mapsto M'$ is bijective and $d'(f(x),f(y))\ge d(x,y)~\forall x,y\in M$ then $M$ compact $\implies M'$ compact Let $(M,d)$ and $(M',d')$ be metric spaces and let $f:M\mapsto M'$ be a bijective function such that $$d'(f(x),f(y))\ge d(x,y)~\forall x,y\in M$$
Is it true that if $M$ is compact then so is $M'$?
I fo... | Yes, your example works. If you're worried, then try going into more detail. How do you know $M$ is compact? (Cite a theorem.) How do you know $M'$ is not compact? (Find a sequence which contains no convergent subsequence, or indeed a non-convergent Cauchy sequence would do!) Are you sure that $d(x, y) \le d'(f(x), f(y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3093485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Sum of two irrational numbers being rational or irrational I am currently doing a project on irrational and transcendental numbers and part of this project requires me to look at sums and products of irrational numbers.
I am aware that the sum of 2 irrational numbers can be rational or irrational but was wondering if a... | No, there is not. If there was, we would know whether $e+\pi$ is rational or not. But, in fact, that's an open problem.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3093684",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Homomorphism $f: \mathbb{Z}_{12} \longrightarrow \mathbb{Z}_{30}$ Suppose that we want to construct a non-surjective homomorphism
$$
f: \mathbb{Z}_{12} \longrightarrow \mathbb{Z}_{30}
$$
Since $\mathbb{Z}_{12}$ is cyclic, $f$ is completely determined from the image of $\overline{1}$ (its generator), $f(\overline{1})$.... | The one requirement for $f(1)$ which must be fulfilled is
$$
0=f(0)=f(12\cdot 1)=12f(1)
$$
Among elements in $\Bbb Z_{30}$, these are exactly the elements which are multiples of $5$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3093778",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
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